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\begin{document}
\title[Traveling waves for the Allen-Cahn relaxation model]{Analytical and numerical investigation of traveling waves for the Allen--Cahn model with relaxation}
\author[C. Lattanzio]{Corrado Lattanzio}
\address[Corrado Lattanzio]{Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, Universit\`a degli Studi dell'Aquila (Italy)}
\email{[email protected]}
\author[C. Mascia]{Corrado Mascia} \address[Corrado Mascia]{Dipartimento di Matematica,
Sapienza Universit\`a di Roma (Italy)}
\email{[email protected]}
\author[R.G. Plaza]{Ram\'on G. Plaza} \address[Ram\'on G. Plaza]{Departamento de Matem\'aticas y Mec\'anica,
IIMAS, Universidad Nacional Aut\'onoma de M\'exico (Mexico)}
\email{[email protected]} \thanks{RGP is grateful to DISIM, University of L'Aquila, and to the MathMods Program (Erasmus Mundus) for their hospitality and financial support in academic visits during the Falls of 2012 and 2013, when this research was carried out.}
\author[C. Simeoni]{Chiara Simeoni} \address[Chiara Simeoni]{Laboratoire de Math\'ematiques J.A. Dieudonn\'e,
Universit\'e Nice Sophia-Antipolis (France)}
\email{[email protected]}
\thanks{This work was partially supported by CONACyT (Mexico) and MIUR (Italy), through the MAE Program for Bilateral Research, grant no.\ 146529 and by the Italian Project FIRB 2012 ``Dispersive dynamics: Fourier Analysis and Variational Methods''.}
\keywords{Allen--Cahn equation; Traveling waves; Nonlinear stability}
\maketitle
\begin{abstract} A modification of the parabolic Allen--Cahn equation, determined by the substitution of Fick's diffusion law with a relaxation relation of Ca\-tta\-neo-Maxwell type, is considered. The analysis concentrates on traveling fronts connecting the two stable states of the model, investigating both the aspects of existence and stability. The main contribution is the proof of the nonlinear stability of the wave, as a consequence of detailed spectral and linearized analyses. In addition, numerical studies are performed in order to determine the propagation speed, to compare it to the speed for the parabolic case, and to explore the dynamics of large perturbations of the front. \end{abstract}
\tableofcontents
\section{Introduction}\label{sect:introduction}
The main topic of this paper is to investigate the dynamical behavior of a scalar variable $u$ subject to a transport mechanism of hyperbolic type coupled with a reaction process, driving the unknown $u$ toward one among two different competing stable states. Restricting the attention to a one-dimensional environment and denoting by $v$ the flux of $u$, standard balance of mass provides the relation \begin{equation}\label{balance}
u_t- v_x = f(u), \end{equation} dictating that the quantity $u$ diffuses with flux $v$ and grows/decays according to the choice of the function $f$. The simplest structure for the reaction term $f$ giving raise to two competing stable states is referred to as a {\it bistable form}, meaning that $f$ is smooth and such that for some $\alpha\in(0,1)$, \begin{equation}\label{bistablef}
\begin{aligned}
&f(0)=f(\alpha)=f(1)=0,
&\qquad &f'(0), f'(1)<0,\quad f'(\alpha)>0,\\
&f(u)>0\textrm{ in }(-\infty,0)\cup(\alpha,1),
&\qquad &f(u)<0\textrm{ in }(0,\alpha)\cup(1,+\infty).
\end{aligned} \end{equation} Equivalently, the function $f$ can be considered as the derivative of a double-well potential with wells centered at $u=0$ and $u=1$. A typical reaction function satisfying \eqref{bistablef} which is often found in the literature is the cubic polynomial \begin{equation}\label{cubicf}
f(u)=\kappa\,u(1-u)(u-\alpha),\qquad\kappa>0,\quad \alpha\in(0,1). \end{equation} Reaction functions of bistable type arise in many models of natural phenomena, such as kinetics of biomolecular reactions \cite{Murr02,Mikh94}, nerve conduction \cite{Lieb67,McKe70}, and electrothermal instability \cite{IDRWZB95}, among others.
To complete the model, an additional relation has to be coupled with \eqref{balance}. The standard approach is based on the use of Fick's diffusion law, which consists in the equality $v=u_x$, so that one ends up with the semilinear parabolic equation \begin{equation}\label{parAC}
u_t = u_{xx} + f(u). \end{equation} Equation \eqref{parAC} has appeared in many different contexts and the nomenclature is not uniform. It is known as the bistable reaction-diffusion equation \cite{FifeMcLe77}, the Nagumo equation in neurophysiological modeling \cite{McKe70,NAY62}, the real Ginzburg--Landau for the variational description of phase transitions \cite{MelSch04}, and the Chafee-Infante equation \cite{ChIn74}, among others. In tribute to S. M. Allen and J. W. Cahn, who proposed it in connection with the motion of boundaries between phases in alloys \cite{AlleCahn79}, we call it the {\it (parabolic) Allen--Cahn equation}.
Equation \eqref{parAC} undergoes the same criticisms received by the standard linear diffusion equation, mainly concerning the unphysical infinite speed of propagation of disturbances (see discussion in \cite{Holm93}). Thus, following the modification proposed by Cattaneo \cite{Catt49} (see also Maxwell \cite{Maxw1867}) for the heat equation, it is meaningful to couple \eqref{balance} with an equation stating that the flux $v$ relaxes toward $u_x$ in a time-scale $\tau>0$, namely \begin{equation*}
\tau v_t + v = u_x, \end{equation*} usually named {\it Maxwell--Cattaneo transfer law} (for a complete discussion on its role and significance in the modeling of heat conduction, see \cite{JosePrez89,JosePrez90}). With this choice, the couple density/flux $(u,v)$ solves the hyperbolic system \begin{equation}\label{relAC}
u_t = v_x + f(u),\qquad
\tau v_t = u_x - v. \end{equation} We are interested in studying the dynamics of solutions to system \eqref{relAC}, which we refer to as the {\it Allen--Cahn model with relaxation}. The corresponding Cauchy problem is determined by initial conditions \begin{equation}\label{relACinitial}
u(x,0)=u_0(x),\qquad v(x,0)=v_0(x), \qquad \, x\in\mathbb{R}. \end{equation}
It is to be observed that we can eliminate the variable $v$ by a procedure known in some references as \textit{Kac's trick} (cf. \cite{Hill97,HaMu01,Kac74}): differentiate the first equation in \eqref{relAC} with respect to $t$, and the second with respect to $x$, to obtain the following scalar second-order equation for the variable $u$, \begin{equation}\label{relAConef}
\tau u_{tt} + (1-\tau f'(u)) u_t = u_{xx} + f(u), \end{equation} which we call the {\it one-field equation} determined by \eqref{relAC}. The initial conditions corresponding to \eqref{relACinitial} read \begin{equation*}
u(x,0)=u_0(x),\qquad u_t(x,0)=f(u_0(x))+v_0'(x),\qquad \, x\in\mathbb{R}. \end{equation*} Notice that equation \eqref{relAConef} formally reduces to \eqref{parAC} in the singular limit $\tau\to 0^+$.
We also observe that if we include a diffusion coefficient $\nu>0$, \begin{equation*}
u_t = v_x + f(u),\qquad
\tau v_t = \nu u_x -v, \end{equation*} then last system can be reduced to the form of \eqref{relAC} under the rescaling $x\mapsto x/\sqrt{\nu}$ and $v\mapsto v/\sqrt{\nu}$. Thus, we can consider the case $\nu = 1$ without loss of generality.
The hyperbolic system \eqref{relAC} can be interpreted as a model for a reaction-diffusion process. An intriguing issue is to compare the properties of the usual parabolic reaction-diffusion equation \eqref{parAC} with the ones of its hyperbolic counterpart \eqref{relAC}. This paper pertains to one of the main hallmarks of the Allen--Cahn equation: the presence of stable heterogeneous structures, describing the interaction between the two stable states. Specifically, we examine traveling wave solutions to \eqref{relAC}, i.e. special solutions of the form \begin{equation}\label{twave}
(u,v)(x,t)=(U,V)(\xi),\qquad\qquad \xi=x-ct,\quad c\in\mathbb{R}, \end{equation} with asymptotic conditions \begin{equation}\label{asymptoticstates}
(U,V)(\pm\infty)=(U_\pm,0),\qquad
\textrm{where}\quad U_-:=0,\quad U_+:=1, \end{equation} with the aim of investigating their existence and stability from both an analytical and numerical point of view.
Existence of traveling waves for systems of the form \eqref{relAC} with monostable/logistic reaction terms has been widely investigated \cite{Hade88,Hade94,GildKers13}. The situation for the bistable case is less explored, even if it is more or less known that, under reasonable assumptions, there exist traveling fronts with a uniquely determined propagation speed. For the sake of completeness, in \S \ref{sect:existence} we give a self-contained proof of the following existence result, which provides also some properties crucial to the stability analysis.
\begin{theorem}\label{thm:existence} Let $f$ be such that \eqref{bistablef} holds and let $\tau$ satisfy \begin{equation}\label{smalltau2}
0 < \tau < \tau_m:=1/\sup\limits_{u\in [0,1]} |f'(u)|. \end{equation} Then, there exists a unique value $c_\ast\in \mathbb{R}$ for which system \eqref{relAC} possesses a traveling wave propagating with speed $c_\ast$ and connecting the state $(0,0)$ to $(1,0)$. Moreover,\par (a) the function $U$ is monotone increasing;\par (b) both components $U$ and $V$ are positive and
converge to their asymptotic states exponentially fast; and,\par (c) the speed $c_\ast$ belongs to the interval $(-1/\sqrt{\tau},1/\sqrt{\tau})$ and depends continuously with respect to $\tau\in(0,\tau_m)$. Moreover, it converges to the speed of the (parabolic) Allen--Cahn equation as $\tau\to 0^+$. \end{theorem}
The smallness assumption \eqref{smalltau2} on the relaxation parameter $\tau$ is not sharp and arises as a consequence of the specific choices of the natural variables $u$ and $v$, which generates some obstruction in the course of the proof. A different choice of unknowns could provide a more general result allowing a weaker requirement on $\tau$.
We observe, however, that condition \eqref{smalltau2} is tantamount to the positivity of the damping coefficient in equation \eqref{relAConef}, a condition which is usually imposed in order to ensure that the solution is positive (hyperbolic equations may have negative solutions even with positive initial conditions; cf. \cite{Hade99}). The latter is an important feature for a density solution, for example. Furthermore, although condition \eqref{smalltau2} may seem a pure mathematical assumption, it relates the relaxation time $\tau$ with the typical time scale
$\tau_{\mathrm{reac}} = \inf |u/f(u)| \sim 1/\sup |f'(u)|$ associated to the reaction.
Passing to the stability issue, as for evolution problems on the whole real line defined by autonomous partial differential equations, invariance with respect to translations determines that any traveling wave belongs to a manifold of solutions of the same type with dimension at least equal to one. Thus, small perturbations of a given front are not expected to decay to the front itself, but to the manifold generated by the traveling wave and, at best, to a specific element of such set. Such property, called {\it orbital stability}, holds for the present case. More precisely, we establish the following
\begin{theorem}\label{thm:nonlinstab} Let $f\in C^3$ be such that \eqref{bistablef} holds and $\tau \in [0,\tau_m)$. Let $(U,V)$ be a traveling wave of \eqref{relAC} satisfying \eqref{asymptoticstates} with speed $c_\ast$. Then, there exists $\varepsilon>0$ such that, for any initial data satisfying $(u_0,v_0) - (U,V) \in H^1(\mathbb{R};\mathbb{R}^2)$ with
$|(u_0,v_0)-(U,V)|_{{}_{H^1}}<\varepsilon$, the solution $(u,v)$ to the Cauchy problem \eqref{relAC}--\eqref{relACinitial} satisfies for any $t>0$ \begin{equation*}
|(u,v)(\cdot,t)-(U,V)(\cdot-c_\ast t+\delta)|_{{}_{H^1}}
\leq C|(u_0,v_0)-(U,V)|_{{}_{H^1}}\,e^{-\theta\,t} \end{equation*} for some shift $\delta\in\mathbb{R}$ and constants $C,\theta>0$. \end{theorem}
This statement is the final outcome of some intermediate fundamental steps which are not detailed at this stage for the sake of simplicity of presentation. Actually, Theorem \ref{thm:nonlinstab} is proved by following a well-estabilished approach based on linearization, spectral analysis, linear and nonlinear stability. All of these steps are developed in a complete way, altogether providing a sound rigorous basis to the stability of propagation fronts for the Allen--Cahn model with relaxation. At this point, we find it suitable to mention recent work by Rottmann-Matthes \cite{Rot11,Rot12}, who proves that spectral stability implies orbital stability of traveling waves for a large class of hyperbolic systems (which includes the Allen-Cahn model with relaxation \eqref{relAC}), and provides numerical evidence of spectral stability with spectral gap for the particular case of system \eqref{relAC} by approximating the spectrum of the linearized operator around the wave with periodic boundary conditions, which reproduce the point and essential spectrum on the unbounded domain accurately (cf. \cite{SS00}). These numerical observations, however, do not constitute a proof of stability. Our contribution is a self-contained study of the dynamics of traveling fronts for the particular model \eqref{relAC} (which warrants note because of its importance in the theory of hyperbolic diffusion), as well as a complete analysis of their stability.
Both the existence and the stability analyses are complemented with a numerical study confirming the theoretical results and providing additional relevant information on what should be expected beyond the boundaries of the proved statements. In the first part, we numerically determine the values of the propagation speeds and discuss their relation with the corresponding value in the case of the standard Allen--Cahn equation. Interestingly enough, the model with relaxation exhibits in some regimes fronts that are faster with respect to their corresponding parabolic ones. At the end of the paper, we consider some numerical simulations relative to perturbations of the traveling front, restricting the attention to the case of standard and perturbed Riemann problems. The outcome is a strong numerical evidence that the domain of attractivity of the wave is wider than what described by the stability result of Theorem \ref{thm:nonlinstab} (see Figure \ref{fig:Random2tot}, detailed description in Section \ref{sect:numerics}). The algorithm used in this part is based on a reformulation of \eqref{relAC}, discretized by using a standard finite-difference method with upwinding of the space derivatives.\\
\begin{figure}
\caption{Random initial datum in $(-\ell,\ell)$, $\ell=25$ (squares). Solution profiles for the Allen--Cahn equation with relaxation at time $t=0.5$ (dot), $t=7.5$ (dash), $t=15$ (continuous). For comparison, in the small window, the solution to the parabolic Allen--Cahn equation. For details, see Section \ref{sect:numerics}. }
\label{fig:Random2tot}
\end{figure}
\noindent \textbf{Plan of the paper.} This work is divided into four more sections. Section \ref{sect:existence} deals with the existence of the propagation fronts together with the estabilishment of their main properties, which are essential for completing the stability arguments. It contains the detailed proof of Theorem \ref{thm:existence}, and it is based on a phase-plane analysis that takes advantage of a specific monotonicity property of the system under consideration. The content of Section \ref{sect:linspecstab} is the spectral analysis of the linearized operator around the front. The main result is Theorem \ref{thm:spectrum}, estabilishing the spectral stability of the wave. The proof is based on a perturbation argument at $\tau=0$, combined with a continuation procedure to show that the same spectral structure holds all along the interval $(0,\tau_m)$. The first part of Section \ref{sect:linnonlinstab} deals with the linear stability property. This is consequence of spectral stability because of the hyperbolic nature of system \eqref{relAC} together with an additional resolvent estimate controling the behavior at large frequencies. With such tools at hand, and following the classical ideas of Sattinger developed for the parabolic setting \cite{Satt76}, it is possible to prove the nonlinear stability theorem, taking advantage of the presence of a spectral gap separating the zero eigenvalue from the rest of the spectrum. That is the content of the final part of the Section. Finally, Section \ref{sect:numerics} is entirely devoted to the numerical approximation of \eqref{relAC}. The principal part of the system is diagonalized and a finite-difference upwind approximation is considered. First, we analyze the Riemann problem connecting the two asymptotic states of the front. Such choice is used to compare the asymptotic speed of propagation with the values determined in Section \ref{sect:existence} and to show the numerical evidence of convergence to the front. Then, as large perturbations of the front, we consider initial data which are randomly chosen with some bias on the values at the left and at the right, mimicking an initial configuration which vaguely resembles the transition from $u=0$ to $u=1$. The large-time convergence to the propagation front is evident from the numerical output.\\
\noindent \textbf{Notations.} We use lowercase boldface roman font to indicate column vectors (e.g., $\mathbf{w}$), and with the exception of the identity matrix $I$ we use upper case boldface roman font to indicate square matrices (e.g., $\mathbf{A}$). Elements of a matrix $\mathbf{A}$ (or vector $\mathbf{w}$) are denoted $A_{ij}$ (or $w_j$, respectively). Linear operators acting on infinite-dimensional spaces are indicated with calligraphic letters (e.g., $\mathcal{L}$ and $\mathcal{T}$). For a complex number $\lambda$, we denote complex conjugation by $\overline{\lambda}$ and denote its real and imaginary parts by $\textrm{\rm Re}\, \lambda$ and $\textrm{\rm Im}\, \lambda$, respectively. Complex transposition of vectors or matrices are indicated by the symbol ${}^*$ (e.g., $\mathbf{w}^*$ and $\mathbf{A}^*$), whereas simple transposition is denoted by the symbol $\mathbf{A}^\top$. For a linear operator $\mathcal{L}$, its formal adjoint is denoted by $\mathcal{L}^*$. Given $m\in\mathbb{N}$, the space $H^m(\mathbb{R};\mathbb{C}^n)$ is composed of vector functions from $\mathbb{R}$ to $\mathbb{C}^n$, where each component belongs to the Sobolev space $H^m(\mathbb{R};\mathbb{C})$. It is endowed with the standard scalar product. Finally, we denote derivatives with respect to the indicated argument by `$'$' (e.g., $f'(u)$, $a'(x)$). Partial or total derivatives with respect to spatial and time variables (e.g. $x$ and $t$) are indicated by lower subscript. For the sake of simplicity, we sometimes use the symbol $\partial$ to indicate the latter when appropriate.
\section{Propagating fronts}\label{sect:existence}
Parabolic Allen--Cahn equation \eqref{parAC} supports traveling waves connecting the stable states $u=0$ and $u=1$. Also, the propagating speed and, up to translations, the wave profile, are unique. For special forms of the reaction function $f$, existence of the wave can be proved by determining explicit formulae for speed and profile. For a general bistable $f$, the proof is based on a phase-plane analysis for the corresponding nonlinear ordinary differential system. Uniqueness arises as consequence of the fact that the heteroclinic orbit linking the asymptotic states is a saddle/saddle connection.
For system \eqref{relAC}, it is not anymore possible to find explicit traveling wave solutions even for $f$ of polynomial type. Phase-plane analysis, instead, is a more flexible approach and it can be applied also in the case of the Allen--Cahn equation with relaxation, as it is shown in the sequel. In Subsection \ref{secnumspeed}, we also tackle the problem of the numerical evaluation of the propagation speed in the case of a third-order polynomial function $f$, analyzing the relation between the velocities of the hyperbolic and the parabolic Allen--Cahn equations for different choices of the parameters $\alpha$ and $\tau$.
\subsection{Existence of the traveling wave} Traveling waves of \eqref{relAC} are special solutions of the form $(u,v)(x,t)=(U,V)(\xi)$ where $\xi=x-ct$ and $c$ is a real parameter. The couple $(U,V)$ is referred to as the {\it profile of the wave} and the value $c$ as its {\it propagation speed}. Here, we are concerned with traveling waves satisfying the asymptotic conditions \eqref{asymptoticstates}, so that the corresponding solution describes how the relaxation system resolves the transition from one stable state to the other. In this respect, the value of the speed $c$ is very significant, since it describes how fast and in which direction the switch from value $u=0$ to value $u=1$ is performed.
Existence of a traveling wave for \eqref{relAC} satisfying \eqref{asymptoticstates} can be deduced by the fact that the ordinary differential equation for the profile, obtained by inserting the \textit{ansatz} $u(x,t)=U(x-ct)$ in the one-field equation \eqref{relAConef}, is convertible into the corresponding equation arising in the case of reaction-diffusion models with density-dependent diffusion. Then, one applies a general result proven by Engler \cite{Engl85}, that relates the existence of traveling wave solutions of reaction-diffusion equations with constant diffusion coefficients to the ones of the density-dependent diffusion coefficient case. Considering such path too tangled, we prefer to give an explicit proof of the existence by dealing directly with the system in the original form \eqref{relAC}.
Substituting the form of the traveling wave solutions, we obtain the system of ordinary differential equations \begin{equation}\label{twode0}
cU_\xi+V_\xi +f(U)=0,\qquad
U_\xi+c\tau V_\xi-V=0, \end{equation} to be complemented with the asymptotic boundary conditions \eqref{asymptoticstates}. The value $c$ in \eqref{twode0} is an unknown and its determination is part of the problem.
\begin{proposition}\label{prop:properties} Assume hypothesis \eqref{bistablef} and let $\tau \in [0,\tau_m)$. If $(U,V)$ is a solution to \eqref{twode0} satisfying the asymptotic conditions \eqref{asymptoticstates}, then \par \textrm{\rm (i)} the velocity $c$ has the same sign of $-\int_{0}^{1} f(u)\,du$;\par \textrm{\rm (ii)} there holds \begin{equation}\label{subchar}
c^2 \tau <1. \end{equation} \end{proposition}
\begin{proof} \textrm{\rm (i)} Multiplying the first equation in \eqref{twode0} by $U_\xi$ and using the second, we obtain \begin{equation*}
\begin{aligned}
0&=c\left|{U_\xi}\right|^2+{V_\xi}{U_\xi}+f(U){U_\xi}\\
&=c\left|{U_\xi}\right|^2
+ ({U_\xi}
+c\tau{V_\xi})_\xi {U_\xi}+f(U){U_\xi}\\
&=c\left|{U_\xi}\right|^2+{U_\xi}{U_{\xi\xi}}
-c\tau ( c{U_\xi}+f(U) )_\xi{U_\xi}
+f(U){U_\xi}\\
&=c(1-\tau f'(U))\left|{U_\xi}\right|^2
+(1-c^2\tau){U_\xi}{U_{\xi\xi}}
+f(U){U_\xi}.
\end{aligned} \end{equation*} Thus, denoting by $F$ a primitive of $f$, there holds \begin{equation}\label{waverelation}
\big( \tfrac{1}{2}(1-c^2\tau)\left|{U_\xi}\right|^2+F(U) \big)_\xi
+c(1-\tau f'(U))\left|{U_\xi}\right|^2=0. \end{equation} Integrating in $\mathbb{R}$, we infer the relation \begin{equation*}
c\int_{\mathbb{R}} (1-\tau f'(U))\left|{U_\xi}\right|^2\,d\xi
=F(0)-F(1)=-\int_{0}^{1} f(u)\,du. \end{equation*} Since $\tau<\tau_m$, then $\tau f'(u)<1$ for any $u$ and part (i) follows.
\textrm{\rm (ii)} The case $c=0$ is obvious. Let us assume $c<0$ (the opposite case being similar). Integrating the equality \eqref{waverelation} in $(-\infty,\xi)$, we get \begin{equation*}
\tfrac{1}{2}(1-c^2\tau)\left|{U_\xi}\right|^2
=F(0)-F(U(\xi))-c\int_{-\infty}^{\xi}(1-\tau f'(U))\left|{U_\xi}\right|^2\,d\xi. \end{equation*} Choosing $\xi$ such that $U(\xi)\in(0,\alpha)$, since $F$ is strictly decreasing in $(0,\alpha)$, the right-hand side is strictly positive and thus \eqref{subchar} holds.
\end{proof}
Condition \eqref{subchar} should be regarded as a \textit{subcharacteristic condition}. Indeed, it has a similar interpretation as the corresponding relation for hyperbolic systems with relaxation: the equilibrium wave velocity cannot exceed the characteristic speed of the perturbed wave equation \eqref{relAConef}.
Thanks to \eqref{subchar}, we are allowed to introduce the independent variable $\eta=(1-c^2\tau)^{-1}\xi$, so that system \eqref{twode0} becomes \begin{equation}\label{twode1}
U_\eta=\phi(U,V):=c\tau f(U)+V,\qquad
V_\eta=\psi(U,V):=-f(U)-cV. \end{equation} Departing from a detailed description of the unstable and stable manifold of the singular points $(0,0)$ and $(1,0)$, respectively, it is possible to show the existence of a saddle/saddle connection between the asymptotic states required by \eqref{asymptoticstates}. The existence result is based on the analysis of the limiting regimes $c\to \pm 1/\sqrt{\tau}$ of the system \eqref{twode1} and on their (monotone) variations for the values in between, using the notion of a {\it rotated vector field} (cf. \cite{Perk93}).
\begin{proof}[Theorem \ref{thm:existence}] 1. The linearization of \eqref{twode1} at $(\bar U,0)$ with $f(\bar U)=0$ is described by the jacobian matrix calculated at $(\bar U,0)$ \begin{equation*}
\frac{\partial(\phi,\psi)}{\partial(u,v)}
=\begin{pmatrix} c\tau f'(\bar U) & \;\;1\\ - f'(\bar U) & \;\; -c \end{pmatrix} \end{equation*} whose determinant is $(1-c^2\tau)f'(\bar U)$. In particular, if $f'(\bar U)<0$, the singular point $(\bar U,0)$ is a saddle. The eigenvalues are the roots of the polynomial \begin{equation*}
p(\mu)=\mu^2+c(1-\tau f'(\bar U))\mu+(1-c^2\tau)f'(\bar U) \end{equation*} and they are given by \begin{equation*}
\mu_\pm(\bar U)=- \tfrac{1}{2}c(1-\tau f'(\bar U))\pm
\tfrac{1}{2}\sqrt{c^2(1-\tau f'(\bar U))^2-4 f'(\bar U)} \end{equation*} with corresponding eigenvectors $\mathbf{r}_\pm(\bar U)=(1,\mu_\pm(\bar U)-c\tau f'(\bar U))^\top$.
For later use, let us note that, as a consequence of \eqref{smalltau2}, \begin{equation}\label{lateruse}
\mu_+(0)-c\tau f'(0)>-\sqrt{\tau}\,f'(0)
\quad\textrm{and}\quad
\mu_-(1)-c\tau f'(1)<\sqrt{\tau}\,f'(1), \end{equation} for $c\in(-1/\sqrt{\tau},1/\sqrt{\tau})$. Indeed, for $f'(\bar U)<0$, there holds \begin{equation*}
p\bigl((c\tau\pm\sqrt{\tau})f'(\bar U)\bigr)
=\sqrt{\tau}\left(\frac{1}{\sqrt{\tau}}\pm c\right)(1-\tau f'(\bar U))f'(\bar U)<0, \end{equation*} so that the values $(c\tau\pm\sqrt{\tau})f'(\bar U)$ belong to the interval $(\mu_-(\bar U),\mu_+(\bar U))$. \vskip.25cm
2. Given $c\in (-1/\sqrt{\tau},1/\sqrt{\tau})$, let us denote by ${\mathbb{U}}_{0}(c)$ the unstable manifold of the singular point $(0,0)$ and by ${\mathbb{S}}_{1}(c)$ the stable manifold of $(1,0)$. Also, let ${\mathbb{U}}_{0}^+(c)$ be the intersection of ${\mathbb{U}}_{0}(c)$ with the strip $[0,\alpha]\times\mathbb{R}$ and let ${\mathbb{S}}_{1}^-(c)$ the intersection of ${\mathbb{S}}_{1}(c)$ with the strip $[\alpha,1]\times\mathbb{R}$. Such sets are graphs of appropriate solutions to the first order equation \begin{equation}\label{traj}
\frac{dV}{dU}=-\frac{f(U)+cV}{c\tau f(U)+V}. \end{equation} Thanks to \eqref{lateruse}, ${\mathbb{U}}_{0}^+(c)$ lies above the graph of the function $v=-\sqrt{\tau} f(u)$ in a neighborhood of $(0,0)$. In addition, for $u\in(0,\alpha)$, there holds \begin{equation*}
(\phi,\psi)\cdot(\sqrt{\tau}\,f'(u),1)\bigr|_{v=-\sqrt{\tau} f(u)}
=-\left(1-c\sqrt{\tau}\right)(1-\tau f'(u))f(u)>0 \end{equation*} showing that no trajectories may trespass the graph of the function $v=-\sqrt{\tau} f(u)$ for $u\in(0,\alpha)$. In particular, the set ${\mathbb{U}}_{0}^+(c)$ lies above the graph $v=-\sqrt{\tau} f(u)$ and hits the line $u=\alpha$ for a given value $v_0(c)\in(0,+\infty)$. Similar considerations show that ${\mathbb{S}}_{1}^-(c)$ stays above the graph $v=\sqrt{\tau} f(u)$ and touches the straight line $u=\alpha$ for a given value $v_1(c)\in(0,+\infty)$.
3. To determine how the unstable/stable manifolds change with the parameter $c$, let us observe that \begin{equation*}
\begin{aligned}
(\phi,\psi,0)^\top\land(\partial_c \phi,\partial_c\psi,0)^\top
&=\det\begin{pmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
c\tau f(U)+V & -f(U)-cV & 0 \\
\tau f(U) & -V & 0
\end{pmatrix}\\
&=\bigl(\tau f(U)^2-V^2\bigr)\mathbf{k}.
\end{aligned} \end{equation*} Thus, the vector field $(\phi,\psi)$ defining \eqref{twode1} rotates clockwise in the region
$\{(u,v)\,:\,v\geq \sqrt{\tau}|f(u)|\}$ as the parameter $c$ increases. As a consequence, the curves describing ${\mathbb{U}}_0^+(c)$ and ${\mathbb{S}}_1^-(c)$ rotate clockwise when $c$ increases and the functions $c\mapsto v_0(c)$ and $c\mapsto v_1(c)$ are, respectively, strictly monotone decreasing and strictly monotone increasing.
4. To conclude the existence of the orbit, we analyze the behavior of \eqref{twode1} in the limiting regimes $c\to \pm 1/\sqrt{\tau}$. For $c=-1/\sqrt{\tau}$, the system reduces to \begin{equation*}
U_\eta=V-\sqrt{\tau} f(U),\qquad
V_\eta=\frac{1}{\sqrt{\tau}}\Big(V-\sqrt{\tau} f(U)\Big). \end{equation*} In particular, all the trajectories lie along straight lines of the form \begin{equation*}
v=\frac{u}{\sqrt{\tau}}+C,\qquad\qquad C\in\mathbb{R}, \end{equation*} and converge, as $t\to -\infty$, to the unique intersection between the invariant straight line and the graph of the function $\sqrt{\tau} f$ (see Fig. \ref{fig:pplaneminus}, depicting the $(U,V)$ plane for the particular case of the cubic reaction function $f$). \begin{figure}\label{fig:pplaneminus}
\end{figure} The unstable manifold ${\mathbb{U}}_{0}(c)$ of the singular point $(0,0)$ is the straight line $v=u/\sqrt{\tau}$, while the center-stable manifold ${\mathbb{S}}_{1}(c)$ of $(1,0)$ is the graph of the function $\sqrt{\tau} f$. In particular, there holds \begin{equation*}
{\mathbb{U}}_{0}(-1/\sqrt{\tau})\bigr|_{U=\alpha}=\alpha(1,1/\sqrt{\tau}\,),\qquad
{\mathbb{S}}_{1}(-1/\sqrt{\tau})\bigr|_{U=\alpha}=\alpha(1, 0), \end{equation*} that gives \begin{equation*}
v_0(-1/\sqrt{\tau})=\alpha/\sqrt{\tau},\qquad v_1(-1/\sqrt{\tau})=0. \end{equation*} The situation for $c=1/\sqrt{\tau}$ is similar, yielding \begin{equation*}
v_0(1/\sqrt{\tau})=0,\qquad
v_1(1/\sqrt{\tau})=(1-\alpha)/\sqrt{\tau}. \end{equation*} The conclusion is at hand, since \begin{equation*}
(v_1-v_0)(-1/\sqrt{\tau})=-\alpha/\sqrt{\tau}<0
<(1-\alpha)/\sqrt{\tau}=(v_1-v_0)(1/\sqrt{\tau}), \end{equation*} implying, together with the monotonicity of $v_0$ and $v_1$, that there exists a unique value $c_\ast$ such that $v_1(c_\ast)=v_0(c_\ast)$, and then system \eqref{twode1} possesses a heteroclinic orbit.
5. To verify the monotonicity of the component $U$, we note that from system \eqref{twode1} one has $U_\eta = c\tau f(U) + V$. Let us suppose that $c > 0$. Assume, by contradiction, that $U_\eta(\eta_*) = 0$ for some $\eta_* \in \mathbb{R}$. Therefore $V_* = - c\tau f(U_*)$, where $V_* = V(\eta_*)$, $U_* = U(\eta_*)$. Since $V$ is positive for all $\eta \in \mathbb{R}$, this implies that $f(U_* ) < 0$ and, necessarily, that $U_* \in (0,\alpha)$. The subcharacteristic condition \eqref{subchar}, however, yields \begin{equation*}
0 < -c \tau f(U_*) < - \sqrt{\tau} f(U_*), \end{equation*} which is a contradiction with the fact that the trajectory $(U,V)$ lies above the graph of the function $v = - \sqrt{\tau} f(u)$ for $u \in (0,\alpha)$. The case $c < 0$ is analogous and a similar argument for $U_* \in (\alpha,1)$ applies. Therefore, $U_\eta$ never changes sign along the trajectory. Since $U$ connects $U(-\infty) = 0$ with $U(+\infty) = 1$, the function is strictly increasing and $U_\eta > 0$ for all $\eta \in \mathbb{R}$. The subcharacteristic condition guarantees that the same statement holds for the original profile in the $\xi$ variable, that is, $U_\xi > 0$.
6. Exponential decay of the profile is a consequence of the hyperbolicity of the non-degenerate end-points $(0,0)$ and $(1,0)$. Indeed, rewriting system \eqref{twode0} (in the original moving variable $\xi$) as \begin{equation*}
U_\xi = (1 - c^2 \tau)^{-1}(c\tau f(U) + V),\qquad
V_\xi = -(1 - c^2 \tau)^{-1}(\tau f(U) + cV), \end{equation*} and linearizing the right hand side around $(\bar U,0)$, one readily notes that its eigenvalues are the same as the eigenvalues of the linearization of \eqref{twode1} at the same point multiplied by a factor $(1 - c^2 \tau)^{-1}$. The positive (unstable) eigenvalue at $(0,0)$ is \begin{equation*}
\mu_u = \tfrac{1}{2}(1 - c^2 \tau)^{-1}\big( -c(1-\tau f'(0))
+ \sqrt{c^2(1-\tau f'(0))^2 - 4 f'(0)}\big) \, > 0, \end{equation*} and the orbit decays to $(0,0)$ as $\xi \to - \infty$ with exponential rate \begin{equation*}
|(U,V)(\xi)| \leq C \exp (\mu_u \xi). \end{equation*} Likewise, the negative (stable) eigenvalue at $(1,0)$ is \begin{equation*}
\mu_s = \tfrac{1}{2}(1 - c^2 \tau)^{-1}\big( -c(1-\tau f'(1))
- \sqrt{c^2(1-\tau f'(1))^2 - 4 f'(1)}\big) \, < 0, \end{equation*} and the orbit decays to $(1,0)$ as $\xi \to + \infty$ with rate \begin{equation*}
|(U,V)(\xi)| \leq C \exp (-|\mu_s| \xi). \end{equation*}
Setting $\nu = \nu(\tau) := \min \{\mu_u, |\mu_s|\} > 0$, we find that \begin{equation*}
\left| \frac{d^j}{d\xi^j}(U-U_\pm,V)(\xi)\right| \leq C \exp (-\nu |\xi|),
\qquad \xi\in\mathbb{R}, \end{equation*} for some constant $C > 0$ and with $j = 0,1,2$.
7. Finally, we have to show continuity of the speed $c_\ast$ with respect to $\tau$, as stated in (c). To this aim, denoting explicitly the dependence on $\tau$, let us consider the function \begin{equation*}
\delta v(c,\tau):=v_0(c,\tau)-v_1(c,\tau) \end{equation*} with $v_0$ and $v_1$ defined at Step 2. For any $\tau$, the value $c_\ast$ is determined implicitly by the equality $\delta v(c,\tau)=0$. Also, smooth dependence with respect to parameters $c$ and $\tau$ of the system \eqref{twode1} implies that the function $\delta v$ is continuous with respect to its variables. Moreover, since $v_0$ is monotone decreasing and $v_1$ monotone increasing as functions of $c$ (see the end of Step 3.), the function $\delta v$ is monotone decreasing with respect to $c$.
Fix $\tau_0\in(0,\tau_m)$ and $\eta>0$. Then, there holds \begin{equation*}
\delta v(c_\ast(\tau_0)+\eta,\tau_0)<\delta v(c_\ast(\tau_0),\tau_0)=0
<\delta v(c_\ast(\tau_0)-\eta,\tau_0) \end{equation*} Since $\delta v$ is continuous, for any $\tau$ in a neighborhood of $\tau_0$, there holds \begin{equation*}
\delta v(c_\ast(\tau_0)+\eta,\tau)<0<\delta v(c_\ast(\tau_0)-\eta,\tau), \end{equation*} which gives, for $\delta v(\tau,c_\ast(\tau))=0$ and the monotonicity of $\delta v$, \begin{equation*}
c_\ast(\tau_0)-\eta<c_\ast(\tau)<c_\ast(\tau_0)+\eta. \end{equation*} The property relative to the limiting behavior as $\tau\to 0$ can be proved in the same way, observing that the dependence of the differential system with respect to $\tau$ is smooth and that the heteroclinic orbit relative to the classical Allen--Cahn case can be obtained by the same procedure.
\end{proof}
\subsection{Numerics of the propagating speed}\label{secnumspeed}
In the case of the Allen--Cahn equation \eqref{parAC} with $f$ given by \eqref{cubicf}, it is possible to determine an explicit form for the speed of the propagation front, namely \begin{equation}\label{parACspeed}
c_\ast^{0}:=\sqrt{\frac{2}{\kappa}}\left(\alpha-\frac12\right), \end{equation} and for the corresponding profile, which is given by a hyperbolic tangent. For the Allen--Cahn system with relaxation, however, finding analogous explicit formulas is awkward if not impossible. Some attempts to derive approximated expressions applying a series expansion method have been performed in \cite{Abdu04,Fahm07,VanGVajr10} with very restricted success.
Here, fixed the reaction strength $\kappa=1$, we address the problem of computing numerically the value of $c_\ast^\tau$, the speed of propagation corresponding to the relaxation parameter $\tau>0$, and we discuss its dependency with respect to the parameters $\alpha, \tau$ and its relation with the limiting value $c_\ast^0$. \begin{figure}
\caption{Graph of the map $\alpha\mapsto c_\ast^\tau$ for $\tau=2$ (dashed line), $\tau=4$ (dotted line), $\tau=8$ (continuous line). The thin straight line corresponds to $c_\ast^{0}$ (parabolic Allen--Cahn equation). }
\label{fig:hetRelcVal}
\end{figure}
To determine reliable approximations of the value $c_\ast^\tau$, we first evaluate numerically the functions $v_0=v_0(c)$ and $v_1=v_1(c)$, defined in the proof of Theorem \ref{thm:existence}. In view of that, we compute the solutions $V_0=V_0(U)$ and $V_1=V_1(U)$ to \eqref{traj} with initial conditions \begin{equation*}
V_0(\delta)=\delta\bigl(\mu_+(0)-c\tau f'(0)\bigr),\qquad
V_1(1-\delta)=-\delta\bigl(\mu_-(1)-c\tau f'(1)\bigr), \end{equation*} for $\delta>0$ small (in the following computations, we actually choose $\delta=10^{-8}$) and we set \begin{equation*}
v_0(c):=V_0(\alpha),\qquad v_1(c):=V_1(\alpha). \end{equation*} As explained in the proof of Theorem \ref{thm:existence}, the function $c\mapsto v_0(c)-v_1(c)$ is monotone decreasing and it has a single zero, so that, by following a standard bisection procedure, we find an approximated value for the unique zero $c_\ast^\tau$ of the difference $v_0-v_1$. Some of the numerically computed speeds for different values of $\alpha$ and $\tau$ can be found in Table \ref{tab:numerspeeds}.
\begin{table}\centering \caption{Numerically computed speeds $c_\ast^\tau=c_\ast^\tau(\alpha)$ for different values of $\alpha$ and $\tau$. The column $\tau=0$ gives the values of the speed for the parabolic Allen--Cahn equation. The presence of parenthesis indicates that condition \eqref{smalltau2} is not satisfied. \label{tab:numerspeeds}}
{\begin{tabular}{@{}r|c|c|c|c|c|c|c|c|c|c|@{}} $\alpha$ & $\tau=$0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline
0.6 & 0.14 & 0.16 & 0.17 & (0.20) & (0.22) & (0.25) & (0.27) & (0.30) & (0.31) \\
0.7 & 0.28 & 0.31 & 0.33 & 0.35 & (0.37) & (0.38) & (0.38) & (0.37) & (0.35) \\
0.8 & 0.42 & 0.44 & 0.46 & 0.46 & 0.45 & (0.43) & (0.41) & (0.38) & (0.35) \\
0.9 & 0.57 & 0.56 & 0.55 & 0.52 & 0.49 & 0.45 & 0.41 & 0.38 & 0.35 \end{tabular}} \end{table}
Applying a variational approach, explicit estimates from below and from above for the value of the speed $c_\ast^\tau$ have been determined by M\'endez \textit{et al.} \cite{MendFortFarj99}. As already noted by these authors, the estimate from below \begin{equation}\label{estbelow}
c_\ast^\tau(\alpha)\geq
c_{{\textrm{low}}}^\tau(\alpha)
:=\frac{\sqrt{2}(\alpha-1/2)}
{\sqrt{(1-\frac{1}{5}(1-2\alpha+2\alpha^2)\tau)^2+\frac12\tau(1-2\alpha)^2}} \end{equation} manifests an excellent agreement with the numerically computed values of the speed.
The value $c_\ast^\tau$ depends on both $\tau$ and $\alpha$. For fixed $\tau$, as a function of $\alpha$ alone, $c_\ast^\tau$ is oddly symmetric with respect to $\alpha=1/2$. Moreover, numerical simulations show that $c_\ast^\tau$ is monotone increasing and $S$-shaped (see Fig. \ref{fig:hetRelcVal}).
\begin{figure}
\caption{Graph of the map $\alpha\mapsto (c_\ast^\tau-c_\ast^0)/|c_\ast^0|$ for $\alpha\in(0.5,1)$ measuring the relative variation of the speed passing from the parabolic Allen--Cahn equation to its hyperbolic counterpart: $\tau=2$ (dashed), $\tau=4$ (dotted), $\tau=6$ (continuous). }
\label{fig:hetRelRela}
\end{figure}
In some regimes of the parameter $\alpha$ (depending on the size of $\tau$), the speed for the hyperbolic model is larger than the one for the parabolic Allen--Cahn equation. Such behavior is different with respect to the {\it damped Allen--Cahn equation}, obtained by solely adding the inertial term $\tau u_{tt}$ to equation \eqref{parAC} \begin{equation*}
\tau u_{tt} + u_t = u_{xx} +f(u), \end{equation*} (see \cite{GallJoly09} and the references therein). Indeed, for such equation, the traveling wave equation can be directly reduced to the one of the parabolic case by a simple rescaling of the independent variable. Such procedure furnishes an explicit relation between the speed of propagation of the fronts with and without the inertial term, that is \begin{equation*}
c^\tau_{\text{damped}} = \frac{c^0_\ast}{\sqrt{1+\tau (c^0_\ast)^2}}. \end{equation*} From this relation, it is evident that such hyperbolic front is always slower with respect to the corresponding parabolic one.
Coming back to the case of the Allen--Cahn equation with relaxation, the discrepancy between $c_\ast^\tau$ and $c_\ast^0$ is described by the function of the
relative variation $\alpha\mapsto (c_\ast^\tau-c_\ast^0)/|c_\ast^0|$, whose behavior is depicted in Fig. \ref{fig:hetRelRela} for different values of $\tau$. For large values of $\tau$, the relative increase of the front can be particularly relevant (as an example, about 150\% for $\tau=6$ and values of $\alpha$ close to $0.5$). The limiting value of the relative variation as $\alpha\to 1/2$ can be approximated by using the estimate \eqref{estbelow}, that gives \begin{equation*}
\lim_{\alpha\to 1/2^+} \frac{c_{\ast}^\tau-c_\ast^0}{|c_\ast^0|}
\approx
\lim_{\alpha\to 1/2^+} \frac{c_{{\textrm{low}}}^\tau-c_\ast^0}{|c_\ast^0|}
=\frac{\tau}{10-\tau} \end{equation*} in good agreement with the numerical values expressed in Fig. \ref{fig:hetRelRela}. It is worth noting the presence of a singularity for $\tau\to 10^-$ that is outside the range of smallness on the parameter $\tau$ that we are considering. We do consider the behavior for large relaxation times $\tau$ a very interesting issue to be analyzed in detail in the future.
Complementary information is provided by the analysis of the value of $c_\ast^\tau$ as a function of $\tau$ for fixed $\alpha$ (see Fig. \ref{fig:hetRel4}). The numerical evidence shows that the speed function has different monotonicity properties depending on the chosen value $\alpha$, passing through the monotone increasing case ($\alpha=0.6$), increasing-decreasing ($\alpha=0.7$ and $0.8$), monotone decreasing ($\alpha=0.9$).
It also worthwhile to observe the relation between the propagation speed $c_\ast^\tau$
and the characteristic speed $1/\sqrt{\tau}$. As stated in Proposition \ref{prop:properties}, $|c_\ast^\tau|$ is always strictly smaller than $1/\sqrt{\tau}$, as it is observed in Fig. \ref{fig:hetRel4}, where the graph of the characteristic speed is depicted by a thick (gray, in the online version) band, lying above the speed curves for all of the values $\alpha$. For large values of $\tau$, the hyperbolic structure of the principal part of \eqref{relAC} becomes crucial and the propagation speed of the front tends to one of the characteristic values $\pm 1/\sqrt{\tau}$ (except for the case of zero speed). \begin{figure}
\caption{Graph of the map $\tau\mapsto c_\ast^\tau(\alpha)$ for different values of $\alpha$: continuous line $\alpha=0.6$, dashed line $\alpha=0.7$, dash-dotted line $\alpha=0.8$, dotted line $\alpha=0.9$. The thick (gray; online version) band corresponds to the graph of the characteristic speed $\tau\mapsto 1/\sqrt{\tau}$; as $\tau$ increases, the speed $c_\ast^\tau(\alpha)$ tends to $1/\sqrt{\tau}$ from below as a consequence of the emergent r\^ole of the principal part of the system \eqref{relAC}.}
\label{fig:hetRel4}
\end{figure}
\section{Linearization and spectral stability}\label{sect:linspecstab}
This section is devoted to establishing both the equations for the perturbation of the traveling wave and the corresponding spectral stability problem. For stability purposes, it is often convenient to recast the system of equations \eqref{relAC} in a moving coordinate frame. For fixed $\tau \in [0,\tau_m)$, let $c := c_*(\tau)$ be the unique wave speed of Theorem \ref{thm:existence}. Rescaling the spatial variables as $x\mapsto x - c t$, we obtain the nonlinear system \begin{equation}\label{relACmove}
\begin{aligned}
u_t &= cu_x + v_x + f(u),\\
\tau v_t &= u_x + c \tau v_x -v.
\end{aligned} \end{equation} From this point on and for the rest of the paper the variable $x$ will denote the moving (galilean) variable $x - ct$. With a slight abuse of notation, traveling wave solutions to the original system \eqref{relAC} transform into stationary solutions $(U,V)(x)$ of the new system \eqref{relACmove} and satisfy the ``profile'' equations \begin{equation}\label{profileq}
cU_x + V_x + f(U)= 0,\qquad
U_x + c\tau V_x - V = 0, \end{equation} together with the asymptotic limits \eqref{asymptoticstates}. Furthermore, the convergence is exponential, \begin{equation}\label{expodecay}
\left| \partial_x^j (U-U_\pm,V)(x)\right| \leq C \exp (-\nu |x|),
\qquad x \in \mathbb{R} \end{equation} for some $C,\nu > 0$ and $j = 0,1,2$. It is to be noted that $U_x, V_x \in H^1(\mathbb{R};\mathbb{R})$. Moreover, by a bootstrapping argument and the second equation in \eqref{profileq}, it is easy to verify that $U_{x}, V_{x} \in H^2(\mathbb{R};\mathbb{R})$.
\subsection{Perturbation equations and the spectral problem}
Consider a solution to the nonlinear system \eqref{relACmove} of the form $(u,v)(x,t) + (U,V)(x)$, being $u$ and $v$ perturbation variables. Upon substitution (and using the profile equations \eqref{profileq}) we arrive at the following nonlinear system for the perturbation: \begin{equation}\label{nlsystw}
\begin{aligned}
u_t &= c u_x + v_x + f(u+U) - f(u),\\
\tau v_t &= u_x + c\tau v_x -v.
\end{aligned} \end{equation} The standard strategy to prove stability of the traveling wave is based on linearizing system \eqref{nlsystw} around the wave. The subsequent analysis can be divided into three steps: the spectral analysis of the resulting linearized operator, the establishment of linear stability estimates for the associated semigroup, and the nonlinear stability under small perturbations to solutions to \eqref{nlsystw}. Thus, we first linearize last system around the profile solutions $(U,V)$. The result is \begin{equation*}
\begin{aligned}
u_t &= c u_x + v_x + f'(U)u,\\
\tau v_t &= u_x + c\tau v_x -v.
\end{aligned} \end{equation*} Specializing these equations to perturbations of the form $e^{\lambda t}(\hat u,\hat v)$, where $\lambda \in \mathbb{C}$ is the spectral parameter and $(\hat u,\hat v)(x)$ belongs to an appropriate Banach space $X$, we obtain a naturally associated spectral problem \begin{equation*}
\begin{aligned}
\lambda \hat u &= c \hat u_x + \hat v_x + f'(U)\hat u,\\
\lambda \tau \hat v &= \hat u_x + c\tau \hat v_x - \hat v.
\end{aligned} \end{equation*} With a slight abuse of notation we denote again $(\hat u, \hat v)^\top = (u,v)^\top \in X$. Henceforth, for each $\tau \in (0,\tau_m)$ we are interested in studying the spectral problem \begin{equation}\label{spectproblem}
\mathcal{L}^\tau \begin{pmatrix}
u\\ v
\end{pmatrix} = \lambda \begin{pmatrix}
u\\ v
\end{pmatrix}, \qquad \begin{pmatrix}
u\\ v
\end{pmatrix} \in D \subset X, \end{equation} where $\mathcal{L}^\tau$ denotes the first order differential operator determined by \begin{equation} \label{defLtau}
\mathcal{L}^\tau = - \mathbf{B}^{-1}\left( \mathbf{A} \, \partial_x + \mathbf{C}(x) \right), \end{equation} with domain $D$ in $X$, and where \begin{equation}\label{defABC}
\mathbf{A}=\begin{pmatrix} -c & -1 \\ -1 & -c\tau \end{pmatrix},\qquad
\mathbf{B}=\begin{pmatrix} 1 & 0 \\ 0 & \tau \end{pmatrix},\qquad
\mathbf{C}(x)=\begin{pmatrix} -a(x) & 0 \\ 0 & 1 \end{pmatrix}, \end{equation} and \begin{equation*}
a(x) := f'(U). \end{equation*} In this analysis we choose the perturbation space as $X = L^2(\mathbb{R};\mathbb{C}^2)$, with dense domain $D = H^1(\mathbb{R};\mathbb{C}^2)$, which corresponds to the study of stability under \textit{localized} perturbations. In this fashion, we obtain a family of closed, densely defined first order operators in $L^2(\mathbb{R};\mathbb{C}^2)$, parametrized by $\tau \in (0,\tau_m)$.
It is to be observed that $\mathcal{L}^\tau$ is not defined for $\tau = 0$, where $\mathbf{B}$ becomes singular. Formally, in the limit $\tau \to 0^+$ system \eqref{nlsystw} can be written (after substitution) as the scalar perturbation equation for the parabolic Allen-Cahn (or Nagumo) front: \begin{equation*}
u_t = u_{xx} + c_0 u_x + f(U^0+u) - f(U^0), \end{equation*}
where $U^0$ denotes the unique (up to translations) traveling wave solution to the parabolic Allen-Cahn equation \eqref{parAC}, traveling with speed $c_0 = c_{|\tau = 0}$. Its linearization leads to the well-studied spectral problem for the operator $\mathcal{L}_0 : H^2(\mathbb{R};\mathbb{C}) \rightarrow L^2(\mathbb{R};\mathbb{C})$, defined by \begin{equation*}
\mathcal{L}_0 u := u_{xx} + c_0 u_x + \overline{a}(x)u \, = \, \lambda u,
\quad u \in H^2(\mathbb{R};\mathbb{C}), \end{equation*} where $\overline{a}(x) = f'(U^0)$. The stability of the parabolic traveling front is a well-known fact: it was first established by Fife and McLeod \cite{FifeMcLe77} using maximum principles. The spectral analysis of the operator $\mathcal{L}_0$ can be found in \cite[pp.128--131]{He81}, and in \cite[Chapter 2]{KaPro13}. The following proposition summarizes the spectral stability properties of the parabolic front.
\begin{proposition}[cf. \cite{He81,KaPro13}]\label{prop:tau0} There exists $\omega_0>0$ such that the spectrum $\sigma(\mathcal{L}_0)$ of the operator $\mathcal{L}_0$ can be decomposed as \begin{equation*}
\sigma(\mathcal{L}_0)=\{0\}\cup\sigma_-^{(0)}, \end{equation*} where $\lambda = 0$ is an (isolated) eigenvalue with algebraic multiplicity equal to one and eigenspace generated by $U^0_x \in L^2(\mathbb{R};\mathbb{C})$, and $\sigma_-^{(0)}$ is contained in the half-space $\{\lambda\in\mathbb{C}\,:\,\textrm{\rm Re}\, \, \lambda\leq -\omega_0\}$. \end{proposition}
Since the operators $\mathcal{L}^\tau$ have been defined on the appropriate spaces, the standard definitions for the resolvent $\rho(\mathcal{L}^\tau)$ and spectrum $\sigma(\mathcal{L}^\tau)$ follow (see \cite{EE87,Kat80} and Section \ref{secfirstorder} below). Our goal is to establish the spectral stability for the family of operators \eqref{defLtau}, for parameter values $\tau \in (0,\tau_m)$, by proving a result analogous to Proposition \ref{prop:tau0}:
\begin{theorem}[Spectral stability]\label{thm:spectrum} For each $\tau \in (0,\tau_m)$, there exists $\omega_0(\tau)>0$ such that the spectrum $\sigma(\mathcal{L}^\tau)$ of the operator $\mathcal{L}^\tau$ can be decomposed as \begin{equation*}
\sigma(\mathcal{L}^\tau)=\{0\}\cup\sigma_-^{(\tau)}, \end{equation*} where $\lambda = 0$ is an (isolated) eigenvalue with algebraic multiplicity equal to one and eigenspace generated by $(U_x,V_x) \in D(\mathcal{L}^\tau)$, and $\sigma_-^{(0)}$ is contained in the half-space $\{\lambda\in\mathbb{C}\,:\,\textrm{\rm Re}\,\lambda\leq -\omega_0(\tau)\}$. \end{theorem}
The approach to prove Theorem \ref{thm:spectrum} is based on rewriting the spectral problem as a first order system with the eigenvalue as a parameter. Then, applying a general theorem for convergence of approximate flows (cf. \cite{PZ1}), we show that the spectral stability for $\tau = 0$ persists for $\tau \sim 0^+$. By a continuation argument, we extend the result to the whole parameter domain $\tau \in (0,\tau_m)$.
\subsection{Reformulation of the spectral problem}\label{secfirstorder}
Set $\tau \in (0,\tau_m)$. Component-wise the spectral problem \eqref{spectproblem} can be written as \begin{equation}\label{evalue}
\begin{aligned}
c{u_x} + {v_x} +(a(x)-\lambda) u &= 0,\\
{u_x} + c \tau {v_x} -(1+\tau\lambda)v &= 0.
\end{aligned} \end{equation} Just like for the original nonlinear Allen-Cahn model with relaxation \eqref{relAC}, the $v$ variable can be eliminated to obtain a second order equation for $u$ (a spectral version of Kac's trick): multiply the first equation by $c\tau$, substract it from the second and differentiate with respect to $x$. The result is the following second order spectral equation, \begin{equation}\label{2ndorderu}
(1-c^2\tau){u_{xx}} + c \big( 1+\tau(2\lambda - a) \big) {u_x}
+ \big( (1+\tau \lambda)(a(x)-\lambda) - c\tau a'(x) \big) u = 0. \end{equation} Following Alexander, Gardner and Jones \cite{AGJ90}, we recast the scalar spectral equation \eqref{2ndorderu} as a first order system of the form \begin{equation}\label{firstorders}
\mathbf{w}_x = \mathbf{A}^\tau (x,\lambda)\mathbf{w}, \end{equation} where $\mathbf{w}=(u,u_x)^\top$ and, for $a=a(x)$ and $a'=a'(x)$, \begin{equation}\label{coeffA}
\mathbf{A}^\tau (x,\lambda) := \frac{1}{1 - c^2 \tau}
\begin{pmatrix} 0 & & 1-c^2\tau \\
c\tau a' + (1+\tau\lambda)(\lambda - a) & & \;c(\tau a -1-2\tau\lambda)
\end{pmatrix}. \end{equation} Observe that the coefficient matrices \eqref{coeffA} can be written as \begin{equation*}
\mathbf{A}^\tau (x,\lambda) = \mathbf{A}^\tau_0(x) + \lambda \mathbf{A}^\tau_1(x) + \lambda^2 \mathbf{A}^\tau_2(x), \end{equation*} where \begin{equation*} \begin{aligned}
\mathbf{A}^\tau_0(x) &:= \mathbf{A}^\tau(x,0) = \frac{1}{1 - c^2 \tau}\begin{pmatrix} 0 & & 1-c^2\tau \\ c\tau a'(x) - a(x) & & \; c(\tau a(x) - 1)
\end{pmatrix},\\
\mathbf{A}^\tau_1(x) &:= \frac{1}{1 - c^2 \tau}\begin{pmatrix} 0 & \;0 \\ 1 - \tau a(x) & \;-2c\tau
\end{pmatrix}, \\
\mathbf{A}^\tau_2(x) &:= \frac{\tau}{1 - c^2 \tau} \begin{pmatrix}
0 & \;0 \\ 1 & \; 0
\end{pmatrix}. \end{aligned} \end{equation*} Since $c = c(\tau)$ is a continuous function of $\tau \in [0,\tau_m)$, we conclude that $\mathbf{A}^\tau (\cdot,\lambda)$ is a function from $(\tau, \lambda) \in [0,\tau_m) \times \mathbb{C}$ to $L^\infty(\mathbb{R};\mathbb{R}^{2\times 2})$, continuous in $\tau$ and analytic (second order polynomial) in $\lambda$.
It is a well-known fact (see \cite{San02,AGJ90} and the references therein) that an alternate but equivalent definition of the spectra and the resolvent sets associated to the spectral problem \eqref{spectproblem} can be expressed in terms of the first order systems \eqref{firstorders}. Consider the following family of linear, closed, densely defined operators \begin{equation} \label{defofTau}
\begin{aligned}
\mathcal{T}^\tau (\lambda) &: \overline{D} = H^1(\mathbb{R};\mathbb{C}^2) \to L^2(\mathbb{R};\mathbb{C}^2),\\ \mathcal{T}^\tau \mathbf{w} &:= \mathbf{w}_x - \mathbf{A}^{\tau}(x,\lambda)\mathbf{w}, \qquad \mathbf{w} \in H^1(\mathbb{R};\mathbb{C}^2),
\end{aligned} \end{equation} indexed by $\tau \in (0,\tau_m)$, parametrized by $\lambda \in \mathbb{C}$ and with domain $\overline{D} = H^1(\mathbb{R};\mathbb{C}^2)$, which is independent of $\lambda$ and $\tau$. With a slight abuse of notation we call $\mathbf{w} \in H^1(\mathbb{R};\mathbb{C}^2)$ an \textit{eigenfunction} associated to the eigenvalue $\lambda \in \mathbb{C}$ provided $\mathbf{w}$ is a bounded solution to the equation \begin{equation*}
\mathcal{T}^\tau (\lambda) \mathbf{w} = 0. \end{equation*} More precisely, for each $\tau \in [0,\tau_m)$ we define the resolvent $\rho$, the point spectrum $\sigma_{\textrm{pt}}$ and the essential spectrum $\sigma_{\textrm{ess}}$ of problem \eqref{evalue} as \begin{equation*}
\begin{aligned}
\rho &= \{\lambda \in \mathbb{C} \, : \, \mathcal{T}^\tau (\lambda) \,\text{ is injective and onto, and }
\mathcal{T}^\tau (\lambda)^{-1} \, \text{is bounded} \, \},\\
\sigma_{\textrm{pt}} &= \{ \lambda \in \mathbb{C} \,: \; \mathcal{T}^\tau (\lambda) \,\text{ is Fredholm with index zero
and non-trivial kernel} \},\\
\sigma_{\textrm{ess}} &= \{ \lambda \in \mathbb{C} \,: \; \mathcal{T}^\tau (\lambda) \,\text{ is either not Fredholm or
with non-zero index} \}, \end{aligned} \end{equation*} respectively. The whole spectrum is $\sigma = \sigma_{\textrm{ess}} \cup \sigma_{\textrm{pt}}$. Since each operator $\mathcal{T}^\tau (\lambda)$ is closed, then $\rho = \mathbb{C} \backslash \sigma$ (cf. \cite{Kat80}). When $\lambda \in \sigma_{\textrm{pt}}$ we call it an eigenvalue, and any element in $\ker \mathcal{T}^\tau(\lambda)$ is an eigenvector.
We call the reader's attention to the fact that, unlike equation \eqref{spectproblem}, the spectral problem formulated in terms of the first order systems \eqref{firstorders} is well defined for $\tau = 0$, with \begin{equation*}
\mathbf{A}^0(x,\lambda) = \begin{pmatrix} 0 & & 1 \\ \lambda - a(x) & & -c_0 \end{pmatrix}, \end{equation*}
and where $c_0 = c(\tau)\bigr|_{\tau = 0}$ is the velocity of the parabolic front.
\begin{remark} \label{remaux} Suppose $\tau \in (0,\tau_m)$. If we substitute $\lambda = -1/\tau$ into \eqref{evalue} we arrive at the equation \begin{equation*}
(1-c^2 \tau)u_x - c(1+\tau a(x))u = 0. \end{equation*} Taking the real part of the $L^2$ product of last equation with $u$ we obtain \begin{equation*}
0 = \int_\mathbb{R} (\underbrace{1+\tau a(x))}_{> 0})|u|^2 \, dx \geq 0, \end{equation*}
inasmuch as $\textrm{\rm Re}\, \langle u,u_x\rangle_{L^2} = 0$, and $\tau < \tau_m =1/\sup |f'|$. This implies that $u = 0$, and hence $v = 0$, showing that $\lambda = -1/\tau$ does not belong to the point spectrum. \end{remark}
\subsubsection{On algebraic and geometric multiplicities}
In the stability of traveling waves literature, it is customary to analyze the spectrum of a differential operator $\mathcal{L}$ of second (or higher) order, for which there is a natural invertible transformation from the kernel of $\mathcal{L}-\lambda$ to the kernel of a first order operator of the form \eqref{defofTau}. In such cases, the matrices \eqref{coeffA} are linear in $\lambda$ and there is a natural correspondence between the Jordan block structures of $\mathcal{L}-\lambda$ and those of the corresponding operators $\mathcal{T} (\lambda)$ (see \cite{San02}). Since we arrived at systems \eqref{firstorders} through a different transformation (Kac's trick), we must show that such property remains in our case. For each $\tau \in (0,\tau_m)$ and $\lambda \in \sigma_{\textrm{pt}}$, we call the mapping, \begin{equation}\label{specKac}
\begin{aligned}
\mathcal{K} &: \ker (\mathcal{L}^\tau - \lambda) \, \to \ker \mathcal{T}^\tau (\lambda), \\
\mathcal{K} \begin{pmatrix}
u \\ v
\end{pmatrix} &:= \begin{pmatrix}
u \\ u_x
\end{pmatrix} = \mathbf{w}, \qquad \; \begin{pmatrix}
u \\ v
\end{pmatrix} \in \ker (\mathcal{L}^\tau - \lambda),
\end{aligned} \end{equation} as the \textit{spectral Kac's transformation}. It is a one-to-one and onto map. Indeed, if $\mathbf{w}_1 = \mathbf{w}_2 \in \ker \mathcal{T}^\tau (\lambda)$ then $u_1 = u_2$ and $\partial_x u_1 = \partial_x u_2$ a.e., and the first equation in \eqref{evalue} yields $\partial_x v_1 = \partial_x v_2$, whereas the second equation implies $v_1 = v_2$ a.e. Thus, $(u_1, v_1)^\top = (u_2, v_2)^\top \in \ker (\mathcal{L}^\tau - \lambda)$. It is onto because for each $\mathbf{w} = (u, u_x)^\top \in \ker \mathcal{T}^\tau (\lambda)$ clearly there exists \begin{equation*} \begin{pmatrix}
u \\ v \end{pmatrix} = \begin{pmatrix}
u \\ (1+\tau \lambda)^{-1} \big( (1-c^2\tau)u_x + c\tau (\lambda - a(x)) u\big) \end{pmatrix} \in \ker (\mathcal{L}^\tau - \lambda), \end{equation*} such that $\mathbf{w} = \mathcal{K}(u,v)^\top$. (Provided, of course, that $\lambda \neq -1/\tau$. But $-1/\tau \notin \sigma_{\textrm{pt}}$ by Remark \ref{remaux}.) Since $v$ satisfies the first equation of \eqref{evalue} we conclude that $v \in H^2(\mathbb{R};\mathbb{C})$ as well.
\begin{proposition}\label{prop:propequiv} Spectral Kac's transformation \eqref{specKac} induces a one-to-one correspondence between Jordan chains. \end{proposition}
\begin{proof} Suppose $(\varphi, \psi)^\top \in \ker (\mathcal{L}^\tau - \lambda)$. This is equivalent to the system \begin{equation}\label{e0}
c{\varphi_x} + {\psi_x} +(a(x)-\lambda) \varphi = 0,\qquad
{\varphi_x} + c \tau {\psi_x} -(1+\tau\lambda)\psi = 0. \end{equation} If we take the next element in a Jordan chain, say $(u,v)^\top \in H^2(\mathbb{R};\mathbb{C}^2)$, solution to \begin{equation*}
(\mathcal{L}^\tau - \lambda)\begin{pmatrix}
u \\ v
\end{pmatrix} = \begin{pmatrix}
\varphi \\ \psi
\end{pmatrix}, \end{equation*} then we obtain the system \begin{equation}\label{e1}
cu_x + v_x +(a(x)-\lambda) u = \varphi,\qquad
u_x + c \tau v_x -(1+\tau\lambda)v = \tau \psi. \end{equation} Multiply the first equation by $c\tau$, substract from the second one, differentiate it, and substitute $v_x$ from the first equation and $\psi_x$ from the second in \eqref{e0}. The result is the following scalar equation \begin{equation}\label{eq2}
\begin{aligned}
&(1-c^2\tau){u_{xx}} + c\big( 1+\tau(2\lambda - a(x))\big){u_x}\\
&\;\;\; +\bigl\{(1+\tau \lambda)(a(x)-\lambda) - c\tau a'(x)\bigr\} u
= (1+ 2\tau \lambda - \tau a(x)) \varphi - 2 c\tau \varphi_x.
\end{aligned} \end{equation} Written as a system for $\mathbf{w}_1 := (u, u_x)^\top$, equation \eqref{eq2} is equivalent to \begin{equation*}
\partial_ x \mathbf{w}_1 - \mathbf{A}^\tau(x,\lambda) \mathbf{w}_1 = \big(\mathbf{A}^\tau_1(x) + 2 \lambda \mathbf{A}^\tau_2(x)\big) \begin{pmatrix}
\varphi \\ \varphi_x
\end{pmatrix}. \end{equation*} Generalizing this procedure, we observe that solutions to \begin{equation*}
(\mathcal{L}^\tau - \lambda)\begin{pmatrix}
u_j \\ v_j
\end{pmatrix} = \begin{pmatrix}
u_{j-1} \\ v_{j-1}
\end{pmatrix}, \end{equation*} are in one-to-one correspondence with solutions to the equation $\mathcal{T}^\tau (\lambda) \mathbf{w}_j = (\partial_\lambda \mathbf{A}^\tau(x,\lambda))\mathbf{w}_{j-1}$, where $w_j$ and $(u_j, v_j)^\top$ are related to each other through Kac's transformation. Hence, a Jordan chain for $\mathcal{L}^\tau-\lambda$ induces a Jordan chain for $\mathcal{T}^\tau (\lambda)$ with same block structure and length.
\end{proof}
Consequently, we have the following definition.
\begin{definition}\label{def:spectT} Assume $\lambda \in \sigma_{\textrm{pt}}$. Its geometric multiplicity (\textit{g.m.}) is the maximal number of linearly independent elements in $\ker \mathcal{T}^\tau (\lambda)$. Suppose $\lambda \in \sigma_{\textrm{pt}}$ has $g.m. = 1$, so that $\ker \mathcal{T}^\tau (\lambda) =$ span $\{\mathbf{w}_0\}$. We say $\lambda$ has algebraic multiplicity (\textit{a.m.}) equal to $m$ if we can solve $\mathcal{T}^\tau (\lambda) \mathbf{w}_j = (\partial_\lambda \mathbf{A}^\tau(x,\lambda)) \mathbf{w}_{j-1}$, for each $j = 1, \ldots, m-1$, with $\mathbf{w}_j \in H^1$, but there is no bounded $H^1$ solution $\mathbf{w}$ to $\mathcal{T}^\tau (\lambda) \mathbf{w} = (\partial_\lambda \mathbf{A}^\tau(x,\lambda)) \mathbf{w}_{m-1}$. For an arbitrary eigenvalue $\lambda \in \sigma_{\textrm{pt}}$ with $g.m.= l$, the algebraic multiplicity is defined as the sum of the multiplicities $\sum_k^l m_k$ of a maximal set of linearly independent elements in $\ker \mathcal{T}^\tau (\lambda) = $ span $\{\mathbf{w}_1, \ldots, \mathbf{w}_l\}$. \end{definition}
Thanks to Proposition \ref{prop:propequiv} and Definition \ref{def:spectT} we readily obtain the following
\begin{corollary} For each $\tau \in [0,\tau_m)$, the sets $\sigma_{\textrm{pt}}$ and $\sigma_{\textrm{pt}}(\mathcal{L}^\tau)$ (the latter defined as the set of complex $\lambda$ such that $\mathcal{L}^\tau-\lambda$ is Fredholm with index zero and has a non-trivial kernel) coincide, with same algebraic and geometric multiplicities. \end{corollary}
\subsection{The (translation invariance) eigenvalue $\lambda=0$}
Here, we prove that $\lambda = 0$ is an eigenvalue of $\mathcal{L}^\tau$ for each $\tau \in (0,\tau_m)$ (the eigenvalue associated to translation invariance of the wave, with eigenfunction $(U_x,V_x)^\top$), and, moreover, that it is simple.
\begin{lemma}\label{lemgm1} For each $\tau \in (0,\tau_m)$, $\lambda = 0$ is an eigenvalue of $\mathcal{L}^\tau$ with geometric multiplicity equal to one, and with eigenspace generated by $(U_x, V_x)^\top \in H^2(\mathbb{R};\mathbb{C}^2)$. \end{lemma}
\begin{proof} Differentiate system \eqref{profileq} with respect to $x$ to verify that $(U_x, V_x)$ is a solution to the spectral system \eqref{evalue} with $\lambda = 0$, that is, \begin{equation}\label{lam0syst}
cU_{xx} + V_x + a(x) V_x = 0, \qquad U_x + c\tau V_{xx} - V_x = 0. \end{equation} Due to exponential decay \eqref{expodecay} of the wave, and solving for $U_{xx}$ and $V_{xx}$ in equations \eqref{lam0syst}, we observe that $(U_x, V_x)^\top \in H^2(\mathbb{R};\mathbb{C}^2) = D(\mathcal{L}^\tau)$. This shows that $(U_x, V_x)^\top$ belongs to $\ker \mathcal{L}^\tau$ for each $\tau \in (0,\tau_m)$. Thus, $\lambda = 0 \in \sigma_{\textrm{pt}}(\mathcal{L}^\tau)$. In view of the equivalence established by Kac's spectral transformation (Proposition \ref{prop:propequiv}), this implies that \begin{equation*}
\mathbf{w}_0 = \begin{pmatrix}
U_x \\ U_{xx}
\end{pmatrix} \in \ker \mathcal{T}^\tau. \end{equation*} To analyze its multiplicity we observe that system \eqref{evalue} with $\lambda = 0$ is equivalent to the following scalar equation (substitute $\lambda = 0$ in \eqref{2ndorderu}): \begin{equation}\label{eq:mult01}
\mathcal{A}u := a_0 u_{xx} + a_1(x) u_x + a_2(x) u = 0, \end{equation} where we have introduced the closed, densely defined auxiliary operator $\mathcal{A} : D(\mathcal{A}) = H^2(\mathbb{R};\mathbb{C}) \to L^2(\mathbb{R};\mathbb{C})$, where $a(x) = f'(U(x))$ as before, and with \begin{equation*}
a_0 = 1-c^2 \tau > 0, \quad a_1(x) = c(1-\tau a(x)), \quad a_2(x) = a(x) - c\tau a'(x). \end{equation*} Let us denote \begin{equation*}
\phi := U_x \in H^2(\mathbb{R};\mathbb{C}). \end{equation*} Clearly, $\phi$ is a bounded solution to \eqref{eq:mult01}, and $\lambda = 0$ is an eigenvalue of $\mathcal{A}$ associated to the eigenfunction $\phi$.
We shall rewrite \eqref{eq:mult01} in self-adjoint form by eliminating the first derivative. To this aim, we introduce the new variable $w$ as follows: \begin{equation}\label{eq:changeuw}
u(x) = z(x) w(x), \qquad z(x) = \exp \Big(\int^x b(y) \, dy \Big). \end{equation} A direct calculation shows that \begin{equation*}
u_x = (w_x + wb)z, \qquad u_{xx} = (w_{xx} + 2bw_x + (b_x + b^2)w)z. \end{equation*} Upon substitution, \begin{equation*}
\mathcal{A}u = \big( a_0 w_{xx} + (a_1(x) + 2a_0b) w_x + (a_0(b_x + b^2) + a_1(x) b + a_2(x))w\big)z. \end{equation*} Choose $b = - a_1(x)/2a_0$. This yields \begin{equation*}
z_x = - \frac{a_1(x)}{2a_0} z, \qquad z(x) = \exp \Big(- \int^x a_1(y)/2a_0 \, dy \Big), \end{equation*} and \begin{equation*}
\mathcal{A} u = \big( \widetilde{\mathcal{A}} w\big) z = 0, \end{equation*} where the self-adjoint operator $\widetilde{\mathcal{A}} : H^2(\mathbb{R};\mathbb{C}) \to L^2(\mathbb{R};\mathbb{C})$ is defined as \begin{equation*}
\widetilde{\mathcal{A}} w := a_0 w_{xx} + h(x) w, \quad
h(x) = a_2(x) - \tfrac{1}{2}a_1'(x) - \tfrac{1}{4}a_0^{-1} a_1(x)^2. \end{equation*} Since $z(x) > 0$ for all $x$, this readily implies that $\mathcal{A} u = 0$ if and only if $w = uz^{-1} \in \ker \widetilde{\mathcal{A}}$.
Upon inspection, we observe that any decaying solution (at $x = \pm \infty$) to $\widetilde{\mathcal{A}} w = 0$ converges to zero exponentially with rate \begin{equation}\label{eq:decayw}
\mp \frac{1}{2a_0}\sqrt{(a_1^{\pm})^2 - 4a_0f'(U^{\pm})} \, . \end{equation} This is true, in particular, for $\varphi = \phi/z$, because of the behavior at $\pm \infty$ of the weight function $z$. In view of these observations, we conclude that there is a one-to-one correspondence between $L^2$ eigenfunctions of $\mathcal{A}$ and $\widetilde{\mathcal{A}}$, determined by the change of variables \eqref{eq:changeuw}.
Now suppose that $u \in H^2$ is a solution to $\mathcal{A} u = 0$. Therefore, it decays at $x = \pm \infty$ with rate \begin{equation*}
- \frac{-a_1^\pm}{2a_0} \mp \frac{1}{2a_0}\sqrt{(a_1^{\pm})^2 - 4a_0f'(U^{\pm})} \, . \end{equation*} This implies that $w = uz^{-1}$ is an $L^2$ solution to $\widetilde{\mathcal{A}} w = 0$ which decays with rate \eqref{eq:decayw}. Hence, the Wronskian determinant of these two solutions, viz. $w \varphi_x - \varphi w_x$ goes to zero when $x \to \pm \infty$ and, moreover, \begin{equation*}
a_0(w \varphi_x - \varphi w_x)_x = a_0(w \varphi_{xx} - \varphi w_{xx}) = h(x)w\varphi - h(x)\varphi w = 0, \end{equation*} implying that $w \varphi_x - \varphi w_x = 0$ for all $x \in \mathbb{R}$. Therefore, $w$ and $\varphi$ (and hence, $u$ and $\phi$) are linearly dependent.
By the equivalence between solutions to the scalar equation \eqref{2ndorderu} and solutions to the first order system \eqref{firstorders}, we have shown that any bounded solution $\mathbf{w}$ to $\mathbf{w}_x = \mathbf{A}^\tau(x,0) \mathbf{w}$ is a multiple of $\mathbf{w}_0$, and consequently $\mathcal{T}^\tau(0)$ is Fredholm with index zero, with non-trivial kernel spanned by $\mathbf{w_0}$. Whence, $\lambda = 0 \in \sigma_{\textrm{pt}}$ with geometric multiplicity equal to one. By Proposition \ref{prop:propequiv}, this implies, in turn, that $\lambda = 0 \in \sigma_{\textrm{pt}}(\mathcal{L}^\tau)$ with geometric multiplicity equal to one, and with associated eigenfunction $(U_x, V_x)^\top$.
\end{proof}
\begin{corollary}
The adjoint equation \begin{equation}\label{adjeq} \mathbf{y}_x = - \mathbf{A}^\tau(x,0)^* \mathbf{y},
\end{equation} has a unique (up to constant multiples) bounded solution $\mathbf{y}_0 = (\zeta, \eta )^\top \in H^1(\mathbb{R};\mathbb{C}^2)$ where $\eta \in H^2(\mathbb{R};\mathbb{C})$ is the unique bounded solution to \begin{equation}\label{eqeta}
\mathcal{A}^* \eta = a_0 \eta_{xx} - a_1(x) \eta_x + (a_2(x) - a_1'(x)) \eta = 0, \end{equation} and $\mathcal{A}^* : H^2(\mathbb{R};\mathbb{C}) \to L^2(\mathbb{R};\mathbb{C})$ denotes the formal adjoint of the auxiliary operator $\mathcal{A}$. \end{corollary}
\begin{proof} Since $\mathcal{T}^\tau(0)$ is Fredholm with index zero and $\ker \mathcal{T}^\tau(0) = \mathrm{span} \{\mathbf{w}_0\}$, by an exponential dichotomies argument (see Remark 3.4 in \cite{San02}), the adjoint equation \eqref{adjeq} has a unique bounded solution $\mathbf{y}_0 \in H^1(\mathbb{R};\mathbb{C}^2)$. Observing that \begin{equation*}
\begin{aligned}
- \mathbf{A}^\tau(x,0)^* = -\mathbf{A}_0^\tau(x)^\top &= (1-c^2\tau)^{-1} \begin{pmatrix}
0 & -(1-c^2 \tau) \\ a(x) - c\tau a'(x) & c(1-\tau a(x))
\end{pmatrix}^\top \\&= a_0^{-1}\begin{pmatrix}
0 & a_2(x) \\ - a_0 & a_1(x)
\end{pmatrix},
\end{aligned} \end{equation*} we arrive at the following system of equations for the components of $\mathbf{y}_0$: \begin{equation*}
\zeta_x = a_0^{-1} a_2(x) \eta, \qquad \eta_x = - \zeta + a_0^{-1} a_1(x) \eta. \end{equation*} Since the coefficients are bounded, by bootstrapping it is easy to verify that $\eta \in H^2(\mathbb{R};\mathbb{C})$. Thus, differentiate the second equation and substitute the first to arrive at \begin{equation*}
a_0 \eta_{xx} - a_1(x) \eta_x + (a_2(x) - a_1'(x)) \eta = 0. \end{equation*} That the left hand side of last equation is $\mathcal{A}^* \eta$ follows from a direct calculation of the formal adjoint.
\end{proof}
\begin{lemma}\label{lemam1} For each $\tau \in (0,\tau_m)$ the algebraic multiplicity of $\lambda = 0 \in \sigma_{\textrm{pt}}(\mathcal{L}^\tau)$ is equal to one. \end{lemma}
\begin{proof} We define the quantity \begin{equation}\label{defGamma} \Gamma := \langle \mathbf{y}_0, \mathbf{A}^\tau_1(x) \mathbf{w}_0 \rangle_{L^2} = \int_{-\infty}^{+\infty} \begin{pmatrix}
\zeta \\ \eta
\end{pmatrix}^* \mathbf{A}_1^\tau(x) \begin{pmatrix}
\phi \\ \phi_x
\end{pmatrix} \, dx. \end{equation} Substituting the expression for $\mathbf{A}_1^\tau$ we obtain \begin{equation*}
\Gamma = a_0^{-1} \int_{-\infty}^{+\infty} \overline{\eta} \big( (1-\tau a(x))\phi - 2c\tau \phi_x \big) \, dx. \end{equation*} Let us suppose that $(u_1, v_1)^\top \in H^1(\mathbb{R};\mathbb{C})$, $(u_1, v_1) \neq 0$, is a non-trivial first element of a Jordan chain for $\mathcal{L}^\tau$ associated to $\lambda = 0$, that is, a solution to \begin{equation*}
\mathcal{L}^{\tau} \begin{pmatrix}
u_1 \\ v_1
\end{pmatrix} = \begin{pmatrix} U_x \\ V_x \end{pmatrix}
\in \ker \mathcal{L}^\tau. \end{equation*} Hence, $(u_1, v_1)^\top$ is a solution to system \eqref{e1} with $\lambda = 0$. By substitution, this is equivalent to equation \eqref{eq2} for $u_1$ and with $\lambda = 0$, namely to \begin{equation*}
\mathcal{A}u_1 = (1-\tau a(x))U_x - 2c\tau U_{xx} = (1-\tau a(x))\phi - 2c\tau \phi_x. \end{equation*} Apply the change of variables \eqref{eq:changeuw} to last equation to obtain \begin{equation*}
\mathcal{A}u_1 = z (\widetilde{\mathcal{A}}w_1) = (1-\tau a(x))\phi - 2c\tau \phi_x, \end{equation*} where $w_1 = u_1 z^{-1}$. Now, since $z$ is real and $\widetilde{\mathcal{A}}$ is self-adjoint, we obtain \begin{equation*}
\Gamma = a_0^{-1} \int_{-\infty}^{+\infty} \overline{z \eta} \widetilde{\mathcal{A}} w_1 \, dx \\
= a_0^{-1} \langle z\eta, \widetilde{\mathcal{A}} w_1 \rangle_{L^2}
= a_0^{-1} \langle \widetilde{\mathcal{A}}(z\eta), w_1 \rangle_{L^2} \, . \end{equation*} Use equation \eqref{eqeta} to compute \begin{equation*}
(z\eta)_{xx} = - a_0^{-1} h(x) z \eta, \end{equation*} yielding $\widetilde{\mathcal{A}}(z\eta) = a_0 (z\eta)_{xx} + h(x)z \eta = 0$. We conclude that $\Gamma = 0$.
Therefore, the contrapositive statement holds true: if $\Gamma \neq 0$ then there exists no non-trivial first element of the Jordan chain. In other words, if we show that $\Gamma \neq 0$ then the algebraic multiplicity of $\lambda = 0$ is equal to one. To compute $\Gamma$ we make the observation that the unique bounded solution to $\mathcal{A}^* \eta = 0$ is precisely $\eta = z^{-2} \phi$. Indeed, by a direct calculation we get \begin{equation*}
\begin{aligned}
\eta_x & = \frac{1}{z^2}(\phi_x + a_0^{-1}a_1(x) \phi), \qquad \text{and,}\\
\eta_{xx} & = \frac{a_0^{-1}a_1(x)}{z^2}(\phi_x + a_0^{-1}a_1(x) \phi)
+ \frac{1}{z^2}(\phi_{xx} + a_0^{-1} a_1'(x) \phi + a_0^{-1}a_1(x) \phi_x).
\end{aligned} \end{equation*} This yields \begin{equation*}
\mathcal{A}^* \eta = \frac{1}{z^2} (a_0 \phi_{xx} + a_1(x) \phi_x + a_2(x) \phi) = \frac{1}{z^2} \mathcal{A}\phi = 0. \end{equation*} Substituting in the expression for $\Gamma$, and since $\phi$ is real, after integration by parts one gets \begin{equation*}
\begin{aligned}
\Gamma &= a_0^{-1} \int_{-\infty}^{+\infty} z^{-2} \phi ((1-\tau a(x))\phi - 2c\tau \phi_x) \, dx\\
&= a_0^{-1} \int_{-\infty}^{+\infty} z^{-2} |\phi|^2 (1-\tau a(x)) \, dx - 2c\tau a_0^{-1} \int_{-\infty}^{+\infty} z^{-3}z_x |\phi|^2 \, dx\\
&= a_0^{-1} (1 + c^2 \tau a_0^{-1}) \int_{-\infty}^{+\infty} |1-\tau a(x)| z^{-2}|\phi|^2 \, dx,
\end{aligned} \end{equation*} because $1 - \tau a(x) > 0$ for all $x$ if $\tau \in (0,\tau_m)$. Since $\phi \not\equiv 0$ a.e. we conclude that $\Gamma > 0$ and the proof is complete.
\end{proof}
\begin{remark} It is well known that the integral \begin{equation*}
\Gamma = \langle \mathbf{y}_0, ((d/d\lambda) \mathbf{A}^\tau(x,\lambda))_{|\lambda = 0} \mathbf{w}_0\rangle_{L^2}, \end{equation*} known as a \textit{Melnikov integral}, decides whether $\lambda = 0$ has higher algebraic multiplicity: $\Gamma$ is proportional, modulo a non-vanishing orientation factor, to the derivative of the Evans function $D'(0)$ at $\lambda = 0$ (see Definition \eqref{defD} below); thus, if $\Gamma \neq 0$ then the algebraic multiplicity is equal to one. See \cite{San02}, Section 4.2.1 for further information. \end{remark}
\subsection{Further properties of the point spectrum}
Next, we use energy estimates to show that there are no purely imaginary eigenvalues different from zero.
\begin{lemma}\label{lemma:imagevalue} If $\lambda$ is an eigenvalue for \eqref{evalue} and $\lambda\in i\mathbb{R}$, then $\lambda=0$. \end{lemma}
\begin{proof} Let $\lambda\in i\mathbb{R}$ be such that equation \eqref{2ndorderu} is satisfied for some function $u$. Applying the transformation \eqref{eq:changeuw} of the proof of Lemma \ref{lemgm1}, i.e. $u(x)=w(x)\,z(x)$ with $z(x)=\exp(-\int^x b)$ and $b=-a_1/2a_0$, equation is transformed into \begin{equation*}
{w_{xx}}+\alpha\lambda\,{w_x}-\beta(x,\lambda)w=0, \end{equation*} with \begin{equation*}
\begin{aligned}
\alpha&=\frac{2c\tau}{1-c^2\tau},\\
\beta(x,\lambda)&=\frac{1}{4a_0^2}\left(a_1(x)^2 -4a_0a_2(x) -2a_0 a_1'(x) \right)
+\frac{(1-\tau\,a(x))}{a_0^2}\lambda+\frac{\tau}{a_0}\,\lambda^2.
\end{aligned} \end{equation*} Multiplying by $\bar w$, we infer the relation \begin{equation*}
\left|w_x\right|^2-\left(\bar w w_x\right)_x
+ \alpha\lambda \bar w w_x +\beta(x,\lambda)|w|^2=0. \end{equation*} Thus, integrating in $\mathbb{R}$ and taking the imaginary part one obtains \begin{equation*}
\int_{\mathbb{R}} \textrm{\rm Im}\,\lambda\bigl\{2c\tau(1-c^2\tau)\textrm{\rm Re}\,\left({w_x}\bar w\right)\,
+\big(1-\tau\,a(x)\big)|w|^2\bigr\}dx=0. \end{equation*} since, by assumption, $\lambda\in i\mathbb{R}$. For $\lambda\neq 0$, thanks to the relation \begin{equation*}
\textrm{\rm Re}\, (\bar w w_x) = \tfrac{1}{2} \big( |w|^2\big)_x, \end{equation*} the previous equality implies $w=0$ a.e. since $1-\tau a(x)>0$.
\end{proof}
\subsection{Consistent splitting and essential spectrum}
Let us look at the asymptotic (constant coefficient) systems derived from \eqref{coeffA} when we take the limit as $x \to \pm \infty$. If we define the positive parameters \begin{equation*}
\begin{aligned}
0 &< \delta_\pm := - \lim_{x \to \pm \infty} a(x) = - \lim_{x \to \pm \infty} f'(U) = - f'(U_\pm)\\
0 &< b_\pm := \lim_{x \to \pm \infty} \bigl(1 - \tau f'(U)\bigr) = 1 + \tau \delta_\pm,
\end{aligned} \end{equation*} for each $\tau \in (0,\tau_m)$, then systems \eqref{firstorders} tend to the constant coefficient asymptotic systems \begin{equation}\label{constcoeff}
\mathbf{w}_x = \mathbf{A}^\tau_\pm (\lambda) \mathbf{w}, \end{equation} where \begin{equation}\label{asymptcoeffA}
\begin{aligned}
\mathbf{A}^\tau_\pm(\lambda) &:= \lim_{x \to \pm \infty} \mathbf{A}^\tau(x,\lambda) \\
&= (1-c^2 \tau)^{-1}\begin{pmatrix}
0 & & 1-c^2 \tau \\ \tau \lambda^2 + \lambda b_\pm + \delta_\pm & &-c(b_\pm + 2 \tau \lambda)
\end{pmatrix}.
\end{aligned} \end{equation} The location of the essential spectrum of problem \eqref{evalue} is determined by systems \eqref{constcoeff}. Let us denote the characteristic polynomial of $\mathbf{A}^\tau_\pm(\lambda)$ as \begin{equation} \label{defcharpol}
\pi_\pm^{(\tau, \lambda)}(\kappa) := \det ( \mathbf{A}^\tau_\pm(\lambda) - \kappa I). \end{equation} Thus, we compute \begin{equation*}
\begin{aligned}
&\det (\kappa I - (1-c^2 \tau) \mathbf{A}^\tau_\pm(\lambda))
= \det \begin{pmatrix} \kappa & & -(1-c^2 \tau) \\
-(\tau \lambda^2 + \lambda b_\pm + \delta_\pm) & &\kappa -c(b_\pm + 2 \tau \lambda)
\end{pmatrix}\\
&\hskip3.25cm
= \kappa^2 + \kappa c (b_\pm + 2 \tau \lambda) - (1-c^2 \tau)(\tau \lambda^2 + \lambda b_\pm + \delta_\pm).
\end{aligned} \end{equation*} Hence, if we assume $\kappa = i \xi$, $\xi \in \mathbb{R}$, is a purely imaginary root of \eqref{defcharpol}, then \begin{equation}\label{disprel}
\xi^2 - ic\xi (b_\pm + 2 \tau \lambda) + (1-c^2\tau)(\tau \lambda^2 + b_\pm \lambda + \delta_\pm) = 0. \end{equation} Equation \eqref{disprel} is the dispersion relation of wave solutions to the constant coefficient equations \eqref{constcoeff}. Its $\lambda$-roots, functions of $\xi \in \mathbb{R}$, define algebraic curves in the complex plane. They bound the essential spectrum on the right as we shall verify. We denote these curves as \begin{equation}\label{algcurves}
\lambda = \lambda_{1,2}^\pm(\xi), \qquad \xi \in \mathbb{R}. \end{equation} It is to be noticed that $\lambda = 0$ does not belong to any of the algebraic curves \eqref{algcurves} inasmuch as \begin{equation*}
\textrm{\rm Re}\, (\xi^2 - ic\xi b_\pm + (1-c^2 \tau) \delta_\pm) = \xi^2 + (1-c^2 \tau) \delta_\pm > 0, \end{equation*} for all $\xi \in \mathbb{R}$.
\subsubsection{Analysis of the dispersion relation}
Fix $0 < \tau < \tau_m$. Suppose that $\lambda(\xi)$ belongs to one of the algebraic curves \eqref{algcurves} and denote $\eta := \textrm{\rm Re}\, \lambda(\xi)$ and $\beta := \textrm{\rm Im}\, \lambda(\xi)$. Taking the real and imaginary parts of \eqref{disprel} yields \begin{equation}\label{realp}
\xi^2 + 2c\tau \xi \beta + (1-c^2\tau)\big( \tau(\eta^2 - \beta^2) + \eta b_\pm + \delta_\pm\big) = 0. \end{equation} \begin{equation}\label{imagp}
\big((1-c^2 \tau) \beta - c\xi \big) \big( b_\pm + 2 \tau \eta\big) = 0. \end{equation} We readily notice that if $\eta = 0$ for some $\xi \in \mathbb{R}$ then from equation \eqref{imagp} we get $\beta = c\xi/(1-c^2 \tau)$, as $b_\pm > 0$. Upon substitution in \eqref{realp} we obtain \begin{equation*}
\xi^2 + \frac{c^2 \tau \xi^2}{1-c^2 \tau} + (1-c^2 \tau) \delta_\pm = 0, \end{equation*} which yields a contradiction with $\delta_\pm > 0$, $\tau > 0$ and the subcharacteristic condition \eqref{subchar}. We conclude that the algebraic curves never cross the imaginary axis: they remain in either the stable or in the unstable complex half plane. Now, from equation \eqref{imagp} we distinguish two cases: \begin{align}
\textrm{either}\quad \eta &= - \frac{b_\pm}{2\tau}, \label{casei}\\
\textrm{or}\quad \beta &= \frac{c\xi}{1-c^2\tau}. \label{caseii} \end{align} Let us first assume \eqref{casei}. Substituting into \eqref{realp} we obtain the equation \begin{equation}\label{eqforbeta}
\tau \beta^2 - \frac{2c\tau \xi}{1-c^2\tau}\,\beta - \delta_\pm
+ \frac{b_\pm^2}{4\tau} - \frac{\xi^2}{1-c^2\tau} = 0. \end{equation} This equation has real solutions $\beta$ provided that \begin{align}
\Delta_1(\xi) &:= \frac{4c^2 \tau^2 \xi^2}{(1-c^2 \tau)^2}
- 4\tau \Big( -\delta_\pm + \frac{b_\pm^2}{4\tau} - \frac{\xi^2}{1-c^2\tau}\Big) \geq 0, \nonumber \\
&\hskip2cm \iff \;\; \xi^2 (1-c^2 \tau)^{-2} + \delta_\pm \geq \frac{b_\pm^2}{4 \tau}. \label{star} \end{align} Secondly, substitute \eqref{caseii} into \eqref{realp}. The result is \begin{equation} \label{eqforeta}
\tau \eta^2 + b_\pm \eta + \delta_\pm + \frac{\xi^2}{(1-c^2\tau)^2} = 0. \end{equation} Last equation has real solutions $\eta$ if and only if \begin{align}
\Delta_2(\xi) &:= b_\pm^2 - 4\tau \Big( \delta_\pm + \frac{\xi^2}{(1-c^2\tau)^2} \Big) \geq 0, \nonumber \\
&\hskip2cm \iff \;\; \xi^2(1-c^2\tau)^{-2} + \delta_\pm \leq \frac{b_\pm^2}{4\tau}. \label{dstar}
\end{align} Then, clearly, from \eqref{star} and \eqref{dstar} we have $\mathrm{sgn}\, \Delta_2 = - \mathrm{sgn}\, \Delta_1$. Observe, however, that by definition, \begin{equation*}
\frac{b_\pm^2}{4\tau} = \frac{(1+\tau \delta_\pm)^2}{4\tau} > \delta_\pm, \end{equation*}
because $(1-\delta_\pm \tau)^2 > 0$ for all $0 < \tau < \tau_m =1/\sup |f'|$, $\delta_\pm = |f'(1)|, |f'(0)|$. Therefore, for small values of $|\xi|$, \eqref{dstar} holds, $\mathrm{sgn}\, \Delta_2 = +1$, and the only algebraic curve solutions $\lambda = \lambda(\xi)$ are \begin{equation}\label{ddstar}
\textrm{\rm Im}\, \lambda(\xi) = \beta(\xi) = \frac{c\xi}{1-c^2 \tau}, \qquad
\textrm{\rm Re}\, \lambda(\xi) = \eta(\xi) = \frac{1}{2\tau} \Big( - b_\pm \pm \sqrt{\Delta_2(\xi)} \Big). \end{equation} Let $\xi^{(0)}_\pm$ be the positive solution to \begin{equation*}
(\xi^{(0)}_\pm)^2 = (1-c^2 \tau)^2 \Big( \frac{b_\pm^2}{4\tau} - \delta_\pm\Big) > 0. \end{equation*} Hence, for each $\xi \in (-\xi^{(0)}_\pm,\xi^{(0)}_\pm)$, condition \eqref{dstar} holds and the algebraic curves are determined by \eqref{ddstar}. Observe that: \begin{itemize}
\item[\textbullet] $\Delta_1(\xi), \Delta_2(\xi) \to 0$,
\item[\textbullet] $\eta(\xi) \to - b_\pm / 2\tau$, $\, \beta(\xi) \to \pm c \xi^{(0)}_\pm/(1-c^2 \tau)$, \end{itemize}
as $|\xi| \uparrow \xi^{(0)}_\pm$.
This behavior guarantees the continuity of the algebraic curves at $|\xi| = \xi^{(0)}_\pm$, as $\beta$ tends to the roots of \eqref{eqforbeta} and $\eta$ tends to \eqref{casei}. Therefore, for $|\xi| \geq \xi^{(0)}_\pm$, $\Delta_1(\xi) \geq 0$ and the solutions $\lambda(\xi)$ are determined uniquely by \begin{equation}\label{dddstar}
\textrm{\rm Im}\, \lambda(\xi) = \beta(\xi) = \frac{c\xi}{1-c^2 \tau} \pm \frac{\sqrt{\Delta_1(\xi)}}{2\tau}, \qquad
\textrm{\rm Re}\, \lambda(\xi) = \eta(\xi) = - \frac{b_\pm}{2\tau}. \end{equation} The algebraic curves \eqref{algcurves} in the case of a cubic reaction \eqref{cubicf}, with $\kappa = 1$ and for the parameter value $\alpha = 3/4$ can be found in Figure \ref{figalgcurves}. To compute them, we approximated the value of the speed $c$ by its lower bound \eqref{estbelow}.
\begin{figure}
\caption{Algebraic Fredholm curves \eqref{algcurves} for systems \eqref{constcoeff} in the case of a cubic nonlinearity \eqref{cubicf} with $\kappa = 1$, $\tau = 1/2$, and unstable state $u = \alpha = 3/4$. The value of the speed $c = c(\tau)$ is approximated by its lower bound \eqref{estbelow}. The curves at $+\infty$, $\lambda_{1,2}^+(\xi)$ are depicted by the solid continuous (blue; online version) curves, whereas the curves at $-\infty$, $\lambda_{1,2}^-(\xi)$, are represented by the dashed (red; online version) curves.}
\label{figalgcurves}
\end{figure}
Finally, notice that, for $|\xi| \leq \xi^{(0)}_\pm$, from equation \eqref{ddstar} we obtain the following bound for the real part of $\lambda$: \begin{equation*}
\begin{aligned}
\textrm{\rm Re}\, \lambda = \eta &= \frac{1}{2\tau} \Big( - b_\pm \pm \sqrt{\Delta_2(\xi)} \Big)
\leq \frac{1}{2\tau} \Big( - b_\pm + \sqrt{b_\pm^2 - 4 \tau \delta_\pm} \Big)\\
&= \frac{1}{2\tau} (-(1+\tau \delta_\pm) + (1- \delta_\pm \tau))
= - \delta_\pm < - \frac{\delta_\pm}{2}.
\end{aligned} \end{equation*}
Likewise, when $|\xi| \geq \xi^{(0)}_\pm$, we have the uniform bound \begin{equation*}
\textrm{\rm Re}\, \lambda = \eta = - \frac{b_\pm}{2\tau} = - \frac{1+\tau \delta_\pm}{2\tau} < - \frac{\delta_\pm}{2}, \end{equation*} for all $0 < \tau < \tau_m$. We have proved the following
\begin{lemma} For all $\tau \in (0, \tau_m)$, there exists a uniform \begin{equation}
\label{defchi0} \chi_0 := \tfrac{1}{2} \min \{\delta_+, \delta_-\} > 0, \end{equation} such that the algebraic curves $\lambda = \lambda_{1,2}^\pm(\xi)$, $\xi \in \mathbb{R}$, solutions to the dispersion relations \eqref{disprel}, satisfy \begin{equation} \label{spectralgapeq} \mathrm{Re}\, \lambda_{1,2}^\pm(\xi) < - \chi_0 < 0, \end{equation} for all $\xi \in \mathbb{R}$. \end{lemma}
\subsubsection{Stability of the essential spectrum}
We define the following open, connected region of the complex plane, \begin{equation}\label{defOmega}
\Omega := \{\lambda \in \mathbb{C} \, : \, \textrm{\rm Re}\, \lambda > - \chi_0\}. \end{equation} It properly contains the unstable complex half plane $\mathbb{C}_+ = \{ \textrm{\rm Re}\, \lambda > 0\}$. Denote ${\mathbb{S}^\tau_\pm}(\lambda)$ and ${\mathbb{U}^\tau_\pm}(\lambda)$ as the stable and unstable eigenspaces of $\mathbf{A}^\tau_\pm(\lambda)$, respectively.
\begin{lemma}\label{lemconsistsplit} For all $\tau \in (0,\tau_m)$, and all $\lambda \in \Omega$, the coefficient matrices $\mathbf{A}^\tau_\pm(\lambda)$ have no center eigenspace and, moreover, \begin{equation*}
\dim {\mathbb{S}^\tau_\pm}(\lambda) = \dim {\mathbb{U}^\tau_\pm}(\lambda) = 1. \end{equation*} \end{lemma}
\begin{proof} Take $\lambda \in \Omega$ and suppose $\kappa = i \xi$, with $\xi \in \mathbb{R}$, is an eigenvalue of $\mathbf{A}_\pm^\tau(\lambda)$. Then $\lambda$ belongs to one of the algebraic curves \eqref{algcurves}. But \eqref{spectralgapeq} yields a contradiction with $\lambda \in \Omega$. Therefore, the matrices $\mathbf{A}_\pm^\tau(\lambda)$ have no center eigenspace.
Since $\Omega$ is a connected region of the complex plane, it suffices to compute the dimensions of ${\mathbb{S}^\tau_\pm}(\lambda)$ and ${\mathbb{U}^\tau_\pm}(\lambda)$ when $\lambda = \eta \in \mathbb{R}_+$, sufficiently large. The characteristic polynomial \eqref{defcharpol} of $\mathbf{A}_\pm^\tau(\lambda)$ is \begin{equation*}
\kappa^2 + \kappa c(b_\pm + 2 \tau \lambda)
- (1 - c^2 \tau)(\tau \lambda^2 + \lambda b_\pm + \delta_\pm) = 0. \end{equation*} Assuming $\lambda = \eta \in \mathbb{R}_+$, the roots are \begin{equation*}
\kappa = - \frac{c}{2}(b_\pm + 2 \tau \eta) \pm \frac{1}{2} \sqrt{c^2 (b_\pm + 2\tau \eta)^2
+ 4(1-c^2\tau)(\tau \eta^2 + \eta b_\pm + \delta_\pm)}. \end{equation*} Clearly, for each $\eta > 0$, one of the roots is positive and the other is negative. This proves the lemma.
\end{proof}
In view of last result, the region $\Omega$ is often called the \textit{region of consistent splitting} \cite{San02}.
\begin{corollary}[Stability of the essential spectrum]\label{lemspectralgap} For each $\tau \in (0,\tau_m)$, the essential spectrum is contained in the stable half-plane. More precisely, \begin{equation*}
\sigma_{\mathrm{ess}} \subset \{\lambda \in \mathbb{C} \, : \, \textrm{\rm Re}\, \, \lambda \leq - \chi_0 < 0\}. \end{equation*} \end{corollary}
\begin{proof} Fix $\lambda \in \Omega$. By exponential dichotomies theory (see \cite{Cop2,San02}), since $\mathbf{A}_\pm^\tau(\lambda)$ are hyperbolic, the asymptotic systems \eqref{firstorders} have exponential dichotomies in $x \in \mathbb{R}_+ = (0,+\infty)$ and in $x \in \mathbb{R}_- = (-\infty,0)$ with respective Morse indices \begin{equation*}
i_+(\lambda) = \dim {\mathbb{U}^\tau_+}(\lambda) = 1, \qquad i_-(\lambda) = \dim {\mathbb{U}^\tau_-}(\lambda) = 1. \end{equation*} This implies (see \cite{Pal1,Pal2} and also \cite{San02}) that the variable coefficient operators $\mathcal{T} ^\tau(\lambda)$ are Fredholm as well, with index \begin{equation*}
\text{ind}\, \mathcal{T} ^\tau(\lambda) = i_+(\lambda) - i_-(\lambda) = 0, \end{equation*} showing that $\Omega \subset \mathbb{C}\backslash \sigma_{\textrm{ess}}$, or equivalently, $\sigma_{\textrm{ess}} \subset \mathbb{C}\backslash \Omega = \{\textrm{\rm Re}\, \lambda \leq - \chi_0 < 0\}$.
\end{proof}
The significance of Corollary \ref{lemspectralgap} is that there is no accumulation of essential spectrum at the eigenvalue $\lambda = 0$, which is an isolated eigenvalue with finite multiplicity. In other words, there is a \textit{spectral gap}.
\subsection{Evans function analysis}
The Evans function (cf. \cite{AGJ90,KaPro13,San02}) is a powerful tool to locate the point spectrum. Thanks to Lemma \ref{lemconsistsplit}, $\Omega$ is the open, connected component of $\mathbb{C} \backslash \sigma_{\textrm{ess}}$ containing the (unstable) right half-plane in which the asymptotic matrices $\mathbf{A}^\tau_{\pm}(\cdot)$ are hyperbolic and the dimensions of their stable ${\mathbb{S}^\tau_\pm}$ (respectively, unstable ${\mathbb{U}^\tau_\pm}$) spaces agree. By spectral separation of ${\mathbb{U}^\tau_\pm}, {\mathbb{S}^\tau_\pm}$, the associated eigenprojections are analytic in $\lambda$ and there exists analytic representations for the bases of subspaces ${\mathbb{S}^\tau_\pm}$ and ${\mathbb{U}^\tau_\pm}$ (by a Kato construction, cf. \cite[pp.99--102]{Kat80}). In our special (low dimensional) case, \begin{equation*}
{\mathbb{S}^\tau_+} = \mathrm{span} \{{\mathbf{w}}_+(\lambda)\}, \qquad
{\mathbb{U}^\tau_-} = \mathrm{span} \{{\mathbf{w}}_-(\lambda)\}, \end{equation*} where ${\mathbf{w}}_\pm(\lambda)$ can be chosen analytic in $\lambda \in \Omega$. The associated Evans function \begin{equation}\label{defD}
D^\tau(\lambda) := \det ({\mathbf{w}}_-(\lambda), \, {\mathbf{w}}_+(\lambda)), \end{equation} is defined to locate non-trivial intersections of the initial conditions ${\mathbf{w}}_+$ which produce solutions to the variable coefficient systems \eqref{firstorders} that decay when $x \to +\infty$, with the initial conditions ${\mathbf{w}}_-$ which produce solutions to \eqref{firstorders} that decay at $x \to - \infty$. The Evans function is not unique, but they all differ by a non-vanishing factor. It is endowed with the following properties: \begin{itemize}
\item[\textbullet] $D^\tau$ is analytic in $\lambda \in \Omega$;
\item[\textbullet] $D^\tau(\lambda) = 0$ if and only if $\lambda \in \sigma_{\textrm{pt}} \cap \Omega$; and,
\item[\textbullet] the order of $\lambda$ as a zero of $D^\tau$ is equal to its algebraic multiplicity \end{itemize} In our case, we end up with a family of Evans functions $D^\tau(\cdot)$ indexed by $\tau \in [0,\tau_m)$ and defined on $\Omega$. It is to be observed that the region $\Omega$ is independent of $\tau$, and that the case when $\tau = 0$ is included in the family. In view of Proposition \ref{prop:tau0}, which guarantees the spectral stability of the parabolic Allen-Cahn front, we have the following
\begin{corollary}\label{corD0} $D^0(\lambda) \neq 0$ for all $\textrm{\rm Re}\, \lambda \geq 0$, $\lambda \neq 0$. Moreover, $\lambda = 0$ is a simple zero of $D^0(\cdot)$. \end{corollary}
In order to establish spectral stability in the regime $\tau \in (0, \tau_m)$, we shall apply a result from Evans function theory (see \cite{PZ1}) which assures that, under suitable structural but rather general conditions, the Evans functions for $\tau > 0$ converge uniformly to the Evans function with $\tau = 0$ in bounded regions of $\lambda \in \Omega$. For that purpose, it will be necessary to show that large $|\lambda|$ values belong to the resolvent set. By analiticity and uniform convergence, the non-vanishing property of $D^0$ persists for $D^\tau$ for each $0 < \tau \ll 1$ sufficiently small. Next, by continuity in $\tau$ of eigenvalues and by Lemma \ref{lemam1} and Lemma \ref{lemma:imagevalue}, we rule out possible crossing of eigenvalues across the imaginary axis as $\tau$ varies within the full set $(0,\tau_m)$, establishing point spectral stability for all values of $\tau$ under consideration.
Therefore, let us consider the family of first order systems \eqref{firstorders} for $\lambda \in \Omega$ and with $\tau$ varying in a compact set $\mathcal{V} : = [0,\tau_1]$, with $\tau_1 < \tau_m$. First, we observe that the coefficients $\mathbf{A}^\tau (\cdot,\lambda)$ are functions of $(\lambda, \tau) \in \Omega \times \mathcal{V}$ into $L^\infty(\mathbb{R};\mathbb{R}^{2 \times 2})$ (the coefficients are bounded), they are analytic in $\lambda \in \Omega$ (second order polynomial in $\lambda$), and continuous in $\tau \in \mathcal{V}$ (this follows from the continuity of the coefficients and of the velocity $c$ in $\tau$). Moreover, in view of Theorem \ref{thm:existence}(c), \begin{equation*} c(\tau) = c_0 + \zeta(\tau), \qquad \zeta(\tau) = o(1) \; \text{as} \, \tau \to 0^+. \end{equation*} Here $c_0$ is, of course, the speed of the traveling wave for the parabolic Allen-Cahn equation (or Nagumo front). Also, notice that from the expressions of the coefficients \eqref{asymptcoeffA} we may write \begin{equation*} \mathbf{A}^\tau_\pm(\lambda) = \mathbf{A}^0_\pm(\lambda) + (1-c^2 \tau)^{-1} \mathbf{Q}^\tau_\pm(\lambda), \end{equation*} where the residual is \begin{equation*}
\begin{aligned}
\mathbf{Q}^\tau_\pm(\lambda) &= \begin{pmatrix} 0 & \; & 0 \\
\tau \lambda^2 + \tau(c^2 -a(x))\lambda +c\tau a'(x) & \; & c\tau(cc_0 - a(x) -2\lambda) - (c-c_0) \end{pmatrix}\\
&= (\lambda^2 + \lambda + 1) O(\tau) + O(|\zeta(\tau)|),
\end{aligned} \end{equation*} so that \begin{equation*}
|\mathbf{Q}^\tau_\pm(\lambda)| \leq O(\tau + |\zeta(\tau)|)(1+|\lambda| + |\lambda|^2). \end{equation*} Thanks to exponential decay of the wave \eqref{expodecay} we conclude that the coefficients $\mathbf{A}^\tau(\cdot,\lambda)$ approach exponentially to its limit coefficients $\mathbf{A}^\tau_\pm(\lambda)$ as $x \to \pm \infty$: \begin{equation*}
|\mathbf{A}^\tau(x,\lambda) - \mathbf{A}^\tau_\pm(\lambda)| \leq C e^{-\nu |x|}, \qquad \text{for} \;\; |x| \to +\infty, \end{equation*} uniformly on compact subsets of $(\lambda,\tau) \in \Omega \times \mathcal{V}$. In addition, by Lemma \ref{lemconsistsplit} the limiting coefficient matrices are hyperbolic with agreeing dimensions of their unstable eigenspaces.
Finally, the geometric separation assumption of Gardner and Zumbrun \cite{GZ98} (namely, that the limits of the spaces ${\mathbb{S}^\tau_\pm}$ and ${\mathbb{U}^\tau_\pm}$ along $\lambda$-rays, $\lambda = r\lambda_0$ as $r \to 0^+$, $\lambda_0 \in \Omega$, are continuous) holds trivially in our case as the matrices $\mathbf{A}^\tau_\pm(0)$ are hyperbolic and the eigenspaces are one-dimensional with uniform spectral separation.
To sum up, we have verified that assumptions (A0)-(A1)-(A2) in \cite[p.894]{PZ1} are satisfied, and the systems \eqref{firstorders} belong to the generic class of equations for which there is convergence of approximate flows \cite[Section 2]{PZ1}. We need to verify one final hypothesis to apply \cite[Proposition 2.4]{PZ1}.
\begin{lemma} \label{lemapprox}
Let $(\lambda,\tau) \in \Omega \times \mathcal{V}$. Then the stable eigenvector ${\mathbf{w}}_+(\lambda)$ of $\mathbf{A}^\tau_+(\lambda)$ and the unstable eigenvector ${\mathbf{w}}_-(\lambda)$ of $\mathbf{A}^\tau_-(\lambda)$ converge as $\tau \to 0^+$ with rate $\eta(\tau) := O(\tau + |\zeta(\tau)|)$ to the stable and unstable eigenvectors of $\mathbf{A}^0_+(\lambda)$ and $\mathbf{A}^0_-(\lambda)$, respectively. Moreover, for all $\tau \in \mathcal{V}$, \begin{equation*}
|(\mathbf{A}^\tau - \mathbf{A}^\tau_\pm) - (\mathbf{A}^0 - \mathbf{A}^0_\pm)| \leq C_1 \eta(\tau) e^{-\tilde \nu |x|}, \end{equation*} as $x \to \pm \infty$ for some constants $C_1, \tilde \nu > 0$, uniformly in compact sets of $\Omega$. \end{lemma} \begin{proof} Let $\gamma$ be a closed rectifiable contour enclosing the stable eigenvalue of $\mathbf{A}^\tau_\pm$. By continuity on $\tau$, we may as well select $\gamma$ such that it encloses the stable eigenvalue of $\mathbf{A}^\tau_\pm$ for each $\tau > 0$ sufficiently small. By compactness of $\bar \gamma$ we have a uniform resolvent bound of the form \begin{equation*}
|(\mathbf{A}^0_+ - z)^{-1}| \leq C, \qquad z \in \gamma. \end{equation*} Thus, expanding, \begin{equation*}
\begin{aligned}
(\mathbf{A}^\tau_+ - z)^{-1} &= (\mathbf{A}^0_+ - z + (1-c^2 \tau)^{-1}\mathbf{Q}^\tau_+)^{-1}\\ &= ((\mathbf{A}^0_+ - z)(I + (\mathbf{A}^0_+ - z)^{-1}(1-c^2\tau)^{-1}\mathbf{Q}^\tau_+))^{-1}\\ &= (I + (\mathbf{A}^0_+ - z)^{-1}(1-c^2\tau)^{-1}\mathbf{Q}^\tau_+)^{-1}(\mathbf{A}^0_+ - z)^{-1}\\
&= ( I + O(|\eta(\tau)|))(\mathbf{A}^0_+ - z)^{-1}.
\end{aligned} \end{equation*}
Here $\eta(\tau)$ depends on $|\mathbf{A}^0_+|$. Therefore, the (one-dimensional) projection $\mathbf{P}^\tau_+$ onto ${\mathbb{S}^\tau_+}$ satisfies the bound \begin{equation*}
\begin{aligned}
|\mathbf{P}^\tau_+(\lambda) - \mathbf{P}^0_+(\lambda)|
&= \left| \frac{1}{2\pi i} \oint_\gamma (z-\mathbf{A}^\tau_+(\lambda))^{-1} \, dz \; - \; \frac{1}{2\pi i}
\oint_\gamma (z-\mathbf{A}^0_+(\lambda))^{-1} \, dz\right|\\
&\leq \frac{1}{2\pi i} \oint_\gamma O(|\eta(\tau)|)|(\mathbf{A}^0_+ -z)^{-1}| \, dz \leq C |\eta(\tau)|,
\end{aligned} \end{equation*}
that is, $\mathbf{P}^\tau_+(\lambda) = \mathbf{P}^0_+(\lambda) + O(|\eta(\tau)|)$, showing that ${\mathbf{w}}_+(\lambda) \to {\mathbf{w}}_+^0(\lambda)$ as $\tau \to 0^+$ with rate $|\eta(\tau)|$ in $\Omega$-neighborhoods of $\lambda$. The same applies to the unstable eigenvectors. The second assertion is an immediate consequence of the exponential decay \eqref{expodecay} of the coefficients.
\end{proof}
\subsection{Resolvent estimates and proof of Theorem \ref{thm:spectrum}}
In order to give a complete proof of Theorem \ref{thm:spectrum}, we establish a general resolvent estimates, based on an approximate diagonalization technique introduced by Mascia and Zumbrun \cite{MasZum02}.
\subsubsection{Resolvent estimates}
Given two real symmetrix matrices $\mathbf{A}, {\mathbf{B}}\in\mathbb{R}^{n\times n}$, a real matrix valued function $x\mapsto \mathbf{C}(x)$ defined for any $x\in\mathbb{R}$, and $\lambda\in\mathbb{C}$, let us consider the {\it resolvent equation} \begin{equation}\label{resolvent}
\mathbf{A}{\mathbf{w}_x}+(\lambda {\mathbf{B}}+\mathbf{C}(x)) \mathbf{w} = \mathbf{f}, \qquad \mathbf{f} \in H^m(\mathbb{R};\mathbb{C}^n), \end{equation} for the unknown $\mathbf{w}\in H^m(\mathbb{R};\mathbb{C}^n)$. Here, the concern is to show that, for appropriate choices of sets $\tilde \Omega\subset\mathbb{C}$, there exists some constant $M$ such that for any solution $\mathbf{w}=\mathbf{w}(\cdot, \,\lambda)$ to \eqref{resolvent} with $\lambda\in \tilde \Omega$, there holds \begin{equation}\label{genres}
|\mathbf{w}(\cdot,\lambda)|_{H^m}\leq M|\mathbf{f}|_{H^m}. \end{equation} The first classical result, consequence of the underlying hyperbolic problem from which \eqref{resolvent} arises, provides an estimate for sets $\tilde \Omega$ composed by numbers with large positive real part.
\begin{proposition}\label{prop:resesti0} Given $m\in\mathbb{N}$, let ${\mathbf{B}}$ be positive definite and $\mathbf{C}=\mathbf{C}(x)$ uniformly bounded in $\mathbb{R}$ together with its derivatives of order $j=1,\dots,m$. Then there exist $L,M>0$ such that for any $\lambda$ with $\textrm{\rm Re}\, \lambda\geq L$ there holds \begin{equation}\label{res0}
|\mathbf{w}(\cdot, \lambda)|_{H^m}\leq \frac{M}{\textrm{\rm Re}\, \lambda}\,|\mathbf{f}|_{H^m}. \end{equation} \end{proposition}
\begin{proof} As a first step, let us consider the case of $L^2$, i.e. $m=0$. We recall that if $\mathbf{A}$ is a constant real symmetrix matrix then \begin{equation*}
\mathbf{w}^* \mathbf{A}{\mathbf{w}_x} = \tfrac{1}{2}\left(\mathbf{w}^* \mathbf{A}\mathbf{w}\right)_x. \end{equation*} Taking the scalar product of \eqref{resolvent} against $\mathbf{w}$ and integrating in $\mathbb{R}$, we get \begin{equation*}
\lambda \langle {\mathbf{w}, \mathbf{B}}\mathbf{w}\rangle_{L^2}+\langle \mathbf{w},
\mathbf{C}(x)\mathbf{w} \rangle_{L^2} = \langle \mathbf{w}, \mathbf{f} \rangle_{L^2}. \end{equation*} Since ${\mathbf{B}}$ is symmetric, the term $\langle \mathbf{w},{\mathbf{B}}\mathbf{w}\rangle_{L^2}$ is real; thus, taking the real part, we infer that there exists some constant $C>0$ such that \begin{equation*}
(\textrm{\rm Re}\, \lambda) \langle \mathbf{w}, \mathbf{B}\mathbf{w}\rangle_{L^2}
\leq C\left(|\mathbf{w}|_{L^2}^2+|\mathbf{f}|_{L^2}^2\right) \end{equation*} having used the standard Young inequality. As a final step, under the hypothesis that ${\mathbf{B}}$ is positive definite, it is possible to absorb the term with $\mathbf{w}$ at the right-hand side into the corresponding term in the left-hand side and deduce \eqref{res0} with $m = 0$.
Next, we may proceed inductively, assuming estimate \eqref{res0} for any $j=0,1,\dots,m-1$. By differentiation of \eqref{resolvent}, we deduce that the function $\mathbf{z}:=d^m \mathbf{w}/dx^m$ solves \begin{equation}\label{resolvent_kth}
\mathbf{A} \mathbf{z}_x + (\lambda {\mathbf{B}}+\mathbf{C}(x)){\mathbf{z}}=\frac{d^m\mathbf{f}}{dx^m}
-\sum_{j=0}^{m-1} \binom{m}{j} \frac{d^{m-j} \mathbf{C}}{dx^{m-j}}\,\frac{d^j \mathbf{w}}{dx^j}. \end{equation} Since the coefficients of the derivatives of $\mathbf{C}$ are assumed to be bounded, as a consequence of \eqref{res0} with $m=0$, we infer the estimate \begin{equation*}
|z(\cdot, \,\lambda)|_{{L^2}}\leq \frac{C}{\textrm{\rm Re}\,\,\lambda}
\left(\left|\frac{d^m\mathbf{f}}{dx^m}\right|_{{L^2}}
+\sum_{j=0}^{m-1} \left|\frac{d^jw}{dx^j}\right|_{{L^2}}\right) \end{equation*} for some $C>0$. Then, the inductive assumption provides the conclusion.
\end{proof}
Next, we turn to the problem of proving \eqref{genres} for sets $\tilde \Omega$ contained in the half plane $\{\textrm{\rm Re}\, \lambda\geq 0\}$ and with sufficiently large modulus. Of course, additional restrictions on the matrices $\mathbf{A}, {\mathbf{B}}, \mathbf{C}$ are needed. Our approach is based on the approximate diagonalization procedure presented in \cite[p.817 and following]{MasZum02}. Specifically, let us consider a change of variable $\mathbf{w}=\mathbf{T}(x,\lambda)\mathbf{v}$ with $\mathbf{T}(x,\lambda)$ in the form \begin{equation*}
\mathbf{T}(x,\lambda)=\mathbf{T}_0(I+\lambda^{-1}\mathbf{T}_1(x)), \end{equation*} with ${\mathbf{T}_0}$ and ${\mathbf{T}_1}={\mathbf{T}_1}(x)$ to be determined. Plugging into \eqref{resolvent} and assuming $\mathbf{A}$ invertible, we deduce the equation solved by the new unknown $\mathbf{v}$ \begin{equation*}
\mathbf{v}_x+\widetilde{\mathbf{A}}(x,\lambda)\mathbf{v}=\tilde{\mathbf{f}}, \end{equation*} where \begin{equation*}
\begin{aligned}
\widetilde{\mathbf{A}}(x,\lambda)
& = (I+\lambda^{-1}{\mathbf{T}_1})^{-1} \Bigl\{\lambda {\mathbf{T}_0}^{-1}\mathbf{A}^{-1}{\mathbf{B}}{\mathbf{T}_0}
+{\mathbf{T}_0}^{-1}\mathbf{A}^{-1}\bigl({\mathbf{B}}{\mathbf{T}_0}{\mathbf{T}_1}+\mathbf{C}{\mathbf{T}_0}\bigr)\\
&\hskip5.0cm +\lambda^{-1}\bigl({\mathbf{T}_0}^{-1}\mathbf{A}^{-1}\mathbf{C}{\mathbf{T}_0}{\mathbf{T}_1}+{\mathbf{T}_1} '\bigr)\Bigr\},\\ \tilde{\mathbf{f}}&=(I+\lambda^{-1}{\mathbf{T}_1}(x))^{-1}{\mathbf{T}_0}^{-1}\mathbf{A}^{-1}\mathbf{f}.
\end{aligned} \end{equation*} The matrix $\widetilde{\mathbf{A}}$ can be represented as \begin{equation*}
\widetilde{\mathbf{A}}(x,\lambda)=\lambda \mathbf{D}_0+{\mathbf{D}_1}+o(1)\qquad\qquad\lambda\to\infty \end{equation*} where the matrices $\mathbf{D}_0$ and ${\mathbf{D}_1}$ are given by \begin{equation}\label{D01form}
\mathbf{D}_0:={\mathbf{T}_0}^{-1}\mathbf{A}^{-1}{\mathbf{B}}{\mathbf{T}_0},\qquad
{\mathbf{D}_1}:=[\mathbf{D}_0,{\mathbf{T}_1}(x)]+{\mathbf{T}_0}^{-1}\mathbf{A}^{-1}\mathbf{C}(x){\mathbf{T}_0}, \end{equation} and $[\mathbf{A},{\mathbf{B}}]=\mathbf{A}{\mathbf{B}}-{\mathbf{B}}\mathbf{A}$. If the matrix $\mathbf{A}^{-1}{\mathbf{B}}$ is diagonalizable, ${\mathbf{T}_0}$ can be chosen so that $\mathbf{D}_0$ is diagonal and ${\mathbf{T}_1}={\mathbf{T}_1}(x)$ is such that the matrix ${\mathbf{D}_1}$ is diagonal (see \cite[Lemma 4.6]{MasZum02}). Moreover, since $[\mathbf{D}_0,\mathbf{T}]$ is equal to zero for any diagonal matrix, the term ${\mathbf{T}_1}$ can be chosen with zero entries in the principal diagonal. With these choices, the special form of the system satisfied by $\mathbf{v}$ can be used to obtain a different form of estimate \eqref{genres}.
\begin{proposition}\label{prop:resesti1} Given $m\in\mathbb{N}$, let the matrices $\mathbf{A}$ ${\mathbf{B}}$ and $\mathbf{C}$ be such that $\mathbf{A}$ is invertible, $\mathbf{A}^{-1}{\mathbf{B}}$ is diagonalizable in $\mathbb{R}$, and $\mathbf{C}=\mathbf{C}(x)$ is uniformly bounded in $\mathbb{R}$ together with all its derivatives of order $j=1,\dots,m$. Let the elements of $\mathbf{D}_0$ and ${\mathbf{D}_1}$, defined in \eqref{D01form}, be denoted by $\mu_0^k$ and $\mu_1^k$, $k=1,\dots,n$. If $\mu_0^k$ and $\textrm{\rm Re}\, \mu_1^k$ have the same sign for any $k$ and \begin{equation}\label{mind1k}
\min_{k=1,\dots,n}\inf_{x\in\mathbb{R}} |\textrm{\rm Re}\, \mu_1^k|>0, \end{equation}
then there exist $R>0$ and $M$ such that the estimate \eqref{genres} holds for any $\lambda\in\tilde\Omega_R:=\{|\lambda|\geq R,\,\textrm{\rm Re}\, \lambda\geq 0\}$. \end{proposition}
\begin{proof} As in the proof of Proposition \ref{prop:resesti0}, we initially consider the $L^2$-case, i.e. $m=0$. With the change of variable $\mathbf{v}=\mathbf{T}(x,\lambda)\mathbf{w}$ where $\mathbf{T}$ has been chosen following the above procedure, we end up with a system for the unknown $\mathbf{v}$, whose $k-$th component solves \begin{equation*}
\frac{dv_k}{dx}+\left(\lambda \mu_{0}^{k}+\mu_{1}^{k}(x)\right)v_k
=o(1)v_k + \tilde{f}_k. \end{equation*} Taking the scalar product against $\bar v_k$, integrating in $\mathbb{R}$, and taking its real part, we deduce \begin{equation*}
\int_{\mathbb{R}}\left(\textrm{\rm Re}\, \lambda\,\mu_{0}^{k}+\textrm{\rm Re}\,\,\mu_{1}^{k}(x)\right)|v_k|^2\,dx
=\textrm{\rm Re}\,\int_{\mathbb{R}}(o(1)v_k+\tilde{f}_k)\bar v_k\,dx. \end{equation*} For any $\lambda$ with positive real part, since $\mu_{0}^{k}$ and $\textrm{\rm Re}\,\,\mu_{1}^{k}(x)$ have the same sign, we obtain \begin{equation*}
\int_{\mathbb{R}}\left(\textrm{\rm Re}\,\lambda\,|\mu_{0}^{k}|+|\textrm{\rm Re}\,\,\mu_{1}^{k}(x)|\right)|v_k|^2\,dx
\leq o(1)|v|_{{L^2}}^2+C|\tilde{f}|_{{L^2}}^2 \end{equation*} for some $C>0$ and then, as a consequence of \eqref{mind1k}, \begin{equation*}
C_0|\mathbf{v}|_{{L^2}}^2\leq o(1)|\mathbf{v}|_{{}_{L^2}}^2+C|\tilde{\mathbf{f}}|_{{L^2}}^2 \end{equation*}
for some $C_0>0$. For $|\lambda|$ sufficiently large, the term $o(1)$ can be controlled by $C_0$, so that we end up with \begin{equation*}
|\mathbf{v}|_{{L^2}}^2\leq C|\tilde{\mathbf{f}}|_{{L^2}}^2. \end{equation*} Finally, since $C$ is uniformly bounded, also $\mathbf{T}$ and ${\mathbf{T}_1}$ are, and the estimate can be brought back to the original variable $\mathbf{w}$.
The case of $m\geq 1$ follows by differentiating \eqref{resolvent} and proceeding as in the proof of Proposition \ref{prop:resesti0}.
\end{proof}
In the case at hand, system \eqref{resolvent} is two-dimensional and defined by the matrices $\mathbf{A}$, ${\mathbf{B}}$ and $\mathbf{C}$ of \eqref{defABC}. Then, the matrix \begin{equation*}
\mathbf{A}^{-1}{\mathbf{B}}=\frac{1}{1-c^2\tau}\begin{pmatrix} -c\tau & \tau \\ 1 & -c\tau \end{pmatrix} \end{equation*} has real eigenvalues $\mu_0^\pm =\pm\sqrt{\tau}(1\mp c\sqrt{\tau})$ and \begin{equation*}
{\mathbf{T}_0}^{-1}\mathbf{A}^{-1}{\mathbf{B}}{\mathbf{T}_0}=\mathbf{D}_0=\textrm{\rm diag}\,(\mu_0^-,\mu_0^+),
\qquad\textrm{where}\quad
{\mathbf{T}_0} =\begin{pmatrix} -\sqrt{\tau} & \sqrt{\tau} \\ 1 & 1 \end{pmatrix}. \end{equation*} Then, straightforward computations give \begin{equation*}
{\mathbf{T}_0}^{-1}\mathbf{A}^{-1}\mathbf{C}(x){\mathbf{T}_0}=\frac{1}{2\sqrt{\tau}}
\begin{pmatrix} -(1-\tau a)/(1-c\sqrt{\tau}) & -(1+\tau a)/(1-c\sqrt{\tau}) \\
(1+\tau a)/(1+c\sqrt{\tau}) & (1-\tau a)/(1+c\sqrt{\tau}) \end{pmatrix}, \end{equation*} (where $a=a(x)$) so that \begin{equation*}
{\mathbf{D}_1}=\textrm{\rm diag}\,(\mu_1^-,\mu_1^+)
=\frac{1}{2\sqrt{\tau}}\textrm{\rm diag}\,\left(-\frac{1-\tau a}{1-c\sqrt{\tau}},\frac{1-\tau a}{1+c\sqrt{\tau}}\right). \end{equation*} Thus, if the function $a=a(x)$ is such that $\sup a<{1}/{\tau}$, then assumption of Proposition \ref{prop:resesti1} holds and the resolvent estimate
\eqref{genres} holds in $\tilde\Omega_R:=\{|\lambda|\geq R,\,\textrm{\rm Re}\,\,\lambda\geq 0\}$ for $R$ sufficiently large.
\subsubsection{Proof of Theorem \ref{thm:spectrum}}
In view of Proposition \ref{prop:resesti1}, take $R > 0$ sufficiently large so that $\sigma_{\textrm{pt}} \cap \Omega \subset \{|\lambda| \leq R\}$, and consider the following compact subset of $\Omega$: \begin{equation*}
\Omega_R := \{\lambda \in \mathbb{C} \, : \, |\lambda| \leq R, \, \textrm{\rm Re}\, \, \lambda \geq - \tfrac{1}{2} \chi_0\}. \end{equation*}
By Lemma \ref{lemapprox}, systems \eqref{firstorders} satisfy the hypotheses of \cite[Proposition 2.4]{PZ1}. Hence, for each $\tau \in \mathcal{V}$ and in a $\Omega_R$-neighborhood of $\lambda$, the local Evans functions $D^\tau(\lambda)$ converge uniformly to $D^0(\lambda)$ in a (possible smaller) neighborhood of $\lambda$ as $\tau \to 0^+$ with rate $|D^\tau(\cdot) - D^0(\cdot)| = O(\eta(\tau)) = O(\tau + |\zeta(\tau)|) \to 0$.
By point spectral stability for $\tau = 0$ (Corollary \ref{corD0}), and by analiticity and uniform convergence, we conclude that $D^\tau(\lambda) \neq 0$ for $\lambda \in \Omega_R$, $\textrm{\rm Re}\, \, \lambda \geq 0$, except only at $\lambda = 0$, and for each $0 \leq \tau \ll 1$ sufficiently small. Hence there exists $\tau_0 \in (0,\tau_m)$ such that point spectral stability holds for each $\tau \in (0,\tau_0)$. Finally, noticing that $\Omega_R$ contains only isolated eigenvalues with finite multiplicity, and by continuous dependence of eigenvalues of Fredholm operators in Banach spaces with respect to its coefficients (cf. \cite{MoZe96}), the eigenvalues $\lambda = \lambda(\tau) \in \sigma_{\textrm{pt}} \cap \Omega_R$ are continuous functions of $\tau \in (0,\tau_m)$. In view of Lemma \ref{lemma:imagevalue}, such eigenvalues may cross the imaginary axis towards the unstable half plane only through the origin. But $\lambda = 0 \in \sigma_{\textrm{pt}}$ is a simple eigenvalue for each $\tau \in (0,\tau_m)$ as proved in Lemma \ref{lemam1}. Hence all point spectrum remains in the stable half plane $\textrm{\rm Re}\, \, \lambda < 0$ for all $\tau \in (0,\tau_m)$. This proves the Theorem.
\section{Decaying semigroup and nonlinear stability}\label{sect:linnonlinstab}
In this Section, we establish the conditions for the generation of a $C_0$-semi\-group of solutions operators for the linearization around the wave, as well as for its asymptotic decaying properties. We also present the proof of nonlinear (orbital) stability, which uses such information in a key way.
\subsection{Generation of the semigroup}
We are now ready to establish that each operator $\mathcal{L}^\tau$ generates a $C_0$ semigroup in $L^2(\mathbb{R};\mathbb{C}^2)$.
\begin{lemma}
\label{lemgenC0} For each $\tau \in (0,\tau_m)$, the operator $\mathcal{L}^\tau : D = H^2(\mathbb{R};\mathbb{C}^2) \to L^2(\mathbb{R};\mathbb{C}^2)$ is the infinitesimal generator of a $C_0$-semigroup, $\{{\mathcal{S}(t)}\}_{t\geq 0}$, satisfying \begin{equation}
\label{quasicontr}
\|{\mathcal{S}(t)}\| \leq e^{\omega t}, \end{equation} for some $\omega = \omega(\tau) \in \mathbb{R}$, all $t \geq 0$. \end{lemma}
Here $\| \cdot \|$ denotes the operator norm.
\begin{proof} First, we note that the domain $D = H^2(\mathbb{R};\mathbb{C}^2)$ is dense in $L^2(\mathbb{R};\mathbb{C}^2)$. Now, for each $\mathbf{u} = (u,v)^\top \in D$: \begin{equation*}
\begin{aligned}
\textrm{\rm Re}\, \, \langle {\mathbf{u}}, \mathcal{L}^\tau {\mathbf{u}}\rangle_{L^2} &= - \textrm{\rm Re}\, \, \langle {\mathbf{u}}, {\mathbf{B}}^{-1}( \mathbf{A} {\mathbf{u}_x} + \mathbf{C}(x){\mathbf{u}}) \rangle_{L^2}\\ &= - \textrm{\rm Re}\, \, \langle {\mathbf{u}}, {\mathbf{B}}^{-1} \mathbf{A} {\mathbf{u}_x}\rangle_{L^2} - \textrm{\rm Re}\, \, \langle {\mathbf{u}}, {\mathbf{B}}^{-1}\mathbf{C}(x) {\mathbf{u}} \rangle_{L^2}\\
&= - \int_\mathbb{R} a(x) |u|^2 \, dx + \tau^{-1} |v|^2_{L^2}\\
&\leq \sup_\mathbb{R} |a(x)| |u|^2_{L^2} + \tau^{-1} |v|^2_{L^2} \leq \omega |{\mathbf{u}}|_{L^2}^2,
\end{aligned} \end{equation*}
with $\omega = \max \{\sup |a(x)|, \tau^{-1}\} > 0$.
Now, thanks to the resolvent estimate \eqref{genres} for any $|\lambda| \geq R$, $\textrm{\rm Re}\, \, \lambda \geq 0$ with $R$ sufficiently large, there are no $L^2$ solutions to $\mathcal{L}^\tau {\mathbf{u}} = \lambda_0 {\mathbf{u}}$ for $\lambda_0 \in \mathbb{R}$, $\lambda_0 > \omega$ and sufficiently large. Thus, for each $\lambda_0 > \omega$ sufficiently large $\mathcal{L} - \lambda_0$ is onto. A direct application of the classical Hille-Yosida theorem \cite{EN00,Pa83} yields the result together with the estimate \eqref{quasicontr}.
\end{proof}
As a consequence of the semigroup properties we have that \begin{equation*}
\frac{d}{dt} ({\mathcal{S}(t)}{\mathbf{u}}) = {\mathcal{S}(t)} \mathcal{L}^\tau {\mathbf{u}} = \mathcal{L}^\tau {\mathcal{S}(t)}{\mathbf{u}}, \end{equation*} for all ${\mathbf{u}} = (u,v)^\top \in D = H^2(\mathbb{R};\mathbb{C}^2)$.
Naturally, the growth rate $\omega$ of estimate \eqref{quasicontr} is not optimal. Actually, $\omega \to +\infty$ as $\tau \to 0^+$, due to the fact that, in the limit, the operator $\mathcal{L}^\tau$ is not defined and becomes singular. The optimal growth rate in the appropriate subspace will be provided by spectral stability. The significance of Lemma \ref{lemgenC0} is simply that, for each fixed $\tau \in (0,\tau_m)$, the operator $\mathcal{L}^\tau$ is the generator of a $C_0$-semigroup. Let us recall the growth bound for a semigroup ${\mathcal{S}(t)}$: \begin{equation*}
\omega_0 = \inf \{\omega \in \mathbb{R} \, : \, \lim_{t \to + \infty} e^{-\omega t}\|{\mathcal{S}(t)}\| \; \text{exists}\}. \end{equation*} We say a semigroup is uniformly (exponentially) stable whenever $\omega_0 < 0$. Let $\mathcal{L}$ be the infinitesimal generator of the semigroup ${\mathcal{S}(t)}$. Its spectral bound is defined as \begin{equation*}
s(\mathcal{L}) = \sup \{ \textrm{\rm Re}\,\, \lambda \, : \, \lambda \in \sigma(\mathcal{L})\}. \end{equation*} Since the spectral mapping theorem --- namely, that, $\sigma({\mathcal{S}(t)}) \backslash \{0\} = e^{t\sigma(\mathcal{L})}$ --- is \textit{not} true in general for $C_0$-semi\-groups (cf. \cite{EN00}), for stability purposes we rely on the Gearhart-Pr\"uss theorem \cite{Gea78,Pr84}, which restricts our attention to semigroups on Hilbert spaces (see also \cite{CrL,EN00}). It states that any $C_0$-semigroup $\{{\mathcal{S}(t)}\}_{t\geq 0}$ on a Hilbert space $H$ is uniformly exponentially stable if and only if its generator satisfies $s(\mathcal{L}) <0$, and the following resolvent estimate holds: \begin{equation*}
\sup_{\textrm{\rm Re}\,\, \lambda > 0} \|(\mathcal{L} - \lambda)^{-1}\| < + \infty. \end{equation*} This task is already substantially completed thanks to the general resolvent estimates of the previous section. It remains to be shown that the estimate holds inside a half circle, with large radius, and on the projected space, which is the content of the proof of Proposition \ref{propues} below.
It is known (see Kato \cite[Remark 6.23, p.184]{Kat80}) that if $\lambda \in \mathbb{C}$ is an eigenvalue of a closed operator $\mathcal{L} : D \subset H \to H$ then $\overline{\lambda}$ is an eigenvalue of $\mathcal{L}^*$ (formal adjoint) with the same geometric and algebraic multiplicities. Also, since $H^2$ and $L^2$ are reflexive Hilbert spaces, $\mathcal{L} : D = H^2 \to L^2$ has a formal adjoint which is also densely defined and closed. Moreover, $\mathcal{L}^{**} = \mathcal{L}$ (cf. \cite[Theorem 5.29, p.168]{Kat80}). Upon these observations we immediately have the following \begin{lemma}
\label{lempre1} $\lambda = 0$ is an isolated, simple eigenvalue of the formal adjoint \begin{equation*}
(\mathcal{L}^\tau)^* : D(\mathcal{L}^\tau) = H^2(\mathbb{R};\mathbb{C}^2) \to L^2(\mathbb{R};\mathbb{C}^2), \end{equation*} and there exists an eigenfunction $(\Psi,\Phi)^\top \in D(\mathcal{L}^\tau)$ such that $(\mathcal{L}^\tau)^* (\Psi,\Phi)^\top = 0$. \end{lemma}
Let us denote the inner product: \begin{equation*}
\Theta := \langle (U_x, V_x), (\Psi,\Phi) \rangle_{L^2} = \int_{-\infty}^{+\infty} \begin{pmatrix}
U_x \\ V_x
\end{pmatrix}^* \begin{pmatrix}
\Psi \\ \Phi
\end{pmatrix} \, dx. \end{equation*} It is not hard to see that $\Theta \neq 0$. Indeed, suppose by contradiction that $\Theta = 0$. Then $(\Psi,\Phi)^\top \in (\ker {\mathcal{L}^\tau})^\perp = {\mathrm{range}}({\mathcal{L}^\tau}^*)$. Hence, there exists $0 \neq (u,v)^\top \in D(\mathcal{L}^\tau) = H^2$ such that $(\mathcal{L}^\tau)^*(u,v)^\top = (\Psi,\Phi)^\top$. Thus, $((\mathcal{L}^\tau)^*)^2(u,v)^\top = 0$, which is a contradiction with $\lambda = 0$ being a simple eigenvalue of $(\mathcal{L}^\tau)^*$. Thus, we may define the Hilbert space $\tilde X \subset L^2(\mathbb{R};\mathbb{C}^2)$ as the range of the spectral projection, \begin{equation*}
{\mathcal{P}}\begin{pmatrix}
u \\ v
\end{pmatrix}:= \begin{pmatrix}
u \\ v
\end{pmatrix} - \Theta^{-1}\langle (u,v),(\Psi,\Phi)\rangle_{L^2}\,\begin{pmatrix}
U_x \\ V_x
\end{pmatrix}. \end{equation*} In this fashion we project out the eigenspace spanned by the single eigenfunction $(U_x, V_x)^\top$. Outside this eigenspace, the semigroup decays exponentially, as we shall see next.
\subsection{Linear decay rates}
We now observe that on a reflexive Banach space, weak and weak$^*$ topologies coincide, and therefore the family of dual operators $\{ {\mathcal{S}(t)}^*\}_{t\geq 0}$, consisting of all the formal adjoints in $L^2$ is a $C_0$-semigroup as well (cf. \cite[p.44]{EN00}). Moreover, the infinitesimal generator of this semigroup is simply $(\mathcal{L}^\tau)^*$ (see \cite[Corollary 10.6]{Pa83}). By semigroup properties we readily have \begin{equation*}
{\mathcal{S}(t)} \begin{pmatrix}
U_x \\ V_x
\end{pmatrix} = \begin{pmatrix}
U_x \\ V_x
\end{pmatrix}, \qquad {\mathcal{S}(t)}^* \begin{pmatrix}
\Psi \\ \Phi
\end{pmatrix}
= \begin{pmatrix}
\Psi \\ \Phi
\end{pmatrix}. \end{equation*} As a result of these properties and the definition of the projector it is easy to verify that \begin{equation*}
{\mathcal{S}(t)} {\mathcal{P}} = {\mathcal{P}} {\mathcal{S}(t)}. \end{equation*} Hence $\tilde X$ is an ${\mathcal{S}(t)}$-invariant closed (Hilbert) subspace of $H^2(\mathbb{R};\mathbb{C}^2)$. So we define the domain \begin{equation*}
\tilde{D} := \{ \mathbf{u} \in D \cap \tilde X \, : \, \mathcal{L}^\tau \mathbf{u} \in \tilde X \} \end{equation*} and the operator \begin{equation*}
\widetilde{\mathcal{L}^\tau} : \tilde D \subset \tilde X \to \tilde X, \end{equation*} \begin{equation*}
\widetilde{\mathcal{L}^\tau} \mathbf{u} := \mathcal{L}^\tau \mathbf{u}, \qquad \mathbf{u} \in \tilde D, \end{equation*} as the restriction of $\mathcal{L}^\tau$ on $\tilde X$. Therefore, $\widetilde{\mathcal{L}^\tau}$ is a closed, densely defined operator on the Hilbert space $\tilde X$. Moreover, we observe that $0 \neq (U_x,V_x)^\top \in \ker {\mathcal{P}}$. Hence, $\lambda = 0 \notin \sigma_{\textrm{pt}}(\widetilde{\mathcal{L}^\tau})$. As a consequence of point spectral stability of $\mathcal{L}^\tau$ we readily obtain \begin{equation*}
\sigma( \widetilde{\mathcal{L}^\tau} ) \subset \{ \lambda \in \mathbb{C} \, : \, \textrm{\rm Re}\, \, \lambda < 0 \}, \end{equation*} and hence the spectral bound of $\widetilde{\mathcal{L}^\tau}$ is strictly negative, $s(\widetilde{\mathcal{L}^\tau}) < 0$. By the above observations, we obtain the following \begin{lemma} The family of operators $\{{\widetilde{\mathcal{S}}(t)}\}_{t\geq 0}$, ${\widetilde{\mathcal{S}}(t)} : \tilde X \to \tilde X$, defined as \begin{equation*}
{\widetilde{\mathcal{S}}(t)} \mathbf{u} := {\mathcal{S}(t)} {\mathcal{P}} \mathbf{u}, \qquad \mathbf{u} \in \tilde X, \;\; t \geq 0, \end{equation*} is a $C_0$-semigroup in the Hilbert space $\tilde X$ with infinitesimal generator $\widetilde{\mathcal{L}^\tau}$. \end{lemma}
\begin{proof}
The semigroup properties are inherited from those of ${\mathcal{S}(t)}$ in $L^2(\mathbb{R};\mathbb{C}^2)$.
That $\widetilde{\mathcal{L}^\tau}$ is the infinitesimal generator follows from Corollary in Section 2.2 of \cite{EN00}, p.61.
\end{proof}
Finally, we apply Gearhart-Pr\"uss theorem.
\begin{proposition}[Uniform exponential stability] \label{propues} For each $\tau \in (0,\tau_m)$ there exist constants $C \geq 1$ and $\theta > 0$ such that \begin{equation}\label{decayest}
|{\widetilde{\mathcal{S}}(t)} {\mathbf{u}}|_{L^2}
\leq C e^{-\theta t}|{\mathbf{u}}|_{L^2}, \qquad {\mathbf{u}} \in \tilde X, \;\; t \geq 0. \end{equation} \end{proposition}
\begin{proof}
In view of the resolvent estimates \eqref{genres}, we can find a radius sufficiently large such that, if $|\lambda| \geq R$ and $\textrm{\rm Re}\, \, \lambda \geq 0$, then \begin{equation*}
\| (\mathcal{L}^\tau - \lambda)^{-1} \|_{L^2 \to L^2} \leq C \end{equation*} for some uniform $C > 0$. Since $\widetilde{\mathcal{L}^\tau} = \mathcal{L}^\tau$ on the subspace $\tilde X \subset L^2(\mathbb{R};\mathbb{C}^2)$, the same estimate applies to $\widetilde{\mathcal{L}^\tau}$ outside that half circle. Inside, however, thanks to (strict) point spectral stability of the operator restricted to $\tilde X$, the resolvent of $\widetilde{\mathcal{L}^\tau}$ is uniformly bounded inside the intersection of any ball of finite radius and $\textrm{\rm Re}\, \, \lambda \geq 0$. We conclude that \begin{equation*}
\sup_{\textrm{\rm Re}\, \, \lambda > 0} \| (\widetilde{\mathcal{L}^\tau}- \lambda)^{-1}\|_{\tilde X \to \tilde X} \leq C, \end{equation*} for some $C > 0$ independent of $\lambda$. In addition, $s(\widetilde{\mathcal{L}^\tau}) < 0$. Thus, a direct application of Gearhart-Pr\"uss theorem to the operator $\widetilde{\mathcal{L}^\tau}$ on the Hilbert space $\tilde X$ implies that the semigroup ${\widetilde{\mathcal{S}}(t)}$ is uniformly exponentially stable, and that estimate \eqref{decayest} holds for some $C \geq 1$ and some $\theta > 0$.
\end{proof}
This result establishes the decaying properties of the linearized operator around the wave, that is, linear stability. The latter can be summarized in the following
\begin{theorem}[Linear stability]\label{thm:linstab} There exists a projection operator ${\mathcal{Q}} = I- {\mathcal{P}}$ with one-dimensional range $\mathrm{span}\{(U_x, V_x)^\top\} \subset L^2(\mathbb{R};\mathbb{C}^2)$ such that for any $t>0$ \begin{equation*}
{\mathcal{S}(t)}{\mathcal{Q}}={\mathcal{Q}} {\mathcal{S}(t)}={\mathcal{Q}}
\qquad\textrm{and}\qquad
\|{\mathcal{S}(t)}(I-{\mathcal{Q}})\|\leq C\,e^{-\theta t} \end{equation*} for some $C, \theta>0$. \end{theorem}
\subsection{Nonlinear stability} \label{secnonlinear} The proof of Theorem \ref{thm:nonlinstab} on nonlinear orbital stability of the traveling fronts for \eqref{relAC} is a consequence of the linear decay estimates combined in a smart way with the standard Duhamel representation formula. The main difficulty stems in the fact that a single traveling front is not isolated as a stationary solution and it belongs to a one-dimensional manifold generated by applying an arbitrary translation in space. This is a common feature of many autonomous evolutive PDEs when considered in the whole space as a consequence of the underlying translation invariance of the corresponding initial value problem.
At the linear level, such feature of the problem is expressed by the membership of $\lambda=0$ to the spectrum of the linearized operator and by the presence of a time-independent projection term into the representation of the solution semigroup. Converting such structure at the nonlinear level amounts in identifying a nonlinear projection operator describing the convergence of a given perturbed initial datum to a translate of the original front. A possible approach is based on the application of the Implicit Function Theorem in Banach spaces and it has been used by Sattinger in the classical paper \cite{Satt76}. For the sake of clarity, we first present here a restyled version of this approach in the framework of Hilbert spaces, as needed in our case, and then apply it to prove Theorem \ref{thm:nonlinstab}.
Let $\mathcal{W}$ be a Hilbert space with norm $|\cdot|_{{}_{\mathcal{W}}}$ and let $B_r(\overline{W})$ be the open ball with center $\overline{W}$ and radius $r$. Let $F$ be a smooth function from $\mathcal{D}\subset\mathcal{W}$ into $\mathcal{W}$ such that $F(\overline{W})=0$ for some $\overline{W}\in \mathcal{D}$. Additionally, let us assume that, for some $r>0$, there holds \begin{equation*}
\{W\in \mathcal{W}\,:\,F(W)=0\}\cap \{|W-\overline{W}|_{{}_{\mathcal{W}}}<r\}=\phi(I) \end{equation*} for some smooth function $\phi\,:\,I\to\mathcal{W}$, $I \subset \mathbb{R}$ an open interval. Without loss of generality, we may assume $0\in I$ and $\phi(0)=\overline{W}$.
Let $W=W(t;W_0)$ be the solution to the abstract Cauchy problem \begin{equation}\label{cauchy2}
\frac{dW}{dt}=F(W), \qquad\qquad W(0)=W_0\in\mathcal{D}. \end{equation} By assumption, there holds $W(t;\phi(\delta))=\phi(\delta)$ for any $t$.
The linearized problem at $\phi(\delta)$ is \begin{equation}\label{lincauchy2}
\frac{dZ}{dt}=dF(\phi(\delta))Z, \qquad\qquad Z(0)=Z_0\in\mathcal{D}. \end{equation} Differentiating with respect to $\delta$ the relation $F(\phi(\delta))=0$ for $\delta\in I$, we infer \begin{equation*}
dF(\phi(\delta))\,\phi'(\delta)=0, \end{equation*} showing that $0\in \sigma\bigl(dF(\phi(\delta))\bigr)$ and that $r(\delta):=\phi'(\delta)$ is a right eigenvector of $dF(\phi(\delta))$. Let us denote by $\ell(\delta)$ the unique left eigenvector of $dF(\phi(\delta))$ such that $\ell(\delta)\cdot r(\delta)=1$. Equivalently, $\ell(\delta)$ can be defined as the unique element in the kernel of the adjoint operator $dF(\phi(\delta))^\ast$ satisfying the normalization condition $\ell(\delta)\cdot r(\delta)=1$. We also set for $\delta\in I$ \begin{equation*}
P(\delta):=r(\delta)\otimes \ell(\delta),\qquad Q(\delta):=I-P(\delta). \end{equation*} In particular, there hold \begin{equation*}
dF(\phi)P=P\,dF(\phi)=0, \qquad\textrm{and}\qquad
dF(\phi)Q=Q\,dF(\phi)=dF(\phi), \end{equation*} where the dependency on $\delta$ has been omitted for shortness. \vskip.25cm
We assume the following hypotheses. \vskip.15cm
{\bf H1.} There exist $C, \theta>0$ such that the solution $Z=Z(t;Z_0,\delta)$ to \eqref{lincauchy2} is such that \begin{equation}\label{lindecay2}
|Q(\delta)Z(t;Z_0,\delta)|\leq Ce^{-\theta t}|Q(\delta)Z_0| \end{equation} for any $Z_0\in\mathcal{D}$. \vskip.15cm
{\bf H2.} The function $\phi$ is differentiable at $\delta=0$ and there exist $C, \delta_0, \gamma>0$ such that \begin{equation}\label{regophi}
|\phi(\delta)-\phi(0)-\phi'(0)\delta|_{{}_{\mathcal{W}}}\leq C\delta^{1+\gamma}, \end{equation}
for $|\delta|<\delta_0$. \vskip.15cm
{\bf H3.} There exist $C, M, \delta_0,\gamma>0$ such that the function $F$ is differentiable at $\phi(\delta)$ for any $\delta\in(-\delta_0,\delta_0)$ and \begin{equation}\label{regoF}
|F(\phi(\delta)+W)-F(\phi(\delta))-dF(\phi(\delta))W|_{{}_{\mathcal{W}}}
\leq C|W|_{{}_{\mathcal{W}}}^{1+\gamma}, \end{equation}
for $|\delta|<\delta_0$ and $|W|_{{}_{\mathcal{W}}}\leq M$.
\begin{theorem}\label{thm:abstract} Assume that hypotheses {\bf H1}, {\bf H2} and {\bf H3} hold. Then there exists $\varepsilon>0$ such that for any $W_0\in B_\varepsilon(\bar W)$ there exists $\delta\in I$ for which the solution $W(t;W_0)$ to \eqref{cauchy2} satisfies \begin{equation}\label{nonlindecay2}
|W(t;W_0)-\phi(\delta)|_{{}_{\mathcal{W}}}\leq C|W_0-\overline{W}|_{{}_{\mathcal{W}}}\,e^{-\theta\,t} \end{equation} for some $C,\theta>0$ \end{theorem}
\begin{proof} Given $W_0\in \mathcal{W}$, let $w_0\in \mathcal{W}$ be such that
$W_0=\overline{W}+\varepsilon w_0$ where $\varepsilon:=|W_0-\overline{W}|_{{}_{\mathcal{W}}}$ and let the solution $W$ to \eqref{cauchy2} be decomposed as \begin{equation*}
W=\phi(\varepsilon\eta)+\varepsilon\,w \end{equation*} with $\eta=\eta(\varepsilon)$ to be determined later, where the function $w$ solves \begin{equation}\label{cauchypert2}
\frac{dw}{dt}=dF(\phi(\varepsilon\eta))w+\varepsilon^\gamma R (\eta,w;\varepsilon),
\quad
w(0)=w_0-\phi'(0)\eta-\varepsilon^\gamma \psi(\eta;\varepsilon) \end{equation} where \begin{equation*}
\begin{aligned}
R (\eta,w;\varepsilon)&:=\varepsilon^{-1-\gamma}
\left\{F(\phi(\varepsilon\eta)+rw)-F(\phi(\varepsilon\eta))-dF(\phi(\varepsilon\eta))rw\right\},\\
\psi (\eta;\varepsilon)&:=\varepsilon^{-1-\gamma}
\left\{\phi(\varepsilon\eta)-\phi(0)-\phi'(0)\varepsilon\eta\right\}.\\
\end{aligned} \end{equation*} Decomposing $w$ as \begin{equation*}
w=\alpha\,\phi'(\varepsilon\eta)+\omega, \qquad\qquad\textrm{where}\qquad
\alpha:=\ell(\varepsilon\eta)\cdot w,\quad \omega:=Q(\varepsilon\eta) w, \end{equation*} and setting \begin{equation*}
S(\eta,\alpha,\omega;\varepsilon):=R(\eta,\alpha\,\phi'(\varepsilon\eta)+\omega;\varepsilon), \end{equation*} the unknowns $\alpha$ and $\omega$ solve \begin{equation}
\left\{\begin{aligned}
\frac{d\alpha}{dt}&=\varepsilon^\gamma \,\ell(\varepsilon\eta)\cdot S(\eta,\alpha,\omega;\varepsilon),\\
\frac{d\omega}{dt}&=dF(\phi(\varepsilon\eta))\omega
+\varepsilon^\gamma Q(\varepsilon\eta) S(\eta,\alpha,\omega;\varepsilon),
\end{aligned}\right. \end{equation} with initial conditions \begin{equation}
\left\{\begin{aligned}
\alpha(0)&=\ell(\varepsilon\eta)\cdot\bigl(w_0-\phi'(0)\eta-\varepsilon^\gamma \psi(\eta;\varepsilon)\bigr),\\
\omega(0)&=Q(\varepsilon\eta)\bigl(w_0-\phi'(0)\eta-\varepsilon^\gamma \psi(\eta;\varepsilon)\bigr).
\end{aligned}\right. \end{equation} Therefore, the following relations hold \begin{equation}\label{alphaomega}
\begin{aligned}
\alpha(t)&=\ell\cdot\bigl(w_0-\phi'(0)\eta\bigr)
-\varepsilon^\gamma\left\{\ell\cdot\psi-\int_0^t \ell\cdot S\,d\tau\right\},\\
\omega(t)&=e^{dF(\phi)t}Q\bigl(w_0-\phi'(0)\eta\bigr)\\
&\hskip2.5cm
-\varepsilon^\gamma\left\{e^{dF(\phi)t}Q\psi-\int_0^t e^{dF(\phi)(t-\tau)}QS\,d\tau\right\}.
\end{aligned} \end{equation} where $\ell=\ell(\varepsilon\eta), Q=Q(\varepsilon\eta), \phi=\phi(\varepsilon\eta), \psi=\psi(\eta;\varepsilon)$ and $S=S(\eta,\alpha,\omega;\varepsilon)$.
Next, we require the value $\eta=\eta(\varepsilon)$ to be such that $\alpha(+\infty)=0$ that is \begin{equation*}
\ell\cdot\bigl(-w_0+\phi'(0)\eta\bigr)
+\varepsilon^\gamma\left\{\ell\cdot\psi -\int_0^{+\infty} \hskip-.25cm \ell\cdot S\,d\tau\right\}=0. \end{equation*} Thus, the triple $(\eta,\alpha,\omega)$ has to be such that \begin{equation}\label{implrel}
\mathcal{F}(\eta,\alpha,\omega;\varepsilon)+\varepsilon^\gamma \mathcal{G}(\eta,\alpha,\omega;\varepsilon) =0 \end{equation} where \begin{equation*}
\begin{aligned}
\mathcal{F}&=\Bigl(\ell\cdot\bigl(-w_0+\phi'(0)\eta\bigr),
\alpha, \omega-e^{dF(\phi)t} Q\bigl(w_0-\phi'(0)\eta\bigr) \Bigr)\\
\mathcal{G}&=\Bigl(\ell\cdot\psi-\int_0^{+\infty} \hskip-.25cm \ell\cdot S\,d\tau ,
\int_t^{+\infty} \hskip-.25cm \ell\cdot S\,d\tau , e^{dF(\phi)t} Q\psi-\int_0^t e^{dF(\phi)(t-\tau)} QS\,d\tau \Bigr).
\end{aligned} \end{equation*} We want to show that, for small $\varepsilon$, the implicit relation \eqref{implrel} defines a function $\varepsilon\mapsto (\eta,\alpha,\omega)$. To prepare for the application of the Implicit function theorem in Banach spaces
(among others, see \cite{AmbrProd95}), let us introduce an appropriate functional setting. Given $\theta>0$ and a Banach space $\mathcal{Y}$ with norm $|\cdot|_{{}_{\mathcal{Y}}}$, set \begin{equation*}
C^0_\theta(\mathbb{R}_+;\mathcal{Y}):=\left\{f\in C^0(\mathbb{R}_+;\mathcal{Y})\,:\,
\sup_{t>0} e^{\theta\,t}|f(t)|_{{}_{\mathcal{Y}}}<+\infty\right\}. \end{equation*} Then, let us consider the Banach space $\mathcal{X}=\mathbb{R}\times C^0_\theta(\mathbb{R}_+;\mathbb{R})\times C^0_\theta(\mathbb{R}_+;\mathcal{W})$ with norm \begin{equation*}
\left\|(\eta,\alpha,\omega)\right\|_{{}_{\mathcal{X}}}
:=|\eta|+\sup_{t>0} e^{\theta\,t}\bigl(|\alpha(t)|+|\omega(t)|_{{}_{\mathcal{W}}}\bigr). \end{equation*} Choosing $\theta$ as in \eqref{lindecay2}, for any $M>0$, the function $\mathcal{F}$
maps the set $\mathcal{X}\times (-\varepsilon,\varepsilon)$ into $\mathcal{X}\cap\{|\eta|\leq M\}$ for $\varepsilon$ sufficiently small, since \begin{equation*}
e^{\theta\,t} |e^{dF(\phi)t} Q\bigl(w_0-\phi'(0)\eta\bigr)|_{{}_{\mathcal{W}}}
\leq C\bigl|Q\bigl(w_0-\phi'(0)\eta\bigr)\bigr|_{{}_{\mathcal{W}}}<+\infty. \end{equation*} Moreover, as a consequence of the estimate \begin{equation*}
\begin{aligned}
e^{\theta\,t}\int_t^{+\infty}
&|S(\eta,\alpha(\tau),\omega(\tau);r)|_{{}_{\mathcal{W}}}\,d\tau \leq C\,e^{\theta\,t}\int_t^{+\infty}
|\alpha(\tau)\phi'(\varepsilon\eta)+\omega(\tau)|_{{}_{\mathcal{W}}}^{1+\gamma}\,d\tau\\
&\leq C\,e^{\theta\,t} \int_t^{+\infty} e^{-(1+\gamma)\theta \tau}\left\{e^{\theta \tau}\left(|\alpha(\tau)|
+|\omega(\tau)|_{{}_{n}}\right)\right\}^{1+\gamma}\,d\tau\\
&\leq C\,e^{-\gamma\theta\,t}\left\|(0,\alpha,\omega)\right\|_{{}_{\mathcal{X}}}^2,
\end{aligned} \end{equation*} also the function $\varepsilon^\gamma\mathcal{G}$ maps $\mathcal{X}\times (-\varepsilon,\varepsilon)$
into $\mathcal{X}\cap\{|\eta|\leq M\}$ for any $M>0$ and for $\varepsilon$ sufficiently small. Moreover, the smoothness of the functions $\ell, Q, \psi, S$ with respect to their arguments guarantees that the map $\mathcal{F}+\varepsilon^\gamma\mathcal{G}$ is differentiable.
For $r=0$, there holds $\mathcal{F}(\eta,\alpha,\omega;0)=\bigl(\ell(0)\cdot w_0-\eta,\alpha,\omega-Q(0) w_0\bigr)$, thus \eqref{implrel} is satisfied if and only if \begin{equation*}
\eta=\ell(0)\cdot w_0,\qquad \alpha=0,\qquad \omega=Q(0)w_0. \end{equation*} In order to apply Implicit Function Theorem, it is sufficient to observe that \begin{equation*}
\frac{\partial\left(\mathcal{F}+\varepsilon^\gamma\mathcal{G}\right)}
{\partial(\eta,\alpha,\omega)}\Bigr|_{\varepsilon=0}
=\frac{\partial \mathcal{F}}{\partial(\eta,\alpha,\omega)}\Bigr|_{\varepsilon=0}
=\begin{pmatrix} 1 &0 &0\\ 0 &I &0\\ 0 &0 &I \end{pmatrix} \end{equation*} since \begin{equation*}
\frac{\partial}{\partial \eta}\ell(\varepsilon\eta)\Bigr|_{\varepsilon=0}
=\frac{\partial}{\partial \eta}Q(\varepsilon\eta)\Bigr|_{\varepsilon=0}=0. \end{equation*} Thus, in a neighborhood of $\varepsilon=0$, there exist a smooth function $\Xi$ with values in a neighborhood of $(\ell(0)\cdot w_0,0,Q(0)w_0)\in \mathcal{X}$ such that \begin{equation}\label{implrel2}
\mathcal{F}+\varepsilon\mathcal{G}=0
\qquad\textrm{if and only if}\qquad
(\eta,\alpha,\omega)=\Xi(r). \end{equation}
The function $\Xi$ is locally bounded, $\|\Xi(r)\|_{\mathcal{X}}\leq C$ for $\varepsilon$ small, and thus \begin{equation*}
|w(t)|=|\alpha(t)\phi'(\varepsilon\eta)+\omega(t)|\leq C\,e^{-\theta\,t}. \end{equation*} Recalling that $W=\phi(\varepsilon\eta)+\varepsilon w$, the decay estimate \eqref{nonlindecay2} follows.
\end{proof}
With Theorem \ref{thm:abstract} at hand, we are able to provide the proof of Theorem \ref{thm:nonlinstab}. For the reader's convenience, let us briefly retrace the path toward nonlinear stability. The traveling wave $(U,V)$ propagates with a specific speed $c$. Thus, considering a reference frame moving with such speed, we obtain the nonlinear system \eqref{nlsystw} for the perturbation variables, for which the Cauchy problem can be written as \begin{equation} \label{relACmove2} \begin{aligned}
\partial_{t}\begin{pmatrix} u\\ v\end{pmatrix}
&= - {\mathbf{B}}^{-1}\left( \mathbf{A} \,\partial_{x}\begin{pmatrix} u\\ v\end{pmatrix}
+\begin{pmatrix} f(U)- f(u+U)\\ v\end{pmatrix}\right),\\ \begin{pmatrix} u\\ v\end{pmatrix}(0) &= \begin{pmatrix} u_0 - U\\ v_0 - V\end{pmatrix} \end{aligned} \end{equation} where $\mathbf{A}$ and $\mathbf{B}$ are defined in \eqref{defABC}, and $(u_0,v_0)$ is the (unperturbed) initial data of Theorem \ref{thm:nonlinstab}. Problem \eqref{relACmove2} corresponds to \eqref{cauchy2} in the general framework previously considered in Theorem \ref{thm:abstract}. Therefore, we are able to make the following identifications:
{ 1.} the Hilbert space $\mathcal{W}$ is $H^1(\mathbb{R};\mathbb{R}^2)$; the steady state $\overline{W}$ is $\overline{W}=0$, and the function $\phi$ is defined by $\phi(\delta):=(U,V)(\cdot+\delta) -(U,V)(\cdot) $ with $\delta\in \mathbb{R}$; observe that for each $\delta \in \mathbb{R}$ fixed, \[ \begin{aligned}
|\phi(\delta)|_{{H^1}}^2 &= \int_\mathbb{R} |(U,V)(\zeta + \delta) - (U,V)(\zeta)|^2 \, d\zeta + \int_\mathbb{R} |(U_x,V_x)(\zeta + \delta) - (U_x,V_x)(\zeta)|^2 \, d\zeta\\
&= \int_\mathbb{R} |(U_x, V_x)(\hat \theta \zeta)|^2 \delta^2 \, d\zeta + \int_\mathbb{R} |(U_{xx}, V_{xx})(\hat \theta \zeta)|^2 \delta^2 \, d\zeta\\
&\leq C_\delta |(U_x,V_x)|_{{H^1}}^2, \end{aligned} \] for some $\hat \theta \in (0,\delta)$, showing that $\phi(\delta) \in \mathcal{W}$.\par { 2.} the linearized equation (corresponding to the one in \eqref{lincauchy2}) is \begin{equation*}
\partial_{t}\begin{pmatrix} u\\ v\end{pmatrix}
= - {\mathbf{B}}^{-1}\Big( \mathbf{A}\,\partial_{x}
+ \mathbf{C}(x) \Big) \begin{pmatrix} u\\ v\end{pmatrix}; \end{equation*} \indent { 3.} the remainder, for which the estimate \eqref{regoF} has to be proved, is \begin{equation*}
\begin{pmatrix} u\\ v\end{pmatrix} \quad \mapsto\quad
\begin{pmatrix} R(U;u) \\ 0 \end{pmatrix}
:=\begin{pmatrix} f(U+u) - f(U) - f'(U)u \\ 0 \end{pmatrix}. \end{equation*} Assuming $f\in C^3$, we next show that hypotheses {\bf H2} and {\bf H3} are verified for the function space $\mathcal{W}=H^1(\mathbb{R};\mathbb{R}^2)$.
First we verify \eqref{regophi}. Denoting by $\Phi$ the couple $(U,V)$, there holds \begin{equation*}
\begin{aligned}
|\phi(\delta) - \phi(0) - \phi'(0)\delta|_{{L^2}}^2 &= \int_{\mathbb{R}}\left|\Phi(x+\delta)-\Phi(x)-\Phi_x(x)\delta\right|^2\,dx\\
&=\delta^2\int_{\mathbb{R}}\left|\int_0^1 \left(\Phi_x(x+\theta\delta)-\Phi_x(x)\right)d\theta\right|^2\,dx\\
&\leq \delta^2\int_{\mathbb{R}}\int_0^1 \left|\Phi_x(x+\theta\delta)-\Phi_x(x)\right|^2 d\theta\,dx\\
&\leq \delta^4\int_{\mathbb{R}}\int_0^1\int_0^1 |\Phi_{xx}|^2 d\eta\,d\theta\,dx,
\end{aligned} \end{equation*} and thus \begin{equation*}
|\phi(\delta) - \phi(0) - \phi'(0)\delta|_{{L^2}} \leq \delta^2 |\Phi_{xx}|_{{L^2}}. \end{equation*} A similar estimate can be obtained by differentiating with respect to $x$, so that \begin{equation*}
|\phi(\delta) - \phi(0) - \phi'(0)\delta|_{{H^1}} \leq \delta^2 |\Phi_{xx}|_{{H^1}}. \end{equation*} To prove estimate \eqref{regoF}, we first observe that \begin{equation*}
\begin{aligned}
R(U;u)&=\left\{\int_0^1 f''(U+\theta u)(1-\theta)\,d\theta\right\} u^2;\\
\partial_{x}R(U;u)
&=\left\{\int_0^1 f'''(U+\theta u)\theta(1-\theta)\,d\theta\right\} u^2\,\partial_{x} u\\
&\quad +2\left\{\int_0^1 f''(U+\theta u)(1-\theta)\,d\theta\right\} u\,\partial_{x} u .
\end{aligned} \end{equation*} Thus, for $u\in H^1(\mathbb{R})$, taking into account the embedding $H^1(\mathbb{R})\subset L^\infty(\mathbb{R})$, there holds \begin{equation*}
|R(U;u)|_{{}_{L^2}}\leq C\,|u|_{{}_{L^2}}^2,\qquad
|\partial_{x} R(U;u)|_{{}_{L^2}}\leq C\,|u|_{{}_{H^1}}^2, \end{equation*} where the constant $C$ depends on $f, U$ and the $L^\infty-$norm of $u$, so that, in particular, estimate \eqref{regoF} holds with $\gamma=1$.
Finally, thanks to Theorem \ref{thm:linstab}, also hypothesis {\bf H1} is verified, so that Theorem \ref{thm:abstract} applies and Theorem \ref{thm:nonlinstab} follows.
\section{Numerical experiments}\label{sect:numerics}
In this Section, we present some numerical experiment on system \eqref{relAC}, based on the observation that it can be rewritten as the weakly coupled semilinear hyperbolic system (a reactive version of the Goldstein--Kac model for correlated random walk): \begin{equation}\label{cineAC}
\left\{\begin{aligned}
\partial_{t} u_--\varrho\partial_{x} u_-
&= \tfrac1{2}\,\tau^{-1}(-u_-+u_+)+\tfrac12 f(u_++u_-),\\
\partial_{t} u_++\varrho\partial_{x} u_+
&= \tfrac1{2}\tau^{-1}(u_--u_+)+\tfrac12 f(u_++u_-),
\end{aligned}\right. \end{equation} where the coefficient $\varrho$ and the unknowns $u_\pm$ are given by \begin{equation*}
\varrho:=1/\sqrt{\tau},\qquad
u_-:=\tfrac{1}{2}\left(u+\varrho^{-1} v\right),\qquad
u_+:=\tfrac{1}{2}\left(u-\varrho^{-1} v\right). \end{equation*} Inverting the equality, we infer the relations $u=u_++u_-$ and $v=\varrho(u_--u_+)$.
Fixed the mesh size $\textrm{\rm dx}>0$, we discretize the space by approximating the first order space derivatives in an upwind fashion. Thus, setting $r_j\approx u_-(j\,\textrm{\rm dx},t)$ and $s_j\approx u_+(j\,\textrm{\rm dx},t)$, we obtain \begin{equation}\label{semidiscrete}
\left\{\begin{aligned}
\frac{dr_j}{dt}&=\frac{\varrho}{\textrm{\rm dx}}\left(r_{j+1}-r_{j}\right)
+ \frac1{2\tau}\left(-r_j+s_j\right)+\frac12 f(r_j+s_j),\\
\frac{ds_j}{dt}&=-\frac{\varrho}{\textrm{\rm dx}}\left(s_{j}-s_{j-1}\right)
+ \frac1{2\tau}\left(r_j-s_j\right)+\frac12 f(r_j+s_j).
\end{aligned}\right. \end{equation} Let us stress that, setting $u_j:=r_j+s_j$ and $v_j:=\varrho(r_j-s_j)$, we infer a semi-discrete version of \eqref{relAC} \begin{equation*}
\left\{\begin{aligned}
\frac{du_j}{dt}&=\frac{1}{2}\varrho\,\textrm{\rm dx}\,\frac{u_{j+1}-2u_j+u_{j-1}}{\textrm{\rm dx}^2}
+\frac{v_{j+1}-v_{j-1}}{2\textrm{\rm dx}}+f(u_j),\\
\frac{dv_j}{dt}&=\frac{1}{2}\varrho\,\textrm{\rm dx}\,\frac{v_{j+1}-2v_j+v_{j-1}}{\textrm{\rm dx}^2}
+\frac{1}{\tau}\left(\frac{u_{j+1}-u_{j-1}}{2\textrm{\rm dx}}-v_j\right),
\end{aligned}\right. \end{equation*} which formally corresponds to \begin{equation*}
\left\{\begin{aligned}
u_t -v_x &=\nu\, u_{xx} u+f(u),\\
\tau v_t - u_x &=\tau\nu\,v_{xx} - v,
\end{aligned}\right.
\qquad\qquad\textrm{where}\quad
\nu:=\tfrac{1}{2}\varrho\,\textrm{\rm dx}, \end{equation*} so that we expect the appearance of a numerical viscosity with strength measured by the parameter $\varrho$.
Next, fixed the time step $\textrm{\rm dt}>0$, we discretize the time derivative in \eqref{semidiscrete} by means of an implicit-explicit approach, leaded by a simplicity criterion suggesting to discretize implicitly only the linear terms \begin{equation*}
\left\{\begin{aligned}
\frac{r_j^{n+1}-r_j^{n}}{\textrm{\rm dt}}&=\frac{\varrho}{\textrm{\rm dx}}\bigl(r_{j+1}^{n+1}-r_{j}^{n+1}\bigr)
+ \frac1{2\tau}\bigl(-r_j^{n+1}+s_j^{n+1}\bigr)+\frac12 f(r_j^{n}+s_j^{n}),\\
\frac{s_j^{n+1}-s_j^{n}}{\textrm{\rm dt}}&=-\frac{\varrho}{\textrm{\rm dx}}\bigl(s_{j}^{n+1}-s_{j-1}^{n+1}\bigr)
+ \frac1{2\tau}\bigl(r_j^{n+1}-s_j^{n+1}\bigr)+\frac12 f(r_j^{n}+s_j^{n}).
\end{aligned}\right. \end{equation*} Fully implicit schemes have been tested with no significant advantage in the approximation, but with a significant increase of the computational time.
Setting \begin{equation*}
\alpha=\varrho\frac{\textrm{\rm dt}}{\textrm{\rm dx}},\qquad \beta=\frac{\textrm{\rm dt}}{2\tau},\qquad f_j^n=f(r_j^n+s_j^n), \end{equation*} and with an upwind discretization of the space derivatives, we end up with \begin{equation}\label{vectSch}
\begin{pmatrix}
\left(1+\beta\right)\mathbb{I}-\alpha\,\mathbb{D}_+ & -\beta\,\mathbb{I} \\
-\beta\,\mathbb{I} &\left(1+\beta\right)\mathbb{I}+\alpha\,\mathbb{D}_- \\
\end{pmatrix}
\begin{pmatrix} r^{n+1} \\ s^{n+1} \end{pmatrix}=
\begin{pmatrix} r^{n}+f^{n}\textrm{\rm dt} /2\\ s^{n}+f^{n}\textrm{\rm dt} /2 \end{pmatrix} \end{equation} where the matrices $\mathbb{I}, \mathbb{D}_\pm$ are given by \begin{equation*}
\mathbb{I}=(\delta_{i,j}),\quad
\mathbb{D}_+=(\delta_{i+1,j}-\delta_{i,j}),\quad
\mathbb{D}_-=(\delta_{i,j}-\delta_{i,j+1}) \end{equation*} (here $\delta_{i,j}$ is the standard Kronecker symbol). The block-matrix in \eqref{vectSch} is invertible, since its spectrum is contained in the complex half plane $\{\lambda\in\mathbb{C}\,:\,\textrm{\rm Re}\,\lambda\geq1\}$ as a consequence of the Ger\v sgorin criterion.
A direct manipulation of \eqref{vectSch} gives \begin{equation}\label{finalSch}
\begin{aligned}
r^{n+1}&=(\mathbb{S}-\alpha^2\mathbb{D}_-\mathbb{D}_+)^{-1}
\Bigl\{[(1+\beta)\mathbb{I}+\alpha\mathbb{D}_-]r^n+\beta s^n\\
&\hskip4.5cm+\tfrac12[(1+2\beta)\mathbb{I}+\alpha\mathbb{D}_-]f^n\textrm{\rm dt}\Bigr\}\\
s^{n+1}&=(\mathbb{S}-\alpha^2\mathbb{D}_+\mathbb{D}_-)^{-1}
\Bigl\{\beta r^n+[(1+\beta)\mathbb{I}-\alpha\mathbb{D}_+]s^n\\
&\hskip4.5cm+\tfrac12[(1+2\beta)\mathbb{I}-\alpha\mathbb{D}_+]f^n\textrm{\rm dt}\Bigr\}
\end{aligned} \end{equation} where $\mathbb{S}$ is the symmetric matrix \begin{equation*}
\mathbb{S}:=(1+2\beta)\mathbb{I}+\alpha(1+\beta)(\mathbb{D}_--\mathbb{D}_+) \end{equation*} To start with, we test the algorithm by analyzing its capability to recover the correct wave speeds $c_\ast$ of the front connecting the stable states $0$ and $1$. Following \cite{LeVYee90}, we introduce an {\it average speed} of the numerical solution at time $t^n$ defined by \begin{equation}\label{numerAve}
c^n=\frac{1}{\textrm{\rm dt}}\mathbf{1}\cdot(u^{n}-u^{n+1})
=\frac{1}{\textrm{\rm dt}}\sum_{j} (u^{n}_{j}-u^{n+1}_{j}) \end{equation} where $\mathbf{1}=(1,\dots,1)$. Indeed, for any differentiable function $\phi$ with asymptotic states $\phi_\pm$ and derivative integrable in $\mathbb{R}$, and for $h\in\mathbb{R}$, there holds \begin{equation*}
\begin{aligned}
\int_{\mathbb{R}} \left(\phi(x+h)-\phi(x)\right)dx
&=h\int_{\mathbb{R}} \int_0^1 \frac{d\phi}{dx}(x+\theta h)dh\,dx\\
&=h\int_0^1\int_{\mathbb{R}} \frac{d\phi}{dx}(x+\theta h)dx\,dh=h[\phi]
\end{aligned} \end{equation*} where $[\phi]:=\phi(+\infty)-\phi(-\infty)$, so that for $h=-c\,\textrm{\rm dt}$, we infer \begin{equation*}
c=\frac{1}{[\phi]\,\textrm{\rm dt}}\int_{\mathbb{R}} \left(\phi(x)-\phi(x-c\,\textrm{\rm dt})\right)dx. \end{equation*} As a test case, we consider the usual cubic function $f(u)=u(u-\alpha)(1-u)$ with $\alpha\in(0,1)$. Our aim is to compare the values for the propagation speed $c_\ast$ as obtained by means of the shooting argument (see Section \ref{sect:existence}) and the one given by calculating \eqref{numerAve} for the solution to the initial-value problem with an increasing datum connecting $0$ and $1$ and computing $c^n$ at a time $t$ so large that stabilization of the speed of propagation of the numerical solution is reached.
To start with, we test three different choices for the couple $(\tau, \alpha)$ for different values of $\textrm{\rm dx}$ and $\textrm{\rm dt}$, where the range of variation of $\tau$ has been chosen so that the condition $\tau\,f'(u)<1$ is satisfied for all the values of the unstable zero $\alpha$.
\begin{table}\centering \caption{Riemann problem with jump at $\ell/2$, $\ell=25$. Relative error for three different cases ($T$ final time): A. $\tau=1$, $\alpha=0.9$, $c_\ast=0.5646$, $T=40$; B. $\tau=2$, $\alpha=0.6$, $c_\ast=0.1737$, $T=30$; C. $\tau=4$, $\alpha=0.7$, $c_\ast=0.3682$, $T=35$. \label{tab:numerspeeds3}}
{\begin{tabular}{@{}r|c|r|r|r|r|r@{}}
&$\textrm{\rm dx}$ &$2^0$ &$2^{-1}$ &$2^{-2}$ &$2^{-3}$ &$2^{-4}$ \\ \hline
&A &0.1664 &0.0787 &0.0325 &0.0091 &0.0018 \\ $\textrm{\rm dt}=10^{-1}$ &B &0.0383 &0.0306 &0.0241 &0.0198 &0.0175 \\
&C &0.1527 &0.1144 &0.0818 &0.0581 &0.0442 \\ \hline
&A &0.1751 &0.0876 &0.0417 &0.0186 &0.0079 \\ $\textrm{\rm dt}=10^{-2}$ &B &0.0275 &0.0196 &0.0128 &0.0084 &0.0061 \\
&C &0.1420 &0.1018 &0.0684 &0.0457 &0.0339 \\ \hline
&A &0.1760 &0.0885 &0.0427 &0.0196 &0.0089 \\ $\textrm{\rm dt}=10^{-3}$ &B &0.0265 &0.0184 &0.0117 &0.0072 &0.0049 \\
&C &0.1411 &0.1006 &0.0670 &0.0441 &0.0321 \end{tabular}} \end{table}
From Table \ref{tab:numerspeeds3}, we note that the smallness of the space mesh $\textrm{\rm dx}$ is more relevant than the corresponding time-step value $\textrm{\rm dt}$.
Requiring to detect the correct speed value with an error that is always less than 5\% of the effective value, we heuristically determine the choice $\textrm{\rm dx}=2^{-3}$ and $\textrm{\rm dt}=10^{-2}$, that will be used for subsequent numerical experiments. For such a choice, we record the results in Table \ref{tab:numerspeeds0} for different choices of $\alpha$ and $\tau=1$ and $\tau=4$ together with the corresponding relative error.
\begin{table}\centering \caption{Final average speed \eqref{numerAve} and relative error with respect to $c_\ast$ given in Sect.\ref{sect:existence} ($N=400$, $\textrm{\rm dx}=0.125$, $\textrm{\rm dt}=0.01$, $\ell=25$, $T=40$). \label{tab:numerspeeds0}}
{\begin{tabular}{@{}r|c|c|c|c@{}}
&$\alpha=0.6$ &$\alpha=0.7$ &$\alpha=0.8$ &$\alpha=0.9$ \\ \hline $\tau=1$ &0.1580 &0.3096 &0.4497 &0.5751 \\
&{\scriptsize 0.0101}
&{\scriptsize 0.0118}
&{\scriptsize 0.0145}
&{\scriptsize 0.0186} \\ \hline $\tau=4$ &0.2102 &0.3533 &0.4337 &0.4825 \\
&{\scriptsize 0.0396}
&{\scriptsize 0.0404}
&{\scriptsize 0.0365}
&{\scriptsize 0.0118} \end{tabular}} \end{table}
Considering different form for matrices $\mathbb{D}_\pm$ giving a second order approximation of the derivatives, such as \begin{equation*}
\mathbb{D}_+=\left(-\tfrac12\delta_{i+2,j}+2\delta_{i+1,j}-\tfrac32\delta_{i,j}\right),\qquad
\mathbb{D}_-=\left(\tfrac32\delta_{i,j}-2\delta_{i,j+1}+\tfrac12\delta_{i,j+2}\right), \end{equation*} the speed approximation gain in accuracy, as reported in Table \ref{tab:secondorder}, that shows an increase of one order.
\begin{table}\centering \caption{Second order in space. Final average speed \eqref{numerAve} and relative error with respect to $c_\ast$ given in Sect.\ref{sect:existence} ($N=400$, $\textrm{\rm dx}=0.125$, $\textrm{\rm dt}=0.01$, $\ell=25$, $T=40$). \label{tab:secondorder}}
{\begin{tabular}{@{}r|c|c|c|c@{}}
&$\alpha=0.6$ &$\alpha=0.7$ &$\alpha=0.8$ &$\alpha=0.9$ \\ \hline $\tau=1$ &0.1560 &0.3052 &0.4421 &0.5630 \\
&{\scriptsize 0.0025}
&{\scriptsize 0.0025}
&{\scriptsize 0.0026}
&{\scriptsize 0.0029} \\ \hline $\tau=4$ &0.2184 &0.3672 &0.4485 &0.4885 \\
&{\scriptsize 0.0022}
&{\scriptsize 0.0025}
&{\scriptsize 0.0034}
&{\scriptsize 0.0004} \end{tabular}} \end{table}
In what follows, we keep considering the previously discussed first order discretization, since we are interested in considering initial data with sharp transitions (as in the case of Riemann problems). In such a case, higher order approximations of the derivatives introduce spurious oscillations that, even being transient, may leads to catastrophic consequences because of the bistable nature of the reaction term.
As a consequence of its capability of correct computations of propagation speeds, we consider the scheme \eqref{finalSch} to be a reliable tools for determining numerically the behavior of the solutions to \eqref{relAC} and we use it to show that the actual domain of attraction of the front is much larger than guaranteed by the nonlinear stability in Theorem \ref{thm:nonlinstab}.
\subsection{Riemann problem} The rigorous result proved in the previous sections guarantees that small perturbations to the propagating front are dissipated, with exponential rate, by the equation. Inspired by the many available results for the parabolic Allen--Cahn equation (starting from the landmark article \cite{FifeMcLe77}), we expect that the front possesses a very large domain of attraction and, specifically, that any bounded initial data $u_0$ such that \begin{equation}\label{decay}
\limsup_{x\to-\infty} u_0(x)<\alpha<\liminf_{x\to+\infty} u_0(x) \end{equation} gives raise to a solution that is asymptotically convergent to a member of the traveling fronts connecting $u=0$ with $u=1$.
\begin{figure}\label{fig:Riemann2}
\end{figure}
To support such conjecture, we perform some numerical experiments choosing the parameters values \begin{equation*}
\tau=4,\qquad \ell=25,\qquad \textrm{\rm dx}=0.125,\qquad \textrm{\rm dt}=0.01. \end{equation*} Moreover, we consider the case $\alpha=1/2$ motivated by the fact that, in such a special case, the profile of the traveling front for the hyperbolic Allen--Cahn equation is stationary and it coincides with the one of the corrresponding original parabolic equation, explicitly given by \begin{equation*}
U(x)=\frac{1}{1+e^{-x/\sqrt{2}}},\qquad
V(x)=\frac{dU}{dx}=\frac{1}{\sqrt{2}}\,\frac{1}{e^{x/\sqrt{2}}+2+e^{-x/\sqrt{2}}} \end{equation*} when normalized by the condition $U(0)=1/2$.
Numerical simulations confirm the decay of the solution to the equilibrium profile (see Figure \ref{fig:Riemann2}, left). When compared with the standard Allen--Cahn equation, it appears evident that the dissipation mechanism of the hyperbolic equation is weaker with respect to the parabolic case (see Figure \ref{fig:Riemann2}, right).
\subsection{Randomly perturbed initial data}
Next, keeping all of the previous parameters fixed, we consider initial data that resemble very roughly the transition from 0 to 1. Namely, we divide the interval $(-\ell,\ell)$ into three parts and we choose a random value in each of these sub-intervals coherently with the requirement \eqref{decay}. Precisely, we choose $u_0(x)$ to be a different random value in $(0,0.5)$ for each $x\in(-\ell,-\ell/3)$, in $(0,1)$ for each $x\in(-\ell/3,\ell/3)$ and in $(0.5,1)$ for each $x\in(\ell/3,\ell)$. \begin{figure}
\caption{Random initial datum in $(-\ell,\ell)$, $\ell=25$ (squares). Solution profiles for the hyperbolic Allen--Cahn equation with relaxation at time $t=0.5$ (dot), $t=7.5$ (dash), $t=15$ (continuous). For comparison, in the small window, the solution to the parabolic Allen--Cahn equation. }
\label{fig:Random1tot}
\end{figure}
The results for both hyperbolic and parabolic Allen--Cahn equation with the same initial datum are illustrated in Fig.\ref{fig:Random1tot}. Convergence to the equilibrium configuration is manifest. It is also worthwhile to note that different level of smoothing produced by the presence/absence of the relaxation parameter $\tau$, measuring the ``hyperbolicity'' of the model.
The transition is even much more robust than what the previous computation shows. With an initial datum $u_0(x)$ given by a random value in $(0,0.9)$ for each $x\in(-\ell,-\ell/3)$, in $(0,1)$ for each $x\in(-\ell/3,\ell/3)$ and in $(0.1,1)$ for each $x\in(\ell/3,\ell)$, we still observe the appearance and formation of the front, as shown in Figure \ref{fig:Random2tot}.
\end{document}
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Continuity equation and vacuum regions { in} compressible flows} \author{Anton\'\i n Novotn\'y$^1$ and Milan Pokorn\' y$^2$} \maketitle
\centerline{$^1$ Institut de Math\'ematiques de Toulon, EA 2134} \centerline{BP20132, 83957 La Garde, France} \centerline{e-mail: {\tt [email protected]}}
\centerline{$^{2}$ Charles University, Faculty of Mathematics and Physics} \centerline{Mathematical Inst. of Charles University} \centerline{Sokolovsk\' a 83, 186 75 Prague 8, Czech Republic} \centerline{e-mail: {\tt [email protected]}} \vskip0.25cm \begin{abstract} We investigate the creation and properties of eventual vacuum regions in the weak solutions of the continuity equation, in general, and in the weak solutions of compressible Navier--Stokes equations, in particular. The main results are based on the analysis of renormalized solutions to the continuity and pure transport equations and their inter-relations which are of independent interest. \end{abstract}
\noindent{\bf MSC Classification:} 76N10, 35Q30
\noindent{\bf Keywords:} compressible Navier--Stokes equations, vacuum regions, renormalized solution, transport equation, continuity equation
\section{Introduction}
{ In this paper we consider evolution of the couple $(\varrho,\vc{u})= (\varrho(t,x),\vc{u}(t,x))$---(density, velocity) of the compressible fluid---over the time interval $I$,} $I=(0,T)$, $T>0$, $t\in \overline I$
in a bounded domain $\Omega\in R^d$, $d\ge 2$, $x\in \Omega$. We concentrate on the question of the creation of vacuum regions $\{x\in\Omega|\varrho(t,x)=0\}$ in this flow. This is one of important open questions in the mathematical fluid mechanics of compressible fluids. It is closely connected to the question of regularity of solutions to the compressible Navier--Stokes equations. If the density is initially bounded away from zero, for weak solutions it is not excluded that the vacuum may appear in finite time.
{ We show that if this happens it must happen in a sense smoothly. More precisely, the measure of the set, where the density may be equal to zero, is continuous in time, or, in the other words, the vacuum (if any) creates and evolutes continuously in time and the vacuum of positive measure cannot appear instantaneously.} The exact formulation of this result is { presented} in Theorem \ref{t2}.
{ More interesting and intriguing is the second result. It translates as follows: Assume that $(\varrho,\vc{u})$ is a (standard) weak solution to the compressible Navier--Stokes equations. Then whatever distributional solution $R$ with a small additional regularity (specified in (\ref{clR-})) of the continuity equation with the same velocity $\vc{u}$ we take (whatever arbitrary its initial data are!), $R$ must develop at any time $t$ a vacuum region $\{x|R(t,x)=0\}$ that includes the vacuum of
$\varrho(t)$, i.e. $\{x|\varrho(t,x)=0\}$ is contained in the vacuum set of the function $R$. This result definitely pleads for a non-existence of vacuum in compressible flows at least in many physically reasonable situations.} The exact formulation of this result is given in Theorem \ref{t3} and its Corollaries \ref{cor1}, \ref{cor2}.
{ On the other hand, it is important to recall that if the velocity field $\vc{u} \in L^2(0,T; W^{1,2}(\Omega;R^d))$ (this is the generic situation for flows of Newtonian fluids with constant viscosities), there is no direct way of constructing solutions $R$ to the continuity equation with the given velocity unless {${\rm div}\, \vc{u}\in L^1(I;L^\infty(\Omega))$---cf. DiPerna--Lions} \cite[Proposition II.1]{DL}. Indeed, existence of solutions to the continuity equation with the transporting velocity fields in spaces $L^1(I;W^{1,p}(\Omega;R^d))$, $p\in [1,\infty)$ only, is, in general, an open problem.}
The conclusions of our paper described above are based on nowadays classical results and techniques for the continuity and transport equation that have been forged within the process of the development of the existence theory for weak solutions to the compressible { Navier--Stokes} equations and recently also for the mixtures of compressible fluids. They are all inspired by the classical regularization technique implemented to the investigation of transport equations with transport coefficients in Sobolev spaces in the seminal work of { DiPerna--Lions} \cite{DL}. (The spaces needed for the results in \cite{DL} are those needed in the Friedrichs lemma about commutators with $\alpha=\infty$, $p=1$, cf. Lemma \ref{l2}.) Some of them are valid only within the functional setting of the transport theory \cite{DL} (namely those dealing with extension of distributional solutions to weak solutions (up to the boundary), time integration of weak or distributional solutions and passage from distributional or weak solutions to renormalized distributional or weak solutions)\footnote{The various notions of solutions used in the above text are rigorously defined in Section { \ref{S2}.}}. They are formulated in Subsection \ref{M2.2} in Theorems \ref{th3} and \ref{th4}. Some of them, namely those valid for the renormalized solutions, must go beyond the transport theory \cite{DL} (in the sense that the transporting velocity belongs still to Sobolev spaces but one requires less summability of the solution then the summability required in \cite{DL}){---in order to get stronger results with respect to the constitutive laws of pressure in the applications to compressible Navier--Stokes equations.} (This is notably the case of Theorems \ref{th1} and \ref{th2} in Subsection \ref{M2.1}). Indeed, all available constructions of weak solutions to the compressible { Navier--Stokes} equations provide a couple $(\varrho,\vc{u})$ which satisfies the continuity equation in the renormalized sense. The latter results are often formulated in the mathematical literature in a particular functional setting applicable to the concrete situation without ambition to full generality, see Lions \cite{L4}, Feireisl \cite{FeBook} and \cite{FeNo_book}, \cite{NoSt} if we limit ourselves to the monographs only. Our aim is to provide generalization and synthesis of the results we need and prove them in their full generality, either for the sake of completeness or if we could not find a reliable exhausting reference.
A new approach to the compactness in the compressible Navier-Stokes equations allowing to treat some other physically different situations then \cite{FeNoPe}, \cite{FeBook}, \cite{FeNo_book} has been introduced by Bresch, Jabin \cite{BrJa}, deriving, in particular a "$\log\log$ estimate" for the Friedrichs type commutator, \cite[Theorem 2.3.6]{BrJa1} in the { DiPerna--Lions} functional framework. This theory does not allow to go beyond the DiPerna--Lions functional setting and seems at the time being so far in-exploitable for our purpose.
Among the main auxiliary questions which has to be answered in order to apply the theory of transport equations
to the compressible fluid dynamics in general, and to the investigation of the vacuum states, in particular, are the following: \begin{enumerate} \item What are the least conditions imposed on the transporting velocity $\vc{u}$ (in terms of Sobolev spaces) and solution $\varrho$ of the continuity equation (in terms of Lebesgue spaces) allowing to pass from renormalized distributional or weak solutions to time integrated weak solutions? The answers to these questions are subject of Theorems \ref{th1} and \ref{th2}. \item What are the least conditions on the couple $(\varrho,\vc{u})$ (in the same functional setting) to pass from distributional solutions of the continuity or pure transport equations to the weak (up to the boundary) solutions (eventually to the renormalized weak solutions), and from distributional or weak solutions to their time integrated counterparts (eventually to the renormalized time integrated counterparts)? The answer to these questions are given in Theorems \ref{th3} and \ref{th4}. \item How are interconnected solutions of pure transport equation and continuity equations? and what does this interconnection imply for the formation of vacuum in the compressible flows? The first question is treated in Theorem \ref{th5}. The last question is object of Theorems \ref{t2}, \ref{t3} and their Corollaries \ref{cor1}, \ref{cor2}. \end{enumerate}
It is to be { noticed that the conditions mentioned in Items 1.-3.} determine in large extend the admissible constitutive laws in the theory of weak solutions to compressible { Navier--Stokes} equations, \cite{FeNoPe}, \cite{FeNo_book}, \cite{Fe2002}, \cite{BrJa}. The usefulness of the subject of Item 4. was firstly discovered in connection with the investigation of weak solutions of systems describing compressible mixtures, see \cite{3MNPZ}, \cite{NoPo}, \cite{AN}, \cite{VWY}.
Our approach is exclusively Eulerian. The Lagrangian approach (dealing with characteristics of the vector field $\vc{u}$ rather than with the transport equation, and translating them afterwards to the Eulerian vocabulary) introduced in seminal paper of Ambrosio \cite{Ambr} allows to extend some results of \cite{DL} (namely those related to existence, uniqueness and passage from distributional or weak to renormalized distributional or weak solutions) to $L^1(I;BV(\Omega;R^d))$ vector fields\footnote{The space $BV(\Omega)$ is the space of functions with bounded variations.} with divergence always in $L^1(I;L^\infty(\Omega))$. It was extended and generalized in several papers by Ambrosio, Crippa, De Lellis \cite{AmbrC}, \cite{CDL} and others. Further deep generalization of this approach consisting in replacing the condition imposed on the divergence of $\vc{u}$ by a weaker condition postulating that "$\vc{u}$ is weakly incompressible" is due to Bianchini, Bonicatto \cite{BIBO}. The latter result (which is essentially about the properties of the flow of the vector field $\vc{u}$) implies as a corollary the uniqueness for the pure transport equation under "weak incompressibility" condition. (It is not without interest, that a stronger form of this corollary can be obtained within the Sobolev functional setting quite easily by the purely Eulerian approach \cite[Proposition 5]{AN}.) In contrast with conservation laws, where the
$BV(\Omega)$ theory found many applications, it { has} not so far appeared to be exploitable in the theory of compressible { Navier--Stokes} equations.
{ The paper is organised as follows. In Section \ref{S2} we introduce various notions of solutions to the continuity and transport equations that will be used in the sequel. Section \ref{M} is devoted to the formulation of the main results, and of the auxiliary results needed for their proofs, which are of independent interest. Theorems \ref{t2} and \ref{t3} (and Corollary \ref{c1} in Subsection \ref{M1}) deal with the properties of vacuum in any renormalized time integrated weak solution of the continuity equation. This implies immediately the same properties of vacuum in any renormalized weak solution to the compressible Navier--Stokes equations. This issue is discussed in Subsection \ref{M3} (see namely Corollary \ref{c1} and Remark \ref{r1}). Theorems \ref{t2}--\ref{t3} and Corollaries \ref{cor1}, \ref{cor2} and \ref{c1} are main results of the paper. Their proofs require a good understanding of the relation between various types of solutions introduced in Section \ref{S2}. This issue of independent interest is treated in Subsection \ref{M2}. The { matters} of time integration of renormalized distributional of weak solutions are treated in Subsection \ref{M2.1} (see Theorems \ref{th1}, \ref{th4}). The passage from distributional to renormalized weak solution is handled in Subsection \ref{M2.2} (see Theorems \ref{th3}, \ref{th5}). The passage from continuity and pure transport equation to a continuity equation is formulated in Subsection \ref{M2.3} (see Theorem \ref{th5}). The remaining part of the paper is devoted to the proof of Theorems (\ref{t2}--\ref{th5}). Section \ref{S4} collects three preliminary classical results whose conclusions will be frequently used throughout the proofs. Section \ref{S5} is devoted to the proof of Theorems \ref{th1}--\ref{th2}, Section \ref{S6} to the proof of Theorems \ref{th3}--\ref{th4} and Section \ref{S7} to the proof of Theorem \ref{th5}. Finally in the last Section we combine the results of Theorems \ref{th1}--\ref{th5} to prove the main theorems: Theorems \ref{t2} and \ref{t3}.}
We finish this section by introducing the functional spaces and some notations. In what follows, we use standard notation for the Lebesgue and Sobolev spaces ($L^p(\Omega)$ and $W^{1,p}(\Omega)$ with the corresponding norms $\|u\|_{L^p(\Omega)}$ and $\|u\|_{W^{1,p}(\Omega)}$, respectively). We do not distinguish the notation for the norms for scalar- and vector-valued functions. However, the vector-valued functions are prin\-ted boldface and we write $\vc{u} \in L^p(\Omega;R^d)$ instead of $\vc{u} \in L^p(\Omega)$, similarly for other functions spaces. For function spaces of time and space dependent function we use the standard notation for the Bochner spaces $L^p(I;L^q(\Omega))$ or $L^p(I;L^q(\Omega;R^d))$, respectively. { We also use the notation $C([0,T];L^p(\Omega))$ for continuous functions on interval [0,T] with values in $L^p(\Omega)$ and $C_{{\rm weak}}([0,T];L^p(\Omega))$ a vector subspace of $L^\infty(0,T;L^p(\Omega))$ of functions continuous on $[0,T]$ with respect to the weak topology of { $L^p(\Omega)$.} More exactly, a function $f:[0,T]\mapsto L^p(\Omega)$ (defined on $[0,T]$) belongs to $C_{{\rm weak}}([0,T];L^p(\Omega))$ iff
$f\in L^\infty(0,T;L^p(\Omega))$ and for all $\eta\in L^{p'}(\Omega)$ the map $\tau\mapsto \intO{f(\tau)\eta}$ is continuous on interval $[0,T]$. For the norms in Bochner spaces we use the function space as full index, as e.g. $\|u\|_{L^{p}(I;L^q(\Omega))}$ or { $\|u\|_{L^p(I;W^{1,q}(\Omega))}$}. Throughout the paper, the constants are denoted by $C$ and their value may change even in the same formula.}
\section{Various notions of solutions to continuity and pure transport equations} \label{S2}
{ The main results of this paper will largely rely on various notions of (weak) solutions to the continuity and pure transport equations and their inter-relations. We shall introduce these notions in this section.}
We consider the equations on the time-space cylinder $Q=I\times\Omega$, $\Omega$ a bounded open set in $R^d$, $d\ge 2$, and $I=(0,T)$, $T>0$ a time interval. The equations read: \begin{enumerate} \item {Continuity equation} \begin{equation}\label{co1} {\partial_t\varrho+{\rm div}\,(\varrho\vc{u})=0\;\mbox{in $(0,T)\times\Omega$}} \end{equation} with initial condition $$ \varrho(0,\cdot)=\varrho_0(\cdot)\;\mbox{in $\Omega$}. $$ \item {Pure transport equation} \begin{equation}\label{tr1} { \partial_t s+\vc{u}\cdot\nabla s=0\;\mbox{in $(0,T)\times\Omega$}} \end{equation} with initial condition $$ s(0,\cdot)=s_0(\cdot)\;\mbox{in $\Omega$}. $$ \end{enumerate}
We shall consider several different notions of solutions to these equations.
\begin{df}[Continuity equation]\label{dfco} Let \begin{equation}\label{covu} {\vc{u}\in L^{1}(I\times\Omega;R^d)},\;{\rm div}\, \vc{u}\in L^1(I\times\Omega). \end{equation} We say that function \begin{equation}\label{cocl1} \varrho\in L^1(I\times\Omega)\;\mbox{such that $\varrho\vc{u}\in L^1(I\times\Omega;R^d)$} \end{equation} is\footnote{ In some cases, it would be enough to assume { $\vc{u}\in L_{\rm loc}^{1}(I\times\Omega;R^d)$}, ${\rm div}\,\vc{u}\in L_{\rm loc}^1(I\times\Omega)$ and condition (\ref{cocl1}) could be weaken to $\varrho\in L_{\rm loc}^1(I\times\Omega)$, such that $\varrho\vc{u}\in L_{\rm loc}^1(I\times\Omega;R^d)$. We do not consider this situation since it is irrelevant from the point of view of the present paper. }: \begin{enumerate} \item {\em Distributional solution} to the continuity equation (\ref{co1}) iff it satisfies (\ref{co1}) in the sense of distributions over the time-space, namely iff \begin{equation}\label{co1.1} \int_0^T \int_\Omega (\varrho \partial_t \varphi + \varrho \vc{u}\cdot \nabla \varphi) \dx \dt = 0 \end{equation} holds for arbitrary $\varphi \in C^\infty_{\rm c}(I\times \Omega)$. \item {\em Weak solution} to the continuity equation (\ref{co1}) iff \begin{equation}\label{co1.2} \mbox{equation (\ref{co1.1}) holds with arbitrary $\varphi \in C^\infty_{\rm c}(I\times \overline\Omega)$.} \end{equation} \item {\em Time integrated distributional solution} to the continuity equation (\ref{co1}) iff $\varrho \in C_{\rm weak}(\overline I;L^1(\Omega))$
and it holds \begin{equation}\label{co1.3} \int_\Omega(\varrho\varphi)(\tau,\cdot) {\, \rm d}x - \int_\Omega (\varrho\varphi) (0,\cdot) {\, \rm d}x = \int_0^\tau \int_\Omega (\varrho \partial_t \varphi + \varrho \vc{u}\cdot \nabla \varphi) \dx \dt \end{equation} for any $\varphi \in C^\infty_c(\overline I\times {\Omega})$ and any $\tau \in\overline I$. \item {\em Time integrated weak solution} to the continuity equation (\ref{co1}) iff $\varrho \in C_{\rm weak}(\overline I;L^1(\Omega))$
and \begin{equation}\label{co1.4} \mbox{equation (\ref{co1.3}) holds with arbitrary $\varphi \in C^\infty_{\rm c}(\overline I\times \overline\Omega)$ and any $\tau \in\overline I$.} \end{equation} \item {\em Renormalized distributional solution} to the continuity equation (\ref{co1}) iff in addition to (\ref{co1.1}), \begin{equation}\label{rco1.1} \int_0^T \int_\Omega \Big(b(\varrho) \partial_t \varphi + b(\varrho) \vc{u}\cdot \nabla \varphi - \big(b'(\varrho)\varrho -b(\varrho)\big) {\rm div}\, \vc{u} \varphi\Big) \dx \dt = 0 \end{equation} holds with all $\varphi \in C^\infty_{\rm c}(I\times \Omega)$ and all renormalizing functions\footnote{Conditions (\ref{ren}), (\ref{covu}) and (\ref{cocl1}) immediately ensure that $b(\varrho)$, $b(\varrho)\vc{u}$ and $(\varrho b'(\varrho)-b(\varrho)){\rm div}\,\vc{u}\in L^1(I\times\Omega)$. As will be seen later, in fact $b(\varrho) \in C([0,T];L^1(\Omega))$, too.} \begin{equation}\label{ren} b \in C^1([0,\infty)),\; b'\in C_c([0,\infty)). \end{equation} \item {\em Renormalized weak solution} to the continuity equation (\ref{co1}) iff in addition to (\ref{co1.2}), \begin{equation}\label{rco1.2} \mbox{equation (\ref{rco1.1}) holds with all $\varphi \in C^\infty_{\rm c}(I\times \overline\Omega)$ and all $b$ in (\ref{ren}).} \end{equation} \item {\em Renormalized time integrated distributional solution} to the continuity equation (\ref{co1}) iff $b(\varrho) \in C_{\rm weak}(\overline I;L^1(\Omega))$ and
in addition to (\ref{co1.3}), \begin{equation}\label{rco1.3} \int_\Omega (b(\varrho)\varphi) (\tau,\cdot) {\, \rm d}x - \int_\Omega (b(\varrho)\varphi)(0,\cdot) {\, \rm d}x= \end{equation} $$ \int_0^T \int_\Omega \Big(b(\varrho) \partial_t \varphi + b(\varrho) \vc{u}\cdot \nabla \varphi - \big(b'(\varrho)\varrho -b(\varrho)\big) {\rm div}\, \vc{u} \varphi\Big) \dx \dt $$ holds with all $\varphi \in C^\infty_{\rm c}(\overline{I}\times \Omega)$, all $\tau\in\overline I$ and all renormalizing functions $b$ in the class (\ref{ren}). \item {\em Renormalized time integrated weak solution} to the continuity equation (\ref{co1}) iff $b(\varrho) \in C_{\rm weak}(\overline I;L^1(\Omega))$and in addition to (\ref{co1.4}), \begin{equation}\label{rco1.4} \begin{aligned} \mbox{equation (\ref{rco1.3}) holds with all $\varphi \in C^\infty_{\rm c}(\overline I\times \overline\Omega)$,} \\ \mbox{all $\tau\in\overline I$ and all $b$ in (\ref{ren}).} \end{aligned} \end{equation} \end{enumerate} \end{df}
{ Due to the presence of term containing $s{\rm div}\,\vc{u}$ in the weak formulation of the pure transport equation, the definition of weak solutions/renormalized weak solutions in this case asks for better summability of the quantity $s$ (compared to the summability required for $\varrho$ expressed through assumption (\ref{cocl1}) in the case of the continuity equation).
\begin{df}[Pure transport equation]\label{dftr} Let $\vc{u}$ satisfy (\ref{covu}). We say that function \begin{equation}\label{trcl1} s\in L^1(I\times\Omega)\;\mbox{such that $s\vc{u}$ and $s{\rm div}\,\vc{u}\in L^1(I\times\Omega)$} \end{equation} is\footnote{ In some cases, it would be enough to assume { $\vc{u}\in L_{\rm loc}^{1}(I\times\Omega;R^d)$}, ${\rm div}\,\vc{u}\in L_{\rm loc}^1(I\times\Omega)$ and condition (\ref{trcl1}) could be weaken to $s\in L_{\rm loc}^1(I\times\Omega)$, such that $s\vc{u}, s {\rm div}\,\vc{u}\in L_{\rm loc}^1(I\times\Omega)$. We do not consider this situation since it is irrelevant from the point of view of the present paper. }: \begin{enumerate} \item {\em Distributional solution} to the pure transport equation (\ref{tr1}) iff it satisfies (\ref{tr1}) in the sense of distributions over the time-space, namely iff \begin{equation}\label{tr1.1} \int_0^T \int_\Omega (s \partial_t \varphi + s \vc{u}\cdot \nabla \varphi+ s{\rm div}\,\vc{u} \varphi) \dx \dt = 0 \end{equation} holds for arbitrary $\varphi \in C^\infty_{\rm c}(I\times \Omega)$. \item {\em Weak solution} to the pure transport equation (\ref{tr1}) iff \begin{equation}\label{tr1.2} \mbox{equation (\ref{tr1.1}) holds with arbitrary $\varphi \in C^\infty_{\rm c}(I\times \overline\Omega)$.} \end{equation} \item {\em Time integrated distributional solution} to the pure transport equation (\ref{tr1}) iff $s \in C_{\rm weak}(\overline I;L^1(\Omega))$
and it holds \begin{equation}\label{tr1.3} \int_\Omega(s\varphi)(\tau,\cdot) {\, \rm d}x - \int_\Omega (s\varphi) (0,\cdot) {\, \rm d}x = \int_0^\tau \int_\Omega (s \partial_t \varphi + s \vc{u}\cdot \nabla \varphi) \dx \dt \end{equation} for any $\varphi \in C^\infty_c(\overline I\times {\Omega})$ and any $\tau \in\overline I$. \item {\em Time integrated weak solution} to the pure transport equation (\ref{tr1}) iff $s \in C_{\rm weak}(\overline I;L^1(\Omega))$
and \begin{equation}\label{tr1.4} \mbox{equation (\ref{tr1.3}) holds with arbitrary $\varphi \in C^\infty_{\rm c}(\overline I\times \overline\Omega)$ and any $\tau \in\overline I$.} \end{equation} \item {\em Renormalized distributional solution} to the pure transport equation (\ref{tr1}) iff in addition to (\ref{tr1.1}), \begin{equation}\label{rtr1.1} \int_0^T \int_\Omega \Big(b(s) \partial_t \varphi + b(s) \vc{u}\cdot \nabla \varphi + b(s) {\rm div}\, \vc{u} \varphi\Big) \dx \dt = 0 \end{equation} holds with all $\varphi \in C^\infty_{\rm c}(I\times \Omega)$ and all renormalizing functions $b$ belonging to class (\ref{ren}). \item {\em Renormalized weak solution} to the pure transport equation (\ref{tr1}) iff in addition to (\ref{tr1.2}), \begin{equation}\label{rtr1.2} \mbox{equation (\ref{rtr1.1}) holds with all $\varphi \in C^\infty_{\rm c}(I\times \overline\Omega)$ and all $b$ in (\ref{ren}).} \end{equation} \item {\em Renormalized time integrated distributional solution} to the pure transport equation (\ref{tr1}) iff $b(\varrho) \in C_{\rm weak}(\overline I;L^1(\Omega))$ and
in addition to (\ref{tr1.3}), \begin{equation}\label{rtr1.3} \int_\Omega (b(s)\varphi) (\tau,\cdot) {\, \rm d}x - \int_\Omega (b(s)\varphi)(0,\cdot) {\, \rm d}x= \end{equation} $$ \int_0^T \int_\Omega \Big(b(s) \partial_t \varphi + b(s) \vc{u}\cdot \nabla \varphi + b(s) {\rm div}\, \vc{u} \varphi\Big) \dx \dt $$ holds with all $\varphi \in C^\infty_{\rm c}(\overline{I}\times \Omega)$, all $\tau\in\overline I$ and all renormalizing functions $b$ in the class (\ref{ren}). \item {\em Renormalized time integrated weak solution} to the pure transport equation (\ref{tr1}) iff $b(s) \in C_{\rm weak}(\overline I;L^1(\Omega))$ and in addition to (\ref{co1.4}), \begin{equation}\label{rtr1.4} \begin{aligned} \mbox{equation (\ref{rtr1.3}) holds with all $\varphi \in C^\infty_{\rm c}(\overline I\times \overline\Omega)$,} \\ \mbox{all $\tau\in\overline I$ and all $b$ in (\ref{ren}).} \end{aligned} \end{equation} \end{enumerate} \end{df} }
\section{Main results}\label{M}
The primal goal of this paper is the investigation of the vacuum formation in the weak solution (density, velocity)---$(\varrho,\vc{u})$---in the compressible Navier--Stokes equations. We shall prove that the volume of eventual vacuum set evolutes continuously in time and, more surprisingly, if there is no vacuum at time $0$ and there is a vacuum of non-zero measure at some time $\tau\in (0,T)$, then any distributional solution $R$ (with certain reasonable summability properties) to the continuity equation (with the same transporting velocity $\vc{u}$)---if it exists---admits at time $\tau$
a larger vacuum set $\{x\in\Omega|R(\tau)=0\}$ than the vacuum set of $\varrho$. This property does not imply absence of vacuum but pleads in favour of the sparseness of the event of creation of vacuum in compressible flows.
All these properties rely exclusively on the properties of continuity and transport equations. We shall therefore formulate them as such in Subsection \ref{M1}, postponing the formulation in the context of Navier--Stokes equations to Subsection \ref{M3}.
The proofs of results in Subsection \ref{M1} rely {essentially} on the properties and inter-relation of various types of weak/renormalized solutions to the continuity and transport equations and their combinations, which are of independent interest. Bits of pieces of some of these results (all of them having ground in the seminal work by DiPerna and Lions \cite{DL}) are non systematically spread through the mathematical literature in several (mostly recent) papers dealing with the existence of weak solutions to the compressible Navier--Stokes equations and compressible mixtures as auxiliary tools, \cite{FeBook}, \cite{FeNo_book}, \cite{FeNoPe}, \cite{NoPo}, \cite{NoSt}, \cite{VWY}. We will state in Subsection \ref{M2} those of these properties needed in this paper in their full generality and provide their detailed proofs.
\subsection{Properties of vacuum in the weak solution of the continuity equation} \label{M1}
The first theorem dealing with vacuum sets in the continuity equation reads.
\begin{thm} \label{t2} Let $\Omega\subset R^d$ be a bounded domain. Let $1\le q,p\le \infty$ and $\vc{u} \in L^{p}(0,T; W^{1,q}(\Omega;R^d))$. Let \begin{equation}\label{gamma+}
0\le\varrho \in C_{{\rm weak}}(\overline I;L^\gamma(\Omega)),\; \gamma>1 \end{equation} be a renormalized time integrated weak solution to the continuity equation (\ref{co1}) with transporting velocity $\vc{u}$ (i.e. it belongs to class (\ref{cocl1}), satisfies equation (\ref{co1.4}) and equation (\ref{rco1.4}) with the renormalizing functions $b$ from (\ref{ren})).
Then the map $t\mapsto s_\varrho(t,\cdot):= { 1_{\{x\in \Omega|\varrho(t,x)=0\}}(\cdot)}$ belongs to $C([0,T];L^r(\Omega))$ with any $1\le r<\infty$ and it is a time integrated renormalized weak solution of the pure transport equation (\ref{tr1}) with transporting velocity $\vc{u}$. In particular, \begin{equation} \label{2.8}
{ |\{x\in \Omega| \varrho(t,x) = 0\}|_d \in C([0,T]).} \end{equation}
In the above $|A|_d$ denotes the $d$-dimensional Lebesgue measure of the set A. \end{thm}
The second theorem about the vacuum issue reads.
\begin{thm} \label{t3} Let $\Omega\subset R^d$ be a bounded Lipschitz domain. Let \begin{equation}\label{qpab} 1\le q,p,\alpha,\beta\le \infty,\; (q,\beta)\neq (1,\infty),\;\frac 1\beta+\frac 1q\le1,\;\frac 1\alpha+\frac 1p\le 1. \end{equation} Let $\varrho$ from {class} (\ref{gamma+}) be a renormalized time integrated weak solution to the continuity equation (\ref{co1}) with transporting velocity $\vc{u} \in L^{p}(0,T; W^{1,q}_0(\Omega;R^d))$ (i.e. it belongs to class (\ref{cocl1}), satisfies equation (\ref{co1.1}) and equation (\ref{rco1.1}) with renormalizing functions $b$ from (\ref{ren})).
Let \begin{equation}\label{clR-} 0\le R\in L^\infty(0,T;L^{\widetilde\gamma}(\Omega))\cap L^\alpha(0,T; L^\beta(\Omega)),\; \widetilde\gamma>1 \end{equation} be a distributional solution to the continuity equation (\ref{co1}) with the same transporting velocity $\vc{u}$.
Then \begin{enumerate} \item Function $R$ belongs to \begin{equation}\label{clR} R\in C_{\rm weak}([0,T];L^{\widetilde\gamma}(\Omega))\cap C([0,T]; L^r(\Omega)),\; 1\le r<\widetilde\gamma \end{equation} and it is a renormalized time integrated weak solution of the continuity equation (\ref{co1}). \item {The map $t\mapsto (s_\varrho R)(t)$} belongs to $C([0,T];L^r(\Omega))$ with any $1\le r<\widetilde\gamma$ and it is a renormalized time integrated weak solution of the continuity equation (\ref{co1}) (with the same transporting velocity). In particular, \begin{equation} \label{clRR} {\int_{\Omega} (s_\varrho R)(t,\cdot) {\, \rm d}x = \int_{\Omega} (s_\varrho R)(0,\cdot) {\, \rm d}x} \end{equation} for all $t \in [0,T]$. \item If further $\varrho(0,\cdot)>0$ a.e. in $\Omega$, then, up to sets of $d$-dimensional Lebesgue measure zero, for all $t\in (0,T]$ $$
\{x\in \Omega| \varrho(t,x) =0\} \subset \{x\in \Omega| R(t,x) =0\}. $$ \end{enumerate} \end{thm}
The second theorem has the following immediate consequences:
\begin{cor}\label{cor1} Let $q$, $p$, $\alpha$, $\beta$ verify conditions (\ref{qpab}) and $\widetilde\gamma,\gamma>1$.
Let $\Omega$, $\varrho$, $\vc{u}$ verify assumptions of Theorem \ref{t3}, where $\varrho(0,x)>0$. (In particular, $\varrho$ is a renormalized time integrated weak solution of the continuity equation (\ref{co1}) with transporting velocity $\vc{u}$.)
Let $\tau\in (0,T)$. Suppose that continuity equation (\ref{co1})
with transporting velocity $\vc{u}$ admits at least one distributional solution $R$
belonging to class (\ref{clR-})
which does not admit in $\Omega$ a vacuum at time $\tau$, i.e. $R(\tau)>0$ a.e. in $\Omega$.
Then $\varrho$ does not admit a vacuum at time $\tau$, i.e. $$
|\{x\in \Omega| \varrho(\tau,x) =0\}|_d=0. $$ \end{cor}
\begin{cor}\label{cor2} Let $q,\alpha,\beta,\gamma,\widetilde\gamma$ verify assumptions of Corollary \ref{cor1} with $p=\infty$.
Let $\Omega$, $\varrho$, $\vc{u}$ verify assumptions of Corollary \ref{cor1}. (In particular, $0\le \varrho$ is a renormalized time integrated weak solution of the continuity equation (\ref{co1}) with transporting velocity $\vc{u}$ and $\varrho(0,x)>0$.) We assume that $\vc{u}$ is time independent, i.e. $\vc{u}=\vc{u}(x)$, { $\vc{u}\in W^{1,q}_0(\Omega;R^d)$}.
Suppose that continuity equation (\ref{co1})
with transporting velocity $\vc{u}$ admits at least one (local in time) distributional solution $R$ on $(0,T')\times\Omega$ with some $T'>0$
belonging to class (\ref{clR-})$_{T=T'}$
which does not admit in $\Omega$ a vacuum at time $\tau\in (0,T')$, i.e. there exists $\tau\in (0,T')$ such that $R(\tau)>0$ a.e. in $\Omega$.
Then $\varrho$ does not admit a vacuum at any time in $[0,T]$, i.e. $$
\forall t\in [0,T],\; |\{x\in \Omega| \varrho(t,x) =0\}|_d=0. $$ \end{cor}
\begin{rmk}\label{rmkd0} \begin{enumerate} \item In practice, if $\vc{u} \in L^{p}(0,T; W^{1,q}(\Omega;R^d)),$ condition (\ref{cocl1}) in Theorems \ref{t2} and \ref{t3} is ensured by assumption \begin{equation}\label{gamma} 1<\gamma\le \infty,\;\frac 1\gamma+\frac 1 q\le 1+\frac 1d. \end{equation} Alternatively, condition (\ref{cocl1}) can be achieved by requiring $\vc{u} \in L^{p}(0,T;$ $ W^{1,q}(\Omega;R^d)),$ $\varrho\in L^\alpha(0,T;L^\beta(\Omega))$, where $p,q,\alpha,\beta$ verifies (\ref{qpab}). In the theory of weak solutions to compressible Navier--Stokes equations, the former setting provides stronger results, cf. Section \ref{M3}.
\item Condition $\vc{u}|_{I\times\partial\Omega}=\vc{0}$ in Theorem \ref{t3} and Corollaries \ref{cor1}, \ref{cor2} can be replaced by
$\vc{u}\cdot\vc n|_{I\times\partial\Omega}=0$. \item We notice that Theorem \ref{t2} holds independently of the boundary condition imposed on $\vc{u}$ at the boundary (since it deals with weak solutions in the sense of Definition \ref{dfco}. This is not the case of Theorems \ref{t3} and Corollaries \ref{cor1}, \ref{cor2}. Nevertheless, they continue to hold if we replace $W_0^{1,q}(\Omega)$ by $W^{1,q}(\Omega)$ provided we suppose that $R$ is { a renormalized time integrated {\rm weak} solution (instead of a renormalized time integrated {\rm distributional} solution).}
Anyway, however, in all these cases the condition $\varrho\vc{u}\cdot\vc n|_{I\times\partial\Omega}=0$ must always be satisfied al least in the weak sense; it is implicitly required in the weak formulation of the equation through the fact that the test functions do not vanish on the boundary. \end{enumerate} \end{rmk}
\subsection{Relations between various types of solutions to continuity and pure transport equations}\label{M2}
The proofs of Theorems \ref{t2} and \ref{t3} are based on the systematic study of relations and properties of the various types of weak solutions to the continuity and pure transport equations and their inter-relations. In this section we formulate the adequate results. They are, indeed, of independent interest.
\subsubsection{Time integration of renormalized distributional/weak solutions} \label{M2.1}
The main message of this subsection is the observation that any renormalized distributional (or weak) solution of the continuity equation/pure transport equation (introduced in Definitions \ref{dfco}--\ref{dftr}) admits---under certain reasonable conditions---a representative that is continuous on the time interval $[0,T]$ with values in $L^1(\Omega)$, and that both continuity/pure transport and renormalized continuity/pure transport equations can be integrated up to the end-points of any time interval $[0,\tau]$, $\tau\in [0,T]$.
\begin{thm}[Continuity equation] \label{th1} Let $\Omega\subset R^d$, $d\ge 2$ be a bounded domain with Lipschitz boundary. Let { $\vc{u}\in L^{p}(I; W^{1,q}(\Omega;R^d))$}, $1\le p,q\le \infty$. Suppose that \begin{equation}\label{t1.1} 0\le \varrho\in L^\infty(I;L^\gamma(\Omega)),\;\gamma>1. \end{equation}
Then the following statements are true: \begin{enumerate} \item If $\varrho$ is a {\em renormalized distributional solution} of the continuity equation with transporting velocity $\vc u$ (i.e. it belongs to class (\ref{cocl1}) and satisfies (\ref{co1.1}), (\ref{rco1.1}) with any renormalizing function $b$ from (\ref{ren})), then
function $\varrho$ and functions $b(\varrho)$ with any $b$ from (\ref{ren}) belong to the class (\ref{clR})$_{\widetilde\gamma=\gamma}$
and $\varrho$ is a {\em renormalized time integrated distributional solution} of the continuity equation with transporting velocity $\vc u$ (i.e. it belongs to class (\ref{cocl1}) and satisfies identities (\ref{co1.3}) and (\ref{rco1.3}) with any renormalizing function $b$ from (\ref{ren})). \item If $\varrho$ is a {\em renormalized weak solution} of the continuity equation with transporting velocity $\vc u$ (i.e. it belongs to class (\ref{cocl1}) and it satisfies equations (\ref{co1.2}), (\ref{rco1.2}) with any $b$ from (\ref{ren})), then function $\varrho$ and functions $b(\varrho)$ with any $b$ from (\ref{ren}) belong to the class (\ref{clR})$_{\widetilde\gamma=\gamma}$ and it is a {\em renormalized time integrated weak solution} of the continuity equation with transporting velocity $\vc u$ (i.e. it belongs to class (\ref{cocl1}) and it satisfies identities (\ref{co1.4}) and (\ref{rco1.4}) with any renormalizing function $b$ in class (\ref{ren})). \item Particularly, in both cases, $\varrho \in C(\overline{I};L^r(\Omega))$, $1\leq r<\gamma$. \end{enumerate} \end{thm}
The same statement holds for the pure transport equation. The theorem reads:
\begin{thm}[Pure transport equation]\label{th2} Let $\Omega$ and $\vc{u}$ satisfy assumptions of Theorem \ref{th1} and let $s$ fulfill (\ref{t1.1}).
Then the following statements are true: \begin{enumerate} \item If $s$ is a {\em renormalized distributional solution} of the pure transport equation with transporting velocity $\vc u$ (i.e. it belongs to class (\ref{trcl1}) and satisfies identities (\ref{tr1.1}), (\ref{rtr1.1})), then $s$ and $b(s)$ with any $b$ from (\ref{ren}) belong to class (\ref{clR})$_{\widetilde\gamma=\gamma}$ and $s$ is a {\em time integrated renormalized distributional solution} of the pure transport equation (i.e. it belongs to class (\ref{trcl1}) and it satisfies identities (\ref{tr1.3}) and (\ref{rtr1.3}) with any renormalizing function $b$ from (\ref{ren})). \item If $s$ is a {\em renormalized weak solution } of the continuity equation with transporting velocity $\vc u$ (i.e. it belongs to class (\ref{trcl1}) and it satisfies identities (\ref{tr1.2}) and (\ref{rtr1.2})), then $s$ and $b(s)$ with any $b$ from (\ref{ren}) belong to class (\ref{clR})$_{\widetilde\gamma=\gamma}$ and $s$
is {\em renormalized time integrated weak solution } (i.e. it belongs to class (\ref{trcl1}) and it satisfies identities (\ref{tr1.4}) and (\ref{rtr1.4}) with any renormalizing function $b$ from (\ref{ren})).
\item Particularly, in both cases, $s \in C(\overline{I};L^r(\Omega))$, $1\leq r<\gamma$. \end{enumerate} \end{thm}
\begin{rmk}\label{remd1} \begin{enumerate} \item Concerning the continuity equation: In practice, if $\vc{u} \in L^{p}(0,T; W^{1,q}(\Omega;R^d)),$ condition (\ref{cocl1}) in Theorem \ref{th1} can be ensured by assumption (\ref{t1.1}) with $\gamma$ from (\ref{gamma}).
If it is so, then the class of admissible renormalizing functions in Theorem \ref{th1} can be extended from (\ref{ren}) to\footnote{Here and in the sequel the exponent $q'$ is the H\"older conjugate exponent for $q$, $q_*$ is the { Sobolev exponent} for $q$ (and $q_*'$ is the H\"older conjugate exponent for $q_*$).} \begin{equation}\label{t1.3} b\in C^1([0,\infty)),\; b(\varrho)\le c(1+s^{\gamma/q_*'}),\; \varrho b'(\varrho)-b(\varrho)\le c(1+\varrho^{\gamma/q'}). \end{equation} This is the setting that allows to get the strongest results in applications to weak solutions to compressible fluids, see Subsection \ref{M3}.
Alternatively, condition (\ref{cocl1}) can be achieved by requiring $\vc{u} \in L^{p}(0,T;$ $ W^{1,q}(\Omega;R^d)),$ $\varrho\in L^\alpha(0,T;L^\beta(\Omega))$, where $p,q,\alpha,\beta$ verify (\ref{qpab}), as mentioned in Remark \ref{rmkd0}. In this case one can take the true condition (\ref{t1.1}) with any $\gamma>1$. Condition (\ref{qpab}) is however more restrictive than (\ref{gamma}) from the point of view of applications to compressible fluids. This setting is merely used only at the level of approximations of underlying compressible systems during the process of construction of weak solutions. Note finally that part of the first two claims of Theorem \ref{th1} hold without the requirement that the solutions is renormalized; i.e., if $\varrho$ is a distributional solution, then under the assumptions of this theorem it is a time integrated distributional solution, similarly in the case of weak solution. On the other hand, Item 3. requires that the solution is renormalized.
\item Concerning the transport equation: In practice, if the transporting velocity $\vc{u} \in L^{p}(0,T;$ $ W^{1,q}(\Omega;R^d)$ and $\varrho\in L^\alpha(0,T; L^\beta(\Omega))$, it is condition (\ref{qpab}) which guarantees satisfaction of condition (\ref{trcl1}) in Theorem \ref{th2}. In this situation the class of admissible renormalizing functions in Theorem \ref{th2} can be extended from (\ref{ren}) to \begin{equation}\label{t1.3+} b\in C^1([0,\infty)),\; b(s)\le c(1+ s^{\gamma/q'}). \end{equation} {Note further that part of the first two claims of Theorem \ref{th2} hold without the requirement that the solutions is renormalized; i.e., if $s$ is a distributional solution, then under the assumptions of this theorem it is a time integrated distributional solution, similarly in the case of weak solution. On the other hand, Item 3. requires that the solution is renormalized.}
\item It appears that condition (\ref{qpab}) coincides with the conditions in the assumptions in the Friedrichs commutator lemma (see Lemma \ref{l2} later) which is the basic tool in the passage from distributional solutions to renormalized distributional solutions. The same condition is needed in the passage from distributional to weak solutions in order to allow the application of the Hardy inequality near the boundary, cf. Theorem \ref{th3} for both features. This makes of the setting (\ref{qpab}) an universal setting convenient for general transport equations (including continuity and pure transport). This setting in the context of general transport equations has been introduced and fully exploited in the seminal DiPerna--Lions' paper \cite{DL}. \end{enumerate} \end{rmk}
\subsubsection{Passage from distributional to renormalized weak solutions} \label{M2.2}
The main message of this section is the observation that, under certain assumptions (which are, in general, slightly stronger than assumptions in the previous section), any distributional solution (time integrated distributional solution) of the continuity equation/pure transport equation (introduced in Definitions \ref{dfco}--\ref{dftr}) is a renormalized weak solution.
\begin{thm} [Continuity equation] \label{th3} Let $\Omega\subset R^d$, $d\ge 2$ be a bounded domain with Lipschitz boundary. Further, let { $\vc{u}\in L^{p}(I; W^{1,q}(\Omega;R^d))$,} $0\le \varrho\in L^\alpha(I;L^\beta(\Omega))$, where $p,q,\alpha,\beta$ satisfy condition (\ref{qpab}).
\begin{enumerate} \item Assume that $\varrho$ is a {\em distributional solution} of the continuity equation with transporting velocity $\vc u$ (i.e. it satisfies (\ref{co1.1})). Then the following statements are true: \begin{description} \item {1.1}
$\varrho$ is a {\rm renormalized distributional solution}, i.e. it satisfies, in addition to equation (\ref{co1.1}), also equation (\ref{rco1.1}) with any renormalizing function $b$ in class (\ref{ren}). \item {1.2} If moreover \begin{equation}\label{w0} \vc u\in L^{p}(I; W^{1,q}_0(\Omega;R^d)), \end{equation} then $\varrho$ is a {\em renormalized weak solution} of the continuity equation, i.e. $\varrho$ satisfies continuity equation (\ref{co1.2}) and its renormalized counterpart (\ref{rco1.2}) with any renormalizing function $b$ belonging to class (\ref{ren}). \end{description} \item Assume that $\varrho$ belongs to class \begin{equation}\label{gamma++} \varrho\in C_{\rm weak}(\overline I; L^\gamma(\Omega))\;\mbox{with some $\gamma>1$} \end{equation} and is a {\em time integrated distributional solution} of the continuity equation with transporting velocity $\vc u$ (i.e. it satisfies (\ref{cocl1}) and (\ref{co1.3})). Then the following statements are true: \begin{description} \item{2.1} Function $\varrho$ belongs to (\ref{clR})$_{\widetilde\gamma=\gamma}$ and functions $b(\varrho)$ with any $b\in (\ref{t1.3})$ belong to class (\ref{clR})$_{\widetilde\gamma=q_*'}$. Moreover, $\varrho$ is a {\it renormalized time integrated distributional solution} and it satisfies
equation (\ref{rco1.3}) with any renormalizing function $b$ belonging to (\ref{t1.3}). \item{2.2} If moreover $\vc{u}$ has zero traces (i.e. $\vc{u}$ satisfies { (\ref{w0})),} then $\varrho$ is a {\rm renormalized time integrated weak solution} of the continuity equation and it satisfies equations (\ref{co1.4}) and (\ref{rco1.4}) with any renormalizing function $b$ belonging to class (\ref{t1.3}). \end{description} \end{enumerate}
\end{thm}
\begin{thm}\label{th4}
Exactly the same statement---only with minor modifications---is va\-lid for the pure transport equation. The modifications are the following: \begin{enumerate} \item In assumptions of Statement 1., equation (\ref{co1.1}) must be replaced by (\ref{tr1.1}), and further:
In Statement 1.1, equation (\ref{rco1.1}) must be replaced by (\ref{rtr1.1}). In Statement 1.2, equations
(\ref{co1.3}), (\ref{rco1.3}) must be replaced (\ref{tr1.3}), (\ref{rtr1.3}) and condition (\ref{t1.3}) by (\ref{t1.3+}). \item In assumptions of Statement 2., equations (\ref{cocl1}) and (\ref{co1.3}) must be replaced by (\ref{trcl1}) and (\ref{tr1.3})
and condition (\ref{t1.3}) by (\ref{t1.3+}), and further:
In Statement 2.1, equations (\ref{rco1.3}) must be replaced by (\ref{rtr1.3}). In Statement 2.2, equations (\ref{co1.4}), (\ref{rco1.4}) must be replaced by (\ref{tr1.4}), (\ref{rtr1.4}) and relation (\ref{clR})$_{\widetilde\gamma=q_*'}$ must be replaced by (\ref{clR})$_{\widetilde\gamma=q'}$. \end{enumerate} \end{thm}
\subsubsection{From pure transport equation to continuity equation} \label{M2.3}
\begin{thm}\label{th5} Let $\Omega$ be a bounded domain with Lipschitz boundary\footnote{As a matter of fact, the assumptions is important only in case of weak solutions. The result dealing with distributional solutions holds for arbitrary domain $\Omega$.}. Suppose that $$ 1\le q,p,\alpha_\varrho,\beta_\varrho,\alpha_s,\beta_s\le \infty,\; (q,\beta_\varrho)\neq (1,\infty),\; (q,\beta_s)\neq (1,\infty), $$ $$
\frac 1{\alpha_\varrho}+ \frac 1{\alpha_s} +\frac 1p\le 1,\; \frac 1{r_\varrho}+ \frac 1{r_s} +\frac 1q\le 1, $$ where $$
r_\varrho\left\{\begin{array}{c} \in [1,\infty)\,\mbox{if $q>1$ and $\beta_\varrho=\infty$}\\ =\beta_\varrho\,\mbox{otherwise} \end{array}\right\}, \;
r_s\left\{\begin{array}{c} \in [1,\infty)\,\mbox{if $q>1$ and $\beta_s=\infty$}\\ =\beta_s\,\mbox{otherwise} \end{array}\right\}. $$ Let $$ \varrho\in L^{\alpha_\varrho}(I;L^{\beta_{\varrho}}(\Omega)),\; s\in L^{\alpha_s}(I;L^{\beta_{s}}(\Omega)),\;{ \vc{u}\in L^p(I;W^{1,q}(\Omega;R^d)).} $$ Then there holds: \begin{enumerate} \item Assume additionally that $$ \frac 1 {t_\varrho} + \frac{1}{t_s} + \frac{1}{p} \leq 1, $$ where $$
t_\varrho\left\{\begin{array}{c} \in [1,\infty)\,\mbox{if $p>1$ and $\alpha_\varrho=\infty$}\\ =\alpha_\varrho\,\mbox{otherwise} \end{array}\right\},\;
t_s\left\{\begin{array}{c} \in [1,\infty)\,\mbox{if $p>1$ and $\alpha_s=\infty$}\\ =\alpha_s\,\mbox{otherwise} \end{array}\right\}. $$ If $\varrho$ is a distributional (resp. weak) solution of the continuity equation (\ref{co1}) and $s$ a distributional (resp. weak) solution of the pure transport equation (\ref{tr1}) with transporting velocity $\vc{u}$, then $\varrho s$ is a renormalized distributional (resp. weak) solution of the continuity equation with the same transporting velocity $\vc{u}$. \item If $\varrho\in C_{\rm weak}(\overline I;L^{\gamma_\varrho}(\Omega))$ is a time integrated distributional (resp. weak) solution of the continuity equation (\ref{co1}) and $s\in C_{\rm weak}(\overline I;L^{\gamma_s}(\Omega))$ a time integrated distributional (resp. weak) solution of the pure transport equation (\ref{tr1}) with transporting velocity $\vc{u}$ (where $1<\gamma_\varrho,\gamma_s\le \infty$, $\frac 1{\gamma_\varrho}+\frac 1{\gamma_s}:=\frac 1\gamma<1$), then $\varrho s\in C(\overline I;L^r(\Omega))$, $1\le r<\gamma$ is a renormalized distributional (resp. weak) solution of the continuity equation with the same transporting velocity $\vc{u}$. \end{enumerate} \end{thm}
\subsection{Application to compressible Navier--Stokes equations}\label{M3}
For simplicity, let us first recall the compressible Navier--Stokes equations in barotropic regime:
\begin{equation} \label{2.9} \begin{aligned} \partial_t \varrho + {\rm div}\, (\varrho \vc{u}) &= 0 \\
\partial_t (\varrho \vc{u}) + {\rm div}\, (\varrho\vc{u} \otimes \vc{u}) + \nabla p(\varrho) & = {\rm div}\,\tn S(\nabla\vc{u})+\varrho \vc{f}\\ \end{aligned} \end{equation} which we consider in $(0,T)\times\Omega$, together with the initial conditions in $\Omega$ \begin{equation} \label{2.10} \varrho(0,\cdot) = \varrho_0, \qquad (\varrho\vc{u})(0,\cdot) = \vc{m}_0 \end{equation} and so called no-slip boundary condition on $(0,T)\times \partial \Omega$ \begin{equation} \label{2.11} \vc{u}(t,x) =\vc{0}. \end{equation} The homogeneous boundary condition (\ref{2.11}) can be replaced by Navier (slip) boundary conditions or by periodic boundary conditions if $\Omega$ is a periodic cell.
In the above, $\tn S$ is the viscous stress tensor, which reads \begin{equation}\label{S} \tn S(\nabla\vc{u})=\mu\Big(\nabla\vc{u}+\nabla\vc{u}^t-\frac 2d{\rm div}\,\vc{u} \tn I\Big)+\lambda{\rm div}\,\vc{u} \tn I. \end{equation} The viscosity coefficients are assumed to be constant: $\mu >0$ and $\lambda \ge 0$. Function $\varrho\mapsto p(\varrho)$ denotes the pressure. One supposes that $$ p\in C^1([0,\infty)). $$
The classical (or strong) solutions, in general, may not exist (we can prove their existence either if the data are smooth and the time interval is sufficiently short or if the data are in some sense additionally sufficiently small). We therefore consider the weak solutions. They are defined as follows:
\begin{df}\label{defws} Let $\varrho_0 \in L^{\gamma}(\Omega)$, $0\le\varrho_0\in L^\gamma(\Omega)$ a.e. in $\Omega$, $\gamma>1$, $r>1$, $(\varrho\vc{u})(0,\cdot) = \vc{m}_0 \in L^1(\Omega;R^d)$ and $\vc{f} \in L^\infty((0,T)\times \Omega;R^d)$. A couple $(\varrho,\vc{u})$ is a renormalized weak solution to the initial boundary value problem (\ref{2.9}--\ref{2.11}) iff: \begin{enumerate} \item The couple $(\varrho,\vc{u})$ belongs to functional spaces $$ 0\le \varrho \in C_{{\rm weak}}(\overline I;L^\gamma (\Omega)),\; \vc{u} \in L^2(I;W^{1,2}_0(\Omega;R^d)),\;p(\varrho)\in L^1(Q), $$ $$ \varrho\vc{u} \in C_{{\rm weak}}(\overline I; L^{r}(\Omega;R^d)), \;\varrho (\vc{u}\otimes \vc{u}), p(\varrho) \in L^1((0,T)\times \Omega; R^{d\times d}). $$ \item $\varrho$ is a time integrated renormalized weak solution to the continuity equation \eqref{2.9}$_1$ with transporting velocity $\vc{u}$. \item The couple $(\varrho,\vc{u})$ verifies the momentum equation \eqref{2.9}$_2$ in the sense of distributions. \end{enumerate} \end{df}
If Navier or periodic conditions are considered, the functional spaces and test functions in the above definition must be accordingly modified, see \cite{FeNo_book}, \cite{NoSt} or \cite{FNPdom}.
\begin{cor} \label{c1} Let $\gamma$ verify condition (\ref{gamma}) with $q=2$ (in particular $\gamma\ge 6/5$ if $d=3$). Then the claims of Theorems \ref{t2} and \ref{t3} (and Corollaries \ref{cor1}, \ref{cor2}) hold for any renormalized weak solution to the compressible Navier--Stokes equations specified in Definition \ref{defws}.
\end{cor}
\begin{rmk}\label{r1} \begin{enumerate} \item Note that renormalized weak solutions to the Navier--Sto\-kes equations with the regularity properties stated above (and, additionally, fulfilling the energy inequality) can be constructed with any of no-slip, Navier (slip) or periodic boundary conditions provided $\gamma>d/2$ and $$ p(0)=0,\;p'(\varrho)\ge a_1\varrho^\gamma-b,\;p(\varrho)\le a_2\varrho^\gamma+b,\;\mbox{with some}\; a_1,a_2,b>0, $$ \cite{FeNoPe} (for monotone pressure), \cite{Fe2002} (for non monotone pressure) and sufficiently regular { domains,} and \cite{NoNo}, \cite{Kuku} or \cite{Poul} for a generalization to Lipschitz domains. \item The above condition for pressure allows pressure functions which are non monotone on a compact portion of $[0,\infty)$. In the case of periodic boundary conditions and provided $\gamma\ge 9/5$, { this condition can be generalized allowing pressure functions non monotone up to infinity and, also, another generalization allows small anisotropic perturbations of the isotropic stress tensor (\ref{S}), see Bresch, Jabin \cite[Theorems 3.1 and 3.2]{BrJa}.} { \item {Theorems \ref{t2}, \ref{t3} and Corollaries \ref{cor1}, \ref{cor2} also apply to a couple $(\varrho,\vc{u})$, where $(\varrho,\vc{u},\vartheta)$---(density, velocity, temperature)---is a weak solution of the full Navier--Stokes--Fourier system,} constructed (according to different definitions of weak solutions under different physical assumptions on constitutive laws and transport coefficients) either in Feireisl \cite[Definition 7.1 and Theorem 7.1]{FeBook} or in \cite[Theorem 3.1]{FeNo_book} or in \cite{FeNoEd}, \cite{FNS}. \item Theorems \ref{t2}, \ref{t3} and Corollaries \ref{cor1}, \ref{cor2} do not, in general, directly apply to a couple $(\varrho,\vc{u})$ of weak solutions of Navier-Stokes equations with degenerate density dependent viscosities unless it cannot be guaranteed that $\vc{u}$ belongs to a Sobolev space of type $L^p(I;W^{1,q}(\Omega;R^d))$. In fact, in this situation, typically, $\nabla\vc{u}$ belongs to a Lebesgue space weighted by a positive power of $\varrho$ (cf. { Bresch}, Desjardins \cite{BrDes}, Mellet, Vasseur \cite{MeVa}, Vasseur, Yu \cite{VY}, Li, Xin \cite{LiXin} for non exhausting relevant references). } \end{enumerate} \end{rmk}
\section{Basic preliminaries}\label{S4}
Let us mention some standard preliminary tools. We shall use several times the theorem on Lebesgue points in the following form. \begin{lem}\label{Lbg} Let $f\in L^1(0,T;L^\gamma(\Omega))$, $1\le\gamma<\infty$. Then there exists $N\subset (0,T)$ of zero Lebesgue measure such that for all $\tau\in (0,T)\setminus N$, { $$ \begin{aligned}
\lim_{h\to 0+} \frac 1h\int_{\tau-h}^\tau \|f(t,\cdot)-f(\tau,\cdot)\|_{L^\gamma(\Omega)}{\rm d}t&\to 0, \\
\lim_{h\to 0+} \frac 1h\int_\tau^{\tau+h} \|f(t,\cdot)-f(\tau,\cdot)\|_{L^\gamma(\Omega)}{\rm d}t&\to 0. \end{aligned} $$ Moreover, if $f\in C_{\rm weak}([0,T];L^\gamma(\Omega))$, then for any $\eta \in L^{\gamma'}(\Omega)$ $$
\forall \tau\in [0,T),\;\intO{f(\tau,\cdot)\eta}= \lim_{h\to 0+}\frac 1h\int_{\tau}^{\tau+h}\Big(\intO{f(t,\cdot)\eta}\Big){\rm d}t $$ and $$
\sup_{\tau\in [0,T]}\|f(\tau,\cdot)\|_{L^\gamma(\Omega)}\le \|f\|_{L^\infty(0,T;L^{\gamma'}(\Omega))}. $$} \end{lem}
We shall also frequently use mollifiers. For the sake of completeness, we recall the basic facts. We denote by $j$ a function on $R^{d}$, $d \geq 1$, satisfying the following requirements:
$
j \in C^{\infty}_c(R^{d}),\; {\rm supp}(j)=B(0,1),
$
$
j(x)=j(-x),\; j \geq 0\;\mbox{on $R^{d}$},\; \int_{R^d} j(x) {\rm d}x=1.
$
Next, for $\epsilon>0$, we denote by $j_\epsilon$ the function
$j_{\epsilon}(x):=\frac{1}{\epsilon^{d}}j(\frac{x}{\epsilon})$.
For a given function $f\in L_{\rm loc}^1(R^{d})$, we finally define mollified $f$ as follows: $ [f]_{\epsilon}:=f*j_{\epsilon} (x)=\int_{R^d}j_{\epsilon}(x-y) f(y){\rm d}y $.
Let us recall the classical properties of these approximations.
\begin{lem}\label{prop1}
\begin{enumerate}
\item
If $1 \leq p < \infty$, then for any $f \in L^{p}(R^{d})$
$$
[f]_\varepsilon \in C^{\infty}(R^{d}) \cap {L^{p}}(R^{d}),\;
\|{[f]_{\epsilon}}\|_{L^{p}(R^d)} \leq \|{f}\|_{L^{p}(R^d)}
$$
and
$$
f_\varepsilon \rightarrow f\;\mbox{ in $L^p(R^d)$.}
$$
\item
If $p=\infty$, then
$$
[f]_\varepsilon \in C^{\infty}(R^{d}) \cap {L^{\infty}}(R^{d}),\;
\|{[f]_{\epsilon}}\|_{L^{\infty}(R^d)} \leq \|{f}\|_{L^{\infty}(R^d)}.
$$
Moreover, if $f$ is uniformly continuous on { $R^d$,} then
$$
[f]_\varepsilon\to f\;\mbox{in $C_b(R^d)$}.
$$
\item Let $1\le p\le\infty$. For all $f\in L^p(R^d)$, $g\in L^{p'}(R^d)$,
$$
\int_{R^d}[f]_\varepsilon g {\, \rm d}x=\int_{R^d} f [g]_\varepsilon {\, \rm d}x.
$$
\end{enumerate}
\end{lem}
The next lemma is the well-known Friedrichs lemma on commutators. It deals with the regularization of the
quantity $\vc{u}\cdot\nabla f$ defined in the sense of distributions as
$$
\vc{u}\cdot\nabla f:={\rm div}(f\vc{u})-f{\rm div}\vc{u}.
$$ The lemma reads. \begin{lem}[Friedrichs commutator lemma] \label{l2} Let $I\subset R$ be an open bounded interval and $f \in L^\alpha(I;L^\beta_{{\rm loc}}(R^d))$, $\vc{u} \in L^p(I;W^{1,q}_{{\rm loc}}(R^d;R^d))$. Let $1\leq q,\beta\leq \infty$, $(q,\beta) \neq (1,\infty)$, $\frac 1q + \frac 1{\beta} \leq 1$, $1\leq \alpha \leq \infty$ and $\frac 1\alpha + \frac 1p \leq 1$. Then $$ [\vc{u} \cdot \nabla f]_\varepsilon - \vc{u} \cdot \nabla [f]_\varepsilon \to 0 $$ strongly in $L^t(I;L^r_{{\rm loc}} (R^d))$, where $$ \frac 1t \geq \frac 1\alpha + \frac 1p,\; t\in [1,\infty) $$ and $$ r\in [1,q) \text{ for } \beta = \infty, \ q \in (1,\infty], $$ while $\frac 1\beta + \frac 1q \leq \frac 1r \leq 1$ otherwise. \end{lem}
\section{Proof of Theorems \ref{th1}--\ref{th2}}\label{S5}
The proof of Theorems { \ref{th1}--\ref{th2} is based} on the following two lemmas. The first lemma deals with distributional (or weak) solutions to conservation laws (\ref{dd}) and claims that their solutions admit, under certain conditions, $C_{\rm weak}([0,T],L^1(\Omega))$-representatives, and can be therefore integrated up to the endpoints of any time interval $[0,\tau]\subset [0,T]$.
\begin{lem}\label{aux3} Let $d \in L^{\infty}(I,L^{\gamma}(\Omega))$, $\gamma >1$ and $\vc{F} \in L^{1}(Q;R^{d})$, $G \in L^{1}(Q) $. \begin{enumerate} \item Suppose that \begin{equation}\label{dd} \partial_{t}d+{\rm div}\, \vc{F}+G=0 \quad \mbox{ in} \quad D'(Q). \end{equation} Then there exists a representative of $d$ such that it belongs to the space $C_{{\rm weak}}([0,T],L^{\gamma}(\Omega))$ and equation (\ref{dd}) can be integrated up to any time $\tau\in (0,T]$, i.e.
$\forall \xi \in C^{1}([0,T])$, $\forall \tau \in (0,T]$ and $\forall \eta \in C_{c}^{1}(\Omega)$, there holds \begin{equation}\label{ddi} \int_{\Omega}d(\tau, x)\xi(\tau)\eta (x){\, \rm d}x-\int_{\Omega}d(0,x)\xi(0)\eta(x){\, \rm d}x= \end{equation} \[\int_{0}^{\tau}\int_{\Omega}\Big(d(t,x)\partial_{t}\xi(t)+\vc{F}(t,x)\cdot\nabla \eta(x)\xi(t) -G(t,x)\xi(t)\eta(x)\Big)\dx \dt.\] \item Suppose that (\ref{dd}) holds up to the boundary, i.e. $$ \int_Q\Big(d\partial_t\varphi +\vc{F}\cdot\nabla\varphi-G\varphi\Big) \dx \dt=0\;\mbox{for all $\varphi\in C^1_c((0,T)\times\overline\Omega)$.} $$ Then there exists a representative of $d$ such that it belongs to the space $C_{{\rm weak}}([0,T],L^{\gamma}(\Omega))$ and equation (\ref{ddi}) holds $\forall \xi \in C^{1}([0,T])$, $\forall \tau \in (0,T]$ and $\forall \eta \in C_{c}^{1}(\overline\Omega)$. \end{enumerate} \end{lem}
\begin{proof} We shall show only Statement 1. of Lemma \ref{aux3}. Statement 2. can be obtained repeating word by word the proof of Statement 1. with minor modifications.
We take in equation (\ref{dd}) test functions $\varphi(t,x)=\psi(t)\eta(x)$, where $\eta\in C^1_c(\Omega)$, and \begin{equation*} \psi(t) =\psi_{\tau,h}^+=
\begin{cases}
\frac{1}{h}t &\text{if $t\in [0,h] $}\\
1 &\text{if $t\in [h,\tau] $}\\
1-\frac{t-\tau}{h} &\text{if $t \in [\tau,\tau+h] $}\\
0 &\text{if $t \in (\tau+h, +\infty). $}
\end{cases} \end{equation*} Under assumptions on $d$, $\vc{F}$ and $G$ it is a folklore to show that this is an admissible test function in equation (\ref{dd}).
We obtain by direct calculation, \begin{equation}\label{ddd} \frac{1}{h}\int_{\tau}^{\tau+h}\int_{\Omega}d(t,x)\eta(x)\dx \dt-\frac{1}{h}\int_{0}^{h}\int_{\Omega}d(t,x)\eta(x)\dx \dt= \end{equation} \[\int_{0}^{\tau+h}\psi (t)\int_{\Omega}\vc{F}(t,x)\cdot \nabla \eta(x)\dx \dt- \int_{0}^{\tau+h}\psi (t)\int_{\Omega}G(t,x)\eta(x)\, {\rm d} t {\, \rm d}x.\] This identity leads to the following observations: \begin{enumerate}
\item According to the theorem on Lebesgue points (cf. Lemma \ref{Lbg}), there is a set $N\subset(0,T)$ of zero Lebesgue measure $|N|=0$, such that for all $\tau\in (0,T)\setminus N$, the limit $h\to 0+$ of the first expression exists. Since the limit of the right hand side as $h\to 0+$ exists as well, we deduce that $$ \forall\eta\in C^1_c(\Omega),\; \lim_{h\to 0+}\frac{1}{h}\int_{0}^{h}\int_{\Omega}d(t,x)\eta(x)\dx \dt:= {\mathfrak{d}}_\eta(0+)\in { R.} $$ The map $C^1_c(\Omega)\ni\eta\to {\mathfrak{d} }_\eta(0+)\in R$ is evidently {linear.} Moreover, since $d \in L^{\infty}(I,L^{\gamma}(\Omega))$, we have estimate $$
\sup_{0<h<T}\Big|\frac{1}{h}\int_{0}^{h}\int_{\Omega}d(t,x)\eta(x)\dx \dt\Big|\le \|d\|_{L^\infty(0,T; L^\gamma(\Omega))}\|\eta\|_{L^{\gamma'}(\Omega))} $$ by virtue of the H\"older inequality. In view of the Riesz representation theorem, we deduce that there exists ${\mathfrak{d}}(0+)\in L^{\gamma}(\Omega)$ such that $$ \forall\eta\in C^1_c(\Omega),\; {\mathfrak{d}}_\eta(0+)=\int_\Omega {\mathfrak{d}}(0+)\eta\, {\rm d}x. $$ \item Now, we take an arbitrary $\tau\in (0,T)$ and calculate limit $h\to 0+$ in equation (\ref{ddd}). We already know that for all $\eta\in C^1_c(\Omega)$ the limits of the second term at the left hand side and the limit of the right hand side exist and belong to $R$. We deduce from this fact that $$ \lim_{h\to 0+}\frac{1}{h}\int_{\tau}^{\tau+h}\int_{\Omega}d(t,x)\eta(x)\dx \dt:={\mathfrak{d}}_\eta(\tau+), $$ where, by the same token as in the previous step, $$ \forall\eta\in C^1_c(\Omega),\; {\mathfrak{d}}_\eta(\tau+)=\int_\Omega{\mathfrak{d}}(\tau+)\eta\, {\rm d}x\;\mbox{with $\mathfrak{d}(\tau+)\in L^{\gamma}(\Omega)$}. $$ \item We test equation (\ref{dd}) by functions $\varphi(t,x)=\psi(t)\eta(x)$, where \begin{equation*} \psi(t) = \psi_{\tau,h}^-
\begin{cases}
\frac{1}{h}t &\text{if $t\in [0,h] $}\\
1 &\text{if $t\in [h,\tau] $}\\
1-\frac{t-\tau+h }{h} &\text{if $t \in [\tau-h,\tau] $}\\
0 &\text{if $t \in (\tau, +\infty). $}
\end{cases} \end{equation*} It reads \begin{equation}\label{ddd!} \frac{1}{h}\int_{\tau-h}^{\tau}\int_{\Omega}d(t,x)\eta(x)\dx \dt-\frac{1}{h}\int_{0}^{h}\int_{\Omega}d(t,x)\eta(x)\dx \dt= \end{equation} \[\int_{0}^{\tau}\psi (t)\int_{\Omega}F(t,x) \nabla \eta(x)\dx \dt- \int_{0}^{\tau}\psi (t)\int_{\Omega}G(t,x)\eta(x)\, {\rm d} t {\, \rm d}x.\]
\item By the same token as in Items 1. and 2. we define ${\mathfrak{d}}_\eta(\tau-)$ and $\mathfrak{d}(\tau-)\in L^{\gamma}(\Omega)$ for all $\tau\in (0,T]$ . Subtracting (\ref{ddd}) and (\ref{ddd!}) and effectuating limit $h\to 0^+$, we obtain $$ \forall \tau \in (0,T), \; \mathfrak{d}(\tau):={\mathfrak{d}(\tau+)}= \mathfrak{d}(\tau-). $$ We define $$ {\mathfrak{d}}(0):=\mathfrak{d}(0+),\quad\mathfrak{d}(\tau):=\mathfrak {d}{(\tau+)},\;\tau\in (0,T),\quad \mathfrak{d}(T):=\mathfrak{d}(T-). $$ We easily verify that $\mathfrak{d}$ satisfies equation (\ref{ddi}).
Subtracting (\ref{ddi}) with $\tau=\tau_1$ and $\tau=\tau_2$, $\tau_1,\tau_2\in [0,T]$ we readily verify that { $$ \forall\eta\in C^1_c(\Omega),\;\mbox{the map } \tau\mapsto\int_{\Omega}{\mathfrak{d}}(\tau)\eta\,{\rm d}x \mbox{ is continuous on } [0,T]. $$} Since $ C^1_c(\Omega)$ is dense in $L^{\gamma'}(\Omega)$, we finally conclude that $$ {\mathfrak{d}}\in C_{\rm weak}([0,T]; L^\gamma(\Omega)). $$ \item According to theorem on Lebesgue points (cf. Proposition \ref{Lbg}), we have $$ d(\tau+)= d(\tau-)=d(\tau)=\mathfrak{d}(\tau)\;\mbox{a.e. in (0,T)}. $$ This completes the proof of the fact that there exists a representative of $d$ such that $d\in C_{\rm weak}([0,T]; L^\gamma(\Omega))$.
\item It remains to show equation (\ref{ddi}). To this end we can repeat the whole procedure consisting of { Items 1.--5.} with test functions $\varphi(t,x)=\psi(t)\xi(t)\eta(x)$, where $\psi=\psi^\pm_{\tau,h}$, $\xi\in C^1([0,T])$ and $\eta\in C^1_c(\Omega)$. \end{enumerate} Lemma \ref{aux3} is thus proved. \end{proof}
The continuity and pure transport equations are particular cases of equations investigated in Lemma \ref{aux3}. If we additionally know that their solutions are renormalized, we can show that they not only belong to the class $C_{\rm weak}([0,T]; L^1(\Omega))$ but even to the class
$C([0,T]; L^1(\Omega))$. { This is subject of the second lemma.}
\begin{lem}\label{aux4} \begin{enumerate} \item Let $\vc{u} \in L^{p}(I,W^{1,q}(\Omega;R^d))$, $1\leq p\le \infty$, $1<q\leq \infty$, \begin{equation}\label{cl1} \varrho\in L^{\infty}(I,L^{\gamma}(\Omega)),\;\gamma>1,\;\frac 1\gamma+\frac 1{q}\le 1+\frac 1d \end{equation} or \begin{equation}\label{cl1+} \varrho \in L^{\infty}(I,L^{\gamma}(\Omega))\cap L^{p'}(I;L^{q'}(\Omega)), \gamma>1. \end{equation} Suppose that $\varrho$ is a renormalized distributional solution of the continuity equation (i.e. it satisfies (\ref{co1.1}), (\ref{rco1.1}) with renormalizing function $b$ in the class (\ref{ren})). Then there exists a representative of $\varrho$ such that $$ \varrho \in C(\overline I;L^r(\Omega)),\; 1\le r<\gamma. $$ \item The same statement, under the same assumptions on $\vc{u}$ and under assumption (\ref{cl1+}) holds for any renormalized distributional solution to the pure transport equation (satisfying (\ref{tr1.1}), (\ref{rtr1.1}) with renormalizing function $b$ in the class (\ref{ren})). \end{enumerate} \end{lem}
\begin{proof} Again, it is enough to prove Statement 1. dealing with the continuity equation. The proof of Statement 2. for the pure transport equation requires only minor modifications and is, therefore, left to the reader as an exercise. It is to be noticed that, due to the presence of term $s{\rm div}\,\vc{u}$ in the weak formulation of the pure transport equation, { Statement 2. is not true under assumption (\ref{cl1}) unless $\gamma \geq q'$. }
Employing Lemma \ref{aux3} (with $d=\varrho$, $\vc{F}=\varrho\vc{u}$, $G=0$) we may suppose that $\varrho\in C_{\rm weak}(\overline I;L^\gamma(\Omega))$.
Since $\varrho$ is a renormalized distributive solution of the continuity equation, it satisfies \begin{equation} \label{eq19} \partial_{t}T_{k}(\varrho)+{\rm div}\,(T_{k}(\varrho)\vc{u})+(\varrho T_{k}'(\varrho)-T_{k}(\varrho)){\rm div}\, \vc{u}=0 \quad \mbox{in} \quad {\cal D}'(Q), \end{equation} where for any $k>1$ $$ T_{k}(\varrho)=kT{\Big(\frac{\varrho}{k}\Big)}\;\mbox{ with $T\in C^{1}([0,\infty))$},
$$ with \begin{equation*} T(s) =
\begin{cases}
s &\text{if $0\leq s\leq 1 $}\\
2 &\text{if $s \geq 3$.}\\
\end{cases} \end{equation*}
According to Lemma \ref{aux3} applied to (\ref{eq19}) with $d:=T_{k}(\varrho)$, $\vc{F}:=T_{k}(\varrho)\vc{u}$ and $G:=(\varrho T_{k}'(\varrho)-T_{k}(\varrho)){\rm div}\, \vc{u}$, there exists \begin{equation}\label{Tmol} \mathcal{T}_{k}(\varrho)\in C_{{\rm weak}}([0,T],L^{p}(\Omega)),\; \forall 1 \leq p < + \infty, \end{equation}
$$ (\mathcal{T}_{k}(\varrho))(t)=T_{k}(\varrho(t))\;\mbox{ a.a. in $\Omega$ for a.a. $t \in (0,T)$}, $$
such that \begin{equation} \label{eq19+} \partial_{t}\mathcal{T}_{k}(\varrho)+{\rm div}\,(\mathcal{T}_{k}(\varrho)\vc{u})+(\varrho \mathcal{T}_{k}'(\varrho)-\mathcal{T}_{k}(\varrho)){\rm div}\, \vc{u}=0 \quad \mbox{in} \quad {\cal D}'(Q). \end{equation}
We can extend $\mathcal{T}_{k}(\varrho)$ by $0$ outside $\Omega$ and regularize it by using standard mollifiers over the space variables. The equation for mollified functions $[\mathcal{T}_{k}(\varrho)]_{\varepsilon}$ reads
\begin{equation} \label{eq20}
\partial_{t}[\mathcal{T}_{k}(\varrho)]_{\varepsilon}+{\rm div}\,([\mathcal{T}_{k}(\varrho)]_{\varepsilon}\vc{u})+\Big[\big(\rho \mathcal{T}_{k}'(\varrho)-\mathcal{T}_{k}(\varrho)\big){\rm div}\, \vc{u}\Big]_\varepsilon=r_{\varepsilon}
\end{equation} a.e. in $$
Q_\varepsilon= I\times\Omega_\varepsilon,\;\Omega_\varepsilon=\{x\in \Omega\,|\,{\rm dist}(x,R^d\setminus\Omega)>\varepsilon\}, $$
where \[ r_\varepsilon:=r_{\varepsilon}(\mathcal{T}_{k}(\varrho),\vc{u})={\rm div}\,([\mathcal{T}_{k}(\varrho)]_{\varepsilon}\vc{u})-{\rm div}\,[\mathcal{T}_{k}(\varrho)\vc{u}]_{\varepsilon}\to 0\;\mbox{as $\varepsilon\to 0$} \] in $L^{p}(I; L^{\widetilde q}( K))$, with any compact $K\subset\Omega$, $\widetilde q <q$ by virtue of the Friedrichs lemma on commutators (cf. Lemma \ref{l2}).
Due to the standard properties of mollifiers $$ \Big[{ \varrho} \mathcal{T}_{k}'(\varrho)-\mathcal{T}_{k}(\varrho){\rm div}\, \vc{u}\Big]_\varepsilon \to \varrho \mathcal{T}_{k}'(\varrho)-\mathcal{T}_{k}(\varrho){\rm div}\, \vc{u} $$ in $L^{p}(I; L^{q}( K))$, $K\subset \Omega$, compact.
On the other hand, since $\mathcal{T}_{k}(\varrho)(t,\cdot))\in L^{r}(\Omega)$ {\em for all} $t \in [0,T], 1\leq r < + \infty$, we get by the same token, in
particular,
\begin{equation}\label{D0}
\forall t \in [0,T] \quad [\mathcal{T}_{k}(\varrho)(t, \cdot)]_{\varepsilon} \to \mathcal{T}_{k}(\varrho)(t, \cdot) \;\mbox{ in $L^{2}(K)$ with any compact $K\subset\Omega$.}
\end{equation}
Moreover, since $\mathcal{T}_{k}(\varrho)\in C_{\rm weak}([0,T]; L^p(\Omega))$, we infer that the mapping $t \mapsto [\mathcal{T}_{k}(\varrho)]_{\varepsilon}(\cdot, x)$ belongs to $C([0,T])$ for all $x \in \Omega_\varepsilon$ and hence $t \mapsto [\mathcal{T}_{k}(\varrho)]_{\varepsilon}^{2}(\cdot, x) \in C[0,T]$ for all $x \in \Omega_\varepsilon$.
Consequently,
\[\Big(t \mapsto \int_{\Omega}[\mathcal{T}_{k}(\varrho)]_{\varepsilon}^{2}(t,x)\eta(x) {\, \rm d}x\Big) \in C([0,T])\] for all $\eta \in C^1_c(\Omega)$ and $0<\varepsilon<{\rm dist}({\rm supp}\,\eta,R^d\setminus\Omega)$. We deduce from estimate \[
\sup_{t \in [0,T]}\int_{\Omega}[\mathcal{T}_{k}(\varrho)]_{\varepsilon}^{2}(t,x)\eta(x) {\, \rm d}x \leq \sup_{t \in [0,T]} \|\mathcal{T}_{k}(\varrho(t,\cdot))\|_{L^{2}(\Omega)}\|\eta\|_{L^{2}(\Omega)} \]
\[\leq\|\mathcal{T}_{k}(\varrho)\|_{L^\infty(0,T;L^{2}(\Omega))}\|\eta\|_{L^{2}(\Omega)}\leq C\]
that the family of maps \begin{equation}\label{maps}
\Big\{t \mapsto \int\limits_{\Omega}[\mathcal{T}_{k}(\varrho)]_{\varepsilon}^{2}(t,x)\eta(x) {\, \rm d}x\,|\, 0<\varepsilon<{\rm dist}({\rm supp}\,\eta, R^d\setminus\Omega)\Big\} \end{equation} is for any $k>1$ and any $\eta \in C^1_c(\Omega)$ equi-bounded in $C([0,T])$.
We multiply (\ref{eq20}) by $2[\mathcal{T}_{k}(\rho)]_{\varepsilon}$, in order to get
\begin{equation}\label{eqD1}
\partial_{t}[\mathcal{T}_{k}(\varrho)]_{\varepsilon}^{2}+{\rm div}\,([\mathcal{T}_{k}(\varrho)]_{\varepsilon}^{2}\vc{u})+[\mathcal{T}_{k}(\varrho)]_{\varepsilon}^{2}{\rm div}\, \vc{u}+
\end{equation}
\[ 2[\mathcal{T}_{k}(\varrho)]_{\varepsilon}\Big[(\varrho \mathcal{T}_{k}'(\varrho)-\mathcal{T}_{k}(\varrho)){\rm div}\, \vc{u}\Big]_\varepsilon= 2[\mathcal{T}_{k}(\varrho)]_{\varepsilon}r_\varepsilon \quad\mbox{ a.e. in $Q_\varepsilon$}. \]
Now, we take $\eta \in C^1_c(\Omega)$, multiply equation (\ref{eqD1}) by $\eta$ and integrate over $\Omega$. We get, after an integration by parts, $$ \partial_{t}\int_\Omega[\mathcal{T}_{k}(\varrho)]_{\varepsilon}^{2}\eta\, {\rm d} x-\int_\Omega[\mathcal{T}_{k}(\varrho)]_{\varepsilon}^{2}\vc{u}\cdot\nabla\eta\, {\rm d}x+ \int_\Omega [\mathcal{T}_{k}(\varrho)]_{\varepsilon}^{2}{\rm div}\, \vc{u}\eta\, {\rm d}x+ $$ $$ \int_\Omega 2[\mathcal{T}_{k}(\varrho)]_{\varepsilon}\Big[\varrho \mathcal{T}_{k}'(\varrho)-\mathcal{T}_{k}(\varrho)]{\rm div}\, \vc{u}\Big]_\varepsilon\eta\, {\rm d}x= \int_\Omega 2[\mathcal{T}_{k}(\varrho)]_{\varepsilon}r_\varepsilon\eta {\, \rm d}x , $$ where $0<\varepsilon<{\rm dist}({\rm supp}\, \eta,R^d\setminus\Omega)$.
We may integrate (\ref{eqD1}) between $t_1,t_2$, where $t_{i}\in [0,T]$, by virtue of Lemma \ref{aux3}, in order to obtain,
\[\Big|\int_{\Omega}[\mathcal{T}_{k}(\varrho)]_{\varepsilon}^{2}(t_2,\cdot)\eta(x) {\, \rm d}x - \int_{\Omega}[\mathcal{T}_{k}(\varrho)]_{\varepsilon}^{2}(t_1,\cdot)\eta(x) {\, \rm d}x\Big| \] \[
\leq C\Big( \|\vc{u}\|_{L^{p}(t_1,t_2; W^{1,q}(\Omega))} +\|r_\varepsilon\|_{L^p(t_1,t_2;L^{\widetilde q}(\Omega))}
\| \eta \|_{C^1(\overline\Omega)}\Big)({t_{2}-t_{1}})^{1/p'}, \] where $C$ may depend on $k$ but is independent of $0<\varepsilon<{\rm dist}({\rm supp}\,\eta, R^d\setminus\Omega)$. The latter inequality shows in view of Lemmas \ref{prop1}, \ref{l2} that the family of maps (\ref{maps}) is for any $k>1$ and $\eta \in C^1_c(\Omega)$ equi-continuous in $C([0,T])$.
Now, we denote $\mathcal{J}(\Omega)\subset C_{c}^{1}({\Omega})$ a countable dense subset of $L^{2}(\Omega)$.
Using Arzel\`a--Ascoli theorem and countability of $\mathcal{J}(\Omega)$ (in order to employ a diagonalization procedure) we may show that there is a subsequence of $\varepsilon \to 0$ and $Z^{(k)}_{\eta}\in C([0,T])$ such that $\forall \eta \in \mathcal{J}({\Omega})$ \[\int_{\Omega}[\mathcal{T}_{k}^{2}(\varrho)(t,x)]_{\varepsilon}\eta(x){\, \rm d}x \mapsto Z^{(k)}_{\eta} \;\mbox{ in} \ C[0,T]\;\mbox{as $\varepsilon\to 0+$}.\] By virtue of (\ref{D0}) \[Z^{(k)}_{\eta}(t)=\int_{\Omega}{\mathcal{T}^2_{k}(\varrho)(t,x)}\eta(x){\, \rm d}x.\]
Now we use density of $\mathcal{J}({\Omega})$ in $L^{2}(\Omega)$ and the uniform bound with respect to $\varepsilon$ of
$\sup\tau\in [0,T]\|[\mathcal{T}_{k}(\varrho)(t,x)]_{\varepsilon}\|_{L^2(\Omega)}$ (cf. the last inequality in Lemma \ref{Lbg} and Item 2. in Lemma \ref{prop1})
to show that \[ \int_{\Omega}[\mathcal{T}_{k}(\varrho)(t,x)]_{\varepsilon}^{2}\eta(x){\, \rm d}x \mapsto \int_{\Omega}{\mathcal{T}^2_{k}(\varrho)(t,x)}\eta(x) {\, \rm d}x \] in $C([0,T])$ for all $\eta \in L^{2}(\Omega)$. In particular, \begin{equation}\label{T2mol} \forall k>1, \;\Big(t \mapsto \int_{\Omega}\mathcal{T}_{k}(\varrho)(t,x)^{2}{\, \rm d}x\Big)\in C([0,T]). \end{equation}
Resuming: According to (\ref{Tmol}) $$ \mathcal{T}_k(\varrho(t'))\to\mathcal{T}_k(\varrho(t))\;\mbox{weakly in $L^2(\Omega)$ as $t'\to t$} $$ and according to (\ref{T2mol}), $$
\|\mathcal{T}_k(\varrho(t'))\|_{L^2(\Omega)}\to\|\mathcal{T}_k(\varrho(t))\|_{L^2(\Omega)}\;\mbox{as $t'\to t$}. $$ Since weak convergence and convergence in norms in $L^2(\Omega)$ imply strong convergence, we have $$ \mathcal{T}_{k}(\varrho)\in C([0,T];L^{2}(\Omega))\;\mbox{ for any $k>1$}. $$
It remains to show that the latter formula implies $\varrho \in C([0,T],L^{r}(\Omega))$, $1\le r<\gamma$. To this end, we write $$
\sup_{t\in[0,T]}\|(\mathcal{T}_{k}(\varrho)-\varrho)(t)\|_{L^{r}(\Omega)}\leq\|T_{k}(\varrho)-\varrho\|_{L^\infty(0,T;L^{r}(\Omega))}, $$ where we have used the last inequality in Lemma \ref{Lbg}. Consequently, for all $t\in [0,T]$, \[
\|(\mathcal{T}_{k}(\varrho)-\varrho)(t)\|^r_{L^{r}(\Omega)}\leq { {\rm ess \, sup}}_{t\in (0,T)}
\int_{\{|\varrho|\geq k\}}2^r|\varrho|^r{\, \rm d}x \] \[
\leq 2^r{ {\rm ess \, sup}}_{t\in (0,T)}\Big[\Big(\int_{\Omega}|\varrho|^{\gamma}{\, \rm d}x\Big)^{\frac{r}{\gamma}}|{\{|\varrho|\geq k\}}|^{\frac{\gamma-r}{\gamma}}\Big], \] where $$
|{\{|\varrho|\geq k\}}|\le\frac 1 k\int_{\{|\varrho|\geq k\}}|\varrho|\, {\rm d} x\le \frac 1k |{\{|\varrho|\geq k\}}|^{1/\gamma'}\|\varrho\|_{L^\gamma(\Omega)}. $$ Whence, $$
\forall t\in [0,T], \; \|\mathcal{T}_{k}(\varrho)-\varrho\|_{L^{r}(\Omega)}\to 0\;\mbox{ as $k\to\infty$}. $$ With this information, writing,
\[\|\varrho(t)-\varrho(t')\|_{L^{r}(\Omega)}\leq \|\varrho(t)-\mathcal{T}_{k}(\varrho)(t)\|_{L^{r}(\Omega)}+\|\mathcal{T}_{k}(\varrho)(t')-\mathcal{T}_{k}(\varrho)(t)\|_{L^{r}(\Omega)}\]
\[+\|\mathcal{T}_{k}(\varrho)(t')-\varrho(t')\|_{L^{r}(\Omega)},
\] we conclude that $\varrho\in C([0,T]; L^r(\Omega))$.
\end{proof}
\section{Proof of Theorems \ref{th3}--\ref{th4}}\label{S6}
It is enough to outline the proof only in the case of Theorem \ref{th3}. The proof of Theorem \ref{th4} follows the same lines.
The proof of Statements 1.1 and 2.1 of Theorem \ref{th3} is based on regularization of the equation via mollifiers, cf. Lemma \ref{prop1}. The regularized equation $$ \partial[\varrho]_\varepsilon+{\rm div}([\varrho]_\varepsilon\vc{u})=r_\varepsilon(\varrho,\vc{u}),\qquad r_\varepsilon(\varrho,\vc{u})={\rm div}([\varrho]_\varepsilon\vc{u})-{\rm div}[\varrho\vc{u}]_\varepsilon $$
{ is satisfied} almost everywhere in $I\times\Omega_\varepsilon$, $\Omega_\varepsilon=\{x\in \Omega\,|\, {\rm dist}(x,R^d\setminus\Omega)>\varepsilon\}$ and can be therefore multiplied by $b'([\varrho]_\varepsilon)$. The Friedrichs commutator lemma (cf. Lemma \ref{l2}) ensures that the term $r_\varepsilon\to 0$ in $L^1(I; L^1_{\rm loc}(\Omega))$. It is the main property which allows to conclude at the first stage for $b$ in class (\ref{ren}), and consequently, for any $b$ in class (\ref{t1.3}), by using a convenient approximation of the function $b$ in class (\ref{t1.3}) and the dominated Lebesgue convergence theorem. This is the standard procedure introduced in the same context in the seminal work \cite{DL}.
Concerning the proof of Statements 1.2 and 2.2 of Theorem \ref{th3}, we shall show solely the { latter.} Furthermore, it is enough to deal only with the "integrability up to $\partial \Omega$" in the case of { Statement 2.2.}
We define a function $\xi_n$ as follows: \begin{equation*} \xi_{n}(x) :=\chi_{n}({\rm dist}\,(x,\partial\Omega)) \end{equation*} with \begin{equation*} \chi_{n}(s) = \chi(ns), \end{equation*} where \begin{equation*} \chi\in C^\infty([0,\infty)),\quad 0\le \chi',\quad \chi(s) =
\begin{cases}
0 &\text{if $0\le s\leq \frac{1}{4} $}\\
1 &\text{if $s \geq \frac{1}{2}$.}\\
\end{cases} \end{equation*}
Recall that ${\rm dist}(\cdot, \partial\Omega)$ is a 1-Lipschitz function.
Notice that it can be deduced from the above \begin{equation*} \xi_n\in C^\infty([0,\infty)),\;\xi_n'(x)\le Cn,\; \xi_{n}(x) =
\begin{cases}
0 &\text{if ${\rm dist}(x,\partial\Omega)\leq \frac{1}{4n} $}\\
1 &\text{if ${\rm dist}(x,\partial\Omega)\geq \frac{1}{2n},$}\\
\end{cases} \end{equation*} with some $C>0$ ($C$ depends on the choice of $\chi$).
We calculate for $\eta \in C^\infty(\overline{\Omega})$ \[\int_{\Omega}\varrho(\tau, x)\psi(\tau)\eta (x){\, \rm d}x-\int_{\Omega}\varrho(0,x)\psi(0)\eta(x){\, \rm d}x-\int_{Q}\varrho(t,x) \partial_{t}\psi(t)\eta(x)\dx \dt\] \[-\int_{Q}\varrho(t,x) \vc{u}(t,x)\cdot \nabla \eta(x) \psi(t)\dx \dt\] \[=\int_{\Omega}\varrho(\tau, x)\psi(\tau)\eta (x)\xi_{n}(x){\, \rm d}x-\int_{\Omega}\varrho(0,x)\psi(0)\eta(x)\xi_{n}(x){\, \rm d}x \] \[ -\int_{Q}\varrho(t,x) \partial_{t}\psi(t)\eta(x)\xi_{n}(x)\dx \dt\] \[-\int_{Q}\varrho(t,x) \vc{u}(t,x)\cdot \nabla \big(\eta(x) \xi_{n}(x)\big)\psi(t)\dx \dt \] \[+\int_{\Omega}\varrho(\tau, x)\psi(\tau)\eta (x)(1-\xi_{n}(x)){\, \rm d}x-\int_{\Omega}\varrho(0,x)\psi(0)\eta(x)(1-\xi_{n}(x)){\, \rm d}x\] \[-\int_{Q}\varrho(t,x) \partial_{t}\psi(t)\eta(x)\big(1-\xi_{n}(x)\big){ \dx \dt } \] \begin{equation}\label{eqxi} -\int_{Q}\psi(t)\varrho(t,x)\vc{u}(t,x)\cdot \nabla \big(\eta(x) (1-\xi_{n}(x))\big)\dx \dt. \end{equation}
We easily verify due to the above formulas for $\xi_n$ that $\eta \xi_{n} \in W_{0}^{1,p}(\Omega)$ with any $1\leq p<+\infty$. Since $C_{{\rm c}}^{1}(\Omega)$ is dense in $W_{0}^{1,p}(\Omega)$, it is an admissible test function for equation (\ref{co1.1}). Consequently, the sum of first four terms at the right hand side (terms containing $\eta \xi_{n} $) is equal to $0$.
To complete the proof we would like to show that the limit $n \to +\infty $ of the sum of the last four terms at the right hand side of identity (\ref{eqxi}) is zero. To this aim we have to assume that all functions are integrable up to the boundary of $\Omega$.
We set $A_{n}:=\{x: {\rm dist}(x,\partial\Omega)\leq \frac{1}{2n}) \}$. Since $\Omega$ is a bounded Lipschitz domain, $|A_{n}|\to 0$. In the sequel, we will systematically use this fact.
We have \begin{enumerate} \item
\[\int_{Q}|\varrho(t,x)\partial_{t}\psi(t)\eta(x)(1-\xi_{n}(x))|\dx \dt\]
\[=\int_{0}^{T}\int_{A_{n}}|\varrho(t,x)\partial_{t}\psi(t)\eta(x)|\dx \dt\]
\[\leq C\|\varrho\|_{L^{\alpha}(0,T;L^\beta(A_n))}\|\partial_{t}\psi\|_{L^{\infty}((0,T))}\|\eta\|_{L^{\infty}(\Omega)}|A_{n}|^{1-\frac{1}{\beta}}\to 0,\quad n\to \infty.\] \item
\[\int_{Q}|\varrho(t,x)\vc{u}(t,x)\cdot \nabla \big(\eta(x)(1-\xi_{n}(x))\big)\psi(t)|\dx \dt\to 0,\quad n \to \infty, \] where we have used the fact that $\vc{u} \in L^{p}(I,W^{1,q}_{0}(\Omega;R^d))$. Indeed,
\[\lim_{n\to \infty}\int_{Q}\big|\varrho(t,x)\vc{u}(t,x)\cdot\nabla\big(\eta(x)(1-\xi_{n}(x))\big)\psi(t)\big|\dx \dt\]
\[\leq \lim_{n\to \infty}\int_{Q}\big|\varrho(t,x)\vc{u}(t,x)\cdot \nabla \eta(x)\big(1-\xi_{n}(x)\big)\psi(t)\big|\dx \dt \] { \[
+\lim_{n\to \infty}\int_{Q}\big|\varrho(t,x)\vc{u}(t,x)\cdot\nabla \xi_{n}(x) \psi(t)\eta(x) \big|\dx \dt\]
\[=\lim_{n\to \infty}\int_{Q}\big|\varrho(t,x)\vc{u}(t,x)\cdot\nabla \xi_{n}(x) \psi(t)\eta(x)\big|\dx \dt\] }
\[\leq\lim_{n\to \infty}C\int_{0}^{T}\int_{A_{n}}\Big|\varrho(t,x)\frac{\vc{u}(t,x)}{{\rm dist}(x,\partial\Omega)}\cdot \nabla {\rm dist}(x,\partial\Omega) \psi(t)\eta(x)\Big|\dx \dt\]
\[\leq \lim_{n\to \infty}C \int_{0}^{T}\Big\|\frac{\vc{u}(t,x)}{{\rm dist}(x,\partial\Omega)}\Big\|_{L^{q}(\Omega)} \|\varrho(t)\|_{L^{\beta}(A_{n})}\|\psi\|_{L^{\infty}((0,T))} \|\eta\|_{L^{\infty}(A_{n})} \, {\rm d} t \]
\[\leq \lim_{n\to \infty}C \int_{0}^{T}\|\varrho(t)\|_{L^{\beta}(A_{n})}\|\psi\|_{L^{\infty}([0,T])} \|\eta\|_{L^{\infty}(A_{n})} \|\nabla \vc{u}\|_{L^{q}(\Omega)} \, {\rm d} t \]
\[\leq \lim_{n\to \infty}C\|\varrho\|_{L^{\alpha}(0,T;L^\beta(A_n))}\|\psi\|_{L^{\infty}((0,T))} \|\eta\|_{L^{\infty}(A_{n})} \|\nabla \vc{u}\|_{L^{p}(0,T;L^q(\Omega;R^{d\times d}))}\] \[=0,\] after employing the Hardy inequality (hence $\Omega$ must have Lipschitz boundary). \end{enumerate} Similarly we treat also the first two integrals over $\Omega${, where we use the fact that $\varrho \in C_{{\rm weak}}(\overline{I};L^\gamma(\Omega))$ and the product $\eta(1-\xi_n)$ is bounded uniformly in $L^\infty(\Omega)$. This finishes the proof of Statement 2.2} and thus Theorem \ref{th3} as well as Theorem \ref{th4} are proved.
\section{Proof of Theorem \ref{th5}}\label{S7}
We present the proof for distributional solutions only. { The case of weak solutions follows more or less the same lines. Due to the fact that $\Omega$ is Lipschitz, we may extend the function $\vc{u}$ to the whole $R^d$ in such a way that it belongs to $L^p(I;W^{1,q}(R^d;R^d))$ and either $\varrho$ or $s$ by zero outside $\Omega$. Then, clearly, the extended $\varrho$ resp. $s$ solve the continuity resp. transport equation in the whole $I\times R^d$ with the transporting velocity the extended $\vc{u}$. We can therefore apply the mollification in $R^d$ and then equations (\ref{4.6}) hold a.e. in $I\times R^d$. Hence we may repeat the whole proof given below in $I\times R^d$.}
Let us start with Statement 1. Since both $\vc{u} \cdot \nabla \varrho$ and $\vc{u} \cdot \nabla s$ fulfill assumptions of the Friedrichs commutator lemma (Lemma \ref{l2}), we see that $[\varrho]_\varepsilon$ and $[s]_\varepsilon$, the corresponding mollifications in the spatial variable satisfy a.e. in $I\times \Omega_\varepsilon$, where $\Omega_\varepsilon$ is defined in the proof of Lemma \ref{aux4}, \begin{equation} \label{4.6} \begin{aligned} \partial_t [s]_\varepsilon + \vc{u} \cdot \nabla [s]_\varepsilon &= r_\varepsilon^1, \\ \partial_t [\varrho]_\varepsilon + {\rm div}\, ([\varrho]_\varepsilon \vc{u}) &= r_\varepsilon^2, \end{aligned} \end{equation} where $r_\varepsilon^1\to 0$ in $L^{\tau_1}(I;L^{\sigma_1}_{{\rm loc}}(\Omega))$, $\sigma_1\in [1,q)$ if $\beta_\varrho=\infty$, $\frac{1}{\sigma_1} \geq \frac{1}{\beta_\varrho}+ \frac{1}{q}$ otherwise, and $\frac{1}{\tau_1} \geq \frac{1}{\alpha_\varrho}+ \frac{1}{p}$, $\tau_1<\infty$. Similarly $r_\varepsilon^2\to 0$ in $L^{\tau_2}(0,T;L^{\sigma_2}(\Omega))$, $\sigma_2\in [1,q)$ if $\beta_s=\infty$, $\frac{1}{\sigma_2} \geq \frac{1}{\beta_s}+ \frac{1}{q}$ otherwise, and $\frac{1}{\tau_2} \geq \frac{1}{\alpha_s}+ \frac{1}{p}$, $\tau_2<\infty$. We may multiply \eqref{4.6}$_1$ by $[V]_\varepsilon$ and \eqref{4.6}$_2$ by $[s_\varrho]_\varepsilon$. Thus, a.e. in { $I\times \Omega_\varepsilon$,} $$ \partial_t ([s]_\varepsilon [\varrho]_\varepsilon) + {\rm div}\,([s]_\varepsilon [\varrho]_\varepsilon \vc{u}) = r_\varepsilon^1 [\varrho]_\varepsilon + r_\varepsilon^2 [s]_\varepsilon, $$ i.e. $$ \int_0^T \int_\Omega \Big([s]_\varepsilon [\varrho]_\varepsilon\partial_t \varphi + [s]_\varepsilon [\varrho]_\varepsilon\vc{u} \cdot \nabla \varphi + (r_\varepsilon^1 [\varrho]_\varepsilon + r_\varepsilon^2 [s]_\varepsilon)\varphi\Big) \dx \dt =0 $$ for all $\varphi \in C^\infty_{\rm c} ((0,T)\times\Omega_\varepsilon)$.
We now intend to let $\varepsilon \to 0+$. We need to verify that the first two terms converge to the corresponding counterparts while the last two terms converge to zero.
First, since the sequence $[s]_\varepsilon$ is bounded in { $L^{\alpha_s}(I;L^{\beta_s}(\Omega_\varepsilon))$,} the term $r_\varepsilon^2 [s]_\varepsilon \to 0$ in $L^1((0,T)\times\Omega)$. Similarly, since $[\varrho]_\varepsilon$ is bounded in the space $L^{\alpha_\varrho}(I;L^{\beta_\varrho}(\Omega_\varepsilon))$, { the other term also goes to zero.}
Next we consider the first and the second term. Indeed, the second term is more restrictive than the first one. Since $[s]_\varepsilon \to s$ in $L^{t_s}(I;L^{r_s}_{{\rm loc}}(\Omega))$, $[\varrho]_\varepsilon \to \varrho$ in $L^{t_\varrho}(I;L^{r_\varrho}_{{\rm loc}}(\Omega))$ and $\vc{u} \in L^p(I;W^{1,q}(\Omega;R^d))$, we easily see $[\varrho]_\varepsilon [s]_\varepsilon \vc{u} \to \varrho s \vc{u}$ in $L^1(I;L^1_{{\rm loc}}(\Omega;R^d))$. This finishes the proof of Statement 1.
In the case of Statement 2 we first proceed as above and verify that $\varrho s$ is a distributional (weak) solution to the continuity equation. Only in the limit passage of $[\varrho]_\varepsilon [s]_\varepsilon \vc{u}$ we have to employ additionally the Sobolev embedding theorem for $\vc{u}$ in the spatial variable together with the $L^\infty$ bound in time for { $[\varrho]_\varepsilon$ and $[s]_\varepsilon$ if some of the exponents is equal to $\infty$, and interpolate these bounds. Next, we apply Theorem \ref{th3}, Statement 1.1,} to see that $\varrho s$ is a renormalized distributional solution to the continuity equation.
Furthermore, since $\varrho s\in C_{{\rm weak}}(\overline{I}; L^\gamma(\Omega))$ and $\gamma >1$, we may employ Theorem { \ref{th1}, Statement 3.,} to verify that $\varrho s \in C(\overline{I};L^r(\Omega))$ for any $1\leq r<\gamma$. Theorem \ref{th5} is proved.
\section{Proof of the main results}\label{S8}
\subsection{Proof of Theorem \ref{t2}}
To proof Theorem \ref{t2} we first use the fact that $\varrho$ is a renormalized time integrated { weak} solution of the transport equation and use $b_{\delta}(\varrho):= \frac{\delta}{\delta + \varrho}$ with $\delta >0$ in the renormalized formulation. As we know that $\varrho \geq 0$ a.e. in $(0,T)\times \Omega$, the function $b_\delta$ is an appropriate renormalizing function\footnote{{ Strictly speaking, function $b_\delta$ does not satisfy the second condition (\ref{ren}). Nevertheless, the map $\varrho\mapsto\varrho b_\delta'(\varrho)-b_\delta(\varrho)$ remains bounded. We can thus take instead of $b_\delta$ a convenient approximation (e.g. $j_\varepsilon*\max\{b_\delta(\cdot+\varepsilon),1/\varepsilon\}$, $\varepsilon\in (0,\delta)$, see Lemma \ref{prop1} for the notation) which satisfies (\ref{ren}), and then let $\varepsilon\to 0$ in order to get (\ref{4.1}).}}. We get \begin{equation} \label{4.1} \begin{aligned} &\int_\Omega \frac{\delta}{\delta +\varrho(t,\cdot)} \varphi(t,\cdot) {\, \rm d}x - \int_\Omega \frac{\delta}{\delta +\varrho(0,\cdot)} \varphi(0,\cdot) {\, \rm d}x -\int_0^t\int_\Omega \frac{\delta}{\delta +\varrho} \partial_t \varphi {\, \rm d}x \, {\rm d}\tau \\ = & \int_0^t\int_\Omega \Big(\frac{\delta}{\delta +\varrho} \vc{u} \cdot \nabla \varphi + \Big(\frac{\delta}{\delta +\varrho}-\frac{\delta \varrho}{(\delta +\varrho)^2}\Big){\rm div}\, \vc{u} \Big){\, \rm d}x \, {\rm d}\tau \end{aligned} \end{equation} for all $\varphi \in C_{\rm c}^\infty([0,T]\times \overline{\Omega})$. We may let $\delta \to 0+$ in \eqref{4.1} to get (we use the Lebesgue dominated convergence theorem; recall that $\frac{\delta}{\delta +\varrho(t,x)} =1$ provided $\varrho(t,x)=0$) \begin{equation} \label{4.3} \begin{aligned} &\int_\Omega s_\varrho(t,\cdot) \varphi(t,\cdot) {\, \rm d}x - \int_\Omega s_\varrho(0,\cdot) \varphi(0,\cdot) {\, \rm d}x - \int_0^t \int_\Omega s_\varrho \partial_t \varphi {\, \rm d}x \, {\rm d}\tau \\
= & \int_0^t \int_\Omega \big(s_\varrho \vc{u}\cdot \nabla \varphi + s_\varrho {\rm div}\, \vc{u} \varphi\big) {\, \rm d}x \, {\rm d}\tau \end{aligned} \end{equation} for all $\varphi$ as above. Here, $s_\varrho$ denotes the characteristic function of the set, where $\varrho=0$. Hence $s_\varrho$ is a time integrated weak solution to the transport equation with the function $\vc{u}$. { Moreover, repeating the argument above with $\widetilde {b}(\varrho):= b\Big(\frac{\delta}{\delta +\varrho}\Big)$, where $b$ belongs to the class (\ref{ren}), we also get that $s_\varrho$ is a renormalized time integrated weak solution.}
Since $\int_\Omega s_\varrho(\tau,\cdot) {\, \rm d}x = |\{x\in \Omega; \varrho(\tau,x)=0\}|_d$, we may subtract equations \eqref{4.3} with $\varphi = 1$ for $t:=\tau_1$ and $t:=\tau_2$ and it is easy to see that $$
\Big|\int_{\tau_1}^{\tau_2} \int_\Omega s_\varrho {\rm div}\, \vc{u} \dx \dt\Big| \to 0 \qquad \text{ for } \tau_1\to \tau_2. $$ Hence $$
|\{x\in \Omega; \varrho(\tau,x)=0\}|_d \in C([0,T]). $$ Note further that repeating the argument to get \eqref{4.3} with a test function only space dependent, we get $s_\varrho \in C_{\rm weak}([0,T];L^r(\Omega))$ for any $1\leq r<\infty$ and thus, by { Lemma \ref{aux4},} $$ s_\varrho \in C([0,T];L^r(\Omega)), \qquad 1\leq r<\infty. $$ The theorem is proved.
\subsection{Proof of Theorem \ref{t3} and Corollaries \ref{cor1}--\ref{cor2}}
The first claim of { Theorem \ref{t3}} is a direct consequence of Theorems \ref{th1} and \ref{th3}. The second claim follows directly from Theorem { \ref{th5}, Statement 2.} The third claim is a direct consequence of formula (\ref{clRR}).
Corollary \ref{cor1} follows immediately from Theorem \ref{t3}.
We now consider Corollary \ref{cor2}. We aim at proving that $|\{x\in \Omega|\varrho(t_0,x)=0\}|_d =0$ for any $t_0 \in (0,T]$. First, for $t_0 \in (0,\tau]$, we define $\widetilde R(t):= R(t-t_0+\tau)$. Since $\vc{u}$ is time independent, the function $\widetilde R$ is a distributional solution to the continuity equation on { $(t_0-\tau,T'+t_0-\tau)$} and we may apply Theorem \ref{t3}, in particular formula (\ref{clRR}) with $\widetilde R$ instead of $R$ and $t_0$ instead of $t$. Hence Corollary \ref{cor2} holds in the time interval $(0,\tau]$. Next we consider { $t_0\in(\tau,2\tau]$.} We redefine $\widetilde R$ as { $\widetilde R(t) :=R(t-\tau)$} and { apply} formula (\ref{clRR}) with $t:=t_0+\tau$ on the {left hand side} and $\tau$ instead of $0$ on { the right hand side. Hence $|\{x\in \Omega|\varrho(t,x)=0\}|_d=0$ for $t \in [0,2\tau]$. Proceeding similarly,
after} finite number of steps we cover the whole interval $(0,T)$. Corollary \ref{cor2} is proved.
Note finally that due to our definition of the weak solution to the compressible Navier--Stokes equations Corollary \ref{c1} follows directly, since all assumptions of Theorems \ref{t2} and \ref{t3}, as well as { of} Corollaries \ref{cor1} and { \ref{cor2},} are fulfilled.
{\bf Acknowledgement:} The work of M. Pokorn\'y was supported by the Czech Science Foundation, grant No. 16-03230S. Significant part of the paper was written during the stay of M. Pokorn\'y at the University of Toulon. The authors acknowledge this support.
{\bf Conflict of interest:} The authors declare that they have no conflict of interest.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Hamiltonian unknottedness of certain monotone Lagrangian tori in $S^2 imes S^2$}
\begin{abstract} \noindent We prove that a monotone Lagrangian torus in $S^2\times S^2$ which suitably sits in a symplectic fibration with two sections in its complement is Hamiltonian isotopic to the Clifford torus. \end{abstract}
\section{Introduction}\label{intro}
The classification of Lagrangian submanifolds in symplectic manifolds up to Lagrangian or Hamiltonian isotopy is an intriguing problem of symplectic topology. While there are many tools for distinguishing Lagrangian submanifolds, actual classification results have been very rare and restricted to special manifolds in dimension $4$. The first circle of results concerns Lagrangian $2$-spheres, in which case the two notions of isotopy coincide: There is a unique $2$-sphere up to Hamiltonian isotopy in $S^2\times S^2$ (Hind~\cite{Hind04}), in $T^*S^2$ and some other Stein surfaces (Hind~\cite{Hind10}), and in certain del Pezzo surfaces (Evans~\cite{jonny}). The second circle of results is contained in A.~Ivrii's PhD thesis~\cite{Ivrii}: It asserts the uniqueness up to Lagrangian isotopy of Lagrangian tori in $\mathbb{R}^4$, $S^2\times S^2$, $\mathbb{CP}^2$, and $T^\ast \mathbb{T}^2$.
Motivated by Ivrii's thesis, we address in this paper the question of {\em Hamiltonian} unknottedness of monotone Lagrangian tori in $S^2\times S^2$.
Recall that a Lagrangian torus is called {\em monotone} if its Maslov class is a positive multiple of its symplectic area class on relative $\pi_2$. The product of the equators in each $S^2$-factor in $S^2\times S^2$ is called the {\em standard Lagrangian torus} $L_{\rm std}$, or the {\em Clifford torus}. This torus is monotone for the standard split symplectic form $\omega_{\rm std}=\ostd\oplus \ostd$. There have been many constructions of monotone Lagrangian tori in $(S^2\times S^2,\omega_{\rm std})$ that are not Hamiltoniian isotopic to $L_{\rm std}$ due to Eliashberg--Polterovich~\cite{ElP}, Chekanov--Schlenk~\cite{CS}, Entov--Polterovich~\cite{EnP}, Biran--Cornea~\cite{BC}, Fukaya--Oh--Ohta--Ono~\cite{FOOO}, and Albers--Frauenfelder~\cite{AF}. Since they were all recently shown to be Hamiltonian isotopic to each other~\cite{Gad,OU}, we will collectively refer to them as the {\em
Chekanov-Schlenk torus} $L_{CS}$.
The following definition is implicit in Ivrii's thesis. Let us call a monotone Lagrangian torus $L$ in $(S^2\times S^2,\omega_{\rm std})$ {\em fibered} if there exists a foliation $\fo{F}$ of $S^2\times S^2$ by symplectic $2$-spheres in the homology class $[pt\times S^2]$ and a symplectic submanifold $\Sigma$ in class $[S^2\times pt]$ with the following properties: \begin{itemize}
\item $\Sigma$ is transverse to the leaves of $\fo{F}$ and is
disjoint from $L$; \item the leaves of $\fo{F}$ intersect $L$ in circles (or not at all). \end{itemize}
Note that each leaf of $\fo{F}$ which intersects the torus $L$ is cut by $L$ into two closed disks glued along $L$. The disks that intersect $\Sigma$ together form a solid torus $T$ with $\partial T=L$.
The first part of Ivrii's thesis now asserts that {\em any
monotone Lagrangian torus in $S^2\times S^2$ is fibered}. In this paper, we prove\footnote{ Let us emphasize that, while inspired by it, the results in this paper are independent of Ivrii's thesis (which is, unfortunately, neither published nor otherwise accessible).}
\begin{thm}[Main Theorem]\label{thm:main} Let $L\subset (S^2\times S^2,\omega_{\rm std})$ be a monotone Lagrangian torus which is fibered by $\fo{F}$ and $\Sigma$. Assume in addition, that there exists a second symplectic submanifold $\Sigma'$ in homology class $[S^2\times pt]$ which is transverse to the leaves of $\fo{F}$, and which is disjoint from $\Sigma$ and $T$. Then $L$ is Hamiltonian isotopic to the standard torus $L_{\rm std}$. \end{thm}
This means that the Chekanov--Schlenk torus $L_{CS}$, or any other exotic monotone Lagrangian torus, cannot possess the additional section $\Sigma'$ required in Theorem~\ref{thm:main}. It also suggests that the classification of monotone Lagrangian tori in $(S^2\times S^2,\omega_{\rm std})$ up to Hamiltonian isotopy may come within reach once we understand better the role of the second section $\Sigma'$.
\begin{remark} It is tempting to conjecture that the Clifford torus and the Chekanov--Schlenk torus are the only monotone Lagrangian tori in $(S^2\times S^2,\omega_{\rm std})$ up to Hamiltonian isotopy. We point out that R.~Vianna~\cite{Via1,Via2} recently constructed infinitely many pairwise Hamiltonian non-isotopic Lagrangian tori in ${\mathbb{C}} P^2$, and conjectured the existence of a monotone Lagrangian tori in $(S^2\times S^2,\omega_{\rm std})$ which is not Hamiltonian isotopic to either $L_{\rm std}$ or $L_{CS}$. \end{remark}
Let us now outline the proof of the main theorem, and in particular explain where the second section is needed. By a \emph{relative
symplectic fibration} on $S^2\times S^2$ we will mean a quintuple $$
\mathfrak{S}=(\fo{F},\omega,L,\Sigma,\Sigma'), $$ as in Theorem~\ref{thm:main}, only with the standard form $\omega_{\rm std}$ replaced by any symplectic form $\omega$ cohomologous to $\omega_{\rm std}$. We will prove (Corollary~\ref{cor:fib}) that for every symplectic fibration $\mathfrak{S}$ with $\omega=\omega_{\rm std}$ there exists a homotopy of relative symplectic fibrations $\mathfrak{S}_t=(\fo{F}_t,\omega_{\rm std},L_t,\Sigma_t,\Sigma_t')$ with fixed symplectic form $\omega_{\rm std}$ such that $\mathfrak{S}_0=\mathfrak{S}$ and $\mathfrak{S}_1=\mathfrak{S}_{std}:=(\fo{F}_{\rm std},\omega_{\rm std},L_{\rm std},S_0,S_\infty)$, where $\mathcal{F}_{\rm std}$ denotes the standard foliation with leaves $\{z\}\times S^2$ and $S_0=S^2\times\{S\}$, $S_\infty=S^2\times\{N\}$ are the standard sections at the south and north pole. Then $L_t$ is an isotopy of monotone Lagrangian tori with respect to $\omega_{\rm std}$ from $L$ to $L_{\rm std}$, which is Hamiltonian by Banyaga's isotopy extension theorem.
A relative symplectic fibration $\mathfrak{S}$ gives rise to a symplectic fibration $p:S^2\times S^2\to\Sigma$ by sending each leaf of $\mathcal{F}$ to its intersection point with $\Sigma$. It determines a symplectic connection whose horizontal subspaces are the symplectic orthogonal complements to the fibres. Parallel transport along closed paths $\gamma:[0,1]\to\Sigma$ gives holonomy maps which are symplectomorphisms of the fibre $p^{-1}(\gamma(0))$ and measure the non-integrability of the horizontal distribution. It is not hard to show that a symplectic fibration $\mathfrak{S}$ with trivial holonomy around all loops is diffeomorphic to $\mathfrak{S}_{\rm std}$, and a theorem of Gromov implies that they are actually homotopic with fixed symplectic form if they both have symplectic form $\omega_{\rm std}$.
Thus, most of the work will go into deforming a given relative symplectic fibration $\mathfrak{S}$ to one with trivial holonomy. After pulling back $\mathfrak{S}$ by a diffeomorphism, we may assume that $(\mathcal{F},L,\Sigma,\Sigma')=(\mathcal{F}_{\rm std},L_{\rm std},S_0,S_\infty)$ (but the symplectic form $\omega$ is nonstandard). In the first step, which takes up Section~\ref{sec:stand}, we make the holonomy trivial near the two sections and near the fibres over the line of longitude through Greenwich in the base, see Figure~\ref{fi1}.
\begin{figure}
\caption{Where the holonomy is trivial after the first step}
\label{fi1}
\end{figure}
In the second step, which takes up most of Section~\ref{sec:killholonomy}, we kill the holonomy along all circles of latitude $C^\lambda$. For this, let $(\lambda,\mu)$ be spherical coordinates on $S^2$, where $\lambda$ denotes the latitude and $\mu$ denotes the longitude. After the first step, the holonomy maps $\phi^\lambda$ along $C^\lambda$ give a loop in ${\rm Symp}(A,\partial A,\ostd)$, the group of symplectomorphisms of the annulus (the sphere minus two polar caps) which equal the identity near the boundary. Since the fundamental group of ${\rm Symp}(A,\partial A,\ostd)$ vanishes, we can contract the loop of inverses $\psi^\lambda=(\phi^\lambda)^{-1}$ and obtain a family of Hamiltonians $H^\lambda_\mu$ which generates the contraction. The closed $2$-form $$
\Omega_H=\omega+d(H^\lambda_\mu d\mu) $$ then defines a symplectic connection with trivial holonomy around all $C^\lambda$. However, $\Omega_H$ need not be symplectic if $\dd{H^\lambda_\mu}{\lambda}$ is large. This can be remedied by the inflation procedure due to Lalonde and McDuff~\cite{LalMcD}. In this procedure, the symplectic form $\omega$ is deformed along a fibre and a section (and $H$ suitably rescaled) in order to make $\Omega_H$ symplectic. However, this process will in general destroy monotonicity of $L_{\rm std}$. In order to keep the Lagrangian torus monotone, we perform the inflation procedure along a fibre and the {\em two} sections $S_0,S_\infty$ in a symmetric way. It is at this point of the proof that we need the existence of a second symplectic section.
Once the holonomy along circles of latitude is trivial, in the third and final step (at the end of Section~\ref{sec:killholonomy}) we deform the symplectic form to the standard form. This finishes the outline of the proof.
\begin{remark} The idea to apply the results of Ivrii's thesis to the Hamiltonian classification of monotone tori in $S^2\times S^2$ originated in 2003 in the first author's discussions with Y.~Eliashberg. However, at the time we did not realize the necessity of a second symplectic section and were puzzled by the apparent contradiction between this result and the existence of an exotic monotone torus in $S^2\times S^2$. This discrepancy was resolved in the second author's PhD thesis~\cite{Sch}, of which this article is a shortened version. \end{remark}
\centerline{\bf Acknowledgement}
We thank Y.~Eliashberg for many fruitful discussions and his continued interest in this work.
\section{Relative symplectic fibrations}\label{fib}
\subsection{Symplectic connections and their holonomy}
Consider a smooth fibration (by which we mean a fibre bundle) $p:M\to B$ and a closed $2$-form $\omega$ on $M$ whose restriction to each fibre $p^{-1}(b)$ is nondegenerate. We will refer to $\omega$ as a {\em symplectic connection} on $M$.\footnote{This terminology differs slightly from the one in~\cite{MS}.} From the next subsection on we will assume $\omega$ to be symplectic, but for now this is not needed.
{\bf Parallel transport. } Since $\omega$ is nondegenerate on the fibres, the $\omega$-orthogonal complements $$
\mathcal{H}_x := (\ker d_xp)^\omega $$ to the tangent spaces of the fibres of $p$ define a distribution of horizontal subspaces $\mathcal{H}$ such that $$
TM = \mathcal{H}\oplus \ker dp. $$ Horizontal lifts of a path $\gamma:[0,1]\to B$ with given initial points in $p^{-1}(\gamma(0))$ give rise to the {\em parallel transport} $$
P_\gamma\colon p^{-1}(\gamma(0)) \to p^{-1}(\gamma(1)) $$ along $\gamma$. Closedness of $\omega$ inplies that $P_\gamma$ is symplectic, i.e. $$
P_\gamma^\ast \omega_{\gamma(1)}=\omega_{\gamma(0)}, $$
where $\omega_b$ denotes the symplectic form $\omega|_{p^{-1}(b)}$.
\begin{comment} {\bf Curvature. } Given two vector fields $X,Y$ on $B$, we define a vertical vector field $R(X,Y)$ on $M$, i.e.~a section of the vertical tangent bundle $\ker dp\to M$, by the formula $$
R(X,Y) := [\tilde{X},\tilde{Y}]^{vert}, $$ where $\tilde{.}$ denotes the horizontal lift and ${.}^{vert}$ denotes the projection to the vertical subbundle $\ker dp$ of $TM$ along $\mathcal{H}$. A short computation shows that the restriction of $R(X,Y)$ to the fibre $p^{-1}(b)$ depends only on the values of $X,Y$ at the point $b\in B$, so $R$ defines a 2-form on the base with values in the vector fields on the fibres. We call $R$ the {\em curvature} of the symplectic connection $\omega$. According to the Frobenius theorem, the curvature measures the non-integrability of the horizontal distribution.
\begin{lemma}[Curvature identity]\label{Pcurvid} Let $\omega$ be a symplectic connection on the fibration $p \colon M \to B$. Then the vertical projection of $$
d\iota_{\tilde{Y}}\iota_{\tilde{X}}\omega -
\iota_{[\tilde{X},\tilde{Y}]}\omega $$ vanishes for all vector fields $X,Y$ tangent to the base. \end{lemma}
\begin{proof} For any 2-form $\alpha$ and vector fields $Y_0,Y_1,Y_2$ we have the following identity: $$d\alpha(Y_0,Y_1,Y_2)=L_{Y_0}(\alpha(Y_1,Y_2))-L_{Y_1}(\alpha(Y_0,Y_2))+L_{Y_2}(\alpha(Y_0,Y_1))$$ $$-\alpha([Y_0,Y_1],Y_2)+\alpha([Y_0,Y_2],Y_1)-\alpha([Y_1,Y_2],Y_0)$$ hence for $\alpha=\omega$, $Y_0=\tilde{v}_1$, $Y_1=\tilde{v}_2$ and $Y_2=v\in VTX$ this gives: $$d\omega(\tv_1,\tv_2,v)=L_{\tv_1}\omega(\tv_2,v)-L_{\tv_2}\omega(\tv_1,v)+L_v\omega(\tv_1,\tv_2)-\omega([\tv_1,\tv_2],v)+\omega([\tv_1,v],\tv_2)-\omega([\tv_2,v],\tv_1)$$ But $[\tv,w]$ is vertical for any vertical $w$. To see this note that the flow $\tilde{\phi}_t$ of $\tv$ is the horizontal lift of the flow $\phi_t$ of $v$ on the base. Hence $\tilde{\phi}_t$ restricted to the fibre $\pi^{-1}(x)$ equals the parallel transport map $P_t \colon \pi^{-1}(x)\to \pi^{-1}(\phi_t(x))$ of the symplectic connection for the path $\left\lbrace \phi_{st}(x)\right\rbrace_{s \in [0,1]}$ in the base. Therefore if $\psi_s$ denotes the flow of $w$ then for fixed $t$, $$\tilde{\phi}_t\circ \psi_s\circ (\tilde{\phi}_t)^{-1}(z)=P_t\circ \psi_s\circ P_t^{-1}(z)$$ remains in the fibre $\pi(z)$ for all $s$. Consequently the vector field $(\tilde{\phi}_t)_\ast w$ is vertical for all $t$. This shows that the Lie-bracket $[\tv,w]$ is also vertical (cf. Remark $6.26$ in \cite{MS} and the discussion before). Then the above reduces to $$d\omega(\tv_1,\tv_2,v)=L_v(\omega(\tv_1,\tv_2))-\omega([\tv_1,\tv_2],v)$$ for all vertical v. Thus $$d\iota_{\tilde{v}_2}\iota_{\tilde{v}_1}\omega-\iota_{\tilde{v}_2}\iota_{\tilde{v}_1}d\omega=\iota_{[\tilde{v}_1,\tilde{v}_2]}\omega \ \text{on} \ VTX.$$ But $\omega$ is closed, hence it reduces to the required form. \end{proof}
Hence, given any two tangent vectors $v,w\in T_bB$ on the base, the curvature vector field $R(v,w)$ on $p^{-1}(b)$ is Hamiltonian with Hamiltonian function
$\omega(\tilde{v},\tilde{w})|_{p^{-1}(b)}$. Note that the vector field determines the Hamiltonian function up to a constant. If the fibres are compact, then we can fix this constant by requiring that the function has average zero with respect to the symplectic volume form on each fibre, and thus view the symplectic curvature as a $2$-form on the base with values in the functions on the fibres. \end{comment}
{\bf Holonomy. } The parallel transport $P_\gamma:p^{-1}(\gamma(0))\to p^{-1}(\gamma(0))$ along a closed curve $\gamma:[0,1]\to B$ is called the {\em holonomy} of $\omega$ along the loop $\gamma$. If $P_\gamma=\mathrm{id}$ for each loop $\gamma$ we say that $\omega$ has {\em trivial holonomy}. In this case parallel transport along any (not necessarily closed) curve depends only on the end points, so we can use parallel transport to define local trivializations of $p:M\to B$.
\begin{remark} There is a natural notion of {\em curvature} of a symplectic connection, see~\cite{MS}. This is a $2$-form on the base with values in the functions on the fibres which measures the nonintegrability of the horizontal distribution. For simply connected base (which is the case of interest to us) the curvature and the holonomy carry the same information, so in this paper we will phrase everything in terms of holonomy. \end{remark}
{\bf From foliations to fibrations. } More generally, we can consider a closed $2$-form $\omega$ on $M$ whose restriction to the leaves of a smooth foliation $\mathcal{F}$ of $M$ is nondegenerate. If all leaves of $\mathcal{F}$ are compact, then the space of leaves is a smooth manifold $B$ and the canonical projection $M\to B$ is a fibration, so we are back in the situation of a symplectic connection as above. Since in our case all leaves will be $2$-spheres, we can switch freely between the terminologies of foliations and fibrations.
\subsection{Fibered Lagrangian tori in $S^2\times S^2$}
Suppose now that $(M,\omega)$ is a symplectic $4$-manifold and $p:M\to B$ is a symplectic fibration over a surface $B$ (i.e., the fibres are symplectic surfaces).
\begin{definition}\label{DLfibered} We say that an embedded $2$-torus $L\subset M$ is \emph{fibered by
$p$} if (see Figure~\ref{f3}) \begin{enumerate} \item $\gamma:=p(L)$ is an immersed loop with transverse
self-intersections which are at most double points; \item $p^{-1}(\gamma(t))\cap L$ is diffeomorphic to a circle if
$\gamma(t)$ is not a double point, and to two disjoint circles if
$\gamma(t)$ is a double point; \item in each of the circles in $p^{-1}(\gamma(t))\cap L$ we can fill
in an embedded disk $D\subset p^{-1}(\gamma(t))$ in the fibre such
that the two disks at a double point are disjoint and all the disks
form a solid torus $T\cong S^1\times D^2$ with $L$ as its boundary. \end{enumerate} \end{definition}
\begin{figure}
\caption{A fibered Lagrangian torus}
\label{f3}
\end{figure}
Suppose now that $L$ is in addition Lagrangian. The following two results are the basis for most of the sequel. The first one states that a fibered Lagrangian torus $L$ is generated by parallel transport along $\gamma$ of the circle in the fibre over a non-double point, see Figure~\ref{f4}.
\begin{figure}
\caption{L is generated by parallel transport}
\label{f4}
\end{figure}
\begin{proposition}\label{PPTlagtorusfibreposition} Let $L\subset M$ be an embedded Lagrangian torus which is fibered by the symplectic fibration $p \colon M \to B$. Then $L$ is invariant under parallel transport along $\gamma=p(L)$ with respect to the symplectic connection $\omega$. \end{proposition}
\begin{proof} At a point $x\in L$ we have the $\omega$-orthogonal splitting $T_xM=\mathcal{H}_x\oplus V_x$, where $V_x:=\ker d_xp$ denotes the tangent space to the fibre. Since $L$ is fibered by $p$, the subspace $T_xL+V_x\subset T_xM$ generated by $T_xL$ and $V_x$ is $3$-dimensional. The condition that $L$ is Lagrangian implies $(T_xL)^\omega=T_xL$, thus $T_xL\cap \mathcal{H}_x = (T_xL)^\omega\cap (V_x)^\omega = (T_xL+V_x)^\omega$ is 1-dimensional. This $1$-dimensional subspace therefore contains the horizontal lift of $\dot\gamma$ through $x$ and the propsition follows. \end{proof}
\begin{remark} Let $N:=p^{-1}(\gamma)$ be the $3$-dimensional submanifold of $M$ formed by the fibres that meet the torus $L$. In the definition of being fibered by $p$ we did not require the torus to be transverse to the fibres of $p$ in $N$. If, however, $L$ is Lagrangian,
then Proposition~\ref{PPTlagtorusfibreposition} shows that we get this property for free. \end{remark}
{\bf Monotone tori in $S^2\times S^2$. } From now on we assume that $$
M = S^2\times S^2 $$ and the symplectic form $\omega$ is cohomologous to the product form $$
\omega_{\rm std}:=\ostd\oplus\ostd, $$ where $\ostd$ is the standard area form on $S^2$ normalized by $\int_{S^2}\ostd=1$. In other words, we require that $$
\int_{S^2\times{\rm pt}}\omega=\int_{{\rm pt}\times S^2}\omega=1. $$ Moreover, we assume that the Lagrangian torus $L$ is {\em monotone}, i.e., its Maslov class $\mu$ (see~\cite{MS}) and its symplectic area satisfy $$
\mu(\sigma)=4\int_\sigma\omega \text{ for all }\sigma\in\pi_2(M,L). $$ Here the monotonicity constant must be equal to $4$ because the class $A=[S^2\times{\rm pt}]\in\pi_2(M)$ has Maslov index $\mu(A)=2\langle
c_1(TM),A\rangle = 2\langle c_1(TS^2),[S^2]\rangle = 4$.
\begin{lemma}\label{Lmonotonetorusembcurve} Let $L\subset (M=S^2\times S^2,\omega)$ be a monotone Lagrangian torus with $\omega$ cohomologous to $\omega_{\rm std}$. Let $p \colon M \to B$ be a symplectic fibration over the surface $B$ such that $L$ is fibered by $p$. Then the loop $\gamma:=p(L)$ is an embedded curve, i.e., it has no double points. \end{lemma}
\begin{proof} Since $L$ is orientable, all its Maslov indices on $\pi_2(M,L)$ are even integers. In view of the monotonicity constant $4$, this implies that the symplectic area of each embedded symplectic disk $D\subset M$ with boundary on $L$ must be a positive multiple of $1/2$. If $\gamma$ had a double point $b$, then the solid torus $T$ from Definition~\ref{DLfibered} would intersect the fibre $p^{-1}(b)$ in two disjoint symplectic disks, which is impossibe because the fibre has symplectic area $1$. \end{proof}
\begin{remark} (a) For a smooth fibration $p:S^2\times S^2\to B$ over a surface $B$,
both the fibres and the base are diffeomorphic to $S^2$. Indeed,
denoting a fibre by $F$, the homotopy exact sequence $\pi_2(F)\to
\pi_2(S^2\times S^2)\to\pi_2(B)$ implies
$\pi_2(F)\cong\pi_2(B)\cong{\mathbb{Z}}$, so $F$ and $B$ must be diffeomorphic
to $S^2$ or ${\mathbb{R}} P^2$. Since by the product formula for the Euler
characteristic $\chi(F)\chi(B)=\chi(S^2\times S^2)=4$, both $F$ and
$B$ must be diffeomorphic to $S^2$.
(b) For a {\em monotone} Lagrangian torus $L$ in $M=S^2\times S^2$,
the third condition in Definition~\ref{DLfibered} is actually a
consequence of the first two. To see this, note first that in the
proof of Lemma~\ref{Lmonotonetorusembcurve} we can rule out the
double point $b$ without reference to the solid torus $T$: By the
Jordan curve theorem, the two circles in $L\cap p^{-1}(b)$ would
bound two disjoint symplectic disks in the fibre $p^{-1}(b)\cong
S^2$, each of area a positive multiple of $1/2$, which again gives the desired
contradiction. Now an orientation of $L$ and a parametrization of
the curve $\gamma\subset B$ induce via horizontal lifts of
$\dot\gamma$ orientations of the circles $L_t:=L\cap p^{-1}(\gamma(t))$,
and we define $T$ as the union of the disks $D_t\subset
p^{-1}(\gamma(t))$ whose oriented boundary is $L_t$. \end{remark}
\subsection{Relative symplectic fibrations of $S^2\times S^2$}
We continue with the manifold $M=S^2\times S^2$ and the generators $$
A=[S^2\times{\rm pt}],\;B=[{\rm pt}\times S^2]\in H_2(M). $$ Now we define the main object of study for this paper.
\begin{definition} \label{DAMSF} A \emph{relative symplectic fibration} on $M=S^2\times S^2$ is a quintuple $\mathfrak{S}=(\mathcal{F},\omega,L,\Sigma,\Sigma')$ where \begin{itemize} \item $\mathcal{F}$ is a smooth foliation of $M$ by $2$-spheres in homology class $B$; \item $\omega$ is a symplectic form on $M$ making the leaves of
$\mathcal{F}$ symplectic with $\omega(A)=\omega(B)=1$; \item $\Sigma,\Sigma'$ are disjoint symplectic submanifolds in class
$A$ which are transverse to all the leaves of $\mathcal{F}$, so in
particular the projection $p \colon M \to \Sigma$ sending each leaf
to its unique intersection point with $\Sigma$ defines a symplectic fibration; \item $L\subset M$ is an embedded monotone Lagrangian torus fibered by $p$; \item $\Sigma'$ is disjoint from the solid torus $T$ with $\partial
T=L$ in Definition~\ref{DLfibered}; \item $\Sigma$ intersects each fibre $p^{-1}(\gamma(t))$ in
the interior of the disk $T\cap p^{-1}(\gamma(t))$. \end{itemize} \end{definition}
Note that for a monotone Lagrangian torus $L$ fibered by $p:M\to B$ there always exist disjoint smooth sections $\Sigma,\Sigma'$ of $p$ with $\Sigma'$ disjoint from the solid torus $T$ and $\Sigma\cap p^{-1}(\gamma)$ contained in the interior of $T$. The crucial condition in Definition~\ref{DAMSF} is that these sections can be chosen to be symplectic.
\begin{definition}\label{DisoAMSF} (a) A {\em homotopy} of relative symplectic fibrations is a smooth 1-parametric family $$
\mathfrak{S}_t=(\mathcal{F}_t,\omega_t,L_t,\Sigma_t,\Sigma_t'),\qquad t\in[0,1] $$ of relative symplectic fibrations.
(b) The group ${\rm Diff}_\mathrm{id}(M)$ of diffeomorphisms $\phi:M\to M$ inducing the identity on the second homology group $H_2(M)$ (and hence on all homology groups) acts on relative symplectic fibrations by push-forward $$
\phi(\mathfrak{S}) := \Bigl(\phi(\mathcal{F}),\phi_\ast
\omega,\phi(L),\phi(\Sigma),\phi(\Sigma')\Bigr). $$ Two relative symplectic fibrations $\mathfrak{S}$ and $\widetilde\mathfrak{S}$ are called \emph{diffeomorphic} if $\widetilde\mathfrak{S}=\phi(\mathfrak{S})$ for a diffeomorphism $\phi$ of $M$ (which then necessarily belongs to ${\rm Diff}_\mathrm{id}(M)$).
(c) Two relative symplectic fibrations $\mathfrak{S}$ and $\widetilde\mathfrak{S}$ on $M$ are called {\em deformation equivalent} if there exists a diffeomorphism $\phi\in{\rm Diff}_\mathrm{id}(M)$ such that $\phi(\mathfrak{S})$ is homotopic to $\widetilde\mathfrak{S}$. \end{definition}
\begin{remark} (a) Note that a diffeomorphism $\phi\in{\rm Diff}_\mathrm{id}(M)$ intertwines the
symplectic connections of $\mathfrak{S}$ and $\phi(\mathfrak{S})$ and their parallel
transports. For example, $\mathfrak{S}$ has trivial holonomy
iff $\phi(\mathfrak{S})$ does.
(b) It is easy to see that deformation equivalence is an equivalence
relation. Moreover, $\mathfrak{S}$ is deformation equivalent to $\widetilde\mathfrak{S}$ iff
there exists a homotopy $\mathfrak{S}_t$ such that $\mathfrak{S}_0=\mathfrak{S}$ and $\mathfrak{S}_1$ is
diffeomorphic to $\widetilde\mathfrak{S}$.
(c) Note that in the above definition nothing is said about the
isotopy class of the diffeomorphism $\phi$. In fact, it is an open
problem whether every $\phi\in{\rm Diff}_\mathrm{id}(M)$ is isotopic to the
identity, so we do not know whether diffeomorphic relative symplectic
fibrations are homotopic in general. However, by a theorem of Gromov
(see Theorem~\ref{G85} below), two diffeomorphic relative symplectic
fibrations {\em with the same symplectic form $\omega_{\rm std}$} are homotopic.
This result will be crucial at the end of the proof of our main
theorem. \end{remark}
\subsection{The standard relative symplectic fibration}
The {\em standard relative symplectic fibration} $\mathfrak{S}_{\rm std}=(\mathcal{F}_{\rm std},\omega_{\rm std},L_{\rm std},S_0,S_\infty)$ of $S^2\times S^2$ consists of the following data: \begin{itemize} \item $\mathcal{F}_{\rm std}$ is the foliation by the fibres $\{z\}\times S^2$ of
the projection $p_1:S^2\times S^2\to S^2$ onto the first factor; \item $\omega_{\rm std}=\ostd\oplus\ostd$ is the standard symplectic form; \item $S_0=S^2\times \{S\}$ and $S_\infty=S^2\times\{N\}$, where
$N,S\in S^2$ are the north and south poles; \item $L_{\rm std}=E\times E$ is the Clifford torus, i.e.~the product of
the equators in the base and fibre; \item $T_{\rm std}=E\times D_{\rm lh}$, where $D_{\rm lh}\subset S^2$ denotes the
lower hemisphere, is the solid torus bounded by $L_{\rm std}$. \end{itemize}
The main goal of this paper will be to deform a given relative symplectic fibration to the standard one (see Theorem~\ref{thm:fib} below). For later use, let us record the relative homology and homotopy groups of the Clifford torus.
\begin{lemma}\label{lem:homology} For the Clifford torus $L_{\rm std}\subset S^2\times S^2$ the second relative homotopy/homology group $$
\pi_2(S^2\times S^2,L_{\rm std})\cong H_2(S^2\times S^2,L_{\rm std})\cong
H_2(S^2\times S^2)\oplus H_1(T^2)$$ is free abelian generated by $$
[S^2\times{\rm pt}],\ [{\rm pt}\times S^2],\ [D_{\rm lh}\times{\rm pt}],\ [{\rm pt}\times D_{\rm lh}]. $$
\end{lemma}
\begin{proof} The long exact sequences of the pair $(M=S^2\times S^2,L=L_{\rm std})$ and the Hurewicz maps yield the commuting diagram \begin{equation*} \begin{CD}
0 @>>> \pi_2(M) @>>> \pi_2(M,L) @>>> \pi_1(L) @>>> 0 \\
@VVV @VV{\cong}V @VVV @VV{\cong}V @VVV \\
H_2(L) @>{0}>> H_2(M) @>>> H_2(M,L) @>>> H_1(L) @>>> 0. \end{CD} \end{equation*} Here the first horizontal map in the lower row is zero because $L$ bounds the solid torus $T_{\rm std}=E\times D_{\rm lh}$ in $S^2\times S^2$, where $E\subset S^2$ denotes the equator. Now the middle vertical map is an isomorphism by the five lemma, and the generators of $H_2(M,L)$ are obtained from the generators $[S^2\times{\rm pt}],\ [{\rm pt}\times S^2]$ of $H_2(M)$ and $[E\times{\rm pt}],\ [{\rm pt}\times E]$ of $H_1(L)$. \end{proof}
\section{Standardisations}\label{sec:stand}
In this section we show how to deform a relative symplectic fibration to make it split (in a sense defined below) near the symplectic sections $\Sigma,\Sigma'$ and near one fibre $F$. In particular, the standardised fibration will have trivial holonomy in these regions. This provides a convenient setup for the discussion in Section~\ref{sec:killholonomy}.
\subsection{Pullback by diffeomorphisms}
In this subsection, we show how to put a relative symplectic fibration $\mathfrak{S}$ into a nicer form via pullback by diffeomorphisms. Note that this is not really changing $\mathfrak{S}$ but just looking at it from a different angle. We will see that using pullbacks we can either standardise all data except the symplectic form $\omega$, or all data except the foliation $\mathcal{F}$. So the nontriviality of a relative symplectic fibration only arises from the interplay of $\omega$ and $\mathcal{F}$, as measured by the holonomy of the corresponding symplectic connection.
In order to establish a clean picture of what can be achieved by pullbacks, we will prove some results in stronger versions than what we actually need in the sequel.
{\bf Fixing the fibration. } We begin with a useful characterisation of diffeomorphisms that are trivial on homology. Recall the generators $A=[S^2\times{\rm pt}]$ and $B=[{\rm pt}\times S^2]$ of $H_2(S^2\times S^2)$.
\begin{lemma}\label{Pdiffrelfib} A diffeomorphism $\phi$ of $S^2\times S^2$ is trivial on homology if and only if it is orientation preserving and satisfies $\phi_\ast(B)=B$. \end{lemma}
\begin{proof} The ``only if'' is clear, so let us prove the ``if''. Let us write $\phi_\ast(A)=mA+nB$ for integers $m,n$. Since $\phi$ is orientation preserving, it preserves the intersection form on $H_2(S^2\times S^2)$ and we obtain \begin{align*}
1 &= \phi_\ast(A)\cdot\phi_\ast(B) = (mA+nB)\cdot B = m, \cr
0 &= \phi_\ast(A)\cdot\phi_\ast(A) = (A+nB)\cdot(A+nB) = 2n. \end{align*} This shows that $\phi_\ast A=A$, so $\phi_\ast$ is the identity on $H_2(S^2\times S^2)$. \end{proof}
Our first normalisation result is
\begin{prop} [Fixing the fibration]\label{prop:fix-F} Let $\mathfrak{S}=(\mathcal{F},\omega,L,\Sigma,\Sigma')$ be a relative symplectic fibration of $M=S^2\times S^2$. Then there exists a diffeomorphism $\phi\in{\rm Diff}_\mathrm{id}(M)$ such that $\phi^{-1}(\mathfrak{S})=(\mathcal{F_{\rm std}},\widetilde\omega,L_{\rm std},S_0,S_\infty)$ for some symplectic form $\widetilde\omega$. \end{prop}
\begin{proof} Consider the fibration $p:M\to\Sigma$ defined by $\mathcal{F}$ and pick an orientation preserving diffeomorphism $u:\Sigma\to S^2$. Then $u\circ p:M\to S^2$ is a fibration by $2$-spheres. Since $\pi_1Diff_+(S^2)$ classifies $S^2$-bundles over $S^2$ and $Diff_+(S^2)$ deformation retracts onto $SO(3)$, there are up to bundle isomorphism precisely two $S^2$-bundles over $S^2$: the trivial one $p_1:S^2\times S^2\to S^2$ and a nontrivial one $X\to S^2$. The total space $X$ of the nontrivial bundle is the blow-up of ${\mathbb{C}} P^2$ at one point, which is not diffeomorphic to $S^2\times S^2$ (e.g.~their intersection forms differ). Thus the nontrivial bundle does not occur, and we conclude that there exists a diffeomorphism $\phi:M\to M$ such the following diagram commutes: \begin{equation*}
\begin{CD}
S^2\times S^2 @>\phi^{-1} >> S^2\times S^2\\
@VV{p }V @VV{ p_1}V\\
\Sigma @>u>> S^2
\end{CD} \end{equation*} Moreover, the restriction of $\phi$ to each fibre is orientation preserving, which implies the $\phi$ itself is orientation preserving. Since the fibres of $p$ and $p_1$ all represent the homology class $B$, it follows that $\phi_*B=B$ and thus $\phi\in{\rm Diff}_\mathrm{id}(M)$ by Lemma~\ref{Pdiffrelfib}. After replacing $\mathfrak{S}$ by $\phi^{-1}(\mathfrak{S})$, we may hence assume that $\mathcal{F}=\mathcal{F}_{\rm std}$ and $p=p_1$.
The section $\Sigma$ of $p_1:S^2\times S^2\to S^2$ can be uniquely parametrized by $z\mapsto(z,f(z))$ for a smooth map $f:S^2\to S^2$. After a preliminary isotopy we may assume that $f(z_0)=S$ equals the south pole $S$ at a base point $z_0\in S^2$. Then $f$ represents a class in $\pi_2(S^2,S)$. Now by Hurewicz's Theorem, $\pi_2(S^2,S))\cong H_2(S^2)$. Since $[\Sigma]=A=[S^2\times{\rm pt}]$, the class of $f$ is trivial in $H_2(S^2)$, and thus in $\pi_2(S^2,S)$, so that $f$ is nullhomotopic. By smooth approximation, we find a smooth homotopy $f_t$ from the constant map $f_0\equiv S$ to $f_1=f$. Now we use (a fibered version of) the isotopy extension theorem to extend the family of embeddings $S^2\times\{S\}\hookrightarrow M$, $(z,S)\mapsto(z,f_t(z))$ to a family of fibre preserving diffeomorphisms $\phi_t:M\to M$ with $\phi_0=\mathrm{id}$. After replacing $\mathfrak{S}$ by $\phi_1^{-1}(\mathfrak{S})$, we may hence assume that $\Sigma=S^2\times\{S\}=S_0$. Now we repeat the same argument with $\Sigma'$ (this time it is even simpler because $S^2\setminus\{S\}$ is contractible) to arrange $\Sigma'=S^2\times\{N\}=S_\infty$.
Now the torus $L$ is fibered by $p_1:S^2\times(S^2\setminus\{N,S\})\to S^2$. By an isotopy of the base $S^2$ we can move the embedded curve $p_1(L)$ to the equator $E\subset S^2$. Let $e(t)$, $t\in{\mathbb{R}}/{\mathbb{Z}}$, be a parametrization of the equator $E$ and consider the loop of embedded closed curves $\Lambda_t:=L\cap p_1^{-1}(e(t))$ in the fibre $S^2$. After a further homotopy we may assume that $\Lambda_0=E$. Pick a smooth family of diffeomorphisms $g_t:S^2\to S^2$, $t\in[0,1]$, such that $g_0=\mathrm{id}$ and $g_t(E)=\Lambda_t$ for all $t$. Moreover, we can arrange that $g_t(N)=N$ and $g_t(S)=S$ for all $t$. Then $g_1$ satisfies $g_1(E)=E$ as well as $g_1(N)=N$ and $g_1(S)=S$. We can alter $g_t$ so that $g_1$ fixes $E$ pointwise. By a theorem of Smale~\cite{Sma}, the group ${\rm Diff}(D^2,\partial D^2)$ of diffeomorphsms of the disk that are the identity near the boundary is contractible. So we can alter $g_t$ further (applying this to the upper and lower hemispheres) so that $g_1=\mathrm{id}$. This may first destroy the conditions $g_t(N)=N$ and $g_t(S)=S$, but they can be reinstalled by a further alteration. Now we again use (a fibered version of) the isotopy extension theorem to extend the embedding $E\times S^2\hookrightarrow M$, $(e(t),w)\mapsto(e(t),g_t(w))$ to a fibre preserving diffeomorphism $\phi:M\to M$ isotopic to the identity. Then $\phi^{-1}(\mathfrak{S})$ has the desired properties and the proposition is proved. \end{proof}
A similar (in fact, simpler) proof yields the following $1$-parametric version of Proposition~\ref{prop:fix-F}.
\begin{prop} [Fixing the fibration -- parametric version]\label{prop:fix-F-par} Let $\mathfrak{S}_t=(\mathcal{F}_t,\omega_t,$ $L_t,\Sigma_t,\Sigma_t')_{t\in[0,1]}$ be a homotopy of relative symplectic fibrations of $M=S^2\times S^2$. Then there exists an isotopy of diffeomorphisms $\phi_t\in{\rm Diff}_\mathrm{id}(M)$ with $\phi_0=\mathrm{id}$ such that $\phi_t^{-1}(\mathfrak{S}_t)=(\mathcal{F}_0,\widetilde\omega_t,L_0,\Sigma_0,\Sigma_0')$ for some family of symplectic forms $\widetilde\omega_t$.
$\square$ \end{prop}
{\bf Fixing the symplectic form. } Our next result is an easy consequence of Moser's and Banyaga's theorems. Since it will be used repeatedly in this paper, let us recall the latter~\cite[Th\'eor\`eme II.2.1]{Ban} for future reference.
\begin{thm}[Banyaga's isotopy extension theorem~\cite{Ban}]\label{thm:Banyaga} Let $(M,\omega)$ be a symplectic manifold and $\psi_t:M\to M$ a smooth isotopy with $\psi_0=\mathrm{id}$ such that each $\psi_t$ is symplectic on a neighbourhood of a compact subset $X\subset M$. Suppose that $\int_{\sigma}\psi_t^*\omega$ is constant in $t$ for each $\sigma\in H_2(M,X)$. Then there exists a symplectic isotopy $\phi_t$ with
$\phi_0=\mathrm{id}$ and $\phi_t|_X=\psi_t|_X$. \end{thm}
\begin{prop} [Fixing the symplectic form -- parametric version]\label{prop:fix-om-par} Let $\mathfrak{S}_t=(\mathcal{F}_t,\omega_t,L_t,\Sigma_t,\Sigma_t')_{t\in[0,1]}$ be a homotopy of relative symplectic fibrations of $M=S^2\times S^2$. Then there exists an isotopy of diffeomorphisms $\phi_t\in{\rm Diff}_\mathrm{id}(M)$ with $\phi_0=\mathrm{id}$ such that $\phi_t^{-1}(\mathfrak{S}_t)=(\widetilde\mathcal{F}_t,\omega_0,L_0,\Sigma_0,\Sigma_0')$ for some family of foliations $\widetilde\mathcal{F}_t$. \end{prop}
\begin{proof} First, Moser's theorem provides an isotopy of diffeomorphisms $\phi_t:M\to M$ with $\phi_0=\mathrm{id}$ such that $\phi_t^*\omega_t=\omega_0$. After replacing $\mathfrak{S}_t$ by $\phi^{-1}(\mathfrak{S}_t)$, we may hence assume that $\omega_t=\omega_0$ for all $t$.
Next, consider the isotopy of submanifolds $X_t:=L_t\amalg\Sigma_t\amalg\Sigma_t'$ of $(M,\omega_0)$. Let us write (using the smooth isotopy extension theorem) $X_t=\psi_t(X_0)$ for diffeomorphisms $\psi_t:M\to M$ with $\psi_0=\mathrm{id}$. Since $L_t$ is Lagrangian and $\Sigma_t\amalg\Sigma_t'$ is symplectic, the Lagrangian and symplectic neighbourhood theorems provide a modification of $\psi_t$ which is symplectic on a neighbourhood of $X_0$.
We claim that the symplectic area $\int_{\sigma_t}\omega_0$ is constant in $t$ for each $\sigma\in H_2(M,X_0)$, where we denote $\sigma_t:=(\psi_t)_*\sigma\in H_2(M,X_t)$. To see this, note that the map $H_2(M,L_0)\to H_2(M,X_0)$ is surjective because $H_1(\Sigma_0\amalg\Sigma_0')=0$. So it suffices to prove the claim for classes $\sigma\in H_2(M,L_0)\cong\pi_2(M,L_0)$. Now recall that the Lagrangian tori $L_t$ are monotone with respect to $\omega_0$. Since the Maslov class $\mu(\sigma_t)$ of $L_t$ is constant in $t$, so is the symplectic area $\int_{\sigma_t}\omega_0$ by monotonicity and the claim is proved.
In view of the claim, $(X_0,\psi_t)$ satisfies the hypotheses of Banyaga's Theorem~\ref{thm:Banyaga}. It follows that the smooth isotopy $\psi_t$ can be altered to a symplectic isotopy $\phi_t$ with $\phi_0=\mathrm{id}$ and $\phi_t(X_0)=X_t$. This is the desired isotopy in the proposition. \end{proof}
A non-parametric version of Proposition~\ref{prop:fix-om-par} is much more subtle and will be discussed in Section~\ref{ss:proof}.
\subsection{Standardisation near a fibre}
Let us pick the point $z_0:=(1,0,0)$ on the equator $E$ in the base, so that $F:=p_1^{-1}(z_0)$ is a fibre of $\mathcal{F}_{\rm std}$ intersecting $L_{\rm std}$ in the equator $E$. The following proposition shows that we can deform every relative symplectic fibration to make the triple $(\mathcal{F},\omega,L)$ standard near $F$.
\begin{prop}[Standardisation near a fibre]\label{prop:fibre-standard} Every relative symplectic fibration $\mathfrak{S}=(\mathcal{F}_{\rm std},\omega,L_{\rm std},\Sigma,\Sigma')$ is homotopic
to a fibration of the form $\widetilde\mathfrak{S}=(\mathcal{F}_{\rm std},\widetilde\omega,L_{\rm std},\widetilde\Sigma,\widetilde\Sigma')$ such that $\widetilde\omega=\omega_{\rm std}$ on a neighbourhood of the fibre $F$. \end{prop}
The proof is given in~\cite[Lemma 3.2.3]{Ivrii}. For convenience, we recall the argument. It is based on two easy lemmas.
\begin{lemma}\label{Pspecialsymp} Let $E$ denote the equator and $D_{\rm lh}$ the lower hemisphere in $S^2\subset \mathbb{R}^3$. Let $\sigma$ be a symplectic form on $S^2$ cohomologous to $\sigma_{\rm std}$ such that $\int_{D_{\rm lh}}\sigma=\frac{1}{2}$. Then there exists an isotopy of diffeomorphisms $h_t:S^2\to S^2$ with $h_0=\mathrm{id}$ such that $h_t(E)=E$ for all $t$ and $h_1^\ast \sigma=\sigma_{\rm std}$. \end{lemma}
\begin{proof} We apply Moser's theorem to $\sigma_t:=(1-t)\sigma_{\rm std}+t\sigma$ to find a diffeotopy $f_t:S^2\to S^2$ with $f_t^*\sigma_t=\sigma_{\rm std}$. Since $\int_{D_{\rm lh}}\sigma_t=1/2$ for all $t$, the $\sigma_{\rm std}$-Lagrangians $f_t^{-1}(E)$ all bound disks of $\sigma_{\rm std}$-area $1/2$. Hence Banyaga's Theorem~\ref{thm:Banyaga} yields $\sigma_{\rm std}$-symplectomorphisms $g_t:S^2\to S^2$ with $g_t(E)=f_t^{-1}(E)$ and $h_t:=f_t\circ g_t$ is the desired diffeotopy. \end{proof}
\begin{lemma}\label{lem:coiso-nbhd}
Let $\omega$ be a symplectic form on $M=S^2\times S^2$ compatible with the standard fibration $p_1:M\to S^2$. Let $\delta\subset S^2$ be an embedded closed arc passing through $z_0$. Then every symplectomorphism $h:(F,\sigma_{\rm std})\to(F,\omega|_F)$ extends to a diffeomorphism $\psi$ between neighbourhoods of $p_1^{-1}(\delta)$ preserving the fibres over $\delta$ and such that $\psi^*\omega=\omega_{\rm std}$. \end{lemma}
\begin{proof} Parallel transport in $N:=p_1^{-1}(\delta)$ with respect to $\omega_{\rm std}$ from $p_1^{-1}(z)$ to $F$ and then with respect to $\omega$ from $F$ to $p_1^{-1}(z)$ yields a fibre preserving diffeomorphism
$\phi:N\to N$ extending $h$ with $\phi^*(\omega|_N)=(\omega_{\rm std})|_N$. By the coisotropic neighbourhood theorem, $\phi$ extends to the desired diffeomorphism $\psi$. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:fibre-standard}]
By Lemma~\ref{Pspecialsymp}, there exists a diffeomorphism
$h:F\to F$ with $h(E)=E$ and $h^*(\omega|_F)=\sigma_{\rm std}$. Let $\delta\subset E$ be an arc in the equator in the base passing though $z_0$. By Lemma~\ref{lem:coiso-nbhd}, the diffeomorphism $h$ extends to a diffeomorphism $\psi$ between neighbourhoods of $p_1^{-1}(\delta)$ preserving the fibres over $\delta$ and such that $\psi^*\omega=\omega_{\rm std}$. Thus the pullback fibration $\psi^*\mathcal{F}_{\rm std}$ and the pullback torus $\psi^{-1}(L_{\rm std})$ coincide over $\delta$ with $\mathcal{F}_{\rm std}$ and $L_{\rm std}$, respectively (the latter holds because $\psi^{-1}(L_{\rm std})$ is obtained by parallel transport of $\{z_0\}\times E$ along $\delta$). Since $h:F\to F$ is isotopic to the identity through diffeomorphisms preserving $E$, we can restrict $\psi$
to a smaller neighbourhood $V=p_1^{-1}(V')$ of $F$ and extend it from there to a diffeomorphism $\phi:M\to M$ preserving $L_{\rm std}$ (and which equals the identity outside a larger neighbourhood of $F$). Then the pullback $\phi^{-1}(\mathfrak{S})$ satisfies $(\phi^*\omega)|_V=\omega_{\rm std}$ and $\phi^{-1}(\mathcal{F})=\mathcal{F}_{\rm std}$ on $p_1^{-1}(\delta)\cap V$.
So far we have just put $\mathfrak{S}$ into a more convenient form by a diffeomorphism, but now we will change it. Note that the fibres of $\mathcal{F}_{\rm std}$ near $p_1^{-1}(\delta)\cap V$ are $C^1$-close to those of $\phi^{-1}(\mathcal{F})$ and therefore symplectic for $\phi^*\omega$. Hence we can deform $\phi^{-1}(\mathcal{F})$ to a foliation $\overline\mathcal{F}$, keeping it $\omega_{\rm std}$-symplectic and fixed on $p_1^{-1}(\delta)\cap V$ and outside $V$, such that $\overline\mathcal{F}=\mathcal{F}_{\rm std}$ on a neighbourhood $U\subset V$ of $F$. Thus we have constructed a homotopy from $\mathfrak{S}$ to $\overline\mathfrak{S}=(\overline\mathcal{F},\phi^*\omega,L_{\rm std},\overline\Sigma,\overline\Sigma')$ such that $(\overline\mathcal{F},\phi^*\omega)=(\mathcal{F}_{\rm std},\omega_{\rm std})$ on $U$. Finally, we apply Proposition~\ref{prop:fix-F-par} to this homotopy outside the set $U$ to replace it by one with fixed foliation $\mathcal{F}_{\rm std}$ (as well as $L_{\rm std}$ which was fixed already). The end point of this homotopy is the desired relative symplectic foliation $\widetilde\mathfrak{S}$. \end{proof}
\begin{remark} Even if in Proposition~\ref{prop:fibre-standard} the sections $\Sigma,\Sigma'$ in $\mathfrak{S}$ are the standard sections $S_0,S_\infty$, this will not be true for the sections in $\widetilde\mathfrak{S}$ unless the original sections were horizontal near $F$. This will be remedied in the following subsection. \end{remark}
\subsection{Standardisation near the sections}
Consider a relative symplectic fibration of the form $\mathfrak{S}=(\mathcal{F}_{\rm std},\omega,L_{\rm std},S_0,S_\infty)$ with the the projections $p_1,p_2:S^2\times S^2$ onto the two factors.
\begin{definition} We say that $\omega$ is {\em split} on a neighbourhood $$
W=(U_F\times S^2)\cup (S^2\times U_0)\cup (S^2\times U_\infty) $$ of $F\cup S_0\cup S_\infty$ if there exist symplectic forms $\sigma_0,\sigma_\infty$ on $S^2$ such that \begin{equation}\label{eq:split} \begin{aligned}
\omega &= p_1^*\sigma_0+p_2^*\sigma_{\rm std}\quad \text{on the set}\quad
W_0=(U_F\times S^2) \cup (S^2\times U_0), \quad\text{and}\cr
\omega &= p_1^*\sigma_\infty+p_2^*\sigma_{\rm std} \quad\text{on the set}\quad
W_\infty=(U_F\times S^2)\cup(S^2\times U_\infty). \end{aligned} \end{equation} \end{definition}
Here the forms $\sigma_0$ and $\sigma_\infty$ may differ, but they agree on $U_F$. Note that if $\omega$ is split, then in particular the sections $S_\infty$ and $S_0$ are horizontal. Moreover, parallel transport of the symplectic connection defined by $\omega$ equals the identity on the region where $\omega$ is split.
The following is the main result of this section.
\begin{prop}[Standardisation near a fibre and the sections]\label{prop:section-standard} Every relative symplectic fibration $\mathfrak{S}=(\mathcal{F}_{\rm std},\omega,L_{\rm std},S_0,S_\infty)$ is homotopic to one of the form $\widetilde\mathfrak{S}=(\mathcal{F}_{\rm std},\widetilde\omega,L_{\rm std},S_0,S_\infty)$ such that $\widetilde\omega$ is split on a neighbourhood $W$ of $F\cup S_0\cup S_\infty$. \end{prop}
The proof of this proposition will occupy the remainder of this section. Standardisation near a symplectic section is more subtle than near a fibre because the section need not be horizontal, so it takes a large deformation to make it symplectically orthogonal to the fibres.
We first consider the local situation in ${\mathbb{R}}^4\cong{\mathbb{C}}^2$ with the standard symplectic form $\Omega_0=dx\wedge dy+du\wedge dv$ in coordinates $z=x+iy$, $w=u+iv$. Let $S=\{w=f(z)\}$ be the graph over the $z$-plane of a smooth function $f=g+ih:{\mathbb{R}}^2\to{\mathbb{R}}^2$ with $f(0)=0$. Orient $S$ by projection onto the $z$-plane. The pullback of $\Omega_0$ under the embedding $F(z)=\bigl(z,f(z)\bigr)$ equals $$
F^*\Omega_0=dx\wedge dy+dg\wedge dh = (1+\det Df)dx\wedge dy. $$ Thus $S$ is symplectic (with the given orientation) iff $$
1+\det Df>0. $$ For a smooth function $\phi:[0,\infty)\to{\mathbb{R}}$ consider the new function $$
\tilde f(z):=\phi(|z|)f(z). $$ We now derive the condition on $\phi$ such that the graph of $\tilde f$ is symplectic. We will see that it suffices to do this for linear maps $f$, so suppose that $f(z)=Az$ for a $2\times 2$ matrix
$A$. We compute for $r:=|z|>0$: \begin{align*}
D\tilde f(z) &= \phi(r)Df(z) +
\phi'(r)f(z)\left(\frac{z}{r}\right)^t = \phi(r)A +
\frac{\phi'(r)}{r}Azz^t \cr
&= A\Bigl(\phi(r)\Id+\frac{\phi'(r)}{r}zz^t\Bigr). \end{align*} Since \begin{align*}
\det\Bigl(\phi(r)\Id+\frac{\phi'(r)}{r}zz^t\Bigr)
&= \det\left(\begin{matrix} \phi+\frac{\phi'}{r}x^2 &
\frac{\phi'}{r}xy \\ \frac{\phi'}{r}xy & \phi+\frac{\phi'}{r}y^2
\end{matrix}\right) \cr
&= \phi^2+\frac{\phi\phi'}{r}(x^2+y^2) = \phi^2+r\phi\phi', \end{align*} we have $\det D\tilde f = (\phi^2+r\phi\phi')\det A$. This proves
\begin{lemma}\label{lem:symp-condition} Let $f(z)=Az$ be a linear function ${\mathbb{R}}^2\to{\mathbb{R}}^2$ with $1+\det A\geq \varepsilon>0$. Let $\phi:[0,\infty)\to{\mathbb{R}}$ be a smooth function with
$\phi(0)=\phi'(0)=0$. Then the graph of $\tilde f(z):=\phi(|z|)f(z)$ is symplectic provided that for all $r>0$, \begin{equation}\label{eq:graph-symp}
0\leq \phi(r)^2+r\phi(r)\phi'(r) < \frac{1}{1-\varepsilon}. \end{equation} \end{lemma}
\begin{lemma}\label{lem:symp-function} For every $0<\varepsilon<1$ and $\delta>0$ there exists a smooth family of nondecreasing functions $\phi_s:[0,\infty)\to[0,1]$, $s\in[0,1]$, satisfying (\ref{eq:graph-symp}) such that $\phi_s(r)=s$ for $r\leq\delta$ and $\phi_s(r)=1$ for $r\geq 2\delta/\sqrt{\varepsilon}$. \end{lemma}
\begin{proof} For $r>0$ define $\psi(r)$ by $\psi(r):=r^2\phi(r)^2$. Then $\psi'=2r(\phi^2+r\phi\phi')$, so (\ref{eq:graph-symp}) is equivalent to $$
\psi'(r)<\frac{2r}{1-\varepsilon}. $$ This will be satisfied if $\psi$ solves the differential equation $$
\psi'(r) = \frac{2r}{1-\varepsilon/4}. $$ Then $\psi(r)=r^2/(1-\varepsilon/4)+c$ for some constant $c$ and $$
\phi^2(r) = \frac{1}{1-\varepsilon/4}+\frac{c}{r^2}. $$ We fix the constant $c$ by $\phi(\delta)=0$ to $c=-\delta^2/(1-\varepsilon/4)$ and obtain $$
\phi^2(r)=\frac{1-\delta^2/r^2}{1-\varepsilon/4}. $$ This is an increasing function with $\phi(\delta)=0$ and $\phi(\gamma)=1$ at the point $\gamma=2\delta/\sqrt{\varepsilon}$. Now observe that if a solution of (\ref{eq:graph-symp}) satisfies $\phi(r_0)\geq 0$ and $\phi'(r_0)\geq 0$ for some $r_0>0$, then we can decrease the slope to $0$ near $r_0$ and extend $\phi$ by $\phi(r)=\phi(r_0)$ for $r\geq r_00$ (or $r\leq r_0$) to a smooth solution of (\ref{eq:graph-symp}). Applying this procedure at $r_0=\delta$ and $r_0=\gamma$ yields the desired function $\phi_0$ for $s=0$. For $s>0$, we obtain $\phi_s$ by smoothing the function ${\rm max}(s,\phi_0)$. \end{proof}
\begin{lemma}\label{lem:symp-standard} Let $\Lambda\subset{\mathbb{R}}^2$ be compact and $(S^\lambda)_{\lambda\in\Lambda}$ be a smooth foliation of a region in $({\mathbb{R}}^4,\Omega_0)$ by symplectic surfaces $S^\lambda$ intersecting the symplectic plane $\{0\}\times{\mathbb{R}}^2$ transversely in $(0,\lambda)$. Then for every neighbourhood $W\subset{\mathbb{R}}^4$ of $\{0\}\times\Lambda$ there exists a neighbourhood $U\subset W$ of $\{0\}\times\Lambda$ and a family of foliations $(S^\lambda_s)_{s\in[0,1],\lambda\in\Lambda}$ with the following properties (see Figure~\ref{fig:Slambda}): \begin{enumerate} \item $S^\lambda_0=S^\lambda$ and $S^\lambda_s=S^\lambda$ outside $W$; \item $S^\lambda_s$ is symplectic and intersects $\{0\}\times{\mathbb{R}}^2$ transversely in $(0,\lambda)$; \item $S^\lambda_1={\mathbb{R}}^2\times\{\lambda\}$ in $U$. \end{enumerate} Moreover, for every $\lambda$ with $S^\lambda={\mathbb{R}}^2\times\{\lambda\}$ in $W$ we have $S^\lambda_s=S^\lambda$ for all $s$. \end{lemma}
\begin{figure}
\caption{The family of foliations $S_s^\lambda$}
\label{fig:Slambda}
\end{figure}
\begin{proof} After shrinking $W$, we may assume that in $W$ each surface can be written as a graph $S^\lambda=\{w=\lambda+f^\lambda(z)\}$ over the $z$-plane with $f^\lambda(0)=0$. After a $C^1$-small perturbation of the surfaces in $W$ (which keeps them symplectic) we may assume that the $f^\lambda$ are linear functions $f^\lambda(z)=A^\lambda z$. Symplecticity implies $\det A^\lambda>-1$. Since $\Lambda$ is compact, there exists an $\varepsilon>0$ with $\det A^\lambda\geq -1+\varepsilon$ in $W$ for all $\lambda$. Moreover, we may assume that the $\varepsilon$-neighbourhood of $\Lambda$ is contained in $W$. Pick $\delta>0$ so small that $2\delta/\sqrt{\varepsilon}<\varepsilon$. Let $\phi_s:[0,\infty)\to[0,1]$,
$s\in[0,1]$, be the functions of Lemma~\ref{lem:symp-function} and define $f^\lambda_s(z):=\phi_{1-s}(|z|)f^\lambda(z)$. By Lemma~\ref{lem:symp-condition}, the graph $S^\lambda_s$ of $f^\lambda_s$ satisfies conditions (i)-(iii) of the proposition, where $U$ is the $\delta$-neighbourhood of $\Lambda$. Note that if $S^\lambda={\mathbb{R}}^2\times\{\lambda\}$ for some $\lambda$, then $f^\lambda(z)\equiv 0$ and thus $S^\lambda_s=S^\lambda$ for all $s$.
It only remains to verify that the surfaces $(S^\lambda_s)_{\lambda\in\Lambda}$ form a foliation for each $s$, or equivalently, that the map $F_s:B^2(\varepsilon)\times\Lambda\to{\mathbb{R}}^4$, $$
F_s(z,\lambda) := \bigl(z,\lambda+f^\lambda_s(z)\bigr) =
\bigl(z,\lambda+\phi_{1-s}(|z|)A^\lambda z\bigr), $$ is an embedding. For injectivity, suppose that $F_s(z,\lambda)=F_s(z',\lambda')$. Then $z=z'$ and
$\lambda-\lambda'=-\phi_{1-s}(|z|)(A^\lambda-A^{\lambda'})z$. This implies $$
|\lambda-\lambda'|\leq\|A^\lambda-A^{\lambda'}\|\;|z| \leq
\varepsilon\|A^\lambda-A^{\lambda'}\|. $$
Since $A^\lambda$ depends smoothly on $\lambda$, there exists a constant $C$ such that $\|A^\lambda-A^{\lambda'}\|\leq C|\lambda-\lambda'|$. For $\varepsilon<1/C$ it follows that $\lambda=\lambda'$. For the immersion property, consider the differential $$
DF_s(z,\lambda) = \left(\begin{matrix} \Id & 0 \\
D_zf^\lambda_s & \Id+B_s
\end{matrix}\right),\qquad B_s = \frac{\partial f^\lambda_s}{\partial\lambda}. $$ This is invertible iff the matrix $$
\Id+B_s = \Id+\phi_{1-s}(|z|)\frac{\partial
A^\lambda}{\partial\lambda}z $$ is invertible. By smoothness in $\lambda$, there exists a constant $C$
with $\|\frac{\partial A^\lambda}{\partial\lambda}z\|\leq C|z|$. Then for $\varepsilon<1/C$ we get $$
\|\phi_{1-s}(|z|)\frac{\partial A^\lambda}{\partial\lambda}z\| \leq C|z| \leq C\varepsilon
<1, $$ which implies invertibility of $\Id+B_s$. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:section-standard}]
We deform the given relative symplectic fibration $\mathfrak{S}=(\mathcal{F}_{\rm std},\omega,L_{\rm std},S_0,S_\infty)$ in 4 steps.
{\em Step 1. }By Proposition~\ref{prop:fibre-standard}, $\mathfrak{S}$ is homotopic to $\widetilde\mathfrak{S}=(\mathcal{F}_{\rm std},\widetilde\omega,L_{\rm std},\widetilde\Sigma,\widetilde\Sigma')$ such that $\widetilde\omega=\omega_{\rm std}$ on a neighbourhood of the fibre $F=\{z_0\}\times S^2$. The sections $\widetilde\Sigma,\widetilde\Sigma'$ intersect the fibre in points $(z_0,q)$ and $(z_0,q')$. After pulling back $\widetilde\mathfrak{S}$ by a symplectomorphism $(z,w)\mapsto(z,g(w))$, where $g:S^2\to S^2$ is a Hamiltonian diffeomorphism preserving the equator and mapping the south pole $S$ to $q$ and the north pole $N$ to $q'$, we may assume in addition that $\widetilde\Sigma\cap F=(z_0,S)$ and $\widetilde\Sigma'\cap F=(z_0,N)$.
{\em Step 2.} Consider the symplectic section $\widetilde\Sigma$. By Lemma~\ref{lem:symp-standard} (with $\Lambda=\{0\}$, $S^0=\widetilde\Sigma$ and $F=\{0\}\times{\mathbb{R}}^2$ in local coordinates), we can deform $\widetilde\Sigma$ such that it agrees with $S_0=S^2\times\{S\}$ near $\widetilde\Sigma\cap F$. Since the section $\widetilde\Sigma$ is isotopic to $S_0$, there exists a diffeomorphism of $S^2\times S^2$, isotopic to the identity and fixed near $F$, mapping $S_0$ to $\widetilde\Sigma$. After pulling back everything by this diffeomorphism, we may assume that $\widetilde\Sigma=S_0$. As in the proof of Proposition~\ref{prop:fibre-standard}, using the symplectic neighbourhood theorem, by pulling back by an isotopy of $S^2\times S^2$ fixed near $F$ we can arrange in addition that $\widetilde\omega=\omega_{\rm std}$ near $\widetilde\Sigma=S_0$ (but the foliation becomes non-standard). The same arguments apply to the other section $\widetilde\Sigma'$. Altogether, we have shown that $\widetilde\mathfrak{S}$ is homotopic to a relative symplectic fibration of the form $\widehat\mathfrak{S}=(\widehat\mathcal{F},\widehat\omega,L_{\rm std},S_0,S_\infty)$ with the following properties: $\widehat\omega=\omega_{\rm std}$ and $\widehat\mathcal{F}=\mathcal{F}_{\rm std}$ near the fibre $F=\{z_0\}\times S^2$, and $\widehat\omega=\omega_{\rm std}$ near the symplectic sections $S_0$ and $S_\infty$.
{\em Step 3. }Next, we adjust the foliation $\widehat\mathcal{F}$ near $S_0\cup S_\infty$. Consider first $S_0$. Take a compact subset $\Lambda\subset S^2\setminus\{z_0\}$ such that $\widehat\mathcal{F}=\mathcal{F}_{\rm std}$ on a neighbourhood of $(S^2\setminus{\rm int}\Lambda)\times S^2$. We identify $\Lambda$ with a subset of $({\mathbb{R}}^2,dx\wedge dy)$, and a neighbourhood of $\Lambda\times\{S\}$ in $S^2\times S^2$ with a neighbourhood $W$ of $\{0\}\times\Lambda$ in $({\mathbb{R}}^4,\Omega_0)$, by a symplectomorphism of the form $(z,w)\mapsto(f(w),g(z))$. Under this identification, $\widehat\mathcal{F}$ corresponds to a symplectic foliation of $W$ transverse to $\{0\}\times\Lambda$ and standard near $\partial\Lambda\times{\mathbb{R}}^2$. By Lemma~\ref{lem:symp-standard}, $\widehat\mathcal{F}$ can be deformed in $W$, keeping it fixed near $\partial\Lambda\times{\mathbb{R}}^2$, to a symplectic foliation that is standard on a neighbourhood $U$ of $\{0\}\times\Lambda$ in ${\mathbb{R}}^4$. Transfering back to $S^2\times S^2$ and performing the same construction near $S_\infty$, we have thus deformed $\widehat\mathfrak{S}$ to a relative symplectic fibration $\overline\mathfrak{S}=(\overline\mathcal{F},\overline\omega,L_{\rm std},S_0,S_\infty)$ satisfying $\overline\omega=\omega_{\rm std}$ and $\overline\mathcal{F}=\mathcal{F}_{\rm std}$ near the set $F\cup S_0\cup S_\infty$. This was the main step. It only remains to deform $\overline\mathcal{F}$ back to $\mathcal{F}_{\rm std}$.
{\em Step 4. }By construction, the foliation $\overline\mathcal{F}$ is obtained in steps 2 and 3 from $\mathcal{F}_{\rm std}$ by a homotopy $\mathcal{F}_t$ with $\mathcal{F}_0=\mathcal{F}$ and $\mathcal{F}_1=\overline\mathcal{F}$ which is fixed outside $V\setminus V_F$, for some neighbourhoods $V$ of $S_0\cup S_\infty$ and $V_F$ of $F$. So we can write $\mathcal{F}_t=\phi_t(\mathcal{F}_{\rm std})$ for diffeomorphisms with $\phi_0=\mathrm{id}$ and $\phi_t=\mathrm{id}$ outside $V\setminus V_F$. Since $\overline\mathcal{F}$ agrees with $\mathcal{F}_{\rm std}$ on a smaller neighbourhood $V_0\cup V_\infty$ of $S_0\cup S_\infty$, the diffeomorphisms $\phi_t$ can be chosen of the form $(z,w)\mapsto(z,f_t(w))$ on $V_0$ and $(z,w)\mapsto(z,g_t(w))$ on $V_\infty$ for diffeomorphisms $f_t,g_t$ of $S^2$. Now the homotopy $\phi_t^{-1}(\overline\mathcal{F})$ connects $\overline\mathcal{F}$ to the symplectic fibration $\phi_1^{-1}\overline\mathfrak{S}=(\mathcal{F}_{\rm std},\phi_1^*\overline\omega,L_{\rm std},S_0,S_\infty)$, where $\phi_1^*\overline\omega$ is split near $F\cup S_0\cup S_\infty$. This concludes the proof of Proposition~\ref{prop:section-standard}. \end{proof}
\begin{remark}\label{rem:section-standard} Replacing Step 4 of the preceding proof by a more careful deformation of the foliation $\overline\mathcal{F}$ (not by diffeomorphisms but keeping it symplectic for $\overline\omega$), we could arrange $\widetilde\omega=\omega_{\rm std}$ near $F\cup S_0\cup S_\infty$ in Proposition~\ref{prop:section-standard}. As the class of split forms is better suited for the modifications in the next section, we content ourselves with making $\widetilde\omega$ split near $F\cup S_0\cup S_\infty$. \end{remark}
\section{Killing the holonomy}\label{sec:killholonomy}
In this section we will deform a relative symplectic fibration to kill all the holonomy and conclude the proof of the main theorem. A crucial ingredient is the inflation procedure from~\cite{LalMcD}.
\subsection{Setup}\label{ss:setup}
Recall that $\mathcal{F}_{\rm std}$ is the foliation on $S^2\times S^2$ given by the fibres of the projection $p_1$ onto the first factor, $S_0=S^2\times \left\lbrace S\right\rbrace$ and $S_\infty=S^2\times \left\lbrace N\right\rbrace$ are the standard sections, $F=p_1^{-1}(z_0)$ is the fibre over the point $z_0=(1,0,0)$, and the Clifford torus $L_{\rm std}=E\times E$ is the product of the equators. In the following, we identify $S_0$ with the base $S^2$ of the projection $p_1$, i.e., we identify $p_1$ with the map $(z,w)\mapsto(z,S)$ sending each fibre to its intersection with $S_0$.
Our starting point is a relative symplectic fibration $\mathfrak{S}=(\mathcal{F}_{\rm std}, \omega, L_{\rm std},S_0,S_\infty)$ as provided by Proposition~\ref{prop:section-standard} such that $\omega$ is split on a neighbourhood $W=(U_F\times S^2)\cup (S^2\times U_0)\cup (S^2\times U_\infty)$ of $F\cup S_0\cup S_\infty$. In particular the sections $S_0,S_\infty$ are horizontal for the symplectic connection. After pulling back $\mathfrak{S}$ by a diffeomorphism of the form $(z,w)\mapsto (\phi(z),w)$ (keeping the same notation), we may replace $U_F$ by the ball $$
B:=\left\lbrace (x,y,z)\in S^2\mid x\geq -1/\sqrt{2}\right\rbrace, $$ so that $\omega$ now satisfies \begin{equation}\label{eq:split2} \begin{aligned}
\omega &= p_1^*\sigma_0+p_2^*\sigma_{\rm std}\quad \text{on the set}\quad
W_0=(B\times S^2) \cup (S^2\times U_0), \quad\text{and}\cr
\omega &= p_1^*\sigma_\infty+p_2^*\sigma_{\rm std} \quad\text{on the set}\quad
W_\infty=(B\times S^2)\cup(S^2\times U_\infty). \end{aligned} \end{equation} Consider the usual spherical coodinates $(\lambda,\mu) \in [-\frac{\pi}{2},\frac{\pi}{2}]\times [0,2\pi]$ on the base $S^2$ centered at $z_0$. Thus $\lambda$ denotes the latitude and $\mu$ the meridian, and $z_0$ lies at $(\lambda,\mu)=(0,0)$; see Figure~\ref{f42}.
\begin{figure}
\caption{Circles of latitude and the set $B$}
\label{f42}
\end{figure}
Denote by $C^\lambda$ the circle of latitude $\lambda$ in the base and by $\phi^\lambda$ the symplectic parallel transport around $C^\lambda$ parametrised by $\mu\in[0,2\pi]$. Since the starting and end points of the parametrisation of $C^\lambda$ are contained in $B$ for all $\lambda$ and the symplectic form $\omega$ equals $p_1^*\sigma_0+p_2^*\sigma_{\rm std}$ over $B$, we can regard $\phi^\lambda$ as living in $Symp(S^2,\ostd)$ for all $\lambda$. Moreover, the maps $\phi^\lambda$ have the following two properties: \begin{enumerate}
\item Since $C^\lambda\subset B$ for all $|\lambda|\geq \frac{\pi}{4}$
and the form $\omega$ is split on $B\times S^2$, we have $\phi^\lambda=\mathrm{id}$ for
$|\lambda|\geq \frac{\pi}{4}$. \item Since $\omega$ is split on $S^2\times(U_0\cup U_\infty)$, each
$\phi^\lambda$ restricts to the identity on $U_0\cup U_\infty$. \end{enumerate} Under stereographic projection $S^2\setminus\{N\}\to{\mathbb{C}}$ from the north pole $N$, the standard symplectic form on $S^2$ corresponds to the form $\ostd=\frac{r}{\pi(1+r^2)^2}dr\wedge d\theta$ in polar coordinates on ${\mathbb{C}}$. We pick a closed annulus $$
A=\left\lbrace z\in \mathbb{C}\;\bigl|\; a\leq |z|\leq
b\right\rbrace\subset{\mathbb{C}}\cong S^2\setminus\{N\} $$ with $a>0$ so small and $b>a$ so large that $\partial A\subset U_0\cup U_\infty$. According to properties (i) and (ii) above, parallel transport along $C^\lambda$ then defines maps $\phi^\lambda \in {\rm Symp}(A,\partial A,\ostd)$ (i.e., symplectomorphisms that equal the identity near $\partial A$, cf.~Appendix~\ref{app:diff}) that
equal the identity for $|\lambda|\geq \frac{\pi}{4}$. In particular, $[-\frac{\pi}{2},\frac{\pi}{2}]\ni\lambda \mapsto \phi^\lambda$ defines a loop in the identity component $Symp_0(A,\partial A,\ostd)$. Consider the loop of inverses $$
\psi^\lambda=(\phi^\lambda)^{-1}. $$ Since $L_{\rm std}=E\times E$ is invariant under parallel transport, the map $\phi^0$, and thus $\psi^0$, preserves the equator $E$.
\subsection{A special contraction}
According to Proposition~\ref{PHE}, the loop $\psi^\lambda$ is contractible in ${\rm Symp}_0(A,\partial A,\ostd)$. However, in order for the inflation procedure below to work, we need a special contraction $\psi^\lambda_s$ with the property that $\psi^0_s(E)=E$ for all $s\in[0,1]$. Here we identify the equator $E$ in $S^2$ via stereographic projection with the circle $E=\left\lbrace
|z|=1\right\rbrace \subset A$.
\begin{prop}\label{Tspeccontr} There exists a smooth contraction $\psi^\lambda_s\in{\rm Symp}_0(A,\partial A,\ostd)$ of the loop $\psi^\lambda$, with $(s,\lambda) \in [0,1]\times [\frac{-\pi}{2},\frac{\pi}{2}]$, such that: \begin{enumerate} \item $\psi^\lambda_0=\mathrm{id}$ and $\psi^\lambda_1=\psi^\lambda$ for all $\lambda$;
\item $\psi_s^\lambda=\mathrm{id}$ for $|\lambda|\geq \frac{\pi}{4}$ and all $s$; \item $\psi^\lambda_s$ is constant in $s$ near $s=0$ and $s=1$; \item $\psi^0_s(E)=E$ for all $s$. \end{enumerate} \end{prop}
\begin{proof} Since the holonomy $\psi^0$ along the equator in the base preserves the equator $E$ in the fibre, Lemma~\ref{L1} provides a path $\alpha(t)\in{\rm Symp}_0(A,\partial A,\ostd)$ from the identity to $\psi^0$ which preserves $E$ for all $t$. We split the loop $\psi^\lambda$ into two paths $\delta_1:=\left\lbrace \psi^\lambda \right\rbrace_{\lambda \in [-\frac{\pi}{2},0]}$ and $\delta_2:=\left\lbrace \psi^\lambda \right\rbrace_{\lambda \in
[0,\frac{\pi}{2}]}$. Using these, we define two loops $\gamma_1:=\delta_1\ast \bar{\alpha}$ and $\gamma_2:=\alpha \ast \delta_2$, where $\ast$ means concatenation of paths and $\bar{\alpha}$ denotes the path $\alpha$ traversed in the opposite direction; see Figure~\ref{f44}. By Proposition~\ref{PHE}, these loops are contractible in ${\rm Symp}_0(A,\partial A,\ostd)$, so we can fill them by half-disks $D_1,D_2$ in ${\rm Symp}_0(A,\partial A,\ostd)$. Gluing these half-disks along $\alpha$ yields a map $\vartheta: D\to {\rm Symp}_0(A,\partial A,\ostd)$ from the unit disk $D\subset{\mathbb{C}}$ which restricts to the loop $\psi^\lambda$ on $\partial D$ (starting and ending at $-i$) and to the path $\alpha$ on the imaginary axis. The composition of $\vartheta$ with the map $$
\eta:[0,1]\times[-\frac{\pi}{2},\frac{\pi}{2}]\to D,\qquad
(s,\lambda)\mapsto (s-1)i+se^{i(2\lambda+\pi/2)} $$ (see Figure~\ref{circlesoflatitude2}) then has properties (i) and (iv) of the proposition. By smoothing and reparametrisation we finally arrange properties (ii) and (iii) to obtain the desired contraction $\psi^\lambda_s$. \begin{figure}
\caption{Construction of the special contraction $\psi^ \lambda_s$}
\label{f44}
\end{figure}
\begin{figure}
\caption{The reparametrisation $\eta$}
\label{circlesoflatitude2}
\end{figure}
\end{proof}
\subsection{A special Hamiltonian function}\label{ConstrH}
Next, we construct a family of time-dependent Hamiltonians generating the contraction $\psi_s^\lambda$ of the previous subsection. We begin with a simple lemma.
\begin{lemma}\label{LformulaHtFt} Let $(M,\omega=d\lambda)$ be an exact symplectic manifold. Let $\phi_t:M\to M$ be a symplectic isotopy starting at $\phi_0=\mathrm{id}$ generated by the time-dependent vector field $X_t$, i.e $\frac{d}{dt}\phi_t=X_t\circ \phi_t$. Then $\iota_{X_t}\omega=dH_t$ for a smooth family of functions $H_t \colon M \to \mathbb{R}$ if and only if $\phi_t^\ast \lambda-\lambda=dF_t$ for a smooth family of functions $F_t \colon M \to \mathbb{R}$. Moreover, $F_t$ and $H_t$ are related by the equations $$
F_t=\int_0^t(H_s+\iota_{X_s}\lambda)\circ \phi_s\,ds,\qquad
H_t=\dot F_t \circ \phi_t^{-1}-\iota_{X_t}\lambda. $$ \end{lemma}
\begin{proof} Assume first that $\iota_{X_t}\omega=dH_t$. Then \begin{gather*} \phi_t^\ast \lambda-\lambda=\int_0^t\frac{d}{ds}(\phi_s^\ast \lambda) ds=\int_0^t\phi_s^\ast (L_{X_s})\lambda ds \cr =\int_0^t \phi_s^\ast (\iota_{X_s}d\lambda+d\iota_{X_s}\lambda)ds=d\int_0^t (H_s+\iota_{X_s}\lambda)\circ \phi_s ds \end{gather*} shows that $\phi_t^\ast \lambda-\lambda=dF_t$ holds for $F_t:=\int_0^t (H_s+\iota_{X_s}\lambda)\circ \phi_s ds$. Conversely, if $\phi_t^\ast \lambda-\lambda=dF_t$, then we differentiate this equation to obtain $$
d\dot F_t = \frac{d}{dt}(\phi_t^\ast \lambda) =
\phi_t^\ast(d\iota_{X_t}\lambda+\iota_{X_t}d\lambda), $$ which shows that $i_{X_t}d\lambda=dH_t$ holds for $H_t := \dot F_t \circ \phi_t^{-1}-\iota_{X_s}\lambda$. \end{proof}
Now let $\psi^\lambda_s\in{\rm Symp}_0(A,\partial A,\ostd)$ be the special contraction from Proposition~\ref{Tspeccontr}. Let $\lambda_{\rm std}=\frac{-1}{2(1+r^2)\pi}d\theta$ be the standard primitive of $\ostd$ (any other primitive would also do). Then for each $(s,\lambda)$ the $1$-form $\alpha_s^\lambda:=(\psi^\lambda_s)^*\lambda_{\rm std}-\lambda_{\rm std}$ on $A$ is closed and vanishes near $\partial A$. So by the relative Poincar\'e lemma, $$
(\psi^\lambda_s)^*\lambda_{\rm std}-\lambda_{\rm std} = dF_s^\lambda $$ for a unique smooth family of functions $F_s^\lambda$ that vanish near the lower boundary component $\partial_-A=\{a\}\times S^1$ of $A$. (We can define $F_s^\lambda(w):=\int_{\gamma_w}\alpha_s^\lambda$ along any path $\gamma_w$ from a base point on $\partial_-A$ to $w$, which does not depend on the path because every loop can be deformed into $\partial_-A$ where $\alpha_s^\lambda$ vanishes.) Note that $F_s^\lambda$ will be constant near the upper boundary component $\partial_+A=\{b\}\times S^1$, where the constant may depend on $s$ and $\lambda$.
By Lemma~\ref{LformulaHtFt}, the family $F^\lambda_s$ is related to a smooth family of Hamiltonians $\widetilde H^\lambda_s$ generating the isotopy $\psi^\lambda_s$ (for fixed $\lambda$) by the formula $$
\widetilde H^\lambda_s = \frac{\partial F^\lambda_s}{\partial s}\circ
(\psi^\lambda_s)^{-1}-\iota_{X^\lambda_s}\lambda_{\rm std}, $$ where $\frac{d}{dt}\psi^\lambda_t=X^\lambda_t\circ
\psi^\lambda_t$. By construction, $\widetilde H_s^\lambda$ vanishes near the lower boundary component $\partial_-A$ of $A$ and it is constant near the upper boundary component $\partial_+A$ (where the constant may vary with $s$ and $\lambda$). Further, since $\psi_s^\lambda$ is constant near its ends in both $s$ and $\lambda$, we have $\widetilde H^\lambda_s=0$ for $|\lambda|\geq \frac{\pi}{4}$ and for $s<2\epsilon$, $s>1-2\epsilon$ with some $\epsilon>0$.
Note that, since $\psi^0_s$ preserves the equator, the Hamiltonian vector field $X^0_s$ is tangent to $E$ for all $s$. So the restriction $\widetilde H^0_s|_E$ is constant for all $s$ and defines a function $\widetilde H_E(s)$. For reasons that will become clear in the next subsection, we wish to modify $\widetilde H$ to make this function vanish. For this, we pick be a smooth cutoff function $\rho:{\mathbb{R}}\to[0,1]$ with $\rho(0)=1$ and support in $[-\frac{\pi}{4},\frac{\pi}{4}]$ and define $$
H_s^\lambda := \widetilde H_s^\lambda - \rho(\lambda)\widetilde H_E(s). $$
Since $H_s^\lambda$ differs from $\widetilde H_s^\lambda$ only by a function of $s$ and $\lambda$, it still has the same Hamiltonian vector field and thus still generates the family $\psi_s^\lambda$. By construction, $H_s^\lambda$ depends only on $s$ and $\lambda$ near the boundary $\partial A$ (with possibly different functions at the two boundary components), $H^\lambda_s=0$ for $|\lambda|\geq \frac{\pi}{4}$ and for $s<2\epsilon$, $s>1-2\epsilon$, and $$
H_s^0|_E=0\quad\text{for all }s. $$ We define in spherical coordinates on the base the squares \begin{align*}
Q &:= \left\lbrace (\mu,\lambda)\in S^2\setminus\left\lbrace
N,S\right\rbrace \;\bigl|\; 2\epsilon \leq \mu \leq 1-2\epsilon,\ |\lambda|\leq
\frac{\pi}{4}\right\rbrace,\cr
\widetilde{Q} &:= \left\lbrace (\mu,\lambda)\in S^2\setminus\left\lbrace
N,S\right\rbrace \;\bigl|\; \epsilon \leq\mu \leq 1-\epsilon, \ |\lambda|\leq
\frac{\pi}{3}\right\rbrace. \end{align*} Note that $Q\subset{\rm int\,}\widetilde Q$ and $\widetilde Q\subset{\rm int\,} B$, where $B$ is the region defined at the beginning of this section over which $\omega$ is split. The family $H_s^\lambda$ constructed above gives rise to a smooth function $$
H \colon \widetilde{Q}\times A \to \mathbb{R},\qquad (\lambda,\mu,w) \mapsto
H^\lambda_\mu(w). $$ Let us write the fibre sphere as $$
S^2={\rm Cap}_{N}\cup A \cup {\rm Cap}_{S}, $$ where ${\rm Cap}_N$ and ${\rm Cap}_S$ denote the northern and southern polar caps, respectively. Then we can extend $H$ first to $\widetilde{Q}\times S^2$ by the corresponding functions of $(\lambda,\mu)$ on the southern and northern polar caps, and then to all of $S^2\times S^2$ by zero outside $\widetilde Q\times S^2$. We still denote the resulting function by $H:S^2\times S^2\to{\mathbb{R}}$. By construction, $H$ has support in $Q\times S^2$, it depends only on $(\lambda,\mu)$
outside $Q\times A$, and $H(0,\mu)|_E\equiv 0$ for all $\mu$, where we denote $H(\lambda,\mu):=H|_{p_1^{-1}(\lambda,\mu)}$.
\subsection{A special symplectic connection}
Recall that we consider a relative symplectic fibration $(\mathcal{F}_{\rm std},\omega,L_{\rm std},S_0,S_\infty)$ such that the symplectic form $\omega$ is split on the set $(B\times S^2)\cup(S^2\times(U_0\cup U_\infty))$. Our current goal is to change the symplectic form $\omega$, in its relative cohomology class in $H^2(S^2\times S^2,L_{\rm std}; \mathbb{R})$, to a form $\omega'$ which has trivial holonomy around the circles of latitude. To explain the idea, consider a circle of latitude $C^\lambda$ (cf.~Figure~\ref{f48}). As the symplectic form is split over $B$, its parallel transport equals the identity along the part of $C^\lambda$ lying within $B$, so the holonomy $\phi^\lambda$ is realised by travelling along the part of $C^\lambda$ outside $B$.
\begin{figure}
\caption{The path $Q\cap C^\lambda$ and $B\cap C^\lambda$}
\label{f48}
\end{figure}
The idea is now to modify $\omega$ to $\omega'$ such that the symplectic connection of $\omega'$ agrees with that of $\omega$ outside $Q\times S^2$ and realises the inverse holonomy $\psi^\lambda$ along $C^\lambda\cap Q$ for all $\lambda$.
For the following computations, let us rename the coordinates $(\lambda,\mu)$ to
$$
x:=\mu\in[0,1],\quad y:=\lambda\in[-\frac{\pi}{3},\frac{\pi}{3}]. $$ Recall that the function $H:S^2\times S^2\to{\mathbb{R}}$ constructed in the previous subsection has support in $Q\times S^2$, where $Q=[2\epsilon,1-2\epsilon]\times [-\frac{\pi}{4},\frac{\pi}{4}]$ in the new coordinates $(x,y)$.
Consider the closed $2$-form $$
\Omega_H=\omega+dH\wedge dx $$ on $Q\times A$, extended by $\omega$ to a form on all of $S^2\times S^2$. Since $\Omega_H$ is vertically nondegenerate, the $\Omega_H$-orthogonal complements to the tangent spaces of the fibres of $p_1$ induce a symplectic connection on $S^2\times S^2$.
\begin{lemma}\label{lem:hol-latitude} (a) The holonomy of $\Omega_H$ along each circle of latitude $C^\lambda$ is trivial.
(b) The closed form $\Omega_H$ vanishes on $L_{\rm std}$ and is relatively cohomologous to $\omega$. \end{lemma}
\begin{proof} (a) Recall that $H$ depends only on $x$ and $y$ outside the set $Q\times A$, so $\Omega_H$ and $\omega$ differ there by the pullback $dH\wedge dx=p_1^*\alpha$ of a $2$-form $\alpha$ from the base. Since adding the pullback of a $2$-form from the base does not change the symplectic connection (because $\iota_v(p_1^*\alpha)=0$ for every vertical vector $v$), the induced connections of $\Omega_H$ and $\omega$ agree outside the set $Q\times A$. Within $Q\times A$ the form $\omega=p_1^*\sigma_0+p_2^*\ostd$ is split, so that its induced connection is flat. The horizontal spaces of the induced connection of $\Omega_H$ are spanned by the horizontal lifts of the coordinate vector fields $\partial_x,\partial_y$. These can be easily seen to be $$
\widetilde\partial_x = \partial_x+X_{H_x^y},\qquad \widetilde\partial_y = \partial_y, $$ where $X_{H_x^y}$ is the Hamiltonian vector field of the Hamiltonian function $H_x^y(w)=H(x,y,w)$ on the annulus $(A,\ostd)$. To see this, let us write $\tilde\partial_x=\partial_x+v_x$ with a vertical vector $v_x$. This is horizontal iff $$
0 = \Omega_H(\widetilde\partial_x,v) = \Omega_H(\partial_x,v) + \Omega_H(v_x,v) =
-dH(v) + \ostd(v_x,v) $$ for all vertical vectors $v$, which just means that $v_x$ is the Hamiltonian vector field of $H_x^y$ with respect to $\ostd$. A similar calculation shows that $v_y=0$.
It follows that the parallel transport of $\Omega_H$ along an interval of latitude $C^\lambda\cap Q\cong [2\epsilon,1-2\epsilon]\times\{y\}$ is the time-$1$ map of the Hamiltonian flow of the time-dependent Hamiltonian $H_s^\lambda$. By construction of $H_s^\lambda$, this is the inverse $\psi^\lambda$ of the holonomy of $\omega$, and thus of $\Omega_H$, along the interval $C^\lambda\setminus Q$. Hence the total holonomy of $\Omega_H$ along each circle of latitude $C^\lambda$ is trivial.
(b) By construction, the horizontal vector field $\widetilde\partial_x=\partial_x+X_{H_x^y}$ is tangent to $L_{\rm std}$. Let $v$ be the vertical vector field along $L_{\rm std}$ given by the positively oriented unit tangent vectors to the equators in the fibres. Since $\Omega_H(\widetilde \partial_x,v)=0$ by definition of horizontality, this shows that $L_{\rm std}$ is Lagrangian for $\Omega_H$. Finally, let us compute the relative homology class of $\Omega_H$ in $H^2(S^2\times S^2,L_{\rm std})$. For this, we evaluate $\Omega_H$ on the generators of $H_2(S^2\times S^2,L_{\rm std})$ in Lemma~\ref{lem:homology}: \begin{align*}
\int_{S^2\times{\rm pt}}\Omega_H &= \int_{{\rm pt}\times S^2}\Omega_H = 1, \cr
\int_{{\rm pt}\times D_{\rm lh}}\Omega_H &= \int_{{\rm pt}\times D_{\rm lh}}\omega = \frac{1}{2}, \cr
\int_{D_{\rm lh}\times{\rm pt}}\Omega_H &= \frac{1}{2} + \int_{(Q\cap\{y\leq
0\})\times\{e\}}dH\wedge dx = \frac{1}{2} +
\int_0^1H(x,0,e)dx = \frac{1}{2}. \end{align*} Here in the last equation $e\in E$ is a base point on the equator in the fibre and we have used the normalisation condition $H(x,0,e) =
H(x,0)|_E\equiv 0$ from the previous subsection. Since $\omega$ takes the same values on these classes by monotonicity of $L_{\rm std}$, this shows that the relative cohomology classes of $\Omega_H$ and $\omega$ agree. \end{proof}
Let us analyse when the form $\Omega_H$ is symplectic. Since it is closed, this is equivalent to the form $\Omega_H\wedge \Omega_H$ being a volume form on $S^2\times S^2$. This is clearly satisfied outside the set $Q\times S^2$ because there $H\equiv 0$. On the set $Q\times S^2$, the form $\omega$ is split of the form $\omega=p_1^*\sigma_0+p_2^\ast \ostd$. We work in the chosen coordinates $x,y$ and write the form on the base as $$
\sigma_0=f(x,y)dx\wedge dy $$ with a positive function $f$. A short computation yields $$
\Omega_H\wedge \Omega_H=\left( 1-\frac{1}{f}\dd{H}{y}\right)
\omega\wedge \omega. $$ So $\Omega_H$ will be symplectic iff \begin{equation*}\label{eq:symp}
1-\frac{1}{f}\dd{H}{y}>0 \end{equation*} everywhere. A priori, this need not be true for the given function $H$, but it can be remedied by the inflation procedure in the next subsection.
\subsection{Inflation}
In this subsection we recall the inflation procedure of McDuff and Lalonde~\cite{LalMcD}, suitably adapted to our situation. Let $f_\sigma, \bar{f}_\tau$ be two smooth nonnegative bump functions on $S^2$, where we think of $f_\sigma$ as living on the fibre sphere and of $\bar{f}_\tau$ as living on the base sphere; see Figure~\ref{fig:inf}. We require that $$
{\rm supp}(f_\sigma)\subset (U_0\cup U_\infty)\setminus A = {\rm
Cap}_S\amalg {\rm Cap}_N, $$ where $U_0,U_\infty$ are the neighbourhoods of $S,N$ over which $\omega$ is split and $A$ is the annulus from the previous subsection, and that $$
\int_{{\rm Cap}_S}f_\sigma \ostd=\int_{{\rm Cap}_N}f_\sigma\ostd=\frac{1}{2}. $$ In particular, $\int_{S^2}f_\sigma\ostd=1$. The function $\bar{f}_\tau$ is required to have support in $\widetilde{Q}$ and satisfy $$
\bar{f}_\tau(x,y)=\bar{f}_\tau(x,-y)
$$ as well as $\bar{f}_\tau|_Q\equiv 1$. We define $$
f_\tau:=\frac{\bar{f}_\tau}{af}\quad\text{with}\quad
a:=\int_{\widetilde{Q}}\frac{\bar{f}_\tau}{f}\sigma_0 =
\int_{\widetilde{Q}}\bar{f}_\tau dx\wedge dy, $$ where $\sigma_0 = f(x,y)dx\wedge dy$ as above. Then $\int_{\widetilde{Q}}f_\tau\sigma_0=\frac{1}{a}\int_{\widetilde{Q}}\bar{f}_\tau dx\wedge dy=1$, which by the symmetry of $\bar{f}_\tau$ implies \begin{equation*}
\int_{\widetilde{Q}\cap \left\lbrace y\geq
0\right\rbrace}f_\tau\sigma_0 =
\frac{1}{a}\int_{\widetilde{Q}\cap\left\lbrace y\geq 0\right\rbrace
}\bar{f}_\tau dx\wedge dy=\frac{1}{2}. \end{equation*}
\begin{figure}
\caption{The functions $f_\sigma$ and $\bar{f}_\tau$}
\label{fig:inf}
\end{figure}
We define the two non-negative $2$-forms $$
\sigma:=f_\sigma \ostd,\qquad \tau:=f_\tau \sigma_0 $$ on $S^2$ and consider the family of $2$-forms on $S^2\times S^2$ \begin{equation}\label{eq:omc}
\omega_c := \frac{1}{c+1}\left( \omega+cp_1^\ast \tau +cp_2^\ast
\sigma \right),\qquad c\geq0. \end{equation}
\begin{lemma}\label{lem:omc} For each $c\geq 0$ the form $\omega_c$ has the following properties:
(a) $\omega_c$ is symplectic and $L_{\rm std}$ is Lagrangian for $\omega_c$;
(b) $\omega_c$ is cohomologous to $\omega_0=\omega$ in $H^2(S^2\times S^2,L_{\rm std};{\mathbb{R}})$;
(c) $\omega_c$ induces the same symplectic connection as $\omega$. \end{lemma}
\begin{proof} (a) First note that $\omega_c$ is closed for all $c\geq 0$ and $$
\omega_c=\frac{1}{c+1}\omega\quad\text{outside }W:=W_0\cup W_\infty, $$ where $W_0,W_\infty$ are the sets from~\eqref{eq:split2} on which $\omega$ is split. On the set $W_0$, \begin{equation}\label{eq:omc1}
\omega_c=\frac{1}{c+1}\Bigl((1+cp_1^\ast f_\tau)p_1^\ast
\sigma_0+(1+cp_2^\ast f_\sigma)p_2^\ast \ostd\Bigr), \end{equation} and therefore \begin{equation}\label{eq:omc2}
\omega_c\wedge \omega_c=\frac{1}{(c+1)^2}(1+c p_1^\ast f_\tau)(1+c
p_2^\ast f_\sigma)\,\omega \wedge \omega>0 \end{equation} because $c, f_\tau, f_\sigma$ are nonnegative. For the set $W_\infty$, we write $\sigma_\infty=g\sigma_0$ for a positive function $g$. Then on $W_\infty$ we have $$
\omega_c=\frac{1}{c+1}\Bigl((p_1^\ast g+cp_1^\ast f_\tau)p_1^\ast
\sigma_0+(1+cp_2^\ast f_\sigma)p_2^\ast \ostd\Bigr), $$ and again positivity of $g$ and nonnegativity of $c,f_\tau,f_\sigma$ implies $$
\omega_c\wedge \omega_c=\frac{1}{(c+1)^2}(p_1^\ast g+c p_1^\ast
f_\tau)(1+c p_2^\ast f_\sigma)\,\omega \wedge \omega>0. $$ This proves that $\omega_c$ is symplectic. The torus $L_{\rm std}$ is Lagrangian for $\omega_c$ because all pullback forms from the base or the fibre vanish on $L_{\rm std}$.
(b) To show that $\omega_c$ is relatively cohomologous to $\omega$, we evaluate it on the basis of $H_2(S^2\times S^2,L_{\rm std})$ from Lemma~\ref{lem:homology}. Using $\int_{S^2}\sigma=\int_{S^2}\tau=1$, $\int_{D_{\rm lh}}\sigma=\int_{{\rm Cap}_S}\sigma=1/2$ and $\int_{D_{\rm lh}}\tau=\int_{\widetilde Q\cap\{y\leq 0\}}\tau=1/2$, we compute with the point $z_0\in E$ on the equator in the base or fibre: \begin{align*}
\int_{{\rm pt}\times S^2}\omega_c &= \frac{1}{c+1} \int_{{\rm pt}\times S^2}
(\omega+cp_2^\ast \sigma) = \frac{1}{c+1}(1+c)=1, \cr
\int_{S^2\times{\rm pt}}\omega_c &= \frac{1}{c+1}\int_{S^2\times{\rm pt}}
(\omega+c p_1^\ast \tau) = \frac{1}{c+1}(1+c)=1, \cr
\int_{{\rm pt}\times D_{\rm lh}}\omega_c &= \frac{1}{c+1}\int_{\{z_0\}\times
D_{\rm lh}} (\omega+cp_2^\ast
\sigma)=\frac{1}{c+1}\Bigl(\frac{1}{2}+\frac{c}{2}\Bigr)=\frac{1}{2}, \cr
\int_{D_{\rm lh}\times{\rm pt}}\omega_c &= \frac{1}{c+1}\int_{D_{\rm lh}\times\{z_0\}}
(\omega+c p_1^\ast \tau)=\frac{1}{c+1}\Bigl(\frac{1}{2}+\frac{c}{2}\Bigr) =
\frac{1}{2}. \end{align*} By monotonicity of $L_{\rm std}$, the form $\omega$ takes the same values on these classes, so $[\omega_c]=[\omega]\in H^2(S^2\times S^2,L_{\rm std};{\mathbb{R}})$.
(c) On the set $W=W_0\cup W_\infty$ the forms $\omega_c$ and $\omega$ are both split, hence both symplectic connections are flat and the horizontal subspaces are the tangent spaces to the other cartesian factor. Outside $W$ we have $\omega_c=\frac{1}{c+1}\omega$ and, since the symplectic complements to the fibres are not affected by scaling of the symplectic form, the symplectic connections of $\omega_c$ and $\omega$ agree here as well. \end{proof}
{\bf A new symplectic connection. } Now recall that the function $H$ from the previous section is a pullback from the base outside the set $S^2\times A$. On the set $S^2\times A$, the function $p_2^\ast f_\sigma$ vanishes and thus $\omega_c=\frac{1}{c+1}\Bigl((1+p_1^*f_\tau)p_1^\ast \sigma_0+p_2^\ast \ostd\Bigr)$. In particular, on this set the restriction of $\omega_c$ to the fibres it is just the standard form $\ostd$ scaled by $\frac{1}{c+1}$. Now the fibrewise Hamiltonian vector field of the rescaled function $\frac{1}{c+1}H$ with respect to $\frac{1}{c+1}\ostd$ equals the fibrewise Hamiltonian vector field $X_{H_x^y}$ of $H$ with respect to $\ostd$. So the horizontal lift of $\partial_x$ with respect to the closed $2$-form \begin{equation}\label{eq:Omc}
\Omega_H^c:=\omega_c+\frac{1}{c+1}dH\wedge dx \end{equation} agrees with its horizontal lift $\partial_x+X_{H_x^y}$ with respect to $\Omega_H$ (see the proof of Lemma~\ref{lem:hol-latitude}), and since the horizontal lift of $\partial_y$ is $\partial_y$ in both cases, we see that $\Omega_H^c$ and $\Omega_H$ define the same symplectic connection for all $c\geq 0$. Moreover, the proof of Lemma~\ref{lem:hol-latitude}(b) shows that $\Omega_H^c$ vanishes on $L_{\rm std}$ and is relatively cohomologous to $\omega_c$, and thus to $\omega$ by Lemma~\ref{lem:omc}.
{\bf Symplecticity. } Again, let us analyse when the form $\Omega_H^c$ is symplectic. Outside $Q\times S^2$, the form $\Omega^c_H$ is just $\omega_c$, which is symplectic by Lemma~\ref{lem:omc}. On the set $Q\times S^2 \subset W_0$, using equations~\eqref{eq:omc1} and~\eqref{eq:omc2} we compute \begin{align*}
\omega_c \wedge \frac{1}{c+1}dH\wedge dx
&= \frac{1}{(c+1)^2}\left(-\dd{H}{y}\right)\left(1+c
p_2^\ast f_\sigma \right)dx\wedge dy\wedge p_2^*\ostd \cr
&= \frac{1}{2f(c+1)^2}\left(-\dd{H}{y}\right)\left(1+c
p_2^\ast f_\sigma \right)\omega\wedge \omega, \cr
\Omega^c_H\wedge \Omega^c_H
&= \omega_c\wedge \omega_c+2\omega_c \wedge \frac{1}{c+1}dH\wedge
dx \cr
&= \frac{1}{(c+1)^2}\left(1+cp_1^\ast f_\tau - \frac{1}{f}\dd{H}{y}
\right)\left(1+cp_2^\ast f_\sigma \right) \omega\wedge \omega. \end{align*} Now $1+cp_2^\ast f_\sigma\geq 1$ for all $c$ by nonnegativity of $c$ and $f_\sigma$. Moreover, by the choice of $f_\tau$ we have $p_1^\ast f_\tau=\frac{1}{af}$ on $Q\times S^2$. Hence $\Omega_H^c$ is symplectic iff \begin{equation*}
1+\frac{1}{f}\left(\frac{c}{a}-\dd{H}{y}\right)>0 \end{equation*} on $Q\times S^2$. But this is satisfied for \begin{equation}\label{eq:c}
c\geq C:=a\,{\rm max}_{Q\times S^2}\left|\frac{\partial H}{\partial y}\right|. \end{equation} We summarize the preceding discussion in
\begin{lemma}\label{lem:OmHc} The closed $2$-form $\Omega_H^c$ vanishes on $L_{\rm std}$, is relatively cohomologous to $\omega_c$ (and thus $\omega$), and has trivial holonomy along each circle of latitude $C^\lambda$ for each $c\geq 0$. Moreover, $\Omega_H^c$ is symplectic for $c\geq C$ given by~\eqref{eq:c}. \end{lemma}
\subsection{Killing the holonomy along circles of latitude}
We denote the $0$-meridian by $m_0:=\left\lbrace (\lambda,\mu) \in S^2\mid \mu=0 \right\rbrace$ in spherical coordinates. Putting the previous subsections together, we can now prove
\begin{proposition}\label{Pkillmonolatitude} Let $\mathfrak{S}=(\mathcal{F}_{\rm std},\omega,L_{\rm std},S_0,S_\infty)$ be a relative symplectic fibration such that $\omega$ is split on a neighbourhood of the fibre $F$ and the sections $S_0,S_\infty$. Then there exists a homotopy of relative symplectic fibrations $\mathfrak{S}_t=(\mathcal{F}_{\rm std},\omega_t,L_{\rm std},S_0,S_\infty)$ with $\mathfrak{S}_0=\mathfrak{S}$ such that the holonomy of $\mathfrak{S}_1$ along the circles of latitude $C^\lambda$ is the identity for all $\lambda$. Moreover, $\omega_1$ is split near the set $(m_0\times S^2)\cup S_0\cup S_\infty$. \end{proposition}
\begin{proof} As explained at the beginning of Section~\ref{ss:setup}, we may assume that $\omega$ is split on a set $(B\times S^2)\cup\bigl((S^2\times (U_0\cup U_\infty)\bigr)$, where the ball $B\subset S^2$ contains the $0$-meridian $m_0$. Let $H$ be the Hamiltonian function constructed in Section~\ref{ConstrH} and let $C$ be the constant defined in~\eqref{eq:c}. For $c\in[0,C]$, let $\omega_c$ be the form defined in~\eqref{eq:omc}. By Lemma~\ref{lem:omc}, $(\mathcal{F}_{\rm std},\omega_c,L_{\rm std},S_0,S_\infty)$ gives a homotopy of relative symplectic fibrations from $\mathfrak{S}$ to $(\mathcal{F}_{\rm std},\omega_C,L_{\rm std},S_0,S_\infty)$. For $t\in[0,1]$, consider the forms $$
\Omega_{tH}^C = \omega_C+\frac{t}{C+1}dH\wedge dx = (1-t)\omega_C+t\,\Omega_H^C $$ as in~\eqref{eq:Omc} (with $H$ replaced by $tH$ and $c$ by $C$). By Lemma~\ref{lem:OmHc} (applied to $tH$), $(\mathcal{F}_{\rm std},\Omega_{tH}^C,L_{\rm std},S_0,S_\infty)$ gives a homotopy of relative symplectic fibrations from $(\mathcal{F}_{\rm std},\omega_C,L_{\rm std},S_0,S_\infty)$ to $\mathfrak{S}_1=(\mathcal{F}_{\rm std},\Omega_H^C,L_{\rm std},S_0,S_\infty)$. By the same lemma, $\mathfrak{S}_1$ has trivial holonomy along all circles of latitude. Hence the concatenation of the previous two homotopies gives the desired homotopy $\mathfrak{S}_t$. For the last assertion, simply observe that by construction all symplectic forms in this homotopy agree with the original split form $\omega$ near $(m_0\times S^2)\cup S_0\cup S_\infty$. \end{proof}
\begin{remark}\label{rem:Ham} The point of departure for the preceding subsections was the standardisation provided by Proposition~\ref{prop:section-standard}. If rather than making the symplectic form $\omega$ split we had made it equal to $\omega_{\rm std}$ near $(m_0\times S^2)\cup S_0\cup S_\infty$ (as suggested in Remark~\ref{rem:section-standard}), then the holonomies $\phi^\lambda$ would lie in the subgroup ${\rm Ham}(A,\partial A,\ostd)\subset{\rm Symp}_0(A,\partial A,\ostd)$ of symplectomorphisms generated by Hamiltonians with compact support in $A\setminus\partial A$ and the whole construction could be performed in that subgroup (which is also contractible). However, since we change the normalisation of the Hamiltonians $H^\lambda_s$ anyway to make them vanish on the equator, we would gain nothing from working in this subgroup. \end{remark}
\subsection{Killing all the holonomy}
Now we will further deform the relative symplectic fibration from the previous subsection to one which has trivial holonomy along {\em all} closed curves in the base. We begin with a simple lemma.
\begin{lemma}\label{Llinearformsinterpol} Let $\omega,\omega'$ be linear symplectic forms on $\mathbb{R}^4$ which define the same orientation and agree on a real codimension one hyperplane $H$. Then $\omega_t:=(1-t)\omega+t\omega'$ is symplectic for all $t\in[0,1]$.
\end{lemma}
\begin{proof} Take a symplectic basis $e_1,f_1,e_2,f_2$ for $\omega$ such that $e_1,f_1,e_2$ is a basis of $H$. Take a vector $f_2'=a_1e_1+b_1f_1+a_2e_2+b_2f_2$ such that $e_1,f_1,e_2,f_2'$ is a symplectic basis for $\omega'$. Since $\omega,\omega'$ induce the same orientation, we have $b_2>0$, and therefore $$
\omega(e_2,f_2')=b_2>0,\qquad \omega'(e_2,f_2)=\frac{1}{b_2}>0. $$ For $\omega_t:=(1-t)\omega+t\omega'$ we find $$
\omega_t\wedge \omega_t=(1-t)^2\omega\wedge\omega+2t(1-t)\omega\wedge
\omega'+t^2\omega'\wedge\omega', $$ and therefore \begin{align*}
\omega_t\wedge\omega_t(e_1,f_1,e_2,f_2')
&= 2(1-t)^2\omega(e_1,f_1)\omega(e_2,f_2')+2t^2\omega'(e_1,f_1)\omega'(e_2,f_2')\cr
&\ \ \ +2t(1-t)\Bigl(\omega(e_1,f_1)\omega'(e_2,f_2')+\omega(e_2,f_2')\omega'(e_1,f_1)
\Bigl) \cr
&> 0. \end{align*} \end{proof}
Recall the definition of the $0$-meridian $m_0$ from the previous subsection.
\begin{proposition}\label{Ptrivholonomy} Let $\mathfrak{S}=(\mathcal{F}_{\rm std},\omega,L_{\rm std},S_0,S_\infty)$ be a relative symplectic fibration which is split near the set $S_0\cup S_\infty\cup(m_0\times S^2)$ and has trivial holonomy around all circles of latitude $C^\lambda$. Then there exists a homotopy of relative symplectic fibrations $\mathfrak{S}_t=(\mathcal{F}_{\rm std},\omega_t,L_{\rm std},S_0,S_\infty)$ with $\omega_0=\omega$ and $\omega_1=\omega_{\rm std}$. \end{proposition}
\begin{proof} The idea of the proof is to use parallel transport along circles of latitude to define a fibre preserving diffeomorphism $\phi$ of $S^2\times S^2$ which pulls back the symplectic form $\omega$ to a form which agrees with the standard form $\omega_{\rm std}$ on $C^\lambda\times S^2$ for all $\lambda$, and then apply Lemma~\ref{Llinearformsinterpol}.
\begin{figure}
\caption{The maps $P^\lambda_\mu$ and the construction of $\phi$}
\label{f410}
\end{figure}
For each $\lambda,\mu$ let $$
P^\lambda_\mu \colon \left\lbrace (\lambda,0) \right\rbrace\times
S^2 \to \left\lbrace (\lambda,\mu) \right\rbrace\times S^2 $$ be the parallel transport of (the symplectic connection on $p_1$ defined by) $\omega$ along the circle of latitude $C^\lambda$ from $(\lambda,0)$ to $(\lambda,\mu)$. Since $\omega$ has trivial holonomy along $C^\lambda$, this does not depend on the path in $C^\lambda$ and is thus well-defined. Note that, due to the fact that $\omega$ is split near $S_0\cup S_\infty$, the map $P^\lambda_\mu$ equals the identity near the north and south poles $N,S$ in the fibre.
We define a fibre preserving diffeomorpism $\phi$ of $S^2\times S^2$ by parallel transport on the left sphere in Figure~\ref{f410} with respect to the standard form $\omega_{\rm std}$ first going backwards along the circle of latitude until we hit the meridian $m_0$, then upwards along $m_0$ until we hit the north pole $N$, then by the identity to the fibre over the north pole of the right sphere, then by parallel transport with respect to $\omega$ along $m_0$, and finally along the circle of latitude to land in the fibre over the original point $(\lambda,\mu)$. Since parallel transport with respect to the symplectic connection $\omega_{\rm std}$ is the identity for all paths, as is parallel transport with respect to $\omega$ along paths in $m_0$ (since $\omega$ is split over $m_0$), we will not explicitly include these maps in the notation. Then we can write the preceding definition in formulas as $$
\phi \colon S^2\times S^2\to S^2\times S^2,\qquad
\Bigl((\lambda,\mu),w\Bigr) \mapsto
\Bigl((\lambda,\mu)\,,\,P^\lambda_\mu(w)\Bigr). $$ Note that $\phi$ is smooth for $(\lambda,\mu)$ near the north and south poles because there $P^\lambda_\mu=\mathrm{id}$. For $z\in S^2$ let us denote by $\omega_z$ the restriction of $\omega$ to the fibre $F_z=\{z\}\times S^2$. We claim that $\phi$ has the following properties: \begin{itemize} \item[(a)] $\phi$ restricts to symplectomorphisms
$(F_z,\ostd)\to(F_z,\omega_z)$ on all fibres; \item[(b)] $\phi$ preserves the Clifford torus $L_{\rm std}$; \item[(c)] $\phi$ equals the identity near $S_0\cup S_\infty\cup(m_0\times
S^2)$; \item[(d)] $\phi$ is isotopic to the identity through fibre preserving
diffeomorphisms $\phi_t$ that preserve $L_{\rm std}$ and equal the
identity near $S_0\cup S_\infty\cup(m_0\times S^2)$. \end{itemize}
For (a), note that $\omega$ restricts to $\ostd$ on the fibre $F_N$ over the north pole (because $\omega$ is split there), so the identity defines a symplectomorphism $(F_N,\ostd)\to (F_N,\omega_N)$. Now (a) follows because parallel transport is symplectic.
Property (b) follows from the fact that $L_{\rm std}$ is given by parallel transport of the equator in the fibre around the equator in the base, hence $P^0_\mu(E)=E$ and thus $$
\phi(\left\lbrace(0,\mu) \right\rbrace\times E) =
\left\lbrace (0,\mu) \right \rbrace \times P^0_\mu(E) =
\left\lbrace (0,\mu) \right \rbrace\times E. $$ Property (c) holds because $\omega$ is split near $S_\infty\cup S_0\cup(m_0\times
S^2)$.
For (d), consider the map $$
P\colon
R:=[-\frac{\pi}{2},\frac{\pi}{2}]\times[0,2\pi]\to{\rm Diff}(A,\partial
A),\qquad (\lambda,\mu)\mapsto P^\lambda_\mu. $$ It maps the boundary $\partial R$ to $\mathrm{id}$ and the interval $\{0\}\times[0,2\pi]$ to the subspace ${\rm Diff}(A,\partial A;E)\subset {\rm Diff}(A,\partial A)$ of diffeomorphisms preserving the equator $E$ (as a set, not pointwise). By Corollary~\ref{cor:Diff}, the loop $[0,2\pi]\ni\mu\mapsto P^0_\mu$ is contractible in ${\rm Diff}(A,\partial A;E)$. Using this and the fact (from Corollary~\ref{TDiff}) that $\pi_2{\rm Diff}(A,\partial A)=0$, we find a contraction of $P$ through maps $P_t:R\to {\rm Diff}(A,\partial A)$ sending $\partial R$ to $\mathrm{id}$ and $\{0\}\times[0,2\pi]$ to ${\rm Diff}(A,\partial A;E)$. (The argument is analogous to the proof of Proposition~\ref{Tspeccontr}, with the $1$-parametric family $\psi^\lambda$ replaced by the $2$-parametric family $P^\lambda_\mu$.) Then $\phi_t((\lambda,\mu),w):=((\lambda,\mu),P_t(\lambda,\mu)(w))$ is the desired isotopy and the claim is proved.
Now we construct the homotopy from $\omega$ to $\omega_{\rm std}$ in two steps. For the first step, let $\phi_t$ be the isotopy in (d) from $\phi_0=\mathrm{id}$ to $\phi_1=\phi$. Then $\phi_t^{-1}(\mathfrak{S})=(\mathcal{F}_{\rm std},\phi_t^*\omega,L_{\rm std},S_0,S_\infty)$ is a homotopy of relative symplectic fibrations from $\mathfrak{S}$ to $(\mathcal{F}_{\rm std},\phi^*\omega,L_{\rm std},S_0,S_\infty)$.
For the second step, note that $\phi^*\omega$ restricts to $\ostd$ on every fibre by property (a). Moreover, since $\phi$ commutes with parallel transport along $C^\lambda$ (with respect to $\omega_{\rm std}$ and $\omega$), the horizontal lifts of vectors tangent to circles of latitude with respect to $\omega_{\rm std}$ and $\phi^*\omega$ agree. Accordingly, $\omega_{\rm std}$ and $\phi^*\omega$ agree on the 3-dimensional subspaces $T_{((\lambda,\mu),w)}(C^\lambda\times S^2)$ in $T_{((\lambda,\mu),w)}(S^2\times S^2)$ for all $((\lambda,\mu),w) \in S^2\times S^2$. Thus, by Lemma \ref{Llinearformsinterpol}, the linear interpolations $\omega_t:=(1-t)\phi^*\omega+t\omega_{\rm std}$ are symplectic for all $t\in[0,1]$.
We claim that $(\mathcal{F}_{\rm std},\omega_t,L_{\rm std},S_0,S_\infty)$ is a relative symplectic fibration for all $t\in[0,1]$. For this, first note that $L_{\rm std}$ is monotone Lagrangian for both $\omega_{\rm std}$ and $\phi^*\omega$: For $\omega_{\rm std}$ this is clear, and for $\phi^*\omega$ it follows from $\phi(L_{\rm std})=L_{\rm std}$ and monotonicity of $L_{\rm std}$ for $\omega$. Hence $L_{\rm std}$ is monotone Lagrangian for $\omega_t$ for all $t$. Next, since $\phi$ preserves fibres as well as the sections $S_0,S_\infty$, the form $\phi^*\omega$ and thus also the form $\omega_t$ is cohomologous to $\omega_{\rm std}$ for all $t$. Finally, since $\phi$ preserves the sections $S_0,S_\infty$, they are symplectic for $\phi^*\omega$ as well as $\omega_{\rm std}$, hence for all $\omega_t$.
The desired homotopy $\omega_t$ is the concatenation of the homotopies constructed in the two steps. This concludes the proof of Proposition~\ref{Ptrivholonomy}. \end{proof}
\begin{remark}\label{rem:trivholonomy} The two steps in the proof of Proposition~\ref{Ptrivholonomy} could have been performed in the opposite order: First homotope $\omega$ to the form $\phi_*\omega_{\rm std}$ which has trivial holonomy along all closed curves in the base, and then homotope $\phi_*\omega_{\rm std}$ to $\omega_{\rm std}$. The latter is then a special case of the more general fact that two symplectic fibrations with conjugate holonomy (e.g., both having trivial holonomy) are diffeomorphic. \end{remark}
\subsection{Proof of the main theorem and some consequences}\label{ss:proof}
We summarize the results of this and the previous section in
\begin{thm}[Classification of relative symplectic fibrations]\label{thm:fib}
Every relative symplectic fibration $\mathfrak{S}=(\mathcal{F},\omega,L,\Sigma,\Sigma')$ on $S^2\times S^2$ is deformation equivalent to $\mathfrak{S}_{\rm std}=(\mathcal{F}_{\rm std},\omega_{\rm std},L_{\rm std},S_0,S_\infty)$. \end{thm}
\begin{proof} By Proposition~\ref{prop:fix-F}, $\mathfrak{S}$ is diffeomorphic to a relative symplectic fibration of the form $\widetilde\mathfrak{S}=(\mathcal{F}_{\rm std},\widetilde\omega,L_{\rm std},S_0,S_\infty)$ for some symplectic form $\widetilde\omega$. Combining Proposition~\ref{prop:section-standard}, Proposition~\ref{Pkillmonolatitude} and Proposition~\ref{Ptrivholonomy}, we find a homotopy from $\widetilde\mathfrak{S}$ to $\mathfrak{S}_{\rm std}$. \end{proof}
The main theorem will be a consequence of Theorem~\ref{thm:fib} and the following theorem of Gromov.
\begin{theorem}[Gromov~\cite{Gro}]\label{G85} Let $\phi \in Symp(S^2\times S^2,\omega_{\rm std})$ act trivially on homology. Then there exists a symplectic isotopy $\phi_t \in Symp(S^2\times S^2,\omega_{\rm std})$ with $\phi_0=\mathrm{id}$ and $\phi_1=\phi$. \end{theorem}
A first consequence is
\begin{cor}\label{cor:fib} Let $\mathfrak{S}=(\mathcal{F},\omega_{\rm std},L,\Sigma,\Sigma')$ be a relative symplectic fibration of $M=S^2\times S^2$, where $\omega_{\rm std}$ is the standard symplectic form. Then there exists a homotopy of relative symplectic fibrations $\mathfrak{S}_t=(\mathcal{F}_t,\omega_{\rm std},L_t,\Sigma_t,\Sigma_t')$ with fixed symplectic form $\omega_{\rm std}$ such that $\mathfrak{S}_0=\mathfrak{S}$ and $\mathfrak{S}_1=(\mathcal{F}_{\rm std},\omega_{\rm std},L_{\rm std},S_0,S_\infty)$. \end{cor}
\begin{proof} By Theorem~\ref{thm:fib}, there exists a homotopy of relative symplectic fibrations $\mathfrak{S}_t=(\mathcal{F}_t,\omega_t,L_t,\Sigma_t,\Sigma_t')$ with $\mathfrak{S}_1=(\mathcal{F}_{\rm std},\omega_{\rm std},L_{\rm std},S_0,S_\infty)$ and a diffeomorphism $\phi$ of $S^2\times S^2$ acting trivially on homology such that $\phi(\mathfrak{S})=\mathfrak{S}_0$. After applying Proposition~\ref{prop:fix-om-par} and modifying $\phi$ accordingly (keeping the same notation), we may assume that $\omega_t=\omega_{\rm std}$ for all $t\in[0,1]$. Then $\phi$ is a symplectomorphism with respect to $\omega_{\rm std}$, so by Gromov's Theorem~\ref{G85} it can be connected to the identity by a family of symplectomorphisms $\phi_t$. Now the concatenation of the homotopies $\phi_t(\mathfrak{S})_{t\in[0,1]}$, and $(\mathfrak{S}_t)_{t\in[0,1]}$ is the desired homotopy with fixed symplectic form $\omega_{\rm std}$. \end{proof}
\begin{proof} [{\bf Proof of the Main Theorem~\ref{thm:main}}] The hypotheses of Theorem~\ref{thm:main} just mean that $\mathfrak{S}=(\mathcal{F},\omega_{\rm std},L,\Sigma,\Sigma')$ is a relative symplectic fibration. By Corollary~\ref{cor:fib}, $\mathfrak{S}$ can be connected to $(\mathcal{F}_{\rm std},\omega_{\rm std},L_{\rm std},S_0,S_\infty)$ by a homotopy of relative symplectic fibrations $\mathfrak{S}_t=(\mathcal{F}_t,\omega_{\rm std},L_t,\Sigma_t,\Sigma_t')$ with fixed symplectic form $\omega_{\rm std}$. In particular, $L_t$ is an isotopy of monotone Lagrangian tori (with respect to $\omega_{\rm std}$) from $L_0=L$ to $L_1=L_{\rm std}$. By Banyaga's isotopy extension theorem, there exists a symplectic isotopy $\phi_t$ with $\phi_0=\mathrm{id}$ and $\phi_t(L)=L_t$ for all $t$ (see the proof of Proposition~\ref{prop:fix-om-par} for the argument, ignoring the symplectic sections). Since $M$ is simply connected, the symplectic isotopy $\phi_t$ is actually Hamiltonian. \end{proof}
Another consequence of Theorem~\ref{thm:fib} is the following result concerning standardisation by diffeomorphisms.
\begin{cor} [Fixing the symplectic form]\label{cor:fix-om} Let $\mathfrak{S}=(\mathcal{F},\omega,L,\Sigma,\Sigma')$ be a relative symplectic fibration of $M=S^2\times S^2$. Then there exists a diffeomorphism $\phi$ of $S^2\times S^2$ acting trivially on homology such that $\phi^{-1}(\mathfrak{S})=(\widetilde\mathcal{F},\omega_{\rm std},L_{\rm std},S_0,S_\infty)$ for some foliation $\widetilde\mathcal{F}$. \end{cor}
\begin{proof} By Theorem~\ref{thm:fib}, there exists a homotopy of relative symplectic fibrations $\mathfrak{S}_t=(\mathcal{F}_t,\omega_t,L_t,\Sigma_t,\Sigma_t')$ with $\mathfrak{S}_1=(\mathcal{F}_{\rm std},\omega_{\rm std},L_{\rm std},S_0,S_\infty)$ and a diffeomorphism $\phi$ of $S^2\times S^2$ acting trivially on homology such that $\phi(\mathfrak{S})=\mathfrak{S}_0$. After applying Proposition~\ref{prop:fix-om-par} and modifying $\phi$ accordingly (keeping the same notation), we may assume that $(\omega_t,L_t,\Sigma_t,\Sigma_t')=(\omega_{\rm std},L_{\rm std},S_0,S_\infty)$ for all $t\in[0,1]$. Then $\phi$ maps $\mathfrak{S}$ to $(\mathcal{F}_0,\omega_{\rm std},L_{\rm std},S_0,S_\infty)$. \end{proof}
In particular, Corollary~\ref{cor:fix-om} implies that every symplectic form $\omega$ on $S^2\times S^2$ which is compatible with a relative symplectic fibration can be pulled back to $\omega_{\rm std}$ by a diffeomorphism $\phi\in{\rm Diff}_\mathrm{id}(M)$. This is a special case of the deep result by Lalonde and McDuff~\cite{LalMcD} that every symplectic form $\omega$ on $S^2\times S^2$ which is cohomologous to the standard form $\omega_{\rm std}$ can be pulled back to $\omega_{\rm std}$ by a diffeomorphism $\phi\in{\rm Diff}_\mathrm{id}(M)$. In fact, the hard part of the proof in~\cite{LalMcD} (using Taubes' correspondence between Seiberg-Witten and Gromov invariants) consists in showing that any such $\omega$ is compatible with a symplectic fibration with a section.
\appendix
\section{Homotopy groups of some diffeomorphism groups}\label{app:diff}
In this appendix we collect some well-known facts about the diffeomorphism and symplectomorphism groups of the disk and annulus. We fix numbers $0<a<b$ and let $$
D:=\left\lbrace z\in \mathbb{C}\;\bigl|\; |z|\leq b \right\rbrace,\qquad
A:=\left\lbrace z \in \mathbb{C}\;\bigl|\; a \leq |z|\leq b \right\rbrace $$ be equipped with the standard symplectic form $\ostd=\frac{r}{\pi(1+r^2)^2}dr\wedge d\theta$ in polar coordinates on ${\mathbb{C}}$ (the precise choice of the symplectic form does not matter because they are all isomorphic up to scaling by Moser's theorem).
We define the following diffeomorphism groups, all equipped with the $C^\infty$ topology: \begin{itemize} \item ${\rm Diff}(D,\partial D)$ the group of diffeomorphisms of the closed
disk $D$ that are equal to the identity in some neighbourhood of the boundary; \item ${\rm Diff}(A,\partial A)$ the group of diffeomorphisms of the closed
annulus $A$ that are equal to the identity in some neighbourhood of
the boundary; \item ${\rm Symp}(A,\partial A,\ostd)\subset {\rm Diff}(A,\partial A)$ the subgroup of
symplectomorphisms of $(A,\ostd)$; \item ${\rm Symp}_0(A,\partial A,\ostd)$ the identity component of
$Symp(A,\partial A,\ostd)$.
\end{itemize}
{\bf Diffeomorphisms. } All results in this appendix are based on the following fundamental theorem of Smale.
\begin{theorem}[Smale~\cite{Sma}]\label{thm:Smale} The group ${\rm Diff}(D,\partial D)$ is contractible. \end{theorem}
With a nondecreasing cutoff function $\rho:{\mathbb{R}}\to[0,1]$ which equals $0$ near $(-\infty,a]$ and $1$ near $[b,\infty)$, we define the {\em
Dehn twist} $$
\phi^D \colon A \to A,\qquad
re^{i\theta}\mapsto re^{i(\theta+2\pi \rho(r))}. $$
\begin{cor}\label{TDiff} All homotopy groups $\pi_i{\rm Diff}(A,\partial A)$ vanish except for the group $\pi_0{\rm Diff}(A,\partial A)=\mathbb{Z}$, which is generated by the Dehn twist $\phi^D$. \end{cor}
\begin{proof}
Restriction of elements in ${\rm Diff}(D,\partial D)$ to the smaller disk $D_a\subset D$ of radius $a$ yields a Serre fibration $$
{\rm Diff}(A,\partial A)\to {\rm Diff}(D,\partial D)\to {\rm Diff}^+(D_a), $$ where ${\rm Diff}^+$ denotes the orientation preserving diffeomorphisms. In view of Smale's Theorem~\ref{thm:Smale}, the long exact sequence of this fibration yields isomorphisms $\pi_i{\rm Diff}^+(D_a)\cong \pi_{i-1}{\rm Diff}(A,\partial A)$ for all $i\geq 1$. Again by Theorem~\ref{thm:Smale}, the long exact sequence of the pair $({\rm Diff}^+(D_a),{\rm Diff}^+(\partial D_a))$ yields isomorphisms $\pi_i{\rm Diff}^+(\partial D_a)\cong\pi_i{\rm Diff}^+(D_a)$ for all $i$. Since $\pi_i{\rm Diff}^+(\partial D_a)\cong \pi_i{\rm Diff}^+(S^1)$ equals ${\mathbb{Z}}$ for $i=1$ and $0$ otherwise, this proves the corollary. \end{proof}
For the following slight refinement of Corollary~\ref{TDiff}, let $E\subset A$ be a circle $\{c\}\times S^1$ for some $c\in(a,b)$.
\begin{cor}\label{cor:Diff} Every smooth loop $(\phi_t)_{t\in[0,1]}$ in ${\rm Diff}(A,\partial A)$ with $\phi_0=\phi_1=\mathrm{id}$ and $\phi_t(E)=E$ for all $t$ can be contracted by a smooth family $\phi_t^s\in{\rm Diff}(A,\partial A)$, $s,t\in[0,1]$, satisfying $\phi_t^0=\phi_0^s=\phi_1^s=\mathrm{id}$, $\phi_t^1=\phi_t$ and $\phi_t^s(E)=E$ for all $s,t$. \end{cor}
\begin{proof} For a point $e\in S^1$ the family of arcs $\phi_t([a,c]\times\{e\})$
starts and ends at $t=0,1$ with the arc $[a,c]\times\{e\}$. This shows that the loop $t\mapsto\phi_t(c,e)$ in $E$ is contractible, hence so is the loop $\phi_t|_E$ in ${\rm Diff}(E)$. Thus we can find a family $\phi_t^s\in{\rm Diff}(A,\partial A)$, $s,t\in[1/2,1]$, satisfying $\phi_0^s=\phi_1^s=\mathrm{id}$, $\phi_t^1=\phi_t$, $\phi_t^s(E)=E$
for all $s,t$, and $\phi_t^{1/2}|_E=\mathrm{id}$ for all $t$. Now apply Corollary~\ref{TDiff} to contract the loops
$\phi_t^{1/2}|_{[a,c]\times S^1}$ in ${\rm Diff}([a,c]\times S^1,\partial[a,c]\times S^1)$ and $\phi_t^{1/2}|_{[c,b]\times S^1}$ in ${\rm Diff}([c,b]\times S^1,\partial[c,b]\times S^1)$. \end{proof}
{\bf Symplectomorphisms. } The following result is an immediate consequence of Corollary~\ref{TDiff} and Moser's theorem.
\begin{proposition}\label{PHE} The groups ${\rm Diff}(A,\partial A)$ and ${\rm Symp}(A,\partial A,\ostd)$ are weakly homotopy equivalent. In particular, ${\rm Symp}_0(A,\partial A,\ostd)$ is contractible. \end{proposition}
\begin{remark}\label{rem:Dehn} The Dehn twist $\phi^D$ defines an element in ${\rm Symp}(A,\partial A,\ostd)$, which according to Proposition~\ref{PHE} generates $\pi_0{\rm Symp}(A,\partial A,\ostd)$. \end{remark}
Finally, we need the following refinement of Proposition~\ref{PHE}. Again, let $E\subset A$ be a circle $\{c\}\times S^1$ for some $c\in(a,b)$.
\begin{lemma}\label{L1} Each $\phi \in {\rm Symp}_0(A,\partial A,\ostd)$ with $\phi(E)=E$ can be connected to the identity by a smooth path $\phi_t \in {\rm Symp}_0(A,\partial A,\ostd)$ satisfying $\phi_t(E)=E$ for all $t\in[0,1]$. \end{lemma}
\begin{proof} After applying Moser's theorem and changing the values of $a,b,c$ (viewing $A$ as a cylinder), we may assume that $\sigma_{\rm std}=dr\wedge d\theta$ and $c=0$. We connect $\phi$ to the identity in 4 steps.
{\em Step 1. }The restriction $f(\theta):=\phi(0,\theta)$ of $\phi$ to $E$ defines an element in the group ${\rm Diff}^+(S^1)$ of orientation preserving diffeomorphisms of the circle. Since this group is path connected, there exists a smooth family $f_t\in{\rm Diff}^+(S^1)$ with $f_0=\mathrm{id}$ and $f_1=f$. This family if generated by the time-dependent vector field $\xi_t$ on the circle defined by $\xi_t(f_t(\theta)):=\dot f_t(\theta)$. Let $H_t:A\to{\mathbb{R}}$ be a smooth family of functions satisfying $H_t(r,\theta) = -r\xi_t(\theta)$ near $E$ and $H_t=0$
near $\partial A$. A short computation shows that the Hamiltonian vector field of $H_t$ agrees with $\xi_t$ on $E$. It follows that the Hamiltonian flow $\psi_t$ of $H_t$ satisfies $\psi_t|_E=f_t$, in particular $\psi_1|_E=f_1=\phi|_E$. Thus $\phi_t:=\phi\circ\psi_t^{-1}$ is a smooth path in ${\rm Symp}_0(A,\partial A,\ostd)$ with $\phi_t(E)=E$
connecting $\phi$ to $\phi_1$ satisfying $\phi_1|_E=\mathrm{id}$. After renaming $\phi_1$ back to $\phi$, we may hence assume that
$\phi|_E=\mathrm{id}$.
{\em Step 2. } Let us write
$\phi(r,\theta)=\bigl(P(r,\theta),Q(r,\theta)\bigr)\in{\mathbb{R}}\times S^1$. Since $\phi|_E=\mathrm{id}$ and $\phi$ is symplectic, the functions $P,Q$ satisfy $$
P(0,\theta)=0,\qquad Q(0,\theta)=\theta,\qquad \frac{\partial P}{\partial
r}(0,\theta)=1. $$ For $s\in(0,1]$ consider the dilations $\tau_s(r,\theta):=(sr,\theta)$ on $A$. Since $\tau_s^*(dr\wedge d\theta)=s\,dr\wedge d\theta$, the maps $\psi_s:=\tau_s^{-1}\circ\phi\circ\tau_s$ are symplectic. Since $$
\psi_s(r,\theta) = \Bigl(\frac{1}{s}P(sr,\theta),Q(sr,\theta)\Bigr)
\xrightarrow[s\to 0]{}\Bigl(r\frac{\partial P}{\partial
r}(0,\theta),Q(0,\theta)\Bigr)
= (r,\theta), $$ the family $\psi_s$ extends smoothly to $s=0$ by the identity (this is a fibered version of the Alexander trick). It follows that for a sufficiently small $\varepsilon>0$ we have a smooth family of symplectic embeddings $\psi_s:A_\varepsilon:=[-\varepsilon,\varepsilon]\times S^1\hookrightarrow A$, $s\in[0,1]$, with $\psi_s(E)=E$, $\psi_0=\mathrm{id}$, and $\psi_1=\phi$. We extend this family to smooth diffeomorphisms $\widetilde\psi_s:A\to A$ with $\widetilde\psi_s=\mathrm{id}$ near $\partial A$ and $\widetilde\psi_1=\phi$. Since $\widetilde\psi_s$ preserves the annuli $A^-:=[a,0]\times S^1$ and $A^+:=[0,b]\times S^1$, it satisfies $\int_{A^\pm}\widetilde\psi_s^*\ostd=\int_{A^\pm}\ostd$ for all $s\in[0,1]$. By Banyaga's Theorem~\ref{thm:Banyaga}, applied to the isotopy $t\mapsto\phi^{-1}\circ\widetilde\psi_{1-t}$ and the set $X:=[a,a+\varepsilon]\times S^1\cup [b-\varepsilon,b]\times S^1\cup A_\varepsilon$ for some possibly smaller $\varepsilon>0$, there exists a symplectic isotopy $\phi_s:A\to A$, $s\in[0,1]$, with $\phi_1=\phi$ and
$\phi_s|_X=\widetilde\psi_s|_X$. In particular, $\phi_s\in{\rm Symp}_0(A,\partial A,\ostd)$ preserves $E$ and $\phi_0|_{A_\varepsilon}=\mathrm{id}$. After renaming $\phi_0$ back to $\phi$, we may hence assume that $\phi=\mathrm{id}$ on an annulus $A_\varepsilon$ around $E$.
{\em Step 3. }
Since $\phi|_{A_\varepsilon}=\mathrm{id}$, it restricts to maps
$\phi|_{A^\pm}\in{\rm Symp}(A^\pm,\partial A^\pm,\ostd)$.
By Proposition~\ref{PHE} and Remark~\ref{rem:Dehn}, $\phi|_{A^\pm}$ can be connected in ${\rm Symp}(A^\pm,\partial A^\pm,\ostd)$ to a multiple $(\phi_\pm^D)^{k_\pm}$ of the Dehn twist on $A^\pm$. Since $\phi$ belongs to the identity component ${\rm Symp}_0(A,\partial A,\ostd)$, it follows that $k_+=-k_-$. Hence we can simultaneously unwind the Dehn twists to connect the map $\psi$ which equals $(\phi_\pm^D)^{k_\pm}$ on $A^\pm$ to the identity by a path $\psi_t$ in ${\rm Symp}_0(A,\partial A,\ostd)$ fixing $E$ (but not restricting to the identity on $E$). Thus $\phi_t:=\phi\circ\psi_t^{-1}$ is a path in ${\rm Symp}_0(A,\partial A,\ostd)$ with $\phi_t(E)=E$ connecting $\phi$ to $\phi_1$ such that
$\phi_1|_{A^\pm}$ belongs to the identity component ${\rm Symp}_0(A^\pm,\partial A^\pm,\ostd)$. Again, we rename $\phi_1$ back to $\phi$.
{\em Step 4. }Finally, we apply Proposition~\ref{PHE} on $A^\pm$ to connect $\phi|_{A^\pm}$ to the identity by a path $\phi_t^\pm$ in ${\rm Symp}_0(A^\pm,\partial A^\pm)$. The maps $\phi_t^\pm$ fit together to a path $\phi_t\in{\rm Symp}_0(A,\partial A,\ostd)$ fixing $E$ that connects $\phi$ to the identity. \end{proof}
\begin{comment}
\section{Killing the flux}
\begin{proposition}\label{PDR} The group $Symp_0(A,\partial A,\ostd)$ deformation retracts onto $Ham(A,\partial A,\ostd)$. \end{proposition} \begin{proof} By Proposition \ref{Fluxa}, given any real number $a$ there exists a canonical symplectomorphism $\phi^a \in Symp_0(A,\partial A,\ostd)$ such that $Flux(\phi^a)=a$. Thus define $$r_t \colon Symp_0(A,\partial A,\ostd) \to Symp_0(A,\partial A,\ostd)$$$$\phi \mapsto \phi^{-tFlux(\phi)}\circ \phi $$ for $t\in [0,1]$. Since composition $\circ$ in $Symp(A,\partial A,\ostd)$ is continuous, to show continuity of $r_t$, it suffices to show that the map $$\phi \mapsto \phi^{-tFlux(\phi)}$$ is continuous. But $\phi^a$ is canonical for $a\in \mathbb{R}$, hence it suffices to show that $\phi \mapsto Flux(\phi)$ is continuous (obviously, then $\phi \mapsto tFlux(\phi)$ is also continuous).\\ $$Flux(\phi)-Flux(\psi)=Flux(\phi \circ \psi^{-1}),$$ and $\phi\circ \psi^{-1}$ is close to id if and only if $\phi$ is close to $\psi$. Hence $Flux$ is continuous, if it is continuous at $id$.\\ Write $\phi(r,\theta)=(R(r,\theta),Q(r,\theta))$ for $\phi \in Symp(A,\partial A,\ostd)$ being $\epsilon$-close to the identity. Then $$Flux(\phi)=\int_\gamma \phi^\ast \lstd-\lstd$$ where $\gamma(t)=((1-t)\frac{1}{2}+2t,0)$ is the path along $\mathbb{R}_+$ which connects the two boundary components.\\
Further $$(\phi^\ast \lstd)_{r,\theta}=\frac{-1}{2\pi (1+R(r,\theta)^2)}\left(\dd{Q}{r}(r,\theta)dr+\dd{Q}{\theta}(r,\theta)d\theta \right)$$ and thus $$|Flux(\phi)|=\mid \int_\gamma\phi^\ast \lstd-\lstd \mid=\mid \int^1_0\phi^\ast \lstd \left( (2-\frac{1}{2})\dd{}{r}\right) dt\mid \leq$$$$ \leq \frac{3}{2}{\rm max} \left| \frac{1}{2\pi (1+R^2)}\dd{Q}{r} \right|\leq \frac{3}{\pi} {\rm max} \left| \dd{Q}{r}\right|\leq {\rm max} \left| \dd{Q}{r}\right|.$$ But $\phi$ is $\epsilon$-close to id thus $\left| \dd{Q}{r}\right|<\epsilon$. This shows the continuity of $Flux$ and that of $r_t$. Now $$r_1 \colon Symp_0(A,\partial A,\ostd) \to Ham(A,\partial A,\ostd)$$ since $Flux(\phi^{-Flux(\phi)}\circ \phi)=0$.\\
Finally $r_t|_{Ham(A,\partial A,\ostd)}=id$, this follows from the fact that $\phi^a=id$ for $a=0$. Thus $r_t$ is indeed a deformation retraction as claimed. \end{proof}
\textbf{Remark}\\ The corresponding statements for the groups $Diff(A_o,\partial A_o)$, $Diff(A_i, \partial A_i)$, $Symp(A_o,\partial A_o,\ostd)$,\\ $Symp(A_i,\partial A_i,\ostd)$, $Symp_0(A_o,\partial A_o,\ostd)$, $Symp_0(A_i,\partial A_i,\ostd)$, $Ham(A_o,\partial A_o,\ostd)$,\\ $Ham(A_i,\partial A_i,\ostd)$ can be proved in exactly the same way. Thus we can replace the annulus $A$ in any of the results above by either $A_o$ or $A_i$.\\
Consider the following lemma, which we need in Appendix D. \begin{lemma}\label{Lisotopyofspecdiff} Let $\phi_0,\phi_1 \in Diff(A,\partial A)$ be of the special form $\phi_i(re^{i\theta})=re^{i(\theta+f_i(r))}$ with $f_i(\frac{1}{2})=2\pi k$ and $f_i(2)=0$ for $k\in \mathbb{Z}$ then there exists a smooth path $\phi_t \in Diff(A,\partial A)$ which connects $\phi_0$ and $\phi_1$. Furthermore if $\phi_0,\phi_1 \in Symp(A,\partial A,\ostd)$ then the $\phi_t \in Symp(A,\partial A,\ostd)$ for all $t$. \end{lemma} \begin{proof} It suffices to write down the linear isotopy $\phi_t(re^{i\theta})=re^{i(\theta+(1-t)f_0(r)+tf_1(r))}$. Then $\phi_t$ is smooth with smooth inverse $\phi_t^{-1}(re^{i\theta})=re^{i(\theta-(1-t)f_0(r)-tf_1(r))}$. Clearly $\phi_t$ restricts to the identity near $\partial A$ due to the fact that $f_0(\frac{1}{2})=f_1(\frac{1}{2})$ and $f_0(2)=f_1(2)$ near $\partial A$. Further by the defining equation it is clear that $\phi_t$ depends smoothly on $t$. Moreover note that a diffeomorphism of this form is always symplectic. This proves the lemma. \end{proof} \end{comment}
\begin{comment}
\section{Note for Martin}
{\bf This can be removed in view of new proof of Theorem~\ref{Tstand}.}
Here is the point in your thesis that i don't understand. Consider a relative symplectic fibration $\mathfrak{S}=(\mathcal{F},\omega,L,\Sigma,\Sigma')$ of $M=S^2\times S^2$. Let $N$ be the union of the leaves containing $L$. This is a $3$-dimensional oriented submanifold intersecting $\Sigma,\Sigma'$ in closed oriented curves $\gamma,\gamma'$ which (positively) bound disks $D\subset\Sigma,D'\subset\Sigma'$. Denote by $a,a'$ the symplectic areas of these disks. They are invariants of the relative symplectic fibration and do not change under pullback by diffeomorphisms. In general, they need not be equal.
Now Theorem 3.5.1 of your thesis contains the statement that, after standardisation near the sections, these areas are equal (they correspond there to the areas of the lower hemispheres in $S_\infty,S_0$ with respect to the same symplectic form $\widetilde\omega$). But i do not see any step in your proof where the original areas $a,a'$ could have changed: The only step that is not pullback by diffeomorphisms is Proposition 3.3.8, but there the leaves are deformed without changing their intersections with the sections $\Sigma,\sigma'$ and so the $a,a'$ will remain the same.
I tried to track down a mistake in your proof, and it seems to be in the proof of Lemma 3.3.10 where you claim that $\pi=p_1$ on a small neighbourhood $S^2\times U_\infty$ of $S_\infty$.
As i understand, the condition $a=a'$ is crucially used in the proof of Proposition 4.2.9. It seems that wihthout this condition the holonomy around a closed curve $\delta$ in the base may not be Hamiltonian. In turn, the Hamilton property is used in the construction of the Hamiltonian $H$ in Section 4.4. Without it, the Hamiltonian may not be zero near the boundary of $\bar Q\times A$ and so we cannot extend it outside.
Do you agree with all this, or am i overlooking something?
If this is indeed false, do you see a way to remedy it?
\section{Note for Kai}
As far as I can see the new proof of Theorem~\ref{Tstand} is correct and has in my opinion the potential to reveal a lot of geometric insight. The reason why I think this is so is the following. Holonomy maps are not problematic in general as along as we can make to symplectic area they ``occupy'' (or where they take place) small (Inflation). But indeed it is true, cf. the proof of Proposition 4.2.9. that the difference between the symplectic area of horizontal discs is given by the Flux of the holonomy map. Your result now gives the opportunity to change freely the area of the disks and Flux provided $A(D)-A(D')=Flux(\phi)$. That way hopefully one can see how an obstruction to the existence of exotic tori may look like. I think of somethink like a holonomy with too much Flux as too violate the above equation. \end{comment}
\end{document}
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\begin{document}
\begin{frontmatter} \title{On the well-posedness of weakly hyperbolic equations with time dependent coefficients} \author[IC]{Claudia Garetto\thanksref{th:grant1}} \author[IC]{Michael Ruzhansky\thanksref{th:grant2}} \thanks[th:grant1]{The first
author was supported by the Imperial College Junior Research Fellowship.}
\thanks[th:grant2]{The second author was supported by the EPSRC Leadership Fellowship EP/G007233/1.} \address[IC]{Department of Mathematics,\\ Imperial College London,\\ 180 Queen's Gate, London SW7 2AZ, UK}
\begin{abstract} In this paper we analyse the Gevrey well-posedness of the Cauchy problem for weakly hyperbolic equations of general form with time-dependent coefficients. The results involve the order of lower order terms and the number of multiple roots. We also derive the corresponding well-posedness results in the space of Gevrey Beurling ultradistributions. \end{abstract} \begin{keyword} Hyperbolic equations, Gevrey spaces, ultradistributions \MSC 35G10\sep 35L30\sep 46F05 \end{keyword} \end{frontmatter}
\section{Introduction}
In this paper we study the well-posedness for weakly hyperbolic equations of higher orders of general form with time-dependent coefficients. Namely, we consider the Cauchy problem \begin{equation} \label{CP} \left\{ \begin{array}{cc} D^m_t u=\sum_{j=0}^{m-1} A_{m-j}(t,D_x)D_t^j u+f(t,x),&\quad (t,x)\in[0,T]\times\mb{R}^n,\\ D^{k-1}_t u(0,x)=g_{k}(x),&\quad k=1,...,m, \end{array} \right. \end{equation} where each $A_{m-j}(t,D_x)$ is a differential operator of order $m-j$ with continuous coefficients only depending on $t$. As usual, $D_t=\frac{1}{{\rm i}}\partial_t$ and $D_x=\frac{1}{{\rm i}}\partial_x$. More precisely, we can write equation \eqref{CP} as \begin{equation} \label{CP1}
D^m_t u=\sum_{j=0}^{m-1}\sum_{|\gamma|=m-j} a_{m-j,\gamma}(t)D_x^\gamma D_t^j u+
\sum_{|\gamma|+j\leq l} a_{m-j,\gamma}(t)D_x^\gamma D_t^j u+f(t,x), \end{equation} where $l$ is the order of lower order terms, $0\leq l\leq m-1$. Concerning the lower order terms, throughout the paper we will only assume that
$a_{m-j,\gamma}(t)\in C[0,T]$ for $|\gamma|+j\leq l$, and that $f\in C([0,T];G^s(\Rn))$ is continuous in $t$ and Gevrey in $x$ of order $s$ appearing in the formulation of the theorems below.
Weakly hyperbolic equations \eqref{CP}, \eqref{CP1} and their special cases have been extensively considered in the literature, see e.g. \cite{B, ColKi:02, ColKi:02-2, DK, I, KS}, to mention only very few, and references therein.
Let $A_{(m-j)}$ denote the principal part of the operator $A_{m-j}$ and let $\tau_k(t,\xi)$, $k=1,...,m$, be the roots of the characteristic equation \[ \tau^m=\sum_{j=0}^{m-1}A_{(m-j)}(t,\xi)\tau^j
\equiv\sum_{j=0}^{m-1}\sum_{|\gamma|=m-j}a_{m-j,\gamma}(t)\xi^\gamma\tau^j. \] We will analyse the following two cases:
\boxed{\bf Case 1} we assume that
\\[0.3cm] the roots $\tau_k(t,\xi)$, $k=1,...,m$, are real-valued and of H\"older class $C^\alpha$, $0<\alpha\le 1$ with respect to $t$; for any $t\in[0,T]$ they either coincide or are all distinct.
\boxed{\bf Case 2} there exists $r=2,...,m-1$ such that \begin{itemize} \item[(i)] the roots $\tau_k(t,\xi)$, $k=1,...,r$, are real-valued, of class $C^\alpha$, $0<\alpha\le 1$ with respect to $t$ and either coincide or are all distinct; \item[(ii)] the roots $\tau_k(t,\xi)$, $k=r+1,...,m$, are real-valued, of class $C^\beta$, $0<\beta\le 1$ with respect to $t$ and are all distinct. \end{itemize}
Before we proceed we note that in the case $\alpha=1$ or $\beta=1$, it is enough to assume Lipschitz regularity for the corresponding roots. This includes the case of weakly hyperbolic equations with smooth coefficients in which case the roots are Lipschitz by Bronshtein's theorem.
In the next section we give the Gevrey well-posedness results for the Cauchy problem \eqref{CP} in Case 1 and Case 2, as well as in the strictly hyperbolic case formulated below in Case 3. In summary, our results will apply to all dimensions and will improve the known Gevrey indices in different settings. First we describe what is known for this problem.
Cauchy problems of such type have been studied in the Gevrey framework by Colombini and Kinoshita in \cite{ColKi:02} but only in the one dimensional case, i.e., $x\in\mb{R}$, and with $f\equiv 0$. In the present paper, we extend the result of \cite{ColKi:02} to any dimension $n\geq 1$, as well as improve the indices for the Gevrey well-posedness (see Remarks \ref{REM:case1} and \ref{REM:case2}). The idea of the proof in \cite{ColKi:02} is to reduce the Cauchy problem
\eqref{CP} to a differential system keeping track of all the derivatives of the solution $u$. The new unknown function contains also the lower order derivatives of $u$ and thus the size of the resulting system is much higher than $m$. Technically, it makes it hard to extend this method to higher dimensions. In this paper we use the pseudo-differential techniques of the reduction of \eqref{CP} to the system. This allows us to keep the size of the system to be equal to $m$ and works equally well in all dimensions. The subsequent estimates can be then improved for several terms in the proof of the energy inequality. Here, we also give results for inhomogeneous equations as well as discuss the well-posedness of the problem \eqref{CP} in the spaces of ultradistributions.
More generally, in dimensions $n\geq 1$, there are a number of results available concerning the problem \eqref{CP}. It was known since a long time (see Ivrii \cite{I} and references therein) that the Cauchy problem for any hyperbolic equation with sufficiently smooth coefficients is well-posed in Gevrey classes $G^s$ with $1\leq s< s_m$ for some $s_m>1$. Subsequently, it was shown by Bronshtein \cite{B} that the equation \eqref{CP} with characteristics of multiplicity $r\leq m$, with coefficients $C^\infty$ in $t$ (and also allowing $G^s$ in $x$), is well-posed in $G^s$ for $1\leq s<1+\frac{1}{r-1}$. This bound is in general sharp but can be improved in particular cases, such as, for example, Case 1 in Theorem \ref{THM:case1}, allowing lower regularity on the coefficients and taking into account the degree of lower order terms. Under the smoothness assumptions on the coefficients, we have $\alpha=\beta=1$ in our assumptions, so that the index $\frac{1}{r-1}$ corresponds to $\frac{\beta}{r-\beta}$ in Theorem \ref{THM:case2}.
When $m=2$, $l=1$ and $r=2$, Colombini, De Giorgi, Jannelli and Spagnolo (see \cite{CDS, CJS}) considered equations \eqref{CP}, \eqref{CP1} with $a_{1,1}(t)\equiv 0$ and $a_{0,2}(t)\in C^\delta[0,T]$, $\delta> 0$. They showed that the Cauchy problem \eqref{CP} is well-posed in $G^s$ provided that $1\leq s<1+\frac{\delta}{2}$. In our setting this is covered by the conditions of Case 1 with $\alpha=\frac{\delta}{2}$, so that the result above is included in Theorem \ref{THM:case1} giving the range $1\leq s<1+\alpha$. They also considered the case of $r=1$ when they proved the well-posedness in $G^s$ for $1\leq s<1+\frac{\delta}{1-\delta}$. In our setting this falls under the assumptions of Case 2 with $\alpha=\frac{\delta}{2}$ and $\beta=\delta$, so that their result is included in Theorem \ref{THM:case2} with the same range for $s$.
In \cite{KS}, Kinoshita and Spagnolo considered the Cauchy problem \eqref{CP} for operators with homogeneous symbols in one-dimension, i.e. assuming that $n=1$ and $a_{m-j,\gamma}(t)\equiv 0$ for $\gamma+j<m$ in \eqref{CP1}. Among other results for such equations, they showed that if $a_{m-j,\gamma}(t)\in C^2[0,T]$, $\gamma+j=m$, and the characteristic roots satisfy \begin{equation}\label{KS} \tau_i(t)^2+\tau_j(t)^2\leq M(\tau_i(t)-\tau_j(t))^2 \quad\textrm{ for } i\not=j, \end{equation} then the Cauchy problem \eqref{CP} is well-posed in the Gevrey space $G^s$ provided that $1\leq s<1+\frac{1}{m-1}$. In our setting, the condition $a_{m-j,\gamma}(t)\in C^2[0,T]$ corresponds to $\alpha=\frac{2}{r}$ and $\beta=1$. Thus, Theorem \ref{THM:case2} implies the well-posedness in the Gevrey space $G^s$ for $1\leq s<1+\min\{\frac{2}{r},\frac{1}{r-1}\}=1+\frac{1}{r-1}$, provided that the equation has multiplicities ($2\leq r\leq m$). In this sense the result of Theorem \ref{THM:case2} improves the $C^2$-coefficients result of \cite{KS}, also allowing any $n\geq 1$ and lower order terms, as well as removing the assumption \eqref{KS} on the roots. {We note that condition \eqref{KS} has been considered earlier in Colombini and Orr\`u in \cite{COr} to prove $C^\infty$ well-posedness in case of analytic coefficients.} Certain improvements have been also observed by Jannelli in \cite{Jan:09}.
We also present the corresponding results for the well-posedness in classes of Gevrey ultradistributions. It is by now well-known that the Cauchy problems for weakly hyperbolic equations even with smooth coefficients do not have to be in general well-posed in the space ${\mathcal D}'(\Rn)$ of distributions, see e.g. Colombini, Jannelli and Spagnolo \cite{CJS} and Colombini, Spagnolo \cite{CS}. In the subsequent paper we will analyse the propagation of singularities for weakly hyperbolic equations, and for such purpose it is necessary to have a framework in which the Cauchy problem would be well-posed. In fact, such a well-posedness result follows directly from the energy estimates that we will establish in the proofs of Theorem \ref{THM:case1} and Theorem \ref{THM:case2}. However, there is still one subtle matter of the definition of the corresponding space of Gevrey ultradistributions. Namely, we will show that one has to take the Beurling Gevrey ultradistributions rather than the Roumieu Gevrey ultradistributions to achieve such results. In general, in the absence of energy inequalities certain conclusions in spaces containing Schwartz distributions are also possible, but such questions will be treated elsewhere.
Furthermore, we complement the weakly hyperbolic analysis by giving the results for strictly hyperbolic equations with coefficients of low regularity. This corresponds to Case 2 above when we take $r=1$. As the equation is strictly hyperbolic, we do not have to distinguish between regularities of simple and multiple roots, so that we can take $\alpha=\beta$ in this case. To summarise, we consider
\boxed{\bf Case 3} we assume that
\\[0.3cm] the roots $\tau_k(t,\xi)$, $k=1,...,m$, are real-valued and of H\"older class $C^\beta$, $0<\beta\le 1$ with respect to $t$; for any $t\in[0,T]$ they are all distinct.
The proof of the corresponding statements will follow by taking the proof of Case 2 and putting $r=1$ and $\alpha=\beta$ at the end.
Finally we note that if the operator in \eqref{CP} is strictly hyperbolic and coefficients are more regular, much more is known. For a detailed analysis of large-time asymptotics of equations \eqref{CP}, \eqref{CP1}, for constant coefficients, we refer to Ruzhansky and Smith \cite{RS}. Here we note that althought \eqref{CP} may be strictly hyperbolic, multiplicities of the full equation (together with lower order terms) may still occur for small frequencies due to the presence of low order terms. Equations with $C^1$-regularity of the coefficients with respect to time have been treated in Matsuyama and Ruzhansky \cite{MR}, while systems with oscillations and more regularity have been analysed in Ruzhansky and Wirth \cite{RW}.
In the sequel, we denote $\langle\xi\rangle=(1+|\xi|^{2})^{1/2}.$
The authors thank Professor T. Kinoshita for useful discussions, and the Daiwa foundation for support.
\numberwithin{equation}{section}
\section{Main results}
From the fact that each $A_{(m-j)}(t,\xi)$ is a polynomial homogeneous of degree $m-j$ in $\xi$ it follows that the roots $\tau_k(t,\xi)$ are positively homogeneous of degree $1$ in $\xi$. Combining this fact with the H\"older regularity it follows in Case 1 that there exists a constant $c>0$ such that \begin{equation} \label{hyp_roots}
|\tau_k(t,\xi)-\tau_k(s,\xi)|\le c|\xi||t-s|^\alpha \end{equation} for $k=1,\ldots,m$, for all $\xi\neq 0$ and $t,s\in[0,T]$.
Throughout the paper, without loss of generality, by relabeling the roots, we can always arrange that they are ordered, so that we will assume that \begin{equation} \label{hyp_roots_2} \tau_1(t,\xi)\le\tau_2(t,\xi)\le\cdots\le\tau_m(t,\xi), \end{equation} for all $t$ \and $\xi$. The index for the H\"older regularity is preserved under such a relabelling. More precisely, in Case 2 we have that \eqref{hyp_roots} is true with exponent $\beta$ when $r+1\le k\le m$ and \begin{multline} \label{hyp_roots_3} \tau_1(t,\xi)\le\tau_2(t,\xi)\le\cdots\le\tau_r(t,\xi)<\tau_{r+1}(t,\xi) \\ <\tau_{r+2}(t,\xi)<\cdots<\tau_m(t,\xi), \end{multline} for all $t$ and $\xi\neq 0$. From the homogeneity in $\xi$ we also have that \begin{equation} \label{strict_hyp}
\tau_{k}(t,\xi)-\tau_{k-1}(t,\xi)\ge c_0|\xi| \end{equation} for some constant $c_0>0$ and all $k=r+1,...,m$, uniformly in $t\in[0,T]$ and $\xi\neq 0$. Throughout the paper we also assume that the roots which coincide have the following uniform property: there exists $c>0$ such that \begin{equation} \label{hyp_coincide}
|\tau_i(t,\xi)-\tau_j(t,\xi)|\le c|\tau_k(t,\xi)-\tau_{k-1}(t,\xi)| \end{equation} for all $1\le i,j,k\le r$, for all $t\in[0,T]$ and $\xi\in\mb{R}^n$. We note that although condition \eqref{hyp_coincide} was not explicitly stated in \cite{ColKi:02}, it is required for their proof also in the case $n=1$ (\cite{Kp}).
We first formulate the results in Gevrey spaces. Throughout the formulations, in inequalities for indices, we will adopt the convention that $\frac{1}{0}=+\infty$. We briefly recall the definition of the space $G^s(\Rn)$, the space of (Roumieu) Gevrey functions. We denote $\mb{N}_0=\mb{N}\cup\{0\}$.
\begin{defn}\label{def_gevrey} Let $s\geq 1$. We say that $f\in C^\infty(\mb{R}^n)$ belongs to the Gevrey class $G^s(\mb{R}^n)$ if for every compact set $K\subset\mb{R}^n$ there exists a constant $C>0$ such that for all $\alpha\in\mb{N}_0^n$ we have the estimate \[
\sup_{x\in K}|\partial^\alpha f(x)|\le C^{|\alpha|+1}(\alpha!)^s. \] \end{defn} We recall that $G^1(\mb{R}^n)$ is the space of analytic functions and that $G^s(\mb{R}^n)\subseteq G^\sigma(\mb{R}^n)$ if $s\le \sigma$. For $s>1$, let
$G^s_0(\mb{R}^n)$ be the space of compactly supported Gevrey functions of order $s$. In the paper we make use of the following Fourier characterisation (see \cite[Theorem 1.6.1]{Rodino:93}), where $\lara{\xi}=(1+|\xi|^2)^{\frac{1}{2}}$. \begin{prop} \label{prop_Fourier} \leavevmode \begin{itemize} \item[(i)] Let $u\in G^s_0(\mb{R}^n)$. Then, there exist constants $c>0$ and $\delta>0$ such that \begin{equation} \label{fou_gey_1}
|\widehat{u}(\xi)|\le c\,\mathrm{e}^{-\delta\lara{\xi}^{\frac{1}{s}}} \end{equation} for all $\xi\in\mb{R}^n$. \item[(ii)] Let $u\in\mathcal{S}'(\mb{R}^n)$. If there exist constants $c>0$ and $\delta>0$ such that \eqref{fou_gey_1} holds then $u\in G^s(\mb{R}^n)$.
\end{itemize} \end{prop}
We now formulate the result for Case 1.
\begin{thm}\label{THM:case1} Let $T>0$ and $0\leq l\leq m-1$. Assume the conditions of Case 1. Then for any $g_k(x)\in G^s(\Rn)$ ($k=1,\ldots,m$), the Cauchy problem \eqref{CP} has a unique global solution $u\in C^m([0,T]; G^s(\Rn))$, provided that \begin{equation}\label{EQ:case1-s} 1\leq s<1+\min\left\{\alpha,\frac{m-l}{l}\right\}. \end{equation} \end{thm}
\begin{rem}\label{REM:case1} In \cite{ColKi:02}, the authors proved that in one dimension $n=1$ and for $f\equiv 0$, the well-posedness in Theorem \ref{THM:case1} holds provided that $1\leq s < 1+\min\left\{\alpha,\frac{m-l}{m-1}\right\}$. Theorem \ref{THM:case1} improves this result by increasing the second factor under the minimum, as well as gives the result for any dimensions and non-zero $f$. \end{rem}
If we observe that $\alpha\leq \frac{m-l}{l}$ is equivalent to $l\leq \frac{m}{\alpha+1}$, we get \begin{cor}\label{lot} Under conditions of Case 1, if the order of lower order terms satisfies $l\leq \frac{m}{2}$, then the well-posedness in Theorem \ref{THM:case1} holds for $1\leq s<1+\alpha$. More precisely, if we assume that $l\leq \frac{m}{\alpha+1}$, then the well-posedness in Theorem \ref{THM:case1} holds provided that $1\leq s<1+\alpha$. \end{cor}
Under assumptions of Case 2, if there are simple roots, sometimes the index in \eqref{EQ:case1-s} can be improved. However, this should not be generally expected as a multiplication of a weakly hyperbolic polynomial by a strictly hyperbolic one should not, in general, improve the well-posedness of the Cauchy problem.
\begin{thm}\label{THM:case2} Let $T>0$, $2\leq r\leq m-1$ and $0\leq l\leq m-1$. Assume the conditions of Case 2. Then for any $g_k(x)\in G^s(\Rn)$ ($k=1,\ldots,m$), the Cauchy problem \eqref{CP} has a unique global solution $u\in C^m([0,T]; G^s(\Rn))$, provided that \begin{equation}\label{EQ:case2-s} 1\leq s<1+\min\left\{\alpha,\frac{\beta}{r-\beta}\right\}. \end{equation} \end{thm}
\begin{rem}\label{REM:case2} In \cite{ColKi:02}, the authors proved that in one dimension $n=1$ and for $f\equiv 0$, the well-posedness in Theorem \ref{THM:case2} holds provided that $1\leq s < 1+\min\left\{\alpha,\frac{\beta}{r-\beta},\frac{m-l}{r-1}\right\}$. Theorem \ref{THM:case2} improves this result by removing the last term under the minimum, as well as applies to all dimensions and non-zero $f$. \end{rem}
We now give a remark about the strictly hyperbolic equations covered by Case 3. We recall that in this case we take $\alpha=\beta$.
\begin{rem}\label{REM:r1statemement} Under the conditions of Case 3, the conclusion of Theorem \ref{THM:case2} holds provided that $$1\le s<1+\frac{\beta}{1-\beta}.$$ See Remark \ref{REM:r1} for the proof. \end{rem}
The result of Theorem \ref{THM:case2} is better than that in Theorem \ref{THM:case1} if there are few multiple roots, or if the order of lower order terms is sufficiently high. In particular and more precisely, it can be easily checked that $\frac{\beta}{r-\beta}\geq \frac{m-l}{l}$ if $r\leq \frac{\beta m}{m-l}$ (where $r$ is the number of multiple roots), or if $l\geq \frac{(r-\beta)m}{r}$ (where $l$ is the order of lower order terms).
It is interesting to observe the implications of Theorem \ref{THM:case2} for equations with at most double roots ($r=2$) in the Cauchy problem \eqref{CP}, \eqref{CP1}, where the coefficients
$a_{m-j,\gamma}$ belong to $C^\delta$ with $0<\delta\le 1$ and $|\gamma|+j=m$. In this case we have $\alpha=\frac{\delta}{2}$ and $\beta=\delta$, and since $\frac{\delta}{2}<\frac{\delta}{2-\delta}$, we obtain
\begin{cor}
Assume that in Case 2, we have $r=2$ (i.e. double roots) and that for $|\gamma|+j=m$ the coefficients satisfy $a_{m-j,\gamma}\in C^\delta[0,T]$, $0< \delta\leq 1$. Then the Cauchy problem \eqref{CP} is well-posed in $C^m([0,T]; G^s(\Rn))$ for $1\leq s<1+\frac{\delta}{2}$. \end{cor}
Finally we observe that all the arguments in the proofs remain valid if the equation \eqref{CP} is pseudo-differential in the $x$-variable:
\begin{rem}\label{REM:local} The results of Theorems \ref{THM:case1} and \ref{THM:case2} apply in the same way for operators in \eqref{CP} that are classical pseudo-differential in $D_x$, if we take $g_k\in G^s_0(\Rn)$, $k=1,\ldots,m$, to be compactly supported.
Also, if the equality in \eqref{EQ:case1-s} or \eqref{EQ:case2-s} is attained, the local well-posedness in Theorems \ref{THM:case1} and \ref{THM:case2} and in the first part of this remark still holds. \end{rem}
Before proceeding with the ultradistributions and with the proof of the theorems above we give some examples. For more examples in one dimension we refer to \cite{ColKi:02}, with the corresponding improvement in indices given by Remarks \ref{REM:case1} and \ref{REM:case2}.
{\bf Example 1.} Let us consider the equation \[ D_t^3u=-a(t)D_t \Delta_x u+L(t,D_x,D_t)u, \]
where $a(t)\ge 0$ belongs to $C^{2\alpha}([0,T])$, $\Delta_x=\partial_{x_1}^2+...+\partial_{x_n}^2$ and $L$ is a differential operator of order $l\le 2$. The corresponding principal symbol is $\tau^3-a(t)|\xi|^2\tau$ with roots
$\tau_1=-\sqrt{a(t)}|\xi|$, $\tau_2=0$ and $\tau_3=\sqrt{a(t)}|\xi|$. According to Theorem \ref{THM:case1} given initial data in $G^s(\mb{R}^n)$ the corresponding Cauchy problem has a unique solution $u\in C^3([0,T]; G^s(\Rn))$ with \[ 1\le s <1+\min\left\{\alpha,\frac{3-l}{l}\right\}. \] Note that the same well-posedness result holds for \[ D_t^3u=\sum_{i=1}^n b_i(t)D_{x_i}D_t^2+L(t,D_x,D_t)u, \] when we assume that the coefficients $b_i$ are real-valued of class $C^{\alpha}$ and the multiplicity is at a point $t_0\in[0,T]$ such that $b_i(t_0)=0$ for all $i=1,...,n$. We can apply Theorem \ref{THM:case1} and as an example of the reordering of the roots in the proof, we relabel the roots of the characteristic polynomial $\tau^3-\sum_i b_i(t)\xi_i\tau^2$ as \[ \tau_1(t,\xi)=\min\left\{\sum_i b_i(t)\xi_i,0\right\},\, \tau_2=0,\, \tau_3(t,\xi)=\max\left\{\sum_i b_i(t)\xi_i,0\right\}. \]
{\bf Example 2.} We study the Cauchy problem \[ D_t^4 u=-(a(t)+b(t))D_t^2\Delta u-a(t)b(t)\Delta^2u,\qquad D_t^j u(0,x)=g_j(x),\, j=0,1,2,3, \]
where we take $a\in C^{2\alpha}[0,T]$, $b\in C^{\beta}[0,T]$ with $a(t)\ge 0$ and $b(t)-a(t)\ge\delta>0$. The roots of the characteristic polynomial are $\tau_1(t,\xi)=-\sqrt{a(t)}|\xi|$, $\tau_2(t,\xi)=+\sqrt{a(t)}|\xi|$, $\tau_3(t,\xi)=-\sqrt{b(t)}|\xi|$ and $\tau_4(t,\xi)=+\sqrt{b(t)}|\xi|$. Hence, $r=2$ and from Theorem \ref{THM:case2} we have well-posedness in $C^4([0,T]; G^s(\Rn))$ with \[ 1\leq s<1+\min\left\{\alpha,\frac{\beta}{2-\beta}\right\}. \] Equations of this type were considered by Colombini and Kinoshita in \cite{ColKi:02-2}, where the well-posedness was proved for $1\leq s<1+\min\{\alpha,\frac{\beta}{2}\}.$ Thus, Theorem \ref{THM:case2} gives an improvement of this result since $\frac{\beta}{2-\beta}\geq \frac{\beta}{2}$. It also extends the one-dimensional version of this equation considered in \cite[Example 3]{ColKi:02}.
Before stating the ultradistributional versions of Theorems \ref{THM:case1} and \ref{THM:case2} we recall a few more facts concerning Gevrey classes and ultradistributions. For more details see Komatsu \cite{K}, or Rodino \cite[Section 1.5]{Rodino:93} for a partial treatment. We first recall the Beurling Gevrey functions.
\begin{defn}\label{def_gevrey2} Let $s\geq 1$. We say that $f\in C^\infty(\mb{R}^n)$ belongs to the Beurling Gevrey class $G^{(s)}(\mb{R}^n)$ if for every compact set $K\subset\mb{R}^n$ and for every constant $A>0$ there exists a constant $C_{A,K}>0$ such that for all $\alpha\in\mb{N}_0^n$ we have the estimate \[
\sup_{x\in K}|\partial^\alpha f(x)|\le C_{A,K} A^{|\alpha|}(\alpha!)^s. \] \end{defn} Analogously to Proposition \ref{prop_Fourier}, we have the following Fourier characterisation, where $G^{(s)}_0(\mb{R}^n)$ denotes the space of compactly supported Beurling Gevrey functions. \begin{prop} \label{prop_Fourie2r} \leavevmode \begin{itemize} \item[(i)] Let $u\in G^{(s)}_0(\mb{R}^n)$. Then, for any $\delta>0$ there exists $C_\delta>0$ such that \begin{equation} \label{fou_gey_2}
|\widehat{u}(\xi)|\le C_\delta\,\mathrm{e}^{-\delta\lara{\xi}^{\frac{1}{s}}} \end{equation} for all $\xi\in\Rn$. \item[(ii)] Let $u\in\mathcal{S}'(\mb{R}^n)$. If for any $\delta>0$ there exists $C_\delta>0$ such that \eqref{fou_gey_2} holds then $u\in G^{(s)}(\mb{R}^n)$. \end{itemize} \end{prop} For $s>1$, the spaces $G^s_0(\Rn)$ and $G^{(s)}_0(\Rn)$ of compactly supported functions can be equipped with natural seminormed topologies, and by $\mathcal{D}'_s(\mb{R}^n)$ and $\mathcal{D}'_{(s)}(\mb{R}^n)$ we denote the spaces of linear continuous functionals on them, respectively. We use the expressions Gevrey Roumieu ultradistributions and Gevrey Beurling ultradistributions for the elements of $\mathcal{D}'_s(\mb{R}^n)$ and $\mathcal{D}'_{(s)}(\mb{R}^n)$, respectively. Let $\mathcal{E}'_s(\mb{R}^n)$ and $\mathcal{E}'_{(s)}(\mb{R}^n)$ be the topological duals of $G^s(\Rn)$ and $G^{(s)}(\Rn)$, respectively. By duality we have $\mathcal{E}'_s(\mb{R}^n)\subset \mathcal{D}'_s(\mb{R}^n)$ and $\mathcal{E}'_{(s)}(\mb{R}^n)\subset \mathcal{D}'_{(s)}(\mb{R}^n)$. We also have $\mathcal{D}'(\mb{R}^n)\subset \mathcal{D}'_s(\mb{R}^n)\subset \mathcal{D}'_{(s)}(\mb{R}^n)$. The Fourier transform of the functionals of $\mathcal{E}'_s(\mb{R}^n)$ and $\mathcal{E}'_{(s)}(\mb{R}^n)$ can be defined in the same way as for the distributions. Then, the following characterisation holds (see \cite{K, LO:09, Rodino:93}):
\begin{prop} \label{prop_ud} A real analytic functional $v$ belongs to $\mathcal{E}'_s(\mb{R}^n)$ if and only if for any $\delta>0$ there exists $C_\delta>0$ such that \[
|\widehat{v}(\xi)|\le C_\delta\,\mathrm{e}^{\delta\lara{\xi}^{\frac{1}{s}}} \] for all $\xi\in\Rn$. Similarly, $v\in \mathcal{E}'_{(s)}(\mb{R}^n)$ if and only if there exist $\delta>0$ and $C>0$ such that \[
|\widehat{v}(\xi)|\le C\,\mathrm{e}^{\delta\lara{\xi}^{\frac{1}{s}}} \] for all $\xi\in\Rn$. \end{prop} We are now ready to state the ultradistributional versions of Theorem \ref{THM:case1} and Theorem \ref{THM:case2}.
\begin{thm}\label{THM:case1u} Let $T>0$ and $0\leq l\leq m-1$. Assume the conditions of Case 1. Then for any $g_k\in \mathcal{E}'_{(s)}(\mb{R}^n)$ ($k=1,\ldots,m$), the Cauchy problem \eqref{CP} has a unique global solution $u\in C^m([0,T];\mathcal{D}'_{(s)}(\mb{R}^n))$, provided that $$ 1\leq s\leq 1+\min\left\{\alpha,\frac{m-l}{l}\right\}. $$ \end{thm} The situation in Case 2 is as follows:
\begin{thm}\label{THM:case2u} Let $T>0$, $2\leq r\leq m-1$ and $0\leq l\leq m-1$. Assume the conditions of Case 2. Then for any $g_k\in \mathcal{E}'_{(s)}(\mb{R}^n)$ ($k=1,\ldots,m$), the Cauchy problem \eqref{CP} has a unique global solution $u\in C^m([0,T]; \mathcal{D}'_{(s)}(\mb{R}^n))$, provided that $$ 1\leq s\leq 1+\min\left\{\alpha,\frac{\beta}{r-\beta}\right\}. $$ \end{thm} It is interesting to note the non-strict inequalities for $s$ in Theorems \ref{THM:case1u} and \ref{THM:case2u} as opposed to strict inequalities for $s$ in Theorems \ref{THM:case1} and \ref{THM:case2}, see also Remark \ref{REM:local}.
Finally, we make a remark about the strictly hyperbolic case with low regularity coefficients. \begin{rem}\label{REM:r1statemement-u} Under the conditions of Case 3, the conclusion of Theorem \ref{THM:case2u} holds provided that $$1\le s<1+\frac{\beta}{1-\beta}.$$ See Remark \ref{REM:r1} for the argument. \end{rem}
\section{Reduction to first order system and preliminary analysis}
We now perform a reduction to a first order system as in \cite{Taylor:81}. Let $\lara{D_x}$ be the pseudo-differential operator with symbol $\lara{\xi}$. The transformation \[ u_k=D_t^{k-1}\lara{D_x}^{m-k}u, \] with $k=1,...,m$, makes the Cauchy problem \eqref{CP} equivalent to the following system \begin{equation} \label{syst_Taylor} D_t\left(
\begin{array}{c}
u_1 \\
\cdot \\
\cdot\\
u_m \\
\end{array}
\right) = \left(
\begin{array}{ccccc}
0 & \lara{D_x} & 0 & \dots & 0\\
0 & 0 & \lara{D_x} & \dots & 0 \\
\dots & \dots & \dots & \dots & \lara{D_x} \\
b_1 & b_2 & \dots & \dots & b_m \\
\end{array}
\right)
\left(\begin{array}{c}
u_1 \\
\cdot \\
\cdot\\
u_m \\
\end{array}
\right)
+ \left(\begin{array}{c}
0 \\
0 \\
\cdot\\
f \\
\end{array}
\right), \end{equation} where \[ b_j=A_{m-j+1}(t,D_x)\lara{D_x}^{j-m}, \] with initial condition \begin{equation} \label{ic_Taylor}
u_k|_{t=0}=\lara{D_x}^{m-k}g_k,\qquad k=1,...,m. \end{equation} The matrix in \eqref{syst_Taylor} can be written as $A+B$ with \[ A=\left(
\begin{array}{ccccc}
0 & \lara{D_x} & 0 & \dots & 0\\
0 & 0 & \lara{D_x} & \dots & 0 \\
\dots & \dots & \dots & \dots & \lara{D_x} \\
b_{(1)} & b_{(2)} & \dots & \dots & b_{(m)} \\
\end{array}
\right), \] where $b_{(j)}=A_{(m-j+1)}(t,D_x)\lara{D_x}^{j-m}$ and \[ B=\left(
\begin{array}{ccccc}
0 & 0 & 0 & \dots & 0\\
0 & 0 & 0& \dots & 0 \\
\dots & \dots & \dots & \dots & 0 \\
b_1-b_{(1)} & b_2-b_{(2)} & \dots & \dots & b_m-b_{(m)} \\
\end{array}
\right). \] It is clear that the eigenvalues of the symbol matrix $A(t,\xi)$ are the roots $\tau_j(t,\xi)$, $j=1,...,m$. By Fourier transforming both sides of \eqref{syst_Taylor} we obtain the system \begin{equation} \label{system_new} \begin{split} D_t V&=A(t,\xi)V+B(t,\xi)V+\widehat{F}(t,\xi),\\
V|_{t=0}(\xi)&=V_0(\xi), \end{split} \end{equation} where $V$ is the $m$-column with entries $v_k=\widehat{u}_k$, $V_0$ is the $m$-column with entries $v_{0,k}=\lara{\xi}^{m-k}\widehat{g}_k$ and \begin{multline*} A(t,\xi)=\left(
\begin{array}{ccccc}
0 & \lara{\xi} & 0 & \dots & 0\\
0 & 0 & \lara{\xi} & \dots & 0 \\
\dots & \dots & \dots & \dots & \lara{\xi} \\
b_{(1)}(t,\xi) & b_{(2)}(t,\xi) & \dots & \dots & b_{(m)}(t,\xi) \\
\end{array}
\right),\\[0.3cm]
b_{(j)}(t,\xi)=A_{(m-j+1)}(t,\xi)\lara{\xi}^{j-m}, \end{multline*} \begin{multline*} B(t,\xi)=\left(
\begin{array}{ccccc}
0 & 0 & 0 & \dots & 0\\
0 & 0 & 0& \dots & 0 \\
\dots & \dots & \dots & \dots & 0 \\
(b_1-b_{(1)})(t,\xi) & \dots & \dots & \dots & (b_m-b_{(m)})(t,\xi) \\
\end{array}
\right),\\[0.3cm] (b_j-b_{(j)})(t,\xi)=(A_{m-j+1}-A_{(m-j+1)})(t,\xi)\lara{\xi}^{j-m}, \end{multline*} \[ \widehat{F}(t,\xi)=\left(\begin{array}{c}
0 \\
0 \\
\vdots\\
\widehat{f}(t,\cdot)(\xi) \\
\end{array}
\right). \] From now on we will concentrate on the system \eqref{system_new}. We collect some preliminary results which will be crucial in the next section. Detailed proofs can be obtained by easily adapting the Lemmas 1, 2, 4 and 5 in \cite[Section 2]{ColKi:02} to our situation. \begin{prop} \label{prop_prelim} Let $\lambda_i\in\mb{R}$, $i=1,...,m$, be distinct and let \begin{equation} \label{def_H} H=\left(
\begin{array}{ccccc}
1 & 1 & 1 & \dots & 1\\
\lambda_1\lara{\xi}^{-1} & \lambda_2\lara{\xi}^{-1} & \lambda_3\lara{\xi}^{-1} & \dots & \lambda_m\lara{\xi}^{-1} \\
\lambda^2_1\lara{\xi}^{-2} & \lambda^2_2\lara{\xi}^{-2} & \lambda^2_3\lara{\xi}^{-2} & \dots & \lambda^2_m\lara{\xi}^{-2} \\
\dots & \dots & \dots & \dots & \dots\\
\lambda^{m-1}_1\lara{\xi}^{-m+1} & \lambda^{m-1}_2\lara{\xi}^{-m+1} & \lambda^{m-1}_3\lara{\xi}^{-m+1} & \dots & \lambda^{m-1}_m\lara{\xi}^{-m+1}\\
\end{array}
\right). \end{equation} Then we have the following properties: \begin{itemize} \item[(i)] $\det H=\lara{\xi}^{-\frac{(m-1)m}{2}}\prod_{1\le j<i\le m}(\lambda_i-\lambda_j)$ and \[ \det (A(t,\xi)-\tau I)=(-1)^m(\tau^m-\sum_{j=0}^{m-1}A_{(m-j)}(t,\xi)\tau^{j}); \] \item[(ii)] the matrix $H^{-1}$ has entries $h_{pq}$ as follows: \[ h_{pq}= (-1)^{q-1}\lara{\xi}^{q-1}\sum_{S^{(m)}_p(m-q)}\lambda_{i_1}\dots\lambda_{i_{m-q}} \biggl(\prod_{i=1,i\neq p}^m (\lambda_i-\lambda_p)\biggr)^{-1}, \] for $1\le q\le m-1$, and \[ h_{pq}=(-1)^{m-1}\lara{\xi}^{m-1}\biggl(\prod_{i=1,i\neq p}^m (\lambda_i-\lambda_p)\biggr)^{-1}, \] for $q=m$, where $$S^{(a)}_b(c)=\{(i_1,...,i_c)\in \mb{N}^c; 1\le i_1<\cdots <i_c\le a, i_k\neq b, 1\le k\le c\}.$$ \item[(iii)] the matrix $H^{-1}A(t,\xi)H$ has entries \[ c_{pq}=(\tau_q-\lambda_q)\frac{\prod_{i=1, i\neq q}^m (\tau_i-\lambda_q)}{\prod_{i=1, i\neq p}^m (\lambda_i-\lambda_p)} \] when $p\neq q$. \item[(iv)] the matrix $H^{-1}B(t,\xi)H$ has entries \[ d_{pq}=(-1)^{m-1}\biggl(\prod_{i=1,i\neq p}^m (\lambda_i-\lambda_p)\biggr)^{-1}g(\lambda_q), \] where $g(\tau)=\sum_{j=0}^{m-1}(A_{m-j}-A_{(m-j)})(t,\xi)\tau^{j}$. \item[(v)] Assume that $\lambda_j\in C^1(\mb{R}_t)$, $j=1,...,m$. The matrix $H^{-1}\frac{d}{dt} H$ has entries \[ e_{pq}=\begin{cases} -\lambda'_p(t)\sum_{i=1, i\neq p}^m\frac{1}{\lambda_i(t)-\lambda_p(t)},& p=q,\\[0.3cm] -\lambda'_q(t)\frac{\prod_{i=1, i\neq p,q}^{m}(\lambda_i(t)-\lambda_q(t))}{\prod_{i=1, i\neq p}^{m}(\lambda_i(t)-\lambda_p(t))},& p\neq q. \end{cases} \] \end{itemize} \end{prop} \begin{pf} We only prove assertions (iii) and (iv) and (v).
(iii) Let $w(\tau)=\sum_{j=0}^{m-1}A_{(m-j)}(t,\xi)\tau^{j}$. Arguing as in the proof of Lemma 5 in \cite{ColKi:02} we have that \begin{multline*} (c_{pq})_{1\le p,q\le m}= \\[0.3cm] H^{-1}\left(
\begin{array}{ccccc}
\lambda_1 & \lambda_2 & \lambda_3 & \dots & \lambda_m\\
\lambda_1^2\lara{\xi}^{-1} & \lambda^2_2\lara{\xi}^{-1} & \lambda^2_3\lara{\xi}^{-1} & \dots & \lambda^2_m\lara{\xi}^{-1} \\
\lambda^3_1\lara{\xi}^{-2} & \lambda^3_2\lara{\xi}^{-2} & \lambda^3_3\lara{\xi}^{-2} & \dots & \lambda^3_m\lara{\xi}^{-2} \\
\dots & \dots & \dots & \dots & \dots \\
w(\lambda_1)\lara{\xi}^{-m+1} & w(\lambda_2)\lara{\xi}^{-m+1} & w(\lambda_3)\lara{\xi}^{-m+1} & \dots & w(\lambda_m)\lara{\xi}^{-m+1}\\
\end{array}
\right). \end{multline*} Assertion (ii) yields \begin{multline*} c_{pq}=\sum_{r=1}^{m-1}h_{pr}\lambda_q^r\lara{\xi}^{-r+1}+h_{pm}\lara{\xi}^{-m+1}f(\lambda_q)\\ =\sum_{r=1}^{m-1}(-1)^{r-1}\sum_{S^{(m)}_p(m-r)}\lambda_{i_1}\dots\lambda_{i_{m-r}}\biggl(\prod_{i=1,i\neq p}^m (\lambda_i-\lambda_p)\biggr)^{-1}\lambda_q^r\\ +(-1)^{m-1}\biggl(\prod_{i=1,i\neq p}^m (\lambda_i-\lambda_p)\biggr)^{-1}f(\lambda_q), \end{multline*} which coincides with formula (25) in \cite{ColKi:02}. The proof continues as in \cite[Lemma 5]{ColKi:02}.
(iv) Let $g(\tau)=\sum_{j=0}^{m-1}(A_{m-j}-A_{(m-j)})(t,\xi)\tau^{j}$. The matrix $H^{-1}B(t,\xi)H$ can be written as \begin{multline*} (d_{pq})_{1\le p,q\le m}= \\[0.3cm] H^{-1}\left(
\begin{array}{ccccc}
0 & 0 & 0 & \dots & 0\\
0 & 0 & 0 & \dots & 0\\
0 & 0 & 0 & \dots & 0 \\
\dots & \dots & \dots & \dots & \dots \\
g(\lambda_1)\lara{\xi}^{-m+1} & g(\lambda_2)\lara{\xi}^{-m+1} & g(\lambda_3)\lara{\xi}^{-m+1} & \dots & g(\lambda_m)\lara{\xi}^{-m+1}\\
\end{array}
\right). \end{multline*} From (ii) we conclude that \begin{multline*} d_{pq}=(-1)^{m-1}\lara{\xi}^{m-1}\biggl(\prod_{i=1,i\neq p}^m (\lambda_i-\lambda_p)\biggr)^{-1}\lara{\xi}^{-m+1}g(\lambda_q)\\ = (-1)^{m-1}\biggl(\prod_{i=1,i\neq p}^m (\lambda_i-\lambda_p)\biggr)^{-1}g(\lambda_q). \end{multline*}
(v) From the definition of $H$ we have that $H^{-1}\frac{d}{dt} H$ is the matrix \[ H^{-1}\left(
\begin{array}{ccccc}
0 & 0 & 0 & \dots & 0\\
\lambda'_1\lara{\xi}^{-1} & \lambda'_2\lara{\xi}^{-1}& \lambda'_3\lara{\xi}^{-1}& \dots & \lambda'_m\lara{\xi}^{-1}\\
(\lambda^2_1)'\lara{\xi}^{-2}& (\lambda^2_2)'\lara{\xi}^{-2} & (\lambda^2_3)'\lara{\xi}^{-2} & \dots & (\lambda^2_m)'\lara{\xi}^{-2} \\
\dots & \dots & \dots & \dots & \dots \\
(\lambda^{m-1}_1)'\lara{\xi}^{-m+1}& (\lambda^{m-1}_2)'\lara{\xi}^{-m+1} & (\lambda^{m-1}_3)'\lara{\xi}^{-m+1} & \dots & (\lambda^{m-1}_m)'\lara{\xi}^{-m+1}\\
\end{array}
\right). \] Hence, making use of the second assertion of this proposition we obtain \begin{multline*} e_{pq}=\sum_{r=2}^{m-1}h_{pr}(r-1)\lambda^{r-2}_q\lambda'_q\lara{\xi}^{-r+1}+h_{pm}(m-1)\lambda^{m-2}_q\lambda'_q \lara{\xi}^{-m+1}\\ =\sum_{r=2}^{m-1}(-1)^{r-1}\lara{\xi}^{r-1}\sum_{S^{(m)}_p(m-r)}\lambda_{i_1}\dots\lambda_{i_{m-r}}\biggl(\prod_{i=1,i\neq p}^m (\lambda_i-\lambda_p)\biggr)^{-1}\lambda^{r-2}_q\lambda'_q\lara{\xi}^{-r+1}\\ +(-1)^{m-1}\lara{\xi}^{m-1}\biggl(\prod_{i=1,i\neq p}^m (\lambda_i-\lambda_p)\biggr)^{-1}(m-1)\lambda^{m-2}_q\lambda'_q \lara{\xi}^{-m+1}\\ =\sum_{r=2}^{m-1}(-1)^{r-1}\sum_{S^{(m)}_p(m-r)}\lambda_{i_1}\dots\lambda_{i_{m-r}}\biggl(\prod_{i=1,i\neq p}^m (\lambda_i-\lambda_p)\biggr)^{-1}\lambda^{r-2}_q\lambda'_q\\ +(-1)^{m-1}\biggl(\prod_{i=1,i\neq p}^m (\lambda_i-\lambda_p)\biggr)^{-1}(m-1)\lambda^{m-2}_q\lambda'_q. \end{multline*} This is the expression for $b_{pq}$ in the proof of Lemma 4 in \cite{ColKi:02}. The proof continues as in \cite[Lemma 4]{ColKi:02}. \end{pf} We now proceed to analyse the roots $\tau_j$. We perform the natural regularisation and separation process, but it will be different under the assumptions of Case 1 or of Case 2. To simplify the notation, although the functions below will depend on $\varepsilon$, for brevity we will write $\lambda_j(t,\xi)$ for $\lambda_j(\varepsilon,t,\xi)$. \begin{prop} \label{prop_roots} Let $\varphi\in C^\infty_{c}(\mb{R})$, $\varphi\ge 0$ with $\int_\mb{R}\varphi(x)\, dx=1$. Under the assumptions of Case 1, let \begin{equation} \label{def_lambdaj} \lambda_j(t,\xi)=(\tau_j(\cdot,\xi)\ast\varphi_\varepsilon)(t)+j\varepsilon^\alpha\lara{\xi}, \end{equation} for $j=1,...,m$ and $\varphi_\varepsilon(s)=\varepsilon^{-1}\varphi(s/\varepsilon)$, $\varepsilon>0$. Then, there exists a constant $c>0$ such that \begin{itemize}
\item[(i)] $|\partial_t\lambda_j(t,\xi)|\le c\,\varepsilon^{\alpha-1}\lara{\xi}$,
\item[(ii)] $|\lambda_j(t,\xi)-\tau_j(t,\xi)|\le c\,\varepsilon^{\alpha}\lara{\xi}$, \item[(iii)] $\lambda_{j}(t,\xi)-\lambda_{i}(t,\xi)\ge \varepsilon^\alpha\lara{\xi}$ for $j>i$, \end{itemize} for all $t,s\in[0,T']$ with $T'<T$ and all $\xi\in\mb{R}^n$. \end{prop} \begin{pf} By definition of convolution, if $R$ is large enough, one has \begin{multline} \label{est_deriv}
|\partial_t\lambda_j(t,\xi)|=\varepsilon^{-1}\int_{-R}^R\tau_j(t-\varepsilon s)\varphi'(s)\, ds\\ =\varepsilon^{-1}\int_{-R}^R(\tau_j(t-\varepsilon s,\xi)-\tau_j(t,\xi))\varphi'(s)\, ds+\varepsilon^{-1}\int_{-R}^R \tau_j(t,\xi)\varphi'(s)\, ds, \end{multline} and, therefore, by \eqref{hyp_roots} we obtain
$|\partial_t\lambda_j(t,\xi)|\le c\varepsilon^{\alpha-1}\lara{\xi}$ for all $t,s\in[0,T']$ and $\xi\in\mb{R}^n$.
The second and third assertions follow immediately from the definition of $\lambda_j$, where we note that in view of \eqref{hyp_roots_2} and the fact that $\varphi\geq 0$ it is enough to observe (iii) for $j-i=1$. \end{pf} \begin{prop} \label{prop_roots_2} Let $\varphi\in C^\infty_{c}(\mb{R})$, $\varphi\ge 0$ with $\int_\mb{R}\varphi(x)\, dx=1$. Under the assumptions of Case 2, let \begin{equation} \label{def_lambdaj_2} \begin{split} \lambda_j(t,\xi)&=(\tau_j(\cdot,\xi)\ast\varphi_\varepsilon)(t)+j\varepsilon^\alpha\lara{\xi},\quad 1\le j\le r,\\ \lambda_j(t,\xi)&=(\tau_j(\cdot,\xi)\ast\varphi_\delta)(t),\quad\qquad r+1\le j\le m, \end{split} \end{equation} for $0<\delta,\varepsilon<1$. Then, there exist constants $c>0$, $c_0>0$ such that \begin{itemize}
\item[(i)] $|\partial_t\lambda_j(t,\xi)|\le c\,\varepsilon^{\alpha-1}\lara{\xi}$ for $j=1,...,r$,
\item[(ii)] $|\lambda_j(t,\xi)-\tau_j(t,\xi)|\le c\,\varepsilon^{\alpha}\lara{\xi}$ for $j=1,...,r$,
\item[(iii)] $\lambda_{j+1}(t,\xi)-\lambda_{j}(t,\xi)\ge \varepsilon^\alpha\lara{\xi}$ for $j=1,...,r-1$,
\item[(iv)] $|\partial_t\lambda_j(t,\xi)|\le c\,\delta^{\beta-1}\lara{\xi}$ for $j=r+1,...,m$,
\item[(v)] $|\lambda_{j}(t,\xi)-\tau_j(t,\xi)|\le c\,\delta^{\beta}\lara{\xi}$ for $j=r+1,...,m$, \item[(vi)] $\lambda_{j+1}(t,\xi)-\lambda_j(t,\xi)\ge c_0\lara{\xi}$ for $j=r,...,m-1$, for
$\varepsilon=\lara{\xi}^{-\gamma}$ with $\gamma\in(0,1)$, $\delta=\lara{\xi}^{-1}$ and $|\xi|$ large enough, \item[(vii)] $\lambda_{j}(t,\xi)-\lambda_i(t,\xi)\ge c_0\lara{\xi}$
for $j=r+1,...,m$, $i=1,...,r$, $\varepsilon=\lara{\xi}^{-\gamma}$ with $\gamma\in(0,1)$, $\delta=\lara{\xi}^{-1}$ and $|\xi|$ large enough, \end{itemize} hold for all $t,s\in[0,T']$ with $T'<T$. \end{prop} \begin{pf} The first three assertions are clear from Proposition \ref{prop_roots} and \eqref{hyp_roots_3}. Assertion (iv) can be proven as in \eqref{est_deriv}. Assertion (v) follows immediately from the $C^\beta$-property of the roots $\tau_j$ when $j=r+1,...,m$. We finally consider the difference $\lambda_{j+1}(t,\xi)-\lambda_j(t,\xi)$. If $j=r+1,...,m-1$ then from the bound from below \eqref{strict_hyp} we obtain the estimate \[ \lambda_{j+1}(t,\xi)-\lambda_j(t,\xi)\ge c_0\lara{\xi} \]
valid for $t\in[0,T']$ and $|\xi|$ large enough. It remains to consider $\lambda_{j+1}(t,\xi)-\lambda_j(t,\xi)$ when $j=r$. Making use of the definition in \eqref{def_lambdaj_2} we can write \begin{multline*} \lambda_{r+1}(t,\xi)-\lambda_r(t,\xi)=\int_\mb{R} \tau_{r+1}(t-\delta s,\xi)\varphi(s)\, ds -\int_\mb{R} \tau_{r}(t-\varepsilon s,\xi)\varphi(s)\, ds-r\varepsilon^\alpha\lara{\xi}\\ = \int_\mb{R} (\tau_{r+1}(t-\delta s,\xi)-\tau_{r+1}(t-\varepsilon s,\xi))\varphi(s)\, ds +\\
+\int_\mb{R} (\tau_{r+1}(t-\varepsilon s,\xi)-\tau_r(t-\varepsilon s,\xi))\varphi(s)\, ds -r\varepsilon^\alpha\lara{\xi}. \end{multline*} Hence, combining \eqref{strict_hyp} with \eqref{hyp_roots} we get \[
\lambda_{r+1}(t,\xi)-\lambda_r(t,\xi)\ge c_0|\xi|-c|\varepsilon-\delta|^\beta|\xi|-r\varepsilon^\alpha\lara{\xi}\ge c_0|\xi|-c|\varepsilon-\delta|^\beta|\xi|-r\varepsilon^\alpha\sqrt{2}|\xi|, \]
for $|\xi|\ge 1$. It follows that for \begin{equation} \label{radius_1}
|\varepsilon-\delta|^\beta\le \frac{c_0}{4c} \Leftrightarrow |\varepsilon-\delta|\le \big({\frac{c_0}{4c}}\big)^{\frac{1}{\beta}} \Leftrightarrow \lara{\xi}^{-\gamma}(1-\lara{\xi}^{-1+\gamma})\le \big({\frac{c_0}{4c}}\big)^{\frac{1}{\beta}} \end{equation} and \begin{equation} \label{radius_2} \varepsilon^\alpha\le \frac{c_0}{4\sqrt{2}r} \Leftrightarrow \varepsilon\le \big({\frac{c_0}{4\sqrt{2}r}}\big)^{\frac{1}{\alpha}} \Leftrightarrow \lara{\xi}^{-\gamma}\le \big({\frac{c_0}{4\sqrt{2}r}}\big)^{\frac{1}{\alpha}} \end{equation} one has \[ \lambda_{r+1}(t,\xi)-\lambda_r(t,\xi)\ge c'_0\lara{\xi}. \]
Assertion (vii) follows from (vi). \end{pf} In the sequel, with abuse of notation, we will still denote the smaller $T'$ in Propositions \ref{prop_roots} and \ref{prop_roots_2} by $T$. \begin{prop} \label{prop_roots_3} The property \eqref{hyp_coincide} holds for the $\lambda_j$'s as well, i.e., \begin{equation} \label{hyp_coincide_2}
|\lambda_i(t,\xi)-\lambda_j(t,\xi)|\le c|\lambda_k(t,\xi)-\lambda_{k-1}(t,\xi)| \end{equation} for all $1\le i,j,k\le r$, for all $t\in[0,T]$ and $\xi\in\mb{R}^n$. \end{prop} \begin{pf} Assume that $i>j$. Hence \[
|\lambda_i(t,\xi)-\lambda_j(t,\xi)|=(\tau_i(\cdot,\xi)-\tau_j(\cdot,\xi))\ast\varphi_\varepsilon(t)+ (i-j)\varepsilon^\alpha\lara{\xi} \] and \[
|\lambda_k(t,\xi)-\lambda_{k-1}(t,\xi)|=(\tau_{k}(\cdot,\xi)-\tau_{k-1}(\cdot,\xi))\ast\varphi_\varepsilon(t)+ \varepsilon^\alpha\lara{\xi}. \] From \eqref{hyp_coincide} and the fact that $\varphi\geq 0$ we get that \begin{multline*}
|\lambda_i(t,\xi)-\lambda_j(t,\xi)|\le c(\tau_{k}(\cdot,\xi)-\tau_{k-1} (\cdot,\xi))\ast\varphi_\varepsilon(t)+(i-j)\varepsilon^\alpha\lara{\xi}\\[0.2cm]
\le c'|\lambda_k(t,\xi)-\lambda_{k-1}(t,\xi)| \end{multline*} holds for all $t\in[0,T]$ and $\xi\in\mb{R}^n$. \end{pf}
\section{Proof in Case 1: Theorem \ref{THM:case1} and Theorem \ref{THM:case1u}}
We first prove Theorem \ref{THM:case1}. It is well-known that the problem \eqref{syst_Taylor}-\eqref{ic_Taylor} is well-posed when $s=1$, see e.g. \cite{Jan:84, Kaj:86}. Hence, we may assume $s>1$. In the case of Theorem \ref{THM:case1} we can also assume that the initial data have compact support. Since weakly hyperbolic equations have the finite speed of propagation property it follows that the solution $u$ is compactly supported in $x$ as well. This observation allows us to proceed with the reduction to a first order system of Section 3.
Let $H(t,\xi)$ be the matrix \eqref{def_H} with entries $\lambda_j(t,\xi)$ as in \eqref{def_lambdaj}. Observe that the approximated roots $\lambda_j$ are distinct for all $\varepsilon>0$. We look for a solution $V$ of the Cauchy problem \eqref{system_new} in the form \begin{equation} \label{VW} V(t,\xi)=\mathrm{e}^{-\rho(t)\lara{\xi}^{\frac{1}{s}}}(\det H)^{-1}HW, \end{equation} where $\rho\in C^1[0,T]$ will be determined in the sequel. By substitution in \eqref{system_new} we obtain \begin{multline*} \mathrm{e}^{-\rho(t)\lara{\xi}^{\frac{1}{s}}}(\det H)^{-1}HD_tW+\mathrm{e}^{-\rho(t)\lara{\xi}^{\frac{1}{s}}}{\rm i}\rho'(t)\lara{\xi}^{\frac{1}{s}}(\det H)^{-1}HW+\\ +{\rm i}\mathrm{e}^{-\rho(t)\lara{\xi}^{\frac{1}{s}}}\frac{\partial_t\det H}{(\det H)^2}HW +\mathrm{e}^{-\rho(t)\lara{\xi}^{\frac{1}{s}}}(\det H)^{-1}(D_tH)W\\ =\mathrm{e}^{-\rho(t)\lara{\xi}^{\frac{1}{s}}}(\det H)^{-1}(A+B)HW + \widehat{F}. \end{multline*} Multiplying both sides of the previous equation by $\mathrm{e}^{\rho(t)\lara{\xi}^{\frac{1}{s}}}(\det H)H^{-1}$ we get \begin{multline*} D_tW+{\rm i}\rho'(t)\lara{\xi}^{\frac{1}{s}}W+{\rm i}\frac{\partial_t\det H}{\det H}W + H^{-1}(D_t H)W= H^{-1}(A+B)HW+\\ +\mathrm{e}^{\rho(t)\lara{\xi}^{\frac{1}{s}}}(\det H) H^{-1}\widehat{F}. \end{multline*} Hence, \begin{multline} \label{energy}
\partial_t |W(t,\xi)|^2=2{\rm Re} (\partial_t W(t,\xi),W(t,\xi))\\
=2\rho'(t)\lara{\xi}^{\frac{1}{s}}|W(t,\xi)|^2+2\frac{\partial_t\det H}{\det H}|W(t,\xi)|^2-2 {\rm Re}(H^{-1}\partial_t HW,W)\\ -2{\rm Im} (H^{-1}AHW,W)-2{\rm Im} (H^{-1}BHW,W) \\ -2{\rm Im} (\mathrm{e}^{\rho(t)\lara{\xi}^{\frac{1}{s}}}(\det H) H^{-1}\widehat{F},W). \end{multline} We proceed by estimating \begin{enumerate} \item $\frac{\partial_t\det H}{\det H}$, \item $\Vert H^{-1}\partial_t H\Vert$, \item $\Vert H^{-1}AH-(H^{-1}AH)^\ast\Vert$, \item $\Vert H^{-1}BH-(H^{-1}BH)^\ast\Vert$. \end{enumerate} \subsection{Estimate of the first term} Proposition \ref{prop_prelim}(i) combined with Proposition \ref{prop_roots} yields the following estimate \begin{multline} \label{est_1}
\biggl|\frac{\partial_t\det H(t,\xi)}{\det H(t,\xi)}\biggr|=\biggl| \frac{\lara{\xi}^{-\frac{(m-1)m}{2}}\partial_t\prod_{1\le j<i\le m} (\lambda_i(t,\xi)-\lambda_j(t,\xi))}{\lara{\xi}^{-\frac{(m-1)m}{2}}
\prod_{1\le j<i\le m}(\lambda_i(t,\xi)-\lambda_j(t,\xi))}\biggr| \\
\le \sum_{1\le j<i\le m}\frac{|\partial_t\lambda_i(t,\xi)-
\partial_t\lambda_j(t,\xi)|}{|\lambda_i(t,\xi)-\lambda_j(t,\xi)|} \le\frac{c_1\varepsilon^{\alpha-1}\lara{\xi}}{\varepsilon^\alpha\lara{\xi}}=c_1\,\varepsilon^{-1}, \end{multline} valid for all $t\in[0,T]$ and $\xi\in\mb{R}^n$. \subsection{Estimate of the second term} \label{second_1} From Proposition \ref{prop_prelim}(v) the entries of the matrix $H^{-1}(t,\xi)\partial_t H(t,\xi)$ can be written as \[ e_{pq}(t,\xi)=\begin{cases} -\partial_t\lambda_p(t,\xi)\sum_{i=1, i\neq p}^m\frac{1}{\lambda_i(t,\xi)-\lambda_p(t,\xi)},& p=q,\\[0.3cm] -\partial_t\lambda_q(t,\xi)\frac{\prod_{i=1, i\neq p,q}^{m}(\lambda_i(t,\xi)-\lambda_q(t,\xi))}{\prod_{i=1, i\neq p}^{m}(\lambda_i(t,\xi)-\lambda_p(t,\xi))},& p\neq q. \end{cases} \] From Proposition \ref{prop_roots} we clearly have that \[
|e_{pp}(t,\xi)|\le c\frac{\varepsilon^{\alpha-1}\lara{\xi}}{\varepsilon^\alpha\lara{\xi}}=c\varepsilon^{-1}. \] To estimate $e_{pq}$ when $q\neq p$ we write \[ \partial_t\lambda_q(t,\xi)\frac{\prod_{i=1, i\neq p,q}^{m}(\lambda_i(t,\xi)-\lambda_q(t,\xi))}{\prod_{i=1, i\neq p}^{m}(\lambda_i(t,\xi)-\lambda_p(t,\xi))} \] as \[ \partial_t\lambda_q(t,\xi)\frac{\prod_{i=1, i\neq p,q}^{m}(\lambda_i(t,\xi)-\lambda_q(t,\xi))}{\prod_{i=1, i\neq p,q}^{m}(\lambda_i(t,\xi)-\lambda_p(t,\xi))(\lambda_q(t,\xi)-\lambda_p(t,\xi))}. \]
Since $$|\lambda_i(t,\xi)-\lambda_q(t,\xi)|\le |\lambda_i(t,\xi)-\lambda_p(t,\xi)|+
|\lambda_p(t,\xi)-\lambda_q(t,\xi)|$$ arguing as in (40) in \cite{ColKi:02} and making use of the estimate \eqref{hyp_coincide_2} we obtain that \[
|e_{pq}(t,\xi)|\le c\frac{\varepsilon^{\alpha-1}\lara{\xi}}{\varepsilon^\alpha\lara{\xi}}=c\varepsilon^{-1}. \] Hence, $\Vert H^{-1}\partial_t H\Vert\le c_2\varepsilon^{-1}$.
\subsection{Estimate of the third term} From Proposition \ref{prop_prelim}(iii) the matrix $H^{-1}AH$ has entries \[ c_{pq}(t,\xi)=(\tau_q(t,\xi)-\lambda_q(t,\xi))\frac{\prod_{i=1, i\neq q}^m (\tau_i(t,\xi)-\lambda_q(t,\xi))}{\prod_{i=1, i\neq p}^m (\lambda_i(t,\xi)-\lambda_p(t,\xi))} \] when $p\neq q$. As in formula (46) in \cite{ColKi:02} we have \begin{multline*}
|\tau_q(t,\xi)-\lambda_q(t,\xi)|\frac{\prod_{i=1, i\neq q}^m |\tau_i(t,\xi)-\lambda_q(t,\xi)|}{\prod_{i=1, i\neq p}^m |\lambda_i(t,\xi)-\lambda_p(t,\xi)|}\\
\le |\tau_q(t,\xi)-\lambda_q(t,\xi)|\frac{\prod_{i=1, i\neq q}^m |\tau_q(t,\xi)-\lambda_q(t,\xi)|+|\tau_i(t,\xi)-\tau_q(t,\xi)|}{\prod_{i=1, i\neq p}^m |\lambda_i(t,\xi)-\lambda_p(t,\xi)|}\\
=\sum_{k=1}^{m-1}|\tau_q(t,\xi)-\lambda_q(t,\xi)|^k\sum_{S^{(m)}_q(m-k)}\frac{|\tau_{i_1}(t,\xi)-\tau_q(t,\xi)|\cdots
|\tau_{i_{m-k}}(t,\xi)-\tau_q(t,\xi)|}{\prod_{i=1, i\neq p}^m |\lambda_i(t,\xi)-\lambda_p(t,\xi)|}\\
+\frac{|\tau_q(t,\xi)-\lambda_q(t,\xi)|^m}{\prod_{i=1, i\neq p}^m |\lambda_i(t,\xi)-\lambda_p(t,\xi)|}. \end{multline*} Proposition \ref{prop_roots} combined with \[
|\tau_{i_k}(t,\xi)-\tau_q(t,\xi)|\le |\tau_{i_k}(t,\xi)-\lambda_{i_k}(t,\xi)|+|\lambda_{i_k}(t,\xi)-\lambda_q(t,\xi)|+|\lambda_q(t,\xi)-\tau_q(t,\xi)|, \]
the property \eqref{hyp_coincide} and the fact that $|\tau_i(t,\xi)-\tau_j(t,\xi)|/|\lambda_i(t,\xi)-\lambda_j(t,\xi)|$ is bounded when $i\neq j$, yields the estimate \[
|c_{pq}(t,\xi)|\le c\sum_{k=1}^{m-1}\varepsilon^{\alpha k}\lara{\xi}^k\sum_{S^{(m)}_q(m-k)}\frac{\lara{\xi}^{m-k-m+1}}{\varepsilon^{\alpha(m-1)-\alpha(m-k)}}+c\,\frac{\varepsilon^{\alpha m}\lara{\xi}^m}{\varepsilon^{\alpha(m-1)}\lara{\xi}^{m-1}}\le c\, \varepsilon^\alpha\lara{\xi}. \] This implies $\Vert H^{-1}AH-(H^{-1}AH)^\ast\Vert\le c_3\varepsilon^\alpha\lara{\xi}$.
\subsection{Estimate of the fourth term} From Proposition \ref{prop_prelim}(iv) we have that $H^{-1}BH$ has entries \[ d_{pq}(t,\xi)=(-1)^{m-1}\biggl(\prod_{i=1,i\neq p}^m (\lambda_i(t,\xi)-\lambda_p(t,\xi))\biggr)^{-1}g(\lambda_q(t,\xi)), \] where $$g(\tau)=\sum_{j=0}^{m-1}(A_{m-j}-A_{(m-j)})(t,\xi)\tau^{j}.$$ Assume that we have lower order terms of order $l$. Then
$$|g(\lambda_q(t,\xi))|\leq C\lara{\xi}^l$$ and by Proposition \ref{prop_roots}(iii) we get \[
|d_{pq}(t,\xi)|\le c\varepsilon^{\alpha(1-m)}\lara{\xi}^{-m+1+l}. \] Hence $\Vert H^{-1}BH-(H^{-1}BH)^\ast\Vert\le c_4\varepsilon^{\alpha(1-m)}\lara{\xi}^{l-m+1}$.
\subsection{Conclusion of the proof} Making use of these four estimates in \eqref{energy} we get \begin{multline} \label{eps_energy}
\partial_t |W(t,\xi)|^2 \\ \le 2(\rho'(t)\lara{\xi}^{\frac{1}{s}}+c_1\varepsilon^{-1}+c_2\varepsilon^{-1}+
c_3\varepsilon^{\alpha}\lara{\xi}+c_4\varepsilon^{\alpha(1-m)}\lara{\xi}^{l-m+1})|W(t,\xi)|^2
\\ + C'\mathrm{e}^{(\rho(t)-\delta_1)\lara{\xi}^{\frac{1}{s}}}|W(t,\xi)|\\ \le (2\rho'(t)\lara{\xi}^{\frac{1}{s}}+C_1\varepsilon^{-1}+C_2\varepsilon^{\alpha}\lara{\xi}+C_3\varepsilon^{\alpha(1-m)}\lara{\xi}^{l-m+1})
|W(t,\xi)|^2+\\
+C'\mathrm{e}^{(\rho(t)-\delta_1)\lara{\xi}^{\frac{1}{s}}}|W(t,\xi)|, \end{multline} where $\delta_1>0$ depends on $f$, in view of Proposition \ref{prop_Fourier}. Set $\varepsilon=\lara{\xi}^{-\gamma}$. By substitution in \eqref{eps_energy} we arrive at comparing the terms \[ \lara{\xi}^\gamma;\quad \lara{\xi}^{-\gamma\alpha+1};\quad \lara{\xi}^{\gamma\alpha(m-1)+l-m+1}. \] Choose $\gamma=\min\{\frac{1}{1+\alpha},\frac{m-l}{\alpha m}\}$. It follows that $$\max\{\gamma, \gamma\alpha(m-1)+l-m+1\}\le -\gamma\alpha+1.$$ Then, if we take $s>0$ such that \begin{multline} \label{form_s} \frac{1}{s}>-\gamma\alpha+1=-\min\biggl\{\frac{1}{1+\alpha},\frac{m-l}{\alpha m}\biggr\}\alpha+1 \\ =-\min\biggl\{\frac{\alpha}{1+\alpha},\frac{m-l}{ m}\biggr\}+1=\max\biggl\{\frac{1}{1+\alpha},\frac{l}{m}\biggr\}, \end{multline} for a suitable decreasing function $\rho$ (for instance $\rho(t)=\rho(0)-\kappa t$ with $\kappa>0$ and $\rho(0)$ to be chosen later) we obtain \begin{multline}\label{EQ:en-est}
\partial_t |W(t,\xi)|^2\le \big(2\rho'(t)\lara{\xi}^{\frac{1}{s}}+C\lara{\xi}^{-\gamma\alpha+1}\big)|W(t,\xi)|^2\\ +
2\mathrm{e}^{\rho(t)\lara{\xi}^{\frac{1}{s}}} \det H(t,\xi) |H^{-1}(t,\xi)||\widehat{F}(t,\xi)||W(t,\xi)|\\
\le\big(2\rho'(t)\lara{\xi}^{\frac{1}{s}}+C\lara{\xi}^{-\gamma\alpha+1}\big)|W(t,\xi)|^2+
C'\mathrm{e}^{(\rho(t)-\delta_1)\lara{\xi}^{\frac{1}{s}}}|W(t,\xi)|. \end{multline}
Note that \eqref{form_s} implies \[ s<\min\biggl\{1+\alpha,\frac{m}{l}\biggr\}=1+\min\biggl\{\alpha, \frac{m-l}{l}\biggr\}. \]
Assuming for the moment that $|W(t,\xi)|\ge 1$, taking $\rho(0)<\delta_1$ we get the energy estimate \begin{equation} \label{energy1}
\partial_t |W(t,\xi)|^2\le \big(2\rho'(t)\lara{\xi}^{\frac{1}{s}}+C\lara{\xi}^{-\gamma\alpha+1}+
C'\mathrm{e}^{(\rho(0)-\delta_1)\lara{\xi}^{\frac{1}{s}}}\big)|W(t,\xi)|^2 \leq 0,
\end{equation} for large enough $|\xi|$ (note that it suffices to consider only large $|\xi|$). Consequently, \eqref{VW} and \eqref{energy1} imply the estimate \begin{multline} \label{last_estimate}
|V(t,\xi)|
=\mathrm{e}^{-\rho(t)\lara{\xi}^{\frac{1}{s}}}\frac{1}{\det H(t,\xi)}|H(t,\xi)||W(t,\xi)|\le \\
\mathrm{e}^{-\rho(t)\lara{\xi}^{\frac{1}{s}}}\frac{1}{\det H(t,\xi)}|H(t,\xi)||W(0,\xi)|=\\
\mathrm{e}^{(-\rho(t)+\rho(0))\lara{\xi}^{\frac{1}{s}}}\frac{\det H(0,\xi)}{\det H(t,\xi)}|H(t,\xi)||H^{-1}(0,\xi)||V(0,\xi)|, \end{multline} where, for $\gamma$ as above, we have \[
\frac{\det H(0,\xi)}{\det H(t,\xi)}|H(t,\xi)||H^{-1}(0,\xi)|\le c\,\varepsilon^{-\alpha\frac{(m-1)m}{2}}=c\lara{\xi}^{\gamma\alpha\frac{(m-1)m}{2}}. \] Hence, \begin{equation} \label{last_estimate2} \left\{ \begin{array}{cc}
|V(t,\xi)|\le c\,\mathrm{e}^{(-\rho(t)+\rho(0))\lara{\xi}^{\frac{1}{s}}}\lara{\xi}^{\gamma\alpha\frac{(m-1)m}{2}}|V(0,\xi)|,\quad & \text{for}\, |W(t,\xi)|\ge 1,\\
|V(t,\xi)|\le c\,\mathrm{e}^{-\rho(t)\lara{\xi}^{\frac{1}{s}}}\lara{\xi}^{\gamma\alpha\frac{(m-1)m}{2}},\quad &\text{for}\, |W(t,\xi)|<1, \end{array} \right. \end{equation} with the second line following directly from \eqref{VW}. The estimate \eqref{last_estimate2} combined with the Fourier characterisations of Proposition \ref{prop_Fourier} yields the statement of Theorem \ref{THM:case1} if we choose $\kappa>0$ small enough. If $s=1+\min\biggl\{\alpha, \frac{m-l}{l}\biggr\}$, we need $\kappa$ to be large enough in \eqref{EQ:en-est}, so that \eqref{last_estimate2} still implies the local in time well-posedness (showing a statement in Remark \ref{REM:local}).
We note that in view of the characterisation in Proposition \ref{prop_ud}, the estimate \eqref{last_estimate2} also yields the statement of Theorem \ref{THM:case1u}. In this case we can also allow the critical case $s=1+\min\biggl\{\alpha, \frac{m-l}{l}\biggr\}$. Indeed, differently from the case of Theorem \ref{THM:case1}, taking $\kappa>0$ to be large enough, we can make sure that the estimate \eqref{energy1} holds, while \eqref{last_estimate2} yields that $V(t,\xi)$ satisfies the estimates of Proposition \ref{prop_ud} for any value of $T$. Because of the presence of the function $\rho$ in \eqref{last_estimate2} the obtained result is in the space of Gevrey Beurling ultradistributions rather than in the space of Gevrey Roumieu ultradistributions.
\section{Proof in Case 2: Theorem \ref{THM:case2} and Theorem \ref{THM:case2u}}
We work on the energy estimate similar to the Case 1. However, the different nature of the approximated roots $\lambda_j(t,\xi)$ yields different estimates for the terms \begin{enumerate} \item $\frac{\partial_t\det H}{\det H}$, \item $\Vert H^{-1}\partial_t H\Vert$, \item $\Vert H^{-1}AH-(H^{-1}AH)^\ast\Vert$, \item $\Vert H^{-1}BH-(H^{-1}BH)^\ast\Vert$. \end{enumerate}
\subsection{Estimate of the first term} Arguing as in \eqref{est_1} we have \begin{multline*}
\biggl|\frac{\partial_t\det H(t,\xi)}{\det H(t,\xi)}\biggr|\le \sum_{1\le j<i\le m}\frac{|\partial_t\lambda_i(t,\xi)-\partial_t\lambda_j(t,\xi)|}{|\lambda_i(t,\xi)-\lambda_j(t,\xi)|}\\
= \sum_{1\le j<i\le r}\frac{|\partial_t\lambda_i(t,\xi)-\partial_t\lambda_j(t,\xi)|}{|\lambda_i(t,\xi)-\lambda_j(t,\xi)|}+\sum_{r+1\le j<i\le m}\frac{|\partial_t\lambda_i(t,\xi)-\partial_t\lambda_j(t,\xi)|}{|\lambda_i(t,\xi)-\lambda_j(t,\xi)|}\\ +\sum_{\substack{1\le j<i\le m,\\ j\le r, i\ge r+1}}
\frac{|\partial_t\lambda_{i}(t,\xi)-\partial_t\lambda_j(t,\xi)|}{|\lambda_{i}(t,\xi)-\lambda_j(t,\xi)|}. \end{multline*}
Proposition \ref{prop_roots_2} yields for $t\in[0,T]$ and $|\xi|$ large enough the following estimate: \[ \begin{split}
\biggl|\frac{\partial_t\det H(t,\xi)}{\det H(t,\xi)}\biggr| & \le c\frac{\varepsilon^{\alpha-1}\lara{\xi}}{\varepsilon^\alpha\lara{\xi}}+c'\frac{\delta^{\beta-1}\lara{\xi}}{c_0\lara{\xi}}+c'' \frac{\varepsilon^{\alpha-1}\lara{\xi}+\delta^{\beta-1}\lara{\xi}}{c_0\lara{\xi}} \\[0.2cm] & \le c_1\max\{\varepsilon^{-1},\delta^{\beta-1}\}. \end{split} \] We note that here we can use Proposition \ref{prop_roots_2}(vi) since we will set $\varepsilon$ and $\delta$ later to be as required.
\subsection{Estimate of the second term} The entries of the matrix $H^{-1}(t,\xi)\partial_t H(t,\xi)$ can be written as \[ e_{pq}(t,\xi)=\begin{cases} -\partial_t\lambda_p(t,\xi)\sum_{i=1, i\neq p}^m\frac{1}{\lambda_i(t,\xi)-\lambda_p(t,\xi)},& p=q,\\[0.3cm] -\partial_t\lambda_q(t,\xi)\frac{\prod_{i=1, i\neq p,q}^{m}(\lambda_i(t,\xi)-\lambda_q(t,\xi))}{\prod_{i=1, i\neq p}^{m}(\lambda_i(t,\xi)-\lambda_p(t,\xi))},& p\neq q. \end{cases} \] Let us start with the case $p=q$. We have \begin{multline*} e_{pp}(t,\xi)=-\partial_t\lambda_p(t,\xi)\sum_{i=1, i\neq p}^r\frac{1}{\lambda_i(t,\xi)-\lambda_p(t,\xi)} \\ -\partial_t\lambda_p(t,\xi)\sum_{i=r+1, i\neq p}^m\frac{1}{\lambda_i(t,\xi)-\lambda_p(t,\xi)}. \end{multline*}
It follows that, for $|\xi|$ large, \[ \begin{split}
|e_{pp}(t,\xi)|&\le c\frac{\varepsilon^{\alpha-1}\lara{\xi}}{\varepsilon^\alpha\lara{\xi}}+ c\frac{\varepsilon^{\alpha-1}\lara{\xi}}{c_0\lara{\xi}},\quad 1\le p\le r,\\
|e_{pp}(t,\xi)|&\le c\frac{\delta^{\beta-1}\lara{\xi}}{c_0\lara{\xi}},\qquad\qquad\quad 1+r\le p\le m. \end{split} \] Hence, \[
|e_{pp}(t,\xi)|\le c'\max\{\varepsilon^{-1},\delta^{\beta-1}\}. \]
When $p\neq q$ we argue as in \cite{ColKi:02} (estimates (38), (39), (40)). In particular, when both $p$ and $q$ belong to $\{1,...,r\}$ we follow the arguments of Subsection \ref{second_1} for the corresponding term in Case 1. We obtain, for $|\xi|$ large enough, \[ \begin{split}
|e_{pq}|&\le c\,\delta^{\beta-1}\varepsilon^{\alpha(1-r)},\qquad 1\le p\le m,\, r+1\le q\le m,\\
|e_{pq}|&\le c\,\varepsilon^{\alpha-1},\qquad r+1\le p\le m,\, 1\le q\le r,\\
|e_{pq}|&\le c\,\varepsilon^{-1},\qquad 1\le p\le r,\, 1\le q\le r. \end{split} \] In conclusion, we get $$\Vert H^{-1}(t,\xi)\partial_t H(t,\xi)\Vert\le c_2\max\{\varepsilon^{-1}, \delta^{\beta-1}\varepsilon^{\alpha(1-r)}\}$$
for $t\in[0,T]$ and $|\xi|$ large enough. \subsection{Estimate of the third term} The matrix $H^{-1}AH$ has entries \[ c_{pq}(t,\xi)=(\tau_q(t,\xi)-\lambda_q(t,\xi))\frac{\prod_{i=1, i\neq q}^m (\tau_i(t,\xi)-\lambda_q(t,\xi))}{\prod_{i=1, i\neq p}^m (\lambda_i(t,\xi)-\lambda_p(t,\xi))}. \]
Arguing as in Case 1, making use of the estimates in Proposition \ref{prop_roots_2} and of the assumption \eqref{hyp_coincide} we obtain, for $|\xi|$ large and $1\le p\le r$, $1\le q\le r$, \begin{multline} \label{form_1}
|\tau_q(t,\xi)-\lambda_q(t,\xi)|\frac{\prod_{i=1, i\neq q}^m |\tau_i(t,\xi)-\lambda_q(t,\xi)|}{\prod_{i=1, i\neq p}^m |\lambda_i(t,\xi)-\lambda_p(t,\xi)|}\\
\le |\tau_q(t,\xi)-\lambda_q(t,\xi)|\frac{\prod_{i=1, i\neq q}^m |\tau_q(t,\xi)-\lambda_q(t,\xi)|+|\tau_i(t,\xi)-\tau_q(t,\xi)|}{\prod_{i=1, i\neq p}^m |\lambda_i(t,\xi)-\lambda_p(t,\xi)|}\\
=\sum_{k=1}^{m-1}|\tau_q(t,\xi)-\lambda_q(t,\xi)|^k\sum_{S^{(m)}_q(m-k)}\frac{|\tau_{i_1}(t,\xi)-\tau_q(t,\xi)|\cdots
|\tau_{i_{m-k}}(t,\xi)-\tau_q(t,\xi)|}{\prod_{i=1, i\neq p}^m |\lambda_i(t,\xi)-\lambda_p(t,\xi)|}\\
+\frac{|\tau_q(t,\xi)-\lambda_q(t,\xi)|^m}{\prod_{i=1, i\neq p}^m |\lambda_i(t,\xi)-\lambda_p(t,\xi)|}\\ \le c\sum_{k=1}^{m-1}\frac{\varepsilon^{\alpha k}\lara{\xi}^k\lara{\xi}^{m-k}}{\varepsilon^{\alpha(r-1)-\alpha(r-k)}\lara{\xi}^{m-1}}+c\frac{\varepsilon^{\alpha m}\lara{\xi}^m}{\varepsilon^{\alpha(r-1)}\lara{\xi}^{m-1}}\le c'\max\{\varepsilon^\alpha, \varepsilon^{\alpha(m-r+1)}\}\lara{\xi}\\ = c'\varepsilon^\alpha\lara{\xi}. \end{multline} If $r+1\le q\le m$ and $1\le p\le r$ then \begin{multline} \label{form_2}
|\tau_q(t,\xi)-\lambda_q(t,\xi)|\frac{\prod_{i=1, i\neq q}^m |\tau_i(t,\xi)-\lambda_q(t,\xi)|}{\prod_{i=1, i\neq p}^m |\lambda_i(t,\xi)-\lambda_p(t,\xi)|} \le c\delta^\beta\lara{\xi}\frac{1}{\varepsilon^{\alpha(r-1)}} \\ =c\delta^\beta\varepsilon^{\alpha(1-r)}\lara{\xi}. \end{multline} If $r+1\le q\le m$ and $1+r\le p\le m$ then \begin{equation} \label{form_3}
|\tau_q(t,\xi)-\lambda_q(t,\xi)|\frac{\prod_{i=1, i\neq q}^m |\tau_i(t,\xi)-\lambda_q(t,\xi)|}{\prod_{i=1, i\neq p}^m |\lambda_i(t,\xi)-\lambda_p(t,\xi)|}\le c\delta^\beta\lara{\xi}\frac{1}{c_0}=c'\delta^\beta\lara{\xi}. \end{equation} Finally, if $1\le q\le r$ and $1+r\le p\le m$ then \begin{equation} \label{form_4}
|\tau_q(t,\xi)-\lambda_q(t,\xi)|\frac{\prod_{i=1, i\neq q}^m |\tau_i(t,\xi)-\lambda_q(t,\xi)|}{\prod_{i=1, i\neq p}^m |\lambda_i(t,\xi)-\lambda_p(t,\xi)|}\le c\varepsilon^\alpha\lara{\xi}\frac{1}{c_0}=c'\varepsilon^\alpha\lara{\xi}. \end{equation} Combining \eqref{form_1} with \eqref{form_2}, \eqref{form_3} and \eqref{form_4} we obtain \[
|c_{pq}(t,\xi)|\le c\max\{\varepsilon^\alpha, \delta^\beta\varepsilon^{\alpha(1-r)}, \delta^\beta\}\lara{\xi}=c\max\{\varepsilon^\alpha, \delta^\beta\varepsilon^{\alpha(1-r)}\}\lara{\xi}. \] Hence,
$$\Vert H^{-1}AH-(H^{-1}AH)^\ast\Vert\le c_3\max\{\varepsilon^\alpha, \delta^\beta\varepsilon^{\alpha(1-r)}\}\lara{\xi}$$ for $t\in[0,T]$ and $|\xi|$ large enough.
\subsection{Estimate of the fourth term} The entries of the matrix $H^{-1}BH$ are given by \[ d_{pq}(t,\xi)=(-1)^{m-1}\biggl(\prod_{i=1,i\neq p}^m (\lambda_i(t,\xi)-\lambda_p(t,\xi))\biggr)^{-1}g(\lambda_q(t,\xi)), \] where $$g(\tau)=\sum_{j=0}^{m-1}(A_{m-j}-A_{(m-j)})(t,\xi)\tau^{j}.$$ Assume that we have lower order terms of order $l$. Then, \[ \begin{split}
|d_{pq}(t,\xi)|\le c\varepsilon^{\alpha(1-r)}\lara{\xi}^{-m+1+l},\quad & 1\le p\le r,\\
|d_{pq}(t,\xi)|\le c\lara{\xi}^{-m+1+l},\quad & r+1\le p\le m,\\ \end{split} \]
and $$\Vert H^{-1}BH-(H^{-1}BH)^\ast\Vert\le c_4\varepsilon^{\alpha(1-r)}\lara{\xi}^{l-m+1}$$ for $t\in[0,T]$ and $|\xi|$ large enough.
\subsection{Conclusion of the proof}
We now make use of the four estimates above in \eqref{energy}. We get, for large $|\xi|$, \begin{multline} \label{eps_energy_2}
\partial_t |W(t,\xi)|^2\le 2(\rho'(t)\lara{\xi}^{\frac{1}{s}}+c_1\max\{\varepsilon^{-1},\delta^{\beta-1}\} +c_2\max\{\varepsilon^{-1},\delta^{\beta-1}\varepsilon^{\alpha(1-r)}\} \\ +
c_3\max\{\varepsilon^\alpha, \delta^\beta\varepsilon^{\alpha(1-r)}\}\lara{\xi}+c_4\varepsilon^{\alpha(1-r)}\lara{\xi}^{l-m+1})|W(t,\xi)|^2
\\ +C'\mathrm{e}^{(\rho(t)-\delta_1)\lara{\xi}^{\frac{1}{s}}}|W(t,\xi)|, \end{multline} where $\delta_1>0$ depends on $f$. Set $\delta=\lara{\xi}^{-1}$ and $\varepsilon=\lara{\xi}^{-\gamma}$. Then we have \begin{multline} \label{est_energy_2}
\partial_t |W(t,\xi)|^2\\ \le \left(2\rho'(t)\lara{\xi}^{\frac{1}{s}} +C\max\{\lara{\xi}^{\gamma},\lara{\xi}^{1-\beta}, \lara{\xi}^{1-\beta-\gamma\alpha(1-r)}, \lara{\xi}^{1-\gamma\alpha}, \lara{\xi}^{-\gamma\alpha(1-r)+l-m+1}\}\right)\cdot\\
\cdot|W(t,\xi)|^2+C'\mathrm{e}^{(\rho(t)-\delta_1)\lara{\xi}^{\frac{1}{s}}}|W(t,\xi)|\\
=(2\rho'(t)\lara{\xi}^{\frac{1}{s}}+C\max\{\lara{\xi}^{\gamma},\lara{\xi}^{1-\beta-\gamma\alpha(1-r)}, \lara{\xi}^{1-\gamma\alpha}, \lara{\xi}^{-\gamma\alpha(1-r)+l-m+1})\}|W(t,\xi)|^2\\ +
C'\mathrm{e}^{(\rho(t)-\delta_1)\lara{\xi}^{\frac{1}{s}}}|W(t,\xi)|. \end{multline} Let \[ \gamma=\min\biggl\{\frac{1}{1+\alpha},\frac{\beta}{\alpha r}, \frac{m-l}{\alpha r}\biggr\}. \] Hence, $\max\{\gamma, {1-\beta-\gamma\alpha(1-r)}, {-\gamma\alpha(1-r)+l-m+1}\}\le 1-\gamma\alpha$ and \[
\partial_t |W(t,\xi)|^2\le \left( 2\rho'(t)\lara{\xi}^{\frac{1}{s}}+C\lara{\xi}^{-\gamma\alpha+1}
\right) |W(t,\xi)|^2 + C'\mathrm{e}^{(\rho(t)-\delta_1)\lara{\xi}^{\frac{1}{s}}}|W(t,\xi)|. \] Let $s>0$ be such that \begin{multline}\label{EQ:formula-s} \frac{1}{s}>-\min\biggl\{\frac{1}{1+\alpha},\frac{\beta}{\alpha r}, \frac{m-l}{\alpha r}\biggr\}\alpha+1 \\ =\max\biggl\{\frac{1}{1+\alpha}, \frac{r-\beta}{r}, \frac{r-m+l}{r}\biggr\}. \end{multline} If $r-m+l>0$, this means that \begin{equation} \label{s_2a} s<\min\biggl\{1+\alpha, \frac{r}{r-\beta}, \frac{r}{r-m+l}\biggr\}=1+\min\biggl\{\alpha,\frac{\beta}{r-\beta},\frac{m-l}{r-m+l}\biggr\}.
\end{equation} We can assume $|W(t,\xi)|\ge 1$ since when $|W(t,\xi)|<1$ we can use \eqref{VW} to directly obtain the estimates as in the second line in \eqref{last_estimate2}. Choosing a suitable decreasing function $\rho$ as in Case 1 we obtain \begin{equation} \label{energy2}
\partial_t |W(t,\xi)|^2\leq 0
\end{equation} for all $t\in[0,T]$ and for $|\xi|$ sufficiently large. If $r-m+l\le 0$ then the last term under the maximum sign in \eqref{EQ:formula-s} is negative, and hence disappears. Hence in this case
\eqref{EQ:formula-s} means that \begin{equation} \label{s_2b} s<1+\min\biggl\{\alpha,\frac{\beta}{r-\beta}\biggr\}. \end{equation} Let us finally show that the inequality \eqref{s_2a} is actually also equivalent to \eqref{s_2b}. Indeed, let us denote $k=m-l$, so that $1\leq k\leq m$. Consequently, for $\beta\leq 1$ one can readily check that we have $\frac{\beta}{r-\beta}\leq \frac{k}{r-k}$, proving the claim.
In analogy to Case 1, by arguing as in \eqref{last_estimate}, we see that \eqref{VW} and \eqref{energy2} imply \begin{equation} \label{last_estimate3}
|V(t,\xi)|\le c\,\mathrm{e}^{(-\rho(t)+\rho(0))\lara{\xi}^{\frac{1}{s}}}\lara{\xi}^{\gamma\alpha\frac{(r-1)r}{2}}|V(0,\xi)|,
\end{equation} for $t\in[0,T]$ and $|\xi|$ large enough. The estimate \eqref{last_estimate3} proves Theorem \ref{THM:case2}. Similarly to Case 1, \eqref{last_estimate3} and Proposition \ref{prop_ud} imply the statement of Theorem \ref{THM:case2u}, also allowing $s=1+\min\biggl\{\alpha,\frac{\beta}{r-\beta}\biggr\}$.
\begin{rem}\label{REM:r1} Assume now that we are under assumptions of Case 3, i.e. the Cauchy problem in consideration is strictly hyperbolic. Analysing the estimates of Case 2 under the assumption of strict hyperbolicity, we will set $r=1$ and repeat the argument first keeping the notation for $\alpha$ and $\beta$ distinguishing them from each other (although, since we are interested in Case 3, we will put $\alpha=\beta$ later). Then, by similar arguments, we readily see that \begin{enumerate}
\item $|\frac{\partial_t\det H}{\det H}|\le c_1\max\{\varepsilon^{\alpha-1},\delta^{\beta-1}\}$, \item $\Vert H^{-1}\partial_t H\Vert\le c_2\max\{\varepsilon^{\alpha-1},\delta^{\beta-1}\}$, \item $\Vert H^{-1}AH-(H^{-1}AH)^\ast\Vert\le c_3\max\{\varepsilon^\alpha,\delta^\beta\}\lara{\xi}$, \item $\Vert H^{-1}BH-(H^{-1}BH)^\ast\Vert\le c_4\lara{\xi}^{-m+1+l}$, \end{enumerate}
for $t\in[0,T]$ and $|\xi|$ large enough. Hence, setting $\delta=\lara{\xi}^{-1}$ and $\varepsilon=\lara{\xi}^{-\gamma}$ in the energy estimate \eqref{eps_energy_2}--\eqref{est_energy_2} we obtain \begin{multline*}
\partial_t |W(t,\xi)|^2\le\left(2\rho'(t)\lara{\xi}^{\frac{1}{s}} +C\max\{\lara{\xi}^{-\gamma\alpha+\gamma},\lara{\xi}^{1-\beta}, \lara{\xi}^{1-\gamma\alpha}, \lara{\xi}^{l-m+1}\}\right)\cdot\\
\cdot|W(t,\xi)|^2+C'\mathrm{e}^{(\rho(t)-\delta_1)\lara{\xi}^{\frac{1}{s}}}|W(t,\xi)|. \end{multline*} Arguing as in Case 2, from $\max\{1-\beta,1-m+l\}\le 1-\gamma\alpha$ we have that $W(t,\xi)$ is of Gevrey order $s$ with \[ \frac{1}{s}>-\min\biggl\{\frac{\beta}{\alpha}, \frac{m-l}{\alpha}\biggr\}\alpha+1=\max\biggl\{{1-\beta}, {1-m+l}\biggr\}=1-\beta. \] This means that $$1\le s<1+\frac{\beta}{1-\beta}.$$ Finally we note that since $m-l\geq 1\geq\beta$, we have in this argument $\gamma=\min\{\frac{\beta}{\alpha}, \frac{m-l}{\alpha}\}= \frac{\beta}{\alpha}.$ Recalling that in Case 3, we actually assume $\alpha=\beta$, we get that in fact $\gamma=1$ (and hence also $\epsilon=\delta$, simplifying the proof of Case 3 compared to that of Case 2, if needed). \end{rem}
\end{document}
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\begin{document}
\title[Sums of CR functions from competing CR structures] {Sums of CR functions from competing CR structures} \author[David E. Barrett]{David E. Barrett} \address{Department of Mathematics\\University of Michigan \\Ann Arbor, MI 48109-1043 USA } \email{[email protected]}
\author[Dusty E. Grundmeier]{Dusty E. Grundmeier} \address{Department of Mathematics \\ Harvard University \\
Cambridge, MA 02138-2901 USA } \email{[email protected]}
\thanks{{\em 2010 Mathematics Subject Classification:} 32V10}
\thanks{The first author was supported in part by NSF grants number DMS-1161735 and DMS-1500142.}
\date{\today}
\begin{abstract} In this paper we characterize sums of CR functions from competing CR structures in two scenarios. In one scenario the structures are conjugate and we are adding to the theory of pluriharmonic boundary values. In the second scenario the structures are related by projective duality considerations. In both cases we provide explicit vector field-based characterizations for two-dimensional circular domains satisfying natural convexity conditions. \end{abstract}
\maketitle
\section{Introduction}\label{S:Intro}
The Dirichlet problem for pluriharmonic functions is a natural problem in several complex variables with a long history going back at least to Amoroso \cite{Amo}, Severi \cite{Sev}, Wirtinger \cite{Wir}, and others. It was known early on that the problem is not solvable for general boundary data, so we may try to characterize the admissible boundary values with a system of tangential partial differential operators. This was first done for the ball by Bedford in \cite{Bed1}; see \S \ref{SS:ball} for details. More precisely, given a bounded domain $\Omega$ with smooth boundary $S$, we seek a system $\mathcal{L}$ of partial differential operators tangential to $S$ such that a function $u\in \mathcal{C}^{\infty}(S,\mathbb C)$ satisfies $\mathcal{L}u=0$ if and only if there exists $U\in \mathcal{C}^{\infty}(\overline{\Omega})$ such that $U|_S = u$ and $\partial \overline{\partial} U =0$. The problem may also be considered locally.
While natural in its own right, this problem also arises in less direct fashion in many areas of complex analysis and geometry. For instance, this problem plays a fundamental role in Graham's work on the Bergman Laplacian \cite{Gra}, Lee's work on pseudo-Einstein structures \cite{Lee}, and Case, Chanillo, and Yang's work on CR Paneitz operators (see \cite{CCY} and the references therein). From another point of view, the existence of non-trivial restrictions on pluriharmonic boundary values points to the need to look elsewhere (such as to the Monge-Amp\`ere equations studied in \cite{BeTa}) for Dirichlet problems solvable for general boundary data.
The pluriharmonic boundary value problem is closely related to the problem of characterizing sums of CR functions from different, competing CR structures; indeed, when the competing CR structures are conjugate then these problems coincide (in simply-connected settings); see Propositions \ref{T:glo-prob} and \ref{T:loc-prob} below. Another natural construction leading to competing CR structures arises from the study of projective duality (see \S \ref{S:proj} or \cite{Bar} for precise definitions).
In each of these two scenarios, we precisely characterize sums of CR functions from the two competing CR structures in the setting of two-dimensional circular domains satisfying appropriate convexity conditions. For conjugate structures we assume strong pseudoconvexity; our result appears as Theorem \ref{T:cir-plh} below. In the projective duality scenario we assume strong convexity (the correct assumption without the circularity assumption would be strong $\mathbb C$-convexity, but these notions coincide in the circular case; see \S \ref{S:pdh}), and the main result appears as Theorem \ref{T:cir-proj} below (with an expanded version appearing later in \S \ref{S:cir-proj}). Our techniques for these two related problems are interconnected to a surprising extent, and the reader will notice that the projective dual scenario actually turns out to have more structure and symmetry.
\begin{restatable}{MThm}{conj} \label{T:cir-plh} Let $S\subset\mathbb C^2$ be a strongly pseudoconvex circular hypersurface. Then there exist nowhere-vanishing tangential vector fields $X,Y$ on $S$ satisfying the following conditions. \refstepcounter{equation}\label{N:XY-cond} \begin{enum} \item If $u$ is a smooth function on a relatively open subset of $S$, then $u$ is CR if and only if $Xu=0$. \label{I:aaa} \item If $u$ is a smooth function on a relatively open subset of $S$, then $u$ is CR if and only if $Y\overline u=0$.\label{I:bbb} \item If $S$ is compact, then a smooth function $u$ on $S$ is a pluriharmonic boundary value (in the sense of Proposition \ref{T:glo-prob} below) if and only if $XXYu=0$. \label{I:cir-plh-glo} \item A smooth function $u$ on a relatively open subset of $S$ is a pluriharmonic boundary value (in the sense of Proposition \ref{T:loc-prob} below) if and only if $XXYu=0=\overline{XXY}u$. \label{I:cir-plh-loc} \end{enum} \end{restatable}
\begin{restatable}{MThm}{proj} \label{T:cir-proj} Let $S\subset\mathbb C^2$ be a strongly convex circular hypersurface. Then there exist nowhere-vanishing tangential vector fields $X,T$ on $S$ satisfying the following conditions. \refstepcounter{equation}\label{N:XT-cond-pre} \begin{enum}
\item If $u$ is a smooth function on a relatively open subset of $S$, then $u$ is CR if and only if $Xu=0$.\label{I:bb-pre} \item If $u$ is a smooth function on a relatively open subset of $S$, then $u$ is dual-CR if and only if $Tu=0$.\label{I:cc-pre} \item If $S$ is compact, then a smooth function $u$ on $S$ is the sum of a CR function and a dual-CR function if and only if $XXTu=0$. \label{I:cir-proj-glo-pre} \item If $S$ is simply-connected (but not necessarily compact), then a smooth function $u$ on $S$ is the sum of a CR function and a dual-CR function if and only if $XXTu=0=TTXu$. \label{I:cir-proj-loc-pre} \end{enum} \end{restatable}
This paper is organized as follows. In \S \ref{S:conj} we focus on the case of conjugate CR structures (the pluriharmonic case). In \S \ref{S:proj} we study the competing CR structures coming from projective duality. In \S \ref{S:pfs} we prove Theorem B, while Theorem A is proved in \S \ref{S:pfs2}. The final section \S \ref{S:furth} includes a discussion of uniqueness issues.
\section{Conjugate structures}\label{S:conj}
\subsection{Results on the ball} \label{SS:ball}
Early work focused on the case of the ball $B^n$ in $\mathbb{C}^n$. In particular, Nirenberg observed that there is no second-order system of differential operators tangent to $S^3$ that exactly characterize pluriharmonic functions (see \S \ref{S:Nir} for more details). Third-order characterizations were developed by Bedford in the global case and Audibert in the local case (which requires stronger conditions). To state these results, we define the tangential operators \begin{align} L_{kl} = z_k \vf{\overline{z}_l}-z_l \vf{\overline{z}_k} && \overline{L_{kl}} = \overline{z}_k \vf{z_l}-\overline{z}_l \vf{z_k} \end{align} for $1\leq k,l \leq n$.
\begin{Thm}[{[Bed1]}]\label{T:Bglobal} Let $u$ be smooth on $S^{2n-1}$, then $$\overline{L_{kl}} \overline{L_{kl}} L_{kl} u=0$$ for $1\leq k,l \leq n$ if and only if $u$ extends to a pluriharmonic function on $B^n$. \end{Thm}
\begin{Thm}[{[Aud]}]\label{T:Alocal} Let $S$ be a relatively open subset of $S^{2n-1}$, and let $u$ be smooth on $S$. Then $$L_{jk} L_{lm} \overline{L_{rs}} u=0=\overline{L_{jk}} \overline{L_{lm}} L_{rs} u$$ for $1\leq j,k,l,m,r,s \leq n$ if and only if $u$ extends to a pluriharmonic function on a one-sided neighborhood of $S$. \end{Thm}
For a treatment of both of these results along with further details and examples, see \S 18.3 of \cite{Rud}.
\subsection{Other results}
Laville \cite{Lav1, Lav2} also gave a fourth order operator to solve the global problem. In \cite{BeFe} Bedford and Federbush solved the local problem in the more general setting where $b\Omega$ has non-zero Levi form at some point. Later in \cite{Bed2}, Bedford used the induced boundary complex $(\partial \overline{\partial})_b$ to solve the local problem in certain settings. In Lee's work \cite{Lee} on pseudo-Einstein structures, he gives a characterization for abstract CR manifolds using third order pseudohermitian covariant derivatives. Case, Chanillo, and Yang study when the kernel of the CR Paneitz operator characterizes CR-pluriharmonic functions (see \cite{CCY} and the references therein).
\subsection{Relation to decomposition on the boundary}\label{SS:conj-bg}
Outside of the proof of Theorem \ref{T:pair} below, all forms, functions, and submanifolds will be assumed $\mathcal{C}^\infty$-smooth.
\begin{Prop} \label{T:glo-prob} Let $S\subset\mathbb C^n$ be a compact connected and simply-connected real hypersurface, and let $\Omega$ be the bounded domain with boundary $S$. Then for $u\colon S\to \mathbb C$ the following conditions are equivalent: \refstepcounter{equation}\label{N:plh-CR-glob} \begin{enum} \item $u$ extends to a (smooth) function $U$ on $\overline\Omega$ that is pluriharmonic on $\Omega$. \label{I:gloext} \item $u$ is the sum of a CR function and a conjugate-CR-function. \label{I:glodecomp} \end{enum} \end{Prop}
\begin{proof} In the proof that \itemref{N:plh-CR-glob}{I:gloext} implies \itemref{N:plh-CR-glob}{I:glodecomp}, the CR term is the restriction to $S$ of an anti-derivative for $\partial U$ on a simply-connected one-sided neighborhood of $S$, and the conjugate-CR term is the restriction to $S$ of an anti-derivative for $\overline\partial U$ on a one-sided neighborhood of $S$ (adjusting one term by a constant as needed).
To see that \itemref{N:plh-CR-glob}{I:glodecomp} implies \itemref{N:plh-CR-glob}{I:gloext} we use the global CR extension result \cite[Thm.\,2.3.2]{Hor} to extend the terms to holomorphic and conjugate-holomorphic functions, respectively; $U$ is then the sum of the extensions. \end{proof}
\begin{Prop} \label{T:loc-prob} Let $S\subset\mathbb C^n$ be a simply-connected strongly pseudoconvex real hypersurface. Then for $u\colon S\to \mathbb C$ the following conditions are equivalent: \refstepcounter{equation}\label{N:plh-CR-loc} \begin{enum} \item \label{I:locext} there is an open subset $W$ of $\mathbb C^n$ with $S\subsetb W$ (with $W$ lying locally on the pseudoconvex side of $S$) so that $u$ extends to a (smooth) function $U$ on $W\cup S$ that is pluriharmonic on $W$. \item \label{I:locdecomp} $u$ is the sum of a CR function and a conjugate-CR-function. \end{enum} \end{Prop}
\begin{proof} The proof follows the proof of Theorem \ref{T:glo-prob} above, replacing the global CR extension result by the Hans Lewy local CR extension result as stated in \cite[Sec.\,14.1, Thm.\,1]{Bog}. \end{proof}
\section{Projective dual structures}\label{S:proj}
\subsection{Projective dual hypersurfaces} \label{S:pdh}
Let $S\subset\mathbb C^n$ be an oriented real hypersurface with defining function $\rho$. $S$ is said to be {\em strongly $\mathbb C$-convex} if $S$ locally equivalent via a projective transformation (that is, via an automorphism of projective space) to a strongly convex hypersurface; this condition is equivalent to either of the following two equivalent conditions: \refstepcounter{equation}\label{N:strCconv-cond} \begin{enum}
\item the second fundamental form for $S$ is positive definite on the maximal complex subspace $H_zS$ of each $T_zS$;
\item the complex tangent (affine) hyperplanes for $S$ lie to one side (the ``concave side") of $S$ near the point of tangency with minimal order of contact. \end{enum}
\begin{Thm}\label{T:gl-Cconv} When $S$ is compact and strongly $\mathbb C$-convex the complex tangent hyperplanes for $S$ are in fact disjoint from the domain bounded by $S$. \end{Thm}
\begin{proof}\, \cite[\S2.5]{APS}. \end{proof}
We note that strongly $\mathbb C$-convex hypersurfaces are also strongly pseudoconvex.
A circular hypersurface (that is, a hypersurface invariant under rotations $z\mapsto e^{i\theta}z$) is strongly $\mathbb C$-convex if and only if it is strongly convex \cite[Prop.\,3.7]{Cer}.
The proper general context for the notion of strong $\mathbb C$-convexity is in the study of real hypersurfaces in complex projective space $\mathbb{CP}^n$ (see for example \cite{Bar} and \cite{APS}).
We specialize now to the two-dimensional case.
\begin{Lem}\label{L:w-def} Let $S\subset\mathbb C^2$ be a compact strongly $\mathbb C$-convex hypersurface enclosing the origin. Then there is a uniquely-determined map \begin{align*} \mapdef{\mathscr D\colon S}{\mathbb C^2\setminus\{0\}}{z}{w(z)=(w_1(z),w_2(z))} \end{align*} satisfying \refstepcounter{equation}\label{N:w-def} \begin{enum} \item $z_1w_1+z_2w_2=1$ on $S$; \label{I:key-rel} \item the vector field \begin{equation*}\label{E:Ydef} Y\eqdef w_2\vf{z_1}-w_1\vf{z_2} \end{equation*}
is tangent to $S$. Moreover, $Y$ annihilates conjugate-CR functions on any relatively open subset of $S$. \label{I:tang(1,0)} \end{enum} \end{Lem}
\begin{proof} It is easy to check that \itemref{N:w-def}{I:key-rel} and \itemref{N:w-def}{I:tang(1,0)} force \begin{align*} w_1(z)&=\frac{\frac{\partial\rho}{\partial z_1}}{z_1\frac{\partial\rho}{\partial z_1}+z_2\frac{\partial\rho}{\partial z_2}}\\ w_2(z)&=\frac{\frac{\partial\rho}{\partial z_2}}{z_1\frac{\partial\rho}{\partial z_1}+z_2\frac{\partial\rho}{\partial z_2}}. \end{align*} establishing uniqueness. Existence follows provided that the denominators do not vanish; but the vanishing of the denominators occurs precisely when the complex tangent line for $S$ at $z$ passes through the origin, and Theorem \ref{T:gl-Cconv} above guarantees that this does not occur under the given hypotheses. \end{proof}
\begin{Remark} It is clear from the proof that the conclusions of Lemma \ref{L:w-def} also hold under the assumption that $S$ is a (not necessarily compact) hypersurface satisfying \begin{equation}\label{E:0spec} \text{no complex tangent line for $S$ passes through the origin.} \end{equation} \end{Remark}
\begin{Remark} Any tangential vector field annihilating conjugate-CR functions will be a scalar multiple of $Y$. \end{Remark}
\begin{Remark} The complex line tangent to $S$ at $z$ is given by \begin{equation}\label{E:Ctan} \{\zeta\in\mathbb C^2\colon w_1(z)\zeta_1+w_2(z)\zeta_2=1\}. \end{equation} \end{Remark}
\begin{Remark}\label{R:ann} The maximal complex subspace $H_zS$ of each $T_zS$ is annihilated by the form $w_1\,dz_1+w_2\,dz_2$. \end{Remark}
\begin{Prop}\label{P:loc-diff} For $S$ strongly $\mathbb C$-convex satisfying \eqref{E:0spec}, the map $\mathscr D$ is a local diffeomorphism onto an immersed strongly $\mathbb C$-convex hypersurface $S^*$, with each maximal complex subspace $H_zS$ of $T_zS$ mapped (non-$\mathbb C$-linearly) by $\mathscr D_z'$ onto the corresponding maximal complex subspace of $H_{w(z)}S^*$. For $S$ strongly $\mathbb C$-convex and compact, $S^*$ is an embedded strongly $\mathbb C$-convex hypersurface and $\mathscr D$ is a diffeomorphism. \end{Prop}
\begin{proof}\, \cite[\S 6]{Bar}, \cite[\S2.5]{APS}. \end{proof}
For $S$ strongly $\mathbb C$-convex satisfying \eqref{E:0spec} we may extend $\mathscr D$ to a smooth map on an open set in $\mathbb C^2$; the extended map $\mathscr D^\star$ will be a local diffeomorphism in some neighborhood $U$ of $S$. We may then define vector fields $\vf{w_1}, \vf{w_2}, \vf{\overline{w_1}}, \vf{\overline{w_2}}$ on $U$ by applying $\left(\left(\mathscr D^\star\right)^{-1}\right)'$ to the corresponding vector fields on $\mathscr D^\star(U)$; these newly-defined vector fields will depend on the choice of the extension $\mathscr D^\star$.
\begin{Lem}\label{L:Vdef} The non-vanishing vector field \begin{equation*} V\eqdef z_2\vf{w_1}-z_1\vf{w_2}\label{E:V-def} \end{equation*} is tangent to $S$ and is independent of the choice of the extension $\mathscr D^\star$. \end{Lem}
\begin{proof} From \itemref{N:w-def}{I:key-rel} we have \begin{align*} 0 &= d(z_1w_1+z_2w_2)\\ &= z_1\,dw_1+z_2\,dw_2 + w_1\,dz_1 + w_2\,dz_2 \end{align*} on $T_zS$.
From Remark \ref{R:ann} we deduce that the null space in $T_z\mathbb C^2$ of $z_1\,dw_1+z_2\,dw_2$ is precisely the maximal complex subspace $H_z S$ of $T_z S$ (and moreover the null space in $\left(T_z\mathbb C^2\right)\otimes\mathbb C$ of of $z_1\,dw_1+z_2\,dw_2$ is precisely $\left(H_z S\right)\otimes\mathbb C$). If we apply $z_1\,dw_1+z_2\,dw_2$ to $V$ we obtain \begin{equation*} z_1\cdot Vw_1+ z_2\cdot Vw_2 = z_1 \cdot z_2 - z_2 \cdot z_1=0 \end{equation*} showing that $V$ takes values in $\left(H_z S\right)\otimes\mathbb C$ and is thus tangential.
If an alternate tangential vector field $\widetilde V$ is constructed with the use of an alternate extension $\widetilde{\mathscr D^\star}$ of $\mathscr D$, then \begin{align*} \widetilde Vw_j&=\pm z_{3-j}=Vw_j\\ \widetilde V\overline w_j &= 0 = V\overline w_j \end{align*} along $S$, so $\widetilde V = V$ along $S$. \end{proof}
\begin{Def}\label{D:dCR} A function $u$ on a relatively open subset of $S$ will be called {\em dual-CR} if $\overline Vu=0$. \end{Def}
\begin{Ex} If $S$ is the unit sphere in $\mathbb C^2$, then $w(z)=\overline z$ and the set of dual-CR functions on $S$ coincides with the set of conjugate-CR functions on $S$. \end{Ex}
The set of dual-CR functions will only rarely coincide with the set of conjugate-CR functions as we see from the following two related results.
\begin{Thm} If $S$ is a compact strongly $\mathbb C$-convex hypersurface in $\mathbb C^2$, then the
set of dual-CR functions on $S$ will coincide with the the set of conjugate-CR functions on $S$ if and only if $S$ is a complex-affine image of the unit sphere. \end{Thm}
\begin{Thm} If $S$ is a strongly $\mathbb C$-convex hypersurface in $\mathbb C^2$, then the
set of dual-CR functions on $S$ will coincide with the the set of conjugate-CR functions on $S$ if and only if $S$ is locally the image of a relatively open subset of the unit sphere by a projective transformation. \end{Thm}
For proofs of these results see \cite{Jen}, \cite{DeTr}, and \cite{Bol}.
\begin{Remark} The constructions of the vector fields $Y$ and $V$ transform naturally under complex-affine mapping of $S$. The construction of the dual-CR structure transforms naturally under projective transformation of $S$. (See for example \cite[\S 6]{Bar}.) \end{Remark}
\begin{Lem}\label{L:YV-rel} Relations of the form \begin{align*} V&= \chi Y + \sigma \overline Y\\ Y&= \kappa V + \xi \overline V \end{align*} hold along $S$ with $\sigma$ and $\xi$ nowhere vanishing. \end{Lem}
\begin{proof} This follows from the following facts: \begin{itemize} \item $V, \overline V, Y$ and $\overline Y$ all take values in the two-dimensional space $\left(H_z S\right)\otimes\mathbb C$; \item $V$ and $\overline V$ are $\mathbb C$-linearly independent, as are $Y$ and $\overline Y$; \item the non-$\mathbb C$-linearity of the map $\mathscr D'_z\colon \left(H_z S\right)\otimes\mathbb C\to \left(H_z S^*\right)\otimes\mathbb C$ (see Proposition \ref{P:loc-diff}). \end{itemize}
\end{proof}
\begin{Lem} If $f_1, f_2$ are CR functions and $g_1, g_2$ are dual-CR functions on a connected relatively open subset $W$ of $S$ with $f_1+g_1=f_2+g_2$, then $g_2-g_1=f_1-f_2$ is constant. \end{Lem}
\begin{proof} From Lemma \ref{L:YV-rel} we deduce that the directional derivatives of $g_2-g_1=f_1-f_2$ vanish in every direction belonging to the maximal complex subspace of $TS$. Applying one Lie bracket we find that in fact all directional derivatives along $S$ of $g_2-g_1=f_1-f_2$ vanish. \end{proof}
\begin{Cor}\label{C:sc} If $W$ is a simply-connected relatively open subset of $S$ and $u$ is a function on $W$ that is locally decomposable as the sum of a CR function and a dual-CR function, then $u$ is decomposable on all of $W$ as the sum of a CR function and a dual-CR function. \end{Cor}
\subsection{Circular hypersurfaces in $\mathbb C^2$} \label{S:cir-proj}
We begin the section with an expanded restatement of the main theorem in the projective setting.
\begin{MThm-rev} \label{T:cir-proj-rev} Let $S\subset\mathbb C^2$ be a strongly ($\mathbb C$-)convex circular hypersurface. Then there exist scalar functions $\phi$ and $\psi$ on $S$ so that the vector fields \begin{subequations}\label{E:aa} \begin{align} X&= V+\phi \overline{V}\\ T&= Y+\psi \overline{Y} \end{align} \end{subequations} satisfy the following conditions. \refstepcounter{equation}\label{N:XT-cond} \begin{enum}
\item If $u$ is a smooth function on a relatively open subset of $S$, then $u$ is CR if and only if $Xu=0$; equivalently, $X$ is a non-vanishing scalar multiple $\alpha \overline Y$ of $\overline Y$.\label{I:bb} \item If $u$ is a smooth function on a relatively open subset of $S$, then $u$ is dual-CR if and only if $Tu=0$; equivalently, $T$ is a non-vanishing scalar multiple $\beta \overline V$ of $\overline V$.\label{I:cc} \item If $S$ is compact, then a smooth function $u$ on $S$ is the sum of a CR function and a dual-CR function if and only if $XXTu=0$. \label{I:cir-proj-glo} \item If $S$ is simply-connected (but not necessarily compact), then a smooth function $u$ on $S$ is the sum of a CR function and a dual-CR function if and only if $XXTu=0=TTXu$. \label{I:cir-proj-loc} \end{enum} \end{MThm-rev}
As we shall see the vector field $X$ in Theorem \ref{T:cir-proj} will also work as the vector field $X$ in Theorem \ref{T:cir-plh}.
\begin{Ex} (Compare \cite{Aud}.) The function $\frac{z_1}{w_2}$ satisfies $XXT\frac{z_1}{w_2}=0$ but is not globally defined. Since $TTX\frac{z_1}{w_2}=2\ne 0$ this function is not locally the sum of a CR function and a dual-CR function. \end{Ex}
Conditions \eqref{E:aa}, \itemref{N:XT-cond}{I:bb} and \itemref{N:XT-cond}{I:cc} uniquely determine $X$ and $T$. See \S \ref{S:uniq} for some discussion of what can happen without condition \eqref{E:aa}.
\section{Proof of Theorem \ref{T:cir-proj}}\label{S:pfs}
To prove Theorem \ref{T:cir-proj} we start by consulting Lemma \ref{L:YV-rel} and note that \eqref{E:aa}, \itemref{N:XT-cond}{I:bb} and \itemref{N:XT-cond}{I:cc} will hold if we set \begin{align*} \alpha&=1/\overline\xi,&
\beta&=1/\overline\sigma,\\
\phi&=\overline\kappa/\overline\xi,&
\psi&=\overline\chi/\overline\sigma; \end{align*} it remains to check \itemref{N:XT-cond}{I:cir-proj-glo} and \itemref{N:XT-cond}{I:cir-proj-loc}.
We note for future reference and the reader's convenience that \begin{align} Xw_1&=z_2& Xw_2&=-z_1\notag\\ \overline Yw_1&=\overline\xi z_2& \overline Yw_2&=-\overline\xi z_1\notag\\ X\overline{w}_1 &= \phi \overline{z}_2 & X \overline{w}_2&=-\phi \overline{z}_1\notag\\ Xz_1&=\overline Yz_1=0& Xz_2&=\overline Yz_2=0\notag\\ X\overline z_1 &= \alpha \overline w_2& X\overline z_2 &= -\alpha \overline w_1\notag\\%needed? Tz_1&=w_2& Tz_2 &= -w_1\label{E:diff-rules}\\ \overline V z_1 &= \overline\sigma w_2& \overline V z_2 &= -\overline\sigma w_1\notag\\ T \overline{z}_1 &= \psi \overline{w}_2 & T\overline{z}_2 &= -\psi \overline{w}_1 \notag\\ Tw_1&=\overline Vw_1=0& Tw_2&=\overline Vw_2=0\notag\\ T\overline w_1 &= \beta z_2& T\overline w_2 &= -\beta z_1.\notag \end{align}
\begin{Lem}\label{L:YVbrack} \begin{align*} [Y,\overline Y] &= \overline\xi \left(z_1 \vf{ z_1}+ z_2 \vf{ z_2}\right)-\xi \left(\overline z_1 \vf{\overline z_1} + \overline z_2 \vf{\overline z_2}\right)
\\ [V,\overline V] &= \overline\sigma \left(w_1 \vf{ w_1}+ w_2 \vf{ w_2}\right) -\sigma \left(\overline w_1 \vf{\overline w_1} + \overline w_2 \vf{\overline w_2}\right).
\end{align*} \end{Lem}
\begin{proof} The first statement follows from \begin{equation*} [Y,\overline Y] = \left(Y\overline w_2\right)\vf{\overline z_1}-\left(Y\overline w_1\right)\vf{\overline z_2} -\left(\overline Y w_2\right)\vf{ z_1}+\left(\overline Y w_1\right)\vf{ z_2} \end{equation*} along with \eqref{E:diff-rules}.
The proof of the second statement is similar. \end{proof}
We note that the assumption that $S$ is circular has not been used so far in this section. We now bring it into play by introducing the real tangential vector field \begin{equation*} R \eqdef i \left( z_1 \vf{z_1}+ z_2 \vf{z_2} -
\overline z_1 \vf{\overline z_1}- \overline z_2 \vf{\overline z_2}\right) \end{equation*} generating the rotations of $z\mapsto e^{i\theta}z$ of $S$.
\begin{Lem} The following hold. \refstepcounter{equation}\label{N:rot-lem1} \begin{enum} \item $\overline\xi=\xi$ \label{I:xir} \item $\overline\sigma=\sigma$ \label{I:sigmar} \item $\overline\alpha=\alpha$ \label{I:alphar} \item $\overline\beta=\beta$ \label{I:betar} \item $R=-i \left( w_1 \vf{w_1}+ w_2 \vf{w_2} -
\overline w_1 \vf{\overline w_1}- \overline w_2 \vf{\overline w_2}\right)$ \label{I:wrot}
\item $[Y,\overline Y]=-i\xi R$ \label{I:YbarY}
\item $[V,\overline V]=i\sigma R$ \label{I:VbarV}
\item $[X,Y]=iR-(Y\alpha)\overline Y$ \label{I:XY*} \end{enum} \end{Lem}
\begin{proof} We start by considering the tangential vector field \begin{equation*} [Y,\overline Y]+i\xi R = (\overline \xi - \xi)\left( z_1 \vf{z_1}+ z_2 \vf{z_2}\right); \end{equation*} if \itemref{N:rot-lem1}{I:xir} fails, then $z_1 \vf{z_1}+ z_2 \vf{z_2}$ is a non-vanishing holomorphic tangential vector field on some non-empty relatively open subset of $S$, contradicting the strong pseudoconvexity of $S$.
To prove \itemref{N:rot-lem1}{I:wrot} we first note from Lemma \ref{L:w-def} that $w\left(e^{i\theta}z\right)=e^{-i\theta} w(z)$; differentiation with respect to $\theta$ yields \itemref{N:rot-lem1}{I:wrot}.
The proof of \itemref{N:rot-lem1}{I:xir} now may be adapted to prove \itemref{N:rot-lem1}{I:sigmar}. \itemref{N:rot-lem1}{I:alphar} and \itemref{N:rot-lem1}{I:betar} follow immediately.
Using Lemma \ref{L:YVbrack} in combination with \itemref{N:rot-lem1}{I:xir} and \itemref{N:rot-lem1}{I:sigmar} we obtain \itemref{N:rot-lem1}{I:YbarY} and \itemref{N:rot-lem1}{I:VbarV}.
From \itemref{N:XT-cond}{I:bb} and \itemref{N:rot-lem1}{I:YbarY} we obtain \itemref{N:rot-lem1}{I:XY}. \end{proof}
\begin{Lem}\label{L:XTbrack} $[X,T]=iR$. \end{Lem}
\begin{proof} On the one hand, \begin{align*} [X,T] &= [V+\phi\overline V,\beta\overline V]\\ &=\left( (V+\phi\overline V)\beta-\beta(\overline V\phi) \right)\overline V + i \beta\sigma R\\ &= \left( (V+\phi\overline V)\beta-\beta(\overline V\phi) \right)\overline V + i R. \end{align*} On the other hand, \begin{align*} [X,T] &= [\alpha \overline Y,Y + \psi\overline Y]\\ &=\left( \alpha(\overline Y\psi)-(Y+\psi\overline Y)\alpha \right)\overline Y + i \alpha \xi R\\ &=\left( \alpha(\overline Y\psi)-(Y+\psi\overline Y)\alpha \right)\overline Y + i R. \end{align*} Since $\overline V$ and $\overline Y$ are linearly independent, it follows that $[X,T]= i R$. \end{proof}
\begin{Lem} The following hold. \refstepcounter{equation}\label{N:Rbrack} \begin{enum} \item $[R,Y]=-2iY$ \label{I:RY} \item $[R,\overline Y]=2i\overline Y$ \label{I:RbY} \item $[R,V]=2iV$ \label{I:RV} \item $[R,\overline V]=-2i\overline V$ \label{I:RbV} \item $[R,X]=2iX$ \label{I:RX} \item $[R,\overline X]=-2i\overline X$ \label{I:RbX} \item $[R,T]=-2iT$ \label{I:RT} \item $[R,\overline T]=2i\overline T$ \label{I:RbT} \item $R\alpha=0$ \label{I:Ralpha} \item $R\beta=0$ \label{I:Rbeta} \end{enum} \end{Lem}
\begin{proof} \itemref{N:Rbrack}{I:RY}, \itemref{N:Rbrack}{I:RbY}, \itemref{N:Rbrack}{I:RV} and \itemref{N:Rbrack}{I:RbV} follow from direct calculation.
For \itemref{N:Rbrack}{I:RT} first note that writing $T=\beta\overline V$ and using \itemref{N:Rbrack}{I:RbV} we see that $[R,T]$ is a scalar multiple of $T$. Then writing \begin{equation*} [R,T] = [R,Y+\psi\overline Y]= -2iY + (\text{multiple of }\overline Y) \end{equation*} we conclude using \eqref{E:aa} that $[R,T]=-2iT$. The proof of \itemref{N:Rbrack}{I:RX} is similar, and \itemref{N:Rbrack}{I:RbX} and \itemref{N:Rbrack}{I:RbT} follow by conjugation.
Using \itemref{N:XT-cond}{I:bb} along with \itemref{N:Rbrack}{I:RbY} and \itemref{N:Rbrack}{I:RX} we obtain \itemref{N:Rbrack}{I:Ralpha}; \itemref{N:Rbrack}{I:Rbeta} is proved similarly. \end{proof}
\begin{Lem}\label{L:XXker} $XXf=0$ if and only if $f=f_1w_1+f_2w_2$ with $f_1, f_2$ CR. \end{Lem}
\begin{proof} From \itemref{N:XT-cond}{I:bb} and \eqref{E:diff-rules} it is clear that $XX\left(f_1w_1+f_2w_2\right)=0$ if $f_1$ and $ f_2$ are CR.
For the other direction, suppose that $XXf=0$. Then setting \begin{align*} f_1&\eqdef z_1f+w_2Xf\\ f_2&\eqdef z_2f-w_1Xf. \end{align*}
it is clear that $f=f_1w_1+f_2w_2$; with the use of \itemref{N:XT-cond}{I:bb} and \eqref{E:diff-rules} it is also easy to check that $f_1$ and $f_2$ are CR. \end{proof}
\begin{Lem}\label{L:proj-1eq} Suppose that $XXTu=0$ so that by Lemma \ref{L:XXker} we may write $Tu=f_1w_1+f_2w_2$ with $f_1, f_2$ CR. Then \begin{equation}\label{E:proj-2op} TTXu = \frac{\partial f_1}{\partial z_1} + \frac{\partial f_2}{\partial z_2}. \end{equation} In particular, $TTXu$ is CR. \end{Lem}
The non-tangential derivatives appearing in \eqref{E:proj-2op} may be interpreted using the Hans Lewy local CR extension result previously mentioned in the proof of Theorem \ref{T:loc-prob}, or else by rewriting them in terms of tangential derivatives (as in the last step of the proof below).
\begin{proof} We have \begin{align*} TTXu &= TXTu + T[T,X]u \\ &= TX\left(f_1w_1+f_2w_2\right)-iTRu && \text{(Lemma \ref{L:XTbrack})}\\ &= T\left(f_1z_2-f_2z_1\right) - iRTu - i[T,R]u && \text{\itemref{N:XT-cond}{I:bb}, \eqref{E:diff-rules} }\\ &= T\left(f_1z_2-f_2z_1\right) - iR\left(f_1w_1+f_2w_2\right) + 2Tu && \text{ \itemref{N:Rbrack}{I:RT} }\\ &= \left(Tf_1\right)z_2 - f_1w_1 - \left(Tf_2\right)z_1 - f_2w_2 \\ &\qquad -i\left(Rf_1\right)w_1 - f_2w_2 - i\left(Rf_2\right)w_2 - f_2 w_2 \\ &\qquad + 2\left(f_1w_1+f_2w_2\right) && \text{\eqref{E:diff-rules}, \itemref{N:rot-lem1}{I:wrot} }\\ &= \left(z_2 T - i w_1 R\right) f_2 - \left( z_1 T + iw_2 R\right) f_2\\ &= \left(z_2 Y - i w_1 R\right) f_2 - \left( z_1 Y + iw_2 R\right) f_2\\ &= \frac{\partial f_1}{\partial z_1} + \frac{\partial f_2}{\partial z_2}. \end{align*} \end{proof}
\begin{Lem}\label{L:spec-2ord} The following hold. \refstepcounter{equation}\label{N:spec-2ord} \begin{enum} \item The operator $XT$ maps CR functions to CR functions. \label{I:XT} \item The operator $XY$ maps CR functions to CR functions. \label{I:XY} \item The operator $TX$ maps dual-CR functions to dual-CR functions. \label{I:TX} \item The operator $\overline{XY}$ maps conjugate-CR functions to conjugate-CR functions. \label{I:bar-XY} \end{enum} \end{Lem}
\begin{proof} To prove \itemref{N:spec-2ord}{I:XT} and \itemref{N:spec-2ord}{I:XY} note that for $u$ CR we have $XTu=XYu=-z_1\frac{\partial u}{\partial z_1}-z_2\frac{\partial u}{\partial z_2}$ which is also CR. The other proofs are similar. \end{proof}
\begin{proof}[Proof of \itemref{N:XT-cond}{I:cir-proj-loc}] To get the required lower bound on the null spaces, it will suffice to show that $XXT$ and $TTX$ annihilate CR functions and dual-CR functions. This follows from \itemref{N:XT-cond}{I:bb} and \itemref{N:XT-cond}{I:cc} along with \itemref{N:spec-2ord}{I:XT} and \itemref{N:spec-2ord}{I:TX}.
For the other direction, if $XXTu=0=TTXu$, then from Lemma \ref{L:proj-1eq} we have a closed 1-form $\omega\eqdef f_2\,dz_1 - f_1\,dz_2$ on $S$ where $f_1$ and $f_2$ are CR functions satisfying $Tu=f_1w_1+f_2w_2$. Since $S$ is simply-connected we may write $\omega=df$ with $f$ CR. Then from \eqref{E:aa} we have \begin{align*} Tf &= Yf\\ &=w_2f_2+w_1f_1\\ &= Tu. \end{align*} Thus $u$ is the sum of the CR function $f$ and the dual-CR function $u-f$. \end{proof}
To set up the proof of the global result \itemref{N:XT-cond}{I:cir-proj-glo} we introduce the form \begin{equation}\label{E:nu-def} \nu\eqdef (z_2\,dz_1-z_1\,dz_2)\wedge dw_1\wedge dw_2 \end{equation} and the $\mathbb C$-bilinear pairing \begin{equation}\label{E:pair-def} \langle\langle \mu,\eta\rangle\rangle \eqdef \int\limits_S \mu\eta\cdot\nu \end{equation} between functions on $S$ (but see Technical Remark \ref{R:tech} below).
\begin{Lem}\label{L:parts} $\langle\langle T\gamma, \eta \rangle\rangle=-\langle\langle \gamma, T\eta \rangle\rangle$. \end{Lem} \begin{proof}
\allowdisplaybreaks \begin{align*} \langle\langle T\gamma, \eta \rangle\rangle+\langle\langle \gamma, T\eta \rangle\rangle &= \int\limits_S T(\gamma\eta)\cdot\nu\\ &= \int\limits_S \iota_T d(\gamma\eta)\cdot\nu\\ &= \int\limits_S d(\gamma\eta)\cdot \iota_T \nu\\ &= \int\limits_S d(\gamma\eta \cdot \iota_T \nu)- \int\limits_S \gamma\eta\cdot d(\iota_T \nu)\\ &= 0- \int\limits_S \gamma\eta\cdot d(\iota_T ((z_2\,dz_1-z_1\,dz_2)\wedge dw_1\wedge dw_2)\\ &= - \int\limits_S \gamma\eta\cdot d((z_2\cdot Tz_1-z_1\cdot Tz_2)\cdot dw_1\wedge dw_2)\\ &\qquad\qquad +\int\limits_S \gamma\eta\cdot d( (z_2\,dz_1-z_1\,dz_2)\cdot Tw_1\wedge dw_2)\\ &\qquad\qquad -\int\limits_S \gamma\eta\cdot d( (z_2\,dz_1-z_1\,dz_2)\wedge dw_1\cdot Tw_2)\\ &= - \int\limits_S \gamma\eta\cdot d((z_2w_2+z_1w_1) \,dw_1\wedge dw_2)+0-0\\ &= - \int\limits_S \gamma\eta\cdot d(dw_1\wedge dw_2)\\ &=0. \end{align*} Here we have quoted \begin{itemize} \item the definition \eqref{E:pair-def} of the pairing $\langle\langle\cdot\rangle\rangle$; \item the Leibniz rule $\iota_T(\varphi_1\wedge\varphi_2)=(\iota_T\varphi_1)\wedge\varphi_2+(-1)^{\deg\varphi_1}\varphi_1\wedge(\iota_T\varphi_2)$ for the interior product $\iota_T$; \item the fact that $S$ is integral for 4-forms; \item Stokes' theorem; \item the rules \eqref{E:diff-rules}; \item the relation \itemref{N:w-def}{I:key-rel}. \end{itemize} \end{proof}
\begin{Thm}\label{T:pair} Let $\mu$ be a CR function on a compact strongly $\mathbb C$-convex hypersurface $S$. Then $\mu=0$ if and only if $\langle\langle \mu, \eta \rangle\rangle =0$ for all dual-CR $\eta$ on $S$. \end{Thm}
\begin{proof}\, \cite[(4.3d) from Theorem 3]{Bar}. (Note also definition enclosing \cite[(4.2)]{Bar}.) \end{proof}
\begin{proof}[Proof of \itemref{N:XT-cond}{I:cir-proj-glo}] Assume that $XXTu=0$. Noting that $S$ is simply-connected, from \itemref{N:XT-cond}{I:cir-proj-loc} it suffices to prove that $TTXu=0$. From Lemma \ref{L:proj-1eq} we know that $TTXu$ is CR. By Theorem \ref{T:pair} it will suffice to show that \begin{equation*} \langle\langle TTXu, \eta \rangle\rangle=0 \end{equation*} for dual-CR $\eta$. But from Lemma \ref{L:parts} we have \begin{align*} \langle\langle TTXu, \eta \rangle\rangle&=- \langle\langle TXu, T\eta \rangle\rangle\\ &=0 \end{align*} as required. \end{proof}
\begin{Remark}\label{R:pd} From symmetry of formulas in Lemma \ref{L:w-def} and \ref{L:Vdef} we have that $X_{S^*}=\mathscr D_* T_{S}, T_{S^*}=\mathscr D_* X_{S}$ and $S^{**}=S$. These facts serve to explain why the formulas throughout this section appear in dual pairs. \end{Remark}
\begin{Tech}\label{R:tech} In \cite{Bar} the pairing \eqref{E:pair-def} applies not to functions $\mu,\nu$ but rather to forms $\mu(z)\,(dz_1\wedge dz_2)^{2/3}$, $\mu(w)\,(dw_1\wedge dw_2)^{2/3}$; the additional notation is important in \cite{Bar} for keeping track of invariance properties under projective transformation but is not needed here.
Note also that \eqref{E:pair-def} coincides (up to a constant) with the pairing (3.1.8) in \cite{APS} with $s=w_1\,dz_1+w_2\,dz_2$. \end{Tech}
\section{Proof of Theorem \ref{T:cir-plh}}\label{S:pfs2}
For the reader's convenience we restate the main theorem in the conjugate setting.
\conj*
It is not possible in general to have $Y=\overline X$.
\begin{Lem}\label{L:prlh-1eq} Suppose that $XXYu=0$ so that by Lemma \ref{L:XXker} we may write $Yu=f_1w_1+f_2w_2$ with $f_1, f_2$ CR. Then \begin{equation}\label{E:plh-2op} \overline{XXY}u = \alpha\left(\frac{\partial f_1}{\partial z_1} + \frac{\partial f_2}{\partial z_2}\right). \end{equation} In particular, $\alpha^{-1}\overline{XXY}u$ is CR. \end{Lem}
\begin{proof} We have \begin{align*} \allowdisplaybreaks \overline{XXY}u &= \overline{XYX}u + \overline{X} [\overline X,\overline Y] u\\ &=\overline{XY} \left( \alpha \left(f_1w_1+f_2 w_2\right) \right) +\overline{X}\left( -iR-(\overline Y\alpha)Y\right) u && \text{\itemref{N:XT-cond}{I:bb}, \itemref{N:rot-lem1}{I:alphar}, \itemref{N:rot-lem1}{I:XY}}\\ &=\overline X\left( \alpha\overline Y \left( f_1w_1+f_2 w_2 \right)\right) -i\overline X R u \\ &=\overline X\left(f_1z_2-f_2 z_1 \right) -iR \overline X u -i[\overline X, R] u && \text{\itemref{N:XT-cond}{I:bb}, \eqref{E:diff-rules}}\\ &= \overline X\left(f_1z_2-f_2 z_1 \right) -iR \left( \alpha \left( f_1w_1+f_2 w_2 \right) \right) +2 \overline X u && \text{\itemref{N:XT-cond}{I:bb}, \itemref{N:Rbrack}{I:RbX}}\\ &= (\overline X f_1)\cdot z_2-f_1\cdot\alpha w_1-(\overline X f_2) \cdot z_1 - f_2 \cdot\alpha w_2\\ &\qquad - i \alpha \left( (Rf_1)\cdot w_1-f_1\cdot(iw_1)+(Rf_2)\cdot w_2-f_2\cdot(iw_2) \right) \\ &\qquad + 2 \alpha \left(f_1w_1+f_2 w_2\right) && \text{\eqref{E:diff-rules}, \itemref{N:Rbrack}{I:Ralpha}, \itemref{N:rot-lem1}{I:wrot}, \itemref{N:XT-cond}{I:bb}}\\ &= (\overline X f_1)\cdot z_2-(\overline X f_2) \cdot z_1
- i \alpha \left( (Rf_1)\cdot w_1+(Rf_2)\cdot w_2 \right)
\\
&= \alpha\left(z_2Y-iw_1R)f_1-(z_1Y+iw_2R)f_2\right)\\ &= \alpha \left( \frac{\partial f_1}{\partial z_1}+ \frac{\partial f_2}{\partial z_2}\right).\\ \end{align*} \end{proof}
\begin{proof}[Proof of \itemref{N:XY-cond}{I:cir-plh-loc}] To get the required lower bound on the null spaces, it will suffice to show that $XXY$ and $\overline{XXY}$ annihilate CR functions and conjugate-CR functions. This follows from \itemref{N:XY-cond}{I:aaa} along with \itemref{N:spec-2ord}{I:XY} and \itemref{N:spec-2ord}{I:bar-XY}.
For the other direction, if $XXYu=0=\overline{XXY}u$, then from Lemma \ref{L:proj-1eq} we have a closed 1-form $\widetilde\omega\eqdef f_2\,dz_1 - f_1\,dz_2$ on the open subset of $S$ where $f_1$ and $f_2$ are CR functions satisfying $Yu=f_1w_1+f_2w_2$. Restricting our attention to a simply-connected subset, we may write $\omega=d f$ with $ f$ CR. Then we have \begin{align*} Y f &=w_2f_2+w_1f_1\\ &= Yu. \end{align*} Thus $u$ is the sum of the CR function $f$ and the conjugate-CR function $u-f$.
The general case follows by localization. \end{proof}
\begin{Lem}\label{L:Ydiv} $\operatorname{div} Y\eqdef \frac{\partial w_2}{\partial z_1} - \frac{\partial w_1}{\partial z_2}$ and $\operatorname{div} \overline Y\eqdef {\frac{\partial \overline w_2}{\partial \overline z_1}} - \overline{\frac{\partial \overline w_1}{\partial \overline z_2}}$ vanish on $S$. \end{Lem}
\begin{proof} Since $S$ is circular, any defining function $\rho$ for $S$ will satisfy $\operatorname{Im} \left( z_1\frac{\partial\rho}{\partial z_1}+z_2\frac{\partial\rho}{\partial z_2} \right) =-\frac{R\rho}{2} = 0$. Adjusting our choice of defining function we may arrange that $z_1\frac{\partial\rho}{\partial z_1}+z_2\frac{\partial\rho}{\partial z_2}\equiv 1$ in some neighborhood of $S$. Then from the proof of Lemma \ref{L:w-def} we have $\frac{\partial w_2}{\partial z_1} - \frac{\partial w_1}{\partial z_2}=\frac{\partial^2 \rho}{\partial z_1 \partial z_2}- \frac{\partial^2 \rho}{\partial z_2 \partial z_1}=0$.
The remaining statement follows by conjugation. \end{proof}
\begin{Lem}\label{L:parts-plh} $\displaystyle{\int\limits_S \left(X\gamma\right)\eta\,\frac{dS}{\alpha} = - \int\limits_S \gamma\left(X\eta\right)\,\frac{dS}{\alpha}}$ \end{Lem} \begin{proof} \begin{align*} \int\limits_S \left(X\gamma\right)\eta\,\frac{dS}{\alpha} &= \int\limits_S \left(\overline Y\gamma\right)\eta\,dS && \text{\itemref{N:XT-cond}{I:bb}}\\ &= -\int\limits_S \gamma\left(\overline Y\eta\right)\,dS && \text{(Lemma \ref{L:Ydiv})}\\ &= - \int\limits_S \gamma\left(X\eta\right)\,\frac{dS}{\alpha} && \text{\itemref{N:XT-cond}{I:bb}} \end{align*} (The integration by parts above may be justified by applying the divergence theorem on a tubular neighborhood of $S$ and passing to a limit.)
\end{proof}
\begin{proof}[Proof of \itemref{N:XY-cond}{I:cir-plh-glo}] Assume that $XXYu=0$. Noting that $S$ is simply-connected, from \itemref{N:XY-cond}{I:cir-plh-loc} it suffices to prove that $\overline{XXY}u=0$. From Lemma \ref{L:proj-1eq} we know that $\alpha^{-1}\overline{XXY}u$ is CR. The desired conclusion now follows from \begin{align*}
\int\limits_S \left|\overline{XXY}u\right|^2\,\frac{dS}{\alpha^2} &=\int\limits_S \alpha^{-1} \overline{XXY}u\cdot {XXY}\overline u\,\frac{dS}{\alpha}\\ &= - \int\limits_S X\left( \alpha^{-1}\overline{XXY}u \right) \cdot {XY}\overline u\,\frac{dS}{\alpha} &&\text{(Lemma \ref{L:parts-plh})}\\ &= - \int\limits_S 0 \cdot {XY}\overline u\,\frac{dS}{\alpha} & & \text{(Lemma \ref{L:prlh-1eq})}\\ &=0. \end{align*}
\end{proof}
\section{Further comments}\label{S:furth}
\subsection{Remarks on uniqueness}\label{S:uniq}
\begin{Prop}\label{T:plh-un} Suppose that in the setting of Theorem \ref{T:cir-proj} we have vector fields $\widetilde X,\widetilde T$ satisfying (suitably-modified) \itemref{N:XT-cond}{I:bb} and \itemref{N:XT-cond}{I:cc}. Then $\widetilde X\widetilde X\widetilde T$ annihilates CR functions and dual-CR functions if and only if there are CR functions $f_1, f_2$ and $f_3$ so that $f_1w_1+f_2w_2$ and $f_3$ are non-vanishing and \begin{align*} \widetilde X&= f_3(f_1w_1+f_2w_2)^2 X\\ \widetilde T&= \frac{1}{f_1w_1+f_2w_2} T. \end{align*} \end{Prop}
\begin{proof} From \itemref{N:XT-cond}{I:bb} and \itemref{N:XT-cond}{I:cc} we have $\widetilde X=\gamma X$, $\widetilde T=\eta T$ with non-vanishing scalar functions $\gamma$ and $\eta$.
Suppose that $\widetilde X\widetilde X\widetilde T$ annihilates CR functions and dual-CR functions. By routine computation we have \begin{equation*} \widetilde X\widetilde X\widetilde T = \gamma^2\eta XXT + \gamma\Big( \left(2\gamma(X\eta)+\eta(X\gamma)\right) XT + (X(\gamma(X\eta))T\Big). \end{equation*}
The operator $\left(2\gamma(X\eta)+\eta(X\gamma)\right) XT + (X(\gamma(X\eta))T$ must in particular annihilate CR functions. But if $f$ is CR, then using Lemma \ref{L:XTbrack} we have \begin{equation*} \Big( \left(2\gamma(X\eta)+\eta(X\gamma)\right) XT + (X(\gamma(X\eta))T\Big)f = \Big( i\left(2\gamma(X\eta)+\eta(X\gamma)\right) R + (X(\gamma(X\eta))T\Big)f \end{equation*} Since $R$ and $T$ are $\mathbb C$-linearly independent and $f$ is arbitrary it follows that we must have \begin{align*} X(\gamma\eta^2)=2\gamma(X\eta)+\eta(X\gamma) &= 0\\ X(\gamma(X\eta) &= 0. \end{align*}
We set $f_3=\gamma\eta^2$ which is CR and non-vanishing. Then the second equation above yields \begin{align*} -f_3 \cdot XX(\eta^{-1}) &= X\left(f_3 \,\eta^{-2}(X\eta)\right)\\ &=X(\gamma(X\eta)\\ &=0 \end{align*} and hence $XX(\eta^{-1})=0$. From Lemma \ref{L:XXker} we have $\eta=\frac{1}{f_1w_1+f_2w_2}$ with $f_1$ and $f_2$ CR. The result now follows.
The converse statement follows by reversing steps. \end{proof}
\begin{Prop} Suppose that in the setting of Theorem \ref{T:cir-plh} we have vector fields $\widetilde X,\widetilde T$ satisfying (suitably-modified) \itemref{N:XY-cond}{I:aaa} and \itemref{N:XY-cond}{I:bbb}. Then $\widetilde X\widetilde X\widetilde Y$ annihilates CR functions and conjugate-CR functions if and only if there are CR functions $f_1, f_2$ and $f_3$ so that $f_1w_1+f_2w_2$ and $f_3$ are non-vanishing and \begin{align*} \widetilde X&= f_3(f_1w_1+f_2w_2)^2 X\\ \widetilde Y&= \frac{1}{f_1w_1+f_2w_2} Y. \end{align*} \end{Prop}
The proof is similar to that of Proposition \ref{T:plh-un}, using \itemref{N:rot-lem1}{I:XY*} in place of Lemma \ref{L:XXker}.
\subsection{Nirenberg-type result}\label{S:Nir}
\begin{Prop}\label{P:Nir-plh}
Given a point $p$ on a strongly pseudoconvex hypersurface $S\subset\mathbb C^2$, any 2-jet at $p$ of a $\mathbb C$-valued function on $S$ is the 2-jet of the restriction to $S$ of a pluriharmonic function on $\mathbb C^2$. \end{Prop}
\begin{proof} After performing a standard local biholomorphic change of coordinates we may reduce to the case where $p=0$ and $S$ is described near 0 by an equation of the form
\[y_2=z_1\overline z_1+O(\|(z_1,x_2)\|)^3.\] The projection $(z_1,x_2+iy_2)\mapsto (z_1,x_2)$ induces a bijection between 2-jets at 0 along $S$ and 2-jets at 0 along $\mathbb C\times\mathbb R$. It suffices now to note that the 2-jet \begin{equation*} A+Bz_1+C\overline{z}_1+Dx_2+E z_1^2+ F \overline{z}_1^2 + G z_1\overline{z}_1 + H z_1 x_2 + I \overline{z}_1 x_2+Jx_2^2 \end{equation*} is induced by the pluriharmonic polynomial \begin{equation*} A+Bz_1+C\overline{z}_1+\frac{D-iG}{2} z_2 + \frac{D+iG}{2} \overline{z}_2+ E z_1^2+ F \overline{z}_1^2 + Hz_1z_2 + I \overline{z}_1\overline{z}_2 + J\overline{z}_2^2. \end{equation*}
\end{proof}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Quantum speedup of classical mixing processes\thanks{This material is based upon work supported by the National Science Foundation under Grant No. 0523866. Part of this research was done while the author was visiting LRI, Universit\'{e} Paris-Sud, Orsay, France.}} \author{Peter C. Richter\thanks{Department of Computer Science, Rutgers University, Piscataway, NJ 08854. {\tt [email protected]}}} \date{} \maketitle
\begin{abstract} Most approximation algorithms for \#P-complete problems (e.g., evaluating the permanent of a matrix or the volume of a polytope) work by reduction to the problem of approximate sampling from a distribution $\pi$ over a large set $\mathcal{S}$. This problem is solved using the {\em Markov chain Monte Carlo} method: a sparse, reversible Markov chain $P$ on $\mathcal{S}$ with stationary distribution $\pi$ is run to near equilibrium. The running time of this random walk algorithm, the so-called {\em mixing time} of $P$, is $O(\delta^{-1} \log 1/\pi_*)$ as shown by Aldous, where $\delta$ is the spectral gap of $P$ and $\pi_*$ is the minimum value of $\pi$. A natural question is whether a speedup of this classical method to $O(\sqrt{\delta^{-1}} \log 1/\pi_*)$, the diameter of the graph underlying $P$, is possible using {\em quantum walks}.
We provide evidence for this possibility using quantum walks that {\em decohere} under repeated randomized measurements. We show: (a) decoherent quantum walks always mix, just like their classical counterparts, (b) the mixing time is a robust quantity, essentially invariant under any smooth form of decoherence, and (c) the mixing time of the decoherent quantum walk on a periodic lattice $\mathbb{Z}_n^d$ is $O(n d \log d)$, which is indeed $O(\sqrt{\delta^{-1}} \log 1/\pi_*)$ and is asymptotically no worse than the diameter of $\mathbb{Z}_n^d$ (the obvious lower bound) up to at most a logarithmic factor. \end{abstract}
\section{Introduction}
\subsection{Markov chain Monte Carlo and quantum walks}
A rich theory has been developed for computing approximate solutions to problems in combinatorial enumeration and statistical physics which are \#P-complete and therefore unlikely to have efficiently computable exact solutions. Among the highlights are randomized polynomial-time approximation schemes for evaluating the permanent of a nonnegative matrix (Jerrum, Sinclair, and Vigoda \cite{JSV}), the volume of a convex polytope (Dyer, Frieze, and Kannan \cite{DFK}), and the partition functions of monomer-dimer and ferromagnetic Ising systems (Jerrum and Sinclair \cite{JS1,JS2}). At the heart of these algorithms (and consuming most of the running time) is a subroutine for approximate sampling from a particular distribution $\pi$ over a large set $\mathcal{S}$. This problem is solved using the {\em Markov chain Monte Carlo} (MCMC) method: a sparse, reversible {\em Markov chain} (stochastic matrix) $P$ on $\mathcal{S}$ with stationary distribution $\pi$ is run from an arbitrary initial state to a random state distributed very close to $\pi$. The time required to guarantee this convergence, or {\em mixing}, is the so-called {\em mixing time} $\tau_{\operatorname{mix}}$. Bounding $\tau_{\operatorname{mix}}$ is typically the chief technical hurdle in analyzing the running time of the overall algorithm. It is reduced to the problem of estimating the spectral gap $\delta$ of $P$ by Aldous' inequality \cite{Ald} \begin{equation}\label{aldous} \delta^{-1} \leq \tau_{\operatorname{mix}} \leq \delta^{-1} \: (1 + \frac{1}{2} \log 1/\pi_*) \end{equation} where $\pi_*$ is the minimum value of $\pi$. Since $\delta$ is generally no easier to estimate directly than $\tau_{\operatorname{mix}}$, we often bound the {\em conductance} $\Phi$ (a geometric parameter of the chain) and appeal to the Cheeger inequality (see e.g., Alon \cite{Alo}): \begin{equation} \frac{1}{2} \Phi^2 \leq \delta \leq 2\Phi \end{equation} Further inequalities (e.g., involving congestion of multicommodity flows through the chain) are often invoked on top of these.
Notice that Aldous' inequality (\ref{aldous}) is tight with respect to the spectral gap. However, it is also somewhat unsatisfactory in that although the diameter of the graph underlying $P$ (the obvious lower bound for sampling from its stationary distribution) scales like $O(\sqrt{\delta^{-1}} \log 1/\pi_*)$, the dependence is $\delta^{-1}$ in Aldous' inequality due to the Gaussian-like spreading behavior of random walks. In MCMC applications, the additional factor of $\sqrt{\delta^{-1}}$ can mean the difference between, say, an $n^3$ and $n^5$ algorithm! Removing it would imply a considerable improvement in both the {\em known} upper bounds (since existing estimates of $\delta$ could be used in conjunction with a sharper inequality than Aldous') and the {\em true} upper bounds (since Aldous' inequality is tight and thus a sharper inequality could only come from a faster approximate sampling method). Thus, a natural question is whether there is a way to modify the standard MCMC method to obtain a speedup to $O(\sqrt{\delta^{-1}} \log 1/\pi_*)$.
This seems unlikely using classical randomized methods: Chen, Lovasz, and Pak \cite{CLP} have shown that we can sometimes speed up mixing by {\em lifting} a Markov chain, but this requires both knowledge of the chain's global structure and its use in solving an NP-hard flow problem to find low-congestion paths along which to ``route'' probability mass efficiently. However, although lifting the chain seems unlikely to be practical, an idea that might work is {\em quantizing} the chain.
Why might a quantized Markov chain, or {\em quantum walk}, help us reach $\pi$ more quickly? Two reasons are: (1) a quantum walk is as simple to realize as its classical counterpart (i.e., it is computable locally and online, unlike a classical lifting of the chain), and (2) there is empirical (and some theoretical) evidence that quantum walks tend to propagate and ``spread'' probability mass across $\mathcal{S}$ in time on the order of the diameter on precisely the same low-dimensional graphs that trip up their classical counterparts. Based on these observations, the possibility of obtaining a quantum speedup for the mixing problem has been pursued by Nayak et al. \cite{NV,ABNVW}, Aharonov et al. \cite{AAKV}, Moore and Russell \cite{MR}, Gerhardt and Watrous \cite{GW}, and Richter \cite{Ric}.
We remark that a quantum speedup theorem of the sort we seek has already been proven for the {\em hitting} problem, in which we search (rather than sample from) the states of a Markov chain: Szegedy \cite{Sze} proved a quadratic quantum speedup for the {\em hitting time} of any symmetric Markov chain, generalizing considerably the celebrated search algorithm of Grover \cite{Gro} and implying quantum speedups for structured search problems such as element distinctness (Ambainis \cite{Amb1}), triangle finding (Magniez, Santha, and Szegedy \cite{MSS}), matrix product verification (Burhman and Spalek \cite{BS}), and group commutativity testing (Magniez and Nayak \cite{MN}). It is this success which inspires us to investigate the possibility of a quantum speedup for the mixing problem with the goal of transferring the speedup to MCMC algorithms.
\subsection{Our contributions}
We present evidence of a possible quantum MCMC speedup to $O(\sqrt{\delta^{-1}} \log 1/\pi_*)$ using quantum walks that {\em decohere} under repeated randomized measurements. Decoherence (in small amounts) was first identified as a way to improve spreading and mixing properties in numerical experiments performed by Kendon and Tregenna \cite{KT} and in analytical estimates by Fedichkin et al. \cite{FST,SF1,SF2}. On the other hand, high rates of decoherence in quantum walks have been shown to degrade mixing properties substantially by the quantum Zeno effect (Alagic and Russell \cite{AR}). For an excellent survey of these and other aspects of decoherent quantum walks, see Kendon \cite{Ken}.
Our technical contributions are as follows: First, we show that for any symmetric Markov chain $P$, we can generate an arbitrarily good approximation to the uniform stationary distribution $\pi$ of $P$ by subjecting the continuous-time quantum walk $U_{ct}(P) = \exp(iP)$ to reasonably ``smooth'' decoherence. Thus, decoherent quantum walks (which are non-unitary) offer a way of circumventing an obstacle first identified by Aharonov et al. \cite{AAKV}, who observed that {\em unitary} quantum walks often converge (in the time-averaged sense) to highly non-uniform distributions.
Second, we show that the optimal mixing time of a decoherent quantum walk is a robust quantity, in that it remains essentially invariant under any sufficiently smooth form of decoherence. In particular, decoherent quantum walks undergoing repeated Cesaro-averaged \cite{AAKV,MR,GW} and Bernoulli/Poisson-averaged \cite{KT,AR} measurements are nearly equivalent in mixing efficiency. The proof applies more generally to a game involving time-dependent Markov chains (not necessarily describing quantum phenomena) and may be of independent interest.
Third, we prove a theorem on threshold mixing of quantum walks on (Cartesian) graph powers in order to show that the decoherent continuous-time quantum walk $U_{ct}(P(G))$ on a periodic lattice $G=\mathbb{Z}_n^d$ (where $P(G)$ denotes the standard Markov chain on $G$) can be used to generate a good approximation to the uniform stationary distribution $\pi$ of $P$ in time $O(n d \log d)$. This upper bound is asymptotically no worse than the diameter of $\mathbb{Z}_n^d$ (the obvious lower bound) up to at most a logarithmic factor and is $O(\sqrt{\delta^{-1}} \log 1/\pi_*)$ for both high-dimensional {\em and} low-dimensional lattices (unlike its classical counterpart). For $d=1$, this proves a conjecture of Kendon and Tregenna \cite{KT} based on numerical experiments.\footnote{In fact this is not quite true: Kendon and Tregenna \cite{KT} conjectured this for the discrete-time Hadamard walk. We prove it for both walks.} For $d \geq 1$, it extends the results of Fedichkin et al. \cite{FST,SF1,SF2} by confirming $O(n)$ and $O(d \log d)$ scaling (suggested by their analytical estimates and numerical experiments in regimes of both high and low decoherence) of the fastest-mixing walk, which they conjectured to be decoherent rather than unitary. We briefly discuss the prospects for extending this result to the discrete-time Grover walk $U_{dt}(P(\mathbb{Z}_n^d))$ \cite{IKK,TFMK,MBSS}.
Previously, mixing speedups had been proven only for the unitary quantum walks of Nayak et al. \cite{NV,ABNVW} and Aharonov et al. \cite{AAKV} (on the cycle) and of Moore and Russell \cite{MR} (on the hypercube). Thus, our work shows that introducing a small amount of decoherence to a quantum walk can simultaneously force convergence to the uniform distribution while preserving a quantum mixing speedup, an advantageous combination for algorithmic applications.\footnote{Alagic and Russell \cite{AR} exhibit a decoherent quantum walk on the hypercube which converges to the uniform distribution and preserves the quantum mixing speedup proven by Moore and Russell \cite{MR}, but in contrast to our work, it is their single (carefully chosen) final measurement that {\em forces} uniform convergence, not the decoherence itself (which only adds ``noise'' that if small enough does not destroy the speedup).}
\section{Preliminaries}
\subsection{Markov chains}
Let $P$ be a {\em Markov chain} (stochastic matrix) on the set
$\mathcal{S}$ ($|\mathcal{S}| = N$) which is {\em irreducible} (strongly connected); then it has a unique distribution $\pi$ which is {\em stationary} (i.e., satisfies $P \pi = \pi$). Moreover, $\pi$ is strictly positive: $\pi_* := \min_x \pi_x$ satisfies $\pi_* > 0$. In particular, if $P$ is symmetric, then $\pi$ is the uniform distribution $u$ (the $N$-dimensional column vector with each component equal to $1/N$). If $G$ is an undirected graph, we denote the standard (``simple random walk'') Markov chain on $G$ by $P(G)$.
A Markov chain $P$ which is both irreducible and {\em aperiodic} (non-bipartite) is by definition {\em ergodic} and satisfies \begin{equation} P^t \rightarrow \pi 1^\dagger = [\pi \cdots \pi] \textrm{ as } t \rightarrow \infty \end{equation} where $1^\dagger$ is the $N$-dimensional row vector with each component equal to one. We can thus define the {\em (threshold) mixing time} \begin{equation}
\tau_{\operatorname{mix}} := \min \{ T : \frac{1}{2}||P^t - \pi 1^\dagger||_1 \leq \frac{1}{2e} \: \forall t \geq T \} \end{equation}
where $|| \cdot ||_1$ is the matrix 1-norm. The mixing is {\em perfect} if $P^t = \pi 1^\dagger$. Let $\delta := 1 - ||P
|_{\pi^\perp}||_2 > 0$ be the {\em spectral gap} of $P$. We say that $P$ is {\em reversible} if the matrix $DPD^{-1}$ is symmetric, where $D$ is the diagonal matrix $D(x,x) = \sqrt{\pi_x}$. A precise statement of Aldous' inequality (\ref{aldous}) is: \begin{theorem}[Aldous \cite{Ald}]\label{ds-ineq} Let $P$ be a reversible, ergodic Markov chain with stationary distribution $\pi$ and spectral gap $\delta$. Then its mixing time satisfies $\delta^{-1} \leq \tau_{\operatorname{mix}} \leq \delta^{-1} \: (1 + \frac{1}{2} \log 1/\pi_*)$. \end{theorem}
We will also use the {\em maximum pairwise column distance}
$d(P) := \max_{x,x'} \frac{1}{2}||P(\cdot,x) - P(\cdot,x')||_1$ to estimate the mixing time. It is related to the matrix 1-norm distance by the inequality: \begin{equation}
\frac{1}{2}||P - \pi 1^\dagger||_1 \leq d(P) \leq ||P - \pi 1^\dagger||_1 \end{equation}
The following propositions (see \cite{Ric} for proofs) can be used to estimate the mixing time of $P$ given a common lower bound on most of the entries in each column. \begin{proposition}\label{dist-ub} If $d(P) \leq \alpha$, then $\tau_{\operatorname{mix}} \leq \lceil \log_{1/\alpha} 2e \rceil$. \end{proposition}
\begin{proposition}\label{entry-lb} If at least $\beta N$ entries in each column of $P$ are bounded below by $\gamma / N$, where $\beta > \frac{1}{2}$ and $\gamma > 0$, then $d(P) \leq 1 - \gamma (1 - 2(1-\beta))$. \end{proposition}
\subsection{Quantum walks}
Heretofore, a {\em quantum walk} is a pair $\langle U, \omega_T \rangle$ where the {\em transition rule} $U$ is a unitary operator acting on a finite-dimensional Hilbert space and the {\em measurement rule} $\omega_T$ is a $T$-parametrized family of probability density functions on $[0,\infty)$ characterizing the (random) time at which a total measurement is performed on the Hilbert space.\footnote{Though repeatedly measured quantum walks generate mixed states, our analysis is simplest without the introduction of density matrices. We refer the reader lacking sufficient background in quantum computing to the excellent text of Kitaev, Shen, and Vyalyi \cite{KSV}.}
The unitary transition rule determines the orbit of a pure quantum state, or {\em wavefunction} ($l_2$-normalized complex vector), just as a Markov chain (or stochastic transition rule) determines the orbit of a classical distribution ($l_1$-normalized nonnegative vector). Let $P$ be the Markov chain used in an MCMC algorithm; in particular, $P$ is reversible. Two natural quantizations of $P$ are: (a) the $\mathcal{S} \times \mathcal{S}$ unitary {\em continuous-time walk} given by $U_{ct}(P) = \exp(iDPD^{-1})$ \cite{FG,AT}, where $H=DPD^{-1}$ is the time-independent {\em Hamiltonian}, and (b) the $\mathcal{S}^2 \times \mathcal{S}^2$ unitary discrete-time {\em Grover/Szegedy walk} given by $U_{dt}(P) = (RS)^2$ \cite{Sze}, where $S$ is the involution $\sum_{x,y \in \mathcal{S}} \ket{x,y} \mapsto \ket{y,x}$, $R$ is the reflection $\sum_{x \in \mathcal{S}} \ketbra{x}{x} \otimes (2 \ketbra{p_x}{p_x} - I)$, and $\ket{p_x}$ is the vector $\sum_{y \in \mathcal{S}} \sqrt{P(y,x)}\ket{y}$. The quantization $U_{ct}(P)$ was used by Childs et al. \cite{CCD} to solve in polynomial time a natural oracle problem for which no classical polynomial-time algorithm exists. The quantization $U_{dt}(P)$ was used by Szegedy \cite{Sze} to prove a quadratic quantum speedup for the hitting time of any symmetric Markov chain.
Measurement collapses the wavefunction to a classical distribution according to the map $\ket{\phi} \mapsto \sum_x \ket{x}
|\braket{x}{\phi}|^2$. For a quantum walk $\langle U, \omega_T
\rangle$, the Markov chain {\em generated by} the quantum walk is given by $\hat{P}_T(y,x) := E_{t \leftarrow \omega_T}[|\braket{y}{U^t|x}|^2]$, where $E$ denotes expected value. We say that the quantum walk {\em threshold-mixes} if the Markov chain it generates mixes in time $O(1)$. Examples of measurement rules from the literature include the point distribution $\delta_T(t) := \delta(t-T)$ where $\delta$ is the delta function \cite{NV,ABNVW}, the uniform distribution $\bar{\mu}_T := \frac{1}{T}\chi_{[0,T]}$ and its discrete-time counterpart $\bar{\nu}_T := \frac{1}{T}\chi_{[0..T-1]}$ where $\chi$ is the characteristic function \cite{AAKV}, the exponential distribution $\tilde{\mu}_T(t) := \frac{1}{T}\exp(-t/T)$, and the geometric distribution $\tilde{\nu}_T(t) := \frac{1}{T} (1-\frac{1}{T})^t$. The exponential and geometric distributions are {\em memoryless} and describe the interarrival time between measurements in a Poisson process with measurements occurring at rate $\lambda=1/T$ and a Bernoulli process with measurements occurring with probability $p=1/T$ at each timestep, respectively. These processes coincide with the decoherence models of Alagic and Russell \cite{AR} and Kendon and Tregenna \cite{KT}, respectively.
Nayak et al. \cite{NV,ABNVW} and Aharonov et al. \cite{AAKV} showed that the so-called {\em Hadamard} walks $\langle U_{Had}, \delta_T \rangle$ and $\langle U_{Had}, \bar{\nu}_T \rangle$ on the cycle $\mathbb{Z}_n$ threshold-mix quadratically faster than the classical random walk, although their definitions of threshold mixing are slightly different than ours. Moore and Russell \cite{MR} showed that the continuous-time walk $\langle U_{ct}(P(G)), \delta_T \rangle$ and the Grover walk $\langle U_{dt}(P(G)), \delta_T \rangle$ on the hypercube $G=\mathbb{Z}_2^d$ mix perfectly and almost perfectly, respectively, in time $T=O(d)$. They also showed that the continuous-time quantum walk on the hypercube with measurement $\bar{\mu}_T$ does not mix to the uniform stationary distribution $\pi$ of $P(\mathbb{Z}_2^d)$ in the limit $T \rightarrow \infty$. Gerhardt and Watrous \cite{GW} showed the same for a continuous-time quantum walk on the symmetric group with measurement $\bar{\mu}_T$. See the survey by Kempe \cite{Kem} for further results on quantum walks.
We remark that there is a nice way to use the quantization $U_{dt}(P)$ to solve the mixing problem in time $O(1/\sqrt{\delta \pi_*})$ which, although prohibitively costly in $\pi_*$, exhibits the desired dependence on $\delta$. Consider the stationary eigenstate $\ket{\tilde{\pi}} := \sum_{x \in \mathcal{S}} \sqrt{\pi_x} \ket{x}\ket{p_x}$ of $U_{dt}(P)$. It is clear that we can retrieve a good approximation to the classical distribution $\pi$ by generating and then measuring a good approximation to $\ket{\tilde{\pi}}$.\footnote{Similarly, we can retrieve a good approximation to $\pi$ by generating and then measuring a good approximation to the ground state $\ket{\pi} = \sum_{x \in \mathcal{S}} \sqrt{\pi_x} \ket{x}$ of the Hamiltonian $H = DPD^{-1}$ \cite{AT}.} Magniez et al. \cite{MNRS} observe that $O(1/\sqrt{\delta})$ steps of phase estimation on $U_{dt}(P)$ enable us to reflect about $\ket{\tilde{\pi}}$. By alternating this reflection with a reflection about $\ket{\tilde{z}} := \ket{z}\ket{p_z}$ where $z$ is the initial walk state (in particular, $\braket{\tilde{z}}{\tilde{\pi}} \geq \sqrt{\pi_*}$), we can generate $\ket{\tilde{\pi}}$ from $\ket{\tilde{z}}$ in time $O(1/\sqrt{\delta \pi_*})$. In fact, this algorithm is described by Magniez et al. \cite{MNRS} as a {\em hitting} algorithm (i.e., generate the {\em unknown} state $\ket{\tilde{z}}$ from the fixed initial state $\ket{\tilde{\pi}}$); the idea of running a quantum hitting algorithm {\em in reverse} as a mixing algorithm was suggested by Childs \cite{Chi}.
\section{Mixing properties of decoherent quantum walks}
\subsection{Two types of convergence}
Let $\hat{P}_T$ be the Markov chain generated by a quantum walk $\langle U, \omega_T \rangle$. Then repeating the quantum walk $T'$ times in succession generates the Markov chain $(\hat{P}_T)^{T'}$. The following lemma (a variant of Theorem 3.4 in Aharonov et al. \cite{AAKV}) and theorem describe the asymptotic behavior of $\hat{P}_T$ and $(\hat{P}_T)^{T'}$ in the limits $T \rightarrow \infty$ and $T' \rightarrow \infty$, respectively. For concreteness we will take $U=U_{ct}$ in this subsection and the next; it is a simple exercise to extend the results to discrete-time walk variants. Although stated explicitly for quantum walks, the results apply to any time-independent quantum dynamics on a finite-dimensional Hilbert space subjected to random destructive measurements.
\begin{lemma}[The limit $T \rightarrow \infty$]\label{single} Let $P$ be a symmetric Markov chain and $\omega_T$ be a family of distributions satisfying $E_{t \leftarrow \omega_T}[e^{i\theta t}] \rightarrow 0$ as $T \rightarrow \infty$ for any $\theta \neq 0$. In the limit $T \rightarrow \infty$, the Markov chain $\hat{P}_T$ generated by the quantum walk $\langle U_{ct}(P), \omega_T \rangle$ approaches the Markov chain $\Pi$ with entries \begin{equation}\label{pimatrix}
\Pi(y,x) := \sum_j \bigl| \sum_{k \in C_j}
\braket{y}{\phi_k}\braket{\phi_k}{x} \bigr|^2 \end{equation} where $\{\lambda_k,\ket{\phi_k}\}$ is the spectrum of $P$ and $\{C_j\}$ is the partition of these indices $k$ obtained by grouping together the $k$ with identical $\lambda_k$. \end{lemma} \begin{proof} Decomposing the quantum walk along spectral components gives us: \begin{equation}
\hat{P}_T(y,x) = E_{t \leftarrow \omega_T}[|\sum_k \braket{y}{\phi_k}
\braket{\phi_k}{x} e^{i \lambda_k t}|^2] \end{equation}
Writing $| \cdot |^2$ as a product of complex conjugates, we obtain: \begin{eqnarray} \hat{P}_T(y,x)
& = & E_{t \leftarrow \omega_T}[(\sum_k \braket{y}{\phi_k} \braket{\phi_k}{x})
(\sum_l \braket{\phi_l}{y} \braket{x}{\phi_l}) e^{i(\lambda_k-\lambda_l)t}]\\
& = & (\sum_k \braket{y}{\phi_k} \braket{\phi_k}{x})(\sum_l \braket{\phi_l}{y}
\braket{x}{\phi_l}) E_{t \leftarrow \omega_T}[e^{i(\lambda_k-\lambda_l)t}] \end{eqnarray} Now by assumption, $E_{t \leftarrow \omega_T}[e^{i(\lambda_k-\lambda_l)t}]$ vanishes as $T \rightarrow \infty$ for all $\lambda_k \neq \lambda_l$, so we have \begin{equation} \hat{P}_T(y,x)
\rightarrow \sum_k \braket{y}{\phi_k} \braket{\phi_k}{x}
(\sum_{l : \theta_l = \theta_k} \braket{\phi_k}{y} \braket{x}{\phi_k})
= \sum_j |\sum_{k \in C_j} \braket{y}{\phi_k} \braket{\phi_k}{x}|^2
= \Pi(y,x) \end{equation} in the limit $T \rightarrow \infty$.\qed \end{proof}
It can be inferred from this lemma that most quantum walks converge to a distribution $\rho$ other than the uniform stationary distribution
$\pi = u$, and that $\rho$ is not even independent of the initial walk state.\footnote{For a walk on the symmetric group $S_n$, Gerhardt and Watrous \cite{GW} showed that $\frac{1}{2}||\Pi - u 1^\dagger||_1
\geq \frac{1}{n}-\frac{1}{n \cdot n!}\binom{2n-2}{n-1}$; for a walk on the hypercube $\mathbb{Z}_2^n$, Moore and Russell \cite{MR} showed that there exists an $\epsilon > 0$ such that $\frac{1}{2}||\Pi - u 1^\dagger||_1 \geq \epsilon$.} There are exceptions to this rule, for example quantum walks with distinct eigenvalues on Cayley graphs of Abelian groups (as observed by Aharonov et al. \cite{AAKV}), but they are not likely to arise in MCMC applications where the Markov chains have little structure. How then are we to sample from $u$ using quantizations of these Markov chains? Here is where decoherence helps.
\begin{theorem}[The limit $T' \rightarrow \infty$]\label{repeated} Let $P$ be a symmetric, irreducible Markov chain and $\omega_T$ be a family of distributions satisfying $E_{t \leftarrow \omega_T}[e^{i\theta t}] \rightarrow 0$ as $T \rightarrow \infty$ for any $\theta \neq 0$. For $T$ sufficiently large (but fixed), the $T'$-repeated quantum walk $\langle U_{ct}(P), \omega_T \rangle$ generates a Markov chain $(\hat{P}_T)^{T'}$ approaching $u 1^\dagger$ in the limit $T' \rightarrow \infty$. \end{theorem} \begin{proof} We need to show that for $T$ sufficiently large, the Markov chain $\hat{P}_T$ is ergodic with uniform stationary distribution.
That the uniform distribution is stationary is clear: each of the
$P_t(y,x) := |\braket{y}{e^{iPt}|x}|^2$ has uniform stationary distribution since the uniform classical state is invariant under unitary quantum operations and under total measurement of the system; thus, any probabilistic combination $\hat{P}_T$ of them has uniform stationary distribution.
To show that $\hat{P}_T$ is ergodic for all sufficiently large $T$, it is sufficient (by Lemma \ref{single}) to prove that $\Pi$ is ergodic. (The latter implies the former because the ergodic matrices form an open subset of the set of stochastic matrices.) Why is $\Pi$ ergodic? Because the $1$-eigenspace of $P$ is precisely the space spanned by $u$, so it follows from Lemma \ref{single} (by consideration of only this nondegenerate eigenspace in the expression (\ref{pimatrix})) that $\Pi(y,x) \geq 1/N^2$ for every $x,y$.
In fact, each of the $P_t$ (and so $\hat{P}_T$ and $\Pi$ as well) is symmetric.\footnote{$\Pi$ is also positive semidefinite: it is the Gram matrix of $\{f_s\}$ with $f_s(kl) := \braket{s}{\phi_k} \braket{\phi_l}{s}$ if $\lambda_k = \lambda_l$, $0$ otherwise.} To see this, write out the Taylor series for $\exp(iPt)$ and note that every positive integer power $P^k$ is symmetric (since $P^2(x,y) = \sum_z P(x,z) \cdot P(z,y) = \sum_z P(y,z) \cdot P(z,x) = P^2(y,x)$). This property will be quite useful in the next subsection: it will allow us to use Theorem \ref{ds-ineq} to relate the spectral gap and the mixing time of $\hat{P}_T$.\qed \end{proof}
\subsection{Invariance of the mixing time}
Consider the quantum walks $\langle U_{ct}(P), \bar{\mu}_T \rangle$ and $\langle U_{ct}(P), \tilde{\mu}_T \rangle$ where $P$ is a symmetric Markov chain. We show that these two quantum walks mix with essentially the same efficiency. The result extends beyond $\bar{\mu}, \tilde{\mu}$ to any pair of measurements $\omega, \omega'$ which are sufficiently smooth or have nontrivial overlap as distributions.
Let $\bar{P}_T$ and $\tilde{P}_T$ be the Markov chains generated by the walks with measurement $\bar{\mu}$ and $\tilde{\mu}$, respectively, and let $\bar{\delta}_T := 1 - ||\bar{P}_T |_{u^\perp}
||_2$ and $\tilde{\delta}_T := 1 - ||\tilde{P}_T |_{u^\perp} ||_2$ be their respective spectral gaps.
\begin{lemma}[Spectral gap inequality]\label{gap} Let $\bar{\delta}_T$ and $\tilde{\delta}_T$ be defined as above. Then for any $k \geq 1$ we have the inequality: \begin{equation} e^{-1}\bar{\delta}_T \leq \tilde{\delta}_T \leq k(1-e^{-k}) \cdot \bar{\delta}_{kT} + 2e^{-k} \end{equation} \end{lemma} \begin{proof} Suppose we want to simulate $\bar{P}_T$ by $\tilde{P}_T$. Scaling the distribution $\bar{\mu}_T$ by $\alpha := 1/e$ allows us to ``fit it inside'' the distribution $\tilde{\mu}_T$ (i.e., $e^{-1} \bar{\mu}_T \leq \tilde{\mu}_T$ pointwise), so we can express $\tilde{\mu}_T$ as the probabilistic combination $\alpha \bar{\mu}_T + (1-\alpha)\nu$ for some distribution $\nu$, so that \begin{equation} \tilde{P}_T = E_{t \leftarrow \tilde{\mu}_T}[P_t] = \alpha E_{t
\leftarrow \bar{\mu}_T}[P_t] + (1-\alpha) E_{t \leftarrow \nu}[P_t]
= \alpha \bar{P}_T + (1-\alpha) Q \end{equation} where $Q$ is stochastic with uniform stationary distribution. It follows that \begin{equation}
||\tilde{P}_T |_{u^\perp} ||_2 \leq 1/e ||\bar{P}_T |_{u^\perp}||_2 +
(1-1/e) ||Q |_{u^\perp}||_2 \end{equation} which implies that $\tilde{\delta}_T \geq 1/e \cdot \bar{\delta}_T$
since $||Q |_{u^\perp}||_2 \leq 1$.
Suppose we want to simulate $\tilde{P}_T$ by $\bar{P}_{kT}$. Then the basic approach is the same, but since the support of $\tilde{\mu}_T$ is not compact we have to be careful. Scaling the distribution $\tilde{\mu}_T$ by $\beta := 1/k$ allows us to fit it inside the distribution $\bar{\mu}_{kT}$ up to the point $t = kT$, and the probability mass in $\tilde{\mu}_T$ past $t = kT$ is only $\Pr_{t \leftarrow \tilde{\mu}_T}[t > kT] = e^{-k}$. So we can write \begin{equation} \tilde{\mu}_T = (1-e^{-k}) \cdot \tilde{\mu}_T^{head} + e^{-k} \cdot \tilde{\mu}_T^{tail} \end{equation} where $\tilde{\mu}_T^{head}$ and $\tilde{\mu}_T^{tail}$ are the conditional distributions of $\tilde{\mu}_T$ such that $t \leq kT$ and $t > kT$, respectively; thus, \begin{equation} \tilde{P}_T = (1-e^{-k}) \cdot \tilde{P}_T^{head} + e^{-k} \cdot \tilde{P}_T^{tail} \end{equation} where $\tilde{P}_T^{head}$ and $\tilde{P}_T^{tail}$ are the expectations of $P_t$ with respect to $\tilde{\mu}_T^{head}$ and $\tilde{\mu}_T^{tail}$, respectively. Since we can fit $\tilde{\mu}_T^{head}$ inside $\bar{\mu}_{kT}$ if we scale it by $1/k$, we can write \begin{equation} \bar{P}_{kT} = \frac{1}{k} \tilde{P}_T^{head} + (1-\frac{1}{k})Q \end{equation} where $Q$ is stochastic with uniform stationary distribution. The above equations yield: \begin{eqnarray} \bar{P}_{kT} & = & \frac{1}{k(1-e^{-k})} (\tilde{P}_T - e^{-k}
\tilde{P}_T^{tail}) + (1-\frac{1}{k})Q \nonumber\\
& = & \frac{1}{k(1-e^{-k})} \tilde{P}_T -
\frac{e^{-k}}{k(1-e^{-k})} \tilde{P}_T^{tail} +
(1-\frac{1}{k}) Q \end{eqnarray} From the triangle inequality, we obtain \begin{equation}
||\bar{P}_{kT} |_{u^\perp}||_2 \leq \frac{1}{k(1-e^{-k})}
||\tilde{P}_T |_{u^\perp}||_2 + \frac{e^{-k}}{k(1-e^{-k})}
||\tilde{P}_T^{tail} |_{u^\perp}||_2 + (1-\frac{1}{k})||Q |_{u^\perp}||_2 \end{equation} and, rearranging terms and simplifying: \begin{equation}
\frac{1}{k(1-e^{-k})} (1 - ||\tilde{P}_T |_{u^\perp}||_2) -
\frac{2e^{-k}}{k(1-e^{-k})} \leq 1 - ||\bar{P}_{kT} |_{u^\perp}||_2 \end{equation} \qed \end{proof}
\begin{theorem}[Equivalence of measurements]\label{equivalence} Let $P$ be a symmetric Markov chain. Then: (a) if the $T'$-repeated quantum walk $\langle U_{ct}(P), \bar{\mu}_T \rangle$ threshold-mixes, then the $T' \cdot O(\log N)$-repeated quantum walk $\langle U_{ct}(P), \tilde{\mu}_T \rangle$ threshold-mixes; (b) if the $T'$-repeated quantum walk $\langle U_{ct}(P), \tilde{\mu}_T \rangle$ threshold-mixes, then the $T' \cdot O(\log T' \log N)$-repeated quantum walk $\langle U_{ct}(P), \bar{\mu}_{T \cdot O(\log T')} \rangle$ threshold-mixes. \end{theorem} \begin{proof} To see (a), note that our assumption implies that $\bar{P}_T$ mixes in time $T'$.
Therefore, $\bar{\delta}_T = \Omega(1/T')$ by Theorem \ref{ds-ineq}, and from Lemma \ref{gap} it follows that $\tilde{\delta}_T = \Omega(1 / T')$. Applying Theorem \ref{ds-ineq} again, we obtain for $\tilde{P}_T$ a mixing time of $O(T' \log N)$.
The proof of (b) is almost as straightforward. Our assumption implies that $\tilde{P}_T$ mixes in time $T'$, so $\tilde{\delta}_T = \Omega(1 / T')$ by Theorem \ref{ds-ineq}. Set $k$ to be the smallest integer for which $\tilde{\delta}_T \geq 3e^{-k}$; in particular, $k = \Theta(\log \tilde{\delta}_T^{-1}) = O(\log T')$. By Lemma \ref{gap}: \begin{equation} \bar{\delta}_{kT} \geq \frac{1}{k(1-e^{-k})} (\tilde{\delta}_T - 2e^{-k}) \geq \frac{1}{k(1-e^{-k})} (e^{-k}) = \Theta(\frac{\tilde{\delta}_T}{\log \tilde{\delta}_T^{-1}}) = \Theta(\frac{1}{T' \log T'}) \end{equation} Applying Theorem \ref{ds-ineq} again, we obtain for $\bar{P}_{kT}$ a mixing time of $O(T' \log T' \log N)$.\qed \end{proof}
It should be readily apparent that this equivalence holds for any two measurement rules with finite expectation and significant overlap for most $T$. We also remark that although the above lemma and theorem are stated in terms of quantum walks, the proofs indicate that they are merely statements about an abstract game involving a collection of symmetric Markov chains $\{P_t\}_{t \geq 0}$ and a $T$-parametrized family of probability measures $\{\omega_T\}$, where we seek to minimize the ``cost function'' $T \cdot T'$.
\section{Quantum speedup for periodic lattices}
\subsection{A near-diameter upper bound}
The classical random walk on the periodic lattice $\mathbb{Z}_n^d$ (with $N = n^d$ vertices and diameter $\lfloor n/2 \rfloor d$) has uniform stationary distribution $\pi = u$ and spectral gap $\delta = \Theta(\min \{ \frac{1}{d}, \frac{1}{n^2} \})$. It threshold-mixes in time $\Theta(n^2 d \log d)$, which is $O(\sqrt{\delta^{-1}} \log 1/\pi_*)$ only when $\mathbb{Z}_n^d$ is quite high-dimensional: in particular, when $d$ is roughly of order $n^4$ or larger. We show that a few repetitions of the continuous-time quantum walk can bring this down to $O(n d \log d)$, which is $O(\sqrt{\delta^{-1}} \log 1/\pi_*)$ for any $d \geq 1$ and $n \geq 2$ and asymptotically no worse than the diameter of $\mathbb{Z}_n^d$ up to at most a logarithmic factor.
First we prove a lemma governing mixing of various decoherent quantum walks on the cycle $\mathbb{Z}_n$.
\begin{lemma}[Mixing on cycles]\label{cycles} Let $\mathbb{Z}_n$ be the cycle on $n \geq 2$ vertices. The continuous-time walks $\langle U_{ct}(P(\mathbb{Z}_n)),\omega_T \rangle$ with measurement $\omega \in \{\delta, \bar{\mu}, \tilde{\mu}\}$ threshold-mix for any $T \in \mathcal{I} := [\frac{2}{3}\cdot\frac{n}{2},\frac{n}{2}]$, and the Hadamard walks $\langle U_{Had}(\mathbb{Z}_n),\omega_T \rangle$ with measurement $\omega \in \{\bar{\nu}, \tilde{\nu}\}$ threshold-mix for any $T \in \mathcal{J} := [\frac{2}{3}\cdot\frac{n}{\sqrt{2}},\frac{n}{\sqrt{2}}]$. \end{lemma} \begin{proof} Consider first the continuous-time walk. To prove that it threshold-mixes with any of the measurements $\omega \in \{\delta, \bar{\mu}, \tilde{\mu}\}$ for any $T \in \mathcal{I}$, it suffices by Proposition \ref{dist-ub} to show that for every $t \in \mathcal{I}' := [\frac{3}{5}\cdot\frac{n}{2}, \frac{4}{5}\cdot\frac{n}{2}]$, $d(P_t)$ is bounded below one by a positive constant, where $P_t$ is the Markov chain generated with measurement $\omega = \delta$. (Indeed, this easily implies that $d(\bar{P}_T)$ and $d(\tilde{P}_T)$ are bounded below one by a smaller positive constant, where $\bar{P}_T$ and $\tilde{P}_T$ are the Markov chains generated with measurement $\omega = \bar{\mu}$ and $\omega = \tilde{\mu}$, respectively.)
Let $\ket{\phi_t}$ and $\ket{\psi_t}$ be the wavefunctions at time $t$ for the continuous-time walks on $\mathbb{Z}$ and $\mathbb{Z}_n$, respectively, starting from the origin (without loss of generality, since $\mathbb{Z}$ and $\mathbb{Z}_n$ are vertex-transitive). Then for each $\bar{y} \in \mathbb{Z}_n$ we have: \begin{equation} \braket{\bar{y}}{\psi_t}
= \sum_{y \equiv \bar{y} \bmod n} \braket{y}{\phi_t} \end{equation} Childs \cite{Chi} shows that $\braket{y}{\phi_t} = (-i)^{y}
J_{y}(t)$, where $J_{y}$ is a Bessel function of the first kind. In particular, for $|y| \gg 1$ the quantity $|J_{y}(t)|$ is (a)
exponentially small in $|y|$ for $t < (1-\epsilon) \cdot |y|$ and (b)
of order $|y|^{-1/2}$ for $t > (1+\epsilon) \cdot |y|$. For every $t < \frac{4}{5} \cdot \frac{n}{2}$, property (a) implies that the only term in the above summand that is non-negligible is the
$\braket{y}{\phi_t}$ with $|y| < \frac{n}{2}$ (call it $\hat{y}$ and note that $\bar{y} \leftrightarrow \hat{y}$ is a 1-1 correspondence), so we can use property (b) to conclude that up to a negligible correction \begin{equation}
|\braket{\bar{y}}{\psi_t}| \approx
|\braket{\hat{y}}{\phi_t}| = \Theta(1/\sqrt{n}) \end{equation}
for every $|\hat{y}| \gg 1$ and $t > (1+\epsilon) \cdot |\hat{y}|$. In particular, the nearly $\frac{3}{5}n$ different $\bar{y}$ with $1
\ll |\hat{y}| \leq \frac{3}{5} \cdot \frac{n}{2}$ satisfy
$|\braket{\bar{y}}{\psi_t}| = \Omega(1/\sqrt{n})$, and therefore $P_t(\bar{y},\bar{0}) = \Omega(1/n)$, for every $t \in \mathcal{I}'$. So by Proposition \ref{entry-lb}, $d(P_t)$ is bounded below one by a positive constant.
For the Hadamard walk, the wavefunction is no longer characterized by Bessel functions, but it retains the same essential asymptotic spreading behavior as its continuous-time counterpart (see Nayak et al. \cite{NV,ABNVW}), and the argument above works with little modification. A caveat is the emergence of a parity problem: if $n$ is even, then $\mathbb{Z}_n$ is bipartite and the wavefunction is supported only on vertices of the same parity at each integer timestep. Hence the Hadamard walk with $\omega = \delta$ threshold-mixes only on vertices of the same parity, but with time-averaged measurement $\omega = \bar{\nu}$ or $\omega = \tilde{\nu}$ parity is broken and threshold-mixing occurs on the entire vertex set.
Although the argument above relies on the asymptotic behavior of the wavefunction as $n \rightarrow \infty$, this is clearly the difficult case: if $n$ is bounded, then it suffices to show only that there exists a time (or a pair of consecutive timesteps, in the case of the Hadamard walk) in which the wavefunction is supported on at least $2/3$ of the vertices.\qed \end{proof}
For the Hadamard walk with measurement $\omega = \tilde{\nu}$, Lemma \ref{cycles} resolves a conjecture of Kendon and Tregenna \cite{KT} based on numerical experiments.
Let $G^d$ denote the $d$th (Cartesian) power of a graph $G$. Examples are the $d$-dimensional standard lattice (the $d$th power of a line) and the $d$-dimensional periodic lattice (the $d$th power of a cycle). The following theorem shows how to extend a threshold-mixing result from $G$ to $G^d$.
\begin{theorem}[Mixing on graph powers]\label{powers} Suppose the continuous-time quantum walk $\langle U_{ct}(P(G)),\delta_T \rangle$ threshold-mixes. Then the $O(\log d)$-repeated walk $\langle U_{ct}(P(G^d)),\delta_{Td} \rangle$ threshold-mixes. \end{theorem} \begin{proof} The Hamiltonian $H' = P(G^d)$ is related to the Hamiltonian $H = P(G)$ by the identity: \begin{equation} H' = \frac{1}{d} \sum_{j=1}^d I^{\otimes (j-1)} \otimes H \otimes
I^{\otimes (d-j)} \end{equation} Since $H'$ commutes with the identity $I$, which can introduce at most a global phase factor to the system, the Markov chain $P'_t$ generated by the walk $\langle U_{ct}(P(G^d)),\delta_t \rangle$ is the $d$th tensor power of the Markov chain $P_{t/d}$ generated by the walk $\langle U_{ct}(P(G)),\delta_{t/d} \rangle$.
By assumption, $d(P_T) \leq \alpha$ for some constant $\alpha < 1$. By Proposition \ref{dist-ub}, we can choose $T' = O(\log d)$ to ensure that: \begin{equation}
\frac{1}{2}||(P_T)^{T'} - u 1^\dagger||_1 \leq d((P_T)^{T'}) \leq \frac{1}{6d^2} \end{equation}
Then at least $n\sqrt[d]{2/3}$ entries in each column of $(P_T)^{T'}$ are bounded below by $\frac{1-1/2d}{n}$, otherwise we would have the contradiction \begin{equation}
\frac{1}{2} ||(P_T)^{T'} - u 1^\dagger||_1
= 1 - \sum_y \min \{ (P_T)^{T'}(y,x), \frac{1}{n} \}
> (1-\sqrt[d]{2/3}) \frac{1}{2d}
\geq \frac{1}{6d^2} \end{equation}
where the first equation uses the identity $\frac{1}{2}||p-q||_1 = 1 - \sum_k \min\{p_k,q_k\}$ for distributions $p, q$ and the last inequality uses simple algebra along with the fact that for any $d \geq 1$: \begin{equation} (1-\frac{1/3}{d})^d \geq \frac{2}{3} \end{equation} Since $(P'_{Td})^{T'} = ((P_T)^{T'})^{\otimes d}$, at least $(n\sqrt[d]{2/3})^d = \frac{2}{3}n^d$ of the entries in each column of $(P'_{Td})^{T'}$ are bounded below by $(\frac{1-1/2d}{n})^d \geq \frac{1}{2 n^d}$. It follows from Proposition \ref{entry-lb} that $(P'_{Td})^{T'}$ threshold-mixes in time $O(1)$.\qed \end{proof}
We have the following corollary for the $d$th power $\mathbb{Z}_n^d$ of the cycle $\mathbb{Z}_n$.
\begin{corollary}[Mixing on periodic lattices]\label{lattices} Let $\mathbb{Z}_n^d$ be the $d$-dimensional periodic lattice with $n \geq 2$ vertices per side. The $O(\log d)$-repeated continuous-time quantum walk $\langle U_{ct}(P(\mathbb{Z}_n^d)), \omega_{nd/2} \rangle$ with measurement $\omega \in \{\delta, \bar{\mu}, \tilde{\mu}\}$ threshold-mixes. \end{corollary} \begin{proof} Combining Lemma \ref{cycles} with Theorem \ref{powers}, we conclude that $\langle U_{ct}(P(\mathbb{Z}_n^d)), \delta_T \rangle$ threshold-mixes for any $T \in [\frac{2}{3}\cdot\frac{nd}{2}, \frac{nd}{2}]$. It is easy to see (cf. Lemma \ref{cycles}) that this is sufficient to imply the stated corollary not only for the measurement $\omega = \delta$ but also for the time-averaged measurements $\bar{\mu}$ and $\tilde{\mu}$.\qed \end{proof}
For $d \geq 1$, this extends the results of Fedichkin et al. \cite{FST,SF1,SF2} by confirming $O(n)$ and $O(d \log d)$ scaling (suggested by their analytical estimates and numerical experiments in regimes of both high and low decoherence) of the fastest-mixing walk, which they conjectured to be decoherent rather than unitary.
\subsection{The Grover walk}
An important question is whether there is a $T'$-repeated Grover walk $\langle U_{dt}(P(\mathbb{Z}_n^d)), \omega_T \rangle$ that threshold-mixes for $T = O(nd)$ and $T' = O(\log d)$. Szegedy \cite{Sze} showed that the {\em phase gap} (minimum nonzero eigenvalue phase from $[-\pi,\pi]$ in absolute value) of $U_{dt}(P)$ is $\Omega(\sqrt{\delta})$ and exploited this property to prove a quadratic quantum speedup for the {\em hitting time} of any symmetric Markov chain $P$. A natural adaptation of his argument to the {\em mixing time} setting would be something like the following: since the phase gap of $U_{dt}(P)$ is $\theta = \Omega(\sqrt{\delta})$, we expect to see by decomposing the action of $U_{dt}(P)$ along spectral components that roughly $O(1/\theta) = O(\sqrt{\delta^{-1}})$ timesteps suffice for the orbit of any initial basis state to ``cover'' the entire state space with sufficient amplitude.
Unfortunately, this argument is incorrect: in fact, the orbit may remain quite localized around the initial basis state. This happens with dramatic effect to the Grover walk on the complete graph $G=K_N$, which mixes in time $T'=\Theta(N)$ (cf. \cite{Ric}) even though the classical random walk on $K_N$ mixes in a single timestep. It also happens to the Grover walk on $\mathbb{Z}_n^d$ for $d=2$, albeit less dramatically \cite{IKK,TFMK,MBSS}.\footnote{For $d=1$ it does not; the Grover walk on $\mathbb{Z}_n$ with measurement $\omega = \bar{\nu}$ mixes perfectly in $T=n$ timesteps.} The probability distribution $p_t$ induced on the vertices of $\mathbb{Z}_n^2$ at time $t \leq n/2$ is primarily localized at the initial basis state \cite{IKK} but has substantial secondary spikes which propogate across the lattice in orthogonal directions \cite{MBSS}. In particular, the standard deviation of $p_t$ appears to grow linearly with $t$ \cite{MBSS} and the mixing time of the Grover walk on $\mathbb{Z}_n^2$ with measurement $\omega = \bar{\nu}$ appears to be $T=O(n)$. In the high-dimensional regime, Moore and Russell \cite{MR} proved that the Grover walk on $\mathbb{Z}_2^d$ with measurement $\omega = \delta$ mixes almost perfectly in time $T=O(d)$. It seems plausible that the decoherent Grover walk on $\mathbb{Z}_n^d$ with measurement $\omega = \bar{\nu}$ mixes as fast asymptotically as the continuous-time walk.
\section{Conclusion and open problems}
We have shown that decoherent quantum walks have the potential to speed up a large class of classical MCMC mixing processes. Exactly how large this class is, and whether a generic quantum mixing speedup to $O(\sqrt{\delta^{-1}} \log 1/\pi_*)$ is possible, remain important open questions. Since it seems that quantum walks can outperform classical random walks in low-dimensional examples and underperform in very high-dimensional examples (such as the complete graph), a hybrid method may work best for generic Markov chains consisting of both low- and high-dimensional substructures. Also worth investigating is whether by randomizing the ``coin'' used in discrete-time quantum walks we can improve mixing on the complete graph and other adversarial examples.
{\bf Acknowledgements.} I would like to thank Mario Szegedy, Viv Kendon, Fr\'{e}d\'{e}ric Magniez, Miklos Santha, Iordanis Kerenidis, and Julia Kempe for useful discussions on quantum walks and Todd Brun for helpful comments on the presentation.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Canonical Nonclassical Hopf-Galois Module Structure of Nonabelian Galois Extensions \ Preliminary Report}
\begin{abstract} Let $ L/K $ be a finite Galois extension of local or global fields in characteristic $ 0 $ or $ p $ with nonabelian Galois group $ G $, and let $ \B $ be a $ G $-stable fractional ideal of $ L $. We show that $ \B $ is free over its associated order in $ K[G] $ if and only if it is free over its associated order in the Hopf algebra giving the canonical nonclassical Hopf-Galois structure on the extension. \end{abstract}
\section{Introduction and Statement of Results}
Throughout let $ L/K $ be a finite Galois extension of fields with nonabelian Galois group $ G $. By the theorem of Greither and Pareigis \begin{itemize} \item the Hopf-Galois structures on $ L/K $ are in bijective correspondence with the regular subgroups of $ \perm{G} $ normalized by $ \lambda(G) $, the image of $ G $ under the left regular embedding, \item the Hopf algebra corresponding to a regular subgroup $ N $ is $ H_{N}=L[N]^{G} $, where $ G $ acts on $ L $ as Galois automorphisms and on $ N $ by conjugation via the embedding $ \lambda $, \item such a Hopf algebra acts on $ L $ by \[ \left( \sum_{\eta \in N} c_{\eta} \eta \right) \cdot x = \sum_{\eta \in N} c_{\eta} \eta^{-1}(1_{G})[x] \hspace{5mm} (c_{\eta} \in L, \; x \in L). \] \end{itemize}
Two examples of such regular subgroups are $ \lambda(G) $ itself and $ \rho(G) $, the image of $ G $ under the right regular embedding. The latter corresponds to the classical structure, with Hopf algebra $ K[G] $ and its usual action on $ L $. Since $ G $ is nonabelian we have $ \lambda(G) \neq \rho(G) $, and the subgroup $ \lambda(G) $ corresponds to a canonical nonclassical Hopf-Galois structure on $ L/K $, whose Hopf algebra we will denote by $ H_{\lambda} $. \\ \\ Our main result is the following:
\begin{thm} \label{thm_main} Let $ L/K $ be a finite nonabelian Galois extension of local fields or global fields with group $ G $, and suppose that $ \B $ is a $ G $-stable fractional ideal of $ L $. Then $ \B $ is free over its associated order in $ K[G] $ if and only if it is free over its associated order in $ H_{\lambda} $. \end{thm}
Note that we make no restriction on the characteristic of $ K $. Some immediate corollaries of this are:
\begin{cor} Suppose that $ L/K $ is an extension of local fields and is at most tamely ramified. Then $ \OL $ is free over its associated order in $ H_{\lambda} $. \end{cor} \begin{proof} In this case $ \OL $ is a free $ \OK[G] $-module by Noether's Theorem, so Theorem \eqref{thm_main} applies. \end{proof}
\begin{cor} Suppose that $ L/K $ is an extension of global fields and is at most tamely ramified. Then $ \OL $ is locally free over its associated order in $ H_{\lambda} $. \end{cor} \begin{proof} The proof of Theorem \eqref{thm_main} does not depend on the fact that $ L $ is a field, so we could replace $ L $ with its completion at some prime $ \p $ of $ \OK $ (a Galois algebra). In this case, for each prime $ \p $ of $ \OK $ we have that $ \OLp $ is a free $ \OKp[G] $-module by Noether's Theorem, so Theorem \eqref{thm_main} applies at each prime, and so $ \OL $ is locally free over its associated order in $ H_{\lambda} $. \end{proof}
\begin{cor} Suppose that $ K = \Q $ and that $ L/K $ is tame and that $ [L:\Q] $ is not divisible by $ 4 $. Then $ \OL $ is free over its associated order in $ H_{\lambda} $. \end{cor} \begin{proof} In this case $ \OL $ is a free $ \Z[G] $-module by Taylor's Theorem, so Theorem \eqref{thm_main} applies. \end{proof}
\begin{cor} Suppose that $ L/K $ is an extension of $ p $-adic fields which is weakly ramified. Then $ \OL $ is free over its associated order in $ H_{\lambda} $. \end{cor} \begin{proof} In this case $ \OL $ is free over its associated order in $ K[G] $ by a theorem of Johnston, so Theorem \eqref{thm_main} applies. \end{proof}
\begin{cor} Suppose that $ G $ is simple. By a result of Byott, $ L/K $ admits exactly two Hopf-Galois structures: the classical structure and the canonical nonclassical structure, and by Theorem \eqref{thm_main} $ \OL $ is either free over its associated order in both of these structures or in neither of them. \end{cor}
(Remember that in all of these we are assuming that $ L/K $ is Galois with nonabelian Galois group $ G $.)
\section{Normal Basis Generators}
In this section we will prove the following theorem:
\begin{thm} \label{thm_NBG} Let $ x \in L $. Then $ x $ is a $ K[G] $-generator of $ L $ if and only if $ x $ is an $ H_{\lambda} $-generator of $ L $. \end{thm}
To do this, for this section only we place ourselves in a slightly more general situation, and adopt the notation used in the proof of the theorem of Greither and Pareigis in Childs: Taming Wild Extensions, Chapter 2. \begin{itemize} \item Let $ N $ be any regular subgroup of $ \perm{G} $ that is stable under the action of $ G $ by conjugation via the left regular embedding $ \lambda $. \item Let $ GL = \mbox{Map}(G,L) $, and let $ \{ u_{g} \mid g \in G \} $ be an $ L $-basis of mutually orthogonal idempotents. That is: \[ u_{g}(\sigma) = \delta_{g,\sigma} \mbox{ for all } g,\sigma \in G. \] \item The group $ N $ acts on $ GL $ by permuting the subscripts of the idempotents $ u_{g} $: \[ \eta \cdot u_{g} = u_{\eta(g)} \mbox{ for any } \eta \in N \mbox{ and } g \in G. \] By extending this action $ L $-linearly, we can view $ GL $ as an $ L[N] $-module. \item As described above, $ G $ acts on $ L[N] $ by acting on $ L $ as Galois automorphisms and on $ N $ by conjugation via $ \lambda $. The group $ G $ also acts on $ GL $ by acting on $ L $ as Galois automorphisms and on the idempotents $ u_{g} $ by left translation of the subscripts. \item We have that $ GL $ is an $ L[N] $-Galois extension of $ L $ and, by Galois descent, we obtain that $ (GL)^{G} $ is an $ L[N]^{G} $-Galois extension of $ K $. Note also that $ L \otimes_{K} L[N]^{G} = L[N] $ and $ L \otimes_{K} (GL)^{G} = GL $. \item Finally, we identify $ (GL)^{G} $ with $ L $ via the isomorphism $ L \xrightarrow{\sim}{} (GL)^{G} $ defined by \[ x \mapsto f_{x} = \sum_{g \in G} g(x) u_{g} \mbox{ for all } x \in L. \] The action of $ L[N]^{G} $ on $ L $ (as given in the statement of the theorem of Greither and Pareigis) is defined via the inverse of this isomorphism. \end{itemize}
With all this notation to hand, we establish two lemmas concerning normal basis generators and then prove Theorem \eqref{thm_NBG}.
\begin{lemma} \label{lem_fixed_generators} An element $ f_{x} \in (GL)^{G} $ is an $ L[N]^{G} $-generator of $ (GL)^{G} $ if and only if it is an $ L[N] $-generator of $ GL $. \end{lemma} \begin{proof} Let $ \{ h_{1}, \ldots ,h_{n} \} $ be a $ K $-basis of $ L[N]^{G} $, and note that this is also an $ L $-basis of $ L[N] $. Suppose first that $ f_{x} $ is an $ L[N]^{G} $ generator of $ (GL)^{G} $. Then the $ K $-span of the elements $ h_{1} \cdot f_{x}, \ldots ,h_{n} \cdot f_{x} $ is $ (GL)^{G} $, so the $ L $-span of these elements is $ L \otimes_{K} (GL)^{G} = GL $. By considering dimensions we see that they must form an $ L $-basis of $ GL $. Conversely, suppose that $ f_{x} $ is an $ L[N] $-generator of $ GL $. Then the elements $ h_{1} \cdot f_{x}, \ldots ,h_{n} \cdot f_{x} $ are linearly independent over $ L $, so they are linearly independent over $ K $, and since $ (GL)^{G} $ is an $ L[N]^{G} $-module they all lie in $ (GL)^{G} $. Considering dimensions again, we conclude that they must form a $ K $-basis of $ (GL)^{G} $. \end{proof}
\begin{lemma} \label{lem_transition_matrix} For $ x \in L $, the element $ f_{x} $ is an $ L[N] $-generator of $ GL $ if and only if the matrix \[ T_{N}(x) = ( \eta(g)[x] )_{\eta \in N, \; g \in G} \] is nonsingular. \end{lemma} \begin{proof} The set $ \{ u_{g} \mid g \in G \} $ is an $ L $-basis of $ GL $. For $ x \in L $ and $ \eta \in N $, we have \begin{eqnarray*} n \cdot f_{x} & = & \eta \cdot \left( \sum_{g \in G} g(x) u_{g} \right) \\ & = & \sum_{g \in G} g(x) u_{\eta(g)} \\ & = & \sum_{g \in G} \eta^{-1}(g)[x] u_{g}, \end{eqnarray*} so the transition matrix from the set $ \{ u_{g} \mid g \in G \} $ to the set $ \{ \eta \cdot f_{x} \mid \eta \in N \} $ is the matrix $ T_{N}(x) $ above, and so $ f_{x} $ is an $ L[N] $-generator of $ GL $ if and only if this matrix is nonsingular. \end{proof}
{\it \noindent Proof of Theorem \eqref{thm_NBG}.} By the theorem of Greither and Pareigis the classical Hopf-Galois structure on $ L/K $ corresponds to the regular subgroup $ \rho(G) $ of $ \perm{G} $ and the canonical nonclassical Hopf-Galois structure corresponds to the regular subgroup $ \lambda(G) $. By Lemma \eqref{lem_fixed_generators}, it is sufficient to show that for a fixed $ x \in L $, the element $ f_{x} $ is an $ L[\lambda(G)] $-generator of $ GL $ if and only if it is an $ L[\rho(G)] $-generator of $ GL $. But for any $ x \in L $, the matrix $ T_{\lambda(G)}(x) $ is row equivalent to the transpose of the matrix $ T_{\rho(G)}(x) $, so the result follows by Lemma \eqref{lem_transition_matrix}. \qed
\section{Three Lemmas}
Henceforth, we will reserve the symbol $ \cdot $ for the action of an element $ h \in H_{\lambda} $ on an element $ x \in L $, viz. $ h \cdot x $, and use brackets for Galois actions and the action of an element $ z \in K[G] $ on an element $ x \in L $, viz. $ z(x) $. In this section we prove three lemmas which we will need in the proof of theorem \eqref{thm_main}. The first of these must be well known but we include it for completeness:
\begin{lemma} \label{lem_dual_basis} Let $ x $ be a $ K[G] $-generator of $ L $ and let $ \{ \widehat{\sigma(x)} \mid \sigma \in G \} $ be the dual basis to $ \{ \sigma(x) \mid \sigma \in G \} $ with respect to the trace form on $ L/K $. Then, for each $ \sigma \in G $, we have $ \widehat{\sigma(x)} = \sigma(\widehat{x}) $. \end{lemma} \begin{proof} For $ \sigma, \tau \in G $ we have: \begin{eqnarray*} \mbox{Tr}_{L/K}(\sigma(\widehat{x})\tau(x)) & = & \sum_{g \in G} g(\sigma(\widehat{x})\tau(x)) \\ & = & \sum_{g \in G} g\sigma(\widehat{x})g\tau(x) \\ & = & \sum_{g \in G} (g\sigma)(\widehat{x})(g\sigma) \sigma^{-1}\tau(x) \\ & = & \sum_{g \in G} (g\sigma)(\widehat{x}\sigma^{-1}\tau(x)) \\ & = & \sum_{g \in G} g(\widehat{x}\sigma^{-1}\tau(x)) \\ & = & \mbox{Tr}_{L/K}(\widehat{x}\sigma^{-1}\tau(x)) \\ & = & \delta_{1,\sigma^{-1}\tau} \\ & = & \delta_{\sigma,\tau}. \end{eqnarray*} \end{proof}
We might view the second lemma as an ``inside out" version of the first:
\begin{lemma} \label{lem_inside_out_trace} Retain the notation of Lemma \eqref{lem_dual_basis}. Then for any $ \sigma, \tau \in G $ we have \[ \sum_{g \in G} \sigma g (\widehat{x}) \tau g (x) = \delta_{\sigma,\tau}. \] \end{lemma} \begin{proof} Enumerate the elements of $ G $ as $ g_{1}, \ldots ,g_{n} $, let $ X $ be the matrix with $ (i,j) $ entry $ ( g_{i}g_{j}(x) ) $, and let $ \widehat{X} $ be the matrix with $ (i,j) $ entry $ ( g_{i}g_{j}(\widehat{x})) $. Then using Lemma \eqref{lem_dual_basis} we have \[ \sum_{k=1}^{n} g_{k} g_{i}(x) g_{k}g_{j}(\widehat{x}) = \delta_{i,j}, \] so $ X^{T} \widehat{X} = I $. But this implies that $ \widehat{X} X^{T} = I $, and the $ (i,j) $ entry of this product is given by \[ \sum_{k=1}^{n} g_{i}g_{k}(\widehat{x}) g_{j} g_{k}(x), \] so this must also equal $ \delta_{i,j} $. \end{proof}
The third lemma tells us how the action of $ H_{\lambda} $ on $ L $ interacts with the action of $ K[G] $:
\begin{lemma} \label{lem_interchange_action} Let $ t \in L $, $ z \in K[G] $ and $ h \in H_{\lambda} $. Then \[ h \cdot z(t) = z(h \cdot t). \] \end{lemma} \begin{proof} The map $ T: L[\lambda(G)] \rightarrow L[\lambda(G)]^{G} = H_{\lambda} $ defined by \[ z \mapsto \sum_{g \in G} \,^{g}\!z \] is $ K $-linear and surjective, so it is sufficient to consider the case where $ h = T(y \lambda(\tau)) $ for some $ y \in L $ and $ \tau \in G $ and $ z=\sigma \in G $. In this case we have: \begin{eqnarray*} \sigma (T(y \lambda(\tau)) \cdot t) & = & \sigma \left( \sum_{g \in G} g(y) \,^{g}\!\lambda(\tau) \cdot t \right)\\ & = & \sigma \left( \sum_{g \in G} g(y) \lambda(g\tau g^{-1}) \cdot t \right)\\ & = & \sigma \left( \sum_{g \in G} g(y) g\tau^{-1} g^{-1} (t) \right) \\ && \mbox{(since } (\lambda(g\tau g^{-1}))^{-1} (1_{G}) = g\tau^{-1} g^{-1} \mbox{)}\\ & = & \sum_{g \in G} \sigma g(y) \sigma g\tau^{-1} g^{-1} (t) \\ & = & \sum_{g \in G} \sigma g(y) \sigma g\tau^{-1} g^{-1} \sigma^{-1} \sigma (t) \\ & = & \sum_{g \in G} \sigma g(y) \,^{ (\sigma g)}\!\lambda(\tau) \cdot \sigma(t) \\ & = & \sum_{g \in G} g(y) \,^{g}\!\lambda(\tau) \cdot \sigma(t) \\ & = & T(y \lambda(\tau)) \cdot \sigma(t), \end{eqnarray*} as claimed. \end{proof}
\section{Proof of the Main Theorem}
Let $ \B $ be a $ G $-stable fractional ideal of $ L $. Write $ \A_{K[G]} $ for the associated order of $ \B $ in $ K[G] $ and $ \A_{\lambda} $ for the associated order of $ \B $ in $ H_{\lambda} $. We shall split the ``if" and ``only if" implications of Theorem \eqref{thm_main} into two separate propositions.
\begin{prop} \label{prop_classical_implies_nonclassical} Suppose that $ x \in \B $ generates $ \B $ as an $ \A_{K[G]} $-module. Then $ x $ generates $ \B $ as a $ \A_{\lambda} $-module. \end{prop} \begin{proof} Since $ x $ generates $ \B $ as an $ \A_{K[G]} $-module, it generates $ L $ as a $ K[G] $-module, so $ \{ \sigma(x) \mid \sigma \in G \} $ is a $ K $-basis of $ L $. By Lemma \eqref{lem_dual_basis}, there exists $ \widehat{x} \in L $ such that $ \{ \sigma(\widehat{x}) \mid \sigma \in G \} $ is the dual basis to $ \{ \sigma(x) \mid \sigma \in G \} $. That is: \[ \sum_{g \in G} g\sigma(\widehat{x})g\tau(x) = \delta_{\sigma,\tau} \mbox{ for all } \sigma,\tau \in G. \] Also, there exist $ a_{1}, \ldots ,a_{n} \in \A_{K[G]} $ such that $ \{ a_{1}(x), \ldots ,a_{n}(x) \} $ is an $ \OK $-basis of $ \B $. For each $ i=1, \ldots, n $, write $ x_{i}=a_{i}(x) $ and define an element $ h_{i} \in L[\lambda(G)] $ by \[ h_{i} = \sum_{g \in G} \left( \sum_{\rho \in G} \rho(x_{i}) g^{-1} \rho(\widehat{x}) \right) \lambda(g). \] For each $ i=1, \ldots ,n $ we make three claims about the element $ h_{i} $: \begin{enumerate} \item $ h_{i} \in L[\lambda(G)]^{G} = H_{\lambda} $ (so it makes sense to let $ h_{i} $ act on an element of $ L $ using the formula given in the theorem of Greither and Pareigis). \item $ h_{i} \cdot x = x_{i} $ (so $ x $ is an $ H_{\lambda} $-generator of $ L $, but we knew this anyway from Theorem \eqref{thm_NBG}). \item $ h_{i} \in \A_{\lambda} $. \end{enumerate} If we can establish these three claims, then it will follow that $ \{ h_{i} \mid i=1, \ldots ,n \} $ is an $ \OK $-basis of $ \A_{\lambda} $ and that $ \B $ is a free $ \A_{\lambda} $-module. \\ \\ To prove (1), let $ \tau \in G $. Then \begin{eqnarray*} \,^{\tau}\! h_{i} & = & \,^{\tau}\! \left( \sum_{g \in G} \left( \sum_{\rho \in G} \rho(x_{i}) g^{-1} \rho(\widehat{x}) \right) \lambda(g) \right) \\ & = & \sum_{g \in G} \tau \left( \sum_{\rho \in G} \rho(x_{i}) g^{-1} \rho(\widehat{x}) \right) \,^{\tau}\! \lambda(g) \\ & = & \sum_{g \in G} \left( \sum_{\rho \in G} \tau\rho(x_{i}) \tau g^{-1} \rho(\widehat{x}) \right) \lambda(\tau g \tau^{-1}) \\ & = & \sum_{g^{\prime} \in G} \left( \sum_{\rho \in G} \tau\rho(x_{i}) (g^{\prime})^{-1} \tau \rho(\widehat{x}) \right) \lambda(g^{\prime}) \\ && \mbox{(writing } g^{\prime} = \tau g \tau^{-1}, \mbox{ so that } \tau g^{-1} = (g^{\prime})^{-1} \tau \mbox{)} \\ & = & \sum_{g \in G} \left( \sum_{\rho \in G} \tau\rho(x_{i}) g^{-1} \tau \rho(\widehat{x}) \right) \lambda(g) \\ && \mbox{(replacing } g^{\prime} \mbox{ by } g \mbox{)} \\ & = & \sum_{g \in G} \left( \sum_{\rho \in G} \rho(x_{i}) g^{-1} \rho(\widehat{x}) \right) \lambda(g) \\ && \mbox{(replacing } \tau \rho \mbox{ by } \rho \mbox{)} \\ & = & h_{i}, \end{eqnarray*} so $ h_{i} \in L[\lambda(G)]^{G} = H_{\lambda} $. \\ \\ Now we know that it makes sense to let $ h_{i} $ act on $ x $, and so we can prove (2): \begin{eqnarray*} h_{i} \cdot x & = & \left( \sum_{g \in G} \left( \sum_{\rho \in G} \rho(x_{i}) g^{-1} \rho(\widehat{x}) \right) \lambda(g) \right) \cdot x \\ & = & \sum_{g \in G} \left( \sum_{\rho \in G} \rho(x_{i}) g^{-1} \rho(\widehat{x}) \right) g^{-1}(x) \\ & = & \sum_{\rho \in G} \rho(x_{i}) \left( \sum_{g \in G} g^{-1} \rho(\widehat{x}) g^{-1}(x) \right) \\ & = & \sum_{\rho \in G} \rho(x_{i}) \mbox{Tr}_{L/K} ( \rho(\widehat{x}) x ) \\ & = & \sum_{\rho \in G} \rho(x_{i}) \delta_{\rho,1} \\ & = & x_{i}. \end{eqnarray*} Finally, we prove (3). It is sufficient to prove that $ h_{i} \cdot x_{j} \in \B $ for each $ j=1, \ldots ,n $. Recall that $ x_{j} = a_{j}(x) $ for some $ a_{j} \in \A_{K[G]} $. Using Lemma \eqref{lem_interchange_action} we have: \begin{eqnarray*} h_{i} \cdot x_{j} & = & h_{i} \cdot a_{j}(x) \\ & = & a_{j} ( h_{i} \cdot x ) \\ & = & a_{j}(x_{i}), \end{eqnarray*} and this lies in $ \B $ since $ x_{i} \in \B $ and $ a_{j} \in \A_{K[G]} $. \\ \\ We have verified all three claims, and so the proof is complete. \end{proof}
The next proposition is the converse of the previous one:
\begin{prop} \label{prop_nonclassical_implies_classical} Suppose that $ x \in \B $ generates $ \B $ as an $ \A_{\lambda} $-module. Then $ x $ generates $ \B $ as an $ \A_{K[G]} $-module. \end{prop} \begin{proof} Since $ x $ generates $ \B $ as an $ \A_{\lambda} $-module, it generates $ L $ as an $ H_{\lambda} $-module, and so by Theorem \eqref{thm_NBG} it generates $ L $ as a $ K[G] $-module. Therefore $ \{ \sigma(x) \mid \sigma \in G \} $ is a $ K $-basis of $ L $ and by Lemma \eqref{lem_dual_basis} there exists $ \widehat{x} \in L $ such that $ \{ \sigma(\widehat{x}) \mid \sigma \in G \} $ is the dual basis to $ \{ \sigma(x) \mid \sigma \in G \} $. Mirroring the proof of Proposition \eqref{prop_classical_implies_nonclassical}, there exist $ h_{1}, \ldots ,h_{n} \in \A_{\lambda} $ such that $ \{ h_{1}\cdot x, \ldots ,h_{n} \cdot x \} $ is an $ \OK $-basis of $ \B $. For each $ i=1, \ldots, n $, write $ x_{i}=h_{i} \cdot x $ and define an element $ a_{i} \in K[G] $ by \[ a_{i} = \sum_{g \in G} \mbox{Tr}_{L/K}(x_{i}g(\widehat{x})) g. \] In this case it is clear that $ a_{i} \in K[G] $, so it makes sense to let $ a_{i} $ act on an element of $ L $, and we only make two claims about $ h_{i} $: \begin{enumerate} \item $ a_{i}(x) = x_{i} $. \item $ a_{i} \in \A_{K[G]} $. \end{enumerate} As in the proof of \eqref{prop_classical_implies_nonclassical}, if we can establish these claims then it will follow that $ \{ a_{i} \mid i=1, \ldots ,n \} $ is an $ \OK $-basis of $ \A_{K[G]} $ and that $ \B $ is a free $ \A_{K[G]} $-module. \\ \\ First we prove (1). We have: \begin{eqnarray*} a_{i}(x) & = & \sum_{g \in G} \mbox{Tr}_{L/K}(x_{i}g(\widehat{x})) g(x) \\ & = & \sum_{g \in G} \sum_{\sigma \in G} \sigma(x_{i}) \sigma g (\widehat{x}) g(x) \\ & = & \sum_{\sigma \in G} \sigma(x_{i}) \sum_{g \in G} \sigma g (\widehat{x}) g(x) \\ & = & \sum_{\sigma \in G} \sigma(x_{i}) \delta_{\sigma,1} \mbox{ (using Lemma \eqref{lem_inside_out_trace}} ) \\ & = & x_{i}. \end{eqnarray*} To prove (2), it is sufficient to prove that $ a_{i}(x_{j}) \in \B $ for each $ j=1, \ldots ,n $. Recall that $ x_{j} = h_{j} \cdot x $ for some $ h_{j} \in \A_{\lambda} $. Using Lemma \eqref{lem_interchange_action} we have: \begin{eqnarray*} a_{i}(x_{j}) & = & a_{i}( h_{j} \cdot x) \\ & = & h_{j} \cdot ( a_{i}(x) ) \\ & = & h_{j} \cdot x_{i}, \end{eqnarray*} and this lies in $ \B $ since $ x_{i} \in \B $ and $ h_{j} \in \A_{\lambda} $. \\ \\ We have verified both the claims, and so the proof is complete. \end{proof}
By combining Propositions \eqref{prop_classical_implies_nonclassical} and \eqref{prop_nonclassical_implies_classical}, we obtain Theorem \eqref{thm_main}
\section{Further Questions and Possible Generalizations} Does assuming that one of $ \A_{K[G]} $ or $ \A_{\lambda} $ is a Hopf order imply that other is too? This might be particularly interesting in the case that $ L/K $ is tame and $ \B = \OL $, since then $ \A_{K[G]} = \OK[G] $, which is certainly a Hopf order. In a similar direction, if $ L/K $ is a Galois extension of $ p $-adic fields and $ p \nmid [L:K] $ then $ \OK[G] $ is a maximal order in $ K[G] $: does this imply that the associated order of $ \OL $ in $ H_{\lambda} $ is also maximal? One way to do this would be to show that it is self dual with respect to some symmetric associative bilinear form, and showing that it is Hopf would certainly suffice for this. \\ \\ I think that some of the nice properties of $ H_{\lambda} $ such as those expressed in Theorem \eqref{thm_NBG} and Lemma \eqref{lem_interchange_action} might boil down to the fact that $ \lambda(G) $ commutes with $ \rho(G) $ inside $ \perm{G} $. Perhaps a similar approach would work for other regular subgroups $ N $ of $ \perm{G} $ that satisfy this condition? In the local case, perhaps it would be sufficient to have some of these nice properties hold modulo $ \p_{K} $ and then argue using Nakayama's lemma?
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Policy with stochastic hysteresis } \author{Georgii Riabov and Aleh Tsyvinski\thanks{Riabov: Institute of Mathematics, NAS of Ukraine; Tsyvinski: Yale University. We thank Fernando Alvarez, Lint Barrage, Job Boerma, Jaroslav Borovi\v{c}ka, Eduardo Faingold, Felix Kubler, Francesco Lippi, Ernest Liu, Erzo Luttmer, Giuseppe Moscarini, Alessandro Pavan, Florian Scheuer, Simon Scheidegger, Michael Sockin, Stefanie Stantcheva, Kjetil Storesletten, Philipp Strack, Ted Temzelides, Pierre Yared, and Nicolas Werquin.}} \maketitle \begin{abstract} The paper develops a general methodology for analyzing policies with path-dependency (hysteresis) in stochastic models with forward looking optimizing agents. Our main application is a macro-climate model with a path-dependent climate externality. We derive in closed form the dynamics of the optimal Pigouvian tax, that is, its drift and diffusion coefficients. The dynamics of the present marginal damages is given by the recently developed functional Itô formula. The dynamics of the conditional expectation process of the future marginal damages is given by a new total derivative formula that we prove. The total derivative formula represents the evolution of the conditional expectation process as a sum of the expected dynamics of hysteresis with respect to time, a form of a time derivative, and the expected dynamics of hysteresis with the shocks to the trajectory of the stochastic process, a form of a stochastic derivative. We then generalize the results. First, we propose a general class of hysteresis functionals that permits significant tractability. Second, we characterize in closed form the dynamics of the stochastic hysteresis elasticity that represents the change in the whole optimal policy process with an introduction of small hysteresis effects. Third, we determine the optimal policy process. \end{abstract} \pagebreak{}
\section{Introduction}
Modern macroeconomics is stochastic and is built on recursive methods. The actions in these models are history dependent but the past is represented by a small number of finite-dimensional state variables. For example, Ljungqvist and Sargent (2018) write that ``finding a recursive way to handle history dependence is a major achievement of the past 35 years and an important methodological theme''. They call these developments an ``imperialistic response of dynamic programming'' to Prescott's (1977) critique of the impossibility of using dynamic programming in government policy design. In this paper, we aim to study environments in which it may in general be difficult or impossible to write a recursive representation. We use the term hysteresis to describe settings in which the trajectory of policies or actions may generally affect the structure of the environment. That is, the effects of policies are path-dependent.
Our main economic application is \textcyr{\cyra} generalization of the macroeconomy-climate model of Golosov, Hassler, Krusell, and Tsyvinski (2014) to the path-dependent climate externalities. There are two main reasons to consider hysteresis in such models. First, there is important recent evidence from climate sciences that a number of climate variables show significant hysteresis behavior. Intergovernmental Panel on Climate Change (IPCC) lists, for example, hysteresis that is present in (1) vegetation change; (2) changes in the ice sheets; (3) ocean acidification, deep ocean warming and associated sea level rise; (4) models of the feedback between the ocean and the ice sheets; and (5) models of the Atlantic meridional overturning circulation (Collins, et al. 2013 in IPCC Fifth Assessment Report -- AR5). While cumulative carbon emission is an important benchmark used in macro-climate models, recent research in climate sciences (further discussed in Section \ref{subsec:Evidence-on-climate}) points to importance of hysteresis in a number of climate variables that constitute a natural science foundation of these models. Moreover, the path-dependency in these models is often rather sophisticated as the exact sequence of the events may matter. The IPCC Special Report further concludes that path dependence of carbon budgets ``remains an important knowledge gap'' (Rogelj et al. 2018). Second, one of the central themes in the recent developments in the economics literature on climate change is incorporating path-dependency. A recent survey by Aghion, et al. (2019) summarizes these developments and argues that path-dependency in which both history and expectations matter is one of the core insights in this literature (see, for example, Acemoglu, Aghion, Bursztyn, and Hemous (2012), Acemoglu, Akcigit, Hanley, and Kerr (2016), Acemoglu, Aghion, Barrage and Hemous (2019), and Aghion, Dechezlepretre, Hemous, Martin, and Van Reenen (2016)). Finally, careful consideration of uncertainty has been another important theme of recent research on economics of climate change. Our application thus puts at the forefront the analysis of an economy where \textcyr{\cyra} forward looking social planner makes emission decisions under uncertainty and faces a path-dependent emissions externality. In contrast to the literature and, specifically to Golosov, Hassler, Krusell, and Tsyvinski (2014), we significantly relax the assumption of how the previous emission choices enter the climate damages. In our setting, the whole path of the emission (hysteresis) matters as opposed to a depreciated stock of the previous emissions. Specifically, we analyze a particular climate hysteresis functional that captures the most important elements of the setting with general hysteresis.
Our main result in this macro-climate application is a closed form characterization of the dynamics of the marginal externality damage which is also the Pigouvian tax that implements the optimal allocation. We start by deriving the first-order conditions for the optimum and showing that the marginal externality damage from emissions is comprised of the marginal contemporaneous damages and the conditional expectation of the cumulative future damages. Both of these terms are path-dependent and their dynamics cannot be derived using the usual Itô formula which applies only to functions of the current state and not the functionals of the trajectories. We use two different sets of tools to provide the dynamics of the present and the expected cumulative damages.
The present marginal damages are already represented as an explicit path-dependent functional and we use recently developed functional Itô calculus for the non-anticipative functionals (Dupire 2009, 2019, Cont and Fournie 2013) to represent this term. The functional Itô formula uses two new concepts of derivatives introduced by Dupire (2009, 2019). The vertical derivative is an analogue of the space derivative and the horizontal derivative is an analogue of the time derivative for the functionals. These derivatives evaluate, respectively, the effects of a discontinuous bump in the underlying trajectory and the effects of a time extension of the trajectory while fixing the terminal position. The Dupire's functional Itô formula then gives the dynamics of the present damages with the drift determined by the horizontal and the second vertical derivative, and the diffusion coefficient given by the vertical derivative of the marginal present damages. Using the notion of the horizontal and vertical derivatives the functional Itô formula allows us to convey the similar intuition as in the case of no hysteresis for which the usual Itô formula is applicable. However, in contrast with the case of no hysteresis path-dependency may lead to significant effects on the dynamics even in the case of absent contemporaneous damages.
The most challenging part of the paper is to characterize the dynamics of the expected future marginal damages represented by the conditional expectation process. The difficulty comes from the fact that with passage of time both the damages themselves and the information set (filtration) are changing. The main new theoretical tool that we develop -- the total derivative formula for conditional expectation processes -- allows us to characterize the dynamics of such processes. Moreover, this formula is broadly applicable to a variety of macroeconomic settings where the conditional expectations processes are central. There are two terms in the semimartingale decomposition of the conditional expectations process that the total derivative formula delivers. The first term can be intuitively thought of as \textcyr{\cyra} time derivative and represents how the conditional expectation of the future marginal damages evolves with respect to time. The second term, given as a conditional expectation of the Malliavin derivative of the cumulative future marginal damages, can be thought of as a stochastic derivative with respect to the underlying process and represents how the conditional expectation changes with the changes in the stochastic process. It is useful to compare the results to the case with no hysteresis. The main difference is that in the case of no hysteresis, the policy does not have future effects. With hysteresis, the contemporaneous effects of the policy may be very small but the expected future effects of the policy may be very large -- small actions today my have significant future consequences. Moreover, these effects change with both time and the stochastic shocks.
Combining the dynamics of the present and the conditional expectation of the future marginal damages, we obtain in closed form the drift and diffusion of the optimal Pigouvian tax correcting the path-dependent externality. The dynamics are given by the semimartingale decomposition of the marginal externality damages which can be though of in terms of the planner changing the tax with the passage of time and with the realization of uncertainty. Alternatively, this can be thought of as determination of what features of the model matter to the first order for the drift and diffusion of the optimal tax. Specifically, those are given by the functional Itô formula and by our total derivative formula for the conditional expectation processes. Even if there are no direct contemporaneous effects, the dynamics of the optimal policy may be significantly impacted by either the past choices or by the future effects. In other words, both the past and the future non-trivially matter.
We then show that one can significantly generalize the application of the climate hysteresis. First, we propose a very general class of hysteresis functionals that allow significant tractability of the analysis of optimal policies. The Fréchet derivatives of these functionals have two sources of variation: an instantaneous influence in the given (current) period and an integral influence of perturbations in previous periods. The assumption is mild and only requires that the derivative of a functional at a given time is absolutely continuous up to that period and may have an atom at the present time. We show that every Fréchet differentiable functional is a pointwise limit of functionals from this class. Importantly, these functionals have an attractive property of conveniently separating the effects of the past and the present. As we are considering a very general environment with path-dependence, it may be difficult or impossible to use dynamic programming to write a recursive representation of the problem. Instead, we directly find the first order conditions for the optimal problem by considering all local variations of the policies. The assumption on the derivative of the hysteresis functional allows us to conveniently separate the first order condition in two parts: the marginal effects of the policy on current period and the conditional expectation of the cumulative future marginal effects of policy paths in the future periods.
Our next goal is to explicitly find the evolution equation for the stochastic hysteresis elasticity. This concept of stochastic hysteresis elasticity is similar to the usual concept of elasticity as it captures the change in the variables following a small change in policy. The difference is that the stochastic elasticity is the change in the whole optimal process. Alternatively, one can think of the stochastic elasticity as the asymptotics of the optimal policy when there is an infinitesimal additional hysteresis. Specifically, we show that the stochastic elasticity is an Itô process and write its explicit semimartingale decomposition. A fundamental property of the solution is that both of the terms -- the present and the expected future marginal effects of policy paths -- in the first order condition for optimality and in the formula for the stochastic elasticity are path-dependent. Similarly to our climate hysteresis application we use the functional Itô formula and the total derivative formula to represent the dynamics of the stochastic elasticity. The semimartingale decomposition of the present and the cumulative marginal effects of policies show that the stochastic elasticity, that is, the change in the optimal policy process following introduction of hysteresis, is an Itô process. The difference with the no-hysteresis case is that now the drift and diffusion coefficients are themselves functionals of the trajectory of the policy. That is, hysteresis leads to the path-dependent stochastic process of the elasticity for which we calculate the drift and diffusion coefficients in the explicit form. Finally, we return to the analysis of the optimal policies with general hysteresis. The evolution of the optimal policy can be analyzed using exactly the same tools as the stochastic elasticity. The explicit form we obtained for the stochastic elasticity thus can be thought of as a characterization of the optimal policy process for small hysteresis, or small hysteresis asymptotics. We use the functional Itô formula to represent the current marginal effects and the total derivative formula to represent the conditional expectation process of the future marginal effects of policies. The drift and diffusion coefficients of the optimal policy are then determined implicitly by the equations having a similar form as the explicit solution for the stochastic elasticity but also depending on the process of the optimal policy itself.
We finally provide several examples of how to calculate the stochastic hysteresis elasticity. All of these examples are straightforward applications of the main semimartingale decomposition formula that we developed.
\subsection*{Literature}
The closest to our work is a sequence of papers by Borovi\v{c}ka, Hansen, and Scheinkman (2014), Borovi\v{c}ka, Hansen, Hendricks, and Scheinkman (2011), and Borovi\v{c}ka and Hansen (2016). They define the concept of the shock elasticity, relate it to impulse responses familiar to macroeconomists, and, importantly, formalize it using Malliavin derivatives. The shock elasticity is an impulse response of the pricing kernel to the marginal perturbation of the underlying stochastic cash flow process. Our work goes beyond their results and delivers the explicit semimartingale decomposition of the evolution of the stochastic elasticity for the path-dependent functionals. In Section \ref{subsec:Relationship-to-the_Clarrk-Ocone} we also provide a connection to the Clark-Ocone formula that features prominently in Borovi\v{c}ka, Hansen, and Scheinkman (2014).
The only paper that we are aware of that uses functional Itô formula in economic settings is an insightful work of Cvitani\'{c}, Possamaï, and Touzi (2017). In the dynamic moral hazard setting where the agent controls the volatility of the process, they show how that without loss of generality the set of admissible contracts can be represented as the path-dependent processes arising from the functional Itô formula. In our environment, both the functional Itô formula and the total derivative formula are applied. We discuss this in more details in Section \ref{subsec:Why-do-we-need-two-formulas}.
The concept of stochastic elasticity is similar to the analysis of the models of investment under uncertainty for the case of small sunk costs or for models with small menu costs. In those models, there is a form of hysteresis where due to the region of inaction, a temporary change may have a permanent effect. The analysis of such models have important implications for the large amounts of inertia that arise in the models of small menu costs such as in Akerlof and Yellen (1985) and Mankiw (1985). Dixit (1991) develops a method of analytical approximation for such hysteresis models. Reis (2006) and Alvarez, Lippi, and Paciello (2011, 2016) derive a similar analytical characterization in the models with inattentive producers. These are essentially the same as our calculation of stochastic elasticity for the environments that can be characterized by Itô formula. A series of papers by Alvarez and Lippi (2019), Alvarez, Le Bihan, and Lippi (2016) and Alvarez, Lippi, and Oskolkov (2020) focus on the cumulative output effects of shock in environments with the menu and adjustment costs frictions. These cumulative output effects are the output impulse response functions\footnote{Similarly, the impulse responses play an important role in Pavan, Segal, and Toikka (2014), Garrett and Pavan (2012), and Bergemann and Strack (2015) in dynamic mechanism design, Makris and Pavan (2017) in the dynamic taxation, and Makris and Pavan (2018) who connect these two literatures. See also a review in Pavan (2017). In general, the recursive formulation of a number of these problems may be challenging and the tools we develop may prove to be useful in those contexts.} that are similar to the future expected marginal effects of policy in our paper. Our analysis can be thought of as generalizing the results to the environments in which there is path-dependence.
An important early paper by Detemple and Zapatero (1991) studies an asset pricing model with habit formation using Malliavin calculus.\footnote{See also Serrat (2001), Bhamra and Uppal (2009) and Cvitanic and Malamud (2009). } Huang and Sockin (2018) study optimal dynamic taxation in continuous time and calculate impulse response to misreporting using Malliavin derivatives. As Sannikov (2014) for the model of moral hazard they connect their analysis to the finance literature on the sensitivity of the ``Greeks'' of the options (see, e.g., Fournié, Lasry, Lebuchoux, Lions, and Touzi 1999). Their analysis relates to our calculation of Malliavin derivatives of the policy process in the last part of Section \ref{subsec:Example Martingale}.
Hysteresis or path-dependence is important in a variety of macroeconomic and other settings. The natural rate of unemployment may directly depend on the previous path of unemployment and, hence, on the path of monetary policy (Blanchard and Summers 1986). Examples of recent explicit focus on path dependency and monetary policy is Berger, Milbradt, Tourre, and Vavra (2018), Gali (2020), and Jorda, Singh, and Taylor (2020). Hysteresis is also central in the models of secular stagnation as in Summers (2014) and Eggertsson, Mehrotra, and Robbins (2019). Adjustment costs in investment generate hysteresis (for example, see a survey in Dixit 1992). Acemoglu, Egorov, and Sonin (2020) exposit a wide variety of political economy models with path-dependence of policies or institutions. Egorov and Sonin (2020) additionally discuss models in which there is non-Markovian dependence on the sequence of policy events. Page (2006) classifies types of history dependence and distinguishes between the path-dependent policies (where the exact sequence of events matters) and the ``phat'' dependent strategies (where the events but not their sequence matters). The models of increasing returns such as Arthur (1989) and David (1985), and a review in Arrow (2000) feature potentially very complicated dependence on the paths.
We now briefly discuss the relationship of the economic environment we consider with mathematical literature. The analysis of deterministic dynamic systems with hysteresis is well established (e.g., Krasnosel'skii and Pokrovskii 2013). In our model, path dependency arises from the optimization of the forward looking agents under uncertainty rather than being directly posed as a property of a dynamic system. The problem that we study also does not fit easily into the standard framework of the infinite-dimensional stochastic optimal control such as Fabbri, Gozzi, and Swiech (2017), see discussion in Section \ref{subsec:Discussion-of-the-optimal control}. The reason is that the hysteresis functional depends on the whole path of the policy. If we treat the policy as a control process we are obliged to take the space of paths (of variable length) as a control space which leads to the analysis of path-dependent HJB equations the understanding of which is in its nascency.
\section{Environment}
We describe an environment of a stochastic problem with forward looking agents and path-dependency.
\subsection{\label{subsec:Baseline-setting}Baseline setting}
Let $w=(w_{t})_{0\leq t\leq T}$ be a Brownian motion.\footnote{All of the results can be straightforwardly extended to more general diffusion processes.} Let the objective of the planner be described by the quadratic loss function: \begin{equation} \max_{c}\mathbb{E}\int_{0}^{T}-\frac{1}{2}\left(c_{t}-w_{t}\right)^{2}dt.\label{eq:Unperturbed problem} \end{equation} Here maximization is taken over all (progressively measurable) policy processes $c:[0,T]\times C([0,T])\to\mathbb{R}.$ That is, policy $c_{t}$ depends on information about $w$ up to moment $t$.
The basic problem is easily solved by interchanging integral and expectation and yields the optimal process $c_{t}^{*}$ : \begin{equation} c_{t}^{*}=w_{t}.\label{eq: Optimal consumption, unperturbed} \end{equation} The optimal policy simply tracks the Brownian motion $w_{t}$.
\subsection{General environment}
Consider a problem where there is an additional path-dependent effect of policy $h_{t}$: \begin{equation} \max_{c}\mathbb{E}\int_{0}^{T}(-\frac{1}{2}\left(c_{t}-w_{t}\right)^{2}-\epsilon h_{t}(c))dt,\label{eq:Perturbed problem} \end{equation} where $h:[0,T]\times C[0,T]\to\mathbb{R}$ is an adapted functional on the space of trajectories (i.e. $h_{t}(c)$ is defined only by the restriction of $c$ on $[0,t]$ ), and $\epsilon$ is a parameter. One can think of the functional $h$ as an additional effect (a cost or a benefit) that depends on the history of policy up to time $t$. The solution of this problem is denoted $c^{\epsilon}$. We define hysteresis as dependence of the effects of policy $h_{t}(c)$ on the path of the previous policies $c_{[0,t]}=c^{t}$ .
\section{Climate externality}
This section generalizes the macro-climate model of Golosov, et al. (2014) to the case where climate externality is path-dependent. Section \ref{subsec:Planner's-problem-Climate} shows how to map some of the key features of that paper to the framework described in the previous section. Section \ref{subsec:Evidence-on-climate} discusses evidence on hysteresis in the climate externality. Section \ref{subsec:Climate-hysteresis-functional} proposes a specific path-dependent climate hysteresis functional that captures most of the insights of the general framework that we develop later in Section \ref{sec:General-setting}. Sections \ref{subsec:First-order-conditions-climate}-\ref{subsec:Dynamics Pigouvian Climate} solve in closed form the dynamics of the marginal climate externality or the Pigouvian tax implementing the optimum, that is, determine its drift and diffusion coefficients.
\subsection{\label{subsec:Planner's-problem-Climate}Planner's problem with a climate externality}
We first note that the structure of the problem (\ref{eq:Perturbed problem}) captures some of the main features of the macroeconomic model with a climate externality of Golosov, et al. (2014). In its essence, a macro-climate model consists of two primary blocks. The first is the specification of the economy which is a standard dynamic stochastic general equilibrium model. The planner maximizes the expected utility of consumption (we assume no discounting) \[ \mathbb{E}\int_{0}^{T}U\left(c_{t}^{\text{cons}}\right)dt, \] subject to the standard feasibility constraint with capital accumulation, where $c_{t}^{\text{cons}}$ is a consumption good. The production function is given by \[ Y_{t}=F\left(K_{t},E_{t},S_{t}\right), \] where $K_{t}$ is capital, $E_{t}$ is energy consumption (measured in carbon emission units), and $S_{t}$ is a climate variable at time $t$. The climate variable is in general a functional $S_{t}=\tilde{S_{t}}\left(E^{t}\right)$ of the path of emissions $E^{t}=E_{\left[0,t\right]}$ (equation (4) in Golosov, et al. (2014) is, in fact, exactly this general formulation). The climate variable $S_{t}$ affects the economy via a damage function $D_{t}\left(S_{t}\right)$ so that the output is reduced multiplicatively $\left(1-D_{t}\left(S_{t}\right)\right)\times\tilde{F}\left(K_{t},E_{t}\right).$ Moreover, the damages are assumed to be exponential $1-D_{t}\left(S_{t}\right)=\exp\left(-\gamma_{t}S_{t}\right)$ for some parameter $\gamma_{t}$ and the utility $U\left(x\right)=\log\left(x\right)$. The essence of the model is how the path of emissions $E^{t}$ translates to damages to the economy. Let us now reduce the model further to focus on this key dimension. Assume that production does not require capital and the feasibility constraint for the economy is then static: \[ c_{t}^{cons}=Y_{t}=e^{-\gamma_{t}S_{t}}F\left(E_{t}\right). \] The planner then maximizes \[ \max_{E_{t}}\mathbb{E}\int_{0}^{T}\left(\tilde{U}\left(E_{t}\right)-\gamma_{t}S_{t}\right)dt, \] which is essentially equivalent to the general problem (\ref{eq:Perturbed problem}). We further assume that $\tilde{U}\left(E_{t}\right)=-\frac{1}{2}\left(E_{t}-w_{t}\right)^{2}$ but this is again done purely for leanness of the model and can be immediately extended.
The key simplification, however, that the literature makes is in simple dependence of the climate variable $S_{t}$ on the path of the emissions. For example, Golosov, et al. (2014)) assume that \[ S_{t}=\int_{0}^{t}d_{s}E_{s}ds, \] where $d_{s}\in\left[0,1\right]$ is a carbon depreciation rate and then also further structure is placed on $d_{s}$. In other words, the climate variable is equal to the stock of the depreciated emissions.
The climate-economy model in its essence reduces to how the previous energy consumption choices and the associated carbon emissions impact today's and future economy. In contrast to the literature, we significantly relax the assumption of how the previous emission choices enter the damages. In our setting, the whole path of the emissions (hysteresis) matters through the functional $h_{t}\left(c^{t}\right)$ as opposed to a depreciated stock of the previous emissions. We, on purpose, stripped down the climate model to focus only on the general path-dependent effects of the emissions and their interaction with the uncertainty.
\subsection{\label{subsec:Evidence-on-climate}Evidence on climate hysteresis}
There are two primary sets of evidence which imply that hysteresis is important in the economic models of climate change.
The first is a large set of recent evidence on importance of hysteresis in climate sciences. The most comprehensive and authoritative source for such research is the Intergovernmental Panel on Climate Change (IPCC). The working group for the physical science basis of the long-term climate change in the Fifth Assessment Report -- AR5 (Collins, et al. 2013), while arguing for the attractiveness of the use of the cumulative carbon emission, notes that a number of climate variables and models show significant hysteresis behavior. These are the models of (1) vegetation change; (2) changes in the ice sheets; (3) ocean acidification, deep ocean warming and associated sea level rise; (4) models of the feedback between the ocean and the ice sheets; and (5) models of the Atlantic meridional overturning circulation. The report also argues that ``the concepts of climate stabilization and targets is that stabilization of global temperature does not imply stabilization for all aspects of the climate system. For example, some models show significant hysteresis behaviour in the global water cycle, because global precipitation depends on both atmospheric CO2 and temperature''. This is also consistent with the study of Zickfield et al. (2012) who argue that, while total cumulative emissions may be an important approximation for many models, there are a number of exceptions where path-dependency is important: among the variables with timescales of several centuries, such as deep ocean temperature and sea level rise, and for the peak responses of atmospheric CO2 or for the surface ocean acidity. We now briefly discuss some other recent evidence of the hysteresis behavior in individual components of the climate system. Eliseev et al. (2014) is a comprehensive study of the permafrost behavior that finds significant evidence of hysteresis especially for the higher concentration of greenhouse gases in the atmosphere. Another important aspect of ice thawing is the release of permafrost carbon which is found to be both very significant and highly path-dependent (Gasser et al. 2018). Garbe, et al. (2020) analysis ``reveals a strong, multi-step hysteresis behaviour of the Antarctic Ice Sheet'' which is potentially reinforced by a number of additional feedback mechanisms. Nordhaus (2019) augments the DICE model with the effects of Greenland ice sheet disintegration and finds that the baseline or the no-policy effects are significantly different when even a simple formulation of the hysteresis is incorporated while the optimal policy results are similar. Boucher et al. (2012) analyzes the response of a number of models to a significant increase in the CO2 concentration and finds that hysteresis is particularly pronounced for the terrestrial (such as long lived soil carbon sinks and vegetation) and marine variables (such as the sea-level rise) as well as in global mean precipitation. Nohara et al. (2013) finds that the hysteresis effects are important for the regional climate change which is notable given, for example, the recent focus on the significant differences in optimal climate policy at the regional level by Hassler, et al. (2020). Wu et al. (2012) describe significant hysteresis in the hydrological cycle that leads to one of the most direct impacts of global warming affecting droughts, floods and water supplies. This is important for the recent economic literature that studies the impact of the floods and the sea level (for example, Bakkensen and Barrage 2017, Barrage and Furst 2019; Hong, Wang, and Young, 2020 where hysteresis may be also compounded with the path-dependency in beliefs). Summarizing, while the cumulative emission and its variants are certainly a central benchmark for the development of the climate-economy models, recent research in climate sciences points to importance of hysteresis with sophisticated trajectory dependence in a number of important climate variables.
The second reason for why hysteresis is important lies in the economics part of the climate-economy models. One of the central themes in the recent advances in this literature has been incorporating path-dependency. A recent survey by Aghion, et al. (2019) summarize these developments: ``The core insight is that technological innovation is a path-dependent process in which history and expectations matter greatly in determining eventual outcome'' leading to ``important implications for climate policy design'' and ``research and knowledge production are path-dependent, deployment of innovations is path-dependent and the incentives for technology adoption create path dependence''. Several papers in the literature develop various parts of these insights. In Acemoglu, Aghion, Bursztyn, and Hemous (2012), dirty technologies have an advantage in the market size and in the initial productivity and, hence, there is path-dependency in the direction of innovation and production. One of the most important findings of Acemoglu, Akcigit, Hanley, and Kerr (2016) is that the nature of innovations in clean or dirty technologies is path dependent. Similarly, there is path-dependence in innovation in the model of the consequences of the shale gas revolution by Acemoglu, Aghion, Barrage and Hemous (2019). Grubb, et al. (2020) and Baldwin, Cai, and Kuralbayeva (2020) develop models of path-dependency due to the costs of switching from the dirty technologies. Aghion, Dechezleprêtre, Hemous, Martin, and Van Reenen (2016) provide extensive empirical evidence of path dependence in innovation in auto industry from aggregate spillovers and from the firm\textquoteright s own innovation history. Meng (2016) finds significant evidence for strong path dependence in energy transition for the U.S. electricity sector over the 20th century, focusing on coal. Fouquet (2016) is a summary of evidence that energy systems are subject to strong and long-lived path dependence due to technological, infrastructural, institutional and behavioral lock-ins. A recent strand of the climate economics literature (Lemoine and Traeger 2014; Lontzek, Cai, Judd, and Lenton 2015; van der Ploeg and de Zeeuw 2018, and Cai and Lontzek 2019) has attempted to incorporate hysteretic effects through the tipping point modeling. Dietz, Rising, Stoerk, and Wagner (2020) synthesize a number of different approaches in this literature into a meta-model and argue for the need to develop sophisticated models of hysteretic behavior.\footnote{Section \ref{subsec:A-tipping-point} further elaborates on an example of the use of our methodology with tipping points.}
Finally, careful consideration of uncertainty such as in Temzelides (2016), Li, Narajabad, and Temzelides (2016), Traeger (2017), Cai and Lontzek (2019), Brock and Hansen (2018), Van den Bremer and van der Ploeg (2018), Barnett, Brock, and Hansen (2020), Giglio, Kelly, and Stroebel (2020), Kotlikoff, Kubler, Polbin, and Scheidegger (2020) and Lemoine (2021) has been another important theme of recent research on economics of climate change.
\subsection{\label{subsec:Climate-hysteresis-functional}Climate hysteresis functional}
In this section, we analyze a particular hysteresis functional that captures the most important elements of the abstract setting that we develop in Section \ref{sec:General-setting} and allows to transparently show how the solution in the general environment works.
Specifically, let \begin{equation} h_{t}\left(c^{t}\right)=g_{t}\left(w^{t}\right)c_{t}+\int_{0}^{t}k_{s,t}\left(w^{t}\right)c_{s}ds,\label{eq: Climate hysteresis functional} \end{equation} where $c_{t}$ denotes the amount of emissions at time $t$.\footnote{Strictly speaking, the functional in this section depends on both the path $c^{t}$ and $w^{t}$ while we consider in Section \ref{sec:General-setting} the functional $h_{t}\left(c^{t}\right)$. The extension to allow for the dependence on the path $w^{t}$ is immediate. Further, for the analysis of the stochastic elasticity in Section \ref{sec:First-order-process}, the underlying optimal process $c_{t}^{*}=w_{t}$ and, hence, this is without loss of generality. } Here, there are two sources of the effects of emissions. The first source is contemporaneous. The amount of emission $c_{t}$ yields climate externality equal to $g_{t}\left(w^{t}\right)$, where $g_{t}$ is a functional of the path of uncertainty $w^{t}$. The second source is the effect of the past emissions $c_{s}$. Each of the emissions in the previous period $c_{s}$ contributes $k_{s,t}\left(w^{t}\right)$ to the climate externality $h_{t}$ at time $t$, where $k_{s,t}$ is a functional of the path of uncertainty $w^{t}$. It is important to note that both of these terms are path-dependent.\footnote{Remark \ref{rem:Integration by parts} in Section \ref{subsec:Class A_t} further discusses motivation for this example as a representation of stochastic integrals.}
We observe that with the functional (\ref{eq: Climate hysteresis functional}) the optimization problem (\ref{eq:Perturbed problem}) becomes intractable for the stochastic control approach. Indeed, two possibilities for such approach would be either to introduce a control $c^{t}$ or to introduce an additional control $x_{t}=h_{t}(c_{t})$. Both approaches lead to complicated infinite-dimensional HJB equations. The first is due to the complicated structure of the control space. The second is due to the path dependent first order condition for the process $x.$\footnote{Of course, for some special cases there is no need to use the theory we develop. In Section \ref{sec:Examples-of-perturbation} we show, whenever possible, how to derive the results using known tools. One interesting class of examples can be solved by extracting a martingale and then using the Clark-Ocone formula. In Section \ref{subsec:Example Cumulative-hysteresis} and Remark \ref{rem:Boulatov} we show how to do this for the case of cumulative hysteresis $h_{t}\left(c_{t}\right)=c_{t}\int_{0}^{t}c_{s}ds$; in Section \ref{subsec:Example Martingale} and Remark \ref{rem:Detemple Zapatero} we show how to do this for when the kernel $k_{s,t}$ is mutliplicative. We further expand on this class of examples in Boulatov, Riabov, and Tsyvinski (2020).}
\subsection{\label{subsec:First-order-conditions-climate}First-order conditions}
We derive the first order conditions of the problem (\ref{eq:Perturbed problem}) with the climate hysteresis functional (\ref{eq: Climate hysteresis functional}) by perturbing the process $c$ by a an adapted process $z$ and computing the derivative in $\nu$ at $\nu=0:$ \begin{align*}
& \partial_{\nu}E\int_{0}^{T}\left(-\frac{1}{2}(c_{t}+\nu z_{t}-w_{t})^{2}-\epsilon\left(g_{t}\left(w^{t}\right)\left(c_{t}+\nu z_{t}\right)+\int_{0}^{t}k_{s,t}\left(w^{t}\right)\left(c_{s}+\nu z_{s}\right)ds\right)\right)dt\bigg|_{\nu=0}=\\
& =E\int_{0}^{T}\left(z_{t}w_{t}-c_{t}z_{t}-\epsilon\left(g_{t}\left(w^{t}\right)z_{t}+\int_{0}^{t}k_{s,t}\left(w^{t}\right)z_{s}ds\right)\right)dt=\\
& =E\int_{0}^{T}z_{t}\left(w_{t}-c_{t}-\epsilon\left(g_{t}\left(w^{t}\right)+\int_{t}^{T}k_{t,s}\left(w^{s}\right)ds\right)\right)dt=0, \end{align*} where the last line is obtained by changing the order of integration. Since $z$ is an arbitrary adapted process we get the first order conditions \begin{equation}
c_{t}=w_{t}-\epsilon\left(g_{t}\left(w^{t}\right)+E\left[\int_{t}^{T}k_{t,s}(w^{s})ds\bigg|\mathcal{F}_{t}\right]\right).\label{eq:FOC emissions} \end{equation} The term \begin{equation}
\Lambda_{t}=g_{t}\left(w^{t}\right)+E\left[\int_{t}^{T}k_{t,s}(w^{s})ds\bigg|\mathcal{F}_{t}\right]\label{eq: Pigouivan tax emissions} \end{equation} plays the central role in Golosov, et al. (2014) and is the marginal externality damage from emissions, it also is equal to the optimal Pigouvian tax that corrects this externality. Specifically, in our setting there are two effects of the marginal change in the emission $c_{t}$. First, there is the immediate damage $g_{t}\left(w^{t}\right)$. Second, there are marginal damages in each future period $s$, given by $k_{t,s}\left(w^{s}\right).$ These damages are then integrated and evaluated as the expectation conditional on the information up to time $t$.
It is important to note the difference with the case with no hysteresis, where the externality is a function $g_{t}\left(w_{t}\right)$ and not a functional of the trajectory $w^{t}$. First, the future expected marginal damage $E\left[\int_{t}^{T}k_{t,s}(w^{s})ds\bigg|\mathcal{F}_{t}\right]$ is absent in that case as the change in emissions today does not have effects on the future. Second, the contemporaneous effects $g_{t}\left(w^{t}\right)$ also may be significantly different from $g_{t}\left(w_{t}\right)$ as it depends now on the path $w^{t}$. Even if emissions have no immediate effects, the past effects and the expected cumulative effects may be very significant. In other words, both the past and the future non-trivially matter.
Our primary interest is in providing the evolution of the marginal externality damage, that is, in the semimartingale decomposition $d\Lambda_{t}=\alpha_{t}\left(w^{t}\right)dt+\beta_{t}\left(w^{t}\right)dw_{t}$. This decomposition can be thought of as the evolution of the marginal externality damage (or the optimal Pigouvian tax) with respect to time $dt$, the drift, and with respect to the realizations of uncertainty $dw_{t}$, the diffusion coefficient. Since both the terms $g_{t}\left(w^{t}\right)$
and $E\left[\int_{t}^{T}k_{t,s}(w^{s})ds\bigg|\mathcal{F}_{t}\right]$ are path-dependent, we cannot use the standard Itô formula as it applies only to the functions and not to the functionals of the path. We proceed by analyzing the present and the future effects separately as the analysis requires the use of different methods.
\subsection{Present effects of the emissions path\label{subsec:Present-effects-Example}}
The present damage $g_{t}\left(w^{t}\right)$ of the current emissions is already a well-defined functional of the path $w^{t}$. We use a recently developed Dupire's functional Itô formula for the non-anticipative functional (Dupire 2009, 2019; Cont and Fournie 2013) to represent this term.
We first introduce two derivatives due to Dupire (2009, 2019): the vertical and the horizontal derivative. These are, respectively, the functional analogues of the space and of the time derivatives. \begin{defn} \textbf{(Vertical Derivative)} Let $D[0,t]$ be the space of càdlàg paths. The vertical derivative of a functional $h:D[0,t]\to\mathbb{R}$ is the limit \[ \partial_{c_{t}}h(c)=\lim_{\epsilon\to0}\frac{h(c+\epsilon e_{t})-h(c)}{\epsilon}, \] where $e_{t}(s)=1_{s=t}.$ \end{defn} Intuitively, this derivative measures an influence on the functional of a (discontinuous) bump to the path at time $t$ (Figure \ref{fig:Dupire's-vertical-derivative}). If the functional is just the function of current realization of the path, this yields the usual derivative $h'\left(c_{t}\right).$
\begin{figure}
\caption{Dupire's vertical derivative: path perturbation}
\label{fig:Dupire's-vertical-derivative}
\end{figure}
\begin{defn} \textbf{(Horizontal derivative)} Let $z$ $:[0,T]\times D[0,T]\to\mathbb{R}$ be an adapted functional on the space of trajectories. The horizontal derivative $\Delta_{t}$ is defined as a limit \[ \Delta_{t}z_{t}(c)=\lim_{\epsilon\to0}\frac{z_{t+\epsilon}(c_{\cdot,\epsilon})-z_{t}(c)}{\epsilon}, \] where $c_{\cdot,\epsilon}$ is an extension of the path $c$ from $[0,t]$ to $[0,t+\epsilon]$ by $c_{s,\epsilon}=c_{t}$ for $t\leq s\leq t+\epsilon.$ \end{defn} Intuitively, this derivative freezes the path of the underlying process at time $t$, extends it for a small time $\epsilon$ and evaluates the functional $z_{t+\epsilon}$ on this extended path (Figure \ref{fig:Dupire's-horizontal-derivative}). If the functional $z_{t}$ is just the function of the value of the path at time $t$, this yields the usual time derivative of $(\partial_{t}z_{t})(c_{t}).$
\begin{figure}
\caption{Dupire's horizontal derivative: path perturbation}
\label{fig:Dupire's-horizontal-derivative}
\end{figure}
If the functional $g_{t}\left(w^{t}\right)$ does not depend on the path of policies and is a function $f\left(w_{t},t\right)$ of only the current realization of uncertainty and of time, we can use the usual Itô formula. For its application, we need to ensure that the first and the second order space derivative $f'_{1}\left(w_{t},t\right)$, $f''_{1}\left(w_{t},t\right)$ and the time derivative $f'_{2}\left(w_{t},t\right)$ exist. When, as is the case in this section, there is hysteresis to apply the functional Itô formula we need to make a similar assumption on the functional $g_{t}\left(w^{t}\right)$ as in the case of it being a function but with different notions of derivatives.
\begin{comment} \begin{assumption} There exists a non-anticipative functional $q:[0,T]\times D[0,T]\to\mathbb{R}$ which is horizontally differentiable, twice vertically differentiable and is such that $q_{t}(c)=\partial_{c_{t}}h_{t}(c).$ In other words, the derivative \textup{$\partial_{c_{t}}h_{t}(c)$} can be extended to a $C_{b}^{1,2}$-functional on the space of cádlág paths. We denote this extension also by $\partial_{c_{t}}h_{t}(c)$. \end{assumption} The next lemma presents Dupire's functional Itô's formula. \begin{lem} \textbf{(Functional Itô's Formula: Dupire, 2009, 2019; Cont and Fournie, 2013)} Let $q:[0,T]\times D[0,T]\to\mathbb{R}$ be a $C_{b}^{1,2}$-functional on the space of cádlág paths. Then the process $(q_{t}(c^{*}))$ is an Itô process with the stochastic differential \begin{equation} dq_{t}\left(c^{*}\right)=\Delta_{t}q_{t}(c^{*})dt+\partial_{c_{t}}q_{t}(c^{*})dc_{t}^{*}+\frac{1}{2}\partial_{c_{t}}^{2}q_{t}(c^{*})(\sigma^{*}(c_{t}^{*}))^{2}dt.\label{eq:Functional Ito-1} \end{equation} \end{lem} The lemma gives a semi-martingale decomposition of the functional $q_{t}$. It is similar in form to the usual Itô's lemma that applies to the functions of the state. However, the time derivative of a function is replaced by the horizontal derivative of the functional $\Delta_{t}$; the first and second space derivatives of the function are replaced by the first and second vertical derivatives $\partial_{c_{t}}$ and $\partial_{c_{t}}^{2}$. Of course, if there is no path-dependency and $q_{t}$ is a function rather than a functional, one recovers the usual Itô's lemma. \end{comment}
\begin{lem} \textbf{(Functional Itô Formula: Dupire, 2009, 2019; Cont and Fournie, 2013)\label{lem:Functional-It=0000F4's-Formula:}} Let $x_{t}$ be an Itô process, $dx_{t}=b_{t}dt+\sigma_{t}dw_{t},$ where the drift coefficient $b_{t}$ and the diffusion coefficient $\sigma_{t}$ are adapted functionals of $w$. Let $g_{t}$ be horizontally differentiable and twice vertically differentiable functional on the space of cádlág paths. Then the process $g_{t}(x^{t})$ is an Itô process with the stochastic differential \begin{equation} dg_{t}\left(x^{t}\right)=\Delta_{t}g_{t}(x^{t})dt+\partial_{c_{t}}g_{t}(x^{t})dx_{t}+\frac{1}{2}\partial_{c_{t}}^{2}g_{t}(x^{t})(\sigma_{t})^{2}dt.\label{eq:Functional Ito} \end{equation} \end{lem} The lemma gives a semimartingale decomposition of the functional $g_{t}$$\left(x^{t}\right)$. It is similar in form to the usual Itô formula that applies to the functions of the state. However, the time derivative of a function is replaced by the horizontal derivative of the functional $\Delta_{t}$; the first and second space derivatives of the function are replaced by the first and second vertical derivatives $\partial_{c_{t}}$ and $\partial_{c_{t}}^{2}$. Of course, if there is no path-dependency and $g_{t}$ is a function rather than a functional, one recovers the usual Itô formula.
Applying functional Itô formula to $g_{t}\left(w^{t}\right)$ we obtain\textbf{ \[ dg_{t}\left(w^{t}\right)=\left(\Delta_{t}g_{t}(w^{t})+\frac{1}{2}\partial_{c_{t}}^{2}g_{t}(w^{t})\right)dt+\partial_{c_{t}}g_{t}(w^{t})dw_{t}. \] }
Summarizing the results in this section: the drift of the dynamics of the present effects of the emissions is determined by the horizontal and the second vertical derivative and the diffusion coefficient is given by the vertical derivative of the present marginal effects of the policy functional. This result both parallels and differs from the case of no hysteresis where, similarly, the first and second order derivatives matter for the dynamics. However, the notion of the derivative is very different as the horizontal and the vertical derivatives measure how hysteresis functionals change with the whole past trajectory. Even if there are no direct contemporaneous effects of the policy, its dependence on the past may lead to significant effects on the dynamics of optimal policy. At the same time, using these different notions of the derivatives the functional Itô formula allows us to convey the similar intuition as in the case of no hysteresis. In particular, uncertainty represents itself in the second vertical derivative of the functional $g_{t}\left(w^{t}\right)$ by affecting the drift and in the first vertical derivative of the functional affecting the diffusion coefficient.
\subsection{\label{subsec:Expected-future-effects}Expected future effects of the emissions paths}
The most challenging part of the paper is to characterize the dynamics of the expected future marginal effects of current emission represented by the conditional expectation process $E\left[\int_{t}^{T}k_{t,s}(w^{s})ds\bigg|\mathcal{F}_{t}\right].$
Let $\xi_{t}=\int_{t}^{T}k_{t,s}(w^{s})ds$. Note that the conditional expectation process $\mathbb{E}\bigg[\xi_{t}|\mathcal{F}_{t}\bigg]$ is in general not a martingale. For example, if $k_{s,t}=1,$ then
$\xi_{t}=T-t$ and we have $\mathbb{E}\bigg[\xi_{t}|\mathcal{F}_{t}\bigg]=T-t$. Also note that $\xi_{t}$ is not adapted in general. The difficulty more broadly comes from the fact that in the conditional expectation both the process $\xi_{t}$ and the filtration $\mathcal{F}_{t}$ are changing with time.
\subsubsection{\label{subsec:Total-derivative-formula}Total derivative formula for conditional expectations}
In this section, we derive a new total derivative formula for conditional expectation processes -- this is the main new theoretical tool that we develop in this paper. A more general statement of the result and a complete proof is given in the Appendix. Here, we state a simpler version of the result and an outline of a heuristic proof. \begin{defn} The Malliavin derivative of the cylindrical functional $f(w)=g(w_{t_{1}},\ldots,w_{t_{k}}),$ where $t_{1}<\ldots<t_{k}$ and $g$ is a smooth function, is defined by $D_{t}f(w)=\sum_{i=1}^{k}\partial_{i}g(w_{t_{1}},\ldots,w_{t_{k}})1_{t\leq t_{i}}.$ This defines a closable linear operator $D:L^{2}(\Omega)\to L^{2}(\Omega\times[0,T]).$ Its closure is also denoted by $D$ and is called the Malliavin derivative operator. \end{defn} Intuitively, a Malliavin derivative is a change in a functional due to a perturbation of the whole path of the process (Figure \ref{fig:Malliavin derivative}).
\begin{figure}
\caption{Malliavin derivative: path perturbation}
\label{fig:Malliavin derivative}
\end{figure}
The next proposition states the total derivative formula, the main technical tool that we develop in this paper. \begin{prop} \label{prop: Total derivative formula} \textbf{(Total derivative formula) }Let an adapted square integrable process $(X_{t})_{0\leq t\leq T}$ be represented as \[
X_{t}=\mathbb{E}[\xi_{t}|\mathcal{F}_{t}]. \] Let $(\xi_{t})_{0\leq t\leq T}$ be a Malliavin differentiable square integrable absolutely continuous process, that is, a process of the form $\xi_{t}=\xi_{0}+\int_{0}^{t}\eta_{s}ds,$ where $\xi$ and $\eta$ may be anticipative. Then, \begin{equation}
dX_{t}=\mathbb{E}[\partial_{t}\xi_{t}|\mathcal{F}_{t}]dt+\mathbb{E}[D_{t}\xi_{t}|\mathcal{F}_{t}]dw_{t},\label{eq:total derivative formula} \end{equation} where $D$ is a Malliavin derivative. \end{prop} \begin{proof} The Clark-Ocone formula gives \[
\xi_{t}=\mathbb{E}\left[\xi_{t}\right]+\int_{0}^{T}\mathbb{E}[D_{r}\xi_{t}|\mathcal{F}_{r}]dw_{r}. \] Taking the conditional expectation leads to \[
X_{t}=\mathbb{E}\left[\xi_{t}\right]+\int_{0}^{t}\mathbb{E}[D_{r}\xi_{t}|\mathcal{F}_{r}]dw_{r}. \] The total derivative is then \begin{align*}
dX_{t} & =\mathbb{E}\left[\partial_{t}\xi_{t}\right]dt+\mathbb{E}[D_{t}\xi_{t}|\mathcal{F}_{t}]dw_{t}+\left(\int_{0}^{t}\mathbb{E}[D_{r}\partial_{t}\xi_{t}|\mathcal{F}_{r}]dw_{r}\right)dt=\\
& =\mathbb{E}[D_{t}\xi_{t}|\mathcal{F}_{t}]dw_{t}+\left(\mathbb{E}\left[\partial_{t}\xi_{t}\right]+\int_{0}^{t}\mathbb{E}[D_{r}\partial_{t}\xi_{t}|\mathcal{F}_{r}]dw_{r}\right)dt. \end{align*} Note now that by the Clark-Ocone formula applied to the derivative $\partial_{t}\xi_{t}$: \[
\partial_{t}\xi_{t}=\mathbb{E}\left[\partial_{t}\xi_{t}\right]+\int_{0}^{T}\mathbb{E}[D_{r}\partial_{t}\xi_{t}|\mathcal{F}_{r}]dw_{r}, \] and taking the conditional expectations: \[
\mathbb{E}[\partial_{t}\xi_{t}|\mathcal{F}_{t}]=\mathbb{E}\left[\partial_{t}\xi_{t}\right]+\int_{0}^{t}\mathbb{E}[D_{r}\partial_{t}\xi_{t}|\mathcal{F}_{r}]dw_{r}. \] The total derivative is then given by (\ref{eq:total derivative formula}). \end{proof}
The lemma provides a time-dependent extension of the Clark-Ocone formula for processes that can be represented as a conditional expectation of an absolutely continuous process.\footnote{See Section \ref{subsec:Relationship-to-the_Clarrk-Ocone} for the more detailed discussion of the relationship of our total derivative formula to the Clark-Ocone formula.} The first term, $\mathbb{E}[\partial_{t}\xi_{t}|\mathcal{F}_{t}]$, can be thought of as a time derivative of this variable and represents how the conditional expectation evolves with respect to time. The second term, $\mathbb{E}[D_{t}\xi_{t}|\mathcal{F}_{t}]$, can be thought of as a stochastic derivative with respect to the underlying process $w_{t}$ and represents how the conditional expectation changes with the changes in $w_{t}$.\footnote{The process $(\xi_{t})$ in Proposition \ref{prop: Total derivative formula}
need not be adapted, and we can understand the process $X_{t}=\mathbb{E}[\xi_{t}|\mathcal{F}_{t}]$ as the adapted projection of the process $\xi$. In Proposition \ref{prop: Total derivative formula} we thus prove that an adapted projection of an absolutely continuous process is necessarily an Itô process. In the Appendix \ref{subsec:Characterization-of-It=0000F4} we show the converse to this clam -- any Itô process can be represented as such projection.}
\subsubsection{Semimartingale decomposition of the expected future effects of policy paths}
We now make mild assumptions on the kernel $k_{t,s}$ -- absolute continuity of the kernel in variable $t$ and Malliavin differentiability as a functional from $w^{s}$ -- and apply the total derivative formula to the expected future marginal damages, $E\left[\int_{t}^{T}k_{t,s}(w^{s})ds\bigg|\mathcal{F}_{t}\right]$: \begin{align*}
dE\left[\int_{t}^{T}k_{t,s}(w^{s})ds\bigg|\mathcal{F}_{t}\right] & =\underset{\text{time derivative}}{\underbrace{\mathbb{E}\bigg[\partial_{t}\int_{t}^{T}k_{t,s}(w^{s})ds\bigg|\mathcal{F}_{t}\bigg]}}dt+\underset{\text{stochastic derivative}}{\underbrace{\mathbb{E}\bigg[D_{t}\int_{t}^{T}k_{t,s}(w^{s})ds\bigg|\mathcal{F}_{t}\bigg]}}dw_{t}=\\
& =\underset{\text{time derivative}}{\underbrace{\mathbb{E}\bigg[\int_{t}^{T}\partial_{t}k_{t,s}(w^{s})ds\bigg|\mathcal{F}_{t}\bigg]-k_{t,t}\left(w^{t}\right)}}dt+\underset{\text{stochastic derivative}}{\underbrace{\mathbb{E}\bigg[\int_{t}^{T}D_{t}k_{t,s}(w^{s})ds\bigg|\mathcal{F}_{t}\bigg]}}dw_{t}. \end{align*} The meaning of this equation is straightforward. The conditional expectation
$dE\left[\int_{t}^{T}k_{t,s}(w^{s})ds\bigg|\mathcal{F}_{t}\right]$ measures the effects of the change in the emissions in period $t$ on the expected future marginal externality $k_{t,s}(w^{s})$ in all periods $s$ ($s\in\left[t,T\right])$. The time evolution of the conditional expectation is given by the analogue of the time derivative. The stochastic evolution of the conditional expectation is given by a stochastic derivative of the cumulative change in all marginal effects of emission in time $t$ on all future periods $s$ with respect to variation in the underlying process $w$.
\subsection{\label{subsec:Dynamics Pigouvian Climate}Dynamics of the marginal externality damage}
We can now collect the results of the previous two sections. We obtain the dynamics of the marginal externality damage \begin{align*}
d\Lambda_{t} & =-\epsilon\left(\Delta_{t}g_{t}(w^{t})+\frac{1}{2}\partial_{c_{t}}^{2}g_{t}(w^{t})+\mathbb{E}\bigg[\int_{t}^{T}\partial_{t}k_{t,s}(w^{s})ds\bigg|\mathcal{F}_{t}\bigg]-k_{t,t}\left(w^{t}\right)\right)dt+\\
& +\left(\partial_{c_{t}}g_{t}(w^{t})+\mathbb{E}\bigg[\int_{t}^{T}D_{t}k_{t,s}(w^{s})ds\bigg|\mathcal{F}_{t}\bigg]\right)dw_{t}. \end{align*} This, we believe, is an important result as it solves in the closed form the dynamics of the evolution of the marginal externality damage or the optimal Pigouvian tax. This formula shows how the policymaker should change the tax with the passage of time and with the realizations of uncertainty -- that is, we determined the drift and the diffusion coefficients of the process of the optimal Pigouvian tax.
A related way to think about this result is that it determines what features of the model matter up to the first order. For the time updating, the functional Itô formula shows that the horizontal and the second order vertical derivative are important and the total derivative formula shows that the ``time derivative'' of the expected cumulative damages are important. For the updating with the movement of uncertainty, the functional Itô formula shows that the vertical derivative and the total derivative formula shows that the ``stochastic derivative'' of the expected cumulative marginal damages are important.
We also immediately obtain the dynamics of the optimal emission policies from equation (\ref{eq:FOC emissions}) in closed form: \begin{align*}
dc_{t} & =-\epsilon\left(\Delta_{t}g_{t}(w^{t})+\frac{1}{2}\partial_{c_{t}}^{2}g_{t}(w^{t})+\mathbb{E}\bigg[\int_{t}^{T}\partial_{t}k_{t,s}(w^{s})ds\bigg|\mathcal{F}_{t}\bigg]-k_{t,t}\left(w^{t}\right)\right)dt+\\
& +\left(1-\epsilon\partial_{c_{t}}g_{t}(w^{t})-\epsilon\mathbb{E}\bigg[\int_{t}^{T}D_{t}k_{t,s}(w^{s})ds\bigg|\mathcal{F}_{t}\bigg]\right)dw_{t}. \end{align*}
We thus extended the result of the macro-climate model of Golosov, et al. (2014) to the case with the path-dependent hysteresis given by (\ref{eq: Climate hysteresis functional}). Note that all of the results here are given in closed form. The next sections show that the main insights derived with this specifications apply to general hysteresis functionals.
\section{\label{sec:General-setting}General hysteresis functionals}
We now return to the analysis of the problem (\ref{eq:Perturbed problem}) for a general hysteresis functional.
\subsection{\label{subsec:Class A_t}The class $\mathcal{A}_{t}$ of path-dependent functionals}
In this section we describe a class of path-dependent functionals $\mathcal{A}_{t}$ which, in a sense, is parallel to continuously differentiable functions of a real-valued argument. It will turn out that this class, while very general, allows significant tractability in the analysis of path-dependent problems. The introduction of this class of functionals is one contribution of our paper. \begin{rem} \label{rem:Discontinuous}Adding a path-dependent functional $h_{t}$ can considerably change the structure of the optimal policy. For example, consider the functional $h_{t}(c)=c_{\frac{t}{2}}$ that describes the effects of the policies at the middle of the time period $\left[0,t\right]$. We show in the appendix that the optimal policy is no longer continuous and has a jump at $t=\frac{T}{2}$. \end{rem} Let us recall that a Fréchet derivative of a functional $h_{t}:C[0,t]\to\mathbb{R}$ at a point $c\in C[0,t]$ is a signed measure $\mu$ on $[0,t]$ such that \[ h_{t}\left(c+z\right)=h_{t}\left(c\right)+\int_{0}^{t}z_{s}d\mu\left(t,s\right)+o\left(\left\Vert z\right\Vert \right),\left\Vert z\right\Vert \rightarrow0. \] The primary difficulty with this formulation is that it features a very general dependence of the measure $\mu$ on time $t$. The next assumption structures this dependence. \begin{assumption} \label{assu:Class A}\textbf{(Class $\mathcal{A}_{t}$)} Suppose the (Fréchet) derivative of the functional $h_{t}$ has an absolutely continuous part and an atom at point $t$: \[ h_{t}(c+z)=h_{t}(c)+\partial_{c_{t}}h_{t}(c)z_{t}+\int_{0}^{t}\delta_{s}h_{t}(c)z_{s}ds+o\left(\left\Vert z\right\Vert \right),\left\Vert z\right\Vert \rightarrow0. \] The family of such functionals is denoted by $\mathcal{A}_{t}.$ \end{assumption} This assumption on the derivative of functionals means that there are two sources of variation for $h_{t}$: an instantaneous influence of a perturbation at the moment $t$ given by an atom and an integral influence of perturbations at previous moments $s\le t$ which is an absolutely continuous process. Assumption \ref{assu:Class A} is mild -- it only requires that the derivative of a functional (which is a measure) is absolutely continuous and has an atom at the present time.
The class $\mathcal{A}_{t}$ contains a variety of functionals: \[ h_{t}(c)=f(c_{t}), \]
which is state-dependent but not path-dependent; \[ h_{t}(c)=f_{t}(c_{t},\int_{0}^{t}g_{t}(c_{s})ds), \] which jointly depends on state $c_{t}$ and the integral influence of the path $\int_{0}^{t}g_{t}(c_{s})ds$ and, moreover, both the joint dependence $f_{t}$ and the effects of the past policies $g_{t}$ depend on time $t$; \[ h_{t}\left(c\right)=f_{t}(\int_{0}^{t}\int_{0}^{t}g_{t}(c_{s},c_{r})dsdr,...) \] where now there is a joint dependence on the past values $c_{s}$ and $c_{r}$ via a repeated integral.
The proposition that follows shows that this assumption covers a very general class of functionals. \begin{prop} \label{claim:Class A is dense}Every Fréchet differentiable functional $g:C[0,T]\to\mathbb{R}$ is a pointwise limit of functionals from the class $\mathcal{A}_{T}.$ \end{prop} \begin{proof} In the appendix. \end{proof} We next explore the nature of the present marginal influence of policy $\partial_{c_{t}}h_{t}$ which is itself a path-dependent functional. The lemma that follows (proven for a more general case in the appendix) connects Dupire's vertical derivative to the derivatives of the functionals in the class $\mathcal{A}_{t}$. \begin{lem} \label{lem: Vertical derivative}Let the functional $h_{t}$ be in the class $\mathcal{A}_{t}$ , then $\partial_{c_{t}}h_{t}(c)$ is the vertical derivative in the Dupire sense. \end{lem} \begin{rem} \label{rem:Integration by parts} We now discuss some additional motivation behind Assumption \ref{assu:Class A}. Consider, for example, hysteresis given by an Itô process: \begin{equation} h_{t}(c)=\int_{0}^{t}b_{s}dc_{s},\label{eq: h as Ito process} \end{equation} where $b$ is an absolutely continuous function. The functional $h_{t}(c)$ in equation (\ref{eq: h as Ito process}) is well-defined if and only if the process $(c_{s})_{0\leq s\leq t}$ is of bounded variation. This is not true for an arbitrary progressively measurable process $(c_{s})_{0\leq s\leq t}$, and we cannot expect that the optimal policy process will be of bounded variation. For example, in our baseline case of Section \ref{subsec:Baseline-setting} $c_{t}^{*}=w_{t}$ and then almost all realizations of $c^{*}$ are of unbounded variation. However, if the coefficient $b$ is absolutely continuous, then we can integrate (\ref{eq: h as Ito process}) by parts and rewrite $h_{t}(c)$ in the form: \[ h_{t}(c)=-\int_{0}^{t}b'_{s}c_{s}ds+b_{t}c_{t}. \] Further, $h_{t}$ is just a linear functional from $c:$ \[ h_{t}(c+z)-h_{t}(c)=b_{t}z_{t}-\int_{0}^{t}b'_{s}z_{s}ds. \] So, $h_{t}\in\mathcal{A}_{t}$ with $\partial_{c_{t}}h(c)=b_{t},$ $\delta_{s}h_{t}(c)=-b_{s}'.$ \end{rem}
\subsection{\label{sec:Optimal-policy}Optimal policy: the first order conditions}
In this section, we derive the first order conditions for the optimal process $c^{\epsilon}$. While in some circumstances one can write a recursive formulation even for a path-dependent problem and then find a Hamilton-Jacobi-Bellman equation, in general it is difficult or impossible to do it. Here, we instead find the first order conditions for optimal policy using a variational method. \begin{prop} \textbf{(The first order conditions for the optimum)} Let the optimal policy process $c^{\epsilon}$ solve (\ref{eq:Perturbed problem}), then \begin{equation}
c_{t}^{\epsilon}=w_{t}-\epsilon\left(\partial_{c_{t}}h_{t}(\left(c^{\epsilon}\right)^{t})+E\left[\int_{t}^{T}\delta_{t}h_{s}(\left(c^{\epsilon}\right)^{s})ds\bigg|\mathcal{F}_{t}\right]\right).\label{eq: FOC, perturbed} \end{equation} \end{prop} \begin{proof} Perturb the process $c$ by $\nu z,$ where $z$ is an adapted process, and compute the derivative in $\nu$ at $\nu=0:$ \begin{align*}
& \partial_{\nu}E\int_{0}^{T}\left(-\frac{1}{2}(c_{t}+\nu z_{t}-w_{t})^{2}-\epsilon h_{t}(c^{t}+\nu z^{t})\right)dt\bigg|_{\nu=0}=\\
& =E\int_{0}^{T}\left(z_{t}w_{t}-c_{t}z_{t}-\epsilon\left(\partial_{c_{t}}h_{t}(c^{t})z_{t}+\int_{0}^{t}\delta_{s}h_{t}(c^{t})z_{s}ds\right)\right)dt=\\
& =E\int_{0}^{T}z_{t}\left(w_{t}-c_{t}-\epsilon\left(\partial_{c_{t}}h_{t}(c^{t})+\int_{t}^{T}\delta_{t}h_{s}(c^{s})ds\right)\right)dt=0. \end{align*} Since $z$ is an arbitrary adapted process, we get the first order conditions (\ref{eq: FOC, perturbed}). The importance of Assumption \ref{assu:Class A} is evident in particular in the third line of the proof. If we did not impose this assumption, then instead of the integral $\int_{t}^{T}\delta_{t}h_{s}(c^{s})ds$ we would have a general measure $\mu_{t}(c^{s},[t,T])$ which can even be discontinuous in $t$. Assumption \ref{assu:Class A} imposes a smooth structure for this measure by requiring that the measure has an absolutely continuous part. \end{proof} This equation (\ref{eq: FOC, perturbed}) has a natural economic meaning. When hysteresis $\epsilon h_{t}$ is present, the solution to the optimal problem ($c_{t}^{*}=w_{t})$ is modified by the two terms. The first term, \[ I_{t}=\epsilon\partial_{c_{t}}h_{t}(c^{\epsilon}), \] is the instantaneous marginal effect of the change in the policy $c_{t}^{\epsilon}$ on functional $h_{t}$. The second term gives the future marginal effects of policy $c_{t}^{\epsilon}$ \[
F_{t}=\epsilon E\left[\int_{t}^{T}\delta_{t}h_{s}((c^{\epsilon})^{s})ds\bigg|\mathcal{F}_{t}\right]. \]
on the values of all future hysteresis functionals $h_{s}\left(\left(c^{\epsilon}\right)^{s}\right)$. For each time $s$ (where $t\le s\le T$), the marginal effect of changing policy in period $t$ is represented by the derivative $\delta_{t}h_{s}(\left(c^{\epsilon}\right)^{s})$ of the functional $h_{s}$ with respect to change in policy at time $t$. These marginal effects are evaluated as a conditional expectation at time $t$ and hence represent the expected future marginal effects. It is useful to think about the term $F_{t}$ as the cumulative impulse response of the change of the policy today on all future hysteresis functionals. This is similar to Alvarez and Lippi (2019), Alvarez, Le Bihan, and Lippi (2016), Alvarez, Lippi, and Oskolkov (2020) and Borovi\v{c}ka, Hansen, Scheinkman (2014), Borovi\v{c}ka, Hansen, Hendricks, and Scheinkman (2011), and Borovi\v{c}ka and Hansen (2016). Note that $F_{t}$ is a conditional expectation process that may non-trivially change with time as both the marginal effects $\delta_{t}h_{s}$ and the filtration $\mathcal{F}_{t}$ changes.
Both terms depend on the path of policy $c_{\left[0,s\right]}^{\epsilon}=(c^{\epsilon})^{s}$. As seen here, the assumption that the functional $h_{t}$ is in the class $\mathcal{A}_{t}$ allows us to conveniently separate the first order condition in two parts: the marginal effects of the policy on current period $I_{t}$ and the conditional expectation of the cumulative future marginal effects of policy paths in the future, $F_{t}$.
\section{Characterizing the general problem}
This section is divided into two main parts. Section \ref{sec:First-order-process} derives a closed-form characterization of the change in the optimal policy when hysteresis is small. Section \ref{subsec:Dynamics-of-optimal} provides a characterization of the optimal policy.\footnote{It may be useful for a reader to also consider an example in the appendix that provides a parallel characterization for the case with no path-dependency.}
\subsection{\label{sec:First-order-process}Stochastic hysteresis elasticity and its dynamics}
In this section we derive a closed-form characterization of the change in the optimal policy when hysteresis is small. We call this first order process a stochastic hysteresis elasticity. We characterize the dynamics of the stochastic elasticity in closed form by providing its semimartingale decomposition which is the main contribution of this section.
\subsubsection{\label{subsec:FOP}Stochastic hysteresis elasticity}
We first formally define the stochastic hysteresis elasticity. This is the first order process that represents the change in optimal policy process $c^{*}$ in response to introduction of an infinitesimal path-dependent hysteresis functional $h_{t}$ . \begin{defn} \textbf{(Stochastic hysteresis elasticity)} Let $c_{t}^{*}$ be a solution to the baseline problem (\ref{eq:Unperturbed problem}) and $c_{t}^{\epsilon}$ be a solution to the problem (\ref{eq:Perturbed problem}), where hysteresis is given by the functional $\epsilon h_{t}$, $\epsilon\rightarrow0$. The first-order process or the stochastic hysteresis elasticity $C_{t}^{h}$ is such that \[ c_{t}^{\epsilon}=c_{t}^{*}+\epsilon C_{t}^{h}+o(\epsilon). \] \end{defn} The stochastic elasticity $C_{t}^{h}$ is the change to the first order in the optimal policy process in response to a small change in the hysteresis $h$. In this sense, it is similar to the usual concept of elasticity but now determines how the whole process $c_{t}$ changes. \begin{comment} In a multi-period and especially in a stochastic model, the concept of elasticity depending on past history is more complicated. Consider a multi-period model first. A change in a tax in period $t$ may potentially affect consumption in all of the future periods. In a discrete $N$-period model, there are therefore $t$ elasticities (how a tax in period $t$ affects consumption at time $t$, at time $t-1,$ and so on). In a stochastic model, there are even more elasticities. A tax in a given history affects consumption in all of the states that precede that history.\footnote{If we had savings in the model, the tax also may affect all of the future levels of consumption.} Rather than focusing on a number of these elasticities, the concept of stochastic elasticity that we propose aims to capture the change in the whole process. \end{comment} {} From now on, to ease notation we drop the dependence of $C_{t}^{h}$ on $h$ and denote it simply by $C_{t}$.
Differentiating (\ref{eq: FOC, perturbed}) with respect to $\epsilon$ at $\epsilon=0$, and recalling that $c_{t}^{0}=c_{t}^{*}=w_{t}$ we find the process $C_{t}$ explicitly \begin{equation}
C_{t}=-\partial_{c_{t}}h_{t}(w^{t})-E\left[\int_{t}^{T}\delta_{t}h_{s}(w^{s})ds\bigg|\mathcal{F}_{t}\right].\label{eq: First order process} \end{equation} This equation is already interesting by itself as it presents the change in the optimal policy plan due to the introduction of the path-dependent effect. Moreover, it is given in closed form.
We now turn to presenting the main result of this section -- showing that $C_{t}$ is an Itô process and, most importantly, writing its explicit semimartingale decomposition. We are seeking a representation \[ dC_{t}=\beta\left(w^{t}\right)dt+\gamma\left(w^{t}\right)dw_{t}, \] where $\beta$ and $\gamma$ are potentially path-dependent. In other words, we want to determine the dynamics of the stochastic elasticity $C_{t}$.
\subsubsection{\label{subsec:Present-effect-of}Present effect of the policy path }
This section provides a semimartingale decomposition of the term $\partial_{c_{t}}h_{t}(c^{*})$ -- the present effect of the path of policies $(c^{*})^{t}=w^{t}$. The principal tool that we use is Dupire's functional Itô formula in Lemma \ref{lem:Functional-It=0000F4's-Formula:} that applies to the non-anticipative functionals of the path. We now make an assumption that allows the use of this formula. \begin{assumption} There exists a non-anticipative functional $q:[0,T]\times D[0,T]\to\mathbb{R}$ which is horizontally differentiable, twice vertically differentiable and is such that $q_{t}(c)=\partial_{c_{t}}h_{t}(c).$ In other words, the derivative \textup{$\partial_{c_{t}}h_{t}(c)$} can be extended to a $C_{b}^{1,2}$-functional on the space of cádlág paths. We denote this extension also by $\partial_{c_{t}}h_{t}(c)$. \end{assumption} Applying the functional Itô formula, and noting that $c_{t}^{*}=w_{t}$ we obtain a semimartingale decomposition of the functional $\partial_{c_{t}}h_{t}(c^{*})$ that depends on the whole history of policy $(c^{*})^{t}=w^{t}$: \[ d(\partial_{c_{t}}h_{t}(c^{*}))=\Delta_{t}\left(\partial_{c_{t}}h_{t}(w^{t})\right)dt+\partial_{c_{t}}\left(\partial_{c_{t}}h_{t}(w^{t})\right)dw_{t}+\frac{1}{2}\partial_{c_{t}}\left(\partial_{c_{t}}^{2}h_{t}(w^{t})\right)dt, \] and gathering the terms we obtain the following lemma. \begin{lem} \label{claim:Dynamics Present}\textbf{(Dynamics of the present effect, $I_{t}$)} The semimartingale decomposition of the present marginal effects, $I_{t}$, of the policy path is given by \begin{align} d(\partial_{c_{t}}h_{t}(c^{*})) & =\left(\Delta_{t}(\partial_{c_{t}}h_{t}(w^{t}))+\frac{1}{2}\partial_{c_{t}}^{3}h_{t}(w^{t})\right)dt+\partial_{c_{t}}^{2}h_{t}(w^{t})dw_{t}.\label{eq: FOP, Dupire part} \end{align} \end{lem} This part of the derivation is already interesting as a stand-alone result. The reason why one can apply the Dupire and Cont-Fournie analysis is that $\partial_{c_{t}}h_{t}(c^{*})$ is already represented as a functional of the path. The decomposition (\ref{eq: FOP, Dupire part}) then has the same intuitive meaning as the standard Itô's formula but now applies to the functional of the past, not the function of the present realization. Note that we are already applying the functional Itô's formula to the marginal effects, that is, to the vertical derivative of the functional $\partial_{c_{t}}h_{t}(w^{t})$. Hence, there are the second, $\partial_{c_{t}}^{2}$, and the third, $\partial_{c_{t}}^{3}$, vertical derivatives as well as the mixed derivative $\Delta_{t}\partial_{c_{t}}$.
\subsubsection{\label{subsec:Semimartingale-Future}Expected future effects of policy paths}
We now make an assumption on the derivative of the functional, $\delta_{t}h_{s}(c^{*})$, that is needed to apply the total derivative formula of Proposition \ref{prop: Total derivative formula}.\footnote{In Section \ref{subsec:Assumptions-on-smoothness}, we further discuss the smoothness assumptions for the functionals $h_{t}$.} \begin{assumption} The derivative $\delta_{t}h_{s}(c^{*})$ is an absolutely continuous in $t$ and is a Malliavin differentiable functional of the path $w^{s}.$ \end{assumption} It follows that the future marginal effect of a given policy path is given by \[ \frac{d}{dt}\int_{t}^{T}\delta_{t}h_{s}(c^{*})ds=-\delta_{t}h_{t}(c^{*})+\int_{t}^{T}\frac{\partial}{\partial t}(\delta_{t}h_{s})(c^{*})ds. \] and its conditional expectation is given by the total derivative formula (\ref{eq:total derivative formula}): \begin{align*}
d\mathbb{E}\bigg[\int_{t}^{T}\delta_{t}h_{s}(c^{*})ds\bigg|\mathcal{F}_{t}\bigg] & =\underset{\text{time derivative}}{\underbrace{\mathbb{E}\bigg[\partial_{t}\int_{t}^{T}\delta_{t}h_{s}(c^{*})ds\bigg|\mathcal{F}_{t}\bigg]}}dt+\underset{\text{stochastic derivative}}{\underbrace{\mathbb{E}\bigg[D_{t}\int_{t}^{T}\delta_{t}h_{s}(c^{*})ds\bigg|\mathcal{F}_{t}\bigg]}}dw_{t}. \end{align*} The meaning of this equation is straightforward. The conditional expectation \[
F_{t}=\mathbb{E}\bigg[\int_{t}^{T}\delta_{t}h_{s}(c^{*})ds\bigg|\mathcal{F}_{t}\bigg] \] describes how a change in policy in period $t$ determines the expected future marginal effects of that policy $\delta_{t}h_{s}(c^{*})$ in all periods $s$ ($s\in\left[t,T\right])$. The time evolution of the conditional expectation is given by the analogue of the time derivative. The stochastic evolution of the conditional expectation is given by a stochastic derivative of the cumulative change in all marginal effects of policy $t$ on all future periods $s$ with respect to variation in the underlying process $w$. Collecting the terms gives us the dynamics of the future effects of policy and shows that it is an Itô process. \begin{lem} \label{lem:Dynamics Future}\textbf{(Dynamics of the future effects, $F_{t}$)} The semimartingale representation of the future marginal effects, $F_{t}$, of policy paths is given by: \begin{align}
& d\mathbb{E}\bigg[\int_{t}^{T}\delta_{t}h_{s}(c^{*})ds\bigg|\mathcal{F}_{t}\bigg]=\nonumber \\
& =\Bigl(-\delta_{t}h_{t}(w^{t})+\int_{t}^{T}\mathbb{E}\left[\frac{\partial}{\partial t}(\delta_{t}h_{s})(w^{s})\bigg|\mathcal{F}_{t}\right]ds\Bigr)dt+\mathbb{E}\bigg[D_{t}\int_{t}^{T}\delta_{t}h_{s}(w^{s})ds\bigg|\mathcal{F}_{t}\bigg]dw_{t}.\label{eq: FOP, total derivative part} \end{align} \end{lem}
\subsubsection{Dynamics of the stochastic elasticity}
We now combine the results of the decompositions of the present effects (\ref{eq: FOP, Dupire part}) and the future effects (\ref{eq: FOP, total derivative part}) of policy and recalling that $c_{t}^{*}=w_{t}$: \begin{align*} dC_{t}= & -\Bigl(\underset{\text{Present effects: functional Itô}}{\underbrace{\Delta_{t}\partial_{c_{t}}h_{t}(w^{t})dt+\frac{1}{2}\partial_{c_{t}}^{3}h_{t}(w^{t})dt+\partial_{c_{t}}^{2}h_{t}(w^{t})dw_{t}}}\Bigr)-\\
& -\Bigl(\underset{\text{Future effects: total derivative}}{\underbrace{-\delta_{t}h_{t}(w^{t})dt+E\left[\int_{t}^{T}\frac{\partial}{\partial t}(\delta_{t}h_{s})(w^{s})ds\bigg|\mathcal{F}_{t}\right]dt+E\left[\int_{t}^{T}D_{t}\left(\delta_{t}h_{s}(w^{s})\right)ds\bigg|\mathcal{F}_{t}\right]dw_{t}}}\Bigr), \end{align*} where in the last term we interchanged the Malliavin derivative and the integral.
Collecting the terms gives the semimartingale decomposition of the first order process $C_{t}$ in closed form. \begin{thm} \label{thm:First order process, main theorem}\textbf{(Dynamics of stochastic elasticity)} The semimartingale decomposition of the stochastic elasticity $C_{t}$ is given by \begin{align}
dC_{t}= & -\Bigl(\Delta_{t}\partial_{c_{t}}h_{t}(w^{t})+\frac{1}{2}\partial_{c_{t}}^{3}h_{t}(w^{t})-\delta_{t}h_{t}(w^{t})+E\left[\int_{t}^{T}\frac{\partial}{\partial t}(\delta_{t}h_{s})(w^{s})ds\bigg|\mathcal{F}_{t}\right]\Bigr)dt-\nonumber \\
& -\Bigl(\partial_{c_{t}}^{2}h_{t}(w^{t})+E\left[\int_{t}^{T}D_{t}\left(\delta_{t}h_{s}(w^{s})\right)ds\bigg|\mathcal{F}_{t}\right]\Bigr)dw_{t}.\label{eq: FOP, main formula} \end{align} \end{thm} This theorem shows that the optimal process $c_{t}^{*}$ changes with the introduction of hysteresis, that is, the additional infinitesimal path-dependent effect of policies. The equation (\ref{eq: FOP, main formula}) gives the first order process in closed form as \[ dC_{t}^{h}=\beta_{t}\left(w^{t}\right)dt+\gamma_{t}\left(w^{t}\right)dw_{t}, \] where $\beta$ and $\gamma$ are path-dependent coefficients.
\subsection{\label{subsec:Dynamics-of-optimal}Dynamics of optimal policy}
We now characterize the dynamics of the optimal policy.
In this section, we assume that $\epsilon=1$, without loss of generality. The first order conditions (\ref{eq: FOC, perturbed}) become \[
w_{t}=c_{t}+\partial_{c_{t}}h_{t}(c^{t})+E\left[\int_{t}^{T}\delta_{t}h_{s}(c^{s})ds\bigg|\mathcal{F}_{t}\right]. \] Let us take the differential of this equation assuming that $c$ is an Itô process, i.e. \[ dc_{t}=\alpha_{t}dt+\beta_{t}dw_{t}. \] The present marginal effects of the policy are differentiated using the functional Itô formula (\ref{eq:Functional Ito}): \[ d\left(\partial_{c_{t}}h_{t}(c^{t})\right)=\Delta_{t}\partial_{c_{t}}h_{t}(c^{t})dt+\partial_{c_{t}}^{2}h_{t}(c^{t})dc_{t}+\frac{1}{2}\partial_{c_{t}}^{3}h_{t}(c^{t})\beta_{t}^{2}dt. \] The future expected marginal effects of the policy are differentiated using the total derivative formula (\ref{eq:total derivative formula}): \begin{align*}
& d\left(E\left[\int_{t}^{T}\delta_{t}h_{s}(c^{s})ds\bigg|\mathcal{F}_{t}\right]\right)=E\left[\partial_{t}\int_{t}^{T}\delta_{t}h_{s}(c^{s})ds\bigg|\mathcal{F}_{t}\right]dt+E\left[\int_{t}^{T}D_{t}\left(\delta_{t}h_{s}(c^{s})\right)ds\bigg|\mathcal{F}_{t}\right]dw_{t}=\\
& =E\left[\int_{t}^{T}\partial_{t}\left(\delta_{t}h_{s}(c^{s})\right)ds\bigg|\mathcal{F}_{t}\right]dt-\delta_{t}h_{t}(c^{t})dt+E\left[\int_{t}^{T}D_{t}\left(\delta_{t}h_{s}(c^{s})\right)ds\bigg|\mathcal{F}_{t}\right]dw_{t}. \end{align*} The differential of the first order conditions is then given by \begin{align*}
& dw_{t}=dc_{t}+\Delta_{t}\partial_{c_{t}}h_{t}(c^{t})dt+\partial_{c_{t}}^{2}h_{t}(c^{t})dc_{t}+\frac{1}{2}\partial_{c_{t}}^{3}h_{t}(c^{t})\beta_{t}^{2}dt+\\
& +E\left[\int_{t}^{T}\partial_{t}\delta_{t}h_{s}(c^{s})ds\bigg|\mathcal{F}_{t}\right]dt-\delta_{t}h_{t}(c^{t})dt+E\left[\int_{t}^{T}D_{t}\left(\delta_{t}h_{s}(c^{s})\right)ds\bigg|\mathcal{F}_{t}\right]dw_{t}. \end{align*} Collecting the terms near $dt$ and $dw_{t}$ we derive dynamics of the optimal policy: \[ \begin{cases}
0=(1+\partial_{c_{t}}^{2}h_{t}(c^{t}))\alpha_{t}+\frac{1}{2}\partial_{c_{t}}^{3}h_{t}(c^{t})\beta_{t}^{2}+\Delta_{t}\partial_{c_{t}}h_{t}(c^{t})+E\left[\int_{t}^{T}\partial_{t}\delta_{t}h_{s}(c^{s})ds\bigg|\mathcal{F}_{t}\right]-\delta_{t}h_{t}(c^{t}),\\
1=(1+\partial_{c_{t}}^{2}h_{t}(c^{t}))\beta_{t}+E\left[\int_{t}^{T}D_{t}\left(\delta_{t}h_{s}(c^{s})\right)ds\bigg|\mathcal{F}_{t}\right], \end{cases} \] and the next theorem characterizes the drift and diffusion coefficients of the optimal policy. \begin{thm} \label{prop:Optimal-policy}(\textbf{Optimal policy}) The dynamics of optimal policy is given by: \begin{align*}
\alpha_{t} & =-\frac{1}{1+\partial_{c_{t}}^{2}h_{t}(c^{t})}\left(\frac{1}{2}\partial_{c_{t}}^{3}h_{t}(c^{t})\beta_{t}^{2}+\Delta_{t}\partial_{c_{t}}h_{t}(c^{t})+E\left[\int_{t}^{T}\partial_{t}\delta_{t}h_{s}(c^{s})ds\bigg|\mathcal{F}_{t}\right]-\delta_{t}h_{t}(c^{t})\right),\\
\beta_{t} & =\frac{1-E\left[\int_{t}^{T}D_{t}\delta_{t}h_{s}(c^{s})ds\bigg|\mathcal{F}_{t}\right]}{(1+\partial_{c_{t}}^{2}h_{t}(c^{t}))}. \end{align*} \end{thm} We can compare this theorem to the results without hysteresis, where there are only contemporaneous effects of the policy $f\left(c_{t}\right)$, the details of which are in the appendix. The diffusion coefficient $\beta_{t}$ is determined by two terms. The first term, in the denominator, contains the vertical derivative of the present marginal effects of the policy $\partial_{c_{t}}\left(\partial_{c_{t}}h_{t}(c^{t})\right)$ and measures the change in the present marginal costs due to a ``bump'' in the trajectory of the costs. This is an analogue of the term $f''\left(c_{t}\right)$
for the case without hysteresis. The second term, in the numerator, is the stochastic derivative $E\left[\int_{t}^{T}D_{t}\delta_{t}h_{s}(c^{s})ds\bigg|\mathcal{F}_{t}\right]$ which measures impact of the change in the trajectory of the stochastic process on the expected future marginal effects in all periods. This term is not present in the case without hysteresis.
The drift coefficient $\alpha_{t}$ is determined by three terms. The first term, in the denominator, is the same relative scaling as in the case of $\beta_{t}$. The second set of terms is the time evolution of the present, $\Delta_{t}\partial_{c_{t}}h_{t}(c^{t})$, and future,
$E\left[\partial_{t}\int_{t}^{T}\delta_{t}h_{s}(c^{s})ds\bigg|\mathcal{F}_{t}\right]$, marginal costs. The analogue of the horizontal derivative $\Delta_{t}\partial_{c_{t}}h_{t}(c^{t})$ for the case without hysteresis would be the time derivative $\partial_{t}f\left(c_{t},t\right)$. The time evolution of the future marginal costs are is in the case without hysteresis. The third term is the quadratic variance of the present marginal costs, $\frac{1}{2}\partial_{c_{t}}^{3}h_{t}(c^{t})\beta_{t}^{2}$. It is a path-dependent analogue of $\frac{1}{2}f'''\left(c_{t}\right)\left(\beta_{t}\right)^{2}$ in the case without hysteresis.
The equations Theorem \ref{prop:Optimal-policy} characterize the drift and diffusion coefficients implicitly in contrast to the explicit form we obtained in Theorem \ref{thm:First order process, main theorem}. The explicit form we obtained for the stochastic elasticity thus can be thought of as a characterization of the optimal process for small hysteresis, or small hysteresis asymptotics. \begin{rem*} We now show that assuming that the marginal effect of policy $\delta_{t}h_{s}$ are in the class $\mathcal{A}_{s}$, we can compute Malliavin derivatives $D_{t}\delta_{t}h_{s}(c^{s})ds$ in Theorem \ref{prop:Optimal-policy} in a more detailed way. \end{rem*} \begin{cor*} Let $\delta_{t}h_{s}$ be in the class $\mathcal{A}_{s}$, then \[ D_{t}\delta_{t}g_{s}(c^{s})=\partial_{c_{t}}\delta_{t}h_{s}(c^{s})D_{t}c_{s}+\int_{t}^{s}\delta_{r}\delta_{t}h_{s}(c^{s})D_{t}c_{r}dr, \] where the tangent process $D_{t}c_{r}$ is given by \begin{equation} \begin{cases} d(D_{t}c_{s})=(D_{t}\alpha_{s})ds+(D_{t}\beta_{s})dw_{s},\ s>t\\ D_{t}c_{t}=\beta_{t}. \end{cases}\label{eq: Tangent process} \end{equation} \end{cor*} \begin{proof} Let us perturb the underlying Brownian motion: $S(w,\epsilon z)_{t}=w_{t}+\epsilon\int_{0}^{t}z_{r}dr,$ $0\leq t\leq T.$ Then by the definition of the Malliavin derivative \[ c_{t}(S(w,\epsilon z))=c_{t}(w)+\epsilon\int_{0}^{t}D_{r}c_{t}(w)z_{r}dr+o(\epsilon), \] where $D_{r}c_{t}$ is a tangent process. Hence, \begin{align*}
& \delta_{t}h_{s}(c^{s}(S(w,\epsilon z)))=\delta_{t}h_{s}(c^{s}(w))+\epsilon\partial_{c_{s}}\delta_{t}h_{s}(c^{s})\int_{0}^{s}D_{r}c_{s}(w)z_{r}dr+\\
& +\epsilon\int_{0}^{s}\delta_{r}\delta_{t}h_{s}(c^{s})\int_{0}^{r}D_{u}c_{r}(w)z_{u}dudr+o(\epsilon)=\\
& =\delta_{t}h_{s}(c^{s}(w))+\epsilon\int_{0}^{s}z_{r}\left(\partial_{c_{s}}\delta_{t}h_{s}(c^{s})D_{r}c_{s}(w)+\int_{r}^{s}\delta_{u}\delta_{t}h_{s}(c^{s})D_{r}c_{u}(w)du\right)dr+o(\epsilon). \end{align*}
The result follows. \end{proof} This characterization is useful as it uses the tangent process which has a particularly simple form. When $c$ is an Itô process, its tangent process is also an Itô process with the drift and diffusion coefficients being the Malliavin derivatives of the drift and diffusion coefficients of the process $c$.
\section{\label{sec:Examples-of-perturbation}Examples}
In this section, we present a number of examples of the hysteresis functionals $h_{t}$ to illustrate how to use our theoretical results. We also provide, whenever possible, an alternative derivation using other tools. Section \ref{subsec:The-user's-guide} is the user's guide that describes the steps needed to apply the theory we developed. The first example, Section \ref{subsec:Example No-hysteresis:-state-dependent}, revisits the case of no hysteresis. In this case, only the present effects of the policy are present and the functional Itô formula reduces to the usual Itô formula. There is no need to use the total derivative formula. The second example, Section \ref{subsec:Example Cumulative-hysteresis}, considers cumulative hysteresis. The effects of the polices are a function of the current policy and the integral of the past policies. For the case of the multiplicative dependence of the effects of the current policies and the cumulative hysteresis, the stochastic elasticity takes a very simple form. The present marginal effects of the policy path only have the horizontal derivative which measures how the cumulant of the past policies changes with time and all of the vertical derivatives are equal to zero. The total derivative formula for the conditional expectation of the future marginal effects only has the stochastic derivative component which itself has a very simple form that measures how lengthy the effects of the stochastic shock are. We expand on this class of examples in Boulatov, Riabov, and Tsyvinski (2020) where we analyze such cases extracting a martingale and not using the main tools of this paper -- the functional Itô formula and the total derivative formula. Our third example, Section \ref{subsec:Example Martingale}, studies hysteresis that depends both on time and the past policies. This example captures three important parts of path-dependent policies. First, there is an effect of the present policy. Second, there is an effect of the hysteresis. The hysteresis is represented by an integral with a kernel that has joint dependence on current time and past policies. That is, path dependency sophisticatedly changes with both the time and evolution of the past polices. Third, there is a joint dependence between the present and the past. For this example, we need to utilize all of the tools developed in the paper. The functional Itô formula gives the evolution of the present marginal effects of the policies in terms of horizontal and vertical derivatives. The total derivative formula straightforwardly gives the evolution of the conditional expectation process in terms of the time and stochastic derivative. We also consider a deterministic setting with hysteresis in Section \ref{subsec: Example Deterministic-hysteresis}. As there is no stochasticity, there is only evolution with respect to time or, rather, the history of policies. We decompose its effects into the effects of the present and the past history. The present marginal effect of policy then evolves as the first-order horizontal and the vertical derivatives. The future marginal effects of policies evolve as their time derivative. In Section \ref{subsec:A-tipping-point} we consider an example with a tipping point in which the time when the stochastic process reaches the maximum becomes a reference point.
\subsection{\label{subsec:The-user's-guide}The user's guide}
In order to apply results from the previous section one needs to calculate a number of derivatives of the functional $(h_{t})_{0\leq t\leq T}.$ In calculation of the first order conditions (\ref{eq: FOC, perturbed}), we need to find a Fréchet derivative of $h_{t}$: \[
h_{t}(c+z)-h_{t}(c)=\partial_{c_{t}}h_{t}(c)z_{t}+\int_{0}^{t}\delta_{s}h_{t}(c)z_{s}ds+o(||z||). \] For the application of Dupire's functional Itô's formula and finding the SDE for the present effects of the path in Section \ref{subsec:Present-effect-of} and equation (\ref{eq: FOP, Dupire part}) we need to calculate the horizontal derivative $\Delta_{t}\partial_{c_{t}}h_{t}(c)$ and two vertical derivatives $\partial_{c_{t}}^{2}h_{t}(c),$ $\partial_{c_{t}}^{3}h_{t}(c).$ For the application of the total derivative formula (\ref{eq: FOP, total derivative part}) and finding the evolution equation for the future effects of the path we need to calculate two types of derivatives of $\delta_{t}h_{s}(c)$: (a) time derivative $\frac{\partial}{\partial t}\delta_{t}h_{s}(c)$, and (b) the Malliavin derivatives $D_{t}\delta_{t}h_{s}(c),$
In other words, we first break the first order condition for the effects of policies of the present and the effects of the past. The dependence of variables in the past implies that any change in a variable affects all the future states. We then use the functional Itô formula to derive the evolution equation for the effect on the present and the total derivative formula for the effects on the future. The total derivative formula also requires calculations of the Malliavin derivatives.
\subsection{\label{subsec:Example No-hysteresis:-state-dependent}No hysteresis: state-dependent perturbations}
We start with the simplest case -- the state-dependent perturbation where $h$ is a function of the current state rather than a functional. Consider the perturbation $h_{t}(c)=f(c_{t}),$ where $f:\mathbb{R}\to\mathbb{R}$ is a smooth function. The list of derivatives becomes the following.
The Fréchet derivative of the functional is just the derivative of $f:$ \[
h_{t}(c+z)-h_{t}(c)=f(c_{t}+z_{t})-f(c_{t})=f'(c_{t})z_{t}+o(||z||). \] Correspondingly, the present marginal effects of policy are given by the regular derivative \[ \partial_{c_{t}}h_{t}(c)=f'(c_{t}), \] and the effects of the past are \[ \delta_{s}h_{t}=0\mbox{ for }s<t. \] In particular, we see that $h_{t}\in\mathcal{A}_{t}$ and $\delta_{s}h_{t}\in\mathcal{A}_{t}.$
We now turn to the analysis of the present effect of the policy path. The horizontal derivative $\Delta_{t}\left(\partial_{c_{t}}h_{t}(c)\right)$ is equal to zero. Indeed, it is defined as a limit \[ \Delta_{t}\partial_{c_{t}}h_{t}(c)=\lim_{\epsilon\to0}\frac{\partial_{c_{t+\epsilon}}h_{t+\epsilon}(c_{\cdot,\epsilon})-\partial_{c_{t}}h_{t}(c)}{\epsilon}, \] where $c_{\cdot,\epsilon}$ is an extension of the path $c$ from $[0,t]$ to $[0,t+\epsilon]$ by $c_{s,\epsilon}=c_{t}$ for $t\leq s\leq t+\epsilon.$ It follows that\footnote{If we had $h_{t}\left(c\right)=f\left(c_{t},t\right)$ the only difference would be that $\Delta_{t}\partial_{c_{t}}h_{t}(c)=\partial_{c_{t},t}^{2}f\left(c_{t},t\right)$ } \[ \Delta_{t}\partial_{c_{t}}h_{t}(c)=\lim_{\epsilon\to0}\frac{f'(c_{t+\epsilon,\epsilon})-f'(c_{t})}{\epsilon}=\lim_{\epsilon\to0}\frac{f'(c_{t})-f'(c_{t})}{\epsilon}=0. \] Vertical differentiation of the present marginal effect of policy is the regular differentiation of $f:$ \[ \partial_{c_{t}}^{2}h_{t}(c)=f''(c_{t}),\ \partial_{c_{t}}^{3}h_{t}(c)=f'''(c_{t}). \]
We now turn to the analysis of the expected future marginal effects. The derivative in $t$ of the future marginal effects in period $s:$ $\delta_{t}h_{s}(c),$ i.e. $\frac{\partial}{\partial t}\delta_{t}h_{s}(c)$ is equal to zero, since $\delta_{t}h_{s}(c)=0:$ \[ \frac{\partial}{\partial t}\delta_{t}h_{s}(c)=0. \] The Malliavin derivative of $\delta_{t}h_{s}(c)$ is also zero, since $\delta_{t}h_{s}(c)=0:$ \[ D_{t}\delta_{t}h_{s}(c)=0. \]
The formula for the dynamics of the stochastic elasticity (\ref{eq: FOP, main formula}) becomes: \begin{align*} dC_{t}= & -\Bigl(\underset{=0}{\underbrace{\Delta_{t}\partial_{c_{t}}h_{t}(w^{t})}dt}+\frac{1}{2}\underset{=f'''\left(w_{t}\right)}{\underbrace{\partial_{c_{t}}^{3}h_{t}(w^{t})}}dt+\underset{=f''\left(w_{t}\right)}{\underbrace{\partial_{c_{t}}^{2}h_{t}(w^{t})}}dw_{t}\Bigr)-\\
& -\Bigl(\underset{=0}{\underbrace{-\delta_{t}h_{t}(w^{t})dt+E\left[\int_{t}^{T}\partial_{t}\delta_{t}h_{s}(w^{s})ds\bigg|\mathcal{F}_{t}\right]dt+E\left[\int_{t}^{T}D_{t}\delta_{t}h_{s}(w^{s})ds\bigg|\mathcal{F}_{t}\right]dw_{t}}}\Bigr), \end{align*} and \[ dC_{t}=-\frac{1}{2}f'''(w_{t})dt-f''(w_{t})dw_{t}. \]
We summarize the result. First, the present effect of the path from the Section \ref{subsec:Present-effect-of} are given by the standard space derivatives. The functional Itô formula reduces to the usual Itô's formula. Second, all of the future effects of the path from are zero, hence, the conditional expectation is zero.
One can, of course, immediately get the result of this section by applying the Itô formula to $f'(c_{t}^{*}),$ see the Appendix.
\subsection{\label{subsec:Example Cumulative-hysteresis}Cumulative hysteresis}
Consider the hysteresis functional $h_{t}(c)=c_{t}\int_{0}^{t}c_{s}ds$. Here, the path-dependence enters cumulatively as the integral of the past realization. The cumulative hysteresis is then multiplicative with the current policy $c_{t}.$
We start with the Fréchet derivative of $h_{t}$ that appears in the first order conditions of the problem (\ref{eq: FOC, perturbed}). It is given by varying the whole path $c_{[0,t]}$ by a variation $z_{[0,t]}$: \[
h_{t}(c+z)-h_{t}(c)=\left(c_{t}+z_{t}\right)\int_{0}^{t}\left(c_{s}+z_{s}\right)ds-c_{t}\int_{0}^{t}c_{s}ds=\int_{0}^{t}c_{t}z_{s}ds+z_{t}\int_{0}^{t}c_{s}ds+o(||z||). \] Correspondingly, the present marginal effects of policy are given by the integral of the path of previous policies \[ \partial_{c_{t}}h_{t}(c)=\int_{0}^{t}c_{s}ds. \] As all previous $c_{s}$ enter identically in the integral, the effects of the past marginally contributes $c_{t}$: \[ \delta_{s}h_{t}=c_{t}\mbox{ for }s<t. \]
In particular, we see that $h_{t}\in\mathcal{A}_{t}$ and $\delta_{s}h_{t}\in\mathcal{A}_{t}.$
The first order process is given by (\ref{eq: First order process}), recalling that $c_{t}^{*}=w_{t}$: \[
C_{t}=-\partial_{c_{t}}h_{t}(c_{[0,t]}^{*})-E\left[\int_{t}^{T}\delta_{t}h_{s}(c_{[0,s]}^{*})ds\bigg|\mathcal{F}_{t}\right]=-\underset{\text{Present effect}}{\underbrace{\int_{0}^{t}w_{s}ds}}-\underset{\text{Future effect}}{\underbrace{E\left[\int_{t}^{T}w_{s}ds\bigg|\mathcal{F}_{t}\right]}}. \] Note that in the future effects it is $\delta_{t}h_{s}(c^{t})=c_{s}^{*}=w_{s}$ that enters as it shows how policy in period $t$ ($t\le s\le T)$ affects future period $s$.
We now turn to calculation of the derivatives needed for the application of the Dupire's functional Itô formula (\ref{eq: FOP, Dupire part}). Recall that the horizontal and two vertical derivatives are needed. The horizontal derivative of the present effects of the path $\partial_{c_{t}}h_{t}(c)$ is simply the time derivative of the integral: \[ \Delta_{t}\left(\partial_{c_{t}}h_{t}(c)\right)=\Delta_{t}\left(\int_{0}^{t}c_{s}^{*}ds\right)=c_{t}^{*}=w_{t}. \] The vertical differentiation of the present effects of the path $\partial_{c_{t}}h_{t}(c)$ is given by bumping the path and is equal to zero: \begin{align*}
& \partial_{c_{t}}\left(\partial_{c_{t}}h_{t}(c)\right)=\partial_{c_{t}}\left(\int_{0}^{t}c_{s}^{*}ds\right)=0,\\
& \partial_{c_{t}}^{3}h_{t}(c)=0. \end{align*}
We now turn to characterizing the expected future effects of policy using the total derivative formula (\ref{eq:total derivative formula}). Since the marginal effect of the policy at time $t$ on time $s$, $\delta_{t}h_{s}\left(w^{s}\right)=w_{s}$, does not depend on time, we get \[ \partial_{t}\left(\delta_{t}h_{s}(w^{s})\right)=0. \] The Malliavin derivative is also simple as it measures the sensitivity of $w^{s}$ to the shock $w_{t}:$ \[ D_{t}\left(\delta_{t}h_{s}\left(w^{s}\right)\right)=D_{t}w_{s}=1, \] and the the stochastic derivative term of the future marginal costs becomes \[
E\left[D_{t}\int_{t}^{T}\delta_{t}h_{s}(w^{s})ds\bigg|\mathcal{F}_{t}\right]=T-t. \] This has a natural interpretation. The stochastic derivative measures the impact of the stochastic shock $dw_{t}$ on the future marginal effects of policies. Those are represented by the integrals $\int_{t}^{T}w_{s}ds$. A shock at time $t$ affects the future marginal effect of policy in each period $s$as $D_{t}w_{s}=1$. Since there are $\left(T-t\right)$ future periods, the shock $dw_{t}$ has the effect $\left(T-t\right)$. In other words, the stochastic shocks for early periods $t$ have longer lasting impact than for the later periods.
We then gather the present and the future terms to determine the dynamics of the first order process in (\ref{eq: FOP, main formula}): \begin{align*} dC_{t}= & -\Bigl(\underset{=w_{t}}{\underbrace{\Delta_{t}\partial_{c_{t}}h_{t}(w^{t})}dt}+\frac{1}{2}\underset{=0}{\underbrace{\partial_{c_{t}}^{3}h_{t}(w^{t})}}dt+\underset{=0}{\underbrace{\partial_{c_{t}}^{2}h_{t}(w^{t})}}dw_{t}\Bigr)-\\
& -\Bigl(\underset{=w_{t}}{\underbrace{-\delta_{t}h_{t}(w^{t})}dt+}\underset{=0}{\underbrace{E\left[\int_{t}^{T}\partial_{t}\delta_{t}h_{s}(w^{s})ds\bigg|\mathcal{F}_{t}\right]}dt+}\underset{=T-t}{\underbrace{E\left[\int_{t}^{T}D_{t}\delta_{t}h_{s}(w^{s})ds\bigg|\mathcal{F}_{t}\right]}dw_{t}}\Bigr), \end{align*}
or \[ dC_{t}=-\left(T-t\right)dw_{t}. \]
\begin{rem}
\label{rem:Boulatov}There are two alternative ways, without using our methodology, to derive the result. The first is to note that $E\left[\int_{t}^{T}w_{s}ds\bigg|\mathcal{F}_{t}\right]=\left(T-t\right)w_{t}$ and then \[ dC_{t}=-w_{t}dt-\left(\left(T-t\right)dw_{t}-w_{t}dt\right)=-\left(T-t\right)dw_{t}. \] The second approach that works in a variety of other circumstances is in extracting a martingale and using the Clark-Ocone formula to provide its explicit characterization in terms of Malliavin derivatives. We can rewrite \[
E\left[\int_{t}^{T}w_{s}ds\bigg|\mathcal{F}_{t}\right]=E\left[\int_{0}^{T}w_{s}ds\bigg|\mathcal{F}_{t}\right]-\int_{0}^{t}w_{s}ds, \] and \[
C_{t}=-\left(\int_{0}^{t}w_{s}ds+E\left[\int_{0}^{T}w_{s}ds\bigg|\mathcal{F}_{t}\right]-\int_{0}^{t}w_{s}ds\right)=-E\left[\int_{0}^{T}w_{s}ds\bigg|\mathcal{F}_{t}\right]. \]
The Clark-Ocone formula gives the representations of the martingale where $M_{t}=E\left[\int_{0}^{T}w_{s}ds\bigg|\mathcal{F}_{t}\right]$ as \[
dM_{t}=E\left[D_{t}\int_{0}^{T}w_{s}ds\bigg|\mathcal{F}_{t}\right]dw_{t}=E\left[\int_{t}^{T}\underset{=1}{\underbrace{D_{t}w_{s}}}ds\bigg|\mathcal{F}_{t}\right]dw_{t}=\left(T-t\right)dw_{t}. \] We expand on this class of examples in Boulatov, Riabov, and Tsyvinski (2020) where we analyze a more general class of environments which can be solved by extracting a martingale and not using the main tools of this paper -- the functional Itô formula and the total derivative formula. \end{rem}
\subsection{\label{subsec:Example Martingale}Hysteresis with time and past dependency}
Consider the functional $h_{t}(c^{t})=h(c_{t},\int_{0}^{t}a_{t,s}c_{s}ds),$ it describes the interaction of the policy at the present moment of time, and average of past values. The averaging is defined by the smooth kernel $a_{t,s}$ which measures how policy in period $s$ affects period $t$. For example, $a_{t,s}=\frac{1}{t}$ corresponds to the usual averaging. Importantly, the kernel depends on current time $t$.
This example captures three important parts of history-dependent policies. First, there is an effect of the present which is represented by the first argument, $c_{t}$. Second, there is an effect of the past which is represented by the second argument, $\int_{0}^{t}a_{t,s}c_{s}ds$. Here, the past enters as the integral of the path of the previous consumptions where the path enters through the kernel $a_{t,s}$ that depends both on the current policy $t$ and the path of previous policies.\footnote{One can easily modify this part to be some more complicated path-dependent object -- for example, $\int_{0}^{t}\alpha\left(c_{\left[0,s\right]}\right)ds$, where $\alpha$ is a functional of the path or to have arbitrary interaction of the past effects; or allow for $h_{t}\left(c\right)=\int_{0}^{t}\int_{0}^{t}g_{t}(c_{s},c_{r})dsdr),...$ where now there is a joint dependence on the past values $c_{s}$ and $c_{r}$ via a repeated integral.} Third, there is a joint dependence $h\left(.,.\right)$ between the present and the past.
The Fréchet derivative of $h_{t}$ that appears in the first order conditions (\ref{eq: FOC, perturbed}) is given by changing the whole path $c^{t}$ by a variation $z^{t}$: \begin{align*}
& h_{t}(c+z)-h_{t}(c)=h\left(c_{t}+z_{t},\int_{0}^{t}a_{t,s}\left(c_{s}+z_{s}\right)ds\right)-h_{t}(c)=\\
& =z_{t}h'_{1}\left(c_{t},\int_{0}^{t}a_{t,s}c_{s}ds\right)+\int_{0}^{t}a_{t,s}h'_{2}\left(c_{t},\int_{0}^{t}a_{t,r}c_{r}dr\right)z_{s}ds+o(||z||). \end{align*} Correspondingly, the present marginal effects of policy are given by the integral of the path of previous policies \[ \partial_{c_{t}}h_{t}(c)=h'_{1}\left(c_{t},\int_{0}^{t}a_{t,s}c_{s}ds\right). \] The marginal effect of policy in time $t$ n the past period $s$ ($s\le t\le T)$ is given by: \[ \delta_{s}h_{t}(c^{t})=a_{t,s}h'_{2}\left(c_{t},\int_{0}^{t}a_{t,r}c_{r}dr\right). \]
In particular, we see that $h_{t}\in\mathcal{A}_{t}$ and $\delta_{s}h_{t}\in\mathcal{A}_{t}.$
The first order condition (\ref{eq: FOC, perturbed}) is \[
w_{t}=c_{t}^{(\epsilon)}+\epsilon h'_{1}\left(c_{t},\int_{0}^{t}a_{t,s}c_{s}ds\right)+\epsilon E\left[\int_{t}^{T}a_{s,t}h'_{2}\left(c_{s},\int_{0}^{s}a_{s,r}c_{r}dr\right)ds\bigg|\mathcal{F}_{t}\right]; \] and the stochastic elasticity (\ref{eq: First order process}) is given by, recalling that $c_{t}^{*}=w_{t}$: \[
C_{t}=-\underset{\text{Present effect}}{\underbrace{h'_{1}\left(w_{t},\int_{0}^{t}a_{t,s}w_{s}ds\right)}}-\underset{\text{Future effect}}{\underbrace{E\left[\int_{t}^{T}a_{s,t}h'_{2}\left(w_{s},\int_{0}^{s}a_{s,r}w_{r}dr\right)ds\bigg|\mathcal{F}_{t}\right]}}. \] Note that in the future effects it is $\delta_{t}h_{s}(c^{t})=a_{s,t}h'_{2}\left(w_{s},\int_{0}^{s}a_{s,r}w_{r}dr\right)$ that enters as the kernel $a_{s,t}$ measures how policy in period $t$ ($t\le s\le T)$ affects future period $s$.
The conditional expectation $E\left[\int_{t}^{T}a_{s,t}h'_{2}\left(w_{s},\int_{0}^{s}a_{s,r}w_{r}dr\right)ds\bigg|\mathcal{F}_{t}\right]$ is intractable for direct computations, as it requires infinite-dimensional integration over the distribution of the process $(w(r)-w(t))_{t\leq r\leq s}.$ However, the semimartingale decomposition is obtained immediately after an application of the total derivative formula (\ref{eq:total derivative formula}).
We now turn to calculation of the derivatives needed for the application of the Dupire's functional Itô formula (\ref{eq: FOP, Dupire part}). These derivatives are used in determining the decomposition of the present effects of the path $\partial_{c_{t}}h_{t}(c)$ in the first order conditions of the perturbed problem (\ref{eq: FOC, perturbed}). Recall that the horizontal and two vertical derivatives are needed.
The horizontal derivative of the present effects of the path $\partial_{c_{t}}h_{t}(c)$ is given by freezing the path at time $t$ and extending it with time: \[ \Delta_{t}\partial_{c_{t}}h_{t}(c)=\Delta_{t}h'_{1}\left(w_{t},\int_{0}^{t}a_{t,s}w_{s}ds\right)=h''_{12}\left(w_{t},\int_{0}^{t}a_{t,s}w_{s}ds\right)\left(a_{t,t}w_{t}+\int_{0}^{t}(\partial_{t}a_{t,s})w_{s}ds\right). \] Vertical differentiation of the present effects of the path $\partial_{c_{t}}h_{t}(c)$ is given by bumping the path. It reduces to the differentiation in the first argument: \begin{align*}
& \partial_{c_{t}}^{2}h_{t}(c)=\partial_{c_{t}}\left(h'_{1}\left(w_{t},\int_{0}^{t}a_{t,s}w_{s}ds\right)\right)=h''_{11}\left(w_{t},\int_{0}^{t}a_{t,s}w_{s}ds\right),\\
& \partial_{c_{t}}^{3}h_{t}(c)=\partial_{c_{t}}^{2}\left(h'_{1}\left(w_{t},\int_{0}^{t}a_{t,s}w_{s}ds\right)\right)=h'''_{111}\left(w_{t},\int_{0}^{t}a_{t,s}w_{s}ds\right). \end{align*}
Note that since we are already finding the derivatives of the marginal effect $\partial_{c_{t}}h_{t}(c)$, the mixed time-derivative and the second and third order space derivatives appear (rather than just the time and first and second order derivatives).
We next calculate the derivatives needed for the application of the total derivative formula (\ref{eq: FOP, total derivative part}) to characterize the conditional expectation of the marginal effects of policy. The time derivative is given by \[ \partial_{t}\left(\delta_{t}h_{s}(w^{s})\right)=\partial_{t}\left(a_{s,t}h'_{2}\left(w_{s},\int_{0}^{s}a_{s,r}w_{r}dr\right)\right)=(\partial_{t}a_{s,t})h'_{2}\left(w_{s},\int_{0}^{s}a_{s,r}w_{r}dr\right). \] The Malliavin derivative is given by \begin{align*}
& D_{t}\left(\delta_{t}h_{s}(w^{s})\right)=D_{t}\left(a_{s,t}h'_{2}\left(w_{s},\int_{0}^{s}a_{s,r}w_{r}dr\right)\right)=\\
& =a_{s,t}h''_{12}\left(w_{s},\int_{0}^{s}a_{s,r}w_{r}dr\right)\underset{=1}{\underbrace{D_{t}w_{s}}}+a_{s,t}h''_{22}\left(w_{s},\int_{0}^{s}a_{s,r}w_{r}dr\right)\underset{=\int_{t}^{s}a_{s,r}D_{t}w_{r}dr=\int_{t}^{s}a_{s,r}dr}{\underbrace{D_{t}\left(\int_{0}^{s}a_{s,r}w_{r}dr\right)}}. \end{align*} Gathering the terms we find the differential of the first order process: \begin{align*} dC_{t} & =-h''_{12}\left(w_{t},\int_{0}^{t}a_{t,s}w_{s}ds\right)\left(a_{t,t}w_{t}+\int_{0}^{t}(\partial_{t}a_{t,s})w_{s}ds\right)dt-\\
& -\left(h''_{11}\left(w_{t},\int_{0}^{t}a_{t,s}w_{s}ds\right)dw_{t}+\frac{1}{2}h'''_{111}\left(w_{t},\int_{0}^{t}a_{t,s}w_{s}ds\right)dt\right)-\\
& -\Bigl(\underset{\delta_{t}h_{t}(w^{t})}{-\underbrace{a_{t,t}h'_{2}\left(w_{t},\int_{0}^{t}a_{t,r}w_{r}dr\right)}}dt+\underset{E\left[\int_{t}^{T}\partial_{t}\delta_{t}h_{s}(w^{s})ds\bigg|\mathcal{F}_{t}\right]}{\underbrace{E\left[\int_{t}^{T}(\partial_{t}a_{s,t})h'_{2}\left(w_{s},\int_{0}^{s}a_{s,r}w_{r}dr\right)ds\bigg|\mathcal{F}_{t}\right]}}dt+\\
& +\underset{E\left[D_{t}\int_{t}^{T}\delta_{t}h_{s}(w^{s})ds\bigg|\mathcal{F}_{t}\right]}{\underbrace{E\left[\int_{t}^{T}(a_{s,t}h''_{12}\left(w_{s},\int_{0}^{s}a_{s,r}w_{r}dr\right)+a_{s,t}h''_{22}\left(w_{s},\int_{0}^{s}a_{s,r}w_{r}dr\right)\int_{t}^{s}a_{s,r}dr)ds\bigg|\mathcal{F}_{t}\right]}}dw_{t}. \end{align*}
\begin{rem} \label{rem:Detemple Zapatero}The presented calculation can be done without the use of the total derivative formula when the occurrence of $t$ under the integral in the conditional expectation can be removed, e.g. when $a_{s,t}=\tilde{g}(t)g(s),$ and a martingale can be extracted. Indeed, then future effects can be represented as \begin{align*}
\tilde{g}(t)E\left[\int_{t}^{T}g(s)h'_{2}\left(w_{s},\int_{0}^{s}a_{s,r}w_{r}dr\right)ds\bigg|\mathcal{F}_{t}\right] & =-\tilde{g}(t)\int_{0}^{t}g(s)h'_{2}\left(w_{s},\int_{0}^{s}a_{s,r}w_{r}dr\right)ds+\tilde{g}(t)M_{t}, \end{align*} where \[
M_{t}=E\left[\int_{0}^{T}g(s)h'_{2}\left(w_{s},\int_{0}^{s}a_{s,r}w_{r}dr\right)ds\bigg|\mathcal{F}_{t}\right]. \] Now the semimartingale decomposition immediately follows from the Itô formula. This type of examples (with exponential functions $g,\tilde{g}$) where studied by Detemple and Zapatero (1991) in the context of asset prices under habit formation. \end{rem}
\subsection{\label{subsec: Example Deterministic-hysteresis}Deterministic hysteresis}
In this section, we consider a deterministic case. Assume that the underlying process shock $\theta$ is deterministic: $d\theta_{t}=b(\theta_{t})dt.$ Then optimal policies $c^{*}$ and $c^{\epsilon}$ are deterministic as well: \[ c_{t}^{*}=\theta_{t}. \]
Our assumption of $h_{t}$ belonging to the class $\mathcal{A}_{t}$ allows us to neatly decompose its effects into those of the effects of the present and the past history \[ c_{t}^{\epsilon}=\theta_{t}-\epsilon\left(\partial_{c_{t}}h_{t}(c^{\epsilon})+\epsilon\int_{t}^{1}\delta_{t}h_{s}(c^{\epsilon})ds\right). \] Differentiating the latter relation in $\epsilon$ at $\epsilon=0$ we get \[ C_{t}=-\left(\partial_{c_{t}}h_{t}(c^{*})+\int_{t}^{1}\delta_{t}h_{s}(c^{*})ds\right). \] Now, apply the functional Itô formula (\ref{eq:Functional Ito}) to $\partial_{c_{t}}h_{t}(c^{*})$ and note that there are no second order vertical derivatives of $\partial_{c_{t}}h_{t}(c^{*})$ due to the absence of stochasticity: \[ d\left(\partial_{c_{t}}h_{t}(c^{*})\right)=\left(\Delta_{t}\partial_{c_{t}}h_{t}(c^{*})+\partial_{c_{t}}^{2}h_{t}(c^{*})b(c_{t}^{*})\right)dt. \] The future marginal effects of policy are also deterministic and we can simply differentiate $\int_{t}^{1}\delta_{t}h_{s}(c^{*})ds$ in time.
\[ dC_{t}=-\bigg(\underset{d\left(\partial_{c_{t}}h_{t}(c^{*})\right)}{\underbrace{\Delta_{t}\partial_{c_{t}}h_{t}(c^{*})+\partial_{c_{t}}^{2}h_{t}(c^{*})b(c_{t}^{*})}}\bigg)dt-\bigg(\underset{d\left(\int_{t}^{1}\delta_{t}h_{s}(c^{*})ds)\right)}{\underbrace{-\delta_{t}h_{t}(c^{*})+\int_{t}^{1}\frac{\partial}{\partial t}\delta_{t}h_{s}(c^{*})ds)}}\bigg)dt. \]
The present effects of the policy $\partial_{c_{t}}h_{t}(c^{*})$ change with time as the horizontal derivative $\Delta_{t}$ and with the movement in the path of the policy as the vertical derivative $\partial_{c_{t}}$. The cumulative future marginal effects of the policy $\int_{t}^{1}\delta_{t}h_{s}(c^{*})ds$ change as the relative difference between the time change in the cumulant of the future marginal effects $\frac{\partial}{\partial t}\delta_{t}h_{s}(c^{*})ds$ relative to today's marginal effect of the policy $\delta_{t}h_{t}(c^{*})$. This case is interesting as a stand alone result that allows to focus on history dependence without stochasticity.
\subsection{\label{subsec:A-tipping-point}A ``tipping point''}
The main premise of the literature on tipping points discussed in \ref{subsec:Evidence-on-climate} is that there is some level of emissions the crossing of which leads to a different behavior of the climate system. For example, upon reaching the threshold the damages become larger or become irreversible. We now show an example where instead of considering a threshold we focus on the time when the climate variable achieves its maximum upto any given period of time.
Let $\theta_{t}$ be the time when the Brownian motion $w$ achieves its maximum over $[0,t]:$ \[ \theta_{t}=\arg\max_{s\in[0,t]}w_{s} \] and it is known that $\theta_{t}$ is a.s. unique. Let $f(x)$ be an absolutely continuous function such that $f(x)=0$ for $x\leq0.$ Consider the objective function \[ E\left[\int_{0}^{T}\left(-\frac{1}{2}(c_{t}-w_{t})^{2}-\int_{0}^{t}f(s-\theta_{t})c_{s}ds\right)dt\right]. \] One can think of this setting as follow. In each period $t$, we consider the time $\theta_{t}$ when the maximal level of $w$ has been achieved -- say, the time of the temperature record. The damages then are counted as the weighted by $f\left(s-\theta_{t}\right)$ emissions from the time of the record. When the new record is achieved, the weighting restarts. This example can be significantly expanded by having the weighting functions change with time or the weighting functions that weigh both the time prior to $\theta_{t}$ and the time after $\theta_{t}$ (a form of highlighting the ``salience'' of the record time) but we chose to present the simple form here. That is, there is a salient ``tipping'' or reference point following which the damages change their behavior.
The first-order conditions give us the optimal policy in a closed form is given by \[
c_{t}=w_{t}-E\left[\int_{t}^{T}f(t-\theta_{s})ds\bigg|\mathcal{F}_{t}\right]. \] The process $\xi_{t}=\int_{t}^{T}f(t-\theta_{s})ds$ is absolutely continuous and square integrable. Indeed, \[ \xi_{t}=\int_{t}^{T}\int_{0}^{t}f'(x-\theta_{s})dxds. \] Let us represent $\xi_{t}$ using the Clark-Ocone formula \[ \xi_{t}=\mathbb{E}\xi_{t}+\int_{0}^{T}g_{t,s}dw_{s}. \] By the total derivative formula of Proposition \ref{prop: Total derivative formula}, we immediately find that $c$ is an Itô process and \begin{equation}
dc_{t}=\bigg(1-g_{t,t}\bigg)dw_{t}-\mathbb{E}[\partial_{t}\xi_{t}|\mathcal{F}_{t}]dt.\label{eq: Tipping point} \end{equation}
\begin{rem} Let us try to get this result using standard methods. We compute the process $c_{t}$ explicitly as a functional of the Wiener process $w.$ At first we find the conditional expectation \begin{align*}
& E[f(t-\theta_{s})|\mathcal{F}_{t}]=E\left[\int_{0}^{\infty}f'(x)1_{x<t-\theta_{s}}dx\bigg|\mathcal{F}_{t}\right]=\\
& =\int_{0}^{t}f'(x)P(\theta_{s}<t-x|\mathcal{F}_{t})dx=\int_{0}^{t}f'(x)P\left(\max_{[0,t-x]}w>\max_{[t-x,s]}w\bigg|\mathcal{F}_{t}\right)dx=\\
& =\int_{0}^{t}f'(x)1_{\max_{[0,t-x]}w>\max_{[t-x,t]}w}P\left(\max_{[0,t-s]}w<z\right)\bigg|_{z=\max_{[0,t-x]}w-w_{t}}dx=\\
& =\int_{0}^{t}f'(x)1_{\max_{[0,t-x]}w>\max_{[t-x,t]}w}\left(2\Phi\left(\frac{\max_{[0,t-x]}w-w_{t}}{\sqrt{t-s}}\right)-1\right)dx. \end{align*} Hence, \begin{align*} c_{t} & =w_{t}-\int_{0}^{t}f'(x)1_{\max_{[0,t-x]}w>\max_{[t-x,t]}w}\int_{t}^{T}\left(2\Phi\left(\frac{\max_{[0,t-x]}w-w_{t}}{\sqrt{t-s}}\right)-1\right)dsdx=\\
& =w_{t}-\int_{0}^{t-\theta_{t}}f'(x)\int_{t}^{T}\left(2\Phi\left(\frac{\max_{[0,t]}w-w_{t}}{\sqrt{t-s}}\right)-1\right)dsdx=\\
& =w_{t}-f(t-\theta_{t})\int_{t}^{T}\left(2\Phi\left(\frac{\max_{[0,t]}w-w_{t}}{\sqrt{t-s}}\right)-1\right)ds. \end{align*} We get the representation of the form \[ c_{t}=w_{t}-f(t-\theta_{t})g(\max_{[0,t]}w-w_{t}). \] Now one can try two standard approaches to get the semimartingale decompostion. The first is to apply the Itô formula. In order to do this, we see that \[ c_{t}=F(t,w_{t},\theta_{t},M_{t}), \] where $M_{t}=\max_{[0,t]}w,$ and $F(t,x,y,z)=x-f(t-y)g(z-x).$ Hence the Itô formula will lead to a semimartingale decomposition that contains terms $d\theta_{t},$ $dM_{t},$ and the sum over jumps of the process $\theta_{t}.$ It is not obvious that the process $c_{t}$ is an Itô process. \end{rem} The second approach is to apply the functional Itô formula. In order to do this the functional $c_{t}$ must be extended to a $C_{b}^{1,2}$-functional (in the sense of Cont and Fournie) on the space of cadlag paths. However, if such extension is possible, vertical derivatives of $c_{t}$ must be equal to zero and the functional Itô formula would lead to semimartingale decomposition $dc_{t}=\alpha_{t}dt,$ which is not the case.
Instead, the application of our methodology give a straightforward and compact answer of the equation (\ref{eq: Tipping point}).
\section{Discussion}
In this section, we discuss some of the more technical issues behind the results.
\subsection{Why do we need both the functional Itô formula and the total derivative formula?\label{subsec:Why-do-we-need-two-formulas}}
The stochastic elasticity $C_{t}^{h}$ is described in \eqref{eq: First order process} via two terms: a vertical derivative $\partial_{c_{t}}h_{t}(c)$ of the perturbation functional $h_{t}$ and the conditional expectation
$\mathbb{E}[\int_{t}^{1}\delta_{t}h_{s}(c^{*})ds|\mathcal{F}_{t}].$ Both are functionals of the path $(c_{s})_{0\leq s\leq t}$ but we need to use different methods to study them.
We assume that the functional $\partial_{c_{t}}h_{t}(c)$ satisfies conditions from Dupire (2019). This is a natural assumption, since $\partial_{c_{t}}h_{t}(c)$ is a well-defined time-dependent functional of the path $(c_{s})_{0\leq s\leq t}$.
However, in the case of the functional $\mathbb{E}[\int_{t}^{1}\delta_{t}h_{s}(c^{*})ds|\mathcal{F}_{t}]$ the same assumption is not generally suitable. The application of the Dupire's functional Itô formula requires that \[
\mathbb{E}\left[\int_{t}^{1}\delta_{t}h_{s}(c^{*})ds|\mathcal{F}_{t}\right]=q_{t}(m), \] where $m$ is a certain semimartingale and $(q_{t})_{t\in[0,T]}$ is smooth in the sense of Dupire (2019) and Cont and Fournie (2013) family of path-dependent functionals. It may be a difficult stand-alone problem even to verify that such representation holds. Consider, for example, the case when \[
\mathbb{E}\left[\int_{t}^{1}\delta_{t}h_{s}(c^{*})ds|\mathcal{F}_{t}\right]=\int_{0}^{t}f_{t}(c_{s}^{*})dc_{s}^{*}, \] and $f_{t}$ is smooth in $t.$ The stochastic integral is defined for almost all realizations of $c^{*}$ only and it is unclear how we can extend it smoothly to all continuous paths.\footnote{Recall that the functional It\textroundcap{o} formula is applicable to functionals defined on a larger space of cádlág paths.}
Instead we assume that the expression under the conditional expectation is smooth in $t$ and then apply the total derivative formula in Proposition \ref{prop: Total derivative formula} to prove directly that $C^{h}$ is an Itô process and to find its semimartingale decomposition as well.
In other words, the functional Itô formula is useful when the functional of the path is already well-defined. When faced with a conditional expectation process such as the one we considered here or that frequently occurs in a variety of other economic problems, the total derivative formula allows to straightforwardly calculate its semimartingale decomposition.
\subsection{\label{subsec:Assumptions-on-smoothness}Assumptions on smoothness of the functionals}
Our results are valid for path-dependent functionals $h=(h_{t})_{0\leq t\leq T}$ such that: (1) $h_{t}\in\mathcal{A}_{t};$ (2) $t\to\partial_{c_{t}}h_{t}$ is horizontally and twice vertically differentiable; (3) $t\to\delta_{t}h_{s}(c^{s})$ is absolutely continuous; (4) $w\to\delta_{t}h_{s}(c^{s}(w))$ is Malliavin differentiable.
The first restriction that $h_{t}$ belongs to the class $\mathcal{A}_{t}$ allows to calculate the first order conditions in a tractable form. The second restriction is a typical condition needed for the functional Itô formula to be valid. Last two conditions are imposed for the ease of presentation and also they are satisfied in all our examples. They are needed for the total derivative formula to be applicable for the process \[
t\to E[\int_{t}^{T}\delta_{t}h_{s}(c^{s})ds|\mathcal{F}_{t}]. \] However, these conditions can be considerably relaxed. It is enough to find an absolutely continuous square integrable process $\xi_{t},$ such that \[
E[\int_{t}^{T}\delta_{t}h_{s}(c^{s})ds|\mathcal{F}_{t}]=E[\xi_{t}|\mathcal{F}_{t}]. \] As a simple example consider the process \[ \delta_{t}g_{s}(c_{s})=w_{t}. \]
This process is not absolutely continuous, however, $E[w_{t}|\mathcal{F}_{t}]=E[w_{T}|\mathcal{F}_{t}]$ and one can take $\xi_{t}=(T-t)w_{T}.$
\subsection{\label{subsec:Relationship-to-the_Clarrk-Ocone}Relationship to the Clark-Ocone formula}
It is instructive to compare Proposition \ref{prop: Total derivative formula} to the well-known Clark-Ocone formula. Recall that $\mathcal{F}=(\mathcal{F}_{t})_{t\in[0,T]}$ is a filtration generated by Wiener process $(w_{t})_{t\in[0,T]}.$
Every $\mathcal{F}$-martingale $(M_{t})_{t\in[0,T]}$ can be represented as a conditional expectation process. Indeed, from the definition of a martingale we get $M_{t}=\mathbb{E}[M_{T}|\mathcal{F}_{t}]$. Conversely, every process of the form \[
Z_{t}=\mathbb{E}[\xi|\mathcal{F}_{t}], \] where $\xi$ is an integrable random variable, is a martingale. This follows from the basic properties of conditional expectations:
\[
\mathbb{E}\left[Z_{t+h}|\mathcal{F}_{t}\right]=\mathbb{E}\left[\mathbb{E}\left[\xi|\mathcal{F}_{t+h}\right]|\mathcal{F}_{t}\right]=\mathbb{E}\left[\xi|\mathcal{F}_{t}\right]=Z_{t}. \]
Moreover, every $\mathcal{F}$-martingale $Z_{t}=\mathbb{E}[\xi|\mathcal{F}_{t}]$ is an Itô process and if $\xi$ is Malliavin differentiable, the Clark-Ocone formula gives \[
Z_{t}=Z_{0}+\int_{0}^{t}\mathbb{E}[D_{s}\xi|\mathcal{F}_{s}]dw_{s}, \] and the differential of $Z:$
\[
dZ_{t}=\mathbb{E}[D_{t}\xi|\mathcal{F}_{t}]dw_{t}. \]
The processes that we consider in the Proposition \ref{prop: Total derivative formula} are of the different type - they are of the form \[
X_{t}=\mathbb{E}[\xi_{t}|\mathcal{F}_{t}] \]
with $\xi=(\xi_{t})_{t\in[0,T]}$ being an absolutely continuous process. Such process are not martingales in general. The simplest example is provided with deterministic non-constant process $\xi$, e.g. if we take $\xi_{t}=t,$ then $X_{t}=t$ and this is obviously not a martingale.
\begin{comment} However, the process $X$ is always an Itô process and its semimartingale representation can be obtained by combining the Clark-Ocone formula with the usual calculus. We now provide a sketch of the argument. For each $t\in[0,1]$ we have
\[
\xi_{t}=\mathbb{E}[\xi_{t}]+\int_{0}^{1}\mathbb{E}[D_{s}\xi_{t}|\mathcal{F}_{s}]dw_{s}. \]
Then,
\[
X_{t}=\mathbb{E}[\xi_{t}]+\int_{0}^{t}\mathbb{E}[D_{s}\xi_{t}|\mathcal{F}_{s}]dw_{s}=\mathbb{E}[\xi_{t}]+\int_{0}^{t}\mathbb{E}[D_{s}\left(\xi_{s}+\int_{s}^{t}\partial_{r}\xi_{r}dr\right)|\mathcal{F}_{s}]dw_{s}= \]
\[
\mathbb{=E}[\xi_{t}]+\int_{0}^{t}\mathbb{E}[D_{s}\xi_{s}|\mathcal{F}_{s}]dw_{s}+\int_{0}^{t}\int_{s}^{t}\mathbb{E}[D_{s}\partial_{r}\xi_{r}|\mathcal{F}_{s}]drdw_{s}= \]
\[
=\mathbb{E}[\xi_{t}]+\int_{0}^{t}\mathbb{E}[D_{s}\xi_{s}|\mathcal{F}_{s}]dw_{s}+\int_{0}^{t}\int_{0}^{r}\mathbb{E}[D_{s}\partial_{r}\xi_{r}|\mathcal{F}_{s}]dw_{s}dr= \] \[
\mathbb{=E}[\xi_{0}]+\int_{0}^{t}\mathbb{E}[D_{s}\xi_{s}|\mathcal{F}_{s}]dw_{s}+\int_{0}^{t}\mathbb{E}[\partial_{r}\xi_{r}]dr+\int_{0}^{t}\int_{0}^{r}\mathbb{E}[D_{s}\partial_{r}\xi_{r}|\mathcal{F}_{s}]dw_{s}dr= \]
\[
=X_{0}+\int_{0}^{t}\mathbb{E}[D_{s}\xi_{s}|\mathcal{F}_{s}]dw_{s}+\int_{0}^{t}(\mathbb{E}[\partial_{r}\xi_{r}]+\int_{0}^{r}\mathbb{E}[D_{s}\partial_{r}\xi_{r}|\mathcal{F}_{s}]dw_{s})dr= \] \[
=X_{0}+\int_{0}^{t}\mathbb{E}[D_{s}\xi_{s}|\mathcal{F}_{s}]dw_{s}+\int_{0}^{t}\mathbb{E}[\partial_{r}\xi_{r}|\mathcal{F}_{r}]dr. \] \end{comment} The Clark-Ocone formula is then a partial case of Proposition \ref{prop: Total derivative formula} corresponding to constant in $t$ process $(\xi_{t})_{t\in[0,T]}$. For the proof and general versions of Proposition \ref{prop: Total derivative formula} we refer to the Appendix.
\subsection{\label{subsec:Discussion-of-the-optimal control}Discussion of the optimal control approach}
A natural question is whether we can use optimal control and dynamic programming to simplify the analysis.
One of the standard approaches for solving optimization problems is the dynamic programming principle. It can be used for maximization of functionals of the type (see see Fabbri, Gozzi, and Swiech 2017 for the exposition of infinitely dimensional problems): \[ V(a)=E\int_{0}^{T}l(t,x^{t},a(t))dt, \] where $a$ is the control process, the process $x$ satisfies certain SDE whose coefficients depend on the control $a$ as well. This framework is not well suited for our problem, as in our case the dependence of the expression under the integral is a path-dependent functional of the policy $c$: \[ V(c)=E\int_{0}^{T}\left(-\frac{1}{2}(c_{t}-w_{t})^{2}-h_{t}(c^{t})\right)dt. \] Hence, to fit the framework of the optimal control we must consider the whole path $c^{t}$ as the value of the control $a(t).$ That is, the control space becomes the set of paths $\cup_{t\in[0,T]}C[0,t].$ This leads to the consideration of the cost functional \[ V(t,x,c^{t})=\max_{a\in C[t,T]}E_{t,x}\int_{t}^{T}\left(-\frac{1}{2}(a_{s}-w_{s-t}-x)^{2}-h_{s}(c^{t}\otimes a^{s})\right)ds. \] The HJB equation then becomes a path-dependent PDE that contains infinite-dimensional optimization over trajectories of the control. Thus the analysis may be as complicated as our original problem. For example, Cosso et. al (2018) use functional Itô calculus to write the path-dependent HJB equation, prove that the value function is a viscosity solution and prove a partial comparison principle.
\section{Conclusion}
Path-dependent policies are a feature of a number of economic models. Doing a lot of sports only when young leads to a very different health outcome than exercising lightly throughout life, even though the cumulative lifetime exercise may be the same. Moreover, the importance of the previous pattern of exercise may change with time. For example, intensively training several times a week is best when young while a steady pattern of light exercise during a week may be preferable when older. Missing a credit card payment twice in a row in a ten year period leads to a very different credit history than missing a payment once every five years. The consequence of the same sequence of non-payments for the credit history may be quite different in a recession versus a boom. These policies are not just a function of the current state or time but depend on the whole trajectory of the past actions. Moreover, the dependency itself is changing with time.
We developed a methodology for analysis of general class of policy with path-dependent effects in an uncertain environment with forward looking agents. The primary difficulty that arises in such models is that the optimizing agents foresee such path dependency and the actions they take incorporate the expectation of the future effects. We show that three ingredients are needed for the analysis of such problems. First, we introduced a general class of path-dependent functionals that allows to tractably write the first order conditions for the problem. Second, the recently developed functional Itô calculus allows to describe the dynamics of the present effect of the past choices. Third, the total derivative formula that we develop in this paper allows to derive the dynamics of the conditional expectation processes of the future effects of today's choices. Our analysis shows that even when policy has very small contemporaneous effects, it may have large effects due to either interaction with the past choices or due to the expected future effects. The effects of the past are represented by the magnitude of horizontal derivatives and higher order vertical derivative. The future expected effects are determined by the magnitude of the time derivative and by the stochastic derivatives. We believe that the methodology we develop in, on purpose, a stark underlying environment will facilitate analysis of a wide class of other economic problems.
\pagebreak{}
\section{Appendix}
\subsection{Remark \ref{rem:Discontinuous} in Section \ref{subsec:Class A_t}}
The objective function (\ref{eq:Perturbed problem}) can be written as follows: \begin{align*}
& \mathbb{E}\int_{0}^{T}(-\frac{1}{2}\left(c_{t}-\theta_{t}\right)^{2}-\epsilon c_{\frac{t}{2}})dt=\mathbb{E}\int_{0}^{T}(-\frac{1}{2}\left(c_{t}-\theta_{t}\right)^{2})dt-\mathbb{E}\int_{0}^{T}\epsilon c_{\frac{t}{2}}dt=\\
& =\mathbb{E}\int_{0}^{T}(-\frac{1}{2}\left(c_{t}-\theta_{t}\right)^{2})dt-2\mathbb{E}\int_{0}^{\frac{T}{2}}\epsilon c_{s}ds=\\
& =\mathbb{E}\int_{0}^{\frac{T}{2}}(-\frac{1}{2}\left(c_{t}-\theta_{t}\right)^{2}-2\epsilon c_{t})dt+\mathbb{E}\int_{\frac{T}{2}}^{T}(-\frac{1}{2}\left(c_{t}-\theta_{t}\right)^{2})dt. \end{align*} The first-order conditions become \[ \begin{cases} c_{t}=\theta_{t}-2\epsilon,\ t<\frac{T}{2}\\ c_{t}=\theta_{t},\ t>\frac{T}{2}, \end{cases} \] which implies that the optimal policy has a jump at $t=\frac{T}{2}$.
\subsection{\label{sec:A-simple-example}An example with contemporaneous policies}
In this section, we develop a simple example of introducing additional effects of policies that can be easily handled by the usual Itô formula. That is, there is no hysteresis and policy only has contemporaneous effects.
Consider an additional effect of policies given by $f(c_{t})$, where $f:\mathbb{R}\to\mathbb{R}$ is a smooth function: \[ \max_{c}\mathbb{E}\int_{0}^{T}(-\frac{1}{2}\left(c_{t}-w_{t}\right)^{2}-\epsilon f(c_{t}))dt, \] and $\epsilon$ is a parameter. The first order conditions are given by \begin{equation} c_{t}^{\epsilon}=w_{t}-\epsilon f'\left(c_{t}^{\epsilon}\right),\label{eq:FOC simple example} \end{equation} and equate the marginal benefit of the policy tracking the process $w_{t}$ with the additional marginal effects $\epsilon f'\left(c_{t}^{\epsilon}\right)$ .
\subsubsection{Stochastic elasticity and its dynamics}
We are particularly interested in the case of $\epsilon\rightarrow0$. One can think about this as a small parameter asymptotics for the problem. Let $C_{t}^{f}$ be a first variation process: \[ c_{t}^{\epsilon}=c_{t}^{*}+\epsilon C_{t}^{f}+o(\epsilon). \] The process $C_{t}^{f}$ represents how optimal policy process changes locally in response to the introduction of a small effect of policy$f$. One can think of $C_{t}^{f}$ as a notion of stochastic elasticity that represents the change in the whole process when policy has additional effects $f$.
Differentiating the first order condition (\ref{eq:FOC simple example}) with respect to $\epsilon$ and evaluating at $\epsilon=0$, we get \[ C_{t}^{f}=f'(c_{t}^{*}). \] Without any restrictions on $f$, the process $C_{t}^{f}$ may be quite arbitrary, for example, discontinuous. However, since we assumed smoothness of $f$, Itô's lemma implies that the process $C_{t}^{f}$ is a semimartingale. Moreover, its decomposition is given as (using also that $c_{t}^{*}=w_{t}$) in closed form: \[ dC_{t}^{f}=\alpha(w_{t})dt+\beta(w_{t})dw_{t}, \] where \begin{align} \alpha(w_{t}) & =\frac{1}{2}f'''(w_{t}),\nonumber \\ \beta(w_{t}) & =f''(w_{t}).\label{eq: FOP state dependent} \end{align}
\subsubsection{Dynamics of optimal policy}
We now turn to characterizing the optimal policy $c_{t}^{\epsilon}$ rather than the first variation process. Let the process $c_{t}^{\epsilon}$ have the form \[ dc_{t}^{\epsilon}=\alpha^{\epsilon}\left(c_{t}^{\epsilon}\right)dt+\beta^{\epsilon}\left(c_{t}^{\epsilon}\right)dw_{t}. \] Applying Itô formula to (\ref{eq:FOC simple example}), we can find the dynamics of the process $c_{t}^{\epsilon}$: \begin{align*}
& dc_{t}^{\epsilon}=dw_{t}-\epsilon\left(f''\left(c_{t}^{\epsilon}\right)dc_{t}^{\epsilon}+\frac{1}{2}f'''\left(c_{t}^{\epsilon}\right)\left(\beta_{t}^{\epsilon}\right)^{2}dt\right),\\
& \alpha_{t}^{\epsilon}dt+\beta_{t}^{\epsilon}dw_{t}=dw_{t}-\epsilon\left(f''\left(c_{t}^{\epsilon}\right)\alpha_{t}^{\epsilon}+\frac{1}{2}f'''\left(c_{t}^{\epsilon}\right)\left(\beta_{t}^{\epsilon}\right)^{2}\right)dt-\epsilon f''\left(c_{t}^{\epsilon}\right)\beta_{t}^{\epsilon}dw_{t}. \end{align*} The drift and diffusion coefficients are then given by collecting terms at $dt$ and $dw_{t}$: \begin{align} \alpha_{t}^{\epsilon}\left(c_{t}^{\epsilon}\right) & =-\epsilon\frac{\frac{1}{2}f'''\left(c_{t}^{\epsilon}\right)\left(\beta_{t}^{\epsilon}\right)^{2}}{1+\epsilon f''\left(c_{t}^{\epsilon}\right)},\nonumber \\ \beta_{t}^{\epsilon}\left(c_{t}^{\epsilon}\right) & =\frac{1}{1+\epsilon f''\left(c_{t}^{\epsilon}\right)}.\label{eq: Coefficients state dependent} \end{align} The coefficients in (\ref{eq: Coefficients state dependent}) are given as a system of coupled equations and themselves depend on the process $c_{t}^{\epsilon}$. This is in contrast to the coefficients of the stochastic elasticity given in (\ref{eq: FOP state dependent}) which are given in closed form as they are evaluated at the process $c_{t}^{*}=c_{t}^{\epsilon=0}=w_{t}.$
The key to tractability in the simple setting of this section is Itô lemma. However, Itô lemma only applies to functions and not the functionals. That is, it does not apply to the path-dependent effects of policies studying which is the main goal of the rest of this paper.
It is useful to summarize these results in terms of how the magnitude of the additional effects influences policy. This parallels the discussion of Dixit (1991), Reis (2006) and Alvarez, Lippi, and Paciello (2011, 2016) who show that in the environments with uncertainty and adjustment frictions the costs up to the fourth order may have first order effects on the dynamics of optimal policies. In our setting, since the additional effects $f$ are smooth, we show that the effects up to the third order, that is the second and the third derivative of $f$ will have the first-order effects. The first and second derivative of the marginal effects (that is, the second and third derivative of $f$) matter for the drift of the optimal policy -- this is the evolution of optimal policy with respect to time. The first derivative of the marginal effects matters for the diffusion coefficients -- this is response of optimal policy to stochastic shocks. Importantly, the dynamics of the optimal policy is not directly influenced by either the past or the future evolution of the policies.
\subsection{\label{subsec:Appendix Vertical-derivatives}Vertical derivatives and the class $\mathcal{A}_{t}$ of functionals}
The functional $h:C[0,t]\to\mathbb{R}$ is said to be in the class $\mathcal{A}_{t}$ if for each path $c\in C[0,t]$ there exists a number $\partial_{c_{t}}h(c)$ and an integrable function $(\delta_{s}h(c))_{s\in[0,t]},$ such that the following asymptotic relation holds for any $c\in C[0,t]:$ \[
h(c+z)=h(c)+\partial_{c_{t}}h(c)\cdot z_{t}+\int_{0}^{t}\delta_{s}h(c)\cdot z_{s}ds+o(||z||),\ z\to0. \] The number $\partial_{c_{t}}h(c)$ is the derivative of $h$ along the value of the path $(c_{s})_{s\in[0,t]}$ at time $t$ (i.e. the derivative along the present value). The function $(\delta_{s}h(c))_{s\in[0,t]},$ represents the integrated influence of the past on the variation of the functional. \begin{lem} Assume that the functional $h\in\mathcal{A}_{t}$ admits a continuous extension to the space $D[0,t]$ of cadlag paths (equipped with the Skorokhod topology). Then the functional $h$ is vertically differentiable at every $c\in C[0,t],$ and the vertical derivative coincides with $\partial_{c_{t}}h(c).$ \end{lem} \begin{proof} Fix $\alpha>0.$ There exists $\delta>0$ such that for any $z\in C[0,t]$
with $||z||\leq\delta,$ \[
\bigg|h(c+z)-h(c)-\partial_{c_{t}}h(c)\cdot z_{t}-\int_{0}^{t}\delta_{s}h(c)\cdot z_{s}ds\bigg|\leq\alpha||z||. \] The function $e_{t}\in D[0,t]$ can be approximated in the Skorokhod topology by continuous functions $z^{(n)},$ such that $0\leq z^{(n)}\leq1,$ $z_{s}^{(n)}=0$ for $s\leq t-\frac{1}{n},$ $z_{t}^{(n)}=1.$ We have for all $\epsilon\leq\delta$ \[
\bigg|h(c+\epsilon z^{(n)})-h(c)-\partial_{c_{t}}h(c)\cdot\epsilon-\epsilon\int_{t-\frac{1}{n}}^{t}\delta_{s}h(c)\cdot z_{s}^{(n)}ds\bigg|\leq\alpha\epsilon||z^{(n)}||=\alpha\epsilon. \] Dividing by $\epsilon$ we get \[
\bigg|\frac{h(c+\epsilon z^{(n)})-h(c)}{\epsilon}-\partial_{c_{t}}h(c)\bigg|\leq\alpha+\int_{t-\frac{1}{n}}^{t}|\delta_{s}h(c)|ds \] Taking $n\to\infty,$ \[
\bigg|\frac{h(c+\epsilon e_{t})-h(c)}{\epsilon}-\partial_{c_{t}}h(c)\bigg|\leq\alpha. \] Since $\alpha>0$ is arbitrary, this proves that the vertical derivative of $h$ exists and is equal to $\partial_{c_{t}}h(c).$ \end{proof}
\subsection{Proof of Claim \ref{claim:Class A is dense}}
We recall that the class $\mathcal{A}_{T}$ consists of functionals $g:C[0,T]\to\mathbb{R}$ such that for all $c,z\in C[0,T]$ \[ g(c+\epsilon z)=g(c)+\epsilon\partial_{c_{t}}g(c)z_{t}+\epsilon\int_{0}^{T}\delta_{s}g(c)z_{s}ds+o(\epsilon),\ \epsilon\to0. \] For any function $c\in C[0,T]$ introduce the transformation \[ (R^{(n)}c)_{t}=e^{-n(T-t)}c_{T}+n\int_{t}^{T}e^{-n(s-t)}c_{s}ds. \] The function $R^{(n)}c$ solves the problem \[ \begin{cases} \frac{dR^{(n)}c}{dt}=-n(c_{t}-(R^{(n)}c)_{t})\\ (R^{(n)}c)_{T}=c_{T} \end{cases} \] Hence, $R^{(n)}c\to c$ uniformly on $[0,T].$ Given any Frechet differentiable functional $g:C[0,T]\to\mathbb{R},$ consider \[ g^{(n)}(c)=g(R^{(n)}c). \] Fix $c\in C[0,T]$ and let the measure $\mu$ be the Frechet derivative of $g$ at $R^{(n)}c.$ \begin{align*} g^{(n)}(c+\epsilon z) & =g(R^{(n)}c+\epsilon R^{(n)}z)=g(R^{(n)})(c)+\epsilon\int_{0}^{T}(R^{(n)}z)_{s}\mu(ds)+o(\epsilon)=\\
& =g^{(n)}(c)+\epsilon\int_{0}^{T}\left(e^{-n(T-s)}z_{T}+n\int_{s}^{T}e^{-n(r-s)}z_{r}dr\right)\mu(ds)+o(\epsilon)=\\
& =g^{(n)}(c)+\epsilon\left(\int_{0}^{T}e^{-n(T-s)}\mu(ds)\right)z_{T}+\epsilon\int_{0}^{T}\left(\int_{0}^{r}e^{-n(r-s)}\mu(ds)\right)z_{r}dr+o(\epsilon) \end{align*} So, $g^{(n)}\in\mathcal{A}_{T}.$ Every differentiable functional is a pointwise limit of functionals from the class $\mathcal{A}_{T}.$
\subsection{Total derivative formula: a general version of Proposition \ref{prop: Total derivative formula}}
Let $\zeta=(\zeta_{t})_{t\in[0,T]}$ be a measurable process. We will say that $\zeta$ is square integrable, if $\mathbb{E}\int_{0}^{T}\zeta_{s}^{2}ds<\infty.$
We say that a measurable stochastic process $\xi=(\xi_{t})_{t\in[0,T]}$ is absolutely continuous, if $\xi_{0}$ is square integrable and there exists a square integrable process $(\zeta_{t})_{t\in[0,T]}$, such that \begin{equation} \xi_{t}=\xi_{0}+\int_{0}^{t}\zeta_{s}ds,\ 0\leq t\leq T.\label{abs_cont_process} \end{equation} Observe that an absolutely continuous process $\xi$ satisfies $\sup_{0\leq t\leq T}\mathbb{E}\xi_{t}^{2}<\infty$ and is a.s. continuous. It is important that an absolutely continuous process need not be adapted. Since the processes under consideration are square integrable and measurable with respect to a Wiener process $w$, their values can be represented as stochastic integrals. Namely, \begin{equation} \xi_{t}=\mathbb{E}\xi_{t}+\int_{0}^{T}g_{t,s}dw_{s},\ \zeta_{t}=\mathbb{E}\zeta_{t}+\int_{0}^{T}h_{t,s}dw_{s},\ 0\leq t\leq T,\label{adapted_derivatives} \end{equation} where $g_{t}=(g_{t,s})_{s\in[0,T]}$ and $h_{t}=(h_{t,s})_{s\in[0,T]}$ are progressively measurable (in $s$) square integrable processes. We need one technical result on the regularity of the process $g_{t}.$ \begin{lem*} For each $t\in[0,T]$ and almost all $s\in[0,T]$ \[ g_{t,s}=g_{0,s}+\int_{0}^{t}h_{r,s}dr\mbox{ a.s.} \] In particular, for almost all $s\in[0,T]$ the process $t\to g_{t,s}$ has a continuous modification. \end{lem*} \begin{proof} Let $u=(u_{s})_{s\in[0,T]}$ be an arbitrary progressively measurable square integrable process. Using Itô's isometry we compute \begin{align*} \mathbb{E}\int_{0}^{T}g_{t,s}u_{s}ds & =\mathbb{E}\int_{0}^{T}g_{t,s}dw_{s}\int_{0}^{T}u_{s}dw_{s}=\mathbb{E}\xi_{t}\int_{0}^{T}u_{s}dw_{s}=\\
& =\mathbb{E}\xi_{0}\int_{0}^{T}u_{s}dw_{s}+\int_{0}^{t}\left(\mathbb{E}\zeta_{r}\int_{0}^{T}u_{s}dw_{s}\right)dr=\\
& =\mathbb{E}\int_{0}^{T}g_{0,s}u_{s}ds+\int_{0}^{t}\left(\mathbb{E}\int_{0}^{T}h_{r,s}u_{s}ds\right)dr=\mathbb{E}\int_{0}^{T}\left(g_{0,s}+\int_{0}^{t}h_{r,s}dr\right)u_{s}ds. \end{align*} Since the latter holds for arbitrary $u$ we deduce that $g_{t}=g_{0}+\int_{0}^{t}h_{r}dr$ as elements of $L^{2}(\Omega\times[0,T]).$ \end{proof} Further we will always deal with a continuous in t modifications of processes $t\to g_{t,s}.$ \begin{prop*}
Let $\eta=(\eta_{t})_{t\in[0,T]}$ be a square integrable progressively measurable process. Then $\eta$ is an Itô process if and only if it can be represented in the form $\eta_{t}=\mathbb{E}[\xi_{t}|\mathcal{F}_{t}]$ for some square integrable absolutely continuous process $\xi=(\xi_{t})_{t\in[0,T]}.$ In this case the semimartingale representation of $\eta$ is given by \[
d\eta_{t}=\mathbb{E}[\zeta_{t}|\mathcal{F}_{t}]dt+g_{t,t}dw_{t}, \] where processes $\zeta$ and $g_{t}$ are determined from \eqref{abs_cont_process}, \eqref{adapted_derivatives}. \end{prop*} \begin{proof} Assume that $\eta$ is an Itô process. Then it can be written in the form \[ \eta_{t}=\eta_{0}+\int_{0}^{t}\alpha_{s}ds+\int_{0}^{t}\beta_{s}dw_{s},\ 0\leq t\leq T. \] Introduce the process \[ \xi_{t}=\eta_{0}+\int_{0}^{T}\beta_{s}dw_{s}+\int_{0}^{t}\alpha_{s}ds. \] Observe that the process $\xi$ is an absolutely continuous process. Since stochastic integrals are martingales, we deduce that \[
\mathbb{E}\left[\xi_{t}|\mathcal{F}_{t}\right]=\eta_{0}+\int_{0}^{t}\alpha_{s}ds+\mathbb{E}\left[\int_{0}^{T}\beta_{s}dw_{s}|\mathcal{F}_{t}\right]=\eta_{0}+\int_{0}^{t}\alpha_{s}ds+\int_{0}^{t}\beta_{s}dw_{s}=\eta_{t}. \]
Conversely, assume that $\eta_{t}=\mathbb{E}[\xi_{t}|\mathcal{F}_{t}]$ with some absolutely continuous process $\xi.$ Related processes $\zeta,g_{t}$ are defined in \eqref{abs_cont_process}, \eqref{adapted_derivatives}. Consider the process \[
M_{t}=\eta_{t}-\eta_{0}-\int_{0}^{t}\mathbb{E}[\zeta_{r}|\mathcal{F}_{r}]dr,\ 0\leq t\leq T. \] We verify that the process $M$ is a martingale. Indeed, for $s<t$ we compute \begin{align*}
\mathbb{E}[M_{t}|\mathcal{F}_{s}] & =\mathbb{E}\left[\eta_{t}-\eta_{0}-\int_{0}^{t}\mathbb{E}[\zeta_{r}|\mathcal{F}_{r}]dr|\mathcal{F}_{s}\right]=\\
& =\mathbb{E}\left[\mathbb{E}[\xi_{t}|\mathcal{F}_{t}]-\eta_{0}-\int_{0}^{t}\mathbb{E}[\zeta_{r}|\mathcal{F}_{r}]dr|\mathcal{F}_{s}\right]=\\
& \mathbb{E}[\xi_{t}|\mathcal{F}_{s}]-\eta_{0}-\int_{s}^{t}\mathbb{E}[\zeta_{r}|\mathcal{F}_{s}]dr-\int_{0}^{s}\mathbb{E}[\zeta_{r}|\mathcal{F}_{r}]dr=\\
& =\mathbb{E}\left[\xi_{s}+\int_{s}^{t}\zeta_{r}dr|\mathcal{F}_{s}\right]-\eta_{0}-\int_{s}^{t}\mathbb{E}[\zeta_{r}|\mathcal{F}_{s}]dr-\int_{0}^{s}\mathbb{E}[\zeta_{r}|\mathcal{F}_{r}]dr=\\
& =\mathbb{E}\left[\xi_{s}|\mathcal{F}_{s}\right]-\eta_{0}-\int_{0}^{s}\mathbb{E}[\zeta_{r}|\mathcal{F}_{r}]dr=\eta_{s}-\eta_{0}-\int_{0}^{s}\mathbb{E}[\zeta_{r}|\mathcal{F}_{r}]dr=M_{s}. \end{align*} As a Wiener martingale, the process $M$ has a representation as a stochastic integral. Since $M_{0}=0$ the representation is of the form: \[ M_{t}=\int_{0}^{t}v_{s}dw_{s} \] for some progressively measurable square integrable process $v=(v_{s})_{s\in[0,T]}.$ To find this process we use Itô's isometry. For arbitrary progressively measurable square integrable process $u=(u_{s})_{s\in[0,T]},$ we have \begin{align*}
& \mathbb{E}\int_{0}^{T}v_{s}u_{s}ds=\mathbb{E}\int_{0}^{T}v_{s}dw_{s}\int_{0}^{T}u_{s}dw_{s}=\mathbb{E}M_{T}\int_{0}^{T}u_{s}dw_{s}=\\
& =\mathbb{E}\left(\xi_{T}-\eta_{0}-\int_{0}^{T}\mathbb{E}[\zeta_{r}|\mathcal{F}_{r}]dr\right)\int_{0}^{T}u_{s}dw_{s}=\mathbb{E}\left(\xi_{T}\int_{0}^{T}u_{s}dw_{s}\right)-\int_{0}^{T}\mathbb{E}\left(\zeta_{r}\int_{0}^{r}u_{s}dw_{s}\right)dr=\\
& =\mathbb{E}\int_{0}^{T}g_{T,s}u_{s}ds-\int_{0}^{T}\mathbb{E}\int_{0}^{r}h_{r,s}u_{s}dsdr=\mathbb{E}\int_{0}^{T}\left(g_{T,s}-\int_{s}^{T}h_{r,s}dr\right)u_{s}ds. \end{align*} From the previous Lemma we deduce \[ v_{s}=g_{T,s}-\int_{s}^{T}h_{r,s}dr=g_{0,s}+\int_{0}^{s}h_{r,s}dr=g_{s,s}. \] Finally, equality \[
\eta_{t}=\eta_{0}+\int_{0}^{t}\mathbb{E}[\zeta_{s}|\mathcal{F}_{s}]ds+M_{t}=\eta_{0}+\int_{0}^{t}\mathbb{E}[\zeta_{s}|\mathcal{F}_{s}]ds+\int_{0}^{t}g_{s.s}dw_{s} \] implies the needed semimartingale representation of $\eta.$ \end{proof} \begin{rem*} For almost all $t\in[0,T]$ the process $\zeta_{t}$ is the time-derivative of the process $\xi_{t}.$ If we assume that $\xi_{t}$ is Malliavin differentiable, then the Clark-Ocone formula implies \[
g_{t,t}=\mathbb{E}\left[D_{t}\xi_{t}|\mathcal{F}_{t}\right]. \] In these terms the result of the theorem can be written as \[
d\mathbb{E}\left[\xi_{t}|\mathcal{F}_{t}\right]=\mathbb{E}\left[\partial_{t}\xi_{t}|\mathcal{F}_{t}\right]dt+\mathbb{E}\left[D_{t}\xi_{t}|\mathcal{F}_{t}\right]dw_{t}, \] which is the motivation behind the term ``total derivative formula''. \end{rem*}
\subsection{\label{subsec:Characterization-of-It=0000F4}Characterization of Itô processes: converse of Proposition \ref{prop: Total derivative formula}}
The total derivative formula states that for any square integrable absolutely continuous process $(\eta_{t})_{t\in[0,T]},$ the process
$\xi_{t}=E[\eta_{t}|\mathcal{F}_{t}]$ is an Itô process. In this section we give the converse statement. \begin{claim*} For any square integrable Itô process $(\xi_{t})_{t\in[0,T]}$ there exists an absolutely continuous process $(\eta_{t})_{t\in[0,T]}$ such that \[
\xi_{t}=E[\eta_{t}|\mathcal{F}_{t}]. \] \end{claim*} \begin{proof} By the definition of the Itô process, there exist adapted square integrable processes $(\alpha_{t})_{t\in[0,T]}$ and $(\beta_{t})_{t\in[0,T]}$ such that \[ \xi_{t}=\xi_{0}+\int_{0}^{t}\alpha_{s}ds+\int_{0}^{t}\beta_{s}dw_{s}. \] We simply define \[ \eta_{t}=\xi_{0}+\int_{0}^{t}\alpha_{s}ds+\int_{0}^{T}\beta_{s}dw_{s}. \] The process $\eta$ is absolutely continuous, and \[
E[\eta_{t}|\mathcal{F}_{t}]=\xi_{0}+\int_{0}^{t}\alpha_{s}ds+E\left[\int_{0}^{T}\beta_{s}dw_{s}\bigg|\mathcal{F}_{t}\right]= \] \[ =\xi_{0}+\int_{0}^{t}\alpha_{s}ds+\int_{0}^{t}\beta_{s}dw_{s}=\xi_{t}. \] \end{proof} \begin{comment} Consider the perturbation $h_{t}(c)=g(c_{t},\int_{0}^{t}c_{s}ds),$where $g:\mathbb{R}^{2}\to\mathbb{R}$ is a smooth function. This is our main example of interest as it captures three important parts of history-dependent policies. First, there is an effect of the present which is represented by the first argument, $c_{t}$. Second, there is an effect of the past which is represented by the second argument, $\int_{0}^{t}c_{s}ds$. Here, the past enters as the integral of the path of the previous consumptions. One can easily modify this part to be, for example, some more complicated path-dependent object -- for example, $\int_{0}^{t}\alpha\left(c_{\left[0,s\right]}\right)ds$, where $\alpha$ is a functional of the path. Third, there is a joint dependence $g$$\left(.,.\right)$ between the present and the past. One can easily modify this part as well, by having higher order interactions via the repeated integrals $h_{t}(c)=g\bigg(c_{t},\int_{0}^{t}c_{s}ds,\int_{0}^{t}\int_{0}^{t}c_{s}c_{s'}dsds',...\bigg)$ to capture even richer interaction between periods $s$ and $s'$, and so on. We now calculate the required derivatives.
We start with the Fréchet derivative of $h_{t}$ that appears in the first order conditions of the perturbed problem (\ref{eq: FOC, perturbed}). It is given by varying the whole path $c_{[0,t]}$ by a variation $z_{[0,t]}$: \[ h_{t}(c+z)-h_{t}(c)=g\bigg(c_{t}+z_{t},\int_{0}^{t}c_{s}ds+\int_{0}^{t}z_{s}ds\bigg)-g\bigg(c_{t},\int_{0}^{t}c_{s}ds\bigg)= \] \[
=g'_{1}\bigg(c_{t},\int_{0}^{t}c_{s}ds\bigg)z_{t}+g'_{2}\bigg(c_{t},\int_{0}^{t}c_{s}ds\bigg)\int_{0}^{t}z_{s}ds+o(||z||)= \]
\[
=g'_{1}\bigg(c_{t},\int_{0}^{t}c_{s}ds\bigg)z_{t}+\int_{0}^{t}g'_{2}\bigg(c_{t},\int_{0}^{t}c_{r}dr\bigg)z_{s}ds+o(||z||). \] Now, bringing \[ \partial_{c_{t}}h_{t}(c)=g'_{1}\bigg(c_{t},\int_{0}^{t}c_{s}ds\bigg),\delta_{s}h_{t}(c)=g'_{2}\bigg(c_{t},\int_{0}^{t}c_{r}dr\bigg),0\leq s\leq t. \]
That is, the present effect of the path is given by the first derivative of $g$, and all of the past effects in any past period period $s$ enter identically as the second derivative of $g$.\footnote{One can easily see how the calculation would be straightforwardly modified if instead of the integral $\int_{0}^{t}c_{r}dr$ (where all of the past terms enter identically) we used a more complicated weighting of the past, say, $\int_{0}^{t}f\left(c_{r},r\right)dr$.}
We now turn to calculation of the derivatives needed for the application of the Dupire's functional Itô's formula (\ref{eq: FOP, Dupire part}). These derivatives are used in determining the decomposition of the present effects of the path $\partial_{c_{t}}h_{t}(c)$ in the first order conditions of the perturbed problem (\ref{eq: FOC, perturbed}). Recall that the horizontal and two vertical derivatives are needed.
The time-derivative (horizontal derivative) of the present effects of the path $\partial_{c_{t}}h_{t}(c)$ is given by freezing the path at time $t$ and extending it with time: \[ \Delta_{t}\partial_{c_{t}}h_{t}(c)=\lim_{\epsilon\to0}\frac{\partial_{c_{t+\epsilon}}h_{t+\epsilon}(c_{\cdot,\epsilon})-\partial_{c_{t}}h_{t}(c)}{\epsilon}= \] \[ =\lim_{\epsilon\to0}\frac{1}{\epsilon}\bigg(g'_{1}\bigg(c_{t},\int_{0}^{t}c_{s}ds+\epsilon c_{t}\bigg)-g'_{1}\bigg(c_{t},\int_{0}^{t}c_{s}ds\bigg)\bigg)=g''_{12}\bigg(c_{t},\int_{0}^{t}c_{s}ds\bigg)c_{t}. \] That is, the time derivative of the present effects of the path $\partial_{c_{t}}h_{t}(c)$ (which we found above to be equal to $g'_{1}(.,.)$) is driven by the increment of the integral $\int_{0}^{t}c_{s}ds$ frozen at time $t$ which is $c_{t}$ multiplied by the cross-partial $g''_{12}$ \[ \Delta_{t}\partial_{c_{t}}h_{t}(c)=g''_{12}\bigg(c_{t},\int_{0}^{t}c_{s}ds\bigg)c_{t}. \] Vertical differentiation of the present effects of the path $\partial_{c_{t}}h_{t}(c)$ is given by bumping the path. It reduces to the differentiation in the first argument: \[ \partial_{c_{t}}^{2}h_{t}(c)=g''_{11}\bigg(c_{t},\int_{0}^{t}c_{s}ds\bigg),\partial_{c_{t}}^{3}h_{t}(c)=g'''_{111}\bigg(c_{t},\int_{0}^{t}c_{s}ds\bigg). \]
Note that since we are already finding the derivatives of the marginal tax $\partial_{c_{t}}$, the mixed time-derivative and the second and third order space derivatives appear (rather than just the time and first and second order derivatives).
We next calculate the derivatives needed for the application of the total derivative formula (\ref{eq: FOP, total derivative part}). These derivatives are needed to provide the decomposition of the expected cumulative marginal taxes in the first order conditions of the perturbed problem (\ref{eq: FOC, perturbed}) -- that is, for determining the evolution of the future effects of the path. Consider each individual term -- an effect of the path of consumption up to time $t$ {[}GEORGII: IS THIS CORRECT WORDING?{]}on the tax in period $s$, $\delta_{t}h_{s}(c)$.
The time derivative of this term is simple. Since $\delta_{t}h_{s}(c)=g'_{2}(c_{s},\int_{0}^{s}c_{r}dr)$ does not depend on $t,$ we get \[ \frac{\partial}{\partial t}\delta_{t}h_{s}(c)=0, \]
so the time evolution of the future effects is then equal \[ \frac{d}{dt}\int_{t}^{1}\delta_{t}h_{s}(c^{*})ds=-\delta_{t}h_{t}(c^{*})=-\delta_{t}h_{t}(c^{*})=-g'_{2}(c_{t}^{*},\int_{0}^{t}c_{r}^{*}dr) \]
{[}GEORGII: CHANGED THE SUBSCRIPTS. NEED TO BE CHECKED{]}The second part of the total derivative formula is determination of the Malliavin derivative as in Section \ref{subsec:Calculating-Malliavin-derivative} and Lemma \ref{lem:Malliavin}. The Fréchet derivative of $\delta_{t}h_{s}$ (which we found above to be equal to $g'_{2}(.,.)$) is given by varying the whole path $c_{[0,s]}$ by a variation $z_{[0,s]}$ \[ \delta_{t}h_{s}(c+z)-\delta_{t}h_{s}(c)=g'_{2}\bigg(c_{s}+z_{s},\int_{0}^{s}c_{r}dr+\int_{0}^{s}z_{r}dr\bigg)-g'_{2}\bigg(c_{s},\int_{0}^{s}c_{r}dr\bigg)= \] \[
=g''_{21}\bigg(c_{s},\int_{0}^{s}c_{r}dr\bigg)z_{s}+\int_{0}^{s}g''_{22}\bigg(c_{s},\int_{0}^{s}c_{u}du\bigg)z_{r}dr+o(||z||). \] This gives us the decomposition of the evolution of $\delta_{t}h_{s}$ in terms of the present effects (that is, how the path up to period $s$ affects period $s$) \[ \partial_{c_{s}}\{\delta_{t}h_{s}(c)\}=g''_{21}\bigg(c_{s},\int_{0}^{s}c_{r}dr\bigg), \]
and the future effects (that is, how the path up to period $s$ affects period $r$), for each $r>s$
\[ \delta_{s}\{\delta_{t}h_{r}(c)\}=g''_{22}\bigg(c_{r},\int_{0}^{r}c_{u}du\bigg). \]
The Malliavin derivative in Lemma \ref{lem:Malliavin} is then given by \[ D_{t}\int_{t}^{1}\left\{ \delta_{t}h_{s}(c^{*})\right\} ds=\int_{t}^{1}\left(\partial_{c_{s}}\left\{ \delta_{t}h_{s}(c^{*})\right\} +\int_{s}^{1}\delta_{s}\left\{ \delta_{t}h_{r}(c^{*})\right\} dr\right)\times D_{t}c_{s}^{*}ds= \] \[ =\int_{t}^{1}\left(g''_{21}\bigg(c_{s}^{*},\int_{0}^{s}c_{u}^{*}du\bigg)+\int_{s}^{1}g''_{22}\bigg(c_{r}^{*},\int_{0}^{r}c_{u}^{*}du\bigg)\right)\times D_{t}c_{s}^{*}ds. \]
The only thing that remains is to find the relevant conditional expectations \[
\mathbb{E}\bigg[\int_{t}^{1}\delta_{t}h_{s}(c^{*})ds\bigg|\mathcal{F}_{t}\bigg]=\mathbb{E}\bigg[\int_{t}^{1}g'_{2}\bigg(c_{s}^{*},\int_{0}^{s}c_{u}^{*}du\bigg)ds\bigg|\mathcal{F}_{t}\bigg] \] and find the evolution of the expected cumulative marginal taxes that the total derivative formula gives \[
d\mathbb{E}\bigg[\int_{t}^{1}\delta_{t}h_{s}(c^{*})ds\bigg|\mathcal{F}_{t}\bigg]=-g'_{2}\bigg(c_{t}^{*},\int_{0}^{t}c_{u}^{*}du\bigg)dt+ \] \begin{equation}
+\bigg(\int_{t}^{1}\mathbb{E}\bigg[g''_{21}\bigg(c_{s}^{*},\int_{0}^{s}c_{u}^{*}ds\bigg)D_{t}c_{s}^{*}|\mathcal{F}_{t}\bigg]ds+\int_{t}^{1}\int_{s}^{1}\mathbb{E}\bigg[g''_{22}\bigg(c_{r}^{*},\int_{0}^{s}c_{u}^{*}du\bigg)D_{t}c_{s}^{*}|\mathcal{F}_{t}\bigg]dsdr\bigg)dw_{t}.\label{eq: Detemple-Zapatero example} \end{equation}
In the Appendix, we show how one can obtain the equation \eqref{eq: Detemple-Zapatero example} using an alternative method and the limitations of such method.
Finally, bringing the terms of the present and the past effects of the path together we get the semimartingale decomposition of the stochastic elasticity: {\footnotesize{} \[ d(u''(c_{t}^{*})C_{t}^{h})= \] \[ =\left(g''_{12}\bigg(c_{t}^{*},\int_{0}^{t}c_{s}^{*}ds\bigg)c_{t}^{*}+g''_{11}\bigg(c_{t}^{*},\int_{0}^{t}c_{s}^{*}ds\bigg)b^{*}(c_{t}^{*})+\frac{1}{2}g'''_{111}\bigg(c_{t}^{*},\int_{0}^{t}c_{s}^{*}ds\bigg)(\sigma^{*}(c_{t}^{*}))^{2}-g'_{2}\bigg(c_{t}^{*},\int_{0}^{t}c_{s}^{*}ds\bigg)\right)dt+ \] \[
+\bigg(g''_{11}\bigg(c_{t}^{*},\int_{0}^{t}c_{s}^{*}ds\bigg)\sigma^{*}(c_{t}^{*})+\int_{t}^{1}\mathbb{E}\bigg[g''_{21}\bigg(c_{s}^{*},\int_{0}^{s}c_{u}^{*}ds\bigg)D_{t}c_{s}^{*}|\mathcal{F}_{t}\bigg]ds+\int_{t}^{1}\int_{s}^{1}\mathbb{E}\bigg[g''_{22}\bigg(c_{r}^{*},\int_{0}^{s}c_{u}^{*}du\bigg)D_{t}c_{s}^{*}|\mathcal{F}_{t}\bigg]dsdr\bigg)dw_{t}. \] } \end{comment} \begin{comment}
\subsection{Proof of Lemma \ref{lem:Malliavin}}
We are first interested in finding the Malliavin derivative at some time $u$ of the marginal tax with respect to $c_{t}$ of the functional $h_{s}$: $D_{u}\delta_{t}h_{s}(c^{*})$.
Consider a perturbation of the Wiener process $w$ by the function $t\to\int_{0}^{t}z_{s}ds:$ \[ w_{t}^{z}=w_{t}+\int_{0}^{t}z_{s}ds. \] Then \[ \delta_{t}h_{s}(c^{*}(w^{z}))-\delta_{t}h_{s}(c^{*}(w))= \]
is comprised of the two terms applied to $\delta_{t}h_{s}$: the instantaneous effect of the present, $\partial_{c_{s}}$, and the sum of effects of the path, $\delta_{r}$, for $r\in\left[0,s\right]$
\[
=\partial_{c_{s}}\delta_{t}h_{s}(c^{*}(w))(c_{s}^{*}(w^{z})-c_{s}^{*}(w))+\int_{0}^{s}\delta_{r}\delta_{t}h_{s}(c^{*}(w))(c_{r}^{*}(w^{z})-c_{r}^{*}(w))dr+o(||z||)= \]
using the definition of the Malliavin derivative $c_{i}^{*}(w^{z})-c_{i}^{*}(w)=\int_{0}^{i}\left(D_{u}c_{i}^{*}\right)z_{u}du$
\[
=\partial_{c_{s}}\delta_{t}h_{s}(c^{*})\int_{0}^{s}\left(D_{u}c_{s}^{*}\right)z_{u}du+\int_{0}^{s}\delta_{r}\delta_{t}h_{s}(c^{*})\int_{0}^{r}\left(D_{u}c_{r}^{*}\right)z_{u}dudr+o(||z||)= \]
changing the order of integration in the second term and collecting the terms containing $z_{u}$
\[
=\int_{0}^{s}\bigg(\left(\partial_{c_{s}}\delta_{t}h_{s}(c^{*})\right)\times\left(D_{u}c_{s}^{*}\right)+\int_{u}^{s}\left(\delta_{r}\delta_{t}h_{s}(c^{*})\right)\times\left(D_{u}c_{r}^{*}\right)dr\bigg)z_{u}du+o(||z||). \]
We get:
\[ D_{u}\delta_{t}h_{s}(c^{*})=\left(\partial_{c_{s}}\delta_{t}h_{s}(c^{*})\right)\times D_{u}c_{s}^{*}+\int_{u}^{s}\left(\delta_{r}\delta_{t}h_{s}(c^{*})\right)\times D_{u}c_{r}^{*}dr. \]
That is, Malliavin derivative at time $u$ (which can be thought of as an effect of the change in the process at time $u$) of the marginal tax $\delta_{t}h_{s}(c^{*})$ is given by the changes in the value at the present, $\partial_{c_{s}}$, and the sum of the effects of the path, $\delta_{r}$, from time $u$ to $s$. Each of this effects are in term multiplied by the Malliavin derivative at time $u$ of the consumption $c_{s}^{*}$ at time $s$.
We now sum these derivatives to account for the cumulative changes in the future: \[ D_{t}\int_{t}^{1}\delta_{t}h_{s}(c^{*})ds=\int_{t}^{1}\bigg(\partial_{c_{s}}\delta_{t}h_{s}(c^{*})D_{t}c_{s}^{*}+\int_{t}^{s}\delta_{r}\delta_{t}h_{s}(c^{*})D_{t}c_{r}^{*}dr\bigg)ds= \] \[ =\int_{t}^{1}\partial_{c_{s}}\delta_{t}h_{r}(c^{*})D_{t}c_{r}^{*}dr+\int_{t}^{1}\int_{r}^{1}\delta_{r}\delta_{t}h_{s}(c^{*})dsD_{t}c_{r}^{*}dr. \]
The conditional expectation of the effects of the future marginal taxes is given by: \[
d\mathbb{E}\bigg[\int_{t}^{1}\delta_{t}h_{s}(c^{*})ds\bigg|\mathcal{F}_{t}\bigg]=\bigg(-\delta_{t}h_{t}(c^{*})+\int_{t}^{1}\mathbb{E}\bigg[\frac{\partial}{\partial t}\delta_{t}h_{s}(c^{*})\bigg|\mathcal{F}_{t}\bigg]ds\bigg)dt+ \] \[
+\bigg(\int_{t}^{1}\mathbb{E}[\partial_{c_{r}}\delta_{t}h_{r}(c^{*})D_{t}c_{r}^{*}|\mathcal{F}_{t}]dr+\int_{t}^{1}\int_{r}^{1}\mathbb{E}[\delta_{r}\delta_{t}h_{s}(c^{*})D_{t}c_{r}^{*}|\mathcal{F}_{t}]dsdr\bigg)dw_{t}. \] \end{comment} \begin{comment}
\subsection{Calculation of the coefficients of the first order process in \eqref{eq:FOP, with beta and gamma}}
By the Itô's formula, \[ d\frac{1}{u''(c_{t}^{*})}=-\frac{u'''(c_{t}^{*})}{(u''(c_{t}^{*}))^{2}}dc_{t}^{*}+\frac{2(u'''(c_{t}^{*}))^{2}-u''''(c_{t}^{*})u''(c_{t}^{*})}{(u''(c_{t}^{*}))^{3}}(\sigma^{*}(c_{t}^{*}))^{2}dt= \] \[ =\bigg(\frac{2(u'''(c_{t}^{*}))^{2}-u''''(c_{t}^{*})u''(c_{t}^{*})}{(u''(c_{t}^{*}))^{3}}(\sigma^{*}(c_{t}^{*}))^{2}-\frac{u'''(c_{t}^{*})}{(u''(c_{t}^{*}))^{2}}b^{*}(c_{t}^{*})\bigg)dt+ \] \[ +\bigg(-\frac{u'''(c_{t}^{*})}{(u''(c_{t}^{*}))^{2}}\sigma^{*}(c_{t}^{*})\bigg)dw_{t} \] The differential of the right-hand side in \eqref{eq: First order process} is \[ \bigg(\Delta_{t}\partial_{c_{t}}h_{t}(c^{*})+\partial_{c_{t}}^{2}h_{t}(c^{*})b^{*}(c_{t}^{*})+\frac{1}{2}\partial_{c_{t}}^{3}h_{t}(c^{*})(\sigma^{*}(c_{t}^{*}))^{2}\bigg)dt+\partial_{c_{t}}^{2}h_{t}(c^{*})\sigma^{*}(c_{t}^{*})dw_{t}+ \] \[
+\bigg(-\delta_{t}h_{t}(c^{*})+\int_{t}^{1}\mathbb{E}\bigg[\frac{\partial}{\partial t}\delta_{t}h_{s}(c^{*})\bigg|\mathcal{F}_{t}\bigg]ds\bigg)dt+ \] \[
+\bigg(\int_{t}^{1}\mathbb{E}[\partial_{c_{r}}\delta_{t}h_{r}(c^{*})D_{t}c_{r}^{*}|\mathcal{F}_{t}]dr+\int_{t}^{1}\int_{r}^{1}\mathbb{E}[\delta_{r}\delta_{t}h_{s}(c^{*})D_{t}c_{r}^{*}|\mathcal{F}_{t}]dsdr\bigg)dw_{t}. \] It follows that \[
dC_{t}^{h}=\beta_{t}dt+\gamma_{t}dw_{t}=d\bigg(\frac{1}{u''(c_{t}^{*})}\bigg(\partial_{c_{t}}h_{t}(c^{*})+\int_{t}^{1}\mathbb{E}[\delta_{t}h_{s}(c^{*})|\mathcal{F}_{t}]ds\bigg)\bigg)= \] \[ =\frac{1}{u''(c_{t}^{*})}\bigg(\bigg(\Delta_{t}\partial_{c_{t}}h_{t}(c^{*})+\partial_{c_{t}}^{2}h_{t}(c^{*})b^{*}(c_{t}^{*})+\frac{1}{2}\partial_{c_{t}}^{3}h_{t}(c^{*})(\sigma^{*}(c_{t}^{*}))^{2}\bigg)dt+\partial_{c_{t}}^{2}h_{t}(c^{*})\sigma^{*}(c_{t}^{*})dw_{t}+ \] \[
+\bigg(-\delta_{t}h_{t}(c^{*})+\int_{t}^{1}\mathbb{E}\bigg[\frac{\partial}{\partial t}\delta_{t}h_{s}(c^{*})\bigg|\mathcal{F}_{t}\bigg]ds\bigg)dt+ \] \[
+\bigg(\int_{t}^{1}\mathbb{E}[\partial_{c_{r}}\delta_{t}h_{r}(c^{*})D_{t}c_{r}^{*}|\mathcal{F}_{t}]dr+\int_{t}^{1}\int_{r}^{1}\mathbb{E}[\delta_{r}\delta_{t}h_{s}(c^{*})D_{t}c_{r}^{*}|\mathcal{F}_{t}]dsdr\bigg)dw_{t}\bigg)+ \] \[
+\bigg(\partial_{c_{t}}h_{t}(c^{*})+\int_{t}^{1}\mathbb{E}[\delta_{t}h_{s}(c^{*})|\mathcal{F}_{t}]ds\bigg)\bigg(\bigg(\frac{2(u'''(c_{t}^{*}))^{2}-u''''(c_{t}^{*})u''(c_{t}^{*})}{(u''(c_{t}^{*}))^{3}}(\sigma^{*}(c_{t}^{*}))^{2}- \] \[ -\frac{u'''(c_{t}^{*})}{(u''(c_{t}^{*}))^{2}}b^{*}(c_{t}^{*})\bigg)dt+\bigg(-\frac{u'''(c_{t}^{*})}{(u''(c_{t}^{*}))^{2}}\sigma^{*}(c_{t}^{*})\bigg)dw_{t}\bigg)- \] \[
-\bigg(\partial_{c_{t}}^{2}h_{t}(c^{*})\sigma^{*}(c_{t}^{*})+\int_{t}^{1}\mathbb{E}[\partial_{c_{r}}\delta_{t}h_{r}(c^{*})D_{t}c_{r}^{*}|\mathcal{F}_{t}]dr+ \] \[
+\int_{t}^{1}\int_{r}^{1}\mathbb{E}[\delta_{r}\delta_{t}h_{s}(c^{*})D_{t}c_{r}^{*}|\mathcal{F}_{t}]dsdr\bigg)\frac{u'''(c_{t}^{*})}{(u''(c_{t}^{*}))^{2}}\sigma^{*}(c_{t}^{*})dt \] Finally, \[
\beta_{t}=\frac{1}{u''(c_{t}^{*})}\bigg(\Delta_{t}\partial_{c_{t}}h_{t}(c^{*})+\partial_{c_{t}}^{2}h_{t}(c^{*})b^{*}(c_{t}^{*})+\frac{1}{2}\partial_{c_{t}}^{3}h_{t}(c^{*})(\sigma^{*}(c_{t}^{*}))^{2}-\delta_{t}h_{t}(c^{*})+\int_{t}^{1}\mathbb{E}[\frac{\partial}{\partial t}\delta_{t}h_{s}(c^{*})|\mathcal{F}_{t}]ds\bigg)+ \] \[
+\bigg(\partial_{c_{t}}h_{t}(c^{*})+\int_{t}^{1}\mathbb{E}[\delta_{t}h_{s}(c^{*})|\mathcal{F}_{t}]ds\bigg)\bigg(\frac{2(u'''(c_{t}^{*}))^{2}-u''''(c_{t}^{*})u''(c_{t}^{*})}{(u''(c_{t}^{*}))^{3}}(\sigma^{*}(c_{t}^{*}))^{2}-\frac{u'''(c_{t}^{*})}{(u''(c_{t}^{*}))^{2}}b^{*}(c_{t}^{*})\bigg)- \] \[
-\frac{u'''(c_{t}^{*})\sigma^{*}(c_{t}^{*})}{(u''(c_{t}^{*}))^{2}}\bigg(\partial_{c_{t}}^{2}h_{t}(c^{*})\sigma^{*}(c_{t}^{*})+\int_{t}^{1}\mathbb{E}[\partial_{c_{r}}\delta_{t}h_{r}(c^{*})D_{t}c_{r}^{*}|\mathcal{F}_{t}]dr+\int_{t}^{1}\int_{r}^{1}\mathbb{E}[\delta_{r}\delta_{t}h_{s}(c^{*})D_{t}c_{r}^{*}|\mathcal{F}_{t}]dsdr\bigg) \] and \[
\gamma_{t}=\frac{1}{u''(c_{t}^{*})}\bigg(\partial_{c_{t}}^{2}h_{t}(c^{*})\sigma^{*}(c_{t}^{*})+\int_{t}^{1}\mathbb{E}[\partial_{c_{r}}\delta_{t}h_{r}(c^{*})D_{t}c_{r}^{*}|\mathcal{F}_{t}]dr+\int_{t}^{1}\int_{r}^{1}\mathbb{E}[\delta_{r}\delta_{t}h_{s}(c^{*})D_{t}c_{r}^{*}|\mathcal{F}_{t}]dsdr\bigg)- \] \[
-\frac{u'''(c_{t}^{*})\sigma^{*}(c_{t}^{*})}{(u''(c_{t}^{*}))^{2}}\bigg(\partial_{c_{t}}h_{t}(c^{*})+\int_{t}^{1}\mathbb{E}[\delta_{t}h_{s}(c^{*})|\mathcal{F}_{t}]ds\bigg) \] \end{comment} \begin{comment}
\subsection{An alternative derivation of \eqref{eq: Detemple-Zapatero example}}
We now show how to recover equation \eqref{eq: Detemple-Zapatero example} using the approach of Detemple and Zapatero (1991).
Represent the conditional expectation as \[
\mathbb{E}\bigg[\int_{t}^{1}g'_{2}\bigg(c_{s}^{*},\int_{0}^{s}c_{u}^{*}du\bigg)ds\bigg|\mathcal{F}_{t}\bigg]=\mathbb{E}\bigg[\int_{0}^{1}g'_{2}\bigg(c_{s}^{*},\int_{0}^{s}c_{u}^{*}du\bigg)ds\bigg|\mathcal{F}_{t}\bigg]- \] \[ -\int_{0}^{t}g'_{2}\bigg(c_{s}^{*},\int_{0}^{s}c_{u}^{*}du\bigg)ds. \] The differential of the second expression is $-g'_{2}(c_{t}^{*},\int_{0}^{t}c_{u}^{*}du)dt.$
The first summand is the martingale. By the Clark-Ocone formula \[
\int_{0}^{1}g'_{2}\bigg(c_{s}^{*},\int_{0}^{s}c_{u}^{*}du\bigg)ds=\mathbb{E}\bigg[\int_{0}^{1}g'_{2}\bigg(c_{s}^{*},\int_{0}^{s}c_{u}^{*}du\bigg)ds\bigg]+\int_{0}^{1}\mathbb{E}\left[D_{r}\int_{0}^{1}g'_{2}\bigg(c_{s}^{*},\int_{0}^{s}c_{u}^{*}du\bigg)ds|F_{r}\right]dw_{r}. \] Then \[
\mathbb{E}\bigg[\int_{0}^{1}g'_{2}\bigg(c_{s}^{*},\int_{0}^{s}c_{u}^{*}du\bigg)ds\bigg|\mathcal{F}_{t}\bigg]s=\mathbb{E}\bigg[\int_{0}^{1}g'_{2}\bigg(c_{s}^{*},\int_{0}^{s}c_{u}^{*}du\bigg)ds\bigg]+\int_{0}^{t}\mathbb{E}\left[D_{r}\int_{0}^{1}g'_{2}\bigg(c_{s}^{*},\int_{0}^{s}c_{u}^{*}du\bigg)ds|F_{r}\right]dw_{r}. \]
The differential is then given by
\[
d\mathbb{E}\bigg[\int_{0}^{1}g'_{2}\bigg(c_{s}^{*},\int_{0}^{s}c_{u}^{*}du\bigg)ds\bigg|\mathcal{F}_{t}\bigg]=\int_{0}^{1}\mathbb{E}\bigg[D_{t}\bigg(g'_{2}\bigg(c_{s}^{*},\int_{0}^{s}c_{u}^{*}du\bigg)\bigg)\bigg|\mathcal{F}_{t}\bigg]dsdw_{t}= \] \[
=\int_{0}^{1}\mathbb{E}\bigg[g''_{12}\bigg(c_{s}^{*},\int_{0}^{s}c_{u}^{*}du\bigg)D_{t}c_{s}^{*}+g''_{22}\bigg(c_{s}^{*},\int_{0}^{s}c_{u}^{*}du\bigg)\int_{0}^{s}D_{t}c_{u}^{*}du\bigg|\mathcal{F}_{t}\bigg]dsdw_{t}= \] \[
=\int_{t}^{1}\mathbb{E}\bigg[g''_{12}\bigg(c_{s}^{*},\int_{0}^{s}c_{u}^{*}du\bigg)D_{t}c_{s}^{*}\bigg|\mathcal{F}_{t}\bigg]dsdw_{t}+\int_{t}^{1}\int_{r}^{1}\mathbb{E}\bigg[g''_{22}\bigg(c_{s}^{*},\int_{0}^{s}c_{u}^{*}du\bigg)D_{t}c_{r}^{*}\bigg|\mathcal{F}_{t}\bigg]dsdrdw_{t}. \] \end{comment}
{}
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\begin{document}
\title{On the Global Solution of Soft $k$-Means}
\author{Feiping~Nie,~\IEEEmembership{Senior Member, IEEE}, Hong~Chen, Rong Wang, and~Xuelong~Li,~\IEEEmembership{Fellow,~IEEE} \thanks{Feiping Nie and Hong Chen are with the School of Computer Science, School of Artificial Intelligence, Optics and Electronics (iOPEN), and the Key Laboratory of Intelligent Interaction and Applications (Ministry of Industry and Information Technology), Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China. (e-mail: [email protected]; [email protected]). (\textit{Corresponding author: Feiping Nie}.)
Rong Wang and Xuelong Li are with the School of Artificial Intelligence, Optics and Electronics (iOPEN), and the Key Laboratory of Intelligent Interaction and Applications (Ministry of Industry and Information Technology), Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China. (e-mail: [email protected]; [email protected]).}}
\markboth{} {Shell \MakeLowercase{\textit{et al.}}: On the Global Solution of Soft $k$-Means}
\maketitle
\begin{abstract} This paper presents an algorithm to solve the Soft $k$-Means problem globally. Unlike Fuzzy $c$-Means, Soft $k$-Means (S$k$M) has a matrix factorization-type objective and has been shown to have a close relation with the popular probability decomposition-type clustering methods, e.g., Left Stochastic Clustering (LSC). Though some work has been done for solving the Soft $k$-Means problem, they usually use an alternating minimization scheme or the projected gradient descent method, which cannot guarantee global optimality since the non-convexity of S$k$M. In this paper, we present a sufficient condition for a feasible solution of Soft $k$-Means problem to be globally optimal and show the output of the proposed algorithm satisfies it. Moreover, for the Soft $k$-Means problem, we provide interesting discussions on stability, solutions non-uniqueness, and connection with LSC. Then, a new model, named Minimal Volume Soft $k$-Means (MVS$k$M), is proposed to address the solutions non-uniqueness issue. Finally, experimental results support our theoretical results. \end{abstract}
\begin{IEEEkeywords} Global solution of Soft $k$-Means, sufficient condition, stability analysis, optimal non-uniqueness. \end{IEEEkeywords}
\IEEEpeerreviewmaketitle
\section{Introduction} \IEEEPARstart{K}{means} \cite{lloyd1982least}, arguably most commonly used techniques for data analysis \cite{wu2008top}, produces a clustering such that the sum of squared error between samples and the mean of their cluster is minimized. For $n$ feature vectors gathered as the $d$-dimensional columns of a matrix $\mathbf{X}\in\mathbb{R}^{d\times n}$, the $k$-Means problem can be written as a matrix factorization problem \cite{bauckhage2015k}: \[
\min_{\mathbf{F},\mathbf{G}\in\{0,1\}^{n,k},\mathbf{G}\mathbb{1}_k = \mathbb{1}_n} \| \mathbf{X} - \mathbf{FG}^\top\|_F^2, \] where $\mathbf{X}\in\mathbb{R}^{d\times n}$ is the data matrix, $\mathbf{F}\in\mathbb{R}^{d\times k}$ is the prototypes, and $\mathbf{G}\in\mathbb{R}^{k\times n}$ is the membership indicator matrix. Several clustering methods have been proposed by relaxing this matrix factorization $k$-Means problem in different aspects. For example, \cite{zha2002spectral,ding2005equivalence} relaxed the structural constraints on indicator matrix $\mathbf{G}$ to an orthogonality constraint; \cite{ding2010convex} changed the constraints on $\mathbf{G}$ in the $k$-Means problem to only require that $\mathbf{G}$ be positive.
Sometimes, dividing data into distinct clusters is too strict, where each data point can only belong to exactly one cluster. In fuzzy clustering, data points can potentially belong to multiple clusters. The probably best known approach to fuzzy clustering is the method of Fuzzy $c$-Means (FCM) proposed by \cite{dunn1973fuzzy}. The objective function of FCM is \[
\min_{\mathbf{F},\mathbf{G}\succeq 0,\mathbf{G}\mathbb{1}_k = \mathbb{1}_n} \sum_{i=1}^n \sum_{j=1}^k g_{ij}^m \| \mathbf{x}_i - \mathbf{f}_j\|_2^2. \] In FCM, membership functions are defined based on a distance function, and membership degrees express proximities of entities to cluster centers (i.e., prototypes $\mathbf{F}$). By choosing a suitable distance function different cluster shapes can be identified. Another approach to fuzzy clustering due to \cite{krishnapuram1993possibilistic} is the possibilistic $c$-Means (PCM) algorithm which eliminates one of the constraints imposed on the search for $c$ partitions leading to possibilistic absolute fuzzy membership values instead of FCM probabilistic relative fuzzy memberships.
Another potentially interesting relaxation of the matrix factorization form $k$-Means is to relax the $\mathbf{G}\in\{0,1\}^{k\times n}$ to $\mathbf{G}\in\mathbb{R}^{k\times n}$. This type of relaxation is popular in submodular optimization and is closely related to the Lov\'{a}sz extension \cite{bach2013learning}. The objective function we concerned is \begin{equation}\label{eq:SfM}
\min_{\mathbf{F},\mathbf{G}\succeq 0,\mathbf{G}\mathbb{1}_k = \mathbb{1}_n} \| \mathbf{X} - \mathbf{FG}^\top\|_F^2, \end{equation} which is the problem we want to solve in this paper and has been termed Soft $k$-Means (S$k$M) in \cite{arora2013similarity}.
Another line of research that is related to Problem \eqref{eq:SfM} is the fuzzy clustering with proportional membership (FCPM) scheme \cite{nascimento2003modeling} in the literature.
The idea of FCPM is to develop explicit mechanisms for data generation from cluster structures, since such a model can provide a theoretical framework for cluster structures found in data. Indeed, the Soft $k$-Means problem is exactly equivalent to the FCPM problem when the fuzzy factor in FCPM is set to zero.
Yet some work \cite{nascimento2000fuzzy,nascimento2003modeling,nascimento2016applying} has been done for solving the Soft $k$-Means problem in Problem \eqref{eq:SfM}, they usually use an alternating minimization scheme or the projected gradient descent method, which cannot guarantee global optimality since the non-convexity of S$k$M.
In this paper, we present a sufficient condition for a feasible solution of Soft $k$-Means problem to be globally optimal and show the output of the proposed algorithm satisfies it. Moreover, for the Soft $k$-Means problem, we provide interesting discussions on stability, solutions non-uniqueness, and connection with LSC. Then, a new model, named Minimal Volume Soft $k$-Means (MVS$k$M), is proposed to address the solutions non-uniqueness issue. Finally, experimental results support our theoretical results.
\textbf{Contributions.} The main contributions of this paper are summarized as follows: \begin{itemize}
\item We report sufficient condition under which $\mathbf{F}$ and $\mathbf{G}$ are a global solution of the Soft $k$-Means problem. As a byproduct, we provide sufficient and necessary conditions for a data matrix to be S$k$Mable.
\item We propose a simple algorithm with computational complexity $\mathcal{O}(nd^2)$. The output of this algorithm satisfies the aforementioned sufficient condition, thus it is a global solution of the Soft $k$-Means problem with theoretical guarantee.
\item Using the sufficient condition, we provide interesting discussions on the stability, optimal non-uniqueness, and connection with Left Stochastic Clustering proposed in \cite{arora2013similarity}. Moreover, we proposed an model, named Minimal Volume Soft $k$-Means to address the optimal non-uniqueness problem.
\end{itemize}
\textbf{Organization.} The rest of this paper is organized as follows. We review some closely related prior work in \Cref{sec:prior}. Notations description and the proposed algorithm to globally solve Soft $k$-Means are put in \Cref{sec:algorithm}. The main theory which gives the sufficient optimality condition is given in \Cref{sec:theory} along with its proof sketch. Several interesting discussions on stability, solutions non-uniqueness, and connection with LSC in \Cref{sec:discuss}. Experimental results on both synthetic dataset and real-world datasets are provided in \Cref{sec:exps}. Finally, conclude the paper in \Cref{sec:conclusion}.
\section{Prior Work}\label{sec:prior}
In this section, we review methods which are closely related to the Soft $k$-Means problem.
\subsection{NMF and $k$-Means} This line of research work includes Convex-NMF, Symmetric-NMF, etc. A note on matrix factorization representation is in \cite{bauckhage2015k}. \cite{zha2002spectral} proposed relaxing the constraints on $\mathbf{G}$ in the $k$-Means optimization problem to an orthogonality constraint: \[
\min_{\mathbf{F},\mathbf{G}\succeq 0,\mathbf{G}^\top\mathbf{G} = \mathbb{I}_{k}} \| \mathbf{X} - \mathbf{FG}^\top\|_F^2. \] \cite{ding2005equivalence} considered the kernelized clustering objective: \[
\min_{\mathbf{G}\succeq 0,\mathbf{G}^\top\mathbf{G} = \mathbb{I}_{k}} \| \mathbf{K} - \mathbf{GG}^\top\|_F^2. \] Then, \cite{ding2010convex} considered changing the constraints on $\mathbf{G}$ in the $k$-Means problem to only require that $\mathbf{G}$ be positive. They explored a number of approximations to the $k$-Means problem that imposed different constraints on $\mathbf{F}$. One such variant that they deemed particularly worthy of further investigation was convex NMF. Convex NMF restricts the columns of $\mathbf{F}$ (the cluster centroids) to be convex combinations of the columns of $\mathbf{X}$: \[ \min_{\mathbf{
W}\succeq 0, \mathbf{G} \succeq 0} \| \mathbf{X} - \mathbf{XWG}^\top\|_F^2. \] \subsection{Fuzzy Proportional Membership Clustering} The fuzzy clustering with proportional membership (FCPM) \cite{nascimento2000fuzzy,nascimento2003modeling,nascimento2016applying}. scheme considers explicit mechanisms for data generation from cluster structures. The objective for general FCPM$_m$ problem is \[ \min_{\mathbf{F}, \mathbf{G}\succeq 0,\mathbf{G}\mathbb{1}_k = \mathbb{1}_n}
\sum_{i=1}^n\sum_{j=1}^k g_{ij}^m \| \mathbf{x}_i - \mathbf{f}_j g_{ij} \|_2^2. \] It is notable that when for $m=0$, FCPM$_0$ is equivalent to Soft $k$-Means problem in Problem \eqref{eq:SfM}. Thus, the algorithm for solving Problem \eqref{eq:SfM} can be used to solve FCPM$_0$.
\subsection{Left Stochastic Clustering} Left Stochastic Clustering (LSC) proposed by \cite{arora2013similarity} provide directly left stochastic similarity matrix factorization. They perform clustering by solving a non-negative matrix factorization problem for the best cluster assignment matrix $\mathbf{G}$. That is, given a matrix $\mathbf{K}$, clustering by finding a scaling factor $c$ and a cluster probability matrix $\mathbf{G}$ that best solve \begin{equation}\label{eq:LSD}
\min_{c\geq 0, \mathbf{G}\succeq 0,\mathbf{G}\mathbb{1}_k = \mathbb{1}_n} \left\| \mathbf{K} - \frac{1}{c}\mathbf{GG}^\top \right\|_F^2. \end{equation} It has been shown that the Problem \eqref{eq:LSD} is closely related to Soft $k$-Means (Theorem \ref{thm:jmlr13prop}). Specifically, if the Gram of given matrix is LSDable then this matrix is S$k$Mable (see details in \Cref{sec:leftsto}).
\section{Notations and Algorithm} \label{sec:algorithm} In this section, we introduce the notations used in the paper and report the proposed algorithm for solving the soft $k$-Means problem. Some remarks just follows. \subsection{Notations}
Throughout this paper, scalars, vectors and matrices are denoted by lowercase letters, boldface lowercase letters and boldface uppercase letters, respectively; for a matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$, $\mathbf{A}^\top$ denotes the transpose of $\mathbf{A}$, $\text{Tr}(\mathbf{A}) = \sum_{i=1}^n a_{ii}$, $\| \mathbf{A} \|_F^2 = \text{Trace}(\mathbf{A}^\top\mathbf{A})$; $\mathbb{1}_n\in\mathbb{R}^n$ denotes vector with all ones; $\|\mathbf{x}\|_0$ denotes the number of non-zero elements; $\|\mathbf{A}\|_{p,q} = \left( \sum_{i=1}^n \|\mathbf{a}\|_p^q \right)^{1/q}$; $\mathbb{I}_n \in \mathbb{R}^{n\times n}$ denotes the identity matrix; $\mathbf{H}_n = \mathbb{I}_{n} - \frac{1}{n}\mathbb{1}_n\mathbb{1}_n^\top$ is the centralization matrix; $\mathbf{A}\succeq 0$ indicates all elements of $\mathbf{A}$ are non-negative; diag$(\mathbf{A})$ is the diagonal elements of $\mathbf{A}$; diag$(\mathbf{v})$ is a matrix with diagonal elements $\mathbf{v}$.
\subsection{Algorithm} In this subsection, we provide an algorithm to solve the soft $k$-Means Problem \eqref{eq:SfM}. In \Cref{sec:mainTheory}, we will show that the output prototype $\mathbf{F}$ and membership $\mathbf{G}$ of Algorithm \ref{alg:global} is a global solution of Problem \eqref{eq:SfM}. \begin{algorithm}[H]
\caption{Solve the Soft $k$-Means Problem \eqref{eq:SfM}}
\label{alg:global}
\begin{algorithmic}[1]
\Require $\mathbf{X}\in\mathbb{R}^{d\times n}$ and the clusters number $k$
\State $\overline{\mathbf{x}} \leftarrow \frac{1}{n}\mathbf{X}\mathbb{1}_n$
\State $\mathbf{U}_{k-1}\leftarrow$ the $(k-1)$-truncated left SVs of $\mathbf{X}$
\State $\mathbf{B} \leftarrow$ the $(k-1)$-truncated EVs of $\mathbf{H}_k$
\State Ensure $r \geq \|\mathbf{U}_{k-1}^\top\mathbf{X} \|_{2,\infty}$
\State $\mathbf{F}\leftarrow r\sqrt{k(k-1)}\mathbf{U}_{k-1}\mathbf{B}^\top + \overline{\mathbf{x}}\mathbb{1}_k^\top$
\State $\mathbf{G}^\top \leftarrow \frac{1}{r\sqrt{k(k-1)}}\mathbf{BU}_{k-1}^\top \mathbf{X} + \frac{1}{k}\mathbb{1}_k\mathbb{1}_n^\top$
\Ensure Prototypes $\mathbf{F} \in \mathbb{R}^{d\times k}$, membership $\mathbf{G} \in \mathbb{R}^{k \times n}$
\label{code:recentEnd}
\end{algorithmic} \end{algorithm}
\newtheorem{remark}{\bf Remark} \begin{remark} Some remarks just follows.
\begin{itemize}
\item The computational complexity of Algorithm \ref{alg:global} is $\mathcal{O}(nkd)$, which is linear with respect to $n$,
which shows that our algorithm is suitable for large-scale data sets.
\item It is notable that, unlike existing method to solve Problem \eqref{eq:SfM}, the newly proposed Algorithm \ref{alg:global} is non-iterative and it is easily implementable with few lines of basic linear algebra manipulations.
While its simpleness, in \Cref{sec:mainRes}, we will show the output of Algorithm \ref{alg:global} is a globally solution of Problem \eqref{eq:SfM}.
\item We should point out that the output of Algorithm \ref{alg:global} is not unique.
The reason is that the $(k-1)$-truncated EVs of $\mathbf{H}_k$, denoted by $\mathbf{B}$, are not unique since the leading $(k-1)$ eigenvalues of $\mathbf{H}_k$ are equal. But the output $\mathbf{F}$ and $\mathbf{G}$ are still optimal, which reveals the solution non-uniqueness issue of the Soft $k$-Means problem.
This issue would be mentioned again in \Cref{remark:proof_sketch_unique} and clearly explained and discussed in \Cref{sec:arbitray}.
\end{itemize}
\end{remark}
\section{Main Theory} \label{sec:mainTheory} \label{sec:theory} In this section, we present our main results on the sufficient condition for $\mathbf{F}$ and $\mathbf{G}$ to globally solve the Problem \eqref{eq:SfM}. Then, we show the output returned by Algorithm \ref{alg:global} satisfies the condition. Finally, the proof sketch for the main result is provided in \Cref{sec:proofSketch}.
\subsection{Our Results}\label{sec:mainRes} \newtheorem{thm}{\bf Theorem} \begin{thm}\label{thm:mainRes}
Let the centralized data $\overline{\mathbf{X}} \in\mathbb{R}^{d\times n}$ be $\mathbf{X}\mathbf{H}_n$.
Let the $(k-1)$-truncated SVD of the centralized data matrix be $\mathbf{U}_{k-1}\mathbf{\Sigma}_{k-1}\mathbf{V}_{k-1}^\top$.
If there exists $\mathbf{S}\in\mathbb{R}^{(k-1)\times k}$ and $\mathbf{G}\in\mathbb{R}^{k\times n}, \mathbf{G}\succeq 0, \mathbf{G}\mathbb{1}_k = \mathbb{1}_n$ satisfy
\[
\mathbf{\Sigma}_{k-1}\mathbf{V}_{k-1}^\top = \mathbf{S}\mathbf{G}^\top.
\]
Then, $\mathbf{F}=\mathbf{U}_{k-1}\mathbf{S} + \frac{1}{n}\mathbf{X}\mathbb{1}_n\mathbb{1}_k^\top$ and $\mathbf{G}$ is a solution of Problem \eqref{eq:SfM}. \end{thm}
\begin{remark}
In Theorem \ref{thm:mainRes}, we report a sufficient condition for $\mathbf{F}$ and $\mathbf{G}$ to be a solution of Problem \eqref{eq:SfM}.
In detail, we require that there exists $\mathbf{S}\in\mathbb{R}^{(k-1)\times k}$ and $\mathbf{G}$ satisfy (a) $\mathbf{\Sigma}_{k-1}\mathbf{V}_{k-1}^\top = \mathbf{S}\mathbf{G}^\top$; (b) $\mathbf{G}\mathbb{1}_k = \mathbb{1}_n$; (c) $\mathbf{G} \succeq 0$.
In the following corollary, we justify the existence of such $\mathbf{F}$ and $\mathbf{G}$ by showing the Algorithm \ref{alg:global} produces
a pair of $\mathbf{F}$ and $\mathbf{G}$ satisfying the conditions in Theorem \ref{thm:mainRes}. \end{remark}
\newtheorem{corollary}{\bf Corollary} \begin{corollary}\label{thm:mainCorollary}
Let $\mathbf{F}^*, \mathbf{G}^*$ be the output of Algorithm \ref{alg:global}. We have
\[
\mathbf{F}^*, \mathbf{G}^* \in \mathop{\arg\min}_{\mathbf{F},\mathbf{G}\succeq 0,\mathbf{G}\mathbb{1}_k = \mathbb{1}_n} \| \mathbf{X} - \mathbf{FG}^\top\|_F^2.
\] \end{corollary}
The Corollary \ref{thm:mainCorollary} justifies the merit of the proposed Algorithm \ref{alg:global}, thus we prove it here. Before the proof of this main corollary, we need following numerical estimation.
\newtheorem{Lemma1}{\bf Lemma} \begin{Lemma1}\label{lem:numerical}
Let $\mathbf{x} \in \mathbb{R}^k$.
If $\mathbb{1}_k^\top\mathbf{x}=0$ and $\|\mathbf{x}\|_2 \leq 1$, then we have $\|\mathbf{x}\|_\infty \leq \frac{\sqrt{k(k-1)}}{k}.$ \end{Lemma1}
\begin{figure*}
\caption{Flowchart of the proof of Theorem \ref{thm:mainRes}.}
\end{figure*}
\begin{proof} of Corollary \ref{thm:mainCorollary}.
We prove by verifying the three conditions in Theorem \ref{thm:mainRes}: (a) $\mathbf{\Sigma}_{k-1}\mathbf{V}_{k-1}^\top=\mathbf{S}\mathbf{G}^\top$;
(b) $\mathbf{G}\mathbb{1}_k = \mathbb{1}_n$;
(c) $\mathbf{G}\succeq 0$.
For (a), using the definition of $\mathbf{F}$, we can see $\mathbf{S} = r\sqrt{k(k-1)}\mathbf{B}^\top$.
Thus,
\[
\begin{aligned}
\mathbf{SG}^\top =& r\sqrt{k(k-1)}\mathbf{B}^\top \left(\frac{\mathbf{B}\mathbf{U}_{k-1}^\top\mathbf{X}}{r\sqrt{k(k-1)}}+\frac{1}{k}\mathbb{1}_k\mathbb{1}_n^\top\right) \\
=& \mathbf{B}^\top\mathbf{B}\mathbf{U}_{k-1}^\top\mathbf{X}+\frac{r\sqrt{k(k-1)}}{k}\mathbf{B}^\top\mathbb{1}_k\mathbb{1}_n^\top \\
=& \mathbf{U}_{k-1}^\top\mathbf{X} = \mathbf{\Sigma}_{k-1}\mathbf{V}_{k-1}^\top,
\end{aligned}
\]
where the last equality holds since $\mathbf{B}^\top\mathbf{B}=\mathbb{I}_{(k-1)}$ and $\mathbb{1}_k^\top\mathbf{B}=0$ since $\mathbf{H}_k \mathbb{1}_k=0$.
For (b), a direct calculation gives
\[
\mathbf{G}\mathbb{1}_k = \frac{\mathbf{X}^\top \mathbf{U}_{k-1} \mathbf{B}^\top \mathbb{1}_k}{r\sqrt{k(k-1)}}+\frac{1}{k}\mathbb{1}_n\mathbb{1}_k^\top\mathbb{1}_k = \mathbb{1}_n,
\]
where we use the fact $\mathbf{G}\mathbb{1}_k = \mathbb{1}_n$.
For (c), let $\mathbf{w}_i$ be the $i$th column of $\mathbf{\Sigma}_{k-1}\mathbf{V}_{k-1}^\top$.
Note that $r \geq \| \mathbf{\Sigma}_{k-1}\mathbf{V}_{k-1}^\top \|_{2,\infty}$ by definition.
Thus $\left\| \frac{\mathbf{B} \mathbf{w}_i}{r}\right\| = \left\| \frac{\mathbf{w}_i }{r} \right\| \leq 1$.
Meanwhile, we note that $\mathbb{1}_k^\top \mathbf{B}\mathbf{w}_i = 0$.
Let $\mathbf{x} \in \mathbb{R}^k$ be $\frac{\mathbf{B}\mathbf{w}_i}{r}$.
We have $\mathbf{x}^\top\mathbb{1}_k = 0$ and $\|\mathbf{x}\|_2\leq 1$.
Using Lemma \ref{lem:numerical}, we conclude that $\left\|\frac{\mathbf{B}\mathbf{w}_i}{r}\right\|_\infty \leq \frac{\sqrt{k(k-1)}}{k}$.
Then we have $\frac{\mathbf{B}\mathbf{w}_i}{r\sqrt{k(k-1)}} + \frac{1}{k}\mathbb{1}_k \succeq 0$,
which directly indicates
$\mathbf{G}^\top = \frac{\mathbf{B}\mathbf{\Sigma}_{k-1}\mathbf{V}_{k-1}^\top}{r\sqrt{k(k-1)}} + \frac{1}{k}\mathbb{1}_k\mathbb{1}_n^\top = \frac{\mathbf{B}\mathbf{W}}{r\sqrt{k(k-1)}} + \frac{1}{k}\mathbb{1}_k\mathbb{1}_n^\top \succeq 0$.
This completes the proof by Theorem \ref{thm:mainRes}. \end{proof}
\begin{remark}\label{remark:proof_sketch_unique}
Corollary \ref{thm:mainCorollary} shows that the output $\mathbf{F}$ and $\mathbf{G}$ by Algorithm \ref{alg:global}
are an optimal solution of Problem \eqref{eq:SfM}. But we note that the solution of Problem \eqref{eq:SfM}
is not unique since for the matrix $\mathbf{B}\in\mathbb{R}^{k\times (k-1)}$ defined in Algorithm \ref{alg:global} as
the $(k-1)$-truncated eigenvectors of centralization matrix $\mathbf{H}_k$ is not unique. Indeed, for any $\mathbf{B}$,
matrix $\mathbf{BR}$ is still $(k-1)$-truncated eigenvectors of centralization matrix $\mathbf{H}_k$, where $\mathbf{R}^\top\mathbf{R}=\mathbb{I}_{(k-1)}$.
That reveals there is non-uniqueness in the solution set of the Problem \eqref{eq:SfM}.
Indeed, we develop a lower bound (\Cref{sec:arbitray}) on the distance between oracle $\mathbf{G}^*$ and the solution of Problem \eqref{eq:SfM} in Frobenius norm. \end{remark}
\begin{remark}\label{remark:intuition}
Here we provide some intuition behind the Algorithm \ref{alg:global}.
For the Soft $k$-Means problem in \eqref{eq:SfM}, we want to find a best fit $k$ simplex constructed by prototype $\mathbf{F}$.
If we have identified the optimal affine span of $\mathbf{F}$, then we can simply enlarge the simplex without change the
objective function value. Thus, we choose the smallest regular $k$ simplex which contain the sphere with $\|\mathbf{X}\|_{2,\infty}$ as
radius. Such a choices gives the $\mathbf{F}$ in Algorithm \ref{alg:global}. \end{remark}
It is interesting to consider the situation when we can get the objective function minimized to zero. Formally, we make a notion of S$k$Mable for data $\mathbf{X}$.
\newenvironment{definition}[1]{\par\noindent\textbf{Definition 1} \it}{} \begin{definition}((S$k$Mable).
We say $\mathbf{X} \in \mathbb{R}^{d\times n}$ is S$k$Mable if
$
\exists \mathbf{F}\in\mathbb{R}^{d\times k}, \mathbf{G}\in\mathbb{R}^{n\times k}:\mathbf{G}\mathbb{1}_k=\mathbb{1}_n
$
such that
$
\mathbf{X}=\mathbf{FG}^\top.
$ \end{definition}
S$k$Mable is closely related to the notion LSDable introduced in \cite{arora2013similarity}. Indeed, if a matrix is LSDable, then it is S$k$Mable (Theorem \ref{thm:jmlr13prop}). We will discuss the relation between them and introduce a more reasonable and more understandable notion TI-LSDable in \Cref{sec:leftsto}.
Following result gives the necessary and sufficient condition for a data matrix $\mathbf{X}$ to be S$k$Mable.
\begin{thm}[Condition of S$k$Mable]\label{thm:sfmable}
For $\mathbf{X}\in\mathbb{R}^{d\times n}$, there exists $\mathbf{F}\in\mathbb{R}^{d\times k}$ and $\mathbf{G}\in\mathbb{R}^{k\times d}, \mathbf{G}\succeq 0,\mathbf{G}\mathbb{1}_k = \mathbb{1}_n$ such that $\mathbf{X}=\mathbf{FG}^\top$ if and only if rank$(\mathbf{X}\mathbf{H}_n) \leq k-1$. \end{thm} The condition shown above is importance since it connects the abstract definition S$k$Kable to an easily checkable condition rank$(\mathbf{XH}_n) \leq k-1$, which will show its usefulness when we analyze the connection between S$k$Mable and LSDable in \Cref{sec:leftsto}.
\subsection{Proof Sketch} \label{sec:proofSketch} In this subsection, we lay out the proof sketch of Theorem \ref{thm:mainRes}.
The strategy is that we first ignore the nonnegative constraint on $\mathbf{G}$ to identify the optimal affine span of $\mathbf{F}$. Then, we enforce the nonnegative constraint to hold by enlarge $\mathbf{F}$ in its affine span. Specifically, we consider following less constrained problem: \begin{equation}\label{eq:lessConstrained}
\min_{\mathbf{F},\mathbf{G}\mathbb{1}_k = \mathbb{1}_n} \| \mathbf{X} - \mathbf{FG}^\top\|_F^2. \end{equation} Note that there is a translation invariance property for above objective function with the constraint $\mathbf{G}\mathbb{1}_k=\mathbb{1}_n$, that is, for any $\mathbf{s} \in \mathbb{R}^d$, we have \[
\|\mathbf{X} - \mathbf{F}\mathbf{G}^\top\|_F^2 = \|\mathbf{X} - \mathbf{s}\mathbb{1}_n^\top - (\mathbf{F}-\mathbf{s}\mathbb{1}_k^\top)\mathbf{G}^\top\|_F^2. \] This property inspire us to build a reformulation of Problem \eqref{eq:lessConstrained} by finding a proper translation $\mathbf{s}$ which makes $\text{rank}(\mathbf{F}) \leq k-1$. In detail, we have following lemma: \begin{Lemma1}\label{eq:transLemma}
Following equivalence can be shown using the translation invariance of Problem \eqref{eq:lessConstrained}:
\[
\min_{\mathbf{F},\mathbf{G}\mathbb{1}_k = \mathbb{1}_n} \| \mathbf{X} - \mathbf{FG}^\top\|_F^2
\Leftrightarrow
\min_{\substack{\mathbf{F},\mathbf{G}\mathbb{1}_k = \mathbb{1}_n \\ \mathbf{s}, \text{rank}(\mathbf{F}) \leq k-1 }} \| \mathbf{X} - \mathbf{s}\mathbb{1}_n^\top- \mathbf{FG}^\top\|_F^2.
\] \end{Lemma1} Thus, we are now interested in solving \begin{equation}\label{eq:rankProblem}
\min_{\substack{\mathbf{F},\mathbf{G}\mathbb{1}_k = \mathbb{1}_n \\ \mathbf{s}, \text{rank}(\mathbf{F}) \leq k-1 }} \| \mathbf{X} - \mathbf{s}\mathbb{1}_n^\top- \mathbf{FG}^\top\|_F^2. \end{equation} The rank deficiency constraint of $\mathbf{F}$ gives us an opportunity to optimize $\mathbf{G}$ first. Specifically, we have following lemma: \begin{Lemma1}\label{lem:optimizedG}
Let $\mathbf{F}\in\mathbb{R}^{d\times k}$ be the prototype matrix satisfying the constraint $\text{rank}(\mathbf{F}) \leq k-1$ and has the thin SVD $\mathbf{F}=\mathbf{U}\mathbf{\Sigma}\mathbf{V}^\top$, where $\mathbf{\Sigma}\in\mathbb{R}^{(k-1)\times(k-1)}$. Let $ \mathbf{\Phi} = \mathbf{V}\mathbf{\Sigma}^{-1}\mathbf{U}^\top \left(\mathbf{X}-\mathbf{s}\mathbb{1}_n^\top\right)$. Define
\[
\mathbf{G}_*^\top = \mathbf{\Phi}
+
\frac{\mathbf{v}_\bot\left(\mathbb{1}_n^\top - \mathbb{1}_k^\top \mathbf{\Phi} \right) }{\mathbb{1}_k^\top\mathbf{v}_\bot },
\]
where $\mathbf{v}_\bot \in \mathbb{R}^k$ satisfies $\|\mathbf{v}_\bot\|_2 = 1$ and $\mathbf{V}^\top \mathbf{v}_\bot = 0$.
Then
\[
\mathbf{G}_* \in \mathop{\arg\min}_{\mathbf{G}\mathbb{1}_k = \mathbb{1}_n} \| \mathbf{X} - \mathbf{s}\mathbb{1}_n^\top- \mathbf{FG}^\top\|_F^2.
\] \end{Lemma1} Plug the $\mathbf{G}_*$ in Lemma \ref{lem:optimizedG} into the objective function of Problem \eqref{eq:rankProblem} and use the easily checkable fact $\mathbf{F}\mathbf{\Phi} = \mathbf{U}\mathbf{U}^\top \left(\mathbf{X}-\mathbf{s}\mathbb{1}_n^\top\right), \mathbf{F}\mathbf{v}_\bot = 0$. We have \[ \begin{aligned}
&\left\| \mathbf{X} - \mathbf{s}\mathbb{1}_n^\top- \mathbf{FG}^\top_*\right\|_F^2 \\
=& \left\| \mathbf{X} - \mathbf{s}\mathbb{1}_n^\top- \mathbf{F}\left(\mathbf{\Phi} +
\frac{\mathbf{v}_\bot\left(\mathbb{1}_n^\top - \mathbb{1}_k^\top \mathbf{\Phi} \right) }{\mathbb{1}_k^\top \mathbf{v}_\bot}\right) \right\|_F^2 \\
=& \left\| \mathbf{X} - \mathbf{s}\mathbb{1}_n^\top- \mathbf{U}\mathbf{U}^\top \left(\mathbf{X}-\mathbf{s}\mathbb{1}_n^\top\right) \right\|_F^2. \end{aligned} \] Then, the Problem \eqref{eq:rankProblem} can be reformulated as \begin{equation}\label{eq:pcaProblem}
\min_{\mathbf{s},\mathbf{U}^\top\mathbf{U}=\mathbb{I}_{(k-1)}} \left\| \left(\mathbb{I}_{d} - \mathbf{U}\mathbf{U}^\top\right)\left( \mathbf{X} - \mathbf{s}\mathbb{1}_n^\top \right) \right\|_F^2, \end{equation} where we omit the variables ($\mathbf{\Sigma}$ and $\mathbf{V}$) since they cannot effect the objective function. Interestingly, the Problem \eqref{eq:pcaProblem} is the objective of the classical principal components analysis in the regression form. Thus the optimal $\mathbf{s}_* = \frac{1}{n}\mathbf{X}\mathbb{1}_n$ and $\mathbf{U}_*$ is the leading $(k-1)$ left singular vectors of matrix $\left(\mathbf{X} - \mathbf{s}_*\mathbb{1}_n^\top\right) = \mathbf{X}\mathbf{H}_n$. Now we have shown $\mathbf{F}=\mathbf{U}^*\mathbf{\Sigma}\mathbf{V}^\top$ and $\mathbf{G}=\mathbf{V}\mathbf{\Sigma}^{-1}\mathbf{U}^{*\top } + \frac{\mathbf{v}_\bot\left(\mathbb{1}_n^\top - \mathbb{1}_k^\top \mathbf{V}\mathbf{\Sigma}^{-1}\mathbf{U}^{*\top } \right) }{\mathbb{1}_k^\top\mathbf{v}_\bot }$ are a solution for the less constrained Problem \eqref{eq:SfM}. But there are still unused degree of freedom in $\mathbf{\Sigma}$ and $\mathbf{V}$. Next, we will make use of these degree of freedom to enforce the constraint $\mathbf{G}\succeq 0$ satisfied.
For ease of notation, let $\overline{\mathbf{X}}=\mathbf{X} - \frac{1}{n}\mathbf{X}\mathbb{1}_n\mathbb{1}_n^\top = \mathbf{X}-\mathbf{s}^*\mathbb{1}_n^\top$ be centralized $\mathbf{X}$ and $\mathbf{\Sigma V}^\top$ be $\mathbf{S}$. We have \[ \begin{aligned}
&\left\| \mathbf{X} - \mathbf{s}^*\mathbb{1}_n^\top - \mathbf{F}\mathbf{G}^\top \right\|_F^2 \\
=& \left\| \overline{\mathbf{X}} - \mathbf{U}^*\mathbf{S}\mathbf{G}^\top \right\|_F^2 \\
=& \left\| \mathbf{U}^*\mathbf{U}^{*\top} \overline{\mathbf{X}} - \mathbf{U}^*\mathbf{S}\mathbf{G}^\top + \left(\mathbb{I}_{d}-\mathbf{U}^*\mathbf{U}^{*\top}\right) \overline{\mathbf{X}} \right\|_F^2 \\
=& \left\| \mathbf{U}^*\mathbf{U}^{*\top} \overline{\mathbf{X}} - \mathbf{U}^*\mathbf{S}\mathbf{G}^\top \right\|_F^2 + \left\| \left(\mathbb{I}_{d}-\mathbf{U}^*\mathbf{U}^{*\top}\right) \overline{\mathbf{X}} \right\|_F^2, \end{aligned} \] where the last equality uses the easily checkable fact \[ \left\langle \mathbf{U}^*(\mathbf{U}^{*\top} \overline{\mathbf{X}} - \mathbf{S}\mathbf{G}^\top), (\mathbb{I}_{d}-\mathbf{U}^*\mathbf{U}^{*\top}) \overline{\mathbf{X}} \right\rangle = 0. \]
Note that $ \left\| \left(\mathbb{I}_{d}-\mathbf{U}^*\mathbf{U}^{*\top}\right) \overline{\mathbf{X}} \right\|_F^2$ is constant with respect to $\mathbf{S}$ and $\left\| \mathbf{U}^*\mathbf{U}^{*\top} \overline{\mathbf{X}} - \mathbf{U}^*\mathbf{S}\mathbf{G}^\top \right\|_F^2 = \left\|\mathbf{U}^{*\top} \overline{\mathbf{X}} - \mathbf{S}\mathbf{G}^\top \right\|_F^2$. Thus, we can solve $\mathbf{S}$ and $\mathbf{G}$ by solving following problem \begin{equation}\label{eq:add_constraint_G}
\min_{\mathbf{S},\mathbf{G}\mathbb{1}_k = \mathbb{1}_n} \left\|\mathbf{U}_{k-1}^\top \overline{\mathbf{X}} - \mathbf{S}\mathbf{G}^\top \right\|_F^2. \end{equation} Indeed, the objective function in Problem \eqref{eq:add_constraint_G} can be minimized to zero with $\mathbf{G}$ has the form mentioned in Lemma
\ref{lem:optimizedG}. That is the reason why $\mathbf{\Sigma V}^\top$ disappeared in Problem \eqref{eq:pcaProblem}, since in that case $\left\| \mathbf{X} - \mathbf{s}^*\mathbb{1}_n^\top - \mathbf{F}\mathbf{G}^\top \right\|_F^2=\left\| \left(\mathbb{I}_{d}-\mathbf{U}^*\mathbf{U}^{*\top}\right) \overline{\mathbf{X}} \right\|_F^2$.
Now we back to Theorem \ref{thm:mainRes}. Note that if there exists $\mathbf{S}\in\mathbb{R}^{(k-1)\times k}$ and $\mathbf{G}$ satisfy (a) $\mathbf{\Sigma}_{k-1}\mathbf{V}_{k-1}^\top = \mathbf{S}\mathbf{G}^\top$; (b) $\mathbf{G}\mathbb{1}_k = \mathbb{1}_n$; (c) $\mathbf{G} \succeq 0$, then the objective function in Problem \eqref{eq:add_constraint_G} can still be minimized to zero. Combining with $\mathbf{U}^*$ minimizing $\left\| \left(\mathbb{I}_{d}-\mathbf{U}^*\mathbf{U}^{*\top}\right) \overline{\mathbf{X}} \right\|_F^2$, that completes the proof of Theorem \ref{thm:mainRes}.
\section{Discussions} \label{sec:discuss} In this section, we discuss the solution non-uniqueness issue and the stability of the soft $k$-Means modal (\Cref{sec:arbitray,sec:stability}) by providing lower and upper bound. Then a new heuristic model, named Minimal Volume Soft $k$-Means, is proposed in \Cref{sec:MVSkM} with an optimization algorithm, which is guaranteed to be descent, to address the non-uniqueness issue. Finally, we proposed the notion of Translation Invariant LSDable towards solving an open problem mentioned in \cite{arora2013similarity}.
\subsection{Stability}\label{sec:stability} As a byproduct of \Cref{sec:proofSketch}, we can analyze the stability of the soft $k$-Means problem by consider the perturbation in data matrix. With \Cref{sec:arbitray}, we know that bound the optimal variables for perturbed data is hopeless. We provide a stability upper bound for the optimal objective value as follows. \begin{thm}[Stability]\label{thm:stability}
Assume additively perturbed data $\widetilde{\mathbf{X}} = \mathbf{X} + \mathbf{E}$.
We have
\[
\|\mathbf{X} - \widetilde{\mathbf{F}}_*\widetilde{\mathbf{G}}_*^\top \|_F^2 \leq 2\|\mathbf{E}\|_F^2 + \|\mathbf{X} - \mathbf{F}_*\mathbf{G}_*^\top \|_F^2.
\] \end{thm}
\begin{remark}
Theorem \ref{thm:sfmable} shows that the objective function value of Soft $k$-Means is stable and up to a constant factor of the oracle
difference $\|\mathbf{E}\|_F^2$. This is definitely a good news.
But for the variables, that is prototype $\mathbf{F}$ and membership $\mathbf{G}$, we cannot expect such a upper bound.
Indeed, even for unperturbed data, we can prove a lower bound on the distance between a solution of Problem \eqref{thm:mainRes} and the oracle membership $\mathbf{G}^*$. \end{remark}
\subsection{Solutions Non-uniqueness}\label{sec:arbitray}
In this section, we report a bad news on the soft $k$-Means model, that is, there is substantial non-uniqueness in its optima. Formally, we provide following lower bound on the distance between its optima and oracle member matrix $\mathbf{G}^*$ in Frobenius norm. \begin{thm}[Lower bound]
Let the best $(k-1)$-rank approximation of $\mathbf{X}$ be $\mathbf{X}_{k-1}$.
For any oracle $\mathbf{G}^*$, there exists $\mathbf{G}$, which is a solution of Problem \eqref{eq:SfM}, such that
\[
\|\mathbf{G} - \mathbf{G}^*\|_F^2 \geq \frac{1}{r\sqrt{k(k-1)}} \| \mathbf{X}_{k-1}\|_F^2.
\] \end{thm}
\begin{remark}
The solutions non-uniqueness issue is mainly caused since
the volume\footnote{We abuse the notion volume here to refer to the volume relative to the affine subspace.} of the polyhedron of $\mathbf{F}$ can be made infinitely
large to minimize the Problem \eqref{eq:add_constraint_G} in the affine span of $\mathbf{F}$,.
We note that this issue has been realized in \cite{nascimento2003modeling} by arguing
that for any $\alpha > 0$, $\alpha\mathbf{f}_i \cdot \frac{1}{\alpha} g_{ij}$ remain unchanged.
Thus $\|\mathbf{f}_i\|$ can tend to infinity without change the objective value.
In the next subsection, we try to address this issue by jointly minimizing the objective function of Soft $k$-Means
and the volume of the $\mathbf{F}$ polyhedron. \end{remark}
\subsection{The Minimal Volume Soft $k$-Means}\label{sec:MVSkM} In this subsection, we report a new model which minimizes the objective function of soft $k$-Means while keep the volume of polyhedron of prototypes $\mathbf{F}$ minimized. While such a problem has been proved to be NP-hard to solve \cite{zhou2002algorithms}, we propose an algorithm that is guaranteed to make objective function value descent. For simplicity, in this subsection, we assume the data is centered, that is, $\mathbf{X}\mathbb{1}_n = 0$.
Since the volume of a polyhedron with affine dimension less than the dimension ambient is always $0$, we consider the volume in the affine space rather than in the ambient space, that is, in $\text{span}(\mathbf{U})$ rather than $\mathbb{R}^d$.
The volume of polyhedron generated by column vector of $\mathbf{F}$ in subspace $\text{span}(\mathbf{U})$ can be written as \[ \text{vol}_{\text{span}(\mathbf{U})}(\mathbf{F}) = \frac{1}{(k-1)!} \det \left[ \begin{array}{c} \mathbf{\Sigma V}^\top \\ \mathbb{1}_k^\top \end{array} \right] \] Without loss of generality, we minimize it within $\log$ and absorb the $1/(k-1)!$ into a hyper-parameter $\lambda$. The problem we concern is \[
\min_{\mathbf{F},\mathbf{G}\succeq 0,\mathbf{G}\mathbb{1}_k = \mathbb{1}_n} \| \mathbf{X} - \mathbf{FG}^\top\|_F^2 + \lambda \log \det \left[ \begin{array}{c} \mathbf{\Sigma V}^\top \\ \mathbb{1}_k^\top \end{array} \right]. \] To solve above problem, we make following observation \begin{Lemma1}
Assume $\mathbf{F}=\mathbf{U\Sigma V}^\top$ such that $\mathbf{F}\mathbb{1}_k = 0$. Then,
\[
\log \det
\left[ \begin{array}{c}
\mathbf{\Sigma V}^\top \\ \mathbb{1}_k^\top \end{array}
\right] = \log(\sqrt{k}) + \sum_{i=1}^{k-1} \log(\sigma_i(\mathbf{F})).
\] \end{Lemma1} Using $\mathbf{X}\mathbb{1}_n = 0$, we can see from \Cref{sec:proofSketch} that the condition $\mathbf{F}\mathbb{1}_k = 0$ satisfied, which leads to a possible interesting problem \[
\min_{\mathbf{F},\mathbf{G}\succeq 0,\mathbf{G}\mathbb{1}_k = \mathbb{1}_n} \| \mathbf{X} - \mathbf{FG}^\top\|_F^2 + \frac{\lambda}{2} \sum_{i=1}^k \log(\sigma_i^2(\mathbf{F})). \] But sadly, above problem approach $-\infty$ when the singular value of $\mathbf{F}$ tends to zero, which is a trivial solution due to the lower unboundedness of logarithm. Thus, we solve following problem with an $\epsilon > 0$ modification: \begin{equation}\label{eq:mvsfm}
\min_{\mathbf{F},\mathbf{G}\succeq 0,\mathbf{G}\mathbb{1}_k = \mathbb{1}_n} \| \mathbf{X} - \mathbf{FG}^\top\|_F^2 + \frac{\lambda}{2} \sum_{i=1}^k \log(\sigma_i^2(\mathbf{F}) + \varepsilon). \end{equation} Inspired by recent research on multi-task learning \cite{nie2018calibrated}, we design an iteratively re-weighted strategy by performing alternating minimization on $\mathbf{F}$ and $\mathbf{G}$. Specifically, we update $\mathbf{F}_{t+1}$ by solving \[
\mathbf{F}_{t+1} = \mathop{\arg\min}_{F} \| \mathbf{X} - \mathbf{F}\mathbf{G}_t^\top \|^2_F + \lambda \text{Tr}\left(\mathbf{D}_t\mathbf{F}^\top\mathbf{F}\right), \] where $\mathbf{D} = \mathbf{V}_t \text{diag}\left( \{ 1/(\sigma_i^2(\mathbf{F}_t)+\varepsilon) \}_{i=1}^{k} \right)$ and $\mathbf{F}_t = \mathbf{U}_t\mathbf{\Sigma}_t\mathbf{V}^\top$. Above subproblem for $\mathbf{F}$ can be solve easily using first-order optimal condition. Then, we update $\mathbf{G}_{t+1}$, by solve following $n$ problem \[
(\mathbf{g}_i)_t = \mathop{\arg\min}_{\mathbf{g}\mathbb{1}_k = 1, \mathbf{g} \succeq 0} \| \mathbf{x}_i - \mathbf{F}_{t+1}\mathbf{g}^\top \|_2^2, \] which is convex and can be solved with Projected Gradient Descent or even with Nesterov's acceleration.
We summarize the procedure to solve the MVS$k$M model in following algorithm.
\begin{algorithm}[H]
\caption{Solve the MVS$k$M Problem}
\label{alg:MVSfM}
\begin{algorithmic}[1]
\Require $\mathbf{X}\in\mathbb{R}^{d\times n}$, $\lambda$, and the clusters number $k$
\State $\mathbf{F}_0 \leftarrow$ random initialized prototypes.
\Repeat
\State $\mathbf{V}_t \leftarrow$ left SV of $\mathbf{F}_t$
\State $\mathbf{D}_t \leftarrow\mathbf{V}_t \text{diag}(\{ 1/(\sigma_i^2(\mathbf{F}_t) + \varepsilon) \}_{i=1}^k) \mathbf{V}_t^\top$
\State $\mathbf{F}_{t+1} \leftarrow \mathbf{X}\mathbf{G}_t(\mathbf{G}_t^\top\mathbf{G}_t + \lambda \mathbf{D}_t)^{-1}$
\State $\displaystyle \mathbf{G}_{t+1} \leftarrow \mathop{\arg\min}_{\mathbf{G}\succeq 0,\mathbf{G}\mathbb{1}_k = \mathbb{1}_n} \| \mathbf{X} - \mathbf{F}_{t+1}\mathbf{G}^\top\|_F^2.$
\Until{converge}
\Ensure Prototypes $\mathbf{F} \in \mathbb{R}^{d\times c}$, membership $\mathbf{G} \in \mathbb{R}^{c \times n}$.
\end{algorithmic} \end{algorithm}
Here, we provide some analysis for above algorithm. For ease of notation, we shall write the objective function as \[
\mathcal{L}(\mathbf{F}, \mathbf{G})=\| \mathbf{X} - \mathbf{F}\mathbf{G}^\top\|_F^2 + \frac{\lambda}{2} \sum_{i=1}^k \log(\sigma_i^2(\mathbf{F}) + \varepsilon). \] Following result shows that our algorithm to solving MVS$k$M is a descent method. \begin{thm}[Descent]\label{thm:mvsfmDescent}
Algorithm \ref{alg:MVSfM} is a descent method, that is,
\[
\mathcal{L}(\mathbf{F}_{t+1}, \mathbf{G}_{t+1}) \leq \mathcal{L}(\mathbf{F}_{t}, \mathbf{G}_{t}).
\] \end{thm} Since the objective function (after $\varepsilon$ modification) of Problem \eqref{eq:mvsfm} is bounded from below ($\log(\sigma_i^2(\mathbf{F}) + \varepsilon) \geq \log(\varepsilon)$), the iterative scheme will finally converge from Theorem \ref{thm:mvsfmDescent}.
\begin{figure*}
\caption{Comparison of solving the Soft $k$-Means problem with the proposed global optimal algorithm (left), alternating minimization (middle), and the Minimal Volume Soft $k$-Means model on synthetic dataset. The synthetic data set is the \texttt{fcmdata.dat} along with MATLAB.}
\label{fig:toy}
\end{figure*}
\subsection{On Left Stochastic Clustering}\label{sec:leftsto} First, we introduce the notion of LSDable appeared in \cite{arora2013similarity}. \newenvironment{definition2}[1]{\par\noindent\textbf{Definition 2} \it}{} \begin{definition2}[(LSDable)
Similarity matrix $\mathbf{K} \in \mathbb{R}^{n\times n}$ is $k$-LSDable if $\exists c \in \mathbb{R}, \mathbf{G} \in \mathbb{R}^{n\times k}: \mathbf{G}\succeq 0,\mathbf{G}\mathbb{1}_k = \mathbb{1}_n$ such that $\mathbf{K} = \frac{1}{c}\mathbf{GG}^\top$. \end{definition2}
LSDable is closely related to the soft $k$-Means problem considered in this paper by following theorem. \begin{thm}[\cite{arora2013similarity}]\label{thm:jmlr13prop}
Assume $\mathbf{K}=\mathbf{X}^\top\mathbf{X}$ is $k$-LSDable. Then the optimal $\mathbf{G}$ in LSD problem is also an optimum in the soft $k$-Means problem of $\mathbf{X}$. \end{thm}
Above theorem indicates that if $\mathbf{X}^\top\mathbf{X}$ is LSDable then it is S$k$Mable, which means that \{LSDable\} $\subseteq$ \{S$k$Mable\}. Ideally, the clustering property of given $\mathbf{X}$ should not change if we translate all points simultaneously with vector $\mathbf{s}\in\mathbb{R}^d$. But for a LSDable matrix, say $\mathbf{X}^\top\mathbf{X}$, the simultaneous translation, then it is $\left(\mathbf{X}+\mathbf{s}\mathbb{1}_n^\top\right)^\top\left(\mathbf{X}+\mathbf{s}\mathbb{1}_n^\top\right)$, may make it not LSDable. To make the notion of LSDable translation invariant, we extend it to following definition.
\newenvironment{definition3}[1]{\par\noindent\textbf{Definition 3} \it}{} \begin{definition3}((Translation Invariant Set).
Define
\[
\mathbb{K}(\mathbf{X}) = \left\{ \left( \mathbf{X} + \mathbf{t}\mathbb{1}_n^\top \right)^\top \left( \mathbf{X} + \mathbf{t}\mathbb{1}_n^\top \right) : \mathbf{t}\in\mathbb{R}^d \right\}.
\] \end{definition3}
\newenvironment{definition4}[1]{\par\noindent\textbf{Definition 4} \it}{} \begin{definition4}((Translation Invariant LSDable).
We say $\mathbf{K}=\mathbf{X}^\top\mathbf{X}$ is Translation Invariant LSDable (TI-LSDable) if there exists
$
\mathbf{K}^\prime \in \mathbb{K}(\mathbf{X})
$ such that $\mathbf{K}^\prime$ is LSDable. \end{definition4}
Indeed, one can easily see \[ \{\text{LSDable}\} \subseteq \{\text{TI-LSDable}\} \subseteq \{\text{S$k$Mable}\}, \] where the first subset notation holds since that if $\mathbf{K}$ is LSDable, it must be TI-LSDable and the second one holds from the fact that if $\mathbf{X}$ is S$k$Mable then $\mathbf{X}+\mathbf{s}\mathbb{1}_n^\top$ is also S$k$Mable..
The problem we concern in this subsection is \begin{equation}\label{eq:TILSD} \min_{\substack{c,\mathbf{G}\succeq 0, \mathbf{G}\mathbb{1}_k = \mathbb{1}_n \\ \mathbf{K}^\prime \in \mathbb{K}(\mathbf{X}) } }
\| \mathbf{K}^\prime - \frac{1}{c}\mathbf{G}^\top\mathbf{G} \|_F^2 \end{equation} The main result in this subsection is closely related to an open problem proposed in \cite{arora2013similarity}, in which it asked for the relation between LSD of unLSDable $\mathbf{X}^\top\mathbf{X}$ and the solution of Soft $k$-Means for $\mathbf{X}$. In the sequel, we provide such a relation but with TI-LSDable rather than LSDable. \begin{thm}[Condition of TI-LSDable]\label{thm:TILSDable}
$\mathbf{K}$ is TI-LSDable if and only if rank$(\mathbf{H}\mathbf{K}\mathbf{H}) \leq k-1$. \end{thm} Theorem \ref{thm:TILSDable} connects the abstract TI-LSDable notions with an easily checkable condition rank$(\mathbf{H}\mathbf{K}\mathbf{H}) \leq k-1$. Using this connection, we have a simple corollary build relation between the TI-LSDable and S$k$Mable. \begin{corollary}[SfMable is TI-LSDable]
Assume $\mathbf{K}=\mathbf{X}^\top\mathbf{X}$. $\mathbf{K}$ is TI-LSDable if and only if $\mathbf{X}$ is S$k$Mable. \end{corollary} \begin{proof}
Note that $\text{rank}(\mathbf{H}_n\mathbf{X}^\top\mathbf{XH}_n) = \text{rank}(\mathbf{XH}_n)$. For the if direction, since $\mathbf{X}$ is S$k$Mable, by Theorem \ref{thm:sfmable}, $\text{rank}(\mathbf{H}_n\mathbf{X}^\top\mathbf{XH}_n) = \text{rank}(\mathbf{XH}_n)\leq k-1$. For the only if direction, since $\mathbf{K}=\mathbf{X}^\top\mathbf{X}$ is TI-LSDable, by Theorem \ref{thm:TILSDable},
$\text{rank}(\mathbf{XH}_n) = \text{rank}(\mathbf{H}_n\mathbf{X}^\top\mathbf{XH}_n) \leq k-1$ \end{proof} \begin{thm}\label{thm:tilsdEqual}
Assume kernel $\mathbf{K}\in\mathbb{R}^{n\times n}$ satisfies rank$(\mathbf{K}) \leq n-1$.
If $\mathbf{G}^*$ is a solution of the TI-LSD in Problem \eqref{eq:TILSD} for kernel $\mathbf{K}$,
then there exists $\mathbf{X}$ such that $\mathbf{K}=\mathbf{X}^\top\mathbf{X}$ and $\mathbf{G}^*$ is also a solution of S$k$M
in Problem \eqref{eq:SfM}. \end{thm} \begin{remark}
In Theorem \ref{thm:tilsdEqual}, we show that for a given kernel $\mathbf{K}=\mathbf{XX}^\top$ satisfying rank$(\mathbf{K})\leq n-1$,
its the optimal membership matrix for the LSD problem in Problem \eqref{eq:LSD} is still an optimal membership
for the Soft $k$-Means problem on $\mathbf{X},$
which gives an answer for the relation between TI-LSD of unLSDable matrix $\mathbf{X}^\top\mathbf{X}$
and the solution of soft $k$-Means for $\mathbf{X}$. \end{remark}
\begin{table*}[]
\centering
\caption{Comparison of S$k$M-Global, S$k$M-AM, and MVS$k$M on Real-world Datasets.}
\label{tab:exps}
\begin{tabular}{@{}cccccccccc@{}}
\toprule
& \multicolumn{3}{c}{MSRCv1} & \multicolumn{3}{c}{ORL Face} & \multicolumn{3}{c}{Numerical Numbers} \\ \cmidrule(l){2-10}
& \multicolumn{1}{c|}{ACC} & \multicolumn{1}{c|}{NMI} & \multicolumn{1}{c|}{Purity} & \multicolumn{1}{c|}{ACC} & \multicolumn{1}{c|}{NMI} & \multicolumn{1}{c|}{Purity} & \multicolumn{1}{c|}{ACC} & \multicolumn{1}{c|}{NMI} & Purity \\ \midrule
\multicolumn{1}{c|}{SkM-AM} & \multicolumn{1}{c|}{0.5714} & \multicolumn{1}{c|}{0.4429} & \multicolumn{1}{c|}{0.5952} & \multicolumn{1}{c|}{0.4100} & \multicolumn{1}{c|}{0.6076} & \multicolumn{1}{c|}{0.4200} & \multicolumn{1}{c|}{0.4775} & \multicolumn{1}{c|}{0.4475} & 0.4780 \\
\multicolumn{1}{c|}{SkM-Global} & \multicolumn{1}{c|}{0.5000} & \multicolumn{1}{c|}{0.4334} & \multicolumn{1}{c|}{0.5476} & \multicolumn{1}{c|}{0.3150} & \multicolumn{1}{c|}{0.5524} & \multicolumn{1}{c|}{0.3425} & \multicolumn{1}{c|}{0.4205} & \multicolumn{1}{c|}{0.4119} & 0.4435 \\
\multicolumn{1}{c|}{MVSkM} & \multicolumn{1}{c|}{\textbf{0.7476}} & \multicolumn{1}{c|}{\textbf{0.6495}} & \multicolumn{1}{c|}{\textbf{0.7476}} & \multicolumn{1}{c|}{\textbf{0.4850}} & \multicolumn{1}{c|}{\textbf{0.7071}} & \multicolumn{1}{c|}{\textbf{0.5050}} & \multicolumn{1}{c|}{\textbf{0.6515}} & \multicolumn{1}{c|}{\textbf{0.6006}} & \textbf{0.6550} \\ \bottomrule
\end{tabular} \end{table*}
\section{Experiments} \label{sec:exps}
In this section, we provide experimental results to back up our theoretical analysis. In detail, we visualize the clustering results with a two dimensional synthetic dataset to demonstrate the difference between alternating minimization \cite{nascimento2016applying}, the proposed Algorithm \ref{alg:global}, and MVS$k$M. Then we conduct experiments on three real-world datasets to perform clustering performance comparison of solving Soft $k$-Means with alternating minimization (S$k$M-AM), the proposed Algorithm \ref{alg:global} (S$k$M-Global) and the MVS$k$M model.
\subsection{Synthetic Dataset} To show the prototype selection and membership assignment difference between solving Soft $k$-Means with alternating minimization (S$k$M-AM), with Algorithm \ref{alg:global} (S$k$M-Global) and the MVS$k$M model, we visualize the clustering result on a synthetic dataset.
In detail, we plot the learned prototypes and visualize the membership assignment from the three comparison methods. The synthetic data is the \texttt{fcmdata.mat} along with the MATLAB.
The clustering results visualization are shown in \Cref{fig:toy}. We make following remarks: \begin{itemize}
\item It is observed that both S$k$M-AM and S$k$M-Global tend to learn the prototypes $\mathbf{F}$ outside of the data convex hull. The reason is that one can always enlarge the simplex with $\mathbf{F}$ as vertices without changing the objective function, which has been mentioned in Remark \ref{remark:intuition}.
\item For the newly proposed model MVS$k$M, with proper trading off between the loss function and the volume regularization,
we can obtain prototypes insider of the data convex hull, which is more reasonable in most cases. \end{itemize} \subsection{Real-world Datasets}
To validate the clustering performance, we conduct experiments on three real-world datasets. They are MSRCv1 \cite{winn2005locus}, ORL Face \cite{samaria1994parameterisation}, and Numerical Numbers \cite{asuncion2007uci}. For the fuzzy membership matrix, we discretize the membership matrix $\mathbf{G}$ by selecting the class with max membership grade for every data point. For every dataset, we follow the preprocessing method in \cite{nie2018multiview} and use the commonly used clustering performance measures: accuracy (ACC), Normalized Mutual Information (NMI), and Purity.
The results are summarized in \Cref{tab:exps} and the best results are marked in bold face. Following remarks can be made: \begin{itemize}
\item It is interesting to see S$k$M-Global cannot outperform S$k$M-AM, while it outputs the global solution
of Problem \eqref{eq:SfM}. The reason is that the solution non-uniqueness issue causes performance degradation.
\item With proper trading off between the loss function and the polyhedron volume, MVS$k$M outperforms both
S$k$M-AM and S$k$M-Global, which validates the motivation of the MVS$k$M model. \end{itemize}
\section{Conclusion}\label{sec:conclusion}
In this paper, we present a sufficient condition for a feasible solution of Soft $k$-Means problem to be globally optimal. Then, we report an algorithm (Algorithm \ref{alg:global}) whose output satisfies this sufficient condition, thus globally solve the Soft $k$-Means problem. Moreover, for the Soft $k$-Means problem, we provide interesting discussions on stability, solutions non-uniqueness, and connection with LSC. Then, we proposed an new clustering model, named Minimal Volume Soft $k$-Means (MVS$k$M), to address the solutions non-uniqueness issue in Soft $k$-Means. Finally, experimental results support our theoretical results.
\appendix \section{Appendix} \subsection{Theorems}
\newtheorem{thmm}{\bf Theorem} \setcounter{thmm}{1} \begin{thmm}[Condition of S$k$Mable]
For $\mathbf{X}\in\mathbb{R}^{d\times n}$, there exists $\mathbf{F}\in\mathbb{R}^{d\times k}$ and $\mathbf{G}\in\mathbb{R}^{k\times d}, \mathbf{G}\succeq 0,\mathbf{G}\mathbb{1}_k = \mathbb{1}_n$ such that $\mathbf{X}=\mathbf{FG}^\top$ if and only if rank$(\mathbf{X}\mathbf{H}_n) \leq k-1$. \end{thmm} \begin{proof}
For the {\em if} direction, choose $\textbf{F}$ and $\textbf{G}$ according to the Algorithm \ref{alg:global}.
Directly calculation gives $\textbf{FG}^\top = \textbf{X}$.
For the {\em only if} direction, note that
\[
\textbf{X} - \frac{1}{k}\textbf{F}\mathbb{1}_k\mathbb{1}_n^\top = \textbf{F}\left(\textbf{G}^\top - \frac{1}{k}\textbf{F}\mathbb{1}_k\mathbb{1}_k^\top\textbf{G}^\top\right) = \textbf{FH}_k\textbf{G}.
\]
Using Lemma \ref{lem:rankXH}, we have
\[
\begin{aligned}
\text{rank}(\textbf{XH}_n) &\leq \text{rank}\left(\textbf{X} - \frac{1}{k}\textbf{F}\mathbb{1}_k\mathbb{1}_n^\top\right)\\
&= \text{rank}\left(\textbf{F}\textbf{H}_k\textbf{G}^\top\right)\\
&\leq \text{rank}(\textbf{H}_k) \\
&= k-1.
\end{aligned}
\] \end{proof}
\begin{thmm}[Stability]
Assume additively perturbed data $\widetilde{\mathbf{X}} = \mathbf{X} + \mathbf{E}$.
We have
\[
\|\mathbf{X} - \widetilde{\mathbf{F}}_*\widetilde{\mathbf{G}}_*^\top \|_F^2 \leq 2\|\mathbf{E}\|_F^2 + \|\mathbf{X} - \mathbf{F}_*\mathbf{G}_*^\top \|_F^2.
\] \end{thmm} \begin{proof}
Let $r_1 = $rank$(\textbf{X})$, $r_2 = $rank$(\widetilde{\textbf{X}})$, and $r_3 = $rank$(\textbf{E})$. Then $r = \max\{r_1, r_2, r_3\}.$
\[
\begin{aligned}
\|\textbf{X} - \widetilde{\textbf{F}}_*\widetilde{\textbf{G}}_*^\top \|_F^2 =& \|\textbf{X} - \widetilde{\textbf{X}} + \widetilde{\textbf{X}} - \widetilde{\textbf{F}}_*\widetilde{\textbf{G}}_*^\top \|_F^2 \\
\leq& \|\textbf{X} - \widetilde{\textbf{X}} \|_F^2 + \| \widetilde{\textbf{X}} - \widetilde{\textbf{F}}_*\widetilde{\textbf{G}}_*^\top \|_F^2 \\
\leq& \|\textbf{E}\|_F^2 + \|\textbf{X} - \textbf{F}_*\textbf{G}_*^\top \|_F^2 + \sqrt{\sum_{i=k}^{r_2}\tilde{\sigma}_i^2} - \sqrt{\sum_{i=k}^{r_1}\sigma_i^2} \\
\overset{(a)}{\leq} & \|\textbf{E}\|_F^2 + \|\textbf{X} - \textbf{F}_*\textbf{G}_*^\top \|_F^2 + \sqrt{\sum_{i=k}^{r}\left(\tilde{\sigma}_i - \sigma_i\right)^2 } \\
\overset{(b)}{\leq}& 2\|\textbf{E}\|_F^2 + \|\textbf{X} - \textbf{F}_*\textbf{G}_*^\top \|_F^2,
\end{aligned}
\]
where $(a)$ holds since $\|a\|-\|b\|\leq\|a-b\|$ and $(b)$ is from Mirsky's inequality (Lemma \ref{lem:mirsky}). \end{proof}
\begin{thmm}[Lower bound]
Let the best $(k-1)$-rank approximation of $\mathbf{X}$ be $\mathbf{X}_{k-1}$.
For any oracle $\mathbf{G}^*$, there exists $\mathbf{G}$, which is a solution of Problem \eqref{eq:SfM}, such that
\[
\|\mathbf{G} - \mathbf{G}^*\|_F^2 \geq \frac{1}{r\sqrt{k(k-1)}} \| \mathbf{X}_{k-1}\|_F^2.
\] \end{thmm} \begin{proof}
Let
\[
\begin{aligned}
\mathbb{A} &= \arg_{\mathbf{G}}\min_{\mathbf{F},\mathbf{G}\succeq 0,\mathbf{G}\mathbb{1}_k = \mathbb{1}_n} \| \mathbf{X} - \mathbf{FG}^\top\|_F^2 \\
\mathbb{B} &= \left\{ \mathbf{G} \left| \mathbf{G}=\frac{1}{r\sqrt{k(k-1)}}\mathbf{B}\mathbf{U}_{k-1}^\top\mathbf{X}+\frac{1}{k}\mathbb{1}_k\mathbb{1}_n^\top \right.\right\}.
\end{aligned}
\]
From Theorem \ref{thm:mainRes}, we know $\mathbb{A}\supseteq \mathbb{B}$. Indeed, for any $\mathbf{G}_1 \in \mathbb{B}$, exists $\mathbf{G}_2 \in \mathbb{B}$ such that
\[
\begin{aligned}
\|\mathbf{G}_1 - \mathbf{G}_2\|_F^2=&\left\| \frac{1}{r\sqrt{k(k-1)}}\mathbf{B}\mathbf{R}\mathbf{U}_{k-1}^\top\mathbf{X}
-\frac{1}{r\sqrt{k(k-1)}}\mathbf{B}\mathbf{U}_{k-1}^\top\mathbf{X} \right\|_F^2 \\
=&\frac{1}{r\sqrt{k(k-1)}} \left\| \mathbf{B}(\mathbf{R}-\mathbb{I})\mathbf{U}_{k-1}^\top\mathbf{X}\right\|_F^2 \\
=&\frac{1}{r\sqrt{k(k-1)}} \left\|(\mathbf{R}-\mathbb{I})\mathbf{U}_{k-1}^\top\mathbf{X}\right\|_F^2 \\
\geq& \frac{2}{r\sqrt{k(k-1)}} \left\|\mathbf{U}_{k-1}^\top\mathbf{X}\right\|_F^2 \\
=& \frac{2}{r\sqrt{k(k-1)}} \left\|\mathbf{X}_{k-1}\right\|_F^2
\end{aligned}
\]
Note that
\[
\begin{aligned}
\max_{\mathbf{G}\in\mathbb{A}} \| \mathbf{G} - \mathbf{G}^* \|_F^2
\geq & \max_{\mathbf{G}\in\mathbb{B}} \| \mathbf{G} - \mathbf{G}^* \|_F^2 \\
\geq & \frac{1}{2}\left(\|\mathbf{G}_1 - \mathbf{G}^*\|_F + \|\mathbf{G}_2 - \mathbf{G}^*\|_F^2 \right) \\
\geq & \frac{1}{2} \|\mathbf{G}_1 - \mathbf{G}_2\|_F^2 \\
\geq & \frac{1}{r\sqrt{k(k-1)}} \left\|\mathbf{X}_{k-1}\right\|_F^2,
\end{aligned}
\]
which completes the proof. \end{proof}
\begin{thmm}[Descent]
Algorithm \ref{alg:MVSfM} is a descent method, that is,
\[
\mathcal{L}(\mathbf{F}_{t+1}, \mathbf{G}_{t+1}) \leq \mathcal{L}(\mathbf{F}_{t}, \mathbf{G}_{t}).
\] \end{thmm} \begin{proof}
Note that
\[
\begin{aligned}
\mathcal{L}(\mathbf{F}_{t+1}, \mathbf{G}_{t})
=&\|\mathbf{X}-\mathbf{F}_{t+1}\mathbf{G}_t\|_F^2 +\lambda \sum_{i=1}^{k-1} \log(\sigma_i^2 (\mathbf{F}_{t+1})+\varepsilon) \\
=& \|\mathbf{X}-\mathbf{F}_{t+1}\mathbf{G}_t\|_F^2 + \lambda(\sum_{i=1}^{k-1} \log(\sigma_i^2 (\mathbf{F}_{t+1})+\varepsilon) \\
&- \sum_{i=1}^{k-1} \log(\sigma_i^2 (\mathbf{F}_{t})+\varepsilon)) + \lambda\sum_{i=1}^{k-1} \log(\sigma_i^2 (\mathbf{F}_{t})+\varepsilon) \\
=& \|\mathbf{X}-\mathbf{F}_{t+1}\mathbf{G}_t\|_F^2 +\lambda \sum_{i=1}^{k-1} \log( \frac{\sigma_i^2 (\mathbf{F}_{t+1})+\varepsilon}{\sigma_i^2 (\mathbf{F}_{t})+\varepsilon}) \\
&+ \lambda\sum_{i=1}^{k-1} \log(\sigma_i^2 (\mathbf{F}_{t})+\varepsilon) \\
\overset{(a)}{\leq}& \|\mathbf{X}-\mathbf{F}_{t+1}\mathbf{G}_t\|_F^2 +\lambda \left( \sum_{i=1}^{k-1} \frac{\sigma_i^2 (\mathbf{F}_{t+1})}{\sigma_i^2 (\mathbf{F}_{t})+\varepsilon} - \sum_{i=1}^{k-1} \frac{\sigma_i^2 (\mathbf{F}_{t})}{\sigma_i^2 (\mathbf{F}_{t})+\varepsilon}\right) \\
&+ \lambda\sum_{i=1}^{k-1} \log\left(\sigma_i^2 (\mathbf{F}_{t})\right) \\
\overset{(b)}{\leq}& \|\mathbf{X}-\mathbf{F}_{t+1}\mathbf{G}_t\|_F^2 +\lambda \text{Tr}(\mathbf{D}_t^\top \mathbf{F}_{t+1}^\top \mathbf{F}_{t+1}) - \lambda \text{Tr}(\mathbf{D}_t^\top \mathbf{F}_{t}^\top \mathbf{F}_{t}) \\
&+ \lambda\sum_{i=1}^{k-1} \log\left(\sigma_i^2 (\mathbf{F}_{t}+\varepsilon)\right) \\
\overset{(c)}{\leq}& \|\mathbf{X}-\mathbf{F}_{t}\mathbf{G}_t\|_F^2 + \lambda\sum_{i=1}^{k-1} \log\left(\sigma_i^2 (\mathbf{F}_{t})+\varepsilon\right) \\
=& \mathcal{L}(\mathbf{F}_{t}, \mathbf{G}_{t}),
\end{aligned}
\]
where (a) uses numerical inequality $\log(x)\leq x - 1$; (b) uses the asymmetric version of von Neumann's trace inequality (Lemma \ref{lem:traceIneq}) which implies
$\sum_{i=1}^k \frac{\sigma_i^2 (\mathbf{F}_{t+1})}{\sigma_i^2 (\mathbf{F}_{t})+\varepsilon} \leq \text{Tr}(\mathbf{D}_t^\top \mathbf{F}_{t+1}^\top \mathbf{F}_{t+1})$ and uses the definition of $\mathbf{D}_t$ which implies $\text{Tr}(\mathbf{D}_t^\top \mathbf{F}_{t}^\top \mathbf{F}_{t})=\sum_{i=1}^{k-1} \frac{\sigma_i^2 (\mathbf{F}_{t})}{\sigma_i^2 (\mathbf{F}_{t})+\varepsilon}$; (c) uses $\|\mathbf{X}-\mathbf{F}_{t+1}\mathbf{G}_t\|_F^2 +\lambda \text{Tr}(\mathbf{D}_t^\top \mathbf{F}_{t+1}^\top \mathbf{F}_{t+1})\leq \|\mathbf{X}-\mathbf{F}_{t}\mathbf{G}_t\|_F^2 +\lambda \text{Tr}(\mathbf{D}_t^\top \mathbf{F}_{t}^\top \mathbf{F}_{t})$.
Leveraging the definition of $\mathbf{G}_{t+1}$, it is easy to see $\mathcal{L}(\mathbf{F}_{t+1}, \mathbf{G}_{t+1}) \leq \mathcal{L}(\mathbf{F}_{t+1}, \mathbf{G}_{t})$, which completes the proof. \end{proof}
\newtheorem{thmmm}{\bf Theorem} \setcounter{thmmm}{6} \begin{thmmm}[Condition of TI-LSDable]
$\mathbf{K}$ is TI-LSDable if and only if rank$(\mathbf{H}\mathbf{K}\mathbf{H}) \leq k-1$. \end{thmmm} \begin{proof}
On the one hand, TI-LSDable $\Rightarrow$ rank$(\mathbf{H}\mathbf{K}\mathbf{H}) \leq k-1$. The reason is that leveraging Theorem \ref{thm:jmlr13prop} we know if $\mathbf{K}$ is TI-LSDable, then $\mathbf{X}$ can be written as $\mathbf{X}=\mathbf{FG}^\top$. Thus, rank$(\mathbf{H}\mathbf{K}\mathbf{H}) \leq k-1$.
On the other hand, if rank$(\mathbf{H}\mathbf{K}\mathbf{H}) \leq k-1$, then there exists $\mathbf{K}^\prime \in \mathbb{K}(\mathbf{X})$ such that $ \mathbf{K}^\prime = c_1\mathbf{P}^\top\mathbf{P}$. We prove this claim as follows. Note that
\[
\mathbf{F} = \mathbf{\bar{F}} + \mathbf{m}\mathbb{1}_k^\top = r\sqrt{k(k-1)}\mathbf{U}_{k-1}\mathbf{B}^\top + \mathbf{m}\mathbb{1}_k^\top.
\]
Then we have
\[
\begin{aligned}
\mathbf{X}^\top\mathbf{X} &= \left( \bar{\mathbf{X}} + \mathbf{m}\mathbb{1}_n^\top \right)^\top \left( \bar{\mathbf{X}} + \mathbf{m}\mathbb{1}_n^\top \right) \\
&= \mathbf{P}^\top\mathbf{F}^\top\mathbf{FP} \\
&= \mathbf{P}^\top\left( \bar{\mathbf{F}} + \mathbf{m}\mathbb{1}_k^\top \right)^\top \left( \bar{\mathbf{F}} + \mathbf{m}\mathbb{1}_k^\top \right)\mathbf{P} \\
&= \mathbf{P}^\top\left( \bar{\mathbf{F}}^\top\bar{\mathbf{F}} + \bar{\mathbf{F}}^\top \mathbf{m}\mathbb{1}_k^\top + \mathbb{1}_k\mathbf{m}^\top \bar{\mathbf{F}} + \|\mathbf{m}\|_2^2 \mathbb{1}_k\mathbb{1}_k^\top \right)\mathbf{P}.
\end{aligned}
\]
According to rank$(\mathbf{H}\mathcal{K}\mathbf{H}) \leq k-1$, let $\mathbf{U}_\bot \in \mathbb{R}^{d\times (d-k+1)}$ be the orthocomplement of $\mathbf{U}_{k-1}$ and $\mathbf{m}=\mathbf{U}_\bot \mathbf{a}$ where $\mathbf{a} \in \mathbb{R}^{d-k+1}$. That is
\[
\begin{aligned}
\mathbf{X}^\top\mathbf{X} &= \mathbf{P}^\top\left( \bar{\mathbf{F}}^\top\bar{\mathbf{F}} + \bar{\mathbf{F}}^\top \mathbf{m}\mathbb{1}_k^\top + \mathbb{1}_k\mathbf{m}^\top \bar{\mathbf{F}} + \|\mathbf{m}\|_2^2 \mathbb{1}_k\mathbb{1}_k^\top \right)\mathbf{P} \\
&= \mathbf{P}^\top\left( \bar{\mathbf{F}}^\top\bar{\mathbf{F}} + \|\mathbf{m}\|_2^2 \mathbb{1}_k\mathbb{1}_k^\top \right)\mathbf{P} \\
&= \mathbf{P}^\top\left( r^2k(k-1)\mathbf{BU}_{k-1}^\top\mathbf{U}_{k-1}\mathbf{B}^\top + \|\mathbf{m}\|_2^2 \mathbb{1}_k\mathbb{1}_k^\top \right)\mathbf{P} \\
&= \mathbf{P}^\top\left( r^2k(k-1)\mathbf{B}\mathbf{B}^\top + \|\mathbf{m}\|_2^2 \mathbb{1}_k\mathbb{1}_k^\top \right)\mathbf{P} \\
&= \mathbf{P}^\top\left( r^2k(k-1)\mathbb{I}_k - r^2(k-1) \mathbb{1}_k\mathbb{1}_k^\top + \|\mathbf{m}\|_2^2 \mathbb{1}_k\mathbb{1}_k^\top \right)\mathbf{P}
\end{aligned}
\]
Let $\|\mathbf{a}\|_2 = \|\mathbf{m}\|_2 = r\sqrt{k-1}$. Then we have
\[
\begin{aligned}
\mathbf{X}^\top\mathbf{X} &= r^2k(k-1)\mathbf{P}^\top\mathbf{P},
\end{aligned}
\]
which completes the proof. \end{proof}
\begin{thmmm}
Assume kernel $\mathbf{K}\in\mathbb{R}^{n\times n}$ satisfies rank$(\mathbf{K}) \leq n-1$.
If $\mathbf{G}^*$ is a solution of the TI-LSD in Problem \eqref{eq:TILSD} for kernel $\mathbf{K}$,
then there exists $\mathbf{X}$ such that $\mathbf{K}=\mathbf{X}^\top\mathbf{X}$ and $\mathbf{G}^*$ is also a solution of S$k$M
in Problem \eqref{eq:SfM}. \end{thmmm} \begin{proof}
Let $\mathbf{M}$ be some kernel that is TI-LSDable. Note that
\[
\begin{aligned}
&\|\mathbf{K} - \frac{1}{c}\mathbf{G}^\top\mathbf{G} \|_F^2\\
=& \|\mathbf{K} - \mathbf{M} + \mathbf{M} - \frac{1}{c}\mathbf{G}^\top\mathbf{G} \|_F^2 \\
= &\|\mathbf{K} - \mathbf{M} \|_F^2 + \|\mathbf{M} - \frac{1}{c}\mathbf{G}^\top\mathbf{G} \|_F^2 + 2\langle \mathbf{K} - \mathbf{M}, \mathbf{M} - \frac{1}{c}\mathbf{G}^\top\mathbf{G} \rangle.
\end{aligned}
\]
Since we will optimize on $\mathbf{G}$ and $\mathbf{M}$ is LSDable, for any $\mathbf{M}$, we can choose $c$ and $\mathbf{G}$ such that $\mathbf{M}=\frac{1}{c}\mathbf{G}^\top\mathbf{G}$. Thus the objective can be written as
\[
\min_{\substack{\mathbf{K}\in\mathbb{K}(\mathbf{X}), \\ \mathbf{M} \text{ is LSDable} }}\|\mathbf{K} - \mathbf{M} \|_F^2.
\]
Assume $\mathbf{M} = \mathbf{Y}^\top\mathbf{Y}$. We always have $\mathbf{Y} = \overline{\mathbf{Y}} + \mathbf{s}\mathbb{1}_n^\top$, where $\mathbf{s} = \frac{1}{n}\mathbf{Y}\mathbb{1}_n$ and $\overline{\mathbf{Y}}\mathbb{1}_n = 0$. Using this decomposition, we rewrite above problem explicitly as
\[
\min_{\substack{ \mathbf{t},\mathbf{s},\overline{\mathbf{Y}} \\ \mathbf{Y}^\top\mathbf{Y} \text{ is LSDable}}}
\|(\mathbf{X}+\mathbf{t}\mathbb{1}_n^\top)^\top(\mathbf{X}+\mathbf{t}\mathbb{1}_n^\top) - (\overline{\mathbf{Y}}+\mathbf{s}\mathbb{1}_n^\top)^\top(\overline{\mathbf{Y}}+\mathbf{s}\mathbb{1}_n^\top) \|_F^2,
\]
where $\mathbf{X}\mathbb{1}_n = 0$.
Note that
\[
\begin{aligned}
&\|(\mathbf{X}+\mathbf{t}\mathbb{1}_n^\top)^\top(\mathbf{X}+\mathbf{t}\mathbb{1}_n^\top) - (\overline{\mathbf{Y}}+\mathbf{s}\mathbb{1}_n^\top)^\top(\overline{\mathbf{Y}}+\mathbf{s}\mathbb{1}_n^\top) \|_F^2 \\
=& \|(\mathbf{X}^\top\mathbf{X}-\overline{\mathbf{Y}}^\top\overline{\mathbf{Y}}) + \mathbb{1}_n(\mathbf{t}^\top \mathbf{X} - \mathbf{s}^\top\overline{\mathbf{Y}}) + (\mathbf{X}^\top\mathbf{t} - \overline{\mathbf{Y}}^\top\mathbf{s})\mathbb{1}^\top \\
&+ (\|\mathbf{t}\|^2 - \|\mathbf{s}\|^2)\mathbb{1}_n\mathbb{1}_n^\top \|_F^2 \\
=& \| \mathbf{X}^\top\mathbf{X}-\overline{\mathbf{Y}}^\top\overline{\mathbf{Y}} \|_F^2 + \| \mathbb{1}_n(\mathbf{t}^\top \mathbf{X} - \mathbf{s}^\top\overline{\mathbf{Y}}) + (\mathbf{X}^\top\mathbf{t} - \overline{\mathbf{Y}}^\top\mathbf{s})\mathbb{1}^\top\\
&+ (\|\mathbf{t}\|^2 - \|\mathbf{s}\|^2)\mathbb{1}_n\mathbb{1}_n^\top \|_F^2 \\
=& \| \mathbf{X}^\top\mathbf{X}-\overline{\mathbf{Y}}^\top\overline{\mathbf{Y}} \|_F^2 + \| \mathbb{1}_n(\mathbf{t}^\top \mathbf{X} - \mathbf{s}^\top\overline{\mathbf{Y}}) + (\mathbf{X}^\top\mathbf{t} - \overline{\mathbf{Y}}^\top\mathbf{s})\mathbb{1}^\top \|_F^2\\
&+ \|(\|\mathbf{t}\|^2 - \|\mathbf{s}\|^2)\mathbb{1}_n\mathbb{1}_n^\top \|_F^2,
\end{aligned}
\]
where the second and third equality hold since $\mathbf{X}\mathbb{1}_n=\overline{\mathbf{Y}}\mathbb{1}_n = 0$.
From rank$(\mathbf{X})\leq n-1$ and rank$(\overline{\mathbf{Y}})\leq k-1$, we can alway choose proper $\mathbf{s}$ and $\mathbf{t}$ such that $\mathbf{Y}^\top\mathbf{Y}$ is LSDable and the second and third term are equal to zero. In detail, we choose $\|\mathbf{s}\|=\|\mathbf{t}\|$ and $\mathbf{X}^\top\mathbf{t}=\overline{\mathbf{Y}}^\top\mathbf{s}=0$. Thus, the original problem becomes
\[
\min_{\text{rank}(\overline{\mathbf{Y}})\leq k-1}
\|(\mathbf{X}^\top\mathbf{X} - \overline{\mathbf{Y}}^\top\overline{\mathbf{Y}} \|_F^2.
\]
The $(k-1)$-truncated SVD of $\mathbf{X}$ is a solution of above problem. Using Theorem \ref{thm:sfmable} and $\mathbf{Y}^\top\mathbf{Y} = \frac{1}{c}\mathbf{G}\mathbf{G}^\top$, the proof completes. \end{proof}
\subsection{Technical Lemmas}
\newtheorem{Lemma2}{\bf Lemma} \begin{Lemma2}
Let $\mathbf{x} \in \mathbb{R}^k$.
If $\mathbb{1}_k^\top\mathbf{x}=0$ and $\|\mathbf{x}\|_2 \leq 1$, then we have $\|\mathbf{x}\|_\infty \leq \frac{\sqrt{k(k-1)}}{k}.$ \end{Lemma2} \begin{proof}
Let $i_\infty$ be the index such that $x_{i_\infty} = \max_{1\leq i \leq k} |x_i|$. Note that
\[
x_{i_\infty}^2 = \left( \sum_{i\neq i_\infty} x_i \right)^2 \leq (k-1)\sum_{i\neq i_\infty}x_i^2 \leq (k-1)(1-x_{i_\infty}^2),
\]
which indicates $x_{i_\infty}^2 \leq \frac{k-1}{k}$. The claim just follows. \end{proof}
\begin{Lemma2}
Following equivalence can be shown using the translation invariance of Problem \eqref{eq:lessConstrained}:
\[
\min_{\mathbf{F},\mathbf{G}\mathbb{1}_k = \mathbb{1}_n} \| \mathbf{X} - \mathbf{FG}^\top\|_F^2
\Leftrightarrow
\min_{\substack{\mathbf{F},\mathbf{G}\mathbb{1}_k = \mathbb{1}_n \\ \mathbf{s}, \text{rank}(\mathbf{F}) \leq k-1 }} \| \mathbf{X} - \mathbf{s}\mathbb{1}_n^\top- \mathbf{FG}^\top\|_F^2.
\] \end{Lemma2} \begin{proof}
Note that,
\[
\begin{aligned}
\min_{\mathbf{F},\mathbf{G}\mathbb{1}_k = \mathbb{1}_n} \| \mathbf{X} - \mathbf{FG}^\top\|_F^2
&=
\min_{\mathbf{s}, \mathbf{F},\mathbf{G}\mathbb{1}_k = \mathbb{1}_n} \| \mathbf{X} - \left( \mathbf{F}- \mathbf{s}\mathbb{1}_k^\top \right)\mathbf{G}^\top\|_F^2\\
&\leq
\min_{\substack{\mathbf{F},\mathbf{G}\mathbb{1}_k = \mathbb{1}_n \\ \mathbf{s}, \text{rank}(F) \leq k-1 }} \| \mathbf{X} - \mathbf{s}\mathbb{1}_n^\top- \mathbf{FG}^\top\|_F^2.
\end{aligned}
\]
Assume that,
\[
\mathbf{F}_*, \mathbf{G}_* = \mathop{\arg\min}_{\mathbf{F},\mathbf{G}\mathbb{1}_k = \mathbb{1}_n}\| \mathbf{X} - \mathbf{FG}^\top\|_F^2.
\]
We have,
\[
\begin{aligned}
\left\|\mathbf{X} - \mathbf{F}_*\mathbf{G}_*^\top\right\|_F^2
&=
\left\|\mathbf{X} - \left(\frac{1}{k}\mathbf{F}_*\mathbb{1}_k\right)\mathbb{1}_k^\top \mathbf{G}_*^\top - \left(\mathbf{F}_* - \frac{1}{k}\mathbf{F}_*\mathbb{1}_k\mathbb{1}_k^\top \right)\mathbf{G}_*^\top \right\|_F^2\\
&=\|\mathbf{X} - \hat{\mathbf{s}}\mathbb{1}_n^\top - \hat{\mathbf{F}}\mathbf{G}_*^\top \|_F^2,
\end{aligned}
\]
where $\hat{\mathbf{s}} := \frac{1}{k}\mathbf{F}_*\mathbb{1}_k, \hat{\mathbf{F}} := \mathbf{F}_* - \frac{1}{k}\mathbf{F}_*\mathbb{1}_k\mathbb{1}_k^\top$.
Note that $\hat{\mathbf{F}} \mathbb{1}_k = \mathbf{F}_*\mathbb{1}_k - \frac{1}{k}\mathbf{F}_*\mathbb{1}_k\mathbb{1}_k^\top\mathbb{1}_k
=\mathbf{F}_*\mathbb{1}_k-\mathbf{F}_*\mathbb{1}_k=0$. Thus, $\text{rank}(\hat{\mathbf{F}}) \leq k-1$ and we have
\[
\begin{aligned}
\min_{\mathbf{F},\mathbf{G}\mathbb{1}_k = \mathbb{1}_n} \| \mathbf{X} - \mathbf{FG}^\top\|_F^2
&=
\left\|\mathbf{X} - \mathbf{F}_*\mathbf{G}_*^\top\right\|_F^2\\
&=\|\mathbf{X} - \hat{\mathbf{s}}\mathbb{1}_n^\top - \hat{\mathbf{F}}\mathbf{G}_*^\top \|_F^2\\
&\geq\min_{\substack{\mathbf{F},\mathbf{G}\mathbb{1}_k = \mathbb{1}_n \\ \mathbf{s}, \text{rank}(F) \leq k-1 }} \| \mathbf{X} - \mathbf{s}\mathbb{1}_n^\top- \mathbf{FG}^\top\|_F^2,
\end{aligned}
\]
which completes the proof. \end{proof}
\begin{Lemma2}
Let $\mathbf{F}\in\mathbb{R}^{d\times k}$ be the prototype matrix satisfying the constraint $\text{rank}(\mathbf{F}) \leq k-1$ and has the thin SVD $\mathbf{F}=\mathbf{U}\mathbf{\Sigma}\mathbf{V}^\top$, where $\mathbf{\Sigma}\in\mathbb{R}^{(k-1)\times(k-1)}$. Let $ \mathbf{\Phi} = \mathbf{V}\mathbf{\Sigma}^{-1}\mathbf{U}^\top \left(\mathbf{X}-\mathbf{s}\mathbb{1}_n^\top\right)$. Define
\[
\mathbf{G}_*^\top = \mathbf{\Phi}
+
\frac{\mathbf{v}_\bot\left(\mathbb{1}_n^\top - \mathbb{1}_k^\top \mathbf{\Phi} \right) }{\mathbb{1}_k^\top\mathbf{v}_\bot },
\]
where $\mathbf{v}_\bot \in \mathbb{R}^k$ satisfies $\|\mathbf{v}_\bot\|_2 = 1$ and $\mathbf{V}^\top \mathbf{v}_\bot = 0$.
Then
\[
\mathbf{G}_* \in \mathop{\arg\min}_{\mathbf{G}\mathbb{1}_k = \mathbb{1}_n} \| \mathbf{X} - \mathbf{s}\mathbb{1}_n^\top- \mathbf{FG}^\top\|_F^2.
\] \end{Lemma2} \begin{proof}
Note that $\mathbf{F} = \mathbf{U}\mathbf{\Sigma}\mathbf{V}^\top = \sum_{i=1}^{k-1}\sigma_i\mathbf{u}_i\mathbf{v}_i^\top$ and
\[
\| \mathbf{X} - \mathbf{s}\mathbb{1}_n^\top- \mathbf{FG}^\top\|_F^2
=
\sum_{i=1}^n \|\mathbf{x}_i - \mathbf{s} - \mathbf{F}\mathbf{g}_i \|_2^2
\]
Then we can write $\mathbf{g}_i = \mathbf{V}\mathbf{a}_i + \mathbf{v}_\bot b_i$, where $\mathbf{a}_i \in \mathbb{R}^{k-1}, b_i \in \mathbb{R}$ and
$\mathbf{v}_\bot \in \mathbb{R}^k,\mathbf{V}^\top\mathbf{v}_\bot=0$. We solve $\mathbf{g}_i$ by solving $\mathbf{a}_i$ and $b_i$.
Note that
\[
\begin{aligned}
\sum_{i=1}^n \|\mathbf{x}_i - \mathbf{s} - \mathbf{F}\mathbf{g}_i \|_2^2
&=
\sum_{i=1}^n \|\mathbf{x}_i - \mathbf{s} - \mathbf{U}\mathbf{\Sigma}\mathbf{V}^\top\mathbf{V}\mathbf{a}_i - \mathbf{U}\mathbf{\Sigma}\mathbf{V}^\top \mathbf{v}_\bot b_i \|_2^2\\
&=
\sum_{i=1}^n \|\mathbf{x}_i - \mathbf{s} - \mathbf{U}\mathbf{\Sigma}\mathbf{a}_i\|_2^2.
\end{aligned}
\]
Then we can obtain the optimal $\mathbf{a}_i$ as $\mathbf{a}_i^* = \mathbf{\Sigma}^{-1}\mathbf{U}^\top(\mathbf{x}_i-\mathbf{s})$.
Using the constraint $\mathbf{G}\mathbb{1}_k=\mathbb{1}_n$, we have
\[
\mathbb{1}_k^\top \mathbf{g}_i = \mathbb{1}_k^\top \mathbf{V}\mathbf{\Sigma}^{-1}\mathbf{U}^\top(\mathbf{x}_i-\mathbf{s}) + \mathbb{1}_k^\top\mathbf{v}_\bot b_i = 1,
\]
which gives the optimal $b_i$ as
\[
b_i = \frac{1 - \mathbb{1}_k^\top \mathbf{V}\mathbf{\Sigma}^{-1}\mathbf{U}^\top(\mathbf{x}_i-\mathbf{s})}{\mathbb{1}_k^\top\mathbf{v}_\bot }.
\]
Then the optimal $\mathbf{g}_i$ writes
\[
\mathbf{g}_i^* = \mathbf{V} \mathbf{\Sigma}^{-1}\mathbf{U}^\top(\mathbf{x}_i-\mathbf{s}) + \frac{1 - \mathbb{1}_k^\top \mathbf{V}\mathbf{\Sigma}^{-1}\mathbf{U}^\top(\mathbf{x}_i-\mathbf{s})}{\mathbb{1}_k^\top\mathbf{v}_\bot } \mathbf{v}_\bot,
\]
which completes the proof. \end{proof}
\begin{Lemma2}
Assume $\mathbf{F}=\mathbf{U\Sigma V}^\top$ such that $\mathbf{F}\mathbb{1}_k = 0$. Then,
\[
\log \det
\left[ \begin{array}{c}
\mathbf{\Sigma V}^\top \\ \mathbb{1}_k^\top \end{array}
\right] = \log(\sqrt{k}) + \sum_{i=1}^{k-1} \log(\sigma_i(\mathbf{F})).
\] \end{Lemma2} \begin{proof}
Using $\det (\mathbf{A}\mathbf{A}^\top)= \det^2 \mathbf{A}$ and $\mathbf{V}^\top\mathbb{1}_k = 0$, we have
\[
\begin{aligned}
\log \det
\left[ \begin{array}{c}
\mathbf{\Sigma V}^\top \\ \mathbb{1}_k^\top \end{array}
\right]
=&
\frac{1}{2} \log \det \left(
\left[ \begin{array}{c}
\mathbf{\Sigma V}^\top \\ \mathbb{1}_k^\top \end{array}
\right]
\left[ \begin{array}{c}
\mathbf{\Sigma V}^\top \\ \mathbb{1}_k^\top \end{array}
\right]^\top
\right) \\
=&
\frac{1}{2} \log \det \left(
\left[ \begin{array}{c c}
\mathbf{\Sigma}^2 & 0 \\ 0 & k \end{array}
\right]
\right) \\
=&
\frac{1}{2}\log \left( k\prod_{i=1}^{k-1}\sigma_i^2(\mathbf{F}) \right) \\
=& \log\sqrt{k} + \sum_{i=1}^{k-1}\log \sigma_i(\mathbf{F}),
\end{aligned}
\]
which completes the proof. \end{proof}
\begin{Lemma2}\label{lem:rankXH}
Given $\mathbf{s}\in\mathbb{R}^d$, we have rank$(\mathbf{XH}_n) \leq$ rank$(\mathbf{X}-\mathbf{s}\mathbb{1}_n^\top)$. \end{Lemma2} \begin{proof}
Leveraging the rank-sum inequality Lemma \ref{lem:rankSum}, we have
\[
\text{rank}\left(\mathbf{X} - \frac{1}{n}\mathbf{X}\mathbb{1}_n\mathbb{1}_n^\top\right)
\leq \text{rank}\left(\mathbf{X} - \mathbf{s}\mathbb{1}_n^\top\right) + \text{rank}\left(\mathbf{s}\mathbb{1}_n^\top - \frac{1}{n}\mathbf{X}\mathbb{1}_n\mathbb{1}_n^\top\right),
\]
where the equality holds only if $\text{range}\left( \mathbf{X} - \mathbf{s}\mathbb{1}_n^\top \right) \cap \text{range}\left(\mathbf{s}\mathbb{1}_n^\top - \frac{1}{n}\mathbf{X}\mathbb{1}_n\mathbb{1}_n^\top\right) = \{0\}$.
Note that
\[
\text{range}\left( \mathbf{X} - \mathbf{s}\mathbb{1}_n^\top \right) \ni \sum_{i=1}^n \frac{\mathbf{x}_i - \mathbf{s}}{n} = \frac{1}{n}\mathbf{X}\mathbb{1}_n - \mathbf{s} \in \text{range}\left(\mathbf{s}\mathbb{1}_n^\top - \frac{1}{n}\mathbf{X}\mathbb{1}_n\mathbb{1}_n^\top\right),
\]
which concludes that the equality cannot hold. Thus
$
\text{rank}\left(\mathbf{XH}_n\right)
< \text{rank}\left(\mathbf{X} - \mathbf{s}\mathbb{1}_n^\top\right) + 1,
$
which completes the proof. \end{proof}
\begin{Lemma2}[\cite{von1937some}]\label{lem:traceIneq}
Assume $\mathbf{A}\in\mathbb{R}^{n\times n}$ and $\mathbf{B}\in\mathbb{R}^{n\times n}$ are symmetric. Then
\[
\sum_{i=1}^n \sigma_{n-i+1}(\mathbf{A})\sigma_i(\mathbf{B}) \leq \text{Tr}(\mathbf{AB})\leq \sum_{i=1}^n \sigma_{i}(\mathbf{A})\sigma_i(\mathbf{B}).
\] \end{Lemma2}
\begin{Lemma2}[\cite{horn2013matrix}, 0.4.5.1]\label{lem:rankSum}
The rank-sum inequality: If $\mathbf{A},\mathbf{B} \in \mathbb{R}^{m\times n}$, then
\[
\text{rank}(\mathbf{A}+\mathbf{B}) \leq \text{rank}(\mathbf{A}) + \text{rank}(\mathbf{B}),
\]
with equality {\em if and only if}
$
(\text{range}\ \mathbf{A}) \cap (\text{range}\ \mathbf{B}) = \{0\}$ and
$
(\text{range}\ \mathbf{A}^\top) \cap (\text{range}\ \mathbf{B}^\top) = \{0\}.
$ \end{Lemma2}
\begin{Lemma2}[Mirsky's inequality, \cite{stewart1998perturbation}]\label{lem:mirsky}
Let $\mathbf{A}\in\mathbb{R}^{n\times m}$. Then
\[
\sum_{i=1}^n \left(\sigma_i(\mathbf{A}+\mathbf{E}) - \sigma_i(\mathbf{A}) \right)^2 \leq \|\mathbf{E}\|_F^2.
\] \end{Lemma2}
\ifCLASSOPTIONcaptionsoff
\fi
\end{document}
|
arXiv
|
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\maketitle
\begin{abstract} This article introduces a novel approach to the mean-field limit of stochastic systems of interacting particles, leading to the first ever derivation of the mean-field limit to the Vlasov-Poisson-Fokker-Planck system for plasmas in dimension~$2$ together with a partial result in dimension~$3$. The method is broadly compatible with second order systems that lead to kinetic equations and it relies on novel estimates on the BBGKY hierarchy. By taking advantage of the diffusion in velocity, those estimates bound weighted $L^p$ norms of the marginals or observables of the system, uniformly in the number of particles. This allows to qualitatively derive the mean-field limit for very singular interaction kernels between the particles, including repulsive Poisson interactions, together with quantitative estimates for a general kernel in $L^2$. \end{abstract}
\section{Introduction}
The rigorous derivation of kinetic models such as the Vlasov-Poisson system from many-particle systems has been a long standing open question, ever since the introduction of the Vlasov-Poisson system in~\cite{Vlasov1,Vlasov2}. While our understanding of the mean-field limit for singular interactions has made significant progress for first-order dynamics, the mean-field limit for second-order systems has remained frustratingly less understood. This article proposed a new approach that is broadly applicable to second-order systems with repulsive interactions and diffusion in velocity. In particular this allows to derive for the first time the Vlasov-Poisson-Fokker-Planck system in dimension higher than one without any truncation or regularizing.
We more precisely consider the classical second-order Newton dynamics \begin{equation} \begin{split} &\frac{d}{dt} X_i(t)=V_i(t),\quad X_i(t=0)=X_i^0,\\ &dV_i(t)=\frac{1}{N}\,\sum_{j\neq i} K(X_i-X_j)\,dt+\sigma\,dW_i,\quad V_i(t=0)=V_i^0, \end{split}\label{Npart} \end{equation} where the $W_i$ are $N$~independent Wiener processes. For simplicity we take the positions $X_i$ on the torus~$\Pi^d$, while the velocities lie in ${\mathbb{R}}^d$. The kernel $K$ models the pairwise interaction between particles and is taken {\em repulsive} throughout this paper, in the basic sense that it derives from a potential $K=-\nabla \phi$ that is even and positive, $\phi\geq 0$.
\begin{remark} For simplicity, we denote $\phi(0)=0$ and $K(0)=0$, even if $\phi$ and $K$ are not continuous at $0$. This notation simplifies the writing in the equation by allowing to sum over all $j$ in~\eqref{Npart} since the term $j=i$ trivially vanishes. \end{remark}
We naturally focus on singular kernels~$K$ with, as a main guiding example, the case of Coulombian interactions \begin{equation}
K=\alpha\,\frac{x}{|x|^d}+K_0(x),\label{coulombian} \end{equation}
with $\alpha>0$ and $K_0$ a smooth correction to periodize $K$. This corresponds to the choice $\phi=\frac{d\,\alpha}{|x|^{d-2}}+\mbox{correction}$ if $d\geq 3$ or $\phi=-\alpha \ln |x|+\mbox{correction}$ if $d=2$.
The Coulombian kernel~\eqref{coulombian} typically model electrostatic interactions between point charges, such as ions or electrons in a plasma when the velocities are small enough with respect to the speed of light. In that setting, diffusion in~\eqref{Npart} may for example represent collisions against a random background, such as the collision of the faster electrons against the background of ions. Such random collisions may also involve some friction in velocity, which we did not include in~\eqref{Npart} but could be added to our method without difficulty. This makes~\eqref{Npart} with~\eqref{coulombian} one of the most classical and important starting point for the modeling of plasmas; we refer in particular to the classical~\cite{Bogo1,Bogo2}.
Coulombian interactions are also a natural scaling in many models. The obvious counterpart to plasmas concerns the Newtonian dynamics of point masses through gravitational interactions. This consists in taking $\alpha<0$ in \eqref{coulombian} and leads to attractive interactions with a negative potential and for this reason cannot be handled with the method presented here.
The system~\eqref{Npart} usually involve a very large number of particles, typically up to $10^{20}-10^{25}$ in plasmas for example. This makes the mean-field limit especially attractive. This is a kinetic, Vlasov-Fokker-Planck equation posed on the limiting 1-particle density $f(t,x,v)$ \begin{equation} \partial_t f+v\cdot \nabla_x f+ (K\star_x \rho)\cdot\nabla_v f=\frac{\sigma^2}{2}\,\Delta_v f \quad \hbox{ with } \quad \rho = \int_{{\mathbb{R}}^d} f dv. \label{vlasov} \end{equation} Well posedness for mean-field kinetic equations such as~\eqref{vlasov} is now reasonably well understood, including for singular Coulombian interactions such as~\eqref{coulombian} in dimension~$d\leq 3$. For the non-diffusive case $\sigma=0$, weak solutions were established in~\cite{Arsenev}, while classical solutions were obtained in dimension~$2$ in~\cite{UkaOka}. The dimension~$3$ case is harder and obtaining classical solutions requires more difficult dispersive argument and were only obtained later in~\cite{LioPer,Pfa,Schaeffer}, see also the more recent~\cite{GasJabPer,HolMio,Loeper, Pal}. In the case with diffusion~$\sigma>0$, we refer to~\cite{Victory} for weak solutions, and to~\cite{Bouchut,De,OnoStr,ReiWec,VicOdw} for classical solutions.
Of course the mean-field scaling is not the only possible scaling on systems such as~\eqref{Npart}. One can in particular mention the likely even more critical Boltzmann-Grad limit, such as obtained in the classical~\cite{Lanford} and the recent major results in \cite{BodGalSaiSim1,BodGalSaiSim2,GalSaiTex,PulSafSim,PulSim}. We note as well that the derivation of macroscopic equations from mesoscopic systems such as~\eqref{vlasov} is another important and challenging questions. For example the passage to the fluid macroscopic system from Vlasov-Poisson-Fokker-Planck has been approached in different low-field (parabolic) or high-field (hyperbolic) regimes depending on the space dimension, see for example \cite{PS,NPS,GNPS,CChP} and references therein.
Mean-field limits have been rigorously derived for general systems, including second order dynamics such as~\eqref{Npart}, in the case of Lipschitz interaction kernels~$K$. We refer to the classical works~\cite{McKean,Sznitman} in the stochastic case and~\cite{BraHep,Dob} for the deterministic case. Uniform in time propagation of chaos has also been obtained in the locally Lipschitz case, notably in a close to convex case in~\cite{BoGuMa} and more recently in a non-convex setting in~\cite{GuLeMo}.
There now exists a large literature on the question of the mean-field limits, see for example the survey in~\cite{Gol,Jareview,JW2}. However in the specific case of second order systems such as~\eqref{Npart}, very little is known. In dimension~$d=1$, the Vlasov-Poisson-Fokker-Planck system was derived in~\cite{GuLe,HaSa}. In dimension~$d\geq 2$, the only results for unbounded interaction kernels were obtained in~\cite{HauJab1,HauJab2}. But those are valid only in the deterministic case~$\sigma=0$, and for only mildly singulars kernels with $|K(x)|\lesssim |x|^{-\alpha}$, $|\nabla K|\lesssim |x|^{-\alpha-1}$ for $\alpha<1$. \cite{JabWan1} derived the mean-field limit with~$K\in L^\infty$ and without extra derivative. Those cannot cover Coulombian interactions, even in dimension~$2$.
More is known for singular interaction kernels~$K$ that are smoothed or truncated at some N-dependent scale~$\varepsilon_N$. In that truncated case, one can mention in particular~\cite{GanVic,GanVic2,VA,Wollman} for the convergence of so-called particle methods. The recent works~\cite{BoePic,Laz,LazPic} in the deterministic case and~\cite{HuLiPi} in the stochastic case considerably extended the results for such truncated kernels and allowed to almost reach the critical physical scale~$\varepsilon_N\sim N^{-1/d}$. One can also mention~\cite{CaChSa} with polynomial cut-off. It is also possible to derive the Vlasov-Poisson system directly from many-particle quantum dynamics such as the Hartree equation, for which we briefly refer to~\cite{GolPau,La,Saf}.
The mean-field limits for first order systems with singular interactions appear to be more tractable. A classical example concerns the dynamics of point vortices or stochastic point vortices where the mean-field limit corresponds to the vorticity formulation of 2d incompressible Euler or Navier-Stokes. The interaction between vortices obey the Biot-Savart law which has the same singularity as the Coulombian kernel in dimension~$2$. In the deterministic case the mean-field limit was classically obtained for example in~\cite{Goodman91,GooHouLow90} or~\cite{Scho95,Scho96} for the 2d Euler and extended to remarkably essentially any Riesz kernels in~\cite{Se}. In the stochastic case, we refer in particular to~\cite{FHM-JEMS,Osada,JabWan2} for the limit to 2d Navier-Stokes, to~\cite{BrJaWa, BreJabWan} for singular attractive kernels, or to \cite{NgRoSe} for multiplicative noise. Uniform in time propagation of chaos was even recently obtained in~\cite{GuiLebMon,RoSe}.
One of the reason second order systems appear more difficult to handle stems from how the structure of the singularity interacts with the distribution of velocities. Because of the term~$K(X_i-X_j)$, the singularity in pairwise interactions is typically localized on collisions~$X_i=X_j$. For first-order systems this corresponds to a point singularity, while for second-order systems the presence of the additional velocity variables makes it into a plane. In that regard, we also note that the derivation of macroscopic system directly from 2nd order dynamics is in fact better understood than the derivation of kinetic equations like~\eqref{vlasov}. We refer to the derivation of incompressible Euler in~\cite{HanIac}, or to the derivation of monokinetic solutions to~\eqref{vlasov} (which are essentially equivalent to a macroscopic system) in~\cite{Se}.
The main argument in our proof is a new quantitative estimate on the so-called marginals of the system through the BBGKY hierarchy. This leads to the propagation of some weighted $L^p$ estimates on the marginals. It implies a weak propagation of chaos in the sense of \cite{Sznitman} but it applies more broadly to initial data that are not chaotic or not close to being chaotic.
Recently new approaches have been introduced to bound marginals on 1st order systems with non-degenerate diffusion: in~\cite{Lacker} using the relative entropy and in~\cite{JaPoSo} using the~$L^2$ norm of the marginals. Both take advantage of the regularizing provided by the diffusion to avoid ``losing'' a derivative in the hierarchy estimates but require interaction kernels~$K$ in~$L^\infty$ for~\cite{JaPoSo} (or in some exponential Orlicz space for ~\cite{Lacker}). Our method combines this general idea with a specific choice of weights for the $L^p$ norms that are propagated. Those weights are based on a total energy reduced to $k$ particles when dealing with the marginal of order~$k$. They allow to take advantage of a further regularizing effect in the hierarchy to only require kernels~$K$ in some $L^p$ with $p>1$.
A direct consequence of our approach is the first ever derivation of the mean-field limit for the repulsive Vlasov-Poisson-Fokker-Planck over a finite time interval. This applies to any chaotic initial data in dimension~$d=2$ and for initial data with more restrictive energy bound in any dimension~$d\geq 3$. We are expecting to extend this derivation in a future work to any chaotic initial data in any dimension~$d\geq 2$ by decomposing appropriately the initial data.
The paper is structured as follows: We start in Section 2 with the notations and main results. We first state our main result, Theorem~\ref{Conv}, that proves the convergence to the Vlasov-Fokker-Planck equation as $N$ tend to infinity followed with a Theorem~\ref{quantitative} proving quantitative estimates for singular kernels in $L^2$. We next introduce Proposition \ref{propLp} which states the explicit propagation of weighted $L^p$ bounds on the marginals. We in particular discuss more thoroughly the limitations and possible extensions of our approach after stating Proposition~\ref{propLp}. Section 3 is devoted to the proof of Proposition~\ref{propLp} and Theorem~\ref{Conv} from the key technical contribution of the article around Lemma~\ref{technicalLemma} and ends with the proof of Theorem~\ref{quantitative}.
\section{Main results}
\subsection{The new result}
We introduce the full $N$-particle joint law of the system $f_N$ which satisfies the Liouville or forward Kolmogorov equation \begin{equation} \begin{split} & \partial_t f_N+\sum_{i=1}^N v_i\cdot\nabla_{x_i} f_N\\ & \hskip1cm +\sum_{i=1}^N \frac{1}{N}\,\sum_{j=1}^N K(x_i-x_j) \cdot\nabla_{v_i} f_N=\frac{\sigma^2}{2}\,\sum_i \Delta_{v_i} f_N, \label{Liouville} \end{split} \end{equation} which is a linear advection-diffusion equation. However the marginals $f_{k,N}$ of $f_N$ will also play a critical role in the analysis. They correspond to the law of $k$ among $N$ particles and are represented through \begin{equation} \begin{split} &f_{k,N}(t,x_1,v_1,\ldots,x_k,v_k) = \\ & \int_{\Pi^{d(N-k)}\times{\mathbb{R}}^{d(N-k)}} f_N(t,x_1,v_1\ldots,x_N,v_N)\,dx_{k+1}\,dv_{k+1}\dots dx_N\,dv_N. \label{marginaldef} \end{split} \end{equation} The question of well-posedness for Eq.~\eqref{Liouville} can be delicate and is separate from the issue of the mean-field limit that we consider here. For this reason, we consider a notion of entropy solution $f_N\in L^\infty({\mathbb{R}}_+\times\Pi^{dN}\times{\mathbb{R}}^{dN})$ to~Eq.~\eqref{Liouville}, that is fully described later in subsection~\ref{Liouvillewellposed}, and to which we impose some Gaussian decay in velocity \begin{equation}
\begin{split}
\sup_{t\leq 1}\int_{\Pi^{dN}\times {\mathbb{R}}^{dN}} e^{\beta\,\sum_{i\leq N} |v_i|^2}\,&f_N\,dx_1\,dv_1\dots dx_N\,dv_N\leq V^N,\\& \mbox{for some}\ \beta>0,\ V>0, \end{split} \label{gaussiandecay} \end{equation} for which we refer to the short discussion in subsection~\ref{Liouvillewellposed}.
Our main is the derivation of the mean-field limit for a broad class of singular kernels.
\begin{theorem} \label{Conv} Assume that that there exists some constant $\theta>0$ s.t. the potential $\phi$ satisfies
\begin{equation}
\int_{\Pi} e^{\theta\,\phi(x)}\,dx < +\infty,\label{expphi} \end{equation} and that \[ K = - \nabla \phi \in L^p ({\Pi}^d) \qquad \hbox{for some}\qquad p>1. \]
Let $f$ be the unique smooth solution to the Vlasov equation~\eqref{vlasov} with initial data $f^0\in C^\infty(\Pi^d\times {\mathbb{R}}^d)$ such that $\int_{\Pi^d\times {\mathbb{R}}^d} f^0\,e^{\beta\,|v|^2}<\infty$. Consider moreover an entropy solution~$f_N$ to~\eqref{Liouville} in the sense of subsection~\ref{Liouvillewellposed} and satisfying~\eqref{gaussiandecay} with initial data $f_N^0\in L^\infty(\Pi^{dN}\times{\mathbb{R}}^{dN})$. Assume that $f_{k,N}^0$ converges weakly in $L^1$ to $(f^0)^{\otimes k}$for each fixed $k$ and that \[
\|f_{k,N}^0\|_{L^\infty(\Pi^{dN}\times{\mathbb{R}}^{dN})}\leq M^k , \] for some $M>0$ and for all $k\leq N$.
Then there exists $T^*$ depending only on $M$, $V$, and $\|K\|_{L^p}$ such that $f_{k,N}$, given by \eqref{marginaldef}, weakly converge to $f_k= f^{\otimes k}$ in $L^q_{loc} ([0,\;T^\star] \times \Pi^{kd} \times {\mathbb{R}}^{kd})$ for any $k$, and any $2<q<\infty$, with $1/q+1/p\le 1$.
\end{theorem}
Our estimates can also provide quantitative rates of convergence though we need to a stronger assumption, namely $K\in L^2$.
\begin{theorem} \label{quantitative} Assume the same conditions and hypotheses of {Theorem \ref{Conv}}, with moreover $p=2$. We also assume that there exists a constant $C$ independent of $N$ and $\varepsilon_N \to 0$ such that
$$\int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} |f_{k,N}^0 - (f^0)^{\otimes k}|^2 e^{\lambda(0) e_k}
\le C^k \varepsilon_N, $$ for all $k$. Then, there exists $T^*$ such that $f_{k,N}$ converges strongly to $f_k$ in $L^2_{loc} ([0,\;T^\star] \times \Pi^{kd} \times {\mathbb{R}}^{kd})$, for any $k$, and we have the following quantitative estimate $$
\sup_{t\le T^\star} \int_{\Pi^{kd} \times {\mathbb{R}}^{kd}} |f_{N,k} - f^{\otimes k}|^2 e^{\lambda(t) e_k} \le {\widetilde C}^k \varepsilon_N , $$ for some $\widetilde C$ independent of $N$. \end{theorem}
In addition to the mean-field limit, Theorem \ref{Conv} implies the weak propagation of chaos in the sense of the famous~\cite{Sznitman}, although with strong conditions on $f_N^0$. Theorem \ref{Conv} also justifies for the first time the convergence to the Vlasov-Poisson-Fokker-Planck in two space dimension. More precisely, we highlight the following result
\begin{corollary} \label{ic} Let $d=2$ and consider the Poisson kernel $K=-\nabla \phi$
with its associated potential $\phi(x)\simeq - \ln |x|$. Then, the convergence properties given by Theorem \ref{Conv} hold true, leading to the Vlasov-Poisson-Fokker-Planck system. \end{corollary}
\subsection{New stability estimates}
Theorem~\ref{Conv} relies on a new approach to estimate on the BBGKY hierarchy solved by the marginals~$f_{k,N}$ which is of significant interest in itself. In general deriving bounds on either the BBGKY or limiting Vlasov hierarchy is complex. We refer for example to~\cite{GolMouRic} for the Vlasov hierarchy, to~\cite{DuerSain} for the study of long-time corrections to mean-field limits. Bounds on the hierarchy are critical for the derivation of collisional models such as the Boltzmann equation, ever since~\cite{Lanford}. Even a partial discussion of the challenges in the collisional setting would go well beyond the scope of this paper and we simply refer again to~\cite{BodGalSai,BodGalSaiSim1,BodGalSaiSim2,GalSaiTex,Kac1956,Lanford,PulSafSim,PulSim}.
The main difficulty in handling the hierarchy consists in the term \begin{equation} \nabla_{v_i} \int_{\Pi^d\times{\mathbb{R}}^d} K(x_i-x_{k+1})\,f_{k+1,N}\,dx_{k+1}\,dv_{k+1},\label{extraterm} \end{equation} as seen in \eqref{hierarchy}, because this introduces the next order marginal~$f_{k+1,N}$ into the equation for $f_{k,N}$. When treated naively as a source term, it leads to a loss of one derivative on each equation of the hierarchy.
However it was recently noticed in~\cite{JaPoSo,Lacker} that one may avoid this loss of derivative in the stochastic case for non-degenerate diffusion: Any $L^2$ estimate then gains an additional~$H^1$ dissipation which can be used to control the loss of one derivative. This idea still appears applicable in the present kinetic context: Even though we only have diffusion in velocity, the derivative in~\eqref{extraterm} is also only on the velocity variable.
Both~\cite{JaPoSo,Lacker} require high integrability on the kernel: $K\in L^\infty$ for~\cite{JaPoSo} and some sort of exponential Orlicz space of the type $\int e^{\lambda\,|K(x)|}\,dx<C$ for~\cite{Lacker}. Paper \cite{Lacker} used relative entropy estimates to prove uniqueness on the BBGKY hierarchy, while \cite{JaPoSo} proved uniqueness on a tree-indexed limiting hierarchy through~$L^2$ bounds. Hence in both case, the corresponding bounds on the marginals was already known uniformly in~$N$ and the challenge was to prove smallness.
This leads to a first key difference with respect to the present approach and to the first critical new idea introduced in this paper. In essence, we note that the integral in~\eqref{extraterm} leads to a regularizing effect that has the same scaling as the convolution at the limit: One has by H\"older estimates that \begin{equation}
\begin{split}
&\left\|\int_{\Pi^d} K(x_i-x_{k+1})\,f(x_1,\ldots,x_{k+1})\,dx_{k+1}\right\|_{L^q(\Pi^{dk})}
\\
&\qquad \qquad\qquad\qquad\qquad\qquad\qquad\leq \|K\|_{L^p(\Pi^d)}\,\|f\|_{L^q(\Pi^{d\,(k+1)})}, \end{split}
\label{regularization}
\end{equation} provided that $1/p+1/q\leq 1$.
Taking advantage of~\eqref{regularization} for singular~$K\in L^p$ with $p$ small naturally leads to try to propagate~$L^q$ norms of the marginals~$f_{k,N}$ for large exponents~$q$; at the opposite of~\cite{JaPoSo,Lacker}. But it also leads to an additional major difficulty, due to the velocity variable in the unbounded space~${\mathbb{R}}^d$ in~\eqref{extraterm}. In fact, trying to use~\eqref{regularization} in~\eqref{extraterm} as is would force the use of a mixed norm $L^q_x L^1_v$ on the marginals. Unfortunately such mixed norms are notoriously ill-behaved on kinetic equations.
Instead a more natural idea, from the point of view of kinetic equations, consists in using some moments or fast decay in velocity. Even if they are less usual for kinetic equations, the use of Gaussian moments is especially attractive in the current case because they are naturally tensorized. For example, one has the extension of~\eqref{regularization} \begin{equation}
\begin{split}
&\int_{\Pi^{dk}\times{\mathbb{R}}^{dk}}e^{|v_1|^2+\ldots+|v_k|^2}\,\left|\int_{\Pi^d\times{\mathbb{R}}^d} K(x_i-x_{k+1})\,f_{k+1,N}\,dx_{k+1}\,dv_{k+1}\right|^q
\\
&\qquad\qquad \leq C_d\, \|K\|_{L^p(\Pi^d)}^q\,\int_{\Pi^{d(k+1)}\times{\mathbb{R}}^{d(k+1)}}e^{|v_1|^2+\ldots+|v_{k+1}|^2}\,{\color{red}|}f_{k+1,N}{\color{red}|^q} \end{split}
\label{regularization2} \end{equation} still provided $1/p+1/q\leq 1$.
However pure Gaussian moments in velocity does not seem to be naturally propagated at the discrete level of the hierarchy, even though they would trivially be propagated on the limiting mean-field equation at least for short time. This leads to the final critical idea of the paper, which is to incorporate the potential energy in the Gaussian: Namely to consider~$e^{\lambda(t)\,e_k}$ instead of a pure Gaussian with \begin{equation}\label{ek}
e_k(x_1,v_1,\ldots,x_k,v_k)=\sum_{i\leq k} (1+|v_i|^2)+\frac{1}{N}\,\sum_{i,j\leq k} \phi(x_i-x_j). \end{equation} We remark that the use dynamical weights argument has been recently developed in \cite{BreJabWan} for first order particle systems with singular kernels. We also note that Proposition~\ref{propLp}, stated below, shows the propagation of weighted $L^q$ bounds on the marginals, without requiring the initial data to be chaotic or close to chaotic as introduced in \cite{Kac1956}. It hence applies to a broader framework than just the mean-field limit. \begin{proposition}\label{propLp}
Let us assume $K\in L^p(\Pi^d)$, for some $p>1$, and define
\[
\lambda(t)=\frac{1}{\Lambda\,(1+t)}, \qquad L = \frac{C}{\lambda(1)^\theta} \|K\|_{L^p}^q , \] for positive constants $\Lambda$, $C$, $\theta$ depending only on $q$, $d$ and $\sigma$ and $q$ and $1/q+1/p \le 1$. Consider a renormalized solution~$f_N$ to~\eqref{Liouville} satisfying~\eqref{gaussiandecay} with initial data $f_N^0\in L^\infty(\Pi^{dN}\times{\mathbb{R}}^{dN})$, satisfying \begin{equation} \begin{split}
& \int_{\Pi^{kd}\times{\mathbb{R}}^{kd}} |f^0_{k,N}|^q\,e^{\lambda(0)\,e_k}\leq F_0^k, \\
& \sup_{t\leq 1} \int_{\Pi^{Nd}\times{\mathbb{R}}^{Nd}} |f_{N}|^q\,e^{\lambda(t)\,e_N}\leq F^N, \end{split} \label{assumptlq} \end{equation} for some $F>0$, $F_0>0$ and $q$ such that $2\leq q<\infty$, with $1/q+1/p\leq 1$. Then, one has that \begin{equation}\label{Estim}
\sup_{t\leq T}\int_{\Pi^{kd}\times{\mathbb{R}}^{kd}} |f_{k,N}|^q\,e^{\lambda(t)\,e_k}\leq 2^{k}\,F_0^k+F^k\,2^{2k-N-1}, \end{equation} where $T$ is given by \[ T\leq \min\left(1, \frac{1}{4\,L\,\max(F_0,F)}\right). \]
\end{proposition}
Proposition~\ref{propLp} shows that the corresponding $L^q$ norm of a marginal at order $k$ behaves like $C^k$ for some constant $C$. This is the expected scaling for propagation of chaos and tensorized marginals $f_k=f^{\otimes k}$.
However Proposition~\ref{propLp} also presents several intriguing features that we want to highlight.
$\bullet$ {\em Vlasov-Poisson-Fokker-Planck in higher dimensions.} Proposition~\ref{propLp} handles just as easily Coulombian interactions in any dimension $d$, and not only dimension~$d=2$ as Theorem~\ref{Conv}. Therefore, as claimed, Proposition~\ref{propLp} already implies a result of propagation of chaos for the Vlasov-Poisson-Fokker-Planck system in any dimension. However, as is, this is not fully satisfying because we cannot take $f^0_N=(f^0)^{\otimes N}$: Assumption~\eqref{assumptlq} cannot hold in such a case as $e^{\lambda(0)\,e_k}$ is not integrable if $K$ is the Poisson kernel in dimension $d>2$. The issue is that by taking $f^0_N=(f^0)^{\otimes N}$, we allow some configurations with high potential energy. And roughly speaking the existence time $T$ in the proposition vanishes as the starting potential energy increases in that case.
$\bullet$ {\em Repulsive potentials.} Proposition~\ref{propLp} does require repulsive potentials $\phi\geq 0$ as this assumption is critical in the proof. The repulsive assumption on the potential only appears to be needed to handle the discrete many-particle system. The extension to non-repulsive settings remains an open problem.
$\bullet$ {\em Extension to the stochastic case of mildly singular kernels}. A special case concerns mildly singular kernels $K$ with $K\in L^p$ for some $p>1$ s.t. $\phi\in L^\infty$. In that situation, by considering $\phi+\|\phi\|_{L^\infty}$ instead of $\phi$, yielding the same interaction kernel~$K$, we can always ensure that $\phi\geq 0$. For example this easily extends for the first time to the stochastic settings the results of~\cite{HauJab1,HauJab2}, that had been obtained only for deterministic second order systems with $|K|\lesssim |x|^{-\alpha}$ for $\alpha<1$.
$\bullet$ {\em Convergence for finite times.} We finally emphasize that Proposition~\ref{propLp}, just as Theorem~\ref{Conv}, holds over a finite time interval, independent of $N$. This may initially appear puzzling since we are dealing with linear equations for any fixed $N$. However because those estimates are essentially independent of $N$, they also extend to the non-linear limiting Vlasov equation. Moreover Proposition~\ref{propLp} includes a propagation of Gaussian moments in velocity over the marginals, from the term $e^{\lambda(t)\,e_k}$ and the definition~\eqref{ek} of $e_k$. The propagation for all times of such moments for Vlasov--Poisson is only known in dimension $d=2$, see~\cite{UkaOka,De}, and dimension~$d=3$, see~\cite{Bouchut,GasJabPer,HolMio,LioPer,OnoStr,Pal,Pfa,ReiWec,Schaeffer,VicOdw} as cited in the introduction; it also requires in dimension~$3$ the use of dispersion estimates that are not present in our proof. As we already noted, Proposition~\ref{propLp} is in fact valid in any dimension which naturally limits it to some given finite time interval.
\subsection{The case of first order system}
While we focus on second order systems, we also emphasize that our method directly applies to first order systems on bounded domains (in a much simpler manner in fact) and provides the mean-field limit under very weak assumptions on the kernel~$K$ again.
Consider in that case \begin{equation} \frac{d}{dt} X_i(t)=\frac{1}{N}\,\sum_{j\neq i} K(X_i-X_j)\,dt+\sigma\,dW_i,,\quad X_i(t=0)=X_i^0, \label{Npart1st} \end{equation} fully on the torus~$\Pi^d$. The mean-field limit is similar to~\eqref{vlasov} \begin{equation} \partial_t f+ (K\star_x f)\cdot\nabla_x f=\frac{\sigma^2}{2}\,\Delta_x f. \label{vlasov1st} \end{equation} Similarly the joint law $f_N(t,x_1,\ldots,x_N)$ solves an appropriately modified Liouville equation \begin{equation}
\partial_t f_N +\sum_{i=1}^N \frac{1}{N}\,\sum_{j=1}^N K(x_i-x_j)\cdot\nabla_{v_i} f_N=\frac{\sigma^2}{2}\,\sum_i \Delta_{x_i} f_N. \label{Liouville1st} \end{equation} Because system~\eqref{Npart1st} does not involve velocities, many technical difficulties in our proofs actually vanish. For example, we do not need anymore to add assumptions such as~\eqref{gaussiandecay}. We do not need either to impose that $K$ derives from a potential and hence do not require assumptions like~\eqref{expphi} either.
We then have the following equivalent of Theorem~\ref{Conv}.
\begin{theorem} \label{Conv1st} Assume that \[ K \in L^p ({\Pi}^d) \quad \hbox{for some}\quad p>1,\qquad (\mbox{div}\, K)_-\in L^\infty(\Pi^d), \] where $x_-$ denotes the negative part of $x$. Let $f$ be the unique smooth solution to the Vlasov equation~\eqref{vlasov1st} with initial data $f^0\in C^\infty(\Pi^d)$. Consider moreover an entropy solution~$f_N$ to~\eqref{Liouville1st}, still in the sense of subsection~\ref{Liouvillewellposed}, with initial data $f_N^0\in L^\infty(\Pi^{dN})$. Assume that $f_{k,N}^0$ converges weakly in $L^1$ to $(f^0)^{\otimes k}$for each fixed $k$ and that \[
\|f_{k,N}^0\|_{L^\infty(\Pi^{dN})}\leq M^k , \]
for some $M>0$ and for all $k\leq N$. Then there exists $T^*$ depending only on $M$, $\|K\|_{L^p}$ and $\|(\mbox{div}\, K)_-\|_{L^\infty}$ such that $f_{k,N}$, given by \eqref{marginaldef}, weakly converge to $f_k= f^{\otimes k}$ in $L^q_{loc} ([0,\;T^\star] \times \Pi^{kd})$ for any $k$, and any $2<q<\infty$, with $1/q+1/p\le 1$. \end{theorem} Because it is not our main focus, we do not give a distinct proof of Theorem~\ref{Conv1st}.
\subsection{Our notion of entropy solution for the hierarchy; the well-posedness of Eq.~\eqref{Liouville}\label{Liouvillewellposed}}
\subsubsection{The definition}
Being non-linear, our estimates cannot be performed on any weak solutions. Moreover, the concept of solution for $f_N$ are carried over the marginals $f_{k,N}$ and not just the joint law $f_N$ so that we also need an appropriate notion of entropy solutions on those marginals.
\noindent {\it The hierarchy for the marginals from the Liouville equation.} From Eq.~\eqref{Liouville}, the $f_{k,N}$ solve the so-called BBGKY hierarchy \begin{equation} \begin{split}
\partial_t f_{k,N}&+\sum_{i=1}^k v_i\cdot\nabla_{x_i} f_{k,N}+\sum_{i\leq k}\,\frac{1}{N}\,\sum_{j\leq k} K(x_i-x_j)\cdot\nabla_{v_i} f_{k,N}\\
& +\frac{N-k}{N} \sum_{i\leq k} \nabla_{v_i} \cdot \int_{\Pi^d\times {\mathbb{R}}^d} f_{k+1,N}\,K(x_i-x_{k+1}) dx_{k+1} dv_{k+1}\\
&=\frac{\sigma^2}{2}\,\sum_{i\leq k} \Delta_{v_i} f_{k,N}.
\end{split}\label{hierarchy} \end{equation} If $f_N$ belongs to $L^\infty$ and verifies~\eqref{gaussiandecay}, then all marginals $f_{k,N}$ belong to $L^\infty_t L^q_{x,v}$ for every $q<\infty$ with similar Gaussian decay. For simplicity, we denote here abstractly $L^q_{x,v}$ any space $L^q(\Pi^{kd}\times{\mathbb{R}}^{kd})$ when there is no confusion about the dimension $k$, as in our case. We also denote by $L^q_{\lambda e_k}$ the weighted $L^q$ space \[
\|f\|_{L^q_{\lambda e_k}}^q=\int_{\Pi^{kd}\times{\mathbb{R}}^{kd}} |f|^q\,e^{\lambda\,e_k}. \]
Since $K\in L^p$, for some $p>1$, then by using a direct H\"older inequality those bounds on the $f_{k,N}$ implies that \[ \int_{\Pi^d\times {\mathbb{R}}^d} f_{k+1,N}\,K(x_i-x_{k+1})\,dx_{k+1}\,dv_{k+1}\in L^\infty_t L^q_{x,v}, \] for all $q<\infty$. This allows to immediately and rigorously derive~\eqref{hierarchy} from Eq.~\eqref{Liouville}.
\noindent {\it Definition of entropy solutions.} We denote the advection component of~\eqref{hierarchy} \begin{equation} \label{L} L_k=\sum_{i\leq k} v_i\cdot\nabla_{x_i}+\frac{1}{N}\,\sum_{i,j\leq k} K(x_i-x_j)\cdot\nabla_{v_i}. \end{equation} The argument above implies that the only difficulties to propagate our estimates in~\eqref{hierarchy} stem from $L_k$. Consequently we define our entropy solution in the following manner: A function~$f_N\in L^\infty([0,\ 1]\times\Pi^{dN}\times {\mathbb{R}}^{dN})$ satisfying~\eqref{gaussiandecay} is an entropy solution iff all marginals~$f_{k,N}$ for $1\leq k\leq N$, as defined by~\eqref{marginaldef}, satisfy that for any $T\in [0,\ 1]$, any $1<q<\infty$ and any $\lambda<\lambda_0$ \begin{equation} \begin{split}
\int_0^T\!\!\int_{\Pi^{dk}\times{\mathbb{R}}^{dk}} &e^{\lambda\,e_k}\,|f_{k,N}|^{q-1}\,\\&\mbox{sign}\,f_{k,N}\,L_k\,f_{k,N}\,dx_1\,dv_1\dots dx_k\,dv_k\,dt\geq 0. \end{split} \label{entropyformal} \end{equation} Inequality~\eqref{entropyformal} is still somewhat formal and should be understood in the following rigorous sense: For some smooth convolution kernel~$K_\varepsilon$, one has that \begin{equation} \begin{split}
\liminf_{\varepsilon\to 0}\int_0^T\!\!\int_{\Pi^{dk}\times{\mathbb{R}}^{dk}}&e^{\lambda\,e_k}\, |K_\varepsilon^{\otimes k}\star f_{k,N}|^{q-1}\, \mbox{sign}(K_\varepsilon^{\otimes k}\star f_{k,N})\\
&K_\varepsilon^{\otimes k}\star\,(L_k\,f_{k,N})\,dx_1\,dv_1\dots dx_k\,dv_k\,dt\geq 0, \label{entropyrigorous} \end{split} \end{equation} where we denote \[ \begin{split}
K_\varepsilon^{\otimes k}\star\,g=\int_{\Pi^{dk}\times{\mathbb{R}}^{dk}} &K_\varepsilon(x_1-y_1,v_1-w_1)\,\dots K_\varepsilon(x_k-y_k,v_k-w_k)\\
&g(y_1,w_1,\ldots,y_k,w_k)\,dy_1\,dw_1\dots dy_k\,dw_k. \end{split} \]
However it is usually more delicate to determine whether any weak solution~$f_N$ in $L^\infty$ and with the bound~\eqref{gaussiandecay} is an entropy solution according to our definition. For linear advection-diffusion equations such as~\eqref{Liouville}, this is usually approached through the notion of renormalized solutions as introduced in~\cite{DL}. In that context,~\eqref{entropyrigorous} is obviously similar to the classical commutator estimate at the basis of many methods for renormalized solutions.
\begin{remark} {\rm 1)} We first remark that~\eqref{entropyformal} is automatically satisfied if we have classical solutions. Indeed $L_k$ is an antisymmetric operator so that we expect it to propagate $L^q$ norms so that if all terms are smooth \[
|f_{k,N}|^{q-1}\, \mbox{sign}\,f_{k,N} \,L_k f_{k,N}=L_k\,|f_{k,N}|^q. \]
\noindent {\rm 2)} We immediately observe that the reduced energy $e_k$ is formally invariant under the advection component of~\eqref{hierarchy}: \[ L_k\,e_k=\frac{2}{N}\,\sum_{i,j\leq k} v_i\cdot \nabla_{x_i} \phi(x_i-x_j)+\frac{2}{N}\,\sum_{i,j\leq k} K(x_i-x_j)\cdot v_i=0, \] since $K=-\nabla_x \phi$. In the same way, we have $L_k\,\Phi(e_k)=0$, for any locally Lipschitz function $\Phi$.
\noindent {\rm 3)} If $K$ is smooth and $f_N$ is a classical solution to~\eqref{Liouville}, we would hence immediately have equality in \eqref{entropyformal}. With $K$ only in $L^p$, it would be straightforward to obtain one entropy solution in the sense defined above, through passing to the limit in a sequence of solutions for a smoother kernel~$K$. \end{remark}
\begin{remark} There exists an extensive literature on renormalized solutions with a comparably large variety of potential assumptions that one may consider. While we cannot do justice to this question in this short discussion, we briefly mention for instance~\cite{Ha} that studies the specific case of the Liouville equation~\eqref{Liouville} for second order systems without diffusion. In the present setting of a constant non-vanishing diffusion, we also refer to~\cite{BoKrRoSh,LeLi,LebLio} that provide broad results of well-posedness for velocity fields in $L^p$.
We in particular note that renormalized solutions apply to the case $K\in L^p$ with $p> 2$ and $f_N$ in $L^\infty$ with $\nabla_{v_i} f_N^{q/2}\in L^2$ for any $q<\infty$ and satisfying the extension of~\eqref{gaussiandecay}, \[ \sup_{t\leq 1}\int_{\Pi^{dN}\times{\mathbb{R}}^{dN}} e^{\lambda_0\,e_k}\,f_N\,dx_1\,dv_1\dots dx_N\,dv_N<\infty. \] The latter estimates are natural for the Liouville~\eqref{Liouville}, as demonstrated by Lemma~\ref{technicallemma} for the case $k=N$ in Section~\ref{proof}. In that situation, all marginals $f_{k,N}$ belong to $L^\infty_t L^q_{x,v}$ for every $q<\infty$ with similar exponential decay in $e_k$ and with as well $\nabla_{v_i} f_{k,N}^{q/2}\in L^r_{t,x,v}$ for any $r<2$. This regularity easily allows to prove that~\eqref{entropyrigorous} holds for $\lambda<\lambda_0$.
We also mention that so-called mild solutions can also offer a natural way to prove~\eqref{entropyrigorous}. We simply refer to~\cite{Bouchut,CS} for such formulations through the Fokker--Planck kernel in whole space, or to~\cite{Clark} or~\cite{De,VicOdw} for periodic conditions. \end{remark}
\subsubsection{\it Strong solutions up to the first collision.} We also emphasize that, in the case of repulsive kernels smooth out of the origin but with singular potentials $\lim_{x\to 0} \phi(x)=+\infty$, a straightforward bound on the energy of the system can easily lead to strong solutions on the many-particle system~\eqref{Npart}, bypassing the need for entropy or renormalized solutions.
Very roughly, if $K\in C^\infty(\Pi^d\setminus\{0\})$, then up to the conditional time of first collision in~\eqref{Npart}, we may write that \[
d\left(\sum_{i=1}^N |V_i|^2+\frac{1}{N}\,\sum_{i\neq j} \phi(X_i-X_j)\right)=\sigma^2\,dt+\sum_{i=1}^N 2\,V_i\cdot dW_i. \] This implies that, with probability $1$, the total energy remains finite if it was so initially. Because $\lim_{x\to 0} \phi(x)=+\infty$, it also implies that collisions almost surely never happen. This argument would in particular apply to the Coulombian case in any dimension~$d\geq 2$.
To conclude this discussion of the well posedness of~\eqref{Liouville} or~\eqref{Npart} for a fixed~$N$, we emphasize the estimates that we described here cannot easily be made uniform in $N$. The previous discussion of the energy bound on the system~\eqref{Npart} for the Coulombian interaction in dimension~$d=2$ is an excellent illustration: If we have the following bound \[
\sum_{i=1}^N |V_i|^2+\frac{1}{N}\,\sum_{i\neq j} \phi(X_i-X_j)\leq E \]
with some large probability on some time interval, and for $\phi(x)=-\log |x|$ then this only proves that for any $i\neq j$ \[
|X_i-X_j|\geq e^{-N\,E}, \] which is indeed finite for any fixed~$N$ but is completely unhelpful when considering the limit $N\to \infty$.
Hence the present discussion remains focused on renormalized solutions for a fixed~$N$. Quantitative approach to renormalized solutions have for example been introduced in~\cite{CrDe}, which are based on the propagation of a sort of $\log$-derivative on the characteristics; see also for example the discussion on Eulerian variants in~\cite{BrJa}. This leads to an interesting and so far mostly fully open question as to whether it would be possible to obtain quantitative bounds that would combine the limit $N\to\infty$ with some regularity estimates on the solution for a fixed~$N$.
\section{Proof of the main results\label{proof}}
\subsection{The BBGKY and Vlasov hierarchies\label{hierarchies}}
Using~\eqref{vlasov}, the tensorized limits $f_k= {\overline f}^{\otimes k}$ satisfy the following Vlasov hierarchy
\begin{equation} \begin{split}
\partial_t f_k &+ \sum_{i=1}^k v_i\cdot\nabla_{x_i} f_k\\ & +\sum_{i=1}^k (K\star \int_{{\mathbb{R}}^d} f dv) \cdot\nabla_{v_i} f _k =\frac{\sigma^2}{2}\,\sum_{i=1}^k \Delta_{v_i} f_k. \end{split}\label{vlasovhierarchy} \end{equation} To avoid repeating the analysis working on \eqref{hierarchy} or \eqref{vlasovhierarchy}, we introduce the generalized hierarchy equation \begin{equation} \begin{split}
& \hspace{-0.5cm} \partial_t F_{k,N}+\sum_{i=1}^k v_i\cdot\nabla_{x_i} F_{k,N}+\sum_{i\leq k}\,\frac{\gamma}{N}\,\sum_{j\leq k} K(x_i-x_j)\cdot\nabla_{v_i} F_{k,N}\\
& +\frac{N-\gamma k}{N}\sum_{i\leq k} \nabla_{v_i} \cdot \int_{\Pi^d\times {\mathbb{R}}^d} F_{k+1,N}\,K(x_i-x_{k+1})dx_{k+1}dv_{k+1}\\
&=\frac{\sigma^2}{2}\,\sum_{i\leq k} \Delta_{v_i} F_{k,N}+R_{k,N}.
\end{split}\label{hierarchyext} \end{equation} Note that Eq.~\eqref{hierarchyext} is exactly Eq.~\eqref{vlasovhierarchy} for $\gamma=0,\; R_{k,N}=0$ and exactly Eq.~\eqref{hierarchy} for $\gamma=1,\; R_{k,N}=0$. In the same spirit we denote \[\begin{split}
&e_{k,\gamma}=\sum_{i\leq k} (1+|v_i|^2)+\frac{\gamma}{N}\,\sum_{i,j\leq k} \phi(x_i-x_j),\\ &L_{k,\gamma}=\sum_{i\leq k} v_i\cdot\nabla_{x_i}+\frac{\gamma}{N}\,\sum_{i,j\leq k} K(x_i-x_j)\cdot\nabla_{v_i} \end{split} \] and observe that we of course still have $L_{k,\gamma}\,e_{k,\gamma}=0$.
The main technical contribution of this section and of the paper is Lemma~\ref{technicalLemma} stated in subsection~\ref{sectechnicalLemma}, which provides estimates for the solutions to \eqref{hierarchy}. We will then use the uniform bound on the $k$-marginals $f_{k,N}$ for the proof of Prop.~\ref{propLp}. Prop~\ref{propLp} allows passing to the limit in the hierarchy~\eqref{hierarchy} and a final use of Lemma~\ref{technicalLemma} leads to prove uniqueness to the limiting hierarchy~\eqref{vlasovhierarchy} to conclude result of Theorem~\ref{Conv}.
\subsection{The key technical lemma}\label{sectechnicalLemma}
We first present the key technical lemma which links the $k$-marginal $L^q_w$ control to the $(k+1)$-marginal $L^q_w$ estimate control. \begin{lemma}
Assume that $K\in L^p(\Pi^d)$, for some $p>1$. There exist some constants $\Lambda$, $C$, $\theta$ depending only on $q$, $d$ and $\sigma$ s.t
\[ \begin{split}
\|F_{k,N}\|_{L^q_{\lambda(t)\,e_k}}^q \leq &\|F_{k,N}(t=0)\|_{L^q_{\lambda(t)\,e_k}}^q\\& +q\,\int_0^t\int |F_{k,N}|^{q-1}\,\mbox{sign}\,F_{k,N}\,R_{k,N}\,e^{\lambda(s)\,e_{k,\gamma}}\,ds \\
&+ k\,\frac{N-\gamma\,k}{N}\,\frac{C}{\lambda^{\theta}(t)}\,\|K\|^q_{L^p}\,\int_0^t \|F_{k+1,N}(s)\|_{L^q_{\lambda(s)\,e_{k+1}}}^q\,ds. \end{split} \]\label{technicalLemma} for any entropy solution $F_{k,N}$ to~\eqref{hierarchyext} in the sense of subsection~\ref{Liouvillewellposed} and satisfying~\eqref{gaussiandecay} with $F_{k,N}\in L^q_{\lambda(t)\,e_{k,\gamma}}$, and for any $2\leq q<\infty$ such that $1/q+1/p\leq 1$, with $\lambda(t)$ defined by $\lambda(t)=\frac{1}{\Lambda\,(1+t)}$.
\end{lemma} \begin{proof} To be made fully rigorous, many calculations in this proof should involve a convolution kernel $K_\varepsilon$, estimating \[
\frac{d}{dt}\int |K_\varepsilon^{\otimes k}\star F_{k,N}|^{q}\,e^{\lambda(t)\,e_{k,\gamma}}, \] and passing to the limit in $\varepsilon\to 0$ while using appropriately the entropy condition~\eqref{entropyrigorous}. For simplicity however, we will only present the corresponding formal calculations.
We hence calculate in a straightforward manner \[ \begin{split}
&\frac{d}{dt}\int |F_{k,N}|^{q}\,e^{\lambda(t)\,e_{k,\gamma}}=q\,\int |F_{k,N}|^{q-1}\,\mbox{sign}\,F_{k,N}\,\partial_t F_{k,N}\,e^{\lambda(t)\,e_{k,\gamma}}\\
&\quad+\lambda'(t)\,\int e_{k,\gamma}\,|F_{k,N}|^{q}\,e^{\lambda(t)\,e_{k,\gamma}}.
\end{split} \] Inserting now in this identity the definition of $\lambda(t)$ and the equation~\eqref{hierarchy} we find \[ \begin{split}
&\frac{d}{dt}\int |F_{k,N}|^{q}\,e^{\lambda(t)\,e_{k,\gamma}}=-q\,\int |F_{k,N}|^{q-1}\,\mbox{sign}\,F_{k,N}\,(L_{k,\gamma} \,F_{k,N})\,e^{\lambda(t)\,e_{k,\gamma}}\\
&\quad +q\,\frac{\sigma^2}{2}\,\int |F_{k,N}|^{q-1}\,\mbox{sign}\,F_{k,N}\,\left(\sum_{i\leq k} \Delta_{v_i}\,F_{k,N}\right)\,e^{\lambda(t)\,e_{k,\gamma}}\\
&\quad-q\,\frac{N-\gamma\,k}{N}\,\sum_{i\leq k}\int |F_{k,N}|^{q-1}\,\mbox{sign}\,F_{k,N}\\
&\qquad\qquad\qquad\nabla_{v_i}\cdot\int K(x_i-x_{k+1})\,F_{k+1,N}\,dx_{k+1}\,dv_{k+1}\,e^{\lambda(t)\,e_{k,\gamma}}\\
&\quad-\Lambda\,\lambda^2(t)\,\int e_{k,\gamma}\,|F_{k,N}|^{q}\,e^{\lambda(t)\,e_{k,\gamma}} +q\,\int |F_{k,N}|^{q-1}\,\mbox{sign}\,F_{k,N}\,R_{k,N}\,e^{\lambda(t)\,e_{k,\gamma}}.
\end{split} \] Note that \[
q\,|F_{k,N}|^{q-1}\,\mbox{sign}\,F_{k,N}\,(L_{k,\gamma}\,F_{k,N})=L_{k,\gamma} \,|F_{k,N}|^q, \] so that by integration by parts, we formally have that \[ \begin{split}
&q\,\int |F_{k,N}|^{q-1}\,\mbox{sign}\,F_{k,N}\,(L_{k,\gamma}\,F_{k,N})\, e^{\lambda(t)\,e_{k,\gamma}}\\
& \qquad = -\int |F_{k,N}|^q\,L_{k,\gamma}\,e^{\lambda(t)\,e_{k,\gamma}}=0. \end{split} \] On the other hand, again by integration by parts \[ \begin{split}
&q\,\frac{\sigma^2}{2}\,\int |F_{k,N}|^{q-1}\,\mbox{sign}\,F_{k,N}\,\left(\sum_{i\leq k} \Delta_{v_i}\,F_{k,N}\right)\,e^{\lambda(t)\,e_{k,\gamma}}\\
&\quad=-q\,(q-1)\,\sum_{i\leq k}\,\frac{\sigma^2}{2}\,\int |F_{k,N}|^{q-2}\,|\nabla_{v_i} F_{k,N}|^2\,e^{\lambda(t)\,e_{k,\gamma}}\\
&\quad-2\,q\,\lambda(t)\,\sum_{i\leq k}\,\frac{\sigma^2}{2}\,\int |F_{k,N}|^{q-1}\,\mbox{sign}\,F_{k,N}\,v_i\cdot\nabla_{v_i} F_{k,N}\,e^{\lambda(t)\,e_{k,\gamma}}. \end{split} \] By Cauchy-Schwartz, since $q\ge 2$, we obtain that \[ \begin{split}
&q\,\frac{\sigma^2}{2}\,\int |F_{k,N}|^{q-1}\,\mbox{sign}\,F_{k,N}\,\left(\sum_{i\leq k} \Delta_{v_i}\,F_{k,N}\right)\,e^{\lambda(t)\,e_{k,\gamma}}\\
&\quad\leq -q\,(q-1)\,\sum_{i\leq k}\,\frac{\sigma^2}{4}\,\int |F_{k,N}|^{q-2}\,|\nabla_{v_i} F_{k,N}|^2\,e^{\lambda(t)\,e_{k,\gamma}}\\
&\qquad+ \frac{q}{q-1}\, \lambda^2\,\frac{\sigma^2}{2} \,\int |F_{k,N}|^q\,\sum_{i\leq k} |v_i|^2\,e^{\lambda(t)\,e_{k,\gamma}}. \end{split} \]
Note that since $\phi\geq 0$, we have that $\sum_{i\leq k} |v_i|^2\leq e_k$ and, therefore, combining all our estimates so far, we deduce that \[ \begin{split}
&\frac{d}{dt}\int |F_{k,N}|^{q}\,e^{\lambda(t)\,e_{k,\gamma}} \leq -q(q-1)\sum_{i\leq k}\,\frac{\sigma^2}{4}\,\int |F_{k,N}|^{q-2}\,|\nabla_{v_i} F_{k,N}|^2\,e^{\lambda(t)\,e_{k,\gamma}}\\
&\ -q\,\frac{N-\gamma\,k}{N}\,\sum_{i\leq k}\int |F_{k,N}|^{q-1}\,\mbox{sign}\,F_{k,N}\\
&\qquad\qquad\qquad\nabla_{v_i}\cdot\int K(x_i-x_{k+1})\,F_{k+1,N}\,dx_{k+1}\,dv_{k+1}\,e^{\lambda(t)\,e_{k,\gamma}}\\
&\ -\frac{\Lambda}{2}\,\lambda^2(t)\,\int e_{k,\gamma}\,|F_{k,N}|^{q}\,e^{\lambda(t)\,e_{k,\gamma}} +q\,\int |F_{k,N}|^{q-1}\,\mbox{sign}\,F_{k,N}\,R_{k,N}\,e^{\lambda(t)\,e_{k,\gamma}},
\end{split} \] provided that $\Lambda\geq \frac{q}{q-1}\,\sigma^2$.
We integrate by parts the second term in the right-hand side to obtain \[ \begin{split}
&\sum_{i\leq k}\int |F_{k,N}|^{q-1}\,\mbox{sign}\,F_{k,N}\\
&\qquad\nabla_{v_i}\cdot\int K(x_i-x_{k+1})\,F_{k+1,N}\,dx_{k+1}\,dv_{k+1}\,e^{\lambda(t)\,e_{k,\gamma}}=RH_1+RH_2,\\ \end{split} \] with \[ \begin{split}
RH_1 = - (q-1)\, \sum_{i\leq k}&\int |F_{k,N}|^{q-2}\nabla_{v_i} F_{k,N}\\
&\times \int K(x_i-x_{k+1})\,F_{k+1,N}\,dx_{k+1}\,dv_{k+1}\,e^{\lambda(t)\,e_{k,\gamma}},\\ \end{split} \] and \[ \begin{split}
RH_2= - 2\lambda(t)\, \sum_{i\leq k}&\int |F_{k,N}|^{q-1}\,\mbox{sign}\,F_{k,N}\,v_i\\
& \times \int K(x_i-x_{k+1})\,F_{k+1,N}\,dx_{k+1}\,dv_{k+1}\,e^{\lambda(t)\,e_{k,\gamma}}. \end{split} \] We perform a straightforward Cauchy-Schwartz inequality on both terms to find that \[ \begin{split}
RH_2
& \leq \frac{\lambda^2(t)}{2}\,\sum_{i\leq k} \int |F_{k,N}|^q\,|v_i|^2\,e^{\lambda(t)\,e_{k,\gamma}}\\
&\quad + \sum_{i\leq k} \int\frac{|F_{k,N}|^{q-2}}{2} \left|\int K(x_i-x_{k+1}) F_{k+1,N}\,dx_{k+1}\,dv_{k+1}\right|^2\,e^{\lambda(t)e_{k,\gamma}}, \end{split} \] and similarly \[ \begin{split}
&RH_1\leq \frac{\sigma^2}{4}\,\sum_{i\leq k} \int |F_{k,N}|^{q-2}\,|\nabla_{v_i} F_{k,N}|^2\,e^{\lambda(t)\,e_{k,\gamma}}\\
&+\frac{(q-1)^2}{\sigma^2}\sum_{i\leq k} \int |F_{k,N}|^{q-2}\left|\int K(x_i\!-\!x_{k+1})F_{k+1,N}dx_{k+1}dv_{k+1}\right|^2e^{\lambda(t)e_{k,\gamma}}. \end{split} \] Note that by H\"older estimates \[ \begin{split}
& \int |F_{k,N}|^{q-2}\,\left|\int K(x_i-x_{k+1})\,\,F_{k+1,N}\,dx_{k+1}\,dv_{k+1}\right|^2\,e^{\lambda(t)\,e_{k,\gamma}} \\
&\qquad\leq \frac{q-2}{q}\,\lambda^2\,\int |F_{k,N}|^q\,e^{\lambda(t)\,e_{k,\gamma}}\\
&\qquad +\frac{2}{q\,\lambda^{q-2}}\,\int e^{\lambda(t)\,e_{k,\gamma}}\,\left|\int K(x_i-x_{k+1})\,\,F_{k+1,N}\,dx_{k+1}\,dv_{k+1}\right|^q. \end{split} \] Therefore, combining together all those terms, we obtain the further estimate \[ \begin{split}
\sum_{i\leq k}\int &|F_{k,N}|^{q-1} \mbox{sign}\left(F_{k,N}\right)\nabla_{v_i}\cdot\int K(x_i\!-\!x_{k+1})\,F_{k+1,N}dx_{k+1}dv_{k+1}e^{\lambda(t)\,e_{k,\gamma}}\\
&\leq \frac{\sigma^2}{4}\,\sum_{i\leq k} \int |F_{k,N}|^{q-2}\,|\nabla_{v_i} F_{k,N}|^2\,e^{\lambda(t)\,e_{k,\gamma}}\\
&\quad+ \lambda^2(t)\,\left(\frac{1}{2}+\frac{(q-2)\,(q-1)^2}{q\,\sigma^2}\right)\, \sum_{i\leq k} \int |F_{k,N}|^q\,(1+|v_i|^2)\,e^{\lambda(t)\,e_{k,\gamma}}\\
&\quad +\frac{2}{q\,\lambda^{q-2}}\left(\frac{1}{2}+\frac{(q-1)^2}{\sigma^2}\right)\\
&\hspace{2cm}\times \sum_{i\leq k}\int e^{\lambda(t)\,e_{k,\gamma}}\left|\int K(x_i-x_{k+1})\,F_{k+1,N}\,dx_{k+1} dv_{k+1}\right|^q. \end{split} \] Hence, provided that \[ \Lambda\geq q\,\left(1+2\,\frac{((q-2)\,q-1)^2}{q\,\sigma^2}\right), \] we obtain that \[ \begin{split}
&\frac{d}{dt}\int |F_{k,N}|^{q}\,e^{\lambda(t)\,e_{k,\gamma}} \\
&\le C_{q,\sigma,d}\,k\frac{N\!-\!\gamma k}{\lambda^{q-2}\,N}\,\int e^{\lambda(t)\,e_{k,\gamma}}\left|\int K(x_1\!-\!x_{k+1})F_{k+1,N}dx_{k+1}dv_{k+1}\right|^q. \\ \end{split} \] At this point
is where we take advantage of the specific structure of the hierarchy. We bound \[ \begin{split}
&\hspace{-1cm} \left|\int K(x_1-x_{k+1})\,F_{k+1,N}\,dx_{k+1}\,dv_{k+1}\right|^q\\
& \leq \left(\int |K(x_1-x_{k+1})|^{q^*} \,e^{-\frac{q^*}{q}\,\lambda(t)\,|v_{k+1}|^2}\,dx_{k+1}\,dv_{k+1}\right)^{q/q^*}\\
&\qquad \times \int |F_{k+1,N}|^q\,e^{\lambda(t)\,|v_{k+1}|^2}\,dx_{k+1}\,dv_{k+1},\\ \end{split} \] which implies \[ \begin{split}
&\hspace{-1cm} \left|\int K(x_1-x_{k+1})\,F_{k+1,N}\,dx_{k+1}\,dv_{k+1}\right|^q\\
& \leq \frac{C_{q,\sigma,d}}{\lambda^{q\,d/2q^*}(t)}\,\|K\|^q_{L^p}\,\int |F_{k+1,N}|^q\,e^{\lambda(t)\,|v_{k+1}|^2}\,dx_{k+1}\,dv_{k+1}, \end{split} \] since $q\geq p^*$. Consequently \[ \begin{split}
&\int e^{\lambda(t)\,e_{k,\gamma}}\,\left|\int K(x_1-x_{k+1})\,\,F_{k+1,N}\,dx_{k+1}\,dv_{k+1}\right|^q\\
& \leq \frac{C_{q,\sigma,d}}{\lambda^{qd/2q^*}(t)}\|K\|^q_{L^p}\int |F_{k+1,N}|^q\,e^{\lambda(t)\,|v_{k+1}|^2+\lambda(t)\,e_{k,\gamma}}\,dx_1dv_1\dots dx_{k+1}dv_{k+1}.
\end{split} \] Note that \[
e_{k+1,\gamma}=e_{k,\gamma}+1+|v_{k+1}|^2+\frac{2\,\gamma}{N}\,\sum_{i\leq k} \phi(x_i-x_{k+1})\geq e_{k,\gamma}+1+|v_{k+1}|^2, \] so that \[ \begin{split}
&\int e^{\lambda(t)\,e_k} \left|\int K(x_i-x_{k+1})\,\,f_{k+1,N}\,dx_{k+1}\,dv_{k+1}\right|^q\\
&\quad \leq \frac{C_{q,\sigma,d}}{\lambda^{qd/2q^*}(t)}\,\|K\|^q_{L^p}\,\int |f_{k+1,N}|^q\,e^{\lambda(t)\,e_{k+1}}\,\,dx_1\,dv_1\dots dx_{k+1}\,dv_{k+1}.
\end{split} \] This finally lets us conclude, as claimed, that \[ \begin{split}
\frac{d}{dt}\int |f_{k,N}|^{q}\,e^{\lambda(t)\,e_{k,\gamma}} & \leq k\,\frac{N-\gamma\,k}{N}\,\frac{C_{q,\sigma},d}{\lambda^{\theta_{q,d}}(t)}\,\|K\|^q_{L^p}\,\int |f_{k+1,N}|^q\,e^{\lambda(t)\,e_{k+1,\gamma}}\\
&\quad +q\,\int |F_{k,N}|^{q-1}\,\mbox{sign}\,F_{k,N}\,R_{k,N}\,e^{\lambda(t)\,e_{k,\gamma}}. \end{split} \] \end{proof}
\subsection{Proof of technical results} \label{conclusion}
We start this subsection with the proof of the Proposition \ref{propLp}. \begin{proof}[Proof of Proposition~\ref{propLp}] From the analysis in subsection~\ref{hierarchies} and the assumptions \eqref{gaussiandecay} and~\eqref{assumptlq} of Proposition~\ref{propLp}, we have that $F_{k,N} = f_{k,N}$ is a renormalized solution to~\eqref{hierarchy} and thus~\eqref{hierarchyext} with $\gamma=1$. Moreover, $f_{k,N}$ satisfies the other assumptions in Lemma \ref{technicalLemma} with $R_{k,N}=0$. Denoting \[
X_k(t)=\int |f_{k,N}|^{q}\,e^{\lambda(t)\,e_k}, \] we hence observe that, by Lemma~\ref{technicalLemma}, we have the coupled dynamical inequality system \[ X_k(t)\leq X_k(0)+k\,L\,\int_0^t X_{k+1}(s)\,ds, \] for any $t\in [0,\ 1]$, where \[
L=\frac{C}{\lambda^\theta(1)}\,\|K\|^q_{L^p}. \] From the assumptions of Proposition~\ref{propLp}, we immediately have that \begin{equation} X_k(t)\leq F_0^k+k\,L\,\int_0^t X_{k+1}(s)\,ds.\label{basicinduction} \end{equation} We now invoke the following simple lemma \begin{lemma}\label{indu}
Consider any sequence $X_k(t)$ satisfying~\eqref{basicinduction} then one has that
\begin{equation}
\begin{split} X_k(t)\leq &\sum_{l=k}^m F_0^l\,L^{l-k}\,t^{l-k}\,\frac{(l-1)!}{(k-1)!\,(l-k)!}\\ &+L^{m+1-k}\,\int_0^t X_{m+1}(s)\,(t-s)^{m-k}\,\frac{m!}{(k-1)!\,(m-k)!}\,ds. \end{split}\label{induction}
\end{equation}
\label{lemsimple}
\end{lemma} Assuming that Lemma~\ref{lemsimple} holds, we use \eqref{induction} up to $m+1=N$ to derive through the assumptions on $f_N$ that \[ \begin{split} X_k(t)\leq &\sum_{l=k}^{N-1} F_0^l\,L^{l-k}\,t^{l-k}\,\frac{(l-1)!}{(k-1)!\,(l-k)!}\\ & \hskip1cm +L^{N-k}\,\int_0^t F^N\,(t-s)^{N-1-k}\,\frac{(N-1)!}{(k-1)!\,(N-1-k)!}\,ds, \end{split} \] that is \begin{equation} \begin{split} &X_k(t)\leq \sum_{l=k}^{N-1} F_0^l\,L^{l-k}\,t^{l-k}\,\frac{(l-1)!}{(k-1)!\,(l-k)!} \\ & \hskip1.5cm +F^N\,L^{N-k}\,t^{N-k}\,\frac{(N-1)!}{(k-1)!\,(N-k)!}.\label{finalinduction} \end{split} \end{equation} Note that \[ \frac{(l-1)!}{(k-1)!\,(l-k)!}=\binom{l-1}{k-1}\leq 2^{l-1}. \] Hence, \eqref{finalinduction} implies that \[ \begin{split}
X_k(t)&\leq \sum_{l=k}^{N} F_0^l\,L^{l-k}\,t^{l-k}\,2^{l-1}+F^N\,L^{N-k}\,t^{N-k}\,2^{N-1}\\
&=2^{k-1}\, F^k_0\,\sum_{l=k}^{N-1} F_0^{l-k}\,2^{l-k}\,L^{l-k}\,t^{l-k}
+F^k 2^{k-1} F^{N-k} \,L^{N-k}\,t^{N-k}\,2^{N-k}\\
&\leq 2^{k-1}\, F_0^k (2- 2^{k-N+1}) + F^k 2^{k-1} 2^{k-N}\\
& \leq F_0^k 2^k + F^k 2^{2k-N-1} \end{split} \] provided that $4\,L\,t\,\max(F_0,F)<1$, which concludes the proof of the proposition. \end{proof}
We finish with the quick proof of Lemma~\ref{lemsimple}. \begin{proof}[Proof of Lemma~\ref{lemsimple}] Taking $m=k$ in \eqref{induction}, we get \[ X_k(t)\leq F_0^k+L\,\int_0^t X_{k+1}(s)\,\frac{k!}{(k-1)!\,(k-k)!}\,ds, \] which is our starting point. Moreover assuming that~\eqref{induction} holds for $m$, we may use~\eqref{basicinduction} to find \[ \begin{split} &X_k(t)\leq \sum_{l=k}^m F_0^l\,L^{l-k}\,t^{l-k}\,\frac{(l-1)!}{(k-1)!\,(l-k)!}\\
&+L^{m+1-k}\,\int_0^t \left(F_0^{m+1}+L\,(m+1)\,\int_0^s X_{m+2}(s)\,ds\right)\\
&\qquad\qquad\qquad\qquad\qquad(t-s)^{m-k}\,\frac{m!}{(k-1)!\,(m-k)!}\,ds. \end{split} \] This yields \[ \begin{split}
X_k(t)\leq &\sum_{l=k}^m F_0^l\,L^{l-k}\,t^{l-k}\,\frac{(l-1)!}{(k-1)!\,(l-k)!}\\
&+L^{m+1-k}\,F_0^{m+1}\,\frac{m!}{(k-1)!\,(m-k)!}\,\int_0^t (t-s)^{m-k}\,ds\\ &+L^{m+2-k} \int_0^t X_{m+2}(r) \int_r^t (t-s)^{m-k}\,ds\,dr \frac{(m+1)!}{(k-1)!\,(m-k)!}\,ds, \end{split} \] or \[ \begin{split}
X_k(t)&\leq \sum_{l=k}^m F_0^l\,L^{l-k}\,t^{l-k}\,\frac{(l-1)!}{(k-1)!\,(l-k)!}\\
&+L^{m+1-k}\,F_0^{m+1}\,\frac{m!}{(k-1)!\,(m+1-k)!}\, t^{m+1-k}\\ &+L^{m+2-k}\int_0^t X_{m+2}(r) (t-r)^{m+1-k}\,ds\,dr \frac{(m+1)!}{(k-1)!\,(m+1-k)!}\,ds, \end{split} \] as claimed. \end{proof}
\subsection{Proof of Theorem \ref{Conv}}
The proof of Theorem~\ref{Conv} follows closely the steps in the proof of Proposition~\ref{propLp}, once appropriate bounds have been derived.
\noindent 1){ \em Uniform bounds on $f_N$ in $L^q_{e_N}$}. First of all, note that from the assumptions of Theorem~\ref{Conv}, we can easily obtain a bound on $f_N^0$ in $L^q_{\lambda^0\,e_N}$ for $\Lambda$ large enough. Indeed \[ \begin{split}
&\int_{\Pi^{dN}\times {\mathbb{R}}^{dN}} |f_N^0|^q\,e^{\lambda^0\,e_N}\\
&\qquad=e^N\,\int_{\Pi^{dN}\times {\mathbb{R}}^{dN}} |f_N^0|^q\,e^{2\,\lambda^0\,\sum_{i\leq N} |v_i|^2}\,e^{\frac{\lambda^0}{N}\,\sum_{i,j\leq N} \phi(x_i-x_j)-\lambda^0\,\sum_{i\leq N} |v_i|^2}. \end{split} \]
We have straightforward $L^r$ estimates on $e^{\frac{\lambda^0}{N}\,\sum_{i,j\leq N} \phi(x_i-x_j)-\lambda^0\,\sum_{i\leq N} |v_i|^2}$ as by H\"older inequality, \[ \begin{split}
\int_{\Pi^{dN}\times {\mathbb{R}}^{dN}} & e^{\frac{r\,\lambda^0}{N}\,\sum_{i,j\leq N} \phi(x_i-x_j)-r\,\lambda^0\,\sum_{i\leq N} |v_i|^2}\\
&=\frac{C^N}{\lambda_0^{N/2}}\,\int_{\Pi^{dN}} e^{\frac{r\,\lambda^0}{N}\,\sum_{i,j\leq N} \phi(x_i-x_j)}\\
& \leq \frac{C^N}{\lambda_0^{N/2}}\,\left(\Pi_{i\leq N} \int_{\Pi^{dN}} e^{r\,\lambda^0\sum_{j\leq N} \phi(x_i-x_j)}\right)^{1/N}\leq \frac{C^N}{\lambda_0^{N/2}}, \end{split} \] from some constant $C$ and by assumption~\eqref{expphi} in Theorem~\ref{Conv}, provided that $r\,\lambda^0\leq 1/\theta$. This implies, again by H\"older inequality \[ \begin{split}
\int_{\Pi^{dN}\times {\mathbb{R}}^{dN}} & |f_N^0|^q\,e^{\lambda^0\,e_N}\leq \frac{C^N}{\lambda_0^{N/2}}\,\int_{\Pi^{dN}\times {\mathbb{R}}^{dN}} |f_N^0|^{r^*\,q}\,e^{2\,r^*\,\lambda^0\,\sum_{i\leq N} |v_i|^2}\\
&\leq \frac{C^N}{\lambda_0^{N/2}}\,\|f_N^0\|_{L^\infty}^{q\,r^*-1}\,\int_{\Pi^{dN}\times {\mathbb{R}}^{dN}} |f_N^0|^{r^*\,q}\,e^{2\,r^*\,\lambda^0\,\sum_{i\leq N} |v_i|^2}. \end{split} \] Using now assumption~\eqref{gaussiandecay}, provided that $2\,r^*\,\lambda_0\leq\beta$, we conclude that \begin{equation}
\int_{\Pi^{dN}\times {\mathbb{R}}^{dN}} |f_N^0|^q\,e^{\lambda^0\,e_N}\leq \left(\frac{C\,V\,M}{\lambda_0}\right)^N,\label{boundfN0} \end{equation} for any $q<\infty$. We now choose any fixed $2<q<\infty$ such that $1/p+1/q<1$ and we remark that the Liouville Eq.~\eqref{Liouville} is included in Eq~\eqref{hierarchyext} for $\gamma=1$, $R_{k,N}=0$, and $k=N$. Thus, we next invoke Lemma~\ref{technicalLemma} for $f_N$ with $k=N$ and $\gamma=1$, to find that $f_N$ solves \[
\frac{d}{dt} \int_{\Pi^{dN}\times {\mathbb{R}}^{dN}} |f_N(t,.,.)|^q\,e^{\lambda(t)\,e_N}\leq 0, \] so that from~\eqref{boundfN0}, we obtain that \[
\sup_{t\leq 1}\int_{\Pi^{dN}\times {\mathbb{R}}^{dN}} |f_N(t,.,.)|^q\,e^{\lambda(t)\,e_N}\leq \left(\frac{C\,V\,M}{\lambda_0}\right)^N. \] This finally implies that there exists some constant $F>0$ such that \begin{equation}\label{fNest}
\sup_{t\leq 1}\int_{\Pi^{dN}\times {\mathbb{R}}^{dN}} |f_N(t,.,.)|^q\,e^{\lambda(t)\,e_N}\leq F^N.
\end{equation}
\noindent 2) {\em Uniform estimates on the marginals and passing the limit in the hierarchy~\eqref{hierarchy}.} First of all we can perform the same bounds on each $f_{k,N}^0$ to find similarly to~\eqref{boundfN0} that
\[
\int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} |f_{k,N}^0|^q\,e^{\lambda^0\,e_k}\leq \left(\frac{C\,V\,M}{\lambda_0}\right)^k.
\]
As a consequence, every assumptions of Proposition~\ref{propLp} hold, and, in particular, assumption~\eqref{assumptlq}. This implies that for some time $T^*>0$, depending only on $V$, $M$, $\|K\|_{L^p}$ and the choice of $q$, we have that
\[
\sup_N\sup_{t\leq T^*}\int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} |f_{k,N}|^q\,e^{\lambda(t)\,e_k}\leq \bar M^k,
\]
for some constant $\bar M$. At this point, we will not need anymore the potential in the reduced energy $e_k$, which was required to handle the $L_k$ operator that vanishes at the limit. For this reason, and since $\phi\geq 0$, we deduce from the previous inequality
\begin{equation}
\sup_N\sup_{t\leq T^*}\int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} |f_{k,N}|^q\,e^{\lambda(T^*)\,\sum_{i\leq k} |v_i|^2}\leq \bar M^k.\label{estfkN} \end{equation} These uniform bounds let us extract a converging subsequence such that all $f_{k,N}$ converge weak-$\star$ to some $\bar f_k$ in $L^\infty([0,\ T^*],\;L^q_{x,v})$ that also satisfies
\begin{equation}
\sup_{t\leq T^*}\int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} |\bar f_{k}|^q\,e^{\lambda(T^*)\,\sum_{i\leq k} |v_i|^2}\leq \bar M^k,\label{estbarfk} \end{equation} where we have used classical convex estimates. We emphasize that for the moment we only have convergence of a subsequence, though we still denote it by $N$ for simplicity. We eventually obtain the convergence of the whole sequence only after the uniqueness of the limit is proved in the next step.
From estimate~\eqref{estfkN}, and since $1/q+1/p\leq 1$, we may simply bound
\[
\left\|\sum_{i\leq k}\frac{1}{N}\sum_{j\leq k} K(x_i-x_j)\cdot\nabla_{v_i} f_{k,N}\right\|_{L^\infty_t L^1_{x,v,loc}}\lesssim \frac{k^2}{N}\,\|K\|_{L^p}\,\|f_{k,N}\|_{L^\infty_t L^q_{x,v}}.
\]
For any fixed $k$, the corresponding term vanishes as $N\to\infty$. Similarly estimate~\eqref{estfkN} allows to pass to the limit
\[
\begin{split}
&\int_{\Pi^d\times{\mathbb{R}}^d} K(x_i-x_{k+1})\,f_{k+1,N}\,dx_{k+1}\,dv_{k+1}\\
&\qquad\longrightarrow \int_{\Pi^d\times{\mathbb{R}}^d} K(x_i-x_{k+1})\,\bar f_{k+1}\,dx_{k+1}\,dv_{k+1}, \end{split}
\]
for the weak-$\star$ topology of $L^\infty([0,\ T^*],\;L^q_{x,v})$. It is straightforward to pass to the limit in the sense of distributions in all other terms of the hierarchy~\eqref{hierarchy} so that we deduce that $\bar f_k$ is a solution to the limiting hierarchy~\eqref{vlasovhierarchy} in the sense of distributions.
We can also easily identify the initial value of $\bar f_k$. From~\eqref{hierarchy} and the bounds derived from~\eqref{estfkN}, we immediately obtain a uniform bound on $\partial_t f_{k,N}$ in $L^\infty_t W^{-1,q}_{x,v,loc}$. By the assumption of Theorem~\ref{Conv}, $f_{k,N}^0$ converges weakly to $(f^0)^{\otimes k}$ so that we have $\bar f_k(t=0)=(f^0)^{\otimes k}$.
\noindent 3) {\em Uniqueness on the limiting hierarchy and conclusion}. We first argue that $\bar f_k$ is automatically a renormalized solution to~\eqref{vlasovhierarchy}. Indeed Eq.~\eqref{vlasovhierarchy} can be seen as a linear advection-diffusion equation with a locally Lipschitz velocity field $(v_1,\ldots,v_k)$ and a remainder \[ \nabla_{v_i}\cdot \int_{\Pi^d\times{\mathbb{R}}^d} K(x_i-x_j)\,\bar f_{k+1}\,dx_{k+1}\,dv_{k+1}, \] that belongs to $L^\infty_t L^q_{x,v}$ with $q>2$ per our prior estimates.
Next we note that since $f$ is a classical solution to the vlasov equation \eqref{vlasov}, the $f^{\otimes k}$ also yield renormalized solutions to the Vlasov hierarchy \eqref{vlasovhierarchy} for every $k\geq 1$. Due to the linearity in terms of the sequence $\{f_k\}_{k \in {\mathbb N}^\star}$ of the Vlasov hierarchy, we get that each $F_k= \bar f_k-f^{\otimes k}$ is also a renormalized solution to the Vlasov Hierarchy~\eqref{vlasovhierarchy} for every $k$. Moreover by the identification above of the initial with zero initial data.
Furthermore, by~\eqref{estbarfk} and the assumption of Gaussian decay on~$f^0$, we have that \begin{equation}
\sup_{t\leq T^*}\int_{\Pi^{kd}\times{\mathbb{R}}^{kd}} |F_k|^q\,e^{\tilde\beta\,\sum_{i\leq k} (1+|v_i|)^2}\leq \tilde M^k, \label{tildeM} \end{equation}
for some~$\tilde\beta$ and some $\tilde M$. Eq.~\eqref{vlasovhierarchy} corresponds to Eq.~\eqref{hierarchyext} in the case $\gamma=0$, where $e_{k,\gamma}$ reduces to $e_{k,0}=\sum_{i\leq k} (1+|v_i|)^2$. Hence, provided we choose some $\tilde \Lambda$, possibly lower than $\Lambda$ we satisfy all assumptions from Lemma~\ref{technicalLemma}.
Denoting by $Y_k = \int |F_k|^qe^{\tilde \lambda(t) e_{k,0}}$, we get for all $k\in {\mathbb N}^\star$ \[ Y_k(t) \le k\,\tilde L\, \int_0^t Y_{k+1} \, ds. \] We can then use Lemma \ref{indu} with $F_0=0$ up to any arbitrary $m$ to show, together with~\eqref{tildeM}, that \begin{equation} \label{epsilon} \begin{split}
Y_k(t) &\le \tilde L^{m+1-k}\,\tilde M^{m+1}\,\int_0^t (t-s)^{m-k}\,\frac{m!}{(k-1)!\,(m-k)!}\,ds\\
&\leq \tilde L^{m+1-k}\,\tilde M^{m+1}\,t^{m+1-k}\,\binom{m}{k-1}\leq 2^k\,\tilde M^k\,(2\,\tilde L\,\tilde M\,t)^{m+1-k}. \end{split} \end{equation} By taking $t<T_0$ with $T_0$ small enough and sending $m$ to $\infty$, we obtain that $Y_k(t)=0$ and hence $\bar f_k=f^{\otimes k}$ on $[0,\ T_0]$. This allows to repeat the argument starting from~$t=T_0$ instead of $t=0$ until we reach the maximum time~$T^*$. This finally allows to conclude as claimed that $\bar f_k=f^{\otimes k}$ over the whole interval $[0,\ T^*]$.
Coming back to our extracted subsequence on $f_{k,N}$, since all such subsequences have the same limit, we have convergence of the whole sequence to the $f^{\otimes k}$ concluding the proof.
\subsection{Proof of Theorem \ref{quantitative}}
The aim of this result is to provide a quantitative estimate between $f_{k,N}$ and $f_k$ that satisfy \eqref{hierarchy} and \eqref{vlasovhierarchy}, respectively, for the tensorized limits $f_k= {\overline f}^{\otimes k}$. First let us note that $F_k^N= f_{k,N} -f_k$ satisfies, \begin{equation} \begin{split} \partial_t F_k^N & + L_k F_k^N \\ & + \frac{N-k}{N} \sum_{i=1}^k \nabla_{v_i} \cdot \int_{\Pi^d \times {\mathbb{R}}^d} F_{k+1}^N K(x_i - x_{k+1}) \, d x_{k+1} d v_{k+1}\\ & =
\frac{\sigma^2}{2} \sum_{i=1}^k \Delta_{v_i} F_{k,N} + R_{k,N}
\end{split} \end{equation} where $L_k $ is defined in \eqref{L}, and \begin{equation} \begin{split} R_{k,N} = &
\sum_{i=1}^k \left[\Big(K\star \int_{{\mathbb{R}}^d} \overline f \Big)(t,x_i) - \frac{1}{N} \sum_{j=1}^kK(x_i-x_j)\right] \cdot \nabla_{v_i} f_k. \\ & - \frac{N-k}{N} \sum_{i=1}^k \nabla_{v_i} \cdot \int_{\Pi^d\times {\mathbb{R}}^d} f_{k+1} K(x_i-x_{k+1}) d x_{k+1} d v_{k+1}. \end{split} \end{equation} We again use Lemma~\ref{technicalLemma} with $q=2$ to deduce \begin{equation} \label{FkN} \begin{split}
\frac{d}{dt} \int_{\Pi^{kd}\times {\mathbb{R}}^{kd}}& |F_{k,N}|^2 e^{\lambda(t) e_{k,\gamma}}
+ \frac{\sigma^2}{4}\sum_{i\le k} \int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} |\nabla_{v_i} F_{k,N}|^2 e^{\lambda(t) e_{k}} \\
& \le k \frac{N- k}{N} \frac{C_{2,\sigma,d}}{\lambda^{\theta_{2,d}}(t)} \|K\|_{L^2}^2
\int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} |F_{k+1,N}|^2 e^{\lambda(t) e_{k+1}}\\
&\quad + \lambda'(t) \int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} e_{k} \, |F_{k,N}|^2 \, e^{\lambda(t) e_{k}}
\\
&\quad + \int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} R_{k,N} F_{k,N} e^{\lambda(t) e_{k}}. \end{split} \end{equation} Note that $R_{k,N}$ may be written as follows \begin{equation} \begin{split} R_{k,N} = &
\sum_{i=1}^k \frac{1}{N} \sum_{j=1}^k \Bigg[(K\star \int_{{\mathbb{R}}^d} \overline f)(t,x_i) - K(x_i-x_j)]\Bigg] \cdot \nabla_{v_i} f_k\\ & - \frac{N-k}{N} \sum_{i=1}^k \Bigg[ \nabla_{v_i} \cdot \int_{\Pi^d\times {\mathbb{R}}^d} f_{k+1} K(x_i-x_{k+1}) d x_{k+1} d v_{k+1}
\\ & \hspace{2.6cm} - \left(K \star \int_{{\mathbb{R}}^d} \overline f \right)(t,x_i)\cdot \nabla_{v_i} f_k\Bigg].
\end{split} \end{equation} Then, using that $f_k= \overline f^{\otimes k}$, we have \[ \begin{split}
\int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} R_{k,N} F_{k,N} e^{\lambda(t) e_{k}}
= \int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} & \frac k N \sum_{i=1}^k \Bigg[ \Big( K\star \int_{{\mathbb{R}}^d} \overline f \Big)(t,x_i) \\&- K(x_i-x_1)\Bigg] \cdot \nabla_{v_i} f_k \, F_{k,N} e^{\lambda(t) e_{k}} , \end{split} \] where we have used the fact that the particles are interchangeable. Integrating by parts with respect to $v_i$ and using Young inequality, we obtain \begin{equation} \label{Rest} \begin{split}
\int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} R_{k,N} F_{k,N} e^{\lambda(t) e_{k}} \le &
\frac{\sigma^2}{4} \frac k N \sum_{i=1}^k \int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} |\nabla_{v_i} F_{k,N}|^2 e^{\lambda(t) e_{k}}
\\ &+\frac{1}{\sigma^2 } \frac k N \sum_{i=1}^k \int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} |\widetilde R_{k,N}^1|^2 e^{\lambda(t) e_{k}} \\
& + \lambda(t) \int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} e_{k} \, |F_{k,N}|^2 \, e^{\lambda(t) e_{k}}
\\ & +\frac{1}{2} \int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} |\widetilde R_{k,N}^2|^2 \, e^{\lambda(t) e_{k}}, \end{split} \end{equation} where \[ \begin{split} \widetilde R_{k,N}^1
&= \Bigg[ \Big(K\star \int_{{\mathbb{R}}^d} \overline f\, dx \Big)(t,x_i) - K(x_i-x_1)\Bigg] \, f_k \,, \\ \widetilde R_{k,N}^2
&= \sum_{i=1}^k \Bigg[ \Big(K\star \int_{{\mathbb{R}}^d} \overline f\, dx \Big)(t,x_i) - K(x_i-x_1)\Bigg] \, f_k \,. \end{split} \]
We observe that
\[\|\widetilde R_{k,N}^i\|^2_{L^2_{\lambda(t) e_k}}
\le
C \, k \int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} |f_{k}|^p e^{\lambda(t) e_{k}} \, ,
\]
with a constant $C$ that does not depend on $k$. We have also used the fact that, in particular, $K \in L^2(\Pi^d)$ and $\overline f \in L^\infty(\Pi^d\times {\mathbb{R}}^d)$.
Then, using \eqref{Estim} and letting $N\to +\infty$, we get $$
\sup_{t\le T^*} \int_{\Pi^{kd}\times {\mathbb{R}}^{kd}}|f_k|^p e^{\lambda(t)e_{k, \gamma}} \le 2^k F_0^k . $$ We can insert this estimate into \eqref{Rest}, for $p= 2$, to derive \[ \begin{split}
\int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} R_{k,N} F_{k,N} e^{\lambda(t) e_{k}} \le &
\frac{\sigma^2}{4} \frac k N \sum_{i=1}^k \int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} |\nabla_{v_i} F_{k,N}|^2 e^{\lambda(t) e_{k}} \\
& + \lambda(t) \int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} e_{k} \, |F_{k,N}|^2 \, e^{\lambda(t) e_{k}} \\ & + C\, k \, 2^k F_0^k . \end{split} \] Once this estimate is incorporated into \eqref{FkN} and using that $\lambda'(t) = - \frac{\lambda(t)}{1+t}$, we can, following the same lines of the proof of Proposition \ref{propLp}, repeat the estimate on the ODE inequality with the extra term coming from the interaction of $F_{k,N}$ with rest term $R_{k,N}$. This provides the conclusion that there exists $T^*$ such that \[ \begin{split}
\sup_{t \le T^\star} \int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} |f_{N,k} - f_k|^2 & e^{\lambda(t) e_{k, \gamma}}
\\ & \le \widetilde C^k \varepsilon_N + \widetilde C^k \int_{\Pi^{kd}\times {\mathbb{R}}^{kd}} |f_{N,k}^0 - f_k^0|^2 e^{\lambda(0) e_{k, \gamma}} , \end{split} \] where $\widetilde C$ is a positive constant that does not depend on $N$, and $\varepsilon_N = O(\varepsilon^N)$, where $\varepsilon < 1$ depends on $T^*$ that is small enough. This expression can be deduced in a similar way as \eqref{epsilon} in the proof of Theorem \ref{Conv}.
We finally emphasize that the quantitative bounds of Theorem \ref{quantitative} would allow to recover the optimal convergence rate in $O(1/N)$ that was recently obtained in~\cite{Lacker}.
\noindent {\bf Acknowledgments.} D. Bresch is partially supported by SingFlows project, grant ANR-18-CE40-0027. P.~E. Jabin is partially supported by NSF DMS Grant 161453, 1908739, 2049020. J. Soler is partially supported by RTI2018-098850-B-I00 from the MICINN-Feder (Spain), PY18-RT-2422 \& B-FQM-580-UGR20 from the Junta de Andalucia (Spain).
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\begin{document}
\title{On the $k$-Means/$k$-Median Cost Function}
\begin{abstract} In this work, we study the $k$-means cost function. Given a dataset $X \subseteq \mathbb{R}^d$ and an integer $k$, the goal of the Euclidean $k$-means problem is to find a set of $k$ centers $C \subseteq \mathbb{R}^d$ such that $\Phi(C, X) \equiv \sum_{x \in X} \min_{c \in C} \norm{x - c}^2$ is minimized. Let $\Delta(X,k) \equiv \min_{C \subseteq \mathbb{R}^d} \Phi(C, X)$ denote the cost of the optimal $k$-means solution. For any dataset $X$, $\Delta(X,k)$ decreases as $k$ increases. In this work, we try to understand this behaviour more precisely. For any dataset $X \subseteq \mathbb{R}^d$, integer $k \geq 1$, and a precision parameter $\varepsilon > 0$, let $\Lp{\varepsilon}$ denote the smallest integer such that $\Delta(X,\Lp{\varepsilon}) \leq \varepsilon \cdot \Delta(X,k)$. We show upper and lower bounds on this quantity. Our techniques generalize for the metric $k$-median problem in arbitrary metric spaces and we give bounds in terms of the {\em doubling dimension} of the metric. Finally, we observe that for any dataset $X$, we can compute a set $S$ of size $O \left(\Lp{\varepsilon/c} \right)$ using {\em $D^2$-sampling} such that $\Phi(S,X) \leq \varepsilon \cdot \Delta(X,k)$ for some fixed constant $c$. Next, we mention some applications of our bounds. First, analysing the pseudo-approximation guarantees of $k$-means++ seeding has been a popular research topic. Our results may be seen as non-trivial addition to the current state of knowledge. Secondly, our bounds imply that any constant approximation algorithm when executed with number of clusters $O \left(\Lp{\varepsilon^2/c} \right)$ gives an {\em $(k, \varepsilon)$-coreset} for the $k$-means problem. This implies that a $D^2$-sampled set of size $O \left(\Lp{\varepsilon^2/c} \right)$ is a $(k, \varepsilon)$-coreset. This is an improvement over similar results of Ackermann \emph{et~al.}\cite{streamkm}. Third, our results also imply that the rate of decrease of $\Delta_k(X)$ with $k$ depends on the {\it intrinsic dimension} of any dataset $X$. Hence the rate at which $\Delta_k(X)$ diminishes may be used to infer the intrinsic dimension of a dataset $X$. We propose a sampling based intrinsic dimension estimator and evaluate it on real and synthetic datasets. \end{abstract}
\section{Introduction}\label{sec:intro} The Euclidean $k$-means problem is one of the most well-studied problems in the clustering literature. The problem is defined in the following manner:
\begin{definition}[$k$-Means problem] Given a dataset $X \subseteq \mathbb{R}^d$ and a positive integer $k$, find a set of $k$ points $C \subseteq \mathbb{R}^d$ (called {\em centers}) such that the cost function $\Phi(C, X) \equiv \sum_{x \in X} \min_{c \in C} D(x, c)^2$ is minimized, where $D(x, c) \equiv \norm{x - c}$. \end{definition}
In the weighted version of the $k$-means problem, there is a weight function $w: X \rightarrow \mathbb{R}^{+}$ and the cost function for the weighted $k$-means problem is defined as $\Phi(C, X, w) \equiv \sum_{x\in X} \min_{c \in C} \left(w(x) \cdot D(x, c)^2 \right)$. Let $\Delta(X,k)$ denote the optimal cost of the $k$-means objective function. That is $\Delta(X,k) \equiv \min_{C \subseteq \mathbb{R}^d,|C|=k} \Phi(C, X)$. In this work, we try to understand the behaviour of $\Delta(X,k)$ as $k$ increases. More specifically, for a small precision parameter $\varepsilon > 0$, we ask: what is the smallest integer $k'$ such that $\Delta(X,k')$ is at most $\varepsilon \cdot \Delta(X,k)$? Note that when $\varepsilon = 1$, $k' = k$ and as $\varepsilon$ becomes smaller, $k'$ should grow. We are interested in understanding the relationship of $k'$ with input parameters such as the size of the dataset $n$, dimension $d$, and $k$. Next, we formally define the quantity $\Lp{\varepsilon}$ for which we obtain bounds in this paper.
\begin{definition} For any dataset $X \subseteq \mathbb{R}^d$, precision parameter $0 < \varepsilon \leq 1$ and positive integer $k$, let $\Lp{\varepsilon}$ denote the smallest integer such that $\Delta(X,(\Lp{\varepsilon}) \leq \varepsilon \cdot \Delta(X,k)$. \end{definition}
We give upper and lower bounds on $\Lp{\varepsilon}$ in terms of the geometric quantities known as the covering and packing numbers (\cite{matousek2002}). These are defined below.
\begin{definition}[Covering number] Let $(\mathbb{X}, D)$ be a metric space and let $0 < \varepsilon \leq 1$. A subset $S$ of $\mathbb{X}$ is said to be an $\varepsilon$-covering set for $\mathbb{X}$ iff for every $x \in \mathbb{X}$, there exists an $s \in S$ such that $D(x, s) \leq \varepsilon$. The minimum cardinality of an $\varepsilon$-covering set of $\mathbb{X}$, if finite, is called the covering number of $\mathbb{X}$ (at scale $\varepsilon$) and is denoted by $\mathbb{N}(\mathbb{X}, \varepsilon)$. \end{definition}
\begin{definition}[Packing number] Let $(\mathbb{X}, D)$ be a metric space and let $0 < \varepsilon \leq 1$. A subset $S$ of $\mathbb{X}$ is said to be an $\varepsilon$-packing set iff for every $x, y \in S$ such that $x \neq y$, we have $D(x, y) \geq \varepsilon$. The maximum cardinality of an $\varepsilon$-packing set of $\mathbb{X}$, if finite, is called the packing number of $\mathbb{X}$ (at scale $\varepsilon$) and is denoted by $\mathbb{P}(\mathbb{X}, \varepsilon)$. \end{definition}
Our bounds on $\Lp{\varepsilon}$ are in terms of $\mathbb{N}(\mathbb{S}^{d-1}, \varepsilon)$ and $\mathbb{P}(\mathbb{S}^{d-1}, \varepsilon)$, where $\mathbb{S}^{d-1}$ denotes a unit sphere in $\mathbb{R}^d$.
The next lemma gives the bounds on $\mathbb{N}(\mathbb{S}^{d-1}, \varepsilon)$ and $\mathbb{P}(\mathbb{S}^{d-1}, \varepsilon)$. The proof of the lemma may be found in Appendix \ref{sec:appen}.
\begin{lemma}[Bounds on $\mathbb{N}(\mathbb{S}^{d-1}, \varepsilon)$ and $\mathbb{P}(\mathbb{S}^{d-1}, \varepsilon)$]\label{lemma:geometric-bounds} Let $\mathbb{S}^{d-1}$ denote a unit sphere in $\mathbb{R}^d$.
Then,
\[\mathbb{N}(\mathbb{S}^{d-1}, \varepsilon) = O\left(\frac{1}{(\varepsilon/8)^{d-1}}\right), \mathbb{P}(\mathbb{S}^{d-1}, \varepsilon) = \Omega\left(\frac{1}{(2\varepsilon)^{d-1}}\right)\]
\end{lemma}
Here is our main result for the Euclidean $k$-means problem.
\begin{theorem}[Main result for $k$-means]\label{thm:main1} Let $\mathbb{S}^{d-1}$ denote a unit sphere in $\mathbb{R}^d$. The following holds for any $0 < \varepsilon \leq 1/8$ and any positive integer $k$: \begin{enumerate} \item For any dataset $X \subseteq \mathbb{R}^d$ with $n$ points, $\Lp{\varepsilon} = O \left( \frac{\mathbb{N}(\mathbb{S}^{d-1}, \sqrt{\frac{\varepsilon}{2}}) \cdot k \cdot \log{n}}{\sqrt{\varepsilon}}\right)$, and
\item There exists a dataset $X \subseteq \mathbb{R}^d$ with $n$ points such that $\Lp{\varepsilon} = \Omega \left( \frac{\mathbb{P}(\mathbb{S}^{d-1}, \sqrt{8 \varepsilon}) \cdot k \cdot \log{n}}{\sqrt{\varepsilon}} \right)$. \end{enumerate} \end{theorem}
Note that a worse upper bound of $O\left( (9d/\varepsilon)^{d/2} \cdot k \cdot \log{n}\right)$ for the Euclidean $k$-means problem was implicit in the work of \cite{streamkm} where as our bound is $O\left((128/\varepsilon)^{(d-1)/2} \cdot k \cdot \log n \right)$. On the lower bound side, this question was open. Next, we show similar bounds for the metric $k$-median problem over arbitrary metrics. We first define the metric $k$-median problem over any metric $(\mathbb{X}, D)$.
\begin{definition}[Metric $k$-Median problem] Let $(\mathbb{X}, D)$ be any metric space. Given $X \subseteq \mathbb{X}$ and an integer $k$, find a set $C \subseteq \mathbb{X}$ of $k$ centers such that the cost function $\Phi(C, X) \equiv \sum_{x \in X} \min_{c \in C} D(x, c)$ is minimised. \end{definition}
We will use $\Delta(X,k)$ to denote the optimal cost of the metric $k$-median problem on dataset $X$ and $\Lp{\varepsilon}$ to denote the smallest integer such that $\Delta(X,\Lp{\varepsilon}) \leq \varepsilon \cdot \Delta(X,k)$. However, here we obtain the bounds in terms of the {\em doubling dimension} of the metric. Let us first define the doubling dimension. The diameter $dia(X)$ of any set $X \subseteq \mathbb{X}$ is defined as $dia(X) = \max_{x, x' \in X}{D(x, x')}$. Given any set $X \subseteq \mathbb{X}$ and $r\in \mathbb{R}^{+}$, a set $\{X_1,X_2,\ldots,X_m\}$ is said to be an $r$-cover of $X$ iff $\cup_i X_i = X$ and for all $1 \leq i \leq m, dia(X_i) \leq r$. Given $X \subseteq \mathbb{X}$ and $r \in \mathbb{R}^{+}$, the covering number of the set $X$ with respect to diameter $r$, denoted by $\lambda(X, r)$, is the size of the $r$-cover of smallest cardinality. We can now define the doubling dimension of any metric $(\mathbb{X}, D)$.
\begin{definition}[Doubling dimension] The doubling dimension of any metric $(\mathbb{X}, D)$ is the smallest integer $d$ such that for every $X \subseteq \mathbb{X}$, $\lambda \left(X, dia(X)/2 \right) \leq 2^d$. \end{definition}
We obtain the above bounds for the metric $k$-median problem in terms of the doubling dimension. Here are the statements of upper and lower bounds that we obtain:
\begin{theorem}[Upper bound for metric $k$-median]\label{thm:metric-median} Let $(\mathbb{X}, D)$ be any metric space with doubling dimension $d$. For any $0 < \varepsilon \leq 1$, any integer $k\geq 1$, and any dataset $X \subseteq \mathbb{X}$ with $n$ points, there exists a set $\xi \subseteq \mathbb{X}$ of size $O \left( \frac{k \cdot \log{n}}{(\varepsilon/8)^d}\right)$ such that $\Phi(\xi, X) \leq \varepsilon \cdot \Delta(X,k)$. \end{theorem}
\begin{theorem}[Lower bound for metric $k$-median] For any $0 < \varepsilon \leq 1/8$, any integer $k\geq 1$, there exists a metric space $(\mathbb{X}, D)$ with doubling dimension $d$ and a dataset $X \subseteq \mathbb{X}$ with $n$ points, such that for any set $\xi \subseteq \mathbb{X}$ with $\Phi(\xi, X) \leq \varepsilon \cdot \Delta(X,k)$, $\xi$ is of size $\Omega \left( \frac{k \cdot \log{n}}{(16\varepsilon)^{d-1}}\right)$. \end{theorem}
Next, we discuss a few applications of our bounds.
\subsection{Applications and related work}
The main applications of our bounds are in understanding the pseudo-approximation behaviour of the $k$-means++ seeding algorithm and coreset constructions for the $k$-means/$k$-median clustering problems. Here, we discuss mainly in terms of the (Euclidean) $k$-means problem, but most of the ideas may be extended for the $k$-median problem in any arbitrary metric.
\subsubsection{Pseudo-approximation of $k$-means++} $k$-means++ seeding is a sampling procedure that is popularly used as a seeding algorithm for the Lloyd's algorithm for $k$-means. The algorithm is given as follows.
\begin{quote}
({\bf $k$-means++ seeding or $D^2$-sampling}): Let $X \subseteq \mathbb{R}^d$. Pick the first center uniformly at random from $X$. After having picked $(i-1)$ centers denoted by $C_{i-1}$, pick a point $x \in X$ to be the $i^{th}$ center with probability proportional to $\min_{c \in C_{i-1}} D(x, c)^2$, where $D(x, c) \equiv \norm{x - c}$. \end{quote}
\cite{ArthurV07} showed that this algorithm gives an $O(\log{k})$-approximation guarantee in expectation. A lot of follow-up research has been done to understand the pseudo-approximation behaviour of this algorithm.
Note that $k$-means++ seeding stops after sampling $k$ centers using $D^2$-sampling\footnote{Given a set $C$ of centers, $D^2$-sampling with respect to $C$ chooses point $x$ with probability proportional to $\min_{c\in C} D(x,c)^2$.}. If one continues to sample centers even after sampling $k$ of them, then do the sampled centers give better than $O(\log{k})$ pseudo-approximation? Pseudo-approximation means that the cost is calculated with respect to the sampled centers, more than $k$ in number, but compared with the optimal solution for $k$ centers. \cite{AggarwalDK09} analysed this behaviour and showed that if one samples $O(k)$ centers, then we get a constant factor pseudo-approximation. \cite{W16} showed that if $\beta k$ centers are sampled for {\em any} constant $\beta > 1$, then we get a constant factor pseudo-approximation in expectation. In a more recent work, \cite{MRS2020} gave improved pseudo-approximation guarantees for $k$-means++ when $k+\Delta$ centers are sampled using $D^2$-sampling, for some $\Delta>0$. Clearly, as the number of centers sampled using $D^2$-sampling increases, the $k$-means cost with respect to the sampled centers will decrease. Let us try to understand this behaviour. One way to formalise this is to find bounds on the number of samples such that the cost becomes at most $\varepsilon$ times the optimal cost with respect to $k$ centers for any $0 < \varepsilon \leq 1$. Combining our bounds with the results of \cite{AggarwalDK09} (i.e, $O(k)$ samples give constant approximation with high probability, we get the following.
\begin{theorem}\label{thm:pseudo}
There is a universal constant $c$ for which the following holds: For any $0 < \varepsilon \leq 1$, positive integer $k$, and any dataset $X \subseteq \mathbb{R}^d$, let $S$ denote a set of centers sampled with $D^2$-sampling such that $|S| = \Omega(\Lp{\varepsilon/c})$. Then, with high probability, $\Phi(S, X) \leq \varepsilon \cdot \Delta(X,k)$. \end{theorem}
\subsubsection{Coresets for $k$-means} Coresets are extremely useful objects in data processing, where a coreset of a large dataset can be thought of as a concise representation of the dataset with respect to the specific data processing task in question. Next, we give the formal definition of a coreset for the $k$-means problem.
\begin{definition}[$(k,\varepsilon)$-coreset] A $(k,\varepsilon)$-coreset of a set $X \subseteq \mathbb{R}^d$ is a set $S \subseteq \mathbb{R}^d$ along with a weight function $w: S \rightarrow \mathbb{R}^{+}$ such that for any set of $k$ centers $C \subseteq \mathbb{R}^d$, we have: \[ (1 - \varepsilon) \cdot \Phi(C, X) \leq \Phi(C, S, w) \leq (1 + \varepsilon) \cdot \Phi(C, X) \] \end{definition}
A lot of work \cite{BadoiuHI02, Har-PeledM04, PeledKushal05, ls10, fl11, fss13} has been done in constructing coresets of small size.\footnote{The size of a coreset is the size of the set $S$ in the definition.} \cite{Har-PeledM04} and \cite{PeledKushal05} had coreset constructions by {\em quantization} of the space and finding points that may ``represent'' more than one point of the given dataset. In some sense, these coreset constructions are more geometric in nature than the more advanced constructions (see \cite{fss13}) and hence, these coresets are also known as ``movement-based'' coresets in \cite{mcc20}.
We define the notion of a movement-based coreset as follows.
\begin{definition}[$(k, \varepsilon)$-movement-based coreset] A $(k, \varepsilon)$-movement-based coreset of a dataset $X \subseteq \mathbb{R}^d$ is a set of points $S \subseteq \mathbb{R}^d$ such that:
\[ \Phi(S, X) \leq \varepsilon \cdot \Delta(X,k). \] \end{definition}
We will show that sampling $O(\Lp{\varepsilon^2/\beta})$ points using $D^2$-sampling gives a $(k,\varepsilon)$-movement-based coreset for some constant $\beta$.
\subsubsection{Estimation of of intrinsic dimension}
The intrinsic dimension of a dataset may be thought of as the minimum number of parameters required to account for the observed properties of the dataset. For most datasets, the {\em extrinsic dimension} (observed dimension) is much larger than the intrinsic dimension. The performance of many data analysis algorithms deteriorates as the dimension increase (popularly known as the curse of dimensionality). However, in many contexts, the data actually lies in a much lower dimensional space. In that case the intrinsic dimension of the dataset is much lower than the extrinsic dimension. The easiest example to consider is a dataset $X \subseteq \mathbb{R}^d$ such that the $X$ sits in a $D \ll d$ dimensional subspace of $\mathbb{R}^d$. In this case the intrinsic dimension is $D$.
Since one of the key algorithmic tools to tackle the high dimensional data is dimensionality reduction, estimating the intrinsic dimension of a given dataset becomes an important task.
There has been a lot of work in developing techniques for estimation of intrinsic dimension of a dataset. \cite{CS16} gave a nice survey on this topic. There are a number of ways to formalize the above intuitive notion of intrinsic dimension. One way to formalize is to say that a high dimensional dataset with extrinsic dimension $d$ has intrinsic dimension $D < d$ if the data sits within a $D$-dimensional sub-manifold. \cite{RL06} gave a dimension estimation technique under the assumption that the data points are uniformly distributed on a $D$-dimensional compact smooth sub-manifold of $\mathbb{R}^{d}$. Their technique involves finding the rate at which the {\em quantization error} diminishes with respect to the rate of the quantizer. Since some of the above terms have not been defined in our current context, let us rephrase them in the context of the $k$-means problem. Essentially, they estimated the intrinsic dimension to be the slope of the log-log plot of $k$ versus $(\Delta(X,k))^{1/2}$. The theoretical justification for such an estimation is that for any regular probability measure over a $D$-dimensional compact manifold, the expectation of the quantization error (i.e., expectation of $(\Delta(X,k))^{1/2}$ for random variable $X$) behaves as $\Theta(k^{-1/D})$. However, such a result is not known for an arbitrary discrete distribution (or an arbitrary set of points). Our results about the behaviour of the $k$-means cost function provides justification for the same estimation method for any arbitrary discrete distribution or even an arbitrary set of points for certain notions of intrinsic dimension similar to that considered by Raginski and Lazebnik. We provide experimental results for estimation of intrinsic dimension on some common datasets.
\subsubsection*{Organization of the Paper} In Section \ref{sec:bounds} and \ref{sec:kmedian}, we prove the bounds for the $k$-means and $k$-median problems. Section \ref{sec:coresets} details the construction of a $k$-means coreset. In Section \ref{sec:intrinsic}, the technique for estimation of intrinsic dimension of a dataset is described and experimental results are given.
\section{Bounds for Euclidean $k$-means}\label{sec:bounds} In this section, we prove the bounds on $\Lp{\varepsilon}$. We do this by using ideas in \cite{PeledKushal05} to reduce the high-dimensional case to a one-dimensional case. We start by discussing the upper and lower bounds for the one-dimensional data.
Let the distance of a point $x$ from a set $S$ be denoted as $D(x,S)=\min_{s \in S}\norm{s - x}$.
\begin{lemma}[Upper bound]\label{lem:1} For any $0 < \varepsilon \leq 1$ and any dataset $X \subseteq \mathbb{R}$ with $n$ points such that it is centered at the origin, there exists a set $S \subseteq \mathbb{R}$ of size $O \left( \sfrac{\log{n}}{\sqrt{\varepsilon}}\right)$ such that $\Phi(S, X) \leq \varepsilon \cdot \Phi(\{0\}, X)$. \end{lemma}
\begin{proof} Let $R = \frac{\sqrt{\Phi(\{0\}, X)}}{n} = \frac{\sqrt{\sum_{x \in X} \norm{x}^2}}{n}$. Let $r = (1 + \sqrt{\varepsilon/2})$ and $t = \lceil \sfrac{\log{n}}{\log{r}} \rceil$. Let \begin{eqnarray*}
\noindent S_1 &=& \{\pm (i \cdot \sqrt{\varepsilon/2} \cdot R) ~|~ 0 \leq i \leq \lfloor \sqrt{2/\varepsilon}\rfloor \} \\
\noindent S_2 &=& \{\pm (r^i \cdot R) ~|~ 0 \leq i \leq t\} \end{eqnarray*} We use $S = S_1 \cup S_2$. Note that for our choice of $t$, there is no point $x \in X$ such that $\norm{x} > r^t \cdot R$. For any point $x \in X$ such that $r^i R \leq \norm{x} < r^{i+1} R$, we have $D(x, S)^2 \leq (\varepsilon/2) (r^i R)^2 \leq (\varepsilon/2) \norm{x}^2$. Also, note that for every $x$ such that $\norm{x} \leq R$, we have $D(x, S)^2 \leq (\varepsilon/2) \cdot R^2$. So, we have: \begin{eqnarray*} &~~&\Phi(S, X)\\ &=& \sum_{x \in X} D(x, S)^2 \\ &=& \sum_{x \in X, \norm{x}\leq R} D(x, S)^2 + \sum_{x \in X, \norm{x} > R} D(x, S)^2\\ &\leq& \sum_{x \in X, \norm{x}\leq R} (\varepsilon/2) R^2 + \sum_{x \in X, \norm{x} > R} (\varepsilon/2) \norm{x}^2\\ &\leq& \varepsilon \cdot \Phi(\{0\}, X). \end{eqnarray*} Finally, note that
$$|S| = |S_1| + |S_2| = O \left( \frac{\log{n}}{\log{(1 + \sqrt{\varepsilon/2})}}\right) = O \left( \frac{\log{n}}{\sqrt{\varepsilon}}\right).$$ This completes the proof of the lemma. \end{proof}
The lower bound instance for the 1-dimensional $k$-means problem below is based on ideas similar to the lower bound instance by \cite{bja}.
\begin{lemma}[Lower bound]\label{lem:2}
For any $0 < \varepsilon < 1/8$, there exists a dataset $X \subseteq \mathbb{R}$ with $n$ points such that any set $S \subseteq \mathbb{R}$ with $\Phi(S, X) \leq \varepsilon \cdot \Phi(\{0\}, X)$ satisfies $|S| = \Omega \left( \sfrac{\log{n}}{\sqrt{\varepsilon}}\right)$. \end{lemma}
\begin{proof} In the dataset that we construct multiple points may be co-located. Let $r = \lceil (1 + \sqrt{32 \varepsilon}) \rceil$. Consider the dataset $X$ described in the following manner: There are $r^{2(t-1)}$ points co-located at $\pm r$, $r^{2(t-2)}$ points co-located at $\pm r^2$, $r^{2(t-3)}$ points co-located at $\pm r^3$, ..., $1$ point located at $\pm r^{t}$.
We can fix the value of $t$ in the above description in terms of $n = |X|$ by noting that: $n = 2 \cdot (1 + r^{2} + ... + r^{2(t-1)})$.
Given this, we have $t = \Omega (\log_r{ ({n(r^2-1)}/{2} + 1)})$. The cost with respect to single center at the origin is given by:
$\Phi(\{0\}, X) = 2 \cdot (r^2 \cdot r^{2(t-1)} + r^4 \cdot r^{2(t-2)} + ... + r^{2t}) = 2t r^{2t}$.
Consider intervals around each of the populated locations. Let $I^{+}_{i} = [r^i(1 - \sqrt{2\varepsilon}), r^i (1 + \sqrt{2\varepsilon})]$ and $I^{-}_{i} = [-r^i(1 + \sqrt{2\varepsilon}), -r^i (1 - \sqrt{2\varepsilon})]$. Note that these intervals are disjoint for our choice of $r$. Consider any set $S$ with less than $t$ points. Note that there will be at least $t$ intervals that do not contain a point from the set $S$. The points located at each of these ``uncovered'' intervals contribute a cost of at least $(2\varepsilon) r^{2t}$. Given this, we have $\Phi(S, X) > t \cdot (2\varepsilon) r^{2t} > \varepsilon \cdot \Phi(\{0\}, X)$. So, for any set $S$ such that $\Phi(S, X) \leq \varepsilon \cdot \Phi(\{0\}, X)$, we have $|S| > t$ which gives
$|S| > t = \Omega(\sfrac{\log{n}}{\log{\lceil(1 + \sqrt{32 \varepsilon}})\rceil})$ which gives the statement of the lemma. \end{proof}
We will now extend these bounds to higher dimensions using the ideas of \cite{PeledKushal05}. We will use $\varepsilon$-covering and $\varepsilon$-packing numbers over the surface of unit spheres crucially in our construction. We now show the upper bound for points in $\mathbb{R}^d$.
\begin{theorem}[Upper bound for Euclidean $k$-means]\label{thm:1} Let $\mathbb{S}^{d-1}$ denote the surface of the unit sphere in $\mathbb{R}^{d}$. For any $0 < \varepsilon \leq 1$, any integer $k\geq 1$, and any dataset $X \subseteq \mathbb{R}^d$ with $n$ points, there exists a set $\xi \subseteq \mathbb{R}^d$ of size $O \left( \frac{\mathbb{N}(\mathbb{S}^{d-1}, \sqrt{\varepsilon/2}) \cdot k \cdot \log{n}}{\sqrt{\varepsilon}}\right)$ such that $\Phi(\xi, X) \leq \varepsilon \cdot \Delta(X,k)$. \end{theorem}
\begin{proof} Let $C = \{c_1,\ldots,c_k\}$ denote the optimal centers for $k$-means on $X$ and let $X_1,\ldots,X_k$ be the optimal clusters. This also means that for all $i$, $c_i$ is the centroid of the point set $X_i$. It will be sufficient to find $k$ sets $\xi_1,\ldots,\xi_k$ such that for all $i$, $\Phi(\xi_i, X_i) \leq \varepsilon \cdot \Phi(\{c_i\}, X_i) = \varepsilon \cdot \Delta({X_i},1)$. Given such sets $\xi_1,\ldots,\xi_k$, let $\xi = \cup_i \xi_i$. Then, we have $\Phi(\xi, X) \leq \sum_{i} \Phi(\xi_i, X_i) \leq \sum_i \varepsilon \cdot \Delta({X_i},1) = \varepsilon \cdot \Delta(X,k)$.
Consider any optimal cluster $X_i$. Let $Y = X_i$ and $c = c_i$. Let points in the set $Y$ be denoted as $y_1,\ldots,y_m$. We will now construct a set $S$ such that $\Phi(S, Y) \leq \varepsilon \cdot \Phi(\{c\}, Y)$.
Consider a unit sphere $\mathbb{S}^{d-1}$ around $c$ and let $R$ denote an $(\sqrt{{\varepsilon}/{2}})$-covering set over $\mathbb{S}^{d-1}$ (note that the size of $R$ is $\mathbb{N}(\mathbb{S}^{d-1}, \sqrt{\varepsilon/2})$). This implies that for any point $r_1 \in \mathbb{S}^{d-1}$, there exists $r_2 \in R$ such that $r_2 \neq r_1$ and $\norm{r_1 - r_2} \leq \sqrt{\varepsilon/2}$. Consider a ``fan'' consisting of $|R|$ lines $F = \{l_1,\ldots, l_{|R|}\}$ connecting $c$ to each of the points in $R$. For any point $y \in Y$, let $y'$ denote the projection of $y$ on the nearest line among $l_1,\ldots,l_{|R|}$. Let $Y'$ denote the set of projected points. For any line $l \in \{l_1,\ldots,l_{|R|}\}$, let $Y_l$ denote the subset of projected points that are on line $l$. We first observe that ``snapping'' the points to the fan $F$ does not cost much. This follows easily from the following simple observation. For all $y \in Y$, we have $\norm{y - y'} \leq \sqrt{\varepsilon/2} \cdot \norm{y - c}$.
This implies that \begin{equation}\label{eqn:1} \mathcal{E} \equiv \sum_{y \in Y} \norm{y - y'}^2 \leq {\varepsilon}/{2} \cdot \sum_{y \in Y} \norm{y - c}^2 = {\varepsilon}/{2} \cdot \Delta(Y,1). \end{equation}
We have: \begin{equation}\label{eqn:2} \Delta(Y,1) = \mathcal{E} + \sum_{l \in F} \Phi(\{c\}, Y_l) \end{equation}
Now, from Lemma~\ref{lem:1}, we know that for every $l \in F$, there exists a set $S_l$ of $O \left(\sfrac{\log{|Y_l|}}{\sqrt{\varepsilon}} \right)$ points on line $l$ such that $\Phi(S_l, Y_l) \leq (\varepsilon/2) \cdot \Phi(\{c\}, Y_l)$. Let $S = \cup_l S_l$. Then we have \begin{eqnarray*} &~~&\Phi(S, Y)\\ &=& \mathcal{E} + \sum_{y \in Y'} D(y, S)^2 \\ &\leq& \mathcal{E} + \sum_{l \in F} \sum_{y \in Y_l} D(y, S_l)^2 \\ &=& \mathcal{E} + \sum_{l \in F} \Phi(S_l, Y_l) \\ &\leq& \mathcal{E} + (\varepsilon/2) \cdot \sum_{l \in F} \Phi(\{c\}, Y_l) \quad \textrm{(using Lemma~\ref{lem:1})}\\ &\leq& \mathcal{E} + (\varepsilon/2) \cdot \Delta(Y,1) \quad \textrm{(using (\ref{eqn:2}))}\\ &\leq& (\varepsilon/2) \cdot \Delta(Y,1) + (\varepsilon/2) \cdot \Delta(Y,1) \quad \textrm{(using (\ref{eqn:1}))}\\ &=& \varepsilon \cdot \Delta(Y,1) \end{eqnarray*}
The size of the set $S$ is $O \left(\frac{\mathbb{N}(\mathbb{S}^{d-1}, \sqrt{\varepsilon/2}) \cdot \log{|Y|}}{\sqrt{\varepsilon}} \right)$.
Repeating the same for all $k$ optimal clusters, we get a set $\xi$ of size $O \left( \frac{\mathbb{N}(\mathbb{S}^{d-1}, \sqrt{\varepsilon/2}) \cdot k \cdot \log{n}}{\sqrt{\varepsilon}}\right)$. \end{proof}
Note that our upper bound is better than the bound of \cite{streamkm} and the improvement may be attributed to reducing the $d$-dimensional case to a $1$-dimensional case. We also give a lower bound below which essentially shows that the factors of $k$, $\log{n}$ and $(1/\varepsilon)^{d/2}$ are unavoidable.
\begin{theorem}[Lower bound for Euclidean $k$-means]\label{thm:2}
For any $0 < \varepsilon < 1/8$, any integer $k \geq 1$, there exists a dataset $X \subseteq \mathbb{R}^d$ with $n$ points such that any set $\xi \subseteq \mathbb{R}^d$ with $\Phi(\xi, X) \leq \varepsilon \cdot \Delta(X,k)$ satisfies $|\xi| = \Omega \left( \frac{\mathbb{P}(\mathbb{S}^{d-1}, \sqrt{8 \varepsilon}) \cdot k \cdot \log{n}}{\sqrt{\varepsilon}} \right)$. \end{theorem}
\begin{proof} Let $c_1,\ldots,c_k \in \mathbb{R}^d$ be $k$ points such that $\forall i \neq j, \norm{c_i - c_j} > n^2$. We will define $k$ sets of points $X_1,\ldots,X_k$, and our dataset will be $X = \cup_i X_i$. Let $R$ denote an $\sqrt{8 \varepsilon}$-packing set over a unit sphere in $\mathbb{R}^d$ (note that the size of $R$ is the packing number and is denoted by $\mathbb{P}(\mathbb{S}^{d-1}, \sqrt{8\varepsilon})$). For any $i$, here is the description of the set $X_i$: Let $\mathbb{S}^{d-1}(c_i)$ denote a unit sphere around $c_i$ and let $R(c_i)$ denote the $(\sqrt{8\varepsilon})$-packing set laid over $\mathbb{S}^{d-1}(c_i)$. Consider a ``fan'' $F$ of lines connecting $c_i$ to each point in the set $R(c_i)$. Each line has $\eta = \sfrac{n}{(k \cdot \mathbb{P}(\mathbb{S}^{d-1}, \sqrt{8 \varepsilon}))}$ points and these points are arranged as in the one-dimensional example of Lemma~\ref{lem:2}.
The analysis is very similar to that in Lemma~\ref{lem:2}. Instead of considering intervals around each populated location, we will consider balls of certain radius. As in Lemma~\ref{lem:2}, we use $r = \lceil (1 + \sqrt{32 \varepsilon}) \rceil$. Also, $t = \Theta \left( \sfrac{\log{\eta}}{\log{r}}\right)$. The populated locations are at distances $r, r^2,\ldots,r^t$ from the $c_i$'s. We have $\Delta({X_i},1) = \Phi(\{c_i\}, X_i) = (2 t r^{2t}) \cdot \mathbb{P}(\mathbb{S}^{d-1}, \sqrt{8 \varepsilon})$ which gives $\Delta(X,k) = \sum_i \Phi(\{c_i\}, X_i) = k \cdot (2 t r^{2t}) \cdot \mathbb{P}(\mathbb{S}^{d-1}, \sqrt{8 \varepsilon})$. Consider balls of radius $\sqrt{2 \varepsilon} \cdot r^j$ around any populated location at a distance $r^j$ from $c_i$. Note that all of these balls are disjoint because of our choice of $r$ and because of the fact that the populated points are defined using an $\sqrt{8 \varepsilon}$-packing set over unit sphere. The number of balls defined is $2kt \cdot \mathbb{P}(\mathbb{S}^{d-1}, \sqrt{8 \varepsilon})$. Consider any set $S$ containing less than $kt \cdot \mathbb{P}(\mathbb{S}^{d-1}, \sqrt{8 \varepsilon})$ points. There are at least $kt \cdot \mathbb{P}(\mathbb{S}^{d-1}, \sqrt{8 \varepsilon})$ balls that do not contain any points from $S$. The cost contribution from the points located in each of these balls is at least $(2 \varepsilon) r^{2t}$. So, $\Phi(S, X) > (2 \varepsilon) \cdot kt \cdot \mathbb{P}(\mathbb{S}^{d-1}, \sqrt{8 \varepsilon}) \cdot r^{2t} > \varepsilon \cdot \Delta(X,k)$. Therefore, any set $\xi$ for which $\Phi(\xi, X) \leq \varepsilon \cdot \Delta(X,k)$ satisfies \begin{align*}
|\xi| = & \Omega(kt \cdot \mathbb{P}(\mathbb{S}^{d-1}, \sqrt{8 \varepsilon})) = \Omega \left( k \cdot \frac{\log{\eta}}{\log{r}} \cdot \mathbb{P}(\mathbb{S}^{d-1}, \sqrt{8 \varepsilon})\right) \\ = & \Omega \left( \frac{\mathbb{P}(\mathbb{S}^{d-1}, \sqrt{8 \varepsilon}) \cdot k \cdot \log{n}}{\sqrt{\varepsilon}}\right) \end{align*} This completes the proof of the theorem. \end{proof}
\subsection{Bounds for Euclidean $k$-median} The Euclidean $k$-median problem is very similar to the Euclidean $k$-means problem except that the cost function is defined using ``sum of distances'' rather than ``sum of squared distances''. Given $X \subseteq \mathbb{R}^d$ and a positive integer $k$, find a set of centers $C \subseteq \mathbb{R}^d$ such that $\Phi(C, X) \stackrel{def}{=} \sum_{x \in X} \min_{c \in C} D(x, c)$ is minimized, where $D(x, y) = \norm{x - y}$. We obtain the following bounds, the details of which we omit here, by replacing the squared Euclidean distances with the Euclidean distances in all the above arguments.
\begin{theorem}[Upper bound for Euclidean $k$-median]\label{thm:med1} For any $0 < \varepsilon \leq 1$, any integer $k\geq 1$, and any dataset $X \subseteq \mathbb{R}^d$ with $n$ points, there exists a set $\xi \subseteq \mathbb{R}^d$ of size $O \left( \frac{\mathbb{N}(\mathbb{S}^{d-1}, \varepsilon/2) \cdot k \cdot \log{n}}{\varepsilon}\right)$ such that $\Phi(\xi, X) \leq \varepsilon \cdot \Delta(X,k)$. \end{theorem}
\begin{theorem}[Lower bound for Euclidean $k$-median]\label{thm:med2}
For any $0 < \varepsilon < 1/8$, any integer $k \geq 1$, there exists a dataset $X \subseteq \mathbb{R}^d$ with $n$ points such that any set $\xi \subseteq \mathbb{R}^d$ with $\Phi(\xi, X) \leq \varepsilon \cdot \Delta(X,k)$ satisfies $|\xi| = \Omega \left( \frac{\mathbb{P}(\mathbb{S}^{d-1}, 4\varepsilon) \cdot k \cdot \log{n}}{\varepsilon} \right)$. \end{theorem}
\section{Bounds for Metric $k$-Median}\label{sec:kmedian} In this section we obtain bounds for $\Lp{\varepsilon}$ for the metric $k$-median problem over any arbitrary metric space $(\mathbb{X}, D)$. Given $X \subseteq \mathbb{X}$ and a positive integer $k$, the metric $k$-median problem asks to find a set $C \subseteq \mathbb{X}$ of $k$ centers such that $\Phi(C, X) \equiv \sum_{x \in X} \min_{c \in C} D(x, c)$ is minimized.
We obtain bounds for $\Lp{\varepsilon}$ in terms of the doubling dimension of the metric. Let us first recall the notion of doubling dimension. Given $X \subseteq \mathbb{X}$, the diameter of the set $X$, $dia(X)$, is defined as $dia(X) = \max_{x, x' \in X}{D(x, x')}$. Given any set $X \subseteq \mathbb{X}$ and $r \in \mathbb{R}^{+}$, a set $\{X_1, X_2,\ldots,X_m\}$ is said to be an $r$-cover of $X$ iff $\cup_i X_i = X$ and for all $1 \leq i \leq m, dia(X_i) \leq r$. Given $X \subseteq \mathbb{X}$ and $r \in \mathbb{R}^{+}$, the covering number of the set $X$ with respect to diameter $r$, denoted by $\lambda(X, r)$, is the size of the $r$-cover of smallest cardinality. We now define the doubling dimension of any metric $(\mathbb{X}, D)$.
\begin{definition}[Doubling dimension] The doubling dimension of any metric $(\mathbb{X}, D)$ is the smallest integer $d$ such that for every $X \subseteq \mathbb{X}$, $\lambda \left(X, dia(X)/2 \right) \leq 2^d$. \end{definition}
The remaining discussion will be with respect to the doubling dimension $d$ of any metric $(\mathbb{X}, D)$. We will use the following lemma for defining our upper bound.
\begin{definition}[$\varepsilon$-covering number] Let $0 < \varepsilon < 1$ be some precision parameter. For any $c \in \mathbb{X}$ and $r \in \mathbb{R}^{+}$, let $M(c, r) = \{x~|~x \in \mathbb{X} \textrm{ and } D(c, x) \leq r\}$. The $\varepsilon$-covering number, denoted using $\gamma_{\varepsilon}$, is defined as $\gamma_{\varepsilon} = \max_{c \in \mathbb{X}, r \in \mathbb{R}^{+}}{(\lambda(M(c, r), \varepsilon \cdot r))}$. \end{definition}
In the next lemma, we give a bound on the $\varepsilon$-covering number for any metric with doubling dimension $d$.
\begin{lemma}\label{lem:met4} Let $(\mathbb{X}, D)$ be any metric space. For any $0 < \varepsilon < 1$, the $\varepsilon$-covering number of $\mathbb{X}$, $\gamma_{\varepsilon} = O \left((4/\varepsilon)^d\right)$. \end{lemma} \begin{proof} The proof follows from the definition of $\lambda$. Consider any $c \in \mathbb{X}$ and any $r \in \mathbb{R}^{+}$. Let $X = M(c, r)$. From the triangle inequality we know that $dia(X) \leq 2r$. Using the fact that the doubling dimension of the metric is $d$, we get $\lambda(X, r) \leq 2^d, \lambda(X, r/2) \leq 2^{2d}$, $\lambda(X, r/2^2) \leq 2^{3d}$ and so on. Let $\sfrac{1}{2^{k+1}} < \varepsilon \leq \sfrac{1}{2^k}$. Then we have $\lambda(X, \varepsilon \cdot r) \leq 2^{(k+2)d}$. This gives us $\lambda(X, \varepsilon \cdot r) = O((4/\varepsilon)^d)$. \end{proof}
We can now give the upper bound in terms of the $\varepsilon$-covering number of the metric.
\begin{theorem}[Upper bound for metric $k$-median] Let $(\mathbb{X}, D)$ be any metric space with doubling dimension $d$. For any $0 < \varepsilon \leq 1$, any integer $k\geq 1$, and any dataset $X \subseteq \mathbb{X}$ with $n$ points, there exists a set $\xi \subseteq \mathbb{X}$ of size $O \left( \frac{k \cdot \log{n}}{(\varepsilon/8)^d}\right)$ such that $\Phi(\xi, X) \leq \varepsilon \cdot \Delta(X,k)$. \end{theorem}
\begin{proof} Let $c_1,\ldots,c_k$ be the optimal $k$ centers and let $X_1,\ldots,X_k$ be the optimal clusters with respect to $c_1,\ldots,c_k$. It is sufficient to show that for all $i$, there exists a set $\xi_i \subseteq X_i$ of size $O \left(\sfrac{\gamma_{\varepsilon} \cdot \log{n}}{\varepsilon} \right)$ such that $\Phi(\xi_i, X_i) \leq \varepsilon \cdot \Delta({X_i},1)$. Let $c = c_i$ and $Y = X_i$. Let $R = {\sum_{y \in Y} D(y, c)}/{|Y|}$ and $t = \Theta \left( \log_{2}{|Y|}\right)$. Consider ``concentric circles'' of radius $R, 2R, 2^2 R,\ldots, 2^t R$ around the center $c$. That is consider the sets $Y_0,\ldots, Y_t$ defined in the following manner. \begin{eqnarray*}
Y_0 &=& \{y \in Y | D(c, y) \leq R\} \quad \textrm{and} \\
Y_{j} &=& \{y \in Y | D(c, y) \leq 2^j \cdot R\} \setminus Y_{j-1} \quad \textrm{for}~ 1\leq j\leq t \end{eqnarray*} First, note that $\cup_{j=0}^{t} Y_j = Y$ due to our choice of $t$. Also, for every $j$, there exists subsets $S_j^{1}, S_j^{2},\ldots, S_j^{\gamma_{\varepsilon/2}}$ such that for all $1 \leq i \leq \gamma_{\varepsilon/2}, dia(S_j^{i}) \leq (\varepsilon/2) \cdot (2^j R)$ and $\cup_{i=1}^{\gamma_{\varepsilon/2}} S_j^{i}= Y_j$. Let $S_j$ be the set of points constructed by picking one point from each of the sets $S_j^{1},\ldots, S_j^{\gamma_{\varepsilon/2}}$ and let $S = \cup_{j=0}^{t} S_j$. We have, \begin{eqnarray*} &~~&\Phi(S, Y)\\
&\leq& \sum_{j=0}^{t} \Phi(S_j, Y_j) \leq \sum_{j=0}^{t} {\varepsilon}/{2} \cdot (2^j R) \cdot |Y_j| \\
&=& \sum_{j=0}^{t} \varepsilon (2^{j-1} R) \cdot |Y_j| \\ &\leq& \sum_{j=0}^{t} \varepsilon \cdot \Phi(\{c\}, Y_j) \leq \varepsilon \cdot \Phi(\{c\}, Y) \leq \varepsilon \cdot \Delta(Y,1) \end{eqnarray*}
The size of the set $S$ is given by $|S| = \gamma_{\varepsilon/2} \cdot t = O(\gamma_{\varepsilon/2} \cdot \log{|Y|})$. Generalizing this for all sets $X_1,\ldots,X_k$, we get that there is a set $\xi \subseteq \mathbb{X}$ of size $O(k \cdot \gamma_{\varepsilon/2} \cdot \log{n})$ such that $\Phi(\xi, X) \leq \varepsilon \cdot \Delta(X,k)$. The theorem follows from the bound on $\gamma_{\varepsilon/2}$ from Lemma~\ref{lem:met4}. \end{proof}
The lower bound below follows trivially from the lower bound for the Euclidean $k$-median problem.
\begin{theorem}[Lower bound for metric $k$-median] For any $0 < \varepsilon \leq 1/8$, any integer $k\geq 1$, there exists a metric space $(\mathbb{X}, D)$ with doubling dimension $d$ and a dataset $X \subseteq \mathbb{X}$ with $n$ points, such that for any set $\xi \subseteq \mathbb{X}$ with $\Phi(\xi, X) \leq \varepsilon \cdot \Delta(X,k)$, $\xi$ is of size $\Omega \left( \frac{k \cdot \log{n}}{(16\varepsilon)^{d-1}}\right)$. \end{theorem}
\section{Coresets for $k$-means} \label{sec:coresets}
We start by recalling the notion of a movement-based coreset, defined as follows.
\begin{definition}[$(k, \varepsilon)$-movement-based coreset] A $(k, \varepsilon)$-movement-based coreset of a dataset $X \subseteq \mathbb{R}^d$ is a set of points $S \subseteq \mathbb{R}^d$ such that $\Phi(S, X) \leq \varepsilon \cdot \Delta(X,k)$.
\end{definition}
We will now see that movement-based coreset is a stronger notion than coreset in the sense that if any dataset $X$ has a $(k, \varepsilon^{O(1)})$-movement-based coreset of size $m$, then it also has a $(k, \varepsilon)$-coreset of size $m$. A $(k, \varepsilon)$-coreset is defined by a set $S$ and a weight function $w$, whereas a movement-based coreset is defined by just a set of points. We will show that for any $(k, \varepsilon)$-movement based coreset $S$, this set along with an appropriately defined weight function (which is dependent just on $S$ and $X$) is also a $(k, \varepsilon)$-coreset. Let $X$ and $Y$ be any set of points. For any point $y \in Y$, we define $\mathbb{N}_X(y)$ to be the set of points from $X$ such that their closest point in set $Y$ is $y$. That is, $\mathbb{N}_X(y) = \{x | x \in X \textrm{ and } \arg\min_{p \in Y} \norm{x - p} = y\}$. We give the proof of the result that a $(k,\varepsilon)$-movement-based coreset implies a $(k,\varepsilon)$-coreset in Appendix \ref{sec:appen2}. We note that this result was implicitly present in \cite{streamkm}.
We have argued that the notion of a movement-based coreset is a stronger notion than a coreset. As far as the existence of such movement-based coresets are concerned, we know from previous discussions that for any dataset $X$, there exists $(k, \varepsilon)$-movement-based coreset of size $\Lp{\varepsilon}$. Moreover, such movement-based coresets may be computed by running any constant factor pseudo-approximation algorithm for $k$-means with $k$ set as $\Lp{O(\varepsilon)}$. As seen in the previous subsection, the $D^2$-sampling algorithm is one such algorithm. Combining Theorem~\ref{thm:pseudo} with Theorem~\ref{thm:coreset-1} given in the Appendix, we get the following result.
\begin{theorem} There is a universal constant $\beta$ for which the following holds: For any $0 < \varepsilon \leq 1$, positive integer $k$ and any dataset $X \subseteq \mathbb{R}^d$, let $S$ denote the a set of centers chosen with $D^2$-sampling from $X$ such that $|S| = \Omega(\Lp{\varepsilon^2/\beta})$. Then, $S$ is a $(k, \varepsilon)$-coreset with high probability. \end{theorem}
\section{Estimation of Intrinsic dimension} \label{sec:intrinsic}
We consider the following three notions for estimation of intrinsic dimension of a dataset.
\begin{enumerate} \item {\it Affine subspace}: Suppose a dataset $X \subseteq \mathbb{R}^D$ sits in an affine subspace of dimension $d < D$. Theorem \ref{thm:main1} suggests that $\Lp{\varepsilon}$ has a dependency of the form $({1}/{\varepsilon})^{d/2}$ over $\varepsilon$. So, if $m$ is the slope of the log-log plot of $\Delta(X,k)$ versus $k$, then $(-2.0)/m$ may be a good estimate of the dimension $d$. Please note that we are making a number of heuristic adjustments while making the previous statement. This estimation technique is a heuristic and Theorem 1 only provides a high-level justification for this heuristic. The only way to have more confidence would be to test the method on real data. This is also true for the two cases discussed below.
\item {\it Doubling Dimension}: Same as in the previous case but now using Theorem \ref{thm:metric-median} instead of Theorem \ref{thm:main1}, we get that $(-1.0)/m$ should provide a good estimate for the doubling dimension of the dataset $X$ where $m$ is the slope of the log-log plot of $\Delta(X,k)$ versus $k$. Here, $\Delta(X,k)$ denotes the $k$-median cost.
\item {\it Covariance Dimension}: Suppose a dataset $X \subseteq \mathbb{R}^D$ does not precisely belong to a $d$-dimensional affine subspace as in the first case but is close to one. This can be made more precise in the following manner: Suppose for simplicity that $X$ is centered around the origin. For an error parameter $\gamma$, the dataset $X$ is said to have a covariance dimension $d_{\gamma}$ if the sum of the first $d_{\gamma}$ eigenvalues of the covariance matrix of $X$ is at least $(1-\gamma)$ times the sum of all eigenvalues. This translates to the fact that the projection of all points to the space orthogonal to the space spanned by the first $d_{\gamma}$ eigenvectors has a very small contribution to the 1-means cost (only a $\gamma$ fraction). So, as in the first case, $\Lp{\varepsilon}$ will have dependency of the form $({1}/{\varepsilon})^{d_{\varepsilon}/2}$ over $\varepsilon$. So, $(-2.0)/m$ should provide a good estimate for the covariance dimension of $X$ where $m$ is the slope of the log-log plot of $\Delta({X},k)$ versus $k$. \end{enumerate}
So, our technique for dimension estimation basically involves estimating the slope of the log-log plot of $k$ versus $\Delta(X,k)$. The main issue in obtaining such a plot is computing $\Delta(X,k)$ which is the optimal $k$-means cost for dataset $X$. This is because $k$-means is an $\mathsf{NP}$-hard problem when $k > 1$. There are a number of good approximation algorithms for the $k$-means problem and we can get an approximate value of $\Delta(X,k)$ for all values of $k$. This leads us to an efficiency issue. The issue is that we will need to repeatedly run the approximation algorithm for different values of $k$. The $k$-means++ seeding algorithm is an interesting approximation algorithm for this context and provides a useful solution for this issue. Note that the $k$-means++ seeding algorithm uses input $k$ only as a termination condition. That is, it continues to sample centers using $D^2$-sampling as long as the number of centers is less than $k$. The approximation analysis says that for any fixed value of $i$, the first $i$ centers sampled by the algorithm provides $O(\log{i})$ approximation to the $i$-means objective (in expectation). So, if $C_i$ denotes the first $i$ centers sampled by the algorithm, then the heuristic we propose simply plots $i$ versus $\Phi(C_i, X)$. Here is the pseudocode that we use to estimate the dimension of the given dataset $X$ when the cost function is sum of squared distances.\footnote{When the cost function is sum of distance as in $k$-median, the last line should return $(-1.0)/m$.} Let $\ell$ be an input parameter that gives better estimates when it is large but at the cost of the running time. Since the datasets on which we perform our experiments have small intrinsic dimension, setting $\ell = 100$ suffices.
\begin{framed} {\tt DimensionEstimate($X, \ell$)}\\ \hspace*{0.1in} - $C_0 \leftarrow \{\}$\\ \hspace*{0.1in} - for $i$ = $1$:$\ell$\\ \hspace*{0.3in} - $D^2$-sample a center $c$ with respect to $C_{i-1}$ \\ \hspace*{0.3in} - $C_i \leftarrow C_{i-1} \cup \{c\}$\\ \hspace*{0.3in} - $Cost[i] \leftarrow \Phi(C_i, X)$\\ \hspace*{0.1in} - Let $m$ be the slope of the best fit line for the curve \\ \hspace*{0.2in} $Cost[i]$ versus $i$ on a log-log plot\\ \hspace*{0.1in} - return($(-2.0)/m$) \end{framed}
We now evaluate our heuristic by testing it on synthetic and real datasets. We first perform experiment on a synthetic dataset with extrinsic dimension $100$ where the points are within a small $d$-dimensional affine subspace with $d$ ranging from $2$ to $7$. We call these six datasets {\tt Affine-2} through {\tt Affine-7}. More specifically, here is how this dataset is generated: First, we pick a random point $\mathbf{x} \in \mathbb{R}^{100}$. We then randomly generate $100000$ points by adding gaussian noise (using $N(0, 1)$) independently to the first $d$ coordinates of $\mathsf{x}$. Figure \ref{fig:1} shows the plot of $Cost[i]$ versus $i$ and the log-log plot for the same. We note that the log-log plot are nearly straight lines and $(-2.0)/slope$ gives a close estimate for the intrinsic dimension. For estimating the dimension, we repeat $10$ times and then report the average value. The dimension estimates for {\tt Affine-2} to {\tt Affine-7} are $\{2.17, 3.27, 3.87, 4.65, 5.33, 6.26\}$.
\begin{figure}
\caption{Figures (a) and (c) show plot of $Cost[i]$ vs $i$ and (b) and (d) show the corresponding log-log plot for the Affine and Swissroll dataset.}
\label{fig:1}
\end{figure}
We also experiment with some of the widely used datasets for dimension estimation. The first of these datasets is the "swissroll" dataset \footnote{\url{http://people.cs.uchicago.edu/~dinoj/manifold/swissroll.html}} comprising of $1600$ points in 3-dimensions located on a manifold shaped like a {\em swissroll}. So, the intrinsic dimension should be $2$. Figure \ref{fig:1} gives the plots for this dataset. The estimate of the dimension given by our technique is $1.8$.
Another dataset used is a dataset generated from a video of a rotating teapot. One frame of this video is considered a data point and there are 100 frames. Since there is only one degree of freedom of motion for the teapot, ideally the dimension of this dataset should be $1$. However, the dimension estimate using our technique is $6.9$ suggesting that lighting, reflection and other parameters may also be important parameters that are not ignored. Another interesting property of the log-log plot for the Teapots dataset that one should note is that the curve is not a straight line as is the case with the previous examples. This suggests that the dimension may depend on the scale at which the data is analysed. At a much more finer scale, the data may have much larger dimension than at a coarser level. The interesting property of our technique is that using the slope at various points in the curve may give dimension at various scales. What is interesting about this observation is that for complex datasets where we have no intuition regarding the intrinsic dimension, the shape of the log-log plot may provide useful insight about the dimensionality of the data at various scales. Another dataset that is similar to the teapots dataset is the hand dataset where the data points are $481$ frames of a video capturing a rotating hand holding a cup. The dimension estimate using our technique is $2.6$. This matches estimates using other techniques. In particular, it matches the estimates of \cite{kegl}.
\begin{figure}
\caption{Figures (a) and (c) show plot of $Cost[i]$ vs $i$ and (b) and (d) show the corresponding log-log plot for the Teapots and Hand dataset.}
\label{fig:2}
\end{figure}
We perform experiments on a dataset where it is not clear what the intrinsic dimension might be. We test our technique on the MNIST test dataset for hand written number "2". The number of data points is around $1000$ and the extrinsic dimension is $784$ ($28 \times 28$ pixels). Our estimate for the intrinsic dimension is $11.4$. \footnote{However, one should mention that the estimate has a high variance with the standard deviation being $2.6$ over 30 runs.} This dimension estimate closely matches the estimate of a number of previous works. The estimate in many previous works are in the range $12$-$14$ (see e.g., \cite{gc16}).
\begin{figure}
\caption{Hand, Teapot, and MNIST(2) datasets}
\label{fig:3}
\end{figure}
\appendix \section{Bounds on $\mathbb{N}(\mathbb{S}^{d-1}, \varepsilon)$ and $\mathbb{P}(\mathbb{S}^{d-1}, \varepsilon)$} \label{sec:appen}
We restate Lemma~\ref{lemma:geometric-bounds} below before giving the proof.
\begin{lemma}[Restatement of Lemma~\ref{lemma:geometric-bounds}] Let $\mathbb{S}^{d-1}$ denote a unit sphere in $\mathbb{R}^d$ and $0 < \varepsilon < 1$. Then \begin{enumerate} \item $\mathbb{N}(\mathbb{S}^{d-1}, \varepsilon) = O\left(\frac{1}{(\varepsilon/8)^{d-1}}\right)$, and \item $\mathbb{P}(\mathbb{S}^{d-1}, \varepsilon) = \Omega\left(\frac{1}{(2\varepsilon)^{d-1}}\right)$. \end{enumerate} \end{lemma}
\begin{proof} We will use the following well known fact to obtain our bounds: \begin{equation}\label{fact:cover-pack}
\mathbb{P}(\mathbb{S}^{d-1}, 2\varepsilon) \leq \mathbb{N}(\mathbb{S}^{d-1}, \varepsilon) \leq \mathbb{P}(\mathbb{S}^{d-1}, \varepsilon) \end{equation}
Let $p$ be any point in $\mathbb{S}^{d-1}$ and let $S(\gamma)$ be the spherical cap over the unit sphere formed by taking the intersection of a ball of radius $\gamma$ at $p$ with the surface of the unit sphere. We will now upper bound the surface area of the spherical cap $S(\gamma)$. Let $A_d(r)$ denote the surface area of a sphere of radius $r$ in a $d$-dimensional Euclidean space. The bound on the surface area can be calculated using the following integral (see Figure~\ref{fig:bounds} for reference): \begin{figure}
\caption{We consider sphere of radius $\gamma$ centered on $p$. $\alpha$ is the angle of the spherical cap at the origin.}
\label{fig:bounds}
\end{figure} \begin{eqnarray*} &~~&S(\gamma)\\ &=& \int_{0}^{\alpha} A_{d-1}(\sin{\theta}) d \theta \\ &=& A_{d-1}(1) \cdot \int_{0}^{\alpha} (\sin{\theta})^{d-2} d\theta \\ &\leq& A_{d-1}(1) \cdot \int_{0}^{\alpha} \theta^{d-2} d \theta = A_{d-1}(1) \cdot \frac{\alpha^{d-1}}{d-1} \end{eqnarray*} Similarly, we can also obtain an upper bound on $S(\gamma)$: \begin{eqnarray*} S(\gamma) &=& \int_{0}^{\alpha} A_{d-1}(\sin{\theta}) d \theta \\ &=& A_{d-1}(1) \cdot \int_{0}^{\alpha} (\sin{\theta})^{d-2} d\theta \\ &\geq& A_{d-1}(1) \cdot \int_{0}^{\alpha} \frac{\theta^{d-2}}{2^{d-2}} d \theta \\ &=& A_{d-1}(1) \cdot \frac{(\alpha/2)^{d-1}}{d-1} \end{eqnarray*} Using the fact that $\sin(\alpha) = \gamma\sqrt{1 - \gamma^2/4}$, we obtain the following bounds on $S(\varepsilon)$ and $S(\varepsilon/2)$ that we will use. \[ S(\varepsilon) \leq A_{d-1}(1)\cdot \frac{(2\varepsilon)^{d-1}}{d-1} \ \ \textrm{and}\ \ S\left(\frac{\varepsilon}{2}\right) \geq A_{d-1}(1) \cdot \frac{(\frac{\varepsilon}{8})^{d-1}}{d-1} \]
Let $N$ be any $\varepsilon$-covering set over the unit sphere with minimal cardinality. Then we have $A_{d}(1) \leq |N| \cdot S(\varepsilon)$ which gives: \begin{eqnarray*}
|N| \geq \frac{A_d(1)}{A_{d-1}(1)} \cdot \frac{d-1}{(2\varepsilon)^{d-1}} = \frac{2 \pi (d-1)}{d} \cdot \frac{1}{(2\varepsilon)^{d-1}}. \end{eqnarray*} Using eqn.(\ref{fact:cover-pack}), we get that $\mathbb{P}(\mathbb{S}^{d-1}, \varepsilon) = \Omega\left(\frac{1}{(2\varepsilon)^{d-1}}\right)$.
Let $P$ be an $\varepsilon$-packing set of maximum cardinality over the unit sphere. Note that balls of radius $(\varepsilon/2)$ centered at points in $P$ do not intersect.
This gives $|P| \cdot S(\varepsilon/2) \leq A_{d}(1)$. So, using lower bound on the surface area of the spherical cap, we obtain: \begin{eqnarray*}
|P| \leq \frac{A_d(1)}{A_{d-1}(1)} \cdot \frac{d-1}{(\varepsilon/8)^{d-1}} = \frac{2 \pi (d-1)}{d} \cdot \frac{1}{(\varepsilon/8)^{d-1}}. \end{eqnarray*} Using eqn.(\ref{fact:cover-pack}), we get that $\mathbb{N}(\mathbb{S}^{d-1}, \varepsilon) = O\left(\frac{1}{(\varepsilon/8)^{d-1}}\right)$. \end{proof}
\section{$(\lowercase{k},\varepsilon)$-movement-based Coreset implies a $(\lowercase{k},\varepsilon)$-coreset} \label{sec:appen2}
\begin{theorem}\label{thm:coreset-1}
Let $X \subseteq \mathbb{R}^d$ be any dataset and $S$ be a $(k, \varepsilon^2/32)$-movement-based coreset of $X$. Let $w: S \rightarrow \mathbb{R}^{+}$ be a weight function defined as follows $\forall s \in S, w(s) = |\mathbb{N}_X(s)|$, where for any point $s \in S$, we define $\mathbb{N}_X(s)$ to be the set of points from $X$ such that their closest point $S$ is $s$. Then, $S$ along with weight function $w$ is a $(k,\varepsilon)$-coreset of $X$. \end{theorem}
\begin{proof} Let $C$ be any set of $k$ centers. For any point $x \in X$, let $c_x$ denote its closest point in the set $C$. Similarly, let $s_x$ denote its closest point in the set $S$. Given this, first we note that for any point $x \in X$, if $D(x, c_x) > D(s_x, c_{s_x})$ we have: \begin{eqnarray*}
|D(x, c_x) - D(s_x, c_{s_x})| &\leq& D(x, c_{s_x}) - D(s_x, c_{s_x}) \\ &\leq& D(x, s_x), \end{eqnarray*} and when $D(x, c_x) \leq D(s_x, c_{s_x})$, we have: \begin{eqnarray*}
|D(x, c_x) - D(s_x, c_{s_x})| &\leq& D(s_x, c_x) - D(x, c_x)\\ &\leq& D(x, s_x) \end{eqnarray*}
So, from the above two inequalities, we get $|D(x, c_x) - D(s_x, c_{s_x})| \leq D(x, s_x)$. Now, for every point $x \in X$, $|D(x, c_x)^2 - D(s_x, c_{s_x})^2|$ equals: \begin{eqnarray*}
&&|D(x, c_x)^2 - D(s_x, c_{s_x})^2|\\
&=& |D(x, c_x) - D(s_x, c_{s_x})| \cdot (D(x, c_x) + D(s_x, c_{s_x})) \\ &\leq& D(x, s_x) \cdot (D(x, c_x) + D(s_x, c_{x})) \\ && \textrm{(since $D(s_x, c_{s_x}) \leq D(s_x, c_x)$ and} \\
&& \textrm{$|D(x, c_x) - D(s_x, c_{s_x})| \leq D(x, s_x)$)} \\ &\leq& D(x, s_x) \cdot (2 \cdot D(x, c_x) + D(x, s_x))\\ && \textrm{(since $D(s_x, c_x) \leq D(x, s_x) + D(x, c_x)$)}\\ &=& 2 \cdot D(x, s_x) \cdot D(x, c_x) + D(x, s_x)^2 \end{eqnarray*} We will use the above inequality to now show the main result.
$|\Phi(C, S, w) - \Phi(C, X)|$ can be upper bounded by: \begin{eqnarray*}
&& \sum_{x \in X} |D(x, c_x)^2 - D(s_x, c_{s_x})^2|\\ &\leq& \sum_{x \in X} 2 \cdot D(x, s_x) \cdot D(x, c_x) + \sum_{x \in X} D(x, s_x)^2 \\ &=& 2 \cdot \sum_{x \in X, \frac{4}{\varepsilon} \cdot D(x, s_x) < D(x, c_x)} D(x, s_x) \cdot D(x, c_x) + \\ && 2 \cdot \sum_{x \in X, \frac{4}{\varepsilon} \cdot D(x, s_x) \geq D(x, c_x)} D(x, s_x) \cdot D(x, c_x)+\\ && \sum_{x \in X} D(x, s_x)^2\\
&\leq & 2 \cdot \sum_{x \in X, \frac{4}{\varepsilon} \cdot D(x, s_x) < D(x, c_x)} (\varepsilon/4) \cdot D(x, c_x)^2 + \\ && 2 \cdot \sum_{x \in X, \frac{4}{\varepsilon} \cdot D(x, s_x) \geq D(x, c_x)} \frac{4}{\varepsilon} \cdot D(x, s_x)^2+\\ && \sum_{x \in X} D(x, s_x)^2\\
&\leq & \frac{\varepsilon}{2} \cdot \sum_{x \in X} D(x, c_x)^2 + \frac{8}{\varepsilon} \cdot \sum_{x \in X} D(x, s_x)^2+\\ && \sum_{x \in X} D(x, s_x)^2\\ &=& \frac{\varepsilon}{2} \cdot \Phi(C, X) + (8/\varepsilon + 1) \cdot \Phi(S, X)\\ &\leq& \frac{\varepsilon}{2} \cdot \Phi(C, X) + \frac{\varepsilon}{2} \cdot \Delta_X(k)\\ &\leq& \varepsilon \cdot \Phi(C, X) \end{eqnarray*}
This implies that $|\Phi(C, S, w) - \Phi(C, X)| \leq \varepsilon \cdot \Phi(C, X)$, which in turn means that $S$ is a $(k, \varepsilon)$-coreset of $X$. This completes the proof of the theorem. \end{proof}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Strongly separable matrices for nonadaptive combinatorial group testing \author{Jinping Fan, Hung-Lin Fu, Yujie Gu, Ying Miao, and Maiko Shigeno} \thanks{J. Fan is with the Department of Policy and Planning Sciences, Graduate School of Systems and Information Engineering, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan (e-mail: [email protected]).} \thanks{H.-L. Fu is with the Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30010, Taiwan (e-mail: [email protected]).} \thanks{Y. Gu is with the Department of Electrical Engineering-Systems, Tel Aviv University, Tel Aviv 6997801, Israel (e-mail: [email protected]).} \thanks{Y. Miao and M. Shigeno are with the Faculty of Engineering, Information and Systems, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan (e-mails: [email protected]; [email protected]).} }
\date{}
\maketitle
\begin{abstract} In nonadaptive combinatorial group testing (CGT), it is desirable to identify a small set of up to $d$ defectives from a large population of $n$ items with as few tests (i.e. large rate) and efficient identifying algorithm as possible. In the literature, $d$-disjunct matrices ($d$-DM) and $\bar{d}$-separable matrices ($\bar{d}$-SM) are two classical combinatorial structures having been studied for several decades. It is well-known that a $d$-DM provides a more efficient identifying algorithm than a $\bar{d}$-SM, while a $\bar{d}$-SM could have a larger rate than a $d$-DM. In order to combine the advantages of these two structures, in this paper, we introduce a new notion of \emph{strongly $d$-separable matrix} ($d$-SSM) for nonadaptive CGT and show that a $d$-SSM has the same identifying ability as a $d$-DM, but much weaker requirements than a $d$-DM. Accordingly, the general bounds on the largest rate of a $d$-SSM are established. Moreover, by the random coding method with expurgation, we derive an improved lower bound on the largest rate of a $2$-SSM which is much higher than the best known result of a $2$-DM. \end{abstract}
\begin{IEEEkeywords} Nonadaptive combinatorial group testing, Disjunct matrices, Strongly separable matrices, Separable matrices \end{IEEEkeywords}
\section{Introduction} \label{sec:1}
Group testing was introduced by Dorfman \cite{RD} in 1940s for a large-scaled blood testing program. The object of this program was testing a large number of blood samples to determine the \textit{defective} (or \textit{positive}) ones. Instead of testing one by one, group testing was proposed to pool all the blood samples into groups and perform a test to each group. If the testing outcome of a group is \textit{positive}, it means that at least one defective is contained in this group. If the testing outcome of a group is \textit{negative}, then this group is considered containing no defective samples. In general, there are two types of group testing models. In an \textit{adaptive} (or \textit{sequential}) setting, the group arrangement of the next test is always based on the previous testing outcomes. In a \textit{nonadaptive} setting, all the group arrangements are designed in advance and all the tests are conducted simultaneously. On the other hand, group testing can be roughly divided into two categories: \textit{probabilistic group testing} (PGT) and \textit{combinatorial group testing} (CGT). In PGT, the defective samples are assumed to follow some probability distribution, while in CGT, the number of defective samples is usually assumed to be no more than a fixed positive integer. In this paper, we focus on \textit{nonadaptive CGT} (or \textit{pooling design}) in the noiseless scenario, which has been studied extensively due to its applications in a variety of fields such as DNA library screening, network security, multi-access communication and so on, see \cite{DH,DH2006,D,KS,ND,W} for example.
A nonadaptive CGT scheme can be represented by a binary matrix whose rows are indexed by the groups and columns are indexed by the items to be tested. Suppose that there are $n$ items among which at most $d$ $(\ll n)$ are positive. Let $M$ be the $t\times n$ testing matrix where the entry $M_j(i)=1$ if the $j$th item is contained in the $i$th test and $M_j(i)=0$ otherwise. Note that each column of $M$ corresponds to an item and each row corresponds to a test. The result of a test is $1$ (positive) if the test contains at least one positive item and $0$ (negative) otherwise. After performing all the $t$ tests simultaneously, one could observe the testing outcome $\vec{r}=(\vec{r}(1),\ldots,\vec{r}(t))\in\{0,1\}^t$ where $\vec{r}(i)$ is the result of the $i$th test. It is easily seen that $\vec{r}$ is the \textit{Boolean sum} of the column vectors of $M$ indexed by all the positive items. The problem of studying nonadaptive CGT is to design the testing matrix $M$ such that all the positives could be identified based on $M$ and the testing outcome $\vec{r}$. The goal is to decrease the computational complexity of the identifying algorithm and to minimize the number of tests needed given the number of items to be tested, or equivalently, to maximize the number of items to be tested given the number of tests, or in other words, to explore the \textit{largest rate} of the testing matrix $M$.
In the literature, \textit{disjunct matrices} and \textit{separable matrices} are two classical combinatorial structures for nonadaptive CGT. Disjunct matrices and separable matrices were first studied by Kautz and Singleton \cite{KS} for file retrieval under the name of \textit{superimposed codes}, and later were extensively investigated under the concepts of \textit{cover-free families} and \textit{union-free families} respectively, see \cite{EFF,EFF1985,FF} for example. The definitions of disjunct matrices and separable matrices could be found in \cite{DH}. \begin{definition}{\rm (\cite{DH})}\label{dm-sm} Let $n,t,d\geq 2$ be integers and $M$ be a binary matrix of size $t\times n$. \begin{enumerate}
\item $M$ is called a \textit{$d$-disjunct matrix}, or briefly $d$-DM, if the Boolean sum of any $d$ column vectors of $M$ does not cover any other one.
\item $M$ is called a \textit{$\bar{d}$-separable matrix}, or briefly $\bar{d}$-SM, if the Boolean sums of $\le d$ column vectors of $M$ are all distinct. \end{enumerate} \end{definition}
It was shown in \cite{DH} that a $d$-DM, as well as a $\bar{d}$-SM, could be utilized in nonadaptive CGT to identify any set of positives with size no more than $d$, but both have their advantages and disadvantages. In general, the computational complexity of the identifying algorithm based on a $d$-DM of size $t\times n$ is $O(tn)$, while that based on a $\bar{d}$-SM of size $t\times n$ is $O(tn^d)$, explicating that a $d$-DM provides a more efficient identifying algorithm than a $\bar{d}$-SM. However, given the number of rows, a $\bar{d}$-SM contains more columns, implying a higher rate, than a $d$-DM. In order to combine the advantages of these two structures, in this paper, we introduce a new notion of \textit{strongly $d$-separable matrix} ($d$-SSM) for nonadaptive CGT which has weaker requirements than $d$-DM but the same identifying ability as $d$-DM. It is also shown that a $d$-SSM has a larger rate than a $d$-DM.
The rest of this paper is organized as follows. In Section \ref{sec:2}, we give the definition of SSM and establish the relationships among SSM, DM and SM. We show that a $d$-SSM could identify any set of up to $d$ positives as efficiently as a $d$-DM. In Section \ref{sec:4}, we first give the general bounds on the largest rate of a $d$-SSM from the known results on DM and SM, and then derive an improved lower bound on the largest rate of a $2$-SSM by the random coding method with expurgation. The conclusion is drawn in Section \ref{sec:5}.
\section{Strongly separable matrices} \label{sec:2}
In this section, we first introduce the notion of $d$-SSM and investigate the relationships among DM, SSM and SM. Next we provide an identifying algorithm based on a $d$-SSM and prove that a $d$-SSM has the same identifying ability as a $d$-DM.
Let $n,t,d$ be integers with $n\ge d\ge 2$, and $M$ be a binary matrix of size $t\times n$. Denote $[t]=\{1,\ldots,t\}$ and $[n]=\{1,\ldots,n\}$. Let ${\cal F}=\{\vec{c}_1,\dots,\vec{c}_n\}\subseteq\{0,1\}^t$ be the set of column vectors of $M$ where $\vec{c}_j=(\vec{c}_j(1),\ldots,\vec{c}_j(t))\in\{0,1\}^t$ for any $j\in[n]$. We say a vector $\vec{c}_j$ \textit{covers} a vector $\vec{c}_k$ if for any $i\in [t]$, $\vec{c}_k(i)=1$ implies $\vec{c}_j(i)=1$.
\begin{definition}\label{ssm}
A $t\times n$ binary matrix $M$ is called a \textit{strongly $d$-separable matrix}, or briefly $d$-{\rm SSM}, if for any ${\cal F}_0\subseteq{\cal F}$ with $|{\cal F}_0|=d$, we have \begin{equation}\label{ssm-1}
\bigcap\limits_{{\cal F}'\in \mathcal{U}({\cal F}_0)}{\cal F}'={\cal F}_0, \end{equation} where \begin{equation}\label{ssm-2}
\mathcal{U}({\cal F}_0)=\Big\{{\cal F}'\subseteq{\cal F}: \bigvee_{\vec{c}\in{\cal F}_0}\vec{c}=\bigvee_{\vec{c}\in{\cal F}'}\vec{c}\Big\}. \end{equation} \end{definition}
\begin{remark}\label{ssm:2}
An equivalent description of Definition \ref{ssm} is as follows. A $t\times n$ binary matrix $M$ is a $d$-SSM if for any ${\cal F}_0,{\cal F}'\subseteq{\cal F}$ with $|{\cal F}_0|=d$, ${\cal F}'\in\mathcal{U}({\cal F}_0)$ implies that ${\cal F}_0\subseteq{\cal F}'$. \end{remark}
\begin{remark}\label{ssm:1}
We call $M$ a $\bar{d}$-SSM if the condition $|{\cal F}_0|=d$ in Definition \ref{ssm} is replaced by $1\le|{\cal F}_0|\le d$. \end{remark}
It is obvious that a $\bar{d}$-SSM is always a $d$-SSM. The following observation shows that the converse also holds.
\begin{lemma}\label{equivalence} A $t\times n$ matrix $M$ is a $d$-SSM if and only if $M$ is a $\bar{d}$-SSM. \end{lemma} \begin{IEEEproof}
It is enough to show the necessity. Assume that $M$ is a $d$-SSM of size $t\times n$, but not a $\bar{d}$-SSM. Let ${\cal F}=\{\vec{c}_1,\dots,\vec{c}_n\}$ be the set of column vectors of $M$. By Definition \ref{ssm} and Remarks \ref{ssm:2} and \ref{ssm:1}, there must exist ${\cal F}_0,{\cal F}'\subseteq{\cal F}$ with $1\le|{\cal F}_0|<d$ such that ${\cal F}'\in \mathcal{U}({\cal F}_0)$ but ${\cal F}_0\not\subseteq{\cal F}'$. Suppose that $|{\cal F}_0|=j$, $1\le j\le d-1$. If ${\cal F}'\setminus{\cal F}_0=\emptyset$, there exists ${\cal F}_1\subseteq{\cal F}\setminus{\cal F}_0$ such that $|{\cal F}_0\cup{\cal F}_1|=d$ since $n\ge d$. Then we have ${\cal F}'\cup{\cal F}_1\in \mathcal{U}({\cal F}_0\cup{\cal F}_1)$ but ${\cal F}_0\cup{\cal F}_1\not\subseteq {\cal F}'\cup{\cal F}_1$, a contradiction to the definition of $d$-SSM. Thus, ${\cal F}'\setminus{\cal F}_0\neq\emptyset$ and $\vec{c}'$ is covered by $\bigvee_{\vec{c}\in{\cal F}_0}\vec{c}$ for any $\vec{c}'\in{\cal F}'\setminus{\cal F}_0$. We discuss $|{\cal F}'\setminus{\cal F}_0|$ based on the following two cases. \begin{enumerate}[1)]
\item If $|{\cal F}'\setminus{\cal F}_0|\ge d-j$, then there exists ${\cal F}_1\subseteq{\cal F}'\setminus{\cal F}_0$ such that $|{\cal F}_0\cup{\cal F}_1|=d$ and ${\cal F}'\in \mathcal{U}({\cal F}_0\cup{\cal F}_1)$. Since ${\cal F}_0\not\subseteq{\cal F}'$, we have ${\cal F}_0\cup{\cal F}_1\not\subseteq {\cal F}'$, a contradiction to the definition of $d$-SSM.
\item If $|{\cal F}'\setminus{\cal F}_0|< d-j$, then $|{\cal F}_0\cup{\cal F}'|<d$. Since $n\ge d$, there exists ${\cal F}_2\subseteq{\cal F}\setminus({\cal F}_0\cup{\cal F}')$ such that $|{\cal F}_0\cup{\cal F}'\cup{\cal F}_2|=d$ and ${\cal F}'\cup{\cal F}_2\in \mathcal{U}({\cal F}_0\cup{\cal F}'\cup{\cal F}_2)$, but ${\cal F}_0\cup{\cal F}'\cup{\cal F}_2\not\subseteq{\cal F}'\cup{\cal F}_2$, also a contradiction. \end{enumerate}
The conclusion follows. \end{IEEEproof}
The relationship between DM and SM was investigated in \cite{CH,DH}.
\begin{lemma}{\rm{(\cite{CH,DH})}}\label{relationship of DM and SM} A $d$-DM is a $\bar{d}$-SM and a $\bar{d}$-SM is a $(d-1)$-DM. \end{lemma}
The following lemma shows that a $d$-SSM lies between a $d$-DM and a $\bar{d}$-SM.
\begin{lemma}\label{relationship2} A $d$-DM is a $d$-SSM and a $d$-SSM is a $\bar{d}$-SM. \end{lemma} \begin{IEEEproof}
We first show that a $d$-DM is a $\bar{d}$-SSM. Let $A$ be a $d$-DM and ${\cal A}$ be the set the column vectors of $A$. By Definition \ref{dm-sm}, for any ${\cal A}_0\subseteq{\cal A}$ with $|{\cal A}_0|=d$, we have $\mathcal{U}({\cal A}_0)=\{{\cal A}_0\}$ where $\mathcal{U}({\cal A}_0)$ is defined by (\ref{ssm-2}). Then we obtain $\bigcap_{{\cal A}'\in \mathcal{U}({\cal A}_0)}{\cal A}'={\cal A}_0$ implying that $A$ is a $d$-SSM according to Definition \ref{ssm}.
Now we prove that a $d$-SSM is a $\bar{d}$-SM. Let $M$ be a $d$-SSM and ${\cal F}$ be the set the column vectors of $M$. If $M$ is not a $\bar{d}$-SM, then by Definition \ref{dm-sm}, there exist distinct ${\cal F}_1,{\cal F}_2\subseteq{\cal F}$ with $1\le |{\cal F}_1|,|{\cal F}_2|\le d$ such that $\bigvee_{\vec{c}\in{\cal F}_1}\vec{c}=\bigvee_{\vec{c}\in{\cal F}_2}\vec{c}$. By (\ref{ssm-2}), we have ${\cal F}_1\in \mathcal{U}({\cal F}_2)$ and ${\cal F}_2\in \mathcal{U}({\cal F}_1)$. Since $M$ is a $d$-SSM, by Lemma \ref{equivalence} we have ${\cal F}_1\subseteq{\cal F}_2$ and ${\cal F}_2\subseteq{\cal F}_1$ which implies ${\cal F}_1={\cal F}_2$, a contradiction to the assumption. \end{IEEEproof}
From Lemma \ref{relationship2} we know that a $d$-DM is always a $d$-SSM. But a $d$-SSM might not be a $d$-DM. To show this, we give an example below.
\begin{example} Let $M$ be a binary matrix of size $7\times 8$ as defined below. \begin{equation*} \begin{aligned} M & =\begin{matrix} \vec{c}_1 \hspace{0.16cm} \vec{c}_2 \hspace{0.175cm} \vec{c}_3 \hspace{0.17cm} \vec{c}_4 \hspace{0.18cm} \vec{c}_5 \hspace{0.2cm} \vec{c}_6 \hspace{0.18cm} \vec{c}_7 \hspace{0.18cm} \vec{c}_8 \\ \begin{bmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 \end{bmatrix} \end{matrix} \end{aligned} \end{equation*} It is easy to check that $M$ is a $2$-SSM and thus a $\bar{2}$-SM by Lemma \ref{relationship2}. But $M$ is not a $2$-DM since $\vec{c}_2$ is covered by $\vec{c}_1\vee\vec{c}_3$. \end{example}
By Lemma \ref{relationship2}, a $d$-SSM, as well as a $\bar{d}$-SM, has weaker requirements than a $d$-DM. However, we will prove that a $d$-SSM could determine any set of positives with size no more than $d$ as efficiently as a $d$-DM.
\begin{theorem}\label{complexity} A $t\times n$ $d$-SSM could identify any set of $\le d$ positives among $n$ items with $t$ tests by applying Algorithm \ref{algorithm 1}, and the computational complexity of Algorithm \ref{algorithm 1} is $O(tn)$. \end{theorem} \begin{IEEEproof}
Denote $[n]$ as $n$ items to be tested. Let $M$ be the testing matrix which is a $d$-SSM of size $t\times n$, and ${\cal F}=\{\vec{c}_1,\vec{c}_2,\ldots,\vec{c}_n\}$ be the set the column vectors of $M$. Suppose that $P_0\subseteq[n]$ is the set of positives with $|P_0|\le d$, and the testing outcome is $\vec{r}\in\{0,1\}^t$. We show that Algorithm \ref{algorithm 1} will output $P_0$ given the input $\vec{r}$, that is, we show $P=P_0$.
Let ${\cal F}_0\subseteq{\cal F}$ be the set of column vectors of $M$ corresponding to $P_0$. Then, $|{\cal F}_0|\le d$ and $\vec{r}=\bigvee_{\vec{c}\in{\cal F}_0}\vec{c}$. According to Algorithm \ref{algorithm 1}, given the input $\vec{r}$, we first remove every column $\vec{c}\in{\cal F}$ that is not covered by $\vec{r}$. Then we obtain a subset ${\cal F}_S=\{\vec{c}_j: j\in S\}\subseteq{\cal F}$, where $S=\{j\in[n]:\vec{c}_j$ is covered by $\vec{r}\}$. It is obvious that ${\cal F}_0\subseteq{\cal F}_S$ and ${\cal F}_S\in\mathcal{U}({\cal F}_0)$ where $\mathcal{U}({\cal F}_0)$ is defined by (\ref{ssm-2}). Next we show how to determine ${\cal F}_0$ from ${\cal F}_S$.
For any $\vec{c}\in{\cal F}_S$, we claim that $\vec{c}\in{\cal F}_0$ if and only if there exists $i\in[t]$ such that $\vec{c}(i)=1$ and $\vec{c}'(i)=0$ for any $\vec{c}'\in{\cal F}_S\setminus\{\vec{c}\}$. To show the necessity, assume that $\vec{c}\in{\cal F}_0$ but there dose not exist $i\in[t]$ such that $\vec{c}(i)=1$ and $\vec{c}'(i)=0$ for any $\vec{c}'\in{\cal F}_S\setminus\{\vec{c}\}$. Then we have ${\cal F}_S\setminus\{\vec{c}\}\in\mathcal{U}({\cal F}_0)$. Since $M$ is a $d$-SSM, we have $\vec{c}\not\in\bigcap_{{\cal F}'\in\mathcal{U}({\cal F}_0)}{\cal F}'={\cal F}_0$, a contradiction to the condition that $\vec{c}\in{\cal F}_0$. To show the sufficiency, assume that there exists $\vec{c}\in{\cal F}_S$ with the property that there exists $i_0\in[t]$ such that $\vec{c}(i_0)=1$ and $\vec{c}'(i_0)=0$ for any $\vec{c}'\in{\cal F}_S\setminus\{\vec{c}\}$, but $\vec{c}\not\in{\cal F}_0$, that is, the item corresponding to $\vec{c}$ is negative. Since ${\cal F}_0\subseteq{\cal F}_S$ and $\vec{r}=\bigvee_{\vec{c}\in{\cal F}_0}\vec{c}$, we must have $\vec{r}(i_0)=0$. Since $\vec{c}(i_0)=1$, it implies that $\vec{c}\not\in{\cal F}_S$ by the previous step of Algorithm \ref{algorithm 1}, a contradiction to the condition that $\vec{c}\in{\cal F}_S$.
Thus, the output $P$ of Algorithm \ref{algorithm 1} is exactly the set of positives $P_0$. The computational complexity of Algorithm \ref{algorithm 1} is $O(nt)$. \end{IEEEproof}
\begin{algorithm}[H]\label{algorithm 1} \caption{SSMIdAlg($\vec{r}$)}
Let $R_0=\{N_1,\ldots,N_{|R_0|}\}\subseteq[t]$ and $R_1=\{J_1,\ldots,J_{|R_1|}\}\subseteq[t]$ be two sets of indices which indicate $\vec{r}(i)=0$ and $\vec{r}(i)=1$ respectively. Clearly, $R_0\cup R_1=[t]$ and $|R_0|+|R_1|=t$.
$S=\{1,2,\ldots,n\}$;\\ $P=\emptyset$;\\
\For{$k=1$ to $|R_0|$}
{ $i=N_k$;\\
\For{$j=1$ to $n$}
{ \If{$\vec{c}_j(i)=1$}
{ $S=S\setminus\{j\}$;
}
}
}
\For{$k=1$ to $|R_1|$}
{ $i=J_k$;\\
\For{$j=1$ to $n$}
{ \If{$j\in S$}
{ \If{$\vec{c}_j(i)=1$ and $\vec{c}_l(i)=0$ for any $l\in S\setminus\{j\}$}
{ $P=P\cup\{j\}$;
}
}
}
}
\eIf{$|P|\le d$}
{ {\bf output} $P$; }
{ {\bf output} ``The set of positives has size at least $d+1$.'' } \end{algorithm}
\section{Bounds for $d$-SSM} \label{sec:4}
In this section, we concentrate on the largest rate of a $d$-SSM. We first provide general bounds for $d$-SSM based on its connections with $d$-DM and $\bar{d}$-SM, and then derive an improved lower bound for $2$-SSM by the random coding method with expurgation, which is much better than the best existing lower bound of $2$-DM.
\subsection{General bounds for $d$-{\rm SSM}} \label{sec:4-1}
Let $n(d,t)$, $f(d,t)$ and $s(\bar{d},t)$ denote the maximum possible number of columns of a $d$-SSM, a $d$-DM and a $\bar{d}$-SM with $t$ rows respectively. Denote their largest rates as \begin{equation*}
\begin{aligned}
R(d) & =\underset{t\to\infty}{\overline{\lim}}\frac{\log_2 n(d,t)}{t},\\
R_D(d) & =\underset{t\to\infty}{\overline{\lim}}\frac{\log_2 f(d,t)}{t},\\
R_S(\bar{d}) & =\underset{t\to\infty}{\overline{\lim}}\frac{\log_2 s(\bar{d},t)}{t}. \end{aligned} \end{equation*}
In the literature, the best known upper and lower bounds of $R_D(d)$ for $d\ge 3$ were proved in \cite{DR,DRR} respectively, and the general bounds of $R_S(\bar{d})$ were derived by Lemma \ref{relationship of DM and SM} and the known results on $R_D(d)$ \cite{DH}. For the case $d=2$, Erd\"{o}s, Frankl and F\"{u}redi \cite{EFF} investigated cover-free families and derived the best known results for $R_D(2)$ in which the lower bound was obtained by the random coding method and the upper bound was obtained by the techniques in extremal combinatorics. In \cite{CS}, Coppersmith and Shearer provided the best known lower bound for $R_S(\bar{2})$ by constructing a $\bar{2}$-union-free family from a deterministic \textit{cancellative} family and a random \textit{weakly union-free} family, and gave the best known upper bound for $R_S(\bar{2})$ by the techniques also in extremal combinatorics.
\begin{theorem}{\rm({\cite{CS,DH,DR,DRR,EFF}})}\label{known bounds of DM,SM} Let $d\ge 2$ be an integer. If $d\to\infty$, then we have \begin{equation*}
\frac{1}{d^2\log_2e}(1+o(1)) \le R_D(d)\le R_S(\bar{d})\le R_D(d-1)\le \frac{2\log_2(d-1)}{(d-1)^2}(1+o(1)) \end{equation*} where $e$ is the base of the natural logarithm. Moreover, \begin{gather*}
0.1814\le R_D(2) \le 0.3219, \\
0.3135\le R_S(\bar{2})\le 0.4998. \end{gather*} \end{theorem}
We remark that the expressions of the general bounds of $R_D(d)$ for any $d\ge 3$ shown in \cite{DR,DRR} are complicated and therefore not stated in this paper. The interested reader may refer to the references therein. The asymptotic version on the bounds of $R_D(d)$ shown in Theorem \ref{known bounds of DM,SM} could also be found in \cite{D}.
By Theorem \ref{known bounds of DM,SM} and Lemma \ref{relationship2}, we immediately have the following results for SSM.
\begin{corollary}\label{general ssm} Let $d\ge 2$ be an integer. Then we have \begin{equation}\label{general bounds of SSM}
\frac{1}{d^2\log_2e}(1+o(1)) \le R(d) \le \frac{2\log_2(d-1)}{(d-1)^2}(1+o(1)) \end{equation} for $d\to\infty$, and \begin{equation}\label{bounds of 2-SSM}
0.1814 \le R(2) \le 0.4998. \end{equation} \end{corollary}
\subsection{An improved lower bound for $2$-{\rm SSM}} \label{sec:4-2}
Inspired by the Kautz-Singleton construction for DM in \cite{KS} which is based on maximum distance separable codes and identity codes, in this part, we provide an improved lower bound of $R(2)$ by the random coding method together with a concatenated construction for SSM based on \textit{strongly separable codes}. For more applications of this method, the interested reader may refer to \cite{AS}.
\begin{theorem}\label{new lower bound of R(2)} $R(2)\ge 0.2213$. \end{theorem}
Before proving Theorem \ref{new lower bound of R(2)}, we do some preparations. Let $q\ge 2$ be an integer and $Q=\{0,1,\ldots,q-1\}$ be an alphabet. A set ${\cal C}=\{\vec{c}_1,\vec{c}_2,\ldots,\vec{c}_n\}\subseteq Q^t$ is called a $(t,n,q)$ code where each $\vec{c}_j$ is called a \textit{codeword}, $t$ is the \textit{length} of the code and $n$ is the \textit{code size}. For a code ${\cal C}\subseteq Q^t$, define the set of the $i$th coordinates of ${\cal C}$ as ${\cal C}(i)=\{\vec{c}(i)\in Q:\ \vec{c}=(\vec{c}(1),\vec{c}(2),\ldots,\vec{c}(t))\in{\cal C}\}$ for any $1\le i\le t$ and define the \textit{descendant code} of ${\cal C}$ as ${\rm desc}({\cal C})={\cal C}(1)\times{\cal C}(2)\times\ldots\times{\cal C}(t).$
In \cite{JCM}, Jiang, Cheng and Miao introduced strongly separable codes in multimedia fingerprinting for the purpose of tracing back to all the traitors in an averaging collusion attack as efficiently as the well-known \textit{frameproof codes} but having a larger code size than frameproof codes.
\begin{definition}{\rm (\cite{JCM})}\label{ssc}
Let ${\cal C}$ be a $(t,n,q)$ code and $d\ge 2$ be an integer. ${\cal C}$ is called a \textit{strongly $\bar{d}$-separable code}, or briefly $\bar{d}$-SSC$(t,n,q)$, if for any ${\cal C}_0\subseteq{\cal C}$ with $1\le|{\cal C}_0|\le d$, we have \begin{equation}\label{ssc-1}
\bigcap\limits_{{\cal C}'\in \mathcal{S}({\cal C}_0)}{\cal C}'={\cal C}_0, \end{equation} where \begin{equation}\label{ssc-2}
\mathcal{S}({\cal C}_0)=\{{\cal C}'\subseteq{\cal C}:\ {\rm desc}({\cal C}')={\rm desc}({\cal C}_0)\}. \end{equation} \end{definition}
\begin{remark}
An equivalent description of Definition \ref{ssc} is as follows. A $(t,n,q)$ code ${\cal C}$ is a $\bar{d}$-SSC if for any ${\cal C}_0,{\cal C}'\subseteq{\cal C}$ with $1\le |{\cal C}_0|\le d$, ${\cal C}'\in\mathcal{S}({\cal C}_0)$ implies that ${\cal C}_0\subseteq{\cal C}'$. \end{remark}
\begin{remark}\label{ssc:2} When $q=2$, an equivalent description of (\ref{ssc-2}) is that $\mathcal{S}({\cal C}_0)=\{{\cal C}'\subseteq{\cal C}: \bigvee_{\vec{c}\in{\cal C}'}\vec{c}=\bigvee_{\vec{c}\in{\cal C}_0}\vec{c}$ and $\bigwedge_{\vec{c}\in{\cal C}'}\vec{c}=\bigwedge_{\vec{c}\in{\cal C}_0}\vec{c}\}.$ \end{remark}
We have the following observation on the relationship between SSM and SSC.
\begin{lemma}\label{lemma 1} If there exists a $d$-SSM of size $t\times n$, then there exists a $\bar{d}$-SSC$(t,n,2)$. \end{lemma} \begin{IEEEproof}
Let $M$ be a $d$-SSM of size $t\times n$ and ${\cal C}$ be the set of column vectors of $M$. It is obvious that ${\cal C}$ is a $(t,n,2)$ code. By (\ref{ssm-2}) and Remark \ref{ssc:2}, for any ${\cal C}_0\subseteq{\cal C}$ with $1\le|{\cal C}_0|\le d$, we have $\mathcal{S}({\cal C}_0)\subseteq \mathcal{U}({\cal C}_0)$. Since $M$ is a $d$-SSM, we have ${\cal C}_0=\bigcap_{{\cal C}'\in \mathcal{U}({\cal C}_0)}{\cal C}'\subseteq \bigcap_{{\cal C}'\in \mathcal{S}({\cal C}_0)}{\cal C}'\subseteq{\cal C}_0$ which yields $\bigcap_{{\cal C}'\in \mathcal{S}({\cal C}_0)}{\cal C}'={\cal C}_0$. Thus ${\cal C}$ is a $\bar{d}$-SSC. \end{IEEEproof}
In \cite{JCM}, Jiang, Cheng and Miao also provided a concatenated construction for $\bar{d}$-SSC$(tq,n,2)$ based on $\bar{d}$-SSC$(t,n,q)$. We show that the $(tq,n,2)$ code they constructed is actually a $tq\times n$ $d$-SSM.
\begin{lemma}\label{lemma 2} If there exists a $\bar{d}$-SSC$(t,n,q)$, then there exists a $d$-SSM of size $tq\times n$. \end{lemma} \begin{IEEEproof} Let ${\cal C}=\{\vec{c}_1,\ldots,\vec{c}_n\}$ be a $\bar{d}$-SSC$(t,n,q)$ on $Q=\{0,1,\ldots,q-1\}$. For each $\vec{c}_j=(\vec{c}_j(1),\ldots,\vec{c}_j(t))$ $\in{\cal C}$, define $\vec{x}_j=(\vec{x}_j^1,\vec{x}_j^2,\ldots,\vec{x}_j^t)\in\{0,1\}^{tq}$ where \begin{equation*}
\vec{x}_j^i=(\vec{x}_j^i(0),\vec{x}_j^i(1),\ldots,\vec{x}_j^i(q-1))\in\{0,1\}^q \end{equation*} for any $1\le i\le t$ and \begin{equation*}
\vec{x}_j^i(k)=\left\{\begin{matrix}
1, & \vec{c}_j(i)=k \\
0, & $otherwise$
\end{matrix}\right. \end{equation*} for any $0\le k\le q-1$. Let ${\cal F}=\{\vec{x}_1,\vec{x}_2,\ldots,\vec{x}_n\}$ be the set of column vectors of a matrix $M$. It is obvious that $M$ is a binary matrix of size $tq\times n$. We show that $M$ is a $d$-SSM.
For any ${\cal F}_0,{\cal F}'\subseteq{\cal F}$ with $|{\cal F}_0|\le d$, let ${\cal C}_0,{\cal C}'\subseteq{\cal C}$ denote the corresponding subsets of codewords to ${\cal F}_0,{\cal F}'$ respectively. Then $|{\cal C}_0|\le d$. If ${\cal F}'\in\mathcal{U}({\cal F}_0)$, we must have ${\cal C}'\in \mathcal{S}({\cal C}_0)$ according to the construction for ${\cal F}$. Since ${\cal C}$ is a $\bar{d}$-SSC, we have ${\cal C}_0\subseteq{\cal C}'$ yielding ${\cal F}_0\subseteq{\cal F}'$. Thus $M$ a $d$-SSM. \end{IEEEproof}
To use Lemma \ref{lemma 2} to derive bounds of SSM, the results on $\bar{d}$-SSC with fixed $q$ and large $t$ is required. However, to the best of our knowledge, there is no known good result for this case in the literature. Therefore, in order to derive the lower bound of $R(2)$ in Theorem \ref{new lower bound of R(2)}, we shall first randomly construct a $\bar{2}$-SSC with fixed small $q$ and large $t$ and then exploit Lemma \ref{lemma 2} to obtain a $2$-SSM. To present the argument more precisely, we need the following concept of \textit{minimal frame}, which was also studied in \cite{JGC}.
\begin{definition}\label{frame} Let ${\cal C}$ be a $(t,n,q)$ code. For any ${\cal C}_0,{\cal C}'\subseteq{\cal C}$, we call ${\cal C}'$ a \textit{frame} of ${\cal C}_0$ if ${\cal C}'\in\mathcal{S}({\cal C}_0)$. Moreover, ${\cal C}'$ is called a \textit{minimal frame} of ${\cal C}_0$ if ${\cal C}'\in\mathcal{S}({\cal C}_0)$ and ${\cal C}'\setminus\{\vec{c}\}\not\in\mathcal{S}({\cal C}_0)$ for any $\vec{c}\in{\cal C}'$. \end{definition}
\begin{lemma}\label{lemma 3}
Let ${\cal C}$ be a $(t,n,q)$ code. If ${\cal C}$ is not a $\bar{d}$-SSC, then there exist ${\cal C}_0\subseteq{\cal C}$ with $1\le |{\cal C}_0|\le d$ and a minimal frame ${\cal C}'$ of ${\cal C}_0$ such that ${\cal C}_0\not\subseteq{\cal C}'$. \end{lemma} \begin{IEEEproof}
If ${\cal C}$ is not a $\bar{d}$-SSC, then by Definition \ref{ssc}, there exist ${\cal C}_0\subseteq{\cal C}$ with $1\le|{\cal C}_0|\le d$ and a frame ${\cal C}'\subseteq{\cal C}$ of ${\cal C}_0$ such that ${\cal C}_0\not\subseteq{\cal C}'$. If ${\cal C}'$ is minimal, then it completes the proof. Otherwise, by Definition \ref{frame}, there must exist a codeword $\vec{c}\in{\cal C}'$ such that ${\cal C}'\setminus\{\vec{c}\}$ is still a frame of ${\cal C}_0$. Consider ${\cal C}'\setminus\{\vec{c}\}$ and repeat the process until it forms a minimal frame of ${\cal C}_0$. \end{IEEEproof}
The following result could be found in \cite{JGC} as well. For readers' convenience, we will give a self-contained proof of it.
\begin{lemma}\label{lemma 4}
Let ${\cal C}$ be a $(t,n,q)$ code. For any ${\cal C}_0\subseteq{\cal C}$ with $1\le|{\cal C}_0|\le d$, the minimal frame of ${\cal C}_0$ has size no more than $td-t+1$. \end{lemma} \begin{IEEEproof}
Suppose that ${\cal C}'$ is a minimal frame of ${\cal C}_0$. We count the number of codewords in ${\cal C}'$ by the order of coordinates. For the first coordinate, by Definition \ref{frame} and (\ref{ssc-2}), there exists ${\cal C}_1\subseteq{\cal C}'$ such that ${\cal C}_1(1)={\cal C}_0(1)$ and $({\cal C}_1\setminus\{\vec{c}\})(1)\neq{\cal C}_0(1)$ for any $\vec{c}\in{\cal C}_1$. It is obvious that $|{\cal C}_1|=|{\cal C}_0(1)|\le d$. Consider the second coordinate, then there exists ${\cal C}_2\subseteq{\cal C}'\setminus{\cal C}_1$ such that $({\cal C}_2\cup{\cal C}_1)(2)={\cal C}_0(2)$ and $({\cal C}_2\cup{\cal C}_1\setminus\{\vec{c}\})(2)\neq{\cal C}_0(2)$ for any $\vec{c}\in{\cal C}_2$. Then we have $|{\cal C}_2|=|{\cal C}_0(2)|-|{\cal C}_1(2)|\le d-1$. Consider ${\cal C}'\setminus({\cal C}_1\cup{\cal C}_2)$ and the $i$th coordinates in a similar way for $3\le i\le t$ until there will be no codewords left, which must occur since ${\cal C}'$ is a minimal frame of ${\cal C}_0$. Then we have $|{\cal C}'|\le d+(d-1)(t-1)=td-t+1$ as desired. \end{IEEEproof}
Now we present the proof of Theorem \ref{new lower bound of R(2)}.
\begin{IEEEproof}[Proof of Theorem \ref{new lower bound of R(2)}] Let $n>3$ and ${\cal C}=\{\vec{c}_1,\vec{c}_2,\ldots,\vec{c}_n\}$ be a collection of vectors of length $t$ where $\vec{c}_j=(\vec{c}_j(1), \vec{c}_j(2),\ldots,\vec{c}_j(t))$ and each $\vec{c}_j(i)$ is chosen uniformly and independently at random from a set $Q=\{0,1,\ldots,q-1\}$ with the probability that
\begin{equation*}
{\rm Pr}(\vec{c}_j(i)=k)=1/q,\ \forall k\in Q \end{equation*} for any $1\le j\le n$ and $1\le i\le t$. The values of $t,n,q$ will be determined later.
For any vector $\vec{c}\in{\cal C}$, $\vec{c}$ is called {\it bad} if there exists ${\cal C}'=\{\vec{c}_0,\vec{c}_1,\ldots,\vec{c}_m\}\subseteq{\cal C}\setminus\{\vec{c}\}$ such that at least one of the following two cases occurs: \begin{enumerate}[(1)] \item there exists some $\vec{c}_i\in{\cal C}'$ such that ${\rm desc}({\cal C}')={\rm desc}(\{\vec{c},\vec{c}_i\})$ with $1\le m\le t$; \item there exists some $\vec{c}_i\in{\cal C}'$ such that ${\rm desc}({\cal C}'\setminus\{\vec{c}_i\})={\rm desc}(\{\vec{c},\vec{c}_i\})$ with $2\le m\le t+1$. \end{enumerate}
Let $\mathcal{T}$ be the collection of all bad vectors of ${\cal C}$ and $\widehat{{\cal C}}={\cal C}\setminus\mathcal{T}$. We claim that all the vectors in $\widehat{{\cal C}}$ are distinct. If not, assume that there exist $\vec{a},\vec{b}\in\widehat{{\cal C}}$ such that $\vec{a}=\vec{b}$. Then, for any $\vec{x}\in\widehat{{\cal C}}\setminus\{\vec{a},\vec{b}\}$, we have ${\rm desc}(\{\vec{a},\vec{x}\})={\rm desc}(\{\vec{b},\vec{x}\})$ which implies that $\vec{a}$ is bad, a contradiction to the assumption. Hence, $\widehat{{\cal C}}$ is a $(t,n-|\mathcal{T}|,q)$ code by regarding each vector in $\widehat{{\cal C}}$ as a codeword.
We further show that $\widehat{{\cal C}}$ is a $\bar{2}$-SSC. If not, by Lemma \ref{lemma 3}, there exists ${\cal C}_0\subseteq\widehat{{\cal C}}$ with $1\le|{\cal C}_0|\le 2$ and a minimal frame ${\cal C}_1\subseteq\widehat{{\cal C}}$ of ${\cal C}_0$ such that ${\cal C}_0\not\subseteq{\cal C}_1$. We discuss the size of ${\cal C}_0$. \begin{enumerate}[1)]
\item If $|{\cal C}_0|=1$, then all the codewords in ${\cal C}_1$ are the same as that in ${\cal C}_0$. Since ${\cal C}_1$ is a minimal frame of ${\cal C}_0$, we have $|{\cal C}_1|=1$ and ${\cal C}_1={\cal C}_0$, a contradiction to the fact that $\widehat{{\cal C}}$ is a $(t,n-|\mathcal{T}|,q)$ code.
\item If $|{\cal C}_0|=2$, let ${\cal C}_0=\{\vec{c},\vec{c}_0\}$.
\begin{enumerate}[2.1)]
\item If $|{\cal C}_0\cap{\cal C}_1|=1$, without loss of generality, assume that ${\cal C}_0\cap{\cal C}_1=\{\vec{c}_0\}$ and $|{\cal C}_1|=m+1$. Then by Lemma \ref{lemma 4}, we have $m\le t$. If $m=0$, we have ${\cal C}_1=\{\vec{c}_0\}$ and ${\rm desc}(\{\vec{c},\vec{c}_0\})={\rm desc}(\{\vec{c}_0\})$, which implies that $\vec{c}=\vec{c}_0$, a contradiction. So, we have $1\le m\le t$. Then ${\cal C}_1$ satisfies case (1) implying that $\vec{c}$ is a bad codeword, a contradiction to the fact that $\widehat{{\cal C}}$ contains no bad codewords.
\item If $|{\cal C}_0\cap{\cal C}_1|=0$, without loss of generality, assume that ${\cal C}_1=\{\vec{c}_1,\vec{c}_2,\ldots,\vec{c}_m\}\subseteq\widehat{{\cal C}}\setminus{\cal C}_0$. Then by Lemma \ref{lemma 4}, we have $m\le t+1$. If $m=1$, we have ${\rm desc}(\{\vec{c},\vec{c}_0\})={\rm desc}(\{\vec{c}_1\})$, which implies that $\vec{c}=\vec{c}_0=\vec{c}_1$, a contradiction. So, we have $2\le m\le t+1$. Then ${\cal C}_1$ satisfies case (2) implying that $\vec{c}$ is a bad codeword, also a contradiction.
\end{enumerate} \end{enumerate}
Thus, $\widehat{{\cal C}}$ is a $\bar{2}$-SSC$(t,n-|\mathcal{T}|,q)$ where $|\mathcal{T}|$ is a random variable due to the random construction of ${\cal C}$. Next we estimate the expected value of $|\mathcal{T}|$.
For any $\vec{c}\in{\cal C}$, let $\mathcal{S}_1(\vec{c})=\{{\cal C}'\subseteq{\cal C}\setminus\{\vec{c}\}:$ $|{\cal C}'|=m+1$ and ${\cal C}'$ satisfies case (1)$\}$ and $\mathcal{S}_2(\vec{c})=\{{\cal C}'\subseteq{\cal C}\setminus\{\vec{c}\}:$ $|{\cal C}'|=m+1$ and ${\cal C}'$ satisfies case (2)$\}$. Then, we have \begin{align*}
{\rm Pr}(\vec{c}\ \rm is\ bad) &={\rm Pr}(|\mathcal{S}_1(\vec{c})|\ge 1\ {\rm or}\ |\mathcal{S}_2(\vec{c})|\ge 1)\\
&\le {\rm Pr}(|\mathcal{S}_1(\vec{c})|\ge 1)+{\rm Pr}(|\mathcal{S}_2(\vec{c})|\ge 1)\\
&\le {\rm E}(|\mathcal{S}_1(\vec{c})|)+{\rm E}(|\mathcal{S}_2(\vec{c})|) \end{align*} where the last inequality is by Markov's inequality, and \begin{equation}\label{expectation} \begin{aligned}
{\rm E}(|\mathcal{T}|) &=n\cdot {\rm Pr}(\vec{c}\ \rm is\ bad)\\
&\le n\cdot ({\rm E}(|\mathcal{S}_1(\vec{c})|)+{\rm E}(|\mathcal{S}_2(\vec{c})|)) \end{aligned} \end{equation} where \begin{align*}
{\rm E}(|\mathcal{S}_1(\vec{c})|) &=\sum_{m=1}^t\binom{n-1}{m+1}\binom{m+1}{1}{\rm Pr}({\rm desc}(\{\vec{c},\vec{c}_i\})={\rm desc}({\cal C}'))\\ &=\sum_{m=1}^t\binom{n-1}{m+1}\binom{m+1}{1}\left((1/q)^{m+1}+(1-1/q)((2/q)^m-(1/q)^m)\right)^t\\ &\le \sum_{m=1}^t (m+1)n^{m+1}\left((2^m-1)q-(2^m-2)\right)^tq^{-(m+1)t}\\ &\le t\cdot \max_{1\le m\le t}\left\{(m+1)n^{m+1}\left((2^m-1)q-(2^m-2)\right)^tq^{-(m+1)t}\right\} \end{align*} and \begin{align*}
{\rm E}(|\mathcal{S}_2(\vec{c})|) &=\sum_{m=2}^{t+1}\binom{n-1}{m+1}\binom{m+1}{1}{\rm Pr}\left({\rm desc}(\{\vec{c},\vec{c}_i\})={\rm desc}({\cal C}'\setminus\{\vec{c}_i\})\right)\\ &=\sum_{m=2}^{t+1}\binom{n-1}{m+1}\binom{m+1}{1}\left((1/q)^{m+1}+(1-1/q)\left((2/q)^m-2(1/q)^m\right)\right)^t\\ &\le \sum_{m=2}^{t+1} (m+1)n^{m+1}\left((2^m-2)q-(2^m-3)\right)^tq^{-(m+1)t}\\ &\le t\cdot \max_{2\le m\le t+1}\left\{(m+1)n^{m+1}\left((2^m-2)q-(2^m-3)\right)^tq^{-(m+1)t}\right\}. \end{align*}
For any $m\ge 1$ and $q\ge 2$, we have $(2^m-1)q-(2^m-2)>(2^m-2)q-(2^m-3)$. If \begin{equation}\label{inequality 1}
t\cdot \max\limits_{1\le m\le t+1}\left\{(m+1)n^{m+1}\left((2^m-1)q-(2^m-2)\right)^tq^{-(m+1)t}\right\}\le 1/3, \end{equation}
then ${{\rm E}}(|\mathcal{S}_1(\vec{c})|)\le1/3$ and ${{\rm E}}(|\mathcal{S}_2(\vec{c})|)\le 1/3$ for any $\vec{c}\in{\cal C}$. According to (\ref{expectation}), we have ${{\rm E}}(|\mathcal{T}|)\le 2n/3$, that is, the expected number of bad codewords in ${\cal C}$ is at most $2n/3$. By the random construction, there exists ${\cal C}$ such that it contains at most $2n/3$ bad codewords which implies that $\widehat{{\cal C}}$, obtained by deleting all the bad codewords from ${\cal C}$, is a $\bar{2}$-SSC with at least $n/3$ codewords. By Lemma \ref{lemma 2}, we can obtain a $2$-SSM with $tq$ rows and at least $n/3$ columns. Thus, \begin{equation}\label{rate-1}
R(2)\ge \underset{t\to\infty}{\overline{\lim}}\frac{\log_2(n/3)}{tq}=\underset{t\to\infty}{\overline{\lim}}\frac{\log_2 n}{tq}. \end{equation}
Now we would like to maximize the lower bound of $R(2)$ in (\ref{rate-1}) under the restriction of (\ref{inequality 1}). It is obvious that (\ref{inequality 1}) is equivalent to that for any $1\le m\le t+1$, \begin{equation}\label{inequality 2}
t(m+1)n^{m+1}\left((2^m-1)q-(2^m-2)\right)^tq^{-(m+1)t}\le 1/3. \end{equation} By taking $\log_2$ on (\ref{inequality 2}) and do some simplifications, we have that for any $1\le m\le t+1$, \begin{equation*}
\frac{\log_2n}{tq}\le\frac{\log_2q}{q}-\frac{\log_2((2^m-1)q-(2^m-2))}{(m+1)q}-\frac{\log_2(3t(m+1))}{tq(m+1)}. \end{equation*} Take $t,n,q$ such that \begin{equation*}
\frac{\log_2n}{tq}=\frac{\log_2q}{q}-\max_{1\le m\le t+1}\left\{\frac{\log_2((2^m-1)q-(2^m-2))}{(m+1)q}-\frac{\log_2(3t(m+1))}{tq(m+1)}\right\}-\frac{\epsilon}{t} \end{equation*} where $\epsilon=o(t)$ is a real number. Then (\ref{inequality 1}) will be established and by (\ref{rate-1}), we have \begin{align*}
R(2) & \ge\underset{t\to\infty}{\overline{\lim}}\frac{\log_2 n}{tq} \\
& =\frac{\log_2 q}{q}-\underset{t\to\infty}{\overline{\lim}}\max_{1\le m\le t+1}\left\{\frac{\log_2((2^m-1)q-(2^m-2))}{(m+1)q}-\frac{\log_2(3t(m+1))}{tq(m+1)}\right\} \\
& \ge\frac{\log_2 q}{q}-\underset{t\to\infty}{\overline{\lim}}\max_{1\le m\le t+1}\left\{\frac{\log_2((2^m-1)q-(2^m-2))}{(m+1)q}\right\} \end{align*} for any $q\ge 2$. To make this lower bound as large as possible, we take $q=4$ and then \begin{align*}
R(2) & \ge \frac{1}{2}-\underset{t\to\infty}{\overline{\lim}}\max\limits_{1\le m\le t+1}\frac{\log_2((2^m-1)4-(2^m-2))}{4(m+1)} \\
& =\frac{1}{2}-\frac{\log_2 22}{16} \\
& \doteq 0.2213 \end{align*} as desired. \end{IEEEproof}
\section{Conclusion} \label{sec:5}
In this paper, we introduced strongly separable matrices for nonadaptive CGT to identify a small set of positives from a large population. We showed that a $d$-SSM has weaker requirements than a $d$-DM, but provides an equally efficient identifying algorithm as a $d$-DM. A general bound on the largest rate of $d$-SSM was established. Besides, by the random coding method with expurgation, we derived an improved lower bound on the largest rate of $2$-SSM which is much higher than the best known result of $2$-DM. The results presented in this paper showed that a $2$-SSM could work better than a $2$-DM for nonadaptive CGT, which makes the research on SSM important. It is of interest to further improve the lower and upper bounds on the largest rate of $2$-SSM and explore the explicit constructions of optimal $2$-SSM. It is also interesting but more challenging to extend the argument for $2$-SSM to general $d$-SSM.
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\keywords{steady state Poisson--Nernst--Planck system, nonlinear Poisson equation, transmission condition, interfacial jump, oscillating coefficients, homogenisation, error corrector, Boltzmann statistics, electro-kinetic, photo-voltaic.}
\subjclass{35B27, 35J60, 78A57, 82B24.}
\begin{abstract} A nonlinear Poisson--Boltzmann equation with transmission boundary conditions at the interface between two materials is investigated. The model describes the electrostatic potential generated by a vector of ion concentrations in a periodic multiphase medium with dilute solid particles.
The key issue is that the interfacial transfer allows jumps and thus discontinuous solutions of the problem. Based on variational techniques, we derive the homogenisation of the discontinuous problem subject to inhomogeneous transmission interface conditions. Moreover, we establish a rigorous residual error estimate up to the first order correction. \end{abstract}
\maketitle
\section{Introduction}\label{sec1}
In this paper, we consider the steady state problem of a nonlinear Poisson--Nernst--Planck (PNP) system, which describes multiple concentrations of charged particles (e.g. ions) subject to a self-consistent electrostatic potential calculated from Poisson's equation. In particular, we shall investigate the PNP model on a multiphase medium. The prototypical multiphase medium in mind consists of an electrolyte medium, which surrounds disjoint solid particles. Such models have numerous applications describing electro-kinetic phenomena in bio-molecular or electro-chemical models, photo-voltaic systems and semiconductors, see e.g. \cite{AMP/10,ACDDJLMTV/04,BFMW/13,Hum/00,LC/06,RMK/12,SB} and references therein. Our specific interests are motivated by models of Li-Ion batteries, see e.g. \cite{Pic/11}.
In order to be able to deal with the nonlinearity of the model, we shall work within an analytic framework, where the PNP system can be equivalently transformed into a scalar semi-linear Poisson--Boltzmann (PB) equation. This is possible, when reaction terms in the charged particle fluxes are omitted and the equations for the concentrations decouple
since the charged particle concentrations are explicitly determined by the corresponding Boltzmann statistics. For references applying linearisation of the PNP equations near the Boltzmann distribution see e.g. \cite{AMP/10,LC/06}.
The major difficulty addressed in this manuscript is the imposed inhomogeneous intermedia transmission boundary condition for the electrostatic field, which complements the PB equation (see \eqref{2.8} below). Thus, the key feature of the presented model is the electric charge transport phenomena over the interfaces at the boundaries of the solid particles. The interfacial transfer shall be described by the Gouy--Chapman-Stern model for electric double layers (EDLs) \cite{Pic/11}. This model proposes a jump of the electrostatic field across the interface (a voltage drop) as well as a current prescribed at the interior boundary of the solid particles.
In the following, we will derive a discontinuous formulation of the PB equation (valid both on the volume occupied by the solid particles and on the surrounding porous space) with inhomogeneous transmission conditions at the interfaces between particles and porous space.
A first aim of this paper is to establish a proper variational setting of the transmission problem, while a second part deals with its rigorous homogenisation. In respect to the later, we emphasise that the averaged effective coefficients of the limit problem represent the macroscopic behaviour of the EDL, which is of primary practical importance.
For reference concerning the classic homogenisation theories, we refer to \cite{Arg/04,BP/89,BLP/11,OSY/92,SP/80,ZKO/94}. The applied methods range from two-scale convergence (see e.g. \cite{All/92}) over Gamma-convergence (see e.g. \cite{DM/06}) to unfolding (see \cite{CDDGZ/12}) and others. While formal methods of averaging are widely used in the literature, their verification in terms of residual error estimates is a hard task.
From the point of view of homogenisation, the principal difficulty of interfacial transmission problems concerns the non-standard boundary conditions with jumps: On the one hand, related jump conditions are inherent for cracks. For models and methods used in crack problems, we refer to \cite{FHLRS/09,HKK/09,KK/00,SP/80} and references therein. From a geometric viewpoint, cracks are open manifolds in the reference domain. Hence, classic Poincare--Friedrichs--Korn inequalities are valid in such situations. In contrast to cracks, the interfaces here are assumed to be closed manifolds disconnecting the reference domain. This difference requires discontinuous versions of Poincare--Friedrichs--Korn inequalities, which are then applied for semi-norm estimates.
On the other hand, the transmission boundary conditions are of Robin type. The homogenisation results known for linear problems with Robin (also called Fourier) conditions are crucially sensitive to the asymptotic rates of the involved homogenisation parameters. This issue concerns the coefficients in the boundary condition (cf. Lemma \ref{lem3.1} below) and the volume fraction of solid particles in periodic cells (cf. Lemma \ref{lem3.2} below), see e.g. \cite{ADH/96,BPC/01,OS/96}.
The literature on homogenisation of transmission problems is very scarce, see e.g. \cite{Hum/00,Orl/12}. The technical challenge of this manuscript is the combination of nonlinearity, discontinuity and Robin type transmission conditions.
In the present work, we homogenise the discontinuous nonlinear PB equation with inhomogeneous interfacial transfer conditions and derive the averaged limit problem. A further major result is the rigorous derivation of the residual error up to the first order correction.
For these purposes, we develop a variational technique based on orthogonal Helmholtz decomposition following the lines of \cite{OSY/92,ZKO/94}. In a periodic cell, we decompose oscillating coefficients (describing the electric permittivity) by using the nontrivial kernel in the space of vector valued periodic functions, which is represented by sums of constant and divergence free (and thus, skew symmetric) vector fields (cf. Lemma \ref{lem3.3}). Employing solutions of appropriately defined discontinuous cell problems, we obtain a regular decomposition of the homogenisation problem (see Theorem \ref{theo3.1}).
A second result establishes the critical rates of the asymptotic behaviour with respect to a homogenisation parameter $\varepsilon\searrow0^+$ for coefficients in the inhomogeneous transmission condition: We find on the one side that the critical rate for the coefficient by interfacial jumps is $\frac{1}{\varepsilon}$. This factor occurs in the discontinuous Poincare inequality (for the norm squared, cf. \eqref{2.19} below) and is thus relevant for a coercivity estimate, which in return contributes to the solvability of the discontinuous problem and the subsequent estimate of the homogenisation error.
On the other side, the critical rate for the flux prescribed at the interior boundary of solid particles is $\varepsilon$. At this rate, the interior boundary flux induces an additional potential, which distributes over the macroscopic domain in the homogenisation limit $\varepsilon\searrow0^+$. If the asymptotic rate is lower than the critical one, then this flux vanishes in the limit. Otherwise, if the asymptotic rate is bigger, then the flux term diverges.
From the above description we summarise the key points of this paper as follows: \begin{itemize} \item the study of inhomogeneous interfacial transfer conditions describing EDL; \item the combination of nonlinear terms, jumps and Robin conditions; \item a variational framework of the transmission problem; \item the performing of the homogenisation procedure with rigorous error estimates; \item the identification of the critical asymptotic rates of the boundary coefficients. \end{itemize}
\noindent\underline{Outline}: In the Sections \ref{sec2.1}, \ref{sec2.2} and \ref{sec2.3}, we first present the problem geometry, the physical and the mathematical model. Section~\ref{sec2.3} establishes moreover the equivalence of the steady-state of the PNP model with the semi-linear Poisson-Boltzmann equation and the existence of a unique solution to the PB equation (see Theorem \ref{theo2.1}).
In Section \ref{sec3}, we consider the homogenisation problem and the residual error estimate. At first, we state three auxiliary Lemmata before stating the main homogenisation Theorem \ref{theo3.1}.
Finally, Section \ref{sec4} provides a brief discussion of the obtained results.
\section{Statment of the Problem}\label{sec2}
We start with the description of the geometry.
\subsection{Geometry}\label{sec2.1}
\\ Let $\omega$ denote the domain occupied by solid particles of general shape (either single or multiple particles), which are located inside the unit cell $\Upsilon=(0,1)^d\subset\mathbb{R}^d$, $d=1,2,3$. We assume that all particles $\omega\subset\Upsilon$ are disjunctively located as well as bounded away from the boundary $\partial\Upsilon$, i.e. $\omega\cap\partial\Upsilon=\emptyset$.
We assume that the boundary $\partial\omega$ is Lipschitz continuous with outer normal vector $\nu =(\nu_1,\dots,\nu_d)^\top$ pointing away from the domain $\omega$. Moreover, we distinguish the positive (outward orientated) surface $\partial\omega^+$ and the negative (inward orientated) surface $\partial\omega^-$ as the faces of the boundary $\partial\omega$, when approaching the boundary $\partial\omega$ from outside, i.e. from $\Upsilon\setminus\omega$ or from the inside, i.e. from $\omega$, respectively. For a two-dimensional example configuration see the illustration in Fig.~\ref{fig1} (a).
\begin{figure}
\caption{Two-dimensional example geometry with one star-shaped particle: (a) the unit cell, (b) the paving and (c) the periodic disjoint domains $\Omega\setminus\partial\omega_\#$.}
\label{fig1}
\end{figure}
In the following, we consider a fixed, small homogenisation parameter $\varepsilon\in\mathbb{R}_+$ and pave $\mathbb{R}^d$ with periodic cells $\Upsilon_p^\varepsilon$ indexed by $p\in\mathbb{N}$. The periodic cells $\Upsilon_p^\varepsilon$ are constructed from $\Upsilon$ in the following way: The position of every spatial point $x=(x_1,\dots,x_d)^\top\in\mathbb{R}^d$ can be decomposed as $$ x=\varepsilon\left\lfloor\frac{x}{\varepsilon}\right\rfloor +\varepsilon\left\{\frac{x}{\varepsilon}\right\}, \qquad\quad \left\lfloor\frac{x}{\varepsilon}\right\rfloor\in\mathbb{Z}^d, \quad \left\{\frac{x}{\varepsilon}\right\}\in\Upsilon, $$ into the integer-valued floor function coordinates $\lfloor\frac{x}{\varepsilon}\rfloor\in\mathbb{Z}^d$ and the fractional coordinates $\{\frac{x}{\varepsilon}\}\in\Upsilon$. We shall then enumerate all possible integer vectors $\lfloor\frac{x}{\varepsilon}\rfloor$ by means of a natural ordering with the index $p\in\mathbb{N}$. According to this index, we associate $\varepsilon\lfloor\frac{x}{\varepsilon}\rfloor$ with the $p$-th cell $\Upsilon_p^\varepsilon$ and $\varepsilon\{\frac{x}{\varepsilon}\} =\varepsilon y$ shall denote the local coordinates in all cells which correspond to $y\in\Upsilon$.
We will denote by $\omega_p^\varepsilon\subset\Upsilon_p^\varepsilon$ the respective solid particles obtained by means of the paving with $\{\frac{x}{\varepsilon}\} =y$ for $y\in\omega$. We note that the rescaling does not change the unit outer normal vector $\nu$.
Evidently, the periodic mapping $x\mapsto y$, $\mathbb{R}^d\mapsto\Upsilon$, is surjective. This construction can be generalised to an arbitrary orthotope $\Upsilon$, see \cite{CDDGZ/12}.
Let $\Omega$ be the reference domain in $\mathbb{R}^d$ with Lipschitz boundary $\partial\Omega$ and denote again the outer normal vector by $\nu$. By reordering the index $p$, it is then possible to account for all solid particles $\omega_p^\varepsilon\subset\Omega$ with the index set $p=1,\dots,N_\varepsilon$, see \cite{CDDGZ/12,Fra/10}. We remark that $N_\varepsilon\sim\varepsilon^{-d}$.
By omitting solid particles which are "too close" to the external boundary $\partial\Omega$, we shall ensure a constant gap with the distance $O(\varepsilon)$ between $\partial\Omega$ and all particles $\omega_p^\varepsilon$. Thus, $\Omega$ is divided into the multiple domains $\omega_\# :=\cup_{p=1}^{N_\varepsilon} \omega_p^\varepsilon$ corresponding to all the solid particles located periodically in the reference domain and the remaining porous space $\Omega\setminus\omega_\#$.
In the following, we shall denote by $\partial\omega_\# =\cup_{p=1}^{N_\varepsilon} \partial\omega_p^\varepsilon$ the union of boundaries $\partial\omega_p^\varepsilon$ and introduce the disjoint multiple domains $$ \Omega\setminus\partial\omega_\# =(\Omega\setminus\omega_\#) \cup\omega_\#, \qquad\quad \partial\omega_\# =\cup_{p=1}^{N_\varepsilon} \partial\omega_p^\varepsilon, \qquad \omega_\# :=\cup_{p=1}^{N_\varepsilon} \omega_p^\varepsilon. $$
Moreover, for functions $\xi$, which are discontinuous over the interface $\partial\omega_\#$, we will denote the jump across the interface by $$
[\![\xi]\!] :=\xi^+-\xi^-, \qquad \xi^\pm :=\xi|_{\partial\omega_\#^\pm}. $$ Here, $\partial\omega_\#^+ =\cup_{p=1}^{N_\varepsilon} (\partial\omega_p^\varepsilon)^+$ summarises the positive faces (orientated towards the interior of the pore space $\Omega\setminus\omega_\#$), and $\partial\omega_\#^- =\cup_{p=1}^{N_\varepsilon} (\partial\omega_p^\varepsilon)^-$ accounts for the negative faces (orientated towards the interior of the solid phase $\omega_\#$).
\subsection{Physical model}\label{sec2.2}
\\ In the heterogeneous domain $\Omega\setminus\partial\omega_\#$, which consist of the particle volumes $\omega_\#$ and the porous space $\Omega\setminus\omega_\#$, we consider the electrostatic potential $\phi$ and $(n+1)$ components of concentrations of charged particles ${c}=({c}_0,\dots,{c}_n)^\top$, $n\ge1$. The physical consistency requires positive concentrations ${c}>0$.
At the external boundary $\partial\Omega$, we shall impose Dirichlet boundary conditions $\phi=\phi^{\rm bath}$ and ${c}={c}^{\rm bath}$ corresponding to a surrounding bath and given by constant values $\phi^{\rm bath}\in\mathbb{R}$ and ${c}^{\rm bath}=({c}^{\rm bath}_0,\dots,{c}^{\rm bath}_n)^\top \in\mathbb{R}^{n+1}_+$. We can then consider the normalised electrostatic potential $\phi-\phi^{\rm bath}$ and concentrations ${c}/{c}^{\rm bath}$ (i.e. ${c_s}/{c_s}^{\rm bath}$ for all $s=0,\dots,n$) and prescribe the
following normalised Dirichlet conditions: \begin{equation}\label{2.1} \phi=0,\qquad {c}=1\qquad\text{on $\partial\Omega$}. \end{equation}
In the following, all further relations will be formulated for the normalised potential and concentrations such that \eqref{2.1} holds.
Let $z_s\in\mathbb{R}$ denote the electric charge of the $s$-th species with concentration ${c}_s$ for $s=0,\dots,n$. For the $n+1$- components of charges particles, we shall assume the following charge-neutrality \begin{equation}\label{2.2} \sum_{s=0}^n z_s =0. \end{equation} A necessary condition for \eqref{2.2} is ${\displaystyle\min_{s\in\{0,\dots,n\}}} z_s <0 <{\displaystyle\max_{s\in\{0,\dots,n\}}} z_s $.
The charge-neutrality assumption \eqref{2.2} implies also the following strong monotonicity property \begin{equation}\label{2.3}
K |\xi|^2\le -\sum_{s=0}^n z_s\xi \exp({-z_s\xi}) \qquad\text{for all $\xi\in\mathbb{R}$}\qquad (K>0), \end{equation} for a constant $K>0$, which follows directly from Taylor expansion with respect to $(-z_s\xi)$.
We consider the following PNP steady-state system consisting of $(n+2)$ nonlinear, homogeneous equations: \begin{subequations}\label{2.4} \begin{align}\label{2.4a} -{\rm div} (\nabla {c}_s^\top D_s)&=0, \qquad s=0,\dots,n,&&\quad\text{in }\omega_\#,\\ \label{2.4b} -{\rm div} \bigl( (\nabla {c}_s +{\textstyle\frac{z_s}{{\kappa}T}} {c}_s \nabla\phi)^\top D_s\bigr) &=0, \qquad s=0,\dots,n,&&\quad\text{in }\Omega\setminus\omega_\#, \end{align} \end{subequations} \begin{subequations}\label{2.5} \begin{align}\label{2.5a} -{\rm div} (\nabla\phi^\top A^\varepsilon) &=0, &&\text{in } \omega_\#,\\ \label{2.5b} -{\rm div} (\nabla\phi^\top A^\varepsilon) -\sum_{s=0}^n z_s {c}_s& =0, && \text{in } \Omega\setminus\omega_\#. \end{align} \end{subequations} In both equations \eqref{2.4}, $D_s\in L^\infty(\Omega)^{d\times d}$, $D_s>0$, $s=0,\dots,n$ denote symmetric and positive definite diffusion matrices, which are in general discontinuous over $\partial\omega_\#$. In \eqref{2.4b}, $\kappa>0$ is the Boltzmann constant, and $T>0$ is the temperature. We remark that the form of \eqref{2.4b} is based on assuming the Einstein relations for the mobilities. Moreover, eq. \eqref{2.4a} models the effect of charges particles being included into the solid particles, which is well known, for instance, for $Li^+$-ions, see e.g. \cite{Pic/11}.
In \eqref{2.5}, $A\in L^\infty(\Upsilon)^{d\times d}$ denotes the symmetric and positive definite matrix of the electric permittivity, which oscillates periodically over cells according to $A^\varepsilon(x) :=A(\{\frac{x}{\varepsilon}\})$ and satisfies \begin{equation}\label{2.6} \begin{split} &A^\top(y)=A(y),\qquad y\in\Upsilon\\
&\underline{K} |\xi|^2\le \xi^\top A(y)\xi \le\overline{K} |\xi|^2 \qquad\forall\xi\in\mathbb{R}^d, y\in\Upsilon, \qquad (0<\underline{K}<\overline{K}). \end{split} \end{equation} The entries of the permittivity matrix $A$ are discontinuous functions in the cell $\Upsilon$ across the interface $\partial\omega$. A typical example considers piecewise constant $A=\sigma_\omega I$ in $\omega$ and $A=\sigma_\Upsilon I$ in $\Upsilon\setminus\omega$, with material parameters $\sigma_\omega>0$ and $\sigma_\Upsilon>0$, where $I$ denotes here the identity matrix in $\mathbb{R}^{d\times d}$. In the following, we denote by $A_{ij}$, $i,j=1,\dots,d$, the matrix entries of $A$.
From a physical point of view, \eqref{2.5a} represents Ohm's law in the solid phase. Moreover, we remark that the equations on $\omega_\#$, i.e. \eqref{2.4a} for ${c}$ and \eqref{2.5a} for $\phi$ are linear while the equations \eqref{2.4b} and \eqref{2.5b} on $\Omega\setminus\omega_\#$ form a coupled nonlinear problem on the porous space.
The modelling of boundary conditions at the interfaces is a delicate issue. For the charge carries fluxes in \eqref{2.4}, we assume homogeneous Neumann conditions \begin{subequations}\label{2.7} \begin{equation}\label{2.7a} (\nabla {c}_s^-)^\top D_s\nu =0, \qquad s=0,\dots,n,\quad\text{on}\quad \partial\omega_\#^-, \end{equation} \begin{equation}\label{2.7b} (\nabla {c}_s^+ +{\textstyle\frac{z_s}{{\kappa}T}} {c}_s^+ \nabla\phi^+)^\top D_s\nu =0, \qquad s=0,\dots,n,\quad\text{on}\quad \partial\omega_\#^+. \end{equation} \end{subequations} For the electrostatic potential in \eqref{2.5}, we suppose the Gouy--Chapman--Stern model for an Electric Double Layer (EDL) by assuming the following inhomogeneous transmission boundary conditions (see \cite{Pic/11}): \begin{subequations}\label{2.8} \begin{align}\label{2.8a} (\nabla\phi^\top A^\varepsilon)^- \nu -{\textstyle\frac{\alpha}{\varepsilon}} [\![\phi]\!] &=\varepsilon g,&&\text{on $\partial\omega_\#^-$},\\ \label{2.8b} -(\nabla\phi^\top A^\varepsilon)^+ \nu +{\textstyle\frac{\alpha}{\varepsilon}} [\![\phi]\!] &=0, &&\text{on $\partial\omega_\#^+$}. \end{align} \end{subequations} Here $\alpha\in\mathbb{R}_+$ and $g\in\mathbb{R}$ are material parameters given at the interface. We note that by summing \eqref{2.8a} and \eqref{2.8b}, we derive the relation \begin{equation}\label{2.9} -[\![\nabla\phi^\top A^\varepsilon]\!] \nu =\varepsilon g, \qquad\qquad\text{on $\partial\omega_\#$}, \end{equation} implying that not only the electric potential $\phi$ but also fluxes $\nabla\phi^\top A^\varepsilon\nu$ are discontinuous functions with jumps across the interface $\partial\omega_\#$.
The asymptotic weights ${\textstyle\frac{1}{\varepsilon}}$ in front of $[\![\phi]\!]$ and $\varepsilon g$ at the right hand side of \eqref{2.8}, which were already mentioned in the introduction, shall be discussed in detail during the below asymptotic analysis as $\varepsilon\searrow0^+$.
We emphasise that the transmission conditions \eqref{2.8} couple the porous phase $\Omega\setminus\omega_\#$ with the solid phase $\omega_\#$ by means of the jump in $[\![\phi]\!]$. In fact, the transmission conditions \eqref{2.8} can be compared with the following two cases of simplified boundary conditions: First, if $\phi$ were continuous across $\partial\omega_\#$, i.e. $[\![\phi]\!]=0$, then \eqref{2.8a} and \eqref{2.8b} would be decoupled into two usual Neumann boundary condition which do not represent the EDL. Second, if $\phi^-$ were known on the solid phase boundary $\partial\omega_\#^-$, then the model would reduced to a model on the porous space $\Omega\setminus\omega_\#$ with the following inhomogeneous Robin (Fourier) boundary condition (see \cite{FK}) \begin{equation*} -(\nabla\phi^\top A^\varepsilon)^+ \nu +{\textstyle\frac{\alpha}{\varepsilon}} \phi^+ ={\textstyle\frac{\alpha}{\varepsilon}} \phi^-, \qquad\text{on $\partial\omega_\#^+$}. \end{equation*} However, the subsequent homogenisation of this alternative model on the porous space $\Omega\setminus\omega_\#$ would nevertheless require a suitable continuation of $\phi^+$ onto $\omega_\#$.
\subsection{Mathematical model}\label{sec2.3}
\\ In the following, we shall amend the state variables with the superscript $\varepsilon$ in order to highlight the dependency on the cell size.
The physical model will be described by the following weak variational formulation of the boundary value problem \eqref{2.1}, \eqref{2.4}--\eqref{2.5}, \eqref{2.7}--\eqref{2.8}: Find an electrostatic potential $\phi^\varepsilon\in H^1(\Omega\setminus\partial\omega_\#)$ and $n+1$ components of charge carrier concentrations ${c}^\varepsilon\in H^1(\Omega\setminus\partial\omega_\#)^{n+1}\cap L^\infty(\Omega\setminus\partial\omega_\#)^{n+1}$ such that the concentrations are positive ${c}^\varepsilon>0$ and satisfy \begin{equation}\label{2.10} \phi^\varepsilon =0,\qquad {c}^\varepsilon =1\quad\text{on $\partial\Omega$}, \end{equation} \begin{multline}\label{2.11} \int_{\Omega\setminus\partial\omega_\#} \bigl(\nabla {c}^\varepsilon_s +\chi_{{}_{\Omega\setminus\omega_\#}} {\textstyle\frac{z_s}{{\kappa}T}}\, {c}^\varepsilon_s\, \nabla\phi^\varepsilon \bigr)^\top D_s \nabla {c}_s \,dx =0,\qquad s=0,\dots,n,\\ \text{for all test-functions ${c}\in H^1(\Omega\setminus\partial\omega_\#)^{n+1}$: ${c}=0$ on $\partial\Omega$}, \end{multline} \begin{multline}\label{2.12} \int_{\Omega\setminus\partial\omega_\#} \!\!\! \bigl( (\nabla\phi^\varepsilon)^\top A^\varepsilon\nabla\phi -\chi_{{}_{\Omega\setminus\omega_\#}} \sum_{s=0}^n z_s {c}^\varepsilon_s \phi \bigr)\,dx +\int_{\partial\omega_\#} \!\!\! {\textstyle\frac{\alpha}{\varepsilon}} [\![\phi^\varepsilon]\!] [\![\phi]\!] \,dS_x\\ =\int_{\partial\omega_\#^-} \varepsilon g \phi^- \,dS_x\qquad\ \text{for all $\phi\in H^1(\Omega\setminus\partial\omega_\#)$: $\phi=0$ on $\partial\Omega$}. \end{multline} Here $\chi_{{}_{\Omega\setminus\omega_\#}}$ denotes the characteristic function of the set $\Omega\setminus\omega_\#$.
\begin{proposition}\label{prop2.1} For strong solutions $(\phi^\varepsilon,{c}^\varepsilon)$, the variational system \eqref{2.10}--\eqref{2.12} and the boundary value problem \eqref{2.1}, \eqref{2.4}--\eqref{2.5}, \eqref{2.7}--\eqref{2.8} are equivalent. \end{proposition}
\begin{proof} The assertion can be verified by usual variational arguments, which we briefly sketch for the sake of the reader.
The variational equations \eqref{2.11} and \eqref{2.12} are derived by multiplying the equations \eqref{2.4}--\eqref{2.5} with test-functions and subsequent integration by parts over $\Omega\setminus\omega_\#$ and $\omega_\#$ due to boundary conditions \eqref{2.1} and \eqref{2.7}--\eqref{2.8}.
In return, given strong solutions $(\phi^\varepsilon,{c}^\varepsilon)$, the boundary value problem \eqref{2.4}--\eqref{2.5}, \eqref{2.7}--\eqref{2.8} is obtained by varying the test-functions $(\phi,{c})$ in \eqref{2.11}, \eqref{2.12} and with the help of the following Green's formulas: By recalling the $\nu$ denotes both the outer normal on $\partial\Omega$ and $\partial\omega$, we have for all $p\in L^2_{\rm div}(\Omega\setminus\partial\omega_\#)^d$ \begin{subequations}\label{13} \begin{multline}\label{13a} \int_{\Omega\setminus\omega_\#} \!\!\! p^\top \nabla v\,dx =-\int_{\Omega\setminus\omega_\#} \!\!\! v\,{\rm div}(p)\,dx -\int_{\partial\omega_\#^+} p^\top v\nu \,dS_x\\ +\int_{\partial\Omega} p^\top v\nu \,dS_x,\qquad \forall\ v\in H^1(\Omega\setminus\omega_\#), \end{multline} \begin{equation}\label{13b} \int_{\omega_\#} \!\!\! p^\top \nabla v\,dx =-\int_{\omega_\#} \!\!\! v\,{\rm div}(p)\,dx +\int_{\partial\omega_\#^-} p^\top v\nu \,dS_x,\qquad \forall v\in H^1(\omega_\#), \end{equation} \end{subequations} which are valid on $\Omega\setminus\omega_\#$ and $\omega_\#$, respectively. Hence, by suming \eqref{13a} and \eqref{13b}, we obtain the Green's formula representation \begin{equation}\label{green} \int_{\Omega\setminus\partial\omega_\#} \!\!\! p^\top \nabla v\,dx =-\int_{\Omega\setminus\partial\omega_\#} \!\!\! v\,{\rm div}(p)\,dx -\int_{\partial\omega_\#} [\![p^\top v]\!]\nu \,dS_x +\int_{\partial\Omega} p^\top v\nu \,dS_x, \end{equation} which holds on the disjoint domain $\Omega\setminus\partial\omega_\#$ for all $p\in L^2_{\rm div}(\Omega\setminus\partial\omega_\#)^d$ and $v\in H^1(\Omega\setminus\partial\omega_\#)$, see e.g. \cite{KK/00}. \end{proof}
The following Proposition \ref{prop2.2} states the crucial observation that introducing Boltzmann statistics allows to decouple the system of the homogeneous equations \eqref{2.11} and derive an equivalent scalar semi-linear Poisson-Boltzmann (PB) equation.
\begin{proposition}\label{prop2.2} The system \eqref{2.10}--\eqref{2.12} it is equivalent to the following nonlinear Poisson-Boltzmann equation: Find $\phi^\varepsilon\in H^1(\Omega\setminus\partial\omega_\#)$ such that \begin{subequations}\label{2.13} \begin{equation}\label{2.13a} \phi^\varepsilon =0\quad\text{on $\partial\Omega$},\\ \end{equation} \begin{multline}\label{2.13b} \int_{\Omega\setminus\partial\omega_\#} \bigl( (\nabla\phi^\varepsilon)^\top A^\varepsilon\nabla\phi -\sum_{s=0}^n z_s e^{- {\textstyle\frac{z_s}{{\kappa}T}}\, \chi_{{}_{\Omega\setminus\omega_\#}}\phi^\varepsilon} \phi \bigr)\,dx\\ +\int_{\partial\omega_\#} \!\!\! {\textstyle\frac{\alpha}{\varepsilon}} [\![\phi^\varepsilon]\!] [\![\phi]\!] \,dS_x =\int_{\partial\omega_\#^-} \varepsilon g \phi^- \,dS_x\\ \text{for all test-functions $\phi\in H^1(\Omega\setminus\partial\omega_\#)$: $\phi=0$ on $\partial\Omega$}, \end{multline} \end{subequations} together with the Boltzmann statistics determining ${c}^\varepsilon$ from $\phi^\varepsilon$, i.e. \begin{equation}\label{2.14} \begin{split} {c}^\varepsilon_s &=\exp\bigl( -{\textstyle\frac{z_s}{{\kappa}T}} \phi^\varepsilon \bigr), \qquad s=0,\dots,n, \quad\text{a.e. on } \Omega\setminus\omega_\#,\\ {c}^\varepsilon_s& \in\mathbb{R}_+,\qquad\qquad\qquad\, s=0,\dots,n,\quad \text{in } \omega_\#. \end{split} \end{equation} \end{proposition}
\begin{proof} Starting with \eqref{2.10}--\eqref{2.12}, we shall first prove the Boltzmann statistics \eqref{2.14} by introducing the entropy variables (the chemical potentials) \begin{equation}\label{2.15} \begin{split} &\mu^\varepsilon_s := \ln {c}^\varepsilon_s,\qquad s=0,\dots,n. \end{split} \end{equation} Then, eq. \eqref{2.11} can be rewritten in terms of \eqref{2.15} in divergence form as \begin{multline}\label{2.16} \int_{\Omega\setminus\partial\omega_\#} {c}^\varepsilon_s \nabla \bigl( \mu^\varepsilon_s +\chi_{{}_{\Omega\setminus\omega_\#}} {\textstyle\frac{z_s}{{\kappa}T}} \phi^\varepsilon \bigr)^\top D_s \nabla {c}_s \,dx =0,\qquad s=0,\dots,n,\\ \text{for all test-functions ${c}\in H^1(\Omega\setminus\partial\omega_\#)^{n+1}$: ${c}=0$ on $\partial\Omega$}. \end{multline} Due to the boundary condition \eqref{2.10}, we have $\phi^\varepsilon =0=\mu^\varepsilon$ on $\partial\Omega$ and the test-function ${c}_s =\mu^\varepsilon_s +\chi_{{}_{\Omega\setminus\omega_\#}} {\textstyle\frac{z_s}{{\kappa}T}} \phi^\varepsilon$ can be inserted into \eqref{2.16}. Hence, by recalling that $D_s$ are symmetric and positive definite matrices and ${c}^\varepsilon>0$, we derive the identity $ \nabla\bigl(\mu^\varepsilon_s +\chi_{{}_{\Omega\setminus\omega_\#}} {\textstyle\frac{z_s}{{\kappa}T}} \phi^\varepsilon\bigr) =0, $ $s=0,\dots,n,$ a.e. in $\Omega\setminus\partial\omega_\#$. Using again the boundary condition \eqref{2.10}, we conclude \begin{equation}\label{2.17} \mu^\varepsilon_s +\chi_{{}_{\Omega\setminus\omega_\#}} {\textstyle\frac{z_s}{{\kappa}T}} \phi^\varepsilon =0,\qquad s=0,\dots,n,\quad \text{a.e. in } \Omega\setminus\omega_\#, \end{equation}
and $\mu^\varepsilon_s$ is an arbitrary constant in $\omega_\#$. This fact together with \eqref{2.15} implies \eqref{2.14}. By substituting the expressions \eqref{2.14} into equation \eqref{2.12} and by using the charge-neutrality \eqref{2.2} on $\omega_\#$, equation \eqref{2.13b} follows directly.
Conversely, the equations \eqref{2.10}--\eqref{2.12} follow evidently from \eqref{2.13} and \eqref{2.14}. This completes the proof. \end{proof}
We remark that the concentrations ${c}^\varepsilon$ in \eqref{2.14} are unique up to fixing the constant positive values within the solid particles $\omega_\#$.
By exploiting Proposition~\ref{prop2.2}, we construct a solution $(\phi^\varepsilon,{c}^\varepsilon)$ for the variational problem \eqref{2.10}--\eqref{2.12} from the scalar problem \eqref{2.13} for the potential $\phi^\varepsilon$. The $n+1$ concentrations ${c}^\varepsilon$ are afterwards explicitly determined by \eqref{2.14}.
\begin{theorem}\label{theo2.1} There exists the unique solution $\phi^\varepsilon$ to the semilinear problem \eqref{2.13} satisfying the following residual estimate \begin{equation}\label{2.18}
\|\nabla \phi^\varepsilon\|_{L^2(\Omega\setminus\partial\omega_\#)}^2 +{\textstyle\frac{1}{\varepsilon}
\|[\![\phi^\varepsilon]\!]\|_{L^2(\partial\omega_\#)}^2
+\|\phi^\varepsilon\|_{L^2(\Omega\setminus\omega_\#)}^2} ={\rm O} (1), \end{equation} which is uniform with respect to $\varepsilon>0$. \end{theorem}
\begin{proof} We first emphasise that for the first two terms on the left hand side of \eqref{2.18} the following discontinuous version of Poincare's inequality for homogeneous Dirichlet condition \eqref{2.13a} holds on the multiple domains $\Omega\setminus\partial\omega_\#$ without interfaces $\partial\omega_\#$ (see e.g. \cite{Hum/00,Orl/12}): \begin{equation}\label{2.19}
K_0 \|\phi^\varepsilon\|_{H^1(\Omega\setminus\partial\omega_\#)}^2
\le\|\nabla \phi^\varepsilon\|_{L^2(\Omega\setminus\partial\omega_\#)}^2 +{\textstyle\frac{1}{\varepsilon}}
\|[\![\phi^\varepsilon]\!]\|_{L^2(\partial\omega_\#)}^2, \quad (K_0>0). \end{equation} Therefore, the lower estimate \eqref{2.19} together with \eqref{2.3} ensures the coercivity of the operator of the problem \eqref{2.13b}.
The main difficulty of the existence proof arises from the unbounded, exponential growth of the nonlinear term in \eqref{2.13b}. While classic existence theorems on quasilinear equations are thus not applicable here, the solution can nevertheless be constructed by a thresholding, see e.g. \cite{LC/06} and references therein for the details.
To derive the estimate \eqref{2.18}, it suffices to insert $\phi=\phi^\varepsilon$ as the test-function in the variational equation \eqref{2.13b} and apply \eqref{2.3} in order to estimate below the nonlinear term at the left hand side of \eqref{2.13b}. Finally the right hand side of \eqref{2.13b} can be estimated by means of the following trace theorem \begin{equation}\label{2.20} \begin{split} &\int_{\partial\omega_\#^-} \varepsilon g \phi^- \,dS_x
\le |g| \|\phi\|_{H^1(\Omega\setminus\partial\omega_\#)}, \end{split} \end{equation} see \cite{BPC/01} for the details. This completes the proof. \end{proof}
We remark that in the following Section \ref{sec3}, we will refine the residual error estimate \eqref{2.18} by means of asymptotic analysis as $\varepsilon\searrow0^+$ and homogenisation.
\section{Homogenisation and residual error estimate}\label{sec3}
We start the homogenisation procedure with three auxiliary cell problems. The first two cell problems serve to expand the inhomogeneous boundary traction $g$ and the volume potential of the variational problem \eqref{2.13} from the porous space $\Omega\setminus\omega_\#$ onto the whole domain $\Omega\setminus\partial\omega_\#$.
The third cell problem is needed to decompose the matrix $A^\varepsilon$ of oscillating coefficients in the cells with respect to small $\varepsilon\searrow0^+$. This procedure will result in a regular asymptotic decomposition of the perturbation problem with a subsequent error estimate of the corrector term.
For a generic cell $\Upsilon$, we introduce the Sobolev space $H^1_\#(\Upsilon)$ of functions which can be extended periodically to $H^1(\mathbb{R}^d)$. This requires matching traces on the opposite faces of $\partial\Upsilon$. Moreover, we shall denote by $H^1_\#(\Upsilon\setminus\partial\omega)$ those periodic functions, which are discontinuous, i.e. allow jumps across the interface $\partial\omega$.
\subsection{Auxiliary results}\label{sec3.1}
\\ We state the first auxiliary cell problem as follows: Find $L\in H^1(\Upsilon\setminus\partial\omega)$ such that \begin{multline}\label{3.1} \int_{\Upsilon\setminus\partial\omega} (\nabla L^\top A\nabla u +L u) \,dy =\int_{\partial\omega^-} u^- \,dS_y\\ \text{for all test-functions } u\in H^1(\Upsilon\setminus\partial\omega). \qquad \end{multline} In view of the homogenisation result stated in Theorem \ref{theo3.1} in Section~\ref{sec3.2} below, the auxiliary problem \eqref{3.1} serves to expand the inhomogeneity of the boundary condition \eqref{2.8a} given by the material parameter $g$ in terms of the weak formulation stated in \eqref{2.13b}.
The existence of a unique solution $L$ in \eqref{3.1} follows via standard elliptic theory from the assumed properties \eqref{2.6} of $A$. With its help, we are able to prove the following result.
\begin{lemma}[The cell boundary-traction problem]\label{lem3.1}
\\ For all test-fucntions $\phi\in H^1(\Omega\setminus\partial\omega_\#)$: $\phi=0$ on $\partial\Omega$ holds the following expansion \begin{equation}\label{3.2} \int_{\partial\omega_\#^-} \varepsilon g \phi^- \,dS_x -\int_{\Omega\setminus\partial\omega_\#}
{\textstyle\frac{|\partial\omega|}{|\Upsilon|}} g\phi\,dx =\varepsilon\, l_1(\phi), \end{equation} where $l_1: H^1(\Omega\setminus\partial\omega_\#)\mapsto\mathbb{R}$ is a linear form satisfying \begin{equation}\label{3.3}
|l_1(\phi)|\le K\|\phi\|_{H^1(\Omega\setminus\partial\omega_\#)}, \qquad (K>0). \end{equation} \end{lemma}
\begin{proof} We apply the auxiliary cell problem \eqref{3.1}. By inserting a constant test-function $u$, we calculate the average value \begin{equation}\label{3.4}
\langle L\rangle_y ={\textstyle\frac{|\partial\omega|}{|\Upsilon|}}, \qquad\text{where}\quad
\langle L\rangle_y :={\textstyle\frac{1}{|\Upsilon|}} \int_{\Upsilon\setminus\partial\omega} L \,dy. \end{equation}
Here, $|\partial\omega|$ and $|\Upsilon|$ denote the Hausdorff measures of the solid particle boundary $\partial\omega$ in $\mathbb{R}^{d-1}$ and of the cell $\Upsilon$ in $\mathbb{R}^d$, respectively.
Subtracting $\int_{\Upsilon\setminus\partial\omega} \langle L\rangle_y u \,dy$ from \eqref{3.1}, we rewrite it equivalently as \begin{multline}\label{3.5} \int_{\partial\omega^-} u^- \,dS_y -\int_{\Upsilon\setminus\partial\omega}
\langle L\rangle_y u \,dy\\ =\int_{\Upsilon\setminus\partial\omega} \bigl(\nabla_{\!y} L^\top A\nabla_{\!y} u +(L -\langle L\rangle_y) (u-\langle u\rangle_y) \bigr) \,dy =:l(u), \end{multline} where we have added to the residuum $l(u)$ the trivial term \[ \int_{\Upsilon\setminus\partial\omega} (L -\langle L\rangle_y) \langle u\rangle_y \,dy=0,\qquad
\langle u\rangle_y :={\textstyle\frac{1}{|\Upsilon|}} \int_{\Upsilon\setminus\partial\omega} u \,dy. \] In the following, we shall apply the discontinuous Poincare inequality \begin{equation}\label{26a} K_1
\|u-\langle u\rangle_y\|_{L^2(\Upsilon\setminus\partial\omega)}
\le \|\nabla_{\!y} u\|_{L^2(\Upsilon\setminus\partial\omega)}
+\|[\![u]\!]\|_{L^2(\partial\omega)},\qquad (K_1>0), \end{equation} and the Trace Theorem \begin{equation}\label{26aa}
\|[\![u]\!]\|_{L^2(\partial\omega)} \le
{\textstyle\frac{K_2}{\sqrt{2}}} \bigl( \|\nabla_{\!y} u \|_{L^2(\Upsilon\setminus\partial\omega)}
+\|u \|_{L^2(\Upsilon\setminus\partial\omega)} \bigr)
\le K_2 \|u \|_{H^1(\Upsilon\setminus\partial\omega)}, \end{equation} with $K_2>0$, which combine to the estimate \begin{equation}\label{26aaa}
\|u-\langle u\rangle_y\|_{L^2(\Upsilon\setminus\partial\omega)}
\le K_3\|u \|_{H^1(\Upsilon\setminus\partial\omega)},\qquad (K_3 =K_1^{-1}(1+K_2)). \end{equation}
By recalling that $A\in L^\infty(\Upsilon)^{d\times d}$ and by applying Cauchy's inequality to the right hand side of \eqref{3.5} and subsequently applying estimate \eqref{26aaa} to $L$ and $u$, we obtain the following estimate \begin{align}\label{26b}
|l(u)| \le&\ \overline{K} \|\nabla L\|_{L^2(\Upsilon\setminus\partial\omega)}
\|\nabla u\|_{L^2(\Upsilon\setminus\partial\omega)}\nonumber
+K_3^2 \|L \|_{H^1(\Upsilon\setminus\partial\omega)} \|u \|_{H^1(\Upsilon\setminus\partial\omega)}\nonumber\\
\le&\ (\overline{K}+K_3^2) \|L \|_{H^1(\Upsilon\setminus\partial\omega)} \|u \|_{H^1(\Upsilon\setminus\partial\omega)} \end{align} with $\overline{K}$ from \eqref{2.6} and $K_3$ from \eqref{26aaa}.
For a proper test-function $\phi(x)$ with $x=\varepsilon\bigl\lfloor\frac{x}{\varepsilon}\bigr\rfloor +\varepsilon\{\frac{x}{\varepsilon}\}$, we insert $u(x,y)=\phi(\varepsilon\bigl\lfloor\frac{x}{\varepsilon}\bigr\rfloor +\varepsilon y)$ into \eqref{3.5} and apply the periodic coordinate transformation $y\mapsto x$, $\Upsilon\mapsto\mathbb{R}^d$, by paving $\mathbb{R}^d$ such that $\{\frac{x}{\varepsilon}\} =y$ (recall Section~\ref{sec2.1}). After observing that $dy\mapsto \varepsilon^{-d} dx$, $dS_y\mapsto \varepsilon^{1-d} dS_x$, $\nabla_y\mapsto \varepsilon\nabla_x$, we also multiply \eqref{3.5} with the constant $g\varepsilon^{d}$ and use \eqref{3.4} in order to derive \begin{equation*} \sum_{p=1}^{N_\varepsilon} \int_{(\partial\omega^\varepsilon_p)^-} \varepsilon g \phi^- \,dS_x -\sum_{p=1}^{N_\varepsilon} \int_{\Upsilon^\varepsilon_p\setminus\partial\omega^\varepsilon_p}
{\textstyle\frac{|\partial\omega|}{|\Upsilon|}} g \phi \,dx\\ =\varepsilon\, l_1(\phi), \end{equation*} which is \eqref{3.2} with the following right hand side term: \begin{equation}\label{26c}
l_1(\phi) :=g \sum_{p=1}^{N_\varepsilon} \int_{\Upsilon^\varepsilon_p\setminus\partial\omega^\varepsilon_p} \bigl( (\varepsilon\nabla_{\!x} L^\varepsilon)^\top A^\varepsilon\nabla_{\!x} \phi +(L^\varepsilon -\langle L\rangle_y) \cdot {\textstyle\frac{1}{\varepsilon}} (\phi-\langle \phi\rangle_y) \bigr) \,dx,
\end{equation} where we denote $L^\varepsilon(x) :=L(\{\frac{x}{\varepsilon}\})$ and $A^\varepsilon(x) :=A(\{\frac{x}{\varepsilon}\})$.
Similarly, the discontinuous Poincare inequality \eqref{26a} and the trace theorem \eqref{26aa} transform, respectively, into \begin{subequations}\label{26aaaa} \begin{equation}\label{26daaaaa} \begin{split} \frac{K_1}{\varepsilon}
\|\phi-\langle \phi\rangle_y \|_{L^2(\Upsilon^\varepsilon_p\setminus\partial\omega^\varepsilon_p)}
\le \| \nabla_{\!x} \phi
\|_{L^2(\Upsilon^\varepsilon_p\setminus\partial\omega^\varepsilon_p)}
+\frac{1}{\sqrt{\varepsilon}} \|[\![\phi]\!]\|_{L^2(\partial\omega^\varepsilon_p)}, \end{split} \end{equation} \begin{equation}\label{26aaaab} \begin{split}
{\frac{1}{\sqrt{\varepsilon}}} \|[\![\phi]\!]
\|_{L^2(\partial\omega^\varepsilon_p)} &\le {\frac{K_2}{\sqrt{2}}} \Bigl(
\|\nabla_{\!x} \phi \|_{L^2(\Upsilon^\varepsilon_p\setminus\partial\omega^\varepsilon_p)}
+{\frac{1}{\varepsilon}} \| \phi
\|_{L^2(\Upsilon^\varepsilon_p\setminus\partial\omega^\varepsilon_p)} \Bigr)\\
& \le K_2 \|\phi \|_{H^1(\Upsilon^\varepsilon_p\setminus\partial\omega^\varepsilon_p)}, \qquad p=1,\dots,N_\varepsilon, \end{split} \end{equation} \end{subequations} which combines to the uniform estimate \begin{equation}\label{26d} \frac{1}{\varepsilon}
\|\phi-\langle \phi\rangle_y\|_{L^2(\Upsilon^\varepsilon_p\setminus\partial\omega^\varepsilon_p)}
\le K_3\|\phi \|_{H^1(\Upsilon^\varepsilon_p\setminus\partial\omega^\varepsilon_p)} \end{equation} with $K_3>0$ from \eqref{26aaa}. We note that the first line of \eqref{26aaaab} expresses the $H^1$-norm by the standard homogeneity argument, see e.g. \cite[Appendix, Lemma~1, p.370]{SP/80}.
Therefore, the estimate \eqref{26b} of $l$ yields the following estimate of $l_1$ \begin{align}
|l_1(\phi)|&\le |g| \sum_{p=1}^{N_\varepsilon} \Bigl(
\overline{K} \|\nabla_y L\|_{L^2(\Upsilon\setminus\partial\omega)}
\|\nabla_x \phi\|_{L^2(\Upsilon^\varepsilon_p\setminus\partial\omega^\varepsilon_p)}\nonumber \\
&\quad +\|L -\langle L\rangle_y\|_{L^2(\Upsilon\setminus\partial\omega)}
\cdot{\frac{1}{\varepsilon}} \| \phi -\langle \phi\rangle_y
\|_{L^2(\Upsilon^\varepsilon_p\setminus\partial\omega^\varepsilon_p)}\Bigr)\nonumber \\
&\le |g| (\overline{K}+K_3^2) \|L \|_{H^1(\Upsilon\setminus\partial\omega)}
\sum_{p=1}^{N_\varepsilon}\|\phi \|_{H^1(\Upsilon^\varepsilon_p\setminus\partial\omega^\varepsilon_p)}. \label{27} \end{align} Here we used \eqref{2.6} and inequalities \eqref{26aaa} for $L$ and \eqref{26d} for $\phi$. Then, \eqref{27} follows \eqref{3.3} with the constant
$K =|g| (\overline{K}+K_3^2) \|L \|_{H^1(\Upsilon\setminus\partial\omega)}$, which completes the proof. \end{proof}
\begin{remark} We remark that Lemma~\ref{lem3.1} justifies not only the a-priori estimate \eqref{2.20}, but also refines it by specifying the limiting asymptotic term as $\varepsilon\searrow0^+$, which consists of the constant potential
${\textstyle\frac{|\partial\omega|}{|\Upsilon|}} g$ distributed uniformly over $\Omega$. \end{remark}
The next auxiliary cell problem studies the asymptotic expansion of a volume force $f\in H^1(\Omega\setminus\partial\omega_\#)$, which is given on the porous space $\Upsilon\setminus\omega$ surrounding the solid particle $\omega\subset\Upsilon$. It will be applied in particular to the nonlinear term in \eqref{2.13b}, i.e. we shall consider the specific volume force $f(x)=-\sum_{s=0}^n z_s \exp({-{\textstyle\frac{z_s}{{\kappa}T}} \phi^0(x)})$ in Theorem~\ref{theo3.1} below.
With $x=\varepsilon\lfloor\frac{x}{\varepsilon}\rfloor +\varepsilon\{\frac{x}{\varepsilon}\}$ (recall Section~\ref{sec2.1}), the following unfolding operator $$ T_\varepsilon: \begin{cases} H^1(\Omega\setminus\partial\omega_\#)\mapsto H^1\bigl((\Omega\setminus\partial\omega_\#) \times(\Upsilon\setminus\partial\omega)\bigr),\\[1mm] (T_\varepsilon f) (x,y) :=f(\varepsilon\bigl\lfloor\frac{x}{\varepsilon}\bigr\rfloor +\varepsilon y), \end{cases} $$ is well defined, see \cite{CDDGZ/12}. For its modification near the boundaries $\partial\Omega$ of non-rectangular domains $\Omega$, see \cite{Fra/10}.
For $x\in\Omega\setminus\partial\omega_\#$, there exists a function $M(x,y)$ piecewisely composed of solutions $M(x,\,\cdot\,)$ of the following $x$-dependent cell problems (compare with \eqref{3.1}): Find $M(x,\,\cdot\,)\in H^1(\Upsilon\setminus\partial\omega)$ such that \begin{multline}\label{3.6} \int_{\Upsilon\setminus\partial\omega} (\nabla_{\!y} M^\top A\nabla_{\!y} u +M u) \,dy =\int_{\Upsilon\setminus\omega} (T_\varepsilon f)\, u \,dy\\ \text{for all test-functions $u\in H^1(\Upsilon\setminus\partial\omega)$}. \end{multline}
\begin{lemma}[Unfolding of the cell volume-force problem]\label{lem3.2}
\\ For all $\phi\in H^1(\Omega\setminus\partial\omega_\#)$: $\phi=0$ on $\partial\Omega$ holds the following expansion \begin{equation}\label{3.7} \int_{\Omega\setminus\omega_\#} f\phi \,dx
-{\textstyle\frac{|\Upsilon\setminus\omega|}{|\Upsilon|}} \int_{\Omega\setminus\partial\omega_\#} f\phi\,dx =\varepsilon\, l_2(\phi), \end{equation} where $l_2: H^1(\Omega\setminus\partial\omega_\#)\mapsto\mathbb{R}$ is a linear form satisfying \begin{equation}\label{3.8}
|l_2(\phi)|\le K\|\phi\|_{H^1(\Omega\setminus\partial\omega_\#)}, \qquad (K>0). \end{equation} \end{lemma}
\begin{proof} By inserting a constant test-function $u$ into the auxiliary cell problem \eqref{3.6}, we obtain the locally averaged value of $M=M(x,y)$ \begin{equation}\label{3.9} \begin{split} &\langle M(x,\,\cdot\,) {\rangle_y}
:={\textstyle\frac{1}{|\Upsilon|}} \int_{\Upsilon\setminus\partial\omega} M \,dy
={\textstyle\frac{1}{|\Upsilon|}} \int_{\Upsilon\setminus\omega} T_\varepsilon f \,dy. \end{split} \end{equation} Moreover, by using the average $\langle T_\varepsilon f {\rangle_y}$, we can expand \begin{equation}\label{3.10} F(x,y) :=(T_\varepsilon f)(x,y) -\langle T_\varepsilon f{\rangle_y},\qquad \langle T_\varepsilon f
{\rangle_y} :={\textstyle\frac{1}{|\Upsilon|}} \int_{\Upsilon\setminus\partial\omega} (T_\varepsilon f)(x,\cdot) \,dy. \end{equation} See \cite{KK/14} for the analysis of expansion \eqref{3.10} in terms of Fourier series. For fixed $x$ the residual $F(x,y)$ has zero average $\langle F\rangle_y =0$ and estimates as \begin{multline}\label{3.11}
\|F(x,\,\cdot\,)
\|_{L^2(\Upsilon\setminus\partial\omega)}
=\|T_\varepsilon f-\langle T_\varepsilon f \rangle_y
\|_{L^2(\Upsilon\setminus\partial\omega)}
\le K_3 \|T_\varepsilon f\|_{H^1(\Upsilon\setminus\partial\omega)} \end{multline} due to the discontinuous Poincare inequality \eqref{26aaa}. By inserting \eqref{3.10} into \eqref{3.9}, we calculate \begin{equation*} \langle M \rangle_y
={\textstyle\frac{1}{|\Upsilon|}} \int_{\Upsilon\setminus\omega} T_\varepsilon f \,dy=
{\textstyle\frac{1}{|\Upsilon|}} \int_{\Upsilon\setminus\omega} F\,dy+
{\textstyle\frac{|\Upsilon\setminus\omega|}{|\Upsilon|}}\, \langle T_\varepsilon f \rangle_y, \end{equation*} and thus derive by using again \eqref{3.10}, i.e. $ \langle T_\varepsilon f \rangle_y=T_\varepsilon f-F$ \begin{equation}\label{3.12}
{\textstyle\frac{|\Upsilon\setminus\omega|}{|\Upsilon|}}\, T_\varepsilon f =\langle M \rangle_y
+{\textstyle\frac{|\Upsilon\setminus\omega|}{|\Upsilon|}}
\Bigl(F -{\textstyle\frac{1}{|\Upsilon\setminus\omega|}} \int_{\Upsilon\setminus\omega} F \,dy\Bigr). \end{equation} After multiplying the identity \eqref{3.12} with $u$ and integrating it over $\Upsilon\setminus\partial\omega$, we subtract it from \eqref{3.6} and rewrite \eqref{3.6} equivalently as \begin{multline}\label{3.13} \int_{\Upsilon\setminus\omega} (T_\varepsilon f)\, u \,dy
-{\textstyle\frac{|\Upsilon\setminus\omega|}{|\Upsilon|}} \int_{\Upsilon\setminus\partial\omega} (T_\varepsilon f)\, u \,dy\\
=-{\textstyle\frac{|\Upsilon\setminus\omega|}{|\Upsilon|}} \int_{\Upsilon\setminus\partial\omega}
\biggl(F -{\textstyle\frac{1}{|\Upsilon\setminus\omega|}} \int_{\Upsilon\setminus\omega} F \,dy\biggr) u \,dy\qquad\qquad\qquad\qquad \\ +\int_{\Upsilon\setminus\partial\omega} \bigl(\nabla_{\!y} M^\top A\nabla_{\!y} u +(M -\langle M \rangle_y) (u-\langle u \rangle_y) \bigr) \,dy =:m(u), \end{multline} where we have added the trivial term $\int_{\Upsilon\setminus\partial\omega} (M -\langle M\rangle_y) \langle u\rangle_y \,dy=0$ and the residuum $m(u)$ shortly denotes the right hand side terms of \eqref{3.13}.
For fixed $x\in\Omega\setminus\partial\omega_\#$, Cauchy's inequality yields for the first term on the right hand side of \eqref{3.13} \begin{multline}\label{3.13a}
\Bigl| \int_{\Upsilon\setminus\partial\omega}
\Bigl(F -{\textstyle\frac{1}{|\Upsilon\setminus\omega|}}
\int_{\Upsilon\setminus\omega} F \,dy\Bigr) u \,dy \Bigr|\\
\le \|F\|_{L^2(\Upsilon\setminus\partial\omega)}
\|u\|_{L^2(\Upsilon\setminus\partial\omega)}
+\sqrt{\textstyle\frac{|\Upsilon|}{|\Upsilon\setminus\omega|}}\,
\|F\|_{L^2(\Upsilon\setminus\omega)}
\|u\|_{L^2(\Upsilon\setminus\partial\omega)}\\
\le \Bigl(1 +\sqrt{\textstyle\frac{|\Upsilon|}{|\Upsilon\setminus\omega|}}\Bigr)
\|F\|_{L^2(\Upsilon\setminus\partial\omega)}
\|u\|_{L^2(\Upsilon\setminus\partial\omega)}. \end{multline} Thus, by applying the estimates \eqref{3.11} and \eqref{3.13a} to $F$ and the discontinuous Poincare inequality \eqref{26aaa} to $M$ and $u$, we estimate $m(u)$ at the right hand side of \eqref{3.13} as \begin{multline}\label{3.14}
|m(u)|\le K_4 \|T_\varepsilon f\|_{H^1(\Upsilon\setminus\partial\omega)}
\|u \|_{L^2(\Upsilon\setminus\partial\omega)}\\
+(\overline{K}+K_3^2) \|M \|_{H^1(\Upsilon\setminus\partial\omega)}
\|u \|_{H^1(\Upsilon\setminus\partial\omega)}, \end{multline}
where $K_4 ={\textstyle\frac{|\Upsilon\setminus\omega|}{|\Upsilon|}}
\Bigl(1 +\sqrt{\textstyle\frac{|\Upsilon|}{|\Upsilon\setminus\omega|}} \Bigr) K_3$ and by recalling $\overline{K}$ from \eqref{2.6} and $K_3$ from \eqref{26aaa}.
Next, we substitute $u=(T_\varepsilon \phi)$ as the test-function in \eqref{3.13} and use the property $T_\varepsilon f\cdot T_\varepsilon \phi =T_\varepsilon (f\phi)$ of the unfolding operator. After applying the periodic coordinate transformation $y\mapsto x$, $\{\frac{x}{\varepsilon}\} =y$ to \eqref{3.13} similar to the proof of Lemma \ref{lem3.1}, we arrive with $T_\varepsilon (f\phi)\mapsto f\phi$ and $T_\varepsilon \phi\mapsto \phi$ at \eqref{3.7} with \begin{multline}\label{3.15} l_2(\phi) :=\sum_{p=1}^{N_\varepsilon} \Bigl[ {\textstyle\frac{1}{\varepsilon}}
{\textstyle\frac{|\Upsilon\setminus\omega|}{|\Upsilon|}} \int_{\Upsilon^\varepsilon_p\setminus\partial\omega^\varepsilon_p}
\Bigl(F^\varepsilon -{\textstyle\frac{1}{|\Upsilon\setminus\omega|}} \int_{\Upsilon\setminus\omega} F \,dy\Bigr) \phi \,dx\\ +\int_{\Upsilon^\varepsilon_p\setminus\partial\omega^\varepsilon_p} \bigl( (\varepsilon\nabla_{\!x} M^\varepsilon)^\top A^\varepsilon\nabla_{\!x} \phi +(M^\varepsilon -\langle M\rangle_y) {\textstyle\frac{1}{\varepsilon}} (\phi-\langle T_\varepsilon \phi\rangle_y) \bigr) \,dx\Bigr], \end{multline} where $F^\varepsilon(x) :=F(x,\{\frac{x}{\varepsilon}\})$ and $M^\varepsilon(x) :=M(x,\{\frac{x}{\varepsilon}\})$. Similarly to \eqref{3.14}, we estimate with $F^\varepsilon(x)=f(x)-\langle T_\varepsilon f\rangle_y(x)$ \begin{equation*} \begin{split}
|l_2(\phi)| \le&\ \sum_{p=1}^{N_\varepsilon} \Bigl[
{\textstyle\frac{|\Upsilon\setminus\omega|}{|\Upsilon|}}
\Bigl(1 \!+\! \sqrt{\textstyle\frac{|\Upsilon|}{|\Upsilon\setminus\omega|}} \Bigr) {\textstyle\frac{1}{\varepsilon}}
\|f-\langle T_\varepsilon f\rangle_y\|_{L^2(\Upsilon^\varepsilon_p\setminus\partial\omega^\varepsilon_p)}
\|\phi\|_{L^2(\Upsilon^\varepsilon_p\setminus\partial\omega^\varepsilon_p)}\\ &+\sup_{x\in\Omega\setminus\partial\omega_\#} \!\! \Bigl\{
\overline{K} \|\nabla_{\!y} M(x,\,\cdot\,)\|_{L^2(\Upsilon\setminus\partial\omega)}
\|\nabla \phi\|_{L^2(\Upsilon^\varepsilon_p\setminus\partial\omega^\varepsilon_p)}\\
&+\|M(x,\,\cdot\,) -\langle M(x,\,\cdot\,)\rangle_y\|_{L^2(\Upsilon\setminus\partial\omega)}
{\textstyle\frac{1}{\varepsilon}} \| \phi -\langle T_\varepsilon\phi\rangle_y
\|_{L^2(\Upsilon^\varepsilon_p\setminus\partial\omega^\varepsilon_p)}\Bigr\}\Bigr],\\ \end{split} \end{equation*} hence, \begin{equation}\label{37a} \begin{split}
|l_2(\phi)| \le&\ K_4 \|f\|_{H^1(\Omega\setminus\partial\omega_\#)}
\|\phi\|_{L^2(\Omega\setminus\partial\omega_\#)}\\ &+(\overline{K}+K_3^2) \sup_{x\in\Omega\setminus\partial\omega_\#}
\|M(x,\,\cdot\,)\|_{H^1(\Upsilon\setminus\partial\omega)}
\|\phi\|_{H^1(\Omega\setminus\partial\omega_\#)}, \end{split} \end{equation} where we have used \eqref{26aaa} for $M(x,\,\cdot\,)$ and \eqref{26d} for $f$ and $\phi$. Thus, \eqref{37a} implies the estimate \eqref{3.8} of the residual term $l_2$ given in \eqref{3.15} with \[
K =K_4 \|f\|_{H^1(\Omega\setminus\partial\omega_\#)} +(\overline{K}+K_3^2) \!\!\! \sup_{x\in\Omega\setminus\partial\omega_\#} \!\!\!
\|M(x,\,\cdot\,)\|_{H^1(\Upsilon\setminus\partial\omega)}. \] This completes the proof. \end{proof}
\begin{remark} We remark that the factor
${\textstyle\frac{|\Upsilon\setminus\omega|}{|\Upsilon|}}$ in \eqref{3.7} reflects the porosity of the cell $\Upsilon$ due to the presence of the solid particles $\omega$.
In our particular geometric setting, we have $|\Upsilon|=1$ and
$|\Upsilon\setminus\omega|=1-|\omega|$, respectively. \end{remark}
The third cell problem considers the solutions of the following system of $d$ linear equations: Find a vector of periodic functions $N=(N_1,\dots,N_d)^\top\in H^1_\#(\Upsilon\setminus\partial\omega)^d$ with componentwise zero average $\langle N \rangle_y =0$ such that \begin{multline}\label{3.16} \int_{\Upsilon\setminus\partial\omega} D(N +y) A\nabla u \,dy +\int_{\partial\omega} \alpha [\![N]\!] [\![u]\!] \,dS_y =0,\\ \text{for all scalar test-functions $u\in H^1_\#(\Upsilon\setminus\partial\omega)$}. \end{multline} Here, $H^1_\#(\Upsilon\setminus\partial\omega)$ denotes the space of periodic $H^1$-functions and $D N(y)\in\mathbb{R}^{d\times d}$ for $y\in\Upsilon\setminus\partial\omega$ stands for the row-wise gradient matrix of the vector $N$, that is $$ D N := \begin{pmatrix}
N_{1,1} & \dots & N_{1,d}\\ \vdots & & \vdots\\ N_{d,1} & \dots & N_{d,d}
\end{pmatrix}, \qquad \text{where}\quad N_{i,j} :={\textstyle\frac{\partial N_i}{\partial y_j}}, \quad i,j=1,\dots,d. $$ Moreover in \eqref{3.16}, $D y = I\in\mathbb{R}^{d\times d}$ yields the identity matrix. The solvability of \eqref{3.16} follows from the symmetry and positive definiteness assumption \eqref{2.6}. The uniqueness of the solution $N$ is provided due to the constraint $\langle N \rangle_y =0$. Indeed, since $N(y)+K$ with an arbitrary constant $K$ solves also \eqref{3.16}, the zero average condition is sufficient (and necessary) to ensure the uniqueness of the solution, see e.g. \cite{OSY/92}. Finally, the solution is smooth locally in $\Upsilon\setminus\partial\omega$.
\begin{remark} We remark in particular, that if $[\![N]\!] =[\![u]\!] =0$ would hold, then the discontinuous cell problem \eqref{3.16} would reduce to a standard, continuous cell problem. \end{remark}
The system \eqref{3.16} is essential to determine the efficient coefficient matrix $A^0$ of the macroscopic model averaged over $\Omega$. In fact, following the lines of \cite{OSY/92,ZKO/94}, we shall establish an orthogonal decomposition of Helmholtz type for the oscillating coefficients $A^\varepsilon$.
The Helmholtz type decomposition is based on the left hand side of \eqref{3.16} defining an inner product $ \langle\!\langle\,\cdot\,,\,\cdot\,\rangle\!\rangle$ in $H^1_\#(\Upsilon\setminus\partial\omega)$. Due to $[\![y]\!] =0$, the variational equation \eqref{3.16} reads as $\langle\!\langle N+y,u\rangle\!\rangle =0$ for all $u\in H^1_\#(\Upsilon\setminus\partial\omega)$, which implies that $N+y$ belongs to the kernel of this topological vector space. Thus, the fundamental theorem of vector calculus (the Helmholtz theorem, see e.g. \cite{ZKO/94}) permits the following representation as sum of a constant matrix $A^0$ and divergence free $B(y)$ fields in $\mathbb{R}^{d\times d}$: \begin{equation}\label{3.23} D \bigl( N(y)+y\bigr) A (y) =A^0 +B(y), \qquad \text{a.e.}\quad y\in \Upsilon\setminus\partial\omega, \end{equation} where $B$ has zero average, i.e. $$
0 =\langle B\rangle_y :={\textstyle\frac{1}{|\Upsilon|}} \int_{\Upsilon\setminus\partial\omega} B(y) \,dy. $$ Thus, we obtain the following lemma:
\begin{lemma}[The cell oscillating-coefficient problem]\label{lem3.3}
\\ The constant matrix of effective coefficients is determined by averaging \begin{equation}\label{3.18} \begin{split} &A^0 :=\bigl\langle D \bigl(N(y) +y\bigr) A\bigr\rangle_y \in \mathbb{R}^{d\times d}. \end{split} \end{equation} Moreover, $A^0$ is a symmetric and positive definite matrix with the entries: \begin{multline}\label{3.19} A^0_{ij} =\Bigl\langle \sum_{k,l=1}^d (N_{i,k} +\delta_{i,k}) A_{kl} (N_{j,l} +\delta_{j,l}) \Bigl\rangle_{\!y}
+{\textstyle\frac{1}{|\Upsilon|}} \int_{\partial\omega} \alpha [\![N_i]\!] [\![N_j]\!] \,dS_y\\ \text{for}\quad i,j=1,\dots,d. \end{multline} For the transformed solution vector $N^\varepsilon(x) :=N(\{\frac{x}{\varepsilon}\})$, which depends only on $\{\frac{x}{\varepsilon}\}$ since the coefficient $A^\varepsilon(x) := A(\{\frac{x}{\varepsilon}\})$ also depends only on $\{\frac{x}{\varepsilon}\}$, the following decomposition holds: \begin{equation}\label{3.17} \begin{split} &D (\varepsilon N^\varepsilon(x) +x) A^\varepsilon(x) =A^0 +\varepsilon B^\varepsilon(x) \quad\text{in $\mathbb{R}^{d\times d}$} \ \text{and} \ \text{a.e.}\; x\in \Omega\setminus\partial\omega_\#. \end{split} \end{equation} The transformed function $B^\varepsilon(x) := B(\{\frac{x}{\varepsilon}\})$ is deduced from the symmetric matrix $B\in L^2_{\rm div}(\Upsilon\setminus\partial\omega)^{d\times d}$
with zero average $\langle B\rangle_y=0$. Its entries $B_{ij}(y)$, $i,j=1,\dots,d$ express divergence free fields (called solenoidal in 3d) obtained by combining the derivatives $\frac{\partial}{\partial y_k}$, $k=1,\dots,d$ of a third-order skew-symmetric tensor $b_{ijk}$ in the following way \begin{equation}\label{3.20} \begin{split} &B_{ij} =\sum_{k=1}^d b_{ijk,k},\qquad b_{ijk} =-b_{ikj}, \;\text{(skew-symmetry)} \quad\text{a.e. on $\Upsilon\setminus\partial\omega$}�. \end{split} \end{equation} It follows in particular from \eqref{3.20} that \begin{equation}\label{3.21} \begin{split} &\sum_{j,k=1}^d b_{ijk}=0, \quad \sum_{j=1}^d B_{ij,j} =0, \quad i=1,\dots,d\quad \text{a.e. on $\Upsilon\setminus\partial\omega$}. \end{split} \end{equation} At the interface the following jump relations hold: \begin{equation}\label{3.22} \begin{split} &[\![B^\varepsilon]\!] =0,\qquad (A^0 +\varepsilon B^\varepsilon)\nu =\alpha [\![N^\varepsilon]\!] \qquad \text{a.e. on $\partial\omega_\#$}. \end{split} \end{equation} \end{lemma}
\begin{proof} The constant values of $A^0$ stated in \eqref{3.18} follow from averaging \eqref{3.23} with $\langle\,\cdot\,\rangle_y$ over $\Upsilon\setminus\partial\omega$ and by using $\langle B\rangle_y=0$. The formula \eqref{3.19} can be checked directly. The symmetry and positive definiteness of $A^0$ follow straightforward from the assumption in \eqref{2.6} of $A$ being symmetric and positive definite. The formulas \eqref{3.20} and \eqref{3.21} describe the fact that the columns of $B$ are divergence free. Inserting the representation \eqref{3.23} into \eqref{3.16} and integrating by parts yields \begin{equation*} \begin{split} 0 &=\int_{\Upsilon\setminus\partial\omega} (A^0+B) \nabla u \,dy +\int_{\partial\omega} \alpha [\![N]\!] [\![u]\!] \,dS_y\\ &=\int_{\partial\omega} \bigl( \alpha [\![N]\!] [\![u]\!] -[\![(A^0+B)\nu\,u]\!] \bigr) \,dS_y \end{split} \end{equation*} due to the second equality in \eqref{3.21}. Then, by choosing test-functions $u\in H^1_\#(\Upsilon\setminus\partial\omega)$ satisfying either $[\![u]\!]=0$ or $[\![u]\!]\not=0$, it follows \begin{equation}\label{3.24} \begin{split} &[\![B]\!] =0,\qquad (A^0 +B)\nu =\alpha [\![N]\!]\qquad \text{a.e.}\ \text{on $\partial\omega$}. \end{split} \end{equation}
Finally, we apply the periodic coordinate transformation $y\mapsto x$, $\Upsilon\mapsto\mathbb{R}^d$, with $y=\{\frac{x}{\varepsilon}\}$ to \eqref{3.23} and \eqref{3.24}. With $\nabla_{\!y}\mapsto \varepsilon\nabla_{\!x}$, we have for the row-wise gradient matrix $D_y N\mapsto \varepsilon D_x N^\varepsilon$ and $B\mapsto \varepsilon B^\varepsilon$. Thus, we arrive at \eqref{3.17} and \eqref{3.22}. The proof is completed. \end{proof}
\subsection{The main Theorem}\label{sec3.2}
Based on the Lemmata \ref{lem3.1}--\ref{lem3.3}, we formulate the main homogenisation result:
\begin{theorem}\label{theo3.1} The homogenisation of the discontinuous nonlinear PB problem under the interfacial transmission conditions \eqref{2.13} yields the following averaged (macroscopic) nonlinear PB problem: Find $\phi^0\in H^1_0(\Omega)$ such that \begin{multline}\label{3.25} \int_\Omega \bigl( (\nabla\phi^0)^\top A^0\nabla\phi
-{\textstyle\frac{|\Upsilon\setminus\omega|}{|\Upsilon|}}\sum_{s=0}^n z_s e^{-{\textstyle\frac{z_s}{{\kappa}T}} \phi^0} \phi \bigr)\,dx
=\int_\Omega {\textstyle\frac{|\partial\omega|}{|\Upsilon|}} g \phi \,dx\\ \text{for all test-functions $\phi\in H^1_0(\Omega)$}. \end{multline} In the limit $\varepsilon\searrow0^+$, the solution $\phi^\varepsilon$ of \eqref{2.13} converges strongly to the first order asymptotic approximation $\phi^1 :=\phi^0 +\varepsilon (\nabla\phi^0)^\top N^\varepsilon$. This corrector term to $\phi^0$ satisfies the residual error estimate (improving \eqref{2.18}): \begin{equation}\label{3.26} \begin{split}
&\|\nabla (\phi^\varepsilon -\phi^1)\|_{L^2(\Omega\setminus\partial\omega_\#)}^{2} +{\textstyle\frac{1}{\varepsilon}}
\|[\![\phi^\varepsilon-\phi^1]\!]\|_{L^2(\partial\omega_\#)}^{2} ={\rm O} (\varepsilon). \end{split} \end{equation} \end{theorem}
\begin{proof} First, we remark that the left hand side of \eqref{3.26} defines a norm in $H^1(\Omega\setminus\partial\omega_\#)$ due to the lower estimate \eqref{2.19}.
Secondly, the unique solution $\phi^0$ of \eqref{3.25} can be establish by following the arguments given in the proof of Theorem~\ref{theo2.1}. Moreover, the solution is smooth inside $\Omega$ by standard arguments of local regularity of weak solutions, see \cite{LC/06} and references therein.
Next, we prove the residual error estimate \eqref{3.26}. Integrating \eqref{3.25} by parts on $\Omega$ yields the strong formulation \begin{equation}\label{47a} -{\rm div} \bigl((\nabla\phi^0)^\top A^0\bigr)
-{\textstyle\frac{|\Upsilon\setminus\omega|}{|\Upsilon|}}\sum_{s=0}^n z_s e^{-{\textstyle\frac{z_s}{{\kappa}T}} \phi^0}
={\textstyle\frac{|\partial\omega|}{|\Upsilon|}} g,\qquad \text{in $\Omega$}. \end{equation} By applying the Green formulas \eqref{13a} and \eqref{13b} in $\Omega\setminus\omega_\#$ and $\omega_\#$, respectively, we have for all $\phi\in H^1(\Omega\setminus\omega_\#)$: $\phi=0$ on $\partial\Omega$ \begin{equation*} \int_{\Omega\setminus\omega_\#} \!\!\! (\nabla\phi^0)^{\!\top} A^0 \nabla \phi\,dx =-\int_{\Omega\setminus\omega_\#} \!\!\! \phi\,{\rm div}\bigl((\nabla\phi^0)^{\!\top} A^0\bigr)\,dx -\int_{\partial\omega_\#^+} (\nabla\phi^0)^{\!\top} A^0 \phi\nu \,dS_x, \end{equation*} and for all $\phi\in H^1(\omega_\#)$: \begin{equation*} \int_{\omega_\#} \!\!\! (\nabla\phi^0)^{\!\top} A^0 \nabla \phi\,dx =-\int_{\omega_\#} \!\!\! \phi\,{\rm div} \bigl((\nabla\phi^0)^{\!\top} A^0\bigr)\,dx +\int_{\partial\omega_\#^-} (\nabla\phi^0)^{\!\top} A^0 \phi\nu \,dS_x. \end{equation*} By summing these two expressions and by using the continuity of $\nabla\phi^0$ across the interface $\partial\omega_\#$, we insert the strong formulation \eqref{47a} into the above right hand sides and rewrite problem \eqref{3.25} in the disjoint domain $\Omega\setminus\partial\omega_\#$ as follows \begin{multline}\label{3.27} \int_{\Omega\setminus\partial\omega_\#} \bigl( (\nabla\phi^0)^\top A^0\nabla\phi
-{\textstyle\frac{|\Upsilon\setminus\omega|}{|\Upsilon|}}\sum_{s=0}^n z_s e^{-{\textstyle\frac{z_s}{{\kappa}T}} \phi^0} \phi \bigr)\,dx\\ +\int_{\partial\omega_\#} (\nabla\phi^0)^\top A^0\nu [\![\phi]\!] \,dS_x =\int_{\Omega\setminus\partial\omega_\#}
{\textstyle\frac{|\partial\omega|}{|\Upsilon|}} g \phi \,dx\\ \text{for all test-functions $\phi\in H^1(\Omega\setminus\partial\omega_\#)$: $\phi=0$ on $\partial\Omega$}. \end{multline}
In the following, we expand the terms in \eqref{3.27} based on the Lemmata~\ref{lem3.1}--\ref{lem3.3}. By applying the decomposition \eqref{3.17} of Lemma~\ref{lem3.3} to the integrand of the first term in the left hand side of \eqref{3.27}, we can represent it as the following sum \begin{equation}\label{3.28} \begin{split} (\nabla\phi^0)^\top A^0 \nabla\phi =&\ (\nabla\phi^0)^\top \bigl( (\varepsilon D N^\varepsilon +I) A^\varepsilon -\varepsilon B^\varepsilon \bigr) \nabla\phi\\ =&\ \Bigl[ \Bigl( \nabla \bigl( \phi^0 +\varepsilon (\nabla\phi^0)^\top N^\varepsilon \bigr) \Bigr)^{\!\top} A^\varepsilon -\varepsilon (N^\varepsilon)^\top D(\nabla\phi^0) A^\varepsilon\\ &\ -(\nabla\phi^0)^\top\varepsilon B^\varepsilon \Bigr] \nabla\phi, \end{split} \end{equation} where we have used that $\bigl[ \nabla \bigl( (\nabla\phi^0)^\top N^\varepsilon \bigr) \bigr]^\top =(\nabla\phi^0)^\top DN^\varepsilon +(N^\varepsilon)^\top D(\nabla\phi^0).$
Next, the integral of the last function on the right hand side of \eqref{3.28} can be integrated by parts by using \eqref{3.20} and \eqref{3.21} to calculate \begin{multline}\label{3.29} -\int_{\Omega\setminus\partial\omega_\#} (\nabla\phi^0)^\top \varepsilon B^\varepsilon \nabla\phi \,dx
=\int_{\Omega\setminus\partial\omega_\#} \sum_{i,j,k=1}^d \phi^0_{,ij} \varepsilon b_{ijk,k}^\varepsilon \phi \,dx\\ +\int_{\partial\omega_\#} \sum_{i,j,k=1}^d \phi^0_{,i} \varepsilon b_{ijk,k}^\varepsilon \nu_j [\![\phi]\!] \,dS_x =-\int_{\Omega\setminus\partial\omega_\#} \sum_{i,j,k=1}^d \phi^0_{,ij} \varepsilon b_{ijk}^\varepsilon \phi_{,k} \,dx\\ +\int_{\partial\omega_\#} \bigl( (\nabla\phi^0)^\top \varepsilon B^\varepsilon \nu -\sum_{i,j,k=1}^d \phi^0_{,ij} b_{ijk}^\varepsilon \nu_k \bigr) [\![\phi]\!] \,dS_x, \end{multline} with $b_{ijk}^\varepsilon(x) :=b_{ijk}(\{\frac{x}{\varepsilon}\})$. Substituting \eqref{3.28} and \eqref{3.29} in \eqref{3.27}, we rewrite it \begin{multline}\label{3.30} \int_{\Omega\setminus\partial\omega_\#} \! \Bigl[ \Bigl( \nabla \bigl( \phi^0 +\varepsilon (\nabla\phi^0)^\top N^\varepsilon \bigr) \Bigr)^{\!\top} A^\varepsilon \nabla\phi
-{\textstyle\frac{|\Upsilon\setminus\omega|}{|\Upsilon|}} \sum_{s=0}^n z_s e^{-{\textstyle\frac{z_s}{{\kappa}T}} \phi^0} \phi \Bigr]\,dx\\ +\int_{\partial\omega_\#} (\nabla\phi^0)^\top (A^0 +\varepsilon B^\varepsilon) \nu [\![\phi]\!] \,dS_x =\int_{\Omega\setminus\partial\omega_\#}
{\textstyle\frac{|\partial\omega|}{|\Upsilon|}} g \phi \,dx\\ +\varepsilon m_{\Omega\setminus\partial\omega_\#} \bigl( D(\nabla\phi^0),\nabla\phi \bigr) +m_{\partial\omega_\#} \bigl( D(\nabla\phi^0),[\![\phi]\!] \bigr), \end{multline} where the bilinear continuous forms are given by \begin{subequations}\label{3.31} \begin{multline}\label{3.31a} m_{\Omega\setminus\partial\omega_\#} \bigl( D(\nabla\phi^0),\nabla\phi \bigr)\\
:=\int_{\Omega\setminus\partial\omega_\#} \bigl( (N^\varepsilon)^\top D(\nabla\phi^0) A^\varepsilon \nabla\phi +\sum_{i,j,k=1}^d \phi^0_{,ij} b_{ijk}^\varepsilon \phi_{,k} \bigr) \,dx, \end{multline}
\begin{equation}\label{3.31b} m_{\partial\omega_\#} \bigl( D(\nabla\phi^0),[\![\phi]\!] \bigr) :=\int_{\partial\omega_\#} \sum_{i,j,k=1}^d \phi^0_{,ij} b_{ijk}^\varepsilon \nu_k [\![\phi]\!] \,dS_x. \end{equation} \end{subequations}
Next, we apply Lemma~\ref{lem3.2} with $f(x)=-\sum_{s=0}^n z_s \exp({-{\textstyle\frac{z_s}{{\kappa}T}} \phi^0(x)})$ and obtain the following representation of the nonlinear term in \eqref{3.30} \begin{multline}
-{\textstyle\frac{|\Upsilon\setminus\omega|}{|\Upsilon|}} \sum_{s=0}^n\int_{\Omega\setminus \partial\omega_\#} z_s e^{-{\textstyle\frac{z_s}{{\kappa}T}} \phi^0(x)}\phi \,dx\\ =-\sum_{s=0}^n \int_{\Omega\setminus \omega_\#} z_s e^{-{\textstyle\frac{z_s}{{\kappa}T}} \phi^0(x)}\phi\,dx +\varepsilon\, l_2(\phi). \end{multline} The boundary integral in \eqref{3.30} can be expanded by using \eqref{3.2} in Lemma~\ref{lem3.1}, i.e. \begin{equation*} \int_{\Omega\setminus\partial\omega_\#}
{\textstyle\frac{|\partial\omega|}{|\Upsilon|}} g\phi\,dx =\int_{\partial\omega_\#^-} \varepsilon g \phi^- \,dS_x -\varepsilon\, l_1(\phi), \end{equation*}
Next, we subtract the equation \eqref{3.30} for $\phi^0$ from the perturbed equation \eqref{2.13b} for $\phi^\varepsilon$ and use the notation $\phi^1 :=\phi^0 +\varepsilon (\nabla\phi^0)^\top N^\varepsilon$. Moreover, for $\phi^1$, we remark that $[\![\phi^0]\!]=0$ at $\partial\omega_\#$. Hence ${\textstyle\frac{\alpha}{\varepsilon}} [\![\phi^1]\!] =\alpha (\nabla\phi^0)^\top [\![N^\varepsilon]\!] =(\nabla\phi^0)^\top (A^0 +\varepsilon B^\varepsilon) \nu$ in view of \eqref{3.22}. Thus, after subtracting \eqref{3.30} from \eqref{2.13b}, we calculate using the above relations \begin{multline}\label{3.32} \int_{\Omega\setminus\partial\omega_\#} \!\!\! \nabla(\phi^\varepsilon -\phi^1)^\top A^\varepsilon\nabla\phi \,dx +\int_{\partial\omega_\#} {\textstyle\frac{\alpha}{\varepsilon}} [\![\phi^\varepsilon -\phi^1]\!] [\![\phi]\!] \,dS_x\\ -\sum_{s=0}^n \int_{{\Omega\setminus\omega_\#}} \!\!\! z_s \bigl( e^{-{\textstyle\frac{z_s}{{\kappa}T}} \phi^\varepsilon} -e^{-{\textstyle\frac{z_s}{{\kappa}T}} \phi^0} \bigr) \phi \,dx =\varepsilon (l_1(\phi)+l_2(\phi))\\ -\varepsilon\, m_{\Omega\setminus\partial\omega_\#} \bigl( D(\nabla\phi^0),\nabla\phi \bigr) -m_{\partial\omega_\#} \bigl( D(\nabla\phi^0),[\![\phi]\!] \bigr). \end{multline}
One difficulty is that $\phi^1$ cannot be substituted as test-function into \eqref{3.32} since $\phi^1\not=0$ at the boundary $\partial\Omega$. For its lifting, we take a cut-off function $\eta_\varepsilon$ supported in a $\varepsilon$-neighborhood of $\partial\Omega$ such that $\eta_\varepsilon=1$ at $\partial\Omega$. Hence, $\nabla\eta_\varepsilon\sim{\textstyle\frac{1}{\varepsilon}}$ and ${\rm supp}(\eta_\varepsilon)\sim\varepsilon$. Due to the assumed $\varepsilon$-gap between $\partial\Omega$ and $\omega_\#$, we remark that ${\rm supp}(\eta_\varepsilon)$ does not intersect $\partial\omega_\#$.
After substitution of $\phi =\phi^\varepsilon -\phi^1_{\eta_\varepsilon}$ with $\phi^1_{\eta_\varepsilon} :=\phi^0 +\varepsilon (1-\eta_\varepsilon) (\nabla\phi^0)^\top N^\varepsilon$ into \eqref{3.32} and by using $[\![\phi^1_{\eta_\varepsilon}]\!] =[\![\phi^1]\!]$, we obtain the equality \begin{align} \int_{\Omega\setminus\partial\omega_\#} & \nabla(\phi^\varepsilon -\phi^1)^\top A^\varepsilon\nabla (\phi^\varepsilon -\phi^1) \,dx +\int_{\partial\omega_\#} {\textstyle\frac{\alpha}{\varepsilon}} [\![\phi^\varepsilon -\phi^1]\!]^2 \,dS_x\nonumber\\ &-\sum_{s=0}^n \int_{\Omega\setminus\omega_\#} z_s \bigl( e^{-{\textstyle\frac{z_s}{{\kappa}T}} \phi^\varepsilon} -e^{-{\textstyle\frac{z_s}{{\kappa}T}} \phi^1_{\eta_\varepsilon}} \bigr) (\phi^\varepsilon -\phi^1_{\eta_\varepsilon}) \,dx\nonumber\\ =& -m_{\eta_\varepsilon} \bigl( \nabla (\phi^\varepsilon -\phi^1),D(\nabla\phi^0)\bigr) -m_{\partial\omega_\#} \bigl( D(\nabla\phi^0), [\![\phi^\varepsilon -\phi^1]\!] \bigr)\nonumber\\ &+\varepsilon\, \widetilde{l}(\phi^\varepsilon -\phi^1_{\eta_\varepsilon}) ,\label{3.33} \end{align} where we introduce the form $m_{\eta_\varepsilon}$ due to the cut-off function as \begin{multline}\label{3.35} m_{\eta_\varepsilon} \bigl( \nabla (\phi^\varepsilon -\phi^1),D(\nabla\phi^0) \bigr)\\ :=\varepsilon \int_{{\rm supp}(\eta_\varepsilon)} \!\!\! \!\!\! \nabla (\phi^\varepsilon -\phi^1)^\top A^\varepsilon \nabla \bigl( \eta_\varepsilon (\nabla\phi^0)^\top N^\varepsilon \bigr) \,dx, \end{multline} and the short notation $\widetilde{l}$ stands for the following terms \begin{equation}\label{3.34} \begin{split} &\widetilde{l}(\phi) :=l_1(\phi)+l_2(\phi) -m_{\Omega\setminus\partial\omega_\#} \bigl( D(\nabla\phi^0),\nabla \phi \bigr) +m^\varepsilon (\phi^0,\phi), \end{split} \end{equation} where the nonlinear form $m^\varepsilon$ in \eqref{3.34} is given by \begin{equation}\label{3.36} m^\varepsilon (\phi^0,\phi) :=\sum_{s=0}^n \int_{\Omega\setminus\omega_\#} \!\!\! z_s e^{-{\textstyle\frac{z_s}{{\kappa}T}} \phi^0} {\textstyle\frac{1}{\varepsilon}} \bigl( 1 -e^{-\varepsilon (1-\eta_\varepsilon) {\textstyle\frac{z_s}{{\kappa}T}} (\nabla\phi^0)^\top N^\varepsilon} \bigr) \phi \,dx. \end{equation} From \eqref{3.36}, it can be estimated uniformly as \begin{equation}\label{3.37} \begin{split}
&\bigl| m^\varepsilon (\phi^0,\phi) \bigr|
\le K \|\nabla \phi\|_{L^2(\Omega\setminus\omega_\#)}
\le K \|\nabla\phi\|_{L^2(\Omega\setminus\partial\omega_\#)}, \qquad (K>0), \end{split} \end{equation} due to the Taylor series $1 -e^{-\varepsilon \xi} =\varepsilon \xi +o(\varepsilon)$ for small $\varepsilon$.
The left hand side of \eqref{3.33} can be estimated from below by applying the coercivity of the matrix $A$ as assumed in \eqref{2.6} and by observing that the third term on the left hand side is nonnegative due to the strict monotonicity of the exponential function. Altogether with \eqref{2.19}, this implies that \begin{multline}\label{3.37a}
K_5 \|\phi^\varepsilon -\phi^1\|_{H^1(\Omega\setminus\partial\omega_\#)}^2
\le \bigl| m_{\eta_\varepsilon} \bigl( D(\nabla\phi^0),
\nabla (\phi^\varepsilon -\phi^1) \bigr) \bigr|\\
+\bigl| m_{\partial\omega_\#} \bigl( D(\nabla\phi^0),
[\![\phi^\varepsilon -\phi^1]\!] \bigr) \bigr|
+\varepsilon |\widetilde{l}(\phi^\varepsilon -\phi^1_{\eta_\varepsilon})|, \end{multline} with $K_5 =K_0(\underline{K} +\alpha)>0$ after recalling $\underline{K}$ from \eqref{2.6} and $K_0$ from \eqref{2.19}.
At this point, we remark that the right-hand side of \eqref{3.37a} is a homogeneous function of degree one with respect to the norm
$\|\phi^\varepsilon -\phi^1\|_{H^1(\Omega\setminus\partial\omega_\#)}$ as the following estimates will prove. Thus, the inequality \eqref{3.37a} implies directly that the norm
$\|\phi^\varepsilon -\phi^1\|_{H^1(\Omega\setminus\partial\omega_\#)}$ is bounded, which reconfirms estimate \eqref{2.18}.
However, the following argument allows to refine the asymptotic residual estimate to obtain \eqref{3.26} as $\varepsilon\searrow0^+$. In particular, we shall estimate the three terms at the right hand side of \eqref{3.37a} and then apply Young's inequality to obtain sums of sufficiently small terms
of order $O(\|\phi^\varepsilon -\phi^1\|^2_{H^1(\Omega\setminus\partial\omega_\#)})$ and constant terms, which will constitute the refined residual estimate.
At first, from the estimates \eqref{3.3}, \eqref{3.8}, \eqref{3.37} and due to the boundedness of the bilinear form \eqref{3.31a} for $\phi\in H^1(\Omega\setminus\partial\omega_\#)$, it follows that \begin{equation}\label{3.38} \begin{split}
&|\widetilde{l}(\phi)|
\le K \|\phi\|_{H^1(\Omega\setminus\partial\omega_\#)}, \qquad (K>0). \end{split} \end{equation} Since $\phi^1_{\eta_\varepsilon} =\phi^1 -\varepsilon \eta_\varepsilon (\nabla\phi^0)^\top N^\varepsilon$, we estimate that \begin{equation}\label{3.39} \begin{split}
&\|\phi^\varepsilon -\phi^1_{\eta_\varepsilon} \|_{H^1(\Omega\setminus\partial\omega_\#)}^2
\le 2\|\phi^\varepsilon -\phi^1\|_{H^1(\Omega\setminus\partial\omega_\#)}^2 +{\rm O}(\varepsilon). \end{split} \end{equation} Therefore, specifically for $\phi=\phi^\varepsilon -\phi^1_{\eta_\varepsilon}$, and by using Young's inequality, it follows from \eqref{3.38} and \eqref{3.39} that \begin{equation}\label{3.40} \begin{split}
&|\widetilde{l}(\phi^\varepsilon -\phi^1_{\eta_\varepsilon})| \le K_6 \bigl(
\|\phi^\varepsilon -\phi^1\|_{H^1(\Omega\setminus\partial\omega_\#)}^2 +1 \bigr),\qquad (K_6>0). \end{split} \end{equation} For $\phi\in H^1(\Omega\setminus\partial\omega_\#)$, by using again Young's inequality and by recalling the properties of the cut-off function $\eta_\varepsilon$ implying
$\int_{{\rm supp}(\eta_\varepsilon)} |\nabla \eta_\varepsilon|^2\,dx ={\rm O}({\textstyle\frac{1}{\varepsilon}})$, we estimate \eqref{3.35} with an arbitrary $t_1\in\mathit{R}_+$ by \begin{equation}\label{3.41} \begin{split}
&\bigl| m_{\eta_\varepsilon} \bigl( \nabla \phi ,D(\nabla\phi^0) \bigr) \bigr| \le \varepsilon t_1 K_7 +{\textstyle\frac{1}{t_1}}
\|\nabla\phi\|_{L^2(\Omega\setminus\partial\omega_\#)}^2,\qquad (K_7>0), \end{split} \end{equation} and the form in \eqref{3.31b} by \begin{equation}\label{3.42} \begin{split}
&\bigl| m_{\partial\omega_\#} \bigl( D(\nabla\phi^0),[\![\phi]\!]
\bigr) \bigr| \le \varepsilon t_2 K_8 +{\textstyle\frac{1}{\varepsilon t_2}}
\|[\![\phi]\!]\|_{L^2(\partial\omega_\#)}^2,\qquad (K_8>0), \end{split} \end{equation} with an arbitrary $t_2\in\mathit{R}_+$. Therefore, by applying the estimates \eqref{3.40}, \eqref{3.41} and \eqref{3.42} with $\phi=\phi^\varepsilon -\phi^1$ to \eqref{3.37a} and for suitable $t_1,t_2$, and $\varepsilon_0>0$ such that $$ 0<K:=K_5 -({\textstyle\frac{1}{t_1}} +{\textstyle\frac{1}{t_2}} ) K_0 -\varepsilon_0 K_6, $$ we conclude \begin{equation*}
K \|\phi^\varepsilon -\phi^1\|_{H^1(\Omega\setminus\partial\omega_\#)}^2 \le \varepsilon (t_1 K_7 +t_2 K_8 +K_6), \end{equation*} for all $\varepsilon<\varepsilon_0$, which yields estimate \eqref{3.26}. This finishes the proof. \end{proof}
\section{Discussion}\label{sec4}
In the following, we shall summarise the main observations concerning the presented results.
\begin{itemize} \item We remark at first that Theorem~\ref{theo3.1}, in particular, implies by standard arguments the weak convergence $\phi^\varepsilon\rightharpoonup\phi^0$ in $H^1(\Omega\setminus\partial\omega_\#)$ and the strong convergence $\phi^\varepsilon\to\phi^0$ in $L^2(\Omega\setminus\partial\omega_\#)$ as $\varepsilon\searrow0^+$, as well as the two-scale convergence and the $\Gamma$-convergence of the solutions.
\item We observe that the first two terms on the right hand side of \eqref{3.37a} express the residual error near $\partial\Omega$ and at $\partial\omega_\#$. These terms are asymptotically of order $O(\sqrt{\varepsilon})$
(as can be see by setting $t_1=O(\varepsilon^{-1/2})=t_2$ in \eqref{3.41} and \eqref{3.42}) and thus constitute the leading order $O(\varepsilon)$ in the residual error estimate \eqref{3.26}.
Therefore, by constructing corrector terms in form of the respective boundary layers, the $O(\varepsilon)$-estimate \eqref{3.26} could be improved to the order $O(\varepsilon^2)$.
\item The factor ${\textstyle\frac{1}{\varepsilon}}$ appears at the jump across interface $\partial\omega_\#$ in the left hand side of microscopic equation \eqref{2.13b}. It is controlled by the coercivity condition \eqref{2.19}. We point out that this term disappears in the homogenisation limit and does not contribute to the macroscopic equation \eqref{3.25}.
\item The factor $\varepsilon$ in front of the inhomogeneous material parameter $g$, which is prescribed at the solid phase boundary $\partial\omega_\#^-$, presents the critical order. After averaging this factor guarantees the presence of the potential
${\textstyle\frac{|\partial\omega|}{|\Upsilon|}} g$ distributed over the homogeneous domain $\Omega$ in \eqref{3.25}.
\item For variable functions $g(\{\frac{x}{\varepsilon}\})$ distributed periodically over the interface $\partial\omega_\#$, the decomposition \begin{equation*} g =\langle g\rangle_y +G,\quad \text{with} \; \langle g\rangle_y
:=\frac{1}{|\partial\omega|} \int_{\partial\omega} g(y)\,dy, \quad \langle G\rangle_y=0, \end{equation*} yields in the limit $\varepsilon\searrow0^+$ that the constant value $\langle g\rangle_y$ replaces $g$ in the averaged problem \eqref{3.25}, see e.g. \cite{CD/88}.
\item The nonlinear term appearing in \eqref{3.25} scales with the porousity coefficient
${\textstyle\frac{|\Upsilon\setminus\omega|}{|\Upsilon|}}$.
\end{itemize}
\noindent {\bf Acknowledgments}. The research results were obtained with the support of the Austrian Science Fund (FWF) in the framework of the SFB F32 "Mathematical Optimization and Applications in Biomedical Sciences" and project P26147-N26. The authors gratefully acknowledge partial support by NAWI Graz.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\begin{frontmatter}
\title{High-order Virtual Element Method \\ on polyhedral meshes}
\author[add1,add2]{L. Beir\~{a}o da Veiga} \ead{[email protected]} \author[add1]{F. Dassi} \ead{[email protected]} \author[add1,add2]{A. Russo} \ead{[email protected]}
\address[add1]{Department of Mathematics and Applications, University of Milano - Bicocca,\\ Via Cozzi 53, I-20153, Milano (Italy)} \address[add2]{IMATI-CNR, 27100 Pavia (Italy)}
\begin{abstract} We develop a numerical assessment of the Virtual Element Method for the discretization of a diffusion-reaction model problem, for higher ``polynomial'' order $k$ and three space dimensions. Although the main focus of the present study is to illustrate some $h$-convergence tests for different orders $k$, we also hint on other interesting aspects such as structured polyhedral Voronoi meshing, robustness in the presence of irregular grids, sensibility to the stabilization parameter and convergence with respect to the order $k$. \end{abstract}
\begin{keyword} Virtual Element Method\sep polyhedral meshes\sep diffusion-reaction problem \MSC[2010] 65N30 \end{keyword}
\end{frontmatter}
\section{Introduction}
The Virtual Element Method (VEM) was introduced in \cite{volley,Hitchhikers} as a generalization of the Finite Element Method (FEM) that allows for very general polygonal and polyhedral meshes, also including non convex and very distorted elements. The VEM is not based on the explicit construction and evaluation of the basis functions, as standard FEM, but on a wise choice and use of the degrees of freedom in order to compute the operators involved in the discretization of the problem. The adopted basis functions are virtual, in the sense that they follow a rigorous definition, include (but are not restricted to) standard polynomials but are \emph{not} computed in practice; the accuracy of the method is guaranteed by the polynomial part of the virtual space. Using such approach introduces other potential advantages, such as exact satisfaction of linear constraints \cite{Stokes:divfree} and the possibility to build easily discrete spaces of high global regularity \cite{Brezzi-Marini:2012,CH-VEM}.
Since its introduction, the VEM has shared a good degree of success and was applied to a large array of problems. We here mention, in addition to the ones above, a sample of papers \cite{Ahmed-et-al:2013,variable-primale,largedefs,nonconforming,Benedetto-VEM-2,Gatica-1,Steklov-VEM,Topology-VEM,vacca2016virtual} and refer to \cite{Arxiv-hp-corner}
for a more complete survey of the existing VEM literature.
Although the construction of the Virtual Element Method for three dimensional problems is accomplished in many papers, at the current level of development very few 3D numerical experiments are available in the literature \cite{Paulino-VEM,Topology-VEM,largedefs}. Moreover, all these tests are limited to the lowest order case ($k=1$).
The objective of this work is to numerically validate, for the first time, the VEM of general order $k$ for three dimensional problems and show that this technology is practically viable also in this case. We consider a simple diffusion-reaction model problem in primal form and follow faithfully the construction in \cite{volley,Ahmed-et-al:2013} and the coding guidelines of \cite{Hitchhikers}.
Although the main focus of the present study is to illustrate some standard $h$-convergence tests for different orders $k$, it also hints on other aspects. In particular, we show some interesting possibilities related to polyhedral Voronoi meshing, we underline the robustness of the method in the presence of irregular grids, we investigate its sensibility to the stabilization parameter and consider also a convergence analysis in terms of the ``polynomial'' order $k$.
The paper is organized as follows. In Section \ref{theVEM} we introduce the model problem and the Virtual Element Method in three dimensions. The review of the method is complete but brief, and we refer to other contributions in the literature for a more detailed presentation of the scheme. Afterwards, in Section \ref{NUMS} an array of numerical tests are shown.
\section{The Virtual Element discretization}\label{theVEM}
In the present section we give a brief overview of the Virtual Element method in three space dimensions for the simple model problem of diffusion-reaction in primal form. More details on the method for this same model and formulation can be found in \cite{volley,Hitchhikers,Ahmed-et-al:2013} while extension to variable coefficients is presented in \cite{variable-primale}. In the following $k$ will denote a positive integer number, associated to the ``polynomial degree'' of the virtual element scheme.
\subsection{Notation}
\newcommand{\myboldmath}[1]{{\boldsymbol{#1}}}
\newcommand{\myboldmath{x}}{\myboldmath{x}} \newcommand{\myboldmath{\alpha}}{\myboldmath{\alpha}} \newcommand{m_\ba}{m_\myboldmath{\alpha}}
\newcommand{{D}}{{D}}
\newcommand{\widehat{\D}}{\widehat{{D}}} \newcommand{\x_\D}{\myboldmath{x}_{D}} \newcommand{h_P}{h_P} \newcommand{{\mathbb{R}}}{{\mathbb{R}}}
In the following, $E$ will denote a polygon and $P$ a polyhedron, while faces, edges and vertices will be indicated by $f$, $e$, and $\nu$ respectively.
If $P$ is a polyhedron in ${\mathbb{R}}^3$, we will denote by $\boldsymbol{x}_P$, $h_P$ and $|P|$ the centroid, the diameter, and the volume of $P$, respectively.
The set of polynomials of degree less than or equal to $s$ in $P$ will be indicated by ${\cal P}_s(P)$.
If $\myboldmath{\alpha}=(\alpha_1,\alpha_2,\alpha_3)$ is a multiindex, we will indicate by $m_\ba$ the scaled monomial
\begin{equation} m_\ba = \left(\dfrac{x-x_P}{h_P}\right)^{\alpha_1} \left(\dfrac{y-y_P}{h_P}\right)^{\alpha_2} \left(\dfrac{z-z_P}{h_P}\right)^{\alpha_3} \end{equation}
and we will denote by ${\cal P}^{{\rm hom}}_{s}(P)$ the the space of scaled monomials of degree exactly (and no less than) $s$:
\begin{equation}
{\cal P}^{{\rm hom}}_{s}(P) = \text{span} \{m_\ba \text{ , } |\myboldmath{\alpha}|=s\} \end{equation}
where $|\myboldmath{\alpha}|=\alpha_1+\alpha_2+\alpha_3$.
The case of a polygon $E\subset{\mathbb{R}}^2$ is completely analogous.
A face $f$ of a polyhedron is treated as a two-dimensional set, using local coordinates $(x,y)$ on the face. Edges of polyhedra and polygons are treated in an analogous way as one-dimensional set.
\subsection{The model problem}
Let $\Omega \subset {\mathbb R}^3$ represent the domain of interest (that we assume to be a polyhedron) and let $\Gamma$ denote a subset of its boundary, that we assume for simplicity to be given by a union of some of its faces. We denote by $\Gamma' = \partial\Omega / \Gamma$.
We consider the simple reaction-diffusion problem \begin{equation} \left\{ \begin{aligned} -\Delta u + u &= f\quad\text{in }\Omega\\
u &= r\quad\text{on }\Gamma \\
\dfrac{\partial u}{\partial n} &= g \quad\text{on }\Gamma' \end{aligned} \right. \end{equation} where $f \in L^2(\Omega)$ and $g \in L^2(\Gamma')$ denote respectively the applied load and Neumann boundary data, and $r \in H^{1/2}(\Gamma)$ is the assigned boundary data function.
The variational form of our model problem reads \begin{equation} \left\{ \begin{aligned} & \textrm{Find } u \in H^1_\Gamma(\Omega) \textrm{ such that } \\ & \int_\Omega \nabla u \cdot \nabla v \: + \: \int_\Omega u v \: = \: \int_\Omega f v + \int_{\Gamma'} g v \qquad \forall v \in H^1_{\Gamma,0}(\Omega) \ , \end{aligned} \right. \end{equation}
where $$
H^1_\Gamma = \big\{ v \in H^1(\Omega) : v|_\Gamma = r \big\} \ , \quad H^1_{\Gamma,0} = \big\{ v \in H^1(\Omega) : v|_\Gamma = 0 \big\}. $$
\subsection{Virtual elements on polygons}
We start by defining the virtual element space on polygons. Given a generic polygon $E$, let the preliminary virtual space $$
\widetilde{V}^k(E) = \Big\{ v \in H^1(E) \cap C^0(E) \: : \: v|_e \in {\cal P}_k(e)\ \forall e \in \partial E, \ \Delta v \in {\cal P}_{k}(E) \Big\} $$ with $e$ denoting a generic edge of the polygon.
For any edge $e$, let the points $\{ \nu_e^i \}_{i=1}^{k-1}$ be given by the $k-1$ internal points of the Gauss-Lobatto integration rule of order $k+1$ on the edge.
We now introduce three sets of linear operators from $\widetilde{V}^k(E)$ into real numbers. For all $v$ in $\widetilde{V}^k(E)$:
\begin{align} \label{dof1} &\bullet\quad\textrm{evaluation of } v(\nu) \ \forall \nu \textrm{ vertex of } E ; \\ \label{dof2} &\bullet\quad\textrm{evaluation of } v(\nu_e^i) \ \forall e \in \partial E, \ i=\{1,2,..,k-1 \} ; \\ \label{dof3} &\bullet\quad\textrm{moments } \int_E v \: p_{k-2} \ \ \forall p_{k-2} \in {\cal P}_{k-2}(E) . \end{align}
The following projector operator $\Pi^\nabla_E \: : \: \widetilde{V}^k(E) \rightarrow {\cal P}_k(E)$ will be useful in the definition of our space and also for computational purposes. For any $v \in \widetilde{V}^k(E)$, the polynomial $\Pi^\nabla_E v \in {\cal P}_k(E)$ is defined by (see \cite{volley}) $$ \left\{ \begin{aligned} & \int_E \nabla (v - \Pi^\nabla_E v) \cdot \nabla p_k = 0 \quad \forall p_k \in {\cal P}_k(E) \\ & \text{for $k=1$:}\quad\sum_{\substack{\nu\,\text{vertex} \\[0.5mm]\text{ of}\,E}} \left(v(\nu) - \Pi^\nabla_E v(\nu)\right) = 0, \\ & \text{for $k\ge2$:}\quad\int_{ E} (v - \Pi^\nabla_E v) = 0 . \end{aligned} \right. $$ Note that, given any $v \in \widetilde{V}^k(E)$ the polynomial $\Pi^\nabla_E v$ only depends on the values of the operators \eqref{dof1}-\eqref{dof2}-\eqref{dof3}. Indeed, an integration by parts easily shows that the values of the above operators applied to $v$ are sufficient to uniquely determine $\Pi^\nabla_E v$ and no other information on the function $v$ is required \cite{volley,Hitchhikers}.
We are now ready to present the two dimensional virtual space: $$ V^k(E) = \Big\{ v \in \widetilde{V}^k(E) \: : \: \int_E v \, q = \int_E (\Pi^\nabla_E v) q \textrm{ for all } q \in {\cal P}_{k-1}^{{\rm hom}}(E)\cup{\cal P}_{k}^{{\rm hom}}(E) \Big\} . $$ It is immediate to check that ${\cal P}_k(E) \subseteq V^k(E) \subseteq \widetilde{V}^k(E)$. Moreover the following lemma holds (the proof can be found in \cite{Ahmed-et-al:2013,variable-primale}).
\begin{lem}\label{lem:dofs1} The operators \eqref{dof1}-\eqref{dof2}-\eqref{dof3} constitute a set of degrees of freedom for the space $V^k(E)$. \end{lem}
Finally note that, since any $p_{k} \in {\cal P}_{k}(E)$ can be written in an unique way as $p_{k} = p_{k-2} + q$, with $p_{k-2} \in {\cal P}_{k-2}(E)$ and $q\in{\cal P}_{k-1}^{{\rm hom}}(E)\cup{\cal P}_{k}^{{\rm hom}}(E)$, it holds \begin{equation}\label{obs-1} \int_E v \, p_{k} = \int_E v \, (p_{k-2} + q) = \int_E v \, p_{k-2} + \int_E (\Pi^\nabla_E v) q \end{equation} for any $p_{k} \in {\cal P}_{k}(E)$. The first term in the right hand side above can be calculated recalling \eqref{dof3} while the second one can be computed directly by integration. This shows that we can actually compute $\int_E v \, p_{k}$ for any $p_{k} \in {\cal P}_{k}(E)$ by using only information on the degree of freedom values of $v$.
\subsection{Virtual elements on polyhedrons}
Let $\Omega_h$ be a partition of $\Omega$ into non-overlapping and conforming polyhedrons. We start by defining the virtual space $V^k$ locally, on each polyhedron $P \in \Omega_h$. Note that each face $f \in \partial P$ is a two-dimensional polygon.
Let the following boundary space \newcommand{{\cal B}^k({\partial P})}{{\cal B}^k({\partial P})} $$
{\cal B}^k({\partial P}) = \Big\{ v \in C^0(\partial P) \: : \: v|_f \in V^k(f) \textrm{ for all } f \textrm{ face of } \partial P \Big\} . $$
The above space is made of functions that on each face are two-dimensional virtual functions, that glue continuously across edges. Recalling Lemma \ref{lem:dofs1}, it follows that the following linear operators constitute a set of degrees of freedom for the space ${\cal B}^k({\partial P})$:
\begin{align} \label{Ldof1} &\bullet\quad\textrm{evaluation of } v(\nu) \ \forall \nu \textrm{ vertex of } P ; \\ \label{Ldof2} &\bullet\quad\textrm{evaluation of } v(\nu_e^i) \ \forall e \textrm{ edge of }\partial P, \ i=\{1,2,..,k-1 \} ; \\ \label{Ldof3} &\bullet\quad\textrm{moments } \int_f v \: p_{k-2} \ \forall p_{k-2} \in {\cal P}_{k-2}(f), \forall f \textrm{ face of } \partial P. \end{align}
Once the boundary space is defined, the steps to follow in order to define the local virtual space on $P$ become very similar to the two dimensional case. We first introduce a preliminary local virtual element space on $P$ $$
\widetilde{V}^k(P) = \Big\{ v \in H^1(P) \: : \: v|_{\partial P} \in {\cal B}^k({\partial P}) , \ \Delta v \in {\cal P}_{k}(P) \Big\} $$ and the ``internal'' linear operators \begin{align} \label{Ldof4} &\bullet\quad\textrm{moments } \int_P v \: p_{k-2} \ \ \forall p_{k-2} \in {\cal P}_{k-2}(P) . \end{align} We can now define the projection operator $\Pi^\nabla_P \: : \: \widetilde{V}^k(P) \rightarrow {\cal P}_k(P)$ by \begin{equation} \left\{ \begin{aligned} & \int_P \nabla (v - \Pi^\nabla_P v) \cdot \nabla p_k = 0 \quad \forall p_k \in {\cal P}_k(P) \\ & \text{for $k=1$:}\quad\sum_{\substack{\nu\,\text{vertex} \\[0.5mm]\text{ of}\,P}} \left(v(\nu) - \Pi^\nabla_P v(\nu)\right) = 0, \\ & \text{for $k\ge2$:}\quad\int_{ P} (v - \Pi^\nabla_P v) = 0 . \end{aligned} \right. \label{eqn:nablaProj} \end{equation}
An integration by parts and observation \eqref{obs-1} show that the projection $\Pi^\nabla_P$ only depends on the operator values \eqref{Ldof1}-\eqref{Ldof2}-\eqref{Ldof3} and \eqref{Ldof4}. Therefore we can define the local virtual space $$ V^k(P) = \Big\{ v \in \widetilde{V}^k(P) \: : \: \int_P v \, q = \int_P (\Pi^\nabla_P v) q \textrm{ for all } q \in {\cal P}_{k-1}^{{\rm hom}}(P)\cup{\cal P}_{k}^{{\rm hom}}(P) \Big\} . $$ The proof of the following lemma mimicks the two dimensional case, see for instance \cite{Ahmed-et-al:2013}.
\begin{lem}\label{lem:dofs2} The operators \eqref{Ldof1}-\eqref{Ldof2}-\eqref{Ldof3} and \eqref{Ldof4} constitute a set of degrees of freedom for the space $V^k(P)$. \end{lem}
It is immediate to verify that ${\cal P}_k(P) \subseteq V^k(P)$, that is a fundamental condition for the approximation properties of the space. Moreover, again due to the observation above, the projection operator $\Pi^\nabla_P : V^k(P) \rightarrow {\cal P}_k(P)$ is computable only on the basis of the degree of freedom values \eqref{Ldof1}-\eqref{Ldof2}-\eqref{Ldof3} and \eqref{Ldof4}. In addition, by following the same identical argument as in \eqref{obs-1} we obtain that $\int_P v p_{k}$ is computable for any $p_{k} \in {\cal P}_{k}(P)$ by using the degrees of freedom. Therefore also the $L^2$ projection operator $\Pi^0_P : V^k(P) \rightarrow {\cal P}_{k}(P)$, defined for any $v \in V^k(P)$ by \begin{equation} \int_P (v - \Pi^0_P v) q_k = 0 \quad \forall q_k \in {\cal P}_k(P) , \label{eqn:zeroProj} \end{equation} is computable by using the degree of freedom values.
Finally, the global virtual space $V^k \subset H^1(\Omega)$ is defined by using a standard assembly procedure as in finite elements. We define $$
V^k = \Big\{ v \in H^1(\Omega) \: : \: v|_P \in V^k(P) \textrm{ for all } P \in \Omega_h \Big\} . $$ The associated (global) degrees of freedom are the obvious counterpart of the local ones introduced above, i.e.
\begin{align} \label{Gdof1} &\bullet\quad\textrm{evaluation of } v(\nu) \ \forall \nu \textrm{ vertex of } \Omega_h\backslash\Gamma ; \\ \label{Gdof2} &\bullet\quad\textrm{evaluation of } v(\nu_e^i) \ \forall e \textrm{ edge of } \Omega_h\backslash\Gamma, \ i=\{1,2,..,k-1 \} ; \\ \label{Gdof3} &\bullet\quad\textrm{moments } \int_f v \: p_{k-2} \ \ \forall p_{k-2} \in {\cal P}_{k-2}(f), \forall f \textrm{ face of } \Omega_h\backslash\Gamma; \\ \label{Gdof4} &\bullet\quad\textrm{moments } \int_P v \: p_{k-2} \ \ \forall p_{k-2} \in {\cal P}_{k-2}(P), \forall P \in \Omega_h . \end{align}
\subsection{Discretization of the problem}\label{sub:probDisc}
We start by introducing the discrete counterpart of the involved bilinear forms. Given any polyhedron $P \in \Omega_h$ we need to approximate the local forms $$ a_P(v,w)=\int_P \nabla v \cdot \nabla w \ , \quad m_P(v,w)=\int_P v \, w . $$ We follow \cite{volley,Hitchhikers}. We first introduce the stabilization form \begin{equation}\label{L:stab} s_P(v,w) = \sum_{i=1}^{N^P_{\textrm{dof}}} \Xi_i(v) \: \Xi_i(w) \quad \forall v,w \in V^k(P) , \end{equation} where $\Xi_i(v)$ is the operator that evaluates the function $v$ in the $i^{th}$ local degree of freedom and $N^P_{\textrm{dof}}$ denotes the number of such local degrees of freedom, see \eqref{Ldof1}-\eqref{Ldof2}-\eqref{Ldof3} and \eqref{Ldof4}.
We then set, for all $v,w \in V^k(P)$, \begin{eqnarray} a_P^h(v,w) &=& \int_P (\nabla \Pi^\nabla_P v) \cdot (\nabla \Pi^\nabla_P w) + h_P \, s_P(v - \Pi^\nabla_P v , w - \Pi^\nabla_P w) , \label{eqn:aForm}\\ m_P^h(v,w) &=& \int_P (\Pi^0_P v) (\Pi^0_P w)
+ |P| \, s_P(v - \Pi^0_P v , w - \Pi^0_P w) \nonumber . \end{eqnarray} The above bilinear forms are consistent and stable in the sense of \cite{volley}. The global forms are given by, for all $v,w \in V^k$, $$ a^h(v,w) = \sum_{P \in \Omega_h} a_P^h(v,w) \ , \quad m^h(v,w) = \sum_{P \in \Omega_h} m_P^h(v,w) . $$ Let now the discrete space with boundary conditions and its corresponding test space $$
V^k_\Gamma = \Big\{ v \in V^k \: : \: v|_\Gamma = r_I \Big\} \ , \quad V^k_0 = \Big\{ v \in V^k \: : \: v|_\Gamma = 0 \Big\} , $$ where $r_I$ is, face by face, an interpolation of $r$ in the virtual space $V^k(f)$.
We can finally state the discrete problem \begin{equation} \left\{ \begin{aligned} & \textrm{Find } u_h \in V^k_\Gamma \textrm{ such that } \\ & a^h(u_h,v_h) \: + \: m^h(u_h,v_h) \: = \: \int_\Omega f_h v_h + \int_{\Gamma'} g_h v_h \qquad \forall v_h \in V^k_0 \ , \end{aligned} \right. \end{equation} where the approximate loading $f_h$ is the $L^2$-projection of $f$ on piecewise polynomials of degree $k$, and where $g_h$ is the $L^2$-projection of $g$ on piecewise polynomials (still of degree $k$) living on $\Gamma'$. Note that all the forms and operators appearing above are computable in terms of the degree of freedom values of $u_h$ and $v_h$.
We close this section by recalling a convergence result. The main argument for the proof can be found in \cite{volley}, while the associated interpolation estimates where shown in \cite{Steklov-VEM} for two dimensions and extended in \cite{Cangiani:apos} to the three dimensional case.
Let now $\{ \Omega_h \}_h$ be a family of meshes, satisfying the following assumption. It exists a positive constant $\gamma$ such that all elements $P$ of $\{ \Omega_h \}_h$ and all faces of $\partial P$ are star-shaped with respect to a ball of radius bigger or equal than $\gamma h_P$; moreover all edges $e \in \partial P$, for all $P \in \{ \Omega_h \}_h$ have length bigger or equal than $\gamma h_P$.
\begin{thm} Let the above mesh assumptions hold. Then, if the data and solution is sufficiently regular for the right hand side to make sense, it holds \begin{equation}\label{H1-conv}
\| u - u_h \|_{H^1(\Omega)} \le C h^{s-1} \Big( |u|_{H^{s}(\Omega_h)}
+ |f|_{H^{s-2}(\Omega_h)} + |g|_{H^{s-3/2}(\Gamma'_h)} \Big) , \end{equation} where $2 \le s \le k+1$, the real $h$ denotes the maximum element diameter size and the constant $C$ is independent of the mesh size. The norms appearing in the right hand side are broken Sobolev norms with respect to the mesh (or its faces). \end{thm}
The above result applies also if $1 \le s<2$, but in that case the regularities on the data $f,g$ need to be modified.
If the domain $\Omega$ is convex (or regular) then under the same assumptions and notations it also holds \cite{variable-primale} \begin{equation}\label{L2-conv}
\| u - u_h \|_{L^2(\Omega)} \le C h^{s} \Big( |u|_{H^{s}(\Omega_h)}
+ |f|_{H^{s-2}(\Omega_h)} + |g|_{H^{s-3/2}(\Gamma'_h)} \Big) . \end{equation} Finally, we note that an extension of the results to more general mesh assumptions could be possibly derived following the arguments for the two-dimensional case shown in \cite{stab:theory}.
In the following numerical tests, we will in particular show the robustness of the method also for quite irregular meshes.
\section{Numerical tests}\label{NUMS}
In this section we collect the numerical results to evaluate the reliability and robustness of the Virtual Element Method in three dimensions.
\subsection{Meshes and error estimators}\label{ssec:mesh}
Before dealing with the numerical examples, we define the domains where we solve the PDEs and the polyhedral meshes which discretize such domains. Moreover, we define the norms that we use to evaluate the error.
\subsubsection{Meshes}\label{sub::mesh}
We consider two different domains: the standard $[0,\,1]^3$ cube, see Figure~\ref{fig:geo} (a), and a truncated octahedron~\cite{patagonGeo}, see Figure~\ref{fig:geo} (b).
\begin{figure}
\caption{(a) Standard cube $[0,\,1]^3$ and (b) the truncated octahedron.}
\label{fig:geo}
\end{figure}
We make different discretizations of such domains by exploiting the c++ library \texttt{voro++}~\cite{voroPlusPlus}. More specifically we will consider the following three mesh types:
\phantom{a}
\noindent \textbf{Random} refers to a mesh where the control points of the Voronoi tessellation are randomly displaced inside the domain. We underline that these kind of meshes are characterized by stretched polyhedrons so the robustness of the VEM will be severely tested.
\phantom{a}
\noindent \textbf{CVT} refers to a Centroidal Voronoi Tessellation, i.e., a Voronoi tessellation where the control points coincides with the centroid of the cells they define. We generate such meshes via a standard Lloyd algorithm~\cite{cvtPaper}. In this case the Voronoi cells are more regular than the ones of the previous case.
\phantom{a}
\noindent \textbf{Structured} refers to meshes composed by structured cubes inside the domain and arbitrary shaped mesh close to the boundary. This mesh is built by considering as control points the vertices of a structured mesh of a cube $\mathcal{C}$ (containing the input geometry $\mathcal{G}$) which are inside $\mathcal{G}$, see Figure~\ref{fig:constr} for a two dimensional example. When we consider the cube $[0,1]^3$, this kind of mesh coincides with a structured mesh composed by cubes. These meshes are really interesting from the computational point of view. Indeed, the VEM local matrices are exactly the same for all cubes inside the domain. It is therefore possible to compute such local matrices \emph{only} once so the computational effort in assembling the stiffness matrix as well as the right hand side is reduced. Moreover, the ensuring scheme may inherit some advantages of structural cubic meshes.
\begin{figure}
\caption{(a) The structured mesh of a square $\mathcal{C}$ which contains the input geometry $\mathcal{G}$, where we highlight the vertices inside $\mathcal{G}$. (b) The final polyhedral Voronoi mesh where we highlight the control points.}
\label{fig:constr}
\end{figure}
\begin{figure}
\caption{Different discretization of the truncated octahedron.}
\label{fig:meshes}
\end{figure}
In Figure~\ref{fig:meshes} we collect an example on the truncated octahedron geometry of all these kinds of meshes. To analyze the error convergence rate, we make a sequence of meshes with decreasing size for each mesh type.
\subsubsection{Error norms}
Let $u$ be the exact solution of the PDE and $u_h$ the discrete solution provided by the VEM. To evaluate how this discrete solution is close to the exact one, we use the local projectors of degree $k$ on each polyhedron $P$ of the mesh, $\Pi^{\nabla}_P\,u_h$ and $\Pi^0_P\,u_h$, defined in~\eqref{eqn:nablaProj} and~\eqref{eqn:zeroProj}, respectively. We compute the following quantities: \begin{itemize} \item\textbf{$\mathbf{H^1}$-seminorm error} $$
e_{H^1} := \sqrt{\sum_{P\in\Omega_h}\left| u - \Pi^{\nabla}_P\,u_h \right|^2_{H^1(P)}}\,, $$ \item\textbf{$\mathbf{L^2}$-norm error} $$
e_{L^2} := \sqrt{\sum_{P\in\Omega_h}\left\| u - \Pi^0_P\,u_h \right\|^2_{L^2(P)}}\,, $$ \item\textbf{$\mathbf{L^{\infty}}$-norm error} $$
e_{L^\infty} := \max_{\nu\in\mathcal{N}}\left|\,u(\nu) - u_h(\nu)\,\right|\,, $$ where $\mathcal{N}$ is the set of all the vertexes and internal edge nodes of the VEM scheme, see Equations~\eqref{Gdof1} and~\eqref{Gdof2}. Since we do not take the $\max$ over all the domain but \emph{only} on some nodes, $e_{L^\infty}$ is an approximation of the true $L^\infty$-norm. Moreover, in this case we can directly compute such quantity without resorting to the projections operators. \end{itemize}
In the following subsections we will present some numerical tests to underline different computational aspects of the method. In all cases the mesh-size parameter $h$ is measured in an averaged sense \begin{equation}\label{h-medio}
h = \left( \frac{|\Omega|}{N_P} \right)^{1/3} , \end{equation} with $N_P$ denoting the number of polyhedrons in the mesh.
\subsection{Test case 1: $h$-analysis for diffusion problem on a cube}
Let us consider the problem \begin{equation} \left\{ \begin{array}{rl} -\Delta u &=\, f\quad\quad\textnormal{in }\Omega\\ u &=\,r \quad\quad\textnormal{on }\Gamma\\ \frac{\partial u}{\partial n} &=\,0 \quad\quad\textnormal{on }\partial\Omega\backslash\Gamma \end{array} \right., \label{eqn:neudiri} \end{equation} where the domain $\Omega$ is the cube $[0,\,1]^3$ and $\Gamma$ is the union of the four faces corresponding to the planes $y=0$, $y=1$, $z=0$ and $z=1$. We choose the right hand side $f$ and $r$ in such a way that the exact solution is $$ u(x,\,y,\,z):=\sin(\pi x)\cos(\pi y)\cos(\pi z)\,. $$ In this example we consider all the three types of discretizations introduced in Subsection~\ref{ssec:mesh}, i.e., Random, CVT and Structured. Note that in this case the last type of discretization becomes a standard structured cubic mesh.
In Figure~\ref{fig:convNeuDiri} we show the resulting graphs and in Tables~\ref{tab:convNeuDiriH1} and~\ref{tab:convNeuDiriL2} we provide the convergence rates. These data show that we achieve the theoretical convergence rate for all the VEM approximation degrees and for each type of meshes, see equations \eqref{H1-conv}, \eqref{L2-conv}.
\begin{table}[!htb] \begin{center}
\begin{tabular}{|l|l|l|l|} \cline{2-4}
\multicolumn{1}{}{}&\multicolumn{3}{|c|}{convergence rates} \\ \hline mesh type &$k=1$ &$k=2$&$k=3$ \\ \hline Structured &1.0344 &2.0543 &3.0125 \\ Random &1.0927 &2.0465 &3.0656 \\ CVT &1.0492 &2.0672 &3.0970 \\ \hline \end{tabular} \end{center} \caption{Test case 1: $H^1$-seminorm convergence rates.} \label{tab:convNeuDiriH1} \end{table}
\begin{table}[!htb] \begin{center}
\begin{tabular}{|l|l|l|l|} \cline{2-4}
\multicolumn{1}{}{}&\multicolumn{3}{|c|}{convergence rates} \\ \hline mesh type &$k=1$ &$k=2$&$k=3$ \\ \hline Structured &1.9763 &3.2551 &4.0372 \\ Random &1.9136 &3.1067 &4.0678 \\ CVT &2.0230 &3.2144 &4.4620 \\ \hline \end{tabular} \end{center} \caption{Test case 1: $L^2$-norm convergence rates.} \label{tab:convNeuDiriL2} \end{table}
\begin{figure}
\caption{Test case 1: $h$-convergence with different meshes.}
\label{fig:convNeuDiri}
\end{figure}
\subsection{Test case 2: $h$-analysis for diffusion-reaction problem on a polyhedron} In this example we consider the problem \begin{equation} \left\{ \begin{array}{rl} -\Delta u + u &=\, f\quad\quad\textnormal{in }\Omega\\ u &=\,r \quad\quad\textnormal{on }\Gamma = \partial\Omega\\ \end{array} \right., \label{eqn:reacDiff} \end{equation} where the domain $\Omega$ is the truncated octahedron, see Figure~\ref{fig:geo} (b), and we choose the right hand side $f$ and $r$ in such a way that the exact solution is $$ u(x,\,y,\,z):=\sin(2xy)\,\cos(z)\,. $$ In this example we analyze the convergence rate for VEM approximation degrees from 1 to 3, and compare the results obtained with the three mesh types described in Subsection~\ref{sub::mesh}.
In Figure~\ref{fig:convReacDiff} we plot the convergence graphs with respect to the total number of degrees of freedom $N_{\textrm{dof}}$.
Considering that (for fixed order $k$ and for fixed mesh family) the mesh-size parameter is expected to behave as $ h \sim N_{\textrm{dof}}^{-1/3}, $ it follows that both error norms behave as expected from the theory (see \eqref{H1-conv} and \eqref{L2-conv}).
Moreover we observe that the error is slightly affected by the shape of the mesh elements. Indeed, the errors associated with the Random mesh are always larger than the ones obtained with a more regular mesh, while the Structured meshes yield the best results (even when compared to the CVT meshes).
\begin{figure}
\caption{Test case 2: $dofs$-convergence with different meshes.}
\label{fig:convReacDiff}
\end{figure}
\subsection{Test case 3: convergence analysis with different $k$}
In this example we consider the convergence with respect to the accuracy degree $k$. We fix the truncated octahedron geometry and we solve the following problem \begin{equation} \left\{ \begin{array}{rl} -\Delta u &=\, f\quad\quad\textnormal{in }\Omega\\ u &=\,r \quad\quad\textnormal{on }\Gamma=\partial \Omega\\ \end{array} \right., \label{eqn:pconvergence} \end{equation} where the right hand side $f$ and the boundary condition $r$ are chosen in such a way that the exact solution is $$ u(x,\,y,\,z):= \sin(\pi x)\,\cos(\pi y)\,\cos(\pi z)\,. $$
The mesh is kept fixed (the CVT mesh of the truncated octahedron composed by 116 polyhedrons) and we rise the polynomial degree $k$ from 1 to 5.
In Figure~\ref{fig:pconvergenceNoStabCase5} we provide the convergence graphs of both $H^1$-seminorm and $L^\infty$-norm. The trend of these errors show an exponential convergence in terms of $k$ and are thus aligned with the existing theory for the two-dimensional case \cite{hp-uniform}.
However, both the $H^1$ and the $L^\infty$ errors show a slight bend in the convergence graphs for $k=5$. This behavior is probably due to the stabilizing matrix~\eqref{L:stab} that should be better devised in order to develop a spectral approximation strategy.
\begin{figure}
\caption{Test case 3: convergence with respect to the order $k$.}
\label{fig:pconvergenceNoStabCase5}
\end{figure}
Although such aspect deserves a deeper study, that is beyond the scopes of the current paper, we propose a novel stabilization strategy which, at least in the present context, cures the problem.
Let $\{ \varphi_i \}_{i=1}^{N^P_{{\rm dof}}}$ represents the canonical basis functions on element $P$, defined by $$ \varphi_i \in V^k(P) \ , \qquad \Xi_j (\varphi_i) = \delta_{ij} \ \textrm{ for } \ j=1,2,..,N^P_{{\rm dof}}, $$ where we refer to \eqref{L:stab} for the notation. Then, a diagonal stabilization form $s_P(\cdot,\cdot)$ should satisfy (see \eqref{eqn:aForm} and \cite{volley,Hitchhikers}) $$ h_P s_P(\varphi_i,\varphi_i) \simeq a_P(\varphi_i,\varphi_i) \qquad i=1,2,..,N^P_{{\rm dof}} $$ in order to mimic the original energy $a_P(\cdot,\cdot)$ of the basis functions. The original form in \eqref{L:stab} corresponds to assuming $a_P(\varphi_i,\varphi_i) \simeq h_P$, that (considering the involved scalings) can be shown to be a reasonable choice with respect to $h_P$ (and thus for moderate $k$). On the other hand, choice \eqref{L:stab} may be less suitable for a higher $k$, especially in three dimensions, where different basis functions may carry very different energies. We therefore propose a simple alternative (still using a diagonal stabilizing form) $$ \widetilde{s}_P(\varphi_i,\varphi_i) = \max \{ h_P , a_P(\Pi^\nabla_P\varphi_i,\Pi^\nabla_P\varphi_i) \} \qquad i=1,2,..,N^P_{{\rm dof}} . $$ Note that at the practical level computing $a_P(\Pi^\nabla_P\varphi_i,\Pi^\nabla_P\varphi_i)$ is immediate, since such term is simply the $i^{th}$ term on the diagonal of the consistency matrix $ a_P ( \Pi^\nabla_P \varphi_i , \Pi^\nabla_P \varphi_j) $, that is already computed.
The choice above, referred in the following as \emph{diagonal recipe}, cures the problem, see Figure~\ref{fig:recipe}. This fact is further confirmed by the data in Table~\ref{tab:recipePendence}. Here, we compute the slopes of these lines at each step and we numerically show that the recipe diagonal stabilization yields better results.
\begin{figure}
\caption{Test case 3: convergence with respect to the order $k$, comparison between the original and the diagonal recipe stabilizations.}
\label{fig:recipe}
\end{figure}
\begin{table}[!htb] \begin{center}
\begin{tabular}{|l|l|l|l|l|} \cline{2-5}
\multicolumn{1}{c}{}&\multicolumn{4}{|c|}{$H^1$ convergence}\\ \hline{2-5} Original &-1.3219 &-1.4065 &-1.3045 &-0.6195\\ \hline Recipe &-1.3218 &-1.4144 &-1.9211 &-1.8977\\ \hline \multicolumn{5}{c}{}\\ \multicolumn{5}{c}{}\\ \cline{2-5}
\multicolumn{1}{c}{}&\multicolumn{4}{|c|}{$L^\infty$ convergence}\\ \hline Original &-1.5704 &-1.6532 &-1.5736 &-1.1455\\ \hline Recipe &-1.5698 &-1.6850 &-1.4805 &-2.3065\\ \hline \end{tabular} \end{center} \caption{Test case 3: slopes at each step for the original and the recipe diagonal stabilization.} \label{tab:recipePendence} \end{table}
\subsection{Test case 4: patch test}
In~\cite{volley,Hitchhikers} it is shown that VEM passes the so-called ``patch test''. If we are dealing with a PDE whose solution is a polynomial of degree $k$ and we use a VEM approximation degree equal to $k$, we recover the ``exact solution'', i.e., the solution up to the machine precision.
We make a patch test for VEM approximation degrees $k$ from 1 up to 5. More specifically, we consider the following PDE \begin{equation} \left\{ \begin{array}{rl} -\Delta u &=\, f\quad\quad\textnormal{in }\Omega\\ u &=\,r \quad\quad\textnormal{on }\Gamma=\partial \Omega\\ \end{array} \right. \label{eqn:patchTest} \end{equation} where the right hand side $f$ and the Dirichlet boundary condition $r$ are chosen in accordance with the exact solution $$ u(x,\,y,\,z):= (x+y+z)^k\,. $$ Since we are not interested in varying the mesh size but \emph{only} the VEM approximation degree, we run the experiments on the same coarse CVT mesh of the truncated octahedron composed by 116 polyhedrons.
In Table~\ref{tab:patch} we collect the results. The errors are close to the machine precision, but for higher VEM approximation degrees they become larger. This fact is natural and stems from the conditioning of the matrices involved in the computation of the VEM solution. Indeed, as in standard FEM, their condition numbers become larger when we consider higher VEM approximation degrees.
\begin{table}[!htb] \begin{center}
\begin{tabular}{|c|c|c|c|} \hline solution degree &$H^1$-seminorm &$L^2$-seminorm &$L^\infty$-norm\\ \hline 1 &5.9775e-12 &6.1919e-13 &1.1479e-12\\ 2 &2.2008e-11 &1.8416e-12 &4.9409e-12\\ 3 &1.0490e-10 &1.1284e-11 &8.5904e-12\\ 4 &3.0959e-10 &1.0039e-10 &2.6197e-11\\ 5 &1.1563e-09 &6.9693e-09 &1.5433e-10\\ \hline \end{tabular} \end{center} \caption{Test case 4: patch tests errors with the mesh CVT of the truncated octahedron composed by 116 elements.} \label{tab:patch} \end{table}
\subsection{Test case 5: stabilizing parameter analysis}
In this example we make an analysis on the stabilizing part of the local stiffness matrix. We slightly modify $a_P^h$ defined in Subsection~\ref{sub:probDisc} by introducing the parameter $\tau\in\mathbb{R}$, i.e., \begin{equation} a_P^h(v,w) = \int_P (\nabla \Pi^\nabla_P v) \cdot (\nabla \Pi^\nabla_P w) + \tau\,h_P \, s_P(v - \Pi^\nabla_P v , w - \Pi^\nabla_P w) . \label{eqn:aFormWithTau} \end{equation} We fix a standard Poisson problem \begin{equation} \left\{ \begin{array}{rl} -\Delta u &=\, f\quad\quad\textnormal{in }\Omega\\ u &=\,0 \quad\quad\textnormal{on }\Gamma=\partial \Omega\\ \end{array} \right., \label{eqn:stabTest} \end{equation} where $\Omega=[0,\,1]^3$ and whose exact solution is $$ u(x,\,y,\,z):= \sin(\pi x)\,\cos(\pi y)\,\cos(\pi z)\,. $$ We exploit the same discretization of $\Omega$, a CVT mesh composed by 1024 polyhedrons, and solve this problem for many values of the parameter $\tau$. More specifically, we consider $\tau = 10^{t}$ for one hundred uniformly distributed values of $t$ in $[-2,2]$ and we compute the errors for the corresponding VEM solutions in the $H^1$-seminorm and $L^\infty$-norm.
\begin{figure}
\caption{Test case 5: (a) $H^1$ error and (b) $L^\infty$ error for varying $\tau$ and different VEM approximation degrees $k$.}
\label{fig:tauAnalysis}
\end{figure}
In Figure~\ref{fig:tauAnalysis} we provide the graphs of both errors, as a function of $\tau$ in a logarithmic scale, for VEM approximation degree $k=1,2,3$ and $4$.
We can observe that the trend of the errors is the same for each $k$: it grows whenever very small or very large choices are taken for the $\tau$ parameter. Although the $L^\infty$-norm seems more sensible than the $H^1$-seminorm, the method appears in general quite robust with respect to the parameter choice. For instance, we show in Table \ref{tab:ratios} the quantities $$ \delta_{e_{H^1}} := \frac{\max_{\tau \in [10^{-1},10]}\left(e_{H_1}\right)}{\min_{\tau\in [10^{-1},10]}\left(e_{H_1}\right)} \ , \qquad \delta_{e_{L^\infty}} := \frac{\max_{\tau \in [10^{-1},10]}\left(e_{L^\infty}\right)}{\min_{\tau\in [10^{-1},10]}\left(e_{L^\infty}\right)} \ , $$ representing the ratios of maximum to minimum error for the parameter range $ [10^{-1},10]$, that corresponds to a factor of 100 between minimum and maximum $\tau$. From this table we can appreciate that the error ratios are all within an acceptable range.
\begin{table}[!htb] \begin{center}
\begin{tabular}{|c|c|c|c|} \hline $k$ &$\delta_{e_{H^1}}$ &$\delta_{e_{L^\infty}}$ \\ \hline 1 &1.4487e+00 &8.9624e+00 \\ 2 &1.7014e+00 &4.9538e+00 \\ 3 &1.3884e+00 &2.0908e+00 \\ 4 &2.6989e+00 &4.8393e+00 \\ \hline \end{tabular} \end{center} \caption{Test case 5: Ratios of the errors for different degrees $k$. Parameter $\tau \in [10^{-1},10]$.} \label{tab:ratios} \end{table}
\begin{center} {\bf Aknowledgments} \end{center} The first and second authors have received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement no. 681162).
\end{document}
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\begin{document}
\title{Büchi automata recognizing sets of reals definable in
first-order logic with addition and order}
\author{Arthur Milchior}
\maketitle \begin{abstract}
This work considers weak deterministic Büchi automata reading encodings of non-negative reals in a fixed base. A Real Number Automaton is an automaton which recognizes all encoding of elements of a set of reals. It is explained how to decide in linear time{} whether a set of reals recognized by a given minimal weak deterministic RNA is $\textsc{FO}[\mathbb R;+,<,1]$-definable{}. Furthermore, it is explained how to compute in quasi-quadratic (respectively, quasi-linear) time an existential (respectively, existential-universal) $\textsc{FO}[\mathbb R;+,<,1]$-formula which defines the set of reals recognized by the automaton. It is also shown that techniques given by Muchnik and by Honkala for automata over vector of natural numbers also works on vector of real numbers. It implies that some problems such as deciding whether a set of tuples of reals $R\subseteq\mathbb R^{d}$ is a subsemigroup of $(\mathbb R^{d},+)$ or is $\textsc{FO}[\mathbb R;+,<,1]$-definable is decidable.
\end{abstract}
\section*{Introduction}
This paper deals with logically defined sets of numbers encoded by weak deterministic Büchi automata. The sets of tuples of integers whose encodings in base $b$ are recognized by a finite automaton are called the $b$-recognizable sets. By \cite{bruyere}, the $b$-recognizable sets of vectors of integers are exactly the sets which are $\fo{\mathbb Z;+,<,V_{b}}$-definable, where $V_{b}(n)$ is the greatest power of $b$ dividing $n$. It was proven in \cite{semenov-theorem,Cobham} that the $\fo{\mathbb N;+}$-definable sets are exactly the sets which are $b$- and $b'$-recognizable for every $b\ge 2$.
The preceding results naturally led to the following problem: deciding whether a finite automaton recognizes a $\fo{\mathbb N;+}$-definable set of $d$-tuples of integers for some dimension $d\in\mathbb N^{>0}$. In the case of dimension $d=1$, the decidability was proven in \cite{Honkala86}. For $d>1$, the decidability was proven in \cite{muchnik}. Another algorithm was given in \cite{Leroux}, which solves this problem in polynomial time. For $d=1$, a quasi linear time algorithm was given in \cite{Sakarovitch}.
\paragraph{} The above-mentioned results about sets of tuples of natural numbers and finite automata have then been extended to results about set of tuples of reals recognized by a Büchi automata. The notion of Büchi automata is a formalism which describes languages of infinite words, also called $\omega$-words. The Büchi automata are similar to the finite automata. The main difference between the two kinds of automata is that finite automata accept finite words which admits runs ending on accepting state, while Büchi automata accepts infinite words which admit runs in which an accepting state appears infinitely often.
One of the main differences between finite automata and Büchi automata is that finite automata can be determinized while deterministic Büchi automata are less expressive than Büchi automata. For example, the language $L_{\mbox{fin }a}$ of words containing a finite number of times the letter $a$ is recognized by a Büchi automaton, but is not recognized by any deterministic Büchi automaton. This statement implies, for example, that no deterministic Büchi automaton recognizes the set of reals of the form $nb^{p}$ with $n\in\mathbb N$ and $p\in\mathbb Z$, that is, the reals which admits no encoding in base $b$ with a finite number of non-0 digits.
Another main difference between the two classes of automata is that the class of languages recognized by finite automata is closed under complement while the class of languages recognized by deterministic Büchi automata is not closed under complement. For example, $L_{\mbox{inf }a}$, the complement of $L_{\mbox{fin }a}$, is recognized by a deterministic Büchi automaton.
A Real Vector Automaton (RVA, See e.g. \cite{weak-R-+-vector}) of dimension $d$ is a Büchi automaton $\mathcal{A}$ of alphabet $\set{0,\dots,b-1}^{d}\cup\set\star$, which recognizes the set of encoding in base $b$ of the elements of a set of vectors of reals. Equivalently, for $w$ an infinite word encoding a vector of dimension $d$ of real $\left(r_{0},\dots,r_{d-1}\right)$, if $w$ is recognized by $\mathcal{A}$, then all encodings $w'$ of $\left(r_{0},\dots,r_{d-1}\right)$ are recognized by $\mathcal{A}$. In the case where the dimension $d$ is 1, those automata are called Real Number Automata (RNA, See e.g. \cite{real}).
The sets of tuples of reals whose encoding in base $b$ is recognized by a RVA are called the $b$-recognizable sets. By \cite{LinArCon}, they are exactly the $\fo{\mathbb R,\mathbb Z;+,<,X_{b},1}$-definable sets. The logic $\fo{\mathbb R,\mathbb Z;+,<,X_{b},1}$ is the first-order logic over reals with a unary predicate which holds over integers, addition, order, the constant one, and the function $X_{b}(x,u,k)$. The function $X_{b}(x,u,k)$ holds if and only if $u$ is equal to some $b^{n}$ with $n\in\mathbb Z$ and there exists an encoding in base $b$ of $x$ whose digit in position $n$ is $k$. That is, $u$ and $x$ are of the form: \begin{eqnarray*}
\begin{array}{lllllllllll}
u=&0&\dots&0&\star&0&\dots&0&1&0&\dots\\
x=& &\dots& &\star& &\dots& &k& &\dots
\end{array} \end{eqnarray*} or of the form: \begin{eqnarray*}
\begin{array}{lllllllllll}
u=&0&\dots&0&1&0&\dots&0&\star&0&\dots \\
x=& &\dots& &k& &\dots& &\star& &\dots
\end{array} \end{eqnarray*} A weak deterministic Büchi automaton is a deterministic Büchi automaton whose set of accepting states is a union of strongly connected components. A set is said to be weakly $b$-recognizable if it is recognized by a weak automaton in base $b$. By \cite{weak-R-+-vector}, a set is $\fo{\mathbb R,\mathbb Z;+,<}$-definable if and only if its set of encodings is weakly $b$-recognizable for all $b\ge2$. The class of weak deterministic Büchi automata is less expressive than the class of deterministic Büchi automata. For example, the language $L_{\mbox{inf }a}$ of words containing an infinite number of $a$ is recognized by a deterministic Büchi automaton but is not recognized by any weak deterministic Büchi automaton. This implies that, for example, no weak deterministic Büchi automaton recognizes the set of reals which are not of the form $nb^{p}$ with $n\in\mathbb N$ and $p\in\mathbb Z$, since those reals are the ones whose encoding in base $b$ contains an infinite number of non-$0$ digits. Furthermore, by \cite{minimal-buchi}, weak deterministic Büchi automata can be efficiently minimized. \paragraph{}
We now recall some results about the above-mentioned logic{}. By \cite{elimFOr}, the logic $\fo{\mathbb R;+,<,1}$ admits quantifier elimination. By \cite[Section 6]{elimination}, the set of reals which are $\fo{\mathbb R;+,<,1}$-definable are the finite union of intervals with rational bounds. Those sets are called the \emph{simple sets}. \subsubsection*{Main results} It is shown that ideas given in \cite{muchnik} and \cite{Honkala86} to create algorithms to decide properties of automata over integers can be adapted to decide properties of RVA. For examples, those ideas are used in Section \ref{sec:method} to give algorithms which decide whether a Büchi automaton recognizes a $\fo{\mathbb R;+,<,1}$-definable set of tuple of reals, a $\fo{\mathbb R,\mathbb Z;+,<}$-definable set of tuple of real or a subsemigroup of $(\mathbb R^{d},+)$ for some $d\in\mathbb N^{>0}$. However, those algorithms are inefficient.
It is then shown in Section \ref{sec:real-aut} that it is decidable in linear time whether a RNA recognizes a $\fo{\mathbb R;+,<,1}$-definable{} set, that is, a{} simple set. This algorithm does not return any false positive on weak deterministic Büchi automata which are not RNA. A false negative is also exhibited and it is explained why this case is more complicated than the case of RNA. A characterization of the minimal weak RNA which recognizes simple sets is also given.
Note that, if an automaton recognizes a{} simple set $R$, that is{} a finite union of intervals, the minimal number of intervals in the union is not polynomially bounded by the number of states of the automaton (this is shown in Example \ref{ex:A2-01}). It is shown in Section \ref{sec:aut->set} that an existential (respectively, existential-universal) $\fo{\mathbb R;+,<,1}$-formula which defines $R$ is computable in quasi-quadratic (respectively quasi-linear) time. \section{Definitions}\label{sec:def} The definitions used in this paper are given in this section. Some basic lemmas are also given. Most definitions are standard. \subsection{Basic Notations}\label{sec:bas} Let $\mathbb N$\index{N@$\mathbb N$}, $\mathbb Z$\index{Z@$\mathbb
Z$}, $\mathbb Q$ \index{Q@$\mathbb Q$} and $\mathbb R$ \index{R} denote the set of non-negative integers, integers, rationals and reals, respectively. For $R\subseteq \mathbb R$, let $R^{\ge 0}$ and $R^{>0}$ denote the set of non-negative and of positive elements of $R$, respectively. Let $\omega$ be the cardinality of $\mathbb N$. For $n\in\mathbb N$, \index{n@$\left[n\right]$ for $n\in\mathbb N$.} let $[n]$ represent $\set{0,\dots,n}$. For $a,b\in\mathbb R$ with $a\le b$, let $[a,b]$ denote the closed interval $\set{r\in\mathbb R\mid a\le r\le b}$, and let $(a,b)$ denote the open interval $\set{r\in\mathbb R\mid a<r<b}$. Similarly, let $(a,b]$ (respectively, $[a,b)$) be the half-open interval equals to the union of $(a,b)$ and of $\set b$ (respectively, $\set a$). For $r\in\mathbb R$ let \index{r@$\floor r$}$\floor r$ be the greatest integer less than or equal to $r$. \subsection{Finite and infinite words} An alphabet is a finite set, its elements are called letters. A finite (respectively infinite) word of alphabet $A$ is a finite (respectively infinite) sequence of letters of $A$. That is, a function from $[n]$ to $A$ for some $n\in\mathbb N$ (respectively from $\mathbb N$ to $A$). A set of finite (respectively infinite) word of alphabet $A$ is called a language (respectively, an $\omega$-language) of alphabet $A$. The empty word is denoted $\epsilon$.
Let $w$ be a word. Let $\length{w}\in\mathbb N\cup\set\omega$ denote the length of $w$. For $v$ a finite word, let $u=vw$ be the concatenation of $v$ and of $w$, that is, the word of length $\length{v}+\length{w}$ such that $u[i]=v[i]$ for $i<\length{v}$ and $u[\length{v}+i]=w[i]$ for $i<\length{w}$. For $n< \length{w}$, let $w[n]$ denote the $n$-th letter of $w$. Let $\prefix w{n}$ denote the \emph{prefix} of $w$ of length $n$, that is, the word $u$ of length $n$ such that $w[i]=u[i]$ for all $i\in[n-1]$. Similarly, let $\suffix w{n}$ denote the suffix of $w$ without its $n$-th first letters, that is, the word $u$ such that $u[i]=w[i+n]$ for all $i\in[n-n]$. Note that $w=\prefix wi\suffix wi$ for all $i<\length{w}$.
Let $L$ be a language of finite word and let $L'$ be either an $\omega$-languages or a language of finite words. Let $LL'$ be the set of concatenations of the words of $L$ and of $L$. For $i\in\mathbb N$, let $L^{i}$ be the concatenations of $i$ words of $L$. Let $L^{*}=\bigcup_{i\in\mathbb N} L^{i}$ and $L^{+}=\bigcup_{i\in\mathbb N^{>0}} L^{i}$. If $L$ is a language which does not contains the empty word, let $L^{\omega}$ be the set of infinite sequences of elements of $L$.
\subsubsection{Encoding of real numbers} Let us now consider the encoding of numbers in an integer base $b\ge2$. Let $\digitSet$ be equal to $[b-1]$, it is the set of digits and let $\digitDotSet=\digitSet\cup\set\star$. The base $b$ is fixed for the remaining of this paper. Two alphabets are considered in this paper: $\digitSet$ and $\digitDotSet$.
\index{.@$\wordToNumber{.}$}\label{def:wordToNumber}\index{Natural
part of a word of $\digitSet^{*}\star\digitSet^{\omega}$} \index{Fractional part of a word of
$\digitSet^{*}\star\digitSet^{\omega}$} Let $\wordToNumber{.}$ denote the function which sends a finite or infinite word of alphabet $\digitDotSet$ to the integer or to the real it represents. Formally, for $w\in\digitSet^{*}$:
\begin{equation*}
\wordToNatural{w}=\sum_{i=0}^{\length{w}-1}b^{\length{w}-1-i}w[i].
\end{equation*}
For $w\in\digitSet^{\omega}$,
\begin{equation*}
\wordToReal{w}=\sum_{i\in\mathbb N}b^{-i-1}w[i].
\end{equation*}
Let $w$ be an $\omega$-word with exactly one $\star$. It is of
the form $w=\natPart{w}\star \fraPart{w}$, with
$\natPart{w}\in\digitSet^{*}$ and
$\fraPart{w}\in\digitSet^{\omega}$. The word $\natPart{w}$ is called the
natural part of $w$ and the $\omega$-word $\fraPart{w}$ is called its
fractional part. Then :
\begin{equation*}
\wordToReal{\natPart{w}\star \fraPart{w}}=\wordToNatural{\natPart{w}}+\wordToFractional{\fraPart{w}}.
\end{equation*}
Finally, $\wordToReal{w}$ is undefined if $w$ contains at least two
letters $\star$.
There is no ambiguity in the definition of $\wordToNatural{\cdot}$
since the four domains of definitions partition
$\left(\digitDotSet\right)^{\omega}$. Note that
$\wordToReal{\natPart{w}}\in\mathbb N$, $\wordToReal{\fraPart{w}}\in[0,1]$ and
$\wordToFractional w=\wordToReal {\natPart{w}}+\wordToReal{\fraPart{w}}$.
Examples of numbers with their base $2$ encodings are now given. \begin{example}
\begin{equation*}
\arraycolsep=0.5pt
\begin{array}{rclp{2mm}rclp{2mm}rclp{2mm}rclp{2mm}rcl}
\wordToReal[2]{(10)^{\omega}}&=&\frac 23 &
&\wordToReal[2]{(01)^{\omega}}&=&\frac13&
&\wordToReal[2]{0(10)^{\omega}}&=&\frac13&
&\wordToReal[2]{0(1)^{\omega}}&=&\frac{1}{2}&
&\wordToReal[2]{1(0)^{\omega}}&=&\frac12
\\
\wordToReal[2]{10}&=&2&
&\wordToReal[2]{1}&=&1&
&\wordToReal[2]{01}&=&1&
&\wordToReal[2]{\epsilon}&=&0&
&\wordToReal[2]{00000}&=&0
\\
\multicolumn{5}{r}{\wordToReal[2]{10\star(10)^{\omega}}}&=&\frac{8}{3}&
&\wordToReal[2]{\realDot0(1)^{\omega}}&=&\frac{1}{2}&
&\multicolumn{5}{r}{\wordToReal[2]{00000\star 1(0)^{\omega}}}&=&\frac{1}{2}.
\end{array}
\end{equation*} \end{example} Some properties of concatenation and encodings of reals are now stated. The proof of the lemma is straightforward from the definition. \begin{lemma}
Note that for all $v\in\digitSet^{*}$, $w\in\digitSet^{\omega}$ and
$a\in\digitSet$:
\begin{equation*}
\begin{array}{rclcrclp{2mm}rcl}
\wordToReal{aw}&=&\frac{a+\wordToReal{w}}{b},&
&\wordToNatural{av}&=&ab^{\length v}+\wordToNatural{v},&
&\wordToReal{w}&=&\wordToReal{0\star w},\\
\wordToNatural{va}&=&b\wordToNatural{v}+a,
&\mbox{ and }
&\wordToReal{av\star w}&=&ab^{\length v}+\wordToReal{v\star w}.
\end{array}
\end{equation*} \end{lemma}
\subsubsection{Encoding of rationals}\label{sec:rati} In this section, some basic facts about rationals are recalled (see e.g. \cite{number-theory-hardy}). The rationals are exactly the numbers which admit encodings in base $b$ of the form $u\star{}vw^{\omega}$ with $u,v\in\digitSet^{*}$ and $w\in\digitSet^{+}$. Rationals of the form $nb^{p}$, with $n\in\mathbb N$ and $p\in\mathbb Z$, admit exactly two encodings in base $b$ without leading 0 in the natural part. If $p<0$, the two encodings are of the form $u\star{}va(b-1)^{\omega}$ and $u\star{}v(a+1)0^{\omega}$, with $u,v\in\digitSet^{*}$ and $a\in[b-2]$. Otherwise, if $p\ge 0$, the two encodings are of the form $ua(b-1)^{q}\star{}(b-1)^{\omega}$ and $u(a+1)0^{q}\star{}0^{\omega}$ with $u\in\digitSet^{*}$, $a\in[b-2]$ and $q\in\mathbb N$. The rationals which are not of the form $n b^{p}$ admit exactly one encoding in base $b$ without leading 0 in the natural part. \subsubsection{Encoding of sets of reals}\label{subsec:rep-set-real} In this section, relations between languages and set of reals are recalled.
Given a language $L$ in $\digitSet^{\omega}$ or in $\digitSet^{*}\star\digitSet^{\omega}$, let \index{$\wordToReal{L}$}$\wordToReal{L}$ be the set of reals admitting an encoding in base $b$ in $L$. The language $L$ is said to be an encoding in base $b$ of the set of reals $\wordToReal L$. Reciprocally, given a set $R\subseteq\mathbb R^{\ge0}$ of reals, $\setRealToLanguage R$ is the set of all encodings in base $b$ of the elements of $R$.
Following \cite{Leroux}, a language $L$ is said to be \emph{saturated}\index{Saturated} if for any number $r$ which admits an encoding in base $b$ in $L$, all encoding in base $b$ of $r$ belongs to $L$. The saturated languages are of the form $\setRealToLanguage R$ for $R\subseteq\mathbb R^{\ge0}$. Note that $\wordToReal{\setRealToLanguage{R}}=R$ for all sets $R\subseteq\mathbb R^{\ge0}$. Note also that $L\subseteq\setRealToLanguage{\wordToReal{L}}$, and the subset relation is an equality if and only if $L$ is saturated. In general, a set of reals may have infinitely many encodings in base $b$. For example, for $I\subseteq\mathbb N$ an arbitrary set, $\set{0,1}^{\omega}\setminus\set{0^{i}1^{\omega}\mid i\in I}$ is an encoding in base 2 of the language of the simple set $[0,1]$. An example of set of reals is now given. \begin{example}
Let $L=\digitSet[2]^{*}0^{\omega}$ and
$L'=\digitSet[2]^{*}0\digitSet[2]^{*}(0^{\omega}+1^{\omega})$. Both
$\wordToReal[2]{L}$ and
$\wordToReal[2]{L'}$ are
$R=\set{\frac{n}{2^{p}}\mid{}n,p\in\mathbb N,n<2^{p}}$, but only $L'$ is
saturated. Therefore $\setRealToLanguage[2]{R}=L'$. \end{example}
\subsection{Deterministic Büchi automata} This paper deals with {Deterministic }Büchi automata. This notion is now defined.
A \emph{{Deterministic }Büchi automaton}\index{{Deterministic }Büchi
automaton} is a 5-tuple $ \mathcal{A}=\autPar{Q}{A}{\delta}{q_0}{F} $, where $Q$ is a finite \emph{set of states}, $A$ is an alphabet, $\delta\subseteq Q\timesA\times Q$ is the \emph{transition
relation}, ${q_0\in Q}$ is the {\emph{initial states}} and $F\subseteq Q$ is the set of \emph{accepting states}. A state belonging to $Q\setminus F$ is said to be a \emph{rejecting
state}.\index{Rejecting state}
An example of deterministic Büchi automaton is now given. This example is used thorough this paper to illustrate properties of Büchi automaton reading set of real numbers. \begin{example}\label{ex:unbounded}
Let $R=\left(\frac13,2\right]\cup\left(\frac83,3\right]\cup\left(\frac{11}3,\infty\right]$. The set of encodings in base 2 of reals of $R$ is
recognized by the automaton pictured in Figure \ref{fig:ex-AR-R>0}.
\begin{figure}
\caption{Automaton $\ARPar{R}$ of Example \ref{ex:unbounded}}
\label{fig:ex-AR-R>0}
\end{figure} \end{example}
{From now on in this paper, all automata are assumed to be
deterministic. The function $\delta$ is implicitly extended on
$Q\times A^{*}$ by $\del{q}{\epsilon}=q$ and
$\del{q}{aw}=\del{\del{q}{a}}{w}$ for $a\inA$ and
$w\inA^{*}$. }
\index{Run of an automaton} Let $\mathcal{A}$ be an automaton and $w$ be an infinite word. A \emph{run}\index{Run} $\pi$ of $\mathcal{A}$ on $w$ is a mapping $\pi:\mathbb N\mapsto Q$ such that $\pi(0)=q_{0}$ and $\del{\pi(i)}{w[i]}={\pi(i+1)}$ for all $i<\length{w}$. Let $inf(\pi)$ be the set of states of $Q$ that occur infinitely often in the run $\pi$. A run $\pi$ on an $\omega$-word is said to be accepting if $inf(\pi)\cap F\ne\emptyset$. Equivalently, the run is accepting if there exists a state $q\in F$ such that there is an infinite number of $i\in\mathbb N$ such that $\pi(i)=q$.
Example \ref{ex:unbounded} is now resumed. \begin{example}
Let $\mathcal{A}$ be the automaton pictured in Figure \ref{fig:ex-AR-R>0}. The run of $\mathcal{A}$ on
$011\star(10)^{\omega}$ is \begin{equation*}
\left(q_{0},q_{0},q_{1},q_{3},(3,\epsilon),(3,1),(3,10),\dots\right)
\end{equation*}
with the two last states repeated infinitely often. The Büchi
automaton $\mathcal{A}$ does not accept $011\star(10)^{\omega}$ since
this run does not contain any accepting state.
The run of $\mathcal{A}$ on $\realDot1^{\omega}$ is $
\left(q_{0},(0,\epsilon),\zuState,\dots\right)$
with the last state repeated infinitely often. The Büchi automaton
$\mathcal{A}$ accepts $\realDot1^{\omega}$ since the accepting state
$\zuState$ appears infinitely often in the run. \end{example}
Let $\mathcal{A}$ be a finite automaton. Let \index{A@$\toInfWord{\mathcal{A}}$}$\toInfWord{\mathcal{A}}$ be the set of infinite words $w$ such that a run of $\mathcal{A}$ on $w$ is accepting. An $\omega$-language is said to be \emph{recognizable}\index{Recognizable
language} if it is recognized by a Büchi automaton. Example \ref{ex:unbounded} is now resumed. \begin{example}
Let $\mathcal{A}$ be the Büchi automaton pictured in Figure \ref{fig:ex-AR-R>0}. It recognizes the language of encodings in
base 2 of the reals of $\left(\frac13,2\right]\cup\left(\frac83,3\right]\cup\left(\frac{11}3,\infty\right]$. It is explained in Example
\ref{ex:construct-RR} how this automaton was computed. \end{example} For $q\in Q$, let $\mathcal{A}_{q}$ \index{A@$\mathcal{A}_{q}$ for
$\mathcal{A}$ an automaton and $q$ a state.} be $\left(Q_{q},A,\delta,q,F_{q}\right)$, where $Q_{q}$\index{Q@$Q_{q}$ for $Q$ a set of states and $q$ a state.} is the set of states of $Q$ accessible from $q$, and $F_{q}=F\cap Q_{q}$.
Note that, if there are no finite word $w$ such that $\del{q_{0}}{w}=q_{0}$, then $Q_{q}\subsetneq Q$ for all $q\ne q_{0}$. Note also that, if $w\inA^{*}$ is such that $\del{q_{0}}{w}=q$ then a word $w'\inA^{\omega}$ is accepted by $\mathcal{A}_{q}$ if and only if $ww'$ is accepted by $\mathcal{A}$.
\subsubsection{Accessibility and recurrent states} When the notions of initial and of accepting states are ignored, an automaton can be considered as a directed labelled graph. Some definitions related to this graph are introduced in this section.
A state $q$ is said to be \emph{accessible}\index{Accessible} from a state $q'$ if there exists a finite non-empty word $w$ such that $\del{q'}{w}={q}$. Following \cite{minimal-buchi}, a state $q$ is said to be \emph{recurrent}\index{Recurrent state} if it is accessible from itself and \emph{transient} otherwise\index{Transient
state}. Transient states are called \emph{trivial} in \cite{Boigelot2007}. The \index{Strongly connected
components}\emph{strongly connected component} of a recurrent state $q$ is the set of states $q'$ such that $q'$ is accessible from $q$ and $q$ is accessible from $q'$. A strongly connected component $C$ is said to be a \index{Leaf}leaf if for all $a\inA$, for all $q\in C$, $\del{q}{a}\in C$. Let $C$ be a strongly connected component. It is said to be a cycle if for each $q\in C$, there exists a unique $\suc{q}\inA$ such that $\del{q}{\suc{q}}\in C$. Example \ref{ex:unbounded} is now resumed. \begin{example}\label{ex:AR-R>0-min}
The transient states of the automaton pictured in Figure
\ref{fig:ex-AR-R>0-min} are $q_{0}$, $q_{1}$, $q_{2}$, $q_{3}$,
$(2,\epsilon)$ and $(3,\epsilon)$. All other states are
recurrent. The cycles are $\set{q_{0}}$, $\set{(0,\epsilon),(0,0)}$,
$\set{(2,0)}$, $\set{(2,1), (2,10)}$, $\set{(3,0)}$ and
$\set{(3,1), (3,10)}$. The strongly connected component which are
not cycles are $\emptyState$, $\inftyState$ and $\zuState$. \end{example}
The following lemma allows to consider recurrent states in any run which is long-enough. \begin{lemma}\label{lem:run-recurrent}
Let $\mathcal{A}$ be a Büchi automaton with $n$ states let $w$ be an
$\omega$-word and let $\pi$ be the run of $\mathcal{A}$ on $w$. Let
$N\subseteq \mathbb N$ be a set of cardinal at least $(n+1)$. Then there is
$i<i'$ belonging to $N$ such that $\pi(i)=\pi(i')$ is a recurrent
state. \end{lemma} \begin{proof}
Since the cardinality of $N$ is greater than the number of state, by
the pigeonhole principle, there exists $i<i'$ belonging to $N$ such
that $\pi(i)=\pi(i')$. Let $w'$ be the factor of $w$ containing the
letters $i+1$ to $i'$, then $\del{\pi(i)}{w'}=\pi(i')=\pi(i)$,
therefore, the state $\pi(i)$ is recurrent, with $i$ belonging to
$N$. \end{proof} \subsubsection{Quotients, Morphisms and Weak Büchi Automata} In this section, the notion of quotient of automata and of morphism of automata are introduced. A class of automata admitting minimal quotient is then introduced.
\begin{definition}[Morphism of Büchi automata,
Quotient]\label{def:morph}\index{Morphism of
automata}\index{Quotient of automata} Let $\mathcal{A}=\autPar QA\delta{q_{0}}F$ and
$\mathcal{A}'=\autPar{Q'}{A}{\delta'}{q_{0}'}{F'}$ be
two Büchi automata over the same alphabet. A surjective function
$\mu:Q\to Q'$ is a \emph{morphism}\index{Morphism of automata} of
Büchi automata if and only if:
\begin{enumerate}
\item $\mu(q_{0})=q'_{0}$,
\item for each $q\in Q$,
$\toInfWord{\mathcal{A}_{q}}\ne\toInfWord{\mathcal{A}_{q'}}$.
\end{enumerate}
The Büchi automaton $\mathcal{A}'$ is said to be a \emph{quotient}
of $\mathcal{A}$ if there exists a morphism from $\mathcal{A}$ to
$\mathcal{A}'$. \end{definition} The notion of minimal Büchi automaton is now introduced. \begin{definition}[Minimal Büchi automaton]\label{Minimal Büchi automaton}
Let $\mathcal{A}=\autPar{Q}{\digitDotSet}{\delta}{q_{0}}{F}$ be
a Büchi automaton. It is said to be minimal if for each distinct
states $q,q'\in Q$,
$\toInfWord{\mathcal A_{q}}=\toInfWord{\mathcal A_{q'}}$. \end{definition} In general, Büchi automata does not admit minimal quotient. A class of Büchi automata admitting minimal quotient is now introduced. \begin{definition}[Weak automata]
Let $\mathcal{A}=\autPar{Q}{\digitDotSet}{\delta}{q_{0}}{F}$ be
a Büchi automaton. It is said to be \emph{weak} if for each
recurrent accepting state $q$ of $\mathcal{A}$, all states of the
strongly connected components of $q$ are accepting.
An $\omega$-language is said to be \emph{weakly
recognizable}\index{Weakly recognizable language} if it is
recognized by a weak Büchi automaton. \end{definition}
The main theorem concerning quotient of weak Büchi automata is now recalled. \begin{theorem}[\cite{minimal-buchi}]
Let $\mathcal{A}$ be a weak Büchi automaton with $n$ states such
that all states of $\mathcal{A}$ are accessible from its initial
state. Let $c$ be the cardinality of $A$. There exists a
minimal weak Büchi automaton $\mathcal{A}'$ such that there exists a
morphism of automaton $\mu$ from $\mathcal{A}$ to
$\mathcal{A}'$. The automaton $\mathcal{A}'$ and the morphism $\mu$
are computable in time $\bigO{n\log(n)c}$ and space $\bigO{nc}$. \end{theorem} It follows easily from Property \eqref{def:morph-trans} that, for all $w\inA^{*}$, $\dell{\mu(q)}{w}=\mu(\del{q}{w})$. Example \ref{ex:unbounded} is now resumed. \begin{example}\label{ex:unbounded-min}
Let $\ARPar{R}$ be the automaton pictured in Figure \ref{fig:ex-AR-R>0}. Its
minimal quotient is pictured in Figure \ref{fig:ex-AR-R>0-min}.
\begin{figure}
\caption{Minimal quotient of automaton $\ARPar{R}$ of
Figure \ref{fig:ex-AR-R>0}}
\label{fig:ex-AR-R>0-min}
\end{figure} \end{example}
The following lemma shows that each strongly connected component of a quotient by a morphism $\mu$ from an automaton $\mathcal{A}$ is the image of a strongly connected component of $\mathcal{A}$. \begin{lemma}\label{lem:scc-morphism-codomain}
Let $\mathcal{A}=\autPar QA\delta{q_{0}}F$ and
$\mathcal{A}'=\autPar{Q'}{\digitSet}{\delta'}{q'_{0}}{F'}$ be two Büchi
automata. Let $\mu$ be a morphism from $\mathcal{A}$ to $\mathcal{A}'$. Let $C'$
be a strongly connected component of $\mathcal{A}'$. There exists a
strongly connected component $C\subseteq Q$ such that $\mu(C)=C'$
and such that, for all $q\in Q\setminus C$ accessible from $C$,
$\mu(q)\not\in C'$. \end{lemma} In order to prove this lemma, two other lemmas are required. \begin{lemma}\label{lem:scc-morphism-codomain-inclusion}
Let $\mathcal{A}$, $\mathcal{A}'$, $C'$ and $\mu$ as in
Lemma \ref{lem:scc-morphism-codomain}. Let $C$ be a strongly connected
component of $\mathcal{A}$. Either $\mu(C)\cap C'=\emptyset$ or $\mu(C)\subseteq C'$. \end{lemma} \begin{proof}
Let us assume that $\mu(C)\cap C'\ne\emptyset$ and let us prove that
$\mu(C)\subseteq C'$. That is, let $q\in C$ and let us prove that
$\mu(q)\in C'$.
Since $\mu(C)\cap C'\ne\emptyset$, there exists
$q'\in\mu(C)\cap C'$. Since $q'\in\mu(C)$, there exists $p\in C$
such that $\mu(p)=q'$. Since $p$ and $q$ belong to the same
strongly connected component, there exists two non-empty finite
words $v$ and $w$ such that $\del{p}{v}=q$ and
$\del{q}{w}=p$. Therefore $\dell{\mu(p)}{v}=\mu(\del{p}{v})=\mu(q)$
and $\dell{\mu(q)}{w}=\mu(\del{q}{w})=\mu(q)$. Therefore $\mu(q')$
is accessible from $\mu(q)$ and $\mu(q')$ is accessible from
$\mu(q)$. Hence $\mu(q)$ belongs to the strongly connected component
of $p$. That is, $\mu(q)$ belongs to $C'$. \end{proof} \begin{lemma}\label{lem:scc-morphism-codomain-follow}
Let $\mathcal{A}$, $\mathcal{A}'$, $C'$ and $\mu$ as in
Lemma \ref{lem:scc-morphism-codomain}. Let $q\in Q$ such that
$\mu(q)\in C'$. There exists a strongly connected component $C$ of
$\mathcal{A}$, accessible from $q$, such that $\mu(C)\subseteq C'$. \end{lemma} \begin{proof}
Since $\mu(q)\in C'$, the state $\mu(q)$ is recurrent, therefore
there exists a non-empty word $w$ such that
$\dell{\mu(q)}{w}=\mu(q)$. Let us prove by induction on $i\in\mathbb N$
that $\mu(\del{q}{w^{i}})=\mu(q)$. The case $i=0$ is trivial, let us
assume that the hypothesis holds for $i\in\mathbb N$ and let us prove that
the induction hypothesis holds for $i+1$. It suffices to see that
\begin{equation*}
\mu(\del{q}{w^{i+1}})=
\dell{\mu(q)}{w^{i+1}}=
\dell{\dell{\mu(q)}{w^{i}}}{w}=
\dell{\mu(q)}{w}=
\mu(q)
\end{equation*}
By Lemma \ref{lem:run-recurrent}, there exists $i\in\mathbb N$ such that
$\del{q}{w^{i}}$ is recurrent. Let $C$ be the strongly connected of
$\del{q}{w^{i}}$. Since $\del{q}{w^{i}}\in C$ and
$\mu(\del{q}{w^{i}})\in C'$, by
Lemma \ref{lem:scc-morphism-codomain-inclusion}, it implies that
$\mu(C)\subseteq C'$. Since $C$ is accessible from $q$ and
$\mu(C)\subseteq C'$, the lemma is satisfied. \end{proof} Lemma \ref{lem:scc-morphism-codomain} is now proven. \begin{proof}[Proof of Lemma \ref{lem:scc-morphism-codomain}]
Let $\mu^{-1}(C')\subseteq Q$ be the set of states $q$ such that
$\mu(q)\in C'$. By definition of morphism, $\mu$ is surjective,
hence $\mu^{-1}(C')$ is not empty. Let $q$ be a state belonging to
$\mu^{-1}(C')$. By Lemma \ref{lem:scc-morphism-codomain-follow}, it
implies that there exists a strongly connected component
$C\subseteq\mu^{-1}(C')$. Since $\mu^{-1}(C')$ is finite, there
exists a strongly connected component $C$ such that no other
strongly connected component of $\mu^{-1}(C')$ is accessible from
$C$. By Lemma \ref{lem:scc-morphism-codomain-follow} it implies that
no state of $\mu^{-1}(C')\setminus C$ is accessible from $C$.
Let us prove that $\mu(C)=C'$. Since, by hypothesis
$\mu(C)\subseteq C'$, it remains to prove that $\mu(C)\supseteq C'$.
Let $q'\in C'$ and let us prove that $q'\in\mu(C)$. Let $q\in C$.
By hypothesis, $\mu(C)\subseteq C'$, therefore $\mu(q)\in C'$. Since
$\mu(q)$ and $q'$ belong to the same strongly connected component
$C'$, there exists a finite word $w$ such that
$\dell{\mu(q)}{w}=q'$. Then
$\mu(\del{q}{w})=\dell{\mu(q)}{w}=q'$. Since
$\mu(\del{q}{w})=q'\in C'$, $\del{q}{w}\in\mu^{-1}(C')$. Since
$\del{q}{w}\in \mu^{-1}(C')$ and $\del{q}{w}$ is accessible from
$q\in C$, by hypothesis on $C$, $\del{q}{w}\in C$. Therefore
$q'=\mu(\del{q}{w})\in\mu(C)$. \end{proof}
\subsection{Logic} The logic{} $\fo{\mathbb R;+,<,1}${} used in this paper {is} introduced in this section. Note that, in order to avoid ambiguity between the mathematical equality and the formal equality of the logic, the symbol $\doteq$ is used in first-order formulas.
Intuitively, $\FO$ stands for first-order. The first parameter $\mathbb R$ means that the (free or quantified) variables are interpreted by real numbers. The $+$ and $<$ symbols mean that the function addition and the binary order relation over reals can be used in formulas. Finally, the last term, $1$, means that the only constant is 1. The logic $\fo{\mathbb R;+,<,1}$ is denoted by $\mathscr{L}$ in \cite{elimFOr}, where it is proved that this logic admits quantifier elimination. In this paper, most results deal with the quantifier-free, the existential fragment and the existential-universal fragment of $\fo{\mathbb R;+,<,1}$ denoted by $\qf{\mathbb R;+,<,1}$, $\ef{\mathbb R;+,<,1}$ and $\sigF{2}{\mathbb R;+,<,1}$ respectively.
In the remaining of the paper, rationals are also used in the formulas. It does not change the expressivity, as all rational constants are $\qf{\mathbb R;+,1}$-definable. Let $\phi\in\fo{\mathbb R;+,<,1}$. The length of $\phi$, denoted by $\length{\phi}$\index{Length of formulas}, is recursively defined as follows: \begin{itemize} \item The lengths of the constant \emph{$\frac{p}{q}$} is
$\log{p+1}+\log{q}$. \item The length of a sum $t_{1}+t_{2}$ is
$1+\length{t_{1}}+\length{t_{2}}$. \item The length of a multiplication by a rational constant
$\frac pqt$ is $\length{\frac pq}+\length{t}$. \item The length of an (in)equality is the sum of the length of the
terms on both side, plus one, that is
$\length{t_{1}<t_{2}}=\length{t_{1}\doteq
t_{2}}=1+\length{t_{0}}+\length{t_{1}}$. \item The length of Boolean combination and of quantification are 1
plus the length of its subterms, that is
$\length{\phi\lor\psi}=\length{\phi\land\psi}=1+\length{\phi}+\length{\psi}$
and
$\length{\exists x.\phi}=\length{\forall
x.\phi}=\length{\neg\phi}=1+\length{\phi}$.
\end{itemize} \subsubsection{First-order definable sets of reals} In this section, notations are introduced for the{} kind{} of sets studied in this paper: the $\fo{\mathbb R;+,<,1}$-definable{} sets.
\label{sec:FU}Following \cite[Section 6]{elimination}, the $\fo{\mathbb R;+,<,1}$-definable sets are called the \index{Simple
sets}\emph{simple sets}. By \cite[Section 6]{elimination}, those sets are the finite union of intervals with rational bounds. It implies that there exists an integer $t_{R}$ such that for all $x,y\ge t_{R}$, $x$ belongs to $R$ if and only if $y$ belongs to $R$. \index{Threshold of a simple set.} The least such integer $t_{R}$ is called the \emph{threshold of $R$}.
Note that every closed and half-closed intervals is the union of an open interval and of singletons, hence it can be assumed that any simple set $R$ is of the form \begin{equation*}
R=\bigcup_{i=0}^{I-1}(\rho_{i,\mathfrak L},\rho_{i,\mathfrak R})\cup\bigcup_{i=0}^{J-1}\set{\rho_{i,\mathfrak S}}, \end{equation*} with $\rho_{i,\mathfrak L},\rho_{i,\mathfrak S}\in\mathbb Q^{\ge0}$ and $\rho_{i,\mathfrak R}\in\mathbb Q^{\ge0}\cup\set\infty$. The $\rho_{i,\mathfrak L}$ are the left bound, the $\rho_{i,\mathfrak R}$ are the right bound and the $\rho_{i,\mathfrak S}$ are the singletons. Without loss of generality, it is assumed that the intervals are disjoint and in increasing order.
Example \ref{ex:unbounded} is now resumed. \begin{example}\label{ex:decom-R-01}
Let $R=\left(\frac13,2\right]\cup\left(\frac83,3\right]\cup\left(\frac{11}3,\infty\right]$ as in Example \ref{ex:unbounded}. Then $t_{R}$ is 4,
$R_{0}=\left(\frac13,1\right]$, $R_{1}=\left[0,1\right]$, and
$R_{2}=R_{3}=\set{0}\cup\left[\frac23,1\right]$. Furthermore, $I=3$,
$J=2$, $\rho_{1,\mathfrak L}=\frac13$, $\rho_{2,\mathfrak R}=2$,
$\rho_{2,\mathfrak{L}}=\frac83$, $\rho_{2,\mathfrak{R}}=3$,
$\rho_{3,\mathfrak{L}}=\frac{11}3$, $\rho_{3,\mathfrak{R}}=\infty$,
$\rho_{1,\mathfrak{S}}=2$ and $\rho_{2,\mathfrak{S}}=3$. \end{example}
\section{Automata reading reals}\label{sec:aut-numb} Automata recognizing encoding of set of reals are considered in this section. The notion of Real Number Automata and of Fractional Number Automata are introduced in Section \ref{sec:RNA-FNA}. Some sets of states of the automata reading encoding of set of reals are considered in Section \ref{sec:setOfState}.
\subsection{Real and Fractional Number Automata}\label{sec:RNA-FNA} In this section, the automata reading saturated languages are considered.
Following \cite{real}, a Büchi automaton of alphabet $\star$ is said to be a Real Number Automaton (RNA) if \index{Real Number
Automaton}\index{RNA- Real Number Automaton}\begin{itemize} \item all words accepted by $\mathcal{A}$ contains exactly one $\star$,
and \item the language $\toInfWord{\mathcal A}$ is saturated. \end{itemize} The Büchi automata pictured in \ref{fig:ex-AR-R>0} and \ref{fig:ex-AR-R>0-min} are RNA. Clearly, the RNAs are the Büchi automata which recognizes saturated languages of $\digitSet^{*}\star\digitSet^{\omega}$. Similarly, the name of Fractional Number Automata (FNA) is given to the Büchi automata of alphabet $\digitSet$ recognizing a saturated language. \index{FNA -
Fractional Number Automaton}\index{Fractional Number Automaton}
A weak {}Büchi automaton which is a RNA or a FNA is said to be a \emph{weak RNA} or a \emph{weak FNA} respectively. An example of FNA is now given. This example shows that the number of intervals required to describe a set is not polynomially bounded by the number of states of the automaton recognizing this set. \begin{example}\label{ex:A2-01}
For every non-negative integer $n$, let $R_{n}$ be
$\set{m2^{-n}\mid m\in[2^{n}]}$. It is the set of reals which admit
an encoding $w$ in base $b$ whose suffixes $\suffix{w}{n}$ are either
equal to $0^{\omega}$ or to $1^{\omega}$. This set can not be
described with less than $2^{n-1}$ intervals and is recognized by
the automaton $\mathcal{A}_{n}$ with $n+3$ states:
\begin{equation*}
\mathcal{A}_{n}=\autPar{\set{q_{i}\mid i\in[n]}\cup\set{q_{n+1,0},q_{n+1,1},\emptyState}}{\digitSet}{\delta}{q_{0}}{\set{q_{n+1,0},q_{n+1,1}}},
\end{equation*}
where the transition function is such that, for $a\in\digitSet[2]$:
\begin{equation*}
\arraycolsep=0.5pt
\begin{array}{rclp{2mm}rclcrcl}
\del{q_{i}}{a}&=&\multicolumn{5}{l}{q_{i+1}\mbox{ for }i\in[n-1],}&
&\del{q_{n}}{a}&=&q_{n+1,a},\\
\del{q_{n+1,a}}{a}&=&q_{n+1,a}&
&\del{q_{n+1,a}}{1-a}&=&\emptyState&\mbox{ and }
&\del{\emptyState}{a}&=&\emptyState.
\end{array}
\end{equation*}
The automaton $\mathcal{A}_{3}$ is pictured in Figure \ref{fig:A2-01}, without
the state $\emptyState$.
\begin{figure}\label{fig:A2-01}
\end{figure} \end{example} \index{$\wordToReal{\mathcal{A}}$} For $\mathcal{A}$ a Büchi automaton of alphabet $\digitSet$ (respectively, $\digitDotSet$). Let $\wordToReal{\mathcal{A}}=\wordToReal{\toInfWord{\mathcal{A}}}$. It is a subset of $[0,1]$ (respectively, $\mathbb R^{\ge0}$). It is said that $\mathcal{A}$ recognizes $\wordToReal{\mathcal{A}}$.
It should be noted that two distinct minimal weak Büchi automata may recognizes the same set of reals. Indeed, they may recognize two distinct language which are two encoding of the same set of reals. At least one of those languages is not saturated. Note however that two distinct minimal RNA or RFA accepts distinct sets of reals.
\subsection{Some sets of states of RNA and of FNA}\label{sec:setOfState} Five sets of states of Büchi automata are of used through this paper. Those sets are introduced and studied in this section.
\begin{definition}[$\emptyStates$, $\zuStates$,
$\inftyStates$, $\natStates$ and $\fraStates$]\label{not:q-empty-01}
Let $\mathcal{A}$ be an automaton over alphabet $\digitDotSet$ or
$\digitSet$.
\begin{itemize}
\item Let $\emptyStates$ \index{QO@$\emptyStates$} be the set of states
$q$ such that $\mathcal{A}_{q}$ recognizes the empty language.
\item Let $\zuStates$ \index{Q01@$\zuStates$} be the set of states
$q$ such that $\mathcal{A}_{q}$ recognizes
$\digitSet^{\omega}=\setRealToLanguage{[0,1]}$.
\item Let $\inftyStates$ \index{Qinfty@$\inftyStates$} be the set of
states $q$ such that $\mathcal{A}_{q}$ recognizes the language
$\digitSet^{*}\star\digitSet^{\omega}=\setRealToLanguage{[0,\infty)}$.
\item Let $\natStates$\index{Qnat@$\natStates$} be the set of states
$q$ such that $\mathcal{A}_{q}$ recognizes a subset of
$\digitSet^{*}\star\digitSet^{\omega}$.
\item Let $\fraStates$\index{Qfra@$\fraStates$} be the set of states
$q$ such that $\mathcal{A}_{q}$ recognizes a subset of
$\digitSet^{\omega}$.
\end{itemize} \end{definition} Example \ref{ex:unbounded-min} is now resumed. \begin{example}
Let $\mathcal{A}$ be the automaton pictured in
Figure \ref{fig:ex-AR-R>0-min}. Let $\emptyState$ be the state
$\del{(2,0)}{1}$, which is not pictured in
Figure \ref{fig:ex-AR-R>0-min}. Then $\zuStates=\set{\zuState[R]}$,
$\inftyStates=\set{\inftyState[R]}$ and
$\emptyStates=\set{\emptyState}$. Furthermore,
$\natStates=\set{q_{0},
q_{1},q_{2},\inftyState[R],\emptyState[R]}$,
its elements are represented in the top row, of
Figure \ref{fig:ex-AR-R>0-min}. Finally,
$\fraStates=\set{(2,\epsilon), (2,0), (0,\epsilon), (0,0),
\zuStates[R],\emptyState[R]}$.
Its elements are pictured in the second row of
Figure \ref{fig:ex-AR-R>0-min}. \end{example}
The following lemma is straightforward from the definition. \begin{lemma}\label{lem:min-empty-infty}
In a minimal weak {}Büchi automaton, the
sets $\emptyStates$, $\zuStates$ and $\inftyStates$ are either
singletons or the empty set. \end{lemma} In a minimal weak {}Büchi automaton $\mathcal{A}$, let $\emptyState$, \index{$\emptyState$} $\zuState$\index{$\zuState$} and $\inftyState$\index{$\inftyState$} denote the only state $q$ such that $\mathcal{A}_{q}$ recognizes the languages $\emptyset$, $\digitSet^{\omega}$ and $\digitSet^{*}\star\digitSet^{\omega}$ respectively. In an automaton of alphabet $\digitSet$, all states belongs to $\fraStates$.
The following lemma states that those five sets are linear time computable. \begin{lemma}\label{lem:comp-set}
Let $\mathcal{A}$ be a Büchi automaton{} with $n$ states{}. Then the sets $\emptyStates$, $\natStates$ and
$\fraStates$ are computable in time $\bigO{n{b}}$. If
$\mathcal{A}$ is weak{}, the sets $\zuStates$ and
$\inftyStates$ are computable in time $\bigO{nb}$. \end{lemma} \begin{proof}
Tarjan's algorithm \cite{Tarjan} can be used to compute the set of
strongly connected component in time $\bigO{{nb}}$, and therefore
the set of recurrent states. Furthermore, it is easy to associate in
linear time to each state its set of predecessors. Let $p_{q}$ be
the number of predecessors of a state $q$.
Let us first explain how to compute the set $\emptyStates$. Note
that $Q\setminus\emptyStates$ is the set of states $q$ such that
$\mathcal{A}_{q}$ accepts some $\omega$-word. Hence
$Q\setminus\emptyStates$ is the smallest set containing all
accepting recurrent states and is closed under taking
predecessors. Therefore $\emptyStates$ is the greatest set which
does not contain the accepting recurrent states and is closed under
taking successor. It can thus be computed by a fixed-point
algorithm. The algorithm is now given.
Two sets $S$ and $S'$ are used by the algorithm. The set $S$
represents $\emptyStates$. The set $S'$ is the set of states of
$\emptyStates$ which must be processed by the fixed-point algorithm.
The algorithm initializes the set $S$ to $Q$ and initializes the set
$S'$ to the empty set. The algorithm runs on each recurrent state
$q$. For each state $q$, if $q$ is accepting, then $q$ is removed
from $S$ and added to $S'$. The algorithm then runs on each element
$q$ of $S'$. For each state $q$, the algorithms removes $q$ from
$S'$ and runs on each predecessors $q'$ of $q$. For each $q'$, if
$q'$ is in $S$, then $q'$ is removed from $S$ and added to
$S'$. Finally, when $S'$ is empty, the algorithm halts and
$\emptyStates$ is the value of $S$.
Let us now consider the computation time of this algorithm. At most
$n$ states are added to $S'$, and each state is added at most
once. For each state $q$ added to $S'$, each of its $c_{q}$
predecessor is considered in constant time. Thus the algorithm runs
in time $\bigO{n+ \sum_{q\in Q}c_{q}}=\bigO{nb}$.
\paragraph{}It is now explained how to compute $\fraStates$ and
$\natStates$. Let $Q_{0}$, $Q_{1}$ and $Q_{2}$ be the set of states
accepting a words with at least 0, 1 and 2 $\star$'s
respectively. Then $\fraStates$ is equal to
$(Q_0\setminus{}Q_{1})\cup\emptyStates$ and $\natStates$ is equal to
$(Q_{1}\setminus{}Q_{2})\cup\emptyStates$. Let us now explain how
to compute the sets $Q_{0}$, $Q_{1}$ and $Q_{2}$. The set $Q_0$ is
the smallest set containing all accepting recurrent states and
closed under taking predecessors. The state $Q_{1}$ is the smallest
set containing the predecessors of $Q_0$ by the letter $\star$
and closed under taking predecessors. Similarly, the set $Q_{2}$ is
the smallest set containing the predecessors of $Q_1$ by the letter
$\star$ and closed under taking predecessors. Those three sets
are computable by a fixpoint algorithm similar to the one computing
$\emptyStates$. It is thus computable in time $\bigO{{nb}}$.
\paragraph{}
Let us now assume that the Büchi automaton $\mathcal{A}$ is weak{}. It is now explained how to compute $\zuStates$
and $\inftyStates$. Note that $\zuStates\subseteq\fraStates$ and
that $\fraStates\setminus\zuStates$ is the set of states
$q\in\fraStates$ such that there exists an $\omega$-word
$w\in\digitSet^{\omega}$ which is not accepted by
$\mathcal{A}_{q}$. Therefore, $\fraStates\setminus\zuStates$ is the
smallest subset of $\fraStates$ containing non-accepting recurrent
state and closed under taking predecessors by $\digitSet$.
Similarly, note that $\inftyStates\subseteq\natStates$ and that
$\natStates\subseteq\inftyStates$ is the set of states $q$ such that
there is an infinite word of the form
$\digitSet^{*}\star\digitSet^{\omega}$ which is not accepted by
$\mathcal{A}_{q}$. Therefore $\natStates\subseteq\inftyStates$ is the
smallest subset of $\natStates$ containing the predecessors of
$\fraStates\setminus\zuStates$ by $\star$ and closed under taking
predecessors by $\digitSet$. The computation of $\natStates$ and of
$\inftyStates$ is thus similar to the computation of $\emptyStates$. \end{proof} This lemma admits the following corollary. \begin{corollary}\label{cor:correct-language}
It is decidable in time $\bigO{nb}$ whether a Büchi automaton with
$n$ states recognizes a subset of
$\digitSet^{*}\star\digitSet^{\omega}$ or of
$\digitSet^{\omega}$. \end{corollary} \begin{proof}
By definition of $\natStates$ (respectively $\fraStates$), the
automaton recognizes a subset of
$\digitSet^{*}\star\digitSet^{\omega}$ (respectively of
$\digitSet^{\omega}$) if and only if its initial state belongs to
$\natStates$ (respectively $\fraStates$). By
Lemma \ref{lem:comp-set}, it is testable in time $\bigO{bn}$. \end{proof}
The following lemma gives a relation between the set of states introduced in Example \ref{not:q-empty-01} and morphisms of automata. \begin{lemma}\label{lem:morphism-sets}
Let $\mathcal{A}=\autPar QA\delta{q_{0}}F$ and
$\mathcal{A}'=\autPar{Q'}{\digitSet}{\delta'}{q'_{0}}{F'}$ be two Büchi
automata. Let $\mu:Q\to Q'$ be a morphism of Büchi automaton. Then
$\mu(\emptyStates)= \emptyStates[\mathcal{A}']$,
$\mu(\fraStates)= \fraStates[\mathcal{A}']$,
$\mu(\inftyStates)= \inftyStates[\mathcal{A}']$,
$\mu(\natStates)= \natStates[\mathcal{A}']$, and
$\mu(\zuStates)= \zuStates[\mathcal{A}']$. \end{lemma} \begin{proof}
The proof is done for the first equality:
$\mu(\emptyStates)= \emptyStates[\mathcal{A}']$. All other cases are
similar. Let $q'\in Q'$, and let us prove that
$q'\in\mu(\emptyStates)\iff{q'\in\emptyStates[\mathcal{A}']}$. It suffices
to see that:
\begin{equation*}
\arraycolsep=0.9pt
\begin{array}{rcllcl}
q'\in\mu(\emptyStates)
&\iff{}&\exists q\in Q.& q\in \emptyStates&\land&\mu(q)=q'\\
&\iff{}&\exists q\in Q.& \toInfWord{\mathcal{A}_{q}}=\emptyset&\land&\mu(q)=q'\\
&\iff{}&\exists q\in Q.& \toInfWord {\mathcal{A}'_{\mu(q)}}=\emptyset&\land&\mu(q)=q'\\
&\iff{}&\exists q\in Q.& \toInfWord {\mathcal{A}'_{q'}}=\emptyset&\land&\mu(q)=q'\\
&\iff{}&&\toInfWord{\mathcal{A}'_{q'}}=\emptyset\\
&\iff{}&&q'\in\emptyStates[\mathcal{A}'].
\end{array}
\end{equation*} \end{proof} The following lemma gives a relation between the set of states introduced in Example \ref{not:q-empty-01} and transitions. All of the results follow easily from the definition of those sets. \begin{lemma}
Let $\mathcal{A}=\autPar{Q}{\digitDotSet}{\delta}{q_{0}}{F}$. Let $q\in Q$
and $a\in\digitDotSet$. Then $\del{q}{a}$ belongs to the sets
indicated in Example \ref{tab:Q-suc}.
\begin{table}[h]
\centering
\begin{tabular}{|l|l|l|}
\hline
If $q$ belongs to & then $\del{q}{a}$, for $a\in\digitSet$, belongs to & and $\del{q}{\star}$ belongs to:\\
\Xhline{4\arrayrulewidth}
$\emptyStates$&$\emptyStates$&$\emptyStates$\\
$\zuStates$&$\zuStates$&$\emptyStates$\\
$\inftyStates$&$\inftyStates$&$\zuStates$\\
$\natStates$&$\natStates$&$\fraStates$\\
$\fraStates$&$\fraStates$&$\emptyStates$\\
\hline
\end{tabular}
\caption{Set of states and transitions}\label{tab:Q-suc}
\end{table} \end{lemma} \section{Three methods to prove decidability of
automata problems}\label{sec:method} In this section, three methods are given. Those methods allow to prove that some problems over automata are decidable.
The method given in Section \ref{subsubsec:Honkala} is based on an algorithm of \cite{Honkala86}, which decide whether an integer automaton recognizes an ultimately periodic set of integer. The method given in Section \ref{subsubsec:muchnik} is based on an algorithm of \cite{muchnik}, which decide whether an automaton reading tuples of integers recognizes a set of tuples of integers which is $\fo{\mathbb N;+}$-definable. Finally, the method given in Section \ref{subsec:efficient} is based on an algorithm of \cite{Sakarovitch}, which decides in linear time whether a minimal automaton recognizes an ultimately periodic set of integer.
Those algorithms are easily adapted to other decision problem on automata reading vectors of integers or of reals. While the methods given in the first two sections are less efficient than the method of the last section, it seems interesting to give those method as they are very general and, as far as the author know, has not yet been given in full generality. For example, it is shown that those methods allow to prove that it is decidable: \begin{itemize} \item whether an automaton recognizes a set of real which is
$\fo{\mathbb R;+,<,1}$-definable or $\fo{\mathbb R,\mathbb Z;+,<}$-definable \item whether the recognized set is a subsemigroup of $(\mathbb R^{d},+)$. \end{itemize}
Finally, the method given in Section \ref{subsec:efficient} is the more efficient method of this section and is the method used in the remaining of this paper. Note that this method leads to proofs which are more complicated than the one needed to apply the two preceding methods.
\subsection{Honkala and brute-force algorithm}\label{subsubsec:Honkala} The method given in this section is based on \cite[Theorem 10]{Honkala86}. Let $\mathcal L$ be a family of regular languages. Conditions about $\mathcal L$ are now given. If a class of language satisfies those condition, it is shown that it is decidable whether an automaton recognizes a language belonging to $\mathcal L$.
Let us assume that there exists a size function $s:\mathcal L\to\mathbb N$, such that: \begin{enumerate} \item \label{honk:s}the number of states of the minimal automaton
recognizing a language $L\in\mathcal L$ is at least $s(L)$. \item\label{honk:compute}a finite superset of $s^{-1}(i)$ is
computable for all $i\in\mathbb N$. \end{enumerate}
In order to decide whether a minimal automaton $\mathcal{A}$ with $n$ states recognizes a language of $\mathcal L$, it suffices to consider the following algorithm: \begin{itemize} \item The algorithm runs on each integer $i\in[n]$, \item for each $i$, the algorithm runs on each language $L\in\mathcal L$
such that $s(L)=i$, by Hypothesis \eqref{honk:compute} it can be done, \item for each $L$, the algorithm constructs the minimal automaton
$\mathcal{A}_{L}$ which recognizes $L$, \item if the minimal automata of $\mathcal{A}$ and of $\mathcal{A}_{L}$ are equal,
the algorithm accepts. \end{itemize} If the algorithm has not accepted, it rejects. \paragraph{} The following proposition shows that this method can be applied to the problem considered in this paper. Note that the algorithm given in the following proof is inefficient. \begin{proposition}\label{prop-honk-simple}
It is decidable whether an automaton recognizes a
$\fo{\mathbb R,\mathbb Z;+,<,1}$-definable set of reals. \end{proposition} The proposition also holds for the more general notion of automata recognizing set of tuples of reals, as defined in \cite{weak-R-+-vector}. \begin{proof}
Let $r\in \mathbb Q$ and $u.vw^{\omega}$ be one of its encoding in base $b$
with $\length{v}$ and $\length{w}$ minimal. Let the pre-periodic
length of $r$ be $\length{u}+1+\length{v}$ and let the periodic
length of $r$ be $\length{w}$.
Let $\mathcal L$ be the class languages which are encodings of
simple sets. Let us first consider Hypothesis \ref{honk:s}.
Let $R$ be a $\fo{\mathbb R,\mathbb Z;+,<,1}$-definable of real. By \cite[Section
2]{elimination}, this logic admits quantifier elimination. Let
$\phi\in\qf{\mathbb R,\mathbb Z;+,<,1}$ be a quantifier-free formula defining $R$.
It can be proven that the number of state of the minimal automaton
recognizing a simple set $R$ is, at least, the maximum of the
pre-periodic lengths and of the periodic lengths of the constant
appearing in $\phi$. Let $s(\phi)$ be the greatest pre-periodic or
periodic length of the constants of $\phi$ and let $s(R)$ be the
minimal $s(\phi)$ for any $\phi\in\qf{\mathbb R,\mathbb Z;+,<,1}$ which defines
$R$.
Let us now consider hypothesis \ref{honk:compute}, let us give an
algorithm which takes as input an integer $i\in N$ and generates all
sets $R$ such that $s(R)=i$. It suffices to remark that, for each
$i\in\mathbb N$, there is a finite number of rationals whose pre-periodic
length and whose periodic length is less than $i$, let $S^{\le i}$
be the set of those rationals. The algorithm runs on each subset $U$
of $S^{\le i}$. For each $U$ the algorithm generates all
$\qf{\mathbb R,\mathbb Z;+,<,1}$-formulas in conjunctive normal form whose
rationals belongs to $S^{\le i}$. Note that there is only a finite
number of such formula up to permutation of the elements of the
conjunctions. Those formulas define all of the sets $R$ such that
$s(R)=i$. \end{proof} \subsection{Muchnik and decidable logic}\label{subsubsec:muchnik} The algorithm given in this section is based on \cite[Theorem 3]{muchnik}.
Let $\mathfrak S$ be a set of subset of $(\mathbb R^{\ge0})^{d}$ be a set of $d$-tuples of non-negative reals. Let $R$ be a $d$-ary symbol representing a set of $d$-tuples of reals. Let us assume that there exists a $\fo{\mathbb R,\mathbb Z;X_{b},+,<,1,R}$-formula $\phi$ which characterizes whether a set $R$ belongs to $\mathfrak S$. It is decidable whether a RVA $\mathcal{A}$ is such that $\wordToReal{\mathcal{A}}\in\mathfrak S$. Indeed, the equivalence between $\fo{\mathbb R,\mathbb Z;X_{b},+,<,1}$ and the Büchi automata is effective, hence it suffices to translate the formula $\phi$ into a Büchi automaton, according to the algorithm of \cite{weak-buchi-r-+}, where $R$ is encoded by the automaton $\mathcal{A}$.
Let us give an example of application of this method. \begin{proposition}
It is decidable whether a Büchi automaton recognizes a set which is a
subsemigroup of $(\mathbb R^{d},+)$. \end{proposition} \begin{proof}
Using the argument given above, it suffices to use the formula:
\begin{eqnarray*}
\forall x_{0},\dots,x_{d-1},
y_{0},\dots,y_{d-1}. \left[ \left(x_{0},\dots,x_{d-1}\right)\in R\land
\left(y_{0},\dots,y_{d-1}\right)\in R\right]\implies{}\\ {\left(x_{0}+y_{0},\dots,x_{d-1}+y_{d-1}\right)\in R}.
\end{eqnarray*} \end{proof}
Note that the method introduced in this section leads to inefficient algorithm. The computation time of those algorithms are, at least, a tower of exponential, whose height is equal to the number of quantifier alternation.
\subsection{The efficient method}\label{subsec:efficient}
The method introduced in this section leads to proofs which are more difficult than the method given in the two preceding sections. However, this method also leads to more efficient algorithms. The method introduced in this section is the one used in Theorems \ref{theo:aut-0-1}{ and} \ref{theo:aut-R}{}. This method is similar to the proofs used in \cite{Leroux} and in \cite{Sakarovitch}.
\begin{proposition}\label{prop:method}
Let $\mathbb L'$ be a class of language and $\mathbb A'$ be a class of weak
{}Büchi automata such that $L\in \mathbb L'$
if and only if it is recognized by a Büchi automaton of $\mathbb A'$.
Let $\mathbb L$ be a class of languages over an alphabet such that there
exists a class $\mathbb A$ of weak {}Büchi
automata such that:
\begin{enumerate}
\item\label{prop:method-algo} it is decidable in time $t(n,b)$
whether a Büchi automaton belongs to $\mathbb A$, for $n$ the number
of states and $b$ the number of letters,
\item\label{prop:method-AR} for each $L\in \mathbb L\cap\mathbb L'$, there exists an
automaton $\mathcal{A}\in\mathbb A$ which recognizes $L$,
\item\label{prop:method-min} the minimal quotient of any automaton
of $\mathbb A$ belongs to $\mathbb A$ and
\item\label{prop:method-L} every language recognized by some automaton
of $\mathbb A$ belongs to $\mathbb L$.
\end{enumerate}
There exists an algorithm $\alpha$, which halts in time
$\bigO{t(n,b)}$, and decides whether a minimal automaton of $\mathbb A'$
recognizes a language of $\mathbb L$. Furthermore, the algorithm
$\alpha$ applied to an automaton belonging to
$\mathbb A'\setminus \mathbb A$ may return a false negative but it may not
return any false positive. \end{proposition} In this paper, $\mathbb A'$ is either the set of FNA or of RNA. Considering the class $\mathbb A'$ allows to restrict the kind of automata studied. The proposition still hold when "weak {}Büchi automata" is replaced by "finite automaton". More generally, a similar proposition can be given as soon as, for each language, there exists a canonical automaton recognizing this language. This requirement is the reason for which this proposition does not hold for non-weak Büchi automata or for non-deterministic automata. \begin{proof}
By Property \eqref{prop:method-algo} there exists an algorithm
$\alpha$ which accepts in time $t(n,l)$ the automata of $\mathbb A$.
Let $\mathcal{A}$ be a weak Büchi automaton and let
$L=\toInfWord{\mathcal{A}}$. Let us prove that if $\alpha$ accepts $\mathcal{A}$
then $L\in\mathbb L$ and if $L\in\mathbb L$ and $\mathcal{A}\in\mathbb A'$ then
$\alpha$ accepts $\mathcal{A}$.
Let us first prove that if $\alpha$ accepts $\mathcal{A}$, then
$L\in \mathbb L$. If $\mathcal{A}$ is accepted by $\alpha$, by Property \eqref{prop:method-algo}, $\mathcal{A}\in\mathbb A$. By Property \eqref{prop:method-L}, since $\mathcal{A}\in\mathbb A$, $L\in\mathbb L$.
Let us now prove that if $L\in\mathbb{L}$ and $\mathcal{A}\in\mathbb A'$ then
$\alpha$ accepts $\mathcal{A}$. Since $\mathcal{A}\in\mathbb A'$, by definition of
$\mathbb{L'}$, $L\in\mathbb{L'}$. Since $L\in\mathbb{L}$ and $L\in\mathbb{L'}$, by
Property \eqref{prop:method-AR}, there exists an automaton
$\mathcal{A}'\in\mathbb A$ which recognizes $L$. Since $\mathcal{A}'$ and $\mathcal{A}$
recognize the same languages and $\mathcal{A}$ is minimal, then $\mathcal{A}$ is
the minimal quotient of $\mathcal{A}'$. Since $\mathcal{A}'$ is the minimal
quotient of an automaton belonging to $\mathbb A$, by Property \eqref{prop:method-min} $\mathcal{A}\in\mathbb A$. Since $\mathcal{A}\in\mathbb A$, by
Property \eqref{prop:method-algo}, the algorithm $\alpha$ accepts the
automaton $\mathcal{A}$. \end{proof}
\section{Automata accepting simple subsets of
$[0,1]$}\label{sec:bounded}
This section deals with automata recognizing simple subsets of $[0,1]$. For each simple set $R$, a weak {}Büchi automaton which recognizes $\toInfWord{R}$ is defined in Section \ref{subsec:bounded-set->aut}. In Section \ref{subsec:char:bounded-aut}, an algorithm is given, which accepts the weak FNAs which recognize simple sets.
\subsection{From sets to automata}\label{subsec:bounded-set->aut} Let $R\subseteq[0,1]$ be a simple set, in this section, it is explained how to compute an automaton $\ARPar{R}$ which recognizes $\toInfWord{R}$. This automaton has the form descried in \cite[Chapter 5]{Boigelot2007}. As seen in Section \ref{sec:FU}, $R$ can be expressed as: \begin{equation*}
\bigcup_{i=0}^{I-1}(\rho_{i,\mathfrak L},\rho_{i,\mathfrak R})\cup\bigcup_{i=0}^{J-1}\set{\rho_{i,\mathfrak S}} \end{equation*} with $\rho_{i,j}\in\mathbb Q$. In this section, it can furthermore be assumed that the $\rho_{i,j}$'s belong to $[0,1]$.
Let $w_{i,j,k}$, with $k$ taking value in $\set{0,(b-1)}$, be the one or two encodings in base $b$ of $\rho_{i,j}$. Let $l$ be an integer such that, for all $i,j,k,i',j',k'$, either $w_{i,j,k}=w_{i',j',k'}$ or $\prefix{w_{i,j,k}}{l}\ne\prefix{w_{i',j',k'}}{l}$. By an easy induction on the number of words $w_{i,j,k}$, such an integer $l$ exists.
Since the $\rho_{i,j}$ are rationals, their encodings in base $b$ are of\index{u@$u_{i,j,k}$}\index{v@$v_{i,j,k}$} the form $u_{i,j,k}v_{i,j,k}^{\omega}$ with $u_{i,j,k}\in\digitSet^{*}$, $v_{i,j,k}\in\digitSet^{+}$ and $k\in\set{0,(b-1)}$. Without loss of generality, let us assume that all $u_{i,j,k}> l$ and that $\length{v_{i,j,k}}$ is minimal. Those two assumptions imply that no $u_{i,j,k}$ is the prefix of a $u_{i',j',k'}$ and that if $u_{i,j,k}=u_{i',j',k'}$ then $v_{i,j,k}=v_{i',j',k'}$.
A weak FNA $\ARPar{R}$ which recognizes $\toInfWord{R}$ is now defined. \begin{definition}[$\ARPar{R}$]\index{AR@$\ARPar{R}$ and $\ARPar{R}'$ - for
a simple set $R\subseteq[0,1]$}
\label{def:AR01}
For $R$ a simple set, let
$\ARPar{R}=\autPar{Q_{R}}{\digitSet}{\delta_{R}}{q_{\epsilon}}{F_{R}}$
where:
\begin{itemize}
\item The set of states $Q_{R}$ is defined as the set of states
$q_{w}$ for $w$ a strict prefixes of $u_{i,j,k}v_{i,j,k}$, plus
two states $\emptyState[R]$ and $\zuState[R]$.
\item The transition function is
\begin{equation*}
\delta_{R}(q,a)=\left\{
\begin{array}{ll}
q&\mbox{ if }q\in\set{\emptyState[R],\zuState[R]},\\
wa&\mbox{ if $q_{w}=q$ and $wa$ is a strict prefix of some $u_{i,j,k}v_{i,j,k}$}\\
u_{i,j,k}&\mbox{ if $q=q_{w}$ and $wa=u_{i,j,k}v_{i,j,k}$},\\
\zuState[R]&\mbox{ if $q=q_{w}$ and $\wordToReal{w a}$ belongs to an
interval $(\rho_{i,\mathfrak L},\rho_{i,\mathfrak R})$}\\
\emptyState[R]&\mbox{ otherwise.}
\end{array}
\right.
\end{equation*}
\item The set $F_{R}$ of accepting states contains $\zuState[R]$ and
the states $q_{w}$ for $w$ some strict prefix of
$u_{i,\mathfrak S,k}v_{i,\mathfrak S,k}$ of length at least
$\length{u_{i,\mathfrak S,k}}$, for some $i$ and $k$.
\end{itemize} \end{definition} The last definition is used to characterize the automaton, not to compute it, hence there is no need to study how to decide whether $\wordToReal{wa}$ belongs to an interval $(\rho_{i,\mathfrak L},\rho_{i,\mathfrak R})$, nor to compute the words $u_{i,j,k}$. Note that the name of the states $\emptyState[R]$ and $\zuState[R]$ are consistent with Section \ref{sec:aut-numb}.
An automaton $\ARPar{R}$ and its minimal quotient are now given. The set $R$ is not the same as the one of Example \ref{ex:decom-R-01} because a subset of $[0,1]$ is now required. \begin{example}\label{ex:auto-R-01}
Let $R=\exRfrac$ as in Example \ref{ex:decom-R-01}. The following table
gives the values associated to those indexes.
\begin{equation*}
\begin{array}[c]{|llll|llll|}
\hline
\rho_{i,j}& \mbox{its encodings in base 2}& u_{i,j,k}& v_{i,j,k}&
\rho_{i,j}& \mbox{its encodings in base 2}& u_{i,j,k}& v_{i,j,k}\\
\Xhline{4\arrayrulewidth}
\frac 14& 001^{\omega}& 001& 1&
\frac13& (01)^{\omega}&0101&01\\
\frac 14& 010^{\omega}& 0100& 0&
\frac23& (10)^{\omega}&101&01\\
\frac5{12}& 01(10)^{\omega}&011&01&
&&&\\
\hline
\end{array}
\end{equation*}
The automaton $\ARPar{R}$ is pictured in Figure \ref{fig:ex-AR-01}, without
the state $\emptyState[R]$, and its minimal quotient is pictured in
Figure \ref{fig:ex-AR-01}.
\begin{figure}
\caption{The automaton $\ARPar{R}$ for $R=\exRfrac$}
\label{fig:ex-AR-01}
\end{figure}
\begin{figure}
\caption{The minimal automaton recognizing $R=\exRfrac$}
\label{fig:ex-AR-01-min}
\end{figure}
\paragraph{}
Intuitively, the states $q_{1}$ and $q_{10}$ are used to read the
binary encoding of $\frac23$. The states $q_{00}$ and $q_{001}$ are
used to read one of the binary encoding of $\frac14$ and $q_{0100}$
is used to read its other encoding. The states $q_{011}$ and
$q_{0111}$ are used to read $\frac{11}{24}$. Finally, the states
$q_{0101}$ and $q_{01010}$ are used to read $\frac13$.
Note that minimization sends the states $q_{011}$ and $q_{10}$ to
the same state, named $q_{10}$. Intuitively, it is because when an
automaton reads a rational $r$ belonging to the boundary of $R$, the
only important information is:
\begin{itemize}
\item the periodic part of the encoding of $r$,
\item whether $r$ belongs to $R$ and
\item whether, $(r-\epsilon,r)$ (respectively, $(r,r+\epsilon)$) is
included in $R$ or is disjoint from $R$, for $\epsilon$ small
enough.
\end{itemize} \end{example}
Let us prove that $\ARPar{R}$ is as expected. \begin{lemma}\label{lem:ar-01-correct}
Let $R\subseteq[0,1]$ be a simple set. The automaton $\ARPar{R}$ recognizes
$\toInfWord{R}$. \end{lemma} \begin{proof}
Let $w\in\digitSet^{\omega}$, and let $r=\wordToReal{w}$. Let us
prove that $\wordToReal{w}\in R$ if and only if $w$ is accepted by
$\ARPar{R}$. Two cases must be considered, depending on whether $r$ is
equal to some $\rho_{i,\mathfrak S}$ for some $i$, or not. In the
second cases, four more cases must be considered, depending on
whether $r$ is equal to some $\rho_{i,\mathfrak L}$, whether $r$ is
equal to some $\rho_{i,\mathfrak R}$, whether $r$ belongs to some
$(\rho_{i,\mathfrak L},\rho_{i,\mathfrak R})$ for some $i$, or
whether $r$ does not satisfy any of those properties.
\begin{itemize}
\item Let us first assume that $r=\rho_{i,\mathfrak S}$ for some
$i$. Then $r\in R$. Let us show that $w$ is accepted by
$\ARPar{R}$. Since $r=\rho_{i,\mathfrak S}$, then $w$ is of the form
$u_{i,\mathfrak S,k}v_{i,\mathfrak S,k}^{\omega}$, hence, by
induction on the prefixes of $w$, all visited states are of the
form $q_{w}$ for $w$ some prefix of
$u_{i,\mathfrak S,k}v_{i,\mathfrak S,k}$. It implies that each
state of the run of $\ARPar{R}$ on $w$ -- apart from the
$\length{u_{i,\mathfrak S,k}}$ first states -- are accepting.
Hence $\ARPar{R}$ accepts $w$.
\item From now on, let us assume that, for all $i$,
$r\ne\rho_{i,\mathfrak S}$. Let us suppose that
$r=\rho_{i,\mathcal L}$ for some $i$. The case $r=\rho_{i,\mathcal R}$ is
similar. In this case, $r\not\in R$. Let us prove that $w$ is not
accepted by $\ARPar{R}$. The word $w$ is of the form
$u_{i,j,k}v_{i,j,k}^{\omega}$, hence by induction on the prefix's
of $w$, each visited state is $q_{w}$ for $w$ a prefix of
$u_{i,j,k}v_{i,j,k}$, and since the $u_{i,j,k}$ are not prefixes
of $u_{i',j',k'}v_{i',j',k'}^{\omega}$, then no accepting state is
visited. Hence $\ARPar{R}$ does not accepts $w$.
\item Let us assume that
$r\in(\rho_{i,\mathfrak L},\rho_{i,\mathfrak R})$. Let us show
that $\ARPar{R}$ accepts $w$. Three cases must be considered, depending
on whether $u_{i,\mathfrak L,k}$ is a prefix of $w$ for some
$k\in\set{0,(b-1)}$, whether $u_{i,\mathfrak R,k}$ is a prefix or
$w$ for some $k\in\set{0,(b-1)}$, or whether none of those words
are prefix of $w$.
\begin{itemize}
\item Let us suppose that $u_{i,\mathfrak L,k}$ is a prefix of
$w$, then there exists a unique 4-tuple $n\in \mathbb N$,
$h<\length{v_{i,\mathfrak L,k}}$, $a<v_{i,\mathfrak L,k}[h]$ and
$w'\in\digitSet^{\omega}$ such that
\begin{equation*}
w=u_{i,\mathfrak L,k}v_{i,\mathfrak L,k}^{n}(\prefix{v_{i,\mathfrak L,k}}{h})aw'.
\end{equation*}
Since $a<v_{i,\mathfrak L,k}[h]$,
$\wordToReal{u_{i,\mathfrak L,k}\prefix{v_{i,\mathfrak
L,k}}{h}aw'}>\rho_{i,\mathfrak L}$.
Since the $u_{i,j,k}$'s are not prefixes of the
$u_{i',j',k'}$'s,
$\wordToReal{u_{i,\mathfrak L,k}\prefix{v_{i,\mathfrak L,k}}{h}aw'}<\rho_{i,\mathfrak L}$. Then, for
$i>\length{u_{i,\mathfrak L,k}v_{i,\mathfrak L,k}^{n}\prefix{v_{i,\mathfrak
L,k}}{h}}$,
the $i$th state is $\zuState[R]$, hence $w$ is accepted by $R$.
\item If $u_{i,\mathfrak R,k}$ is a prefix of $w$, the proof is
similar by symmetry.
\item Let us now assume that there are no $j$ and $k$ such that
$u_{i,j,k}$ is a prefix of $w$. Let us consider the longest
prefix $p$ of $w$ and $u_{i,j,k}$. The $\length{p}$ first steps
of the run of $\ARPar{R}$ on $w$ are the states $q_{v}$ for $v$ some
prefix of $p$. The following steps are $\zuState[R]$, hence
$\ARPar{R}$ accepts $w$.
\end{itemize}
\item In all other cases, the proof is similar to the preceding
case, replacing $\zuState[R]$ by $\emptyState[R]$.
\end{itemize} \end{proof}
\subsection{Characterization of automata recognizing simple
sets}\label{subsec:char:bounded-aut} The main theorem of this paper concerning subsets of $[0,1]$ is now stated. \begin{theorem}\label{theo:aut-0-1}
It is decidable in time $\bigO{nb}$ whether a minimal FNA over the
alphabet $\digitSet$ with $n$ states recognizes a simple set. \end{theorem}
The proof of this theorem uses Proposition \ref{prop:method}. In order to use this proposition, a set $\mathbb{A}_{F,\mathcal{S}}$ of automata is now introduced. Four lemmas are then proved, which corresponds to the four properties of Proposition \ref{prop:method}. \begin{definition}[$\mathbb{A}_{F,\mathcal{S}}$]\index{A@$\mathbb{A}_{F,\mathcal{S}}$}\label{def:BAF}
Let $\mathbb{A}_{F,\mathcal{S}}$ be the set of weak {}Büchi
automata $\mathcal{A}=(Q,\digitSet,\delta,q_{0},F)$ such that, for each
strongly connected component
$C\subseteq \fraStates\setminus(\zuStates\cup{}\emptyStates)$, there
exists $\beta_{<,C}$ and $\beta_{>,C}$, two states of
$\zuStates\cup{}\emptyStates$, such that, for all $q\in C$:
\begin{enumerate}
\item\label{def:BAF-cycle} $C$ is a
cycle,
\item\label{def:BAF-cycle>} for all $a>a_{q}$, $\delta(q,a)$ is
$\beta_{>,C}$ and
\item\label{def:BAF-cycle<} for all $a<a_{q}$, $\delta(q,a)$ is
$\beta_{<,C}$.
\end{enumerate} \end{definition}
Property \eqref{def:BAF-cycle>} implies that $\zuStates\cup\emptyStates$ is not empty.
Example \ref{ex:auto-R-01} is resumed in order to show the construction of the preceding lemma. \begin{example}\label{ex:auto-R-01-aut}
Let $R=\left(\frac13,2\right]\cup\left(\frac83,3\right]\cup\left(\frac{11}3,\infty\right]
$
and $\mathcal{A}$ be the minimal automaton of Figure
\ref{fig:ex-AR-01-min}. Let us first consider the recurrent states
$q$ equal to $1$, $00$, $10$, $0101$ or to $01010$. The integer $n$,
the sequence of letters $a_{0},\dots,a_{n-1}$, the states
$\beta_{<,C}$ and $\beta_{>,C}$ associated to each of those states
$q$ are given in the following table and the Boolean $\beta_{=,C}$
which is true if and only if $C$ is composed of accepting states.
The two last columns of the table show the language of infinite
words recognized by the automaton $\mathcal{A}_{q}$, and the set of
reals recognized by this state.
\begin{equation*}
\begin{array}[c]{lllllllll}
\hline
q& n& a_{0}&a_{1}&\beta_{<,C}&\beta_{=,C}&\beta_{>,C}& \toInfWord{\mathcal{A}_{q}}& \wordToReal{\mathcal{A}_{q}}\\
\Xhline{4\arrayrulewidth}
1& 2& 0 & 1 & \emptyState & \true & \emptyState & (01)^{\omega}&\left\{\frac13\right\}\\
10& 2& 1 & 0 & \emptyState & \true & \emptyState & (10)^{\omega}&\left\{\frac23\right\}\\
00& 1& 1 & & \emptyState & \true & \emptyState & (1)^{\omega}&\left\{1\right\}\\
0101& 2& 0 & 1 & \zuState & \false & \emptyState & 0(10)^{\omega}+(10)^{*}0(0+1)^{\omega}&\left[0,\frac13\right)\\
01010& 2& 1 & 0 & \zuState & \false & \emptyState & (10)^{\omega}+(10)^{*}0(0+1)^{\omega}&\left[0,\frac23\right)\\
\zuState[R]&&&& &&& (0+1)^{\omega}&\left[0,1\right]\\
\emptyState[R]&&&& &&& \emptyset&\emptyset\\
\hline
\end{array}
\end{equation*}
\paragraph{}
Transient states $010$, $01$, $0$ and $\epsilon$ are now
considered. One has:
\begin{equation*}
\arraycolsep=0.5pt
\begin{array}{ccccccccccccccc}
\wordToReal{\mathcal{A}_{010}} &=&\frac{0+\wordToReal{\mathcal{A}_{\zuState[R]}}}{2}&\cup&\frac{1+\wordToReal{\mathcal{A}_{0101}}}{2}&=&\frac{0+\left[0,1\right]}{2} &\cup&\frac{1+\left[0,\frac13\right)}{2}
&=&\left[0,\frac23\right),\\
\wordToReal{\mathcal{A}_{01}} &=&\frac{0+\wordToReal{\mathcal{A}_{010}}}{2} &\cup&\frac{1+\wordToReal{\mathcal{A}_{10}}}{2} &=&\frac{0+\left[0,\frac23\right)}{2} &\cup&\frac{1+\set{\frac23}}{2} &
=&\left[0,\frac13\right)\cup\set{\frac56},\\
\wordToReal{\mathcal{A}_{0}} &=&\frac{0+\wordToReal{\mathcal{A}_{00}}}{2} &\cup&\frac{1+\wordToReal{\mathcal{A}_{01}}}{2} &=&\frac{0+\left\{1\right\}}{2} &\cup&\frac{1+\left\{\left[0,\frac13\right)\cup\set{\frac56}\right\}}{2}&=
&\left[\frac12,\frac23\right)\cup\set{\frac{11}{12}},\\
\wordToReal{\mathcal{A}_{\epsilon}}&=&\frac{0+\wordToReal{\mathcal{A}_{0}}}{2} &\cup&\frac{1+\wordToReal{\mathcal{A}_{1}}}{2} &=&\frac{0+\left[\frac12,\frac23\right)\cup\set{\frac{11}{12}}}{2}&\cup&\frac{1+\set{\frac13}}{2}
&=&\left(\frac13,2\right]\cup\left(\frac83,3\right]\cup\left(\frac{11}3,\infty\right] .
\end{array}
\end{equation*} \end{example} It is now proven that the set $\mathbb{A}_{F,\mathcal{S}}$ satisfies Property \eqref{prop:method-algo} of Proposition \ref{prop:method}. \begin{lemma}\label{lem:dans-baf}
It is decidable in time $\bigO{nb}$ whether a weak {}Büchi automaton
with $n$ states belongs to $\mathbb{A}_{F,\mathcal{S}}$. \end{lemma} \begin{proof}
Tarjan's algorithm \cite{Tarjan} can be used to compute the set of
strongly connected component in time $\bigO{nb}$ and thus the set of
recurrent state. By Lemma \ref{lem:comp-set}, the sets $\emptyStates$
and $\zuStates$ are computable in linear time. The algorithm runs on
each strongly connected component distinct from $\zuStates$ and from
$\emptyStates$.
It is now explained how the algorithm checks whether Property
\eqref{def:BAF-cycle} is satisfied by the automaton. The algorithm
runs on each $q\in Q$. The algorithm keeps a counter $c_{q}$,
initialized to 0, of the number of letters $a\in\digitSet$ such that
$\del{q}{a}\in C$. For each $q$, the algorithm runs on each letter
$a\in\digitSet$. For each $a$, the algorithm tests whether
$\del{q}{a}\in C$, and if it is the cases, $c_{q}$ is
incremented. If $c_{q}\ne 1$ the algorithm rejects.
It is now explained how the algorithm checks whether Property
\eqref{def:BAF-cycle>} is satisfied by the automaton. Checking
Property \eqref{def:BAF-cycle<} is done similarly. The algorithm
runs on each $q\in C$. If $a_{q}>0$ then:
\begin{itemize}
\item if $\beta_{<,C}$ is not set, then $\beta_{<,C}$ is set to
$\del{q}{0}$.
\item otherwise, let us assume that $\beta_{<,C}$ is set. The
algorithm runs on each $0\le a<a_{q}$. For each $a$, if
$\del{q}{a}$ is different from $\beta_{<,C}$, then the algorithm
rejects.
\end{itemize}
If the algorithm has not rejected, it accepts. \end{proof} It is now proven that the set $\mathbb{A}_{F,\mathcal{S}}$ satisfies Property \eqref{prop:method-AR} of Proposition \ref{prop:method}. \begin{lemma}\label{lem:AR-in-BAF}
Let $R\subseteq [0,1]$ be a simple set. The automaton $\ARPar{R}$ belongs
to $\mathbb{A}_{F,\mathcal{S}}$. \end{lemma} \begin{proof}
It suffices to see that the recurrent states
$q\not\in\emptyStates\cup \zuStates$ of $\ARPar{R}$ are of the form
$u_{i,j,k}w$ with $w$ a prefix of $v_{i,j,k}$. For each prefix $w$
of length $l$ of $v_{i,j,k}$, the digit $\suc{q_{u_{i,j,k}w}}$ is
$v_{i,j,k}[l]$, thus Property \eqref{def:BAF-cycle} holds. The state
$\beta_{<,C}$ (respectively, $\beta_{>,C}$) is $\emptyState$ if
$u_{i,j,k}0^{\omega}\not\in R$ and $\zuState$ otherwise, thus
Property \eqref{def:BAF-cycle>} (respectively \ref{def:BAF-cycle<})
holds. Then the conditions of Definition \ref{def:BAF} are
satisfied. \end{proof}
It is now proven that the set $\mathbb{A}_{F,\mathcal{S}}$ satisfies Property \eqref{prop:method-min} of Proposition \ref{prop:method}. \begin{lemma}\label{lem:quotient-BAF}
The minimal quotients of automata of $\mathbb{A}_{F,\mathcal{S}}$ belong to $\mathbb{A}_{F,\mathcal{S}}$. \end{lemma} \begin{proof}
Let $\mathcal{A}=\autPar{Q}{\digitSet}{\delta}{q_{0}}{F}$ be an
automaton belonging to $\mathbb{A}_{F,\mathcal{S}}$ and let its minimal quotient be
$\mathcal{A}'=\autPar{Q'}{\digitSet}{\delta'}{q'_{0}}{F'}$. Let
$\mu$ be the morphism from $\mathcal{A}$ to $\mathcal{A}'$. Let us
show that $\mathcal{A}'$ belongs to $\mathbb{A}_{F,\mathcal{S}}$. Let $C'$ be a component
included in $\mathcal{A}'$, distinct from
$\emptyStates[\mathcal{A}']$ and from $\zuStates[\mathcal{A}']$. By
Lemma \ref{lem:scc-morphism-codomain}, there exists a strongly
connected component $C$ of $\mathcal{A}$ such that $\mu(C)=C'$ and
such that, for all $q\in Q\setminus C$ accessible from $C$,
$\mu(q)\not\in C'$.
Let us first show that Property \eqref{def:BAF-cycle} is satisfied by
$\mathcal{A}'$. Let $q'\in C'$ and let us prove that there exists
exactly one letter $\suc{q'}$ such that $\dell{q'}{\suc{q'}}\in C'$.
Since $q'$ is recurrent, at least one such letter exists. It remains
to prove the unicity. Let us assume that there exists two letters
$a_{0}$ and $a_{1}$ such that $\dell{q'}{a_{0}}\in C'$ and
$\dell{q'}{a_{1}}\in C'$ and let us prove that $a_{0}=a_{1}$. Since
$\mu(C)=C'$, there exists $q\in C$ such that $\mu(q)=q'$. Since
$\mathcal{A}\in\mathbb{A}_{F,\mathcal{S}}$, there exists exactly one letter $\suc{q}$ such
that $\del{q}{\suc{q}}\in C$. It suffices to prove that
$a_{0}=\suc{q}=a_{1}$. Let us prove that $a_{0}=\suc{q}$, the other
case is similar. Since
$\mu(\del{q}{a_{0}})=\dell{\mu(q)}{a_{0}}=\dell{q'}{a_{0}}\in C'$
and since $\del{q}{a_{0}}$ is accessible from $C$, by hypothesis
about $C$, $\del{q}{a_{0}}\in C$. Since $\del{q}{a_{0}}\in C$, by
Property \eqref{def:BAF-cycle} applied to $\mathcal A$, by definition of
$\suc{q}$, $a_{0}=\suc{q}$.
It is now proven that $\mathcal{A}'$ satisfies Property \eqref{def:BAF-cycle>}. The case of Property \eqref{def:BAF-cycle<} is
similar. Let $q'_{1}$ and $q'_{2}$ be two states of $C'$, and let
$a_{1}>\suc{q'_{1}}$ and $a_{2}>\suc{q'_{2}}$. Since $\mu(C)=C'$,
there exists $q_{1},q_{2}\in C$ such that $\mu(q_{1})=q'_{1}$ and
$\mu(q_{2})=q'_{2}$. Let $q_{1}$ and $q_{2}$ be those two
antecedents, note that $\suc{q_{1}}=\suc{q'_{1}}$ and
$\suc{q_{2}}=\suc{q'_{2}}$. Since $\mathcal{A}\in\mathbb{A}_{F,\mathcal{S}}$, by Property applied to
$\mathcal A$ \ref{def:BAF-cycle>}
$\del{q_{1}}{a_{1}}=\del{q_{2}}{a_{2}}$, hence
$\dell{q'_{1}}{a_{1}}=\dell{q'_{2}}{a_{2}}$. \end{proof} It is now proven that the set $\mathbb{A}_{F,\mathcal{S}}$ satisfies Property \eqref{prop:method-L} of Proposition \ref{prop:method}. \begin{lemma}\label{lem:BAF->pour}
The automata of $\mathbb{A}_{F,\mathcal{S}}$ recognize simple subsets of $[0,1]$. \end{lemma} Note that it is not required in this lemma that the Büchi automaton is a FNA. In order to prove this lemma, another lemma is now required. Its proof is straightforward from Definition \ref{def:BAF}. \begin{lemma}\label{lem:BAF->pour:aux}
Let $\mathcal{A}$ be a Büchi automaton of $\mathbb{A}_{F,\mathcal{S}}$ and let $q$ be a state of
$\mathcal{A}$. The automaton $\mathcal{A}_{q}$ belongs to $\mathbb{A}_{F,\mathcal{S}}$. \end{lemma}
\begin{proof}[Proof of Lemma \ref{lem:BAF->pour}]
The proof is by induction on the number $n$ of states of
$\mathcal{A}$. Two cases must be considered, depending on whether
the initial state of the automaton is recurrent or not.
Let us first suppose that the initial state is transient. For each
$a\in\digitSet$, by Lemma \ref{lem:BAF->pour:aux}, the automaton
$\mathcal{A}_{\delta(q_{0},a)}$ belongs to $\mathbb{A}_{F,\mathcal{S}}$. The automaton
$\mathcal{A}_{\delta(q_{0},a)}$ has less than $n$ states and belongs to
$\mathbb{A}_{F,\mathcal{S}}$, hence, by induction hypothesis, it recognizes a set $R_{a}$ of
the form
$\bigcup_{i}^{I_a-1}(\rho_{i,\mathfrak L,a},\rho_{i,\mathfrak R,a})\cup\bigcup_{i=0}^{J_a-1}\set{\rho_{i,\mathfrak S,a}}$. The
set $\wordToReal{\mathcal{A}}$ is then equal to:
\begin{equation*}
\bigcup_{a=0}^{b-1}\frac{a+R_{a}}{b}\\
=\bigcup_{a=0}^{b-1}\bigcup_{i=0}^{I_a-1}\left(\frac{a+\rho_{i,\mathfrak L,a}}b,\frac{a+\rho_{i,\mathfrak R,a}}b\right)\cup\bigcup_{i=0}^{J_a-1}\left\{\frac{a+\rho_{i,\mathfrak S,a}}b\right\},
\end{equation*}
where the $\rho_{i,j}$ belongs to $\mathbb Q\cap[0,1]$. Hence
$\wordToReal{\mathcal{A}}$ is a simple set.
\paragraph{}
Let us now assume that the initial state is recurrent. Three cases
must be considered depending on whether $q_{0}$ belongs to
$\emptyStates$, to $\zuStates$ or to neither of those sets. Let us
first assume that $q_{0}\in{}\emptyStates$, the case
$q_{0}\in{}\inftyStates$ is similar. The automaton recognizes
$\emptyset$ by definition of $\emptyStates$, thus is simple.
Let us suppose that $q_{0}\not\in\emptyStates\cup\zuStates$ and
let $C$ be the strongly connected component of $q_{0}$. By
Property \eqref{def:BAF-cycle} of Definition \ref{def:BAF}, there
exists a sequence $a_{0},\dots,a_{n-1}$ such that
$\del{q_{0}}{a_{0}\dots a_{n-1}}=q_{0}$. Let
$c_{q_{0}}$ be the real represented by
$\left(a_{0}\dots a_{n'-1}\right)^{\omega}$ that is
$\frac{\sum_{i=0}^{n'-1}a_{i}b^{n'-i-1}}{b^{n'}-1}$. The number
$c_{q_{0}}$ is rational, and its length is
$\bigO{\log(b)n}$. Let $w\in\digitSet^{\omega}$ and
$x=\wordToReal{w}$. It is then clear from Definition \ref{def:BAF},
that $\mathcal{A}$ accepts $w$ if and only if, either $x=c_{q_{0}}$
and $C\subseteq F$, either $x<c_{q_{0}}$ and
$\beta_{<,C} \in \fraStates$, or similarly $x>c_{q_{0}}$ and
$\beta_{>,C} \in \fraStates$. \end{proof}
Theorem \ref{theo:aut-0-1} can now be proven. \begin{proof}
It suffices to use Proposition \ref{prop:method} with $\mathbb A$ being the set
of automata $\mathbb{A}_{F,\mathcal{S}}$, $\mathbb A'$ being the set of FNA, and Lemmas
\ref{lem:dans-baf}, \ref{lem:AR-in-BAF}, \ref{lem:quotient-BAF} and
\ref{lem:BAF->pour}. \end{proof} Note that the algorithm given in the proof of Theorem \ref{theo:aut-0-1} returns no false positive even when it is applied to a Büchi Automaton which is not a FNA. The author conjecture that there exists no false negative.
\section{Simple subsets of $\mathbb R^{\ge0}$}\label{sec:real-aut} In the preceding section, the problem studied in this paper was solved on $[0,1]$. The general problem is solved in this section using the notations and lemmas of Section \ref{sec:bounded}.
In Section \ref{subsec:set->aut}, given a simple set $R$, a weak {}automaton $\ARPar{R}$ is constructed, which recognizes $R$. In Section \ref{subsec:char:aut}, an algorithm is given, which takes as input a weak {}automaton of alphabet $\digitDotSet$ and decides whether it recognizes a simple set.
\subsection{From sets to automata}\label{subsec:set->aut} Let us fix in this section a simple set $R\subseteq\mathbb R^{\ge0}$. Since $R$ is a simple set, there exists a least integer $t_{R}\in\mathbb N^{\ge0}$ such that $[t_{R},\infty)$ is either a subset of $R$ or is disjoint from $R$. For all $i\in\mathbb N$, let $R_{i}$ denote $\set{x\in[0,1]\mid x+i\in R}$, and let $\ARPar{{i}}=\autPar{Q_{i}}{\digitSet}{\delta_{i}}{q_{0,i}}{F_{i}}$ be the minimal automaton accepting $\wordToReal{R_{i}}$. By Example \ref{ex:unbounded} is now resumed. \begin{example}\label{ex:unbounded-follow}
Let $R=\left(\frac13,2\right]\cup\left(\frac83,3\right]\cup\left(\frac{11}3,\infty\right]$ as in Example \ref{ex:unbounded}. Then $t_{R}$ is 4,
$R_{0}=\left(\frac13,1\right]$, $R_{1}=\left[0,1\right]$, and
$R_{2}=R_{3}=\set{0}\cup\left[\frac23,1\right]$. \end{example}
In this section, a weak RNA $\ARPar{R}$ which recognizes $\toInfWord{R}$ is constructed. The part of $\ARPar{R}$ which reads the fractional parts of the reals is based on the construction of Section \ref{subsec:bounded-set->aut}. The formal definition of $\ARPar{R}$ is now given. \begin{definition}[$\ARPar{R}$]\index{AR@$\ARPar{R}$ - for
$R\subseteq\mathbb R^{\ge0}$}\label{def:AR-FU}
Let $R\subsetneq[0,\infty)$ be a simple non-empty set. Note that
$t_{R}>0$. Let $\ARPar{R}$ be the automaton
$(Q,\digitDotSet,\delta,q_{0},F)$ where:
\begin{itemize}
\item the set $Q$ of state contains:
\begin{itemize}
\item a state $q_{i}$ for all $i\in[t_{R}-1]$,
\item a state $\emptyState[R]$,
\item a state $\inftyState[R]$ if $[T_{R},\infty)\subseteq R$,
\item a state $(i,q)$ for all $i\in[t_{R}-1]$, and for each state
$q\in Q_{i}\setminus \emptyState[\mathcal{A}_{i}]$.
\end{itemize}
\item The initial state is $q_{0}$.
\item The accepting states are $\zuState$ and $(i,q)$ for all
accepting states $q$ of $\mathcal{A}_{i}$.
\item The transition function is such that, for all $i\in[t_{R}-1]$,
$a\in\digitSet$:
\begin{itemize}
\item $\delta(q,a)=q$ for $q$ being $\zuState$, $\inftyState[R]$
or $\emptyState[R]$,
\item $\delta(q,\star)=\emptyState[R]$ for $q$ being $\zuState$ or
$\emptyState[R]$,
\item $\delta(\inftyState[R],\star)$ is $\zuState[R]$,
\item $\delta(q_{i},a)$ is $bi+a$ if $bi+a< t_{R}$,
\item $\delta(q_{i},a)$ is $\inftyState[R]$ if $bi+a\ge t_{R}$
and if $[t_{R},\infty)\subseteq R$,
\item $\delta(q_{i},a)$ is $\emptyState[R]s$ if $bi+a\ge t_{R}$
and if $[t_{R},\infty)\cap R=\emptyset$,
\item $\delta(q_{i},\star)=(i,q_{0,i})$,
\item $\delta((i,q),a)$ is $(i,\delta_{i}(q,a))$ for
$q\in\mathcal{A}_{i}$, if $\delta_{i}(q,a)\not\in\emptyStates[\mathcal{A}_{i}]$
\item $\delta((i,q),a)$ is $\emptyState$ for $q\in\mathcal{A}_{i}$, if
$\delta_{i}(q,a)\in\emptyStates[\mathcal{A}_{i}]$ and
\item $\delta((i,q),\star)$ is $\emptyState[R]$.
\end{itemize}
\end{itemize} \end{definition} An example of automaton $\mathcal{A}_{R}$ is now given, resuming Example \ref{ex:unbounded-follow}. \begin{example}\label{ex:construct-RR}
Let $R=\left(\frac13,2\right]\cup\left(\frac83,3\right]\cup\left(\frac{11}3,\infty\right]$, as in Example \ref{ex:unbounded-follow}. The automaton
$\ARPar{R}$ is pictured in Figure \ref{fig:ex-AR-R>0}, without the non
accepting state $\emptyState$. Its minimal quotient is pictured in
Figure \ref{fig:ex-AR-R>0-min}. \end{example}
Let us now show that $\ARPar{R}$ is as expected. \begin{proposition}\label{prop:ar-recognizes-R-unbound}
Let $R\subsetneq\mathbb R^{\ge0}$ be a simple non-empty set. The automaton
$\ARPar{R}$ recognizes $\toInfWord{R}$. \end{proposition} \begin{proof}
Let $\natPart{w}\in\digitSet^{*}$, $\fraPart{w}\in\digitSet^{\omega}$ and
$w=\natPart{w}\star{}\fraPart{w}$. Let $\natPart{r}=\wordToNatural{\natPart{w}}$,
$\fraPart{r}=\wordToReal{\fraPart{w}}$ and $r=\wordToReal{w}=\natPart{r}+\fraPart{r}$. Let us prove that
$\wordToReal{w}\in R$ if and only if $w$ is accepted by $\ARPar{R}$.
By an easy induction on the length of $\natPart{w}$,
$\del{q_{0}}{\natPart{w}}$ is $q_{\natPart{w}}$ if
$\natPart{r}<t_{R}$, otherwise it is $\inftyState[R]$ if
$[t_{R},\infty)\subseteq R$ and it is $\emptyState$ otherwise. Two
cases are considered, depending on whether $\natPart{r}<t_{R}$ or
whether $\natPart{r}\ge t_{R}$.
\begin{itemize}
\item Let us first assume that $\natPart{r}<t_{R}$. Then
$\delta(0,\natPart{w})=q_{\natPart{w}}$ and thus
$\delta(0,\natPart{w}\star{})=(\natPart{r},q_{0,\natPart{r}})$. By
Lemma \ref{lem:ar-01-correct}, $\mathcal{A}_{\natPart{r}}$ recognizes
$R_{\natPart{r}}$, hence $\fraPart{w}$ is accepted by
$(\ARPar{R})_{(\natPart{r},q_{0,\natPart{r}})}$ if and only if
$\fraPart{r}\in \ARPar{{\natPart{r}}}$. Hence $w$ is accepted by $\ARPar{R}$
if and only if $r\in R$.
\item Let us now assume that $\natPart{r}\ge t_{R}$. Let us assume
that $[t_{R},\infty)\subseteq R$, the case
$[t_{R},\infty)\cap R=\emptyset$ is similar. Since
$r_{i}\ge t_{R}$, then $r\in R$. By definition of $\ARPar{R}$,
$\del{q_{0}}{\natPart{w}}=\inftyState[R]$, therefore
$\del{q_{0}}{\natPart{w}\star}=\zuState$. It follows that
each state of the run of $\ARPar{R}$ on $w$ is $\zuState$, apart from
the $\length{\natPart{w}\star}$ first ones. Since furthermore
$\zuState$ is an accepting state, $\ARPar{R}$ accepts $w$.
\end{itemize} \end{proof}
\subsection{Characterization of automata recognizing simple
sets}\label{subsec:char:aut} The first main theorem of this paper is now given. \begin{theorem}\label{theo:aut-R}
It is decidable in time $\bigO{nb}$ whether a minimal weak
{}Büchi RNA with $n$ states recognizes a simple set. \end{theorem} In order to simplify the proof, this theorem is reduced to a simpler case given in the following proposition. \begin{proposition}\label{prop:aut-R}
It is decidable in time $\bigO{nb}$ whether a minimal automaton with
$n$ states recognizes a simple set different from $\emptyset$ and from
$[0,\infty)$. \end{proposition} As in Section \ref{subsec:char:bounded-aut}, a set of automata $\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$ is now introduced. Four lemmas are then proved, which corresponds to the four properties of Proposition \ref{prop:method}. \begin{definition}[$\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$]\index{A@$\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$}\label{def:BAR}
Let $\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$ be the set of weak {}Büchi automata
$\mathcal{A}=\autPar{Q}{\digitDotSet}{\delta}{q_{0}}{F}$ such that:
\begin{enumerate}
\item\label{BAR-bar} The automaton $\mathcal{A}$ satisfies the properties of
Definition \ref{def:BAF}.
\item\label{BAR-non-trivial} There exists an accepting and a
rejecting strongly connected component, accessible from the initial
state, belonging to $\fraStates$.
\item\label{BAR-q0-0} $\del{q_{0}}{0}=q_{0}$.
\item\label{BAR-empty-singleton} The set $\emptyStates$ contains
exactly one recurrent state. Let $\emptyState$ denotes its only
state.
\item\label{BAR-inf-singleton} The set $\inftyStates$ contains at
most one recurrent element. If $\inftyStates$ contains one recurrent
element, let $\inftyState$ denote this only element.
\item\label{BAR-q0-a}
$\del{q_{0}}{a}\neq_{0}$ for all $0<a<b$.
\item\label{BAR-infty-or-empty} If $\inftyState$ exists, then
$\del{q}{a}\ne\emptyState$ for all
$q\in\natStates\setminus\set\emptyState$ and $a\in\digitSet$.
\item\label{BAR-recurrent-nat}\label{BAR-last} Let $q$ be a natural
recurrent state.
The state $q$ is either $\emptyState$, $\inftyState$ or $q_{0}$.
\end{enumerate} \end{definition} Note that the properties of Definition \ref{def:BAF} only consider the states of $\fraStates$. Therefore, there is no trouble to state that those properties are satisfied on some subset of an automaton closed under the function $\delta$. Let us show that $\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$ admits the properties of Proposition \ref{prop:method}. It is now proven that the set $\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$ satisfies Property \eqref{prop:method-algo} of Proposition \ref{prop:method}. \begin{lemma}\label{lem:dans-BAR}
It is decidable in time $\bigO{nb}$ whether a weak{} Büchi automaton
$\mathcal{A}$ with $n$ states belong to $\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$. \end{lemma} \begin{proof}
It suffices to check that all properties of Definition \ref{def:BAR} are
testable in time $\bigO{nb}$. To check Property \eqref{BAR-bar}, it
suffices to use the algorithm of Lemma \ref{lem:dans-baf}.
The set of states accessible from $q_{0}$ is easily computed by a
fixed-point algorithm in time $\bigO{nb}$. Using Tarjan's
algorithm~\cite{Tarjan}, the set of strongly connected component are
computable in time $\bigO{nb}$. By Lemma \ref{lem:comp-set}, the sets
$\emptyStates$ and $\inftyStates$ are computable in time
$\bigO{nb}$. It easily follows that testing whether $\inftyState$
exists is testable in time $\bigO{nb}$. Since the set of recurrent
states, the states $\emptyState$ and $\inftyState$, the set of
states accessible from $q_{0}$, the sets $\emptyStates$,
$\inftyStates$, $\natStates$ and $\fraStates$ are computed, it is
trivial to test the seven last properties in time $\bigO{nb}$. \end{proof} It is now proven that the set $\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$ satisfies Property \eqref{prop:method-AR} of Proposition \ref{prop:method}. \begin{lemma}\label{lem:AR-in-BAR}
Let $R\subsetneq[0,\infty)$ be a simple non-empty set. The
automaton $\ARPar{R}$ belongs to $\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$. \end{lemma} \begin{proof}
By Lemma \ref{lem:AR-in-BAF} and Lemma \ref{lem:quotient-BAF}, all
automata $\mathcal{A}_{i}$ belongs to $\mathbb{A}_{F,\mathcal{S}}$. Since Property \eqref{BAR-bar} only consider the states of $\fraStates$, that is the
states of the form $(i,q)$ for $q\in Q_{i}$, then $\mathcal{A}_{R}$
satisfies Property \eqref{BAR-bar}.
Property \eqref{BAR-non-trivial} is now considered. Since $R$ is
neither the empty set nor $[0,\infty)$, there exists a word
belonging to $\digitSet^{*}\star\digitSet^{\omega}$ which is
rejected by $\mathcal{A}_{R}$ and a word belonging to
$\digitSet^{*}\star\digitSet^{\omega}$ which is accepted by
$\mathcal{A}_{R}$. Therefore $\mathcal{A}_{R}$ satisfies Property \eqref{BAR-non-trivial}.
Property \eqref{BAR-q0-0} is now considered. Recall that the
threshold of $R$, $t_{R}$ is positive. Thus $b0+0<t_{R}$ and by
construction of $\mathcal{A}_{R}$, $\del{q_{0}}{0}=q_{b0+0}=q_{0}$,
therefore $\mathcal{A}_{R}$ satisfies \ref{BAR-q0-0}.
The automaton $\mathcal{A}_{R}$ satisfy Properties
\ref{BAR-empty-singleton}, \ref{BAR-inf-singleton} and
\ref{BAR-infty-or-empty} by construction.
Property \eqref{BAR-q0-a} is now considered. Let $a>0$,
$q\in\natStates$ and let us prove that $\del{q}{a}\ne q_{0}$. Two
cases must be considered, depending on whether $t_{R}\ge a$ or
whether $a<t_{R}$. Let us first assume that $t_{R}\ge a$. Note that
$b0+a\ge t_{R}$. By construction of $\mathcal{A}_{R}$, $\del{q_{0}}{a}$ is
either $\inftyState[R]$ or is $\emptyState[R]$, which are distinct
from $q_{\emptyState}$. Finally, let us assume that $a<t_{R}$. By
construction of $\mathcal{A}_{R}$ $\del{q_{0}}{a}=q_{b0+a}=q_{a}\ne q_{0}$,
therefore $\mathcal{A}_{R}$ satisfy Property \eqref{BAR-q0-a}.
Property \eqref{BAR-recurrent-nat} is now considered. By construction
of $\mathcal{A}_{R}$, there are at most three recurrent states in
$\natStates$: the initial state, the state
$\emptyState[R]\in\emptyStates[\mathcal{A}_{R}]$, and potentially
$\inftyState[R]\in\inftyStates[\mathcal{A}_R]$. Therefore $\mathcal{A}_{R}$
satisfy Property \eqref{BAR-recurrent-nat}. \end{proof}
It is now proven that the set $\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$ satisfies Property \eqref{prop:method-min} of Proposition \ref{prop:method}. \begin{lemma}\label{lem:quotient-BAR}
The minimal quotient of automata of $\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$ belong to $\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$. \end{lemma} \begin{proof}
Let $\mathcal{A}=\autPar{Q}{\digitDotSet}{\delta}{q_{0}}{F}$ belonging
to $\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$ and let
$\mathcal{A}'=\autPar{Q'}{\digitDotSet}{\delta'}{q'_{0}}{F'}$ be a quotient
of $\mathcal{A}$ by a morphism $\mu$. It is now shown that $\mathcal{A}'$ belongs
to $\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$. Each of the \ref{BAR-last} propositions of
Definition \ref{def:BAR} are considered separately.
Property \eqref{BAR-bar} is first considered. By
Lemma \ref{lem:quotient-BAF}, the set of automata satisfying Property \eqref{BAR-bar} is closed under quotient. Since $\mathcal{A}$ satisfies
Property \eqref{BAR-bar}, and since $\mathcal{A}'$ is a quotient of $\mathcal{A}$,
$\mathcal{A}'$ satisfies Property \eqref{BAR-bar}.
Property \eqref{BAR-non-trivial} is now considered. It is now proven
that there exists an accepting recurrent state $q'$ accessible from
the initial state of $\mathcal{A}'$. The case of a non-accepting
state is similar. By Property \eqref{BAR-non-trivial}, there exists an
accepting recurrent state $q$ accessible from $q_{0}$, in
$\fraStates$. Since $q$ is accessible from $q_{0}$, there exists a
finite word such that $\del{q_{0}}{w}=q$. Since $q$ is a recurrent,
there exists a non-empty word $v$ such that $\del{q}{v}=q$. Since
$q$ is recurrent and accepting, it does not belongs to
$\emptyStates$. Since $q$ is fractional, since $\mathcal{A}$ accepts
a subset of $\digitSet^{*}\star\digitSet^{\omega}$, and since
$q\not\in\emptyStates$, then
$w\in\digitSet^{*}\star\digitSet^{*}$. Note that
$\dell{\mu(q)}{v}=\mu(\del{q}{v})=\mu(q)$, therefore $\mu(q)$ is
recurrent. By Lemma \ref{lem:morphism-sets}, $\mu(q)$ is
fractional. Furthermore
$\dell{q'_{0}}{w}=\dell{\mu(q_{0})}{w}=\mu(\del{q_{0}}{w})=\mu(q)$,
therefore $q$ is accessible from $q'_{0}$. Since $q$ is an
accepting recurrent state of $\fraStates[\mathcal{A}']$, accessible
from $q'_{0}$, $\mathcal{A}'$ satisfies Property \eqref{BAR-non-trivial}.
Property \eqref{BAR-q0-0} is now considered. Since
$\dell{q'_{0}}{0}=\dell{\mu(q_{0})}{0}=
\mu(\del{q_{0}}{0})=\mu(q_{0})=q'_{0}$,
the automaton $\mathcal{A}'$ satisfies Property \eqref{BAR-q0-0}.
Properties \ref{BAR-empty-singleton} and \ref{BAR-inf-singleton} are
now considered. By Lemma \ref{lem:min-empty-infty}, since
$\mathcal{A}'$ is minimal, there is at most one state in
$\inftyStates[\mathcal{A}']$ and in
$\emptyStates[\mathcal{A}']$. Furthermore
$\del{q_{0}}{\star\star}$ belongs to
$\emptyStates[\mathcal{A}']$, hence $\emptyStates[\mathcal{A}']$ is
not empty. Hence $\mathcal{A}'$ satisfies Properties
\ref{BAR-empty-singleton} and \ref{BAR-inf-singleton}.
Property \eqref{BAR-q0-a} is now considered. Let $0<a<b$, it is now
proven that $\dell{q'_{0}}{a}\ne q'_{0}$. By Property \eqref{BAR-q0-a}, $\del{q_{0}}{a}\ne q_{0}$. By
Lemma \ref{lem:run-recurrent}, there exists $i\in\mathbb N^{>0}$ such that
$\del{q_{0}}{a^{i}}$ is recurrent. Let $q=\del{q_{0}}{a^{i}}$. By
Property \eqref{BAR-recurrent-nat}, $q$ is either $q_{0}$,
$\emptyState$ or $\inftyState$. The three cases are considered
separately.
\begin{itemize}
\item It is first assumed that $q=q_{0}$, in this case, all of the
states $\del{q_{0}}{a^{i}}$ are in the same strongly connected
component. And since $\del{q_{0}}{a}\ne q_{0}$, Property \eqref{BAR-recurrent-nat} implies that $\del{q_{0}}{a}$ is either
$\emptyState$ or $\inftyState$. The case where $\del{q_{0}}{a}$ is
$\emptyState$ is now considered. The other case is similar. Since
$\emptyState$ and $q_{0}$ are in the same strongly connected
component, $q_{0}$ is accessible from $\emptyState$. Therefore
$\mathcal{A}$ recognizes the empty language. Having $\mathcal{A}$ recognizing
the empty language contradicts Property \eqref{BAR-non-trivial}.
\item It is now assumed that $q = \emptyState$, the case
$q=\inftyState$ is similar. By Property \eqref{BAR-empty-singleton},
since $q=\emptyState$, $\mu(q)\in\emptyStates[\mathcal{A}']$.
Furthermore,
$\mu(q)=\mu(\del{q_{0}}{a^{i}})=\dell{\mu(q_{0})}{a^{i}}=\dell{q'_{0}}{a^{i}}$.
Note that $\dell{q'_{0}}{a^{i}}=\inftyState[\mathcal{A}']\ne q'_{0}$.
Since $\dell{q'_{0}}{a^{i}}\ne q'_{0}$ it follows that
$\dell{q'_{0}}{a}\ne q'_{0}$. Therefore, $\mathcal{A}'$ satisfies
Property \eqref{BAR-q0-a}.
\end{itemize}
Property \eqref{BAR-infty-or-empty} is now considered. Let us assume
that $\inftyState[\mathcal{A}']$ exists. Let $q'\in Q'$ and
$a\in\digitSet$ such that
$\dell{q'}{a}=\emptyState[\mathcal{A}']$. It must be proven that
$q'\not\in\natStates[\mathcal{A}']\setminus\set{\emptyState[\mathcal{A}']}$. Since
the automaton is minimal, the strongly connected component of
$\inftyState[\mathcal{A}']$ is $\set{\inftyState[\mathcal{A}']}$.
By Lemma \ref{lem:scc-morphism-codomain}, there exists a strongly
connected component $C$ in $\mathcal{A}$ such that
$\mu(C)=\set{\inftyState[\mathcal{A}']}$. Let $\inftyState$ be a
state of $C$, since $\mathcal{A}_{\inftyState}$ and
$\mathcal{A}'_{\mu(\inftyState)}=\mathcal{A}'_{\inftyState[\mathcal{A}']}$
recognizes the same language, and since
$\inftyState[\mathcal{A}']\in\inftyStates[\mathcal{A}']$ then
${\inftyState}\in \inftyStates$. Since $\inftyState$ belongs to a
strongly connected component, it is recurrent. Since $\inftyState$
is a recurrent state belonging to $\inftyState$, by Property \eqref{BAR-infty-or-empty}, $\del{q}{a}\ne\emptyState$ for all
$q\in\natStates\setminus\set{\emptyState}$ and $a\in\digitSet$. By
definition of morphism, there exists a state $q\in Q$ such that
$\mu(q)=q'$. By Lemma \ref{lem:morphism-sets}, since
$q'\in\natStates[\mathcal{A}']$, $q\in\natStates$. Note that
$\mu(\del{q}{a})=\dell{\mu(q)}{a}=\dell{q'}{a}=\emptyState[\mathcal{A}']\in\emptyStates[\mathcal{A}']$.
By Lemma \ref{lem:morphism-sets}, since
$\mu(\del{q}{a})\in\emptyStates[\mathcal{A}']$,
$\del{q}{a}\in\emptyStates$. Since $\del{q}{a}\in\emptyStates$, by
Property \eqref{BAR-empty-singleton}, $\del{q}{a}=\emptyState$. It
implies that $q\not\in\natStates\setminus\set{\emptyState}$. By
Lemma \ref{lem:morphism-sets}, it implies that
$q'\not\in\natStates[\mathcal{A}']\setminus\set{\emptyState[\mathcal{A}']}$. Therefore,
$\mathcal{A}'$ satisfies Property \eqref{BAR-infty-or-empty}.
Finally, Property \eqref{BAR-recurrent-nat} is now considered. Let
$q'$ be a natural recurrent state of $\mathcal{A}'$. It must be proven that
$q'$ is either $q_{0}$, $\emptyState[\mathcal{A}']$ or
$\inftyState[\mathcal{A}']$. The state $q'$ is natural, by
Lemma \ref{lem:morphism-sets}, its antecedents by $\mu$ are natural.
The state $q'$ is recurrent hence, by
Lemma \ref{lem:scc-morphism-codomain}, it admits a recurrent
antecedent $q$. By Property \eqref{BAR-recurrent-nat} applied to
$\mathcal{A}$, either $q=q_{0}$, $q=\emptyState$ or $q=\inftyState$. The
three cases are considered separately.
\begin{itemize}
\item The case $q=q_{0}$ is first considered, in this case, clearly,
$q'=q'_{0}$.
\item The case $q=\emptyState$ is now considered. As proven above,
it implies that $\mu(q)=\emptyState[\mathcal{A}']$.
\item The case where $q=\inftyState$ is similar to the preceding
case.
\end{itemize}
Therefore, $\mathcal{A}'$ satisfies Property \eqref{BAR-recurrent-nat}. \end{proof} It is now proven that the set $\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$ satisfies Property \eqref{prop:method-L} of Proposition \ref{prop:method}. \begin{lemma}\label{BAR->pour}
The automata of $\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$ recognize simple sets. \end{lemma} In order to prove this lemma, another lemma is first introduced. It implies that the set $R$ recognized by an automaton $\mathcal{A}\in\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$ with $n$ states is such that $[0,b^{n})$ is either a subset of $R$ or is disjoint from $R$. \begin{lemma}\label{lem:cycle-N}
Let $\mathcal{A}\in\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$ be an automaton with $n$ states recognizing a set
$R$. If $\mathcal{A}$ contains a state $\inftyState$, as in
Definition \ref{def:BAR}, then $(b^{n-1},\infty)\subseteq R$, otherwise
$(b^{n-1},\infty)\cap R=\emptyset$. \end{lemma} \begin{proof}
Two cases must be considered, depending on whether the state
$\inftyState$ exists. Let us assume that the state $\inftyState$
exists, the other case is similar. Let $x>b^{n-1}$ and let
$0^{c}\natPart{w}\star \fraPart{w}$, be one of its encoding in base
$b$, with $c\in\mathbb N$ and $\natPart{w}[0]\ne0$. Let us prove that $\mathcal A$
accepts $0^{c}\natPart{w}\star \fraPart{w}$
Note that since $x>b^{n-1}$, it implies that the length of
$\natPart{w}$ is at least $n$. For $i\le\length{\natPart{w}}$, let
$q_{i}^{I}=\del{q_{0}}{0^{c}\prefix{\natPart{w}}{i}}$. By Lemma
\ref{lem:run-recurrent}, there exists $0\le i'<i\le n$ such that
$q^{I}_{i}$ is recurrent. By Property \eqref{BAR-recurrent-nat} of
Definition \ref{def:BAR}, the only natural recurrent states of
$\mathcal{A}$ are $\inftyState$, $\emptyState$ and $q_{0}$. By
Properties \ref{BAR-q0-a} and \ref{BAR-infty-or-empty},
$q^{I}_{i}\ne q_{0}$. Since $\inftyState$ exists, by Property \eqref{BAR-infty-or-empty} then ${q^{I}_{i}}\ne\emptyState$. Since
$q^{I}_{i}$ is either $\inftyState$, $\emptyState$, or $q_{0}$,
since ${q^{I}_{i}}\ne\emptyState$ and since
$q^{I}_{i}\ne q_{0}$ $q^{I} _{i}=q_{\inftyState}$. It follows
that $\del{q^{I}_{i}}{\natPart{w}}= \inftyState$, and then, for all
$j\in\mathbb N$,
$\del{q^{I}}{0^{c}\natPart{w}\star\prefix{\fraPart{w}}{j}}=
\zuState$.
Therefore $\mathcal{A}$ accepts $0^{c}\natPart{w}\star \fraPart{w}$. \end{proof} Example \ref{BAR->pour} is now proven \begin{proof}[Proof of Example \ref{BAR->pour}]
Let $\mathcal{A}\in \mathbb{A}_{\R^{\ge0,\mathcal{S}}}$ with $n$ states and let
$R=\wordToReal{\mathcal{A}}$. Let us prove that $R$ is simple. By
Lemma \ref{lem:cycle-N}, it suffices to prove that $R\cap[0,b^{n-1})$ is
simple. In order to do this, it suffices to prove that $R_{i}$ is
simple for all $i\in[b^{n-1}-1]$.
By Property \eqref{BAR-bar} of Definition \ref{def:BAR},
$\mathcal{A}_{\del{q_{0}}{w\star}}\in\mathbb{A}_{F,\mathcal{S}}$ for all
$w\in\digitSet^{*}$. Note that the difference between $R_{i}$ and
$R'_{i}=\bigcup_{j\in\mathbb N}\wordToReal{\mathcal{A}_{\del{q_{0}}{0^{j}w\star}}}$
is a subset of $\set{0,1}$. Therefore, it suffices to prove that
$R'_{i}$ is simple. Note that, since $\del{q_{0}}{0}=q_{0}$, all
$\mathcal{A}_{\del{q_{0}}{0^{j}w\star}}$ are equals. Therefore the
infinite union $R'_{i}$ of simple sets is a simple set. That is,
$R'_{i}$ is a simple set. \end{proof} Proposition \ref{prop:method} is now proven. Its proof is similar to the proof of Theorem \ref{theo:aut-0-1}. \begin{proof}[Proof of Proposition \ref{prop:method}]
It suffices to use Proposition \ref{prop:method} with $\mathbb A$ being the set
of automata $\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$, $\mathbb A'$ being the set of RNA, and Lemmas
\ref{lem:dans-BAR}, \ref{lem:AR-in-BAR}, \ref{lem:quotient-BAR} and
\ref{BAR->pour}. \end{proof} Theorem \ref{theo:aut-R} is now proven. \begin{proof}[Proof of Theorem \ref{theo:aut-R}]
The algorithm to decide whether a minimal weak
{}Büchi RNA $\mathcal{A}$ recognizes a simple set
is now given. The algorithm first checks whether Property \eqref{BAR-non-trivial} of Definition \ref{def:BAR} holds. If it does not
hold, then the automaton recognizes the empty language or
$\digitSet^{*}\star\digitSet^{\omega}$, in which cases the
algorithm accepts. If Property \eqref{BAR-non-trivial} holds, then
the automaton recognizes a non empty strict subset of
$\digitSet^{*}\star\digitSet^{\omega}$ and therefore the problem
is reduced to the problem considered in
Proposition \ref{proposition:bounded-construct}. It thus suffices to apply
the algorithm of Proposition \ref{proposition:bounded-construct} to $\mathcal{A}$
and to return the result of this algorithm. \end{proof}
The algorithm of Theorem \ref{theo:aut-R} takes as input a RNA and runs in time $\bigO{nb}$. It should be noted that it is not known whether it is decidable in time $\bigO{nb}$ whether an automaton is a RNA. However, as for the algorithm of Theorem \ref{theo:aut-0-1}, if the algorithm of Theorem \ref{theo:aut-R} is applied to a weak {}Büchi automaton which is not a real automaton, the algorithm returns no false positive. An example of false negative is now given. Let $L$ be the language described by the regular expression: \begin{equation*}
\left(00\right)^{*}\left(01+2\right)\digitSet[3]^{*}\star
\digitSet[3]^{\omega}. \end{equation*} This language is recognized by the automaton of Figure \ref{fig:false-neg}. Note that $\wordToReal{L}=[1,\infty)$, which is a simple set. However, since Property \eqref{BAR-q0-0} is not satisfied by the automaton of Figure \ref{fig:false-neg}, the algorithm of Theorem \ref{theo:aut-R} does not accept this automaton. \begin{figure}
\caption{An automaton which recognizes $[1,\infty)$ and is refused by
the algorithm of Theorem \ref{theo:aut-R}.}
\label{fig:false-neg}
\end{figure}
\section{From automata to simple set}\label{sec:aut->set} In this section, it is explained, given a weak {}Büchi automaton recognizing a simple set $R$, how to compute a first-order formula which defines $R$. The exact theorem is now stated.
\begin{theorem}\label{theo:bounded-construct}
Let $\mathcal{A}$ be a be a minimal RNA with $n$ states, over the alphabet
$\digitSet$, which recognizes a simple set. There exists two
formulas which define $\wordToReal{\mathcal{A}}$:
\begin{itemize}
\item a $\ef{\mathbb R;+,<,1}$-formula computable in time
$\bigO{n^{2}b\log(nb)}$ and
\item a $\sigF{2}{\mathbb R;+,<,1}$-formula computable in time
$\bigO{nb\log(nb)}$.
\end{itemize} \end{theorem}
In order to prove this theorem, a more general proposition is introduced. This proposition shows that the condition that $\mathcal{A}$ is a RNA can be replaced by the condition $\mathcal{A}\in\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$. \begin{proposition}\label{proposition:bounded-construct}
Let $\mathcal{A}\in\mathbb{A}_{F,\mathcal{S}}$ be a minimal automaton. There exists two formulas
which define $\wordToReal{\mathcal{A}}$:
\begin{itemize}
\item a $\ef{\mathbb R;+,<,1}$-formula computable in time
$\bigO{n^{2}b\log(nb)}$ and
\item a $\sigF{2}{\mathbb R;+,<,1}$-formula computable in time
$\bigO{nb\log(nb)}$.
\end{itemize} \end{proposition}
In order to prove this proposition, a technical lemma is first introduced. \begin{lemma}\label{lem:C}
Let $A\in\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$ be minimal with $n$ states, $\natPart{w}\in\digitSet^{*}$,
$\fraPart{w}\in\digitSet^{\omega}$ and $\mathcal{Q}\subseteq Q$ containing
exactly one state of each strongly connected component. Then, there
exists $s\in[n]$ such that
$\del{q}{\natPart{w}\star(\prefix{\fraPart{w}}s)}\in \mathcal{Q}$. \end{lemma} \begin{proof}
Let
$q_{i}^{F}=\del{q_{0}}{\natPart{w}\star(\prefix{\fraPart{w}}i)}$
for any $i\in\mathbb N$. By Lemma \ref{lem:run-recurrent}, there exists
$0\le i<i'\le n$ such that $q_{i}^{F}$ is recurrent. Let $C$ be
the strongly connected component of $q_{i}^{F}$. By Property \eqref{BAR-recurrent-nat} of Definition \ref{def:BAR}, there are three kinds
of strongly connected components in the fractional part of a minimal
automaton of $\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$: the singleton $\set{\emptyState}$, the
singleton $\set{\zuState}$, and the cycles. Three cases must be
considered depending on the kind of strongly connected components
that is $C$. In the two first cases, if $q^{F}_{i}$ is
$\emptyState$ or $\zuState$ then $q^{F}_{i}\in\mathcal{Q}$ and it
suffices to take $s=i$. Otherwise, if $C$ is a cycle, then
$\set{q^{F}_{j}\mid i\le j<i'}=C$, therefore there exists an
integer $i\le s<i'$ such that $q^{F}_{s}\in C$. \end{proof}
Proposition \ref{proposition:bounded-construct} can now be proven. \begin{proof}
Let $R=\wordToReal{\mathcal{A}}$. Since the automaton $\mathcal{A}$
belongs to $\mathbb{A}_{\R^{\ge0,\mathcal{S}}}$, the notations of Definition \ref{def:BAR} can now be
used, and therefore the notations of Definition \ref{def:BAF} can also
be used. Recall that all strongly connected components are cycles,
apart from $\setq_{0}$, $\set\inftyState$, $\set\zuState$ and
$\set\emptyState$. Furthermore, for each $q\in\fraStates$ in a cycle
$C$, the digit $\suc{q}$ is the only one such that
$\del{q}{\suc{q}}\in C$. In this proof, it is assumed that each
state has an integer index between $0$ and $n-1$. For $q$ a state
with index $i$ and $p$ a variable, the atomic first-order formula
$p\doteq q$ is an abbreviation for $p\doteq i$. Let
$\mathcal{Q}\subseteq Q$ be a set as in Lemma \ref{lem:C}. Note
that the states $\emptyState$, $\zuState$ and $\inftyState$ all
belong to $\mathcal{Q}$ if they exists.
It is first explained how to compute an existential formula
$\phi(x)$ which defines $R$ in time
$\bigO{n^{2}\log(n)b\log(b)}$. As seen in Lemma \ref{lem:cycle-N},
the formula which defines $R$ is either $\phi(x)\lor x\ge b^{n-1}$
if $\inftyState$ exists, or $\phi(x)$ otherwise. At the end of the
proof, it is explained how to decrease the time by adding universal
quantifiers.
For any real $0\le x<b^{n-1}$, let $\natPart{x}\in\mathbb N$ and
$x^{F}\in[0,1]$ be numbers such that
$x=\natPart{x}+\fraPart{x}$. If $x\in\mathbb N^{>0}$, then the pair
$\left(\natPart{x},\fraPart{x}\right)$ is either
$\left(\floor{\natPart{x}},x-\floor{\natPart{x}}\right)$ or $\left(x-1,1\right)$.
Otherwise, the pair $\left(\natPart{x},\fraPart{x}\right)$ is
$\left(\floor{\natPart{x}},x-\floor{\natPart{x}}\right)$. Let
$w_{\natPart{x}}\in\digitSet^n$ be an encoding in base $b$ of $\natPart{x}$ of
length $n$. Since $\natPart{x}<b^{n-1}$, such an encoding exists. Let
$w_{\fraPart{x}}\in\digitSet^{\omega}$ be an encoding in base $b$ of
$\fraPart{x}$. Let
$q^{I}_{x,i}=\del{q_{0}}{\prefix{w_{\natPart{x}}}{i}}$ for all
$i\in[n]$ and
$q^{F}_{x,i}=\del{q_{0}}{w_{\natPart{x}}\star(\prefix{w_{\fraPart{x}}}
i)}$ for all $i\in\mathbb N$.
\paragraph{}
The formula $\phi\left(x\right)$ which defines $\wordToReal{\ARPar{R}}$ is
defined as the conjunction of two subformulas. Intuitively, the
first formula, $\phi_{I}\left(p^{I},\natPart{x}\right)$ considers
the run on $\natPart{w}$ and the second formula,
$\fraPart{\phi}\left(p^{F},\fraPart{x}\right)$, considers the run
on $\fraPart{w}$.
Let us assume that there exists a formula
$\phi_{I}\left(p^{I},\natPart{x}\right)$, of size
$\bigO{n^{2}b\log\left(nb\right)}$, such that, if
$\natPart{x_{n}}<b^{n-1}$, the formula holds if and only if $p^{I}$ is the
index of $\fraPart{w_{x}}$ and if $\fraPart{x}\in\mathbb N$. Let us assume
that there exists a formula
$\fraPart{\phi}\left(p^{F},\fraPart{x}\right)$, of size
$\bigO{n^{2}b\log(nb)}$, which accepts $\fraPart{x}\in[0,1]$ if and
only if $\mathcal{A}_{p^{I}}$ accepts an encoding of $\fraPart{x}$. Then it
suffices to take $\phi(x)$ to be the formula:
\begin{equation*}
\arraycolsep=0.5pt
\begin{array}{llll}
\phi(x)=\exists \natPart{x},\fraPart{x},p^{I},p^{F}.
& x\doteq \natPart{x}+\fraPart{x}
\land \natPart{x}<b^{n-1}\land
\fraPart{x}\in[0,1]
\\
&\land\bigvee_{q\in \natStates}\left(p^{I}\doteq{}q\land p^{F}\doteq{}\del{q}{\star}\right)
\\
&\land
\phi_{I}\left(p^{I},\natPart{x}\right)
\land
\fraPart{\phi}\left(p^{F},\fraPart{x}\right).
\end{array}
\end{equation*}
\paragraph{}
The formula $\phi_{I}\left(p^{I},\natPart{x}\right)$ is now
defined. A sequence $\left(\natPart{x_{i}}\right)_{i\in[n]}$ of
variables is existentially quantified in this formula. The variable
$\natPart{x_{i}}$ is intended to be interpreted by the value
$\floor{\frac{\natPart{x}}{b^{n-i}}}$, its encoding in base $b$ is
$\prefix{w_{\natPart{x}}}{i}$. A sequence
$\left(p_{i}^{I}\right)_{i\in[n]}$ of variables is existentially
quantified. They are used to encode the $n$ steps of the run of
$\mathcal{A}$ on $\natPart{w}$. More precisely, the variable
$p^{I}_{i}$ is meant to be interpreted by the index of
$q_{i}^{I}$. Note that if $\natPart{x_{0}}=0$, since
$\natPart{x_{i+1}}=b \natPart{x_{i}}+w_x[i]$ for $i\in[n-1]$, an easy
induction shows that $\natPart{x_{i}}\in\mathbb N$ for all $0\le i\le n$.
Let
us assume that there exists a $\qf{\mathbb R;+,<,1}$-formula
$\psi^{I}\left(p^{I}_{i},\natPart{x_{i}},p^{I}_{i+1},x_{i+1}^{I}\right)$,
of size $\bigO{nb\log(nb)}$, which asserts that if $p_{i}$ is the
index of a state $q$, and if $\natPart{x_{i+1}}= b\natPart{x_{i}}+a$ for
some $a\in\digitSet$, then $p_{i+1}$ is the index of $\del{q}{a}$.
The formula $\phi_{I}\left(p^{I},\natPart{x}\right)$ can then be
taken to be the $\ef{\mathbb R;+,<,1}$-formula of length
$\bigO{n^{2}b\log\left(nb\right)}$:
\begin{equation*}
\arraycolsep=0.5pt
\begin{array}{ll}
\phi_{I}\left(p^{I},\natPart{x}\right)
=
\exists \left(p_{i}^{I},\natPart{x_{i}}\right)_{0\le{i}\le{n}}.&
\natPart{x_{0}}\doteq0\land p_{0}^{I}\doteq q_{0}
\\&\land
\bigwedge_{i=0}^{n-1}\natPart{\psi}\left(p^{I}_{i},\natPart{x_{i}},p^{I}_{i+1},\natPart{x_{i+1}}\right)
\\&\land \natPart{x_{n}}\doteq \natPart{x}\land p_{n}^{I}\doteq p^{I}.
\end{array}
\end{equation*}
Formally, the formula
$\natPart{\psi}\left(p^{I}_{i},\natPart{x_{i}},p^{I}_{i+1},\natPart{x_{i+1}}\right)$
can be taken to be:
\begin{equation*}
\natPart{\psi}\left(
\begin{array}{l}
p^{I}_{i},\natPart{x_{i}},\\p^{I}_{i+1},\natPart{x_{i+1}}
\end{array}
\right) =
\bigwedge_{q\in{\natStates}}\bigwedge_{a\in\digitSet}
\left[
\begin{array}{l}
\left(
\natPart{x_{i+1}}\doteq b\natPart{x_{i}}+a\land p^{I}_{i}\doteq
q\right)\implies{}\\{p^{I}_{i+1}\doteq \del{q}{a}}
\end{array}
\right]
.
\end{equation*}
No notations introduced during the construction of the formula
$\phi_{I}\left(p^{I},\natPart{x}\right)$, are used in the
remaining of the proof.
\paragraph{}
The formula $\fraPart{\phi}\left(p^{F},\fraPart{x}\right)$ is now
defined. Let $s$ be the smallest integer such that
$q^{F}_{x,s}\in \mathcal{Q}$, by Lemma \ref{lem:C}, such an integer
exists. The formula
$\fraPart{\phi}\left(p^{F},\fraPart{x}\right)$ is the conjunction
of two subformulas. The first formula,
$\fraPart{\phi_{1}}\left(p,y,p^{F},\fraPart{x}\right)$, considers
the run of $\mathcal{A}$ on the first $s$ letters of $\fraPart{w_{x}}$. The
second formula $\fraPart{\phi_{2}}(p,y)$ considers the end of the
run, on $\suffix{w_x}{s}$, beginning at the state
$q_{x,s}\in\mathcal{Q}$.
Two variables $p$ and $y$ are existentially quantified. They are
meant to be interpreted by $p^{F}_{s}$ and
$\suffix{\fraPart{w_{x}}}{s}$ respectively. Assume that there is a
$\ef{\mathbb R;+,<,1}$-formula,
$\fraPart{\phi_{1}}\left(p,y,p^{F},\fraPart{x}\right)$, of length
$\bigO{n^{2}b\log(b)}$, which states that, given $p^{F}$,
$\fraPart{x}$'s, the variables $p$, $y$ are interpreted as stated
above. Let us also assume that there exists a
$\qf{\mathbb R;+,<,1}$-formula $\fraPart{\phi_{2}}(p,y)$, of length
$\bigO{n^{2}\log(b)}$, which states that $\mathcal{A}_{q}$ accepts
an encoding of $y$, where $p$ is the index of the state $q$. Then
the formula $\fraPart{\phi}$ can be taken to be the
$\ef{\mathbb R;+,<,1}$-formula of length $\bigO{n^{2}b\log(nb)}$:
\begin{eqnarray*}
\fraPart{\phi}\left(p^{F},\fraPart{x}\right)
=
\exists
p,y.
\fraPart{\phi_{1}}\left(p,y,p^{F},\fraPart{x}\right)
\land
\fraPart{\phi_{2}}(p,y).
\end{eqnarray*}
\paragraph{}
The formula,
$\fraPart{\phi_{1}}\left(p,y,p^{F},\fraPart{x}\right)$ is now
constructed. As in the construction of
$\fraPart{\phi}\left(p^{F},\fraPart{x}\right)$, two sequences of
variables are existentially quantified to encode some suffix of
$\fraPart{w}$ and to encode a part of the run of the automaton on
$w$. The sequence $\left(\fraPart{x_{i}}\right)_{i\in[n]}$ is
existentially quantified. The variable $\fraPart{x_{i}}$ is meant to
be interpreted by $\wordToReal{\suffix{w_{\fraPart{x}}}{i}}$. It
represents the real that must be read at the $i$-th step of the run
after the $\star$. It follows that $\fraPart{x_{0}}=\fraPart{x}$,
$\fraPart{x_{s}}=y$ and that $\fraPart{x_{i}}$ is equal to
$bx_{i-1}-w_{\fraPart{x}}[i]$. The sequence
$\left(p_{i}^{F}\right)_{i\in[n]}$ of variables is
existentially quantified. It is used to encode the first $(n+1)$ steps
of the run after the $\star$ part of the run. More precisely, the
variable $p^{F}_{i}$ is meant to be interpreted by the indexes
of $q_{i}^{F}$. A third sequence of variables,
$\left(s_{i}\right)_{i\in[n]}$, is existentially quantified. The
variable $s_{i}$ is meant to be interpreted by $0$ if $i< s$ and $1$
otherwise. Note that $s_{0}=0$ as $s\ge 0$. Those variables allows
to know the value of $s$.
Let
$\psi^{F}\left(p,y,p^{F}_{i},\fraPart{x_{i}},s_{i},p^{F}_{i+1},\fraPart{x_{i+1}},s_{i+1}\right)$
be $\qf{\mathbb R;+,<,1}$-formula of length $\bigO{nb\log(nb)}$ which
states that, given $p^{F}_{i},\fraPart{x_{i}},s_{i}$, the
variables $p^{F}_{i+1}$, $\fraPart{x_{i+1}}$ and $s_{i+1}$ are
correctly interpreted, and furthermore if $i=s$ -- that is if
$p^{F}_{i}\in\mathcal{Q}$ and for all $j<i$,
$p^{F}_{j}\not\in\mathcal{Q}$ -- then the variables $p$ and $y$
are correctly interpreted. The formula
$\fraPart{\phi_{1}}\left(p,y,p^{F},\fraPart{x}\right)$ can then
be expressed as the $\ef{\mathbb R;+,<,1}$-formula of length
$\bigO{n^{2}b\log(nb)}$:
\begin{equation*}
\arraycolsep=0.5pt
\fraPart{\phi_{1}}\left(\begin{array}{l}p,y,\\p^{F},\fraPart{x}\end{array}\right)
=
\begin{array}{ll}
\exists(p^{F}_{i},\fraPart{x_{i}},s_{i})_{i\in[n]}.
&p^{F}\doteq p^{F}_{0}\land \fraPart{x}\doteq \fraPart{x_{0}}\land
\\&\bigwedge_{i=0}^{n-1}\psi^{F}\left(
\begin{array}{l}
p,y,p^{F}_{i},\fraPart{x_{i}},\\s_{i},p^{F}_{i+1},\fraPart{x_{i+1}},s_{i+1}
\end{array}
\right).
\end{array}
\end{equation*}
The formula
$\fraPart{\psi}\left(p,y,p^{F}_{i},\fraPart{x_{i}},s_{i},p^{F}_{i+1},\fraPart{x_{i+1}},s_{i+1}\right)$
is now given. It is the $\qf{\mathbb R;+,<,1}$-formula of length
$\bigO{nb\log(nb)}$:
\begin{equation*}
\arraycolsep=0.5pt
\fraPart{\psi}\left(\begin{array}{llll}p^{F}_{i},&\fraPart{x_{i}},&s_{i},&p\\p^{F}_{i+1},&\fraPart{x_{i+1}},&s_{i+1}&y\end{array}\right)=
\begin{array}{l}
\left\{\left(s_{i+1}\doteq1\right)\iff{\left(s_{i}\doteq1\lor\bigvee_{q\in\mathcal{Q}}p^{F}_{i}\doteq{q}\right)}\right\}
\land\\
\left\{\bigvee_{q\in{Q}}\bigvee_{a=0}^{b-1} \left[
\begin{array}{rcl c rcll}
p^{F}_{i}&\doteq{}&q&\land&\fraPart{x_{i}}&\in&\left[\frac{a}{b},\frac{a+1}{b}\right]&
\land\\
p^{F}_{i+1}&\doteq{}&\delta(q,a)&\land& \fraPart{x_{i+1}}&\doteq{}&b\left(\fraPart{x_{i}}-\frac{a}{b}\right)&
\end{array}
\right]\right\}\land\\
\left\{\left[ s_{i+1}\doteq 1\land s_{i}\doteq0\right]\implies
{\left[p\doteq p^{F}_{i}\land y\doteq \fraPart{x_{i}}\right]}\right\}.
\end{array}
\end{equation*}
\paragraph{}
The formula $\fraPart{\phi_{2}}(t,y)$ is now constructed. It is a
disjunction, which states that there exists $q\in C$, such that
$q_{x,s}=q$, and such that $\mathcal{A}_{q_{x,s}}$ accepts an encoding of
$y=\fraPart{x_{s}}$. By definition of $s$, $q_{x,s}\in\mathcal{Q}$.
Let us assume that, for each $q\in\mathcal{Q}$ in a strongly connected
component $C$, there exists a $\qf{\mathbb R;+,<,1}$-formula $\xi_{q}(y)$
of length $\bigO{\log(b)\card C}$, where $\card C$ is the cardinal
of $C$, which states that $\mathcal{A}_{q}$ accepts an encoding of
$y$. Then, $\fraPart{\phi_{2}}(p,y)$ can be taken as the
$\qf{\mathbb R;+,<,1}$-formula of length $\bigO{n\log(nb)}$:
\begin{equation*}
\fraPart{\phi_{2}}(p,y)=\bigvee_{q\in\mathcal{Q}}p=q\land \xi_{q}(y).
\end{equation*}
Let us now construct the formula $\xi_{q}(y)$. Trivially,
$\xi_{\zuState}(y)$ can be taken to be $\true$ and
$\xi_{\emptyState}(y)$ can be taken to be $\false$. They are
constant size formula. It remains to construct the formulas
$\xi_{q}(y)$ for $q$ in a cycle $C$. Let $s_{q}$ be the value
$\suc{q}$ defined as in the proof of Lemma \ref{lem:BAF->pour}, for
the automaton $\mathcal{A}_{q}$. As shown in the proof of Lemma
\ref{lem:BAF->pour}, the length of $\suc{q}$ is
$\bigO{\log(b)\card C}$ and the automaton $\mathcal{A}_{q}$
recognizes a set, which is a union of $[0,\suc{q})$,
$\set{\suc{q}}$, and $(\suc{q},1]$, and which is defined by a
formula $\xi_{q}$ of length $\bigO{\log(b)\card C}$.
\paragraph{}
It is now explained how to transform the $\ef{\mathbb R;+,<,1}$-formula
$\phi(x)$ of length $\bigO{n^{2}b\log\left(nb\right)}$ into an
equivalent $\sigF{2}{\mathbb R;+,<,1}$-formula of length
$\bigO{nb\log(nb)}$. Let us assume that there exists
$\phi^{\prime[0,1]}_{1}\left(p,y,p^{F},\fraPart{x}\right)$ and
$\phi^{\prime\mathbb N}\left(p^{I},\natPart{x}\right)$, two
$\sigF{2}{\mathbb R;+,<,1}$-formulas of length $\bigO{nb\log(nb)}$,
equivalent to
$\fraPart{\phi_{1}}\left(p,y,p^{F},\fraPart{x}\right)$ and to
$\phi_{I}\left(p^{I},\natPart{x}\right)$ respectively. It thus
suffices to replace the two formulas
$\fraPart{\phi_{1}}\left(p,y,p^{F},\fraPart{x}\right)$ and
$\phi_{I}\left(p^{I},\natPart{x}\right)$ in $\phi(x)$ by their
equivalent smaller formulas.
In order to construct a $\sigF{2}{\mathbb R;+,<,1}$-formula of length
$\bigO{nb\log(nb)}$ equivalent to
$\fraPart{\phi_{1}}\left(p,y,p^{F},\fraPart{x}\right)$ or to
$\phi_{I}\left(p^{I},\natPart{x}\right)$, it suffices to replace
their last conjunctions by universal quantifications. The formula
$\fraPart{\phi_{1}}\left(p,y,p^{F},\fraPart{x}\right)$ is
equivalent to the following $\sigF{2}{\mathbb R;+,<,1}$-formula of length
$\bigO{nb\log(nb)}$:
\begin{equation*}
\fraPart{\phi_{1}}\left(
\begin{array}{l}
p,y,\\p^{F},\fraPart{x}
\end{array}
\right)=
\begin{array}{ll}
\exists
(p^{F}_{i},\fraPart{x_{i}},s_{i})_{i\in[n]}. p^{F}\doteq p^{F}_{0}\land \fraPart{x}\doteq \fraPart{x_{0}}\land\\
\forall \arraycolsep=0.5pt
\begin{array}{lll}
\rho,&\xi,&\gamma,\\\rho',&\xi',&\gamma'
\end{array}.
\left\{
\begin{array}{l}
\left[\bigvee_{i=0}^{n-1}\left(
\arraycolsep=0.5pt
\begin{array}{llllll}
\rho=p^{F}_{i}&\land& \xi=\fraPart{x_{i}}&\land&\gamma=\suc{i}&\land\\
\rho'=p^{F}_{i+1}&\land& \xi'=\fraPart{x_{i+1}}&\land& \gamma'=\suc{i+1}&
\end{array}
\right)
\right]
\\\implies{}
\fraPart{\psi}\left(
\arraycolsep=0.5pt
\begin{array}{llll}
\rho,&\xi,&\gamma,&t,\\\rho',&\xi',&\gamma',&y
\end{array}
\right)
\end{array}
\right\}.
\end{array}
\end{equation*}
Similarly, the formula
$\phi_{I}\left(p^{I}_{0},\natPart{x_{0}}\right)$ is equivalent to
the $\sigF{2}{\mathbb R;+,<,0}$-formula of length $\bigO{nb\log(nb)}$:
\begin{equation*}
\phi_{I}\left(p^{I},\natPart{x}\right)=
\exists \left(p^{I},\natPart{x_{i}}\right)_{0\le{i}\le{n}}.
\begin{array}{ll}
\natPart{x_{0}}\doteq0\land p_{0}^{I}\doteq q_{0}\\
\land \natPart{x_{n}}\doteq \natPart{x}\land p_{n}^{I}\doteq p^{I}
\land\\
\forall
\arraycolsep=0.5pt
\begin{array}{ll}
\rho,&\xi,\\\rho',&\xi'
\end{array}.
\left\{
\begin{array}{l}
\left[\bigvee_{i=0}^{n-1}\left(
\arraycolsep=0.5pt
\begin{array}{ll}
\rho\doteq p^{I}_{i}\land&\xi\doteq \natPart{x_{i}}\land\\
\rho'\doteq p^{I}_{i+1}\land&\xi'\doteq \natPart{x_{i+1}}\\
\end{array}
\right)\right]\\\implies\fraPart{\psi}\left(\rho,\xi,\rho',\xi'\right)
\end{array}
\right\}.
\end{array}
\end{equation*} \end{proof}
Theorem \ref{theo:bounded-construct} can now be proven. \begin{proof}[Proof of Theorem \ref{theo:bounded-construct}]
The algorithm is exactly the same than the algorithm of
Proposition \ref{proposition:bounded-construct}. It suffices to prove that
the algorithm of Proposition \ref{proposition:bounded-construct} can be
applied to $\mathcal{A}'$, that is, that $\mathcal{A}'\in\mathbb{A}_{F,\mathcal{S}}$.
Let $L=\toInfWord{\mathcal{A}}$ and $R=\wordToReal{L}$. Since $\mathcal{A}$ is fractional,
then $L$ is also fractional, hence $L=\toInfWord R$. By
Lemma \ref{lem:ar-01-correct}, $L$ is also recognized by $\ARPar{R}$ as in
Definition \ref{def:AR01}. By Lemma \ref{lem:AR-in-BAF}, $\ARPar{R}\in\mathbb{A}_{F,\mathcal{S}}$ and
by Lemma \ref{lem:quotient-BAF}, its minimal quotient $\mathcal{A}''$ belongs
to $\mathbb{A}_{F,\mathcal{S}}$. Since $\mathcal{A}$ and $\ARPar{R}$ recognizes the same language,
$\mathcal{A}''$ is also the minimal quotient of $\mathcal{A}$, therefore
$\mathcal{A}''=\mathcal{A}'$ and $\mathcal{A}'\in\mathbb{A}_{F,\mathcal{S}}$. \end{proof}
\section{Conclusion} In this paper, we proved that it is decidable in linear time whether a weak Büchi Real Number Automaton $\mathcal{A}$ reading a set of real number $R$ recognizes a finite union of intervals. It is proved that a quasi-linear sized existential-universal formula defining $R$ exists. And that a quasi-quadratic existential formula defining $R$ also exists.
The theorems of this paper lead us to consider two natural generalization. We intend to adapt the algorithm of this paper to similar problems for automata reading vectors of reals instead of automata reading reals. We also would like to solve a similar problem, deciding whether an RNA accepts a $\fo{\mathbb R,\mathbb Z;+,<}$-definable set of reals. Solving this problem would also solve the problem of deciding whether an automaton reading natural number, beginning by the most-significant digit, recognizes an ultimately-periodic set. Similar problems has already been studied, see e.g \cite{DBLP:journals/corr/cs-CC-0309052,DBLP:journals/fuin/LacroixRRV12} and seems to be difficult.
The author thanks Bernard Boigelot, for a discussion about the algorithm of Theorem \ref{theo:bounded-construct}, which led to a decrease of the computation time.
\printindex
\end{document}
|
arXiv
|
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\selectlanguage{english}
\title{A fast Mixed Model B-splines algorithm } \author{Martin P. Boer
\\ Biometris WUR \\ Wageningen \\
The Netherlands
\\
{\tt [email protected]}} \date{\today}
\maketitle
\section*{Abstract} A fast algorithm for B-splines in mixed models is presented. B-splines have local support and are computational attractive, because the corresponding matrices are sparse. A key element of the new algorithm is that the local character of B-splines is preserved, while in other existing methods this local character is lost. The computation time for the fast algorithm is linear in the number of B-splines, while computation time scales cubically for existing transformations. \\
\section{Introduction}\label{sec:intro} Penalized regression using B-splines can be computationally efficient, because of the local character of B-splines. The corresponding linear equations are sparse and can be solved quickly. However, the main problem is to find the optimal value for the penalty parameter. A good way to approach this problem is to use mixed models and restricted maximum likelihood \citep[REML;][]{Patterson1971}. Several methods have been proposed to transform the original penalized B-spline model to a mixed model \citep{Currie2002,Lee2011}. A problem with existing transformations to mixed models is that the local character of the B-splines is lost, which reduces the computational efficiency. For relatively small datasets this is not a major issue. However, for long time series, for example with measurements every five minutes for several months, the computational efficiency becomes quite important.
In this paper I present a new transformation to a mixed model. This model is closely related to the transformation proposed by \cite{Currie2002}. However, the computation time in the transformation of \cite{Currie2002} increases cubically in the number of B-splines, while for the new transformation the computation time increases linearly in time, using sparse matrix algebra \citep{Furrer2010}. One of the key elements of the proposed algorithm is that the transformation preserves the local character of B-splines, and all the equations can be solved quickly.
The paper is organized as follows. In Section~\ref{sec:Bsplines} relevant information about B-splines is given. In Section~(\ref{sec:PsplinesMixedModels}) first the P-spline model \citep{Eilers1996} is described and the transformation to mixed models by \cite{Currie2002} is stated. The new transformation is presented, and details for a sparse mixed model formulation are given. In Section~\ref{sec:Rcode)} the R-code is briefly described and a comparison is made between the computation time of the new method and the transformation by \cite{Currie2002}.
\section{B-splines}\label{sec:Bsplines} In this preliminary section, a few relevant details about B-splines are given. For a detailed overview of B-splines, see for example \cite{Boor1978} and \cite{Hastie2009}. B-splines have local support. This property is important and can speedup calculations considerably. To illustrate the idea of the local support, see Figure~1, with quadratic (i.e.\ degree $q=2$) B-splines. Throughout the paper we will assume equal distance between the splines, denoted by $h$. In the example presented in Figure~1 the distance is unity, $h=1$. The number of B-splines will be denoted by $m$. The first quadratic B-spline, $B_{1,2}(x)$ is zero outside the interval $[-2,1]$. The last one, $B_{12,2}(x)$, is zero outside the interval $[9,12]$. So, for this example, there are $m=12$ quadratic B-splines which define the B-spline basis for the domain $[x_{\text{min}},x_{\text{max}}]=[0,10]$ of interest.
The second derivative of B-splines is given by \citep{Boor1978}: \begin{equation}
h^2 B^{''}_{j,q}(x) = B_{j,q-2}(x) - 2 B_{j+1,q-2}(x) + B_{j+2,q-2}(x) \;. \label{eqn:Bsderiv} \end{equation} The second derivative is illustrated in Figure~1. The red curves are second-order derivatives of quartic (fourth-degree) B-splines. As can be seen from equation~(\ref{eqn:Bsderiv}) it can represented as a linear combination of three quadratic B-splines.
\section{P-splines and Mixed Models}\label{sec:PsplinesMixedModels} In this section we will give a brief description of P-splines \citep{Eilers1996}. Let $n$ be the number of observations. Suppose the variable $\mathbf{y} = (y_1,\ldots,y_n)'$ depends smoothly on the variable $\mathbf{x} = (x_1,\ldots,x_n)'$. Let $\mathbf{B} =(B_{1,2}(\mathbf{x}),\ldots,B_{m,2}(\mathbf{x}))$ be a $n \times m$ matrix, and $\mathbf{a} = (a_1,a_2,\ldots,a_m)'$ be a vector of regression coefficients. Then the following objective function to be minimized can be defined: \begin{equation}\label{eqn:Psplines}
S(\mathbf{a}) =
(\mathbf{y} - \mathbf{B} \mathbf{a})'(\mathbf{y} - \mathbf{B} \mathbf{a})
+ \lambda \; \mathbf{a}' \mathbf{D}' \mathbf{D} \mathbf{a}
\;, \end{equation} where $\lambda>0$ is a penalty or regularization parameter, and $\mathbf{D}$ is an $(m-2) \times m$ second-order difference matrix \citep[see e.g.][]{Eilers1996}.
\cite{Currie2002} showed that equation~(\ref{eqn:Psplines}) can be reformulated as a mixed model: \begin{equation}\label{eqn:mixedmodel}
\mathbf{y} = \mathbf{X b} + \mathbf{Z u} + \mathbf{e} \;, \quad
\mathbf{u} \backsim N(\mathbf{0},\frac{1}{\lambda}\mathbf{Q}^{-1} \sigma^2) \;,
\quad \mathbf{e} \backsim N(\mathbf{0},\mathbf{I}\sigma^2) \;, \end{equation} where $\mathbf{X}$ and $\mathbf{Z}$ are design matrices, $\mathbf{Q}$ is a precision matrix, $\mathbf{b}=(b_0,b_1)'$ are the fixed effects, $\mathbf{u} = (u_1,u_2,\ldots,u_{m-2})'$ the random effects, $\mathbf{e}$ is the residual error, and $\sigma^2$ is the residual variance.
\cite{Currie2002} used the following transformation: \begin{equation}\label{eqn:currietransform}
\mathbf{a} = \mathbf{G} \mathbf{b} + \mathbf{D}' (\mathbf{D} \mathbf{D}')^{-1} \mathbf{u} \; , \end{equation} where $\mathbf{G}$ is an $m \times 2$ matrix with columns $\mathbf{g}_0 = (1,1,\ldots,1)'$ and $\mathbf{g}_1 = (1,2,\ldots,m)'$. This transformation gives the following expressions for the design matrices and the precision matrix: \begin{equation}\label{eqn:Z_Currie}
\mathbf{X} = \mathbf{B} \mathbf{G} \;, \quad
\mathbf{Z} = \mathbf{B} \mathbf{D}' (\mathbf{D} \mathbf{D}')^{-1} \;, \quad
\mathbf{Q} = \mathbf{I} \;. \end{equation}
The mixed model equations \citep{Henderson1963} corresponding to equation~(\ref{eqn:mixedmodel}) are given by: \begin{equation}\label{eqn:mme}
\begin{pmatrix}
\mathbf{X'X} & \mathbf{X'Z} \\
\mathbf{Z'X} & \mathbf{Z'Z} + \lambda \mathbf{Q}
\end{pmatrix}
\begin{pmatrix}
\widehat{\mathbf{b}} \\ \widehat{\mathbf{u}}
\end{pmatrix} =
\begin{pmatrix}
\mathbf{X'y} \\ \mathbf{Z'y}
\end{pmatrix} \;. \end{equation} The coefficient matrix $\mathbf{C}_{\lambda}$ in equation~(\ref{eqn:mme}) is given by: \begin{equation}\label{eqn:mme_coefmatrix}
\mathbf{C}_{\lambda} =
\begin{pmatrix}
\mathbf{X'X} & \mathbf{X'Z} \\
\mathbf{Z'X} & \mathbf{Z'Z} + \lambda \mathbf{Q}
\end{pmatrix} \;. \end{equation} This coefficient matrix is dense, since the local character of the B-splines has been destroyed by equation~(\ref{eqn:Z_Currie}). This implies that the computation complexity for solving equation~(\ref{eqn:mme}) is $\mathcal{O}(m^3)$.
The following transformation preserves the local character of the B-splines: \begin{equation}\label{eqn:mmb_transform}
\mathbf{a} = \mathbf{G} \mathbf{b} + \mathbf{D}' \mathbf{u} \;. \end{equation} Figure~1 illustrates the underlying idea of this transformation. The quadratic B-splines basis consists of $m=12$ B-splines. This quadratic B-spline basis is transformed to a second-order derivative quartic B-splines basis of $m-2=10$ B-splines, plus a parameter for intercept $b_0$ and linear trend $b_1$. The second-order derivative quartic B-splines can be constructed from quadratic B-splines by second-order differencing. Using the new transformation the design and precision matrices are given by: \begin{equation}\label{eqn:mmb_model}
\mathbf{X} = \mathbf{B} \mathbf{G} \;, \quad
\mathbf{Z} = \mathbf{B} \mathbf{D}' \;, \quad
\mathbf{Q} = \mathbf{D} \mathbf{D}' \mathbf{D} \mathbf{D}' \;. \end{equation} Let us refer to equations~(\ref{eqn:mixedmodel}) and~(\ref{eqn:mmb_model}) as a Mixed Model of B-splines (MMB), since it uses the B-splines directly as building blocks for the mixed model. The matrix $\mathbf{Z}'\mathbf{Z}+\lambda \mathbf{Q}$ has bandwidth $4$. This implies that $\mathbf{C}_\lambda$ is sparse and computation complexity has been reduced to $\mathcal{O}(m)$. An efficient way to calculate the REML profile log likelihood \citep{Gilmour1995, Crainiceanu2004,Searle2009} is given by the following four steps :
\begin{enumerate}
\item Sparse Cholesky factorization \citep{Furrer2010}: $\mathbf{C}_\lambda = \mathbf{U}_{\lambda} \mathbf{U}^{'}_{\lambda}$, where $\mathbf{U}_{\lambda}$ is an upper-triagonal matrix.
\item Forward-solve and back-solve \citep{Furrer2010}, with $\mathbf{w}$ a vector of length $m$:
\begin{equation}
\mathbf{U}_\lambda \mathbf{w} = \begin{pmatrix}
\mathbf{X'y} \\ \mathbf{Z'y}
\end{pmatrix} \;,
\quad
\mathbf{U}^{'}_{\lambda}
\begin{pmatrix}
\widehat{\mathbf{b}} \\ \widehat{\mathbf{u}}
\end{pmatrix} = \mathbf{w} \;.
\end{equation}
\item Calculate $\hat{\sigma}^2$ \citep{Johnson1995}, $p$ is the dimension of the fixed effects:
\begin{equation}
\hat{\sigma}^2 =
\frac{\mathbf{y}'\mathbf{y}
-\widehat{\mathbf{b}}'\mathbf{X}' \mathbf{y}
-\widehat{\mathbf{u}}'\mathbf{Z}' \mathbf{y}
}{n-p} \;.
\end{equation}
\item REML log profile likelihood \citep{Gilmour1995, Crainiceanu2004,Searle2009}:
\begin{equation}
L(\lambda) = -\frac{1}{2} \left( 2 \log |\mathbf{U}_\lambda|
- (m-p) \log \lambda
+ (n-p) \log \hat{\sigma}^2
+ C \right) \;,
\end{equation}
where $C$ is a constant: $C = n-p - \log |\mathbf{Q}|$. \end{enumerate} A one-dimensional optimization algorithm can be used to find the maximum for $L(\lambda)$. The computation time is linear in $m$.
\section{R-package MMBsplines}\label{sec:Rcode)} An R-package, {\bf MMBsplines}, is available at GitHub:
\url{https://github.com/martinboer/MMBsplines.git}.
The sparse matrix calculations are done with the {\bf spam} package \cite{Furrer2010}. The B-splines are constructed with {\tt splineDesign()} of the {\bf splines} library.
The following example code sets some parameter values and runs the simulations: {\small \begin{verbatim}
nobs = 1000; xmin = 0; xmax = 10
set.seed(949030)
sim.fun = function(x) { return(3.0 + 0.1*x + sin(2*pi*x))}
x = runif(nobs, min = xmin, max = xmax)
y = sim.fun(x) + 0.5*rnorm(nobs) \end{verbatim}} A fit to the data on a small grid can be obtained as follows, using $m=100$ quadratic B-splines: {\small \begin{verbatim}
obj = MMBsplines(x, y, xmin, xmax, nseg = 100)
x0 = seq(xmin, xmax, by=0.01)
yhat = predict(obj, x0)
ylin = predict(obj, x0, linear = TRUE)
ysim = sim.fun(x0) \end{verbatim}}
Figure~\ref{fig:sim10days} shows the result, with $\lambda_{\text{max}}=1.33$.
\begin{figure}\label{fig:sim10days}
\end{figure} The Currie and Durban transformation can be run by setting the {\tt sparse } argument to {\tt FALSE}: \begin{verbatim}
obj = MMBsplines(x, y, xmin, xmax, nseg = 100, sparse = FALSE) \end{verbatim}} For $m=100$, as in Figure~\ref{fig:sim10days}, the differences in computation time are small. If we increase the length of the simulated time series, with a fixed stepsize $h=0.1$, the advantage of the MMB-splines method becomes clear, see Figure~3. As expected the Currie and Durban transformation computation time increases cubical in the number of B-splines, computatation time for MMB-splines is linear in $m$.
\begin{figure}
\caption{{\small Comparison of computation times. The computation time for the Currie and Durban transformation is cubical in the number of B-splines. The computation time for MMB-splines is linear in the number of B-splines.}}
\end{figure}
\section{Conclusion}\label{sec:Discussion)} The MMB-splines method presented in this paper seems to be an attractive way to use B-splines in mixed models. The method was only presented for quadratic splines, but also cubical or higher-degree B-splines could have been used. Other generalizations are also possible, for example extension to multiple penalties \citep{Currie2002} or multiple dimensions \citep{Rodriguez-Alvarez2014}.
\section*{Acknowledgments} I am indebted to Hugo van den Berg for useful comments on earlier drafts. I would also like to thank Paul Eilers, for explaining to me the local character of B-splines, and many valuable discussions. I would like to thank Cajo ter Braak and Willem Kruijer for valuable discussions and corrections of earlier versions of the paper.
\end{document}
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arXiv
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\begin{document}
\title{Causal screening in dynamical systems}
\begin{abstract}
Many classical algorithms output graphical representations of causal
structures by testing conditional independence among a set of random
variables. In dynamical systems, local independence can be used analogously
as a testable implication of the underlying data-generating process.
We suggest some inexpensive methods for causal screening which provide
output with a sound causal interpretation under the assumption of ancestral
faithfulness. The popular model class of linear Hawkes processes is used
to provide an example of a dynamical causal model.
We argue
that for sparse causal graphs the output will often be close to complete.
We give
examples of this framework and apply it to a challenging biological system. \end{abstract}
\section{INTRODUCTION}
Constraint-based causal learning is computationally and statistically challenging. There is a large literature on learning structures that are represented by directed acyclic graphs (DAGs) or margina\-li\-za\-tions thereof (see \cite{handbookGraphical2019} for references). The fast causal inference algorithm \citep[FCI,][]{spirtes1993} provides in a certain sense maximally informative output \citep{zhang2008}, but at the cost of using a large number of conditional independence tests \citep{colombo2012}. To reduce the computational cost, other methods provide output which has a sound causal interpretation, but may be less informative. Among these are the anytime FCI \citep{spirtesAnytime2001} and RFCI \citep{colombo2012}. A recent algorithm, ancestral causal inference \citep[ACI,][]{magliacaneACI2016}, aims to learn only the directed part of the underlying graphical structure which allows for a sound causal interpretation even though some information is lost.
In this paper, we describe some simple methods for learning causal structure in dynamical systems represented by stochastic processes. Many authors have described frameworks and algorithms for learning structure in systems of time series, ordinary differential equations, stochastic differential equations, and point processes. However, most of these methods do not have a clear causal interpretation when the observed processes are part of a larger system and most of the current literature is either non-causal in nature, or requires that there are no unobserved processes.
Analogously to testing conditional independence when learning DAGs, one can use tests of local independence in the case of dynamical systems. \cite{eichler2013}, \cite{meek2014}, and \cite{mogensenUAI2018} propose algorithms for learning graphs that represent local independence structures. We show empirically that we can recover features of their graphical learning target using considerably fewer tests of local independence. First, we suggest a learning target which is easier to learn, though still conveys useful causal information, analogously to ACI \citep{magliacaneACI2016}. Second, the proposed algorithm is only guaranteed to provide a supergraph of the learning target and this also reduces the number of local independence tests drastically. A central point is that our proposed methods retain a causal interpretation in the sense that absent edges in the output correspond to implausible causal connections.
\cite{meek2014} suggests learning a directed graph to represent a causal dynamical system and gives a learning algorithm which we will describe as a {\it simple screening algorithm} (Section \ref{ssec:simpScreen}). We show that this algorithm can be given a sound interpretation under a weaker faithfulness assumption than that of \cite{meek2014}. We also provide a simple interpretation of the output of this algorithm and we show that similar screening algorithms can give comparable results using considerably fewer tests of local independence.
All proofs are provided in the supplementary material.
\begin{figure*}
\caption{ Top: Example data from a four-dimensional
Hawkes process. Bottom: The corresponding intensities. The time
axis is aligned between the two plots.}
\label{fig:data}
\caption{ Left: The causal graph (see Section
\ref{ssec:dynCauModel}) of a four-dimensional Hawkes process.
Right: Learning output of standard approach (see Section
\ref{sec:hawkes}) when 3 is
unobserved. When 3 is unobserved, 2 is predictive of 4 and vice
versa
(heuristically, more events in process 2 indicate more events in 3
which in
turn
indicates
more events in 4). However, they are not causally connected and
using
local independence one can learn that 2 is not a parent of 4. This
is
important to predict what would happen under interventions in the
system as the right-hand graph indicates that an intervention on 2
would
change the distribution of 4 even though this is not the case as
$g^{\alpha 2} = 0$ for $\alpha\in \{1,3,4\}$.}
\label{fig:causal}
\caption{ Subfigure \ref{fig:data} shows data generated
from the system in \ref{fig:causal} (left). Until the first event all
intensities are constant (equal to $\mu_\alpha$ for the
$\alpha$-process). The first event occurs in process 3. We see that
$g^{23}$, $g^{33}$, and $g^{43}$ are different from zero as encoded by
the graph in \ref{fig:causal} (left).
Therefore the event makes the intensity processes of 2, 3, and 4 jump,
making new events in these processes more likely in the immediate
future (\ref{fig:data}, bottom).}
\label{fig:fig1}
\end{figure*}
\section{HAWKES PROCESSES} \label{sec:hawkes}
Local independence can be defined in a wide range of discrete-time and continous-time dynamical models (e.g., point processes \citep{didelez2000}, time series \citep{eichler2012}, and diffusions \citep{ mogensenUAI2018}. See also \cite{commenges2009}), and the algorithmic results we present apply to all these classes of models. However, the causal interpretation will differ between these model classes, and we will use the {\it linear
Hawkes
processes} to exemplify the framework. \cite{laubHawkes2015} give an
accessible introduction to this continuous-time model class and \cite{linigerThesis}, \cite{bacry2015}, and \cite{daleyVere} provide more background. Hawkes processes have also been studied in the machine learning community in recent years \citep{zhou2013, zhou2013b, luo2015, xu2016, etesami2016, achab2017, tan2018, xu2018, trouleau2019}. It is important to note that these papers all consider the case of full observation, i.e., every coordinate process is observed. In causal systems that are not fully observed that assumption may lead to false conclusions (see Figure \ref{fig:causal}). Our work addresses the learning problem without the assumption of full observation, hence there can be unknown and unobserved confounding processes.
On a filtered probability space, $(\Omega, \mathcal{F}, (\mathcal{F}_t), \text{P})$, we consider an $n$-dimensional multivariate point process, $X = (X^1, \ldots, X^n)$. $\mathcal{F}_t$ is a filtration, i.e., a nondecreasing family of $\sigma$-algebras, and it represents the information which is available at a specific time point. Each coordinate process $X^\alpha$ is described by a sequence of positive, stochastic event times $T_1^\alpha, T_2^\alpha, \ldots$ such that $T_j^\alpha > T_i^\alpha$ almost surely for $j >i$. We let $V = \{1,\ldots,n\}$. This can also be formulated in terms of a counting process, $N$, such that $N_s^\alpha = \sum_i \mathds{1}_{(T_i \leq s)} $, $\alpha\in V$. There exists so-called {\it intensity
processes}, $\lambda = (\lambda^1,\ldots,\lambda^n)$, such that
\[ \lambda_t^\alpha = \lim_{h\rightarrow 0} \frac{1}{h} \text{P}(N_{t+h}^\alpha - N_t^\alpha = 1 \mid \mathcal{F}_t) \]
and the intensity at time $t$ can therefore be thought of as describing the probability of a jump in the immediate future after time $t$ conditionally on the history until time $t$ as captured by the $\mathcal{F}_t$-filtration. In a linear Hawkes model, the intensity of the $\alpha$-process, $\alpha\in V$, is of the simple form
\begin{align*} \lambda_t^\alpha &= \mu_\alpha + \sum_{\gamma \in V} \int_{0}^{t} g^{\alpha\gamma}(t-s)\ \mathrm{d} N_s^\gamma \\ & = \mu_\alpha + \sum_{\gamma\in V} \sum_{i:T_i^\gamma < t} g^{\alpha\gamma}(t-T_i^\gamma) \end{align*}
\noindent where $\mu_\alpha \geq 0$ and the function $g^{\alpha\gamma} : \mathbb{R}_{+} \rightarrow \mathbb{R}$ is nonnegative for all $\alpha,\gamma \in V$. From the above formula, we see that if $g^{\beta\alpha} = 0$, then the $\alpha$-process does not enter directly into the intensity of the $\beta$-process and we will formalize this observation in subsequent sections. The intensity processes determine how the Hawkes process evolves and if $g^{\beta\alpha} = 0$ then the $\alpha$-process does not directly influence the evolution of the $\beta$-process (it may of course have an indirect influence which is mediated by other processes). Figure \ref{fig:data} provides an example of data from a linear Hawkes process and an illustration of its intensity processes.
\begin{figure*}
\caption{ Left: A causal graph on nodes $V =
\{\alpha,\beta,\gamma,\delta,\epsilon, \phi \}$. Right: The
corresponding
parent
graph on nodes $O = \{\alpha,\delta,\epsilon\}$. Note that causal
graphs
and parent graphs may contain cycles. The parent graph does not contain
information on the confounder process $\phi$ as it only encodes `causal
ancestors'. One can also {\it marginalize} the causal graph to obtain a
{\it directed mixed graph} from which one can read off the parent
graph (see the supplementary material).}
\label{fig:paG}
\end{figure*}
\subsection{A DYNAMICAL CAUSAL MODEL} \label{ssec:dynCauModel}
We will in this section define what we mean by a {\it dynamical causal model} in the case of a linear Hawkes process and also define a graph $(V,E)$ which represents the causal structure of the model. The node set $V$ is the index set of the coordinate processes of the multivariate Hawkes process, thus identifying each node with a coordinate process. If we first consider the case where $X = (X_1,\ldots,X_n)$ is a multivariate random variable, it is common to define a {\it causal} model in terms of a DAG, $\mathcal{D}$, and a structural causal model \citep{pearl2009, petersElements2017} by assuming that there exists functions $f_i$ and error terms $\epsilon_i$ such that
\[ X_i = f_i(X_{\text{pa}_\mathcal{D}(X_i)}, \epsilon_i) \]
for $i = 1,\ldots,n$. The causal assumption amounts to assuming that the functional relations are stable under interventions. This idea can be transferred to dynamical systems (see also \cite{roysland2012, mogensenUAI2018}). In the case of a linear Hawkes process as described above, we can consider intervening on the $\alpha$-process and force events to occur at the deterministic times $t_1, \ldots, t_k$, and at these times only. In this case, the causal assumption amounts to assuming that the distribution of the intervened system is governed by the intensities
\begin{align*} \lambda_t^\beta = \mu_\beta &+ \int_{0}^t g^{\beta\alpha}(t-s)\ \mathrm{d} \bar{N}_s^\alpha \\ &+ \sum_{\gamma\in V \setminus \{\alpha\}} \int_{0}^t g^{\beta\gamma}(t-s)\ \mathrm{d} N_s^\gamma \end{align*}
for all $\beta \in V\setminus \{\alpha\}$ and where $\bar{N}_t^\alpha = \sum_{i
= 1}^{k} \mathds{1}_{(t_i \leq t)}$. We will not go into a discussion of the existence of these interventional stochastic processes. The above is a {\it hard} intervention in the sense that the $\alpha$-process is fixed to be a deterministic function of time. Note that one could easily imagine other types of interventions such as {\it soft} interventions where the intervention process, $\alpha$, is not deterministic. One can also extend this to interventions on more than one process. It holds that $N_{t + h}^\beta - N_t^\beta \sim \Pois(\lambda_t^\beta \cdot h)$ in the limit $h\rightarrow 0$, and we can think of this as a simulation scheme in which we generate the points in one small interval in accordance to some distribution depending on the history of the process. As such the intensity describes a structural causal model at infinitesimal time steps and the $g^{\alpha\beta}$-functions are in a causal model stable under interventions in the sense that they also describe how the intervention process $\bar{N}^\alpha$ enters into the intensity of the other processes.
We use the set of functions $\{g^{\beta\alpha}\}_{\alpha,\beta\in
V}$ to define the {\it causal graph} of the Hawkes process. A {\it graph} is a pair $(V,E)$ where $V$ is a set of nodes and $E$ is a set of edges between these nodes. We assume that we observe the Hawkes process in the time interval $J = [0,T]$, $T\in \mathbb{R}$. The causal graph has node set $V$ (the index set of the coordinate processes) and the edge $\alpha\rightarrow\beta$ is in the causal graph if and only if $g^{\beta\alpha}$ is not identically zero on $J$. We call this graph {\it
causal} as it is defined using $\{g^{\beta\alpha}\}_{\alpha,\beta \in V}$
which is a set of mechanisms assumed stable under interventions, and this causal assumption is therefore analogous to that of a classical structural causal model as briefly introduced above.
\subsection{PARENT GRAPHS}
In principle, we would like to recover the causal graph, $\mathcal{D}$, using local independence tests. Often, we will only have partial observation of the dynamical system in the sense that we only observe the processes in $O\subsetneq V$. We will then aim to learn the {\it parent graph} of $\mathcal{D}$ on nodes $O$.
\begin{defn}[Parent graph]
Let $\mathcal{D} = (V,E)$ be a causal graph and let $O\subseteq V$. The
{\it parent graph} of
$\mathcal{D}$ on nodes $O$ is the graph $(O,F)$ such that for
$\alpha,\beta\in O$, the edge
$\alpha\rightarrow \beta$ is in $F$ if and only if the edge
$\alpha\rightarrow\beta$ is in the causal graph or there is
a path $\alpha
\rightarrow \delta_1 \rightarrow \ldots
\rightarrow \delta_k \rightarrow \beta$ in the causal graph such that
$\delta_1,\ldots,\delta_k \notin O$, for some $k>0$ .
\label{def:paG} \end{defn}
We denote the parent graph of the causal graph by $\mathcal{P}_O(\mathcal{G})$, or just $\mathcal{P}(\mathcal{G})$ if the set $O$ used is clear from the context. In applications, a parent graph may provide answers to important questions as it tells us the causal relationships between the observed nodes. A similar idea was applied in DAG-based models by \cite{magliacaneACI2016}, though that paper describes an exact method and not a screening procedure. In large systems, it can easily be infeasible to learn the complete independence structure of the observed system, and we propose instead to estimate the parent graph which can be done efficiently. In the supplementary material, we give another characterization of a parent graph. Figure \ref{fig:paG} contains an example of a causal graph and a corresponding parent graph.
\subsection{LOCAL INDEPENDENCE}
Local independence has been studied by several authors and in different classes of continuous-time models as well as in time series \citep{aalen1987,
didelez2000, didelez2008, eichler2010}. We give an abstract definition of local independence, following the exposition by \cite{mogensenUAI2018}.
\begin{defn}[Local independence]
Let $X$ be a multivariate stochastic process and let $V$ be an index set of
its coordinate processes. Let $\mathcal{F}_t^D$ denote the complete and
right-continuous version of the $\sigma$-algebra
$\sigma(\{X_s^\alpha: s\leq t,\alpha \in D \})$, $D \subseteq V$. Let
$\lambda$ be a
multivariate stochastic
process (assumed to be integrable and c\`adl\`ag) such that its coordinate
processes are
indexed by $V$. For
$A,B,C\subseteq V$, we say that $X^B$ is $\lambda$-locally independent of
$X^A$ given $X^C$ (or simply $B$ is $\lambda$-locally independent of $A$
given $C$) if the process
\[
t \mapsto \text{E}(\lambda_t^\beta \mid \mathcal{F}_t^{C\cup A})
\]
has an $\mathcal{F}_t^C$-adapted version for all $\beta\in B$. We write
this as
$A\not\rightarrow_\lambda B \mid C$, or simply $A\not\rightarrow B \mid C$. \end{defn}
In the case of Hawkes processes, the intensities will be used as the $\lambda$-processes in the above definition. \cite{didelez2000}, \cite{mogensenUAI2018}, and \cite{mogensen2018} provide technical details on the definition of local independence. Local independence can be thought of as a dynamical system analogue to the classical conditional independence. It is, however, asymmetric which means that $A\not\rightarrow B\mid C$ does not imply $B\not\rightarrow A\mid C$. This is a natural and desirable feature of an independence relation in a dynamical system as it helps us distinguish between the past and the present. It is important to note that by testing local independences we can obtain more information about the underlying parent graph than by simply assuming full observation and fitting a model to the observed data (see Figure \ref{fig:causal}).
\subsubsection{Local Independence and the Causal Graph}
To make progress on the learning task, we will in this subsection describe the link between the local independence model and the causal graph.
\begin{defn}[Pairwise Markov property \citep{didelez2008}]
We say that a local independence model satisfies the {\it pairwise
Markov
property} with respect to a directed graph, $\mathcal{D} = (V,E)$, if
the absence
of the edge $\alpha
\rightarrow\beta$ in $\mathcal{D}$ implies $\alpha
\not\rightarrow_\lambda \beta
\mid V\setminus \alpha$ for all $\alpha,\beta\in V$. \end{defn}
We will make the following technical assumption throughout the paper. In applications, the functions $g^{\alpha\beta}$ are often assumed to be of the below type (\cite{laubHawkes2015}).
\begin{ass}
Assume that $N$ is a multivariate Hawkes process and that we observed $N$
over the interval $J = [0,T]$ where $T > 0$. For all $\alpha,\beta\in V$,
the function $g^{\beta\alpha}: \mathbb{R}_+ \rightarrow \mathbb{R}$ is
continuous and $\mu_\alpha > 0$.
\label{ass:gFunc} \end{ass}
A version of the following result was also stated by \cite{eichlerHawkes2017} but no proof was given and we provide one in the supplementary material. If $\mathcal{G}_1 = (V,E_1)$ and $\mathcal{G}_2 = (V,E_2)$ are graphs, we say that $\mathcal{G}_1$ is a {\it proper subgraph} of $\mathcal{G}_2$ if $E_1 \subsetneq E_2$.
\begin{prop}
The local independence model of a linear Hawkes process satisfies the
pairwise Markov property with respect to the causal graph of the process
and no proper subgraph of the causal graph has the property.
\label{prop:pairCau} \end{prop}
\section{GRAPH THEORY AND INDEPENDENCE MODELS}
A {\it graph} is a pair $(V,E)$ where $V$ is a finite set of nodes and $E$ a finite set of edges. We will use $\sim$ to denote a generic edge. Each edge is between a pair of nodes (not necessarily distinct), and for $\alpha,\beta\in V$, $e \in E$, we will write $\alpha\overset{e}{\sim} \beta$ to denote that the edge $e$ is between $\alpha$ and $\beta$. We will in particular consider the class of {\it
directed graphs} (DGs) where between each pair of nodes $\alpha,\beta\in
V$ one has a subset of the edges $\{\alpha\rightarrow\beta, \alpha\leftarrow\beta\}$, and we say that these edges are {\it directed}.
Let $\mathcal{G}_1 = (V,E_1)$ and $\mathcal{G}_2 = (V,E_2)$ be graphs. We say that $\mathcal{G}_2$ is a {\it
supergraph} of $\mathcal{G}_1$, and write $\mathcal{G}_1 \subseteq \mathcal{G}_2$, if $E_1\subseteq
E_2$. For a graph $\mathcal{G} = (V,E)$ such that $\alpha,\beta \in V$, we write $\alpha\rightarrow_\mathcal{G} \beta$ to indicate that the directed edge from $\alpha$ to $\beta$ is contained in the edge set $E$. In this case we say that $\alpha$ is a {\it parent} of $\beta$. We let $\text{pa}_\mathcal{G}(\beta)$ denote the set of nodes in $V$ that are parents of $\beta$. We write $\alpha\not\rightarrow_\mathcal{G} \beta$ to indicate that the edge is {\it not} in $E$. Earlier work allowed loops, i.e., self-edges $\alpha\rightarrow\alpha$, to be either present or absent in the graph \citep{meek2014,mogensenUAI2018, mogensen2018}. We assume that all loops are present, though this is not an essential assumption.
A {\it walk} is a finite sequence of nodes, $\alpha_i \in V$, and edges, $e_i\in E$, $\langle \alpha_1, e_1, \alpha_2, \ldots ,\alpha_k, e_k, \alpha_{k+1} \rangle$ such that $e_i$ is between $\alpha_i$ and $\alpha_{i+1}$ for all $i = 1,\ldots,k$ and such that an orientation of each edge is known. We say that a walk is {\it nontrivial} if it contains at least one edge. A {\it
path} is a walk such that no node is repeated. A {\it
directed} path from $\alpha$ to $\beta$ is a path such that all edges are directed and point in the direction of $\beta$.
\begin{defn}[Trek, directed trek]
A {\it trek} between $\alpha$ and $\beta$ is a (nontrivial) path $\langle
\alpha,
e_1,\ldots,e_k,\beta\rangle$ with no colliders \citep{foygelHalftrek2012}.
We say that a trek
between
$\alpha$ and $\beta$ is {\it directed} from $\alpha$ to $\beta$ if $e_k$
has a head at $\beta$. \end{defn}
We will formulate the following properties using a general {\it independence
model}, $\mathcal{I}$, on $V$. Let $\mathbb{P}(\cdot)$ denote the power set of some set. An independence model on $V$ is simply a subset of $\mathbb{P}(V) \times \mathbb{P}(V)\times \mathbb{P}(V)$ and can be thought of as a collection of independence statements that hold among the processes/variables indexed by $V$. In subsequent sections, the independence models will be defined using the notion of local independence. In this case, for $A,B,C\subseteq V$, $A\not\rightarrow_\lambda B\mid C$ is equivalent to writing $\langle A,B\mid C\rangle \in \mathcal{I}$ in the abstract notation, and we use the two interchangeably. We do not require $\mathcal{I}$ to be symmetric, i.e., $\langle A,B\mid C\rangle \in \mathcal{I}$ does not imply $\langle B,A\mid C\rangle \in \mathcal{I}$. In the following, we also use $\mu$-separation which is a ternary relation and a dynamical model (and asymmetric) analogue to $d$-separation or $m$-separation.
\begin{defn}[$\mu$-separation]
Let $\mathcal{G} = (V,E)$ be a DMG, and let $\alpha,\beta\in V$ and $C\subseteq
V$.
We say that a (nontrivial) walk from $\alpha$ to $\beta$, $\langle
\alpha,
e_1,\ldots,e_k, \beta\rangle$, is $\mu$-connecting given $C$ if $\alpha
\notin C$, the edge $e_k$ has a head at $\beta$, every collider on the
walk
is in $\text{an}(C)$ and no noncollider is in $C$. Let $A,B,C \subseteq V$.
We
say that $B$ is $\mu$-separated from $A$ given $C$ if there is no
$\mu$-connecting walk from any $\alpha\in A$ to any $\beta\in B$ given
$C$.
In this case, we write
$\musep{A}{B}{C}$, or $\musepG{A}{B}{C}{\mathcal{G}}$ if we wish to emphasize
the
graph to
which the statement relates. \end{defn}
More graph-theoretical definitions and references are given in the supplementary material.
\begin{defn}[Global Markov property]
We say that an independence model $\mathcal{I}$ satisfies the {\it global Markov
property} with respect to a DG, $\mathcal{G} = (V,E)$, if $\musepG{A}{B}{C}{\mathcal{G}}$
implies $\langle A,B \mid C\rangle \in \mathcal{I}$ for all $A,B,C \subseteq V$. \end{defn}
From Proposition \ref{prop:pairCau}, we know that the local independence model of a linear Hawkes process satisfies the pairwise Markov property with respect to its causal graph, and using the results in \cite{didelez2008} and \cite{mogensenUAI2018} it also satisfies the global Markov property with respect to this graph.
\begin{defn}[Faithfulness]
We say that $\mathcal{I}$ is {\it faithful} with respect to a DG, $\mathcal{G} = (V,E)$,
if $\langle A,B \mid C\rangle \in \mathcal{I}$ implies $\musepG{A}{B}{C}{\mathcal{G}}$ for
all $A,B,C \subseteq V$.
\label{def:faith} \end{defn}
\section{NEW LEARNING ALGORITHMS}
In this section, we state a very general class of algorithms which is easily seen to provide sound causal learning and we describe some specific algorithms. We throughout assume that there is some underlying, true DG, $\mathcal{D}_0 = (V,E)$, describing the causal model and we wish to output $\mathcal{P}_O(\mathcal{D}_0)$. However, this graph is not in general identifiable from the local independence model. In the supplementary material, we argue that for an equivalence class of parent graphs, there exists a unique member of the class which is a supergraph of all other members. Denote this unique graph by $\bar{\mathcal{D}}$. Our algorithms will output supergraphs of $\bar{\mathcal{D}}$, and the output will therefore also be supergraphs of the true parent graph.
We assume that we are in the `oracle case', i.e., have access to a local independence oracle that provides the correct answers. We will say that an algorithm is {\it sound} if it in the oracle case outputs a supergraph of $\bar{\mathcal{D}}$ and that it is {\it complete} if it outputs $\bar{\mathcal{D}}$. We let $\mathcal{I}^O$ denote the local independence model restricted to subsets of $O$, i.e., this is the observed part of the local independence model. We provide algorithms that are guaranteed to be sound, but only complete in particular cases. Naturally, one would wish for completeness as well. However, complete algorithms can easily be computationally infeasible whereas sound algorithms can be very inexpensive \citep[e.g., ][]{mogensenUAI2018}. We think of these sound algorithms as {\it screening procedures} as they rule out some causal connections, but do not ensure completeness.
\subsection{ANCESTRAL FAITHFULNESS}
Under the faithfulness assumption, every local independence implies $\mu$-separation in the graph. We assume a weaker, but similar, property to show soundness. For learning marginalized DAGs, weaker types of faithfulness have also been explored, see \cite{zhangRobust2008,
zhalamaSATweakerAssump,
zhalamaDS2017}.
\begin{defn}[Ancestral faithfulness]
Let $\mathcal{I}$ be an independence model and let
$\mathcal{D}$ be a DG. We say that $\mathcal{I}$ satisfies {\it ancestral
faithfulness} with respect to $\mathcal{D}$
if for every $\alpha,\beta\in V$ and $C\subseteq V\setminus \{\alpha\}$,
$\ind{\alpha}{\beta}{C} \in \mathcal{I}$ implies that there is no
$\mu$-connecting directed path from $\alpha$
to
$\beta$ given $C$ in $\mathcal{D}$. \end{defn}
Ancestral faithfulness is a strictly weaker requirement than faithfulness. We conjecture that local independence models of linear Hawkes processes satisfy ancestral faithfulness with respect to their causal graphs. Heuristically, if there is a directed path from $\alpha$ to $\beta$ which is not blocked by any node in $C$, then information should flow from $\alpha$ to $\beta$, and this cannot be `cancelled out' by other paths in the graph as the linear Hawkes processes are self-excitatory, i.e., no process has a dampening effect on any process. This conjecture is supported by the so-called {\it Poisson cluster representation} of a linear Hawkes process (see \cite{jovanovic2015}).
\subsection{SIMPLE SCREENING ALGORITHMS} \label{ssec:simpScreen}
As a first step in describing a causal screening algorithm, we will define a very general class of learning algorithms that simply test local independences and sequentially remove edges. It is easily seen that under the assumption of ancestral faithfulness every algorithm in this class gives sound learning in the oracle case. The {\it complete} DG on nodes $V$ is the DG with edge set $\{\alpha\rightarrow\beta \mid \alpha,\beta\in V\}$.
\begin{defn}[Simple screening algorithm]
We say that a learning algorithm is a {\it simple screening algorithm} if
it starts from a complete DG on nodes $O$ and removes an edge $\alpha
\rightarrow \beta$ only if a conditioning set $C \subseteq O\setminus
\{\alpha\}$ has been found such that $\langle \alpha,\beta \mid C\rangle
\in \mathcal{I}^O$. \end{defn}
The next results describe what can be learned from absent edges in the output of a simple screening algorithm.
\begin{prop}
Assume that $\mathcal{I}$ satisfies ancestral faithfulness with respect
to
$\mathcal{D}_0 = (V,E)$. The
output
of any simple screening algorithm is sound in the oracle case.
\label{prop:soundScreen} \end{prop}
\begin{cor}
Assume ancestral faithfulness of $\mathcal{I}$ with respect to $\mathcal{D}_0$ and
let $A,B,C\subseteq O$. If
every
directed path from $A$
to $B$ goes through $C$ in the output graph of a simple screening
algorithm, then every directed path from
$A$ to $B$ goes through $C$ in $\mathcal{D}_0$.
\label{cor:soundPath} \end{cor}
\begin{cor}
If there is no directed path from $A$ to $B$ in the output graph, then
there is no directed path from $A$ to $B$ in $\mathcal{D}_0$. \end{cor}
\subsection{PARENT LEARNING}
In the previous section, it was shown that if edges are only removed when a separating set is found the output is sound under the assumption of ancestral faithfulness. In this section we give a specific algorithm. The key observation is that we can easily retrieve structural information from a rather small subset of local independence tests.
Let $\mathcal{D}^t$ denote the output from Subalgorithm \ref{subalgo:trekStep} (see below). The following result shows that under the assumption of faithfulness, $\alpha\rightarrow_{\mathcal{D}^t} \beta$ if and only if there is a directed trek from $\alpha$ to $\beta$ in $\mathcal{D}_0$.
\begin{prop}
There is no directed trek from $\alpha$ to $\beta$ in
$\mathcal{D}_0$ if
and only if
$\musepG{\alpha}{\beta}{\beta}{\mathcal{D}_0}$.
\label{prop:trekSep} \end{prop}
Note that above, $\beta$ in the conditioning set represents the $\beta$-past while the other $\beta$ represents the present of the $\beta$-process. While there is no distinction in the graph, this interpretation follows from the definition of local independence and the global Markov property. We will refer to running first Subalgorithm \ref{subalgo:trekStep} and then Subalgorithm \ref{subalgo:paStep} (using the output DG from the first as input to the second) as the causal screening (CS) algorithm. Intuitively, Subalgorithm \ref{subalgo:paStep} simply tests if a candidate set (the parent set) is a separating set and other candidate sets could be chosen.
\begin{prop}
The CS algorithm is a simple screening algorithm. \end{prop}
It is of course of interest to understand under what conditions the edge $\alpha\rightarrow\beta$ is guaranteed to be removed by the CS algorithm when it is not in the underlying target graph. In the supplementary material we state and prove a result describing one such condition.
\begin{subalgorithm}
\SetKwInOut{Input}{input}
\SetKwInOut{Output}{output}
\Input{a local independence oracle for $\mathcal{I}^O$}
\Output{a DG on nodes $O$}
initialize $\mathcal{D}$ as the complete DG on $O$\;
\ForEach{$(\alpha,\beta)\in V\times V$}{
\If{$\alpha \not\rightarrow_{\lambda} \beta\mid
\beta$}{delete
$\alpha
\rightarrow \beta$ from $\mathcal{D}$\;
}
}
\Return $\mathcal{D}$
\newline
\caption{Trek step}
\label{subalgo:trekStep}
\end{subalgorithm}
\begin{subalgorithm}
\SetKwInOut{Input}{input}
\SetKwInOut{Output}{output}
\Input{a local independence oracle for $\mathcal{I}^O$ and a DG, $\mathcal{D} =
(O,E)$}
\Output{a DG on nodes $O$}
\ForEach{$(\alpha,\beta)\in V\times V$ such that
$\alpha\rightarrow_\mathcal{D}\beta$}{
\If{\normalfont $\alpha \not\rightarrow_{\lambda} \beta\mid
\text{pa}_\mathcal{D}(\beta) \setminus \{\alpha\}$}{delete
$\alpha
\rightarrow \beta$ from $\mathcal{D}$\;
}
}
\Return $\mathcal{D}$
\newline
\caption{Parent step}
\label{subalgo:paStep}
\end{subalgorithm}
\subsection{ANCESTRY PROPAGATION}
In this section, we describe an additional step which propagates ancestry by reusing the output of Subalgorithm \ref{subalgo:trekStep} to remove further edges. This comes at a price as one needs faithfulness to ensure soundness. The idea is similar to ACI \citep{magliacaneACI2016}.
\begin{subalgorithm}
\SetKwInOut{Input}{input}
\SetKwInOut{Output}{output}
\Input{a DG, $\mathcal{D} =
(O,E)$}
\Output{a DG on nodes $O$}
initialize $E_r = \emptyset$ as the empty edge set\;
\ForEach{$(\alpha,\beta, \gamma)\in V\times V \times V$ such that
$\alpha,\beta,\gamma$ are all distinct}{
\If{$\alpha \rightarrow_\mathcal{D} \beta$,
$\beta\not\rightarrow_\mathcal{D}\alpha$, $\beta
\rightarrow_\mathcal{D}
\gamma$, and
$\alpha
\not\rightarrow_\mathcal{D} \gamma$}{update $E_r = E_r \cup \{\beta
\rightarrow
\gamma \}$\;
}
}
Update $\mathcal{D} = (V, E\setminus E_r)$\;
\Return $\mathcal{D}$
\newline
\caption{Ancestry propagation}
\label{subalgo:ancProp1} \end{subalgorithm}
In ancestry propagation, we exploit the fact that any trek between $\alpha$ and $\beta$ (such that $\gamma$ is not on this trek) composed with the edge $\beta\rightarrow\gamma$ gives a directed trek from $\alpha$ to $\gamma$. We only use the trek between $\alpha$ and $\beta$ `in one direction', as a directed trek from $\alpha$ to $\beta$. In Subalgorithm \ref{subalgo:ancProp2} (supplementary material), we use a trek between $\alpha$ and $\beta$ twice when possible, at the cost of an additional test.
We can construct an algorithm by first running Subalgorithm \ref{subalgo:trekStep}, then Subalgorithm \ref{subalgo:ancProp1}, and finally Subalgorithm \ref{subalgo:paStep} (using the output of one subalgorithm as input to the next). We will call this the CSAPC algorithm. If we use Subalgorithm \ref{subalgo:ancProp2} (in the supplementary material) instead of Subalgorithm \ref{subalgo:ancProp1}, we will call this the CSAP.
\begin{prop}
If $\mathcal{I}$ is faithful with respect to $\mathcal{D}_0$, then CSAP and
CSAPC both
provide sound
learning.
\label{prop:soundAncProp} \end{prop}
\begin{figure*}
\caption{Comparison of number of tests used. For each level of sparsity
(number of edges in
true graph), we generated 500 graphs, all on
5
nodes. The number of tests required quickly rises for dFCI and CA
while CS spends no more than $2\cdot 5(5 -1)$ tests. The output of
dFCI and CA is not considerably more informative as measured by the
mean number of excess edges: CS 0.96, dFCI
0.07, CA
0.81 (average over all levels of sparsity).}
\label{fig:sub1}
\caption{Mean number of excess edges in output graphs
for
varying
numbers of edges (bidirected and directed) in the true graph (all
graphs are on 10 nodes), not
counting loops.}
\label{fig:sub2}
\caption{Comparison of performance.}
\label{fig:test}
\end{figure*}
\section{APPLICATION AND SIMULATIONS}
When evaluating the performance of a sound screening algoritm, the output graph is guaranteed to be a supergraph of the true parent graph, and we will say that edges that are in the output but not in the true graph are {\it excess edges}. For a node in a directed graph, the {\it indegree} is the number of directed edges adjacent with and pointed into the node, and the {\it outdegree} is the number of directed edges adjacent with and pointed away from the node.
One should note that all our experiments are done using an {\it oracle test}, i.e., instead of using real or synthetic data, the algorithms simply query an oracle for each local independence and receive the correct answer. This tests whether or not an algorithm can give good results using an efficient testing strategy (i.e., a low number of queries to the oracle) and therefore it evaluates the algorithms. This approach separates the algorithm from the specific test of local independence and evaluates only the algorithm. As such this is highly unrealistic as we would never have access to an oracle with real data, however, we should think of these experiments as a study of efficiency. The oracle approach to evaluating graphical learning algorithms is common in the DAG-based case, see \cite{spirtes2010} for an overview.
Also note that the comparison is only made with other constraint-based learning algorithms that can actually solve the problem at hand. Learning methods that assume full observation (such as the Hawkes methods mentioned in Section \ref{sec:hawkes}) would generally not output a graph with the correct interpretation even in the oracle case (see the example in Figure \ref{fig:causal}).
\subsection{C. ELEGANS NEURONAL NETWORK}
Caenorhabditis elegans is a roundworm in which the network between neurons has been mapped completely \citep{varshneyCelegans2011}. We apply our methods to this network as an application to a highly complex network. It consists of 279 neurons which are connected by both non-directional {\it gap junctions} and directional chemical synapses. We will represent the former as an unobserved process and the latter as a direct influence which is consistent with the biological system \citep{varshneyCelegans2011}. From this network, we sampled subnetworks of 75 neurons each (details in the supplementary material) and computed the output of the CS algorithm. These subsampled networks had on average 1109 edges (including bidirected edges representing unobserved processes, see the supplementary material) and on average 424 directed edges. The output graphs had on average 438 excess edges which is explained by the fact that there are many unobserved nodes in the graphs. To compare the output to the true parent graph, we computed the rank correlation between the indegrees of the nodes in the output graph and the indegrees of the nodes in the true parent graph, and similarly for the outdegree (indegree correlation: 0.94, outdegree correlation: 0.52). Finally, we investigated the method's ability to identify the observed nodes of highest directed connectivity (i.e., highest in- and outdegrees). The neuronal network of c. elegans is inhomogeneous in the sense that some neurons are extremely highly connected while others are only very sparsely connected. We considered the 15 nodes of highest indegree/outdegree (out of the 75 observed nodes). On average, the CS algorithm placed 13.4 (in) and 9.2 (out) of these 15 among the 15 most connected nodes.
From the output of the CS algorithm, we can find areas of the neuronal network which mediates information from one area to another, e.g., using Corollary \ref{cor:soundPath}.
\subsection{COMPARISON OF ALGORITHMS}
In this section we compare the proposed causal screening algorithms with previously published algorithms that solve similar problems. \cite{mogensenUAI2018} propose two algorithms, one of which is sure to output the correct graph when an oracle test is available. They note that this complete algorithm is computationally very expensive and adds little extra information, and therefore we will only consider their other algorithm for comparison. We will call this algorithm {\it
dynamical} FCI (dFCI) as it resembles FCI \citep{mogensenUAI2018}. dFCI actually solves a harder learning problem (see details in the supplementary material), however, it is computationally infeasible for many problems.
The Causal Analysis (CA) algorithm of \cite{meek2014} is a simple screening algorithm and we have in this paper argued that it is sound for learning the parent graph under the weaker assumption of ancestral faithfulness. Even though this algorithm uses a large number of tests, it is not guaranteed to provide complete learning as there may be inseparable nodes that are not adjacent \citep{mogensenUAI2018, mogensen2018}.
For the comparison of these algorithms, two aspects are important. As they are all sound, one aspect is the number of excess edges. The other aspect is of course the number of tests needed. The CS and CSAPC algorithms use at most $2n(n-1)$ tests and empirically the CSAP uses roughly the same number as the two former. This makes them feasible in large graphs. The quality of their output is dependent on the sparsity of the true graph, though the CSAP and CSAPC algorithms can deal considerably better with less sparse graphs (Subfigure \ref{fig:sub2}).
\section{DISCUSSION}
We suggested inexpensive constraint-based methods for learning causal structure based on testing local independence. An important observation is that local independence is asymmetric while conditional independence is symmetric. In a certain sense, this may help when constructing learning algorithms as there is no need of something like an `orientation phase' as in the FCI. This facilitates using very simple methods to give sound causal learning as we do not need the independence structure in full to give interesting output. Simple screening algorithms may be either adaptive or nonadaptive. We note that nonadaptive algorithms may be more robust to false conclusions from statistical tests of local independence.
The amount of information in the output of the screening algorithms depends on the sparsity of the true graph. However, even in examples with very little sparsity interesting structural information can be learned.
We showed that the proposed algorithms have a computational advantage over previously published algorithms within this framework. This makes it feasible to consider causal learning in large networks with unobserved processes. We obtained this gain in efficiency in part by outputting only the directed part of the causal structure. This means that we may be able to answer structural questions, but not questions relating to causal effect estimation.
\subsubsection*{Acknowledgments}
This work was supported by VILLUM FONDEN (research grant 13358). We thank Niels Richard Hansen and the anonymous reviewers for their helpful comments that improved this paper.
{\centering \huge Supplementary material}
This supplementary material contains additional graph theory, results, and definitions, as well as the proofs of the main paper.
\section{GRAPH THEORY}
In the main paper, we introduce the class of DGs to represent causal structures. One can represent marginalized DGs using the larger class of DMGs. A {\it
directed mixed graph} (DMG) is a graph such that any pair of nodes $\alpha,\beta\in V$ is joined by a subset of the edges $\{\alpha\rightarrow\beta,\alpha\leftarrow\beta,\alpha\leftrightarrow\beta\}$.
We say that edges $\alpha\rightarrow\beta$ and $\alpha\leftarrow\beta$ are {\it directed}, and that $\alpha\leftrightarrow\beta$ is {\it bidirected}. We say that the edge $\alpha \rightarrow \beta$ has a {\it head} at $\beta$ and a {\it tail} at $\alpha$. $\alpha\leftrightarrow\beta$ has heads at both $\alpha$ and $\beta$. We also introduced a walk $\langle \alpha_1, e_1, \alpha_2, \ldots ,\alpha_n, e_n, \alpha_{n+1} \rangle$. We say that $\alpha_1$ and $\alpha_{n+1}$ are endpoint nodes. A nonendpoint node $\alpha_i$ on a walk is a {\it collider} if $e_{i-1}$ and $e_i$ both have heads at $\alpha_i$, and otherwise it is a {\it noncollider}. A cycle is a path $\langle\alpha, e_1, \ldots, \beta\rangle$ composed with an edge between $\alpha$ and $\beta$. We say that $\alpha$ is an {\it ancestor} of $\beta$ if there exists a directed path from $\alpha$ to $\beta$. We let $\text{an}(\beta)$ denote the set of nodes that are ancestors of $\beta$. For a node set $C$, we let $\text{an}(C) = \cup_{\beta\in C} \text{an}(\beta)$. By convention, we say that a trivial path (i.e., with no edges) is directed and this means that $C\subseteq \text{an}(C)$.
For DAGs $d$-separation is often used for encoding independences. We use the analogous notion of $\mu$-separation which is a generalization of $\delta$-separation \cite{didelez2000, didelez2008, meek2014, mogensen2018}.
We use the class of DGs to represent the underlying, data-generating structure. When only parts of the causal system is observed, the class of DMGs can be used to represent marginalized DGs \cite{mogensen2018}. This can be done using {\it latent projection} \cite{vermaEquiAndSynthesis, mogensen2018} which is a map that for a DG (or more generally, for a DMG), $\mathcal{D} = (V,E)$, and a subset of observed nodes/processes, $O\subseteq V$, provides a DMG, $m(\mathcal{D}, O)$, such that for all $A,B,C \subseteq O$,
\[ \musepG{A}{B}{C}{\mathcal{D}} \Leftrightarrow \musepG{A}{B}{C}{m(\mathcal{D},
O)}. \]
See \cite{mogensen2018} for details on this graphical marginalization. We say that two DMGs, $\mathcal{G}_1 = (V,E_1), \mathcal{G}_2 = (V,E_2)$, are {\it Markov
equivalent} if
\[ \musepG{A}{B}{C}{\mathcal{G}_1} \Leftrightarrow \musepG{A}{B}{C}{\mathcal{G}_2}, \]
for all $A,B,C \subseteq V$, and we let $[\mathcal{G}_1]$ denote the Markov equivalence class of $\mathcal{G}_1$. Every Markov equivalence class of DMGs has a unique {\it maximal element} \cite{mogensen2018}, i.e., there exists $\mathcal{G} \in [\mathcal{G}_1]$ such that $\mathcal{G}$ is a supergraph of all other graphs in $[\mathcal{G}_1]$.
For a DMG, $\mathcal{G}$, we will let $D(\mathcal{G})$ denote the {\it directed part} of $\mathcal{G}$, i.e., the DG obtained by deleting all bidirected edges from $\mathcal{G}$.
\begin{prop}
Let $\mathcal{D} = (V,E)$ be a DG, and let $O\subseteq V$. Consider $\mathcal{G}
=
m(\mathcal{D}, O)$. For $\alpha,\beta\in O$ it holds that $\alpha\in
\text{an}_\mathcal{D}(\beta)$ if and only if $\alpha\in \text{an}_{D(\mathcal{G})}(\beta)$.
Furthermore, the directed part of $\mathcal{G}$ equals the parent graph of
$\mathcal{D}$ on nodes $O$, i.e., $D(\mathcal{G}) = \mathcal{P}_O(\mathcal{D})$. \end{prop}
\begin{proof}
Note first that $\alpha\in
\text{an}_\mathcal{D}(\beta)$ if and only if $\alpha\in
\text{an}_\mathcal{G}(\beta)$ \cite{mogensen2018}. Ancestry is only defined by
the directed
edges, and it follows that $\alpha\in
\text{an}_\mathcal{G}(\beta)$ if and only if $\alpha\in
\text{an}_{D(\mathcal{G})}(\beta)$. For the second statement, the definition of the
latent
projection gives that there is a directed edge from $\alpha$ to $\beta$
in
$\mathcal{G}$ if and only if there is a directed path from $\alpha$ to $\beta$
in
$\mathcal{D}$ such that no nonendpoint node is in $O$. By definition,
this
is the parent graph, $\mathcal{P}_O(\mathcal{D})$. \end{proof}
In words, the above proposition says that if $\mathcal{G}$ is a marginalization (done by latent projection) of $\mathcal{D}$, then the ancestor relations of $\mathcal{D}$ and $D(\mathcal{G})$ are the same among the observed nodes. It also says that our learning target, the parent graph, is actually the directed part of the latent projection on the observed nodes. In the next subsection, we use this to describe what is actually identifiable from the induced independence model of a graph.
\subsection{MAXIMAL GRAPHS AND PARENT GRAPHS}
Under faithfulness of the local independence model and the causal graph, we know that the maximal DMG is a correct representation of the local independence structure in the sense that it encodes exactly the local independences that hold in the local independence model. From the maximal DMG, one can use results on equivalence classes of DMGs to obtain every other DMG which encodes the observed local independences \citep{mogensen2018} and from this graph one can find the parent graph as simply the directed part. However, it may require an infeasible number of tests to output such a maximal DMG. This is not surprising, seeing that the learning target encodes this complete information on local independences.
Assume that $\mathcal{D}_0 = (V,E)$ is the underlying causal graph and that $\mathcal{G}_0 = (O,F), O\subseteq V$ is the marginalized graph over the observed variables, i.e., the latent projection of $\mathcal{D}_0$. In principle, we would like to output $\mathcal{P}(\mathcal{D}_0) = D(\mathcal{G}_0)$, the directed part of $\mathcal{G}_0$. However, no algorithm can in general output this graph by testing only local independences as Markov equivalent DMGs may not have the same parent graph. Within each Markov equivalence class of DMGs, there is a unique maximal graph. Let $\bar{\mathcal{G}}$ denote the maximal graph which is Markov equivalent of $\mathcal{G}_0$. The DG $D(\bar{\mathcal{G}})$ is a supergraph of $D(\mathcal{G}_0)$ and we will say that a learning algorithm is complete if it is guaranteed to output $D(\bar{\mathcal{G}})$ as no algorithm testing local independence only can identify anything more than the equivalence class.
\section{COMPLETE LEARNING}
The CS algorithm provides sound learning of the parent graph of a general DMG under the assumption of ancestral faithfulness. For a subclass of DMGs, the algorithm actually provides complete learning. It is of interest to find sufficient graphical conditions to ensure that the algorithm removes an edge $\alpha\rightarrow\beta$ which is not in the true parent graph. In this section, we state and prove one such condition which can be understood as `the true parent set is always found for unconfounded processes'. We let $\mathcal{D}$ denote the output of the CS algorithm.
\begin{prop}
If $\alpha \not\rightarrow_{\mathcal{G}_0} \beta$ and there is no $\gamma\in
V\setminus\{\beta\}$ such that $\gamma\leftrightarrow_{\mathcal{G}_0}\beta$,
then
$\alpha\not\rightarrow_\mathcal{D}\beta$. \end{prop}
\begin{proof}
Let $\mathcal{D}_1, \mathcal{D}_2, \ldots, \mathcal{D}_N$ denote the
DGs
that are constructed when running the algorithm by sequentially
removing
edges, starting from the complete DG, $\mathcal{D}_1$. Consider a
connecting walk
from
$\alpha$ to $\beta$ in $\mathcal{G}_0$. It must be of the form $\alpha \sim
\ldots
\sim \gamma
\rightarrow \beta$, $\gamma\neq \alpha$. Under ancestral faithfulness,
the edge $\gamma \rightarrow\beta$ is in $\mathcal{D}$,
thus $\gamma\in\text{pa}_{\mathcal{D}_i}(\beta)$ for all $\mathcal{D}_i$ that
occur during the algorithm, and therefore when $\langle \alpha, \beta
\mid
\text{pa}_{\mathcal{D}_i}(\beta) \setminus \{\alpha\} \rangle$ is tested, the
walk is closed. Any walk
from $\alpha$ to $\beta$ is of this form, thus also closed, and we have
that $\musep{\alpha}{\beta}{\text{pa}_{\mathcal{D}_i}(\beta)}$ and therefore
$\langle \alpha,\beta \mid \text{pa}_{\mathcal{D}_i}(\beta) \setminus
\{\alpha\}\rangle \in \mathcal{I}$. The
edge $\alpha
\rightarrow_{\mathcal{D}_i} \beta$ is removed and thus absent in the
output
graph, $\mathcal{D}$. \end{proof}
\begin{comment}
We can strengthen the above result in the following way.
\begin{prop} Assume $\alpha\not\rightarrow_{\mathcal{G}_0} \beta$ and that $\mathcal{I}^O$ is faithful with respect to $\mathcal{G}_0$. If every walk from $\alpha$ to $\beta$ has a noncollider, $\gamma$, such that there exists a directed trek from $\gamma$ to $\beta$, then $\alpha \not \rightarrow_\mathcal{D}\beta$. \end{prop}
\begin{proof} Consider a walk, $\omega$, from $\alpha$ to $\beta$ and let $\gamma$ denote the noncollider such that there is directed trek from $\gamma$ to $\beta$. Let $\mathcal{D}_i$ denote the current graph in the algorithm when testing $\langle \alpha,\beta\mid \text{pa}_{\mathcal{D}_i}(\beta) \setminus\{\alpha\}\rangle \in \mathcal{I}$. If $\gamma\in \text{pa}_{\mathcal{D}_i}(\beta)$, then the walk is closed. If $\gamma\notin \text{pa}_{\mathcal{D}_i}(\beta) \setminus \{\alpha\}$, then there exist $j < i$ such that $\langle \gamma,\beta \mid \text{pa}_{\mathcal{D}_j}(\beta) \setminus \{\gamma\}\rangle \in \mathcal{I}$. Using faithfulness, it follows that $\musep{\gamma}{\beta}{ \text{pa}_{\mathcal{D}_j}(\beta) \setminus \{\gamma\}}$. This means that the subwalk between $\gamma$ and $\beta$ is closed given a superset of $\text{pa}_{\mathcal{D}_i}(\beta)$. Consider the subwalk between $\gamma$ and $\beta$. If it is not closed by $\text{pa}_{\mathcal{D}_i}(\beta) \setminus \{\alpha\}$, then there is some noncollider on the subwalk is not in this conditioning set, but is in $\text{pa}_{\mathcal{D}_j}(\beta) \setminus\{\gamma\}$ for $j < i$. Let $\phi$ be the such noncollider which is the closest to $\beta$ (along the walk). Then every node between $\phi$ and $\beta$ must be colliders, or else the walk is closed. The colliders are all directedly trek-connected to $\beta$ and in $\text{pa}_{\mathcal{D}_j}(\beta)$ for $j > i$. However, this is a contraction as then the edge $\phi \rightarrow \beta$ would never have been removed (there is a directed trek from $\phi$ to $\beta$ since $\phi$ is in a $\text{pa}_{\mathcal{D}_j}(\beta)$-set after Subalgorithm 1). \end{proof}
\end{comment}
\section{ANCESTRY PROPAGATION}
We state Subalgorithm \ref{subalgo:ancProp2} here.
\setcounter{algocf}{3}
\begin{subalgorithm}
\SetKwInOut{Input}{input}
\SetKwInOut{Output}{output}
\Input{a local independence oracle for $\mathcal{I}^O$ and a DG, $\mathcal{D} =
(O,E)$}
\Output{a DG on nodes $O$}
initialize $E_r = \emptyset$ as the empty edge set\;
\ForEach{$(\alpha,\beta, \gamma)\in V\times V \times V$ such that
$\alpha,\beta,\gamma$ are all distinct}{
\If{$\alpha \sim_\mathcal{D} \beta$, $\beta \rightarrow_\mathcal{D}
\gamma$, and
$\alpha
\not\rightarrow_\mathcal{D} \gamma$}{
\If{$\langle \alpha,\gamma\mid \emptyset\rangle \in
\mathcal{I}^O$}{update
$E_r = E_r \cup \{\beta
\rightarrow
\gamma \}$\;}
}
}
Update $\mathcal{D} = (V, E\setminus E_r)$\;
\Return $\mathcal{D}$
\newline
\caption{Ancestry propagation}
\label{subalgo:ancProp2} \end{subalgorithm}
Composing Subalgorithm \ref{subalgo:trekStep}, Subalgorithm \ref{subalgo:ancProp2}, and Subalgorithm \ref{subalgo:paStep} is referred to as the causal screening, ancestry propagation (CSAP) algorithm. If we use Subalgorithm \ref{subalgo:ancProp1} instead of Subalgorithm \ref{subalgo:ancProp2}, we call it the CSAPC algorithm (C for cheap as this does not entail any additional independence tests compared to CS).
\section{APPLICATION AND SIMULATIONS}
In this section, we provide some additional details about the c. elegans neuronal network and the simulations.
\subsection{C. ELEGANS NEURONAL NETWORK}
For each connection between two neurons a different number of synapses are present (ranging from 1 to 37). We only consider connections with more than 4 synapses when we define the true underlying network. When sampling the subnetworks, highly connected neurons were sampled with higher probability to avoid a fully connected subnetwork when marginalizing.
\subsection{COMPARISON OF ALGORITHMS}
As noted in the main paper, the dFCI algorithm solves a strictly harder problem. By using the additional graph theory in the supplementary material, we can understand the output of the dFCI algorithm as a supergraph of the maximal DMG, $\bar{\mathcal{G}}$. There is also a version of the dFCI which is guaranteed to output not only a supergraph of $\bar{\mathcal{G}}$, but the graph $\bar{\mathcal{G}}$ itself. Clearly, from the output of the dFCI algorithm, one can simply take the directed part of the output and this is a supergraph of the underlying parent graph.
\section{PROOFS}
In this section, we provide the proofs of the result in the main paper.
\begin{proof}[Proof of Proposition \ref{prop:pairCau}]
Let $\mathcal{D}$ denote the causal graph. Assume first that
$\alpha\not\rightarrow_\mathcal{D}\beta$. Then $g^{\beta\alpha}$ is
identically zero over the observation interval, and it follows directly
from the
functional form of $\lambda_t^\beta$ that
$\alpha\not\rightarrow\beta\mid
V\setminus \{\alpha\}$. This shows that the local independence model
satisfies the pairwise Markov property with respect to $\mathcal{D}$.
If instead $g^{\beta\alpha} \neq 0$ over $J$, there exists $r\in J$
such
that $g^{\beta\alpha}(r) \neq 0$. From continuity of $g^{\beta\alpha}$
there exists a compact interval of positive measure, $I\subseteq
J$, such that $\inf_{s\in I}(g^{\beta\alpha}(s)) \geq
g_{\min}^{\beta\alpha}$
and $g_{\min}^{\beta\alpha}
> 0$. Let $i_0$ and $i_1$ denote the endpoints of this interval, $i_0 <
i_1$. We consider now the events
\begin{align*}
D_k = (N_
{T - i_0}^\alpha - N_
{T - i_1}^\alpha = k, N_T^\gamma = 0 \text{ for all } \gamma \in
V\setminus \{\alpha\})
\label{eq:setsDk}
\end{align*}
\noindent $k\in \mathbb{N}_0$. Then under Assumption \ref{ass:gFunc},
for
all $k$
\[
\lambda_T^\beta \mathds{1}_{D_k} \geq \mathds{1}_{D_k}
\int_I
g^{\beta\alpha}(T-s)\ \mathrm{d}
N_s^\alpha \geq
g_{\min}^{\beta\alpha}\cdot k \cdot \mathds{1}_{D_k}.
\]
\noindent Assume for contradiction that $\beta$ is locally independent
of
$\alpha$ given $V\setminus \{\alpha\}$. Then $\lambda_T^\beta =
\text{E}(\lambda_T^\beta \mid
\mathcal{F}_T^{V}) = \text{E}(\lambda_T^\beta \mid
\mathcal{F}_T^{V\setminus
\{\alpha\}})$ is constant on $\cup_k D_k$ and furthermore $\text{P}(D_k)
>
0$ for
all $k$. However, this contradicts the above inequality when $k
\rightarrow
\infty$. \end{proof}
\begin{proof}[Proof of Proposition \ref{prop:soundScreen}]
Let $\mathcal{D}$ denote the DG which is output by the algorithm. We
should
then show that $\mathcal{P}(\mathcal{D}_0) \subseteq \mathcal{D}$.
Assume
that $\alpha
\rightarrow_{\mathcal{P}(\mathcal{D}_0)} \beta$. In this case, there is
a
directed path from $\alpha$ to $\beta$ in $\mathcal{D}_0$ such that no
nonendpoint node
on this directed walk is in $O$ (the
observed coordinates). Therefore for any $C\subseteq
O\setminus
\{\alpha\}$ there exists a directed $\mu$-connecting walk from $\alpha$
to
$\beta$ in $\mathcal{D}_0$ and by ancestral faithfulness it follows
that
$\ind{\alpha}{\beta}{C} \notin \mathcal{I}$. The algorithm starts from the
complete
directed graph, and the above means that the directed edge from
$\alpha$
to $\beta$ will not be
removed. \end{proof}
\begin{proof}[Proof of Corollary \ref{cor:soundPath}]
Consider some directed path from $\alpha$ to $\beta$ in $\mathcal{D}_0$
on
which no node is in $C$. Then there is also a directed path from
$\alpha$
to $\beta$ on which no nodes is in $C$ in the graph
$\mathcal{P}(\mathcal{D}_0)$, and therefore also in the output graph
using
Proposition \ref{prop:soundScreen}. \end{proof}
\begin{proof}[Proof of Proposition \ref{prop:trekSep}]
Assume that there is a $\mu$-connecting walk from $\alpha$ to $\beta$
given
$\{\beta\}$. If this walk has no colliders, then it is a directed trek,
or
can be reduced to one.
Otherwise, assume that $\gamma$ is the collider which is the closest to
the
endpoint
$\alpha$. Then $\gamma \in \text{an}(\beta)$, and composing the subwalk from
$\alpha$ to $\gamma$ with the directed path from $\gamma$ to $\beta$
gives
a directed trek, or it can be reduced to one. On the other hand, assume
there is a directed trek
from
$\alpha$ to
$\beta$. This is $\mu$-connecting from $\alpha$ to $\beta$ given
$\{\beta\}$. \end{proof}
\begin{proof}[Proof of Proposition \ref{prop:soundAncProp}]
Assume $\beta\rightarrow_{\mathcal{P}(\mathcal{D}_0)} \gamma$.
Subalgorithms
\ref{subalgo:trekStep} and \ref{subalgo:paStep} are both
simple screening algorithms, and they will not remove this edge. Assume
for
contradiction that
$\beta\rightarrow\gamma$ is removed by Subalgorithm
\ref{subalgo:ancProp1}.
Then
there
must exist $\alpha\neq \beta,\gamma$ and a directed trek from $\alpha$
to
$\beta$ in $\mathcal{D}_0$. On this directed trek, $\gamma$ does not
occur
as this would imply
a directed trek either from $\alpha$ to $\gamma$ or from $\beta$ to
$\alpha$, thus implying $\alpha\rightarrow_\mathcal{D}\gamma$ or
$\beta\rightarrow_\mathcal{D}\alpha$, respectively ($\mathcal{D}$ is
the
output graph of Subalgorithm \ref{subalgo:trekStep}). As $\gamma$ does not
occur on the trek, composing this trek with the edge $\beta
\rightarrow \gamma$ would give a directed trek from $\alpha$ to
$\gamma$.
By faithfulness, $\langle \alpha,\gamma\mid \gamma\rangle \notin \mathcal{I}$,
and
this is a contradiction as $\alpha\rightarrow\gamma$ would not have
been
removed during Subalgorithm 1.
We consider instead CSAP. Assume for contradiction that
$\beta\rightarrow\gamma$ is removed during Subalgorithm
\ref{subalgo:ancProp2}. There exists in $\mathcal{D}_0$ either a
directed
trek from $\alpha$
to $\beta$ or a directed trek from $\beta$ to $\alpha$. If $\gamma$ is
on
this trek, then $\gamma$ is not $\mu$-separated from $\alpha$ given the
empty set (recall that there are loops at all nodes, therefore also at
$\gamma$), and using faithfulness we conclude that $\gamma$ is not on
this
trek. Composing it with the edge $\beta\rightarrow\gamma$ would give a
directed trek from $\alpha$ to $\gamma$ and using faithfulness we
obtain a
contradiction. \end{proof}
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title[Single Parameter Inference of Non-sparse Logistic Regression Models]{Single Parameter Inference of Non-sparse Logistic Regression Models}
\author[1]{\fnm{Yanmei} \sur{Shi}}\email{[email protected]} \author*[1]{\fnm{Qi} \sur{Zhang}}\email{[email protected]}
\affil[1]{Institute of Mathematics and Statistics, Qingdao University, 308 Ningxia Road, Shinan District, Qingdao, Shandong, China}
\abstract{ This paper infers a single parameter in non-sparse logistic regression models. By transforming the null hypothesis into a moment condition, we construct the test statistic and obtain the asymptotic null distribution. Numerical experiments show that our method performs well. }
\keywords{Logistic models \sep Non-sparse \sep Single parameter hypothesis test \sep Moment condition}
\pacs[MSC Classification]{62F03, 62F35, 62J15}
\maketitle
\section{Introduction}\label{sec1} The logistic regression models have been widely used in finance and genetics analysis, which increasingly rely on high-dimensional observations. In other words, the dimension $p$ is high, and the sample size $n$ is relatively small, i.e. $n\rightarrow \infty$ and $p/n\rightarrow \infty$, therefore, modeling, inference, and prediction become more challenging than in traditional environments.
Hypothesis test and confidence intervals in high-dimensional generalized linear models have been widely studied. \cite{2014On} constructed confidence intervals and statistical tests for single or low-dimensional components of regression coefficients. \cite{Ning0A2017} proposed a general framework for hypothesis testing and confidence intervals for low-dimensional components based on general penalized M-estimators.
\cite{2019Optimal} constructed a debiased estimator based on Lasso estimator and consistently established its asymptotic normality for future observations of arbitrary high dimensions.
In the logistic regression models, \cite{Sur2019} studied the likelihood ratio test under $p/n\rightarrow k$ for some $k<\frac{1}{2}$.
\cite{Shi2021} focused on the logistic link and imposed certain stringent assumptions.
\cite{2021RongMa} constructed a test statistic for testing the global null hypothesis using a generalized low-dimensional projection for bias correction.
\cite{Guo2021} proposed a novel bias-corrected estimator through linearization and variance enhancement techniques.
The above methods are sensitive to the sparsity assumption, which leads to the easy loss of error control when this assumption is violated. Statistical inference in non-sparse linear models has been studied extensively. \cite{LuLin2011} proposed semiparametric re-modeling and inference method. \cite{LuFeng2011} introduced a simulation-based procedure to reformulate a new model, with no need to estimate high-dimensional nuisance parameter. \cite{Dezeure2017} proposed a residual and wild bootstrap methodology for individual and simultaneous inference. By transforming the null hypothesis into a testable moment condition, \cite{2018Significance} proposed an asymptotically sparse CorrT method to solve the single-parameter testing problem. By convolving the variables from the two samples and combining the moment method, \cite{2016Two} conducted the homogeneity test of the global parameters in two populations. \cite{2016Linear} further extended this moment method to test linear functionals of the regression parameters, and proposed Modified Dantzig Selector (MDS) to estimate model parameters. \cite{2022TESTABILITY} developed uniform and essentially uniform nontestability which identified a collection of alternatives such that the power of any test was at most equal to the nominal size.
In this paper, we consider single parameter significance test in high-dimensional non-sparse logistic regression models, which is of great importance in practice, and is a prerequisite to statistical analysis.
For example, we study the effect of a treatment/drug on response after controlling for the impact of high-dimensional non-sparse genetic markers.
This problem of statistical inference has not been solved in the existing literature.
First, we linearize the regression function based on the logistic Lasso estimator.
Then, the approximate linear model is reconstructed according to the hypothesis, which is transformed into a testable moment condition. Finally, we use MDS estimators to construct the test statistics and prove the asymptotic null distribution and power property. Besides its applicability in logistic regression, this method can be extended to other nonlinear regression models.
The remainder of this paper is organized as follows. In Section \ref{section 2}, We present a significance test method for single parameter in non-sparse logistic regression model, and introduce a new moment construction method.
Section \ref{section 3} shows the size and power properties of the proposed test.
Section \ref{Numerical Examples} shows the numerical experiments and compares them with the results of another advanced method.
\section{Single parameter significance test} \label{section 2}
\subsection{Notations}
For a vector $V\in \mathbb{R}^{k}$, $v_{i}$ represents the $i$-th element of $V$. $\|V\|_{\infty}=\max\limits_{1\leq i \leq k}\vert v_{i}\vert$ and $\|V\|_{0}=\sum\limits_{i=1}^{k}\mathrm{I}(v_{i}\neq 0)$, where $\mathrm{I}( \cdot )$ denotes the indicator function.
For matrix $A$, its $(i,j)$ entry is denoted by $A_{i,j}$, and the $i$-th row is denoted by $A_{i}$.
For two sequences $a_{n}, b_{n} >0$, $a_{n}\asymp b_{n}$ means that there exist constants $C_{1}, C_{2} >0$ such that $\forall n$, $a_{n}\leq C_{1}b_{n}$ and $b_{n} \leq C_{2}a_{n}$.
\subsection{Model and hypothesis}
We consider the non-sparse logistic regression model: \begin{equation} \label{1.1} y_{i}=f(\beta^{T}X_{i})+\varepsilon_{i},i=1,2,...,n \tag{2.1} \end{equation}
where $f(u)=e^{u}/(1+e^{u})$, and $\beta=(\beta_{*}, \theta_{*})\in\mathbb{R}^{p}$ is a non-sparse regression vector with single parameter $\beta_{*}$ and redundant parameter $\theta_{*}\in\mathbb{R}^{p-1}$.
The observations are i.i.d. samples $(X_{i},y_{i})\in \mathbb{R}^{p}\times \{0,1\}$ for $i=1,2,...,n$, and $y_{i}\mid X_{i} \thicksim Bernoulli(f(\beta^{T}X_{i}))$ independently for each $i=1,2,...,n$. We assume $X_{i}\sim N(0,\Sigma)$. In fact, this result can be extended to sub-Gaussian distribution.
The $\varepsilon=(\varepsilon_{1}, \varepsilon_{2},..., \varepsilon_{n})^{'}\in\mathbb{R}^{n}$ is the error term, which is not correlated with $X=(X_{1},X_{2},...,X_{n})^{'}\in\mathbb{R}^{n\times p}$.
In this paper, we focus on the significance test of single parameter $\beta_{*}$, i.e. \begin{equation}\label{1.2} H_{0}: \beta_{*}=\beta_{0}, \ \ versus \ \ H_{1}:\beta_{*}\neq\beta_{0}. \tag{2.2} \end{equation} where $\beta_{0}$ is a given value.
As a preliminary, we first give an estimator of the global parameter $\beta$. For technical reasons, we split the samples into two independent subsets $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$. The $\mathcal{L}_{1}$-regularized M-estimator $\hat{\beta}$ of $\beta$ is obtained from $\mathcal{D}_{1}$: \begin{equation}\label{M-estimator}
\hat{\beta}=\arg\min\limits_{\beta}\{\frac{1}{n}\sum\limits_{i=1}^{n}[-y_{i}\beta^{T}X_{i}+log(1+e^{\beta^{T}X_{i}})]+\lambda\|\beta\|_{1}\}, \tag{2.3} \end{equation} which is the minimizer of a penalized log-likelihood function with $\lambda\asymp\sqrt{\frac{logp}{n}}$. Although $\hat{\beta}$ can achieve the optimal rate of convergence \citep{Sahand2012A, 2012Estimation}, it's not suitable to construct confidence intervals and hypotheses test directly because of its biases.
In the following, we reconstruct the regression model based on $\hat{\beta}$ and the samples from $\mathcal{D}_{2}$.
We consider the Taylor expansion of $f(u_{i})$ at $\hat{u}_{i}$ for $u_{i}=\beta^{T}X_{i}$ and $\hat{u}_{i}=\hat{\beta}^{T}X_{i}$, \begin{equation}\label{Taylor expansion} f(u_{i})=f(\hat{u}_{i})+\dot{f}(\hat{u}_{i})(u_{i}-\hat{u}_{i})+Re_{i}, \ i=1,2,...,n, \tag{2.4} \end{equation} where $\dot{f}(\cdot)$ is the derivative of $f(\cdot)$, and $Re(i)$ is the reminder term. Plugging \eqref{Taylor expansion} into \eqref{1.1}, we have \begin{equation}\label{moxingchongxie} y_{i}-f(\hat{\beta}^{T}X_{i})+\dot{f}(\hat{\beta}^{T}X_{i})X_{i}^{T}\hat{\beta}-Re_{i}=\dot{f}(\hat{\beta}^{T}X_{i})X_{i}^{T}\beta+\varepsilon_{i}, \ i=1,2,...,n. \tag{2.5} \end{equation} We can treat $y_{i}-f(\hat{\beta}^{T}X_{i})+\dot{f}(\hat{\beta}^{T}X_{i})X_{i}^{T}\hat{\beta}-Re_{i}$ as a new response variable $y_{new,i}$, and $Y_{new}=(y_{new,1},y_{new,2},...,y_{new,n})^{T}\in \mathbb{R}^{n}$, whereas $\dot{f}(\hat{\beta}^{T}X_{i})X_{i}$ as the new covariate $X_{new,i}=(z_{i},W_{i}^{T})^{T}$ with $z_{i}\in\mathbb{R}$ and $W_{i}\in\mathbb{R}^{p-1}$. Consequently, $\beta$ can be considered as the regression coefficient of this approximate linear model. Then \eqref{moxingchongxie} is transformed into \begin{equation}\label{moxingchongxie1} y_{new,i}=X_{new,i}^{T}\beta+\varepsilon_{i}, \ i=1,2,...,n. \tag{2.6} \end{equation}
Since the null hypothesis is $H_{0}: \beta_{*}=\beta_{0}$, the above equation can be rewritten as: \begin{equation}\label{moxingchongxie2} y_{new,i}=z_{i}\beta_{*}+W_{i}^{T}\theta_{*}+\varepsilon_{i}, \ i=1,2,...,n,\tag{2.7} \end{equation} where $Z=(z_{1},z_{2},...,z_{n})^{'}\in\mathbb{R}^{n}$ and $W=(W_{1},W_{2},...,W_{n})^{'}\in\mathbb{R}^{n\times(p-1)}$ are the design matrices.
Subtracting $z_{i}\beta_{0}$ from both sides of model \eqref{moxingchongxie2}, we build the following reconstructed model \begin{equation}\label{chonggoumoxing11111} y_{new,i}-z_{i}\beta_{0}=z_{i}\gamma_{*}+W_{i}^{T}\theta_{*}+\varepsilon_{i}, \ i=1,2,...,n, \tag{2.8} \end{equation} where $\gamma_{*}=\beta_{*}-\beta_{0}$ is of main interest, and the original $H_{0}$ in \eqref{1.2} is equivalent to \begin{equation}\label{chonggoumoxing22222} H_{0}:\gamma_{*}=0. \tag{2.9} \end{equation} Thus, we define a pseudo-response $V=Y_{new}-Z\beta_{0}$ and a pseudo-error $e=Z\gamma_{*}+\varepsilon$, which satisfy that \begin{equation}\label{VWe} V=W\theta_{*}+e, \tag{2.10} \end{equation}
Since $X$ and $\varepsilon$ are unrelated, $W$ and $\varepsilon$ are unrelated. Therefore, when the null hypothesis is true, we have $E(e^{T}W)=E(\varepsilon^{T}W)=0$. Otherwise, $e$ and $W$ may be linear dependent through $z$, which is caused by the confounding effects of $W$ and $Z$. Next, we establish a linear correlation model between $Z$ and $W$: \begin{equation}\label{xianxingxiangguanmoxing} Z=W\pi+U, \tag{2.11} \end{equation} where $\pi\in\mathbb{R}^{p-1}$ is an unknown regression coefficient vector and $U\in\mathbb{R}^{n}$ is the error term, internally independent of each other, which follows Gaussian distribution with zero mean, and $U$ is uncorrelated to $(V, \ W)$. It is worth mentioning that we assume that $\pi$ is sparse to decouple the correlation between $Z$ and $W$.
We consider the correlation between $e$ in \eqref{VWe} and $U$ in \eqref{xianxingxiangguanmoxing}: \begin{equation}\label{liangwuchaxiangguanxing} E(U^{T}e)=E(U^{T}Z\gamma_{*}+U^{T}\varepsilon)=E(U^{T}U)\gamma_{*}. \tag{2.12} \end{equation} Therefore, the original test problem in \eqref{1.2} is equivalent to \begin{equation}\label{jianyanwenti} H_{0}: E((V-W\theta_{*})^{T}(Z-W\pi))=0 \ \ versus \ \ H_{1}: E((V-W\theta_{*})^{T}(Z-W\pi))\neq0. \tag{2.13} \end{equation} Since $\pi$ is sparse, consistent estimator $\breve{\pi}$ is easy to obtain. However, it is difficult to obtain the consistent estimator of non-sparse parameter $\theta_{*}$. For any estimator $\check{\theta}_{*}$ of $\theta_{*}$, we have \begin{equation} E((V-W\breve{\theta}_{*})^{T}(Z-W\breve{\pi}))\rightarrow E((V-W\theta_{*})^{T}(Z-W\pi))+E((Z-W\pi)^{T}W(\theta_{*}-\breve{\theta}_{*})). \notag \end{equation} In the above equation, $\breve{\theta}_{*}$ is a function of $(V,W)$, while $U$ is uncorrelated to $(V,W)$, so $Z-W\pi$ and $W(\theta_{*}-\breve{\theta}_{*})$ are uncorrelated. Then \begin{equation} E((Z-W\pi)^{T}W(\theta_{*}-\breve{\theta}_{*}))=E(Z-W\pi)^{T}E(W(\theta_{*}-\breve{\theta}_{*}))=0. \notag \end{equation} Therefore, \begin{equation} E((V-W\breve{\theta}_{*})^{T}(Z-W\breve{\pi}))\rightarrow E((V-W\theta_{*})^{T}(Z-W\pi)). \notag \end{equation} The inner product structure in \eqref{jianyanwenti} alleviates the reliance on a good estimator of $\theta_{*}$. We will estimate the unknown parameters $\pi$ and $\theta_{*}$ in the next subsection.
\subsection{Modified Dantzig Selector} \label{section 2.2}
MDS is used to estimate the unknown parameter $\theta_{*}$ and error variance $\sigma_{e}^{2}$ simultaneously,
\begin{equation}\tag{2.14} \label{theta_tilde} \begin{split}
\tilde{\theta}_{*}=&\arg\min\limits_{\theta_{*}\in\mathbb{R}^{p-1}} \|\theta_{*}\|_{1} \notag \\
s.t.& \ \|W^{T}(V-W\theta_{*})\|_{\infty}\leq \eta\rho_{1}\sqrt{n}\|V\|_{2} \notag \\
&V^{T}(V-W\theta_{*})\geq \rho_{0}\rho_{1}\|V\|_{2}^{2}/2 \notag \\ &\rho_{1} \in [\rho_{0}, 1] \notag, \end{split} \end{equation} where $\rho_{1}=\sigma_{e}/ \sqrt{E(v_{1})^{2}}$, and $\rho_{0} \in (0,1)$ is a lower bound for this ratio. $\eta\asymp \sqrt{n^{-1}log p}$ is a tuning parameter.
Similarly, the estimator $\tilde{\pi}\in \mathbb{R}^{p-1}$ of $\pi$ is \begin{equation} \tag{2.15} \label{pij_tilde} \begin{split}
\tilde{\pi}=&\arg\min\limits_{\pi\in\mathbb{R}^{p-1}} \|\pi\|_{1} \notag \\
s.t.& \|W^{T}(Z-W\pi)\|_{\infty}\leq\eta\rho_{2}\sqrt{n}\|Z\|_{2} \notag \\
&Z^{T}(Z-W\pi)\geq \rho_{0}\rho_{2}\|Z\|_{2}^{2}/2 \notag \\ & \rho_{2}\in[\rho_{0},1] \notag, \end{split} \end{equation} where $\rho_{2}=\sigma_{u}/\sqrt{E(z_{1})^{2}}$.
\subsection{Test statistic} \label{ section 2.3}
By plugging in the estimators $\tilde{\pi}$ and $\tilde{\theta}_{*}$, we construct the following test statistic
\begin{equation}\label{jianyantongjiliang} T_{n}= n^{-\frac{1}{2}}\hat{\sigma}_{e}^{-1}(Z-W\tilde{\pi})^{T}(V-W\tilde{\theta}_{*}) \tag{2.16}, \end{equation}
where $\hat{e}=V-W\tilde{\theta}_{*}$ and $\hat{\sigma}_{e}=\|V-W\tilde{\theta}_{*}\|_{2}/\sqrt{n}$. Obviously, under the null hypothesis and the sparsity assumption of $\pi$, we have \begin{equation} T_{n}=n^{-\frac{1}{2}}\hat{\sigma}_{e}^{-1}(Z-W\tilde{\pi})^{T}(V-W\tilde{\theta}_{*}) = \Delta\cdot\hat{\sigma}_{e}^{-1}+n^{-\frac{1}{2}}\hat{\sigma}_{e}^{-1}U^{T}\hat{e} \notag, \end{equation} where $\Delta=n^{-\frac{1}{2}}(\pi-\tilde{\pi})^{T}W^{T}\hat{e}$, and we can proof that
$\Delta\cdot\hat{\sigma}_{e}^{-1}=o_{p}(1)$. So the statistical properties of $T_{n}$ is determined by $n^{-\frac{1}{2}}U^{T}\hat{e}\hat{\sigma}_{e}^{-1}=n^{-\frac{1}{2}}\sum\limits_{i=1}^{n}u_{i}\hat{e}_{i}\hat{\sigma}_{e}^{-1}$.
Under the null hypothesis, $U$ is uncorrelated of $(V, W)$, while $\tilde{\theta}_{*}$ is completely dependent on $(V,W)$, so $\hat{e}$ is also only related to $(V,W)$. Therefore, $U$ and $\hat{e}$ are independent. Because of this independence, we have \begin{align}
E[n^{-\frac{1}{2}}\sum\limits_{i=1}^{n}u_{i}\hat{e}_{i}\hat{\sigma}_{e}^{-1}\mid\hat{e}_{i}]=&\frac{1}{\|\hat{e}\|_{2}}\sum\limits_{i=1}^{n}\hat{e}_{i}E(u_{i})=0 \notag, \\ Var[n^{-\frac{1}{2}}\sum\limits_{i=1}^{n}u_{i}\hat{e}_{i}\hat{\sigma}_{e}^{-1}\mid\hat{e}_{i}] =&n^{-1}\hat{\sigma}_{e}^{-2}\sum\limits_{i=1}^{n}\hat{e}_{i}^{2}Var(u_{i}) =E(u_{1}^{2}) \notag. \end{align}
Therefore, according to the Gaussianity of $U$, the distribution of $n^{-\frac{1}{2}}\sum\limits_{i=1}^{n}u_{i}\hat{e}_{i}\hat{\sigma}_{e}^{-1}$ conditional on $\{\hat{e}_{i}\},i=1,2,...,n$ is $N(0,Q)$ and $Q=E(u_{1}^{2})$. That is, \begin{equation} n^{-\frac{1}{2}}\hat{\sigma}_{e}^{-1}\hat{e}^{T}U\mid \hat{e}\sim N(0, Q ), \notag \end{equation} where $Q$ is unknown, which we want to replace with a natural estimator $\hat{Q}=\frac{1}{n}\sum\limits_{i=1}^{n}\hat{u}_{i}^{2}$.
\section{Theoretical results} \label{section 3} \subsection{Size property}
We now turn our attention to the property of the test, which is imposed under extremely weak conditions. \begin{assumption} \label{assumption 2.1} Consider the model \eqref{1.1}. Suppose that the following hold: \\
(i) there exist constants $c$, $d$ $\in (0,+\infty)$ such that the eigenvalues of covariance matrix $\Sigma$ lie in $(c,d)$;\\ (ii) $\pi$ is sparse, which means $s_{\pi}=o(\sqrt{n/log^{3}p})$, where
$s_{\pi}={\|\pi\|_{0}}$. \\
\end{assumption} Assumption \ref{assumption 2.1} is reasonably weak.
Assumption \ref{assumption 2.1}(i) is a common condition imposed in high-dimensional literature.
Assumption \ref{assumption 2.1}(ii) imposes a sparsity condition on the regression coefficient vector $\pi$, rather than on $\beta$ or $\Sigma$ of the model \eqref{1.1}, which shows that the following conclusions are robust to dense models. Then we provide the following result for $T_{n}$. \begin{theorem} \label{theorem 2.1}
Let Assumption \ref{assumption 2.1} be hold, when $n, p\rightarrow \infty$ with $logp=o(\sqrt{n})$, then under null hypothesis,
\begin{align} \label{test}
P(\vert T_{n}\vert>\hat{Q}^{\frac{1}{2}}\Phi^{-1}(1-\alpha/2))\rightarrow\alpha, \forall\alpha\in(0,1)
\tag{3.1},
\end{align}
where $\Phi^{-1}(1-\alpha/2)$ is the $1-\alpha/2$ quantile of standard normal distribution. \end{theorem} Theorem \ref{theorem 2.1} shows that $T_{n}$, under the null hypothesis, converges to $N(0,\hat{Q})$. Hence, a test with nominal size $\alpha\in(0,1)$ rejects null hypothesis if and only if $\vert T_{n}\vert>\hat{Q}^{\frac{1}{2}}\Phi^{-1}(1-\alpha/2)$.
In particular, the test is robust to dense $\theta_{*}$, in the sense that even under dense $\theta_{*}$, our procedure does not generate false positive results.
Instead of an inference on the basis of an estimator, it is a direct statistical conclusion on the basis of a null hypothesis.
At the same time, we can construct confidence sets for $\beta_{*}$ even when the nuisance parameter $\theta_{*}$ is non-sparse.
\begin{corollary} \label{corollary} Let Assumption \ref{assumption 2.1} be hold and $1-\alpha$ be the nominal coverage level. We define
\begin{align} \label{confidence interval}
\mathcal{C}_{1-\alpha/2}:=\{\beta: \vert T_{n}\vert \leq \hat{Q}^{\frac{1}{2}}\Phi^{-1}(1-\alpha/2)\} \tag{3.2},
\end{align}
which has the exact coverage asymptotically:
\begin{align} \label{coverage} \lim\limits_{n,p\rightarrow\infty}P(\beta_{*}\in\mathcal{C}_{1-\alpha})=1-\alpha \tag{3.3}.
\end{align} \end{corollary}
\subsection{Power property} To evaluate the power property of the test, we consider the following test problem:
\begin{align} \label{H1} H_{0}:\beta_{*}=\beta_{0} \ \ versus \ \ H_{1}:\beta_{*}=\beta_{0}+h \tag{3.4},
\end{align}
where $h$ is a given constant.
It is clear that the difficulty in distinguishing $H_{0}$ from $H_{1}$ depends on $h$.
\begin{assumption} \label{assumption 2.2}
Let Assumption \ref{assumption 2.1} be hold. In addition, suppose
\\
(i) $\|\theta_{*}\|_{0}=o(\sqrt{n}/logp)$; \\
(ii) there exist constants $\delta \ and\ \kappa_{1} \in (0,+\infty)$ such that $E\vert \varepsilon \vert^{2+\delta}<\kappa_{1}$.
\end{assumption} Assumption \ref{assumption 2.2} is relatively mild. The sparsity condition of $\theta_{*}$ is used to guarantee the asymptotic power of high-dimensional tests in Assumption \ref{assumption 2.2}(i), which implies the sparsity of the model, and it is consistent with the traditional test \citep{2013Two,2014On}.
Assumption \ref{assumption 2.1}(ii) is a regular moment condition.
Then we provide the following result for $T_{n}$.
\begin{theorem} \label{theorem 2.2}
Let $H_{1}$ in \eqref{H1} and Assumption \ref{assumption 2.2} be hold. When $n, p\rightarrow \infty$, with $logp=o(\sqrt{n})$, then there exist constants $K_{1},K_{2}>0$ depending only on the constants in Assumption \ref{assumption 2.2} such that, whenever
\begin{align} \vert \sigma_{u}^{2}(\beta_{*}-\beta_{0})\vert\geq \sqrt{n^{-1}logp}(K_{1}\vert\beta_{*}-\beta_{0}\vert+K_{2}), \notag
\end{align}
where $\sigma_{u}^{2}=E(u^{2})$, the test is asymptotically powerful, $i.e.$
\begin{align}
P(\vert T_{n}\vert>\hat{Q}^{\frac{1}{2}}\Phi^{-1}(1-\alpha/2))\rightarrow 1, \notag \ {\forall} \alpha \in (0,1).
\end{align}
\end{theorem}
Theorem \ref{theorem 2.2} establishes the power property of the proposed test under the sparse model.
\section{Numerical Examples} \label{Numerical Examples} In this section, we evaluate the proposed method in the finite sample setting by observing its behavior in both simulated and real data. \subsection{Simulation Examples}\label{section 4.1}
We consider model \eqref{1.1}. In all simulations, we set $n=200$, $p=500$ and the nominal size is 5\%. The rejection probabilities are based on 100 repetitions. For application purposes, we recommend choosing the tuning parameters as $\eta=0.5\sqrt{\frac{log p}{n}}$ and $\rho_{0}=0.01$, which are commonly used options, and we will demonstrate in our simulations that it provides good results.
For the test problem \eqref{1.2}, without loss of generality, we consider the test of the first component of the parameter, i.e. \begin{equation} \label{simulatetest}
H_{0}: \beta_{1}=\beta_{1}^{0} \ versus \ H_{1}: \beta_{1}=\beta_{1}^{0}+h \tag{4.1}, \end{equation}
where $\beta_{1}^{0}$ is a given constant.
We show the results for three different Gaussian designs as follows. \\ (1) (Toeplitz) Here we consider the standard Toeplitz design where the rows of $X$ are drawn as an i.i.d random draws from a multivariate Gaussian distribution $N(0,\Sigma_{X})$, with covariance matrix $(\Sigma_{X})_{i,j}=0.4^{\vert i-j \vert}$. \\ (2) (Noncorrelation) Here we consider uncorrelated design where the rows of $X$ are i.i.d draws from $N(0,\Sigma_{X})$, where $(\Sigma_{X})_{i,j}$ is 1 for $i=j$ and is 0 for $i\neq j$. \\ (3) (Equal correlation) Here we consider a non-sparse design matrix with equal correlation among the features. Namely, the rows of $X$ are i.i.d draws from $N(0,\Sigma_{X})$, where $(\Sigma_{X})_{i,j}$ is 1 for $i=j$ and is 0.01 for $i\neq j$.
Let $s=\|\beta\|_{0}$ denotes model sparsity.
To show the size property of our method for dense model, we vary $s$ from $s=10$ to extremely large $s=p$. For sparsity $s$, we set the model parameters as $\beta_{j}=\frac{3}{\sqrt{p}}$, $1\leq j \leq s$ and $\beta_{j}=0$, $j>s$.
We compare our method with the generalized low-dimensional projection (LDP) method for bias correction \citep{2021RongMa}.
The size results are collected in Table \ref{Table 1},
where we can clearly see that the LDP method does not have the size property in the dense model, that is, the Type I error probabilities are much higher than the nominal level $\alpha$. This indicates that the LDP method fails to dense models. Conversely, when the sparsity of the model is equal to $s=p$, the Type I error probability of our method remains stable. That is true even if we change the correlation among the features. \begin{table}[h] \centering \begin{center} \begin{minipage}{\textwidth} \caption{Size properties of LDP and our method}\label{Table 1}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}lcc|cc|cc@{\extracolsep{\fill}}} \toprule & \multicolumn{2}{@{}c@{}}{Toeplitz} & \multicolumn{2}{@{}c@{}}{Noncorrelation} & \multicolumn{2}{@{}c@{}}{Equal correlation} \\ \cmidrule{2-3}\cmidrule{4-5}\cmidrule{6-7} Method & LDP & Ours & LDP & Ours & LDP & Ours \\ \midrule s=10 & 0.70 & 0.09 & 0.66 & 0.05 & 0.61 & 0.05 \\ s=20 & 0.69 & 0.02 & 0.69 & 0.03 & 0.65 & 0.03 \\ s=50 & 0.72 & 0.02 & 0.70 & 0.02 & 0.79 & 0.05 \\ s=100 & 0.81 & 0.03 & 0.72 & 0.03 & 0.77 & 0.05 \\ s=n & 0.82 & 0.05 & 0.78 & 0.07 & 0.89 & 0.05 \\ s=p & 0.90 & 0.04 & 0.89 & 0.04 & 0.86 & 0.07 \\ \bottomrule \end{tabular*} \end{minipage} \end{center} \end{table}
For the first parameter component $\beta_{1}=\frac{3}{\sqrt{p}}$, we construct its $1-\alpha$ confidence intervals for different sparsity levels, and obtain the coverage probabilities (CP) based on 100 repetitions. According to Theorem \ref{theorem 2.1}, the asymptotic distribution of $T_{n}$ is $N(0,\hat{Q})$. Also by the analysis in Section \ref{ section 2.3}, we have \begin{align} T_{n}\rightarrow n^{-\frac{1}{2}}\hat{\sigma}_{e}^{-1}U^{T}\hat{e}\rightarrow N(0,\hat{Q}), \notag \end{align}
By inverting the solution $|T_{n}| \leq \hat{Q}^{\frac{1}{2}}\Phi^{-1}(1-\alpha/2)$, the $1-\alpha$ confidence interval of the parameter $\beta_{1}$ can be obtained as
\begin{align} [\beta_{1}^{0}-\frac{n^{\frac{1}{2}}\hat{Q}^{\frac{1}{2}}\Phi^{-1}(1-\alpha/2)\hat{\sigma}_{e}+\hat{U}^{T}(W(\theta_{*}-\tilde{\theta}_{*})+\varepsilon)}{\hat{U}^{T}Z}, \notag\\ \beta_{1}^{0}+\frac{n^{\frac{1}{2}}\hat{Q}^{\frac{1}{2}}\Phi^{-1}(1-\alpha/2)\hat{\sigma}_{e}-\hat{U}^{T}(W(\theta_{*}-\tilde{\theta}_{*})+\varepsilon)}{\hat{U}^{T}Z}]. \notag \end{align}
The results for confidence intervals (CI), lengths and CP are collected in Table \ref{Table 222}. \begin{table}[h] \begin{center} \begin{minipage}{\textwidth} \caption{Confidence intervals, lengths and coverage probabilities}\label{Table 222}
\begin{tabular*}{\textwidth}{@{\extracolsep{\fill}}lccc|ccc|ccc@{\extracolsep{\fill}}} \toprule
& \multicolumn{3}{c}{Toeplitz} & \multicolumn{3}{c}{Noncorrelation} & \multicolumn{3}{c}{Equal correlation} \\ \cmidrule{2-4}\cmidrule{5-7}\cmidrule{8-10}
Sparsity & CI & Length & \multicolumn{1}{c|}{CP} & CI & Length & \multicolumn{1}{c|}{CP} & CI & Length & CP \\ \hline s=10 & (-0.2,0.5) & 0.7 & 94\% & (-0.3,0.3) & 0.6 & 95\% & (-0.3,0.4) & 0.7 & 95\% \\ s=20 & (-0.2,0.4) & 0.6 & 94\% & (-0.2,0.4) & 0.6 & 96\% & (-0.2,0.5) & 0.7 & 95\% \\ s=50 & (-0.2,0.5) & 0.7 & 93\% & (-0.3,0.3) & 0.6 & 98\% & (0.3,0.3) & 0.6 & 95\% \\ s=100 & (-0.4,0.2) & 0.6 & 97\% & (0,0.5) & 0.5 & 94\% & (-0.4,0.2) & 0.6 & 95\% \\ s=n & (-0.1,0.6) & 0.7 & 99\% & (-0.2,0.3) & 0.5 & 95\% & (-0.3,0.3) & 0.6 & 91\% \\ s=p & (-0.4,0.2) & 0.6 & 94\% & (-0.2,0.4) & 0.6 & 95\% & (0,0.6) & 0.6 & 95\% \\ \bottomrule \end{tabular*} \end{minipage} \end{center} \end{table}
In addition, Theorem \ref{theorem 2.2} gives the power property of the test under sparse models ($\|\theta_{*}\|_{0}=o(\sqrt{n}/logp)$). For simplicity, we observe the power property only for $s=3$.
The data is generated by the same model as in Table \ref{Table 1}, except that the true value of $\beta_{1}=\frac{3}{\sqrt{p}}+h$. The results are collected in Figure \ref{111}, which presents full power curves with various values of $h$. Therefore, the far left presents Type I error ($h=0$) whereas other points on the curves correspond to Type II error ($h\neq 0$).
We clearly observe that our method outperforms LDP by providing firm Type I error and reaching full power quickly.
Therefore, our proposed method provides a robust and more broadly applicable alternative to the existing inference process, achieving better error control. \begin{figure}
\caption{Power curves of competing methods under different settings of design matrix}
\label{111}
\end{figure}
\subsection{Real Data}\label{section 4.2} We illustrate our proposed method by analyzing "Lee Silverman voice treatment" (LSVT) voice rehabilitation dataset \citep{Athanasios2014Objective}. Vocal performance degradation is a common symptom for the vast majority of Parkinson's disease (PD) subjects. The current study aims to investigate the potential of automatically assessing sustained vowel articulation as “acceptable” (a clinician would allow persisting in speech treatment) or “unacceptable” (a clinician would not allow persisting in speech treatment). We first standardized the data. The complete data includes 309 dysphonia measures, where each produces a single number per phonation, resulting in a design matrix of size $126\times 309$. There are no missing entries in the design matrix. This is a high-dimensional logistic regression problem with $n=126$ and $p=309$. We try to determine "which of the originally computed dysphonia measures matter in this problem."
Results are reported in Table \ref{Table3}. Therein we report the significant variables identified using our approach and LDP that affect the assessments of speech experts, respectively.
In addition to the above 11 dysphonia measures, the LDP method selects 98 measures as significant variables. \begin{center} \begin{table}[]\centering \caption{Significant variables selected by our method and the LDP method} \label{Table3} \begin{tabular}{ccc} \hline
& Dysphonia Measure & Number \\ \hline Ours & $x_{3}$, $x_{18}$, $x_{37}$, $x_{97}$, $x_{100}$, $x_{111}$, $x_{115}$, $x_{229}$, $x_{230}$, $x_{231}$, $x_{265}$ & 11 \\ LDP & the above + $x_{4}$, $x_{4}$, $x_{6}$, $x_{7}$, $x_{8}$, $x_{9}$... & 109 \\ \hline \end{tabular} \end{table} \end{center}
We divide the 126 samples into two parts, in which the first 100 samples are used as the training set and the last 26 samples are used as the testing set. The significant variables selected by the two methods are used to fit the logistic regression model on the training set. The logistic regression model obtained by our method is
\begin{align} \hat{f}_{Ours}=\frac{e^{-1.8+52.8x_{3}+...+0.45x_{231}-25.73x_{265}}}{1+e^{-1.8+52.8x_{3}+...+0.45x_{231}-25.73x_{265}}}. \notag \end{align} And LDP's logistic regression model is \begin{align} \hat{f}_{LDP}=\frac{e^{-171.4-918.1x_{3}+...-14563x_{264}+30204x_{265}}}{1+e^{-171.4-918.1x_{3}+...-14563x_{264}+30204x_{265}}}. \notag \end{align}
The predicted values $y_{i}\mid X_{i}\sim Bernoulli(\hat{f}), i=1,2,...,26$, which are shown in Table \ref{Table4}.
\begin{center} \begin{table}[]\label{Table4} \centering \caption{The predicted values of our method and the LDP method} \label{Table4} \begin{tabular}{cccccccccccccc} \hline Measure & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ \hline Original Value & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ Ours & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ LDP & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 \\ \hline Measure & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 & 26 \\ \hline Original Value & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ Ours & 0 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ LDP & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\ \hline \end{tabular} \end{table} \end{center}
According to the prediction results in Table \ref{Table4}, the prediction accuracies of our method and LDP method are 0.73 and 0.62, respectively. This shows that our method is more accurate. Such finding would indicate that this dataset likely does not follow a sparse model and that previous method was reporting false positives. In conclusion, our method identifies the 11 most representative significant variables, greatly simplifies the fitting model and presents more accurate results than existing methods. This finding provides a reference for improving the effectiveness of automatic rehabilitation speech assessment tools.
\section{Conclusion} This paper considers the inference of single parameter in high-dimensional non-sparse logistic models. We first find the linearization of the regression model, and then construct the test statistics based on the moment method, which incorporates the null hypothesis. The proposed procedure is proved to have tight Type I error control even in the dense model. Our test also has desirable power property. Our test reaches full power quickly when the model is indeed sparse. It is worth mentioning that the method used in this paper can be extended to sub-Gaussian distribution design and other high dimensional generalized linear models. For these reasons, our method greatly complements existing literature. \backmatter \section*{Acknowledgments} This work was supported by National Social Science Fund project of China [21BTJ045].
\section*{Supplementary information} \textbf{Supplement of "Single Parameter Inference of Non-sparse Logistic Regression Models".} The detailed proofs about the asymptotic distribution of test statistics are given. In addition, we also give detailed proofs of the power property of the test. Technical lemmas are also proved in the supplement.
\end{document}
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\begin{document}
\title{Upper Bounds on the average eccentricity of Graphs of Girth $6$ and $(C_4$, $C_5)$-free Graphs}
\author{Alex Alochukwu, Peter Dankelmann (University of Johannesburg)}
\maketitle
\begin{abstract} Let $G$ be a finite, connected graph. The eccentricity of a vertex $v$ of $G$ is the distance from $v$ to a vertex farthest from $v$. The average eccentricity of $G$ is the arithmetic mean of the eccentricities of the vertices of $G$. We show that the average eccentricity of a connected graph $G$ of girth at least six is at most $\frac{9}{2} \lceil \frac{n}{2\delta^2 - 2\delta+2} \rceil + 7$, where $n$ is the order of $G$ and $\delta$ its minimum degree. We construct graphs that show that whenever $\delta-1$ is a prime power, then this bound is sharp apart from an additive constant. For graphs containing a vertex of large degree we give an improved bound. We further show that if the girth condition on $G$ is relaxed to $G$ having neither a $4$-cycle nor a $5$-cycle as a subgraph, then similar and only slightly weaker bounds hold. \end{abstract} Keywords: average eccentricity; eccentricity; eccentric mean; total eccentricity index;
minimum degree; girth \\[5mm] MSC-class: 05C12
\section{Introduction}
Let $G$ be a connected graph. The {\em eccentricity} $e(v)$ of a vertex $v$ is the distance from $v$ to a vertex farthest from $v$, i.e., $e_G(v) = \max_{w \in V(G)}d_G(v,w)$, where $V(G)$ denotes the vertex set of $G$ and $d_G(v,w)$ is the usual distance between $v$ and $w$. The {\em average eccentricity} ${\rm avec}(G)$ of $G$ is defined as the arithmetic mean of the eccentricities of its vertices, i.e., ${\rm avec}(G) = \frac{1}{n} \sum_{v \in V(G)} e_G(v)$, where $n$ is the order of $G$. The average eccentricity was introduced under the name {\em eccentric mean} by Buckley and Harary \cite{BucHar1990}, but it attracted major attention only after its first systematic study in \cite{DanGodSwa2004}. One of the basic results in this paper determined the maximum average distance of a connected graph of given order:
\begin{theorem} {\rm \cite{DanGodSwa2004}} \label{theo:path-maximises-avec} If $G$ is a connected graph of order $n$, then \[
{\rm avec}(G) \leq \frac{1}{n} \left \lfloor \frac{3n^2}{4} -\frac{n}{2} \right \rfloor, \] with equality if and only if $G$ is a path. \end{theorem}
Several bounds on the average eccentricity have been found since. For example for graphs of given order and size \cite{AliDanMorMukSwaVet2018, TanZho2012}, and for maximal planar graphs \cite{AliDanMorMukSwaVet2018}. Several relations between the average eccentricity and other graph parameters, for example independence number \cite{DanMuk2014, DanOsa2019, Ili2012}, domination number \cite{DanMuk2014, DanOsa2019, DuIli2013, DuIli2016, Du2017}, clique number \cite{DasMadCanCev2017, Ili2012}, chromatic number \cite{TanWes2019}, proximity \cite{MaWuZha2012} and Wiener index \cite{DarAliKlaDas2018} have been explored. Bounds on the average eccentricity of the strong product of graphs were given in \cite{CasDan2019}.
The natural question if the bound in Theorem \ref{theo:path-maximises-avec} can be improved for graphs whose minimum degree is greater than $1$ was answered in the affirmative in \cite{DanGodSwa2004}, where it was shown that if $G$ is a graph of order $n$ and minimum degree $\delta$, then \begin{equation} \label{eq:bound-avec-given-n-delta} {\rm avec}(G) \leq \frac{9n}{4(\delta+1)} + \frac{15}{4}, \end{equation} and this inequality is best possible apart from a small additive constant. Further results relating the average eccentricity of a graph to its vertex degrees are known. Bounds on the average eccentricity of trees of given order and maximum degree were given in \cite{Ili2012}. Trees with given degree sequence that minimise or maximise the average eccentricity were determined in \cite{SmiSzeWan2016}. For relations between average eccentricity and Randi\'{c} index see \cite{LiaLiu2012}. An upper bound on the average eccentricity in terms of order, size and first Zagreb index was given in \cite{DasMadCanCev2017}.
It was observed in \cite{DanMukOsaRod2019} that the upper bound \eqref{eq:bound-avec-given-n-delta} can be improved for triangle-free graphs and for graphs not containing four-cycles. The aim of this paper is to further pursue the idea of improving \eqref{eq:bound-avec-given-n-delta} for graphs not containing certain subgraphs. In this paper we give upper bounds on the average eccentricity of graphs of girth at least $6$, and of graphs containing neither $4$-cycles nor $5$-cycles, in terms of order, minimum degree and maximum degree.
The notation we use is as follows. We denote the vertex set and edge set of a graph $G$ by $V(G)$ and $E(G)$, respectively, and $n(G)$ stands for the order of $G$, i.e., for the number of vertices of $G$.
By ${\rm deg}_G(v)$ we mean the degree of $v$, i.e., the number of vertices adjacent to $v$. The largest of the eccentricities of the vertices of $G$ is called the {\em diameter} of $G$ and denoted by ${\rm diam}(G)$..
For $k\in \mathbb{Z}$, we denote the set of vertices at distance exactly $k$ and at most $k$ from a vertex $v$ by $N_k(v)$ and $N_{\leq k}(v)$, respectively. If $uv$ is an edge of $G$, then $N_{\leq k}(uv)$ is the set $N_{\leq k}(u) \cup N_{\leq k}(v)$. The $k$-th power of $G$, denoted by $G^k$, is the graph with the same vertex set as $G$ in which two vertices are adjacent if their distance is not more than $k$.
The {\em line graph} of a graph $G$ is the graph $L$ whose vertex set is $E(G)$, with two vertices of $L$ being adjacent in $L$ if, as edges of $G$, they share a vertex.
A {\em matching} of $G$ is a set of edges in which no two edges share a vertex. The vertex set $V(M)$ of a matching $M$ is the set of vertices incident with an edge in $M$. The distance $d_G(e_1, e_2)$ between two edges $e_1$ and $e_2$ is the smallest of the distances between a vertex incident with $e_1$ and a vertex incident with $e_2$. (Note that in general this is not equal to the distance in the line graph of $G$.) If $M$ is a set of edges, then the distance $d(e,M)$ between an edge $e$ and $M$ is the smallest of the distances between $e$ and the edges in $M$.
If $A\subseteq V(G)$, then we write $G[A]$ for the sugbgraph of $G$ induced by $A$.
By $C_n$ we mean the cycle on $n$ vertices. We say a graph is $C_k$-free if it does not contain $C_k$ as a (not necessarily induced) subgraph. A graph is $(C_4,C_5)$-free if it contains neither $C_4$ nor $C_5$ as a subgraph. The girth of a graph $G$ is the length of a smallest cycle of $G$.
\section{Preliminary results}
In this section we present some results which will be needed for the proof or our main theorems.
If $v$ and $w$ are two adjacent vertices of a graph of girth at least $6$, then the sets of vertices at distance at most two from $v$ or $w$, respectively, in $G-vw$ are disjoint if $G$ has girth at least $6$, hence we have the following well-known result (see for example \cite{AloDan-manu}).
\begin{lem}[\cite{AloDan-manu}] \label{la:neighbourhood-in-C4-free-bipartite} Let $G$ be a graph of girth at least $6$ and minimum degree $\delta$. If $v$ and $w$ are adjacent vertices of $G$, then
$$|N_{\leq 2}(vw)| \geq 2(\delta^2 - \delta + 1). $$ \end{lem}
It was shown in \cite{AloDan-manu} that if we relax the girth condition to $G$ having neither $4$-cycles nor $5$-cycles (so triangles are permitted),
then a slightly weaker bound on $|N_{\leq 2}(u) \cup N_{\leq 2}(w)|$ holds.
\begin{lem}[\cite{AloDan-manu}] \label{lowC4C5} Let $G$ be a $(C_{4}, C_{5})$-free graph with minimum degree $\delta \geq 3$. If $v$ and $w$ are adjacent vertices of $G$, then
\[ |N_{\leq 2}(vw)|
\geq \left\{ \begin{array}{cc}
2\delta^{2} - 5\delta + 5 & \textrm{if $\delta$ is even}, \\
2\delta^{2} - 5\delta + 7 & \textrm{if $\delta$ is odd}.
\end{array} \right. \] \end{lem}
We also require bounds on the number of vertices within distance three of a vertex of large degree.
\begin{lem}[\cite{AloDan-manu}] \label{la:N3} Let $G$ be a graph of girth $6$, minimum degree $\delta \geq 3$ and maximum degree $\Delta$. If $v$ is a vertex of degree $\Delta$, then
\[ |N_{\leq 3}(v)| \geq \Delta \delta + (\delta-1)\sqrt{\Delta(\delta-2)} + \frac{3}{2}. \] \end{lem}
\begin{lem}[\cite{AloDan-manu}] \label{la:N3-for-C4-C5-free} Let $G$ be a $(C_4, C_5)$-free graph of minimum degree $\delta \geq 3$ and maximum degree $\Delta$. If $v$ is a vertex of degree $\Delta$, then
\[ |N_{\leq 3}(v)| \geq \Delta (\delta-1) + (\delta-2)\sqrt{\Delta(\delta-3)} + \frac{3}{2}. \] \end{lem}
Let $G$ be a connected graph with a weight function $c: V (G) \rightarrow \mathbb{R}^{\geq 0}$. Then the eccentricity of $G$ with respect to $c$ is defined by \[ EX_c(G) = \sum_{v\in V(G)} c(v) e_G(v). \] If the total weight of the vertices of $G$ is strictly greater than $0$, we define the average eccentricity of $G$ with respect to $c$ by \[ {\rm avec}_c(G) = \frac{\sum_{v\in V(G)} c(v) e_G(v)}{\sum_{v\in V(G)} c(v)}. \] We usually denote the total weight of the vertices of $G$ by $N$. Hence, if $N>0$, we have ${\rm avec}_c(G) = \frac{EX_c(G)}{N}$.
\begin{lem}[\cite{DanGodSwa2004}] \label{la:weighted-avec-maximise-by-path} Let $G$ be a connected, weighted graph with a weight function $c; V(G) \rightarrow \mathbb{R}^{\geq 0}$. Let $N = \sum_{v\in V(G)} c(v)$. If $c(v) \geq 1$ for all $v\in V(G)$, then \[ {\rm avec}_c(G) \leq {\rm avec}(P_{\lceil N \rceil}). \] \end{lem}
\section{Bounds in terms of order and minimum degree}
In this section we present the first two of our main results: upper bounds on the average eccentricity of graphs of girth at least six and of $(C_4, C_5)$-free graphs in terms of order and minimum degree. The basic proof strategy follows that used in \cite{DanMukOsaRod2019}.
\begin{theorem} \label{theo:avec-mindegree-girth6} Let $G$ be a connected graph of order $n$, minimum degree $\delta \geq 3$ and girth at least $6$. Then \[ {\rm avec}(G) \leq
\frac{9}{2} \Big\lceil \frac{n}{\delta^*} \Big\rceil + 8, \] where $\delta^* = 2\delta^2-2\delta+2$. \end{theorem}
\begin{proof} We first find a matching $M$ of $G$ as follows. Choose an arbitrary edge $e_1$ of $G$ and let $M= \{e_1\}$. If there exists an edge $e_2$ of $G$ at distance exactly $5$ from $M$, then let $M=\{e_1, e_2\}$. If there exists an edge $e_3$ at distance $5$ from $M$ then let $M=\{e_1, e_2, e_3\}$. Repeat this step, i.e., successively add edges at distance $5$ from $M$ until, after $k$ steps say, each edge of $G$ is within distance $4$ of $M$. Let $M=\{e_1, e_2,\ldots, e_k\}$.
The sets $N_{\leq 2}(e_i)$ are pairwise disjoint for $i=1,2,\ldots,k$. For $i=1,2,\ldots, k$ let $T(e_i)$ be a spanning tree of $N_{\leq 2}(e_i)$ that contains $e_i$ and preserves the distances to $e_i$. Since the sets $N_{\leq 2}(e_i)$ are pairwise disjoint, the trees $T(e_i)$ are vertex disjoint, so the union $\bigcup_{i=1}^k T(e_i)$ forms a subforest $T_1$ of $G$. It follows from the construction of $M$ that for every $i \in \{2,3,\ldots,k\}$ there exists an edge $f_i$ in $G$ joining a vertex in $T(e_i)$ to a vertex in $T(e_j)$ for some $j$ with $1 \leq j <i$. Hence $T_2:=T_1 + \{f_2, f_3,\ldots, f_k\}$ is a subtree of $G$. Now every vertex of $G$ is within distance $5$ from some vertex of $V(M)$. Hence we can extend $T_2$ to a spanning tree $T$ of $G$ that preserves the distances to a nearest vertex in $V(M)$. Since the average eccentricity of any spanning tree of $G$ is not less than the average eccentricity of $G$, it suffices to show that \begin{equation} \label{eq:bound-on-avec-T} {\rm avec}(T) \leq
\frac{9}{2} \Big\lceil \frac{n}{\delta^*} \Big\rceil + 8. \end{equation} For every vertex $u \in V (T)$ let $u_M$ be a vertex in $V(M)$ closest to $u$ in $T$. The tree $T$ can be thought of as a weighted tree, where each vertex has weight exactly $1$. Informally speaking, we now move the weight of every vertex to the closest vertex in $V(M)$. More precisely, we define a weight function $c: V(T) \rightarrow \mathbb{R}^{\geq 0}$ by
\[ c(v) = | \{u \in V(T) \ | \ u_M = v\}|. \] Since $d_T(x, x_M) \leq 5$ for all $x\in V(G)$, we have \begin{eqnarray}
|{\rm avec}_c(T) - {\rm avec}(T)|
& = & \big| \frac{1}{n} \sum_{u\in V(T)} c(u) e_T(u) - \frac{1}{n}\sum_{v\in V(T)} e_T(v) \big|
\nonumber \\
& = & \big| \frac{1}{n}\sum_{v\in V(T)} e_T(v_M) - \frac{1}{n}\sum_{v\in V(T)} e_T(v) \big|
\nonumber \\
& \leq & \frac{1}{n} \sum_{v\in V(T)} \big| e_T(v_M) - e_T(v) \big|
\nonumber \\
&\leq & \frac{1}{n}\sum_{v \in V(T)} d_T(v_M,v) \nonumber \\
& \leq & 5. \label{eq:avec(T)-vs-avec_c(T)} \end{eqnarray} Note that $c(u) = 0$ if $u \notin V(M)$ and $\sum_{v\in V(G)} c(v) =n$, where $n$ is the order of $G$. We consider the line graph $L$ of $T$ and define a new weight function $\overline{c}$ on $V(L) = E(T)$ by \[ \overline{c}(uv) = \left\{ \begin{array}{cc}
c(u) + c(v) & \textrm{if $uv\in M$,} \\
0 & \textrm{if $uv \notin M$}. \end{array} \right. \] Let $uv \in M$. For each vertex $x \in N_{\leq 2}(uv)$, we have $x_M \in \{u, v\}$. Hence, by Lemma \ref{la:neighbourhood-in-C4-free-bipartite} it follows that for all $uv \in M$, \begin{equation} \label{eq:lower-bound-on-N2(e)-girth6}
\overline{c}(uv) = c(u) + c(v) \geq |N_{\leq 2}(uv)|
\geq \delta^*. \end{equation} We now bound the difference between ${\rm avec}_c(T)$ and ${\rm avec}_{\overline{c}}(L)$. If $x$ and $y$ are vertices of $T$, and $e_x, e_y$ are edges of $T$ incident with $x$ and $y$, respectively, then it is easy to prove that $d_T(x, y) \leq d_L(e_x, e_y) + 1$ and consequently $e_T(x) \leq e_L(e_x)+1$. Since the weight of $c$ is concentrated entirely in the vertices in $V(M)$, we have \begin{eqnarray*} \sum_{v \in V(T)} c(v) e_T(v)
& = & \sum_{uv \in M} c(u)e_T(u) + c(v) e_T(v) \\
& \leq & \sum_{uv \in M} \overline{c}(uv) (e_L(uv)+1) \\
& = & \big(\sum_{uv \in M} \overline{c}(uv) e_L(uv) \big) + n. \end{eqnarray*} Division by $n$ now yields \begin{equation} \label{eq:avecT-vs-avecL} {\rm avec}_c(T) \leq {\rm avec}_{\overline{c}}(L) + 1. \end{equation} Now, if the distance $d_T(e_i, e_j)$ between two matching edges $e_i, e_j \in M$ equals five, then $d_L(e_1, e_2) \leq 6$. By the construction of $M$, every edge $e_i \in M$ with $i>1$ is thus adjacent in $L^6$ to an edge $e_j \in M$ with $j<i$. It follows that $L^6[M]$ is connected. Moreover, we have for all pairs $e, f \in M$ that \[ d_L(e,f) \leq 6d_{L^6[M]}(e,f). \] Now, for every edge $e$ of $T$ there exists an edge $f \in M$ such that $d_L(e, f) \leq 5$. It follows that for every $f \in M$ we have \[ e_L(f) \leq 6e_{L^6[M]}(f) + 5, \] and thus \begin{equation} \label{eq:avec(L)-vs-avec(L6)} {\rm avec}_{\overline{c}}(L) \leq 6 {\rm avec}_{\overline{c}}(L^6[M]) +5. \end{equation}
To normalise the weights of the vertices of $L^6[M]$, we now define the new weight function $\overline{c}'$ by $\overline{c}'(e) = \frac{\overline{c}(e)}{\delta^*}$ for all $e\in M$. Clearly, \begin{equation} \label{eq:avec-equal-for-both-weights} {\rm avec}_{\overline{c}'}(G)
= \frac{\sum_{v \in V(T)} \overline{c}'(v) e_{L^[M}}{\sum_{v \in V(T)} \overline{c}'(v)}
= \frac{\sum_{v \in V(T)} \overline{c}(v) e_{L^[M}}{\sum_{v \in V(T)} \overline{c}(v)}
= {\rm avec}_{\overline{c}}(G) \end{equation} Observe that $\overline{c}'(e) \geq 1$ for all $e\in M$ by \eqref{eq:lower-bound-on-N2(e)-girth6} and that $\sum_{v \in V(T)} \overline{c}'(v) = \frac{n}{\delta^*}$.
We thus have by Lemma \ref{la:weighted-avec-maximise-by-path} \begin{equation} \label{eq:bound-on-avec-L6[m]} {\rm ave}_{\overline{c}'}(L^6[M])
\leq \frac{3}{4} \Big\lceil \frac{n}{\delta^*} \Big\rceil - \frac{1}{2}. \end{equation} From \eqref{eq:avec(T)-vs-avec_c(T)}, \eqref{eq:avecT-vs-avecL}, \eqref{eq:avec(L)-vs-avec(L6)}, \eqref{eq:avec-equal-for-both-weights} and \eqref{eq:bound-on-avec-L6[m]} we obtain \begin{eqnarray*} {\rm avec}(T) & \leq & {\rm avec}_c(T)+5 \\
& \leq & {\rm avec}_{\overline{c}}(L) + 6 \\
& \leq & 6 \; {\rm avec}_{\overline{c}}(L^6[M]) + 11 \\
& \leq & 6\Big( \frac{3}{4} \Big\lceil \frac{n}{\delta^*} \Big\rceil
- \frac{1}{2} \Big) + 11 \\
& = & \frac{9}{2} \Big\lceil \frac{n}{\delta^*} \Big\rceil + 8, \end{eqnarray*} which is \eqref{eq:bound-on-avec-T}, as desired. \end{proof}
We now show that the bound in Theorem \ref{theo:avec-mindegree-girth6} is sharp apart from an additive constant whenever $\delta-1$ is a prime power. This holds even if we restrict ourselves to a subclass of graphs of girth at least six, to $C_4$-free bipartite graphs.
\begin{theorem}\label{theo:sharpness-example-for-bip-C4-free-avec-mindegree} Let $\delta \in \mathbb{N}$ such that $\delta - 1 $ is a prime power. Then there exists an infinite family of bipartite $C_4$-free graphs $G$ of order $n$ and minimum degree $\delta$ such that \[
{\rm avec}(G) \geq \frac{9n}{2\delta^*} - 5. \] where $\delta^* = 2\delta^2 - 2\delta + 2$. \end{theorem}
\begin{proof} Given $\delta$, let $q=\delta-1$. Then $q$ is a prime power. Our construction is based on the graph $H_q$ first constructed by Reimann \cite{Rei1958}. Let $GF(q)$ be the finite field of order $q$. Consider the $3$-dimensional vector space $GF(q)^3$, i.e., the set of all triples of elements of $GF(q)^3$. For $i=1,2$ let $V_i$ be the set of all $i$-dimensional subspaces of $GF(q)^3$. Now $H_q$ is defined as the bipartite graph with partite sets $V_1$ and $V_2$, where two vertices $v_1\in V_1$ and $v_2 \in V_2$ are adjacent if and only if $v_1$ is a subspace of $v_2$. It is easy to verify that $H_{q}$ has $2(q^2+q+1)$ vertices, has diameter three, is $(q+1)$-regular, and that $H_{q}$ does not contain any $4$-cycles.
Let $\ell \in \mathbb{N}$ with $\ell$ even, and let $uv$ be an edge of $H_{q}$. Let $H^{1}$ and $H^{\ell}$ be disjoint copies of $H_{q}$, and let $H^{2}, H^{3}, \ldots , H^{\ell-1}$ be disjoint copies of $H_{q}-uv$. Let $G_{\delta,\ell}$ be the graph obtained from the union of $H^{1}, H^{2}, \ldots H^{\ell}$ by adding the edges $v^{(t)}u^{(t+1)}$ for every $t \in \{1,2,\ldots,\ell-1\}$ where $u^{(t)}$ and $v^{(t)}$ are the vertices of $H_{t}$ corresponding to the vertices $u$ and $v$, respectively, of $H_{q}$.
Clearly, $G_{\delta,\ell}$ is bipartite and $C_4$-free, so its girth is at least six. Its minimum degree is $\delta$. Since $\delta=q+1$, the order $n$ of $G_{\delta, \ell}$ is \[ n = 2\ell (q^{2} + q + 1) = 2\ell (\delta^{2} - \delta + 1) = \ell \delta^*. \] In order to bound the average eccentricity of $G_{\delta,\ell}$ from below, choose vertices $u^*$ of $H^1$ and $v^*$ of $H^{\ell}$ with $d(u^*,v^1)=d(u^{\ell},v^*)=3$. Since $H_{q}$ has girth at least $6$, the distance between $u^{(i)}$ and $v^{(i)}$ in $H^i$ is at least $5$ for $i=2,3,\ldots,\ell-1$. It is easy to verify that in fact ${\rm diam}(H^i)=5$ for $i=2,3,\ldots,\ell-1$. Hence ${\rm diam}(G^{*}_{\delta,\ell}) =d(u^*,v^*) = 6\ell - 5 = \frac{3n}{\delta^{2} - \delta + 1} - 5$. If $w \in V(H^i)$, then $ e(w) = d(w,v^*) \geq d(v^i,v^*) = 6(\ell-i)-2$ if $i \leq \frac{\ell}{2}$, and $ e(w) = d(w,u^*) \geq d(u^i,v^*) = 6(i-1)-2$ if $i > \frac{\ell}{2}$. Hence \begin{eqnarray*} EX(G_{\delta,\ell}) & = & \sum_{i=1}^{\ell/2} \sum_{w\in V(H^i)} e(w)
+ \sum_{i=\ell/2 +1}^{\ell} \sum_{w\in V(H^i)} e(w) \\
& \geq & \sum_{i=1}^{\ell/2} \delta^* \big[ 6(\ell-i)-2 \big]
+ \sum_{i=\ell/2 + 1}^{\ell} \delta^* \big[ 6(i-1)-2 \big] \\
& = & {\delta}^* (\frac{9}{2}\ell^2-5\ell). \end{eqnarray*} Since $n= \ell\delta^*$, division by $n$ yields that \[ {\rm avec}(G_{\delta,\ell}) \geq
\frac{\delta^* (\frac{9}{2}\ell^2-5\ell)}{\ell \delta^* }
= \frac{9}{2}\ell - 5
=\frac{9n}{2\delta^*} - 5, \] as desired. \end{proof}
If we relax the condition on $G$ to have girth at least six to $G$ being $(C_4, C_5)$-free, we obtain a bound very similar to Theorems \ref{theo:avec-mindegree-girth6}. We omit the proof as it is almost identical to that of Theorems \ref{theo:avec-mindegree-girth6}.
\begin{theorem} \label{theo:avec-mindegree-C4-C5-free} Let $G$ be a connected $(C_4, C_5)$-free graph of order $n$ and minimum degree $\delta \geq 3$. Then \[ {\rm avec}(G) \leq
\frac{9}{2} \Big\lceil \frac{n}{\delta^\circ } \Big\rceil + 8, \] where $\delta^\circ = 2\delta^2-5\delta+5$ if $\delta$ is even, and $\delta^\circ = 2\delta^2-5\delta+7$ if $\delta$ is odd. \end{theorem}
We do not know if the bound in Theorem \ref{theo:avec-mindegree-C4-C5-free} is sharp. But since $\lim_{\delta \rightarrow \infty} \frac{\delta^*}{\delta^{\circ}} = 1$, it is clear from Theorem \ref{theo:sharpness-example-for-bip-C4-free-avec-mindegree} that for large $\delta$ the coefficient of $n$ in the bound is close to being optimal.
\section{Bounds in terms of order, minimum degree and maximum degree}
We now show that the bound in Theorem \ref{theo:avec-mindegree-girth6} can be improved if $G$ contains a vertex of large degree. The proof of this bound follows broadly that of Theorem \ref{theo:avec-mindegree-girth6}, and also borrows ideas from \cite{DanOsa-manu}, but several modifications and additional arguments are required.
\begin{theorem} \label{theo:avec-mindegree-maxdegree-girth6} Let $G$ be a connected graph of order $n$, minimum degree $\delta \geq 3$, maximum degree at least $\Delta$ and girth at least $6$. Then \[ {\rm avec}(G) \leq \frac{n-\Delta^*}{2\delta^*} \, \frac{9n+3\Delta^*}{n} + 21, \] where $\delta^*=2(\delta^2-\delta+1)$ and $\Delta^*= \Delta \delta + (\delta-1)\sqrt{\Delta(\delta-2)} +\frac{3}{2}$. \end{theorem}
\begin{proof} Let $v_1$ be a vertex of degree $\Delta$ and let $e_1$ be an edge incident with $v_1$. We first find a matching $M$ of $G$ as follows. Let $M= \{e_1\}$. If there exists an edge $e_2$ with $d_G(e_1, e_2)=6$, add $e_2$ to $M$.
Assume that $M=\{e_1, e_2,\ldots,e_{i-1}\}$. If there exists an edge $e_i$ satisfying \\ (i) $d_G(e_i,e_1)\geq 6$, \\
(ii) $\min \{d_G(e_i,e_j) \ | \ j = 2,3,\ldots,i-1\} \geq 5$, and \\ (iii) we have equality in (i) or (ii) or both, \\ then add $e_i$ to $M$. We repeat this process until, after $k$ steps say, every edge not in $M_0\cup \{e_1\}$ is within distance $5$ of $e_1$, or within distance $4$ of an edge in $M_0$. Let $M=\{e_1, e_2,\ldots, e_k\}$.
The sets $N_{\leq 3}(e_1)$ and $N_{\leq 2}(e_i)$ for $i=2,3,\ldots,k$ are pairwise disjoint. Let $T(e_1)$ be a spanning tree of $N_{\leq 3}(e_1)$ that contains $e_1$ and preserves the distances to $e_1$. For $i=2,3,\ldots, k$ let $T(e_i)$ be a spanning tree of $N_{\leq 2}(e_i)$ that contains $e_i$ and preserves the distances to $e_i$. Then the trees $T(e_i)$, $i=1,2,\ldots,k$, are vertex disjoint, so the union $T(e_1) \cup \bigcup_{i=2}^k T(e_i)$ forms a subforest $T_1$ of $G$. It follows from the construction of $M$ that for every $i \in \{2,3,\ldots,k\}$ there exists an edge $f_i$ in $G$ joining a vertex in $T(e_i)$ to a vertex in $T(e_j)$ for some $j$ with $1 \leq j <i$. Hence $T_2:=T_1 + \{f_2, f_3,\ldots, f_k\}$ is a subtree of $G$. We extend $T_2$ to a spanning tree $T$ of $G$ that preserves the distances to a nearest vertex in $V(M)$. In $T$ every vertex is within distance $6$ from some vertex in $V(M)$. Since the average eccentricity of any spanning tree of $G$ is not less than the average eccentricity of $G$, it suffices to show that \begin{equation} \label{eq:bound-on-avec-T-max-degree} {\rm avec}(T) \leq \frac{n-\Delta^*}{2\delta^*} \, \frac{9n+3\Delta^*}{n} + 21. \end{equation} For every vertex $u \in V (T)$ let $u_M$ be a vertex in $V(M)$ closest to $u$ in $T$. We may assume that $u_M$ is a vertex incident with $e_i$ whenever $u\in V(T(e_i))$. As in the proof of Theorem \ref{theo:avec-mindegree-girth6} we define a weight function $c: V(T) \rightarrow \mathbb{R}^{\geq 0}$ by
\[ c(v) = | \{u \in V(T) \ | \ u_M = v\}|. \] Now $d_T(x, x_M) \leq 6$ for all $x\in V(G)$. The same arguments as in the proof of Theorem \ref{theo:avec-mindegree-girth6} show that \begin{equation} \label{eq:avec(T)-vs-avec_c(T)-maxdegree} {\rm avec}(T) \leq {\rm avec}_c(T) + 6. \end{equation} We consider the line graph $L$ of $T$ and define a new weight function $\overline{c}$ on $V(L) = E(T)$ by \[ \overline{c}(uv) = \left\{ \begin{array}{cc}
c(u) + c(v) & \textrm{if $uv\in M$,} \\
0 & \textrm{if $uv \notin M$}. \end{array} \right. \] As in the proof of Theorem \ref{theo:avec-mindegree-girth6}, we have \begin{equation} \label{eq:avec(T)-vs-avec(L)} {\rm avec}_c(T) \leq {\rm avec}_{\overline{c}}(L) +1. \end{equation} Let $H$ be the graph obtained from $L^6[M]$ by joining $e_1$ to every edge $e_i \in M$ for which $d_L(e_1,e_i)\leq 7$. Such edges $e_i$ exist since by the construction of $M$ we have $d_T(e_1, e_2)=6$ and thus $d_L(e_1, e_2) \leq 7$. Essentially the same argument as in the proof of Theorem \ref{theo:avec-mindegree-girth6} shows that $H$ is connected. \\ Let $e,f \in M$ and let $P$ be a shortest path from $e$ to $f$ in $H$ of length $\ell$ say. First assume that $P$ does not pass through $e_1$. Then each edge of $P$ yields a path in $L$ of length $6$, so $P$ yields a path from $e$ to $f$ of length at most $6\ell$. Now assume that $P$ passes through $e_1$. Then each edge on $P$ not incident with $e_1$ yields a path of length at most $6$ in $L$, while each edge of $P$ incident with $e_1$ yields a path of length at most $7$ in $L$. Since $P$ has at most two edges incident with $e_1$, $P$ yields a path of length at most $6\ell+2$. Hence \begin{equation} \label{eq:avec(L)-vs-avec(H)} d_L(e,f) \leq 6d_{H}(e,f) +2. \end{equation} Now, for every edge $f \in E(T)$ there exists an edge $g \in M$ such that $d_L(f,g) \leq 6$. Hence $e_L(e) \leq 6 \, {\rm avec}_{\overline{c}}H(e) + 8$ for every $e \in M$, and thus \begin{equation} \label{eq:avec(L)-vs-avec(H)} {\rm avec}_{\overline{c}}(L) \leq 6 {\rm avec}_{\overline{c}}(H) + 8. \end{equation} As in \eqref{eq:lower-bound-on-N2(e)-girth6} we have \begin{equation} \label{eq:lower-bound-weight-of-ei} \overline{c}(e_2), \overline{c}(e_3), \ldots, \overline{c}(e_k)
\geq \delta^*. \end{equation} By Lemma \ref{la:N3} we have \begin{equation} \label{eq:lower-bound-weight-of-e1}
\overline{c}(e_1) \geq |N_{\leq 3}(v_1)| \geq \Delta^*. \end{equation} Since $\sum_{e\in M} \overline{c}(e) = \sum_{v \in V(T)} c(v) = n$, we have
$|M| \leq \frac{n}{\delta^*}$. We now modify the weight function $\overline{c}$ to obtain a new weight function $\overline{c}'$. We define \[ \overline{c}'(e_i) = \left\{ \begin{array}{cc}
\frac{\overline{c}(e_i) -\Delta^* + \delta^*}{\delta^*} & \textrm{if $i=1$,} \\[1mm]
\frac{\overline{c}(e_i)}{\delta^*} & \textrm{if $i\geq 2$.}
\end{array} \right. \] Clearly, $\sum_{v\in V(H[M])} \overline{c}'(v) = \frac{n-\Delta^*+\delta^*}{\delta^*}=:N^*$. Hence \begin{eqnarray} \label{eq:express-avec-as -linear-combination-1} {\rm avec}_{\overline{c}'}(H)
& = & \frac{ EX_{\overline{c}'}(H)}{N^*} \nonumber \\
& = & \frac{ \frac{1}{\delta^*} \Big[ \sum_{u\in M} \overline{c}(u) e_{H}(u)
- (\Delta^* - \delta^*) e_{H}(e_1) \Big]}{N^*} \nonumber \\
& = &
\frac{ EX_{\overline{c}}(H) - (\Delta^* - \delta^*)e_H(e_1)}{n-\Delta^* + \delta^*}
\nonumber \\
& = & \frac{n}{n-\Delta^* + \delta^*} {\rm avec}_{\overline{c}}(H)
- \frac{\Delta^* - \delta^*}{n-\Delta^* + \delta^*} e_H(e_1). \end{eqnarray} Rearranging yields \begin{equation} \label{eq:express-avec-as -linear-combination-2} {\rm avec}_{\overline{c}}(H)
= \frac{n-\Delta^* + \delta^*}{n} {\rm avec}_{\overline{c}'}(H)
+ \frac{\Delta^* - \delta^*}{n} e_H(e_1). \end{equation} We now bound the two terms on the right hand side of \eqref{eq:express-avec-as -linear-combination-2} separately. Note that $\overline{c}'(e_i) \geq 1$ for all $e_i \in M$. Applying Lemma \ref{la:weighted-avec-maximise-by-path} we obtain \[ {\rm avec}_{\overline{c}'}(H) \leq {\rm avec}(P_{\lceil N^* \rceil})
= \frac{3}{4} \lceil N^* \rceil - \frac{1}{2}. \] Now $\lceil N^* \rceil
= \lceil \frac{n-\Delta^* + \delta^*}{\delta^*} \rceil
< \frac{n-\Delta^*}{\delta^*} + 2$. Hence \begin{equation} \label{eq:bound-on-avec-H} {\rm avec}_{\overline{c}'}(H) < \frac{3(n-\Delta^*)}{4\delta^*} + 1. \end{equation}
To bound $e_H(e_1)$ note that $H$ has order $|M|$. Now
$|M| = \sum_{e \in M} 1 \leq \sum_{e \in M} \overline{c}'(e_i)
= \frac{n-\Delta^*+\delta^*}{\delta^*}$. Hence \begin{equation} \label{eq:bound-on-eccentricity-of-e1}
e_H(e_1) \leq |M|-1 = \frac{n-\Delta^*+\delta^*}{\delta^*} - 1
= \frac{n-\Delta^*}{\delta^*}. \end{equation} From \eqref{eq:express-avec-as -linear-combination-2}, \eqref{eq:bound-on-avec-H} and \eqref{eq:bound-on-eccentricity-of-e1} we get, after some calculations, \begin{eqnarray} {\rm avec}_{\overline{c}}(H)
& < & \frac{n-\Delta^* + \delta^*}{n} \Big( \frac{3(n-\Delta^*)}{4\delta^*} + 1
\Big) + \frac{\Delta^* - \delta^*}{n} \frac{n-\Delta^*}{\delta^*} \nonumber \\
& = & \frac{n-\Delta^*}{4\delta^*} \, \frac{3n+\Delta^*}{n}
+ \frac{3n-3\Delta^* + 4\delta^*}{4n} \nonumber \\
& \leq & \frac{n-\Delta^*}{4\delta^*} \, \frac{3n+\Delta^*}{n} + 1.
\label{eq:bound-on-avec-H-2} \end{eqnarray} Applying the inequalities \eqref{eq:avec(T)-vs-avec_c(T)-maxdegree}, \eqref{eq:avec(T)-vs-avec(L)}, \eqref{eq:avec(L)-vs-avec(H)} and \eqref{eq:bound-on-avec-H-2} we obtain \begin{eqnarray*} {\rm avec}(T) & \leq & {\rm avec}_{c}(T) + 6 \\
& \leq & {\rm avec}_{\overline{c}}(L) + 7 \\
& \leq & 6\, {\rm avec}_{\overline{c}}(H) + 15 \\
& < & \frac{n-\Delta^*}{2\delta^*} \, \frac{9n+3\Delta^*}{n} + 21, \end{eqnarray*} as desired. \end{proof}
The following Theorem demonstrates that the bound in Theorem \ref{theo:avec-mindegree-maxdegree-girth6} is sharp if $\delta-1$ is a prime power, except for an additive term $O(\sqrt{\Delta})$.
\begin{theorem}[\cite{AloDan-manu}] \label{theo:cage-construction} Let $\delta, k \in \mathbb{N}$ such that $\delta-1$ is a prime power and $k\geq 7$. Then there exist a bipartite, $C_4$-free graph $G^{\delta,k}$ of minimum degree $\delta$, maximum degree $\Delta = \frac{(q^k-1)(q^{k-1}-1)}{(q^2-1)(q^2-q)} - \frac{1}{q}$, where $q= \delta-1$, whose order $n_{\delta,k}$ satisfies \[ \Delta^* \leq n_{\delta,k} \leq \Delta^* + 2 \sqrt{\Delta(\delta-2)} + \frac{1}{2}. \] \end{theorem}
We make use of the fact that the graph in Theorem \ref{theo:cage-construction} has diameter at least $3$, which is easy to check from the construction (see \cite{AloDan-manu}). In the proof of Theorem \ref{theo:sharpness-example-min-max-degree} we make use of a graph $G$, which was first described in \cite{AloDan-manu}.
\begin{theorem} \label{theo:sharpness-example-min-max-degree} Let $\delta\in \mathbb{N}$ be such that $\delta-1$ is a prime power. Then there exist infinitely many connected graphs $G$ of minimum degree $\delta$ and girth $6$ with \[ {\rm avec}(G) > \frac{n-\Delta^*}{2\delta^*} \, \frac{9n+3\Delta^*}{n}
- O(\sqrt{\Delta}), \] where $\Delta$ is the maximum degree and $n$ the order of $G$. \end{theorem}
\begin{proof} Let $q:=\delta-1$, so $q$ is a prime power. Let $k, \ell\in \mathbb{N}$ with $\ell$ even and $\ell$ sufficiently large. Consider the graph $G^{\delta,k}$ in Theorem \ref{theo:cage-construction} and let $u^1$ be a vertex of degree $\Delta$, and let $v^1$ be a vertex at distance three from $u^1$.
As in the construction of the graph $G_{\delta,\ell}$ in Theorem \ref{theo:sharpness-example-for-bip-C4-free-avec-mindegree} let $H^2, H^3,\ldots,H^{\ell}$ be isomorphic to $H_q$, but let $H^1$ be the graph $G^{\delta,k}$. Denote the resulting graph by $G$. It is easy to verify that $G$ has minimum degree $\delta$ and maximum degree $\Delta$, and that its diameter is $d(u_1,v_{\ell}) = 6\ell-3$. For the order $n$ of $G$ we have
\begin{equation} \label{eq:order-of-sharpness-example-for-max-degree} n = n(G^{\delta,k}) + (\ell-1) n(H_{q})
= n_{\delta,k} + (\ell-1) \delta^*. \end{equation} We now bound the average eccentricity of $G$ from below. For $i \in \{2,3,\ldots,\ell\}$ let $V(i)$ be the vertex set of $H^i$, and for $i=1$ let $V(1)$ be a set of $\delta^*$ vertices of $H^1$. Let $x \in V(i)$. If $i \leq \frac{1}{2}\ell$, then \[ e_G(x) \geq d_G(x, v^{\ell}) \geq d_G(v^i, v^{\ell}) = 6(\ell+1-i)-8, \] and if $i > \frac{1}{2}\ell$, then \[ e_G(x) \geq d_G(x, u^1) \geq d_G(u^i, u^1) = 6i-8. \] The $(n_{\delta,k} - \delta^*$ vertices in $V(H^1)-V(1)$ have eccentricity at least $6\ell -8$. Hence, \begin{eqnarray*} EX(G) & = &
\big(\sum_{i=1}^{\ell/2} + \sum_{i=\ell/2+1}^{\ell} \big) \sum_{x \in V(i)} e_G(x)
+ \sum_{x \in V(H^1)-V(1)} e_G(x) \\
& \geq &
\Big(\sum_{i=1}^{\ell/2} \delta^* [ 6(\ell +1-i) - 8] \Big)
+ \Big( \sum_{i=\ell/2 + 1}^{\ell} \delta^* [6i- 8] \Big)
+ (n_{\delta,k} - \delta^*) (6\ell-8) \\
& = & ( \frac{9}{2} \ell^2 - 5 \ell) \delta^*
+ (n_{\delta,k} - \delta^*)(6\ell-8). \end{eqnarray*} Now $\ell= \frac{n-n_{\delta,k}}{\delta^*} + 1$ by \eqref{eq:order-of-sharpness-example-for-max-degree}. Substituting this and dividing by $n$ yields, after simplification, \begin{eqnarray*} {\rm avec}(G) &\geq& \frac{n-n_{\delta,k}}{2\delta^*} \, \frac{9n+3n_{\delta,k}}{n}
- 2 + \frac{3\delta^*}{2n} \\
&>& \frac{n-n_{\delta,k}}{2\delta^*} \, \frac{9n+3n_{\delta,k}}{n}
- 2. \end{eqnarray*} Now let $\varepsilon = n_{\delta,k} - \Delta^*$. Replacing $n_{\delta,k}$ by $\Delta^*+\varepsilon$ in the above lower bound, we obtain \begin{eqnarray*} {\rm avec}(G)
&>& \frac{n-\Delta^*-\varepsilon}{2\delta^*} \, \frac{9n+3\Delta^*+3\varepsilon}{n}
- 2 \\
&=& \frac{n-\Delta^*}{2\delta^*} \, \frac{9n+3\Delta^*}{n}
- \frac{\varepsilon}{2\delta^* n}(6n + 6 \Delta^* + 3\varepsilon) - 2. \end{eqnarray*} Since $6n + 6 \Delta^* + 3\varepsilon \leq 12n$, and since $0 \leq \varepsilon \leq 2\sqrt{\Delta(\delta-2)} + \frac{1}{2}$ by Theorem \ref{theo:cage-construction} we have, for constant $\delta$ and large $n$ and $\Delta$, \[ {\rm avec}(G) > \frac{n-\Delta^*}{2\delta^*} \, \frac{9n+3\Delta^*}{n}
- O(\sqrt{\Delta}), \] as desired. \end{proof}
Theorem \ref{theo:avec-mindegree-maxdegree-girth6} generalises Theorem \ref{theo:avec-mindegree-girth6} in the sense that it implies (by setting $\Delta=\delta$) a bound that differs from Theorem \ref{theo:avec-mindegree-girth6} only by having a weaker additive constant.
As in the previous section, a bound slightly weaker than that in Theorem \ref{theo:avec-mindegree-maxdegree-girth6} holds for all $(C_4, C_5)$-free graphs. We omit the proof, which is very similar to the proof of Theorem \ref{theo:avec-mindegree-maxdegree-girth6}.
\begin{theorem} \label{theo:avec-mindegree-maxdegree-C4-C5-free} Let $G$ be a connected $(C_4, C_5)$-free graph of order $n$, minimum degree $\delta \geq 3$ and maximum degree $\Delta$. Then \[ {\rm avec}(G) \leq \frac{n-\Delta^\circ}{2\delta^\circ} \, \frac{9n+3\Delta^\circ}{n} + 21, \] where $\delta^\circ = 2\delta^2-5\delta+5$ if $\delta$ is even, $\delta^\circ = 2\delta^2-5\delta+7$ if $\delta$ is odd, and $\Delta^\circ= \Delta (\delta-1) + (\delta-2)\sqrt{\Delta(\delta-3)} +\frac{3}{2}$. \end{theorem}
We do not know if the bound in Theorem \ref{theo:avec-mindegree-maxdegree-C4-C5-free} is sharp.
\end{document}
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\begin{document}
\title[Duality theory and characterizations of optimal solutions for ...]{\textbf{Duality theory and characterizations of optimal solutions for a class of conic linear problems}}
\author[Nick Dimou]{Nick Dimou$^*$}
\footnote[0]{$^*$Prospective student; recently completed undergraduate studies at the National and Kapodistrian University of Athens, Department of Mathematics, Athens, Greece.\par Work was completed at the Game Theory Lab, School of Mathematical Sciences in Tel Aviv University, while a visiting student. Invitation was given by professor Eilon Solan of Tel Aviv University while still being an undergraduate student. All research facilities were provided by the same department. Besides professor Eilon Solan, for this enlightening academic opportunity that he offered and his extremely significant help and guidance, the author would also like to thank Panayotis Mertikopoulos, Alexander Shapiro and Rakesh Vohra for their comments and recommendations.} \footnote[0]{\textit{email address}: [email protected]}
\subjclass[2020]{90C05, 90C46, 49N15}
\keywords{Duality theory, conic linear programming, optimal solutions, complementarity conditions, Farkas Lemma, strict feasibility}
\begin{abstract} For a primal-dual pair of conic linear problems that are described by convex cones $S\subset X$, $T\subset Y$, bilinear symmetric objective functions $\langle\cdot,\cdot\rangle_X$, $\langle\cdot,\cdot\rangle_Y$ and a linear operator $A:X\rightarrow Y$, we show that the existence of optimal solutions $x^*\in S$, $y^*\in T$ that satisfy $Ax^*=b$ and $A^Ty^*=c$ eventually comes down to the consistency and solvability of the problems $min\langle z,z\rangle_Y,\;z\in\{Ax-b:x\in S\}$ and $ min\langle w,w\rangle_X,\; w\in\{A^Ty-c:y\in T\}$. Assuming that these two problems are consistent and solvable, strong duality theorems as well as geometric and algebraic characterizations of optimal solutions are obtained via natural generalizations of the Farkas' Lemma without a closure condition. Some applications of the main theory are discussed in the cases of continuous linear programming and linear programming in complex space. \end{abstract}
\maketitle
\section{Introduction} Optimization problems of the subsequent form have been profoundly studied and strong duality theorems under different frameworks have been acquired: \begin{center} \begin{equation*}{(P')\;\;\;\;\;\;\;\;} \begin{array}{ll}
min\;\; \langle c,x\rangle_X\\
s.t.\;\; Ax-b\in T\\ \;\;\;\;\;\;\;\;x\in S
\end{array} \end{equation*} \end{center} \begin{center} \begin{equation*}{(D')\;\;\;\;\;\;\;\;} \begin{array}{ll}
max\;\; \langle y,b\rangle_Y \\
s.t.\;\; -A^Ty+c\in S^*\\ \;\;\;\;\;\;\;\;\;\;\;y\in T^*
\end{array}
\end{equation*} \end{center} where the objective functions are usually defined over pairings $(X,X')$ and $(Y,Y')$. Such primal-dual pairs of problems are called conic linear problems when the objective functions are bilinear, $A:X\rightarrow Y$ is a linear map and $S,T$ are convex cones \cite{3}. The above pair of problems is often characterized by many authors as a generalized form of the following primal-dual pair of conic optimization problems to the infinite-dimensional case: \begin{center} \begin{equation*}{(P^*)\;\;\;\;\;\;\;\;} \begin{array}{ll}
min\;\; \langle c,x\rangle_X\\
s.t.\;\; Ax=b\\ \;\;\;\;\;\;\;\;x\in S
\end{array} \end{equation*} \end{center} \begin{center} \begin{equation*}{(D^*)\;\;\;\;\;\;\;\;} \begin{array}{ll}
max\;\; \langle y,b\rangle_Y \\
s.t.\;\; -A^Ty+c\in S^*\\ \;\;\;\;\;\;\;\;\;\;\;y\in T
\end{array}
\end{equation*} \end{center} Here the objecive functions usually represent inner products defined over a product of finite Euclidean spaces and the sets $S,$ $T$ are (polyhedral) convex cones.\par For problems of both forms that are defined over finite-dimensional vector spaces, a rather complete theory of duality has been developed (see e.g. \cite{5, 12}). The infinite-dimensional case is not yet considered complete, however numerous results regarding strong duality have been obtained under general frameworks and assumptions. These assumptions over the feasible sets are known as \textit{constraint qualifications} (CQs) in the relevant literature, and in comparison to the standard linear programming (LP) case, they are more than often necessary for a zero duality gap to exist. The most well-known, sufficient of these constraint qualifications turns out to be that of \textit{strict feasibility}, or else known as \textit{Slater} CQ (or \textit{Slater's} CQ), which assumes that a feasible solution on the (relative) interior of the corresponding cone exists. In particular the following known result for the pair of problems $\{(P^*),(D^*)\}$ in the finite-dimensional case can be readily found in any textbook of semidefinite programming (for example \cite{5, 16}):\\ \\ \textbf{Theorem.}
\textit{Suppose that there exists $y^*\in T^*$ such that $-A^Ty^*+c\in int(S^*)$. Then $val(P^*)=val(D^*)$ and there exists optimal feasible $x^*\in S$ such that $Ax^*=b$.}\\
\par
Various CQs have been proved to guarantee a zero duality gap between the primal and dual conic linear problems in the infinite dimensional case; Some early results are given by Kretschmer \cite{8} in 1961, where he treated problems of the form $\{(P'),(D')\}$ defined over paired vector spaces and closed convex cones with respect to the weak topologies through subconsistency and subvalues. A wide review of duality theory in infinite-dimensional linear programming was later given by Anderson \cite{1} and Nash and Anderson \cite{2}. A very interesting connection between conic linear problems and sensitivity analysis was also made by Shapiro \cite{15}. In his work strong duality results were obtained under the subdifferentiality of the optimal value function $v(y):=inf\{\langle c,x\rangle :x\in S,$ $Ax-y\in T\}$ in $b$. Results in a framework and problem setup similar to \cite{8} were recently given by Khanh et al. \cite{7}.\par
However, in all the above, strong duality results for the pair of conic linear problems $\{(P'),(D')\}$ are obtained in topological frameworks where a Slater CQ, when it does guarantee strong duality, does not necessarily guarantee the existence of optimal solutions $x^*\in S$, $y^*\in T$ that satisfy $Ax^*=b$ and $A^Ty^*=c$ respectively. In other words, while the complementarity conditions (slackness) \begin{equation}
\langle y^*,Ax^*-b\rangle_Y =0\;\;\mbox{and\;\;} \langle c-A^Ty^*,x^*\rangle_X =0 \end{equation} \textit{strongly} hold (that is there exist $x^*\in S$ such that $Ax^*=b$) for the pair $\{(P^*),(D^*)\}$, they do not necessarily \textit{strongly} hold for the primal-dual pair of conic linear problems $\{(P'),(D')\}$.\par
In this paper we show that in a certain class of conic linear problems, under strict feasibility the complementarity conditions (slackness) (1.1) hold, and therefore strong duality between $(P')$ and $(D')$ also holds. In fact we show something stronger; if there exists an optimal solution $x^*$ such that it belongs to int$S$, then whether $Ax^*-b$ belongs to the interior of $T$ or not, there always exists feasible $y^*$ such that $A^Ty^*=c$; and the same goes for a feasible solution of $(D')$ in int$T^*$. That is we show that (1.1) \textit{strongly} holds for both the primal and the dual under some specific conditions.\par
This particular class of conic linear problems is defined over a more general topological framework than that of conic convex programming problems of the form $(P^*),$ $(D^*)$, since the objective functions are not necessarily inner products (nor continuous), the vector spaces $X$, $Y$ are not necessarily Euclidean spaces (or subsets) and the convex cones $S,$ $T$ are not necessarily closed nor polyhedral. It should be mentioned that conic linear problems treated here belong to the wider class of problems that appear in previous work (\cite{7, 8, 15}) and therefore similar duality results are already known.\par The key to our approach resides on the fact that the basic properties that constitute this class of conic linear problems are such that the proof of zero duality gap eventually comes down to the requirement that the two subsequent problems are consistent and solvable (that is that they have optimal feasible solutions):
\begin{equation}
min\langle z,z\rangle_Y,\;z\in\{Ax-b:x\in S\}\;\;\;\mbox{and\;\;\;} min\langle w,w\rangle_X,\; w\in\{A^Ty-c:y\in T\} \end{equation} In fact, it can be easily shown that (1.2) can be replaced be three other conditions, the alternations that involve the max$\langle z,z\rangle_Y$ and min$\langle w,w\rangle_X$ (that we shall call condition (1.2)a), min$\langle z,z \rangle_Y$ and max$\langle w,w\rangle_X$ (condition (1.2)b), and max$\langle z,z\rangle_Y$ and max$\langle w,w\rangle_X$ (condition (1.2)c), of the objective functions of these two convex sets respectively.\par Since our aim is to prove the existence of points $x^*$ and $y^*$ such that $Ax^*=b$ and $A^Ty^*=c$ it is only logical that our approach should involve some kind of theorems similar to the Farkas alternative in real space. Indeed, for the proof of the two main results we use simple generalized versions of the Farkas' Lemma which hold without a closure assumption, in comparison to the trivial case of (LP) and other similar linear problems that require a closeness property for strong duality to hold. Instead, these generalized forms of the Farkas' Lemma derive from properties that constitute the class of conic linear problems treated, and from simple extended hyperplane theorems.\par
The rest of the paper is organized as follows: in Section 2 the framework and class of conic linear problems are defined, the main results are formulated and some basic notations are given. The main section of this paper, Section 3, includes the proofs of the two key theorems as well as some other useful results regarding solutions that satisfy (1.1) and the consistency of some subsets of the feasible sets. In Section 4 some applications of the main theory are discussed in the case of linear programming in complex space. Lastly, in Section 5 we obtain a simple result regarding continuous linear programming problems which can be characterized as complementary to the strong duality results given by Grinold \cite{6} and Levinson \cite{9}.\\ \\
\section{A class of conic linear problems}
In this paper we deal with conic linear problems of the following form: \begin{center} \begin{equation*}{(P)\;\;\;\;\;\;\;\;} \begin{array}{ll}
min\;\; \langle c,x\rangle_X\\
s.t.\;\; Ax-b\in T^*\\ \;\;\;\;\;\;\;\;x\in S
\end{array} \end{equation*} \end{center} \begin{center} \begin{equation*}{(D)\;\;\;\;\;\;\;\;} \begin{array}{ll}
max\;\; \langle y,b\rangle_Y\\
s.t.\;\; -A^Ty+c\in S^*\\ \;\;\;\;\;\;\;\;\;\;\;y\in T
\end{array}
\end{equation*} \end{center} \par Similarly to the linear restrictions that appear in conic programming theory $S,T$ are convex cones$^1$\footnote[0]{$^1$ A set $S\subset X$ is a convex cone iff $\forall\lambda\in[0,1]:$ $\lambda S+(1-\lambda)S\subset S$ and $\forall\mu\in[0,\infty):$ $\mu S\subset S$.}, subsets of vector spaces $X$ and $Y$ respectively and $S^*,T^*$ are their respective positive dual cones$^2$\footnote[0]{$^2$ The positive dual cone of $S$ is the convex set $S^*:=\{x^*\in X:\langle x^*,x\rangle_X\geq0\;\forall x\in S\}$.}. The objective functions $\langle\cdot,\cdot\rangle_X$, $\langle\cdot,\cdot\rangle_Y$ defined over $X\times X$ and $Y\times Y$ respectively, are bilinear and symmetric (although symmetry is not always necessary in the following). Here $A:X\rightarrow Y$ is a linear operator and $A^T:Y\rightarrow X$ is its adjoint$^3$\footnote[0]{$^3$ Some authors also denote the adjoint by $A^*$.}, $b\in Y$ and $c\in X$. In addition, the spaces $X,Y$ are not necessarily assumed to be Euclidean or Hilbert spaces, nor do the objective functions represent specifically defined inner products. Alike \cite{15}, the topologies corresponding to the vector spaces (if $X,\;Y$ are equipped with any) are abstract and the space $X$ is ``large" enough, such that the adjoint of the linear operator exists. That is, either the linear mapping $A$ is assumed continuous (usually represented by a matrix function), or space $X$ is ``large" enough such that ``for every $y\in Y$ there exists a unique $x^*\in X$ such that $\langle y,Ax\rangle =\langle x^*,x\rangle$ for every $x\in X$", so that the adjoint can be accordingly defined and $\langle y,Ax\rangle=\langle A^Ty,x\rangle$ for every $x\in S$ and every $y\in T$.\par
A noticable difference between the two structures of the restrictions of dual problems $(P'),(D')$ and $(P),(D)$ lies on the switching of the cones $T$ and $T^*$. Although this alternation might seem confusing to the reader that is familiar with the known work on duality theory of conic programming, it does not affect the gist of the following analysis nor the main purposes of the paper.\\ \par The class of conic linear problems that is treated in the above form is characterized by the following three assumptions:\\ \\ \\ \textbf{(F1)} Convex cones $S,\;T$ are solid, that is int$S\neq\O$, int$T\neq\O$.\\ \textbf{(F2)} The positive-definiteness condition is met for the objective functions in the sets $(C_{A}-b)\times(C_{A}-b)$, $(D_{A}-c)\times(D_{A}-c)$, where $C_{A}:=\{Ax:x\in X\}\subset Y$ and $D_{A}:=\{A^{T}y:y\in Y\}\subset X$, that is $\langle z,z\rangle_Y > 0$ for every $0\neq z\in C_{A}-b$, $\langle w,w \rangle_X > 0$ for every $0\neq w\in D_{A}-c$.\\ \textbf{(F3)} The objective functions $\langle\cdot,\cdot\rangle_Y$, $\langle\cdot,\cdot\rangle_X$ acquire a minimum value in the sets $(C_A-b)\times(C_A-b)$, $(D_A-c)\times(D_A-c)$ respectively.\\ \\ Assumption \textbf{(F1)} means that we only deal with essential non-trivial conic linear problems. It is also essential for strictly feasible solutions to exist. Assumption \textbf{(F2)} generalizes the case of the objective functions being inner products, and \textbf{(F3)} implies that the closure of the convex cones $S$, $T$ and the continuity of the objective functions are not required.\par
Hypothesis \textbf{(F3)} essentially translates into the significant remark we made on the previous senction, that is that the existence of optimal solutions $x^*\in S$, $y^*\in T$ such that $Ax^*=b$ and $A^Ty^*=c$ eventually comes down to the consistency and solvability of the problems $min\langle z,z\rangle_Y,\;z\in\{Ax-b:x\in S\}$ and $ min\langle w,w\rangle_X,\; w\in\{A^Ty-c:y\in T\}$. In addition, it seems to be the most obscure of the three judging by the fact that the first two are met more than often, compared to \textbf{(F3)}, in the relevant literature. However, the latter is also met under some generic hypotheses and known results in variational analysis (e.g. see \cite{14} Theorem 1.9). Moreover, in the framework of \cite{5} \textbf{(F2)} and \textbf{(F3)} are also true for the closure of the sets $C_A$, $D_A$, therefore it is seen that the case of $S,T$ being closed overlaps already known results in finite (real) Euclidean spaces.\\ \par
It must be pointed out that, due to the symmetric form of the primal-dual pair and the linear symmetry that appears in the proofs of the generalized forms of the Farkas' Lemma, this last assumption may be replaced by the acquirement of the maximum instead of the minimum, respectively. What this essentially means is that if assumptions \textbf{(F1), (F2)} hold as well as just one of the four different variations of assumption \textbf{(F3)} (conditions (1.2), (1.2)a, (1.2)b, (1.2)c), then the same main results presented here hold. In what follows we only treat with the case where the minimization of the two problems is assumed, in order to avoid repeating results.\par
Before we state the two main results of this paper, we give some basic notations that are used throughout the next sections:\\ \\ \par We denote by $\mathcal{S}(P):=\{x\in S:Ax-b\in T^*\}$ the set of all feasible solutions of the primal problem $(P)$. Respectively, we denote by $\mathcal{S}(D):=\{y\in T:-A^Ty+c\in S^*\}$ the set of all feasible solutions of the dual problem $(D)$. The optimal value of the primal problem $(P)$ is defined by $v(P):=inf\{\langle c,x\rangle_X :Ax-b\in T^*$, $x\in S\}$ and the optimal value of the dual problem $(D)$ is defined by $v(D):=sup\{\langle y,b\rangle_Y : -A^Ty+c\in S^*$, $y\in T\}$, when of course the respective feasibility sets are non-empty (we define $v(P):=\infty$, $v(D):=-\infty$ if the problems have no feasible solutions). The following weak duality relation holds by the standard minimax duality: \begin{equation}
v(P)\geq v(D) \end{equation} \par In addition we define the sets $\tilde{\mathcal{S}}(P):=\{x\in\mathcal{S}(P):\langle c,x\rangle_X<\infty\}$ and $\tilde{\mathcal{S}}(D):=\{y\in\mathcal{S}(D):\langle y,b\rangle_Y>-\infty\}$ (we tend to use these sets of feasible solutions in order to avoid ``unsavory" optimization cases). We lastly denote by $\mathcal{S}^*(P)$ and $\mathcal{S}^*(D)$ the sets of all optimal feasible solutions of problems $(P)$ and $(D)$ respectively, that is the sets of all feasible solutions for which the minimum and maximum value is attained respectively.\par Note that a problem might have an optimal value while not having an optimal solution (the interested reader may look at problem (4.10) in \cite{5}). \par We shall also from now on skip the indicators $``X",\;``Y"$ of the objective functions, and we will write $\langle \cdot,\cdot \rangle$ without provoking any confusion regarding the space that this is defined over.\\ \\
We now state the two main results that are to be obtained in the next section. Under assumptions \textbf{(F1)-(F3)} the following is true: \begin{theorem} If the problems $(P)$, $(D)$ have optimal solutions $x^*$, $y^*$ respectively such that $x^*\in$ int$S$, $y^*\in$ int$T$ and the optimal values are finite, then $v(P)=v(D)$. In fact, there exist optimal solutions $\hat{x}\in S$, $\hat{y}\in T$ of $(P)$, $(D)$ respectively such that $A\hat{x}=b$ and $A^T\hat{y}=c$.\\ \end{theorem} Now assume the following: \textbf{(F4)} The sets $X\smallsetminus$int$S$ and $Y\smallsetminus$int$T$ are cones (not necessarily convex).\\ \par Let the sets $\hat{\mathcal{S}}(D):=\{y\in$ int$T:-A^Ty+c\in S^*,-A^Ty\in S^*,\langle y,b\rangle>-\infty\}$, $\hat{\mathcal{S}}(P):=\{x\in$ int$S:Ax-b\in T^*,Ax\in T^*,\langle x,c\rangle<\infty\}$. If \textbf{(F4)} also holds for the above class of conic linear problems, then the following is true: \begin{theorem}
If $\hat{\mathcal{S}}(D)\neq\O$, $\hat{\mathcal{S}}(P)\neq\O$, $\tilde{\mathcal{S}}(P)\smallsetminus\hat{\mathcal{S}}(P)\neq\O$, $\tilde{\mathcal{S}}(D)\smallsetminus\hat{\mathcal{S}}(D)\neq\O$ and if $v(P),\;v(D)$ are finite, then $v(P)=v(D)$. In fact, there exist optimal solutions $\hat{x}\in S$, $\hat{y}\in T$ of $(P)$, $(D)$ respectively such that $A\hat{x}=b$ and $A^T\hat{y}=c$. \\ \end{theorem} Even though assumption \textbf{(F4)} ``shortens" the main class that is treated in this paper, it essentially releases us from the necessity of the existence of optimal solutions for both dual problems when specific feasible solutions can be found. The boundeness conditions for the optimal values and the upper and lower boundeness for the values of certain feasible solutions in the formulations of the two results could as well be omitted when their existence is known, while their presence insinuates that a class of functions that take infinite values over the cones $S,T$ is not excluded from this particular framework.\\ \par Now, one can straight away identify the geometric and algebraic characterizations that occur from the above results regarding the existence of optimal solutions:\\ \par The algebraic characterization has a clear explanation as it directly occurs from both Theorems that there exist points $x^*\in S$, $y^*\in T$ that are solutions to the linear ``systems" $Ax=b$, $A^Ty=c$. The geometric characterizations is justified by the converse direction. To be more specific, if there exist optimal solutions of the dual problems with finite optimal values, but the two systems $Ax=b$, $x\in S$ and $A^Ty=c$, $y\in T$ do not have a solution, then every optimal solution of the two problems belongs to the boundary of the corresponding convex cone (when of course such set is non-empty). Same goes with Theorem 2.2., for in the case where the two systems have no feasible solutions, but the optimal values exist and are finite, then the subsets $\hat{\mathcal{S}}(P):=\{x\in$ int$S:Ax-b\in T^*$, $Ax\in T^*,\langle x,c\rangle<\infty\}$, $\hat{\mathcal{S}}(D):=\{y\in$ int$T:-A^Ty+c\in S^*$, $-A^Ty\in S^*,\langle y,b\rangle>-\infty\}$ of feasible solutions are empty. In other words, by knowing that the linear ``systems" $Ax=b$, $A^Ty=c$ have no solutions in $S$ and $T$ respectively, then we instantly know in which ``geometric" place to ``look for" the optimal solutions.\\ \\ \section{Duality theory and characterization of optimal solutions} In this section the main results of the paper are proved. Theorem 2.1 derives from the following two key theorems, which can be characterized as dual due to their formulation and initial hypothesis. \begin{theorem}
If $\mathcal{S}^*(D)\;\cap$ int$T\neq\O$ and $v(D)<+\infty$ then there exists $\hat{x}\in\mathcal{S}(P)$ such that $A\hat{x}=b$. \end{theorem} \begin{theorem}
If $\mathcal{S}^*(P)\;\cap$ int$S\neq\O$ and $v(P)>-\infty$ then there exists $\hat{y}\in\mathcal{S}(D)$ such that $A^T\hat{y}=c$. \end{theorem} Here we only prove Theorem 3.1; the proof of Theorem 3.2 is completely analogous, as are the proofs of the lemmas that are used in the following analysis.\par In what follows we only deal with the class of conic linear problems of Section 2, that is the convex cones $S$, $T$ and the objective functions satisfy assumptions \textbf{(F1)-(F3)}.\\ \begin{lemma}
Let $C$ be a convex subset of $X$ such that the positive-definiteness condition is met for the objective function over $C-b$ for some $b\in X\smallsetminus C$ and let $c\in C$. Then the following are equivalent:\\
\begin{itemize}
\item[(i)] $\langle c-b,c-b\rangle\leq\langle x,x\rangle$ $\forall x\in C-b$
\item[(ii)] $\langle c-b,x-c\rangle\geq 0$ $\forall x\in C$
\end{itemize} \end{lemma} \begin{proof}
(i)$\Rightarrow$(ii): Let $x\in C$ and
$\lambda\in (0,1)$, then $c+\lambda(x-c)\in C$. We now have:
\begin{equation*}
\langle c-b+\lambda(x-c),c-b+\lambda(x-c)\rangle - \langle c-b,c-b\rangle = 2\lambda\langle c-b,x-c\rangle + \lambda^2\langle x-c,x-c\rangle
\end{equation*}
due to the linearity and symmetry of $\langle\cdot,\cdot\rangle$. By (i) $2\langle c-b,x-c\rangle + \lambda\langle x-c,x-c\rangle\geq 0$, hence for $\lambda=1/n$, $n\in\mathbb{N}$ we obtain (ii) as $\lambda\rightarrow 0$.\par
(ii)$\Rightarrow$(i): Let $x\in C$. Then
\begin{equation*}
\langle x-b,x-b\rangle - \langle c-b,c-b\rangle = \langle x-c,x-c\rangle + 2\langle c-b,x-c\rangle\Rightarrow\langle x-b,x-b\rangle - \langle c-b,c-b\rangle\geq 0
\end{equation*}
by (ii) and the positive-definiteness condition.\\ \end{proof} \begin{proposition}
Let $C$ be a convex subset of $X$ and let $b\in X\smallsetminus C$ such that the positive-definiteness condition is met for the objective function over $C-b$. If $\langle\cdot,\cdot\rangle$ acquires a minimum value on $C-b$ then there exists $\alpha\in X$ such that $\langle\alpha,b\rangle< \langle\alpha,x\rangle$ $\forall x\in C$. In fact $\alpha\in C-b$. \end{proposition} \begin{proof} Let $\gamma\in C$ such that $\langle\gamma-b,\gamma-b\rangle\leq\langle x-b,x-b\rangle$ $\forall x\in C$. Then by Lemma 3.3 and the positive-definiteness condition: \begin{equation*}
\langle\gamma-b,x\rangle\geq\langle\gamma-b,\gamma\rangle > \langle\gamma-b,b\rangle \end{equation*} since $\gamma\neq b$. The inequality $\langle\alpha,b\rangle< \langle\alpha,x\rangle$ holds for $\alpha:=\gamma-b\in C-b$.\\ \end{proof} We now obtain the following generalized form of Farkas' Lemma:\\ \begin{theorem}(1st generalized form of Farkas' Lemma)
Let $S\subset X$ be a convex cone, $S^*$ its positive dual and $b\in Y$. Then exactly one of the following problems has at least one solution:\\ \begin{equation*} (I) \left. \begin{array}{ll}
Ax=b\\
x\in S \end{array} \right., \;\;\; (II) \left. \begin{array}{ll}
(-A)^Ty\in S^*\\
\langle y,-b\rangle <0\\
y\in Y \end{array} \right.
\end{equation*} \end{theorem} \begin{proof}
Suppose that both $(I)$ and $(II)$ have solutions. Let $\hat{x}\in S$ be a solution of $(I)$ and $\hat{y}\in Y$ be a solution of $(II)$. Then:
\begin{equation*}
(-A)^T\hat{y}\in S^*\Rightarrow\langle (-A)^T\hat{y},x\rangle\geq0\;\;\; \forall x\in S\Rightarrow \langle (-A)^T\hat{y},\hat{x}\rangle\geq 0\Rightarrow\langle \hat{y},-A\hat{x}\rangle\geq 0\Rightarrow \langle\hat{y},-b\rangle\geq 0
\end{equation*}
which is a contradiction.\par
Now assume that $(I)$ does not have a solution. Then $b\not\in C_A=\{Ax:x\in S\}\subset Y$. Since $S$ is a convex cone and $A$ is linear, the set $C_A$ is also a convex cone. Therefore Proposition 3.4 can be applied for $C_{A}$ in space $Y$ as \textbf{(F2)} and \textbf{(F3)} hold; there exists $\alpha\in -C_A+b$ such that $\langle\alpha,-b\rangle<\langle\alpha,-Ax\rangle$ $\forall x\in S$. Let $x\in S$, then $\lambda x\in S$ $\;\forall\lambda\in(0,\infty)$ hence $\langle\alpha,-b\rangle/\lambda<\langle\alpha,-Ax\rangle$. Since $x\in S$ is arbitrary, for $\lambda=n\in\mathbb{N}$ we obtain
\begin{equation*}
\langle\alpha,-Ax\rangle\geq 0\;\; \forall x\in S\Rightarrow\langle -A^T\alpha,x\rangle\geq 0\;\; \forall x\in S\Rightarrow -A^T\alpha\in S^*
\end{equation*}
The inequality $\langle\alpha,-b\rangle<0$ holds by definition ($0\in S$), therefore $\alpha\in -C_A+b\subset Y$ is a solution of $(II)$.\\ \end{proof} The proof of Theorem 3.1 now follows: \begin{proof}\textit{(Theorem 3.1)} Suppose by contradiction that a feasible solution $x\in S$ of $(P)$ such that $Ax=b$ does not exist. By Theorem 3.5 the following problem has a solution: \begin{equation*}
(II^*) \left. \begin{array}{ll}
(-A)^Ty\in S^*\\
\langle y,-b\rangle <0\\
y\in Y \end{array} \right. \end{equation*} Suppose that for every solution $\hat{y}$ of $(II^*)$ there exists $(y^*,\lambda)\in(\mathcal{S}^*(D)\;\cap$\;int$T)\times(0,\infty)$ such that $y^*+\lambda\hat{y}\in$ int$T$. Then $y^*+\lambda\hat{y}\in\mathcal{S}(D)$. Indeed: \begin{equation*}
\langle-A^Ty^*+c,x\rangle\geq 0 \mbox{ and}\;\langle-A^T\hat{y},x\rangle\geq0\;\;\forall x\in S\Rightarrow \langle -A^T(y^*+\lambda\hat{y})+c,x\rangle\geq 0\;\;\forall x\in S \end{equation*} Then $\langle y^*+\lambda\hat{y},b\rangle=\langle y^*,b\rangle+\lambda\langle\hat{y},b\rangle>\langle y^*,b\rangle$, which is a contradiction since $y^*$ is optimal and $v(D)<\infty$.\par Therefore there exists a solution $\hat{y}$ of $(II^*)$ such that $\forall (y^*,\lambda)\in(\mathcal{S}^*(D)\;\cap$ int$T)\times(0,\infty)$ $y^*+\lambda\hat{y}\in Y\smallsetminus$int$T$. For $\lambda=1/n$, $n\in\mathbb{N}$ we obtain $y^*\in Y\smallsetminus$int$T$ for every $y^*\in\mathcal{S}^*(D)\;\cap$ int$T$ as $\lambda\rightarrow 0$ since $Y\smallsetminus$int$T$ is closed (with respect to the corresponding topology of $Y$).\\ \end{proof} The proof of Theorem 3.2 is completely analogous to the above, and derives from the generalized form of the Farkas' Lemma, which is similarly formulated and proved in the topological vector space $Y$ as follows:\\ \begin{theorem}(2nd generalized form of Farkas' Lemma) Let $T\subset Y$ be a convex cone, $T^*$ its positive dual and $c\in X$. Then exactly one of the following problems has at least one solution:\\ \begin{equation*} (I') \left. \begin{array}{ll}
A^Ty=c\\
y\in T \end{array} \right., \;\;\; (II') \left. \begin{array}{ll}
Ax\in T^*\\
\langle x,c\rangle <0\\
x\in X \end{array} \right.
\end{equation*} \end{theorem}
For the purposes of this paper, the proof is omitted (we leave the formulations and proofs of the corresponding Lemma 3.3 and Proposition 3.4 to the reader).\par By Theorems 3.1 and 3.2 we obtain the following corollary which is the first of the two main results of this paper: \begin{corollary}\textit{(Theorem 2.1)}
If both $(P)$ and $(D)$ have optimal solutions $x^*$, $y^*$ such that $x^*\in$ int$S$, $y^*\in$ int$T$, and $v(P),\;v(D)$ are finite then there exist feasible solutions $\hat{x}\in S$, $\hat{y}\in T$ that satisfy $A\hat{x}=b$, $A^T\hat{y}=c$ and therefore $v(P)=v(D)$ and $\hat{x},$ $\hat{y}$ are optimal.\\ \end{corollary} Similarly to Theorem 2.1, the second main result of this paper, Theorem 2.2, results from two key theorems when \textbf{(F4)} holds: \begin{theorem}
If $\hat{\mathcal{S}}(D)=\{y\in$ int$T:-A^Ty+c\in S^*,-A^Ty\in S^*,\langle y,b\rangle>-\infty\}\neq\O$, $\tilde{\mathcal{S}}(D)\smallsetminus\hat{\mathcal{S}}(D)\neq\O$ and $v(D)<\infty$ then there exists $\hat{x}\in\mathcal{S}(P)$ such that $A\hat{x}=b$. \end{theorem} \begin{theorem}
If $\hat{\mathcal{S}}(P)=\{x\in$ int$S:Ax-b\in T^*,Ax\in T^*,\langle x,c\rangle<\infty\}\neq\O$, $\tilde{\mathcal{S}}(P)\smallsetminus\hat{\mathcal{S}}(P)\neq\O$ and $v(P)>-\infty$ then there exists $\hat{y}\in\mathcal{S}(D)$ such that $A^T\hat{y}=c$.\\ \end{theorem} Again, the proof of Theorem 3.9 is omitted and only that of Theorem 3.8 is given in order to avoid superfluous repeated methods and contiguous results. We will need the following Lemma: \begin{lemma}
Let $S\subset X$, $T\subset Y$. Then at most one of the following two problems has a solution:
\begin{equation*} (I) \left. \begin{array}{ll}
\langle Ax-b,y\rangle\leq 0\;\;\forall y\in T\\
x\in S \end{array} \right., \;\;\; (II) \left. \begin{array}{ll}
A^Ty\in S^*\\
\langle y,b\rangle <0\\
y\in T \end{array} \right.
\end{equation*} \end{lemma} \begin{proof}
Suppose that both $(I)$ and $(II)$ have a solution. Let $\hat{x}\in S$ be a solution of $(I)$ and $\hat{y}\in T$ be a solution of $(II)$. Then
\begin{equation*}
\langle A\hat{x}-b,y\rangle\leq 0\;\;\forall y\in T\Rightarrow\langle A\hat{x},\hat{y}\rangle\leq\langle b,\hat{y}\rangle< 0
\end{equation*}
while $\langle A\hat{x},\hat{y}\rangle=\langle\hat{x},A^T\hat{y}\rangle\geq 0$, which is a contradiction.\\ \end{proof} We can now prove Theorem 3.8: \begin{proof}\textit{(Theorem 3.8)} Suppose by contradiction that a feasible solution $x\in S$ of $(P)$ such that $Ax=b$ does not exist. By Theorem 3.5 the following problem has a solution: \begin{equation*}
(II^*) \left. \begin{array}{ll}
(-A)^Ty\in S^*\\
\langle y,-b\rangle <0\\
y\in Y \end{array} \right. \end{equation*} The set $\hat{\mathcal{S}}(D)$ is non-empty, therefore the feasible set $Ax-b\in T^*$, $x\in S$ is non-empty. Due to the relation $T^*\subset($int$T)^*$, $Ax-b\in($int$T)^*$, $x\in S$ has a solution. Hence there exists $x\in S$ such that $\langle (-A)x-(-b),y\rangle\leq 0$ $\forall y\in$ int$T$. By Lemma 3.10 for the sets $S,$ int$T$ the following does not have a solution: \begin{equation*} (II) \left. \begin{array}{ll}
(-A)^Ty\in S^*\\
\langle y,-b\rangle <0\\
y\in \mbox{int}T \end{array} \right. \end{equation*} Therefore problem $(III)$ has a solution, where problem $(III)$ is defined by: \begin{equation*} (III) \left. \begin{array}{ll}
(-A)^Ty\in S^*\\
\langle y,-b\rangle <0\\
y\in Y\smallsetminus\mbox{int}T \end{array} \right. \end{equation*} Note that if $\hat{y}\in Y\smallsetminus$int$T$ is a solution of $(III)$ then $\lambda\hat{y}$ is also a solution of the problem $\forall\lambda\in (0,\infty)$ due to \textbf{(F4)}.\par Now suppose that for every solution $\hat{y}$ of $(III)$ there exists a feasible solution $y^*\in T$ such that $\hat{y}+y^*\in$ int$T$ and $\langle y^*,b\rangle>-\infty$. Then $\hat{y}+y^*$ is also a feasible solution. Indeed, we have \begin{equation*}
\langle-A^Ty^*+c,x\rangle\geq 0 \mbox{ and}\;\langle-A^T\hat{y},x\rangle\geq0\;\;\forall x\in S\Rightarrow \langle -A^T(y^*+\hat{y})+c,x\rangle\geq 0\;\;\forall x\in S \end{equation*} For solutions $\{n\hat{y}\in Y\smallsetminus$int$T:n\in\mathbb{N}\}$, where $\hat{y}$ is a solution of $(III)$, we obtain $v(D)\rightarrow\infty$ as $n\rightarrow\infty$, which is a contradiction.\par Therefore there exists a solution $\hat{y}\in Y\smallsetminus$int$T$ of $(III)$ such that $\forall y\in\tilde{\mathcal{S}}(D)$: $\hat{y}+y\in Y\smallsetminus$int$T$. Let $y^*\in\hat{\mathcal{S}}(D)\subset\tilde{\mathcal{S}}(D)$. Then $-A^Ty^*\in S^*$, hence $\langle -A^Tny^*+c,x\rangle\geq 0$ $\forall x\in S$ and $\forall n\in\mathbb{N}$. Alike the above we obtain that $ny^*\in\tilde{\mathcal{S}}(D)$, $\forall y^*\in\hat{\mathcal{S}}(D)$ and $\forall n\in\mathbb{N}$.\par Since $Y\smallsetminus$int$T$ is a cone and $\hat{y}+ny^*\in Y\smallsetminus$int$T$ $\forall y^*\in\hat{\mathcal{S}}(D)$, $\forall n\in\mathbb{N}$ we obtain that $\hat{y}/n+y^*\in Y\smallsetminus$int$T$ $\forall y^*\in\hat{\mathcal{S}}(D)$, $\forall n\in\mathbb{N}$. As $n\rightarrow\infty$ we get $y^*\in Y\smallsetminus$int$T$ $\forall y^*\in\hat{\mathcal{S}}(D)$, which is a contradiction. \end{proof} Again, the proof of Theorem 3.9 is similar to the above, and the dual formulation of Lemma 3.10 is used (the reader may complement the unnecessary details). Theorem 2.2 is obtained as a corollary of the two results:\\ \begin{corollary}\textit{(Theorem 2.2)} If $\hat{\mathcal{S}}(P)\neq\O$, $\hat{\mathcal{S}}(D)\neq\O$, $\tilde{\mathcal{S}}(P)\smallsetminus\hat{\mathcal{S}}(P)\neq\O$, $\tilde{\mathcal{S}}(D)\smallsetminus\hat{\mathcal{S}}(D)\neq\O$ and the optimal values $v(P),v(D)$ are finite, then there exist feasible solutions $\hat{x}\in S$, $\hat{y}\in T$ such that $A\hat{x}=b$, $A^T\hat{y}=c$ respectively and therefore $v(P)=v(D)$ and $\hat{x}$, $\hat{y}$ are optimal.\\ \\ \end{corollary}
\section{Linear programming in Complex space}
The application of the main results in the trivial case of linear programming in real space is obvious and equivalent results are already known; it suffices to look at any textbook on linear programming and duality theory.\par We will particularly work with conic linear problems that are formulated in complex space and discuss some cases where the main results can be put in an application. Firstly, let the following primal-dual pair of problems: \begin{center} \begin{equation*}{(P)\;\;\;\;\;\;\;\;} \begin{array}{ll}
min\;\; Re(c,z)\\
s.t.\;\; Az-b\in T^*\\ \;\;\;\;\;\;\;\;z\in S
\end{array} \end{equation*} \end{center} \begin{center} \begin{equation*}{(D)\;\;\;\;\;\;\;\;} \begin{array}{ll}
max\;\; Re(w,b)\\
s.t.\;\; -A^*w+c\in S^*\\ \;\;\;\;\;\;\;\;w\in T
\end{array} \end{equation*} \end{center}
Here the objective function is bilinear, symmetrical and continuous, and is defined by $Re(z,c)=Re(z\cdot c)=Re(z^*c)=Re(\displaystyle\sum_{i=1}^{m}\bar{z_i}c_i)$. The sets $S,T$ are defined by \begin{equation*}
S:=\{z\in\mathbb{C}^{m}:|argz|\leqq\alpha\},\;\;\; T:=\{w\in\mathbb{C}^{n}:|argw|\leqq\beta\} \end{equation*} where $\alpha$, $\beta$ are real m and n-vectors in $(0,\frac{\pi}{2})e$ respectively, where $e$ are the m and n-vectors with all coordinates equal to $1$. Note that the sets $S,T$ are closed convex cones and that their positive duals are defined by \begin{equation*}
S^*=\{z\in\mathbb{C}^{m}:|argz|\leqq\ \frac{\pi}{2}e-\alpha\},\;\;\; T^*=\{w\in\mathbb{C}^{n}:|argw|\leqq \frac{\pi}{2}e-\beta\} \end{equation*} Therefore the dual problems can also be writen as \begin{center} \begin{equation*}{(P)\;\;\;\;\;\;\;\;} \begin{array}{ll}
min\;\; Re(z,c)\\
s.t.\;\; \lvert argz\rvert\leqq\alpha\\ \lvert arg(Az-b)\rvert\leqq\frac{\pi}{2}e-\beta
\end{array}
\end{equation*} \end{center} \begin{center} \begin{equation*}{(D)\;\;\;\;\;\;\;\;} \begin{array}{ll}
max\;\; Re(w,b)\\
s.t.\;\; \lvert argw\rvert\leqq\beta\\ \lvert arg(-A^*w+c)\rvert\leqq\frac{\pi}{2}e-\alpha
\end{array}
\end{equation*} \end{center} In addition, the linear operator $A$ is a $n\times m$ complex matrix and $A^*$ is its conjugate transpose.\\ \par Strong duality for the above pair of problems was first proved by Levinson \cite{10}.: \begin{theorem}
If $(P),(D)$ have feasible solutions, then they have optimal solutions $\hat{z}\in S$, $\hat{w}\in T$ respectively and $v(P)=v(D)$. \end{theorem}
\par
Conditions \textbf{(F1), (F2)} hold by definition, while the case of assumption \textbf{(F3)} is slightly more complicated and, in agreement to the previous discussion in Sections 1 and 2, is the one that essentially needs to hold in order for the main theorems to be applied. Let us see the following example:\\ \par Let $\alpha\in(0,\frac{\pi}{2})e$, $\beta\in(0,\frac{\pi}{2})e$, $S=\{z\in\mathbb{C}^{m}:0\leqq argz\leqq\alpha\},\; T=\{w\in\mathbb{C}^{n}:0\geqq argw\geqq-\beta\}$ and all elements $a_{ij}$ of the matrix $A$ satisfy: \begin{equation*}
-\frac{\pi}{2}\leq arg(a_{ij})\leq0 \end{equation*}
Then, the objective function $Re(\cdot,\cdot)$ is lower level-bounded over the sets $C_A-b$, $D_A-c$ and as a result \textbf{(F3)} is true (\cite{14} Theorem 1.9) and Theorems $2.1$, $2.2$ can be applied. In particular, we can easily identify the geometric properties of the optimal solutions and get the longed for characterizations analyzed in the previous section:\\ \par By Theorem $2.1$ if $0<argz<\alpha$ and $0>argw>-\beta$ (for a least one coordinate) then there exist solutions $z^*\in S$, $w^*\in T$ such that $Az=b$ and $A^*w=c$. \par Now assume that the linear problems $\{Az=b$, $0\leqq argz\leqq\alpha\}$ and $\{A^*w=c$, $0\geqq argw\geqq-\beta\}$ have no solutions. Then, again by Theorem 2.1 we instantly obtain that the optimal solutions $\hat{z},\hat{w}$ belong to the boundary of their corresponfing cones, that is $arg\hat{z}\equiv\alpha$ and $arg\hat{w}\equiv -\beta$ (when the optimal solutions are not real numbers). Therefore, since we already know that the optimal solutions exist when both problems are feasible (Theorem 4.1) and that the two linear systems have no solutions in $S,T$, significant information regarding the geometrical location of these optimal solutions is immediately acquired; when they are not real numbers, they lie on the boundary of the argument cones in their corresponding finite-dimensional complex planes. Note that since \textbf{(F1)-(F3)} hold for cones that are not necessarily closed, this example also holds for cones $S$ and $T$ in which their elements might have coordinates whose argument strictly belongs to $(0,\alpha_j)$ (to $(-\beta_i,0)$ respectively).\par The latter analysis for this case of complex linear problems however does not work if our goal is to apply the second main result. Obviously, this is due to the fact that all elements of $A$ have a positive real part. One should therefore work in a different setup. For instance, one could work with matrix $A$ such that $|arg(a_{ij})|\leqq\frac{\pi}{2}+min\{\alpha,\beta\}_{k^*}$, $\forall i,j$, with $b>0$ and $c\in S^*$ (why?).\\
\par The geometric characterization of the optimal solutions of dual problems $(P),(D)$ obtained can have further appliances, such as in game theory. For example, if one works with complex matrix games, as they are defined and analyzed in \cite{4}, Theorem 2.1 implies whether or not the players $I$ and $II$ choose optimal strategies that lie on the boundary of their respective compact strategy sets. Of course, as seen from the proof of the minimax theorem in \cite{4}, the closed cones $S,T$ have a stricter geometrical structure, as they are defined by \begin{equation*}
S=\{z\in\mathbb{C}^m:|argz|\leqq\alpha,\;\displaystyle\sum_{i=1}^{m}Im(z_i)=0\},\;\;T=\{w\in\mathbb{C}^n:|argw|\leqq\beta,\;\displaystyle\sum_{j=1}^{n}Im(w_j)=0\} \end{equation*} \par Therefore the main results can readily be applied in the case of finite-dimensional linear programs under the three original assumptions, where duality theory already exists and combutational methods for the solution of the complex linear systems $Az=b$, $A^*w=c$ are known. These results however may be applied in the case of infinite linear programming problems in complex space as well, where the above linear systems turn out to be harder to solve.\par For instance, consider the space of all square-summable complex sequences $\ell ^2(\mathbb{C})$ with objective function $Re(z,w)=Re(\displaystyle\sum_{i=1}^{\infty}z_i\bar{w_i})$. Let $b,c\in\ell^2(\mathbb{C})$, $A\in\mathcal{M}(\ell^2(\mathbb{C}))$ and the convex cones: \begin{equation*}
S:=\{z\in\ell^2(\mathbb{C}):|argz|\leqq\alpha\},\;T:=\{w\in\ell^2(\mathbb{C}):|argw|\leqq\beta\} \end{equation*} where $\alpha\in(0,\frac{\pi}{w})e$, $\beta\in(0,\frac{\pi}{2})e$. We assume that the linear operator $A$ and its adjoint $A^*$ are well defined, that is: \begin{equation*} \displaystyle\sum_{i=1}^{\infty}\displaystyle\sum_{j=1}^{\infty}{z_i}\bar{a_{ij}}\bar{w_j}=\displaystyle\sum_{i=1}^{\infty}\displaystyle\sum_{j=1}^{\infty}\bar{z_i}a_{ij}w_j<\infty\;\; \forall z\in S,\;\; \forall w\in T \end{equation*} Conditions \textbf{(F1),\;(F2)} and \textbf{(F4)} hold be definition, while the case of \textbf{(F3)} is again more sensitive. However, when working with specific subsets of the feasible cones and certain cases of the linear operators $A,A^*$, such as in the previous example, the third assumption holds by the known theory of minimum attainment (see \cite{9}). Therefore Theorem 2.1 can be applied. Note that solutions of the problems $\{Az=b$, $z\in S\}$ and $\{A^*w=c$, $w\in T\}$ are not so easily found by similar combutational methods as in the finite case (e.g. with Gauss elimination). Hence the results obtained in Section 3 are instantly more significant and useful for the infinite case.\par Since some regular and strong duality theorems for conic linear problems in infinite-dimensional Hilbert spaces have already been obtained (see for instance \cite{1, 2}), the existence of optimal solutions can result in valuable mathematical details regarding both the geometric properties of such solutions, that is whether or not they lie on the boundary of the cones $S,T$, as well as the existence and form of solutions of the systems $\{Az=b$, $z\in S\}$ and $\{A^*w=c$, $w\in T\}$.\\ \\
\section{Continuous linear programming} Another interesting application takes place in continuous linear programming. Here problems of the subsequent forms are discussed:
\begin{center} \begin{equation*}{(P^*)\;\;\;\;\;\;\;\;} \begin{array}{ll}
min\;\; \mathop{\mathlarger{\int_0^T}} x(t)c(t)dt \\
s.t.\;\;x(t)B(t)\geqq b(t)+ \mathop{\mathlarger{\int_t^T}}x(s)K(s,t)ds\;\;\;\;\;\;f.a.e.\;\;t\in[0,T]\\
x(t)\geqq 0\;\; f.a.e.\;\;t\in[0,T]\\
\\
\end{array}
\end{equation*} \end{center} \begin{center} \begin{equation*}{(D^*)\;\;\;\;\;\;\;\;} \begin{array}{ll}
max\;\; \mathop{\mathlarger{\int_0^T}} y(t)b(t)dt \\
s.t.\;\;B(t)y(t)\leqq c(t)+ \mathop{\mathlarger{\int_0^t}}K(s,t)y(s)ds\;\;\;\;\;\;f.a.e.\;\;t\in[0,T]\\
y(t)\geqq 0\;\; f.a.e.\;\;t\in[0,T]\\
\\
\end{array}
\end{equation*} \end{center} where $c(t),b(t)$ are bounded and Lebesgue measurable m and n-vectors (functions) respectively, $B(t)$ is an $m\times n$ bounded and Lebesgue measurable matrix, $K(s,t)$ is an $m\times n$ bounded and Lebesgue measurable matrix which is equal to 0 for $s>t$, the functions $x(t),y(t)$ are bounded and Lebesgue measurable, $T$ is finite and $f.a.e.$ $t\in[0,T]$ stands for \textit{``for almost every"} $t\in[0,T]$, that is the inequalities that apart the feasible regions hold for every $t\in[0,T]\smallsetminus U$, where $U\subset[0,T]$ has a Lebesgue measure of zero.\par Problems of the above form have been discussed in \cite{6, 9} and strong duality theorems have been acquired under specific hypotheses. The following algebraic assumptions are always made:\\ \begin{equation} \{z:B(t)z\leqq0,z\geqq0\}=\{0\}\;\;\forall t\in[0,T] \end{equation} \begin{equation}
B(t)\geqq0,\;K(s,t)\geqq0,\;c(t)\geqq0\;\;\forall s,t\in[0,T] \end{equation} while the continuity of these functions (almost everywhere in $[0,T]$) is also required for no duality gap between $(P^*)$ and $(D^*)$. In particular, the following strong duality theorem is known:
\begin{theorem}[\cite{6} Theorem 3.4, \cite{9} Theorem 3] If (5.1) and (5.2) hold and if $B(t),\;b(t),\;c(t),\;K(s,t),\;$ are continuous at almost all $t$ in $[0,T]$ and almost all $s,\;t$ in $[0,T]\times[0,T]$ respectively, then $v(P^*)=v(D^*)$. \end{theorem} \par
Here we work with the conves cones: \begin{equation*}
S:=(L_{\infty}^{+}[0,T])^m,\;\;\;T:=(L_{\infty}^{+}[0,T])^n \end{equation*} where $L_{\infty}^{+}[0,T]$ is the space of all almost everywhere positively valued on $[0,T]$ bounded Lebesgue measurable functions. Strong duality results regarding continuous linear problems of similar form with these exact feasible convex cones are given in \cite{15}. \par The linear operators $A:X\rightarrow Y$, $A^T:Y\rightarrow X$ are defined by: \begin{equation*}
(Ax)(t):= x(t)B(t)-\mathop{\mathlarger{\int_t^T}}x(s)K(s,t)ds,\;\;t\in[0,T] \end{equation*} \begin{equation*}
(A^Ty)(t):=B(t)y(t)-\mathop{\mathlarger{\int_0^t}}K(s,t)y(s)ds,\;\;t\in[0,T] \end{equation*} We can see that these are well defined under our initial hypothesis. Also for the bilinear and symmetric objective function $\langle z(t),w(t)\rangle = \mathop{\mathlarger{\int_0^T}}z(t)w(t)dt$ the following is true: $\langle (Az)(t),w(t)\rangle=\langle z(t),(A^Tw)(t)\rangle$. Indeed, it suffices to show that:\\ \begin{equation*} \mathop{\mathlarger{\int_0^T}}\left(\mathop{\mathlarger{\int_t^T}}z(s)K(s,t)ds\right)w(t)dt=\mathop{\mathlarger{\int_0^T}}z(t)\left(\mathop{\mathlarger{\int_0^t}}K(s,t)w(s)ds\right)dt \end{equation*}
This is true by the Fubini Theorem (see \cite{6}, Proposition 1.3).\par Now \textbf{(F1),(F2),(F4)} hold by definition for almost every $t\in[0,T]$ (it is easy to notice that the \textit{``for almost every"} does not affect the gist of the main results - the reader may fill in the appropriate details). Condition \textbf{(F3)} is the one that discommodes us once again. In contrast to the previous example, it is not wise to work with $L_{\infty}^{+}[0,T]$ (or a subspace $L_{k}^{+}[0,T]$ for some $k\in\mathbb{N}$) in order to prove level-boundness and/or apply Theorems of the attainment of minimum, as it was done previously. This is mainly because the set $\{f(t)\in (L_{\infty}^{+}[0,T])^m: |f(t)|\leqq N, t\in[0,T]\}$ for some $N\in\mathbb{R}$ is not bounded. Therefore we would either have to work in specific subspaces of $L_{\infty}^{+}[0,T]$, that is with specific spaces - types - of bounded Lebesgue measurable functions, or with suitable matrices $K(s,t)$, $B(t)$ and vectors $b(t)$, $c(t)$, as it was effectively done in the previous section.\par For the main purposes of this section we shall not deal with the analysis of cases under which this condition is met; for the result that follows we will assume that \textbf{(F3)} is true.\\ \par The next Theorem can be characterized as partially complementary to Theorem 5.1:\\
\begin{theorem} Let $\hat{x}(t)$ be a feasible solution of $(P^*)$ and $\hat{y}(t)$ be a feasible solution of $(D^*)$ such that $\hat{x}(t)>0$ and $\hat{y}(t)>0$ f.a.e.\;\;$t\in[0,T]$. Assume that the optimal values $v(P^*)$, $v(D^*)$ exist and are bounded, assumption (F3) holds and that either $(i)$ or $(ii)$ of the following is satisfied:\\ \\ (i) $B(t)\leqq0$, $K(s,t)\geqq0$ $\forall s,t\in[0,T]$ and $b\in T^*$\\ (ii) $B(t)\geqq0$, $K(s,t)\leqq0$ $\forall s,t\in[0,T]$ and $-c\in S^*$\\ \\ Then there exist optimal solutions $x^*(t)$ of $(P^*)$ and $y^*(t)$ of $(D^*)$ such that $x^*(t)B(t)= b(t)+ \int_t^T x^*(s)K(s,t)ds$ and $B(t)y^*(t)= c(t)+ \int_0^t K(s,t)y^*(s)ds$,\;\;\;f.a.e. $t\in[0,T]$, and $v(P^*)=v(D^*)$. \end{theorem} \begin{proof}
Assume that the first condition $(i)$ is met (the proof for condition $(ii)$ is completely analogous). Then, since $B(t)\leqq0$ and $K(s,t)\geqq0$ for $s,t\in[0,T]$, the set $\hat{\mathcal{S}}(D^*)=\{y\in$ int$T:-A^Ty+c\in S^*,-A^Ty\in S^*,\langle y,b\rangle>-\infty\}$, where the linear operator $A^T$ is defined as above, is nonempty for almost every $t\in[0,T]$. Therefore, by Theorem 3.8 there exists $x^*(t)\in\mathcal{S}(P^*)$ such that $Ax^*=b$, that is $x^*(t)B(t)= b(t)+ \int_t^T x^*(s)K(s,t)ds$ for almost every $t\in[0,T]$. Now, since $b\in T^*$, Theorem 3.9 may also be applied, hence there also exists $y^*(t)\in\mathcal{S}(D^*)$ such that $A^Ty^*=c$, that is $B(t)y^*(t)= c(t)+ \int_0^t K(s,t)y^*(s)ds$,\;\;\;f.a.e. $t\in[0,T]$. The strong duality relation $v(P^*)=v(D^*)$ follows. \end{proof} Note that for the proof the continuity of $B(t),\;b(t),\;c(t),\;K(s,t)$ is not assumed in comparison to Theorem 5.1, although it may be (partially) needed in order for \textbf{(F3)} to hold.
\end{document}
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\begin{document}
\title{Binary codes that do not preserve primitivity}
\begin{abstract} A code $X$ is not primitivity preserving if there is a primitive list $\mathbf w \in \lists X$ whose concatenation is imprimitive. We formalize a full characterization of such codes in the binary case in the proof assistant Isabelle/HOL. Part of the formalization, interesting on its own, is a description of $\{x,y\}$-interpretations of the square $xx$ if $\abs y \leq \abs x$. We also provide a formalized parametric solution of the related equation $x^jy^k = z^\ell$. \end{abstract}
\section{Introduction} Consider two words ${\tt abba}$ and ${\tt b}$. It is possible to concatenate (several copies of) them as ${\tt b}\cdot {\tt abba} \cdot {\tt b}$, and obtain a power of a third word, namely a square ${\tt bab}\cdot {\tt bab}$ of ${\tt bab}$. In this paper, we completely describe all ways how this can happen for two words, and formalize it in Isabelle/HOL.
The corresponding theory has a long history. The question can be formulated as solving equations in three variables of the special form $W(x,y) = z^\ell$ where the left hand side is a sequence of $x$'s and $y$'s, and $\ell \geq 2$. The seminal result in this direction is the paper by R. C. Lyndon and M.-P. Sch\"utzen\-ber\-ger \cite{lyndon1962} from 1962, which solves in a more general setting of free groups the equation $x^jy^k = z^\ell$ with $2 \leq j,k,\ell$. It was followed, in 1967, by a partial answer to our question by A. Lentin and M.-P. Sch\"{u}tzenberger \cite{lentin}.
Complete characterization of monoids generated by three words was provided by L. G. Budkina and Al. A. Markov in 1973 \cite{budkina}. The characterization was later, in 1976, reproved in a different way by Lentin's student J.-P. Spehner in his Ph.D. thesis \cite{spehner}, which even explicitly mentions the answer to the present question. See also a comparison of the two classifications by T. Harju and D. Nowotka \cite{terodirk}. In 1985, the result was again reproved by E. Barbin-Le Rest and M. Le Rest \cite{lerest}, this time specifically focusing on our question. Their paper contains a characterization of binary interpretations of a square as a crucial tool. The latter combinatorial result is interesting on its own, but is very little known. In addition to the fact that, as far as we know, the proof is not available in English, it has to be reconstructed from Th\'eor\`eme 2.1 and Lemme 3.1 in \cite{lerest}, it is long, technical and little structured, with many intuitive steps that have to be clarified. It is symptomatic, for example, that Ma\v nuch \cite{Manuch} cites the claim as essentially equivalent to his desired result but nevertheless provides a different, shorter but similarly technical proof.
This complicated history makes the topic a perfect candidate for formalization. The proof we present here naturally contains some ideas of the proof from \cite{lerest} but is significantly different. Our main objective was to follow the basic methodological requirement of a good formalization, namely to identify claims that are needed in the proof and formulate them as separate lemmas and as generally as possible so that they can be reused not only in the proof but also later. Moreover, the formalization naturally forced us to consider carefully the overall strategy of the proof (which is rather lost behind technical details of published works on this topic). Under Isabelle's pressure we eventually arrived at a~hopefully clear proof structure which includes a simple, but probably innovative use of the idea of ``gluing'' words.
The analysis of the proof is therefore another, and we believe the most important contribution of our formalization, in addition to the mere certainty that there are no gaps in the proof.
In addition, we provide a complete parametric solution of the equation $x^ky^j = z^\ell$ for arbitrary $j$, $k$ and $\ell$, a classification which is not very difficult, but maybe too complicated to be useful in a mere unverified paper form.
The formalization presented here is an organic part of a larger project of formalization of combinatorics of words (see an introductory description in \cite{itp2021}). We are not aware of a similar formalization project in any proof assistant. The existence of the underlying library, which in turn extends the theories of ``List'' and ``HOL-Library.Sublist'' from the standard Isabelle distribution, critically contributes to a smooth formalization which is getting fairly close to the way a~human paper proof would look like, outsourcing technicalities to the (reusable) background. We accompany claims in this text with names of their formalized counterparts.
\section{Basic facts and notation}\label{sec:notation}
Let $\Sigma$ be an arbitrary set. Lists (i.e. finite sequences) $[x_1,x_2,\dots,x_n]$ of elements $x_i \in \Sigma$ are called \emph{words} over $\Sigma$. The set of all words over $\Sigma$ is usually denoted as $\Sigma^*$, using the Kleene star. A notorious ambivalence of this notation is related to the situation when we consider a set of words $X \subset \Sigma^*$, and are interested in lists over $X$. They should be denoted as elements of $X^*$. However, $X^*$ usually means something else (in the theory of rational languages), namely the set of all words in $\Sigma^*$ generated by the set $X$. To avoid the confusion, we will therefore follow the notation familiar from the formalization in Isabelle, and write $\lists X$ instead, to make clear that the entries of an element of $\lists X$ are themselves words. In order to further help to distinguish words over the basic alphabet from lists over a set of words, we shall use boldface variables for the latter. In particular, it is important to keep in mind the difference between a letter $a$ and the word $[a]$ of length one, the distinction which is usually glossed over lightly in the literature on combinatorics on words. The set of words over $\Sigma$ generated by $X$ is then denoted as $\gen X$.
The (associative) binary operation of concatenation of two words $u$ and $v$ is denoted by $u \cdot v$ We prefer this algebraic notation to the Isabelle's original \verb|@|. Moreover, we shall often omit the dot as usual. If~$\mathbf u = [x_1,x_2,\ldots, x_n] \in \lists X$ is a list of words, then we write $\concat \mathbf u$ for $x_1\cdot x_2 \cdots x_n$. We write $\varepsilon$ for the empty list, and $u^k$ for the concatenation of $k$ copies of $u$ (we use $u^{\verb|@|}k$ in the formalization). We write $u \leq_p v$, $u <_p v$, $u \leq_s v$, $u <_s v$, and $u \leq_f v$ to denote that $u$ is a \emph{prefix}, a \emph{strict prefix}, \emph{suffix}, \emph{strict suffix} and \emph{factor} (that is, a contiguous sublist) respectively. A word is \emph{primitive} if it is nonempty and not a power of a shorter word. Otherwise, we call it \emph{imprimitive}. Each nonempty word $w$ is a power of a unique primitive word $\rho\, w$, its \emph{primitive root}. A nonempty word $r$ is a \emph{periodic root} of a word $w$ if $w \leq_p r \cdot w$. This is equivalent to $w$ being a prefix of the right infinite power of $r$, denoted $r^\omega$. Note that we deal with finite words only, and we use the notation $r^\omega$ only as a convenient shortcut for ``a sufficiently long power of $r$''. Two words $u$ and $v$ are \emph{conjugate}, we write $u \sim v$, if $u = rq$ and $v=qr$ for some words $r$ and $q$. Note that conjugation is an equivalence whose classes are also called \emph{cyclic words}. A word $u$ is \emph{cyclic factor} of $w$ if it is a factor of some conjugate of $w$. A set of words $X$ is a \emph{code} if its elements do not satisfy any nontrivial relation, that is, they are a basis of a free semigroup. For two element set $\{x,y\}$, this is equivalent to $xy\neq yx$, and/or to $\rho\, x \neq \rho\, y$. An important characterization of a semigroup $S$ of words to be free is the \emph{stability condition} which is the implication $u,v,uz,zv \in S \Longrightarrow z \in S$. The longest common prefix of $u$ and $v$ is denoted by $u \wedge_p v$. If $\{x,y\}$ is a (binary) code, then $(x \cdot w) \wedge_p (y \cdot w') = xy\wedge_p yx$ for any $w,w'\in \gen{\{x,y\}}$ sufficiently long. We explain some elementary facts from combinatorics on words in more detail in the Appendix~\ref{sec:appendix}.
\section{Main theorem} Let us introduce the central definition of the paper. \begin{definition}
We say that a set $X$ of words is \emph{primitivity preserving} if there is no word $\mathbf w \in \lists X$ such that
\begin{itemize}
\item $\abs \mathbf w \geq 2$;
\item $\mathbf w$ is primitive; and
\item $\concat \mathbf w$ is imprimitive.
\end{itemize} \end{definition}
Note that our definition does not take into account singletons $\mathbf w = [x]$. In particular, $X$ can be primitivity preserving even if some $x \in X$ is imprimitive. Nevertheless, in the binary case, we will also provide some information about the cases when one or both elements of the code have to be primitive.
In \cite{Mitrana1997}, V. Mitrana formulates the primitivity of a set in terms of morphisms, and shows that $X$ is primitivity preserving if and only if it is the minimal set of generators of a ``pure monoid'', cf. \cite[p. 276]{BerstelCodes}. This brings about a wider concept of morphisms preserving a given property, most classically squarefreeness, see for example a characterization of squarefree morphisms over three letters by M. Crochemore \cite{Crochemore1982}.
The target claim of our formalization is the following characterization of words witnessing that a binary code is not primitivity preserving:
\begin{theorem}[\texttt{bin\_imprim\_code}] \label{th:Theorem1} Let $B = \{x,y\}$ be a code that is not primitivity preserving.
Then there are integers $j \geq 1$ and $k \geq 1$, with $k = 1$ or $j = 1$, such that the following conditions are equivalent for any
$\mathbf w \in \lists B$ with $\abs {\mathbf w} \geq 2$:
\begin{itemize}
\item $\mathbf w$ is primitive, and $\concat \mathbf w$ is imprimitive
\item $\mathbf w$ is conjugate with $[x]^j[y]^k$.
\end{itemize}
Moreover, assuming $\abs y \leq \abs x$,
\begin{itemize}
\item if $j \geq 2$, then $j=2$ and $k=1$, and both $x$ and $y$ are primitive;
\item if $k \geq 2$, then $j=1$ and $x$ is primitive.
\end{itemize} \end{theorem}
\begin{proof} Let $\mathbf w$ be a word witnessing that $B$ is not primitivity preserving. That is, $\abs{\mathbf w} \geq 2$, $\mathbf w$ is primitive, and $\concat \mathbf w$ is imprimitive. Since $[x]^j[y]^k$ and $[y]^k[x]^j$ are conjugate, we can suppose, without loss of generality, that $\abs y \leq \abs x$.
First, we want to show that $\mathbf w$ is conjugate with $[x]^j[y]^k$ for some $j,k \geq 1$ such that $k = 1$ or $j = 1$.
Since $\mathbf w$ is primitive and of length at least two, it contains both $x$ and $y$. If it contains one of these letters exactly once, then the desired from $[x]^j [y]^k$ up to conjugation is guaranteed, with $j = 1$ or $k = 1$. Therefore, the difficult part is to show that no primitive $\mathbf w$ with $\concat \mathbf w$ imprimitive can contain both letters at least twice. This is the main task of the rest of the paper, which is finally accomplished by Theorem \ref{th:main} claiming that words that contain at least two occurrences of $x$ are conjugate with $[x,x,y]$.
To complete the proof of the first part of the theorem, it remains to show that $j$ and $k$ do not depend on $\mathbf w$. This follows from Lemma~\ref{le:jk_unique}.
Note that the imprimitivity of $\concat \mathbf w$ induces the equality $x^jy^k = z^\ell$ for some $z$ and $\ell \geq 2$. The already mentioned seminal result of Lyndon and Sch\"utzenberger shows that $j$ and $k$ cannot be simultaneously at least two, since otherwise $x$ and $y$ commute. For the same reason, considering its primitive root, the word $y$ is primitive if $j \geq 2$. Similarly, $x$ is primitive if $k \geq 2$. The primitivity of $x$ when $j = 2$ is a part of Theorem \ref{th:main}. \qed \end{proof}
We start by giving a complete parametric solution of the equation $x^jy^k = z^\ell$ in the following theorem. This will eventually yield, after the proof of Theorem \ref{th:Theorem1} is completed, a full description of not primitivity preserving binary codes. Since the equation is mirror symmetric, we omit symmetric cases by assuming $|y| \leq |x|$.
\newcommand*{\thead}[1]{\multicolumn{1}{c|}{\bfseries #1}} \newcommand*{\theadlast}[1]{\multicolumn{1}{c}{\bfseries #1}}
\begin{theorem}[\texttt{LS\_parametric\_solution}] \label{thm:xjykzl_solution}
Let $\ell \geq 2$, $j,k \geq 1$ and $|y| \leq |x|$.
The equality $x^jy^k = z^\ell$ holds if and only if one of the following cases takes place:
\begin{enumerate}[A.] \setlength\itemsep{0.5em}
\item \label{A} There exists a word $r$, and integers $m,n,t \geq 0$ such that $$mj+nk = t \ell ,$$ and
\begin{align*}
x = r^m, \quad y = r^n, \quad z = r^t;
\end{align*}
\item \label{B} $j = k = 1$ and there exist non-commuting words $r$ and $q$,
and integers $m,n \geq 0$ such that $$m+n+1 = \ell ,$$ and
$$x = (rq)^mr, \quad y = q(rq)^{n}, \quad z = rq;$$
\item \label{C} $j = \ell = 2$, $k = 1$ and there exist non-commuting words
$r$ and $q$ and an integer $m \geq 2$ such that
$$x = (rq)^m r, \quad y = qrrq, \quad z = (rq)^mrrq;$$
\item \label{D} $j = 1$ and $k \geq 2$ and there exist non-commuting words
$r$ and $q$ such that \\
$$x = (qr^k)^{\ell-1}q, \quad y = r, \quad z = qr^k;$$
\item \label{E} $j = 1$ and $k \geq 2$ and there are non-commuting words
$r$ and $q$, an integer $m \geq 1$ such that
$$x = (qr(r(qr)^m)^{k - 1})^{\ell - 2}qr(r(qr)^m)^{k - 2}rq, \quad y = r(qr)^m, \quad z = qr(r(qr)^m)^{k - 1}.$$
\end{enumerate}
\end{theorem}
\begin{proof} If $x$ and $y$ commute, then all three words commute, hence they are a power of a common word. A length argument yields the solution \ref{A}.
Assume now that $\{x,y\}$ is a code. Then no pair of words $x$, $y$ and $z$ commutes. We have shown in the overview of the proof of Theorem \ref{th:Theorem1} that $j = 1$ or $k = 1$
by the Lyndon-Sch\"utzenberger theorem.
The solution is then split into several cases.
\noindent \emph{Case 1}: $j = k = 1$. \\
Let $m$ and $r$ be such that $z^mr = x$ with $r$ a strict prefix of $z$.
By setting $z = rq$, we obtain the solution \ref{B} with $n = \ell - m -1$.
\noindent \emph{Case 2}: $j \geq 2, k = 1$.\\
Since $|y| \leq |x|$ and $\ell \geq 2$, we have
$$2|z| \leq |z^\ell| = |x^j| + |y| < 2|x^j|,$$
so $z$ is a strict prefix of $x^j$.
As $x^j$ has periodic roots both $z$ and $x$, and $z$ does not commute with $x$, the Periodicity lemma implies
\[
|x^j| < |z| + |x|.
\]
That is, $z = x^{j-1}u$, $x^j = zv$ and $x = uv$ for some nonempty words $u$ and $v$.
As $v$ is a prefix of $z$, it is also a prefix of $x$. Therefore, we have
\[
x = uv = vu'
\]
for some word $u'$.
This is a well known conjugation equality which implies $u = rq$, $u' = qr$ and $v = (rq)^nr$ for some words $r$, $q$ and an integer $n \geq 0$.
We have
\[
j|x| + |y| = |x^jy| = |z^\ell| = \ell(j-1)|x| + \ell|u|,
\]
and thus $|y| = (\ell j-\ell-j)|x| + \ell|u|$.
Since $|y| \leq |x|$, $|u| > 0$, $j \geq 2$, and $\ell \geq 2$,
it follows that $\ell j - \ell - j = 0$, which implies $j = l = 2$.
We therefore have $x^2y = z^2$ and $x^2 = zv$, hence $vy = z$.
Combining $u = rq$, $u' = qr$, and $v = (rq)^nr$
with $x = vu'$, $z = x^{j-1}u = xu = vu'u$, and $vy = z$, we obtain the solution \ref{C} with $m = n+1$. The assumption $\abs y \leq \abs x$ implies $m \geq 2$.
\noindent\emph{Case 3}: $j = 1, k \geq 2, y^k {\leq}_s z$. \\
We have $z = qy^k$ for some word $q$.
Noticing that $x = z^{\ell-1}q$ yields the solution \ref{D}.
\noindent\emph{Case 4}: $j = 1, k \geq 2, z <_s y^k$. \\
This case is analogous to the second part of Case 2.
Using the Periodicity lemma, we obtain $uy^{k-1} = z$, $y^k = vz$, and $y = vu$ with nonempty $u$ and $v$.
As $v$ is a suffix of $z$, it is also a suffix of $y$, and we have
\[
y = vu = u'v
\]
for some $u'$.
Plugging the solution of the last conjugation equality, namely
$u' = rq$, $u = qr$, $v = (rq)^nr$,
into $y = u'v$, $z = uy^{k-1}$ and $z^{\ell-1} = xv$ gives the solution \ref{E} with $m = n + 1$.
Finally, the words $r$ and $q$ do not commute since $x$ and $y$, which are generated by $r$ and $q$, do not commute.
The proof is completed by a direct verification of the opposite implication.
\qed
\end{proof}
We now show that, for a given not primitivity preserving binary code, there is a unique pair of exponents $(j,k)$ such that $x^jy^k$ is imprimitive.
\begin{lemma}[\texttt{LS\_unique}] \label{le:jk_unique} Let $B = \{x,y\}$ be a code. Assume $j,k,j',k' \geq 1$. If both $x^jy^k$ and $x^{j'}y^{k'}$ are imprimitive, then $j = j'$ and $k = k'$. \end{lemma} \begin{proof} Let $z_1,z_2$ be primitive words and $\ell,\ell' \geq 2$ be such that \begin{equation}\label{eq:both_imprim} x^jy^k = z_1^\ell \quad \text{ and } \quad x^{j'}y^{k'} = z_2^{\ell'}. \end{equation} Since $B$ is a code, the words $x$ and $y$ do not commute. We proceed by contradiction.
\noindent \emph{Case 1}: First, assume that $j = j'$ and $k \neq k'$.\\
Let, without loss of generality, $k < k'$. From \eqref{eq:both_imprim} we obtain $z_1^\ell y^{k' - k} = z_2^{\ell'}$. The case $k' - k \geq 2$ is impossible due to the Lyndon-Sch\"utzenberger theorem. Hence $k' - k = 1$. This is another place where the formalization triggered a simple and nice general lemma (easily provable by the Periodicity lemma) which will turn out to be useful also in the proof of Theorem \ref{th:main}. Namely, the lemma \verb|imprim_ext_suf_comm| claims that if both $uv$, and $uvv$ are imprimitive, then $u$ and $v$ commute. We apply this lemma to $u = x^jy^{k-1}$ and $v = y$, obtaining a contradiction with the assumption that $x$ and $y$ do not commute.
\noindent \emph{Case 2.} The case $k = k'$ and $j \neq j'$ is symmetric to Case 1.
\noindent \emph{Case 3.} Let finally $j \neq j'$ and $k \neq k'$. The Lyndon-Sch\"utzenberger theorem implies that either $j$ or $k$ is one, and similarly either $j'$ or $k'$ is one. We can therefore assume that $k = j' = 1$ and $k',j \geq 2$. Moreover, we can assume that $\abs y \leq \abs x$. Indeed, in the opposite case, we can consider the words $y^kx^j$ and $y^{k'}x^{j'}$ instead, which are also both imprimitive.
Theorem~\ref{thm:xjykzl_solution} now allows only the case \ref{C} for the equality $x^jy = z_1^\ell$. We therefore have $j = \ell = 2$ and $x = (rq)^mr$, $y = qrrq$ for an integer $m \geq 2$ and some non-commuting words $r$ and $q$. Since $y = qrrq$ is a suffix of $z_2^\ell$, this implies that $z_2$ and $rq$ do not commute. Consider the word $x \cdot qr = (rq)^mrqr$, which is a prefix of $xy$, and therefore also of $z_2^\ell$. This means that $x \cdot qr$ has two periodic roots, namely $rq$ and $z_2$, and the Periodicity lemma implies that $\abs {x \cdot qr}< \abs{rq} + \abs {z_2}$. Hence $x$ is shorter than $z_2$. The equality $xy^{k'} = z_2^{\ell'}$, with $\ell' \geq 2$, now implies on one hand that $rqrq$ is a prefix of $z_2$, and on the other hand that $z_2$ is a suffix of $y^{k'}$. It follows that $rqrq$ is a factor of $(qrrq)^k$. Hence $rqrq$ and $qrrq$ are conjugate, which is possible only if $r$ and $q$ commute, a fact not difficult to prove, see Appendix~\ref{sec:appendix}. This is a contradiction. \qed \end{proof}
The rest of the paper, and therefore also of the proof of Theorem~\ref{th:Theorem1}, is organized as follows. In Section~\ref{sec:interpretace}, we introduce a general theory of interpretations, which is behind the main idea of the proof, and apply it to the (relatively simple) case of a binary code with words of the same length. In Section~\ref{sec:square} we characterize the unique disjoint extendable $\{x,y\}$-interpretation of the square of the longer word $x$. This is a result of independent interest, and also the cornerstone of the proof of Theorem~\ref{th:Theorem1} which is completed in Section~\ref{sec:main} by showing that a word containing at least two $x$'s witnessing that $\{x,y\}$ is not primitivity preserving is conjugate with $[x,x,y]$.
\section{Interpretations and the main idea}\label{sec:interpretace} Let $X$ be a code, let $u$ be a factor of $\concat \mathbf w$ for some $\mathbf w \in \lists X$. The natural question is to decide how $u$ can be produced as a factor of words from $X$, or, in other words, how it can be interpreted in terms of $X$. This motivates the following definition. \begin{definition} \label{def:interpretation} Let $X$ be a set of words over $\Sigma$. We say that the triple $(p,s,\mathbf w) \in \Sigma^*\times \Sigma^* \times \lists X$ is an \emph{$X$-interpretation} of a word $u \in \Sigma^*$ if \begin{itemize}
\item $\mathbf w$ is nonempty;
\item $p \cdot u \cdot s = \concat \mathbf w$;
\item $p <_p \hd \mathbf w$ and
\item $s <_s \last \mathbf w$. \end{itemize} \end{definition} The definition is illustrated by the following figure, where $\mathbf w = [w_1,w_2,w_3,w_4]$: \[ \begin{tikzpicture}[thick]
\draw (-2.5,-.2) rectangle (2.5,.2); \node at (0,0) {$u$};
\draw (-3,.2) rectangle (-1,.6); \node at (-2,.4) {$w_1$};
\draw (-1,.2) rectangle (0,.6); \node at (-.5,.4) {$w_2$};
\draw (0,.2) rectangle (2.2,.6); \node at (1.1,.4) {$w_3$};
\draw (2.2,.2) rectangle (3.6,.6); \node at (2.9,.4) {$w_4$};
\draw[dotted] (-3,-.2) rectangle (-2.5,.2); \node at (-2.75,0) {$p$};
\draw[dotted] (2.5,-.2) rectangle (3.6,.2); \node at (3.05,0) {$s$}; \end{tikzpicture} \] The first condition of the definition motivates the notation $p\, u\, s \sim_{\mathcal I} \mathbf w$ for the situation when $(p, s, \mathbf w)$ is an $X$-interpretation of $u$.
\begin{remark} For sake of historical reference, we remark that our definition of $X$-interpretation differs from the one used in \cite{lerest}. Their formulation of the situation depicted by the above figure would be that $u$ is interpreted by the triple $(s', w_2 \cdot w_3, p')$ where $p\cdot s' = w_1$ and $p'\cdot s = w_4$. This is less convenient for two reasons. First, the decomposition of $w_2 \cdot w_3$ into $[w_2,w_3]$ is only implicit here (and even ambiguous if $X$ is not a code). Second, while it is required that the the words $p'$ and $s'$ are a prefix and a suffix, respectively, of an element from $X$, the identity of that element is left open, and has to be specified separately. \end{remark}
If $u$ is a nonempty element of $\gen X$ and $u = \concat \mathbf u$ for $\mathbf u \in \lists X$, then the $X$-interpretation $\varepsilon\, u\, \varepsilon \sim_{\mathcal I} \mathbf u$ is called \emph{trivial}. Note that the trivial $X$-interpretation is unique if $X$ is a code.
As nontrivial $X$-interpretations of elements from $\gen X$ are of particular interest, the following two concepts are useful. \begin{definition}
An $X$-interpretation $p\, u\, s \sim_{\mathcal I} \mathbf w$ of $u = \concat \mathbf u$ is called
\begin{itemize}
\item \emph{disjoint} if $\concat \mathbf w' \neq p \cdot \concat \mathbf u'$ whenever $\mathbf w' \leq_p \mathbf w$ and $\mathbf u' \leq_p \mathbf u$.
\item \emph{extendable} if $p \leq_s w_p$ and $s \leq_p w_s$ for some elements $w_p, w_s \in \gen X$.
\end{itemize} \end{definition}
Note that a disjoint $X$-interpretation is not trivial, and that being disjoint is relative to a chosen factorization $\mathbf u$ of $u$ (which is nevertheless unique if $X$ is a code).
The definitions above are naturally motivated by \textbf{the main idea} of the characterization of sets $X$ that do not preserve primitivity, which dates back to Lentin and Sch\"utzenberger \cite{lentin}. If $\mathbf w$ is primitive, while $\concat \mathbf w$ is imprimitive, say $\concat \mathbf w = z^k$, $k\geq 2$, then the shift by $z$ provides a nontrivial and extendable $X$-interpretation of $\concat \mathbf w$. (In fact, $k-1$ such nontrivial interpretations). Moreover, the following lemma, formulated in a more general setting of two words $\mathbf w_1$ and $\mathbf w_2$, implies that the $X$-interpretation is disjoint if $X$ is a code. \begin{lemma}[\texttt{shift\_interpret}, \texttt{shift\_disjoint}]\label{lem:disjoint_interp}
Let $X$ be a code. Let $\mathbf w_1, \mathbf w_2 \in \lists X$ be such that $z \cdot \concat \mathbf w_1 = \concat \mathbf w_2 \cdot z$ where $z \notin \gen X$. Then
$z \cdot \concat \mathbf v_1 \neq \concat \mathbf v_2$, whenever $\mathbf v_1 \leq_p \mathbf w_1^n$ and $\mathbf v_2 \leq_p \mathbf w_2^n$, $n\in \mathbb N$.
In particular $\concat \mathbf u$ has a disjoint extendable $X$-interpretation for any prefix $\mathbf u$ of $\mathbf w_1$. \end{lemma} The excluded possibility is illustrated by the following figure. \[ \begin{tikzpicture}[thick, scale = .85]
\draw (2,-.5) rectangle (8,0); \node at (5,-.25) {$\concat \mathbf w_1$};
\draw[gray] (8,-.5) rectangle (14,0); \node at (11,-.25) {$\concat \mathbf w_1$};
\draw (0,0) rectangle (6,.5); \node at (3,.25) {$\concat \mathbf w_2$};
\draw (6,0) rectangle (12,.5); \node at (9,.25) {$\concat \mathbf w_2$};
\draw[dotted] (0,-.5) rectangle (2,0); \node at (1,-.25) {$z$};
\draw[dotted] (12,0) rectangle (14,.5); \node at (13,.25) {$z$};
\draw [dashed] (7,.7) -- (7,-.7);
\draw [thin] (2,-.5) to[bend right = 10] node[below] {$\concat \mathbf v_1$} (7, -.5);
\draw [thin] (0,.5) to[bend left = 7] node[above] {$\concat \mathbf v_2$} (7, .5); \end{tikzpicture} \] \begin{proof}
First, note that $z \cdot \concat \mathbf w_1^n = \concat \mathbf w_2^n \cdot z$ for any $k$. Let $\mathbf w_1^n = \mathbf v_1\cdot \mathbf v_1'$ and
$\mathbf w_2^n= \mathbf v_2\cdot \mathbf v_2'$. If $z \cdot \concat \mathbf v_1 = \concat \mathbf v_2$, then also $\concat \mathbf v_2' \cdot z = \concat \mathbf v_1'$.
This contradicts $z \notin \gen X$ by the stability condition.
An extendable $X$-interpretation of $\mathbf u$ is induced by the fact that $\concat \mathbf u$ is covered by $\concat (\mathbf w_2 \cdot \mathbf w_2)$.
The interpretation is disjoint by the first part of the proof. \qed \end{proof} In order to apply the above lemma to the imprimitive $\concat \mathbf w = z^k$ of a primitive $\mathbf w$, set $\mathbf w_1= \mathbf w_2 = \mathbf w$. The assumption $z \notin \gen X$ follows from the primitivity of $\mathbf w$: indeed, if $z = \concat \mathbf z$, with $\mathbf z \in \lists X$, then $\mathbf w = \mathbf z^k$ since $B$ is a code.
We first apply the main idea to a relatively simple case of nontrivial $\{x,y\}$-interpretation of the word $x \cdot y$ where $x$ and $y$ are of the same length. \begin{lemma}[\texttt{uniform\_square\_interp}]\label{lem:xy_interp}
Let $B = \{x,y\}$ be a code with $\abs x = \abs y$. Let $p\ (x\cdot y)\ s \sim_{\mathcal I} \mathbf v$ be a nontrivial $B$-interpretation.
Then $\mathbf v = [x,y,x]$ or $\mathbf v = [y,x,y]$ and $x\cdot y$ is imprimitive. \end{lemma} \begin{proof} From $p \cdot x \cdot y \cdot s = \concat \mathbf v$, it follows, by a length argument, that $\abs \mathbf v$ is three. A straightforward way to proof the claim is to consider all eight possible candidates. In each case, it is then a routine few line proof that shows that $x = y$, unless $\mathbf v = [x,y,x]$ or $\mathbf v = [y,x,y]$, which we omit. In the latter cases, $x \cdot y$ is a nontrivial factor of its square $(x \cdot y)\cdot (x \cdot y)$, which yields the imprimitivity of $x \cdot y$. \qed \end{proof} The previous (sketch of the) proof nicely illustrates on a small scale the advantages of formalization. It is not necessary to choose between a tedious elementary proof for sake of completeness on one hand, and the suspicion that something was missed on the other hand (leaving aside that the same suspicion typically remains even after the tedious proof). A bit ironically, the most difficult part of the formalization is to show that $\mathbf v$ is indeed of length three, which needs no further justification in a human proof.
We have the following corollary which is a variant of Theorem \ref{th:main}, and also illustrates the main idea of its proof. \begin{lemma}[\texttt{bin\_imprim\_not\_conjug}]\label{lem:not_conjugate}
Let $B = \{x,y\}$ be a binary code with $\abs x = \abs y$. If $\mathbf w \in \lists B$ is such that $\abs {\mathbf w} \geq 2$, $\mathbf w$ is primitive, and $\concat \mathbf w$ is imprimitive,
then $x$ and $y$ are not conjugate. \end{lemma} \begin{proof}
Since $\mathbf w$ is primitive and of length at least two, it contains both letters $x$ and $y$. Therefore, it has either $[x,y]$ or $[y,x]$ as a factor. The imprimitivity of $\concat \mathbf w$ yields a nontrivial $B$-interpretation of $x \cdot y$, which implies that $x \cdot y$ is not primitive by Lemma \ref{lem:xy_interp}.
Let $x$ and $y$ be conjugate, and let $x = r \cdot q$ and $y = q \cdot r$. Since $x \cdot y = r \cdot q \cdot q \cdot r$ is imprimitive, also
$r \cdot r \cdot q \cdot q$ is imprimitive. Then $r$ and $q$ commute by the theorem of Lyndon and Sch\"utzenberger, a contradiction with $x \neq y$. \qed \end{proof}
\section{Binary interpretation of a square}\label{sec:square} Let $B = \{x,y\}$ be a code such that $\abs y \leq \abs x$. In accordance with the main idea, the core technical component of the proof is the description of the disjoint extendable $B$-interpretations of the square $x^2$. This is a very nice result which is relatively simple to state but difficult to prove, and which is valuable on its own. As we mentioned already, it can be obtained from Th\'eor\`eme 2.1 and Lemme 3.1 in \cite{lerest}.
\begin{theorem}[\texttt{square\_interp\_ext.sq\_ext\_interp}] \label{thm:sq_interp} Let $B = \{x,y\}$ be a code such that $\abs y \leq \abs x$, both $x$ and $y$ are primitive, and
$x$ and $y$ are not conjugate.
Let $p\, (x\cdot x)\, s \sim_{\mathcal I} \mathbf w$ be a disjoint extendable $B$-interpretation. Then \begin{align*}
\mathbf w &= [x,y,x], &
s \cdot p &= y, & p \cdot x &= x \cdot s. \end{align*} \end{theorem}
In order to appreciate the theorem, note that the definition of interpretation implies \[p \cdot x \cdot x \cdot s = x \cdot y \cdot x,\]
hence $x \cdot y \cdot x = (p \cdot x)^2$. This will turn out to be the only way how primitivity may not be preserved if $x$ occurs at least twice in $\mathbf w$. Here is an example with $x = \verb|01010|$ and $y = \verb|1001|$: \[ \begin{tikzpicture}[thick]
\foreach \x/\p in {0/{\tt 0},1/{\tt 1},2/{\tt 0},3/{\tt 1},4/{\tt 0},5/{\tt 1},6/{\tt 0},7/{\tt 0},8/{\tt 1},9/{\tt 0},10/{\tt 1},11/{\tt 0},12/{\tt 1},13/{\tt 0}}
\node (p\x) at (\x*.45,0) {\p};
\draw (p0.south west) rectangle (p4.north east);
\draw (p4.south east) rectangle (p8.north east);
\draw (p8.south east) rectangle (p13.north east);
\foreach \x/\p in {0/{\tt 0},1/{\tt 1},2/{\tt 0},3/{\tt 1},4/{\tt 0},5/{\tt 1},6/{\tt 0},7/{\tt 0},8/{\tt 1},9/{\tt 0},10/{\tt 1},11/{\tt 0},12/{\tt 1},13/{\tt 0}}
\node (q\x) at (\x*.45,-.45) {\p}; \draw (p2.south west) rectangle (q6.south east); \draw (p6.south east) rectangle (q11.south east); \draw[dotted] (p0.south west) rectangle (q2.south west); \draw[dotted] (p11.south east) rectangle (q13.south east);
\end{tikzpicture} \]
\begin{proof} By the definition of a disjoint interpretation, we have $p\cdot x\cdot x \cdot s = \concat \mathbf w$, where $p \neq \varepsilon$ and $s \neq \varepsilon$. A length argument implies that $\mathbf w$ has length at least three. Since a primitive word is not a nontrivial factor of its square, we have $\mathbf w = [\hd \mathbf w] \cdot [y]^k \cdot [\last \mathbf w]$, with $k \geq 1$. Since the interpretation is disjoint, we can split the equality into $p \cdot x = \hd \mathbf w \cdot y^m \cdot u$ and $x \cdot s = v \cdot y^\ell \cdot \last \mathbf w$, where $y = u \cdot v$, both $u$ and $v$ are nonempty, and $k = \ell + m + 1$.
We want to show $\hd \mathbf w = \last \mathbf w = x$ and $m = \ell = 0$.
The situation is mirror symmetric so we can solve cases two at a time.
If $\hd \mathbf w = \last \mathbf w = y$, then powers of $x$ and $y$ share a factor of length at least $\abs x + \abs y$. Since they are primitive, this implies that they are conjugate, a contradiction. The same argument applies when
$\ell \geq 1$ and $\hd \mathbf w = y$ (if $m \geq 1$ and $\last \mathbf w = y$
respectively). Therefore, in order to prove $\hd \mathbf w = \last \mathbf w = x$, it remains to exclude the case $\hd \mathbf w = y$, $\ell = 0$ and $\last \mathbf w = x$ ($\last \mathbf w = y$, $m = 0$ and $\hd \mathbf w = x$ respectively).
This is covered by one of the technical lemmas that we single out:
\begin{lemma}[\texttt{pref\_suf\_pers\_short}]\label{lem:aux1} Let $x \leq_p v \cdot x$, $x \leq_s p \cdot u \cdot v \cdot u$ and $\abs x > \abs{v \cdot u}$ with $p \in \gen{\{u,v\}}$. Then $u \cdot v = v \cdot u$. \end{lemma}
This lemma indeed excludes the case we wanted to exclude, since the conclusion implies that $y$ is not primitive. We skip the proof of the lemma here and make instead an informal comment. Note that $v$ is a period root of $x$. In other words, $x$ is a factor of $v^\omega$. Therefore, with the stronger assumption that $v \cdot u \cdot v$ is a factor of $x$, the conclusion follows easily by the familiar principle that $v$ being a factor of $v^\omega$ ``synchronizes'' primitive roots of $v$. Lemma~\ref{lem:aux1} then exemplifies one of the virtues of formalization, which makes it easy to generalize auxiliary lemmas, often just by following the most natural proof and checking its minimal necessary assumptions.
Now we have $\hd \mathbf w = \last \mathbf w = x$, hence $p \cdot x = x \cdot y^m \cdot u$ and $x \cdot s = v \cdot y^\ell \cdot x$. The natural way to describe this scenario is to observe that $x$ has both the (prefix) period root $v \cdot y^\ell$, and the suffix period root $y^m \cdot u$. Using again Lemma~\ref{lem:aux1}, we exclude situations when $\ell = 0$ and $m \geq 1$ ($m = 0$ and $\ell \geq 1$ resp.). It therefore remains to deal with the case when both $m$ and $\ell$ are positive. We divide this into four lemmas according to the size of the overlap the prefix $v\cdot y^\ell$ and the suffix $y^m\cdot u$ have in $x$. More exactly, the cases are: \begin{itemize}
\item $\abs{v\cdot y^\ell} + \abs{y^m\cdot u} \leq \abs x$
\item $\abs x < \abs{v\cdot y^\ell} + \abs{y^m\cdot u} \leq \abs x + \abs u$
\item $\abs x + \abs u < \abs{v\cdot y^\ell} + \abs{y^m\cdot u} < \abs x + \abs {u \cdot v}$
\item $\abs x + \abs {u\cdot v} \leq \abs{v\cdot y^\ell} + \abs{y^m\cdot u}$ \end{itemize}
and they are solved by an auxiliary lemma each. The first three cases yield that $u$ and $v$ commute, the first one being a straightforward application of the Periodicity lemma. The last one is also straightforward application of the ``synchronization'' idea. It implies that $x \cdot x$ is a factor of $y^\omega$, a contradiction with the assumption that $x$ and $y$ are primitive and not conjugate. Consequently, the technical, tedious part of the whole proof is concentrated in lemmas dealing with the second, and the third case (see lemmas {\verb|short_overlap|} and {\verb|medium_overlap|} in the theory \verb|Binary_Square_Interpretation.thy|). The corresponding proofs are further analyzed and decomposed into more elementary claims in the formalization, where further details can be found.
This completes the proof of $\mathbf w = [x,y,x]$. A byproduct of the proof is the description of words $x$, $y$, $p$ and $s$. Namely, there are non-commuting words $r$ and $t$, and integers $m$, $k$ and $\ell$ such that \begin{align*}
x &= (rt)^{m+1}\cdot r, & y &= (tr)^{k+1}\cdot (rt)^{\ell+1}, & p &= (rt)^{k+1}, & s &= (tr)^{\ell+1}\,. \end{align*}
The second claim of the present theorem, that is, $y = s \cdot p$ is then equivalent to $k = \ell$, and it is an easy consequence of the assumption that the interpretation is extendable. \qed
\end{proof}
\section{The witness with two $x$'s} \label{sec:main}
In this section, we characterize words witnessing that $\{x,y\}$ is not primitivity preserving and containing at least two $x$'s.
\begin{theorem}[\texttt{bin\_imprim\_longer\_twice}]\label{th:main}
Let $B = \{x,y\}$ be a code such that $\abs y \leq \abs x$. Let $\mathbf w \in \lists \{x,y\}$ be a primitive word which contains $x$ at least twice
such that $\concat \mathbf w$ is imprimitive.
Then $\mathbf w \sim [x,x,y]$ and both $x$ and $y$ are primitive. \end{theorem}
We divide the proof in three steps.
\subsubsection{The core case.} We first prove the claim with two additional assumptions which will be subsequently removed. Namely, the following lemma shows how the knowledge about the $B$-interpretation of $x \cdot x$ from the previous section is used. The additional assumptions are displayed as items. \begin{lemma}[\texttt{bin\_imprim\_primitive}] \label{lem:all_primitive}
Let $B = \{x,y\}$ be a code with $\abs y \leq \abs x$ where
\begin{itemize}
\item both $x$ and $y$ are primitive,
\end{itemize}
and let $\mathbf w \in \lists B$ be primitive such that $\concat \mathbf w$ is imprimitive, and
\begin{itemize}
\item $[x,x]$ is a cyclic factor of $\mathbf w$.
\end{itemize}
Then $\mathbf w \sim [x,x,y]$. \end{lemma} \begin{proof}
Choosing a suitable conjugate of $\mathbf w$, we can suppose, without loss of generality, that $[x,x]$ is a prefix of $\mathbf w$.
Now, we want to show $\mathbf w = [x,x,y]$.
Proceed by contradiction and assume $\mathbf w \neq [x,x,y]$.
Since $\mathbf w$ is primitive, this implies $\mathbf w \cdot [x,x,y] \neq [x,x,y] \cdot \mathbf w$.
By Lemma~\ref{lem:not_conjugate}, we know that $x$ and $y$ are not conjugate.
Let $\concat \mathbf w = z^k$, $2 \leq k$ and $z$ primitive. Lemma~\ref{lem:disjoint_interp} yields a disjoint extendable $B$-interpretation of $(\concat \mathbf w)^2$. In particular, the induced disjoint extendable $B$-interpretation of the prefix $x \cdot x$ is of the form $p\, (x \cdot x)\, s \sim_{\mathcal I} [x,y,x]$ by Theorem~\ref{thm:sq_interp}: \[ \begin{tikzpicture}[very thick, scale = .85]
\draw[dotted,thin] (0,-.5) rectangle (2,0); \node at (1,-.25) {$z$};
\draw (2,-.5) rectangle (8,0); \node at (5,-.25) {};
\draw[thin] (2,-.5) rectangle (3.5,0); \node at (2.75,-.25) {$x$};
\draw[thin] (3.5,-.5) rectangle (5,0); \node at (4.25,-.25) {$x$};
\draw[thin] (1.5,0) rectangle (3,.5); \node at (2.25,.25) {$x$};
\draw[thin] (4,0) rectangle (5.5,.5); \node at (4.75,.25) {$x$};
\node at (3.5,.25) {$y$};
\draw (0,0) rectangle (6,.5); \node at (3,.25) {};
\draw (6,0) rectangle (12,.5); \node at (9,.25) {$\concat \mathbf w$};
\draw (8,-.5) rectangle (14,.0); \node at (11,-.25) {$\concat \mathbf w$};
\draw [thin] (1.5,-.5) to[bend right = 45] node[below] {$p$} (2, -.5);
\draw [thin] (5,-.5) to[bend right = 45] node[below] {$s$} (5.5, -.5);
\draw [thin] (3,.5) to[bend left = 45] node[above] {$s$} (3.5, .5);
\draw [thin] (3.5,.5) to[bend left = 45] node[above] {$p$} (4, .5); \end{tikzpicture} \] Let $\mathbf p$ be the prefix of $\mathbf w$ such that $\concat \mathbf p \cdot p = z$. Then \begin{equation*}
\concat (\mathbf p \cdot [x,y]) = z \cdot (x \cdot p), \quad \concat [x,x,y] = (x \cdot p)^2, \quad
\concat \mathbf w = z^k, \end{equation*} and we want to show $z = xp$, which will imply $\concat ([x,x,y]\cdot \mathbf w) = \concat (\mathbf w \cdot [x,x,y])$, hence $\mathbf w = [x,x,y]$ since $\{x,y\}$ is a code, and both $\mathbf w$ and $[x,x,y]$ are primitive.
Again, proceed by contradiction, and assume $z \neq xp$. Then, since both $z$ and $x\cdot p$ are primitive, they do not commute. We now have two binary codes, namely $\{\mathbf w,[x,x,y]\}$ and $\{z,xp\}$. The following two equalities, \eqref{eq:lcp1} and \eqref{eq:lcp2} exploit the fundamental property of longest common prefixes of elements of binary codes mentioned in Section \ref{sec:notation}. In particular, we need the following lemma:
\begin{lemma}[\texttt{bin\_code\_lcp\_concat}]\label{lem:lcp} Let $X = \{u_0,u_1\}$ be a binary code, and let $\mathbf z_0,\mathbf z_1 \in \lists X$ be such that $\concat \mathbf z_0$ and $\concat \mathbf z_1$ are not prefix-comparable. Then \[(\concat \mathbf z_0) \wedge_p (\concat \mathbf z_1) = \concat (\mathbf z_0 \wedge_p \mathbf z_1) \cdot (u_0 \wedge u_1).\] \end{lemma} See Appendix \ref{sec:appendix} for more comments on this property. Denote $\alpha_{z,xp} = z \cdot xp \wedge_p xp \cdot z$. Then also $\alpha_{z,xp} = z^k \cdot (xp)^2 \wedge_p (xp)^2 \cdot z^k$. Similarly, let $\alpha_{x,y} = x \cdot y \wedge_p y \cdot x$. Then Lemma \ref{lem:lcp} yields \begin{equation} \begin{aligned} \label{eq:lcp1}
\alpha_{z,xp} &= \concat (\mathbf w \cdot [x,x,y]) \wedge_p \concat ([x,x,y] \cdot \mathbf w) = \\
&= \concat (\mathbf w \cdot [x,x,y] \wedge_p [x,x,y] \cdot \mathbf w) \cdot \alpha_{x,y} \end{aligned} \end{equation} and also \begin{equation}
\begin{aligned}\label{eq:lcp2}
z \cdot \alpha_{z,xp} = &\concat (\mathbf w \cdot \mathbf p \cdot [x,y]) \wedge_p \concat (\mathbf p\cdot[x,y] \cdot \mathbf w) = \\
= &\concat (\mathbf w \cdot \mathbf p \cdot [x,y] \wedge_p \mathbf p\cdot[x,y] \cdot \mathbf w) \cdot \alpha_{x,y}. \end{aligned} \end{equation} Denote \begin{align*}
\mathbf v_1 &= \mathbf w \cdot [x,x,y] \wedge_p [x,x,y] \cdot \mathbf w, & \mathbf v_2 &= \mathbf w \cdot \mathbf p \cdot [x,y] \wedge_p \mathbf p\cdot[x,y] \cdot \mathbf w. \end{align*}
From \eqref{eq:lcp1} and \eqref{eq:lcp2} we now have
$ z \cdot \concat \mathbf v_1 = \concat \mathbf v_2$. Since $\mathbf v_1$ and $\mathbf v_2$ are prefixes of some $\mathbf w^n$, we have
a contradiction with Lemma \ref{lem:disjoint_interp}.
\qed \end{proof}
\subsubsection{Dropping the primitivity assumption.} We first deal with the situation when $x$ and $y$ are not primitive. A natural idea is to consider the primitive roots of $x$ and $y$ instead of $x$ and $y$. This means that we replace the word $\mathbf w$ with $\R \mathbf w$, where $\R$ is the morphism mapping $[x]$ to $[\rho\, x]^{e_x}$ and $[y]$ to $[\rho\, y]^{e_y}$ where $x = (\rho\, x)^{e_x}$ and $y = (\rho\, y)^{e_y}$. For example, if $x = abab$ and $y = aa$, and $\mathbf w = [x,y,x] = [abab,aa,abab]$, then $\R \mathbf w = [ab,ab,a,a,ab,ab]$.
Let us check which hypotheses of Lemma \ref{lem:all_primitive} are satisfied in the new setting, that is, for the code $\{\rho\,x,\rho\,y\}$ and the word $\R \mathbf w$. The following facts are not difficult to see.
\begin{itemize}
\item $\concat \mathbf w = \concat (\R \mathbf w)$;
\item if $[c,c]$, $c\in \{x,y\}$, is a cyclic factor $\mathbf w$, then $[\rho\,c,\rho\,c]$ is a cyclic factor of $\R \mathbf w$.
\end{itemize} The next required property: \begin{itemize}
\item if $\mathbf w$ is primitive, then $\R \mathbf w$ is primitive; \end{itemize}
deserves more attention. It triggered another little theory of our formalization which can be found in locale \verb|sings_code|. Note that it fits well into our context, since the claim is that $\R$ is a primitivity preserving morphism, which implies that its image on the singletons $[x]$ and $[y]$ forms a primitivity preserving set of words, see theorem \verb|code.roots_prim_morph|.
Consequently, the only missing hypothesis preventing the use of Lemma \ref{lem:all_primitive} is $\abs y \leq \abs x$ since it may happen that $\abs {\rho\, x} < \abs{\rho\,y}$. In order to solve this difficulty, we shall ignore for a while the length difference between $x$ and $y$, and obtain the following intermediate lemma.
\begin{lemma}[\texttt{bin\_imprim\_both\_squares}, \texttt{bin\_imprim\_both\_squares\_prim}]\label{lem:two_squares}
Let $B = \{x,y\}$ be a code, and let $\mathbf w \in \lists B$ be a primitive word such that $\concat \mathbf w$ is imprimitive.
Then $\mathbf w$ cannot contain both $[x,x]$ and $[y,y]$ as cyclic factors. \end{lemma} \begin{proof}
Assume that $\mathbf w$ contains both $[x,x]$ and $[y,y]$ as cyclic factors.
Consider the word $\R \mathbf w$ and the code $\{\rho\, x,\rho\, y\}$. Since $\R \mathbf w$ contains both $[\rho\,x,\rho\,x]$ and $[\rho\,y,\rho\,y]$, Lemma \ref{lem:all_primitive}
implies that $\R \mathbf w$ is conjugate either with the word $[\rho\,x,\rho\,x,\rho\,y]$ or with $[\rho\,y,\rho\,y,\rho\,x]$, which is a contradiction with the assumed presence of both squares.
\qed \end{proof}
\subsubsection{Concluding the proof by gluing.}\label{sec:gluing} It remains to deal with the existence of squares. We use an idea that is our main innovation with respect to the proof from \cite{lerest}, and contributes significantly to the reduction of length of the proof, and hopefully also to its increased clarity. Let $\mathbf w$ be a list over a set of words $X$. The idea is to choose one of the words, say $u \in X$, and to concatenate (or ``glue'') blocks of $u$'s to words following them. For example, if $\mathbf w = [u,v,u,u,z,u,z]$, then the resulting list is $[uv,uuz,uz]$. This procedure is in the general case well defined on lists whose last ``letter'' is not the chosen one and it leads to a new alphabet $\{u^i\cdot v \mid v \neq u\}$ which is a code if and only if $X$ is. This idea is used in an elegant proof of the Graph lemma (see \cite{itp2021} and \cite{Berstel1979}). In the binary case, which is of interest here, if $\mathbf w$ in addition does not contain a square of a letter, say $[x,x]$, then the new code $\{x\cdot y, y\}$ is again binary. Moreover, the resulting glued list $\mathbf w'$ has the same concatenation, and it is primitive if (and only if) $\mathbf w$ is. Note that gluing is in this case closely related to the Nielsen transformation $y \mapsto x^{-1}y$ known from the theory of automorphisms of free groups.
Induction on $\abs \mathbf w$ now easily leads to the proof of Theorem \ref{th:main}. \begin{proof}[of Theorem \ref{th:main}]
If $\mathbf w$ contains $y$ at most once, then we are left with the equation $x^j\cdot y = z^\ell$, $\ell \geq 2$. The equality $j = 2$ follows from the Periodicity lemma, see Case 2 in the proof of Theorem \ref{thm:xjykzl_solution}.
Assume for contradiction that $y$ occurs at least twice in $\mathbf w$. Lemma \ref{lem:two_squares} implies that at least one square, $[x,x]$ or $[y,y]$ is missing as a cyclic factor.
Let $\{x',y'\} = \{x,y\}$ be such that, that $[x',x']$ is not a cyclic factor of $\mathbf w$. We can therefore perform the gluing operation, and obtain a new, strictly shorter word $\mathbf w' \in \lists \{x' \cdot y', y'\}$. The longer element $x'\cdot y'$ occurs at least twice in $\mathbf w'$, since the number of its occurrences in $\mathbf w'$ is the same as the number of occurrences of $x'$ in $\mathbf w$, the latter word containing both letters at least twice by assumption.
Moreover, $\mathbf w'$ is primitive, and $\concat \mathbf w' = \concat \mathbf w$ is imprimitive. Therefore, by induction on $\abs \mathbf w$, we have $\mathbf w' \sim [x'\cdot y',x' \cdot y', y']$. In order to show that this is not possible we can successfully reuse the lemma \verb|imprim_ext_suf_comm| mentioned in the proof of Lemma \ref{le:jk_unique}, this time for $u = x'y'x'$ and $v = y'$. The words $u$ and $v$ do not commute because $x'$ and $y'$ do not commute. Since $uv$ is imprimitive, the word $uvv \sim \concat \mathbf w'$ is primitive. \qed \end{proof}
This also completes the proof of our main target, Theorem \ref{th:Theorem1}.
\section{Additional notes on the formalization}
The formalization is a part of an evolving combinatorics on words formalization project. It relies on its backbone session, called CoW, a version of which is also available in the Archive of Formal Proofs \cite{Combinatorics_Words-AFP}. This session covers basics concepts in combinatorics on words including the Periodicity lemma. An overview is available in~\cite{itp2021}.
The evolution of the parent session CoW continued along with the presented results and its latest stable version is available at our repository \cite{CoW_gitlab_v1_6}. The main results are part of another Isabelle session CoW\_Equations, which, as the name suggests, aims at dealing with word equations. We have greatly expanded its elementary theory \verb|Equations_Basic.thy| which provides auxiliary lemmas and definitions related to word equations. Noticeably, it contains the definition \verb|factor_interpretation| (Definition~\ref{def:interpretation}) and related facts.
Two dedicated theories were created: \verb|Binary_Square_Interpretation.thy| and \verb|Binary_Code_Imprimitive.thy|. The first contains lemmas and locales dealing with $\{x,y\}$-interpretation of the square $xx$ (for $\abs y \leq \abs x$), culminating in Theorem~\ref{thm:sq_interp}. The latter contains Theorems~\ref{th:Theorem1}~and~\ref{th:main}.
Another outcome was an expansion of formalized results related to the Lyndon-Sch\"utzenberger theorem. This result, along with many useful corollaries, was already part of the backbone session CoW, and it was newly supplemented with the parametric solution of the equation $x^jy^k = z^\ell$, specifically Theorem~\ref{thm:xjykzl_solution} and Lemma~\ref{le:jk_unique}. This formalization is now part of CoW\_Equations in the theory \verb|Lyndon_Schutzenberger.thy|.
Similarly, the formalization of the main results triggered a substantial expansion of existing support for the idea of gluing as mentioned in Section~\ref{sec:gluing}. Its reworked version is now in a separate theory called \verb|Glued_Codes.thy| (which is part of the session CoW\_Graph\_Lemma).
Let us give a few concrete highlights of the formalization. A very useful tool, which is part of the CoW session, is the \verb|reversed| attribute. The attribute produces a symmetrical fact where the symmetry is induced by the mapping \isakw{rev}, i.e., the mapping which reverses the order of elements in a list. For instance, the fact stating that if $p$ is a prefix of $v$, then $p$ a prefix of $v \cdot w$, is transformed by the reversed attribute into the fact saying that if $s$ is suffix of $v$, then $s$ is a suffix of $w \cdot v$. The attribute relies on ad hoc defined rules which induce the symmetry. In the example, the main reversal rule is
\begin{quote}
\ {\isachardoublequoteopen}{\isacharparenleft}{\kern0pt}rev\ u\ {\isasymle}p\ rev\ v{\isacharparenright}{\kern0pt}\ {\isacharequal}{\kern0pt}\ u{\isasymle}s\ v{\isachardoublequoteclose} \end{quote} \noindent The attribute is used frequently in the present formalization. For instance, Figure \ref{isa:reversed} shows the formalization of the proof of Cases 1 and 2 of Theorem~\ref{le:jk_unique}. Namely, the proof of Case 2 is smoothly deduced from the lemma that deals with Case 1, avoiding writing down the same proof again up to symmetry.
\begin{figure}
\caption{Using the \texttt{reversed} attribute to solve symmetric cases.}
\caption{Methods \texttt{primitivity\_inspection}, \texttt{list\_inspection} and \texttt{mismatch}.}
\caption{Highlights from the formalization in Isabelle/HOL.}
\label{isa:reversed}
\end{figure}
To be able to use this attribute fully in the formalization of main results, it needed to be extended to be able to deal with elements of type \texttt{{\isacharprime}a\ list list}, as the constant \verb|factor_interpretation| is of the function type over this exact type. The new theories of the session CoW\_Equations contain almost 50 uses of this attribute.
The second highlight of the formalization is the use of simple but useful proof methods. The first method, called \verb|primitivity_inspection|, is able to show primitivity or imprimitivity of a given word.
Another method named \verb|list_inspection| is used to deal with claims that consist of straightforward verification of some property for a set of words given by their length and alphabet. For instance, this method painlessly concludes the proof of lemma \verb| bin_imprim_both_squares_prim|. The method divides the goal into eight easy subgoals corresponding to eight possible words. All goals are then discharged by \verb|simp_all|.
The last method we want to mention is \verb|mismatch|. It is designed to prove that two words commute using the property of a binary code mentioned in Section \ref{sec:notation} and explained in the Appendix \ref{sec:appendix}. Namely, if a product of words from $\{x,y\}$ starting with $x$ shares a prefix of length at least $\abs{xy}$ with another product of words from $\{x,y\}$, this time starting with $y$, then $x$ and $y$ commute. Examples of usage of the attribute \texttt{reversed} and all three methods are given in Figure~\ref{isa:reversed}.
\section*{Acknowledgments}
The authors acknowledge support by the Czech Science Foundation grant GA\v CR 20-20621S.
\appendix
\section{Background results in combinatorics on words} \label{sec:appendix}
One of the main advantages of formalization is that it allows to tune the level of detail without compromising precision and correctness. In the present paper we tried to keep the exposition on the level which should make all claims convincing for a researcher in combinatorics on words. This Appendix can help readers who seek more information about the folklore and other well known results.
Most ideas in combinatorics on words are related to periodicity. A word $w$ has a periodic root $r$ if it is a prefix of repeated occurrences of $r$. This can be expressed as $w \leq_p r^\omega$, or equivalently, and using finite words only, as $w \leq_p r \cdot w$, provided $r$ is nonempty. Note that the periodic root $r$ of $w$ need not be primitive, but it is always possible to consider the corresponding primitive root $\rho\, r$, which is also a periodic root of $w$. Note that any word has infinitely many periodic roots since we allow $r$ to be longer than $w$. Nevertheless, a word can have more than one period even if we consider only periods shorter than $|w|$. Such a possibility is controlled by the Periodicity lemma, often called the Theorem of Fine and Wilf (see \cite{FW1965}):
\begin{lemma}[\texttt{per\_lemma\_comm}] \label{perlem}
If $w$ has a period $u$ and $v$, i.e., $w \leq_p uw$ and $w \leq_p vw$, with $|u|+|v| - \gcd(|u|,|v|) \leq |w|$, then $uv = vu$. \end{lemma}
Usually, the weaker test $|u| + |v| \leq |w| $ is sufficient to indicate that $u$ and $v$ commute.
Conjugation $u \sim v$ of two words is defined by the existence of two words $r$ and $q$ such that $u = rq$ and $v = qr$. Two words are conjugate if they are the same up to rotation, therefore the conjugacy class is naturally seen as a word understood cyclically. A word $z$ is then called a ``cyclic factor'' of $u$ if it is a factor of some conjugate of $u$, which is equivalent to being a factor of the square of $u\cdot u$ if $|z| \leq |u|$. Importantly, conjugation $u \sim v$ is also characterized as follows: \begin{lemma}[\texttt{conjugation}]\label{conjugation}
If $uz = zv$ for nonempty $u$, then there exists words $r$ and $q$ and an integer $k$ such that
\[
u = rq, \quad v = qr \quad \text{ and } \quad z = (rq)^kr.
\] \end{lemma}
We have said that $w$ has a periodic root $r$ if it is a prefix of $r^\omega$. If $w$ is a factor, not necessarily a prefix, of $r^\omega$, then it has a periodic root which is a conjugate of $r$. In particular, if $\abs u = \abs v$, then $u \sim v$ is equivalent to $u$ and $v$ being mutually factors of a power of the other word.
Commutation of two words, that is, equality $u \cdot v = v \cdot u$, is characterized as follows: \begin{lemma}[\texttt{comm}]\label{commutation}
If $xy = yx$ if and only if $x = t^k$ and $y = t^m$ for some word $t$ and some integers $k,m \geq 0$. \end{lemma} Since every nonempty word has a (unique) primitive root, the word $t$ above can be chosen primitive ($k$ or $m$ can be chosen $0$ if $x$ or $y$ is empty).
We mention that the given characterizations of conjugation and commutation (in a slightly expanded form) are called ``the first theorem of Lyndon and Sch\"utzenberger'' and ``the second theorem of Lyndon and Sch\"utzenberger'' respectively in \cite{AlloucheShallit}, while we reserve the term ``the theorem of Lyndon and Sch\"u\-tzen\-ber\-ger'' to the following fact often used in the paper: \begin{theorem}[\texttt{Lyndon\_Schutzenberger}] \label{thm:LS}
If $x^jy^k = z^\ell$ with $j \geq 2$, $k \geq 2$ and $\ell \geq 2$, then the words $x$, $y$ and $z$ commute. \end{theorem}
A crucial property of a primitive word $t$ is that it cannot be a nontrivial factor of its own square. For a general word $u$, the equality $u\cdot u = p \cdot u \cdot s$ with nonempty $p$ and $s$ implies that all three words $p$, $s$, $u$ commute, that is, have a common primitive root $t$. This can be seen by writing $u = t^k$, and noticing that the presence of a nontrivial factor $u$ inside $uu$ can be obtained exclusively by a shift by several $t$'s. This is the ``synchronization'' idea mentioned in the paper.
One of the typical applications of this idea is the fact that if $w$ has a periodic root $r$, and at the same time $r$ is a suffix of $w$, then $w$ and $r$ commute. Another application is the proof of the fact that $qrqr$ cannot be a factor of a power of $qrrq$ unless $q$ and $r$ commute, used in the proof of Lemma \ref{le:jk_unique}. This can be seen as follows. If $qrqr$ is a factor of $(qrrq)^\omega$, then $qrrq \sim qrqr$ since the words are of the same length. This means that also $qrrq$ is a factor of $(qr)^\omega$. Synchronizing the prefix $qr$ of $qrrq$ within $(qr)^\omega$ we obtain that the rest of the word, the word $rq$, is a prefix of $qr$, and the conclusion follows.
Let $x$ and $y$ be two words that do not commute. The longest common prefix of $xy$ and $yx$ is denoted $\alpha$. Let $c_x$ and $c_y$ be the letter following $\alpha$ in $xy$ and $yx$ respectively. A crucial property of $\alpha$ is that it is a prefix of any sufficiently long word in $\gen{\{x,y\}}$. Moreover, if $\mathbf w = [u_1,u_2,\ldots,u_n] \in \lists \{x,y\}$ is such that $\concat \mathbf w$ is longer than $\alpha$, then $\alpha\cdot [c_x]$ is a prefix of $\concat \mathbf w$ if $u_1 = x$ and
$\alpha\cdot [c_y]$ is a prefix of $\concat \mathbf w$ if $u_1 = y$. That is why the length of $\alpha$ is sometimes called ``the decoding delay'' of the binary code $\{x,y\}$. Note that the property indeed in particular implies that $\{x,y\}$ is a code, that is, it does not satisfy any nontrivial relation. It is also behind our method \verb|mismatch|. Finally, using this property, the proof of Lemma \ref{lem:lcp} is straightforward.
\end{document}
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\begin{document}
\normalem
\title{Control limitations from distributed sensing:\ theory and Extremely Large Telescope application}
\begin{abstract} We investigate performance bounds for feedback control of distributed plants where the controller can be centralized (i.e.~it has access to measurements from the whole plant), but sensors only measure differences between neighboring subsystem outputs. Such ``distributed sensing'' can be a technological necessity in applications where system size exceeds accuracy requirements by many orders of magnitude. We formulate how distributed sensing generally limits feedback performance robust to measurement noise and to model uncertainty, without assuming any controller restrictions (among others, no ``distributed control'' restriction). A major practical consequence is the necessity to cut down integral action on some modes. We particularize the results to spatially invariant systems and finally illustrate implications of our developments for stabilizing the segmented primary mirror of the European Extremely Large Telescope. \end{abstract}
\noindent {\footnotesize Keywords:\newline distributed detection, performance limitations, robustness, integral control, telescopes, distributed control.}
\section{Introduction}\label{sec:Intro}
The massive availability of sensors and actuators in our environment is an opportunity to address large-scale problems by exploiting their interaction. The control of interacting localized subsystems (distributed plants) has drawn tremendous interest in the last decades, covering e.g.~agreement (consensus) in computer networks \cite{Tsitsiklis93}, synchronization of dynamical systems \cite{SujitSum,Synch} or collective robotic task solving \cite{BulloBook}. A defining property is the information sharing between subsystems --- information content, and interconnection topology. Common distinctions are reference-following \cite{BeardOnSats} vs.~autonomous coordination \cite{Synch} and centralized vs.~distributed control~\cite{B2Ptac11,Dahleh-Bamieh,Castro2002,Gorinevsky2008,Langbort2005,Stewart2003}. In centralized control, each local action is a function of measurements all over the system. The distributed control paradigm \cite{Dahleh-Bamieh,Castro2002,Langbort2005,Stewart2003,Gorinevsky2008} imposes a localized coupling in closed-loop: each local action depends on neighboring subsystem outputs only.
This paper shows how, ahead of the controller choice, structural restrictions on the \emph{sensing architecture} of a distributed plant can fundamentally constrain the performance of feedback. Specifically, we consider systems which only sense \emph{differences between neighboring subsystem outputs}. Unlike in distributed control, we allow the resulting measurements to be used in any --- in particular, centralized --- control computation. We call this ``local relative sensing'' or ``distributed sensing''. It is motivated by applications in which communication capabilities allow to quickly broadcast all measurements and control signals --- questioning a priori restrictions on controller structure --- but sensor technology does not allow accurate enough absolute measurements over the entire plant. This occurs in multi-scale problems, where accuracy requirements and plant size differ by many orders of magnitude.
The setting is inspired by our study of primary mirror stabilization for the European Extremely Large Telescope (EELT)~\cite{EELTreport}. We therefore propose indicative analytical results --- for a general case and for 1-degree-of-freedom spatially invariant systems --- followed by an illustration on this case study. We focus on two concerns. First, how distributed sensing influences the sensor noise vs.~disturbance rejection tradeoff, using the sensitivity transfer functions of classical linear control theory~\cite{Astrom2000,Astrom-Murray}. Second, how measurement model errors affect robustness. A major concrete consequence is the necessity to cut down integral action on some modes.
Effects of noise and perturbations in distributed systems are examined in various ways in the literature. The authors of~\cite{B2Ptac11,Barooah1,Barooah2} restrict not only sensing, but also control, to local relative coupling (this is \emph{distributed control}). The authors of \cite{BarHesp1} study, in a static setting, bounds on the reconstruction of absolute position with respect to a leader from noisy local relative measurements. Optimal controllers for \emph{spatially invariant} plants are investigated in \cite{Dahleh-Bamieh}; for a locally coupled plant, the optimal gains decay exponentially as a function of distance between actuated and measured subsystems. This supports the use of distributed control, for which \cite{B2Ptac11} investigates performance limitations on a benchmark spatially invariant system. The robustness issue, that we first raised in \cite{EELTreport}, has been observed numerically with $\mu$-analysis for the segmented mirror application \cite{MacMyn2009}. Segmented mirror stabilization has been investigated by a few teams associated to Extremely Large Telescope projects \cite{SXJ2009,MacMartin2003,MacMyn2009}. In \cite{SXJ2009} the distributed sensing issue is put aside by assuming absolute measurements.
The paper is organized as follows. Section~\ref{sec:TheorA} formalizes distributed sensing and gives two motivating examples: a benchmark vehicle-chain problem and segmented mirror stabilization. Section~\ref{sec:TheorB} formulates how sensor noise (\ref{ssec:Noise}) and model errors (\ref{ssec:Robust}) induce performance limitations. Section~\ref{sec:TheorD} particularizes to 1-degree-of-freedom spatially invariant systems. Section~\ref{sec:EELT} illustrates our point on EELT primary mirror stabilization.
\noindent \textbf{Notation:} We write $i=\sqrt{-1}$ the imaginary unit. The element on row $j$, column $k$ of matrix $C \in \mathbb{C}^{l \times m}$ is denoted $(C)_{j,k}$. $C^T$ and $C^*$ respectively denote transpose and complex conjugate transpose of $C$, and $\otimes$ the Kronecker product of two matrices. We denote $c \in \mathbb{C}^l$ a column vector, $\Vert c \Vert = \sqrt{ \sum_k \vert c_k \vert^2 }$ its Euclidean norm. $I_m \in \mathbb{R}^{m \times m}$ is the identity matrix and $\mathbf{1}_m \in \mathbb{R}^m$ the column-vector of all ones. We interpret $s+C = s\, I_m + C$ if $s \in \mathbb{C}$ and $C \in \mathbb{C}^{m \times m}$. For $D \in \mathbb{C}^{m \times m}$ diagonal and $f$ a scalar function, $Y=f(D) \in \mathbb{C}^{m \times m}$ is diagonal with $(Y)_{k,k} = f((D)_{k,k})$ for all $k$.
\section{Distributed sensing models}\label{sec:TheorA}
\begin{figure}
\caption{Schematic representation of distributed sensing}
\label{fig:modelA}
\end{figure}
We consider a Laplace-domain model (see Fig.~\ref{fig:modelA}) \begin{eqnarray} \label{eq:mod1} y(s) & = & G(s) \; [\,u(s) + d(s)\,]\\ \label{eq:mod2} z(s) & = & [\, B + \Delta \,] \; y(s) + n(s)\\ \label{eq:mod3} u(s) & = & -C(s) \; z(s) \end{eqnarray} to represent $M \gg 1$ coupled $N$-dimensional subsystems. Components $kN$+$1$ to $(k$+$1)N$ of $y(s), u(s), d(s) \in \mathbb{C}^{N_y}$ denote outputs, inputs and disturbances of subsystem $k$ in the Laplace domain, with $N_y$ = $NM$. We assume that the plant governed by $G(s)$ is stable. Output $z(s) \in \mathbb{C}^{N_z}$ is obtained through the static map $[B+\Delta] \in \mathbb{R}^{N_z \times N_y}$, where $B$ is the nominal sensor behavior and $\Delta$ a sensor model error. Each sensor measurement is corrupted by zero-mean independent identical Gaussian white noise, represented by $n(s)$ with covariance matrix $\sigma^2\, I_{N_z}$. For ease of presentation we assume $N_z \geq N_y$. The purpose of controller $C(s) \in \mathbb{C}^{N_y\times N_z}$ is to reject disturbances $d(s)$ from $y$. Importantly, we do not restrict the controller (\ref{eq:mod3}) to be distributed, i.e.~we allow $C(s)$ to be a full matrix. We also allow the disturbances on different subsystems to be correlated, by investigating how a general vector $d(s)$ affects the controlled plant. This differs from e.g.~\cite{B2Ptac11,Barooah2} which examine $y$ for a given disturbance distribution (and controller).
The central element of our investigation is \emph{local relative measurement}. Let $q_k = \{kN$+$1, kN$+$2,...,(k$+$1)N\}$.
\newline \textbf{Definition 1:} $\;B$ gives (unit-gain) \emph{relative measurements} between subsystem outputs if for each $l \in \{1,...,N_z\}$ there exist $q_j,q_k$ such that
\begin{eqnarray} \nonumber 1. && (B)_{l,m} = 0 \;\;\; \text{for } m \notin q_j \cup q_k \\ \nonumber 2. && {\textstyle \sum_{m \in q_j} \;\;} \vert(B)_{l,m}\vert \;=\; {\textstyle \sum_{m \in q_j} \;\;} (B)_{l,m} \\ \nonumber && =\; {\textstyle \sum_{m \in q_k} \;\;} \vert(B)_{l,m}\vert \;=\; -{\textstyle \sum_{m \in q_k} \;\;} (B)_{l,m} \; =\; 1 \; . \end{eqnarray} That means, each row $l$ of $B$ measures the difference between a convex combination of outputs of subsystem $j$ and a convex combination of outputs of subsystem $k$. For $N=1$, $B^T$ would be the \emph{oriented incidence matrix} of some graph $\Gamma_B$, where subsystems are nodes and sensors are edges; for $N>1$, $B^T$ has the interpretation of a generalized incidence matrix, with matrix-valued weights on each edge~\cite{BarHesp1}. The (generalized) Laplacian matrix of $\Gamma_B$ is $L=B^T B$. If $(L)_{l,m} \neq 0$ for some $l \in q_j$ and $m \in q_k$, then subsystems $j$ and $k$ are connected in $\Gamma_B$.
\newline \textbf{Definition 2:} A spatial structure $\mathcal{S}$ of dimension $\gamma \in \mathbb{N}$ associates a position $p(k) \in \mathbb{R}^{\gamma}$ to each subsystem $k$ such that $\Vert p(k)-p(l) \Vert \geq 1$ for $l \neq k$.
\newline \textbf{Definition 3:} Given a spatial structure and a fixed spatial range $\rho \geq 1$, measurement map $B$ gives \emph{local relative measurements of range $\rho$} if it gives relative measurements and it only connects in $\Gamma_B$ subsystems for which $\Vert p(k)-p(l) \Vert \leq \rho$. We call this \emph{distributed sensing}.\newline Many decentralized control settings associate a local measurement to each subsystem (graph node). With distributed sensing in contrast, \emph{measurements are the result of interactions between subsystems (graph edges).}
\textbf{Remark 1:} Local sensing has no meaning if it is not relative. Sensors giving ``absolute'' e.g.~positions actually physically measure positions with respect to a common (``central'') reference physically shared among all sensors. Absolute measurements thus correspond to centralized sensing. This is also acknowledged in the robotics community, distinguishing local$\cong$onboard from global$\cong$offboard sensors, see e.g.~\cite[Chapter 3]{BraunlBook}.
We study disturbance rejection limitations due to distributed sensing with $\tfrac{\rho}{M} \ll 1$. In the following two applications this arises as $M$ increases with the size of a large-scale plant while $\rho$ is limited by sensor technology.
\subsection{Vehicle chain application}\label{ssec:Cars}
A basic objective of e.g.~automated highway driving is to maintain constant inter-vehicle distance in a chain~\cite{SwaroopThesis}. Centralized sensing, typically the Global Positioning System (GPS), determines vehicle positions with respect to a common reference. GPS accuracy --- limited by atmospheric effects to a few meters --- is remarkable on the global scale, but likely insufficient to avoid collisions on a crowded highway. In local relative sensing, each vehicle directly measures the distance to its neighboring vehicles, easily up to centimeter-accuracy.
Define vehicle configuration $(y)_k = s_k - r_k \in \mathbb{R}$ with $s_k$ the coordinate of vehicle $k$ along the road and $r_k$ its desired coordinate, typically $r_k = k\, r_d$ in a moving frame with $r_d$ the desired distance between vehicle centers. Vehicles are controlled ($u$) and disturbed ($d$) by forces. Local relative sensors compare relative positions of consecutive vehicles, $(z)_k = s_{k+1}-s_k-r_d = (y)_{k+1} - (y)_k$, so $(B)_{k,k+1}=-(B)_{k,k}=1$ $\forall k$ and all other $(B)_{k,j}=0$ (path interconnection). This topology follows from a spatial structure $p(k) = k \in \mathbb{R}$ with $1<\rho<2$.
This benchmark problem has been studied before. String stability~\cite{SwaroopThesis} restricts its attention to a perturbation on the first vehicle and studies how it propagates along the chain, in absence of noise. \cite{B2Ptac11} imposes distributed \emph{control}, i.e.~each vehicle relies on local sensors only. Instead, we allow each vehicle to use information gathered by all sensors. By leaving the controller free, we also allow antisymmetric coupling, which is termed ``mistuned control'' in \cite{Barooah1} and improves disturbance rejection.
\subsection{Segmented mirror application}\label{ssec:Telescope}
\begin{figure}
\caption{The EELT primary mirror made of 984 segments.}
\label{fig:M1}
\end{figure} \begin{figure}
\caption{Left: actuation and sensing architecture. Position actuators (PACT) operate the segment's piston, tip and tilt degrees of freedom. Edge sensors (ES) measure local relative displacement of the segments. Right: points $(\mathsf{h}_{k1},\mathsf{h}_{k2},\mathsf{h}_{k3})$ used to define segment configuration.}
\label{fig:segment}
\end{figure}
This paper has been inspired by our involvement \cite{EELTreport,EELTlaunch} in designing a controller to stabilize the segmented primary mirror of the European Extremely Large Telescope (EELT, see~\cite{Gilmozzi2007,ESOweb}). This telescope, run by the European Organization for Astronomical Research in the Southern Hemisphere (ESO), will offer unprecedented optical observation capabilities thanks to its primary mirror (M1) of world-record $42\un{m}$ diameter~\cite{Gilmozzi2007,ESOweb}. Its construction has started in 2012.
The scientific objective requires the light wave reflected by M1 to differ from an ideal one by less than $10$~nm root-mean-squared, after a linear filter that models adaptive optics corrections elsewhere in the telescope\footnote{See \cite{EELTreport,EELTlaunch} for more about this ``wavefront error''.}. One-piece mirrors meeting this accuracy under disturbances are currently limited to $\sim 9 \un{m}$ diameters. M1 is therefore composed of $M=984$ hexagonal segments of $0.7\un{m}$ edge length, see Fig.~\ref{fig:M1}, which are actively stabilized. The huge scale factor between the $10$~nm accuracy requirement and the $42$~m mirror size makes it technologically impossible to rely on segment displacement measurements with respect to a common reference. $N_z=5604$ ``edge sensors'' (ES) therefore measure the relative displacement, perpendicular to mirror surface, of adjacent segment edges (see Fig.~\ref{fig:segment}).
Disregarding in-plane motions of the segments (which have no optical effect), we consider $N=3$ degrees of freedom corresponding to piston, tip and tilt (PTT) of each segment i.e.~for a telescope pointing to zenith: vertical position and rotation around two horizontal axes; so $y \in \mathbb{R}^{2952}$. We write the model after a coordinate change from the nominal mirror shape to a horizontal reference plane. Denoting $\pi_k$ the plane in $\mathbb{R}^3$ which contains mirror segment $k$, define $(y_{3k-2},y_{3k-1},y_{3k})$ the height w.r.t.~reference plane of 3 points $\mathsf{h}_{k1},\mathsf{h}_{k2},\mathsf{h}_{k3} \in \pi_k$ positioned as depicted on Fig.~\ref{fig:segment}, right. Then the height of each point on segment $k$ is a convex combination of $(y_{3k-2},y_{3k-1},y_{3k})$. Each edge sensor measures the difference between two such convex combinations and thus fits Definition 1 of relative sensing. The spatial structure can be defined by $p(k)$= position of segment $k$'s center in the hexagonal lattice, and $0.7\sqrt{3}\text{m}< \rho < 0.7\cdot 3\text{m}$.
To control its 3 degrees of freedom, each segment is supported by 3 position actuators (PACT) which move perpendicularly to the mirror surface and whose force commands make up $u \in \mathbb{R}^{2952}$. The small displacements allow for a linear model $G(s)$, discussed in Section \ref{sec:EELT}. Wind force is the strongest varying part in $d$, with characteristic frequencies below a few $\tfrac{1}{10}$Hz, and is spatially correlated among the segments. A second main component of $d$ are quasi-static disturbances: thermal effects and gravity on the moving mirror induce deformations of very low temporal frequency but high amplitude ($\simeq 1\un{mm}$). Noise $n$ is modeled as white Gaussian with $1 \un{nm}/\sqrt{\text{Hz}}$ power distribution for each sensor.
\subsection{Geometrical considerations on distributed sensing}\label{ssec:Interpretation}
Relative sensing reflects signal space invariance: sensors are insensitive to a common deviation of all system outputs since $B \, y = B \, ( y + \alpha \mathbf{1}_{N_y})$, for any $\alpha \in \mathbb{R}$. This {\it signal space} invariance \cite{Sarlette2009a}, involving $y$, should not be confused with the {\it spatial} invariance of \cite{B2Ptac11,Dahleh-Bamieh}, involving translations along spatial index $k$.
\newline\emph{Local} relative sensing can be viewed as measuring a spatial derivative: $\frac{y_k - y_l}{\Vert p(k)-p(l) \Vert}$ is the standard Euler discretization of the derivative of $y$ in direction $p(k)-p(l)$, evaluated at $\tfrac{p(k)+p(l)}{2}$. The approximation holds for $y$ varying on characteristic spatial scales much larger than $\Vert p(k)-p(l) \Vert$. For large $M$, an analogy with PDEs can therefore give insight for feedback design, see e.g.~\cite{Barooah1,Barooah2,Sarlette2009a}, although a rigorous link is tricky to establish~\cite{CurtainCrit}. The strong sensitivity to particular model errors $\Delta$ which we highlight in Section~\ref{ssec:Robust}, is analog to the drastic changes in PDE properties under small perturbations that change the dominant spatial derivatives.
\section{Distributed sensing limits performance}\label{sec:TheorB}
\subsection{Sensitivity and spectral graph properties}\label{ssec:Noise}
Like in classical control theory, the sensitivity and the complementary sensitivity are key transfer functions to capture the performance limitations of the feedback system. We take $\Delta=0$ but $n \neq 0$ in (\ref{eq:mod2}). Since $B\, B^T \, B$ has the same range and kernel as $B$ and there is no incentive to assign any controller gain to pure noise, we can write $C(s) = C_a(s) \, B B^T =: C_b(s) \, B^T$ in (\ref{eq:mod3}) such that the closed-loop system (\ref{eq:mod1})-(\ref{eq:mod3}) becomes ($s$ dropped) \begin{equation}\label{eq:Ranged} y = [\,I_{N_y}+G\,C_b\,L\,]^{-1} \; [\,G\,d \, + \, G\,C_b\,B^T\,n \,]
\end{equation} where $L=B^T B$. Write eigendecomposition $L = Q\,\Lambda\,Q^T$ with $Q$ orthogonal and eigenvalues $\lambda_k := (\Lambda)_{k,k}$ ordered as $\lambda_1 \geq \lambda_2 \geq ... \geq \lambda_{N_y} \geq 0$. Defining $\modal{y} = Q^T y$ and $K_b(s) = Q^T\, G(s) \, C_b(s) \, Q$ in (\ref{eq:Ranged}) we get $\modal{y} = [\,I_{N_y}+K_b\,\Lambda\,]^{-1} \; [\,Q^T\,G\, d + K_b\,Q^T\,B^T\,n \,]$.
Since we impose no restriction on the controller, we can view $K_b(s)$ as a freely tunable matrix transfer function\footnote{Issues like pole-zero cancellation and controller realizability, not specific to distributed sensing, are thereby ignored. They reduce to their SISO counterpart if $Q^T G(s) Q$ is diagonal in some ($s$-independent) basis, which occurs e.g.~when $G(s)$ is a scalar multiple of the identity matrix.}. Moreover, disregarding (temporarily) the singular modes of $\Lambda$, view $K(s)=K_b(s) \Lambda \in \mathbb{C}^{N_y \times N_y}$ as a tunable loop transfer function. Then (\ref{eq:Ranged}) becomes \begin{equation}\label{eq:Modes}
\modal{y}(s) = [\,I_{N_y}+K(s) \,]^{-1} \, \delta(s) + [\,I_{N_y}+K(s) \,]^{-1} K(s) \,\nu(s) \end{equation} where $\nu(s) := \Lambda^{-1}\,Q^T\,B^T\,n(s) \in \mathbb{C}^{N_y}$ is rescaled noise and $\delta(s) := Q^T\,G(s)\, d(s)$ is the open-loop deviation of $\modal{y}(s)$ induced by $d(s)$.
In the single-input / single-output case, expression (\ref{eq:Modes}) formulates the classical tradeoff between sensitivity $S = \frac{1}{1+K(s)}$ and complementary sensitivity $T = \frac{K(s)}{1+K(s)} = 1-S$. The specificity of distributed sensing is how $\nu$ relates to $n$, namely how the spectrum of the Laplacian $L$ scales the effect of measurement noise.
\newline
\textbf{Lemma 1:} \emph{If $\,n(s)$ is distributed as a zero-mean Gaussian of covariance $\,\sigma^2\, I_{N_z}$, then $\nu(s)$ is distributed as a zero-mean Gaussian of covariance $\sigma^2\, \Lambda^{-1}$.}
\newline \textbf{Proof:} Write the singular value decomposition $B = U \Sigma V^T$ where $U \in \mathbb{R}^{N_z \times N_y}$ has orthonormal columns, $V \in \mathbb{R}^{N_y\times N_y}$ is orthogonal and $\Sigma \in \mathbb{R}^{N_y \times N_y}$ diagonal. $L=B^TB$ allows to identify $V=Q$ and $\Sigma = \Lambda^{1/2}$. Thus $\nu = \Lambda^{-1}\,Q^T Q \Sigma U^T\, n = \Lambda^{-1/2}\,U^T\,n$. As $U^T n$ is an orthonormal projection of $n$ on some subspace, it has a zero-mean Gaussian distribution with covariance matrix $\sigma^2\, I_{N_y}$. Gain $1/\sqrt{\lambda_k}$ on $(U^T n)_{k}$ multiplies its $\sigma^2$-variance by $1/\lambda_k$.
$\square$
Unobservable modes, for which $\lambda_k = 0$, appear with infinite noise gain in (\ref{eq:Modes}) unless they have zero gain in $K(s)$ --- the only actual possibility. As $L\, \mathbf{1}_{N_y} = 0$, there is at least one unobservable mode. The following proposition shows how distributed sensing implies that, for $M \gg 1$, many other modes $k$ have $\lambda_k \ll 1$.
\newline
\textbf{Proposition 1:} \emph{Consider distributed sensing with a given spatial structure of dimension $\gamma$, range $\rho \ge 1$, and let $D_m$ the maximal number of sensors connected to a given subsystem. Then for any (small) $c,\,N_c > 0$, there exists a (large enough) number of subsystems $M$ such that: $L=B^T B$ has at least $N_c$ eigenvalues $\lambda_k$ smaller than $c$, for any model of distributed sensing among $M$ subsystems according to $\rho$, $\gamma$, and $D_m$.}
\newline \textbf{Proof:} A few computations on eigenvalue bounds yield: if there are $N_c$ orthonormal vectors $y_i \in \mathbb{R}^{N_y}$ for which $\Vert B \, y_i \Vert \leq b = \tfrac{c}{N_c}$, then $L=B^T B$ has at least $N_c$ eigenvalues smaller than $c$. We explicitly build such $y_i$.\newline Given $N_c$, bounds on $D_m$, $\rho$ and $\gamma$ for distributed sensing, and any $\beta > 0$, the spatial structure ensures that for sufficiently large $N_y$, any compatible distributed sensing system can be partitioned into $N_c$ groups of subsystems such that:
\begin{itemize} \item[(i)] each group contains $N_y/N_c$ outputs and \item[(ii)] the maximum number of sensors connecting a subsystem of group $j$ with a subsystem of group $k \neq j$, over all group pairs $(j,k)$, is bounded by $\beta \, N_y$. \end{itemize}
We build the $y_i$ by assigning the same value $\in \{\, \tfrac{+1}{\sqrt{N_y}},\, \tfrac{-1}{\sqrt{N_y}} \,\}$ to all the output components of a same group; see Fig.~\ref{fig:NewFig} for a schematic illustration. Without loss of generality, assume $N_c$ to be a power of $2$. Then $N_c$ orthogonal such $y_i$ can be built by associating $+$ or $-$ signs to the groups according to the elements of a Walsh code, as used in CDMA communication (see e.g.~\cite{CDMAex}). This code includes the vector $y_0$ where the output is a multiple of $\mathbf{1}_{N_y}$ (unobservable, $\Vert L \, y_0 \Vert = 0$). For the other $y_i$, it assigns $-$ signs to precisely $N_c/2$ groups. Then from (ii), $B\, y_i$ has at most $\beta N_y \, \tfrac{N_c(N_c-1)}{4}$ nonzero components, of value $\pm 2/\sqrt{N_y}$, such that $\Vert B \, y_i \Vert \leq \sqrt{\beta N_c (N_c-1)}$. Thus taking the initial $\beta \leq c^2 / (N_c^3 \, (N_c-1))$ yields the result.
$\square$
\begin{figure}
\caption{Poorly observable configuration $y_i$ constructed for the proof of Proposition 1, in a 1D context with $N_y=20$, $N_c=4$, $\beta = 1/20$, assuming the sensing range extends to direct neighbors only. Each bar represents $(y_i)_k$ for a given $k$, with $y$ proportional to vertical position and subsystem index $k$ varying along the horizontal direction; $(z)_5$, $(z)_{10}$ and $(z)_{15}$ are the only measurements on the figure that differ from $0$.}
\label{fig:NewFig}
\end{figure}
The bound $D_m$ avoids that an arbitrarily large number of sensors can be used to perform the same measurement; this would indeed statistically improve the signal-to-noise ratio in a way that is not practically meaningful. The proof constructs poorly observable configurations by concerted deviations of large parts of the distributed system (e.g.~large subareas of the telescope mirror) inducing discontinuities that affect only a few sensors, see Fig.~\ref{fig:NewFig}. These are not the only ``poorly observable'' deformations. For settings with more structure, like in \cite{BarHesp1} or Section \ref{sec:TheorD}, one constructs even less observable deformations that have a different interpretation. Prop.1 focuses on worst-case modes, in contrast to \cite{B2Ptac11,Barooah2} which consider a mean over all modes. This is particularly relevant if, like in the EELT application, unfavorable $\delta$ are dominant. Indeed, for this application, a major disturbance source is wind force, which naturally features correlations on long spatial scales.
\subsection{Robustness to sensing model errors}\label{ssec:Robust}
We consider measurement model uncertainty $\Delta$ with \begin{equation}\label{eqRob:Unc}
(\Delta)_{k,l} \in [\,-\varepsilon\, (B)_{k,l} \,,\, +\varepsilon\, (B)_{k,l} \,] \quad \text{for all } k,l \end{equation} unknown, i.e.~each component of $B$ is (independently) subject to a relative uncertainty $\varepsilon$, for some given $\varepsilon \ll 1$. This uncertainty does not change the interconnection graph $\Gamma_B$.
Repeating the development of Section \ref{ssec:Noise} with $n=0$ and $\Delta \neq 0$ yields, similarly to \eqref{eq:Modes}, \begin{equation}\label{eqRob:Modes}
\modal{y}(s) = [\,I_{N_y}+K(s) \,[I_{N_y}+\Phi] \,]^{-1} \, \delta(s) \end{equation} where $\Phi = \Lambda^{-1}\, Q^T \, B^T \,\Delta\, Q = \Lambda^{-1/2}\, U^T \Delta Q$, with singular value decomposition $B = U \Lambda^{1/2} Q^T$ as in Lemma 1. Note that in general $\Lambda$ is singular and the correct result is obtained by taking its pseudo-inverse in the definition of $\Phi$, i.e.~treating the lines $N_0+1$ to $N_y$ of $\Phi$ equal to zero, where $N_0$ is the index of the last nonzero eigenvalue of $\Lambda$. (By construction the columns $N_0+1$ to $N_y$ of $K$ equal zero.) For robust stability, the zeros of $\text{det}(\, I_{N_y}+K(s) \,[I_{N_y}+\Phi] \,)$ must have negative real part.
{\asedit We next analyze which properties on $\Phi$ are bad for stability. By standard matrix properties, $\text{det} (\, I_{N_y}+K(s) \,[I_{N_y}+\Phi] \,) = \text{det}(\, I_{N_0}+\bar{K}(s) \,[I_{N_0}+\bar{\Phi}] \,)$, where $\bar{K}$ (resp.$\bar{\Phi}$) contains the $N_0$ first columns and rows of $K$ (resp.$\Phi$). If further $K(s)$ is stable, we have for $s$ with positive real part: $\;\text{det}(I_{N_0}+\bar{K}(s)[I_{N_0}+\bar{\Phi}]) =0 \;$ $\Leftrightarrow$ $\; \;$ $\;\text{det}(\bar{K}(s)^{-1} + I_{N_0} +\bar{\Phi}) = 0 \;$. Consider e.g.~the hypothetical case of a disturbance $\Phi=\Phi_b$ defined by \begin{equation}\label{eq:ApproxPhi}
(\Phi_b)_{k,l} = \; \left\{\begin{array}{ll} 0 & \text{for } (k,l) \neq (b,b) \\ \phi_b < 0 & \text{for } (k,l) = (b,b) \end{array} \right. \; . \end{equation} We will approach this situation with an explicit construction later. Then we get:}
\newline
\textbf{Proposition 2:} \emph{If there exists a real $s>0$ for which $\bar{K}(s)^{-1}$ exists and has entries bounded by $\varepsilon_b < 1/N_0$, and a sensing model error $\Delta$ can lead to a disturbance $\Phi = \Phi_b$ as in \eqref{eq:ApproxPhi} with $|\phi_b| > 1+N_0\,\varepsilon_b$, then the system is not robustly stable.}
\newline \textbf{Proof:} {\asedit With $\Phi = \beta \Phi_b$ and $\beta$ varying from $0$ to $1$, the Gershgorin disks containing the eigenvalues of $\bar{K}(s)^{-1} + I_{N_0} +\bar{\Phi}$ all remain within the positive right-half plane, except for the row/column $b$ which moves from fully within positive to fully within negative right-half plane. By continuity, the corresponding eigenvalue must pass through zero, i.e.~$\text{det}(\, I_{N_y}+K(s) \,[I_{N_y}+\Phi_\beta] \,)$ has a zero with positive real $s$.}
$\square$
More precise results can be given if reasonable structures are assumed for $K(s)$. For instance, if $K(s)$ is diagonal and positive for real positive $s$ (modal negative feedback) then robust stability requires $(\bar{K}(s))_{b,b} \;<\; 1\,/\,\vert 1 + \phi_b\vert$ for all real positive $s$.
{\asedit The point is that, given a nominal model with distributed sensing, a $\Phi_b$ with significant value of $\phi_b$ can be constructed from very small uncertainties in $\Delta$.} Indeed, the particular error $\forall k,l$\, : \begin{equation}\label{eq:ZeMethod1}
(\Delta)_{k,l} = -\text{sign}[(U)_{k,b}] \cdot \text{sign}[(Q)_{l,b}] \cdot \varepsilon\,\vert(B)_{k,l}\vert \end{equation} yields a maximally negative $(\Phi)_{b,b} = - \tfrac{\varepsilon}{\sqrt{\lambda_b}}\, \sum_{k,l}\; \vert (B)_{k,l}\vert \cdot \vert (U)_{k,b} \vert \cdot \vert (Q)_{l,b} \vert \, =:\phi_b$. {\asedit Here small values of $\lambda_b$ can lead to a strongly amplified model error, in particular a $\vert \phi_b \vert >1$ despite retaining $\varepsilon$-small uncertainty on each component of $\Delta$.} Numerical investigations on practical examples show that often with (adaptions of) \eqref{eq:ZeMethod1}, for some of the largest $b\leq N_0$ we can obtain: $(\Phi)_{b,b}$ largely dominates all the other elements of $\Phi$, so that $\Phi \approx \Phi_b$ given by \eqref{eq:ApproxPhi} for our purposes. {\asedit This shows that distributed sensing makes us dangerously close to the conditions of Proposition 2.
}
\newline {\asedit \textbf{Consequence:} \emph{From \eqref{eq:Modes}, a limitation on $K(s)$ as expressed by Prop.2 implies limited disturbance rejection. Most strikingly, integral control cannot be used for all modes since this would imply arbitrarily small $\bar{K}(s)^{-1}$ as $s$ approaches $0$. In modal control, poorly observable modes have poor static disturbance rejection as a tradeoff for robustness.}
} \newline Note that a disturbance like \eqref{eq:ZeMethod1} does not retain the symmetry owing to relative sensing, i.e.~$\Delta$ does not satisfy \begin{equation}\label{eqRob:gain}
{\textstyle \sum_l} \; (\Delta)_{k,l} \, = \, 0 \quad \forall k \, . \end{equation} It is by breaking the symmetry of relative sensing that small errors can create large disturbances, analogously to changing the type of a PDE. In contrast, uncertainties that preserve the relative sensing symmetry (such as uncertainties in each sensor's gain) are easily shown to be much less detrimental to robust stability. However, many physical situations can plausibly lead to model uncertainties that violate \eqref{eqRob:gain}. In the EELT example (see Fig.~\ref{fig:segment}), each sensor consists of two parts fixed on adjacent segment edges and its measured value depends on the 3-dimensional relative motion of these parts. As a result of mirror curvature, a displacement $y \rightarrow y + \alpha \mathbf{1}_{N_y}$ with $\alpha>0$ brings the segments closer together in the mirror plane. This relative displacement along an unmodeled degree of freedom slightly affects the measurement, although nominally it should not. Even with a flat mirror, mechanical constraints could induce systematic deformations of the segments or of their supporting structure, thus affecting the measurement through misalignments of sensor pairs even if the modeled $y$-difference does not change. Residual sensitivity to absolute output also seems plausible e.g.~for relative pressure or temperature sensors. All such errors can break the relative sensing symmetry and thereby severely limit the performance of the feedback system.
\section{Spatially invariant systems}\label{sec:TheorD}
Even though they do not require spatial invariance, the performance limitations described in the previous section take a simpler form under that additional hypothesis because they lead to insightful expressions in the spatiotemporal frequency domain. We refer the reader to \cite{Dahleh-Bamieh,B2Ptac11,Gorinevsky2008,Castro2002} for system analysis of (LTSI) linear time-invariant and spatially invariant systems.
The hypothesis of spatial invariance requires that all subsystems are equal and repeat an identical interconnection pattern. It \emph{restricts the controller}\footnote{Although, optimal control solutions for spatially invariant plants yield spatially invariant controllers \cite{Dahleh-Bamieh}.}, but it does not impose distributed control. We take $N = 1$ (1-degree-of-freedom subsystems) for simplicity, but we allow spatial invariance on a $\gamma$-dimensional toroidal structure. The system is then completely decoupled in spatiotemporal frequency domain, so we use the Fourier transform instead of the singular value decomposition; the real/orthogonal matrices of Section \ref{ssec:Noise} then become complex/unitary, with no added difficulty.
Our notation indexes subsystems in the toroidal spatial structure by $k=(k_1,k_2,...,k_\gamma) \in \mathcal{D} := \{1,...,M_1\} \times \{1,...,M_2\} \times ... \times \{1,...,M_\gamma\}$ with $M_1 \, M_2 \,...\, M_\gamma = M$. Localized measurement allows subsystems $j$ and $k$ to be connected through $B$ only if their indices are close, e.g.~if $\vert (j_a-k_a)\text{mod}(M_a) \vert \leq \rho'$ for all $a\in \{ 1,2,...,\gamma\}$, with $\rho'$ related to $\rho$. We denote $\; \mathcal{L} = \{\; l \in \mathcal{D} \; : \; \exists \text{ sensor comparing subsystems } k \text{ and } (k+l)\text{mod}(M_1,...,M_\gamma)\;,\; \forall k \; \}$. We write $D$ the number of sensors involving one given agent, such that e.g.~$N_z/M=D/2$.
Under spatial Fourier transform, the closed-loop equation (\ref{eq:Modes}) decouples into a set of (spatial) modes \begin{equation}\label{eq:modcahat} \modal{y}_\xi(s) = \frac{1}{1 + K_\xi(s)} \, \delta_\xi + \frac{K_\xi(s)}{1 + K_\xi(s)} \, \nu_\xi \, , \end{equation} indexed by spatial frequency $\xi=(\xi_1,\xi_2,...,\xi_\gamma)$, with $\xi_j \in \{\, \tfrac{2k\pi}{M_j} : k=0,1,...,M_j-1 \, \}$. The following result holds, assuming that each sensor measures the difference between two output values; i.e.~each row of $B$ contains one element $+1$, one element $-1$ and the rest zeros.
\newline
\textbf{Lemma 2:} \emph{If $\,n(s)$ is distributed as a zero-mean Gaussian of covariance $\,\sigma^2\, I_{N_z}$, then $\nu_\xi(s)$ is distributed as a zero-mean Gaussian of variance $\; \sigma^2 / \lambda_\xi \;$ where \begin{equation}\label{eq:SInoiseNu} \; \lambda_\xi = 2\; {\textstyle \, \sum_{l \in \mathcal{L}}} \; \sin(l^T\, \xi/2)^2 \;\; \text{ for each } \xi\,. \end{equation} For any local interconnection structure, given $c \ll 1$ there are at least $w_c := \Pi_{j=1}^\gamma \, (2\text{\emph{floor}}(\tfrac{c\,M_j}{\sqrt{2D}\gamma\rho'\pi})+1)$ frequencies $\xi$ for which $\nu_\xi$ has variance at least $\sigma^2/c^2$.}
\newline \textbf{Proof.} Lemma 1 remains valid with the Fourier transform replacing the SVD. Then we have $\lambda_\xi = B_\xi^* \, B_\xi$, where $B_\xi \in \mathbb{C}^{D/2}$ is the $\xi$-component of the spatial Fourier transform of $B$. Computing the latter analytically yields \eqref{eq:SInoiseNu}. Bound $w_c$, which is not tight for $\gamma > 1$, is obtained by counting all $\xi=(\tfrac{2k_1\pi}{M_1},...,\tfrac{2k_\gamma\pi}{M_\gamma})$ for which $\tfrac{k_j}{M_j} \notin (c_1,\,1-c_1) \; \forall j$, then redefining $c=\sqrt{2D} \gamma \rho' \pi c_1$.
$\square$
The detrimental low $\lambda_\xi$ occur for frequencies $\xi$ close to $0$ modulo $2\pi$, i.e.~low spatial frequencies, corresponding to deformation modes of large characteristic length. Their poor observability can be understood as $B$ measuring a spatial derivative of the deformation \cite{Sarlette2009a}; see also Fig.~\ref{fig:LongvsShortrange}. \begin{figure}
\caption{Schematic representation of eigenvectors of $L$ for a 1-dimensional SI plant, similarly to Fig.~\ref{fig:NewFig}: (a) $\xi \approx 0$, long-range deformation; (b) $\xi=\pi$, short-range.}
\label{fig:LongvsShortrange}
\end{figure}
Regarding model errors, we can explicitly compute $\Phi$ if we restrict also $\Delta$ to be spatially invariant. This does not give as strong possibilities as for Proposition 2, but it still highlights a robustness issue.
\newline
\textbf{Lemma 3:} \emph{For a spatially invariant uncertainty $\Delta$ as in \eqref{eqRob:Unc}, the possible $\Phi$ in an analog of \eqref{eqRob:Modes} diagonalized in spatial Fourier modes, are diagonal and include all $\beta \bar{\Phi}$ with $\beta \in [-1,1] \subset \mathbb{R}$ and \begin{equation}\label{eq:Fourrob} (\bar{\Phi})_{\xi,\xi} = \frac{\varepsilon \;\;{\textstyle \, \sum_{l \in \mathcal{L}}} \; \sin(l^T\, \xi/2)^2 \,\cdot\, \tfrac{i}{\tan(l^T\, \xi/2)}}{\phantom{\varepsilon \;}{\textstyle \, \sum_{l \in \mathcal{L}}} \; \sin(l^T\, \xi/2)^2 \phantom{ \,\cdot\, \tfrac{i}{\tan(l^T\, \xi/2)}}} \;\; \text{ for each } \xi\,. \end{equation}}
\newline \textbf{Proof.} We have $\Phi = \Lambda^{-1} [Q^* B^T (Q \otimes I)] \, [(Q^* \otimes I) \Delta Q]$ where $Q$ encodes spatial Fourier transform. Then $[Q^* B^T (Q \otimes I)]$ is a vertical concatenation of $D/2$ diagonal matrices: each one contains Fourier components $2i\, e^{-i l^T\xi/2}\, \sin(l^T \xi/2)$ of all $\xi$ for a given $l \in \mathcal{L}$ (avoid double counting). Taking `bad' $(\Delta)_{m,n} = \beta \varepsilon (B)_{m,n}$ for all $m,n$, we get $[(Q^* \otimes I) \Delta Q]$ a horizontal concatenation of $D/2$ diagonal matrices with Fourier components $2\, e^{i l^T\xi/2}\, \cos(l^T \xi/2)$.
$\square$
Since all $l$ are restricted to small values by local sensing, for a low-frequency $\xi$ every term of the sum in the numerator is multiplied by a large factor $i\, / \, \tan(l^T\, \xi/2)$ with respect to the sum in the denominator, so the model error can have a dominant effect. Like for the general case, a `bad' $\Delta$ is one that changes the `meaning' of the measurement, i.e.~that violates \eqref{eqRob:gain}. `Bad' $\Delta$ have the following effect on robustness margins that account for other uncertainties in the system (e.g.~neglected dynamics, controller sampling / quantization, spatial invariance approximation).
\newline
\textbf{Proposition 3:} \emph{Consider that robustness margins must be ensured under all spatially invariant measurement model errors $\Delta$ as in (\ref{eqRob:Unc}). Then the Nyquist plots of the nominal system ($\Delta=0$), at each spatial frequency $\xi$, must avoid enlarged ``exclusion zones'' as depicted on the right of Fig.\ref{fig:PhaseMargin}. In particular, for any given $\varepsilon,\kappa>0$ and fixed lattice dimension $\gamma$, there exist critical network sizes $M_{c,(i)}$ and $M_{c,(ii)}$ such that\newline (i) For all $M>M_{c,(i)}$, the set not to be encircled for nominal closed-loop stability, analog to $-1$ in a SISO system, has points in a $\kappa$-neighborhood of the origin for some $\xi \neq 0$.\newline (ii) For all $M>M_{c,(ii)}$, the exclusion zone to ensure a phase margin $\phi_m=\kappa$ crosses the imaginary axis for some $\xi \neq 0$.}
\newline \textbf{Proof.} Encirclement of $-1$ by the Nyquist plot of $K_\xi(1+\beta \bar{\Phi}_\xi)$ --- for $\bar{\Phi}$ as in Lemma 3 and $\beta \in [-1,1]$ --- is equivalent to encirclement by $K_\xi$ of the set $\mathcal{S}_\xi = \{ -1/(1+\beta (\bar{\Phi})_{\xi,\xi}) \; : \; \beta \in [-1,1] \}$. The latter set is an arc, extending by an angle $\theta=2\text{arctan}(\vert \bar{\Phi}_{\xi,\xi}\vert)$ on each side of the point $-1$, on a circle of radius $0.5$ centered at $-0.5 \in \mathbb{C}$. The exclusion zone to robustly guarantee a gain (resp.~phase) margin $g_m$ (resp.~$\phi_m$) is obtained by rotating this arc around $0 \in \mathbb{C}$ by all angles in $[-\phi_m,\phi_m]$ (resp.~scaling this arc towards $0$ by all ratios in $[1,g_m]$), see Fig.~\ref{fig:PhaseMargin}. A large enough system allows low enough $\xi$, such that $\vert \bar{\Phi}_{\xi,\xi}\vert$ defined by \eqref{eq:Fourrob} can get arbitrarily large, for any given $\varepsilon$. Then $\mathcal{S}_\xi$ tends towards a closed circle tangent to the imaginary axis.
$\square$
\begin{figure}
\caption{In dotted (left) and light (right) red: zones to exclude for Nyquist plot of $K_\xi(s)$ to have (top) gain margin $g_m$ and (bottom) phase margin $\phi_m$. Left: no/negligible sensing uncertainty, $K_\xi(s)$ must avoid circling $-1$. Right: ``bad'' sensing error $\beta \bar{\Phi}$ from Lemma 3, $K_\xi(s)$ must avoid circling an arc of circle (thick blue) that approaches $0 \in \mathbb{C}$ up to $2\text{arctan}(1/\vert \bar{\Phi}_{\xi,\xi}\vert)$. [Plot values: $g_m=2$, $\phi_m=\pi/4$, $M=100$, $\varepsilon=0.05$ and $\xi=\tfrac{2\pi}{M}$, 1D nearest-neighbor coupling.]}
\label{fig:PhaseMargin}
\end{figure}
Proposition 3 shows how the spatially invariant measurement errors of Lemma 3, although not directly destabilizing, strongly diminish the robustness margins of an integral controller on low spatial frequencies $\xi$. This indicates that the system is particularly sensitive to dynamical model errors affecting those modes $\xi$: the combination of $\Delta$ with phase \& gain margins creates a kind of `trap' around the origin: the zone to avoid has an angular extension from $\sim \pi/2-\phi_m$ through $\pi$ to $\sim 3\pi/2+\phi_m$ and its inner radius can be arbitrarily close to $0$ for all angles in $[\tfrac{\pi}{2}\!-\!\phi_m,\,\tfrac{\pi}{2}\!+\!\phi_m] \cup [\tfrac{3\pi}{2}\!-\!\phi_m,\,\tfrac{3\pi}{2}\!+\!\phi_m]$. The following case study shows how, when spatial invariance is not imposed, worse situations as described in Section \ref{sec:TheorB} can indeed occur in practice.
\section{Case study: European Extremely Large Telescope primary mirror stabilization}\label{sec:EELT}
The ideas worked out in this paper are inspired by our study of the EELT primary mirror controller for and with ESO, which included review, proposal \& tuning of controllers, fault detection and isolation, specifications,... \cite{EELTreport}. We now show that the EELT system features the performance limitations studied above, and illustrate how they affect controller design. This supports the practical relevance of our observations. More details about the controller are given in the appendix.
\noindent\textbf{Model details:} We refer to Section \ref{ssec:Telescope} for a general modeling in the distributed sensing framework. The general model \eqref{eq:mod1} allows to include slight differences among segments and dynamical interactions through vibrations of the mirror-supporting structure (``backstructure''). Dynamics are however dominated by a static component plus mechanical oscillation of the actuators, such that we take $G(s)$ block-diagonal, with $3 \times 3$ blocks \begin{equation}\label{eq:teldyns}
G_0(s) = ( \, s^2 J + sb_a\, I_3 + k_a\, I_3 \, )^{-1} \, k \end{equation} where $k_a \in \mathbb{R}$ is actuator stiffness, $b_a \in \mathbb{R}$ gives (very small) dissipation, $J \in \mathbb{R}^{3 \times 3}$ represents segment+actuator inertia matrix. Characteristic values for this mass-spring-damper subsystem are oscillation frequencies around $50$~Hz, with $\sim 1\%$ modal damping. The dominance of low frequency disturbances and the difficulty of high-frequency plant characterization (e.g.~backstructure vibrations) motivate a low-bandwidth model \& controller. We recall that the latter must seek an optimal tradeoff in presence of Gaussian white sensor noise $n$ with covariance matrix $\sigma^2 \, I_{5604}$ and $\sigma = 1$~nm/$\sqrt{\text{Hz}}$. An ES model uncertainty of the form \eqref{eqRob:Unc} is expected, with $\varepsilon$ up to $0.01$ and possibly all-independent components (see paragraph after \eqref{eqRob:gain}). We recall the main disturbances in $d$: \newline $\bullet$ Mounting/wear errors and deformations due to gravity acting differently as the mirror moves, are static but large (micro- to millimeters), and mostly expected to be spatially uncorrelated. \newline $\bullet$ Wind force is concentrated on low temporal frequencies, but it features strong long-range correlations in space, i.e. predominantly low spatial frequency.
\noindent\textbf{Limitations from distributed sensing:} Figure~\ref{fig:M1_singularvalues} shows the singular values $\lambda_k^{1/2}$ of $B$, in logarithmic scale. The last four modes are unobservable, $\lambda_k = 0$ for $k = 2949,...,2952$ (thus $N_0=2948$); they consist of full mirror translation, rotations and ``defocus'', and have to be addressed by other controllers based on wavefront sensing (see e.g.~\cite{EELTlaunch}). Among the observable modes, some indeed have very small $\lambda_k$ \textbf{[Section~\ref{ssec:Noise}, Prop.~1]} --- e.g.~the last 260 modes have a $\lambda_k < 0.01 \, \lambda_1$. This makes the performance limitations as expressed in Section \ref{ssec:Noise} relevant. The corresponding deformation modes (rows of $Q$, as $B^T B = Q \Lambda Q^T$) feature oscillations of long characteristic length, unfortunately where wind is strongest; the modes of the first $\lambda_k$ in contrast wildly oscillate from one segment to its neighbor. Regarding robustness, $\varepsilon = 0.01$ is insufficient to exclude all dangerous $\Delta$ \textbf{[Section~\ref{ssec:Robust}, Prop.~2]}: applying the procedure described in \eqref{eq:ZeMethod1} to $b=2948$, we get a $\Phi$ with one (exact) eigenvalue at $-9.7$. Thus large $K(s)$ --- e.g.~controllers with an integral term --- would bear the danger of instability, unless we can further reduce model uncertainty by a factor of at least $9.7$. This analysis agrees with the observations of~\cite{MacMyn2009}.
\begin{figure}
\caption{Singular values of $B$, i.e.~the diagonal elements $\lambda_k^{1/2}$ of $\Lambda^{1/2}$, in logarithmic scale. The last 4 modes are unobservable ($\lambda_k = 0$ for $k=2949$ to $2952$, beyond plot limits).}
\label{fig:M1_singularvalues}
\end{figure}
\noindent\textbf{General controller design:} We start from integral control $K_I/s$ to reject the expected low-frequency disturbances, but have to add integrator leakage $A_I$, yielding $(sI+A_I)^{-1} K_I$, to limit $\vert K(s) \vert$ for robustness. A double-pole is added to further damp the high-frequency resonance peaks due to the low damping of the actuator oscillator in \eqref{eq:teldyns}. This finally gives \begin{equation}\label{eq:GeneralController}
C(s) = -(sI+A_I)^{-1} K_I \; \frac{1}{(s/p+1)^2} \, . \end{equation} The last factor is a scalar, with fixed $p = 2 \pi\,20\un{rad/s}$. For $K_I,A_I$ we have analyzed two options: a centralized controller, based on diagonalization of the system into spatial eigenmodes \textbf{[Section \ref{sec:TheorB}]}; and a distributed controller, tuned in spatial frequency domain \textbf{[Section \ref{sec:TheorD}]}. The tradeoff between robust stability \textbf{[Section~\ref{ssec:Robust}]} and disturbance rejection \textbf{[Section~\ref{ssec:Noise}]} makes integrator leakage an essential tuning parameter.
The centralized controller is analyzed in the \emph{modal basis}, obtained from the SVDecomposition of $B$ \textbf{[Section \ref{sec:TheorB}]} by $\modal{y} = Q^T y$ (``deformation modes''), $Q^T u,\, U^T z$. Both $K_I$ and $A_I$ are taken diagonal in this basis, writing their components $K_I(k)$ and $A_I(k)$, with $k=1,2,...,2952$ in order of decreasing $\lambda_k$. Due to the distributed sensing, small $\lambda_k$ --- which indeed correspond to mirror deformations of large characteristic length \textbf{[Lemma 2]} --- are sensitive both to noise \textbf{[Section~\ref{ssec:Noise}]} and to model uncertainties \textbf{[Section~\ref{ssec:Robust}]}, and this limits the control performance, especially the properties of its disturbance rejection sensitivity $S(s)=[I_{N_y}+K(s)]^{-1}$ at steady-state $s=0$ \textbf{[Eq.~\eqref{eq:Modes}]}. Figure~\ref{fig:modal_SS_error} shows the steady-state disturbance rejection $\vert (S(0))_{k,k} \vert \approx \left\vert \dfrac{1}{1+\sqrt{\lambda_k} K_I(k)/A_I(k)} \right\vert$ resulting from our robust modal tuning. For modes $k$ where leakage was not necessary $\vert (S(0))_{k,k} \vert = 0$; but for the modes of lowest $\lambda_k$ rejection of static errors becomes illusory \textbf{[Consequence of Prop.2 together with \eqref{eq:Modes}]}: e.g. $\vert (S(0))_{2948,2948} \vert \approx 1/(1+0.0017*5.7/0.7) = 0.986$. \begin{figure}
\caption{DC gain of disturbance-to-output transfer function, $\vert (S(0))_{k,k} \vert$, with centralized control in modal basis.}
\label{fig:modal_SS_error}
\end{figure}
Because the performance limitations are independent from the controller design, one might wonder whether a distributed controller --- relying on local information only --- can attain the same performance as a centralized one. Benefits of distributed control include local communication requirements, arguably better robustness to failures, and a design method that builds directly on the physical distributed spatial structure, rather than on mathematically diagonalizing a large system matrix. We therefore design a controller where $(u)_k$ depends only on measurements made at the edges of segment $k$ and of its immediate neighbors. We further approximate the mirror as invariant w.r.t.~translations from one segment to another (after all our basic model is, except at the mirror boundaries). This allows us to use the LTSI (Linear Time- and Space-Invariance) method described in \cite{Dahleh-Bamieh,Stein2005,Gorinevsky2008} for controller tuning in spatiotemporal frequency domain \textbf{[Section \ref{sec:TheorD}} extended to block-MIMO: one $3\times 3$ transfer matrix per spatial frequency]. \begin{figure}
\caption{Spatial Fourier modes in a linear spatial structure with two degrees of freedom: piston and tilt. At low spatial frequency, a high-amplitude piston mode ($\mathbf{a}$) gives the same ES signal as a low-amplitude tilt mode ($\mathbf{b}$).}
\label{fig:locestmodes}
\end{figure} The performance limitations owing to distributed sensing are illustrated by considering how a local controller would estimate the overall mirror deformation from local measurements \textbf{[Lemma 2]}. Indeed, the similar local sensor values can indicate global deformations of very different magnitude, depending on how they have to be extrapolated. Figure \ref{fig:locestmodes} depicts the most extreme case, where in a linear spatial structure, situations $\mathbf{a}$ and $\mathbf{b}$ would give the same ES signals, while the individual segments in $\mathbf{a}$ are displaced much more than in $\mathbf{b}$.
Properties of the distributed controller \emph{for an LTSI model} of the mirror \textbf{[Section \ref{sec:TheorD}]} are further illustrated on the steady-state disturbance rejection sensitivity $S(\xi;s\!\!=\!\!0) = [I_3+K_\xi(s\!\!=\!\!0)]^{-1}$ \textbf{[Eq.\eqref{eq:modcahat}} extended to block-MIMO], the LTSI equivalent of Fig.\ref{fig:modal_SS_error}. Tip and tilt Fourier modes are all sufficiently observable in the two-dimensional structure to allow zero leakage; lower observability at high spatial frequencies however requires to limit $K_I^*$. For piston, $K_I^*$ is limited at low to moderate spatial frequencies and, most notably, leakage is necessary at low spatial frequencies \textbf{[Lemma 3]} and induces large $S$, as shown on Fig.\ref{fig:LTSI_tuning_analysis}a. Figure~\ref{fig:LTSI_tuning_analysis}b shows the maximum real part of closed-loop poles of the MIMO system with uncertainties. The tradeoff resulting from distributed sensing is visible: at low spatial frequencies, where leakage limits $S$-performance, the closed-loop poles are pushed against the robustness limit. \begin{figure}
\caption{LTSI approximation properties of the distributed controller, on piston motion. See text for details.}
\label{fig:LTSI_tuning_analysis}
\end{figure}
\noindent\textbf{Simulations:} We have simulated the controllers on a model of the full telescope composed of 20000 states extracted from a comprehensive finite element model. The model is subjected to a representative wind profile and to $1 \un{nm}/\sqrt{\text{Hz}}$ sensor noise. The limitations induced by distributed sensing \textbf{[Section \ref{sec:TheorB}]} do show up. When setting leakage $A_I$ to zero, simulations with `bad' $\Delta$ constructed from \eqref{eq:ZeMethod1} indeed feature unstable behavior, both for centralized \textbf{[Prop.2]} and distributed controllers \textbf{[Lemma 3]}. {\asedit And on the nominal system (with $\Delta=0$), choosing a constant closed-loop gain $K_I(i) \sqrt{\lambda_i}$ over all modes causes so much noise amplification that closed-loop performance is worse than in open-loop. This illustrates the reality of distributed sensing performance limitations.}
Proper control design does improve however the performance of the feedback system --- {\asedit within the limits imposed by distributed sensing}. The stability of the full simulation model, under a controller designed from simplified models (modal, LTSI), indicates the presence of sufficient margins for dynamic model uncertainties \textbf{[Prop.3]}. We can estimate simulated disturbance rejection by taking the ratio of two simulation results, one with controller and one in open loop, with the same disturbance (wind) input. Figure \ref{fig:simu} shows the result for two representative deformation modes. The curves are in good agreement with the predicted $\vert (S(s))_{i,i} \vert^2$ under centralized control. (The irregularity of the graph at high frequencies just reflects that the denominator (disturbance input) of the ratio is nearly zero.) For the large-scale deformation mode 2900, the distributed controller performs slightly worse than the centralized one; the latter already does not reject much, due to its large $A_I(i)$ and low $K_I(i) \sqrt{\lambda_i}$. Overall, both controllers achieve similarly good rejection on small spatial scales and bad rejection on large spatial scales. In the EELT implementation, adaptive optics is added to the distributed sensing-based controller in order to reject the strong low spatial frequency disturbances and reach the overall $10$~nm precision requirement. \begin{figure*}\label{fig:simu}
\end{figure*}
The simulations confirm that, under distributed sensing, the larger freedom offered by centralized control --- information transmission over large distances --- does not allow for significant improvement over the distributed controller. In agreement with similar theoretical investigation in \cite{Dahleh-Bamieh}, properly designed centralized controller gains are naturally ``distributed'': the gain from measurement at sensor $j$ to input at segment $i$ quickly decreases with the distance between $j$ and $i$, see Fig.~\ref{fig:locality}. \begin{figure}
\caption{Gain given to sensors $j=1...5604$ in the centralized control input of segment $i$ (fixed to a typical value), as a function of distance to segment $i$.}
\label{fig:locality}
\end{figure}
\noindent\textbf{Remarks:} Spatiotemporal correlations also show up in model approximations. To avoid exciting actuator-backstructure dynamics at $x\un{Hz}$, a rule of thumb is to impose a closed-loop bandwidth below $\tfrac{x}{10}\un{Hz}$. From a finite-element model of the full telescope, low temporal frequency vibration modes also have low \emph{spatial} frequency. As a result, vibration modes significantly coupled to deformation modes $i=1$ to 2692 have frequencies above $\sim 30\un{Hz}$, allowing $3\un{Hz}$ control bandwidth. Only deformation modes $i>2693$ are coupled to lower temporal frequency vibrations --- down to $\sim 3\un{Hz}$ --- and should thus be restricted to $0.3\un{Hz}$ bandwidth. It is remarkable that these bandwidth restrictions coincide with restrictions imposed for noise limitation.
\section{Conclusion}\label{sec:conclusion}
This paper highlights how important limitations for linear control of distributed plants arise directly from \emph{local relative sensing} technology, even in the absence of any (e.g.~communication, decentralization) restrictions on the controller. We specifically show how distributed sensing can severely degrade the tradeoff between disturbance rejection and robustness to noise and to model uncertainties. The viewpoint allows correlations among subsystems and among disturbances. We illustrate our developments on the European Extremely Large Telescope's segmented primary mirror controller. The simulations suggest that with distributed sensing, a ``distributed controller'', computing each action from local measurements, achieves performance sensibly equal to an unrestricted centralized controller. Given distributed controllers' apparent adequacy for locally coupled distributed systems (see also \cite{Dahleh-Bamieh}), an exact study of the interplay between distributed sensing and structural limitations on the controller (distributed, leader-follower,...) could be interesting for future work. The role of \emph{relative} sensing in this context has recently attracted attention, see e.g.~\cite{Sarlette2009a,B2Ptac11}. ``Boundary conditions'' reminiscent of PDEs can also play an important role~\cite{Meurer11,Langbort2005}.
The paper more generally highlights the role of different sensor types for controlling distributed systems. A controller that relies on absolute measurements (``centralized sensing'') needs a common physical reference and achieves control performance uniform over all spatial frequencies. In multi-scale problems, the measurement accuracy attainable in this way may be insufficient and localized relative sensing is then used. But this seems to follow a waterbed effect: it improves control on spatial small-range signals at the expense of degradation on long-range signals. Centralized and distributed sensing are thus different spatial filters whose adequacy depend on the objective. The final EELT performance is obtained by combining the control loop of Section \ref{sec:EELT}, based on local edge sensors, and an adaptive optics controller based on a centralized wavefront-sensing. This suggests to combine sensors on a hierarchy of spatial scales for challenging distributed systems applications.
\section*{Acknowledgments}
The authors thank C.~Bastin at U.Li\`ege and M.~Dimmler, B.~Sedghi, T.~Erm, B.~Bauvir from ESO for joint work on the EELT application. P.~Kokotovic is acknowledged for insightful comments on an early version of the manuscript. This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Program, initiated by the Belgian State, Science Policy Office. AS was an FNRS postdoctoral researcher at U.Li\`ege during part of this work.
\section*{Appendix: design of the controllers}
\paragraph*{Centralized controller} Modulo approximating the high-frequency actuator resonances, both $G(s)$ and $B$ are diagonal (concatenated with appropriate all-zero matrices) in the modal basis $\modal{y} = Q^T y,\,Q^T u,\, U^T z$, so taking $K_I$ and $A_I$ diagonal in this same basis seems justified. The unobservable modes cannot be controlled, so $K_I(k) = 0$ for $k=2949$ to $2952$. The remaining components $1,2,...,N_0=2948$ are tuned as follows.
\newline First we discard leakage. For low $k$, the noise is negligible in Eq.~\eqref{eq:Modes}. Disturbances are divided by a given value on all modes by taking the same $(K)_{k,k} = K_0$, requiring \begin{equation}\label{eq:Kival} K_I(k) = K_0 / \sqrt{\lambda_k} \,, k = 1,2,...,2692\,. \end{equation} That is, less observable modes need larger $K_I(k)$. For large $k$, noise amplification makes \eqref{eq:Kival} a poor choice. We therefore must impose $(K)_{k,k} = K_1 \sqrt{\lambda_k}$ for $k>2692$ (where $\sqrt{\lambda_k} < 10 \sqrt{\lambda_1}$), i.e.~decreasing closed-loop gain, which translates into $K_I(k) = K_1$ constant.
Our simulations take $K_0= 14.4$~rad/s, $K_1=5.7$~rad/s.
\newline Then we tune $A_I(k)$ with the approximate procedure around Proposition 2: for each $k$, we build a `bad' $\Delta$ according to \eqref{eq:ZeMethod1}, and assume $\Phi \approx \Phi_k$ as in \eqref{eq:ApproxPhi}. Writing the low-frequency approximation $y = G\cdot K\cdot z \approx 1 \cdot \tfrac{K_I}{s+A_I} \cdot z$ in time domain and replacing $z$ yields $\tfrac{d}{dt} (\modal{y})_k = - \left(K_I(k)\sqrt{\lambda_k}(1+\phi_k)+A_I(k) \right)\, (\modal{y})_k \, =: - \mu_k\; (\modal{y})_k \,.$ We hence impose $A_I(k) = \max\{p_0 - K_I(k)\sqrt{\lambda_k}(1+\phi_k)\}$ to get $-\mu_k \leq -p_0 = -0.1$~Hz. This results in nonzero leakage for $k\geq 2846$, with maximum $A_I(N_0) = 0.7$~rad/s. A numerical computation confirms that the exact closed-loop eigenvalues with this $\Delta$ are below $-p_0$.
\paragraph*{Distributed controller} To gain more insight, we factorize the controller into two steps. First each segment computes a local estimate $\hat{y}_k \in \mathbb{R}^3$ of its own configuration from the ES measurements at its boundaries. Then a local controller computes $(u)_k \in \mathbb{R}^3$ from the estimates $\hat{y}_j$ of $k$ and of its immediate neighbors. Thus $(u)_k$ depends only on measurements made at the edges of segment $k$ and of its immediate neighbors.
\newline Controller structure, plant, and measurement operator are all invariant from one segment to another, except for the mirror's boundary. Our spatial invariance assumption (i) approximates the mirror as infinite or periodic and (ii) imposes spatially invariant controller parameters. The system then decouples into one $3$-dimensional (piston,tip,tilt)-system per $2$-dimensional spatial frequency. We adapt the LTSI method of \cite{Dahleh-Bamieh,Stein2005,Gorinevsky2008} from SISO- to MIMO-subsystems in a straightforward way.
\newline
The local estimation $\hat{y}_k$ is computed by pseudo-inverting, at each segment, the $12\times 3$ local matrix linking the 12 sensors at its edges to its own 3 degrees of freedom only. This amounts to segment $k$ assuming that its neighbors are perfectly positioned and only itself is displaced. That is of course not correct, so the local estimator acts like a first spatial filter\footnote{Other spatial filters can be envisioned --- central estimation is one, for other options see \cite{MacMyn2005} --- but none will alleviate the limitations imposed by distributed sensing.}, reflecting the difficulties of distributed sensing described with Fig.~\ref{fig:locestmodes}. As a consequence, the chosen estimation filter will
\begin{itemize}\tightlist \item[-] couple piston, tip and tilt (i.e.~it estimates a tilt component in $\hat{y}$ for a pure piston deformation $y$); \item[-] underestimate piston (resp.~tip,tilt) dominated deformations of low (resp.~high) spatial frequency; \item[-] slightly overestimate piston (tip,tilt) dominated deformations of high (low) spatial frequency. \end{itemize}
We denote the output-to-estimation transfer function $(\hat{y})_{\xi} = H^*(\xi,s)\, (y)_{\xi}$, such that $H^*(\xi,s)$ is a $3 \times 3$ transfer function matrix for each $\xi$ and $s$. Our static plant approximation makes $H^*$ static, so it is trivially spatiotemporally factorized, facilitating tuning (see \cite{Gorinevsky2008}).
\newline
Restricting the controller \eqref{eq:GeneralController} to local coupling, spatial invariance, and symmetry w.r.t.~axes reversal, we get
\newline $ K_I^*(\xi) = k_{\alpha} + k_{\beta} \cos(\xi_1) + k_{\gamma} \cos(\xi_2) + k_{\delta} \cos(\xi_1\text{-}\xi_2)$ \newline $ A_I^*(\xi) = a_{\alpha} + a_{\beta} \cos(\xi_1) + a_{\gamma} \cos(\xi_2) + a_{\delta} \cos(\xi_1\text{-}\xi_2)$
\newline where $k_i, \, a_i$, $i \in \{\alpha,\beta,\gamma,\delta \}$, are $3\times3$ matrices to tune. The LTSI tuning is performed in two steps. First, we approximate that $H^*$ is diagonal, i.e.~it leaves piston, tip and tilt uncoupled. This allows to directly apply the SISO tuning method of \cite{Stein2005}, using a large but linear optimization setting that minimizes leakage under robust stability and disturbance rejection constraints. We set the latter according to expected $d$ and $n$ on spatiotemporal frequencies and impose robustness as for centralized control: closed-loop eigenvalues $\leq -p_0 = -0.1$~Hz with any model uncertainties $\Delta$. The spatial invariance approximation requires to include extra robustness, so we increase $\varepsilon$ to $2.5\%$. In a second step, we add a stability constraint for the nominal MIMO system ($H^*$ not diagonal, $\Delta=0$). This yields a set of nonlinear constraints. The resulting nonlinear optimization is solved locally, starting from the solution of the first step. Simulating the resulting controller on the exact system confirms that intended properties hold despite the LTSI approximation in its design. Thus the system with less than $40$ segments diameter and a hole in the middle (see Fig.~\ref{fig:M1}) can be approximated as spatially invariant for controller design when its coupling is sufficiently local.
\end{document}
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\begin{document}
\title{\LARGE \bf
Counter-examples in first-order optimization: a constructive approach} \thispagestyle{empty} \pagestyle{empty}
\begin{abstract}
While many approaches were developed for obtaining worst-case complexity bounds for first-order optimization methods in the last years, there remain theoretical gaps in cases where no such bound can be found. In such cases, it is often unclear whether no such bound exists (e.g., because the algorithm might fail to systematically converge) or simply if the current techniques do not allow finding them.
In this work, we propose an approach to automate the search for cyclic trajectories generated by first-order methods. This provides a constructive approach to show that no appropriate complexity bound exists, thereby complementing approaches providing sufficient conditions for convergence. Using this tool, we provide ranges of parameters for which the famous Polyak heavy-ball, Nesterov accelerated gradient, inexact gradient descent, and three-operator splitting algorithms fail to systematically converge, and show that it nicely complements existing tools searching for Lyapunov functions. \end{abstract}
\section{Introduction}
\label{sec:intro}
In the last years, first-order optimization methods (or algorithms) have attracted a lot of attention due to their practical success in many applications, including in machine learning (see, e.g.,~\cite{bottou2007tradeoffs}). Theoretical foundations for those methods played a crucial role in this success, e.g., by enabling the development of momentum-type methods (see, e.g.,~\cite{polyak_gradient_1963, nesterov1983method}). Formally, we consider the optimization problem
\begin{equation}
x_\star \triangleq \arg\min_{x\in \mathbb R^d} f(x) \tag{OPT}\label{eq:opt}
\end{equation}
for a function $f$ belonging to a class of functions $\mathcal F$ (e.g., the set of convex functions, or the set of strongly convex and smooth functions, etc.). Classical first-order optimization methods for solving this problem include \emph{gradient descent}~(GD), \emph{Nesterov accelerated gradient method}~(NAG)~\cite{nesterov1983method}, and the \emph{heavy-ball method}~(HB)~\cite{polyak_gradient_1963}.
These families of algorithms are parametrized: for example, GD is parametrized by a step-size $\gamma$ and HB is parametrized by both a step-size~$\gamma$ and a momentum parameter~$\beta$.
We generically denote by $\mathcal A$ any such method, for a specific choice of its parameters.
For a given class of function $\mathcal{F}$ and an algorithm $\mathcal{A}$, we typically aim at answering the question
\begin{center}
\fbox{\parbox{0.48\textwidth}{
\begin{center}
Does $\mathcal{A}$ converge on every function of $\mathcal{F}$ \\ to their respective minimum?
\end{center}
}}
\end{center}
Common examples of function classes $\mathcal F$ {include} the set $\mathcal F_{\mu,L}$ of $\mu$-strongly convex and $L$-smooth functions, and the set $\mathcal Q_{\mu,L}$ of $\mu$-strongly convex and $L$-smooth \textit{quadratic} functions, for $\mu, L\geq 0$.
This type of analysis, requiring results to hold on every function of a given class $\mathcal{F}$ is commonly referred to as \emph{worst-case analysis} and is the most popular paradigm for the analysis of optimization algorithms, see, e.g.,~\cite{nesterov1983method,dvurechensky2021first,bubeck2015convex,d2021acceleration,chambolle2016introduction}.
In this context, a very successful technique for proving worst-case convergence consists in looking for a decreasing sequence (called \textit{Lyapunov sequence}~\cite{lyapunov1992general,Kalman1960a,Kalman1960b}) of expressions $V_t$ of the iterates $x_t$, i.e.~such that
\begin{equation}
\forall f \in \mathcal{F},~ \forall t,~ \forall x_t, \quad V_{t+1}((x_s)_{s\leq t+1}) \leq V_t((x_s)_{s \leq t}), \label{eq:lyap}
\end{equation}
where some quantity of interest is upper-bounded by~ $V_T((x_s)_{s \leq T})$ as $T$ goes to infinity.
For instance, when studying GD with step-size $1/L$ on the class $\mathcal{F}_{0, L}$ of $L$-smooth convex functions, we prove that $\forall f \in\mathcal{F}_{0, L}$, $ (t+1)(f(x_{t+1}) - f(x_\star))+ \tfrac{1}{2}\|x_{t+1}-x_\star\|^2 \leq t (f(x_{t})-f(x_\star)) + \tfrac{1}{2}\|x_{t}-x_\star\|^2$.
Therefore, $V_t((x_s)_{s \leq t}) = t (f(x_{t})-f(x_\star)) + \tfrac{1}{2}\|x_{t}-x_\star\|^2$ defines a decreasing sequence, and $f(x_t)-f(x_\star) \le V_t((x_s)_{s \leq t})/t \le V_0(x_0)/t $, proving convergence of this method on this class of functions.
Due to the simplicity of the underlying proofs, the Lyapunov approach is particularly popular, e.g., for NAG~\cite{nesterov1983method,beck2009fast}, and HB~\cite{ghadimi2014global}. See~\cite{bansal2019potential,d2021acceleration} for surveys on this topic.
\textbf{Necessary condition for worst-case convergence.}~While finding a decreasing Lyapunov sequence guarantees convergence, not finding one does not guarantee anything: there may still exist a Lyapunov sequence, that the current analysis was not able to capture, or the method could converge without the existence of such Lyapunov sequence.
Establishing that a method
\emph{provably does not admit a worst-case convergence analysis} is therefore critical for avoiding spending an indefinite amount of time and effort searching for a non-existent convergence guarantee.
The existence of a \textit{cycle} for the algorithm on a particular function means that it diverges on that function: in other words, the absence of cycle on all functions is a necessary condition for worst-case convergence.
Moreover, a cycle can be observed after only a \textit{finite} number of steps of the algorithm, while observing the divergence of a non periodic sequence is difficult or impossible. Overall, this makes the search for cycles a computationally practical way of proving divergence.
In order to discover cycles, we rely on computer-assisted worst-case analysis. \emph{Performance estimation problems} (PEP~\cite{drori2014performance, taylor2017smooth}) provide a systematic approach to obtain convergence guarantees, including the search for appropriate Lyapunov arguments. Some packages (especially \texttt{Pesto}~\cite{taylor2017performance} and \texttt{Pepit}~\cite{goujaud2022pepit}) automate these tasks. We formulate cycle discovery as a minimization problem that can be cast in a PEP, and rely on the \texttt{Pepit} package to solve it.
{\textbf{Examples:}} We demonstrate the applicability of our method on several examples.
In particular, the case of HB illustrates the potential of our methodology. In fact, the search for the step-size $\gamma$ and momentum $\beta$ parameters leading to the fastest worst-case convergence over $\mathcal F_{\mu, L}$ is still an open problem, and the existence of parameters resulting in an accelerated rate remains a lingering question.
Indeed,~\cite{lessard2016analysis} exhibits a smooth and strongly convex function on which HB cycles, for parameters $\gamma$ and $\beta$ optimizing the worst-case guarantee on $\mathcal Q_{\mu,L}$.
On the other hand,~\cite{ghadimi2014global} obtains a worst-case convergence on $\mathcal F_{\mu, L}$ for other parameters, but without acceleration. Recently,~\cite{upadhyaya2023automated} proposes a very general procedure to find Lyapunov sequences and extended the region of parameters $\gamma$ and $\beta$ HB provably converges on, leveraging PEPs. However, outside this region of the parameter space, the question of the convergence of the HB method remains open in the absence of a proof of divergence. For this example, our approach demonstrates that a cycle exists for almost all parameters for which no Lyapunov is known.
{\textbf{Summary of contributions:}}
This paper proposes a systematic approach to prove that no worst-case certificate of convergence can be obtained for a given algorithm $\mathcal{A}$ on a class~$\mathcal{F}$. To do so, we establish the existence of a function in $\mathcal{F}$ over which $\mathcal{A}$ cycles.
We illustrate our approach by applying it to three famous first-order optimization algorithms, namely HB, NAG, inexact gradient descent with relatively bounded error. We further showcase the applicability of the approach to more general types of problems by studying the three-operator splitting method for monotone inclusions. For each method, we describe the set of parameters for which it is known to converge and the ones where we establish the existence of a cycle. In the first three examples, our approach enables to fill the gap: we show the existence of cycles for all parametrizations not known to result in convergence.
{\textbf{Organization:}}
The rest of the paper is organized as follows. In \Cref{sec:definitions}, we introduce the concept of a stationary algorithm and formally define a cycle. In \Cref{sec:searching_for_cycles}, we present our methodology to discover cycles, relying on PEP. Finally, in \Cref{sec:examples}, we provide the numerical results.
\section{Definitions and notations}
\label{sec:definitions}
\noindent
\begin{table}[t]
\centering
\begin{tabular}{ll}
\toprule
Notation & Corresponding object \\
\midrule
$\mathcal A$ & Generic algorithm \\
$A$ & Update function of the algorithm $\mathcal A$ \\
\eqref{eq:hb} & Heavy-ball \\
\eqref{eq:nag} & Nesterov accelerated gradient \\
\eqref{eq:igd} & Inexact GD\\
\eqref{eq:tos} & Three operator splitting\\
$\beta, \gamma $ & Algorithm parameters\\
$(x_t)_t$ & Sequence of iterates generated by $\mathcal A$ \\
$x_*$ & Optimal point \\
$V$ & Lyapunov function \\
$f$ & Objective function \\
$\mathcal F$ & Generic class of functions \\
$\mathcal F_{\mu, L}$ & Class of $L$-smooth and $\mu$-strongly convex \\& functions \\
$\mathcal Q_{\mu, L}$ & Class of $L$-smooth and $\mu$-strongly convex \\ &quadratic functions \\
$\ell $& Order of the algorithm $\mathcal A$ \\
$ K $ & Length of the considered cycle \\
$\mathcal O^{(f)} $ & Generic oracle applied on $f$ \\
$u, F, G$ & Linearization variables (after SPD lifting)\\
$d$ & Dimension\\
$s_K$ & Score \\
\bottomrule
\end{tabular}
\end{table}
In this section, we consider a subclass of first-order methods, tailored for our analysis. It is chosen to ensure the periodicity of an algorithm that cycles once (see \Cref{prop:finite_horizon_cycle}). The class reduces to ``\emph{p-stationary canonical linear iterative optimization algorithms}'' (p-SCLI, see~\cite[Definition~1]{arjevani2016lower}) when the dependency to the previous iterates and gradients is linear which is a particular case of ``\emph{fixed-step first-order methods}'' (\emph{FSFOM}, see in~\cite[Definition~4]{taylor2017smooth}).
Here, we consider \emph{stationary first-order methods (SFOM)}, whose iterates are defined as a fixed function of a given number of lastly observed iterates, as well as output of some oracles called on those iterates.
Examples of such oracles include gradients, approximate gradients, function evaluations, proximal step, exact line-search, Frank-Wolfe-type steps (see~\cite{taylor2017exact,goujaud2022pepit} for lists of oracles that can be handled using PEPs).
The oracles we use depend on the setting under consideration.
\begin{leftbot}
\begin{Def}[Stationary first-order method (SFOM)]
\label{def:SFOM}
A method $\mathcal{A}$ is called order-$\ell$ \emph{stationary first-order method} if there exists a deterministic first-order oracle $\mathcal{O}^{(f)}$ and a function $A$ such that the sequence generated on the function $f$ verifies $\forall t\geq\ell$,
\begin{equation}
x_{t} = A((x_{t-s}, \mathcal{O}^{(f)}(x_{t-s}))_{s\in\llbracket 1, \ell \rrbracket}). \tag{SFOM} \label{eq:sfom}
\end{equation}
\end{Def}
\end{leftbot}
For any given function of interest $f$ and any initialization $(x_t)_{t\in\llbracket 0, \ell -1 \rrbracket}$, an order-$\ell$ \eqref{eq:sfom} $\mathcal{A}$ iteratively generates a sequence $(x_t)_{t\in\mathbb{N}}$ that we denote $\mathcal{A}(f, (x_t)_{t\in\llbracket 0, \ell -1 \rrbracket})$.
Definition~\ref{def:SFOM} above is very similar to the definition of a general first-order method. However, the key assumption here is that the operation $A$ does not depend on the iteration counter~$t$: the algorithm is \textit{stationary}.
While this assumption is restrictive, many first-order methods are of the form \eqref{eq:sfom}, including (but not limited to): GD, HB~\cite{polyak_gradient_1963} and NAG~\cite{Nest03a} with constant step-sizes. On the other hand, any strategy involving decreasing step-size (e.g. for GD), or increasing momentum parameter (e.g. for NAG on $\mathcal F_{0,L}$ as in \cite{nesterov1983method}) are not in the scope of this definition.
Note that the aforementioned examples use the first-order oracle $\mathcal{O}^{(f)}(x)\triangleq (\nabla f(x), f(x))$, although our methodology applies beyond this simple setting, as previously discussed. As an example, \ref{subsec:tos} considers an algorithm relying on the resolvent (or proximal operation).
Stationarity is essential for being able to prove existence of a cyclical behavior in a finite number of steps. Next, we define a cyclic sequence.
\begin{leftbot}
\begin{Def}[Cycle]
For any positive integer $K\geq2$, a sequence $(x_t)_{t \geq 0}$ is said to be $K$-cyclic if $\forall t \geq 0, x_t = x_{t+K}$.
A sequence $x$ is said to be cyclic if there exists $K\geq2$ such that $x$ is $K$-cyclic.
\end{Def}
\end{leftbot}
For any given order-$\ell$ \eqref{eq:sfom} $\mathcal{A}$, and any function class $\mathcal{F}$, we want to address the question
\begin{center}
\fbox{\parbox{0.48\textwidth}{
\begin{center}
Does there exist a function $f \in \mathcal{F}$ and an initialization $(x_t)_{t\in\llbracket 0, \ell -1 \rrbracket}$ such that $\mathcal{A}(f, (x_t)_{t\in\llbracket 0, \ell -1 \rrbracket})$ is cyclic?
\end{center}
}}
\end{center}
\begin{Ex}
\textit{
In~\cite[Equation 4.11]{lessard2016analysis}, {the authors} answer positively to this question by providing a cycle of length~3, on the class $\mathcal{F}_{\mu, L}$ with $(\mu,L)=(1,25)$, and for $\mathcal{A}$ the heavy-ball method with step-size $\gamma = (\frac{2}{\sqrt{L} + \sqrt{\mu}})^2$ and momentum parameter $\beta = (\frac{\sqrt{L} - \sqrt{\mu}}{\sqrt{L} + \sqrt{\mu}})^2$. Those parameters are natural candidates, that correspond to the limit of the step-size and momentum in Chebychev acceleration~\cite{flanders1950numerical,lanczos1952solution,young1953richardson}, and result in an acceleration for quadratic functions.
}
\end{Ex}
In \Cref{sec:examples}, we extend this result to more parameters.
\section{Searching for cycles}
\label{sec:searching_for_cycles}
In this section, we show how to find cyclic trajectories.
\subsection{Motivation}
Finding diverging trajectories for an algorithm $\mathcal A$ might be challenging. We thus focus on cycles, as they allow to focus on a finite sequences of iterates only. Indeed, for an SFOM, once we observe the cycle to be repeated once, we can easily extrapolate: this same cycle is repeated again and again. This statement is formalized in the following proposition.
\begin{leftbot}
\begin{Prop}
\label{prop:finite_horizon_cycle}
Let $\mathcal{A}$ be a order-$\ell$ \eqref{eq:sfom}, and $(x_t)_{t\in\mathbb{N}}$ be any sequence generated by $\mathcal{A}$.
Then the sequence $(x_t)_{t\in\mathbb{N}}$ is cyclic if and only if there exists $K\geq2$ such that $\forall t \in \llbracket 0, \ell-1 \rrbracket, x_t = x_{t+K}$.
\end{Prop}
\end{leftbot}
\begin{proof}{}
Let $\mathcal{A}$ be a order-$\ell$ \eqref{eq:sfom}, and $(x_t)_{t\in\mathbb{N}}$ be any sequence generated by $\mathcal{A}$. The method $\mathcal A$ is cyclic if and only if there exists $K\geq 2$, such that the translated sequence $(\Tilde{x}_t)_{t\in\mathbb{N}}:= (x_{t+K})_{t\in\mathbb{N}}$, is identical to $({x}_t)_{t\in\mathbb{N}}$. \Cref{prop:finite_horizon_cycle} states that those two sequences are identical if and only if their $\ell$ first terms are. It is clear that if the sequences $x$ and $\Tilde{x}$ are identical, their $\ell$ first terms also are. Reciprocally, let's assume that their $\ell$ first terms are identical and let's introduce the function $f$ and associated oracles $\mathcal{O}^{(f)}$ defined such that $x = \mathcal{A}(f, (x_t)_{t\in\llbracket 0, \ell -1 \rrbracket})$.
Then $\forall t \geq 0, x_{t} = A((x_{t-s}, \mathcal{O}^{(f)}(x_{t-s}))_{s\in\llbracket 1, \ell \rrbracket})$.
In particular $\forall t \geq 0$
\begin{align*}
x_{t+K} & = A((x_{t+K-s}, \mathcal{O}^{(f)}(x_{t+K-s}))_{s\in\llbracket 1, \ell \rrbracket}),
\end{align*}
thereby reaching
\begin{align*}
\Tilde{x}_{t} & = A((\Tilde{x}_{t-s}, \mathcal{O}^{(f)}(\Tilde{x}_{t-s}))_{s\in\llbracket 1, \ell \rrbracket}).
\end{align*}
Consequently, $\tilde x = \mathcal{A}(f, (\Tilde{x}_t)_{t\in\llbracket 0, \ell -1 \rrbracket}) $, and since the $\ell$ first terms of $x$ and $\Tilde{x}$ are identical, \\
$\Tilde{x} = \mathcal{A}(f, (\Tilde{x}_t)_{t\in\llbracket 0, \ell -1 \rrbracket}) = \mathcal{A}(f, (x_t)_{t\in\llbracket 0, \ell -1 \rrbracket}) = x$.
\end{proof}
\subsection{Approach}
We now present the approach used to search for cycles, based on performance estimation problems (PEPs)~\cite{drori2014performance,taylor2017smooth}. We consider an algorithm $\mathcal{A}$, a function $f$ and initial points $(x_t)_{t\in\llbracket 0, \ell -1 \rrbracket}$, and run $\mathcal{A}$ on $f$ starting on $(x_t)_{t\in\llbracket 0, \ell -1 \rrbracket}$. This generates the sequence $x = \mathcal{A}(f, (x_t)_{t\in\llbracket 0, \ell -1 \rrbracket})$. For any positive integer~$K$, we then define the non-negative score
\begin{equation*}
s_K(\mathcal{A}, f, (x_t)_{t\in\llbracket 0, \ell -1 \rrbracket}) = \sum_{t=0}^{\ell -1} \|x_t - x_{t+K}\|^2.
\end{equation*}
From \Cref{prop:finite_horizon_cycle}, this score is identically zero if and only if $\mathcal{A}$ cycles on $f$ when starting from $(x_t)_{t\in\llbracket 0, \ell -1 \rrbracket}$.
This suggests that one can search for cycles of length $K$ by minimizing the score $s_K(\mathcal{A}, f, (x_t)_{t\in\llbracket 0, \ell -1 \rrbracket})$ w.r.t.~the function $f$ and the initialization $(x_t)_{t\in\llbracket 0, \ell -1 \rrbracket}$.
Observe that fixed points of $\mathcal{A}$, that correspond to cycles of length 1, also cancel this score. Our goal is to search for cycles of length at least $K\geq2$, that entail that the algorithm diverges for a particular function and initialization. As any convergent algorithm must admit the optimizer of $f$ as fixed point, we have to exclude fixed points. To do so, we add the constraint that the two first iterates are far from each other. In most cases of interest, making this constraint can be done without loss of generality due to the homogeneity of the underlying problems.
We arrive to the following formulation:
\begin{equation}
\left|
\begin{array}{cc}
\underset{d \geq 1, f \in\mathcal{F}, x\in\left(\mathbb{R}^d\right)^{\mathbb{N}}}{\text{minimize }} & \sum_{t=0}^{\ell -1} \|x_t - x_{t+K}\|^2 \\
\text{subject to } &
\left\{
\begin{array}{c}
x = \mathcal{A}(f, (x_t)_{t\in\llbracket 0, \ell -1\rrbracket}) \\
\|x_1 - x_0\|^2 \geq 1.
\end{array}
\right.
\end{array}
\right.
\tag{$\mathcal{P}$} \label{eq:problem}
\end{equation}
As we see in the next sections, this problem can be used to answer the question of interest by testing the nullity of the solution of \eqref{eq:problem}.
As is, \eqref{eq:problem} looks intractable due to the minimization over the infinite-dimensional space $\mathcal{F}$ and its non-convexity. This can be handled using the techniques proposed in~\cite{taylor2017smooth, taylor2017exact}, developed for PEP. It consists in reformulating \eqref{eq:problem} into a semi-definite program (SDP) using interpolation / extension properties for the class $\mathcal{F}$, together with SDP lifting.
Indeed, \eqref{eq:problem} does not fully depend on $f$, but only on $\mathcal{O}^{(f)}(x_t)$ where $t \in \llbracket 0 ;K+\ell-2\rrbracket$.
By introducing the variables $\mathcal{O}_{ t} \triangleq \mathcal{O}^{(f)}(x_t)$, we can replace the constraint $x = \mathcal{A}(f, (x_t)_{t\in\llbracket 0, \ell -1\rrbracket})$ of \eqref{eq:problem} by
\begin{equation*}
\left\{
\begin{array}{ccll}
x_{\ell} & = & A(( & \hspace{-.3cm} x_{\ell -s}, \mathcal{O}_{ \ell -s})_{s\in\llbracket 1, \ell \rrbracket}), \\
& \vdots && \\
x_{K + \ell - 1} & = & A(( & \hspace{-.3cm} x_{K+\ell - 1-s}, \mathcal{O}_{ K+\ell - 1-s})_{s\in\llbracket 1, \ell \rrbracket}),
\end{array}
\right.
\end{equation*}
\noindent and minimize over the finite dimensional variables $(\mathcal{O}_{t})_{0 \leq t \leq K+\ell-2}$ instead of $f$, under the constraint that there exists a function $f\in\mathcal{F}$ that interpolates those values, i.e.~that verifies $\mathcal{O}^{(f)}(x_t) = \mathcal{O}_{t}$ for all $t \in \llbracket 0 ;K+\ell-2\rrbracket$. For some classes $\mathcal F$, those interpolation property are equivalent to tractable inequalities, as in the following example.
\begin{Ex}[$L$-smooth convex functions]\label{ex:ic_smooth}
\textit{
If the oracles are only the gradients and the function values of the objective function $f$, denoting $f_i \triangleq f(x_i)$ and $g_i \triangleq \nabla f(x_i)$ (i.e.~$\mathcal{O}_{i}\triangleq (g_i, f_i)$), the interpolation conditions of $\mathcal{F}_{0, L}$ are provided in~\cite{taylor2017smooth} as
\begin{equation}
\forall i, j,~ f_i \geq f_j + \left< g_j, x_i - x_j \right> + \tfrac{1}{2L}\|g_i - g_j\|^2. \tag{IC} \label{eq:cni}
\end{equation}
}
\end{Ex}
This function class is considered in three of the four examples under consideration in the next section (HB, NAG, inexact GD).
However, the methodology described in this paper applies to many other classes beyond $\mathcal F_{\mu,L}$, see for instance \cite[\href{https://pepit.readthedocs.io/en/0.2.1/api/functions_and_operators.html}{Function classes}]{goujaud2022pepit}: an example of such a class is used in the fourth example of the following section. Each class considered must be described by its interpolation conditions, similar to \eqref{eq:cni}. Other examples of known interpolations conditions are provided in \cite[Th. 3.4-3.6]{taylor2017exact}, \cite[Cor.1\&2]{dragomir2021optimal}, \cite[Th.1]{guille2022gradient} \cite[Th. 2.6]{goujaud2022optimal}.
The key ingredient for a class to enter in the scope of this paper, is that its interpolation conditions are expressed as a degree~2 polynomial in $x_t$ and $\mathcal{O}_t$, and that a given variable is not involved both in a monomial of degree 1 and one of degree~2, as in \eqref{eq:cni}.
Then, the SDP lifting part consists in introducing a Gram matrix~$G$ \cite[Theorem~5]{taylor2017smooth} of vectors among $x_t$ and $\mathcal{O}_t$ that are involved in degree 2 monomials, so that those quadratic expressions of $x_t$ and $\mathcal{O}_t$ are then expressed linearly in term of $G\succeq 0$. Thereby, the problem can be cast a standard SDP.
In the case where the oracle is $\mathcal{O}_t = (g_t, f_t)$,
and the class of interest $\mathcal{F}$ is the class of $L$-smooth convex functions $\mathcal{F}_{0, L}$, the objective and all the constraints of \eqref{eq:problem} are written linearly in terms of $(f_t)_t$ and quadratically in terms of $(x_t, g_t)_t$. Therefore, we define $G$ as the Gram matrix of $(x_t, g_t)_t$ and $F$ as a vector storing all the values $f_t$ leading to an SDP reformulation of the problem. See, e.g.,~\cite[Section 2]{goujaud2022pepit} for a detailed derivation on a simple example.
Setting $u \triangleq (G, F)$, \eqref{eq:problem} is generally rewritten, under above mentioned key ingredients, as
\begin{equation}
\left|
\begin{array}{cc}
\underset{u}{\text{minimize }} & \left< u, v_{obj} \right> \\
\text{subject to } &
\left\{
\begin{array}{c}
\left< u, v_1 \right> \geq 0 \\
\ldots \\
\left< u, v_n \right> \geq 0 \\
\left< u, v_{\text{aff}} \right> \geq 1 \\
u \in \mathcal{C}.
\end{array}
\right.
\end{array}
\right.
\tag{SDP-$\mathcal{P}$} \label{eq:sdp_problem}
\end{equation}
The objective is linear, as well as the first $n$ constraints.
The affine constraint $\left< u, v_{\text{aff}} \right> \geq 1 $ enables to discard the trivial solution $u=0$ and corresponds in \eqref{eq:problem} to the constraint $\|x_1 - x_0\|^2 \geq 1$.
Finally, the constraint $u\in\mathcal{C}$ corresponds to the constraint $G \succeq 0$. $\mathcal{C}$ is then a closed convex semi-cone.
By definition, if there exists a feasible vector $u$ such that the objective of \eqref{eq:sdp_problem} is zero, then it describes a cycle.
Moreover \eqref{eq:sdp_problem} is convex and efficiently solvable (due to the existence of a Slater point~\cite[Theorem 6]{taylor2017smooth}).
In the next sections, we numerically apply this methodology through the \texttt{Pepit} python package~\cite{goujaud2022pepit} which takes care about the tractable reformulations of \eqref{eq:problem} into~\eqref{eq:sdp_problem}. \eqref{eq:sdp_problem} is then solved using through a standard solver~\cite{mosek} to determine the infimum value of $\left< u, v_{\text{obj}} \right>$ over the feasible set of \eqref{eq:sdp_problem}. The next theorem allows to conclude about the existence of cycles.
\begin{leftbot}
\begin{Th}
\label{thm:min=inf}
Assuming the infimum value of \eqref{eq:problem} to be 0, then there exists a cycle.
\end{Th}
\end{leftbot}
\begin{proof}{}
By equivalence between~\eqref{eq:problem} and~\eqref{eq:sdp_problem}, we assume that there exists a sequence of feasible vectors $u_i$ with $\langle u_i, v_{\text{obj}} \rangle \rightarrow 0$.
The constraint $\left< u_i, v_{\text{aff}} \right> \geq 1$, guarantees that none of the~$u_i$ is equal to 0.
Considering any norm (all equivalent to each other in finite dimension) and projecting $u_i$ on the associated sphere defines $s_i \triangleq \tfrac{u_i}{\|u_i\|}$. The other $n$ \textit{linear} constraints still hold after the scaling and $s_i \in \mathcal{C}$ since $\mathcal{C}$ is a semi-cone.
Moreover, the norms of all $u_i$ are lower bounded by the distance from 0 to the affine hyperplan $\left\{ w | \left< w, v_{\text{aff}} \right> = 1 \right\}$. Hence $|\left< s_i, v_{\text{obj}} \right>| = \tfrac{|\left< u_i, v_{\text{obj}} \right>|}{\|u_i\|} \rightarrow 0$.
Finally by compacity of the sphere, there exists a subsequent limit $s$ of the sequence $(s_i)_i$ and by continuity of the linear operator $\left< \cdot , v_{\text{obj}} \right>$, $\left< s , v_{\text{obj}} \right>=0$.
We conclude that $s$ is a vector the objective reaches 0 on, that verifies all the constraints of \eqref{eq:sdp_problem} but the affine one.
Note the affine constraint only aimed at discarding the trivial solution 0 in a linear way (for solver purpose), and that $s$ is not 0. Then $s$ describes a cycle.
\end{proof}
\section{Application to four different \texorpdfstring{\eqref{eq:sfom}s}{SFOMs}}
\label{sec:examples}
In this section we illustrate this methodology on four examples: heavy-ball (HB), Nesterov accelerated gradient (NAG), gradient descent (GD) with inexact gradients, and three-operator splitting (TOS). For each, we apply the methodology proposed in \Cref{sec:searching_for_cycles}. The code is available in the public github repository \url{https://github.com/bgoujaud/cycles}. Since ingredients are the same as those of classical PEPs, we also use the python package \texttt{Pepit}~\cite{goujaud2022pepit}. We perform a grid search over the spaces of \textit{parameters of interest}~$\Omega$, described in the respective subsections.
We compare the parameter region where Lyapunov functions can be obtained with the region in which we establish that the method cannot have a guaranteed worst-case convergence (due to the existence of cycles). More precisely, in \Cref{fig:hb,fig:nag,fig:igd,fig:tos} below, green regions corresponds to parameter choices for which the methods converge (existence of a Lyapunov function, found using the code provided with~\cite{taylor2018lyapunov}). Conversely, in the red regions, our methodology establishes that the method cycles on at least one function of $\mathcal{F}$.
In short, the algorithms converge in the green regions and do not converge in the worst-case in the red ones.
Note that some parameters for which the algorithm $\mathcal A$ admits a worst-case convergence guarantee could theoretically exist outside the green region: indeed, in~\cite{taylor2018lyapunov}, the authors do not guarantee that they necessarily find convergence.
Similarly, parameters for which $\mathcal A$ does \textit{not} admit a worst-case convergence guarantee could theoretically exist outside of the red region: indeed
\eqref{eq:problem} is defined for a fixed cycle length $K$, and we therefore run it several times with different values of~$K$. Longer cycles are therefore not detected. Moreover, the non-existence of cycles does not necessarily imply that the algorithm always converges.
Interestingly, in practice, we observe on~\Cref{fig:nag,fig:igd} that the set $\Omega$ of parameters of interest is completely filled by the union of those 2 regions and that it is almost the case on~\Cref{fig:hb} (it may have been if we had searched for cycles of all lengths). As a consequence, we fully characterize for which tuning the algorithms admit a guaranteed worst-case convergence. On the contrary, there remains a significant gap between the red and the green regions in our last example, see~\Cref{fig:tos}.
\subsection{Heavy-ball}
The HB algorithm, as introduced by~\cite{polyak_gradient_1963}, corresponds to the following update, for a step-size parameter $\gamma$ and a momentum parameter $\beta$:
\begin{equation}
x_{t+1} = x_t + \beta (x_t - x_{t-1}) - \gamma \nabla f(x_t). \tag{HB} \label{eq:hb}
\end{equation}
Therefore \eqref{eq:hb} is an order-2 \eqref{eq:sfom}.
\begin{figure}\label{fig:hb}
\end{figure}
\textbf{Set $\Omega_{\mathrm{HB}}$ of parameters of interest:} HB converges on the set $\mathcal Q_{0,L}$ if and only if the parameters $\gamma, \beta$ verify $0 \leq \gamma \leq {2(1+\beta)}/{L} \leq {4}/{L}$~\cite{polyak_gradient_1963}. Note this condition enforces $-1 \leq \beta \leq 1$. Moreover, we restrict to $\beta \geq 0$ as $\beta<0$ is not an interesting setting (slowing down convergence with respect to GD). Therefore, we limit our analysis to this set of parameters.
\textbf{Interpretation.} The red area in \Cref{fig:hb} shows parameters where cycles of length $K \in \llbracket2; 25\rrbracket$ are found by our methodology.
The red color intensity indicates the length of the shortest cycle.
A striking observation is that the space $\Omega_{\mathrm{HB}}$ is almost filled by the union of the red area and green one (where Lyapunov functions exist). Thereby, for almost all values of the parameters, we have a definitive answer on the existence of a certificate of convergence in the worst-case. That being said, there exists a small unfilled region in the top left corner (see the zoom on \Cref{fig:hb}) In this region, we do not know how HB behaves, and whether it accelerates. However, adding longer cycle length may enable to obtain cycles in that area. Indeed we considered only cycles of length $K \le 25$, for computational reasons.
\subsection{Nesterov accelerated gradient}
\begin{figure}\label{fig:nag}
\end{figure}
NAG (also known as the \textit{fast gradient method}) was introduced by~\cite{Nest03a} and corresponds to the following update, for a step-size parameter~$\gamma$ and a momentum parameter $\beta$:
\begin{equation}
\left \lbrace \begin{array}{ccl}
y_t &= & x_t + \beta (x_t - x_{t-1}), \\
x_{t+1} & =& y_t - \gamma \nabla f(y_t).
\end{array}\right.
\tag{NAG} \label{eq:nag}
\end{equation}
\eqref{eq:nag} is also written as follows
$y_{t+1} = (1 + \beta) (y_t - \gamma \nabla f(y_t)) - \beta (y_{t-1} - \gamma \nabla f(y_{t-1}))$,
and is therefore also an order-2 \eqref{eq:sfom}.
\textbf{Set $\Omega_{\mathrm{NAG}}$ of parameters of interest:} As for HB, we consider the set of $\beta$ and $\gamma$ for which~\eqref{eq:nag} converges on $\mathcal Q_{0,L}$. This corresponds to considering all $\beta, \gamma$ verifying $0\leq\beta\leq 1$ and $0\leq{\gamma}\leq \tfrac{2}{L}\tfrac{1 + \beta}{1+2\beta}$.
\textbf{Interpretation:} \eqref{eq:nag} is known to converge, with an accelerated rate, on $\mathcal F_{\mu,L}$, for the tuning $(\gamma,\beta)= (\frac{1}{L}, \tfrac{\sqrt{L} - \sqrt{\mu}}{\sqrt{L} + \sqrt{\mu}})$, that optimizes the convergence rate on $\mathcal Q_{\mu,L}$. For this reason,~\eqref{eq:nag} is considered to be more ``robust'' than HB.
\Cref{fig:nag} shows that \eqref{eq:nag} admits a Lyapunov function for almost any parameters in $\Omega_{\mathrm{NAG}}$.
Moreover, our methodology does not detect any set of parameters at which a cycle of length $K \in \llbracket 2; 25\rrbracket$ exists, apart on the boundary $\{(\gamma, \beta), \gamma= \tfrac{2}{L}\tfrac{1 + \beta}{1+2\beta}\}$. On the boundary, cycles of length 2 are observed, whose existences are theoretically verified on one-dimensional quadratic functions. This illustrates the robustness of our methodology when very few cycles exist.
\subsection{Inexact gradient method}
Next, we consider the inexact gradient method, parameterized by $\gamma$ and $\varepsilon$ and the update
\begin{equation}
\begin{array}{rl}
\text{Get } \mathcal{O}(x_t)\ = & d_t \ \text{ such that } \|d_t - \nabla f (x_t)\| \leq \varepsilon \|\nabla f (x_t)\|, \\
x_{t+1} \ = & x_t - \gamma d_t.
\end{array}
\tag{IGD} \label{eq:igd}
\end{equation}
\eqref{eq:igd} is thus an~\eqref{eq:sfom} of order 1.
\begin{figure}\label{fig:igd}
\end{figure}
\textbf{Set $\Omega_{\mathrm{IGD}}$ of parameters of interest:} Since the exact gradient method converges only for $\gamma < \frac{2}{L}$, we only consider such steps-sizes. Moreover, $\varepsilon \geq 1$ allows $d_t = 0$ and thereby does not make much sense. This motivates considering the set $\Omega_{\mathrm{IGD}}=\{(\gamma, \varepsilon) \in [0; \frac{2}{L}]\times [0,1]\}$.
\textbf{Interpretation:} We search for cycles of length $K \in \llbracket 2; 25\rrbracket$ and use color intensity to show the minimal cycle length. \eqref{eq:igd} is known to converge for any $\gamma \leq \frac{2}{L(1 + \varepsilon)}$~(see~\cite{de2020worst,gannot2022frequency}). \Cref{fig:igd} shows that the complementary of this region of convergence is completely filled by parameters allowing cycles, showing that no other parameters values than the known ones allow obtaining worst-case convergence of \eqref{eq:igd}.
\subsection{Three-operator splitting}\label{subsec:tos}
\begin{figure}\label{fig:tos}
\end{figure}
The three-operator splitting (TOS) method, introduced by \cite{davis2017three}, aims at solving the \emph{inclusion problem} $0 \in Ax + Bx + \partial f(x)$, where $A$ is a monotone operator, $B$ is a co-coercive operator and $\partial f$ denotes the differential of the smooth (strongly) convex function $f$.
It corresponds to the following update, for a step-size parameter $\gamma$, a smoothing parameter $\alpha$ and an update parameter $\beta$:
\begin{equation}
\left \lbrace
\begin{array}{ccl}
x_{t+1} & = & J_{\alpha B} (w_t), \\
y_{t+1} & = & J_{\alpha A} \left(2 x_{t+1} - w_t - \frac{\gamma}{\beta} \nabla f(x_{t+1})\right), \\
w_{t+1} & = & w_t - \beta (x_{t+1} - y_{t+1}),
\end{array}\right.
\tag{TOS} \label{eq:tos}
\end{equation}
where $J_{O}$ denotes the resolvent of the operator $O$, i.e. $J_O = (I + O)^{-1}$.
Note that \eqref{eq:tos} is therefore an order-1 \eqref{eq:sfom}.
\textbf{Set $\Omega_{\mathrm{TOS}}$ of parameters of interest:} When considering $A$, $B$ and $\nabla f$ to be linear symmetric and co-diagonalisable operators, the set of convergence of \eqref{eq:tos} is $\Omega_{\mathrm{TOS}} = \left\{ (\gamma, \beta) \in \left[0, \frac{2}{L}\right] \times [0, 2] \right\}$, which we therefore consider.
\textbf{Interpretation:} We search for cycles of length $K\in\llbracket 2, 25\rrbracket$ and use color intensity to show the minimal cycle length. Interestingly, there is a gap between the green region and the red ones. Unlike for \eqref{eq:hb}, it seems that increasing the length of the cycle does not help covering this gap and shows that some algorithms might have no Lyapunov function while not cycling. Understanding the behavior of \eqref{eq:tos} in the grey region is therefore still an open question.
\section{Conclusions}
\label{sec:conclusions}
\textbf{Summary.}
This work proposes a systematic approach for finding counter-examples to convergence of first-order methods, bringing a complementary tool to the existing systematic techniques for finding convergence guarantees (that include certifications through the existence of Lyapunov functions). Our approach is based on now classical tools and techniques used in the field of first-order optimization and a few existing packages~\cite{goujaud2022pepit,taylor2017performance} allows for straightforward implementations of our methodology.
\textbf{Discussion and Future works.}
While our analysis complements the Lyapunov one, existence of a Lyapunov function or existence of a cycle are not the only 2 options. In this section, we discuss the eventuality that an algorithm diverges on at least one function in a class, without resulting in cycles.
Indeed, the sequence of iterates produced by an algorithm on a given function may diverge by (i)~tending to infinity, (ii)~simply growing unbounded, or even (iii)~showing a chaotic behavior, while staying in a compact set.
Understanding if those three divergence cases might occur on functions $f\in \mathcal F$ while \textit{no} function $f\in \mathcal F$ results in a cycle, is thus an interesting open question:
\begin{center}
\fbox{\parbox{0.48\textwidth}{
\begin{center}
If the class $\mathcal{F}$ is connected, the update $A$ of an algorithm $\mathcal{A}$ and the oracle $\mathcal{O}$ are continuous, if $\mathcal{A}$ converges on one function of $\mathcal{F}$ and diverges on another one, is there necessarily one function of $\mathcal{F}$ which $\mathcal{A}$ cycles on?
\end{center}
}}
\end{center}
An interesting example is $\mathcal{A}$ being GD with step-size 1 on the $1+\rho$-smooth function $f_\rho$, such that $f_\rho(x)$ is equal to:
\begin{equation*}
\left\{
\begin{array}{ll}
\tfrac{\rho}{3} |x|^3 + \tfrac{1-\rho}{2} x^2 + \frac{(\rho-1)^3}{6\rho^2} & \text{if } |x| \leq 1, \\
-\tfrac{\rho}{3} |x|^3 + \tfrac{1+\rho}{2} x^2 - 2 \rho |x|+ \frac{(\rho-1)^3 + 4\rho^3}{6\rho^2} & \text{if } 1 \leq |x| \leq \tfrac{3}{2}, \\
\tfrac{1}{2}x^2 + \tfrac{\rho}{4} |x| + \frac{4(\rho-1)^3+11\rho^3}{24\rho^2} & \text{if } \tfrac{3}{2} \leq |x|,
\end{array}
\right.
\end{equation*}
for $\rho \in [0, 4]$.
From any point in the interval $(1, \infty)$, the next iterate is in $[-1, 0]$.
Similarly, starting in the interval $(-\infty, -1)$, the second iterate is in $[0, 1]$.
Those 2 intervals are stable and, by symmetry of the function, the dynamics in those 2 intervals are themselves symmetric.
Note that for any $x\in[0, 1]$, $f'(x) = \rho x^2 + (1-\rho)x$, leading to the dynamic $x_{t+1} = x_t- 1\times \nabla f_\rho(x_t)= \rho x_t (1-x_t)$, known as \emph{logistic map}.
The behavior of this dynamic is highly dependent on the value of $\rho$. On the first hand, for $\rho< 3$, $\mathcal A(f_\rho, x_0)$ converges for any $x_0$. On the other hand, for almost all values of $\rho$ close enough to 4, this dynamic is chaotic.
Note however, that for any $\rho_0<4$, there exists $\rho > \rho_0$ such that Gradient descent with step-size 1 cycles on $f_\rho$.
Therefore, on the class $\left\{f_\rho, \rho\in [0, 4]\right\}$ we have: functions over which $\mathcal A$ converges, functions over which it diverges because it is chaotic, but also functions over which it cycles. This example thus shows that those behaviors can co-exist, but does not provide an answer to the open question.
\textsc{Acknowledgments.}{\small{}The authors thank Margaux Zaffran for her feedbacks and fruitful discussions and her assistance in making plots. The work of B. Goujaud and A. Dieuleveut is partially supported by ANR-19-CHIA-0002-01/chaire SCAI, and Hi!Paris. A.~Taylor acknowledges support from the European Research Council (grant SEQUOIA 724063). This work was partly funded by the French government under management of Agence Nationale de la Recherche as part of the ``Investissements d’avenir'' program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA Institute).}
\end{document}
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\begin{document}
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\def{\cyss Yu}{{\tencyss\cyracc Yu}}
\centerline{\bbf Compact 16-dimensional planes. An update}
\par
\hspace{160pt} {by Helmut R. \sc Salzmann}
\par
\begin{abstract} This paper is an addition to the book \cite{cp} on {\it Compact projective planes\/}. Such planes, if connected and finite-dimensional, have a point space of topological dimension 2, 4, 8, or 16, the classical example in the last case being the projective closure of the affine plane over the octonion algebra. The final result in the book (which was published 20 years ago) is a complete description of all planes admitting an automorphism group of dimension at least 40. Newer results on 8-dimensional planes have been collected in \cite{sz1}. Here, we present a classification of 16-dimensional planes with a group of dimension ${\hskip-3pt\ge}35$, provided the group does not
fix exactly one flag, and we prove several further theorems, among them criteria for a
connected group of automorphisms to be a Lie group. My sincere thanks are due to
Hermann H\"ahl for many fruitful discussions. \end{abstract} \par
{\Bf 1. Introduction} \par
The last section {\it ``Principles of classification''\/} of the treatise {\it Compact projective planes\/} \cite{cp} consists of a detailed programme how to determine all sufficiently homogeneous compact $16$-dimensional topological planes. Here, we shall survey and amplify the results that have been achieved in the meantime. For $2$- and $4$-dimensional planes see \cite{cp} \S\S\ 3 and 7; newer developments in the $8$-dimensional case are summarized in \cite{sz1}. It is an open problem if the dimension of a compact projective plane is necessarily finite; the point space of any finite dimensional compact plane has necessarily dimension $0$ or $2^m$ with $m{\,\in\,}\{1,2,3,4\}$,\; see \cite{cp} 41.10 and 54.11. \par
The classical $16$-dimensional plane ${\cal O}$ is the projective closure of the affine plane over the (bi-associative) division algebra ${\fam=\Bbbfam O}$ of the (real) octonions; for the properties of ${\fam=\Bbbfam O}$ see 2.5 below or \cite{cp} \S\,1, cf. also \cite{eb} and \cite{es}. The automorphism group $\mathop{{\rm Aut}}{\cal O}$ is isomorphic to the $78$-dimensional exceptional simple Lie group ${\rm E}_6(-26)$, it is transitive on the set of quadrangles of ${\cal O}$ (cf. \cite{cp} 17.\,6,\,7,\,10). In fact, ${\cal O}$ is the only compact $16$-dimensional plane admitting a point-transitive group ${\fam=\ssfam \usuDelta}$ of automorphisms (L\"owen \cite{lw1}). Such a group ${\fam=\ssfam \usuDelta}$ necessarily contains the compact flag-transitive elliptic motion group of ${\cal O}$, see \cite{cp} 63.8. \par
Let ${\cal P}{\,=\,}(P,{\frak L})$ be a compact projective plane with a point space $P$ of (covering) dimension $\dim P{\,=\,}16$. Taken with the compact-open topology, the automorphism group ${\fam=\ssfam \usuSigma}{\,=\,}\mathop{{\rm Aut}}{\cal P}$ of all continuous collineations is a locally compact transformation group of $P$ of finite dimension $\dim{\fam=\ssfam \usuSigma}$ with a countable basis, see \cite{cp} 44.3 and 83.26, and $\dim{\fam=\ssfam \usuSigma}{\,\ge\,}n$ if, and only if, ${\fam=\ssfam \usuSigma}$ contains a euclidean $n$-ball (\cite{cp} 93.\,5 and 6). In all known cases ${\fam=\ssfam \usuSigma}$ is even a Lie group, see also 2.3 below. \par
Homogeneity of ${\cal P}$ can best be expressed by the size of ${\fam=\ssfam \usuSigma}$ measured by its dimension. The most homogeneous planes ${\cal P}$ are known explicitly: if $\dim{\fam=\ssfam \usuSigma}{\,\ge\,}40$, then ${\cal P}$ is the projective closure of an affine plane coordinatized by a {\it mutation\/} ${\fam=\Bbbfam O}_{(t)}{\,=\,}({\fam=\Bbbfam O},+,\circ)$ of the octonions, where $c{\hskip1pt\circ}z{\,=\,}t{\cdot}cz{\,+\,}(1{-}t){\cdot}zc$ with $t{\,>\,}\frac{1}{2}$; either $t{\,=\,}1$, $\,{\fam=\Bbbfam O}_{(t)}{\,=\,}{\fam=\Bbbfam O}$, and ${\cal P}{\,\cong\,}{\cal O}$, or $\dim{\fam=\ssfam \usuSigma}{\,=\,}40$ and ${\fam=\ssfam \usuSigma}$ fixes the line at infinity and some point on this line (\cite{cp} 87.7). \par
The ultimate goal is to describe all pairs $({\cal P},{\fam=\ssfam \usuDelta})$, where ${\fam=\ssfam \usuDelta}$ is a {\it connected\/} closed subgroup of $\mathop{{\rm Aut}}{\cal P}$ and $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}b$ for a suitable bound $b$ in the range $27{\,\le\,}b{\,<\,}40$. This bound varies with the structure of ${\fam=\ssfam \usuDelta}$ and the configuration ${\cal F}_{\fam=\ssfam \usuDelta}$ of the fixed elements (points and lines) of ${\fam=\ssfam \usuDelta}$. In some cases, only ${\fam=\ssfam \usuDelta}$ can be determined, but there seems to be no way to find an explicit description of all the corresponding planes. \par
{\Bf 2. Background} \par
In the following, ${\cal P}$ will always denote a compact $16$-dimensional topological projective plane with point space $P$ and line space ${\frak L}$, if not stated otherwise. Note that line pencils ${\frak L}_p{\,=\,}\{L{\,\in\,}{\frak L} \mid p{\,\in\,}L\}$ are homeomorphic to lines. Hence the dual of ${\cal P}$ is also a compact $16$-dimensional plane. \par
{\bf Notation} is standard and agrees with that in the book \cite{cp}. A {\it flag\/} is an incident point-line pair. A topological plane with a $2$-dimensional point space will be called {\it flat\/}. For a locally compact group ${\fam=\ssfam \usuGamma}$ and a closed subgroup ${\fam=\ssfam \usuDelta}$ the coset space $\{{\fam=\ssfam \usuDelta}\gamma\mid\gamma{\,\in\,}{\fam=\ssfam \usuGamma}\}$ will be denoted by ${\fam=\ssfam \usuGamma}/{\fam=\ssfam \usuDelta}$, its dimension $\dim{\fam=\ssfam \usuGamma}{-}\dim{\fam=\ssfam \usuDelta}$ by ${\fam=\ssfam \usuGamma}{:\hskip2pt}{\fam=\ssfam \usuDelta}$. As in \cite{cp} 94.1, the fact that the topological groups ${\fam=\ssfam \usuGamma}$ and ${\fam=\ssfam \usuDelta}$ are {\it locally\/} isomorphic\ will sometimes be symbolized by ${\fam=\ssfam \usuGamma}{\,\circeq\,}{\fam=\ssfam \usuDelta}$. As customary, $\Cs{\fam=\ssfam \usuDelta}{\fam=\ssfam \usuUpsilon}$ or just~$\Cs{}{\fam=\ssfam \usuUpsilon}$ is the centralizer of ${\fam=\ssfam \usuUpsilon}$ in ${\fam=\ssfam \usuDelta}$; the center $\Cs{}{\fam=\ssfam \usuDelta}$ is usually denoted by ${\fam=\ssfam Z}$. Distinguish between the connected component ${\fam=\ssfam \usuDelta}^{\hskip-1.5pt1}$ and the commutator subgroup ${\fam=\ssfam \usuDelta}'$ of ${\fam=\ssfam \usuDelta}$. A {\it Levi complement\/} of the radical $\sqrt{\fam=\ssfam \usuDelta}$ is a maximal semi-simple
subgroup of ${\fam=\ssfam \usuDelta}$. If $M^{\fam=\ssfam \usuGamma}{\,=\,}M$, then ${\fam=\ssfam \usuGamma}|_M$ is the group induced by ${\fam=\ssfam \usuGamma}$ on $M$. The configuration of all fixed points and fixed lines of a subset ${\fam=\ssfam \usuXi}{\,\subseteq\,}{\fam=\ssfam \usuDelta}$ will be denoted by ${\cal F}_{\fam=\ssfam \usuXi}$. We write ${\fam=\ssfam \usuDelta}_{[A]}$ for the subgroup of all {\it axial} collineations in ${\fam=\ssfam \usuDelta}$ with axis $A$\; (i.e., of collineations fixing $A$ pointwise) and, dually, ${\fam=\ssfam \usuDelta}_{[c]}$ for the subgroup of collineations with center $c$. Let ${\fam=\ssfam \usuDelta}_{[c,A]}{\,=\,}{\fam=\ssfam \usuDelta}_{[c]}{\kern 2pt {\scriptstyle \cap}\kern 2pt}{\fam=\ssfam \usuDelta}_{[A]}$ and put ${\fam=\ssfam \usuDelta}_{[C,A]}{\,=\,}\bigcup_{c\in C}{\fam=\ssfam \usuDelta}_{[c,A]}$. A subgroup ${\fam=\ssfam \usuGamma}{\,\le\,}{\fam=\ssfam \usuDelta}$ is called {\it straight\/}, if each point orbit $x^{\fam=\ssfam \usuGamma}$ is contained in some line; by a result of Baer \cite{ba}, the group ${\fam=\ssfam \usuGamma}$ is then contained in ${\fam=\ssfam \usuDelta}_{[c,A]}$ for some center~$c$ and axis $A$, or ${\cal F}_{{\fam=\ssfam \usuGamma}}$ is a Baer subplane (see 2.2). An element $\gamma$ is straight, if it generates a straight cyclic group $\langle \gamma\rangle$, and $\gamma$ is said to be {\it planar\/}, if ${\cal F}_{\hskip-1pt\gamma}$ is a subplane. \par
{\bf 2.1 Topology.} {\it Each line $L$ of ${\cal P}$ is homotopy equivalent $(\simeq)$ to an $8$-sphere ${\fam=\Bbbfam S}_8$\/}. \\ This fundamental theorem is due to L\"owen \cite{Lw}, see also \cite{cp} 54.11. If $L$ is a manifold, then $L$ is homeomorphic ($\approx)$ to ${\fam=\Bbbfam S}_8\,$ (\cite{cp} 52.3). In this case, the spaces $P$ and ${\frak L}$ are homeomorphic (Kramer \cite{kr2}). So far, no example with $L{\,\not\approx\,}{\fam=\Bbbfam S}_8$ has been found. Furthermore, lines have the following properties: \\ {\it Each line $\,pq\,$ is a locally homogeneous and locally contractible homology manifold, and it has domain invariance\/}, see \cite{cp} 51.12, 54.10, and 51.21. Moreoer, $pq\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{p,q\}{\,\simeq\,}{\fam=\Bbbfam S}_7$ by \cite{cp} 52.5 combined with 54.11. \\ The Lefschetz fixed point theorem implies that each homeomorphism $\phi{\,:\,}P{\,\to\,}P$ has a fixed point. By duality, {\it each automorphism of ${\cal P}$ fixes a point and a line\/}, see \cite{cp} 55.\,19,\,45. \par
{\bf 2.2 Baer subplanes.} {\it Each $8$-dimensional $closed$ subplane ${\cal B}$ of ${\cal P}$ is a Baer subplane\/}, i.e., each point of ${\cal P}$ is incident with a line of ${\cal B}$ (and dually, each line of ${\cal P}$ contains a point of ${\cal B}$), see \cite{sz3} \S~\hskip-3pt3 or \cite{cp} 55.5 for details. By a result of L\"owen \cite{lw2}, any two closed Baer subplanes of ${\cal P}$ have a point and a line in common. This is remarkable because a finite Pappian plane of order $q^2$ is a union of $q^2{-}q{+}1$ disjoint Baer subplanes, see \cite{sz3} 2.5. Generally, $\langle {\cal M}\rangle$ will denote the smallest {\it closed\/} subplane of ${\cal P}$ containing the set ${\cal M}$ of points and lines. We write ${\cal B}{\,\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$ if ${\cal B}$ is a Baer subplane and ${\cal B}{\,\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\!}{\cal P}$ if $\,8\vert\dim{\cal B}$. \par
{\bf 2.3 Groups.} {\it Any {\rm connected} subgroup ${\fam=\ssfam \usuDelta}$ of ${\fam=\ssfam \usuSigma}{\,=\,}\mathop{{\rm Aut}}{\cal P}$ with $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}27$ is a Lie group\/}, see \cite{psz}. For $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}35$, the result has been proved in \cite{cp} 87.1. If $\dim{\fam=\ssfam \usuSigma}{\,\ge\,}29$, then ${\fam=\ssfam \usuSigma}$ itself is a Lie group, cf. \cite{sz4}. In particular, ${\fam=\ssfam \usuDelta}$ is then either semi-simple, or ${\fam=\ssfam \usuDelta}$ has a central torus subgroup or a minimal normal subgroup ${\fam=\ssfam \usuTheta}{\,\cong\,}{\fam=\Bbbfam R}^k$, see \cite{cp} 94.26. \par
{\bf 2.4 Coordinates.} Let ${{\frak e}}{\,=\,}(o,e,u,v)$ be a (non-degen\-erate) quadrangle in ${\cal P}$. Then the affine subplane ${\cal P}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt uv$ can be coordinatized with respect to the {\it frame\/} ${\frak e}$ by a so-called {\it ternary field\/} $K_\tau$, where $K$ is homeomorphic to an affine line, the affine point set is written as $K{\times}K$, lines are given by an equation $y{\,=\,}\tau(s,x,t)$ or $x{\,=\,}c$ (verticals), they are denoted by $[\hskip1pt s,t\hskip1pt]$ or $[\hskip1pt c\hskip1pt]$, respectively. Parallels are either vertical or both have the same {\it``slope''\/} $\hskip-2pt s$. The axioms of an affine plane can easily~be translated into properties of $\tau$, see \cite{cp} \S\hskip2pt22. Write $e{\,=\,}(1,1)$ and put $x{\,+\,}t{\,=\,}\tau(1,x,t)$. In the classical case $\tau(s,x,t){\,=\,}s\hskip1pt x+t$. A~ternary field $K_\tau$ is called a {\it Cartesian field\/} if $(K,+)$ is a group and if identically $\tau(s,x,t){\,=\,}s{\raise1pt\hbox{$\scriptscriptstyle\bullet$}}x+t$ with multiplication $s{\raise1pt\hbox{$\scriptscriptstyle\bullet$}}x{\,=\,}\tau(s,x,0)$; equivalently, the maps $(x,y)\mapsto(x,y{+}t)$ form a transitive group of translations. Any ternary field $K_\tau$ such that $K$ is homeomorphic to ${\fam=\Bbbfam R}^8$ and $\tau{\,:\,} K^3{\,\to\,} K$ is continuous yields a compact projective plane ${\cal P}$,
cf. \cite{cp} 43.6. \par
{\bf 2.5 Octonions.} The division algebra ${\fam=\Bbbfam O}$ of the octonions can be defined as the real vector space ${\fam=\Bbbfam H}^2$ with the multiplication $(a,b)(x,y){\,=\,}(ax{-}y\overline b, xb{+}\overline ay)$. Put $\overline{(a,b)}{\,=\,}(\overline a,-b)$. {\it Conjugation\/} $c{\,\mapsto\,}\overline c$ is an anti-automorphism of ${\fam=\Bbbfam O}$;
as usual, {\it norm\/} and {\it orthogonality\/} are given by $\|c\|^2{\,=\,}c\overline c$ and
$c{\,\perp\,}d{\,\Leftrightarrow\,}\|c{+}d\|{\,=\,}\|c{-}d\|$. Sometimes it is convenient to write $\ell{\,=\,}(0,1)$ and ${\fam=\Bbbfam O}{\,=\,}{\fam=\Bbbfam H}{+}\ell{\fam=\Bbbfam H}$. Any two elements of ${\fam=\Bbbfam O}$ are contained in an associative subfield: ${\fam=\Bbbfam O}$ is {\it bi-associative\/}. The center of ${\fam=\Bbbfam O}$ consists of the real field ${\fam=\Bbbfam R}{\,\subset\,}{\fam=\Bbbfam C}{\times\{0\}}$; if $c$ belongs to the orthogonal complement ${\fam=\Bbbfam R}^\perp$, then $c$ is called a {\it pure\/} element. An arbitrary octonion $c$ will also be written in the form $c{\,=\,}c_0{\,+\,}{\frak z}$ with $c_0{\,=\,}{\rm re}\,c{\,=\,}\frac{1}{2}(c{\,+\,}\overline c)$. The group ${\fam=\ssfam \usuGamma}{\,=\,}\mathop{{\rm Aut}}{\fam=\Bbbfam O}$ is the $14$-dimensional compact simple Lie group ${\rm G}_2$, it is transitive on the unit sphere ${\fam=\Bbbfam S}_6$ in ${\fam=\Bbbfam R}^\perp$, and ${\fam=\ssfam \usuGamma}_{\hskip-1.5pt i}$ is transitive on ${\fam=\Bbbfam S}_5{\,\subseteq\,}{\fam=\Bbbfam C}^\perp$\ (\cite{cp} 11.31--35). Obviously, $\lambda{\,:\,}(x,y){\,\mapsto\,}(x,-y){\,\in\,}{\fam=\ssfam \usuGamma}$, and easy verification shows~that $$\Cs{\fam=\ssfam \usuGamma}\lambda{\,=\,}\{(x,y){\,\mapsto\,}(x^a,y^ab)\mid a,b{\,\in\,} {\fam=\Bbbfam H}{\,\hskip2pt{\scriptstyle \land}\hskip2pt\,} a\overline a{\,=\,}b\overline b{\,=\,}1\}{\,\cong\,}\SO4{\fam=\Bbbfam R}\,.$$ \break Each involution in ${\fam=\ssfam \usuGamma}$ is conjugate to $\lambda$ \ (see \cite{cp} 11.31d). Note that the commutator subgroup of ${\fam=\Bbbfam H}^{\times}$ is ${\fam=\Bbbfam H}'{\,=\,} \{c{\,\in\,}{\fam=\Bbbfam H}\mid c\overline c{\,=\,}1\}{\,\cong\,}\Spin3{\fam=\Bbbfam R}{\,\cong\,}\SU2{\fam=\Bbbfam C}{\,\cong\,}\U1{\fam=\Bbbfam H}$.
\par
{\bf 2.6 Stiffness} refers to the fact that the dimension of the stabilizer ${\fam=\ssfam \usuLambda}$ of a (non-degenerate) quadrangle ${\frak e}$ cannot be very large. The group ${\fam=\ssfam \usuLambda}$ can be interpreted as the automorphism group of the ternary field $K_\tau$ defined with respect to ${\frak e}$. The classical plane is coordinatized by the octonion algebra ${\fam=\Bbbfam O}$; in this case ${\fam=\ssfam \usuLambda}$ is the $14$-dimensional exceptional compact simple Lie group ${\rm G}_2{\,\cong\,}\mathop{{\rm Aut}}{\fam=\Bbbfam O}$ for {\it each\/} choice of ${\frak e}$, cf. \cite{cp} 11.\,30--35. In the general case, let ${\fam=\ssfam \usuLambda}$ denote the connected component of ${\fam=\ssfam \usuDelta}_{\frak e}$. Then the following holds: \par \quad (a) {\it ${\fam=\ssfam \usuLambda}{\,\cong\,}{\rm G}_2$ and ${\cal F}_{\fam=\ssfam \usuLambda}$ is flat, or $\dim{\fam=\ssfam \usuLambda}{\,<\,}14$ \/}\ (\cite{sz6},\,\cite{cp} 83.\,23,\,24), \par \quad (\^a) {\it ${\fam=\ssfam \usuLambda}{\,\cong\,}{\rm G}_2$ or $\dim{\fam=\ssfam \usuLambda}{\,\le\,}11$\/}\ (\cite{B1} 4.1), \par \quad (b) {\it if ${\cal F}_{\fam=\ssfam \usuLambda}$ is a Baer subplane, then ${\fam=\ssfam \usuDelta}_{\frak e}$ is compact and $\dim{\fam=\ssfam \usuLambda}{\,\le\,}7$\/}\ (\cite{cp} 83.6), \par \quad (\^b) {\it if ${\cal F}_{\fam=\ssfam \usuLambda}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$ and ${\fam=\ssfam \usuLambda}$ is a Lie group, then ${\fam=\ssfam \usuLambda}{\,\cong\,}\Spin3{\fam=\Bbbfam R}$ or $\dim{\fam=\ssfam \usuLambda}{\,\le\,}1$\/}\ (\cite{cp} 83.22), \par \quad (c) {\it if there exists a subplane ${\cal B}$ with ${\cal F}_{\fam=\ssfam \usuLambda}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal B}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$, then ${\fam=\ssfam \usuLambda}$ is compact\/} \ (\cite{sz6} 2.2, \cite{cp} 83.9), \par \quad (\^c) {\it if ${\fam=\ssfam \usuLambda}$ contains a pair of commuting involutions, then ${\fam=\ssfam \usuDelta}_{\frak e}$ is compact\/}\ (\cite{cp} 83.10), \par \quad (d) {\it if ${\fam=\ssfam \usuLambda}$ is compact or semi-simple or if ${\cal F}_{\fam=\ssfam \usuLambda}$ is connected, then ${\fam=\ssfam \usuLambda}{\,\cong\,}{\rm G}_2$ or $\dim{\fam=\ssfam \usuLambda}{\,\le\,}10$\/}\par \hskip30pt (\cite{gs} XI.9.8, \cite{sz6} 4.1, \cite{B1} 3.5), \par \quad (e) {\it if ${\fam=\ssfam \usuLambda}$ is compact, or if $\dim{\cal F}_{\fam=\ssfam \usuLambda}{\,=\,}4$, or if ${\fam=\ssfam \usuLambda}$ is a Lie group and ${\cal F}_{\fam=\ssfam \usuLambda}$ is connected, then \par \hskip30pt ${\fam=\ssfam \usuLambda}{\,\cong\,}{\rm G}_2$ or ${\fam=\ssfam \usuLambda}{\,\cong\,}\SU3{\fam=\Bbbfam C}$ or $\dim{\fam=\ssfam \usuLambda}{\,<\,}8$\/}\
(\cite{sz6} 2.1; \cite{sz7}, \cite{B1} 3.5; \cite{B2}), \par \quad (\^e) {\it if ${\fam=\ssfam \usuLambda}$ is a compact Lie group, then ${\fam=\ssfam \usuLambda}{\,\cong\,}{\rm G}_2,\; \SU3{\fam=\Bbbfam C},\; \SO4{\fam=\Bbbfam R}$, or $\dim{\fam=\ssfam \usuLambda}{\,\le\,}4$\/}\,\ (\cite{sz6} 2.1), \par \quad (f) {\it if ${\fam=\ssfam A}$ is the automorphism group of a locally compact $8$-dimensional {\emph double loop}, then \par\hskip30pt $\dim{\fam=\ssfam A}{\,\le\,}16$\/}\ (\cite{bd}; for the notion of a double loop cf. \cite{gs} XI. \,\S\S\,1,\,8, and 9). \par
{\tt Remark.} The proof of \cite{cp} 83.10 refers to 83.9, which has been stated only for {\it connected\/} groups, but 83.8 can be used instead. \par
{\bf 2.7 Dimension formula.} By \cite{hal} or \cite{cp} 96.10, the following holds for the action of ${\fam=\ssfam \usuDelta}$ on $P$ or on any closed ${\fam=\ssfam \usuDelta}$-invariant subset $M$ of $P$, and for any point $a{\,\in\,}M$: $$\dim{\fam=\ssfam \usuDelta}{\,=\,}\dim{\fam=\ssfam \usuDelta}_a{\,+\,}\dim a^{\fam=\ssfam \usuDelta} \quad {\rm or} Ê\quad \dim a^{\fam=\ssfam \usuDelta}{\,=\,}{\fam=\ssfam \usuDelta}{\,:\,}{\fam=\ssfam \usuDelta}_a\,.$$ \par
{\bf 2.8 Corollary: Fixed configurations.} {\it If $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}27$, then ${\fam=\ssfam \usuDelta}$ fixes at most a triangle or, up to duality, some collinear points and one further line\/}. \\ Thus, there are only $9$ possibilities for ${\cal F}_{\fam=\ssfam \usuDelta}$, again up to duality: \\ (a) $\emptyset$, \quad (b) $\{W\}$, $W{\in\,}{\frak L}$, \quad (c) {\it flag\/} $\{v,W\}$, $v{\hskip1pt\in\hskip1pt}W$, \quad (d) $\{o,W\}$, $o{\hskip1pt\notin\hskip1pt}W$, \quad (e) $\{u,v,uv\}$,\\ (f) $\{u,v,w,...,uv\}$,\quad (g) {\it double flag\/} $\{u,v,ov,uv\}$,\quad (h) $\{u,v,w,...,uv,ov\}$,\quad (i) {\it triangle\/}. \\ These cases will be treated separately. \par
{\bf 2.9 Reflections and translations.} {\it Let $\sigma$ be a reflection with axis $W$ and center~$c$ in the connected group ${\fam=\ssfam \usuDelta}$, and let ${\fam=\ssfam T}$ denote the group of translations in ${\fam=\ssfam \usuDelta}$ with axis~$W$. If $W^{\fam=\ssfam \usuDelta}{\,=\,}W$ and $\dim c^{\fam=\ssfam \usuDelta}{\,=\,}k{\,>\,}0$, then $\,\sigma^{\fam=\ssfam \usuDelta}\sigma{\,=\,}{\fam=\ssfam T}{\,\cong\,}{\fam=\Bbbfam R}^k$, $\,\tau^\sigma{\,=\,}\tau^{-1}$ for each $\tau{\,\in\,}{\fam=\ssfam T}$, and $k$ is even\/}. This improves \cite{cp} 61.19b; a detailed {\tt proof} can be found in \cite{sz2} Lemma 2. \par
{\bf 2.10 Involutions.} Each involution $\iota$ is either a reflection or it is {\it planar\/} (${\cal F}_{\hskip-1.5pt\iota}\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}{\cal P}$), see \cite{cp} 55.29. Commuting involutions with the same fixed point set are identical (\cite{cp} 55.32). Let ${\fam=\Bbbfam Z}_2^{\hskip3pt r}{\,\cong\,}{\fam=\ssfam \usuPhi}{\,\le\,}\mathop{{\rm Aut}}{\cal P}$ and $\dim P{\,=\,}2^m$. Then $r{\,\le\,}m{+}1$; if ${\fam=\ssfam \usuPhi}$ is generated by reflections, then $r{\,\le\,}2$, see \cite{cp} 55.34(c,b). If $r{\,\ge\,}m \ ({\,>\,}2)$, then ${\fam=\ssfam \usuPhi}$ contains a reflection (a planar involution), cf. \cite{cp} 55.34(d,b). Any torus group in $\mathop{{\rm Aut}}{\cal P}$ has dimension at most $m$,\, see \cite{cp} 55.37, for $m{\,=\,}4$ in particular, $\rm rk\,{\fam=\ssfam \usuDelta}{\,\le\,}4$, see also~\cite{cs}. The orthogonal group $\SO5{\fam=\Bbbfam R}$ cannot act non-trivially on ${\cal P}\,$ (\cite{cp} 55.40). \par
{\bf 2.11 Lemma.} {\it If ${\cal F}_{\fam=\ssfam \usuDelta}{\,\ne\,}\emptyset$, then ${\fam=\ssfam \usuDelta}$ has no subgroup ${\fam=\ssfam \usuPhi}{\,\cong\,}(\SO3{\fam=\Bbbfam R})^2$\/}. \par
{\tt Proof.} Each of the two simple factors of ${\fam=\ssfam \usuPhi}$ contains $3$ pairwise commuting conjugate involutions. Let them be denoted by $\alpha_\mu$ and $\beta_\nu$, respectively. If one of these triples consists of reflections, their centers $c_\mu$ form a triangle. Each fixed line of ${\fam=\ssfam \usuDelta}$ is incident with one of the centers $c_\mu$; on the other hand the $c_\mu$ are contained in a ${\fam=\ssfam \usuPhi}$-orbit. This contradiction shows that ${\fam=\ssfam \usuDelta}$ fixes no line, dually, ${\fam=\ssfam \usuDelta}$ has no fixed point. Hence the $\alpha_\mu$ and the $\beta_\nu$ are planar. Similarly, all products $\alpha_\mu\beta_\nu$ are planar, but the group $\langle \alpha_\mu,\beta_\nu\mid \mu,\nu{\,=\,}1,2,3\rangle{\,\cong\,}({\fam=\Bbbfam Z}_2)^4$ must contain a reflection by \cite{cp} 55.34(d) or 2.10. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\tt Remark.} The assumption ${\cal F}_{\fam=\ssfam \usuDelta}{\,\ne\,}\emptyset$ in Lemma 2.11 is indispensable, even in the case of $8$-dimensional planes. In fact, the stabilizer of the real subplane of the classical quaternion plane ${\cal P}_{\fam=\Bbbfam H}$ contains $\mathop{{\rm Aut}}{\cal P}_{\fam=\Bbbfam R}{\times}\mathop{{\rm Aut}}{\fam=\Bbbfam H}{\,\cong\,}\SL3{\fam=\Bbbfam R}{\times}\SO3{\fam=\Bbbfam R}$\: (cf. \cite{cp} 13.6). By \cite{cp}~ 86.31, the stabilizer of a quaternion subplane of ${\cal O}$ has a subgroup $\SL3{\fam=\Bbbfam H}$. Hence the stabilizer of a real subplane contains $\SL3{\fam=\Bbbfam R}$ and $\mathop{{\rm Aut}}{\fam=\Bbbfam O}{\,=\,}{\rm G}_2$ and their direct product (use 2.5). \par
{\bf 2.12 Homologies.} {\it If $W$ is the translation axis of a $16$-dimensional {\rm translation} plane, then any group ${\fam=\ssfam \usuDelta}_{[z,A]}$ of homologies with center $z{\,\in\,}W$ has dimension at most $1\,;$ more precisely, ${\fam=\ssfam \usuDelta}_{[z,A]}$ is discrete or an extension of $e^{{\fam=\Bbbfam R}}$ by a finite group\/} \ (\cite{bh} Th.\,2 and \cite{cp} 25.4). {\it In general a connected homology group of a compact connected projective plane is two-ended\/}: {\it it is either compact or a direct product of $e^{{\fam=\Bbbfam R}}$ with a compact group\/}, cf. \cite{cp}~61.2.
\par
{\bf 2.13 Lemma.} {\it Each commutative connected subgroup ${\fam=\ssfam H}{\,\le\,}\mathop{{\rm Aut}}{\cal P}$ fixes a point or a line$;$ if ${\cal P}$ is flat, then ${\fam=\ssfam H}$ fixes a point {\rm and} a line\/}. \par
{\tt Proof by induction on $\dim{\cal P}$.} Let $1\kern-2.5pt {\rm l}{\,\ne\,}\eta{\,\in\,}{\fam=\ssfam H}$ and $a^\eta{\,=\,}a$. Then $a^{\fam=\ssfam H}{\,=\,}a$ or $a^{\fam=\ssfam H}$ is contained in a fixed line of ${\fam=\ssfam H}$ or
$\langle a^{\fam=\ssfam H}\rangle{\,=\,}{\cal E}$ is a subplane of ${\cal P}$. In the last case, $\eta|_{\cal E}{\,=\,}1\kern-2.5pt {\rm l}$ and ${\cal E}{\,<\,}{\cal P}$. By induction, ${\fam=\ssfam H}$ fixes some element of ${\cal E}$, cf. also \cite{gs} XI.10.19. If $\dim P{\,=\,}2$, then the lines are homeomorphic to a circle (\cite{cp} 42.10 or 51.29). It suffices to show that ${\fam=\ssfam H}$ has a fixed point, by duality there is also a fixed line. Suppose that
$\forall_{\hskip-1pt x{\in}P}\;x^{\fam=\ssfam H}{\,\ne\,}x$. Then $\eta|_{a^{\fam=\ssfam H}}{\,=\,}1\kern-2.5pt {\rm l}$,\; $a^{\fam=\ssfam H}$ is contained in some line $L$, and ${\fam=\ssfam H}$ is transitive on $L$\; (or the endpoint of an orbit in $L$ would be fixed). Thus $\eta$ is an axial collineation with axis $L$ and the unique center of $\eta$ is ${\fam=\ssfam H}$-invariant. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
From \cite{cp} 71.8,10 and 72.1--4 we need the following results: \par
{\bf 2.14 Semi-simple groups of a $4$-dimensional subplane.} {\it A semi-simple group ${\fam=\ssfam \usuUpsilon}$ of automorphisms of a $4$-dimensional plane ${\cal D}$ is almost simple. If ${\fam=\ssfam \usuUpsilon}$ is compact and $\dim{\fam=\ssfam \usuUpsilon}{\,=\,}3$, the action of ${\fam=\ssfam \usuUpsilon}$ on the point space is equivalent to the standard action on the classical complex plane, and ${\cal F}_{\fam=\ssfam \usuUpsilon}{\,=\,}\emptyset$ or ${\cal F}_{\fam=\ssfam \usuUpsilon}{\,=\,}\{a,W\}$ with $a{\,\notin\,}W$. If $\dim{\fam=\ssfam \usuUpsilon}{\,>\,}3$, then ${\cal D}$ is classical and the action is the standard one$;$ in particular, ${\fam=\ssfam \usuUpsilon}$ does not fix a flag or exactly one line\/}. \par
{\bf Richardson's classification} \cite{cp} 96.34 of compact groups acting on the sphere ${\fam=\Bbbfam S}_4$ plays a fundamental r\^ole in the study of $8$-dimensional planes: \par
($\dagger$) {\it If a compact connected group ${\fam=\ssfam \usuPhi}$ acts effectively on the $4$-sphere $S$, and if ${\fam=\ssfam \usuPhi}$ is a Lie group or if ${\fam=\ssfam \usuPhi}$ has an orbit of dimernsion ${>\,}1$, then $({\fam=\ssfam \usuPhi},S)$ is equivalent to the obvious standard action of a subgroup of $\SO5{\fam=\Bbbfam R}$ on ${\fam=\Bbbfam S}_4$ or ${\fam=\ssfam \usuPhi}{\,\cong\,}\SO3{\fam=\Bbbfam R}$ has no fixed point on $S$\/}. \par
Only fragmentary results are known for compact transformation groups on ${\fam=\Bbbfam S}_8$. The following will be needed, see \cite{cp} 96.\,35 and 36. \par
{\bf 2.15 Compact groups on ${\fam=\Bbbfam S}_8$.} (a) {\it Any non-trivial action of a group ${\fam=\ssfam \usuGamma}{\,\cong\,}{\rm G}_2$ on the sphere ${\fam=\Bbbfam S}_8$ is equivalent to the action of ${\rm G}_2$ on ${\fam=\Bbbfam O}{\kern 2pt {\scriptstyle \cup}\kern 2pt}\{\infty\}$ as group of automorphisms of ${\fam=\Bbbfam O}\,;$ in particular, ${\fam=\ssfam \usuGamma}$ fixes a circle, all other orbits are $6$-spheres\/}. \\ (b) {\it Any non-trivial action of a group ${\fam=\ssfam \usuOmega}{\,\cong\,}\Spin7{\fam=\Bbbfam R}$ on ${\fam=\Bbbfam S}_8$ has a fixed point $\infty$ and is linear on ${\fam=\Bbbfam R}^8{\,=\,}{\fam=\Bbbfam S}_8\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{\infty\}\,;$ either the action is faithful, ${\fam=\ssfam \usuOmega}$ has exactly $2$ fixed points, all other orbits are $7$-spheres, and stabilizers ${\fam=\ssfam \usuOmega}_x$ are isomorphic to ${\rm G}_2$, or ${\fam=\ssfam \usuOmega}$ induces the group $\SO7{\fam=\Bbbfam R}$, there is a circle of fixed points, and all other orbits are $6$-spheres\/}. \par
{\bf 2.16 The complex group of type G$\hskip-1pt_2$.} {\it If ${\cal F}_{\fam=\ssfam \usuDelta}{\,\ne\,}\emptyset$, then ${\fam=\ssfam \usuDelta}{\,\not\hskip1pt\cong\,}{\rm G}_2^{{\fam=\Bbbfam C}}$\/}. \par
{\tt Proof.} Suppose that ${\fam=\ssfam \usuDelta}{\,\cong\,}{\rm G}_2^{{\fam=\Bbbfam C}}{\,=\,}\mathop{{\rm Aut}}({\fam=\Bbbfam O}{\hskip1pt\otimes\hskip1pt}{\fam=\Bbbfam C})$ is the complexification of its maximal compact subgroup ${\fam=\ssfam \usuGamma}{=\,}{\rm G}_2$. Then all involutions of ${\fam=\ssfam \usuDelta}$ are conjugate by \cite{cp} 11.31(d) and 93.10(a), and no involution can act trivially on the fixed line $W$ since ${\fam=\ssfam \usuDelta}$ is strictly simple. The centralizer of a given involution $\lambda$ contains a subgroup ${\fam=\ssfam \usuPhi}{\,\cong\,}\SO3{\fam=\Bbbfam R}$, see 2.5. If ${\fam=\ssfam \usuPhi}$ would contain a reflection, then even three pairwise commuting conjugate ones whose centers form a triangle in some ${\fam=\ssfam \usuPhi}$-orbit, and ${\cal F}_{\fam=\ssfam \usuDelta}{\,\subseteq\,}{\cal F}_{\fam=\ssfam \usuPhi}{\,=\,}\emptyset$. Consequently, each involution in ${\fam=\ssfam \usuDelta}$ is planar and the involutions in ${\fam=\ssfam \usuPhi}$ induce planar involutions on ${\cal F}_\lambda$. We conclude that $W{\,\approx\,}{\fam=\Bbbfam S}_8$ is a manifold, cf. \cite{cp} 55.6. According to 2.15, the action of ${\fam=\ssfam \usuGamma}$ on $W$ is equivalent to~the action of $\mathop{{\rm Aut}}{\fam=\Bbbfam O}$ on ${\fam=\Bbbfam O}{\kern 2pt {\scriptstyle \cup}\kern 2pt}\{\infty\}$, and the fixed points of ${\fam=\ssfam \usuGamma}$ on $W$ form a circle $C$. Note that ${\fam=\ssfam \usuGamma}$ is even a maximal subgroup of ${\fam=\ssfam \usuDelta}$\: (\cite{cp} 94.34). If $c{\,\in\,}C$, then $\dim{\fam=\ssfam \usuDelta}_c{\,\ge\,}20$ and ${\fam=\ssfam \usuGamma}{\,<\,}{\fam=\ssfam \usuDelta}_c{\,=\,}{\fam=\ssfam \usuDelta}$. The same argument, applied to the pencils ${\frak L}_c$ with $c{\,\in\,}C$ instead of $W$, shows that ${\cal F}_{\fam=\ssfam \usuDelta}$ is a flat subplane, but this contradicts Stiffness. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
The next lemma will be needed repeatedly: \par
{\bf 2.17 Stabilizer of a triangle.} {\it Suppose that the connected subgroup $\nabla$ of ${\fam=\ssfam \usuDelta}$ fixes the triangle $a,u,v$ and that $\nabla$ is transitive on $S{\,:=\,}uv\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{u,v\}$. For $w{\,\in\,}S$, let ${\fam=\ssfam \usuPsi}$ and ${\fam=\ssfam \usuPhi}$ be maximal compact subgroups of $\nabla$ and of ${\fam=\ssfam \usuOmega}{\,=\,}\nabla_{\hskip-2pt w}$, respectively, such that ${\fam=\ssfam \usuPhi}{\,\le\,}{\fam=\ssfam \usuPsi}$. Then\/{\rm:}} \par \quad(a) {\it ${\fam=\ssfam \usuPhi}$ and ${\fam=\ssfam \usuPsi}$ have equally many almost simple factors\/}, \par \quad(b) {\it $0{\,<\,}{\fam=\ssfam \usuPsi}'{:}{\fam=\ssfam \usuPhi}'{\,<\,}8$ and ${\fam=\ssfam \usuPhi}'{\,\ne\,}1\kern-2.5pt {\rm l}$\/},\: $\dim{\fam=\ssfam \usuPhi}'{\,\ne\,}10$, \par \quad(c) {\it If $\dim{\fam=\ssfam \usuPhi}'{\,=\,}3$, then ${\fam=\ssfam \usuPsi}'{\,\cong\,}\U2{\fam=\Bbbfam H}{\,\cong\,}\Spin5{\fam=\Bbbfam R}$\/}, \par \quad(\^c) {\it If $\dim{\fam=\ssfam \usuPhi}'{\,=\,}6$, then ${\fam=\ssfam \usuPsi}'{\,\cong\,}\U2{\fam=\Bbbfam H}{\,\cdot}\U1{\fam=\Bbbfam H}$\/}, \par \quad(d) {\it If $\dim{\fam=\ssfam \usuPhi}'{\,=\,}8$, then ${\fam=\ssfam \usuPsi}'{\,\cong\,}\SU4{\fam=\Bbbfam C}{\,\cong\,}\Spin6{\fam=\Bbbfam R}$\/}, see also \cite{sz5} Lemma (5), \par \quad(e) {\it If ${\fam=\ssfam \usuPhi}'{\,\cong\,}{\rm G}_2$, then ${\fam=\ssfam \usuPsi}'{\,\cong\,}\Spin7{\fam=\Bbbfam R}$\/}, \par \quad(f) {\it If ${\fam=\ssfam \usuPhi}'$ is almost simple and $\dim{\fam=\ssfam \usuPhi}'{\,>\,}14$, then ${\fam=\ssfam \usuPsi}'{\,\cong\,}\Spin8{\fam=\Bbbfam R}$\/}. \par
{\tt Proof.} Note that $S$ is homotopy equivalent ($\simeq$) to ${\fam=\Bbbfam S}_7$. By the Mal'cev-Iwasawa theorem \cite{cp} 93.10, $\nabla{\,\simeq\,}{\fam=\ssfam \usuPsi}$ and ${\fam=\ssfam \usuOmega}{\,\simeq\,}{\fam=\ssfam \usuPhi}$; therefore the exact homotopy sequence (\cite{cp} 96.12) for the action of $\nabla$ on $S$ can be written in the form $$\dots\to\pi_{q{+}1}S\to\pi_q{\fam=\ssfam \usuPhi}\to\pi_q{\fam=\ssfam \usuPsi}\to\pi_qS\to\pi_{q{-}1}{\fam=\ssfam \usuPhi}\to\dots,\quad q{\,\ge\,}1\,.\leqno{(*)}$$ From \cite{cp} 94.31(c) it follows that $\pi_q{\fam=\ssfam \usuPsi}{\,\cong\,}\pi_q{\fam=\ssfam \usuPsi}'$ and $\pi_q{\fam=\ssfam \usuPhi}{\,\cong\,}\pi_q{\fam=\ssfam \usuPhi}'$ for $q{\,>\,}1$. We have \\ $\pi_qS{\,=\,}0\; (q{\,<\,}7),\: \pi_7S{\,\cong\,}{\fam=\Bbbfam Z}$ (\cite{sp} 7.5.6 or \cite{br} II.16.4),\: $\pi_8S$ is finite (${\cong\,}{\fam=\Bbbfam Z}_2$),\: see \cite{sp} 9.7.7. For the same range of the dimension $q$ (and beyond), the homotopy groups $\pi_q$ of the al\-most simple compact Lie groups ${\fam=\ssfam K}$ are known, cf. the references leading up to Theorem 94.36 in \cite{cp}. This theorem states that $\pi_2{\fam=\ssfam K}{\,=\,}0$ and $\pi_3{\fam=\ssfam K}{\,\cong\,}{\fam=\Bbbfam Z}$ for all compact simple Lie groups. For $\dim{\fam=\ssfam K}{\,\le\,}52$ in particular, $\pi_4\U n{\fam=\Bbbfam H}{\,\cong\,}{\fam=\Bbbfam Z}_2$,\: $\pi_4{\fam=\ssfam K}{\,=\,}0$ in all other cases, and $\pi_6{\fam=\ssfam K}{\,\le\,}{\fam=\Bbbfam Z}_{12}$. Furthermore $\pi_5\SU n{\fam=\Bbbfam C}{\,\cong\,}{\fam=\Bbbfam Z}$ for $n{\,>\,}2$,\: $\pi_5{\fam=\ssfam K}{\,\cong\,}{\fam=\Bbbfam Z}_2$ for ${\fam=\ssfam K}{\,\cong\,}\Spin n{\fam=\Bbbfam R}\hskip6pt (n{=}3,5)$ or $\U n{\fam=\Bbbfam H}\hskip6pt (n{\ge}1)$,\: $\pi_5{\fam=\ssfam K}{\,=\,}0$ for all other groups ${\fam=\ssfam K}$. Note also $\pi_7\SO3{\fam=\Bbbfam R}{\,\cong\,}{\fam=\Bbbfam Z}_2$,\: $\pi_7{\fam=\ssfam K}{\,\cong\,}{\fam=\Bbbfam Z}$ for ${\fam=\ssfam K}{\,\cong\,}\SU n{\fam=\Bbbfam C}\hskip6pt (n{>}3)$,\: $\Spin n{\fam=\Bbbfam R}\hskip6pt (n{>\hskip1pt}4,\, n{\ne}8)$, or $\U n{\fam=\Bbbfam H}\hskip6pt (n{>}1)$, $\pi_7\Spin8{\fam=\Bbbfam R}{\,\cong\,}{\fam=\Bbbfam Z}^2$, and $\pi_7{\fam=\ssfam K}{\,=\,}0$ for all other groups ${\fam=\ssfam K}$. Recall from the last part of 2.10 that each non-trivial action of $\U2{\fam=\Bbbfam H}{\,\cong\,}\Spin5{\fam=\Bbbfam R}$ is faithful. \\ (a) is an immediate consequence of $\pi_3{\fam=\ssfam \usuPhi}{\,\cong\,}\pi_3{\fam=\ssfam \usuPsi}$. \\ (b) If ${\fam=\ssfam \usuPhi}'{\,=\,}{\fam=\ssfam \usuPsi}'$, then $(*)$ yields $\pi_7{\fam=\ssfam \usuPhi}'{\,\cong\,}\pi_7{\fam=\ssfam \usuPsi}'\to{\fam=\Bbbfam Z}\to\pi_6{\fam=\ssfam \usuPhi}'$, but this contradicts finiteness of $\pi_6{\fam=\ssfam \usuPhi}'$. In the case ${\fam=\ssfam \usuPhi}'{\,=\,}1\kern-2.5pt {\rm l}$, step (a) implies ${\fam=\ssfam \usuPsi}'{\,=\,}1\kern-2.5pt {\rm l}$, which is impossible by the first part of~(b). Because $\pi_1{\fam=\ssfam \usuPhi}{\,\cong\,}\pi_1{\fam=\ssfam \usuPsi}$, again by $(*)$, the torus factors of ${\fam=\ssfam \usuPhi}$ and ${\fam=\ssfam \usuPsi}$ have the same dimension (cf. the structure theorem \cite{cp} 31(c) for compact Lie groups). From $w^\nabla{\,\cong\,}S$ follows ${\fam=\ssfam \usuPsi}'{:}{\fam=\ssfam \usuPhi}'{\,=\,}\dim{\fam=\ssfam \usuPsi}{-}\dim{\fam=\ssfam \usuPhi}{\,<\,}\nabla{:\hskip1pt}{\fam=\ssfam \usuOmega}{\,=\,}\dim w^\nabla{\,=\,}8$. If $\dim{\fam=\ssfam \usuPhi}'{\,=\,}10$, then ${\fam=\ssfam \usuPhi}'{\,\cong\,}\U2{\fam=\Bbbfam H}$, and $(*)$ implies $\pi_4{\fam=\ssfam \usuPsi}'{\,\cong\,}{\fam=\Bbbfam Z}_2$, but the groups $\U n{\fam=\Bbbfam H}$ are the only groups ${\fam=\ssfam K}$ with $\pi_4{\fam=\ssfam K}{\,\cong\,}{\fam=\Bbbfam Z}_2$ and $\dim\U3{\fam=\Bbbfam H}{\,=\,}21$ is too large. \\ (c) Part of $(*)$ reads \ $0\to\pi_4{\fam=\ssfam \usuPhi}'{\,=\,}\pi_4\U1{\fam=\Bbbfam H}{\,=\,}{\fam=\Bbbfam Z}_2\to\pi_4{\fam=\ssfam \usuPsi}'\to0$. The groups $\U n{\fam=\Bbbfam H}$ are the only groups ${\fam=\ssfam K}$ with $\pi_4{\fam=\ssfam K}{\,\ne\,}0$. The dimension bounds in (b) and 2.10 imply ${\fam=\ssfam \usuPsi}'{\,\cong\,}\U2{\fam=\Bbbfam H}$. \\ (\^c) By assumption ${\fam=\ssfam \usuPhi}'{\,\circeq\,}(\SU2{\fam=\Bbbfam C})^2$. Hence ${\fam=\ssfam \usuPsi}'$ is a product of two almost simple factors. From $(*)$ we obtain $\pi_5{\fam=\ssfam \usuPsi}'{\,\cong\,}\pi_5{\fam=\ssfam \usuPhi}'{\,\cong\,}{\fam=\Bbbfam Z}_2^{\hskip3pt2}$. It follows that each factor of ${\fam=\ssfam \usuPsi}'$ is a group $\U n{\fam=\Bbbfam H}$, $n{\,\ge\,}1$. We have $6{\,<\,}\dim{\fam=\ssfam \usuPsi}'{\,\le\,}13$, thus ${\fam=\ssfam \usuPsi}'{\,\cong\,}\U2{\fam=\Bbbfam H}{\hskip2pt\cdot\hskip1pt}\U1{\fam=\Bbbfam H}$, because a factor $\SO3{\fam=\Bbbfam R}$ would contain planar involutions and $\U2{\fam=\Bbbfam H}$ cannot fix a triangle in a Baer subplane. \\ (d) The homotopy sequence $(*)$ yields $\pi_5{\fam=\ssfam \usuPsi}'{\cong\,}\pi_5\SU3{\fam=\Bbbfam C}{\,\cong\,}{\fam=\Bbbfam Z}$. The groups $\SU n{\fam=\Bbbfam C}$ are the only groups ${\fam=\ssfam K}$ with $\pi_5{\fam=\ssfam K}{\,\cong\,}{\fam=\Bbbfam Z}$, and $\SU5{\fam=\Bbbfam C}$ is too large. \\ (e) We have $\pi_5{\fam=\ssfam \usuPsi}'\cong\pi_5{\rm G}_2{\,=\,}0$ by $(*)$ and hence ${\fam=\ssfam \usuPsi}'{\,\cong\,}\Spin7{\fam=\Bbbfam R}$ by (b), the fact that ${\rm G}_2{\,<\,}{\fam=\ssfam \usuPsi}$, and the last part of 2.10. \\ (f) follows from $\pi_5{\fam=\ssfam \usuPhi}'{\,\cong\,}\pi_5{\fam=\ssfam \usuPsi}'$,\: $\pi_5\SU4{\fam=\Bbbfam C}{\,\cong\,}{\fam=\Bbbfam Z}$,\: $\pi_5\Spin7{\fam=\Bbbfam R}{\,\cong\,}0$,\: $\pi_5\U3{\fam=\Bbbfam H}{\,\cong\,}{\fam=\Bbbfam Z}_2$,\: $\pi_5\SU5{\fam=\Bbbfam C}{\,\cong\,}{\fam=\Bbbfam Z}$,\: $\pi_5\Spin8{\fam=\Bbbfam R}{\,=\,}0$,\: $\SU5{\fam=\Bbbfam C}{\,:\,}\SU4{\fam=\Bbbfam C}{\,>\,}7$, and $\dim\nabla{\,\le\,}30$. \par
{\bf 2.18 Corollary.} {\it If $\nabla$ is transitive on the complement $D{\,=\,}P\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt(au\kern 2pt {\scriptstyle \cup}\kern 2pt av\kern 2pt {\scriptstyle \cup}\kern 2pt uv)$ of a triangle, then $\nabla'{\,\cong\,}\Spin8{\fam=\Bbbfam R}$\/}. \par
{\tt Proof.} We use the same notation as in 2.17. If $\nabla$ is transitive on $D$, then $\nabla$ is a Lie group by \cite{cp} 53.2, and ${\fam=\ssfam \usuOmega}{\,=\,}\nabla_{\hskip-1pt w}$ is transitive on $V{\,=\,}av\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a,v\}$. Choose $c{\,\in\,}V$, and let ${\fam=\ssfam \usuLambda}$ be a maximal compact subgroup of ${\fam=\ssfam \usuOmega}_c$. Stiffness implies that ${\fam=\ssfam \usuLambda}'$ is one of the groups $\Spin3{\fam=\Bbbfam R}$, $\SO4{\fam=\Bbbfam R}$, $\SU3{\fam=\Bbbfam C}$, or ${\rm G}_2$. Applying 2.17 to the action of ${\fam=\ssfam \usuOmega}$ on $V$, the possible groups ${\fam=\ssfam \usuPhi}'$ can be determined. If $\dim{\fam=\ssfam \usuLambda}'{\,\ge\,}8$, then ${\fam=\ssfam \usuPhi}'{\,\cong\,}\Spin n{\fam=\Bbbfam R}$ with $n{\,\in\,}\{6,7\}$, and then 2.17(f) shows ${\fam=\ssfam \usuPsi}'{\,\cong\,}\Spin8{\fam=\Bbbfam R}$. In the case $\dim{\fam=\ssfam \usuLambda}'{\,=\,}3$ we obtain $\dim{\fam=\ssfam \usuPhi}'{\,=\,}10$, which is excluded by 2.17(b). If ${\fam=\ssfam \usuLambda}'{\,\cong\,}\SO4{\fam=\Bbbfam R}$, then 2.17(\^c) shows
${\fam=\ssfam \usuPhi}'{\,\cong\,}\U2{\fam=\Bbbfam H}{\hskip2pt\cdot\hskip1pt}\U1{\fam=\Bbbfam H}$. Now $13{\,<\,}\dim{\fam=\ssfam \usuPsi}'{\,\le\,}20$, and ${\fam=\ssfam \usuPsi}'$ is an almost direct product ${\fam=\ssfam \usuXi}{\hskip1pt\cdot}{\fam=\ssfam \usuUpsilon}$, where ${\fam=\ssfam \usuXi}{\,\cong\,}{\fam=\ssfam \usuUpsilon}{\,\cong\,}\U2{\fam=\Bbbfam H}$ and ${\fam=\ssfam \usuXi}{\,\triangleleft\,}{\fam=\ssfam \usuPhi}'$. According to 2.10, the torus rank of $\nabla$ is $4$; hence ${\fam=\ssfam \usuPsi}'{\,=\,}{\fam=\ssfam \usuPsi}$,\, ${\fam=\ssfam \usuPhi}{\,=\,}{\fam=\ssfam \usuPhi}'$, and $\dim{\fam=\ssfam \usuPhi}{\,=\,}13$. By construction ${\fam=\ssfam \usuPhi}{\,=\,}{\fam=\ssfam \usuPsi}_w$, and $\dim w^{\fam=\ssfam \usuUpsilon}{\,=\,}\dim w^{\fam=\ssfam \usuPsi}{\,=\,}{\fam=\ssfam \usuPsi}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuPhi}{\,=\,7}$.
Therefore ${\fam=\ssfam \usuXi}|_{w^{\fam=\ssfam \usuUpsilon}}{\,=\,}1\kern-2.5pt {\rm l}$ and ${\cal F}_{{\fam=\ssfam \usuXi}_c}{\,=\,}\langle a,c,w^{\fam=\ssfam \usuUpsilon}\rangle{\,=\,}{\cal P}$, but $\dim{\fam=\ssfam \usuXi}_c{\,>\,}2$, a contradiction. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\Bf 3. No fixed elements} \par
The Hughes planes form a one-parameter family of planes admitting a fixed element free group action. They can be characterized as follows: ${\cal P}$ has a ${\fam=\ssfam \usuSigma}$-invariant Baer subplane ${\cal H}$ such that ${\fam=\ssfam \usuSigma}$ induces on ${\cal H}$~the full automorphism group $\PSL3{\fam=\Bbbfam H}$, see \cite{cp} \S\hskip2pt86 or \cite{sz3} 3.21. If ${\cal P}$ is a proper Hughes plane, then $\dim{\fam=\ssfam \usuSigma}{\,=\,}36$ and ${\fam=\ssfam \usuSigma}$ is transitive on the set of flags of the {\it outer\/} subgeometry consisting of the points and lines not belonging to ${\cal H}$, see \cite{cp} 86.5. Sometimes it is convenient to consider~the~classical plane together with the stabilizer of a Baer subplane also as a Hughes plane. Other characterizations of the Hughes planes are given in \cite{pw1} and \cite{sz5}. \par
{\bf 3.0.} {\it If ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\emptyset$ and $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}23$, then ${\fam=\ssfam \usuDelta}$ is a Lie group\/}. \par
{\tt Proof.} (a) Suppose that ${\fam=\ssfam \usuDelta}$ is not a Lie group. Then $\dim{\fam=\ssfam \usuDelta}{\,<\,}27$ and there are arbitrarily small compact, $0$-dimensional central subgroups ${\fam=\ssfam N}$ such that ${\fam=\ssfam \usuDelta}/{\fam=\ssfam N}$ is a Lie group, cf. \cite{cp} 93.8. Choose a fixed point~$x$ of some element $\zeta{\,\in\,}{\fam=\ssfam N}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{1\kern-2.5pt {\rm l}\}$.
From $\zeta|_{x^{\fam=\ssfam \usuDelta}}{\,=\,}1\kern-2.5pt {\rm l}$ and ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\emptyset$ it follows that ${\cal D}{\,=\,}\langle p^{\fam=\ssfam \usuDelta}\rangle$ is a proper connected subplane.
Put ${\fam=\ssfam \usuDelta}|_{\cal D}{\,=\,}{\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$. By Stiffness, $\dim{\fam=\ssfam K}{\,\le\,}14$,\; ${\cal D}$ is not flat, $\dim{\cal D}{\,\ge\,}4$ and $\dim{\fam=\ssfam K}{\,\le\,}8$. Hence ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\ge\,}15$, and ${\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$ is a Lie group by \cite{cp} 71.2 and \cite{pw3}. We may assume, therefore, that ${\fam=\ssfam N}{\,\le\,}{\fam=\ssfam K}$. If ${\cal D}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$, then $\dim{\fam=\ssfam K}{\,<\,}8$
by Stiffness, and $\dim{\fam=\ssfam \usuDelta}|_{\cal D}{\,\ge\,}16$. The same is true, if $\dim{\cal D}{\,=\,}4$: in this case ${\cal D}$ is classical (\cite{cp} 72.8), $\mathop{{\rm Aut}}{\cal D}$ has no subgroup of dimension $15$, and ${\fam=\ssfam \usuDelta}/{\fam=\ssfam K}{\,\cong\,}\PSL3{\fam=\Bbbfam C}$. If $\dim{\fam=\ssfam K}{\,=\,}8$, then ${\fam=\ssfam K}^1{\,\cong\,}\SU3{\fam=\Bbbfam C}$,\, ${\fam=\ssfam \usuDelta}/{\fam=\ssfam K}^1$ is a finite covering of ${\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$, and ${\fam=\ssfam \usuDelta}$ would be a Lie group. \\ (b) Suppose first that ${\cal D}{\,=\,}{\cal F}_{\fam=\ssfam N}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$. Then ${\fam=\ssfam K}$ is compact and
\cite{sz1} 2.1 applies to ${\fam=\ssfam \usuDelta}|_{\cal D}$: either ${\cal D}$ is the classical quaternion plane, ${\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$ is the elliptic or hyperbolic motion group of ${\cal D}$, and $2{\,\le\,}\dim{\fam=\ssfam K}{\,\le\,}5$, or ${\cal D}$ is a Hughes plane, ${\fam=\ssfam \usuDelta}'/{\fam=\ssfam K}{\,\cong\,}\SL3{\fam=\Bbbfam C}$, and $\dim{\fam=\ssfam K}{\,\ge \,}6$. \\ (c) In the first case, the motion group ${\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$ is covered by a subgroup ${\fam=\ssfam \usuUpsilon}{\hskip1pt\circeq\,}\U3({\fam=\Bbbfam H},r)$ of ${\fam=\ssfam \usuDelta}$, see \cite{cp}~94.27. The representation of ${\fam=\ssfam \usuUpsilon}$ on ${\fam=\ssfam K}/{\fam=\ssfam N}$ shows that ${\fam=\ssfam K}{\,\le\,}\Cs{}{\fam=\ssfam \usuUpsilon}$. We may assume that ${\fam=\ssfam K}$ is connected, since $\dim{\fam=\ssfam K}^1{\fam=\ssfam \usuUpsilon}{\,=\,}\dim{\fam=\ssfam \usuDelta}$ and ${\fam=\ssfam \usuDelta}$ is connected. Moreover, the center ${\fam=\ssfam Z}$ of ${\fam=\ssfam K}$ has positive dimension: because ${\fam=\ssfam K}$ is not a Lie group, the claim follows from the structure of compact groups as described in \cite{cp} 93.11. The stabilizer ${\fam=\ssfam \usuPi}{\,=\,}{\fam=\ssfam \usuDelta}_p$ of a non-absolute point $p{\,\in\,}{\cal D}$ fixes also the polar $L$ of $p$, and ${\fam=\ssfam \usuPi}/{\fam=\ssfam K}{\,\circeq\,}\U2{\fam=\Bbbfam H}{\times}\U1{\fam=\Bbbfam H}$. In particular, ${\fam=\ssfam \usuPi}$ is compact and $\dim{\fam=\ssfam \usuPi}{\,\ge\,}15$. Choose $s{\,\in\,}L\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt{\cal D}$ and $z{\,\in\,}ps\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{p,s\}$, and note that ${\fam=\ssfam \usuPi}_s\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam K}{\,=\,}1\kern-2.5pt {\rm l}$ because $\langle {\cal D},s\rangle{\,=\,}{\cal P}$. Hence ${\fam=\ssfam \usuPi}_s$ and ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuPi}_{s,z})^1$ are Lie groups. From \cite{cp} 96.11 it follows that ${\fam=\ssfam \usuPi}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuPi}_s,\, {\fam=\ssfam \usuPi}_s{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuLambda}{\,\le\,}7$, and $\dim{\fam=\ssfam \usuLambda}{\,>\,}0$. As ${\fam=\ssfam N}{\,\le\,}{\fam=\ssfam Z}{\,\le\,}\Cs{}{\fam=\ssfam \usuDelta}{\,\le\,}\Cs{}{\fam=\ssfam \usuLambda}$, we conclude that $\langle s^{\fam=\ssfam Z},p,z\rangle{\,\le\,}{\cal F}_{\fam=\ssfam \usuLambda}$, and ${\cal F}_{\fam=\ssfam \usuLambda}$ is a connected subplane; in fact, ${\cal F}_{\fam=\ssfam \usuLambda}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$\; (otherwise ${\fam=\ssfam Z}$ would be a Lie group by \cite{cp} 32.21 and 71.2). Stiffness implies ${\fam=\ssfam \usuLambda}{\,\le\,}\Spin3{\fam=\Bbbfam R}$. The involution $\iota{\,\in\,}{\fam=\ssfam \usuLambda}$ induces
an involution $\overline\iota{\,=\iota|_{\cal D}\,}$ on ${\cal D}$, and ${\cal F}_{\fam=\ssfam \usuLambda}{\,=\,}{\cal F}_\iota$. If $\overline\iota$ is planar, then ${\cal D}\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal F}_\iota{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal F}_\iota$, and the lines of ${\cal F}_\iota$ are $4$-spheres (see \cite{cp} 53.15 and \cite{sz3} 3.7). If $\overline\iota$ is a reflection, then the axis $A$ of $\overline\iota$ is a $4$-sphere contained in a line of ${\cal F}_\iota$.
From \cite{cp} 51.20 it follows that the lines of ${\cal F}_\iota$ are manifolds, and then
$H{\,=\,}L\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal F}_\iota{\,\approx\,}{\fam=\Bbbfam S}_4$. \\ (d) The structure theorem \cite{cp} 93.11 for compact Lie groups together with \cite{cp} 93.19 shows that ${\fam=\ssfam K}$ is commutative or a product of a commutative central group ${\fam=\ssfam A}$ with a factor ${\fam=\ssfam \usuOmega}{\,\cong\,}\Spin3{\fam=\Bbbfam R}$. In the latter case $\dim{\fam=\ssfam \usuPi}{\,\ge\,}17$ and ${\fam=\ssfam \usuLambda}{\,\cong\,}\Spin3{\fam=\Bbbfam R}$. As $\mathop{{\rm Aut}}{\fam=\ssfam \usuOmega}{\,\cong\,}\SO3{\fam=\Bbbfam R}$, the involution $\iota{\,\in{\fam=\ssfam \usuLambda}\,}$ centralizes ${\fam=\ssfam K}$, and $H^{\fam=\ssfam K}{\,=\,}H$. Therefore the compact connected group ${\fam=\ssfam K}$ acts freely on $s^{\fam=\ssfam K}{\,\subset\,}H{\,\approx\,}{\fam=\Bbbfam S}_4$, and Richardson's theorem (\cite{cp} 96.34 or $(\dagger)$) implies that ${\fam=\ssfam K}$ is a Lie group contrary to the assumption. \\ (e) In the second case mentioned in step (b) or if ${\cal D}{\,<\,}{\cal F}_{\fam=\ssfam N}$, the plane ${\cal F}_{\fam=\ssfam N}$ is a Hughes plane, ${\fam=\ssfam \usuDelta}$ has a subgroup ${\fam=\ssfam \usuUpsilon}{\cong\,}\SL3{\fam=\Bbbfam C}$, and $\dim{\fam=\ssfam K}{\,\in\,}\{6,7\}$. By \cite{cp} 86.33, there exists a ${\fam=\ssfam \usuDelta}$-invariant $4$-dimensional classical complex subplane ${\cal C}$. If $\dim{\cal D}{\,=\,}4$, we will change the notation and put ${\cal D}{\,=\,}{\cal F}_{\fam=\ssfam N}$. By Stiffness, ${\fam=\ssfam K}$ is compact, and ${\fam=\ssfam K}$ acts freely on the set of points outside~${\cal D}$. The structure of the compact Lie group ${\fam=\ssfam K}/{\fam=\ssfam N}$ implies that ${\fam=\ssfam \usuUpsilon}{\,\le\,}\Cs{}{\fam=\ssfam K}$. As proved in \cite{cp}~86.34, the center $\langle \zeta\rangle{\,\cong\,}{\fam=\Bbbfam Z}_3$ of ${\fam=\ssfam \usuUpsilon}$ acts effectively on ${\cal D}$ and fixes each element of ${\cal C}$. Consider a point $z$ which is not incident with a line of ${\cal C}$. Because ${\cal D}$ is a Baer subplane, there is a unique line $L$ of ${\cal D}$ with $z{\,\in\,}L$, and $L^\zeta{\,\ne\,}L$. The stabilizer ${\fam=\ssfam \usuUpsilon}_{\!z}$ fixes each point of $z^{\fam=\ssfam K}{\,\subset\,}L$ and of $z^{\zeta{\fam=\ssfam K}}{\,\subset\,}L^\zeta$. Both orbits of $z$ are homeomorphic to ${\fam=\ssfam K}$; their dimension is at least $6$. Hence $\langle z^{\fam=\ssfam K},z^{\zeta{\fam=\ssfam K}}\rangle{\,=\,}{\cal P}$ and ${\fam=\ssfam \usuUpsilon}_{\!z}{\,=\,}1\kern-2.5pt {\rm l}$. It follows that $z^{\!{\fam=\ssfam \usuUpsilon}}$ is open in $P$, and ${\fam=\ssfam \usuDelta}$ would be a Lie group by \cite{cp} 53.2. \par (f) Only the following possibility remains:\, ${\cal D}{\,=\,}{\cal F}_{\fam=\ssfam K}{\,=\,}{\cal F}_{\fam=\ssfam N}$ is the classical complex plane, ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,=\,}16$, and ${\fam=\ssfam \usuDelta}$ has a subgroup ${\fam=\ssfam \usuUpsilon}{\,\circeq\,}\SL3{\fam=\Bbbfam C}$ inducing the full automorphism group on ${\cal D}$\: (see \cite{cp} 94.27). All involutions in ${\fam=\ssfam \usuUpsilon}$ are conjugate, they are either planar or reflections of~${\cal P}$. There are lines which do not intersect the point set of ${\cal D}$; dually some points are not incident with a line of ${\cal D}$.\\ (g) If $\iota{\,\in\,}{\fam=\ssfam \usuUpsilon}$ is a reflection of ${\cal P}$, then for $L{\,\in\,}{\cal D}$ the translation group with axis $L$ is isomorphic to ${\fam=\Bbbfam R}^4$. Each translation $\tau$ is uniquely determined by its axis and a
pair $p,p^\tau$ with $p^\tau{\,\ne\,}p{\,\in\,}{\cal D}$. As ${\fam=\ssfam K}|_{\cal D}{\,=\,}1\kern-2.5pt {\rm l}$, it follows that ${\fam=\ssfam K}{\,\le\,}\Cs{}\tau$. The almost simple group ${\fam=\ssfam \usuUpsilon}$ is generated by all these translations. Hence ${\fam=\ssfam \usuUpsilon}{\,\le\,}\Cs{}{\fam=\ssfam K}$. Choose a point $z$ such that $\forall_{\hskip-1pt L{\hskip1pt\in\hskip1pt}{\cal D}}\;z{\,\notin\,}L$. If $\dim{\fam=\ssfam \usuUpsilon}_{\!z}{\,=\,}0$, then $z^{\fam=\ssfam \usuDelta}$ is open in $P$, and ${\fam=\ssfam K}$ would be Lie group by \cite{cp} 53.2. Therefore $\dim{\fam=\ssfam \usuUpsilon}_{\!z}{\,>\,}0$. Moreover, $\langle z^{\fam=\ssfam K}\rangle$ is a subplane, or else $z^{\fam=\ssfam K}$ is contained in some line $L{\,=\,}L^{\fam=\ssfam K}{\,\in\,}{\cal D}$. From ${\fam=\ssfam K}{\,\le\,}\Cs{}{\fam=\ssfam \usuUpsilon}$
it follows that ${\fam=\ssfam \usuUpsilon}_{\hskip-2pt z}|_{\langle z^{\fam=\ssfam K}\rangle}{\,=\,}1\kern-2.5pt {\rm l}$ and $\langle z^{\fam=\ssfam K}\rangle{\,<\,}{\cal P}$, in fact, $\langle z^{\fam=\ssfam K}\rangle\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}{\cal P}$\: (if not, then
${\fam=\ssfam K}|_{\langle z^{\fam=\ssfam K}\rangle}$ is a Lie group, and we may assume that
${\fam=\ssfam N}|_{\langle z^{\fam=\ssfam K}\rangle}{\,=\,}1\kern-2.5pt {\rm l}$, but then $z^{\fam=\ssfam K}{\,\subseteq\,}{\cal F}_{\fam=\ssfam N}$ contrary the choice of $z$). By Stiffness, ${\fam=\ssfam \usuUpsilon}_{\hskip-2pt z}$ is a compact Lie group, and an involution $\iota{\,\in\,}{\fam=\ssfam \usuUpsilon}_{\!z}$ would be planar. \\ (h) Suppose next that $\iota$ is planar and that ${\fam=\ssfam K}{\,\le\,}\Cs{}{\fam=\ssfam \usuUpsilon}$. Then
$\langle {\cal D},{\cal F}_\iota\rangle{\,=\,}{\cal P}$ because $\iota|_{\cal D}{\,\ne\,}1\kern-2.5pt {\rm l}$ and ${\cal D}{\,\not\le\,}{\cal F}_\iota$. Hence ${\fam=\ssfam K}$ acts effectively on ${\cal F}_\iota$. We have ${\fam=\ssfam \usuGamma}{\,=\,}\Cs{\fam=\ssfam \usuUpsilon}\iota{\,\cong\,}\GL2{\fam=\Bbbfam C}$ and $\dim{\fam=\ssfam \usuGamma}{\fam=\ssfam K}{\,=\,}15$. Stiffness
implies $\dim{\fam=\ssfam \usuGamma}{\fam=\ssfam K}|_{{\cal F}_\iota}{\,=\,}h{\,\ge\,}12$. According to a theorem of Priwitzer \cite{pw3},\, ${\fam=\ssfam \usuGamma}{\fam=\ssfam K}$ induces a Lie group on ${\cal F}_\iota$, but
${\fam=\ssfam K}{\,\cong\,}{\fam=\ssfam K}|_{{\cal F}_\iota}$ and ${\fam=\ssfam K}$ is not a Lie group. Therefore ${\fam=\ssfam \usuUpsilon}$ has a non-trivial representation on the Lie algebra of ${\fam=\ssfam K}/{\fam=\ssfam N}$, and \cite{cp} 95.10 shows that ${\fam=\ssfam \usuUpsilon}{\,\cong\,}\SL3{\fam=\Bbbfam C}$. \\ (i) Again by Stiffness, the almost simple commutator group ${\fam=\ssfam \usuGamma}'{\,\cong\,}\SL2{\fam=\Bbbfam C}$ induces
on ${\cal F}_\iota$ a group $\PSL2{\fam=\Bbbfam C}{\,\cong\,}\SO3{\fam=\Bbbfam C}$. Note that $\iota|_{\cal D}$ is a reflectiom of
${\cal D}$. The line $L$ of ${\cal P}$ such that $L\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal D}$ is the axis of $\iota|_{\cal D}$ intersects
${\cal F}_\iota$ in a line $S{\,=\,}L\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal F}_\iota$, and $S^{{\fam=\ssfam \usuGamma}'}{\,=\,}S$. An involution of a compact subgroup ${\fam=\ssfam \usuPhi}{\,\cong\,}\SO3{\fam=\Bbbfam R}$ of ${\fam=\ssfam \usuGamma}'|_{{\cal F}_\iota}$ cannot have the axis $S$\: (use \cite{cp} 55.35 and simplicity of ${\fam=\ssfam \usuPhi}$). Hence such an involution fixes a Baer subplane of ${\cal F}_\iota$. Consequently, $S{\,\approx\,}{\fam=\Bbbfam S}_4$, see \cite{cp} 53.15 and \cite{sz3} 3.7. The set $L\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal D}{\,\subseteq\,}S$ is ${\fam=\ssfam \usuGamma}'$-invariant. From Richardson's theorem \cite{cp}~96.34 it follows that the fixed points of ${\fam=\ssfam \usuPhi}$ in $S\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt{\cal D}$ form a circle $C$. The group ${\fam=\ssfam N}$ is in the center of ${\fam=\ssfam \usuDelta}$, we have ${\fam=\ssfam N}{\,\le\,}\Cs{}{\fam=\ssfam \usuPhi}$ and $C^{\fam=\ssfam N}{\,=\,}C$.
By \cite{MZ} 6.1 Th.\,3, the induced group ${\fam=\ssfam N}|_C$ is a Lie group, and we may assume that
${\fam=\ssfam N}|_C{\,=\,}1\kern-2.5pt {\rm l}$. Now ${\cal D}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal F}_{\fam=\ssfam N}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$, and Stiffness implies that ${\fam=\ssfam K}$ is compact, but then ${\fam=\ssfam \usuUpsilon}$ acts trivially on ${\fam=\ssfam K}/{\fam=\ssfam N}$\: (use \cite{cp} 93.19 and 94.31(c)\,). This contradicts step (h). \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\bf 3.1 Semi-simple groups.} {\it If ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\emptyset$ and ${\fam=\ssfam \usuDelta}$ is semi-simple, and if $\dim{\fam=\ssfam \usuDelta}{\,>\,}24$, then ${\cal P}$ is a Hughes plane\/}. \par
{\tt Proof.} (a) {\it The assertion is true, if the center ${\fam=\ssfam Z}{\,=\,}\Cs{}{\fam=\ssfam \usuDelta}$ contains an element $\zeta{\,\ne\,}1\kern-2.5pt {\rm l}$\/}. \\ There is some point $x$ such that $x^\zeta{\,=\,}x\,$ (see 2.1) and
$\zeta|_{x^{\fam=\ssfam \usuDelta}}{\,=\,}1\kern-2.5pt {\rm l}$. As $\,{\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\emptyset$, the orbit $x^{\fam=\ssfam \usuDelta}$ is not contained in a line, and ${\cal D}{\,=\,}\langle x^{\fam=\ssfam \usuDelta}\rangle$ is a connected proper subplane. Put
${\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}{\,=\,}{\fam=\ssfam \usuDelta}|_{\cal D}{\,=\,}{\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$ and ${\fam=\ssfam \usuLambda}{\,=\,}{\fam=\ssfam K}^1$. If ${\cal D}$ is flat, then ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuLambda}{\,\le\,}8$, $\dim{\fam=\ssfam \usuLambda}{\,\le\,}14$, and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}22$. Suppose that $\dim{\cal D}{\,=\,}4$. Then \cite{cp} 71.8 shows that $\dim{\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}{\,\le\,}8$ or ${\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}{\,\cong\,}\PSL3{\fam=\Bbbfam C}$. In the first case, Stiffness would imply $\dim{\fam=\ssfam \usuDelta}{\,<\,}22$. Therefore ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuLambda}{\,=\,}16$, and $\dim{\fam=\ssfam \usuLambda}{\,>\,}8$. For any point $z$ outside ${\cal D}$ on a line of ${\cal D}$ it follows that $\dim z^{\fam=\ssfam \usuLambda}{\,\le\,}8$ and ${\fam=\ssfam \usuLambda}_z{\,\ne\,}1\kern-2.5pt {\rm l}$. Consequently, $\langle {\cal D},z\rangle$ is a Baer subplane, and 2.6(c,e) imply that ${\fam=\ssfam \usuLambda}$ is compact and that $\dim{\fam=\ssfam \usuLambda}{\,\le\,}8$, a contradiction. Hence ${\cal D}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$, $\,{\fam=\ssfam \usuLambda}$ is a compact connected normal, hence semi-simple group, the structure theorem \cite{cp} 93.11 shows that ${\fam=\ssfam \usuLambda}$ is a Lie group, Stiffness implies ${\fam=\ssfam \usuLambda}{\,\le\,}\Spin3{\fam=\Bbbfam R}$, and $\dim{\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}{\,>\,}21$. Now \cite{sz1} 2.1 shows that $\dim{\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}{\,=\,}35$ and ${\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}{\,\cong\,}\PSL3{\fam=\Bbbfam H}$ as claimed. \\ (b) Thus we may assume that ${\fam=\ssfam \usuDelta}$ has trivial center. If a proper simple factor ${\fam=\ssfam \usuGamma}$ of ${\fam=\ssfam \usuDelta}$ contains a reflection with a center $c$, then the complement ${\fam=\ssfam \usuLambda}$ of ${\fam=\ssfam \usuGamma}$ in ${\fam=\ssfam \usuDelta}$ acts trivially on the subplane ${\cal D}{\,=\,}\langle c^{\fam=\ssfam \usuGamma}\rangle{\,=\,}{\cal D}^{\fam=\ssfam \usuDelta}$. As before, ${\cal D}$ is a Baer subplane, and then ${\fam=\ssfam \usuLambda}{\,\cong\,}\Spin3{\fam=\Bbbfam R}$, and ${\fam=\ssfam \usuDelta}$ would have a center ${\fam=\ssfam Z}{\,\ne\,}1\kern-2.5pt {\rm l}$. \\ (c) Assume that ${\fam=\ssfam Z}{\,=\,}1\kern-2.5pt {\rm l}$ and that $\beta$ is a planar involution in a proper simple factor ${\fam=\ssfam \usuGamma}$ of ${\fam=\ssfam \usuDelta}$ of minimal dimension. Let ${\fam=\ssfam \usuDelta}{\,=\,}{\fam=\ssfam \usuGamma}{\times}{\fam=\ssfam \usuPsi}$ and consider the action of ${\fam=\ssfam \usuPsi}$ on the Baer plane ${\cal B}{\,=\,}{\cal F}_\beta$. If ${\fam=\ssfam \usuPsi}$ fixes some point $x$ in ${\cal B}$, then
$x^{\fam=\ssfam \usuGamma}{\,\ne\,}x$, $\,{\fam=\ssfam \usuPsi}|_{x^{\fam=\ssfam \usuGamma}}{\,=\,}1\kern-2.5pt {\rm l}$, $\,x^{\fam=\ssfam \usuGamma}$ is not contained in a line, and ${\cal E}{\,=\,}\langle x^{\fam=\ssfam \usuGamma}\rangle{\,=\,}{\cal E}^{\fam=\ssfam \usuDelta}{\,<\,}{\cal P}$. Now $\dim{\fam=\ssfam \usuPsi}{\,\le\,}14$,
$\,\dim{\fam=\ssfam \usuGamma}{\,>\,}8$, $\,\dim{\cal E}{\,>\,}2$, and $\dim{\fam=\ssfam \usuGamma}{\,\le\,}\dim{\fam=\ssfam \usuPsi}{\,\le\,}8$ by Stiffness, a contradiction. Hence ${\fam=\ssfam \usuPsi}$ has no fixed elements in ${\cal B}$ and ${\fam=\ssfam \usuPsi}{\,\cong\,}{\fam=\ssfam \usuPsi}|_{\cal B}$, since ${\fam=\ssfam Z}{\,=\,}1\kern-2.5pt {\rm l}$ and a non-trivial kernel would be isomorphic to $\Spin3{\fam=\Bbbfam R}$; moreover, ${\fam=\ssfam \usuPsi}{\,\not\cong\,}\SL3{\fam=\Bbbfam C}$, and \cite{sz1} 2.1 implies that ${\cal B}{\,=\,}(B,{\frak B})$ is the classical quaternion plane and that ${\fam=\ssfam \usuPsi}$ contains one of the $3$ motion groups. Each orbit of ${\fam=\ssfam \usuPsi}$ on $B$ is either open or a sphere of dimension $7$ or $5$, see \cite{cp} 13.17 and 18.32 or \cite{sz1} 1.15. As $\beta$ is not central, there is some conjugate $\beta^\gamma{\,\ne\,}\beta$ in ${\fam=\ssfam \usuGamma}$. By \cite{cp} 55.38 or \cite{lw2}, the Baer subplanes ${\cal B}$ and ${\cal B}^\gamma$ have a common point $c$, and $c^{\fam=\ssfam \usuPsi}{\,\subseteq\,}B{\kern 2pt {\scriptstyle \cap}\kern 2pt}B^\gamma$. Consequently, $B{\,=\,}B^\gamma$ for each $\gamma{\,\in\,}{\fam=\ssfam \usuGamma}$. The simple group ${\fam=\ssfam \usuGamma}$
is generated by its conjugacy class $\beta^{\hskip1pt{\fam=\ssfam \usuGamma}}$, and ${\fam=\ssfam \usuGamma}|_{\cal B}{\,=\,}1\kern-2.5pt {\rm l}$. From the stiffness result 2.6(\^b) it follows that ${\fam=\ssfam \usuGamma}{\,\cong\,}\Spin3{\fam=\Bbbfam R}$ has a non-trivial center contrary to the assumption. Hence ${\fam=\ssfam \usuDelta}$ is strictly simple. \\ (d) Even without assuming ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\emptyset$, the theorem is true for semi-simple groups ${\fam=\ssfam \usuDelta}$ of dimension ${>}28\,$ (see \cite{pw1}). Therefore it suffices to exclude the groups $\POpr8({\fam=\Bbbfam R},r)$, $0{\,\le\,}r{\,\le\,}4$, and the complex group ${\rm G}_2^{{\fam=\Bbbfam C}}{\,=\,}\mathop{{\rm Aut}}({\fam=\Bbbfam O}{\hskip1pt\otimes\hskip1pt}{\fam=\Bbbfam C})$. For $r{\,<\,}4$, the orthogonal groups contain a subgroup $\SO5{\fam=\Bbbfam R}$ and cannot act on ${\cal P}$ by 2.10. \\ (e) A maximal compact subgroup ${\fam=\ssfam \usuOmega}$ of $\POpr8({\fam=\Bbbfam R},4)$ is isomorphic to $(\SO4{\fam=\Bbbfam R})^2/\langle -1\kern-2.5pt {\rm l}\rangle$. Each factor $\SO4{\fam=\Bbbfam R}$ has a subgroup $\SO3{\fam=\Bbbfam R}{\,>\,}{\fam=\Bbbfam Z}_2^{\hskip2pt2}$, such that the elements of the latter group are represented by diagonal matrices with entries $\pm1$. By 2.10, the product ${\fam=\ssfam \usuPhi}{\,=\,}{\fam=\Bbbfam Z}_2^{\hskip2pt2}{\times{\fam=\Bbbfam Z}_2^{\hskip2pt2}}$ contains a reflection $\alpha$, and each conjugate of $\alpha$ in ${\fam=\ssfam \usuPhi}$ is also a reflection. Consequently, there are exactly $3$ reflections in ${\fam=\ssfam \usuPhi}$, and 2.10 implies that the central involution $\omega{\,\in\,}{\fam=\ssfam \usuOmega}$ is planar. Stiffness 2.6(\^b) shows that ${\fam=\ssfam \usuOmega}/\omega{\,\cong\,}(\SO3{\fam=\Bbbfam R})^4{\,>\,}{\fam=\Bbbfam Z}_2^{\hskip2pt8}$ acts effectively on ${\cal F}_\omega$, but this contradicts 2.10. \\ (f) The real form ${\fam=\ssfam \usuGamma}{=\,}{\rm G}_2$ is a maximal compact subgroup of $\Gamma\hskip-4.6pt\Gamma{\,=\,}{\rm G}_2^{{\fam=\Bbbfam C}}$. By 2.5, the centralizer of any involution contains a subgroup $\SO4{\fam=\Bbbfam R}$. Hence ${\fam=\ssfam \usuGamma}$ has a subgroup ${\fam=\Bbbfam Z}_2^{\hskip2pt3}$, and then 2.\,10 and 5 imply that each involution $\beta{\,\in\,}\Gamma\hskip-4.6pt\Gamma$ is planar, and that $\Cs{\fam=\ssfam \usuGamma}\beta{\,\cong\,}\SO4{\fam=\Bbbfam R}$. Consequently, the centralizer of $\beta$ in $\Gamma\hskip-4.6pt\Gamma$ induces on ${\cal F}_\beta$ a group ${\fam=\ssfam X}{\,\cong\,}(\SO3{\fam=\Bbbfam C})^2$, and this action is effective by Stiffness. Inspection of \cite{sz1} 7.3 shows that ${\fam=\ssfam X}$ fixes a non-incident pair $\{o,W\}$ in ${\cal F}_\beta$. It follows that each involution in ${\fam=\ssfam X}$ is planar. On the other hand, ${\fam=\ssfam X}$ has a subgroup ${\fam=\Bbbfam Z}_2^{\hskip2pt4}$ and must contain a reflection by 2.10. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\tt Remark.} {\it If ${\fam=\ssfam \usuDelta}$ is semi-simple, ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\emptyset$ and $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}21$, then ${\fam=\ssfam \usuDelta}$ is a Lie group\/}. \par
{\tt Proof.} It suffices to show that a maximal compact subgroup ${\fam=\ssfam \usuPhi}$ of ${\fam=\ssfam \usuDelta}$ is semi-simple, cf. \cite{cp} 93.\hskip1pt10,11, or that the center ${\fam=\ssfam Z}$ of ${\fam=\ssfam \usuDelta}$ is a Lie group. Suppose that $1\kern-2.5pt {\rm l}{\,\ne\,}\zeta{\,\in\,}{\fam=\ssfam Z}$, and let $x$ be a fixed point of $\zeta$. Then $x^{\fam=\ssfam \usuDelta}$ is not contained in a line, and ${\cal D}{\,=\,}\langle x^{\fam=\ssfam \usuDelta}\rangle{\,\le\,}{\cal F}_\zeta{\,<\,}{\cal P}$. Put
${\fam=\ssfam \usuDelta}^{\hskip-1.5pt\ast}{\,=\,}{\fam=\ssfam \usuDelta}|_{\cal D}{\,=\,}{\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$ and ${\fam=\ssfam K}^1{\,=\,}{\fam=\ssfam \usuLambda}$; both ${\fam=\ssfam \usuDelta}^{\hskip-1.5pt\ast}$ and ${\fam=\ssfam \usuLambda}$ are semi-simple. Note that ${\cal D}$ is connected, and write $\dim{\cal D}{\,=\,}d$. If $d{\,=\,}2$, then ${\fam=\ssfam \usuDelta}^{\hskip-1.5pt\ast}{\le\,}\SL3{\fam=\Bbbfam R}$ and ${\fam=\ssfam \usuLambda}{\,\cong\,}{\rm G}_2$ or $\dim{\fam=\ssfam K}{\,\le\,}10$\: (see 2.6(d)\,). Hence $\dim{\fam=\ssfam \usuDelta}^{\hskip-1.5pt\ast}{\,=\,}8$ and ${\fam=\ssfam \usuPhi}$ is semi-simple. If $d{\,=\,}4$, then $5{\,\le\,}\dim{\fam=\ssfam \usuLambda}{\,\le\,}8$,\: $\dim{\fam=\ssfam \usuDelta}^{\hskip-1.5pt\ast}{\,\ge\,}13$, and then ${\fam=\ssfam \usuDelta}^{\hskip-1.5pt\ast}{\,\cong\,}\PSL3{\fam=\Bbbfam C}$,\: (see \cite{sz15} 5.6, \cite{cp} 71.8, or 2.14 above). By Stiffness, ${\fam=\ssfam \usuLambda}{\,\cong\,}\SU3{\fam=\Bbbfam C}$ or $\dim{\fam=\ssfam \usuLambda}{\,=\,}6$. If ${\cal D}{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal B}{\,<\,}{\cal P}$ for some (closed) subplane ${\cal B}$, then ${\fam=\ssfam \usuLambda}$ is compact by Stiffness, and ${\fam=\ssfam \usuPhi}$ is semi-simple. We may assume, therefore, that ${\cal D}$ is a maximal subplane of ${\cal P}$, in particular, ${\cal D}{\,=\,}{\cal F}_\zeta$. Choose a line $L{\,\in\,}{\cal D}$ and a point $p{\,\in\,}L\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt{\cal D}$. Then ${\fam=\ssfam K}_p{\,=\,}1\kern-2.5pt {\rm l}$, so that ${\fam=\ssfam \usuDelta}_p$ acts faithfully on ${\cal D}$. Moreover, $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}16{+}6$ and $\dim{\fam=\ssfam \usuDelta}_p{\,\ge\,}10$. Hence ${\cal D}$ is the cšassical complex plane, see \cite{cp} 72.8. For two distinct points $a,b{\,\in\,}{\cal D}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt L$, the stabilizer ${\fam=\ssfam \usuGamma}{=\,}{\fam=\ssfam \usuDelta}_{p,a,b}$ fixes each point of the orbit $p^{\langle \zeta\rangle}$ and of the line $ab\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal D}$, and ${\fam=\ssfam \usuGamma}$ is contained in a group ${\fam=\Bbbfam C}{\rtimes}{\fam=\Bbbfam C}^{\times}$. As $\dim{\fam=\ssfam \usuGamma}{\,\ge\,}2$, the group ${\fam=\ssfam \usuGamma}$ is not compact. It follows that ${\cal F}{\,=\,}{\cal F}_{\fam=\ssfam \usuGamma}$ is a $4$-dimensional subplane ${\ne\,}{\cal D}$. Note that both ${\cal D}$ and ${\cal F}$ are ${\fam=\ssfam Z}$-invariant. The induced groups
${\fam=\ssfam Z}|_{\cal D}{\,=\,}{\fam=\ssfam Z}/{\fam=\ssfam N}$ and ${\fam=\ssfam Z}|_{\cal F}{\,=\,}{\fam=\ssfam Z}/{\fam=\ssfam \usuXi}$ are Lie groups by \cite{cp} 71.2, and ${\fam=\ssfam N}\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam \usuXi}$ acts trivially on $\langle {\cal D},{\cal F}\rangle$, which coincides with ${\cal P}$ by maximality of ${\cal D}$. Consequently, ${\fam=\ssfam N}\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam \usuXi}{\,=\,}1\kern-2.5pt {\rm l}$ and ${\fam=\ssfam Z}$ itself is a Lie group, cf. \cite{cp} 94.3. Finally let ${\cal D}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$. By Stiffness, ${\fam=\ssfam \usuLambda}$ is compact and semi-simple, hence ${\fam=\ssfam \usuLambda}{\,\cong\,}\Spin3{\fam=\Bbbfam R}$ or ${\fam=\ssfam \usuLambda}{\,=\,}1\kern-2.5pt {\rm l}$. Thus $\dim{\fam=\ssfam \usuDelta}^{\hskip-1.5pt\ast}{\,\ge\,}18$, and we may assume that $\dim{\fam=\ssfam \usuDelta}{\,<\,}27$. Then \cite{sz1} 2.1 implies ${\fam=\ssfam \usuDelta}^{\hskip-1.5pt\ast}{\,\cong\,}\PU3({\fam=\Bbbfam H},r)$, and ${\fam=\ssfam \usuPhi}$ is semi-simple. \hglue 0pt plus 1filll $\scriptstyle\square$
\par
{\bf 3.2 Normal torus.} {\it Assume that ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\emptyset$ and that $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}23$, so that ${\fam=\ssfam \usuDelta}$ is a Lie group. If ${\fam=\ssfam \usuDelta}$ has a normal torus subgroup ${\fam=\ssfam \usuTheta}$, then ${\cal P}$ is a Hughes plane\/}. \par
{\tt Proof.} The torus ${\fam=\ssfam \usuTheta}$ is even central (\cite{cp} 93.19), hence it cannot contain a reflection, and there is a planar involution $\iota$ in the center of ${\fam=\ssfam \usuDelta}$. Let
${\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}{\,=\,}{\fam=\ssfam \usuDelta}|_{{\cal F}_\iota}{\,=\,}{\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$.
By Stiffness, $\dim{\fam=\ssfam K}{\,\le\,}3$. If ${\fam=\ssfam \usuTheta}|_{{\cal F}_\iota}{\,\ne\,}1\kern-2.5pt {\rm l}$, then ${\cal F}_\iota$ has a ${\fam=\ssfam \usuDelta}$-invariant Baer subplane and $\dim{\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}{\,\le\,}17$\: (see \cite{cp} 86.35 or use \cite{cp} 83.11). Hence ${\fam=\ssfam \usuTheta}{\,\trianglelefteq\,}{\fam=\ssfam K}^1$ and ${\fam=\ssfam K}^1{\,\not\cong\,}\Spin3{\fam=\Bbbfam R}$. Therefore ${\fam=\ssfam K}^1{\,=\,}{\fam=\ssfam \usuTheta}$ and $\dim{\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}{\,>\,}21$; moreover, ${\fam=\ssfam \usuDelta}$ fixes no element in ${\cal F}_\iota$. Now \cite{sz1} 2.1 implies that $\dim{\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}{\,=\,}35$ and ${\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}{\,\cong\,}\PSL3{\fam=\Bbbfam H}$ as claimed. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\bf 3.3 Normal vector group.} {\it If ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\emptyset$ and if ${\fam=\ssfam \usuDelta}$ has a {\rm(}minimal\,{\rm)} normal vector subgroup ${\fam=\ssfam \usuTheta}$, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}23$\/}. \par
{\tt Proof.} By Lemma 2.13, the group ${\fam=\ssfam \usuTheta}$ fixes some element, say a point $a$, and the assumption ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\emptyset$ implies that $\langle a^{\fam=\ssfam \usuDelta}\rangle{\,=\,}{\cal D}$ is a
connected subplane of ${\cal P}$; moreover, ${\fam=\ssfam \usuTheta}|_{\cal D}{\,=\,}1\kern-2.5pt {\rm l}$ and ${\cal D}{\,<\,}{\cal P}$. Put
${\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}{\,=\,}{\fam=\ssfam \usuDelta}|_{\cal D}{\,=\,}{\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$ and note that ${\fam=\ssfam \usuTheta}{\,\le\,}{\fam=\ssfam \usuLambda}{\hskip1pt:=\,}{\fam=\ssfam K}^1$; in particular, ${\fam=\ssfam \usuLambda}$ is not compact,
$\dim{\cal D}\hskip1pt|\hskip1pt4$, and $\dim{\fam=\ssfam \usuLambda}{\,\le\,}10$ by 2.6(d). If ${\cal D}$ is flat, then $\dim{\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}{\,\le\,}8$ and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}18$. Hence we may assume that $\dim{\cal F}_{\fam=\ssfam \usuLambda}{\,=\,}4$. Stiffness 2.6(e) yields $\dim{\fam=\ssfam \usuLambda}{\,\le\,}7$. As ${\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}$ fixes no element in ${\cal D}$, it follows from \cite{cp} 71.\,4 and 8 or 2.14 that ${\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}$ is semi-simple, even almost simple, and that $\dim{\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}{\,\le\,}8$ or ${\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}{\,\cong\,}\PSL3{\fam=\Bbbfam C}$. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\tt Remarks.} Cf. also \cite{gs} XI.10.19. Under the assumptions of 3.3, the invariant subplane ${\cal D}$ is uniquely determined. $\langle {\cal D},z\rangle{\,=\,}{\cal P}$ for each point $z{\,\notin\,}{\cal D}$ \ (or ${\fam=\ssfam \usuLambda}$ would be compact by 2.6(c)\,). Hence ${\fam=\ssfam \usuLambda}$ acts freely on the set of points outside ${\cal D}$. There is a covering group ${\fam=\ssfam \usuPsi}$ of ${\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}$ in ${\fam=\ssfam \usuDelta}$, see \cite{cp} 94.27. As in \cite{sz2} Th.\,1(e) it follows that the involutions in ${\fam=\ssfam \usuPsi}$ are planar or ${\fam=\ssfam \usuLambda}{\,\le\,}\Cs{}{\fam=\ssfam \usuPsi}$. Problem: is there a plane with a group ${\fam=\ssfam \usuDelta}{\,\cong\,}\SL3{\fam=\Bbbfam C}{\ltimes}{\fam=\Bbbfam C}^3$\, and ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\emptyset$\,? \par
{\Bf 4. Exactly one fixed element} \par
Throughout this section, assume that ${\fam=\ssfam \usuDelta}$ fixes a unique line $W$ and no point. In the classical octonion plane, ${\fam=\ssfam \usuSigma}_W$ has a subgroup ${\fam=\ssfam \usuDelta}{\,\cong\,}\Spin{10}({\fam=\Bbbfam R},1){\ltimes}{\fam=\Bbbfam R}^{16}$ of dimension $61$,\, cf. \cite{cp} 15.6; in all other cases $\dim{\fam=\ssfam \usuDelta}{\,<\,}40$ by \cite{cp} 87.7. In \cite{sz8} it has been shown that ${\cal P}$ is a translation plane whenever $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}35$. \par
{\bf 4.0.} {\it If ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\{W\}$ and if $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}23$, then ${\fam=\ssfam \usuDelta}$ is a Lie group\/}. \par
{\tt Proof.} (a) There exist arbitrarily small compact central subgroups ${\fam=\ssfam N}{\,\le\,}{\fam=\ssfam \usuDelta}$ of dimension $0$ such that ${\fam=\ssfam \usuDelta}/{\fam=\ssfam N}$ is a Lie group, cf. \cite{cp} 93.8. If ${\fam=\ssfam N}$ acts freely on $P{\smallsetminus}W$, then each stabilizer ${\fam=\ssfam \usuDelta}_x$ with $x{\,\notin\,}W$ is a Lie group because ${\fam=\ssfam \usuDelta}_x{\kern 2pt {\scriptstyle \cap}\kern 2pt}{\fam=\ssfam N}{\,=\,}1\kern-2.5pt {\rm l}$. By \cite{cp} 51.\hskip1pt6 and 8 and 52.12, the one-point compactification $X$ of $P{\smallsetminus}W$ is homeomorphic to the quotient space $P/W$, and $X$ is a Peano continuum (i.e., a continuous image of the unit interval); moreover, $X$ is homotopy equivalent to ${\fam=\Bbbfam S}_{16}$, and $X$ has Euler characteristic $\chi(X){\,=\,}2$. According to a theorem of L\"owen \cite{lw3}, these properties suffice for ${\fam=\ssfam \usuDelta}$ to be a Lie group. \\ (b) Suppose now that $x^\zeta{\,=\,}x$ for some $x{\,\notin\,}W$ and some $\zeta{\,\in\,}{\fam=\ssfam N}{\smallsetminus}1\kern-2.5pt {\rm l}$. By assumption, $x^{\fam=\ssfam \usuDelta}$ is not contained in a line and hence generates a ${\fam=\ssfam \usuDelta}$-invariant subplane ${\cal D}{\,=\,}\langle x^{\fam=\ssfam \usuDelta}\rangle{\,\le\,}{\cal F}_\zeta{\,<\,}{\cal P}$. Put
${\fam=\ssfam \usuDelta}^{\hskip-1pt*}{\,=\,}{\fam=\ssfam \usuDelta}|_{\cal D}{\,=\,}{\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$. If ${\cal D}$ is flat, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}6{+}14$ by Stiffness; similarly, $\dim{\cal D}{\,=\,}4$ implies $\dim{\fam=\ssfam \usuDelta}{\,\le\,}12{+}8$. Hence ${\cal D}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$,\: ${\fam=\ssfam K}$ is compact, $\dim{\fam=\ssfam K}{\,<\,}8$,\: $\dim{\fam=\ssfam \usuDelta}^{\hskip-1pt*}{\,\ge\,}16$,\: ${\fam=\ssfam \usuDelta}^{\hskip-1pt*}$ is a Lie group\, (\cite{pw3}), and we may assume that $1\kern-2.5pt {\rm l}{\,\ne\,}{\fam=\ssfam N}{\,\le\,}{\fam=\ssfam K}$. If even $\dim{\fam=\ssfam \usuDelta}^{\hskip-1pt*}{\,>\,}16$, then \cite{sz1} 1.10 shows that ${\cal D}$ is the classical quaternion plane, because ${\fam=\ssfam \usuDelta}^{\hskip-1pt*}$ does not fix a flag. \\ (c) If ${\fam=\ssfam \usuDelta}$ is not transitive on $S{\,=\,}W\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal D}$, then there are points $u,v,w{\,\in\,}S$ such that ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuDelta}_{u,v,w}{\,\le\,}9$. Choose a line $L$ of ${\cal D}$ in the pencil ${\frak L}_v\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{W\}$ and a point $z{\,\in\,}L\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt{\cal D}$, so that ${\fam=\ssfam \usuDelta}_z\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam K}{\,=\,}1\kern-2.5pt {\rm l}$ and ${\fam=\ssfam \usuDelta}_z$ acts faithfully on ${\cal D}$. In particular, ${\fam=\ssfam \usuDelta}_z$ fixes $L$ and ${\fam=\ssfam \usuDelta}_z$ is a Lie group. Moreover, $\dim z^{{\fam=\ssfam \usuDelta}_L}{\,<\,}8$\: (or else ${\fam=\ssfam K}$ would be a Lie group by \cite{cp} 53.2). Therefore ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuDelta}_z{\,\le\,}3{+}4{+}7{\,=\,}14$ and ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuDelta}_{z,u,w})^1$ satisfies $\dim{\fam=\ssfam \usuLambda}{\,\ge\,}3$. Equality is possible only if ${\fam=\ssfam \usuDelta}$ is triply transitive on $V{\,=\,}v^{\fam=\ssfam \usuDelta}$. Next, it will be shown that ${\cal F}_{\fam=\ssfam \usuLambda}$ is connected. If $\dim{\fam=\ssfam K}{\,<\,}7$, then ${\cal D}$ is classical and ${\fam=\ssfam \usuLambda}$ fixes all points of a circle in $S$ containing $u,v,w$. By \cite{cp} 55.32, the compact group ${\fam=\ssfam K}$ has no subgroup ${\fam=\Bbbfam T}^2$. Hence ${\fam=\ssfam K}^1$ is commutative or an almost direct product of ${\fam=\ssfam \usuOmega}{\,\circeq\,}\SO3{\fam=\Bbbfam R}$ with a commutative group.
From \cite{cp} 93.19 it follows that ${\fam=\ssfam \usuLambda}$ acts trivially on the commutative factor ${\fam=\ssfam A}$ of ${\fam=\ssfam K}^1$. If $\dim{\fam=\ssfam K}{\,>\,}3$, then $z^{\fam=\ssfam A}{\,\subseteq\,}{\cal F}_{\fam=\ssfam \usuLambda}$ has positive dimension. In any case, $\dim{\cal F}_{\fam=\ssfam \usuLambda}{\,\ge\,}2$. As ${\fam=\ssfam N}$ acts freely on $L\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt{\cal D}$ and
groups of planes of dimension ${\le\,}4$ are Lie groups (cf. \cite{cp} 32.21 and 71.2), it
follows that ${\cal F}_{\fam=\ssfam \usuLambda}{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$. Stiffness implies ${\fam=\ssfam \usuLambda}{\,\cong\,}\Spin3{\fam=\Bbbfam R}$. Consequently, ${\fam=\ssfam \usuDelta}_z$ is doubly transitive on $V\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{v\}{\,\approx\,}{\fam=\Bbbfam R}^3$, the group $\nabla{\,=\,}({\fam=\ssfam \usuDelta}_{z,u})^1$ is $6$-dimensional, and
$\nabla|_V{\,\cong\,}e^{\fam=\Bbbfam R}{\hskip1pt\cdot\hskip1pt}\SO3{\fam=\Bbbfam R}$.
Now $\dim{\fam=\ssfam \usuLambda}|_V{\,=\,}1$, but the almost simple group ${\fam=\ssfam \usuLambda}$ does not have a $1$-dimensional factor groi§p. \\ (d) The previous step shows that ${\fam=\ssfam \usuDelta}$ is transitive on $S{\,=\,}W\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal D}$. Hence $S{\,\approx\,}{\fam=\Bbbfam S}_4$, and ${\fam=\ssfam \usuDelta}$ has a subgroup {\tencyss\cyracc Yu}${\,\cong\,}\U2{\fam=\Bbbfam H}{\,\cong\,}\Spin5{\fam=\Bbbfam R}$, see \cite{cp} 53.2,\;96.19--22,\;55.40, and 94.27. The central involution $\sigma$ of {\tencyss\cyracc Yu} induces on ${\cal D}$ a reflection with axis $S$. As its center is not fixed by ${\fam=\ssfam \usuDelta}$, it follows from \cite{cp} 61.13 that ${\fam=\ssfam \usuDelta}^{\hskip-1.5pt*}$ contains a transitive translation group with axis~$S$. Therefore ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\ge\,}18$ and ${\cal D}$ is classical. By the last part of 2.10, the involution $\sigma$ is even a reflection of ${\cal P}$, and $\sigma^{\fam=\ssfam \usuDelta}\sigma{\,=\,}{\fam=\ssfam T}{\,\cong\,}{\fam=\Bbbfam R}^8$ is normal in ${\fam=\ssfam \usuDelta}$.
The group ${\fam=\ssfam \usuDelta}|_{\cal D}$ is contained in $\SL2{\fam=\Bbbfam H}{\hskip1pt\cdot\hskip1pt}{\fam=\Bbbfam H}^{\times}{\ltimes}{\fam=\Bbbfam R}^8$.
Either ${\fam=\ssfam \usuDelta}|_S$ is compact, or ${\fam=\ssfam \usuDelta}$ has a subgroup ${\fam=\ssfam \usuUpsilon}{\,\cong\,}\SL2{\fam=\Bbbfam H}$. In the first case, ${\fam=\ssfam \usuDelta}|_S{\,=\,}${\tencyss\cyracc Yu}$|_S{\,\cong\,}\SO5{\fam=\Bbbfam R}$. For $v,w{\,\in\,}S$, the stabilizer ${\fam=\ssfam \usuDelta}_v$ fixes a second point $u{\,\in\,}S$, and ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuDelta}_{u,v,w}{\,=\,}4{+}3$. Choose a line $L$ of ${\cal D}$ in the pencil ${\frak L}_v\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{W\}$, a point $z{\,\in\,}L\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt{\cal D}$, and put ${\fam=\ssfam \usuGamma}{\,=\,}{\fam=\ssfam \usuDelta}_{L,w}$. Then ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuGamma}{\,\le\,}11$, and ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuGamma}_{\hskip-2pt z})^1$ satisfies $\dim{\fam=\ssfam \usuLambda}{\,\ge\,}4$.
Again ${\fam=\ssfam \usuLambda}$ is a Lie group since ${\fam=\ssfam N}$ acts freely on $L\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt{\cal D}$. As ${\cal D}$ is the classical quaternion plane, any collineation which fixes $3$ collinear points of ${\cal D}$ even fixes all points of a circle. Hence ${\cal F}_{\fam=\ssfam \usuLambda}$ is connected. By \cite{cp} 32.21 and 71.2, we have $z^{\fam=\ssfam N}{\,\subseteq\,}{\cal F}_{\fam=\ssfam \usuLambda}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$. Now Stiffness would imply $\dim{\fam=\ssfam \usuLambda}{\,\le\,}3$ contrary to what has been stated before. \\ (e) In the case $\SL2{\fam=\Bbbfam H}{\,\cong\,}{\fam=\ssfam \usuUpsilon}{\,\le\,}{\fam=\ssfam \usuDelta}$, choose $z{\hskip1pt\in\hskip1pt}W\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt{\cal D}$ and $p{\,\notin\,}{\cal D}\kern 2pt {\scriptstyle \cup}\kern 2pt W$\hskip-2pt. As ${\fam=\ssfam K}$ acts freely outside~${\cal D}$,\; \cite{cp} 53.2 implies $\dim z^{\fam=\ssfam \usuDelta}{\,<\,}8$ and $\dim p^{\fam=\ssfam \usuDelta}{\,<\,}16$. Consequently, ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuDelta}_{p,z})^1$ has positive dimension, and ${\fam=\ssfam \usuLambda}$ is a Lie group because ${\fam=\ssfam \usuLambda}\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam K}{\,=\,}1\kern-2.5pt {\rm l}$. Moreover, $\langle p^{\fam=\ssfam N},z^{\fam=\ssfam N}\rangle{\,\le\,}{\cal F}_{\fam=\ssfam \usuLambda}{\,<\,}{\cal P}$. The orbit $p^{\fam=\ssfam N}$ is contained in a unique ``inner'' line $L$ of ${\cal D}$ intersecting $W$ in a point $v$. Hence the line $pz$ does not belong to ${\cal D}$; it intersects ${\cal D}$ in a unique inner point $a$, and $av{\,\ne\,}L,W$. By definition, ${\fam=\ssfam \usuLambda}$ fixes $L,v,$ and $a$ and hence $3$ distinct lines of the quaternion plane ${\cal D}$ in the pencil ${\frak L}_v$. Therefore ${\cal F}_{\fam=\ssfam \usuLambda}$ is connected. Again $\dim{\cal F}_{\fam=\ssfam \usuLambda}{\,>\,}4$,\; ${\cal F}_{\fam=\ssfam \usuLambda}{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$, and ${\fam=\ssfam \usuLambda}$ is isomorphic to a subgroup of $\Spin3{\fam=\Bbbfam R}$. In particular, ${\fam=\ssfam \usuLambda}$ is compact and there exists an involution $\lambda{\,\in\,}{\fam=\ssfam \usuLambda}$ with ${\cal F}_\lambda{\,=\,}{\cal F}_{\fam=\ssfam \usuLambda}$. If $\dim{\fam=\ssfam \usuDelta}{\,=\,}23$, then a maximal compact subgroup is isomorphic to {\tencyss\cyracc Yu} and ${\fam=\ssfam \usuDelta}$ is a Lie group. Therefore $\dim{\fam=\ssfam \usuDelta}{\,>\,}23$,\; $\dim{\fam=\ssfam \usuLambda}{\,>\,}1$, and ${\fam=\ssfam \usuLambda}{\,\cong\,}\Spin3{\fam=\Bbbfam R}$. Obviously
$\overline\lambda{\,=\,}\lambda|_{\cal D}{\,\ne\,}1\kern-2.5pt {\rm l}$,\; $\overline\lambda$ is not conjugate to the central involution $\sigma$ of {\tencyss\cyracc Yu}, and Richardson's theorem $(\dagger)$ shows that ${\fam=\ssfam \usuLambda}$ does not fix a $2$-sphere in $S{\,=\,}W\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal D}$. Therefore $\overline\lambda$ is a reflection with center $v$ and some axis $au\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal D}$. Up to conjugation, $a^{\hskip-1.5pt{\fam=\ssfam \usuUpsilon}}{=\,}a$. As ${\fam=\ssfam \usuUpsilon}_{\hskip-2pt v}$ is (doubly) transitive on $S\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{v\}$, the dual of \cite{cp} 61.19 shows that the elation group ${\fam=\ssfam \usuDelta}^{\hskip-1.5pt*}_{[v,av]}$ is transitive. Consequently ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,=\,}27$, and then $\dim{\fam=\ssfam \usuLambda}{\,\ge\,}5$, a contradiction. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\bf 4.1 Theorem.} {\it If ${\fam=\ssfam \usuDelta}$ is transitive on $W$, then ${\cal P}$ is classical\/}. \par
{\tt Proof.} By \cite{cp} 42.8 and 44.3, the space $W$ is locally contractible and ${\fam=\ssfam \usuDelta}|_W$ is a locally compact transformation group with a countable basis. Hence \cite{HK} Cor.\,5.5 applies and $W$ is a topological manifold, in fact, $W{\,\approx\,}{\fam=\Bbbfam S}_8$, see \cite{cp} 52.3.
From \cite{cp} 96.\,19,\,21, and~22 it follows that ${\fam=\ssfam \usuDelta}|_W$ has a subgroup $\SO9{\fam=\Bbbfam R}$;
it is covered by an almost simple subgroup ${\fam=\ssfam \usuPhi}{\,\le\,}{\fam=\ssfam \usuDelta}$\, (cf. \cite{cp} 94.27). The last part
of 2.10 implies that ${\fam=\ssfam \usuPhi}{\,\cong\,}\Spin9{\fam=\Bbbfam R}$. The central involution $\sigma{\,\in\,}{\fam=\ssfam \usuPhi}$ is a
reflection with axis $W$ and some center $c$. By assumption $c^{\fam=\ssfam \usuDelta}{\,\ne\,}c$, and 2.9 shows that the translation group $\sigma^{\fam=\ssfam \usuDelta}\sigma{\,=\,}{\fam=\ssfam T}$ has dimension $\dim c^{\fam=\ssfam \usuDelta}{\,>\,}0$. As ${\fam=\ssfam \usuDelta}$ is transitive on $W$, each group ${\fam=\ssfam \usuDelta}_{[z,W]}$ with $z{\,\in\,}W$ has the same dimension, and then ${\fam=\ssfam T}{\,\cong\,}{\fam=\Bbbfam R}^{16}$ by \cite{sz11} or \cite{cp} 61.13, and $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}52$. \hglue 0pt plus 1filll $\scriptstyle\square$ \\ For a related result see \cite{cp} 81.19.
\par
Without the restriction $|{\cal F}_{\fam=\ssfam \usuDelta}|{\,=\,}1$, Priwitzer \cite{pw2}, \cite{pw1} has treated semi-simple groups of dimension $\ge29$. She proved in particular: \par
{\bf 4.2 Theorem.} {\it If ${\cal P}$ is not classical, if ${\fam=\ssfam \usuDelta}$ is semi-simple, ${\cal F}_{\fam=\ssfam \usuDelta}{\,\ne\,}\emptyset$, and $\dim{\fam=\ssfam \usuDelta}{\,>\,}28$, then ${\fam=\ssfam \usuDelta}{\,\cong\,}\Spin9({\fam=\Bbbfam R},r)$ with $r{\,\le\,}1$, the central involution of ${\fam=\ssfam \usuDelta}$ is a reflection, and ${\cal F}_{\fam=\ssfam \usuDelta}$ is a non-incident point-line pair\/}. \par
{\bf 4.3 Semi-simple groups.} {\it If ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}W$ and if ${\fam=\ssfam \usuDelta}$ is semi-simple, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}16$ or $\dim{\fam=\ssfam \usuDelta}{\,=\,}21$ and ${\fam=\ssfam \usuDelta}$ is almost simple or ${\fam=\ssfam \usuDelta}$ may be a product of at most $7$ factors isomorphic to the simply connected covering group of $\SL2{\fam=\Bbbfam R}$. If $\dim{\fam=\ssfam \usuDelta}{\,>\,}13$, then ${\fam=\ssfam \usuDelta}$ is a Lie group\/}, see \cite{szp} for a {\tt proof}. \par
{\bf 4.4 Normal torus.} {\it Suppose that ${\fam=\ssfam \usuDelta}$ is a Lie group, that $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}18$, and that ${\fam=\ssfam \usuDelta}$ has a normal torus subgroup ${\fam=\ssfam \usuTheta}$. Then the fixed elements of the involution $\iota{\,\in\,}{\fam=\ssfam \usuTheta}$ form a ${\fam=\ssfam \usuDelta}$-invariant classical quaternion plane ${\cal B}{\,=\,}{\cal F}_\iota$\/}. \par
{\tt Proof.} ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\{W\}$ implies that $\iota$ is not a reflection. Put ${\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}{\,=\,}{\fam=\ssfam \usuDelta}|_{\cal B}{\,=\,}{\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$, and note that ${\cal F}_{{\fam=\ssfam \usuDelta}^{\hskip-1pt\ast}}{\,=\,}\{W\}$. Stiffness shows ${\fam=\ssfam K}^1{\,\cong\,}\Spin3{\fam=\Bbbfam R}$ or $\dim{\fam=\ssfam K}{\,\le\,}1$. If ${\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}$ is semi-simple or has a normal torus, in particular, if ${\fam=\ssfam \usuTheta}^*{\,\ne\,}1\kern-2.5pt {\rm l}$, then ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\le\,}13$ by \cite{sz1} 3.\hskip1pt1,2, and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}16$. Hence ${\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}$ has a normal vector subgroup, ${\fam=\ssfam \usuTheta}^*{\,=\,}1\kern-2.5pt {\rm l}$, $\,{\fam=\ssfam \usuTheta}{\,\trianglelefteq\,}{\fam=\ssfam K}$, and ${\fam=\ssfam \usuTheta}{\,=\,}{\fam=\ssfam K}^1$. From \cite{sz1} 3.3 it follows that ${\cal B}$ is classical or $\dim{\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}{\,\le\,}16$. In the second case, $\dim{\fam=\ssfam \usuDelta}{\,<\,}18$. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\tt Remark.} Even if ${\fam=\ssfam \usuDelta}$ induces on ${\cal B}$ the full affine group or if ${\fam=\ssfam \usuDelta}$ is an {\it affine Hughes group\/} (i.e., the $28$-dimensional stabilizer of an interior line in a proper Hughes plane), it seems to be difficult to describe all the corresponding planes, cf. \cite{sz1} 3.2 Remark. \par
{\bf 4.5 Normal vector group.} {\it If ${\fam=\ssfam \usuTheta}{\,\cong\,}{\fam=\Bbbfam R}^t$ is a minimal normal subgroup of ${\fam=\ssfam \usuDelta}$ and if $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}33$, then the connected component ${\fam=\ssfam T}$ of the translation group ${\fam=\ssfam \usuDelta}_{[W,W]}$ is a vector group of dimension at least $2$. Moreover, ${\fam=\ssfam \usuTheta}{\,\le\,}{\fam=\ssfam T}$, or $t{\,=\,}8$ and $\dim{\fam=\ssfam T}{\,\ge\,}7$, or $t{\,=\,}12$\/}\, (\cite{sz10}). \par
{\tt Remarks on the proof.} (a) {\it If a one-parameter subgroup ${\fam=\ssfam \usuPi}{\,\le\,}{\fam=\ssfam \usuTheta}$ is straight, then ${\fam=\ssfam \usuPi}{\,\le\,}{\fam=\ssfam T}$\/}. As ${\fam=\ssfam \usuPi}$ is not compact, ${\cal F}_{\fam=\ssfam \usuPi}$ is not a Baer subplane, and ${\fam=\ssfam \usuPi}{\,\le\,}{\fam=\ssfam \usuDelta}_{[z]}$ for some center $z$. In fact, $z{\,\in\,}W$ and ${\fam=\ssfam \usuPi}{\,\le\,}{\fam=\ssfam \usuDelta}_{[z,W]}$, because $z^{\fam=\ssfam \usuDelta}{\,\ne\,}z$ and ${\fam=\ssfam \usuPi}^{\fam=\ssfam \usuDelta}$ is contained in the commutative group ${\fam=\ssfam \usuTheta}$. From $z^{\fam=\ssfam \usuDelta}{\,\ne\,}z$ it follows also that ${\fam=\ssfam T}$ is commutative, ${\fam=\ssfam \usuPi}{\,<\,}{\fam=\ssfam T}$ and $\dim{\fam=\ssfam T}{\,>\,}1$. \\ (b) {\it If some one-parameter subgroup ${\fam=\ssfam \usuPi}{\,\le\,}{\fam=\ssfam \usuTheta}$ is not straight and if $t{\,<\,}8$, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}32$\/}. Some orbit $b^{\fam=\ssfam \usuPi}$ is not contained in a line, and $\langle b^{\fam=\ssfam \usuPi}\rangle$ is a subplane of ${\cal P}$. Choose $\rho{\,\in\,}{\fam=\ssfam \usuPi}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{1\kern-2.5pt {\rm l}\}$, put ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuDelta}_{b,\rho})^1$,
and note that ${\fam=\ssfam \usuLambda}{\,\le\,}\Cs{}{\fam=\ssfam \usuPi}$. Hence ${\fam=\ssfam \usuLambda}|_{\langle b^{\fam=\ssfam \usuPi}\rangle}{\,=\,}1\kern-2.5pt {\rm l}$, and the stiffness result 2.6(e) shows that ${\fam=\ssfam \usuLambda}{\,\cong\,}{\rm G}_2$ or $\dim{\fam=\ssfam \usuLambda}{\,\le\,}8$. The dimension formula yields $$\dim{\fam=\ssfam \usuDelta}{\,=\,}\dim b^{\fam=\ssfam \usuDelta}{\,+\,}\dim{\fam=\ssfam \usuDelta}_b{\,=\,}\dim b^{\fam=\ssfam \usuDelta}{\,+\,}\dim\rho^{{\fam=\ssfam \usuDelta}_b}{\,+\,} \dim{\fam=\ssfam \usuLambda}{\,\le\,}16{\,+\,}t{\,+\,}\dim{\fam=\ssfam \usuLambda}\,.$$ In the second case $\dim{\fam=\ssfam \usuDelta}{\,<\,}32$. If ${\fam=\ssfam \usuLambda}{\,\cong\,}{\rm G}_2$, then ${\fam=\ssfam \usuLambda}{\,\le\,}\Cs{}{\fam=\ssfam \usuTheta}$, since ${\fam=\ssfam \usuPi}^{\fam=\ssfam \usuLambda}{\,=\,}{\fam=\ssfam \usuPi}$ and ${\fam=\ssfam \usuLambda}$ has irreducible representations only in dimensions $7,14$ or larger. It follows that $\langle b^{\fam=\ssfam \usuTheta}\rangle$ is contained in the flat plane ${\cal F}_{\fam=\ssfam \usuLambda}$. Note that ${\fam=\ssfam \usuTheta}_b{\,\le\,}{\fam=\ssfam \usuDelta}_{b,\rho}$, that ${\fam=\ssfam \usuTheta}_{\hskip-.5pt b}^{\hskip1pt1}{\,\le\,}{\fam=\ssfam \usuLambda}$, and that ${\fam=\ssfam \usuLambda}$ is compact. Hence ${\fam=\ssfam \usuTheta}_{\hskip-.5pt b}^{\hskip1pt1}{\,=\,}1\kern-2.5pt {\rm l}$, $\,\dim{\fam=\ssfam \usuTheta}_b{\,=\,}0$, $\,\dim b^{\fam=\ssfam \usuTheta}{\,=\,}t{\,\le\,}2$, and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}32$ as claimed. \\ (c) For each $t{\,\ge\,}8$ the proof in \cite{sz10} uses the irreducible representation of ${\fam=\ssfam \usuDelta}$ on ${\fam=\ssfam \usuTheta}$. The arguments are different for distinct values of $t$. They are rather involved and shall not be reproduced here. \par
{\bf 4.6 Translation group.} {\it If ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\{W\}$, if ${\fam=\ssfam \usuDelta}$ has a normal vector subgroup, and if $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}35$, then the translation group ${\fam=\ssfam T}$ is transitive\/}\, (\cite{sz8}). \par
{\tt Remarks on the proof.} (a) ${\fam=\ssfam T}$ contains a minimal normal subgroup ${\fam=\ssfam \usuTheta}{\,\cong\,}{\fam=\Bbbfam R}^t$ of ${\fam=\ssfam \usuDelta}$. In a first step, we show that $t{\,\ge\,}6$. Choose some point $a{\,\notin\,}W$ and put ${\fam=\ssfam \usuGamma}{\,=\,}({\fam=\ssfam \usuDelta}_a)^1$. For $\rho,\rho'$ on one-parameter groups in ${\fam=\ssfam \usuTheta}$ with different centers, the group ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuGamma}_{\hskip-2pt\rho,\rho'})^1$ fixes a connected subplane and Stiffness applies. We have $19{\,\le\,}\dim{\fam=\ssfam \usuGamma}{\,\le\,}2t{\,+\,}\dim{\fam=\ssfam \usuLambda}{\,\le\,}2t{\,+\,}14$ and $t{\,>\,}2$. If ${\fam=\ssfam \usuLambda}{\,\cong\,}{\rm G}_2$, then ${\fam=\ssfam \usuTheta}{\kern 2pt {\scriptstyle \cap}\kern 2pt}\Cs{}{\fam=\ssfam \usuLambda}{\,\cong\,}{\fam=\Bbbfam R}^2$. Hence $t{\,\ge\,}9$ or $\dim{\fam=\ssfam \usuLambda}{\,\le\,}8$ and $t{\,\ge\,}6$. \\ (b) Suppose that $t{\,=\,}6$. Then ${\fam=\ssfam \usuLambda}{\,\not\hskip1pt\cong\,}\SU3{\fam=\Bbbfam C}$ and $\dim{\fam=\ssfam \usuLambda}{\,<\,}8$. Consequently $\dim\rho^{\fam=\ssfam \usuGamma}{\,=\,}6$, $\,\rho^{\fam=\ssfam \usuGamma}$ is open in
${\fam=\ssfam \usuTheta}$ for each $\rho$, and ${\fam=\ssfam \usuGamma}$ is transitive on ${\fam=\ssfam \usuTheta}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{1\kern-2.5pt {\rm l}\}$. Similarly, ${\fam=\ssfam \usuGamma}$ is doubly transitive on the $5$-dimensional projective space ${\rm P}{\fam=\ssfam \usuTheta}$ consisting of the one-dimensional subspaces of ${\fam=\ssfam \usuTheta}$. Put $\tilde{\fam=\ssfam \usuGamma}{\,=\,}{\fam=\ssfam \usuGamma}|_{\fam=\ssfam \usuTheta}{\,=\,}{\fam=\ssfam \usuGamma}/{\fam=\ssfam K}$, and note that $\langle a^{\fam=\ssfam \usuTheta}\rangle{\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\hskip-3pt}{\cal P}$. From \cite{cp} 96.17 it follows that $\tilde{\fam=\ssfam \usuGamma}$ is isomorphic to $\PSL3{\fam=\Bbbfam C}$. Therefore $\dim{\fam=\ssfam K}{\,=\,}3$ and $\langle a^{\fam=\ssfam \usuTheta}\rangle{\,=\,}{\cal B}$ is a Baer subplane. \cite{sz9} Th.~3.3 or \cite{sz1} 7.3 implies that ${\cal B}$ is the classical quaternion plane. Hence $\tilde{\fam=\ssfam \usuGamma}$ would be contained in ${\fam=\Bbbfam H}^{\times}{\cdot\,}\SL2{\fam=\Bbbfam H}$, which is impossible. \\ (c) Similarly, for each $t{\,<\,}16$ a contradiction can be derived from $\dim{\fam=\ssfam T}{\,<\,}16$ and a detailed analysis of the representaion of ${\fam=\ssfam \usuDelta}$ on ${\fam=\ssfam \usuTheta}$ in combination with the consequences of the assumptions on ${\fam=\ssfam \usuDelta}$ as a group of automorphisms of ${\cal P}$. The complicated proofs in \cite{sz8} shall not be repeated here. \par
{\bf 4.7 Lemma.} {\it Suppose that ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\{W\}$ and that $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}35$. If ${\fam=\ssfam \usuTheta}{\,\cong\,}{\fam=\Bbbfam R}^t$ is a minimal normal subgroup of ${\fam=\ssfam \usuDelta}$ and if ${\fam=\ssfam \usuTheta}{\,\le\,}{\fam=\ssfam T}$, then $8{\,\le\,}t\equiv0\bmod2$\/}. \par
{\tt Remark.} The claim is contained in steps (1)--(10) of the proof in \cite{sz8}; note that these steps do not use the assumption $\dim{\fam=\ssfam T}{\,<\,}16$. \par
{\bf 4.8 Translation complement.} {\it Under the assumptions of {\rm 4.6}, the plane ${\cal P}$ is the classical octonion plane or a complement ${\fam=\ssfam \usuGamma}{\,=\,}{\fam=\ssfam \usuDelta}_a$ of the translation group ${\fam=\ssfam T}$ contains one of the groups $\SL2{\fam=\Bbbfam H}{\,\cdot\,}\SU2{\fam=\Bbbfam C}$ or $\SU4{\fam=\Bbbfam C}{\,\cdot\,}{\rm H}$ with ${\rm H}{\,=\,}\SU2{\fam=\Bbbfam C}$ or $\SL2{\fam=\Bbbfam R}$, and $\dim{\fam=\ssfam \usuDelta}{\,=\,}35$\/}. \par
{\tt Remarks.} A proof of Theorem 4.8 is due to H\"ahl \cite{Ha1}, who also determined all planes such that ${\fam=\ssfam \usuGamma}$ has a factor $\SU4{\fam=\Bbbfam C}$, see \cite{Ha2} and 4.12 below. By different arguments we will show the following weaker result: \par
{\bf 4.9 Proposition.} {\it Suppose that ${\cal P}$ is not classical. Under the assumptions of {\rm 4.7} and {\rm 4.8}, either ${\fam=\ssfam \usuTheta}{\,\cong\,}{\fam=\Bbbfam R}^8$ and ${\fam=\ssfam \usuGamma}'{\,\cong\,}{\fam=\Bbbfam H}'{\cdot\hskip1pt}\SL2{\fam=\Bbbfam H}$, or ${\fam=\ssfam \usuTheta}{\,=\,}{\fam=\ssfam T}$ and ${\fam=\ssfam \usuGamma}'$ has a proper semi-simple factor ${\fam=\ssfam B}$ acting faithfully and irreducibly on some $8$-dimensional subspace ${\fam=\ssfam \usuXi}{\,<\,}{\fam=\ssfam T}$\/}. \par
{\tt Proof.} (a) We may assume that $\dim{\fam=\ssfam \usuDelta}{\,<\,}38$ and $19{\,\le\,}\dim{\fam=\ssfam \usuGamma}{\,\le\,}21$. In fact, coordinatizing quasi-fields of all {\it translation planes\/} with a group of dimension at least $38$ have been determined by H\"ahl, see \cite{cp} 82.28 and \cite{Ha3}. There are two large families of such quasi-fields, the so-called generalized mutations and the perturbations of the octonion algebra~${\fam=\Bbbfam O}$. In the first case, it follows from \cite{cp} 82.29 that the full automorphism group ${\fam=\ssfam \usuSigma}$ of the corresponding plane fixes a flag. Now let $\rho$ be a homeomorphism of the interval $[0,\infty)$ with $1^\rho{\,=\,}1$. The multiplication of the {\it perturbation\/} ${\fam=\Bbbfam O}^{(\rho)}{\,=\,}({\fam=\Bbbfam O},+,\circ)$ is defined by
$$c{\,\circ\,}z{\,=\,}c\hskip1pt z_0{\,+\,}\|c\|^{\rho-1}\hskip1pt c\hskip1.5pt{\frak z} \quad {\rm and} \quad 0{\,\circ\,}z{\,=\,}0\,.$$ Obviously, each automorphism of ${\fam=\Bbbfam O}$ is also an automorphism of ${\fam=\Bbbfam O}^{(\rho)}$. The plane over ${\fam=\Bbbfam O}^{(\rho)}$ admits even a subgroup ${\fam=\ssfam \usuPhi}{\,\cong\,}\Spin7{\fam=\Bbbfam R}$ fixing the points $(0)$ and $(\infty)$ on $W$, see \cite{cp} 82.5(a). It follows that $(0)$ and $(\infty)$ are fixed points of the full automorphism group if the plane is not classical. Thus in both cases ${\cal F}_{\fam=\ssfam \usuDelta}{\,\ne\,}\{W\}$. Because of 4.6, we may also assume that ${\fam=\ssfam \usuGamma}$ contains a one-parameter group ${\fam=\ssfam P}$ of homologies. \par (b) Of the cases $t{\,\in\,}\{10,12,14\}$ of Lemma 4.7 the last one is quite easy: the commutator group ${\fam=\ssfam \usuGamma}'$ is semi-simple by \cite{cp} 95.6, moreover, $17{\,\le\,}\dim{\fam=\ssfam \usuGamma}'{\,\le\,}20$. According to \cite{cp} 95.10, the group ${\fam=\ssfam \usuGamma}'$ is not almost simple, and ${\fam=\ssfam \usuGamma}'{\,=\,}{\fam=\ssfam A}{\fam=\ssfam B}$ is a product of an almost simple factor ${\fam=\ssfam A}$ of minimal dimension and the semi-simple centralizer ${\fam=\ssfam B}{\,=\,}{\fam=\ssfam \usuGamma}'{\hskip-2pt\kern 2pt {\scriptstyle \cap}\kern 2pt}\Cs{}{\fam=\ssfam A}$, and $\dim{\fam=\ssfam B}{\,\ge\,}10$. Clifford's Lemma \cite{cp} 95.5 implies that ${\fam=\ssfam B}$ acts faithfully and irreducibly on ${\fam=\Bbbfam R}^7$. Now ${\fam=\ssfam B}$ is a group of type ${\rm G}_2$ and dimension $14$,\: ${\fam=\ssfam \usuGamma}{:\,}{\fam=\ssfam \usuGamma}'{\,<\,}2$,\: $\dim{\fam=\ssfam \usuGamma}'{\,>\,}17$, and then $\dim{\fam=\ssfam A}{\,=\,}6$ and ${\fam=\ssfam A}{\,\circeq\,}\SL2{\fam=\Bbbfam C}$. Again by Clifford's Lemma, ${\fam=\ssfam A}$ acts faithfully and irreducibly
on ${\fam=\Bbbfam R}^s$ with $s|14$, but this contradicts \cite{cp} 95.10 and shows that $t{\,\ne\,}14$. \\
(c) Similarly, $t{\,=\,}10$ is impossible: note that ${\fam=\ssfam \usuGamma}$ acts faithfully on ${\fam=\ssfam \usuTheta}$. The only almost simple group in the dimension range for ${\fam=\ssfam \usuGamma}'$ admitting a $10$-dimensional irreducible representation is $\SO5{\fam=\Bbbfam C}$, but this group contains $\SO5{\fam=\Bbbfam R}$ which cannot act on ${\cal P}$, see 2.10. Again ${\fam=\ssfam \usuGamma}'{\,=\,}{\fam=\ssfam A}{\fam=\ssfam B}$ is a product of proper factors with $\dim{\fam=\ssfam B}{\,\ge\,}10$. Clifford's Lemma implies that ${\fam=\ssfam B}$ acts faithfully and irreducibly on ${\fam=\Bbbfam R}^5$, and then ${\fam=\ssfam B}$ is a group $\Opr5({\fam=\Bbbfam R},r)$ with $r{\,>\,}0$. The fixed elements in ${\fam=\ssfam \usuTheta}$ of a suitable subgroup of ${\fam=\ssfam B}$ form an ${\fam=\ssfam A}$-invariant $2$-dimensional vector space on which ${\fam=\ssfam A}$ acts faithfully, $\dim{\fam=\ssfam A}{\,=\,}3$ and $\dim{\fam=\ssfam \usuGamma}'{\,=\,}13$ contrary to the hypothesis. \\ (d) The case $t{\,=\,}12$ is more complicated. An almost simple group ${\fam=\ssfam \usuGamma}'$ would be locally isomorphic to $\Sp4{\fam=\Bbbfam C}$ and has no irreducible representation on ${\fam=\Bbbfam R}^{12}$. Therefore ${\fam=\ssfam \usuGamma}'{\,=\,}{\fam=\ssfam A}{\fam=\ssfam B}$, where ${\fam=\ssfam A}$ is a factor of minimal dimension and $10{\,\le\,}\dim{\fam=\ssfam B}{\,\le\,}17$. By complete reducibility, ${\fam=\ssfam \usuTheta}$ has a ${\fam=\ssfam \usuGamma}'$-invariant complement ${\fam=\ssfam X}$ in ${\fam=\ssfam T}$. Suppose that $a^{\fam=\ssfam X}$ is contained in a line $av, \ v{\,\in\,}W$. Then $v^{{\fam=\ssfam \usuGamma}'}{\hskip-2pt=\,}v{\,\ne\,}v^\zeta$ for some $\zeta$ in the center of ${\fam=\ssfam \usuGamma}$, and ${\fam=\ssfam \usuGamma}'$ fixes the triangle $a,v,v^\zeta$. For $\tau{\,\in\,}{\fam=\ssfam X}$, the centralizer ${\fam=\ssfam \usuLambda}{\,=\,}{\fam=\ssfam \usuGamma}'{\hskip-2pt\kern 2pt {\scriptstyle \cap}\kern 2pt}\Cs{}\tau$ fixes also the point $a^{\tau^\zeta}$, hence a quadrangle, $\dim{\fam=\ssfam \usuLambda}{\,\ge\,}13$, and Stiffness yields
${\fam=\ssfam \usuLambda}{\,\cong\,}{\rm G}_2$ and ${\fam=\ssfam \usuLambda}|_{\fam=\ssfam X}{\,=\,}1\kern-2.5pt {\rm l}$, but $\dim{\cal F}_{\fam=\ssfam \usuLambda}{\,=\,}2$. \\ (e) Consequently, $\langle a^{\fam=\ssfam X}\rangle{\,=\,}{\cal F}$ is a subplane, and ${\cal F}{\,<\,}{\cal P}$\, (or ${\fam=\ssfam \usuGamma}'$ would act faithfully on ${\fam=\ssfam X}{\,\cong\,}{\fam=\Bbbfam R}^4$).
Put ${\fam=\ssfam \usuGamma}'|_{\cal F}{\,=\,}{\fam=\ssfam \usuGamma}'/{\fam=\ssfam K}$, and note that ${\fam=\ssfam \usuGamma}'|_{\cal F}{\,\le\,}\SL4{\fam=\Bbbfam R}$. If $\dim{\cal F}{\,=\,}4$, then Stiffness implies $\dim{\fam=\ssfam K}{\,\le\,}8$ and ${\fam=\ssfam \usuGamma}'{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,>\,}8$, which is impossible. Hence ${\cal F}{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$,\, ${\fam=\ssfam A}{\,=\,}{\fam=\ssfam K}{\,\cong\,}\SU2{\fam=\Bbbfam C}$,\, $\dim{\fam=\ssfam B}{\,\ge\,}14$, and ${\fam=\ssfam B}{\,\cong\,}\SL4{\fam=\Bbbfam R}$. If $\tau{\,\in\,}{\fam=\ssfam T}$ is a translation of ${\cal P}$ such that $a^\tau{\,\in\,}{\cal F}$, then $a^{\tau{\fam=\ssfam X}}{\,\subseteq\,}{\cal F}^{\fam=\ssfam X}{\,=\,}{\cal F}$ and ${\cal F}^\tau{\,=\,}\langle a^{\fam=\ssfam X}\rangle^\tau{\,=\,}\langle a^{\tau{\fam=\ssfam X}}\rangle{\,=\,}{\cal F}$. Therefore ${\cal F}$ is a translation plane, $\dim\mathop{{\rm Aut}}{\cal F}{\,\ge\,}23$, and ${\cal F}$ is the classical quaternion plane by \cite{cp} 81.9 or 84.27. (In fact, an $8$-dimensional plane is classical, if it has an automorphism group of dimension ${\ge\,}19$, see \cite{sz1} 1.10.) It follows that ${\fam=\ssfam B}{\,\cong\,}\SL2{\fam=\Bbbfam H}$, a contradiction. This proves that ${\fam=\ssfam \usuTheta}{\,\cong\,}{\fam=\Bbbfam R}^8$ or ${\fam=\ssfam \usuTheta}{\,=\,}{\fam=\ssfam T}$. \\ (f) {\it If ${\fam=\ssfam \usuGamma}$\,acts {\rm faithfully} and irreducibly on ${\fam=\ssfam \usuTheta}{\,\cong\,}{\fam=\Bbbfam R}^8$, then ${\fam=\ssfam \usuGamma}'{\,=\,}{\fam=\ssfam A}{\fam=\ssfam B}$ with ${\fam=\ssfam A}{\,\cong\,}{\fam=\Bbbfam H}'$ and ${\fam=\ssfam B}{\,\cong\,}\SL2{\fam=\Bbbfam H}$\/}. {Proof:} The group ${\fam=\ssfam \usuGamma}'$ is semi-simple by \cite{cp} 95.6, and again $17{\,\le\,}\dim{\fam=\ssfam \usuGamma}'{\,\le\,}20$. If ${\fam=\ssfam \usuGamma}'$ is almost simple, then $\dim{\fam=\ssfam \usuGamma}'{\,=\,}20$. The list \cite{cp} 95.10 of irreducible representations shows that ${\fam=\ssfam \usuGamma}'{\,\cong\,}\Sp4{\fam=\Bbbfam C}{\,\cong\,}\Spin5{\fam=\Bbbfam C}$. The central involution in ${\fam=\ssfam \usuGamma}'$ is a reflection with center $a$ because ${\fam=\ssfam \usuGamma}$ has no fixed point on $W$. The involution $\omega{\,=\,}{\rm diag}\,(1,1,-1,-1)$ is either planar or a reflection with some center $u{\,\in\,}W$, and ${\fam=\ssfam \usuOmega}{\,=\,}{\fam=\ssfam \usuGamma}'\kern 2pt {\scriptstyle \cap}\kern 2pt\Cs{}\omega{\,\cong\,}(\Sp2{\fam=\Bbbfam C})^2{\,\cong\,}(\SL2{\fam=\Bbbfam C})^2$. If ${\cal F}_\omega{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$, then ${\fam=\ssfam \usuOmega}$ induces on $W\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal F}_\omega{\,\approx\,}{\fam=\Bbbfam S}_4$ a group $({\rm P})\SL2{\fam=\Bbbfam C}{\times}\SO3{\fam=\Bbbfam C}$, and a maximal compact subgroup ${\fam=\ssfam \usuPhi}$ of ${\fam=\ssfam \usuOmega}$ acts as $\SU2{\fam=\Bbbfam C}{\times}\SO3{\fam=\Bbbfam R}$ or as $(\SO3{\fam=\Bbbfam R})^2$ in contradiction to Richardson's Theorem \cite{cp} 96.34. If $\omega{\,\in\,}{\fam=\ssfam \usuDelta}_{[u,av]}$, then ${\fam=\ssfam \usuOmega}$ fixes the triangle $a,u,v$, and \cite{cp} 81.8 implies that ${\fam=\ssfam \usuOmega}$ has a compact subgroup of codimension at most $2$, but ${\fam=\ssfam \usuOmega}{\,:\,}{\fam=\ssfam \usuPhi}{\,=\,}6$. Hence ${\fam=\ssfam \usuGamma}'{\,=\,}{\fam=\ssfam A}{\fam=\ssfam B}$ is a product of an almost simple factor ${\fam=\ssfam A}$ of minimal dimension and the semi-simple centralizer ${\fam=\ssfam B}{\,=\,}{\fam=\ssfam \usuGamma}'{\hskip-2pt\kern 2pt {\scriptstyle \cap}\kern 2pt}\Cs{}{\fam=\ssfam A}$, and $\dim{\fam=\ssfam B}{\,\ge\,}10$. By Clifford's Lemma \cite{cp} 95.5, ${\fam=\ssfam B}$ acts faithfully and irreducibly on a subspace ${\fam=\ssfam \usuXi}$ of ${\fam=\ssfam \usuTheta}$ of dimension $4$ or~$8$. In the second case, ${\fam=\ssfam A}{\,\cong\,}{\fam=\Bbbfam H}'$ and ${\fam=\ssfam B}{\,\cong\,}\SL2{\fam=\Bbbfam H}$. If ${\fam=\ssfam \usuXi}{\,\cong\,}{\fam=\Bbbfam R}^4$, then ${\fam=\ssfam B}$ is isomorphic to $\Sp4{\fam=\Bbbfam R}$ or $\SL4{\fam=\Bbbfam R}$. A suitable subgroup of ${\fam=\ssfam B}$ fixes a one-parameter subgroup of ${\fam=\ssfam \usuXi}$ and of each ${\fam=\ssfam B}$-invariant complement of ${\fam=\ssfam \usuXi}$ in ${\fam=\ssfam \usuTheta}$, and ${\fam=\ssfam A}$ acts on a $2$-dimensional subspace of ${\fam=\ssfam \usuTheta}$. Consequently, ${\fam=\ssfam A}{\,\cong\,}\SL2{\fam=\Bbbfam R}$ and ${\fam=\ssfam B}{\,\cong\,}\SL4{\fam=\Bbbfam R}$. If $a^{\fam=\ssfam \usuXi}{\,\subseteq\,}av$ with $v{\,\in\,}W$, then $v^{\fam=\ssfam B}{\,=\,}v{\,\ne\,}v^{\fam=\ssfam A}$, and ${\fam=\ssfam B}$ fixes a triangle $a,u,v$. By \cite{cp} 81.8, a maximal compact subgroup ${\fam=\ssfam \usuPhi}$ of ${\fam=\ssfam B}$ has codimension ${\fam=\ssfam B}{\,:\,}{\fam=\ssfam \usuPhi}{\,\le\,}2$, but ${\fam=\ssfam \usuPhi}{\,\cong\,}\SO4{\fam=\Bbbfam R}$ is much too small. Thus ${\fam=\ssfam \usuXi}$ contains translations $\xi,\eta$ with different centers, the stabilizer ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam B}{\fam=\ssfam P})_{\hskip-1pt a^\xi,\hskip1pt a^\eta}$ fixes a quadrangle, and Stiffness implies ${\fam=\ssfam \usuLambda}{\,\cong\,}\SU3{\fam=\Bbbfam C}$, but then ${\fam=\ssfam \usuLambda}$ is not contained in $\SL4{\fam=\Bbbfam R}$. \par
(g) Suppose now that $\tilde{\fam=\ssfam \usuGamma}{\,=\,}{\fam=\ssfam \usuGamma}|_{\fam=\ssfam \usuTheta}{\,=\,}{\fam=\ssfam \usuGamma}/{\fam=\ssfam K}$ with ${\fam=\ssfam K}{\,\ne\,}1\kern-2.5pt {\rm l}$. Then $\langle a^{\fam=\ssfam \usuTheta}\rangle{\,\le\,}{\cal F}_{\fam=\ssfam K}$ is a Baer subplane since $z^{\fam=\ssfam \usuGamma}{\ne\,}z$ for each $z{\,\in\,}W$. Either ${\fam=\ssfam K}{\,\cong\,}\Spin 3{\fam=\Bbbfam R}{\,\cong\,}{\fam=\Bbbfam H}'$ or $\dim{\fam=\ssfam K}{\,\le\,}1$. In the second case, $\dim\tilde{\fam=\ssfam \usuGamma}{\,\ge\,}18$ and $\dim\mathop{{\rm Aut}}{\cal F}_{\fam=\ssfam K}{\,\ge\,}26$. By \cite{cp} 84.27 the plane ${\cal F}_{\fam=\ssfam K}$ is the classical quaternion plane. Hence $\tilde{\fam=\ssfam \usuGamma}$ is contained in ${\fam=\Bbbfam H}^{\times}{\cdot\,}\SL2{\fam=\Bbbfam H}$ and ${\fam=\ssfam \usuGamma}$ has a subgroup $\SL2{\fam=\Bbbfam H}$. Moreover, $\dim(\tilde{\fam=\ssfam \usuGamma}\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\Bbbfam H}'){\,\ge\,}2$ and ${\fam=\Bbbfam H}'{\,\le\,}{\fam=\ssfam \usuGamma}$. Consequently ${\fam=\ssfam \usuGamma}'{\,\cong\,}{\fam=\Bbbfam H}'{\cdot\,}\SL2{\fam=\Bbbfam H}$ \par (h) Only the case that ${\fam=\ssfam \usuGamma}$ acts irreducibly on ${\fam=\ssfam \usuTheta}{\,=\,}{\fam=\ssfam T}{\,\cong\,}{\fam=\Bbbfam R}^{16}$ remains. If ${\fam=\ssfam \usuGamma}'$ is almost simple, then $\dim{\fam=\ssfam \usuGamma}'{\,=\,}20$, and ${\fam=\ssfam \usuGamma}'$ is isomorphic to $\SO5{\fam=\Bbbfam C}$ or to its covering group $\Sp4{\fam=\Bbbfam C}$. In the first case, ${\fam=\ssfam \usuGamma}$ would contain $\SO5{\fam=\Bbbfam R}$ contrary to 2.10. In the second case, ${\fam=\ssfam T}$ is a direct sum of two subspaces on which ${\fam=\ssfam \usuGamma}'$ acts in the natural way (see \cite{cp} 95.\,6 and 10), and ${\fam=\ssfam \usuGamma}{\,=\,}{\fam=\ssfam \usuGamma}'{\fam=\ssfam P}$ would not be irreducible on ${\fam=\ssfam T}$. Therefore ${\fam=\ssfam \usuGamma}'{\,=\,}{\fam=\ssfam A}{\fam=\ssfam B}$ is again a product of proper factors such that ${\fam=\ssfam A}$ has minimal dimension, and $10{\,\le\,}\dim{\fam=\ssfam B}{\,\le\,}17$. By Clifford's Lemma, ${\fam=\ssfam B}$ acts
faithfully and irreducibly on a subspace ${\fam=\ssfam \usuXi}$ of ${\fam=\ssfam T}$ of dimension $k|16$. \\ (i) In the case $k{\,=\,}4$, we have either ${\fam=\ssfam B}{\,\cong\,}\Sp4{\fam=\Bbbfam R}$ and $\dim{\fam=\ssfam A}{\,\ge\,}8$ or ${\fam=\ssfam B}{\,\cong\,}\SL4{\fam=\Bbbfam R}$ and $\dim{\fam=\ssfam A}{\,=\,}3$. First, let $\dim{\fam=\ssfam B}{\,=\,}10$. The centralizer ${\fam=\ssfam \usuOmega}$ of the involution $\omega{\,=\,}{\rm diag}(1,1,-1,-1){\,\in\,}{\fam=\ssfam B}$ contains ${\fam=\ssfam A}{\hskip.5pt\cdot\hskip.5pt}(\SL2{\fam=\Bbbfam R})^2$. If $\omega$ is a reflection, then ${\fam=\ssfam \usuOmega}$ fixes a triangle $a,u,v$. By \cite{cp} 81.8, a maximal compact subgroup ${\fam=\ssfam \usuPhi}$ of ${\fam=\ssfam \usuOmega}$ satisfies ${\fam=\ssfam \usuOmega}{\,:\,}{\fam=\ssfam \usuPhi}{\,\le\,}2$, but ${\fam=\ssfam \usuPhi}{\,:\,}{\fam=\ssfam A}{\,\le\,}2$ and ${\fam=\ssfam \usuOmega}{\,:\,}{\fam=\ssfam A}{\,=\,}6$. Hence ${\cal F}_\omega{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$ and ${\fam=\ssfam \usuOmega}$ induces on ${\cal F}_\omega$ a group of dimension at least $14$; moreover, ${\fam=\ssfam T}\kern 2pt {\scriptstyle \cap}\kern 2pt\Cs{}\omega{\,\cong\,}{\fam=\Bbbfam R}^8$ and ${\cal F}_\omega$ is a translation plane. Therefore ${\cal F}_\omega$ is the classical quaternion plane, cf. \cite{cp} 81.9.
It follows that ${\fam=\ssfam \usuOmega}|_{{\cal F}_\omega}{\,\cong\,}\SL2{\fam=\Bbbfam H}$, but ${\fam=\ssfam A}$ is a proper factor of~${\fam=\ssfam \usuOmega}$. Consequently, $\dim{\fam=\ssfam B}{\,=\,}15$ and ${\fam=\ssfam \usuGamma}$ has a subgroup ${\fam=\ssfam \usuUpsilon}{=\,}{\fam=\ssfam B}{\fam=\ssfam P}{\,\cong\,}\GL4{\fam=\Bbbfam R}$ acting on ${\fam=\ssfam \usuXi}$. If $a^{\fam=\ssfam \usuXi}{\,\subseteq\,}av$ with $v{\,\in\,}W$, then $v^{\fam=\ssfam \usuUpsilon}{=\,}v{\,\ne\,}v^{\fam=\ssfam A}$, and ${\fam=\ssfam \usuUpsilon}$ fixes a triangle $a,u,v$, but the maximal compact subgroup $\SO4{\fam=\Bbbfam R}$ of ${\fam=\ssfam \usuUpsilon}$ is too small (cf. \cite{cp} 81.8). Thus ${\fam=\ssfam \usuXi}$ contains tranlations $\xi,\eta$ with different centers, the stabilizer ${\fam=\ssfam \usuLambda}{\,=\,}{\fam=\ssfam \usuUpsilon}_{\hskip-1pt a^\xi,\hskip1pt a^\eta}$ fixes a quadrangle, and Stiffness implies ${\fam=\ssfam \usuLambda}{\,\cong\,}\SU3{\fam=\Bbbfam C}$, an obvious contradiction. Hence $k{\,\ge\,}8$. \\ (j) Suppose next that $k{\,=\,}16$. As ${\fam=\ssfam A}{\,\le\,}\Cs{}{\fam=\ssfam B}$, it follows from \cite{cp} 95.4 that ${\fam=\ssfam A}{\,\cong\,}{\fam=\Bbbfam H}'$. Therefore $14{\,\le\,}\dim{\fam=\ssfam B}{\,\le\,}17$, and ${\fam=\ssfam B}$ acts irreducibly on ${\fam=\Bbbfam H}^4$. The list \cite{cp} 95.10 shows that ${\fam=\ssfam B}$ is not almost simple. Clifford's Lemma implies that each proper factor ${\fam=\ssfam N}$ of ${\fam=\ssfam B}$ is contained in $\SL2{\fam=\Bbbfam H}$, in particular, ${\fam=\ssfam N}$ has torus rank $\rm rk\,{\fam=\ssfam N}{\,\le\,}2$, and ${\fam=\ssfam B}$ is composed of factors ${\fam=\Bbbfam H}'$,\, \ $\U2({\fam=\Bbbfam H},r)$, and $\SL2{\fam=\Bbbfam H}$, but then the conditions on rank and dimension cannot be satisfied simultaneously. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
Based on his previous papers \cite{Lh1} and \cite{Lh2}, L\"owe \cite{Lh} explicitly determined not only the planes of the first case in 4.9, but all translation planes admitting a group locally isomorphic to $\SL2{\fam=\Bbbfam H}$. \par
{\bf 4.10 Construction.} Choose a {\it continuous\/} map $\mu{\,:\,}{\fam=\Bbbfam H}'{\,\to\,}e^{{\fam=\Bbbfam R}}$. Write $\eta^*{\,=\,}(\overline\eta^{\hskip1.5pt t})^{-1}$ and put
${\fam=\ssfam \usuPsi}{\,=\,} \bigl\{\bigl({\eta^* \atop }{ \atop \eta}\bigr){\,\big|\,}\eta{\,\in\,}\SL2{\fam=\Bbbfam H} \bigr\}$. For $h{\,\in\,}{\fam=\Bbbfam H}$, let $L_h{\,=\,}\{(x,hx)\mid x{\,\in\,}{\fam=\Bbbfam O}\}$. Then $E{\,=\,}\{({\fam=\Bbbfam H},\ell{\fam=\Bbbfam H}){\,\subset\,}{\fam=\Bbbfam O}^2\}$ and
the {\it ``lines''\/} $L_{h^\mu h}$ with $h{\,\in\,}{\fam=\Bbbfam H}'$ together with all their ${\fam=\ssfam \usuPsi}$-images
form a spread ${\cal S}_\mu$. Obviously ${\fam=\ssfam \usuPsi}$ acts on the translation plane ${\cal P}_\mu$
defined by this spread, see \cite{Lh} for proofs.
\par
Let again ${\fam=\ssfam \usuGamma}{\,=\,}{\fam=\ssfam \usuDelta}_a$ denote a complement of the translation group, and consider the subgroup ${\fam=\ssfam \usuOmega}{\,=\,}{\rm S}{\fam=\ssfam \usuGamma}{\,=\,}\{\gamma{\,\in\,}{\fam=\ssfam \usuGamma}{\,\le\,}\GL{16}{\fam=\Bbbfam R}\mid \det\gamma{\,=\,}1\}$. In our terminology, L\"owe's main result \cite{Lh} 3.8 can be stated as follows: \par
{\bf 4.11 \,$\SL2{\fam=\Bbbfam H}$-planes.} {\it Let ${\cal P}$ be a non-classical translation plane. If ${\fam=\ssfam \usuDelta}$ contains a subgroup locally isomorphic to $\SL2{\fam=\Bbbfam H}$, then ${\cal P}$ is isomorphic to ${\cal P}_\mu$, where $\mu{\,:\,}{\fam=\Bbbfam H}'{\,\to\,}e^{{\fam=\Bbbfam R}}$ is continuous and not constant. There are $3$ possibilities\/}: \par
(1) {\it If $\mu$ depends only on the real part of its argument, then ${\fam=\ssfam \usuOmega}$ is an almost direct product of ${\fam=\ssfam \usuPsi}{\,\cong\,}\SL2{\fam=\Bbbfam H}$ and ${\fam=\ssfam H}{\,=\,}\{(x,y){\,\to\,}(hx,hy)\mid h{\,\in\,}{\fam=\Bbbfam H}'\}$. In particular, $\dim{\fam=\ssfam \usuSigma}{\,=\,}35$\/}. \par
(2) {\it If $\mu$ depends only on the absolute value of the complex part of its argument, then
${\fam=\ssfam \usuOmega}$ is an almost direct product of ${\fam=\ssfam \usuPsi}{\,\cong\,}\SL2{\fam=\Bbbfam H}$ and
${\fam=\ssfam A}{\,=\,}\{(x,y){\,\to\,}(bx,cy)\mid b,c{\,\in\,}{\fam=\Bbbfam H}'{\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\Bbbfam C}}\}{\,\cong\,}{\fam=\Bbbfam T}^2$. In particular, $\dim{\fam=\ssfam \usuSigma}{\,=\,}34$\/}. \par
(3) {\it In all other cases, ${\fam=\ssfam \usuOmega}$ is isomorphic to ${\fam=\ssfam \usuPsi}{\,\cong\,}\SL2{\fam=\Bbbfam H}$ or to an almost direct product of ${\fam=\ssfam \usuPsi}$ and $\SO2{\fam=\Bbbfam R}$. Hence $32{\,\le\,}\dim{\fam=\ssfam \usuSigma}{\,\le\,}33$\/}. \par
{\bf 4.12 \,$\SU4{\fam=\Bbbfam C}$-planes.} As mentioned above, H\"ahl \cite{Ha2} has described explicitly all translation planes admitting a semi-simple group ${\fam=\ssfam \usuUpsilon}{\,=\,}{\fam=\ssfam \usuPhi}{\fam=\ssfam \usuPsi}$ with ${\fam=\ssfam \usuPhi}{\,\cong\,}\SU4{\fam=\Bbbfam C}$ and ${\fam=\ssfam \usuPsi}{\,\in\,}\{\SU2{\fam=\Bbbfam C},\SL2{\fam=\Bbbfam R}\}$. Only some facts shall be stated; for the full theorem the reader is referred to \cite{Ha2}. \par
(1) {\it Let the translation group ${\fam=\ssfam T}$ be written in the form ${\fam=\Bbbfam C}^4{\,{\otimes}_{{\fam=\Bbbfam C}}\,}{\fam=\Bbbfam C}^2$. The group ${\fam=\ssfam \usuPhi}$ acts in the standard way on the left factor and trivially on the right one; ${\fam=\ssfam \usuPsi}$ acts only on the right factor. The subspaces ${\fam=\Bbbfam C}^4{\,{\otimes}_{{\fam=\Bbbfam C}}\,}z$,\: $z{\,\in\,}{\fam=\Bbbfam C}^2\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{0\}$, are affine lines\/}. \par
(2) {\it With ${\fam=\Bbbfam C}^2{\hskip1pt=\,}{\fam=\Bbbfam H}$, the map
$h{\,{\otimes}_{{\fam=\Bbbfam C}}\,}(x,y){\,\mapsto\,}h{\,{\otimes}_{{\fam=\Bbbfam H}}\,}(x,y)$ yields an isomorphism ${\fam=\ssfam T}{\,\cong\,}{\fam=\Bbbfam H}^2{{\otimes}_{{\fam=\Bbbfam H}}\,}{\fam=\Bbbfam H}^2$. Let $\U2{\fam=\Bbbfam H}{\,=\,}${\tencyss\cyracc Yu}${\,<\,}{\fam=\ssfam \usuPhi}$. The {\tencyss\cyracc Yu}-invariant subspaces of ${\fam=\ssfam T}$ are exactly $U_q{\,=\,}{\fam=\Bbbfam H}^2{\,{\otimes}_{{\fam=\Bbbfam H}}\,}(1,q)$ and $U_\infty{\,=\,}{\fam=\Bbbfam H}^2{\,{\otimes}_{{\fam=\Bbbfam H}}\,}(0,1)$\/}. \par
(3) {\it Let $\sigma{\,:\,}(-1,1){\,\to\,}\{c{\,\in\,}{\fam=\Bbbfam C}\mid c\overline c{\,=\,}1\}$ be any continuous map {\rm(}which satisfies $\sigma(-s){\,=\,}\sigma(s)$ if ${\fam=\ssfam \usuPsi}$ is compact{\rm)}. Then $\sigma$ selects a ${\fam=\ssfam \usuUpsilon}\!$-invariant spread among the subspaces described in\/} (1) {\it and\/} (2) {\it containing the affine lines $U_q$, where $q$ has the form $si{\,+\,}tj\sigma(s){\,\in\,}{\fam=\Bbbfam H}$, $s^2{\,+\,}t^2{\,=\,}1$ and $s,t{\,\in\,}(-1,1)$, their ${\fam=\ssfam \usuUpsilon}\!$-images, and the lines $U_c$, $c{\,\in\,}{\fam=\Bbbfam C}\kern 2pt {\scriptstyle \cup}\kern 2pt\{\infty\}$\/}. \par
(4) {\it Each translation plane admitting an action of the group ${\fam=\ssfam \usuUpsilon}$ can be obtained in this way. The plane is classical if, and only if, $\sigma$ is constant; in all other cases $\dim{\fam=\ssfam \usuSigma}{\,=\,}35$\/}. \par
{\Bf 5. Exactly two fixed elements} \par
For comparison with later results, \cite{cp} 87.4 shall be restated here: \par
{\bf Proposition.} {\it Assume that ${\cal P}$ is not classical. If $\dim{\fam=\ssfam \usuDelta}{\,=\,}h{\,\ge\,}36$, and if ${\fam=\ssfam \usuDelta}$ has a minimal normal vector subgroup ${\fam=\ssfam \usuTheta}{\,\cong\,}{\fam=\Bbbfam R}^t$, then there are only the following possibilities\/}: \par
$$\begin{cases} {\fam=\ssfam \usuTheta}{\,\le\,}{\fam=\ssfam \usuDelta}_{[a,W]}, \ a{\,\notin\,}W \begin{cases} {\fam=\ssfam \usuDelta}'{\,\cong\,}\Spin9({\fam=\Bbbfam R} ,r),\; r{\,\le\,}1 \hskip8pt and \hskip8pt h{\,=\,}37 \cr v^{\fam=\ssfam \usuDelta}{\,=\,}v{\,\in\,}W \hskip8pt and \hskip8pt h{\,\le\,}38 \cr \end{cases} \cr {\fam=\ssfam \usuTheta}{\,\le\,}{\fam=\ssfam T}{\,=\,}{\fam=\ssfam \usuDelta}_{[W,W]} \begin{cases} t{\,>\,}8 \begin{cases} h{\,=\,}36 \cr {\fam=\ssfam T}{\,\cong\,}{\fam=\Bbbfam R}^{16} \hskip8pt and \hskip8pt h{\,=\,}37 \cr \end{cases}\cr t{\,=\,}8 \Rightarrow {\fam=\ssfam \usuTheta}{\,=\,}{\fam=\ssfam \usuDelta}_{[v,W]} \hskip8pt or \hskip8pt {\fam=\ssfam T}{\,\cong\,}{\fam=\Bbbfam R}^{16} \cr t{\,<\,}8 \Rightarrow {\fam=\ssfam \usuTheta}{\,\le\,}{\fam=\ssfam \usuDelta}_{[v,W]},\; v{\,\in\,}W\;. \cr \end{cases} \end{cases}$$ \par
The case $h{\,\ge\,}40$ has already been described in the introduction, for $h{\,=\,}39$ it follows from the Proposition that ${\cal F}_{\fam=\ssfam \usuDelta}$ is a flag. If a semi-simple group ${\fam=\ssfam \usuDelta}$ of the octonion plane ${\cal O}$ fixes a flag, then ${\fam=\ssfam \usuDelta}$ is contained in $\Spin8{\fam=\Bbbfam R}$ and hence fixes even a triangle. Priwitzer's Theorem 4.2 shows that ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\{a,W\}$ with $a{\,\notin\,}W$, if ${\fam=\ssfam \usuDelta}$ is a semi-simple group of a non-classical plane, ${\cal F}_{\fam=\ssfam \usuDelta}{\,\ne\,}\emptyset$, and $\dim{\fam=\ssfam \usuDelta}{\,>\,}28$. \par
{\Bf A. Fixed flag} \par
{\bf 5.0.} {\it If ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\{v,W\}$ is a flag, and if $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}22$, then ${\fam=\ssfam \usuDelta}$ is a Lie group\/}. \par
This can be {\tt proved} in a similar way as 4.0. Steps (a) and the first part of step (b) are the same as before. In the present case the weaker assumption on $\dim{\fam=\ssfam \usuDelta}$ yields
$\dim{\fam=\ssfam \usuDelta}|_{\cal D}{\,\ge\,}15$, and ${\cal D}$ is no longer known explicitly. Choose a line $L$ of ${\cal D}$ in ${\frak L}_v\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{W\}$ and points $p{\,\in\,}L,\,z{\,\in\,}W$ outside ${\cal D}$. Put ${\fam=\ssfam \usuGamma}{\,=\,}{\fam=\ssfam \usuDelta}_L$ and ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuGamma}_{\hskip-2pt p,z})^1$.
Recall that ${\fam=\ssfam \usuDelta}|_{\cal D}{\,=\,}{\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$. If ${\fam=\ssfam K}$ is not a Lie group, then $\dim p^{\fam=\ssfam \usuGamma},\, \dim z^{\fam=\ssfam \usuGamma}{<\,}8$ by \cite{cp} 53.2, and
$\dim{\fam=\ssfam \usuLambda}{\,\ge\,}22{-}4{-}2{\cdot}7{\,>\,}3$. The line $pz$ intersects ${\cal D}$ in a unique point $a$. If $\dim{\fam=\ssfam K}{\,\le\,}3$, then $\dim{\fam=\ssfam \usuDelta}|_{\cal D}{\,\ge\,}19$ and ${\cal D}$ is classical (cf. \cite{cp} 84.28). The group ${\fam=\ssfam \usuLambda}$ fixes the distinct lines $L,W,av$ and hence a connected set of lines in the pencil ${\frak L}_v$. In particular, ${\cal F}_{\fam=\ssfam \usuLambda}$ is connected. If $\dim{\fam=\ssfam K}{\,>\,}3$, then ${\fam=\ssfam K}^1$ has a connected commutative factor ${\fam=\ssfam A}$ and ${\fam=\ssfam \usuLambda}$ acts trivially on ${\fam=\ssfam A}$\: (see the arguments in step (c) in the proof of 4.0). Hence $p^{\fam=\ssfam A}{\,\subseteq\,}{\cal F}_{\fam=\ssfam \usuLambda}$ and again ${\cal F}_{\fam=\ssfam \usuLambda}$ is connected. As ${\fam=\ssfam N}$ acts freely on $L\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt{\cal D}$, it follows from \cite{cp} 32.21 and 71.2 that $\dim{\cal F}_{\fam=\ssfam \usuLambda}{\,>\,}4$ and ${\cal F}_{\fam=\ssfam \usuLambda}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$. Now ${\fam=\ssfam \usuLambda}{\,\cong\,}\Spin3{\fam=\Bbbfam R}$ contrary to the statement above. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\bf 5.1 Lie.} {\it If ${\fam=\ssfam \usuDelta}$ fixes a flag $\{v,W\}$ and no line other than $W\hskip-3pt$, if ${\fam=\ssfam \usuDelta}$ is semi-simple, and if $\dim{\fam=\ssfam \usuDelta}{\,>\,}14$, then ${\fam=\ssfam \usuDelta}$ is a Lie group\/}.
\par
The first part of the {\tt Proof} is identical with step (a) and the beginning of step (b) in the proof of 4.0. Let ${\fam=\ssfam \usuDelta}|_{\cal D}{\,=\,}{\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$ and ${\fam=\ssfam \usuLambda}{\,=\,}{\fam=\ssfam K}^1$\hskip-2pt. Note that ${\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$ and ${\fam=\ssfam \usuLambda}$ are semi-simple. If $\dim{\cal D}{\,=\,}2$, then ${\fam=\ssfam \usuDelta}{\,=\,}{\fam=\ssfam \usuLambda}$ by \cite{cp}~33.8, and $\dim{\fam=\ssfam \usuLambda}{\,\le\,}14$ by Stiffness; if $\dim{\cal D}{\,=\,}4$, then ${\fam=\ssfam \usuDelta}{\,:\,}{\fam=\ssfam K}{\,\le\,}3$ and $\dim{\fam=\ssfam \usuLambda}{\,\le\,}8$ by 2.14 and 2.6(e). Finally, if ${\cal D}{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$, then ${\fam=\ssfam \usuDelta}{\,:\,}{\fam=\ssfam K}{\,\le\,}11$ by \cite{sz1} 7.3, $\,{\fam=\ssfam \usuLambda}$ is compact and semi-simple, hence a Lie group (cf. \cite{cp} 93.11 and 94.20), and $\dim{\fam=\ssfam \usuLambda}{\,\le\,}3$ by 2.6(\^b). Thus $\dim{\fam=\ssfam \usuDelta}{\,\le\,}14$ or ${\fam=\ssfam N}{\,=\,}1\kern-2.5pt {\rm l}$. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\bf 5.2 Semi-simple groups.} {\it If ${\cal F}_{\fam=\ssfam \usuDelta}$ is a flag $\{v,W\}$ and if ${\fam=\ssfam \usuDelta}$ is semi-simple, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}21${\rm;} equality is possible only if each compact subgroup of ${\fam=\ssfam \usuDelta}$ is trivial. If ${\fam=\ssfam \usuDelta}$ is almost simple, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}16$\/}. \par
The following {\tt proof} does not make full use of the assumption ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\{v,W\}$ but works also in the case ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\{u,v,W{=\hskip2pt}uv\}$, see 6.2. \\ (a) {\it Each involution in ${\fam=\ssfam \usuDelta}$ is planar\/}: if $\sigma$ is a reflection, then $\sigma^{\fam=\ssfam \usuDelta}\sigma$ would be a normal vector subgroup by 2.9 or its dual. \\ (b) {\it If there exists a proper ${\fam=\ssfam \usuDelta}$-invariant subplane ${\cal E}$, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}14$\/}. This follows in the same way as in the proof of 5.1; for discrete ${\cal E}$, it is an immediate consequence of Stiffness. \\ (c) Suppose now and in the next steps that $\dim{\fam=\ssfam \usuDelta}{\,>\,}14$. Then ${\fam=\ssfam \usuDelta}$ does not contain a central involution, and ${\fam=\ssfam \usuDelta}$ is a Lie group by 5.1. \\ (d) Let ${\fam=\ssfam \usuGamma}$ be an almost simple factor of ${\fam=\ssfam \usuDelta}$ and denote the product of the other factors by~${\fam=\ssfam \usuPsi}$. In steps (e--h),\, ${\fam=\ssfam \usuGamma}$ is assumed to be a proper factor of minimal dimension. \\
(e) If ${\fam=\ssfam \usuGamma}$ contains an involution $\iota$, then ${\fam=\ssfam \usuPsi}|_{{\cal F}_\iota}{\,=\,}{\fam=\ssfam \usuPsi}/{\fam=\ssfam K}$ has dimension ${\fam=\ssfam \usuPsi}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\le\,}11$ by \cite{sz1} 7.3, the semi-simple group ${\fam=\ssfam K}$ is compact, and $\dim{\fam=\ssfam K}{\,\le\,}3$ by Stiffness. Either $\dim{\fam=\ssfam \usuGamma}{\,=\,}3$ and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}17$, or $\dim{\fam=\ssfam K}{\,=\,}0$,\, $\dim{\fam=\ssfam \usuPsi}{\,\ne\,}11$, and $6{\,\le\,}\dim{\fam=\ssfam \usuGamma}{\,\le\,}\dim{\fam=\ssfam \usuPsi}{\,\in\,}\{6,8,10\}$. In any case,\, $\dim{\fam=\ssfam \usuDelta}{\,\le\,}20$. \\ (f) If ${\fam=\ssfam \usuGamma}$ does not contain an involution, in particular, if $\dim{\fam=\ssfam \usuDelta}{\,>\,}20$, then ${\fam=\ssfam \usuGamma}$ is isomorphic to the simply connected covering group ${\fam=\ssfam \usuOmega}$ of $\SL2{\fam=\Bbbfam R}$, since ${\fam=\ssfam \usuOmega}$ is the only almost simple Lie group without involution. \\ (g) {\it If ${\fam=\ssfam \usuGamma}$\hskip2pt is straight and if $\dim{\fam=\ssfam \usuDelta}{\,>\,}14$, then ${\fam=\ssfam \usuGamma}$\hskip2pt is a group of translations and the center is a fixed point of ${\fam=\ssfam \usuDelta}$\/}: By Baer's theorem, either ${\fam=\ssfam \usuGamma}$ is a group of axial collineations and center and axis of ${\fam=\ssfam \usuGamma}$ are fixed elements, or ${\cal D}{\,:=\,}{\cal F}_{\fam=\ssfam \usuGamma}{\,=\,}{\cal D}^{\fam=\ssfam \usuDelta}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$, and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}14$ by step (b). \\ (h) {\it If no factor of ${\fam=\ssfam \usuDelta}$ has a compact subgroup other than $\{1\kern-2.5pt {\rm l}\}$, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}21$\/}. \\ In fact, ${\fam=\ssfam \usuDelta}$ is a product of factors ${\fam=\ssfam \usuGamma}_{\hskip-1.5pt\nu}{\,\cong\,}{\fam=\ssfam \usuOmega}$. At most two factors ${\fam=\ssfam \usuGamma}_{\hskip-1.5pt\nu}$ consist of translations (or they would be commutative by \cite{cp} 23.13). Either $\langle x^{\fam=\ssfam \usuDelta}\rangle$ is a proper subplane for some point $x$ and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}14$ by (b), or there are factors ${\fam=\ssfam \usuGamma}_{\hskip-1.5pt\nu}$ such that $\langle x^{{\fam=\ssfam \usuGamma}_1}\rangle$ is a non-flat subplane (use \cite{cp} 33.8) and ${\cal H}{\,=\,}\langle x^{{\fam=\ssfam \usuGamma}_1{\fam=\ssfam \usuGamma}_2}\rangle{\;\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\!}{\cal P}$. In the latter case, put
${\fam=\ssfam X}{\,=\,}{\fam=\ssfam \usuPsi}_1\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam \usuPsi}_2$. Then ${\fam=\ssfam X}_x|_{\cal H}{\,=\,}1\kern-2.5pt {\rm l}$ and ${\fam=\ssfam X}_x$ is compact by 2.6(b), hence ${\fam=\ssfam X}_x{\,=\,}1\kern-2.5pt {\rm l}$ and $\dim{\fam=\ssfam X}{\,\le\,}15$\, (${\fam=\ssfam X}$ being a product of $3$-dimensional factors).\\ (i) From now on assume that $\dim{\fam=\ssfam \usuDelta}{\,>\,}20$ and that ${\fam=\ssfam \usuGamma}$ contains an involution $\iota$, but is not necessarily of minimal dimension. Step (e) implies ${\fam=\ssfam \usuPsi}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\le\,}11$ and $\dim{\fam=\ssfam \usuGamma}{\,\ge\,}8$, and this is true for each possible choice of ${\fam=\ssfam \usuGamma}$. Consequently $\dim{\fam=\ssfam K}{\,=\,}0$, and then
$\dim{\fam=\ssfam \usuGamma}{\,\ge\,}10$. In the case $\dim{\fam=\ssfam \usuPsi}{\,=\,}11$, no factor of ${\fam=\ssfam \usuPsi}$ contains an involution and each factor of ${\fam=\ssfam \usuPsi}$ is isomorphic to ${\fam=\ssfam \usuOmega}$, but then $\dim{\fam=\ssfam \usuPsi}{\,|\,}9$. Hence $\dim{\fam=\ssfam \usuGamma}{\,>\,}10$ and $\dim{\fam=\ssfam \usuGamma}{\,\ge\,}14$. Again ${\fam=\ssfam \usuPsi}$ is a product of factors isomorphic to ${\fam=\ssfam \usuOmega}$ and $\dim{\fam=\ssfam \usuPsi}{\,\le\,}9$. The kernel of the action of ${\fam=\ssfam \usuPsi}$ on ${\cal F}_\iota$ is compact and hence trivial (cf. 2.6(b)\,). As $\rm rk\,{\fam=\ssfam \usuGamma}{\hskip1pt>\hskip1pt}1$, some other involution $\iota'{\,\in\,}{\fam=\ssfam \usuGamma}$ commutes with $\iota$, and ${\fam=\ssfam \usuPsi}$ acts also effectively on ${\cal F}_{\hskip-1pt\iota'}$. Suppose that ${\cal F}_{\hskip-1pt\iota,\iota'}$ is a subplane. Then ${\fam=\ssfam \usuPsi}$ is even almost effective on ${\cal F}_{\hskip-1pt\iota,\iota'}$ by the stiffness result 2.6(c), and 2.14 implies
$\dim{\fam=\ssfam \usuPsi}{\,\le\,}3$,\, $\dim{\fam=\ssfam \usuGamma}{\,\ge\,}18$, and then $\dim{\fam=\ssfam \usuGamma}{\,\ge\,}20$. If $\overline{\iota'}{\,=\,}\iota'|_{{\cal F}_\iota}$ is a reflection, then ${\fam=\ssfam \usuPsi}$ fixes the center $a{\,\notin\,}W$ of $\overline{\iota'}$\, (up duality) or its axis $av$ and center $u{\,\in\,}W$. In the first case,
${\fam=\ssfam \usuPsi}|_{a^{\fam=\ssfam \usuGamma}}{\,=\,}1\kern-2.5pt {\rm l}$,\, $\langle a ^{\fam=\ssfam \usuGamma}\rangle{\,=\,}{\cal P}$ by step~(b), and ${\fam=\ssfam \usuPsi}{\,=\,}1\kern-2.5pt {\rm l}$; in the second case, ${\fam=\ssfam \usuPsi}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuPsi}_a{\,\le\,}4$,\, $u^{\fam=\ssfam \usuGamma}{\,=\,}u^{\fam=\ssfam \usuDelta}{\,\ne\,}u$,\, ${\cal E}{\,=\,}\langle a^{\fam=\ssfam \usuGamma},u^{\fam=\ssfam \usuGamma}\rangle{\,=\,}{\cal E}^{\fam=\ssfam \usuGamma}{\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\!}{\cal P}$ by 2.14,
${\fam=\ssfam \usuPsi}_{\hskip-1pt a}|_{\cal E}{\,=\,}1\kern-2.5pt {\rm l}$, ${\fam=\ssfam \usuPsi}_{\hskip-1pt a}$ is compact, $\dim{\fam=\ssfam \usuPsi}_{\hskip-1pt a}{\,=\,}0$, and $\dim{\fam=\ssfam \usuPsi}{\,\le\,}3$. Again, $\dim{\fam=\ssfam \usuGamma}{\,\ge\,}20$. \\ (j) If $\dim{\fam=\ssfam \usuGamma}{\,\ge\,}30$, Priwitzer's theorem \cite{pw1} shows the following: $\dim{\fam=\ssfam \usuGamma}{\,=\,}36$ and ${\fam=\ssfam \usuGamma}$ fixes a non-incident point-line pair, or ${\fam=\ssfam \usuGamma}{\,\cong\,}\SL3{\fam=\Bbbfam H}$ and ${\cal F}_{\fam=\ssfam \usuGamma}{\,=\,}\emptyset$, or the plane is classical; in the last case ${\cal F}_{\fam=\ssfam \usuGamma}$ is not a flag, since a maximal semi-simple subgroup in the stabilizer of a
flag is isomorphic to $\Spin8{\fam=\Bbbfam R}$. The case $|{\cal F}_{\fam=\ssfam \usuDelta}|{\,=\,}3$ is also covered by Theorem 6.2. Therefore, only groups ${\fam=\ssfam \usuGamma}$ with $\dim{\fam=\ssfam \usuGamma}{\,<\,}30$ have to be discussed.
2.16 shows that ${\fam=\ssfam \usuGamma}{\,\not\hskip1pt\cong\,}{\rm G}_2^{\hskip1pt{\fam=\Bbbfam C}}$. According to step (c),\, ${\fam=\ssfam \usuGamma}$ does not contain a central involution. By 2.10 and 2.11, neither $\SO5{\fam=\Bbbfam R}$ nor $(\SO3{\fam=\Bbbfam R})^2$ can be a subgroup of~${\fam=\ssfam \usuGamma}$. These facts exclude the group $\SO5{\fam=\Bbbfam C}$ of dimension $20$ and leave only the following $9$ possibilities: $\PU3({\fam=\Bbbfam H},r)$ and $\PSp6{\fam=\Bbbfam R}$ of type ${\rm C}_3$,\, ${\rm(\hskip-.5pt P\hskip-1pt)}\SU5({\fam=\Bbbfam C},r)$ of type ${\rm A}_4$. For each of these groups a contradiction will be derived. \\ (k) The symplectic group contains $3$ pairwise commuting conjugate involutions. Such an involution $\iota$ must be planar, its centralizer ${\fam=\ssfam \usuUpsilon}$ is locally isomorphic to
$\SL2{\fam=\Bbbfam R}{\times}\Sp4{\fam=\Bbbfam R}$. Stiffness implies $\dim{\fam=\ssfam \usuUpsilon}|_{{\cal F}_\iota}{\,=\,}13$, which contradicts \cite{sz1} 7.3, see also \cite{cp} 84.18. Similarly, the compact group $\PU3{\fam=\Bbbfam H}$ contains a planar involution $\iota$ such that ${\fam=\ssfam \usuUpsilon}{\,=\,}\Cs{}\iota$ is locally isomorphic to $\U2{\fam=\Bbbfam H}{\times}{\fam=\Bbbfam H}'{\,=\,}\Spin5{\fam=\Bbbfam R}{\times}\Spin3{\fam=\Bbbfam R}$. By \cite{cp} 96.13 or by Richardson's classification of compact (Lie) groups on ${\fam=\Bbbfam S}_4$\, (see \cite{cp} 96.34), the $10$-dimensional factor of ${\fam=\ssfam \usuUpsilon}$ is transitive on $W{\kern 2pt {\scriptstyle \cap}\kern 2pt}{\cal F}_\iota{\,\approx\,}{\fam=\Bbbfam S}_4$ and cannot fix the point~$v$. Finally, let ${\fam=\ssfam \usuGamma}{\,\cong\,}\PU3({\fam=\Bbbfam H},1)$. Then ${\fam=\ssfam \usuGamma}$ has $3$ pairwise commuting involutions $\alpha,\beta$, and $\gamma{\,=\,}\alpha\beta$ such that $\alpha$ and $\beta$ are conjugate and hence are planar. The centralizer ${\fam=\ssfam \usuUpsilon}{\,=\,}\Cs{}\beta$ is locally isomorphic to $\U2({\fam=\Bbbfam H},1){\times}{\fam=\Bbbfam H}'$, and
$\dim{\fam=\ssfam \usuUpsilon}|_{{\cal F}_\beta}{\,\le\,}11$ by \cite{sz1} 4.1 or 7.3. Therefore the compact factor of ${\fam=\ssfam \usuUpsilon}$ acts trivially on ${\cal F}_\beta$, and $\gamma$ is also planar. The group $\U2{\fam=\Bbbfam H}{\,<\,}\Cs{}\gamma$ acts faithfully on ${\cal F}_\gamma$; for the same reason as before, it cannot fix a flag. \\ ($\ell$) Lastly, let $\dim{\fam=\ssfam \usuGamma}{\,\ge\,}24$. The compact group ${\rm(P\hskip-.8pt)}\SU5{\fam=\Bbbfam C}$ has two conjugacy classes of pairwise commuting (diagonal) involutions consisting of $5$ and of $10$ elements respectively, but this contradicts 2.10. If $r{\,>\,}0$, then some diagonal planar involution $\iota$ has a centralizer
${\fam=\ssfam \usuUpsilon}{\,\cong\,}\SU4({\fam=\Bbbfam C},r{-}1)$; on the other hand, $\dim{\fam=\ssfam \usuUpsilon}|_{{\cal F}_\iota}{\,\le\,}11$ by \cite{sz1} 7.3. This contradiction shows that ${\fam=\ssfam \usuGamma}$ is not of type ${\rm A}_4$. \\ (m) {\tt Conclusion.} In steps (h--$\ell$), the following has been shown: If $\dim{\fam=\ssfam \usuDelta}{\,>\,}20$ and if ${\fam=\ssfam \usuGamma}$ is an almost simple factor of ${\fam=\ssfam \usuDelta}$, then ${\fam=\ssfam \usuGamma}$ does not contain an involution, and $\dim{\fam=\ssfam \usuDelta}{\,=\,}21$; if $\dim{\fam=\ssfam \usuDelta}{\,\le\,}20$ and ${\fam=\ssfam \usuDelta}$ is almost simple, then $\dim{\fam=\ssfam \usuDelta}{\,\ne\,}20$ and hence $\dim{\fam=\ssfam \usuDelta}{\,\le\,}16$. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\bf 5.3 Normal torus.} {\it If ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\{v,W\}$ is a flag, if ${\fam=\ssfam \usuDelta}$ is a Lie group with a normal subgroup ${\fam=\ssfam \usuTheta}{\,\cong\,}{\fam=\Bbbfam T}$, and if $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}18$, then there exists a ${\fam=\ssfam \usuDelta}$-invariant Baer subplane which is a known translation plane\/}. \par
{\tt Proof.} Note that ${\fam=\ssfam \usuTheta}{\,\le\,}\Cs{}{\fam=\ssfam \usuDelta}$\, (see \cite{cp} 93.19). The involution $\iota{\,\in\,}{\fam=\ssfam \usuTheta}$ cannot be a reflection, and ${\cal F}_{\hskip-1.5pt\iota}{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$. Write
${\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}{\,=\,}{\fam=\ssfam \usuDelta}|_{{\cal F}_{\hskip-1pt\iota}}{\,=\,}{\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$. Obviously, ${\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}$ fixes exactly $v$ and $W$. If ${\fam=\ssfam \usuTheta}$ acts non-trivially on ${\cal F}_\iota$, then ${\fam=\ssfam \usuDelta}{\,:\,}{\fam=\ssfam K}{\,\le\,}11$ by \cite{sz1} 4.2, Stiffness gives $\dim{\fam=\ssfam K}{\,\le\,}3$, and $\dim{\fam=\ssfam \usuDelta}$ would be too small. Hence ${\fam=\ssfam \usuTheta}{\,\le\,}{\fam=\ssfam K}$,\: ${\fam=\ssfam K}^1{\,\not\hskip1pt\cong\,}\Spin3{\fam=\Bbbfam R}$,\: $\dim{\fam=\ssfam K}{\,=\,}1$, and $\dim{\fam=\ssfam \usuDelta}^{\hskip-2pt\ast}{\,\ge\,}17$. As $v$ and $W$ are fixed, ${\cal F}_\iota$ is not a Hughes pane, and the claim follows from \cite{sz1} 1.10. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
Recall from 2.3 that we put ${\fam=\ssfam \usuSigma}{\,=\,}\mathop{{\rm Aut}}{\cal P}$. The following has been proved in L\"uneburg's dissertation \cite{Lb}: \par
{\bf 5.4 Large groups.} {\it Assume that $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}38$, that ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\{v,W\}$ is
a flag, and that ${\fam=\ssfam \usuUpsilon}$ is a maximal semi-simple subgroup of ${\fam=\ssfam \usuDelta}$. Then there are $3$ possibilities\/}:\par \quad(a) {\it ${\cal P}$ or its dual is a translation plane\/}, \par \quad(b) {\it ${\fam=\ssfam \usuUpsilon}{\,\cong\,}\Spin7{\fam=\Bbbfam R}$,\: $\dim{\fam=\ssfam \usuSigma}{\,=\,}38$, and ${\fam=\ssfam \usuDelta}_{[v,W]}{\,\cong\,}{\fam=\Bbbfam R}^8$\/} \ (\cite{Lb} V Satz), \par \quad(c) {\it ${\fam=\ssfam \usuUpsilon}{\,\cong\,}{\rm G}_2$ is a maximal compact subgroup of ${\fam=\ssfam \usuDelta}$\/}; {\it if $\dim{\fam=\ssfam \usuDelta}{\,=\,}39$, then ${\fam=\ssfam \usuDelta}_{[v,W]}{\,\cong\,}{\fam=\Bbbfam R}^8$ and,\par \hskip30pt up to duality, ${\fam=\ssfam \usuDelta}$ is transitive on ${\frak L}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt{\frak L}_v$ and hence on $W\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{v\}$\/} \par \hskip30pt (\cite{Lb} VI Kor.\:5, Satz\:5, and Kor.\:3). \par
{\tt Remarks.} All {\it translation\/} planes with $\dim{\fam=\ssfam \usuSigma}{\,\ge\,}38$ have been determined by H\"ahl, see \cite{cp} 82.\hskip1pt28,\hskip1pt29 and \cite{Ha6}, \cite{Ha3}. No substantial progress has been made in the meantime. \par
A few additional results can be obtained in case (c) of the preceding theorem: \par
{\bf 5.5 Proposition.} {\it Suppose that $\dim{\fam=\ssfam \usuDelta}{\,=\,}39$, that ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\{v,W\}$ is a flag, and that neither ${\cal P}$ nor its dual is a translation plane. Let ${\fam=\ssfam \usuUpsilon}$ denote a Levi complement of $\sqrt{\fam=\ssfam \usuDelta}$, choose $a{\,\notin\,}W$, and put ${\fam=\ssfam \usuGamma}{\hskip1pt=\,}({\fam=\ssfam \usuDelta}_a)^1$. Then\/} \par \quad(a) {\it ${\cal F}_{\fam=\ssfam \usuUpsilon}$ is isomorphic to the classical real plane ${\cal P}_{\fam=\Bbbfam R}$\/}, \par \quad(b) {\it if $a^{\hskip-1pt{\fam=\ssfam \usuUpsilon}\hskip-2.5pt}{\,=\,}a$, then $\dim a^{\fam=\ssfam \usuDelta}{\,\ge\,}15$\/}, \par \quad(c) {\it ${\fam=\ssfam \usuDelta}$ is transitive on $P\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt W$ and, dually, on ${\frak L}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt{\frak L}_v$, or
${\fam=\ssfam \usuGamma}_{\hskip-1pt[u]}{\,=\,}1\kern-2.5pt {\rm l}$ for each $u{\,\in\,}W\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{v\}$\/}; \par \hskip30pt {\it if ${\fam=\ssfam \usuDelta}_{[a]}{\,\ne\,}1\kern-2.5pt {\rm l}$, then $\dim a^{\fam=\ssfam \usuDelta}{\,=\,}15$ and ${\fam=\ssfam \usuDelta}_{[a]}$ does not contain an involution\/}, \par \quad(d) {\it if $a^{\hskip-1pt{\fam=\ssfam \usuUpsilon}\hskip-2.5pt}{\,=\,}a$, then $a^{\fam=\ssfam \usuDelta}$ is open in $P$ or $\dim{\fam=\ssfam \usuDelta}_{[v,av]}{\,=\,}1$\/}. \par
{\tt Proof.} (a) The representation of ${\fam=\ssfam \usuUpsilon}{\,\cong\,}{\rm G}_2$ on the Lie algebra ${\frak l}\hskip1pt\sqrt{\fam=\ssfam \usuDelta}$ together with \cite{cp} 95.10 shows that $\dim\Cs{\fam=\ssfam \usuDelta}{\hskip-2pt{\fam=\ssfam \usuUpsilon}}{\,=\,}4$. By \cite{cp} 96.35 the fixed points of ${\fam=\ssfam \usuUpsilon}$ on $W$ form a circle $S$. Dually, ${\fam=\ssfam \usuUpsilon}$ fixes a one-dimensional set of lines in each pencil ${\frak L}_s$ with $s{\,\in\,}S$. Therefore ${\cal F}_{\fam=\ssfam \usuUpsilon}$ is a flat subplane. Stiffness implies that the connected component of $\Cs{}{\fam=\ssfam \usuUpsilon}$ acts effectively on ${\cal F}_{\fam=\ssfam \usuUpsilon}$. According to \cite{cp} 33.9(a), either ${\cal F}_{\fam=\ssfam \usuUpsilon}$ is classical, or $\Cs{}{\fam=\ssfam \usuUpsilon}$ fixes exactly a non-incident point-line pair. \\ (b) Note that ${\fam=\ssfam \usuUpsilon}$ is a Levi complement of ${\fam=\ssfam P}{\,=\,}\sqrt{\fam=\ssfam \usuGamma}$. Choose $b{\,\notin\,}W\hskip-2pt\kern 2pt {\scriptstyle \cup}\kern 2pt av$, and consider a minimal ${\fam=\ssfam \usuGamma}_{\hskip-1pt b}$-invariant subgroup ${\fam=\ssfam \usuXi}{\,\cong\,}{\fam=\Bbbfam R}^s$ of ${\fam=\ssfam \usuDelta}_{[v,W]}{\,\cong\,}{\fam=\Bbbfam R}^8$. As ${\fam=\ssfam \usuGamma}$ does not act irreducibly on ${\fam=\ssfam \usuDelta}_{[v,W]}$, we have $s{\,<\,}8$. For $c{\,\in\,}a^{\fam=\ssfam \usuXi}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a\}$, the action of ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuGamma}_{\hskip-1pt b,c})^1$ on ${\fam=\ssfam \usuXi}$ shows ${\fam=\ssfam \usuLambda}{\,\not\cong\,}{\rm G}_2$. The stiffness result 2.6(e) implies $\dim{\fam=\ssfam \usuLambda}{\,\le\,}8$ and $\dim{\fam=\ssfam \usuGamma}_{\hskip-1pt b}{\,\le\,}15$. Now let ${\fam=\ssfam \usuDelta}{\,:\,}{\fam=\ssfam \usuGamma}{\,=\,}\dim a^{\fam=\ssfam \usuDelta}{\,=\,}k$, and choose $b{\,\in\,}a^{\fam=\ssfam \usuDelta}$. Then $39{\,=\,}\dim{\fam=\ssfam \usuDelta}{\,\le\,}2k{\,+\,}\dim{\fam=\ssfam \usuGamma}_{\hskip-1pt b}{\,\le\,}2k{\,+\,}15$. Consequently $k{\,\ge\,}12$ and $\dim{\fam=\ssfam \usuGamma}{\,\le\,}27$. We have ${\fam=\ssfam \usuGamma}{\,=\,}{\fam=\ssfam \usuUpsilon}{\fam=\ssfam P}$ with $\dim{\fam=\ssfam P}{\,<\,}14$. The representation of ${\fam=\ssfam \usuUpsilon}$ on ${\frak l}\hskip1pt{\fam=\ssfam P}$ shows $\dim{\fam=\ssfam P}{\,=\,}7{\,+\,}\dim({\fam=\ssfam P}\kern 2pt {\scriptstyle \cap}\kern 2pt\Cs{}{\fam=\ssfam \usuUpsilon}){\,\le\,}7{\,+\,}3$ and $\dim{\fam=\ssfam \usuGamma}{\,\le\,}24$. \\ (c) Suppose that $\dim a^{\fam=\ssfam \usuDelta}{\,=\,}15$ and that ${\fam=\ssfam \usuGamma}_{\hskip-1pt[u]}{\,\ne\,}1\kern-2.5pt {\rm l}$. Put $L{\,=\,}av$ and ${\fam=\ssfam E}{\,=\,}{\fam=\ssfam \usuDelta}_{[v,L]}$. By \cite{cp} 61.20, we have $e{\,=\,}\dim{\fam=\ssfam E}{\,=\,}\dim u^{{\fam=\ssfam \usuDelta}_L}$ and $h{\,=\,}\dim{\fam=\ssfam T}_{\hskip-1pt[u]}{\,=\,}\dim L^{{\fam=\ssfam \usuDelta}_u}$. We may assume that ${\fam=\ssfam \usuUpsilon}{\,\le\,}\nabla{\,:=\,}{\fam=\ssfam \usuGamma}_{\hskip-1.5pt u}$, since $u^{\fam=\ssfam \usuDelta}{\,=\,}W\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{v\}$ according to 5.4(c). It will turn out that $e{\,>\,}0$. If ${\fam=\ssfam \usuDelta}_L{\,\le\,}{\fam=\ssfam \usuDelta}_u$ and if $u^\delta{\,\ne\,}u$, then $L^\delta{\,\ne\,}L$, and ${\fam=\ssfam \usuGamma}_{\hskip-2pt L^\delta}$ fixes $a, u, L^\delta, u^\delta$ and, hence, a quadrangle, but $\dim{\fam=\ssfam \usuGamma}{\,=\,}24$ and $\dim{\fam=\ssfam \usuGamma}_{\hskip-2pt L^\delta}{\,>\,}14$, a contradiction to Stiffness. Therefore $u^{{\fam=\ssfam \usuDelta}_L}{\,\ne\,}u$. Analogously, $L^{{\fam=\ssfam \usuDelta}_u}{\,\ne\,}L$ and $h{\,>\,}0$. By assumption $e,h{\,<\,}8$, and the action of ${\fam=\ssfam \usuUpsilon}$ implies $e,h{\,\in\,}\{1,7\}$. Moreover, $\dim u^{\fam=\ssfam \usuGamma}{\,\le\,}e{\,=\,}\dim u^{\fam=\ssfam E}{\,\le\,}\dim u^{{\fam=\ssfam \usuDelta}_{[L]}}{\,\le\,}\dim u^{\fam=\ssfam \usuGamma}$. Therefore $24{\,=\,}\dim{\fam=\ssfam \usuGamma}{\,=\,}\dim\nabla{\,+\,}e$ and $\dim\nabla{\,\ge\,}17$. On the other hand, $\dim\nabla{\,\le\,}e{\,+\,}h{\,+\,}14$ and $10{\,\le\,}2e{\,+\,}h$. In the same way it follows that $10{\,\le\,}e{\,+\,}2h$. Therefore $e{\,=\,}h{\,=\,}7$ and $\dim\nabla{\,=\,}17$. Note that ${\fam=\ssfam \usuUpsilon}$ and $\nabla$ act irreducibly on ${\fam=\ssfam E}$ and on ${\fam=\ssfam T}_{\hskip-1pt[u]}$. The action of ${\fam=\ssfam \usuUpsilon}$ on $\mathop{\strut{\frak l}}\nabla$ shows that ${\fam=\ssfam M}{\,=\,}\nabla\kern 2pt {\scriptstyle \cap}\kern 2pt\Cs{}{\fam=\ssfam \usuUpsilon}$ satisfies $\dim{\fam=\ssfam M}{\,=\,}3$. On the other hand, ${\fam=\ssfam M}$ induces on ${\fam=\ssfam E}$ and on ${\fam=\ssfam T}_{\hskip-1pt[u]}$ groups of dimension at most $1$,\: see \cite{cp} 95.10. Hence ${\fam=\ssfam M}$ has a non-trivial subgroup which acts trivially on $\langle a^{{\fam=\ssfam T}_{[u]}},L^{\fam=\ssfam E}\rangle{\,=\,}{\cal P}$. This contradiction proves the first part of the claim. The second part follows from \cite{cp} 61.20 and 2.9 above. \\ (d) If $a^{\fam=\ssfam \usuDelta}$ is not open, then $\dim{\fam=\ssfam \usuGamma}{\,=\,}24$. By step (c), ${\fam=\ssfam \usuGamma}$ does not contain a homology with axis $av$. Representation of ${\fam=\ssfam \usuUpsilon}$ on the Lie algebra $\mathop{\strut{\frak l}}{\fam=\ssfam \usuGamma}$ shows that $\dim\Cs{\fam=\ssfam \usuGamma}{\hskip-2pt{\fam=\ssfam \usuUpsilon}}{\,\ge\,}3$. Put again ${\fam=\ssfam E}{\,=\,}{\fam=\ssfam \usuDelta}_{[v,av]}$ and $e{\,=\,}\dim{\fam=\ssfam E}$. As has been stated in 2.15(a), ${\fam=\ssfam \usuUpsilon}$ acts on ${\fam=\ssfam \usuXi}{\,=\,}{\fam=\ssfam \usuDelta}_{[v,W]}{\,\cong\,}{\fam=\Bbbfam R}^8$ in the same way as $\mathop{{\rm Aut}}{\fam=\Bbbfam O}$. In this action the centralizer of ${\fam=\ssfam \usuUpsilon}$ in $\mathop{{\rm Aut}}{\fam=\ssfam \usuXi}$ is $2$-dimensional. We have ${\fam=\ssfam \usuGamma}\kern 2pt {\scriptstyle \cap}\kern 2pt\Cs{}{\fam=\ssfam \usuXi}{\,\le\,}{\fam=\ssfam E}$. If $e{\,=\,}0$, then ${\fam=\ssfam E}{\,=\,}1\kern-2.5pt {\rm l}$, and ${\fam=\ssfam \usuGamma}$embeds into $\mathop{{\rm Aut}}{\fam=\ssfam \usuXi}$, so that $\dim\Cs{\fam=\ssfam \usuGamma}{\hskip-2pt{\fam=\ssfam \usuUpsilon}}{\,\le\,}2$. Hence $e{\,>\,}0$. As $\dim\{z{\,\in\,}W\mid z^{\hskip-1.5pt{\fam=\ssfam \usuUpsilon}}{\,=\,}z\}{\,=\,}1$, it follows that $e{\,\in\,}\{1,7\}$. If $e{\,=\,}7$, then ${\fam=\ssfam \usuUpsilon}$ acts irreducibly on ${\fam=\ssfam E}$ and ${\fam=\ssfam E}\kern 2pt {\scriptstyle \cap}\kern 2pt\Cs{\fam=\ssfam \usuGamma}{\hskip-2pt{\fam=\ssfam \usuUpsilon}}{\,=\,}1\kern-2.5pt {\rm l}$. Again this would imply $\dim\Cs{\fam=\ssfam \usuGamma}{\hskip-2pt{\fam=\ssfam \usuUpsilon}}{\,\le\,}2$. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\bf Conjecture.} {\it If $\dim{\fam=\ssfam \usuDelta}{\,=\,}39$ and if ${\cal F}_{\fam=\ssfam \usuDelta}$ is a flag, then ${\cal P}$ or its dual is a translation plane\/}. \par
{\Bf B. Non-incident fixed elements} \par
The following has been proved in \cite{sz12}; for semi-simple groups see also 4.2: \par
{\bf 5.6 Theorem.} {\it If ${\fam=\ssfam \usuDelta}$ fixes exactly one line $W$ and one point $a{\,\notin\,}W$, and if $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}35$, then ${\fam=\ssfam \usuDelta}$ contains a group $\Spin9({\fam=\Bbbfam R},r)$ with $r{\,\le\,}1$ and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}37$, or ${\fam=\ssfam \usuDelta}$ is triply transitive on~$W$ and ${\cal P}$ is the classical Moufang plane\/}. \par
{\bf 5.7 Almost simple groups.} {\it If $28{\,\le\,}\dim{\fam=\ssfam \usuDelta}{\,<\,}36$, $\,{\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\{a,W\},\: a{\,\notin\,}W$, and if ${\fam=\ssfam \usuDelta}$ is almost simple, then ${\fam=\ssfam \usuDelta}$ is isomorphic to one of the groups $\Spin8({\fam=\Bbbfam R},r),\, r{\,=\,}1,3$ or to some covering of the anti-unitary group {\tencyss\cyracc Ya}${=\,}\SaU4{\,\circeq\,}\Opr8({\fam=\Bbbfam R},2)$ and $\dim{\fam=\ssfam \usuDelta}{\,=\,}28$\/}. \par
{\tt Proof.} Because of Priwitzer's results 4.2, it suffices to consider groups of dimension~$28$. From 2.16 it follows that ${\fam=\ssfam \usuDelta}{\,\not\hskip1pt\cong\,}{\rm G}_2^{{\fam=\Bbbfam C}}$. Hence ${\fam=\ssfam \usuDelta}$ is locally isomorphic to an orthogonal group $\Opr8({\fam=\Bbbfam R},r)$. Recall from 2.10 and 2.11 that ${\fam=\ssfam \usuDelta}$ has no subgroup $\SO5{\fam=\Bbbfam R}$ or $(\SO3{\fam=\Bbbfam R})^2$. This excludes all groups ${\rm(P)}\Opr8({\fam=\Bbbfam R},r)$. Each
central involution $\zeta{\,\in\,}{\fam=\ssfam \usuDelta}$ is a reflection (or $\dim{\fam=\ssfam \usuDelta}|_{{\cal F}_\zeta}{\,=\,}28$ and $a^{\fam=\ssfam \usuDelta}{\,\ne\,}a$). If $r$ is even, then $\Spin8({\fam=\Bbbfam R},r)$ has a center $({\fam=\Bbbfam Z}_2)^2$ and hence fixes a triangle; in particular, $r{\,\ne\,}0$. Note that the double covering $\Spin8({\fam=\Bbbfam R},r)$ of $\Opr8({\fam=\Bbbfam R},r)$ is not simply connected for $r{\,\ge\,}2$. In each case, let $\tilde{\fam=\ssfam \usuDelta}$ denote the simply connected covering group of ${\fam=\ssfam \usuDelta}$. If $r{\,\ge\,}2$, the center of ${\fam=\ssfam \usuDelta}$ consists of homologies with axis $W\,$\: (or some stabilizer ${\fam=\ssfam \usuDelta}_p$ would fix a quadrangle and $({\fam=\ssfam \usuDelta}_p)^1{\,\cong\,}{\rm G}_2$ by Stiffness, but then $r{\,\le\,}1$). \\ (a) First let $r{\,=\,}2$. The group $\Opr8({\fam=\Bbbfam R},2){\,\circeq}${\tencyss\cyracc Ya} has center {\rm Cs}\,{\tencyss\cyracc Ya}${\,\cong\,} {\fam=\Bbbfam Z}_2$ by \cite{cp} 94.32(f),\hskip1pt33, and 95.10. A maximal compact subgroup ${\fam=\ssfam K}$ of {\tencyss\cyracc Ya} is isomorphic to $\U4{\fam=\Bbbfam C}$ and satisfies ${\fam=\ssfam K}'{\,\cong\,}\Spin6{\fam=\Bbbfam R}$. Thus {\tencyss\cyracc Ya} and its coverings are possible candidates for a group of automorphisms of a compact plane. \\ (b) If $r{\,=\,}3$, then a maximal compact subgroup of $\tilde{\fam=\ssfam \usuDelta}$ is isomorphic to $\Spin5{\fam=\Bbbfam R}{\times}\Spin3{\fam=\Bbbfam R}$, and $\Cs{}{\tilde{\fam=\ssfam \usuDelta}}{\,\cong\,}({\fam=\Bbbfam Z}_2)^2$. Hence ${\fam=\ssfam \usuDelta}$ is not simply connected. Assume now that ${\fam=\ssfam \usuDelta}$ has a compact subgroup {\tencyss\cyracc Yu}${\times}{\fam=\ssfam \usuOmega}{\,=\,}\Spin5{\fam=\Bbbfam R}{\times}\SO3{\fam=\Bbbfam R}$, and choose commuting involutions $\alpha,\beta{\,\in\,}{\fam=\ssfam \usuOmega}$, they are planar (as in the proof of 2.11). By \cite{cp} 96.13, the factor {\tencyss\cyracc Yu} acts transitively on $S{\,=\,}W{\kern 2pt {\scriptstyle \cap}\kern 2pt}{\cal F}_\beta$, and the central involution $\sigma$ of
{\tencyss\cyracc Yu} induces a reflection on ${\cal F}_\beta$, moreover, $\alpha|_S{\,=\,}1\kern-2.5pt {\rm l}$ because $\alpha$ fixes some point $z{\,\in\,}S$ and {\tencyss\cyracc Yu} is transitive on $S$. According to \cite{cp} 55.32, commuting involutions with the same fixed point set are equal.
Therefore $\alpha|_{{\cal F}_\beta}{\,=\,}\sigma|_{{\cal F}_\beta}$, $\,\alpha\sigma|_{{\cal F}_\beta}{\,=\,}1\kern-2.5pt {\rm l}$, $\,\alpha\sigma{\,=\,}\beta$, and $\sigma{\,=\,}\alpha\beta{\,\in\,}{\fam=\ssfam \usuOmega}$, a contradiction. \\ (c) For $r{\,=\,}4$, a maximal compact subgroup of $\tilde{\fam=\ssfam \usuDelta}$ is isomorphic to $(\Spin4{\fam=\Bbbfam R})^2{\,\cong\,}(\Spin3{\fam=\Bbbfam R})^4$, and $\Cs{}{\tilde{\fam=\ssfam \usuDelta}}{\,\cong\,}({\fam=\Bbbfam Z}_2)^4$. Consequently ${\fam=\ssfam \usuDelta}$ is simple or a double covering of ${\rm(P)}\Opr8({\fam=\Bbbfam R},4)$, but then ${\fam=\ssfam \usuDelta}$ has a subgroup $(\SO3{\fam=\Bbbfam R})^2$. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\tt Remarks.} Suppose that ${\fam=\ssfam \usuDelta}$ satisfies the conditions of 5.7 and that ${\fam=\ssfam \usuDelta}$ acts on the classical octonion plane ${\cal O}$. Then ${\fam=\ssfam \usuDelta}$ is contained in the affine group ${\fam=\ssfam A}{\,=\,}{\fam=\ssfam \usuSigma}_{a,W}$ of ${\cal O}$. From \cite{cp} 15.6 it follows that ${\fam=\ssfam A}'{\,\cong\,}\Spin{10}({\fam=\Bbbfam R},1)$, and $\dim{\fam=\ssfam A}'{\,=\,}45$. Hence ${\fam=\ssfam \usuDelta}$ intersects a maximal compact subgroup $\Spin9{\fam=\Bbbfam R}$ of ${\fam=\ssfam A}'$ in a group of dimension at least $19$. This excludes all orthogonal groups of index $r{\,>\,}1$\: (see\cite{cp} 94.33). Consequently ${\fam=\ssfam \usuDelta}{\,\cong\,}\Spin8({\fam=\Bbbfam R},1)$. For each $z{\,\in\,}W$ a Levi complement of $\sqrt{{\fam=\ssfam A}_z}$ in ${\fam=\ssfam A}$ is a compact group $\Spin8{\fam=\Bbbfam R}$. Hence $z^{\fam=\ssfam \usuDelta}{\,\ne\,}z$, and ${\fam=\ssfam \usuDelta}$ has no fixed point other than $a$. No other example for the situation of 5.7 has been found as yet. \par
{\bf 5.8 Semi-simple groups.} {\it If $\dim{\fam=\ssfam \usuDelta}{\,=\,}28$,\: $\,{\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\{a,W\}$ with $a{\,\notin\,}W$, and if ${\fam=\ssfam \usuDelta}$ is semi-simple, then ${\fam=\ssfam \usuDelta}$ is almost simple\/}. \par
{\tt Proof.} Suppose first that ${\fam=\ssfam \usuDelta}$ has exactly two almost simple factors. The list \cite{cp} 94.33 shows that there are only the following two cases: \\ (a) $\:{\fam=\ssfam \usuDelta}$ has a factor ${\fam=\ssfam \usuPsi}{\,\cong\,}\Sp4{\fam=\Bbbfam C}$ with a maximal compact subgroup ${\fam=\ssfam \usuPhi}{\,\cong\,}\U2{\fam=\Bbbfam H}$ and an $8$-dimensional factor ${\fam=\ssfam \usuGamma}$ of type ${\rm A}_2$. The central involution $\sigma{\,\in\,}{\fam=\ssfam \usuPsi}$ is a reflection in ${\fam=\ssfam \usuDelta}_{[a,W]}$ because ${\fam=\ssfam \usuPsi}/\langle \sigma\rangle{\,\cong\,}\SO5{\fam=\Bbbfam C}$ cannot act on a Baer subplane. Let $\alpha{\,=\,}\bigl({1 \atop }{ \atop -1}\bigr){\,\in\,}{\fam=\ssfam \usuPhi}$. The centralizer $\Cs{}\alpha$ contains a group ${\fam=\ssfam \usuGamma}{\cdot\hskip1pt}(\Sp2{\fam=\Bbbfam C})^2$ of dimension $20$. Stiffness and \cite{cp} 83.26 imply that $\alpha$ is not planar. Therefore $\alpha$ is a reflection, its center is some point $u{\,\in\,}W$, and $u^{\fam=\ssfam \usuGamma}{\,=\,}u{\,\ne\,}u^{\fam=\ssfam \usuPsi}$ by the assumption on ${\cal F}_{\fam=\ssfam \usuDelta}$. In fact, $u^{\fam=\ssfam \usuPhi}{\,\ne\,}u$ because $\alpha$ is conjugate to $\alpha\sigma$ in ${\fam=\ssfam \usuPhi}$ and the center of $\alpha\sigma$ is incident with the axis of $\alpha$. Now $\dim u^{\fam=\ssfam \usuPhi}{\,\ge\,}4$
by \cite{cp} 96.13. Note that ${\fam=\ssfam \usuGamma}|_{u^{\fam=\ssfam \usuPsi}}{\,=\,}1\kern-2.5pt {\rm l}$. From 2.17(b) it follows that ${\fam=\ssfam \usuGamma}$ is not transitive on $H{\,=\,}au\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a,u\}$, and there is some $b{\,\in\,}H$ such that ${\fam=\ssfam \usuGamma}_{\hskip-1.5pt b}{\,\ne\,}1\kern-2.5pt {\rm l}$. Hence ${\cal F}_{{\fam=\ssfam \usuGamma}_{\hskip-,5pt b}}{\,= \langle a,b,u^{\fam=\ssfam \usuPsi}\rangle\,}{\,=\,}{\cal B}$ is a Baer subplane of ${\cal P}$, and ${\cal B}^{\fam=\ssfam \usuPsi}{\,=\,}{\cal B}$ since ${\fam=\ssfam \usuGamma}{\,\le\,}\Cs{}{\fam=\ssfam \usuPsi}$. Moreover, ${\fam=\ssfam \usuPsi}$ acts effectively on ${\cal B}$ and fixes $a$ and $W$, but $\dim(\mathop{{\rm Aut}}{\cal B})_{a,W}{\,\le\,}19$, see again \cite{cp} 83.26. Thus, case (a) is impossible. \\ (b) $\:{\fam=\ssfam \usuDelta}$ is a product of two $14$-dimensional factors of type ${\rm G}_2$. The compact form is excluded by Stiffness. A maximal compact subgroup of ${\rm G}{\,=\,}{\rm G}_2(2)$ is isomorphic to $\SO4{\fam=\Bbbfam R}$. Lemma 2.11 implies that at least one of the two factors of ${\fam=\ssfam \usuDelta}$ is the simply connected covering group ${\fam=\ssfam \usuGamma}$ of ${\rm G}$, the other factor ${\fam=\ssfam \usuPsi}$ is locally isomorphic to ${\rm G}$. Stroppel's theorem \cite{str} 4.5\, (cf. \cite{sz1} 2.2) shows that neither ${\fam=\ssfam \usuGamma}$ nor ${\fam=\ssfam \usuPsi}$ contains a planar involution. As both factors have torus rank $2$, the group ${\fam=\ssfam \usuDelta}$ contains a group ${\fam=\Bbbfam Z}_2^{\hskip2pt4}$ generated by reflections. This contradicts 2.10 and proves that ${\fam=\ssfam \usuDelta}$ has more than two factors. \\ If ${\fam=\ssfam \usuDelta}$ is properly semi-simple, only the following possibilities remain: \\ (c) $\:{\fam=\ssfam \usuDelta}$ has a factor ${\fam=\ssfam \usuPsi}{\,\circeq\,}\SL3{\fam=\Bbbfam C}$ and some factors of dimension $3$ or $6$. There are $3$ pairwise commuting conjugate involutions in ${\fam=\ssfam \usuPsi}$ corresponding to diagonal elements in $\SL3{\fam=\Bbbfam C}$. If they were reflections, their centers would be ${\fam=\ssfam \usuPsi}$-conjugate, but one would have center~$a$, which is fixed. Hence there is a planar involution $\beta{\,\in\,}{\fam=\ssfam \usuPsi}$ such that $\SL2{\fam=\Bbbfam C}{\,\le\,}\Cs{\fam=\ssfam \usuPsi}\beta$, and $\Cs{\fam=\ssfam \usuDelta}\beta$ contains a semi-simple $18$-dimensional group inducing on ${\cal F}_\beta$ a semi-simple group ${\fam=\ssfam X}$ of dimension at least $15$\: (by Stiffness). Therefore, ${\cal F}_\beta$ is isomorphic to the classical quaternion plane, see \cite{sz9} 3.3 or \cite{sz1} 4.4. The group ${\fam=\ssfam X}$ is contained in a maximal semi-simple subgroup ${\fam=\ssfam B}$ of $(\mathop{{\rm Aut}}{\cal F}_\beta)_{a,W}$. We have $\dim{\fam=\ssfam B}{\,=\,}18$ and ${\fam=\ssfam B}$ has a $10$-dimensional compact subgroup {\cyss Yu}${\,\cong\,}\U2{\fam=\Bbbfam H}$. It follows that ${\fam=\ssfam \usuPhi}{\,=\,}{\fam=\ssfam X}\kern 2pt {\scriptstyle \cap}\kern 2pt${\cyss Yu}\ has dimension $\ge\!7$. If ${\fam=\ssfam \usuPhi}{\,<\,}${\cyss Yu}, then {\cyss Yu}\ acts trans\-itively and almost effectively on the coset space {\cyss Yu}$/{\fam=\ssfam \usuPhi}$, and \cite{cp} 96.13(a) would imply {\cyss Yu}${\hskip-1pt:\hskip0pt}{\fam=\ssfam \usuPhi}{\,\ge\,}4$. Consequently, ${\fam=\ssfam X}$ has a subgroup $\U2{\fam=\Bbbfam H}$, but this is impossible because ${\fam=\ssfam X}$ is a product of almost simple factors of dimension $3$ or $6$. \\ (d) $\:{\fam=\ssfam \usuDelta}$ has a factor ${\fam=\ssfam \usuPsi}$ of type ${\rm A}_3$, a factor ${\fam=\ssfam \usuUpsilon}$ of type ${\rm C}_2$, and a factor ${\fam=\ssfam \usuOmega}$ of dimension~$3$. Several combinations are excluded by 2.10 and 11; the rank condition, in particular, implies that at least one of the factors has an infinite center, cf. \cite{cp} 94.32(e) and 33. In any case, $\rm rk\,{\fam=\ssfam \usuPsi}{\,\ge\,}2$ and ${\fam=\ssfam \usuPsi}$ contains a non-central involution $\alpha$ centralizing a $6$-dimensional semi-simple subgroup ${\fam=\ssfam \usuGamma}$ of ${\fam=\ssfam \usuPsi}$. If $\alpha$ is a reflection, its center is some point $u{\,\in\,}W$\: (or else $\langle \alpha^{\fam=\ssfam \usuPsi}\rangle{\,=\,}{\fam=\ssfam \usuPsi}$ would act trivially on $W$), and $u^{{\fam=\ssfam \usuUpsilon}{\fam=\ssfam \usuOmega}}{\,=\,}u{\,\ne\,}u^{\fam=\ssfam \usuPsi}$. As ${\fam=\ssfam \usuPsi}$ acts almost effectively on $u^{\fam=\ssfam \usuPsi}$ and contains a compact subgroup of dimension ${\!\ge\,}6$, it follows from \cite{cp} 96.13 that $\dim u^{\fam=\ssfam \usuPsi}{>\,}2$. Let $b{\,\in\,}au\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a,u\},\ {\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuUpsilon}{\fam=\ssfam \usuOmega})_b$, and
${\cal F}{\,=\,}\langle u^{\fam=\ssfam \usuPsi},a,b\rangle$. Then $\dim{\fam=\ssfam \usuLambda}{\,\ge\,}5$,\: ${\fam=\ssfam \usuLambda}|_{\cal F}{\,=\,}1\kern-2.5pt {\rm l}$, and ${\cal F}{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$, which contradicts Stiffness. If $\alpha$ is planar, then ${\fam=\ssfam \usuGamma}{\fam=\ssfam \usuUpsilon}{\fam=\ssfam \usuOmega}$ induces on ${\cal F}_\alpha$ a semi-simple group ${\fam=\ssfam X}$ of dimension
at least 16. Again ${\cal F}_\alpha$ is the classical quaternion plane (cf. \cite{sz1} 4.4) and ${\fam=\ssfam X}$ is
contained in $\SL2{\fam=\Bbbfam H}{\,\cdot}\mathop{{\rm Aut}}{\fam=\Bbbfam H}$. As $\SL2{\fam=\Bbbfam H}$ has no proper subgroup of codimension ${\le\,}2$, it follows that $\SL2{\fam=\Bbbfam H}{\,\le\,}{\fam=\ssfam X}$, but ${\fam=\ssfam X}$ is a product of factors of smaller dimension. Hence case (d) is impossible. \\ (e) $\:{\fam=\ssfam \usuDelta}$ has a factor ${\fam=\ssfam \usuPsi}$ of type ${\rm G}_2$ and a factor ${\fam=\ssfam \usuUpsilon}$ of type ${\rm A}_2$. Stiffness implies that ${\fam=\ssfam \usuPsi}$ is not compact. A maximal compact subgroup ${\fam=\ssfam \usuPhi}$ of ${\fam=\ssfam \usuPsi}$ is isomorphic to $\SO4{\fam=\Bbbfam R}$ or to $\Spin4{\fam=\Bbbfam R}$, and there is an involution $\alpha{\,\in\,}{\fam=\ssfam \usuPhi}$ such that ${\fam=\ssfam \usuPhi}{\,\le\,}\Cs{\fam=\ssfam \usuPsi}\alpha{\,<\,}{\fam=\ssfam \usuPsi}$. As in the previous step, $\alpha$ is not planar, and
$\alpha|_W{\,\ne\,}1\kern-2.5pt {\rm l}$ because $\langle \alpha^{\fam=\ssfam \usuPsi}\rangle{\,=\,}{\fam=\ssfam \usuPsi}{\,\not\le\,}{\fam=\ssfam \usuDelta}_{[W]}$. Therefore $\alpha$ has some center $u{\,\in\,}W$. Choose $b{\,\in\,}au\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a,u\}$ and let ${\fam=\ssfam \usuLambda}{\,=\,}(\Cs{}{\fam=\ssfam \usuPsi})_{\hskip-.5pt b}^{\,1}$. Then $\dim{\fam=\ssfam \usuLambda}{\,\ge\,}6$ and $\dim{\cal F}_{\fam=\ssfam \usuLambda}{\,\le\,}4$, but then ${\fam=\ssfam \usuPsi}$ cannot act on ${\cal F}_{\fam=\ssfam \usuLambda}$. This excludes case (e). \\ (f) $\:{\fam=\ssfam \usuDelta}$ has a factor ${\fam=\ssfam \usuUpsilon}$ of type ${\rm C}_2$, all other factors have dimension at most $10$. Put ${\fam=\ssfam \usuPsi}{\,=\,}(\Cs{\fam=\ssfam \usuDelta}{\hskip-2pt{\fam=\ssfam \usuUpsilon}})^1$. Then $\dim{\fam=\ssfam \usuPsi}{\,=\,}18$. If ${\fam=\ssfam \usuPsi}$ fixes a point or a line other than $a$ or $W$, then there is some point $u{\,\in\,}W$ such that
$u^{\fam=\ssfam \usuPsi}{\,=\,}u{\,\ne\,}u^{\fam=\ssfam \usuUpsilon}$ and ${\fam=\ssfam \usuPsi}|_{u^{\fam=\ssfam \usuUpsilon}}{\,=\,}1\kern-2.5pt {\rm l}$. If $b{\,\in\,}au\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a,u\}$, we have $\dim{\fam=\ssfam \usuPsi}_b{\,\ge\,}10$; the fixed elements of ${\fam=\ssfam \usuPsi}_b$ form a subplane ${\cal E}{\,<\,}{\cal P}$,
and ${\fam=\ssfam \usuPsi}_b|_{\cal E}{\,=\,}1\kern-2.5pt {\rm l}$. Hence ${\cal E}$ is flat by Stiffness, ${\cal E}^{\fam=\ssfam \usuUpsilon}{\,=\,}{\cal E}$,
and $\dim({\fam=\ssfam \usuUpsilon}|_{\cal E}){\,=\,}10$. This contradiction shows that that $\langle x^{\fam=\ssfam \usuPsi}\rangle$ is a subplane whenever $a{\,\ne\,}x{\,\notin\,}W$. The r\^oles of ${\fam=\ssfam \usuUpsilon}$ and ${\fam=\ssfam \usuPsi}$ can be interchanged, and $u^{\hskip-1pt{\fam=\ssfam \usuUpsilon}}{\,=\,}u$ implies successively $u^{\fam=\ssfam \usuPsi}{\,\ne\,}u$,\:
${\fam=\ssfam \usuUpsilon}|_{u^{\fam=\ssfam \usuPsi}}{\,=\,}1\kern-2.5pt {\rm l}$,\: $\dim{\fam=\ssfam \usuUpsilon}_{\hskip-1pt b}{\,\ge\,}2$,\:
${\cal F}{\,=\,}{\cal F}_{{\fam=\ssfam \usuUpsilon}_{\hskip-1pt b}}{\,<\,}{\cal P}$,\: ${\fam=\ssfam \usuUpsilon}_{\hskip-1pt b}|_{\cal F}{\,=\,}1\kern-2.5pt {\rm l}$, and ${\cal F}^{\fam=\ssfam \usuPsi}{\,=\,}{\cal F}$. If ${\cal F}$ is flat, then $\dim{\fam=\ssfam \usuPsi}{\,\le\,}4{+}11$\: (use Stiffness and
${\fam=\ssfam \usuUpsilon}_{\hskip-1pt b}|_{\cal F}{\,=\,}1\kern-2.5pt {\rm l}$). Similarly, $\dim{\cal F}{\,=\,}4$ implies $\dim{\fam=\ssfam \usuPsi}{\,\le\,}8{+}8$. Therefore ${\cal F}{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$, and $\dim{\fam=\ssfam \usuPsi}|_{\cal F}{\,=\,}18$\: (again because
${\fam=\ssfam \usuUpsilon}_{\hskip-1pt b}|_{\cal F}{\,=\,}1\kern-2.5pt {\rm l}$). Now ${\cal F}$ is classical by \cite{sz9} 3.3 or
\cite{sz1} 4.4, and ${\fam=\ssfam \usuPsi}|_{\cal F}$ contains a group $\SL2{\fam=\Bbbfam H}$ contrary to the assumption in case (f). It follows that $\langle x^{\fam=\ssfam \usuUpsilon}\rangle$ and $\langle x^{\fam=\ssfam \usuPsi}\rangle$ are subplanes of ${\cal P}$, and $\langle x^{\fam=\ssfam \usuUpsilon}\rangle{\,<\,}{\cal P}$, since
${\fam=\ssfam \usuPsi}_{\hskip-1pt x}|_{\langle x^{\fam=\ssfam \usuUpsilon}\rangle}{\,=\,}1\kern-2.5pt {\rm l}$ and $\dim{\fam=\ssfam \usuPsi}_{\hskip-1pt x}{\,\ge\,}2$. Consequently, $\dim x^{\hskip-1pt{\fam=\ssfam \usuUpsilon}}{\,\le\,}8$ and $\dim{\fam=\ssfam \usuUpsilon}_{\hskip-1.5pt x}{\,\ge\,}2$. Hence $\langle x^{\fam=\ssfam \usuPsi}\rangle$ is also a proper subplane, $\dim x^{\fam=\ssfam \usuPsi}{\,\le\,}8$, and $\dim{\fam=\ssfam \usuPsi}_{\hskip-1pt x}{\,\ge\,}10$. Stiffness implies that ${\cal E}{\,=\,}\langle x^{\fam=\ssfam \usuUpsilon}\rangle$ is even flat, but $\dim(\mathop{{\rm Aut}}{\cal E})_{a,W}{\,\le\,}4$. \\ (g) Last case: all factors of ${\fam=\ssfam \usuDelta}$ have dimension at most $8$. Then there are exactly two $8$-dimensional factors ${\fam=\ssfam \usuUpsilon}_{\hskip-1.5pt1},\!{\fam=\ssfam \usuUpsilon}_{\hskip-1.5pt2}$. An involution $\sigma{\,\in\,}\Cs{}{\fam=\ssfam \usuDelta}$ is a reflection with axis $W$: if ${\cal F}_\sigma{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$} }{\cal P}$, then
$\dim{\fam=\ssfam \usuDelta}|_{{\cal F}_\sigma}{\,<\,}20$ contrary to Stiffness. Moreover, each involution $\iota$ in one of the factors is a reflection: if $\iota$ is planar, denote the product of the other factors by ${\fam=\ssfam \usuPsi}$ and
put ${\fam=\ssfam \usuPsi}|_{{\cal F}_\iota}{\,=\,}{\fam=\ssfam \usuPsi}/{\fam=\ssfam K}$. Then ${\fam=\ssfam \usuPsi}{\hskip0pt:\hskip1pt}{\fam=\ssfam K}{\,<\,}20$,\: $\dim{\fam=\ssfam K}{\,>\,}0$, and ${\fam=\ssfam K}^1$ is semi-simple. Stiffness implies ${\fam=\ssfam K}^1{\,\cong\,}\Spin3{\fam=\Bbbfam R}$, and the involution $\sigma{\,\in\,}{\fam=\ssfam K}^1$ acts trivially on ${\cal F}_\iota$; on the other hand, $\sigma$ is in the center of ${\fam=\ssfam \usuDelta}$, but then $\sigma{\,\in\,}{\fam=\ssfam \usuDelta}_{[a,W]}$. It follows that none of the factors has a subgroup isomorphic to $\SO3{\fam=\Bbbfam R}$. Suppose that ${\fam=\ssfam \usuDelta}$ has a factor ${\fam=\ssfam \usuUpsilon}{\circeq\,}\SU3({\fam=\Bbbfam C},r)$. Then ${\fam=\ssfam \usuUpsilon}$ has a subgroup ${\fam=\ssfam \usuPhi}{\,\cong\,}\SU2{\fam=\Bbbfam C}$. The involution $\alpha{\,\in\,}{\fam=\ssfam \usuPhi}$ is a reflection. As $\langle \alpha^{\fam=\ssfam \usuUpsilon}\rangle{\,=\,}{\fam=\ssfam \usuUpsilon}$, the axis of $\alpha$ is a line $av$, and $\dim v^{\fam=\ssfam \usuUpsilon}{\,>\,}1$. Choose $c{\,\in\,}av\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a,v\}$ and put ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuPsi}_c)^1$. Then ${\cal F}_{\fam=\ssfam \usuLambda}{\,=\,}\langle a,c,v^{\fam=\ssfam \usuUpsilon}\rangle$ and $\dim{\fam=\ssfam \usuLambda}{\,\ge\,}12$. This contradicts Stiffness and proves that the ${\fam=\ssfam \usuUpsilon}_{\hskip-1.5pt\nu}$ are isomorphic to the simply connected covering group of $\SL3{\fam=\Bbbfam R}$. This possibility will be excluded in the next steps. \\ (h) Assume that ${\fam=\ssfam \usuDelta}$ has a factor ${\fam=\ssfam \usuPhi}{\,\cong\,}\SU2{\fam=\Bbbfam C}$. Then ${\fam=\ssfam \usuUpsilon}_{\hskip-1.5pt1}$ and ${\fam=\ssfam \usuPhi}$ have the reflection $\sigma{\,\in\,}{\fam=\ssfam \usuDelta}_{[a,W]}$ in common. Consequently, ${\fam=\ssfam \usuUpsilon}_{\hskip-1.5pt1}{\fam=\ssfam \usuPhi}$ contains a subgroup ${\fam=\ssfam \usuOmega}{\,\cong\,}\SO4{\fam=\Bbbfam R}{\,>\,}\SO3{\fam=\Bbbfam R}$. Each involution $\beta{\,\in\,}\SO3{\fam=\Bbbfam R}$ is planar. Put ${\fam=\ssfam \usuPsi}{\,=\,}(\Cs{}\beta)^1$, note that ${\fam=\ssfam \usuOmega}\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam \usuPsi}{\,\cong\,}{\fam=\Bbbfam T}^2$, and consider
$\overline{\fam=\ssfam \usuPsi}{\,=\,}{\fam=\ssfam \usuPsi}|_{{\cal F}_\beta}{\,=\,}{\fam=\ssfam \usuPsi}/{\fam=\ssfam K}$. We have $\dim{\fam=\ssfam \usuPsi}{\,\ge\,}2{+}17$,\: $\dim\overline{\fam=\ssfam \usuPsi}{\,\ge\,}16$, and $\overline{\fam=\ssfam \usuPsi}$ has a normal torus subgroup. According to \cite{sz1} 7.3,\: ${\cal F}_\beta$ is the classical quaternion plane, $\overline{\fam=\ssfam \usuPsi}$ is contained in ${\fam=\Bbbfam H}^{\times}{\cdot\,}\SL2{\fam=\Bbbfam H}$, and $\dim(\overline{\fam=\ssfam \usuPsi}\kern 2pt {\scriptstyle \cap}\kern 2pt\SL2{\fam=\Bbbfam H}){\,\ge\,}12$, but such a group $\overline{\fam=\ssfam \usuPsi}$ does not exist. Therefore ${\fam=\ssfam \usuDelta}$ has no compact factor. If $\SL2{\fam=\Bbbfam C}{\,\cong\,}{\fam=\ssfam \usuGamma}{\,\triangleleft\,}{\fam=\ssfam \usuDelta}$, analogous arguments yield a group $\overline{\fam=\ssfam \usuPsi}$ of dimension at least $13$. Again ${\cal F}_\beta$ is classical, and ${\fam=\ssfam \usuUpsilon}_{\hskip-1pt2}$ maps onto a subgroup $\SL3{\fam=\Bbbfam R}$ of $\SL2{\fam=\Bbbfam H}$, since ${\fam=\ssfam \usuUpsilon}_{\hskip-1pt2}$ has no faithful linear representation, cf. \cite{cp} 95.9. In particular,
$\sigma|_{{\cal F}_\beta}{\,=\,}1\kern-2.5pt {\rm l}$, which is absurd. \\ (i) Steps (g,h) imply that there are $4$ factors ${\fam=\ssfam \usuGamma}_{\hskip-1.5pt\nu}{\,\circeq\,}\SL2{\fam=\Bbbfam R}$. For similar reasons as before, ${\fam=\ssfam \usuUpsilon}_{\hskip-1pt1}{\fam=\ssfam \usuUpsilon}_{\hskip-1pt2}$ has a subgroup ${\fam=\ssfam \usuOmega}{\,\cong\,}\SO4{\fam=\Bbbfam R}$ containing a planar involution $\beta$. Let again ${\fam=\ssfam \usuPsi}{\,=\,}(\Cs{}\beta)^1$ ${\,\cong\,}{\fam=\Bbbfam T}^2{\cdot}${\Large{$\!\times\!$}}$_\nu{\fam=\ssfam \usuGamma}_\nu$ and
$\overline{\fam=\ssfam \usuPsi}{\,=\,}{\fam=\ssfam \usuPsi}|_{{\cal F}_\beta}$. As before, ${\cal F}_\beta$ is classical and $\overline{\fam=\ssfam \usuPsi}$ is a subgroup of ${\fam=\Bbbfam H}^{\times}{\cdot\,}\SL2{\fam=\Bbbfam H}$. None of the groups $\overline{\fam=\ssfam \usuGamma}_{\hskip-1pt\nu}$ is simply connected, each ${\fam=\ssfam \usuGamma}_{\hskip-1pt\nu}$ maps onto a group of rank $1$. Hence the rank of $\overline{\fam=\ssfam \usuPsi}$ is at least $5$, but this contradicts 2.10 or ${\fam=\ssfam \usuPsi}{\,\le\,}{\fam=\Bbbfam H}^{\times}{\cdot\,}\SL2{\fam=\Bbbfam H}$. \hglue 0pt plus 1filll $\scriptstyle\square$
\par
{\bf 5.9 Normal torus.} {\it If $\dim{\fam=\ssfam \usuDelta}{\,>\,}28$ and $\,{\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\{a,W\},\: a{\,\notin\,}W$,
if ${\fam=\ssfam \usuDelta}$ has a normal subgroup ${\fam=\ssfam \usuTheta}{\,\cong\,}{\fam=\Bbbfam T}$, and if ${\fam=\ssfam \usuDelta}$ is doubly transitive on
some proper subset $V{\,\subset\,}W$, then $V{\,\approx\,}{\fam=\Bbbfam S}_6$,\, ${\fam=\ssfam \usuDelta}|_V{\,\cong\,}\POpr8({\fam=\Bbbfam R},1)$ is even triply transitive, and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}30$\/}. \par
{\tt Proof.} If even $\dim{\fam=\ssfam \usuDelta}{\,>\,}30$, it has been shown in \cite{sz5} that ${\cal P}$ is a Hughes plane (including the classical plane). Groups of dimension ${\ge}35$ have been dealt with in 5.6. We may suppose, therefore, that $\dim{\fam=\ssfam \usuDelta}{\,<\,}35$; in the cases $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}31$ a few arguments in \cite{sz5} will be simplified. Note that $v^{\fam=\ssfam \usuDelta}{\,=\,}W$ implies $\dim{\fam=\ssfam \usuDelta}{\,>\,}36$, see \cite{cp} 96.\,19\hskip1pt--22 and use ${\fam=\ssfam \usuTheta}$. \\ (a) {\it ${\fam=\ssfam \usuTheta}{\,\le\,}{\fam=\ssfam \usuDelta}_{[a,W]}$ if, and only if, ${\fam=\ssfam \usuDelta}$ has no subgroup ${\fam=\ssfam \usuGamma}{\,\cong\,}{\rm G}_2$\/}: in fact, ${\cal F}_{\fam=\ssfam \usuGamma}$ would be a flat subplane, which does not admit
a torus group of homologies. If, on the other hand, ${\fam=\ssfam \usuTheta}|_W{\,\ne\,}1\kern-2.5pt {\rm l}$, then there is some point $x$ such that ${\cal F}_{{\fam=\ssfam \usuDelta}_x}$ is a connected subplane, $\dim{\fam=\ssfam \usuDelta}_x{\,>\,}12$, and Stiffness yields ${\fam=\ssfam \usuGamma}{\,:=\,}({\fam=\ssfam \usuDelta}_x)^1{\,\cong\,}{\rm G}_2$. \\ (b) {\it If $v^{\fam=\ssfam \usuDelta}{\,=\,}V{\,\subseteq\,}W$, then $\dim V{\,=:\,}k{\,>\,}4$\/}: let ${\fam=\ssfam \usuLambda}$ denote the connected component of the stabilizer of points $u,v,w{\,\in\,}V$ and $c{\,\in\,}S{\,:=\,}av\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a,v\}$. The dimension formula yields $$28{\,<\,}\dim{\fam=\ssfam \usuDelta}{\,\le\,}3k{\,+\,}8{\,+\,}\dim{\fam=\ssfam \usuLambda}\,,\hskip5pt 21{\,\le\,}3k{\,+\,}\dim{\fam=\ssfam \usuLambda}{\,\le\,}3k{\,+\,}14\,,\hskip5pt{\rm and} \ k{\,\ge\,}3\,.$$
If ${\fam=\ssfam \usuTheta}|_W{\,\ne\,}1\kern-2.5pt {\rm l}$, then ${\fam=\ssfam \usuDelta}$ has a subgroup ${\fam=\ssfam \usuGamma}{\,\cong\,}{\rm G}_2$ by step (a), and 2.15 implies $z^{\fam=\ssfam \usuGamma}{\,\approx\,}{\fam=\Bbbfam S}_6$ for some $z{\,\in\,}V$ and $k{\,\ge\,}6$.
If ${\fam=\ssfam \usuTheta}|_W{\,=\,}1\kern-2.5pt {\rm l}$, however, then ${\fam=\ssfam \usuTheta}$ induces a group of homologies on ${\cal F}_{\fam=\ssfam \usuLambda}$,\: $\dim{\cal F}_{\fam=\ssfam \usuLambda}{\,\ge\,}4$,\: $\dim{\fam=\ssfam \usuLambda}{\,\le\,}8$ by Stiffness, and then $3k{\,>\,}12$. \\
(c) Put $\tilde{\fam=\ssfam \usuDelta}{\,=\,}{\fam=\ssfam \usuDelta}|_V{\,=\,}{\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$. {\it The kernel ${\fam=\ssfam K}$ has dimension $\dim{\fam=\ssfam K}{\,<\,}8$, and $22{\,\le\,}\dim\tilde{\fam=\ssfam \usuDelta}{\,\le\,}34$\/}: note that $\langle a,c,V\rangle{\,=\,}{\cal P}$, so that ${\fam=\ssfam K}$ acts freely on $S$. If $\dim{\fam=\ssfam K}{\,=\,}8$, then each orbit $c^{\fam=\ssfam K}$ is open in $S$ by \cite{cp} 96.11(a), and ${\fam=\ssfam K}$ is transitive on $S$. The space $S$ is homotopy equivalent to ${\fam=\Bbbfam S}_7$. A maximal compact subgroup ${\fam=\ssfam \usuPhi}$ of ${\fam=\ssfam K}$ is a product of a torus and some almost simple Lie groups ${\fam=\ssfam \usuPsi}_\nu$, and $\pi_3{\fam=\ssfam \usuPsi}_\nu{\,\cong\,}{\fam=\Bbbfam Z}$, see \cite{cp} 94.\,31(c) and 36. The exact homotopy sequence \cite{cp} 96.12 shows that $\pi_q{\fam=\ssfam \usuPhi}{\,\cong\,}\pi_q{\fam=\Bbbfam S}_7$ for each $q{\,\ge\,}1$. Consequently ${\fam=\ssfam \usuPhi}$ is trivial, but $\pi_7{\fam=\Bbbfam S}_7{\,\cong\,}{\fam=\Bbbfam Z}$. This contradiction proves the claim; see also 2.17(b). \\ (d) {\it $V$ is compact\/}: if not, then $\tilde{\fam=\ssfam \usuDelta}$ has a normal subgroup ${\fam=\ssfam N}{\,\cong\,}{\fam=\Bbbfam R}^k$ acting sharply transitive on $V$, and $\tilde{\fam=\ssfam \usuDelta}$ is an extension of ${\fam=\ssfam N}$ by a transitive linear group, cf. \cite{cp} 96.16. We use the notation of step~(b), and we write $\nabla{\,=\,}{\fam=\ssfam \usuDelta}_{v,u}$. Suppose first that $k{\,\le\,}6$. Then ${\rm G}_2$ is not a subgroup of~${\fam=\ssfam \usuDelta}$,\, ${\fam=\ssfam \usuDelta}{\hskip1pt:}\nabla{\,=\,}2k$,\, $\nabla$ fixes a line of the vector space ${\fam=\Bbbfam R}^k$ or ${\fam=\Bbbfam C}^3$,\, ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuLambda}{\,<\,}2k{+}2{+}8$,\, ${\fam=\ssfam \usuLambda}{\,\not\hskip1pt\cong\,}\SU3{\fam=\Bbbfam C}$,\, $\dim{\fam=\ssfam \usuLambda}{\,\le\,}6$ by 2.6(\^e), and $\dim{\fam=\ssfam \usuDelta}{\,<\,}28$ contrary to the assumption. \\ Next, let $k{\,=\,}7$. Then ${\fam=\ssfam \usuDelta}_v$ induces a transitive, hence irreducible, linear group on ${\fam=\ssfam N}$, and ${\fam=\ssfam \usuDelta}_v$ has a subgroup ${\fam=\ssfam X}{\,\cong\,}{\rm G}_2$ by \cite{cp} 96.~\hskip-3pt16\,--22 and 94.27. By step (a) there is some point $x$ such that ${\cal E}{\,=\,}\langle a,x^{\fam=\ssfam \usuTheta},W\rangle$ is a connected subplane, and
${\fam=\ssfam \usuDelta}_x|_{\cal E}{\,=\,}1\kern-2.5pt {\rm l}$. Stiffness implies $\dim{\fam=\ssfam \usuDelta}_x{\,\le\,}14$ and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}30$.
There are two possibilities: ${\fam=\ssfam \usuDelta}_v|_V$ is isomorphic either to $e^{{\fam=\Bbbfam R}}{\hskip1pt\cdot\hskip1pt}\SO7{\fam=\Bbbfam R}$ or to $e^{{\fam=\Bbbfam R}}{\hskip1pt\cdot\hskip1pt}{\rm G}_2$. As $k$ is odd, ${\fam=\ssfam \usuTheta}{\,\le\,}{\fam=\ssfam K}$. In the first case, ${\fam=\ssfam \usuDelta}{\,:\,}{\fam=\ssfam K}{\,=\,}29$,\, $\dim{\fam=\ssfam K}{\,=\,}1$,\, ${\fam=\ssfam K}^1{\,=\,}{\fam=\ssfam \usuTheta}$, and $\nabla{\,\cong\,}\Spin6{\fam=\Bbbfam R}{\hskip1pt\cdot\hskip1pt}{\fam=\ssfam \usuTheta}$ is a compact subgroup of ${\fam=\ssfam \usuDelta}$. Consequently, $\nabla$ fixes the points of a $1$-dimensional subspace of~${\fam=\Bbbfam R}^7$, the orbit $c^\nabla$ is not open in $av$, and ${\fam=\ssfam \usuLambda}{\,=\,}\nabla_{\hskip-2pt c}$ has dimension $>8$, but this contradicts 2.6(\^e). In the second case, ${\fam=\ssfam \usuDelta}{\,:\,}{\fam=\ssfam K}{\,=\,}22$,\, $\dim{\fam=\ssfam K}{\,=\,}7$,\, ${\fam=\ssfam \usuTheta}{\,\triangleleft\,}{\fam=\ssfam K}$, the representation of ${\fam=\ssfam X}$ on $\mathop{\strut{\frak l}}{\fam=\ssfam K}$ shows ${\fam=\ssfam K}{\,\le\,}\Cs{}{\fam=\ssfam X}$, and
${\fam=\ssfam K}|_{{\cal F}_{\fam=\ssfam X}}$ would be too large. \\
Finally, if $k{\,=\,}8$, then ${\fam=\ssfam \usuDelta}_v|_V$ contains one of the groups $\SO8{\fam=\Bbbfam R}$, $\SU4{\fam=\Bbbfam C}$, or $\U2{\fam=\Bbbfam H}$, again by \cite{cp} 96.~\hskip-3pt16\,--22. In the first case, $\rm rk\,{\fam=\ssfam \usuDelta}{\,=\,}5$ contrary to 2.10. In the second case, there exists a subgroup ${\fam=\ssfam \usuPhi}{\,\cong\,}\SU4{\fam=\Bbbfam C}{\,\cong\,}\Spin6{\fam=\Bbbfam R}$ in ${\fam=\ssfam \usuDelta}$, see \cite{cp} 94.27. The central involution $\sigma{\,\in\,}{\fam=\ssfam \usuPhi}$ is not planar\, (or
${\fam=\ssfam \usuPhi}|_{{\cal F}_\sigma}{\,\cong\,}\SO6{\fam=\Bbbfam R}$), and $\sigma$ is a reflection with center $v$ or axis $av$. By double transitivity, the elation group ${\fam=\ssfam \usuDelta}_{[v,av]}$ is transitive (cf. \cite{cp} 61.19(b)\hskip1pt), and so is each group ${\fam=\ssfam \usuDelta}_{[z,az]}$ with $z{\,\in\,}W$. It follows that ${\fam=\ssfam \usuDelta}$ is transitive on $W$ and has a subgroup $\Spin9{\fam=\Bbbfam R}$, but then $\rm rk\,{\fam=\ssfam \usuDelta}{\,\le\,}4$ implies that ${\fam=\ssfam \usuDelta}$ does not have a normal torus. The third case can be excluded by the same arguments. \\ (e) We can now prove the assertions. All doubly transitive transformation groups have been determined by Tits 1955, cf. \cite{cp} 96.16\hskip1pt--18 for results and a sketch of the proof. Consider the effective transformation group $(\tilde{\fam=\ssfam \usuDelta},V)$ defined in step (c). The space $V$ is compact if, and only if, $\tilde{\fam=\ssfam \usuDelta}$ is a strictly simple Lie group; in this case $V$ is a projective space or a sphere, and $\tilde{\fam=\ssfam \usuDelta}$ is the corresponding projective or hyperbolic group. In the given dimension range $5{\,\le\,}\dim V{\,<\,}8$ and $22{\,\le\,}\dim\tilde{\fam=\ssfam \usuDelta}{\,\le\,}34$ there are exactly the following possibilities: $(\PSL4{\fam=\Bbbfam C},{\rm P}_3{\fam=\Bbbfam C})$,\, $(\PSU5({\fam=\Bbbfam C},1),{\fam=\Bbbfam S}_7)$, and $(\POpr8({\fam=\Bbbfam R},1),{\fam=\Bbbfam S}_6)$.
In the first case, the assumption on~${\fam=\ssfam \usuDelta}$ implies $\dim{\fam=\ssfam \usuDelta}{\,>\,}\dim\tilde{\fam=\ssfam \usuDelta}{\,=\,}30$. Choose $c$ as before, $u{\,\in\,}V$, and $w$ on the line $uv$ in the projective space ${\rm P}_3{\fam=\Bbbfam C}$. Let ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuDelta}_{c,u,w})^1$. By step (a), the central torus ${\fam=\ssfam \usuTheta}$ induces a group of homologies on ${\cal F}_{\fam=\ssfam \usuLambda}$, and $\dim{\cal F}_{\fam=\ssfam \usuLambda}{\,>2\,}$. Stiffness yields $\dim{\fam=\ssfam \usuLambda}{\,\le\,}8$, and the dimension formula gives $\dim{\fam=\ssfam \usuDelta}{\,\le\,}2{\cdot}6{\,+\,}2{\,+\,}8{\,+\,}\dim{\fam=\ssfam \usuLambda}{\,\le \,}30$, a contradiction. \\ If $V{\,\approx\,}{\fam=\Bbbfam S}_7$, then ${\fam=\ssfam \usuDelta}$ has a subgroup ${\fam=\ssfam \usuUpsilon}{\,\circeq\,}\SU5({\fam=\Bbbfam C},1)$ and a compact subgroup ${\fam=\ssfam \usuPhi}{\,\cong\,}\SU4{\fam=\Bbbfam C}$. The kernel ${\fam=\ssfam K}$ satisfies $5{\,\le\,}\dim{\fam=\ssfam K}{\,\le\,}7$, and the representation of ${\fam=\ssfam \usuUpsilon}$ on ${\frak l}\hskip1pt{\fam=\ssfam K}$ shows that ${\fam=\ssfam K}{\,\le\,}\Cs{}{\fam=\ssfam \usuUpsilon}$. There are $6$ conjugate planar (diagonal) involutions in ${\fam=\ssfam \usuPhi}$. Let $\beta$ be one of these. At most $3$ of the others induce reflections on ${\cal F}_\beta$. Hence ${\cal F}_{\beta,\hskip.6pt\beta'}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal F}_\beta$ for some $\beta'$. As ${\fam=\ssfam K}$ acts freely on a non-trivial point orbit in ${\cal F}_{\beta,\hskip.6pt\beta'}$, this would imply $5{\,\le\,}\dim{\fam=\ssfam K}{\,\le\,}4$. Therefore $V{\,\approx\,}{\fam=\Bbbfam S}_6$, and ${\fam=\ssfam \usuDelta}$ contains $\Spin8({\fam=\Bbbfam R},1)$ and has a subgroup ${\rm G}_2$. As in step (d), it follows that $\dim{\fam=\ssfam \usuDelta}{\,\le\,}30$. According to \cite{cp} 96.18, the group ${\fam=\ssfam \usuDelta}$ is triply transitive on $V$. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\tt Remark.} In the classical plane, ${\fam=\ssfam \usuSigma}'_{a,W}{\,\cong\,}\Spin{10}({\fam=\Bbbfam R},1)$ contains ${\fam=\ssfam \usuDelta}{\,\cong\,}\Spin8({\fam=\Bbbfam R},1){\times}\SO2{\fam=\Bbbfam R}$. For any $v{\,\in\,}W$, the compact group $\Spin8{\fam=\Bbbfam R}$ is a maximal semi-simple subgroup of ${\fam=\ssfam \usuSigma}_{a,v,W}$. Therefore ${\fam=\ssfam \usuDelta}'$ has no fixed point on $W$, and ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\{a,W\}$. No other plane with such an action of ${\fam=\ssfam \usuDelta}$ is known. \par
{\bf 5.10 Normal torus.} {\it If $\,{\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\{a,W\}, a{\,\notin\,}W$, and
if ${\fam=\ssfam \usuDelta}$ has a normal subgroup ${\fam=\ssfam \usuTheta}{\,\cong\,}{\fam=\Bbbfam T}$, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}30$\/}. \\ For $\dim{\fam=\ssfam \usuDelta}{\,>\,}30$, it has been proved in \cite{sz5} that ${\cal P}$ is a Hughes plane, and then ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\emptyset$. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\bf 5.11 Normal vector groups.} {\it Assume that ${\fam=\ssfam \usuDelta}$ is not semi-simple and that ${\cal P}$ is not a Hughes plane. If $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}33$, then ${\fam=\ssfam \usuDelta}$ has a minimal normal subgroup ${\fam=\ssfam \usuTheta}{\,\cong\,}{\fam=\Bbbfam R}^t$. In the cases $1{\,<\,}t{\,\ne\,}8,\,12$, the group ${\fam=\ssfam \usuTheta}$ consists of elations with common axis or common center. If $t{\,=\,}8$ or $12$, then either ${\fam=\ssfam \usuTheta}$ or some other minimal normal subgroup $\tilde{\fam=\ssfam \usuTheta}$ has this property\/}. \par
This has been proved in \cite{sz10} Theorem A and Propositions 8 and 9. The arguments are rather involved, they shall not be indicated here. \par
{\bf 5.12 Corollary.} {\it If $\,{\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\{a,W\}, a{\,\notin\,}W$, and if ${\fam=\ssfam \usuDelta}$ has a minimal normal vector subgroup ${\fam=\ssfam \usuTheta}$ of dimension $t{\,>\,}1$, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}32$\/}.
\par
{\Bf 6. Two fixed points} \par
In the classical octonion plane, ${\fam=\ssfam \usuSigma}_{u,v}$ is an extension of a transitive translation group by $(e^{{\fam=\Bbbfam R}})^2{\cdot\,}\Spin8{\fam=\Bbbfam R}$, cf. \cite{cp} 17.13. Throughout this section let ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\langle u,v\rangle$. \par
{\bf 6.0.} {\it If ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\langle u,v\rangle$ and if $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}18$, then ${\fam=\ssfam \usuDelta}$ is a Lie group\/}. \par
{\tt Proof.} (a) Suppose that ${\fam=\ssfam \usuDelta}$ is not a Lie group. As in the proof of 4.0, it follows from \cite{lw3} that $x^\zeta{\,=\,}x$ for some $x{\,\notin\,}W$ and some $\zeta{\,\ne\,}1\kern-2.5pt {\rm l}$ in a compact central subgroup ${\fam=\ssfam N}$ of ${\fam=\ssfam \usuDelta}$ such that ${\fam=\ssfam \usuDelta}/{\fam=\ssfam N}$ is a Lie group. By hypothesis, the orbit $x^{\fam=\ssfam \usuDelta}$ is not contained in a line. Therefore ${\cal D}{\,=\,}\langle x^{\fam=\ssfam \usuDelta}\rangle{\,\le\,}{\cal F}_\zeta$ is a proper connected subplane.
Put ${\fam=\ssfam \usuDelta}^{\hskip-1.5pt*}{=\,}{\fam=\ssfam \usuDelta}|_{\cal D}{\,=\,}{\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$. If ${\cal D}$ is flat, then ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\le\,}4$,\: $\dim{\fam=\ssfam K}{\,=\,}14$,\: ${\fam=\ssfam \usuDelta}^{\hskip-1.5pt*}{\,\approx\,}{\fam=\Bbbfam R}^4$, and ${\fam=\ssfam K}$ is a maximal compact subgroup of ${\fam=\ssfam \usuDelta}$. Hence ${\fam=\ssfam K}$ is connected by the Mal'cev-Iwasawa theorem \cite{cp} 93.10, and ${\fam=\ssfam K}{\,\cong\,}{\rm G}_2$ would be a Lie group. If $\dim{\cal D}{\,=\,}4$, then ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\le\,}8$ and $\dim{\fam=\ssfam K}{\,\le\,}8$ by Stiffness. Consequently ${\cal D}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$,\: ${\fam=\ssfam K}$ is compact and $\dim{\fam=\ssfam K}{\,\le\,}7$ by Stiffness. According to Lemma 6.0$'$ below, ${\fam=\ssfam \usuDelta}^{\hskip-1.5pt*}$ is a Lie group. We may assume, therefore, that ${\fam=\ssfam N}{\,\le\,}{\fam=\ssfam K}$. Choose elements $w,L$ in ${\cal D}$ with $w{\,\in\,}uv\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{u,v\}$ and $L{\,\in\,}{\frak L}_v\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{uv\}$, let $z{\,\in\,}L\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt{\cal D}$, put ${\fam=\ssfam \usuGamma}{\,=\,}{\fam=\ssfam \usuDelta}_{L,w}$ and ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuGamma}_{\hskip-1.5pt z})^1$. Then ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuGamma}{\,\le\,}8$,\: $\dim{\fam=\ssfam \usuGamma}{\,\ge\,}10$ and ${\fam=\ssfam K}{\,\le\,}{\fam=\ssfam \usuGamma}$. From \cite{cp} 53.2 it follows that ${\fam=\ssfam \usuGamma}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuLambda}{\,=\,}\dim z^{\fam=\ssfam \usuGamma}{\,<\,}8$ and $\dim{\fam=\ssfam \usuLambda}{\,\ge\,}3$. Moreover, ${\fam=\ssfam \usuLambda}$ is a Lie group, since ${\fam=\ssfam \usuLambda}\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam K}{\,\le\,}{\fam=\ssfam K}_z{\,=\,}1\kern-2.5pt {\rm l}$. If $\dim{\fam=\ssfam K}{\,=\,}0$, then \cite{sz1} 1.10 implies that $\dim{\fam=\ssfam \usuDelta}^{\hskip-2pt*}{\,>\,}18$, and ${\cal D}$ is the classical qaternion plane. It follows that a maximal compact subgroup of ${\fam=\ssfam \usuDelta}$ is locally isomorphic to $(\Spin3{\fam=\Bbbfam R})^3$, and ${\fam=\ssfam \usuDelta}$ would be a Lie group. \\ (b) Thus $\dim{\fam=\ssfam K}{\,>\,}0$. We need to show that ${\cal F}_{\fam=\ssfam \usuLambda}$ is connected. By \cite{cp} 55.32, the group ${\fam=\ssfam K}$ does not contain a pair of commuting involutions. Hence ${\fam=\ssfam K}^1$ is a product of a commutative connected group ${\fam=\ssfam A}$ with at most one almost simple factor ${\fam=\ssfam \usuOmega}{\,\cong\,}\Spin3{\fam=\Bbbfam R}$, see the structure of compact groups as stated in \cite{cp} 93.11. Obviously, ${\fam=\ssfam A}$ is normal in ${\fam=\ssfam \usuDelta}$, and \cite{cp} 93.19 implies that ${\fam=\ssfam A}$ is in the center of ${\fam=\ssfam \usuLambda}{\hskip.5pt\cdot\hskip.5pt}{\fam=\ssfam A}$. Now ${\fam=\ssfam A}{\,\approx\,}z^{\fam=\ssfam A}{\,\subseteq\,}{\cal F}_{\fam=\ssfam \usuLambda}{\,<\,}{\cal P}$, and ${\cal F}_{\fam=\ssfam \usuLambda}$ is indeed connected except possibly if ${\fam=\ssfam K}^1{\,\cong\,}{\fam=\ssfam \usuOmega}$. \\ (c) Suppose first that ${\fam=\ssfam K}^1{\,\not\cong\,}{\fam=\ssfam \usuOmega}$. Then $z^{\fam=\ssfam N}{\,\subset\,}{\cal F}_{\fam=\ssfam \usuLambda}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$ by \cite{cp} 32.21 and 71.2, and ${\fam=\ssfam \usuLambda}{\,\cong\,}\Spin3{\fam=\Bbbfam R}$. It follows that $\dim{\fam=\ssfam \usuGamma}{\,=\,}10$,\; ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuGamma}{\,=\,}8$, and $\dim{\fam=\ssfam \usuDelta}{\,=\,}18$.
The involution $\iota{\,\in\,}{\fam=\ssfam \usuLambda}$ induces a reflection on ${\cal D}$; in fact, $\iota|_{\cal D}{\,\ne\,}1\kern-2.5pt {\rm l}$ because $z^\iota{\,=\,}z{\,\notin\,}{\cal D}$. Hence ${\fam=\ssfam \usuLambda}$ acts faithfully on ${\cal D}$, and
$\iota|_{\cal D}$ cannot be planar by the stiffness result \cite{sz1} 1.5(4), see also \cite{cp} 83.11. The
axis of $\iota|_{\cal D}$ is $S{\,=\,}uv\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal D}$\; (since $w^{\fam=\ssfam \usuLambda}{\,=\,}w$), the center is some point $a$. As ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuGamma}{\,=\,}8$, we have $\dim w^{\fam=\ssfam \usuDelta}{\,=\,}4$ for each choice of $w$, and $w^{\fam=\ssfam \usuDelta}{\,=\,}S\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{u,v\}$ is a manifold by \cite{HK} Cor.\,5.5. Therefore $S{\,\approx\,}{\fam=\Bbbfam S}_4$, and $uv{\,\approx\,}{\fam=\Bbbfam S}_8$. \\ (d) Let ${\fam=\ssfam \usuPi}{\,=\,}({\fam=\ssfam \usuDelta}_z)^1{\,=\,}({\fam=\ssfam \usuDelta}_{L,z})^1$. Then ${\fam=\ssfam \usuPi}_w{\,=\,}{\fam=\ssfam \usuLambda}$,\: $\dim{\fam=\ssfam \usuPi}{\,=\,}7$, and ${\fam=\ssfam \usuPi}$ is transitive on $S\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{u,v\}$. Consequently ${\fam=\ssfam \usuPi}/{\fam=\ssfam \usuLambda}{\,\approx\,}{\fam=\Bbbfam H}^{\times}$ and ${\fam=\ssfam \usuPi}'\cong\Spin4{\fam=\Bbbfam R}$,\, (use the exact homotopy sequence \cite{cp} 96.12 together with 94.36). One of the central involutions of ${\fam=\ssfam \usuPi}'$ induces on ${\cal D}$ a reflection with axis $L\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal D}$. Denote this involution by $\rho$. We will show that $\rho$ is a reflection of ${\cal P}$ with axis $L$ and center~$u$. If not, then $L\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal D}{\,\subset\,}{\cal F}_{\hskip-2pt\rho}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$, and $L\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal F}_{\hskip-2pt\rho}$ is a manifold (${\approx\;}{\fam=\Bbbfam S}_4$) by \cite{cp} 92.16. In fact, $L\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal D}{\,=\,}L\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal F}_{\hskip-2pt\rho}$, see \cite{cp} 51.20, \cite{du} XVI,\,6.2(2), but $\rho$ fixes $z{\,\notin\,}{\cal D}$ and even each point of $z^{\fam=\ssfam N}$. As $\dim L^{\fam=\ssfam \usuDelta}{\,=\,}4$, we conclude from 2.9 that $\rho^{\fam=\ssfam \usuDelta}\rho{\,\cong\,}{\fam=\Bbbfam R}^4$ consists of translations with center $u$. Interchanging the r\^oles of $u$ and $v$, we find that also the translation group ${\fam=\ssfam T}_{\hskip-1pt[v]}$ is isomorphic to ${\fam=\Bbbfam R}^4$. Therefore the translation group ${\fam=\ssfam T}$ induces a transitive group of translations on ${\cal D}$. Its complement contains ${\fam=\ssfam \usuPi}$. Hence $\dim{\fam=\ssfam \usuDelta}^{\hskip-2pt*}{\,\ge\,}8{+}7$ and $\dim{\fam=\ssfam K}{\,\le\,}3$. \\ (e) If $\dim{\fam=\ssfam K}{\,=\,}1$, then ${\cal D}$ is a near-field plane (cf. \cite{sz1} 1.10), its automorphism group is described in \cite{cp} 82.24. We conclude that ${\fam=\ssfam \usuDelta}_a$ has a unique normal subgroup ${\fam=\ssfam \usuPhi}{\,\cong\,}\Spin4{\fam=\Bbbfam R}$. It follows that ${\fam=\ssfam \usuPi}'{\,=\,}{\fam=\ssfam \usuPhi}$, and $z^{\fam=\ssfam \usuPhi}{\,=\,}z$. Analogously, there is a point $y{\,\in\,}au\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt{\cal D}$ such that $y^{\fam=\ssfam \usuPhi}{\,=\,}y$, but the reflection in ${\fam=\ssfam \usuPi}'_{[L]}$ does not fix $y$. This contradiction shows that $\dim{\fam=\ssfam K}{\,\ge\,}2$. \\ (f) Note that each involution of ${\fam=\ssfam \usuPi}'$ is central. Therefore ${\fam=\ssfam \usuPi}'{\,\le\,}{\fam=\ssfam \usuDelta}_a$ and the radical ${\fam=\ssfam \usuGamma}{\,=\,}\sqrt{{\fam=\ssfam \usuDelta}_a}$ is $4$-dimensional. If $\dim{\fam=\ssfam K}{\,=\,}2$, then complete reducibility shows ${\fam=\ssfam \usuPi}'{\,\le\,}\Cs{}{\fam=\ssfam \usuGamma}/{\fam=\ssfam N}$ and hence ${\fam=\ssfam \usuPi}'{\,\le\,}\Cs{}{\fam=\ssfam \usuGamma}$. The same is true in the case $\dim{\fam=\ssfam K}{\,=\,}3$ because ${\fam=\ssfam K}/{\fam=\ssfam N}$ is a torus. Thus ${\fam=\ssfam \usuPi}'$ is the unique Levi complement of ${\fam=\ssfam \usuGamma}$, but this is impossible by the argument in step (e). \\ (g) Finally let ${\fam=\ssfam K}^1{\,\cong\,}{\fam=\ssfam \usuOmega}$. Again $\dim{\fam=\ssfam \usuDelta}^{\hskip-2pt*}{\,\ge\,}15$, and \cite{sz1} 5.1-4 implies that ${\fam=\ssfam \usuDelta}^{\hskip-2pt*}$ contains a transitive group of translations, their center will be chosen as the point $v$. In particular, $L\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal D}{\,\approx\,}{\fam=\Bbbfam S}_4$. If $\dim{\fam=\ssfam \usuLambda}{\,>\,}3$, if ${\fam=\ssfam \usuLambda}$ is solvable, or if ${\fam=\ssfam \usuLambda}{\,\circeq\,}\SL2{\fam=\Bbbfam R}$, then ${\fam=\ssfam P}{\,=\,}(\Cs{\fam=\ssfam \usuLambda}{\fam=\ssfam \usuOmega})^1{\,\trianglelefteq\,}{\fam=\ssfam \usuLambda}$ has positive dimension, since ${\fam=\ssfam \usuLambda}/\Cs{\fam=\ssfam \usuLambda}{\fam=\ssfam \usuOmega}{\,\le\,}\mathop{{\rm Aut}}{\fam=\ssfam \usuOmega}{\,\cong\,}\SO3{\fam=\Bbbfam R}$. It follows that $z^{\fam=\ssfam P}{\subset\,}{\cal F}_{\fam=\ssfam P}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$, and ${\fam=\ssfam P}{\,\cong\,}{\fam=\Bbbfam T}$ or ${\fam=\ssfam P}{\,\cong\,}\Spin3{\fam=\Bbbfam R}$. Hence there exists a planar involution $\rho$ in the center of ${\fam=\ssfam \usuLambda}{\fam=\ssfam \usuOmega}$. In the case ${\cal D}\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal F}_\rho{\,=\,}{\cal C}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal D},{\cal F}_\rho$, the group ${\fam=\ssfam \usuOmega}$ would act trivially on ${\cal C}$, which contradicts Stiffness. Consequently $\rho$ induces on ${\cal D}$ a reflection with axis $uv\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal D}{\,\subset\,}{\cal F}_\rho$, and then $S{\,=\,}L\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal F}_\rho{\,\approx\,}{\fam=\Bbbfam S}_4$, again by \cite{cp} 92.16 and 51.20. According to $(\dagger)$, the action of ${\fam=\ssfam \usuOmega}$ on $S$ is equivalent to the action of ${\fam=\Bbbfam H}'$ on ${\fam=\Bbbfam H}$ by multiplication, $S/{\fam=\ssfam \usuOmega}{\,=\,}I$ is an interval, and ${\fam=\ssfam N}$ acts trivially on $I$. Therefore $z^{\fam=\ssfam \usuOmega}{\,=\,}z^{{\fam=\ssfam \usuOmega}{\fam=\ssfam N}}$, but $({\fam=\ssfam \usuOmega}{\fam=\ssfam N})_z{\,=\,}1\kern-2.5pt {\rm l}$, a contradiction. \\ (h) Only the cases ${\fam=\ssfam K}'{\,\cong\,}{\fam=\ssfam \usuOmega}{\,\circeq\,}{\fam=\ssfam \usuLambda}$ remain. If ${\fam=\ssfam \usuLambda}$ is simply connected, then ${\fam=\ssfam \usuOmega}$ centralizes the involution $\rho{\,\in\,}{\fam=\ssfam \usuLambda}$. Again
$\rho|_{\cal D}$ is a reflection, and one may reason as in step (g). Similarly, $\SO3{\fam=\Bbbfam R}{\,\cong\,}{\fam=\ssfam \usuLambda}{\,\le\,}\Cs{}{\fam=\ssfam \usuOmega}$ is impossible. Therefore ${\fam=\ssfam \usuLambda}{\fam=\ssfam \usuOmega}{\,\cong\,}\SO4{\fam=\Bbbfam R}$ contains only one involution and a different argument is needed. We will show that ${\fam=\ssfam \usuLambda}{\fam=\ssfam K}$ is a maximal compact subgroup of the connected group ${\fam=\ssfam \usuXi}{\,=\,}{\fam=\ssfam \usuDelta}_{w,L}$. In fact, ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuXi}{\,\le\,}8$,\; $\dim{\fam=\ssfam \usuXi}{\,\ge\,}10$,\;${\fam=\ssfam \usuXi}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuLambda}{\,=\,}\dim z^{\fam=\ssfam \usuXi}{\,<\,}8$,\;
$\dim{\fam=\ssfam \usuXi}{\,=\,}10$,\; $\dim w^{\fam=\ssfam \usuDelta}{\,=\,}\dim L^{{\fam=\ssfam \usuDelta}_w}{\,=\,}4$ for each choice of $w$ and $L$, both orbits are simply connected, hence ${\fam=\ssfam \usuXi}$ is connected and so is ${\fam=\ssfam \usuXi}|_{\cal D}$. Recall from step (g) that ${\fam=\ssfam \usuXi}|_{\cal D}$ contains a transitive group ${\fam=\ssfam \usuTheta}{\,\approx\,}{\fam=\Bbbfam R}^4$ of translations\; (note that each compact subgroup of ${\fam=\ssfam \usuTheta}$ is trivial by \cite{cp} 55.28). We have
${\fam=\ssfam \usuLambda}{\,\cong\,}{\fam=\ssfam \usuLambda}|_{\cal D}{\,<\,}{\fam=\ssfam \usuXi}|_{\cal D}$ and ${\fam=\ssfam \usuXi}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,=\,}7$.
Consequently ${\fam=\ssfam \usuXi}|_{\cal D}{\,\cong\,}{\fam=\ssfam \usuLambda}{\ltimes}{\fam=\ssfam \usuTheta}$, and ${\fam=\ssfam \usuLambda}{\fam=\ssfam K}$ is a compact subgroup of ${\fam=\ssfam \usuXi}$ of maximal dimension. By the Mal'cev-Iwasawa theorem \cite{cp} 93.10 the group ${\fam=\ssfam \usuLambda}{\fam=\ssfam K}$ is connected and coincides with the Lie group ${\fam=\ssfam \usuLambda}{\fam=\ssfam \usuOmega}$. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\bf 6.0$'$ Lemma.} {\it Suppose that ${\fam=\ssfam \usuDelta}$ is a group of automorphisms of an $8${\rm -dimensional} plane~${\cal P}$. If ${\fam=\ssfam \usuDelta}$ fixes $2$ distinct points $u,v$ and exactly one line $uv$, and if $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}8$, then ${\fam=\ssfam \usuDelta}$ is a Lie group\/}. \par
{\tt Proof.} If ${\fam=\ssfam \usuDelta}/{\fam=\ssfam N}$ is a Lie group, but the compact central subgroup ${\fam=\ssfam N}$ is not, then the arguments of 4.0 show that for some $x{\,\notin\,}uv$ the orbit $x^{\fam=\ssfam \usuDelta}$ generates
a proper connected sublane ${\cal D}$. Put ${\fam=\ssfam \usuDelta}|_{\cal D}{\,=\,}{\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$ and apply the stiffness results \cite{sz1} 1.5 to ${\fam=\ssfam K}$. If ${\cal D}$ is flat, then ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\le\,}4$ and $\dim{\fam=\ssfam K}{\,\le\,}3$ \ (see also \cite{cp} 83.12). Hence ${\cal D}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$ and
$\dim{\fam=\ssfam K}{\,\le\,}1$. As ${\fam=\ssfam \usuDelta}|_{\cal D}$ is a Lie group by \cite{cp} 71.2, we may assume ${\fam=\ssfam N}{\,\le\,}{\fam=\ssfam K}$. In the case $\dim{\fam=\ssfam K}{\,=\,}0$ it follows fom \cite{cp} 73.22 that ${\fam=\ssfam \usuDelta}$ does not have two fixed points in ${\cal D}$. Therefore $\dim{\fam=\ssfam K}{\,=\,}1$. Choose a line $L$ of ${\cal D}$ in ${\frak L}_v\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{uv\}$ and points $w{\,\in\,}uv\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal D}$ and $z{\,\in\,}L\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt{\cal D}$. Put ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuDelta}_{z,w})^1$. Then $\dim z^{{\fam=\ssfam \usuDelta}_L}{\,<\,}4$ by \cite{cp} 53.2, and $\dim{\fam=\ssfam \usuLambda}{\,>\,}0$. Let ${\fam=\ssfam \usuXi}{\,=\,}{\fam=\ssfam K}^1{\fam=\ssfam N}$. Then \cite{cp} 93.19 implies ${\fam=\ssfam \usuLambda}{\,\le\,}\Cs{}{\fam=\ssfam \usuXi}$. Therefore $z^{\fam=\ssfam \usuXi}{\,\subseteq\,}{\cal F}_{\fam=\ssfam \usuLambda}{\,<\,}{\cal P}$,\: ${\cal F}_{\fam=\ssfam \usuLambda}$ is connected, but then ${\fam=\ssfam \usuXi}$ would be a Lie group by \cite{cp} 32.21 and 71.2. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\bf 6.1 Lie.} {\it If ${\fam=\ssfam \usuDelta}$ is semi-simple of dimension $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}14$, then ${\fam=\ssfam \usuDelta}$ is a Lie group\/}. \par
This can be {\tt proved} in the same way as 5.1. The sharper bound is obtained because ${\rm G}_2$ is a Lie group, and because ${\cal D}{\,=\,}{\cal D}^{\fam=\ssfam \usuDelta}{\,<\,}{\cal P}$ implies
$\dim{\fam=\ssfam \usuDelta}|_{\cal D}{\,\le\,}10$ instead of $11$, see \cite{cp} 7.3. {\tt Remark.} The result is also true if ${\fam=\ssfam \usuDelta}$ fixes more than $2$ collinear points but only one line. \par
{\bf 6.2 Semi-simple groups.} {\it If $|{\cal F}_{\fam=\ssfam \usuDelta}|{\,=\,}3$ and ${\fam=\ssfam \usuDelta}$ is semi-simple, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}20$ or, conceivably, ${\fam=\ssfam \usuDelta}$ is a product of $7$ factors each of which is isomorphic to the simply connected covering group ${\fam=\ssfam \usuOmega}$ of $\SL2{\fam=\Bbbfam R}$. If ${\fam=\ssfam \usuDelta}$ is almost simple, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}16$\/}. \par
{\tt Remark.} Results and proof are the same as in the case that ${\cal F}_{\fam=\ssfam \usuDelta}$ is a flag, see 5.2. \par
{\bf 6.3 Normal torus.} {\it Suppose that ${\fam=\ssfam \usuDelta}$ has a one-dimensional compact connected normal subgroup ${\fam=\ssfam \usuTheta}$. If $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}18$, then there exists a ${\fam=\ssfam \usuDelta}$-invariant near-field plane ${\cal H}{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$$;$ either $\dim{\fam=\ssfam \usuDelta}{\,=\,}18$, or ${\cal H}$ is classical and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}20$\/}. \par
{\tt Proof.} As ${\fam=\ssfam \usuDelta}$ is a Lie group, ${\fam=\ssfam \usuTheta}{\,\cong\,}{\fam=\Bbbfam T}$ and the involution $\iota{\,\in\,}{\fam=\ssfam \usuTheta}$ is in the center of ${\fam=\ssfam \usuDelta}$. Hence $\iota$ is planar.
Put ${\cal H}{\,=\,}{\cal F}_\iota$ and ${\fam=\ssfam \usuDelta}|_{\cal H}{\,=\,}{\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$. If
${\fam=\ssfam \usuTheta}|_{\cal H}{\,\ne\,}1\kern-2.5pt {\rm l}$, Stiffness and \cite{sz1} 5.3 imply $\dim{\fam=\ssfam \usuDelta}{\,\le\,}9{+}3$. Hence ${\fam=\ssfam \usuTheta}{\,\le\,}{\fam=\ssfam K}$,\: ${\fam=\ssfam K}^1{\,\not\hskip1pt\cong\,}\Spin3{\fam=\Bbbfam R}$,\: $\dim{\fam=\ssfam K}{\,=\,}1$, and ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\ge\,}17$. All planes ${\cal H}$ with a group of dimension at least $17$ are described in \cite{sz1} 1.10. Only in the case of the near-field planes ${\fam=\ssfam \usuDelta}$ fixes two distinct points. Either ${\cal H}$ is classical, or ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,=\,}17$ and $\dim{\fam=\ssfam \usuDelta}{\,=\,}18$. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\tt Remark.} If $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}17$, then ${\fam=\ssfam \usuDelta}$ induces on ${\cal H}$ a group of dimension ${\ge\,}16$. Such planes are translation planes, they have been determined explicitly by H\"ahl, cf. \cite{sz1} 3.3. \par
Without assumption on ${\cal F}_{\fam=\ssfam \usuDelta}$, the following has been proved in \cite{sz10}: \par
{\bf 6.4 Theorem.} {\it If ${\fam=\ssfam \usuDelta}$ has a normal vector subgroup and if $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}33$, then ${\fam=\ssfam \usuDelta}$ fixes some element, say a line $W$, and ${\fam=\ssfam \usuDelta}$ has a minimal normal subgroup ${\fam=\ssfam \usuTheta}{\,\cong\,}{\fam=\Bbbfam R}^t$ consisting of axial collineations with common axis $W$. Either ${\fam=\ssfam \usuTheta}{\,\le\,}{\fam=\ssfam \usuDelta}_{[a,W]}$ is a group of homologies and $t=1$, or ${\fam=\ssfam \usuTheta}$ is contained in the translation group ${\fam=\ssfam T} = {\fam=\ssfam \usuDelta}_{[W,W]}$\/}. \par
The next result has been proved in \cite{HS} Theorem~1 under the assumption $\dim{\fam=\ssfam \usuDelta}{\,>\,}33$\,: \par
{\bf 6.5 Theorem.} {\it If $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}34$, then the group ${\fam=\ssfam T}$ of translations in ${\fam=\ssfam \usuDelta}$ satisfies $\dim{\fam=\ssfam T}{\,\ge\,}15$. Either ${\fam=\ssfam \usuDelta}$ has a subgroup ${\fam=\ssfam \usuUpsilon}{\,\cong\,}\Spin7{\fam=\Bbbfam R}$ and $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}36$, or ${\fam=\ssfam T}$ is transitive, $\dim{\fam=\ssfam \usuDelta}{\,=\,}34$, and there exists a maximal semi-simple subgroup ${\fam=\ssfam \usuUpsilon}{\,\cong\,}\Spin6{\fam=\Bbbfam R}$ of ${\fam=\ssfam \usuDelta}$\/}. \par
{\bf 6.6 Normal vector subgroup.} {\it If ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\{u,v,W\}$ and if $\dim{\fam=\ssfam \usuDelta}{\,=\,}33$, then the translation group ${\fam=\ssfam T}$ is transitive\/}. \par
{\tt Proof.} In steps (a--r), the weaker assumption $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}33$ suffices; the exact value
of $\dim{\fam=\ssfam \usuDelta}$ is needed only in steps (s--v). \\ (a) From 6.\,2 and 3 it follows that 6.4 applies. Suppose that ${\fam=\ssfam \usuDelta}_{[a]}{\,\ne\,}1\kern-2.5pt {\rm l}$ for some $a{\,\notin\,}W$. Then ${\fam=\ssfam \usuDelta}{\,=\,}{\fam=\ssfam \usuDelta}_{[a]}{\ltimes}{\fam=\ssfam T}$ by \cite{cp} 61.20, and $a^{\fam=\ssfam \usuDelta}{\,=\,}a^{\fam=\ssfam T}$ is not contained in a line. Hence ${\fam=\ssfam T}$ is a vector group ${\fam=\Bbbfam R}^k$, and each subgroup ${\fam=\ssfam T}_{\hskip-1pt[z]}$ with $z{\,\in\,}W$ is connected by \cite{cp} 61.9. Let $1\kern-2.5pt {\rm l}{\,\ne\,}\tau{\,\in\,}{\fam=\ssfam T}_{\hskip-1pt[z]}$ with $z{\,\ne\,}u,v$, and put $\nabla{\,=\,}({\fam=\ssfam \usuDelta}_a)^1$,\: ${\fam=\ssfam \usuLambda}{\,=\,}(\nabla_{\hskip-2.5pt a^\tau}\hskip-2pt)^1$. Then $$33{\,\le\,}\dim{\fam=\ssfam \usuDelta}{\,=\,}{\fam=\ssfam \usuDelta}{:}\nabla{\,+\,}\nabla{:\hskip1pt}{\fam=\ssfam \usuLambda}{\,+\,}\dim{\fam=\ssfam \usuLambda}{\,\le\,} 2k{\,+\,}\dim{\fam=\ssfam \usuLambda}{\,\le\,}2k{\,+\,}14\quad{\rm and}\quad k{\,\ge\,}10\,.$$ By the stiffness result 2.6(e), either ${\fam=\ssfam \usuLambda}{\,\cong\,}{\rm G}_2$ or $\dim{\fam=\ssfam \usuLambda}{\,\le\,}8$. In the first case ${\fam=\ssfam T}{\kern 2pt {\scriptstyle \cap}\kern 2pt}\Cs{}{\fam=\ssfam \usuLambda}$ acts effectively on the flat plane ${\cal F}_{\fam=\ssfam \usuLambda}$ and $\dim({\fam=\ssfam T}{\kern 2pt {\scriptstyle \cap}\kern 2pt}\Cs{}{\fam=\ssfam \usuLambda}){\,\le\,}2$. As ${\fam=\ssfam \usuLambda}$ fixes $\tau$ and each non-trivial representation of ${\fam=\ssfam \usuLambda}$ on ${\fam=\ssfam T}$ has dimension $7$ or $14$, it follows that $k{\,\ge\,}15$. In the second case, $k{\,\ge\,}13$ and $\dim{\fam=\ssfam T}_{\hskip-1pt[z]}{\,\ge\,}5$ for each $z{\,\in\,}W$. \\ (b) {\it The elements of a minimal normal subgroup ${\fam=\ssfam \usuTheta}$ of ${\fam=\ssfam \usuDelta}$, ${\fam=\Bbbfam R}^t{\,\cong\,}{\fam=\ssfam \usuTheta}{\,\le\,}{\fam=\ssfam T}$, have center $u$ or~$v$, say ${\fam=\ssfam \usuTheta}{\,\le\,}{\fam=\ssfam T}_{\hskip-1pt[v]}$, or ${\fam=\ssfam T}$ is transitive and ${\cal P}$ is classical\/}. In fact, $t{\,\le\,}8$\: (or $L{\,\in\,}{\frak L}_v\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{W\}$ implies $1\kern-2.5pt {\rm l}{\,\ne\,}{\fam=\ssfam \usuTheta}_L{\,=\,} {\fam=\ssfam \usuTheta}\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam T}_{\hskip-1.5pt L}{\,=\,}{\fam=\ssfam \usuTheta}\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam T}_{\hskip-1pt[v]}{\,\triangleleft\,}{\fam=\ssfam \usuDelta}$, and ${\fam=\ssfam \usuTheta}$ would not be minimal). If some $\tau{\,\in\,}{\fam=\ssfam \usuTheta}$ has a center $z{\,\ne\,}u,v$, then $17{\,\le\,}\dim\nabla{\,\le\,}t{\,+\,}\dim{\fam=\ssfam \usuLambda}$,\: $\dim{\fam=\ssfam \usuLambda}{\,>\,}8$, ${\fam=\ssfam \usuLambda}{\,\cong\,}{\rm G}_2$,\: $t{\,>\,}2$,\: ${\fam=\ssfam \usuLambda}$ acts non-trivially on ${\fam=\ssfam \usuTheta}$, and $t{\,>\,}7$. By minimality, ${\fam=\ssfam \usuDelta}$ induces an irreducible group $\tilde{\fam=\ssfam \usuDelta}$ on ${\fam=\ssfam \usuTheta}{\,\cong\,}{\fam=\Bbbfam R}^8$, and $\tilde{\fam=\ssfam \usuDelta}'$ is semi-simple and properly contains ${\rm G}_2$. The list \cite{cp} 95.10 shows that $\tilde{\fam=\ssfam \usuDelta}'$ has a subgroup $\Spin7{\fam=\Bbbfam R}$. This simply connected group can be lifted to a subgroup ${\fam=\ssfam \usuUpsilon}$ of ${\fam=\ssfam \usuDelta}$, see \cite{cp} 94.27. The central involution $\sigma{\,\in\,}{\fam=\ssfam \usuUpsilon}$ inverts the elements of ${\fam=\ssfam \usuTheta}$ and hence fixes $z$. Therefore $\sigma$ is a reflection with axis $W$, the center of $\sigma$ will be denoted by $a$. From 2.9 and step (a) it follows that $\sigma^{\fam=\ssfam \usuDelta}\sigma{\,=\,}{\fam=\ssfam T}{\,\cong\,}{\fam=\Bbbfam R}^k$ with $k{\,\ge\,}14$ and $a^{\fam=\ssfam \usuDelta}{\,=\,}a^{\fam=\ssfam T}$. Moreover, ${\fam=\ssfam \usuUpsilon}$ acts faithfully on ${\fam=\ssfam T}_{\hskip-1pt[u]}$ and ${\fam=\ssfam T}_{\hskip-1pt[v]}$, but then ${\fam=\ssfam T}$ is transitive, and ${\cal P}{\,\cong\,}{\cal O}$ by \cite{cp} 81.17. \\ (c) {\it If ${\fam=\ssfam \usuDelta}$ has a subgroup ${\fam=\ssfam \usuUpsilon}{\,\cong\,}\Spin7{\fam=\Bbbfam R}$, then $\dim{\fam=\ssfam T}{\,\ge\,}15$\/}. By the last part of 2.10, the central involution $\sigma{\,\in\,}{\fam=\ssfam \usuUpsilon}$ is a reflection. If $\sigma{\,\in\,}{\fam=\ssfam \usuDelta}_{[W]}$, steps (a,b) show transitivity of ${\fam=\ssfam T}$. If $\sigma$ has center $u$ or $v$, say $\sigma{\,\in\,}{\fam=\ssfam \usuDelta}_{[v]}$, the dual of 2.9 implies that ${\fam=\ssfam \usuUpsilon}$ acts effectively on ${\fam=\ssfam T}_{\hskip-1pt[v]}$, and then ${\fam=\ssfam T}_{\hskip-1pt[v]}{\,\cong\,}{\fam=\Bbbfam R}^8$ by \cite{cp} 95.10. It follows that ${\fam=\ssfam \usuDelta}$ induces on ${\frak L}_u$ a trans\-itive group ${\fam=\ssfam \usuDelta}/{\fam=\ssfam \usuDelta}_{[u]}$. Either ${\fam=\ssfam \usuUpsilon}$ is contained in a subgroup of ${\fam=\ssfam \usuDelta}$ of type ${\rm D}_4$,\: $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}36$, and the claim follows from 6.5, or ${\fam=\ssfam \usuDelta}/{\fam=\ssfam \usuDelta}_{[u]}$ is contained in $e^{{\fam=\Bbbfam R}}{\cdot\hskip1pt}\Spin7{\fam=\Bbbfam R}{\hskip1pt\ltimes}{\fam=\Bbbfam R}^8$ and has dimension ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuDelta}_{[u]}{\,\le\,}30$. In the latter case $\dim{\fam=\ssfam \usuDelta}_{[u]}{\,\ge\,}3$. According to \cite{cp} 61.20, the connected component ${\fam=\ssfam \usuXi}$ of ${\fam=\ssfam \usuDelta}_{[u]}$ is isomorphic to ${\fam=\ssfam \usuDelta}_{[u,xv]}{\ltimes}{\fam=\ssfam T}_{\hskip-1pt[u]}$ and $\dim{\fam=\ssfam T}_{\hskip-1pt[u]}{\,>\,}0$. Note that the homology group ${\fam=\ssfam \usuDelta}_{[u,xv]}$ acts freely on ${\fam=\ssfam T}_{\hskip-1pt[u]}$. By 2.15(b), the fixed points of ${\fam=\ssfam \usuUpsilon}$ on the axis of $\sigma$ form a circle~$C$. Suppose that ${\fam=\ssfam \usuXi}{\,\le\,}\Cs{\hskip-2pt}{\fam=\ssfam \usuUpsilon}$. Then ${\fam=\ssfam \usuXi}$ acts effectively on $C$, and Brouwer's theorem \cite{cp} 96.30 would imply $\dim{\fam=\ssfam \usuDelta}_{[u]}{\,\le\,}2$. Consequently ${\fam=\ssfam \usuUpsilon}$ induces on the Lie algebra $\mathop{\strut{\frak l}}{\fam=\ssfam \usuXi}$ a group $\SO7{\fam=\Bbbfam R}$ and $\dim{\fam=\ssfam \usuXi}{\,\ge\,}7$. Hence $\dim{\fam=\ssfam T}_{\hskip-1pt[u]}{\,\ge\,}\frac{1}{2}\dim{\fam=\ssfam \usuXi}{\,\ge\,}4$, the action of ${\fam=\ssfam \usuUpsilon}$ on ${\fam=\ssfam T}_{\hskip-1pt[u]}$ is not trivial by 2.15, and then $\dim{\fam=\ssfam T}_{\hskip-1pt[u]}{\,\ge\,}7$. In particular: \\ (c$'$) {\it If the central involution of ${\fam=\ssfam \usuUpsilon}$ is a reflection with center $v$, then ${\fam=\ssfam T}_{\hskip-1pt[v]}$ is transitive\/}. \\ {\bf Change of notation.} From now on ${\fam=\ssfam \usuTheta}{\,\cong\,}{\fam=\Bbbfam R}^t$ will denote a minimal $\nabla$-invariant subgroup of ${\fam=\ssfam T}_{\hskip-1pt[v]}$, where $\nabla{\,=\,}({\fam=\ssfam \usuDelta}_a)^1$ for a fixed $a{\,\notin\,}W$; normality of ${\fam=\ssfam \usuTheta}$ in ${\fam=\ssfam \usuDelta}$ will not be needed. Throughout, let $w{\,\in\,}S{\,:=\,}W\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{u,v\}$ and ${\fam=\ssfam \usuOmega}{\,=(\nabla_{\hskip-1.5pt w})^1\,}$. \\ (d) {\it If $t{\,=\,}1$ and if ${\fam=\ssfam \usuDelta}$ has a subgroup ${\fam=\ssfam \usuGamma}{\,\cong\,}{\rm G}_2$, then ${\fam=\ssfam T}$ is transitive\/}. \\ Consider the radical ${\fam=\ssfam P}{\,=\,}\sqrt{\fam=\ssfam \usuDelta}$ and note that \cite{HS} Lemma~4 combined with ${\fam=\ssfam \usuTheta}{\,\cong\,}{\fam=\Bbbfam R}$ yields $\dim{\fam=\ssfam P}{\,\le\,}19$. Suppose that ${\fam=\ssfam \usuGamma}$ is a Levi complement of ${\fam=\ssfam P}$ in ${\fam=\ssfam \usuDelta}$. Then ${\fam=\ssfam \usuDelta}{\,=\,}{\fam=\ssfam P}{\fam=\ssfam \usuGamma}$,\: $\dim{\fam=\ssfam P}{\,=\,}19$, and ${\fam=\ssfam P}$ is transitive on the affine point set $P\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt W$. Thus, up to conjugacy, $a^{\fam=\ssfam \usuGamma}{\,=\,}a$ and ${\fam=\ssfam \usuGamma}{\,\le\,}\nabla$. Stiffness and 2.15(a) imply that ${\cal F}_{\fam=\ssfam \usuGamma}$ is a flat subplane. Consequently, $\dim\Cs\nabla{\fam=\ssfam \usuGamma}{\,\le\,}2{\,<\,}\break\dim(\nabla\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam P}){\,=\,}3$. Therefore ${\fam=\ssfam \usuGamma}$ acts non-trivially on $\nabla\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam P}$, and $\dim(\nabla\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam P}){\,\ge\,}7$, a contradiction. Hence ${\fam=\ssfam \usuGamma}$ is properly contained in a maximal semi-simple subgroup ${\fam=\ssfam \usuPsi}$ of ${\fam=\ssfam \usuDelta}$. By Stiffness, $\Cs{}{\fam=\ssfam \usuGamma}$ acts almost effectively on ${\cal F}_{\fam=\ssfam \usuGamma}$, the induced group is solvable by \cite{cp} 33.8. Because of 2.16, it follows that ${\fam=\ssfam \usuPsi}$ is an almost simple orthogonal group containing ${\fam=\ssfam \usuUpsilon}{\,\cong\,}\Spin7{\fam=\Bbbfam R}$. Step (c) implies that $\dim{\fam=\ssfam T}{\,\ge\,}15$ and $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}36$. Hence $\dim\nabla{\,\ge\,}21$ and $\dim\nabla_{\hskip-2pt c,w}{\,\ge\,}12$ for $a{\,\ne\,}c{\,\in\,}a^{\fam=\ssfam \usuTheta}$. By Stiffness $(\nabla_{\hskip-2pt c,w})^1{\,\cong\,}{\rm G}_2$, and we may assume that ${\fam=\ssfam \usuGamma}{\,\le\,}\nabla$. If ${\fam=\ssfam \usuUpsilon}{\,\le\,}\nabla$, the central involution $\sigma{\,\in\,}{\fam=\ssfam \usuUpsilon}$ is a reflection with axis $av$, since ${\fam=\ssfam \usuTheta}{\,\le\,}\Cs{}{\fam=\ssfam \usuUpsilon}$, and~(c$'$) shows that ${\fam=\ssfam T}_{\hskip-1pt[u]}{\,\cong\,}{\fam=\Bbbfam R}^8$. The group ${\fam=\ssfam T}_{\hskip-1pt[v]}$ is a product of ${\fam=\ssfam \usuTheta}{\,=\,}{\fam=\ssfam \usuTheta}^{\fam=\ssfam \usuUpsilon}$ and a $7$-dimensional factor; hence ${\fam=\ssfam T}$ is transitive in this case. From $a^{\fam=\ssfam \usuUpsilon}{\,\ne\,}a$ we will derive a contradiction. First let ${\fam=\ssfam \usuUpsilon}_{\hskip-1pt[v]}{\,=\,}\langle \sigma\rangle$ as in step (c$'$). Then ${\fam=\ssfam T}_{\hskip-1pt[v]}$ is transitive and, up to conjugacy, the axis of $\sigma$ contains the point $a$. Now ${\fam=\ssfam \usuGamma}{\,\le\,}\nabla\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam \usuUpsilon}{\,=\,}{\fam=\ssfam \usuUpsilon}_{\hskip-2pt a}$, but $a^{\fam=\ssfam \usuUpsilon}{\,\approx\,}{\fam=\Bbbfam S}_6$ by 2.15, and ${\fam=\ssfam \usuUpsilon}_{\hskip-1.5pt a}{\,\cong\,}\Spin6{\fam=\Bbbfam R}$. The case that the central involution $\sigma$ of ${\fam=\ssfam \usuUpsilon}$ has center $u$ can be dealt with is analogously. \\ (e) {\it If $t{\,=\,}1$ and if ${\fam=\ssfam \usuDelta}$ has no subgroup ${\rm G}_2$, then ${\fam=\ssfam T}$ is transitive\/}. \\ Let $a{\,\ne\,}c{\,\in\,}a^{\fam=\ssfam \usuTheta}$. Then $\dim\nabla_{\hskip-2pt c}{\,\ge\hskip1pt}16$, and 2.6(e) implies ${\fam=\ssfam \usuLambda}{\,=\,}(\nabla_{\hskip-2pt c,w})^1{\,\cong\,}\SU3{\fam=\Bbbfam C}$. Con\-sequently, $a^{\fam=\ssfam \usuDelta}$ is open in $P$,\: $\nabla_{\hskip-2pt c}$ is transitive on $S$, and a maximal compact subgroup ${\fam=\ssfam \usuUpsilon}$ of $\nabla_{\hskip-2pt c}$ is isomorphic to $\SU4{\fam=\Bbbfam C}{\,\cong\,}\Spin6{\fam=\Bbbfam R}$\: (see 2.17(d)\,). The central involution $\sigma{\,\in\,}{\fam=\ssfam \usuUpsilon}$ is a reflection with axis $av$ and center $u$,\: $\sigma{\,\in\,}\nabla_{\hskip-1.5pt[u]}$. As $(av)^{\fam=\ssfam \usuDelta}{\,\ne\,}av$, it follows from the dual of 2.9 that $\sigma^{\fam=\ssfam \usuDelta}\sigma{\,=\,}{\fam=\ssfam T}_{\hskip-1pt[u]}$ is a vector group. The representation of ${\fam=\ssfam \usuUpsilon}$ on ${\fam=\ssfam T}_{\hskip-1pt[u]}$ is faithful. Therefore ${\fam=\ssfam T}_{\hskip-1pt[u]}{\,\cong\,}{\fam=\Bbbfam R}^8$, and ${\fam=\ssfam \usuDelta}$ induces on ${\frak L}_v\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{W\}$ a doubly transitive group ${\fam=\ssfam \usuDelta}/{\fam=\ssfam \usuDelta}_{[v]}$, so that ${\fam=\ssfam \usuXi}{\,=\,}{\fam=\ssfam \usuDelta}_{av}/{\fam=\ssfam \usuDelta}_{[v]}$ is a trans\-itive linear group and ${\fam=\ssfam \usuXi}'$ is semi-simple (cf. \cite{cp} 95.6 and 96.16). As $\nabla_{\hskip-2pt c}\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam \usuDelta}_{[v]}{\,\le\,}{\fam=\ssfam \usuDelta}_{[v,av]}{\,=\,}1\kern-2.5pt {\rm l}$, we have ${\fam=\ssfam \usuUpsilon}{\,\cong\,}\SU4{\fam=\Bbbfam C}{\,\le\,}{\fam=\ssfam \usuXi}'$. Moreover $\Cs{\fam=\ssfam \usuXi}{\fam=\ssfam \usuUpsilon}{\,\le\,}{\fam=\Bbbfam C}^{\times}$, and ${\fam=\ssfam \usuXi}'$ is almost simple. If ${\fam=\ssfam \usuUpsilon}{\,<\,}{\fam=\ssfam \usuXi}'$, then \cite{cp} 96.10 would show that ${\rm G}_2{\,<\,}\Spin7{\fam=\Bbbfam R}{\,<\,}{\fam=\ssfam \usuXi}$ or $\dim{\fam=\ssfam \usuXi}'{\,\ge\,}28$; the first possibility is excluded by the assumption in step (e), in the second case $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}36$ and theorem 6.5 would imply ${\fam=\ssfam \usuXi}'{\,\cong\,}\Spin7{\fam=\Bbbfam R}$. Hence ${\fam=\ssfam \usuXi}{\,\le\,}{\fam=\Bbbfam C}^{\times}{\cdot\,}\SU4{\fam=\Bbbfam C}$,\: $16{\,\le\,}\dim{\fam=\ssfam \usuXi}{\,\le\,}17$, and $\dim{\fam=\ssfam \usuDelta}_{[v]}{\,\ge\,}8$. Either ${\fam=\ssfam \usuDelta}_{[v]}{\,=\,}{\fam=\ssfam T}_{\hskip-1pt[v]}$ is transitive, or ${\fam=\ssfam \usuDelta}_{[v]}$ contains a homology with axis~$au$. As $\dim(au)^{\fam=\ssfam \usuDelta}{\,=\,}8$, the dual of 2.9 implies again ${\fam=\ssfam T}_{\hskip-1pt[v]}{\,\cong\,}{\fam=\Bbbfam R}^8$. \\ (f) {\it If $t{\,=\,}2$ and if $\dim{\fam=\ssfam \usuLambda}{\,<\,}8$ for each stabilizer of a quadrangle, then
${\fam=\ssfam T}$ is transitive\/}.\break {\tt Proof.} Let $a{\,\ne\,}c{\,\in\,}a^{\fam=\ssfam \usuTheta}$, and put ${\fam=\ssfam \usuLambda}{\,=\,}(\nabla_{\hskip-2pt c,w})^1$. Then ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuLambda}{\,\le\,}26$,\: $\dim{\fam=\ssfam \usuLambda}{\,=\,}7$,\: $\dim\nabla{\,=\,}17$,\: $\nabla_{\hskip-2pt c}$ is transitive on $S$ and on $au\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a,u\}$, moreover, $a^{\fam=\ssfam \usuDelta}{\,=\,}P\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt W$. Consequently, ${\fam=\ssfam \usuDelta}_{au}$ is doubly transitive on $au\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{u\}$. As mentioned in 5.9(e), all such actions are known (Tits 1955, cf. in particular \cite{Vl} Satz 1). As $\nabla_{\hskip-2pt c}$ acts faithfully on $au$ and $\dim\nabla_{\hskip-2pt c}{\,=\,}15$, it follows that $\nabla_{\hskip-2pt c}{\,\cong\,}\SL2{\fam=\Bbbfam H}$\: (see \cite{cp} 96.10 or \cite{Vl} and note that $\nabla_{\hskip-2pt c}$ is not compact). A maximal compact subgroup ${\fam=\ssfam \usuGamma}$ of $\nabla_{\hskip-2pt c}$ is isomorphic to $\U2{\fam=\Bbbfam H}{\,\cong\,}\Spin5{\fam=\Bbbfam R}$, and the central involution $\sigma$ of ${\fam=\ssfam \usuGamma}$ is a reflection with axis $av$\: (use the last part of 2.10). Therefore $\sigma^{\fam=\ssfam \usuDelta}\sigma{\,=\,}{\fam=\ssfam T}_{\hskip-1pt[u]}{\,\cong\,}{\fam=\Bbbfam R}^8$. The group ${\fam=\ssfam \usuDelta}/{\fam=\ssfam \usuDelta}_{[v]}$ induced by ${\fam=\ssfam \usuDelta}$ on the pencil ${\frak L}_v\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{W\}$ is also doubly trans\-itive, and $\nabla_{\hskip-2pt c}{\,\le\,}{\fam=\ssfam \usuUpsilon}{\,:=\,}{\fam=\ssfam \usuDelta}_{av}/{\fam=\ssfam \usuDelta}_{[v]}$. As ${\fam=\ssfam \usuTheta}{\,\le\,}{\fam=\ssfam \usuXi}{\,:=\,}{\fam=\ssfam \usuDelta}_{[v]}$, we have $15{\,\le\,}\dim{\fam=\ssfam \usuUpsilon}{\,\le\,}23$. By \cite{Vl} either $\dim{\fam=\ssfam \usuUpsilon}'{\,\le\,}18$ or ${\fam=\ssfam \usuUpsilon}'{\,\cong\,}\Sp4{\fam=\Bbbfam C}$. The second possibility can be excluded however, since $\SL2{\fam=\Bbbfam H}$ is not contained in $\Sp4{\fam=\Bbbfam C}$\: (this follows easily from the complete reducibility of the adjoint representation of the smaller group on the Lie algebra of the larger one and the fact that each representation of $\SL2{\fam=\Bbbfam H}$ in dimension $<6$ is trivial. It is also a consequence of the far more general results of Tits \cite{Ti} p.160/61). Hence $\dim{\fam=\ssfam \usuUpsilon}{\,\le\,}19$ and $\dim{\fam=\ssfam \usuXi}{\,\ge\,}6$. If ${\fam=\ssfam \usuXi}$ contains a non-trivial homology, then ${\fam=\ssfam T}_{\hskip-1pt[v]}$ is transitive by \cite{cp} 61.20. Assume now that ${\fam=\ssfam \usuXi}{\,=\,}{\fam=\ssfam T}_{\hskip-1pt[v]}$
has dimension $\dim{\fam=\ssfam \usuXi}{\,<\,}8$. Then ${\fam=\ssfam \usuXi}{\,\le\,}\Cs{}\nabla_{\hskip-2pt c}{\,\le\,}\Cs{}{\fam=\ssfam \usuLambda}$\: (since ${\fam=\ssfam \usuTheta}{\,<\,}{\fam=\ssfam \usuXi}$), and ${\fam=\ssfam \usuLambda}|_{a^{\fam=\ssfam \usuXi}}{\,=\,}1\kern-2.5pt {\rm l}$, but then ${\fam=\ssfam \usuLambda}$ would be trivial. \\ (g) {\it If $t{\,=\,}2$, then ${\fam=\ssfam \usuDelta}$ is transitive on the affine point space $P\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{W\}$\/}. \\ If not, then $\dim\nabla_{\hskip-2pt c}{\,>\,}15$,\: ${\fam=\ssfam \usuLambda}{\,\cong\,}\SU3{\fam=\Bbbfam C}$,\: $\dim\nabla{\,=\,}18$, and $\nabla_{\hskip-2pt c}$ is transitive on $S$ and on $au\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a,u\}$. Consequently, ${\fam=\ssfam \usuDelta}_{au}$ is doubly transitive on $au\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{u\}$. The exact homotopy sequence shows $\nabla\hskip.5pt'_{\hskip-3pt c}{\,\cong\,}\SU4{\fam=\Bbbfam C}$, see \cite{sz5} Lemma (5) or 2.17(d). The central involution in $\nabla\hskip.5pt'_{\hskip-3pt c}$ is a reflection with axis $av$, and ${\fam=\ssfam T}_{\hskip-1pt[u]}{\,\cong\,}{\fam=\Bbbfam R}^8$ by the dual of 2.9. It follows that $\nabla$ induces on ${\fam=\ssfam T}_{\hskip-1pt[u]}$ a subgroup of ${\fam=\Bbbfam C}^{\times}{\cdot\hskip1pt}\SU4{\fam=\Bbbfam C}$. In particular, $\nabla{\hskip1pt:\hskip1pt}\nabla_{\hskip-2pt[au]}{\,\le\,}17$ and ${\fam=\ssfam \usuDelta}_{[v,au]}{\,\ne\,}1\kern-2.5pt {\rm l}$. As ${\fam=\ssfam \usuTheta}{\,<\,}{\fam=\ssfam T}_{\hskip-1pt[v]}$ and ${\fam=\ssfam \usuLambda}$ acts non-trivially on ${\fam=\ssfam T}_{\hskip-1pt[v]}$, we have even $\dim{\fam=\ssfam T}_{\hskip-1pt[v]}{\,=\,}8$, and ${\fam=\ssfam T}$ would be transitive contrary to the assumption. \\ (h) {\it If $t{\,=\,}2$ and $\nabla$ is transitive on $S{\,=\,}uv\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{u,v\}$, then ${\fam=\ssfam T}$ is transitive\/}.\\ The {\tt proof} is similar to that in step (g). Because of step (f), we may assume that $(\nabla_{\hskip-2pt w})'{\,=\,}{\fam=\ssfam \usuLambda}{\,\cong\,}\SU3{\fam=\Bbbfam C}$ for $w{\,\in\,}S$. Then a maximal compact subgroup ${\fam=\ssfam \usuPhi}$ of $\nabla$ is isomorphic to $\SU4{\fam=\Bbbfam C}$, see \cite{sz5} Lemma (5) or 2.17(d). Again the central involution of ${\fam=\ssfam \usuPhi}$ is a reflection in $\nabla_{\hskip-1pt[u,av]}$, and ${\fam=\ssfam T}_{\hskip-1pt[u]}{\,\cong\,}{\fam=\Bbbfam R}^8$ by 2.9. The group ${\fam=\ssfam \usuUpsilon}{\,=\,}{\fam=\ssfam \usuDelta}_{av}/{\fam=\ssfam \usuDelta}_{[v]}$ induced by ${\fam=\ssfam \usuDelta}_{av}$ on the pencil ${\frak L}_v\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{W\}$ acts faithfully on ${\fam=\ssfam T}_{\hskip-1pt[u]}$. If ${\fam=\ssfam T}_{\hskip-1pt[v]}{\,<\,}{\fam=\ssfam \usuDelta}_{[v]}$, then ${\fam=\ssfam T}_{\hskip-1pt[v]}{\,\cong\,}{\fam=\Bbbfam R}^8$ by step (g) and \cite{cp} 61.20. Hence we may assume that ${\fam=\ssfam \usuDelta}_{[v]}{\,=\,}{\fam=\ssfam T}_{\hskip-1pt[v]}$. Representation theory as summarized in \cite{cp} 95.10 shows that ${\fam=\ssfam \usuUpsilon}{\,\le\,}{\fam=\Bbbfam C}^{\times}{\rtimes\,}\SU4{\fam=\Bbbfam C}$ or ${\fam=\ssfam \usuUpsilon}'{\,\cong\,}\Spin7{\fam=\Bbbfam R}$. In the first case $\dim{\fam=\ssfam \usuUpsilon}{\,\le\,}17$ and $\dim{\fam=\ssfam T}_{\hskip-1pt[v]}{\,=\,}8$, in the second case $\dim{\fam=\ssfam \usuUpsilon}{\,\le\,}22$ and $\dim{\fam=\ssfam T}_{\hskip-1pt[v]}{\,\in\,}\{3,4\}$, but then ${\fam=\ssfam T}_{\hskip-1pt[v]}{\,\le\,}\Cs{}{\fam=\ssfam \usuPhi}$ and ${\cal F}_{\fam=\ssfam \usuLambda}{\,\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\hskip-3pt}{\cal P}$, which contradicts Stiffness. \\ (i) {\it If $t{\,=\,}2$, then the translation group ${\fam=\ssfam T}$ is transitive\/}. \\ Because of steps (f--h), only the following situation must be considered: $\dim\nabla{\,=\,}17$, for some $w{\,\in\,}S$ the orbit $w^\nabla$ has dimension $\nabla{:}\nabla_{\hskip-2pt w}{\,<\,}8$,\: ${\fam=\ssfam \usuOmega}{\,=\,}(\nabla_{\hskip-2.5pt w})^1$ satisfies $\dim{\fam=\ssfam \usuOmega}{\,=\,}10$, and ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuOmega}_c)^1{\,\cong\,}\SU3{\fam=\Bbbfam C}$ for $a{\,\ne\,}c{\,\in\,}a^{\fam=\ssfam \usuTheta}$. Put
${\fam=\ssfam P}{\,=\,}\sqrt{\fam=\ssfam \usuOmega}$, so that ${\fam=\ssfam \usuOmega}{\,=\,}{\fam=\ssfam P}{\fam=\ssfam \usuLambda}$. As ${\fam=\ssfam \usuLambda}|_{\fam=\ssfam \usuTheta}{\,=\,}1\kern-2.5pt {\rm l}$, the radical ${\fam=\ssfam P}$ is transitive on ${\fam=\ssfam \usuTheta}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{1\kern-2.5pt {\rm l}\}$,\: ${\fam=\ssfam P}/{\fam=\ssfam P}_{\hskip-2pt c}{\,\approx\,}{\fam=\Bbbfam C}^{\times}$,\: $\dim{\fam=\ssfam P}_{\hskip-2pt c}{\,=\,}0$, and ${\fam=\ssfam P}_{\hskip-2pt c}$ is compact by 2.6(\^c). Therefore ${\fam=\ssfam P}_{\hskip-2pt c}$ is finite and ${\fam=\ssfam P}$ is a finite covering of the cylinder group. Hence ${\fam=\ssfam P}$ contains an involution $\sigma$, and $\sigma$ is a reflection (or else ${\fam=\ssfam \usuLambda}$ would act trivially on $W\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal F}_\sigma$), the axis of $\sigma$ is $W$\: (since $w^\sigma{\,=\,}w$), and $\sigma{\,\in\,}{\fam=\ssfam \usuDelta}_{[a]}$. Now 2.9 and step (g) imply $\dim{\fam=\ssfam T}{\,=\,}16$. \\ (j) {\it $t{\,\ne\,}3$\/}. Assume the contrary. Choose $w{\,\in\,}S$ and consider the plane ${\cal B}{\,=\,}\langle a^{\fam=\ssfam \usuTheta},u,w\rangle$ of dimension ${\ge\,}8$. Suppose first that ${\cal B}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$. By Stiffness, the connected component ${\fam=\ssfam \usuOmega}$ of $\nabla_{\hskip-2pt w}$ induces on ${\fam=\ssfam \usuTheta}$ a subgroup $\overline{\fam=\ssfam \usuOmega}{\,\le\,}\GL3{\fam=\Bbbfam R}$ of dimension at least $6$, and there is a subgroup ${\fam=\ssfam \usuXi}{\,\le\,}{\fam=\ssfam \usuOmega}$ such that the fixed elements of ${\fam=\ssfam \usuXi}$ in ${\fam=\ssfam \usuTheta}$ form a $2$-dimensional subgroup\footnote{\ The group $\overline{\fam=\ssfam \usuOmega}$ fixes a proper subspace of ${\fam=\ssfam \usuTheta}$ or $\overline{\fam=\ssfam \usuOmega}$ is irreducible and contains $\SL3{\fam=\Bbbfam R}$. In any case, the stabilizer ${\fam=\ssfam X}$ in $\overline{\fam=\ssfam \usuOmega}$ of some $2$-dimensional subspace ${\fam=\ssfam H}{\,<\,}{\fam=\ssfam \usuTheta}$ has dimension ${>}4$, but
$\dim{\fam=\ssfam X}|_{\fam=\ssfam H}{\,\le\,}4$.}. It follows that ${\cal B}\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal F}_{\fam=\ssfam \usuXi}{\hskip4pt\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal B}$. For a suitable $c{\,\in\,}a^{\fam=\ssfam \usuTheta}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a\}$ the group ${\fam=\ssfam \usuLambda}{\,=\,}(\nabla_{\hskip-2pt c,w})^1$ is compact by \cite{cp} 83.9. Moreover, $\dim{\fam=\ssfam \usuLambda}{\,\ge\,}6$, and 2.6(\^e) shows ${\fam=\ssfam \usuLambda}{\,\cong\,}\SO4{\fam=\Bbbfam R}$. Now
${\fam=\ssfam \usuLambda}|_{\cal B}{\,\cong\,}{\fam=\ssfam \usuLambda}|_{\fam=\ssfam \usuTheta}{\,\cong\,}\SO3{\fam=\Bbbfam R}$, but then $c^{\fam=\ssfam \usuLambda}{\,\ne\,}c$. Therefore ${\cal B}{\,=\,}{\cal P}$ and $\nabla_{\hskip-2pt w}$ acts faithfully on ${\fam=\ssfam \usuTheta}$. Consequently, the connected components of $\nabla_{\hskip-2pt w}$ and $\GL3{\fam=\Bbbfam R}$ are isomorphic, and $\dim{\fam=\ssfam \usuLambda}{\,=\,}6$. As $\SO3{\fam=\Bbbfam R}$ is a maximal subgroup of $\SL3{\fam=\Bbbfam R}$ (see \cite{cp} 94.34), it follows that a maximal compact subgroup of ${\fam=\ssfam \usuLambda}$ is trivial or a torus; hence $\pi_q{\fam=\ssfam \usuLambda}{\,=\,}0$ for $q{\,>\,}1$. Moreover, $\dim\nabla_{\hskip-2pt c}{\,=\,}14$ and $\nabla_{\hskip-2pt c}$ is transitive on $S$. Let ${\fam=\ssfam \usuPhi}$ be a maximal compact subgroup of $\nabla_{\hskip-2pt c}$. The exact homotopy sequence (cf. \cite{cp} 96.12) implies $\pi_3{\fam=\ssfam \usuPhi}{\,\cong\,}\pi_3{\fam=\ssfam \usuLambda}{\,=\,}0$ and $\pi_7{\fam=\ssfam \usuPhi}{\,\cong\,}\pi_7S{\,\cong\,}{\fam=\Bbbfam Z}$. From the first condition it follows by \cite{cp} 94.36 that ${\fam=\ssfam \usuPhi}$ has no almost simple factor, so that ${\fam=\ssfam \usuPhi}$ is a torus group (\cite{cp} 94.31(c)\,), but then ${\fam=\ssfam \usuPhi}$ cannot satisfy the second condition. \\ (k) {\tt Remark.} The arguments in step (j) do not use the fact that ${\fam=\ssfam \usuTheta}^\nabla{\,=\,}{\fam=\ssfam \usuTheta}$. They show therefore that there is no ${\fam=\ssfam \usuOmega}$-invariant $3$-dimensional subgroup of ${\fam=\ssfam \usuTheta}$. Similarly, only the ${\fam=\ssfam \usuOmega}$-invariance is used in steps ($\ell$), (m), and (n). \\ ($\ell$) {\it If $t{\,=\,}4$ and if ${\cal B}{\,=\,}\langle a^{\fam=\ssfam \usuTheta},u,w\rangle{\:\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$, then ${\fam=\ssfam T}$ is transitive\/}. \\ Assume first that ${\fam=\ssfam \usuOmega}{\,=\,}(\nabla_{\hskip-2pt w})^1$ acts irreducibly on ${\fam=\ssfam \usuTheta}$, and put
${\fam=\ssfam \usuOmega}^*{\,=\,}{\fam=\ssfam \usuOmega}|_{\fam=\ssfam \usuTheta}{\,=\,}{\fam=\ssfam \usuOmega}/{\fam=\ssfam K}$. Stiffness implies $\dim{\fam=\ssfam K}{\,=\,}3$ or $\dim{\fam=\ssfam K}{\,\le\,}1$. Therefore $\dim{\fam=\ssfam \usuOmega}^*{\,\ge\,}6$ and even $\dim{\fam=\ssfam \usuOmega}^*{'}{\,\ge\,}6$ (see \cite{cp} 95.6). It follows that ${\fam=\ssfam \usuOmega}^*{'}$ is isomorphic to one of the groups $\Sp4{\fam=\Bbbfam R}$,\: $\SL2{\fam=\Bbbfam C}$, or $\Opr4({\fam=\Bbbfam R},r)$, taken in its standard action on ${\fam=\ssfam \usuTheta}$. In each case there is a subset ${\fam=\ssfam \usuXi}{\,\subset\,}{\fam=\ssfam \usuOmega}$ such that the fixed point set ${\fam=\ssfam \usuTheta}\kern 2pt {\scriptstyle \cap}\kern 2pt\Cs{}{\fam=\ssfam \usuXi}$ is isomorphic to ${\fam=\Bbbfam R}^2$\: (this is obvious for the complex group; in the other cases ${\fam=\ssfam \usuXi}$ may be chosen as preimage of a $1$-dimensional torus). Now ${\cal F}_{\fam=\ssfam \usuXi}\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal B}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal B}$,\: ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuOmega}_c)^1{\,\cong\,}\SO4{\fam=\Bbbfam R}$ by 2.6(c,\^e), and $\dim{\fam=\ssfam \usuOmega}^*{\,\le\,}7$. Hence ${\fam=\ssfam K}^1{\,\cong\,}\Spin3{\fam=\Bbbfam R}$. As ${\fam=\ssfam \usuLambda}$ has a subgroup $\SO3{\fam=\Bbbfam R}$, there is a group ${\fam=\ssfam \usuPhi}{\,\cong\,}{\fam=\Bbbfam Z}_2^{\hskip2pt4}$ in ${\fam=\ssfam \usuOmega}$, and ${\fam=\ssfam \usuPhi}$ contains at least one reflection $\sigma$ by 2.10. In fact, $\sigma{\,\in\,}{\fam=\ssfam \usuDelta}_{[a,W]}$, since $\sigma{\,\in\,}\nabla_{\hskip-2pt w}$. We have $\dim{\fam=\ssfam \usuOmega}{\,\le\,}10$ and $\dim a^{\fam=\ssfam \usuDelta}{\,=\,}{\fam=\ssfam \usuDelta}{:}\nabla{\,\ge\,}15$. Finally 2.9 shows that $\sigma^{\fam=\ssfam \usuDelta}\sigma{\,=\,}{\fam=\ssfam T}{\,\cong\,}{\fam=\Bbbfam R}^{16}$. \\Hence we may suppose that there is a proper ${\fam=\ssfam \usuOmega}$-invariant subgroup ${\fam=\ssfam H}{\,<\,}{\fam=\ssfam \usuTheta}$, and $\dim{\fam=\ssfam H}{\,<\,}3$ by Remark (k). Again there exists a subplane ${\cal F}{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal B}$, and for some $c{\,\in\,}a^{\fam=\ssfam H}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a\}$ Stiffness implies ${\fam=\ssfam \usuLambda}{\,\cong\,}\SU3{\fam=\Bbbfam C}$, but then
${\fam=\ssfam \usuLambda}|_{\fam=\ssfam \usuTheta}{\,=\,}1\kern-2.5pt {\rm l}$, which is impossible. \\ (m) {\it If $t{\,=\,}4$ and if ${\cal B}{\,=\,}\langle a^{\fam=\ssfam \usuTheta},u,w\rangle{\,=\,}{\cal P}$, then ${\fam=\ssfam T}$ is transitive\/}. \\ ${\cal B}{\,=\,}{\cal P}$ implies that ${\fam=\ssfam \usuOmega}$ acts effectively on ${\fam=\ssfam \usuTheta}$. If this action is irreducible, then ${\fam=\ssfam \usuOmega}'$ is isomorphic to $\Sp4{\fam=\Bbbfam R}$ or to $\SL4{\fam=\Bbbfam R}$. In both cases the central involution $\omega{\,\in\,}{\fam=\ssfam \usuOmega}'$ is a reflection, since otherwise
$\dim{\fam=\ssfam \usuOmega}'|_{{\cal F}_\omega}{\,\le\,}7$ by \cite{cp} 83.17. Again $\omega{\,\in\,}{\fam=\ssfam \usuDelta}_{[a,W]}$ and ${\fam=\ssfam T}{\,=\,}\omega^{\fam=\ssfam \usuDelta}\omega$ has even dimension ${\fam=\ssfam \usuDelta}{:}\nabla$, see 2.9. If $\dim{\fam=\ssfam \usuOmega}'{\,=\,}10$, then $\dim\nabla{\,\le\,}19$ and $\dim{\fam=\ssfam T}{\,\ge\,}14$,\: $\dim{\fam=\ssfam T}_{\hskip-1pt[w]}{\,\ge\,}6$. As $\omega$ inverts each $\tau{\,\in\,}{\fam=\ssfam T}$, the group ${\fam=\ssfam \usuOmega}'$ acts faithfully on ${\fam=\ssfam T}_{\hskip-1pt[w]}$, and then \cite{cp} 95.10 shows $\dim{\fam=\ssfam T}_{\hskip-1pt[w]}{\,=\,}8$ and ${\fam=\ssfam T}{\,\cong\,}{\fam=\Bbbfam R}^{16}$. Now let ${\fam=\ssfam \usuOmega}'{\,\cong\,}\SL4{\fam=\Bbbfam R}$. Then ${\fam=\ssfam \usuOmega}$ has a subgroup ${\fam=\ssfam \usuPhi}{\,\cong\,}\SO3{\fam=\Bbbfam R}$, and ${\fam=\ssfam \usuPhi}$ fixes some $c{\,\in\,}a^{\fam=\ssfam \usuTheta}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a\}$, so that ${\fam=\ssfam \usuPhi}{\,<\,}{\fam=\ssfam \usuLambda}{\,=\,}{\fam=\ssfam \usuOmega}_c^{\,1}$. Stiffness properties 2.6(\^c,\^e) imply ${\fam=\ssfam \usuLambda}{\,\cong\,}\SO4{\fam=\Bbbfam R}$, but $\dim{\fam=\ssfam \usuLambda}{\,\ge\,}11$. Suppose now that there is an ${\fam=\ssfam \usuOmega}$-invariant proper subgroup ${\fam=\ssfam H}$ of ${\fam=\ssfam \usuTheta}$. From Remark (k) it follows that $\dim{\fam=\ssfam H}{\,\le\,}2$. Choose $c{\,\in\,}a^{\fam=\ssfam H}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a\}$. Then $\dim{\fam=\ssfam \usuLambda}{\,\ge\,}9-\dim{\fam=\ssfam H}$. We will show that ${\fam=\ssfam \usuLambda}$ is compact. If ${\fam=\ssfam H}{\,\cong\,}{\fam=\Bbbfam R}$ and ${\fam=\ssfam \usuLambda}$ is transitive on ${\fam=\ssfam \usuTheta}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt{\fam=\ssfam H}$, then the linear group ${\fam=\ssfam \usuLambda}$ is also transitive on ${\fam=\ssfam \usuTheta}/{\fam=\ssfam H}{\,\cong\,}{\fam=\Bbbfam R}^3$; hence ${\fam=\ssfam \usuLambda}$ contains $\SO3{\fam=\Bbbfam R}$ and ${\fam=\ssfam \usuLambda}{\,\cong\,}\SU3{\fam=\Bbbfam C}$ by 2.6(\^c,\^e). In all other cases, there are some points $d{\,\in\,}a^{\fam=\ssfam \usuTheta}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a\}$ and $d'{\,\in\,}a^{\fam=\ssfam \usuTheta}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt a^{\fam=\ssfam H}$ such that $\dim{\fam=\ssfam \usuLambda}_d{\,\ge\,}5$,\: $d'{\,\notin\,}{\cal F}_{{\fam=\ssfam \usuLambda}_d}$, and ${\fam=\ssfam \usuLambda}_{d,d'}{\,\ne\,}1\kern-2.5pt {\rm l}$. Therefore ${\cal F}_{{\fam=\ssfam \usuLambda}_d}{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal F}_{{\fam=\ssfam \usuLambda}_{d,d'}}{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$. Again ${\fam=\ssfam \usuLambda}{\,\cong\,}\SU3{\fam=\Bbbfam C}$ by Stiffness, but then the action of ${\fam=\ssfam \usuLambda}$ would be trivial. \\ (n) {\it If $t{\,=\,}5$, then ${\fam=\ssfam T}$ is transitive\/}. In fact, the group ${\fam=\ssfam \usuOmega}$ acts effectively on ${\fam=\ssfam \usuTheta}$, since $\langle a^{\fam=\ssfam \usuTheta},u,w\rangle{\,=\,}{\cal P}$. If this action is irreducible, then \cite{cp} 95.10 shows that ${\fam=\ssfam \usuOmega}'{\,\cong\,}\Opr5({\fam=\Bbbfam R},r)$; the case $r{\,=\,}0$ is excluded by 2.10. Consider a subgroup ${\fam=\ssfam \usuPhi}{\,\cong\,}\SO3{\fam=\Bbbfam R}$ in ${\fam=\ssfam \usuOmega}$, a point $c{\,\in\,}a^{\fam=\ssfam \usuTheta}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a\}$ such that $c^{\fam=\ssfam \usuPhi}{\,=\,}c$, and ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuOmega}_c)^1$. Stiffness 2.6(\^c,\^e) implies ${\fam=\ssfam \usuLambda}{\,\cong\,}\SO4{\fam=\Bbbfam R}$\: (note that each action of $\SU3{\fam=\Bbbfam C}$ on ${\fam=\ssfam \usuTheta}$ is trivial). Consequently $r{\,=\,}1$, and ${\fam=\ssfam \usuOmega}$ has a subgroup ${\fam=\ssfam X}{\,\cong\,}\SO4({\fam=\Bbbfam R},1)$. The group ${\fam=\ssfam X}$ fixes some point $d{\,\in\,}a^{\fam=\ssfam \usuTheta}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a\}$. Now ${\fam=\ssfam X}$ is compact by Stiffness 2.6(\^c), but this is not true. Hence ${\fam=\ssfam \usuTheta}$ has a minimal ${\fam=\ssfam \usuOmega}$-invariant subgroup ${\fam=\ssfam H}{\,\cong\,}{\fam=\Bbbfam R}^s,\: s{\le}2$ or $s{\,=\,}4$. In the first case, let $c{\,\in\,}a^{\fam=\ssfam H}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a\}$. Because any action of $\SU3{\fam=\Bbbfam C}$ on ${\fam=\ssfam \usuTheta}$ is trivial, stiffness property 2.6(e) implies $\dim{\fam=\ssfam \usuOmega}_c{\,=\,}7$ and $s{\,=\,}2$. It follows that $\dim\nabla_{\hskip-2pt c}{\,=\,}15$, and $\dim{\fam=\ssfam \usuOmega}{\,=\,}9$. Consider the subplane ${\cal E}{\,=\,}\langle a^{\fam=\ssfam H},u,w\rangle{\,=\,}{\cal E}^{\fam=\ssfam \usuOmega}$. If ${\cal E}{\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\hskip-3pt}{\cal P}$ and
${\fam=\ssfam \usuOmega}|_{\cal E}{\,=\,}{\fam=\ssfam \usuOmega}/{\fam=\ssfam K}$, then $\dim{\fam=\ssfam K}{\,\le\,}3$ and ${\fam=\ssfam \usuOmega}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\ge\,}6$; on the other hand,
${\fam=\ssfam \usuOmega}|_{\cal E}{\,\cong\,}{\fam=\ssfam \usuOmega}|_{\fam=\ssfam H}$ is a subgroup of the $4$-dimensional group $\GL2{\fam=\Bbbfam R}$. Therefore $\dim{\cal E}{\,=\,}4$ and
$\dim{\fam=\ssfam \usuOmega}|_{\cal E}{\,\le\,}2$\: (see \cite{cp} 71.7). Hence $\dim{\fam=\ssfam K}{\,\ge\,}7$ and ${\fam=\ssfam K}^1{\,=\,}{\fam=\ssfam \usuLambda}$. Choose any $d{\,\in\,}a^{\fam=\ssfam \usuTheta}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt a^{\fam=\ssfam H}$. Then $\dim{\fam=\ssfam \usuLambda}_d{\,\ge\,}2$,\: $\langle {\cal F}_{\fam=\ssfam \usuLambda},d\rangle{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$, and Stiffness implies that ${\fam=\ssfam \usuLambda}$ is compact, but then the action of ${\fam=\ssfam \usuLambda}$ on ${\fam=\ssfam \usuTheta}$ is completely reducible, and ${\fam=\ssfam \usuLambda}$ acts effectively on a complement ${\fam=\Bbbfam R}^3$ of ${\fam=\ssfam H}$ in ${\fam=\ssfam \usuTheta}$, which is impossible. Thus $s{\,=\,}4$, and ${\fam=\ssfam T}$ is transitive by the previous steps. \\ (o) {\it If $t{\,=\,}6$ and ${\fam=\ssfam \usuOmega}$ acts irreducibly on ${\fam=\ssfam \usuTheta}$, then ${\fam=\ssfam T}$ is transitive\/}. {\tt Proof:} ${\fam=\ssfam \usuOmega}'$ is semi-simple by \cite{cp} 95.6, and $\dim{\fam=\ssfam \usuOmega}'{\,>\,}6$. Clifford's Lemma implies that ${\fam=\ssfam \usuOmega}'$ is almost simple or isomorphic to a direct product $\SL2{\fam=\Bbbfam R}{\times}\SL3{\fam=\Bbbfam R}$. In the second case any involution $\beta$ in the larger factor ${\fam=\ssfam B}$ is planar, and ${\fam=\ssfam \usuGamma}{:=\,}\Cs{\fam=\ssfam B}\beta{\,\cong\,}\SL2{\fam=\Bbbfam R}$. The fixed elements of ${\fam=\ssfam \usuGamma}$ on ${\fam=\ssfam \usuTheta}$ form a $2$-dimensional subgroup. As ${\fam=\ssfam \usuGamma}$ is not compact, $\dim{\cal F}_{\fam=\ssfam \usuGamma}{\,=\,}4$ and ${\cal F}_{\fam=\ssfam \usuGamma}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal F}_\beta{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$, but then ${\fam=\ssfam \usuGamma}$ would be compact by 2.6(c). Now it follows from \cite{cp} 95.10 that $\dim{\fam=\ssfam \usuOmega}'{\,=\,}8$ or $\dim{\fam=\ssfam \usuOmega}'{\,\ge\,}15$. In the latter case $9{\,\le\,}\dim{\fam=\ssfam \usuOmega}_c{\,\ne\,}14$ in contradiction to 2.6(e). Hence $9{\,\le\,}\dim{\fam=\ssfam \usuOmega}{\,\le\,}10$. If $\dim{\fam=\ssfam \usuOmega}{\,=\,}10$, then $\dim a^{\fam=\ssfam \usuDelta}{\,\ge\,}15$,\: ${\fam=\ssfam \usuOmega}{\,\cong\,}{\fam=\Bbbfam C}^{\times}{\cdot\hskip1pt}\SU3({\fam=\Bbbfam C},r)$,\: ${\fam=\ssfam \usuOmega}$ contains a central involution $\alpha{\,\in\,}\nabla_{\hskip-1pt[a]}$, and 2.9 shows that ${\fam=\ssfam T}{\,\cong\,}{\fam=\Bbbfam R}^{16}$. If $\dim{\fam=\ssfam \usuOmega}{\,=\,}9$ for each choice of $w$, then $\nabla$ is transitive on $S$,\: $\dim\nabla{\,=\,}17$,\; $a^{\fam=\ssfam \usuDelta}$ is open in $P$, a maximal compact subgroup ${\fam=\ssfam \usuPhi}$ of ${\fam=\ssfam \usuOmega}$ satisfies $\dim{\fam=\ssfam \usuPhi}'{\,\in\,}\{3,8\}$, and 2.17 implies that $\nabla$ must have a compact subgroup $\U2{\fam=\Bbbfam H}$ or $\SU4{\fam=\Bbbfam C}$. These possibilities will be discussed in the next two steps. \\
(p) Suppose that $\nabla$ contains ${\fam=\ssfam \usuUpsilon}{\,\cong\,}\U2{\fam=\Bbbfam H}$, but not $\SU4{\fam=\Bbbfam C}$. The central involution $\sigma$ of ${\fam=\ssfam \usuUpsilon}$ acts trivially on ${\fam=\ssfam \usuTheta}$ and ${\fam=\ssfam \usuUpsilon}|_{\fam=\ssfam \usuTheta}{\,\cong\,}\SO5{\fam=\Bbbfam R}$, moreover, $\sigma$ is a reflection with axis $av$, and ${\fam=\ssfam T}_{\hskip-1pt[u]}$ has positive dimension. In fact, ${\fam=\ssfam T}_{\hskip-1pt[u]}{\,\cong\,}{\fam=\Bbbfam R}^8$ because the representation of ${\fam=\ssfam \usuUpsilon}$ on ${\fam=\ssfam T}_{\hskip-1pt[u]}$ is faithful or since $\dim(av)^{\fam=\ssfam \usuDelta}{\,=\,}8$. Note that $\nabla\kern 2pt {\scriptstyle \cap}\kern 2pt\Cs{}{\fam=\ssfam T}_{\hskip-1pt[u]}{\,\le\,}{\fam=\ssfam \usuDelta}_{[v,au]}$. If ${\fam=\ssfam \usuDelta}_{[v,au]}{\,\ne\,}1\kern-2.5pt {\rm l}$, then ${\fam=\ssfam T}_{\hskip-1pt[v]}$ is transitive because $\dim(au)^{\fam=\ssfam \usuDelta}{\,=\,}8$, and ${\fam=\ssfam T}{\,\cong\,}{\fam=\Bbbfam R}^{16}$. Thus we may assume that $\nabla$ acts effectively on ${\fam=\ssfam T}_{\hskip-1pt[u]}$; as ${\fam=\ssfam \usuUpsilon}{<\,}\nabla$, the action is also irreducible. Hence the commutator group $\nabla'$ is semi-simple, and then \cite{cp} 94.33 shows $\nabla'{\,\cong\,}\SL2{\fam=\Bbbfam H}$. Recall that ${\fam=\ssfam \usuOmega}'$ is an almost simple group of dimension~$8$, and note that ${\fam=\ssfam \usuOmega}'$ acts effectively on ${\fam=\ssfam T}_{\hskip-1pt[u]}{\,\cong\,}{\fam=\Bbbfam R}^8$. Let ${\fam=\ssfam X}{\,=\,}{\fam=\ssfam T}_{\hskip-1pt[u]}\kern 2pt {\scriptstyle \cap}\kern 2pt\Cs{}{\fam=\ssfam \usuOmega}'$. Either ${\fam=\ssfam X}$ is trivial and ${\fam=\ssfam \usuOmega}'$ acts irreducibly on ${\fam=\ssfam T}_{\hskip-1pt[u]}$, or ${\fam=\ssfam X}{\,\cong\,}{\fam=\Bbbfam R}^2$ by complete reducibiliy and \cite{cp} 95.10,\; ${\cal C}{\,=\,}\langle a^{\fam=\ssfam X},v,w\rangle$ is a $4$-dimensional subplane,
${\fam=\ssfam \usuOmega}'|_{\cal C}{\,=\,}1\kern-2.5pt {\rm l}$, and ${\fam=\ssfam \usuOmega}'{\,\cong\,}\SU3{\fam=\Bbbfam C}$ by Stiffness. In the first case the centralizer of ${\fam=\ssfam \usuOmega}'|_{{\fam=\ssfam T}_{\hskip-1pt[u]}}$ is ${\fam=\Bbbfam R}^{\times}$\: (see \cite{cp} 95.10), but the centralizer of the action of $\nabla$ on ${\fam=\ssfam T}_{\hskip-1pt[u]}$ is ${\fam=\Bbbfam H}^{\times}$, a contradiction. In the second case $\SU3{\fam=\Bbbfam C}$ would be contained in $\U2{\fam=\Bbbfam H}$, which is not true. \\
(q) Thus $\nabla'{\,\cong\,}\SU4{\fam=\Bbbfam C}$ and $\nabla'|_{\fam=\ssfam \usuTheta}{\,\cong\,}\SO6{\fam=\Bbbfam R}$. Again the central involution in $\nabla'$ is a reflection in $\nabla_{\hskip-1.5pt[u]}$ and ${\fam=\ssfam T}_{\hskip-1pt[u]}{\,\cong\,}{\fam=\Bbbfam R}^8$. Let ${\fam=\ssfam \usuGamma}{\,=\,}{\fam=\ssfam \usuDelta}_{av}$ and consider the action of $\nabla'$ on (the additive group of) the Lie algebra ${\frak l}\hskip1pt{\fam=\ssfam \usuGamma}$. As ${\fam=\ssfam \usuTheta}^{\hskip-1pt\nabla}{\,=\,}{\fam=\ssfam \usuTheta}$ and $\dim{\fam=\ssfam \usuGamma}{\,=\,}25$, the group ${\fam=\ssfam \usuXi}{\,=\,}{\fam=\ssfam \usuGamma}\kern 2pt {\scriptstyle \cap}\kern 2pt\Cs{}\nabla'$ is $4$-dimensional. Put ${\fam=\ssfam \usuXi}\kern 2pt {\scriptstyle \cap}\kern 2pt\Cs{}{\fam=\ssfam T}_{\hskip-1pt[u]}{\,=\,}{\fam=\ssfam X}$, and note that ${\fam=\ssfam X}^{\nabla'}{\,=\,}{\fam=\ssfam X}$ and that ${\fam=\ssfam \usuXi}/{\fam=\ssfam X}{\,\le\,}{\fam=\Bbbfam C}^{\times}$\; (see \cite{cp} 95.10). By definition, ${\fam=\ssfam X}{\,\le\,}{\fam=\ssfam \usuDelta}_{[v]}$ and $\dim{\fam=\ssfam X}{\,\ge\,}2$. Either ${\fam=\ssfam X}{\,\le\,}{\fam=\ssfam T}_{\hskip-1pt[v]}$, or ${\fam=\ssfam X}$ contains a homology and \cite{cp} 61.20 implies ${\fam=\ssfam T}_{\hskip-1pt[v]}{\,\cong\,}{\fam=\Bbbfam R}^8$. In the first case, the action of $\nabla'$ on ${\fam=\ssfam \usuTheta}$ shows that ${\fam=\ssfam X}\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam \usuTheta}{\,=\,}1\kern-2.5pt {\rm l}$. Hence ${\fam=\ssfam X}{\fam=\ssfam \usuTheta}{\,=\,}{\fam=\ssfam T}_{\hskip-1pt[v]}{\,\cong\,}{\fam=\Bbbfam R}^8$ and ${\fam=\ssfam T}$ is transitive.\\ (r) If $t{\,=\,}6$ and ${\fam=\ssfam T}$ is not transitive, then steps (k--o) imply that there exists a minimal ${\fam=\ssfam \usuOmega}$-invariant subgroup ${\fam=\ssfam H}{\,<\,}{\fam=\ssfam \usuTheta}$ of dimension ${\,\le\,}2$. As in step (n), let $a{\,\ne\,}c{\,\in\,}a^{\fam=\ssfam H}$. Then representation on ${\fam=\ssfam \usuTheta}$ shows that ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuOmega}_c)^1{\,\not\cong\,}\SU3{\fam=\Bbbfam C}$, and Stiffness 2.6(e) implies $\dim{\fam=\ssfam \usuLambda}{\,=\,}7$. Hence ${\fam=\ssfam H}{\,\cong\,}{\fam=\Bbbfam R}^2$,\; $\dim\nabla_{\hskip-2pt c}{\,=\,}15$, and ${\fam=\ssfam \usuXi}{\,=\,}(\nabla_{\hskip-2pt c})^1$ is transitive and faithful on $S$. By 2.17(\^c) it follows that ${\fam=\ssfam \usuXi}$ has a compact subgroup ${\fam=\ssfam \usuPsi}{\,\cong\,}\U2{\fam=\Bbbfam H}$, but ${\fam=\ssfam \usuXi}$ is not compact. The central involution $\sigma{\,\in\,}{\fam=\ssfam \usuPsi}$ is a reflection with axis $av$. The dual of 2.9 shows $\dim{\fam=\ssfam T}_{\hskip-1pt[u]}{\,>\,}0$, and then ${\fam=\ssfam T}_{\hskip-1pt[u]}{\,\cong\,}{\fam=\Bbbfam R}^8$ since ${\fam=\ssfam \usuPsi}$ acts faithfully on ${\fam=\ssfam T}_{\hskip-1pt[u]}$. Consequently, ${\fam=\ssfam \usuXi}$ is an irreducible subgroup of $\GL8{\fam=\Bbbfam R}$ and ${\fam=\ssfam \usuXi}'$ is semi-simple. \cite{cp} 94.33 and 95.10 imply ${\fam=\ssfam \usuXi}{\,\cong\,}\SL2{\fam=\Bbbfam H}$, but $c^{\hskip.5pt{\fam=\ssfam \usuXi}}{\,=\,}c$ and each representation of $\SL2{\fam=\Bbbfam H}$ on ${\fam=\ssfam \usuTheta}{\,\cong\,}{\fam=\Bbbfam R}^6$ with a fixed point is trivial. This is impossible. \\ (s) {\it If $\dim{\fam=\ssfam \usuDelta}{\,=\,}33$ and $t{\,=\,}7$, then ${\fam=\ssfam T}$ is transitive\/}:\: The action of $\nabla$ on ${\fam=\ssfam \usuTheta}$ is irreducible, because ${\fam=\ssfam \usuTheta}$ is a minimal $\nabla$-invariant group, and $\dim\nabla{\,<\,}28$, since $\dim\nabla{\fam=\ssfam \usuTheta}{\,\le\,}33$.
By \cite{cp} 95.\:5,\:6, and 10, the group $(\nabla|_{\fam=\ssfam \usuTheta})'$ is isomorphic to ${\rm G}_2$ or locally isomorphic to $\Spin7({\fam=\Bbbfam R},r),\: r{\le}3$, and \cite{cp} 94.27 shows that
$(\nabla|_{\fam=\ssfam \usuTheta})'$ is covered by a subgroup ${\fam=\ssfam \usuUpsilon}{\,\le\,}\nabla$. In any case it will turn out that $\nabla$ contains a group ${\fam=\ssfam \usuGamma}{\,\cong\,}{\rm G}_2$. Suppose that $\dim{\fam=\ssfam \usuUpsilon}{\,=\,}21$. Then $\dim{\fam=\ssfam \usuOmega}{\,\ge\,}13$. Let ${\fam=\ssfam H}{\,\cong\,}{\fam=\Bbbfam R}^s$ be a minimal ${\fam=\ssfam \usuOmega}$-invariant subgroup of~${\fam=\ssfam \usuTheta}$. According to {Remark} (k),
we have $s{\,\le\,}2$ or $s{\,\ge\,}6$. Choose $c{\,\in\,}a^{\fam=\ssfam H}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a\}$ and put ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuOmega}_c)^1$. In the first case $\dim{\fam=\ssfam \usuLambda}{\,\ge\,}11$, and Stiffness 2.6(d) implies ${\fam=\ssfam \usuLambda}{\,\cong\,}{\rm G}_2$. If $\dim{\fam=\ssfam H}{\,=\,}6$, then ${\fam=\ssfam \usuOmega}$ acts effectively and irreducibly on ${\fam=\ssfam H}$, and ${\fam=\ssfam \usuOmega}'$ is almost simple by Clifford's Lemma \cite{cp} 95.5. The list \cite{cp} 95.10 of representations shows that $\dim{\fam=\ssfam \usuOmega}'{\,\ge\,}15$. Consequently $\dim{\fam=\ssfam \usuLambda}{\,>\,}8$. By 2.6(e) again ${\fam=\ssfam \usuLambda}{\,\cong\,}{\rm G}_2$, and $\nabla$ has indeed a subgroup ${\fam=\ssfam \usuGamma}$ as claimed. The representation of ${\fam=\ssfam \usuGamma}$ on (the additive group of) the Lie algebra ${\frak l}\hskip1pt{\fam=\ssfam \usuDelta}$ is completely reducible, and ${\fam=\ssfam \usuDelta}{\,:\,}{\fam=\ssfam \usuGamma}{\fam=\ssfam \usuTheta}{\,=\,}12$. The only representation of ${\rm G}_2$ in dimension ${<\,}14$ is the natural one as $\mathop{{\rm Aut}}{\fam=\Bbbfam O}$ on the space of pure octonions. Therefore ${\fam=\ssfam P}{\,=\,}(\Cs{}{\fam=\ssfam \usuGamma})^1$ satisfies $\dim{\fam=\ssfam P}{\,=\,}5$, but ${\fam=\ssfam P}$ acts faithfully on the flat plane ${\cal F}_{\fam=\ssfam \usuGamma}$, fixes $u$ and $v$, and hence has dimension ${\le\,}4$, a contradiction. \\ (t) {\bf Corollary.} {\it If $\dim{\fam=\ssfam \usuDelta}{\,=\,}33$ and $z{\,\in\,}\{u,v\}$, then $\dim{\fam=\ssfam T}_{\hskip-1pt[z]}{\,=\,}0$ or ${\fam=\ssfam T}_{\hskip-1pt[z]}{\,\cong\,}{\fam=\Bbbfam R}^8$\/}. \\ (u) {\it If $\dim{\fam=\ssfam \usuDelta}{\,=\,}33$ and if $\dim{\fam=\ssfam \usuDelta}_{[u]}{\,>\,}0$ or $\nabla_{\hskip-1pt[u]}{\,\ne\,}1\kern-2.5pt {\rm l}$, then ${\fam=\ssfam T}$ is transitive\/}. \\ This is an immediate consequence of \cite{cp} 61.20, the assumption that $W$ is the only fixed line, and Corollary (t). \\ Only the case that $\nabla$ acts faithfully and irreducibly on ${\fam=\ssfam \usuTheta}{\,\cong\,}{\fam=\Bbbfam R}^8$ has still to be considered. Put ${\fam=\ssfam \usuGamma}{\,=\,}{\fam=\ssfam \usuDelta}_{au}$ and note that $\nabla{\,\le\,}{\fam=\ssfam \usuGamma}$ and $\dim{\fam=\ssfam \usuGamma}{\,=\,}25$. \\ (v) {\it ${\fam=\ssfam \usuGamma}{:\,}{\fam=\ssfam \usuGamma}_{\hskip-1pt[u]}{\,\le\,}22$,\: $\dim{\fam=\ssfam \usuGamma}_{\hskip-1pt[u]}{\,>\,}0$, and ${\fam=\ssfam T}$ is transitive\/}. \\
The group ${\fam=\ssfam \usuUpsilon}{=\,}{\fam=\ssfam \usuGamma}|_{\fam=\ssfam \usuTheta}{\,=\,}{\fam=\ssfam \usuGamma}/{\fam=\ssfam \usuGamma}_{\hskip-1pt[u]}$ is an irreducible subgroup of $\GL8{\fam=\Bbbfam R}$. Suppose that ${\fam=\ssfam \usuGamma}{:\,}{\fam=\ssfam \usuGamma}_{\hskip-1pt[u]}{\,>\,}20$. Then $\dim{\fam=\ssfam \usuUpsilon}'{\,>\,}18$, and Clifford's Lemma \cite{cp} 95.6 implies that ${\fam=\ssfam \usuUpsilon}'$ is almost simple and irreducible (cf. \cite{HS} proof, step 18) for details). The list \cite{cp} 95.10 of representations shows ${\fam=\ssfam \usuUpsilon}'{\hskip1pt\cong\,}\Spin7({\fam=\Bbbfam R},r)$ with $r{\,=\,}0,3$; in particular, $\dim{\fam=\ssfam \usuUpsilon}'{\,=\,}21$ and $\dim{\fam=\ssfam \usuUpsilon}{\,\le\,}22$. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
The planes of 6.5 with $\dim{\fam=\ssfam \usuDelta}{\,>\,}34$ have been described in \cite{HS}: \par
{\bf 6.7 Cartesian planes.} {\it A plane ${\cal P}$ satisfies the conditions of {\rm 6.5} with $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}35$ if, and only if, ${\cal P}$ can be coordinatized by a topological Cartesian field $({\fam=\Bbbfam O},+,\diamond)$ defined as follows\/}: {\it let $({\fam=\Bbbfam R},+,\ast,1)$ be an arbitrary real Cartesian field with unit element $1$ such that identically $(-r){\ast}s{\,=\,}-(r{\ast}s)$, and let $\rho{\,:\,}[0,\infty){\,\approx\,}[0,\infty)$ be any homeomorphism with $\rho(1){\,=\,}1$. Write each octonion in the form $x{\,=\,}x_0{\,+\,}{\frak x}$ as in $2.5$, and put
$$s\diamond x{\,=\,}|s|^{-1}s\hskip1.5pt(|s|{\ast\hskip1pt}x_0+\rho(|s|){\hskip1pt\cdot\hskip1pt}{\frak x}) \ {\rm for} \ s\ne 0, \ 0\diamond x{\,=\,}0\,.$$\/}\par
Recall that ${\fam=\ssfam \usuSigma}{\,=\,}\mathop{{\rm Aut}}{\cal P}$.\quad (a) {\it If $\dim{\fam=\ssfam \usuSigma} = 39$, then ${\cal P}$ is a translation plane\/}. \par (b) {\it The plane ${\cal P}$ is a translation plane if\/, and only if\/, it can be coordinatized by a quasi-field ${\fam=\Bbbfam O}_{\textstyle\diamond}$ where $*$ is the ordinary multiplication of the reals. In this case $\dim{\fam=\ssfam \usuSigma}=39$ if\/, and only if\/, $\rho$ is a multiplicative homomorphism$;$ otherwise $\dim{\fam=\ssfam \usuSigma}=38$\/}.\par (c) {\it If ${\cal P}$ is not a translation plane, then $\dim{\fam=\ssfam \usuSigma}{\,=\,}38$ if, and only if, ${\cal P}$ can be coordinatized by a Cartesian field ${\fam=\Bbbfam O}_{\textstyle\diamond}$ where
$r{\ast}s{\,=\,}rs\hskip6pt(s{\,\ge\,}0)$ and $r{*}s{\,=\,}|r|^{\gamma\,} rs\hskip6pt(s{\,<\,}0)$ for some $\gamma{\,>\,}0$ and $\rho:[0,\infty) \to [0,\infty)$ is a multiplicative homomorphism\/}. \par For the cases $\dim{\fam=\ssfam \usuSigma}{\,=\,}37$ the reader is referred to the last part of \cite{HS}. \par
Proofs will not be repeated here. There seems to be little chance to improve these results. \par
{\Bf 7. Collinear fixed points} \par
Assume in this section that ${\fam=\ssfam \usuDelta}$ fixes more than $2$ points but only one line. Some results will be stronger than in the previous section. Write ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\{u,v,w,...,W\}$. \par
{\bf 7.0 Lie.} {\it If ${\fam=\ssfam \usuDelta}$ fixes at least $3$ distinct points $u,v,w$ and exactly one line $W$, and if $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}15$, then ${\fam=\ssfam \usuDelta}$ is a Lie group\/}. \par
(a) The {\tt proof} is similar to that of 6.0, and the same notation will be used. Assume that ${\fam=\ssfam N}$ is not a Lie group. If ${\cal D}{\,=\,}\langle x^{\fam=\ssfam \usuDelta}\rangle$ is flat, then 2.6(\^a) implies $\dim{\fam=\ssfam \usuDelta}{\,\le\,}3{+}11$, otherwise ${\fam=\ssfam K}{\,\cong\,}{\rm G}_2$ would be a maximal compact subgroup. If $\dim{\cal D}{\,=\,}4$, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}6{+}8$, again by Stiffness. Hence ${\cal D}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$, the kernel ${\fam=\ssfam K}$ is compact, and $\dim{\fam=\ssfam K}{\,\le\,}7$. \\ (b) Choose a line $L{\,\in\,}{\cal D}, \ v{\,\in\,}L$, a point $z{\hskip1pt\in\hskip1pt}L\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt{\cal D}$, and let ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuDelta}_z)^1$. According to Lemma 7.0$'$ below, ${\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$ is a Lie group, and so is ${\fam=\ssfam \usuDelta}/({\fam=\ssfam K}\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam N})$. Therefore we may assume that ${\fam=\ssfam N}{\,\le\,}{\fam=\ssfam K}$. We have ${\fam=\ssfam K}_z{\,=\,}1\kern-2.5pt {\rm l}$\; because $\langle {\cal D},z\rangle{\,=\,}{\cal P}$. Hence ${\fam=\ssfam \usuLambda}\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam N}{\,=\,}1\kern-2.5pt {\rm l}$ and ${\fam=\ssfam \usuLambda}$ is a Lie group. Moreover, ${\fam=\ssfam \usuDelta}_L{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuLambda}{\,<\,}8$, or ${\fam=\ssfam K}$ would be a Lie group by \cite{cp} 53.2. Consequently $\dim{\fam=\ssfam \usuLambda}{\,\ge\,}4$. \\ (c) If $\dim{\fam=\ssfam K}{\,=\,}0$, then ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,=\,}15$ and ${\cal D}$ is the classical quaternion plane (see \cite{sz1} 5.\,1,3, and~5). In this case, ${\fam=\ssfam \usuDelta}$ fixes a connected subset of $uv\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal D}$ pointwise, and ${\cal F}_{\fam=\ssfam \usuLambda}$ is a connected subplane of dimension at most $4$. By \cite{cp} 55.4 the plane $\langle z^{\fam=\ssfam N},u,w\rangle{\,\le\,}{\cal F}_{\fam=\ssfam \usuLambda}$ is connected, and then ${\fam=\ssfam N}$ is a Lie group by \cite{cp} 32.21 and 71.2. \\ (d) As in the proof of 6.0, the connected component ${\fam=\ssfam K}^1$ is a product of a commutative connected group ${\fam=\ssfam A}$ with at most one almost simple factor ${\fam=\ssfam \usuOmega}{\,\cong\,}\Spin3{\fam=\Bbbfam R}$, and ${\fam=\ssfam A}$ is in the centralizer of ${\fam=\ssfam \usuLambda}$. Either ${\fam=\ssfam K}^1{\,\cong\,}{\fam=\ssfam \usuOmega}$ or $z{\,\ne\,}z^{\fam=\ssfam A}{\,\subseteq\,}{\cal F}_{\fam=\ssfam \usuLambda}$ and ${\cal F}_{\fam=\ssfam \usuLambda}$ is connected. In the second case ${\fam=\ssfam N}$ would again be a Lie group. \\ (e) If ${\fam=\ssfam K}^1{\,\cong\,}{\fam=\ssfam \usuOmega}$, then ${\fam=\ssfam \usuLambda}/\Cs{\fam=\ssfam \usuLambda}{\fam=\ssfam \usuOmega}{\,\le\,}\mathop{{\rm Aut}}{\fam=\ssfam \usuOmega}{\,\cong\,}\SO3{\fam=\Bbbfam R}$ and $\dim\Cs{\fam=\ssfam \usuLambda}{\fam=\ssfam \usuOmega}{\,\ge\,}1$; in other words, ${\fam=\ssfam \usuLambda}$ has a connected subgroup ${\fam=\ssfam P}$ which centralizes ${\fam=\ssfam \usuOmega}$ and ${\fam=\ssfam K}{\,\approx\,}z^{\fam=\ssfam K}{\,\subseteq\,}{\cal F}_{\fam=\ssfam P}{\,\ge\,}{\cal F}_{\fam=\ssfam \usuLambda}$. Therefore ${\cal F}_{\fam=\ssfam P}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$ and ${\fam=\ssfam P}$ is compact. In particuar, there is an involution $\rho{\,\in\,}{\fam=\ssfam P}$ such that ${\cal F}_{\hskip-2pt\rho}{\,=\,}{\cal F}_{\fam=\ssfam P}$. As $z^\rho{\,=\,}z$, the
induced map $\overline\rho{\,=\,}\rho|_{\cal D}$ is either a Baer involution or a reflection. \\ (f) If $\overline\rho$ is planar, i.e., if ${\cal D}\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal F}_{\hskip-2pt\rho}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal D},{\cal F}_{\hskip-2pt\rho}$, then the lines of ${\cal D}$ are $4$-spheres (cf. \cite{cp} 53.10 or \cite{sz3} 3.7) and so are the lines of ${\cal F}_{\hskip-2pt\rho}$. Richarson's theorem $(\dagger)$ implies that the action of ${\fam=\ssfam \usuOmega}$ on $L\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal F}_{\hskip-2pt\rho}$ is equivalent to the standard action of $\SU2{\fam=\Bbbfam C}$ on ${\fam=\Bbbfam S}_4$. Hence ${\fam=\ssfam \usuOmega}$ cannot fix the $2$-sphere $L\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal F}_{\overline\rho}$. This contradiction shows that $\overline\rho$ is a reflection with axis $W\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal D}$ and some center $a$. \\ (g) In the latter case, the lines of ${\cal D}$ are again homeomorphic to ${\fam=\Bbbfam S}_4$; this follows from the action of ${\fam=\ssfam \usuDelta}$ on ${\cal D}$: if $L\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal D}$ is not a manifold, then ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuDelta}_L{\,<\,}4$ by \cite{cp} 53.2 \ (cf. also \cite{HK} 5.5), and ${\fam=\ssfam \usuDelta}_L{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\ge\,}9$. On the other hand, ${\fam=\ssfam \usuDelta}_a{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,<\,}7$, or $L\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal D}$ would be a manifold by
\cite{sz15}~$(\ast\ast)$. Therefore $\dim a^{{\fam=\ssfam \usuDelta}_L}{\,=\,}3$, and $\overline\rho^{{\fam=\ssfam \usuDelta}_L}\overline\rho$ is a $3$-dimensional translation group ${\fam=\ssfam \usuTheta}$ of ${\cal D}$, but $\dim{\fam=\ssfam \usuTheta}$ is even by 2.9. \\
(h) The lines of ${\cal K}{\,=\,}{\cal F}_{\hskip-2pt\rho}$ are also $4$-spheres, in fact, $W\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal D}{\,=\,}W\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal K}$ and $S{\,=\,}L\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal K}{\,\approx\,}{\fam=\Bbbfam S}_4$. This is a consequence of \cite{cp} 92.16 and 51.20, see the proof of 4.0. \\ (i) Now Richarson's theorem $(\dagger)$ can be applied to the action of ${\fam=\ssfam \usuOmega}$ on $S$. It follows that $S\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal D}{\,=\,}\{a,v\}$ and that ${\fam=\ssfam \usuOmega}$ acts freely on $S\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a,v\}$. The orbit space $S/{\fam=\ssfam \usuOmega}$ is an interval and the compact group ${\fam=\ssfam N}$ in $\Cs{}{\fam=\ssfam \usuOmega}$ induces the identity on this interval. Hence $z^{{\fam=\ssfam \usuOmega}{\fam=\ssfam N}}{\,=\,}z^{\fam=\ssfam \usuOmega}$, but $({\fam=\ssfam \usuOmega}{\fam=\ssfam N})_z{\,=\,}1\kern-2.5pt {\rm l}$. This contradiction completes the proof. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\bf 7.0$'$ Lemma.} {\it Suppose that ${\fam=\ssfam \usuDelta}$ is a group of automorphisms of an $8${\rm -dimensional} plane~${\cal P}$. If ${\fam=\ssfam \usuDelta}$ fixes at least $3$ distinct points $u,v,w$ and exactly one line $W$, and if $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}8$, then ${\fam=\ssfam \usuDelta}$ is a Lie group\/}. \par
The {\tt proof} is similar to the previous one but less involved. If ${\fam=\ssfam \usuDelta}/{\fam=\ssfam N}$ is a Lie group, but the compact central subgroup ${\fam=\ssfam N}$ is not, then the arguments of 4.0 show that for some $x{\,\notin\,}W$ the orbit $x^{\fam=\ssfam \usuDelta}$ generates a proper connected subplane. Put ${\cal D}{\,=\,}\langle x^{\fam=\ssfam \usuDelta},u,v,w\rangle$ and
${\fam=\ssfam \usuDelta}|_{\cal D}{\,=\,}{\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$, and apply the stiffness results \cite{sz1} 1.5 to ${\fam=\ssfam K}$. If ${\cal D}$ is flat, then ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\le\,}3$ and $\dim{\fam=\ssfam K}{\,\le\,}3$ \ (see also \cite{cp} 83.12); if ${\cal D}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$, then ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\le\,}6$ and $\dim{\fam=\ssfam K}{\,\le\,}1$ \ (\cite{cp} 83.11). \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\bf 7.1.} {\it If ${\fam=\ssfam \usuDelta}$ is semi-simple of dimension $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}14$, then ${\fam=\ssfam \usuDelta}$ is a Lie group\/}, cf. 6.1. \par
{\bf 7.2 Semi-simple groups.} {\it If ${\fam=\ssfam \usuDelta}$ is semi-simple with more than $2$ collinear fixed points, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}18$\/}. \par
{\tt Proof.} (a) Suppose that $\dim{\fam=\ssfam \usuDelta}{\,>\,}14$. Then each involution in ${\fam=\ssfam \usuDelta}$ is planar, there is no ${\fam=\ssfam \usuDelta}$-invariant proper subplane, and the center ${\fam=\ssfam Z}$ of ${\fam=\ssfam \usuDelta}$ does not contain an involution, cf.~5.2. Moreover, ${\fam=\ssfam Z}$ acts freely on the affine point set, because $x^\zeta{\,=\,}x$ and $\zeta{\,\in\,}{\fam=\ssfam Z}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{1\kern-2.5pt {\rm l}\}$ imply $\langle x^{\fam=\ssfam \usuDelta}\rangle{\,\le\,}{\cal F}_\zeta$. \\ (b) Let ${\fam=\ssfam \usuGamma}$ be an almost simple factor of ${\fam=\ssfam \usuDelta}$, and put ${\fam=\ssfam \usuPsi}{\,=\,}(\Cs{\fam=\ssfam \usuDelta}{\fam=\ssfam \usuGamma})^1$. In steps (c--g), assume that ${\fam=\ssfam \usuGamma}$ is a proper factor of minimal dimension. \\ (c) {\it If ${\fam=\ssfam \usuGamma}$ contains an involution $\iota$, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}18$\/}: consider
${\fam=\ssfam \usuPsi}|_{{\cal F}_\iota}{\,=\,}{\fam=\ssfam \usuPsi}/{\fam=\ssfam K}$ and ${\fam=\ssfam \usuLambda}{\,=\,}{\fam=\ssfam K}^1$. According to \cite{sz1} 5.2,
we have ${\fam=\ssfam \usuPsi}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\le\,}9$, or ${\fam=\ssfam \usuPsi}/{\fam=\ssfam K}$ is locally isomorphic to $\Opr5({\fam=\Bbbfam R},2)$ and ${\fam=\ssfam \usuPsi}$ has a subgroup $\SO3{\fam=\Bbbfam R}$ by step (a). The factor ${\fam=\ssfam \usuLambda}$ of ${\fam=\ssfam \usuDelta}$ is trivial or isomorphic to $\Spin3{\fam=\Bbbfam R}$\, (see 2.6(\^b)\hskip1pt), but the latter is excluded by (a). If $\dim{\fam=\ssfam \usuPsi}{\,=\,}9$, then $\dim{\fam=\ssfam \usuGamma}{\,=\,}3$ and $\dim{\fam=\ssfam \usuDelta}{\,=\,}12$; if $\dim{\fam=\ssfam \usuPsi}{\,<\,}9$, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}16$. In the case $\dim{\fam=\ssfam \usuGamma}{\,=\,}\dim{\fam=\ssfam \usuPsi}{\,=\,}10$ both factors would contain $\SO3{\fam=\Bbbfam R}$, but this contradicts
Lemma 2.11. Hence $\dim{\fam=\ssfam \usuGamma}{\,\le\,}8$ and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}18$. \\
(d) Suppose that $\{1\kern-2.5pt {\rm l}\}$ is the only compact subgroup of ${\fam=\ssfam \usuDelta}$. Then each factor of
${\fam=\ssfam \usuDelta}$ is isomorphic to the simply connected covering group ${\fam=\ssfam \usuOmega}$ of $\SL2{\fam=\Bbbfam R}$.
If such a factor ${\fam=\ssfam \usuGamma}$ is straight, then Baer's theorem \cite{ba} implies
${\fam=\ssfam \usuGamma}$ consists of translations, because ${\fam=\ssfam \usuGamma}$ is not planar and does not consist of
homologies by \cite{cp} 61.2, cf. also 5.2(g). All translations in
${\fam=\ssfam \usuDelta}$ have the same center (or the translation group would be commutative). Hence there
are at most two straight factors. \\ (e) For each $x{\,\notin\,}W$ and each factor ${\fam=\ssfam \usuGamma}$ of ${\fam=\ssfam \usuDelta}$ the orbit $x^{{\fam=\ssfam \usuGamma}\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam Z}}$ is non-trivial by step (a), and $x^{\fam=\ssfam \usuGamma}{\,\ne\,}x$. Consequently ${\cal E}{\,=\,}{\cal E}(x,{\fam=\ssfam \usuGamma}){\,=\,}\langle x^{{\fam=\ssfam \usuGamma}\hskip-2pt},u,v,w\rangle$ is a connected subplane, and ${\cal E}$ is not flat because ${\fam=\ssfam \usuGamma}$ acts non-trivially on ${\cal E}$\, (see \cite{cp} 3.8). If ${\cal E}{\,\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\hskip-3pt}{\cal P}$, then
${\fam=\ssfam \usuPsi}_{\hskip-1pt x}|_{\cal E}{\,=\,}1\kern-2.5pt {\rm l}$,\, ${\fam=\ssfam \usuPsi}_{\hskip-1pt x}$ is compact by 2.6(b), and then ${\fam=\ssfam \usuPsi}_{\hskip-1pt x}{\,=\,}1\kern-2.5pt {\rm l}$,\, $\dim{\fam=\ssfam \usuPsi}{\,\le\,}15$, and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}18$. We may assume, therefore, that each plane ${\cal E}(x,{\fam=\ssfam \usuGamma})$ is $4$-dimensional. \\ (f) Choose a factor ${\fam=\ssfam \usuGamma}$ which is not straight. If $L$ is a line in ${\cal E}$ through $z{\,\in\,}\{u,v,w\}$, then $\dim{\fam=\ssfam \usuGamma}_{\hskip-2pt L}{\,>\,}0$ by the assumption in step~(e). From $L^{\fam=\ssfam \usuPsi}{\,=\,}{\frak L}_z\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt W$ it follows that ${\fam=\ssfam \usuGamma}_{\hskip-2pt L}$ and hence ${\fam=\ssfam \usuGamma}$ consists of translations with center $z$, and ${\fam=\ssfam \usuGamma}$ would be straight against our assumption. Consequently there are lines $K{\,=\,}xu$ and $L{\,=\,}xv$ such that $\dim K^{\fam=\ssfam \usuPsi}\!,\:\dim L^{\fam=\ssfam \usuPsi}{\,<\,}8$ and ${\fam=\ssfam \usuPsi}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuPsi}_{\hskip-1pt x}{\,\le\,}14$. \\ (g) Consider a factor $\hat{\fam=\ssfam \usuGamma}{\,\ne\,}{\fam=\ssfam \usuGamma}$, the product ${\fam=\ssfam X}{\,=\,}{\fam=\ssfam \usuPsi}\kern 2pt {\scriptstyle \cap}\kern 2pt\hat{\fam=\ssfam \usuPsi}$ of the remaining factors, and the corresponding planes ${\cal E},\,\hat{\cal E}{\,=\,}{\cal E}(x,\hat{\fam=\ssfam \usuGamma})$, where $x$ is chosen as in step (f) such that ${\fam=\ssfam X}{\hskip1pt:\hskip1pt}{\fam=\ssfam X}_x{\,\le\,}14$. By 2.14, we have ${\cal E}^{\hskip1pt\hat{\fam=\ssfam \usuGamma}}{\,\ne\,}{\cal E}$,\; $x^{\hat{\fam=\ssfam \usuGamma}}{\,\not\subseteq\,}{\cal E}$,\; $\hat{\cal E}{\,\ne\,}{\cal E}$, and ${\cal F}{\,=\,}\langle {\cal E},\hat{\cal E}\rangle{\,\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\!}{\cal P}$.
Obviously, ${\fam=\ssfam X}_x|_{\cal F}{\,=\,}1\kern-2.5pt {\rm l}$,\: ${\fam=\ssfam X}_x$ is compact by 2.6(b), and then ${\fam=\ssfam X}_x{\,=\,}1\kern-2.5pt {\rm l}$. Now $\dim{\fam=\ssfam X}{\,\le\,}12$ because ${\fam=\ssfam X}$ is a product of $3$-dimensional factors, and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}18$. \\ (h) Corollary. {\it If $\dim{\fam=\ssfam \usuDelta}{\,>\,}18$, and if ${\fam=\ssfam \usuDelta}$ is not almost simple, then all factors of minimal dimension are isomorphic to the simply connected covering group ${\fam=\ssfam \usuOmega}$ of $\SL2{\fam=\Bbbfam R}$ and there is at least some other factor\/}. \\ (i) Assume from now on that $\dim{\fam=\ssfam \usuDelta}{\,>\,}18$, and choose a factor ${\fam=\ssfam \usuGamma}{\,\not\hskip1pt\cong\,}{\fam=\ssfam \usuOmega}$ of ${\fam=\ssfam \usuDelta}$. Then $\dim{\fam=\ssfam \usuGamma}{\,\ge\,}6$ and ${\fam=\ssfam \usuGamma}$ contains a planar involution $\iota$. The arguments in step (c) show that either $\dim\hskip-.5pt{\fam=\ssfam \usuPsi}{\,\le\,}9$, or ${\fam=\ssfam \usuPsi}{\,\circeq\,}\Opr5({\fam=\Bbbfam R},2)$ has a subgroup $\SO3{\fam=\Bbbfam R}$. In both cases $\dim{\fam=\ssfam \usuGamma}{\,\ge\,}10$. Note that this is true for each admissible choice of ${\fam=\ssfam \usuGamma}$. Consequently, ${\fam=\ssfam \usuDelta}$ has no factors of dimension $6$ or~$8$. If $\dim{\fam=\ssfam \usuGamma}{\,=\,}10$, then ${\fam=\ssfam \usuGamma}$ contains $\SO3{\fam=\Bbbfam R}$ because of step (a), and Lemma 2.11 implies $\dim{\fam=\ssfam \usuPsi}{\,\le\,}9$. There is another involution $\iota'{\,\in\,}{\fam=\ssfam \usuGamma}$ which commutes with $\iota$, and $\iota'$ fixes a subplane ${\cal C}{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal F}_\iota$, or it induces a reflection on ${\cal F}_\iota$ with some center $a{\,\notin\,}W$. In the first case, the stiffness result \cite{sz1} 1.5(4) shows that ${\fam=\ssfam \usuPsi}$ acts almost effectively on ${\cal C}$, and
$\dim{\fam=\ssfam \usuPsi}{\,\le\,}3$ by 2.14; in the second case, $a^{\fam=\ssfam \usuPsi}{\,=\,}a$,\: ${\fam=\ssfam \usuPsi}|_{a^{\fam=\ssfam \usuGamma}}{\,=\,}1\kern-2.5pt {\rm l}$,\: $\langle a^{\fam=\ssfam \usuGamma}\rangle{\,=\,}\langle a^{\fam=\ssfam \usuDelta}\rangle{\,=\,}{\cal P}$, and ${\fam=\ssfam \usuPsi}{\,=\,}1\kern-2.5pt {\rm l}$. Hence we have even $\dim{\fam=\ssfam \usuGamma}{\,\ge\,}16$. By Priwitzer's results \cite{pw1} (see 5.2(j)\,), we may assume that $\dim{\fam=\ssfam \usuGamma}{\,\le\,}30$. This proves: \\ (j) {\it If $\dim{\fam=\ssfam \usuDelta}{\,>\,}18$, then ${\fam=\ssfam \usuDelta}$ is almost simple or a product of an almost simple group with ${\fam=\ssfam \usuOmega}$\/}. \\ (k) The almost simple groups ${\fam=\ssfam \usuGamma}$ with $\dim{\fam=\ssfam \usuGamma}{\,\ge\,}20$ have been dealt with in 5.2 steps (j--l) under the assumption that ${\fam=\ssfam \usuGamma}$ fixes a flag (and possibly further elements). In each case a contradiction has been obtained. Thus, only the possibility ${\fam=\ssfam \usuGamma}{\,\cong\,}{\rm (P)}\SL3{\fam=\Bbbfam C}$ and ${\fam=\ssfam \usuDelta}{\,=\,}{\fam=\ssfam \usuGamma}{\fam=\ssfam \usuOmega}$ remains. An involution $\iota{\,\in\,}{\fam=\ssfam \usuGamma}$ is the center of a subgroup ${\fam=\ssfam \usuUpsilon}{\,\cong\,}\SL2{\fam=\Bbbfam C}$ of ${\fam=\ssfam \usuGamma}$, and ${\fam=\ssfam \usuUpsilon}$ acts almost effectively on ${\cal F}_\iota$. A maximal compact subgroup ${\fam=\ssfam \usuPhi}{\,\cong\,}\SU2{\fam=\Bbbfam C}$ of ${\fam=\ssfam \usuUpsilon}$ induces on ${\cal F}_\iota$ a group $\SO3{\fam=\Bbbfam R}$, and the
involutions of ${\fam=\ssfam \usuPhi}|_{{\cal F}_\iota}$ are planar. Hence ${\cal F}_\iota$ has lines homeomorphic to ${\fam=\Bbbfam S}_4$, and Richardson's theorem$(\dagger)$ implies that ${\cal E}{\,=\,}{\cal F}_{\iota,{\fam=\ssfam \usuPhi}}$ is flat.
By Stiffness, ${\fam=\ssfam \usuOmega}$ acts almost effectively on ${\cal E}$, and ${\fam=\ssfam \usuOmega}|_{\cal E}$ is almost simple;
on the other hand, ${\fam=\ssfam \usuOmega}|_{\cal E}$ is solvable by \cite{cp} 33.8. This contradiction completes the proof. \hglue 0pt plus 1filll $\scriptstyle\square$
\par
{\bf 7.3 Normal torus.} {\it If ${\fam=\ssfam \usuDelta}$ has a one-dimensional compact connected normal subgroup ${\fam=\ssfam \usuTheta}$, and if $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}15$, then there exists a ${\fam=\ssfam \usuDelta}$-invariant classical quaternion subplane ${\cal H}$, and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}16$\/}. \par
{\tt Proof.} ${\fam=\ssfam \usuDelta}$ is a Lie group by 7.0, hence ${\fam=\ssfam \usuTheta}{\,\cong\,}{\fam=\Bbbfam T}$ is a torus.
As in 6.3, the involution $\iota{\,\in\,}{\fam=\ssfam \usuTheta}$ is in the center of ${\fam=\ssfam \usuDelta}$, and
${\cal H}{\,:=\,}{\cal F}_\iota{\,=\,}{\cal H}^{\fam=\ssfam \usuDelta}$ is a Baer subplane. Put again
${\fam=\ssfam \usuDelta}|_{\cal H}{\,=\,}{\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$. If ${\fam=\ssfam \usuTheta}|_{\cal H}{\,\ne\,}1\kern-2.5pt {\rm l}$, then it
follows from \cite{sz1} 7.3 and Stiffness that $\dim{\fam=\ssfam \usuDelta}{\,\le\,}7{+}3$.
Therefore ${\fam=\ssfam \usuTheta}{\,\le\,}{\fam=\ssfam K}$,\: ${\fam=\ssfam K}^1{\,\not\cong\,}\Spin3{\fam=\Bbbfam R}$,\: $\dim{\fam=\ssfam K}{\,=\,}1$,
and $14{\,\le\,}{\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\le\,}15$. Now the claim is a consequence of \cite{sz1} 7.3 or \cite{sz2} Th.~5. \hglue 0pt plus 1filll $\scriptstyle\square$
\par
{\bf 7.4 Theorem.} {\it If $\dim {\fam=\ssfam \usuDelta} \ge 32$ and ${\fam=\ssfam \usuDelta}$ has\/} ({\it at least\/}) {\it $3$ fixed points, then ${\fam=\ssfam \usuDelta}$ contains a trans\-itive translation group ${\fam=\ssfam T}$. Either $\dim {\fam=\ssfam \usuDelta} = 32$ and a maximal semi-simple subgroup ${\fam=\ssfam \usuPsi}$ of ${\fam=\ssfam \usuDelta}$ is isomorphic to $\SU 4 {\fam=\Bbbfam C}$, or $\dim {\fam=\ssfam \usuDelta} \ge 37$ and ${\cal P} \cong {\cal O}$\/}. \par
This has been {\tt proved} in \cite{sz14}; the result seems to be best possible. Translation planes with a group locally isomorphic to $\SU4{\fam=\Bbbfam C}$ have been studied in a long paper by H\"ahl \cite{Ha4} without any assumption on the existence of a fixed point on the translation axis: \par
{\bf 7.5 Theorem.} {\it Let ${\fam=\ssfam \usuSigma}$ be the automorphism group of a translation plane with translation axis $W$. If ${\fam=\ssfam \usuSigma}$ has a subgroup ${\fam=\ssfam \usuUpsilon}{\,\circeq\,}\SU4{\fam=\Bbbfam C}$ with a unique fixed point $o{\,\notin\,}W$, then ${\fam=\ssfam \usuUpsilon}$ is normal in the connected component of ${\fam=\ssfam \usuSigma}_o$ or ${\cal P}$ is the classical octonion plane\/}. \par
{\tt Remark.} Specific non-classical examples have been given in 4.12. \par
{\Bf 8. Fixed double flag} \par
Throughout this section, ${\fam=\ssfam \usuDelta}$ fixes exactly $2$ points and $2$ lines, i.e., ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\langle u,v,av\rangle$ is a {\it double flag\/}. In the classical plane ${\cal O}$ a maximal semi-simple subgroup ${\fam=\ssfam \usuPhi}$ of ${\fam=\ssfam \usuSigma}_{u,v,av}$ is isomorphic to $\Spin8{\fam=\Bbbfam R}$ and fixes even a triangle. \par
{\bf 8.0 Semi-simple Lie groups.} {\it If ${\fam=\ssfam \usuDelta}$ is semi-simple, ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\langle u,v,av\rangle$ is a double flag, and $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}26$, then ${\fam=\ssfam \usuDelta}$ is a Lie group\/}. \par
{\tt Proof.} By \cite{psz} or 2.3 it suffices to consider the case $\dim{\fam=\ssfam \usuDelta}{\,=\,}26$. Write ${\fam=\ssfam \usuDelta}{\,=\,}{\fam=\ssfam \usuGamma}{\fam=\ssfam \usuPsi}$, where ${\fam=\ssfam \usuGamma}$ is an almost simple factor of {\it maximal\/} dimension. Note that $\dim{\fam=\ssfam \usuGamma}{\,\ge\,}8$ since $\dim{\fam=\ssfam \usuDelta}{\,\equiv\,}2\bmod3$. \par (a) ${\fam=\ssfam \usuDelta}$ does not contain a reflection $\sigma$ with center $v$ or axis $uv$\: (or else $\sigma^{\fam=\ssfam \usuDelta}\sigma$ would be a commutative normal subgroup, see 2.9). Therefore each involution in ${\fam=\ssfam \usuDelta}$ is either planar or a reflection in ${\fam=\ssfam \usuDelta}_{[u,av]}$, and one of two commuting involutions is planar, see 2.10. Moreover, any non-central involution $\iota$ of an almost simple factor ${\fam=\ssfam X}$ is planar: if $\iota$ is a reflection, then $\langle \iota^{\fam=\ssfam X}\rangle{\,=\,}{\fam=\ssfam X}{\,\le\,}{\fam=\ssfam \usuDelta}_{[u,av]}$, and ${\fam=\ssfam X}$ is compact or two-ended by \cite{cp} 61.2, hence a compact Lie group of rank $1$. If ${\fam=\ssfam X}{\,\cong\,}\SO3{\fam=\Bbbfam R}$, the involutions in ${\fam=\ssfam X}$ are planar; if ${\fam=\ssfam X}$ is simply connected, the only involution in ${\fam=\ssfam X}$ is central. \par (b) {\it If ${\fam=\ssfam \usuUpsilon}$ is a semi-simple group in the centralizer of a planar involution $\beta$, then~$\dim{\fam=\ssfam \usuUpsilon}{\le13}$\/}:\: Consider the induced group
${\fam=\ssfam \usuUpsilon}|_{{\cal F}_\beta}{\,=\,}{\fam=\ssfam \usuUpsilon}/{\fam=\ssfam K}$. From \cite{sz1} 6.1 it follows that ${\fam=\ssfam \usuUpsilon}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\le\,}10$. On the other hand, ${\fam=\ssfam K}^1{\,=\,}{\fam=\ssfam \usuLambda}{\,\triangleleft\,}{\fam=\ssfam \usuUpsilon}$ is compact and semi-simple, and then ${\fam=\ssfam \usuLambda}$ is a Lie group (cf. \cite{cp} 93.11), and Stiffness implies ${\fam=\ssfam \usuLambda}{\,\cong\,}\Spin3{\fam=\Bbbfam R}$ or ${\fam=\ssfam \usuLambda}{\,=\,}1\kern-2.5pt {\rm l}$. \\ (c) Suppose that $\dim{\fam=\ssfam \usuGamma}{\,=\,}20$. Then ${\fam=\ssfam \usuGamma}{\,\cong\,}\Sp4{\fam=\Bbbfam C}$\; (because $\PSp4{\fam=\Bbbfam C}$ contains $\SO5{\fam=\Bbbfam C}{\,>\,}\SO5{\fam=\Bbbfam R}$) and $\beta{\,=\,}{\rm diag}(1,1,-1,-1)$ is planar. We have $\Cs{}\beta{\,\ge\,}(\SL2{\fam=\Bbbfam C})^2{\cdot}{\fam=\ssfam \usuPsi}$, and $\dim\Cs{}\beta$ would be too large. \\ (d) If $\dim{\fam=\ssfam \usuGamma}{\,=\,}16$, then ${\fam=\ssfam \usuGamma}{\,\cong\,}{\rm(P\hskip-1pt)}\SL3{\fam=\Bbbfam C}$ and ${\rm diag}(1,-1,-1)$ corresponds to a planar involution $\beta$ containing $\SL2{\fam=\Bbbfam C}{\hskip1pt\cdot{\fam=\ssfam \usuPsi}}$ in its centralizer, and $\dim\Cs{}\beta$ is again too large. \\ (e) Each group ${\fam=\ssfam \usuGamma}$ of type ${\rm A}_3$ contains a planar involution $\beta$ centralizing a $6$-dimensional semi-simple subgroup of ${\fam=\ssfam \usuGamma}$. This is easy in the cases $\SU4{\fam=\Bbbfam C}$,\, $\SL2{\fam=\Bbbfam H}$,\, $\SL4{\fam=\Bbbfam R}$ and its universal covering group; it is less obvious for the groups related to $\SU4({\fam=\Bbbfam C},r)$, $r{\,=\,}1,2$, which are not necessarily Lie groups, see 8.2(j) Case 4) below. Once again this contradicts step (b). \\ (f) If ${\fam=\ssfam \usuGamma}$ is the compact group ${\rm G}_2$, then ${\fam=\ssfam \usuPsi}$ would be solvable by Ê\cite{cp} 33.8. In the case ${\fam=\ssfam \usuGamma}{\,\circeq\,}{\rm G}_2(2)$ a maximal compact subgroup ${\fam=\ssfam \usuPhi}$ of ${\fam=\ssfam \usuGamma}$ is isomorphic to $\SO4{\fam=\Bbbfam R}$ or to $(\Spin3{\fam=\Bbbfam R})^2$, and ${\fam=\ssfam \usuPhi}{\fam=\ssfam \usuPsi}$ would centralize a planar involution in contradiction to step (b). \\ (g) {\tt Corollary.} {\it All factors of ${\fam=\ssfam \usuDelta}$ have dimension at most $10$\/}. \\ (h) {\it If $\dim{\fam=\ssfam \usuGamma}{\,=\,}10$, then $ {\fam=\ssfam \usuDelta}$ is a Lie group\/}. Suppose that $x^{\fam=\ssfam \usuGamma}{\,=\,}x$ for some point $x{\,\notin\,}av\kern 2pt {\scriptstyle \cup}\kern 2pt uv$. Then $x^{\fam=\ssfam \usuDelta}{\,=\,}x^{\fam=\ssfam \usuPsi}$,\:
${\cal D}{\,:=\,}\langle x^{\fam=\ssfam \usuPsi}\rangle{\,=\,}{\cal D}^{\fam=\ssfam \usuDelta}$, and ${\fam=\ssfam \usuGamma}|_{\cal D}{\,=\,}1\kern-2.5pt {\rm l}$. Now ${\cal D}$ is flat by Stiffness, but then $\dim{\fam=\ssfam \usuDelta}$ would be too small. Dually each fixed line of ${\fam=\ssfam \usuGamma}$ contains either $u$ or $v$. As $\dim{\fam=\ssfam \usuDelta}_{[u,av]}{\,\le\,}8$, we have ${\fam=\ssfam \usuGamma}{\,\ne\,}{\fam=\ssfam \usuGamma}_{\hskip-2pt[u]}$. If ${\fam=\ssfam \usuGamma}{\,=\,}{\fam=\ssfam \usuGamma}_{\hskip-2pt[v]}$, then the dual of \cite{cp} 61.20(a) shows that ${\fam=\ssfam \usuGamma}{\,\cong\,}{\fam=\ssfam \usuGamma}_{\hskip-2pt[au]}{\ltimes}{\fam=\ssfam \usuGamma}_{\hskip-2pt[uv]}$ is not almost simple. Hence there is a point $p{\,\notin\,}av\kern 2pt {\scriptstyle \cup}\kern 2pt uv$ such that ${\cal E}{\,=\,}\langle p^{\fam=\ssfam \usuGamma}\rangle$ is a subplane, and ${\cal E}{\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\hskip-3pt}{\cal P}$, or else
$\dim{\fam=\ssfam \usuGamma}|_{\cal E}{\,\le\,}6$. As ${\fam=\ssfam \usuPsi}_{\hskip-1pt p}|_{\cal E}{\,=\,}1\kern-2.5pt {\rm l}$, Stiffness shows that ${\fam=\ssfam \usuPsi}_{\hskip-1pt p}$ is compact and $\dim{\fam=\ssfam \usuPsi}_{\hskip-1pt p}{\,\le\,}7$. Therefore $\dim p^{\fam=\ssfam \usuPsi}{>\,}8$, \: $\langle p^{\fam=\ssfam \usuPsi}\rangle{\,=\,}{\cal P}$, and ${\fam=\ssfam \usuGamma}_{\hskip-2pt p}{\,=\,}1\kern-2.5pt {\rm l}$. It follows that ${\cal E}{\,=\,}{\cal P}$,\: ${\fam=\ssfam \usuPsi}_{\hskip-1pt p}{\,=\,}1\kern-2.5pt {\rm l}$, and $\dim p^{\fam=\ssfam \usuPsi}{\,=\,}16$. According to \cite{cp} 96.11(a) the orbit $p^{\fam=\ssfam \usuPsi}$ is open in $P$, and then ${\fam=\ssfam \usuPsi}$ is a Lie group by \cite{cp} 53.2. The center of ${\fam=\ssfam \usuGamma}$ contains arbitrarily small compact subgroups ${\fam=\ssfam N}$ such that ${\fam=\ssfam \usuGamma}/{\fam=\ssfam N}$ is a Lie group (cf. \cite{cp} 93.8.) For each $\nu{\,\in\,}{\fam=\ssfam N}$ there exists a unique $\hat\nu{\,\in\,}{\fam=\ssfam \usuPsi}$ with $p^\nu{\,=\,}p^{\hat\nu}$. Because ${\fam=\ssfam N}{\,\le\,}\Cs{}{\fam=\ssfam \usuDelta}$, the map $(\nu{\,\mapsto\,}\hat\nu){\,:\,}{\fam=\ssfam N}{\,\to\,}{\fam=\ssfam \usuPsi}$ is a continuous homomorphism, ${\fam=\ssfam N}$ is a Lie group, and so are ${\fam=\ssfam \usuGamma}$~and~${\fam=\ssfam \usuDelta}$. \\ (i) {\it If $\dim{\fam=\ssfam \usuGamma}{\,=\,}8$, then ${\fam=\ssfam \usuDelta}$ is a Lie group\/}. The first arguments are similar to those in step~(h), but not quite the same. If $x^{\fam=\ssfam \usuGamma}{=\,}x{\,\notin\,}av\kern 2pt {\scriptstyle \cup}\kern 2pt uv$ and
${\cal D}{\,=\,}\langle x^{\fam=\ssfam \usuPsi}\rangle$, then ${\fam=\ssfam \usuGamma}|_{\cal D}{\,=\,}1\kern-2.5pt {\rm l}$ and $\dim{\cal D}{\,\le\,}4$. Let $y{\,\in\,}{\cal D}$ such that ${\fam=\ssfam \usuLambda}{\,=\,}{\fam=\ssfam \usuDelta}_{x,y}$ fixes a quadrangle. It follows that ${\fam=\ssfam \usuDelta}{\,:\,}{\fam=\ssfam \usuLambda}{\,\le\,}8$ and $\dim{\fam=\ssfam \usuLambda}{\,\le\,}14$. Thus $\dim{\fam=\ssfam \usuDelta}$ would be too small. Dually $L^{\fam=\ssfam \usuGamma}{\,=\,}L{\,\in\,}{\frak L}$ implies $L{\,\in\,}{\frak L}_u\kern 2pt {\scriptstyle \cup}\kern 2pt{\frak L}_v$. If ${\fam=\ssfam \usuGamma}{\,=\,}{\fam=\ssfam \usuGamma}_{\hskip-1.5pt[u]}$, then ${\fam=\ssfam \usuGamma}$ is compact by \cite{cp} 61.2 and transitive on $uv\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{u,v\}$, which is impossible. If ${\fam=\ssfam \usuGamma}{\,=\,}{\fam=\ssfam \usuGamma}_{\hskip-2pt[v]}$, then ${\fam=\ssfam \usuGamma}{\,\ne\,}{\fam=\ssfam \usuGamma}_{\hskip-2pt[v,uv]}$ \ (note that ${\fam=\ssfam \usuGamma}$ contains an involution and apply \cite{cp} 55.28). Therefore ${\fam=\ssfam \usuGamma}{\,\cong\,}{\fam=\ssfam \usuGamma}_{\hskip-2pt[au]}{\ltimes}{\fam=\ssfam \usuGamma}_{\hskip-2pt[uv]}$ by \cite{cp} 61.20, and ${\fam=\ssfam \usuGamma}$ would not be almost simple.
Again there is a point $p$ such that ${\cal E}{\,=\,}\langle p^{\fam=\ssfam \usuGamma}\rangle{\,\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\!}{\cal P}$. We have ${\fam=\ssfam \usuPsi}_{\hskip-1pt p}|_{\cal E}{\,=\,}1\kern-2.5pt {\rm l}$ and $\dim{\fam=\ssfam \usuPsi}_{\hskip-1pt p}{\,\ge\,}2$ because $\dim{\fam=\ssfam \usuPsi}{\,=\,}18$. Hence ${\cal E}$ is a Baer subplane, $\dim{\fam=\ssfam \usuPsi}_{\hskip-1pt p}{\,\le\,}7$ by Stiffness, $\dim p^{\fam=\ssfam \usuPsi}{\,\ge\,}11$, and $\langle p^{\fam=\ssfam \usuPsi}\rangle{\,=\,}{\cal P}$. The center of ${\fam=\ssfam \usuDelta}$ contains a compact subgroup ${\fam=\ssfam \usuXi}{\,<\,}{\fam=\ssfam \usuPsi}$ such that ${\fam=\ssfam \usuPsi}/{\fam=\ssfam \usuXi}$ is a Lie group. Obviously
${\fam=\ssfam \usuPsi}_{\hskip-1pt p}|_{p^{{\fam=\ssfam \usuGamma}{\fam=\ssfam \usuXi}}}{\,=\,}1\kern-2.5pt {\rm l}$,\; ${\cal E}{\,<\,}{\cal P}$, and $\langle p^{{\fam=\ssfam \usuGamma}{\fam=\ssfam \usuXi}}\rangle{\,=\,}{\cal E}$. As $\langle p^{\fam=\ssfam \usuPsi}\rangle{\,=\,}{\cal P}$, it follows that $({\fam=\ssfam \usuGamma}{\fam=\ssfam \usuXi})_p{\,=\,}1\kern-2.5pt {\rm l}$ and $\dim p^{{\fam=\ssfam \usuGamma}{\fam=\ssfam \usuXi}}{\,=\,}8$. Now $p^{{\fam=\ssfam \usuGamma}{\fam=\ssfam \usuXi}}$ is an open point set in ${\cal E}$, and ${\fam=\ssfam \usuGamma}{\fam=\ssfam \usuXi}$ is a Lie group by \cite{cp} 53.2. Consequently ${\fam=\ssfam \usuGamma}$ and ${\fam=\ssfam \usuPsi}$ are Lie groups. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\bf 8.1 Lemma.} {\it If ${\fam=\ssfam \usuDelta}$ is almost simple, and if ${\cal F}_{\fam=\ssfam \usuDelta}$ is a double flag, then each non-central involution in ${\fam=\ssfam \usuDelta}$ is planar\/}, see step (a) in the previous {\tt proof}. \par
{\bf 8.2 Semi-simple groups.} {\it If ${\cal F}_{\fam=\ssfam \usuDelta}$ is a double flag and ${\fam=\ssfam \usuDelta}$ is a
semi-simple Lie group, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}24$\/}. \par
Because of the possible existence of a reflection in ${\fam=\ssfam \usuDelta}_{[u,av]}$, the {\tt proof} differs in several steps from the proof of 7.2; accordingly, the result is much weaker. By Priwitzer's Theorem 4.2, we may assume $\dim{\fam=\ssfam \usuDelta}{\,\le\,}28$. \\ (a) {\it If there exists a ${\fam=\ssfam \usuDelta}$-invariant proper subplane ${\cal F}$, then
$\dim{\fam=\ssfam \usuDelta}{\,\le\,}18$\/}: write ${\fam=\ssfam \usuDelta}|_{\cal F}{\,=\,}{\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$, and put ${\fam=\ssfam \usuLambda}{\,=\,}{\fam=\ssfam K}^1$ and $\dim{\cal F}{\,=\,}d$. Note that ${\cal F}_{\fam=\ssfam \usuDelta}$ need not be contained in ${\cal F}$. If $d{\,\le\,}2$, then ${\fam=\ssfam \usuDelta}{\,=\,}{\fam=\ssfam K}$ or ${\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$ is solvable by \cite{cp} 33.8, hence ${\fam=\ssfam \usuDelta}{\,=\,}{\fam=\ssfam \usuLambda}$ and $\dim{\fam=\ssfam \usuLambda}{\,<\,}14$ by Stiffness. For $d{\,=\,}4$, there are $3$ subcases: if ${\cal F}{\,=\,}{\cal F}_{\fam=\ssfam K}$, then $u,v,av{\,\in\,}{\cal F}$,\; ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\le\,}3$ by 2.14, and $\dim{\fam=\ssfam K}{\,\le\,}8$. If ${\cal F}{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal F}_{\fam=\ssfam K}{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$, then $v{\,\in\,}{\cal F}$ or $v$ is incident with a unique line $L{\,=\,}L^{\fam=\ssfam \usuDelta}{\,\in\,}{\cal F}$, ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\le\,}8$ by \cite{cp} 71.8 and $\dim{\fam=\ssfam K}{\,\le\,}7$. If ${\cal F}_{\fam=\ssfam K}{\,=\,}{\cal P}$, then ${\fam=\ssfam K}{\,=\,}1\kern-2.5pt {\rm l}$ and ${\fam=\ssfam \usuDelta}$ is a subgroup of the $16$-dimensional group $\PSL3{\fam=\Bbbfam C}$. Finally, let $d{\,=\,}8$. Either $u,v,av{\,\in\,}{\cal F}$,\; ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\le\,}10$ by \cite{sz1} 6.1, and ${\fam=\ssfam \usuLambda}$ is compact and semi-simple, hence a Lie group contained in $\Spin3{\fam=\Bbbfam R}$, or ${\cal F}_{\fam=\ssfam K}{\,=\,}{\cal P}$,\, ${\fam=\ssfam K}{\,=\,}1\kern-2.5pt {\rm l}$, and ${\fam=\ssfam \usuDelta}$ acts faithfully on ${\cal F}$. Each exterior fixed element is incident with a unique interior element, and this is ${\fam=\ssfam \usuDelta}$-invariant. From \cite{sz1} 7.3 it follows that $\dim{\fam=\ssfam \usuDelta}{\,\le\,}13$, or ${\cal F}\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal F}_{\fam=\ssfam \usuDelta}$ is a non-incident point-line pair and ${\cal F}$ is the classical quaternion plane. In the latter case $\dim{\fam=\ssfam \usuDelta}{\,\le\,}18$. (This is best possible: the stabilizer ${\fam=\ssfam X}$ of an exterior flag in a proper Hughes plane fixes a non-incident interior point-line pair and hence a double flag in ${\cal P}$; the group ${\fam=\ssfam X}$ has a semi-simple subgroup ${\fam=\ssfam \usuDelta}$ of dimension $18$.) This proof shows also the following: \\ (b) Corollary. {\it If $u,v,av{\,\in\,}{\cal F}$, in particular, if ${\cal F}_{\fam=\ssfam \usuXi}{\,=\,}{\cal F}_{\hskip-.7pt{\fam=\ssfam \usuXi}}^{\hskip2pt{\fam=\ssfam \usuDelta}}{\,<\,}{\cal P}$ for some set ${\fam=\ssfam \usuXi}{\,\subseteq\,}{\fam=\ssfam \usuDelta}$, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}13$. Hence each central involution in ${\fam=\ssfam \usuDelta}$ is a reflection with center $u$ and axis $av$ whenever $\dim{\fam=\ssfam \usuDelta}{\,>\,}13$\/} (see 2.\,9 and 10). \\ (c) Let ${\fam=\ssfam \usuGamma}$ be a proper almost simple factor of ${\fam=\ssfam \usuDelta}$ and put ${\fam=\ssfam \usuPsi}{\,=\,}(\Cs{\fam=\ssfam \usuDelta}{\fam=\ssfam \usuGamma})^1$. Assume first that ${\fam=\ssfam \usuGamma}$ has $minimal$ dimension. If
${\fam=\ssfam \usuGamma}$ contains a planar involution $\iota$, we write ${\fam=\ssfam \usuPsi}|_{{\cal F}_\iota}{\,=\,}{\fam=\ssfam \usuPsi}/{\fam=\ssfam K}$. According to \cite{sz1} 6.1, we have ${\fam=\ssfam \usuPsi}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\le\,}10$. Either ${\fam=\ssfam K}$ is discrete or ${\fam=\ssfam K}^1{\,\cong\,}\Spin3{\fam=\Bbbfam R}$. In the latter case, $\dim{\fam=\ssfam \usuGamma}{\,=\,}3$ by minimality, $\dim{\fam=\ssfam \usuPsi}{\,\le\,}13$, and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}16$. \\ (d) Suppose from now on that $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}20$. If ${\fam=\ssfam \usuDelta}$ has two distinct factors ${\fam=\ssfam \usuGamma}_{\!\nu}{\,\cong\,}\Spin3{\fam=\Bbbfam R}$, then step (b) implies that the central involution $\iota_\nu{\,\in\,}{\fam=\ssfam \usuGamma}_{\!\nu}$ is a reflection in ${\fam=\ssfam \usuDelta}_{[u,av]}$. As $\iota_1\iota_2{\,=\,}\iota_2\iota_1$, it follows from 2.10 that $\iota_1{\,=\,}\iota_2{\,=\,}\iota$, and ${\fam=\ssfam \usuGamma}_{\!1}{\fam=\ssfam \usuGamma}_{\!2}{\,\cong\,}\SO4{\fam=\Bbbfam R}$ has a subgroup ${\fam=\ssfam \usuPhi}{\,\cong\,}\SO3{\fam=\Bbbfam R}$. Any involution $\beta{\,\in\,}{\fam=\ssfam \usuPhi}$ is distinct from $\iota$, hence planar, and the arguments of step (c) show that $\dim({\fam=\ssfam \usuPsi}_1\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam \usuPsi}_2){\,\le\,}13$,\: $\dim{\fam=\ssfam \usuDelta}{\,<\,}20$ contrary to the assumption. More generally, the same reasoning shows that at most one $3$-dimensional factor has a non-trivial torus subgroup. (Note that a subgroup ${\fam=\Bbbfam T}^2{\,<\,}{\fam=\ssfam \usuGamma}_{\!1}{\fam=\ssfam \usuGamma}_{\!2}$ contains a planar involution.)\\ (e) {\it If ${\fam=\ssfam \usuGamma}$\hskip1pt is straight, then ${\fam=\ssfam \usuGamma}{\,\cong\,}\Spin3{\fam=\Bbbfam R}$ is a compact group of homologies in ${\fam=\ssfam \usuDelta}_{[u,av]}$, or ${\fam=\ssfam \usuGamma}$ is con\-tained in the translation group ${\fam=\ssfam T}{\,=\,}{\fam=\ssfam \usuDelta}_{[v,uv]}$ and ${\fam=\ssfam \usuGamma}$ has no compact subgroup other~than~$\{1\kern-2.5pt {\rm l}\}$. In particular, $\dim{\fam=\ssfam \usuGamma}{\,=\,}3$. At most one straight factor of ${\fam=\ssfam \usuDelta}$ is compact, at most two consist of translations\/}. By Baer's theorem, either ${\cal F}_{\fam=\ssfam \usuGamma}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$ is a ${\fam=\ssfam \usuDelta}$-invariant subplane, which contradicts steps (a,b), or center and axis of ${\fam=\ssfam \usuGamma}$ are fixed elements of ${\fam=\ssfam \usuDelta}$. If ${\fam=\ssfam \usuGamma}{\,\le\,}{\fam=\ssfam \usuDelta}_{[u]}$, then ${\fam=\ssfam \usuGamma}$ is compact by the second part of 2.12, and $6{\,\ne\,}\dim{\fam=\ssfam \usuGamma}{\,<\,}8$; if ${\fam=\ssfam \usuGamma}{\,\le\,}{\fam=\ssfam \usuDelta}_{[v]}$ and hence ${\fam=\ssfam \usuGamma}{\,\le\,}{\fam=\ssfam T}$, use \cite{cp} 55.28. \\ (f) {\it If ${\fam=\ssfam \usuDelta}$ is a product of $3$-dimensional factors, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}21$\/}: By the assumption in step~(d), there are at least $7$ such factors; at most $3$ of them can be straight, see step~(e). Hence there is a point $x$ and a factor ${\fam=\ssfam \usuGamma}$ such that ${\cal E}{\,=\,}\langle x^{\fam=\ssfam \usuGamma},u,v,av\rangle$ is a subplane, ${\cal E}$~is not flat by \cite{cp}~3.8, and
${\fam=\ssfam \usuPsi}_{\hskip-1pt x}|_{\cal E}{\,=\,}1\kern-2.5pt {\rm l}$. If ${\cal E}{\,\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\!}{\cal P}$, then Stiffness implies $\dim{\fam=\ssfam \usuPsi}{\,\le\,}16{+}3$. As $\dim{\fam=\ssfam \usuPsi}{\,\equiv\,}0{\,\bmod\,}3$, it follows that $\dim{\fam=\ssfam \usuPsi}{\,\le\,}18$. We may assume, therefore, that $\dim{\cal E}{\,=\,}4$. If possible, choose ${\fam=\ssfam \usuGamma}$ as the factor of torus rank $1$. Otherwise let $\hat{\fam=\ssfam \usuGamma}$ denote the straight compact factor if such a factor exists. In the second case, ${\cal G}{\,=\,}\langle x^{{\fam=\ssfam \usuGamma}\hskip1pt\hat{\fam=\ssfam \usuGamma}},u,v,av\rangle{\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\hskip-3pt}{\cal P}$ because $\hat{\fam=\ssfam \usuGamma}$ consists of homologies; in the first case, ${\cal E}^{\hat{\fam=\ssfam \usuGamma}}{\,\ne\,}{\cal E}$ for some factor $\hat{\fam=\ssfam \usuGamma}$ by step (b), so that again ${\cal G}{\,\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\hskip-3pt}{\cal P}$. Let ${\fam=\ssfam X}{\,=\,}{\fam=\ssfam \usuPsi}\kern 2pt {\scriptstyle \cap}\kern 2pt\hat{\fam=\ssfam \usuPsi}$
be the product of the remaining factors. Then ${\fam=\ssfam X}_x|_{\cal G}{\,=\,}1\kern-2.5pt {\rm l}$,\: ${\fam=\ssfam X}_x$ is compact, and then ${\fam=\ssfam X}_x{\,=\,}1\kern-2.5pt {\rm l}$ by the choice of ${\fam=\ssfam \usuGamma}$ and $\hat{\fam=\ssfam \usuGamma}$. Consequently, $\dim{\fam=\ssfam X}{\,<\,}16$, and (f) is proved. \\ (g) {\it If some almost simple factor ${\fam=\ssfam \usuGamma}\!$ of ${\fam=\ssfam \usuDelta}$ has dimension $\dim{\fam=\ssfam \usuGamma}{\,\in\,}\{6,8\}$, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}26$\/}; {\it moreover, $\dim{\fam=\ssfam \usuGamma}{\,=\,}6$ or $\dim{\fam=\ssfam \usuPsi}{\,<\,}19$\/}. By step (e), ${\fam=\ssfam \usuGamma}$ is not straight. Hence there is a point $x$ such that ${\cal E}{\,=\,}\langle x^{\fam=\ssfam \usuGamma},u,v,av\rangle$ is a subplane. 2.14 implies ${\cal E}{\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\hskip-3pt}{\cal P}$ (note that ${\fam=\ssfam \usuGamma}$ acts non-trivially on~${\cal E}$). Again
${\fam=\ssfam \usuPsi}_{\hskip-1pt x}|_{\cal E}{\,=\,}1\kern-2.5pt {\rm l}$,\: $\dim{\fam=\ssfam \usuPsi}_{\hskip-1pt x}{\,\le\,}3$, and $\dim{\fam=\ssfam \usuPsi}{\,\le\,}16{+}3$. In the case of equality, ${\cal E}$ is a Baer subplane, $\dim{\fam=\ssfam \usuPsi}_{\hskip-1pt x}{\,=\,}3$,\: $\dim x^{\fam=\ssfam \usuPsi}{\,=\,}16$ and $x^{\fam=\ssfam \usuPsi}$ is open in $P$, see \cite{cp} 96.11. By construction, $x^\psi{\,\in\,}{\cal E}{\,\Leftrightarrow\,}{\cal E}^\psi{\,=\,}{\cal E}$ for $\psi{\,\in\,}{\fam=\ssfam \usuPsi}$. Put ${\fam=\ssfam \usuXi}{\,=\,}{\fam=\ssfam \usuPsi}_{\cal E}$. Then $x^{\fam=\ssfam \usuXi}$ is an open point set in ${\cal E}$, and ${\fam=\ssfam \usuXi}/{\fam=\ssfam \usuPsi}_{\hskip-1pt x}{\,\approx\,}x^{\fam=\ssfam \usuXi}$\: (cf. \cite{cp} 96.9). Consequently, $\dim{\fam=\ssfam \usuXi}{\,=\,}8{+}3$. If $\dim{\fam=\ssfam \usuGamma}{\,>\,}6$, then
$\dim{\fam=\ssfam \usuGamma}{\fam=\ssfam \usuXi}|_{\cal E}{\,\ge\,}8{+}11{-}3$, but the stabilizer of a double flag in ${\cal E}$ has dimension ${\le\,}15$ by \cite{cp} 83.26. Hence $\dim{\fam=\ssfam \usuGamma}{\,=\,}6$ or $\dim{\fam=\ssfam \usuPsi}{\,\le\,}18$. More can be shown, using the structure of ${\fam=\ssfam \usuDelta}$: \\ (h) {\it If ${\fam=\ssfam \usuDelta}$ has an almost simple factor ${\fam=\ssfam \usuGamma}$ of dimension $6$, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}24$\/}. \\ Suppose that $\dim{\fam=\ssfam \usuDelta}{\,>\,}24$. Then $\dim{\fam=\ssfam \usuPsi}{\,=\,}19$. This case has been discussed in the previous step, and the same notation will be used. Step (b) shows that ${\fam=\ssfam \usuGamma}$ contains a unique involution, and this is a reflection. Consequently, ${\fam=\ssfam \usuGamma}{\,\cong\,}\SL2{\fam=\Bbbfam C}$,\:
${\fam=\ssfam \usuGamma}$ is not a group of homologies by \cite{cp} 61.2, and ${\fam=\ssfam \usuGamma}|_{av}{\,\cong\,}\SO3{\fam=\Bbbfam C}$. Moreover, ${\cal E}{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$,\; $x^{\fam=\ssfam \usuXi}$ is open in the point space of ${\cal E}$, and ${\fam=\ssfam \usuXi}$ has an open orbit on $av\kern 2pt {\scriptstyle \cap}\kern 2pt{\cal E}$. Therefore the lines of ${\cal E}$ are manifolds homeomorphic to ${\fam=\Bbbfam S}_4$, see \cite{cp} 53.2. According to Richardson's theorem ($\dagger$), a maximal compact subgroup ${\fam=\ssfam \usuPhi}$ of ${\fam=\ssfam \usuGamma}$ fixes a circle $C{\,\subset\,}av{\kern 2pt {\scriptstyle \cap}\kern 2pt}{\cal E}$. Consider the $11$-dimensional group
${\fam=\ssfam X}{\,=\,}{\fam=\ssfam \usuPhi}{\fam=\ssfam \usuXi}|_{\cal E}$, and note that $C^{\fam=\ssfam X}{\,=\,}C$. Let $a,b{\,\in\,}C\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{v\}$. Then $\dim{\fam=\ssfam X}_{a,b}{\,\ge\,}9$. On the other hand, $\dim{\fam=\ssfam X}_{a,b}{\,\le\,}7$ by the stiffness result \cite{sz15} (*) or \cite{sz1} 1.5(6). \\ (i) {\it If some almost simple factor ${\fam=\ssfam \usuGamma}\!$ of ${\fam=\ssfam \usuDelta}$ has dimension $\dim{\fam=\ssfam \usuGamma}{\,=\,}8$, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}24$\/}. \\ Let $\dim{\fam=\ssfam \usuDelta}{\,>\,}24$. Step (g) implies $\dim{\fam=\ssfam \usuPsi}{\,\in\,}\{17,18\}$. If ${\fam=\ssfam \usuPsi}$ is an almost direct product ${\fam=\ssfam A}{\fam=\ssfam B}$ with $\dim{\fam=\ssfam B}{\,>\,}10$ and $\rm rk\,{\fam=\ssfam B}{\,\ge\,}2$, then ${\fam=\ssfam B}$ contains a planar involution $\beta$, and
$\dim{\fam=\ssfam A}{\fam=\ssfam \usuGamma}|_{{\cal F}_\beta}{\,\le\,}10$ by \cite{sz1} 6.1. The kernel of the action of ${\fam=\ssfam A}{\fam=\ssfam \usuGamma}$ on ${\cal F}_\beta$ is a compact normal subgroup of ${\fam=\ssfam A}{\fam=\ssfam \usuGamma}$ of positive dimension ${\le\,}3$, hence it coincides with ${\fam=\ssfam A}$,\: ${\cal F}_{\fam=\ssfam A}{\,=\,}{\cal F}_\beta$, and ${\fam=\ssfam A}{\,\cong\,}\Spin3{\fam=\Bbbfam R}$ by Stiffness. Now ${\cal F}_{\hskip-2pt{\fam=\ssfam A}}^{\hskip1pt{\fam=\ssfam B}}{\,=\,} {\cal F}_{\fam=\ssfam A}$ is ${\fam=\ssfam \usuDelta}$-invariant in contradiction to step (b). \\ Next, let ${\fam=\ssfam \usuDelta}$ be~a product of two $8$-dimensional almost simple factors ${\fam=\ssfam \usuGamma}\hskip-1pt,\, \hat{\fam=\ssfam \usuGamma}$ and one $10$-dimensional factor~${\fam=\ssfam \usuOmega}$. Each of these factors has a $3$-dimensional compact subgroup. If some factor contains a planar involution $\iota$\: (in particular, if the factor has rank $2$, or if it has a subgroup $\SO3{\fam=\Bbbfam R}$), then the product of the other two factors of ${\fam=\ssfam \usuDelta}$ induces on ${\cal F}_\iota$ a semi-simple group of dimension at least $16$, but this contradicts \cite{sz1} 6.1. Hence each of the $3$ factors of ${\fam=\ssfam \usuDelta}$ has a maximal compact subgroup isomorphic to $\Spin3{\fam=\Bbbfam R}$. The central involutions of these subgroups coincide with a unique reflection $\sigma{\,\in\,}{\fam=\ssfam \usuDelta}_{[u,av]}$. There is some point $x$ such that ${\cal E}{\,=\,}\langle x^{\hat{\fam=\ssfam \usuGamma}},u,v,av\rangle{\:\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\!}{\cal P}$. Let ${\fam=\ssfam \usuLambda}$ be the connected component of $({\fam=\ssfam \usuGamma}{\fam=\ssfam \usuOmega})_x$, and note that $\dim{\fam=\ssfam \usuLambda}{\,\ge\,}2$ and that
${\fam=\ssfam \usuLambda}|_{\cal E}{\,=\,}1\kern-2.5pt {\rm l}$. By Stiffness, ${\fam=\ssfam \usuLambda}{\,\cong\,}\Spin3{\fam=\Bbbfam R}$, and ${\fam=\ssfam \usuLambda}$ is contained in a maximal compact subgroup ${\fam=\ssfam \usuPhi}{\,\cong\,}\SO4{\fam=\Bbbfam R}$ of ${\fam=\ssfam \usuGamma}{\fam=\ssfam \usuOmega}$. It follows that ${\fam=\ssfam \usuLambda}$ is one of the two normal factors of ${\fam=\ssfam \usuPhi}$. Consequently, ${\fam=\ssfam \usuLambda}$ is contained in ${\fam=\ssfam \usuGamma}$ or in ${\fam=\ssfam \usuOmega}$, but ${\fam=\ssfam \usuLambda}$ is planar, a contradiction. \\ The case of two $8$-dimensional factors ${\fam=\ssfam \usuGamma}\hskip-1pt,\:\hat{\fam=\ssfam \usuGamma}$ and factors ${\fam=\ssfam A}_\nu,\: \nu{=}0,1,2$ of dimension $3$ can be dealt with similarly: by step (d), at most one of the ${\fam=\ssfam A}_\nu$, say ${\fam=\ssfam A}_2$, can have positive rank. Because of \cite{sz1} 6.1, the fixed point set of ${\fam=\ssfam A}_\nu$ is at most $4$-dimensional. By Baer's theorem, there is some point $x$ such that
${\cal E}{\,=\,}\langle x^{{\fam=\ssfam A}_1{\fam=\ssfam A}_2},u,v,av\rangle$ is a subplane and that ${\fam=\ssfam A}_1{\fam=\ssfam A}_2$
induces a $6$-dimensional group on ${\cal E}$. From 2.14 it follows that $\dim{\cal E}{\,>\,}4$, and
${\cal E}{\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\hskip-3pt}{\cal P}$. Put ${\fam=\ssfam \usuXi}{\,=\,}{\fam=\ssfam \usuGamma}\hat{\fam=\ssfam \usuGamma}{\fam=\ssfam A}_0$. Then $\dim{\fam=\ssfam \usuXi}{\,=\,}19$,\:
$\dim{\fam=\ssfam \usuXi}_x{\,=\,}3$,\: ${\fam=\ssfam \usuXi}_x|_{\cal E}{\,=\,}1\kern-2.5pt {\rm l}$, and ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuXi}_x)^1{\,\cong\,}\Spin3{\fam=\Bbbfam R}$ is contained in a maximal compact subgroup ${\fam=\ssfam \usuPhi}$ of ${\fam=\ssfam \usuXi}$. Again ${\fam=\ssfam \usuPhi}{\,\cong\,}\SO4{\fam=\Bbbfam R}$. For the same reason as before, this is impossible. The only remaining case that ${\fam=\ssfam \usuPsi}$ is a product of six $3$-dimensional factors can be excluded by the arguments of step (f). \\ (j) {\it If ${\fam=\ssfam \usuDelta}$ has a $10$-dimensional factor ${\fam=\ssfam \usuGamma}$, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}24$\/}. \\ We may assume that $25{\,\le\,}\dim{\fam=\ssfam \usuDelta}{\,\le\,}28$ and that there is no factor of dimension $6$ or $8$. If ${\fam=\ssfam \usuGamma}$ contains two commuting involutions, then there is a planar involution
$\beta{\,\in\,}{\fam=\ssfam \usuGamma}$, $\dim{\fam=\ssfam \usuPsi}|_{{\cal F}_\beta}{\,\le\,}10$ by \cite{sz1} 6.1, and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}23$. Therefore ${\fam=\ssfam \usuGamma}$ is isomorphic to the simply connected covering group of $\Opr5({\fam=\Bbbfam R},2)$, and the unique central involution $\sigma{\,\in \,}{\fam=\ssfam \usuGamma}$ is a reflection in ${\fam=\ssfam \usuDelta}_{[u,av]}$. \\ Case 1) There are exactly two almost simple factors ${\fam=\ssfam A},{\fam=\ssfam \usuUpsilon}{\,\ne\,}{\fam=\ssfam \usuGamma}$, where $\dim{\fam=\ssfam \usuUpsilon}{\,\in\,}\{14,15\}$ and $\rm rk\,{\fam=\ssfam \usuUpsilon}{\,=\,}2$. Then ${\fam=\ssfam \usuUpsilon}$ contains a planar involution
$\iota$, and $\dim{\fam=\ssfam \usuGamma}{\fam=\ssfam A}|_{{\cal F}_\iota}{\,\le\,}10$ by \cite{sz1} 6.1. Hence
${\fam=\ssfam A}|_{{\cal F}_\iota}{\,=\,}1\kern-2.5pt {\rm l}$ and ${\fam=\ssfam A}{\,\cong\,}\Spin3{\fam=\Bbbfam R}$ by Stiffness. The involution
$\alpha{\,\in\,}{\fam=\ssfam A}$ is planar, and ${\fam=\ssfam \usuGamma}{\fam=\ssfam \usuUpsilon}|_{{\cal F}_\alpha}$ would be too large. \\ Case 2) ${\fam=\ssfam \usuDelta}$ has at least two $3$-dimensional factors, ${\fam=\ssfam A}$ and ${\fam=\ssfam B}$. At most one of
them consists of homologies with axis $av$\: (see steps (d,e)\,). Let ${\fam=\ssfam A}|_{av}{\,\ne\,}1\kern-2.5pt {\rm l}$.
Then there is a point $x$ such that $\langle x^{\fam=\ssfam A},u,v,av\rangle$ is a subplane, and so is
${\cal E}{\,=\,}\langle x^{{\fam=\ssfam A}{\fam=\ssfam B}},u,v,av\rangle$. If $\dim{\cal E}{\,<\,}8$, then 2.14 shows that
${\fam=\ssfam A}{\fam=\ssfam B}|_{\cal E}$ is almost simple. By assumption ${\fam=\ssfam A}|_{\cal E}{\,\ne\,}1\kern-2.5pt {\rm l}$. Hence
${\fam=\ssfam B}|_{\cal E}{\,=\,}1\kern-2.5pt {\rm l}$,\: ${\cal E}{\,\le\,}{\cal F}_{\fam=\ssfam B}$,\:
${\cal F}_{\hskip-1pt{\fam=\ssfam B}}^{\hskip1pt{\fam=\ssfam \usuDelta}}{\,=\,}{\cal F}_{\fam=\ssfam B}{\,<\,}{\cal P}$, and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}13$ by Corollary (b). Therefore ${\cal E}{\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\hskip-4pt}{\cal P}$. Put ${\fam=\ssfam X}{\,=\,}({{\fam=\ssfam A}{\fam=\ssfam B}})^1$.
Then ${\fam=\ssfam X}_x|_{\cal E}{\,=\,}1\kern-2.5pt {\rm l}$,\: $\dim{\fam=\ssfam X}{\,\le\,}16{+}3$, and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}25$. This takes care of all possibilities except the following ones: ${\fam=\ssfam \usuPsi}$ is an almost simple group of dimension $16$, or $\dim{\fam=\ssfam \usuPsi}{\,=\,}15$ and ${\fam=\ssfam \usuPsi}$ is almost simple or a product of $3$-dimensional factors. \\ Case 3) ${\fam=\ssfam \usuPsi}$ is isomorphic to (P\hskip-1pt)$\hskip-1pt\SL3{\fam=\Bbbfam C}$. There are $3$ pairwise commuting involutions conjugate to $\beta{\,=\,}{\rm diag}(-1,-1,1)$; they are necessarily planar, and $(\Cs{\fam=\ssfam \usuPsi}\beta)'{\,\cong\,}\SL2{\fam=\Bbbfam C}$. Hence $(\Cs{\fam=\ssfam \usuDelta}\beta)'$ induces on ${\cal F}_\beta$ a $16$-dimensional semi-simple group in contradiction to \cite{sz1} 6.1. \\ Case 4) Similarly, the groups ${\fam=\ssfam \usuPsi}$ of type ${\rm A}_3$ can be dealt with by exhibiting a suitable semi-simple group ${\fam=\ssfam \usuUpsilon}$ in the centralizer of a planar involution $\beta$. \\ \begin{center}
\begin{tabular}{|c|c|c|c|} \hline ${\fam=\ssfam \usuPsi}$ & $\beta$ & ${\fam=\ssfam \usuUpsilon}$& ${\fam=\ssfam \usuPsi}/\langle -1\rangle$ \\ \hline $\SU4{\fam=\Bbbfam C}$ & $\:{\rm diag}(-1,-1,1,1)\:$ & $(\SU2{\fam=\Bbbfam C})^2$ & $\SO6{\fam=\Bbbfam R}$ \\ $\SL2{\fam=\Bbbfam H}$ & ${\rm diag}(-1,1)$ & $(\SU2{\fam=\Bbbfam C})^2$ & $\Opr6({\fam=\Bbbfam R},1){>}\SO5{\fam=\Bbbfam R}$ \\ \hline $\SU4({\fam=\Bbbfam C},1)$ & ${\rm diag}(-1,-1,1,1)$ & $\:\SU2{\fam=\Bbbfam C}{\times}\SL2{\fam=\Bbbfam R}\:$& $\SaU3$ \\ $\SU4({\fam=\Bbbfam C},2)$ & ${\rm diag}(-1,-1,1,1)$ & $(\SU2{\fam=\Bbbfam C})^2$ & $\Opr6({\fam=\Bbbfam R},2)$ \\ \hline $\SL4{\fam=\Bbbfam R}$ & ${\rm diag}(-1,-1,1,1)$ & $(\SL2{\fam=\Bbbfam R})^2$ & $\Opr6({\fam=\Bbbfam R},3){>}(\SO3{\fam=\Bbbfam R})^2$ \\ $\tilde{\rm SL}_4{\fam=\Bbbfam R}$ & $\beta{\,\in\,}\Spin4{\fam=\Bbbfam R}$ & $\Spin4{\fam=\Bbbfam R}$ & \\ \hline \end{tabular}\end{center} Here $\tilde{\fam=\ssfam \usuPsi}$ denotes the simply connected covering group of ${\fam=\ssfam \usuPsi}$. Recall from 2.\,10, and 11 that ${\fam=\ssfam \usuDelta}$ has no subgroup $\SO5{\fam=\Bbbfam R}$ or $(\SO3{\fam=\Bbbfam R})^2$. Hence ${\fam=\ssfam \usuPsi}$ cannot be a proper factor group of $\SU4{\fam=\Bbbfam C},\: \SL4{\fam=\Bbbfam R}$, or $\SL2{\fam=\Bbbfam H}$. (Note that $\{\hskip1pt({a \atop } { \atop \raise2pt\hbox{${\scriptstyle b}$}})\mid a,b{\,\in\,}{\fam=\Bbbfam H}'\,\} {\,\le\,}\SL2{\fam=\Bbbfam H}$.) For ${\fam=\ssfam \usuPsi}{\,\cong\,}\SU4({\fam=\Bbbfam C},1)$, the involution $\beta$ is contained in $\SU3{\fam=\Bbbfam C}$ and corresponds to a planar involution in each factor group of ${\fam=\ssfam \usuPsi}$ as well as in each covering group; the centralizer of such an involution contains a semi-simple group locally isomorphic to $\SU2{\fam=\Bbbfam C}{\times}\SU2({\fam=\Bbbfam C},1){\,\cong\,}\SU2{\fam=\Bbbfam C}{\times}\SL2{\fam=\Bbbfam R}$. Similarly, if ${\fam=\ssfam \usuPsi}{\,\cong\,}\SU4({\fam=\Bbbfam C},2)$, then $\beta{\,\in\,}(\SU2{\fam=\Bbbfam C})^2$ commutes with the reflection $-1\kern-2.5pt {\rm l}$; hence $\beta$ corresponds to a planar involution in each factor group and in each covering group of ${\fam=\ssfam \usuPsi}$, centralizing a $6$-dimensional compact semi-simple group. \\ Case 5) All factors of ${\fam=\ssfam \usuDelta}$ other than ${\fam=\ssfam \usuGamma}$ are $3$-dimensional. Recall that ${\fam=\ssfam \usuGamma}$ is the simply connected covering group of $\Opr5({\fam=\Bbbfam R},2)$. This case
is more complicated. Choose ${\fam=\ssfam A}$ and ${\fam=\ssfam B}$ among the $3$-dimensional factors and a line $ux$ such that $(ux)^{\fam=\ssfam A}{\,\ne\,}ux$ and ${\cal E}{\,=\,}\langle x^{{\fam=\ssfam A}{\fam=\ssfam B}},u,v,av\rangle{\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\hskip-4pt}{\cal P}$ as in Case 2). Let again ${\fam=\ssfam X}{\,=\,}(\Cs{}{\fam=\ssfam A}{\fam=\ssfam B})^1$. If $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}25$,
then $\dim{\fam=\ssfam X}{\,=\,}19$,\: ${\fam=\ssfam X}_x|_{\cal E}{\,=\,}1\kern-2.5pt {\rm l}$,\: $\dim{\fam=\ssfam X}_x{\,\le\,}3$,\: $\dim x^{\fam=\ssfam X}{\,=\,}16$, and $(vx)^{\fam=\ssfam X}$ is open in the pencil ${\frak L}_v$ by \cite{cp} 96.11. As $x{\,\notin\,}av$ is an arbitrary point of $ux\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{u\}$, it follows that ${\fam=\ssfam \usuDelta}$ is transitive on ${\frak S}{\,=\,}{\frak L}_v\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{av,uv\}$. The exact homotopy sequence (\cite{cp} 96.12) will be applied to this action. Let $L$ be some line in ${\frak S}$, and consider the part $$\dots\to\pi_7{\fam=\ssfam \usuDelta}\to\pi_7{\frak S}\to\pi_6{\fam=\ssfam \usuDelta}_L\to\dots$$ of the homotopy sequence. Note that ${\frak S}$ is homotopy equivalent ($\simeq$) to ${\fam=\Bbbfam S}_7$ by 2.1. Choose maximal compact subgroups ${\fam=\ssfam \usuPhi}$ in ${\fam=\ssfam \usuDelta}$ and ${\fam=\ssfam K}$ in ${\fam=\ssfam \usuDelta}_L$ such that ${\fam=\ssfam K}{\,\le\,}{\fam=\ssfam \usuPhi}$. Then ${\fam=\ssfam \usuDelta}{\,\simeq\,}{\fam=\ssfam \usuPhi}$ and ${\fam=\ssfam \usuDelta}_L{\,\simeq\,}{\fam=\ssfam K}$ by the Mal'cev-Iwasawa theorem \cite{cp} 93.10, and we get
an exact sequence $$\dots\to\pi_7{\fam=\ssfam \usuPhi}\to\pi_7{\fam=\Bbbfam S}_7\to\pi_6{\fam=\ssfam K}\to\dots.$$
It is well-known that $\pi_7{\fam=\Bbbfam S}_7{\,\cong\,}{\fam=\Bbbfam Z}$\: (see, e.g., \cite{sp} 7.5.6 or \cite{br} II.16.4). A maximal compact subgroup of ${\fam=\ssfam \usuGamma}$ is isomorphic to $\Spin3{\fam=\Bbbfam R}$, hence homeomorphic to ${\fam=\Bbbfam S}_3$. At most one of the $3$-dimensional factors has a non-trivial maximal compact subgroup, it is homeomorphic to ${\fam=\Bbbfam S}_3$ or to ${\fam=\Bbbfam S}_1$. If $k$ is odd, then all homotopy groups $\pi_q{\fam=\Bbbfam S}_k$ with $q{\,\ne\,}k$ are finite (\cite{sp} 9.7.7) and $\pi_q{\fam=\Bbbfam S}_1{\,=\,}0$ for $q{\,\ne\,}1$ (\cite{sp} 7.2.12); in fact, $\pi_7{\fam=\Bbbfam S}_3{\,\cong\,}{\fam=\Bbbfam Z}_2$ and $\pi_6{\fam=\Bbbfam S}_3{\,\cong\,}{\fam=\Bbbfam Z}_{12}$. Therefore both $\pi_7{\fam=\ssfam \usuPhi}$ and $\pi_6{\fam=\ssfam K}$ are finite. Exactness would force $\pi_7{\fam=\Bbbfam S}_7$ to be finite, a contradiction. This completes the proof of step (j) and shows that ${\fam=\ssfam \usuDelta}$ is almost simple or has some almost simple factor of dimension ${\hskip-2pt\ge\,}14$. \\ (k) {\it If each almost simple factor of ${\fam=\ssfam \usuDelta}$ has dimension $3$ or $14$, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}20$\/}. \\ By step (f), we may assume that $\dim{\fam=\ssfam \usuGamma}{=\,}14$, and then $\rm rk\,{\fam=\ssfam \usuGamma}{=\,}2$. Lemma 8.1 implies that ${\fam=\ssfam \usuGamma}$ contains a planar involution $\iota$. For the compact group ${\fam=\ssfam \usuGamma}$ of type ${\rm G}_2$ it has been stated in 2.5 that $\Cs{\fam=\ssfam \usuGamma}\iota$ is semi-simple and $\dim\Cs{\fam=\ssfam \usuGamma}\iota{\,=\,}6$. Now let ${\fam=\ssfam \usuGamma}$ be a non-compact group of type ${\rm G}_2$. If ${\fam=\ssfam \usuGamma}$ is strictly simple, then ${\fam=\ssfam \usuGamma}$ has a subgroup ${\fam=\ssfam \usuPhi}{\,\cong\,}\SO4{\fam=\Bbbfam R}$; by Lemma 8.1 we may choose $\iota$ as the central involution of ${\fam=\ssfam \usuPhi}$; the double covering group of ${\fam=\ssfam \usuGamma}$ contains $\tilde{\fam=\ssfam \usuPhi}{\,\cong\,}(\Spin3{\fam=\Bbbfam R})^2$, and the central involution of one of the two factors of $\tilde{\fam=\ssfam \usuPhi}$ can be taken as $\iota$, so that in each case $\dim\Cs{\fam=\ssfam \usuGamma}\iota{\,\ge\,}6$. Now the following lemma shows $\dim{\fam=\ssfam \usuPsi}{\,\le\,}6$ and the assertion is proved. \\ ($\ell$) {\tt Lemma.} {\it If $\iota$ is a \emph{planar} involution in ${\fam=\ssfam \usuGamma}$ and if ${\fam=\ssfam X}$ is a
maximal semi-simple subgroup of $\Cs{\fam=\ssfam \usuGamma}\iota$, then $\dim{\fam=\ssfam X}{\fam=\ssfam \usuPsi}|_{{\cal F}_\iota}{\,\le\,}10\,$\/} (by \cite{sz1} 6.1) {\it and $\dim{\fam=\ssfam X}{\,+\,}\dim{\fam=\ssfam \usuPsi}{\,\le\,}10{+}3$\/}. \\ (m) {\it If each almost simple factor of ${\fam=\ssfam \usuDelta}$ has dimension $15,16$, or $3$, then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}22$\/}. \\ In fact, if $\dim{\fam=\ssfam \usuGamma}{\,\in\,}\{15,16\}$, then there is a planar involution $\iota{\,\in\,}{\fam=\ssfam \usuGamma}$ such that $\Cs{\fam=\ssfam \usuGamma}\iota$ contains a $6$-dimensional semi-simple group, see cases 3) and 4) of step (j). Lemma ($\ell$) implies $\dim{\fam=\ssfam \usuDelta}{\,\le\,}16{\,+\,}\dim{\fam=\ssfam \usuPsi}{\,\le\,}22$. \\ (n) {\it If ${\fam=\ssfam \usuDelta}$ has an almost simple factor ${\fam=\ssfam \usuGamma}\!$ of dimension $20$, then ${\fam=\ssfam \usuDelta}{\,\cong\,}\Sp4{\fam=\Bbbfam C}$\/}. \\ By the last part of 2.10, the simple group $\SO5{\fam=\Bbbfam C}$ cannot act on a plane. Its covering group $\Spin5{\fam=\Bbbfam C}{\,\cong\,}\Sp4{\fam=\Bbbfam C}$ contains a planar involution $\iota{\,=\,}{\rm diag}(-1,-1,1,1)$ centralized by the $12$-dimensional semi-simple group $(\Sp2{\fam=\Bbbfam C})^2{\,\cong\,}(\SL2{\fam=\Bbbfam C})^2$. Hence ${\fam=\ssfam \usuDelta}{\,=\,}{\fam=\ssfam \usuGamma}$ by Lemma ($\ell$). (In fact, the group $(\SL2{\fam=\Bbbfam C})^2$ cannot act on ${\cal F}_\iota$ and case (n) is impossible). \\ (o) {\it Suppose that ${\fam=\ssfam \usuDelta}$ has an almost simple factor ${\fam=\ssfam \usuGamma}$ of type ${\rm B}_3$. Then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}24$\/}. \\ By 2.\,10 and 11, none of the simple groups $\Opr7({\fam=\Bbbfam R},r)$ can act on ${\cal P}$. The central involution of the compact group ${\fam=\ssfam \usuGamma}{\,\cong\,}\Spin7{\fam=\Bbbfam R}$ is a reflection with axis $av$, and the action of ${\fam=\ssfam \usuGamma}$ on $av\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{v\}$ is equivalent to the linear action of $\SO7{\fam=\Bbbfam R}$ on ${\fam=\Bbbfam R}^8$, see 2.15. Hence ${\fam=\ssfam \usuGamma}$ fixes a circle on $av$. For $z{\,\in\,}uv\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{u,v\}$, Stiffness implies that
${\fam=\ssfam \usuGamma}_{\hskip-2pt z}{\,=\,}{\fam=\ssfam \usuLambda}{\,\cong\,}{\rm G}_2$, that ${\cal F}_{\fam=\ssfam \usuLambda}$ is a flat subplane, and that ${\fam=\ssfam \usuPsi}{\,=\,}\Cs{}{\fam=\ssfam \usuGamma}$ acts almost effectively on ${\cal F}_{\fam=\ssfam \usuLambda}$. Now ${\fam=\ssfam \usuPsi}|_{{\cal F}_{\fam=\ssfam \usuLambda}}$ is solvable by \cite{cp} 33.8. Hence~${\fam=\ssfam \usuDelta}{\,=\,}{\fam=\ssfam \usuGamma}$. \\
Next, let ${\fam=\ssfam \usuGamma}{\,\cong\,}\Spin7({\fam=\Bbbfam R},1)$. As ${\fam=\ssfam \usuGamma}$ contains a $3$-torus, there are (at least)
$6$ non-central planar involutions in ${\fam=\ssfam \usuGamma}$. By the covering homomorphism, they are mapped
to involutions in $\Opr7({\fam=\Bbbfam R},1)$, up to conjugacy to diagonal matrices with entries $\pm1$ and determinant $1$. In the centralizer of each of these matrices there is a semi-simple group of dimension ${\!\ge\,}9$, which is covered by a semi-simple group in the centralizer of a planar involution in ${\fam=\ssfam \usuGamma}$. Consequently, Lemma ($\ell$) shows $9{\,+\,}\dim{\fam=\ssfam \usuPsi}{\,\le\,}13$,\: $\dim{\fam=\ssfam \usuPsi}{\,\le\,}3$, and $\dim{\fam=\ssfam \usuDelta}{\,\in\,}\{21,24\}$. The same arguments apply in the cases $r{\,>\,}1$ even if there are more possibilities than just a double covering of $\Opr7({\fam=\Bbbfam R},r)$. \\
(p) {\it If ${\fam=\ssfam \usuDelta}$ has an almost simple factor ${\fam=\ssfam \usuGamma}$ of type ${\rm C}_3$, then $\dim{\fam=\ssfam \usuDelta}{\,=\,}21$ and ${\fam=\ssfam \usuDelta}{\,=\,}{\fam=\ssfam \usuGamma}$\/}. \\ The planar involution ${\rm diag}(-1,-1,1)$ in $\U3({\fam=\Bbbfam H},r)$ has a semi-simple centralizer $\U2{\fam=\Bbbfam H}{\times}{\fam=\Bbbfam H}'$ of dimension $13$. From Lemma ($\ell$) it follows that ${\fam=\ssfam \usuDelta}{\,=\,}{\fam=\ssfam \usuGamma}$, and the claim holds for the $4$ groups (P\hskip-.5pt)$\hskip-2pt\U3({\fam=\Bbbfam H},r)$ with $r{\,=\,}0,1$. The symplectic group ${\fam=\ssfam \usuGamma}{\,=\,}\Sp6{\fam=\Bbbfam R}$ has a maximal compact subgroup $\U3{\fam=\Bbbfam C}$. The involution $\iota{\,=\,}{\rm diag}(-1,-1,1){\,\in\,}\SU3{\fam=\Bbbfam C}$ is planar, its centralizer contains the semi-simple group ${\fam=\ssfam \usuUpsilon}{\,=\,}\Sp4{\fam=\Bbbfam R}{\times}\Sp2{\fam=\Bbbfam R}$ of dimension $13$. The element $\iota$ is mapped to a non-central involution in $\PSp6{\fam=\Bbbfam R}$, and it belongs to each of the infinitely many covering groups of ${\fam=\ssfam \usuGamma}$ because $\SU3{\fam=\Bbbfam C}$ is simply connected. In any case, the centralizer of $\iota$ is locally isomorphic to ${\fam=\ssfam \usuUpsilon}$. Again $\dim{\fam=\ssfam \usuUpsilon}{\,+\,}\dim{\fam=\ssfam \usuPsi}{\,\le\,}13$ and ${\fam=\ssfam \usuDelta}{\,\circeq\,}{\fam=\ssfam \usuGamma}$. \\ (q) Each group ${\fam=\ssfam \usuGamma}{\,\circeq\,}\SU5({\fam=\Bbbfam C},r)$ contains a non-central and hence planar involution corresponding to $\iota{\,=\,}{\rm diag(-1,-1,-1,-1,1)}$, and $\iota$ is centralized by the $15$-dimensional group ${\fam=\ssfam \usuUpsilon}{\,=\,}\SU4({\fam=\Bbbfam C},r{-}1)$, but planarity implies $\dim{\fam=\ssfam \usuUpsilon}{\,\le\,}10$. Analogously, the simply connected covering group of $\SL5{\fam=\Bbbfam R}$ can be excluded. \\ (r) Only one possibility remains: ${\fam=\ssfam \usuGamma}{\,=\,}{\fam=\ssfam \usuDelta}$ is an almost simple group of dimension $28$, in fact, an orthogonal group of type D$\hskip-1.5pt_4$, see \cite{cp} 94.33 and 2.16 above. From 2.10 and 2.11 it follows that ${\fam=\ssfam \usuDelta}$ is a proper covering of a group $\POpr8({\fam=\Bbbfam R},r)$ different from $\Opr8({\fam=\Bbbfam R},r)$. By step (b) there is at most one central involution. As $\Spin8{\fam=\Bbbfam R}$ has center $({\fam=\Bbbfam Z}_2)^2$, the group ${\fam=\ssfam \usuDelta}$ is not compact and $r{\,\ne\,}0$. If $r{\,=\,}4$, a maximal compact subgroup of the simply connected covering group $\tilde{\fam=\ssfam \usuDelta}$ is isomorphic to $(\Spin4{\fam=\Bbbfam R})^2{\,\cong\,}(\Spin3{\fam=\Bbbfam R})^4$. Hence each non-trivial element in the center of ${\fam=\ssfam \usuDelta}$ is an involution, and ${\fam=\ssfam \usuDelta}$ would contain $(\SO3{\fam=\Bbbfam R})^2$ contrary to 2.11. \\ (s) The cases $0{\,<\,}r{\,<\,}3$ can be excluded by similar arguments as in step (p):\: ${\fam=\ssfam \usuDelta}$ contains the simply connected double covering ${\fam=\ssfam \usuPhi}{\,=\,}\Spin6{\fam=\Bbbfam R}$ of $\SO6{\fam=\Bbbfam R}$. Choose a $1$-torus ${\fam=\ssfam \usuTheta}{\,<\,}{\fam=\ssfam \usuPhi}$ which is disjoint from the kernel of the covering map $\kappa{\,:\,}{\fam=\ssfam \usuDelta}{\,\to\,}\POpr8({\fam=\Bbbfam R},r)$. Up to conjugation, the involution $\iota{\,\in\,}{\fam=\ssfam \usuTheta}$ is mapped onto ${\rm diag}(-1,-1,1,...,1){\,\in\,}\Opr8({\fam=\Bbbfam R},r)$. Hence $\Cs{\fam=\ssfam \usuDelta}\iota$ contains a $15$-dimensional semi-simple group ${\fam=\ssfam \usuUpsilon}{\,\circeq\,}\Opr6({\fam=\Bbbfam R},r)$, but $\dim{\fam=\ssfam \usuUpsilon}{\,\le\,}10$ by \cite{sz1} 6.1. \\ (t) Finally, let ${\fam=\ssfam \usuDelta}$ be a double covering of $\Opr8({\fam=\Bbbfam R},3)$, and consider the covering map of a $2$-torus ${\fam=\ssfam \usuTheta}$ in the subgroup $\Spin5{\fam=\Bbbfam R}$ of ${\fam=\ssfam \usuDelta}$ into $\SO5{\fam=\Bbbfam R}$. Up to conjugation, some non-central (and hence planar) involution $\iota{\,\in\,}{\fam=\ssfam \usuDelta}$ is contained in ${\fam=\ssfam \usuTheta}$ and is mapped onto ${\rm diag}(-1,-1,1,1,1){\,\in\,}\SO5{\fam=\Bbbfam R}$, and $\Cs{\fam=\ssfam \usuDelta}\iota$ is locally isomorphic to $\Opr6({\fam=\Bbbfam R},3)$, again a contradiction. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\bf 8.3 Normal torus.} {\it Suppose that ${\cal F}_{\fam=\ssfam \usuDelta}$ is a double flag and that ${\fam=\ssfam \usuDelta}$ has a one-dimensional compact connected normal subgroup ${\fam=\ssfam \usuTheta}{\,\triangleleft\,}{\fam=\ssfam \usuDelta}$. Then $\dim{\fam=\ssfam \usuDelta}{\,\le\,}30$\/}. \par
{\tt Proof.} (a) We may assume that $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}27$. Then ${\fam=\ssfam \usuDelta}$ is a Lie group by 2.3, and ${\fam=\ssfam \usuTheta}$ is a central torus. If ${\fam=\ssfam \usuLambda}{\,<\,}{\fam=\ssfam \usuDelta}$ and if ${\cal F}_{\fam=\ssfam \usuLambda}$ is a subplane, then ${\fam=\ssfam \usuTheta}$ acts non-trivially on ${\cal F}_{\fam=\ssfam \usuLambda}$\: (or else ${\cal F}_{\fam=\ssfam \usuLambda}{\,\le\,}{\cal F}_{\fam=\ssfam \usuTheta}{\,=\,}{\cal F}_{\hskip-1.5pt{\fam=\ssfam \usuTheta}}^{\hskip1pt{\fam=\ssfam \usuDelta}}{\,<\,}{\cal P}$ and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}18$ by Stiffness). Hence ${\cal F}_{\fam=\ssfam \usuLambda}$ is not flat, $\dim{\cal F}_{\fam=\ssfam \usuLambda}{\,\ge\,}4$,\: $\dim{\fam=\ssfam \usuLambda}{\,\le\,}8$,\: ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuLambda}{\,\le\,}8{+}16$, and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}32$. In fact, ${\fam=\ssfam \usuTheta}{\,\le\,}{\fam=\ssfam \usuDelta}_{[av]}$: let $a,c{\,\in\,}K{\,=\,}av\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{v\}$ and $z{\,\in\,}S{\,=\,}uv\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{u,v\}$, and consider ${\fam=\ssfam \usuGamma}{\,=\,}{\fam=\ssfam \usuDelta}_{a,c}$ and ${\fam=\ssfam \usuLambda}{\,=\,}{\fam=\ssfam \usuGamma}_{\hskip-2pt z}$. If $a{\,\ne\,}c{\,\in\,}a^{\fam=\ssfam \usuTheta}$, then ${\fam=\ssfam \usuGamma} {\,= \,}{\fam=\ssfam \usuDelta}_a$ and ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuLambda}{\,\le\,}16$ contrary to the assumption. \\ (b) Suppose that $\dim{\fam=\ssfam \usuDelta}{\,=\,}32$. Then ${\fam=\ssfam \usuDelta}_z$ is doubly transitive on $K$,\: ${\fam=\ssfam \usuGamma}$ is transitive on $S$,\: $\dim{\fam=\ssfam \usuGamma}{\,=\,}16$, and ${\fam=\ssfam \usuLambda}{\,\cong\,}\SU3{\fam=\Bbbfam R}$. If ${\fam=\ssfam \usuPhi}$ is a maximal compact subgroup of ${\fam=\ssfam \usuGamma}$, then ${\fam=\ssfam \usuPhi}'{\,\cong\,}\SU4{\fam=\Bbbfam C}$ by 2.17(d). Now ${\fam=\ssfam \usuTheta}{\fam=\ssfam \usuPhi}'{\,\le\,}{\fam=\ssfam \usuPhi}$ and ${\fam=\ssfam \usuPhi}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuLambda}{\,=\,}8$, but then ${\fam=\ssfam \usuPhi}$ would be transitive on $S$, which is impossible. Therefore $\dim{\fam=\ssfam \usuGamma}{\,<\,}16$ and $\dim{\fam=\ssfam \usuDelta}{\,<\,}32$. \\ (c) Let $\dim{\fam=\ssfam \usuDelta}{\,=\,}31$. Then $\dim{\fam=\ssfam \usuGamma}{=\,}15$ by step (b), and ${\fam=\ssfam \usuDelta}$ is doubly transitive on~$K$\: (notation as in (a)\,). Put $\nabla{\,=\,}{\fam=\ssfam \usuDelta}_a$ and
$\hat\nabla{\,=\,}\nabla|_K{\,\cong\,}\nabla/{\fam=\ssfam N}$. The kernel ${\fam=\ssfam N}{\,=\,}{\fam=\ssfam \usuDelta}_{[av]}$ consists of homologies with center $u$ and ${\fam=\ssfam \usuTheta}{\,\trianglelefteq\,}{\fam=\ssfam N}$ by step (a). Let ${\fam=\ssfam K}$ be a maximal compact connected subgroup of ${\fam=\ssfam N}$. Then ${\fam=\ssfam N}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\le\,}1$ by \cite{cp} 61.2, and ${\fam=\ssfam K}$ has torus rank $\rm rk\, {\fam=\ssfam K}{\,=\,}1$\; (use \cite{cp} 55.35). Therefore ${\fam=\ssfam K}{\,=\,}{\fam=\ssfam \usuTheta}$,\: $\dim{\fam=\ssfam N}{\,\le\,}2$, and $21{\,\le\,}\nabla{\hskip1pt:\hskip1pt}{\fam=\ssfam N}{\,<\,}23$. The structure of doubly transitive groups has been determined by Tits, cf. \cite{cp} 96.16. As $K$ is not compact, $\hat\nabla$ is a transitive subgroup of $\GL8{\fam=\Bbbfam R}$; these groups are described explicitly in \cite{Vl}, see also \cite{cp} 96.\,19--22. In particular, $\hat\nabla{\,=\,}{\fam=\ssfam H}\hat{\fam=\ssfam \usuUpsilon}$ is a product of an almost simple Lie group $\hat{\fam=\ssfam \usuUpsilon}$ and a subgroup ${\fam=\ssfam H}$ of its centralizer; moreover ${\fam=\ssfam H}{\,\le\,}{\fam=\Bbbfam H}^{\times}$, a maximal compact subgroup of $\hat{\fam=\ssfam \usuUpsilon}$ is transitive on the $7$-sphere formed by the rays in ${\fam=\Bbbfam R}^8$, and $16{\,<\,}\dim\hat{\fam=\ssfam \usuUpsilon}{\,<\,}23$. In this dimension range, only three almost simple groups have an $8$-dimensional irreducible representation, viz. the groups $\Sp4{\fam=\Bbbfam C}$,\: $\Spin7{\fam=\Bbbfam R}$, and $\Spin7({\fam=\Bbbfam R},3)$, see \cite{cp} 96.10. The last one does not act transitively on ${\fam=\Bbbfam S}_7$, the other two are simply connected. By \cite{cp} 94.27 it follows that $\nabla$ has a subgroup ${\fam=\ssfam \usuUpsilon}$ which is isomorphic to one of $\Sp4{\fam=\Bbbfam C}$ or $\Spin7{\fam=\Bbbfam R}$. First, let ${\fam=\ssfam \usuUpsilon}$ be compact. Then $\nabla{\,\cong\,}e^{{\fam=\Bbbfam R}}{\cdot}\Spin7{\fam=\Bbbfam R}{\cdot\hskip.5pt}{\fam=\ssfam \usuTheta}$ and $\nabla_{\hskip-2pt c}{\,=\,}{\fam=\ssfam \usuLambda}{\times}{\fam=\ssfam \usuTheta}$ with ${\fam=\ssfam \usuLambda}{\,\cong\,}{\rm G}_2$\: (see 2.15). Now ${\cal F}_{\fam=\ssfam \usuLambda}$ is a flat subplane and ${\fam=\ssfam \usuTheta}$ acts as a group of homologies on ${\cal F}_{\fam=\ssfam \usuLambda}$. This contradicts \cite{cp} 32.17 and shows that ${\fam=\ssfam \usuUpsilon}{\,\cong\,}\Sp4{\fam=\Bbbfam C}$. The fixed elements of the involution $\iota{\,=\,}(-1,-1,1,1)$ on $av$ form a $4$-sphere. Hence ${\cal F}_\iota{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$. Choose $c$ and $z$ as before, but in ${\cal F}_\iota$. Then ${\cal F}_{\fam=\ssfam \usuLambda}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal F}_\iota$, and Stiffness 2.6(c,\^e) implies ${\fam=\ssfam \usuLambda}{\,\cong\,}\SU3{\fam=\Bbbfam C}$. This group, however, is not contained in the maximal compact subgroup $\U2{\fam=\Bbbfam H}$ of ${\fam=\ssfam \usuUpsilon}$. Alternatively, $\Cs{\fam=\ssfam \usuUpsilon}\iota$ has a subgroup $(\SL2{\fam=\Bbbfam C})^2{\,\cong\,}(\Sp2{\fam=\Bbbfam C})^2$ contrary to \cite{sz1} 6.1. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\bold Remark.} {\it Under the assumptions of {\rm8.3}, suppose that $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}20$. Then ${\fam=\ssfam \usuDelta}$ is a Lie group or the center ${\fam=\ssfam Z}$ of ${\fam=\ssfam \usuDelta}$ consists of homologies in ${\fam=\ssfam \usuDelta}_{[u,av]}$\/}. \par
{\tt Proof.} (a) Assume that $a^{\fam=\ssfam Z}{\,\ne\,}a$ and that ${\fam=\ssfam \usuDelta}$ is not a Lie group. Let ${\fam=\ssfam N}$ be a compact $0$-dimensional normal subgroup such that ${\fam=\ssfam \usuDelta}/{\fam=\ssfam N}$ is a Lie group\; (cf. \cite{cp} 93.8). Put $\nabla{\,=\,}{\fam=\ssfam \usuDelta}_a$ and note that ${\fam=\ssfam \usuTheta}{\,\le\,}{\fam=\ssfam Z}$ and that
$\nabla|_{a^{\fam=\ssfam Z}}{\,=\,}1\kern-2.5pt {\rm l}$. There is some point $p{\,\in\,}au\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a,u\}$ such that $p^{\fam=\ssfam \usuTheta}{\,\ne\,}p$\; (or else ${\fam=\ssfam \usuTheta}{\,\le\,}{\fam=\ssfam \usuDelta}_{[v,au]}$, which contradicts $a^{\fam=\ssfam Z}{\,\ne\,}a$). The group ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuDelta}_p)^1$ fixes $a^{\fam=\ssfam Z}$, hence a quadrangle,
${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuLambda}{\,\le\,}16$, and ${\cal E}{\,:=\,}{\cal F}_{\fam=\ssfam \usuLambda}{\,=\,}{\cal E}^{\fam=\ssfam \usuTheta}$ is a connected proper subplane. From ${\fam=\ssfam \usuTheta}|_{\cal E}{\,\ne\,}1\kern-2.5pt {\rm l}$ it follows that $\dim{\cal E}{\,\ge\,}4$\; (use \cite{cp} 32.21{b} and 17). \\
(b) If $\dim{\cal E}{\,=\,}4$, then ${\fam=\ssfam N}|_{\cal E}{\,=\,}{\fam=\ssfam N}/{\fam=\ssfam \usuXi}$ is a Lie group by \cite{cp} 71.2, and ${\fam=\ssfam \usuXi}{\,\ne\,}1\kern-2.5pt {\rm l}$. Hence ${\cal E}{\,\le\,}{\cal F}_{\fam=\ssfam \usuXi}{\,<\,}{\cal P}$, and ${\cal F}_{\fam=\ssfam \usuXi}$ is ${\fam=\ssfam \usuDelta}$-invariant. We have ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}\nabla{\,\le\,}8$,\; $\nabla{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuLambda}{\,=\,}\dim p^\nabla$, and $p^\nabla{\,\subseteq\,}{\cal F}_{\fam=\ssfam \usuXi}$. Consequently ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuLambda}{\,\le\,}12$ and $\dim{\fam=\ssfam \usuLambda}{\,\le\,}8$. If ${\cal F}_{\fam=\ssfam \usuXi}{\,=\,}{\cal E}$, then $\dim p^\nabla{\,\le\,}2$ and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}18$; if ${\cal F}_{\fam=\ssfam \usuXi}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$ and $\dim{\fam=\ssfam \usuLambda}{\,=\,}8$, then ${\fam=\ssfam \usuLambda}{\,\cong\,}\SU3{\fam=\Bbbfam C}$
and the lines of ${\cal F}_{\fam=\ssfam \usuXi}$ are $4$-spheres, but then ${\fam=\ssfam \usuLambda}$ cannot act on ${\cal F}_{\fam=\ssfam \usuXi}$, see $(\dagger)$. Therefore $\dim{\fam=\ssfam \usuDelta}{\,<\,}20$. If ${\cal E}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$ and ${\fam=\ssfam \usuLambda}\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam N}{\,=\,}1\kern-2.5pt {\rm l}$, then ${\fam=\ssfam \usuLambda}$ is a Lie group and $\dim{\fam=\ssfam \usuLambda}{\,\le\,}3$ by Stiffness, hence $\dim{\fam=\ssfam \usuDelta}{\,\le\,}16{+}3{\,<\,}20$. If $1\kern-2.5pt {\rm l}{\,\ne\,}\zeta{\,\in\,}{\fam=\ssfam \usuLambda}\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam N}$, then ${\cal E}{\,=\,}{\cal F}_\zeta{\,=\,}{\cal E}^{\fam=\ssfam \usuDelta}$ and ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuLambda}{\,\le\,}8$. In this case $\dim{\fam=\ssfam \usuDelta}{\,\le\,}15$. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
The following theorems have been proved in \cite{hs}: \par
{\bf 8.4.} {\it If $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}33$ and if ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\langle u,v,av\rangle\,}$ is a double flag, then the translation group ${\fam=\ssfam T}{\hskip1pt=\,}{\fam=\ssfam \usuDelta}_{[v,uv]}$ is transitive and ${\fam=\ssfam \usuPhi}{\,=\,}({\fam=\ssfam \usuDelta}_a)'{\,\cong\,}\Spin8{\fam=\Bbbfam R}$. In particular, $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}36$. If $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}38$, then the plane is classical\/}. \par
{\bf 8.5 Distorted octonions.} {\it Let $({\fam=\Bbbfam R},+,\ast,1)$ be a topological Cartesian field with unit element $1$ such that $\:(-r){\,\ast\,}s{\,=\,} {\,-\,} (r{\,\ast\,}s){\,=\,}r{\,\ast\,}(-s)$ holds identically. Define a new multiplication on the octonion algebra $({\fam=\Bbbfam O},+, \: )$ by
$ a {\,\circ\,} x {\,=\,} {|a|{\ast}|x|\,|ax|^{-1}}\, a\hskip1pt x $ for $a,x{\,\ne\,} 0$ and $0{\,\circ\,}x {\,=\,} a{\,\circ\,}0 {\,=\,} 0$. Then the {\emph distorted octonions} $({\fam=\Bbbfam O},+,\circ)$ form a topological Cartesian field\/}. \par
{\bf 8.6 Theorem.} {\it A plane ${\cal P}$ can be coordinatized by distorted octonions if, and only if, ${\cal P}$ has the properties of Theorem\/} 8.4. \par
{\Bf 9. Fixed triangle} \par
Let ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\langle a,u,v\rangle$ be a triangle. If ${\cal P}$ is a translation plane with translation axis $uv$, then ${\fam=\ssfam \usuDelta}$ has a compact subgroup ${\fam=\ssfam \usuPhi}$ of codimension ${\fam=\ssfam \usuDelta}{\,:\,}{\fam=\ssfam \usuPhi}{\,\le\,}2$, see H\"ahl \cite{Ha5}, \cite{cp} 81.8. There seems to be no way to extend this basic result to general $16$-dimensional planes; cf. 9.1, however. \par
{\bf 9.0 Lie.} {\it If ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\langle a,u,v\rangle$ is a triangle, and if $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}26$, then ${\fam=\ssfam \usuDelta}$ is a Lie group\/}. \par
By \cite{psz}, only the case $\dim{\fam=\ssfam \usuDelta}{\,=\,}26$ requires a {\tt proof.} Suppose that ${\fam=\ssfam \usuDelta}$ is not a Lie group, and consider the action of ${\fam=\ssfam \usuDelta}$ on the sides $S_\nu$ of the triangle (without the vertices).
If ${\fam=\ssfam \usuDelta}$ is transitive on $S_\nu$, then ${\fam=\ssfam \usuDelta}|_{S_\nu}{\,=\,}{\fam=\ssfam \usuDelta}/{\fam=\ssfam K}_\nu$ is a Lie group, see \cite{cp} 53.2. Suppose that this happens for two distinct indices $\mu,\nu$. It follows that ${\fam=\ssfam K}_\nu$ is a Lie group, since ${\fam=\ssfam K}_\mu\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam K}_\nu{\,=\,}1\kern-2.5pt {\rm l}$, and then ${\fam=\ssfam \usuDelta}$ itself is a Lie group by \cite{cp} 94.3(d). For the same reason, all orbits on two sides of the triangle have dimension~${<\,}8$, and $\dim x^{\fam=\ssfam \usuDelta}{\,\le\,}14$ for each point $x{\,\notin\,}au{\kern 2pt {\scriptstyle \cup}\kern 2pt}av{\kern 2pt {\scriptstyle \cup}\kern 2pt}uv$. Put ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuDelta}_x)^1$. Then ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuLambda}{\hskip1pt\le\hskip1pt}14$ and $\dim{\fam=\ssfam \usuLambda}{\,>\,}11$. Stiffness implies ${\fam=\ssfam \usuLambda}{\,\cong\,}{\rm G}_2$, and representation of ${\fam=\ssfam \usuLambda}$ on the Lie algebra ${\frak l}\hskip1pt{\fam=\ssfam \usuDelta}$ shows $\dim\Cs{}{\fam=\ssfam \usuLambda}{\,\in\,}\{5,12\}$. Again by Stiffness, $\Cs{}{\fam=\ssfam \usuLambda}$ acts almost effectively on the flat plane ${\cal F}_{\fam=\ssfam \usuLambda}$. This contradicts the assumption on ${\cal F}_{\fam=\ssfam \usuDelta}$. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\bf Remark.} {\it If ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\langle a,u,v\rangle$ is a triangle, if $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}19$, and if $x{\,\notin\,}au{\kern 2pt {\scriptstyle \cup}\kern 2pt}av{\kern 2pt {\scriptstyle \cup}\kern 2pt}uv$, then ${\fam=\ssfam \usuDelta}_x$ is a Lie group\/}. \par
{\tt Proof.} We may assume that ${\fam=\ssfam \usuDelta}$ is not a Lie group. Again there exist arbitrarily small compact central subgroups ${\fam=\ssfam N}{\,\le\,}{\fam=\ssfam \usuDelta}$ of dimension $0$ such that ${\fam=\ssfam \usuDelta}/{\fam=\ssfam N}$ is a Lie group, see \cite{cp} 93.8. We will show that ${\fam=\ssfam N}$ acts freely on $P\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt (au{\kern 2pt {\scriptstyle \cup}\kern 2pt}av{\kern 2pt {\scriptstyle \cup}\kern 2pt}uv)$. Suppose that $a,u,v,x$ is a non-degenerate quadrangle, and that $x^\zeta{\,=\,}x$ for some $\zeta{\,\in\,}{\fam=\ssfam N}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{1\kern-2.5pt {\rm l}\}$. Then
$\zeta|_{\langle x^{\fam=\ssfam \usuDelta}\rangle}{\,=\,}1\kern-2.5pt {\rm l}$ and ${\cal D}{\,=\,}\langle x^{\fam=\ssfam \usuDelta}, a,u,v\rangle{\,=\,}{\cal D}^{\fam=\ssfam \usuDelta}$ is a proper subplane; in fact, ${\cal D}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$\; (or else $\dim x^{\fam=\ssfam \usuDelta}{\,\le\,}4$, and ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuDelta}_x)^1$ would have dimension $\dim{\fam=\ssfam \usuLambda}{\,>\,}14$, which contradicts Stiffness).
Put ${\fam=\ssfam \usuDelta}|_{\cal D}{\,=\,}{\fam=\ssfam \usuDelta}/{\fam=\ssfam K}$. By \cite{sz1} 1.7 we have ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\le\,}11$ and $\dim{\fam=\ssfam K}{\,\ge\,}8$, but the stiffness property 2.6(b) implies $\dim{\fam=\ssfam K}{\,\le\,}7$. This contradiction shows that ${\fam=\ssfam \usuDelta}_x{\kern 2pt {\scriptstyle \cap}\kern 2pt}{\fam=\ssfam N}{\,=\,}1\kern-2.5pt {\rm l}$. \hglue 0pt plus 1filll $\scriptstyle\square$
\par
{\bold 9.1 Semi-simple groups.} {\it If ${\fam=\ssfam \usuDelta}$ is semi-simple, if ${\cal F}_{\fam=\ssfam \usuDelta}$ is a triangle $a,u,v$, and if $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}28$, then ${\fam=\ssfam \usuDelta}{\,\cong\,}\Spin8{\fam=\Bbbfam R}$\/}. \par
{\tt Proof.} (a) Choose a point $e$ such that $a,e,u,v$ is a non-degenerate quadrangle, and consider the connected component ${\fam=\ssfam \usuLambda}$ of ${\fam=\ssfam \usuDelta}_e$. Then ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuLambda}{\,\le\,}16$,\: $\dim{\fam=\ssfam \usuLambda}{\,>\,}11$, and ${\fam=\ssfam \usuLambda}{\,\cong\,}{\rm G}_2$ by Stiffness. In particular, $\dim{\fam=\ssfam \usuDelta}{\,\le\,}30$. \\ (b) {\it ${\fam=\ssfam \usuDelta}$ is almost simple\/}. If not, then some proper factor ${\fam=\ssfam \usuGamma}$ of ${\fam=\ssfam \usuDelta}$ contains an isomorphic copy $\hat{\fam=\ssfam \usuLambda}$ of ${\fam=\ssfam \usuLambda}$. Repeated application of 2.15(a) and Stiffness show that ${\cal F}_{\hat{\fam=\ssfam \usuLambda}}$ is a flat subplane and that ${\fam=\ssfam \usuPsi}{\,=\,}(\Cs{\fam=\ssfam \usuDelta}{\fam=\ssfam \usuGamma})^1$ acts effectively on ${\cal F}_{\hat{\fam=\ssfam \usuLambda}}$. By \cite{cp} 33.8 the group ${\fam=\ssfam \usuPsi}$ is solvable, in fact $\dim{\fam=\ssfam \usuPsi}{\,\le\,}2$\: (see \cite{cp} 32.10, cf. also 33.10); hence ${\fam=\ssfam \usuPsi}{\,=\,}1\kern-2.5pt {\rm l}$ and ${\fam=\ssfam \usuDelta}{\,=\,}{\fam=\ssfam \usuGamma}$, a contradiction. \\ (c) If $\dim{\fam=\ssfam \usuDelta}{\,=\,}30$, then a maximal compact subgroup of ${\fam=\ssfam \usuDelta}$ is locally isomorphic to $\SU4{\fam=\Bbbfam C}$ and does not contain ${\rm G}_2$. Hence $\dim{\fam=\ssfam \usuDelta}{\,=\,}28$, and ${\fam=\ssfam \usuDelta}$ is a group of type ${\rm D}_4$ by 2.16. From 2.10 and ${\fam=\ssfam \usuLambda}{\,<\,}{\fam=\ssfam \usuDelta}$ it follows that ${\fam=\ssfam \usuDelta}{\,\cong\,}\Spin8({\fam=\Bbbfam R},r)$ with $r{\,\le\,}1$. Suppose that $r{\,=\,}1$. Then ${\fam=\ssfam \usuDelta}$ has a maximal compact subgroup ${\fam=\ssfam \usuPhi}{\,\cong\,}\Spin7{\fam=\Bbbfam R}$ such that ${\fam=\ssfam \usuLambda}$ is contained in ${\fam=\ssfam \usuPhi}$. Note that ${\fam=\ssfam \usuLambda}$ is even maximal in ${\fam=\ssfam \usuPhi}$\; (if not, then the action of ${\fam=\ssfam \usuLambda}$ on ${\frak l}{\fam=\ssfam \usuPhi}$ shows that ${\fam=\ssfam X}{\,=\,}\Cs{\fam=\ssfam \usuPhi}{\fam=\ssfam \usuLambda}$ is $7$-dimensional and ${\fam=\ssfam \usuPhi}{\,=\,}{\fam=\ssfam \usuLambda}{\fam=\ssfam X}$, but then ${\fam=\ssfam \usuLambda}$ would be normal in ${\fam=\ssfam \usuPhi}$). The central involution $\sigma{\,\in\,}{\fam=\ssfam \usuPhi}$ is a reflection, say with axis $uv$. Therefore ${\fam=\ssfam \usuDelta}$ acts effectively on $av$. Let $c{\,=\,}eu\kern 2pt {\scriptstyle \cap}\kern 2pt av$ and ${\fam=\ssfam \usuGamma}{\,=\,}({\fam=\ssfam \usuDelta}_c)^1$. Each orbit of ${\fam=\ssfam \usuPhi}$ on $av\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a,v\}$ is a $7$-sphere (cf. 2.15), and ${\fam=\ssfam \usuPhi}$ acts linearly on $av\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{v\}{\,\approx\,}{\fam=\Bbbfam R}^8$, see \cite{cp} 96.36. The stabilizer ${\fam=\ssfam \usuPhi}_c{\,=\,}{\fam=\ssfam \usuLambda}{\,\cong\,}{\rm G}_2$ is a maximal compact subgroup of ${\fam=\ssfam \usuGamma}$\; (use the Mal$'$cev-Iwasawa theorm). We have ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuGamma}{\,\ge\,}\dim c^{\fam=\ssfam \usuPhi}$ and ${\fam=\ssfam \usuGamma}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuLambda}{\,\in\,}\{6,7\}$. Stiffness implies $\dim(\Cs{\fam=\ssfam \usuGamma}{\fam=\ssfam \usuLambda}){\,\le\,}1$. As the action of ${\fam=\ssfam \usuLambda}$ on (the vector space underlying) the Lie algebra ${\frak l}\hskip1pt{\fam=\ssfam \usuGamma}$ is completely reducible, it follows from \cite{cp} 95.10 that $\dim{\fam=\ssfam \usuGamma}{\,=\,}21$. Hence $\dim c^{\fam=\ssfam \usuDelta}{\,=\,}7$, and then $c^{\fam=\ssfam \usuDelta}{\,=\,}c^{\fam=\ssfam \usuPhi}$. In particular, ${\fam=\ssfam \usuDelta}_c{\,=\,}{\fam=\ssfam \usuGamma}$ is connected. It follows that the map $\kappa{\,:\,}\delta{\,\mapsto\,}c^\delta$ on ${\fam=\ssfam \usuDelta}$ induces a homeomorphism ${\fam=\ssfam \usuDelta}/{\fam=\ssfam \usuGamma}{\,\approx\,}c^{\fam=\ssfam \usuPhi}{\,\approx\,}{\fam=\Bbbfam S}_7$, see \cite{cp} 96.9(a). Moreover, ${\fam=\ssfam \usuLambda}{\,<\,}{\fam=\ssfam \usuGamma}$ implies that ${\fam=\ssfam \usuGamma}$ is neither almost simple nor semi-simple. Consequently, $\sqrt{\fam=\ssfam \usuGamma}{\,=\,}{\fam=\ssfam \usuXi}{\,\cong\,}{\fam=\Bbbfam R}^7$ and ${\fam=\ssfam \usuGamma}{\,=\,}{\fam=\ssfam \usuXi}{\rtimes}{\fam=\ssfam \usuLambda}$ is given by the action of $\mathop{{\rm Aut}}{\fam=\Bbbfam O}$ on the pure octonions. The same arguments may be applied to $b{\,=\,}ev\kern 2pt {\scriptstyle \cap}\kern 2pt au$ instead of $c$. In particular, $b^{\fam=\ssfam \usuDelta}{\,=\,}b^{\fam=\ssfam \usuPhi}{\,=\,}S{\,\approx\,}{\fam=\Bbbfam S}_7$. The vector group ${\fam=\ssfam \usuXi}$ acts freely on $S$ because each stabilizer ${\fam=\ssfam \usuXi}_s$ wwith $s{\,\in\,}S$ fixes a quadrangle and is compact according to step (a). By the open mapping theorem \cite{cp} 51.19 or 96.11, each orbit $s^{\fam=\ssfam \usuXi}$ is open in $S$. Hence $b^{\fam=\ssfam \usuXi}{\,=\,}S$ and $\xi{\,\mapsto\,}b^\xi{\,:\,}{\fam=\ssfam \usuXi}{\,\to\,}S$ is a homeomorphism, but obviously this is not true. \hglue 0pt plus 1filll $\scriptstyle\square$
\par
{\bold Remark.} {\it If ${\fam=\ssfam \usuDelta}$ is semi-simple and ${\cal F}_{\fam=\ssfam \usuDelta}{\,=\,}\langle a,u,v\rangle$ is a triangle, then $\dim{\fam=\ssfam \usuDelta}{\,\ne\,}25$\/}. \par
{\tt Proof.} Assume that $\dim{\fam=\ssfam \usuDelta}{\,=\,}25$. This number being odd, ${\fam=\ssfam \usuDelta}$ has either a $15$-dimensional factor or at least one factor of dimension $3$. \\ (a) In the first case, ${\fam=\ssfam \usuDelta}{\,=\,}{\fam=\ssfam \usuGamma}{\fam=\ssfam \usuPsi}$ has exactly two factors. Let $\dim{\fam=\ssfam \usuGamma}{\,=\,}15$. If ${\fam=\ssfam \usuGamma}$ is transitive on $au\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a,u\}{\,\ni\,}x$, then ${\fam=\ssfam \usuPsi}_{\hskip-1pt x}$ is a group of homologies with axis $au$, and so is ${\fam=\ssfam \usuPsi}$ because ${\fam=\ssfam \usuPsi}$ is almost simple, but then $\dim{\fam=\ssfam \usuPsi}{\,\le\,}8$, a contradiction. Analogously, ${\fam=\ssfam \usuGamma}$ has an orbit $y^{\fam=\ssfam \usuGamma}{\subset\,}av\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a,v\}$ of dimension ${<\,}8$. Hence there is a point $p{\,\notin\,}au\kern 2pt {\scriptstyle \cup}\kern 2pt av\kern 2pt {\scriptstyle \cup}\kern 2pt uv$ such that ${\fam=\ssfam \usuGamma}_{\hskip-2pt p}{\,\ne\,}1\kern-2.5pt {\rm l}$, and ${\cal E}{\,=\,}{\cal F}_{{\fam=\ssfam \usuGamma}_{\hskip-1pt p}}$ is a proper ${\fam=\ssfam \usuPsi}$-invariant subplane. Consequently, $\dim p^{\fam=\ssfam \usuPsi}{\,\le\,}8$ and $\dim{\fam=\ssfam \usuPsi}_{\hskip-1pt p}{\,\ge\,}2$. It follows that
${\cal H}{\,=\,}{\cal F}_{{\fam=\ssfam \usuPsi}_{\hskip-1pt p}}{<\,}{\cal P}$, and ${\fam=\ssfam \usuGamma}|_{\cal H}$ is too large. \\ (b) Suppose that ${\fam=\ssfam \usuDelta}$ has a $10$-dimensional almost simple factor ${\fam=\ssfam \usuPsi}$. Then its complement ${\fam=\ssfam \usuGamma}{\,=\,}(\Cs{}{\fam=\ssfam \usuPsi})^1$ has dimension $15$. In step (a), the fact that ${\fam=\ssfam \usuGamma}$ is almost simple has been used only at the very end. Therefore there is again a subplane
${\cal H}{\,=\,}{\cal H}^{\fam=\ssfam \usuGamma}{<\,}{\cal P}$. Put ${\fam=\ssfam \usuGamma}|_{\cal H}{\,=\,}{\fam=\ssfam \usuGamma}/{\fam=\ssfam K}$. If ${\cal H}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$, then ${\fam=\ssfam K}$ is compact and semi-simple, hence a Lie group, and ${\fam=\ssfam K}{\,\cong\,}\Spin3{\fam=\Bbbfam R}$ by Stiffness. On the other hand, ${\fam=\ssfam \usuGamma}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\le\,}9$, see \cite{sz9} 6.1 or \cite{sz1} 7.3. Hence $\dim{\cal H}{\,\le\,}4$, and then ${\fam=\ssfam \usuGamma}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\le\,}3$ by 2.14. Now Stiffness implies ${\fam=\ssfam K}{\,\cong\,}{\rm G}_2$, which is impossible. \\ (c) If ${\fam=\ssfam \usuDelta}$ has an almost simple factor ${\fam=\ssfam \usuGamma}$ of dimension $16$, then a maximal compact subgroup of ${\fam=\ssfam \usuGamma}$ is locally isomorphic to $\SU3{\fam=\Bbbfam C}$, and 2.17 implies that ${\fam=\ssfam \usuGamma}$ cannot be transitive on one of the sides of the fixed triangle (without its vertices). Hence there is again a point $p$ such that ${\fam=\ssfam \usuGamma}_{\hskip-2pt p}{\,\ge\,}2$ and ${\cal E}{\,=\,}{\cal F}_{{\fam=\ssfam \usuGamma}_{\hskip-1pt p}}$ is a proper ${\fam=\ssfam \usuPsi}$-invariant subplane. Now
$\dim p^{\fam=\ssfam \usuPsi}{\,\le\,}8$, and $ {\cal H}{\,=\,}{\cal F}_{{\fam=\ssfam \usuPsi}_{\hskip-1pt p}}{<\,}{\cal P}$, but then $\dim{\fam=\ssfam \usuGamma}|_{\cal H}{\,\le\,}11$ by \cite{sz9} 6.1 or \cite{sz1} 1.7. \\ (d) The case that ${\fam=\ssfam \usuDelta}$ has two $8$-dimensional factors is similar but somewhat more complicated. Put $D{\,=\,}P\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt(au\kern 2pt {\scriptstyle \cup}\kern 2pt av\kern 2pt {\scriptstyle \cup}\kern 2pt uv)$, let $\dim{\fam=\ssfam \usuGamma}{\,=\,}8$, and denote the product of the other factors by ${\fam=\ssfam \usuPsi}$. Then $\dim{\fam=\ssfam \usuPsi}{\,=\,}17$,\, ${\fam=\ssfam \usuPsi}_{\hskip-1pt q}{\,\ne\,}1\kern-2.5pt {\rm l}$ for $q{\,\in\,}D$, and ${\cal D}{\,=\,}{\cal F}_{{\fam=\ssfam \usuPsi}_{\hskip-.5pt q}}{\,=\,}{\cal D}^{\hskip.5pt{\fam=\ssfam \usuGamma}}$ is a proper subplane. Either ${\fam=\ssfam \usuGamma}$ is sharply transitive on ${\cal D}\kern 2pt {\scriptstyle \cap}\kern 2pt D$, or ${\fam=\ssfam \usuGamma}_{\hskip-2pt p}{\,\ne\,}1\kern-2.5pt {\rm l}$ for some $p$ and ${\cal E}{\,=\,}{\cal F}_{{\fam=\ssfam \usuGamma}_{\hskip-1pt p}}{\,=\,}{\cal E}^{\fam=\ssfam \usuPsi}{\,<\,}{\cal P}$. In the first case, let ${\fam=\ssfam \usuPhi}$ be a maximal compact subgroup of ${\fam=\ssfam \usuGamma}$. Note that $\dim{\fam=\ssfam \usuPhi}{\,<\,}8$ and that ${\cal D}\kern 2pt {\scriptstyle \cap}\kern 2pt D$ is homotopy equivalent to ${\fam=\Bbbfam S}_3^{\;2}$. It follows that ${\fam=\ssfam \usuPhi}'{\,\circeq\,}\Spin3{\fam=\Bbbfam R}$, and the exact homotopy sequence \cite{cp} 96.12 yields $0\to\pi_3{\fam=\ssfam \usuPhi}'\to\pi_3{\fam=\Bbbfam S}_3^{\;2}{\hskip1pt=\hskip1pt}{\fam=\Bbbfam Z}^2\to0$, a contradiction. Hence ${\cal E}{\,<\,}{\cal P}$. For each possible dimension
of ${\cal E}$ it turns out that ${\fam=\ssfam \usuPsi}$ is too large: write ${\fam=\ssfam \usuPsi}|_{\cal E}{\,=\,}{\fam=\ssfam \usuPsi}/{\fam=\ssfam K}$. If $\dim{\cal E}{\,\le\,}4$, then ${\fam=\ssfam \usuPsi}{\,=\,}{\fam=\ssfam K}$\; (see \cite{cp} 33.8 and 2.14 above), but $\dim{\fam=\ssfam K}{\,\le\,}14$ by Stiffness; if ${\cal E}{\;\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$, then $\dim{\fam=\ssfam K}{\,\le\,}3$ and ${\fam=\ssfam \usuPsi}{\hskip1pt:\hskip1pt}{\fam=\ssfam K}{\,\ge\,}14$, which contradicts \cite{sz1} 1.7. \\
(e) Only one possibility remains: ${\fam=\ssfam \usuDelta}$ has a factor ${\fam=\ssfam \usuGamma}$ of type ${\rm G}_2$, a factor ${\fam=\ssfam \usuPsi}$ of dimension $8$, and a $3$-dimensional factor ${\fam=\ssfam \usuOmega}$. If ${\fam=\ssfam \usuGamma}$ is compact, then ${\cal F}_{\fam=\ssfam \usuGamma}$ is flat and ${\fam=\ssfam \usuPsi}{\,\cong\,}{\fam=\ssfam \usuPsi}|_{{\cal F}_{\fam=\ssfam \usuGamma}}$ would be trivial. Hence ${\fam=\ssfam \usuGamma}{\,\circeq\,}{\rm G}_2(2)$. In a similar way as before, this leads to a contradiction: $({\fam=\ssfam \usuGamma}{\fam=\ssfam \usuOmega})_p{\,\ne\,}1\kern-2.5pt {\rm l}$, and there is a subplane ${\cal E}{\,=\,}{\cal E}^{\fam=\ssfam \usuPsi}{\,<\,}{\cal P}$. If ${\fam=\ssfam \usuPsi}$ is sharply transitive on ${\cal E}\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt(au\kern 2pt {\scriptstyle \cup}\kern 2pt av\kern 2pt {\scriptstyle \cup}\kern 2pt uv)$, then a maximal compact subgroup ${\fam=\ssfam \usuPhi}$ of ${\fam=\ssfam \usuPsi}$ is homeomorphic to $({\fam=\Bbbfam H}')^2$ by the Mal'cev-Iwasawa theorem, but $\dim{\fam=\ssfam \usuPhi}{\,\in\,}\{3,4,8\}$. Hence there is some point $q{\,\in\,}{\cal E}$ such that ${\fam=\ssfam \usuPsi}_{\hskip-1pt q}{\,\ne\,}1\kern-2.5pt {\rm l}$, and ${\fam=\ssfam \usuGamma}$ acts almost faithfully on
${\cal H}{\,=\,}{\cal F}_{{\fam=\ssfam \usuPsi}_{\hskip-1pt q}}$, but $\dim{\fam=\ssfam \usuGamma}|_{\cal H}{\,\le\,}11$. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\bf 9.2 Lie.} {\it Assume that ${\fam=\ssfam \usuDelta}$ fixes a triangle $a,u,v$\/}. \\ (A) {\it If ${\fam=\ssfam \usuDelta}$ is semi-simple and if $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}22$, then ${\fam=\ssfam \usuDelta}$ is a Lie group\/}. \\
(B) {\it If ${\fam=\ssfam \usuDelta}$ has a $1$-dimensional compact normal subgroup ${\fam=\ssfam \usuTheta}$ and if
$\dim{\fam=\ssfam \usuDelta}{\,\ge\,}18$, then ${\fam=\ssfam \usuDelta}$ is a Lie group and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}23$\/}. \\ (C) {\it If ${\fam=\ssfam \usuDelta}$ has a minimal normal vector subgroup ${\fam=\ssfam \usuTheta}{\,\cong\,}{\fam=\Bbbfam R}^t$ and if $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}22$, then ${\fam=\ssfam \usuDelta}$ is a Lie group\/}. \par
Part (A) {\tt Proof.} Because of the previous Remark and 9.0 we may assume that $\dim{\fam=\ssfam \usuDelta}{\,\le\,}24$. Suppose that ${\fam=\ssfam \usuDelta}$ is not a Lie group, und use the same notation as in 9.0. Let ${\fam=\ssfam \usuGamma}$ be a proper factor of ${\fam=\ssfam \usuDelta}$ of minimal dimension and denote the product of the other factors by ${\fam=\ssfam \usuPsi}$. Recall from the proof of 9.0 and the Remark added to 9.0 that ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuDelta}_p{\,\le\,}14$ and that ${\fam=\ssfam \usuDelta}_p\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam N}{\,=\,}1\kern-2.5pt {\rm l}$ for any point $p{\,\notin\,}au\kern 2pt {\scriptstyle \cup}\kern 2pt av\kern 2pt {\scriptstyle \cup}\kern 2pt uv$. \\
(a) If $p^{\fam=\ssfam \usuGamma}{\,=\,}p$, then ${\fam=\ssfam \usuGamma}|_{p^{\fam=\ssfam \usuDelta}}{\,=\,}1\kern-2.5pt {\rm l}$. The action of ${\fam=\ssfam N}$ on ${\cal F}_{\fam=\ssfam \usuLambda}$ implies ${\fam=\ssfam \usuLambda}{\,\not\cong\,}{\rm G}_2$. Hence $\dim{\fam=\ssfam \usuDelta}_p{\,\le\,}11$ and $\dim p^{\fam=\ssfam \usuDelta}{\,\ge\,}11{\,>\,}8$, but then ${\fam=\ssfam \usuGamma}{\,=\,}1\kern-2.5pt {\rm l}$. Therefore $p^{\fam=\ssfam \usuGamma}{\,\ne\,}p$. \\ (b) {\it $\dim{\fam=\ssfam \usuGamma}{\,\ge\,}6$\/}: we have ${\cal E}{\,=\,}\langle p^{{\fam=\ssfam \usuGamma}{\fam=\ssfam N}},a,u,v\rangle\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\!{\cal P}$ by \cite{cp} 32.21 and 71.2.
Obviously, ${\fam=\ssfam \usuPsi}_p|_{\cal E}{\,=\,}1\kern-2.5pt {\rm l}$ and ${\fam=\ssfam \usuPsi}_p$ is a Lie group. Hence $\dim{\fam=\ssfam \usuPsi}_p{\,\le\,}3$,\; $\dim{\fam=\ssfam \usuPsi}{\,\le\,}17$, and $\dim{\fam=\ssfam \usuGamma}{\,\ge\,}5$. \\ (c) Suppose in steps (c--f) that $\dim{\fam=\ssfam \usuGamma}{\,\ge\,}23$. Then
$\dim{\fam=\ssfam \usuDelta}{\,=\,}24$, or $\dim{\fam=\ssfam \usuGamma}{\,=\,}8$,\; $\dim{\fam=\ssfam \usuPsi}{\,=\,}15$, and ${\fam=\ssfam \usuPsi}$ is almost simple. In both cases, $\dim{\fam=\ssfam \usuGamma}{\,>\,}6$. \\ (d) If $\dim{\fam=\ssfam \usuGamma}{\,=\,}8$, then again ${\cal E}{\,=\,}\langle p^{{\fam=\ssfam \usuGamma}{\fam=\ssfam N}},a,u,v\rangle\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\!{\cal P}$
and ${\fam=\ssfam \usuPsi}_p|_{\cal E}{\,=\,}1\kern-2.5pt {\rm l}$. Stiffness 2.6(\^b) shows that $\dim{\fam=\ssfam \usuPsi}_p{\,\in\,}\{1,3\}$,\, ${\cal E}{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}\,}{\cal P}$, and $\langle p^{\fam=\ssfam \usuPsi}\rangle{\,=\,}{\bold{\cal P}}$. Consequently, $({\fam=\ssfam \usuGamma}{\fam=\ssfam N})_p{\,=\,}1\kern-2.5pt {\rm l}$,\, $p^{\fam=\ssfam \usuGamma}$ is open in the point space of ${\cal E}$, and ${\fam=\ssfam N}$ would be a Lie group by \cite{cp} 53.2 contrary to the assumption. \\ (e) {\it ${\fam=\ssfam \usuDelta}$ is almost simple\/}. The only other possibility is $\dim{\fam=\ssfam \usuGamma}{\,=\,}10$. In this case ${\fam=\ssfam \usuPsi}$ is a Lie group of type ${\rm G_2}$, and ${\fam=\ssfam \usuPsi}$ is simple or a twofold covering of a simple group. Each non-central involution of ${\fam=\ssfam \usuPsi}$ acts non-trivially on all three sides of the fixed triangle. Hence ${\fam=\ssfam \usuPsi}$ contains a planar involution $\beta$, and
${\fam=\ssfam \usuGamma}|_{{\cal F}_\beta}{\,\ne\,}1\kern-2.5pt {\rm l}$, but then $\dim{\fam=\ssfam \usuGamma}{\,\le\,}9$ by \cite{sz9} 6.1. \\ (f) Because of step (e), the group ${\fam=\ssfam \usuDelta}$ maps onto $\PSU5({\fam=\Bbbfam C},r)$ with $r{\,=\,}1$ or $r{\,=\,}2$, and ${\fam=\ssfam \usuDelta}$ has a subgroup ${\fam=\ssfam \usuPhi}{\times}{\fam=\ssfam X}$, where ${\fam=\ssfam \usuPhi}{\,\cong\,}\SU3{\fam=\Bbbfam C}$ and ${\fam=\ssfam X}{\,\cong\,}\SU2({\fam=\Bbbfam C},2{-}r)$. If $\iota$ denotes the involution ${\rm diag}(1,1,1,-1,-1)$, then ${\fam=\ssfam \usuPhi}{\times}{\fam=\ssfam X}{\,\le\,}\Cs{\fam=\ssfam \usuDelta}\iota$ and $\dim\Cs{\fam=\ssfam \usuDelta}\iota{\,\le\,}9{+}4$. Consequently, $\iota$ is not in the center of ${\fam=\ssfam \usuDelta}$ and $\iota$ acts non-trivially on the sides of the fixed triangle.
Therefore $\iota$ is planar. As $\dim{\fam=\ssfam \usuPhi}{\,=\,}8$, Stiffness implies that ${\fam=\ssfam \usuPhi}|_{{\cal F}_\iota}{\,\ne\,}1\kern-2.5pt {\rm l}$. An involution $\kappa{\,\in\,}{\fam=\ssfam \usuPhi}$ is not in the center of ${\fam=\ssfam \usuPhi}$ and induces a Baer involution on ${\cal F}_\iota$ for analogous reasons as those which proved the planarity of $\iota$. Therefore ${\cal C}{\,=\,}{\cal F}_{\iota,\kappa}$ is a $4$-dimensional subplane. Note that ${\fam=\ssfam N}$ acts freely on ${\cal C}$. By \cite{cp} 53.2, the group ${\fam=\ssfam N}$ is a Lie group and so is ${\fam=\ssfam \usuDelta}$. \\ (g) Now let $\dim{\fam=\ssfam \usuDelta}{\,=\,}22$. Then ${\fam=\ssfam \usuDelta}$ has no almost simple factor of dimension $16$, or else $\dim{\fam=\ssfam \usuGamma}{\,=\,}6$, both ${\fam=\ssfam \usuGamma}$ and ${\fam=\ssfam \usuPsi}$ are Lie groups, and so is ${\fam=\ssfam \usuDelta}{\,=\,}{\fam=\ssfam \usuGamma}{\fam=\ssfam \usuPsi}$. (Note that the maximal compact subgroups of ${\fam=\ssfam \usuGamma}$ and ${\fam=\ssfam \usuPsi}$ are almost simple, and use the approximation theorem \cite{cp} 93.8 together with 93.11.) If ${\fam=\ssfam \usuPsi}$ is almost simple of type ${\rm G}_2$, then ${\fam=\ssfam \usuPsi}{\,\circeq\,}{\rm G}_2(2)$. As in step (e), a non-central involution $\beta{\,\in\,}{\fam=\ssfam \usuPsi}$ is planar. There is a compact $6$-dimensional group ${\fam=\ssfam \usuPhi}{\,\le\,}\Cs{\fam=\ssfam \usuPsi}\beta$, and
$\dim{\fam=\ssfam \usuGamma}{\fam=\ssfam \usuPhi}|_{{\cal F}_\beta}{\,\ge\,}8{+}3$. This contradicts \cite{sz1} 7.3. \\ (h) Thus $\dim{\fam=\ssfam \usuGamma}{\,=\,}6$ and ${\fam=\ssfam \usuPsi}$ is a product of two almost simple factors ${\fam=\ssfam X}$ and ${\fam=\ssfam \usuUpsilon}$, where $\dim{\fam=\ssfam \usuUpsilon}{\,\i\,}\{8,10\}$. Again a non-central involutiom $\beta{\,\in\,}{\fam=\ssfam \usuUpsilon}$
is planar; moreover, $\dim{\fam=\ssfam X}{\,\ge\,}6$, and $\dim{\fam=\ssfam \usuGamma}{\fam=\ssfam X}{\fam=\ssfam N}|_{{\cal F}_\beta}{\,\ge\,}12$, but then ${\fam=\ssfam N}$ is a Lie group by \cite{pw3}. \\
Part (B) {\tt Proof.} (a) Suppose that ${\fam=\ssfam \usuDelta}$ is not a Lie group. As in the proof of 9.0, all orbits on two sides of the fixed triangle have dimension~${<\,}8$, and $\dim p^{\fam=\ssfam \usuDelta}{\,\le\,}14$ for each point $p{\,\notin\,}au{\kern 2pt {\scriptstyle \cup}\kern 2pt}av{\kern 2pt {\scriptstyle \cup}\kern 2pt}uv$. Put ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuDelta}_p)^1$. By the approximation theorem \cite{cp} 93.8, there is a compact $0$-dimensional central subgroup ${\fam=\ssfam N}{\,\triangleleft\,}{\fam=\ssfam \usuDelta}$ such that ${\fam=\ssfam \usuDelta}/{\fam=\ssfam N}$ is a Lie group. Note that ${\fam=\ssfam \usuTheta}{\,\le\,}\Cs{}{\fam=\ssfam \usuDelta}$. If $p^{\fam=\ssfam \usuTheta}{\,=\,}p$, then ${\cal F}_{\fam=\ssfam \usuTheta}{\,=\,}{\cal F}_{\hskip-1pt{\fam=\ssfam \usuTheta}}^{\:{\fam=\ssfam \usuDelta}}{\,<\,}{\cal P}$ and ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuDelta}_p{\,=\,}\dim p^{\fam=\ssfam \usuDelta}{\,\le\,}\dim{\cal F}_{\fam=\ssfam \usuTheta}{\,=\,}d{\,\in\,}\{0,2,4,8\}$. Either $d{\,\le\,}2$,\, ${\fam=\ssfam \usuTheta}{\,\triangleleft\,}{\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuDelta}_p)^1{\,\not\cong\,}{\rm G}_2$,\, $\dim{\fam=\ssfam \usuLambda}{\,\le\,}11$, and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}13$, or $d{\,\ge\,}4$,\; $\dim{\fam=\ssfam \usuLambda}{\,\le\,}8$, and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}16$. If $p^{\fam=\ssfam \usuTheta}{\,\ne\,}p$, then ${\fam=\ssfam \usuTheta}$ acts non-trivially on the subplane ${\cal E}{\,=\,}\langle p^{{\fam=\ssfam \usuTheta}{\fam=\ssfam N}},a,u,v \rangle$, ${\cal E}$ is not flat by \cite{cp} 32.17, and
$\dim{\cal E}{\,=\,}4$ or ${\cal E}{\,\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\!}{\cal P}$. In the first case, ${\fam=\ssfam N}|_{\cal E}{\,=\,}{\fam=\ssfam N}/{\fam=\ssfam K}$ is a Lie group by \cite{cp} 71.2, and ${\fam=\ssfam K}{\,\ne\,}1\kern-2.5pt {\rm l}$. Hence ${\cal F}_{\fam=\ssfam K}{\,=\,}{\cal F}_{\fam=\ssfam K}^{\hskip2pt{\fam=\ssfam \usuDelta}}{\,<\,}{\cal P}$,\, $p^{\fam=\ssfam \usuDelta}{\,\subseteq\,}{\cal F}_{\fam=\ssfam K}$,\, ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuDelta}_p{\,\le\,}8$,
${\fam=\ssfam \usuDelta}_p|_{\cal E}{\,=\,}1\kern-2.5pt {\rm l}$,\, $\dim{\fam=\ssfam \usuDelta}_p{\,\le\,}8$, and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}16$.
In the second case, we have ${\fam=\ssfam N}_p|_{\langle p^{\fam=\ssfam \usuDelta}\rangle}{\,=\,}1\kern-2.5pt {\rm l}$. Therefore ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuDelta}_p{\,=\,}\dim p^{\fam=\ssfam \usuDelta}{\,\le\,}8$ or ${\fam=\ssfam \usuDelta}_p\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam N}{\,=\,}1\kern-2.5pt {\rm l}$ and ${\fam=\ssfam \usuDelta}_p$ is a Lie group. Stiffness implies $\dim{\fam=\ssfam \usuDelta}_p{\,\le\,}7$ or $\dim{\fam=\ssfam \usuDelta}_p{\,\le\,}3$, respectively, and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}17$. \\ (b) If ${\fam=\ssfam \usuDelta}$ is transitive on the complement of the triangle, then ${\fam=\ssfam \usuDelta}'{\,\cong\,}\Spin8{\fam=\Bbbfam R}$ by Corollary 2.18, $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}28$, ${\fam=\ssfam \usuDelta}$ is a Lie group, the torus rank $\rm rk\,{\fam=\ssfam \usuDelta}$ is $4$, and ${\fam=\ssfam \usuDelta}$ cannot have a normal torus subgroup. Hence there is some point $p$ such that $p,a,u,v$ form a quadrangle and ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuDelta}_p{\,<\,}16$. Put ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuDelta}_p)^1$ and
${\cal E}{\,=\,}{\cal F}_{\fam=\ssfam \usuLambda}$. If ${\fam=\ssfam \usuTheta}|_{\cal E}{\,=\,}1\kern-2.5pt {\rm l}$, then ${\fam=\ssfam \usuTheta}{\,\le\,}{\fam=\ssfam \usuLambda}$, ${\cal E}{\,\le\,}{\cal F}{\,=\,}{\cal F}_{\fam=\ssfam \usuTheta}{\,=\,}{\cal F}^{\fam=\ssfam \usuDelta}{\,<\,}{\cal P}$, ${\fam=\ssfam \usuDelta}{\hskip1pt:\hskip1pt}{\fam=\ssfam \usuDelta}_p{\,=\,}\dim p^{\fam=\ssfam \usuDelta}{\,\le\,}8$, ${\fam=\ssfam \usuLambda}{\,\not\cong\,}{\rm G}_2$,
$\dim{\fam=\ssfam \usuLambda}{\,\le\,}11$, and $\dim{\fam=\ssfam \usuDelta}{\,\le\,}19$. If ${\fam=\ssfam \usuTheta}|_{\cal E}{\,\ne\,}1\kern-2.5pt {\rm l}$, however, then
${\cal E}{\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\!}{\cal P}$ or ${\fam=\ssfam \usuTheta}|_{\cal E}$ is a Lie group by \cite{cp} 32.21 and 71.2, and
${\fam=\ssfam \usuTheta}|_{\cal E}{\,\cong\,}{\fam=\Bbbfam T}$. From \cite{cp} 32.17 it follows that ${\cal E}$ is not flat, and $\dim{\cal E}$ is at least $4$. Stiffness implies $\dim{\fam=\ssfam \usuLambda}{\,\le\,}8$ and $\dim{\fam=\ssfam \usuDelta}{\,<\,}24$. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
Part (C) {\tt Proof.} (a) As in part (A) and with the same notation, $\dim p^{\fam=\ssfam \usuDelta}{\,\le\,}14$ for each point $p{\,\notin\,}au{\kern 2pt {\scriptstyle \cup}\kern 2pt}av{\kern 2pt {\scriptstyle \cup}\kern 2pt}uv$. Select $p$ in such a way that $p^{\fam=\ssfam \usuTheta}{\,\ne\,}p$ and choose a one-parameter subgroup ${\fam=\ssfam \usuPi}{\,\le\,}{\fam=\ssfam \usuTheta}$ and an element $\rho{\,\in\,}{\fam=\ssfam \usuPi}$ with $p^\rho{\,\ne\,}p$. The action of ${\fam=\ssfam \usuDelta}$ on ${\fam=\ssfam \usuTheta}$ being linear, we have ${\fam=\ssfam \usuTheta}_p{\,\le\,}{\fam=\ssfam \usuGamma}{\,=\,}{\fam=\ssfam \usuDelta}_{p,\rho}{\,\le\,}\Cs{}{\fam=\ssfam \usuPi}{\fam=\ssfam N}$ and $\dim{\fam=\ssfam \usuGamma}{\,\ge\,}\dim{\fam=\ssfam \usuDelta}{\hskip1pt-\hskip1pt}14{\hskip1pt-\hskip1pt}t$. As $p^{\fam=\ssfam \usuPi}{\,\ne\,}p$, the plane ${\cal E}{\,=\,}\langle p^{{\fam=\ssfam \usuPi}{\fam=\ssfam N}},a,u,v\rangle$ is a connected subplane. By the Remark added to 9.0, the central subgroup ${\fam=\ssfam N}$ satisfies ${\fam=\ssfam \usuDelta}_p\kern 2pt {\scriptstyle \cap}\kern 2pt{\fam=\ssfam N}{\,=\,}1\kern-2.5pt {\rm l}$. Therefore ${\fam=\ssfam \usuDelta}_p$ and ${\fam=\ssfam \usuGamma}$ are Lie groups and ${\fam=\ssfam N}$ acts effectively on ${\cal E}$. Because of \cite{cp} 32.21 and 71.2 it follows that ${\cal E}{\hskip1pt\le\!\!\!\raise2pt\hbox{$\scriptscriptstyle\bullet$}\;\hskip-3pt}{\cal P}$. Note that
${\fam=\ssfam \usuGamma}|_{\cal E}{\,=\,}1\kern-2.5pt {\rm l}$. Stiffness shows that ${\fam=\ssfam \usuGamma}$ is compact. In particular, ${\fam=\ssfam \usuTheta}_p$ is compact and hence ${\fam=\ssfam \usuTheta}_p{\,=\,}1\kern-2.5pt {\rm l}$. Moreover, ${\fam=\ssfam \usuGamma}^1{\,\le\,}\Spin3{\fam=\Bbbfam R}$ and $\dim{\fam=\ssfam \usuGamma}{\,\le\,}3$. This proves the claim for $t{\,<\,}5$. If $t{\,=\,}5$, then $\dim\rho^{{\fam=\ssfam \usuDelta}_p}{\,=\,}{\fam=\ssfam \usuDelta}_p{:\hskip1pt}{\fam=\ssfam \usuGamma}{\,=\,}5$ and $\dim{\fam=\ssfam \usuDelta}_p{\,=\,}8$. As ${\fam=\ssfam \usuTheta}_p{\,=\,}1\kern-2.5pt {\rm l}$, there is no restriction for the choice of $\rho$. Hence ${\fam=\ssfam \usuDelta}_p$ is transitive on ${\fam=\ssfam \usuTheta}\sm1\kern-2.5pt {\rm l}$, but a transitive linear group on ${\fam=\Bbbfam R}^5$ contains $\SO5{\fam=\Bbbfam R}$ and has dimension at least $11$, see \cite{Vl} Satz~1 or use \cite{cp} 96.19--22. \\ (b) Let ${\fam=\ssfam \usuLambda}{\,=\,}({\fam=\ssfam \usuDelta}_p)^1$ and consider a minimal ${\fam=\ssfam \usuLambda}$-invariant subgroup ${\fam=\ssfam H}{\,\cong\,}{\fam=\Bbbfam R}^s$ of ${\fam=\ssfam \usuTheta}$. According to step (a), we may assume that $s{\,>\,}5$. By definition, ${\fam=\ssfam \usuLambda}$ induces an irreducible re\-presentation $\overline{\fam=\ssfam \usuLambda}$ on ${\fam=\ssfam H}$; the kernel of this representation is contained in~${\fam=\ssfam \usuGamma}$. Consequently $\dim{\fam=\ssfam \usuLambda}{\,\ge\,}\dim{\fam=\ssfam \usuDelta}{\hskip1pt-\hskip1pt}14{\,\ge\,}8$ and $\dim\overline{\fam=\ssfam \usuLambda}{\,\ge\,}5$. A Levi complement $\overline{\fam=\ssfam \usuUpsilon}$ in $\overline{\fam=\ssfam \usuLambda}$ is the image of a maximal semi-simple subgroup ${\fam=\ssfam \usuUpsilon}$ of ${\fam=\ssfam \usuLambda}$, see \cite{cp} 94.27. \\ (c) In the case $s{\,=\,}6$ and $\dim{\fam=\ssfam \usuLambda}{\,>\,}8$ it follows that $\dim\rho^{\fam=\ssfam \usuLambda}{\,=\,}6$ for each choice of $\rho$ in~${\fam=\ssfam H}$. Hence $\overline{\fam=\ssfam \usuLambda}$ is a transitive linear group of ${\fam=\Bbbfam R}^6$, and ${\fam=\ssfam \usuLambda}$ is transitive on the $5$-sphere consisting of the rays in~${\fam=\Bbbfam R}^6$. From \cite{cp} 96.\hskip1pt19,21,22 and 94.27 we conclude that ${\fam=\ssfam \usuUpsilon}{\,\cong\,}\SU3{\fam=\Bbbfam C}$\; (since $\dim{\fam=\ssfam \usuUpsilon}{\,\le\,}14$). In particular $\rm rk\,{\fam=\ssfam \usuLambda}{\,\ge\,}2$, and ${\fam=\ssfam \usuLambda}$ contains two commuting planar involutions $\alpha,\beta$ such that $\dim{\cal F}_{\alpha,\beta}{\,=\,}4$. As ${\fam=\ssfam N}$ acts freely on ${\cal F}_{\alpha,\beta}$,\; ${\fam=\ssfam N}$ would be a Lie group by \cite{cp} 71.2, a contradiction to our assumption. More generally, the last arguments prove the following \\ (d) {\tt Lemma.} {\it If there exists a pair of commuting involutions in ${\fam=\ssfam \usuDelta}_p$, then ${\fam=\ssfam \usuDelta}$ is a Lie group\/}. \\ (e) The possibility $s{\,=\,}6$ and $\dim{\fam=\ssfam \usuLambda}{\,=\,}8$ will be postponed to step (h). If $s{\,=\,}7$, then Clifford's Lemma \cite{cp} 95.5 implies that $\overline{\fam=\ssfam \usuUpsilon}$ is almost simple of dimension $\dim\overline{\fam=\ssfam \usuUpsilon}{\,>3}$, and representation theory \cite{cp} 95.10 shows that $\overline{\fam=\ssfam \usuUpsilon}$ is a group of type ${\rm G_2}$ and torus rank 2. Hence ${\fam=\ssfam \usuDelta}$ is a Lie group by Lemma (d). \\ (f) Similarly, the other prime numbers can dealt with: if $s{\,=\,}11{\;\rm or\;}13$, then $\overline{\fam=\ssfam \usuUpsilon}{\,\circeq\,}\SO s{\fam=\Bbbfam R}$\; (again by \cite{cp} 95.10), and $\overline{\fam=\ssfam \usuUpsilon}$ is by far too large. \\ (g) For compound $s$ the group $\overline{\fam=\ssfam \usuUpsilon}$ may be properly semi-simple. If $s$ is even, complex representations are possible, and we can only infer $\dim\overline{\fam=\ssfam \usuUpsilon}{\,\ge\,}3$. The kernel ${\fam=\ssfam K}{\,=\,}\Cs{\fam=\ssfam \usuUpsilon}{\fam=\ssfam H}$ of the action of ${\fam=\ssfam \usuUpsilon}$ on ${\fam=\ssfam H}$ is contained in ${\fam=\ssfam \usuGamma}$; hence ${\fam=\ssfam K}$ is compact and $\dim{\fam=\ssfam K}{\,\le\,}3$. In fact, ${\fam=\ssfam K}^1{\,\cong\,}\Spin3{\fam=\Bbbfam R}$ or ${\fam=\ssfam K}$ is finite (note that ${\fam=\ssfam K}$ is semi-simple). As proper covering groups of $\SL2{\fam=\Bbbfam R}$ do not have a faithful linear representation, each factor of $\overline{\fam=\ssfam \usuUpsilon}$ has positive torus rank. The same holds for the factors of ${\fam=\ssfam \usuUpsilon}$. Because of Lemma (d) we may assume that $\overline{\fam=\ssfam \usuUpsilon}$ is almost simple and that ${\fam=\ssfam K}$ is finite. If $\rm rk\,{\fam=\ssfam \usuUpsilon}{\,>\,}1$ or if ${\fam=\ssfam \usuUpsilon}$ has a subgroup $\SO3{\fam=\Bbbfam R}$, Lemma (d) applies also. Thus only the cases $\overline{\fam=\ssfam \usuUpsilon}{\,\circeq\,}\SL2{\fam=\Bbbfam C},\ \SL3{\fam=\Bbbfam R}$, and $\Sp4{\fam=\Bbbfam R}$ need further consideration. \\
(h) In the first case, ${\fam=\ssfam \usuUpsilon}{\,\cong\,}\SL2{\fam=\Bbbfam C}$ contains a central involution $\iota$, which is planar because it fixes $p$. Let ${\fam=\ssfam \usuUpsilon}^*{\,=\,}{\fam=\ssfam \usuUpsilon}|_{{\cal F}_\iota}$. By Stiffness, ${\fam=\ssfam \usuUpsilon}^*{\,\ne\,}1\kern-2.5pt {\rm l}$, and then $\dim{\fam=\ssfam \usuUpsilon}^*{\,=\,}6$; on the other hand, the stiffness property \cite{sz1} 1.5(1) implies $\dim{\fam=\ssfam \usuUpsilon}^*{\,\le\,}4$. For similar reasons, $\dim{\fam=\ssfam \usuUpsilon}{\,\ne\,}8$: the central involution $\kappa$ of the simply connected two-fold covering ${\fam=\ssfam \usuUpsilon}$ of $\SL3{\fam=\Bbbfam R}$ is again planar, Stiffness shows
${\fam=\ssfam \usuUpsilon}|_{{\cal F}_\kappa}{\,\cong\,}\SL3{\fam=\Bbbfam R}$, but $\dim{\fam=\ssfam \usuUpsilon}|_{{\cal F}_\kappa}{\,\le\,}4$ by \cite{sz1} 1.5(1). Finally, let $\dim{\fam=\ssfam \usuUpsilon}{\,=\,}10$. Each faithful linear representation $\overline{\fam=\ssfam \usuUpsilon}$ of ${\fam=\ssfam \usuUpsilon}$ is isomorphic to ${\rm(P)}\Sp4{\fam=\Bbbfam R}$\; (see\cite{cp} 95,10), and $\U2{\fam=\Bbbfam C}$ is a maximal compact subgroup of $\Sp4{\fam=\Bbbfam R}$. By assumption, ${\fam=\ssfam \usuUpsilon}$ is a finite covering of $\overline{\fam=\ssfam \usuUpsilon}$. Hence $\rm rk\,{\fam=\ssfam \usuUpsilon}{\,=\,}2$, and Lemma (d) applies. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\bold Corollary.} {\it If ${\cal F}_{\fam=\ssfam \usuDelta}$ is a triangle and if $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}22$, then ${\fam=\ssfam \usuDelta}$ is a Lie group\/}. \par
{\tt Proof.} There is a compact $0$-dimensional central subgroup ${\fam=\ssfam N}$ such that ${\fam=\ssfam \usuDelta}/{\fam=\ssfam N}$ is a Lie group, and ${\fam=\ssfam \usuDelta}/{\fam=\ssfam N}$ is semi-simple or has a central torus ${\fam=\ssfam T}$ or a minimal normal vector group ${\fam=\ssfam \usuXi}$, see 2.3. In the first case, each commutative connected normal subgroup ${\fam=\ssfam A}{\,\triangleleft\,}{\fam=\ssfam \usuDelta}$ maps onto the identity of ${\fam=\ssfam \usuDelta}/{\fam=\ssfam N}$. Hence ${\fam=\ssfam A}{\,\le\,}{\fam=\ssfam N}$, and ${\fam=\ssfam A}$ is trivial, in other words, ${\fam=\ssfam \usuDelta}$ is semi-simple and (A) applies. In the other two cases, the connected component ${\fam=\ssfam \usuTheta}$ of the pre-image of ${\fam=\ssfam T}$ or ${\fam=\ssfam \usuXi}$ satisfies the assumptions in (B) or (C) respectively. \hglue 0pt plus 1filll $\scriptstyle\square$ \par
{\bf 9.3.} {\it If $\dim{\fam=\ssfam \usuDelta}{\,=\,}30$ and if ${\fam=\ssfam \usuDelta}$ fixes a triangle, then ${\fam=\ssfam \usuDelta}{\,=\,}{\fam=\ssfam P}{\times}{\fam=\ssfam \usuPhi}$, where ${\fam=\ssfam \usuPhi}{\,\cong\,}\Spin8{\fam=\Bbbfam R}$ and ${\fam=\ssfam P}$ is a product of two one-parameter groups of homologies with distinct centers\/}. \par
{\tt Proof.} Let $c{\,\in\,}av\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{a,v\}$ and $z{\,\in\,}S{\,=\,}uv\hskip-1pt$\raise1pt\hbox{$\scriptstyle\setminus$}$\hskip-1pt\{u,v\}$, and put ${\fam=\ssfam \usuGamma}{=\,}{\fam=\ssfam \usuDelta}_c$ and ${\fam=\ssfam \usuLambda}{\,=\,}{\fam=\ssfam \usuGamma}_{\hskip-1.5pt z}$. Then $\dim z^{\fam=\ssfam \usuGamma}{\,=\,}8$ and ${\fam=\ssfam \usuGamma}$ is transitive on $S$. As ${\fam=\ssfam \usuLambda}{\,\cong\,}{\rm G}_2$, it follows by 2.17(e) that ${\fam=\ssfam \usuGamma}{\,\cong\,}e^{{\fam=\Bbbfam R}}{\times}\Spin7{\fam=\Bbbfam R}$. Similarly, 2.17(f) shows that a maximal compact subgroup ${\fam=\ssfam \usuPhi}$ of ${\fam=\ssfam \usuDelta}$ is isomorphic to $\Spin8{\fam=\Bbbfam R}$. The $2$-dimensional radical ${\fam=\ssfam P}{\,=\,}\sqrt{\fam=\ssfam \usuDelta}$ acts effectively on the flat plane ${\cal F}_{\fam=\ssfam \usuLambda}$, and ${\fam=\ssfam P}{\,\cong\,}{\fam=\Bbbfam R}^2$ by \cite{cp} 33.10. Therefore
${\fam=\ssfam P}_{\hskip-2pt z}{\,\le\,}\Cs{}{\fam=\ssfam \usuGamma}$,\: ${\fam=\ssfam P}_{\hskip-2pt z}|_{z^{\fam=\ssfam \usuGamma}}{\,=\,}1\kern-2.5pt {\rm l}$, and ${\fam=\ssfam P}_{\hskip-2pt z}{\,\le\,}{\fam=\ssfam \usuDelta}_{[a,uv]}$. Analogously, ${\fam=\ssfam P}_{\hskip-2pt c}{\,\le\,}{\fam=\ssfam \usuDelta}_{[u,av]}$. \hglue 0pt plus 1filll $\scriptstyle\square$
\par
\break {\Bf 10. Summary} \par
The entries in the table below have the following meaning:\quad \par
\begin{tabular}{l l} $\dim{\fam=\ssfam \usuDelta}\ge b \Rightarrow {\cal P}$ is known, & $\dim{\fam=\ssfam \usuDelta}\ge b' \Rightarrow {\cal P}$ is a translation plane, \\ $\dim{\fam=\ssfam \usuDelta}\ge b''\Rightarrow{\cal P}$ is a Cartesian plane,\hspace{30pt} & $\dim{\fam=\ssfam \usuDelta}\ge b^*\Rightarrow {\cal P}$ is a Hughes plane, \\ $\dim{\fam=\ssfam \usuDelta}\ge c \Rightarrow {\cal P}$ is classical (Moufang), & $\dim{\fam=\ssfam \usuDelta}\le d$, \ $\dim{\fam=\ssfam \usuDelta}\ge g \Rightarrow {\fam=\ssfam \usuDelta}$ known. \end{tabular} \par
\begin{center}
\begin{tabular}{|c||c|c|c|c||l|} \hline ${\cal F}_{\fam=\ssfam \usuDelta}$ & ${\fam=\ssfam \usuDelta}$ $s$-$s$ & ${\fam=\Bbbfam T}\triangleleft {\fam=\ssfam \usuDelta}$ & ${\fam=\Bbbfam R}^t\triangleleft{\fam=\ssfam \usuDelta}$ & ${\fam=\ssfam \usuDelta}$ arbitrary & References \\ \hline $\emptyset$ & $b^*{=}25$ & $b^*{=}23$ & $d{\,\le}23$ & $b{\,=}25$\quad$c{\,=}37$ & 3.1,2,3,\;\cite{cp}\,86.35 \\ $\{W\}$ & $d{\,=\,}21\hskip5pt$ & $4.4\hskip3pt^{1)}$ & $b'{\,=}35$ & $b{\,=}35$\quad$c{\,=}36$ & 4.3,6,8,11,12 \\ flag & $d{\,=}21\,^{2)}$ & $5.3\hskip3pt^{1)}$ & ? & $b{\,=}40$\quad$c{\,=}41$ & 5.2,\,\cite{cp}\,87.7 \\ \hline $\{o,W\}$ & $g{\,=}29 \ $ & $d{\,=}30$ & $d{\,=\,}32\,^{7)}$ &
$c{\,=}38$ & 4.2,\,5.\,6/12 \\ $\langle u,v\rangle$ & $d{\,=}21\,^{2)}$ & $d{\,=}20 \ $ & $b{\,=}35\hskip3pt^{3)} $ & $b{\,=}35$\quad $b'{=}39$ & 6.2,3,\,5/7 \\ $\langle u,v,w\rangle$ & $d{\,=}18 \ $ & $d{\,=}16 \ $ &$g{\,=}32\quad c{\,=}33$ & $b'{=}32$\quad$c{\,=}33 $ & 7.2,3,4 \\ \hline $\langle u,v,ov\rangle$ & $d{\,=}24\,^{4)}$ & $d{\,=}30 \ $ & $b''{=}33 \ $ & $b{\,=}33$\quad $c{\,=}38$ & 8.2,3,\hskip1pt4/6 \\ $\langle o,u,v\rangle$ & $g{\,=}28 \ $ & $d{\,=}23 \ $ & ? & $d{\,=}30$\quad$g{\,=}30$ & 9.\,1,\hskip1pt2,\hskip1pt3 \\ arbitrary & $b^*{=}29\;^{5)}$ & $b^*{=}31$ & $b{\,=}35\,^{6)}$ & $b{\,=}40$\quad$c{\,=}41$ & \cite{sz5},\,\cite{cp}\,87.7 \\ \hline \end{tabular} \end{center} \par \begin{tabular}{l l l l}
\quad $^{1)}$ & $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}18\Rightarrow\exists_{{\cal B}{\,\cong\,}{\cal H}}\,{\cal B}^{\fam=\ssfam \usuDelta}{\,=\,}{\cal B}{\,\!<\!\!\!\raise1.3pt\hbox{$\scriptscriptstyle\bullet$}}{\cal P}$ & \quad $^{2)}$ & $\dim{\fam=\ssfam \usuDelta}{\,\le\,}16$ {\it if ${\fam=\ssfam \usuDelta}$ is almost simple\/} \cr \quad $^{3)}$ & $\dim{\fam=\ssfam \usuDelta}{\,\in\,}\{33,34\}\Rightarrow{\cal P}\ is\ transl.\ plane$ & \quad $^{4)}$ & {\it if ${\fam=\ssfam \usuDelta}$ is a Lie group\/} \cr \quad $^{5)}$ & or $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}36$ & \quad $^{6)}$ & {\it or ${\cal F}_{\fam=\ssfam \usuDelta}$ is a flag\/} \cr \quad $^{7)}$ & {\it if $t{\,\ne\,}1$\/} & \cr \end{tabular} \par
{\bf Lie groups.} {\it If ${\fam=\ssfam \usuTheta}$ denotes a compact conected $1$-dimensional subgroup and if $\dim{\fam=\ssfam \usuDelta}{\,\ge\,}k$, then ${\fam=\ssfam \usuDelta}$ is a Lie group\/}. \begin{center}
\begin{tabular}{|c||c|c|c|c||l|} \hline ${\cal F}_{\fam=\ssfam \usuDelta}$ & ${\fam=\ssfam \usuDelta}$ $s$-$s$ & ${\fam=\ssfam \usuTheta}\triangleleft {\fam=\ssfam \usuDelta}$ & ${\fam=\Bbbfam R}^t\triangleleft{\fam=\ssfam \usuDelta}$ & ${\fam=\ssfam \usuDelta}$ arbitrary & References \\ \hline $\emptyset$ & $21$ & && 23 & 3.1, 3.0 \\ $\{W\}$ & $14$ & && 23 & 4.3, 4.0 \\ flag & $15$ & && 22 & 5.1, 5.0 \\ \hline $\{o,W\}$ &26& && 27 & \cite{psz}\hskip1pt({\bold a}), \cite{psz} \\ $\langle u,v\rangle$ & $14$ & && 18 & 6.1, 6.0 \\ $\langle u,v,w\rangle$ & $14$ & && 15 & 7.1, 7.0 \\ \hline $\langle u,v,ov\rangle$ & $26$ & $20\,^{1)}$ && 27 & 8.0, \cite{psz} \\ $\langle o,u,v\rangle$ & 22 & $18\,^{\ \ }$ & 22 & 22 & 9.\hskip1pt2 \\ arbitrary & 26 & && 27& \cite{psz} \\ \hline \end{tabular} \end{center} \par \begin{tabular}{l l l l}
\quad $^{1)}$ & or $\Cs{}{\fam=\ssfam \usuDelta}|_{av}{\,=\,}1\kern-2.5pt {\rm l}$\; (8.3 Remark) \cr \end{tabular} \par
\qquad Helmut R. Salzmann \par \qquad Mathematisches Institut \par \qquad Auf der Morgenstelle 10 \par \qquad D-72076 T\"ubingen \par \qquad [email protected]
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\begin{frontmatter}
\title{Solving the triharmonic equation over multi-patch domains \\ using isogeometric analysis}
\author[lnz]{Mario Kapl\corref{cor}} \ead{[email protected]}
\author[slo1,slo2]{Vito Vitrih} \ead{[email protected]}
\address[lnz]{Johann Radon Institute for Computational and Applied Mathematics, \\Austrian Academy of Sciences, Linz, Austria}
\address[slo1]{IAM and FAMNIT, University of Primorska, Koper, Slovenia}
\address[slo2]{Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia}
\cortext[cor]{Corresponding author}
\begin{abstract} We present a framework for solving the triharmonic equation over bilinearly parameterized planar multi-patch domains by means of isogeometric analysis. Our approach is based on the construction of a globally $C^2$-smooth isogeometric spline space which is used as discretization space. The generated $C^2$-smooth space consists of three different types of isogeometric functions called patch, edge and vertex functions. All functions are entirely local with a small support, and {\color{black} numerical examples indicate that they are well-conditioned}. The construction of the functions is simple and works uniformly for all multi-patch configurations. While the patch and edge functions are given by a closed form representation, the vertex functions are {\color{black} obtained by computing the null space of a small system of linear equations}. Several examples demonstrate the potential of our approach for solving the triharmonic equation. \end{abstract}
\begin{keyword}
isogeometric analysis, triharmonic equation, geometric continuity, $C^2$-continuity,
multi-patch domain
\MSC 65D17 \sep 65N30 \sep 68U07 \end{keyword}
\end{frontmatter}
\section{Introduction}
In isogeometric analysis (IGA), which was introduced by Hughes et al. \cite{HuCoBa04}, standard CAD functions for describing the geometry, such as polynomial splines or NURBS, are also used {\color{black} for the numerical simulation} of partial differential equations (PDEs), cf. \cite{ANU:9260759,CottrellBook,HuCoBa04}. IGA provides the possibility to solve high order PDEs by using standard Galerkin discretization, see e.g.~\cite{BaDe15, TaDe14}, but which requires isogeometric spline spaces of high smoothness. In case of $4$-th order PDEs, such as the biharmonic equation \cite{BaDe15, CoSaTa16, KaBuBeJu16, KaViJu15, TaDe14}, the Kirchhoff-Love shell problem \cite{ benson2011large, kiendl-bazilevs-hsu-wuechner-bletzinger-10,kiendl-bletzinger-linhard-09, KiHsWuRe15, NgZhZh17}, or the Cahn-Hilliard equation \cite{gomez2008isogeometric, LiDeEvBoHu13}, $C^{1}$-smooth isogeometric functions are needed. {\color{black} Furthermore, $C^1$-smooth isogeometric functions are also needed for plane problems of first strain gradient elasticity \cite{gradientElast2011, KhakaloNiiranenC1} and for a locking-free reformulation of Reissner-Mindlin plates \cite{ReissnerMindlin2015}.} In order to solve $6$-th order PDEs, such as the triharmonic equation \cite{BaDe15, KaVi17b, KaVi17a, TaDe14}, the phase-field crystal equation \cite{BaDe15, Gomez2012}, the Kirchhoff plate model based on the Mindlin's gradient elasticity theory {\color{black} \cite{KhakaloNiiranenC2, Niiranen2016}}, or the gradient-enhanced continuum damage model \cite{GradientDamageModels}, even $C^{2}$-smooth functions are required. In particular for the case of $6$-th order PDEs, these problems have been mainly considered so far for single-patch domains or simple closed surfaces, where the required smoothness of an isogeometric functions is directly obtained by the smoothness of the underlying spline space. In case of multi-patch domains, the construction of $C^{s}$-smooth ($s\geq 1$) isogeometric spline spaces defined on multi-patch domains is linked to the concept of geometric continuity of multi-patch surfaces (cf. \cite{HoLa93, Pe02}). More precisely, an isogeometric function is $C^{s}$-smooth on a multi-patch domain if and only if its graph surface over the multi-patch domain is $G^{s}$-smooth (cf. \cite{Pe15, KaViJu15}). The design of $C^s$-smooth isogeometric spline spaces {\color{black} over multi-patch domains} is the task of recent research, see e.g. \cite{BeMa14, BlMoVi17, CoSaTa16, KaBuBeJu16, KaSaTa17a, KaSaTa17c, KaViJu15, Pe15-2, mourrain2015geometrically, NgPe16, ToSpHu17b,ToSpHu17} for $s=1$ and e.g. \cite{KaVi17a, KaVi17b, KaVi17c, ToSpHiHu16} for $s=2$.
This work focuses on solving the triharmonic equation over bilinearly parameterized planar multi-patch domains by using IGA. To our knowledge this problem was handled for the first time in \cite{KaVi17b, KaVi17a}. There, a basis of the entire space of $C^2$-smooth isogeometric functions is generated. The construction is based on the concept of minimal determining sets (cf.~\cite{LaSch07}) for the involved spline coefficients and requires the symbolic {\color{black} computation of the null space} of a large (global) system of linear equations. Further disadvantages of this approach are the following: The resulting functions which are defined across the common interfaces possess in general large supports along one or more interfaces. The method is restricted to isogeometric spline functions of bidegree~$(p,p)$ with $p=5,6$ and regularity $r=2$ within the single patches. Moreover, the presented examples of solving the triharmonic equation were restricted to one particular level of $h$-refinement.
{\color{black} Two further constructions of $C^2$-smooth spline functions over multi-patch domains are \cite{KaVi17c, ToSpHiHu16}, but both methods have not been applied so far to solve $6$th order {\color{black} PDEs}. In~\cite{KaVi17c}, $C^2$-smooth spline spaces over the class of so-called bilinear-like two-patch parameterizations, which contains the subclass of bilinear two-patch geometries, were considered. There, the dimension of this space was analyzed and an explicit basis construction was presented, which will serve as a basis for our construction in the multi-patch case. In~\cite{ToSpHiHu16}, a polar spline framework is developed to construct $C^2$-smooth isogeometric spaces which is based on a special construction in the vicinity of the polar point to ensure $C^2$-smoothness also there.
Beside multi-patch {\color{black} quadrangular} domains, triangulations have been used to generate $C^2$-smooth (or even smoother) spline spaces over complex domains. The book~\cite{LaSch07} gives an overview of different techniques to model such smooth spline spaces, and provides a detailed bibliography on this topic. There, also the concept of minimal determining sets is recalled, which is a common strategy to generate a basis of a smooth spline space over a given triangulation. The minimal determining set implicitly describes a basis of the null space of the homogeneous linear system obtained by the corresponding smoothness conditions. We will use this concept for the construction of those basis functions which will be defined in the neighborhood of a vertex of the multi-patch domain.} {\color{black} Some more recent constructions of $C^2$-smooth spline spaces on triangulations are e.g. \cite{DavydovYeo2013, Groselj2016, LycheMuntingh2014, Speleers2012, Speleers2013}.}
The present paper improves and extends the approach~\cite{KaVi17b, KaVi17a} in several directions. Instead of constructing the entire space of $C^2$-smooth isogeometric functions, which has a complex structure, a simpler subspace~$\mathcal{W}_{0h}$ is generated. The subspace~$\mathcal{W}_{0h}$ maintains the full approximation properties of the entire space and is defined as the direct sum of spaces corresponding to the single patches, edges and vertices. For each of these spaces the construction of the basis functions is simple and leads to basis functions which possess small supports {\color{black} and} can be described by explicit formulae {\color{black} or by computing the null space of a small system of linear equations.} {\color{black} Furthermore, the numerical examples indicate that the generated basis functions are well-conditioned}. The basis construction of the single spaces is based on and extends the explicit {\color{black} construction in~\cite{KaVi17c}, and can be applied} for any degree $p \geq 5$ and any regularity $2\leq r \leq p-3$ {\color{black} at the inner knots} within the single patches. {\color{black} Moreover,} the construction of the space~$\mathcal{W}_{0h}$ works uniformly for all possible multi-patch configurations. In contrast to \cite{KaVi17b, KaVi17a}, the triharmonic equation is solved on several bilinearly parameterized multi-patch domains for different levels of $h$-refinement, where the numerical results show the potential of our approach.
The remainder of the paper is organized as follows. Section~\ref{sec:modelProblem} introduces the model problem which is studied in this work, i.e., solving the triharmonic equation over bilinear multi-patch domains by means of IGA. This requires the use of a discretization space consisting of globally $C^2$-smooth isogeometric functions. Section~\ref{sec:C2smoothspaces} recalls the concept of $C^2$-smooth isogeometric spline spaces and summarizes the explicit construction~\cite{KaVi17c} for the case of two patches which serves as a basis for the multi-patch case. In Section~\ref{sec:C2smoothtestfunctions}, we describe the construction of the discretization space for solving the triharmonic equation. This space is a subspace of the entire space of globally $C^2$-smooth isogeometric spline spaces and is defined as the direct sum of subspaces of three different types called patch, edge and vertex subspaces. The potential of our method for solving the triharmonic equation is demonstrated on the basis of several examples in Section~\ref{sec:triharmonic_examples}, where amongst others the convergence rates and condition numbers obtained under $h$-refinement are numerically studied. Finally, we conclude the paper.
\section{The model problem} \label{sec:modelProblem}
We introduce the model problem which will be considered throughout the paper. The goal is to solve a particular sixth-order partial differential equation, namely the triharmonic equation with homogeneous boundary conditions of order 2.
\subsection{The triharmonic equation}
Let $\Omega = \cup_{\ell=1}^P \Omega^{(\ell)}$ be a planar multi-patch domain. We have to find the function $u : \Omega \rightarrow \mathbb R$ which solves
for $f \in H^{0}(\Omega)$ the equation \begin{equation} \label{eq:triharmonic_problem}
\triangle^{3} u(\ab{x}) = - f(\ab{x}), \quad \ab{x} \in \Omega,
\end{equation}
with the boundary conditions \begin{equation} \label{eq:triharmonic_problem_boundary}
u(\ab{x}) = \frac{\partial u}{\partial \ab{n}}(\ab{x}) = \triangle u(\ab{x}) = 0 , \quad \ab{x} \in \partial \Omega.
\end{equation} Using the weak formulation of \eqref{eq:triharmonic_problem} and \eqref{eq:triharmonic_problem_boundary} we have to find $u \in \mathcal{V}_{0}$, with \[
\mathcal{V}_{0} = \{ v \in H^2(\Omega) : \triangle v \in H^1(\Omega) \mbox{ and } v(\ab{x}) = \frac{\partial v}{\partial \ab{n}}(\ab{x}) = \triangle v(\ab{x}) = 0 \mbox{ for }\ab{x} \in \partial \Omega \}, \] such that
\begin{equation} \label{eq:weak_triharmonic}
\int_{\Omega} \nabla\left(\triangle u(\ab{x})\right) \cdot \nabla \left( \triangle v(\ab{x}) \right) \mathrm{d}\ab{x} = \int_{\Omega} f(\ab{x}) v(\ab{x}) \mathrm{d}\ab{x},
\end{equation}
where $\cdot$ denotes the standard inner product, is satisfied for all $v \in \mathcal{V}_{0}$, cf. \cite{BaDe15, TaDe14}.
In order to discretize problem~\eqref{eq:weak_triharmonic} by applying Galerkin projection, a finite dimensional function space $\mathcal{W}_{0h} \subseteq \mathcal{V}_{0}$ is
required. Assume that we have such a space $\mathcal{W}_{0h}$ with a basis $\{w_{i}\}_{i \in I_h}$, where $I_h= \{1, 2, \ldots, \dim \mathcal{W}_{0h} \}$. Then, we have to find \begin{equation*}
u_{h}(\ab{x})=\sum_{i \in I_h}
c_{i} w_{i}(\ab{x}), \quad c_{i} \in \mathbb R, \end{equation*} which solves the system of equations \begin{equation*}
\int_{\Omega} \nabla\left(\triangle u_{h}(\ab{x}) \right) \cdot \nabla \left( \triangle v_{h}(\ab{x})\right) \mathrm{d}\ab{x} = \int_{\Omega} f(\ab{x}) v_{h}(\ab{x})
\mathrm{d}\ab{x}
\end{equation*} for all $v_{h} \in \mathcal{W}_{0h}$. This results in a system of linear equations \[ S\ab{c}=\ab{f} \]for the unknown coefficients $\ab{c} = (c_{i})_{i \in I_h}$, where the elements of the matrix $S = (s_{i,j} )_{i,j \in I_h}$ and the elements of the right-hand side vector $\ab{f} = (f_i)_{i \in I_h}$ are given by \begin{equation} \label{eq:sij}
s_{i,j}=\int_{\Omega} \nabla\left(\triangle w_{i}(\ab{x}) \right) \cdot \nabla \left(\triangle w_{j}(\ab{x}) \right) \mathrm{d}\ab{x}
\quad {\rm and} \quad f_{i}= \int_{\Omega} f(\ab{x}) w_{i}(\ab{x})
\;\mathrm{d}\ab{x}.
\end{equation}
In this work, we will follow the isogeometric approach to solve the triharmonic equation. For this purpose, we will construct an isogeometric space $\mathcal{W}_{0h} \subseteq \mathcal{V}_{0}$ and an associated basis $\{w_{i}\}_{i \in I_h}$, see Section~\ref{sec:C2smoothtestfunctions}. Beside the fulfillment of the homogeneous boundary conditions~\eqref{eq:triharmonic_problem_boundary}, the generated basis functions~$w_i$ will be $C^2$-smooth, since $C^1$-smoothness is not enough to ensure that $w_{i} \in \mathcal{V}_{0}$.
\subsection{Using the isogeometric approach}
We describe the isogeometric approach to compute the elements in \eqref{eq:sij}. We assume that the planar multi-patch domain $\Omega$ consists of \begin{itemize}
\item $P$ patches~$\Omega^{(\ell)}$, $\ell=1,2,\ldots, P$, with $P \in \mathbb N$ and $P \geq 2$,
\item $E$ non-boundary common edges $\Gamma^{(s)}$, $s=1,2,\ldots,E$, and
\item $V$ inner and boundary vertices $\bfm{v}^{(\rho)}$ of valency $\bar{\nu}_{\rho} \geq 3$, $\rho=1,2,\ldots,V$.
\footnote{In this work, a boundary vertex of valency two is not considered as a vertex $\ab{v}^{(\rho)}$.}{} \end{itemize} In addition, we assume that \begin{itemize} \item the deletion of any vertex does not split $\Omega$ into subdomains, whose union would be unconnected, \item {\color{black} all subdomains $\Omega^{(\ell)}$ are strictly convex quadrangular patches, whose interiors are mutually disjoint,} \item any two patches~$\Omega^{(\ell)}$ and $\Omega^{(\ell')}$ have either an empty intersection, possess exactly one common vertex or share the whole common edge, and \item each patch $\Omega^{(\ell)}$ is parameterized by a bilinear, bijective and regular geometry mapping~$\ab{F}^{(\ell)}$, \begin{align*}
\ab{F}^{(\ell)}: [0,1]^{2} \rightarrow \mathbb R^{2}, \quad
\bb{\xi}^{(\ell)} =(\xi^{(\ell)}_1,\xi^{(\ell)}_2) \mapsto
(F^{(\ell)}_1,F^{(\ell)}_2)=
\ab{F}^{(\ell)}(\bb{\xi}^{(\ell)}), \quad \ell \in \{1, 2,\ldots,P\}, \end{align*} such that $\Omega^{(\ell)} = \ab{F}^{(\ell)}([0,1]^{2})$, see Fig.~\ref{fig:geommetryToOmega}. \end{itemize}
\begin{figure}
\caption{The multi-patch domain $\Omega= \cup_{\ell=1}^{P} \Omega^{(\ell)}$ with the corresponding geometry mappings $\bfm{F}^{(\ell)}$, $\ell=1,2,\ldots,P$. }
\label{fig:geommetryToOmega}
\end{figure}
Let $J^{(\ell)}$ be the Jacobian of $\ab{F}^{(\ell)}$ and let \begin{equation*}
{\color{black} K^{(\ell)}(\bb{\xi}^{(\ell)})} = \left(J^{(\ell)}(\bb{\xi}^{(\ell)})\right)^{-T} \left(J^{(\ell)}(\bb{\xi}^{(\ell)})\right)^{-1} |\det J^{(\ell)}(\bb{\xi}^{(\ell)})|. \end{equation*} {Furthermore, let $W^{(\ell)}_i = w_i $} $\circ \ab{F}^{(\ell)}$, $i \in I_{h}$. Then, we compute the elements in \eqref{eq:sij} patch-wise by \begin{equation*}
s_{i,j} = \sum_{\ell=1}^P s^{(\ell)}_{i,j} \; \mbox{ and } \quad f_{i} = \sum_{\ell=1}^P f^{(\ell)}_{i}, \end{equation*} where \begin{align*}
s^{(\ell)}_{i,j} = & \int_{[0,1]^{2}} \nabla \left( \frac{1}{|\det J^{(\ell)}(\bb{\xi}^{(\ell)})|} \,
\nabla \cdot \left( {\color{black} K^{(\ell)}(\bb{\xi}^{(\ell)})} \, \nabla W^{(\ell)}_{i}(\bb{\xi}^{(\ell)})\right) \right)
\cdot \nonumber \\
& \left( {\color{black} K^{(\ell)}(\bb{\xi}^{(\ell)})} \, \nabla \left( \frac{1}{|\det J^{(\ell)}(\bb{\xi}^{(\ell)})|} \,
\nabla \cdot \left( {\color{black} K^{(\ell)}(\bb{\xi}^{(\ell)})} \,\nabla W^{(\ell)}_{j}(\bb{\xi}^{(\ell)})\right) \right) \right)
\, \mathrm{d}\bb{\xi}^{(\ell)}, \end{align*} and \begin{equation*}
f^{(\ell)}_{i} = \int_{[0,1]^{2}} f(\ab{F}^{(\ell)}(\bb{\xi}^{(\ell)})) W^{(\ell)}_{i}(\bb{\xi}^{(\ell)}) |\det J^{(\ell)}(\bb{\xi}^{(\ell)})| \; \mathrm{d}\bb{\xi}^{(\ell)}, \end{equation*} cf. \cite{BaDe15, KaVi17a}.
\section{$C^2$-smooth isogeometric spline spaces} \label{sec:C2smoothspaces}
In Section~\ref{sec:C2smoothtestfunctions}, the isogeometric discretization space~$\mathcal{W}_{0h}$ will be generated as a subspace of the space of $C^2$-smooth isogeometric spline functions on $\Omega$. Before, we recall the concept of $C^2$-smooth isogeometric spline spaces, cf.~\cite{KaVi17b, KaVi17c}, and adapt the notations appropriately.
\subsection{The space of $C^2$-smooth isogeometric spline functions}
In order to define the space of $C^2$-smooth isogeometric spline functions on $\Omega$, we need some additional definitions and notations. {\color{black} Let $p \geq 5$, $k \in \mathbb N_{0}$ and for $k\geq 1$ let $2 \leq r \leq p-3$. Moreover let $h=\frac{1}{k+1}$.}
We denote by $\mathcal{S}_{h}^{p,r}([0,1])$ the univariate spline space on the interval~$[0,1]$ of degree~$p$ and regularity $C^r$ possessing the open knot vector \begin{equation*} (\underbrace{0,0,\ldots,0}_{(p+1)-\mbox{\scriptsize times}}, \underbrace{\textstyle \tau_1,\tau_1,\ldots ,\tau_1}_{(p-r) - \mbox{\scriptsize times}}, \underbrace{\textstyle \tau_2,\tau_2,\ldots ,\tau_2}_{(p-r) - \mbox{\scriptsize times}},\ldots, \underbrace{\textstyle \tau_k,\tau_k,\ldots ,\tau_k}_{(p-r) - \mbox{\scriptsize times}}, \underbrace{1,1,\ldots,1}_{(p+1)-\mbox{\scriptsize times}}), \end{equation*} where the $k$ different inner knots $\tau_j$, $j \in \{1,2,\ldots, k\}$, are equally distributed, i.e., $\tau_j = \frac{j}{k+1}=jh$. Let $N_{i}^{p,r}$, $i=0,1,\ldots,p+k(p-r)$, be the B-splines of the spline space $\mathcal{S}^{p,r}_{h}([0,1])$, and let $\mathcal{S}^{p,r}_{h}([0,1]^2)$ be the bivariate tensor-product spline space on the unit-square~$[0,1]^2$ spanned by the B-splines $N^{p,r}_{i,j} = N^{p,r}_{i}N^{p,r}_{j}$, $i,j=0,1,\ldots,p + k (p-r)$. Note that $h$ is the mesh-size of the spline spaces~$\mathcal{S}^{p,r}_{h}([0,1])$ and $\mathcal{S}^{p,r}_{h}([0,1]^2)$. {\color{black} In addition, in case of $k=0$ (i.e. $h=1$), the spaces $\mathcal{S}^{p,r}_{1}([0,1])$ and $\mathcal{S}^{p,r}_{1}([0,1]^2)$ are for any $r$ just the corresponding spaces of polynomials of degree~$p$ and bidegree~$(p,p)$, respectively.} Below, we assume that the number of inner knots satisfies $k \geq \frac{9-p}{p-r-2}$, which implies $h \leq \frac{p-r-2}{7-r} $.
Recall that the geometry mappings $\ab{F}^{(\ell)}$, $\ell = 1,2,\ldots, P $, are bilinear parameterizations, which also implies that $\ab{F}^{(\ell)} \in \mathcal{S}_{h}^{p,r}([0,1]^2) \times \mathcal{S}^{p,r}_{h}([0,1]^2)$. Then, the space of globally $C^{2}$-smooth isogeometric spline functions on $\Omega$ (with respect to the spline space~$\mathcal{S}_{h}^{p,r}([0,1]^2)$) is defined as \begin{equation*}
\mathcal{V}_h = \left\{ \phi \in C^{2}(\Omega) : \; \phi |_{\Omega^{(\ell)}} \in {\mathcal{S}_{ h}^{p,r}([0,1]^{2})} \circ (\ab{F}^{(\ell)})^{-1}, \; \ell \in \{1,2, \ldots, P\} \right\}. \end{equation*} The graph surface $\ab{\Sigma}:[0,1]^2 \to \Omega \times \mathbb R$ of an isogeometric function $\phi \in \mathcal{V}_h$ is given patch-wise by the graph surface patches $$ \ab{\Sigma}^{(\ell)}(\bfm{\xi}^{(\ell)}) = \left(\ab{F}^{(\ell)}(\bfm{\xi}^{(\ell)}), g^{(\ell)}(\bfm{\xi}^{(\ell)}) \right)^T, \quad g^{(\ell)} \in {\mathcal{S}_{ h}^{p,r}([0,1]^2)},
\quad \ell=1,2,\ldots,P, $$ with \begin{equation} \label{eq:g_ell} g^{(\ell)}(\bfm{\xi}^{(\ell)})= \phi \circ \ab{F}^{(\ell)}(\bfm{\xi}^{(\ell)}) = \sum_{i=0}^{p+k(p-r)} \sum_{j=0}^{p+k(p-r)} d^{(\ell)}_{i,j} N_{i,j}^{p,r} (\bfm{\xi}^{(\ell)}), \quad d^{(\ell)}_{i,j} \in \mathbb R. \end{equation}
\begin{figure}
\caption{Considering two neighboring patches~$\Omega^{(\ell)}$ and $\Omega^{(\ell)}$, we can always assume (without loss of generality) that the two corresponding geometry mappings $\ab{F}^{(\ell)}$ and $\ab{F}^{(\ell')}$ are parameterized as shown.}
\label{fig:situation_two_patches}
\end{figure}
The functions in $\mathcal{V}_h$ can be characterized by using the concept of geometric continuity (cf. \cite{Pe15, KaViJu15}): \emph{An isogeometric function $\phi$ belongs to the space $\mathcal{V}_h$ if and only if for all neighboring patches $\Omega^{(\ell)}$ and $\Omega^{(\ell')}$ sharing an interface~$\Gamma^{(s)} = \Omega^{(\ell)} \cap \Omega^{(\ell')}$ (where $s \in \{1,2,\ldots,E\}$), the two graph surface patches $\ab{\Sigma}^{(\ell)}$ and $\ab{\Sigma}^{(\ell')}$ meet at the common interface $\Gamma^{(s)}$ with $G^2$ continuity.} Since the geometry mappings $\ab{F}^{(\ell)}$ and $\ab{F}^{(\ell')}$ are given in advance, the $G^2$ continuity conditions for the graph surface patches~$\ab{\Sigma}^{(\ell)}$ and $\ab{\Sigma}^{(\ell')}$ lead to conditions for the spline functions $g^{(\ell)}$ and $g^{(\ell')}$, which determine again linear constraints on the spline coefficients $d_{i,j}^{(\ell)}$ and $d_{i,j}^{(\ell')}$. These conditions were studied in \cite{KaVi17c} for the class of so-called bilinear-like $G^2$ geometries, which includes the class of bilinearly parameterized geometries. Let us shortly recall the conditions for the two neighboring patches~$\Omega^{(\ell)}$ and $\Omega^{(\ell')}$. For the sake of simplicity, we can always reparameterize (if needed) the two geometry mappings $\ab{F}^{(\ell)}$ and $\ab{F}^{(\ell')}$ to have the situation as given in Fig.~\ref{fig:situation_two_patches}, i.e., $$ \color{black} \ab{F}^{(\ell)}(0,{\xi_2})= \ab{F}^{(\ell')}(0,{\xi_2}),\quad \xi_2 =\xi_2^{(\ell)}=\xi_2^{(\ell')}\in[0,1]. $$ To simplify the notation, let us denote the common interface $\Gamma^{(s)}$ in this section by $\Gamma$ and let \begin{equation*} \label{eq:alphaLRbar}
\bar{\alpha}_{\Gamma}^{(\tau)}(\xi) = \det [D_{\xi_1^{(\tau)}}\ab{F}^{(\tau)}(0,\xi), D_{\xi}\ab{F}^{(\tau)}(0,\xi)] , \;
\alpha^{(\tau)}_\Gamma(\xi) = \gamma_1(\xi) \bar{\alpha}^{(\tau)}_\Gamma(\xi),\; \widehat{\alpha}^{(\tau)}_\Gamma(\xi) = \gamma_2(\xi) \bar{\alpha}_\Gamma^{(\tau)}(\xi), \end{equation*} for $\tau\in \{\ell,\ell'\}$ and \[ \bar{\beta}_\Gamma(\xi) = \det[ D_{\xi_1^{(\ell)}}\ab{F}^{(\ell)}(0,\xi) , D_{\xi_1^{(\ell')}}\ab{F}^{(\ell')}(0,\xi) ], \quad
\beta_\Gamma(\xi) = \gamma_1(\xi) \bar{\beta}_\Gamma (\xi) \] for $\gamma_i: [0,1]\to \mathbb R,\; i=1,2.$ Note that $\bar{\alpha}^{(\ell)}_{\Gamma}$ and $\bar{\alpha}^{(\ell')}_{\Gamma}$ are linear polynomials with $\bar{\alpha}^{(\ell)}_{\Gamma} < 0$ and $\bar{\alpha}^{(\ell')}_{\Gamma} > 0$, respectively, and $\bar{\beta}_{\Gamma}$ is a quadratic polynomial. We can write the function $\beta_\Gamma$ also as \begin{equation*} \label{eq:beta}
\beta_\Gamma(\xi) = \alpha_\Gamma^{(\ell)}(\xi) \beta_\Gamma^{(\ell')}(\xi) - \alpha_\Gamma^{(\ell')}(\xi) \beta_\Gamma^{(\ell)}(\xi), \end{equation*} where $\beta_\Gamma^{(\ell)}$, $\beta_\Gamma^{(\ell')}: [0,1] \rightarrow \mathbb R$ are given as \begin{equation*} \label{eq:betaS}
\beta_\Gamma^{(\tau)}(\xi) = \frac{D_{\xi_1^{(\tau)}} \ab{F}^{(\tau)}(0,\xi) \cdot D_{\xi}\ab{F}^{(\tau)}(0,\xi)}{||D_{\xi}\ab{F}^{(\tau)}(0,\xi)||^{2}}, \quad \tau \in \{\ell,\ell'\}. \end{equation*} Moreover let \begin{equation*} \label{eq:eta}
\eta_\Gamma(\xi) = 2 \gamma_2(\xi) ( \alpha_\Gamma^{(\ell)})'(\xi) \alpha_\Gamma^{(\ell')}(\xi) \beta_\Gamma(\xi), \end{equation*} \begin{equation*} \label{eq:theta}
\theta_\Gamma(\xi)= 2 \gamma_2(\xi) \left(\alpha_\Gamma^{(\ell)}(\xi) (\beta_\Gamma^{(\ell)})'(\xi) - (\alpha_\Gamma^{(\ell)})'(\xi) \beta_\Gamma^{(\ell)}(\xi)\right) \alpha_\Gamma^{(\ell')}(\xi) \beta_\Gamma(\xi). \end{equation*} Then, we have: \emph{$\phi \in \mathcal{V}_h$ if and only if \begin{equation} \label{eq: C0}
g^{(\ell)}(0,\xi) = g^{(\ell')}(0,\xi), \end{equation} \begin{equation} \label{eq: C1}
\alpha_\Gamma^{(\ell')}(\xi) D_{\xi_1^{(\ell)}} g^{(\ell)}(0,\xi) - \alpha_\Gamma^{(\ell)}(\xi) D_{\xi_1^{(\ell')}} g^{(\ell')}(0,\xi) + \beta_\Gamma(\xi) D_{\xi} g^{(\ell)}(0,\xi) = 0, \end{equation} and \begin{equation} \label{eq: C2}
\widehat{\alpha}_\Gamma^{(\ell)}(\xi) w_\Gamma(\xi) + \eta_\Gamma(\xi) D_{\xi_1^{(\ell)}} g^{(\ell)}(0,\xi) + \theta_\Gamma(\xi) D_{\xi}g^{(\ell)}(0,\xi) = 0, \end{equation} where \begin{align*}
w_\Gamma(\xi) =& \;( \alpha_\Gamma^{(\ell)}(\xi))^{2} D_{\xi_1^{(\ell')}\xi_1^{(\ell')}} g^{(\ell')}(0,\xi) - ( (\alpha_\Gamma^{(\ell')}(\xi))^{2} D_{\xi_1^{(\ell)} \xi_1^{(\ell)}} g^{(\ell)}(0,\xi) \nonumber\\[-0.3cm]
& \label{eq:w} \\[-0.3cm]
& + 2 \alpha_\Gamma^{(\ell')}(\xi) \beta_\Gamma(\xi) D_{\xi_1^{(\ell)} \xi} g^{(\ell)}(0,\xi) + (\beta_\Gamma(\xi))^{2} D_{\xi \xi} g^{(\ell)}(0,\xi) ) . \nonumber \end{align*} }
Note that condition~\eqref{eq: C0} guarantees that $\phi$ is $C^{0}$-smooth, condition~\eqref{eq: C1} additionally ensures that $\phi$ is $C^{1}$-smooth, and condition~\eqref{eq: C2} finally implies that $\phi$ is $C^{2}$-smooth.
\begin{rem} Below, we choose $\gamma_2(\xi) = 1$, implying $\widehat{\alpha}^{(\tau)} = \bar{\alpha}^{(\tau)} $, $\tau \in \{\ell, \ell'\}$. In addition, we select $\gamma_1(\xi) = c_1 \in \mathbb R$ such that \[
|| \alpha_\Gamma^{(\ell)}+1 ||^2_{L^2} + || \alpha_\Gamma^{(\ell')}-1 ||^2_{L^2} \] is minimized, cf. \cite{KaSaTa17c}. \end{rem}
\subsection{The two-patch case} \label{subsec:two-patch-case}
In this subsection we restrict ourselves to the two-patch case~$\Omega = \Omega^{(\ell)} \cup \Omega^{(\ell')}$ for two neighboring patches~$\Omega^{(\ell)}$ and $\Omega^{(\ell')}$ having the common interface~$\Gamma^{(s)} = \Omega^{(\ell)} \cap \Omega^{(\ell')}$. Without loss of generality, we can assume that the two geometry mappings~$\ab{F}^{(\ell)}$ and $\ab{F}^{(\ell')}$ are parameterized as in Fig.~\ref{fig:situation_two_patches}. We recall now the construction of a $C^2$-smooth isogeometric spline space $\widetilde{\mathcal{W}}_h \subseteq \mathcal{V}_h$, which was described in \cite{KaVi17c}, by using now adapted notations. The subspace $\widetilde{\mathcal{W}}_h$ is advantageous compared to the entire space~$\mathcal{V}_h$, since its basis construction is simpler and works uniformly for all possible configurations. In addition, it was numerically demonstrated in \cite{KaVi17c} that already the subspace~$\widetilde{\mathcal{W}}_h$ possesses optimal approximation properties. For a detailed investigation of the spaces~$\widetilde{\mathcal{W}}_h$ and $\mathcal{V}_h$ we refer to~\cite{KaVi17c}.
The space~$\widetilde{\mathcal{W}_{h}}$ is the direct sum of three subspaces, i.e., $$ \widetilde{\mathcal{W}}_h = \mathcal{\widetilde{W}}_{h;\Omega^{(\ell)}} \oplus \mathcal{\widetilde{W}}_{h;\Omega^{(\ell')}} \oplus \mathcal{\widetilde{W}}_{h; \Gamma^{(s)}}. $$ The subspaces $\mathcal{\widetilde{W}}_{h;\Omega^{(\ell)}}$ and $\mathcal{\widetilde{W}}_{h;\Omega^{(\ell')}}$ are given by \begin{align*}
\mathcal{\widetilde{W}}_{h;\Omega^{(\tau)}} &= \Span \{ \widetilde{\phi}_{\Omega^{(\tau)};i,j} |\; i=3,4,\ldots,p+k(p-r),\; j=0,1,\ldots, p+k(p-r)\},\quad \tau \in \{\ell,\ell'\}, \end{align*}
with the functions \begin{equation} \label{eq:PhiOmega} \widetilde{\phi}_{\Omega^{(\tau)};i,j}(\bfm{x}) = \begin{cases}
(N_{i,j}^{p,r}\circ (\ab{F}^{(\tau)})^{-1})(\bfm{x}) \; \mbox{ if }\f \, \bfm{x} \in\Omega^{(\tau)}, \\ 0 \quad \mbox{ if }\f \, \bfm{x} \in \Omega \backslash \Omega^{(\tau)}. \end{cases} \end{equation}
In order to define the subspace~$\mathcal{\widetilde{W}}_{h;\Gamma^{(s)}}$, we need some additional definitions. Let \begin{equation} \label{eq:Mj}
M_0(\xi) = \sum_{i=0}^2 N_i^{p,r}(\xi) , \;
M_1(\xi) = \frac{ h}{p} \left( N_1^{p,r}(\xi) + 2 N_2^{p,r}(\xi) \right), \;
M_2(\xi) = \frac{h^2}{p(p-1)} N_2^{p,r}(\xi), \end{equation} and let \begin{equation*} \label{eq:ni}
n_0 = \dim\left( \mathcal{S}_{h}^{p,r+2}([0,1])\right),\;
n_1 = \dim \left( \mathcal{S}_{ h}^{p-d_\alpha,r+1}([0,1])\right),\;
n_2 = \dim \left( \mathcal{S}_{h}^{p-2d_\alpha,r}([0,1])\right), \end{equation*} where $d_\alpha = \max\left( \deg(\alpha_{\Gamma^{(s)}}^{(\ell)}), \deg(\alpha_{\Gamma^{(s)}}^{(\ell')})\right) \in \{0,1\}$. The space $\widetilde{\mathcal{W}}_{h;\Gamma^{(s)}}$ is given by \begin{equation*} \label{eq:edgeSubspaceTwoPatch}
\widetilde{\mathcal{W}}_{h;\Gamma^{(s)}} = \Span \{ \widetilde{\phi}_{\Gamma^{(s)};i,j} | \; i=0,1,2, \; j=0,1,\ldots,n_i-1\}, \end{equation*} with the functions \begin{equation} \widetilde{\phi}_{\Gamma^{(s)};i,j}(\bfm{x}) = \begin{cases}
(g_{\Gamma^{(s)}; i,j}^{(\ell)} \circ (\ab{F}^{(\ell)})^{-1})(\bfm{x}) \; \mbox{ if }\f \, \bfm{x} \in\Omega^{(\ell)}, \\[0.15cm]
(g_{\Gamma^{(s)}; i,j}^{(\ell')} \circ (\ab{F}^{(\ell')})^{-1} )(\bfm{x}) \; \mbox{ if }\f \, \bfm{x} \in\Omega^{(\ell')}, \end{cases}
\label{eq:basisFunctionsGenericEdge} \end{equation} where \begin{align} \label{eq:basisFunctionsGenericG} \scalebox{0.93}{$ g_{\Gamma^{(s)}; 0,j}^{(\tau)} (\bfm{\xi}^{(\tau)})$} & \scalebox{0.93}{$= N_j^{p,r+2}(\xi_2^{(\tau)}) M_0(\xi_1^{(\tau)}) + \beta_{\Gamma^{(s)}}^{(\tau)}(\xi_2^{(\tau)}) (N_j^{p,r+2})'(\xi_2^{(\tau)}) M_1(\xi_1^{(\tau)}) $} \nonumber \\ & \; + \left( \beta^{(\tau)}_{\Gamma^{(s)}}(\xi_2^{(\tau)})\right)^2 (N_j^{p,r+2})''(\xi_2^{(\tau)}) M_2(\xi_1^{(\tau)}),\nonumber \\ \scalebox{0.93}{$ g_{\Gamma^{(s)}; 1,j}^{(\tau)} (\bfm{\xi}^{(\tau)})$} & \scalebox{0.93}{$ = \displaystyle \frac{p}{ h} \left( {\alpha}^{(\tau)}_{\Gamma^{(s)}}(\xi_2^{(\tau)}) {N}_j^{p-d_{{\alpha}},r+1}(\xi_2^{(\tau)}) M_1(\xi_1^{(\tau)}) \right. $} \\ & \left. \; + 2\, {\alpha}^{(\tau)}_{\Gamma^{(s)}} (\xi_2^{(\tau)})\beta^{(\tau)}_{\Gamma^{(s)}}(\xi_2^{(\tau)}) ({N}_j^{p-d_{{\alpha}},r+1})'(\xi_2^{(\tau)}) M_2(\xi_1^{(\tau)})\right), \nonumber \\ \scalebox{0.93}{$
g_{\Gamma^{(s)}; 2,j}^{(\tau)} (\bfm{\xi}^{(\tau)})$} & \scalebox{0.93}{$= \displaystyle \frac{p(p-1)}{h^2} \left( {\alpha}^{(\tau)}_{\Gamma^{(s)}}(\xi_2^{(\tau)})\right)^2 {N}_j^{p-2d_{{\alpha}},r}(\xi_2^{(\tau)}) M_2(\xi_1^{(\tau)}) $} \nonumber, \end{align} for $\tau \in \{\ell,\ell'\} $. \begin{rem}
The functions in \eqref{eq:basisFunctionsGenericG} are scaled in comparison to the ones in \cite{KaVi17c}. \end{rem} The following proposition gives an estimate for the support of the function~$g_{\Gamma^{(s)}; i,j}^{(\tau)}$, and will be needed later. \begin{prop} \label{thm:mainC4} Let $d= \dim\left( \mathcal{S}_{h}^{p,r}([0,1])\right) = p + k(p-r) +1$. The functions $ g_{\Gamma^{(s)}; i,j}^{(\tau)}$, $j=0,1,\ldots,n_i-1,\, i =0,1,2,\, \tau \in \{\ell,\ell'\},$
can be represented as \begin{equation} \label{eq:defgljS}
g_{\Gamma^{(s)}; i,j}^{(\tau)} (\bfm{\xi}^{(\tau)}) = \sum_{m=0}^2 \;
\sum_{n=\max (0,i+j-m)}^{\min(d-1,d-n_i+j-i+m) } d^{(\tau)}_{\Gamma^{(s)}; m,n} N_{m,n}^{p,r} (\bfm{\xi}^{(\tau)}), \qquad d^{(\tau)}_{\Gamma^{(s)}; m,n}\in \mathbb R. \end{equation} \end{prop} \begin{proof}
See \ref{app:proofProposition}. \end{proof} In the next section we will use the $C^2$-smooth isogeometric functions for the two-patch case to construct a $C^2$-smooth isogeometric spline space~$\mathcal{W}_{0h}$ for the multi-patch case.
\section{$C^2$-smooth discretization space $\mathcal{W}_{0h}$} \label{sec:C2smoothtestfunctions}
A $C^2$-smooth discretization space $\mathcal{W}_{0h}$ will be constructed which can be used for solving the triharmonic equation \eqref{eq:triharmonic_problem} with homogeneous boundary conditions \eqref{eq:triharmonic_problem_boundary}, see Section~\ref{sec:triharmonic_examples}. This space will be a subspace of $\mathcal{V}_h$ or more precisely of the space~$\mathcal{V}_{0h}$ given by \begin{equation*} \mathcal{V}_{0h} = \{ \phi \in \mathcal{V}_h : \; \phi(\ab{x}) = \frac{\partial \phi}{\partial \ab{n}}(\ab{x}) = \triangle \phi(\ab{x}) = 0 , \quad \ab{x} \in \partial \Omega \}, \end{equation*} which contains all $C^2$-smooth functions on $\Omega$ fulfilling the homogeneous boundary conditions~\eqref{eq:triharmonic_problem_boundary}.
\subsection{Structure of the space~$\mathcal{W}_{0h}$}
The discretization space~$\mathcal{W}_{0h}$ is the direct sum of smaller subspaces corresponding to the single patches~$\Omega^{(\ell)}$, edges~$\Gamma^{(s)}$ and vertices $\ab{v}^{(\rho)}$, i.e., \begin{equation} \label{eq:Wh}
\mathcal{W}_{0h} = \left(\bigoplus_{\ell=1}^P \mathcal{W}_{0h;\Omega^{(\ell)}}\right) \oplus \left(\bigoplus_{s=1}^E \mathcal{W}_{0h;\Gamma^{(s)}}\right)
\oplus \left(\bigoplus_{\rho=1}^V \mathcal{W}_{0h;\bfm{v}^{(\rho)}}\right). \end{equation} This decomposition is a common strategy to generate smooth spline spaces, e.g. \cite{KaSaTa17c, KaVi17b}. The construction of the single subspaces will be presented in the following subsections and will be based on functions from the subspaces $\mathcal{\widetilde{W}}_{h;\Omega^{(\ell)}}$ and $\mathcal{\widetilde{W}}_{h;\Gamma^{(s)}}$ for the two-patch case in Section~\ref{subsec:two-patch-case}.
\subsection{The patch subspace~$\mathcal{W}_{0h;\Omega^{(\ell)}}$}
Let $\ell \in \{1, 2, \ldots, P \}$. We denote by $\phi_{\Omega^{(\ell)};i,j}: \Omega \to \mathbb R$, ${\color{black} i,j=0,1,\ldots,p+k(p-r)}$, the functions \begin{equation} \label{eq:defphiOmega}
\phi_{\Omega^{(\ell)};i,j} (\bfm{x}) =
\begin{cases}
\widetilde{ \phi}_{\Omega^{(\ell)};i,j} (\bfm{x})\; \mbox{ if }\f \, \bfm{x} \in\Omega^{(\ell)} , \\[0.15cm]
0\quad \mbox{otherwise}, \end{cases} \end{equation} with $\widetilde{\phi}_{\Omega^{(\ell)};i,j}$ given in \eqref{eq:PhiOmega}, {\color{black} and }define the patch subspace $\mathcal{W}_{0h; \Omega^{(\ell)}}$ as \begin{equation*} \label{eq:spaceW0hOmega}
\mathcal{W}_{0h; \Omega^{(\ell)}} = \Span \{ \phi_{\Omega^{(\ell)};i,j} |\; i,j=3,4,\ldots,p+k(p-r)-3 \}. \end{equation*} \begin{lem} \label{lem:patchspace} We have \[ \mathcal{W}_{0h; \Omega^{(\ell)}} \subseteq \mathcal{V}_{0h}. \] \end{lem} \begin{pf} By \eqref{eq:defphiOmega}, the functions~$\phi_{\Omega^{(\ell)};i,j}$, $i,j=3,4,\ldots,p+k(p-r)-3$, possess a support \[ \mbox{supp}(\phi_{\Omega^{(\ell)};i,j}) \subseteq \Omega^{(\ell)}, \] they are clearly $C^2$-smooth on $\Omega^{(\ell)}$, and have vanishing values, gradients and Hessians on $\partial \Omega^{(\ell)}$. This implies that $\phi_{\Omega^{(\ell)};i,j} \in \mathcal{V}_{0h}$. \qed \end{pf}
\subsection{The edge subspace~$\mathcal{W}_{0h;\Gamma^{(s)}}$}
Let $s \in \{1,2, \ldots , E\}$ and let $\ell,\ell' \in \{1,2, \ldots, P\}$, $\ell \neq \ell'$, be the corresponding indices of the two patches such that $\Gamma^{(s)} = \Omega^{(\ell)} \cap \Omega^{(\ell')}$. Without loss of generality, we can assume that the two geometry mappings~$\ab{F}^{(\ell)}$ and $\ab{F}^{(\ell')}$ are parameterized as in Fig.~\ref{fig:situation_two_patches}. Otherwise, suitable linear reparameterizations of the two patches can be applied to fulfill this situation.
We denote by $\phi_{\Gamma^{(s)};i,j}: \Omega \to \mathbb R, \; i=0,1,2,\; j=0,1,\ldots, n_i-1$, the functions \begin{equation} \label{eq:defphiGamma}
\phi_{\Gamma^{(s)};i,j} (\bfm{x}) =
\begin{cases}
\widetilde{ \phi}_{\Gamma^{(s)};i,j} (\bfm{x})\; \mbox{ if }\f \, \bfm{x} \in\Omega^{(\ell)} \cup \Omega^{(\ell')} , \\[0.15cm]
0\quad otherwise, \end{cases} \end{equation} with $\widetilde{\phi}_{\Gamma^{(s)};i,j}$ given in \eqref{eq:basisFunctionsGenericEdge}. Then, the edge subspace~$\mathcal{W}_{0h;\Gamma^{(s)}}$ is defined as \begin{equation*} \label{eq:spaceW0hGamma}
\mathcal{W}_{0h;\Gamma^{(s)}} = \Span \{\phi_{\Gamma^{(s)};i,j}| \;\; j=5-i,6-i,\ldots,n_i+i-6;\; i=0,1,2\}. \end{equation*} \begin{lem} \label{lem:edgespace} It holds that \[ \mathcal{W}_{0h; \Gamma^{(s)}} \subseteq \mathcal{V}_{0h}. \] \end{lem} \begin{proof} Let $ i=0,1,2$ and $j=5-i,6-i,\ldots, n_i+i-6$. By \eqref{eq:defphiGamma}, the functions $\phi_{\Gamma^{(s)};i,j}$ possess a support
\[
\mbox{supp}(\phi_{\Gamma^{(s)};i,j}) \subseteq \Omega^{(\ell)} \cup \Omega^{(\ell')}.
\]
Furthermore, it was shown in~\cite{KaVi17c}, that the functions $\phi_{\Gamma^{(s)};i,j}$ are $C^2$-smooth on~$\Omega^{(\ell)} \cup \Omega^{(\ell')}$. Since
Proposition~\ref{thm:mainC4} ensures that the functions $ \phi_{\Gamma^{(s)};i,j}$ have vanishing values, gradients and Hessians on $\partial (\Omega^{(\ell)} \cup \Omega^{(\ell')} )$, we obtain $\phi_{\Gamma^{(s)};i,j} \in \mathcal{V}_{0h}$. \end{proof}
\subsection{The vertex subspace $\mathcal{W}_{0h;\bfm{v}^{(\rho)}}$} We consider an inner or boundary vertex~$\bfm{v}^{(\rho)}$, $\rho \in \{1,2, \ldots, V \}$, possessing the valency $\bar{\nu}_{\rho} \geq 3$. We define $\nu_{\rho}$ as \[ \nu_{\rho} = \begin{cases}
\bar{\nu}_{\rho}, \; \mbox{ if }\bfm{v}^{(\rho)}\mbox{ is an inner vertex} ,\\ \bar{\nu}_{\rho}-1, \; \mbox{ if }\bfm{v}^{(\rho)}\mbox{ is a boundary vertex} . \end{cases} \] For the sake of simplicity, we relabel the patches containing the vertex $\ab{v}^{(\rho)}$ in counterclockwise order by $\Omega^{(0)}, \Omega^{(1)}, \ldots, \Omega^{(\nu_{\rho}-1)}$. Furthermore, we assume without loss of generality that the corresponding geometry mappings $\ab{F}^{(\ell)}$, $\ell=0,1,\ldots,\nu_\rho -1$, are parameterized as shown in Fig.~\ref{fig:situation_multi_patches}, which assures that $$
\bfm{F}^{(0)}(\bfm{0}) = \bfm{F}^{(1)}(\bfm{0}) = \cdots = \bfm{F}^{(\nu_\rho -1)}(\bfm{0}) = \bfm{v}^{(\rho)}. $$ \begin{figure}
\caption{The geometry mappings~$\ab{F}^{(\ell)}$ of the patches~$\Omega^{(\ell)}$, $\ell =0, 1,\ldots, \nu_{\rho} -1$, which contain the vertex~$\bfm{v}^{(\rho)}$, can be always reparameterized as shown.}
\label{fig:situation_multi_patches}
\end{figure} Moreover, we relabel the common interface of every two-patch subdomain $ \Omega^{(\ell)} \cup \Omega^{(\ell+1)}, \, \ell=0,1,\ldots,\nu_\rho -1, $ by $\Gamma^{(\ell+1)}$. In case of an inner vertex~$\bfm{v}^{(\rho)}$, we consider the upper index~$\ell$ of $\Omega^{(\ell)}$ and $\Gamma^{(\ell)}$ modulo~$\nu_{\rho}$, and in case of a boundary vertex~$\bfm{v}^{(\rho)}$, we denote by $\Gamma^{(0)}$ the edge of $\Omega^{(0)}$ corresponding to $\ab{F}^{(0)}([0,1]\times \{0\})$, and by $\Gamma^{(\nu_{\rho})}$ the edge of $\Omega^{(\nu_{\rho}-1)}$ corresponding to $\ab{F}^{(\nu_{\rho}-1)}(\{0 \} \times [0,1])$. In addition, we denote by $\bar{\bfm{\xi}}^{(\ell)} $ the pair of parameters $\bar{\bfm{\xi}}^{(\ell)} = ({\xi}_2^{(\ell)},\xi_1^{(\ell)})$.
The idea is to construct the vertex subspace $\mathcal{W}_{0h;\bfm{v}^{(\rho)}}$ as the space of functions which can be represented by suitable linear combinations of functions $\phi_{\Omega^{(\ell)};i,j}$, ${\color{black} 0 \leq i,j \leq 2},\, 0\leq\ell\leq {\nu}_\rho-1,$ and of functions $\phi_{\Gamma^{(\ell)};i,j}$, $0\leq i\leq 2,\, 0\leq j\leq 4-i,\, 0 \leq\ell\leq \bar{\nu}_\rho-1$. Note that none of these functions are contained in any of the spaces $\mathcal{W}_{0h;\Omega^{(\ell)}}$ and $\mathcal{W}_{0h;\Gamma^{(s)}}$. {\color{black} Furthermore, these are exactly those functions~$\phi_{\Omega^{(\ell)};i,j}$ and $\phi_{\Gamma^{(\ell)};i,j}$, which are involved in the continuity constraints at the vertex, since they can possess nonzero spline coefficients (with respect to the representation~\eqref{eq:g_ell}), which are affected by the $C^2$-continuity conditions of more than one edge~$\Gamma^{(\ell)}$. {\color{black} These corresponding spline coefficients are the ones in the grey region in Fig.~\ref{fig:supports} }}. Recall that the functions $\phi_{\Gamma^{(\ell)};i,j}$ are $C^2$-smooth on the two-patch subdomain~$\Omega^{(\ell-1)} \cup \Omega^{(\ell)}$.
For each patch $\Omega^{(\ell)}$, $\ell=0,1,\ldots, {\nu}_{\rho} -1$, we define the function $f_{\ell} : [0,1]^2 \rightarrow \mathbb R$ as \[ f_{\ell} (\bfm{\xi}^{(\ell)}) = f_{\ell}^{\Gamma^{(\ell)}} (\bfm{\xi}^{(\ell)}) + f_{\ell}^{\Gamma^{(\ell+1)}} (\bfm{\xi}^{(\ell)}) - f_{\ell}^{\Omega^{(\ell)}} (\bfm{\xi}^{(\ell)}), \] where the functions $f_{\ell}^{\Gamma^{(\ell)}} , f_{\ell}^{\Gamma^{(\ell+1)}}, f_{\ell}^{\Omega^{(\ell)}} : [0,1]^2 \to \mathbb R$ are given by {\color{black} \begin{align*}
f_{\ell}^{\Gamma^{(\ell)}} (\bfm{\xi}^{(\ell)}) & = \sum_{i=0}^2 \sum_{j=0}^{4-i} a^{\Gamma^{(\ell)}}_{i,j} \,
g_{\Gamma^{(\ell); i,j}}^{(\ell)} (\bar{\bfm{\xi}}^{(\ell)}) , \nonumber \\
f_{\ell}^{\Gamma^{(\ell+1)}} (\bfm{\xi}^{(\ell)}) & = \sum_{i=0}^2 \sum_{j=0}^{4-i} a^{\Gamma^{(\ell+1)}}_{i,j} \,
g_{\Gamma^{(\ell+1); i,j}}^{(\ell)} (\bfm{\xi}^{(\ell)}), \\ \label{eq:functionsonpatch}
f_{\ell}^{\Omega^{(\ell)}} (\bfm{\xi}^{(\ell)}) & = \sum_{i=0}^2 \sum_{j=0}^{2} a^{(\ell)}_{i,j}
N_{i,j}^{p,r} (\bfm{\xi}^{(\ell)}) , \nonumber \end{align*} } with $a^{\Gamma^{(\ell)}}_{i,j}, a^{\Gamma^{(\ell+1)}}_{i,j}, a_{i,j}^{(\ell)} \in \mathbb R$. Furthermore, we define the function $\phi_{\bfm{v}^{(\rho)}} : \Omega \to \mathbb R$ as \begin{equation} \label{eq:defphiXi}
\phi_{\bfm{v}^{(\rho)}} (\bfm{x}) =
\begin{cases}
(f_{\ell} \circ (\ab{F}^{(\ell)})^{-1})(\bfm{x}) \; \mbox{ if }\f \, \bfm{x} \in \Omega^{(\ell)},\; \ell=0,1,\ldots,{\nu}_\rho -1, \\ 0 \quad \mbox{ otherwise}. \end{cases} \end{equation}
The idea for the construction of the function $\phi_{\bfm{v}^{(\rho)}} $ is as follows. On each patch $\Omega^{(\ell)}$, $\ell=0,1,\ldots, {\nu}_{\rho} -1$, the function~$ \phi_{\bfm{v}^{(\rho)}} $ is determined by the spline function~$f_{\ell}$, where the sum of the functions $f_{\ell}^{\Gamma^{(\ell)}}$ and $f_{\ell}^{\Gamma^{(\ell+1)}}$ should ensure $C^2$-smoothness across the interfaces~$\Gamma^{(\ell)}$ and $\Gamma^{(\ell+1)}$, and the function~$f_{\ell}^{\Omega^{(\ell)}}$ is used to subtract those B-splines $N_{i,j}^{p,r}$ (with respect to the spline space $\mathcal{S}^{p,r}_{h}([0,1]^2)$), which have been added twice, see Fig.~\ref{fig:supports}.
\begin{figure}\label{fig:supports}
\end{figure}
Clearly, not any choice of the coefficients~$a^{\Gamma^{(\ell)}}_{i,j}$ and $a_{i,j}^{(\ell)}$, $ \ell=0,1,\ldots,{\nu}_\rho-1$, guarantees $\phi_{\bfm{v}^{(\rho)}} \in \mathcal{V}_h$ or as needed in our case even~$\phi_{\bfm{v}^{(\rho)}} \in \mathcal{V}_{0h}$. The following lemma characterizes when the function~$\phi_{\bfm{v}^{(\rho)}}$ belongs to the space~$\mathcal{V}_{0h}$: \begin{lem} \label{lem:C2smoothvertex}
$\phi_{\bfm{v}^{(\rho)}} \in \mathcal{V}_{0h}$ if the corresponding functions $f_{\ell}^{\Gamma^{(\ell)}}$,
$f_{\ell}^{\Gamma^{(\ell+1)}}$, and $f_{\ell}^{\Omega^{(\ell)}}$, $\ell =0,1, \ldots,$ $\nu_{\rho}-1$, satisfy
\begin{equation} \label{eq:system1}
\partial_{\xi_1^{(\ell)}}^i
\partial_{\xi_2^{(\ell)}}^j
\left( f_{\ell}^{\Gamma^{(\ell+1)}} - f_{\ell}^{\Gamma^{(\ell)}} \right) (\bfm{0}) = 0, \quad
{\color{black} 0 \leq i,j \leq 2}, \end{equation}
and
\begin{equation} \label{eq:system2}
\partial_{\xi_1^{(\ell)}}^i
\partial_{\xi_2^{(\ell)}}^j
\left( f_{\ell}^{\Gamma^{(\ell+1)}} - f_{\ell}^{\Omega^{(\ell)}} \right) (\bfm{0}) = 0, \quad
0 \leq i,j \leq 2, \end{equation} and in case of a boundary vertex~$\bfm{v}^{(\rho)}$, additionally \begin{equation} \label{eq:system3}
a_{i,j}^{\Gamma^{(0)}} = 0 , \mbox{ and } a_{i,j}^{\Gamma^{({\nu}_{\rho})}} = 0, \quad 0 \leq i \leq 2, \, 0\leq j\leq 4-i, \end{equation} and \begin{equation} \label{eq:system4}
a_{i,j}^{(\ell)} = 0 , \quad {\color{black} 0 \leq i,j \leq 2, \, 0 \leq \ell \leq \nu_{\rho}-1}. \end{equation} \end{lem} \begin{proof} By \eqref{eq:defphiXi}, the function~$\phi_{\bfm{v}^{(\rho)}}$ possesses a support
\[
\mbox{supp}({\phi_{\bfm{v}^{(\rho)}}}) \subseteq \cup_{\ell=0}^{{\nu}_{\rho} -1} \Omega^{(\ell)}.
\] Equations~\eqref{eq:system1} and \eqref{eq:system2} ensure that the coefficients~$a^{\Gamma^{(\ell)}}_{i,j}$ and $a_{i,j}^{(\ell)}$ are well-defined, which implies that the
function~$\phi_{\bfm{v}^{(\rho)}}$ is well-defined. The function~$\phi_{\bfm{v}^{(\rho)}}$ is now $C^2$-smooth across the interfaces~$\Gamma^{(\ell)}$, since its values,
gradients and Hessians along the interfaces~$\Gamma^{(\ell)}$ are given by
\[
\phi_{\bfm{v}^{(\rho)}}(\Gamma^{(\ell)}) = \sum_{i=0}^2 \sum_{j=0}^{4-i} a^{\Gamma^{(\ell)}}_{i,j} \, \phi_{\Gamma^{(\ell)};i,j}(\Gamma^{(\ell)}),
\]
\[
\nabla \phi_{\bfm{v}^{(\rho)}}(\Gamma^{(\ell)}) = \sum_{i=0}^2 \sum_{j=0}^{4-i} a^{\Gamma^{(\ell)}}_{i,j} \, \nabla \phi_{\Gamma^{(\ell)};i,j}(\Gamma^{(\ell)})
\]
and
\[
\mbox{Hess} (\phi_{\bfm{v}^{(\rho)}})(\Gamma^{(\ell)}) = \sum_{i=0}^2 \sum_{j=0}^{4-i} a^{\Gamma^{(\ell)}}_{i,j} \, \mbox{Hess}(\phi_{\Gamma^{(\ell)};i,j})(\Gamma^{(\ell)}),
\] respectively, Finally, we obtain $\phi_{\bfm{v}^{(\rho)}} \in \mathcal{V}_{0h}$, since Proposition~\ref{thm:mainC4} and equations~\eqref{eq:system3} {\color{black} and \eqref{eq:system4}} (in case of a boundary vertex ~$\bfm{v}^{(\rho)}$) ensure that the function $\phi_{\bfm{v}^{(\rho)}}$ has vanishing values, gradients and Hessians already on the boundary of the multi-patch subdomain $\cup_{\ell=0}^{\nu_{\rho}-1} \Omega^{(\ell)}$. \end{proof}
The equations~\eqref{eq:system1} and \eqref{eq:system2}, and additionally equations~\eqref{eq:system3} {\color{black} and \eqref{eq:system4}} in case of a boundary vertex~$\bfm{v}^{(\rho)}$, form a homogeneous linear system \begin{equation} \label{eq:whole_system}
H^{(\rho)} \ab{a}^{(\rho)} = \ab{0}, \end{equation} where $\ab{a}^{(\rho)}$ is the vector of all involved coefficients $a_{i,j}^{\Gamma^{(\ell)}}$ and $a_{i,j}^{(\ell)}$. Any basis of the null space (i.e., the kernel) of the matrix~$H^{(\rho)}$, determines $ \dim (\ker H^{(\rho)})$ linearly independent functions~$\phi_{\bfm{v}^{(\rho)}} \in \mathcal{V}_{0h}$, which will be denoted by $\phi_{\bfm{v}^{(\rho)};m}$, $m=1,2,\ldots, \dim (\ker H^{(\rho)})$. One possible strategy is to find a basis by constructing minimal determining sets (cf.~\cite{BeMa14, LaSch07}) for the unknown coefficients of the homogeneous linear system~\eqref{eq:whole_system}. In our examples in Section~\ref{sec:triharmonic_examples}, we use the minimal determining set algorithm introduced in \cite[Section 6.1]{KaVi17a}, which works well and yields well-conditioned functions, {\color{black} cf. Examples~\ref{ex:example1} and \ref{ex:example2}}.
Finally, the vertex subspace $\mathcal{W}_{0h;\bfm{v}^{(\rho)}}$ is defined as \begin{equation*}
\mathcal{W}_{0h;\bfm{v}^{(\rho)}} = \Span \{ \phi_{\bfm{v}^{(\rho)};m}\, |\; m = 1,2,\ldots, \dim (\ker H^{(\rho)}) \}. \end{equation*}
\begin{lem} \label{lem:vertexspace} We have \[
\mathcal{W}_{0h;\bfm{v}^{(\rho)}} \subseteq \mathcal{V}_{0h}. \] \end{lem} \begin{proof} Recall \eqref{eq:defphiXi}. The functions $\phi_{\bfm{v}^{(\rho)};m}$, $m=1,2,\ldots, \dim (\ker H^{(\rho)})$, are constructed in such a way that they satisfy $\phi_{\bfm{v}^{(\rho)};m} \in \mathcal{V}_{0h}$. \end{proof}
\begin{rem} A further possible way for the computation of suitable vertex subspaces could be the extension of the method~\cite{KaSaTa17c} proposed for the case of $C^1$-smooth isogeometric functions to our case of $C^2$-smooth isogeometric functions. In~\cite{KaSaTa17c}, the vertex subspace is defined by globally $C^1$-smooth functions which are $C^2$-smooth at the vertex. However, the extension of this approach to our case would require globally $C^2$-smooth functions which have to be $C^4$-smooth at the vertex. \end{rem}
\subsection{The space~$\mathcal{W}_{0h}$} Recall that the space~$\mathcal{W}_{0h}$ is the direct sum~\eqref{eq:Wh}. \begin{thm} \label{thm:discretizationspace} It holds that \[
\mathcal{W}_{0h} \subseteq \mathcal{V}_{0h}, \] and the collection of functions \begin{align} \label{eq:setsOfFunctions} &\phi_{\Omega^{(\ell)};i,j} , \quad i,j=3,4,\ldots,p+k(p-r)-3,\; \ell=1,2,\ldots,P, \nonumber \\ &\phi_{\Gamma^{(s)};i,j} , \quad i=0,1,2,\; {\color{black} j=5-i,6-i,\ldots,n_{i}+i-6,} \;s=1,2,\ldots,E,\\ &\phi_{\bfm{v}^{(\rho)};m}, \quad m=1,2,\ldots, \dim (\ker H^{(\rho)}), \; \rho=1,2,\ldots,V, \nonumber \end{align} forms a basis of the space $\mathcal{W}_{0h}$. \end{thm} \begin{proof} $\mathcal{W}_{0h} \subseteq \mathcal{V}_{0h}$ is a direct consequence of Lemma~\ref{lem:patchspace}, \ref{lem:edgespace} and \ref{lem:vertexspace}, and the definition of the space~$\mathcal{W}_{0h}$, see~\eqref{eq:Wh}. {\color{black} By construction, the collection of functions~\eqref{eq:setsOfFunctions} spans the space~$\mathcal{W}_{0h}$, and all functions are linearly independent. The latter property follows directly from the following tree facts. First, the functions $\phi_{\Omega^{(\ell)};i,j}$, $\phi_{\Gamma^{(s)};i,j}$ and $\phi_{\bfm{v}^{(\rho)};m}$ are linearly independent in their particular sets. Second, the selected functions~$\phi_{\Omega^{(\ell)};i,j}$ do not have a common set of nonzero coefficients with the corresponding functions~$\phi_{\Gamma^{(s)};i,j}$ and $\phi_{\bfm{v}^{(\rho)};m}$ with respect to spline representation~\eqref{eq:g_ell}. Third, the functions~$\phi_{\bfm{v}^{(\rho)};m}$ are linear combinations only of functions~ $\phi_{\Omega^{(\ell)};i,j}$ and $\phi_{\Gamma^{(s)};i,j}$, which are not contained in any of the spaces~$\mathcal{W}_{0h;\Omega^{(\ell)}}$ and $\mathcal{W}_{0h;\Gamma^{(s)}}$.} \end{proof}
\begin{rem}
The functions $\phi_{\Omega^{(\ell)};i,j}$, $\phi_{\Gamma^{(s)};i,j}$ and $\phi_{\bfm{v}^{(\rho)};m}$ are called patch, edge and vertex functions, respectively. All these functions
possess a small local support, and are obtained by {\color{black} computing the null space of} a small system of linear equations and/or by simple explicit formulae.
{\color{black} The patch functions~$\phi_{\Omega^{(\ell)};i,j}$ are just the ``standard'' isogeometric functions
{\color{black} whose supports are contained} in one patch {\color{black} only}. The small, local supports of
the edge and vertex functions are contained in two or in at least two patches, respectively. More precisely, the edge functions~$\phi_{\Gamma^{(s)};i,j}$ {\color{black} have their supports contained}
in a small
region across the common interface, and the vertex functions~$\phi_{\bfm{v}^{(\rho)};m}$ possess a support in the vicinity of the vertex. While, the edge functions interpolate
values and specific first and second derivatives along the common interface, cf.~\cite{KaVi17c}, the vertex functions are just built up from functions~$\phi_{\Omega^{(\ell)};i,j}$
and $\phi_{\Gamma^{(s)};i,j}$, which are not contained in any patch subspace~$\mathcal{W}_{0h;\Omega^{(\ell)}}$ and in any edge subspace~$\mathcal{W}_{0h;\Gamma^{(s)}}$, respectively.}
By means of interpolation, {\color{black} the edge and vertex functions, or more precisely, their} spline functions $\phi_{\Gamma^{(s)};i,j} \circ \ab{F}^{(\ell)}$ and
$\phi_{\bfm{v}^{(\rho)};m} \circ \ab{F}^{(\ell)}$ can be represented as {\color{black} a linear combination of} the spline functions $\phi_{\Omega^{(\ell)};i,j} \circ \ab{F}^{(\ell)}$,
{\color{black} i.e.} with respect to the spline representation~\eqref{eq:g_ell} (compare e.g., \cite{KaVi17c}). \end{rem}
\begin{ex} \label{ex:functions} We consider the three-patch domain~(a) visualized in Fig.~\ref{fig:example1}~(first row). The space~$\mathcal{W}_{0h}$ is defined as \[ \mathcal{W}_{0h} = \left(\bigoplus_{\ell=1}^3 \mathcal{W}_{0h;\Omega^{(\ell)}}\right) \oplus \left(\bigoplus_{s=1}^3 \mathcal{W}_{0h;\Gamma^{(s)}}\right)
\oplus \left(\bigoplus_{\rho=1}^4 \mathcal{W}_{0h;\bfm{v}^{(\rho)}}\right) \] with the vertices $\bfm{v}^{(1)}=(\frac{17}{3},2)$, $\bfm{v}^{(2)}=(\frac{35}{4},\frac{15}{7})$, $\bfm{v}^{(3)}=(\frac{13}{3},4)$ and $\bfm{v}^{(4)}=(5,0)$, and the edges $\Gamma^{(1)}=\Omega^{(1)} \cap \Omega^{(2)}$, $\Gamma^{(2)}=\Omega^{(2)} \cap \Omega^{(3)}$ and $\Gamma^{(3)}=\Omega^{(3)} \cap \Omega^{(1)}$. For $p=5$, $r=2$ and $h=\frac{1}{6}$, the dimensions of the single subspaces are given by \[ \dim \mathcal{W}_{0h;\Omega^{(\ell)}} = 225, \mbox{ } \dim \mathcal{W}_{0h;\Gamma^{(s)}} =6 , \mbox{ } \dim \mathcal{W}_{0h;\bfm{v}^{(1)}} = 16 \mbox{ and } \dim \mathcal{W}_{0h;\bfm{v}^{(\rho)}} = 3, \] for $s,\ell=1,2,3$ and $\rho=2,3,4$. Furthermore, the functions of the edge space $\mathcal{W}_{0h;\Gamma^{(1)}}$ and the functions of the vertex spaces $\mathcal{W}_{0h;\bfm{v}^{(1)}}$ and $\mathcal{W}_{0h;\bfm{v}^{(2)}}$ are shown in Fig.~\ref{fig:ex_functions_edge} and Fig.~\ref{fig:ex_functions_vertex}, respectively. {\color{black} Recall that the functions of the edge spaces are determined by the explicit representation~\eqref{eq:basisFunctionsGenericG}, and that the functions of the vertex spaces are defined via appropriate bases of the null spaces of the corresponding homogeneous linear systems~\eqref{eq:whole_system}, which are computed by means of the minimal determining set algorithm~\cite[Section 6.1]{KaVi17a}.}
\begin{figure}\label{fig:ex_functions_edge}
\end{figure}
\begin{figure}\label{fig:ex_functions_vertex}
\end{figure} \end{ex}
\begin{rem} For the sake of simplicity we restricted ourselves to the case of bilinearly parameterized multi-patch domains. The construction of the space~$\mathcal{W}_{0h}$ and of its basis should be extendable in a straightforward way to the class of bilinear-like geometries~\cite{KaVi17c}. However, the construction and the study of bilinear-like geometries themselves are limited to the case of two-patch domains~\cite{KaVi17c} so far. But an extension to the case of multi-patch domains is of vital interest for the future research. \end{rem}
\section{Solving the triharmonic equation -- Examples} \label{sec:triharmonic_examples}
We present several examples to demonstrate the potential of our approach for solving the triharmonic equation over bilinear multi-patch domains.
\begin{ex} \label{ex:example1} We consider the three bilinearly parameterized multi-patch domains given in Figure~\ref{fig:example1}~(first row), which possess extraordinary vertices of valency~$3$, $5$ or $6$ {\color{black} and describe a triangular, pentagonal and hexagonal domain, respectively.} For all three domains (a)-(c), we construct nested isogeometric spline spaces $\mathcal{W}_{0h}$ of degree~$p=5$ and regularity~$r=2$ for the mesh-sizes $h=\frac{1}{k+1}$, $k \in \{3,7,15,31 \}$. Note that for the case of $h=\frac{1}{4}$, the construction of the space $\mathcal{W}_{0h}$ has to be slightly modified. More precisely, the vertex subspace $\mathcal{W}_{0h;\bfm{v}^{(\rho)}}$ is constructed without the use of the functions~$\phi_{\Gamma^{(\ell)};0,4}$. Instead, these functions are added to the corresponding edge subspaces~$\mathcal{W}_{0h;\Gamma^{(\ell)}}$ after subtracting suitable linear combinations of functions $\phi_{\Gamma^{(\ell)};i,j}$, $0 \leq i \leq 2$, $0 \leq j \leq \min(4-i,3)$ to obtain functions~$\widehat{\phi}_{\Gamma^{(\ell)};0,4}$ which have vanishing values, gradients and Hessians on $\partial (\Omega^{(\ell-1)} \cup \Omega^{(\ell)} )$.
We solve the triharmonic equation~\eqref{eq:triharmonic_problem} with the homogeneous boundary conditions~\eqref{eq:triharmonic_problem_boundary} over the domains~(a)-(c) for right side functions~$f$ obtained by the exact solutions \[ \footnotesize
u_{a}(\ab{x}) = \left(\frac{1}{20} x_2 (\frac{12 x_1}{13}-x_2)(\frac{120 - 12 x_1}{7} -x_2) \right)^3, \] \[ \footnotesize
u_{b}(\ab{x}) = \left(\frac{1}{20000} (\frac{121 +8 x_1}{15} - x_2)( \frac{7 x_1}{2} + x_2)x_2(\frac{52}{3} - \frac{13 x_1}{6} +
x_2)( \frac{31}{2} - \frac{9 x_1}{11} - x_2) \right)^3 \] and \[ \footnotesize
u_{c}(\ab{x}) = \left( \frac{1}{200000}(\frac{55- 5 x_1}{2} - x_2)(\frac{1799}{160} - \frac{7 x_1}{64} - x_2)(\frac{652 +78 x_1}{61} - x_2)
(-\frac{18 x_1}{11} - x_2)x_2(\frac{5 x_1}{3} -10 - x_2)\right)^3,
\] see Fig.~\ref{fig:example1}~(second row). The resulting relative $H^{i}$-errors, $i=0,1,2,3$, are visualized in Fig.~\ref{fig:example1}~(third row) and indicate convergence rates
of order $\mathcal{O}(h^{6-i})$ in the corresponding norms.~\footnote{Note that for the spaces~$\mathcal{W}_{0h}$ the norms $|| \cdot ||_{H^3(\Omega)}$ and
$||\nabla \triangle (\cdot) ||_{L^2(\Omega)}$ are equivalent.}{} {\color{black} Furthermore, Fig.~\ref{fig:example1}~(fourth row) shows the resulting condition numbers $\kappa$ of the stiffness matrices~$S$ by using diagonally scaling (cf. \cite{Br95}) and by employing no preconditioner. In case of the non-preconditioned stiffness matrices, the errors are slightly higher, but for both cases} the estimated growth rates are of order $\mathcal{O}(h^{-6})$, which demonstrate that the constructed basis functions are well-conditioned.
\begin{figure}
\caption{Solving the triharmonic equation over different multi-patch domains~$\Omega$ (cf. Example~\ref{ex:example1}).}
\label{fig:example1}
\end{figure} \end{ex}
\begin{ex} \label{ex:example2} We consider the bilinearly parameterized five-patch domain with four extraordinary vertices of valency~$3$, which is visualized in Fig.~\ref{fig:example2} (first row). {\color{black} For the mesh-sizes $h=\frac{1}{k+1}$, $k \in \{3,7,15,31 \}$, nested isogeometric spline spaces $\mathcal{W}_{0h}$ of degree~$p=5,6$ and regularity~$r=2$ (for $p=5,6$) and $r=3$ (for $p=6$) are generated. As in Example~\ref{ex:example1}, the construction of the space~$\mathcal{W}_{0h}$ has to be slightly changed for the case $p=5$, $r=2$ and $h=\frac{1}{4}$. The resulting spaces} are used to solve the triharmonic equation~\eqref{eq:triharmonic_problem} with the homogeneous boundary conditions~\eqref{eq:triharmonic_problem_boundary}. We use for testing the right side function~$f$ which is obtained by the exact solution \[
u(\ab{x}) = (\frac{1}{20000}x_2 (\frac{405}{8} - \frac{27 x_1}{8} - x_2)(\frac{425}{38} + \frac{4 x_1}{19} - x_2)(\frac{23 x_1}{3} -
x_2))^3 , \] see Fig.~\ref{fig:example2} (first row). {\color{black} The resulting relative $H^i$-errors are of order $\mathcal{O}(h^{p+1-i})$,} and the estimated growth rates of the diagonally scaled stiffness matrices~$S$ are of order~$\mathcal{O}(h^{-6})$. {\color{black} As in Example~\ref{ex:example1}, we also present the condition numbers of the non-preconditioned stiffness matrices~$S$, see Fig.~\ref{fig:example2}, which are again slightly higher than for the preconditioned case (i.e. using diagonal scaling) but still seems to grow of order~$\mathcal{O}(h^{-6})$. This indicates again that the constructed basis functions are well-conditioned.} \begin{figure}
\caption{Solving the triharmonic equation over the given multi-patch domain~$\Omega$ (cf. Example~\ref{ex:example2}).}
\label{fig:example2}
\end{figure} \end{ex}
\section{Conclusion} \label{sec:conclusion}
We described a method for solving the triharmonic equation over bilinearly parameterized planar multi-patch domains. The presented approach is based on the concept of IGA and uses as discretization space~$\mathcal{W}_{0h}$ a space of globally $C^{2}$-smooth isogeometric functions. The discretization space~$\mathcal{W}_{0h}$ is the span of three different types of {\color{black} basis} functions called patch, edge and vertex functions. All of these functions possess a simple representation with small local supports, can be uniformly generated for all possible multi-patch configurations, {\color{black} and numerical examples indicate {\color{black} that} they are well-conditioned}. The numerical results obtained by solving the triharmonic equation over different bilinear multi-patch domains using $h$-refinement demonstrate the potential of our approach.
The paper leaves several open questions which are worth to study. A first possible topic for future research could be the study of a priori error estimates for the triharmonic equation over multi-patch domains under $h$-refinement (similar to the ones in \cite{TaDe14} for single patch domains), and the theoretical investigation of the approximation properties of the discretization space~$\mathcal{W}_{0h}$. {\color{black} Another topic could be the detailed study of the dimension of the space~$\mathcal{W}_{0h}$ to get an explicit dimension formula. In \cite{KaVi17b}, the case of the entire $C^2$-smooth space~$\mathcal{V}_h$ was investigated, and the obtained formula there provides an upper bound for the dimension of~$\mathcal{W}_{0h}$. Like in~\cite{KaVi17b} for the case of~$\mathcal{V}_h$, the dimension of the space~$\mathcal{W}_{0h}$ is just the sum of the dimensions of the single subspaces (i.e. patch, edge and vertex subspaces). While the numbers of basis functions for the patch subspaces~$\mathcal{W}_{0h;\Omega^{(\ell)}}$ and for the edge subspaces~$\mathcal{W}_{0h;\Gamma^{(s)}}$ are explicitly given, the computation of the numbers of basis functions for the vertex subspaces~$\mathcal{W}_{0h;\bfm{v}^{(\rho)}}$ still deserves further investigation.
Moreover, one could consider} further $6$-th order PDEs for which the use of the discretization space~$\mathcal{W}_{0h}$ could be suitable, since these problems require functions of $C^2$-smoothness. Possible examples are the Kirchhoff plate model based on the Mindlin's gradient elasticity theory~\cite{Niiranen2016}, the Phase-field crystal equation \cite{BaDe15, Gomez2012} and the gradient-enhanced continuum damage model~\cite{GradientDamageModels}. The extension of our approach to more general multi-patch domains, such as e.g., bilinear-like planar domains, shells or volumetric domains could be considered, too.
\paragraph*{\bf Acknowledgment}
{\color{black} The authors wish to thank the anonymous reviewers for their comments that helped to improve the paper.} V.~Vitrih was partially supported by the Slovenian Research Agency (research program P1-0285). This support is gratefully acknowledged.
\appendix
\section{Proof of Proposition~\ref{thm:mainC4}} \label{app:proofProposition}
The proof will be mainly based on the concept of blossoming. Let $q \in \mathcal{S}_{h}^{p,r}([0,1])$, and let $t_0^{p,r}, t_1^{p,r},\ldots,t_{2p+1+k(p-r)}^{p,r}$ be the corresponding knots of the spline space $\mathcal{S}_{h}^{p,r}([0,1])$. Then there exists a unique function $\mathcal{Q}^{p,r}:{\color{black} \mathbb R^p \to \mathbb R}$, called the \emph{blossom} of $q$, which is symmetric, multi-affine and fulfill $\mathcal{Q}^{p,r}(\xi,\xi,\ldots,\xi) = q(\xi)$. These properties imply that the control points of $q$ can be written as $$ d_\iota = \mathcal{Q}^{p,r}(t_{\iota+1}^{p,r},t_{\iota+2}^{p,r},\ldots,t_{\iota+p}^{p,r}), \quad \iota=0,1,\ldots,p+k(p-r). $$ Blossoming is a simple approach, which can be used amongst others to perform knot insertion for a spline function or to multiply two spline functions. For more details about the concept of blossoming we refer to e.g.~\cite{GO03, Ra89, Se93}.
The following two lemmas will be needed. \begin{lem} \label{lem:knotInsertion} Let $N_j^{p,r+1}(\xi) = \sum_{ \iota=0}^{p+k(p-r)} \widetilde{d}_\iota N_\iota^{p,r}(\xi)$. Then $ \widetilde{d}_\iota = 0 $ for $ \iota<j.$ \end{lem} \begin{pf} Let $d_\iota$ be control points of $N_j^{p,r+1} \in \mathcal{S}_{h}^{p,r+1}([0,1])$, i.e., $d_\iota = \delta_{j,\iota}$. Moreover let $ \widetilde{d}_\iota $ denote control points of $N_j^{p,r+1}$ represented in the space $\mathcal{S}_{h}^{p,r}([0,1])$. Then (see e.g. \cite{GO03}) $$
\widetilde{d}_\iota = \mathcal{Q}^{p,r+1} (t_{\iota+1}^{p,r},\ldots, t_{\iota+p}^{p,r}). $$ Since $$
{d}_\iota = \mathcal{Q}^{p,r+1} (t_{\iota+1}^{p,r+1},\ldots, t_{\iota+p}^{p,r+1})
\quad {\rm and} \quad
t_{\iota+p}^{p,r+1} \geq t_{\iota+p}^{p,r}, $$ it follows that $\widetilde{d}_\iota = \sum_{m \leq \iota} c_m d_m$, $c_m \in \mathbb R$, which implies $\widetilde{d}_\iota = 0$ for $\iota<j$. \qed \end{pf} \begin{lem} \label{lem:multiplication} Let $(\omega_0 (1-\xi) + \omega_1 \xi) \, N_j^{p-1,r}(\xi) = \sum_{ \iota=0}^{p+k(p-r)} \widehat{d}_\iota N_\iota^{p,r}(\xi)$. Then $ \widehat{d}_\iota = 0 $ for $ \iota<j.$ \end{lem} \begin{pf} Let $d_\iota$ denote control points of $ N_j^{p-1,r} \in \mathcal{S}_{h}^{p-1,r}([0,1])$, i.e., $d_\iota = \delta_{j,\iota}$, and let $\mathcal{Q}^{p-1,r}$ denote its blossom. Moreover let $\widehat{d}_\iota$ denote the control points of $(\omega_0 (1-\xi) + \omega_1 \xi) \, N_j^{p-1,r}(\xi) \in \mathcal{S}_{h}^{p,r}([0,1])$. Then (see e.g. \cite{GO03}) $$
\widehat{d}_\iota = \frac{1}{p} \sum_{m=1}^p \mathcal{Q}^{p-1,r} (t_{\iota+1}^{p,r},\ldots,t_{\iota+m-1}^{p,r} ,t_{\iota+m+1}^{p,r},\ldots, t_{\iota+p}^{p,r})
\left(\omega_0 (1-t_{\iota+m}^{p,r}) + \omega_1 t_{\iota+m}^{p,r} \right). $$ We have to prove that $\widehat{d}_\iota = \sum_{n \leq \iota} c_n d_n$, $c_n \in \mathbb R$. Since $d_\iota= \mathcal{Q}^{p-1,r} (t_{\iota+1}^{p-1,r},\ldots,t_{\iota+p}^{p-1,r})$ and $t_{\iota+p}^{p,r} \leq t_{\iota+p}^{p-1,r}$, it follows that $ \mathcal{Q}^{p-1,r} (t_{\iota+1}^{p,r},\ldots,t_{\iota+m-1}^{p,r} ,t_{\iota+m+1}^{p,r},\ldots, t_{\iota+p}^{p,r}) $ does not involve $d_n$, $n>\iota$. Therefore $\widehat{d}_\iota $ is independent of $d_n$, $n>\iota$, implying $\widehat{d}_\iota = 0$ for $ \iota<j$. \qed \end{pf}
Proof of Proposition~\ref{thm:mainC4}: Recall \eqref{eq:basisFunctionsGenericG}. We first observe that the first summation in \eqref{eq:defgljS} follows directly from \eqref{eq:Mj}. It remains to prove that the only nonzero coefficients $d_{\Gamma^{(s)};m,n}^{(\tau)}$ might be the ones with $n \geq \max(0, i+j-m)$ and $n \leq \min(d-1,d-n_i+j-i+m)$.
The lower bound follows immediately by using $$ \left( N_j^{p,r}(\xi) \right) ' = \sum_{\iota=j-1}^j d_\iota \, N_\iota^{p-1,r-1}(\xi), \quad
\left( N_j^{p,r}(\xi) \right) '' = \sum_{\iota=j-2}^j c_\iota \, N_\iota^{p-2,r-2}(\xi),
$$
and by Lemma~\ref{lem:knotInsertion} and Lemma~\ref{lem:multiplication}. The upper bound can be shown by first considering the function
$$
\widehat{g}^{(\tau)}_{\Gamma^{(s)};i,j}(\xi_1^{(\tau)},\xi_2^{(\tau)}) = g^{(\tau)}_{\Gamma^{(s)};i,n_i-1-j} (\xi_1^{(\tau)},1-\xi_2^{(\tau)}), $$ which possesses again the lower bound $n \geq \max(0, i+j-m)$ for possible nonzero coefficients $\widehat{d}_{\Gamma^{(s)};m,n}^{(\tau)}$. This directly implies the upper bound $n \leq \min(d-1,d-n_i+j-i+m)$ for possible nonzero coefficients $d_{\Gamma^{(s)};m,n}^{(\tau)}$ of the function~$g^{(\tau)}_{\Gamma^{(s)};i,j}$. \qed
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\begin{center}
{\Large\bf Relations among spheroidal\\ and spherical harmonics}
Raybel Garc{\'i}a-Ancona\\ Jo\~{a}o Morais\\ R. Michael Porter
\today \end{center}
\begin{abstract}
A \textit{contragenic} function in a domain $\Omega\subseteq\mathbb{R}^3$ is
a reduced-quaternion-valued (i.e\ the last coordinate function is
zero) harmonic function, which is orthogonal in
$L_2(\Omega)$ to all monogenic functions and their conjugates. The
notion of contragenicity depends on the domain and thus is not a
local property, in contrast to harmonicity and monogenicity. For
spheroidal domains of arbitrary eccentricity, we relate standard
orthogonal bases of harmonic and contragenic functions for one
domain to another via computational formulas. This permits us to
show that there exist nontrivial contragenic functions common to the
spheroids of all eccentricities. \end{abstract}
\noindent \textbf{Keywords:} spherical harmonics, spheroidal harmonics, quaternionic analysis, monogenic function, contragenic function.
\noindent \textbf{Classification:} 30G35; 33D50
\section{Introduction}
In certain physical problems in nonspherical domains, it has been found convenient to replace the classical solid spherical harmonics with harmonic functions better adapted to the domain in question. For example, spheroidal harmonics are used in \cite{Hot} for modeling potential fields around the surface of the earth.
A systematic analysis of harmonic functions on spheroidal domains was initiated by Szeg\"o \cite{Szego1935}, followed by Garabedian \cite{Garabedian} who produced orthogonal bases with respect to certain natural inner products associated to prolate and oblate spheroids, among them the $L^2$-Hilbert space structures on the interior and on the boundary of the spheroid. Some aspects of the generation of harmonic functions which are orthogonal in the region exterior to a prolate spheroid were considered in \cite{MoraisNguyenKou2016} and generalized recently in \cite{Morais_Habilitation2018}.
The main question which interests us is to relate systems of harmonic functions associated with the spheroid $\Omega_\mu$ (defined in \eqref{eq:Omegamu} below) to those associated with the unit ball $\Omega_0$.
Our starting point is a fundamental formula for spheroidal harmonics which was worked out in the short but beautiful paper \cite{BBS} and is discussed thoroughly in Chapter 22 of the monumental text \cite{Hot}. In classical books such as \cite{Hobson1931,MorseFeshbach1953,NikiforovUvarov1988}, these expansions in terms of these bases are used separately without specifying relations between them.
We complete the above formulas by relating different systems of harmonic functions associated with spheroids of different eccentricity. While the manipulation of the coefficients is essentially algebraic, it must be borne in mind that we are dealing with continuously varying families of function spaces which are determined by integration over varying domains.
This study is then extended to include the contragenic functions, which are those harmonic functions orthogonal to both the monogenic functions and the antimonogenic functions in the domain under consideration. In \cite{GMP} a short table of contragenic polynomials was provided, which included some which did not depend on the parameter describing the eccentricity of the spheroid. Such polynomials are thus contragenic for all spheroids. Our main result, Theorem \ref{th:intersection}, describes the intersection of the spaces of contragenic functions.
\section{Background on spheroidal harmonics}
As a preliminary to the discussion of monogenic and contragenic functions on spheroids, we establish the basic facts for harmonics in this and the next section. Consider the family of coaxial spheroidal domains $\Omega_\mu$, scaled so that the major axis is of length 2: \begin{equation}\label{eq:Omegamu}
\Omega_\mu =
\{x \in \mathbb{R}^3 |\ x_0^2 + \frac{x_1^2 + x_2^2}{e^{2\nu}} < 1 \}, \end{equation} where $\nu\in\mathbb{R}$ and where following the notation in \cite{GMP} the parameter $\mu=(1-e^{2\nu})^{1/2}$ will be useful in later formulas. The equations relating the Cartesian coordinates of a point $x=(x_0,x_1,x_2)$ in $\Omega_\mu$ to spheroidal coordinates $(u,v,\phi)$ are \begin{equation}\label{eq:prolatecoords}
x_0 = \mu \cos u \cosh v, \ x_1 = \mu \sin u \sinh v \cos \phi, \
x_2 = \mu \sin u \sinh v \sin \phi, \end{equation} where in the case of the prolate spheroid ($\nu<0$) the coordinates range over $u\in[0,\pi]$, $v\in[0,\operatorname{arctanh} e^\nu]$, $\phi\in[0,2\pi)$ and the eccentricity is $0<\mu<1$, while for the oblate spheroid ($\nu>0$) we have $u\,\in\,[0,\pi]$ and $v\in[0,\operatorname{arctanh} e^\nu]$, $\phi\in[0,2\pi)$ and $\mu$ is imaginary, $\mu/i>0$. The spheroids reduce to the unit ball $\Omega_0$ for $\nu=0$. In many other treatments of spheroidal functions, which discuss the two (confocal) families separately, the ball is not represented. See \cite{GMP} for a discussion of this question.
In terms of the coordinates \eqref{eq:prolatecoords}, the {\it solid
spheroidal harmonics} are \begin{equation} \label{eq:prolatepreharmonics}
\harorigsol{n,m}{\pm}[\mu](x) = \harorig{n,m}[\mu](u,v) \, \Phi_m^\pm(\phi), \end{equation} where \begin{equation}
\Phi_m^+(\phi)=\cos (m\phi), \quad \Phi_m^-(\phi)=\sin (m\phi) \end{equation} and for $\mu\not=0$, \begin{equation} \label{eq:prolatepreharmonicsq}
\harorig{n,m}[\mu](u,v) = \frac{(n-m)!}{2^n(1/2)_n}\mu^n
P_{n}^{m}(\cos u) P_{n}^{m}(\cosh v) . \end{equation}
Here $P_n^m$ are the associated Legendre functions of the first kind \cite[Ch. III]{Hobson1931} of degree $n$ and order $m$, and
the (rising) Pochhammer symbol is $(a)_n=a(a+1)\cdots(a+n-1)$ with $(a)_0=1$ by convention. To avoid repetition, we state once and for all that $\harorigsol{n,m}{-}[\mu]$ is only defined for $m\ge1$, i.e.\ $\harorigsol{n,0}{-}[\mu]$ is expressly excluded from all statements of theorems.
It was shown in \cite{GMP} that with the scale factor which has been included in \eqref{eq:prolatepreharmonicsq}, the $\harorigsol{n,m}{\pm}[\mu]$ are polynomials in $(x_0,x_1,x_2)$ which are normalized so that the limiting case $\mu\to0$ gives the classical {\it solid spherical harmonics}, \begin{equation*}
\harorigsol{n,m}{\pm}[0](x) = |x|^nP_n^m(x_0/|x|) \Phi_m^\pm(\phi). \end{equation*} It is known from \cite{Garabedian} that while the $\harorigsol{n,m}{\pm}[\mu]$ are mutually orthogonal with respect to the Dirichlet norm on $\Omega_\mu$, the closely related functions, which we will call the \textit{Garabedian spheroidal harmonics}, \begin{equation} \label{eq:prolateharmonics}
\hargarabsol{n,m}{\pm}[\mu](x) =
\frac{\partial}{\partial x_0} \harorigsol{n+1,m}{\pm}[\mu](x) \end{equation} form an orthogonal basis for $\mathcal{H}_2(\Omega_\mu)$, the linear subspace of real-valued harmonic functions in $L_2(\Omega_\mu)$. This property makes the $\hargarabsol{n,m}{\pm}[\mu]$ of greater interest for many considerations. The corresponding boundary Garabedian harmonics $\hargarab{n,m}[\mu]$ in $\Omega_\mu$ are characterized by the relation \begin{equation} \label{eq:Vhat}
\hargarabsol{n,m}{\pm}[\mu](x) = \hargarab{n,m}[\mu](u,v) \,\Phi_m^\pm(\phi). \end{equation} We recall \cite{MoraisGuerlebeck2012} that for spherical harmonics, there is a formula analogous to Appell differentiation of monomials, \begin{align} \label{eq:V[0]}
\frac{\partial}{\partial x_0} \harorigsol{n+1,m}{\pm}[0](x)
= (n+m+1) \harorigsol{n,m}{\pm}[0](x). \end{align} However, $\hargarabsol{n,m}{\pm}[\mu]$ is not so simply related to $\harorigsol{n,m}{\pm}[\mu]$ for $\mu\not=0$, as was explained in \cite{Morais4}. We examine such relations in the next section.
\section{Conversions among orthogonal spheroidal harmonics and spherical harmonics}
\subsection{Garabedian harmonics expressed by classical harmonics}
As mentioned in the Introduction, it is of interest to express the orthogonal basis of harmonic functions for one spheroid $\Omega_\mu$ in terms of those for another spheroid. It is natural to use the unit ball $\Omega_0$ as a point of reference, which will be the case in the first results. We begin the calculation of the coefficients for the relationships among the various classes of harmonic functions by presenting various known formulas in a uniform manner. For $n\ge0$, consider the rational constants \begin{equation} \label{eq:cUU} \cUU{n,m,k} = \frac{ (1/2)_{n-k}\, (n+m-2k+1)_{2k}}{ (-4)^k (1/2)_{n}\, k! } \end{equation} for $0\le m\le n$, $0\le2k\le n$, and let $\cUU{n,m,k}=0$ otherwise. In the present notation, the main result of \cite{BBS} may be expressed as follows (i.e.\ the factor $\cscale{m,n} = (n-m)!/(2n-1)!!$ has been incorporated into \eqref{eq:cUU}). \begin{prop}[\cite{BBS}] \label{prop:BBS} Let $n\ge0$ and $0\le m\le n$. Then \begin{equation*}
\harorig{n,m}[\mu] = \sum_{0\le2k\le n-m} \cUU{n,m,k} \mu^{2k}\,
\harorig{n-2k,m}[0]. \end{equation*} \end{prop}
An important characteristic of this relation is that the same coefficients $\cUU{n,m,k}$ work for the ``$+$'' and ``$-$'' cases (cosines and sines) and, strikingly, for all values of $\mu$. By \eqref{eq:prolatepreharmonics}, an equivalent form of expressing Proposition \ref{prop:BBS} is \begin{equation} \label{eq:UqfromUq}
\harorigsol{n,m}{\pm}[\mu] = \sum_{0\le2k\le n-m}
\cUU{n,m,k} \mu^{2k}\, \harorigsol{n-2k,m}{\pm}[0]. \end{equation}
Since $\partial/\partial x_0$ in \eqref{eq:prolateharmonics} is a linear operator, \eqref{eq:UqfromUq} gives automatically the corresponding result for the Garabedian harmonics,
\begin{align} \label{eq:VV}
\hargarabsol{n,m}{\pm}[\mu] &=
\sum_{0\le2k\le n-m+1} \cVV{n,m,k} \mu^{2k} \,\hargarabsol{n-2k,m}{\pm}[0], \end{align} where $\cVV{n,m,k}=\cUU{n+1,m,k}$. This in turn gives via \eqref {eq:V[0]} the following expression in terms of the spherical harmonics: \begin{coro} \label{coro:VfromU} Let $n\ge0$ and $0\le m\le n$. Then \begin{align*}
\hargarab{n,m}[\mu]
&= \sum_{0\le2k\le n-m+1} \cUV{n,m,k} \mu^{2k}\, \harorig{n-2k,m}[0] , \end{align*} where \[\cUV{n,m,k}=(n+m-2k+1)\cVV{n,m,k}. \]
\end{coro}
The coefficients \begin{align*}
\cVUmu{n,m,k}
&= \frac{ (n+m+1)! \,(1/2)_{n-2k+1}}{4^k(n+m-2k)!(1/2)_{n+1}} \end{align*}
give a similar expression for the Garabedian basic harmonics $\hargarabsol{n,m}{\pm}[\mu]$ in terms of the standard harmonics $\harorigsol{n,m}{\pm}[\mu]$ for the same spheroid, rather than in terms of $\harorig{n,m}[0]$:
\begin{theo} [\cite{Morais4}] \label{th:VfromU} Let $n\ge0$ and $0\le m\le n$. Then \begin{equation*} \displaystyle \hargarab{n,m}[\mu] = \sum_{0\le 2k\le n-m}
\cVUmu{n,m,k} \mu^{2k}\, \harorig{n-2k,m}[\mu]. \end{equation*} \end{theo}
In \cite{BBS} the inverse relation of \eqref{eq:UqfromUq} was also derived, expressing $\harorigsol{n,m}{\pm}[0]$ in terms of $\harorigsol{n,m}{\pm}[\mu]$, via \begin{equation} \label{eq:UbacktoU}
\harorig{n,m}[0] = \sum_{0\le k\le n-m}
\cUU{n,m,k}^0 \mu^{2k} \, \harorig{n-2k,m}[\mu], \end{equation} where the coefficients can be written as \begin{equation} \label{eq:UbacktoUcoef}
\cUU{n,m,k}^0 =
\frac{ 4^{n-2k}(2n-4k+1)(n-k)!(m+n)!(1/2)_{n-2k}}
{ k!(2n-2k+1)!(n+m-2k)!}, \end{equation} again independent of $\mu$. In consequence, applying the operator $\partial/\partial x_0$ and using \eqref{eq:V[0]}, we have the following result. \begin{prop}\label{prop:UfromV} Let $n\geq0$ and $0\leq m\leq n$. Then \[ \harorig{n,m}[0] =
\sum_{0\leq2k\leq n-m}\cUV{n,m,k}^0\mu^{2k} \,\hargarab{n-2k,m}[\mu], \] where \begin{equation*}
\cUV{n,m,k}^0 = \dfrac{\cUU{n+1,m,k}^0}{n+m+1}. \end{equation*} \end{prop}
The inverse relation for Theorem \ref{th:VfromU} is a much simpler formula, given as follows:
\begin{coro} [\cite{Morais4}] For $n\geq0$ and $0\leq m \leq n$, \[ \harorig{n,m}[\mu] = \frac{1}{n+m+1} \hargarab{n,m}[\mu]
+ \frac{n+m}{4n^2-1} \mu^2 \,\hargarab{n-2,m}[\mu]. \] \end{coro} This uses the convention $\hargarab{n-2,m}[\mu]=0$ when $m>n$; i.e. \begin{align*}
\harorig{n,n-1}[\mu] &= \frac{1}{2n} \hargarab{n,n-1}[\mu],\\
\harorig{n,n}[\mu] &= \frac{1}{2n+1} \hargarab{n,n}[\mu]. \end{align*}
\subsection{Conversion among Garabedian harmonics}
The preceding subsection does not include the inverse relation of \eqref{eq:VV} of the form \begin{equation} \label{eq:VVinv}
\hargarab{n,m}[0] = \sum_{0\leq2k\leq n-m} \cVV{n,m,k}^0 \mu^{2k}\hargarab{n-2k,m}[\mu]. \end{equation} Instead of deriving it directly, we verify first the following remarkable conversion formula, which relates the spheroidal harmonics associated with $\Omega_{\mu}$ to those associated with any other $\Omega_{\widetilde{\mu}}$. Write \begin{equation*} \cmumut{n,m,k}=
\dfrac{(n+m+1)!(1/2)_{n-2k+2}}{4^{k}k!(n+m-2k+1)! (1/2)_{n-k+2}} \end{equation*} when $0 \leq 2k \leq n-m+2$, otherwise $\cmumut{n,m,k}=0$.
\begin{theo}\label{th:cVVmu}
Let $n\geq0$, $0\leq m\leq n$, and let
$\mu,\widetilde{\mu} \in [0,1) \cup i\mathbb{R}^{+}$ such that $\mu\neq0$. The
coefficients $\cVVmu{n,m,k}[\widetilde{\mu},\mu]$ in the
relation \[ \hargarab{n,m}[\widetilde{\mu}] = \sum_{0\leq2k\leq n-m}
\cVVmu{n,m,k}[\widetilde{\mu},\mu]\,\hargarab{n-2k,m}[\mu] \] are given by \[ \cVVmu{n,m,k}[\widetilde{\mu},\mu] = {}_2F_1(-k,-n+k-3/2;-n-1/2;(\widetilde{\mu}/\mu)^2)
\, \cmumut{n,m,k}\,\mu^{2k}, \] with $_2F_1$ denoting the classical Gaussian hypergeometric function. \end{theo}
\begin{proof}
We begin by replacing $\mu$ with $\widetilde{\mu}$ in Corollary
\ref{coro:VfromU} and substituting the terms on the right-hand side
according to Proposition \ref{prop:UfromV}. By linear independence
of the harmonic basis elements, it follows that \begin{equation}\label{eq:sumw}
\cVVmu{n,m,k}[\widetilde{\mu},\mu] = \mu^{2k}\sum_{l=0}^{k}\cUV{n,m,l}\cUV{n-2l,m,k-l}^0
\left(\frac{\widetilde{\mu}}{\mu}\right)^{2l} \end{equation} in which we note that all terms are real valued. Using reductions such as $(2n-4k+3)(1/2)_{n-2k+1}=2(1/2)_{n-2k+2}$ and recalling $0\le l\le k$, one easily sees that \begin{align*}
\cUV{n,m,l} &= \frac{ (1/2)_{n-l+1}(n+m-2l+1)_{2l+1}} {(-4^l)l!(1/2)_{n+1}}, \\
\cUV{n-2l,m,k-l}^0 &= \frac{2\cdot4^{n-2k+1}(n+m-2l)!(n-k-l+1)!(1/2)_{n-2k+2} }
{ (k-l)! (2n-2k-2l+3)! (n+m-2k+1)! }. \end{align*} Therefore the product can be expressed as \[ \cUV{n,m,l} \cUV{n-2l,m,k-l}^0 = \cmumut{n,m,k} \chypgeom{n,k,l} \] where \begin{align*}
\chypgeom{n,k,l} &= \frac{ 2\cdot 4^{n-2k+1}(n+m+1)!(n-k-l+1)!(1/2)_{n-2k+2} }
{ (-4^l)l!(k-l)! (2n-2k-2l+3)! (n+m-2k+1)! } \\
&= \frac{ (-k)_l(-n+k-3/2)_l }{ l!(-n-1/2)_l} \end{align*} is the coefficient in the polynomial $ {}_2F_1(-k,-n+k-3/2;-n-1/2;(\widetilde{\mu}/\mu)^2) =
\sum_{l=0}^k \chypgeom{n,k,l} (\widetilde{\mu}/\mu)^{2l}$. \end{proof}
\begin{coro}\label{coro:proplim} For each $n\geq0$, $0\leq m\leq n$, the limits \begin{equation*}
\lim_{\widetilde{\mu}\rightarrow0} \cVVmu{n,m,k}[\widetilde{\mu},\mu], \quad
\lim_{\mu\rightarrow0} \cVVmu{n,m,k}[\widetilde{\mu},\mu] \end{equation*} exist and are given, respectively, by \begin{equation*}
\cVVmu{n,m,k}[0,\mu] = (n+m+1)\cUV{n,m,k}^0\mu^{2k}, \quad
\cVVmu{n,m,k}[\widetilde{\mu},0] = \dfrac{\cUV{n,m,k}}{n+m-2k+1}\widetilde{\mu}^{2k}. \end{equation*} \end{coro}
\begin{proof} We may write \eqref{eq:sumw} as \begin{align*}
\cVVmu{n,m,k}[\widetilde{\mu},\mu] &=
\sum_{l=1}^{k-1} \cUV{n,m,l} \cUV{n-2l,m,k-l}^0 \, \mu^{2(k-l)}\widetilde{\mu}^{2l} \\
& \quad\ \ +\cUV{n,m,k}\cUV{n-2k,m,0}^0\widetilde{\mu}^{2k}
+ \cUV{n,m,0}\cUV{n,m,k}^0\mu^{2k} \end{align*} and then simply take $\mu=0$ or $\widetilde{\mu}=0$ to obtain the desired limit. \end{proof}
Referring to \eqref{eq:VVinv}, we have \[ \cVV{m,n,k}^0= \frac{\cUV{n,m,k}}{(n+m-2k+1)}. \]
\section{Application to orthogonal monogenic and contragenic functions}
The standard bases for spheroidal harmonics have their counterparts for the spaces of orthogonal monogenic polynomials taking values in $\mathbb{R}^3$. Monogenic functions are defined by considering $\mathbb{R}^3$ as the real linear subspace of the quaternions $\mathbb{H} = \{\sum_{i=0}^3 x_ie_i\colon\ x_i \in \mathbb{R}\}$ for which the last coordinate $x_3$ vanishes. (Quaternionic multiplication is defined, as usual, so that $e_1^2 = e_2^2 = e_3^2 = -1$ and $e_1 e_2 = e_3 = - e_2 e_1$, $e_2 e_3 = e_1 = - e_3 e_2$, $e_3 e_1 = e_2 = - e_1 e_3$.) For background on quaternionic analysis in $\mathbb{R}^3$, see \cite{Delanghe2007,Joao2009,
MoraisGuerlebeck2012,MoraisGuerlebeck22012,MoraisAG}. A function $f\colon\Omega_\mu\to\mathbb{R}^3$ is \textit{monogenic} when it is annihilated by the quaternionic differential operator $ \partial = \partial/\partial x_0+e_1\partial/\partial x_1+e_2\partial/\partial x_2$ acting from the left. The \textit{basic
spheroidal monogenic polynomials} are constructed \cite{Morais4,Morais5} as \begin{equation} \label{def:monog}
\monog{n,m}{\pm}[\mu] = \overline{\partial}(\harorigsol{n+1,m}{\pm}[\mu]), \end{equation} where $ \overline{\partial} = \partial/\partial x_0 - e_1\partial/\partial x_1- e_2\partial/\partial x_2$. This is analogous to the definition \eqref{eq:prolateharmonics} for harmonic polynomials. $\monog{n,m}{\pm}[\mu]$ is monogenic because $\partial\overline{\partial}$ is equal to the Laplacian operator. We continue with the convention that $m\ge1$ when the ``-'' sign appears in a superscript.
\begin{theo} [\cite{Morais4,Morais5}] \label{th:sphmonogformula}
For all $n\geq0$, the basic spheroidal monogenic polynomial
\eqref{def:monog} is equal to \begin{align*}
\monog{n,0}{+}[\mu] = \hargarabsol{n,0}{+}[\mu] -
\frac{1}{n+2} \big( \hargarabsol{n,1}{+}[\mu] e_1 +
\hargarabsol{n,1}{-}[\mu] e_2 \big) \end{align*} for $m=0$, and \begin{align*}
\monog{n,m}{\pm}[\mu] &=
\hargarabsol{n,m}{\pm}[\mu] + \Bigl[(n+m+1) \hargarabsol{n,m-1}{\pm}[\mu] -
\frac{1}{n+m+2} \hargarabsol{n,m+1}{\pm}[\mu] \Bigr]\frac{e_1}{2} \nonumber\\
& \phantom{\quad \hargarabsol{n,m}{\pm}[\mu]\ } \mp \Bigl[(n+m+1) \hargarabsol{n,m-1}{\mp}[\mu] +
\frac{1}{n+m+2} \hargarabsol{n,m+1}{\mp}[\mu] \Bigr] \frac{e_2}{2} \end{align*} for $1\leq m\leq n+1$.
The polynomials $\monog{n,m}{\pm}[\mu]$
are orthogonal in $L^2(\Omega_\mu)$, i.e.\ in the sense of the scalar
product defined by \[ \langle f,g\rangle_{[\mu]} = \int_{\Omega_\mu} {\rm Sc}(\overline{f}g)\,dV. \] \end{theo}
\subsection{Bases for monogenics in distinct spheroids}
Analogously to \eqref{eq:VV} and \eqref{eq:VVinv}, we now express $\monog{n,m}{\pm}[\mu]$ in terms of the spherical monogenics $\monog{n,m}{\pm}[0]$.
\begin{theo}\label{th:monogenics}
For $n\geq0$ and $0\leq m \leq n+1$, \begin{align*}
\monog{n,m}{\pm}[\mu] =& \sum_{0\le 2k\le n-m+1}
\cVV{n,m,k} \mu^{2k} \monog{n-2k,m}{\pm}[0],\\
\monog{n,m}{\pm}[0] =& \sum_{0\le 2k\le n-m+1}
\cVV{n,m,k}^0 \mu^{2k} \monog{n-2k,m}{\pm}[\mu],\\
\monog{n,m}{\pm}[\widetilde{\mu}] =& \sum_{0\leq2k\leq n-m+1}
\cVV{n,m,k}[\widetilde{\mu},\mu]\, \monog{n-2k,m}{\pm}[\mu], \end{align*} where $\cVV{n,m,k}$, $\cVV{n,m,k}^0$, and $\cVV{n,m,k}[\mu,\widetilde{\mu}]$ are
as in the previous section. \end{theo}
\begin{proof} Fix a value of $\mu$. Note that for given $n$, the
collections $\{\displaystyle\monog{k,m}{\pm}[0]\colon\ k\le n,\ 0\le m\le k\}$ and
$\{\displaystyle\monog{k,m}{\pm}[\mu]\colon\ k\le n,\ 0\le m\le k\}$ are bases for the same
linear space, namely the monogenic $\mathbb{R}^3$-valued polynomials in the
variables $(x_0,x_1,x_2)$ of degree $\le n$. Therefore there must
exist real coefficients $a^\pm_{k}$ such that
$\monog{n,m}{+}[\mu]=\sum_k\sum_m a^+_k\monog{n,k}{+}[0] +\sum_k\sum_m
a^-_k\monog{n,k}{-}[0]$.
By Theorem \ref{th:sphmonogformula}, the scalar part of this equation
expresses the spheroidal harmonics $\hargarabsol{n,m}{\pm}[\mu]$ as a
linear combination of the spherical harmonics
$\hargarabsol{k,m}{\pm}[0]$. By the uniqueness of the representation
\eqref{eq:VV} we have that $a^\pm_{k} = \cVV{n,m,k} \mu^{2k}$. The
second formula follows by the same reasoning, and then the relationship
between $\monog{n,m}{\pm}[\mu]$ and $\monog{n,m}{\pm}[\widetilde{\mu}]$ is a
consequence of the fact that by Theorem \ref{th:cVVmu} the matrix
$(\cVV{n,m,k}[\widetilde{\mu},\mu])_{n,k}$ is essentially the product of
$(\cVV{n,m,k}\widetilde{\mu}^{2k})_{n,k}$ and the inverse of
$(\cVV{n,m,k}^0\mu^{2k})_{n,k}$. \end{proof}
\subsubsection{Spheroidal ambigenic polynomials} \textit{Antimonogenic} functions (quaternionic conjugates of monogenics, i.e.\ annihilated by $\overline{\partial}$) are generally not studied independently, since their properties may be obtained by taking the conjugate of facts about monogenic functions. For example, the basic antimonogenic polynomials satisfy essentially the same relation as given in Theorem \ref{th:monogenics}, \[ \antimonog{n,m}{\pm}[\mu] =
\sum_{0\leq2k\leq n-m} \cVV{n,m,k}[\mu,\widetilde{\mu}]\, \antimonog{n-2k,m}{\pm}[\widetilde{\mu}]. \] However, the subspace of the $\mathbb{R}^3$-valued harmonic functions generated by the monogenic and antimonogenic functions together, that is, the \textit{ambigenic} functions \cite{Alvarez}, is of interest.
An ambigenic function is not represented uniquely as a sum of a monogenic and an antimonogenic function because one may add and subtract a \textit{monogenic constant}, that is, a function which is simultaneously monogenic and antimonogenic.
A collection of ambigenic polynomials denoted $\{\ambibasic{n,m}{\pm,\pm}[\mu]\}$ was constructed in \cite{GMP} and shown to be a basis of $2n(n+3)+3$ elements for the ambigenic polynomials of degree no greater than $n$, mutually orthogonal in $L^2(\Omega_\mu)$. For our purposes we will only need the particular ambigenic functions \begin{equation} \label{eq:defambi}
\ambig{n,m}{\pm}[\mu] = 2\operatorname{Vec} \monog{n,m}{\pm}[\mu]
= \monog{n,m}{\pm}[\mu] - \antimonog{n,m}{\pm}[\mu], \end{equation} where $q=\operatorname{Sc} q + \operatorname{Vec} q$ denotes the decomposition of a quaternionic quantity into its scalar and vector parts. It is simple to verify that for fixed $\mu$, the $\ambig{n,m}{\pm}[\mu]$ are linearly independent.
\subsection{Relations among contragenic functions for distinct spheroids}
The notion of contragenic harmonic functions was introduced in \cite{Alvarez}, arising from the previously unobserved fact that in contrast to $\mathbb{C}$-valued or $\mathbb{H}$-valued functions, there exist $\mathbb{R}^3$-valued harmonic functions which are not ambigenic. Thus a function is called \textit{contragenic} for a given domain $\Omega$ when it is orthogonal in $L^2(\Omega)$ to all monogenic and antimonogenic functions in $\Omega$. In contrast to monogenicity and antimonogenicity, this is not a local property and therefore cannot be characterized in general by direct application of any differential operator. It is of interest to have a basis for the contragenic functions, in order to express an arbitrary harmonic function in a calculable way as a sum of an ambigenic function and a contragenic function. In the following, we will write \begin{align*}
\mathcal{N}_*^{(n)}[\mu] =\ &
\{\parbox[t]{.6\textwidth}{polynomials
of degree $\le n$ in $x_0,x_1,x_1$ which are
orthogonal in $L_2(\Omega_\mu)$ to all ambigenic
functions in $\Omega_\mu\}$,} \end{align*} for $n\ge 1$ (nonzero constant harmonic functions are never contragenic, so we will have no use for $\mathcal{N}_*^{(0)}[\mu]=\{0\}$), and we have the successive orthogonal complements \[ \mathcal{N}^{(n)}[\mu] = \mathcal{N}_*^{(n)}[\mu] \ominus \mathcal{N}_*^{(n-1)}[\mu], \] which are composed of polynomials of degree precisely $n$. Thus $\mathcal{N}_*^{(n)}[\mu] =\bigoplus_{k=1}^n\mathcal{N}^{(k)}[\mu]$ and there is a Hilbert space orthogonal decomposition $\mathcal{N}_*[\mu] =\bigoplus_{k=1}^\infty\mathcal{N}^{(k)}[\mu]$ of the full collection of contragenic functions in $L^2(\Omega_\mu)$. The following explicit construction of a basis of the $\mathcal{N}^{(n)}[\mu]$, using as building blocks the scalar components of the monogenic functions, can be found in \cite{GMP}. Write \begin{align}
\normrat{n,0}[\mu]=& \ 1, \nonumber \\
\normrat{n,m}[\mu] =& \left( \frac{1}{(n+m+1)_2}
\frac{ \|\hargarabsol{n,m+1}{+}[\mu]\|_{[\mu]}}
{ \|\hargarabsol{n,m-1}{+}[\mu] \|_{[\mu]}}\right)^2 \label{eq:normrat} \end{align} for $1\le m\le n-1$, and $\normrat{n,m}[\mu]=0$ for $m\ge n$ since then $\hargarabsol{n,m}{\pm}[\mu]=0$ (this definition involves a slight modification of the notation in \cite{GMP}), where integration over the ellipsoid gives explicitly \begin{equation*}\label{eq:harqnorms}
\|\hargarabsol{n,m}{\pm}[\mu]\|_{[\mu]}^2 =
(1+\delta_{0,m})\normconst{n,m} \pi \mu^{2n+3}
\int_{1}^{\frac{1}{\mu}}P_{n}^{m}(t)P_{n+2}^{m}(t)\,dt. \end{equation*} Here $\delta_{m,m'}$ is the Kronecker symbol and \[ \normconst{n,m} =
\frac{ (n+m+1) (n+m+1)!(n-m+2)!} {2^{2n+1} (1/2)_{n+1}(1/2)_{n+2} } . \]
\begin{defi} \label{def:Basic_contragenics} For all $n\ge1$, the \textit{basic contragenic polynomials} $\contra{n,m}{\pm}[\mu]$ associated to $\Omega_\mu$ are
\begin{align*}
\contra{n,0}{+}[\mu] =& -\ambig{n,0}{+}[\mu] e_3 \end{align*} for $m=0$, and
\begin{align*}
\contra{n,m}{\pm}[\mu] =
\frac{1}{2}\big(
\mp(\normrat{n,m}[\mu]+1)\ambig{n,m}{\pm}[\mu]
+ (\normrat{n,m}[\mu]-1)\ambig{n,m}{\mp}[\mu] e_3 \big)
\end{align*} for $1\leq m\leq n-1$, where $\ambig{n,m}{\pm}[\mu]$ are defined by \eqref{eq:defambi}. \end{defi}
In \cite{GMP} it was shown that $\{\contra{n,m}{\pm}[\mu]\colon\ 0\le m< n-1\}$ is an orthonormal basis for $\mathcal{N}^{(n)}[\mu]$, and that the harmonic polynomials of degree $\le n$ in $\Omega_\mu$ decompose as orthogonal direct sums of the ambigenic and contragenic polynomials of degree $\le n$. With the further notation \begin{align*}
\angPsi{+,m}{\pm} &= \Phi_m^{\pm}(\phi)e_1 \pm
\Phi_m^{\mp}(\phi)e_2, \\
\angPsi{-,m}{\pm} &= \Phi_m^{\pm}(\phi)e_1 \mp
\Phi_m^{\mp}(\phi)e_2, \end{align*} which satisfy the obvious relations $\angPsi{+,m}{\pm}e_3=\pm\angPsi{+,m}{\mp}$, $\angPsi{-,m}{\pm}e_3=\mp\angPsi{-,m}{\mp}$, $e_1\hargarabsol{n,m}{\pm}[\mu]+e_2\hargarabsol{n,m}{\mp}[\mu]= \hargarab{n,m}\angPsi{\pm,m}{\pm}[\mu]$, $e_1\hargarabsol{n,m}{\pm}[\mu]-e_2\hargarabsol{n,m}{\mp}[\mu]= \hargarab{n,m}\angPsi{\mp,m}{\pm}[\mu]$ (where the $\hargarab{n,m}[\mu]$ are given by \eqref{eq:Vhat}), the definitions give us almost immediately that \begin{align}
\ambig{n,0}{+}[\mu] &=
\frac{-2}{n+2} \hargarab{n,1}[\mu] \angPsi{+,1}{+},\nonumber\\
\ambig{n,m}{\pm}[\mu] &=
(n+m+1)\hargarab{n,m-1}[\mu] \angPsi{-,m-1}{\pm} \nonumber\\
&\ \ -\dfrac{1}{n+m+2}\hargarab{n,m+1}[\mu] \angPsi{+,m+1}{\pm}, \label{eq:ambiformula} \end{align} \begin{align}
\contra{n,0}{+}[\mu] &=
\frac{2}{n+2} \hargarab{n,1}[\mu]\angPsi{+,1}{-},\nonumber\\
\contra{n,m}{\pm}[\mu] &= (n+m+1)\normrat{n,m}[\mu]
\hargarab{n,m-1}[\mu] \angPsi{-,m-1}{\mp} \nonumber \\
&\ \ + \dfrac{1}{n+m+2}\hargarab{n,m+1}[\mu]
\angPsi{+,m+1}{\mp}, \label{eq:contraformula} \end{align} where $1 \le m \leq n-1$.
Adding and subtracting instances of \eqref{eq:ambiformula} and \eqref{eq:contraformula} gives by cancellation decompositions of the harmonic polynomials $\hargarab{n,m}\angPsi{+,m}{\pm}$ and $\hargarab{n,m}\angPsi{-,m}{\pm}$ as the sum of a contragenic and an ambigenic:
\begin{lema}\label{lem:VZA} Let $n\geq1$ and $1\leq m \leq n+1$. Then
\begin{align*}
\hargarab{n,m-1}[\mu]\angPsi{-,m-1}{\pm} =
\dfrac{1}{(n+m+1)(\normrat{n,m}[\mu]+1)}\big(&\contra{n,m}{\mp}[\mu]
+ \ambig{n,m}{\pm}[\mu] \big), \end{align*} and \begin{align*}
\hargarab{n,m+1}[\mu]\angPsi{+,m+1}{\pm} =
\dfrac{n+m+2}{ \normrat{n,m}[\mu]+1}\big(&\contra{n,m}{\mp}[\mu]
- \normrat{n,m}[\mu]\ambig{n,m}{\pm}[\mu] \big) . \end{align*} \end{lema}
The definition of contragenic function does not imply that an $L^2$-function which belongs to the space $\mathcal{N}_*^{(n)}[\widetilde{\mu}]$ should also be in $\mathcal{N}_*^{(n)}[\mu]$ when $\widetilde{\mu}\neq\mu$, because the notion of orthogonality is different for different spheroids. In other words, we may not expect a formula like ``$\contra{n,m}{\pm}[\widetilde{\mu}]=\sum z_{n,m,k}[\widetilde{\mu},\mu]\contra{n-2k,m}{\pm}[\mu]$.'' The following result will enable us to give many examples for which $\contra{n,m}{\pm}[\widetilde{\mu}]\not\in\mathcal{N}_*^{(n)}[\mu]$ for $m\geq1$. However, it also shows that the intersection of all of the $\mathcal{N}_*^{(n)}[\mu]$ is nontrivial, giving what may be called {\it universal contragenic
functions} in the context of spheroids.
We will use the coefficients \begin{align}
\cZZmu{n,0,k}{C}[\widetilde{\mu},\mu] =& \
\dfrac{n-2k+2}{n+2}\cVVmu{n,1,k}[\widetilde{\mu},\mu], \nonumber\\
\cZZmu{n,m,k}{C}[\widetilde{\mu},\mu] =&\ \left\{\begin{array}{ll}
\dfrac{\normrat{n,m}[\widetilde{\mu}]+1}{\normrat{n-2k,m}[\mu]+1}
\cVVmu{n,m,k}[\widetilde{\mu},\mu],\ \quad & 0\le2k\le n-m-1,\\[2ex]
\dfrac{\normrat{n,m}[\widetilde{\mu}]}{\normrat{n-2k,m}[\mu]+1}
\cVVmu{n,m,k}[\widetilde{\mu},\mu], & n-m\le2k\le n-m+1;
\end{array}\right. \nonumber \\
\cZZmu{n,m,k}{A}[\widetilde{\mu},\mu] =&\ \left\{\begin{array}{ll}
\dfrac{\normrat{n,m}[\widetilde{\mu}]-\normrat{n,m}[\mu]}{\normrat{n-2k,m}[\mu]+1}
\cVVmu{n,m,k}[\widetilde{\mu},\mu], & 0\le2k\le n-m-1,\\[2ex]
\dfrac{\normrat{n,m}[\widetilde{\mu}] }{\normrat{n-2k,m}[\mu]+1}
\cVVmu{n,m,k}[\widetilde{\mu},\mu], & n-m\le2k\le n-m+1;
\end{array}\right. \label{eq:CZZmu} \end{align} ($1 \le m \le n-1$) to express the decomposition of contragenics for one spheroid in terms of contragenics and ambigenics of any other.
\begin{prop} \label{prop:contragenicrelations}
Let $n\geq1$. Then \begin{align*}
\contra{n,0}{+}[\widetilde{\mu}] &= \sum_{0\leq 2k\leq n-1}
\cZZmu{n,k}{C}[\widetilde{\mu},\mu]\contra{n-2k,0}{}[\mu]; \end{align*}
and for $1\leq m\leq n-1$, \begin{equation*}
\contra{n,m}{\pm}[\widetilde{\mu}]=\sum_{0\leq 2k\leq n-m+1}
\big( \cZZmu{n,m,k}{C}[\widetilde{\mu},\mu]\contra{n-2k,m}{\pm}[\mu] +
\cZZmu{n,m,k}{A}[\widetilde{\mu},\mu] \ambig{n-2k,m}{\pm}[\mu] \big). \end{equation*}
\end{prop}
\begin{proof}
Apply Theorem \ref{th:cVVmu} to the first formula of
\eqref{eq:contraformula} with $\widetilde{\mu}$ in place of $\mu$ to obtain that \begin{align*}
\contra{n,0}{+}[\widetilde{\mu}] = \frac{2}{n+2} \sum_{0\leq 2k\leq n-1}
\cVVmu{n,1,k}[\widetilde{\mu},\mu]\hargarab{n-2k,1}[\mu] \angPsi{+,1}{-} , \end{align*} which after another application of \eqref{eq:contraformula} reduces to the first statement. In the same way, for $m\ge1$, \begin{align}
\contra{n,m}{\pm}[\widetilde{\mu}] =&
\ (n+m+1)\normrat{n,m}[\widetilde{\mu}] \sum_{0\leq 2k\leq n-m+1}
\cVVmu{n,m-1,k}[\widetilde{\mu},\mu]\hargarab{n-2k,m-1}[\mu]
\angPsi{-,m-1}{\pm} \nonumber\\
& \ + \dfrac{1}{n+m+2} \sum_{0\leq 2k\leq n-m-1}
\cVVmu{n,m+1,k}[\widetilde{\mu},\mu]\hargarab{n-2k,m+1}[\mu]
\angPsi{+,m+1}{\pm}. \label{eq:Zsum} \end{align}
We observe from the definitions leading to Proposition \ref{prop:UfromV} that \begin{align*}
\cUV{n,m-1,l}\,\cUV{n-2l,m-1,k-l}^0
= \dfrac{n+m-2k+1}{n+m+1}\cUV{n,m,l}\,\cUV{n-2l,m,k-l}^0, \end{align*} so \eqref{eq:sumw} tells us that \begin{align*} \dfrac{n+m+1}{n+m-2k+1}\cVVmu{n,m-1,k}[\widetilde{\mu},\mu] &= \cVVmu{n,m,k}[\widetilde{\mu},\mu] \\ &= \dfrac{n+m-2k+2}{n+m+2}\cVVmu{n,m+1,k}[\widetilde{\mu},\mu]. \end{align*} From this and Lemma \ref{lem:VZA} we have that \begin{align*}
(n+m+1) \cVVmu{n,m-1,k}[\widetilde{\mu},\mu] \hargarab{n-2k,m-1}[\mu]
& \angPsi{-,m-1}{\pm} \\
= \frac{1}{\normrat{n-2k,m}[\mu]+1} \cVV{n,m,k}[\widetilde{\mu},\mu]
(\contra{n-2k,m}{\mp}&[\mu]+\ambig{n-2k,m}{\pm}[\mu] ), \end{align*} and \begin{align*}
\frac{1}{n+m+2}\cVVmu{n,m+1,k}[\widetilde{\mu},\mu] \hargarab{n-2k,m+1}[\mu]
& \angPsi{+,m+1}{\pm} \\
= \frac{1}{\normrat{n-2k,m}[\mu]+1} \cVV{n,m,k}[\widetilde{\mu},\mu]
(\contra{n-2k,m}{\mp}& [\mu]- \normrat{n-2k,m}[\mu]\ambig{n-2k,m}{\pm}[\mu] ). \end{align*} Inserting these two relations into the respective sums of \eqref{eq:Zsum} gives the desired result. \end{proof}
Proposition \ref{prop:contragenicrelations} provides us with some information about the intersection of the spaces of contragenic functions up to degree $n$.
\begin{theo}\label{th:intersection}
Let $n\geq1$. The following statements hold: \begin{enumerate} \item[(i)] $\contra{n,0}{+}[\mu]\in\mathcal{N}_*^{(n)}[0]$ for all $\mu$;
\item[(ii)]
$\contra{n,m}{\pm}[\mu] \notin N_*^{(n)}[0]$
when $\mu\neq0$ and $1\le m\le n-1$. \end{enumerate} \end{theo}
\begin{proof} The first statement is an immediate consequence of the
first formula of Proposition \ref{prop:contragenicrelations}.
Now consider a basic element $\contra{n,m}{\pm}[\mu]$ of
$\mathcal{N}_*^{(n)}[\mu]$, with $\mu\not=0$ and $1\leq m\leq n-1$. A
particular instance of the second formula of Proposition
\ref{prop:contragenicrelations} is \[ \contra{n,m}{\pm}[\mu]=\sum_{0\leq 2k\leq n-m+1}
\big( \cZZmu{n,m,k}{C}[\mu,0] \contra{n-2k,m}{\pm}[0] +
\cZZmu{n,m,k}{A}[\mu,0]\ambig{n-2k,m}{\pm}[0] \big). \] Suppose that $\contra{n,m}{\pm}[\mu]\in\mathcal{N}_*^{(n)}[0]$. Then since the right hand side is orthogonal to all $\Omega_0$-ambigenics, \[ \sum_{0\leq 2k\leq n-m+1}
\cZZmu{n,m,k}{A}[\mu,0]\ambig{n-2k,m}{\pm}[0]=0, \] and so by the linear independence, $\cZZmu{n,m,k}{A}[\mu,0]=0$ for all $k$. The case in \eqref{eq:CZZmu} where $2k$ is $n-m$ or $n-m+1$ tells us that $\normrat{n,m}[\mu]=0$, which is manifestly false by \eqref{eq:normrat}. Consequently, $\contra{n,m}{\pm}[\mu]\not\in\mathcal{N}_*^{(n)}[0]$ as claimed. \end{proof}
Note that Theorem \ref{th:intersection} does not assert that $\contra{n,0}{+}[\mu]$ lies in the top-level slice $\mathcal{N}^{(n)}[0]$ of $\mathcal{N}_*^{(n)}[0]$.
\begin{coro} Let $n\geq1$. Then \[ \dim \bigcap_{\mu\in[0,1)\cup\mathbb{R}^+}\mathcal{N}_*^{(n)}[\mu] \ge n. \] \end{coro}
\begin{proof}
The result is an immediate consequence of the fact that Theorem
\ref{th:intersection} is applicable to arbitrary $\mu$, so the intersection contains a fixed $n$-dimensional subspace of
$\mathcal{N}_*^{(n)}[0]$. \end{proof}
It also follows from Theorem \ref{th:intersection} that the common intersection $\mathcal{N}_0=\bigcap\mathcal{N}_*[\mu]$ of the full spaces of contragenic functions on spheroids is infinite dimensional, containing all of the contragenic polynomials $\contra{n,m}{+}[\mu]$ for which $m=0$. It seems likely that these contragenic polynomials have special characteristics because of their simpler structure, cf.\ \eqref{eq:contraformula}. This phenomenon is not yet fully understood. Further questions relating to the exact relations among the spaces $\mathcal{N}_*^{(n)}[\mu]$ still remain open. If the method of the proof of Theorem \ref{th:intersection} is applied to linear combinations of the $\contra{n,m}{\pm}[\mu]$ instead of just to these generators individually, transcendental equations related to \eqref{eq:normrat} appear. It is not yet known how the angles between the orthogonal complements of the mode-$0$ subspace $\mathcal{N}_0^{n}[0]$ in $\mathcal{N}_*^{(n)}[\mu]$, or of their union $\mathcal{N}_0[0]$ in $\mathcal{N}[\mu]$, vary with $\mu$.
\end{document}
|
arXiv
|
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\date{} \title{Multi-part cross-intersecting families}
\begin{abstract}
Let $\mathcal{A}\subseteq{[n]\choose a}$ and $\mathcal{B}\subseteq{[n]\choose b}$ be two families of subsets of $[n]$, we say $\mathcal{A}$ and $\mathcal{B}$ are cross-intersecting if $A\cap B\neq \emptyset$ for all $A\in\mathcal{A}$, $B\in\mathcal{B}$. In this paper, we study cross-intersecting families in the multi-part setting. By characterizing the independent sets of vertex-transitive graphs and their direct products, we determine the sizes and structures of maximum-sized multi-part cross-intersecting families. This generalizes the results of Hilton's~\cite{H77} and Frankl--Tohushige's~\cite{FT92} on cross-intersecting families in the single-part setting.
\noindent{\it Keywords:} extremal set theory, cross-intersecting families, Erd\H{o}s--Ko--Rado theorem, vertex-transitive graph.
\noindent {{\it AMS subject classifications\/}: 05D05.}
\end{abstract}
\section{Introduction}
Let $[n]$ be the standard $n$-element set. For an integer $0\leq k\leq n$, denote ${[n]\choose k}$ as the family of all $k$-element subsets of $[n]$. A family $\mathcal{F}$ is said to be $intersecting$ if $A\cap B\neq \emptyset$, for any $A,B\in\mathcal{F}$. The celebrated Erd\H{o}s--Ko--Rado theorem~\cite{EKR} says that if $\mathcal{F}\subseteq {[n]\choose k}$ is an intersecting family with $1\leq k\leq \frac{n}{2}$, then \begin{equation*}
|\mathcal{F}|\leq {{n-1}\choose {k-1}}, \end{equation*} and if $n>2k$, the equality holds if and only if every subset in $\mathcal{F}$ contains a fixed element.
Because of its fundamental status in extremal set theory, the Erd\H{o}s--Ko--Rado theorem has numerous extensions in different ways. One of the directions is to study $cross$-$t$-$intersecting$ families: Denote $2^{[n]}$ as the $power~set$ of $[n]$, let $\mathcal{A}_i\subseteq 2^{[n]}$ for each $1\leq i\leq m$, $\mathcal{A}_1,\mathcal{A}_2,\ldots,\mathcal{A}_m$ are said to be cross-t-intersecting, if $|A\cap B|\geq t$ for any $A\in\mathcal{A}_i$ and $B\in\mathcal{A}_j$, $i\neq j$. Especially, we say $\mathcal{A}_1,\mathcal{A}_2,\ldots,\mathcal{A}_m$ are cross-intersecting if $t=1$.
Hilton~\cite{H77} investigated the cross-intersecting families in ${[n]\choose k}$, and proved the following inequality: \begin{thm}\emph{(\cite{H77})}\label{h77} Let $\mathcal{A}_1,\mathcal{A}_2,\ldots, \mathcal{A}_m$ be cross-intersecting families in ${[n]\choose k}$ with $n\geq 2k$. Then \begin{flalign*}
\sum\limits_{i=1}^{m}|\mathcal{A}_i|\leq\left\{\begin{array}{ll}{n\choose k},&~~~\text{if}~m\leq\frac{n}{k};\\ m\cdot{{n-1}\choose {k-1}},&~~~\text{if}~m\geq\frac{n}{k}.\end{array}\right. \end{flalign*} \end{thm}
In the same paper, Hilton also determined the structures of $\mathcal{A}_i$'s when the equality holds. Since then, there have been many extensions about Theorem~\ref{h77}. Borg~\cite{B09} gave a simple proof of Theorem~\ref{h77}, and generalized it to labeled sets~\cite{B08}, signed sets~\cite{BL10} and permutations~\cite{B10}. Using the results of the independent number about vertex-transitive graphs, Wang and Zhang~\cite{WZ11} extended this theorem to general symmetric systems, which comprise finite sets, finite vector spaces and permutations, etc.
Hilton and Milner~\cite{HM67} also dealt with pairs of cross-intersecting families in ${[n]\choose k}$ when neither of the two families is empty: \begin{thm}\emph{(\cite{HM67})}\label{hm67}
Let $\mathcal{A},\mathcal{B}\subseteq {[n]\choose k}$ be non-empty cross-intersecting families with $n\geq 2k$. Then $|\mathcal{A}|+|\mathcal{B}|\leq {n\choose k}-{{n-k}\choose k}+1$. \end{thm}
This result was generalized by Frankl and Tokushige~\cite{FT92} to the case when $\mathcal{A}$ and $\mathcal{B}$ are not necessarily in the same $k$-uniform subfamily of $2^{[n]}$: \begin{thm}\emph{(\cite{FT92})}\label{FT92}
Let $\mathcal{A}\subseteq {[n]\choose a}$ and $\mathcal{B}\subseteq {[n]\choose b}$ be non-empty cross-intersecting families with $n\geq a+b$, $a\leq b$. Then $|\mathcal{A}|+|\mathcal{B}|\leq {n\choose b}-{{n-a}\choose b}+1$. \end{thm}
In~\cite{WZ13}, Wang and Zhang generalized Theorem~\ref{FT92} to cross-t-intersecting families. Recently, using shifting techniques, Frankl and Kupavskii~\cite{FK17} gave another proof of the result of Wang and Zhang for the case when $\mathcal{A},\mathcal{B}\subseteq{[n]\choose k}$.
As another direction, the multi-part extension of the Erd\H{o}s--Ko--Rado problem was introduced by Frankl~\cite{F96}, in connection with a similar result of Sali~\cite{Sali92}. For an integer $p\geq 1$ and positive integers $n_1,\ldots, n_p$, take $[\sum_{i\in[p]}n_i]$ to be the ground set. Then this ground set can be viewed as the disjoint union of $p$ parts $\bigsqcup_{i\in[p]}S_i$, where $S_1=[n_1]$ and $S_i=\{1+\sum_{j\in[i-1]}n_j,\ldots,\sum_{j\in[i]}n_j\}$ for $2\leq i\leq p$. More generally, denote $2^{S_i}$ as the $power~ set$ of $S_i$, for sets $A_1\in 2^{S_1},\ldots, A_p\in 2^{S_p}$, let $\bigsqcup_{i\in[p]} A_i$ be the subset of $\bigsqcup_{i\in[p]}S_i$ with $A_i$ in the $i$-th part, and for families $\mathcal{F}_1\subseteq 2^{S_1},\ldots,\mathcal{F}_p\subseteq 2^{S_p}$, let $\prod_{i\in[p]}\mathcal{F}_i=\{\bigsqcup_{i\in[p]} A_i:~A_i\in \mathcal{F}_i\}$. Then consider $k_1\in[n_1],\ldots, k_p\in[n_p]$, we denote $\prod_{i\in[p]}{[n_i]\choose k_i}$ as the family of all subsets of $\bigsqcup_{i\in[p]}S_i$ which have exactly $k_i$ elements in the $i$-th part. Therefore, families of the form $\mathcal{F}\subseteq \prod_{i\in [p]}{[n_i]\choose k_i}$ can be viewed as the natural generalization of $k$-uniform families to the multi-part setting. Similarly, a multi-part family is intersecting if any two sets of this family intersect in at least one of the $p$ parts.
Frankl proved that for any integer $p\geq 1$, any positive integers $n_1,\ldots,n_p$ and $k_1,\ldots,k_p$ satisfying $\frac{k_1}{n_1}\leq\ldots\leq\frac{k_p}{n_p}\leq\frac{1}{2}$, if $\mathcal{F}\subseteq\prod_{i\in[p]}{[n_i]\choose k_i}$ is a multi-part intersecting family, then \begin{equation*}
|\mathcal{F}|\leq\frac{k_p}{n_p}\cdot\prod_{i\in[p]}{n_i\choose k_i}. \end{equation*} This bound is sharp, for example, it is attained by the following family: \begin{equation*} \{A\in{[n_p]\choose k_p}:~i\in A,\text{~for~some~}i\in[n_p]\}\times\prod_{i\in[p-1]}{[n_i]\choose k_i}. \end{equation*}
Recall that the $Kneser~graph$ $KG_{n,k}$ is the graph on the vertex set ${[n]\choose k}$, with $A,B\in{[n]\choose k}$ forming an edge if and only if $A\cap B=\emptyset$. And an intersecting subfamily of ${[n]\choose k}$ corresponds to an independent set in $KG_{n,k}$. Hence an intersecting subfamily of $\prod_{i\in[p]}{[n_i]\choose k_i}$ corresponds to an independent set in the graph (direct) product $KG_{n_1,k_1}\times\ldots\times KG_{n_p,k_p}$. Therefore, Frankl's result can be viewed as a consequence of the general fact that $\alpha(G\times H)=\max\{\alpha(G)|H|,\alpha(H)|G|\}$ for vertex-transitive graphs $G$ and $H$, which was proved by Zhang in~\cite{zhang2012independent}.
Recently, Kwan, Sudakov and Vieira~\cite{KSV18} considered a stability version of the Erd\H{o}s--Ko--Rado theorem in the multi-part setting. They determined the maximum size of the non-trivially intersecting multi-part family when all the $n_i$'s are sufficiently large. This disproved a conjecture proposed by Alon and Katona, which was also mentioned in~\cite{Katona2017}.
In this paper, we extend Theorem~\ref{h77} and Theorem~\ref{FT92} to the $multi$-$part$ version. For any subset $S\subseteq[n]$, denote $\bar{S}$ as the complementary set of $S$ in $[n]$. Moreover, let $\mathcal{F}$ be a family of subsets of $[n]$, for any subfamily $\mathcal{A}\subseteq\mathcal{F}$, denote ${\mathcal{A}}_{\mathcal{F}}=\{B\in\mathcal{F}:A\cap B=\emptyset\text{ for some }A\in\mathcal{A}\}$.
Our main results are as follows.
\begin{thm}\label{main0} Given a positive integer $p$, let $n_1,n_2,\ldots,n_p$ and $k_1,k_2,\ldots,k_p$ be positive integers satisfying $n_i\geq 2k_i$ for all $i\in[p]$. Let $X=\prod_{i\in[p]}{[n_i]\choose k_i}$ and $\mathcal{A}_1,\mathcal{A}_2,\ldots,\mathcal{A}_m$ be cross-intersecting families over $X$ with $\mathcal{A}_1\neq \emptyset$. Then \begin{flalign}\label{bound1}
\sum\limits_{i=1}^{m}|\mathcal{A}_i|\leq\left\{\begin{array}{ll}|X|,&~~~\text{if}~m\leq\min\limits_{i\in[p]}\frac{n_i}{k_i};\\m\cdot M,&~~~\text{if}~m\ge\min\limits_{i\in[p]}\frac{n_i}{k_i},\end{array}\right. \end{flalign} where $M=\max_{i\in[p]}{n_i-1\choose k_i-1}\prod_{j\ne i}{n_j\choose k_j}$. Furthermore, the bound is attained if and only if one of the following holds: \begin{itemize}
{\item[(i)] $m<\min_{i\in[p]}\frac{n_i}{k_i}$ and $\mathcal{A}_1=X$, $\mathcal{A}_2=\cdots=\mathcal{A}_m=\emptyset$;}
{\item[(ii)] $m>\min_{i\in[p]}\frac{n_i}{k_i}$ and $\mathcal{A}_1=\cdots=\mathcal{A}_m=I$, where $I$ is a maximum intersecting family in $X$;}
{\item[(iii)] $m=\min_{i\in[p]}\frac{n_i}{k_i}$ and $\mathcal{A}_1,\ldots,\mathcal{A}_m$ are as in $(i)$ or $(ii)$, or there exists a non-empty set $S_1\subseteq\{s\in[p]:~\frac{n_s}{k_s}=2\}$ and $\mathcal{F}=\prod_{s\in S_1}{[n_s]\choose k_s}$ such that
\begin{flalign}\label{formula}
\begin{array}{l}
\mathcal{A}_1=(\mathcal{A}\sqcup(\mathcal{E}\cup \mathcal{E}_{\mathcal{F}}))\times\prod\limits_{s\in [p]\setminus S_1}{[n_s]\choose k_s}~\text{and}~
\mathcal{A}_2=(\mathcal{A}\sqcup(\mathcal{E}'\cup\mathcal{E}'_{\mathcal{F}}))\times\prod\limits_{s\in [p]\setminus S_1}{[n_s]\choose k_s}
\end{array}
\end{flalign}
for some $\mathcal{A}$, $\mathcal{E}$, $\mathcal{E}'\subseteq\mathcal{F}$, where $\mathcal{A}=\{A_1,\ldots,A_{w_0}\}$ satisfying $2w_0< |\mathcal{F}|$ and $A_i\ne \bar{A}_j$ for all $i\ne j\in [w_0]$, $\mathcal{E}\sqcup\mathcal{E}'=\{E_1,\ldots,E_v\}$ and $\mathcal{E}_\mathcal{F}\sqcup\mathcal{E}'_\mathcal{F}=\{\bar{E}_1,\ldots,\bar{E}_v\}$ satisfying $2(v+w_0)=\prod_{s\in S_1}{n_s\choose k_s}$ and $\sqcup_{j=1}^v\{E_j,\bar{E}_j\}=\mathcal{F}\setminus(\mathcal{A}\sqcup \mathcal{A}_\mathcal{F})$.
}
\end{itemize} \end{thm}
\begin{rem}\label{rem1} In \cite{WZ11}, the authors proved a similar result (Theorem 2.5 in \cite{WZ11}) for general connected symmetric systems. Actually, Theorem \ref{main0} can be viewed as an application of the method involved to obtain Theorem 2.5 in \cite{WZ11}. But different from the general case, Theorem \ref{main0} determines all the exact structures when the bound in (\ref{bound1}) is attained. \end{rem}
\begin{thm}\label{main1} For any $p\geq 2$, let $n_i, t_i, s_i$ be positive integers satisfying $n_i\geq s_i+t_i+1$, $2\leq s_i,t_i\leq \frac{n_i}{2}$ for every $i\in[p]$ and $n_i\leq\frac{7}{4}n_j$ for all distinct $i,j\in [p]$. If $\prod_{i\in [p]}{n_i\choose s_i}\geq \prod_{i\in [p]}{n_i\choose t_i}$ and $\mathcal{A}\subseteq\prod_{i\in [p]}{[n_i]\choose t_i},~\mathcal{B}\subseteq\prod_{i\in [p]}{[n_i]\choose s_i}$ are non-empty cross-intersecting families, then \begin{equation}\label{eq02}
|\mathcal{A}|+|\mathcal{B}|\leq \prod_{i\in [p]}{n_i\choose s_i}-\prod_{i\in [p]}{{n_i-t_i}\choose s_i}+1, \end{equation} and the bound is attained if and only if the following holds: \begin{itemize}
{\item[(i)] $\prod_{i\in [p]}{n_i\choose s_i}\geq \prod_{i\in [p]}{n_i\choose t_i}$, $\mathcal{A}=\{A\}$ and $\mathcal{B}=\{B\in \prod_{i\in [p]}{[n_i]\choose s_i}:B\cap A\neq\emptyset\}$ for some $A\in \prod_{i\in [p]}{[n_i]\choose t_i}$;}
{\item[(ii)] $\prod_{i\in [p]}{n_i\choose s_i}= \prod_{i\in [p]}{n_i\choose t_i}$, $\mathcal{B}=\{B\}$ and $\mathcal{A}=\{A\in \prod_{i\in [p]}{[n_i]\choose t_i}:B\cap A\neq \emptyset\}$ for some $B\in \prod_{i\in [p]}{[n_i]\choose s_i}$.}
\end{itemize} \end{thm}
\begin{rem}\label{condition_of_main1} The restrictions $s_i, t_i\leq \frac{n_i}{2}$ for every $i\in[p]$ and $n_i\leq\frac{7}{4}n_j$ for all distinct $i,j\in [p]$ in Theorem~\ref{main1} are necessary.
When $s_i, t_i\leq \frac{n_i}{2}$ is violated, for example, taking $n_1=n_2=18$, $(s_1,t_1)=(15,2)$ and $(s_2,t_2)=(2,3)$, we have $s_1>\frac{n_1}{2}$ and ${n_1\choose s_1}\cdot{n_2\choose s_2}={n_1\choose t_1}\cdot{n_2\choose t_2}$. Set $\mathcal{A}=\{A_1\}\times{[n_2]\choose t_2}$ for some 2-subset $A_1\subseteq [n_1]$ and $\mathcal{B}=\{B_1\in {[n_1]\choose s_1}:B_1\cap A_1\neq\emptyset\}\times{[n_2]\choose s_2}$. Then we have $|\mathcal{A}|+|\mathcal{B}|>{n_1\choose s_1}\cdot{n_2\choose s_2}-{{n_1-t_1}\choose s_1}\cdot{{n_2-t_2}\choose s_2}+1$.
As for the restriction $n_i\leq\frac{7}{4}n_j$, the constant $\frac{7}{4}$ here might not be tight, but the quantities of $n_i,n_j$ for distinct $i,j\in[p]$ need to be very close. For example, taking $n_1=5,n_2=12$ and $(s_1,t_1)=(s_2,t_2)=(2,2)$, we have $n_2>\frac{7}{4}n_1$ and ${n_1\choose s_1}\cdot{n_2\choose s_2}={n_1\choose t_1}\cdot{n_2\choose t_2}$. Similarly, set $\mathcal{A}=\{A_1\}\times{[n_2]\choose t_2}$ for some 2-subset $A_1\subseteq [n_1]$ and $\mathcal{B}=\{B_1\in {[n_1]\choose s_1}:B_1\cap A_1\neq\emptyset\}\times{[n_2]\choose s_2}$. Then we have $|\mathcal{A}|+|\mathcal{B}|>{n_1\choose s_1}\cdot{n_2\choose s_2}-{{n_1-t_1}\choose s_1}\cdot{{n_2-t_2}\choose s_2}+1$.
The families $\mathcal{A}$ and $\mathcal{B}$ we constructed here are closely related to imprimitive subsets of ${[n_1]\choose t_1}\times{[n_2]\choose t_2}$, which we will discuss later in Section 2.2 and Section 4. \end{rem}
We shall introduce some results about the independent sets of vertex-transitive graphs and their direct products in the next section, and prove Theorem~\ref{main0} in Section 3, Theorem~\ref{main1} in Section 4. In Section 5, we will conclude the paper and discuss some remaining problems. For the convenience of the proof, if there is no confusion, we will denote $\prod_{i\in[p]}A_i$ as the subset $\bigsqcup_{i\in[p]}A_i\subseteq \bigsqcup_{i\in[p]}S_i$ in the rest of the paper.
\section{Preliminary results}
\subsection{Independent sets of vertex-transitive graphs}
Given a finite set $X$, for every $A\subseteq X$, denote $\bar{A}=X\setminus A$. For a simple graph $G=G(V,E)$, denote $\alpha(G)$ as the independent number of $G$ and $I(G)$ as the set of all maximum independent sets of $G$. For $v\in V(G)$, define the neighborhood $N_G(v)=\{u\in V(G):(u,v)\in E(G)\}$. For a subset $A\subseteq V(G)$, write $N_G(A)=\{b\in V(G):(a,b)\in E(G)$ for some $a\in A\}$ and $N_G[A]=A\cup N_G(A)$, if there is no confusion, we denote them as $N(A)$ and $N[A]$ for short respectively.
A graph $G$ is said to be vertex-transitive if its automorphism group $\Gamma(G)$ acts transitively upon its vertices. As a corollary of the ``No-Homomorphism'' lemma for vertex-transitive graphs in~\cite{albertson1985homomorphisms}, Cameron and Ku~\cite{CK03} proved the following theorem.
\begin{thm}\label{CK}\emph{(\cite{CK03})}
Let $G$ be a vertex-transitive graph and $B$ a subset of $V(G)$. Then any independent set $S$ in $G$ satisfies that $\frac{|S|}{|V(G)|}\leq\frac{\alpha(G[B])}{|B|}$, equality implies that $|S\cap B|=\alpha(G[B])$. \end{thm}
Using the above theorem, Zhang~\cite{zhang2011primitivity} proved the following result. \begin{lem}\emph{(\cite{zhang2011primitivity})}\label{subgraph independent set}
Let $G$ be a vertex-transitive graph, and $A$ be an independent set of $G$, then $\frac{|A|}{|N_G[A]|}\leq\frac{\alpha(G)}{|G|}$. Equality implies that $|S\cap N_G[A]|=|A|$ for every $S\in I(G)$, and in particular $A\subseteq S$ for some $S\in I(G)$. \end{lem}
An independent set $A$ in $G$ is said to be imprimitive if $|A|<\alpha(G)$ and $\frac{|A|}{|N[A]|}=\frac{\alpha(G)}{|G|}$, and $G$ is called IS-imprimitive if $G$ has an imprimitive independent set. Otherwise, $G$ is called IS-primitive. Note that a disconnected vertex-transitive graph $G$ is always IS-imprimitive. Hence IS-primitive vertex-transitive graphs are all connected.
The following inequality about the size of an independent set $A$ and its non-neighbors $\bar{N}[A]$ is crucial for the proof of Theorem~\ref{main0}.
\begin{lem}\label{lemmaimprimitive} Let $G$ be a vertex transitive graph, and let $A$ be an independent set of $G$. Then \begin{flalign}
|A|+\frac{\alpha(G)}{|G|}|\bar{N}[A]|\leq\alpha(G).\nonumber \end{flalign}
Equality holds if and only if $A=\emptyset$ or $|A|=\alpha(G)$ or $A$ is an imprimitive independent set. \end{lem} For the integrity of the paper, we include the proof here. In~\cite{WZ11}, Wang and Zhang proved the same inequality for a more generalized combinatorial structure called $symmetric$ $system$ (see~\cite{WZ11}, Corollary 2.4).
\begin{proof}
If $A=\emptyset$ or $A=\alpha(G)$, the equality trivially holds. Suppose $0<|A|<\alpha(G)$, and let $B$ be a maximal independent set in $\bar{N}[A]$, then $|B|=\alpha(\bar{N}(A))$. Clearly, $A\cup B$ is also an independent set of $G$, thus we have $|A|+|B|\leq\alpha(G)$. By Theorem $\ref{CK}$, we obtain that $\frac{|B|}{|\bar{N}[A]|}\ge\frac{\alpha(G)}{|G|}$. Therefore, \begin{flalign}
|A|+\frac{\alpha(G)}{|G|}|\bar{N}[A]|\leq|A|+|B|\leq\alpha(G),\nonumber \end{flalign}
the equality holds when $\alpha(G)=|A|+\frac{\alpha(G)}{|G|}|\bar{N}[A]|=|A|+\frac{\alpha(G)}{|G|}(|G|-|N[A]|)$, which leads to $\frac{|A|}{|N[A]|}=\frac{\alpha(G)}{|G|}$, i.e., $A$ is an imprimitive independent set. \end{proof}
Let $X$ be a finite set, and $\Gamma$ a group acting transitively on $X$. Then $\Gamma$ is said to be primitive on $X$ if it preserves no nontrivial partition of $X$. A vertex-transitive graph $G$ is called primitive if the automorphism $\text{Aut}(G)$ is primitive on $V(G)$. To show the connection between the primitivity and the IS-primitivity of a vertex-transitive graph $G$, Zhang (see Proposition 2.4 in \cite{zhang2011primitivity}) proved that if $G$ is primitive, then it must be IS-primitive. As a consequence of this result, Wang and Zhang~\cite{WZ11} derived the IS-primitivity of the Kneser graph.
\begin{prop}\emph{(\cite{WZ11})}\label{imprimitive Kneser graph} The Kneser graph $KG_{n,k}$ is IS-primitive except for $n=2k\ge 4$. \end{prop}
In order to deal with the multi-part case, we also need the results about the independent sets in direct products of vertex-transitive graphs. Let $G$ and $H$ be two graphs, the direct product $G\times H$ of $G$ and $H$ is defined by \begin{flalign*} V(G\times H)=V(G)\times V(H), \end{flalign*} and \begin{flalign*} E(G\times H)=\{[(u_1,v_1),(u_2,v_2)]:(u_1,u_2)\in E(G)\text{ and }(v_1,v_2)\in E(H)\}. \end{flalign*}
Clearly, $G\times H$ is a graph with $\text{Aut}(G)\times\text{Aut}(H)$ as its automorphism group. And, if $G,H$ are vertex-transitive, then $G\times H$ is also vertex-transitive under the actions of $\text{Aut}(G)\times\text{Aut}(H)$. We say the direct product $G\times H$ is MIS-normal (maximum-independent-set-normal) if every maximum independent set of $G\times H$ is a preimage of an independent set of one factor under projections.
In \cite{zhang2012independent}, Zhang obtained the exact structure of the maximal independent set of $G\times H$.
\begin{thm}\emph{(\cite{zhang2012independent})}\label{tensor product} Let $G$ and $H$ be two vertex-transitive graphs with $\frac{\alpha(G)}{|G|}\ge\frac{\alpha(H)}{|H|}$. Then \begin{flalign}
\alpha(G\times H)=\alpha(G)|H|,\nonumber \end{flalign} and exactly one of the following holds: \begin{itemize} \item[(i)] $G\times H$ is MIS-normal;
\item[(ii)] $\frac{\alpha(G)}{|G|}=\frac{\alpha(H)}{|H|}$ and one of $G$ or $H$ is IS-imprimitive;
\item[(iii)] $\frac{\alpha(G)}{|G|}>\frac{\alpha(H)}{|H|}$ and $H$ is disconnected. \end{itemize} \end{thm}
In fact if $\frac{\alpha(G)}{|G|}=\frac{\alpha(H)}{|H|}$ and $A$ is an imprimitive independent set of $G$, then for every $I\in I(H)$, it is easy to see that $S=(A\times V(H))\cup (\bar{N}[A]\times I)$ is an independent set of $G\times H$ with size $\alpha(G)|H|$.
Zhang \cite{zhang2011primitivity} also investigated the relationship between the graph primitivity and the structures of the maximum independent sets in direct products of vertex-transitive graphs.
\begin{thm}\emph{(\cite{zhang2011primitivity})}\label{normal-imprimitive} Suppose $G\times H$ is MIS-normal and $\frac{\alpha(H)}{|H|}\leq\frac{\alpha(G)}{|G|}$. If $G\times H$ is IS-imprimitive, then one of the following two possible cases holds:
\begin{itemize}
\item[(i)] $\frac{\alpha(G)}{|G|}=\frac{\alpha(H)}{|H|}$ and one of them is IS-imprimitive or both $G$ and $H$ are bipartite;
\item[(ii)] $\frac{\alpha(G)}{|G|}>\frac{\alpha(H)}{|H|}$ and $G$ is IS-imprimitive.
\end{itemize} \end{thm}
As an application of Theorem~\ref{tensor product} and Theorem~\ref{normal-imprimitive}, Geng et al.~\cite{geng2012structure} showed the MIS-normality of the direct products of Kneser graphs.
\begin{thm}\emph{(\cite{geng2012structure})}\label{imprimitive product kneser graph} Given a positive integer $p$, let $n_1,n_2,\ldots,n_p$ and $k_1,k_2,\ldots,k_p$ be $2p$ positive integers with $n_i\ge 2k_i$ for $1\leq i\leq p$. Then the direct product of the Kneser graph $$\text{KG}_{n_1,k_1}\times\text{KG}_{n_2,k_2}\times\cdots\times\text{KG}_{n_p,k_p}$$ is MIS-normal except that there exist $i,j$ and $\ell$ with $n_i=2k_i\ge 4$ and $n_j=2k_j$, or $n_i=n_j=n_{\ell}=2$. \end{thm}
\subsection{Nontrivial independent sets of part-transitive bipartite graphs}
For a bipartite graph $G(X,Y)$ with two parts $X$ and $Y$, an independent set $A$ is said to be non-trivial if $A\nsubseteq X$ and $A\nsubseteq Y$. $G(X,Y)$ is said to be part-transitive if there is a group $\Gamma$ acting transitively upon each part and preserving its adjacency relations. Clearly, if $G(X,Y)$ is part-transitive, then every vertex of $X~(Y)$ has the same degree, written as $d(X)~(d(Y))$. We use $\alpha(X,Y)$ and $I(X,Y)$ to denote the size and the set of the maximum-sized nontrivial independent sets of $G(X,Y)$, respectively.
Let $G(X,Y)$ be a non-complete bipartite graph and let $A\cup B$ be a nontrivial independent set of $G(X,Y)$, where $A\subseteq X$ and $B\subseteq Y$. Then $A\subseteq X\setminus N(B)$ and $B\subseteq Y\setminus N(A)$, which implies \begin{equation*}
|A|+|B|\leq \max{\{|A|+|Y|-|N(A)|, |B|+|X|-|N(B)|\}}. \end{equation*} So we have \begin{equation}\label{eq03}
\alpha(X,Y)=\max{\{|Y|-\epsilon(X), |X|-\epsilon(Y)\}}, \end{equation}
where $\epsilon(X)=\min\{|N(A)|-|A|: A\subseteq X, N(A)\neq Y\}$ and $\epsilon(Y)=\min\{|N(B)|-|B|: B\subseteq Y, N(B)\neq X\}$.
We call $A\subseteq X$ a fragment of $G(X,Y)$ in $X$ if $N(A)\neq Y$ and $|N(A)|-|A|=\epsilon(X)$, and we denote $\mathcal{F}(X)$ as the set of all fragments in $X$. Similarly, we can define $\mathcal{F}(Y)$. Furthermore, denoting $\mathcal{F}(X,Y)=\mathcal{F}(X)\cup\mathcal{F}(Y)$, we call an element $A\in \mathcal{F}(X,Y)$ a $k$-fragment if $|A|=k$. And we call a fragment $A\in\mathcal{F}(X)$ trivial if $|A|=1$ or $A=X\setminus N(b)$ for some $b\in Y$. Since for each $A\in \mathcal{F}(X)$, $Y\setminus N(A)$ is a fragment in $\mathcal{F}(Y)$. Hence, once we know $\mathcal{F}(X)$, $\mathcal{F}(Y)$ can also be determined.
Let $X$ be a finite set, and $\Gamma$ a group acting transitively on $X$. If $\Gamma$ is imprimitive on $X$, then it preserves a nontrivial partition of $X$, called a block system, each element of which is called a block. Clearly, if $\Gamma$ is both transitive and imprimitive, there must be a subset $B\subseteq X$ such that $1<|B|<|X|$ and $\gamma(B)\cap B=B$ or $\emptyset$ for every $\gamma \in \Gamma$. In this case, $B$ is called an imprimitive set in $X$. Furthermore, a subset $B\subseteq X$ is said to be $semi$-$imprimitive$ if $1<|B|<|X|$ and for each $\gamma\in \Gamma$ we have $\gamma(B)\cap B=B$, $\emptyset$ or $\{b\}$ for some $b\in B$.
The following theorem (cf. \cite[Theorem~1.12]{J85}) is a classical result on the primitivity of group actions. \begin{thm}\emph{(\cite{J85})}\label{primitivity} Suppose that a group $\Gamma$ transitively acts on $X$. Then $\Gamma$ is primitive on $X$ if and only if for each $a\in X$, $\Gamma_a$ is a maximal subgroup of $\Gamma$. Here $\Gamma_a=\{\gamma\in\Gamma:\gamma(a)=a\}$, the stabilizer of $a\in X$. \end{thm}
Noticing the similarities about families that are cross-t-intersecting or cross-Sperner, Wang and Zhang \cite{WZ13} proved the following theorem about $\alpha(G(X,Y))$ and $I(X,Y)$ of a special kind of part-transitive bipartite graphs.
\begin{thm}\emph{(\cite{WZ13})}\label{key00}
Let $G(X,Y)$ be a non-complete bipartite graph with $|X|\leq|Y|$. If $G(X,Y)$ is part-transitive and every fragment of $G(X,Y)$ is primitive under the action of a group $\Gamma$. Then $\alpha(X,Y)=|Y|-d(X)+1$. Moreover, \begin{itemize}
{\item[(1)] If $|X|<|Y|$, then $X$ has only 1-fragments;}
{\item[(2)] If $|X|=|Y|$, then each fragment in $X$ has size 1 or $|X|-d(X)$ unless there is a semi-imprimitive fragment in $X$ or $Y$.} \end{itemize} \end{thm}
To deal with multi-part cross-intersecting families, we introduce the following variation of Theorem \ref{key00}.
\begin{thm}\label{key01}
Let $G(X,Y)$ be a non-complete bipartite graph with $|X|\leq|Y|$. If $G(X,Y)$ is part-transitive under the action of a group $\Gamma$. Then \begin{equation}\label{eq04}
\alpha(X,Y)=\max{\{|Y|-d(X)+1, |A'|+|Y|-|N(A')|, |B'|+|X|-|N(B')|\}}, \end{equation}
where $A'$ and $B'$ are minimum imprimitive subsets of $X$ and $Y$ respectively. By minimum, here we mean that $|N(A')|-|A'|=\min{\{|N(A)|-|A|: A\in X~(\text{or~} Y) \text{~is~imprimitive}\}}$. \end{thm}
For the proof of Theorem~\ref{key01}, we need the following two lemmas from \cite{WZ13}.
\begin{lem}\emph{(\cite{WZ13})}\label{lem01}
Let $G(X,Y)$ be a non-complete bipartite graph. Then, $|Y|-\epsilon(X)=|X|-\epsilon(Y)$, and \begin{itemize} {\rm\item[(i)] $A\in \mathcal{F}(X)$ if and only if $(Y\setminus N(A))\in \mathcal{F}(Y)$ and $N(Y\setminus N(A))=X\setminus A$;} {\rm\item[(ii)] $A\cap B$ and $A\cup B$ are both in $\mathcal{F}(X)$ if $A$, $B\in \mathcal{F}(X)$, $A\cap B\neq \emptyset$ and $N(A\cup B)\neq Y$.} \end{itemize} \end{lem}
\begin{lem}\emph{(\cite{WZ13})}\label{lem02} Let $G(X,Y)$ be a non-complete and part-transitive bipartite graph under the action of a group $\Gamma$. Suppose that $A\in\mathcal{F}(X,Y)$ such that $\emptyset\neq\gamma(A)\cap A\neq A$ for some $\gamma\in\Gamma$. Define $\phi: \mathcal{F}(X,Y)\rightarrow\mathcal{F}(X,Y)$, \begin{flalign*} \phi (A)=\left\{\begin{array}{ll}Y\setminus N(A),&~~~\text{if}~A\in\mathcal{F}(X);\\X\setminus N(A),&~~~\text{if}~A\in\mathcal{F}(Y).\end{array}\right. \end{flalign*}
If $|A|\leq|\phi(A)|$, then $A\cup\gamma(A)$ and $A\cap\gamma(A)$ are both in $\mathcal{F}(X,Y)$. \end{lem}
\begin{rem}\label{balanced} As a direct consequence of Lemma~\ref{lem01}, a maximum-sized nontrivial independent set in $G(X,Y)$ is of the form $A\sqcup(Y\setminus N(A))$ for some $A\in \mathcal{F}(X)$ or $B\sqcup(X\setminus N(B))$ for some $B\in \mathcal{F}(Y)$. Therefore, in order to address our problems, it suffices to determine $\mathcal{F}(X)$ $(\text{or~}\mathcal{F}(Y))$.
Meanwhile, for the mapping $\phi$ in Lemma~\ref{lem02}, we have $\phi^{-1}=\phi$ and $|A|+|\phi(A)|=\alpha(X,Y)$. When $|A|=|\phi(A)|$, we call the fragment $A$ balanced. Thus, all balanced fragments have size $\frac{1}{2}\alpha(X,Y)$. \end{rem}
\begin{proof}[\bf{Proof of Theorem~\ref{key01}}]
The same as the original proof of Theorem \ref{key00} in \cite{WZ13}, we apply Lemma~\ref{lem02} repeatedly. For any $A_0\in\mathcal{F}(X,Y)$ satisfying $|A_0|\leq|\phi(A_0)|$, if there exists $\gamma\in\Gamma$ such that $\emptyset\neq\gamma(A_0)\cap A_0\neq A_0$, then by Lemma~\ref{lem02} we have: (1) $A_0\cap\gamma(A_0)\in\mathcal{F}(X,Y)$ or (2) $\gamma(A_0)\cap A_0=\emptyset$ or $\gamma(A_0)\cap A_0=A_0$ for any $\gamma\in\Gamma$.
For case (1), denote \begin{flalign*}
A_1=\left\{\begin{array}{ll}A_0\cap\gamma(A_0), &~\text{if}~|A_0\cap\gamma(A_0)|\leq|\phi(A_0\cap\gamma(A_0))|; \\ \phi(A_0\cap\gamma(A_0)), &~\text{~~~~~~~~~~~~~~~otherwise};\end{array}\right. \end{flalign*} and consider the primitivity of $A_1$, i.e., whether there is a $\gamma'\in\Gamma$ such that $\emptyset\neq\gamma'(A_1)\cap A_1\neq A_1$ or not.
For case (2), if $|A_0|\neq1$, according to the definition, $A_0$ is an imprimitive set of $X$ (or $Y$). Otherwise, $|A_0|=1$, which means $\mathcal{F}(X,Y)$ contains a singleton.
By doing these procedures repeatedly, after $r$ $(0\leq r\leq |A_0|-1)$ steps, we have a fragment $A_r\in\mathcal{F}(X,Y)$ such that $A_r$ is either a singleton or an imprimitive set. Hence, we have
$$\alpha(X,Y)=\max{\{|Y|-d(X)+1, |X|-d(Y)+1, |A'|+|Y|-|N(A')|, |B'|+|X|-|N(B')|\}},$$
where $A'$ and $B'$ are minimum imprimitive subsets of $X$ and $Y$ respectively. Noticing that $|Y|\geq|X|$ and $d(X)|X|=d(Y)|Y|$, we have $d(X)=d(Y)|Y|/|X|\geq d(Y)$. Therefore,
$$|Y|-|X|=d(X)|X|/d(Y)-|X|=(d(X)-d(Y))|X|/d(Y)\geq d(X)-d(Y),$$
which implies that $|X|-d(Y)+1\leq|Y|-d(X)+1$. Finally we have
$$\alpha(X,Y)=\max{\{|Y|-d(X)+1, |A'|+|Y|-|N(A')|, |B'|+|X|-|N(B')|\}}.$$ \end{proof}
\section{Proof of Theorem~\ref{main0}}
Throughout this section, for any nonempty subset $S\subseteq[p]$ and $A=\prod_{i\in S}A_i\in\prod_{i\in S}{[n_i]\choose k_i}$, denote $\bar{A}=\prod_{i\in S}\bar{A}_i$. Before we start the proof of Theorem~\ref{main0}, we introduce the following proposition about the direct product of Kneser graphs.
\begin{prop}\label{note1} Given a positive integer $p$, let $n_1,n_2,\ldots,n_p$ and $k_1,k_2,\ldots,k_p$ be positive integers with $n_i\ge 2k_i$ for $1\leq i\leq p$. Let $G=\prod_{i\in[p]}{KG_{n_i,k_i}}$. Then $G$ is IS-imprimitive if and only if there exists an $i\in[p]$ such that $n_i=2k_i\ge 4$ or there exist distinct $i,j\in[p]$ such that $n_i=n_j=2$ and $k_i=k_j=1$. \end{prop}
\begin{proof} Note that if the Kneser graph $KG_{n,k}$ is disconnected, then $n=2k\geq 4$ and $KG_{n,k}$ is bipartite. Thus by Proposition \ref{imprimitive Kneser graph}, $KG_{2k,k}$ is IS-imprimitive for all $k\geq 2$. Moreover, since $\chi({KG_{n,k}})=n-2k+2$ for all $n\geq 2k$ (Lov\'{a}sz-Kneser Theorem, see \cite{Lovasz78}), we know that if $KG_{n,k}$ is bipartite, then $n=2k\geq 2$. Now we use induction on the number of factors $p$.
If $p=2$, let $G_1=KG_{n_1,k_1}$, $G_2=KG_{n_2,k_2}$, and $G=G_1\times G_2$. W.l.o.g., assume that $\frac{\alpha(G_1)}{|G_1|}\ge\frac{\alpha(G_2)}{|G_2|}$. Then, by Theorem \ref{tensor product}, (i) $G_1\times G_2$ is MIS-normal, or (ii) $\frac{\alpha(G_1)}{|G_1|}=\frac{\alpha(G_2)}{|G_2|}$ and one of $G_1$ and $G_2$ is IS-imprimitive, or (iii) $\frac{\alpha(G_1)}{|G_1|}>\frac{\alpha(G_2)}{|G_2|}$ and $G_2$ is disconnected. For case (i), by Theorem \ref{normal-imprimitive}, at least one factor of $G$ is IS-imprimitive or both $G_1$ and $G_2$ are bipartite. Noticed that $KG_{2,1}$ is IS-primitive, therefore, either there exists an $i\in [2]$ such that $n_i=2k_i\ge 4$ or there exist distinct $i,j\in[2]$ such that $n_i=n_j= 2k_i=2k_j=2$. For cases (ii) and (iii), since $G$ is not MIS-normal, by Theorem \ref{imprimitive product kneser graph}, at least one of $G_1$ and $G_2$ is IS-imprimitive. Thus the proposition holds when $p=2$.
Suppose the proposition holds when the number of factors is $p-1$. Set $G'_1=\prod_{i=1}^{p-1}KG_{n_i,k_i}$ and $G'_2=KG_{n_p,k_p}$, by Theorem \ref{normal-imprimitive}, at least one factor of $G'_1$ and $G'_2$ is IS-imprimitive or both $G'_1$ and $G'_2$ are bipartite. If $G_1'$ is IS-imprimitive, by the induction hypothesis, there exists an $i'\in[p-1]$ such that $n_{i'}=2k_{i'}\ge 4$ or there exist distinct $i',j'\in[p-1]$ such that $n_{i'}=n_{j'}=2k_{i'}=2k_{j'}=2$. If $G_2'$ is IS-imprimitive, then $n_p=2k_p\geq 4$. Otherwise, both $G'_1$ and $G'_2$ are IS-primitive and bipartite. Thus, for $G'_2$, we have $n_p=2k_p=2$. For $G'_1$, since $\chi(G'_1)\cdot\alpha(G'_1)\ge |V(G'_1)|$, we know that there exists $i'\in[p-1]$ such that $n_{i'}=2k_{i'}=2$ by Lemma \ref{tensor product}. This completes the proof. \end{proof}
The idea of the proof for Theorem~\ref{main0} is similar to that for general connected symmetric systems in \cite{WZ11}. Since $\prod_{i=1}^{p}\text{KG}_{n_i,k_i}$ is a vertex transitive graph, by Lemma \ref{lemmaimprimitive}, we can prove the bound (\ref{bound1}). Then, through a careful analysis, we can obtain the structure of all imprimitive independent sets of this graph. This leads to the unique structures of $\mathcal{A}_1$ and $\mathcal{A}_2$ in $(\ref{formula})$.
\begin{proof}[\bf{Proof of Theorem~\ref{main0}}] Define a graph $G$ on the vertex set $X=\prod_{s\in[p]}{[n_s]\choose k_s}$ with $A,B\in X$ forming an edge in $G$ if and only if $A\cap B=\emptyset$. Therefore, $G$ is the direct product of Kneser graphs $\text{KG}_{n_1,k_1}\times\cdots\times\text{KG}_{n_p,k_p}$.
Assume that $2\leq\frac{n_1}{k_1}\leq\frac{n_2}{k_2}\leq\ldots\leq\frac{n_p}{k_p}$, then $\frac{|G|}{\alpha(G)}=\frac{n_1}{k_1}$ by Theorem~\ref{tensor product}. Following the notations of Borg in \cite{B09,B10,BL10}, write $\mathcal{A}^\ast_i=\{A\in\mathcal{A}_i|A\cap B\ne\emptyset\text{ for any }B\in\mathcal{A}_i\}$, $\hat{\mathcal{A}}_i=\mathcal{A}_i\setminus\mathcal{A}_i^\ast$, $\mathcal{A}^\ast=\bigcup_{i=1}^m\mathcal{A}_i^{\ast}$, $\hat{\mathcal{A}}=\bigcup_{i=1}^m\hat{\mathcal{A}}_i$. Note that $\bar{N}_{G}[\mathcal{A}]=\{B\in X|A\cap B\ne\emptyset,\text{ for any }A\in\mathcal{A}\}$ for $\mathcal{A}\subseteq X$, it is easy to show that $\mathcal{A}^\ast$ is an intersecting family and $\hat{\mathcal{A}}\subseteq\bar{N}_{G}[\mathcal{A}^\ast]$. It follows that $\mathcal{A}_i\cap\mathcal{A}_j\subseteq\mathcal{A}_i^\ast\cap\mathcal{A}_j^\ast$ from the definition, therefore $\hat{\mathcal{A}}_i\cap\hat{\mathcal{A}}_j=\emptyset$ for $i\ne j$, and $|\hat{\mathcal{A}}|=\sum_{i=1}^m|\hat{\mathcal{A}}_i|$. Thus by Lemma~\ref{lemmaimprimitive} we have \begin{align*}\label{imprimitive equation}
\sum\limits_{i=1}^m|\mathcal{A}_i|&=\sum\limits_{i=1}^m|\hat{\mathcal{A}}_i|+\sum\limits_{i=1}^m|\mathcal{A}_i^\ast|\leq|\hat{\mathcal{A}}|+m|\mathcal{A}^\ast|\leq|\bar{N}_{G}[\mathcal{A}^\ast]|+m|\mathcal{A}^\ast|\nonumber\\
&=\frac{|G|}{\alpha(G)}(\frac{\alpha(G)}{|G|}|\bar{N}_{G}[\mathcal{A}^\ast]|+|\mathcal{A}^\ast|)+(m-\frac{|G|}{\alpha(G)})|\mathcal{A}^\ast|\\
&\leq |G|+(m-\frac{|G|}{\alpha(G)})|\mathcal{A}^\ast|=|G|+(m-\frac{n_1}{k_1})|\mathcal{A}^\ast|.\nonumber \end{align*}
If $m<\frac{n_1}{k_1}$, then $\sum_{i=1}^m|\mathcal{A}_i|\leq|G|$, and the equality implies $\mathcal{A}^\ast=\emptyset$. Thus $\mathcal{A}_i=\hat{\mathcal{A}_i}$ for every $i\in[m]$, and this yields that the graph $G$ is a disjoint union of the induced subgraph $G[\mathcal{A}_i]'s$. And by the cross-intersecting property, each $G[\mathcal{A}_i]$ is a connected component of $G$. Since $G$ is connected when $\frac{n_s}{k_s}>2$ for all $s\in[p]$ and $m\geq 2$, we know that one of $\mathcal{A}_i$ is $X$ and the rest are empty sets, as case (i).
If $m>\frac{n_1}{k_1}$, then $\sum_{i=1}^m|\mathcal{A}_i|\leq m\alpha(G)$, and the equality implies that $\mathcal{A}_1^\ast=\ldots=\mathcal{A}_m^\ast=\mathcal{A}^\ast$, $|\mathcal{A}^\ast|=\alpha(G)$, as case (ii).
If $m=\frac{n_1}{k_1}$, then $\sum_{i=1}^m|\mathcal{A}_i|\leq|X|$, and the equality implies that $\mathcal{A}_1^\ast=\ldots=\mathcal{A}_m^\ast=\mathcal{A}^{\ast}$ and $\frac{\alpha(G)}{|G|}|\bar{N}_{G}[\mathcal{A}^\ast]|+|\mathcal{A}^\ast|=\alpha(G)$. By Lemma \ref{lemmaimprimitive}, we know that $|\mathcal{A}^\ast|=0$, or $|\mathcal{A}^\ast|=\alpha(G)$, or $\mathcal{A}^\ast$ is an imprimitive independent set of $G$. In the last case, $\hat{\mathcal{A}}_1,\ldots,\hat{\mathcal{A}}_m$ are cross-intersecting families and form a partition of $\bar{N}_{G}[\mathcal{A}^\ast]$. In order to determine the structures of the maximum-sized cross-intersecting families in this case, we shall characterize the imprimitive independent set of $G$.
\begin{claim}\label{imprimitive independent set} Let $\mathcal{F}=\prod_{s\in S}{[n_s]\choose k_s}$ and $X'=\prod_{s\in[p]\setminus S}{[n_s]\choose k_s}$, where $S=\{s\in [p]:~\frac{n_s}{k_s}=2\}$. If $\mathcal{A}^\ast$ is an imprimitive independent set of $G$, then $\mathcal{A}^\ast=\mathcal{A}\times X'$, where $\mathcal{A}\subseteq\mathcal{F}$ is a non-maximum intersecting family. \end{claim}
According to Proposition \ref{note1}, $G$ is IS-imprimitive if and only if there exists an $i\in S$ such that $n_i=2k_i\ge 4$ or there exist distinct $i,j\in S$ such that $n_i=n_j=2$ and $k_i=k_j=1$. Thus, with the assumptions in this claim, $S\neq \emptyset$ and $S=\{i_0\}$ if and only if $n_{i_0}=2k_{i_0}\geq 4$ for some $i_0\in[p]$. W.l.o.g., assume that $S=[s_0]$, where $s_0=|S|$. Under this circumstance, $m=\frac{n_1}{k_1}=2$.
Divide $\mathcal{A}^\ast$ into $u$ disjoint parts $\{C_i\times \mathcal{D}_i\}_{i=1}^{u}$, where $C_i=C_{i,1}\times\ldots\times C_{i,s_0}\in\mathcal{F}$, $\mathcal{D}_i\subseteq X'$ for all $i\in [u]$ and $C_i\ne C_j$ for any $i\ne j\in[u]$. Since $N_G(C_i\times \mathcal{D}_i)=\bar{C_i}\times\mathcal{D}_i'$, where $\mathcal{D}'_i=\{A\in X': A\cap D_i=\emptyset\text{~for~some~} D_i\in\mathcal{D}_i\}$, we know that $N_G[C_i\times \mathcal{D}_i]\cap N_G[C_j\times \mathcal{D}_j]=\emptyset$ for all $i\ne j\in[u]$. Meanwhile, $C_i\times \mathcal{D}_i\cap N_G(C_j\times \mathcal{D}_j)=\emptyset$ for all $i\ne j\in[u]$. Otherwise, assume that there exists $T_1\times T_2\in C_i\times\mathcal{D}_i\cap N_G(C_j\times\mathcal{D}_j)$, for some $T_1\in\mathcal{F}$ and $T_2\in X'$. Thus we have $T_1\times T_2\cap C_j\times D_j=\emptyset$, for some $D_j\in\mathcal{D}_j$, which contradicts the fact that $\mathcal{A}^\ast$ is an intersecting family.
By projecting $G$ onto the last $p-s_0$ factors, we obtain a graph $G'$ with vertex set $X'$ such that $A,B\in X'$ form an edge in $G'$ if and only if $A,B$ are disjoint. Consider the cross-intersecting families $\{\mathcal{D}_i,\bar{N}_{G'}(\mathcal{D}_i)\}$ in $X'$, since $|\{\mathcal{D}_i,\bar{N}_{G'}(\mathcal{D}_i)\}|=2<\frac{n_{s_0+1}}{k_{s_0+1}}$, by case (i), we know that \begin{flalign}
|\mathcal{D}_i|+|\bar{N}_{G'}(\mathcal{D}_i)|=|\mathcal{D}_i|+|X'|-|N_{G'}(\mathcal{D}_i)|\leq|X'|,\nonumber \end{flalign}
thus we have $|\mathcal{D}_i|\leq|N_{G'}(\mathcal{D}_i)|$, and $|C_i\times \mathcal{D}_i|=|\mathcal{D}_i|\leq |N_{G'}(\mathcal{D}_i)|=|N_G(C_i\times \mathcal{D}_i)|$. Therefore \begin{flalign*}
\frac{|\mathcal{A}^\ast|}{|N_G[\mathcal{A}^\ast]|}=\frac{\sum_{i\in[u]}|C_i\times\mathcal{D}_i|}{\sum_{i\in[u]}|N_G[C_i\times\mathcal{D}_i]|}\leq\frac{1}{2}=\frac{\alpha(G)}{|G|}=\frac{k_1}{n_1}, \end{flalign*} and the equality holds if and only if for all $i\in[u]$, $\mathcal{D}_i=X'$ or $\bar{N}_{G'}(\mathcal{D}_i)=X'$. Since $\mathcal{D}_i\ne\emptyset$, we have $\mathcal{A}^\ast=\bigsqcup_{i=1}^{u}C_i\times X'=\mathcal{A}\times X'$. Recall that $\frac{n_s}{k_s}>2$ for all $s>s_0$, hence $C_i\cap C_j\neq \emptyset$ for any $i\neq j\in[u]$. Therefore, by the imprimitivity of $\mathcal{A}^\ast$, $\mathcal{A}^\ast$ is a non-maximum independent set of $G$, thus $\mathcal{A}\subseteq\mathcal{F}$ is a non-maximal intersecting family and the claim holds.
For every intersecting family $\mathcal{A}\subseteq\mathcal{F}$, since $\frac{n_s}{k_s}=2$ for all $s\in {S}$, then $\mathcal{A}=\{A_1,A_2,\ldots,A_w\}\times\prod_{s\in S\setminus S'}{[n_s]\choose k_s}$ for some nonempty subset $S'\subseteq S$, where $\{A_1,\ldots,A_w\}\subseteq\prod_{s\in S'}{[n_s]\choose k_s}$ satisfying $A_i\neq \bar{A}_j$ for all $i\neq j\in [w]$. In particular, if $\mathcal{A}$ is a maximum intersecting family, we can obtain that $\bigsqcup_{j=1}^{w}\{A_j,\bar{A}_j\}=\prod_{s\in S'}{[n_s]\choose k_s}$ and $2w=\prod_{s\in S'}{n_s\choose k_s}$.
Therefore, $\mathcal{A}^\ast=\{A_1,A_2,\ldots,A_{w_0}\}\times\prod_{s\in S\setminus S_1}{[n_s]\choose k_s}\times X'$ and $N_G(\mathcal{A}^\ast)=\{\bar{A}_1,\bar{A}_2,\ldots,\bar{A}_{w_0}\}\times\prod_{s\in S\setminus S_1}{[n_s]\choose k_s}\times X'$, for some positive integer $w_0<\frac{\prod_{s\in S_1}{n_s\choose k_s}}{2}$ and nonempty subset $S_1\subseteq S$.
From the structure of the imprimitive independent set $\mathcal{A}^\ast$, we know that \begin{flalign*} \bar{N}_G[\mathcal{A}^\ast]=\{E_1,\bar{E}_1,E_2,\bar{E}_2,\ldots,E_v,\bar{E}_v\}\times\prod\limits_{s\in S\setminus S_1}{[n_s]\choose k_s}\times X', \end{flalign*} where $\emptyset\neq\{E_1,\ldots,E_v\}\subseteq\prod_{s\in S_1}{[n_s]\choose k_s}$, and $\bigsqcup_{j=1}^{w_0}\{A_j,\bar{A}_j\}\sqcup\bigsqcup_{j=1}^v\{E_j,\bar{E}_j\}=\prod_{s\in S_1}{[n_s]\choose k_s}$.
Since $E_j\times\prod_{s\in S\setminus S_1}{[n_s]\choose k_s}\times X'$ and $\bar{E}_j\times\prod_{s\in S\setminus S_1}{[n_s]\choose k_s}\times X'$ must be contained in the same one of $\hat{\mathcal{A}_1}$, $\hat{\mathcal{A}_2}$, we have \begin{flalign} \hat{\mathcal{A}}_1&=(\mathcal{E}\cup\tilde{\mathcal{E}})\times \prod\limits_{s\in S\setminus S_1}{[n_s]\choose k_s}\times X',\nonumber\\ \hat{\mathcal{A}}_2&=(\mathcal{E}'\cup\tilde{\mathcal{E}}')\times \prod\limits_{s\in S\setminus S_1}{[n_s]\choose k_s}\times X',\nonumber \end{flalign} where $\mathcal{E}\sqcup\mathcal{E}'=\{E_1,\ldots,E_v\}$ and $\tilde{\mathcal{E}}\sqcup\tilde{\mathcal{E}}'=\{\bar{E}_1,\ldots,\bar{E}_v\}$. Here we denote $\tilde{\mathcal{E}}=\{\bar{E}_{i_1},\ldots,\bar{E}_{i_l}\}$ if $\mathcal{E}=\{E_{i_1},\ldots,E_{i_l}\}\subseteq\prod_{s\in S_1}{[n_s]\choose k_s}$, for some subset $\{i_1,\ldots,i_l\}\subseteq[v]$.
Finally, to sum up, \begin{flalign} \mathcal{A}_1&=\mathcal{A}^\ast\sqcup\hat{\mathcal{A}}_1=(\mathcal{A}\times X')\sqcup ((\mathcal{E}\cup\tilde{\mathcal{E}})\times \prod\limits_{s\in S\setminus S_1}{[n_s]\choose k_s}\times X')\nonumber,\\ \mathcal{A}_2&=\mathcal{A}^\ast\sqcup\hat{\mathcal{A}}_2=(\mathcal{A}\times X')\sqcup ((\mathcal{E}'\cup\tilde{\mathcal{E}}')\times \prod\limits_{s\in S\setminus S_1}{[n_s]\choose k_s}\times X')\nonumber. \end{flalign}
\end{proof}
\section{Proof of Theorem \ref{main1}}
Throughout this section, we denote $S_n$ as the symmetric group on $[n]$ and $S_C$ as the symmetric group on $C$ for $C\subseteq [n]$. For each $i\in[p]$, let $X_i$ be a finite set, then for each family $\mathcal{A}\subseteq \prod_{i\in[p]}X_i$, we denote $\mathcal{A}|_i$ as the projection of $\mathcal{A}$ onto the $i$-th factor.
For the proof of Theorem~\ref{main1}, we need the following proposition obtained by Wang and Zhang in \cite{WZ13}.
\begin{prop}\emph{(\cite{WZ13})}\label{fragment2}
Let $G(X,Y)$ be a non-complete bipartite graph with $|X|=|Y|$ and $\epsilon(X)=d(X)-1$, and let $\Gamma$ be a group part-transitively acting on $G(X,Y)$. If each fragment of $G(X,Y)$ is primitive and there are no $2$-fragments in $\mathcal{F}(X,Y)$, then every nontrivial fragment $A\in \mathcal{F}(X)$ (if there exists) is balanced (see Remark~\ref{balanced}), and for each $a\in A$, there is a unique nontrivial fragment $B$ such that $A\cap B=\{a\}$. \end{prop}
The proof of Theorem~\ref{main1} is divided into two parts: Firstly, we prove the bound (\ref{eq02}). Consider a non-complete bipartite graph defined by the multi-part cross-intersecting family. Through discussions about the primitivity of group $\prod_{i=1}^{p}S_{n_i}$ and careful evaluations about $|\mathcal{A}|+|\mathcal{Y}|-|N(\mathcal{A})|$, the bound (\ref{eq02}) follows from Theorem \ref{key01}. Secondly, based on a characterization of all nontrivial fragments in this bipartite graph, we determine all the structures of $\mathcal{A}$ and $\mathcal{B}$ when the bound (\ref{eq02}) is attained.
\begin{proof}[\bf{Proof of Theorem~\ref{main1}}]
With the assumptions in the theorem, we define a bipartite graph $G(\mathcal{X},\mathcal{Y})$ with $\mathcal{X}=\prod_{i=1}^{p}{[n_i]\choose t_i}$ and $\mathcal{Y}=\prod_{i=1}^{p}{[n_i]\choose s_i}$. For $A=\prod_{i=1}^{p}{A_i}\in \mathcal{X}$ and $B=\prod_{i=1}^{p}{B_i}\in \mathcal{Y}$ ($A_i\in {[n_i]\choose t_i}$ and $B_i\in {[n_i]\choose s_i}$, for every $1\leq i\leq p$), $(A,B)$ forms an edge in $G(\mathcal{X},\mathcal{Y})$ if and only if $A\cap B=\emptyset$, i.e., $A_i\cap B_i=\emptyset$ for each $1\leq i\leq p$.
It can be easily verified that $\prod_{i=1}^{p}S_{n_i}$ acts transitively on $\mathcal{X}$ and $\mathcal{Y}$, respectively, and preserves the property of cross-intersecting. Thus we have $d(\mathcal{X})=|N(A)|$ for each $A\in \mathcal{X}$, and $d(\mathcal{Y})=|N(B)|$ for each $B\in \mathcal{Y}$. Since, for each $A=\prod_{i=1}^{p}{A_i}\in \mathcal{X}$, \begin{equation*} N(A)=\{B=\prod_{i=1}^{p}{B_i}\in \mathcal{Y}:~A_i\cap B_i=\emptyset\text{ for each }1\leq i\leq p\}=\prod_{i=1}^{p}{[n_i]\setminus A_i\choose s_i}, \end{equation*}
we have $d(\mathcal{X})=|N(A)|=\prod_{i=1}^{p}{{n_i-t_i}\choose s_i}$. Similarly, $d(\mathcal{Y})=|N(B)|=\prod_{i=1}^{p}{{n_i-s_i}\choose t_i}$.
By Theorem~\ref{key01}, we obtain that \begin{equation*}
\alpha(\mathcal{X},\mathcal{Y})=\max{\{|\mathcal{Y}|-d(\mathcal{X})+1, |\mathcal{A}'|+|\mathcal{Y}|-|N(\mathcal{A}')|, |\mathcal{B}'|+|\mathcal{X}|-|N(\mathcal{B}')|\}}, \end{equation*} where $\mathcal{A}'$ and $\mathcal{B}'$ are minimum imprimitive subsets of $\mathcal{X}$ and $\mathcal{Y}$ respectively. Therefore, in order to estimate $\alpha(\mathcal{X},\mathcal{Y})$ accurately, more discussions about the sizes and the structures of the imprimitive subsets of $\mathcal{X}$ and $\mathcal{Y}$ are necessary.
\begin{claim}\label{imprimitive subset} Let $\mathcal{A}$ and $\mathcal{B}$ be imprimitive subsets of $\mathcal{X}$ and $\mathcal{Y}$ respectively, then \begin{align*} \mathcal{A}&=\prod_{i\in T_1}{\{A_i,\bar{A_i}\}}\times\prod_{i\in T_2}{\{A_i\}}\times\prod_{i\in [p]\setminus(T_1\sqcup T_2)}{[n_i]\choose t_i},~\text{for some disjoint $T_1, T_2\subseteq [p]$},\\ \mathcal{B}&=\prod_{i\in R_1}{\{B_i,\bar{B_i}\}}\times\prod_{i\in R_2}{\{B_i\}}\times\prod_{i\in [p]\setminus(R_1\sqcup R_2)}{[n_i]\choose s_i},~\text{for some disjoint $R_1, R_2\subseteq [p]$}, \end{align*} where $A_i\in {[n_i]\choose t_i}$, $B_i\in {[n_i]\choose s_i}$, $T_1\sqcup T_2\neq\emptyset$, $R_1\sqcup R_2\neq\emptyset$ and $T_2,R_2\neq[p]$. Furthermore, for each $i\in T_1$, $n_i=2t_i$ and for each $i\in R_1$, $n_i=2s_i$. \end{claim}
If $\Gamma=\prod_{i=1}^{p}S_{n_i}$ is imprimitive on $\mathcal{X}$, then from the definition we know that $\Gamma$ preserves a nontrivial partition $\{\mathcal{X}_j\}_{j=1}^{L}$ of $\mathcal{X}$. By projecting $\mathcal{X}_j$ to the $i$-th factor, we can obtain that $\bigsqcup_{j=1}^{L}(\mathcal{X}_j|_{i})=\mathcal{X}|_i={[n_i]\choose t_i}$ and $\Gamma|_{i}=S_{n_i}$ preserving this partition of $[n_i]\choose t_i$.
It is well known that for each $A_i\in {[n_i]\choose t_i}$, the stabilizer of $A_i$ is isomorphic to $S_{t_i}\times S_{n_i-t_i}$, which is a maximal subgroup of $S_{n_i}$ if $2t_i\neq n_i$ (see e.g. \cite{NB06}). Then by Theorem~\ref{primitivity}, we obtain that $S_{n_i}$ is primitive on $[n_i]\choose t_i$ unless $2t_i=n_i$, which means for the factors with $2t_i\neq n_i$ the partition $\bigsqcup_{j=1}^{L}(\mathcal{X}_j|_{i})$ of ${[n_i]\choose t_i}$ must be a trivial partition. Thus for each $j\in L$, $\mathcal{X}_j|_{i}$ is either a singleton in ${[n_i]\choose t_i}$, or $\mathcal{X}_j|_{i}={[n_i]\choose t_i}$.
When $2t_i=n_i$, it can be easily verified that the only imprimitive subset of $[n_i]\choose t_i$ has the form $\{A_i,\bar{A_i}\}$. Therefore, for the factors with $2t_i=n_i$, the partition $\bigsqcup_{j=1}^{L}(\mathcal{X}_j|_{i})$ of ${[n_i]\choose t_i}$ is either a trivial partition, or each partition block has the form $\mathcal{X}_j|_{i}=\{A_{i,j},\bar{A}_{i,j}\}$ for some $A_{i,j}\in {[n_i]\choose t_i}$.
Since each imprimitive subset of $\mathcal{X}$ can be seen as a block of a nontrivial partition of $\mathcal{X}$, we have $\mathcal{A}=\mathcal{X}_j$ for some $j\in [L]$. From the analysis above, we know that $\mathcal{A}|_i=\{A_i\}$ or $\{A_i,\bar{A_i}\}$ for some $A_i\in {[n_i]\choose t_i}$, or $\mathcal{A}|_i={[n_i]\choose t_i}$. Therefore, set $T_1\subseteq [p]$ such that for all $i\in T_1$, $2t_i=n_i$ and $\mathcal{A}|_i=\{A_i,\bar{A_i}\}$ for some $A_i\in {[n_i]\choose t_i}$; set $T_2\subseteq [p]$ such that for all $i\in T_1$, $\mathcal{A}|_i$ is a singleton, finally, we have \begin{equation*} \mathcal{A}=\prod_{i\in T_1}{\{A_i,\bar{A_i}\}}\times\prod_{i\in T_2}{\{A_i\}}\times\prod_{i\in [p]\setminus(T_1\sqcup T_2)}{[n_i]\choose t_i}. \end{equation*} The proof for the imprimitive subsets of $\mathcal{Y}$ is the same as that of $\mathcal{X}$. Thus, the claim holds.
By Claim~\ref{imprimitive subset}, we know that for the imprimitive subsets $\mathcal{A}$ and $\mathcal{B}$ above \begin{equation*}
|\mathcal{A}|=2^{|T_1|}\cdot\prod_{i\in [p]\setminus(T_1\sqcup T_2)}{n_i \choose t_i}~\text{and}~|\mathcal{B}|=2^{|R_1|}\cdot\prod_{i\in [p]\setminus(R_1\sqcup R_2)}{n_i \choose s_i}. \end{equation*} And since \begin{align*} N(\mathcal{A})&=\{B\in \mathcal{Y}:~A\cap B=\emptyset\text{ for some }A\in\mathcal{A}\}\\ &=\prod_{i\in T_1}{({A_i\choose s_i}\sqcup{\bar{A}_i\choose s_i})}\times\prod_{i\in T_2}{{[n_i]\setminus A_i}\choose s_i}\times\prod_{i\in [p]\setminus(T_1\sqcup T_2)}{[n_i]\choose s_i},\\ N(\mathcal{B})&=\{A\in \mathcal{X}:~A\cap B=\emptyset\text{ for some }B\in\mathcal{B}\}\\ &=\prod_{i\in R_1}{({B_i\choose t_i}\sqcup{\bar{B}_i\choose t_i})}\times\prod_{i\in R_2}{{[n_i]\setminus B_i}\choose t_i}\times\prod_{i\in [p]\setminus(R_1\sqcup R_2)}{[n_i]\choose t_i},\\ \end{align*} we have \begin{align*}
|N(\mathcal{A})|=2^{|T_1|}\cdot\prod_{i\in T_1}{{\frac{n_i}{2}}\choose s_i}\cdot\prod_{i\in T_2}{{n_i-t_i}\choose s_i}\cdot\prod_{i\in [p]\setminus(T_1\sqcup T_2)}{n_i\choose s_i},\\
|N(\mathcal{B})|=2^{|R_1|}\cdot\prod_{i\in R_1}{{\frac{n_i}{2}}\choose t_i}\cdot\prod_{i\in R_2}{{n_i-s_i}\choose t_i}\cdot\prod_{i\in [p]\setminus(R_1\sqcup R_2)}{n_i\choose t_i}. \end{align*}
Now we can estimate quantities $|\mathcal{A}'|+|\mathcal{Y}|-|N(\mathcal{A}')|$ and $|\mathcal{B}'|+|\mathcal{X}|-|N(\mathcal{B}')|$.
\begin{claim}\label{size estamitae}
With the assumptions in the theorem, for all imprimitive subsets $\mathcal{A}\subseteq\mathcal{X}$ and $\mathcal{B}\subseteq\mathcal{Y}$, $|\mathcal{Y}|-d(\mathcal{X})+1> |\mathcal{A}|+|\mathcal{Y}|-|N(\mathcal{A})|$, and $|\mathcal{Y}|-d(\mathcal{X})+1> |\mathcal{B}|+|\mathcal{X}|-|N(\mathcal{B})|$. \end{claim}
We prove the claim by estimating the difference directly. Denote \begin{align*}
&~D_1=|N(\mathcal{A})|-|\mathcal{A}|-d(\mathcal{X})+1\text{~and}\\
D_2&=|\mathcal{Y}|-|\mathcal{X}|+|N(\mathcal{B})|-|\mathcal{B}|-d(\mathcal{X})+1 \end{align*}
to be the differences between $|\mathcal{Y}|-d(\mathcal{X})+1$ and, respectively, $|\mathcal{A}|+|\mathcal{Y}|-|N(\mathcal{A})|$ and $|\mathcal{B}|+|\mathcal{X}|-|N(\mathcal{B})|$. Set $d_1=\frac{D_1}{|N(\mathcal{A})|}$, $d_2=\frac{D_2}{|\mathcal{X}|}$.
Then, we have $d_1=1-\beta_1-\beta_2+\theta$, $d_2=\delta+\eta_0\cdot(1-\eta_1-\eta_2)+\theta'$, where $\theta=|N(\mathcal{A})|^{-1}$, $\delta=\frac{|\mathcal{Y}|-|\mathcal{X}|}{|\mathcal{X}|}$, $\eta_0=\frac{|N(\mathcal{B})|}{|\mathcal{X}|}$, $\theta'=|\mathcal{X}|^{-1}$, $\beta_1=\frac{|\mathcal{A}|}{|N(\mathcal{A})|}$, $\beta_2=\frac{d(\mathcal{X})}{|N(\mathcal{A})|}$, $\eta_1=\frac{|\mathcal{B}|}{|N(\mathcal{B})|}$, and $\eta_2=\frac{d(\mathcal{X})}{|N(\mathcal{B})|}$.
Since ${n_i\choose t_i}\cdot{{n_i-t_i}\choose s_i}={n_i\choose s_i}\cdot{{n_i-s_i}\choose t_i}$ for each $i\in [p]$, we have $1/{{n_i-t_i}\choose s_i}={n_i\choose t_i}/({n_i\choose s_i}\cdot{{n_i-s_i}\choose t_i})$ for each $i\in[p]$. This yields that \begin{align*}
&~~~~\beta_1=\prod_{i\in [p]}\frac{{n_i \choose t_i}}{{n_i\choose s_i}}\cdot\prod_{i\in T_1\sqcup T_2}\frac{1}{{{n_i-s_i}\choose t_i}},~~\beta_2=\frac{1}{2^{|T_1|}}\cdot\prod_{i\in [p]\setminus(T_1\sqcup T_2)}\prod_{j=0}^{s_i-1}(1-\frac{t_i}{n_i-j}),\\
\eta_1&=\prod_{i\in [p]}\frac{{n_i \choose s_i}}{{n_i\choose t_i}}\cdot\prod_{i\in R_1\sqcup R_2}\frac{1}{{{n_i-t_i}\choose s_i}},~~\eta_2=\prod_{i\in [p]}\frac{{n_i \choose s_i}}{{n_i\choose t_i}}\cdot\frac{1}{2^{|R_1|}}\cdot\prod_{i\in [p]\setminus(R_1\sqcup R_2)}\prod_{j=0}^{t_i-1}(1-\frac{s_i}{n_i-j}). \end{align*}
By the assumptions, we know that $n_i\geq s_i+t_i+1\geq5$, $\prod_{i\in [p]}\frac{{n_i \choose t_i}}{{n_i\choose s_i}}\leq 1$ and ${{n_i-s_i}\choose t_i}\geq {\lceil\frac{n_i}{2}\rceil\choose t_i}\geq \frac{n_i}{2}$. Since $T_1\sqcup T_2\neq\emptyset$, $R_1\sqcup R_2\neq\emptyset$ and $T_2,R_2\neq[p]$, we can obtain \begin{align*}
\beta_1&\leq\prod_{i\in T_1\sqcup T_2}\frac{1}{{{n_i-s_i}\choose t_i}}\leq\max\limits_{i\in (T_1\sqcup T_2)}{\{(\frac{2}{n_i+2})^{|T_1|}\cdot(\frac{2}{n_i})^{|T_2|}\}},\\
&~\beta_2\leq\frac{1}{2^{|T_1|}}\cdot\max\limits_{i\in[p]\setminus (T_1\sqcup T_2)}\{(1-\frac{4n_i-6}{n_i(n_i-1)})^{p-(|T_1|+|T_2|)}\}, \end{align*} and \begin{align*}
\eta_1&\leq(1+\delta)\cdot\prod_{i\in R_1\sqcup R_2}\frac{1}{{{n_i-t_i}\choose s_i}}\leq(1+\delta)\cdot\max\limits_{i\in (R_1\sqcup R_2)}{\{(\frac{2}{n_i+2})^{|R_1|}\cdot(\frac{2}{n_i})^{|R_2|}\}},\\
&~~~~~~~~~\eta_2\leq(1+\delta)\cdot\frac{1}{2^{|R_1|}}\cdot\max\limits_{i\in[p]\setminus (R_1\sqcup R_2)}\{(1-\frac{4n_i-6}{n_i(n_i-1)})^{p-(|R_1|+|R_2|)}\}. \end{align*} This leads to \begin{flalign*} \beta_1+\beta_2\leq\left\{\begin{array}{ll}1-\min\limits_{i\neq j\in[p]}\{\frac{6}{n_i}-\frac{2}{n_i-1}-\frac{2}{n_j}\}, &~\text{if}~T_2\neq\emptyset;\\ \frac{1}{2}-\min\limits_{i\neq j\in[p]}\{\frac{3}{n_i}-\frac{1}{n_i-1}-\frac{2}{n_j+2}\}, &~\text{otherwise};\end{array}\right. \end{flalign*} and \begin{flalign*} \frac{\eta_1+\eta_2}{1+\delta}\leq\left\{\begin{array}{ll}1-\min\limits_{i\neq j\in[p]}\{\frac{6}{n_i}-\frac{2}{n_i-1}-\frac{2}{n_j}\}, &~\text{if}~R_2\neq\emptyset;\\ \frac{1}{2}-\min\limits_{i\neq j\in[p]}\{\frac{3}{n_i}-\frac{1}{n_i-1}-\frac{2}{n_j+2}\}, &~\text{otherwise}.\end{array}\right. \end{flalign*} Since $5\leq n_i\leq\frac{7}{4} n_j$ for all distinct $i,j\in[p]$, thus we have $\beta_1+\beta_2,\frac{\eta_1+\eta_2}{1+\delta}\leq 1$. Therefore, \begin{align*} &~~~~~~~~~~~~~~~~~~~~~~~~~d_1=1-\beta_1-\beta_2+\theta> 1-\beta_1-\beta_2\geq0,\\ d_2&=\delta+\eta_0\cdot(1-\eta_1-\eta_2)+\theta'=\delta\cdot(1-\eta_0\cdot\frac{\eta_1+\eta_2}{1+\delta})+\eta_0\cdot(1-\frac{\eta_1+\eta_2}{1+\delta})+\theta'>0. \end{align*} Thus, the claim holds.
For each pair of non-empty cross-intersecting families $(\mathcal{A},\mathcal{B})\in 2^{\mathcal{X}}\times2^{\mathcal{Y}}$, $\mathcal{A}\cup \mathcal{B}$ forms a nontrivial independent set of $G(\mathcal{X},\mathcal{Y})$. Therefore, by Claim~\ref{size estamitae}, the inequality~(\ref{eq02}) holds.
To complete the proof, we need to characterize all the nontrivial fragments in $\mathcal{F}(\mathcal{X})$. As a direct consequence of Claim~\ref{size estamitae}, every fragment of $G(\mathcal{X},\mathcal{Y})$ is primitive. Hence, by Theorem~\ref{key00}, when $\prod_{i\in [p]}{n_i \choose t_i}<\prod_{i\in [p]}{n_i\choose s_i}$, $\mathcal{X}$ has only $1$-fragments.
When $\prod_{i\in [p]}{n_i \choose t_i}=\prod_{i\in [p]}{n_i\choose s_i}$, suppose there are nontrivial fragments in $\mathcal{F}(\mathcal{X})$. W.l.o.g., assume that $\mathcal{S}$ is a minimal-sized nontrivial fragment in $\mathcal{X}$. By Theorem~\ref{key00}, $\mathcal{S}$ is semi-imprimitive. Since for any two different elements $A,B\in\mathcal{X}$, $|N(A)\cap N(B)|<\prod_{i\in [p]}{{n_i-t_i}\choose s_i}-1$. Therefore, there are no $2$-fragments in $\mathcal{F(\mathcal{X})}$. By Proposition~\ref{fragment2}, $\mathcal{S}$ is balanced.
Now we are going to prove the non-existence of such $\mathcal{S}$ by analyzing its size and structure, which will yield that $\mathcal{X}$ also has only $1$-fragments when $\prod_{i\in [p]}{n_i \choose t_i}=\prod_{i\in [p]}{n_i\choose s_i}$.
For each $A=\prod_{i\in [p]}A_i\in \mathcal{S}$, let $\Gamma_A=\prod_{i\in [p]}(S_{A_i}\times S_{\bar{A}_i})$, $\Gamma_{\mathcal{S}}=\{\sigma\in \Gamma:~\sigma(\mathcal{S})=\mathcal{S}\}$ and $\Gamma_{A,\mathcal{S}}=\{\sigma\in \Gamma_A:~\sigma(\mathcal{S})=\mathcal{S}\}$. We claim that there exists a subset $C\in \mathcal{S}$ such that $\Gamma_{C}\neq \Gamma_{C,\mathcal{S}}$. Otherwise, for any two different subsets $B,B'\in \mathcal{S}$, we have $\Gamma_{B}=\Gamma_{B,\mathcal{S}}$ and $\Gamma_{B'}= \Gamma_{B',\mathcal{S}}$. Since $\Gamma_{B,\mathcal{S}}$ and $\Gamma_{B',\mathcal{S}}$ are both subgroups of $\Gamma_{\mathcal{S}}$, we have $\langle\Gamma_{B},\Gamma_{B'}\rangle$ is a subgroup of $\Gamma_{\mathcal{S}}$. Let $T\subseteq [p]$ be the factors where $B'_i=B_i~(\text{or}~\bar{B}_i~\text{if}~2t_i=n_i)$, write $$\Gamma_B=\prod_{i\in T}(S_{B_i}\times S_{\bar{B}_i})\times\prod_{i\in [p]\setminus T}(S_{B_i}\times S_{\bar{B}_i}),$$ then we have, $$\Gamma_{B'}=\prod_{i\in T}(S_{B_i}\times S_{\bar{B}_i})\times\prod_{i\in [p]\setminus T}(S_{B'_i}\times S_{\bar{B}'_i}).$$ Since $\langle S_{B_i}\times S_{\bar{B}_i}, S_{B'_i}\times S_{\bar{B}_i'}\rangle=S_{n_i}$ for each $B'_i\ne B_i~(\text{and}~B'_i\ne\bar{B}_i~\text{if}~2t_i=n_i)$, we have $$\langle\Gamma_{B},\Gamma_{B'}\rangle=\prod_{i\in T}(S_{B_i}\times S_{\bar{B}_i})\times\prod_{i\in [p]\setminus T}S_{n_i}.$$ Therefore, for some fixed $B\in\mathcal{S}$, $\Gamma_{\mathcal{S}}$ contains $\prod_{i\in T'}(S_{B_i}\times S_{\bar{B}_i})\times\prod_{i\in [p]\setminus T'}S_{n_i}$ as a subgroup, where
$$T'=\{i|i\in[p],\text{ such that }A_i=B_i~(\text{or }\bar{B}_i\text{ if }2t_i=n_i)\text{ for all }A\in\mathcal{S}\}.$$
When $T'=\emptyset$, we have $\Gamma_{\mathcal{S}}=\prod_{i\in [p]}S_{n_i}$, thus $\mathcal{S}=\mathcal{X}$, yielding a contradiction. When $T'\neq \emptyset$, if $|T'|=1$, w.l.o.g., taking $T'=\{1\}$, we have $(S_{B_1}\times S_{\bar{B}_1})\times\prod_{i\in [p]\setminus \{1\}}S_{n_i}\subseteq\Gamma_{\mathcal{S}}$. Therefore, since $\mathcal{S}\neq\mathcal{X}$, from the definition of $T'$ we have $$\mathcal{S}=\{B_1\}\times\prod_{i\in [p]\setminus \{1\}}{[n_i]\choose t_i}, \text{~or~} S=\{B_1,\bar{B}_1\}\times\prod_{i\in [p]\setminus \{1\}}{[n_i]\choose t_i}\text{~when $2t_1=n_1$}.$$
In both cases, $|\mathcal{S}|<\frac{\alpha(\mathcal{X},\mathcal{Y})}{2}$. If $|T'|\geq 2$, we have $$\mathcal{S}\subseteq\{B_{i_0}\}\times\prod_{i\in [p]\setminus \{i_0\}}{[n_i]\choose t_i}, \text{~or~} S\subseteq\{B_{i_0},\bar{B}_{i_0}\}\times\prod_{i\in [p]\setminus \{i_0\}}{[n_i]\choose t_i}\text{~when $2t_{i_0}=n_{i_0}$},$$
for some $i_0\in T'$. Therefore, when $T'\neq \emptyset$, we always have $|\mathcal{S}|<\frac{\alpha(\mathcal{X},\mathcal{Y})}{2}$, which contradicts the fact that $\mathcal{S}$ is balanced. Hence, the existence of $C$ is guaranteed.
By Proposition~\ref{fragment2} we have that $[\Gamma_{C}:\Gamma_{C,\mathcal{S}}]$, the index of $\Gamma_{C,\mathcal{S}}$ in $\Gamma_{C}$, equals 2. Now let $\Gamma_{C,\mathcal{S}}[C_i]$ be the projection of $\Gamma_{C,\mathcal{S}}$ onto $S_{C_i}$, $\Gamma_{C,\mathcal{S}}[C_i]$ must be a subgroup of $S_{C_i}$ of index no greater than 2. Thus $\Gamma_{C,\mathcal{S}}[C_i]=S_{C_i}$ or $A_{C_i}$. Since $\Gamma_C=\prod_{i\in [p]}(S_{C_i}\times S_{\bar{C}_i})$, we know that $\Gamma_{C,\mathcal{S}}=\prod_{i\in [p]\setminus\{j\}}(S_{C_i}\times S_{\bar{C}_i})\times(A_{C_{j}}\times S_{\bar{C}_{j}})$ or $\prod_{i\in [p]\setminus\{j\}}(S_{C_i}\times S_{\bar{C}_i})\times(S_{C_{j}}\times A_{\bar{C}_{j}}$), for some $j\in [p]$.
Since for all $i\in [p]$, $t_i=|B_i\cap C_i|+|B_i\cap \bar{C}_i|$ for each pair $B,C\in \mathcal{S}$. If $|B_i\cap C_i|>1$, let $s,t\in B_i\cap C_i$, then the transposition $(s~t)$ fixes both $C_i$ and $B_i$. Taking $i=j$, the semi-imprimitivity of $\mathcal{S}$ implies that $(s~t)\in \Gamma_{C,\mathcal{S}}|_{S_{C_{j}}\times S_{\bar{C}_{j}}}$. This yields $\Gamma_{C,\mathcal{S}}|_{S_{C_{j}}\times S_{\bar{C}_{j}}}=S_{C_{j}}\times A_{\bar{C}_{j}}$. From this process it follows that, for each $B\in \mathcal{S}$, there exists at most one of $|B_j\cap C_j|$ and $|B_j\cap \bar{C}_j|$ to be greater than $1$. Note that if $B_j\in\bar{C}_j$, then $S_{C_j}$ and $S_{B_j}$ fix both $C_j$ and $B_j$, i.e., $S_{C_j}\times S_{B_j}\subseteq \Gamma_{C,\mathcal{S}}|_{S_{C_{j}}\times S_{\bar{C}_{j}}}$. Since $\Gamma_{C,\mathcal{S}}|_{S_{C_{j}}\times S_{\bar{C}_{j}}}=A_{C_{j}}\times S_{\bar{C}_{j}}$ or $S_{C_{j}}\times A_{\bar{C}_{j}}$, and neither $A_{C_{j}}\times S_{\bar{C}_{j}}$ nor $S_{C_{j}}\times A_{\bar{C}_{j}}$ contains $S_{C_j}\times S_{B_j}$. Therefore, we obtain that $|B_j\cap C_j|=1$ for each $B\in \mathcal{S}$, or $|B_j\cap C_j|=t_j-1$ for each $B\in \mathcal{S}$.
We claim that for both cases, $\mathcal{S}$ can not be balanced.
Suppose $|B_j\cap C_j|=1$ for each $B\in \mathcal{S}$. W.l.o.g., assume $B_j\cap C_j=\{1\}$ for some $B\in \mathcal{S}$. From the semi-imprimitivity of $\mathcal{S}$, we know that for all $\gamma\in \Gamma,~\gamma(\mathcal{S})\cap\mathcal{S}=\emptyset, ~\mathcal{S}$ or $\{A\}$ for some $A\in \mathcal{S}$. Thus $(\gamma(\mathcal{S})\cap\mathcal{S})|_j=\emptyset,~\mathcal{S}|_j$ or $\{A_j\}$ for some $A_j\in {[n_j]\choose t_j}$. If $t_j>2$, then $|B_j\cap \bar{C}_j|\geq 2$, so $\Gamma_{C,\mathcal{S}}|_{S_{C_{j}}\times S_{\bar{C}_{j}}}=A_{C_{j}}\times S_{\bar{C}_{j}}$. On the other hand, we can find distinct $s,t\in C_j$ such that $(1~s~t)(B_j)=B_j\setminus\{1\}\cup\{s\}\in \mathcal{S}|_j$ since $(1~s~t)\in A_{C_j}$. Then $(1~s)(\mathcal{S}|_j)$ has more than one element of $\mathcal{S}|_j$, therefore $(1~s)\in \Gamma_{C,\mathcal{S}}|_{S_{C_{j}}\times S_{\bar{C}_{j}}}$. This contradiction proves that $t_j=2$. Thus $\mathcal{S}|_j=\mathcal{C}=\{A_j\in{[n_j]\choose 2}:~1\in A_j\}$. Otherwise, w.l.o.g., assume $C_j=\{1,2\}$ and there exists $B\in \mathcal{S}$ such that $B_j\cap C_j=\{2\}$. Since $\Gamma_{C,\mathcal{S}}|_{S_{C_{j}}\times S_{\bar{C}_{j}}}=A_{C_{j}}\times S_{\bar{C}_{j}}$ or $S_{C_{j}}\times A_{\bar{C}_{j}}$, we have $\mathcal{C}\subseteq \mathcal{S}|_j$ and $\mathcal{C'}=\{A_j\in{[n_j]\choose 2}:~2\in A_j\}\subseteq\mathcal{S}|_j$. Thus $\mathcal{S}|_j=\mathcal{C}\cup\mathcal{C'}$. This yields $\Gamma_{C,\mathcal{S}}|_{S_{C_{j}}\times S_{\bar{C}_{j}}}=S_{C_{j}}\times S_{\bar{C}_{j}}$, leading to a contradiction.
Suppose now $|B_j\cap C_j|=t_j-1>1$ for each $B\in \mathcal{S}$. Similarly, we can prove that $n_j-t_j=2$, which contradicts $n_j\geq s_j+t_j+1$ and $2\leq s_j$, $t_j\leq \frac{n}{2}$. Therefore, for each $B\in \mathcal{S}$, $|B_j\cap C_j|=1$.
From the analysis above, we know that for each $B\in \mathcal{S}$, $B_j=\{1,b\}$ for some $b\in[n_j]$. Thus, for each $B\in \mathcal{S}$, we have $\Gamma_{B,\mathcal{S}}|_{S_{B_{j}}\times S_{\bar{B}_{j}}}=A_{B_{j}}\times S_{\bar{B}_{j}}$, and $\Gamma_{B,\mathcal{S}}=\prod_{i\in[p]\setminus \{j\}}(S_{B_i}\times S_{\bar{B}_i})\times(A_{B_{j}}\times S_{\bar{B}_{j}})$ since $[\Gamma_{B}:\Gamma_{B,\mathcal{S}}]=2$. Therefore $\Gamma_{\mathcal{S}}$ contains $$\langle\Gamma_{B,\mathcal{S}},\text{~for~all~}B\in \mathcal{S}\rangle=\prod_{i\in T''}(S_{C_i}\times S_{\bar{C}_i})\times\prod_{i\in [p]\setminus (T''\cup\{j\})}S_{n_i}\times S_{[n_j]\setminus\{1\}}$$
as a subgroup, where $T''=\{i|i\in[p],\text{ such that }B_i=C_i~(\text{or }\bar{C}_i\text{ if }2t_i=n_i)\text{ for all }B\in\mathcal{S}\}$. Similarly, by arguing the structure of $\mathcal{S}$, if $T''\neq\emptyset$, we can prove that $|\mathcal{S}|<\frac{\alpha(\mathcal{X},\mathcal{Y})}{2}$. Thus we have $T''=\emptyset$ and $\mathcal{S}=\prod_{i\in[p]\setminus\{j\}}{[n_i]\choose t_i}\times \mathcal{C}$.
Since $\mathcal{S}$ is balanced, $\prod_{i\in [p]}{n_i \choose t_i}=\prod_{i\in [p]}{n_i\choose s_i}$ and $|\mathcal{S}|=\prod_{i\in[p]\setminus\{j\}}{n_i\choose t_i}\cdot (n_j-1)$, we have \begin{equation}\label{eq05} 2\prod_{i\in[p]\setminus\{j\}}{n_i\choose t_i}\cdot (n_j-1)=\prod_{i\in[p]\setminus\{j\}}{n_i\choose t_i}\cdot{n_j\choose 2}-\prod_{i\in[p]\setminus\{j\}}{{n_i-s_i}\choose t_i}\cdot{{n_j-s_j}\choose 2}+1, \end{equation} which means $n_j$ must be an integral zero of the following function \begin{align*} H(x)=(1-a_0)\cdot x^2-(5-a_0\cdot(2s_j+1))\cdot x+(2b_0+4-a_0\cdot (s_j^2+s_j)), \end{align*} where $a_0=\prod_{i\in [p]\setminus \{j\}}\frac{{{n_i-s_i}\choose t_i}}{{n_i\choose t_i}}$ and $b_0=\prod_{i\in [p]\setminus \{j\}}{{n_i\choose t_i}^{-1}}$. Since $n_j\geq3+s_j$ and $2\leq s_j\leq \frac{n_j}{2}$, by Vieta's formulas for quadratic polynomials, there is no such $n_j$ satisfying $H(n_j)=0$ when $s_j\geq 3$. Hence $\mathcal{S}=\prod_{i\in[p]\setminus\{j\}}{[n_i]\choose t_i}\times \mathcal{C}$ is a nontrivial balanced fragment of $\mathcal{X}$ if and only if $t_j=s_j=2$ and equation~(\ref{eq05}) holds. Using the fact that $\frac{{{n_i-s_i}\choose t_i}}{{n_i\choose t_i}}\leq (1-\frac{s_i}{n_i})(1-\frac{s_i}{n_i-1})$ and the assumption $n_i\leq \frac{7}{4}n_j$ for distinct $i,j\in[p]$, it can be easily verified that the LHS of equation~(\ref{eq05}) is strictly less than the RHS when $s_j=2$. Therefore, $\mathcal{S}$ can not be balanced.
This completes the proof. \end{proof}
\section{Concluding remarks}
In this paper we have investigated two multi-part generalizations of the cross-intersecting theorems. Our main contribution is determining the maximal size and the corresponding structures of the families for both trivially and nontrivially (with the non-empty restriction) cross-intersecting cases.
The method we used for the proof was originally introduced by Wang and Zhang in~\cite{WZ11}, which was further generalized to the bipartite case in~\cite{WZ13}. This method can deal with set systems, finite vector spaces and permutations uniformly. It is natural to ask whether we can extend the single-part cross-intersecting theorems for finite vector spaces and permutations to the multi-part case. It is possible for permutations when considering the case without the non-empty restriction, and we believe it is also possible for finite vector spaces. But when it comes to the case where the families are non-empty, as far as we know, there is still no result for finite vector spaces and permutations.
For single-part families $\mathcal{A}$ and $\mathcal{B}$, it is natural to define cross-t-intersecting as $|A\cap B|\geq t$ for each pair of $A\in \mathcal{A}$ and $B\in \mathcal{B}$. But for multi-part families, when defining cross-t-intersecting between two families, the simple extension of the definition for single-part case can be confusing. Therefore, a reasonable definition and related problems for multi-part cross-$t$-intersecting families are also worth considering.
\end{document}
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\begin{document}
\title{Simultaneous elastic shape optimization for a domain splitting in bone tissue engineering}
\begin{abstract} This paper deals with the simulateneous optimization of a subset ${\mathcal{O}}_0$ of some domain $\Omega$ and its complement ${\mathcal{O}}_1 = \Omega \setminus \overline {\mathcal{O}}_0$ both considered as separate elastic objects subject to a set of loading scenarios. If one asks for a configuration which minimizes the maximal elastic cost functional both phases compete for space since elastic shapes usually get mechanically more stable when being enlarged. Such a problem arises in biomechanics where a bioresorbable polymer scaffold is implanted in place of lost bone tissue and in a regeneration phase new bone tissue grows in the scaffold complement via osteogenesis. In fact, the polymer scaffold should be mechanically stable to bear loading in the early stage regeneration phase and at the same time the new bone tissue grown in the complement of this scaffold should as well bear the loading. Here, this optimal subdomain splitting problem with appropriate elastic cost functionals is introduced and existence of optimal two phase configurations is established for a regularized formulation. Furthermore, based on a phase field approximation a finite element discretization is derived. Numerical experiments are presented for the design of optimal periodic scaffold microstructure. {\textbf{Keywords:} elastic shape optimization, phase field model, homogenization, bone microstructure} \end{abstract}
\section{Introduction}
In this paper we investigate an elastic shape optimization with two competing objects ${\mathcal{O}}_0$ and ${\mathcal{O}}_1$ which result from a splitting of a given domain $\Omega\subset\mathbb{R}^d$, i.e. ${\mathcal{O}}_1 = \Omega \setminus \overline {\mathcal{O}}_0$, where $d=3$ turns out to be the interesting and application relevant case. Both objects obey a constitutive law of linearized elasticity with in general different elasticity tensors and corresponding boundary conditions. The stored elastic energy of an object is a measure for its elastic rigidity. Hence, for each object we take into account a cost functional which is a monotone decreasing function of the stored elastic energies for a set of loading scenarios induced by prescribed boundary displacements. As the total objective functional we then consider the maximum of the resulting cost function values for the two objects.
Typically, the optimization of a single object ${\mathcal{O}} \subset \Omega$ is complemented by a volume constraint or a volume penalty \cite{Al04}. Otherwise, usually
${\mathcal{O}}=\Omega$ would be maximally rigid and thus optimal. An equality constraint for the volume can either be ensured by a Cahn--Hilliard-type $H^{-1}$-gradient flow \cite{ZhWa07} or a Lagrange-multiplier ansatz \cite{AlJoTo03,BoCh03,LiKoHu05}. An inequality constraint $|{\mathcal{O}}|\leq V$ has been implemented in \cite{WaWaGu03, WaZhDi04, GuZhWa05} using a Lagrange multiplier approach. In our setup the two objects simultaneously compete for space and we explicitly ask for an optimal distribution of space to the two objects.
Such a class of shape optimization problems is generically ill-posed and one observes the onset of microstructures along a minimizing sequence of the the cost functional which are associated with a weak but not strong convergence of the characteristic functions of the elastic objects along a minimizing sequence. To avoid this ill-posedness we replace the void by some weak material and add a perimeter penalty to the objective functional (cf. \cite{AmBu93} for a scalar problem). Alternatively, one might relax the problem, explicitly allowing for microstructures and considering a quasi--convexification of the integrand of the cost by taking the infimum over all possible microstructures. For this relaxation rank-$d$ sequential laminates are known to be optimal for compliance minimization for a single load scenario \cite{AlBoFr97}. Even in the multiple load scenario sequential laminates of possible higher rank are optimal.
To compute approximate solutions of elastic shape optimization problems there are different alternatives to an explicit discretization of the objects to be optimized. An implicit representation of shapes via level sets \cite{Al04,AlBoFr97,AlGoJo04,AlJoTo02} can be used and combined with a topological optimization \cite{AlJoMa04}. In \cite{AlDa14}, Allaire\,et\,al. studied the optimization of multiphase materials with a regularization of the shape derivative in the level set context. An implicit description of shapes via phase fields---the approach also employed in this paper---is both analytically and numerically attractive. Phase-field models originated in the physical description of multiphase materials, where the integrant of a chemical bulk energy has two minima corresponding to the two phases and an additional diffusive interfacial energy encodes the preference for a small material interface. This approach has been employed to elastic shape optimization by Wang and Zhou \cite{ZhWa07} and Blanck et al. \cite{BlGa14, BlGa16}. Guo\,et\,al.\ \cite{GuZhWa05} described the characteristic function of the object by the concatenation of a smoothed Heaviside function with a level-set function, where the smoothed Heaviside function acts like a phase-field profile. Wei and Wang \cite{WeWa09} encoded the object via a piecewise constant level set function closely related to the phase-field approach.
Our set-up is similar to the study of two-phase materials, where one phase is for example a good electrical, but poor heat conductor, and vice-versa for the other phase. The problem of finding a microstructure that maximizes the sum of both heat and electrical conductivity in this setting has been studied in \cite{Torquato:2002gh}. In particular, it has been conjectured that in this scalar case, domains bounded by periodic minimal surfaces (for example the Schwarz P surface) are optimal \cite{Torquato:2004ey}. Further analysis in \cite{Silvestre:2007hn}, however, casts doubt on this conjecture.
Motivated by an application from biomechanics and medicine, in the present work we consider an elasticity state equation, as opposed to a scalar problem, and more general objective functions where multiple load cases can be combined.
The paper is organized as follows. In Section~\ref{sec:bio} we discuss the biomechanical application of an optimal design of a polymer scaffolds for bone tissue engineering. Then the associated state equations are described in Section~\ref{sec:state} and in Section~\ref{sec:cost} we derive a suitable cost functional for the simultaneous optimization of both phases. An existence result for a regularized problem with soft instead of void material outside the actual objects is given in Section~\ref{sec:exist}. Section~\ref{sec:bone} deals with the actual application of optimal polymer scaffold microstructures and in Section~\ref{sec:fem} we derive a phase field approximation and discretize it based on a finite element approach. Finally, in Section~\ref{sec:num} numerical results are presented for different material properties of the two phases and different sets of loading scenarios and Section~\ref{sec:conclusions} provides some conclusions and mentions a number of open questions.
\section{A biomechanical optimization problem}\label{sec:bio} As an application of simultaneous two phase optimization let us consider the optimal design of polymer scaffolds for bone tissue engineering. Globally, bone loss due to trauma, osteoporosis, or osteosarcoma comprises a major reason for disability. To this day, autograft, i.e., a graft of bone tissue from a different place in the same body, remains the gold standard for large scale bone loss. This might be impossible for example due to donor site morbidity and limited availability. Therefore, a number of substitutes are being explored \cite{Bhatt12, Campana14}. Among these substitutes, polymer scaffolds that function as a tissue expander (creating initial void space and allowing for tissue in-growth) show tremendous potential for bone regeneration \cite{Teo:2015fu, Goh:2015jy, Schuckert:2008hg,Schantz:2006ka}.
An ideal scaffold must satisfy a number of different criteria, apart from the requirement of biocompatibility. In the initial phase of regeneration, the scaffold must provide adequate mechanical stability and the appropriate mechano-biological signal to promote osteogenesis. After new bone is formed and functional bridging through the bone defect is achieved, the scaffold should be completely resorbed allowing for a restoration of the original skeleton. With the advent of easily accessible additive manufacturing technology one possibility for such bone scaffolds are porous structures made from bioresorbable polymers, for example polycaprolactone (PCL) \cite{Poh16} (cf. the examples of printer polymer scaffolds in Figure \ref{fig:PrintedScaffolds}). Over the long regeneration time scale of more than one year, in-vivo evidence shows that such polymers degrade via bulk-erosion, that is, they lose molecular weight (and therefore mechanical stability) without a significant change in the occupied volume \cite{Lam09}, before finally being completely resorbed.
Usually, the implantation of a PCL scaffold is accompanied by a metal implant to provide further stability (see \cite{Henkel13} for illustrations of such procedures in an ovine model). It would, however, be advantageous if such metal implants were not necessary and the implanted structure, together with the regrown bone tissue, were capable of bearing the occurring mechanical loading during the regeneration time. These loading conditions depend on the stresses acting at the implant site. For example, if a section of the tibia is to be replaced, they would consist of unilateral compression (due to the weight of the patient) and shear (due to torsion). Optimization procedures to design microstructures for such implants with the goal of balancing mechanical stability of the scaffold and bone tissue regeneration have been explored for example in \cite{Adachi06}. Furthermore, as it is suggested by the optimality conjecture for competing phases \cite{Torquato:2004ey}, designs for bone scaffolds based on periodic minimal surfaces are under consideration \cite{Kapfer:2011kz}. Currently, the structure of the printed polymer strands still limits the achievable microstructures (cf. Figure \ref{fig:PrintedScaffolds}).
In the beginning of the regeneration process, the polymer scaffold alone has to be able to withstand the given loading conditions. Later in the regeneration process, the regrown bone tissue (which due to the effect of bulk erosion can only grow in the space that was \emph{not} occupied by the scaffold) has to bear this mechanical loading.
Thus, we are led to a shape optimization, where we simultaneously optimize the shape of the polymer and its complement which will be occupied by bone generated via osteogenesis. Thereby, we focus solely on this optimization problem and not on the dynamical process of the osteogenesis and the dissolving of the polymer. Furthermore, we assume that the polymer scaffold forms a spatially homogeneous microstructure and ask for the optimal shape of the polymer phase in a fundamental cell of the scaffold with affine period boundary conditions.
\begin{figure}
\caption{Three different examples of 3D printed polymer scaffolds based on periodic minimal surface designs (from left to right, Schwarz P, Gyroid, and Schwarz D) with a unit cell size of approximately $1\, mm^3$. Pictures courtesy of D.~Valainis.}
\label{fig:PrintedScaffolds}
\end{figure}
\section{State equation} \label{sec:state} We consider a domain $\Omega\subset \mathbb{R}^d$ ($d\geq2$) with Lipschitz boundary, which is split up in two subdomains ${\mathcal{O}}_0$ and ${\mathcal{O}}_1$ ($\overline {\mathcal{O}}_0 \cup \overline {\mathcal{O}}_1 = \overline \Omega$ and ${\mathcal{O}}_0 \cap {\mathcal{O}}_1 = \emptyset$) with corresponding characteristic functions $\chi_0$ and $\chi_1$, respectively. At first, we suppose the boundary of both domains to be Lipschitz and take into account displacement $u^m: {\mathcal{O}}_m \to \mathbb{R}^d$, with a decomposition \begin{align} u^m = \hat u^m + \tilde u^m\,, \end{align} where $\hat u^m \in H^{1,2}({\mathcal{O}}_m)^d$ is fixed and $\tilde u^m$ lies in a closed subspace $\mathcal{V}^m$ of $H^{1,2}({\mathcal{O}}_m)^d$. In explicit $\hat u^m$ and the choice of $\mathcal{V}^m$ determine the encountered boundary condition, e.g. Dirichlet or period boundary condition (see Section~\ref{sec:bone}) on $\partial {\mathcal{O}}^m \cap \partial \Omega$ and Neumann boundary conditions on $\partial {\mathcal{O}}^m \cap \Omega$. Here, we assume that $\mathcal{V}^m$ is such that Korn's inequality holds for displacements $u^m\in \mathcal{V}^m$, i.e. there exists a constant $C_K$ such that \begin{align*}
\|u^m\|_{L^2({\mathcal{O}}^m)} \leq C_K \|\varepsilon[u^m]\|_{L^2({\mathcal{O}}^m)}\,, \end{align*} where the strain tensor $\varepsilon[u]$ is given as $\frac12 (Du^T+Du)$ with $Du$ defining the Jacobian of the displacement $u$. Now, we take into account linearized elasticity and consider the elasticity tensors $C^m = (C^m_{ijkl})_{i,j,k,l = 1,\ldots, d}$ of both subdomains ($m=0,1$). For simplicity we assume that the two materials are isotropic and thus determined by the Lam{\'e}-Navier parameters $\mu^m>0$ and $\lambda^m>0$, i.e. $C \varepsilon[u^m] : \varepsilon[u^m] = 2 \mu \varepsilon[u^m] : \varepsilon[u^m] + \lambda \mathrm{div}(u^m) \mathrm{div}(u^m)$. A generalization allowing for anisotropic materials is straightforward. Then, the associated energy of a displacement $u\in H^{1,2}({\mathcal{O}}_m)^d$ is given by \begin{align}\label{eq:energy} E^m[\chi^m,u] = \frac12 \int_\Omega \chi^m \, C^m \varepsilon[u] : \varepsilon[u] {\,\mathrm{d}} x \,, \end{align} where $C^m A:B = \sum_{i,j,k,l} C^m_{ijkl} A_{ij} B_{kl}$ for $A,B \in \mathbb{R}^{d,d}$. Thereby, $C \varepsilon[u]$ is the stress tensor associated with the strain tensor $\varepsilon[u]$. Using Korn's inequality it is easy to see that there exists a unique displacement $u^m$ on ${\mathcal{O}}_m$ which minimizes the energy $E[\chi^m,\cdot]$. The corresponding weak form of the Euler Lagrange equations \begin{align}\label{eq:state} 0 = \partial_{u}E^m[\chi^m,u^m](\phi) = \int_\Omega \chi^m \, C^m \varepsilon[u^m] : \varepsilon[\phi] {\,\mathrm{d}} x \end{align} for all $\phi \in \mathcal{V}^m$ is the state equation for $u^m$ on ${\mathcal{O}}_m$. To express the dependence of $u^m$ on the shape of the subdomain ${\mathcal{O}}_m$ we also write $u^m[\chi^m]$.
\section{Cost functional} \label{sec:cost} As already discussed in the introduction the stored elastic energy of an object is a measure for its elastic rigidity. We assume that a set of different boundary conditions, encoded via $\hat u^m_l$ for $l=1, \ldots, L$, reflects typical loading conditions of the subdomain ${\mathcal{O}}_m$. For the sake of simplicity we consider them to be independent of the subdomain. Thus, we take into account a continuous function $g: \mathbb{R}^d \to \mathbb{R}$, which is supposed to be monoton decreasing in each argument and define \begin{align*} J^m[\chi^m] := g(E^m_{1}, \ldots, E^m_{d}) \end{align*} as the cost associated with the set of loading conditions, where $E^m_{l} := E^m[\chi^m, u^m_l[\chi^m]]$ is the stored elastic energy of the equilibrium solution $u^m_l$ of \eqref{eq:state} corresponding to the prescribed $\hat u^m_l$. Aiming for an optimization of the expected value of the total energy we would have to choose \begin{align*} g(E_1,\ldots, E_L) = - (E_1 + \ldots + E_L)\,. \end{align*} In this paper we take into account \begin{align*} g(E_1,\ldots, E_L) = \left(\sum_{l=1,\ldots,L} (E_l)^{-p} \right)^{\frac1p}\,. \end{align*} For $p\to\infty$ the resulting cost converges to the maximal inverse total energy $$\max_{l=1,\ldots,L} (E_l)^{-1} = (\min_{l=1,\ldots,L} E_l)^{-1}$$ and thus represents a worst case optimization problem where solely the loading scenario with the smallest stored elastic energy is taken into account.
Now, the objective functional associated with the domain splitting $\chi^0 =\chi$ and $\chi^1=1-\chi$ for a characteristic function $\chi$ can be defined as \begin{align}\label{eq:totalcost} J[\chi] = \max \left(J^0[\chi],J^1[1-\chi]\right)\,. \end{align} This objective functionals reflects the competition of both subdomains aiming to increase their rigidity via domain enlargement with significant payoff in the cost $J^m[\chi^m]$. Due to this competition no volume constraint or penalty is needed to formulate the shape optimization problem.
\section{A hard--soft approximation and a perimeter regularization} \label{sec:exist} It is advantageous to approximate the characteristic function $\chi^m$ for each object ${\mathcal{O}}_m$ by $\chi^m + \delta (1-\chi^m)$ for some small constant $\delta >0$. In explicit, instead of considering the elasticity problem on ${\mathcal{O}}_m$ we take into account an elasticity problem on the domain $\Omega$. To obtain a well-posed optimization problem, the void phase on the complementary set $\Omega \setminus \overline {\mathcal{O}}_m$ is replaced by a very soft phase. Consequently we consider $u^m$, $\hat u^m$, and $\tilde u^m$ as functions in $H^{1,2}(\Omega)^d$ and $\mathcal{V}^m$ as a subspace of $H^{1,2}(\Omega)^d$.
Then, Korn's inequality is assumed to hold on this extended function space. Furthermore, taking into account characteristic functions $\chi\in BV(\Omega, \{0,1\})$ we add a penalty given by the perimeter of the subdomains $\eta |D\chi|(\Omega)$ for some $\eta>0$. Then, we obtain the following straightforward existence theorem for a minimizer. \begin{Theorem}[Existence of an optimal subdomain splitting]\label{thm:exist} For $\delta >0$, given displacements $\hat u^m_l$ with $m=0,1$ and $l=1,\ldots, L$, elastic energies \begin{align}\label{eq:energydelta} E^{m,\delta}[\chi,u] = \frac12 \int_\Omega (\chi + \delta (1-\chi)) \, C^m \varepsilon[u] : \varepsilon[u] {\,\mathrm{d}} x \end{align} for $m=0,1$, and objective functional \begin{align}\label{eq:totalcostPer}
J^\eta[\chi] = \max \left(J^0[\chi],J^1[1-\chi]\right) + \eta |D\chi|(\Omega) \end{align} there exists a characteristic function $\chi$ which minimizes $J[\cdot]$ over all the admissible characteristic functions in $BV(\Omega, \{0,1\})$ where $\tilde u^m_l[\chi^m]$ is the unique minimizer of $E^{m,\delta}[\chi^m,\cdot + \hat u^m_l[\chi]]$ over all displacements in $H^{1,2}(\Omega)^d$ with $\chi^0=\chi$ and $\chi^1=1-\chi$. \end{Theorem} \begin{proof} For fixed $\chi \in BV(\Omega, \{0,1\})$ a direct application of the Lax-Milgram theorem combined with Korn's inequality ensures the existence of unique minimizers $u^0[\chi]$ and $u^1[1-\chi]$ of the energy $E^{0,\delta}[\chi,\cdot]$ in $\mathcal{V}^0 + \hat u^0$ and $E^{1,\delta}[\chi,\cdot]$ in $\mathcal{V}^1 + \hat u^1$ , respectively.
Now, we consider a minimizing sequence $(\chi_k)_{k=1,\ldots}$ of the objective functional $J^\eta$ in $BV(\Omega, \{0,1\})$. Due to the perimeter term in the objective functional the $\chi_k$ are uniformly bounded in $BV(\Omega, \{0,1\})$. Hence, there is a weakly converging subsequence, which we denote for simplicity again by $(\chi_k)_{k=1,\ldots}$ and which converges to some $\chi \in BV(\Omega, \{0,1\})$. By the definition of the energies $E^{m,\delta}$ the corresponding minimizing displacements $u^m_{l,k}$ for $m=0,1$, $l=1,\ldots, L$ and $k=1,\ldots$ are uniformly bounded in $H^{1,2}(\Omega)^d$ with uniformly bounded energies $E^{m,\delta}[\chi^m_k,u^m_{l,k}[\chi^m_k]]$. Thus, for $m=0,1$ and $l=1,\ldots, L$ there is subsequences, again denoted by $(u^m_{l,k})_{k=1,\ldots}$, which converge weakly in $H^{1,2}(\Omega)^d$. Using the compact embedding of $BV(\Omega)$ in $L^1$ and Lebesgue's dominated convergence theorem, one obtains the $\Gamma$-convergence of the functionals $E^{m,\delta}[\chi^m_k,\cdot + \hat u^m_l[\chi_k]]$ to $E^{m,\delta}[\chi^m,\cdot + \hat u^m_l[\chi_k]]$ for $k\to \infty$ in the $H^{1,2}$-topology. As a direct consequence of this and the equi-coerciveness of the elastic energies, $E^{m,\delta}[\chi^m_k,\cdot + u^m_l[\chi_k]]$ converges to $E^{m,\delta}[\chi^m,\cdot + u^m_l[\chi]]$. Finally, the continuity of $g$ and the $\max$ function implies the lower semi-continuity of the objective functional $J^\eta$. Thus the claim holds. \end{proof}
For a similar proof in the case of nonlinear elastic shape optimization and a phase field approach instead of an approach with characteristic functions in $BV$ we refer to \cite{PeRuWi12}.
\section{Optimal microstructured polymer scaffold} \label{sec:bone} In the context of the biomechanical application described in Section~\ref{sec:bio} we consider a microstructured scaffold. The spatial scale of the microstructure is thereby determined by the 3D printer technology and biological considerations, such as the nutrient supply via blood vessel of a minimum thickness. Hence, we are led to the problem of an optimal domain splitting described in Sections \ref{sec:state} and \ref{sec:cost}. In explicit, we consider $\Omega=[0,1]^d$ as the fundamental cell of the polymer scaffold. We take into account prescribed affine displacements $\hat u^m_l$ with a symmetrized strain tensor $\epsilon[u^m_l] = A_l$ with $A_l \in R^{d,d}_{sym} \cap GL(d)$ and choose the subspace \begin{align*} \mathcal{V}^m = \mathcal{V} :=
H^1_{\#}(\Omega,\mathbb{R}^d) = \left\{ \displacementPeriodic{} \in H^{1,2}(\Omega,\mathbb{R}^d) \, : \, \displacementPeriodic{} \text{ periodic on } \Omega \,,\,\int_\Omega \displacementPeriodic{} {\,\mathrm{d}} x = 0 \right\}\,. \end{align*} Then, the elastic energy of a displacement $u^m_l = \hat u^m_l + \tilde u^m_l$ is given by \begin{align*} E^m[\chi^m,u^m_l] = \frac12 \int_\Omega \chi^m \, C^m \left( A_l + \varepsilon[u]\right) : \left( A_l + \varepsilon[u)]\right) {\,\mathrm{d}} x \,. \end{align*} Let us remark that it is well-known from the theory of elastic homogenization \cite{Al02} that the homogenized elasticity tensor $C^m_{*}$ of the resulting microstructure is uniquely described by \begin{align}\label{eq:homogenzedTensor}
C^m_{*} B : B
= \min_{\displacementPeriodic{m} \in \mathcal{V}}
\int_\Omega \chi^m C^m \; \left(B + \varepsilon[\displacementPeriodic{m}]\right) \; : \; \left(B + \varepsilon[\displacementPeriodic{m}]\right) {\,\mathrm{d}} x \end{align} for all $B\in R^{d,d}_{sym} \cap GL(d)$.
\section{Phase field approximation and finite element discretization} \label{sec:fem} A direct numerical treatment of the characteristic function $\chi$ or an explicit parametric description of the subdomains ${\mathcal{O}}_0$ and ${\mathcal{O}}_1$ is algorithmically quite demanding. Hence, we replace the characteristic function $\chi$ by a phase-field function $v:\Omega \to\mathbb{R}$ of Modica--Mortola type. Then the associated phase field energy functional is given by \begin{equation*}
L^\varepsilon[v]:=\frac12\int_\Omega \varepsilon|\nabla v|^2+\tfrac1\varepsilon\Psi(v)\,{\,\mathrm{d}} x\,, \end{equation*} where $\varepsilon$ describes the width of the diffused interface between the two subdomains (\emph{cf}\ \cite{PeRuWi12}). Here, we set $\Psi(v):=\frac{9}{16}(v^2-1)^2$
with two minima at $v=-1$ and $v=1$ and replace the perimeter $|D\chi|(\Omega)$ by the phase field energy $L^\varepsilon[v]$. In the limit $\varepsilon\to0$, the phase field $v$ leads to a clear separation between two pure phases $-1$ and $1$
and $L^\varepsilon$ is $\Gamma$-converging to the perimeter functional $|D\chi|(\Omega)$ of both faces in the domain $\Omega$ \cite{Br02}.
For the numerical discretization in 3D ($d=3$) we use a cuboid mesh, i.e. the unit cube $\Omega$ is uniformly divided into $(N - 1)^3$ cuboid elements with $N^3$ nodes. On this mesh we define the space $\mathcal{V}_h$ of piecewise trilinear, continuous functions and consider discrete phase fields $v\in \mathcal{V}_h$ and discrete displacement $u^m_h \in \mathcal{V}_{h}^3$. In analogy to the continuous case we then restrict to space of discrete, affine periodic functions. Furthermore, the elastic energies are approximated by a tensor product Simpson quadrature. Concerning the solver, the average value conditions on $u^m_h$ are imposed via a Lagrange multiplier approach. The corresponding linear systems for the elasticity problems are solved using the conjugate gradient method with diagonal preconditioning. The actual shape optimization problem in the unknown phase field $v_h$ is solved using the IPOPT package \cite{WaechterBiegler2006}. To implement the periodicity we identify the nodal values of the discrete phase field and the discrete displacements on corresponding pairs of nodes. To deal with the translational invariance of the phase field description of the subdomains -- indeed if $v$ is optimal, then the periodically extended $v(\cdot + \xi)$ is also optimal for all $\xi$ -- we fix the center of mass of the phase field $v$ taking into account the additional constraints $\int_\Omega \chi \, v_h (x_i - \tfrac12) {\,\mathrm{d}} x = 0$ for $ i=1,2,3$.
\section{Numerical results} \label{sec:num} \definecolor{activeLoadsShapeOptBones}{rgb}{0.95, 0.95, 0.95} In this section we present computational results for optimal microstructures in 3D and their dependence on material and model parameters. Furthermore, in a conceptual study we investigate realistic material parameters for bone material and a bioresorbable polymer.
The computational results are obtained on mesh with $65^3$ or $33^3$ vertices, where we use a prolongation of the optimal phase field on $17^3$ mesh as the initialization. On this coarser mesh random values in the interval $[-1,1]$ are used to initialize the phase field. For the phase field parameter we choose $\epsilon=2h$ where $h$ is the grid size.
To take into account compression and shear modes in the cost functional we investigate different sets of load scenarios based on the following $6$ affine displacements $\displacementAffine{}_i (x) = \displacementAffineDerivativeDirection{}{i} x$ as boundary data with $A_{11} = \beta e_1^T e_1$, $A_{22} = \beta e_2^T e_2$, $A_{33} = \beta e_3^T e_3$, $A_{12} = \beta (e_1^T e_2 + e_2^T e_1)$, $A_{13} = \beta (e_1^T e_3 + e_3^T e_1)$, and $A_{23} = \beta (e_2^T e_3 + e_3^T e_2)$ for $\{e_1, e_2, e_3 \}$ being the canonical bases in $\mathbb{R}^3$. We take into account the parameters $\beta = - 0.25$, $\eta= 2$, and $\delta=10^{-4}$.
Given an effective elasticity tensor $C_*$ the components $C_*^{iiii}= \beta^{-2} C_* A_{ii} : A_{ii}$ ($i=1,2,3$) represent compressive stresses caused by corresponding compressive strains, whereas the components $C_*^{ijij}= \beta^{-2} C_* A_{ij} : A_{ij}$ ($i,j=1,2,3,\; i\neq j$) represent shear strains induced shear stresses. If not indicated elsewise, we always consider $p=2$ in the definition of the weight function $g$.
\paragraph{Equal material parameters.} In Fig. \ref{fig:equalmaterial} we consider equal material parameters, i.e. $(E^0,\nu^0) = (E^1,\nu^1) = (10,0.25)$, where $E^m$ is the Young's modulus ($E^m = \frac{\mu^m ( 3 \lambda^m + 2 \mu^m}{\lambda^m + \mu^m}$) and $\nu^m$ the Poisson ratio ($\nu^m = \frac{\lambda^m}{\lambda^m + \mu^m}$). Three different load scenarios are compared: three compression modes ($C_*^{1111}, \, C_*^{2222},\, C_*^{3333}$), two compression modes combined with a single shear mode ($C_*^{1111}, \, C_*^{2222},\, C_*^{2323}$), and one compression mode combined with two shear modes ($C_*^{1111}, \, C_*^{1212},\, C_*^{1313}$). For both subdomains identical loads are taken into account. We observe an almost equal volume for both subdomains in the optimal configurations. In all cases the interface between the two subdomains are of the same topology as the Schwarz P surface, a periodic minimal surface representing a local minimizer of the perimeter functional. But there are significant differences in the components of the objective functional, where always those entries of the effective elasticity tensor present in the objective functional indicate a substantially stronger stiffness.
In the literature \cite{Torquato:2004ey,Silvestre:2007hn} the subdomain splitting associated with the Schwarz P surface as the interface has been investigated concerning its optimality in the context of a PDE constraint optimization. On this background, we compute an optimal phase field representing a discrete minimizer of the perimeter functional as a numerical approximation of the Schwarz P surface. For this configuration we computed the entries of the effective elasticity tensor and observe significantly different values $C_*^{iiii}= 2.7811$ ($i=1,2,3$) and $C_*^{ijij}= 2.481$ ($i,j=1,2,3,\; i\neq j$) compared to the optimizer in the load scenario based on three compressions. Furthermore, the phase field area functional $L^\varepsilon$ differs by approximately $3\%$. \begin{figure}
\caption{ Comparison of optimal micro-structures and relevant induced components of the effective elasticity tensors for different load scenarios indicated above. In the top row we depict the subdomains on the fundamental cell of the microstructure and below a $3\times3\times3$ composition pronouncing the periodicity. Those components of the tensor which are part of the corresponding objective functional are highlighted in grey.}
\label{fig:equalmaterial}
\end{figure}
Next, for the scenario with three shear loads ($C_*^{1212},\, C_*^{1313},\, C_*^{2323}$) we successively increase the parameter $\eta$ in front of the perimeter functional ($\eta =2,\,4,\,10$). For small $\eta$ we obtain a laminate type optimal configuration, whereas for larger $\eta$ the interface is again similar to the Schwarz P surface as shown in Figure \ref{fig:ShapeOptBonesPerimeter}. On the intermediate range of the parameter $\eta$ we obtain a optimal microstructure with an interface similar to a gyroid minimal surface \cite{Kapfer:2011kz}.
\begin{figure}
\caption{Optimal microstructures for different values of the perimeter parameter $\eta$ (From left to right: $
\eta = 2, 4, 10$).}
\label{fig:ShapeOptBonesPerimeter}
\end{figure}
Furthermore, we investigate the impact of the choice of the weight function $g$ on the optimal splitting and the associated stiffness moduli. In Figure \ref{fig:p} we show in the load scenario with two compression loads and one shear load the relevant entries of the effective elasticity tensor. For increasing $p$ we observe a successive balancing of the different components of the objective functional, in particular the largest component $C^{2222}_{*}$ of the effective elasticity tensor is slightly deccreasing while the smallest compenent $C^{1111}_{*}$ is slightly increasing.
\begin{figure}
\caption{ For different values of $p$ stiffness moduli of the optimal subdomain splitting are depicted.}
\label{fig:p}
\end{figure}
\paragraph{Varying Young modulus.} Next we study the influence of the Youngs modulus and consider $E^0 = 20,40,80,160,320$, whereas $(E^1,\nu^1) = (10,0.25)$ and $\nu^0 = 0.25$. Furthermore, for the perimeter parameter we choose $\eta = 1$. We observe that the subdomain with increasing Young modulus is getting successively thinner in the optimal domain splitting and the relative decrease in stiffness of the other material has to compensated by a higher volume fraction. Figure \ref{fig:young} shows results obtained for different load scenarios.
\begin{figure}\label{fig:young}
\end{figure}
\paragraph{Realistic material parameters for polymer and bone.} Real bone is substantially stiffer than the bioresorbable polymer with a $15$ times larger Youngs modulus and the Poisson ratio of $\nu^B = 0.1$ compared to the Poisson ratio $\nu^P = 0.3$ for the polymer. Figure \ref{fig:bp} shows the optimal bone and polymer subdomains together with a plot of the von Mises stresses on the boundary of the corresponding subdomains in the fundamental cell. Here, again the case of one compression load and two shear loads is taken into account.
\begin{figure}
\caption{ Optimal bone and polymer micro-structures with color coded von Mises stresses using a log scaled color value in HSV model.}
\label{fig:bp}
\end{figure}
\begin{figure}
\caption{ An optimal domain decompositions in 2D in case of the hard--soft approximation with $\delta=10^{-4}$ is depicted for a load scenario with two different loads corresponding to two compression modes ($C^{1111}_*$ and $C^{2222}_*$). A block of $3\,x\,3$ cells is plotted with the two subdomains in white and black together with a color plot of the von Mises stresses. }
\label{fig:2D}
\end{figure}
\begin{Remark}
Let us briefly comment on the 2D case ($d=2$). Figure~\ref{fig:2D} shows the numerical result for a scenario with two uniaxial compression loads.
In the optimized shape configuration we obtain diamond shaped regions of both subdomains which meet at the tips of the diamonds. In the context of our hard--soft approximation introduced in Section~\ref{sec:exist} this is a mechanically admissible configuration. For a hard--void shape optimization model and two uniaxial compression loads in vertical and horizontal direction no mechanically favourable splitting of the unit square $[0,1]^2$ into two subdomains is possibly.
Indeed, a uniaxial load requires a truss with non vanishing interior connecting the components of the boundary opposite in the loading direction.
A truss configuration simultaneously in horizontal and vertical direction for both subdomains is thus topologically impossible.
\end{Remark}
\section{Conclusions} \label{sec:conclusions} We considered the problem of designing an optimal periodic microstructure for a domain splitting problem in shape optimization. The setting of this article is motivated by the biomechanical application of designing optimal scaffolds for bone regeneration, where both the scaffold as well as the regenerated bone (which can grow only in the space not occupied by the scaffold material) need to be stable individually. The numerical method presented here is able to treat a general cost functional whose input is given by the effective moduli of an elastic material occupying either the optimized domain or its complement.
The case of maximizing the compressive moduli in the three coordinate axes can be compared to the scalar problem of maximizing for example isotropic heat as well as electrical conduction in a two phase material where in each phase one parameter is large and the other is small \cite{Torquato:2002gh}. Our simulations suggest that also in this 3d-elasticity setting a domain separation by a periodic minimal surface appears not to be optimal but with already comparable small cost, which was verified for the scalar case in \cite{Silvestre:2007hn}.
The optimization for other load cases, in particular a compression in one direction combined with a shear in the two directions orthogonal to the compressive load, yields optimal structures that are very clearly distinct from minimal surfaces. Since this is the most physiologically relevant case, as the typical loading condition on major long bones is compression and torsion, we note that it might be possible to further improve scaffold designs based on minimal surfaces (see Figure \ref{fig:PrintedScaffolds} and \cite{Kapfer:2011kz}), which are currently considered for medical practice.
A number of open issues remain. So far, we investigated solely spatially homogeneous microstructure and optimized the periodic scaffold on the fundamental cell. When considering realistic patient specific implant geometries and corresponding boundary conditions one has to exploit the inherent multiscale nature of the problem and design an optimal microstructure with spatially varying microscopic shapes. Furthermore, a major drawback of our current approach is that some significant physiological conditions are not considered in this work. A scaffold design as seen in Figure \ref{fig:bp} may be optimal from a purely mechanical point of view, but the very low porosity would seriously impede vascularization and therefore prevent the regeneration of bone matrix. Thus, in future study, quantities like the effective diffusivity in the pores should be considered (as it is done for example done in \cite{Adachi06}). The uncertainty in translating a given scaffold design into an additively manufactured scaffold can be observed in Figure \ref{fig:PrintedScaffolds}, where strands and layers stemming from the printing process are clearly visible. A quantification of the uncertainty in the resulting effective elastic moduli due to both periodic as well as random fluctuations in the final product thus seems necessary as well.
\end{document}
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\begin{document}
\date{} \title{On the convergence of solutions for SPDEs under perturbation of the domain
ootnote{The author acknowledges the support provided by NSFs of
China (No.11271013) and the Fundamental Research Funds for the
Central Universities, HUST: 2012QN028.}
\begin{abstract} We concern the effect of domain perturbation on the behaviour of stochastic partial differential equations subject to the Dirichlet boundary condition. Under some assumptions, we get an estimate for the solutions under changes of the domain.\\ \textbf{Keywords}: Stochastic partial differential equation; Domain perturbation; Convergence of solutions; \end{abstract}
\section{Introduction} \quad\, Stochastic partial differential equations (SPDEs) have a broad spectrum of application including natural sciences and economics. The purpose of this article is to study the behavior of solution for stochastic partial differential equations with Dirichlet boundary condition under the singular domain perturbations, which means that change of variables is not possible on these domains. Under property conditions, we show how solutions of stochastic differential equations behave as a sequence of domains $\Omega_n$ converges to an open set $\Omega$ in a certain sense. The motivation to study domain perturbation comes from various sources. The main ones include shape optimization, solution structure of nonlinear problems and numerical analysis.\par
Domain perturbation or sometimes referred to as ``perturbation of the boundary" for boundary value problems is a special topic in perturbation problems. The main characteristic is that the operators and the nonlinear term live in differential spaces which lead to the solution of differential equation live in differential spaces. Domain perturbation appears to be a simple problem if we are only interested in smooth perturbation of the domain. This is because we could perform a change of variables to consider the perturbed problems in a fixed domain and only perturb the coefficients. Hence, it turns back to a standard perturbation problem and we may apply standard techniques such as the implicit function theorem, the Liapunov-Schmidt method and the transversality theorem. Nevertheless, difficulties arrive when we perform a change of variables and standard tools are not enough(see \cite{D.H}). When a change of variables is not possible, domain perturbation is even more challenging.\par
The fundamental question in domain perturbation is to look at how solutions behave upon varying domains. In particular we would like to know when the solutions converge and what the limit problem is. There have many papers concern on this topic, which main under the condition of Mosco convergence. For elliptic equations case see \cite{Daners,JA} and references therein. In \cite{JA} the author give a sufficient condition on domains which guarantee the spectrum behaves continuously. The work of \cite{Daners} prove the converge of solution for elliptic equations subject to Dirichlet boundary condition. For parabolic and evolution equation, we recommend \cite{D.Daners,D} and so on. In \cite{D.Daners} the author concern domain perturbation for non-autonomous parabolic equations under the assumption of mosco converge. With such a assumption we have that the condition of mosco converge is equivalent to the strong convergence of pseudo resolvent operators for Dirivhlet promble. Under the assumption of mosco converge, the author of \cite{D.Daners} get the result of convergence of solutions for both linear and semilinear parabolic initial value problems subject to Dirichlet boundary boundary condition as well as persistence of periodic solutions under domain perturbation. There also some other papers about invariant manifolds under the domain perturbation see \cite{JE,Varchon,PSN}, which main concern the converge of invariant manifolds under the perturbation of the domain.\par For Drichlet problems, the strong convergence of pseudo resolvent operators is equivalent to Mosco convergence(see \cite{D.Daners}, Theorem 5.2.4 or \cite{Daners}, Theorem 3.3). In this paper, we take the condition of strong convergence of pseudo resolvent operators relate to the domain perturbation. Compare with the Mosco condition, it is more convenient and effective for proving the convergence of solution for partial differential equations and stochastic differential equations under perturbation of the domain. \par The remainder of this paper is organized as follows: In Section 2, we will review some basic properties of infinitesimal generator and its semigroups, and existence and unique of solution to stochastic partial differential equation. The result on the converge of solution for stochastic differential equation under the perturbation is described in section $3$\,.
\section{Preliminaries} Let $H$ be an infinite dimensional separable Hilbert space with norm
$\|\cdot\|$\,. Let the sectorial operator $A: D(A)\rightarrow H$ be a self-adjoint positive linear operator with a compact resolvent. Then the spectrum of $A$ is real. We denote its spectrum by \begin{eqnarray*} \sigma(A)=\{\lambda_{n}\}_{n=1}^{\infty}, \quad 0<c\leq\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n}\leq\cdots, \end{eqnarray*} and an associated orthonormal family of eigenfunctions by $\{\phi_{n}\}_{n=1}^{\infty}$. Since $A$ is a sectorial operator, $-A$ is the infinitesimal generator of a analytic semigroup, which is denoted by \begin{eqnarray*} e^{-At}=\frac{1}{2\pi i}\int_{\gamma}(\lambda I+ A)^{-1}e^{\lambda t}d\lambda , \end{eqnarray*} where $\gamma$ is a contour in the resolvent set of $-A$. Since $A$ is a self-adjoint operator, the formula above is equivalent to \begin{eqnarray*} e^{-At}u=\sum_{n=1}^{\infty}e^{-\lambda_{n}t}(u,\phi_{n})\phi_{n}. \end{eqnarray*} By the definition $e^{-At}$, we can easily get the following estimate \begin{eqnarray*}
\|e^{-At}\|_{L(H, H)}\leq e^{-\lambda_{1}t}\leq 1 \end{eqnarray*} for $t\geq 0$\,, which implies that $e^{-At}$ is an analytic contraction semigroup.
Consider the nonlinear stochastic partial differential equation \begin{eqnarray}\label{e2.1} \left\{\begin{array}{ll} du+Audt=f(u)dt+g(u)dw(t)\,,&\quad in \quad D\times(0, T]\\ u=0\,,&\quad on \quad \partial D\times(0, T]\\
u(0)=u_{0}\,,&\quad in \quad D \end{array}\right. \end{eqnarray} for $t\in[0,T]$\,. Here $u\in H$\,, $A$ is a sectorial operator, which will be discussed later, $W(t)$ is the standard $\mathbb{R}$-valued Wiener process on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$\,. For the drift coefficients $f(u): H\rightarrow H$ and diffusion coefficients $g(u):H\rightarrow H$, we adopt the following assumptions throughout this paper.\\ $(\mathbf{A.1})$\quad There exists a constant $k_{2}>0$ such that for any $u, v\in H$, $t\in[0,T]$, \begin{eqnarray*}
\|f(u)-f(v)\|^{2}+\|g(u)-g(v)\|^{2}\leq k_{2}\|u-v\|^{2}. \end{eqnarray*} Notice that $(\mathbf{A.1})$ implies there exists a constant $k_{1}> 0$ such that \begin{eqnarray*}
\|f(u)\|^{2}+\|g(u)\|^{2}\leq k_{1}(1+\|u\|^{2}) \end{eqnarray*} for any $u\in H$, $t\in[0,T]$.
Now we introduce the definition of solution to Eq.(\ref{e2.1}) and the existence and uniqueness of solution, Both of them are taken from \cite{DZ}.
\begin{definition}[Mild solution]\label{d2.1} An $H$-valued predictable process $u(t)$ is called a mild solution of Eq.(\ref{e2.1}) if for any $t\in [0, T]$ \begin{eqnarray}\label{s1} u(t)=e^{-At}u_{0}+\int_{0}^{t}e^{-A(t-s)}f(u(s))ds+\int_{0}^{t}e^{-A(t-s)}g(u(s))dw(s) \end{eqnarray} \end{definition}
Let $X_{T}$ denote the set of all continuous $\mathcal
{F}_{t}$-adapted processes valued in $H$ for $0\leq t\leq T$ such that $E \sup\limits_{0\leq t\leq T}\|u\|^{2}<\infty$. Then $X_{T}$
is a Banach space under the norm \begin{eqnarray*} \|u\|_{T}=E \sup\limits_{0\leq t\leq T}\|u\|^{2}. \end{eqnarray*} Define an operator $\Gamma$ in $X_{T}$ as follows \begin{eqnarray}\label{e2.2} \Gamma u(t)=e^{-At}u_{0}+\int_{0}^{t}e^{-A(t-s)}f(u(s))ds+\int_{0}^{t}e^{-A(t-s)}g(u(s))dw(s), \end{eqnarray} for $u\in X_{T}$. It is easy to prove that the operator $\Gamma$ is well defined and Lipschitz continuous in $X_{T}$. Then by the contraction mapping principle, it is easy to prove the existence and unique of mild solution for the Eq.(\ref{e2.1}) is the following
\begin{theorem}\label{t2.4} Suppose the condition $(A.1)$ holds true, and $u_{0}$ be a $\mathcal {F}_{0}$-measurable random field such that
$E\|u_{0}\|^{2}<\infty$. Then the initial-boundary value problem for the Eq.(\ref{e2.1}) has a unique mild solution $u(t)$ which is a continuous adapted process in $H$ such that $u\in L^{2}(\Omega; C([0, T]; H))$ and $$E \sup\limits_{0\leq t\leq T}\|u\|^{2}\leq C(1+E\|u_{0}\|^{2})$$ for some constant $C>0$. \end{theorem}
\section{Solution under perturbation of the domain} In this section, we consider the following perturbation equation of Eq.(\ref{e2.1}) \begin{eqnarray}\label{e3.2} \left\{\begin{array}{ll} du^{\epsilon}+A_{\epsilon}u^{\epsilon}dt=f^{\epsilon}(u^{\epsilon})dt+g^{\epsilon}(u^{\epsilon})dw(t)\,,&\quad in \quad D^{\epsilon}\times(0, T],\\ u^{\epsilon}=0\,,&\quad on \quad \partial D^{\epsilon}\times(0, T],\\
u^{\epsilon}(0)=u^{\epsilon}_{0}\,,&\quad in \quad D^{\epsilon} \end{array}\right. \end{eqnarray} for $\epsilon>0$, where $A_{\epsilon}: D(A_{\epsilon})\subset H^{\epsilon}\rightarrow H^{\epsilon}$ is a self-adjoint positive linear operator on a Hilbert space $H^{\epsilon}$ with norm
$\|\cdot\|_{\epsilon}$, and $u_{0}^{\epsilon}$ be a $\mathcal {F}_{0}$-measurable random field such that
$E\|u_{0}^{\epsilon}\|^{2}<\infty$. We also assume that the nonlinear terms $f^{\epsilon}:H^{\epsilon}\rightarrow H^{\epsilon}$ and $g^{\epsilon}:H^{\epsilon}\rightarrow H^{\epsilon}$ satisfy (A.1), which guarantees the existence and unique of mild solution to the Eq.(\ref{e3.2}). By Theorem \ref{t2.4}, for each ${\epsilon}>0$, there is an $H$-valued continuous $\mathcal {F}_{t}$-adapted process $u^{\epsilon}(t)$ such that \begin{eqnarray}\label{s2} u^{\epsilon}(t)=e^{-A_{\epsilon}t}u^{\epsilon}_{0}+\int_{0}^{t}e^{-A_{\epsilon}(t-s)}f(u^{\epsilon}(s))ds+\int_{0}^{t}e^{-A_{\epsilon}(t-s)}g(u^{\epsilon}(s))dw(s) \end{eqnarray} for any $t\in [0,T]$ and $u^{\epsilon}\in L^{2}(\Omega; C([0, T]; H^{\epsilon}))$.
Note that the solutions value in different function spaces $H^{\epsilon}$ for different ${\epsilon}$. To deal with domain perturbation, we assume there exist bound linear operators $\mathbf{P}$ and $\mathbf{Q}$ such that \begin{eqnarray*} \mathbf{P}:H\rightarrow H^{\epsilon}, \quad \mathbf{Q}:H^{\epsilon}\rightarrow H\,,\quad Q\circ P=I, \end{eqnarray*} \begin{eqnarray*}
\|\mathbf{P}\|_{\mathcal{L}(H, H^{\epsilon})}\leq 2, \quad \|\mathbf{Q}\|_{\mathcal{L}( H^{\epsilon}, H)}\leq 2, \end{eqnarray*} and \begin{eqnarray*}
\|\mathbf{P}u\|_{H^{\epsilon}}\rightarrow
\|u\|_{H}, \quad\mbox{as}~~ \epsilon\rightarrow 0 \end{eqnarray*} for all $u\in H$.
To derive the solution of Eq.(\ref{e3.2}) converges to the solution of Eq.(\ref{e2.1}), we also impose the following hypotheses\\ $(\mathbf{H.1})$ For $A$ and $A_{\epsilon}$, we assume \begin{eqnarray*}
\|A_{\epsilon}^{-1}\mathbf{P}-\mathbf{P}A^{-1}\|_{\mathcal{L}(H, H^{\epsilon})}=\tau(\epsilon) \rightarrow 0 \quad as \quad \epsilon\rightarrow 0. \end{eqnarray*}
$(\mathbf{H.2})$ We assume that the nonlinear terms
$g^{\epsilon}\,,f^{\epsilon}:H^{\epsilon}\rightarrow H$ for $0\leq \epsilon \leq\epsilon_{0}$,
satisfy:
\begin{itemize}
\item $f^{\epsilon}$ and $g^{\epsilon}$ approximate $f$ and $g$
in the following sense, \begin{eqnarray*}
\sup\limits_{u\in H}\|f^{\epsilon}(\mathbf{P}u)-
\mathbf{P}f(u)\|^{2}_{H^{\epsilon}}=\tau_{1}(\epsilon)\rightarrow 0,\quad
\mbox{as}~~ \epsilon\rightarrow 0. \end{eqnarray*} \begin{eqnarray*}
\sup\limits_{u\in H}\|g^{\epsilon}(\mathbf{P}u)-
\mathbf{P}g(u)\|^{2}_{H^{\epsilon}}=\tau_2(\epsilon)\rightarrow 0,\quad \mbox{as}~~ \epsilon\rightarrow 0\,. \end{eqnarray*}
\item $f$ and $f^{\epsilon}$ have the uniformly bounded support, that is
\begin{eqnarray*} Supp f\subset D_{R}=\{u\in H: \|u\|_{H}\leq R\} \end{eqnarray*}
\begin{eqnarray*} Supp f^{\epsilon}\subset D_{R}=\{u^{\epsilon}\in H^{\epsilon}: \|u^{\epsilon}\|_{H^{\epsilon}}\leq R\} \end{eqnarray*}
\end{itemize} $(\mathbf{H.3})$ For initial value $u_{0}$ and $u_{0}^{\epsilon}$, we assume
\begin{eqnarray*} E\|u_{0}^{\epsilon}-\mathbf{P}u_{0}\|^{2}_{H^{\epsilon}}=\tau_{0}(\epsilon)\rightarrow 0,\quad \mbox{as}~~ \epsilon\rightarrow 0. \end{eqnarray*}
By the condition $(\mathbf{H.1})$ we have the following result, which concerns the relationship of spectrum between $A$ and $A_{\epsilon}$ (see \cite{JE}). \begin{lemma}\label{l3.1} If $K_{0}$ is a compact set of the complex plane with $K_{0}\subset \rho(-A)$, the resolvent set of $A$, and hypothesis $\mathbf{(H.1)}$ is satisfied, then there exists $\epsilon_{0}(K_{0})>$ such that $K_{0}\subset\rho(-A_{\epsilon})$ for all $0<\epsilon\leq\epsilon_{0}(K_{0})$. Moreover, we have the estimates
$$\|(\lambda I-A_{\epsilon})^{-1}\|_{\mathcal{L}(H^{\epsilon}, H^{\epsilon})}\leq C(K_{0})\,$$ for all $\lambda\in K_{0},\, 0<\epsilon\leq\epsilon_{0}(K_{0})$. \end{lemma}
The result implies the upper semi-continuity of the spectrum, that is, if $\lambda_{\epsilon}\in \sigma(A_{\epsilon})$ and $\lambda_{\epsilon}\rightarrow\lambda$ then $\lambda\in\sigma(A)$. Also we have the resolvent operator estimate as follows (see \cite{JE}). \begin{lemma}\label{l3.2} Let the condition $\mathbf{(H.1)}$ be satisfied, if $\lambda\in\rho(-A)$ and $\epsilon$ is small enough so that $\lambda\in\rho(-A_{\epsilon})$, we have
$$\|(\lambda+A_{\epsilon})^{-1}\mathbf{P}-\mathbf{P}(\lambda+A)^{-1}\|_{\mathcal{L}(H,H^{\epsilon})}\leq C(\epsilon, \lambda)\tau(\epsilon)\rightarrow 0, \quad \mbox{as}~~ \epsilon\rightarrow 0.$$ \end{lemma}
As we all known, the relationship between resolvent operator and semigroup is denoted by \begin{eqnarray}\label{r2} e^{-At}=\frac{1}{2\pi i}\int_{\gamma}(\lambda I+ A)^{-1}e^{\lambda t}d\lambda \end{eqnarray} where $\gamma$ is the boundary of
$\Sigma_{-a,\phi}=\{\lambda\in \mathbb{C}:|arg(\lambda+a)|\leq \pi-\phi\}\subset \rho(-A)$,\, $\phi\in (0, \frac{\pi}{2})$.
For simply we take the $a=0,\, \phi=\frac{\pi}{4}$. Then we have $$\gamma=\gamma_{1}\cup\gamma_{2}
=\{re^{-i\frac{3\pi}{4}}: 0\leq r<\infty\}\cup \{re^{i\frac{3\pi}{4}}: 0\leq r<\infty\} $$ and $C(\epsilon, \lambda)\leq 6$ for all $\lambda\in \Sigma_{0,\frac{\pi}{4}}$. From Lemma \ref{l3.2} we have the following estimate. \begin{lemma}\label{l3.3} Let $(\mathbf{H.1})$ be satisfied. Then we have
$$\|e^{-A_{\epsilon}t}\mathbf{P}-\mathbf{P}e^{-At}\|_{\mathcal{L}(H, H^{\epsilon})}\leq\frac{C}{r}\tau(\epsilon) \rightarrow 0,~~\mbox{as}~ \epsilon\rightarrow 0$$ for any $t\in [r,T]$, here $r>0$. \end{lemma} \begin{proof} By (\ref{r2}) and Lemma \ref{l3.2}, we can estimate \begin{eqnarray*}
&&\|e^{-A_{\epsilon}t}\mathbf{P}-\mathbf{P}e^{-At}\|_{\mathcal{L}(H, H^{\epsilon})}\\
&=&\|\frac{1}{2\pi i}\int_{\gamma}(\lambda I+
A_{\epsilon})^{-1}\mathbf{P}e^{\lambda t}d\lambda-\frac{1}{2\pi i}\int_{\gamma}\mathbf{P}(\lambda I+ A)^{-1}e^{\lambda t}d\lambda\|_{\mathcal{L}(H, H^{\epsilon})}\\
&\leq&C |\int_{\gamma_{1}\cup\gamma_{2}}\tau(\epsilon)e^{\lambda t}d\lambda|.
\end{eqnarray*} For $\lambda\in\gamma_{1}\cup\gamma_{2}$, we compute $|e^{\lambda t}|=|e^{rte^{-i\frac{3\pi}{4}}}|=e^{-\frac{\sqrt{2}}{2}rt}$ with $0\leq r\leq +\infty$. Then we have
$$\|e^{-A_{\epsilon}t}\mathbf{P}-\mathbf{P}e^{-At}\|_{\mathcal{L}(H, H^{\epsilon})}\leq C\tau(\epsilon)\int_{0}^{+\infty}e^{-\frac{\sqrt{2}}{2}rt}dr\leq\frac{C}{t}\tau(\epsilon)\,.$$ Hence by $(\mathbf{H.2})$ \begin{eqnarray*}
\|e^{-A_{\epsilon}t}\mathbf{P}-\mathbf{P}e^{-At}\|_{\mathcal{L}(H, H^{\epsilon})}\leq\frac{C}{r}\tau(\epsilon)\rightarrow 0,~~\mbox{as}~~ \epsilon\rightarrow 0 \end{eqnarray*} for any $t\in[r,T]$, here $r>0$. \end{proof}
Now we state and prove our main result as the following. \begin{theorem}\label{t3.1} Suppose the conditions $(\mathbf{H.1})$ to
$(\mathbf{H.3})$ hold true. Then we have \begin{eqnarray*} E\sup\limits_{0\leq t\leq T}\|u^{\epsilon}(t)-\mathbf{P}u(t)\|_{H^{\epsilon}}^{2}\leq \frac{C(T,R)(r^{2}+r+\tau_{0}(\epsilon)+\tau_{1}(\epsilon)+\tau(\epsilon))}{1-C(T)k_{2}}.
\end{eqnarray*} In particular, \begin{eqnarray*} E\sup\limits_{0\leq t\leq T}\|u^{\epsilon}(t)-\mathbf{P}u(t)\|_{H^{\epsilon}}^{2}\rightarrow 0, \end{eqnarray*} when we first let $\epsilon\rightarrow 0$ and then $r\rightarrow 0$. \end{theorem} \begin{proof} From equation (\ref{s1}) and equation (\ref{s2}), we have \begin{eqnarray*}
&&E\sup\limits_{0\leq t\leq T}\|u^{\epsilon}(t)-\mathbf{P}u(t)\|_{H^{\epsilon}}^{2}\\
&=&E\sup\limits_{0\leq t\leq T}\|e^{-A_{\epsilon}t}u^{\epsilon}_{0}-\mathbf{P}e^{-At}u_{0} +\int_{0}^{t}e^{-A_{\epsilon}(t-s)}f^{\epsilon}(u^{\epsilon})-\mathbf{P}e^{-A(t-s)}f(u)ds\\
&&+\int_{0}^{t}e^{-A_{\epsilon}(t-s)}g^{\epsilon}(u^{\epsilon})-\mathbf{P}e^{-A(t-s)}g(u)dw(s)\|_{H^{\epsilon}}^{2}\\
&\leq&3E\sup\limits_{0\leq t\leq T}\|e^{-A_{\epsilon}t}u^{\epsilon}_{0}-\mathbf{P}e^{-At}u_{0}\|_{H^{\epsilon}}^{2}\\
&&+3E\sup\limits_{0\leq t\leq T}\|\int_{0}^{t}e^{-A_{\epsilon}(t-s)}f^{\epsilon}(u^{\epsilon})-\mathbf{P}e^{-A(t-s)}f(u)ds\|_{H^{\epsilon}}^{2}\\
&&+3E\sup\limits_{0\leq t\leq T}\|\int_{0}^{t}e^{-A_{\epsilon}(t-s)}g^{\epsilon}(u^{\epsilon})-\mathbf{P}e^{-A(t-s)}g(u)dw(s)\|_{H^{\epsilon}}^{2}\\ &=:&3I_{1}+3I_{2}+3I_{3} \end{eqnarray*} Next we will estimate $I_{1}$, $I_{2}$ and $I_{3}$ respectively. Fix $r$ sufficient small. For $I_{1}$ we have
\begin{eqnarray*} I_{1}&\leq &E\sup\limits_{r\leq t\leq T}\|e^{-A_{\epsilon}t}u^{\epsilon}_{0}-\mathbf{P}e^{-At}u_{0}\|_{H^{\epsilon}}^{2}
+E\sup\limits_{0\leq t\leq r}\|e^{-A_{\epsilon}t}u^{\epsilon}_{0}-\mathbf{P}e^{-At}u_{0}\|_{H^{\epsilon}}^{2} \end{eqnarray*} with \begin{eqnarray*}
&&E\sup\limits_{r\leq t\leq T}\|e^{-A_{\epsilon}t}u^{\epsilon}_{0}-\mathbf{P}e^{-At}u_{0}\|_{H^{\epsilon}}^{2}\\
&\leq& 2E\sup\limits_{r\leq t\leq T}\|e^{-A_{\epsilon}t}u_{0}^{\epsilon}-e^{-A_{\epsilon}t}\mathbf{P}u_{0}\|_{H^{\epsilon}}^{2}
+2E\sup\limits_{r\leq t\leq T}\|e^{-A_{\epsilon}t}\mathbf{P}u_{0}-\mathbf{P}e^{-At}u_{0}\|_{H^{\epsilon}}^{2}\\
&\leq& CE\sup\limits_{r\leq t\leq T}\|u_{0}^{\epsilon}-\mathbf{P}u_{0}\|_{H^{\epsilon}}^{2}+C\frac{\tau(\epsilon)}{r}\\ &\leq&C\tau_{0}(\epsilon)+C\frac{\tau(\epsilon)}{r}
\end{eqnarray*} and \begin{eqnarray*}
&&E\sup\limits_{0\leq t\leq r}\|e^{-A_{\epsilon}t}u^{\epsilon}_{0}-\mathbf{P}e^{-At}u_{0}\|_{H^{\epsilon}}^{2}\\\
&\leq&\!\!\!3E\sup\limits_{0\leq t\leq r}\|e^{-A_{\epsilon}t}u_{0}^{\epsilon}-u_{0}^{\epsilon}\|_{H^{\epsilon}}^{2}\!\!+3E\|u_{0}^{\epsilon}-\mathbf{P}u_{0}\|_{H^{\epsilon}}^{2}
\!\!+3E\sup\limits_{0\leq t\leq r}\|\mathbf{P}u_{0}-\mathbf{P}e^{-At}u_{0}\|_{H^{\epsilon}}^{2}\\
&\leq&\!\!\!CE\sup\limits_{0\leq t\leq r}\|e^{-A_{\epsilon}t}-I\|_{{\mathcal{L}(H^{\epsilon}, H^{\epsilon})}}^{2}+CE\|u_{0}^{\epsilon}-\mathbf{P}u_{0}\|_{H^{\epsilon}}^{2}
+CE\sup\limits_{0\leq t\leq r}\|e^{-At}-I\|_{H}^{2}\\
&\leq&\!\!\!CE\sup\limits_{0\leq t\leq r}\|e^{-A_{\epsilon}t}-I\|_{{\mathcal{L}(H^{\epsilon}, H^{\epsilon})}}^{2}+C\tau_{0}(\epsilon)+CE\sup\limits_{0\leq t\leq r}\|e^{-At}-I\|_{{\mathcal{L}(H, H)}}^{2}, \end{eqnarray*} where the contraction of $e^{-At}$, Lemma \ref{l3.3} and $(\mathbf{H.3})$ are used. Therefore
\begin{eqnarray*} I_{1}&\leq&C\big(E\sup\limits_{0\leq t\leq r}\|e^{-A_{\epsilon}t}-I\|_{{\mathcal{L}(H^{\epsilon}, H^{\epsilon})}}^{2}+E\sup\limits_{0\leq t\leq r}\|e^{-At}-I\|_{{\mathcal{L}(H, H)}}^{2}\big)\\ &&+C(\frac{\tau(\epsilon)}{r}+\tau_{0}(\epsilon)\big). \end{eqnarray*} For
$I_{2}$ we have \begin{eqnarray*} I_{2}&\leq &2E\sup\limits_{0\leq t\leq T}\|\int_{0}^{t}e^{-A_{\epsilon}(t-s)}(f^{\epsilon}(u^{\epsilon})-\mathbf{P}f(u))\|_{H^{\epsilon}}^{2}\\
&&+2E\sup\limits_{0\leq t\leq T}\|\int_{0}^{t}(e^{-A_{\epsilon}(t-s)}\mathbf{P}-\mathbf{P}e^{-A(t-s)})f(u)ds\|_{H^{\epsilon}}^{2}\\
&\leq & 4E\sup\limits_{0\leq t\leq T}\|\int_{0}^{t}e^{-A_{\epsilon}(t-s)}(f^{\epsilon}(u^{\epsilon})-f^{\epsilon}(\mathbf{P}u))ds\|_{H^{\epsilon}}^{2}\\
&&+ 4E\sup\limits_{0\leq t\leq T}\|\int_{0}^{t}e^{-A_{\epsilon}(t-s)}(f^{\epsilon}(\mathbf{P}u)-\mathbf{P}f(u))ds\|_{H^{\epsilon}}^{2}\\
&&+2E\sup\limits_{0\leq t\leq T}\|\int_{0}^{t}(e^{-A_{\epsilon}(t-s)}\mathbf{P}-\mathbf{P}e^{-A(t-s)})f(u)ds\|_{H^{\epsilon}}^{2}\\
&\leq &4T^{2}k_{2}E\sup\limits_{0\leq t\leq T}\|u^{\epsilon}-\mathbf{P}u\|_{H^{\epsilon}}^{2}+4T^{2}\|f^{\epsilon}(\mathbf{P}u)-\mathbf{P}f(u)\|_{H^{\epsilon}}^{2}\\
&&+2E\sup\limits_{0\leq t\leq T}\|\int_{0}^{t}(e^{-A_{\epsilon}(t-s)}\mathbf{P}-\mathbf{P}e^{-A(t-s)})f(u)ds\|_{H^{\epsilon}}^{2}.
\end{eqnarray*} Denote $I_{21}=E\sup\limits_{0\leq t\leq T}\|\int_{0}^{t}(e^{-A_{\epsilon}(t-s)}\mathbf{P}-\mathbf{P}e^{-A(t-s)})f(u)ds\|_{H^{\epsilon}}^{2}$. Then we have \begin{eqnarray*} I_{21}
&=&E\sup\limits_{0\leq t\leq T}\|\int_{t-r}^{t}(e^{-A_{\epsilon}(t-s)}\mathbf{P}-\mathbf{P}e^{-A(t-s)})f(u)ds\\
&&+\int_{0}^{t-r}(e^{-A_{\epsilon}(t-s)}\mathbf{P}-\mathbf{P}e^{-A(t-s)})f(u)ds\|_{H^{\epsilon}}^{2}\\
&\leq& 2E\sup\limits_{0\leq t\leq T}\|\int_{t-r}^{t}(e^{-A_{\epsilon}(t-s)}\mathbf{P}-\mathbf{P}e^{-A(t-s)})f(u)ds\|_{H^{\epsilon}}^{2}\\
&&+2E\sup\limits_{0\leq t\leq T}\|\int_{0}^{t-r}(e^{A_{\epsilon}(t-s)}\mathbf{P}-\mathbf{P}e^{-A(t-s)})f(u)ds\|_{H^{\epsilon}}^{2}\\
&\leq& C(R,T)r^{2}+C(R,T)\frac{\tau(\epsilon)^2}{r^2}. \end{eqnarray*} Hence we obtain \begin{eqnarray*} I_{2}&\leq&4T^{2}k_{2}E\sup\limits_{0\leq t\leq T}\|u^{\epsilon}-\mathbf{P}u\|_{H^{\epsilon}}^{2}+4T^{2}\tau_{1}(\epsilon)\\ &&+C(R,T)r^{2}+C(R,T)\frac{\tau(\epsilon)^2}{r^2}\,. \end{eqnarray*} For $I_{3}$ we have \begin{eqnarray*} I_{3}&\leq&
CE\int_{0}^{T}\|e^{-A_{\epsilon}(t-s)}g^{\epsilon}(u^{\epsilon})-\mathbf{P}e^{-A(t-s)}g(u)\|_{H^{\epsilon}}^{2}ds\\ &\leq&
CE\int_{0}^{T}\|e^{-A_{\epsilon}(t-s)}(g^{\epsilon}(u^{\epsilon})-\mathbf{P}g(u))\|_{H^{\epsilon}}^{2}ds\\
&&+CE\int_{0}^{T}\|(e^{-A_{\epsilon}(t-s)}\mathbf{P}-\mathbf{P}e^{-A(t-s)})g(u)\|_{H^{\epsilon}}^{2}ds\\ &\leq &
CE\int_{0}^{T}\|e^{-A_{\epsilon}(t-s)}(g^{\epsilon}(u^{\epsilon})-g^{\epsilon}(\mathbf{P}u))\|_{H^{\epsilon}}^{2}ds\\ &&+
CE\int_{0}^{T}\|e^{-A_{\epsilon}(t-s)}(g^{\epsilon}(\mathbf{P}u)-\mathbf{P}g(u))\|_{H^{\epsilon}}^{2}ds\\
&&+CE\int_{0}^{T}\|(e^{-A_{\epsilon}(t-s)}\mathbf{P}-\mathbf{P}e^{-A(t-s)})g(u)\|_{H^{\epsilon}}^{2}ds\\
&\leq &CTk_{2}E\sup\limits_{0\leq t\leq T}\|u^{\epsilon}-\mathbf{P}u\|_{H^{\epsilon}}^{2}+CT\tau_{2}(\epsilon)+CI_{31}, \end{eqnarray*} where
\begin{eqnarray*} I_{31}=E\int_{0}^{T}\|(e^{-A_{\epsilon}(t-s)}\mathbf{P}-\mathbf{P}e^{-A(t-s)})g(u)\|_{H^{\epsilon}}^{2}ds. \end{eqnarray*} Let $l=t-s$. Note that $t\geq s$ and $0\leq t\leq T$. Then we have
\begin{eqnarray*} I_{31}&=&E\int_{0}^{t}\|(e^{-A_{\epsilon}l}\mathbf{P}-\mathbf{P}e^{-Al})g(u)\|_{H^{\epsilon}}^{2}dl\\
&=&E\int_{0}^{r}\|(e^{-A_{\epsilon}l}\mathbf{P}-\mathbf{P}e^{-Al})g(u)\|_{H^{\epsilon}}^{2}dl+
E\int_{r}^{t}\|(e^{-A_{\epsilon}l}\mathbf{P}-\mathbf{P}e^{-Al})g(u)\|_{H^{\epsilon}}^{2}dl\\ &\leq& C(T,R)r+C(T,R)\frac{\tau(\epsilon)}{r}. \end{eqnarray*} A combination of
$I_1$, $I_2$ and $I_3$, we finally get \begin{eqnarray*} &&E\sup\limits_{0\leq t\leq T}\|u^{\epsilon}(t)-\mathbf{P}u(t)\|_{H^{\epsilon}}^{2}\\
&\leq& C(T)k_{2}E\sup\limits_{0\leq t\leq T}\|u^{\epsilon}(t)-\mathbf{P}u(t)\|_{H^{\epsilon}}^{2}\\ &+&C(T,R)(r^{2}+r+\tau_{0}(\epsilon)+\tau_{1}(\epsilon))+\frac{\tau(\epsilon)^2}{r^2}+\frac{\tau(\epsilon)}{r}).
\end{eqnarray*} We can choose a sufficiently small $T$ such that $C(T)k_{2}<1$, thus \begin{eqnarray*} &&E\sup\limits_{0\leq t\leq T}\|u^{\epsilon}(t)-\mathbf{P}u(t)\|_{H^{\epsilon}}^{2}\\&&\leq \frac{C(T,R)(r^{2}+r+\tau_{0}(\epsilon)+\tau_{1}(\epsilon)+\frac{\tau(\epsilon)^2}{r^2}+\frac{\tau(\epsilon)}{r})}{1-C(T)k_{2}}.
\end{eqnarray*} In particular, \begin{eqnarray*} E\sup\limits_{0\leq t\leq T}\|u^{\epsilon}(t)-\mathbf{P}u(t)\|_{H^{\epsilon}}^{2}\rightarrow 0 \end{eqnarray*} as $\epsilon\rightarrow 0$ and then $r\rightarrow 0$. \end{proof}
\end{document}
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\begin{document}
\title{$p$-adic Distance, Finite Precision and Emergent Superdeterminism: A Number-Theoretic Consistent-Histories Approach to Local Quantum Realism} \begin{abstract} Although the notion of superdeterminism can, in principle, account for the violation of the Bell inequalities, this potential explanation has been roundly rejected by the quantum foundations community. The arguments for rejection, one of the most substantive coming from Bell himself, are critically reviewed. In particular, analysis of Bell's argument reveals an implicit unwarranted assumption: that the Euclidean metric is the appropriate yardstick for measuring distances in state space. Bell's argument is largely negated if this yardstick is instead based on the alternative $p$-adic metric. Such a metric, common in number theory, arises naturally when describing chaotic systems which evolve precisely on self-similar invariant sets in their state space. A locally-causal realistic model of quantum entanglement is developed, based on the premise that the laws of physics ultimately derive from an invariant-set geometry in the state space of a deterministic quasi-cyclic mono-universe. This dynamically invariant self-similar subset is locally homeomorphic to $\mathbb Z_2 \times \mathbb R$ where $\mathbb Z_2$ denotes the set of 2-adic integers and $\mathbb R$ denotes a state-space trajectory, or history. Based on this, the notion of a complex Hilbert vector is reinterpreted in terms of an uncertain selection from a finite sample space of states, leading to a novel form of `consistent histories' based on number-theoretic properties of the transcendental cosine function. For example, for a Mach-Zehnder experiment with phase angle $\phi$, histories where $\cos \phi$ and $\phi/\pi$ are describable by a finite number of bits are, by number theory, almost always mutually incompatible, all supplementary variables being equal. This leads to novel realistic interpretations of position/momentum non-commutativity, EPR, the Bell Theorem and the Tsirelson bound. In this inherently holistic theory - neither conspiratorial, retrocausal, fine tuned nor nonlocal - superdeterminism is not invoked by fiat but is emergent from these `consistent histories' number-theoretic constraints. Because of finite experimental precision, experimenters have no direct control on whether $\cos \phi$ is finitely describable or not. Hence, Bell inequalities are violated without constraining experimenter free will. Quantum decoherence is described by chaotic riddled-basin dynamics, leading to a natural clustering of trajectories ('measurement eigenstates') on the invariant set. The algebraically closed complex Hilbert Space and associated Schr\"{o}dinger/Dirac equation arise in the singular limit when a fractal parameter $N$ goes to infinity. Invariant set theory provides new perspectives on many of the contemporary problems at the interface of quantum and gravitational physics, and, if correct, may signal the end of particle physics beyond the Standard Model.
\end{abstract}
\tableofcontents
\section{Introduction}
The recent experiments by Shalm et al \cite{Shalm} on entangled photon pairs appear to have nailed the coffin on local realism, i.e. have comprehensively ruled out putative theories of quantum physics which are both locally causal and deterministic. However, this conclusion is predicated on an assumption which is impossible to test with this type of experiment: that the experimental measurement settings are physically independent of any of the properties of the entangled particles. This is variously referred to as the `Free Choice' or `Free Will' assumption, the `No Conspiracy' assumption, or, more neutrally, the `Measurement Independence' assumption. A theory in which this assumption is violated is referred to as `superdeterministic'. Most physicists who work in the field of either quantum foundations or quantum information theory feel that this assumption is self evident and are consequently almost unanimous in rejecting superdeterministic approaches to fundamental physics. Reasons for rejection include implausible conspiracies, an inability to do science, violation of experimenter free will, or unacceptable fine tuning. In this paper I will explain why I believe the arguments against superdeterminism are unconvincing, and will propose a new theory of quantum physics - invariant set theory - which is not superdeterministic by fiat, but from which plausible superdeterminism is emergent. Although locally causal and realistic, the model violates the Bell inequalities as does quantum theory. It has none of the objectionable properties typically associated with a superdeterministic theory.
To make such a claim I need some `new meat': \begin{itemize}
\item Firstly and most importantly, the developments in this paper are based around the use of the $p$-adic metric. Number theory provides us with precisely two norm-based metrics: the Euclidean and the $p$-adic \cite{Katok}. It is proposed that both play central but distinct roles in physics: the former in space-time, the latter in state space. Use of the $p$-adic metric in state space is consistent with a physical theory where states evolve \emph{precisely} on certain self-similar dynamically invariant geometries in state space. This can arise if the corresponding laws of physics are ultimately derived from such state-space geometries.
\item Secondly, a representation of the multiplicative group of $2^N$th roots of unity is described in terms of cyclic permutations of $2^N$-element bit strings. Based on this, complex Hilbert vectors (and tensor products) are interpreted in terms of uncertain selections of bits from sets of $2^N$-element sample spaces.
\item Thirdly, the notion of finite experimental precision is exploited. This, together with the items above, ensures that invariant set theory is consistent with experimenter free will. An interplay between the Euclidean metric in space-time and the $p$-adic metric in state space will be used to demonstrate why the violation of the Bell inequalities is robust and requires no special precision on the part of the experimenter, in setting experimental parameters. The notion of finite precision is actually not so new; it has already been shown to be capable of nullifying the Kochen-Specker theorem \cite{Meyer:1999}.
\end{itemize}
My primary motivation for this work was less about resolving the Bell Theorem, and more about exploring novel possibilities for synthesising quantum and gravitational physics - ones that contrast with conventional approaches based on quantum field theory - which might help resolve contemporary problems at the interface of quantum and gravitational physics: the dark universe, black hole information loss, the 2nd law of thermodynamics, space-time singularities and so on. Some speculative remarks on these issues are addressed towards the end of the paper.
The structure of this paper is provided in the contents listing above. Most of the technical discussion is relegated to Appendices. No prior knowledge about the $p$-adic metric is assumed.
\section{Why is Superdeterminism so Disliked?} \label{Dislike}
In this Section I review the reasons why superdeterminism has been largely rejected by the quantum foundations community and argue why I find these reasons unconvincing. In laying out these arguments, my purpose is not to change sceptical minds, but rather to try to persuade readers that might otherwise be completely dismissive about the concept of superdeterminism, to read on.
Following the CHSH version of the Bell Theorem, Alice and Bob make measurements on some entangled physical system. The measurement apparatus available to each experimenter has two possible settings, referred to as 0 and 1. For each setting, the measurement has two possible outcomes, also 0 and 1. If $x$ and $y$ refer to the settings, and $a$ and $b$ to the outcomes, then an experiment over multiple measurements determines conditional frequencies of occurrence $p(a,b|x,y)$. A realistic theory attempts to explain these frequencies by assuming that $a$ and $b$ are determined both by $x$ and $y$ and some supplementary information, represented by the generic variable $\lambda$, over which the experimenters do not have control. In this way we can write \begin{equation} \label{probs}
p(a,b|x,y) = \sum_{\lambda} p(a,b|x,y, \lambda) p(\lambda|x,y) \end{equation}
This equation has been written in a general probabilistic form, commonplace in modern accounts of the Bell Theorem (e.g. \cite{Brunner}). This allows for probabilistic as well as deterministic models. In the discussions below, I will be discussing a strictly deterministic model, so that either $p(a,b|x,y, \lambda)=1$ or $p(a,b|x,y, \lambda)=0$. Indeed, as discussed in Section \ref{Dynamics}, determinism is critically important in this model, unlike in more conventional hidden-variable models. By contrast, $p(\lambda| x,y)$ denotes some non-trivial probability distribution, defined by frequentism on finite sample spaces $\Lambda_{xy}$.
The Measurement Independence assumption states that $\lambda$ is not correlated with $x$ and $y$, so that \begin{equation} \label{mi}
p(\lambda|x,y)=p(\lambda) \end{equation} or equivalently that $\Lambda_{xy}=\Lambda$, independent of $x$ and $y$. As first recognised by Bell himself, if (\ref{mi}) is violated in some putative hidden-variable theory, then this theory will not necessarily violate the Bell inequality, even if the theory is locally causal and deterministic. A theory in which (\ref{mi}) does not hold is said to be superdeterministic.
However, with one notable exception \cite{tHooft:2015b, tHooft:2015} discussed in Section \ref{tHooft}, contemporary researchers are largely unequivocal about the correctness of (\ref{mi}). Wiseman and Cavalcanti \cite{WisemanCavalcanti} summarise the prevailing view: \begin{quote} This temptation [to reject (\ref{mi})] should vanish if the reader thinks through what it would actually mean to explain away Bell correlations through the real (not just in-principle) failure of free choice. There is no general theory that does this. If such a theory did exist, it would require a grand conspiracy of causal relationships leading to results in precise agreement with quantum mechanics, even though the theory itself would bear no resemblance to quantum mechanics. Moreover, it is hard to imagine why it should only be in Bell experiments that free choices would be significantly influenced by causes relevant also to the observed outcomes. [R]ather, every conclusion based upon observed correlations, scientific or casual, would be meaningless because the observers's method would always be suspect. It seems to us that any such theory would be about as plausible, and appealing, as, belief in ubiquitous alien mind-control. \end{quote} This latter point is also emphasised by Ara\'{u}jo \cite{Araujo} who refers to (\ref{mi}) as the `no-conspiracy' condition: \begin{quote} I think [(\ref{mi})] is necessary to even do science, because if it were not possible to probe a physical system independently of its state, we couldn't hope to be able to learn what its actual state is. It would be like trying to find a correlation between smoking and cancer when your sample of patients is chosen by a tobacco company. \end{quote} Shalm et al \cite{Shalm} also highlight the supposedly convoluted nature of physical reality if (\ref{mi}) is violated: \begin{quote} If [(\ref{mi})] is not true, then a hidden variable could predict the chosen settings in advance and use that information to produce measurement outcomes that violate a Bell inequality \end{quote}
Bell \cite{Bellb}, in trying to get to the heart of the issue, advanced what I consider to be the most important argument against superdeterminism. As part of this, Bell shows that experimenter free will is not itself the fundamental issue - though I will discuss human free will in Section \ref{free}. In particular, Bell replaces Alice and Bob with mechanical pseudo-random number generators. For these machines, the output is extraordinarily sensitive to minute variations of the initial conditions. In particular, Bell imagines that $x$ and $y$ are set equal to 0 if the millionth bit of some input variable is a 0 and set equal to 1 if the millionth bit of the same input variable is a 1. That is to say, fixing $x$ fixes something about this input variable. Bell then writes \cite{Bellb}: \begin{quote} But this peculiar piece of information is unlikely to be the vital piece for any distinctively different purpose, i.e. it is otherwise rather useless. With a physical shuffling machine, we are unable to perform the analysis to the point of saying just what peculiar feature of the input is remembered by the output. But we can quite reasonably assume that it is not relevant for other purposes. In this sense the output of such a device is indeed a sufficiently free variable for the purpose at hand. For this purpose the assumption [(\ref{mi})] is then true enough, and the [Bell] theorem follows. \end{quote}
These arguments may seem compelling. However, below I attempt to provide critiques of all of them, leaving Bell's argument for last.
In my view, it seems illogical to say that if a superdeterministic theory did exist, it would require a grand conspiracy of causal relationships. The word `conspiracy' implies something secret or covert. If a plausible superdeterministic theory of quantum physics were somehow to be formulated, then by definition the causal structure of the universe would be laid open for all to see and understand. There would be no conspiracies by definition. Referring to (\ref{mi}) as a `no-conspiracy' condition is therefore prejudicial.
Moreover, if a plausible superdeterministic theory were able to be formulated, then it also seems to me unlikely that it is only in Bell experiments that `free choices would be influenced by causes relevant also to the observed outcomes'. Rather, this situation would become the norm where all non-classical measurements are made, i.e. where the quantum mechanical observables are non-commutative. Far from being special, this situation may be completely ubiquitous in quantum physics. Does that make the problem worse? Far from it, as I attempt to show below.
The notion that if a superdeterministic theory did exist, its results would have to be `in agreement with quantum mechanics even though the theory would bear no resemblance to quantum mechanics', could be a merit rather than a drawback. Results from General Relativity agree well with Newtonian Gravity in the appropriate limit, even though General Relativity bears no resemblance to Newtonian theory. Given the ongoing problems synthesising quantum and gravitational physics, some might view the formulation of a deterministic locally causal (\emph{a fortiori} geometric) theory, whose results agree with quantum theory but which bears no resemblance to quantum theory, to be something that is sorely needed! This is my view, at least.
And before worrying about `alien mind control' I argue in Section \ref{Gene} that by rejecting superdeterminism \emph{a priori}, we may, ironically, have been subverted in our thinking about the Bell Theorem by something just as insidious but rather closer to home - gene control!
Shalm et al's notion that a hidden variable could somehow predict the chosen settings in advance and use that information to ensure measurement statistics are quantum theoretic, is rather anthropomorphic. It conjures up a picture of a Laplacian demon, furiously computing the future before it happens, and then rearranging things to ensure the quantum theoretic result always obtains. If this is how the world works, it would indeed be bizarre. However, if the demon could predict the chosen settings in advance, then ordinary determinism (never mind superdeterminism) would be in deep trouble. Suppose $\lambda$ is a set of Cauchy data on a spacelike hypersurface in the past of the event where Alice chooses $x$. If the demon could use $\lambda$ to predict Alice's choice, and somehow chalk it up for her to see, then Alice would merely have to choose differently to the predicted choice to create a logical paradox. There is no paradox because, even if the laws of physics are deterministic, they are not predictably so. From a practical perspective one could put it like this: if the laws of physics are sufficiently chaotic, any reliable prediction would require a computational capability exceeding that of the whole universe. Since our putative demon would necessarily be a subset of the universe, his limited computational capability prevents such reliable prediction. These ideas can be expressed more formally in terms of the G\"{o}del-Turing theorem: there are deterministic systems whose properties are generically not algorithmic at all. Fractals, the limiting states at $t=\infty$ arising from certain chaotic systems and to which we return below, provide a beautiful geometric manifestation of the G\"{o}del-Turing theorem \cite{Blum} \cite{Dube:1993}. (For this reason, there is actually no need to keep the supplementary variables $\lambda$ `hidden' in a model which encodes the unpredictability of the real world.) In summary, Shalm et al's argument against superdeterminism assumes properties which such a theory need not have.
The argument that we can't do science when the system being measured is not fully independent of the system that performs the measurement is reminiscent of the argument that it is not possible to do objective science on the topic of consciousness, since we, the investigators, are conscious beings and therefore not fully independent of the topic we investigate. However, given the extraordinary advances in the science of consciousness in the last decade or so (e.g. using Functional MRI), this argument does not appear to be sound. In the case of consciousness, it merely indicates that we have to be careful when designing and analysing experiments which probe our and other animals' consciousness, and I would say the same about analysis of experiments which probe quantum physics. Fortunately, there is a way to analyse data without potential self-referential limitations. That is to say, nature has given us the capability to study causal relationships where the system and the measuring apparatus have become decoupled. It is the classical limit. Hence, if we want to do science where we are sure that classical notions of measurement independence hold, then we must do it in the classical limit (where observables commute). In any case, to say that we can't do science when not in the classical limit is countered explicitly by the theory discussed below, which are being based on (what I think are) sound scientific principles.
In my view, the most important argument against superdeterminism was provided by Bell and this needs more extensive analysis. In Bell's example, the number `one million' is there to emphasise that the output of the pseudo-random number machine depends on something arbitrarily small (the logic works \emph{a fortiori} if we replace `one million' by `one trillion'). Since the supplementary variables $\lambda$ in principle describe all that is going on in the universe other than these bits - the moons of Jupiter for example\footnote{In \cite{Bellb} Bell notes: `In this matter of causality, it is a great inconvenience that the real world is given to us only once. We cannot know what would have happened had something been different. We cannot repeat an experiment changing just one variable; the hands of the clock will have moved and the moons of Jupiter.' A perturbation which changes just one variable is therefore a perturbation in state space, not in space-time.} - then Bell is drawing attention to our intuition that surely the moons of Jupiter couldn't care less whether that millionth bit here on Earth was a 0 or a 1. Implicit in Bell's argument is that if for some strange reason you think that they do care about the millionth digit, then surely they won't care about the trillionth digit (and son on)! Hence, if we posit some superdeterministic theory of quantum physics, then we need some argument to explain why the moons do `care', no matter how small the effect that sets $x$ and $y$. Maybe we could suppose that, for some theoretical reason, parts of the state space of the universe contains gaps or lacunae: regions where, for some theoretical reason, no space-time trajectories, or `histories', can exist. Maybe by flipping the value of the millionth digit, but keeping fixed the supplementary variables (including the moons of Jupiter), the state of the universe somehow falls into one of these lacunae. However, be this as it may, this doesn't address Bell's concern that in such a theory, the condition of being in a lacuna or otherwise would be sensitive to arbitrarily small perturbations (i.e. to the value of the millionth digit of the random number generator input). Any theory describing this state of affairs would be implausibly fine tuned. We require theories of physics to be structurally stable, with gross properties which are not sensitive to arbitrarily small perturbations. After all, as Bell pointed out, the experimenter's hands tremble when she set the dials on her instruments. A that requires the hands to be completely steady and the values of experimental parameters precise is simply inconsistent with observation. Finely-tuned models, which appear to be implicit in all causal description of the Bell inequalities \cite{WoodSpekkens}, are not acceptable. Game, set and match to Bell? No!
\section{Rethinking the Notion of State-Space Distance} \label{metric}
There is a subtle but important issue in Bell's argument above: perhaps the single most important single issue in this paper. When we describe some perturbation as `small' what do we mean? Suppose, in a particular run of Bell's pseudo-random number generator, the millionth digit was a 0 (and hence, in the consequent experiment, $x=0$). Label the real world in which these events transpire by $U$, and imagine a counterfactual world $U'$, identical to $U$ in all respects (including the position of the moons of Jupiter) except that the millionth digit was a 1 rather than a 0. Is the distance between $U$ and $U'$ small (say at the time the input variable was being inputted to the pseudo-random number generator)? Certainly in terms of the familiar Euclidean metric, the distance between $U$ and $U'$ can indeed be considered small (and can be made smaller still by making the output of the machine sensitive to the trillionth rather than millionth input bit).
In assessing the distance between $U$ and $U'$, we are considering distances in state space (recall Footnote 1 above). In physics, we typically use the Euclidean metric to define distances not only in space-time but also in state space. Hence, when the philosopher David Lewis made the following superficially incontrovertible statement in his seminal paper on causation \cite{Lewis}: \begin{quote} We may say that one world is closer to actuality than another if the first resembles the actual world more than the second does. \end{quote}
Lewis is implicitly assuming closeness is synonymous with smallness of the Euclidean metric. Mathematically, the Euclidean metric is ultimately a number-theoretic concept: given numbers $x$ and $y$, the Euclidean distance between them is $|x-y|$ where $|\ldots|$ represents the absolute value (or Euclidean norm). However, in number theory, there does exist another norm-induced metric, called the $p$-adic metric (e.g. \cite{Katok}), where $p$ is typically a prime number. Moreover, by Ostrowski's theorem \cite{Katok}, the $p$-adic metric is the only norm-induced alternative to the Euclidean metric. As will be discussed below (see also Fig \ref{cantor}), Lewis' intuition is false if the notion of closeness is defined $p$-adically.
Before outlining why, it is worth briefly discussing an analogy which illustrates the importance of using the correct metric. At first sight, Penrose's impossible triangle (Fig 4-7 of \cite{Penrose:2016}) might suggest something incomprehensible about physical space. However, this conclusion is only reached because we visualise the triangle in 2-dimensional Euclidean space and therefore imagine that parts of the triangle that are close in the Euclidean 2D metric, are physically close. However, this is not so. As is well known, the object is constructible in 3-dimensional Euclidean space, whence one pair of sides that appear to meet at a vertex, do not. The ends of these sides are far apart in the more physically relevant Euclidean 3D metric. In the case of the impossible triangle, use of the wrong metric could perhaps lead us into thinking that space is weirder than it really is.
The $p$-adic metric is very commonly used in number theory: \begin{quote}
`We [number theorists] tend to work as much $p$-adically as with the reals and complexes nowadays, and in fact it it best to consider all at once.' (Andew Wiles, personal communication 2015.)
\end{quote} Physicists use real and complex numbers in equal measure, but, so far, not the $p$-adics\footnote{There have been attempts to use $p$-adics in physics, see. e.g. \cite{Volovich, Khrennikov}. However, in these approaches, the $p$-adic metric was used to describe Plankian scales in space-time, and not as a fundamental yardstick in state space. Moreover, so-called $p$-adic integers were not singled out as having special ontological status (see below). The particular use of $p$-adic numbers in this paper is, I believe, new}. However, if there exist two and only two fundamentally inequivalent metrics in mathematics, why would physics only make use of one of them and completely shun the other? Is there any overriding reason to use the Euclidean metric in state space? I am not aware of one. Are there good physical reasons for using the Euclidean metric in space-time and the $p$-adic metric in state space? I believe there are.
A brief review of $p$-adic numbers is given in Appendix \ref{padic}. An even briefer introduction is given here. Given a distance function on the rationals $\mathbb Q$, we can complete $\mathbb Q$ based on equivalence classes of Cauchy sequences of rationals. Completion with respect to the Euclidean metric produces the familiar real numbers $\mathbb R$, whilst completion with respect to the $p$-adic metric produces a number system, the field of $p$-adic numbers $\mathbb Q_p$, which is quite different to $\mathbb R$. In my view, the most physical way to understand the differences between these number systems is through their links to geometry. $\mathbb R$ provides the basic building block to analyse Euclidean geometries. By contrast, a subset $\mathbb Z_p$ of $\mathbb Q_p$, known as the set of $p$-adic integers, is homeomorphic to self-similar Cantor sets $C(p)$ with $p$ iterated pieces, i.e. to fractal geometries (see Fig \ref{cantor}a with $p=2$, the value for $p$ used in this paper). We close the circle with physics by noting that fractals play a central role in describing the asymptotic structure of certain deterministic chaotic dynamical systems \cite{Strogatz}, one of the most famous examples being the Lorenz attractor \cite{Lorenz:1963}. Locally, these fractals describe a Cantor set's worth of trajectories, or histories as they are commonly described in quantum theory (I use both words below). A schematic illustration of part of $C(2) \times \mathbb R$ is given in Fig \ref{cantor}b.
To understand the key property of the $p$-adic metric needed to counter Bell's argument against superdeterminism, consider two points $a$ and $b$ which both lie on $C(p)$. Then the $p$-adic distance between $a$ and $b$ can be made as small as one likes by bringing $a$ and $b$ sufficiently close together. (This is related to the fact that $a$ and $b$ can both be represented by elements of $\mathbb Z_p$.) However, consider a third point $c$ which does not lie on $C(p)$, but, let's say, in one of the lacunae between pieces of $C(p)$. Then the $p$-adic distance between $a$ and $c$, or between $b$ and $c$, cannot be made smaller than $p$, no matter how small the Euclidean distance is between $a$ and $c$ or between $b$ and $c$ \footnote{There is a subtlety here. There are many real numbers which have no correspondence in $\mathbb Q_p$ and hence whose $p$-adic distance to the invariant set is undefined (but hence not small). However, $\mathbb Q_p$ can be extended to an algebraically complete field $\mathbb C_p$, isomorphic to the field $\mathbb C$ of complex numbers \cite{Robert}. This suggests that one should start with a state space which is inherently complex, such as that in twistor theory \cite{Penrose:1967}, and then consider an embedded dynamically invariant subset with a topological structure associated with the corresponding ring of integers of $\mathbb C_2$ which is $2$-adically distant from the rest of state space.} Hence, distance measured using the $p$-adic metric provides a direct and natural way to determine the ontology of putative states of a deterministic system which makes an ontological distinction between those states lying on some fractal invariant subset of state space, and those not lying on this invariant set. The theory that is developed below is predicated on the notion that whilst the Euclidean metric is the metric of choice in space time, the 2-adic metric is the metric of choice in state space.
\begin{figure}
\caption{a) Some iterates of the Cantor Ternary Set $C(2)$, itself equal to the intersection of all such iterates and homeomorphic to the set of 2-adic integers $\mathbb Z_2 \subset \mathbb Q_2$. The points $a$ and $b$ both belong to $C(2)$ but the point $c$ does not. Both the Euclidean and the 2-adic distance between $a$ and $b$ is small. However, even though the Euclidean distance between $a$ and $c$ is smaller than between $a$ and $b$, the 2-adic distance between $a$ and $c$ is not smaller than between $a$ and $b$. Hence $b$ is 2-adically closer to $a$ than is $c$. b) Two trajectories or histories of the $i$th iterate $C_i(2) \times \mathbb R$ of $C(2) \times \mathbb R$. Zooming into one trajectory reveals two trajectories of $C_{i+1}(2) \times \mathbb R$ and zooming into one of these reveals two trajectories of $C_{i+2}(2) \times \mathbb R$, and so on.}
\label{cantor}
\end{figure}
What has this got to do with the Bell argument against superdeterminism? Let us suppose that the universe $U$ at any particular time can be considered the state of a deterministic dynamical system evolving \emph{precisely} on a fractal invariant set $I_U$ \cite{Palmer:2009a}, locally of the form $C(2) \times \mathbb R $ in cosmological state space\footnote{In Appendix \ref{Dirac} we show that the trajectories, or histories, are in fact discretised in the time direction. Hence, strictly, we should write $C(p) \times \mathbb Q$ rather than $C(2) \times \mathbb R$.} \cite{Palmer:2014}. There are a number of cosmological implications for such a supposition, discussed in Section \ref{Gravity}. Such a model of the universe, although deterministic, is not classical. The states of classical dynamical systems, being based on differential or difference equations of motion, evolve from arbitrary states in state space, and therefore do not generically lie \emph{precisely} on their asymptotic invariant-set attractors except in the (classically unattainable) limit $t=\infty$. As mentioned above, from a $p$-adic perspective, the difference between not lying on $I_U$ and lying on $I_U$ is never small, even for arbitrarily large $t$ (and even if the difference appears very small from a Euclidean perspective). That is to say, from a $p$-adic perspective, lying on $I_U$ precisely at $t=\infty$ is a singular limit \cite{Berry}. The question posed at the beginning of this section can be posed as follows: does the counterfactual world $U'$ where the millionth input bit has been flipped, lie on $I_U$ or not? If it does not, it cannot be considered close to $U$, whether it is the millionth or the trillionth digit that has been flipped. At this stage in the paper we do not have enough information to say whether $U'$ lies on $I_U$ or not. We need to make contact with quantum physics, and complex Hilbert vectors in particular.
Before concluding this Section, it can be noted that many of the methods of analysis (algebra, calculus, Fourier transforms, and Lie group theory - the bread and butter of theoretical physics) can be applied to the set of $p$-adic numbers \cite{Robert}. In essence, $p$-adic numbers are to fractal geometry as real numbers are to Euclidean geometry.
\section{Number-Theoretic Consistent Histories and EPR} \label{interfere}
In this Section, a novel realistic `consistent-histories' interpretation of complex Hilbert vectors is developed. Consider a standard Mach-Zehnder interferometric experiment. Let $x=0$ if the experimenter chooses to perform a momentum measurement (see Fig \ref{MachZehnder}a) and $x=1$ if she chooses to perform a position measurement (see Fig \ref{MachZehnder}b). \begin{figure}
\caption{a) A momentum experiment. b) A position experiment (obtained by removing the second half-silvered mirror) in the single photon Mach-Zehnder apparatus.}
\label{MachZehnder}
\end{figure}
According to quantum theory, the input state vector $|a\rangle$ is transformed by the non-commuting unitary operators \be U=\bp \; 1 &1 \\ 1& -1 \; \ep; \ \ \ V=\bp \; 1 &0 \\ 0& e^{i\phi} \; \ep \nonumber
\ee When $x=0$, the transformation is given by \be \label{momentum}
|a\rangle \stackrel{UVU}{\mapsto} \cos\frac{\phi}{2}|a\rangle + \sin\frac{\phi}{2} |\cancel a\rangle \ee and the probability of detection by the two detectors, is equal to $\cos^2 \phi/2$ and $\sin^2\phi/2$ respectively. When $x=1$, the transformation is given by \be \label{position}
|a\rangle \stackrel{VU}{\mapsto} \frac{1}{\sqrt 2}(|a\rangle + e^{i \phi} |\cancel a\rangle) \ee and the probability of detection by either of the detectors, and hence the frequency of detection by either detector, is equal to $1/2$. The experimenter is of course free to set $\phi$ and choose between $x=0$ and $x=1$ as she likes. However, necessarily $\phi$ can only be set to finite precision - the experimenter's hands tremble \cite{Bellb} and so cannot have complete control on all the bits that describe $\phi$. This will be important in what is described below.
Recall that in classical physics, unit vectors in a real Hilbert space provide a natural quantity to represent the uncertain state of a system when - e.g. when describing some future state. For example, let $\mathbf i$ denote a unit vector representing the event $a$: it rains somewhere in London sometime this coming Saturday. The orthogonal unit vector $\mathbf j$ can therefore represent the event $\cancel a$: it rains nowhere in London at any time this coming Saturday. By Pythagoras's theorem, \be \label{realhilbert} \mathbf{v}(P_a)=\sqrt{P_a}\; \mathbf i+\sqrt{P_{\cancel a}} \;\mathbf j \nonumber \ee also has unit norm, for any probability assignment $P_a$ that it will rain somewhere in London this coming Saturday, and $P_{\cancel a}$ that it won't. Hence, in this case, $\mathbf{v}(P_a)$ represents the uncertain state of London's weather this coming Saturday, where the probability that it will rain equals $P_a$. A best estimate of $P_a$ is given by an ensemble weather forecast for Saturday \cite{Palmer:2006}. Such a forecast may typically comprise 50 integrations of a numerical weather forecast model - itself encoding the Navier-Stokes equations - each individual forecast being made from slightly different starting conditions (consistent with the fact that available weather observations only define the starting conditions imperfectly). Hence $P_a=q/50$ and $P_{\cancel a}=1-q/50$ where $q \le 50$ denotes the number of individual forecasts which predict rain over London on Saturday. Since each weather forecast defines a trajectory of the classical equations of motion, $P_a$ is defined by frequentism over the space of 50 weather trajectories which lie, approximately, on Earth's weather attractor in meteorological state space. We can assume the real weather follows one of these trajectories - though which one is uncertain until we have observed Saturday's actual weather.
By analogy, we can interpret the real Hilbert vector (\ref{momentum}) in terms of an uncertain selection from some finite string \be \label{string} \{a_1 \; a_2 \; a_3 \ldots a_{2^N}\} \ee where $a_i \in \{a, \cancel a\}$, $N$ is a parameter to be described below, and a fraction $\cos^2 \theta/2$ of the $2^N$ elements are $a\;$s, the remaining $\sin^2 \theta/2\; 2^N$ elements being $\cancel a\;$s. Each element of the string labels a state-space trajectory; $a_i=a$ if a photon is detected by $D_a$, $a_i=\cancel a$ if the photon is detected by $D_{\cancel a}$. By definition, $\cos^2 \theta/2$ and hence $\cos \theta$ is describable by a finite number of bits. Hereafter, I will use the phrase `finitely describable' to mean `describable by a finite number of bits'. It will be convenient to order the bits of (\ref{string}) so that the first $\cos^2 \theta/2$ of the $2^N$ elements are $a\;$s.
The complex Hilbert vector (\ref{position}) can be interpreted in a similarly realistic way. First let $\theta=\pi/2$ so that the first half of the elements of (\ref{string}) are $a\;$s. Then define a cyclical permutation operator $\zeta$ \be \label{zeta} \zeta \{a_1 \; a_2 \; a_3 \ldots a_{2^N}\}= \{a_{2^N} \; a_1 \; a_2 \ldots a_{2^{N-1}}\} \ee so that $\zeta^{2^N}$ is equal to the identity operator, i.e. $\zeta$ is a representation of a $2^N$th root of unity. We can therefore write \be \label{exp} \zeta= e^ {2\pi i /2^N} \ee as an equivalent expression for the operator $\zeta$. In this way, we will define (\ref{position}) as an uncertain selection of element from the bit string \be \zeta^n \{a_1 \; a_2 \; a_3 \ldots a_{2^N}\} \ee where $n=2^N \phi/2\pi$.
In order to avoid disrupting the flow of this Section too much, I summarise below a number of important points to be made about these representations of (\ref{momentum}) and (\ref{position}), leaving more detailed discussion to Sections \ref{measurement}, \ref{Gravity} and Appendix \ref{Hilbert}.
\begin{itemize}
\item At a more fundamental level, this symbolic labelling of trajectories reflects the fact that each trajectory is attracted to one of two distinct clusters on $I_U$ - labelled $a$ and $\cancel a$ - through a deterministic chaotic (and hence nonlinear) process. These clusters therefore correspond to measurement eigenstates. The parameter $N$ is linked to this chaotic process, reflecting the number of fractal iterates needed to evolve to one cluster or the other. In Section \ref{Gravity} we speculate that this clustering of state-space trajectories is a manifestation of the phenomenon of gravity.
\item By the discussion above, a momentum measurement on $I_U$ is therefore associated with a finitely describable $\cos \phi$. A position measurement on $I_U$ is associated with a finitely describable $\phi/\pi$.
\item The chaotic procedure which determines which particular element $a_i$ of (\ref{string}) is selected, will be sensitively dependent on the phase angle $\phi$, even though the probability of selecting $a_i$ is independent of $\phi$. The selection procedure is periodic in $\phi$.
\item The bit-string representations of (\ref{momentum}) and (\ref{position}) are examples of bit-string representations of the general single-qubit Hilbert vector $
\cos \frac{\theta}{2}|a\rangle
+e^{i\phi} \sin \frac{\theta}{2} |\cancel{a}\rangle $ which is invariant under an $SU(2)$ transformation and hence reflects the underlying isotropy of space. As such, the underlying symbolic labelling also reflects this isotropy. and is spontaneously broken by the clustering procedure mentioned in the bullet above. A general $m$-qubit state is represented by $m$ partially correlated bit strings.
\end{itemize}
We now come to a number-theoretic result which is central to this paper: \\ \\ \textbf{Theorem} \cite{Niven, Jahnel:2005}. It is impossible for both $\phi/\pi$ and $\cos \phi$ to be simultaneously finitely describable unless $\phi=0$, $\pi/2$, $\pi$, $3\pi/2 \ldots$. \\ \\ An elementary proof is given in Appendix \ref{number}. \\ \\ A direct consequence of this theorem is the generic non-commutativity of position and momentum measurements - the essence of quantum theory \cite{Griffiths}. Suppose we perform a momentum measurement with the Mach-Zehnder interferometer. Consider the following counterfactual question: Even though in reality the experiment was a momentum measurement, could it have been a position measurement? In asking this counterfactual question we will imagine that the supplementary variables $\lambda$ - including the moons of Jupiter - are held fixed. Now because in reality the first measurement was a momentum measurement, the set of allowed orientations $\phi$ - corresponding to the set of histories on $I_U$ - are those where $\cos \phi$ is finitely describable. Assuming $\phi$ isn't precisely one of the four exceptions (very unlikely if $2^N \gg 4$), we can scan through all the values of $\phi$ where $\cos \phi$ is finitely describable, and we will \emph{never} find one where $\phi/\pi$ is finitely describable. That is to say, a counterfactual state of the universe corresponding to a position measurement does not and cannot lie on the invariant set $I_U$. Now suppose a second actual measurement with the interferometer was a position measurement, i.e. where $\phi/\pi$ is finitely describable. Then a second counterfactual momentum measurement, i.e. where $\cos \phi$ is finitely describable, necessarily lies off $I_U$. This is turn means that the order of the two measurements (the first momentum, the second position) could not be reversed - the measurements are non-commutative, consistent with the Uncertainty Principle.
The finite describability of $\cos \phi$ or $\phi/\pi$ delineates two distinct types of invariant set structure (`momentum structure' and `position structure'). Now as Bell noted, experimenters hands tremble - they only have direct control on the leading bits of $\phi$ and therefore have no direct control, when they set the dials on their instruments, on whether $\cos \phi$ or $\phi/\pi$ is finitely describable. That is to say, whilst the experimenter is completely free to set as many or as few of the bits she can control as she likes, she cannot directly control whether $\cos \phi$ or $\phi/\pi$ is finitely describable. Finite precision is an important consistency condition for this theory: if experimenters could control all the bits of $\phi$, then they would be able to set $\phi/\pi$ to be finitely describable in a situation where a momentum measurement was being made (or $\cos \phi$ to be finitely describable when a position measurement was being made). This inconsistency is reminiscent of the inconsistency described in Section \ref{Dislike} which would arise if the Laplacian demon could predict the future. In both cases the inconsistency is avoided by the consequences of finite precision (in not being able to control whether $\cos \phi$ or $\phi/\pi$ is finitely describable in the first instance, or not being able to predict the future in the second case).
Also crucial to the consistency of this picture is the use of the $p$-adic metric. In conventional physical theory, it is an irrelevance whether or not $\cos \phi$ is finitely describable: the standard Euclidean distance between a universe where $\cos \phi$ is finitely describable and one where it is not, can be made as small as one likes. Since we require our theories to be structurally stable, a conventional theory where these differences mattered would be considered unacceptably fine tuned because the differences would be destroyed by arbitrarily Euclidean-small noise. However, as discussed, if the counterfactual world $U'$ where $\cos \phi$ is not finitely describable lies off the invariant set, it cannot be $p$-adically close to the real world $U$ where $\cos \phi$ is finitely describable (no matter how close these worlds appear from a Euclidean perspective). As such, one cannot perturb the system off the invariant set with $p$-adic-small amplitude noise. That is to say, the finitely describability of $\cos \phi$ is structurally stable to $p$-adic noise.
As Bell has pointed out, the no-go theorems of quantum physics can tolerate a considerable amount of hand trembling. Now hand trembling occurs in space-time, not in state space. We can imagine that as the hand is on the dial, the actual $\phi\;$ realised at any time varies slightly from one time to the next, where `slightly' is meant in the sense that the Euclidean distance between any two $\phi\;$s is small. However, no amount of such trembling need violate the state-space constraint that all elements of the time series of $\cos \phi$s realised are finitely describable when the hand trembles on the dial. As such, the counterfactual universe $U'$ where $\cos \phi$ is not finitely describable will never be encountered, even momentarily, by any such hand trembling. Bell noted that quantification of hand trembling `would require careful epsilonics'. As this discussion shows, such epsilonics must carefully distinguish space-time epsilonics, where the measure of distance is Euclidean, from state-space epsilons where the measure of distance is $p$-adic. There is a subtle interplay between these different metrics when unravelling these matters.
Referring to a state-space trajectory as a history, then invariant set theory can be viewed as a realistic 'consistent histories' theory, although at a technical level this is quite different to the conventional notion of Consistent Histories \cite{Griffiths}. For example, here the word 'consistent' has the precise number-theoretic meaning discussed above: if a trajectory with finitely describable $\cos \phi$ lies on $I_U$, then a trajectory with the same supplementary variables $\lambda$ but with finitely describable $\phi/\pi$ is not a consistent history and does not lie on $I_U$.
This number-theoretic notion of consistency in turn implies that the sample space $\Lambda_{x=0}$ of supplementary variables $\lambda$ associated with finitely describable $\cos \phi$ must be disjoint from the sample space $\Lambda_{x=1}$ of supplementary variables associated with finitely describable $\phi/\pi$. Hence, $\rho(\lambda | x) \ne \rho(\lambda)$, violating (\ref{mi}). A key point is that this is not superdeterminism by fiat (a criticism levelled at other superdeterministic approaches), rather, it is emergent superdeterminism; emergent from this number-theoretic consistent histories approach to complex Hilbert vectors.
Now if $\phi_1/\pi$ and $\phi_2/\pi$ are both finitely describable, then so is $(\phi_1 - \phi_2)/\pi$. By the theorem of Appendix \ref{number}, ignoring the four exceptions, $\cos (\phi_1-\phi_2)/2$ is not finitely describable. Hence, from the identity \be \label{noadd} \frac{1}{2}(e^{i \phi_1}+e^{i \phi_2})=e^{i \frac{\phi_1+\phi_2}{2}} \cos \frac{\phi_1-\phi_2}{2} \ee on the field of complex numbers $\mathbb C$, the multiplicative operators $e^{i \phi}$ in (\ref{exp}) cannot be additive. Fundamentally, it is this property that makes invariant set theory holistic. Without additivity, it is not permissible to break up a system into sub-units. Despite this, it may be computationally convenient to extend the set of angles $\phi/\pi$ from those that are finitely describable to all angles on the whole circle, allowing multiplication and addition of $e^{i\phi}$. Similarly, from a computational point of view we may wish to extend the finitely allowable $\cos \phi$ to the full set of reals on $-1 \le \cos \phi \le 1$. This completion gives us us the algebraically closed complex Hilbert Space of quantum theory. However, no ontological significance should be given to the extended set of states so formed: by construction, they are there solely for computational convenience. Because the invariant set operators $e^{i\phi}$ are not additive for any finite $N$, they only become algebraic at the singular \cite{Berry} limit $N=\infty$. In Appendix \ref{Dirac} we show how the relativistic form of the Schr\"{o}dinger equation, the Dirac equation, can be viewed as a dynamical evolution equation in invariant set theory, in the singular limit $N=\infty$.
The analysis provides a novel realistic interpretation of the EPR experiment (one which has nothing to do with Einstein-Rosen bridges \cite{MaldacenaSusskind}, therefore suggesting that ER$\ne$EPR). In the EPR experiment, if Alice measures the position of her particle, then the position of the second particle is determined. However, by the discussion above, it is not the case that that Alice, having actually measured position, might have measured momentum. Hence, it is not the case that the second particle must be prepared for the possibility that Alice might have measured momentum and therefore must have a well-defined momentum.
The invariant set concept is holistic. If a point in cosmological state space does not lie on $I_U$, then nothing about the universe associated with this point is real. It doesn't matter a jot that the two particles exist on opposite sides of the universe and do not interact. Any counterfactual perturbation that leads Alice to measure momentum, takes the whole universe, including the moons of Jupiter, off $I_U$. In this sense the moons of Jupiter do care what experiment is conducted here on Earth. However, nowhere do we need to invoke a breakdown of determinism or local causality to explain this: the information about whether $U$ is a state of physical reality is encoded in the holistic but locally causal invariant set.
As discussed, the number-theoretic incommensurateness of $\phi/\pi$ and $\cos \phi$ encodes the uncertainty principle. In this sense, this account of EPR does not reveal any inconsistency with the uncertainty principle - indeed invariant set theory provides a rational explanation for it. However, because the state space of quantum theory is an algebraically complete vector space (consistent with the singular limit at $N=\infty$), quantum theory itself cannot discriminate between physically real states on the invariant set, and physically unreal states off the invariant set; its square-integrable functional form is too coarse-grain to provide such discrimination. This is essentially why quantum theory can never be considered a realistic theory. In this sense, I agree with EPR that the wave function does not and cannot provide a complete description of physical reality.
These issues will be revisited in a more precise way when discussing the Bell Theorem in Section \ref{BellTheorem}.
\section{Probability \emph{vs} Frequency of Outcome} \label{measurement}
Above, we have interpreted the Hilbert vectors (\ref{momentum}) and (\ref{position}) as an uncertain selection from a sample space of neighbouring trajectories on $I_U$ in state space. However, in quantum theory the squared amplitudes of these Hilbert vectors also describe frequencies of occurrence of sequences of outcomes of similarly-prepared experiments in our unique space-time (i.e. relative to one particular history). The relationship between probability and frequency of occurrence can often be conceptually problematic in physical theory \cite{Wallace}; but not so with fractal geometry.
To understand this, I need to outline how self-similar structure manifests itself on $I_U$. Fig \ref{magglass}, starting at $t=t_0$ at the bottom of the Figure, shows what appears to be a single trajectory or history, i.e. an element of $C_i(2) \times \mathbb R$ for some iterate number $i$. However, looking through the magnifying glass the trajectory in fact comprises $2^N$ trajectories - shown (and reminsicent of DNA) with a compact helical structure (c.f. Appendix \ref{Dirac}). Between $t_1$ and $t_2$ these trajectories diverge chaotically into two state space clusters (described in more detail in Section \ref{Dynamics}) and the labelling above describes whether or not a trajectory evolves to cluster $a$ or cluster $\cancel a$ as a result of this chaotic evolution. The parameter $N$ describes the number of iterates of $C(2)$ that it takes to describe the clustering process between $t_1$ and $t_2$ and defines the strength of the chaotic dynamics that operates during the clustering procedure (see Sections \ref{Dynamics} and \ref{Gravity}). The physical reason for such instability can be related to the phenomenon of decoherence: on different nearby trajectories, the system interacts in different ways with its environment. As these differences grow, the trajectories diverge more from one another. In Section \ref{Dynamics}, I relate these clusters to measurement eigenstates and In Section \ref{Gravity}, I speculate that the clustering process is a manifestation of the phenomenon of gravity.
\begin{figure}
\caption{A schematic illustration of state-space trajectories or histories (each a cosmological space-time projected into 2D) on the invariant set $I_U$, here represented locally as $C(2) \times \mathbb{R}$. Periods of chaotic evolution (between $t_1$ and $t_2$ and between $t_3$ and $t_4$) correspond to what would conventionally be referred to as periods of decoherence as the system interacts nonlinearly with its environment. Here the state-space clusters $a$, $\cancel a$, $b$ and $\cancel b$, to which the trajectories on $I_U$ are attracted, correspond to eigenstates of the relevant observable in standard quantum theory but are associated with deterministic riddled basin dynamics in invariant set theory (see Section \ref{Dynamics}). A given trajectory, under $N$-iterate magnification, reveals a neighbourhood comprising $2^N$ trajectories, consistent with the self-similar structure of $C(2)$.}
\label{magglass}
\end{figure}
If we take one of the trajectories of $C_{i+N}(2) \times \mathbb R$ (e.g. between $t_2$ and $t_3$), by self similarity it can also be seen to comprise a set of $2^N$ trajectories (of $C_{i+2N}(2) \times \mathbb R$) which undergoes a further period of chaotic evolution, between $t_3$ and $t_4$. And so on and so on.
A point $X \in C(p)$ can be represented by the $p$-adic integer $...x_3x_2x_1.$ where $x_i \in \{0,1,2, \ldots p-1\}$. Here $x_i$ defines the segment of the $i$th iterate of $C(p)$ in which $X$ lies. $C(p)$ comes with a natural measure: the Haar Measure. With respect to this measure, the probability that $x_i$ equals any of the digits in $\{0,1,2, \ldots p-1\}$ is the same and equal to $1/p$. The following theorem relates probability to frequency of occurrence.
$\mathbf{Theorem}$ \cite{Ruban} Let $X$ be a typical element of a Cantor set $C(p)$, i.e. an element drawn randomly with respect to the Haar measure. Then with probability one, the frequency of occurrence of any of the digits $\{0,1,2, \ldots p-1\}$ in the expansion for $X$ is equal to $1/p$.
Hence, by Ruban's theorem with $p=2$, if the probability of a typical trajectory being attracted to the $a$ cluster is equal to $q/2^N$, then the frequency of occurrence of the $a$ cluster in a long sequence of similarly-prepared experiments (i.e. instability/cluster pairs) in any one trajectory is also $q/2^N$. The relationship between probability and frequency of occurrence is straightforward in a fractal setting.
\section{The Bell Theorem} \label{BellTheorem}
The discussion on EPR in Section \ref{interfere} is relevant to the Bell Theorem. As above, suppose Alice chooses between $x=0$ or $x=1$ and Bob $y=0$ or $y=1$. A pair of values $(x,y)$ imply a pair of spin measurements on pairs of entangled particles, where the measuring apparatuses have relative orientation $\theta_{xy}$. If $x$ and $y$ are considered as points on $\mathbb S^2$, then $\theta_{xy}$ is the angular distance between $x$ and $y$. In Appendix \ref{Hilbert}we extend the analysis in Section \ref{interfere} to show how to interpret the tensor-product Hilbert state \be
|\psi_{ab}\rangle= \gamma_0 |a\rangle |b \rangle + \gamma_1 e^{i \chi_1} |a\rangle |\cancel{b} \rangle + \gamma_2 e^{i \chi_2}|\cancel{a}\rangle |b \rangle + \gamma_3 e^{i \chi_3}|\cancel{a}\rangle |\cancel{b} \rangle, \ee where $\gamma_i, \chi_i \in \mathbb R$ and $\gamma_0^2+\gamma_1^2+\gamma_2^2+\gamma_3^2=1$, as an uncertain selection of a pair of elements $\{a_i, b_i\}$ from the bit strings \begin{align} S_a&=\{a_1 \; a_2 \ldots a_{2^N}\} \nonumber \\ S_b&=\{b_1 \; b_2 \ldots b_{2^N}\} \end{align} where $a_i \in \{a, \cancel a\}$, $b_i \in \{\cancel b\}$. This interpretation is only possible if $\gamma^2_i$ and $\chi_i/\pi$ are finitely describable. That is to say, the mapping from bit-string space to Hilbert Space is an injection. As discussed in Appendix \ref{Hilbert}, this implies $\cos \theta_{xy}$ must be finitely describable.
Suppose Alice and Bob choose some particular pair $(x, y)$, so that $\{\lambda, x, y\}$ describes a universe on $I_U$. Then, based on the finite describability of $\cos \theta_{xy}$ the following can be shown: \begin{itemize} \item The counterfactual universe $\{\lambda, 1-x, 1-y\}$ also lies on $I_U$ - this history is consistent with $\{\lambda, x, y\}$ as far as invariant set theory is concerned. That is to say, if $\cos \theta_{xy}$ is finitely describable, then so too is $\cos \theta_{(1-x)(1-y)}$. \item The counterfactual universes $\{\lambda, x, 1-y\}$ and $\{\lambda, 1-x, y\}$ (almost certainly) do not lie on $I_U$ - these histories are inconsistent with $\{\lambda, x, y\}$ as far as invariant set theory is concerned. That is to say, if $\cos \theta_{xy}$ is finitely describable, then $\cos \theta_{(1-x)y}$ and $\cos \theta_{x(1-y)}$ are not finitely describable. \item Let $z =x+y \mod 2$. Then, the set $\Lambda_{z=0}$ of supplementary variables $\lambda$ consistent with $z=0$ must be disjoint from the set $\Lambda_{z=1}$ of supplementary variables $\lambda$ consistent with $z=1$. \end{itemize}
The reasons for these conclusions combine geometry and number theory. Fig \ref{fig:CHSH} shows, separately for $\Lambda_{z=0}$ and $\Lambda_{z=1}$, the choices $x=0$, $x=1$, $y=0$ and $y=1$ represented as four points on the 2-sphere. The lines represent great circles and are solid if $\cos \theta_{xy}$ can be finitely described, and dashed otherwise. The left-hand figure corresponds to the sample $\Lambda_{z=}0$ and hence measurements where either $x=0$ and $y=0$, or $x=1$ and $y=1$. Here $\cos \theta_{00}$ and $\cos \theta_{11}$ are finitely describable, and $\cos \theta_{01}$ and $\cos \theta_{10}$ not. The reason why the latter are not finitely describable is discussed in Appendix \ref{CHSH} and makes use of the theory of Pythagorean triples - specifically that there are no Pythagorean Triples $\{a,b,c\}$ where $c$ is a power of 2.The right-hand figure corresponds to the sample $\Lambda_{z=1}$ and hence measurements where either $x=0$ and $y=1$, or $x=1$ and $y=0$. Here $\cos \theta_{01}$ and $\cos \theta_{10}$ are finitely describable, and $\cos \theta_{00}$ and $\cos \theta_{11}$ not. Of conceptual importance is the implication that the \emph{precise} position of the four points cannot be absolutely identical in the left and right-hand figures. However, to within \emph{any} finite experimental precision, the position of the four points can be considered identical. Like quantum interferometry, this violation of the Bell inequality relies on finite experimental precision. As discussed in Section \ref{interfere}, these results are robust to $p$-adic noise and hence structurally stable.
The CHSH inequality is \be \label{CHSH}
|C(0,0)+C(0,1)+C(1,0)-C(1,1)| \le 2 \ee where
\be C(x,y)=p(a=b|x,y)-p(a \ne b|x,y) \ee Experimentally, each correlation is determined by a separate sub-ensemble of particles: let's say the first correlation is determined on Monday, the second on Tuesday and so on. Then, based on the results above, in any experiment which seeks to test the CHSH inequality, the supplementary variables $\lambda$ must necessarily be drawn from the two disjoint samples: $\Lambda_{z=0}$ (from which Monday and Thursday's sub-ensembles are drawn) and $\Lambda_{z=1}$ (from which Tuesday and Wednesday's sub-ensembles are drawn).
For supplementary variables drawn from $\Lambda_{z=0}$ (left hand figure), the bit-string construction discussed in Appendix \ref{Hilbert} provides us with four bit strings, $S_0(x=0)$, $S_0(x=1)$, $S_0(y=0)$, $S_0(y=1)$, each of length $2^N$, such that \begin{itemize} \item Each of $S_0(x=0)$, $S_0(x=1)$, $S_0(y=0)$ and $S_0(y=1)$ has an equal number of 0s and 1s. \item The correlation between $S_0(x=0)$ and $S_0(y=0)$ is equal to $-\cos \theta_{00}$, by construction finitely describable. \item The correlation between $S_0(x=1)$ and $S_0(y=1)$ is equal to $-\cos \theta_{11}$, by construction finitely describable. \item Because $\cos \phi_{01}$ and $\cos \phi_{10}$ are not finitely describable when $\Lambda_{z=0}$, the correlations between $S_0(x=0)$ and $S_0(y=1)$, and between $S_0(x=1)$ and $S_0(y=0)$ do not correspond to anything physically realisable on $I_U$ - they do not correspond to correlations on entangled particle measurements in an experiment to test the CHSH inequality. \end{itemize}
Similarly, for supplementary variables drawn from $\Lambda_{z=1}$ (right-hand figure) we similarly have four bit strings, $S_1(x=0)$, $S_1(x=1)$, $S_1(y=0)$, $S_1(y=1)$, independent of the $S_0$ bit strings above, such that \begin{itemize} \item Each of $S_1(x=0)$, $S_1(x=1)$, $S_1(y=0)$, $S_1(y=1)$ has an equal number of 0s and 1s. \item The correlation between $S_1(x=0)$ and $S_1(y=1)$ is equal to $-\cos \theta_{01}$ by construction finitely describable. \item The correlation between $S_1(x=1)$ and $S_1(y=0)$ is equal to $-\cos \theta_{10}$, by construction finitely describable. \item Because $\cos \phi_{00}$ and $\cos \phi_{11}$ are not finitely describable when $\Lambda_{z=1}$, the correlations between $S_1(x=0)$ and $S_0(y=0)$, and between $S_1(x=1)$ and $S_0(y=1)$ do not correspond to anything physically realisable on $I_U$ - they do not correspond to correlations on entangled particle measurements in an experiment to test the CHSH inequality. \end{itemize}
\begin{figure}
\caption{Alice and Bob's choices $x=0$, $x=1$, $y=0$ and $y=1$ associated with measurement options for a CHSH experiment, shown schematically as four points on the 2-sphere. The lines between these points actually represent great circles where the angular distance between $x$ and $y$ is $\theta_{xy}$. Where the lines are solid (dashed), the corresponding cosines of the angular distances are (are not) finitely describable for number-theoretic reasons (see Appendix \ref{CHSH}). In a), corresponding to the sample space $\Lambda_{z=0}$ of supplementary variables, where $z=x+y \mod 2$, $\cos \theta_{00}$ and $\cos \theta_{11}$ are finitely describable. In b), corresponding to the disjoint sample space $\Lambda_{z=1}$ of supplementary variables, $\cos \theta_{01}$ and $\cos \theta_{10}$ are finitely describable. This means that the \emph{precise} positions of the $x$ and $y$ points in the left- and right-hand panels are not identical. However, to within the necessarily finite precision of the measurement orientations corresponding to the $x$ and $y$ button pushes, these points can be treated in any practical sense as if they are identical. In both figures, the cosines of the angular distances between $x=0$ and $x=1$, and between $y=0$ and $y=1$ are finitely describable because it is always possible to measure a particle pressing $x$ and then re-inject the particle into the measuring device pressing button $1-x$ (and similarly for $y$ and $1-y$). }
\label{fig:CHSH}
\end{figure}
Since $C(x,y)=-\cos \theta_{xy}$ on $I_U$, invariant set theory violates the Bell inequality as does quantum theory. Formally, it can do this because of superdeterminism: $\Lambda_{z=0}$ is disjoint from $\Lambda_{z=1}$. However, this is not \emph{ad hoc} superdeterminism, but superdeterminism emergent from our number-theoretic/fractal geometric approach to consistent histories. This raises an important point. The need to mention here (and in Section \ref{interfere}) the reality or otherwise of certain counterfactual worlds may seem puzzling to some. After all, modern accounts of Bell's theorem (e.g. \cite{Brunner}) make no mention at all about the notion of counterfactual reality - as such the notion may seem an irrelevance. Why, then, does counterfactuality seem to play such a crucial role in the invariant set model's evasion of Bell inequalities? The answer is that, as mentioned, in the invariant set model, superdeterminism does not arise by fiat, it is a consequence of deeper number-theoretic/fractal geometric principles. Presenting (\ref{mi}) as axiomatic (or, at least, not acknowledging that it might arise, or fail, from something deeper) is again prejudicial, because it immediately suggests that any failure of (\ref{mi}) is \emph{ad hoc}. Although modern accounts of Bell's theorem are very compact, I believe that this compactness is a hindrance to the emergence of physically plausible alternatives to quantum theory.
The transcendental property of the cosine function is essential if invariant set theory is to violate the CHSH inequality. For example, suppose the invariant set's correlations were given instead by some $F(\theta)$, where $F$ was a polynomial with finitely describable coefficients. Then if both $F(\theta_1)$ and $F(\theta_2)$ were finitely describable, so too would be $F(\theta_1+\theta_2)$. In this case, the argument for disjoint sample spaces $\Lambda_{z=0}$ and $\Lambda_{z=1}$ would fail. In turn, the model would have to be constrained by the CHSH inequality (i.e. be essentially classical) and would therefore be inconsistent with experiment. With such polynomial functions, it would also be possible to create sub-systems, and the essential holism of the model would also fail.
It is well known that it is possible to concoct `superquantum' theories where the CHSH inequality is violated more that does quantum theory \cite{Popescu,Tsirelson}. Could one concoct a type of invariant set theory in which the CHSH inequalities are maximally violated? The answer depends on whether there exist other transcendental functions $T$ (in addition to the cosine function) such that if $T(\theta_1)$ and $T(\theta_2)$ are finitely describable, then $T(\theta_1+\theta_2)$ is almost certainly not, and where $T(0)=1$, $T(\pi/2)=0$, $T(\pi)=-1$. The failure to find an alternative suitable transcendental correlation function would provide a new approach to understanding the Tsirelson bound.
\section{Random Bits and Free Will} \label{free}
As discussed, if $U$ is a universe on $I_U$ where $x=0$ and $y=0$, then the counterfactual universe $U'$ with the same supplementary variables $\lambda$, but where $x=1$ and $y=0$, does not lie on $I_U$. Let us suppose that $x$ is set by Bell's pseudo-random number generator, i.e. the value of the millionth bit in $U$ is a 0, and the value of the millionth bit in $U'$ is a 1. If $U$ lies on $I_U$ at the time $x$ is set, then it also lies on $I_U$ at the earlier time the million-digit variable is input to the pseudo random number generator. This is true, not because of any notion of retrocausality \cite{Price:1997}, because the notion of dynamical invariance is necessarily independent of time: if a state lies on the invariant set now, it \emph{always} has lain on it, and \emph{always} will lie on it. Similarly, if a state does not lie on $I_U$ now it \emph{never} has lain on it and \emph{never} will lie on it. In particular $U'$ did not lie on $I_U$ when the million-digit variable with its last counterfactually-flipped bit was input to the pseudo-random number generator.
The reason, then, why Bell's argument about physical randomisers may be incorrect is that in this invariant set theoretic approach, the difference between $U$ and $U'$ at the time the million-digit variable was input to the pseudo-random number generator is not (2-adically) small. Indeed, reducing the bit which sets $x$ from a millionth to a trillionth will not affect the 2-adic distance of $U'$ from $I_U$ one jot (even though it makes the Euclidean distance smaller). Bell \cite{Bellb} recognised that his arguments were not water tight, commenting: \begin{quote} Of course it might be that these reasonable ideas about physical randomisers are just wrong- for the purposes at hand. A theory may appear in which such conspiracies inevitably occur, and these conspiracies may then seem more digestible than the non-localities of other theories. When that theory is announced, I will not refuse to listen, either on methodological or other grounds. \end{quote} As discussed above, there are no conspiracies in invariant set theory: there is nothing secretive about the invariant set, even though many of its properties may be non-computable.
In contemporary attempts to test (\ref{mi}), $x$ and $y$ are often set by for-all-practical-purposes random bits rather than direct experimenter choices e.g. \cite{Shalm} extracted bits from the movie \emph{Back to the Future}. Exactly the same arguments as used above apply to these movie bits. Hence, if a certain bit of the movie determines $x$, then flipping the bit keeping the supplementary variables $\lambda$ fixed, takes the whole state of the universe off the invariant set to a non-ontological state of physical unreality. If $\{\lambda, x=0\}$ denotes a state of the universe on $I_U$, where the relevant movie bit from \emph{Back to the Future} is a 0, then $\{\lambda, x=1\}$ denotes the state $U'$ which does not lie on $I_U$. As before, consistent with the holism of the theory, none of the components of $U'$, such as the moons of Jupiter, have physical reality. Their reality (and that of all the clusters of galaxies in distant parts of the universe) can be obliterated at an instant by just counterfactually flipping that one movie bit - as discussed, $p$-adically this bit flip is not a small-amplitude perturbation. The fact that these moons and clusters are spacelike separated from the place where the counterfactual bit flip occurs is a complete irrelevance because the counterfactual bit flip violates a global state-space constraint (that states of reality lie on a global invariant set). This discussion can be thought of as an illustration of the Takens embedding theorem in dynamical systems theory \cite{Takens:1981}: the whole invariant set can be constructed from a sufficiently long time series of just one component of the invariant set, even if that component is spatially localised and energetically insignificant.
Suppose at the last minute, the experimenters decide that $x$ and $y$ will be determined by bits from \emph{Jaws} rather than \emph{Back to the Future}. Can we still say that counterfactually flipping the bit from \emph{Back to the Future} will cause the state of the universe to move off the invariant set? No we can't, because now the bit from \emph{Back to the Future} is no longer the determinant to a future experiment involving non-commutative operators. Hence, we can no longer invoke the deterministic invariance argument that because counterfactually flipping $x=0$ to $x=1$ takes the universe off the invariant set, counterfactually flipping the \emph{Back to the Future} bit must necessarily do the same.
As discussed, Bell used randomisers as a way to avoid discussing the contentious metaphysics of human free will. However, fundamentally, there are no new issues to discuss if we replace these randomisers with human brains. Human cognition can be just as susceptible to pseudo-random (but ultimately deterministic) processes inside the brain, as does Bell's pseudo-random number generator. These arise not least because of the extreme slenderness of human axons, making the protein transistors which amplify electric signals propagating along these axons, subject to thermal noise \cite{RollsDeco}. Indeed this susceptibility, born out of a need for the brain with its hundred-billion neurons to be outstandingly energy efficient (it operates with the power needed to light a domestic light bulb), could be what makes us creative \cite{PalmerOShea}. If the decision to measure $x=0$ rather than $x=1$ depends sensitively on the action of one of these protein transistors, then the consequences for lying on the invariant set in the counterfactual world where a particular ion failed to activate a particular protein transistor, are no different to the pseudo-random number bits and movie bits that have been discussed above. As has been argued by such eminent philosophers as Thomas Hobbes, David Hulme and John Stuart Mill \cite{Kane}, our sense of free will merely reflects an absence of constraints preventing us from doing what we want to do. From this point of view, neither determinism nor indeed superdeterminism is an impediment to free will. Positing superdeterminism as a constraint on free will is simplistic and unnecessarily restrictive on the class of superdeterministic theories.
In Appendix \ref{PBR}, I discuss the PBR Theorem \cite{Pusey} from this superdeterministic perspective (where violation of Measurement Independence is traded for violation of Preparation Independence).
I do not think it will ever be possible to test the Measurement Independence assumption directly in these Bell test experiments. We have to find other ways. These may involve experimental studies, or direct cosmological observations, where the effects of neither quantum nor gravitational physics are negligible. We briefly address such matters below.
\section{Fooled by Our Genes?} \label{Gene}
Let me address a question I have been asked by one eminent quantum foundations expert. Why would nature be so incredibly devious to lead us, a seemingly intelligent species capable of some astonishing achievements in both the arts and the sciences, to be so comprehensively fooled into thinking that the world around us is not locally real, when it actually is?
Actually I do not believe nature is the least bit devious. Rather, I believe we have been fooled by our genes. Consider our first experiences as human beings. A baby in the cot sees a colourful toy. For one or more reasons, she is genetically programmed to be attracted to such colourful objects, and so instinctively wants to explore the toy further. To do this, she has to get a part of her body (typically mouth and/or hands) in close proximity to the toy. In so doing, she implicitly learns a fundamental fact about the nature of physical space: closeness is synonymous with smallness of Euclidean distance. If this sense of spatial awareness is the first thing we learn as humans, it may be the hardest thing to let go of, when, in later life, we come to explore more abstract spaces, notably state spaces. Hence, when the philosopher David Lewis says that one world is closer to actuality than another if the first resembles the actual world more than the second does (c.f. Section \ref{metric}), I believe Lewis is inadvertently misapplying intuitive ideas learned in the cot.
Ironic then that a theoretically sound class of theory has been rejected, supposedly because of concerns about subversion from effects worse than alien mind control, but in fact because our minds have been comprehensively subverted by something much closer to home - gene control!
\section{Invariant Set Theory \emph{vs} Cellular Automaton Theory} \label{tHooft}
It was mentioned in Section \ref{Dislike} that there is a notable exception to the community's rejection of superdeterminism \cite{tHooft:2015b, tHooft:2015}. 't Hooft argues that it is possible to formulate a classical model of the real world which underpins quantum physics. He proposes a realistic deterministic cellular automaton dynamical system in place of the Schršdinger equation. This system is presumed to describe the evolution of the world with respect to in a unique `ontological basis'. The types of counterfactual worlds discussed above are deemed unrealistic simply because, by fiat, the corresponding basis is not ontological.
There is a fundamental theoretical difference between cellular automaton theory and invariant set theory. As a model of quantum physics, cellular automaton theory requires two separate elements: a deterministic (Schr\"{o}dinger-like) dynamical evolution equation, $D$, and a constraint on possible initial conditions $I$. In classical theories (which includes 'tHooft's theory), the dynamical laws of evolution and the initial conditions from which future states evolve, are largely independent of one another. For example, the classical Lorenz equations \cite{Lorenz:1963} can be integrated from any point in the Euclidean state space $\mathbb R^3$ of the model. In classical theory, we can constrain the initial state to some region of state space, e.g. $X \ge 0$, but such a constraint must be applied independently of, and in addition to, the dynamical equations themselves. To constrain initial conditions 't Hooft considers the special nature of the early universe \cite{tHooft:2015}: \begin{quote}
\ldots even in a superdeterministic world, contradictions with Bell's theorem would ensue if it would be legal to consider a change of one or a few bits in the beables describing Alice's world, without making any modifications in Bob's world. \ldots [However,] it is easy to observe that, certainly in the distant past, the effects of such a modification would be enormous and it may never be compatible with a simple low-entropy Big Bang \ldots Thus, we can demand in our theory that a modification of just a few beables in Alice's world without any changes in Bob's world is fundamentally illegal. This is how an ontological deterministic model can `conspire' to violate Bell's theorem.
\end{quote}
There are similarities with the arguments above. However, notwithstanding the fact that the reasons for the low-entropy of the early universe remain controversial \cite{Penrose:2010}, 'tHooft's model raises a conceptual problem: Why has nature chosen to constrain both $D$ to be Schr\"{o}dinger like, \emph{and} at the same time, only permitted a single initial state, the ontological initial state? Of course without the constraint on $I$, the Bell inequalities would not be violated, and without the constraint on $D$, we would observe basic quantum phenomena like black body radiation. Hence, both these constraints are necessary to account the world around us. But why both? There seems to be something unnecessarily complex and theoretically perplexing that we have to invoke two seemingly separate and independent constraints to arrive at a description of the observed world.
In invariant theory, $D$ and $I$ are not independent. They are both subservient to the underpinning fractal geometry of the invariant set. That is to say, consistent with invariant set theory not being a classical theory, there aren't two separate constraints, but only one: the geometry of $I_U$. For this reason, invariant set theory is conceptually a simpler theory than cellular automaton theory. Indeed, rather than relying on the low entropy of the early universe, as discussed in Section \ref{Gravity}, it is possible that the low entropy of the early universe may in fact be derivable from properties of the invariant set.
\section {Riddled Basins, The Schr\"{o}dinger Equation as a Singular Limit, and Why Stochasticity Fails.} \label{Dynamics}
The discussion so far has mostly been about the kinematics of the invariant set. How do we describe dynamical evolution? To frame the problem, let us use the canonical experimental situation where a system is prepared by a device with a knob for varying the state of the system produced and a release button for releasing the system, a transformation device for transforming the state (and a knob to vary the transformation), and a measuring apparatus for measuring the state (with a knob to vary what is measured) which outputs classical information. See Fig \ref{Hardy} (from \cite{Hardy:2004}).
\begin{figure}
\caption{A schematic of a typical situation where a system is prepared in some initial state and subject to a transformation and measurement. From \cite{Hardy:2004}}
\label{Hardy}
\end{figure}
In invariant set theory, this experimental situation is described graphically by one of the trajectories in Fig \ref{magglass} which, let's say, belongs to the $a$ cluster at time $t_2$ (i.e. quantum mechanically, is prepared in the $|a\rangle$ state at $t_2$) and belongs to either the $b$ cluster or the $\cancel b$ cluster at time $t_3$ (i.e. quantum mechanically, reduces to the $|b\rangle$ or $| \cancel b \rangle$ state at $t_3$). This description is consistent with the notion that in invariant set theory the most primitive expression of the laws of physics is a description of the geometry of the invariant set $I_U$ in state space.
In invariant set theory, dynamical evolution associated with divergence and clustering are associated with deterministic processes. It is known that unpredictable evolution towards multiple attractors can be described deterministically by so-called riddled-basin dynamics \cite{Alexander:1992}. This arises when a chaotic oscillator is coupled to a model with multiple potential wells. With such coupling, the basin boundaries can become fractally intertwined and it is generally unpredictable whether the system lies in one basin of attraction or another other. Chaotic evolution can be based on nonlinear $p$-adic mappings, much like the logistic maps on $\mathbb R$ \cite{WoodcockSmart}. Here is enough to utilise a simple Bernoulli or binary shift map - a type of logistic map - on the 2-adic bits representing a state on $I_U$. This shift map would generate the divergence of trajectories shown in Fig \ref{magglass}, and the grouping into the two clusters $b$ and $\cancel b$ is then consistent with two riddled basins of attraction on the invariant set. The parameter $N$ describes the number of iterations of the binary shift needed to lead to evolution to a cluster. With $N$ large, the fate of an individual trajectory at $t_2$ is unpredictable. This evolution is reminiscent of the notion of `objective reduction' in quantum physics. However, such a description would be misleading because invariant set theory does not assume any fundamental ontological significance to the superposed Hilbert state, and hence, as such, no `reduction' is actually taking place. Rather, the riddled-basin procedure combines a quasi-linear instability process, which separates nearby trajectories on $I_U$ due to way in which neighbouring histories interact with their environment (i.e. decohere) and with a nonlinear clustering process. In Section \ref{Gravity} I speculate that the clustering (or 'clumping') of trajectories in state space is a manifestation of the phenomenon we call gravity.
In the literature, objective representations of the measurement procedure have been described by stochastic mathematics \cite{Pearle:1976, Ghirardi, Percival:1995}. However, in this framework, conventional stochastic dynamics would completely destroy the properties of the invariant set which allow it to violate the Bell Theorem in a locally causal context. This may at first sight seem surprising given that stochasticity makes no difference to the interpretation of the Bell inequalities for conventional hidden-variable models, if the dynamics of these models is deterministic or stochastic \cite{Bell}. However, here, if the measure-zero nature of the invariant set is stochastically smeared out onto the full measure of the Euclidean space in which $I_U$ is embedded, then the model will no longer have the property that a small (in the Euclidean sense) perturbation can take a point on the invariant set off it.
The preparation procedure can be viewed in a similar way. As a result of the preparation procedure, the system, at the time of preparation, lies in one of the state-space clusters associated with preparation ($a$ and $\cancel a$). When the system is released, it emerges from the preparation procedure with (say) the property `I have originated from cluster $a$'.
In quantum theory, the transformation from $|a\rangle$ to either $|b\rangle$ or $|\cancel b \rangle$ is given by the Schr\"{o}dinger equation. In Appendix \ref{Dirac} we show how the simplest form of the Dirac equation arises in invariant set theory in the singular limit \cite{Berry} $N= \infty$. For finite $N$, time evolution of the bit strings $S$ are effected by the cyclical permutation operator $\zeta$ described in (\ref{zeta}). Because $\zeta^{2^N} S_b= S_b$, this evolution is inherently oscillatory in time. A key physical relationship associated with the Schr\"{o}dinger equation is between the frequency of this oscillation and the mass and hence energy of the associated particle: $E=\hbar \omega$. With energy having an expression in terms of space-time curvature through the field equations of general relativity, this can be seen as an example of a fundamental relationship \be \label{generalisedequation} \parbox{2.5in}{Expression for a property of the locally Euclidean space-time geometry $\mathcal M_U$} \ \ = \ \ \ \parbox{2.5in}{Expression for a property of the locally $p$-adic state-space geometry $I_U$ } \ee with Planck's constant providing the basic constant of nature that translates properties of state-space geometry into properties of space-time geometry. We discuss further possible implications of this relationship in Section \ref{Gravity}.
In quantum theory, the Schr\"{o}dinger equation provides the dynamical underpinning for the transformation phase of Fig \ref{Hardy}. The equation has of course proven exceptionally accurate. Hence, rather than add extra terms to this equation, invariant set theory provides restrictions on the equation by not allowing the full Hilbert Space of states but only an algebraically open subset (where complex phases are finitely describable multiples of $\pi$ and where space-time orientations have finitely describable cosines). This restriction does not destroy the Schr\"{o}dinger equation's key property of describing conservation of probability (just as the classical Liouville equation still describes conservation of probability when state-space trajectories are restricted to a fractal attractor). The precise linearity of the Schr\"{o}dinger equation reflects this conservation of probability, and this linearity is not in any way perturbed by the presence of the highly nonlinear riddled-basin dynamics discussed above (just as the linearity of the Liouville equation in classical physics is not challenged by underpinning nonlinear dynamical evolution).
This suggests a new interpretation of the Schr\"{o}dinger equation: as a computationally powerful tool, but one whose functional analytic form is too coarse grained to be able to distinguish ontological and non-ontological states, and therefore too coarse grained to describe quantum ontology. Partial analogies arise in classical physics where it is computationally convenient to represent properties of some fundamentally discrete system, e.g. molecules of a fluid, using variables drawn from the reals, e.g. the fluid variables of the Navier-Stokes equation. However, there is a crucial difference between this classical situation and the one proposed here. In classical physics we use the Euclidean metric in both space time and state space. Hence the continuum approximation is arbitrarily good if the mean free path of molecules between collisions is small enough. However, in invariant set theory, with its use of the $p$-adic metric as a yardstick of distance in state space, this is no longer true. In the latter framework at the ontological level, it does matter whether we are dealing with the original discrete variables or the more computationally convenient continuum variables, no matter how large the (finite) parameter $N$. In particular, a realistic theory based on continuum variables must necessarily be nonlocal in order to violate the Bell inequalities.
In this sense, one can interpret the quantum potential of Bohmian theory as a coarse-grained $L^2$ representation of the invariant set in configuration space. Again, with such a realistic representation, the Bohmian quantum potential does not have the fine-grained ontological properties of the invariant set and therefore Bohmian theory has to be explicitly nonlocal.
\section{Quantum Gravity and the Dark Universe - The End of Particle Physics?} \label{Gravity}
The primary motivation for developing the model above was based on a growing belief that the fundamental impediment to synthesising quantum and gravitational physics is in fact quantum theory \cite{Penrose:1976}. If a causal geometric model of quantum physics can be developed, it may, superdeterministic or not, stand a much greater chance of being synthesised with Einstein's causal geometric model of gravity, than does quantum theory. Here I wish to suggest some possible implications of invariant set theory for such a synthesis. The remarks below are, of course, very speculative.
As discussed, the geometric basis of the proposed superdeterministic model is of a fractal invariant set in the state space of the universe. Such invariant sets (e.g. the Lorenz attractor), comprise just one indefinitely long trajectory wrapped up in a compact set. By comparison then, $I_U$ comprises the trajectory of a mono-universe (i.e. not a multiverse) evolving over multiple epochs on a similar compact set. From this perspective, the sample space of neighbouring trajectories over which probabilities are defined, merely define instances of our universe at earlier or later epochs. There are no `Many Worlds' as in the Everett interpretation - just one. Whether the universe is open or closed is currently an unresolved question; it appears to be on the borderline. The theory proposed here suggests an underlying finite and hence closed universe.
Continuing the notion of finiteness, although the discussion so far has assumed that $I_U$ is precisely fractal, there is nothing in the theory that would prevent $I_U$ from being a sufficiently complex finite limit cycle (that approximates well a fractal). That is to say, it could be that the self-similarity discussed above, only persists to some large but finite number of iterations.
By treating the geometry of $I_U$ as a primitive expression of the laws of physics, then, as suggested by (\ref{generalisedequation}), we can expect that the pseudo Riemannian geometry of our space time is influenced by the fractal geometry of $I_U$. That is to say, the geometry of space time should be partially influenced by the geometry of neighbouring state-space space-times (i.e. our universe at later or earlier epochs). Could this provide an explanation of dark matter in our space time? That is to say, could it be that what we call dark matter in our space-time, is merely ordinary matter on neighbouring space-time trajectories on $I_U$, whose influence on our space time arises from the particular geometric form of $I_U$?
Indeed, following (\ref{generalisedequation}), perhaps the curvature of space-time is intimately linked to the clustering of trajectories on $I_U$ as shown in Fig \ref{magglass}. This would suggest, consistent with earlier speculations by Penrose \cite{Penrose:1989} and Di\'{o}si \cite{Diosi:1989}), that gravity should itself be an intrinsically decoherent process. An experimental confirmation of this would provide a strong indication that the conventional quantum field theory approach to quantum gravity is misguided. Now this clustering is certainly a nonlinear process, and therefore would not occur if differences in space-time energy were sufficiently small. This idea can be made quantitative. If $E_G$ denote the gravitational self energy of the difference between mass/energy distributions in two space-times \cite{Penrose:2004}, then these space-times can be considered gravitationally indistinct (over a time interval $\tau$) if \be \int_{\tau} E_G dt < \hbar \ee Two space-times that are sufficiently similar would then be gravitationally indistinct. One can speculate that two space-times which differ merely by vacuum fluctuations would be gravitationally indistinct in this sense. This would imply that vacuum fluctuations do not couple to gravity and this can help explain why the cosmological constant is not 120 orders of magnitude bigger than it is. The value that the cosmological constant does take may be the cosmological consequence of (\ref{generalisedequation}), together with the generic and ubiquitous divergence of trajectories on $I_U$ (shown in Fig \ref{magglass}). This provides a new proposal for the origin of dark energy in $\mathcal M_U$.
If the ideas above are correct, there will be no quantum field theoretic excitation (i.e. `particle') associated with either dark matter or dark energy. Indeed there will additionally be no such particle as a graviton. Invariant set theory, if correct, implies a limit to the ability of particle physics to explain everything we see in the world around us and indeed may signal the end of particle physics beyond the Standard Model. However, the Standard Model has been extraordinarily successful in explaining non-gravitational physics. What role does that model have in invariant set theory? I would argue that the gauge groups of the Standard Model define the state space in which $I_U$ is embedded. From this perspective it would be conceptually wrong to imagine gravity as some extension of the Standard Model's gauge-group structure, e.g. as some kind of Yang-Mills theory. Rather we should think of gravity as the geometric phenomenon of clustering or clumping of histories on the non-trivial fractal measure-zero geometry in the space spanned by the degrees of freedom of the Standard Model's gauge group.
One of the reasons for developing a so-called quantum theory of gravity is that it will eliminate the occurrence of space-time singularities. Equation (\ref{generalisedequation}), whereby neighbouring state-space trajectories influence our space-time, could also eliminate singularities by smearing out what would otherwise be a delta function in curvature in our space time (corresponding to the singular limit at $N=\infty$), into a finite gaussian-like function (at finite but large $N$) with support on some neighbourhood of histories in state space.
The notion of a fractal invariant set has many interesting implications for the perplexing question of time asymmetry and the Second Law of Thermodynamics. The growth of entropy from the early phase of a cosmological epoch would, as in standard chaotic dynamics, be associated with the generic divergence of trajectories shown in Fig \ref{magglass}. What causes the low entropy at the beginning of a cosmological epoch? In invariant set theory, it must be associated with state-space convergence of trajectories in the late phase of the previous epoch (and hence nothing to do with inflation). In conventional nonlinear dynamics, such convergence is typically associated with dissipation. However, here we would need to find something more fundamental. Penrose \cite{Penrose:2010} has speculated that such state-space convergence would generically be associated with the formation of black holes - more generally of collapse towards a final-time singularity (a quasi-singularity from the paragraph above). It is this process of state-space convergence that gives rise to the lacunae which allow invariant set theory to violate Bell inequalities in the laboratory today.
These ideas in turn have implications and new perspectives on the problem of black-hole information loss: in invariant set theory information is not lost, but is compressed to the extent that the information, like vacuum fluctuations, no longer has a distinct gravitational signature. The information reacquires its gravitational signature during the divergence phase when entropy is again increasing.
As mentioned, these remarks are speculative. However, they illustrate the fact that the development of a superdeterminstic theory of quantum physics may open up some very new ways of thinking about some of the deepest problems of contemporary fundamental physics.
\section{Conclusions}
It is time for superdeterminism and the corresponding Measurement Independence assumption to be reappraised. Visceral arguments should be replaced with more logical and reasoned arguments. A key part of this reappraisal is the role the $p$-adic metric plays as the yardstick of choice in state space. The implications are potentially enormous, not only for our understanding of quantum physics, but even more for resolving many of the conceptual difficulties that seem to be creating a crisis of understanding in contemporary foundational physics.
Following the Shalm et al experiments, media headlines proclaimed that the Einstein/Bohr debate has finally been settled in favour of Bohr. The theory described in this paper provides a meeting ground for both Einstein and Bohr's views and suggests that the the result of the debate may in fact be a dead heat.
\appendix \section{$p$-adic Integers and Cantor Sets} \label{padic}
By way of introduction to the $p$-adic numbers, consider the sequence \be \{1, 1.4, 1.41., 1.414, 1.4142, 1.41421 \ldots\} \nonumber
\ee where each number is an increasingly accurate rational approximation to $\sqrt 2$. As is well known, this is a Cauchy sequence relative to the Euclidean metric $d(a,b)=|a-b|$, $a$, $b \in \mathbb{Q}$.
Surprisingly perhaps, the sequence \be \{1, 1+2, 1+2+2^2, 1+2+2^2+2^3, 1+2+2^2+2^3+2^4, \ldots\}
\ee is also a Cauchy sequence, but with respect to the ($p=2$) $p$-adic metric $d_p(a,b)=|a-b|_p$ where \be
|x|_p=\left \{ \begin{array} {ll} p^{-\textrm{ord}_p x} &\textrm{if } x \ne 0 \\ 0 &\textrm{if } x=0 \end{array} \right. \ee and \be \textrm{ord}_p x= \left \{ \begin{array}{ll} \textrm{the highest power of \emph p which divides \emph x, if } x \in \mathbb Z \\ \textrm{ord}_p a - \textrm{ord}_p b, \textrm{ if } x=a/b, \ \ a,b \in \mathbb Z, \ b \ne 0 \end{array} \right. \ee Hence, for example \be d_2(1+2+2^2, 1+2)=1/4,\ \ d_2(1+2+2^2+2^3, 1+2+2^2)=1/8 \ee Just at $\mathbb{R}$ represents the completion of $\mathbb{Q}$ with respect to the Euclidean metric, so the $p$-adic numbers $\mathbb{Q}_p$ represent the completion of $\mathbb{Q}$ with respect to the $p$-adic metric. A general $p$-adic number can be written in the form \be \sum_{k=-m}^{\infty} a_k p^k \ee where $a_{-m} \ne 0$ and $a_k \in \{0,1,2, \ldots, p-1\}$. The so-called $p$-adic integers $\mathbb Z_p$ are those $p$-adic numbers where $m=0$.
It is hard to sense any physical significance to $\mathbb Z_p$ and the $p$-adic metric from the definition above. However, they acquire relevance in invariant set theory by virtue of their association with fractal geometry. In particular, the map $F_2: \mathbb Z_2 \rightarrow C(2)$ \be F_2: \sum_{k=0}^{\infty} a_k2^k \mapsto \sum_{k=0}^{\infty} \frac{2a_k}{3^{k+1}} \textrm{ where } a_k \in \{0,1\} \ee is a homeomorphism \cite{Robert}, implying that every point of the Cantor ternary set can be represented by a 2-adic integer. More generally, \be F_p: \sum_{k=0}^{\infty} a_kp^k \mapsto \sum_{k=0}^{\infty} \frac{2a_k}{(2p-1)^{k+1}} \textrm{ where } a_k \in \{0,1, \ldots p-1\} \ee is a homeomorphism between $\mathbb Z_p$ and $C(p)$ To understand the significance of the $p$-adic metric, consider two points $a, b \in C(p)$. Because $F_p$ is a homeomorphism, then as $d(a,b) \rightarrow 0$, so too does $d_p(\bar a, \bar b)$ where $F(\bar a)=a, \ F(\bar b)=b$. On the other hand, suppose $a \in C(p)$, $b \notin C(p)$. By definition, if $b \notin C(p)$, then $\bar b \notin \mathbb Z_p$. Let us assume that $b \in \mathbb Q$ (see Footnote 3 above). Then $\bar b \in \mathbb Q_p$. This implies that $d_p(\bar a,\bar b) \ge p$. Hence, $d(a,b) \ll 1 \centernot \implies d_p(\bar a, \bar b) \ll 1$. In particular, it is possible that $d_p(\bar a, \bar b) \gg 0$, even if $d(a,b) \ll 0$. From a physical point of view, a perturbation which seems insignificantly small with respect to the (intuitively appealing) Euclidean metric, may be unrealistically large with respect to the $p$-adic metric, if the perturbation takes a point on $C(p)$ and perturbs it off $C(p)$. The $p$-adic metric recognises the primal ontological property of lying on the invariant set. The Euclidean metric, by contrast, does not.
Let $g(x, x')$ denote the pseudo-Riemannian metric on space-time, where $x, x' \in \mathcal M_U$. By contrast let $g_p(\mathcal M_U, \mathcal M_U')$ denote a corresponding metric in $U$'s state space, transverse to the state-space trajectories. As above, we suppose that if $\mathcal M_U \in I_U$, then $g_p(\mathcal M_U, \mathcal M_U') \rightarrow 0$ only in the $p$-adic sense, ie. only if $\mathcal M_U' \in I_U$.
\section{A Key Number-Theoretic Property of the Cosine Function} \label{number} A key number theorem for this paper is this: \\ \\ $\mathbf{Theorem}$\cite{Niven, Jahnel:2005}. Let $\phi/\pi \in \mathbb{Q}$. Then $\cos \phi \notin \mathbb{Q}$ except when $\cos \phi =0, \pm 1/2, \pm 1$. \\ \\ $\mathbf{Proof}$. Assume that $2\cos \phi = a/b$ where $a, b \in \mathbb{Z}, b \ne 0$ have no common factors. Since \be \label{cosidentity} 2\cos 2\phi = (2 \cos \phi)^2-2 \ee then \be 2\cos 2\phi = \frac{a^2-2b^2}{b^2}
\ee Now $a^2-2b^2$ and $b^2$ have no common factors, since if $p$ were a prime number dividing both, then $p|b^2 \implies p|b$ and $p|(a^2-2b^2) \implies p|a$, a contradiction. Hence if $b \ne \pm1$, then the denominators in $2 \cos \phi, 2 \cos 2\phi, 2 \cos 4\phi, 2 \cos 8\phi \dots$ get bigger without limit. On the other hand, if $\phi/\pi=m/n$ where $m, n \in \mathbb{Z}$ have no common factors, then the sequence $(2\cos 2^k \phi)_{k \in \mathbb{N}}$ admits at most $n$ values. Hence we have a contradiction. Hence $b=\pm 1$ and $\cos \phi =0, \pm1/2, \pm1$. \\ \\ If, moreover, $\phi/\pi$ is describable by a finite number of bits, then $\cos \phi =0, \pm 1$
\section{A Realistic Interpretation of Complex Hilbert Vectors} \label{Hilbert} \subsection{One Qubit} Extending the discussion in Section \ref{interfere}, we view the Hilbert vector \be \label{1qubit1}
|\psi_{a}(\theta, \phi)\rangle=
\cos \frac{\theta}{2}|a\rangle
+e^{i\phi} \sin \frac{\theta}{2} |\cancel{a}\rangle \ee as an uncertain element $a_i \in \{a, \cancel a\}$ selected from the bit string $S_a(\theta, \phi)=\{a_1, a_2, \ldots a_{M}\}$. By this is meant that there is a deterministic procedure which selects an element of $S_a(\theta, \phi)$ but this procedure is sufficiently unpredictable as to be unknowable (Section \ref{Dynamics}). If the qubit is associated with a pure state, then $M=2^N$ where $N$ is related to the number of fractal iterates needed to evolve to a state-space cluster. The probability of selecting any particular element $a_i$ is independent of $i$. As discussed in the main body of the text, the additive form of (\ref{1qubit1}) results from Pythagoras' theorem.
Let us start with $\phi=0$. Then \be S_a(\theta, 0)=\{\underbrace{a\;a\;a\ldots a}_{M\cos^2\theta/2}\ \underbrace{\cancel a\;\cancel a\;\cancel a\ldots \cancel a}_{M\sin^2\theta/2}\} \ee where $\cos^2 \theta/2$ is a rational number of the form $M'/M$ for integer $0 \le M'\le M$. For a pure state, then in addition to being rational, $\cos^2 \theta/2$ and hence $\cos \theta$ must be finitely describable.
To define $S_a(\theta,\phi)$ where $\phi \ne 0$, let \be \label{zetan} \zeta \{a_1\; a_2\; \ldots a_{M}\}=\{a_M \;a_1 \; a_2\; \ldots a_{M-1}\}. \ee Since $\zeta^M \{a_1\; a_2\; \ldots a_{M}\}= \{a_1\; a_2\; \ldots a_{M}\}$ we can treat $\zeta$ as an operator representation of an $M$th root of unity and write $\zeta=e^{2\pi i /M}$. Using this (\ref{1qubit1}) is defined as an uncertain element of \be S(\theta, \phi)=\zeta^{M''}S_a(\theta, 0) \ee for integer $M''$, where $\phi=2\pi M''/M$. When $M=2^N$ then $\phi/\pi$ must be finitely describable. Since the selection procedure is assumed chaotic, the uncertain element selected will be sensitive to the ordering of bit-string elements. Hence, for example, the chaotic procedure will not in general select the same element when applied to the bit string $S_a(\theta, 0)$ as to $\zeta S_a(\theta, 0)$. The degree of unpredictability of the selection procedure is at a maximum for $\theta=\pi/2$ when $S_a$ contains equal numbers of $a\;$s and $\cancel a\;$s, and at a minimum when $\theta=0$ or $\pi$ (where $\zeta$ has no impact on the bit string).
As discussed in the main body of the text, for computational convenience, the values at which $\cos^2 \theta/2$ is defined can be extended into the reals, and the $M$th roots of unity can be considered embedded into the complexes. Here it is claimed that such extensions result in the legendary conceptual problems of quantum theory.
\subsection{Two Qubits}
In quantum theory, the general tensor-product form for a 2-qubit Hilbert state is given by \be \label{2qubit3}
|\psi_{ab}\rangle= \gamma_0 |a\rangle |b \rangle + \gamma_1 e^{i \chi_1} |a\rangle |\cancel{b} \rangle + \gamma_2 e^{i \chi_2}|\cancel{a}\rangle |b \rangle + \gamma_3 e^{i \chi_3}|\cancel{a}\rangle |\cancel{b} \rangle, \ee where $\gamma_i, \chi_i \in \mathbb R$ and $\gamma_0^2+\gamma_1^2+\gamma_2^2+\gamma_3^2=1$. Equation (\ref{2qubit3}) can be written in two equivalent forms. The first is \be \label{2qubit1}
|\psi_{ab}\rangle=
\cos \frac{\theta_1}{2}|a\rangle
|\psi_b(\theta_2, \phi_2)\rangle
+e^{i\phi_1} \sin \frac{\theta_1}{2} |\cancel{a}\rangle |\psi_b(\theta_3,\phi_3)\rangle \ee where \begin{align} \label{gamma} \gamma_0&=\cos\frac{\theta_1}{2}\cos\frac{\theta_2}{2} \ \ \ \ \gamma_1=\cos\frac{\theta_1}{2} \sin\frac{\theta_2}{2} \ \ \ \ \gamma_2=\sin\frac{\theta_1}{2}\cos\frac{\theta_3}{2} \ \ \ \ \gamma_3=\sin\frac{\theta_1}{2}\sin\frac{\theta_3}{2} \nonumber \\ \chi_1&=\phi_2\ \ \ \ \chi_2=\phi_1 \ \ \ \ \ \chi_3=\phi_1+\phi_3 \end{align} The second is \be \label{2qubit2}
|\psi_{ab}\rangle=
\cos \frac{\theta_6}{2} |\psi_a(\theta_4, \phi_4)\rangle |b\rangle+ e^{i \phi_6} \sin \frac {\theta_6}{2}
|\psi_a(\theta_5, \phi_5)\rangle |\cancel b\rangle \ee where \begin{align} \label{firsttosecond} \cos \frac{\theta_1}{2} \cos \frac{\theta_2}{2} &= \cos \frac{\theta_4}{2} \cos \frac{\theta_6}{2} \nonumber \\ \sin \frac{\theta_1}{2} \cos \frac{\theta_3}{2} &= \sin \frac{\theta_4}{2} \cos \frac{\theta_6}{2} \nonumber \\ \cos \frac{\theta_1}{2} \sin \frac{\theta_2}{2} &= \cos \frac{\theta_5}{2} \sin \frac{\theta_6}{2} \nonumber \\ \sin \frac{\theta_1}{2} \sin \frac{\theta_1}{2} &= \sin \frac{\theta_5}{2} \cos \frac{\theta_6}{2} \nonumber \\ \phi_1&=\phi_4 \nonumber \\ \phi_2&=\phi_6 \nonumber \\ \phi_1+\phi_3&=\phi_5+\phi_6 \nonumber \\ \end{align}
Let us start by assuming all $\phi_i=0$. Then in invariant set theory, a pure Hilbert state of the form (\ref{2qubit1}) is considered an uncertain selection of some particular pair of elements $\{a_i, b_i\}$ from the two bit strings \begin{align} \label{twobs} S_a&=\{a_1 \; a_2 \ldots a_{2^N}\} \nonumber \\ S_b&=\{b_1 \; b_2 \ldots b_{2^N}\} \end{align} where $a_i \in \{a, \cancel a\}$, $b_i \in \{b, \cancel b\}$, where all $\cos^2 \theta_i/2$ are finitely describable, and where, as above, the probability of selecting $\{a_i, b_i\}$ is independent of $i$. Using the form of (\ref{2qubit1}) and Fig \ref{tensor} as guidance, these bit strings are defined as follows. With reference to the first line in Fig \ref{tensor}a, the first $2^N \cos^2 \theta_1/2$ elements of $S_a$ are $a\;$s, and the rest $\cancel a\;$s. With reference to the second line in Fig \ref{tensor}a, the first $2^N \cos^2 \theta_1/2 \; \cos^2 \theta_2/2$ elements of $S_b$ are $b\;$s; the next $2^N \cos^2 \theta_1/2 \; \sin^2 \theta_2/2$ elements are $\cancel b\;$s; the next $2^N\sin^2 \theta_1/2 \; \cos^2 \theta_3/2$ elements are $b\;$s and the final $2^N\sin^2 \theta_1/2 \; \sin^2 \theta_3/2$ elements are $\cancel b\;$s. Based on this we can assert that the uncertain selection $\{a_i, b_i\}$ corresponds to the Hilbert vector (\ref{2qubit1}) with $\phi_1=\phi_2=\phi_3=0$.
\begin{figure}
\caption{A schematic illustration of the injective correspondence between (\ref{twobs}) and (\ref{2qubit3}) where phase angles are set to zero. In both a) and b) the top line refers to $S_a$, the bottom line to $S_b$. Solid lines illustrate sub-strings of bits which are either $a\;$s or $b\;$s and ordered differently in a) and b). Dashed lines refer to sub-strings of elements which are either $\cancel a\;$s or $\cancel b\;$s. Hence, for example, panel b) indicates that the first $2^N\; \cos^2 \theta_4/2\; \cos^2\theta$ elements of $S_a$ and $S_b$ are $a\;$s and $b\;s$ respectively. Despite different ordering, panels a) and b) are equivalent in terms of correlations between $a$ and $b$ (and other pairs of) elements because of the relations (\ref{firsttosecond}). Panel a) shows the correspondence between (\ref{twobs}) and (\ref{2qubit1}). Panel b) shows the correspondence between (\ref{twobs}) and (\ref{2qubit2}). (\ref{2qubit1}) and (\ref{2qubit2}) are equivalent forms of the tensor product Hilbert state (\ref{2qubit3}). The role of phase angles is described in the text.}
\label{tensor}
\end{figure}
Now if we use the relationships (\ref{firsttosecond}) it is easily seen that the probability of selecting $a$ from $S_a$ and $b$ from $S_b$ in Fig \ref{tensor}a is the same as in Fig \ref{tensor}b (as are the probabilities for any other pairs of elements). These probabilities are the same as implied by the Born rule on (\ref{2qubit3}) with $\chi_i=0$. For example, the probability that $a_i=a$ and $b_i=\cancel b$ is equal to $\cos^2 \theta_1/2\; \sin^2 \theta_2=\cos^2 \theta_5/2\; \sin^2 \theta_6$ which is equal to $\gamma^2_1$ from (\ref{gamma}) and (\ref{firsttosecond}).
The three phase degrees of freedom are introduced through the cyclical permutation operators $\zeta$. Note that we can cyclically permute the two strings $\{a_i\}$ and $\{b_i\}$ together without affecting the correlations between $S_a$ and $S_b$. Similarly, from Fig \ref{tensor}a, one can cyclically permute the first $\cos^2 \theta_1/2$ elements of $S_b$, or the final $\sin^2 \theta_1/2$ elements of $S_b$, without affecting the correlations between $S_a$ and $S_b$. Similarly, from Fig \ref{tensor}b, one can cyclically permute the first $\cos^2 \theta_6/2$ elements of $S_a$, or the final $\sin^2 \theta_6/2$ elements of $S_a$, without affecting the correlations between $S_a$ and $S_b$. As before, it is important to note that because of the linkage to cyclical permutations, the corresponding phase angles must be rational multiples of $\pi$.
Now consider the special case where $\theta_1=\pi/2$, $\theta_2=\pi-\theta_3$ (or equivalently, $\theta_6=\pi/2$, $\theta_5=\pi-\theta_4$). Then the correlation between $S_a$ and $S_b$ is equal to $-\cos \theta_2=-\cos \theta_4$ and consistent with quantum theoretic correlations of measurement outcomes on the Bell state \be
\frac{|a\rangle|b\rangle+ | \cancel{a} \rangle | \cancel{b} \rangle}{\sqrt 2} \ee where $\theta_2=\theta_4$ denotes the relative orientation of Alice and Bob's measurement apparatuses, with finitely describable cosine. The finite describability of the cosine of the relative orientation of Alice and Bob's measuring apparatus is fundamental in invariant set theory's account of the Bell Theorem.
It is a simple matter to use the principle of induction to extend the construction described in this Section to link general $m$-qubit tensor-product Hilbert states (with rational squared amplitudes) to families of $m$ (generically partially correlated) bit strings.
\section{CHSH} \label{CHSH}
Let $x=0$, $x=1$, $y=0$, $y=1$ represent four distinct, randomly chosen points on $\mathbb S_2$ (corresponding to four distinct randomly chosen directions in physical space). Join together each pair of points by a great circle. Let $\theta_{xy}$ denote the angular distance between any $x$ point and any $y$ point.
Let $\alpha$ denote the relative angle between the two possible measuring directions $x=0$ and $x=1$ of Alice's measuring apparatus, and let $\beta$ denote the corresponding angle between $y=0$ and $y=1$ of Bob's apparatus. Suppose Alice chooses $x=0$. It is always possible for Alice to send a particle which she has just measured along direction $x=0$, back into the instrument to be measured in the $x=1$ direction. This corresponds to a simple (single-qubit) measurement where the input state has been prepared along $x=0$ and the measurement taken along $x=1$ (or \emph{vice versa} if Alice instead chose $x=1$). This means, according to invariant set theory, it must be the case that $\cos \alpha$ and $\cos \beta$ are finitely describable.
Now consider a spherical triangle comprising any three of the four distinct points $x=0$, $x=1$, $y=0$ and $y=1$. For definiteness, consider the triangle whose vertices are $x=0$, $x=1$ and $y=0$. The cosine of the angular distance the side joining $x=0$ and $x=1$ is equal to $\cos \alpha$ and therefore is finitely describable. Suppose Alice and Bob choose $x=0$ and $y=0$ so that $\cos \theta_{00}$ is also finitely describable. Then, according to the cosine rule for spherical triangles \be \label{cosinerule} \cos \theta_{01}=\cos \theta_{00} \cos \alpha + \sin \theta_{00} \sin \alpha \cos \gamma \ee where $\gamma$ is the angle subtended at $x=1$. If $\gamma=0$ then we have the usual co-planar arrangement of angles. By the theory of Pythagorean triples, if $\cos \theta_{00}$ is finitely describable, then $\sin \theta_{00}$ is not finitely describable (there are no Pythagorean triples $\{a,b,c\}$ where $c$ is a power of $2$). Similarly, if $\cos \alpha$ is finitely describable, then $\sin \alpha$ is not. Since $\theta_{00}$ and $\alpha$ are essentially independent angles (reflecting the independence and finite precision of Alice and Bob's apparatuses), we can assume that the product $\sin \theta_{00} \sin \alpha$ is not in general finitely describable. Hence $\cos \theta_{01}$ is the sum of a term which is finitely describable, and a term which isn't. Hence $\cos\theta_{01}$ is not finitely describable. With $\gamma$ a further independently chosen angle, one can assume that this result holds for non-zero $\gamma$ too.
Hence, with $z=x+y \mod 2$, in any CHSH experiment the sample space $\Lambda_{z=0}$ of supplementary variables from which $C(0,0)$ and $C(1,1)$ are computed (based on finitely describable $\cos \theta_{00}$ and $\cos \theta_{11}$) must be disjoint from the sample space $\Lambda_{z=1}$ from which $C(0,1)$ and $C(1,0)$ are computed (based on finitely describable $\cos \theta_{01}$ and $\cos \theta_{10}$). Hence, the position of the four points in Fig \ref{fig:CHSH}a is not identical to that in Fig \ref{fig:CHSH}b. This is not physically inconsistent because the measurement orientations can only be set to some finite precision.
\section{The Dirac Equation in the Singular Limit $N=\infty$} \label{Dirac}
In this Appendix we discuss the simplest form of the Dirac equation \be \label{Dirac1} i\hbar \gamma_{\mu}\; \partial_{\mu}\psi - mc \psi = 0 \ee for a particle of mass $m$ in order to describe dynamical evolution between preparation and measurement in invariant set theory.
As is well known, the Dirac 4-spinor can be written as 2 Weyl 2-spinors. We will associate the latter with fields of $2^N$-element bit strings $S_a(\mathbf x, t)$, $S_b(\mathbf x, t)$. Consider a frame where the particle is at rest at $\mathbf x=0$. Let \begin{align} \label{evolution1} S_a(0, t) &= \zeta^n S_a(0,0) \nonumber \\ S_b(0, t) &= \zeta^{-n} S_b(0,0) \end{align} where \be \label{n} n= \frac{2^N mc^2}{2 \pi \hbar} \ t \ee From (\ref{zetan}) we can write \be \zeta^n \equiv e^{2 \pi i n/2^{N}}= e^{i \omega t} \ee where $\hbar \omega=mc^2$. As before, the set $\{e^{i \omega t}\}$ of time evolution operators is isomorphic to the multiplicative group of complex phases $\phi$ where $\phi / 2\pi$ is describable by $N-1$ bits. With $E=mc^2$, then \be \label{E} E=\hbar \omega \ee and with $E$ a source of space-time curvature, this iconic equation of quantum mechanics can be interpreted is a manifestation of (\ref{generalisedequation}), linking space-time geometry to the periodic state-space geometry of $I_U$. A key property of (\ref{evolution1}) is the granularity of time. From (\ref{n}), the unit $\Delta t$ of granularity is given by \be \Delta t = \frac{2 \pi \hbar}{2^N mc^2} \ee the consequences of which will be developed elsewhere.
Writing \be \psi(t)=\bp S_a(0, t) \\ S_b(0, t) \ep \ee then the evolution equation (\ref{evolution1}) can be written as \be \label{evolution2} \psi(t)= \bp e^{i \omega t} & 0 \\ 0 & e^{-i \omega t} \ep \psi(0) \ee For any finite $N$, $\{e^{i \omega t}\}$ is not closed under addition (see (\ref{noadd})). As discussed in the main body of the text, this is considered a desirable property of invariant set theory, making it counterfactually incomplete. In the singular limit $N=\infty$, $e^{i \omega t}$ can be identified with the familiar complex exponential function, in which case, $e^{i \omega \Delta t} \approx 1+ i \omega \Delta t$ for small $\Delta t$ and \be i\partial_t \ e^{i \omega t}+ \omega e^{i \omega t}=0 \nonumber \ee Because the limit is singular, the derivative is undefined for any finite $N$, no matter how big. In this sense, the Dirac equation for a particle at rest, \be \label{Dirac1} i\hbar \gamma_0\; \partial_t \psi - mc^2 \psi = 0 \ee corresponds to the singular limit of (\ref{evolution1}) at $N=\infty$. In (\ref{Dirac1}), $\psi$ is to be considered some abstract but computationally powerful `wavefunction', lying in complex Hilbert Space. From the perspective of invariant set theory, (\ref{evolution1}) and (\ref{n}) should be considered more fundamental, but less computationally powerful, than (\ref{Dirac1}).
In a non-rest frame, we must generalise (\ref{evolution1}) to include spatial variations and the full set of Dirac gamma matrices are needed to be relativistically invariant. Quantities such as \be \gamma_i \bp S_a\\ S_b\ep = \bp 0 & \sigma_i \\ -\sigma_i & 0 \ep \bp S_a\\ S_b\ep =\bp \sigma_i S_b \\ -\sigma_i S_a \ep \ee are straightforwardly defined by considering the Pauli matrices as operators on $S_a$ and $S_b$. For example \begin{align} \sigma_2 S_b &= \bp 0 & -i \\ i & 0 \ep
\{b_1\;b_2\ldots b_{2^{N-1}}\} \| \{b_{2^{N-1}+1}\; b_{2^{N-1}+2}\ldots b_{2^N}\} \\ &=
\zeta^{2^{N-3}}\{\cancel b_{2^{N-1}+1}\; \cancel b_{2^{N-1}+2}\ldots \cancel b_{2^N}\} \| \zeta^{2^{N-3}}\{b_1\;b_2\ldots b_{2^{N-1}}\} \end{align}
where $\|$ denotes the concatenation operator. Full details will be given elsewhere.
\section{The PBR Theorem} \label{PBR}
The recent PBR theorem \cite{Pusey} is a no-go theorem casting doubt on $\psi$-epistemic theories (where the quantum state is presumed to represent information about some underlying physical state of the system). Unlike CHSH where Alice and Bob each choose measurement orientations A or B, here Alice and Bob, by each choosing $0$ or $1$, prepare a quantum system in one of four input states to some quantum circuit: $|\psi_0\rangle |\psi_0\rangle$, $|\psi_0\rangle |\psi_1\rangle$, $|\psi_1\rangle |\psi_0\rangle$ or $|\psi_1\rangle |\psi_1\rangle$, where \begin{align}
|\psi_0\rangle&=\cos \frac{\theta}{2} |0\rangle + \sin \frac{\theta}{2} |1\rangle \nonumber \\
|\psi_1\rangle&=\cos \frac{\theta}{2} |0\rangle - \sin \frac{\theta}{2} |1\rangle \nonumber \end{align} In addition to the parameter $\theta$, the circuit contains two phase angles $\alpha$ and $\beta$; as discussed below, the phase angle $\alpha$ most closely plays the role of the phase angle $\phi$ in the Mach-Zehnder interferometer in Section \ref{interfere}. The output states of the quantum circuit are characterised as `$\mathrm{Not}\; 00$', `$\mathrm{Not}\; 01$', `$\mathrm{Not}\; 10$' and `$\mathrm{Not}\; 00$'. The $\alpha$ and $\beta$ are chosen to ensure that (according to quantum theory), if Alice and Bob's input choices are $\{IJ\}$ where $I,J\in\{0,1\}$, then the probability of `$\mathrm{Not}\; IJ$' is equal to zero. However, if physics is governed by some underpinning $\psi$-epistemic theory, then, so the argument goes, at least occasionally the measuring device will be uncertain as to whether, for example, the input state was prepared using $00$ and $01$ and on these occasions it is possible that an outcome `$ \mathrm{Not} \; 01$' is observed when the state was prepared as $01$, contrary to quantum theory (and experiment). How does Invariant Set Theory, which is indeed a $\psi$-epistemic theory, avoid this problem?
Working through the algebra, it is found that the probabilities of various outcomes are trigonometric functions of $\alpha-\beta$, $\alpha-2\beta$, $\beta$ and $\theta$. For example, if Alice and Bob chose $00$, then, according to quantum theory (and therefore experiment), the probability of obtaining the outcome `$\mathrm{Not}\; 01$' is equal to \be \label{XXX} X= \cos^4 \frac{\theta}{2} + \sin^4 \frac{\theta}{2} +2 \cos^2 \frac{\theta}{2} \sin^2 \frac{\theta}{2}\cos (\alpha-2\beta) \ne 0 \ee On the other hand, if Alice and Bob chose $01$, then the probability of obtaining the outcome `$\mathrm{Not}\; 01$' would be equal to \be \label{ZZZ} Z=X-4 \cos^2 \frac{\theta}{2} \sin^2 \frac{\theta}{2}-4 \cos^3 \frac{\theta}{2} \sin \frac{\theta}{2}\cos(\alpha-\beta)-4 \cos \frac{\theta}{2} \sin^3 \frac{\theta}{2}\cos\beta=0 \ee The key point is that $X$ contains the trigonometric term $\cos(\alpha-2\beta)$, whilst $Z$ contains the terms $\cos(\alpha-\beta)$ and $\cos \beta$. Now one can clearly find values for $\alpha$, $\beta$ and $\theta$ such that $X$ is described by $N$ bits. That is to say, for large enough $N$, Invariant Set Theory can predict the quantum theoretic probability of outcome `$\mathrm{Not}\;01$' when Alice and Bob chose 00. However, in general it is impossible to find values for these angles such that $X$ and $Z$ are \emph{simultaneously} describable by $N$ bits. The number-theoretic argument is exactly that used to negate the Bell Theorem. For example, if $\cos (\alpha-2\beta)$ and $\cos \beta$ are describable by a finite number of bits, then $\cos(\alpha-\beta)=\cos(\alpha-2\beta) \cos \beta+ \sin(\alpha-2\beta) \sin \beta$ is not. This means that if in reality Alice and Bob chose $00$ in preparing a particular quantum system and the outcome was `$\mathrm{Not}\; 01$' , then there is no counterfactual world on $I_U$ where Alice and Bob chose $01$ in preparing the same quantum system, and the outcome was again `$\mathrm{Not}\; 01$'. That is to say, it is not the case that $Z=0$ for this counterfactual experiment - rather, $Z$ is undefined. Conversely, if in reality Alice and Bob chose 01, then there exist values for $\alpha$, $\beta$ and $\theta$ such that $Z$ is described by $N$ bits and equal to zero (to within experimental accuracy) and `$\mathrm{Not}\; 01$' is not observed.
Let $\{\alpha_X, \beta_X, \theta_X\}$ denote a set of angles such that $X$ is describable by $N$ bits, and $\{\alpha_Z, \beta_Z, \theta_Z \}$ a set of angles such that $Z$ is describable by $N$ bits, i.e. these correspond to experiments on $I_U$. Now, as before, we can find values such that the differences $\alpha_Z-\alpha_X$, $\beta_Z-\beta_X$ and $\theta_Z-\theta_X$ are each smaller than the precision by which these angles can be set experimentally. Hence, Invariant Set Theory can readily account for pairs of experiments performed sequentially with seemingly identical parameters, the first where Alice and Bob choose $00$ and the outcome is sometimes `$\mathrm{Not}\; 01$', and the second where Alice and Bob choose $01$ and the outcome is never `$\mathrm{Not}\; 01$'. That is to say, the Invariant Set Theoretic interpretation of the PBR quantum circuit reveals no inconsistency with experiment. Even though Invariant Set Theory is $\psi$-epistemic, the holistic structure of the invariant set $I_U$ ensures that the measuring device will never be uncertain as to whether, for example, the input state was prepared using $00$ and $01$.
It was shown above that Invariant Set Theory evades the Bell theorem by violating the Measurement Independence assumption. Here it has been shown that Invariant Set Theory evades the PBR theorem by violating an equivalent Preparation Independence assumption. As before, this does not conflict at all with the experimenter's sense of free will. Neither does it imply fine-tuning with respect to the physically relevant $p$-adic metric on $I_U$.
\end{document}
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\begin{document}
\title{Jordan and Jordan Higher All-derivable Points of Some Algebras} \abstract
In this paper, we characterize Jordan derivable mappings in terms
of Peirce decomposition and determine Jordan all-derivable points
for some general bimodules. Then we generalize the results to the
case of Jordan higher derivable mappings. An immediate application
of our main results shows that for a nest $\mathcal{N}$ on a Banach
$X$ with the associated nest algebra $alg\mathcal{N}$, if there exists
a non-trivial element in $\mathcal{N}$ which is complemented in $X$,
then every $C\in alg\mathcal{N}$ is a Jordan all-derivable point of
$L(alg\mathcal{N}, B(X))$ and a Jordan higher all-derivable point of $L(alg\mathcal{N})$. \
{\sl Keywords} : Jordan all-derivable point, Jordan derivation, Jordan higher all-derivable point, Jordan higher derivation; nest algebra\
{\sl 2010 AMS classification} : Primary 47L35; Secondly 16W25 \
\section{Introduction}\
Let $\mathcal{A}$ be a unital algebra and $\mathcal{M}$ be a unital $\mathcal{A}$-bimodule. We denote $C(\mathcal{A},\mathcal{M})=\{M\in \mathcal{M}: AM=MA~ \textrm{for~ every}~A\in \mathcal{A}\}$ and $L(\mathcal{A},\mathcal{M})$ the set of all linear mappings from $\mathcal{A}$ to $\mathcal{M}$. When $\mathcal{M}=\mathcal{A}$, we relabel $L(\mathcal{A},\mathcal{M})$ as $L(\mathcal{A})$. Let $\delta\in L(\mathcal{A},\mathcal{M})$. $\delta$ is called a \textit{derivation} if $\delta(AB)=\delta(A)B+A\delta(B)$ for all $A,B\in \mathcal{A}$; it is a \textit{Jordan derivation} if $\delta(AB+BA)=\delta(A)B+A\delta(B)+\delta(B)A+B\delta(A)$ for all $A,B\in \mathcal{A}$; it is a \textit{generalized derivation} if there exists an $M_{\delta}\in C(\mathcal{A},\mathcal{M})$ such that $\delta(AB)=\delta(A)B+A\delta(B)-M_{\delta}AB$ for all $A,B\in \mathcal{A}$. For any fixed $M\in \mathcal{M}$, each mapping of the form $\delta_{M}(A)=MA-AM$ for every $A\in \mathcal{A}$ is called an \textit{inner derivation}. Clearly each inner derivation is a derivation and each derivation in a Jordan derivation. But the converse is not true in general. The questions of characterizing derivations and Jordan derivations have received considerable attention from several authors, who revealed the relations among derivations, Jordan derivations as well as inner derivations (see for example \cite{CHDM,semiprime,prime,Johnson,LU2,Moore,JDOTA}, and the references therein).
In general there are two directions in the study of the local actions of derivations of operator algebras. One is the well known local derivation problem (see for example \cite{local3,local2,local1,Zhu6,Zhu5}). The other is to study conditions under which derivations of operator algebras can be completely determined by the action on some subsets of operators (see for example \cite{CHDM,Chebotar, LI1,Hou2, ZHOU, ZHUJUN2}). A mapping $\delta\in L(\mathcal{A},\mathcal{M})$ is called a \textit{Jordan derivable mapping at $C\in \mathcal{A}$} if $\delta(AB+BA)=\delta(A)B+A\delta(B)+\delta(B)A+B\delta(A)$ for all $A,B\in \mathcal{A}$ with $AB=C$. It is obvious that a linear mapping is a Jordan derivation if and only if it is Jordan derivable at all points. It is natural and interesting to ask the question whether or not a linear mapping is a Jordan derivation if it is Jordan derivable only at one given point. If such a point exists, we call this point a Jordan all-derivable point. To be more precise, an element $C\in \mathcal{A}$ is called a \textit{Jordan all-derivable point} of $L(\mathcal{A},\mathcal{M})$ if every Jordan derivable mapping at $C$ is a Jordan derivation. It is quite surprising that there do exist Jordan all-derivable points for some algebras. An and Hou \cite{Hou} show that under some mild conditions on unital prime ring or triangular ring $\mathcal{A}$, $I$ is a Jordan all-derivable point of $L(\mathcal{A})$. Jiao and Hou \cite{Hou3} study Jordan derivable mappings at zero point on nest algebras. Zhao and Zhu \cite{ZHUJUN} prove that $0$ and $I$ are Jordan all-derivable points of the triangular algebra. In \cite{submit}, the authors study some derivable mappings in the generalized matrix algebra $\mathcal{A}$, and show that $0$, $P$ and $I$ are Jordan all-derivable points, where $P$ is the standard non-trivial idempotent. In \cite{ZHUJUN}, Zhao and Zhu prove that every element in the algebra of all $n\times n$ upper triangular matrices over the complex field $\mathbb{C}$ is a Jordan all-derivable point. In Section 2, we give some general characterizations of Jordan derivable mappings, which will be used to determine Jordan all-derivable points for some general bimodules.
Let $\mathcal{A}$ be a unital algebra and $\mathbb{N}$ be the set of non-negative integers. A sequence of mappings $\{d_i\}_{i\in\mathbb{N}}\in L(\mathcal{A})$ with $d_0=I_\mathcal{A}$ is called a \textit{higher derivation} if $d_{n}(AB)=\sum_{i+j=n}d_i(A)d_j(B)$ for all $A,B\in \mathcal{A}$; it is called a \textit{Jordan higher derivation} if $d_{n}(AB+BA)=\sum_{i+j=n}(d_i(A)d_j(B)+d_i(B)d_j(A))$ for all $A,B\in \mathcal{A}$. With the development of derivations, the study of higher and Jordan higher derivations has attracted much attention as an active subject of research in operator algebras, and the local action problem ranks among in the list. A sequence of mappings $\{d_i\}_{i\in\mathbb{N}}\in L(\mathcal{A})$ with $d_0=I_\mathcal{A}$ is called \textit{Jordan higher derivable at $C\in \mathcal{A}$} if $d_{n}(AB+BA)=\sum_{i+j=n}(d_i(A)d_j(B)+d_i(B)d_j(A))$ for all $A,B\in \mathcal{A}$ with $AB=C$. An element $C\in \mathcal{A}$ is called a \textit{Jordan higher all-derivable point} if every sequence of Jordan higher derivable mappings at $C$ is a Jordan higher derivation. In Section 3, we generalize the results in Section 2 to the case of Jordan higher derivable mappings. Meanwhile, we find the connection between Jordan all-derivable points (all-derivable points, S-Jordan all derivable points, respectively) and Jordan higher all-derivable points (higher all-derivable points, S-Jordan higher all-derivable points, respectively). We also discuss the automatic continuity property of (Jordan) higher derivations.
Let $X$ be a complex Banach space and $B(X)$ be the set of all bounded linear operators on $X$. For any non-empty subset $L\subseteq X$, $L^\perp$ denotes its annihilator, that is, $L^\perp=\{f\in X^*: f(x)=0~\mathrm{for}~\mathrm{all}~x\in L\}$. By a \textit{subspace lattice} on $X$, we mean a collection $\mathcal{L}$ of closed subspaces of $X$ with (0) and $X$ in $\mathcal{L}$ such that for every family $\{M_r\}$ of elements of $\mathcal{L}$, both $\cap M_r$ and $\vee M_r$ belong to $\mathcal{L}$. For a subspace lattice $\mathcal{L}$ of $X$, let alg$\mathcal{L}$ denote the algebra of all operators in $B(X)$ that leave members of $\mathcal{L}$ invariant. A totally ordered subspace lattice is called a \textit{nest}. If $\mathcal{L}$ is a nest, then alg$\mathcal{L}$ is called a \textit{nest algebra}, see \cite{NEST} for more on nest algebras. When $X$ is a separable Hilbert space over the complex field $\mathbb{C}$, we change it to $H$. In a Hilbert space, we disregard the distinction between a closed subspace and the orthogonal projection onto it. An immediate but noteworthy application of our main result shows that for a nest $\mathcal{N}$ on a Banach $X$ with the associated nest algebra $alg\mathcal{N}$, if there exists a non-trivial element in $\mathcal{N}$ which is complemented in $X$, then every $C\in alg\mathcal{N}$ is a Jordan all-derivable point of $L(alg\mathcal{N}, B(X))$ and a Jordan higher all-derivable point of $L(alg\mathcal{N})$.
\section{Jordan derivable mappings}\
We start with Peirce decomposition of algebras and its bimodules.
Let $\mathcal{A}$ be a unital algebra and $\mathcal{M}$ be a unital $\mathcal{A}$-bimodule. For any idempotent $E_1\in \mathcal{A}$, let $E_2=I-E_1$. For $i,j\in\{1,2\},$ define $\mathcal{A}_{ij}=E_i\mathcal{A}E_j$, which gives the Peirce decomposition of $\mathcal{A}: A=A_{11}+A_{12}+A_{21}+A_{22}.$ Similarly, we define $\mathcal{M}_{ij}=E_i\mathcal{M}E_j$. We say $\mathcal{A}_{ij}$ is \textit{left faithful} with respect to $\mathcal{M}$ if for any $M\in \mathcal{M}$, the condition $M\mathcal{A}_{ij}=\{0\}$ implies $ME_i=0$ and $\mathcal{A}_{ij}$ is \textit{right faithful} with respect to $\mathcal{M}$ if the condition $\mathcal{A}_{ij}M=\{0\}$ implies $E_jM=0.$ We say $\mathcal{A}_{ij}$ is \textit{faithful} with respect to $\mathcal{M}$ if it is both left faithful and right faithful. In this paper, we will always use the notations $P=E_1$ and $Q=E_2=I-E_1$ for convenience.
In this section, we will assume $\mathcal{A}$ is a unital algebra over a field $\mathbb{F}$ of characteristic not equal to $2$ and $|\mathbb{F}|\geq4$, $\mathcal{M}$ is a unital $\mathcal{A}$-bimodule and $\mathcal{A}$ has a non-trivial idempotent $P=E_1\in \mathcal{A}$ such that the corresponding Peirce decomposition has the following property: Every element of $A_{11}$ is a linear combination of invertible elements of $A_{11}$ and every element of $A_{22}$ is a linear combination of invertible elements of $A_{22}$. Algebras satisfying these assumptions include all finite-dimensional unital algebras over an algebraically closed field and all unital Banach algebras.
For any $A,B\in \mathcal{A}$, define $A\circ B=AB+BA$; similarly, for any $A\in \mathcal A$ and $M\in \mathcal M$, define $A\circ M=AM+MA.$ For $A,B, D, E\in \mathcal{A}$, we say any $\delta \in L(\mathcal A, \mathcal M)$ \textit{differentiates} $A\circ B$ if $\delta(A\circ B)=\delta(A)\circ B+A\circ \delta(B)$ ; we say $\delta $ \textit{differentiates} $A\circ B+ C\circ D$ if $\delta (A\circ B+ C\circ D)=\delta (A)\circ B+A\circ \delta (B)+\delta (D)\circ E+D\circ \delta (E)$. We see that a mapping $\delta\in L(\mathcal{A},\mathcal{M})$ is a Jordan derivation if and only if $\delta $ differentiates $A\circ B$ for all $A,B\in \mathcal{A}$, and $\delta$ is Jordan derivable at $C\in \mathcal{A}$ if and only if $\delta $ differentiates $A\circ B$ for all $A,B\in \mathcal{A}$ with $AB=C$.
The following proposition is elementary, we omit the proof.
\begin{proposition}\label{0000} Let $\mathcal{V}$ be a vector space over a field $\mathbb{F}$ with
$|\mathbb{F}|>n$. For any fixed $v_i\in \mathcal{V}, i=0, 1, \cdots , n$, define $p(t)=\sum_{i=0}^n v_i t^i $ for $t\in \mathbb{F}$. If $p(t)=0$ has at least $n+1$ distinct solutions in $\mathbb{F}$, then $v_i=0, \ i=0, 1, \cdots , n$. \end{proposition}
A simple application of Proposition $2.1$ yields the following proposition, which will be used repeatedly in this paper.
\begin{proposition}\label{0001} Suppose $A, B, D, E, K, L \in \mathcal{A}$ and $\delta\in L(\mathcal{A},\mathcal{M})$.
(a) If $\delta $ differentiates $(tA+B)\circ (tD+E)$ for at least three $t\in \mathbb{F}$, then $\delta $ differentiates $A\circ D$, $B\circ E$, and $ A\circ E+B\circ D$; in particular, if $A=0$ then $\delta $ differentiates $B\circ D$.
(b) If $\delta $ differentiates $A\circ (tD+E)+B\circ (tK+L)$ for at least two $t\in \mathbb{F}$, then
$\delta $ differentiates $A\circ D+B\circ K$ and $A\circ E+B\circ L$. \end{proposition}
Now we characterize Jordan-derivable mappings in terms of Peirce decomposition as follows.
\begin{theorem}\label{0002} For any $C\in\mathcal{A}$ such that $C_{21}=0$, if $\Delta\in L(\mathcal{A},\mathcal{M})$ is Jordan-derivable at $C$, then there exists a $\delta\in L(\mathcal{A},\mathcal{M})$ such that $\Delta-\delta$ is an inner derivation and the following hold:
(a)~~$\delta(P)A_{12}=A_{12}\delta(Q)$ for any $A_{12}\in \mathcal{A}_{12}$.\
(b)~~$A_{12}\delta(A_{12})=\delta(A_{12})A_{12}=0$ for any $A_{12}\in \mathcal{A}_{12}$.\
(c)~~$\delta(\mathcal{A}_{11})\subset \mathcal{M}_{11}$, $\delta(\mathcal{A}_{22})\subset \mathcal{M}_{22}$.
If $\mathcal{A}_{12}$ is left faithful, then
(d)~~$\delta(P)\in C(\mathcal{A}_{11},\mathcal{M})$.
(e)~~$\delta|_{\mathcal{A}_{11}}$ is a generalized derivation from $\mathcal{A}_{11}$ to $\mathcal{M}_{11}$.
If $\mathcal{A}_{12}$ is right faithful, then
(f)~~$\delta(Q)\in C(\mathcal{A}_{22},\mathcal{M})$.
(g)~~$\delta|_{\mathcal{A}_{22}}$ is a generalized derivation from $\mathcal{A}_{22}$ to $\mathcal{M}_{22}$. \end{theorem}
\begin{proof} Let $M=P\Delta(Q)Q-Q\Delta(Q)P$ and define $\delta(A)=\Delta(A)-(MA-AM)$ for every $A\in \mathcal{A}$. Then $\delta$ is Jordan-derivable at any $G\in \mathcal{A}$ if and only if $\Delta$ is Jordan-derivable at $G$; moreover $\delta(Q)\in \mathcal{M}_{11}+\mathcal{M}_{22}$ by direct computation. Write $C= C_{11}+C_{12}+C_{22}$. Fix any $A_{11}\in\mathcal{A}_{11}$ that is invertible in $\mathcal{A}_{11}$ with $A_{11}^{-1}\in \mathcal{A}_{11}$ and $Z_{22}, W_{22}\in \mathcal{A}_{22}$ such that $Z_{22}W_{22}=C_{22}$. Note that we can take any $W_{22}$ that is invertible in $\mathcal{A}_{22}$ with $W_{22}^{-1}\in \mathcal{A}_{22}$ and $Z_{22}=C_{22}W_{22}^{-1}$ to satisfy $Z_{22}W_{22}=C_{22}$. For any $0\neq t\in \mathbb{F}$, $s\in \mathbb{F}$, and $A_{12}\in \mathcal{A}_{12}$, a routine computation yields $$[A_{11}+t(sA_{11}A_{12}+Z_{22})][(A_{11}^{-1}C-sA_{12}W_{22})+t^{-1}W_{22}]=C.$$ Since $\delta$ is Jordan derivable at C, $\delta $ differentiates $$[A_{11}+t(sA_{11}A_{12}+Z_{22})]\circ [(A_{11}^{-1}C-sA_{12}W_{22})+t^{-1}W_{22}].$$ Thus $\delta $ differentiates $[A_{11}+t(sA_{11}A_{12}+Z_{22})]\circ [t(A_{11}^{-1}C-sA_{12}W_{22})+W_{22}].$ By Proposition \ref{0001}$(a)$, we get $(i)$ $\delta $ differentiates $A_{11}\circ W_{22}$, \ \ $(ii)$ $\delta $ differentiates $(sA_{11}A_{12}+Z_{22})\circ (A_{11}^{-1}C-sA_{12}W_{22})$, and $(iii)$ $\delta $ differentiates $A_{11}\circ (A_{11}^{-1}C-sA_{12}W_{22})+(sA_{11}A_{12}+Z_{22})\circ W_{22}.$
By $(i)$, we get \begin{eqnarray}\delta (A_{11})\circ W_{22}+A_{11}\circ \delta (W_{22})=\delta (A_{11}\circ W_{22})=0 \label{2000} \end{eqnarray}
By $(ii)$ and Proposition \ref{0001}$(a)$, we have $\delta $ differentiates $(A_{11}A_{12})\circ (A_{12}W_{22})$, i.e. \begin{eqnarray} \delta (A_{11}A_{12})\circ (A_{12}W_{22}) +(A_{11}A_{12})\circ \delta (A_{12}W_{22}) =\delta [(A_{11}A_{12})\circ (A_{12}W_{22})]=0\label {2002} \end{eqnarray}
By $(iii)$ and Proposition \ref{0001}$(b)$, we have \begin{eqnarray*} &0=\delta [A_{11}\circ (-A_{12}W_{22})+(A_{11}A_{12})\circ W_{22}] =\delta (A_{11})\circ (-A_{12}W_{22}) +A_{11}\circ \delta (-A_{12}W_{22}) \\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\delta (A_{11}A_{12})\circ W_{22}+(A_{11}A_{12})\circ \delta (W_{22}) \end{eqnarray*}
Thus \begin{eqnarray} \delta (A_{11}A_{12})\circ W_{22}+(A_{11}A_{12})\circ \delta (W_{22}) -\delta (A_{11})\circ (A_{12}W_{22}) - A_{11}\circ \delta (A_{12}W_{22})=0 \label {2003} \end{eqnarray} Since $\delta (Q)\in \mathcal M_{11}+\mathcal M_{22}$, $A_{11}\circ \delta (Q)\in \mathcal M_{11}$. Setting $W_{22}=Q$ in Eq. $(2.1)$ gives \begin{eqnarray*} \delta (A_{11})\circ Q+A_{11}\circ \delta (Q)=0 \end{eqnarray*} Thus $A_{11}\circ \delta (Q)=\delta (A_{11})\circ Q=0$. It follows $\delta (A_{11})Q=Q\delta (A_{11})=0.$ Hence $\delta (A_{11})\in \mathcal M_{11},$ and $\delta (A_{11})\circ W_{22}=0$. By Eq. $(2.1)$ again, we get $A_{11}\circ \delta (W_{22})=0$; in particular, $P\circ \delta (W_{22})=0$. It follows that $\delta (W_{22})\in \mathcal M_{22}$,
which proves $(c)$.
Taking $A_{11}=P$ in Eq. $(2.3)$ yields \begin{eqnarray} \delta (A_{12})\circ W_{22}+A_{12}\circ \delta (W_{22}) -\delta (P)\circ (A_{12}W_{22}) - P\circ \delta (A_{12}W_{22})=0 \label {2004} \end{eqnarray} Multiplying $P$ from both sides of Eq. $(2.4)$ gives $P \delta (A_{12}W_{22})P=0$. In particular, \begin{eqnarray} P \delta (A_{12})P=0 \label {2005} \end{eqnarray} Multiplying $P$ from the left of Eq. $(2.4)$ and applying Eq. $(2.5)$, we get \begin{eqnarray} P\delta (A_{12}) W_{22}+A_{12}\delta (W_{22}) -\delta (P)A_{12}W_{22} - P\delta (A_{12}W_{22})=0 \end{eqnarray} Setting $W_{22}=Q$ in Eq. $(2.6)$ and combining with Eq. $(2.5)$ leads to \begin{eqnarray} A_{12}\delta (Q)=\delta (P)A_{12} \label {2006} \end{eqnarray} This proves $(a)$.
Taking $A_{11}=P$ and $W_{22}=Q$ in Eq. $(2.2)$, we get $A_{12}\circ \delta (A_{12})=0$, i.e. \begin{eqnarray} A_{12}\delta (A_{12})+\delta (A_{12})A_{12}=0 \label {2007} \end{eqnarray} Multiplying $P$ from the left of Eq. $(2.8)$ and applying Eq. $(2.5)$, yields $A_{12}\delta (A_{12})=0$; which gives $\delta (A_{12})A_{12}=0$, when applied to Eq. $(2.8)$. This proves $(b)$.
Taking $W_{22}=Q$ in Eq. $(2.3)$ yields \begin{eqnarray} \delta (A_{11}A_{12})\circ Q+(A_{11}A_{12})\circ \delta (Q) -\delta (A_{11})\circ A_{12} - A_{11}\circ \delta (A_{12})=0 \label {2008} \end{eqnarray} Multiplying $Q$ from both sides of Eq. $(2.9)$ gives $Q\delta (A_{11}A_{12})Q=0$. In particular, \begin{eqnarray} Q \delta (A_{12})Q=0 \label {2009} \end{eqnarray} Multiplying $Q$ from the right of Eq. $(2.9)$ and applying Eq. $(2.10)$ gives \begin{eqnarray*} \delta (A_{11}A_{12})Q+A_{11}A_{12}\delta (Q) -\delta (A_{11})A_{12} - A_{11}\delta (A_{12})Q=0 \end{eqnarray*} Combining this with Eq. $(2.5)$ yields \begin{eqnarray} \delta (A_{11}A_{12})Q= \delta (A_{11})A_{12} + A_{11}\delta (A_{12}) - A_{11}A_{12}\delta (Q) \label {2010} \end{eqnarray} Replacing $A_{11}$ with $A_{11}U_{11}$ in Eq. $(2.11)$ gives \begin{eqnarray} \delta (A_{11}U_{11}A_{12})Q= \delta (A_{11}U_{11})A_{12} + A_{11}U_{11}\delta (A_{12}) - A_{11}U_{11}A_{12}\delta (Q) \label {2011} \end{eqnarray} On the other hand, applying Eq. $(2.11)$ twice gives \begin{align}
\delta (A_{11}U_{11}A_{12})Q &= A_{11}\delta (U_{11}A_{12}) + \delta (A_{11})U_{11}A_{12} - A_{11}U_{11}A_{12}\delta (Q) \nonumber \\ & = A_{11}\delta (U_{11}A_{12})Q + \delta (A_{11})U_{11}A_{12} - A_{11}U_{11}A_{12}\delta (Q) \nonumber \\ & = A_{11}[\delta (U_{11})A_{12} + U_{11}\delta (A_{12}) - U_{11}A_{12}\delta (Q)] \nonumber \\ &~~~~ + \delta (A_{11})U_{11}A_{12} - A_{11}U_{11}A_{12}\delta (Q) \label {2012} \end{align} By Eqs. $(2.12)$, $(2.13)$, and $(2.7)$, we have \begin{eqnarray*} \delta (A_{11}U_{11})A_{12}= [\delta (A_{11})U_{11} + A_{11}\delta (U_{11}) - \delta (P)A_{11}U_{11}]A_{12} \end{eqnarray*} If $\mathcal A_{12}$ is left faithful, \begin{eqnarray} \delta (A_{11}U_{11})= \delta (A_{11})U_{11} + A_{11}\delta (U_{11}) - \delta (P)A_{11}U_{11} \label {2013} \end{eqnarray} Taking $U_{11}=P$ in Eq. $(2.14)$ gives $A_{11}\delta (P) = \delta (P)A_{11}$, that is, $\delta (P)\in C(\mathcal A_{11}, \mathcal M)$. This proves $(d)$ and now $(e)$ follows directly from Eq. $(2.14)$.
Since $\delta(P)A_{12}=A_{12}\delta(Q)$ for any $A_{12}\in \mathcal{A}_{12}$, we have $A_{12}\delta(Q)A_{22}=\delta(P)A_{12}A_{22}=A_{12}A_{22}\delta(Q)$, then faithfulness of $\mathcal{A}_{12}$ leads to $\delta(Q)A_{22}=A_{22}\delta(Q)$, that is,
$\delta(Q)\in C(\mathcal{A}_{22},\mathcal{M})$. This proves $(f)$.
By Eqs. $(2.6)$ and $(2.10)$, \begin{eqnarray} P\delta (A_{12}W_{22})=\delta (A_{12}) W_{22}+A_{12}\delta (W_{22}) -\delta (P)A_{12}W_{22} \end{eqnarray} Replacing $W_{22}$ with $V_{22}W_{22}$ in Eq. $(2.15)$ gives \begin{eqnarray} P\delta (A_{12}V_{22}W_{22})=\delta (A_{12}) V_{22}W_{22}+A_{12}\delta (V_{22}W_{22}) -\delta (P)A_{12}V_{22}W_{22} \end{eqnarray} On the other hand, applying Eq. $(2.15)$ twice gives \begin{align} P\delta (A_{12}V_{22}W_{22})&=\delta (A_{12}V_{22}) W_{22}+A_{12}V_{22}\delta (W_{22}) -\delta (P)A_{12}V_{22}W_{22} \nonumber \\ &=P\delta (A_{12}V_{22}) W_{22}+A_{12}V_{22}\delta (W_{22}) -\delta (P)A_{12}V_{22}W_{22} \nonumber \\ &=[\delta (A_{12}) V_{22}+A_{12}\delta (V_{22}) -\delta (P)A_{12}V_{22}]W_{22} \nonumber \\ &~~~~+A_{12}V_{22}\delta (W_{22}) -\delta (P)A_{12}V_{22}W_{22} \end{align} By Eqs. $(2.16)$, $(2.17)$, and $(2.7)$, \begin{eqnarray*} A_{12}\delta (V_{22}W_{22})=A_{12}[\delta (V_{22})W_{22}+V_{22}\delta (W_{22}) -\delta (Q)V_{22}W_{22}] \end{eqnarray*} Since $\mathcal A_{12}$ is left faithful, \begin{eqnarray} \delta (V_{22}W_{22})=\delta (V_{22})W_{22}+V_{22}\delta (W_{22}) -\delta (Q)V_{22}W_{22} \end{eqnarray} This proves $(g)$. \end{proof}
Suppose $\mathcal{B}$ is an algebra containing $\mathcal{A}$ and shares the same identity with $\mathcal{A}$, then $\mathcal{B}$ is an $\mathcal{A}$-bimodule with respect to the multiplication and addition of $\mathcal{B}$. Let $\mathcal{T}_{\mathcal{A}}= \{A\in \mathcal A: A_{21}=0\}$.
The following proposition is contained in \cite[Theorem 3.3]{PAN}, we include a proof here for completeness.
\begin{proposition}\label{2.4} Suppose $\mathcal{A}_{12}$ is faithful to $\mathcal{B}$, $C(\mathcal{T}_{\mathcal{A}},\mathcal{B})=\mathbb{F}I$, and $B\in \mathcal B$. If $T_{12}BT_{12}=0$ for every $T_{12}\in \mathcal{A}_{12}$, then $QBP=0$. \end{proposition} \begin{proof} Suppose $T_{12}BT_{12}=0$ for every $T_{12}\in \mathcal{A}_{12}$. For any non-zero $A_{12},T_{12}\in\mathcal{A}_{12}$, we have $T_{12}BT_{12}=0$, $A_{12}BA_{12}=0$ and $(A_{12}+T_{12})B(A_{12}+T_{12})=0$. It follows that \begin{eqnarray} A_{12}BT_{12}+T_{12}BA_{12}=0.\label{2023} \end{eqnarray} For any $A_{11}\in \mathcal{A}_{11}$, replacing $A_{12}$ in Eq. $(2.19)$ with $A_{11}A_{12}$ gives \begin{eqnarray} A_{11}A_{12}BT_{12}+T_{12}BA_{11}A_{12}=0.\label{2024} \end{eqnarray} Multiplying $A_{11}$ from the left of Eq. $(2.19)$ gives \begin{eqnarray} A_{11}A_{12}BT_{12}+A_{11}T_{12}BA_{12}=0.\label{2025} \end{eqnarray} By Eq. (\ref{2024}) and Eq. (\ref{2025}), we have \begin{eqnarray*} T_{12}BA_{11}A_{12}=A_{11}T_{12}BA_{12}. \end{eqnarray*} Since $A_{12}$ is arbitrary and $\mathcal{A}_{12}$ is faithful, we have \begin{eqnarray} T_{12}BA_{11}=A_{11}T_{12}BP.\label{2026} \end{eqnarray} Similarly, we have \begin{eqnarray} A_{22}QBT_{12}=QBT_{12}A_{22}.\label{2027} \end{eqnarray} Let $\widetilde{B}=T_{12}BP-QBT_{12}$. It follows from Eqs. (\ref{2023}), (\ref{2026}) and (\ref{2027}) that $\widetilde{B}$ commutes with $A_{12}$, $A_{11}$ and $A_{22}$, that is, $\widetilde{B}\in C(\mathcal{T}_{\mathcal{A}},\mathcal{B})$. Hence there exists a $k\in \mathbb{F}$ such that $\widetilde{B}=kI$. It follows $T_{12}BP=kP$. Now $T_{12}BT_{12}=0$ leads to $kT_{12}=0$. Hence $k=0$ and $T_{12}BP=0$. Since $T_{12}$ is arbitrary and $\mathcal{A}_{12}$ is faithful, we have $QBP=0$. \end{proof}
\begin{theorem}\label{2.5} Suppose $\mathcal{A}_{12}$ is faithful to $\mathcal{B}$ and $C(\mathcal{T}_{\mathcal{A}},\mathcal{B})=\mathbb{F}I$. If $\delta\in L(\mathcal{A},\mathcal{B})$ is Jordan derivable at $C\in \mathcal{T}_{\mathcal{A}}$ then
$\delta |_{\mathcal{T}_{\mathcal{A}}}$ is a derivation from $\mathcal{T}_{\mathcal{A}}$ to $\mathcal{B}$. \end{theorem} \begin{proof} Substracting an inner derivation from $\delta$ if necessary, we can assume $\delta$ satisfies the properties of Theorem $2.3$. Thus, for any $A_{12}$ and $T_{12}$ in $\mathcal{A}_{12}$, we have $\delta(A_{12})A_{12}=0$, $\delta(T_{12})T_{12}=0$ and $\delta(A_{12}+T_{12})(A_{12}+T_{12})=0$. It follows that $\delta(A_{12})T_{12}+\delta(T_{12})A_{12}=0.$ Multiplying $T_{12}$ from the left we obtain $T_{12}\delta(A_{12})T_{12}=0$. Since $T_{12}$ is arbitrary, $Q\delta(A_{12})P=0$, by Proposition $2.4$. This, together with Eqs. $(2.5)$ and $(2.10)$, yields $\delta(A_{12})\in \mathcal{B}_{12}$.
For any $A_{11}\in \mathcal A_{11}$ and $A_{22}\in \mathcal A_{22}$, by Theorem $2.3$ $\delta (A_{11})\in \mathcal B_{11}$ and $\delta (A_{22})\in \mathcal B_{22}$.
Since $\delta(P)\in\mathcal{B}_{11}$ and $\delta(Q)\in\mathcal{B}_{22}$, by Theorem $2.3$ $\delta(I)=\delta(P)+\delta(Q)$ commutes with $\mathcal{A}_{11}$, $\mathcal{A}_{12}$ and $\mathcal{A}_{22}$, whence $\delta(I)\in C(\mathcal{T}_{\mathcal{A}},\mathcal{B})$. Thus $\delta(I)=\lambda I$. By the fact that $\delta$ is Jordan derivable at $C$, we have $\delta(IC+CI)=\delta(I)C+I\delta(C)+C\delta(I)+\delta(C)I$, which implies $\lambda C=0$. If $C\neq0$, $\lambda=0$. Hence $\delta(P)=\delta(Q)=0$. If $C=0$, then the fact that $A_{12}A_{11}=0$ holds for every $A_{11}\in \mathcal{A}_{11}$ and $A_{12}\in \mathcal{A}_{12}$ implies that $\delta(A_{11}A_{12})=\delta(A_{12})A_{11}+A_{12}\delta(A_{11})+A_{11}\delta(A_{12})+\delta(A_{11})A_{12}=A_{11}\delta(A_{12})+\delta(A_{11})A_{12}$, which together with faithfulness of $\mathcal{A}_{12}$ leads to $\delta(A_{11}U_{11})=\delta(A_{11})U_{11}+A_{11}\delta(U_{11})$ for every $A_{11}, U_{11}\in \mathcal{A}_{11}$. Comparing with Eq. $(2.14)$, we have that $\delta(P)=\delta(Q)=0$.
To see $\delta |_{\mathcal{T}_{\mathcal{A}}}$ is a derivation, it suffices to show that for any $A_{ij}, A_{kl}\in \mathcal{T}_{\mathcal{A}}$ $$\delta(A_{ij}A_{kl})=\delta(A_{ij})A_{kl}+A_{ij}\delta(A_{kl}).$$ We will label each case as Case $(ij,kl)$. Since $\delta(A_{11})\in\mathcal{B}_{11}$, $\delta(A_{12})\in\mathcal{B}_{12}$, and $\delta(A_{22})\in\mathcal{B}_{22}$, we only need to check cases for $j=k$. There are only 4 cases.
Case $(11,11)$ follows from Eq. $(2.14)$.
Case $(11,12)$ follows from Eq. $(2.11)$.
Case $(12,22)$ follows from Eq. $(2.15)$.
Case $(22,22)$ follows from Eq. $(2.18)$. \end{proof}
\begin{corollary}\label{0002} Suppose $\mathcal{A}_{12}$ is faithful to $\mathcal{B}$, $\mathcal{A}_{21}=\{0\}$, and $C(\mathcal{A},\mathcal{B})=\mathbb{F}I$. If $\delta\in L(\mathcal{A},\mathcal{B})$ is Jordan derivable at $C\in \mathcal A$ then $\delta $ is a derivation. In particular, every $C\in \mathcal A$ is a Jordan all-derivable point of $L(\mathcal A, \mathcal B)$ and every Jordan derivation is a derivation. \end{corollary}
As a consequence of Corollary \ref{0002}, similar to \cite[Theorem 4.4]{PAN} we have
\begin{theorem}\label{0003} Let $\mathcal{L}$ be a subspace lattice on a Banach space $X$ and $\mathcal{A}=alg\mathcal{L}$. Suppose there exists a non-trivial idempotent $P\in \mathcal{A}$ such that $ran(P)\in \mathcal{L}$ and $PB(X)(I-P)\subseteq \mathcal{A}$. If $\delta\in L(\mathcal{A},B(X))$ is Jordan derivable at $C\in \mathcal A$ then $\delta $ is a derivation. In particular, every $C\in \mathcal A$ is a Jordan all-derivable point of $L(\mathcal A, B(X))$. \end{theorem}
\begin{proof} We will apply Corollary \ref{0002} with $\mathcal{B}=B(X)$. Let $Q=I-P$. The condition $ran(P)\in \mathcal{L}$ implies $\mathcal{A}_{21}=Q\mathcal{A}P=\{0\}$. The condition $PB(X)Q\subseteq \mathcal{A}$ implies $\mathcal{A}_{12}=PB(X)Q$ is faithful. To see $C(\mathcal{A},B(X))=\mathbb{C}I$, take any $B\in C(\mathcal{A},B(X))$, from $BP=PB$ we get $PBQ=0$. From $BQ=QB$, we have $QBP=0$. Thus $B=B_{11}+B_{22}$. For any $x\in ran(P)$ and $f\in X^*$, $x\otimes f Q\in \mathcal{A}_{12}$. It follows from $Bx\otimes f Q=x\otimes f Q B$ that $B_{11}x\otimes f Q=x\otimes f Q B_{22}$, which leads to $B_{11}x\in \mathbb{C}x$. Since $x\in ran(P)$ is arbitrary, it follows $B_{11}=kP$ for some $k\in \mathbb{C}$. Hence $x\otimes f(kQ-B_{22})=0$, and we have $B_{22}=kQ$ and $B=kI$. Now the conclusion follows from Corollary \ref{0002}. \end{proof}
As an immediate but noteworthy application of Theorem \ref{0003}, we have the following corollary which generalizes the main result in \cite{ZHUJUN}.
\begin{corollary} Let $\mathcal{N}$ be a nest on a Banach space $X$ and $\mathcal{A}=alg\mathcal{N}$. Suppose there exists a non-trivial idempotent $P\in \mathcal{A} $ such that $ran(P) \in \mathcal{N}$. If $\delta\in L(\mathcal{A},B(X))$ is Jordan derivable at $C\in \mathcal A$ then $\delta $ is a derivation. In particular, every $C\in \mathcal A$ is a Jordan all-derivable point of $L(\mathcal A, B(X))$. \end{corollary} \begin{proof} Let $Q=I-P$. Then $PB(X)Q\subseteq \mathcal{A}$. Now applying Theorem \ref{0003} completes the proof. \end{proof} For an algebra $\mathcal{A}$ and a left $\mathcal{A}$-module $\mathcal{M}$,
we call a subset $\mathcal{B}$ of $\mathcal{A}$ \emph{separates} $\mathcal{M}$
if for every $M\in \mathcal{M}$, $\mathcal{B}M={0}$ implies $M=0$. Let $[\mathcal A_{11}, \mathcal A_{11}]=\{A_{11}B_{11}-B_{11}A_{11}: \ A_{11}, B_{11}\in \mathcal A_{11}\}$.
\begin{theorem} \label{0008} Suppose $\mathcal{A}_{12}$ is faithful to $\mathcal{B}$, $C(\mathcal{T}_{\mathcal{A}},\mathcal{B})=\mathbb{F}I$, and $[\mathcal A_{11}, \mathcal A_{11}]$ separates $\mathcal B_{12}$. If $\delta \in L(\mathcal A, \mathcal B)$ is Jordan derivable at some $C\in \mathcal A_{11}+\mathcal A_{12}$ then $\delta $ is a derivation. In particular, every $C\in \mathcal A_{11}+\mathcal A_{12}$ is a Jordan all-derivable point of $L(\mathcal A, \mathcal B).$ \end{theorem} \begin{proof} Let $C\in \mathcal A_{11}+\mathcal A_{12}$ and $\delta\in L(\mathcal{A},\mathcal{B})$ be Jordan derivable at $C$. Substracting an inner derivation from $\delta$ if necessary, we can assume $\delta$ satisfies the properties of Theorem $2.3$. Let $Q=I-P$ then $QC=0$. By Theorem $2.3$ and Proposition \ref{2.4}, $\delta (\mathcal A_{11})\subseteq \mathcal B_{11}$, $\delta (\mathcal A_{22})\subseteq \mathcal B_{22}$, and $\delta (\mathcal A_{12})\subseteq \mathcal B_{12}$; moreover, $\delta (I)=\delta (P)=\delta (Q)=0.$ For any $t\in \mathbb F$, $A_{11}\in \mathcal A_{11}$ that is invertible in $\mathcal A_{11}$ with $A_{11}^{-1}\in \mathcal A_{11}$, and $A_{21}\in \mathcal A_{21}$, clearly $A_{11}(A_{11}^{-1}C+tA_{21})=C$. Since $\delta $ is Jordan derivable at $C$, $\delta $ differentiates $A_{11}\circ (A_{11}^{-1}C+tA_{21})$. By Proposition $2.2$, \begin{eqnarray} \delta (A_{11}\circ A_{21}) = \delta (A_{11})\circ A_{21}+A_{11}\circ \delta (A_{21}) \end{eqnarray} Multiplying $Q$ from the right of Eq. $(2.24)$ gives \begin{eqnarray} A_{11}\delta (A_{21})Q = \delta (A_{21}A_{11})Q \end{eqnarray} For any $U_{11}\in \mathcal A_{11}$, by Eq. $(2.25)$ we get \begin{eqnarray*} A_{11}U_{11}\delta (A_{21})Q = \delta (A_{21}A_{11}U_{11})Q \end{eqnarray*} On the other hand, applying Eq. $(2.25)$ twice gives \begin{eqnarray*} U_{11}A_{11}\delta (A_{21})Q = U_{11}\delta (A_{21}A_{11})Q = \delta (A_{21}A_{11}U_{11})Q \end{eqnarray*} It follows $[A_{11}U_{11}-U_{11}A_{11}]\delta (A_{21})Q=0$. Since $[\mathcal A_{11}, \mathcal A_{11}]$ separates $\mathcal B_{12}$, $P\delta (A_{21})Q=0$. Multiplying $Q$ from the left of Eq. $(2.25)$ gives $Q\delta (A_{21}A_{11})Q = 0$. In particular, $Q\delta (A_{21})Q = 0$. Multiplying $P$ from both sides of Eq. $(2.24)$ and setting $A_{11}=P$ leads to $P\delta (A_{21})P = 0$. Thus $\delta (A_{21})\in \mathcal B_{21}$.
To see $\delta $ is a derivation, it suffices to show that for any $A_{ij}\in \mathcal A_{ij}, A_{kl}\in \mathcal A_{kl}$ $$\delta(A_{ij}A_{kl})=\delta(A_{ij})A_{kl}+A_{ij}\delta(A_{kl})$$ We will again label each case as Case $(ij,kl)$. Since $\delta(A_{ij})\in\mathcal{B}_{ij}$, for all $i, j=1, 2,$ we only need to check cases for $j=k$. There are 8 cases.
Case $(11,11)$ follows from Eq. $(2.14)$.
Case $(11,12)$ follows from Eq. $(2.11)$.
Case $(12,22)$ follows from Eq. $(2.15)$.
Case $(22,22)$ follows from Eq. $(2.18)$.
Case $(21,11)$ follows from Eq. $(2.24)$. \\ It remains to show Cases $(12, 21)$, $(21, 12)$, and $(22, 21)$.
For any $s, t \in \mathbb F$, a routine computation shows $(P+sA_{12})[t(A_{21}-sA_{12}A_{21})-sA_{12}+C+Q]=C$. Since $\delta $ is Jordan derivable at $C$, $\delta $ differentiates
$(P+sA_{12})\circ [t(A_{21}-sA_{12}A_{21})-sA_{12}+C+Q].$ By Proposition $2.2$, $\delta $ differentiates $(P+sA_{12})\circ (A_{21}-sA_{12}A_{21})$. Applying Proposition $2.2$ again , we see
$\delta $ differentiates $P\circ (-A_{12}A_{21})+A_{12}\circ A_{21}$. Case $(11,11)$ implies $\delta $ differentiates $P\circ (-A_{12}A_{21})$, It follows that $\delta $ differentiates $A_{12}\circ A_{21},$ i.e. \begin{eqnarray} \delta (A_{12}\circ A_{21}) = \delta (A_{12})\circ A_{21}+A_{12}\circ \delta (A_{21}) \end{eqnarray} Multiplying $P$ from both sides of Eq. $(2.26)$ gives Case $(12, 21)$ and multiplying $Q$ from both sides of Eq. $(2.26)$ gives Case $(21, 12)$.
Applying Case $(21, 12)$, we obtain \begin{eqnarray} \delta (A_{22}A_{21}A_{12})=\delta (A_{22}A_{21})A_{12}+A_{22}A_{21}\delta (A_{12}) \end{eqnarray}
Using Cases $(22, 22)$ and $(21, 12)$, we have \begin{eqnarray} & \delta (A_{22}A_{21}A_{12}) =\delta (A_{22})A_{21}A_{12}+A_{22}\delta (A_{21}A_{12})\nonumber \\ &~~~~~~~~~~= \delta (A_{22})A_{21}A_{12}+A_{22}\delta (A_{21})A_{12}+A_{22}A_{21}\delta (A_{12}) \end{eqnarray}
By $(2.27)$ and $(2.28)$, $\delta (A_{22}A_{21})A_{12} = \delta (A_{22})A_{21}A_{12}+A_{22}\delta (A_{21})A_{12}.$ Since $\mathcal A_{12}$ is faithful, we get $\delta (A_{22}A_{21})= \delta (A_{22})A_{21}+A_{22}\delta (A_{21}),$ completing the proof of Case $(21, 11)$. \end{proof}
\begin{corollary} Suppose $H$ is a Hilbert space and $C\in B(H)$ such that $\ker (C)\neq 0$ or $\ker (C^{\ast})\neq 0.$ If $\delta \in L(B(H), B(H))$ is Jordan derivable at $C$ then $\delta $ is a derivation. In particular, $C$ is a Jordan all-derivable point of $L(B(H), B(H)).$ \end{corollary} \begin{proof} If $\ker (C^{\ast})\neq 0$, then there exists a proper orthogonal projection $P\in B(H)$ such that $ran(C)\subseteq PH$. Let $Q=I-P$ then $QC=0$. Take $\mathcal A=\mathcal B =B(H)$, the one can check that all hypotheses of Theorem $2.9$ are satisfied and the conclusions follow.
If $\ker (C)\neq 0$, we can define $\delta ^{\ast}\in L(B(H), B(H))$
by $\delta ^{\ast}(A)=(\delta(A^{\ast}))^{\ast}$ for every $A\in B(H)$.
Since $\delta$ is Jordan derivable at $C$, we have $\delta ^{\ast}$ is
Jordan derivable at $C^{\ast}$. Now by the argument in the first paragraph
we have $\delta ^{\ast}$ is a derivation, and in turn $\delta$ is a derivation.
This completes the proof. \end{proof}
\section{Jordan higher derivable mappings}\
In this section, we assume that $\mathcal{A}$ is an algebra over a field $\mathbb{F}$ of characteristic zero. Before stating our main results in this section, we first need a proposition that characterizes Jordan higher derivations in terms of Jordan derivations. Since the proof is similar to the proof of \cite[Theorem 2.5]{Higher}, we omit it here.
\begin{proposition}\label{0005} Let $\mathcal{A}$ be an algebra, $\{d_i\}_{i\in\mathbb{N}}$ be a sequence of mappings on $\mathcal{A}$ with $d_0=I_\mathcal{A}$ and $\{\delta_i\}_{i\in\mathbb{N}}$ be the a sequences of (Jordan) derivations on $\mathcal{A}$ with $\delta_0=0$. If the following recursive relation holds: \begin{eqnarray*} nd_n=\sum_{k=0}^{n-1}\delta_{k+1}d_{n-1-k} \end{eqnarray*} for $n\geq1$, then $\{d_i\}_{i\in\mathbb{N}}$ is a (Jordan) higher derivation. \end{proposition}
Let $\mathcal R(\mathcal A)$ be a relation on $\mathcal A$, i.e. $\mathcal R(\mathcal A)$ is a nonempty subset of $\mathcal A \times \mathcal A$. We say $\delta\in L(\mathcal{A},\mathcal{M})$ is \textit{derivable on} $\mathcal R(\mathcal A)$ if $\delta(AB)=\delta(A)B+A\delta(B)$ for all $(A,B)\in \mathcal R(\mathcal{A})$. We say $\delta\in L(\mathcal{A},\mathcal{M})$ is \textit{Jordan derivable on} $\mathcal R(\mathcal A)$ if $\delta(AB+BA)=\delta(A)B+A\delta(B)+\delta(B)A+B\delta(A)$ for all $(A,B)\in \mathcal R(\mathcal{A})$. A sequence of mappings $\{d_i\}_{i\in\mathbb{N}}\in L(\mathcal{A})$ with $d_0=I_\mathcal{A}$ is called \textit{higher derivable on} $\mathcal R(\mathcal A)$ if $d_{n}(AB)=\sum_{i+j=n}d_i(A)d_j(B)$ for all $(A,B)\in \mathcal R(\mathcal{A})$. A sequence of mappings $\{d_i\}_{i\in\mathbb{N}}\in L(\mathcal{A})$ with $d_0=I_\mathcal{A}$ is called \textit{Jordan higher derivable on} $\mathcal R(\mathcal A)$ if $d_{n}(AB+BA)=\sum_{i+j=n}(d_i(A)d_j(B)+d_i(B)d_j(A))$ for all $(A,B)\in \mathcal R(\mathcal{A})$. We say $\mathcal R(\mathcal A)$ is \textit{(Jordan) derivational} for $L(\mathcal{A},\mathcal{M})$ if every (Jordan) derivable mapping on $\mathcal R(\mathcal A)$ is a (Jordan) derivation. We say $\mathcal R(\mathcal A)$ is \textit{(Jordan) higher derivational} for $L(\mathcal{A})$ if every (Jordan) higher derivable mapping on $\mathcal R(\mathcal A)$ is a (Jordan) higher derivation.
\begin{remark} \emph{The above definitions allow us to unify some of the notions in the literature. For example, in literature, there are two definitions of Jordan derivable mappings, one is what we use in this paper (see for example \cite{chen, ZHUJUN} and references therein), and the other (see for example \cite{Hou, Hou3, ZHUJUN1}) is what we call S-Jordan derivable mappings (S stands for standard). A mapping $\delta\in L(\mathcal{A},\mathcal{M})$ is called a \textit{S-Jordan derivable mapping at $C\in \mathcal{A}$} if $\delta(AB+BA)=\delta(A)B+A\delta(B)+\delta(B)A+B\delta(A)$ for all $A,B\in \mathcal{A}$ with $AB+BA=C$. An element $C\in \mathcal{A}$ is called a \textit{S-Jordan all-derivable point} if every S-Jordan derivable mapping at $C$ is a Jordan derivation. A sequence of mappings $\{d_i\}_{i\in\mathbb{N}}\in L(\mathcal{A})$ with $d_0=I_\mathcal{A}$ is called \textit{S-Jordan higher derivable at $C\in \mathcal{A}$} if $d_{n}(AB+BA)=\sum_{i+j=n}(d_i(A)d_j(B)+d_i(B)d_j(A))$ for all $A,B\in \mathcal{A}$ with $AB+BA=C$. An element $C\in \mathcal{A}$ is called a \textit{S-Jordan higher all-derivable point} if every sequence of S-Jordan higher derivable mappings at $C$ is a Jordan higher derivation. The above two notions of Jordan derivable mappings at $C$ are special case of Jordan derivable mappings on $\mathcal R(\mathcal A)$, where $\mathcal R(\mathcal A)=\{(A, B)\in \mathcal A \times \mathcal A: AB=C\}$ and $\mathcal R(\mathcal A)=\{(A, B)\in \mathcal A \times \mathcal A: AB+BA=C\}$, respectively.} \end{remark}
\begin{theorem} \label{0006} If $\mathcal{A}$ is an algebra such that $\mathcal R(\mathcal A)$ is (Jordan) derivational for $L(\mathcal{A})$, then $\mathcal R(\mathcal A)$ is (Jordan) higher derivational. \end{theorem}
\begin{proof} First, suppose $\mathcal R(\mathcal A)$ is Jordan derivational and $\{d_i\}_{i\in\mathbb{N}}$ is a sequence of mappings in $L(\mathcal{A})$ Jordan higher derivable on $\mathcal R(\mathcal A)$. Let $\delta_1=d_1$ and $\delta_n=nd_n-\sum_{k=0}^{n-2}\delta_{k+1}d_{n-1-k}$ for every $n(\geq2)\in \mathbb{N}.$ We will show $\{\delta_i\}_{i\in\mathbb{N}}$ is a sequence of Jordan derivations, and in turn $\{d_i\}_{i\in\mathbb{N}}$ is a Jordan higher derivation by Proposition \ref{0005}. We prove by induction.
When $n=1$, since $\mathcal R(\mathcal A)$ is Jordan derivational, we have that $\delta_1$ is a Jordan derivation.
Now suppose $\delta_k$ is defined as above and is a Jordan derivation for $k\leq n$. For $(A,B)\in \mathcal R(\mathcal{A})$, we have \begin{eqnarray*} &&\delta_{n+1}(A\circ B)=(n+1)d_{n+1}(A\circ B)-\sum_{k=0}^{n-1}\delta_{k+1}d_{n-k}(A\circ B)\nonumber\\ &&~~~~~~~~~~~~~~~=(n+1)\sum_{k=0}^{n+1}\{d_k(A)\circ d_{n+1-k}(B)\} -\sum_{k=0}^{n-1}\delta_{k+1}\sum_{l=0}^{n-k}\{d_l(A)\circ d_{n-k-l}(B)\}.\nonumber\\ \end{eqnarray*} By induction we have \begin{eqnarray*} &&\delta_{n+1}(A\circ B) =\sum_{k=0}^{n+1}kd_k(A)\circ d_{n+1-k}(B)+\sum_{k=0}^{n+1}d_k(A)\circ (n+1-k)d_{n+1-k}(B)\nonumber\\ &&~~~~~~~~~~~~~-\sum_{k=0}^{n-1}\sum_{l=0}^{n-k}\{\delta_{k+1}(d_l(A))\circ d_{n-k-l}(B)+d_l(A)\circ \delta_{k+1}(d_{n-k-l}(B))\}.\nonumber\\ \end{eqnarray*} Set \begin{eqnarray*} &&K_1=\sum_{k=0}^{n+1}kd_k(A)\circ d_{n+1-k}(B)-\sum_{k=0}^{n-1}\sum_{l=0}^{n-k}\delta_{k+1}(d_l(A))\circ d_{n-k-l}(B),\\ &&K_2=\sum_{k=0}^{n+1}d_k(A)\circ (n+1-k)d_{n+1-k}(B)-\sum_{k=0}^{n-1}\sum_{l=0}^{n-k}d_l(A)\circ \delta_{k+1}(d_{n-k-l}(B)).\\ \end{eqnarray*} Then $\delta_{n+1}(A\circ B)=K_1 +K_2$. Let us compute $K_1$ and $K_2$. If we put $r=k+l$ in the summation $\sum_{k=0}^{n-1}\sum_{l=0}^{n-k}$, then we may write it as $\sum_{r=0}^{n}\sum_{0\leq k\leq r, k\neq n}$. Hence \begin{eqnarray*} K_1=\sum_{k=0}^{n+1}kd_k(A)\circ d_{n+1-k}(B)-\sum_{r=0}^{n}\sum_{0\leq k\leq r, k\neq n}\delta_{k+1}(d_{r-k}(A))\circ d_{n-r}(B). \end{eqnarray*} Putting $r+1$ instead of $k$ in the first summation, we have \begin{eqnarray*} &&K_1+\sum_{k=0}^{ n-1}\delta_{k+1}(d_{n-k}(A))\circ B\\ &&~~~=\sum_{r=0}^{n}(r+1)d_{r+1}(A)\circ d_{n-r}(B)-\sum_{r=0}^{n-1}\sum_{k=0}^r\delta_{k+1}(d_{r-k}(A))\circ d_{n-r}(B)\\ &&~~~=\sum_{r=0}^{n-1}\{(r+1)d_{r+1}(A)-\sum_{k=0}^r\delta_{k+1}(d_{r-k}(A))\}\circ d_{n-r}(B)+(n+1)d_{n+1}(A)\circ B. \end{eqnarray*} By our assumption $(r+1)d_{r+1}(A)=\sum_{k=0}^r\delta_{k+1}(d_{r-k}(A))$ for $r=0, \ldots, n-1,$ we obtain \begin{eqnarray*} &&K_1=(n+1)d_{n+1}(A)\circ B-\sum_{k=0}^{ n-1}\delta_{k+1}(d_{n-k}(A))\circ B=\delta_{n+1}(A)\circ B.\\ \end{eqnarray*} Similary, we may deduce that \begin{eqnarray*} &&K_2=(n+1)A\circ d_{n+1}(B)-\sum_{k=0}^{ n-1}A\circ \delta_{k+1}(d_{n-k}(B))=A\circ \delta_{n+1}(B).\\ \end{eqnarray*} Therefore, $\delta_{n+1}$ is Jordan derivable on $\mathcal R(\mathcal A)$. Since $\mathcal R(\mathcal A)$ is Jordan derivational, we have that $\delta_{n+1}$ is a Jordan derivation.
Similarly, we can prove the case when $\mathcal R(\mathcal A)$ is assumed to be derivational by changing ``$\ \circ $" to the normal multiplication of $\mathcal A$. \end{proof}
Recall that a mapping $\delta\in L(\mathcal{A},\mathcal{M})$ is called \textit{derivable at $C\in \mathcal{A}$} if $\delta(AB)=\delta(A)B+A\delta(B)$ for all $A,B\in \mathcal{A}$ with $AB=C$. An element $C\in \mathcal{A}$ is called an \textit{all-derivable point} if every derivable mapping at $C$ is a derivation. A sequence of mappings $\{d_i\}_{i\in\mathbb{N}}\in L(\mathcal{A})$ with $d_0=I_\mathcal{A}$ is called \textit{higher derivable at $C\in \mathcal{A}$} if $d_{n}(AB)=\sum_{i+j=n}d_i(A)d_j(B)$ for all $A,B\in \mathcal{A}$ with $AB=C$. An element $C\in \mathcal{A}$ is called a \textit{higher all-derivable point} if every sequence of higher derivable mappings at $C$ is a higher derivation.
\begin{remark} \emph{Several authors (see for example \cite{Hou5,Hou6, Hou4, LI7,LI5,LU1, Hou2,Zhu2,Zhu3,Zhu4,ZHUJUN2}) investigate derivable mappings at 0, invertible element, left (right) separating point, non-trivial idempotent, and the unit $I$ on certain algebras. By Theorem \ref{0006}, we can generalize these results to the higher derivation case. Many authors also study (S-)Jordan derivable mappings (see for example \cite{ Hou, chen,Hou3,ZHUJUN1,ZHUJUN}) at these points. Theorems \ref{0006} also allow us to generalize these results to the (S-)Jordan higher derivation case. } \end{remark}
Combining Theorem \ref{0006} with Corollary \ref{0002}, we have
\begin{corollary}\label{0007} Suppose $\mathcal{A}$, $\mathcal{B}$ are as in Corollary \ref{0002} with $\mathcal{B}=\mathcal{A}$. Then every $C\in \mathcal{A}$ is a Jordan higher all-derivable point. \end{corollary}
Combining Theorem \ref{0006} with Theorem \ref{0003}, we have
\begin{corollary} Let $\mathcal{L}$ be a subspace lattice on a Banach space $X$ and $\mathcal{A}=alg\mathcal{L}$. If there exists a non-trivial idempotent $P\in \mathcal{A}$ such that $ran(P)\in \mathcal{L}$ and $PB(X)(I-P)\subseteq \mathcal{A}$, then every $C \in \mathcal{A}$ is a Jordan higher all-derivable point. \end{corollary}
Combining Theorem \ref{0006} with \cite[Theorem 3.3]{PAN}, we have
\begin{corollary} Suppose $\mathcal{A}$, $\mathcal{B}$ are as in Corollary \ref{0002} with $\mathcal{B}=\mathcal{A}$. Then every $0\neq C\in \mathcal{A}$ is a higher all-derivable point . \end{corollary} We say that $W$ in an algebra $\mathcal{A}$ is a left (or right) separating point of $\mathcal{A}$ if $WA=0$ (or $AW=0$) for $A \in \mathcal{A}$ implies $A=0.$ In \cite[Remark 1]{LI2}, the authors point out that if every Jordan derivation on a unital Banach algebra $\mathcal{A}$ is a derivation, then every linear mapping on $\mathcal{A}$ which is derivable at an arbitrary left or right separating point of $\mathcal{A}$ is a derivation. Together with Theorem \ref{0006}, we may generalize this result to the higher derivation case.
\begin{theorem} Let $\mathcal{A}$ be a unital Banach algebra such that every Jordan derivation on $\mathcal{A}$ is a derivation. Suppose that $W$ in $\mathcal{A}$ is a left or right separating point. If $D=(d_i)_{i\in\mathbb{N}}$ is a family of linear mappings higher derivable at $W$, then $D=(d_i)_{i\in\mathbb{N}}$ is a higher derivation. \end{theorem}
For any non-zero vectors $x\in X$ and $f\in X^*$, the rank one operator $x\otimes f$ is defined by $x\otimes f(y)=f(y)x$ for $y\in X$.
\begin{lemma}\label{3002} If $\mathcal{A}$ is a norm-closed subalgebra of $B(X)$ such that $\vee \{x: x\otimes f\in \mathcal A\}=X$ and $\wedge \{\emph{ker}(f): x\otimes f\in \mathcal A\}=(0)$, then every derivation $\delta$ from $\mathcal{A}$ into $B(X)$ is bounded. \end{lemma} \begin{proof} By the closed graph theorem, it is sufficient to show if $A_n\rightarrow A$ and $\delta(A_n)\rightarrow B$, as $n\rightarrow \infty$, then $\delta(A)= B$.
For any $x\otimes f, \ y\otimes g \in \mathcal{A}$, since \begin{align*} \delta(x\otimes f A_n y\otimes g)&=f(A_ny)\delta(x\otimes g)\\ &=x\otimes f\delta(A_n y\otimes g)+\delta(x\otimes f)(A_n y\otimes g)\\ &=x\otimes f(\delta(A_n)y\otimes g+A_n \delta(y\otimes g))+\delta(x\otimes f)(A_n y\otimes g),\\ \end{align*} we have \begin{eqnarray} (x\otimes f)\delta(A_n)(y\otimes g)=f(A_ny)\delta(x\otimes g)-(x\otimes f) A_n \delta(y\otimes g)-\delta(x\otimes f)(A_n y\otimes g).\label{3001} \end{eqnarray} Taking limit in (\ref{3001}) yields \begin{eqnarray*} (x\otimes f) B (y\otimes g)=(x\otimes f)\delta(A)(y\otimes g). \end{eqnarray*} Hence $f(By)=f(\delta(A))$. Thus $\delta(A)= B$. \end{proof}
By \cite{higher2}, if $\{d_i\}_{i\in\mathbb{N}}$ is a Jordan higher derivation on an algebra $\mathcal{A}$, then there is a sequence $\{\delta_i\}_{i\in\mathbb{N}}$ of Jordan derivations on $\mathcal{A}$ such that \begin{eqnarray*} d_n=\sum_{i=1}^n\left(\sum_{\sum_{j=1}^i r_j=n}\left(\prod_{j=1} ^ i \frac{1}{r_j+\cdots r_i} \right)\delta_{r_1}\ldots\delta_{r_i} \right), \end{eqnarray*} where the inner summation is taken over all positive integers $r_j$ with $\sum_{j=1}^i r_j=n$. This together with Lemma \ref{3002} leads to the following Theorem.
\begin{theorem}\label{3003} If $\mathcal{A}$ is a norm-closed subalgebra of $B(X)$ such that $\vee \{x: x\otimes f\in \mathcal A\}=X$ and $\wedge \{\emph{ker}(f): x\otimes f\in \mathcal A\}=(0)$, then every Jordan higher derivation on $alg\mathcal{L}$ is bounded. \end{theorem} \begin{proof} Since every Jordan derivation on $\mathcal{A}$ is a derivation by \cite[Theorem 4.1]{LI7}. \end{proof}
For a subspace lattice $\mathcal{L}$ of a Banach space $X$ and for $E\in \mathcal{L}$, define $$E_-=\vee\{F\in \mathcal{L}: F\nsupseteq E\}.$$ Put $$\mathcal{J}(\mathcal{L})=\{K\in \mathcal{L}: K\neq(0)~\mathrm{and}~K_-\neq X\}.$$
\begin{remark} \emph{ It is well known (see \cite{ORO}) that $x\otimes f\in \textrm{alg}\mathcal{L}$ if and only if there exists some $K\in \mathcal{J}(\mathcal{L})$ such that $x\in K$ and $f\in K_-^\perp$. It follows that if a subspace lattice $ \mathcal{L}$ satisfies $\vee\{K: K\in \mathcal{J}(\mathcal{L})\}=H$ and $\wedge\{K_-: K\in \mathcal{J}(\mathcal{L})\}=(0)$, then $alg\mathcal{L}$ satisfies the hypothesis of Theorem \ref{3003}. Such subspace lattices include completely distributive subspace lattices, $\mathcal{J}$-subspace lattices, and subspace lattices with $H_-\neq H$ and $(0)_+\neq (0)$. Recall that (see \cite{SRL}), a subspace lattice $\mathcal{L}$ is called \textit{completely~distributive} if $L=\vee\{E\in \mathcal{L}: E_-\ngeq L\}$ and $L=\wedge\{E_-: E\in \mathcal{L}~\mathrm{and}~E\nleq L\}$ for all $L\in \mathcal{L}$. It follows that completely distributive subspace lattices satisfy the conditions $\vee\{K: K\in \mathcal{J}(\mathcal{L})\}=H$ and $\wedge\{K_-: K\in \mathcal{J}(\mathcal{L})\}=(0)$. A subspace lattice $\mathcal{L}$ is called a \textit{$\mathcal{J}$-subspace lattice} on $H$ if $\vee\{K: K\in \mathcal{J}(\mathcal{L})\}=H$, $\wedge\{K_-: K\in \mathcal{J}(\mathcal{L})\}=(0)$, $K\vee K_-=H$ and $K\wedge K_-=(0)~\mathrm{for~any}~K\in \mathcal{J}(\mathcal{L})$.} \end{remark}
On October 22, 2011, in the Third Operator Theory and Operator Algebras Conference of China, the first author reported main results of the paper.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Incidence-free sets and edge domination in incidence graphs}
\begin{abstract}
A set of edges $\Gamma$ of a graph $G$ is an edge dominating set if every edge of $G$ intersects at least one edge of $\Gamma$, and the edge domination number $\gamma_e(G)$ is the smallest size of an edge dominating set. Expanding on work of Laskar and Wallis, we study $\gamma_e(G)$ for graphs $G$ which are the incidence graph of some incidence structure $D$, with an emphasis on the case when $D$ is a symmetric design. In particular, we show in this latter case that determining $\gamma_e(G)$ is equivalent to determining the largest size of certain incidence-free sets of $D$. Throughout, we employ a variety of combinatorial, probabilistic
and geometric techniques, supplemented with tools from spectral graph theory.
\end{abstract}
\noindent {\bf Keywords:} edge domination, design, incidence structure, matching, incidence-free sets. \\
{\bf MSC2020 Classification:} 05B05, 05C70.
\section{Introduction}
Roughly speaking, the area of \textit{extremal combinatorics} centres around problems which ask how large or small a given combinatorial object can be under certain restrictions. For example, Mantel's theorem states that the maximum number of edges of an $n$-vertex triangle-free graph is $\floor{n^2/4}$. In this paper, the extremal questions we consider are domination-type problems, which roughly involves studying the minimum number of vertices or edges one needs to ``cover'' a graph $G$. Below we recall the basic definitions for domination, and we refer the interested reader to the book by Haynes, Hedetniemi, and Slater~\cite{haynes2013fundamentals} for more on this subject. \\
A \emph{dominating set} in a graph $G = (V,E)$ is a set of vertices $S \subseteq V$ such that each vertex is either contained in $S$, or has a neighbour in $S$.
The size of the smallest dominating set is called the \emph{domination number} of $G$ and is denoted as $\gamma(G)$. A large body of work is dedicated to studying $\gamma(G)$ as well as its many variants such as roman domination \cite{cockayne2004roman} and total domination \cite{henning2013total}.
A related concept is an \emph{edge dominating set} in a graph $G=(V,E)$.
This is a subset $\Gamma \subseteq E$ of edges, such that for each edge $e \in E$, there exists an edge $e' \in \Gamma$ with $e' \cap e \neq \emptyset$.
We note that this could be equivalently defined as a dominating set in the line graph of $G$.
The \emph{edge domination number} of $G$ is defined as the size of the smallest edge dominating set, and will be denoted by $\gamma_e(G)$. Again a large body of work is dedicated to studying $\gamma_e(G)$, see for example \cite{horton1993minimum,gavril}.\\
In this paper we study domination type problems for graphs which come from incidence structures. An \emph{incidence structure} is a pair $D = (\mathcal P,\mathcal B)$, such that $\mathcal B$ is a collection of subsets of $\mathcal P$.
We say that $P \in \mathcal P$ and $B \in \mathcal B$ are \emph{incident} if $P \in B$.
The elements of $\mathcal P$ and $\mathcal B$ are called \emph{points} and \emph{blocks} respectively.
We say that $D$ is of \emph{type} $(v,b,k,r,\lambda)$ if
\begin{itemize}
\item $|\mathcal P| = v$, and $|\mathcal B| = b$,
\item every block is incident with $k$ points, every point is incident with $r$ blocks,
\item $\lambda \ge 1$ is the maximum number such that there exist two distinct points $P$ and $Q$ which have $\lambda$ blocks incident with both $P$ and $Q$.
\end{itemize}
The convention $\lambda\ge 1$ is somewhat non-standard and is adopted to exclude trivial incidence structures. From these conditions, we have $v r = b k$ and $(v-1)\lambda \geq r(k-1)$.
If $v=b$ (which implies $r=k$), and if $\lambda$ is also the maximum number of points incident with two distinct blocks, then we call $D$ a \emph{symmetric incidence structure}, abbreviated SIS, of type $(v,k,\lambda)$.
If any two distinct points of an incidence structure are incident with exactly $\lambda$ blocks, then we call $D$ a $(v,k,\lambda)$-\emph{design}, and in this case
\begin{align}
\label{EqDesigns}
v r = b k, && (v-1)\lambda = r(k-1).
\end{align}
Fisher's inequality states that in this case we always have $v \leq b$.
In case of equality, i.e.\ if $D$ is a design with $v=b$, then any two blocks of $D$ intersect in exactly $\lambda$ points, and $D$ is called a \emph{symmetric design}. We refer the reader to \cite{colbourndinitz} for proofs of these statements and other background on designs.
The \emph{incidence graph} $I(D)$, also called the \emph{Levi graph}, of an incidence structure $D=(\mathcal P,\mathcal B)$ is the bipartite graph with bipartition $\mathcal P$ and $\mathcal B$, where a point $P$ and a block $B$ are adjacent if and only if they are incident. \\
It turns out that the problem of determining the domination number of $I(D)$ is closely related to studying the smallest blocking sets and covers in $D$. Partially because of this, there have been several papers in recent years investigating the domination number of incidence graphs of block designs and other incidence structures.
See, for example, \cite{goldberg, tang} for incidence graphs of designs, \cite{hegernagy} for incidence graphs of projective planes, and \cite{hegerhernandezlucas} for incidence graphs of generalised quadrangles.
Some work for edge domination numbers of incidence graphs of projective planes was done by Laskar and Wallis \cite{laskar}, but outside of this very little is known about edge domination numbers of incidence graphs. Our goal for this paper is to expand upon this literature by focusing on the following problem:
\begin{center}
\textit{Given an incidence structure $D$, determine the edge domination number of its incidence graph $I(D)$.}
\end{center} For convenience's sake, we will denote this quantity by $\gamma_e(D)$ instead of $\gamma_e(I(D))$. \\
We prove a number of general bounds for $\gamma_e(D)$ for incidence structures throughout this paper. We highlight one such result below, which shows that for symmetric designs $D$, determining $\gamma_e(D)$ is equivalent to determining the maximum size of certain incidence-free sets.
\begin{df}\label{def:incidencefree}
Let $X \subseteq \mathcal P$ and $Y \subseteq \mathcal B$ be such that there are no incidences between points in $X$ and blocks in $Y$. In this case the pair $(X,Y)$ will be called \emph{incidence-free}. If $|X| = |Y|$ we will say that the pair is \emph{equinumerous} and refer to $|X|$ as its size.
\end{df}
We note that the problem of determining the size of large incidence-free subsets of incidence structures is a well-studied problem, see for example \cite{dewinterschillewaertverstraete,mattheuspavesestorme,mubayiwilliford,stinson}. As such, the following connection to $\gamma_e(D)$ and incidence-free sets is of particular interest.
\begin{thm}
\label{ThmMainIntro}
Let $D=(\mathcal P,\mathcal B)$ be a symmetric $(v,k,\lambda)$-design with $k\ge 100$. Then
\[\gamma_e(D)=v-\alpha,\]
where $\alpha$ is the maximum size of an equinumerous incidence-free pair $(X,Y)$.
\end{thm}
The bound $k\ge 100$ can easily be improved, but we do not optimise this value since, in particular, \Cref{EqDesigns} implies there exist only a finite number of symmetric $(v,k,\lambda)$-designs with $k < 100$. It is easy to show that \Cref{ThmMainIntro} can not be strengthened to hold for arbitrary SIS's, but a somewhat analogous statement does hold if one assumes some extra mild conditions on an SIS; see \Cref{ThmSISMatching} for more on this.
\subsection{Organisation}
The rest of the paper is organised as follows.
The first half of the paper is dedicated to general techniques and results, with preliminaries established in \Cref{SecPrel}, lower bounds in \Cref{sec:lowerbounds}, upper bounds in \Cref{SecConstructions}, and a proof of \Cref{ThmMainIntro} in \Cref{sec:incidencefreeishall}.
The last half of the paper, \Cref{SecParticularClasses}, establishes bounds on specific classes of incidence structures, both by applying results from the first half of the paper and by utilising algebraic structures and symmetries particular to the individual incidence structures. Some specific classes of designs we look at include those coming from projective planes and Hadamard matrices, and in many of these cases we establish asymptotically sharp bounds. Our examples cover almost all known basic constructions of symmetric designs (see \cite[Part II Chapter 6.8]{colbourndinitz}), with the notable exception of symmetric designs coming from difference sets, as there seems to exist no uniform way of establishing effective bounds in this case. The problem of solving this remaining case, as well as a number of other open problems, is discussed in \Cref{SecConclusion}.
\section{Preliminaries}
\label{SecPrel}
The following key observation allows us to translate upper bounds for sizes of incidence-free sets into lower bounds for $\gamma_e(D)$. This will be the driving force behind almost all of the lower bounds throughout this paper.
\begin{lm}\label{EdgeDomToIncidenceFree}
If $D = (\mathcal P,\mathcal B)$ is an incidence structure with $|\mathcal P|=v$ and $|\mathcal B| = b$, then there is an incidence-free set $(X,Y)$ with $|X| \geq v-\gamma_e(D)$ and $|Y| \geq b-\gamma_e(D)$.
\end{lm}
\begin{proof}
Let $\Gamma$ be an edge dominating set of size $\gamma_e(D)$. Let $X\subseteq \mathcal P$ denote the set of points which are not contained in any edge of $\Gamma$, and similarly define $Y\subseteq \mathcal B$. Note that $(X,Y)$ is incidence-free by definition of $\Gamma$ being an edge dominating set. Since there are at most $|\Gamma|$ points of $\mathcal P$ in an edge of $\Gamma$, we have
\[|X|\ge |\mathcal P|-|\Gamma|=v-\gamma_e(D),\]
and the same argument gives the desired bound for $Y$.
\end{proof}
To construct small edge dominating sets, we rely on matchings in $I(D)$. Recall that a \emph{matching} in a graph $G=(V,E)$ is a subset $M \subseteq E$ such that each vertex of $G$ lies in at most one edge of $M$.
A matching is called \emph{maximal} if it cannot be extended to a larger matching.
It is not hard to see that a matching is maximal if and only if it is an edge dominating set, and an observation of Yannakakis and Gavril~\cite{gavril} implies that $\gamma_e(G)$ is equal to the smallest size of a maximal matching of $G$.
We say that a matching $M$ of $G=(V,E)$ \emph{covers} a subset $S$ of vertices if each element of $S$ is contained in an element of $M$. A \emph{perfect matching} is a matching that covers all vertices exactly once. Note that perfect matchings are maximal matchings and edge dominating sets. A standard but important fact about matchings that we need is the following easy consequence of Hall's theorem, where we recall that a graph is \textit{biregular} if it has a bipartition $U\cup V$ such that every vertex within $U$ has the same degree and every vertex within $V$ has the same degree.
\begin{lm}\label{LmBiregular}
If $G$ is a biregular graph with bipartition $U\cup V$ such that $|U|\le |V|$, then $G$ has a matching which covers $U$.
\end{lm}
We can use this to show when a trivial upper bound on $\gamma_e(D)$ is tight for designs.
\begin{lm}
\label{LmLargeRepNumber}
If $D = (\mathcal P,\mathcal B)$ is an incidence structure with $|\mathcal P|=v$, then
\[\gamma_e(D)\le v.\]
If $D$ is a $(v,k,\lambda)$-design, then equality holds if and only if $r \geq v$.
\end{lm}
\begin{proof}
To prove the upper bound, we can assume without loss of generality that every point $P\in \mathcal P$ is incident to at least one block $B_p\in \mathcal B$. Then $\set{(P,B_P)}_{P\in \mathcal P}$ is an edge dominating set of $I(D)$ of size $v$, which shows $\gamma_e(D) \leq v$. From now on we suppose $D$ is a design.
First consider the case $r\ge v$. Let $\Gamma$ be a smallest edge dominating set of $I(D)$. If every point in $\mathcal P$ is contained in an edge of $\Gamma$, then $\gamma_e(D)=|\Gamma|\ge |\mathcal P|=v$. Thus we may assume there exists some point $P$ not contained in an edge of $\Gamma$. Since $\Gamma$ is an edge dominating set, this implies that all of the $r$ blocks incident to $P$ are contained in an edge of $\Gamma$. Thus $\gamma_e(D)\ge r\ge v$, proving the `if' statement.
Now consider the case $r<v$, and let $\mathcal B_P$ denote the set of $r$ blocks incident to a point $P \in \mathcal P$.
The incidence graph of $(\mathcal P \setminus \set P, \mathcal B_P)$ is biregular, hence it has a matching $\Gamma'$ covering $\mathcal B_P$ by \Cref{LmBiregular}. Since $|\Gamma'|=r<v$, one can find a set of edges $\Gamma$ which contains $\Gamma'$ and is such that each point of $\mathcal P\setminus \set P$ is contained in exactly one edge of $\Gamma$ (namely, by arbitrarily adding to $\Gamma$ an edge containing each point not covered by $\Gamma'$). This is an edge dominating set of size $v-1$, completing the proof of the if and only if statement.
\end{proof}
We note that the equality case of \Cref{LmLargeRepNumber} partially motivates our focus on studying symmetric incidence structures where the number of points is comparable to the number of blocks. However, even in this setting the trivial upper bound of \Cref{LmLargeRepNumber} is often close to the true value. This is perhaps not too surprising given that the incidence graphs of designs are `expanding' in the sense of Thomason \cite[Section 3.3]{thomason}. Roughly speaking, this means that small sets of points are incident with a relatively large number of blocks and vice versa. This implies that edges are `well-spread' and do not concentrate on small sets of vertices. It is thus reasonable to think that in order to dominate all edges, we need a fairly large number of them. This is indeed the case as we will see.
\section{Lower bounds}\label{sec:lowerbounds}
To start this section, we will give a lower bound on the number of blocks incident with at least one point of a certain point set.
For designs, this can be done using eigenvalue techniques, as was done by Haemers \cite[Corollary 5.3]{haemers}. We will revisit this idea in \Cref{SubsecSemibiplanes}.
This same result can also be proved using simple counting techniques.
Stinson \cite[Theorem 3.1]{stinson} and De Winter, Schillewaert, Verstraëte \cite[Theorem 3]{dewinterschillewaertverstraete} independently bound the largest number of non-incident points and blocks in a projective plane.
Later, Elvey Price, Adib Surani, Zhou \cite[Lemma 3]{elveyprice} gave a combinatorial proof for Haemers' bound for symmetric designs.
However, their arguments can be generalised in a straightforward way to hold for all incidence structures of type $(v,b,k,r,\lambda)$.
For the sake of completeness, we give such a proof.
\begin{df}
A \emph{maximal arc (of order n)} in an incidence structure of type $(v,b,k,r,\lambda)$ is a non-empty set of points $S$ such that every block intersects $S$ in either 0 or $n$ points, for some integer $n$.
We call a maximal arc \emph{trivial} if it consists of one point, all points, or the complement of a block\footnote{The complement of a block $B$ is a maximal arc if and only if every other block intersects $B$ in a constant number of points.
This is the case in symmetric designs.}.
\end{df}
Suppose that $D$ is a $(v,k,\lambda)$-design, and that $S$ is a maximal arc of order $n$ in $D$.
Fix a point $P \in S$.
Then there are $r$ blocks incident to $P$, each containing $n-1$ other points of $S$.
Since each point of $S \setminus \set P$ is incident with $\lambda$ blocks incident to $P$, this implies that $|S| = 1 + r \frac{n-1}\lambda$, or equivalently that the order of a maximal arc $S$ equals $1+\frac{|S|-1}{r}\lambda$.
Now take a point $Q \notin S$.
Consider the set $\sett{(P,B) \in S \times \mathcal B}{P,Q \in B}$.
By performing a double count on this set, we see that there are $(1 + r \frac{n-1} \lambda) \frac \lambda n = r - \frac{r-\lambda}n$ blocks incident with $Q$ containing $n$ points of $S$.
So, if $T$ denotes the set of blocks not incident with any point of $S$, then any point is incident with either 0 or $\frac{r-\lambda}n$ blocks of $T$.
We call $T$ the \emph{dual arc} of $S$.
\begin{lm}
\label{LemBlocksThroughPoints}
Let $D = (\mathcal P,\mathcal B)$ be an incidence structure of type $(v,b,k,r,\lambda)$.
For any set $S \subseteq \mathcal P$, there are at least $\frac{r^2|S|}{r + (|S| - 1)\lambda}$ blocks which intersect $S$.
Equality holds in designs if and only if $S$ is a maximal arc of order $1 + \frac{|S|-1} r \lambda$.
\end{lm}
\begin{proof}
Let $x$ be the number of blocks intersecting $S$, and let $n_i$ be the number of blocks intersecting $S$ in exactly $i$ points.
Then the so-called standard equations tell us that
\begin{align*}
\sum_{i>0} n_i = x, &&
\sum_{i>0} i n_i = r|S|, &&
\sum_{i>0} i^2 n_i \leq |S|(r + (|S|-1) \lambda),
\end{align*}
by performing a double count on $\sett{ (P,B) \in S \times \mathcal B}{P \in B}$ and $\sett{ (P,Q,B) \in S \times S \times \mathcal B}{ P,Q \in B}$. Using these equations we find
\begin{align*}
0 & \leq \sum_{i>0} \left(i - \left( 1 + \frac{|S|-1} r \lambda \right) \right)^2 n_i \\
&= \sum_{i>0} i^2 n_i - 2 \left( 1 + \frac{|S|-1} r \lambda \right) \sum_{i>0}i n_i + \left( 1 + \frac{|S|-1} r \lambda \right)^2 \sum_{i>0} n_i \\
& \leq |S|(r + (|S|-1) \lambda) - 2 \left( 1 + \frac{|S|-1} r \lambda \right) r|S| + \left( 1 + \frac{|S|-1} r \lambda \right)^2 x \\
\iff & \left( 1 + \frac{|S|-1} r \lambda \right)^2 x \geq 2 \left( 1 + \frac{|S|-1} r \lambda \right) r |S| - |S|(r + (|S|-1) \lambda) \\
\iff & \,\,x \geq \frac{r|S|}{1 + \frac{|S|-1} r \lambda} = \frac{r^2|S|}{r + (|S|-1) \lambda}.
\end{align*}
Equality holds if and only if $n_i$ is only non-zero for $i = 0$ and $i = 1 + \frac{|S|-1} r \lambda$ and any two points of $S$ determine $\lambda$ blocks, which is always true in designs.
This is equivalent to $S$ being a maximal arc of order $1 + \frac{|S|-1} r \lambda$.
\end{proof}
We can do a bit better for small values of $|S|$, which we will need later on.
\begin{lm}
\label{LmIneqCap}
Let $D = (\mathcal P,\mathcal B)$ be an incidence structure of type $(v,b,k,r,\lambda)$.
For any set $S \subseteq \mathcal P$, there are at least $r|S| - \lambda \binom{|S|}{2}$ blocks which intersect $S$.
Equality holds in designs if and only if no three (or more) points of $S$ are incident with the same block.
\end{lm}
\begin{proof}
We can recycle the definition of $x$, $n_i$ and the inequalities above to compute
\begin{align*}
0 &\leq \sum_{i > 0}(i-1)(i-2)n_i \\
& = \sum_{i>0} i^2 n_i - 3 \sum_{i>0}i n_i + 2 \sum_{i>0} n_i \\
& \leq |S|(r+(|S|-1)\lambda)-3r|S|+2x \\
&\Leftrightarrow x \geq r|S|-\lambda\binom{|S|}{2}
\end{align*}
In case of equality, we see that only $n_1$ and $n_2$ can be non-zero, which implies the last statement.
\end{proof}
One can check that Lemma \ref{LmIneqCap} yields a better bound than Lemma \ref{LemBlocksThroughPoints} if and only if $|S| < 1 + \frac r \lambda $. In fact, a set of points with no three in a block can have at most $1 + r/\lambda$ points, and equality holds if and only if it is a maximal arc of order 2. In that case, both bounds coincide.
We note that in \Cref{sec:incidencefreeishall} we will similarly want to estimate the number of blocks incident with a small set of points. There we will use a greedy approach, which would work equally well to prove the lemma above. \\
We will now use the first bound combined with \Cref{EdgeDomToIncidenceFree} to obtain lower bounds on the edge domination number of designs.
We restrict ourselves to designs instead of general incidence structures both for ease of presentation and the fact that we do not need the general bounds in the remainder of this text.
\begin{prop}
\label{PropUpperBoundIF}
Let $D = (\mathcal P,\mathcal B)$ be a $(v,k,\lambda)$-design and let $(X,Y)$ be an incidence-free pair satisfying $v - |X| = b - |Y|$. Then
\begin{enumerate}[(1)]
\item \( \displaystyle |X| \leq r \frac {k-r-2 + \sqrt{(r-k)^2 + 4(r-\lambda)}}{2\lambda} + 1 \).
Equality holds if and only if $X$ is a maximal arc of order $\displaystyle \frac{k - r + \sqrt{(r-k)^2 + 4(r-\lambda)}}2$ and $Y$ is its dual arc.
\item If $D$ is a symmetric design, then \( \displaystyle |X| \leq k \frac{\sqrt{k-\lambda} - 1}{\lambda} + 1 \).
Equality holds if and only if $X$ is a maximal arc of order $\sqrt{k-\lambda}$ and $Y$ is its dual arc.
\end{enumerate}
\end{prop}
\begin{proof}
(1) Since $Y$ contains no blocks incident to $X$, there are at most $b - |Y| = v- |X|$ blocks incident with a point in $X$.
By Lemma \ref{LemBlocksThroughPoints}, this implies that
\[
\frac{r^2 |X|}{r + (|X|-1)\lambda} \leq v - |X|.
\]
This gives us an equation of the form $f(|X|) \leq 0$, where $f$ is a quadratic polynomial with positive leading coefficient.
Therefore, $|X|$ is at most the largest root of $f$.
Calculating this root is tedious, but straightforward, and yields the bound from the Proposition, relying on \Cref{EqDesigns}.
Now suppose that equality holds.
Then we attain equality in the bound from \Cref{LemBlocksThroughPoints}, which implies that $X$ is a maximal arc.
In this case, one can calculate from the size of $X$ that it is a maximal arc of order $(k - r + \sqrt{(r-k)^2 + 4(r-\lambda)}) / 2$.
Necessarily, $Y$ must consist of all blocks missing $X$, i.e.\ $Y$ must be the dual arc of $X$.
(2) This follows immediately from (1) using $k=r$.
\end{proof}
\begin{crl}
\label{CrlDesignDomIneqs}
Let $D=(\mathcal P,\mathcal B)$ be a $(v,k,\lambda)$-design, then
\begin{enumerate}
\item[(1)] $ \displaystyle\gamma_e(D) \geq r \frac{r + k - \sqrt{ (r-k)^2 + 4(r-\lambda)}}{2 \lambda}$.
Equality holds if and only if $D$ has a maximal arc of order $\displaystyle \frac{k - r + \sqrt{(r-k)^2 + 4(r-\lambda)}}2$.
\item[(2)] If $D$ is a symmetric design, then $\gamma_e(D) \geq \displaystyle k\frac{k - \sqrt{k-\lambda}} \lambda$.
Equality holds if and only if $D$ has a maximal arc of order $\sqrt{k-\lambda}$.
\end{enumerate}
\end{crl}
\begin{proof}
By \Cref{EdgeDomToIncidenceFree}, there exists an incidence-free pair $(X,Y)$ with $|X| \geq v-\gamma_e(D)$, and $|Y| \geq b-\gamma_e(D)$.
Hence, we can find subsets $X' \subseteq X$ and $Y' \subseteq Y$ of size $v-\gamma_e(D)$ and $b-\gamma_e(D)$ respectively.
Note that $(X',Y')$ is necessarily also incidence-free.
Plugging this into \Cref{PropUpperBoundIF} yields the desired bound.
Vice versa, suppose that $D$ has a maximal arc $X$ of order $n = (k - r + \sqrt{(r-k)^2 + 4(r-\lambda)})/2$.
Let $Y$ denote its dual arc.
Consider the induced subgraph $H$ of $I(D)$ on $(\mathcal P \setminus X) \cup (\mathcal B \setminus Y)$.
Then every vertex in $\mathcal P \setminus X$ has degree $r - \frac{r-\lambda}n$, and every vertex in $Y$ has degree $k-n$, i.e.\ $H$ is biregular.
Furthermore, from equality in \Cref{PropUpperBoundIF}, it follows that $|\mathcal P \setminus X| = |\mathcal B \setminus Y|$.
Thus, $H$ is a regular bipartite graph, and hence has a perfect matching by \Cref{LmBiregular}.
Since there are no edges between $X$ and $Y$ in $I(D)$, this matching is an edge dominating set of size $v - |X|$.
It is straightforward to check that $v-|X|$ equals the right-hand side of the above bound.
\end{proof}
\begin{rmk}
\label{RmkBoundDomSIS}
\begin{enumerate}
\item Recall that for a symmetric $(v,k,\lambda)$-design we have $r=k$ and $v = k(k-1)/\lambda + 1$, so (2) can be restated as \[\gamma_e(D) \geq v - \frac{k\sqrt{k-\lambda}-k}{\lambda}-1.\] This indeed shows that $\gamma_e(D)$ is equal to $v$ up to lower order terms.
\item Laskar and Wallis \cite{laskar} showed that if $D$ is a symmetric $(n^2+n+1,n+1,1)$ design one has $\gamma_e(D) > \half (n^2+3n)$. Our results improve this to $\gamma_e(D) \geq n^2-n\sqrt{n}+2n-\sqrt{n}+1$.
\item One could try to improve the bounds above by studying the existence of maximal arcs in designs. This is however not feasible to do in general: even for the class of projective planes defined over a finite field $\mathbb F_q$, which are a special type of symmetric $(q^2+q+1,q+1,1)$-designs, the existence of maximal arcs depends on the parity of $q$ as we will see later. Improvements are thus only possible by restricting oneself to particular classes of designs.
\end{enumerate}
\end{rmk}
\section{Constructing large incidence-free sets}\label{SecConstructions}
In view of Theorem~\ref{ThmMainIntro}, upper bounding $\gamma_e(D)$ when $D$ is a symmetric design is equivalent to constructing large incidence-free pairs. Here we discuss two general techniques for achieving this goal.
\subsection{A probabilistic construction}
In a symmetric $(v,k,\lambda)$-design, one can greedily construct sets $X\subseteq \mathcal P,\ Y\subseteq \mathcal B$ such that no point in $X$ is contained in a block of $Y$ and such that $|X|=|Y|\ge v/(k+1)\approx k/\lambda$. One can improve this bound somewhat through a simple probabilistic argument, which is based on a similar proof by De Winter, Schillewaert, and Verstraëte \cite[Theorem 6]{dewinterschillewaertverstraete}.
\begin{prop}\label{PropProbability}
Let $D=(\mathcal P,\mathcal B)$ be an SIS of type $(v,k,\lambda)$ with $\lambda\le \frac{k}{64\log k}$.
Then there exists an equinumerous incidence-free pair whose size is at least $\frac{k \log k}{2 \lambda}$.
\end{prop}
\begin{proof}
Let $X\subseteq \mathcal P$ be a random set of points obtained by including each point in $\mathcal P$ independently with probability $p=\frac{\log k}{2k}$, and let $Y\subseteq \mathcal B$ consist of the blocks which do not contain any points of $X$. Note that by construction no point in $X$ is contained in a block of $Y$, so it suffices to show that with positive probability $|X|,|Y|\ge \frac{k\log k}{\lambda}$.
First observe that \begin{equation}\mathbb{E}[|X|]=p v\ge \frac{\log k}{2k}\cdot \frac{k(k-1)}{\lambda}\ge \frac{k \log k}{4\lambda } \ge 16,\label{EqXExpectation}\end{equation}
where the second to last step used $k\ge 2$, which is implicit from the bounds $1\le \lambda\le \frac{k}{64\log k}$, and the last step used $\lambda\le \frac{k}{64\log k}$. Since $|X|$ is a binomial random variable, by the Chernoff bound we have
\begin{equation}\label{EqXConcentration}
\Pr\left[|X|\le \frac{1}{2}\mathbb{E}[|X|]\right]\le e^{-\mathbb{E}[|X|]/8}< \frac12.
\end{equation}
For $B\in \mathcal B$, let $A_B$ denote the event that $B\in Y$. Then
\[\Pr[A_B]=(1-p)^k\ge e^{-kp/(1-p)}\ge k^{-1/4} ,\]
where this first inequality follows from exponentiating the inequality $ \log(1+x)\ge \frac{x}{1+x}$, and the last used $1-p\ge 1/2$. As $v \geq \frac{k(k-1)}{\lambda}+1$, we conclude that
\begin{equation}\mathbb{E}[|Y|]=\sum_{B\in \mathcal B} \Pr[A_B]\ge \frac{k\log k}{2\lambda}.\label{EqYExpectation}\end{equation}
Note that for any $B,B'\in \mathcal B$ (possibly non-distinct), we have
\[\Pr[A_B\cap A_{B'}]\le (1-p)^{2k-\lambda}=(1-p)^{-\lambda} \Pr[A_B]\Pr[A_{B'}]\le e^{\lambda p/(1-p)}\Pr[A_B]\Pr[A_{B'}]\le e^{1/32} \Pr[A_B]\Pr[A_{B'}],\]
where the second inequality used $\log(1+x)\ge \frac{x}{1+x}$ as before, and the second used $\lambda p\le 1/64$ and $1-p\ge 1/2$ by hypothesis. Combining these two results gives
\[\mathrm{Var}(|Y|)=\sum_{B,B'\in \mathcal B} \Pr[A_B\cap A_{B'}]-\Pr[A_B]\Pr[A_B']\le \left(e^{1/32}-1\right) \mathbb{E}[|Y|]^2\le .04 \mathbb{E}[|Y|]^2.\]
Using this, Chebyshev's inequality and denoting $Z = |Y|$ for readability, we have
\begin{equation}
\Pr \left(Z\le \frac{1}{2} \mathbb{E}[Z] \right)\le
\Pr \left[ |Z-\mathbb{E}[Z]| \geq \half \mathbb{E}[Z] \right] \le
\frac{\mathrm{Var}(Z)}{\frac{1}{4}\mathbb{E}[Z]^2}\le \frac{4}{25}.\label{EqYConcentration}
\end{equation}
By \eqref{EqXExpectation}, \eqref{EqXConcentration}, \eqref{EqYExpectation} and \eqref{EqYConcentration} we conclude with positive probability we have $|X|,|Y|\ge \frac{k \log k}{2\lambda } $, and hence there exists some pair of sets $X,Y$ of at least this size such that no point in $X$ is contained in a block of $Y$.
Taking subsets of $X,Y$ of size $\min\{|X|,|Y|\}\ge \frac{k \log k}{2\lambda }$ gives the result.
\end{proof}
For symmetric designs, this probabilistic construction is suboptimal for proving asymptotically sharp bounds on the edge domination number, and it is reasonable to think that more advanced techniques might provide better constructions.
However, \cite[Problem 7]{dewinterschillewaertverstraete} suggests that improving this result to $k^{1+\varepsilon}/\lambda$ for any $\varepsilon > 0$ might be hard when $\lambda$ is small, as it is related to an open and difficult conjecture by Erd\H{o}s on $C_4$-free graphs and its generalisation to $K_{2,t}$-free graphs.
This result (and its limitations mentioned above) motivates the need to look at particular classes of symmetric designs in order to do better, which is exactly what we will do in Section~\ref{SecParticularClasses}.
\subsection{From polarities}
In this section we describe a technique to construct equinumerous incidence-free pairs in symmetric designs with special kinds of symmetries, such as the Hadamard designs and those corresponding to projective planes over finite fields.
\begin{df}
A \emph{polarity} of a symmetric design $D = (\mathcal P,\mathcal B)$ is a bijection $\rho: \mathcal P \cup \mathcal B \to \mathcal P \cup \mathcal B$, such that
\begin{enumerate}
\item $\rho$ maps points to blocks and blocks to points,
\item $\rho$ preserves incidence: for each pair $(P,B) \in \mathcal P \times \mathcal B$ it holds that $P\in B$ if and only if $\rho(B) \in \rho(P)$,
\item $\rho^2$ is the identity map.
\end{enumerate}
A point or block is \emph{absolute} if it is incident with its image under $\rho$.
\end{df}
Given a symmetric design $D = (\mathcal P, \mathcal B)$ with a polarity $\rho$, define its
\emph{polarity graph} $R(D,\rho)$ as the graph with $\mathcal P$ as vertices, where two vertices $P$ and $Q$ are adjacent if and only if $P \in \rho(Q)$. Note that each absolute point will give rise to a loop in this graph. With this we can translate the problem of finding equinumerous incidence-free pairs to finding cocliques in the polarity graph, which we record in the following observation.
\begin{lm}\label{LemIncidencFreeFromPolarity}
If $C$ is a coclique in $R(D,\rho)$, then $(C,\rho(C))$, where $\rho(C) := \sett{\rho(P)}{P \in C}$, is an equinumerous incidence-free pair.
\end{lm}
We count loops as edges, which implies that a coclique in the polarity graph contains no absolute points.
\section{Incidence-free sets in symmetric designs}\label{sec:incidencefreeishall}
In this section we prove \Cref{ThmMainIntro}, which says that for symmetric designs $D$, the edge domination number $\gamma_e(D)$ is equal to $v-\alpha$ where $\alpha$ is the size of a largest equinumerous incidence-free pair $(X,Y)$. The fact that $\gamma_e(D)$ is at least this quantity follows from \Cref{EdgeDomToIncidenceFree}, so it remains to construct an edge dominating set of this size. We do this through the following.
\begin{prop}\label{PropHallPair}
Let $D=(\mathcal P,\mathcal B)$ be a symmetric $(v,k,\lambda)$-design with $k\ge 100$. If $(X,Y)$ is an equinumerous incidence-free pair, then there is a perfect matching between $\mathcal P \setminus X$ and $\mathcal B \setminus Y$ in $I(D)$.
\end{prop}
The perfect matching guaranteed by \Cref{PropHallPair} gives an edge dominating set of size $v-|X|$, so choosing a largest equinumerous incidence-free pair gives the upper bound of \Cref{ThmMainIntro}. Thus to prove \Cref{ThmMainIntro}, it only remains to prove \Cref{PropHallPair}. We do this in the next subsection.
\subsection{\texorpdfstring{Proof of \Cref{PropHallPair}}{Proof of Proposition 5.1}}
For the remainder of this subsection, we fix the following assumptions and notation:
\begin{itemize}
\item $D = (\mathcal P,\mathcal B)$ is a symmetric $(v,k,\lambda)$-design with $k \geq 100$;
\item $(X,Y)$ is an equinumerous incidence-free pair in $D$;
\item $G$ will denote the induced subgraph of $I(D)$ obtained by removing $X$ and $Y$ from its vertex set;
\item $\alpha := |X| = |Y|$.
\end{itemize}
We note that under the assumption $k\ge 100$, we have
\begin{equation}\label{EqAlpha}
\alpha\le \frac{k^2}{10\lambda}.
\end{equation}
Indeed, by \Cref{PropUpperBoundIF} we have $\alpha\le \frac{k^{3/2}}{\lambda} \le \frac{k^2}{10\lambda}$, where this last step used $k\ge 100$.
We will use the following easy consequence of Hall's theorem to show the existence of perfect matchings in $G$; see e.g.\ \cite{ehrenhorg} for details of its proof. Here $N_{G'}(S)$ denotes the set of vertices which are adjacent to some vertex of $S$ in a graph $G'$.
\begin{lm}\label{LmHallVariant}
Let $G'$ be a bipartite graph on $U\cup V$ with $|U|=|V|$. If every $S\subseteq U$ and $T\subseteq V$ with $|S|,|T|\le \ceil{\half |U|}$ satisfies $|N_{G'}(S)|\ge |S|$ and $|N_G(T)|\ge |T|$, then $G'$ has a perfect matching.
\end{lm}
We first show that \Cref{LmHallVariant} is satisfied whenever $S\subseteq \mathcal P \setminus X$ is relatively large. For this, we observe that by definition of $D$ being a symmetric $(v,k,\lambda)$-design, the graph $I(D)$ is $K_{2,\lambda+1}$-free, and hence all of its subgraphs are as well. By using the famous result by K\H{o}v\'ari, S\'os, and Tur\'an~\cite{kovarisosturan}, which upper bounds the maximum number of edges that a $K_{s,t}$-free bipartite graph can have, we immediately get the following.
\begin{lm}
For any $S \subseteq \mathcal P$ and $T \subseteq \mathcal B$, we have
\[e(S,T) \leq \sqrt{\lambda}(|S|-1)\sqrt{|T|} + |T|,\]
where $e(S,T)$ denotes the number of edges in $I(D)$ between $S$ and $T$.
\end{lm}
This gives the following.
\begin{lm}\label{LemKST}
If $S\subseteq \mathcal P\setminus X$ satisfies \[\lambda (|S|+\alpha)\le \left(k -1-\frac{\alpha}{|S|}\right)^2,\] then $|N_{G}(S)|\ge |S|$.
\end{lm}
\begin{proof}
Let $S$ be as in the lemma statement and set $T=N_{I(D)}(S)$. If $|T|\ge |S|+\alpha$, then
\[|N_{G}(S)|=|T\setminus Y|\ge |T|-|Y|=|T|-\alpha\ge |S|.\]
Thus we may assume for contradiction that $|T|<|S|+\alpha$.
By the preceding lemma, we find
\[k|S|=e(S,T)\le \sqrt{\lambda}(|S|-1)\sqrt{|T|}+|T|<\sqrt{\lambda}|S|\sqrt{|S|+\alpha}+|S|+\alpha.\]
By dividing both sides by $|S|$, we see that the above is equivalent to
\[\sqrt{\lambda(|S|+\alpha)}>k-1-\frac{\alpha}{|S|}.\]
This contradicts our choice of $S$, so we conclude the result.
\end{proof}
\begin{crl}\label{CrlLarge}
If $S\subseteq \mathcal P\setminus X$ satisfies
\[\alpha \le |S|\le \frac{(k-2)^2}{\lambda}-\alpha,\]
then $|N_{G}(S)|\ge |S|$.
\end{crl}
Note that the number of vertices in each part of $G$ is exactly $\frac{k(k-1)}{\lambda}+1-\alpha$, so this corollary shows that Hall's condition is satisfied for almost all sets of size at least $\alpha$.
\begin{proof}
Let $S$ be as in the hypothesis of the statement and assume for contradiction that
\[\lambda (|S|+\alpha)> \left(k -1-\frac{\alpha}{|S|}\right)^2\ge (k-2)^2,\]
where this last inequality used $|S|\ge \alpha$. This contradicts our assumption on $S$. We conclude that the hypothesis of \Cref{LemKST} is satisfied, and hence $|N_{G}(S)|\ge |S|$.
\end{proof}
\begin{crl}\label{CrlMedium}
If $S\subseteq \mathcal P\setminus X$ satisfies
\[\frac{2\alpha}{k}\le |S|\le \alpha,\]
then $|N_{G}(S)|\ge |S|$.
\end{crl}
\begin{proof}
Let $S$ be as in the hypothesis of the statement and assume for contradiction that
\[\lambda (|S|+\alpha)> \left(k -1-\frac{\alpha}{|S|} \right)^2\ge \left(\half k-1\right)^2 \ge \frac{1}{5} k^2,\]
where the second inequality used $\frac{\alpha}{|S|} \le \half k$, and the last inequality used $k\ge 100$. Because $|S|\le \alpha$, this implies $2\lambda \alpha > \frac{1}{5} k^2$, and this does not hold by \eqref{EqAlpha}. We conclude that the hypothesis of \Cref{LemKST} is satisfied, and hence $|N_{G}(S)|\ge |S|$.
\end{proof}
Note that up to this point, we have not used the hypothesis that $(X,Y)$ is incidence-free. We use this to deal with sets of size at most $\frac{2\alpha}{k}$. The following lemma can be seen as a variation of \Cref{LmIneqCap}.
\begin{lm}\label{LmSmall}
Every $S\subseteq \mathcal P\setminus X$ has
\[|N_{G}(S)|\ge \max_{s\le |S|} \frac{\lambda \alpha s}{k} -\lambda \binom s 2.\]
\end{lm}
\begin{proof}
Fix any vertex $P\in \mathcal P\setminus X$. Observe that each vertex in $X$ has $\lambda$ common neighbours with $P$ in $I(D)$, and that these neighbours necessarily lie in $\mathcal B\setminus Y$ since $X$ and $Y$ are incidence-free. Thus
\[\lambda \alpha=\lambda |X|\le e_{I(D)}(N_{G}(P), X)\le k|N_{G}(P)|,\]
where this last step used that each vertex of $I(D)$ has degree $k$. We conclude that $|N_{G}(P)|\ge \frac{\lambda \alpha}{k}$ for all $P\in \mathcal P\setminus X$.
Now given any $S\subseteq \mathcal P\setminus X$, let $S'\subseteq S$ be a subset of size $s$. By an inclusion-exclusion argument, we see
\[|N_{G}(S)|\ge |N_{G}(S')|\ge \sum_{P\in S'} |N_{G}(P)|-\sum_{P,Q\in S'} |N_{G}(P)\cap N_{G}(Q)|\ge \frac{\lambda \alpha s}{k}-\lambda \binom s 2,\]
where this last step used the previous observation and that every two distinct vertices in $S'$ have at most $\lambda$ common neighbours in $G$. The above inequality holds for any $s\le |S|$, giving the result.
\end{proof}
We now have everything in place to prove our main result for this subsection.
\begin{proof}[Proof of \Cref{PropHallPair}]
If $\alpha=0$ then $G=I(D)$ is a regular bipartite graph, which has a perfect matching by \Cref{LmBiregular}. Thus we may assume $\alpha>0$. We now show that $G$ has a perfect matching by using \Cref{LmHallVariant}, and by the symmetry of the problem, it suffices to prove $|N_{G}(S)|\ge |S|$ whenever $S\subseteq \mathcal P\setminus X$ with $|S|\le \ceil{\half |\mathcal P\setminus X|}$.
We first claim that $\frac{(k-2)^2}{\lambda}-\alpha\ge \ceil{\half |\mathcal P \setminus X|}$. Indeed, recall that $|\mathcal P|=\frac{k(k-1)}{\lambda}+1\le \frac{k^2}{\lambda}$. Thus to prove the claim it suffices to prove
\[2 \frac{(k-2)^2}{\lambda}-2\alpha\ge \frac{k^2}{\lambda}+1-\alpha,\]
where the extra $1$ stems from upper bounding the ceiling function. This is equivalent to showing
\[\alpha \le \frac{k^2-8k+8-\lambda}{\lambda}.\]
Note that $k^2-8k+8-\lambda \ge (k-9)k\ge \frac{1}{10} k^2$ since $k\ge \lambda$ and $k\ge 100$. Thus the inequality above follows from \eqref{EqAlpha}, proving the claim.
With this claim, Corollaries~\ref{CrlLarge} and \ref{CrlMedium} imply $|N_{G}(S)|\ge |S|$ for $\frac{2\alpha}{k}\le |S|\le \ceil{\half |\mathcal P\setminus X|}$. If $\lambda\ge 2$ and $|S|\le \frac{2\alpha}{k}$, then Lemma~\ref{LmSmall} with $s=1$ implies $|N_{G}(S)|\ge |S|$, proving the result in this case.
If $\lambda=1$ and $2\le |S|<\frac{2\alpha}{k}$, then $|S| \leq \ceil{\frac{2 \alpha}k-1}$, and Lemma~\ref{LmSmall} with $s=2$ implies $|N_{G}(S)|\ge \ceil{\frac{2 \alpha}k-1} \geq |S|$. Finally, if $\lambda =1$ and $S=\{P\}$, we claim there is at least one block in $N_{I(D)}(P) \setminus Y$. Indeed, $X$ is non-empty since $\alpha>0$, and any $Q\in X$ is contained in a unique block $B$ with $P$ since $\lambda=1$.
We can not have $B\in Y$ since $(X,Y)$ is incidence-free, so $B\in N_{I(D)}(P) \setminus Y$. We conclude that the conditions of \Cref{LmHallVariant} are satisfied in all cases, giving the result.
\end{proof}
\subsection{Adapting the proof to symmetric incidence structures}
As mentioned in the introduction, \Cref{ThmMainIntro} does not hold for SIS's in general. Indeed, if $D$ is the disjoint union of two symmetric designs $(\mathcal P_1,\mathcal B_1)$ and $(\mathcal P_2,\mathcal B_2)$, then taking $X=\mathcal P_1$ and $Y=\mathcal B_2$ shows that such a result can not hold in general. Counterexamples continue to exist even if one assumes $I(D)$ is connected, as we will see in \Cref{RmkConnectedCounterexample}.
This being said, there were only a few crucial instances where we used the properties of a design. Firstly, we used \Cref{PropUpperBoundIF} to show that $\alpha$ is small in comparison to $v$. Secondly, in \Cref{LmSmall} we used that in a design every two points have at least $\lambda$ common neighbours in order to show small sets satisfied Hall's condition. In this subsection, we will replace these consequences by making assumptions on the size of $X$ and a minimum degree condition in $G$, respectively.
In addition to this, we will require that the incidence structure is somewhat close to a symmetric design in the sense that $v$ is comparable to $\frac{k^2}{\lambda}$. The construction indicated above to obtain an SIS from disjoint unions of symmetric designs can be iterated an arbitrarily number of times. The edge density of the resulting SIS will be very small, and in this case the upper bound in \Cref{PropUpperBoundIF} will be useless. For this reason we will require that $v = c\frac{k^2}{\lambda}$ for some constant $c > 0$.
All of these conditions will allow us to construct small edge dominating sets whenever the incidence structure is not a symmetric design in \Cref{SubsecSemibiplanes}. The theorem we state will not be best possible, but suffices for our purposes in this paper and allows us to recycle the proofs in the previous subsection.
\begin{thm}
\label{ThmSISMatching}
Let $D=(\mathcal P,\mathcal B)$ be an SIS of type $(v,k,\lambda)$ where $v \leq \frac{9}{5}\frac{k^2}{\lambda}$ and $k \geq 100$.
If $(X,Y)$ is an equinumerous incidence-free pair with $|X|\leq \frac{k^2}{10\lambda}$ and every point or block not in $X \cup Y$ is incident with at least $\frac{2|X|}{k}$ blocks or points not in $X \cup Y$, then
\[\gamma_e(D)\le v-|X|.\]
\end{thm}
\begin{proof}
The proof is nearly identical to that of \Cref{ThmMainIntro}, so we omit some of the redundant details. As before, it suffices to show there is a perfect matching between $\mathcal P \setminus X$ and $\mathcal B \setminus Y$ in $I(D)$.
Let $G$ denote the induced subgraph of $I(D)$ after removing the vertices of $X \cup Y$ and define $\alpha := |X| = |Y|$. It is not difficult to show that the proofs of \Cref{CrlLarge} and \Cref{CrlMedium} continue to hold in this setting by our assumptions on $\alpha$ and $k$. Instead of \Cref{LmSmall}, we find
\begin{align}\label{mindegree}
|N_{G}(S)|\ge \max_{s\le |S|} \frac{2 \alpha s}{k} -\lambda \binom s 2
\end{align}
for every $S \subset \mathcal P \setminus X$ by the minimum degree condition.
Then we prove again the claim that $\frac{(k-2)^2}{\lambda}-\alpha \geq \ceil{\half|\mathcal P \setminus X|}$. This will follow from the inequality
\[2 \frac{(k-2)^2}{\lambda}-2\alpha\ge \frac{9k^2}{5\lambda}-\alpha,\]
using the assumption $|\mathcal P| = v \leq \frac{9k^2}{5\lambda}$. Equivalently
\[\alpha \leq \frac{11k^2-80k+80}{10\lambda}.\]
Since $\alpha \leq \frac{k^2}{10\lambda}$ by assumption, we see that the claim is satisfied for $k \geq 100$.
Now we conclude the proof by observing that for all sets $S \subseteq \mathcal P \setminus X$ with $\frac{2\alpha}{k} \leq |S| \leq \half |\mathcal P \setminus X|$ we have $N_{G}(S) \geq |S|$ by the analogues of \Cref{CrlLarge} and \Cref{CrlMedium}. For $|S| \leq \frac{2\alpha}{k}$, we see that the minimum degree condition provides $N_{G}(S) \geq |S|$ by setting $s = 1$ in \Cref{mindegree}.
\end{proof}
\section{Application to particular classes of designs}\label{SecParticularClasses}
\subsection{Projective point-subspace designs}
In this section we will discuss the edge domination number of the design of points and $k$-spaces in the projective space $\textnormal{PG}(n,q)$, with $0 < k < n$. By $k$-spaces we refer to subspaces of projective dimension $k$. The $0$-, $1$-, and $(n-1)$-spaces are referred to as points, lines and hyperplanes. This design is symmetric only when $k = n-1$.
Denote the number of points in $\textnormal{PG}(n,q)$ by
\[
\theta_n = \frac{q^{n+1}-1}{q-1}.
\]
The smallest case is the design of points and lines in the Desarguesian projective plane. In this setting, we can construct large incidence-free sets from polarities.
Assign coordinates $(x_0,\dots,x_n) \in \mathbb F_q^{n+1} \setminus \set{(0,\dots,0)}$ to the points in $\textnormal{PG}(n,q)$, where $(x_0,\dots,x_n)$ and $(y_0,\dots,y_n)$ are considered as the same point if they are scalar multiples of each other.
For each projective point $P = (a_0,\dots,a_n)$, let $P^\perp$ denote the hyperplane with equation $a_0 X_0 + \dots + a_n X_n = 0$.
Then mapping a point $P$ to the hyperplane $P^\perp$ and vice versa, determines a polarity $\perp$ of $\textnormal{PG}(n,q)$.
When $n=2$, the following lower bounds on the independence number of the polarity graph $R(D,\rho)$ are known.
\begin{thm}[{\cite{mubayiwilliford,mattheuspavesestorme}}]\label{cocliquepolaritygraph}
Let $D$ be the design of points and lines in $\textnormal{PG}(2,q)$, $q = p^h$ where $p$ prime and $h \geq 1$, and let $\rho$ be the polarity described above.
Then the polarity graph $R(D,\rho)$ has a coclique of size at least $c q\sqrt{q} - O(q)$,
where
\[
c = \begin{cases}
1 & \text{for $p=2$ and $h$ even}, \\
\frac 1 {\sqrt2} & \text{for $p=2$ and $h$ odd}, \\
\frac 1 2 & \text{for $p$ odd and $h$ even}, \\
0.19 & \text{for $p$ odd and $h$ odd}.
\end{cases}
\]
\end{thm}
\begin{crl}\label{corprojplane}
Let $D$ be the design of points and lines in $\textnormal{PG}(2,q)$, $q \geq 101$. Then
\[q^2-q\sqrt{q}-O(q) \leq \gamma_e(D) \leq q^2-cq\sqrt{q}-O(q),\]
where $c$ is defined as in \Cref{cocliquepolaritygraph}.
\end{crl}
\begin{proof}
The lower bound follows from \Cref{CrlDesignDomIneqs}. The upper bound follows from \Cref{LemIncidencFreeFromPolarity} and \Cref{cocliquepolaritygraph}, giving us the incidence-free pair, and \Cref{ThmMainIntro} turning it into an edge-dominating set of the wanted size.
\end{proof}
Going via the polarity graph is actually a detour when $p = 2$. It is known that $\textnormal{PG}(2,q)$ has a maximal arc of order $n$ if and only if $q = 2^h$ and $n$ divides $q$ \cite{denniston,ball}.
So starting from a maximal arc
of order $\sqrt{q}$ or $\sqrt{q/2}$, depending on the parity of $h$, we can combine our earlier observations on arcs
and the corresponding dual arcs with \Cref{LmBiregular} to improve the result in \Cref{corprojplane} for even $q$. The
improvement only lies in the omission of the restriction on $q$, the actual value of $\gamma_e(D)$ will be the same.
We state the sharp result for even $h$ for completeness.
\begin{crl}
For all $q = 2^{2h}$ we have
\[\gamma_e(D) = q^2-q\sqrt{q}+\sqrt{q}+1.\]
\end{crl}
A similar result holds in the more general design of points and hyperplanes in $\textnormal{PG}(n,q)$, $n \geq 2$. A large incidence-free set was found by De Winter, Schillewaert, and Verstra\"ete \cite[\S 4.2]{dewinterschillewaertverstraete}.
\begin{thm}
\label{ResIncidenceFreeHyperplanes}
There exists an equinumerous incidence-free pair $(X,Y)$ of sets of points and hyperplanes respectively in $\textnormal{PG}(n,q)$ of size $\Theta (q^{\frac{n+1}2})$.
\end{thm}
\begin{crl}
Let $D$ be the design of points and hyperplanes in $\textnormal{PG}(n,q)$, $n \geq 2$.
Then
\[
\gamma_e(D) = \theta_n - \Theta_q (q^{\frac{n+1}2}).
\]
\end{crl}
\begin{proof}
This follows from \Cref{CrlDesignDomIneqs} and the preceding result combined with \Cref{ThmMainIntro}.
\end{proof}
In other projective point-subspace designs, it is not difficult to determine the edge domination number exactly.
\begin{prop}
Let $D$ be the design of points and $k$-spaces in $\textnormal{PG}(n,q)$, $1 \leq k < n-1$, and $n \geq 3$.
Then
$$\gamma_e(D) =
\begin{cases}
\theta_n - q & \text{if } k=1, \\
\theta_n & \text{otherwise.}
\end{cases}$$
\end{prop}
\begin{proof}
If $1 < k < n-1$, then the number of $k$-spaces each point is incident to equals
\[
r = \prod_{i=1}^{k} \frac{q^{n-k+i}-1}{q^i-1} > \frac{q^{n+1}-1}{q-1} = v.
\]
By Lemma \ref{LmLargeRepNumber}, $\gamma_e(D) = v = \theta_n$.
Now suppose $k=1$.
First we show that $\gamma \geq \theta_n - q$.
One can see that
\[
\theta_n - (q+1) < (q+1) \theta_{n-1} - \binom{q+1}{2}
\]
if $n \geq 3$.
Therefore, by Lemma \ref{LmIneqCap} every set $S$ of $q+1$ points is incident with a set of lines $T$, where $|T| > \theta_n - (q+1)$. Now suppose for the sake of contradiction that we have an edge dominating set $\Gamma$ with $|\Gamma| \leq \theta_n-(q+1)$. Then there are at least $q+1$ points not covered by an edge of $\Gamma$, so that the lines with which they are incident should all be covered by an edge. However, the number of such lines is more than $\theta_n - (q+1) = |\Gamma|$, so this is impossible.
We can describe a construction that yields edge dominating sets of $I(D)$ of size $\theta_n - q$.
Consider an incident point-line pair $(x,\ell)$.
Let $S$ denote the points not on $\ell$ and $T$ denote the set of lines intersecting $\ell \setminus \{x\}$ in a point (so not $\ell$ itself).
Then $|S| = |T| = \theta_n-(q+1)$. The subgraph of $I(D)$ induced on $(S,T)$ is $q$-regular and hence has a perfect matching due to \Cref{LmBiregular}. By adding the edge $(x,\ell)$, we find a maximal matching in $I(D)$ of size $\theta_n-q$.
\end{proof}
\subsection{Symmetric designs from Hadamard matrices}
A \emph{Hadamard matrix} is an $n \times n$-matrix $M$ with only $1$ and $-1$ as entries such that $M^t M = n I_n$.
We refer the reader to \cite[Chapter 4]{ioninshrikhande} for a treatise of the subject.
There are several constructions of symmetric designs from Hadamard matrices.
\subsubsection{Hadamard designs and Paley matrices}
Let $\mathbf 1$ denote the all-one column vector.
If a $4u \times 4u$ Hadamard matrix $M$ is normalised, i.e.\ of the form
\[
M = \begin{pmatrix} 1 & \mathbf 1^t \\ \mathbf 1 & N \end{pmatrix},
\]
then we can replace every $-1$ in $N$ with a 0. Equivalently, if we define $J$ to be the all-one matrix of the appropriate size (which will be clear from context), then we consider the matrix $\half(N+J)$.
This yields the incidence matrix of a symmetric $2-(4u-1,2u-1,u-1)$ design.
Such a design is called a \emph{Hadamard design}.
The next bound follows directly from
\Cref{CrlDesignDomIneqs}(2).
\begin{lm}
Let $D$ be a $(4u-1,2u-1,u-1)$ Hadamard design.
Then
\[
\gamma_e(D) \geq 4u - 2 \sqrt u - \frac 1 {\sqrt u + 1}.
\]
\end{lm}
We consider the following construction due to Paley \cite{paley} of a Hadamard matrix.
Consider the field $\mathbb F_q$ with $q$ odd.
Define the quadratic character of $\mathbb F_q$ as
\[
\chi(x) = \begin{cases}
0 & \text{if } x = 0, \\
1 & \text{if $x$ is a non-zero square}, \\
-1 & \text{if $x$ is not a square},
\end{cases}
\]
Let $Q$ denote the Jacobstahl matrix of $\mathbb F_q$, i.e.\ the rows and columns of $Q$ are indexed by the elements of $\mathbb F_q$, and $Q_{x,y} = \chi(x-y)$.
Suppose that $q \equiv 1 \pmod 4$.
Then the matrix
\[
M = \begin{pmatrix}
0 & \mathbf 1^t \\
\mathbf 1 & Q
\end{pmatrix} \otimes
\begin{pmatrix}
1 & 1 \\ 1& -1
\end{pmatrix} +
I \otimes \begin{pmatrix}
1 & -1 \\ -1 & -1
\end{pmatrix}
\]
is a Hadamard matrix, where $\otimes$ denotes the tensor product.
We can interpret this matrix as follows.
Say that the the first row and column of $\begin{pmatrix}
0 & \mathbf 1^t \\
\mathbf 1 & Q
\end{pmatrix}$
are indexed by $\infty$.
The other rows and columns are naturally indexed by $\mathbb F_q$.
Say that the rows and columns of the $2 \times 2$-matrices are indexed by $1$ and $-1$, in that order.
Define $\infty - \infty = 0$, $\infty - x = x - \infty = \infty$ for all $x \in \mathbb F_q$, and $\chi(\infty) = 1$.
Then
\[
M((x,i),(y,j)) = \begin{cases}
1 & \text{if $x=y$ and } (i,j)=(1,1), \\
-1 & \text{if $x=y$ and } (i,j) \neq (1,1), \\
\chi(x-y) & \text{if $x\neq y$ and } (i,j) \neq (-1,-1), \\
- \chi(x-y) & \text{if $x \neq y$ and } (i,j)=(-1,-1).
\end{cases}
\]
If we scale the second the second row and the second column with a factor $-1$, then we obtain a Hadamard matrix of the form $\begin{pmatrix}
1 & \mathbf 1^t \\ \mathbf 1 & N
\end{pmatrix}$, and therefore $\frac 1 2 (N+J)$ is the incidence matrix of a Hadamard design.
We denote this design as $HD(q)$.
It is a $(2q+1,q,\frac{q-1}2)$ design.
The point set is $(\mathbb F_q \times \set{1,-1}) \cup \set \infty$.
For each point $P$ there is a block $B_P$, with
\begin{align*}
B_\infty &= \mathbb F_q \times \set {-1}, \\
B_{(x,1)} &= \sett{(y,i)}{\chi(x-y)=1, i=\pm 1} \cup \set{(x,1)}, \\
B_{(x,-1)} &= \sett{(y,i)}{\chi(y-x)=i} \cup \set \infty.
\end{align*}
\begin{lm}
Let $C$ be a clique in the Paley graph of order $q \geq 101$.
Then $I(HD(q))$ has an edge dominating set of size $2q+1 - |C|$.
\end{lm}
\begin{proof}
Consider the map $\rho$ which maps a point $P$ to the block $B_P$, and vice versa the block $B_P$ to the point $P$.
As $N$ is symmetric, for any two points $P$ and $Q$, it holds that $P \in Q^\rho$ if and only if $Q \in P^\rho$.
In other words, $\rho$ is a polarity of the design.
The cocliques in the polarity graph of $HD(q)$ are exactly the sets $K \times \set {-1}$, with $K$ a clique in the Paley graph of order $q$, and $\set \infty$.
Hence, if $C$ is a clique in the polarity graph, then the statement of the lemma follows from \Cref{ThmMainIntro} and \Cref{LemIncidencFreeFromPolarity}.
\end{proof}
If $q$ is a square, then it is known that $\mathbb F_{\sqrt{q}}$ is a clique of maximal size in the Paley graph of order $q$. We thus obtain the following corollary.
\begin{crl}
If $q \geq 101$ is a square, then $\gamma_e(HD(q)) = 2q - \Theta (\sqrt{q})$.
\end{crl}
\subsubsection{Menon designs and symmetric Bush-type Hadamard matrices}
If a $4u \times 4u$ Hadamard matrix $M$ has constant row sum, then it is called regular.
In that case $u$ must be square, say $u =h^2$, and $\frac 1 2 (J+M)$ and $\frac 1 2 (J-M)$ are incidence matrices of symmetric designs, called \emph{Menon designs}.
These designs have parameters $2$-$(4h^2,2h^2+\varepsilon h,h^2 + \varepsilon h)$ with $\varepsilon = \pm 1$.
By Lemma \ref{LemBlocksThroughPoints}(2), the edge domination number of such a design is at least $4h^2 - 2h$ if $\varepsilon = -1$, and $4h^2 - 2h + 2 - \frac{2}{h+1}$ if $\varepsilon = 1$.
A regular $4h^2\times 4h^2$ Hadamard matrix $M$ is \emph{of Bush type} if it is a block matrix where all the blocks are $2h \times 2h$-matrices, all the diagonal blocks are all-one matrices, and all the other blocks have constant row and column sum 0.
If such a matrix $M$ is symmetric, then $\frac 1 2 (J-M)$ is the incidence matrix of Menon design with $\varepsilon = -1$, which has a polarity without absolute points, and where the vertices of the polarity graph can be partitioned into cocliques of size $2h$.
Note that each such coclique consists of $2h$ points, with $2h$ blocks not incident with these points (namely their image under the polarity).
\begin{prop}
If $D$ is a Menon design associated with a symmetric Hadamard matrix of Bush type, it is a $(4h^2,2h^2-h,h^2 - h)$-design and $\gamma_e(D) = 4h^2 - 2h$.
\end{prop}
\begin{proof}
Applying \Cref{CrlDesignDomIneqs}(2) to this set of parameters gives the lower bound, while the incidence-free set of size $2h$ described above provides the upper bound. We note that this set is a maximal arc by \Cref{PropUpperBoundIF}(2) and so indeed gives rise to an edge dominating set as seen before.
\end{proof}
There are constructions of infinite families of symmetric Bush-type Hadamard matrices. One such family can be found in \cite{muzychuckxiang}.
Another interesting family are the symplectic symmetric designs, which Kantor \cite{kantor1985} proved to be one of the only two infinite families of 2-transitive symmetric designs with $v>2k$.
\subsection{Semi-biplanes}\label{SubsecSemibiplanes}
There are only a very limited number of explicit symmetric designs with $\lambda=2$.
These designs are known as \emph{biplanes}.
However, we can relax the definition to obtain the so-called semi-biplanes, of which there are several known infinite families.
They were introduced by Hughes \cite{hughes}.
\begin{df}
A \emph{semi-biplane} is a symmetric incidence structure $D = (\mathcal P,\mathcal B)$ such that
\begin{enumerate}
\item for every two distinct points $P$ and $Q$, there are either 0 or 2 blocks incident with $P$ and $Q$,
\item for every two distinct blocks $B$ and $C$, $|B \cap C|$ equals either 0 or 2,
\item the incidence graph $I(D)$ is connected.
\end{enumerate}
\end{df}
Given a semi-biplane, there exist numbers $v$ and $k$ such that $|\mathcal P|= |\mathcal B| = v$, and every point or block is incident with exactly $k$ blocks or points respectively.
The parameters $(v,k)$ are called the \emph{order} of the semi-biplane, and a semi-biplane of order $(v,k)$ is denoted as $\sbp(v,k)$.
The \emph{collinearity graph} of an incidence structure $(\mathcal P,\mathcal B)$ is the graph with vertex set $\mathcal P$, where two distinct points $P$ and $Q$ are adjacent if and only if there is some block incident with $P$ and $Q$.
The collinearity graph of a $\sbp(v,k)$ is regular of degree $\binom k 2$.
If we denote by $N$ the point-block incidence matrix of a $\sbp(v,k)$, and by $A$ the adjacency matrix of its collinearity graph, then
\[
N N^t = k I_v + 2 A.
\]
\subsubsection{Divisible semi-biplanes}
We call a semi-biplane $D = (\mathcal P,\mathcal B)$ \emph{divisible} if its collinearity graph is a complete multipartite graph.
Another way to formulate is that the relation $\sim$ defined on $\mathcal P$ by
\[
P \sim Q \iff P=Q \text{ or $P$ and $Q$ are not collinear}
\]
is an equivalence relation.
It is not difficult to show that every equivalence class contains the same number of elements, say $d$, and that $v = \binom k 2 + d$.
We would like to give a lower bound on the edge domination number of the incidence graph of a divisible semi-biplane.
As before, such a bound follows from an upper bound on the size of an equinumerous incidence-free pair.
One way to bound this size is by using \Cref{LemBlocksThroughPoints}.
Another way is by using an eigenvalue bound from spectral graph theory.
For designs, this yields the same bound, but for divisible semi-biplanes, this is no longer the case.
Both approaches yield lower bounds on $\gamma_e(D)$ that look roughly like $v - \frac{k \sqrt k}2$, but the spectral approach yields a significantly simpler expression.
\begin{lm}[{\cite[Theorem 5.1]{haemers}} Expander mixing lemma]
\label{LmExpanderMixing}
Let $G$ be a $k$-regular bipartite graph, with bipartition $L$ and $R$.
Take sets $S \subseteq L$ and $T \subseteq R$.
Suppose that the second largest eigenvalue of the adjacency matrix of $G$ is $\lambda_2$.
Then
\[
\left( e(S,T) - \frac{k}{|R|} |S| |T| \right)^2 \leq \lambda_2^2 |S| |T| \left( 1 - \frac{|S|}{|L|} \right)\left( 1 - \frac{|T|}{|R|} \right).
\]
\end{lm}
\begin{prop}
\label{PropEML}
Let $D$ denote an SIS of type $(v,k,\lambda)$.
Let $\lambda_2$ denote the second largest eigenvalue of $I(D)$.
Then
\[
\gamma_e(D) \geq \frac k {\lambda_2 + k} v.
\]
\end{prop}
\begin{proof}
Take an edge dominating set of size $\gamma := \gamma_e(D)$.
Then there exist sets $S$ and $T$ of blocks and points respectively of size $v - \gamma$ with no incidences between them.
Apply the expander mixing lemma in $I(D)$ to $S$ and $T$.
This tells us that
\[
\left( \frac k v (v-\gamma)^2 \right)^2 \leq \lambda_2^2 (v-\gamma)^2 \left( \frac \gamma v \right)^2.
\]
Taking square roots, this reduces to
\[
k (v-\gamma) \leq \lambda_2 \gamma,
\]
which gives the desired equality.
\end{proof}
\begin{rmk}
Observe that if $N$ is the incidence matrix of an SIS of type $(v,k,\lambda)$, then $vk = \mathrm{Tr}(NN^t) \leq k^2 + (v-1)\lambda_2$. This shows that $\lambda_2 = \Omega(\sqrt{k})$. On the other hand, we can have $\lambda_2 = k$ when the SIS is the union of two disjoint symmetric designs. This shows that the edge domination number of an SIS could lie anywhere between $\left(1-\Omega \left(\frac{1}{\sqrt{k}} \right) \right)v$ and $\half v$. For the symmetric designs we have seen before, we observe that the truth is closer to the higher end. This is again related to the fact that incidence graphs of symmetric designs are `expanding', this time expressed by the fact that $\lambda_2$ is small compared to $k$.
\end{rmk}
\begin{lm}
\label{LmSpectrumSbp}
Let $D$ be a divisible $\sbp(v,k)$ and write $d = v - \binom k 2$.
Then the spectrum of the adjacency matrix of $I(D)$ equals
$$\{\pm k, \pm \sqrt{k}^{(\frac v d (d-1))}, \pm \sqrt{k-2d}^{(\frac vd -1)}\}.$$
\end{lm}
\begin{proof}
Let $N$ denote the incidence matrix of $D$.
It suffices to compute the spectrum of $N N^t$.
The adjacency matrix of the collinearity graph of $D$ equals $A = (J-I)_{\frac v d} \otimes J_d$.
Since $(J-I)_{\frac v d}$ has spectrum $(\frac v d - 1)^{(1)}, -1^{(\frac v d -1)}$, and $J_d$ has spectrum $d^{(1)}, 0^{(d-1)}$, the spectrum of $N N^t = k I + 2 A$ equals \[
\{{k^2}, k^{(\frac v d (d-1))}, (k-2d)^{(\frac v d - 1)}\}.\]
It is well-known that if $N$ is a square matrix, then $\lambda$ is an eigenvalue of $NN^t$ with multiplicity $m$ if and only if $\sqrt \lambda$ and $- \sqrt \lambda$ are eigenvalues of $\begin{pmatrix}
O & N \\ N^t & O
\end{pmatrix}$, both with multiplicity $m$.
\end{proof}
\begin{rmk}
The lemma implies that $d \leq k/2$ and thus $v \leq k^2/2$.
This also has an easy combinatorial proof, see Wild \cite[Result 2]{wild81}.
\end{rmk}
\begin{crl}\label{CrlDivisibleSBP}
Let $D$ be a divisible $\sbp(v,k)$. Then
\[
\gamma_e(D) \geq v - \frac{v}{\sqrt k + 1}
\]
\end{crl}
\begin{proof}
This follows directly from \Cref{LmSpectrumSbp} and \Cref{PropEML}.
\end{proof}
\subsubsection{Divisible semi-biplanes from projective planes}
A classical construction of a divisible $\sbp(v,k)$ is given by Hughes, Leonard, and Wilson \cite{hughes}.
Take an involution $\varphi$ of $\textnormal{PG}(2,q)$, i.e.\ an incidence-preserving map of order two.
Then either $\varphi$ is a \emph{perspectivity}, which means it fixes a point, called the \emph{centre} of $\varphi$, and fixes a line pointwise, called the \emph{axis} of $\varphi$; or $\varphi$ is a \emph{Baer involution}, i.e.\ it fixes a Baer subplane.
Take as point set the points $P$ of $\textnormal{PG}(2,q)$ not fixed by $\varphi$, where we consider $P$ and $P^\varphi$ as equivalent, i.e.\ $\mathcal P = \sett{ \set{P,P^\varphi}}{P \neq P^\varphi}$.
For each line $l$ such that $l^\varphi \neq l$ define the block $B_l = \sett{\set{P,P^\varphi}}{P \in l, \, P \neq P^\varphi}$.
Then $B_l = B_{l^\varphi}$.
The biplane is given by $(\mathcal P,\mathcal B)$ with $\mathcal B = \sett{B_l}{l \neq l^\varphi}$.
An easy counting argument shows that a perspectivity of order 2 in $\textnormal{PG}(2,q)$ is an \emph{elation} (i.e.\ centre and axis are incident) if $q$ is even, and a \emph{homology} (i.e.\ the centre and axis are not incident) if $q$ is odd.
It is not difficult to verify that two perspectivities of order 2 in $\textnormal{PG}(2,q)$ must be projectively equivalent, and all Baer involutions are projectively equivalent as well.
This allows us to give a more concrete description of the semi-biplanes described above. For more details on involutions of projective planes, we refer to \cite[page 30]{dembowksi68}.
{\bf Case 1.} $q$ is even, $\varphi$ is an elation.
Then we can give $\textnormal{PG}(2,q)$ coordinates such that $\varphi:(x_0,x_1,x_2) \mapsto (x_0,x_1,x_2+x_0)$.
The axis of $\varphi$ is the line $X_0=0$, and the centre is $(0,0,1)$.
Every point not on the axis has a unique representation $(1,x_1,x_2)$.
We denote this point as $(x_1,x_2)$.
Every line not through the centre has a unique equation of the form $X_2 = m X_1 + b$.
We denote this line as $l_{m,b}$.
Then $(x_1,x_2)^\varphi = (x_1,x_2+1)$ and $l_{m,b}^\varphi = l_{m,b+1}$.
If we identify each point and line with its image under $\varphi$, then $(x_1,x_2)$ and $l_{m,b}$ are incident if and only if $x_2 + m x_1 + b \in \set{0,1}$.
Inspired by Mubayi and Williford \cite{mubayiwilliford}, we give the following construction of a small edge dominating set.
\begin{prop}
Let $q \geq 128$ be an even prime power, and let $D$ be the $\sbp(q^2/2,q)$ arising from an elation in $\textnormal{PG}(2,q)$.
Then
\[
\frac{q^2}{2}-\frac{q^2}{2(\sqrt{q}+1)} \leq \gamma_e(D) \leq \begin{cases}
\frac {q^2} 2 - \frac{q \sqrt q}4 & \text{if $q$ is a square}, \\
\frac {q^2} 2 - \frac{q \sqrt q}{4\sqrt 2} & \text{otherwise.}
\end{cases}
\]
\end{prop}
\begin{proof}
The lower bound follows from \Cref{CrlDivisibleSBP}, so we focus on the upper bound.
Use the notation for points and lines of this semi-biplane as described above.
Suppose that $q=2^h$.
Let $\omega$ be a primitive element of $\mathbb F_q$.
Then $1, \omega, \dots, \omega^{h-1}$ is an $\mathbb F_2$-basis of $\mathbb F_q$.
For each $x \in \mathbb F_q$, let $(x^{(0)}, \dots, x^{(h-1)})$ denote its coordinate vector with respect to this basis, i.e.\ $x = \sum_{i=0}^{h-1} x^{(i)} \omega^i$.
Define $f = \lfloor \frac h 2 \rfloor - 1$, and let $F$ denote $\mathbb F_2$-span of $1, \omega, \dots, \omega^f$.
If we take $x_1$ and $x_2$ in $F$, then $(x_1 x_2)^{(h-1)} = 0$, since $2 f < h-1$.
Let $X$ denote the set of all points $(x_1,x_2)$ with $x_1 \in F$ and $x_2^{(h-1)} = 0$.
Let $Y$ denote the set of all blocks $l_{m,b}$ with $m \in F$ and $b_{h-1} = 1$.
Then $(x_2 + m x_1 + b)^{(h-1)} = 1$.
In particular, $x_2 + m x_1 + b \notin \set{0,1}$, which implies that no point of $X$ is incident with a block of $Y$.
Since each point has two coordinate representatives, and likewise for the blocks, we find an equinumerous incidence-free pair of size
\[
|X| = |Y| = \frac{|F| \frac q 2}2
= \frac{2^{f+1} q}4
= \frac{2^{\lfloor \frac h 2 \rfloor} q}4
= \begin{cases}
\frac{q \sqrt q}4 & \text{if $h$ is even,} \\
\frac{q \sqrt q}{4\sqrt 2} & \text{if $h$ is odd.}
\end{cases}
\]
Thus, all that is left to show that $(X,Y)$ satisfy the conditions of \Cref{ThmSISMatching} with $v=q^2/2$, $k = q$ and $\alpha = |X|$ as above.
First off, we have $v = q^2/2$ and $q \geq 128$, which is larger than $9/5(q^2/2)$ and $100$ respectively. Secondly, we have indeed that $|X|\leq q^2/20$ for $q \geq 128$ as $|X| \leq q\sqrt{q}/4$. Finally, take a point $(x_1,x_2)$.
For every value of $m$, there is a unique $b$ with $(x_1,x_2) \in l_{m,b}$, hence $(x_1,x_2)$ lies on at most $|F| \leq \sqrt q$ lines in $Y$. Equivalently, $(x_1,x_2)$ has at least $q-\sqrt{q} \geq 2|X|/k$ neighbours in $\mathcal B \setminus Y$.
\end{proof}
{\bf Case 2.} $q$ is odd, $\varphi$ is a homology.
We can choose coordinates such that $\varphi: (x_0,x_1,x_2) \mapsto (-x_0,x_1,x_2)$.
The axis of $\varphi$ is the line $X_0=0$, the centre of $\varphi$ is $(1,0,0)$.
Similar to the previous case, we represent each point of the semi-biplane as $(x_1,x_2) \neq (0,0)$, where $(x_1,x_2)$ and $(-x_1,-x_2)$ are considered to be the same point.
Every line in $\textnormal{PG}(2,q)$ distinct from $X_0=0$ that misses $(1,0,0)$ has a unique equation of the form $X_0 = a X_1 + b X_2$, $(a,b) \neq (0,0)$.
Denote this line as $l_{a,b}$.
Then $l_{a,b}^\varphi = l_{-a,-b}$.
The lines in the biplane are of the form $l_{a,b}$ with $(x_1,x_2) \in l_{a,b}$ if and only if $a x_1 + b x_2 = \pm 1$.
\begin{prop}
Let $q \geq 11$ be odd, and let $D$ be the $\sbp((q^4-1)/2,q^2)$ arising from a homology in $\textnormal{PG}(2,q^2)$.
Then
\[
\frac{q^4-1}{2}-\frac{q^4-1}{2q+1}\leq \gamma_e(D) \leq \frac{q^4-1}2 - q\frac{q^2-1}4.
\]
\end{prop}
\begin{proof}
The lower bound again follows from \Cref{CrlDivisibleSBP}.
Let $\zeta$ denote a primitive $(q+1)$st root of unity in $\mathbb F_{q^2}$.
Consider the sets
\begin{align*}
X &= \sett{(x_1,x_2)}{ x_1 \in \mathbb F_q, \, x_2^{q-1} \in \set{\zeta^0, \dots, \zeta^{(q-1)/2} } } \\
Y &= \sett{l_{a,b}}{ a \in \mathbb F_q, \, b^{q-1} \in \set{\zeta^1, \dots, \zeta^{(q+1)/2} }}
\end{align*}
If you take a point $(x_1,x_2) \in X$ and a line $l_{a,b} \in Y$, then $a x_1 \in \mathbb F_q$ and $(b x_2)^{q-1} \in \set{\zeta^1, \dots, \zeta^q}$.
In particular, $(b x_2)^{q-1} \notin \set{0,1}$, which implies that $b x_2 \notin \mathbb F_q$.
Therefore, $a x_1 + b x_2 \notin \mathbb F_q$, hence $a x_1 + b x_2 \neq \pm 1$.
Thus, $(X,Y)$ is an equinumerous incidence-free pair.
To calculate the size of $X$, note that there are $q$ choices for $x_1$, $\frac{q+1}2$ choices for $x_2^{q-1}$, so $\frac{(q-1)(q+1)}2$ choices for $x_2$. Since $(x_1,x_2)$ and $(-x_1,-x_2)$ are the same point and either both or neither are in $X$, we conclude that $|X| = |Y| = \frac{q(q^2-1)}4$.
To finish the proof, we check that the conditions of \Cref{ThmSISMatching} are met.
Using that $q \geq 11$, the only non-trivial condition to check is that every point outside of $X$ is incident with at least $\frac 2 {q^2} q \frac{q^2-1}4 = \frac{q^2-1}{2q}$ lines outside of $Y$ and vice versa.
By the symmetry of the situation, we only check the condition for points outside of $X$.
So take a point $(x_1,x_2) \notin X$.
First suppose that $x_2 \neq 0$.
Then for every $a \notin \mathbb F_q$, the lines $l_{a,\frac{\pm 1 - ax_1}{x_2}}$ are lines outside of $Y$ incident with $(x_1,x_2)$.
So $(x_1,x_2)$ is incident with at least $q^2-q > \frac{q^2-1}{2q}$ lines outside of $Y$.
Now suppose that $x_2 = 0$.
Then $x_1 \neq 0$, and every line $l_{\frac{\pm 1}{x_1},b}$ is incident with $(x_1,x_2)$.
There are $q^2 - \frac{q^2-1}2 = \frac{q^2+1}2$ values of $b$ such that $b^{q-1} \notin \set{\zeta^1, \dots, \zeta^{(q+1)/2}}$, so $(x_1,x_2)$ is incident with at least $\frac{q^2+1}2 \geq \frac{q^2-1}{2q}$ lines outside of $Y$.
\end{proof}
\begin{rmk}
The semi-biplanes with $\varphi$ a homology and $q$ an odd non-square prime power are left untreated.
If $\mathbb F_q$ has a large subfield, a similar construction yields a fairly large incidence-free pair.
However, if $\mathbb F_q$ does not have a large subfield, one would need some new ideas.
\end{rmk}
{\bf Case 3.} $\varphi$ is a Baer involution in $\textnormal{PG}(2,q^2)$.
After choosing appropriate coordinates, $\varphi: (x_0,x_1,x_2) \mapsto (x_0^q,x_1^q,x_2^q)$.
The semi-biplane then consists of the points $(x_0,x_1,x_2)$ not fixed by $\varphi$, which we identify with $(x_0^q,x_1^q,x_2^q)$, and the blocks $l_{a,b,c}$ with $(a,b,c) \neq (a^q,b^q,c^q)$ containing the points $(x_0,x_1,x_2)$ satisfying $a x_0 + b x_1 + c x_2 = 0$ or $a x_1^q + b x_1^q + c x_2^q = 0$.
Note that $l_{a,b,c} = l_{a^q,b^q,c^q}$.
\begin{prop}
Let $q\geq 11$ be a prime power, and let $D$ denote the $\sbp(q(q^3-1)/2,q^2)$ arising from a Baer involution in $\textnormal{PG}(2,q^2)$.
Then
\[
\frac{q^4-q}{2}-\frac{q^4-q}{2q+1} \leq \gamma_e(D) \leq
\begin{cases}
q\frac{q^3-1}2 - \frac{q-1}2 \frac{q^2+2q-1}2 & \text{if } q \equiv 3 \pmod 4, \\
q\frac{q^3-1}2 - (q^2-1) \floor{ \frac q 4 } & \text{otherwise.}
\end{cases}
\]
\end{prop}
\begin{proof}
The lower bound follows from \Cref{CrlDivisibleSBP}.
Let $\zeta$ be a primitive $(q+1)$st root of unity in $\mathbb F_{q^2}$.
Define $f = \floor{\frac{q+1}4}$.
Consider the sets
\begin{align*}
X & = \sett{ (x_0,x_1,x_2) }{ (x_0, x_1) \in \textnormal{PG}(1,q), \, x_2^{q-1} \in \set{\zeta, \dots, \zeta^f} }, \\
Y & = \sett{ l_{a,b,c} }{ (a, b) \in \textnormal{PG}(1,q), \, c^{q-1} \in \set{\zeta^{f+1}, \dots, \zeta^{2f}} }.
\end{align*}
Take $(x_0,x_1,x_2) \in X$ and $l_{a,b,c} \in Y$.
Then $a x_0 + b x_1 = a x_0^q + b x_1^q \in \mathbb F_q$, $x_2^{q-1} = \zeta^i$ with $1 \leq i \leq f$, and $c^{q-1} = \zeta^j$ with $f+1 \leq j \leq 2f$.
Therefore, $(c x_2)^{q-1} = \zeta^{i+j}$ with $f+1 \leq i+j \leq 3f$.
On the other hand, since $\zeta^q = \zeta^{-1}$, $(x_2^q c)^{q-1} = \zeta^{j-i}$ with $1 \leq j-i \leq 2f -1$.
Thus, $(c x_2)^{q-1}$ and $(c x_2^q)^{q-1}$ are not equal to 1.
This means that $c x_2$ and $c x_2^q$ are not in $\mathbb F_q$, which implies that $a x_0 + b x_1 + c x_2$ and $a x_0^q + b x_1^q + c x_2^q$ are not in $\mathbb F_q$, so definitely not equal to 0.
Hence, no point of $X$ lies on a block of $Y$.
For every $(q+1)$st root of unity $\zeta^i$, there are $q-1$ elements $x_2$ with $x_2^{q-1} = \zeta^i$, which in fact form a multiplicative coset of $\mathbb F_q^*$.
Hence, $|X| = (q+1)(q-1)f$.
Note that we do not count any points twice under the equivalence $(x_0,x_1,x_2) = (x_0^q,x_1^q,x_2^q)$, since $(x_0,x_1) \in \textnormal{PG}(1,q)$ implies that $(x_0^q,x_1^q) = (x_0,x_1)$, and $x_2^{q-1} \in \set{\zeta, \dots, \zeta^f}$ implies that $(x_2^q)^{q-1} \in \set{\zeta^q, \dots, \zeta^{q+1-f}}$.
For the set $Y$, things are a little more delicate.
If $f = \frac{q+1}4$, i.e.\ $q \equiv 3 \pmod 4$, we must beware that if $c^{q-1} = \zeta^{2f}$, then $l_{a,b,c}$ and $l_{a,b,c^q}$ define the same line, but $(c^q)^{q-1}$ also equals $\zeta^{2f}$.
Thus, in this case, the size of $Y$ equals $(q+1)((f-1)(q-1) + \frac{q-1}2) = (q^2-1)\frac{q-1}4$.
Consider the blocks $l_{a,1,0}$ with $a \notin \mathbb F_q$.
If $(x_0,x_1,x_2) \in X$, then $a x_0 + x_1$ cannot be zero, since $a x_0 \notin \mathbb F_q$ and $x_1 \in \mathbb F_q$.
There are $\frac{q^2-q}2$ such blocks.
Define $Y' = Y \cup \sett{l_{a,1,0}}{a \notin \mathbb F_q }$.
Then $|Y'| = |X| - \frac{q-1}2$.
So we can delete $\frac{q-1}2$ points from $X$ to obtain a subset $X'$.
Then $(X', Y')$ is an equinumerous incidence-free pair.
To finish the proof, one can again check that the conditions of \Cref{ThmSISMatching} are satisfied. Apart from some elementary inequalities, one needs to observe that every point $(x_0,x_1,x_2)$ in $\mathcal P \setminus X$ is contained in at most $q+1$ blocks of $Y$. First suppose that $x_2=0$ so that we can write it as $(x_0,1,0)$ with $x_0 \notin \mathbb F_q$. Then it cannot be incident with a block in $Y$ as otherwise $ax_0+b=0$ or $ax_0^q+b=0$ leads to the contradiction that $x_0 \in \mathbb F_q$. Now suppose that $x_2 \neq 0$. Choose $(a,b) \in \textnormal{PG}(1,q)$, then $ax_0+bx_1+cx_2 = 0$ has a unique solution for $c$. Hence for this choice of $(a,b)$, there is at most one $l_{a,b,c} \in Y$ incident with the point $(x_0,x_1,x_2)$.
\end{proof}
\subsubsection{Semi-biplanes from binary affine spaces}
In this subsection, we consider another large family of semi-biplanes. They are no longer divisible, but we can still find lower bounds on the edge-domination number, emphasising the flexibility of the approach based on eigenvalues. In one particular member of this family of semi-biplanes, we will construct a large equinumerous incidence-free pair in an SIS that will not give rise to an edge dominating set.
This shows the necessity of some extra conditions in \Cref{ThmSISMatching} when compared with \Cref{PropHallPair}. \\
Consider the $n$-dimensional affine space over $\mathbb F_2$, denoted $\textnormal{AG}(n,2)$.
Give coordinates to the points.
The \emph{weight} of a point is the number of coordinate positions in which the point has a non-zero entry.
Let $W$ and $\mathcal P$ denote the sets of points of odd and even weight, respectively.
Consider a set $S \subseteq W$ of size $k$ such that
\begin{itemize}
\item the (affine) span of $S$ is $W$,
\item $S$ does not fully contain any (affine) plane of $\textnormal{AG}(n,2)$.
\end{itemize}
Define $\mathcal B = \sett{y + S}{y \in W}$.
Then $D = (\mathcal P,\mathcal B)$ is an $\sbp(2^{n-1},k)$, see \cite{wild95}.
\begin{lm}
Let $\mathcal H$ denote the set of all affine hyperplanes of $W$.
Then the spectrum of the adjacency matrix of $I(D)$ equals the multiset
\[
\set{\pm k} \cup \sett{k-2|H \cap S|}{H \in \mathcal H}.
\]
\end{lm}
\begin{proof}
As in \Cref{LmSpectrumSbp}, let $N$ denote the incidence matrix of the semi-biplane, and $A$ the adjacency matrix of its collinearity graph.
Then $N N^t = k I + 2 A$, and the spectrum of the adjacency matrix of $I(D)$ can be derived from the spectrum of $N N^t$.
Denote $S^+ = \sett{ s+t }{ s,t \in S, \, s \neq t }$.
The planes in $\textnormal{AG}(n,2)$ are exactly the sets of four distinct points whose total sum is the zero vector.
Since $S$ does not contain any plane, $s+t$ uniquely determines $\set{s,t} \subseteq S$, and $|S^+| = \binom k 2$.
Furthermore, two distinct points $x,z \in \mathcal P$ are collinear if and only if $x, z \in y + S$ for some $y \in W$.
This is equivalent to $x = y + s$ and $z = y + t$ for some $y \in W$, and some $s,t \in S$.
This again is equivalent to $x + s = z + t$ for some $s,t \in S$, since this immediately implies that $x+s=z+t \in W$.
The last statement is equivalent to $x + z \in S^+$.
In conclusion, $x$ and $z$ are collinear if and only if $x + z \in S^+$.
We can now use \cite[\S 7.1]{brouwervanmaldeghem} to conclude that the spectrum of $A$ equals the multiset
\[
\set{ \binom k 2 } \cup \sett{ 2 |H \cap S^+| - |S^+| }{ H \text{ a hyperplane of $\mathcal P$ through } \mathbf 0}.
\]
Consider the standard inner product $x \cdot y = \sum_i x_i y_i$ on $\textnormal{AG}(n,2)$.
Every hyperplane $H$ in $\mathcal P$ through $\mathbf 0$ is of the form $X \cdot a = 0$ for some $a \in \mathcal P \setminus \set {\mathbf 0,\mathbf 1}$.
Then $H$ is an $(n-2)$-space in $\textnormal{AG}(n,2)$.
There are three hyperplanes in $\textnormal{AG}(n,2)$ through $H$, namely $H_0$ with equation $X \cdot a = 0$, $H_1$ with equation $X \cdot (a + \mathbf 1) = 0$, and $\mathcal P$.
Then $H_0 \cap W$ and $H_1 \cap W$ are parallel hyperplanes of $W$, partitioning the points.
Take a point $x$ in $S^+$.
There exist unique $s$ and $t$ in $S$ such that $x = s+t$.
Then
\[
x \notin H \iff x \cdot a = 1 \iff s \cdot a \neq t \cdot a.
\]
Thus, $|S^+ \setminus H|$ equals the number of ways to choose an element of $s \in S \cap H_0$ and an element $t \in S \cap H_1$.
If $|S \cap H_0| = m$, then $|S \cap H_1| = k-m$.
Hence,
\[
2 |H \cap S^+ | - |S^+| = |S^+| - 2 |H \setminus S^+| = \binom k 2 - 2 m (k-m)
\]
This gives an eigenvalue
\[
k + 2 \left( \binom k 2 - 2m(k-m) \right)
= k + k(k-1) - 4m(k-m) = k^2 - 4m(k-m)
= (k-2m)^2
\]
of $N N^t$.
This gives us eigenvalues $k-2m$ and $-(k-2m)$ of the adjacency matrix of $I(D)$.
Note that if $k - 2 |S \cap H_0| = k-2m$, then $k - 2 |S \cap H_1| = -(k-2m)$.
Lastly, the eigenvalue $\binom k 2$ of $A$, gives us the eigenvalue $k^2$ of $N N^t$, hence the eigenvalues $\pm k$ of the adjacency matrix of $I(D)$.
This proves the statement of the lemma.
\end{proof}
The following lower bound on the edge domination number of $I(D)$ follows directly from \Cref{PropEML}.
\begin{lm}
Let $W$ denote the hyperplane of odd-weight points in $\textnormal{AG}(n,2)$, and let $S$ be a subset spanning $W$, not containing a plane.
Denote the associated $\sbp(2^{n-1},k)$ by $D$.
Let $m$ denote the maximum number of points of $S$ contained in a hyperplane of $W$.
Then
\[
\gamma_e (D) \geq \frac k m 2^{n-2}.
\]
\end{lm}
\begin{rmk}\label{RmkConnectedCounterexample}
Let $S$ be the set of weight one vectors. In this $\sbp(2^{n-1},n)$, we can construct a large incidence-free set by taking $X$ to be the set of even weight vectors with weight at most $\floor{n/2}-1$ while we take $Y$ to be the set of blocks $y + S$, where $y$ runs over the odd weight vectors with weight at least $\ceil{n/2}+1$.
By taking subsets, we can easily find a large equinumerous incidence-free pair of sets, but it will not give rise to a small edge dominating set in its complement. This is easily seen, as a point corresponding to a vector of even weight at least $\ceil{n/2}+2$ has no neighbours in $\mathcal B \setminus Y$. This example reaffirms that an assumption like the minimum degree condition in \Cref{ThmSISMatching} is necessary, even though $I(D)$ is connected.
\end{rmk}
\section{Conclusion}\label{SecConclusion}
In this paper, we studied the edge domination number $\gamma_e(D)$ of incidence structures $D$ through its connections with maximal matchings and incidence-free sets.
In almost all families we studied, we saw that $\gamma_e(D) = (1-o(1)) v$, so the interesting quantity to study is $v - \gamma_e(D)$.
Using a combination of probabilistic, combinatorial and geometric techniques, supplemented with tools from spectral graph theory, we made headway on bounding the edge domination number of various designs, often obtaining sharp bounds on $v - \gamma_e(D)$ up to a constant factor. Nevertheless, many problems remain open, and we state a few of them here.
\begin{prob}
What is the edge domination number for non-Desarguesian projective planes? Can one construct large incidence-free pairs?
\end{prob}
Various classes of symmetric designs are constructed using difference sets \cite[Section 6.8]{colbourndinitz}. Quite a few of them are difference sets in cyclic groups.
\begin{prob}
Is it possible to construct incidence-free pairs in a `uniform' way for symmetric designs coming from difference sets? That is, without resorting to the specific structure of the difference set, but only using properties of the group?
\end{prob}
If $G$ is a group with difference set $S$, one can translate this problem to finding so-called cross-intersecting independent sets in the (directed) Cayley graph $C(G,S)$ where we have an arc between two distinct elements $x$ and $y$ if $xy^{-1} \in S$. \\
The following problem asks if we can improve our general lower bound from \Cref{PropProbability}.
\begin{prob}
Can we always find equinumerous incidence-free pairs of size $k^{1+\varepsilon}/\lambda$ in a symmetric $(v,k,\lambda)$-design for small $\lambda$?
\end{prob}
It seems plausible that such incidence-free pairs should always exist for symmetric designs, but we do not necessarily think it should hold for SIS's in general. Finding an example of such an SIS (if it exists) would also be of significant interest.
\end{document}
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\begin{document}
\title[Dynamics of the horocycle flow for homogeneous foliations ]{Remarks on the dynamics of the horocycle flow for homogeneous foliations by hyperbolic surfaces}
\author[F. Alcalde Cuesta]{Fernando Alcalde Cuesta} \address{GeoDynApp - ECSING group (Spain)} \email{[email protected]}
\author[F. Dal'Bo]{Fran\c{c}oise Dal'Bo} \address{Institut de Recherche Math\'ematiques de Rennes \\
Universit\'e de Rennes 1 \\ F-35042 Rennes (France)} \email{[email protected]} \date{\today}
\subjclass[2010]{
37D40, 37C85, 57R30}
\dedicatory{Dedicated to Pierre Molino with admiration}
\begin{abstract} This article is a first step towards the understanding of the dyna\-mics of the horocycle flow on foliated manifolds by hyperbolic surfaces. This is motivated by a question formulated by M. Mart\'{\i}nez and A. Verjovsky on the minimality of this flow assuming that the "natural" affine foliation is minimal too. We have tried to offer a simple presentation, which allows us to update and shed light on the classical theorem proved by G. A. Hedlund in 1936 on the minimality of the horocycle flow on compact hyperbolic surfaces. Firstly, we extend this result to the product of $PSL(2,\mathbb{R})$ and a Lie group $G$, which places us within the homogeneous framework investigated by M. Ratner. Since our purpose is to deal with non-homogeneous situations, we do not use Ratner's famous Orbit-Closure Theorem, but we give an elementary proof. We show that this special situation arises for homogeneous Riemannian and Lie foliations, reintroducing the foliation point of view. Examples and counter-examples take an important place in our work, in particular, the very instructive case of the solvable manifold $T^3_A$. Our aim in writing this text is to offer to the reader an accessible introduction to a subject that was intensively studied in the algebraic setting, although there still are unsolved geometric problems. \end{abstract}
\maketitle
\section{Introduction and motivation} \label{Smot}
In this paper, we start by focusing our attention on the following subgroups $$ U = \{ \ \matrice{1}{t}{0}{1} \ / \ t \in \mathbb{R} \ \} \quad \mbox{and} \quad B = \{ \ \matrice{\lambda}{t}{0}{\lambda^{-1}} \ / \ t \in \mathbb{R} , \lambda \in \mathbb{R}^+_\ast \ \} $$ of the group $PSL(2,\mathbb{R}) = SL(2,\mathbb{R}) / \{\pm Id\}$. We also consider a connected Lie group $G$ and the natural right actions of $U$ and $B$ on the product $PSL(2,\mathbb{R}) \times G$ where every element of $PSL(2,\mathbb{R})$ acts trivially on the second factor $G$. We discuss the minimality of the right actions of $U$ and $B$ induced on the left quotient $X = \Gamma \backslash PSL(2,\mathbb{R}) \times G$ by a cocompact discrete subgroup of $PSL(2,\mathbb{R}) \times G$. Recall that an action is said to be {\em minimal} if all the orbits are dense.
In the case where $G$ is trivial,
assuming $\Gamma$ is torsion-free, the quotient $X = \Gamma \backslash PSL(2,\mathbb{R})$ becomes the unit tangent bundle $T^1S$ to the compact hyperbolic surface $S = \Gamma \backslash \mathbb{H}$ obtained from the Poincar\'e half-plane $\mathbb{H}$. In 1936, G. A. Hedlund \cite{H} proved that the horocycle flow on $X$ is minimal (for an elementary proof, see \cite{G2}). In our context, this theorem can be reformulated as follows:
\begin{namedtheorem}[Hedlund] Let $\Gamma$ be a discrete torsion-free cocompact subgroup of $PSL(2,\mathbb{R})$. Then the right a on $X = \Gamma \backslash PSL(2,\mathbb{R})$ is minimal. \end{namedtheorem}
\noindent On the contrary, if $X$ is not compact, M. Kulikov \cite{Ku} constructed an infinitely generated Fuchsian group without non-empty $U$-minimal sets. In the case of non uniform lattices of $PSL(2,\mathbb{R})$, like the modular subgroup $PSL(2,\mathbb{Z})$, the $U$-orbits are dense or periodic. Actually, it is known from \cite{D} that the $U$-action on $X$ is minimal if and only if $X$ is compact.
When $G$ is not trivial, even assuming $X$ is compact, the $U$-action may be non minimal. This is the case for example when $G = PSL(2,\mathbb{R})$ and $\Gamma$ is the product of two cocompact Fuchsian groups. However, in this setting, we prove the following criterion:
\begin{itheorem} \label{thm1} Let $G$ be a connected Lie group and $\Gamma$ be a discrete cocompact subgroup of $PSL(2,\mathbb{R}) \times G$. Then the right $U$-action on $X = \Gamma \backslash PSL(2,\mathbb{R}) \times G$ is minimal if and only if the corresponding $PSL(2,\mathbb{R})$-action is minimal. \end{itheorem}
\noindent Our proof of Theorem~\ref{thm1} does not use Ratner's famous Orbit-Closure Theorem \cite{R}, see also \cite{G2} and \cite{KSS} for an overview. In fact, some ideas will be applied in a non-homogeneous context.
In the second part of this paper, we adopt a foliation point of view, which is natural in the previous context. For any connected Lie group $G$, the horizontal foliation of $PSL(2,\mathbb{R})\times G$ by the fibres of the projection on the second factor $G$ is invariant by the action of $\Gamma$ and so induces a foliation on $X = \Gamma \backslash PSL(2,\mathbb{R})\times G$ whose leaves are the orbits of the right $PSL(2,\mathbb{R})$-action. In fact, this action gives rise to a {\em $G$-Lie foliation} as defined in \cite{G} and \cite{Mo2}. As stated in a theorem by E. F\'edida \cite{F}, such a foliation is characterised as follows. Given a discrete group $\Gamma$ acting freely and properly discontinuously on a smooth manifold $\widetilde{M}$, a group homomorphism $h : \Gamma \to G$ and a locally trivial smooth fibration $\rho : \widetilde{M} \to G$ with connected fibres that is $\Gamma$-equivariant (i.e. $\rho(\gamma x)=h(\gamma)\rho(x)$ for all $\gamma \in \Gamma$ and for all $x \in \widetilde{M}$), the foliation $\widetilde{\mathcal{F}}$ by the fibres of $\rho$ induces a foliation $\mathcal{F}$ of $M = \Gamma \backslash \widetilde{M}$, called {\em $G$-Lie foliation}, whose leaves are quotients of the fibres of $\rho$ by the kernel of $h$.
Assume $\widetilde{M}$ is a connected Lie group $H$ equipped with a surjective morphism $\rho : H \to G$ and $\Gamma$ is a cocompact discrete subgroup of $H$. Like before, we obtain a $G$-Lie foliation on the homogeneous manifold $M = \Gamma \backslash H$ whose leaves are the orbits of the right action of the kernel $K$ of $\rho$. Namely, they are diffeomorphic to $K \cap \Gamma \backslash K$. Given a compact subgroup $K_0$ of $K$, we can modify this construction by considering $\widetilde{M} = H/K_0$ and $M = \Gamma \backslash H / K_0$. According to \cite{G1}, any $G$-Lie foliation constructed by this method is called {\em homogeneous}.
Let $\mathcal{F}$ a $G$-Lie foliation on a compact manifold $M$. When the leaves of $\mathcal{F}$ are equipped with a complete Riemann metric induced by a Riemann metric on $M$, we can define the {\em unit tangent bundle} $X = T^1\mathcal{F}$ of $\mathcal{F}$ as the vector bundle whose fibre $T^1_x \mathcal{F}$ at $x \in M$ is the unit tangent space $T^1_x L_x$ to the leaf $L_x$ passing through $x$. We say $\mathcal{F}$ is a $G$-Lie foliation by {\em hyperbolic surfaces} if the leaves of $\mathcal{F}$ are two-dimensional and the manifold $M$ is endowed with a complete Riemannian metric
whose restriction to each leaf
has hyperbolic conformal type. Actually, according to
\cite{C} and \cite{V}, we can assume (up to multiplication by a continuous function) that
each restriction has constant negative curvature, namely each leaf is a hyperbolic surface. Once each leaf $L$ has a hyperbolic structure, its unit tangent bundle $T^1 L$ becomes diffeomorphic to the quotient of $PSL(2,\mathbb{R})$ by a discrete torsion-free subgroup. The transitive smooth right $PSL(2,\mathbb{R})$-action on $T^1 L$ extends to a leafwise smooth continuous right $PSL(2,\mathbb{R})$-action on $X = T^1\mathcal{F}$. Notice that the unit tangent bundles of the leaves of a $G$-Lie foliation $\mathcal{F}$ are always the leaves of a $G$-Lie foliation of $X = T^1\mathcal{F}$ which has the same transverse structure than $\mathcal{F}$. If $\mathcal{F}$ is a foliation by hyperbolic surfaces, this foliation is given by the continuous right $PSL(2,\mathbb{R})$-action described above.
In the situation described in Theorem~\ref{thm1}, the homogeneous manifold $X = \Gamma \backslash PSL(2,\mathbb{R}) \times G$ is the unit tangent bundle of the homogeneous $G$-Lie foliation $\mathcal{F}$ by hyperbolic surfaces on $M = \Gamma \backslash PSL(2,\mathbb{R})/PSO(2,\mathbb{R}) \times G$ which is obtained where $H = PSL(2,\mathbb{R}) \times G$, $K = PSL(2,\mathbb{R})$ is the kernel of the second projection $p_2 : PSL(2,\mathbb{R}) \times G \to G$ and $K_0 = PSO(2,\mathbb{R})$ is the compact stabiliser of $z=i$ for the $PSL(2,\mathbb{R})$-action on $\mathbb{H}$.
Thus, from Theorem 1, we can derive the following generalisation of Hedlund's Theorem in the spirit of the work of M. Mart\'{\i}nez and A. Verjovsky \cite{MV} on which we comment below:
\begin{itheorem} \label{thm2}
Let $X = T^1\mathcal{F}$ be the unit tangent bundle of a homogeneous $G$-Lie foliation $\mathcal{F}$ by hyperbolic surfaces of a compact manifold. If $\mathcal {F}$ is minimal, then the right $U$-action on $X$ is minimal. \end{itheorem}
\noindent Note, however, that there are $G$-Lie foliations which are not homogeneous \cite{HMM}. A natural question arises when we replace $G$ with the quotient $G/G_0$ by a closed Lie subgroup $G_0$: does Theorem~\ref{thm2} remains valid for these more general foliations? They are {\em transversely homogeneous foliations} \cite{Bu} whose structure can be described in a similar way to that of the $G$-Lie foliations. If $G_0$ is compact, we can construct by averaging a left-invariant Riemannian metric on $G$ that is also invariant by the right action of $G_0$. Then the distance between two $PSL(2,\mathbb{R})$-orbits in $X = \Gamma \backslash PSL(2,\mathbb{R}) \times G/G_0$ remains locally constant and therefore the right $PSL(2,\mathbb{R})$-action on $X$ defines a {\em Riemannian foliation} according to \cite{Mo1} and \cite{Mo2}. As in the Lie case, the homogeneous manifold $X = \Gamma \backslash PSL(2,\mathbb{R}) \times G/G_0$ is the unit tangent bundle of a homogeneous Riemannian foliation $\mathcal{F}$ constructed on the compact manifold $M = \Gamma \backslash PSL(2,\mathbb{R})/PSO(2,\mathbb{R}) \times G/G_0$. Using Molino's theory \cite{Mo1}, we extend Theorem~\ref{thm2} to this context in Corollary~\ref{corthm2}.
In the third part of this paper, we show that, on the contrary, Theorem~\ref{thm2} does not hold for general transversely homogeneous foliations where $G_0$ is a non-compact closed Lie subgroup of $G$:
\begin{itheorem} \label{thm3} There are minimal transversely homogeneous foliations on compact manifolds such that neither the $U$-action nor the $B$-action on its unit tangent bundles are minimal. Moreover, there is such an example admitting a unique $B$-minimal set which is not $U$-minimal. \end{itheorem}
As mentioned before, the problem of generalising Hedlund's Theorem for compact foliated manifolds by hyperbolic surfaces has been discussed by M. Mart\'{\i}nez and A. Verjovsky in several versions of their article \cite{MV}. Theorems~\ref{thm1}~and~\ref{thm2} give an affirmative answer to the initial conjecture for homogeneous Lie foliations, also valid for homogeneous Riemannian foliations, while Theorem~\ref{thm3} gives a negative answer in the transversely homogeneous setting. Nevertheless, according to the second version of \cite{MV}, the question can be reformulated in the following way:
\begin{namedquestion}[Mart\'{\i}nez-Verjovsky] Let $X = T^1\mathcal{F}$ be the unit tangent bundle of a compact foliated manifold whose leaves are hyperbolic surfaces. Is it true that the right $U$-action on $X$ is minimal if and only if the right $B$-action is minimal? \end{namedquestion}
\noindent We complete the paper with some comments on this question.
\section{Proof of Theorem~\ref{thm1}} \label{Section1}
Let $G$ be a connected Lie group. Let $\Gamma$ be a discrete subgroup of the Lie group $H = PSL(2,\mathbb{R}) \times G$ acting on $H$ by left translation. We denote by $p_1$ and $p_2$ the first and second projection of $H = PSL(2,\mathbb{R}) \times G$ onto $PSL(2,\mathbb{R})$ and $G$ respectively. Any subgroup $F$ of $PSL(2,\mathbb{R})$ acts on the quotient $X = \Gamma \backslash PSL(2,\mathbb{R}) \times G$ by right translation $$
\Gamma (f,g)f' = \Gamma (ff', g) $$ for all $(f,g) \in H$ and for all $f' \in F$. In the following, we will replace $F$ with $U$, $B$ or $PSL(2,\mathbb{R})$. By duality, the right $F$-action on $X$ is minimal if and only if the action of $\Gamma$ on the quotient $PSL(2,\mathbb{R})/F \times G$ by left translation is minimal. In particular, the right $PSL(2,\mathbb{R})$-action on $X$ is minimal if and only if $\overline{p_2(\Gamma)} = G$. If the right $F$-action is minimal, then $p_1(\Gamma)$ acts minimally on $PSL(2,\mathbb{R})/F$ and $\overline{p_2(\Gamma)} = G$. For $F = B$, we prove:
\begin{proposition} \label{charactB} Let $G$ be a connected Lie group and let $\Gamma$ be a discrete subgroup of $H = PSL(2,\mathbb{R}) \times G$. Then the right $B$-action on $X$ is minimal if and only if the following two properties hold: \begin{list}{\labelitemi}{\leftmargin=5pt}
\item[(i)] $p_1(\Gamma)$ acts minimally on $PSL(2,\mathbb{R})/B$,
\item[(ii)] $\overline{p_2(\Gamma)} = G$, or equivalently the right $PSL(2,\mathbb{R})$-action on $X$ is minimal. \end{list} \end{proposition}
\noindent The proof of Proposition~\ref{charactB} uses hyperbolic geometry. Let $\mathbb{H} = \{ z \in \mathbb{C} / Im \, z > 0 \}$ be the Poincar\'e half-plane equipped with the hyperbolic distance $d$. The action of $PSL(2,\mathbb{R})$ on $\mathbb{H}$ by isometries extends to a $PSL(2,\mathbb{R})$-action on its boundary $\partial \mathbb{H} = \mathbb{R} \cup \{\infty\}$. This action is conjugated to the right $PSL(2,\mathbb{R})$-action on $PSL(2,\mathbb{R})/B$. Since $SL(2,\mathbb{R})$ acts transitively on $\mathbb{R}^2 - \{0\}$ and $U$ is the stabiliser of the vector $e_1 = (1,0)$, the homogeneous manifolds $PSL(2,\mathbb{R})/U$ and $PSL(2,\mathbb{R})/B$ are diffeomorphic to the linear space $E = \mathbb{R}^2 - \{0\}/\{\pm Id\}$ and the projective line $\mathbb{R}P^1$ respectively. Before we prove Proposition~\ref{charactB}, we state the following key lemma:
\begin{lemma} \label{keylemma} Let $\{ f_n\} _{n \geq 0}$ be a sequence of elements of $PSL(2,\mathbb{R})$. If for some $z \in \mathbb{H}$, there are points $\xi^+$ and $\xi^-$ in $\partial \mathbb{H}$ such that $$ \lim_{n \to +\infty} f_n(z) = \xi^+ \quad \mbox{and} \quad \lim_{n \to +\infty} f_n^{-1}(z) = \xi^-, $$ then for every point $\xi \neq \xi^-$ in $\mathbb{H} \cup \partial \mathbb{H}$, we have: $$ \lim_{n \to +\infty} f_n(\xi) = \xi^+. $$ \end{lemma}
\begin{proof} For each point $\xi \in \mathbb{H}$, we have $\lim_{n \to +\infty} f_n(\xi) = \xi^+$ since $d(f_n(\xi),f_n(z)) = d(\xi,z)$. For $\xi \neq \xi^-$ in $\partial \mathbb{H}$, we choose $\xi' \in \partial \mathbb{H}$ different from $\xi$ and $\xi^-$and a geodesic $\alpha : \mathbb{R} \to \mathbb{H}$ joining $\xi$ to $\xi'$, that is, $\xi= \lim_{t \to -\infty} \alpha(t)$ and $\xi' = \lim_{t \to +\infty} \alpha(t)$. If we denote by $\alpha_n = f_n {\scriptstyle \circ} \alpha$ the geodesic joining $f_n(\xi)$ to $f_n(\xi')$, then $d(f_n^{-1}(z), \alpha(t)) = d(z,\alpha_n(t))$ for all $t \in \mathbb{R}$. Since $\lim_{n \to +\infty} f_n^{-1}(z) = \xi^-$ and $\xi^-$ is different from $\xi$ and $\xi'$, we have $\lim_{n \to +\infty} d(z,\alpha_n(t)) = + \infty$ for all $t \in \mathbb{R}$. It follows that the sequence of geodesics $\alpha_n$ converges to a point $\zeta \in \partial \mathbb{H}$. This implies that $\lim_{n \to +\infty} f_n(\alpha(t)) = \zeta$ for all $t \in R$. Now, since $\alpha(t)$ belongs to $\mathbb{H}$, we have $\lim_{n \to +\infty} f_n(\alpha(t)) = \xi^+$ and hence $\lim_{n \to +\infty} f_n(\xi) = \zeta = \xi^+$. \end{proof}
\begin{proof}[Proof of Proposition~\ref{charactB}] By duality, it is enough to prove the action of $\Gamma$ on $\partial \mathbb{H} \times G$ is minimal when $p_1(\Gamma)$ acts minimally on $\partial \mathbb{H}$ and $\overline{p_2(\Gamma)} = G$. This second condition allows us to choose a non stationary sequence $\{ g_n\} _{n \geq 0}$ in $p_2(\Gamma)$ that converges to the identity element $1$ of $G$. Then there is a sequence $\{ f_n\} _{n \geq 0}$ in $PSL(2,\mathbb{R})$ such that $\gamma_n = (f_n,g_n) \in \Gamma$ for all $n \geq 0$. Since $\Gamma$ is discrete, this sequence $\{ f_n\} _{n \geq 0}$ is not bounded. Thus, without loss of generality, we can assume that the sequences $\{ f_n(z)\} _{n \geq 0}$ and $\{ f_n^{-1}(z)\} _{n \geq 0}$ converge to some points $\xi^+$ and $\xi^-$ in $\partial \mathbb{H}$ for some $z \in \mathbb{H}$. For each point $\xi \neq \xi^-$ in $\partial \mathbb{H}$, we deduce from the key lemma~\ref{keylemma} that $$ (\xi^+,g) = \lim_{n \to +\infty} (f_n(\xi),g_ng) = \lim_{n \to +\infty} \gamma_n(\xi,g) \in \overline{\Gamma(\xi,g)} $$ for all $g \in G$. More generally, assuming that $\xi \neq f(\xi^-)$ for some $f \in p_1(\Gamma)$ and replacing $\gamma_n$ with $\gamma'\gamma_n(\gamma')^{-1}$ where $\gamma' = (f,g') \in \Gamma$, we have: $$ (f(\xi^+),g) = \lim_{n \to +\infty} (ff_n(f^{-1}(\xi)),g'g_n(g')^{-1}g) = \lim_{n \to +\infty} \gamma'\gamma_n(\gamma')^{-1}(\xi,g) \in \overline{\Gamma(\xi,g)}. $$ Thus, if $\xi \in \partial \mathbb{H}$ does not belong to the orbit $p _1(\Gamma)\xi^-$, then $\overline{p_1(\Gamma)\xi^+}\times \{g\} \subset \overline{\Gamma(\xi,g)}$. Using the minimality of the action of $p_1(\Gamma)$ on $\partial \mathbb{H}$, we get $\partial \mathbb{H} \times \{g\} \subset \overline{\Gamma(\xi,g)}$ for all $g \in G$. Now, since $\overline{p_2(\Gamma)} = G$, it follows that $\overline{\Gamma(\xi,g)} = \partial \mathbb{H} \times G$. Finally, assume that $\xi = f(\xi^-)$ for some $f \in p_1(\Gamma)$. Since the $p_1(\Gamma)$ acts minimally on $\partial \mathbb{H}$ and contains unbounded sequences like $\{ f_n\} _{n \geq 0}$, either $p_1(\Gamma)$ is dense in $PSL(2,\mathbb{R})$ or $p_1(\Gamma)$ is a Fuchsian group of first kind (i.e having $\partial \mathbb{H}$ as limit set). This implies that there exists $\gamma'= (f',g') \in \Gamma$ such that the sequence $(f')^k(\xi^+)$ converges to a point $\xi' \notin p_1(\Gamma)\xi^-$ when $k$ goes to $+\infty$ and $(f')^k(\xi^-) \neq \xi$ for all $k \geq 0$. So the sequence $(\gamma')^k = ((f')^k,(g')^k ) \in \Gamma$ verifies: $$ \lim_{n \to +\infty} (\gamma')^k\gamma_n(\gamma')^{-k}(\xi,g) = ((f')^k(\xi^+), g) $$ and therefore $(\xi',g)$ belong to $\overline{\Gamma(\xi,g)}$. Since $\xi' \notin p_1(\Gamma)\xi^-$, according to the previous step, $\Gamma(\xi',g)$ is dense in $\partial \mathbb{H} \times G$ and hence $\Gamma(\xi,g)$ is also dense. \end{proof}
Theorem~\ref{thm1} really concerns cocompact discrete subgroups. Before we deal with this case, let us introduce the notion of semi-parabolic element of the Lie group $PSL(2,\mathbb{R})\times G$. Thus, we say that $(f,g) \in PSL(2,\mathbb{R})\times G$ is {\em semi-parabolic} if $f$ is conjugated in $PSL(2,\mathbb{R})$ to an element $u \neq Id$ in $U$. The existence of semi-parabolic elements in $\Gamma$ is related to the behaviour of the right $D$-action on $X$ where
$$ D = \{ \ \matrice{\lambda}{0}{0}{\lambda^{-1}} \ / \lambda > 0 \ \} \quad \mbox{and} \quad D^+ = \{ \ \matrice{\lambda}{0}{0}{\lambda^{-1}} \ / \lambda > 1 \ \} $$ are the diagonal group and its strictly positive cone.
\begin{lemma} \label{splemma} If $\Gamma$ contains a semi-parabolic element, then there are divergent positive semi-orbits with respect the right $D^+$-action on $X$. \end{lemma}
\begin{proof} Assume that $\Gamma$ contains a semi-parabolic element $\gamma = (fuf^{-1},g)$ where $u \in U - \{Id\}$, $f \in PSL(2,\mathbb{R})$ and $g \in G$. Given $g' \in G$, we set $x = \Gamma(f,g') \in X$ and we prove that $xD^+$ diverges. Suppose on the contrary that the sequence $\{xd_n\}_{n \geq 0}$ converges for some non-bounded sequence
$\{d_n\}_{n \geq 0}$ in $D^+$. Put $$ d_n = \matrice{\lambda_n}{0}{0}{\lambda_n^{-1}} $$ such that $\lambda _n \to +\infty$. Also write $$ u = \matrice{1}{t}{0}{1} $$ with $t \neq 0$. By hypothesis, there exists a sequence $\{\gamma_n\}_{n\geq 0}$ in $\Gamma$ such that $\gamma_n(f,g')d_n$ converge to some element $(f'',g'')$ in $H$. Notice that $$ \gamma_n(f,g')d_n = \gamma_n\gamma^{-1}\gamma(f,g')d_n = \gamma_n\gamma^{-1}(fud_n,gg') = \gamma_n\gamma^{-1}(fd_nd_n^{-1}ud_n,gg') $$ and $$ \lim_{n \to +\infty} d_n^{-1}ud_n = \lim_{n \to +\infty} \matrice{1}{t\lambda_n^{-2}}{0}{1} = Id. $$ We deduce that the sequence $\gamma_n\gamma^{-1}(fd_n,gg')$ also converges to $(f'',g'')$. Now, since $$ \gamma_n\gamma^{-1}(fd_n,gg') = \gamma_n\gamma^{-1}\gamma_n^{-1} \Big( \gamma_n(f,g')d_n\Big) (Id,(g')^{-1}g g') $$ and $$ \lim_{n \to +\infty} \gamma_n(f,g')d_n = (f'',g''), $$ it follows that $\gamma_n\gamma^{-1}\gamma_n^{-1}$ converges to $(Id,g''(g')^{-1}g^{-1} g'(g'')^{-1})$ in $H$. Since $\Gamma$ is discrete, for $n$ large enough, we have $p_1(\gamma_n\gamma^{-1}\gamma_n^{-1}) = Id$ and therefore $u = Id$ contradicting the hypothesis. \end{proof}
Let us assume $X$ is compact. From Lemma~\ref{splemma}, we have immediately:
\begin{proposition} \label{semiparabolic}
If $X = \Gamma \backslash PSL(2,\mathbb{R}) \times G$ is compact, then $\Gamma$ does not contain semi-parabolic elements. \qed \end{proposition}
\noindent Before we reformulate Proposition~\ref{charactB} in the cocompact case, let us recall the following classification lemma:
\begin{speciallemma} \label{lemmaD} Let $\Delta$ be a subgroup of $PSL(2,\mathbb{R})$ and denote by $\oclosure{\Delta}$ the connected component of the identity of its closure $\overline{\Delta}$. If $\Delta$ is neither discrete nor dense, then $\oclosure{\Delta}$ is conjugated to $PSO(2,\mathbb{R})$ or a Lie subgroup of $B$. \qed \end{speciallemma}
\begin{proposition} \label{cocompactB} Let $\Gamma$ be a cocompact discrete subgroup of $H = PSL(2,\mathbb{R}) \times G$. Denote by $X$ the compact quotient by left translation. Then the right $B$-action on $X$ is minimal if and only if $\overline{p_2(\Gamma)} = G$, or equivalently if the right $PSL(2,\mathbb{R})$-action on $X$ is minimal. \end{proposition}
\begin{proof} We have only to prove the \lq if\rq~part. Now, according to Proposition~\ref{charactB}, it is enough to show that $\Delta = p_1(\Gamma)$ of $PSL(2,\mathbb{R})$ acts minimally on $\partial \mathbb{H}$. We distinguish two cases, depending on whether this group is discrete or not.
If $\Delta = p_1(\Gamma)$ is discrete, the surface $\Delta\backslash \mathbb{H}$ is compact because $X$ is compact too. It follows that any orbit of $p_1(\Gamma)$ in $\mathbb{H}$ accumulates to its full boundary $\partial \mathbb{H}$. In other words, the limit set of $\Delta$ is equal to $\partial \mathbb{H}$. Since the limit set of any non-elementary Fuchsian group is minimal, we deduce that the action of $\Delta$ on $\partial \mathbb{H}$ is minimal.
If $\Delta = p_1(\Gamma)$ is non-discrete but dense, then the action of $\Delta$ on $\partial \mathbb{H}$ is still minimal. Otherwise, according to Classification Lemma~\ref{lemmaD} and using the fact that $\Delta$ normalises $\oclosure{\Delta}$, we deduce that $\Delta$ is conjugated to a subgroup of $PSO(2,\mathbb{R})$ or $B$. Since $X$ is compact, the first case is excluded. Assuming $\Delta \subset fBf^{-1}$ for some $f \in PSL(2,\mathbb{R})$, we have $[\Delta,\Delta ] \subset fUf^{-1}$ and therefore $\Delta$ is abelian as a consequence of Proposition~\ref{semiparabolic}. It follows that $\Delta$ is conjugated to a subgroup of $D$, which contradicts the compactness of $X$. \end{proof}
\begin{proof}[Proof of Theorem~\ref{thm1}] According to Proposition~\ref{cocompactB}, it is enough to prove that if the right $B$-action is minimal, then the right $U$-action is minimal too. By compactness of $X$, the $U$-action has a non-empty minimal set $\mathcal{M}$. Let us prove $\mathcal{M}$ is $B$-invariant so that $\mathcal{M} = X$ and the right $U$-action minimal.
Let $h = (f,g)$ be an element of $H = PSL(2,\mathbb{R}) \times G$ such that $x = \Gamma h \in \mathcal{M}$. Since $\overline{xU} = \mathcal{M}$, there are elements $\gamma_n = (\gamma_{1n},\gamma_{2n}) \in \Gamma$ and $$u_n = \matrice{1}{t_n}{0}{1} \in U $$ with $t_n \to +\infty$ such that $$
\lim_{n \to +\infty} \gamma_n (f,g)u_n = \lim_{n \to +\infty} (\gamma_{1n}fu_n,\gamma_{2n}g) = (f,g) = h $$ If we write $f_n = f^{-1}\gamma_{1n}fu_n$, $g_n = g^{-1}\gamma_{2n}g$, and $h_n = (f_n,g_n)$, the sequence $$hh_n = (ff_n,gg_n) = \gamma_nhu_n$$ converges to $h$ so that $$\lim_{n \to +\infty} h_n = \lim_{n \to +\infty} (f_n,g_n) = (Id,e).$$ Notice that the sequence $\{\gamma_{1n}\}_{n \geq 0}$ does not admit any convergent subsequence because $t_n \to +\infty$. On the other hand, since $hh_n = \gamma_nhu_n$ represents the class $xu_n$ in the orbit $xU$, the element $h_n = (f_n,g_n) \in H$ belongs to the set \begin{eqnarray*} H_\mathcal{M} = \{ h' \in H / \mathcal{M} h' \cap \mathcal{M} \neq \emptyset \} \end{eqnarray*} having the following properties:
\begin{lemma} \label{invset} The set $H_\mathcal{M}$ is a closed subset of $H = PSL(2,\mathbb{R}) \times G$ which is invariant under the right and left $U$-actions on $H$. \end{lemma}
\begin{proof} Let $h'_n \in H_\mathcal{M}$ be a sequence that converges to some element $h' \in H$. By definition, for any $n \in \mathbb{N}$, there is $x_n = \Gamma h_n \in \mathcal{M}$ such that $x_n h'_n = \Gamma h_nh'_n \in \mathcal{M}$. By compactness of $\mathcal{M}$ and replacing the sequence with some subsequence if necessary, we may assume that the sequence $x_n$ converges to a class $x=\Gamma h$ in $\mathcal{M}$. Then $x h' = \lim_{n \to +\infty} x_nh'_n \in \mathcal{M}$ and hence $h' \in H_\mathcal{M}$.
Let us prove $H_\mathcal{M}$ is invariant under the right and left $U$-actions on $PSL(2,\mathbb{R}) \times G$. Indeed, since $\mathcal{M} u^{-1} = \mathcal{M}$, we have: $$ \mathcal{M} uh \cap \mathcal{M} = \mathcal{M} u^{-1}uh \cap \mathcal{M} = \mathcal{M} h \cap \mathcal{M} \neq \emptyset $$ for all $u \in U$ and for all $h \in H_\mathcal{M}$. Likewise, we have: $$ \mathcal{M} hu \cap \mathcal{M} = (\mathcal{M} h \cap \mathcal{M} u^{-1})u = (\mathcal{M} h \cap \mathcal{M} )u \neq \emptyset $$ proving the right invariance. \end{proof}
Returning to the proof of Theorem~\ref{thm1}, we have:
\begin{lemma} \label{hn} There exists $k \in \mathbb{N}$ such that $f_n \notin B$ for $n \geq k$. \end{lemma}
\begin{proof} Let us assume on the contrary that for every $k \in \mathbb{N}$, there exists $n_k \geq k$ such that $f_{n_k} \in B$. Then $f^{-1}\gamma_{1n_k}f = f_{n_k}u_{n_k}^{-1} \in B$ and hence $\gamma_{1n_k} \in fBf^{-1}$. It follows that $[\gamma_{1n_k},\gamma_{1n_{k'}}] \in \Gamma \cap fUf^{-1}$ for all $k,k' \geq 0$. But according to Proposition~\ref{semiparabolic}, $\Gamma$ does not contain semi-parabolic elements and therefore $[\gamma_{1n_k},\gamma_{1n_{k'}}] = Id$. Then there exists $u \in U$ such that $$ f^{-1}\gamma_{1n_k}f = u\matrice{\lambda_{n_k}}{0}{0}{\lambda_{n_k}^{-1}}u^{-1} $$ for all $k \geq 0$. Since the sequence $\{\gamma_{1n_k}\}_{k \geq 0}$ does not converge, the sequence $\{\lambda_{n_k}\}_{k \geq 0}$ is not bounded, which is impossible because the matrices $f_{n_k} = f^{-1}\gamma_{1n_k}f u_{n_k}$ converge to $Id$ and hence the vectors $f_{n_k} e_1 = (\lambda_{n_k}, 0)$ converge to $e_1 = (1,0)$. \end{proof}
To conclude, let us put $$ f_n = \matrice{a_n}{b_n}{c_n}{d_n} $$ where $c_n \neq 0$ according to Lemma~\ref{hn}. For every $\alpha \in \mathbb{R}^\ast_+$, take $$ u'_n = \matrice{1}{\frac{\alpha - a_n} {c_n}}{0}{1} \quad \mbox{ and } \quad u''_n = \matrice{1}{-\frac{1}{\alpha}(b_n+d_n \frac{\alpha - a_n}{c_n})}{0}{1} $$ in $U$. From Lemma~\ref{invset}, as $h_n = (f_n,g_n) \in H_\mathcal{M}$, we have: $$ u'_nh_nu''_n = (u'_nf_nu''_n,g_n) = (\matrice{\alpha}{0}{c_n}{\alpha^{-1}},g_n) \in H_\mathcal{M} $$ Since $\lim_{n \to +\infty} c_n = 0$ and $\lim_{n \to +\infty} g_n = e$ , we deduce that $$ (\matrice{\alpha}{0}{0}{\alpha^{-1}},e) \in H_\mathcal{M}. $$ This means that $$ \mathcal{M}_\alpha = \mathcal{M} \matrice{\alpha}{0}{0}{\alpha^{-1}} \cap \mathcal{M} \neq \emptyset $$ for all $\alpha \in \mathbb{R}^\ast$. Since $$ \matrice{\alpha}{0}{0}{\alpha^{-1}} \matrice{1}{t}{0}{1} = \matrice{1}{\alpha^2 t}{0}{1} \matrice{\alpha}{0}{0}{\alpha^{-1}}, $$ the set $\mathcal{M} _\alpha$ is a $U$-invariant closed subset of $\mathcal{M}$. By minimality, we have $\mathcal{M}_\alpha = \mathcal{M}$ and therefore $\mathcal{M}$ is $D$-invariant, i.e. $$ \mathcal{M} \matrice{\alpha}{0}{0}{\alpha^{-1}} = \mathcal{M} $$ for all $\alpha \in \mathbb{R}^\ast_+$. So $\mathcal{M}$ is also $B$-invariant and hence $\mathcal{M} = X$ from Proposition~\ref{cocompactB}. \end{proof}
In the particular case where $G$ is trivial, we have just given a simple proof of Hedlund's Theorem, which is essentially the one that Ghys gave in \cite{G2}. We illustrate the general situation with two examples:
\begin{examples} \label{examplesLie} (i)
According to Theorem C of \cite{B}, if $G = PSL(2,\mathbb{R})$, then $H = PSL(2,\mathbb{R}) \times PSL(2,\mathbb{R})$ admits discrete uniform subgroups $\Gamma$. If $\Gamma$ is irreducible, then $p_1(\Gamma)$ and $p_2(\Gamma)$ are dense in $PSL(2,\mathbb{R})$. In particular, the natural right $PSL(2,\mathbb{R})$-action on $X = \Gamma \backslash H$ is minimal. From Theorem~\ref{thm1}, the natural right $U$-action is minimal too.
\noindent (ii) In \cite{BGSS}, the authors proved that any torsion-free cocompact Fuchsian group $\Gamma$ can be realised as a dense subgroup of $G = SO(3,\mathbb{R})$. Let $h$ be an injective representation of $\Gamma$ into $SO(3,\mathbb{R})$ and consider the free and properly discontinuous action of $\Gamma$ on $H = PSL(2,\mathbb{R}) \times SO(3,\mathbb{R})$ given by $\gamma.(f,g) = (\gamma f, h(\gamma)g)$ for all $\gamma \in \Gamma$ and for all $(f,g) \in H$. This allows us to see $\Gamma$ as a cocompact discrete subgroup of $H$. Since $h(\Gamma)$ is dense in $SO(3,\mathbb{R})$, by applying Theorem~\ref{thm1}, we conclude that the natural right $U$-action on $X = \Gamma \backslash H$ is minimal. \end{examples}
\section{Proof of Theorem~\ref{thm2}} \label{Section2}
Let $G$ be a connected Lie group and let $\mathfrak{g}$ be its Lie algebra. Right $PSL(2,\mathbb{R})$-actions on homogeneous manifolds $X = \Gamma \backslash PSL(2,\mathbb{R}) \times G$ are examples of smooth $G$-Lie foliations. This type of foliations has been classically defined using smooth foliated cocycles with values in $G$ or smooth differential $1$-forms with values in $\mathfrak{g}$, see \cite{G}, \cite{Mo1} and \cite{Mo2}. However, in our context, it is more convenient to use the following criterion as definition:
\begin{theorem}[\cite{F}] \label{Fedida} A smooth foliation $\mathcal{F}$ on a compact connected manifold $M$ is a $G$-Lie foliation if and only if there are \begin{list}{\labelitemi}{\leftmargin=5pt}
\item[(i)] a discrete group $\Gamma$ acting freely and properly discontinuously on a manifold $\widetilde{M}$,
\item[(ii)] a group homomorphism $h : \Gamma \to G$,
\item[(iii)] a $\Gamma$-equivariant locally trivial smooth fibration $\rho : \widetilde{M} \to G$ with connected fibres,
\end{list} such that $M = \Gamma \backslash \widetilde{M}$ and $\mathcal{F}$ is induced by the foliation $\widetilde{\mathcal{F}}$ of $\widetilde{M}$ whose leaves are the fibres of $\rho$. The group $\Gamma$ is called the {\em holonomy group} of $\mathcal{F}$. \end{theorem}
\noindent Assume that the leaves of $\mathcal{F}$ are $2$-dimensional. Given a complete Riemannian metric $g_0$ on $M$, $\mathcal{F}$ is said to be a foliation {\em by hyperbolic surfaces} if the restriction of $g_0$ to each leaf has hyperbolic conformal type. Actually, according to the Uniformisation Theorem of \cite{C} and \cite{V} which remains valid for any foliation by hyperbolic surfaces, there exists a (leafwise smooth) continuous function $u : M \to \mathbb{R}$ such that the restriction of the conformal Riemann metric $g = ug_0$ to each leaf has constant negative curvature equal to $-1$. Then each leaf $L$ is the quotient of the Poincar\'e half-plane $\mathbb{H}$ by the action of a discrete torsion-free subgroup $\Gamma_L$ of $PSL(2,\mathbb{R})$. Since $PSL(2,\mathbb{R})$ acts freely and transitively on $T^1 \mathbb{H}$, the unit tangent bundle $T^1 L$ is diffeomorphic to $\Gamma_L \backslash PSL$. The natural smooth right $PSL$-action on $T^1 L \cong \Gamma_L \backslash PSL$ extends to continuous global $PSL$-action on $T^1\mathcal{F}$. The {\em foliated horocycle and geodesic flows} on $T^1\mathcal{F}$ are defined by the corresponding $U$-action and $D$-action, which coincide with the usual geodesic and horocycle flow on $T^1 L$ in restriction to each leaf $L$.
In the case of the $G$-Lie foliations, we have also the following additional property:
\begin{proposition} \label{principal} Let $\mathcal{F}$ be a $G$-Lie foliation by hyperbolic surfaces of a compact connected manifold $M$. Then the developing map $\rho$ is trivial, so $\widetilde{M}$ is homeomorphic to a product $L \times G$. Moreover, the homeomorphism becomes a diffeomorphism if and only if $\mathcal{F}$ admits a smooth uniformisation. \end{proposition}
\begin{proof} Firstly, by replacing $\widetilde{M}$ and $G$ with the universal coverings of $M$ and $G$, we can assume that $\widetilde{M}$ and $G$ are simply connected. Furthermore, since the second homotopy group of the Lie group $G$ is trivial \cite{Ca}, we can use the homotopy sequence of $\rho$ to deduce that the fibre $L$ is also simply connected and hence $L = \mathbb{H}$. Then the natural right $PSL(2,\mathbb{R})$-action on $X = T^1\mathcal{F}$ lifts to a free and proper $PSL(2,\mathbb{R})$-action on $\widetilde{X} =T^1 \widetilde{\mathcal{F}}$ whose orbits are diffeomorphic to the unit tangent bundles of the fibres of $\rho$. It follows that $\widetilde{X}$ is a continuous principal $PSL(2,\mathbb{R})$-bundle over $G$, which becomes smooth if and only if $\mathcal{F}$ has a smooth uniformisation. By construction, the bundle map $\widetilde{\rho} : \widetilde{X} \to G$ is the developing map of the $G$-Lie foliation on $X$ whose leaves are the $PSL(2,\mathbb{R})$-orbits.
On the other hand, since the structure group $PSL(2,\mathbb{R})$ retracts by deformation on the stabiliser $PSO(2,\mathbb{R})$ of $z=i$ in $\mathbb{H}$, the $PSL(2,\mathbb{R})$-bundle $\widetilde{X}$ admits a reduction to $PSO(2,\mathbb{R})$. This means that there exists a continuous principal $PSO(2,\mathbb{R})$-bundle $P$ over $G$ such that $\widetilde{X}$ is isomorphic to the continuous principal $PSL(2,\mathbb{R})$-bundle associated to $P$, that is, $\widetilde{X}$ is homeomorphic to the quotient of $P \times PSL(2,\mathbb{R})$ by the diagonal $PSO(2,\mathbb{R})$-action that is given by $(p,f)r = (pr,r^{-1}f)$ for all $(p,f) \in P \times PSL(2,\mathbb{R})$ and all $r \in PSO(2,\mathbb{R})$.
Finally, let us recall that principal $PSO(2,\mathbb{R})$-bundles over $G$ are classified by the Euler class in the integer cohomology group $H^2(G,\mathbb{Z})$, see for example \cite{BT}. Actually, according to the universal coefficient theorem (see also \cite{BT}), this group $H^2(G,\mathbb{Z}) = Hom(H_2(G,\mathbb{Z}),\mathbb{Z}) \oplus Ext(H_1(G,\mathbb{Z}),\mathbb{Z})$ is trivial because the homotopy groups $\pi_1(G)$ and $\pi_2(G)$ are trivial. Briefly, the principal $PSO(2,\mathbb{R})$-bundle $P$ is trivial, so there is a homeomorphism $\varphi : P \to PSO(2,\mathbb{R}) \times G$ which is equivariant for the natural right $PSO(2,\mathbb{R})$-actions. By sending each $PSO(2,\mathbb{R})$-orbit represented by $(p,f) \in P \times PSL(2,\mathbb{R})$ with $\phi(p) = (r,g)$ to the point $\Phi((p,f)PSO(2,\mathbb{R})) = (rf,g)$ in $PSL(2,\mathbb{R}) \times G$, we obtain a well-defined $PSL(2,\mathbb{R})$-equivariant homeomorphism $\Phi : \widetilde{X} \to PSL(2,\mathbb{R}) \times G$ such that $\widetilde{\rho} = p_2 {\scriptstyle \circ} \Phi$. Now, by passing to the quotient by the corresponding $PSO(2,\mathbb{R})$-action, $\Phi$ induces a homeomorphism $\overline{\Phi} : \widetilde{M} \to \mathbb{H} \times G$ such that $\rho = p_2 {\scriptstyle \circ} \overline{\Phi}$. From the previous discussion, it is also clear that $\overline{\Phi}$ is a diffeomorphism if and only if $\mathcal{F}$ admits a smooth uniformisation. \end{proof}
Now, we restrict our attention to the notion of homogeneous $G$-Lie foliation as defined in the introduction and illustrated by Examples~\ref{examplesLie}. Recall that a $G$-Lie foliation $\mathcal{F}$ on a compact manifold $M$ is said to be {\em homogeneous} if there are a connected Lie group $H$ equipped with a surjective morphism $\rho : H \to G$, a compact subgroup $K_0$ of the kernel $K$ of $\rho$ and a cocompact discrete subgroup $\Gamma$ of $H$ such that $M$ is diffeomorphic to $\Gamma \backslash H / K_0$ and $\mathcal{F}$ is conjugated to the foliation induced by the right $K$-action on $H$. In the case where $\mathcal{F}$ is a two-dimensional foliation by hyperbolic surfaces, we can assume $K = PSL(2,\mathbb{R})$ and $K_0 = PSO(2,\mathbb{R})$.
\begin{proposition} \label{homogeneous}Let $\mathcal{F}$ be $G$-Lie foliation by hyperbolic surfaces of a compact connected manifold $M$. Then the following conditions are equivalent: \begin{list}{\labelitemi}{\leftmargin=5pt}
\item[(i)] The foliation $\mathcal{F}$ is homogeneous.
\item[(ii)] The right $PSL(2,\mathbb{R})$-action on $X=T^1 \mathcal{F}$ is conjugated to the natural right $PSL(2,\mathbb{R})$-action on some quotient of the Lie group $H = PSL(2,\mathbb{R}) \times G$ by a cocompact discrete subgroup.
\item[(iii)] Up to conjugation by a diffeomorphism between $\widetilde{M}$ and $\mathbb{H} \times G$, the holonomy group $\Gamma$ acts diagonally on $\mathbb{H} \times G$, that is, $\gamma.(z,g) = (\gamma(z),\rho(\gamma)g)$ for all $\gamma \in \Gamma$ and for all
$(z,g) \in \mathbb{H} \times G$. \end{list} \end{proposition}
\begin{proof} As in the proof of Proposition~\ref{principal}, we are assuming that $\widetilde{M}$ and $G$ are simply connected. We also keep the notation just described. Now we prove the proposition through the following cycle of implications:
\noindent $(i) \Rightarrow (ii)$ According to a result of H. Cartan, see for example \cite{S}, the Lie algebra $\mathfrak{h}$ of the Lie group $H$ split into the direct sum $\mathfrak{h} = \mathfrak{sl}(2,\mathbb{R}) \oplus \mathfrak{g}$ of the Lie algebras of $PSL(2,\mathbb{R})$ and $G$. Then the simply connected Lie group $\widetilde{H}$ integrating $\mathfrak{h}$ split into the product $\widetilde{PSL}(2,\mathbb{R}) \times G$ where $\widetilde{PSL}(2,\mathbb{R})$ is the universal covering of $PSL(2,\mathbb{R})$. Moreover, the fundamental group of $H$ is isomorphic to the fundamental group of $PSL(2,\mathbb{R})$. If follows that $H$ is isomorphic to $PSL(2,\mathbb{R}) \times G$.
\noindent $(ii) \Rightarrow (iii)$ By hypothesis, the action of the holonomy group $\Gamma$ on $\widetilde{X}$ is conjugated to the action of some discrete cocompact subgroup of $H = PSL(2,\mathbb{R}) \times G$. Then the $\Gamma$-action on $\widetilde{M}$ is conjugated to a diagonal action on $H / PSO(2,\mathbb{R}) \cong \mathbb{H} \times G$.
\noindent $(iii) \Rightarrow (i)$ Assume the holonomy group $\Gamma$ acts diagonally on the universal covering $\widetilde{M}$. Then, up to conjugation by a diffeomorphism between $\widetilde{M}$ and $\mathbb{H} \times G$, the $\Gamma$-action on $\widetilde{X} = T^1\widetilde{\mathcal{F}}$ is conjugated to the natural left action of a discrete cocompact subgroup of $H = PSL(2,\mathbb{R}) \times G$ and therefore $X = T^1\mathcal{F}$ becomes diffeomorphic to the corresponding quotient of $H$, endowed with the natural $PSL(2,\mathbb{R})$-action. \end{proof}
In \cite{HMM}, G. Hector, S. Matsumoto and G. Meigniez constructed an example of minimal $PSL(2,\mathbb{R})$-Lie foliation by hyperbolic surfaces which is not homogeneous. Comparing with Propositions~\ref{principal}~and~\ref{homogeneous}, the universal covering $\widetilde{M}$ is diffeomorphic to $\mathbb{H} \times PSL(2,\mathbb{R})$, but he holonomy group $\Gamma$ does not act diagonally. However, in the homogeneous setting, Theorem~\ref{thm2} can be immediately deduced as a corollary of Proposition~\ref{homogeneous} and Theorem~\ref{thm1}:
\begin{proof}[Proof of Theorem~\ref{thm2}] Let $\mathcal{F}$ be a $G$-Lie foliation of a compact connected manifold $M$ whose leaves are hyperbolic surfaces. The natural right $PSL(2,\mathbb{R})$-action on the unit tangent bundle $X = T^1 \mathcal{F}$ is minimal if and only if $\mathcal{F}$ is minimal because they have the same holonomy representation $h : \Gamma \to G$. Assuming $\mathcal{F}$ is homogeneous and using Proposition~\ref{homogeneous}, we can apply Theorem~\ref{thm1} to deduce that $PSL(2,\mathbb{R})$-minimality and $U$ minimality are equivalent on $X = T^1 \mathcal{F}$. \end{proof}Ê
As we already mentioned in the introduction, when we replace the Lie group $G$ with the quotient $G/G_0$ by a compact Lie subgroup $G_0$, we obtain an example of Riemannian foliation where the distance between two leaves (deduced from a left-invariant Riemannian metric on $G$ that is also invariant by the right $G_0$-action) remains locally constant. In general, a foliation $\mathcal{F}$ is said to be {\em Riemannian} when the distance between two leaves verifies this property, see \cite{G}, \cite{Mo1} and \cite{Mo2}.
\begin{examples} \label{examplesRiemann} (i) Consider the $PSL(2,\mathbb{R})$-Lie foliation constructed in Examples \ref{examplesLie}.(i) by quotienting the Lie group $H = PSL(2,\mathbb{R}) \times PSL(2,\mathbb{R})$ by an irreducible cocompact discrete subgroup $\Gamma$. Assuming $\Gamma$ torsion-free, the $\Gamma$-action on \mbox{$\mathbb{H} \times \mathbb{H}$} $\cong H / PSO(2,\mathbb{R}) \times PSO(2,\mathbb{R})$ is free and proper, so the horizontal foliation of $\mathbb{H} \times \mathbb{H}$ induces a minimal Riemannian foliation $\mathcal{F}$ on the quotient manifold $M = \Gamma \backslash \mathbb{H} \times \mathbb{H}$. The foliation $\mathcal{F}$ lifts to a minimal $PSL(2,\mathbb{R})$-Lie foliation $\mathcal{F}_T$ on $E_T = \Gamma \backslash \mathbb{H} \times PSL(2,\mathbb{R})$ defined by the representation of $\Gamma$ onto the dense subgroup $p_2(\Gamma)$ of $PSL(2,\mathbb{R})$. Notice that $E_T$ is a principal $PSO(2,\mathbb{R})$-bundle on $M$ whose elements are positively-oriented orthonormal frames for the normal bundle to the foliation. If $\Gamma$ is the product of two torsion-free cocompact Fuchsian groups, we have again a Riemannian foliation $\mathcal{F}$ on $M = \Gamma \backslash \mathbb{H} \times \mathbb{H}$, but the lifted foliation on $E_T = \Gamma \backslash \mathbb{H} \times PSL(2,\mathbb{R})$ is not longer minimal (since the leaves closures are parametrised by the compact manifold $p_2(\Gamma) \backslash PSL(2,\mathbb{R})$).
\noindent (ii) According to the construction given in \cite{BGSS}, let $h$ be an injective group homomorphism of a torsion-free discrete subgroup $\Gamma$ of $PSL(2,\mathbb{R})$ into $SO(3,\mathbb{R})$ such that $\overline{h(\Gamma)} = SO(3,\mathbb{R})$. As observed in Examples \ref{examplesLie}.(ii), the horizontal foliation of $H = PSL(2,\mathbb{R}) \times SO(3,\mathbb{R})$ induces a minimal $SO(3)$-Lie foliation on the quotient of $H$ by the image of the injective group homomorphism $i : \Gamma \to PSL(2,\mathbb{R}) \times SO(3,\mathbb{R})$ deduced from $h$. Thus $\Gamma$ acts freely and properly on the product $\mathbb{H} \times S^2$ and the quotient manifold $M = \Gamma \backslash \mathbb{H} \times S^2$ admits a minimal Riemannian foliation $\mathcal{F}$, which can be directly defined by the suspension of the representation of $\Gamma$ as a group of orientation-preserving isometries of $S^2$. As before, the foliation $\mathcal{F}$ lifts to a minimal $SO(3,\mathbb{R})$-Lie foliation $\mathcal{F}_T$ on $E_T = \Gamma \backslash \mathbb{H} \times SO(3,\mathbb{R})$ defined by the representation $h : \Gamma \to SO(3,\mathbb{R})$.
In both examples, the leaves of $\mathcal{F}$ are dense hyperbolic planes and cylinders. By replacing the unit tangent bundle $X = T^1 \mathcal{F}$ by $X_T = T^1 \mathcal{F}_T = \Gamma \backslash PSL(2,\mathbb{R}) \times G$ where $G = PSL(2,\mathbb{R})$ or $G = SO(3,\mathbb{R})$, we can derive $U$-minimality on $X$ from $U$-minimality on $X_T$. The same strategy can be applied to general Riemannian foliations by using Molino's theory \cite{Mo1} and more specifically the following important result: \end{examples}
\begin{namedstheorem}[Molino] If $\mathcal{F}$ is a smooth Riemannian foliation of a compact connected manifold, then $\mathcal{F}$ lifts to a smooth foliation $\mathcal{F}_T$ on the transverse orthonormal frame bundle $E_T$ of $\mathcal{F}$ such that \begin{list}{\labelitemi}{\leftmargin=5pt}
\item[(i)] the closures of the leaves of $\mathcal{F}_T$ are the fibres of a locally trivial smooth fibration $\pi_T : E_T \to B_T$;
\item[(ii)] there is a Lie group $G$ such that $\mathcal{F}_T$ induces a $G$-Lie foliation with dense leaves \hspace*{-8pt} on each fibre of $\pi_T$. \end{list} \end{namedstheorem}
\noindent Let us explain how to construct the lifted foliation $\mathcal{F}_T$. Assume $\mathcal{F}$ is given by foliated charts $\varphi _ i : U_i \to P_i \times T_i$ from open subsets $U_i$ that covers $M$ to the product of open discs $P_i$ and $T_i$ in $\mathbb{R}^p$ and $\mathbb{R}^q$ respectively. If we can endow each local transversal $T_i$ with a Riemannian metric $g_i$ that is invariant by the changes of chart, the foliation $\mathcal{F}$ is Riemannian. From this local point of view, it is clear that each canonical projection $\pi_i = p_2 {\scriptstyle \circ} \varphi_i : U_i \to T_i$ becomes a Riemannian submersion, so the lifted foliation $\mathcal{F}_T$ is defined by the projection
$\pi_{i_\ast} : E_T |_{U_i} = p_T^{-1}(U_i) \to E_i$ where $p_T : E_T \to M$ is the bundle map and $E_i$ is the orthonormal frame $O(q,\mathbb{R})$-bundle over $T_i$. By construction, if $\mathcal{F}$ is a foliation by hyperbolic surfaces, then $\mathcal{F}_T$ is also a foliation by hyperbolic surfaces. As in Examples~\ref{examplesRiemann}, the $U$-minimality problem for Riemannian foliations by hyperbolic surfaces can be reduced to the simpler case of Lie foliations by hyperbolic surfaces:
\begin{proposition} \label{thmriemann} Let $\mathcal{F}$ be a minimal Riemannian foliation by hyperbolic surfaces of a compact connected manifold $M$. Let $X = T^1\mathcal{F}$ and $X_T = T^1 \mathcal{F}_T$ be the unit tangent bundles of $\mathcal{F}$ and $\mathcal{F}_T$. For $F =U$, $B$ or $PSL(2,\mathbb{R})$, if the right $F$-action on $X_T$ is minimal, then the right $F$-action on $X$ is minimal. \end{proposition}
\begin{proof} We first see that, under the conditions above, $\mathcal{F}_T$ is a minimal $G$-Lie foliation by hyperbolic surfaces. Let $L_T$ be any fibre of the basic fibration $\pi_T : E_T \to B_T$, and let
$\mathcal{F}_T |_{\textstyle L_T}$ be the $G$-Lie foliation induced by $\mathcal{F}_T$ on $L_T$. Any closed subset
$C \subset L_T$ saturated by $\mathcal{F}_T |_{\textstyle L_T}$ is also a closed subset of $E_T$ saturated by $\mathcal{F}_T$. Since $O(q,\mathbb{R})$ is compact, its image $p_T(C)$ is a closed subset of $M$ saturated by $\mathcal{F}$. Now, since $\mathcal{F}$ minimal, we have $p_T(C) = M$ and hence the fibre $F_T$ projects on the whole manifold $M$. In other words, $F_T = E_T$ and $B_T$ reduces to one point. Thus, according to Molino's theorem, $\mathcal{F}_T$ is a minimal $G$-Lie foliation. By construction, its unit tangent bundle $X_T = T^1 \mathcal{F}_T$ is a $O(q,\mathbb{R})$-principal bundle over $X = T^1 \mathcal{F}$ and the right $PSL(2,\mathbb{R})$-action on $X$ is induced by the right $PSL(2,\mathbb{R})$-action on $X_T$. Finally, if the right $F$-action on $X_T$ is minimal for $F =U$, $B$ or $PSL(2,\mathbb{R})$, then the right $F$-action on $X$ is minimal too. \end{proof}
A minimal Riemannian foliation $\mathcal{F}$ is homogeneous if and only if the lifted foliation $\mathcal{F}_T$ is homogeneous. In this case, using Proposition~\ref{thmriemann}, we obtain the following corollary of Theorem~\ref{thm2}:
\begin{corollary} \label{corthm2}
Let $X = T^1\mathcal{F}$ be the unit tangent bundle of a minimal Rieman\-nian foliation $\mathcal{F}$ by hyperbolic surfaces of a compact manifold $M$.
Assume $\mathcal{F}$ is homogeneous. Then the right $U$-action on $X$ is minimal. \end{corollary}
\noindent In fact, any minimal Riemannian foliation is transversely homogeneous, like the transversely hyperbolic and transversely elliptic foliations described in Examples~\ref{examplesRiemann}, see \cite{G}. Now it is a natural question to ask if the generalisation of Hedlund's theorem holds for transversely homogeneous foliations.
\section{Proof of Theorem~\ref{thm3}} \label{Section3}
In this section, we prove that Theorem~\ref{thm2} fails when we consider a transversely homogeneous foliation instead a $G$-Lie foliation. We start by exhibiting a first example of transversely projective counter-example:
\begin{example} \label{firstexample} Let $\Gamma$ be a torsion-free discrete subgroup of $PSL(2,\mathbb{R})$. Consider its diagonal action on $PSL(2,\mathbb{R}) \times \partial \mathbb{H}$ given by $\gamma(f,\xi) = (\gamma f, \gamma(\xi))$ for all $\gamma \in \Gamma$ and for all $(f,\xi) \in PSL(2,\mathbb{R}) \times \partial \mathbb{H}$. If $\Gamma$ is cocompact, then $\Gamma$ acts minimally on $\partial \mathbb{H}$ and hence the right $PSL(2,\mathbb{R})$-action on $X = \Gamma \backslash PSL(2,\mathbb{R}) \times \partial \mathbb{H}$ is minimal. However, the right $B$-action is not minimal because the dual $\Gamma$-action on $\partial \mathbb{H} \times \partial \mathbb{H}$ is not minimal. More precisely, the diagonal set $\Delta$ consisting of all pairs $(\xi,\xi)$ is a non-trivial $\Gamma$-invariant closed subset of $\partial \mathbb{H} \times \partial \mathbb{H}$. This means that neither Proposition~\ref{charactB}, nor Proposition~\ref{cocompactB} can be extended to this more general context. In fact $\Delta$ is the unique non-empty $\Gamma$-minimal subset of $\partial \mathbb{H} \times \partial \mathbb{H}$. By duality, $$\mathcal{M} = \{ \, \Gamma \Big( \pm\!\matrice{a}{b}{c}{d}, \frac a c \Big) \, / \, \pm\!\matrice{a}{b}{c}{d}\in PSL(2,\mathbb{R}) \, \}$$
is the unique non-empty $B$-minimal subset of $X$, proving the first part of Theorem~\ref{thm3}. However, we have the following result:
\begin{proposition} \label{U-minimalset}
The set $\mathcal{M}$ is the unique non-empty $U$-minimal subset of $X$.
\end{proposition}
\begin{proof}
According to Hedlund's theorem, the $U$-action on $\Gamma \backslash PSL(2,\mathbb{R})$ is minimal. By duality, the $\Gamma$-action on $E = \mathbb{R}^2 - \{0\}/\{\pm Id\}$ is minimal too. This implies that the diagonal $\Gamma$-action on the subset
$$ \mathcal{K} = \{ \, (v,\xi) \in E \times \partial \mathbb{H} \, / \mbox{ $v$ is collinear to $\vect{\xi}{1}$ if $\xi \neq \infty$ and collinear to $\vect{1}{0}$ if $\xi = \infty$ } \}
$$
is minimal. Coming back to $X$ and using again duality, we obtain that $\mathcal{M}$ is $U$-minimal. Let $\gamma_1$ and $\gamma_2$ two hyperbolic isometries in $\Gamma$ generating a Schottky group, that is, the fundamental group of a pair of pants. Since $\Gamma$ acts minimal on $E$, for each point
$(v,\xi) \in E \times \partial \mathbb{H}$, there exists $(v_1,\xi_1) \in \Gamma(v,\xi)$ such that $\gamma_1 v_1 = \lambda_1 v_1$ with $|v_1| > 1$.
Moreover, since $\gamma_1$ and $\gamma_2$ have no common fixed points, for some sequences $\{ p_n \}_{n \geq 0}$ and $\{ q_n \}_{n \geq 0}$ in $\mathbb{Z}$, we have:
$$
\lim_{n \to + \infty} \gamma_2^{p_n} \gamma_1^{q_n} v_1 = v_2 \quad
\mbox{ and } \quad
\lim_{n \to + \infty} \gamma_2^{p_n} \gamma_1^{q_n} \xi_1 = \xi_2
$$
where $\gamma_2 v_2 = \lambda_2 v_2$ with $|v_2| > 1$ and $v_2$ is collinear to $\vect{\xi_2}{1}$ or $\vect{1}{0}$. It follows that
$\overline{\Gamma(v,\xi)} \cap \mathcal{K} \neq \emptyset$ and hence $\overline{\Gamma(v,\xi)} = \mathcal{K}$.
\end{proof}
Notice also that the dynamics of the $B$-action on $X - \mathcal{M}$ are related to the dynamics of the geodesic flow on $\Gamma \backslash PSL(2,\mathbb{R})$ since each point $(\xi^-,\xi^+) \in \partial \mathbb{H} \times \partial \mathbb{H} -\Delta$ represents a geodesic in $\mathbb{H}$. Like in Examples~\ref{examplesRiemann}, $\Gamma$ acts freely and properly discontinuously on $\mathbb{H} \times \partial \mathbb{H}$, so the horizontal foliation of $\mathbb{H} \times \partial \mathbb{H}$ induces a foliation $\mathcal{F}$ on the quotient manifold $M = \Gamma \backslash \mathbb{H} \times \partial \mathbb{H}$ whose unit tangent bundle is $X$. \end{example}
\begin{proposition} \label{Bfirstexample} The transversely homographic foliation $\mathcal{F}$ is defined by a locally free $B$-action whose orbits are dense hyperbolic planes and cylinders. \end{proposition}
\begin{proof} By construction, since $\partial \mathbb{H}$ is identified to homogeneous space $PSL(2,\mathbb{R})/B$, $\mathcal{F}$ is a minimal transversely homographic foliation whose leaves are dense hyperbolic planes and cylinders. To prove that they are the orbits of a smooth $B$-action on $M$, we use an idea of Mart\'{\i}nez and Verjovsky from \cite{MV}. Indeed, the bundle map $\pi : X = T^1 \mathcal{F} \to M$ becomes a diffeomorphism from the unique $B$-minimal set $\mathcal{M}$ onto the quotient manifold $$M = \{ \, \Gamma \big(\frac{ai+b}{ci+d},\frac a c \big) \, / \, \pm\!\matrice{a}{b}{c}{d}\in PSL(2,\mathbb{R}) \,\}.$$ It follows that $\mathcal{F}$ is defined by a locally free $B$-action, which is conjugated to the $B$-action on $\mathcal{M}$. \end{proof}
We are now interested to provide another counter-example (locally modelled by $PSL(2,\mathbb{R}) \times \mathbb{R}$) having a non-trivial $B$-minimal set which is not $U$-minimal. Although the construction is classical, see \cite{GS}, we recall some details. Thus, any matrix $$ A = \matrice{a}{b}{c}{d} \in SL(2,\mathbb{Z}) $$ defines an orientation-preserving automorphism of the torus $T^2 = \mathbb{R}^2/\mathbb{Z}^2$. The Lie group automorphism $(z,t) \in T^2 \times \mathbb{R} \mapsto (A(z),t+1) \in T^2 \times \mathbb{R}$ generates a free and properly discontinuous $\mathbb{Z}$-action on the product $T^2 \times \mathbb{R}$. Its orbit space is a compact $3$-manifold $T^3_A$ admitting a natural structure of fibre bundle over $S^1 = \mathbb{R} / \mathbb{Z}$. In fact, we consider only the hyperbolic case where $tr \, A >2$ and hence $A$ has two real eigenvalues $\lambda > 1$ and $1/ \lambda < 1$.
\begin{lemma} \label{eigenvectors} If $A$ is hyperbolic, then the eigenvectors $u$ and $v$ associated to the eigenvalues $\lambda > 1$ and $1/ \lambda < 1$ generate two different eigenlines with irrational slope. \end{lemma}
\begin{proof} Assume on the contrary that $w =(p,q)$ is an eigenvector of $A$ where $p,q \in Z$ are relatively prime (including the cases where $p=0$ and $q=1$ or $p=1$ and $q=0$). Then there exists $w' = (p',q') \in \mathbb{Z}$ such that $pq' - qp' = 1$ and then the matrix $$ B = \matrice{p}{p'}{q}{q'} \in SL(2,\mathbb{Z}) $$ satisfies $Be_1= w$ and $Be_2 = w'$, This implies that $e_1$ is an eigenvector of $B^{-1}AB$ so that $B^{-1}AB$ is an upper triangular matrix. Since $B^{-1}AB$ belongs to $SL(2,\mathbb{Z})$, we have $\pm B^{-1}AB \in U$. Then the eigenvalues of $A$ are equal to $\lambda = \pm 1$, which contradicts the hyperbolicity of $A$. \end{proof}
\begin{example} \label{secondexample} The foliation of $\mathbb{R}^2$ by parallel $u$-lines induces a minimal flow on $T^2$. The product of this foliation with the vertical factor defines a $2$-dimension foliation of $T^2 \times \mathbb{R}$ which is invariant by the $\mathbb{Z}$-action described above. Thus, by passing to the quotient, we obtain a foliation $\mathcal{F}$ of $T^3_A$ whose leaves are planes and cylinders. Indeed, according to \cite{BR} and denoting by $\pi$ the projection from $\mathbb{R}^2$ onto $T^2$, for each point $(x,y) \in \mathbb{Q}^2$, there is a positive integer $p \geq 1$ such that $A^p\pi(x,y) = \pi(x,y)$, and so $\mathcal{F}$ contains infinitely many cylindrical leaves. We will see that all the leaves are hyperbolic surfaces. Denote by ${\rm Aff}_+(\R)$ the group of orientation-preserving affine transformations of $\mathbb{R}$, which is isomorphic to $B$. Let $\Gamma$ be the discrete subgroup of $SL(2,\mathbb{R}) \times {\rm Aff}_+(\R)$ generated by
\begin{eqnarray} T_1(x,y,t) & = &( x+1,y,t) \label{t1} \\ T_2(x,y,t) & = & (x,y+1,t) \label{t2} \\ h_A(x,y,t) & = & (A \vect{x}{y}, t+1). \label{hA} \end{eqnarray}
acting on $\mathbb{R}^3 = \mathbb{R}^2 \times \mathbb{R}$. The foliated manifold $T^3_A$ is the quotient of $\mathbb{R}^3$ by the action of $\Gamma$, so is endowed with a complete affine structure. Let $u$ and $v$ be the eigenvectors of $A$ verifying $Au = \lambda u$ and $Av = \lambda^{-1} v$. Assume $det(u|v) = 1$. By changing the canonical affine frame $(0,e_1,e_2,e_3)$ by $(0,u,v,e_3)$ in $\mathbb{R}^3$, the transformations (\ref{t1}), (\ref{t2}) and (\ref{hA}) can be written as follows: \begin{eqnarray} T_1(x',y',t') & = &( x'+a', y'+b',t') \label{T1} \\ T_2(x',y',t') & = & (x'+c',y'+d',t') \label{T2} \\ h_A(x',y',t') & = & (\lambda x', \lambda^{-1}y', t'+1) \label{HA} \end{eqnarray} where $u = (d',-b')$ and $v = (c',-a')$. Thus, from Lemma~\ref{eigenvectors}, the entries $a'$ and $c'$, and the entries $b'$ and $d'$ are linearly independent over $\mathbb{Z}$. In fact, the universal covering $\mathbb{R}^3$ of $T^3_A$ can be identified with the product $\mathbb{H} \times \mathbb{R}$ by sending each point $(x',y',t') \in \mathbb{R}^3$ to the point $(z',y') = (x' + i \lambda^{t'},y') \in \mathbb{H} \times \mathbb{R}$. In this model, the transformations (\ref{T1}), (\ref{T2}) and (\ref{HA}) can be written \begin{eqnarray} T_{1\ast}(z',y') & = &( z'+a', y'+b') \label{T*1} \\ T_{2\ast}(z',y') & = & (z'+c',y'+d') \label{T*2} \\ h_{A\ast} (z',y') & = & (\lambda z', \lambda^{-1}y') \label{H*A} \end{eqnarray} Moreover, the foliation $\mathcal{F}$ lifts to the horizontal foliation of $\mathbb{H} \times \mathbb{R}$. \end{example}
\begin{proposition} [\cite{GS}] \label{GS}
The transversely affine foliation $\mathcal{F}$ is defined by a locally free $B$-action whose orbits are dense hyperbolic planes and cylinders. \end{proposition}
\begin{proof} Firstly, we remark that $\mathcal{F}$ admits a affine transverse structure because the $\Gamma$-action on $\mathbb{H} \times \mathbb{R}$ defined by (\ref{T*1}), (\ref{T*2}) and (\ref{H*A}) induces an affine action on the $\mathbb{R}$-factor generated by \begin{eqnarray} \overline{T}_{1\ast}(y' )& = & y'+b' \label{T'1} \\ \overline{T}_{2\ast}(y') & = & y' + d' \label{T'2} \\ \overline{h}_{A\ast} (y') & = & \lambda^{-1}y' \label{H'A} \end{eqnarray} We know that the leaves of $\mathcal{F}$ are planes or cylinders. Since $b'$ and $d'$ are linear independent over $\mathbb{Z}$, $\overline{T}_{1\ast}$ and $\overline{T}_{2\ast}$ generate a dense subgroup of translations of $\mathbb{R}$ and hence all leaves are dense. On the other hand, there is a natural right $B$-action on $\mathbb{H} \times \mathbb{R}$ where each element $$ \matrice{\sqrt \alpha}{\beta / \sqrt \alpha}{0}{1 / \sqrt \alpha} $$ of $B$ acts by homographies on the first factor $\mathbb{H}$ sending $z$ to $\alpha z + \beta$, and trivially on the second factor $\mathbb{R}$. Since this free $B$-action commutes with the $\Gamma$-action, it induces a locally free $B$-action on $T^3_A$ whose orbits are just the leaves of $\mathcal{F}$. \end{proof}
\begin{remark} The group law $(x',y',t')(x'',y'',t'') = (x'+\lambda^{t'}x'', y' +\lambda^{-t'}y'', t'+t'')$ defines a group structure on $\mathbb{R}^3$ that becomes a Lie group isomorphic to the solvable Lie group $Sol^3$. Each horizontal leaf $\mathbb{H} \times \{y'\}$ is the orbit of any point $(x',y',t')$ by the right $B$-action determined by the inclusion $$ i\big( \matrice{\sqrt \alpha}{\beta / \sqrt \alpha}{0}{1 / \sqrt \alpha} \big) = (\beta,0,\frac{log \, \alpha}{log \, \lambda}) $$ of $B$ as closed subgroup of $Sol^3$. The orbits of the corresponding $U$-action are the horizontal $x'$-lines in $\mathbb{R}^3$, which correspond to the parallel $u$-lines before changing the affine frames. \end{remark}
\begin{proof}[Proof of Theorem~\ref{thm3}] By construction, the unit tangent bundle $X = T^1\mathcal{F}$ is the quotient of $T^1 \mathbb{H} \times \mathbb{R}$ by the $\Gamma$-action generated by (\ref{T*1}), (\ref{T*2}), and (\ref{H*A}). By duality, the right $B$-action on $X = T^1\mathcal{F}$ has the same dynamics as the $\Gamma$-action on $\partial \mathbb{H} \times \mathbb{R}$ generated by the transformations \begin{eqnarray} T_{1\ast} (\xi,y') & = &( \xi+a', y'+b') \label{T1*} \\ T_{2\ast} (\xi,y') & = & (\xi+c',y'+d') \label{T2*} \\ h_{A\ast} (\xi,y') & = & (\lambda \xi, \lambda^{-1}y') \label{HA*} \end{eqnarray} We first observe that $\{\infty\} \times \mathbb{R}$ is a closed $\Gamma$-invariant subset of $\partial \mathbb{H} \times \mathbb{R}$. Minimality and uniqueness arise from $\lim_{n \to +\infty} h_{A\ast}^n (\xi,y') = (\infty,0)$ for all $(\xi,y') \in (\partial \mathbb{H} -\{0\}) \times \mathbb{R}$ and $\overline{\Gamma(0,y')} = \partial \mathbb{H} \times \mathbb{R}$ for all $y' \in \mathbb{R}$. Therefore, there is an unique minimal set $\mathcal{M}$ for the right $B$-action on $X = T^1\mathcal{F}$, obtained as the $\Gamma$-quotient of the pre-image of $\{\infty\} \times \mathbb{R}$ by the canonical projection of $T^1 \mathbb{H} \times \mathbb{R}$ onto $\partial \mathbb{H} \times \mathbb{R}$. But the closure of each $U$-orbit reduces to a toroidal fibre of the bundle structure of $T^3_A$ over $S^1$. \end{proof}
An important difference between this example and all the previous ones is that the discrete subgroup $\Gamma$ of $PSL(2,\mathbb{R}) \times {\rm Aff}_+(\R)$ projects onto a subgroup $p_1(\Gamma)$ of $PSL(2,\mathbb{R})$ which is neither discrete, nor dense. Moreover, by construction, the $\Gamma$-action induced on $(\partial \mathbb{H} - \{\infty\} ) \times \mathbb{R}$ is conjugated to the action of the group of affine transformations of $\mathbb{R}^2$ generated by the linear automorphism $A$ and the translations $t_1(x,y) = (x+1,y)$ and $t_2(x,y) = (x,y+1)$. Thus, the $B$-action induced on $X - \mathcal{M}$ is dual to the $\mathbb{Z}$-action on $T^2$ generated by $A$, whose topological dynamics have been carefully described by R. Adler \cite{A}.
It follows that there are $B$-orbits which are dense in $X$, and others whose closures are not manifolds.
\section{Final comments} \label{SFC}
As we already mentioned, Example~\ref{firstexample} shows that neither Proposition~\ref{charactB}, nor Proposition~\ref{cocompactB} are valid in the non-Riemannian case. Nevertheless, even if Hedlund's Theorem cannot be generalised, the question formulated by Mart\'{\i}nez and Verjovsky remains open: {\em is it true that the horocycle flow on the unit tangent bundle $X=T^1 \mathcal{F}$ of a minimal foliation $\mathcal{F}$ of a compact manifold $M$ by hyperbolic surfaces is minimal if and only the $B$-action is minimal?} Example~\ref{secondexample} proves that this conjecture cannot be strengthened by establishing an equivalence between $U$-minimal and $B$-minimal sets. As proved in Propositions~\ref{Bfirstexample}~and~\ref{GS}, Examples~\ref{firstexample}~and~ \ref{secondexample} are defined by locally free $B$-actions.
In \cite{MV}, Mart\'{\i}nez and Verjovsky have reformulated their conjecture as follows: {\em is it true that for any compact manifold foliated by dense hyperbolic surfaces, either the foliation is defined by a $B$-action or the $U$-action on $X$ is minimal?} Notice that the non-homogeneous $PSL(2,\mathbb{R})$-Lie foliation constructed in \cite{HMM} (as well as any Riemannian foliation) cannot be defined by a $B$-action, since it admits a transverse invariant volume, according to Proposition~3.1 of \cite{P}. It is an open question to know if the $U$-action is minimal or not.
Progress on this issue is interesting but very restricted in the non-homogeneous case. We place in an appendix some results, which are related to those of Sections~\ref{Section1} in this more general context.
\section*{Appendix: $U$-minimality for some non-homogeneous foliations}
Let us introduce now the group of orientation-preserving $C^r$-diffeomorphisms ${\rm Diff}_+^r(F)$ of some orientable $C^r$-manifold $F$, $0 \leq r \leq +\infty$ or $r = \omega$, and give some remarks for the case where $M$ is a compact manifold obtained as the quotient of $\mathbb{H} \times F$ by a subgroup $\Gamma \subset PSL(2,\mathbb{R}) \times {\rm Diff}_+^r(F)$ acting freely and properly discontinuously on $\mathbb{H} \times F$. Like in Section~\ref{Section1}, we denote by $p_1$ and $p_2$ the first and second projection of $PSL(2,\mathbb{R}) \times {\rm Diff}_+^r(F)$ onto $PSL(2,\mathbb{R})$ and ${\rm Diff}_+^r(F)$ respectively. Recall that $M$ admits a foliation $\mathcal{F}$ induced by the horizontal foliation of $\mathbb{H} \times F$ and $\mathcal{F}$ is minimal if and only if $p_2(\Gamma)$ acts minimally on $F$. Denote by $p$ and $q$ the canonical projections $$p : PSL(2,\mathbb{R}) \times F \to \partial \mathbb{H} \times F = PSL(2,\mathbb{R}) / B \times F$$ and $$q : PSL(2,\mathbb{R}) \times F \to X = \Gamma \backslash PSL(2,\mathbb{R}) \times F$$ corresponding to the natural right $B$-action and left $\Gamma$-action on $PSL(2,\mathbb{R}) \times F$.
The first result generalises Example~\ref{secondexample}:
\begin{aproposition} \label{prop5.3} If $p_1(\Gamma)$ is solvable, then the natural right $B$-action on $X$ is not minimal. More precisely, there is a $B$-minimal set homeomorphic to $M$. \end{aproposition}
\begin{proof} Since $p_1(\Gamma)$ is solvable, but it is not included in $PSO(2,\mathbb{R})$ by the compactness of $M$, this group fixes a point $\xi \in \partial \mathbb{H}$. Then $Z = \{\xi\} \times F$ is a $B$-minimal closed subset of $\partial \mathbb{H} \times F$ because $p_2(\Gamma)$ acts minimally on $F$. It follows that $\widetilde{Y} = p^{-1}(Z)$ is a $\Gamma$-invariant and $B$-invariant closed subset of $PSL(2,\mathbb{R}) \times F$, which is homeomorphic to $\mathbb{H} \times F$. Clearly, we deduce that $\widetilde{Y}$ projects onto a $B$-minimal closed set $Y = q(\widetilde{Y}) \subset X$, which is homeomorphic to $M$. \end{proof}
Suppose now that $p_1(\Gamma)$ is not solvable, so $p_1(\Gamma)$ is discrete cocompact or dense. In particular, its action on $\partial \mathbb{H}$ is minimal. Assuming $F$ compact and $\Gamma$ torsion-free, we have the following result:
\begin{aproposition} \label{prop5.5} Assume that $F$ is compact, $\Gamma$ is torsion-free, and $p_1(\Gamma)$ is not solvable. If $p_2$ is not injective, then
the natural right $U$-action on $X$ is minimal. \end{aproposition}
\begin{proof} Assume the projection $p_2$ is not injective. Since the kernel $N$ is normalised by $p_1(\Gamma)$, the group $p_1(\Gamma)$ is discrete cocompact and $N$ is not cyclic. It follows that $\mathcal{F}$ admits leaves which are not homeomorphic to the plane or the cylinder. Moreover, the foliated manifold $M$ is the quotient of $\mathbb{H} \times F$ by the diagonal action $\gamma(z,y) = (\gamma_1(z),\gamma_2(y))$ where $\gamma_1$ is an element of the cocompact discrete subgroup $p_1(\Gamma)$ of $PSL(2,\mathbb{R})$ and $\gamma_2$ is the corresponding element of ${\rm Diff}_+^r(F)$. So $\mathcal{F}$ is obtained as suspension of the representation $h : \gamma_1 \in p_1(\Gamma) \mapsto \gamma_2 \in {\rm Diff}_+^r(F)$.
Now, let us prove that the corresponding $U$-action is minimal. By duality, it is enough to prove that $\Gamma$ acts minimally on the product $E \times F$ where $E = \mathbb{R}^2- \{0\} / \{ \pm Id\}$.
Let $v$ be an element of $E$ such that $\gamma_1 v = \lambda_1 v$ for some $\gamma_1 \in N$ with $|\lambda_1| \neq 1$. Since $N$ acts minimally on $\partial \mathbb{H}$, it is known \cite{DL} that $\overline{Nv} = E$. It follows $\overline{\Gamma(v,y)}$ contains $E \times \{y\}$ for all $y \in F$. Using the minimality of the action of $p_2(\Gamma)$ on $F$, we deduce that $\overline{\Gamma(v,y)} = E \times F$ for all $y \in F$. Indeed, for each point $(w,z) \in E \times F$, there is a sequence $\{ \gamma_n \}_{n \geq 0} = \{(\gamma_{1n},\gamma_{2n})\}_{n \geq 0}$ in $\Gamma$ such that $z = \lim_{n \to +\infty} \gamma_{2n}(y)$. Since $(\gamma_{1n}^{-1}w, y) \in E \times \{ y \} \subset \overline{\Gamma(v,y)}$, we have: $$ (w,z) = \lim_{n \to +\infty} (w, \gamma_{2n}(y)) = \lim_{n \to +\infty} \gamma_n(\gamma_{1n}^{-1}w, y) \in \overline{\Gamma(v,y)}. $$ Finally, since $p_1(\Gamma)$ is discrete cocompact, given any point $(w,z) \in E \times F$, there is another sequence $\{ \gamma'_n \}_{n \geq 0} = \{(\gamma'_{1n},\gamma'_{2n})\}_{n \geq 0}$ in $\Gamma$ such that $v = \lim_{n \to +\infty} \gamma'_{1n}w$. By compactness of $F$, extracting a subsequence if necessary, we may assume that $\gamma'_{2n}(y)$ converges to a point $y' \in F$.Thus $$ \lim_{n \to +\infty}\gamma'_n(w,y) = \lim_{n \to +\infty} ((\gamma'_{1n}w, \gamma'_{2n}(y)) = (v,y') $$ Since $\overline{\Gamma(v,y')} \subset \overline{\Gamma(w,y)}$ and $\overline{\Gamma(v,y')} = E \times F$, we obtain that $\overline{\Gamma(w,y)} = E \times F$. \end{proof}
\subsection*{Note added in proof} A Hedlund's theorem for foliations by hyperbolic surfaces which admit a leaf that contains an essential loop without holonomy has been announced by the authors in collaboration with M. Mart\'{\i}nez and A. Verjovsky \cite{ADMV} after the submission of this paper.
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\begin{document}
\title[regularity of the interface of the Stefan problem] {On the regularity of the interface of a thermodynamically consistent two-phase Stefan problem with surface tension}
\author[J.~Pr\"uss]{Jan Pr\"uss} \address{Institut f\"ur Mathematik,
Martin-Luther-Universit\"at Halle-Wittenberg,
60120 Halle, Germany} \email{[email protected]}
\author[Y. Shao]{Yuanzhen Shao} \address{Department of Mathematics,
Vanderbilt University,
Nashville, TN 37240, USA} \email{[email protected]}
\author[G. Simonett]{Gieri Simonett} \address{Department of Mathematics\\
Vanderbilt University \\
Nashville, TN~37240, USA} \email{[email protected]}
\thanks{The research of the third author was partially supported by NSF DMS-1265579.} \subjclass[2010]{35R35, 82C26, 35K55, 35B65} \keywords{free boundary problems, phase transitions, the Stefan problem, regularity of moving interfaces, real analytic solutions, maximal regularity, the implicit function theorem}
\begin{abstract} We study the regularity of the free boundary arising in a thermodynamically consistent two-phase Stefan problem with surface tension by means of a family of parameter-dependent diffeomorphisms, $L_p$-maximal regularity theory, and the implicit function theorem. \end{abstract} \maketitle
\section{\bf Introduction}
The main objective of this article is to develop a technique relying on a family of parameter-dependent diffeomorphisms, maximal regularity theory, and the implicit function theorem to prove regularity of moving interfaces occurring in free boundary problems. As an application, we prove that the moving interface in a thermodynamically consistent two-phase Stefan problem is jointly $C^k$-smooth in time and space, for $k\in \mathbb{N}\cup \{\infty,\omega\}$ with $\omega$ being the symbol of real analyticity, as long as several physical quantities, that is, the coefficients of the heat conductivity, kinetic undercooling, and the free energy, enjoy appropriate regularity assumptions.
The idea of establishing regularity of solutions to differential equations by means of the implicit function theorem in conjunction with a translation argument was first introduced by S.B.~Angenent in \cite{Ange88} to prove analyticity of the free boundary in one dimensional porous medium equations, and has proved itself a useful tool later in many publications. See for example \cite{Ange902,EscPruSim0302, EscSim96, Lunar95, PruSim07}. More precisely, to study the regularity of the solution to a differential equation, one introduces parameters representing translation in space and time into the solution to the given differential equation. Then one studies the parameter-dependent equation satisfied by this transformed solution. The implicit function theorem yields the smooth dependence upon the parameters of the solution to the parameter-dependent problem. This regularity property is then inherited by the original solution. An advantage of this technique is reflected by its power to prove analyticity of solutions to differential equations, which cannot be attained through the classical method of bootstrapping.
A well-known approach to free boundary problems is to transform the original problem with a moving boundary, or separating interface, which we denote by $\Gamma(t)$, into one with a fixed reference manifold $\Sigma$ by means of the Hanzawa transformation, see \cite{Han81}. Then the problem of establishing the regularity of the free boundary $\Gamma(t)$ is transferred into establishing the regularity of the height function parameterizing $\Gamma(t)$ over $\Sigma$. However, applying the aforementioned translation technique to the height function on the surface $\Sigma$ causes an essential challenge, considering for instance the usual translation $(t,x)\mapsto(t+\lambda,x+\mu)$, because of the global nature of these translations. Hence we desire an alternative that only shifts the variables ``locally".
The idea of localizing translations was first introduced by J.~Escher, J.~Pr\"uss and G.~Simonett in \cite{EscPruSim03} to study regularity of solutions to elliptic and parabolic equations in Euclidean space. The basic building block of \cite{EscPruSim03} consists of rescaling translations by some cutoff function. This technique was later generalized in Y.~Shao \cite{ShaoPre} to introduce a family of parameter-dependent diffeomorphisms acting on functions or tensor fields on Riemannian manifolds by means of a smooth atlas in order to study the regularity of solutions to geometric evolution equations. But in view of the physical quantities in the bulk phases adjacent to $\Gamma(t)$, e.g., the temperature function in the case of phase transitions, or the velocity and the pressure field in the case of two-phase fluids, we need introduce a localized translation not only on the fixed reference surface $\Sigma$, but also in a neighborhood of $\Sigma$. This adds one more degree of complexity to the aforementioned technique for geometric flows. In Section~3, we will build up a complete theory of parameter-dependent diffeomorphisms for free boundary problems.
Free boundary problems form an important field of applied analysis. They deal with solving partial differential equations in a given domain, a part of whose boundary is a priori unknown. That portion of the boundary is called the free boundary or the moving boundary. In addition to the standard boundary conditions that are needed in order to solve the prescribed partial differential equations, an additional condition must be imposed at the free boundary. One then seeks to determine both the free boundary and the solution of the differential equation. This field has drawn great attention over decades due to its applications to physics, chemistry, medicine, material science, and so forth.
Great progress has been seen in the studies of regularity of free boundaries during the past half century. Many mathematicians have made contributions to the theory of free boundary problems, among them H.W.~Alt, H.~Berestycki, L.A.~Caffarelli, A.~Friedman, D.~Kinderlehrer, L.~Nirenberg, J.~Spruck, G.~Stampacchia \cite{AltCaf81,AltCafFdm84, BerCaf90, Caf77, Caf87, KinNir77, KinNir78, KinNirSpr79, KinSta80}, E.~DiBenedetto \cite{Dib82}. We also refer the reader to \cite{FmdRei88,KinSta80} for a historical account of the field up to the 1980s.
The Stefan problem, arguably, is the most studied free boundary problem, with over 1200 mathematical publications devoted to the topic. It was first introduced in 1889 by J.~Stefan. We refer the reader to the books \cite{Mei92, Rub71, Vis96} for further information. The Stefan problem describes phase transitions in liquid-solid systems and accounts for heat diffusion and exchange of latent heat in a homogeneous medium, wherein the liquid and solid phases are separated by a closed moving interface $\Gamma(t)$. The basic physical law governing this process is conservation of energy. To be more precise, let $\Omega\subset \mathbb{R}^{m+1}$ be a bounded domain of class $C^{2}$, $m\geq 1$. $\Omega$ is occupied by a material that can undergo phase changes: at time $t$, phase $i$ occupies the subdomain $\Omega_i(t)$ of $\Omega$, respectively, with $i=1,2.$ We assume that $\partial \Omega_1(t)\cap\partial \Omega=\emptyset$; this means that no {\em boundary contact} can occur. The closed compact hypersurface $\Gamma(t):=\partial \Omega_1(t)\subset \Omega$ forms the interface between the phases. Then the {\em Stefan problem with surface tension}, or the \emph{Stefan problem with Gibbs-Thomson correction}, can be formulated as follows: \begin{equation} \label{S1: Stefan-0} \left\{\begin{aligned} \partial_t \vartheta-d\Delta \vartheta&=0 &&\text{in}&&\Omega\setminus\Gamma(t),\\ \partial_{\nu_{\partial\Omega}}\vartheta&=0 &&\text{on}&&\partial \Omega, \\ V(t)-[\![d\partial_{\nu_\Gamma}\vartheta]\!]&=0 &&\text{on}&&\Gamma(t),\\ \vartheta+\sigma \mathcal{H} &=0 &&\text{on}&&\Gamma(t), \\
\vartheta(0)=\vartheta_0, \quad \Gamma(0) &=\Gamma_0. && \end{aligned}\right. \end{equation} Here
$\vartheta(t)=\vartheta_1(t)\chi_{\Omega_1(t)} + \vartheta_2(t)\chi_{\Omega_2(t)}$, where $\vartheta_i$ denotes the relative temperature distribution in phase $i$,
$\nu_\Gamma(t)$ the outer normal field of $\partial\Omega_1(t)$,
$V(t)$ the normal velocity of $\Gamma(t)$,
$\mathcal{H}(t)=\mathcal{H}(\Gamma(t))=-{\rm div}_{\Gamma(t)} \nu_\Gamma(t)/m$ the mean curvature of $\Gamma(t)$, and
$[\![v]\!]=v_2|_{\Gamma(t)}-v_1|_{\Gamma(t)}$ the jump of a quantity $v$ across $\Gamma(t)$. The sign of the mean curvature $\mathcal{H}$ is chosen to be negative at a point $x\in\Gamma$ if $\Omega_1\cap \mathbb{B}(x,r)$ is convex for some sufficiently small $r>0$. Thus if $\Omega_1$ is a ball of radius $R$ then $\mathcal{H}=-1/R$ for its boundary $\Gamma$. The function $d$ agrees with a positive constant $d_i$ in $\Omega_i(t)$, which stand for the heat conductivities in different phases.
The condition that $V(t)=[\![d\partial_{\nu}\vartheta]\!]$ is caused by the law of conservation of energy and is usually called the {\em Stefan condition}.
If a pair $(\vartheta,\Gamma)$ solves the system of equations (1.1),
$\Gamma$ is called the free boundary to the Stefan problem. In the case that the condition $\vartheta+\sigma \mathcal{H} =0$ is replaced by \begin{equation} \label{S1: GT} \vartheta =0 \quad\text{on}\quad\Gamma(t), \end{equation} i.e., if $\sigma=0$, the resulting problem is usually referred to as the \emph{classical Stefan problem}. Here we mention the monographs by B.~Chalmers~\cite[Chapter~1]{Cha77}, P.~Hartman~\cite{Har73}, A.~Visintin~\cite{Vis96}, and the research papers by G.~Caginalp~\cite{Cag86}, M.E.~Gurtin \cite{Gur86}-\cite{Gur88-2}, J.S.~Langer~\cite{Lan80}, W.W.~Mullins and R.F.~Sekerka~\cite{MuSe63, MuSe64}, A. Visintin~\cite{Vis93}, where this law has been motivated and derived based on various mathematical and physical principles. As has been explained in \cite[Section 1]{PruSimZac12}, see also \cite{PruSimPre}, condition \eqref{S1: GT} can be understood as a first order approximation of the relation \begin{equation} \label{S1: GT-consistent} [\![\psi(\theta)]\!]+\sigma \mathcal{H}=0\quad\text{on}\quad \Gamma(t) \end{equation} around the melting temperature, where $\theta$ is the absolute temperature and $\psi$ is the free energy of the system.
In the absence of one of $\vartheta_i$, that is, one of $\vartheta_i$ is identically equal to some constant, the problem is called the one-phase Stefan problem, otherwise it is alluded to as the two-phase Stefan problem.
In the one-dimensional case, different aspects of the \emph{classical Stefan problem} were extensively studied before the 1980s. In higher dimensions, global existence and uniqueness of weak solutions was first established by S.L.~Kamenomostskaja in \cite{Kam61}, see also A.~Friedman \cite{Fmd68}. Existence of local classical solutions was obtained by E.I.~Hanzawa \cite{Han81} and A.M.~Me\v{i}rmanov \cite{Mei80}, provided that the initial data are smooth enough and satisfy some higher order compatibility conditions.
In the case of the one-phase {\em classical Stefan problem}, regularity of the free boundary for weak solutions is studied in \cite{Caf77,Caf78,FmdKin74,KinNir77,KinNir78}. The formulation of the problem as a parabolic variational inequality was initiated by G.~Duvaut in \cite{Duv73}. After that, it has been applied in \cite{FmdKin74} to get the Lipschitz continuity of the free boundary under some conditions on the initial geometry and the given data. Later in a paper of L.A.~Caffarelli \cite{Caf77}, the author proved that the free boundary is $C^1$ in space and time and the temperature is $C^2$ up to the free boundary near a density point for the states set, namely the solid phase. In particular, if the free boundary is Lipschitz continuous then it is actually $C^1$. In \cite{Caf78}, the author identified conditions to guarantee a locally Lipschitz free boundary. Almost at the same time, D.~Kinderlehrer and L.~Nirenberg \cite{KinNir77} established via the partial hodograph and Legendre transformations in conjunction with a variational inequality argument smoothness of the free boundary as well as the solution. The conditions in \cite{KinNir77} can be verified by the results in \cite{Caf77}. In a subsequent paper \cite{KinNir78}, the same authors showed under some integral conditions by the same technique that the free boundary and the solution are analytic in the space variables and in the second Gevrey class for the time variable. Regularization of the free boundary for large time was obtained by H.~Matano in \cite{Mat82}, i.e., any weak solution eventually becomes smooth. Continuity of the temperature distribution for weak solutions is proved in \cite{CafFmd79}.
Regularity of the free boundary for weak solutions or viscosity solutions has been explored in \cite{AthCaf96,AthCafSal98,Koc98} for the multi-dimensional two-phase Stefan problem based on a non-degeneracy condition. The non-degeneracy condition states, roughly speaking, that the heat fluxes are not vanishing simultaneously on the free boundary, which is automatically satisfied when the free boundary is regular enough, say belongs to the H\"older class of order more than $1$. Assuming that the free boundary is locally Lipschitz continuous, it was proven in \cite{AthCaf96} that the solution is classical, i.e., the free boundary and the viscosity solution are $C^1$. Later the author showed that the non-degeneracy condition can be replaced by some flatness condition. In the presence of the non-degeneracy condition, H.~Koch verified that the $C^1$ free boundary and the temperature are in fact smooth. Through a different approach by imposing a mild regularity condition on the initial data, it was shown by J.~Pr\"uss, J.~Saal and G.~Simonett in \cite{PruSim07} that the free boundary and the temperature are analytic under the assumption that the free boundary can be expressed as the graph of some function. Continuity of the temperature distribution for weak solutions was obtained in \cite{CafEva83,Dib80,Dib82,Zie82}.
Although the Stefan problem with the {\em Gibbs-Thomson correction} \eqref{S1: GT} has been around for many decades, only few analytical results concerning existence of solutions can be found in the literature. A.~Friedman and F.~Reitich \cite{FmdRei91} considered the case with small surface tension $0<\sigma\ll 1$ and linearized the problem about $\sigma=0$. Assuming the existence of a smooth solution for the case $\sigma=0$, that is, for the {\em classical Stefan problem}, the authors proved existence and uniqueness of a weak solution for the \textit{linearized} problem and then investigated the effect of small surface tension on the shape of $\Gamma(t)$.
Existence of long time weak solutions was first established by Luckhaus~\cite{Luc90}, using a discretized problem and a capacity-type estimate for approximating solutions. The weak solutions obtained have a sharp interface, but are highly non-unique. See also R\"oger~\cite{Roe04}, and Almgren and Wang~\cite{AlWa00}. Existence of classical solutions, but without uniqueness, was proved by E.V.~Radkevich in \cite{Rad91}. In A.M. Me\v{i}rmanov~\cite{Mei94}, the way in which a spherical ball of ice in a supercooled fluid melts down was investigated. It was presented by J.~Escher, J.~Pr\"uss and G.~Simonett in \cite{EscPruSim0302} that there exists a unique local strong solution and the free boundary immediately regularizes to be analytic in space and time provided that the initial data satisfy some mild regularity assumptions. This result is based on the assumption that the free boundary is given by the graph of some function. In J.~Pr\"uss and G.~Simonett~\cite{PruSim08}, linearized stability and instability of equilibria was investigated. Nonlinear stability results were obtained by M.~Had{\v{z}}i{\'c}, Y.~Guo~\cite{Had12,HadGuo10} and J.~Pr\"uss, G.~Simonett and R.~Zacher~\cite{PruSimZac12}.
In this paper we consider a general model for phase transitions, formulated in \cite{PruSimZac12}, that is thermodynamically consistent, see \cite{Gur07} and \cite{Ish06} for related work. It involves the thermodynamic quantities of absolute temperature, free energy, internal energy, and entropy, and is complemented by constitutive equations for the free energies and the heat fluxes in the bulk regions. An important assumption is that there be no entropy production on the interface. In particular, the interface is assumed to carry no mass and no energy except for surface tension.
To be more precise, we choose $\Omega$, $\Omega_i(t)$ and $\Gamma(t)$ as in \eqref{S1: Stefan-0}. By the thermodynamically consistent two-phase {\em Stefan problem with surface tension}, possibly with kinetic undercooling, we mean the following problem: find a family of closed compact hypersurfaces $\{\Gamma(t)\}_{t\geq0}$ contained in $\Omega$ and an appropriately smooth function $\theta:\mathbb{R}_+\times\bar{\Omega}\rightarrow\mathbb{R}$ such that \begin{equation} \label{stefan} \left\{\begin{aligned} \kappa (\theta)\partial_t \theta-{\rm div}(d(\theta)\nabla \theta)&=0 &&\text{in}&&\Omega\setminus\Gamma(t),\\ \partial_{\nu_\Omega} \theta &=0 &&\text{on}&&\partial \Omega, \\ [\![\theta]\!]&=0 &&\text{on}&&\Gamma(t),\\ [\![\psi(\theta)]\!]+\sigma \mathcal{H} &=\gamma(\theta) V &&\text{on}&&\Gamma(t), \\ [\![d(\theta)\partial_{\nu_\Gamma} \theta]\!] &=(l(\theta)-\gamma(\theta)V) V &&\text{on}&&\Gamma(t),\\ \theta(0)=\theta_0, \quad \Gamma(0)&=\Gamma_0. && \end{aligned}\right. \end{equation} Here $\theta$ denotes the (absolute) temperature. Several quantities are derived from the free energies $\psi_i(\theta)$ as follows: \begin{itemize} \item
$\epsilon_i(\theta)= \psi_i(\theta)+\theta\eta_i(\theta)$, the internal energy in phase $i$, \item
$\eta_i(\theta) =-\psi_i^\prime(\theta)$, the entropy, \item
$\kappa_i(\theta)= \epsilon^\prime_i(\theta)=-\theta\psi_i^{\prime\prime}(\theta)>0$, the heat capacity, \item $l(\theta)=\theta[\![\psi^\prime(\theta)]\!]=-\theta[\![\eta(\theta)]\!]$, the latent heat. \end{itemize} Furthermore, $d_i(\theta)>0$ denotes the coefficient of heat conduction in Fourier's law,
$\gamma(\theta)\geq0$ the coefficient of kinetic undercooling. As is commonly done, we assume that there exists a unique (constant) {\em melting temperature} $\theta_m$, characterized by the equation $[\![\psi(\theta_m)]\!]=0.$ Finally, system \eqref{stefan} is to be completed by constitutive equations for the free energies $\psi_i$ in the bulk phases $\Omega_i(t)$.
In the sequel we drop the index $i$, as there is no danger of confusion; we just keep in mind that the coefficients depend on the phases. The temperature is assumed to be continuous across the interface, as indicated by the condition $[\![\theta]\!]=0$ in \eqref{stefan}. However, the free energy and the heat conductivities depend on the respective phases, and hence the jumps $[\![\psi(\theta)]\!]$, $[\![\kappa(\theta)]\!]$, $[\![\eta(\theta)]\!]$, $[\![d(\theta)]\!]$ are in general non-zero at the interface. In this paper we assume that the coefficient of surface tension is constant.
In this paper, we will prove the following regularity result for $k\in \mathbb{N}\cup \{\infty,\omega\}$. \begin{theorem} \label{S1: main theorem: gamma=0} {\rm ($\gamma \equiv 0$).} Let $p>m+3$, $\gamma=0$, $\sigma>0$. Suppose that $d_i\in C^{k+2}(0,\infty)$, $\psi_i\in C^{k+3}(0,\infty)$ for $i=1,2$ such that $$\kappa_i(u)=-u\psi_i^{\prime\prime}(u)>0,\quad d_i(u)>0,\quad u\in(0,\infty).$$ Assume the {\em regularity conditions} $$\theta_0\in W^{2-2/p}_p(\Omega\setminus\Gamma_0)\cap C(\bar{\Omega}),\quad \theta_0>0, \quad \Gamma_0\in W^{4-3/p}_p,$$ the {\em compatibility conditions} $$ \partial_{\nu_\Omega}\theta_0=0,\quad [\![\psi(\theta_0)]\!]+\sigma {\mathcal{H}}(\Gamma_0)=0, \quad [\![d(\theta_0)\partial_{\nu_{\Gamma_0}} \theta_0]\!]\in W^{2-6/p}_p(\Gamma_0),$$ and the {\em well-posedness condition} $$\quad l(\theta_0)\neq0\quad\mbox{on}\; \Gamma_0.$$ Then there exists a unique $L_p$-solution $(u,\Gamma)$ for the Stefan problem with surface tension \eqref{stefan} on some possibly small but nontrivial time interval $J=[0,T]$, and $$
\mathcal{M}:=\bigcup_{t\in(0,T)}\{\{t\}\times\Gamma(t)\} $$ is a $C^k$-manifold in $\mathbb{R}^{m+2}$. In particular, each manifold $\Gamma(t)$ is $C^k$ for $t\in (0,T)$. \end{theorem} The result in the presence of kinetic undercooling reads as \begin{theorem} \label{S1: main theorem: gamma>0} {\rm ($\gamma > 0$).} Let $p>m+3$, $\sigma>0$. Suppose that $d_i, \gamma \in C^{k+2}(0,\infty)$,
$\psi_i\in C^{k+3}(0,\infty)$ for $i=1,2$ such that $$\kappa_i(u)=-u\psi_i^{\prime\prime}(u)>0,\quad d_i(u)>0,\quad \gamma(u)>0,\quad u\in(0,\infty).$$ Assume the {\em regularity conditions} $$\theta_0\in W^{2-2/p}_p(\Omega\setminus\Gamma_0)\cap C(\bar{\Omega}),\quad \theta_0>0,\quad \Gamma_0\in W^{4-3/p}_p,$$ and the {\em compatibility conditions} \begin{equation*} \begin{split} \partial_{\nu_\Omega}\theta_0=0, \;\; \big([\![\psi(\theta_0)]\!]+\sigma \mathcal{H}(\Gamma_0)\big)\big(l(\theta_0)-[\![\psi(\theta_0)]\!]-\sigma \mathcal{H}(\Gamma_0)\big)=\gamma(\theta_0)[\![d(\theta_0)_{\Gamma_0} \theta_0]\!]. \end{split} \end{equation*} Then there exists a unique $L_p$-solution $(\theta,\Gamma)$ for the Stefan problem with surface tension \eqref{stefan} on some possibly small but nontrivial time interval $J=[0,T]$, and $$ \mathcal{M}:=\bigcup_{t\in(0,T)}\{\{t\}\times\Gamma(t)\} $$ is a $C^k$-manifold in $\mathbb{R}^{m+2}$. In particular, each manifold $\Gamma(t)$ is $C^k$ for $t\in (0,T)$. \end{theorem}
\begin{remark} \label{S1: main RMK} (a) For $k\in \mathbb{N}\cup\{\infty\}$, under the conditions in Theorems~\ref{S1: main theorem: gamma=0} and \ref{S1: main theorem: gamma>0}, we can show that the temperature satisfies $$\theta \in C^k(((0,T)\times\Omega)\setminus \mathcal{M}). $$ See Remark~\ref{S5: Proof of main RMK} for a justification. However, in the case $k=\omega$, additional work is needed to establish the interior analyticity of $\theta$ in the bulk phases. In order to keep this already long paper at a reasonable length, we will refrain from establishing this here.
\\ (b) According to \cite[Theorem 3.9]{PruSimZac12} we obtain a solution of \eqref{stefan} in the state manifold $\mathcal{SM_\gamma}$ on a maximal interval of existence $[0,t_*)$. The regularity assertions of Theorem~\ref{S1: main theorem: gamma=0} and Theorem~\ref{S1: main theorem: gamma>0}, respectively, then hold true on $[0,t_*)$. \qed \end{remark}
\textbf{Notations:} Throughout this paper, we always assume that \begin{mdframed} \begin{itemize}
\item $E$ denotes a finite dimensional Banach space. \item $m+3<p<\infty$, and $s\geq 0$, unless stated otherwise. \end{itemize} \end{mdframed}
Given two Banach spaces $X,Y$, the notation $\mathcal{L}(X,Y)$ stands for the set of all bounded linear operators from $X$ to $Y$, and $\mathcal{L}{\rm{is}}(X,Y)$ denotes the set of all bounded linear isomorphisms from $X$ to $Y$.
\\
For any topological sets $U$ and $V$, $U\subset\subset V$ means that $\bar{U}\subset \mathring{V}$ with $\bar{U}$ compact.
\section{\bf Parameterization over a fixed interface}
Let $\Omega\subset\mathbb{R}^{m+1}$ be a bounded domain with boundary $\partial\Omega$ of class $C^2$, and suppose that $\Gamma_0\subset \Omega$ is a closed embedded hypersurface of class $C^2$, that is, a $C^2$-manifold which is the boundary of a bounded domain $\Omega_1(0)\subset \Omega$. Set $\Omega_2(0):=\Omega \setminus \bar{\Omega}_1(0)$. Following \cite{PruSimZac12}, we may approximate $\Gamma_0$ by an $m$-dimensional real analytic compact closed oriented reference hypersurface $(\Sigma,g)$, with $g$ the tangential metric induced by the Euclidean metric $g_{m+1}$, in the sense that there exists a function $h_0\in C^2(\Sigma,(-a,a))$ such that the map \begin{align*} \Lambda_{h_0}:\Sigma\rightarrow \mathbb{R}^{m+1}:\quad [\mathsf{p}\mapsto \mathsf{p}+h_0(\mathsf{p}) \nu_\Sigma(\mathsf{p})] \end{align*} is a diffeomorphism from $\Sigma$ onto $\Gamma_0$, where $\nu_\Sigma$ is the outer normal of $\Sigma$. The positive constant $a$ depends on the inner and outer ball conditions of $\Sigma$. It is well-known that $\Sigma$ admits a $a$-tubular neighborhood ${\sf T}_a$, which means that the map \begin{align*} \Lambda:\Sigma\times (-a,a)\rightarrow\mathbb{R}^{m+1}:\hspace*{.5em}(\mathsf{p},r)\mapsto \mathsf{p}+r \nu_\Sigma(\mathsf{p}) \end{align*} is a diffeomorphism from the fiber bundle $\Sigma\times (-a,a)$ onto ${{\rm{im}}}(\Lambda):={\sf T}_a$. For sufficiently small $a$, $\bar{{\sf T}}_a\subset\Omega$. $\Sigma$ bounds a domain $\Omega_1$, and we set $\Omega_2=\Omega\setminus \bar{\Omega}_1$. The inverse $\Lambda^{-1}$ can be decomposed as \begin{align*} \Lambda^{-1}:{\sf T}_a \rightarrow \Sigma\times(-a,a): \hspace*{.5em} z\mapsto (\Pi(z),d_\Sigma(z)), \end{align*}
where $\Pi(z)$ is the metric projection of $z$ onto $\Sigma$ and $d_\Sigma(z)$ denotes the signed distance from $z$ to $\Sigma$ such that $|d_\Sigma(z)|={\rm dist}(z,\Sigma)$ and $d_\Sigma(z)<0$ iff $z\in \Omega_1$. It follows from the inverse function theorem that the maps $\Pi$ and $d_\Sigma$ are both real analytic.
\\ We may use the map $\Lambda$ to parameterize the unknown free boundary $\Gamma(t)$ over $\Sigma$ by a height function $h(t):\Sigma\to \mathbb{R}$ via \begin{align*} \Gamma(t):=\{ \mathsf{p}+h(t,\mathsf{p})\nu_\Sigma(\mathsf{p}),\quad \mathsf{p}\in\Sigma\},\quad t\geq 0, \end{align*}
for small $t\geq 0$, at least. $\Gamma(t)$ bounds a bounded region $\Omega_1(t)$. Put $\Omega_2(t)=\Omega\setminus \bar{\Omega}_1(t)$. Choosing an auxiliary function $\zeta\in \mathcal{D}((-2a/ 3,2a/ 3),[0,1])$ such that $\zeta|_{(-a/ 3,a/ 3)}\equiv 1$, we may extend the above diffeomorphism onto $\bar{\Omega}$ via \begin{align*} \Xi_h(t,z)=z+\zeta(d_\Sigma(z))h(t,\Pi(z))\nu_\Sigma(\Pi(z))=:z+\Upsilon(h)(t,z). \end{align*} We have transformed the time varying regions $\Omega_i(t)$ to the fixed domains $\Omega_i$. This is the direct mapping method, also known as Hanazawa transformation. By means of this transformation, we obtain the following transformed problem for $\vartheta(t,z):=\theta(t,\Xi_h(t,z))$: \begin{align} \label{S2: transf Stefan} \left\{\begin{aligned} \kappa(\vartheta)\partial_t \vartheta+\mathcal{A}(\vartheta,h)\vartheta&=\kappa(\vartheta)\mathcal{R}(h)\vartheta &&\text{in}&&\Omega\setminus\Sigma,\\ \partial_{\nu_{\Omega}}\vartheta&=0 &&\text{on}&&\partial\Omega,\\ [\![\vartheta]\!]&=0 &&\text{on}&&\Sigma,\\ [\![\psi(\vartheta)]\!]+\sigma{\mathcal{H}}(h)&=\gamma(\vartheta)\beta(h)\partial_t h &&\text{on}&& \Sigma, \\ \{l(\vartheta)-\gamma(\vartheta)\beta(h)\partial_t h \}\beta(h)\partial_t h +\mathcal{B}(\vartheta,h)\vartheta&=0 &&\text{on}&& \Sigma,\\ \vartheta(0)=\vartheta_0, \quad h(0)&=h_0. && \end{aligned}\right. \end{align} Here $\mathcal{A}(\vartheta,h)$ and $\mathcal{B}(\vartheta,h)$ denote the transformations of $-{\rm div}(d\nabla )$ and $-[\![d\partial_{\nu_\Gamma}]\!]$, respectively. Moreover, ${\mathcal{H}}(h) $ stands for the mean curvature of the hypersurface $$\Gamma_h := \Lambda_h ( \Sigma).$$
The term $\beta(h)\partial_t h$ represents the normal velocity $V$ with $\beta(h):=(\nu_\Sigma|\nu_\Gamma(h))$, where $\nu_\Gamma(h)$ is the outer normal field of $\Gamma_h$, and $$\mathcal{R}(h)\vartheta :=\partial_t \vartheta -\partial_t \theta\circ \Xi_h.$$ It is shown in \cite{PruSimZac12} that $\nu_\Gamma(h)=\beta(h) (\nu_\Sigma -\alpha(h))$, and
$$\beta(h)=(1+|\alpha(h)|^2)^{-1/2},\hspace{1em} \mathcal{R}(h)\vartheta=(\nabla \vartheta|{[I + \nabla\Upsilon(h)^{\sf T}]}^{-1}\partial_t\Upsilon(h) ). $$ Here with the Weingarten map $L_\Sigma=-\nabla_\Sigma \nu_\Sigma$. We have \begin{align*} \alpha(h):=M_0(h)\nabla_\Sigma h,\quad \text{where}\quad M_0(h)&:=(I- h L_\Sigma)^{-1}. \end{align*} With $\partial_\nu:= \partial_{\nu_{\Sigma}}$, the operator $\mathcal{B}(\vartheta,h)$ becomes \begin{align*}
\mathcal{B}(\vartheta,h)\vartheta&= -[\![d(\theta)\partial_{\nu_\Gamma} \theta]\!]\circ\Xi_h=-([\![d(\vartheta)(I-M_1(h))\nabla \vartheta]\!]|\nu_\Gamma)\\
&= -\beta(h)([\![d(\vartheta)(I-M_1(h))\nabla \vartheta]\!]|\nu_\Sigma -\alpha(h))\\ &= -\beta(h)[\![d(\vartheta)\partial_{\nu} \vartheta]\!]
+\beta(h)([\![d(\vartheta)\nabla \vartheta]\!]|(I-M_1(h))^{\sf T}\alpha(h)), \end{align*} where $M_1(h):=[(I+\nabla\Upsilon(h)^{\sf T})^{-1}\nabla\Upsilon(h)^{\sf T}]^{\sf T}$, and finally \begin{align*}
\mathcal{A}(\vartheta,h)\vartheta= & -{\rm div}( d(\theta)\nabla \theta)\circ\Xi_h= -((I-M_1(h))\nabla|d(\vartheta)(I-M_1(h))\nabla \vartheta)\\
= & -d(\vartheta)\Delta \vartheta + d(\vartheta)[M_1(h)+M_1^{\sf T}(h)-M_1(h)M_1^{\sf T}(h)]:\nabla^2 \vartheta\\
&-d^\prime(\vartheta)|(I-M_1(h))\nabla \vartheta|^2
+ d(\vartheta)((I-M_1(h)):\nabla M_1(h)|\nabla \vartheta). \end{align*} We recall that, for matrices $A,B\in\mathbb{R}^{n\times n}$, $ A:B=\sum_{i,j=1}^n a_{ij}b_{ij}=\text{tr}\,(AB^{\sf T}) $ denotes the inner product.
We set \begin{align*} &\mathbb{E}_{1}(J):=\{\vartheta\in H^1_p(J;L_p(\Omega))\cap L_p(J; H^2_p(\Omega\setminus\Sigma)): [\![\vartheta]\!]=0, \partial_{\nu_\Omega}\vartheta=0\},\\
&\mathbb{E}_{2}(J):= \begin{cases} W^{3/2-1/2p}_p(J; L_p(\Sigma))\cap W^{1-1/2p}_p(J; H^2_p(\Sigma))\cap L_p(J; W^{4-1/p}_p(\Sigma)),\\ \hspace{27em} \gamma\equiv 0,\\ W^{2-1/2p}_p(J;L_p(\Sigma))\cap L_p(J;W^{4-1/p}_p(\Sigma)),\hspace{9.3em} \gamma>0, \end{cases}\\ &\mathbb{E}(J):=\mathbb{E}_{1}(J)\times\mathbb{E}_{2}(J), \end{align*} that is, $\mathbb{E}(J)$ denotes the solution space for \eqref{S2: transf Stefan}.
Similarly, we define \begin{align} \label{S2: def of Fj} \left\{\begin{aligned} &\mathbb{F}_{1}(J):=L_p(J;L_p(\Omega)),\\ &\mathbb{F}_{2}(J):=W^{1-1/2p}_p(J;L_p(\Sigma))\cap L_p(J; W^{2-1/p}_p(\Sigma)),\\ &\mathbb{F}_{3}(J):=W^{1/2-1/2p}_p(J;L_p(\Sigma))\cap L_p(J; W^{1-1/p}_p(\Sigma)),\\ &\mathbb{F}_{4}:=[W^{2-2/p}_p(\Omega\setminus\Sigma)\cap C(\bar{\Omega})]\times W^{4-3/p}_p(\Sigma). \end{aligned}\right. \end{align} A left subscript zero means vanishing time trace at $t=0$, whenever it exists. So for example $\prescript{}{0}{\mathbb{E}_{2}(J)}=\{h\in \mathbb{E}_{2}(J):\; h(0)=\partial_t h(0)=0\}$ for $p>3$.
\\ Whenever $(J)$ is replaced by $(J,U)$, or $(J,U;E)$ for any set $U$, e.g., in $\mathbb{E}(J)$, it always means that in the corresponding spaces the original spatial domain is replaced by $U$ and the functions become $E$-valued in the latter case. For example, \begin{align*} \mathbb{F}_2(J,{\sf T}_a;E):=W^{1-1/2p}_p(J;L_p({\sf T}_a,E))\cap L_p(J;W^{2-1/p}_p({\sf T}_a,E)). \end{align*}
We equip $\Sigma$ with a normalized real analytic atlas $\mathfrak{A}=(\mathsf{O}_{\kappa},\varphi_{\kappa})_{\kappa\in\frak{K}}$, where $\frak{K}$ is a finite index set, in the sense that $\varphi_{\kappa}(\mathsf{O}_{\kappa})=\mathsf{Q}^{m}:=(-1,1)^m$, for every $\kappa$. Let $\psi_{\kappa}:=[\varphi_{\kappa}]^{-1}$. Then we may endow $\Omega$ with a real analytic atlas $\mathfrak{A}_{\frak{e}}=(\mathsf{O}_{\frak{e},\kappa},\varphi_{\frak{e},\kappa})_{\kappa\in \frak{K}_{\frak{e}}}$ with $\frak{K}_{\frak{e}}:=\frak{K}\cup\{\kappa_1,\kappa_2\}$ by the following construction. When $\kappa\in\frak{K}$ \begin{align*} \mathsf{O}_{\frak{e},\kappa}:={\sf T}_{a,\kappa}:=\Lambda({\mathsf{O}_{\kappa}\times(-a,a)}),\hspace*{1em} \varphi_{\frak{e},\kappa}(z):=(\varphi_{\kappa}\circ\Pi(z),d_\Sigma(z)),\hspace*{.5em}z\in\mathsf{O}_{\frak{e},\kappa}. \end{align*} Let $\mathsf{Q}_a:=\mathsf{Q}^{m}\times(-a,a)$ and $$\psi_{\frak{e},\kappa}:=[\varphi_{\frak{e},\kappa}]^{-1}:\mathsf{Q}_a \rightarrow \mathsf{O}_{\frak{e},\kappa}: \quad[(x,y)\mapsto \psi_{\kappa}(x)+y\nu_{\Sigma}(\psi_{\kappa}(x))]$$ with $x\in\mathsf{Q}^{m}$ and $y\in(-a,a)$. For $\kappa\notin\frak{K}$, we set $$ (\mathsf{O}_{\frak{e},\kappa_i},\varphi_{\frak{e},\kappa_i}):=(\mathsf{O}_{\frak{e},\kappa_i},{\rm{id}}_{\mathsf{O}_{\frak{e},\kappa_i}}),\quad i=1,2, $$ where $\mathsf{O}_{\frak{e},\kappa_i}:=\Omega_i\setminus \bar{{\sf T}}_{a/6}$ for $i=1,2$. It is immediate from our construction that the atlas $\mathfrak{A}_{\frak{e}}$ is real analytically compatible with the Euclidean structure of $\Omega$. Moreover, all the transition maps are $C^\infty$-continuous and have bounded derivatives, which we refer to as $BC^\infty$-continuous.
\\ A family $(\pi_\kappa)_{\kappa\in\frak{K}}$ is called a localization system subordinate to $\mathfrak{A}$ if \begin{center} $\pi_\kappa\in\mathcal{D}(\mathsf{O}_{\kappa},[0,1])$ and $(\pi_{\kappa}^2)_{\kappa\in\frak{K}}$ is a partition of unity subordinate to $(\mathsf{O}_{\kappa})_{\kappa\in\frak{K}}$. \end{center} Based on \cite[Lemma~3.2]{Ama13}, $\Sigma$ admits a localization system $(\pi_\kappa)_{\kappa\in\frak{K}}$ subordinate to $\mathfrak{A}$. Using the cut-off function $\zeta$ introduced above, we may construct a new localization system $(\pi_{\frak{e},\kappa})_{\kappa\in\frak{K}_{\frak{e}}}$ subordinate to $\mathfrak{A}_{\frak{e}}$ as follows: \begin{itemize} \item[(L1)] For any $\kappa\in\frak{K}$, $\pi_{\frak{e},\kappa}\in \mathcal{D}(\mathsf{O}_{\frak{e},\kappa},[0,1])$ is defined by \begin{align*} \pi_{\frak{e},\kappa}(z):= \begin{cases} \zeta(d_\Sigma(z))\pi_\kappa(\Pi(z)), \hspace{1em}& z\in \mathsf{O}_{\frak{e},\kappa},\\ 0, &z\notin \mathsf{O}_{\frak{e},\kappa}. \end{cases} \end{align*}
\item[(L2)] For $i=1,2$, $\pi_{\frak{e},\kappa_i}\in C^\infty(\bar{\mathsf{O}}_{\frak{e}_i,\kappa_i},[0,1])$ is defined by \begin{align*} \pi_{\frak{e},\kappa_i}(z)= \begin{cases} \displaystyle \frac{1-\zeta(d_\Sigma(z))}{\sqrt{(1-\zeta(d_\Sigma(z)))^2+\zeta^2(d_\Sigma(z))}}, \hspace{1em}& z\in \Omega_i\cap{\sf T}_a,\\ 1, &z\in \bar{\Omega}_i\setminus{\sf T}_a. \end{cases} \end{align*} \end{itemize} $(\pi^2_{\frak{e},\kappa})_{\kappa\in\frak{K}_{\frak{e}}}$ forms a localization system subordinate to $(\mathsf{O}_{\frak{e},\kappa})_{\kappa\in\frak{K}_{\frak{e}}}$.
We put \begin{align*} \mathbb{X}_{\frak{e},\kappa}:= \begin{cases} \mathbb{R}^{m+1},\hspace*{1em}&\kappa\in\frak{K},\\ \mathsf{O}_{\frak{e},\kappa_i}, &i=1,2. \end{cases}
\end{align*} For any finite dimensional Banach space $E$, the maps $\mathcal{R}^c_{\frak{e},\kappa}$ and $\mathcal{R}_{\frak{e},\kappa}$ are defined by \begin{align*} \mathcal{R}^c_{\frak{e},\kappa}: L_{1,loc}(\Omega,E)\rightarrow L_{1,loc}(\mathbb{X}_{\frak{e},\kappa},E):\hspace*{.5em}u \mapsto \psi^{\ast}_{\frak{e},\kappa}\pi_{\frak{e},\kappa} u, \quad \kappa\in \frak{K}_{\frak{e}}, \end{align*} and \begin{align*} \mathcal{R}_{\frak{e},\kappa}:L_{1,loc}(\mathbb{X}_{\frak{e},\kappa},E)\rightarrow L_{1,loc}(\Omega,E): \hspace*{.5em} u\mapsto \pi_{\frak{e},\kappa}\varphi^{\ast}_{\frak{e},\kappa} u, \quad \kappa\in \frak{K}_{\frak{e}}. \end{align*} Here and in the following it is understood that a partially defined and compactly supported map is automatically extended over the whole base manifold by identifying it to be zero outside its original domain. Moreover, let \begin{align*} \mathcal{R}^c_{\frak{e}}: L_{1,loc}(\Omega,E)\rightarrow \prod\limits_{\kappa\in\frak{K}_{\frak{e}}}L_{1,loc}(\mathbb{X}_{\frak{e},\kappa},E):\hspace*{.5em}u \mapsto (\mathcal{R}^c_{\frak{e},\kappa} u)_{\kappa\in\frak{K}_{\frak{e}}}, \end{align*} and \begin{align*} \mathcal{R}_{\frak{e}}:\prod\limits_{\kappa\in\frak{K}_{\frak{e}}} L_{1,loc}(\mathbb{X}_{\frak{e},\kappa},E)\rightarrow L_{1,loc}(\Omega,E): \hspace*{.5em} (u_{\kappa})_{\kappa}\mapsto \sum\limits_{{\kappa\in\frak{K}_{\frak{e}}}}\pi_{\frak{e},\kappa}\varphi^{\ast}_{\frak{e},\kappa} u. \end{align*}
On the manifold $\Sigma$, similar maps $\mathcal{R}^c_{\kappa}$, $\mathcal{R}^c$, $\mathcal{R}_{\kappa}$, and $\mathcal{R}$ are defined in terms of $\pi_\kappa$, $\varphi_{\kappa}$, and $\psi_{\kappa}$. See \cite{ShaoPre}.
\subsection{Function spaces} For any open subset $U\subset\mathbb{R}^n$, the Banach space $BC^{k}(U,E)$ is defined by \begin{align*}
BC^{k}(U,E):=(\{u\in C^k(U,E):\|u\|_{k,\infty}<\infty \},\|\cdot\|_{k,\infty}). \end{align*}
The closed linear subspace $BU\!C^k(U,E)$ of $BC^{k}(U,E)$ consists of all functions $u\in BC^{k}(U,E)$ such that $\partial^{\alpha}u$ is uniformly continuous for all $|\alpha|\leq k$. Moreover, \begin{align*} BC^{\infty}(U,E):=\bigcap_{k}BC^k(U,E)=\bigcap_{k}BU\!C^k(U,E). \end{align*} It is a Fr\'echet space equipped with the natural projective topology.
\\ For $0<s<1$, $0<\delta\leq\infty$ and $u\in E^U$, the seminorm $[\cdot]^\delta_{s,\infty}$ is defined by \begin{align*}
[u]^{\delta}_{s,\infty}:=\sup_{h\in(0,\delta)^n}\frac{\|u(\cdot+h)-u(\cdot)\|_{\infty}}{|h|^s}, \hspace*{1em}[\cdot]_{s,\infty}:=[\cdot]^{\infty}_{s,\infty}. \end{align*} Let $k<s<k+1$. The \textbf{H\"older} space $BC^s(U,E)$ is defined as \begin{align*}
BC^s(U,E):=(\{u\in BC^k(U,E):\|u\|_{s,\infty}<\infty \},\|\cdot\|_{s,\infty}), \end{align*}
where $\|u\|_{s,\infty}:=\|u\|_{k,\infty}+\max_{|\alpha|=k}[\partial^{\alpha} u]_{s-k,\infty}$.
\\ In order to have a general theory that is applicable to situations other than the Stefan problem,
we also introduce the \textbf{little H\"older} space of order $s\geq 0$, which is defined by \begin{center} $bc^s(U,E):=$ the closure of $BC^{\infty}(U,E)$ in $BC^s(U,E)$. \end{center} By \cite[formula~(11.13), Corollary~11.2, Theorem~11.3]{AmaAr}, we have \begin{align*} bc^k(U,E)=BU\!C^k(U,E), \end{align*} and for $k<s<k+1$ \begin{center}
$u\in BC^s(U,E)$ belongs to $bc^s(U,E)$ iff $\lim\limits_{\delta\rightarrow 0}[\partial^{\alpha}u]^{\delta}_{s-[s],\infty}=0$, \hspace{1em} $|\alpha|=[s]$. \end{center} The spaces $\mathfrak{F}^s(\Sigma,E)$ with $\mathfrak{F}\in \{bc,BC,W_p,H_p\}$ are defined in terms of the smooth atlas $\mathfrak{A}$, that is, $u\in\mathfrak{F}^s(\Sigma,E)$ iff $\psi^{\ast}_{\kappa} u:= u \circ \psi_{\kappa} \in \mathfrak{F}^s(\mathsf{Q}^{m},E)$ for all $\kappa\in\frak{K}$. See \cite{Ama13} and \cite{ShaoPre}.
\\ Throughout, for any finite index set $\mathbb{A}$ and Banach spaces $E_\alpha$, $\alpha\in\mathbb{A}$, it is understood that the space $\mathbb{E}:=\prod\limits_{\alpha\in\mathbb{A}}E_\alpha$ is equipped with the maximum norm. For $\mathfrak{F}\in \{bc,BC,BU\!C,W_p,H_p\}$, we put $$\boldsymbol{\mathfrak{F}}^s_{\frak{e}}:=\prod\limits_{\kappa\in\frak{K}_{\frak{e}}}{\mathfrak{F}}^s(\mathbb{X}_{\frak{e},\kappa},E),
\quad \boldsymbol{\mathfrak{F}}^s:=\prod\limits_{\kappa\in\frak{K}}{\mathfrak{F}}^s_{\kappa},$$ where ${\mathfrak{F}}^s_{\kappa}=\mathfrak{F}^s(\mathbb{R}^m,E)$. \begin{prop} \label{S2: retraction} Suppose that $\frak{B}=\mathfrak{F}$ when $\mathfrak{F}\in \{bc, W_p,H_p\}$, or $\frak{B}\in \{ BC, BU\!C\}$ when $\mathfrak{F}=BC$. Then \begin{itemize} \item[(a)] $\mathcal{R}$ is a retraction from $\boldsymbol{\frak{B}}^s$ onto $\mathfrak{F}^s(\Sigma,E)$ with $\mathcal{R}^c$ as a coretraction. Moreover, the case $s<0$ and $\mathfrak{F}\in \{W_p,H_p\}$ is also admissible. \item[(b)] $\mathcal{R}_{\frak{e}}$ is a retraction from $\boldsymbol{\frak{B}}^s_{\frak{e}}$ onto $\mathfrak{F}^s(\Omega,E)$ with $\mathcal{R}^c_{\frak{e}}$ as a coretraction.
\end{itemize} \end{prop} \begin{proof} (a) The assertions are special cases of \cite[Theorem~9.3]{AmaAr} and \cite[Propositions~2.1, 2.2]{ShaoPre}.
\\ (b) For $\kappa\in\frak{K}$, the boundedness of the maps $\mathcal{R}_{\frak{e},\kappa}$ and $\mathcal{R}^c_{\frak{e},\kappa}$ follows from the proofs in \cite[Section~6, 7]{Ama13} and \cite[Section~12]{AmaAr}. The continuity of the maps $\mathcal{R}^c_{\frak{e},\kappa_i}$ and $\mathcal{R}_{\frak{e},\kappa_i}$ is straightforward.
\end{proof} Let $J:=[0,T]$ for some $T>0$. Due to the temporal independence of the above retraction-coretraction systems, we readily infer that \begin{prop} \label{S2: retraction-involving time} Let $r\geq 0$. Suppose that $\frak{C}\in\{C,W_p\}$, and $\frak{B}=\mathfrak{F}$ when $\mathfrak{F}\in \{bc, W_p,H_p\}$, or $\frak{B}\in \{ BC, BU\!C\}$ when $\mathfrak{F}=BC$. Then \begin{itemize} \item[(a)] $\mathcal{R}$ is a retraction from $\frak{C}^r(J;\boldsymbol{\frak{B}}^s)$ onto $\frak{C}^r(J;\mathfrak{F}^s(\Sigma,E))$ with $\mathcal{R}^c$ as a coretraction. Moreover, the case $s<0$ and $\mathfrak{F}\in \{W_p,H_p\}$ is also admissible. \item[(b)] $\mathcal{R}_{\frak{e}}$ is a retraction from $\frak{C}^r(J;\boldsymbol{\frak{B}}^s_{\frak{e}})$ onto $\frak{C}^r(J;\mathfrak{F}^s(\Omega,E))$ with $\mathcal{R}^c_{\frak{e}}$ as a coretraction.
\end{itemize} \end{prop}
\section{\bf Parameter-dependent diffeomorphisms}
\subsection{\bf Model diffeomorphisms in Euclidean space}
Given $z_c\in {\sf T}_{a/3,{\kappa_c}}\setminus\Sigma \subset\Omega$ for some ${\kappa_c}\in \frak{K}$,
let $$(x_c,y_c)=(\varphi_{{\kappa_c}}\circ\Pi(z_c),d_\Sigma(z_c)).$$ We may choose a sufficiently small positive constant $\varepsilon_0$ such that \begin{align*} \mathbb{B}^m(x_c,5\varepsilon_0)\subset\mathsf{Q}^{m},\hspace*{1em} \mathbb{B}(y_c,5\varepsilon_0)\subset(-1)^i(0,a/3). \end{align*} Let $\mathbb{B}_{\varepsilon,\alpha}:=\mathbb{B}^m(x_c,\varepsilon)\times(-\alpha,\alpha)$. We pick several auxiliary functions in the following manner: \begin{itemize}
\item $\varpi \in\mathcal{D}(-13a/18,13a/18),[0,1])$, and $\varpi|_{(-2a/3,2a/3)}\equiv 1$.
\item $\chi_m\in\mathcal{D}(\mathbb{B}^m(x_c,2\varepsilon_0),[0,1])$, and $\chi_m|_{\mathbb{B}^m(x_c,\varepsilon_0)}\equiv 1$.
\item $\chi\in\mathcal{D}(\mathbb{B}(y_c,2\varepsilon_0),[0,1])$, and $\chi|_{\mathbb{B}(y_c,\varepsilon_0)}\equiv 1$.
\item $\varsigma_{\frak{e}}\in\mathcal{D}(\psi_{{\kappa_c}}(\mathbb{B}_{5\varepsilon_0,17a/18}),[0,1])$, and $\varsigma_{\frak{e}}|_{\psi_{{\kappa_c}}(\mathbb{B}_{4\varepsilon_0,8a/9})}\equiv 1$. We set $\varsigma:=\varsigma_{\frak{e}}|_{\Sigma}$ \end{itemize} Then we can introduce a localized translation on $\mathsf{Q}_a$ as follows \begin{align*} \theta_{\mu,\eta}:(x,y)\mapsto (x+\chi_m(x)\varpi(y)\mu, y+\chi_m(x)\chi(y)\eta),\hspace*{1em} (\mu,\eta)\in\mathbb{B}(0,r_0), \end{align*} with $\mu\in\mathbb{R}^m$, $\eta\in\mathbb{R}$ and sufficiently small $r_0>0$. Henceforth, $\mathbb{B}(x_0,r_0)$ always denotes the ball with radius $r_0$ centered at $x_0$ in ${\mathbb{R}}^n$. The dimension $n$ of the ball is not distinguished as long as it is clear from the context.
\\ The related partial translations in horizontal/vertical directions can be defined separately as \begin{align} \label{S3: sep translations} \begin{cases} \theta_\mu: (x,y)\mapsto (x+\chi_m(x)\varpi(y)\mu, y),\\ \bar{\theta}_\eta: (x,y)\mapsto (x, y+\chi_m(x)\chi(y)\eta). \end{cases} \end{align} For $r_0$ small and every $(\mu,\eta)\in\mathbb{B}(0,r_0)$ \begin{align} \label{S3: comp of tme} \theta_{\mu,\eta}=\theta_\mu\circ \bar{\theta}_\eta. \end{align} It is not hard to show that for sufficiently small $r_0$, we have $\theta_{\mu,\eta}(\mathbb{B}_{3\varepsilon_0,7a/9})\subset \mathbb{B}_{3\varepsilon_0,7a/9}$ for any $(\mu,\eta)\in\mathbb{B}(0,r_0)$. \eqref{S3: sep translations} implies that \begin{align} \label{S3: pull-back&push-forward} \theta^{\ast}_{\mu,\eta}=\bar{\theta}^{\ast}_{\eta}\circ\theta^{\ast}_{\mu},\quad \text{and}\quad\theta^{\mu,\eta}_\ast:=[\theta_{\mu,\eta}]_\ast=\theta^\mu_\ast\circ\bar{\theta}^\eta_\ast. \end{align} Since $\theta_\mu$ and $\bar{\theta}_\eta$ are the restrictions of truncated translations in the sense of those defined in \cite{EscPruSim03} in horizontal and vertical direction, respectively, the readers should have no difficulty convincing themselves that all the results in \cite{EscPruSim03} are at our disposal for functions defined on an open subset $U\subset \mathbb{R}^{m+1}$ and the transformations $\theta^{\ast}_{\mu}$ and $\bar{\theta}^{\ast}_{\eta}$. Taking advantage of these considerations, we will state some properties of $\theta^{\ast}_{\mu,\eta}$ for later use in the following with brief proofs.
Hereafter, we assume $O\subset U\subset\mathbb{R}^{m+1}$ are open, and $O$ contains $\bar{\mathbb{B}}_{3\varepsilon_0,7a/9}$. We conclude from $\theta_{\mu,\eta}(U)\subset U$ that $\theta^{\ast}_{\mu,\eta}:\mathfrak{F}^s(U,E)\rightarrow E^{U}$.
In the rest of this section, we always assume the following, unless mentioned otherwise, \begin{mdframed} $$1<p<\infty,\quad n,l\in\mathbb{N}_0,\quad k\in\mathbb{N}_0\cup\{\infty,\omega\}.$$ \end{mdframed} \begin{prop} \label{S3: U-continuity} Suppose that $\mathfrak{F}\in\{bc, BU\!C, W_p, H_p\}$ and $u\in C^{n+k}(O,E)\cap \mathfrak{F}^s(U,E)$ with $s\in [0,n]$ when $\mathfrak{F}\in\{bc, W_p, H_p\}$, or with $s=n$ when $\mathfrak{F}=BU\!C$. Then for sufficiently small $r_0$, \begin{itemize} \item[(a)] $\theta_{\mu,\eta}\in{{\rm{Diff}^{\hspace{.1em}\infty}}}(U)$, $(\mu,\eta)\in\mathbb{B}(0,r_0).$ \item[(b)] $\theta^{\ast}_{\mu,\eta}\in \mathcal{L}{\rm{is}}(\mathfrak{F}^s(U,E))$, $[\theta^{\ast}_{\mu,\eta}]^{-1}=\theta^{\mu,\eta}_\ast$, $(\mu,\eta)\in\mathbb{B}(0,r_0). $ Moreover, there exists a constant $M>0$ such that
$$\|\theta^{\ast}_{\mu,\eta}\|_{\mathcal{L}(\mathfrak{F}^s(U,E))}\leq M,\hspace*{1em} (\mu,\eta)\in\mathbb{B}(0,r_0).$$ \item[(c)] $[(\mu,\eta)\mapsto \theta^{\ast}_{\mu,\eta} u]\in C^k(\mathbb{B}(0,r_0),\mathfrak{F}^s(U,E)).$ \end{itemize} \end{prop} \begin{proof} (a) The statement is immediate from \eqref{S3: pull-back&push-forward} and \cite[Proposition~2.2]{EscPruSim03}.
\\ (b) We infer from (a) that $$\theta^{\ast}_{\mu,\eta}\circ\theta_{\ast}^{\mu,\eta}=\theta_{\ast}^{\mu,\eta} \circ \theta^{\ast}_{\mu,\eta} ={\rm{id}}_{\mathfrak{F}^s(U,E)}.$$ \cite[Proposition~2.4]{EscPruSim03} implies that for $r$ small enough $$\theta_\mu^*, \bar{\theta}_\eta^*\in \mathcal{L}{\rm{is}}(\mathfrak{F}^s(U,E)),\quad (\mu,\eta)\in\mathbb{B}(0,r_0).$$ Moreover, there exists a constant $M>0$ such that
$$\|\theta_\mu^*\|_{\mathcal{L}(\mathfrak{F}^s(U,E))} + \|\bar{\theta}_\eta^*\|_{\mathcal{L}(\mathfrak{F}^s(U,E))} \leq M,\quad (\mu,\eta)\in\mathbb{B}(0,r_0).$$ The assertion now follows from \eqref{S3: pull-back&push-forward}.
\\ (c) It is a simple matter to see that the assertion holds for the transformation $\theta^{\ast}_{\mu}$ and $\bar{\theta}^{\ast}_{\eta}$, respectively, namely that we have \begin{align} \label{S3: C^k-1} [\mu\mapsto \theta^{\ast}_{\mu} u]\in C^k(\mathbb{B}(0,r_0),\mathfrak{F}^s(U,E)),\hspace{1em} [\eta\mapsto \bar{\theta}^{\ast}_{\eta} u]\in C^k(\mathbb{B}(0,r_0),\mathfrak{F}^s(U,E)) \end{align}
with $\partial^{\alpha}_{\mu}[\theta^{\ast}_{\mu} u]=\chi_m^{|\alpha|}\varpi^{|\alpha|}[\theta^{\ast}_{\mu}\partial^{\alpha}_{x}u]$, and $\partial^{\beta}_{\eta}[\bar{\theta}^{\ast}_{\eta} u]=\chi_m^{\beta}\chi^\beta[\bar{\theta}^{\ast}_{\eta}\partial^{\beta}_{y}u]$. Then we can obtain from \eqref{S3: pull-back&push-forward}, \eqref{S3: C^k-1}, the point-wise multiplier theorem in \cite[Section~3.3.2]{Trie78}, and induction that the map \begin{align*}
(\mu,\eta)\mapsto\partial^{(\alpha,\beta)}_{(\mu,\eta)}[\theta^{\ast}_{\mu,\eta} u]=\chi_m^{|\alpha|+\beta}\chi^\beta\varpi^{|\alpha|}[\theta^{\ast}_{\mu,\eta}\partial^{(\alpha,\beta)}_{(x,y)}u],\hspace*{1em} |\alpha|+\beta\leq k, \end{align*}
is separately $C^{k-|\alpha|-\beta}$-continuous into the space $\mathfrak{F}^s(U,E)$. So the case $k\in\mathbb{N}_0\cup\{\infty\}$ follows immediately from a well-known fact in multi-variable calculus.
When $k=\omega$, the proof is basically the same as that of \cite[Proposition~3.2]{EscPruSim03}. \end{proof} Let $$
\mathcal{A}:=\sum\limits_{|\alpha|\leq l}a_\alpha \partial^\alpha, $$ be an $l$-th order linear differential operator on $U$ with coefficients $a_\alpha:U\rightarrow \mathcal{L}(E)$, where $\alpha\in\mathbb{N}_0^{m+1}$. We set $\tilde{\mathcal{A}}_{\mu,\eta}:=\theta^{\ast}_{\mu,\eta}\mathcal{A}\theta^{\mu,\eta}_\ast$. \begin{theorem} \label{S3: U-diff op} Suppose that $a_\alpha\in C^{n+k}(O,\mathcal{L}(E))\cap BU\!C^n(U,\mathcal{L}(E))$. Then \begin{align*} [(\mu,\eta)\mapsto \tilde{\mathcal{A}}_{\mu,\eta}]\in C^k(\mathbb{B}(0,r_0), \mathcal{L}(\mathfrak{F}^{s+l}(U,E),\mathfrak{F}^s(U,E))), \end{align*} where $\mathfrak{F}\in\{bc,BU\!C,W_p,H_p\}$, and $s\in [0,n]$. \end{theorem} \begin{proof} By modifying the proof of \cite[Proposition~4.1]{EscPruSim03} in an obvious way, one may verify the case of constant coefficients. Now the assertion is a direct consequence of point-wise multiplication results, and Proposition~\ref{S3: U-continuity}(c). \end{proof}
\subsection{\bf Parameter-dependent diffeomorphisms on $\Omega$ and $\Omega_i$}
By means of $\theta_{\mu,\eta}$, we are now in a position to introduce a family of localized translations on $\Omega$ and $\Omega_i$, respectively, by \begin{align*} \Theta_{\mu,\eta}(z)= \begin{cases} \psi_{\frak{e},{\kappa_c}}\circ\theta_{\mu,\eta}\circ\varphi_{\frak{e},{\kappa_c}}(z),\hspace*{1em} &z\in\mathsf{O}_{\frak{e},{\kappa_c}},\\ z, &z\in \Omega\setminus\mathsf{O}_{\frak{e},{\kappa_c}}. \end{cases} \end{align*} Thanks to Proposition~\ref{S3: U-continuity}(a), $\Theta_{\mu,\eta}\in{{\rm{Diff}^{\hspace{.1em}\infty}}}(\Omega)\cap {{\rm{Diff}^{\hspace{.1em}\infty}}}(\Omega_i)$ is evident.
\\ There is a universal extension operator $\mathscr{E}_i\in \mathcal{L}(\mathfrak{F}^s(\Omega_i,E), \mathfrak{F}^s(\Omega,E))$.
The restriction operator from $\mathfrak{F}^s(\Omega,E)$ to $\mathfrak{F}^s(\Omega_i,E)$ is denoted by $\mathscr{R}_i$. Then $\mathscr{R}_i$ is a retraction from $\mathfrak{F}^s(\Omega,E)$ onto $\mathfrak{F}^s(\Omega_i,E)$ with $\mathscr{E}_i$ as a coretraction. When $u\in \mathfrak{F}^s(\Omega\setminus\Sigma,E)$, it is understood that $\mathscr{E}_i u:= \mathscr{E}_i (u|_{\Omega_i})$.
\\ For $u\in E^{\Omega}$, its pull-back and push-forward by $\Theta_{\mu,\eta}$ can be explicitly expressed as $$ {{\Theta}^{\ast}_{\mu,\eta}}u={\varphi^{\ast}_{\frak{e},{\kappa_c}}}{\theta^{\ast}_{\mu,\eta}}{\psi^{\ast}_{\frak{e},{\kappa_c}}}(\varsigma_{\frak{e}} u)+(1_{\Omega}-{\varsigma_{\frak{e}}})u, \quad {{\Theta}^{\mu,\eta}_{\ast}}u={\varphi^{\ast}_{\frak{e},{\kappa_c}}}{\theta_{\ast}^{\mu,\eta}}{\psi^{\ast}_{\frak{e},{\kappa_c}}}(\varsigma_{\frak{e}} u)+(1_{\Omega}-{\varsigma_{\frak{e}}})u.$$ Analogously, given $u\in E^{\Omega_i}$ of $u$, we have \begin{align} \label{S3: Omega-Omega_i} {{\Theta}^{\ast}_{\mu,\eta}}u=\mathscr{R}_i\circ{{\Theta}^{\ast}_{\mu,\eta}}\circ \mathscr{E}_i u, \quad {{\Theta}^{\mu,\eta}_{\ast}}u=\mathscr{R}_i\circ{{\Theta}^{\mu,\eta}_{\ast}}\circ \mathscr{E}_i u. \end{align} The construction of $\Theta_{\mu,\eta}$ implies that ${\Theta}^{\ast}_{\mu,\eta}(E^{\Omega_i})\subset E^{\Omega_i}$, and likewise for ${\Theta}^{\mu,\eta}_{\ast}$. Suppose that $\mathfrak{F}\in\{bc,BC, W_p,H_p\}$. We define \begin{align*} \mathfrak{F}^{s,\Omega}_{\rm{cp}}:=\{u\in \mathfrak{F}^s(\Omega,E): {\rm{supp}}(u)\subset \psi_{\frak{e},{\kappa_c}}(\bar{\mathbb{B}}_{5\varepsilon_0,17a/18})\}, \end{align*} and \begin{align*} \mathfrak{F}^{s}_{\rm{cp}}:=\{u\in \mathfrak{F}^s(\mathbb{R}^{m+1},E): {\rm{supp}}(u)\subset \bar{\mathbb{B}}_{5\varepsilon_0,17a/18}\}. \end{align*} The analogue of \cite[Lemma~3.1]{ShaoPre} is at our disposal, i.e., it holds that \begin{lem} \label{S3: main lem} $\varphi^{\ast}_{\frak{e},{\kappa_c}}\in \mathcal{L}{\rm{is}}(\mathfrak{F}^s_{\rm{cp}}, \mathfrak{F}^{s,\Omega}_{\rm{cp}})$ with $[\varphi^{\ast}_{\frak{e},{\kappa_c}}]^{-1}=\psi^{\ast}_{\frak{e},{\kappa_c}}$. \end{lem} \begin{proof} Since \cite[(3.2)]{ShaoPre} still holds with $\nabla$ and $g$ denoting the usual gradient and Euclidean metric in $\mathbb{R}^{m+1}$, respectively, the proof is essentially the same as that of \cite[Lemma~3.1]{ShaoPre}. \end{proof} By means of Lemma~\ref{S3: main lem}, with only minor changes to the proofs in \cite[Section~3]{ShaoPre}, one can show that $\Theta_{\mu,\eta}$ inherits all the properties of its counterpart therein.
\begin{prop} \label{S3: Omega-continuity} Suppose that $\mathfrak{F}\in\{bc,BC, W_p,H_p\}$. Then \begin{align*} {\Theta}^{\ast}_{\mu,\eta}\in \mathcal{L}{\rm{is}}(\mathfrak{F}^s(\Omega,E))\cap \mathcal{L}{\rm{is}}(\mathfrak{F}^s(\Omega_i,E)),\hspace*{1em}[{\Theta}^{\ast}_{\mu,\eta}]^{-1}={\Theta}^{\mu,\eta}_{\ast},\hspace*{1em} (\mu,\eta)\in\mathbb{B}(0,r_0). \end{align*} Moreover, there exists a constant $M>0$ such that \begin{align*}
\|{\Theta}^{\ast}_{\mu,\eta}\|_{\mathcal{L}(\mathfrak{F}^s(\Omega,E))} + \|{\Theta}^{\ast}_{\mu,\eta}\|_{\mathcal{L}(\mathfrak{F}^s(\Omega_i,E))}\leq M,\hspace*{1em} (\mu,\eta)\in\mathbb{B}(0,r_0). \end{align*} \end{prop} \begin{proof} The proof is almost the same as that of \cite[Proposition~3.3]{ShaoPre}. For the reader's convenience, we will present a proof based on Lemma~\ref{S3: main lem} for the case of $\Omega_i$, and the other case follows in a similar manner. The identity $${\Theta}^{\ast}_{\mu,\eta}\circ{\Theta}^{\mu,\eta}_{\ast}={\Theta}^{\mu,\eta}_{\ast}\circ{\Theta}^{\ast}_{\mu,\eta}={\rm{id}}_{\mathfrak{F}^s(\Omega,E)}\cap{\rm{id}}_{\mathfrak{F}^s(\Omega_i,E)}$$ is obvious from the definitions of ${\Theta}^{\ast}_{\mu,\eta}$ and ${\Theta}^{\mu,\eta}_{\ast}$ and $\theta_{\mu,\eta}$. In the formula $$ {{\Theta}^{\ast}_{\mu,\eta}}u=\varphi^{\ast}_{\frak{e},{\kappa_c}}{\theta^{\ast}_{\mu,\eta}}\psi^{\ast}_{\frak{e},{\kappa_c}}(\varsigma_{\frak{e}} u)+(1_{\Omega}-{\varsigma_{\frak{e}}})u ,$$ it can easily verified that the multiplication operators $\varsigma_{\frak{e}}$ and $(1_{\Omega}-{\varsigma_{\frak{e}}})$ are uniformly bounded, i.e.,
$$\|\varsigma_{\frak{e}}\|_{\mathcal{L}(\mathfrak{F}^s(\Omega,E))}+ \|(1_{\Omega}-{\varsigma_{\frak{e}}})\|_{\mathcal{L}(\mathfrak{F}^s(\Omega,E))}\leq C_1 $$ for some constant $C_1$. Lemma~\ref{S3: main lem} implies that for some constant $C_2$
$$\|\varphi^{\ast}_{\frak{e},{\kappa_c}}\|_{\mathcal{L}(\mathfrak{F}^{s}_{{\rm{cp}}}, \mathfrak{F}^{s,\Omega}_{\rm{cp}})}+\|\psi^{\ast}_{\frak{e},{\kappa_c}}\|_{\mathcal{L}(\mathfrak{F}^{s,\Omega}_{\rm{cp}}, \mathfrak{F}^{s}_{{\rm{cp}}})} \leq C_2.$$ It follows from Proposition~\ref{S3: U-continuity}(b) that there is a uniform bound $C_3$ such that
$$\|\theta^{\ast}_{\mu,\eta}\|_{\mathcal{L}(\mathfrak{F}^{s}_{{\rm{cp}}})}\leq C_3. $$ The statement then is a direct consequence of the open mapping theorem and \eqref{S3: Omega-Omega_i}. \end{proof}
\subsection{\bf Time-dependence} Let $J=[0,T]$, $T>0$. Assume that $I\subset\mathring J$ is an open interval and $t_c\in I$ is a fixed point. Choose $\varepsilon_{0}$ so small that $\mathbb{B}(t_c,3\varepsilon_0)\subset\subset I$. Pick an auxiliary function \begin{align*}
\xi\in\mathcal{D}(\mathbb{B}(t_c,2\varepsilon_{0}),[0,1])\hspace{1em}\text{ with }\hspace{.5em} \xi|_{\mathbb{B}(t_c,\varepsilon_{0})}\equiv{1}. \end{align*} The localized temporal translation is defined by \begin{align*} \varrho_{\lambda}(t):=t+\xi(t)\lambda,\hspace{1em} t\in J \text{ and } \lambda\in{\mathbb{R}}. \end{align*} For $v:J\times\mathsf{Q}_a\rightarrow E$, the parameter-dependent diffeomorphism can be expressed as \begin{align} \label{S3: Vlme} \tilde{v}_{\lambda,\mu,\eta}(t,\cdot):=\theta^{\ast}_{\lambda,\mu,\eta} v(t,\cdot)=\tilde{T}_{\mu,\eta}(t){\varrho}^{\ast}_{\lambda} v(t,\cdot), \hspace{1em} (\lambda,\mu,\eta)\in\mathbb{B}(0,r_0), \end{align} where $\tilde{T}_{\mu,\eta}(t):=\theta^{\ast}_{\xi(t)\mu,\xi(t)\eta}$ for $t\in J$.
\\ Given $v:J\times\mathsf{Q}^{m}\rightarrow E$, analogously, setting $\tilde{T}_{\mu}(t):=\theta^{\ast}_{\xi(t)\mu}$, we can define \begin{align} \label{S3: Vlm} \tilde{v}_{\lambda,\mu}(t,\cdot):=\theta^{\ast}_{\lambda,\mu} v(t,\cdot)=\tilde{T}_{\mu}(t){\varrho}^{\ast}_{\lambda} v(t,\cdot), \hspace{1em} (\lambda,\mu)\in\mathbb{B}(0,r_0). \end{align} It is understood that $\theta_\mu$ in \eqref{S3: Vlm} is restricted on the hyperplane $y=0$, appearing as a special case of \eqref{S3: sep translations}.
\\ For any $u\in J\times\Omega\rightarrow E$, or $u\in J\times\Omega_i\rightarrow E$, we define $$u_{\lambda,\mu,\eta}(t,\cdot):={\Theta}^{\ast}_{\lambda,\mu,\eta} u(t,\cdot):={T}_{\mu,\eta}(t){\varrho}^{\ast}_{\lambda} u(t,\cdot), \hspace*{1em} (\lambda,\mu,\eta)\in\mathbb{B}(0,r_0).$$ Here ${T}_{\mu,\eta}(t):=\Theta^*_{\xi(t)\mu,\xi(t)\eta}$ for $t\in J$. By Proposition~\ref{S3: Omega-continuity}, for every $t$, ${T}_{\mu,\eta}(t)$ is invertible.
\\ Let ${\Theta}_{\ast}^{\lambda,\mu,\eta}:=[{\Theta}^{\ast}_{\lambda,\mu,\eta}]^{-1}$. Note that $u_{\lambda,\mu,\eta}(0,\cdot)=u(0,\cdot)$.
The following lemma strengthens \cite[Lemma~5.1]{EscPruSim03}. \begin{lem} \label{S3: involve time} Suppose that $[(\mu,\eta)\mapsto f(\mu,\eta)]\in C^k+j(\mathbb{B}(0,r_0),X)$ for $j=0,1$ and some Banach space $X$. Let $F(\mu,\eta)(t):=f(\xi(t)(\mu,\eta))$ with $t\in J$. Then \begin{align*} [(\mu,\eta)\mapsto F(\mu,\eta)]\in C^k(\mathbb{B}(0,r_0),C^j(J,X)). \end{align*} \end{lem} \begin{proof} To economize the notations, we let $\upsilon:=(\mu,\eta)$. The case $j=0$ is proved in \cite[Lemma~5.1]{EscPruSim03}. We will only treat the case $k=\omega$ and $j=1$. The remaining cases follow in a similar way.
\\ Given $\upsilon_0\in \mathbb{B}(0,r_0)$, for every $\upsilon\in \mathbb{B}(\upsilon_0,r)$ with $r$ small enough $$f(\upsilon)=\sum\limits_\beta \frac{1}{\beta !} \partial^\beta f(\upsilon_0) (\upsilon-\upsilon_0)^\beta,\quad \beta\in\mathbb{N}_0^{m+1}, $$ and there exist constants $M,R$ such that
$$\|\partial^\beta f(\upsilon)\| \leq M \frac{\beta !}{R^{|\beta|}},\hspace*{1em} \upsilon\in\mathbb{B}(\upsilon_0,r),\quad \beta\in \mathbb{N}_0^{m+1}.$$ The constants $r,M,R$ depend continuously on $\upsilon_0$.
Thus we can find uniform constants $r,M,R$ for $\bar{\mathbb{B}}^{m+1}(0,|\upsilon_0|)$. For each $t\in J$ and $\upsilon\in\mathbb{B}(\upsilon_0,r)$, the following series \begin{align*}
\sum\limits_\beta \frac{\xi^{|\beta|}(t)}{\beta !} \partial^\beta f(\xi(t)\upsilon_0) (\upsilon-\upsilon_0)^\beta \end{align*} converges in $X$ and represents $F(\upsilon)(t)$. The temporal derivatives can be computed as follows. For $\beta\neq 0$, \begin{align*}
&\quad \|\frac{d}{dt}[\xi^{|\beta|}(t)\partial^\beta f(\xi(t)\upsilon_0)]\|_{C(J,X)}\\
& \leq \| \xi^\prime(t) |\beta|\{\xi^{|\beta|-1}(t)\partial^\beta f(\xi(t)\upsilon_0) +\xi^{|\beta|}(t)\sum\limits_{j=1}^{m+1} \partial^{\beta+e_j} f(\xi(t)\upsilon_0)(\upsilon_0)_j \} \|_{C(J,X)}\\
&\leq M\|\xi \|_{1,\infty} (|\beta|\frac{\beta!}{R^{|\beta|}} + r_0 \sum\limits_{j=1}^{m+1} \frac{(\beta+e_j)!}{R^{|\beta|+1}})\leq M^\prime \frac{\beta !}{(R/C)^{|\beta|}} \end{align*} for some sufficiently large $C$. Now the assertion follows right away. \end{proof}
Suppose that $\mathsf{O}_\frak{e} \subset\mathbb{R}^{m+1}$ is an open subset containing $\psi_{\frak{e},{\kappa_c}}(\bar{\mathbb{B}}_{3\varepsilon_0,7a/9})$. Put $\mathsf{O}_{\frak{e}_i}:=\mathsf{O}_\frak{e}\cap \Omega_i$.
\\ When $k\in\mathbb{N}_0\cup\{\infty\}$, we say $u\in C^k(\mathsf{O}_{\frak{e}_i},E)\cap\mathfrak{F}^s(\Omega_i,E)$ if $\mathscr{E}_i u\in C^{k}(\mathsf{O}_\frak{e},E)\cap\mathfrak{F}^s(\Omega,E)$. By convention, $u\in C^\omega(\mathsf{O}_{\frak{e}_i},E)\cap\mathfrak{F}^s(\Omega_i,E)$ means that $u$ has an analytic extension $u^e\in C^{\omega}(\mathsf{O}_\frak{e},E)\cap\mathfrak{F}^s(\Omega,E)$.
\begin{prop} \label{S3: Omega-Tue} Suppose that $(S,V)\in\{(\mathsf{O}_\frak{e},\Omega), (\mathsf{O}_{\frak{e}_i},\Omega_i) \}$.
\begin{itemize} \item[(a)] Suppose that $u\in C^{n+k+j}(S,E)\cap{\mathfrak{F}}^{s}(V,E)$ for $j=0,1$, where either $s\in [0,n]$ if $\mathfrak{F} \in\{bc,W_p,H_p\}$, or $s=n$ if $\mathfrak{F}=BC$. Here $k\in \mathbb{N}_0\cup\{\infty\}$ if $(S,V)= (\mathsf{O}_{\frak{e}_i},\Omega_i)$. Then we have $$[(\mu,\eta)\mapsto {T}_{\mu,\eta} u]\in C^k(\mathbb{B}(0,r_0),C^j(J;\mathfrak{F}^s(V,E))).$$
\item[(b)] Suppose that $\mathcal{A}=\sum\limits_{|\alpha|\leq l} a_\alpha\partial^\alpha$ with $a_\alpha\in C^{n+k+j}(S,\mathcal{L}(E))\cap BC^n(V,\mathcal{L}(E))$ for $j=0,1$. Then $$[(\mu,\eta)\mapsto {T}_{\mu,\eta}\mathcal{A}{T}^{-1}_{\mu,\eta}]\in C^k(\mathbb{B}(0,r_0),C^j(J;\mathcal{L}(\mathfrak{F}^{s+l}(V,E),\mathfrak{F}^{s}(V,E)))),$$ for $s\in[0,n]$ if ${\mathfrak{F}}\in\{BC,W_{p},H_{p}\}$, or $s\in[0,n)$ if $\mathfrak{F}=bc$. \end{itemize} \end{prop} \begin{proof} (a) Modifying the proof of \cite[Theorem~3.4]{ShaoPre} as in that of Proposition~\ref{S3: Omega-continuity}, one can show by means of Proposition~\ref{S3: U-continuity}(c) that $$[(\mu,\eta)\mapsto {\Theta}^{\ast}_{\mu,\eta} u]\in C^k(\mathbb{B}(0,r_0),\mathfrak{F}^s(V,E)). $$ Set $X=\mathfrak{F}^s(V,E)$, $f(\mu,\eta)={\Theta}^{\ast}_{\mu,\eta} u$. Now (a) is a direct consequence of Lemma~\ref{S3: involve time}.
\\
(b) When $(S,V)=(\mathsf{O}_{\frak{e}_i},\Omega_i)$, let $\mathcal{A}_i:=\sum\limits_{|\alpha|\leq l}( a_\alpha^e)\partial^\alpha$ on $\Omega$. Then $${\mathcal{A}}_{\mu,\eta}= \mathscr{R}_i \circ {\Theta}^{\ast}_{\mu,\eta}\mathcal{A}_i \Theta^{\mu,\eta}_\ast \circ \mathscr{E}_i.$$ The atlas $\mathfrak{A}_{\frak{e}}$ is real analytically compatible with the Euclidean structure. Similar to the proof of \cite[Proposition~3.6]{ShaoPre}, by Theorem~\ref{S3: U-diff op}, we can show that $$[(\mu,\eta)\mapsto {\mathcal{A}}_{\mu,\eta}:={\Theta}^{\ast}_{\mu,\eta}\mathcal{A}{\Theta}^{\mu,\eta}_{\ast}]\in C^k(\mathbb{B}(0,r_0), \mathcal{L}(\mathfrak{F}^{s+l}(V,E),\mathfrak{F}^s(V,E))).$$ Set $X=\mathcal{L}(\mathfrak{F}^{s+l}(V,E),\mathfrak{F}^s(V,E))$, and $f(\mu,\eta)={\mathcal{A}}_{\mu,\eta}$. The assertion follows by Lemma~\ref{S3: involve time}. \end{proof} \begin{remark} \label{S3: rl-differentiability} Given any Banach space $X$, suppose that $u\in C^{n+k}(I,X)\cap{\mathfrak{F}}^s(J,X)$, where either $s\in [0,n]$ if $\mathfrak{F} \in\{bc,W_p,H_p\}$, or $s=n$ if $\mathfrak{F}=BC$. Following the proofs in \cite[Section~3]{EscPruSim03}, we can verify that $$[\lambda\mapsto {\varrho}^{\ast}_{\lambda} u]\in C^k(\mathbb{B}(0,r_0),\mathfrak{F}^s(J,X)). $$ \qed \end{remark}
The localized translation $\Theta_{\mu,\eta}$, being restricted to $\Sigma$, induces a family of diffeomorphism $\{{\Theta}^{\ast}_{\mu}: \mu\in\mathbb{B}(0,r_0)\}$ on function spaces over $\Sigma$, which was introduced in \cite{ShaoPre}. For the reader's convenience, we will briefly state some of its properties herein. \begin{align*} \Theta_\mu(q)= \begin{cases} \psi_{{\kappa_c}}(\theta_\mu(\varphi_{{\kappa_c}}(q))), \hspace{1em}&q\in \mathsf{O}_{{\kappa_c}},\\ q, &q\in \Sigma\setminus \mathsf{O}_{{\kappa_c}}. \end{cases} \end{align*} It is evident that $\Theta_\mu \in{{\rm{Diff}^{\hspace{.1em}\infty}}}(\Sigma)$ for $\mu\in\mathbb{B}(0,r_0)$ with sufficiently small $r_0$. Given $u \in E^\Sigma$, ${\Theta}^{\ast}_{\mu}$ and ${\Theta}^{\mu}_{\ast}$ can be explicitly expressed as \begin{align*} {\Theta}^{\ast}_{\mu} u=\varphi^{\ast}_{{\kappa_c}} \theta^{\ast}_{\mu} \psi^{\ast}_{{\kappa_c}}(\varsigma u)+(1_\Sigma-{\varsigma})u, \hspace{1em}{\Theta}^{\mu}_{\ast} u=\varphi^{\ast}_{{\kappa_c}} \theta_{\ast}^{\mu} \psi^{\ast}_{{\kappa_c}}(\varsigma u)+(1_\Sigma-{\varsigma})u, \end{align*} respectively. Given $u\in E^{J\times\Sigma}$, setting ${T}_{\mu}(t):=\Theta^*_{\xi(t)\mu}$, we define \begin{align*} u_{\lambda,\mu}(t,\cdot):={\Theta}^{\ast}_{\lambda,\mu} u(t,\cdot)={T}_{\mu}(t){\varrho}^{\ast}_{\lambda} u(t,\cdot),\hspace{1em} (\lambda,\mu)\in\mathbb{B}(0,r_0). \end{align*} Let ${\Theta}_{\ast}^{\lambda,\mu}:=[{\Theta}^{\ast}_{\lambda,\mu}]^{-1}$.
\\
A linear operator $\mathcal{A}:C^\infty(\Sigma,E)\rightarrow E^\Sigma$ is called a linear differential operator of order $l$ on $\Sigma$ if in every local chart $(\mathsf{O}_{\kappa},\varphi_{\kappa})$ with $\kappa\in\frak{K}$, there exists some linear differential operator defined on $\mathsf{Q}^{m}$ \begin{align*}
\mathcal{A}_\kappa(x,\partial):=\sum\limits_{|\alpha|\leq l}a^\kappa_\alpha(x)\partial^\alpha,\hspace*{.5em}\text{ with }a^\kappa_\alpha\in \mathcal{L}(E)^{\mathsf{Q}^{m}}, \end{align*} called the local representation of $\mathcal{A}$ in $(\mathsf{O}_{\kappa},\varphi_{\kappa})$, such that for any $u\in C^\infty(\Sigma,E)$ \begin{align} \label{S3:local exp} \psi^{\ast}_{\kappa}(\mathcal{A}u)=\mathcal{A}_{\kappa}(\psi^{\ast}_{\kappa} u), \hspace*{1em}\kappa\in\frak{K}. \end{align} \begin{prop} \label{S3：M-reg}
Let $\mathsf{O}:=\mathsf{O}_\frak{e}\cap\Sigma$. \begin{itemize} \item[(a)] Suppose that $u\in C^{n+k+j}(\Sigma,E)\cap \mathfrak{F}^s(\Sigma,E)$ for $j=0,1$, where either $s\in[0,n]$ if $\mathfrak{F}\in\{bc,W_p,H_p\}$, or $s=n$ if $\mathfrak{F}=BC$. Then we have $$[\mu\mapsto{T}_{\mu} u]\in C^k(\mathbb{B}(0,r_0),C^j(J;\mathfrak{F}^s(\Sigma,E))).$$
\item[(b)]
Suppose that $\mathcal{A}$ is a linear differential operator on $\Sigma$ of order $l$ satisfying that for all $|\alpha|\leq l$ and $\kappa\in\frak{K}$, $a_\alpha^\kappa \in BC^n(\mathsf{Q}^{m},\mathcal{L}(E))$ and $a_\alpha^{\kappa_c} \in C^{n+k+j}(\varphi_{{\kappa_c}}(\mathsf{O}),\mathcal{L}(E))$ for $j=0,1$. Then for $s\in [0,n]$ if $\mathfrak{F}=BC$, or $s\in[0,n)$ if $\mathfrak{F}=bc$, or $s\in (-\infty,n]$ if $\mathfrak{F}\in\{W_p,H_p\}$ $$[\mu \mapsto {T}_{\mu}\mathcal{A}{T}^{-1}_{\mu}]\in C^k(\mathbb{B}(0,r_0),C^j(J;\mathcal{L}(\mathfrak{F}^{s+l}(\Sigma,E),\mathfrak{F}^s(\Sigma,E)))).$$ \end{itemize} \end{prop} \begin{proof} The proofs for (a) and the case $s\geq 0$ in (b) are given in \cite[Section~3]{ShaoPre}. We will only treat the case $k=\omega$ when $s<0$, the other cases follow similarly. Firstly, we show that for any $s\in (-\infty,0)$ and $\mathfrak{F}_p\in \{W_p,H_p\}$ \begin{align*} [\mu \mapsto \theta^{\ast}_{\mu} \partial_j \theta_{\ast}^{\mu}]\in C^\omega(\mathbb{B}(0,r_0), \mathcal{L}(\mathfrak{F}^{s+1}_p(\mathbb{R}^m,E),\mathfrak{F}^s_p(\mathbb{R}^m,E))). \end{align*} (i) $s\leq -1$. On account that with $p^\prime$ denoting the H\"older duality of $p$, \begin{align*} \partial_j\in \mathcal{L}(\mathfrak{F}^{s+1}_p(\mathbb{R}^m,E),\mathfrak{F}^s_p(\mathbb{R}^m,E)), \hspace{.5em}\text{ and }\hspace{.5em}\mathfrak{F}^s_p(\mathbb{R}^m,E)=(\mathfrak{F}^{-s}_{p^\prime}(\mathbb{R}^m,E))^\prime , \end{align*} for every $u\in \mathfrak{F}^{s+1}_p(\mathbb{R}^m,E)$ and $v\in \mathfrak{F}^{-s}_{p^\prime}(\mathbb{R}^m,E)$, we have \begin{align*}
&\quad\langle \theta^{\ast}_{\mu} \partial_j \theta_{\ast}^{\mu} u, v \rangle:=- \langle u, \theta^{\ast}_{\mu}\partial_j[\theta_{\ast}^{\mu} v |\det(D(\theta_\mu)^{-1})|]|\det(D\theta_\mu)|\rangle\\ &=-\langle u, \theta^{\ast}_{\mu}\partial_j\theta_{\ast}^{\mu} v \theta^{\ast}_{\mu}(\det(D(\theta_\mu)^{-1}))\det(D\theta_\mu) \rangle
-\langle u, v \theta^{\ast}_{\mu}\partial_j [\det(D(\theta_\mu)^{-1})] \det(D\theta_\mu)\rangle. \end{align*} Here $\langle \cdot,\cdot \rangle$ denotes the duality pairing from $\mathfrak{F}^s_p(\mathbb{R}^m,E)\times\mathfrak{F}^{-s}_{p^\prime}(\mathbb{R}^m,E)$ to $\mathbb{R}$. By \cite[Proposition~4.1]{EscPruSim03}, we have \begin{align*} [\mu\mapsto \theta^{\ast}_{\mu}\partial_j\theta_{\ast}^{\mu}]\in C^\omega (\mathbb{B}(0,r_0), \mathcal{L}(\mathfrak{F}^{-s}_{p^\prime}(\mathbb{R}^m,E),\mathfrak{F}^{-s-1}_{p^\prime}(\mathbb{R}^m,E))). \end{align*} It is a simple matter to check that $\theta^{\ast}_{\mu}(\det(D(\theta_\mu)^{-1}))\det(D\theta_\mu)=1$. One may compute \begin{align*} &\quad-\theta^{\ast}_{\mu}\partial_j [\det(D(\theta_\mu)^{-1})] \det(D\theta_\mu)\\ &=-\theta^{\ast}_{\mu}[\partial_j [\det(D(\theta_\mu)^{-1})] \theta_{\ast}^{\mu}\det(D\theta_\mu)]=\theta^{\ast}_{\mu}\partial_j\theta_{\ast}^{\mu} \det(D\theta_\mu) \theta^{\ast}_{\mu}\det(D(\theta_\mu)^{-1}). \end{align*} We immediately have for all $k\in\mathbb{N}_0$ that \begin{align*} [\mu\mapsto \det(D\theta_\mu)]\in C^\omega(\mathbb{B}(0,r_0), BC^{k+1}(\mathbb{R}^m)). \end{align*} It follows again from \cite[Proposition~4.1]{EscPruSim03} that \begin{align*} [\mu\mapsto \theta^{\ast}_{\mu}\partial_j\theta_{\ast}^{\mu} \det(D\theta_\mu)]\in C^\omega(\mathbb{B}(0,r_0), BC^k(\mathbb{R}^m)). \end{align*} On the other hand, we obtain \begin{align*} [\mu\mapsto \theta^{\ast}_{\mu} \det(D(\theta_\mu)^{-1})]=1/\det(D\theta_\mu)\in C^\omega(\mathbb{B}(0,r_0), BC^k(\mathbb{R}^m)). \end{align*} Therefore, for every $u\in \mathfrak{F}^{s+1}_p(\mathbb{R}^m,E)$ and $v\in \mathfrak{F}^{-s}_{p^\prime}(\mathbb{R}^m,E)$, \begin{align*} [\mu\mapsto \langle \theta^{\ast}_{\mu} \partial_j \theta_{\ast}^{\mu} u, v \rangle] \in C^\omega(\mathbb{B}(0,r_0)). \end{align*} Now \cite[Proposition~1]{Browd62} implies that \begin{align*} [\mu\mapsto \theta^{\ast}_{\mu}\partial_j\theta_{\ast}^{\mu} ] \in C^\omega(\mathbb{B}(0,r_0), \mathcal{L}(\mathfrak{F}^{s+1}_p(\mathbb{R}^m,E), \mathfrak{F}^s_p(\mathbb{R}^m,E))). \end{align*} (ii) $s\in (-1,0)$. \cite[Proposition~4.1]{EscPruSim03} and (i) show that \begin{align*} [\mu\mapsto \theta^{\ast}_{\mu}\partial_j\theta_{\ast}^{\mu} ] \in & C^\omega(\mathbb{B}(0,r_0), \mathcal{L}(\mathfrak{F}^{s+2}_p(\mathbb{R}^m,E), \mathfrak{F}^{s+1}_p(\mathbb{R}^m,E)))\\ &\cap C^\omega(\mathbb{B}(0,r_0), \mathcal{L}(\mathfrak{F}^s_p(\mathbb{R}^m,E), \mathfrak{F}^{s-1}_p(\mathbb{R}^m,E))). \end{align*} Thus for any $\mu_0\in \mathbb{B}(0,r_0)$, there exists some constants $r_i$, $M_i$ and $R_i$ with $i=1,2$ such that for all $\mu\in \mathbb{B}^m(\mu_0,r_i)$ \begin{align} \label{S3: Cauchy est}
\| \frac{\partial^\alpha}{\partial \mu^\alpha} [\theta^{\ast}_{\mu}\partial_j\theta_{\ast}^{\mu}]\|_{X_i} \leq M_i\frac{\alpha !}{R_i^{|\alpha|}},\hspace{1em} \alpha\in \mathbb{N}_0^m. \end{align} Here $X_1:=\mathcal{L}(\mathfrak{F}^{s+2}_p, \mathfrak{F}^{s+1}_p)$ and $X_2:=\mathcal{L}(\mathfrak{F}^s_p, \mathfrak{F}^{s-1}_p)$. It follows from interpolation theory that $[\mu\mapsto \theta^{\ast}_{\mu}\partial_j\theta_{\ast}^{\mu} ] \in C^\infty(\mathbb{B}(0,r_0), \mathcal{L}(\mathfrak{F}^{s+1}_p(\mathbb{R}^m,E), \mathfrak{F}^s_p(\mathbb{R}^m,E)))$. Indeed, we have for $h=1,2,\cdots,m$ and any $\mu\in \mathbb{B}(0,r_0)$
$$ \| \frac{\theta^*_{\mu+t e_h} \partial_j \theta_*^{\mu+t e_h} -\theta^{\ast}_{\mu}\partial_j\theta_{\ast}^{\mu}}{t} -\frac{\partial}{\partial \mu_h} \theta^{\ast}_{\mu}\partial_j\theta_{\ast}^{\mu} \|_{X_i} \to 0, \quad t\to 0,\quad i=1,2.$$ By interpolation theory, the above limits converge when $X_i$ is replaced by $X:=\mathcal{L}(\mathfrak{F}^{s+1}_p(\mathbb{R}^m,E), \mathfrak{F}^s_p(\mathbb{R}^m,E))$. Continuity, or even $C^\infty$-smoothness, of $\theta^{\ast}_{\mu}\partial_j\theta_{\ast}^{\mu}$ in $X$ can be verified analogously.
\\ Similar Cauchy estimate to \eqref{S3: Cauchy est} holds for $X$ by interpolation theory, i.e., for any $\mu_0\in \mathbb{B}(0,r_0)$, there exists some constants $r=\min r_i$, $M=\max M_i$ and $R=\min R_i$ such that for all $\mu\in \mathbb{B}^m(\mu_0,r)$ \begin{align*}
\|\frac{\partial^\alpha}{\partial \mu^\alpha} [\theta^{\ast}_{\mu}\partial_j\theta_{\ast}^{\mu}]\|_{\mathcal{L}(\mathfrak{F}^{s+1}_p, \mathfrak{F}^s_p)} \leq M\frac{\alpha !}{R^{|\alpha|}},\hspace{1em} \alpha\in \mathbb{N}_0^m. \end{align*} It is well-known that this estimate implies that \begin{align*} [\mu\mapsto \theta^{\ast}_{\mu}\partial_j\theta_{\ast}^{\mu} ] \in C^\omega(\mathbb{B}(0,r_0), \mathcal{L}(\mathfrak{F}^{s+1}_p(\mathbb{R}^m,E), \mathfrak{F}^s_p(\mathbb{R}^m,E))). \end{align*}
(iii) For $s\in (-\infty,n]$, in view of the proofs for \cite[Proposition~4.1, Theorem~4.2]{EscPruSim03}, we thus infer that for any linear differential operator $\tilde{\mathcal{A}}=\sum_{|\alpha|\leq l}a_\alpha \partial^\alpha$, if $a_\alpha\in BC^n(\mathbb{R}^m,\mathcal{L}(E))\cap C^{n+k+j}(O,\mathcal{L}(E))$ for $O:=\varphi_{\kappa_c}(\mathsf{O})$,
then \begin{align*} [\mu\mapsto \theta^{\ast}_{\mu}\tilde{\mathcal{A}}\theta_{\ast}^{\mu}] \in C^\omega (\mathbb{B}(0,r_0), \mathcal{L}(\mathfrak{F}^{s+l}_p(\mathbb{R}^m,E), \mathfrak{F}^s_p(\mathbb{R}^m,E))). \end{align*} A similar proof to (i) shows that \cite[Lemma~3.1]{ShaoPre} still holds for Sobolev-Slobodeckii and Bessel potential spaces of negative order. Indeed, let $$\mathfrak{F}^{s,\Sigma}_{\rm{cp}}:=\{u\in \mathfrak{F}^s(\Sigma,E): {{\rm{supp}}}(u)\subset\psi_{{\kappa_c}}(\bar{\mathbb{B}}(x_c,5\varepsilon_0)) \}, $$ and $$\mathfrak{F}^{s,\mathbb{R}^m}_{\rm{cp}}:=\{u\in \mathfrak{F}^s(\mathbb{R}^m,E): {{\rm{supp}}}(u)\subset \bar{\mathbb{B}}(x_c,5\varepsilon_0) \} . $$
Pick $\bar{\mathbb{B}}(x_c,5\varepsilon_0)\subset\mathring{U}\subset\subset\mathsf{Q}^{m}$ with $U$ closed, and $\tau\in \mathcal{D}(\mathring{U},[0,1])$ with $\tau|_{\bar{\mathbb{B}}(x_c,5\varepsilon_0)}\equiv 1$. For $r> 0$, given any $u\in W^r_{p,{\rm{cp}}}$ and $v\in W^{-r}_{p^\prime}(\mathbb{R}^m,E)$, we have \begin{align*}
|\langle u, \varphi^{\ast}_{{\kappa_c}} v\rangle_\Sigma |&= \|\langle \psi^{\ast}_{{\kappa_c}} u, \tau v \sqrt{|\det G|}\rangle|\\
&\leq M \|\psi^{\ast}_{{\kappa_c}} u\|_{W^r_p(\mathbb{R}^m)} \|v\|_{W^{-r}_{p^\prime}(\mathbb{R}^m)}
\leq M \|u\|_{W^r_p(\Sigma)} \|v\|_{W^{-r}_{p^\prime}(\mathbb{R}^m)}. \end{align*} Here $G$ is the local matrix expression of the metric $g$, and
$\langle \cdot,\cdot \rangle_\Sigma$ denotes the duality pairing from $\mathfrak{F}^r_p(\Sigma,E)\times\mathfrak{F}^{-r}_{p^\prime}(\Sigma,E)$ to $\mathbb{R}$. The ultimate line follows from the point-wise multiplier theorem, see \cite[Section~9]{Ama13}, and a similar assertion to \cite[Lemma~3.1]{ShaoPre}. Thus the constant $M$ is independent of $u$ and $v$. It implies that for $r\in\mathbb{R}$ $$\varphi^{\ast}_{{\kappa_c}}\in \mathcal{L}{\rm{is}}(\mathfrak{F}^{r,\mathbb{R}^m}_{\rm{cp}}, \mathfrak{F}^{r,\Sigma}_{\rm{cp}}),\quad \text{ with }\quad [\varphi^{\ast}_{{\kappa_c}}]^{-1}=\psi^{\ast}_{{\kappa_c}}. $$ Modifying the proof \cite[Proposition~3.6]{ShaoPre} in an obvious way and applying Lemma \ref{S3: involve time}, we have proved the statement of (b). \end{proof} Recall that $\partial_\nu=\partial_{\nu_\Sigma}$. \begin{prop} \label{S3: normal der} ${T}_{\mu}\partial_\nu {T}^{-1}_{\mu,\eta} =\partial_\nu$ on $H^1_p(\Omega_i)$. \end{prop} \begin{proof} It suffices to show that ${\Theta}^{\ast}_{\mu}\partial_\nu {\Theta}^{\mu,\eta}_{\ast}=\partial_\nu$. On account of $$\psi^{\ast}_{\kappa}(\partial_\nu u)=\psi^{\ast}_{\kappa}(\nu_\Sigma \cdot \nabla u)=\partial_{m+1} \psi^{\ast}_{\kappa} \mathscr{E}_i u$$ for any $u\in H^1_p(\Omega_i)$, one readily computes \begin{align*} &\quad {\Theta}^{\ast}_{\mu}\partial_\nu {\Theta}^{\mu,\eta}_{\ast} u = {\Theta}^{\ast}_{\mu} \partial_\nu \varphi^{\ast}_{\frak{e},{\kappa_c}} \theta_{\ast}^{\mu} \psi^{\ast}_{\frak{e},{\kappa_c}} \varsigma_{\frak{e}} \mathscr{E}_i u +{\Theta}^{\ast}_{\mu} \partial_\nu (1-\varsigma_{\frak{e}}) u\\ &=\varphi^{\ast}_{{\kappa_c}}\theta^{\ast}_{\mu}\psi^{\ast}_{{\kappa_c}} \varsigma \partial_\nu \varphi^{\ast}_{\frak{e},{\kappa_c}} \theta_{\ast}^{\mu} \psi^{\ast}_{\frak{e},{\kappa_c}} \varsigma_{\frak{e}} \mathscr{E}_i u + (1-\varsigma)\partial_\nu \varphi^{\ast}_{\frak{e},{\kappa_c}} \theta_{\ast}^{\mu} \psi^{\ast}_{\frak{e},{\kappa_c}} \varsigma_{\frak{e}} \mathscr{E}_i u +\partial_\nu (1-\varsigma_{\frak{e}}) u\\ &=\varphi^{\ast}_{{\kappa_c}}\theta^{\ast}_{\mu}\psi^{\ast}_{{\kappa_c}} \varsigma \varphi^{\ast}_{{\kappa_c}} \partial_{m+1} \theta_{\ast}^{\mu} \psi^{\ast}_{\frak{e},{\kappa_c}} \varsigma_{\frak{e}} \mathscr{E}_i u + (1-\varsigma)\varphi^{\ast}_{{\kappa_c}}\partial_{m+1} \theta_{\ast}^{\mu} \psi^{\ast}_{\frak{e},{\kappa_c}} \varsigma_{\frak{e}} \mathscr{E}_i u +\partial_\nu (1-\varsigma_{\frak{e}}) u\\ &=\varphi^{\ast}_{{\kappa_c}}\theta^{\ast}_{\mu}\psi^{\ast}_{{\kappa_c}} \varsigma \varphi^{\ast}_{{\kappa_c}} \theta_{\ast}^{\mu} \partial_{m+1} \psi^{\ast}_{\frak{e},{\kappa_c}} \varsigma_{\frak{e}} \mathscr{E}_i u +\varphi^{\ast}_{{\kappa_c}}\theta^{\ast}_{\mu}\psi^{\ast}_{{\kappa_c}}(1-\varsigma)\varphi^{\ast}_{{\kappa_c}}\theta_{\ast}^{\mu} \partial_{m+1} \psi^{\ast}_{\frak{e},{\kappa_c}} \varsigma_{\frak{e}} \mathscr{E}_i u\\ &\quad
+\partial_\nu (1-\varsigma_{\frak{e}}) u\\ &=\varphi^{\ast}_{{\kappa_c}} \partial_{m+1} \psi^{\ast}_{\frak{e},{\kappa_c}} \varsigma_{\frak{e}} \mathscr{E}_i u +\partial_\nu (1-\varsigma_{\frak{e}}) u=\partial_\nu u. \end{align*} In the above, we have used the fact that ${\Theta}^{\ast}_{\mu} \partial_\nu (1-\varsigma_{\frak{e}}) u=(1-\varsigma_{\frak{e}}) u$. \end{proof}
Recall definitions~\eqref{S3: Vlme} and \eqref{S3: Vlm}. The following proposition shows that the space $\mathbb{E}(J)$ is invariant under the transformation ${\Theta}^{\ast}_{\lambda,\mu,\eta}$ and ${\Theta}^{\ast}_{\lambda,\mu}$, respectively. \begin{prop} \label{S3: Omega-invariant spaces} $[(\vartheta,h)\mapsto ({\Theta}^{\ast}_{\lambda,\mu,\eta} \vartheta,{\Theta}^{\ast}_{\lambda,\mu} h)]\in\mathcal{L}{\rm{is}}(\mathbb{E}(J))\cap \mathcal{L}{\rm{is}}(\prescript{}{0}{\mathbb{E}(J)})$ with $(\lambda,\mu,\eta)\in\mathbb{B}(0,r_0)$. Moreover, there exist some $B_{\lambda,\mu,\eta}$, $B_{\lambda,\mu}$ satisfying \begin{align*} \begin{cases} [(\lambda,\mu,\eta)\mapsto B_{\lambda,\mu,\eta}]\in C^\omega(\mathbb{B}(0,r_0), \mathcal{L}(\mathbb{E}_{1}(J),\mathbb{F}_{1}(J))),\\ [(\lambda,\mu)\mapsto B_{\lambda,\mu}]\in C^\omega(\mathbb{B}(0,r_0),\mathcal{L}(\mathbb{E}_{2}(J),\mathbb{F}_{2}(J))), \end{cases} \end{align*} such that \begin{align*} \begin{cases} \partial_t [\vartheta_{\lambda,\mu,\eta}]=(1+\xi^\prime\lambda){\Theta}^{\ast}_{\lambda,\mu,\eta} \partial_t \vartheta + B_{\lambda,\mu,\eta}(\vartheta_{\lambda,\mu,\eta}),\\ \partial_t[h_{\lambda,\mu}]=(1+\xi^{\prime}\lambda){\Theta}^{\ast}_{\lambda,\mu} \partial_t h +B_{\lambda,\mu}(h_{\lambda,\mu}). \end{cases} \end{align*} In particular, $B_{\lambda,0,0}=0$ and $B_{\lambda,0}=0$. \end{prop} \begin{proof}
(i) Following the proof of \cite[Proposition~2.4]{EscPruSim03} and interpolation theory, we can show that for any Banach space $X$, ${\varrho}^{\ast}_{\lambda}\in{\mathcal{L}{\rm{is}}}(\mathfrak{F}(I,X))$ and $[{\varrho}^{\ast}_{\lambda}]^{-1}=\varrho^{\lambda}_{\ast}$ with ${\mathfrak{F}}\in\{BC^s,W^s_p\}$. In particular, there exists $M>0$ such that $\|{\varrho}^{\ast}_{\lambda}\|_{\mathcal{L}(\mathfrak{F}(I,X))}\leq M$ for $\lambda\in{\mathbb{B}}$. A similar estimate as in \cite[Lemma~8.3]{EscPruSim03} by using the intrinsic norms of Besov spaces reveals that $$[(u,\rho)\mapsto(\tilde{u}_{\lambda,\mu,\eta},\tilde{\rho}_{\lambda,\mu}] \in \mathcal{L}{\rm{is}}(\mathbb{E}_1(J,\mathbb{R}^{m+1};E)\times\mathbb{E}_2(J,\mathbb{R}^m;E)).$$ (ii) Observe that \begin{align} \label{S3: time trace} ({\Theta}^{\ast}_{\lambda,\mu,\eta} \vartheta,{\Theta}^{\ast}_{\lambda,\mu} h)(0)=(\vartheta,h)(0),\hspace{1em} \partial_t({\Theta}^{\ast}_{\lambda,\mu,\eta} \vartheta,{\Theta}^{\ast}_{\lambda,\mu} h)(0)=\partial_t(\vartheta,h)(0), \end{align} whenever the corresponding derivative exists.
For any $z\in\Omega\setminus \mathsf{O}_{\frak{e},{\kappa_c}}$, ${T}_{\mu,\eta} \vartheta(z)=\vartheta(z)$, we conclude that $\partial_{\nu_{\Omega}}{T}_{\mu,\eta} \vartheta=0$. For every $t\in J$, $\vartheta(t)\in BC(\Omega)$. It follows from Proposition~\ref{S3: Omega-continuity} that ${T}_{\mu,\eta}(t) \vartheta(t)\in BC(\Omega)$, which implies that $[\![{T}_{\mu,\eta}(t)\vartheta(t) ]\!]=0$. Moreover, for $ (\mu,\eta)\in\mathbb{B}(0,r_0)$ \begin{align*}
&\quad \|{T}_{\mu,\eta} \vartheta\|_{\mathbb{E}_{1}(J)}=\sum_{i}\|\mathscr{R}_i\varphi^{\ast}_{\frak{e},{\kappa_c}}\tilde{T}_{\mu,\eta} \psi^{\ast}_{\frak{e},{\kappa_c}}(\varsigma_{\frak{e}} \mathscr{E}_i \vartheta) + (1-\varsigma_{\frak{e}}) \vartheta \|_{\mathbb{E}_{1}(J)}\\
&\leq M\sum_i [\|\tilde{T}_{\mu,\eta}\psi^{\ast}_{\frak{e},{\kappa_c}}(\varsigma_{\frak{e}} \mathscr{E}_i \vartheta) \|_{\mathbb{E}_1(J,\mathbb{R}^{m+1};E)}+\|(1-\varsigma_{\frak{e}})\mathscr{E}_i\vartheta\|_{\mathbb{E}_{1}(J)}]\leq M \|\vartheta\|_{\mathbb{E}_{1}(J)}. \end{align*} The last line follows from (i), Lemma~\ref{S3: main lem} and point-wise multiplier theorem, see \cite[Section~3.3.2]{Trie78}. In virtue of \cite[Lemma~3.1]{ShaoPre}, one can show similarly that $$[h\mapsto {T}_{\mu} h]\in \mathcal{L}(\mathbb{E}_{2}(J)),\quad \mu\in\mathbb{B}(0,r_0).$$ Then it is a direct consequence of the open mapping theorem that $$[(\vartheta,h)\mapsto ({\Theta}^{\ast}_{\lambda,\mu,\eta} \vartheta,{\Theta}^{\ast}_{\lambda,\mu} h)]\in\mathcal{L}{\rm{is}}(\mathbb{E}(J)).$$ By \eqref{S3: time trace}, the statement $[(\vartheta,h)\mapsto ({T}_{\mu,\eta} \vartheta,{T}_{\mu} h)]\in\mathcal{L}{\rm{is}}(\prescript{}{0}{\mathbb{E}(J)})$ is immediate.
\\ (iii) The temporal derivative of $\vartheta_{\lambda,\mu,\eta}$ can be computed as follows. \begin{align*} &\quad \partial_t [\vartheta_{\lambda,\mu,\eta}]= (1-\varsigma_{\frak{e}})\partial_t[{\varrho}^{\ast}_{\lambda} \vartheta] + \sum_i \mathscr{R}_i\varphi^{\ast}_{\frak{e},{\kappa_c}}\partial_t[\tilde{T}_{\mu,\eta} \psi^{\ast}_{\frak{e},{\kappa_c}}(\varsigma_{\frak{e}} {\varrho}^{\ast}_{\lambda} \mathscr{E}_i\vartheta)]\\ &=(1-\varsigma_{\frak{e}})(1+\xi^{\prime} \lambda){\varrho}^{\ast}_{\lambda} \partial_t\vartheta +\sum_i \{ \mathscr{R}_i\varphi^{\ast}_{\frak{e},{\kappa_c}}\tilde{T}_{\mu,\eta} \psi^{\ast}_{\frak{e},{\kappa_c}}(\varsigma_{\frak{e}} \partial_t[{\varrho}^{\ast}_{\lambda} \mathscr{E}_i \vartheta])\\ &\quad +\mathscr{R}_i \sum\limits_{j=1}^m \varphi^{\ast}_{\frak{e},{\kappa_c}}(\xi^\prime \chi_m\varpi \mu_j \tilde{T}_{\mu,\eta}\partial_j[\psi^{\ast}_{\frak{e},{\kappa_c}}\varsigma_{\frak{e}}{\varrho}^{\ast}_{\lambda} \mathscr{E}_i \vartheta] )\\ &\quad +\mathscr{R}_i\varphi^{\ast}_{\frak{e},{\kappa_c}}(\xi^\prime \chi_m\chi\eta \tilde{T}_{\mu,\eta}\partial_{m+1}[\psi^{\ast}_{\frak{e},{\kappa_c}}\varsigma_{\frak{e}}{\varrho}^{\ast}_{\lambda} \mathscr{E}_i \vartheta]) \}\\ &=(1+\xi^\prime\lambda){\Theta}^{\ast}_{\lambda,\mu,\eta} \partial_t\vartheta + B_{\lambda,\mu,\eta}(\vartheta_{\lambda,\mu,\eta}). \end{align*} The last line follows from the definition of ${\Theta}^{\ast}_{\lambda,\mu,\eta}$ and the equality \begin{align*} &\quad \mathscr{R}_i \varphi^{\ast}_{\frak{e},{\kappa_c}}\tilde{T}_{\mu,\eta} \psi^{\ast}_{\frak{e},{\kappa_c}}(\varsigma_{\frak{e}} \partial_t[{\varrho}^{\ast}_{\lambda} \mathscr{E}_i \vartheta])+(1-\varsigma_{\frak{e}})(1+\xi^{\prime} \lambda){\varrho}^{\ast}_{\lambda} \partial_t\vartheta\\ &=\mathscr{R}_i\varphi^{\ast}_{\frak{e},{\kappa_c}}\tilde{T}_{\mu,\eta} \psi^{\ast}_{\frak{e},{\kappa_c}}[\varsigma_{\frak{e}}(1+\xi^\prime\lambda){\varrho}^{\ast}_{\lambda} \partial_t \mathscr{E}_i \vartheta]+(1-\varsigma_{\frak{e}})(1+\xi^{\prime} \lambda){\varrho}^{\ast}_{\lambda} \partial_t\vartheta\\ &=(1+\xi^\prime\lambda){\Theta}^{\ast}_{\lambda,\mu,\eta} \partial_t\vartheta. \end{align*} Given $u\in H^1_p(\Omega\setminus\Sigma)$, the term $B_{\lambda,\mu,\eta}$ can be explicitly written as \begin{align*} B_{\lambda,\mu,\eta}(u)&=\sum_i \{ \sum\limits_{j=1}^m \mathscr{R}_i\varphi^{\ast}_{\frak{e},{\kappa_c}}(\xi^\prime \chi_m\varpi \mu_j \tilde{T}_{\mu,\eta}\partial_j[\psi^{\ast}_{\frak{e},{\kappa_c}}\varsigma_{\frak{e}}{T}^{-1}_{\mu,\eta} \mathscr{E}_i u] )\}\\ &\quad+\mathscr{R}_i\varphi^{\ast}_{\frak{e},{\kappa_c}}(\xi^\prime \chi_m\chi\eta \tilde{T}_{\mu,\eta}\partial_{m+1}[\psi^{\ast}_{\frak{e},{\kappa_c}}\varsigma_{\frak{e}}{T}^{-1}_{\mu,\eta} \mathscr{E}_i u])\\ &=\sum_i \{ \sum\limits_{j=1}^m \mathscr{R}_i\xi^\prime \varphi^{\ast}_{\frak{e},{\kappa_c}}\chi_m \varphi^{\ast}_{\frak{e},{\kappa_c}}\varpi \mu_j {T}_{\mu,\eta}\mathcal{A}^j{T}^{-1}_{\mu,\eta} \mathscr{E}_i u \\ &\quad +\mathscr{R}_i \xi^\prime \varphi^{\ast}_{\frak{e},{\kappa_c}}\chi_m \varphi^{\ast}_{\frak{e},{\kappa_c}}\chi\eta {T}_{\mu,\eta}\mathcal{A}^{m+1}{T}^{-1}_{\mu,\eta} \mathscr{E}_i u \}. \end{align*} Here $\mathcal{A}^j$ with $j=1,\cdots,m+1$ are first order linear differential operators defined on $\Omega\setminus \Sigma$ compactly supported in ${\sf T}_{a,{\kappa_c}}$ satisfying $\psi^{\ast}_{\frak{e},{\kappa_c}}\mathcal{A}^j u=\varsigma_{\frak{e}}\partial_j \psi^{\ast}_{\frak{e},{\kappa_c}} u$ for all $u\in H^1_p(\Omega\setminus\Sigma)$. By means of Proposition~\ref{S3: Omega-Tue}(b) and the real analytic compatibility of the atlas $\mathfrak{A}_{\frak{e}}$ with the Euclidean structure, it is not hard to check that \begin{align} \label{S3: analyticity of Blme} [(\lambda,\mu,\eta)\mapsto B_{\lambda,\mu,\eta}]\in C^\omega(\mathbb{B}(0,r_0), C^1(J;\mathcal{L}(W^{s+1}_p(\Omega\setminus\Sigma),W^s_p(\Omega\setminus\Sigma)))). \end{align} In \cite[Proposition~3.10]{ShaoPre}, it is shown that \begin{align*} \partial_t[h_{\lambda,\mu}]=(1+\xi^{\prime}\lambda){\Theta}^{\ast}_{\lambda,\mu} \partial_t h +B_{\lambda,\mu}(h_{\lambda,\mu}) \end{align*} with $B_{\lambda,0}=0$ and \begin{align} \label{S3: analyticity of Blm} [(\lambda,\mu)\mapsto B_{\lambda,\mu}]\in C^\omega(\mathbb{B}(0,r_0),C^1(J;\mathcal{L}( W^{s+1}_p(\Sigma), W^s_p(\Sigma)) )). \end{align} This completes the proof. \end{proof} With minor modification of the above proof, we immediately have the following similar result. \begin{prop} \label{S3: Omega: invariant spaces 2} For $(\lambda,\mu,\eta)\in\mathbb{B}(0,r_0)$ with sufficiently small $r_0>0$, $$[(f,g,q)\mapsto (f_{\lambda,\mu,\eta}, g_{\lambda,\mu}, q_{\lambda,\mu})]\in\mathcal{L}{\rm{is}}(\prod_{i=1}^3 \mathbb{F}_i(J),\prod_{i=1}^3 \mathbb{F}_i(J)),$$ and $$[(f,g,q)\mapsto (f_{\lambda,\mu,\eta}, g_{\lambda,\mu}, q_{\lambda,\mu})]\in\mathcal{L}{\rm{is}}(\prod_{i=1}^3 \prescript{}{0}{\mathbb{F}}_i(J),\prod_{i=1}^3 \prescript{}{0}{\mathbb{F}_i}(J)).$$ \end{prop} The main theorem of this section is \begin{theorem} \label{S3: main thm} Suppose that $$(\vartheta,h)\in BC(J\times \Omega,E)\times BC(J \times \Sigma,E) .$$ Then $$(\vartheta,h) \in C^k(\mathring{J}\times {\sf T}_{a/3}\!\setminus\!\Sigma,E) \times C^k(\mathring{J}\times \Sigma, E)$$ iff for any $(t_c,z_c) \in \mathring{J}\times {\sf T}_{a/3}\!\setminus\!\Sigma$, there exists some $r_0=r_0(t_c,z_c)>0$ and a corresponding family of parameter-dependent diffeomorphisms $\{({\Theta}^{\ast}_{\lambda,\mu,\eta},{\Theta}^{\ast}_{\lambda,\mu}):(\lambda,\mu,\eta)\in\mathbb{B}(0,r_0)\}$ such that \begin{align*} [(\lambda,\mu,\eta)\mapsto (\vartheta_{\lambda,\mu,\eta},h_{\lambda,\mu})]\in C^k(\mathbb{B}(0,r_0), BC(J\times \Omega,E)\times BC(J \times \Sigma,E)). \end{align*} \end{theorem} \begin{proof} The proof follows in a similar manner to that of \cite[Theorem~3.5]{ShaoPre}. \end{proof}
\section{\bf Regularity of solutions to the Stefan Problem}
Throughout the rest of this paper, we assume that $\sigma>0$ and $d_i, \gamma \in C^{k+2}(0,\infty) $, $\psi_i\in C^{k+3}(0,\infty)$ for $i=1,2$ with $k\in\mathbb{N}\cup \{\infty,\omega\}$ such that $$\kappa_i(u)=-u\psi_i^{\prime\prime}(u)>0,\quad d_i(u)>0,\quad u\in(0,\infty),$$ and $$\gamma(u)>0,\quad u\in(0,\infty),\quad \text{ or }\quad \gamma\equiv 0. $$ Let $J:=[0,T]$. We define \begin{align*} &\mathbb{G}_{2}(J):=H^1_p(J; W^{-1/p}_p(\Sigma))\cap L_p(J; W^{2-1/p}_p(\Sigma))\\ &\mathbb{G}_{3}(J):=H^1_p(J; W^{-1-1/p}_p(\Sigma))\cap L_p(J; W^{1-1/p}_p(\Sigma)), \end{align*} and \begin{equation*} \begin{aligned} &\mathbb{C}_{1}(J):=C(J\times\bar{\Omega})\cap C(J;C^1(\bar{\Omega}_i))\\ &\mathbb{C}_{2}(J):=C(J;C^3(\Sigma))\cap C^1(J;C(\Sigma)). \end{aligned} \end{equation*} It is shown in \cite[Section~3]{PruSimZac12} that \begin{equation} \label{S5: embed of ea,eb} \mathbb{E}_{1}(J)\hookrightarrow \mathbb{C}_{1}(J),\hspace{1em} \mathbb{E}_{2}(J)\hookrightarrow \mathbb{C}_{2}(J). \end{equation} Observe that the constants in these embeddings blow up as $T\to 0$, however, they are uniform in $T$ if one considers the space $\prescript{}{0}{\mathbb{E}(J)}$!
\subsection{\bf Linearization at a real analytic temperature and the initial interface} Suppose that $\hat{z}_0=(\hat{\vartheta}_0,\hat{h}_0)\in\mathbb{F}_{4}$ satisfies $ \hat{\vartheta}_0>0$.
Given any $\varepsilon>0$, by the Weierstrass approximation theorem, we can find some
$$\vartheta_A\in C^\omega(\bar{\Omega}),\quad \|\vartheta_A-\hat{\vartheta}_0\|_{BC(\Omega)}\leq \varepsilon.$$ For sufficiently small $\varepsilon$, $\vartheta_A>0$, and when $\gamma\equiv 0$, $l(\vartheta_A)\neq 0$. Let \begin{align*} &\kappa_A(x)=\kappa(\vartheta_A(x)),\hspace{1em}d_A(x)=d(\vartheta_A(x)),\hspace{1em} l_A(x)=l(\vartheta_A(x)),\hspace{1em} \sigma_0=\frac{\sigma}{m},\\ &l_1(t,\cdot)=[\![\psi^\prime (e^{\Delta_\Sigma t}\vartheta_A) ]\!],\hspace{1em} \gamma_1(t,\cdot)=\gamma(e^{\Delta_\Sigma t}\vartheta_A). \end{align*}
For a function $\vartheta\in \mathbb{E}_{1}(J)$, we do not distinguish $\vartheta|_\Sigma$ from $\vartheta$ if the choice is self-evident from the context. Here $\Delta_\Sigma$ denotes the Laplace-Beltrami operator on $\Sigma$. Similarly, we define \begin{align*} &\hat{\kappa}_0(x)=\kappa(\hat{\vartheta}_0(x)),\hspace{1em}\hat{d}_0(x)=d(\hat{\vartheta}_0(x)),\hspace{1em} \hat{l}_0(x)=l(\hat{\vartheta}_0(x)),\hspace{1em}\hat{\gamma}_0=\gamma(\hat{\vartheta}_0),\\ &\hat{l}_1(t,\cdot)=[\![\psi^\prime (e^{\Delta_\Sigma t}\hat{\vartheta}_0) ]\!],\hspace{1em} \hat{\gamma}_1(t,\cdot)=\gamma(e^{\Delta_\Sigma t}\hat{\vartheta}_0). \end{align*} For $z=(\vartheta,h)\in\mathbb{E}(J)$, we define \begin{align*}
\notag&F(z)=(\kappa_A-\kappa(\vartheta))\partial_t \vartheta +(d(\vartheta)-d_A)\Delta \vartheta -d(\vartheta)M_2(h):\nabla^2 \vartheta \\
\notag&\quad\quad \quad +d^\prime(\vartheta)|(I-M_1(h))\nabla \vartheta|^2 -d(\vartheta)(M_3(h)| \nabla \vartheta) +\kappa(\vartheta)\mathcal{R}(h)\vartheta,\\ \notag&G(z)= -([\![\psi(\vartheta)]\!] + \sigma {\mathcal{H}}(h) ) +l_1 \vartheta +\sigma_0\Delta_\Sigma h +(\gamma(\vartheta)\beta(h)-\gamma_1)\partial_t h,\\
\notag & Q(z)=[\![ (d(\vartheta)-d_A)\partial_\nu \vartheta ]\!] +(l_A -l(\vartheta))\partial_t h -([\![d(\vartheta)\nabla \vartheta]\!]| M_4(h)\nabla_\Sigma h)\\ &\quad\quad\quad +\gamma(\vartheta)\beta(h)(\partial_t h)^2. \end{align*} Here we have \begin{align*} M_2(h)&=M_1(h)+M_1^{\sf T}(h)-M_1(h)M_1^{\sf T}(h),\\ M_3(h)&=(I-M_1(h)):\nabla M_1(h), \\ M_4(h)&=(I-M_1(h))^{\sf T}M_0(h). \end{align*} Employing the above notations and splitting into the principal linear part and a nonlinear part, we arrive at the following formulation of problem \eqref{S2: transf Stefan} \begin{equation} \label{S5: Stefan-linear/nonlinear} \left\{\begin{aligned} \kappa_A \partial_t \vartheta -d_A \Delta \vartheta &=F(\vartheta,h)
&&\text{in}&&\Omega\setminus\Sigma,\\ \partial_{\nu_\Omega} \vartheta &=0
&&\text{on}&&\partial \Omega,\\ [\![\vartheta]\!]&=0 &&\text{on}&&\Sigma,\\ l_1(t )\vartheta + \sigma_0\Delta_\Sigma h- \gamma_1(t)\partial_t h &=G(\vartheta,h)
&&\text{on}&&\Sigma,\\ \mbox{}l_A \partial_t h -[\![d_A(x)\partial_\nu \vartheta]\!]&=Q(\vartheta,h)
&&\text{on}&&\Sigma,\\ \vartheta(0)=\hat{\vartheta}_0,\ h(0)&=\hat{h}_0.&& \end{aligned}\right. \end{equation} We assume that $\hat{z}_0$ further satisfies the compatibility conditions ($\mathcal{CC}$), that is, when $\gamma\equiv 0$ $$\partial_{\nu_\Omega}\hat{\vartheta}_0=0,\quad l_1(0)\hat{\vartheta}_0+\sigma_0\Delta_\Sigma \hat{h}_0=G(\hat{z}_0), \quad Q(\hat{z}_0)+[\![ d_A\partial_{\nu} \hat{\vartheta}_0]\!]\in W^{2-6/p}_p(\Sigma),$$ and when $\gamma> 0$ $$ \partial_{\nu_\Omega}\hat{\vartheta}_0=0,\quad l_A l_1(0)\hat{\vartheta}_0 +l_A\sigma_0\Delta_\Sigma \hat{h}_0 -\gamma_1(0)[\![ d_A\partial_{\nu} \hat{\vartheta}_0]\!]=\gamma_1(0)Q(\hat{z}_0)+l_A G(\hat{z}_0).$$ In the definition of $(G(\hat{z}_0),Q(\hat{z}_0))$, it is understood that $\partial_t h(0)$ is replaced by \begin{align} \label{S5: replace h_t} \partial_t h(0)= \left\{\begin{aligned}
&\frac{1}{\hat{l}_0} ([\![\hat{d}_0 \partial_\nu \hat{\vartheta}_0]\!] -([\![\hat{d}_0 \nabla \hat{\vartheta}_0]\!]|M_4(\hat{h}_0)\nabla_\Sigma \hat{h}_0 )), &&\gamma\equiv 0,\\ &\frac{1}{\beta(\hat{h}_0)\hat{\gamma}_0}([\![\psi(\hat{\vartheta}_0)]\!] +\sigma{\mathcal{H}}(\hat{h}_0)), &&\gamma>0. \end{aligned}\right. \end{align} When $\gamma\equiv 0$, we also need to impose the {\em well-posedness condition} $l(\hat{\vartheta}_0)\neq 0$ on $\Gamma_0$.
\subsection{\bf Regularity of a special solution} In this section, we prove the analyticity of the solution to a linearized Stefan problem with initial data $\hat{z}_0$, that is, we consider regularity of solutions to the following equation. \begin{align} \label{S5: reducing initial data} \left\{\begin{aligned} \kappa_A \partial_t \vartheta -d_A \Delta \vartheta &=0 &&\text{in}&& \Omega\setminus\Sigma,\\ \partial_{\nu_\Omega}\vartheta &=0 &&\text{on}&& \partial\Omega,\\ [\![\vartheta]\!]&=0 &&\text{on}&& \Sigma,\\ l_1 \vartheta+\sigma_0 \Delta_\Sigma h - \gamma_1 \partial_t h &=g_1 &&\text{on}&& \Sigma,\\ l_A \partial_t h -[\![ d_A\partial_\nu \vartheta]\!] &=q_1 &&\text{on}&& \Sigma,\\ \vartheta(0)=\hat{\vartheta}_0,\hspace{1em} h(0)&=\hat{h}_0. \end{aligned}\right. \end{align} Here $g_1:=e^{\Delta_\Sigma t} G(\hat{z}_0)$ and $q_1:=e^{\Delta_\Sigma t} Q(\hat{z}_0)$. Recall definition \eqref{S2: def of Fj}. We set \begin{align*} &\mathbb{F}(J)=\{ (f,g,q,(\vartheta_0,h_0))\in \mathbb{F}_{1}(J)\times\mathbb{F}_{2}(J)\times\mathbb{F}_{3}(J)\times\mathbb{F}_{4}:\\ &\quad\quad\quad (f,g,q,(\vartheta_0,h_0))\text{ satisfies the linear compatibility conditions }(\mathcal{LCC})\}, \end{align*} where $(f,g,q,(\vartheta_0,h_0))$ is said to satisfy the linear compatibility conditions $(\mathcal{LCC})$ if
\begin{align*} & \partial_{\nu_\Omega}\vartheta_0=0,\hspace{.5em} l_1(0)\vartheta_0+\sigma_0\Delta_\Sigma h_0=g(0), \hspace{.5em} q(0)+[\![ d_A\partial_{\nu} \vartheta_0]\!]\in W^{2-6/p}_p(\Sigma),\hspace{1.7em}\gamma\equiv 0,\\ & \partial_{\nu_\Omega}\vartheta_0=0,\hspace{.5em} l_A l_1(0)\vartheta_0 +l_A\sigma_0\Delta_\Sigma h_0 -\gamma_1(0)[\![ d_A\partial_{\nu} \vartheta_0]\!]=\gamma_1(0)q(0)+l_A g(0), \hspace{0.2em}\gamma> 0. \end{align*} We equipped $\mathbb{F}(J)$ with the following norm \begin{align*}
\| (f,g,q,(\vartheta_0,h_0)) \|_{\mathbb{F}(J)}:=&\|f\|_{\mathbb{F}_{1}(J)}+\|g\|_{\mathbb{F}_{2}(J)} +\|q\|_{\mathbb{F}_{3}(J)} +\|(\vartheta_0, h_0)\|_{\mathbb{F}_{4}}\\
& + (1-\text{sgn}(\gamma))\|q(0)+[\![ d_A\partial_{\nu} \vartheta_0]\!]\|_{W^{2-6/p}_p(\Sigma)}. \end{align*}
(i) Regularity of $g_1$ and $q_1$.
\\ Case 1: $\gamma\equiv 0$. It is not hard to check that \begin{align} \label{S5: reg of k,d,gam,l} l_1, \gamma_1\in \mathbb{F}_{2}(J),\quad
\kappa_A\in W^1_p(\Omega\setminus\Sigma),\hspace{.5em} d_A\in W^{2-2/p}_p(\Omega\setminus\Sigma),\hspace{.5em} l_A\in W^{2-3/p}_p(\Sigma). \end{align} Indeed, the first term can be obtained as follows.
$\vartheta_A|_\Sigma\in W^{2-3/p}_p(\Sigma)$ and the $L_p$-maximal regularity of $\Delta_\Sigma$ imply that \begin{align*} e^{\Delta_\Sigma t} \vartheta_A \in H^1_p(J; W^{-1/p}_p(\Sigma))\cap L_p(J; W^{2-1/p}_p(\Sigma)). \end{align*} By \cite[Proposition~3.2]{MeySch12}, we infer that $e^{\Delta_\Sigma t} \vartheta_A \in \mathbb{F}_{2}(J).$ Now the first term in \eqref{S5: reg of k,d,gam,l} follows from the regularity of $l,\gamma$. Similar to \eqref{S5: reg of k,d,gam,l}, one checks that \begin{align} \label{S5: reg of k,d,gam,l-0} \hat{l}_1, \hat{\gamma}_1\in \mathbb{F}_{2}(J), \hspace{.5em}\hat{\kappa}_0\in W^1_p(\Omega\setminus\Sigma),\hspace{.5em}\hat{d}_0\in W^{2-2/p}_p(\Omega\setminus\Sigma),\hspace{.5em} \hat{l}_0\in W^{2-3/p}_p(\Sigma). \end{align} In \cite{Shao13}, an analysis of the structure of the mean curvature operator ${\mathcal{H}}(h)$ and $\beta(h)$ is obtained, which shows that ${\mathcal{H}}(h)$ is a rational function in the height function $h$ and its spatial derivatives up to second order, while $\beta(h)$ is a rational function in $h$ and its first order spatial derivatives. On account of the embeddings $W^{3-3/p}_p(\Sigma)\hookrightarrow C(\Sigma)$ and $ W^{2-3/p}_p(\Sigma)\hookrightarrow C(\Sigma)$, we conclude from \cite[Theorem~2.8.3]{Trie78} and a similar argument to \cite[Proposition~2.7]{ShaoSim13} that $W^{3-3/p}_p(\Sigma)$ and $W^{2-3/p}_p(\Sigma)$ are multiplication algebras under point-wise multiplication. Since $$[x\mapsto x^a]\in C^{\omega}((0,\infty)),\quad a\in\mathbb{R},$$ well-known results for substitution operators for Sobolev-Slobodeckii spaces imply that \begin{align} \label{S5: reg of H & beta} \beta(\hat{h}_0) \in W^{3-3/p}_p(\Sigma),\quad {\mathcal{H}}(\hat{h}_0)\in W^{2-3/p}_p(\Sigma). \end{align} \eqref{S5: reg of k,d,gam,l}-\eqref{S5: reg of H & beta} then imply \begin{align} \label{S5: reg of G(z0)} G(\hat{z}_0)\in W^{2-3/p}_p(\Sigma), \end{align} and a similar argument yields \begin{align} \label{S5: reg of Q(z0)} Q(\hat{z}_0)\in W^{1-3/p}_p(\Sigma). \end{align} Together with the $L_p$-maximal regularity of $\Delta_\Sigma$ and \cite[Proposition~3.2]{MeySch12}, we conclude from \eqref{S5: reg of G(z0)} and \eqref{S5: reg of Q(z0)} that \begin{align} \label{S5: reg of g_1,q_1} g_1\in\mathbb{G}_{2}(J)\hookrightarrow \mathbb{F}_{2}(J),\hspace{1em} q_1\in\mathbb{G}_{3}(J)\hookrightarrow \mathbb{F}_{3}(J). \end{align} Case 2: $\gamma>0$. Based on the discussion in (i), it suffices to show the regularity of $\partial_t h(0)$, which is defined in \eqref{S5: replace h_t}. As illustrated in (i), we have $$\beta(\hat{h}_0)\hat{\gamma}_0, ([\![\psi(\hat{\vartheta}_0)]\!] +\sigma{\mathcal{H}}(\hat{h}_0))\in W^{2-3/p}_p(\Sigma).$$ It implies that $\partial_t h(0)\in W^{2-3/p}_p(\Sigma)$. It yields the desired results, i.e., $$g_1\in \mathbb{F}_{2}(J),\hspace{1em} q_1\in \mathbb{F}_{3}(J).$$ In virtue of condition ($\mathcal{CC}$) and the definitions of $g_1$ and $q_1$, one checks that condition ($\mathcal{LCC}$) is at our disposal. Therefore, all the compatibility conditions in \cite[Theorems~3.3, 3.5]{PruSimZac12} are satisfied, and then there exists a unique solution $z^* =(\vartheta^* ,h^* )\in\mathbb{E}(J)$ to the linear system \eqref{S5: reducing initial data}.
\begin{remark} The compatibility condition $[\![d(\theta_0)\partial_{\nu_{\Gamma_0}} \theta_0]\!]\in W^{2-6/p}_p(\Gamma_0)$ in Theorem~\ref{S1: main theorem: gamma=0} implies that
$$[\![ \hat{d}_0 \partial_\nu \hat{\vartheta}_0]\!] - ([\![ \hat{d}_0 \nabla \hat{\vartheta}_0]\!]| M_4(\hat{h}_0) \nabla_\Sigma \hat{h}_0) \in W^{2-6/p}_p(\Sigma) , $$ which is equivalent to
$$([\![ \hat{d}_0 \nabla \hat{\vartheta}_0]\!]| \nu_\Sigma - M_4(\hat{h}_0) \nabla_\Sigma \hat{h}_0) \in W^{2-6/p}_p(\Sigma) .$$ From the above discussion, we infer that $$[\![ \nabla \hat{\vartheta}_0]\!] \in W^{2-6/p}_p(\Sigma, \mathbb{R}^{m+1}) . $$ Now based on this observation and \eqref{S5: reg of k,d,gam,l-0}, we conclude that $$Q(\hat{z}_0)+[\![ d_A\partial_{\nu} \hat{\vartheta}_0]\!]\in W^{2-6/p}_p(\Sigma). $$ The other two conditions in conditions ($\mathcal{CC}$), i.e., when $\gamma\equiv 0$ $$\partial_{\nu_\Omega}\hat{\vartheta}_0=0,\quad l_1(0)\hat{\vartheta}_0+\sigma_0\Delta_\Sigma \hat{h}_0=G(\hat{z}_0),$$ and when $\gamma> 0$ $$ \partial_{\nu_\Omega}\hat{\vartheta}_0=0,\quad l_A l_1(0)\hat{\vartheta}_0 +l_A\sigma_0\Delta_\Sigma \hat{h}_0 -\gamma_1(0)[\![ d_A\partial_{\nu} \hat{\vartheta}_0]\!]=\gamma_1(0)Q(\hat{z}_0)+l_A G(\hat{z}_0).$$ can be easily obtained from the remaining compatibility conditions in Theorems~\ref{S1: main theorem: gamma=0} and \ref{S1: main theorem: gamma>0}. Similarly, the compatibility conditions in Theorems~\ref{S1: main theorem: gamma=0} and \ref{S1: main theorem: gamma>0} can also be concluded from conditions ($\mathcal{CC}$). By \cite[Theorems~3.1, 3.2]{PruSimZac12}, problem \eqref{S5: Stefan-linear/nonlinear} with initial data $\hat{z}_0$ has a unique $L_p$-solution on some possibly small but nontrivial interval $J:=[0,T]$, which is denoted by $\hat{z}=(\hat{\vartheta},\hat{h})$. \qed \end{remark}
Next, we apply the parameter-dependent diffeomorphisms ${\Theta}^{\ast}_{\lambda,\mu,\eta}$ and ${\Theta}^{\ast}_{\lambda,\mu}$ to show the analyticity of the solution $z^* $. We will use the following useful fact that for any time-independent map $\mathcal{N}$ acting on $\mathfrak{F}^s(\Omega_i,E)$ $${\Theta}^{\ast}_{\lambda,\mu,\eta}\mathcal{N}={T}_{\mu,\eta}\mathcal{N}{T}^{-1}_{\mu,\eta} {\Theta}^{\ast}_{\lambda,\mu,\eta} ,$$ and a similar result also holds for ${\Theta}^{\ast}_{\lambda,\mu}$. By Proposition~\ref{S3: Omega-invariant spaces}, we have \begin{align*} \partial_t [\vartheta^*_{\lambda,\mu,\eta}]&=(1+\xi^\prime\lambda){\Theta}^{\ast}_{\lambda,\mu,\eta} \partial_t\vartheta^* + B_{\lambda,\mu,\eta}(\vartheta^*_{\lambda,\mu,\eta})\\ &=(1+\xi^\prime\lambda){\Theta}^{\ast}_{\lambda,\mu,\eta} (d_A/\kappa_A)\Delta \vartheta^* +B_{\lambda,\mu,\eta}(\vartheta^*_{\lambda,\mu,\eta})\\
&=(1+\xi^\prime\lambda) (d_{A,\lambda,\mu,\eta}/\kappa_{A,\lambda,\mu,\eta}){T}_{\mu,\eta}\Delta{T}^{-1}_{\mu,\eta} \vartheta^*_{\lambda,\mu,\eta}+B_{\lambda,\mu,\eta}(\vartheta^*_{\lambda,\mu,\eta}), \end{align*} and either by Proposition~\ref{S3: normal der} \begin{align*} \partial_t [h^*_{\lambda,\mu}]&=(1+\xi^{\prime}\lambda){\Theta}^{\ast}_{\lambda,\mu} \partial_t h^* +B_{\lambda,\mu}(h^*_{\lambda,\mu})\\ &=(1+\xi^{\prime}\lambda){\Theta}^{\ast}_{\lambda,\mu} [(q_1+[\![d_A\partial_\nu \vartheta^* ]\!])/l_A] +B_{\lambda,\mu}(h^*_{\lambda,\mu})\\ &=(1+\xi^{\prime}\lambda)\{(q_{1,\lambda,\mu}/l_{A,\lambda,\mu})+[\![(d_{A,\lambda,\mu}/l_{A,\lambda,\mu}) \partial_\nu \vartheta^*_{\lambda,\mu}]\!] \} +B_{\lambda,\mu}(h^*_{\lambda,\mu}), \end{align*} when $\gamma\equiv 0$, or when $\gamma>0$, we have \begin{align*} \partial_t [h^*_{\lambda,\mu}]&=(1+\xi^{\prime}\lambda){\Theta}^{\ast}_{\lambda,\mu} [(l_1 \vartheta^* +\sigma_0\Delta_\Sigma h^* -g_1 )/\gamma_1] +B_{\lambda,\mu}(h^*_{\lambda,\mu})\\ &=(1+\xi^{\prime}\lambda)(l_{1,\lambda,\mu} \vartheta^*_{\lambda,\mu} +\sigma_0{T}_{\mu}\Delta_\Sigma{T}^{-1}_{\mu} h^*_{\lambda,\mu}- g_{1,\lambda,\mu} )/( \gamma_{1,\lambda,\mu}) +B_{\lambda,\mu}(h^*_{\lambda,\mu}). \end{align*} We define a map $\Phi:\mathbb{E}(J)\times\mathbb{B}(0,r_0)\rightarrow \mathbb{F}(J)$: $((\vartheta,h),(\lambda,\mu,\eta))\mapsto$ \begin{align*} \begin{cases} \kappa_{A,\lambda,\mu,\eta}\partial_t \vartheta -(1+\xi^\prime\lambda) d_{A,\lambda,\mu,\eta}{T}_{\mu,\eta}\Delta{T}^{-1}_{\mu,\eta} \vartheta-\kappa_{A,\lambda,\mu,\eta} B_{\lambda,\mu,\eta}(\vartheta) \hspace{2em} &\text{in }\quad \Omega\setminus\Sigma,\\ (1+{\rm sgn}(\gamma) \xi^{\prime}\lambda)(l_{1,\lambda,\mu} \vartheta +\sigma_0{T}_{\mu}\Delta_\Sigma{T}^{-1}_{\mu} h-g_{1,\lambda,\mu})-\gamma_{1,\lambda,\mu}\partial_t h\\ \quad\quad +\gamma_{1,\lambda,\mu} B_{\lambda,\mu}(h) &\text{on }\quad\Sigma,\\ l_{A,\lambda,\mu}\partial_t h-(1+\xi^{\prime}\lambda)(q_{1,\lambda,\mu}+[\![d_{A,\lambda,\mu} \partial_\nu \vartheta]\!]) -l_{A,\lambda,\mu}B_{\lambda,\mu}(h) &\text{on }\quad\Sigma,\\ \vartheta(0)-\hat{\vartheta}_0 &\text{in }\quad\Omega\setminus\Sigma,\\ h(0)-\hat{h}_0 &\text{on }\quad\Sigma. \end{cases} \end{align*} Setting $z^*_{\lambda,\mu,\eta}=(\vartheta^*_{\lambda,\mu,\eta},h^*_{\lambda,\mu})$, note that $\Phi(z^*_{\lambda,\mu,\eta},(\lambda,\mu,\eta))=0$ for all $(\lambda,\mu,\eta)\in\mathbb{B}(0,r_0)$.
(ii) We need to show that $\Phi$ actually maps into $\mathbb{F}(J)$. For simplification, we set $\Phi(z,(\lambda,\mu,\eta))=(f_2,g_2,q_2,(\vartheta_0,h_0))^{\sf T}$. It is obvious that based on trace theory of anisotropic Sobolev-Slobodeckii spaces, see \cite[Section~2]{DenPruZac08}, \begin{align*} (\vartheta_0, h_0) \in \mathbb{F}_{4}. \end{align*} By means of Propositions~\ref{S3: Omega-Tue} and \ref{S3: Omega-invariant spaces}, we infer that $f_2 \in \mathbb{F}_{1}(J)$. For the regularity of the next two terms, we split into two cases as before. When $\gamma\equiv 0$, observe that $\Delta_\Sigma \in \mathcal{L}(\mathbb{E}_{2}(J),\mathbb{F}_{2}(J))$. Proposition~\ref{S3: Omega: invariant spaces 2}, \eqref{S5: reg of k,d,gam,l} and \eqref{S5: reg of g_1,q_1} yield \begin{align} \label{S5: reg of trans of g_1,q_1,l_1,gam_1} \gamma_{1,\lambda,\mu}\, ,\, l_{1,\lambda,\mu}\, ,\, g_{1,\lambda,\mu}\in \mathbb{F}_{2}(J), \hspace{1em} q_{1,\lambda,\mu}\in\mathbb{F}_{3}(J). \end{align} Taking into consideration that $\partial_t h\in \mathbb{F}_{3}(J)$ and the fact that $\mathbb{F}_{2}(J),\mathbb{F}_{3}(J)$ are multiplication algebras, we conclude from Propositions~\ref{S3：M-reg},~\ref{S3: Omega-invariant spaces} and \eqref{S5: reg of trans of g_1,q_1,l_1,gam_1} that \begin{align*} g_2 \in \mathbb{F}_{2}(J),\hspace{1em} q_2\in \mathbb{F}_{3}(J). \end{align*} When $\gamma>0$, since $\partial_t h\in\mathbb{F}_{2}(J)$, the desired regularity results clearly hold true. It remains to show condition ($\mathcal{LCC}$).
\\ Case 1: $\gamma\equiv 0$. It is immediate that $\partial_{\nu_\Omega} \vartheta_0 = \partial_{\nu_\Omega}(\vartheta(0)-\hat{\vartheta}_0)=0$. One checks that \begin{align*} &\quad l_1(0)\vartheta_0+\sigma_0\Delta_\Sigma h_0 = l_1(0)(\vartheta(0)-\hat{\vartheta}_0)+\sigma_0\Delta_\Sigma (h(0)-\hat{h}_0)\\ &=l_{1,\lambda,\mu}(0)\vartheta(0)+\sigma{T}_{\mu}(0)\Delta_\Sigma {T}^{-1}_{\mu}(0)h(0)-g_{1,\lambda,\mu}(0)=g_2(0), \end{align*} by recalling $u_{\lambda,\mu}(0,\cdot)=u(0,\cdot)$ for any $u\in E^\Sigma$, and \begin{align*} q_2(0)+[\![ d_{A,\lambda,\mu }(0) \partial_\nu \vartheta_0]\!]&= l_{A,\lambda,\mu }(0)\partial_t h(0) -q_{1,\lambda,\mu }(0)-[\![ d_A \partial_\nu \vartheta(0)]\!] \\ &\quad -l_{A,\lambda,\mu }(0)B_{\lambda,\mu}(h(0)) +[\![ d_A\partial_\nu (\vartheta(0)- \hat{\vartheta}_0)]\!]\\ &=l_{A}\partial_t h(0)-q_{1}(0) -l_{A}B_{\lambda,\mu}(h(0))-[\![ d_A\partial_\nu \hat{\vartheta}_0]\!]. \end{align*} By the discussion in (ii), $q_{1}(0) +[\![ d_A\partial_\nu \hat{\vartheta}_0]\!]\in W^{2-6/p}_p(\Sigma)$. Since $W^{2-6/p}_p(\Sigma)$ is a multiplication algebra, \eqref{S3: analyticity of Blm}, \eqref{S5: reg of k,d,gam,l} and \cite[formula~(3.1)]{PruSimZac12} imply that \begin{align*} q_2(0)+[\![ d_{A,\lambda,\mu }(0) \partial_\nu \vartheta_0]\!]\in W^{2-6/p}_p(\Sigma). \end{align*} Case 2: $\gamma>0$. \begin{align*} &\quad l_A l_1(0)\vartheta_0 + l_A\sigma_0 \Delta_\Sigma h_0 -\gamma_1(0)[\![ d_A \partial_\nu \vartheta_0 ]\!]\\ &=l_{A,\lambda,\mu}(0) l_{1,\lambda,\mu}(0)(\vartheta(0)-\hat{\vartheta}_0)+ l_{A,\lambda,\mu}(0)\sigma_0 {T}_{\mu}(0)\Delta_\Sigma {T}^{-1}_{\mu}(0)(h(0)-\hat{h}_0)\\ &\quad -\gamma_{1,\lambda,\mu}(0)[\![ d_{A,\lambda,\mu} \partial_\nu (\vartheta(0)-\hat{\vartheta}_0) ]\!]\\ &=l_A l_1(0) \vartheta(0) + l_A\sigma_0 \Delta_\Sigma h(0) -\gamma_1(0)[\![ d_A \partial_\nu \vartheta(0) ]\!]
-l_A g_1(0) -\gamma_1(0)q_1(0)\\ &=l_A g_2(0) +\gamma_1(0)q_2(0). \end{align*} Therefore $\Phi(z,(\lambda,\mu,\eta))\in\mathbb{F}(J)$.
(iii) Let $w=(u,\rho)\in\mathbb{E}(J)$. The Fr\`echet derivative of $\Phi$ with respect to $(\vartheta,h)$ at $(z^* ,0)$ is clearly given by \begin{align*} D_1 \Phi(z^* ,0)w= \left\{\begin{aligned} &\kappa_A\partial_t u -d_A\Delta u &&\text{in}&& \Omega\setminus\Sigma,\\ &l_1 u +\sigma_0\Delta_\Sigma\rho -\gamma_1\partial_t \rho &&\text{on}&& \Sigma,\\ &l_A \partial_t \rho -[\![d_A\partial_\nu u]\!] &&\text{on}&& \Sigma,\\ &u(0) &&\text{in}&& \Omega\setminus\Sigma,\\ &\rho(0) &&\text{on}&& \Sigma. \end{aligned}\right. \end{align*} It is obvious that \begin{align*}
D_1 \Phi(z^* ,0)w\in \mathbb{F}_{1}(J)\times\mathbb{F}_{2}(J)\times\mathbb{F}_{3}(J)\times\mathbb{F}_{4}. \end{align*} Condition ($\mathcal{LCC}$) can be verified as in (ii) by using \cite[formula~(3.1)]{PruSimZac12} in the case $\gamma\equiv 0$. We can deduce from \cite[Theorems~3.3, 3.5]{PruSimZac12} that \begin{align*} D_1 \Phi(z^* ,0) \in \mathcal{L}{\rm{is}}(\mathbb{E}(J),\mathbb{F}(J)). \end{align*}
\\ (iv) Regularity of $[(\lambda,\mu)\mapsto (g_{1,\lambda,\mu},q_{1,\lambda,\mu})]$. We express $g_{1,\lambda,\mu}$ as \begin{align*} g_{1,\lambda,\mu}&={\Theta}^{\ast}_{\lambda,\mu} e^{\Delta_\Sigma t}G(\hat{z}_0)={\Theta}^{\ast}_{\lambda,\mu}(c+\Delta_\Sigma)e^{\Delta_\Sigma t}(c+\Delta_\Sigma)^{-1}G(\hat{z}_0)\\ &={T}_{\mu} (c+\Delta_\Sigma){T}^{-1}_{\mu} {\Theta}^{\ast}_{\lambda,\mu} e^{\Delta_\Sigma t}(c+\Delta_\Sigma)^{-1}G(\hat{z}_0). \end{align*} For sufficiently large $c$, it is well-known that $(c+\Delta_\Sigma)$ is an isomorphism from $W^{s+2}_p(\Sigma)$ to $W^s_p(\Sigma)$ for any $s\in \mathbb{R}$. We consider the solution to \begin{align} \label{S5: eq of g_1} \partial_t \rho+ \Delta_\Sigma \rho=0, \quad \rho(0)=(c+\Delta_\Sigma)^{-1} G(\hat{z}_0). \end{align} $\hat{\rho}:=e^{\Delta_\Sigma t}(c+\Delta_\Sigma)^{-1}G(\hat{z}_0)$ is the unique solution to \eqref{S5: eq of g_1}. Furthermore, Proposition~\ref{S3: Omega-invariant spaces} shows that $\hat{\rho}_{\lambda,\mu}$ satisfies \begin{align*} \partial_t [\hat{\rho}_{\lambda,\mu}]&= (1+\xi^\prime \lambda){\Theta}^{\ast}_{\lambda,\mu} \partial_t \hat{\rho} + B_{\lambda,\mu}(\hat{\rho}_{\lambda,\mu})\\ &= -(1+\xi^\prime \lambda){T}_{\mu}\Delta_\Sigma {T}^{-1}_{\mu} \hat{\rho}_{\lambda,\mu} + B_{\lambda,\mu}(\hat{\rho}_{\lambda,\mu}). \end{align*} We define the map $\Phi_g: \mathbb{X}_2(J) \times \mathbb{B}(0,r_0) \rightarrow L_p(J;W^{2-1/p}_p(\Sigma))\times W^{4-3/p}_p(\Sigma)$ by \begin{align*} \Phi_g(\rho,(\lambda,\mu))=\binom{\partial_t \rho +(1+\xi^\prime \lambda){T}_{\mu}\Delta_\Sigma {T}^{-1}_{\mu} \rho -B_{\lambda,\mu}(\rho)}{\rho(0)-(c+\Delta_\Sigma)^{-1} G(\hat{z}_0)}. \end{align*} Here $\mathbb{X}_2(J):=H^1_p(J; W^{2-1/p}(\Sigma))\cap L_p(J; W^{4-1/p}(\Sigma))$. Note that $\Phi_g(\hat{\rho}_{\lambda,\mu}, (\lambda,\mu))=(0,0)^{\sf T}.$ By the $L_p$-maximal regularity of $\Delta_\Sigma$, we immediately have \begin{align*} D_1\Phi_g(\hat{\rho},0)\in \mathcal{L}{\rm{is}}(\mathbb{X}_2(J), L_p(J;W^{2-1/p}_p(\Sigma))\times W^{4-3/p}_p(\Sigma)). \end{align*} We define a bilinear and continuous map \begin{align*} T: C(J;\mathcal{L}(W^{4-1/p}_p(\Sigma), W^{3-1/p}_p(\Sigma)))\times \mathbb{X}_2(J)\rightarrow L_p(J;W^{2-1/p}_p(\Sigma)) \end{align*} by $(B,u)\mapsto [t\mapsto B(t)u(t)]$. Since $T$ is real analytic, by \eqref{S3: analyticity of Blm}, we get \begin{align*} [(u,(\lambda,\mu))\mapsto B_{\lambda,\mu}(u)]\in C^\omega (\mathbb{X}_2 \times \mathbb{B}(0,r_0), L_p(J;W^{2-1/p}_p(\Sigma))). \end{align*} In virtue of Proposition~\ref{S3：M-reg}(b) and the above bilinear map argument, one gets \begin{align*} \Phi_g \in C^\omega (\mathbb{X}_2 \times \mathbb{B}(0,r_0), L_p(J;W^{2-1/p}_p(\Sigma))\times W^{4-3/p}_p(\Sigma)). \end{align*} By the implicit function theorem, there exists some $\mathbb{B}(0,r_1)\subset\mathbb{B}(0,r_0)$ such that \begin{align*} [(\lambda,\mu)\mapsto {\Theta}^{\ast}_{\lambda,\mu} e^{\Delta_\Sigma t}(c+\Delta_\Sigma)^{-1}G(\hat{z}_0)] \in C^\omega(\mathbb{B}(0,r_1), \mathbb{X}_2). \end{align*} Without loss of generality, we may assume that $r_1=r_0$. In view of Proposition~\ref{S3：M-reg}(b), we have that for all $s\in\mathbb{R}$ \begin{align*} [\mu\mapsto {T}_{\mu} (c+\Delta_\Sigma){T}^{-1}_{\mu}]\in C^{\omega}(\mathbb{B}(0,r_0), C^1(J; \mathcal{L}(W^{s+2}_p(\Sigma), W^s_p(\Sigma)))). \end{align*} The above bilinear map argument and the embedding $\mathbb{G}_{2}(J)\hookrightarrow \mathbb{F}_{2}(J)$ yield \begin{align} \label{S5: ana of g_1-trans} [(\lambda,\mu)\mapsto g_{1,\lambda,\mu}] \in C^{\omega} (\mathbb{B}(0,r_0), \mathbb{F}_{2}(J)). \end{align} Following a similar discussion, one obtains \begin{align} \label{S5: ana of q_1-trans} [(\lambda,\mu)\mapsto q_{1,\lambda,\mu}] \in C^{\omega} (\mathbb{B}(0,r_0), \mathbb{F}_{3}(J)). \end{align}
\\ (v) Regularity of the map $\Phi$. It follows from Proposition~\ref{S3: Omega-Tue} that \begin{align} \label{S5: ana of ka,d-trans} [(\lambda,\mu,\eta)\mapsto (\kappa_{A,\lambda,\mu,\eta}, d_{A,\lambda,\mu,\eta})] \in C^k (\mathbb{B}(0,r_0), C^1(J; BC(\Omega))^2). \end{align} Proposition~\ref{S3：M-reg} implies that \begin{align} \label{S5: ana of l,d-trans} [(\lambda,\mu)\mapsto (l_{A,\lambda,\mu}, d_{A,\lambda,\mu})] \in C^k (\mathbb{B}(0,r_0), C^1(J; BC^1(\Sigma))^2). \end{align} Similarly, we have \begin{align} \label{S5: ana of l_1,ga_1-trans} [(\lambda,\mu)\mapsto (l_{1,\lambda,\mu}, \gamma_{1,\lambda,\mu})] \in C^k (\mathbb{B}(0,r_0), (C^1(J; BC(\Sigma)) \cap C(J; BC^2(\Sigma)))^2). \end{align} By Proposition~\ref{S3: Omega-Tue}, one concludes that $$[(\mu,\eta)\mapsto {T}_{\mu,\eta} \Delta {T}^{-1}_{\mu,\eta}]\in C^\omega (\mathbb{B}(0,r_0), C^1(J; \mathcal{L}(H^2_p(\Omega\setminus \Sigma), L_p(\Omega))) ).$$ Combined with \eqref{S5: ana of ka,d-trans}, this yields $$[(\lambda,\mu,\eta)\mapsto d_{A,\lambda,\mu,\eta}{T}_{\mu,\eta} \Delta {T}^{-1}_{\mu,\eta}] \in C^k (\mathbb{B}(0,r_0), \mathcal{L}(\mathbb{E}_{1}(J),\mathbb{F}_{1}(J)) ).$$ It follows again from \eqref{S5: ana of ka,d-trans} that $$[(\vartheta, (\lambda,\mu,\eta))\mapsto \kappa_{A,\lambda,\mu,\eta}\partial_t \vartheta]\in C^k (\mathbb{E}_{1}(J)\times\mathbb{B}(0,r_0), \mathbb{F}_{1}(J)).$$ Together with Proposition~\ref{S3: Omega-invariant spaces}, the above discussion shows that \begin{align*} [(\vartheta,(\lambda,\mu,\eta))& \mapsto \kappa_{A,\lambda,\mu,\eta}\partial_t \vartheta -(1+\xi^\prime\lambda) d_{A,\lambda,\mu,\eta}{T}_{\mu,\eta}\Delta{T}^{-1}_{\mu,\eta} \vartheta-B_{\lambda,\mu,\eta}(\vartheta)]\\ & \in C^k(\mathbb{E}_{1}(J)\times\mathbb{B}(0,r_0),\mathbb{F}_{1}(J)). \end{align*} Applying \eqref{S5: ana of g_1-trans}-\eqref{S5: ana of l_1,ga_1-trans}, one can check the regularity of all the other entries of $\Phi$, and thus $$\Phi\in C^k(\mathbb{E}(J)\times\mathbb{B}(0,r_0),\mathbb{F}(J)).$$ Now the implicit function theorem yields \begin{align*} [(\lambda,\mu,\eta)\mapsto z^*_{\lambda,\mu,\eta}]\in C^k(\mathbb{B}(0,r_0), \mathbb{E}(J)). \end{align*} As a conclusion of Theorem~\ref{S3: main thm}, we obtain \begin{align*} z^* \in C^k(\mathring{J}\times {\sf T}_{a/3}\!\setminus\!\Sigma) \times C^k (\mathring{J}\times\Sigma). \end{align*} To attain the regularity of $\vartheta^* $ in $\Omega\setminus \bar{{\sf T}}_{a/6}$, we study the linear equations \begin{equation*}
\left\{\begin{aligned} \kappa_1(\vartheta_A) \partial_t \vartheta -d_1(\vartheta_A)\Delta\vartheta&=0 &&\text{in}&&\Omega_1\setminus\bar{{\sf T}}_{a/6}\\ \vartheta&=\vartheta^* &&\text{on}&&\partial_1{\sf T}_{a/6}\\ \vartheta(0)&=\hat{\vartheta}_0 &&\text{in}&&\Omega_1\setminus\bar{{\sf T}}_{a/6}, \end{aligned}\right. \end{equation*} and \begin{equation*}
\left\{\begin{aligned} \kappa_2(\vartheta_A) \partial_t \vartheta -d_2(\vartheta_A)\Delta\vartheta&=0 &&\text{in}&&\Omega_2\setminus\bar{{\sf T}}_{a/6}\\ \partial_{\nu_\Omega} \vartheta &=0 &&\text{on}&&\partial \Omega \\ \vartheta&=\vartheta^* &&\text{on}&&\partial_2{\sf T}_{a/6}\\ \vartheta(0)&=\hat{\vartheta}_0 &&\text{in}&&\Omega_2\setminus\bar{{\sf T}}_{a/6}, \end{aligned}\right. \end{equation*} where $\partial_i{\sf T}_{a/6}:=\Lambda(\Sigma,(-1)^ia/6)$. Since $\vartheta^*$ is analytic on $\partial_i{\sf T}_{a/6}$, we can obtain the regularity of $\vartheta^*$ in these two domains by means of the parameter-dependent diffeomorphism defined in \cite{EscPruSim03}, the results in \cite[Section~8]{DenHiePru03}, and the implicit function theorem as above.
To sum up, we have established the analyticity of the solution $z^* $, i.e., \begin{align} \label{S5: ana of z*} z^* \in C^k(\mathring{J}\times \Omega\setminus\Sigma) \times C^k (\mathring{J}\times\Sigma). \end{align}
\subsection{\bf Reduction to zero initial data} Recall that $\hat{z}=(\hat{\vartheta},\hat{h})$ is the unique $L_p$ solution to problem \eqref{S5: Stefan-linear/nonlinear} with initial data $\hat{z}_0$. We set $\bar{\vartheta}=\hat{\vartheta}-\vartheta^*$, $\bar{h}=\hat{h}-h^*$. Then $\bar{z}=(\bar{\vartheta},\bar{h})$ is the unique solution to the following equation. \begin{equation} \label{S5: equation-zero intial} \left\{\begin{aligned} \kappa_A \partial_t \vartheta -d_A\Delta \vartheta&=F(\vartheta+\vartheta^*,h+h^*) &&\text{in}&&\Omega\setminus\Sigma, \\ \partial_{\nu_\Omega} \vartheta&=0 &&\text{on}&&\partial \Omega,\\ [\![\vartheta]\!]&=0 &&\text{on}&& \Sigma,\\ l_1 \vartheta + \sigma_0 \Delta_\Sigma h- \gamma_1\partial_t h &=\bar{G}(\vartheta,h;\vartheta^* ,h^* ) &&\text{on}&& \Sigma,\\ l_A \partial_t h -[\![d_A\partial_\nu \vartheta]\!] &=\bar{Q}(\vartheta,h; \vartheta^* ,h^* ) &&\text{on}&& \Sigma,\\ \vartheta(0)&=0 &&\text{in}&&\Omega\setminus\Sigma, \\ h(0)&=0 &&\text{on}&& \Sigma. \end{aligned}\right. \end{equation} Here we have set \begin{equation*} \begin{split} &\bar{G}(\vartheta,h;\vartheta^* ,h^*)=G(\vartheta+\vartheta^* ,h+h^* )-e^{\Delta_\Sigma t}G(\hat{z}_0),\\ &\bar{Q}(\vartheta,h;\vartheta^* ,h^*) = Q(\vartheta+\vartheta^* ,h+h^* )-e^{\Delta_\Sigma t}Q(\hat{z}_0).\\ \end{split} \end{equation*} Note that $\bar{G}(0,0;\hat{\vartheta}_0,\hat{h}_0)=\bar{Q}(0,0;\hat{\vartheta}_0,\hat{h}_0)=0$ by construction, which ensures time trace zero at $t=0$.
Let $z^*_{\lambda,\mu,\eta}:=(\vartheta^*_{\lambda,\mu,\eta},h^*_{\lambda,\mu})$. As in Section~5.1, we compute the temporal derivative of $\bar{z}_{\lambda,\mu,\eta}:=(\bar{\vartheta}_{\lambda,\mu,\eta},\bar{h}_{\lambda,\mu})$ as follows. \begin{align*} \partial_t[\bar{\vartheta}_{\lambda,\mu,\eta}]&=(1+\xi^\prime \lambda)(d_{A,\lambda,\mu,\eta}/\kappa_{A,\lambda,\mu,\eta}){T}_{\mu,\eta}\Delta{T}^{-1}_{\mu,\eta} \bar{\vartheta}_{\lambda,\mu,\eta } +B_{\lambda,\mu,\eta}(\bar{\vartheta}_{\lambda,\mu,\eta})\\ &\quad +F_{\lambda,\mu,\eta}(\bar{z}_{\lambda,\mu,\eta},z^*_{\lambda,\mu,\eta})/\kappa_{A,\lambda,\mu,\eta}, \end{align*} and either \begin{align*} \partial_t[\bar{h}_{\lambda,\mu}]=(1+\xi^\prime \lambda)[\![ (d_{A,\lambda,\mu}/l_{A,\lambda,\mu})\partial_\nu \bar{\vartheta}]\!] + B_{\lambda,\mu}(\bar{h}_{\lambda,\mu})+\bar{Q}_{\lambda,\mu}(\bar{z}_{\lambda,\mu,\eta},z^*_{\lambda,\mu,\eta})/l_{A,\lambda,\mu}, \end{align*} when $\gamma\equiv 0$, or when $\gamma>0$ we have \begin{align*} \partial_t[\bar{h}_{\lambda,\mu}]&=(1+\xi^\prime \lambda)(l_{1,\lambda,\mu}\bar{\vartheta}_{\lambda,\mu} +\sigma_0{T}_{\mu}\Delta_\Sigma {T}^{-1}_{\mu} \bar{h}_{\lambda,\mu})/\gamma_{1,\lambda,\mu} +B_{\lambda,\mu}(\bar{h}_{\lambda,\mu})\\ &\quad -\bar{G}_{\lambda,\mu}(\bar{z}_{\lambda,\mu,\eta},z^*_{\lambda,\mu,\eta})/\gamma_{1,\lambda,\mu}. \end{align*} For sufficiently small $r_0$, $\Theta_{\mu,\eta}(z)=z$ for $(\mu,\eta)\in \mathbb{B}(0,r_0)$ and $z\in\Omega\setminus{\sf T}_{a/3}$. Hence, $(\zeta\circ d_\Sigma)_{\lambda,\mu,\eta}=\zeta\circ d_\Sigma$. In virtue of $(\zeta\circ d_\Sigma)_{\lambda,\mu,\eta}(x)=0$ for $x\notin\Omega\setminus{\sf T}_{2a/3}$, and $(h\circ \Pi)_{\lambda,\mu,\eta}(x)=(h_{\lambda,\mu}\circ \Pi)(x)$ for $x\in {\sf T}_{2a/3}$, one readily verifies that \begin{align*} {\Theta}^{\ast}_{\lambda,\mu,\eta} \Upsilon(h)&=(\zeta\circ d_\Sigma)_{\lambda,\mu,\eta}(h_{\lambda,\mu}\circ\Pi)_{\lambda,\mu,\eta}(\nu_\Sigma\circ \Pi)_{\lambda,\mu,\eta}\\ &= (\zeta\circ d_\Sigma)(h_{\lambda,\mu}\circ\Pi)(\nu_\Sigma\circ \Pi)_{\lambda,\mu,\eta}. \end{align*} In the above expressions, for $z=(\vartheta,h)$, \begin{align*} F_{\lambda,\mu,\eta}(z,z^{*}_{\lambda,\mu,\eta})&=(1+\xi^\prime \lambda){\Theta}^{\ast}_{\lambda,\mu,\eta} F({\Theta}_{\ast}^{\lambda,\mu,\eta}(z+z^*_{\lambda,\mu,\eta}))\\ &= (\kappa_{A,\lambda,\mu,\eta}-\kappa(\vartheta+\vartheta^*_{\lambda,\mu,\eta}))[\partial_t(\vartheta+\vartheta^*_{\lambda,\mu,\eta})-B_{\lambda,\mu,\eta}(\vartheta+\vartheta^*_{\lambda,\mu,\eta})]\\ &\quad + (1+\xi^\prime \lambda)\{(d(\vartheta+\vartheta^*_{\lambda,\mu,\eta})-d_{A,\lambda,\mu,\eta}){T}_{\mu,\eta}\Delta {T}^{-1}_{\mu,\eta}(\vartheta+\vartheta^*_{\lambda,\mu,\eta})\\ &\quad -d(\vartheta+\vartheta^*_{\lambda,\mu,\eta})M_{2,\lambda,\mu,\eta}(h+h^*_{\lambda,\mu}):{T}_{\mu,\eta}\nabla^2 {T}^{-1}_{\mu,\eta} (\vartheta+\vartheta^*_{\lambda,\mu,\eta})\\
&\quad +d^\prime(\vartheta+\vartheta^*_{\lambda,\mu,\eta})|(I- M_{1,\lambda,\mu,\eta}(h +h^*_{\lambda,\mu})){T}_{\mu,\eta}\nabla {T}^{-1}_{\mu,\eta} (\vartheta+\vartheta^*_{\lambda,\mu,\eta})|^2\\
&\quad -d(\vartheta+\vartheta^*_{\lambda,\mu,\eta})(M_{3,\lambda,\mu,\eta}(h +h^*_{\lambda,\mu})| {T}_{\mu,\eta}\nabla {T}^{-1}_{\mu,\eta} (\vartheta+\vartheta^*_{\lambda,\mu,\eta}))\\ &\quad + \kappa(\vartheta+\vartheta^*_{\lambda,\mu,\eta})\mathcal{R}_{\lambda,\mu,\eta}({h}+h^*_{\lambda,\mu})(\vartheta+\vartheta^*_{\lambda,\mu,\eta})\}, \end{align*} with \begin{align*} \begin{split} \Upsilon_{\lambda,\mu,\eta}(h)&={\Theta}^{\ast}_{\lambda,\mu,\eta} \Upsilon({\Theta}_{\ast}^{\lambda,\mu} h)=(\zeta\circ d_\Sigma) (h\circ \Pi) (\nu_\Sigma \circ\Pi)_{\lambda,\mu,\eta}\\ M_{1,\lambda,\mu,\eta}(h)&={\Theta}^{\ast}_{\lambda,\mu,\eta} M_1({\Theta}_{\ast}^{\lambda,\mu} h)\\ &=[(I+ ({T}_{\mu,\eta}\nabla{T}^{-1}_{\mu,\eta} \Upsilon_{\lambda,\mu,\eta}(h))^{\sf T})^{-1} ({T}_{\mu,\eta}\nabla{T}^{-1}_{\mu,\eta} \Upsilon_{\lambda,\mu,\eta}(h))^{\sf T}]^{\sf T},\\ M_{2,\lambda,\mu,\eta}(h)&={\Theta}^{\ast}_{\lambda,\mu,\eta} M_2({\Theta}_{\ast}^{\lambda,\mu} h)\\ &=M_{1,\lambda,\mu,\eta}(h)+(M_{1,\lambda,\mu,\eta}(h))^{\sf T} -M_{1,\lambda,\mu,\eta}(h)(M_{1,\lambda,\mu,\eta}(h))^{\sf T},\\ M_{3,\lambda,\mu,\eta}(h)&={\Theta}^{\ast}_{\lambda,\mu,\eta} M_3({\Theta}_{\ast}^{\lambda,\mu} h)=(I-M_{1,\lambda,\mu,\eta}(h)): {T}_{\mu,\eta}\nabla{T}^{-1}_{\mu,\eta} M_{1,\lambda,\mu,\eta}(h),\\ \mathcal{R}_{\lambda,\mu,\eta}(h)\vartheta&={\Theta}^{\ast}_{\lambda,\mu,\eta} \mathcal{R}({\Theta}_{\ast}^{\lambda,\mu} h){\Theta}_{\ast}^{\lambda,\mu,\eta} \vartheta\\
&=({T}_{\mu,\eta}\nabla{T}^{-1}_{\mu,\eta} \vartheta |(I+({T}_{\mu,\eta}\nabla{T}^{-1}_{\mu,\eta} \Upsilon_{\lambda,\mu,\eta}(h))^{\sf T})^{-1}\\ &\quad[\partial_t \Upsilon_{\lambda,\mu,\eta}(h)- B_{\lambda,\mu,\eta}(\Upsilon_{\lambda,\mu,\eta}(h))])/(1+\xi^\prime \lambda), \end{split} \end{align*} and \begin{align*} \bar{G}_{\lambda,\mu}(z,&z^*_{\lambda,\mu,\eta})=(1+\xi^\prime \lambda){\Theta}^{\ast}_{\lambda,\mu} \bar{G}({\Theta}_{\ast}^{\lambda,\mu}(z+z^*_{\lambda,\mu,\eta}))\\ &=\{ \gamma(\vartheta+\vartheta^*_{\lambda,\mu})\beta_{\lambda,\mu}(h+h^*_{\lambda,\mu})-\gamma_{1,\lambda,\mu}\}[\partial_t({h}+h^*_{\lambda,\mu})-B_{\lambda,\mu}({h}+h^*_{\lambda,\mu})]\\ &\quad +(1+\xi^\prime \lambda)\{-([\![\psi(\vartheta+\vartheta^*_{\lambda,\mu})]\!]+\sigma{\mathcal{H}}_{\lambda,\mu}(h+h^*_{\lambda,\mu}))+l_{1,\lambda,\mu}(\vartheta+\vartheta^*_{\lambda,\mu})\\ &\quad +\sigma_0 {T}_{\mu}\Delta_\Sigma{T}^{-1}_{\mu} ({h}+h^*_{\lambda,\mu}) -g_{1,\lambda,\mu} \}, \end{align*} with ${\mathcal{H}}_{\lambda,\mu}={\Theta}^{\ast}_{\lambda,\mu} \mathcal{H}{\Theta}_{\ast}^{\lambda,\mu} ,$ and \begin{align*} \begin{split} M_{0,\lambda,\mu}(h)&={\Theta}^{\ast}_{\lambda,\mu} M_0({\Theta}_{\ast}^{\lambda,\mu} h)=(I-h L_{\Sigma,\lambda,\mu})^{-1},\hspace{1em}\\ \alpha_{\lambda,\mu}(h)&={\Theta}^{\ast}_{\lambda,\mu} \alpha({\Theta}_{\ast}^{\lambda,\mu} h)=M_{0,\lambda,\mu}(h){T}_{\mu}\nabla{T}^{-1}_{\mu} h,\\
\beta_{\lambda,\mu}(h)&={\Theta}^{\ast}_{\lambda,\mu} \beta({\Theta}_{\ast}^{\lambda,\mu} h)=(1+|\alpha_{\lambda,\mu}(h)|^2)^{-1/2}, \end{split} \end{align*} and \begin{align*} \bar{Q}_{\lambda,\mu}&(z ,z^*_{\lambda,\mu,\eta})=(1+\xi^\prime \lambda){\Theta}^{\ast}_{\lambda,\mu} \bar{Q}({\Theta}_{\ast}^{\lambda,\mu}(z+z^*_{\lambda,\mu,\eta}))\\ &=\gamma(\vartheta+\vartheta^*_{\lambda,\mu})\beta_{\lambda,\mu}(h+h^*_{\lambda,\mu})[\partial_t({h}+h^*_{\lambda,\mu})-B_{\lambda,\mu}({h}+h^*_{\lambda,\mu})]^2/(1+\xi^\prime \lambda)\\ &\quad+(l_{A,\lambda,\mu}-l(\vartheta+\vartheta^*_{\lambda,\mu}))[\partial_t({h}+h^*_{\lambda,\mu})-B_{\lambda,\mu}({h}+h^*_{\lambda,\mu})]\\ &\quad+(1+\xi^\prime \lambda)\{[\![(d(v+\bar{\vartheta}_{\lambda,\mu})-d_{A,\lambda,\mu})\partial_\nu (\vartheta+\vartheta^*_{\lambda,\mu})]\!]-q_{1,\lambda,\mu}\\
&\quad-( [\![d(\vartheta+\vartheta^*_{\lambda,\mu}){T}_{\mu}\nabla {T}^{-1}_{\mu}(\vartheta+\vartheta^*_{\lambda,\mu})]\!] | M_{4,\lambda,\mu}(h+h^*_{\lambda,\mu}){T}_{\mu}\nabla_\Sigma {T}^{-1}_{\mu}({h}+h^*_{\lambda,\mu})) \} \end{align*} with $M_{4,\lambda,\mu}(h)={\Theta}^{\ast}_{\lambda,\mu} M_4({\Theta}_{\ast}^{\lambda,\mu} h)=(I -M_{1,\lambda,\mu}(h ))^{\sf T} M_{0,\lambda,\mu}(h)$.
\\ Consider the map $\Phi:\prescript{}{0}{\mathbb{E}(J)}\rightarrow \prescript{}{0}{\mathbb{F}(J)}$: $((\vartheta,h),(\lambda,\mu,\eta))\mapsto$ \begin{align*} \begin{cases} \kappa_{A,\lambda,\mu,\eta}\partial_t \vartheta -(1+\xi^\prime\lambda) d_{A,\lambda,\mu,\eta}{T}_{\mu,\eta}\Delta{T}^{-1}_{\mu,\eta} \vartheta-\kappa_{A,\lambda,\mu,\eta}B_{\lambda,\mu,\eta}(\vartheta)\hspace{3em} &\\ \quad\quad -F_{\lambda,\mu,\eta}(z ,z^*_{\lambda,\mu,\eta}) &\text{in}\hspace{.5em} \Omega\!\setminus\!\Sigma,\\
(1+{\rm sgn}(\gamma) \xi^{\prime}\lambda)(l_{1,\lambda,\mu} \vartheta +\sigma_0{T}_{\mu}\Delta_\Sigma{T}^{-1}_{\mu} h)-\gamma_{1,\lambda,\mu}\partial_t h \\ \quad\quad +\gamma_{1,\lambda,\mu} B_{\lambda,\mu}(h) - \bar{G}_{\lambda,\mu}(z,z^*_{\lambda,\mu,\eta}) &\text{on}\hspace{.5em} \Sigma,\\ l_{A,\lambda,\mu}\partial_t h-(1+\xi^{\prime}\lambda)( [\![d_{A,\lambda,\mu} \partial_\nu \vartheta]\!]) -l_{A,\lambda,\mu}B_{\lambda,\mu}(h)-\bar{Q}_{\lambda,\mu}(z ,z^*_{\lambda,\mu,\eta}) &\text{on}\hspace{.5em} \Sigma, \end{cases} \end{align*} where $z=(\vartheta,h)$. Note that $\Phi(\bar{z}_{\lambda,\mu},(\lambda,\mu))=0$ for all $(\lambda,\mu)\in\mathbb{B}(0,r_0)$. It is understood that $\prescript{}{0}{\mathbb{F}(J)}:=\mathbb{F}_{1}(J)\times\prescript{}{0}{\mathbb{F}_{2}(J)}\times\prescript{}{0}{\mathbb{F}_{3}(J)}$.
\\ (i) First, we shall check that $\Phi$ actually maps into $\prescript{}{0}{\mathbb{F}(J)}$. Functions in $\prescript{}{0}{\mathbb{F}(J)}$ automatically satisfy ($\mathcal{LCC}$). One can check the temporal traces without difficulty. Indeed, recalling that $z^*_{\lambda,\mu,\eta}(0,\cdot)=z^*(0,\cdot)$, we have \begin{align*} &\bar{G}_{\lambda,\mu}(z,z^*_{\lambda,\mu,\eta})(0)={\Theta}^{\ast}_{\lambda,\mu} G(z^*)(0)-g_1(0)=0,\\ &\bar{Q}_{\lambda,\mu}(z,z^*_{\lambda,\mu,\eta})(0)={\Theta}^{\ast}_{\lambda,\mu} Q(z^*)(0)-q_1(0)=0. \end{align*} It suffices to check regularity of $F_{\lambda,\mu,\eta}(z ,z^*_{\lambda,\mu,\eta})$, $\bar{G}_{\lambda,\mu}(z,z^*_{\lambda,\mu,\eta})$ and $\bar{Q}_{\lambda,\mu}(z,z^*_{\lambda,\mu,\eta})$, which will become clear in our argument for the regularity of $\Phi$.
\\ It follows from well-known results for substitution operators for Sobolev spaces and \eqref{S5: ana of z*} that \begin{align*} [(\vartheta,(\lambda,\mu,\eta))\mapsto \kappa(\vartheta+\vartheta^*_{\lambda,\mu,\eta})]\in C^k (\prescript{}{0}{\mathbb{E}_{1}(J)}\times\mathbb{B}(0,r_0), C(J, BC(\Omega))). \end{align*} Analogous statements hold for $d(\vartheta+\vartheta^*_{\lambda,\mu,\eta})$, $d^\prime(\vartheta+\vartheta^*_{\lambda,\mu,\eta})$, $l(\vartheta+\vartheta^*_{\lambda,\mu,\eta})$, $\gamma(\vartheta+\vartheta^*_{\lambda,\mu,\eta})$ and $\psi(\vartheta+\vartheta^*_{\lambda,\mu,\eta})$ as well.
\\ Adopting the notation in Section~2, we introduce an extension operator $$e_\Pi:\mathbb{E}_{2}(J)\rightarrow \mathbb{E}_2(J,{\sf T}_a):\quad h\mapsto h\circ\Pi.$$ $((\pi_\kappa\circ\Pi)^2)_{\kappa\in\frak{K}}$ forms a partition of unity for ${\sf T}_a$. For $\mathfrak{F}\in\{W_p,H_p\}$, we define \begin{align*} \begin{split} &\mathcal{R}_\Pi^c: \mathfrak{F}^s({\sf T}_a, E)\rightarrow \prod\limits_{\kappa\in\frak{K}}\mathfrak{F}^s({\sf T}_{a,\kappa},E):u\mapsto (u\pi_\kappa\circ\Pi )_\kappa,\\ &\mathcal{R}_\Pi:\prod\limits_{\kappa\in\frak{K}}\mathfrak{F}^s({\sf T}_{a,\kappa},E)\rightarrow \mathfrak{F}^s({\sf T}_a, E): (u_\kappa)_\kappa\mapsto \sum\limits_{\kappa\in\frak{K}} u_\kappa\pi_\kappa\circ\Pi . \end{split} \end{align*} Then $\mathcal{R}_\Pi$ is a retraction with $\mathcal{R}_\Pi^c$ as a coretraction. By using this retraction-coretraction system, it is a simple matter to check that \begin{align*} e_\Pi\in \mathcal{L}(\mathbb{E}_{2}(J),\mathbb{E}_2(J,{\sf T}_a)). \end{align*} Extending $\mathcal{V}:=(\zeta\circ d_\Sigma)(\nu_\Sigma\circ \Pi)$ to be identically zero outside ${\sf T}_a$, it belongs to $BC^\infty(\Omega,\mathbb{R}^{m+1})\cap C^\omega({\sf T}_{a/3},\mathbb{R}^{m+1})$. Proposition~\ref{S3: Omega-Tue}(a) implies $$[(\lambda,\mu,\eta)\mapsto \mathcal{V}_{\lambda,\mu,\eta}]\in C^\omega(\mathbb{B}(0,r_0), C^1(J, BC^k(\Omega,\mathbb{R}^{m+1})))$$ for all $k\in\mathbb{N}_0$. Proposition~\ref{S3：M-reg} and \eqref{S5: ana of z*} then yield \begin{align*} [(h,(\lambda,\mu))\mapsto \Upsilon_{\lambda,\mu,\eta}({h}+h^*_{\lambda,\mu})]\in C^\omega(\prescript{}{0}{\mathbb{E}_{2}(J)}\times\mathbb{B}(0,r_0), \mathbb{E}_2(J,\Omega;\mathbb{R}^{m+1})). \end{align*} Because $\mathbb{F}_j(J,\Omega)$ with $j=2,3$ and $W^{1-1/2p}_p(J;H^1_p(\Omega))\cap L_p(J; W^{3-1/p}_p(\Omega))$ are Banach algebras, we verify via Proposition~\ref{S3: Omega-Tue}(b) that \begin{align} \label{S5: ana of M_j} [(h,(\lambda,\mu,\eta))\mapsto M_{j,\lambda,\mu,\eta}({h}+h^*_{\lambda,\mu})]\in C^\omega(\prescript{}{0}{\mathbb{E}_{2}(J)}\times\mathbb{B}(0,r_0), \mathbb{F}_2(J,\Omega;\mathbb{R}^{(m+1)^2})) \end{align} for $j=1,2,3$. Proposition~\ref{S3: Omega-Tue}(b) and \eqref{S5: ana of z*} lead to $$[(\vartheta,(\lambda,\mu,\eta)) \mapsto{T}_{\mu,\eta}\nabla{T}^{-1}_{\mu,\eta} (\vartheta+\vartheta_{\lambda,\mu,\eta})]\in C^\omega(\prescript{}{0}{\mathbb{E}_{1}(J)}\times\mathbb{B}(0,r_0), \mathbb{F}_3(J,\Omega_i;\mathbb{R}^{m+1})). $$ In virtue of Proposition~\ref{S3: Omega-Tue}(b) and \eqref{S3: analyticity of Blme}, we get \begin{align*} [(z,(\lambda,\mu,\eta))\mapsto \mathcal{R}_{\lambda,\mu,\eta}(h+h^*_{\lambda,\mu})(\vartheta+\vartheta^*_{\lambda,\mu,\eta})]\in C^\omega(\prescript{}{0}{\mathbb{E}(J)}\mathbb{B}(0,r_0), \mathbb{F}_3(J,\Omega_i)). \end{align*} Now it is immediate that $$[(z,(\lambda,\mu))\mapsto F_{\lambda,\mu,\eta}(z,z^*_{\lambda,\mu})]\in C^k(\prescript{}{0}{\mathbb{E}(J)}\times\mathbb{B}(0,r_0), \mathbb{F}_{1}(J)).$$ Since $\mathbb{C}_{2}(J)$ is a multiplication algebra, we have \begin{align*} [(h,(\lambda,\mu))\mapsto M_{0,\lambda,\mu}(h+h^*_{\lambda,\mu})]\in C^\omega(\prescript{}{0}{\mathbb{E}_{2}(J)}\times\mathbb{B}(0,r_0), \mathbb{C}_2(J;T^1_1\Sigma)). \end{align*} Proposition~\ref{S3：M-reg} and point-wise multiplier results on $\Sigma$ then imply that \begin{align*} [(h,(\lambda,\mu))\mapsto \alpha_{\lambda,\mu}(h+h^*_{\lambda,\mu})]\in C^\omega(\prescript{}{0}{\mathbb{E}_{2}(J)}\times\mathbb{B}(0,r_0), \mathbb{F}_2(J;T\Sigma)). \end{align*} Combined with the properties for substitution operators for Sobolev-Slobodeckii spaces, it yields \begin{align*} [(h,(\lambda,\mu))\mapsto \beta_{\lambda,\mu}(h+h^*_{\lambda,\mu})]\in C^\omega(\prescript{}{0}{\mathbb{E}_{2}(J)}\times\mathbb{B}(0,r_0), \mathbb{F}_{2}(J)). \end{align*} It is shown in \cite{Shao13} that in every coordinate patch $(\mathsf{O}_{\kappa},\varphi_{\kappa})$, the local expression of the mean curvature operator reads as \begin{align*} ({\mathcal{H}}(h))_\kappa=\beta(h) \frac{P_\kappa(h,\partial_j h, \partial_{jk}h)}{R_\kappa(h,\partial_j h)}. \end{align*} Here $P_\kappa$ is a polynomial in $h$ and its derivatives up to second order, and $R_\kappa$ is a polynomial in $h$ and its first order derivatives. Both have $BC^\infty \cap C^\omega$-coefficients. We again use the fact that $\mathbb{F}_{2}(J)$ is a Banach algebra. Following a similar argument in the same reference, we get \begin{align*} [(h,(\lambda,\mu))\mapsto {\mathcal{H}}_{\lambda,\mu}(h+h^*_{\lambda,\mu})]\in C^\omega(\prescript{}{0}{\mathbb{E}_{2}(J)}\times\mathbb{B}(0,r_0), \mathbb{F}_{2}(J)). \end{align*} By \eqref{S5: ana of M_j} and trace theorems of anisotropic Sobolev-Slobodeckii spaces, we infer that \begin{align*}
[(h,(\lambda,\mu))\mapsto M_{1,\lambda,\mu}(h+h^*_{\lambda,\mu})|_\Sigma]\in C^\omega(\prescript{}{0}{\mathbb{E}_{2}(J)}\times\mathbb{B}(0,r_0), \mathbb{F}_3(J;\mathbb{R}^{(m+1)^2}). \end{align*} It follows from the point-wise multiplier theorem in \cite[Section~9]{Ama13} that \begin{align*} [(h,(\lambda,\mu))\mapsto M_{4,\lambda,\mu}(h+h^*_{\lambda,\mu})]\in C^\omega(\prescript{}{0}{\mathbb{E}_{2}(J)}\times\mathbb{B}(0,r_0), \mathbb{F}_3(J;\mathbb{R}^{(m+1)^2}). \end{align*} To sum up, we conclude that \begin{align*} \begin{split} [(z,(\lambda,\mu))\mapsto \bar{G}_{\lambda,\mu}(z,z^*_{\lambda,\mu})]&\in C^k(\prescript{}{0}{\mathbb{E}(J)}\times\mathbb{B}(0,r_0), \mathbb{F}_{2}(J)),\\ [(z,(\lambda,\mu))\mapsto \bar{Q}_{\lambda,\mu}(z,z^*_{\lambda,\mu})]&\in C^k(\prescript{}{0}{\mathbb{E}(J)}\times\mathbb{B}(0,r_0), \mathbb{F}_{3}(J)). \end{split} \end{align*} Taking into account all the above discussion, this leads to \begin{align*} \Phi\in C^k(\prescript{}{0}{\mathbb{E}(J)}\times\mathbb{B}(0,r_0), \prescript{}{0}{\mathbb{F}(J)}). \end{align*}
\\ (ii) We look at the Fr\'echet derivative of $\Phi$ with respect to $\prescript{}{0}{\mathbb{E}(J)}$ at $(\bar{z},0)$. To this end, we find it more convenient to consider the following non-linear maps. \begin{align*}
\hat{F}(z)=&(\hat{\kappa}_0-\kappa(\vartheta))\partial_t \vartheta +(d(\vartheta)-\hat{d}_0)\Delta \vartheta -d(\vartheta)M_2(h):\nabla^2 \vartheta \\
&+d^\prime(\vartheta)|(I-M_1(h))\nabla \vartheta|^2 -d(\vartheta)(M_3(h)| \nabla \vartheta) +\kappa(\vartheta)\mathcal{R}(h)\vartheta,\\
\hat{G}(z)=& -([\![\psi(\vartheta)]\!] + \sigma {\mathcal{H}}(h) ) +\hat{l}_1 \vartheta +\sigma_0\Delta_\Sigma h +(\gamma(\vartheta)\beta(h)-\hat{\gamma}_1)\partial_t h,\\
\hat{Q}(z)=&[\![ (d(\vartheta)-\hat{d}_0)\partial_\nu \vartheta ]\!] +(\hat{l}_0 -l(\vartheta))\partial_t h -([\![d(\vartheta)\nabla \vartheta]\!]| M_4(h)\nabla_\Sigma h)\\
&+\gamma(v)\beta(h)(\partial_t h)^2, \end{align*} i.e., replacing $\vartheta_A$ by $\hat{\vartheta}_0$ in the definition of $F$, $G$ and $H$. With only slight modification of (i), we immediately obtain \begin{align*} (\hat{F},\hat{G},\hat{Q})\in C^k(\mathbb{E}(J),\mathbb{F}_{1}(J)\times\mathbb{F}_{2}(J)\times\mathbb{F}_{3}(J)). \end{align*} Recall that $\hat{z}=\bar{z}+z^*$. Letting $w=(u,\rho)$, one computes \begin{align*} D_1\Phi(\bar{z},0)w= \left\{\begin{aligned} &\hat{\kappa}_0 \partial_t u -\hat{d}_0 \Delta u -\hat{F}^\prime(\hat{z})w && \text{in} && \Omega\setminus\Sigma, \\ &\hat{l}_1 u +\sigma_0 \Delta_\Sigma \rho -\hat{\gamma}_1\partial_t\rho - \hat{G}^\prime(\hat{z})w && \text{on}&& \Sigma,\\ &\hat{l}_0 \partial_t\rho -[\![ \hat{d}_0 \partial_\nu u]\!] -\hat{Q}^\prime(\hat{z})w && \text{on}&& \Sigma. \end{aligned}\right. \end{align*} A moment of reflection shows that we have the liberty to exchange the coefficients $(\kappa_A,d_A,l_A)$, used in the definition of $\Phi$, by $(\hat{\kappa}_0,\hat{d}_0, \hat{l}_0)$. These quantities are only used for the principal linearization, and are then subtracted off in the nonlinearities.
We split $D_1\Phi(\bar{z},0)$ into two parts. Let $\hat{\beta}_0:=\beta(\hat{h}_0)$. Define $\mathbb{L}(\hat{\vartheta}_0):\prescript{}{0}{\mathbb{E}(J)}\rightarrow \prescript{}{0}{\mathbb{F}(J)}$ by \begin{align*} \mathbb{L}(\hat{\vartheta}_0)w= \left\{\begin{aligned} &\hat{\kappa}_0 \partial_t u -\hat{d}_0 (I-M_2(\hat{h}_0)): \nabla^2 u && \text{in}&& \Omega\setminus\Sigma, \\ &\hat{l}_1 u +\sigma \mathcal{H}^\prime(\hat{h}_0) \rho -\hat{\gamma}_1\beta^\prime(\hat{h}_0)e^{\Delta_\Sigma t}(\partial_t \hat{h}(0))\rho\\ &\quad -\hat{\gamma}_1 \hat{\beta}_0\partial_t\rho-\gamma^\prime(e^{\Delta_\Sigma t}\hat{\vartheta}_0)\hat{\beta}_0 e^{\Delta_\Sigma t}(\partial_t \hat{h}(0))u && \text{on} && \Sigma,\\
&\hat{l}_0 \partial_t\rho -([\![ \hat{d}_0 \nabla u]\!]| \nu_\Sigma- M_4(\hat{h}_0)\nabla_\Sigma \hat{h}_0) && \text{on}&& \Sigma. \end{aligned}\right. \end{align*} and $\mathbb{K}(\hat{z}):\prescript{}{0}{\mathbb{E}(J)}\rightarrow \prescript{}{0}{\mathbb{F}(J)}$ by $$\mathbb{K}(\hat{z})=:(\mathbb{K}_1,\mathbb{K}_2,\mathbb{K}_3)^{\sf T}:=\mathbb{L}(\hat{\vartheta}_0)w-D_1\Phi(\hat{z},0).$$ \begin{theorem} Let $J=[0,T]$. Then $\mathbb{L}(\hat{\vartheta}_0)\in \mathcal{L}{\rm{is}}(\mathbb{E}(J),\mathbb{F}(J))$. In particular, $\mathbb{L}(\hat{\vartheta}_0)\in \mathcal{L}{\rm{is}}(\prescript{}{0}{\mathbb{E}(J)},\prescript{}{0}{\mathbb{F}(J)})$. \end{theorem} \begin{proof} We first solve the following model problem for the case $\gamma\equiv 0$. Suppose that $$\kappa_0 \in BU\!C(\mathbb{R}^m \times \mathbb{R}_+),\quad d_0\in BU\!C^1(\mathbb{R}^m),\quad a_0\in BU\!C^1(\mathbb{R}^m;\mathbb{R}^m) ,\quad \kappa_0,d_0>0. $$ Moreover, $l_0\in W^{2-6/p}_p(\mathbb{R}^m)$, $l_2\in \mathbb{F}_2(J,\mathbb{R}^m)$ satisfy $l_2 l_0>0$. The differential operators $-P(x,y):\nabla^2$ and $-S(x):\nabla^2_x$ are uniformly elliptic. For any given $(f,g,q,(u_0,\rho_0))\in \mathbb{F}(J,\mathbb{R}^m \times \mathbb{R}_+)$, \begin{align*} \left\{\begin{aligned} \kappa_0(x,y) \partial_t u(x,y) - P(x,y): \nabla^2 u(x,y) &=f(x,y)
&&\text{in}&& \mathbb{R}^m \times \mathbb{R}_+ ,\\ l_2(t,x) u(x) + S:\nabla^2_x \rho(x) &=g(x)
&&\text{on}&&\mathbb{R}^m,\\ l_0(x) \partial_t \rho(x) + d_0(x)\partial_\nu u(x) +a_0(x)\cdot\nabla_x u(x) &=q(x)
&&\text{on}&&\mathbb{R}^m,\\ u(0)=u_0,\quad \rho(0)&=\rho_0,&& \end{aligned}\right. \end{align*} admits a unique solution $(u,\rho)\in \mathbb{E}(J,\mathbb{R}^m \times \mathbb{R}_+)$. We identify $\mathbb{R}^m$ with $\mathbb{R}^m\times\{0\}$. Here $x\in\mathbb{R}^m,y\in\mathbb{R}_+$, and the space $\mathbb{F}(J,\mathbb{R}^m \times \mathbb{R}_+)$ is defined by replacing $\Omega\setminus\Sigma$ and $\Sigma$ by $\mathbb{R}^m \times \mathbb{R}_+$ and $\mathbb{R}^m$, respectively. The space $\mathbb{E}(J,\mathbb{R}^m \times \mathbb{R}_+)$ is defined analogously. Similar problems have been considered in \cite{DenPruZac08}. It suffices to check the Lopatinskii-Shapiro condition (${\bf LS}$) and the asymptotic Lopatinskii-Shapiro condition (${\bf LS}^+_\infty$) defined therein. For simplicity, we assume all the coefficients to be constants and set $$P(\xi):=\xi_e^{\sf T}P\xi_e,\quad S(\xi):=\xi^{\sf T}S\xi, \quad P_{m+1}:=P_{m+1,m+1} $$ and $P_m:=(P_{m+1, 1},\cdots,P_{m+1,m})$ with $\xi\in \mathbb{R}^m$ and $\xi_e:=(\xi^{\sf T},0)^{\sf T}\in\mathbb{R}^{m+1}$.
(${\bf LS}$) is satisfied, if for any $\xi\in\mathbb{R}^m$ and $\lambda\in \bar{\mathbb{C}}_+$ with $|\xi|+|\lambda|\neq 0$, the following ordinary differential equation in $\mathbb{R}_+$ \begin{align*} \left\{\begin{aligned} (\kappa_0\lambda +P(\xi)+2iP_m\cdot\xi\partial_y - P_{m+1}\partial^2_y) v(y) &=0,
&& y>0 ,\\ l_2 v(0) - S(\xi) \delta &=0, \\ l_0 \lambda \delta - (i a_0\cdot\xi +d_0(x)\partial_y ) v(0) &=0,\\ \end{aligned}\right. \end{align*} has a unique solution $(v,\delta)\in C_0(\mathbb{R}_+;\mathbb{C})\times \mathbb{C}$. It is clear that the only stable solution to the first line is $$\displaystyle v(y)=e^{\mu y}v(0), \quad \mu:=\frac{\sqrt{-(P_m\cdot\xi)^2+P_{m+1}\kappa_0\lambda+P_{m+1}P(\xi)}+iP_m\cdot\xi}{ P_{m+1}}$$ in the case $\lambda\notin \mathbb{R}$, or, $v\equiv 0$, otherwise. We conclude from the second and third lines that $$[i S(\xi)a_0\cdot\xi +S(\xi)\mu d_0 -l_2 l_0 \lambda]\delta=0.$$
It implies that $\delta=0$, $v\equiv 0$. (${\bf LS}^+_\infty$) is satisfied, if for any $\xi\in\mathbb{R}^m$ and $\lambda\in \bar{\mathbb{C}}_+$ with $|\xi|+|\lambda|\neq 0$, \begin{align*} \left\{\begin{aligned} (\kappa_0\lambda +P(\xi)+2iP_m\cdot\xi\partial_y - P_{m+1}\partial^2_y) v(y) &=0,
&& y>0 ,\\ l_2 v(0) - S(\xi) \delta &=0, \\ (i a_0\cdot\xi +d_0(x)\partial_y ) v(0) &=0,\\ \end{aligned}\right. \end{align*}
and $|\xi|=1$, $\lambda\in\bar{\mathbb{C}}_+\setminus\{0\}$ \begin{align*} \left\{\begin{aligned} (\kappa_0\lambda - P_{m+1}\partial^2_y) v(y) &=0,
&& y>0 ,\\ l_2 v(0) - S(\xi) \delta &=0, \\ l_0 \lambda \delta +d_0(x)\partial_y v(0) &=0,\\ \end{aligned}\right. \end{align*}
and $|\xi|=1$, $\lambda\in\bar{\mathbb{C}}_+\setminus\{0\}$ \begin{align*} \left\{\begin{aligned} (\kappa_0\lambda - P_{m+1}\partial^2_y) v(y) &=0,
&& y>0 ,\\ l_2 v(0)- S(\xi) \delta &=0, \\ d_0(x)\partial_y v(0) &=0,\\ \end{aligned}\right. \end{align*} admit unique solutions $(v,\delta)\in C_0(\mathbb{R}_+;\mathbb{C})\times \mathbb{C}$. One can check in an analogous manner that stable solutions to those equations are trivial. For general coefficients, the problem can be proved by a perturbation argument as in \cite{DenHiePru03}. The rest of the proof now follows from similar arguments to those of \cite[Theorems~3.3, 3.5]{PruSimZac12}. \end{proof}
In order to prove that $D_1\Phi(\bar{z},0)$ is an isomorphism, we need to control the norm $\| \mathbb{K}(\hat{z}_0)\|_{\mathcal{L}(\prescript{}{0}{\mathbb{E}(J)},\prescript{}{0}{\mathbb{F}(J)})}$. To this end, we first compute several derivatives related to $\mathbb{K}$ explicitly. \begin{align*} \hat{F}^\prime(z)w=&(\hat{\kappa}_0- \kappa(\vartheta))\partial_t u -\kappa^\prime(\vartheta)u \partial_t \vartheta +(d(\vartheta)-\hat{d}_0)\Delta u +d^\prime(\vartheta)u\Delta \vartheta\\ & -d^\prime(\vartheta)u M_2(h):\nabla^2 \vartheta -d(\vartheta)M_2^\prime(h)\rho:\nabla^2 \vartheta -d(\vartheta)M_2(h):\nabla^2 u\\
& -d^\prime(\vartheta)u (M_3(h)|\nabla \vartheta) -d(\vartheta)(M_3^\prime(h)\rho | \nabla \vartheta)-d(\vartheta)(M_3(h)| \nabla u)\\
& +2d^\prime(\vartheta)((I-M_1(h)\nabla \vartheta|(I-M_1(h))\nabla u -M_1^\prime(h)\rho\nabla \vartheta)\\
& +d^{\prime\prime}(\vartheta)u |(I-M_1(h))\nabla \vartheta|^2
+\kappa(\vartheta)(\nabla u| (I+\nabla\Upsilon(h)^{\sf T})^{-1}\partial_t\Upsilon(h)) \\
& -\kappa(\vartheta)(\nabla \vartheta| (I+\nabla\Upsilon(h)^{\sf T})^{-1}\nabla\Upsilon(\rho)^{\sf T}(I+\nabla\Upsilon(h)^{\sf T})^{-1}\partial_t\Upsilon(h))\\
&+\kappa(\vartheta)(\nabla \vartheta| (I+\nabla\Upsilon(h)^{\sf T})^{-1}\partial_t\Upsilon(\rho)) + \kappa^\prime(\vartheta)u \mathcal{R}(h)\vartheta , \end{align*} and \begin{align*} \hat{G}^\prime(z)w=& -([\![\psi^\prime(\vartheta)]\!]u+\sigma {\mathcal{H}}^\prime(h)\rho) +\hat{l}_1 u +\sigma_0\Delta_\Sigma \rho +(\gamma(\vartheta)\beta(h)-\hat{\gamma}_1)\partial_t\rho \\ & +(\gamma^\prime(\vartheta)u\beta(h) -\gamma(\vartheta)\beta^\prime(h)\rho)\partial_t h, \end{align*} and \begin{align*} \hat{Q}^\prime(z)w=& [\![d^\prime(\vartheta)\partial_\nu \vartheta ]\!]u +[\![(d(\vartheta)-\hat{d}_0)\partial_\nu u ]\!] +(\hat{l}_0-l(\vartheta))\partial_t \rho -l^\prime(\vartheta)u \partial_t h\\
&-([\![d^\prime(\vartheta)\nabla \vartheta]\!]u| M_4(h)\nabla_\Sigma h)
-([\![d(\vartheta)\nabla u ]\!]| M_4(h)\nabla_\Sigma h) \\
&-([\![d(\vartheta)\nabla \vartheta ]\!]| M_4(h)\nabla_\Sigma \rho)
-([\![d(\vartheta)\nabla \vartheta ]\!]| M_4^\prime(h)\rho \nabla_\Sigma h) \\ & +\gamma^\prime(\vartheta)u \beta(h)[\partial_t h]^2 +\gamma(\vartheta) \beta^\prime(h)\rho [\partial_t h]^2 +2\gamma(\vartheta)\beta(h) \partial_t h \partial_t \rho. \end{align*} The derivatives of $M_0(h)$, $\alpha(h)$, $\beta(h)$ and ${\mathcal{H}}(h)$ are given by \begin{align*} & M_0^\prime(h)\rho=\rho M_0(h)L_\Sigma M_0(h),\hspace{1em} \alpha^\prime(h)\rho=M_0(h)\nabla_\Sigma \rho +\rho M_0(h)L_\Sigma M_0(h)\alpha(h),\\
& \beta^\prime(h)\rho=-\beta^3(h)(\alpha(h)|M_0(h)\nabla_\Sigma \rho +\rho M_0(h)L_\Sigma\alpha(h)),\\ & {\mathcal{H}}^\prime(h)\rho=\beta(h)\{ {{\rm{tr}}}[M_0^\prime(h)\rho(L_\Sigma +\nabla_\Sigma \alpha(h))] +{{\rm{tr}}}[M_0(h)\nabla_\Sigma \alpha^\prime(h)\rho]\\
&\quad -2\beta(h)\beta^\prime(h)\rho(M_0(h)\alpha(h)| [\nabla_\Sigma\alpha(h)]\alpha(h) ) -\beta^2(h)(M_0^\prime(h)\rho\alpha(h)|[\nabla_\Sigma \alpha(h)]\alpha(h) )\\
&\quad -\beta^2(h)(M_0(h)\alpha^\prime(h)\rho| [\nabla_\Sigma \alpha(h)]\alpha(h) )
-\beta^2(h)(M_0(h)\alpha(h)| [\nabla_\Sigma \alpha^\prime(h)\rho]\alpha(h) )\\
&\quad -\beta^2(h)(M_0(h)\alpha(h)| [\nabla_\Sigma\alpha(h)]\alpha^\prime(h)\rho) \}/m +\beta^\prime(h)\rho ({\mathcal{H}}(h)/\beta(h)). \end{align*} See \cite[formula~(32)]{PruSim13} for a justification for the last equality.
We will use the following lemma frequently in the sequel. \begin{lem} \label{S5: F2-lem} There exists a constant $C_0$ independent of $T$ such that \begin{itemize} \item[(a)] For all $(v_1,v_2)\in \mathbb{F}_j(J)\times \prescript{}{0}{\mathbb{F}_j(J)}$ and $j=2,3$, \begin{align*}
\|v_1 v_2\|_{\mathbb{F}_j(J)}\leq C_0(\|v_1\|_{C(J\times\Sigma)}+\|v_1\|_{\mathbb{F}_j(J)})\|v_2\|_{\mathbb{F}_j(J)}. \end{align*}
\item[(b)] For all $(v_1,v_2)\in \prescript{}{0}{\mathbb{F}_j(J)}\times \prescript{}{0}{\mathbb{F}_j(J)}$ and $j=2,3$, \begin{align*}
\|v_1 v_2\|_{\prescript{}{0}{\mathbb{F}_j(J)}}\leq C_0 \|v_1\|_{\prescript{}{0}{\mathbb{F}_j(J)}}\|v_2\|_{\prescript{}{0}{\mathbb{F}_j(J)}}. \end{align*}
\item[(c)] For all $(v_1,v_2)\in \mathbb{F}_{3}(J)\times \prescript{}{0}{\mathbb{F}_{2}(J)}$ \begin{align*}
\|v_1 v_2\|_{\mathbb{F}_{3}(J)}\leq C_0 \|v_1\|_{\mathbb{F}_{3}(J)} \|v_2\|_{\prescript{}{0}{\mathbb{F}_{2}(J)}}. \end{align*} \end{itemize} \end{lem} \begin{proof} (a) The case $j=2$ is shown in \cite[Lemma~5.5(b)]{PruSim09}. For the reader's convenience, we will nevertheless include a proof herein. The case $j=3$ follows in a similar way. By the retraction-coretraction system defined in Section~2, it suffices to show the estimates for functions on $\mathsf{Q}^{m}$, i.e., we assume $(v_1,v_2)\in \mathbb{F}_j(J,\mathsf{Q}^{m})\times \prescript{}{0}{\mathbb{F}_j(J,\mathsf{Q}^{m})}$. We equip $\mathbb{F}_2(J,\mathsf{Q}^{m})$ with the norm: \begin{align*} \begin{split}
&\|v\|_{\mathbb{F}_2(J,\mathsf{Q}^{m})}= \|v\|_{W^{1-1/2p}_p(J;L_p(\mathsf{Q}^{m}))} +\|v\|_{L_p(J;W^{2-1/p}_p(\mathsf{Q}^{m}))},\\
&\|v\|_{L_p(J;W^{2-1/p}_p(\mathsf{Q}^{m}))}=\|v\|_{L_p(J;H^1_p(\mathsf{Q}^{m}))} +\sum\limits_{j=1}^m \| \langle \partial_j v \rangle_{W^{1-1/p}_p(\mathsf{Q}^{m})}\|_{L_p(J)}. \end{split} \end{align*} Here $\langle \cdot \rangle_{W^{1-1/p}_p(J;X)}$ is the Slobodeckii seminorm of the space $W^{1-1/p}_p(J;X)$ for a Banach space $X$. Then, \begin{align*}
&\|v_1 v_2\|_{W^{1-1/2p}_p(J;L_p(\mathsf{Q}^{m}))}\leq C^\prime \{ \|v_1\|_{W^{1-1/2p}_p(J;L_p(\mathsf{Q}^{m}))}\|v_2\|_{BC(J\times\mathsf{Q}^{m})}\\
&\quad + (\int_J\int_J \|v_1(s)(v_2(t)- v_2(s))\|^p_{L_p(\mathsf{Q}^{m})}\frac{1}{|t-s|^{1/2+p}}\, dt\, ds )^{1/p} \} \\
&\leq C^\prime\{ \|v_1\|_{W^{1-1/2p}_p(J;L_p(\mathsf{Q}^{m}))}\|v_2\|_{BC(J\times\mathsf{Q}^{m})}+\|v_1\|_{BC(J\times\mathsf{Q}^{m})}\langle v_2 \rangle_{W^{1-1/2p}_p(J;L_p(\mathsf{Q}^{m}))} \}. \end{align*} Similarly, one computes \begin{align*}
&\quad \|v_1 v_2\|_{L_p(J;W^{2-1/p}_p(\mathsf{Q}^{m}))}\leq C^\prime\{ \|v_1\|_{L_p(J;W^{2-1/p}_p(\mathsf{Q}^{m}))}\|v_2\|_{C(J,BC^1(\mathsf{Q}^{m}))}\\
&+\sum\limits_{j=1}^m[\int_J (\int_{\mathsf{Q}^{m}}\!\int_{\mathsf{Q}^{m}} |v_1(t,x)(\partial_j v_2(t,x)-\partial_j v_2(t,y) |^p \frac{1}{|x-y|^{m-1+p}}\,dx\, dy)\, dt ]^{1/p}\\
&+\sum\limits_{j=1}^m[\int_J (\int_{\mathsf{Q}^{m}}\!\int_{\mathsf{Q}^{m}} |\partial_j v_1(t,x)(v_2(t,x)-v_2(t,y) |^p \frac{1}{|x-y|^{m-1+p}}\,dx\, dy)\, dt ]^{1/p} \}. \end{align*} We immediately have
\begin{align*}
&\quad\int_J (\int_{\mathsf{Q}^{m}}\!\int_{\mathsf{Q}^{m}} |v_1(t,x)(\partial_j v_2(t,x)-\partial_j v_2(t,y) |^p \frac{1}{|x-y|^{m-1+p}}\,dx\, dy)\, dt\\
&\leq \|v_1\|_{BC(J\times\mathsf{Q}^{m})}^p\| v_2\|_{L_p(J,W^{2-1/p}_p(\mathsf{Q}^{m}))}^p. \end{align*} The remaining estimate can be carried out as follows. \begin{align}
&\notag \quad \int_{\mathsf{Q}^{m}}\!\int_{\mathsf{Q}^{m}} |\partial_j v_1(t,x)(v_2(t,x)-v_2(t,y) |^p \frac{1}{|x-y|^{m-1+p}}\,dx\, dy\\
&\notag \leq \int_{\mathsf{Q}^{m}}\!\int_{\mathsf{Q}^{m}} |\partial_j v_1(t,x)|^p (\int\limits_0^1 |(\nabla v_2(t,x+\tau(y-x))|(y-x))|\, d\tau)^p\frac{1}{|x-y|^{m-1+p}}\,dx\, dy\\
&\notag\leq C^\prime\int_{\mathsf{Q}^{m}}|\partial_j v_1(t,x)|^p \int_{\mathsf{Q}^{m}} \frac{|x-y|^p}{|x-y|^{m-1+p}}\,dy \, dx \|\nabla v_2(t)\|_{BC(\mathsf{Q}^{m})}^p\\ \label{S5: est 2}
&\leq C^\prime\|v_1(t)\|_{H^1_p(\mathsf{Q}^{m})}^p\| v_2(t)\|_{BC^1(\mathsf{Q}^{m})}^p. \end{align} Combining these discussions yields \begin{align*}
\|v_1 v_2\|_{L_p(J;W^{2-1/p}_p(\mathsf{Q}^{m}))}&\leq C^\prime(\|v_1\|_{BC(J\times\mathsf{Q}^{m})}+\|v_1\|_{L_p(J;W^{2-1/p}_p(\mathsf{Q}^{m}))})\\
&\quad(\|v_2\|_{C(J;BC^1(\mathsf{Q}^{m}))}+\|v_2\|_{L_p(J;W^{2-1/p}_p(\mathsf{Q}^{m})})). \end{align*} Using the fact that the embedding constant of $\prescript{}{0}{\mathbb{F}_2(J,\mathsf{Q}^{m})}\hookrightarrow C(J;BC^1(\mathsf{Q}^{m}))$ is independent of $T$ yields the asserted result.
\\ (b) is an immediate consequence of (a) and the fact that the embedding constant of $\prescript{}{0}{\mathbb{F}_2(J,\mathsf{Q}^{m})}\hookrightarrow C(J;BC(\mathsf{Q}^{m}))$ is independent of $T$.
\\ (c) Suppose that $(v_1,v_2)\in \mathbb{F}_3(J,\mathsf{Q}^{m})\times \prescript{}{0}{\mathbb{F}_2(J,\mathsf{Q}^{m})}$. Then \begin{align*}
&\|v_1 v_2\|_{W^{1/2-1/2p}_p(J;L_p(\mathsf{Q}^{m}))}\leq \|v_1\|_{W^{1/2-1/2p}_p(J;L_p(\mathsf{Q}^{m}))}\|v_2\|_{BC(J\times\mathsf{Q}^{m})}\\
&\quad + (\int_J\int_J \|v_1(s)(v_2(t)- v_2(s))\|^p_{L_p(\mathsf{Q}^{m})}\frac{1}{|t-s|^{1/2+p/2}}\, dt\, ds )^{1/p}. \end{align*} By \cite[Lemma~2.5]{MeySch12}, the comments below \cite[formula~(3.5)]{MeySch12}, an analogue of \cite[diagram~(6.12)]{PruSim07} with $BU\!C$ replaced by $C^s$, we obtain the embedding result $$\prescript{}{0}{\mathbb{F}_2(J,\mathsf{Q}^{m})}\hookrightarrow C^s(J;BC(\mathsf{Q}^{m}))$$ for some $s>1/2-1/2p$ with an embedding constant $M=M(s)$ uniform in $T$. Thus, we infer that \begin{align*}
&\quad \int_J\int_J \|v_1(s)(v_2(t)- v_2(s))\|^p_{L_p(\mathsf{Q}^{m})}\frac{1}{|t-s|^{1/2+p/2}}\, dt\, ds\\
&\leq C\int_J \|v_1(s)\|^p_{L_p(\mathsf{Q}^{m})} \int_J \frac{1}{|t-s|^{1/2+p/2-sp}}\, dt\, ds \|v_2\|^p_{C^s(J;BC(\mathsf{Q}^{m}))}\\
&\leq C^\prime \|v_1(s)\|^p_{L_p(J;L_p(\mathsf{Q}^{m}))} \|v_2\|_{C^s(J;BC(\mathsf{Q}^{m}))}^p. \end{align*} By an analogous estimate as in (a), see in particular \eqref{S5: est 2}, one obtains
$$ \|v_1 v_2\|_{L_p(J;W^{1-1/p}_p(\mathsf{Q}^{m}))}\leq C^\prime \|v_1\|_{L_p(J;W^{1-1/p}_p(\mathsf{Q}^{m}))} \|v_2\|_{C(J;BC^1(\mathsf{Q}^{m}))}.$$ \end{proof} Note that the multiplication constant $C_0$ in (b) blows up as $T\to 0$ if $\prescript{}{0}{\mathbb{F}_j(J)}$ is replaced by $\mathbb{F}_j(J)$.
We write \begin{align*} \mathbb{K}_2(\hat{z})w =& \hat{l}_1 u- [\![\psi^\prime(\hat{\vartheta})]\!]u +(\gamma(\hat{\vartheta})-\hat{\gamma}_1)\beta(\hat{h})\partial_t\rho +\hat{\gamma}_1(\beta(\hat{h})-\hat{\beta}_0)\partial_t\rho\\ & -\sigma( {\mathcal{H}}^\prime(\hat{h})- {\mathcal{H}}^\prime(\hat{h}_0))\rho +(\gamma(\hat{\vartheta})\beta^\prime(\hat{h})\partial_t \hat{h} - \hat{\gamma}_1\beta^\prime(\hat{h}_0)e^{\Delta_\Sigma t}(\partial_t \hat{h}(0)))\rho\\ & +(\gamma^\prime(\hat{\vartheta})\beta(\hat{h})\partial_t \hat{h}-\gamma^\prime(e^{\Delta_\Sigma t}\hat{\vartheta}_0)\hat{\beta}_0 e^{\Delta_\Sigma t}(\partial_t \hat{h}(0)))u. \end{align*} Let $\varepsilon$ sufficiently small be fixed. Recall that $$\hat{l}_1 u-[\![\psi^\prime(\hat{\vartheta})]\!]u=([\![\psi^\prime(e^{\Delta_\Sigma t}\hat{\vartheta}_0)-\psi^\prime(\hat{\vartheta})]\!])u.$$ Due to fact that $[\![\psi^\prime(e^{\Delta_\Sigma t}\hat{\vartheta}_0)-\psi^\prime(\hat{\vartheta})]\!]\in \prescript{}{0}{\mathbb{F}_{2}(J)}$ and Lemma~\ref{S5: F2-lem}(b), by making $T$ small enough, we can achieve that $$
\|\hat{l}_1 u-[\![\psi^\prime(\hat{\vartheta})]\!]u\|_{\mathbb{F}_{2}(J)}\leq \varepsilon \|u\|_{\mathbb{F}_{2}(J)}. $$
The arguments for the remaining terms in $\mathbb{K}_2(\hat{z})w$ are similar. Thus given any $\varepsilon>0$, for $T$ small enough, we have
$$\|\mathbb{K}_2(\hat{z}) w\|_{\mathbb{F}_{2}(J)}\leq \varepsilon\|w\|_{\prescript{}{0}{\mathbb{E}(J)}}. $$
One can obtain an analogous assertion for
$\|\mathbb{K}_3(\hat{z})\|_{\mathcal{L}(\prescript{}{0}{\mathbb{E}(J)},\prescript{}{0}{\mathbb{F}_{3}(J)})}$. Indeed,
$$\| [\![ (d(\vartheta)-\hat{d}_0) \partial_\nu u ]\!]\|_{\mathbb{F}_{3}(J)} \leq C_0 \|(d(\vartheta)-\hat{d}_0)\|_{\prescript{}{0}{\mathbb{F}_{3}(J)}} \|[\![\partial_\nu u]\!]\|_{\prescript{}{0}{\mathbb{F}_{3}(J)}}. $$ Similar estimates also apply to
$$\| (\hat{l}_0- l(\hat{\vartheta})) \partial_t \rho\|_{\mathbb{F}_{3}(J)}, \|( [\![ d(\hat{h})\nabla u ]\!]| M_4(\hat{h})\nabla \hat{h} ) -( [\![ \hat{d}_0 \nabla u ]\!]| M_4(\hat{h}_0)\nabla \hat{h}_0 )\|_{\mathbb{F}_{3}(J)}. $$ It follows from Lemma~\ref{S5: F2-lem}(c) that
$$ \|[\![ d^\prime(\hat{\vartheta}) \partial_\nu \hat{\vartheta} ]\!] u\|_{\mathbb{F}_{3}(J)} \leq C_0 \|[\![ d^\prime(\hat{\vartheta}) \partial_\nu \hat{\vartheta} ]\!]\|_{\mathbb{F}_{3}(J)} \|u\|_{\prescript{}{0}{\mathbb{F}_{2}(J)}}.$$ The remaining terms in $\mathbb{K}_3(\hat{z})w$ can be estimates in an analogous way.
For $\|\mathbb{K}_1(\hat{z}) \|_{\mathcal{L}(\prescript{}{0}{\mathbb{E}(J)},\mathbb{F}_{1}(J))}$, one verifies by direct computation that \begin{align*}
\|\mathbb{K}_1(\hat{z})w \|_{\mathbb{F}_{1}(J)}=&\|\hat{F}^\prime(\hat{z})w+ \hat{d}_0 M_2(\hat{h}_0):\nabla^2 u\|_{\mathbb{F}_{1}(J)}\\
&\leq C_2\|w\|_{\mathbb{E}(J)}+C_3\|w\|_{\mathbb{C}_1(J)\times\mathbb{C}_{2}(J)}. \end{align*} Here the constants $C_2$ and $C_3$ tend to zero as $T\to 0$.
\begin{prop} Let $p>m+3$, $\sigma>0$. Suppose that $d_i\in C^2(0,\infty)$, $\gamma,\psi_i\in C^3(0,\infty)$. Then there exists some constant $\tau_0$ such that given any $T\leq \tau_0$, on $J=[0,T]$, we have \begin{align*}
\|\mathbb{L}^{-1}(\hat{\vartheta}_0)\|_{\mathcal{L}(\prescript{}{0}{\mathbb{F}(J)},\prescript{}{0}{\mathbb{E}(J)})}\|\mathbb{K}^\prime(\hat{z}_0)\|_{\mathcal{L}(\prescript{}{0}{\mathbb{E}(J)},\prescript{}{0}{\mathbb{F}(J)})}\leq 1/2. \end{align*} \end{prop} It follows from a Neuman series argument that $$D_1\Phi(\bar{z},0)\in \mathcal{L}{\rm{is}}(\prescript{}{0}{\mathbb{E}(J)},\prescript{}{0}{\mathbb{F}(J)}).$$ Employing now the implicit function theorem, we attain $$[(\lambda,\mu,\eta)\mapsto \bar{z}_{\lambda,\mu,\eta}]\in C^k (\mathbb{B}(0,r_0),\prescript{}{0}{\mathbb{E}(J)}) .$$ It follows then from Theorem~\ref{S3: main thm} that $$\bar{z} \in C^k(\mathring{J}\times {\sf T}_{a/3}\setminus\Sigma)\times C^k(\mathring{J}\times\Sigma).$$ \begin{remark} \label{S5: Proof of main RMK} By a similar argument to the proof given at the end of Section~4.2, we can show that $$\bar{z} \in C^k(\mathring{J}\times \Omega\setminus\Sigma) \times C^k (\mathring{J}\times\Sigma). $$ Together with \eqref{S5: ana of z*}, we thus have $$\hat{z} \in C^k(\mathring{J}\times \Omega\setminus\Sigma) \times C^k (\mathring{J}\times\Sigma). $$ For $k\in\mathbb{N}\cup\{\infty\}$, since the Hanzawa transformation is $C^\infty$-smooth, the above assertion implies that the solution $(\theta, \Gamma)$ to \eqref{stefan} has the same regularity.
\\ But when $k=\omega$, analyticity of the temperature $\theta$, in general, cannot be attained by applying the Hanzawa transformation. \qed \end{remark}
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{On the definition of solution to the total variation flow} \author{Juha Kinnunen and Christoph Scheven} \maketitle
\abstract{We show that the notions of weak solution to the total
variation flow based on the Anzellotti pairing and the variational
inequality coincide under some restrictions on the boundary data.
The key ingredient in the argument is a duality result for the total
variation functional, which is based on an approximation of the
total variation by area-type functionals.} \section{Introduction} This paper discusses the total variation flow
\[
\partial_tu-\Div\bigg(\dfrac{Du}{|Du|}\bigg)=0
\quad\mbox{on $\Omega_T=\Omega\times (0,T)$,}
\] where $\Omega$ is a bounded domain in ${\mathds{R}}^n$ and $T>0$. For this nonlinear parabolic equation we refer to the monograph by Andreu, Caselles and Mazon \cite{Andreu-Caselles-Mazon:book}. The total variation flow can be seen as the limiting case of the parabolic $p$-Laplace equation
\[
\partial_tu-\Div(|Du|^{p-2}Du)=0, \qquad 1<p<\infty,
\]
as $p\to1$. A Sobolev space is the natural function space in the existence and regularity theories for a weak solution to the parabolic $p$-Laplace equation, see the monograph by DiBenedetto \cite{DiBe1993}. The corresponding function space for the total variation flow is functions of bounded variation and in that case the weak derivative of a function is a vector valued Radon measure. A standard definition of weak solution to the parabolic $p$-Laplace equation is based on integration by parts, but it is not immediately clear what is the corresponding definition of weak solution to the total variation flow. One possibility is to apply the so-called Anzellotti pairing \cite{Anzellotti:1984}. This approach has been applied for the total variation flow, for example, in the monograph by Andreu, Caselles and Maz\'on \cite{Andreu-Caselles-Mazon:book}.
For the parabolic $p$-Laplace equation, it is also possible to consider solutions to the parabolic variational inequality \[
\tfrac1p\int\hspace{-0.6em}\int_{\Omega_T}|\nabla u|^p\dxt -\int\hspace{-0.6em}\int_{\Omega_T}u\partial_t\varphi\dxt
\le\tfrac1p\int\hspace{-0.6em}\int_{\Omega_T}|\nabla u-\nabla\varphi|^p\dxt
\]
for every $\varphi\in C^\infty_0(\Omega_T)$.
The variational approach goes back to Lichnewsky and Temam \cite{LicT78}, who employed an analogous concept in the case of the time-dependent minimal surface equation. Wieser \cite{Wie87} showed that the variational approach gives the same class of weak solutions as the standard definition. Moreover, he introduced a more general class of quasiminimizers related to parabolic problems. The variational inequality related to the total variation flow is of the form \[
\int_0^T\|Du(t)\|(\Omega)\dt
-\int\hspace{-0.6em}\int_{\Omega_T}u\partial_t\varphi\dxt
\le\int_0^T\|D(u-\varphi)(t)\|(\Omega)\dt
\]
for every $\varphi\ C^\infty_0(\Omega_T)$, where the total variation $\|Du(t)\|(\Omega)$ is a Radon measure for almost every $t\in(0,T)$.
A distinctive feature is that the variational definition is based on
total variation instead of weak gradient.
There are advantages in both approaches. For example, semigroup theory can be applied for the Anzellotti pairing and the direct methods in the calculus of variations can be applied in theory of parabolic variational integrals with linear growth.
Initial and boundary value problems to the total variation flow
have been studied by Andreu, Ballester, Caselles and Maz{\'o}n \cite{AndBCM01,ABCM} and by Andreu, Caselles, D\'iaz and Maz\'on \cite{Andreu-Caselles-Diaz-Mazon:2002}. They have shown that a unique solution exists to the problem \[
\begin{cases}
\partial_tu-\Div\bigg(\dfrac{Du}{|Du|}\bigg)=0&\mbox{in }\Omega\times(0,T),\\
u(x,t)=f(x)&\mbox{on }\partial\Omega\times(0,T),\\
u(x,t)=u_o(x)&\mbox{in }\Omega,
\end{cases} \] where $\Omega$ is a bounded Lipschitz domain in ${\mathds{R}}^n$, $f\in L^1(\partial\Omega)$ and $u_0\in L^1(\Omega)$. The case of homogeneous boundary data $f=0$ is discussed in \cite{Andreu-Caselles-Diaz-Mazon:2002} and \cite{ABCM} discusses inhomogeneous time-independent boundary data. The Neumann problem for the total variation flow has
been studied in \cite{AndBCM01}. The concepts of solution discussed in \cite{ABCM} are more general than the concept of variational solution considered in this work. A variational approach to existence and uniqueness questions has been discussed by B\"ogelein, Duzaar and Marcellini \cite{BDM}, see also \cite{BoegelDuzSchev:2016}, and for the corresponding obstacle problem by B\"ogelein, Duzaar and Scheven \cite{BoegelDuzSchev:2015}. A necessary and sufficient condition for continuity of a variational solution has been proved by DiBenedetto, Gianazza and Klaus \cite{DiBeGiaKla2017}. Gianazza and Klaus \cite{GiaKlaus2019} showed that variational solutions to the Cauchy-Dirichlet problem for the total variation flow are obtained as the limit as $p\to1$ of variational solutions to the corresponding problem for the parabolic $p$-Laplace equation. See also B\"ogelein, Duzaar, Sch\"atzler and Scheven \cite{BDSS:2019}.
Our main result in Theorem \ref{thm:main} below shows that the notions of weak solution to the total variation flow based on the Anzellotti pairing and the variational inequality coincide under natural assumptions. We consider weak solutions to a Cauchy-Dirichlet problem for the total variation flow, which can formally be written as
\begin{equation}
\label{Cauchy-Dirichlet}
\begin{cases}
\partial_tu-\Div\bigg(\dfrac{Du}{|Du|}\bigg)=0&\mbox{in
}\Omega_T,\\
u=u_o&\mbox{on }\partial_{\mathcal{P}}\Omega_T.
\end{cases}
\end{equation}
For this problem, we consider the appropriate definitions of weak solution with time-dependent boundary values,
see Definition \ref{def:Mazon-solution} and Definition \ref{def:var-sol} below.
It is relatively straight forward to show that a weak solution to the total variation flow is a variational solution.
This question has been studied in the context of metric measure spaces in \cite{Gorny-Mazon:2021}.
However, it is much more challenging to prove that a variational solution is a weak solution. The key ingredient is a duality result for the total variation functional in Theorem \ref{thm:TV-duality}. This is based on an approximation of the total variation by area-type functionals, see Theorem \ref{thm:area-duality}.
\section{Preliminaries}
\subsection{Functions of bounded variation}
Throughout this article, we consider a bounded Lipschitz domain $\Omega\subset{\mathds{R}}^n$. We will prescribe Dirichlet boundary values on $\partial\Omega$ in form of \emph{solid boundary values}. In the stationary case, this means that we choose an open Lipschitz domain $\Omega^\ast\subset{\mathds{R}}^n$ with $\Omega\Subset\Omega^\ast$, consider the Dirichlet data \begin{equation}
\label{bdry-data-assum}
u_o\in \W^{1,1}(\Omega^\ast)\cap\L^2(\Omega^\ast), \end{equation} and restrict ourselves to functions that agree with $u_o$ almost everywhere with respect to the Lebesgue measure $\Ln$ on $\Omega^\ast\setminus\Omega$.
We point out that in the parabolic setting,
we will consider time-dependent boundary data as in \eqref{bdry-data-parabolic}, which satisfies \eqref{bdry-data-assum} on almost every time slice. The space $\BV(\Omega^\ast)$ is defined as the space of functions
$u\in L^1(\Omega^\ast)$ for which the distributional derivative $Du$ is given by a finite vector-valued Radon measure on $\Omega^\ast$. By $\|Du\|$ we denote the total variation measure of $Du$, which is defined by \begin{equation*}
\|Du\|(A):=\sup\bigg\{\sum_{i=1}^\infty |Du(A_i)|\colon
A_i \mbox{ are pairwise disjoint Borel sets with }
A=\bigcup_{i=1}^\infty A_i\bigg\} \end{equation*} for every Borel set $A\subset\Omega^\ast$, cf. \cite[Def.~1.4]{AmbrosioFuscoPallara}. For the Dirichlet problem, we consider the function class \begin{equation*}
\BV_{u_o}(\Omega):=\{u\in \BV(\Omega^\ast)\colon u=u_o \mbox{
a.e. in }\Omega^\ast\setminus\Omega\}. \end{equation*} For $u\in\BV_{u_o}(\Omega)$, we write $D^au$ and $D^su$ for the absolutely continuous and the singular part of $Du$ with respect to the Lebesgue measure $\Ln$ and, moreover, $\nabla u$ for the Radon-Nikodym derivative of $Du$ with respect to the Lebesgue measure. With this notation, we have the decomposition \begin{equation}\label{decomposition-Du}
Du=D^au+D^su=\nabla u\Ln+D^su \end{equation} for every $u\in\BV_{u_o}(\Omega)$. From \cite[Thm. 3.88]{AmbrosioFuscoPallara} we know that on a bounded Lipschitz domain $\Omega\subset{\mathds{R}}^n$, there exist bounded inner and outer trace operators \begin{equation*}
T_\Omega: \BV(\Omega)\to L^1(\partial\Omega)
\qquad\mbox{and}\qquad
T_{{\mathds{R}}^n\setminus\overline\Omega}: \BV({\mathds{R}}^n\setminus\overline\Omega)\to L^1(\partial\Omega). \end{equation*} With these trace operators, we have the following extension result for $\BV$-functions. \begin{lemma}[{\cite[Cor. 3.89]{AmbrosioFuscoPallara}}] \label{lem:ambrosio-et-al}
Let $\Omega\subset{\mathds{R}}^n$ be an open set with bounded Lipschitz
boundary, $u\in \BV(\Omega)$ and
$v\in\BV({\mathds{R}}^n\setminus\overline\Omega)$. Then the function
\begin{equation*}
w(x)=
\begin{cases}
u(x),& \mbox{for }x\in\Omega,\\
v(x),& \mbox{for }x\in{\mathds{R}}^n\setminus\overline\Omega,
\end{cases}
\end{equation*}
belongs to $\BV({\mathds{R}}^n)$, and its derivative is given by the measure
\begin{equation*}
Dw= Du+Dv+\big(T_{{\mathds{R}}^n\setminus\overline\Omega}v-T_\Omega
u\big)\nu_\Omega\mathcal{H}^{n-1}\hspace{0.1em}\mbox{\LARGE$\llcorner$}\hspace{0.05em}
\partial\Omega,
\end{equation*}
where $\nu_\Omega$ denotes the generalized outer unit normal to
$\Omega$. In the above formula, we interpret $Du$ and $Dv$ as
vector-valued measures on the entire ${\mathds{R}}^n$ that are concentrated in
$\Omega$ and in ${\mathds{R}}^n\setminus\overline\Omega$,
respectively. \end{lemma}
We apply this lemma with the boundary values $u_o$ as in \eqref{bdry-data-assum} in place of $v$. This is possible because we can extend the boundary values to a function $u_o\in W^{1,1}({\mathds{R}}^n)$ without changing the boundary condition.
\subsection{Parabolic function spaces}\label{sec:parabolic-spaces}\label{sec:spaces}
A map $v\colon [0,T]\to X$ into a Banach space $X$ is called \emph{Bochner measurable} or \emph{strongly measurable} if it can be approximated by simple functions
$v_k\colon [0,T]\to X$ in the sense $\|v_k(t)-v(t)\|_X\to0$ for a.e. $t\in [0,T]$ as $k\to\infty$. A simple function is of the form \[ v_k(t)=\sum_{i=1}^N v^{(i)}\chi_{E_i}(t) \] for $v^{(1)},\ldots,v^{(N)}\in X$ and pairwise disjoint measurable sets $E_1,\ldots,E_N\subset[0,T]$. For $1\le p\le\infty$, we write $L^p(0,T;X)$ for the space of equivalence classes of Bochner measurable functions
$v\colon [0,T]\to X$ with $\|v(t)\|_X\in L^p([0,T])$.
For non-separable Banach spaces, the assumption of Bochner measurability often turns out to be too strong. For maps into non-separable dual spaces $X=X_0'$, we use the following weaker condition. A function $v\colon [0,T]\to X_0'$ is called weakly$\ast$-measurable if the map $[0,T]\ni t\mapsto \langle v(t),\varphi\rangle\in {\mathds{R}}$ is measurable for every $\varphi\in X_0$, where $\langle\cdot,\cdot\rangle$ denotes the dual pairing between $X_0'$ and $X_0$. Using this concept of measurability, we introduce the weak$\ast$-Lebesgue space \begin{equation}\label{def-weak-Lebesgue}
\Lw{p}(0,T;X_0'):=
\bigg\{v\colon [0,T]\to X_0'\ \bigg|\,
\begin{array}{l}
v\mbox{ is weakly$*$-measurable with }\\
t\mapsto \|v(t)\|_{X_0'}\in L^p([0,T])
\end{array}
\bigg\}\,, \end{equation} with $1\le p\le\infty$. The weak$\ast$-Lebesgue space with exponent $p=\infty$ naturally occurs in the case of the dual space $W^{-1,\infty}(\Omega)=[W^{1,1}_0(\Omega)]'$, since \begin{equation}\label{parabolic-dual-space}
\big[L^1(0,T;W^{1,1}_0(\Omega))\big]'
=
\Lw\infty(0,T;W^{-1,\infty}(\Omega)), \end{equation} cf. \cite[Sect. VII.4]{Theory-of-lifting}. Moreover, from \cite[Remark 3.12]{AmbrosioFuscoPallara} we know that $\BV(\Omega^\ast)$ is a dual space with a separable pre-dual $X_0$, whose elements take the form $g-\Div G$ with $g\in C^0_0(\Omega^\ast)$ and $G\in C^0_0(\Omega^\ast,{\mathds{R}}^n)$.
We will consider the Cauchy-Dirichlet problem for the
total variation flow with time-dependent boundary data satisfying
\begin{equation}
\label{bdry-data-parabolic}
u_o\in L^1(0,T;W^{1,1}(\Omega))\cap C^0([0,T];L^2(\Omega)).
\end{equation} The natural solution space for this problem
is the weak$\ast$-Lebesgue space \begin{equation*}
\Lw1(0,T;\BV_{u_o}(\Omega))
:=
\big\{ u\in \Lw1(0,T;\BV(\Omega^\ast))\,\colon
u=u_o\mbox{ a.e. in $(\Omega^\ast\setminus\Omega)_T$}
\big\}. \end{equation*} The next lemma states that weak$\ast$-measurability implies measurability of the total variation functional. \begin{lemma}\label{lem:TV-measurable}
For $u\in\Lw1(0,T;\BV_{u_o}(\Omega))$, the total variation
$\|Du(t)\|(\overline\Omega)$ depends measurably on $t\in(0,T)$. \end{lemma}
\begin{proof} The total variation of the open set $\Omega^\ast$ is given by \begin{equation}\label{def:tot-var}
\| Du(t)\| (\Omega^\ast)
:=
\sup\bigg\{\int_{\Omega^\ast} u(t) \Div\zeta\dx\ \bigg|\ \zeta\in C^1_0(\Omega^\ast, {\mathds{R}}^n),\,
\|\zeta\|_{L^\infty(\Omega^\ast)}\le 1\bigg\}, \end{equation} cf. \cite[Prop. 3.6]{AmbrosioFuscoPallara}. The integrals in the supremum depend measurably on time by definition of the weak$\ast$-measurability. Since the supremum is taken over a separable set, the supremum is measurable as well. Therefore, \begin{equation*}
\|Du(t)\|(\overline\Omega)
=
\|Du(t)\|(\Omega^\ast)
-
\int_{\Omega^\ast\setminus\Omega}|\nabla u_o(t)|\dx \end{equation*} depends measurably on $t\in(0,T)$. \end{proof}
\subsection{The area functional} For a parameter $\mu\ge0$, we consider the area functional \begin{equation}\label{def:areafcnl}
\mathcal A^{(\mu)}_{u_o}(u):=\int_{\Omega}\sqrt{\mu^2+|\nabla u|^2}\dx
+\|D^su\|(\overline\Omega) \end{equation} where $u\in\BV_{u_o}(\Omega)$. The limit case $\mu=0$ corresponds to the total variation functional, i.e. \[
\mathcal A_{u_o}^{(0)}(u)=\|Du\|(\overline\Omega) \] for every $u\in\BV_{u_o}(\Omega)$. We point out that the functional $\mathcal A^{(\mu)}_{u_o}$ depends on the prescribed boundary function $u_o$. More precisely, since $\Omega$ is a bounded Lipschitz domain, Lemma~\ref{lem:ambrosio-et-al} gives the decomposition \begin{equation*}
Du= Du\hspace{0.1em}\mbox{\LARGE$\llcorner$}\hspace{0.05em}\Omega+Du_o\hspace{0.1em}\mbox{\LARGE$\llcorner$}\hspace{0.05em}(\Omega^\ast\setminus\overline\Omega)
+\big(T_{{\mathds{R}}^n\setminus\overline\Omega}u_o-T_\Omega u\big)\nu_\Omega\mathcal{H}^{n-1}\hspace{0.1em}\mbox{\LARGE$\llcorner$}\hspace{0.05em}
\partial\Omega \end{equation*} for every $u\in\BV_{u_o}(\Omega)$. Therefore, the last term in~\eqref{def:areafcnl} can be expressed as \begin{align*}
\|D^su\|(\overline\Omega)
&=
\|D^su\|(\Omega)+\|D^su\|(\partial\Omega)\\
&=
\|D^su\|(\Omega)+\int_{\partial\Omega}|T_\Omega u-T_{{\mathds{R}}^n\setminus\overline\Omega}u_o|\,\d \mathcal{H}^{n-1}, \end{align*} which implies \begin{equation*}
\mathcal A^{(\mu)}_{u_o}(u)=\int_{\Omega}\sqrt{\mu^2+|\nabla u|^2}\dx
+\|D^su\|(\Omega)+\int_{\partial\Omega}|T_\Omega u-T_{{\mathds{R}}^n\setminus\overline\Omega}u_o|\,\d \mathcal{H}^{n-1}. \end{equation*}
The following approximation result for $\BV$-functions will be useful for us.
\begin{lemma}[Strict interior approximation]\label{lem:interior-approximation}
Let $\Omega\subset{\mathds{R}}^n$ be a bounded Lipschitz domain and
$u_o\in W^{1,1}(\Omega^\ast)\cap L^2(\Omega^\ast)$. For every
$u\in\BV_{u_o}(\Omega)$ and $\mu\in[0,1]$, there exists a sequence of functions $u_i\in
u_o+C^\infty_0(\Omega)$, $i\in{\mathds{N}}$, with $u_i\to u$ in $L^2(\Omega)$ and $\mathcal{A}_{u_o}^{(\mu)}(u_i)\to \mathcal{A}_{u_o}^{(\mu)}(u)$
as $i\to\infty$. \end{lemma} \begin{proof}
With the mollification operator $M_\varepsilon$ defined in
\cite[Sect. 5]{BDSS:2019} we define
\[
u_i=u_o+M_{\varepsilon_i}(u-u_o),
\qquad i\in{\mathds{N}},
\]
for some sequence $\varepsilon_i\downarrow0$ as $i\to\infty$. From \cite[Lemma 5.1]{BDSS:2019}
we infer $u_i\to u$ in $L^2(\Omega)$ and
$\mathcal A_{u_o}^{(1)}(u_i)\to \mathcal A_{u_o}^{(1)}(u)$ as $i\to\infty$. For the other parameters $\mu\in[0,1)$, the
asserted convergence follows from
the Reshetnyak continuity theorem \cite[Thm. 2.39]{AmbrosioFuscoPallara}. \end{proof}
\section{The notion of weak solution by Anzellotti pairing}
For any vector field $z\in L^\infty(\Omega,{\mathds{R}}^n)$ with $\Div z\in L^2(\Omega)$, there exists a uniquely determined outer normal trace $[z,\nu]\in L^\infty(\partial\Omega)$ with \[
\big\|[z,\nu]\big\|_{L^\infty(\partial\Omega)}\le\|z\|_{L^\infty(\Omega)} \] and \begin{equation}
\label{property-normal-trace}
\int_\Omega w\Div z\dx+\int_\Omega z\cdot\nabla w\dx
=
\int_{\partial\Omega}[z,\nu]w\,\d\mathcal{H}^{n-1} \end{equation} for every $w\in W^{1,1}(\Omega)\cap L^2(\Omega)$, see \cite[Prop. C.4]{Andreu-Caselles-Mazon:book}. We use the normal trace for the following version of an Anzellotti pairing, which is tailored for the Dirichlet problem. \begin{definition}
For any $u\in\BV_{u_o}(\Omega)\cap L^2(\Omega^\ast)$ and
$z\in L^\infty(\Omega,{\mathds{R}}^n)$ with
$\Div z\in L^2(\Omega)$, we define the Anzellotti pairing of
$z$ and $Du$ as the distribution
\begin{equation*}
(z,Du)_{u_o}(\varphi):=-\int_{\Omega}\Div z\,u\varphi\dx
-\int_{\Omega}z\cdot\nabla\varphi \,u\dx
+\int_{\partial\Omega}[z,\nu] u_o\varphi\,\d\mathcal{H}^{n-1}
\end{equation*}
for every $\varphi\in C^\infty_0(\Omega^\ast)$. \end{definition}
It turns out that this distribution is a measure.
\begin{lemma}\label{lem:anzellotti-bound}
For any $u\in\BV_{u_o}(\Omega)\cap L^2(\Omega^\ast)$ and
$z\in L^\infty(\Omega,{\mathds{R}}^n)$ with
$\Div z\in L^2(\Omega)$, the pairing $(z,Du)_{u_o}$ defines a
Radon measure on $\overline\Omega$, and we have \begin{equation}\label{Anzellotti-bound}
\big|(z,Du)_{u_o}(\overline\Omega)\big|\le \|z\|_{L^\infty(\Omega)}\|Du\|(\overline\Omega). \end{equation} \end{lemma} Before giving the proof, we state a variant of the preceding estimate that involves the area functional instead of the total variation. To this end, we note that
for any vectors $z,v\in{\mathds{R}}^n$ with $|z|\le 1$ and $\mu>0$, we have the Fenchel-type inequality \begin{equation}\label{Fenchel-pointwise}
|z\cdot v|\le \sqrt{\mu^2+|v|^2}-\mu\sqrt{1-|z|^2}. \end{equation} This inequality can be verified by a straightforward calculation or by noting that
$f_\mu^*(z)=-\mu\sqrt{1-|z|^2}$ is the convex conjugate function of
$f_\mu(v)=\sqrt{\mu^2+|v|^2}$ and recalling the general Fenchel inequality $|z\cdot v|\le f_\mu(z)+f_\mu^\ast(v)$. We note that equality in~\eqref{Fenchel-pointwise} holds if and only if
$z=v(\mu^2+|v|^2)^{-1/2}$. The Fenchel-type estimate above leads to an estimate for the Anzellotti pairing.
\begin{lemma}\label{lem:Fenchel-Anzellotti}
For every $\mu\in(0,1]$, every
$u\in\BV_{u_o}(\Omega)\cap L^2(\Omega^\ast)$ and every $z\in L^\infty(\Omega,{\mathds{R}}^n)$
with $\|z\|_{L^\infty}\le1$ and $\Div z\in L^2(\Omega)$ we have
\begin{equation}\label{Anzellotti-Fenchel-bound}
\big|(z,Du)_{u_o}(\overline\Omega)\big|
\le
\A_{u_o}^{(\mu)}(u)-\mu\int_\Omega \sqrt{1-|z|^2}\dx.
\end{equation} \end{lemma}
\begin{proof}[Proof of Lemmas~\ref{lem:anzellotti-bound} and~\ref{lem:Fenchel-Anzellotti}]
Let $u\in\BV_{u_o}(\Omega)\cap L^2(\Omega^\ast)$.
Lemma~\ref{lem:interior-approximation} provides us with a sequence
of approximating functions
$u_i\in u_o+C^\infty_0(\Omega)$, $i\in{\mathds{N}}$, that converges strictly to $u$, i.e. $u_i\to u$ in $L^2(\Omega)$ and
\begin{equation}\label{strict-convergence}
\|\nabla u_i\|_{L^1(\Omega)}\to \|Du\|(\overline\Omega)
\end{equation}
as $i\to\infty$.
For every test function $\varphi\in
C^\infty_0(\Omega^\ast)$ we obtain
\begin{align}\label{approximation-anzellotti}
(z,Du)_{u_o}(\varphi)
&=
-\int_{\Omega}\Div z\,u\varphi\dx
-\int_\Omega z\cdot\nabla\varphi\, u\dx
+\int_{\partial\Omega}[z,\nu] u_o\varphi\,\d\mathcal{H}^{n-1}\\\nonumber
&=
\lim_{i\to\infty}\bigg(-\int_{\Omega}\Div z\,u_i\varphi\dx
-\int_\Omega z\cdot\nabla\varphi\, u_i\dx\bigg)
+\int_{\partial\Omega}[z,\nu] u_o\varphi\,\d\mathcal{H}^{n-1}\\\nonumber
&=
\lim_{i\to\infty}\int_{\Omega} z\cdot\nabla u_i\,\varphi\dx,
\end{align}
where in the last step, we applied~\eqref{property-normal-trace} with $w=u_i\varphi$ and the fact $u_i=u_o$ on $\partial\Omega$ in the sense
of traces. By~\eqref{strict-convergence}, we deduce
\begin{align*}
|(z,Du)_{u_o}(\varphi)|
&\le
\lim_{i\to\infty}\|\nabla
u_i\|_{L^1(\Omega)}\|z\|_{L^\infty(\Omega)}\|\varphi\|_{C^0(\overline\Omega)}\\
&=\|Du\|(\overline\Omega)\|z\|_{L^\infty(\Omega)}\|\varphi\|_{C^0(\overline\Omega)}.
\end{align*}
This implies that $(z,Du)_{u_o}$ defines a measure on
$\overline\Omega$ that satisfies~\eqref{Anzellotti-bound}. This
completes the proof of Lemma~\ref{lem:anzellotti-bound}. For the
proof of Lemma~\ref{lem:Fenchel-Anzellotti}, we observe that
according to Lemma~\ref{lem:interior-approximation}, the sequence of
approximating functions $u_i\in u_o+C^\infty_0(\Omega)$, $i\in{\mathds{N}}$, has the property
$ \mathcal{A}_{u_o}^{(\mu)}(u_i)\to\mathcal{A}_{u_o}^{(\mu)}(u)$ as $i\to\infty$ for every $\mu\in(0,1]$.
We use \eqref{approximation-anzellotti} with a cut-off
function $\varphi\in C^\infty_0(\Omega^\ast)$ with $\varphi\equiv1$
on $\Omega$. Estimating the last integrand in
\eqref{approximation-anzellotti}
by means of~\eqref{Fenchel-pointwise},
we arrive at
\begin{align*}
\big|(z,Du)_{u_o}(\overline\Omega)\big|
&=\big|(z,Du)_{u_o}(\varphi)\big|\\
&\le
\lim_{i\to\infty}\int_{\Omega} \sqrt{\mu^2+|\nabla u_i|^2}\dx
-
\mu\int_{\Omega} \sqrt{1-|z|^2}\dx\\
&=
\mathcal{A}_{u_o}^{(\mu)}(u)
-
\mu\int_{\Omega} \sqrt{1-|z|^2}\dx.
\end{align*}
This completes the proof of Lemma~\ref{lem:Fenchel-Anzellotti}. \end{proof}
Next, we state an elementary identity for the Anzellotti pairings that will frequently be used in the proofs that follow. \begin{lemma}\label{lem:comparison-pairings}
Let $z\in L^\infty(\Omega,{\mathds{R}}^n)$ be with $\Div z\in
L^2(\Omega)$ and $v\in\BV_{u_o}(\Omega)\cap L^2(\Omega)$. Then we
have
\begin{equation*}
(z,Dv)_{u_o}(\overline\Omega)
-
\int_\Omega z\cdot\nabla u_o\dx
=
\int_\Omega \Div z\, (u_o-v) \dx.
\end{equation*} \end{lemma}
\begin{proof}
We use first the definition of the Anzellotti
pairing and then property~\eqref{property-normal-trace}
of the normal trace with
$w=u_o$ in order to have
\begin{align*}
(z,Dv)_{u_o}(\overline\Omega)
&=
-\int_\Omega v\Div z \dx
+\int_{\partial\Omega}[z,\nu]u_o\,\d\mathcal{H}^{n-1}\\
&=
-\int_\Omega \Div z\,(v-u_o) \dx
-\int_\Omega u_o\Div z \dx
+\int_{\partial\Omega}[z,\nu]u_o\,\d\mathcal{H}^{n-1}
\\
&=
-\int_\Omega \Div z\,(v-u_o) \dx
+
\int_\Omega z\cdot\nabla u_o\dx. \qedhere
\end{align*} \end{proof}
We apply the following definition of weak solution.
\begin{definition}[Weak solution]
\label{def:Mazon-solution}
Assume that $u_o\in L^1(0,T;W^{1,1}(\Omega))\cap
C^0([0,T];L^2(\Omega))$.
We say that a function $u\in \Lw1(0,T;BV_{u_o}(\Omega))\cap
C^0([0,T];L^2(\Omega))$ with $\partial_tu\in L^2(\Omega_T)$ is
a weak solution of~\eqref{Cauchy-Dirichlet}
if $u(0)=u_o(0)$ and if there exists a vector field $z\in L^\infty(\Omega_T,{\mathds{R}}^n)$ with
$\|z\|_{L^\infty}\le1$, $\Div z=\partial_tu$ in $\Omega_T$
in the sense of distributions and
\begin{equation}\label{mazon-TV}
\|Du(t)\|(\overline\Omega)
+\int_{\Omega\times\{t\}}\partial_tu(u-v)\dx
=
(z(t),Dv)_{u_o}(\overline\Omega)
\end{equation}
for every $v\in\BV_{u_o(t)}(\Omega)\cap L^2(\Omega)$ and a.e. $t\in(0,T)$. \end{definition}
We recall an equivalent way to formulate the preceding concept of solution that has already been observed in \cite[Thm. 1]{Andreu-Caselles-Diaz-Mazon:2002}.
\begin{lemma}\label{lem:Mazon-2}
A map $u\in \Lw1(0,T;BV_{u_o}(\Omega))\cap
C^0([0,T];L^2(\Omega))$ with $\partial_tu\in L^2(\Omega_T)$ and
$u(0)=u_o(0)$ is
a weak solution of~\eqref{Cauchy-Dirichlet} in the sense of Definition~\ref{def:Mazon-solution}
if and only if there exists a vector field
$z\in L^\infty(\Omega_T,{\mathds{R}}^n)$ with
\begin{equation}\label{1-harmonic-flow}
\Div z=\partial_tu \qquad\mbox{in $\Omega_T$}
\end{equation}
in the sense of distributions, for which
\begin{equation}\label{maximal-pairing}
\|z\|_{L^\infty(\Omega_T)}\le 1
\qquad\mbox{and}\qquad
\big(z(t),Du(t)\big)_{u_o}(\overline\Omega)
=
\|Du(t)\|(\overline\Omega)
\end{equation}
hold true for a.e. $t\in(0,T)$. \end{lemma}
\begin{remark}
The condition~\eqref{maximal-pairing} for the vector field $z$ can
be interpreted as an analogue of the identity $z=\frac{Du}{|Du|}$
for $\BV$-functions. In this sense, equation~\eqref{1-harmonic-flow}
is the generalization of the differential equation
\eqref{Cauchy-Dirichlet}$_1$ to the $\BV$-setting. \end{remark}
\begin{proof}[Proof of Lemma~\ref{lem:Mazon-2}]
If $u$ is a weak solution of~\eqref{Cauchy-Dirichlet} in the sense
of Definition~\ref{def:Mazon-solution}, we simply choose $v=u(t)$
in~\eqref{mazon-TV} to deduce~\eqref{maximal-pairing}.
For the other direction, assume that $u\in \Lw1(0,T;BV_{u_o}(\Omega))\cap
C^0([0,T];L^2(\Omega))$, with $\partial_tu\in L^2(\Omega_T)$, and
$u(0)=u_o(0)$
and that there exists a vector field
$z\in L^\infty(\Omega_T,{\mathds{R}}^n)$ with the properties in~\eqref{1-harmonic-flow}
and~\eqref{maximal-pairing}. For
$v\in\BV_{u_o(t)}(\Omega)\cap L^2(\Omega)$, we apply
Lemma~\ref{lem:comparison-pairings}, once with $v$ and once with
$u(t)$, to obtain
\begin{align*}
\big(z(t),Dv\big)_{u_o}(\overline\Omega)
&=
\int_\Omega z(t)\cdot\nabla u_o(t)\dx
+
\int_{\Omega\times\{t\}} \partial_tu (u_o-v) \dx\\
&=
\big(z(t),Du(t)\big)_{u_o}(\overline\Omega)
+
\int_{\Omega\times\{t\}} \partial_tu (u-v) \dx\\
&=
\|Du(t)\|(\overline\Omega)
+
\int_{\Omega\times\{t\}} \partial_tu (u-v) \dx,
\end{align*}
for a.e. $t\in(0,T)$.
In the last line, we used~\eqref{maximal-pairing}. This proves that $u$ is a
solution in the sense of Definition~\ref{def:Mazon-solution}. \end{proof}
\section{The concept of variational solution}
The following notion of solution of the Cauchy-Dirichlet problem~\eqref{Cauchy-Dirichlet} is based on the variational approach by Lichnewsky and Temam \cite{LicT78}.
\begin{definition}[Variational solution]\label{def:var-sol}
Assume that the initial and
boundary values satisfy $u_o\in L^1(0,T;W^{1,1}(\Omega))\cap
C^0([0,T];L^2(\Omega))$.
A function $u\in \Lw1(0,T;BV_{u_o}(\Omega))\cap C^0([0,T];L^2(\Omega))$
is called a variational solution of~\eqref{Cauchy-Dirichlet} if
\begin{align}\label{var-sol-TVF}
\int_0^T\|Du(t)\|(\overline\Omega)\dt
&\le
\int\hspace{-0.6em}\int_{\Omega_T}\partial_tv(v-u)\dxt
+\int_0^T\|Dv(t)\|(\overline\Omega)\dt\\\nonumber
&\quad
-\tfrac12\int_{\Omega}|v(T)-u(T)|^2\dx
+\tfrac12\int_{\Omega}|v(0)-u_o(0)|^2\dx
\end{align}
holds true for every $v\in\Lw1(0,T;\BV_{u_o}(\Omega))\cap
C^0([0,T];L^2(\Omega))$ with $\partial_tv\in L^2(\Omega_T)$.
\end{definition}
In the following, we will consider variational solutions with the additional property $\partial_tu\in L^2(\Omega_T)$, as it is required in the notion of weak solution in the sense of Definition~\ref{def:Mazon-solution}. In this case, the variational inequality can also be considered separately on the time slices.
\begin{lemma}\label{lem:var-sol-slicewise}
Assume that $u_o\in L^1(0,T;W^{1,1}(\Omega))\cap C^0([0,T];L^2(\Omega))$.
A function $u\in \Lw1(0,T;BV_{u_o}(\Omega))\cap C^0([0,T];L^2(\Omega))$,
with $\partial_tu\in L^2(\Omega_T)$,
is a variational solution of~\eqref{Cauchy-Dirichlet} in the sense of
Definition~\ref{def:var-sol} if and only if
\begin{equation}\label{var-sol-TVF-slicewise}
\|Du(t)\|(\overline\Omega)
\le
\int_{\Omega\times\{t\}}\partial_tu(v-u)\dx
+\|Dv\|(\overline\Omega)
\end{equation}
for a.e. $t\in(0,T)$ and every $v\in\BV_{u_o(t)}(\Omega)\cap
L^2(\Omega)$ and if $u$ attains the initial values in the sense
\begin{equation}\label{initial values}
u(0)=u_o(0)\qquad\mbox{in }L^2(\Omega).
\end{equation}
\end{lemma}
\begin{remark}
Condition~\eqref{var-sol-TVF-slicewise} can be reformulated in
terms of the subdifferential
\begin{equation*}
\partial\Phi(u)=\big\{w\in L^2(\Omega)\colon
\Phi(u)+\langle w,v-u\rangle\le \Phi(v)\mbox{ for every $v\in L^2(\Omega)$}\big\}
\end{equation*}
of the functional $\Phi:L^2(\Omega)\to{\mathds{R}}$, defined by
\begin{equation*}
\Phi(u)=
\begin{cases}
\|D\bar u\|(\overline\Omega),
&\mbox{if }
\bar u\in\BV_{u_o}(\Omega)\cap L^2(\Omega^\ast),\\[2ex]
\infty,&\mbox{if }\bar u\in L^2(\Omega^\ast)\setminus\BV_{u_o}(\Omega).
\end{cases}
\end{equation*}
Here, the function $\bar u$ denotes the extension of $u$ by
$u_o$ to $\Omega^\ast\setminus\Omega$.
By definition of the subdifferential,
the variational inequality~\eqref{var-sol-TVF-slicewise} can be
reformulated as
\begin{equation*}
-\partial_tu(t)\in \partial\Phi(u(t))
\qquad\mbox{for a.e. }t\in(0,T).
\end{equation*}
\end{remark}
\begin{proof}[Proof of Lemma~\ref{lem:var-sol-slicewise}]
Assume that the map $u$ satisfies~\eqref{var-sol-TVF-slicewise} and~\eqref{initial values} and let
$v\in\Lw1(0,T;\BV_{u_o}(\Omega))\cap
C^0([0,T];L^2(\Omega))$ with $\partial_tv\in L^2(\Omega_T)$ be an
arbitrary comparison function in~\eqref{var-sol-TVF}.
The function $v(t)$, for a.e. $t\in(0,T)$, is admissible in~\eqref{var-sol-TVF-slicewise}.
Integrating the resulting
inequalities over time, we deduce
\begin{align*}
\int_0^T\|Du(t)\|(\overline\Omega)\dt
&\le
\int\hspace{-0.6em}\int_{\Omega_T}\partial_tu(v-u)\dxt
+\int_0^T\|Dv(t)\|(\overline\Omega)\dt.
\end{align*}
Since $\partial_tu,\partial_tv\in L^2(\Omega_T)$ and $u(0)=u_o(0)$ by~\eqref{initial values},
an integration by parts implies
\begin{align*}
\int\hspace{-0.6em}\int_{\Omega_T}\partial_tu(v-u)\dxt
&=
\int\hspace{-0.6em}\int_{\Omega_T}\partial_tv(v-u)\dxt\\
&\quad
-\tfrac12\int_{\Omega}|v(T)-u(T)|^2\dx
+\tfrac12\int_{\Omega}|v(0)-u_o(0)|^2\dx.
\end{align*}
Combining the two preceding formulae, we obtain~\eqref{var-sol-TVF},
which proves that $u$ is a variational solution
to~\eqref{Cauchy-Dirichlet}.
\noindent
For the opposite direction, we start with a variational solution
$u\in \Lw1(0,T;\BV_{u_o}(\Omega))\cap C^0([0,T];L^2(\Omega))$
with $\partial_tu\in L^2(\Omega_T)$.
We begin with the observation that for any $v\in\BV(\Omega)\cap
L^2(\Omega)$,
the extension
\begin{equation}\label{extension-v}
\bar v(x,t)
=
\begin{cases}
v(x),&x\in\Omega,\\[0.6ex]
u_o(x,t),&x\in\Omega^\ast\setminus\Omega,
\end{cases}
\end{equation}
defines a function $v\in\Lw1(0,T;\BV_{u_o}(\Omega))\cap
C^0([0,T];L^2(\Omega^\ast))$ with
$\partial_tv=0$ in $\Omega_T$. This follows by applying
Lemma~\ref{lem:ambrosio-et-al} separately on the time slices.
For a cut-off function in time
$\zeta\in C^\infty([0,T])$ with $\zeta(T)=0$
and the extension $\bar v$ defined above, we consider
\begin{equation*}
w=u+\zeta(t)(\bar v-u).
\end{equation*}
We note that this function is
admissible as comparison function in~\eqref{var-sol-TVF}, since the
properties of $u$ and $\bar v$ imply
$w\in\Lw1(0,T;\BV_{u_o}(\Omega))\cap
C^0([0,T];L^2(\Omega))$ and $\partial_tw\in L^2(\Omega_T)$.
By convexity of the total variation functional
\begin{equation*}
\|Dw(t)\|(\overline\Omega)
\le
(1-\zeta(t))\|Du(t)\|(\overline\Omega)+\zeta(t)\|D\bar v(t)\|(\overline\Omega)
\end{equation*}
for a.e. $t\in(0,T)$, we obtain
\begin{align*}
\int_0^T\zeta(t)\|Du(t)\|(\overline\Omega)\dt
&\le
\int\hspace{-0.6em}\int_{\Omega_T}\partial_t\big(u+\zeta(v-u)\big)(v-u)\zeta\dxt
+\int_0^T\zeta(t)\|D\bar v(t)\|(\overline\Omega)\dt\\
&\qquad+
\tfrac12\int_\Omega \big|(1-\zeta(0))u(0)+\zeta(0)v-u_o(0)\big|^2\dx.
\end{align*}
Here, we also used the fact $\zeta(T)=0$, which ensures
that no integral over the time slice at the final
time occurs in the variational inequality. Integrating by
parts, the integral involving
the time derivative can be rewritten as
\begin{align*}
&\int\hspace{-0.6em}\int_{\Omega_T}\partial_t\big(u+\zeta(v-u)\big)(v-u)\zeta\dxt\\
&\qquad=
\int\hspace{-0.6em}\int_{\Omega_T}\partial_tu (v-u)\zeta\dxt
+
\int\hspace{-0.6em}\int_{\Omega_T}\big(\zeta'\zeta|v-u|^2+\zeta^2
\tfrac12\partial_t|v-u|^2\big)\dxt\\
&\qquad=
\int\hspace{-0.6em}\int_{\Omega_T}\partial_tu (v-u)\zeta\dxt
-
\tfrac12\int_\Omega\zeta^2(0)|v-u(0)|^2\dx.
\end{align*}
Combining the preceding formulae, we arrive at
\begin{align}\label{var-ineq-cutoff}
&\int_0^T\zeta(t)\|Du(t)\|(\overline\Omega)\dt
\le
\int\hspace{-0.6em}\int_{\Omega_T}\partial_tu (v-u)\zeta\dxt
+
\int_0^T\zeta(t)\|D\bar v(t)\|(\overline\Omega)\dt\\\nonumber
&\qquad\qquad+
\tfrac12\int_\Omega \big|(1-\zeta(0))u(0)+\zeta(0)v-u_o(0)\big|^2\dx
-
\tfrac12\int_\Omega\zeta^2(0)|v-u(0)|^2\dx.
\end{align}
Our first goal is to show that the initial values are
attained. To this end, we observe that
an approximation argument implies the above inequality also for the
characteristic function $\zeta=\chi_{[0,\tau]}$, for any
$\tau\in(0,T)$. This gives
\begin{align*}
\int_0^\tau\|Du(t)\|(\overline\Omega)\dt
&\le
\int\hspace{-0.6em}\int_{\Omega_\tau}\partial_tu (v-u)\dxt
+
\int_0^\tau \|D\bar v(t)\|(\overline\Omega)\dt\\\nonumber
&\qquad+
\tfrac12\int_\Omega \big(|v-u_o(0)|^2-|v-u(0)|^2\big)\dx,
\end{align*}
for every $\tau\in(0,T)$. Since $\partial_tv=0$ in $\Omega_T$, an
integration by parts gives
\begin{align*}
\int\hspace{-0.6em}\int_{\Omega_\tau}\partial_tu (v-u)\dxt
&=
-\tfrac12\int\hspace{-0.6em}\int_{\Omega_\tau}\partial_t|v-u|^2\dxt\\
&=
\tfrac12\int_\Omega |v-u(0)|^2\dx
-\tfrac12\int_\Omega |v-u(\tau)|^2\dx.
\end{align*}
Combining the two preceding formulae, we arrive at
\begin{align}\label{var-ineq-local}
&\tfrac12\int_\Omega |v-u(\tau)|^2\dx
+
\int_0^\tau\|Du(t)\|(\overline\Omega)\dt\\\nonumber
&\qquad\le
\int_0^\tau \|D\bar v(t)\|(\overline\Omega)\dt
+
\tfrac12\int_\Omega |v-u_o(0)|^2\dx,
\end{align}
for any $v\in\BV(\Omega)\cap L^2(\Omega)$ and $\tau\in(0,T)$.
For a given $\varepsilon>0$, we choose $u_{o,\varepsilon}\in C^\infty_0(\Omega)$
with $\|u_{o,\varepsilon}-u_o(0)\|_{L^2(\Omega)}\le\varepsilon$ and apply the
preceding estimate with $v=u_{o,\varepsilon}$. Discarding the
second integral on the left-hand side of~\eqref{var-ineq-local},
we have
\begin{align*}
\tfrac12\int_\Omega|u_{o,\varepsilon}-u(\tau)|^2\dx
\le
\int_0^\tau\|D\bar u_{o,\varepsilon}(t)\|(\overline\Omega)\dt
+
\tfrac12\int_\Omega|u_{o,\varepsilon}-u_o(0)|^2\dx,
\end{align*}
which implies
\begin{align}\label{L2-bound-initial}
\tfrac14\int_\Omega|u_o(0)-u(\tau)|^2\dx
&\le
\tfrac12\int_\Omega|u_{o,\varepsilon}-u(\tau)|^2\dx
+
\tfrac12\int_\Omega|u_{o,\varepsilon}-u_o(0)|^2\dx\\\nonumber
&\le
\int_0^\tau\|D\bar u_{o,\varepsilon}(t)\|(\overline\Omega)\dt
+
\int_\Omega|u_{o,\varepsilon}-u_o(0)|^2\dx\\\nonumber
&\le
\int_0^\tau\|D\bar u_{o,\varepsilon}(t)\|(\overline\Omega)\dt
+
\varepsilon^2.
\end{align}
Using Lemma~\ref{lem:ambrosio-et-al}, we estimate the last
integral by
\begin{equation*}
\int_0^\tau \|D\bar u_{o,\varepsilon}(t)\|(\overline\Omega)\dt
\le
\tau\int_\Omega|\nabla u_{o,\varepsilon}|\dx
+
\int_0^\tau
\int_{\partial\Omega}|T_{{\mathds{R}}^n\setminus\overline\Omega}u_o(t)|\d\mathcal{H}^{n-1}\dt \to0
\end{equation*}
as $\tau\downarrow0$.
Letting first $\tau\downarrow0$ and then
$\varepsilon\downarrow0$ in~\eqref{L2-bound-initial}, we conclude that
\begin{equation*}
\int_\Omega|u_o(0)-u(0)|^2\dx
=
\lim_{\tau\downarrow0}\int_\Omega|u_o(0)-u(\tau)|^2\dx
=0,
\end{equation*}
which implies the assertion $u(0)=u_o(0)$.
It remains to show~\eqref{var-sol-TVF-slicewise}. For a cut-off
function $\zeta\in C^\infty_0((0,T))$,
inequality~\eqref{var-ineq-cutoff} and the fact that $u(0)=u_o(0)$ imply
\begin{equation*}
\int_0^T\zeta(t)\|Du(t)\|(\overline\Omega)\dt
\le
\int\hspace{-0.6em}\int_{\Omega_T}\partial_tu (v-u)\zeta\dxt
+
\int_0^T\zeta(t)\|D\bar v(t)\|(\overline\Omega)\dt
\end{equation*}
for every $v\in\BV(\Omega)\cap L^2(\Omega)$, where $\bar v$ is
defined by \eqref{extension-v}.
For a given time $s\in(0,T)$ and $0<\delta<\min\{s,T-s\}$,
we use this estimate with the
cut-off function $\zeta(t)=\frac1\delta\phi(\frac{s-t}\delta)$,
where $\phi\in C^\infty_0((-1,1))$ denotes a standard mollifier.
By letting $\delta\downarrow0$, we infer
\begin{align*}
\|Du(s)\|(\overline\Omega)
&\le
\int_{\Omega\times\{s\}}\partial_tu (v-u)\dx
+
\|D\bar v(s)\|(\overline\Omega)
\end{align*}
for a.e. $s\in(0,T)$ and every $v\in\BV(\Omega)\cap L^2(\Omega)$.
This implies the remaining assertion~\eqref{var-sol-TVF-slicewise}
and completes the proof of Lemma~\ref{lem:var-sol-slicewise}.
\end{proof}
\section{Equivalence of variational and weak solutions}
In this section we prove the equivalence of the two concepts of solution that have been introduced in Definition~\ref{def:Mazon-solution} and Definition~\ref{def:var-sol}, respectively. The precise statement of the result is the following.
\begin{theorem}\label{thm:main}
A function $u\in\Lw1(0,T;\BV_{u_o}(\Omega))\cap C^0([0,T];L^2(\Omega))$
with $\partial_tu\in L^2(\Omega_T)$
is a variational solution of~\eqref{Cauchy-Dirichlet} if and only if
it is a weak solution of~\eqref{Cauchy-Dirichlet}. \end{theorem}
In Subsection~\ref{sec:weakisvar} we show that a weak solution is a variational solution. The proof of the converse claim is presented in the remaining three subsections. The key step is an elliptic duality result for the total variation functional in Subsection~\ref{sec:duality-TV}. This will be established as a stability result by approximating the total variation by area-type functionals in Subection~\ref{sec:area-duality}. Finally, in Subsection~\ref{sec:varisweak} we complete the proof of the claim that a variational solution is a weak solution.
\subsection{Weak solutions are variational solutions} \label{sec:weakisvar}
Assume that $u\in\Lw1(0,T;\BV_{u_o}(\Omega))\cap C^0([0,T];L^2(\Omega))$,
with $\partial_tu\in L^2(\Omega_T)$, is a weak solution according to Definition~\ref{def:Mazon-solution}. Let $z\in
L^\infty(\Omega_T)$ with $\|z\|_{L^\infty}\le1$ be the vector field
that is provided by Definition~\ref{def:Mazon-solution}. For
a.e. $t\in(0,T)$ and any
$v\in\BV_{u_o(t)}(\Omega)\cap L^2(\Omega)$, we use~\eqref{mazon-TV} and~\eqref{Anzellotti-bound} to
deduce
\begin{align*}
\|Du(t)\|(\overline\Omega)
+\int_{\Omega\times\{t\}}\partial_tu(u-v)\dx
=
(z(t),Dv)_{u_o}(\overline\Omega)
\le
\|Dv\|(\overline\Omega)
\end{align*}
for a.e. $t\in(0,T)$. This means that the variational
inequality~\eqref{var-sol-TVF-slicewise} is satisfied on a.e. time
slice, and Lemma~\ref{lem:var-sol-slicewise} implies
that $u$ is a variational solution according to Definition~\ref{def:var-sol}.
\subsection{An auxiliary result for the area functional} \label{sec:area-duality}
The following approximation result for the area functional \eqref{def:areafcnl} will be applied in the proof of Theorem~\ref{thm:TV-duality}.
\begin{theorem}\label{thm:area-duality}
Let $f\in W^{-1,\infty}(\Omega)\cap L^2(\Omega)$, $u_o\in
W^{1,1}(\Omega)\cap L^2(\Omega)$, $\mu>0$, and $\lambda\in{\mathds{R}}$ be
given. Assume that
$u\in\BV_{u_o}(\Omega)\cap L^2(\Omega^\ast)$ is a minimizer of the functional
\begin{equation*}
\Psi(v)=\mathcal{A}_{u_o}^{(\mu)}(v)+\int_{\Omega}\big(\tfrac\lambda2|v|^2+f(v-u_o)\big)\dx
\end{equation*}
in the space $\BV_{u_o}(\Omega)\cap L^2(\Omega^\ast)$.
Then the vector field
\begin{equation*}
z=\frac{\nabla u}{\sqrt{\mu^2+|\nabla u|^2}}
\end{equation*}
satisfies
\begin{equation}\label{div=f}
\Div z=\lambda u+f \qquad\mbox{in $\Omega$}
\end{equation}
in the sense of distributions and we have the
estimate
\begin{align}\label{dual-problem-area}
&\mathcal{A}_{u_o}^{(\mu)}(u)+\int_{\Omega}\big(\tfrac\lambda2|u|^2+f(u-u_o)\big)\dx\\\nonumber
&\qquad\le
\int_\Omega z\cdot\nabla u_o\dx
+
\tfrac\lambda2\int_\Omega|u_o|^2\dx
+
\mu\int_\Omega\sqrt{1-|z|^2}\dx.
\end{align}
\end{theorem}
\begin{proof}
For the proof of~\eqref{div=f}, we test the minimality of $u$
with the comparison map $v_r=u-r\varphi\in\BV_{u_o}(\Omega)\cap L^2(\Omega^\ast)$,
where $\varphi\in C^\infty_0(\Omega)$ and $r>0$. We apply the fact
that $Du$ and $Dv_r$ have the same singular parts. This implies
\begin{align*}
&\int_{\Omega}\sqrt{\mu^2+|\nabla u|^2}\dx
+\int_{\Omega}\big(\tfrac\lambda2|u|^2+f(u-u_o)\big)\dx\\
&\qquad\le
\int_{\Omega}\sqrt{\mu^2+|\nabla u-r\nabla\varphi|^2}\dx
+
\int_{\Omega}\big(\tfrac\lambda2|u-r\varphi|^2+f(u-r\varphi-u_o)\big)\dx,
\end{align*}
and, after dividing by $r>0$,
\begin{align*}
\int_{\Omega}f\varphi\dx
&\le
\frac1r\int_{\Omega}\Big(\sqrt{\mu^2+|\nabla u-r\nabla\varphi|^2}-
\sqrt{\mu^2+|\nabla u|^2}\Big)\dx\\
&\quad+
\frac\lambda2\frac1r\int_{\Omega}\big(|u-r\varphi|^2-|u|^2\big)\dx.
\end{align*}
Letting $r\downarrow0$ on the right-hand side, we deduce
\begin{align*}
\int_{\Omega}f\varphi\dx
&\le
\int_\Omega\frac{\partial}{\partial
r}\Big|_{r=0}\sqrt{\mu^2+|\nabla u-r\nabla\varphi|^2}\dx
+
\frac\lambda2\int_\Omega\frac{\partial}{\partial
r}\Big|_{r=0}\,|u-r\varphi|^2\dx\\
&=-\int_\Omega\frac{\nabla
u\cdot\nabla\varphi}{\sqrt{\mu^2+|\nabla u|^2}}\dx
-
\lambda\int_\Omega u\varphi\dx.
\end{align*}
Note that it is allowed to differentiate under the integrals because
in the first case,
the derivative of the integral is dominated by $|\nabla\varphi|\in
L^1(\Omega)$, and in the second integral, it is bounded by
$2(|u|+|\varphi|)|\varphi|\in L^1(\Omega)$.
In view of the definition of $z$, we have shown
that
\begin{align*}
\int_{\Omega}(\lambda u+f)\varphi\dx
\le
-\int_\Omega z\cdot \nabla\varphi\dx
\end{align*}
holds true for every $\varphi\in C^\infty_0(\Omega)$.
Since the same estimate holds with $-\varphi$ instead of $\varphi$,
the opposite inequality holds as well. This proves $\Div z=\lambda u+f$ in
the distributional sense in $\Omega$.
Next, we use $w_r=u+r(u_o-u)\in\BV_{u_o}(\Omega)\cap
L^2(\Omega^\ast)$, for $r>0$, as a comparison function for the minimizer
$u$.
Since
$D^sw_r=(1-r)D^su$, we obtain
\begin{align*}
&\int_{\Omega}\sqrt{\mu^2+|\nabla u|^2}\dx
+\|D^su\|(\overline\Omega)
+\int_{\Omega}\big(\tfrac\lambda2|u|^2+f(u-u_o)\big)\dx\\
&\quad\le
\int_{\Omega}\sqrt{\mu^2+|\nabla w_r|^2}\dx
+(1-r)\|D^su\|(\overline\Omega)
+\int_{\Omega}\big(\tfrac\lambda2|w_r|^2+(1-r)f(u-u_o)\big)\dx.
\end{align*}
Rearranging the terms and dividing by $r>0$, we deduce
\begin{align*}
&\|D^su\|(\overline\Omega)+\int_{\Omega}f(u-u_o)\dx\\
&\qquad\le
\frac1r\int_{\Omega}\big(\sqrt{\mu^2+|\nabla
w_r|^2}-\sqrt{\mu^2+|\nabla u|^2}\big)\dx
+
\frac\lambda2\frac1r\int_{\Omega}\big(|w_r|^2-|u|^2\big)\dx.
\end{align*}
Passing to the limit $r\downarrow0$, we have
\begin{align*}
&\|D^su\|(\overline\Omega)+\int_{\Omega}f(u-u_o)\dx\\
&\qquad\le
\int_{\Omega}\frac{\partial}{\partial r}\Big|_{r=0}\sqrt{\mu^2+|\nabla
w_r|^2}\dx
+
\frac\lambda2\int_{\Omega}
\frac{\partial}{\partial r}\Big|_{r=0}|w_r|^2\dx\\
&\qquad=
\int_{\Omega}\frac{\nabla u}{\sqrt{\mu^2+|\nabla
u|^2}}\cdot(\nabla u_o-\nabla u)\dx
+
\lambda\int_\Omega u(u_o-u)\dx.
\end{align*}
Here, it is legitimate to differentiate under the integral because
the derivative of the integrands are dominated by $|\nabla u_o-\nabla
u|\in L^1(\Omega)$ and $2(|u|+|u_o|)|u_o-u|\in L^1(\Omega)$,
respectively.
By Young's inequality,
\begin{align*}
\lambda\int_\Omega u(u_o-u)\dx
\le
\tfrac\lambda2\int_\Omega \big(|u_o|^2-|u|^2\big)\dx.
\end{align*}
Combining the two preceding estimates and
recalling the definition of $z$, we arrive at
\begin{align}\label{euler-area-1}
&\int_{\Omega}\frac{|\nabla u|^2}{\sqrt{\mu^2+|\nabla
u|^2}}\dx+\|D^su\|(\overline\Omega)+\int_{\Omega}\big(\tfrac\lambda2|u|^2+f(u-u_o)\big)\dx\\\nonumber
&\qquad\qquad\le
\int_{\Omega} z\cdot\nabla u_o\dx
+\tfrac\lambda2\int_\Omega |u_o|^2\dx.
\end{align}
For the first integrand on the left-hand side, we have the identity
\begin{align*}
\frac{|\nabla u|^2}{\sqrt{\mu^2+|\nabla u|^2}}
=
\sqrt{\mu^2+|\nabla u|^2}-\mu\sqrt{1-|z|^2}.
\end{align*}
This corresponds to the equality case in~\eqref{Fenchel-pointwise}. Thus~\eqref{euler-area-1} can be rewritten as
\begin{align}\label{euler-area-2}
&\int_{\Omega}\sqrt{\mu^2+|\nabla u|^2}\dx+\|D^su\|(\overline\Omega)
+\int_{\Omega}\big(\tfrac\lambda2|u|^2+f(u-u_o)\big)\dx\\\nonumber
&\qquad\le
\int_{\Omega} z\cdot\nabla u_o\dx
+
\tfrac\lambda2\int_\Omega |u_o|^2\dx
+
\mu\int_\Omega\sqrt{1-|z|^2}\dx,
\end{align}
which is the asserted estimate \eqref{dual-problem-area}.
\end{proof}
\subsection{A duality result for the total variation functional} \label{sec:duality-TV}
The following duality result will be applied in the proof of Theorem~\ref{thm:main}. More general duality results for problems with linear growth have been established in \cite{Beck-Schmidt:2015}. Here, we give a simple proof for a special case. The argument applies an approximation process given by Theorem~\ref{thm:area-duality}.
\begin{theorem}\label{thm:TV-duality}
Let $f\in W^{-1,\infty}(\Omega)\cap L^2(\Omega)$ with
$\|f\|_{W^{-1,\infty}}\le 1$ and $u_o\in W^{1,1}(\Omega^\ast)\cap
L^2(\Omega^\ast)$. Then we have
\begin{align}\label{dual-problem-TV}
\inf_{u\in\BV_{u_o}(\Omega)\cap L^2(\Omega^\ast)}
\bigg(\|Du\|(\overline\Omega)+\int_{\Omega}f(u-u_o)\dx\bigg)
=
\max_{z\in S^\infty_f(\Omega)}\int_\Omega z\cdot\nabla u_o\dx,
\end{align}
where $S^\infty_f(\Omega)=\{z\in
L^\infty(\Omega,{\mathds{R}}^n)\colon \|z\|_{L^\infty}\le1\ \mbox{and}\ \Div z=f\}$. \end{theorem}
\begin{proof} Let $u\in\BV_{u_o}(\Omega)\cap L^2(\Omega^\ast)$ and $z\in S^\infty_f(\Omega)$.
By Lemma~\ref{lem:comparison-pairings} and~\eqref{Anzellotti-bound} we have
\begin{align*}
\int_\Omega z\cdot\nabla u_o\dx
=
(z,Du)_{u_o}(\overline\Omega)+\int_\Omega f(u-u_o)\dx
\le
\|Du\|(\overline\Omega)+\int_\Omega f(u-u_o)\dx.
\end{align*}
Taking the supremum on the left-hand side and the infimum on the
right, we infer
\begin{align}\label{dual-problem-easy-estimate}
\sup_{z\in S^\infty_f(\Omega)}\int_\Omega z\cdot\nabla u_o\dx
\le
\inf_{u\in\BV_{u_o}(\Omega)\cap L^2(\Omega^\ast)}
\bigg(\|Du\|(\overline\Omega)+\int_{\Omega}f(u-u_o)\dx\bigg).
\end{align}
In order to conclude the opposite inequality,
we construct a minimizer of the functional
\begin{align*}
\Psi_\mu(u):=\mathcal{A}_{u_o}^{(\mu)}(u)+\frac\mu2\int_\Omega |u|^2\dx+\int_\Omega f_\mu(u-u_o)\dx
\end{align*}
in the space $\BV_{u_o}(\Omega)\cap\L^2(\Omega^\ast)$ for
$\mu\in(0,1)$, where $f_\mu:=(1-\mu)f$.
Since every $u\in\BV_{u_o}(\Omega)\cap\L^2(\Omega^\ast)$ can be
strictly approximated by functions $u_i\in (u_o+W^{1,1}_0(\Omega))\cap
L^2(\Omega)$, $i\in{\mathds{N}}$, in the sense of Lemma~\ref{lem:interior-approximation},
we have
\begin{align*}
\bigg|\int_\Omega f_\mu(u-u_o)\dx\bigg|
&=
\lim_{i\to\infty}\bigg|\int_\Omega f_\mu(u_i-u_o)\dx\bigg|\\[0.7ex]
&\le
\lim_{i\to\infty}\|f_\mu\|_{W^{-1,\infty}}\|\nabla u_i-\nabla u_o\|_{L^1(\Omega)}\\[0.7ex]
&\le
(1-\mu)\big(\|Du\|(\overline\Omega)+\|\nabla u_o\|_{L^1(\Omega)}\big)
\end{align*}
for every $u\in\BV_{u_o}(\Omega)\cap\L^2(\Omega^\ast)$. This implies the lower
bound
\begin{align}
\label{coercive}
\Psi_\mu(u)
&\ge
\|Du\|(\overline\Omega)+\tfrac\mu2\|u\|_{L^2(\Omega)}^2-\bigg|\int_\Omega
f_\mu(u-u_o)\dx\bigg|\\\nonumber
&\ge
\mu\|Du\|(\overline\Omega)
+
\tfrac\mu2\|u\|_{L^2(\Omega)}^2
-
(1-\mu)\|\nabla u_o\|_{L^1(\Omega)}.
\end{align}
We deduce that $\Psi_\mu$ is coercive on the space
$\BV_{u_o}(\Omega)\cap L^2(\Omega^\ast)$. Since, moreover, $\mathcal{A}_{u_o}^{(\mu)}$ is
convex, the direct method of the calculus of variations yields the
existence of a minimizer $u_\mu\in\BV_{u_o}(\Omega)\cap
L^2(\Omega^\ast)$ of $\Psi_\mu$.
We define the vector field
\begin{equation*}
z_\mu=\frac{\nabla u_\mu}{\sqrt{\mu^2+|\nabla u_\mu|^2}},
\end{equation*}
where $\nabla u_\mu$ denotes the Lebesgue density of the absolutely
continuous part of $Du_\mu$.
Theorem~\ref{thm:area-duality} with $\lambda=\mu$ and $f$ replaced
by $f_\mu$ implies that
\begin{equation}\label{Div-z-mu}
\Div z_\mu =\mu u_\mu + f_\mu
\qquad\mbox{in }\Omega
\end{equation}
in the sense of distributions, as well as the estimate
\begin{align*}
&\mathcal{A}_{u_o}^{(\mu)}(u_\mu)
+\int_{\Omega}\big(\tfrac\mu2|u_\mu|^2+f_\mu(u_\mu-u_o)\big)\dx\\\nonumber
&\qquad\le
\int_{\Omega} z_\mu\cdot\nabla u_o\dx
+
\tfrac\mu2\int_\Omega |u_o|^2\dx
+\mu\int_\Omega\sqrt{1-|z_\mu|^2}\dx.
\end{align*}
Since
\[
\mathcal{A}_{u_o}^{(\mu)}(u_\mu)\ge\|Du_\mu\|(\overline\Omega)\ge(1-\mu)\|Du_\mu\|(\overline\Omega),
\]
the left-hand
side can be bounded from below in terms of the infimum
in~\eqref{dual-problem-TV}. More precisely, we have
\begin{align}
\label{upper-bound-inf-1}
&(1-\mu)\inf_{u\in\BV_{u_o}(\Omega)\cap L^2(\Omega^\ast)}
\bigg(\|Du\|(\overline\Omega)+\int_{\Omega}f(u-u_o)\dx\bigg)\\\nonumber
&\qquad\le
\int_{\Omega} z_\mu\cdot\nabla u_o\dx
+
\tfrac\mu2\int_\Omega |u_o|^2\dx
+
\mu\int_\Omega\sqrt{1-|z_\mu|^2}\dx
\end{align}
for every $\mu\in(0,1)$. Since $\|z_\mu\|_{L^\infty}\le1$ for
every $\mu\in(0,1)$, we can find a sequence $\mu_i\downarrow0$ and a
limit vector field $z_\ast\in L^\infty(\Omega,{\mathds{R}}^n)$ with
\begin{equation}\label{weak-star-z-mu}
z_{\mu_i}\wsto z_\ast
\qquad\mbox{weakly$\ast$ in $L^\infty(\Omega,{\mathds{R}}^n)$, as $i\to\infty$.}
\end{equation}
Using estimate~\eqref{coercive} and the minimality of $u_\mu$, we
infer the bound
\begin{align*}
\tfrac{\mu_i}2\|u_{\mu_i}\|_{L^2(\Omega)}^2
&\le
\Psi_{\mu_i}(u_{\mu_i})+(1-\mu_i)\|\nabla u_o\|_{L^1(\Omega)}\\
&\le
\Psi_{\mu_i}(u_o)+\|\nabla u_o\|_{L^1(\Omega)}\\
&\le
\int_\Omega\Big(\sqrt{1+|\nabla u_o|^2}+|u_o|^2+|\nabla u_o|\Big)\dx.
\end{align*}
This implies that the sequence of functions $\sqrt{\mu_i}u_{\mu_i}$, $i\in{\mathds{N}}$, is bounded
in $L^2(\Omega)$, and we get
\begin{equation*}
\mu_i\int_\Omega u_{\mu_i}\varphi\dx
\le \mu_i\|u_{\mu_i}\|_{L^2(\Omega)}\|\varphi\|_{L^2(\Omega)}
\to0
\end{equation*}
as $i\to\infty$, for every $\varphi\in
C^\infty_0(\Omega)$.
We use this together with the convergence~\eqref{weak-star-z-mu} to
pass to the limit in~\eqref{Div-z-mu}, which implies
$\Div z_\ast=f$ in $\Omega$, in the sense of distributions. Since
\[
\|z_\ast\|_{L^\infty}\le\liminf_{i\to\infty}\|z_{\mu_i}\|_{L^\infty}\le1,
\]
we infer $z_\ast\in S^\infty_f(\Omega)$.
Next, we use the convergence~\eqref{weak-star-z-mu}
to pass to the limit $i\to\infty$
in~\eqref{upper-bound-inf-1} and arrive at
\begin{align*}
&\inf_{u\in\BV_{u_o}(\Omega)\cap L^2(\Omega^\ast)}
\bigg(\|Du\|(\overline\Omega)+\int_{\Omega}f(u-u_o)\dx\bigg)\\
&\qquad\qquad\le
\int_\Omega z_\ast\cdot\nabla u_o\dx
\le
\sup_{z\in S^\infty_f(\Omega)}\int_\Omega z\cdot\nabla u_o\dx\\
&\qquad\qquad\le
\inf_{u\in\BV_{u_o}(\Omega)\cap L^2(\Omega)}
\bigg(\|Du\|(\overline\Omega)+\int_{\Omega}f(u-u_o)\dx\bigg).
\end{align*}
For the last inequality, we recall~\eqref{dual-problem-easy-estimate}.
We conclude that we have an equality throughout, and in
particular, the supremum above is attained. This completes the proof
of~\eqref{dual-problem-TV}. \end{proof}
\subsection{Variational solutions are weak solutions} \label{sec:varisweak}
In this subsection, we complete the proof of Theorem~\ref{thm:main}.
To this end, assume that
$u\in \Lw1(0,T;\BV_{u_o}(\Omega))\cap
C^0([0,T];L^2(\Omega))$ is a variational solution
of~\eqref{Cauchy-Dirichlet}
with $\partial_tu\in L^2(\Omega_T)$.
Lemma~\ref{lem:var-sol-slicewise} implies that the initial values
are attained in the sense $u(0)=u_o(0)$ and
that the slicewise variational inequality
\begin{equation}\label{var-ineq-TV-slicewise}
\|Du(t)\|(\overline\Omega)
\le
\int_{\Omega\times\{t\}}\partial_tu(v-u)\dx
+\|Dv\|(\overline\Omega)
\end{equation}
holds true for every $v\in\BV_{u_o(t)}(\Omega)\cap
L^2(\Omega^\ast)$ and a.e. $t\in(0,T)$.
For the proof of~\eqref{mazon-TV}, we begin with the observation that the variational inequality~\eqref{var-ineq-TV-slicewise}
implies $\partial_tu\in \Lw\infty(0,T; W^{-1,\infty}(\Omega))$ with
\begin{equation}\label{dtu-dual-est}
\esssup_{t\in(0,T)}\|\partial_tu(t)\|_{W^{-1,\infty}(\Omega)}\le 1.
\end{equation}
In order to prove this claim, we consider an arbitrary $\varphi\in
L^1(0,T;W^{1,1}_0(\Omega))$ and use $v=u(t)-\varphi(t)$
as a comparison function in
the variational inequality~\eqref{var-ineq-TV-slicewise}. After integrating
over $t\in(0,T)$, we obtain the bound
\begin{align*}
\int\hspace{-0.6em}\int_{\Omega_T}\partial_tu\varphi\dxt
&\le
\int_0^T\Big(\big\|Du(t)-D\varphi(t)\big\|(\overline\Omega)-\|Du(t)\|(\overline\Omega)\Big)\dt\\
&\le
\int\hspace{-0.6em}\int_{\Omega_T}|\nabla\varphi|\dxt.
\end{align*}
The preceding estimate implies
$$
\partial_tu\in \big[L^1(0,T;W^{1,1}_0(\Omega))\big]'=
\Lw\infty(0,T;W^{-1,\infty}(\Omega)),
$$
together with~\eqref{dtu-dual-est}.
Next, we note that the variational inequality
\eqref{var-ineq-TV-slicewise} corresponds to a minimization
property of
$u(t)\in\BV_{u_o(t)}(\Omega)\cap
L^2(\Omega^\ast)$. More precisely, for a.e. $t\in(0,T)$, the
function $u(t)$ is a minimizer of the functional
\begin{equation*}
\Psi(v)=\|Dv\|(\overline\Omega)+\int_{\Omega\times\{t\}}\partial_tu(v-u_o)\dx
\end{equation*}
in the space $\BV_{u_o(t)}(\Omega)\cap L^2(\Omega^\ast)$.
In view of~\eqref{dtu-dual-est}, Theorem~\ref{thm:TV-duality}
is applicable with the choice
$f=\partial_tu(t)\in W^{-1,\infty}(\Omega)\cap L^2(\Omega)$, for
a.e. $t\in(0,T)$.
Theorem~\ref{thm:TV-duality} implies
\begin{equation}\label{duality-slicewise}
\|Du(t)\|(\overline\Omega)
+\int_{\Omega\times\{t\}}\partial_tu(u-u_o)\dx
=
\max_{\tilde z\in S^\infty_{\partial_tu(t)}(\Omega)}\,
\int_{\Omega\times\{t\}} \tilde z\cdot\nabla u_o\dx
\end{equation}
for a.e. $t\in(0,T)$. Our next goal is to show that there is a
vector field $z\in L^\infty(\Omega_T,{\mathds{R}}^n)$ such that $z(t)\in
L^\infty(\Omega,{\mathds{R}}^n)$
realizes the maximum in~\eqref{duality-slicewise} for a.e. $t\in(0,T)$.
By definition of the maximum, for a.e. $t\in(0,T)$ we can choose
a vector field $z_\ast(t)\in L^\infty(\Omega,{\mathds{R}}^n)$ with
$\|z_\ast(t)\|_{L^\infty(\Omega)}\le1$ and
$\Div z_\ast(t)=\partial_tu(t)$ in $\Omega$, such that
\begin{equation}
\label{choice-z}
\int_{\Omega\times\{t\}} z_\ast\cdot\nabla u_o\dx
=
\max_{\tilde z\in S^\infty_{\partial_tu(t)}(\Omega)}\,
\int_{\Omega\times\{t\}} \tilde z\cdot\nabla u_o\dx.
\end{equation}
However, at this stage we can not rule out the possibility that
$t\mapsto z_\ast(t)$ is not measurable.
In this case, we need to replace $z_\ast$
by a measurable vector field $z\in L^\infty(\Omega_T,{\mathds{R}}^n)$. To
show the existence of such a vector field,
we identify elements in $L^\infty(\Omega_T,{\mathds{R}}^n)$ with bounded linear
functionals on $L^1(\Omega_T,{\mathds{R}}^n)$.
Let us consider the subspace
\begin{equation*}
\mathcal{W}=\big\{ r\nabla u_o+\nabla\varphi\ \colon\,
r\in{\mathds{R}},\
\varphi\in C^\infty_0(\Omega_T)\big\}\subset L^1(\Omega_T,{\mathds{R}}^n).
\end{equation*}
For a given element $V\in\mathcal{W}$, we choose $r\in{\mathds{R}}$ and
$\varphi\in C^\infty_0(\Omega_T)$ with $V=r\nabla
u_o+\nabla\varphi$. For a.e. $t\in(0,T)$, we have
\begin{align*}
\int_{\Omega\times\{t\}} z_\ast\cdot V\dx
&=
r\int_{\Omega\times\{t\}} z_\ast\cdot \nabla u_o\dx
+
\int_{\Omega\times\{t\}} z_\ast\cdot \nabla\varphi\dx
\\
&=
r\max_{\tilde z\in S^\infty_{\partial_tu(t)}(\Omega)}\,
\int_{\Omega\times\{t\}} \tilde z\cdot\nabla u_o\dx
-
\int_{\Omega\times\{t\}} \partial_tu\,\varphi\dx.
\end{align*}
For the last identity, we used the choice of $z_\ast(t)$ according
to \eqref{choice-z} and the fact that $\Div z_\ast(t)=\partial_tu(t)$.
We observe that the maximum in the last line
depends measurably on time because it coincides with a measurable
function by~\eqref{duality-slicewise},
cf. Lemma~\ref{lem:TV-measurable}.
We conclude that the
left-hand side of the preceding formula depends measurably on time
as well. Moreover, because of $\|z_\ast(t)\|_{L^\infty}\le 1$
for a.e. $t\in(0,T)$, we have
\begin{equation}\label{bound-z-star-slicewise}
\bigg|\int_{\Omega\times\{t\}} z_\ast\cdot V\dx\bigg|
\le
\int_{\Omega\times\{t\}}|V|\dx
\end{equation}
for a.e. $t\in(0,T)$.
Therefore, we may define a linear functional
\begin{equation*}
\ell: \mathcal{W}\to{\mathds{R}},
\quad
V\mapsto \int_0^T\bigg(\int_{\Omega\times\{t\}}z_\ast\cdot V\dx\bigg)\dt.
\end{equation*}
Integrating estimate~\eqref{bound-z-star-slicewise} over time,
we infer the bound
\begin{equation*}
|\ell(V)|\le \|V\|_{L^1(\Omega_T)}
\quad\mbox{for every $V\in\mathcal{W}$.}
\end{equation*}
This means that $\ell$ is a bounded linear functional on
$\mathcal{W}$ with
$\|\ell\|_{\mathcal{W}'}\le1$. By the Hahn-Banach theorem, there
exists an extension $L\in [L^1(\Omega_T,{\mathds{R}}^n)]'$ with
$L|_{\mathcal{W}}=\ell$ and
\begin{equation*}
\|L\|_{[L^1(\Omega_T,{\mathds{R}}^n)]'}=\|\ell\|_{\mathcal{W}'}\le 1.
\end{equation*}
The Riesz representation theorem yields a vector field $z\in
L^\infty(\Omega_T,{\mathds{R}}^n)$ with
\begin{equation}\label{bound-z-one}
\|z\|_{L^\infty(\Omega_T)}=\|L\|_{[L^1(\Omega,{\mathds{R}}^n)]'}\le1
\end{equation}
and
\begin{equation}\label{Riesz}
\int\hspace{-0.6em}\int_{\Omega_T}z\cdot V\dxt
=
L(V)
=
\int_0^T\bigg(\int_{\Omega\times\{t\}}z_\ast\cdot V\dx\bigg)\dt
\end{equation}
for every $V\in\mathcal{W}$. We exploit this identity in two
ways. First, we choose $V=\nabla\varphi\in\mathcal{W}$, where
$\varphi\in C^\infty_0(\Omega_T)$, and deduce
\begin{equation*}
\int\hspace{-0.6em}\int_{\Omega_T}z\cdot\nabla\varphi\dxt
=
\int_0^T\bigg(\int_{\Omega\times\{t\}}z_\ast\cdot \nabla\varphi\dx\bigg)\dt
=
-\int\hspace{-0.6em}\int_{\Omega_T}\partial_tu \,\varphi\dxt,
\end{equation*}
which means $\Div z=\partial_tu$ in $\Omega_T$, in the sense of
distributions. In view of~\eqref{bound-z-one}, we infer $z(t)\in
S^\infty_{\partial_tu(t)}(\Omega)$ for a.e. $t\in(0,T)$.
Second, the choice $V=\nabla u_o\in\mathcal{W}$
in~\eqref{Riesz} implies
\begin{align*}
\int\hspace{-0.6em}\int_{\Omega_T}z\cdot \nabla u_o\dxt
&=
\int_0^T\bigg(\int_{\Omega\times\{t\}}z_\ast\cdot \nabla
u_o\dx\bigg)\dt\\
&=
\int_0^T\bigg(\max_{\tilde z\in S^\infty_{\partial_tu(t)}(\Omega)}\,
\int_{\Omega\times\{t\}} \tilde z\cdot\nabla u_o\dx\bigg)\dt\\
&\ge
\int\hspace{-0.6em}\int_{\Omega_T}z\cdot \nabla u_o\dxt,
\end{align*}
where we used~\eqref{choice-z} and the fact $z(t)\in
S^\infty_{\partial_tu(t)}(\Omega)$ for a.e. $t\in(0,T)$.
We deduce that the last inequality must be an identity, which implies
\begin{align*}
\int_{\Omega\times\{t\}}z\cdot \nabla u_o\dx
=
\max_{\tilde z\in S^\infty_{\partial_tu(t)}(\Omega)}\,
\int_{\Omega\times\{t\}} \tilde z\cdot\nabla u_o\dx
\end{align*}
for a.e. $t\in(0,T)$. Hence, we have found the desired vector field
$z\in L^\infty(\Omega_T,{\mathds{R}}^n)$ such that $z(t)\in
S^\infty_{\partial_tu(t)}(\Omega)$ realizes the
maximum in~\eqref{duality-slicewise} for
a.e. $t\in(0,T)$. Equation~\eqref{duality-slicewise} implies the identity
\begin{align*}
\|Du(t)\|(\overline\Omega)
+\int_{\Omega\times\{t\}}\partial_tu(u-u_o)\dx
=
\int_{\Omega\times\{t\}} z\cdot\nabla u_o\dx
\end{align*}
for a.e. $t\in(0,T)$.
For the proof of~\eqref{mazon-TV},
it remains to replace $u_o$ by an arbitrary
$v\in\BV_{u_o(t)}(\Omega)\cap L^2(\Omega^\ast)$ in the preceding identity.
This can be done with the help of
Lemma~\ref{lem:comparison-pairings}. Since
$\Div z(t)=\partial_tu(t)$, Lemma~\ref{lem:comparison-pairings} implies
\begin{equation*}
\int_{\Omega\times\{t\}} z\cdot\nabla u_o\dx
=
(z(t),Dv)_{u_o}(\overline\Omega)
+
\int_{\Omega\times\{t\}} \partial_tu(v-u_o)\dx.
\end{equation*}
Combining the two preceding identities, we arrive at
\begin{align*}
\|Du(t)\|(\overline\Omega)
+\int_{\Omega\times\{t\}}\partial_tu(u-v)\dx
=
(z(t),Dv)_{u_o}(\overline\Omega)
\end{align*}
for a.e. $t\in(0,T)$ and every
$v\in\BV_{u_o(t)}(\Omega)\cap L^2(\Omega^\ast)$, which is
the assertion~\eqref{mazon-TV}.
Therefore, the function $u$ is a weak solution
of~\eqref{Cauchy-Dirichlet} in the sense of Definition~\ref{def:Mazon-solution}. This completes
the proof of Theorem~\ref{thm:main}.
\noindent Juha Kinnunen, Aalto University, Department of Mathematics, P.O. Box 11100, FI-00076 Aalto, Finland. Email: [email protected]
\noindent Christoph Scheven, Fakult\"at f\"ur Mathematik, Universit\"at Duisburg-Essen, 45117 Essen, Germany. Email: [email protected]
\noindent KEY WORDS AND PHRASES: Parabolic variational integral, total variation flow, Cauchy-Dirichlet problem.
\noindent 2010 MATHEMATICS SUBJECT CLASSIFICATION: 35K67, 35K20, 35K92.
\end{document}
|
arXiv
|
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Gaussian-state theory of two-photon imaging} \date{\today} \author{Baris I. Erkmen} \email{[email protected]} \author{Jeffrey H. Shapiro} \affiliation{Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA}
\begin{abstract} Biphoton states of signal and idler fields---obtained from spontaneous parametric downconversion (SPDC) in the low-brightness, low-flux regime---have been utilized in several quantum imaging configurations to exceed the resolution performance of conventional imagers that employ coherent-state or thermal light. Recent work---using the full Gaussian-state description of SPDC---has shown that the same resolution performance seen in quantum optical coherence tomography and the same imaging characteristics found in quantum ghost imaging can be realized by classical-state imagers that make use of phase-sensitive cross correlations. This paper extends the Gaussian-state analysis to two additional biphoton-state quantum imaging scenarios: far field diffraction-pattern imaging; and broadband thin-lens imaging. It is shown that the spatial resolution behavior in both cases is controlled by the nonzero phase-sensitive cross correlation between the signal and idler fields. Thus, the same resolution can be achieved in these two configurations with classical-state signal and idler fields possessing a nonzero phase-sensitive cross correlation. \end{abstract} \pacs{42.30.Va, 42.50.Ar, 42.50.Dv} \maketitle
\section{Introduction}
Spontaneous parametric downconversion (SPDC), with a continuous-wave nondepleting pump, produces signal and idler fields that are in a maximally-entangled, zero-mean jointly Gaussian state \cite{Shapiro:Gaussian,Brambilla:SPDC}. When the downconverter is operated in its low-brightness, low-flux regime, so that at most one signal-idler photon pair is emitted during an observation interval, this state reduces to the superposition of a predominant multimode vacuum state plus a weak biphoton component \cite{Kim}. Biphoton illumination is at the heart of several quantum imaging configurations, including quantum optical coherence tomography (OCT) \cite{Abouraddy}, quantum ghost imaging \cite{Pittman}, quantum holography \cite{Abouraddy:Holography} and quantum lithography \cite{Dangelo}. These systems offer performance advantages over conventional optical imagers, which employ coherent-state or thermal-state sources, that have traditionally been ascribed to the entanglement between the biphoton's signal and idler components. We have shown---using Gaussian-state analysis and phase-sensitive coherence theory---that the advantages in quantum OCT and ghost imaging predominantly stem from the phase-sensitive cross correlation between the signal and idler fields, rather than their entanglement {\it per se} \cite{ErkmenShapiro:PCOCT,ErkmenShapiro:GhostImaging2}. Furthermore, because a pair of classical-state fields \cite{classicalstates} can also have nonzero phase-sensitive cross correlation, most of the advantages seen in these biphoton-state imagers are also attainable with {\em classical} phase-sensitive sources, but conceivably (and conveniently) at much higher photon flux and without the need for single-photon counters.
In this paper we will study Fourier-plane and thin-lens imaging of a transmission mask using jointly-Gaussian source states. These states encompass the biphoton state, thermal states (used in conventional low-coherence imaging) and coherent states (used in conventional coherent imaging). Previous theoretical and experimental work has shown that biphoton-state illumination of a transmission mask yields a far-field diffraction (Fourier-plane) pattern that is factor-of-two compressed relative to that produced by coherent plane-wave illumination of the mask \cite{Dangelo}. In addition, it has been claimed that imaging a transmission mask using broadband, spatially-incoherent biphoton-state illumination and a finite-diameter thin lens will yield a point-spread function that is a factor of two narrower than that obtained with quasimonochromatic spatially-incoherent thermal-state illumination \cite{Shih:QI}. As noted above for quantum OCT and ghost imaging, these biphoton-state resolution enhancements in Fourier-plane and thin-lens imaging have been ascribed to signal-idler entanglement. We will use Gaussian-state analysis, together with phase-sensitive coherence theory, to develop a unified---and generalized---understanding of the classical and quantum regimes of these imaging configurations. In particular, we will show that, once again, the benefits ascribed to entanglement are in fact due to phase-sensitive coherence at the source, and therefore are obtainable with classical-state sources with phase-sensitive coherence. Furthermore, we will show that the narrowing of the point-spread function in the thin-lens imaging configuration is much less than a factor of two even at the theoretical upper limit of SPDC bandwidth, and that it is marginal for typical SPDC bandwidths.
The remainder of this paper is organized as follows. In Sec.~\ref{CH6:CoherenceTheory} we present the phase-insensitive and phase-sensitive coherence theory results that are relevant to Fourier-plane imaging. In Sec.~\ref{CH6:FarFieldDiffraction} we use the Sec.~\ref{CH6:CoherenceTheory} results for quasimonchromatic light to determine the far-field diffraction properties of Gaussian-state source fields after they have illuminated a source-plane transmission mask. Section~\ref{CH6:BroadbandImaging} presents our analysis of thin-lens imaging, in which we determine the effect of the source's bandwidth on the point-spread functions resulting from phase-insensitive and phase-sensitive Gaussian-state illumination. In doing so we pay particular attention to frequency-dependent propagation effects that did not enter into our quasimonochromatic treatment of Fourier-plane imaging. Finally, in Sec.~\ref{CH6:discussion}, we summarize the results we have obtained for the Fourier-plane and thin-lens imaging configurations, highlighting the role played by phase-sensitive coherence.
\section{Second-order coherence propagation} \label{CH6:CoherenceTheory}
The imaging configurations we shall consider later acquire the far-field diffraction pattern or the transverse image of a transmission mask placed at the source's output plane. In both cases linear optical elements---a transmission mask, a lens, and a polarizing beam splitter---plus free-space propagation lie between the source and image-acquisition planes. Because the images are acquired via photocurrent correlation, their properties are determined by fourth-order correlations of the detected fields. To determine these correlation functions, it is necessary to propagate the fourth-order correlation function from the source plane to the appropriate image-acquisition planes. Fortunately, zero-mean Gaussian source states are completely determined by their phase-insensitive and phase-sensitive, second-order auto- and cross-correlation functions. Therefore, we need only consider second-order coherence transfer.
Let $\hat{E}_{z}(\rhovec,t) e^{-i \omega_{0}t}$ denote a scalar, $z$-propagating, positive-frequency field operator with center-frequency $\omega_{0}$, and $\sqrt{\text{photons}/\text{m}^2 \text{s}}$ units. The commutators for the baseband envelope at a fixed transverse plane are given by \cite{Yuen:Part_I} \begin{align} [ \hat{E}_{z}(\rhovec_{1},t_{1}),\hat{E}_{z}(\rhovec_{2},t_{2})] & =0 \label{commutator1}\\[5pt] [ \hat{E}_{z}(\rhovec_{1},t_{1}),\hat{E}_{z}^{\dagger}(\rhovec_{2},t_{2})] & =\delta(\rhovec_{2} - \rhovec_{1}) \delta(t_{2} - t_{1})\,, \label{commutator2} \end{align} where $\delta(\cdot)$ is the impulse function.
Free-space paraxial propagation is governed by the Huygens-Fresnel principle \cite{Yuen:Part_I}, which states that the baseband field operator at $z=L$ is related to the field operator at $z=0$ by the following superposition integral: \begin{eqnarray} \hat{E}_{L}(\rhovec,t) &=& \int \! \frac{d\Omega}{2\pi}\, \int\! d \rhovec'\, e^{-i \Omega t} \hat{\mathcal{E}}_{0}\big(\rhovec',\Omega \big) \nonumber \\[.05in] &\times& h_{L}(\rhovec-\rhovec',\omega_{0}+\Omega) , \label{FS:prop} \end{eqnarray} where \begin{equation} \hat{\mathcal{E}}_{z}(\rhovec,\Omega) \equiv \int \! dt \: \hat{E}_{z} (\rhovec,t) e^{i \Omega t} \label{FTR:FieldOperator} \end{equation} is the Fourier transform of the baseband-envelope field operator and \begin{equation}
h_{L}(\rhovec,\omega) \equiv \frac{\omega}{i 2 \pi cL} e^{i \omega(L+ |\rhovec|^2/2L)/c}\,, \label{HFGreensFunction} \end{equation} is the Huygens-Fresnel Green's function for paraxial diffraction at frequency $\omega$ with $c$ being the vacuum light speed.
For Eqs.~\eqref{FS:prop}--\eqref{HFGreensFunction} to be consistent with Eq.~\eqref{commutator2}, the integrals in Eqs.~\eqref{FS:prop} and \eqref{FTR:FieldOperator} must all be over infinite limits. This poses no difficulty for the spatial integration, which can be taken over the entire $z=0$ plane, but there is a problem with the frequency integration. Because $\hat{E}_z(\rhovec,t)$ is a positive-frequency field operator, the lower limit of integration in Eq.~\eqref{FS:prop} should be $-\omega_0$. But, if we impose this propagation condition we do \em not\/\rm\ preserve the commutator in Eq.~\eqref{commutator2}, i.e., assuming that this commutator applies at $z=0$ and employing the diffraction integral does not recover the delta-function commutator at $z=L$. In almost all quantum optics situations---both theoretical and experimental---we can circumvent this issue as follows. If the source at $z=0$ only excites frequencies that are within some bandwidth $\pm \Omega_0$ about $\omega_0$ and the measurements performed at $z=L$ are only sensitive to frequencies within that bandwidth, then so long as $\Omega_0 < \omega_0$, we can allow the frequency lower limit in Eq.~\eqref{FS:prop} to be $-\infty$. In the quasimonochromatic cases to be considered below the preceding condition will be satisfied. However, we will impose the finite lower limit on the frequency integral when we address thin-lens imaging at the ultimate theoretical limit of broadband SPDC. In this case, the relevant commutators are \begin{align} [ \hat{\mathcal{E}}_{z}(\rhovec_{1},\Omega_{1}),\hat{\mathcal{E}}_{z}(\rhovec_{2},\Omega_{2})] & =0 \label{OmegaCommutator1}\\[5pt] [ \hat{\mathcal{E}}_{z}(\rhovec_{1},\Omega_1),\hat{\mathcal{E}}_{z}^{\dagger}(\rhovec_{2},\Omega_{2})] & =2\pi \delta(\rhovec_{2} - \rhovec_{1}) \delta(\Omega_{2} - \Omega_{1})\,. \label{OmegaCommutator2} \end{align}
The non-Hermitian baseband field operator $\hat{E}_{z}(\rhovec,t)$ has two second-order correlation functions, namely the (normally-ordered) phase-insensitive correlation function \begin{equation} K^{(n)}_{z} (\rhovec_{1},\rhovec_{2}, t_{2}-t_{1}) \equiv \langle \hat{E}_{z}^{\dagger}(\rhovec_1,t_1) \hat{E}_{z}(\rhovec_2,t_2) \rangle\,, \label{PIScorr} \end{equation} and the phase-sensitive correlation function \begin{equation} K^{(p)}_{z} (\rhovec_{1},\rhovec_{2}, t_{2}-t_{1}) \equiv \langle \hat{E}_{z}(\rhovec_1,t_1) \hat{E}_{z}(\rhovec_2,t_2) \rangle\,, \label{PScorr} \end{equation} in which we have assumed, for simplicity, that the baseband field operator is in a complex-stationary state, viz., the correlation functions depend on the time difference $t_{2}-t_{1}$, but not on the absolute times.
We will find it convenient and insightful to work with the frequency spectra associated with Eqs.~\eqref{PIScorr} and \eqref{PScorr}, defined as the Fourier transforms, \begin{equation} S_{z}^{(x)} (\rhovec_{1}, \rhovec_{2}, \Omega) \equiv \int\!d\tau \: K_{z}^{(x)}(\rhovec_{1}, \rhovec_{2}, \tau) e^{i \Omega \tau}\,, \label{CH6:Spectra} \end{equation} for $x = n,p $. The phase-insensitive and phase-sensitive correlation spectra at $z=L$ can be expressed in terms of the correlation spectra at $z=0$ by evaluating Eq.~\eqref{CH6:Spectra} for the propagated field operators, via Eq.~\eqref{FS:prop}, which yields the following phase-insensitive spectrum, \begin{eqnarray} \lefteqn{S_{L}^{(n)} (\rhovec_{1}, \rhovec_{2}, \Omega) = \iint d\rhovec_{1}'\, d\rhovec_{2}' \, S_{0}^{(n)} (\rhovec_{1}', \rhovec_{2}', \Omega)} \nonumber \\[5pt] &\times& h_{L}^{*}(\rhovec_{1}-\rhovec_{1}',\omega_{0} + \Omega) h_{L}(\rhovec_{2}-\rhovec_{2}',\omega_{0} + \Omega)\,, \label{PIS:spectrum} \end{eqnarray} and the following phase-sensitive spectrum, \begin{eqnarray} \lefteqn{S_{L}^{(p)} (\rhovec_{1}, \rhovec_{2}, \Omega) = \iint d\rhovec_{1}'\, d\rhovec_{2}' \, S_{0}^{(p)} (\rhovec_{1}', \rhovec_{2}', \Omega)} \nonumber \\[5pt] &\times& h_{L}(\rhovec_{1}-\rhovec_{1}',\omega_{0} - \Omega) h_{L}(\rhovec_{2}-\rhovec_{2}',\omega_{0} + \Omega)\,, \label{PS:spectrum} \end{eqnarray} where $*$ in Eq.~\eqref{PIS:spectrum} denotes complex conjugation. Note that the phase-insensitive correlation spectrum is a monochromatic equation, i.e., the frequency dependence is $\omega_{0} + \Omega$ on both sides of the equality, whereas the phase-sensitive spectrum is a bichromatic equation involving $\omega_{0} \pm \Omega$. This difference occurs because complex-stationary phase-insensitive correlations in time have uncorrelated frequency components, whereas complex-stationary phase-sensitive correlation functions in time have nonzero (phase-sensitive) correlations between frequency components with equal and opposite detunings from the field's center frequency \cite{ErkmenShapiro:PSCoherenceThy}.
Consider a quasimonochromatic state in which the preceding spectra are only nonzero for $| \Omega | /\omega_{0} \ll 1$. It follows that the Huygens-Fresnel principle simplifies to \begin{equation} \hat{E}_{L}(\rhovec,t) \!=\! \int\! d\rhovec'\, \hat{E}_{0}\big(\rhovec',t-L/c \big) h_{L}(\rhovec-\rhovec',\omega_{0}) , \label{FS:propQM} \end{equation} and the propagation Green's function in Eqs.~\eqref{PIS:spectrum} and \eqref{PS:spectrum} become $h_{L}(\rhovec, \omega_{0} \pm \Omega) \approx h_{L}(\rhovec,\omega_{0})$. Let us now review the far-field propagation regime for this quasimonochromatic situation. We will assume that the field at the $z=0$ plane is in a zero-mean state with cross-spectrally pure, Schell-model correlation spectra \cite{coherence_separability} given by \begin{align} S_{0}^{(n)}(\rhovec_1, \rhovec_2,\Omega) &\!=\! T^{*}\!(\rhovec_1) T(\rhovec_2) G^{(n)}(\rhovec_2\!-\!\rhovec_1) S^{(n)} (\Omega), \label{SM0:PIS} \\[5pt] S_{0}^{(p)}(\rhovec_1, \rhovec_2,\Omega) & \!=\! T(\rhovec_1) T(\rhovec_2) G^{(p)}(\rhovec_2\!-\!\rhovec_1) S^{(p)}(\Omega).\!\label{SM0:PS} \end{align}
With no loss of generality, we require $|T(\rhovec)| \leq 1$, so that it may be regarded as a (possibly complex-valued) spatial attenuation of an optical field operator in a homogenous and stationary state with separable phase-insensitive spectrum $S^{(n)}(\Omega) G^{(n)}(\rhovec_{2}-\rhovec_{1})$ and phase-sensitive spectrum $S^{(p)}(\Omega) G^{(p)}(\rhovec_{2}-\rhovec_{1})$. This spatial attenuation will become the transmission mask to be imaged when we turn our attention to the Fourier-plane and thin-lens imaging configurations. For now, however, it is convenient to lump this mask together with the source.
Our primary interest is in sources with narrow $G^{(x)}_{0}(\rhovec)$, for $x=n,p$, such that $T(\rhovec)$ does not vary appreciably within a (phase-insensitive or phase-sensitive) coherence area. For this case, we may approximate the source correlation spectra as follows: \begin{align}
S_{0}^{(n)}(\rhovec_1, \rhovec_2,\Omega) & \approx \bigl | T(\rhovec_{s} )\bigr |^2 G^{(n)}(\rhovec_d) S^{(n)}(\Omega), \label{SM02:PIS}\\[5pt] S_{0}^{(p)}(\rhovec_1, \rhovec_2,\Omega) &\approx T^{2} ( \rhovec_s ) G^{(p)}(\rhovec_d) S^{(p)}(\Omega) \,,\label{SM02:PS} \end{align} in terms of the sum coordinate $\rhovec_{s} \equiv (\rhovec_{2} + \rhovec_{1}) /2 $ and the difference coordinate $\rhovec_{d} \equiv \rhovec_{2} - \rhovec_{1}$. This approximation simplifies the subsequent analytic treatment considerably, without significant impact on the fundamental physics. The $z=L$ spectra then become \begin{align}
\lefteqn{S_{L}^{(n)}(\rhovec_{1}, \rhovec_{2}, \Omega) = \frac{\omega_{0}^{2} S^{(n)}(\Omega) }{(2 \pi c L)^2} \, e^{i \omega_{0} \rhovec_{s}\cdot \rhovec_{d}/cL}} \nonumber \\[5pt] &&\times \int \!d\rhovec_{s}' \int\! d\rhovec_{d}' \, e^{-i \omega_{0} (\rhovec_{s}\cdot \rhovec_{d}' + \rhovec_{d} \cdot \rhovec_{s}')/cL}\, e^{i \omega_{0} \rhovec_{s}' \cdot \rhovec_{d}'/cL} \nonumber \\[5pt] && \times |T(\rhovec_{s}')|^2 G^{(n)}(\rhovec_{d}'), \label{PIS:CoherencePropagation} \end{align} and \begin{align}
S_{L}^{(p)} (\rhovec_{1}, &\rhovec_{2}, \Omega) = \frac{-\omega_{0}^{2} S^{(p)}(\Omega)}{(2 \pi cL)^2} \, e^{i \omega_{0} \left (2 L^{2} + |\rhovec_{s}|^2+ |\rhovec_{d}|^2/4 \right )/cL} \nonumber \\[5pt] &\times \int \!d\rhovec_{s}' \int \!d\rhovec_{d}' \, e^{-i \omega_{0} (2 \rhovec_{s}\cdot \rhovec_{s}' + \rhovec_{d} \cdot \rhovec_{d}'/2)/cL} \nonumber \\[5pt] & \times e^{i \omega_{0} (|\rhovec_{s}'|^2+ |\rhovec_{d}'|^2/4)/cL} \, T^2(\rhovec_{s}') G^{(p)}(\rhovec_{d}'), \label{PS:CoherencePropagation} \end{align} respectively, where the quasimonochromatic assumption has permitted our approximating the frequency-dependent leading coefficients by their values at the center frequency \cite{ApproxValidation}.
Let $\rho_{0}$ denote the coherence radius of the source, which we shall assume to be the same for both phase-insensitive and phase-sensitive correlations, i.e., $\rho_0$ is the radius within which $G^{(x)}(\rhovec)$, for $x=n,p$, differ appreciably from zero. Also, let $a_{0} \gg \rho_0$ denote the transverse radius of $|T(\rhovec)|^{2}$, which will be the photon-flux density radius of the source's state just after the transmission mask in our imaging configurations. In far field phase-insensitive correlation propagation, which applies when $\omega_{0} a_{0} \rho_{0}/2cL \ll 1$, the phase term $e^{i \omega_{0} \rhovec_{s}' \cdot \rhovec_{d}'/cL}$ can be neglected in Eq.~\eqref{PIS:CoherencePropagation}, and we find that \begin{eqnarray} S_{L}^{(n)} (\rhovec_{1}, \rhovec_{2}, \Omega) &=& \frac{\omega_{0}^{2} S^{(n)}(\Omega)}{(2 \pi cL)^2} e^{i \omega_{0} \rhovec_{s}\cdot \rhovec_{d}/cL} \nonumber \\[5pt] &\times& \mathcal{T}_{n}\!\left (\frac{\omega_{0} \rhovec_{d}}{cL} \right ) \mathcal{G}^{(n)}\!\left (\frac{\omega_{0} \rhovec_{s}}{cL} \right )\,, \label{VCZ:PIS} \end{eqnarray} where $\mathcal{T}_{n}(\kvec)$ and $\mathcal{G}^{(n)}(\kvec)$ are the $2$-D Fourier transforms, \begin{align}
\mathcal{T}_{n}(\kvec) & \equiv \int\!{d}\rhovec' e^{-i \kvec \cdot \rhovec'} |T(\rhovec')|^{2}\,, \label{Tn}\\[5pt] \mathcal{G}^{(n)} (\kvec) &\equiv \int\! {d}\rhovec' e^{-i \kvec \cdot \rhovec'} G^{(n)} (\rhovec')\,. \end{align} The Fourier-transform duality between the source-plane and the far-field phase-insensitive correlation spectra---seen in Eq.~\eqref{VCZ:PIS}---is the well known van Cittert-Zernike theorem for phase-insensitive correlation propagation \cite{Mandel}. A similar duality is present between the source-plane and the far-field phase-sensitive correlation spectra, but the far-field regime---in which the quadratic phase terms of the integrand in Eq.~\eqref{PS:CoherencePropagation} become negligible---corresponds to $\omega_{0} a_{0}^{2}/2cL \ll 1$, which is more stringent than the far-field condition for the phase-insensitive case. In this regime, we find that \begin{eqnarray}
S_{L}^{(p)} (\rhovec_{1}, \rhovec_{2}, \Omega) &=& \frac{-\omega_{0}^{2} S^{(p)}(\Omega)}{(2 \pi cL)^2} \,e^{i \omega_{0} \left ( 2 L^{2} + |\rhovec_{s}|^2+ |\rhovec_{d}|^2/4 \right)/cL} \nonumber \\[5pt] &\times&\mathcal{T}_{p}\left (\frac{2 \omega_{0} \rhovec_{s} }{cL} \right ) \mathcal{G}^{(p)}\! \left ( \frac{\omega_{0} \rhovec_{d}}{2cL} \right ), \label{VCZ:PS} \end{eqnarray} gives the far-field phase-sensitive correlation spectrum, with \begin{equation} \mathcal{T}_{p}(\kvec) \equiv \int\!{d}\rhovec' e^{-i \kvec \cdot \rhovec'} T^{2}(\rhovec') \end{equation} and \begin{equation} \mathcal{G}^{(p)} (\kvec) \equiv \int \!{d}\rhovec' e^{-i \kvec \cdot \rhovec'} G^{(p)} (\rhovec')\,. \end{equation} By analogy to the phase-insensitive case, we refer to the Fourier transform relation in Eq.~\eqref{VCZ:PS} as the van Cittert-Zernike theorem for phase-sensitive coherence propagation.
To conclude our review of far-field coherence propagation it is worth emphasizing the similarities and differences between Eqs.~\eqref{VCZ:PIS} and \eqref{VCZ:PS}. The source-plane transmission mask, $T(\rhovec)$, has been assumed to be a slowly-varying and broad function in comparison to the rapidly decaying $G^{(x)}(\rhovec)$. Thus, for $x = n, p$, Fourier-transform duality implies that $\mathcal{T}_{x}(\kvec)$, decays more rapidly than $\mathcal{G}^{(x)}(\kvec)$. Therefore, the far-field phase-sensitive correlation function consists of a narrow function of $\rhovec_{s}$ multiplying a broad function of $\rhovec_{d}$, whereas the corresponding phase-insensitive correlation function consists of a narrow function of $\rhovec_{d}$ times a broad function of $\rhovec_{s}$. Owing to this difference, point pairs on the transverse plane in the far field that are symmetrically disposed about the origin, viz., points satisfying $|\rhovec_{s}| \approx 0$, have appreciable phase-sensitive correlation. The phase-insensitive correlation, however, is highest between point pairs that are in close proximity on the transverse plane, i.e., point pairs obeying $|\rhovec_{d}| \approx 0$. In addition, if we evaluate the correlations at a single transverse point, i.e., when $\rhovec_{d} = {\bf 0}$, we find that the phase-insensitive correlation traces out the broad envelope $\mathcal{G}^{(n)}(\omega_0\rhovec_s/cL)$, whereas the phase-sensitive correlation traces out the narrow function $\mathcal{T}_{p}(2\omega_0\rhovec_s/cL)$, a property we shall make use of in the following section. Finally, it is relevant to emphasize that Eq.~\eqref{VCZ:PS} is a general property of phase-sensitive coherence propagation that applies regardless of whether the source state is classical or nonclassical \cite{ErkmenShapiro:PSCoherenceThy,ErkmenShapiro:GhostImaging2}.
\section{Two-Photon Diffraction Pattern Imaging} \label{CH6:FarFieldDiffraction}
Consider the experimental setup, shown in Fig.~\ref{TPI:FFdiff}, whose purpose is to obtain the far-field diffraction pattern of a source-plane transmission mask of (possibly complex-valued) field transmissivity $T(\rhovec)$. An experiment using this setup---which we will explain in detail shortly---was reported in \cite{Dangelo} as a proof-of-principle demonstration of biphoton-state quantum lithography. That experiment exhibited a factor-of-two compression in the fringe pattern produced using a two-slit transmission mask as compared to what was obtained with conventional (coherent-state) illumination at the same wavelength of that two-slit mask. The observed fringe-pattern compression was therefore interpreted as the expected result for quantum lithography using a pair of entangled photons \cite{QuantLithFootnote}. Our aim in this section is to show that it is phase-sensitive coherence, not entanglement {\it per se}, that is responsible for this fringe-pattern compression. In particular, we will find that classical phase-sensitive light and biphoton-state light yield \em identical\/\rm\ images, except for the image produced with the classical-state source being embedded in a prominent featureless background that is absent for the case of biphoton illumination. Thus, the fringe-pattern compression previously ascribed to the entangled nature of the signal and idler photons is in fact due to their having a phase-sensitive cross correlation, which is a property that classical states can also possess. Signal-idler entanglement, which in the Gaussian-state framework is a stronger-than-classical phase-sensitive cross correlation between the signal and idler fields, is responsible for dramatically improving the contrast of this image when the illumination is broadband.
\begin{figure}
\caption{(Color online) Imaging the far-field diffraction pattern of a transmission mask. PBS, polarizing beam splitter.}
\label{TPI:FFdiff}
\end{figure}
Now let us flesh out the details of the preceding assertions within the context of the Fig.~\ref{TPI:FFdiff} setup. The source will be taken to be frequency-degenerate type-II phase-matched SPDC with a continuous-wave pump. It generates paraxial, $z$-propagating signal ($S$) and idler ($I$) fields in orthogonal polarizations with common center frequency $\omega_{0}$. Their positive-frequency field operators will be denoted as $\hat{E}_{S}(\rhovec,t)e^{-i \omega_{0}t}$ and $\hat{E}_{I}(\rhovec,t)e^{-i \omega_{0}t}$. With a nondepleting plane-wave pump and ignoring boundary effects due to the nonlinear crystal's finite cross-section, these two output fields are in a zero-mean, jointly Gaussian state that is homogeneous and stationary. Moreover, the signal and idler will then have identical fluorescence spectra and maximum phase-sensitive cross correlation, but no phase-sensitive autocorrelation or phase-insensitive cross correlation \cite{Shapiro:Gaussian,Brambilla:SPDC}. It follows that after passing through the transmission mask the jointly Gaussian state is fully determined by the Schell-model autocorrelation spectra \cite{coherence_separability} \begin{equation} S_{m,m}^{(n)}(\rhovec_1, \rhovec_2,\Omega)= S_{0}^{(n)}(\rhovec_1, \rhovec_2,\Omega)\,, \label{PIS:Source} \end{equation} for $m = S, I$, and the phase-sensitive cross-correlation spectrum \begin{equation} S_{S,I}^{(p)}(\rhovec_1, \rhovec_2,\Omega) = S_{0}^{(p)}(\rhovec_1, \rhovec_2,\Omega)\,, \label{PS:Source} \end{equation} where $S_{0}^{(n)}$ and $S_{0}^{(p)}$ are given by Eqs.~\eqref{SM02:PIS} and \eqref{SM02:PS} respectively, and we shall assume that they satisfy the quasimonochromatic condition.
In the Fig.~\ref{TPI:FFdiff} setup, the signal and idler fields both propagate over an $L\,$m free-space path---assumed to be sufficiently long to satisfy the far-field condition, $\omega_0a_0^2/2cL \ll 1$, for the phase-sensitive source---and then are separated by a polarizing beam splitter such that they impinge on separate pinhole detectors, each centered on the transverse-plane coordinate $\rhovec$ with respect to its optical axis. Because linear transformations of zero-mean Gaussian states are still zero-mean Gaussian \cite{Shapiro:Gaussian}, and because free-space diffraction is a linear transformation, we need only determine the second-order moments at the detection planes to determine the joint state of $\hat{E}_{1}(\rhovec,t)$ and $\hat{E}_{2}(\rhovec,t)$, which denote the far-field propagated field operators of the signal and the idler respectively. Thus, quasimonochromatic paraxial diffraction into the far field results in phase-insensitive autocorrelation spectra given by Eq.~\eqref{VCZ:PIS} and a phase-sensitive cross-correlation spectrum given by Eq.~\eqref{VCZ:PS} \cite{AutovsCross}.
The two pinhole photodetectors are assumed to have identical parameters: quantum efficiency $\eta$; photosensitive area $A$; and current-pulse output $qh_B(t)$ from detection of a single photon, where $q$ is the electron charge and $\int\!dt\,h_B(t) = 1$. Then, the time-average photocurrent cross-correlation at the detection planes has an ensemble average \cite{ShapiroSun,ErkmenShapiro:GhostImaging2}
\begin{equation}
C(\rhovec) = \frac{1}{T} \int_{-T/2}^{T/2} {d}t \, \langle \hat{\imath}_{1}(t) \hat{\imath}_{2}(t) \rangle\,, \label{coinc} \end{equation} in which the equal-time photocurrent cross correlation is given by \begin{multline} \langle \hat{\imath}_{1}(t) \hat{\imath}_{2}(t) \rangle = q^2 \eta^{2} A^{2} \int\! \int \! du_{1} du_{2} \\[5pt] \times \langle \hat{E}_{1}^{\dagger}(\rhovec,u_{1}) \hat{E}_{2}^{\dagger}(\rhovec,u_{2}) \hat{E}_{1}(\rhovec,u_{1}) \hat{E}_{2}(\rhovec,u_{2}) \rangle \\[5pt] \times h_{B}(t-u_{1}) h_{B}(t-u_{2})\,. \label{IntCorr} \end{multline} Here, we have approximated the integrals over the pinhole detectors' photosensitive regions by the value of the integrand at $\rhovec$ times $A^{2}$. For our Gaussian-state $\hat{E}_{1}$ and $\hat{E}_{2}$, the fourth-order field moment in Eq.~\eqref{IntCorr} reduces to a sum of products of second-order correlation functions by virtue of the Gaussian moment-factoring theorem \cite{Mandel,ShapiroSun}. This procedure simplifies the photocurrent cross-correlation expression to \begin{equation}
C(\rhovec) = C_{0}(\rhovec) + C_{p} \left | \mathcal{G}^{(p)} ( \mathbf{0} ) \: \mathcal{T}_{p}\left (\frac{2 \omega_{0} \rhovec }{cL} \right ) \right | ^{2}\,. \label{TPI:corr} \end{equation} Here, \begin{eqnarray} C_{0}(\rhovec) &=& \biggl [ \frac{\omega_{0}^{2} q \eta A}{4 \pi^2 c^2L^2} \, \mathcal{T}_{n}( \mathbf{0} )\, \mathcal{G}^{(n)}\! \left (\frac{\omega_{0} \rhovec}{cL} \right ) \nonumber \\[5pt] &\times& \int_{-\omega_{0}}^{\infty} \! \frac{d\Omega}{2\pi} \, S^{(n)}(\Omega)\, \int_{-\infty}^{\infty} \! dt \, h_{B}(t) \biggr ]^{2} \label{Co} \end{eqnarray} is a non-image-bearing background, which is broad and featureless owing to ${\mathcal{G}}^{(n)}({\boldsymbol k})$ being the Fourier transform of the narrow spatial-domain correlation function $G^{(n)}(\rhovec)$. The second term in Eq.~\eqref{TPI:corr} is the image-bearing term. Its constant factor is \begin{equation} C_{p} = \left ( \frac{\omega_{0}^{2} q \eta A}{4 \pi^{2} c^2L^{2}}\right )^{\!2} \! \left [ \bigl \lvert \mathcal{F}^{-1} \{ S^{(p)}(\Omega)\} \bigr \rvert ^{2} \!\star h_{B} \star \overleftarrow{h}_{\!\!B} \right ]_{t=0},\label{Cp} \end{equation}
where $\mathcal{F}^{-1}\{ \cdot \}$ denotes the inverse Fourier transform of the bracketed term [see Eq.~\eqref{CH6:Spectra} for our Fourier transform sign convention], $\star$ denotes convolution, and $\overleftarrow{h}_{\!\!B}$ represents the time-reversed impulse response. From Eq.~\eqref{TPI:corr}, we see that the image-bearing term is proportional to $| \mathcal{T}_{p}(2 \omega_{0} \rhovec / cL)|^{2}$, which is the far-field diffraction pattern of the {\em square} of the mask's field transmissivity $T(\rhovec)$.
Let us compare the imaging characteristics of this imager to those of a conventional classical imager that utilizes a coherent-state beam to illuminate the mask, and a single (scanning) pinhole detector located in the far field that records the diffraction pattern. If we assume the field impinging on the transmission mask is a monochromatic plane wave at center frequency $\omega_{0}$ and with photon-flux density $I_{0}$, the field just after the mask is in the coherent state satisfying \begin{equation} \hat{E}_{0}(\rhovec, t) \lvert \sqrt{I_{0}} T(\rhovec)\rangle = \sqrt{I_{0}} T(\rhovec) \lvert \sqrt{I_{0}} T(\rhovec)\rangle\,. \end{equation} Because free-space propagation is a multimode beam splitter relation \cite{Yuen:Part_I}, the detection-plane field operator $\hat{E}_{1}(\rhovec,t)$ is also in a coherent state, whose eigenfunction is determined by substituting $\sqrt{I_{0}} T(\rhovec)$ into the classical Huygens-Fresnel diffraction integral, i.e., Eq.~\eqref{FS:prop} with the field operator replaced by the coherent-state eigenfunction.
We shall assume the path length $L$ satisfies the far-field condition, $\omega_{0} a_{0}^{2}/2 cL \ll 1$, for coherent-state diffraction \cite{NearvsFar:Coherent}, so that the quadratic phase term in the Huygens-Fresnel Green's function becomes negligible and the mean photocurrent becomes \begin{equation}
\langle \hat{\imath}(t) \rangle = \frac{\omega_{0}^{2} q \eta A}{4 \pi^{2} c^2L^{2}} \, I_{0} \int_{-\infty}^{\infty} dt \, h_{\!B}(t) \, \left | \mathcal{T}_{c}\left (\frac{\omega_{0} \rhovec}{cL} \right ) \right |^{2}\,, \end{equation} where there is no background and the image term is given by, \begin{equation} \mathcal{T}_{c}(\kvec) \equiv \int_{\mathbb{R}^{2}} d\rhovec' \, e^{-i \kvec \cdot \rhovec'} T(\rhovec')\,. \end{equation}
If $T(\rhovec)$ only takes values zero or one---as was the case for the two-slit transmission mask employed in \cite{Dangelo}---then $T^{2}(\rhovec) = T(\rhovec)$ and the biphoton source yields a far-field diffraction pattern proportional to $| \mathcal{T}_{p}(2 \omega_{0} \rhovec / cL)|^{2}$, whereas the coherent diffraction pattern is proportional to $| \mathcal{T}_{p}(\omega_{0} \rhovec / cL)|^{2}$. Thus, the far-field pattern observed with the biphoton source is spatially compressed by a factor of two relative to that obtained when the coherent-state source is employed, which is why the biphoton case has been said to beat the classical resolution limit. However, it is worth re-emphasizing that Eq.~\eqref{TPI:corr} is true for both classical and quantum Gaussian-state sources whose signal and idler fields have a nonzero phase-sensitive cross correlation. Consequently, it is the phase-sensitive coherence of the source---and {\em not} the entanglement of the biphoton---that is the fundamental cause for the factor-of-two compression in the far-field diffraction pattern as compared to what is obtained using a coherent-state plane wave. Finally, it is important to note that unless $T^2(\rhovec) \propto T(\rhovec)$, as we have above when $T(\rhovec)$ is a zero-one function, then the diffraction pattern acquired from phase-sensitive sources will be distorted relative to what will be obtained with a coherent-state field.
Utilizing phase-insensitive Gaussian-state light in the Fig.~\ref{TPI:FFdiff} configuration does not result in a photocurrent cross correlation containing a diffraction-pattern image. This imaging failure occurs because, from Eq.~\eqref{VCZ:PIS}, the equal-position correlation in the Fourier plane traces out $\mathcal{G}_{0}^{(n)}(\omega_0 \rhovec/cL)$, which does not contain any information about the transmission mask $T(\rhovec)$. However, by modifying the Fig.~\ref{TPI:FFdiff} setup to have one detector scan $-\rhovec$, while the other scans $\rhovec$, then the photocurrent cross correlation will contain an image of $| \mathcal{T}_{n}(2 \omega_{0} \rhovec / cL )|^{2}$. For a zero-one function transmission mask, this phase-insensitive imager also achieves the factor-of-two pattern compression in comparison with the image formed using a coherent-state plane wave. In this regard we note that \em \/\rm\ phase-insensitive Gaussian-state sources are classical \cite{ErkmenShapiro:GhostImaging2}.
Thus far we have considered only the image-bearing term in Eq.~\eqref{TPI:corr}. Now we will address the image contrast. For simplicity, we will assume that $T(\rhovec)$ is real valued. In addition, we restrict ourselves to an observation region ${\cal{R}}$ that encompasses the image-bearing term in Eq.~\eqref{TPI:corr}, and we define the contrast as \begin{equation} {\cal{C}} \equiv \frac{\max_{\cal{R}}[C(\rhovec)] - \min_{\cal{R}}[C(\rhovec)]}{C_0({\bf 0})}\,, \end{equation} so that the numerator yields the dynamic range of the image-bearing term in the photocurrent correlation $C(\rhovec)$, while the denominator is the featureless background.
Here we compare the contrast from classical and quantum sources that have identical autocorrelation spectra and the maximum phase-sensitive cross correlation allowed in classical and quantum physics, respectively. When the source is in a classical Gaussian state, whose autocorrelation spectrum---just before the transmission mask---is $\mathcal{G}^{(n)}(\kvec) S^{(n)}(\Omega)$, the maximum magnitude for the phase-sensitive cross-correlation spectrum is equal to the autocorrelation spectrum \cite{Shapiro:Gaussian}, i.e., the Gaussian state with maximum classical phase-sensitive cross correlation satisfies \begin{equation}
|S^{(p)}(\Omega) \mathcal{G}^{(p)}(\kvec)| = S^{(n)}(\Omega) \mathcal{G}^{(n)}(\kvec). \label{PS:classical} \end{equation} Taking the phase of this phase-sensitive spectrum to be zero, and recalling that $\int dt\, h_{B} (t) = 1$, the contrast with classical phase-sensitive Gaussian-state sources can be written in the form \begin{equation} {\cal{C}}^{(c)} = {\cal{C}}_s^{(c)}{\cal{C}}_t^{(c)}, \end{equation} where the spatial ($s$) factor is given by \begin{equation}
\mathcal{C}^{(c)}_{s} = \frac{\max_{\kvec}[ |\mathcal{T}_{p}(\kvec)|^{2} ] - \min_{\kvec}[ |\mathcal{T}_{p}(\kvec)|^{2} ]}{\mathcal{T}_{n}^{2}(\mathbf{0})} \leq 1\,, \end{equation} with equality if $T(\rhovec)$ is real, so that \begin{equation} \mathcal{C}^{(c)} = \mathcal{C}_{t}^{(c)} = \frac{ \left [ \bigl \lvert \mathcal{F}^{-1} \{ S^{(n)}(\Omega)\} \bigr \rvert ^{2}\!\star h_{B} \star \overleftarrow{h}_{\!\!B} \right ]_{t=0}}{\Bigl ( \int\! d\Omega\, S^{(n)}(\Omega)/2\pi \Bigr )^{2}} \end{equation} for such masks. For analytical convenience, let us take the spectral part of the phase-insensitive autocorrelation function to be Gaussian with $e^{-2}$-attenuation baseband bandwidth $2/T_{0}$, i.e., \begin{equation} S^{(n)}(\Omega) = e^{-T_{0}^{2} \Omega^{2}/2}\,\sqrt{2 \pi T_{0}^{2}}\,, \label{Sn:Gaussian} \end{equation} and let us take the baseband impulse response $h_B(t)$ to be a Gaussian with $e^{-2}$-attenuation time duration $T_d$, viz., \begin{equation} h_B(t) = e^{-8t^2/T_d^2} \sqrt{8/\pi T_d^2} \,. \label{Hb:Gaussian} \end{equation} With these assumptions, we find that the classical contrast for a real-valued mask is \begin{equation} \mathcal{C}^{(c)} = \frac{1} { \sqrt{1 + (T_{d}/2 T_{0})^{2}}}\,, \end{equation} which is approximately unity for narrowband sources that satisfy $T_{d} \ll T_{0}$. On the other hand, in the broadband limit $T_{d} \gg T_{0}$, we have \begin{equation} \mathcal{C}^{(c)} \approx 2 T_{0} / T_{d} \ll 1\,, \end{equation} so the contrast is severely degraded in this case.
Now consider a nonclassical Gaussian state with the maximum phase-sensitive cross correlation. In the low-brightness regime, i.e., when $S^{(n)}(\Omega) \mathcal{G}_{0}^{(n)}(\kvec) \ll 1$, the maximum phase-sensitive cross-correlation spectrum is approximately \cite{ErkmenShapiro:GhostImaging2} \begin{equation}
|S^{(p)}(\Omega) \mathcal{G}^{(p)}(\kvec)| \approx \sqrt{S^{(n)}(\Omega) \mathcal{G}^{(n)}(\kvec)}, \end{equation} which is much higher, in this limit, than the classical maximum given by Eq.~\eqref{PS:classical}. Taking the phase of this correlation to be zero, the contrast is found to factor into the product of spatial and temporal terms, with $\mathcal{C}^{(q)}_{s} =\mathcal{C}^{(c)}_{s}$, and the temporal term given by \begin{equation} \mathcal{C}^{(q)}_{t} = \frac{ \left [ \bigl \lvert \mathcal{F}^{-1} \{ \sqrt{S^{(n)}(\Omega)} \} \bigr \rvert ^{2}\!\star h_{B} \star \overleftarrow{h}_{\!\!B} \right ]_{t=0}}{\mathcal{G}^{(n)}(\mathbf{0}) \left ( \int\! d\Omega\, S^{(n)}(\Omega)/2\pi \right )^{2}}\,. \end{equation} Once again using Eq.~\eqref{Sn:Gaussian} for the fluorescence spectrum and Eq.~\eqref{Hb:Gaussian} for the baseband current filter, we obtain \begin{equation} \mathcal{C}^{(q)} = 2 / \mathcal{G}^{(n)}(\mathbf{0}) S^{(n)}(0) \sqrt{1 + T_{d}^{2} / 2 T_{0}^{2}}\, \end{equation} for real-valued $T(\rhovec)$. Here, the narrowband contrast $\mathcal{C}^{(q)} = 2/ \mathcal{G}^{(n)}(\mathbf{0}) S^{(n)}(0)$ is very high because of the low-brightness condition, and even for broadband fields the contrast, \begin{equation} \mathcal{C}^{(q)} = 2\sqrt{2} T_{0} / T_{d} \mathcal{G}^{(n)}(\mathbf{0}) S^{(n)}(0) \,, \end{equation} may be high. In particular, in the biphoton regime, wherein $\mathcal{G}^{(n)}(\mathbf{0}) T_{d} \ll 1$ (low flux as well as low brightness) also prevails, very high contrast is predicted in this broadband limit \cite{LowFluxCond}, which is in agreement with the background-free diffraction pattern reported in \cite{Dangelo}. Therefore, low-brightness quantum Gaussian-state fields have a contrast advantage over classical phase-sensitive Gaussian-state fields when the phase-sensitive cross correlation is measured via a photocurrent correlation measurement, and the biphoton state yields images with negligible background even when it is a broadband state.
In summary, in this section we have studied a cornerstone proof-of-principle experiment for quantum optical lithography by applying Gaussian-state analysis and the coherence theory results from Sec.~\ref{CH6:CoherenceTheory} to the propagation of classical and quantum phase-sensitive cross correlations. Our analysis has shown that the {\em only} performance difference between using a biphoton-state source and a classical phase-sensitive Gaussian-state source in the Fig.~1 setup is the diffraction-pattern image's contrast. The resolution improvement seen with a biphoton source is entirely due to the diffraction properties of the phase-sensitive cross correlation between the signal and idler fields, hence it is also achievable with a pair of classical Gaussian-state fields with phase-sensitive cross correlation. However, low-brightness quantum sources achieve higher contrast than classical sources, which permits imaging with broader bandwidth quantum sources. Finally, the broadband biphoton state yields very high contrast images, which is the reason why biphoton-state quantum lithography experiments have yielded background-free diffraction-pattern images~\cite{Dangelo}.
\section{Two-Photon Thin-Lens Imaging} \label{CH6:BroadbandImaging}
Let us now consider using an optical source with low spatial coherence and a thin lens to image a transmission mask placed at the source plane, as depicted in Fig.~\ref{TPI:NFlens}. Primary attention in our analysis of this experimental setup will be given to the resolution limitations imposed by the finite aperture of the lens, as previous work has claimed that a factor-or-two resolution improvement accrues when a broadband biphoton source is employed \cite{Shih:QI}.
As in the previous section, we shall assume an SPDC source that generates zero-mean Gaussian-state signal and idler beams whose phase-insensitive correlation spectra at the exit plane of the transmission mask are given by Eq.~\eqref{PIS:Source} and whose phase-sensitive cross-correlation spectrum at that plane is given by Eq.~\eqref{PS:Source}. The optical fields first propagate through a $d_{1}$-m-long free-space path according to Eq.~\eqref{FS:prop}. A thin lens of radius $R$ and focal length $f$ embedded in an otherwise opaque screen is placed on this plane. This thin lens will be assumed to be entirely free of chromatic aberration over the frequency range of interest, i.e., each frequency component of the impinging field is multiplied by $\text{circ}(|\rhovec|/R) e^{-i \omega|\rhovec|^{2}/2cf}$, where $\omega$ is a passband frequency centered around $\omega_{0}$ and the circle function is \begin{equation}
\text{circ}(x) = \begin{cases} 1, & |x| \leq 1, \\ 0, & \text{otherwise.} \end{cases} \end{equation} Finally the field at the exit plane of the lens propagates $d_{2}\,$m in free space to reach the image plane, which is defined by the lens maker's formula $d_{1}^{-1} + d_{2}^{-1} = f^{-1} $. The image-plane signal and idler fields are separated by a polarizing beam splitter, after which each illuminates a pinhole photodetector located at transverse coordinate $\rhovec$ relative to its optical axis. The resulting photocurrents are then correlated to obtain the same fourth-order field measurement given in Eqs.~\eqref{coinc} and \eqref{IntCorr} in terms of the image-plane field operators $\hat{E}_{1}(\rhovec,t)$ and $\hat{E}_{2}(\rhovec,t)$.
\begin{figure}
\caption{(Color online) Two-photon thin-lens imaging of a transmission mask. PBS, polarizing beam splitter.}
\label{TPI:NFlens}
\end{figure}
The overall mapping from the source-plane field operators to the image-plane field operators is linear, and therefore $\hat{E}_{1}(\rhovec,t)$ and $\hat{E}_{2}(\rhovec,t)$ are in a zero-mean jointly-Gaussian state. With the simplifying assumption that the detectors have electrical bandwidths much broader than the source spectra \cite{BBAssumption}, we can combine and reduce Eqs.~\eqref{coinc} and \eqref{IntCorr} for the biphoton state to obtain \begin{align} C(\rhovec) & \!\approx\! q^{2} \eta^{2} A^{2} \langle \hat{E}_{1}^{\dagger}(\rhovec,t) \hat{E}_{2}^{\dagger}(\rhovec,t) \hat{E}_{1}(\rhovec,t) \hat{E}_{2}(\rhovec,t) \rangle \\[5pt]
& \!=\! q^{2} \eta^{2} A^{2} \, \bigl [ K_{1,1}^{(n)}(\rhovec,\rhovec,0) K_{2,2}^{(n)}(\rhovec,\rhovec,0) \nonumber \\[5pt] &+ | K_{1,2}^{(p)}(\rhovec,\rhovec,0) |^{2}\bigr ] \,, \label{IntCorr:GaussianStates} \end{align} where \begin{align} K_{m,\ell}^{(n)} (\rhovec, \rhovec, \tau) & \equiv \langle \hat{E}_{m}^{\dagger}(\rhovec,t) \hat{E}_{\ell}(\rhovec,t+\tau) \rangle, \\[5pt] K_{m,\ell}^{(p)} (\rhovec, \rhovec, \tau) &\equiv \langle \hat{E}_{m}(\rhovec,t) \hat{E}_{\ell}(\rhovec,t+\tau) \rangle, \end{align} for $m,\ell = 1,2$, and Eq.~\eqref{IntCorr:GaussianStates} follows from the Gaussian moment-factoring theorem \cite{Mandel,ShapiroSun}. Furthermore, as we have determined in the previous section, for maximally-entangled Gaussian states with low-brightness and low-flux, i.e., the biphoton state, the second term in \eqref{IntCorr:GaussianStates} is much stronger than the first, permitting the approximation \begin{equation}
C(\rhovec) \approx q^{2} \eta^{2} A^{2} | K_{1,2}^{(p)}(\rhovec,\rhovec,0) |^{2}\,. \label{TPI:NFImage} \end{equation} Therefore, for biphotons a photocurrent correlation measurement with broadband detectors is a means for measuring the squared magnitude of the phase-sensitive cross correlation between the image-plane field operators \cite{ShapiroSun}. In this section we demonstrate that the interesting biphoton-state results predicted for this imaging configuration are a consequence of the phase-sensitive cross correlation, and the photocurrent correlation does not play a role beyond facilitating its measurement. Hence, in the remainder of this section we shall bypass this photocurrent correlation measurement and focus directly on the phase-sensitive and phase-insensitive cross correlations between the image-plane field operators.
The frequency-domain image-plane field operators are given by the following linear transformation of the frequency-domain source-plane field operators, \begin{equation} \hat{\mathcal{E}}_{m}(\rhovec,\Omega) = \int\! {d}\rhovec' \, h(\rhovec,\rhovec', \omega_{0} + \Omega) \hat{\mathcal{E}}_{\ell}(\rhovec', \Omega) + \hat{\mathcal{L}}_{m}(\rhovec,\Omega)\,, \label{PSF:Freq} \end{equation} for $(\ell,m) = (S,1), (I,2)$, where $\hat{\mathcal{L}}_{m}(\rhovec,\Omega)$ is an auxiliary vacuum-state operator such that $\hat{E}_{m}(\rhovec,t)$ satisfies the free-field commutators given in Eqs.~\eqref{commutator1} and \eqref{commutator2}. The point-spread function $h(\rhovec,\rhovec', \omega)$, found from the Huygens-Fresnel principle and the lens transfer function, is given by \begin{equation} h(\rhovec,\rhovec', \omega) = \mathcal{H}\bigl ( r(\rhovec,\rhovec'), \omega/ \omega_{0} \bigr ) e^{i\phi(\rhovec,\rhovec',\omega)}, \label{H:lens} \end{equation} where \begin{equation}
r(\rhovec,\rhovec') \equiv \frac{\omega_{0} R}{cd_{1}} | d_{1} \rhovec/d_{2} + \rhovec' |\,, \label{r} \end{equation} and \begin{equation} \mathcal{H}(r,\xi) \equiv \frac{- \omega_{0}^{2} R^{2} \xi^{2}}{4 \pi c^{2} d_{1} d_{2}} \frac{2 J_{1}(r \xi)}{r \xi}\,, \label{H:lensAmp} \end{equation} with $2 J_{1}(x)/x$ for $x\geq 0$ being the well-known Airy function. The phase term in Eq.~\eqref{H:lens} is \begin{equation}
\phi(\rhovec,\rhovec',\omega) \!= \omega \left (d_{1} + d_{2} + |\rhovec|^{2}/2 d_{2} + | \rhovec'|^{2}/2 d_{1} \right )/c, \label{H:lensPhase} \end{equation} which incorporates the group delay arising from the $(d_{1} + d_{2})$-m propagation, and the parabolic phases at the source and image planes that are associated with the diffraction process.
Because our concern is with the resolution limit imposed by the lens having a finite radius $R$, we will further simplify our analysis by assuming spatially-incoherent source statistics and appropriate focusing at the source plane to compensate for the parabolic phase in Eq.~\eqref{H:lensPhase}. These assumptions simplify the phase-insensitive autocorrelation functions and the phase-sensitive cross-correlation function of the $I_{0}\,$photons/m$^{2}$s signal and idler fields---given in Eqs.~\eqref{SM02:PIS} and \eqref{SM02:PS}---to \begin{eqnarray}
S_{0}^{(n)}(\rhovec_1, \rhovec_2,\Omega) &=& | T (\rhovec_{1} ) |^2 \bigl [ 2\pi c / (\omega_{0}+\Omega)\bigr ]^{2} I_0 \nonumber \\[5pt] &\times& \delta(\rhovec_2-\rhovec_1) \, s^{(n)}(\Omega) \label{PISBB} \end{eqnarray} and \begin{eqnarray}
\lefteqn{S_{0}^{(p)}(\rhovec_1, \rhovec_2,\Omega) = e^{-i \omega_{0} |\rhovec_{1}|^{2}/c d_{1}} \,T^{2}(\rhovec_1 )} \nonumber \\[5pt] &\times& \bigl [ (2\pi c)^{2} / (\omega_{0}^{2} - \Omega^{2} )\bigr ] I_0\delta(\rhovec_2-\rhovec_1)\, s^{(p)}(\Omega)\,, \label{PSBB} \end{eqnarray} respectively, where $s^{(x)}(\Omega)/2\pi$, for $x=n,p$, are normalized (unity area) spectra \cite{CohArea}. Evaluating the phase-insensitive autocorrelations and the phase-sensitive cross correlation of the two image-plane fields at equal spatial coordinates (relative to their optical axes) and at equal times, yields \begin{equation}
K_{m,m}^{(n)}(\rhovec,\rhovec,0) = \int_{\mathbb{R}^{2}} \!d\rhovec' \, |T(\rhovec')|^2 g_{n} \bigl (r(\rhovec,\rhovec') \bigr)\,, \label{PIS:detplanecorr} \end{equation} for $m=1,2$, and \begin{eqnarray}
K_{1,2}^{(p)}(\rhovec,\rhovec,0) &=& e^{i \omega_{0} (2 d_{1} + 2 d_{2} + |\rhovec|^{2}/d_{2})/c} \nonumber \\[5pt] &\times& \int_{\mathbb{R}^{2}} \! d\rhovec' \, T^{2}(\rhovec') g_{p} \bigl (r(\rhovec,\rhovec') \bigr)\,. \label{PS:detplanecorr} \end{eqnarray}
The point-spread function in the superposition integral involving $|T(\rhovec)|^{2}$ is \begin{eqnarray}
g_{n}(r) &\equiv& \frac{(2\pi c)^{2}I_0}{\omega_{0}^{2}} \int_{-\omega_{0}}^{\infty} \frac{d\Omega}{2 \pi} s^{(n)}(\Omega)\nonumber \\[5pt] &\times& |\mathcal{H}(r ,1+\Omega/\omega_{0})|^{2} / (1 + \Omega/\omega_{0})^{2} \,, \label{PIS:PSF} \end{eqnarray} and it determines the phase-insensitive autocorrelation functions. Likewise, the point-spread function in the superposition integral involving $T^{2}(\rhovec)$ is \cite{IntegralLimit} \begin{eqnarray}
\lefteqn{g_{p}(r) \equiv \frac{(2\pi c)^{2}I_0}{\omega_{0}^{2}} \int_{-\omega_{0}}^{\omega_{0}} \frac{d\Omega}{2 \pi} s^{(p)}(\Omega)} \nonumber \\[5pt] &\times& \mathcal{H}(r,1+\Omega/\omega_{0}) \mathcal{H}(r,1-\Omega/\omega_{0}) / (1-\Omega^{2}/\omega_{0}^{2}),\, \label{PS:PSF} \end{eqnarray} and it controls the phase-sensitive cross-correlation function. Therefore, apart from an unimportant parabolic phase factor, the most important difference between phase-insensitive and phase-sensitive coherence propagation---insofar as two-photon thin-lens imaging is concerned---is the frequency coupling between $\pm \Omega/\omega_{0}$ that is present in Eq.~\eqref{PS:PSF} but is absent from Eq.~\eqref{PIS:PSF}. In the quasimonochromatic limit, however, this coupling becomes insignificant, because \begin{equation} 1 \pm \Omega/\omega_{0} \approx 1\,, \end{equation} so that $g_{n}(r) = g_{p}(r)$ prevails whenever $s^{(n)}(\Omega) = s^{(p)}(\Omega) $. Thus the quasimonochromatic point-spread function for the phase-insensitive correlation is identical to its quasimonochromatic phase-sensitive counterpart.
\begin{figure}
\caption{(Color online) Comparison of the imaging point-spread functions for phase-insensitive (PIS) and phase-sensitive (PS) correlations when $W/\omega_{0}=0.25$, and when the imaging source is quasimonochromatic (QM). The normalizing coefficient is $\kappa \equiv I_{0} \omega_{0}^{2} R^{4} / 4 c^{2} d_{1}^{2} d_{2}^{2}$.}
\label{PSNFKernel:1}
\label{PSNFKernelLog:1}
\label{Plot:BB1}
\end{figure}
\begin{figure}
\caption{(Color online) Comparison of the imaging point-spread functions for phase-insensitive (PIS) and phase-sensitive (PS) correlations at the ultimate theoretical limit of SPDC bandwidth ($W/\omega_{0} = 1$), and the quasimonochromatic (QM) limit. The normalizing coefficient is $\kappa \equiv I_{0} \omega_{0}^{2} R^{4}/4 c^{2} d_{1}^{2} d_{2}^{2}$.}
\label{PSNFKernel:4}
\label{PSNFKernelLog:4}
\label{Plot:BB2}
\end{figure}
However, $g_{n}(r)$ and $g_{p}(r)$ begin to differ as the bandwidth of the source increases. Suppose that the normalized phase-insensitive and phase-sensitive source spectra are both taken to be flat over a $2W$-bandwidth window, i.e., \begin{equation}
s^{(n)}(\Omega) = s^{(p)}(\Omega) = \begin{cases} \pi/W, & |\Omega| < W, \\ 0, & \text{otherwise.} \end{cases} \end{equation} Substituting this expression into Eqs.~\eqref{PIS:PSF} and \eqref{PS:PSF} permits us to express the point-spread functions as the following dimensionless integrals, \begin{equation}
g_{n}(r) = \frac{I_{0} \omega_{0}^{3} R^{4} }{2 c^{2} d_{1}^{2} d_{2}^{2} W} \int_{-W/\omega_{0}}^{W/\omega_{0}} {d}u \, \frac{J_{1}^{2} \bigl(r (1+u) \bigr) }{r^{2}} \,, \label{PIS:nodimen} \end{equation} and \begin{eqnarray}
g_{p}(r) &=& \frac{I_{0} \omega_{0}^{3} R^{4} }{2 c^{2} d_{1}^{2} d_{2}^{2} W} \int_{-W/\omega_{0}}^{W/\omega_{0}} {d}u \nonumber \\[5pt] &\times& \frac{J_{1}\bigl (r (1+u) \bigr ) }{r } \frac{J_{1}\bigl (r (1-u) \bigr ) }{r }\,. \label{PS:nodimen} \end{eqnarray} In the quasimonochromatic limit, in which $W/\omega_{0} \ll 1$ holds, both point-spread functions simplify to \begin{equation} g_{n}(r) = g_{p}(r) = \frac{I_{0} \omega_{0}^{2} R^{4} }{4 c^{2} d_{1}^{2} d_{2}^{2}} \Bigl ( \frac{2 J_{1}(r)}{r} \Bigr )^{2}\,. \label{QM:PSF} \end{equation} Hence with a quasimonochromatic source there is no difference between the image of a real-valued transmission mask acquired with phase-insensitive (thermal) illumination or phase-sensitive (classical or quantum) illumination.
The point-spread functions for broader bandwidth sources are plotted in Figs.~\ref{Plot:BB1} and \ref{Plot:BB2} at two different $W$ values. The $W= \omega_{0}/4$ phase-sensitive point-spread function, shown in Fig.~\ref{Plot:BB1}, represents imaging performance for unusually broadband SPDC \cite{ODonnell}. The $W=\omega_{0}$ phase-sensitive point-spread function, plotted in Fig.~\ref{Plot:BB2}, represents imaging performance at the ultimate theoretical limit of SPDC bandwidth. The point-spread functions in these figures show that the peak amplitude of the phase-insensitive function increases to \begin{equation} g_n(0) = \frac{I_{0} \omega_{0}^{2} R^{4} }{4 c^{2} d_{1}^{2} d_{2}^{2}} \left ( 1 + \frac{W^{2}}{3 \omega_{0}^{2}} \right )\,, \end{equation} whereas that of the phase-sensitive point-spread function attenuates to \begin{equation} g_p(0) = \frac{I_{0} \omega_{0}^{2} R^{4} }{4 c^{2} d_{1}^{2} d_{2}^{2}} \left ( 1 - \frac{W^{2}}{3 \omega_{0}^{2}} \right )\,, \end{equation}
relative to the peak amplitude in the quasimonochromatic limit as the source bandwidth increases. The $(1+u)^{2}$ factor multiplying the frequency-resolved Airy patterns in Eq.~\eqref{PIS:nodimen}, where $|u|<W/\omega_{0}\leq 1$, is responsible for the increase in $g_n(0)$ with increasing source bandwidth. This scaling favors the {\em blue}-detuned frequency contributions to the phase-insensitive point-spread function. So, because of the quadratic scaling, the average of the amplitude increase for $u>0$ versus the amplitude decrease for $u<0$ is greater than one. Thus, the peak value of $g_n(0)$ increases with increasing source bandwidth. On the other hand, the scaling for the frequency-resolved Airy patterns in the integrand of Eq.~\eqref{PS:nodimen} is $(1-u)(1+u) = 1-u^{2}$, implying that {\em all} detuned frequencies are attenuated, which results in $g_p(0)$ decreasing as the source bandwidth increases.
Figures~\ref{Plot:BB1}(a) and \ref{Plot:BB2}(a) indicate that some narrowing of the main lobe occurs for both the broadband phase-sensitive and phase-insensitive point-spread functions, with the latter suffering from a slower-decaying tail. From a practical perspective this main-lobe narrowing behavior is of little interest. Taking the resolution to be set by the first zero in the point-spread function, Fig.~\ref{Plot:BB1}(b) shows that the resolution benefit offered by broadband phase-sensitive as compared to quasimonochromatic (phase-sensitive or phase-insensitive) imaging is merely a factor of $1.14$. From this figure we also note that the tail of the phase-insensitive point-spread function traces the envelope of the oscillations in the phase-sensitive point-spread function, so no appreciable loss of resolution results from the former's slowly-decaying tail. Even at the $W = \omega_{0}$ ultimate bandwidth limit, the resolution improvement offered by the broadband phase-sensitive point-spread function---as found from Fig.~\ref{Plot:BB2}(b)---is only a factor of $1.38$. In this limit, the tail of the broadband phase-insensitive point-spread function falls off somewhat---but not dramatically---slower than the envelope of the phase-sensitive point-spread function's oscillations.
\begin{figure}
\caption{(Color online) Image-plane spot diameters for different frequency components of a broadband point source.}
\label{TPI:LensDiffFreq}
\end{figure}
The difference between the behavior of the phase-insensitive and phase-sensitive point-spread functions as a function of the source bandwidth deserves closer examination to understand the underlying physics. Recall that a source generating a complex-stationary baseband field around a center frequency $\omega_{0}$ is a superposition of monochromatic field components which have phase-insensitive autocorrelations at each frequency and phase-sensitive cross correlations between frequencies that sum to $2 \omega_{0}$. Thus, the phase-insensitive correlation, measured at a given spatiotemporal coordinate $(\rhovec,t)$, is a superposition of all the different autocorrelations at detunings $\Omega$ over the fluorescence bandwidth of the source. On the other hand, the phase-sensitive correlation measured at $(\rhovec,t)$ is a superposition of all the {\em cross correlations} between frequency components detuned by $\pm \Omega$, over the phase-sensitive bandwidth of the source. Now, consider a point source at the source-plane that emits signal and idler fields that have nonzero phase-insensitive autocorrelations and phase-sensitive cross correlation. From Eq.~\eqref{PSF:Freq}, signal and idler frequency components at $\omega = \omega_{0} + \Omega$ will yield image-plane spots with common radius $c d_{2}/(\omega_{0} +\Omega) R$. Thus, as shown in Fig.~\ref{TPI:LensDiffFreq}, lower frequency components produce broader spots on the image plane than do higher frequency components. The phase-insensitive autocorrelation---of either image-plane field---measured by scanning a point detector on the transverse plane, therefore decays slowly as $|\rhovec|$ increases, because of the large spots from the lower frequencies. However, this slowly-decaying tail does not cause a significant increase in the point-spread function's width, because the quadratic weighting coefficient in Eq.~\eqref{PIS:nodimen} accentuates blue-detuned frequencies and attenuates those that are red detuned. On the other hand, if we are measuring the phase-sensitive cross correlation between the signal and idler fields \cite{AutovsCross:2}, we are in effect measuring the superposition of the cross correlations between the $\omega_{0}+ \Omega $ signal-field component and the $\omega_{0}- \Omega $ idler-field component, where $\Omega \in [-\omega_{0}, \omega_{0}]$. For $\Omega>0$, the former yields a narrow spot of radius $c d_{2}/(\omega_{0} +\Omega) R$, and the latter yields a broad spot of radius $c d_{2}/(\omega_{0} -\Omega) R$. Because the phase-sensitive cross correlation is given by their product, however, the narrower radius from the higher frequency determines the radius within which there is appreciable phase-sensitive coherence. Furthermore, this coherence radius is symmetric in $\Omega$, so, as the phase-sensitive bandwidth of the source increases, the width of the image-plane phase-sensitive point-spread function decreases. However, the weighting coefficient $1-u^{2}$ in Eq.~\eqref{PS:nodimen} counteracts this advantage by attenuating the frequency contributions with higher detunings, such that the net reduction in the main lobe's width is very small.
Notice that we have made no reference to the classical or quantum nature of the source in explaining the physics governing the point-spread functions' width. Thus, this effect is entirely a consequence of phase-sensitive versus phase-insensitive source correlations and scalar paraxial diffraction theory, both of which are valid in the classical and quantum theories of light. The quantum nature of the fields, therefore, does not play a role in determining the resolution capabilities of thin-lens correlation imaging, regardless of whether the source has phase-sensitive or phase-insensitive coherence. However, for particular measurement schemes, nonclassical field states may offer contrast advantages akin to those found in the previous section for diffraction-pattern imaging. In particular, in this section we have determined that the phase-sensitive correlation differs from its phase-insensitive counterpart only in the broadband limit. Thus, if we opt to utilize a photocurrent correlation measurement, then the contrast will be significantly better when the broadband fields' state is maximally-entangled (nonclassical) and has low-brightness, which encompasses the biphoton state.
\section{Discussion} \label{CH6:discussion}
SPDC with vacuum-state inputs generates signal and idler fields in a zero-mean jointly-Gaussian state, with nonzero phase-insensitive autocorrelations and a phase-sensitive cross correlation that fully determine their joint state. When the output state is driven to the low-brightness, low-flux limit, this Gaussian state becomes equivalent to a dominant vacuum state plus a weak biphoton contribution, in which the biphoton wave function equals the phase-sensitive cross correlation between the signal and idler fields. On the other hand, classical imagers have traditionally utilized optical sources in thermal states or coherent states, both of which are Gaussian states but have only nonzero phase-insensitive correlations. Hence, quantum imaging experiments that rely on biphoton sources, as well as conventional classical imaging configurations, can be unified and generalized by studying the imaging characteristics of Gaussian-state sources. Furthermore, such states are fully characterized by their first and second moments, and are closed under linear transformations on the field operators. So, imaging configurations utilizing Gaussian-state sources, linear optical elements and free-space propagation can be fully understood in both the classical and quantum regimes by tracking the evolution of the first and second moments of the fields from the source plane to the detection planes.
A particularly relevant distinction that has been overlooked in most previous work is the phase-sensitive nature of the correlation between the two photons in a biphoton state, as opposed to the phase-insensitive correlation that is present between thermal-state fields. Phase-sensitive coherence has propagation characteristics that differ from those of phase-insensitive coherence. Furthermore, complex-stationary phase-sensitive correlations have cross-frequency couplings that are not present in complex-stationary phase-insensitive correlations. Distinctions such as these often underlie the interesting observations and theoretical predictions in quantum imaging. However, phase-sensitive coherence is not exclusive to nonclassical states (such as the biphoton). Classical Gaussian states (random mixtures of coherent states) may very well have nonzero phase-sensitive correlations, and those features in quantum imaging that stem from the phase-sensitive coherence between the two photons in a biphoton state can be replicated with classical phase-sensitive sources, as we have previously demonstrated for optical coherence tomography and ghost imaging \cite{ErkmenShapiro:PCOCT,ErkmenShapiro:GhostImaging2}.
In this paper we continued to distinguish the truly quantum phenomena in quantum imaging theory and experiments from the phase-sensitive coherence phenomena that can be exploited both in the classical and quantum regimes. Toward this end, we performed Gaussian-state analyses of two significant experimental configurations in biphoton-state quantum imaging. In Sec.~\ref{CH6:FarFieldDiffraction} we showed that the factor-of-two spatial compression in the far-field diffraction pattern of a transmission mask placed at the source plane is precisely due to the phase-sensitive cross correlation between the signal and idler fields, in {\em both} the classical and quantum regimes. Indeed, the only significant difference---insofar as this experiment is concerned---between phase-sensitive classical and quantum sources is the image contrast when photocurrent correlation measurements are employed. Narrowband classical Gaussian states can achieve acceptable contrast, but the contrast degrades severely when the source is broadband. On the other hand, with low-brightness quantum Gaussian states that are maximally-entangled, the contrast is high for both narrowband and broadband sources. Note that the strength of the background in the signature may be a relevant factor in determining whether a classical or quantum source is more desirable for a particular application. For example, in photolithographic applications, in which extraneous noise may be eliminated by virtue of operation in a controlled environment, a biphoton-state source in combination with a two-photon absorber at the detection plane generates an optical image with no background, whereas a classical phase-sensitive source yields significant background that requires postdetection processing prior to etching. Hence the contrast advantage offered by the biphoton state---which cannot be replicated by classical phase-sensitive light---is a desirable feature in this case.
In Sec.~\ref{CH6:BroadbandImaging} we compared thin-lens imaging of a source plane transmission mask using incoherent phase-insensitive light to the same imaging arrangement using phase-sensitive light. When the sources are narrowband (quasimonochromatic), the point-spread functions of the two cases turn out to be identical, yielding no resolution difference between the various source possibilities. As the source bandwidth increases, the point-spread functions for the phase-insensitive and phase-sensitive correlation functions become narrower, with the phase-insensitive point-spread function developing a more slowly decaying tail. The differences between the two cases stem from complex-stationary source statistics, and frequency-dependent free-space diffraction. Once again, the biphoton state facilitates a high-contrast image and a convenient measurement apparatus (coincidence counting) for detecting phase-sensitive correlation, but it is not responsible for the physics governing the changes to the point-spread functions.
Although Sec.~\ref{CH6:FarFieldDiffraction} concentrated on a biphoton proof-of-principle experiment for quantum lithography, the driving motivation for quantum optical lithography is the $N$-fold improvement in etching resolution that is predicted for a system using $N00N$ states \cite{Boto}. The $N00N$ state is an equal-weight superposition of two pure states: an $N$-photon signal field and a vacuum-state idler, plus a vacuum-state signal field and an $N$-photon idler. The $N00N$ state is nonclassical; its $P$ representation in terms of coherent states is not a proper probability density. The $N=2$ case can be achieved with biphoton states, viz., the output of a 50-50 beam splitter when the two inputs are the signal and idler fields from SPDC operating in the low-brightness, low-flux regime. Unfortunately, generating $N00N$ states for $N>2$ has proven challenging. Thus far the interference fringes for the $N=3$ and $N=4$ cases have been demonstrated in proof-of-principle experiments \cite{Mitchell:3003state,Walther:4004state}, showing factors of $3$ and $4$ fringe compression respectively, and efforts to generate higher orders continue. $N00N$ states with $N>2$ are not Gaussian states or any limiting form of Gaussian states, because their second-order moments do not determine the state. Therefore, the Gaussian-state analysis presented in this paper does not generalize to $N00N$ states with $N >2$. As a result, to better appreciate the fundamental physics that leads to improved resolution with these sources, it is of great interest to develop a unifying coherence theory for higher-order moments of continuous field operators, and perform an analysis for these moments to determine whether the advantages observed with these states are truly due to their nonclassical nature or due to a measurement of a $2N$th-order moment of the field operator.
The analyses presented in Secs.~\ref{CH6:FarFieldDiffraction} and \ref{CH6:BroadbandImaging} reveal that the physics governing the resolution improvement in Fourier-plane and thin-lens imaging are {\em different}. Specifically, the improvement in resolution that is observed in the Fourier-plane measurement is due to the difference in paraxial propagation of phase-sensitive and phase-insensitive correlations, and is valid in the quasimonochromatic regime as well as in the broadband regime. Furthermore, to observe this effect with classical fields, it is preferable to utilize narrowband sources. On the other hand, the marginal improvement in resolution observed in Sec.~\ref{CH6:BroadbandImaging} is a {\em strictly} broadband effect that manifests itself in complex-stationary phase-sensitive and phase-insensitive correlation functions. The two experiments capitalize on different properties of phase-sensitive coherence and therefore they are not experiments demonstrating equivalent physical principles.
In conclusion, we have presented a unified Gaussian-state analysis of two transverse imaging configurations, one that images the far-field diffraction pattern of a source-plane transmission mask, and one that performs thin-lens imaging of that source-plane transmission mask. We have shown that the far-field diffraction patterns obtained with classical phase-sensitive Gaussian-state light and nonclassical Gaussian-state light with low-brightness---such as the biphoton---differ only in contrast, viz., the fringe compression is a classical phenomenon owing to the far-field diffraction of phase-sensitive coherence. In the second experiment, we have demonstrated that the cross-frequency coupling in complex-stationary broadband phase-sensitive light---whether classical or quantum---leads to a slightly narrower point-spread function than that obtained with quasimonochromatic phase-sensitive or phase-insensitive light. However, because of the enormous bandwidth that is necessary to observe any appreciable change in the point-spread function, contrary to what is stated in \cite{Shih:QI}, there is no practical advantage to be gained from broadband operation in this image acquisition configuration.
\end{document}
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arXiv
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\begin{document}
\title{Branching-Time Model Checking Gap-Order Constraint Systems (Extended
Version)}
\begin{abstract}
We consider the model checking problem for Gap-order Constraint Systems (GCS) w.r.t.\ the branching-time temporal logic CTL, and in particular its fragments \text{EG}\ and \text{EF}. GCS are nondeterministic infinitely branching processes described by evolutions of integer-valued variables, subject to Presburger constraints of the form $x-y\ge k$, where $x$ and $y$ are variables or constants and $k\in\mathbb{N}$ is a non-negative constant. We show that \text{EG}\ model checking is undecidable for GCS, while \text{EF}\ is decidable. In particular, this implies the decidability of strong and weak bisimulation equivalence between GCS and finite-state systems.
\end{abstract}
\section{Introduction}
Counter machines \cite{Min1967} extend a finite control-structure with unbounded memory in the form of counters that can hold arbitrarily large integers (or natural numbers), and thus resemble basic programming languages. However, almost all behavioral properties, e.g., reachability and termination, are undecidable for counter machines with two or more counters \cite{Min1967}. For the purpose of formal software verification, various formalisms have been defined that approximate counter machines and still retain the decidability of some properties. E.g., Petri nets model weaker counters that cannot be tested for zero, and have a decidable reachability problem \cite{May1984}.
Gap-order constraint systems \cite{Rev1993,FR1996,Boz2012,BP2012} are another model that approximates the behavior of counter machines. They are nondeterministic infinitely branching processes described by evolutions of integer-valued variables, subject to Presburger constraints of the form $x-y\ge k$, where $x$ and $y$ are variables or constants and $k\in\mathbb{N}$ is a non-negative constant. Unlike in Petri nets, the counters can be tested for zero, but computation steps still have a certain type of monotonicity that yields a decidable reachability problem. In fact, control-state reachability is decidable even for the more general class of constrained multiset rewriting systems \cite{AD2006}.
\paragraph{Previous work.} Beyond reachability, several model checking problems have been studied for GCS and related formalisms. The paper \cite{Cer1994} studies Integral Relational Automata (IRA), a model that is subsumed by GCS, that allows only constraints of the form $x\ge y$ or $x > y$, where $y$ and $x$ are variables or constants. It is shown that \text{CTL}\ model checking of IRA is undecidable, even for the restriction \text{RCTL}, that forbids next-state modalities. In contrast, model checking IRA remains decidable for the existential and universal fragments of \text{CTL*}. Models of equal expressivity include the monotonicity constraint systems (MCS) of \cite{Ben2009} and $(\mathbb{Z},<,=)$-automata in \cite{DD2007}. Demri and D'Souza \cite{DD2007} show that satisfiability and model checking \text{LTL}\ is decidable and PSPACE-complete.
Bozzelli and Pinchinat \cite{Boz2012,BP2012} study the more general model of gap-order constraint systems (GCS), which strictly extend the models mentioned above. They show that model checking GCS is decidable and PSPACE-complete for the logic \text{EGCCTL*}, but undecidable for \text{AGCCTL*}, which are the existential and universal fragments of \text{CTL*}, respectively, extended with gap constraints as atomic propositions. Moreover, satisfiability is PSPACE-complete for both these fragments. \text{EGCCTL*}\ and \text{AGCCTL*}\ are not dual, since gap constraints are not closed under negation. Moreover, they are orthogonal to the fragments \text{EF}\ and \text{EG}\ considered in this paper, which allow nesting of negation and the operator $\OP{EF}$ (resp. $\OP{EG}$). Checking fairness (the existence of infinite runs where a variable has a fixed value infinitely often) and thus termination (the non-existence of infinite runs), and also strong termination (the existence of a bound on the length of all runs) are decidable in polynomial space \cite{BP2012,Boz2012}. An important ingredient for these results are effectively constructible under-approximations of the set of GCS runs induced by a given sequence of transitions, which preserve enabledness (Thm.~2 in \cite{BP2012}). This comes at the cost of losing information about the induced runs. In particular, it is impossible to recover (a representation of) the exact set of runs induced by a sequence of transitions from its approximation.
\paragraph{Our contribution.} We study the decidability of model checking problems for GCS with fragments of computation-tree logic (CTL), namely \text{EG}\ and \text{EF}\ (see e.g.~\cite{Esp1997}).
We first show that \text{EG}-model checking is undecidable, even for the weaker model of IRA \cite{Cer1994}. On the other hand, model checking GCS with respect to \text{EF}\ remains decidable. This positive result is based on the observation that one can use boolean combinations of gap constraints to represent the sets of variable valuations satisfying a given \text{EF}\ formula, and that such a representation can be effectively computed in a bottom-up fashion. An immediate consequence is that checking strong and weak bisimulation equivalence is decidable between GCS and finite-state systems.
\section{Gap-Order Constraint Systems}
Let $\mathbb{Z}$ and $\mathbb{N}$ denote the sets of integers and non-negative integers.
A \emph{labeled transition system} (LTS) is described by a triple $T=(V,\mathit{Act},\step{})$ where $V$ is a (possibly infinite) set of states, $\mathit{Act}$ is a finite set of action labels and $\longrightarrow\,\subseteq V\times \mathit{Act}\times V$ is the labeled transition relation. We use the infix notation $s\step{a}s'$ for a transition $(s,a,s')\in\ \step{}$, in which case we say $T$ makes an \emph{$a$-step} from $s$ to $s'$. For a set $S\subseteq V$ of states and $a\in\mathit{Act}$ we define the set of $a$-predecessors by
$\Pre{a}{S} = \{s'|s'\step{a}s\in S\}$. We write $\step{}^*$ for the transitive and reflexive closure of $\step{}$ and let $Pre^*(S) = \{s'|s'\step{}^*s\in S\}$.
We fix a finite set ${\it Var}$ of \emph{variables} ranging over the integers and a finite set ${\it Const}\subseteq\mathbb{Z}$ of constants. Let ${\it Val}$ denote the set of variable \emph{valuations} $\nu:{\it Var}\to\mathbb{Z}$. To simplify the notation, we will extend the domain of valuations to constants, where they behave as the identity, i.e., $\nu(c)=c$ for all $c\in \mathbb{Z}$.
\begin{definition}[Gap Constraints]
A \emph{gap clause} over (${\it Var},{\it Const}$)
is an inequation of the form
\begin{equation}
(x-y\ge k)
\end{equation}
where $x,y\in {\it Var}\cup {\it Const}$ and $k\in \mathbb{Z}$.
A clause is called \emph{positive} if $k\in\mathbb{N}$.
A (positive) gap \emph{constraint} is a finite conjunction of (positive) gap clauses.
A \emph{gap formula} is an arbitrary boolean combination of gap clauses.
A valuation $\nu:{\it Var}\to\mathbb{Z}$ satisfies the clause $\Con{C}:(x-y)\ge k$
(write $\nu\models \Con{C}$) if it respects the prescribed inequality. That is,
\begin{equation}
\nu\models (x-y)\ge k \iff \nu(x)-\nu(y)\ge k.
\end{equation}
We define the satisfiability of arbitrary gap formulae inductively in the usual
fashion and write
${\it Sat}(\varphi) = \{\nu\in {\it Val}\ |\ \nu\models\varphi\}$
for the set of valuations that satisfy the formula $\varphi$. In particular, a valuation
satisfies a gap constraint iff it satisfies all its clauses.
A set $S\subseteq {\it Val}$ of valuations is called \emph{gap definable} if there is a gap formula
$\varphi$ with $S={\it Sat}(\varphi)$. \end{definition}
We will consider processes whose states are described by valuations and whose dynamics is described by stepwise changes in these variable valuations, according to positive gap constraints.
Let ${\it Var}'=\{x' \,|\, x\in {\it Var}\}$ be the set of primed copies of the variables. These new variables are used to express constraints on how values can change when moving from one valuation to another: $x'$ is interpreted as the next value of variable $x$. A \emph{transitional} gap clause (-constraint, -formula) is a gap clause (-constraint, -formula) with variables in ${\it Var}\cup {\it Var}'$. The combination $\nu\oplus\nu':{\it Var}\cup {\it Var}'\to\mathbb{Z}$ of two valuations $\nu,\nu':{\it Var}\to \mathbb{Z}$ maps variables $x\in{\it Var}$ to $\nu\oplus\nu'(x)=\nu(x)$ and $x'\in{\it Var}'$ to $\nu\oplus\nu'(x')=\nu'(x)$.
Transitional gap clauses can be used as conditions on how valuations may evolve in one step. For instance, $\nu$ may change to $\nu'$ only if $\nu\oplus\nu'\models\varphi$ for some gap clause $\varphi$.
\begin{definition}
A \emph{Gap-Order Constraint System} (GCS) is given by
a finite set of \emph{positive} transitional gap constraints
together with a labeling function.
Formally, a GCS is a tuple $\mathcal{G}=({\it Var},{\it Const},\mathit{Act},\Delta,\lambda)$ where ${\it Var},{\it Const},\mathit{Act}$ are
finite sets of variables, constants and action symbols, $\Delta$ is a finite set of
\emph{positive} transitional gap constraints over $({\it Var},{\it Const})$ and
$\lambda:\Delta\to\mathit{Act}$ is a labeling function.
Its operational semantics is given by an infinite LTS with states ${\it Val}$ where
\begin{equation}
\nu\step{a}\nu' \iff \nu\oplus\nu'\models\Con{C}
\end{equation}
for some constraint $\Con{C}\in\Delta$ with $\lambda(\Con{C})=a$.
For a set $M\subseteq {\it Val}$ of valuations we write $\Pre{\Con{C}}{M}$ for the set
$\{\nu\ |\ \exists \nu'\in M.\, \nu\oplus\nu'\models\Con{C}\}$
of $\Con{C}$-predecessors. \end{definition}
Observe that a positive gap constraint $(x-0 \ge 0)\ \land\ (0-x\ge 0)$ is satisfied only by valuations assigning value $0$ to variable $x$. Similarly, one can test if a valuation equates two variables. Also, it is easy to simulate a finite control in a GCS using additional variables.\footnote{
In fact, \cite{BP2012,Boz2012} consider an equivalent notion of GCS that
explicitly includes a finite control.} What makes this model computationally non-universal is the fact that we demand \emph{positive} constraints: while one can easily demand an increase or decrease of variable $x$ by \emph{at least} some offset $k\in\mathbb{N}$, one cannot demand a difference of \emph{at most} $k$ (nor exactly $k$).
\begin{example}
\label{ex:countdown}
Consider the GCS with variables $\{x,y\}$ and single constant $\{0\}$ with two constraints $\Delta=\{\Con{C}X,\Con{C}Y\}$ for which $\lambda(\Con{C}X)=a$ and $\lambda(\Con{C}Y)=b$. \begin{align}
\Con{C}X = &((x-x'\ge 1) \land\ (y'-y\ge 0)\ \land\ (y-y'\ge 0)\ \land\ (x'-0\ge 0))\\
\Con{C}Y = &((y-y'\ge 1)\ \land\ (x'-x\ge 0)\ \land\ (y'-0\ge 0)). \end{align} This implements a sort of lossy countdown where every step strictly decreases the tuple $(y,x)$ lexicographically: $\Con{C}X$ induces $a$-steps that decrease $x$ while preserving the value of $y$ and $\Con{C}Y$ induces $b$-steps that increase $x$ arbitrarily but have to decrease $y$ at the same time. The last clauses in both constraints ensure that $x$ and $y$ never change from a non-negative to a negative value.
\end{example} In the sequel, we allow ourselves to abbreviate constraints for the sake of readability. For instance, the constraint $\Con{C}X$ in the previous example could equivalently be written as $(x>x'\ge 0)\ \land\ (y=y')$.
\section{Branching-Time Logics for GCS}
We consider (sublogics of) the branching-time logic CTL over processes defined by gap-order constraint systems, where atomic propositions are gap clauses. The denotation of an atomic proposition $\mathcal{C} = (x-y\ge k)$ is $\DEN{\mathcal{C}} = Sat(\mathcal{C})$, the set of valuations satisfying the constraint. Well-formed CTL formulae are inductively defined by the following grammar, where $\Con{C}$ ranges over the atomic propositions and $a \in \mathit{Act}$ over the action symbols. \begin{equation}
\label{eq:grammar}
\psi ::= \Con{C} \ \ |\ \ true\ \ |\ \ \lnot \psi\ \ |\ \ \psi \land \psi\ \ |\ \ \exnext{a}\psi
\ \ |\ \ \OP{EF}\psi\ \ |\ \ \OP{EG}\psi\ \ |\ \ \OP{E}(\psi \OP{U} \psi) \end{equation} To define the semantics, we fix a GCS $\mathcal{G}$. Let ${\it Paths}^\omega(\nu_0)$ be the set of infinite derivations \begin{equation}
\pi = \nu_0\step{a_0}\nu_1\step{a_1}\nu_2\dots \end{equation} of $\mathcal{G}$ starting with valuation $\nu_0\in {\it Val}$ and let $\pi(i)=\nu_i$ denote the $i$-th valuation $\nu_i$ on $\pi$. Similarly, we write ${\it Paths}^*(\nu_0)$ for the set of finite derivations starting from $\nu_0$. The denotation of formulae, with respect to the fixed GCS $\mathcal{G}$, is defined in the standard way. \begin{align}
\DEN{\Con{C}} &= Sat(\Con{C})\\
\DEN{{\it true}} &= {\it Val}\\
\DEN{\lnot \psi} &= {\it Val}\setminus \DEN{\psi}\\
\DEN{\psi_1\land\psi_2} &= \DEN{\psi_1}\cap\DEN{\psi_2}\\
\DEN{\exnext{a}\psi} &= \Pre{a}{\DEN{\psi}}\\
\DEN{\OP{EF}\psi} &= \{\nu\ |\ \exists \pi \in Paths^*(\nu).\ \exists i\in\mathbb{N}.\ \pi(i)\in\DEN{\psi}\}\\
\DEN{\OP{EG}\psi} &= \{\nu\ |\ \exists \pi \in Paths^\omega(\nu).\ \forall i\in\mathbb{N}.\ \pi(i)\in\DEN{\psi}\}\\
\DEN{\OP{E}(\psi_1 \OP{U} \psi_2)} &= \{\nu\ |\ \exists \pi\in Paths^*(\nu).\ \exists i\in\mathbb{N}.
\pi(i)\in\DEN{\psi_2} \land
\forall j<i. \pi(j)\in\DEN{\psi_1}\} \end{align} We use the usual syntactic abbreviations ${\it false} = \lnot {\it true}$, $\psi_1\lor\psi_2 = \lnot(\lnot\psi_1\land\lnot\psi_2)$.
The sublogics $\text{EF}$ and $\text{EG}$ are defined by restricting the grammar \eqref{eq:grammar} defining well-formed formulae: \text{EG}\ disallows subformulae of the form $\OP{E}(\psi_1 \OP{U}\psi_2)$ and $\OP{EF}\psi$ and in \text{EF}, no subformulae of the form $\OP{E}(\psi_1 \OP{U} \psi_2)$ or $\OP{EG}\psi$ are allowed. The \emph{Model Checking Problem} is the following decision problem.
\begin{tabular}{ll}
{\sc Input:} &A GCS $G=({\it Var},Const,\mathit{Act},\Delta,\lambda)$, a valuation $\nu:{\it Var}\to\mathbb{Z}$\\
&and a formula $\psi$.\\
{\sc Question:} &$\nu\models \psi$? \end{tabular}
Cerans \cite{Cer1994} showed that general \text{CTL}\ model checking is undecidable for gap-order constraint systems. This result holds even for restricted \text{CTL}\ without \emph{next} operators $\exnext{a}$. In the following section, we show a similar undecidability result for the fragment \text{EG}. On the other hand, model checking GCS with the fragment \text{EF}\ turns out to be decidable; cf.\ Section~\ref{sec:EF}.
\section{Undecidability of EG Model Checking}
\label{sec:EG}
\begin{theorem} \label{thm:eg}
The model checking problem for \text{EG}\ formulae over GCS is undecidable. \end{theorem}
\begin{proof}
By reduction from the halting problem of deterministic $2$-counter Minsky Machines (2CM).
$2$-counter machines consist of a deterministic finite control, including a designated halting
control-state ${\it halt}$ and two integer counters that can be incremented and decremented
by one and tested
for zero. Checking if such a machine reaches the halting state
from an initial configuration with control-state ${\it init}$ and
counter values $x_1=x_2=0$ is undecidable \cite{Min1967}.
Given a 2CM $M$, we will construct a GCS together with an initial valuation $\nu_0$ and a \text{EG}\
formula $\psi$ such that $\nu_0\models\psi$ iff $M$ does not halt.
First of all, observe that we can simulate a finite control of $n$ states using one additional
variable $state$ that will only ever be assigned values from $1$ to $n$.
To do this, let $[p]\le n$ be the index of state $p$ in an arbitrary enumeration of the state
set. Now, a transition $p\step{}q$ from state $p$ to $q$ introduces the constraint
$(state=[p]\land state' = [q])$. We will abbreviate such constraints by $(p\step{}q)$ in the
sequel and simply write $p$ to mean the clause $(state=[p])$.
We use two variables $x_1,x_2$ to act as integer counters. Zero-tests can then directly be
implemented as constraints $(x_1=0)$ or $(x_2=0)$.
It remains to show how to simulate increments and decrements by exactly $1$.
Our GCS will use two auxiliary variables $y,z$ and a new state ${\it err}$.
We show how to implement increments by one; decrements can be done analogously.
Consider the $x_1$-increment $p\step{x_1=x_1+1}q$ that takes the 2CM from state $p$ to $q$ and
increments the counter $x_1$. The GCS will simulate this in two steps, as depicted in
Figure~\ref{fig:eg-inc} below.
\begin{figure}
\caption{Forcing faithful simulation of $x_1$-increment.
All steps contain the additional constraint $x_2'=x_2$, which is not shown,
to preserve the value of the other counter $x_2$.
}
\label{fig:eg-inc}
\end{figure}
The first step can arbitrarily increment $x_1$ and will remember (in variable $y$) the old value
of $x_1$. The second step does not change any values and just moves to the new control-state.
However, incrementing by more than one in the first step enables an extra move to the error state
${\it err}$ afterwards. This error-move is enabled if one can assign a value to variable $z$ that is
strictly in between the old and new value of $x_1$, which is true iff the increment in step 1
was not faithful.
The incrementing transition of the 2CM is thus translated to the following three constraints.
\begin{align}
&(p\step{}to_q)\land (x_1'>x_1=y') \land (x_2'=x_2)\\
&(to_q\step{}q)\land (x_1'=x_1)\land (x_2'=x_2)\\
&(to_q\step{}{\it err})\land (y<z'<x_1).
\end{align}
If we translate all operations of the 2CM into the GCS formalism as indicated above,
we end up with an over-approximation of the 2CM that allows runs that faithfully
simulate runs in the 2CM but also runs which `cheat' and possibly
increment or decrement by more than one and still don't go to state ${\it err}$ in the following step.
We enforce a faithful simulation of the 2CM by using the formula that is to be checked, demanding
that the error-detecting move is never enabled.
The GCS will only use a unary alphabet $\mathit{Act}=\{a\}$ to label constraints. In particular,
observe that the formula $\exnext{a}err$ holds in every configuration which can move to state
${\it err}$ in one step. Now, the \text{EG}\ formula
\begin{equation}
\psi= \OP{EG}(\lnot halt\ \land\ \lnot\, \exnext{a}{\it err})
\end{equation}
asserts that there is an infinite path which never visits state ${\it halt}$ and
along which no step to state ${\it err}$ is ever enabled.
This means $\psi$ is satisfied by valuation $\nu_0=\{state=[init], x_1=x_2=y=z=0\}$ iff there is
a faithful simulation of the 2CM from initial state ${\it init}$ with both counters set to $0$
that never visits the halting state.
Since the 2CM is deterministic, there is only one way
to faithfully simulate it and hence $\nu_0\models\psi$ iff the 2CM does not halt.
Notice that the constructed GCS is in fact an IRA \cite{Cer1994},
since it only uses gap constraints of the form $x>y$ or $x=y$.
\qed \end{proof}
\section{Decidability of EF Model Checking}
\label{sec:EF}
Let us fix sets ${\it Var}$ and ${\it Const}$ of variables and constants, respectively. We will use an alternative characterization of gap constraints in terms of \emph{monotonicity graphs} \footnote{These were called \emph{Graphose Inequality Systems} in \cite{Cer1994}
and \emph{gap-graphs} in \cite{Rev1993}. }, which are finite graphs with nodes ${\it Var}\cup {\it Const}$. Monotonicity graphs are used to represent sets of variable valuations. We show that so represented sets are effectively closed under all logical connectors allowed in \text{EF}, and one can thus evaluate a formula bottom up.
\begin{definition}[Monotonicity Graphs]
\label{def:MGs}
A \emph{monotonicity graph} (MG) over $({\it Var}, {\it Const})$ is a finite, directed graph $M=(V,E)$ with
nodes $V={\it Var}\cup {\it Const}$ and in which each edge in $E$ carries a
\emph{weight} in $\mathbb{Z}\cup\{-\infty,\infty\}$.
The \emph{degree} of $M$ is the
largest $k\in\mathbb{N}$ such that there is an edge with weight $-k$ in $M$
or $0$ if no edge has weight in $\mathbb{Z}\setminus\mathbb{N}$.
The degree of a set $\{M_0,M_1,\dots,M_j\}$ of MG is defined as the
maximal degree of any MG $M_k$ in the set.
A valuation $\nu:{\it Var}\to\mathbb{Z}$ satisfies $M$
(write $\nu\models M$) if for every edge
$(x\step{k}y)$ it holds that
$\nu(x)-\nu(y)\ge k$.
Let ${\it Sat}(M)$ denote the set of
valuations satisfying $M$.
A set $S\subseteq {\it Val}$ is \emph{MG-definable} if there is a finite set
$\{M_0,M_1,\dots,M_j\}$ of MG such that
\begin{equation}
S=\bigcup_{0\le i\le j} Sat(M_i)
\end{equation}
and called $\text{MG}^n$-definable if there is such a set of MG with degree
$\le n$. We write $\textit{MG}$ and $\textit{MG}^n$ for the classes of
$\text{MG}$- and $\text{MG}^n$-definable sets respectively.
For a monotonicity graph $M$,
we write $M(x,y) \in\{-\infty,\infty\}\cup\mathbb{Z}$ for the least upper bound
of the cumulative weight of all paths from node $x$ to node $y$.
Note that this is $-\infty$ if there is no such path.
The \emph{closure} $|M|$ is the unique complete MG with edges $x\step{M(x,y)}y$ for all $x,y \in {\it Var}\cup {\it Const}$. \end{definition} The following lemma and definition state some basic properties of monotonicity graphs that can easily be verified; see \cite{Cer1994}.
\begin{lemma} \label{lem:mg-basics}
\
\begin{enumerate}
\item ${\it Sat}(M)=\emptyset$ holds for any monotonicity graph $M$ that contains an edge
with weight $\infty$
or a cycle with positive weight sum.
\item $|M|$ is polynomial-time computable from $M$ and ${\it Sat}(M) = {\it Sat}(|M|)$.
\item If we fix sets ${\it Var},{\it Const}$ of variables and constants then
for any gap constraint $\Con{C}$ there is a unique monotonicity graph
$M_{\Con{C}}$,
containing an edge $x\step{k}y$ iff there is a clause $x-y\ge k$ in $\Con{C}$.
Moreover, ${\it Sat}(M_{\Con{C}})={\it Sat}(\Con{C})$.
\end{enumerate} \end{lemma} The last point of this lemma states that monotonicity graphs and gap constraints are equivalent formalisms. We call a MG \emph{positive} if it has degree $0$. Positive MG are equivalent to positive gap constraints. We thus talk about \emph{transitional} monotonicity graphs over $({\it Var},{\it Const})$ as those with nodes ${\it Var}\cup {\it Var}'\cup {\it Const}$. We further define the following operations on MG. \begin{definition} \label{def:mg-ops}
Let $M,N$ be monotonicity graphs over ${\it Var}, {\it Const}$ and $V\subseteq {\it Var}$.
\begin{itemize}
\item The \emph{restriction} $M|_V$ of $M$ to $V$ is the maximal subgraph of $M$ with nodes
$V\cup {\it Const}$.
\item The \emph{projection} ${\it Proj}(M,V) = |M|_V$ is the restriction of $M$'s closure to
$V$.
\item The \emph{intersection} $M\otimes N$ is the MG that contains an edge $x\step{k}y$ if
$k$ is the maximal weight of any edge from $x$ to $y$ in $M$ or $N$.
\item The \emph{composition} $G\circ M$ of a \emph{transitional} MG $G$ and $M$ is obtained
by consistently renaming variables in $M$ to their primed copies, intersecting the
result with $G$ and projecting to ${\it Var}\cup {\it Const}$.
$G\circ M:= {\it Proj}(M_{[{\it Var}\mapsto {\it Var}']}\otimes G, {\it Var})$.
\end{itemize} \end{definition} These operations are surely computable in polynomial time. The next lemma states important properties of these operations; see also \cite{Cer1994,BP2012}. \begin{lemma} \label{lem:mg-ops}
\
\begin{enumerate}
\item ${\it Sat}({\it Proj}(M,V)) = \{\nu|_V : \nu\in {\it Sat}(M)\}$.
\item ${\it Sat}(M\otimes N) = {\it Sat}(M)\cap {\it Sat}(N)$
\item ${\it Sat}(G\circ M) = \{\nu\ |\ \exists \nu'\in {\it Sat}(M).\ \nu\oplus\nu'\in {\it Sat}(G)\} = \Pre{G}{M}$.
\item If $M$ has degree $n$ and $G$ is a transitional MG of degree $0$,
then $G\circ M$ has degree $\le n$.
\end{enumerate} \end{lemma}
\begin{example}
\label{ex:mg}
The monotonicity graph on the left below corresponds to the contraint $\Con{C}X$ in Example~\ref{ex:countdown}. On the right we see its closure (where edges with weight $-\infty$ are omitted). Both have degree $0$. \begin{center}
\begin{minipage}[c]{.4\textwidth}
\centering
\begin{tikzpicture}[scale=1.0,node distance=2cm]
\node(x) [mgnode] {$x$};
\node(x')[mgnode,right of=x] {$x'$};
\node(y) [mgnode,below of=x] {$y$};
\node(y')[mgnode,right of=y] {$y'$};
\node (0)[mgnode] at ($(x)!0.5!(y')$) {$0$};
\path[->] (x) edge node[label] {$1$} (x');
\path[->] (y') edge[bend left=10] node[auto,label,pos=0.45] {$0$} (y);
\path[->] (y) edge[bend left=10] node[auto,label,pos=0.45] {$0$} (y');
\path[->] (x') edge node[auto,label] {$0$} (0);
\end{tikzpicture}
\end{minipage}
\begin{minipage}[c]{.4\textwidth}
\centering
\begin{tikzpicture}[scale=1.0,node distance=2cm]
\node(x) [mgnode] {$x$};
\node(x')[mgnode,right of=x] {$x'$};
\node(y) [mgnode,below of=x] {$y$};
\node(y')[mgnode,right of=y] {$y'$};
\node (0)[mgnode] at ($(x)!0.5!(y')$) {$0$};
\path[->] (x) edge node[label] {$1$} (x');
\path[->] (x) edge node[label,swap] {$1$} (0);
\path[->] (y') edge[bend left=10] node[auto,label,pos=0.45] {$0$} (y);
\path[->] (y) edge[bend left=10] node[auto,label,pos=0.45] {$0$} (y');
\path[->] (x') edge node[auto,label] {$0$} (0);
\end{tikzpicture}
\end{minipage} \end{center}
Let us compute the $\Con{C}X$-predecessors of the set $S= \{\nu\ |\ \nu(x)>\nu(y)=0\}$ which is characterized by the single MG on the right below. \begin{center}
\begin{minipage}[c]{.25\textwidth}
\centering
\begin{tikzpicture}[scale=1.0,node distance=2cm]
\node (0)[mgnode] at (1,0) {$0$};
\node(x) at (0,1) [mgnode] {$x$};
\node(y) [mgnode] at (0,-1){$y$};
\path[->] (0) edge[bend left=10] node[auto,label,pos=0.45] {$0$} (y);
\path[->] (y) edge[bend left=10] node[auto,label,pos=0.45] {$0$} (0);
\path[->] (x) edge node[auto,label] {$2$} (0);
\path[->] (x) edge node[swap,label] {$2$} (y);
\end{tikzpicture}
\end{minipage}
\begin{minipage}[c]{.25\textwidth}
\centering
\begin{tikzpicture}[scale=1.0,node distance=2cm]
\node (0)[mgnode] at (0,0) {$0$};
\node(x) [mgnode] at (-1,1) {$x$};
\node(x')[mgnode] at (1,1) {$x'$};
\node(y) [mgnode] at (-1,-1) {$y$};
\node(y')[mgnode] at (1,-1) {$y'$};
\path[->] (x) edge node[label] {$1$} (x');
\path[->] (y') edge[bend left=10] node[below,label,pos=0.45] {$0$} (y);
\path[->] (y) edge[bend left=10] node[above,label,pos=0.45] {$0$} (y');
\path[->] (x') edge node[swap,label] {$1$} (0);
\path[->] (0) edge[bend left=10] node[right,label,pos=0.45] {$0$} (y');
\path[->] (y') edge[bend left=10] node[left,label,pos=0.45] {$0$} (0);
\end{tikzpicture}
\end{minipage}
\begin{minipage}[c]{.25\textwidth}
\centering
\begin{tikzpicture}[scale=1.0,node distance=2cm]
\node (0)[mgnode] at (0,0) {$0$};
\node(x) at (1,1) [mgnode] {$x$};
\node(y) [mgnode] at (1,-1){$y$};
\path[->] (0) edge[bend left=10] node[auto,label,pos=0.45] {$0$} (y);
\path[->] (y) edge[bend left=10] node[auto,label,pos=0.45] {$0$} (0);
\path[->] (x) edge node[swap,label] {$1$} (0);
\end{tikzpicture}
\end{minipage} \end{center} If we rename variables $x$ and $y$ to $x'$ and $y'$ and intersect the result with $M_{\Con{C}X}$ we get the MG in the middle. We project into $Var\cup Const$ by computing the closure and restricting the result accordingly. This leaves us with the MG on the left, which characterizes the set
$\Pre{\Con{C}X}{S}=\{\nu\ |\ \nu(x)\ge 2\ \land\ \nu(y)=0\}$ as expected.
\end{example}
We have seen how to construct a representation of the $\Con{C}$-predecessors $\Pre{\Con{C}}{S}$ and thus $\Pre{a}{S}$ for a MG-definable set $S$, gap constraint $\Con{C}$ and action $a\in\mathit{Act}$. The next lemma is a consequence of Lemma~\ref{lem:mg-basics}, point 3 and asserts that we can do the same for complements. \begin{lemma}
\label{lem:neg}
The class of MG-definable sets is effectively closed under complementation. \end{lemma} \begin{proof}
By Lemma~\ref{lem:mg-basics} we can interpret a finite set of MG $\mathcal{M} = \{M_0,M_1,\dots,M_k\}$ as
a gap formula in DNF. One can then use De Morgan's laws to propagate negations to atomic
propositions, which are gap clauses of the form $x-y\ge k$. The negation is
then expressible as $x-y < k$, which is equivalent to $y-x > -k$ and thus
to the gap clause $y-x\ge -k+1$. It remains to bring the formula into DNF again,
which can then be described by finitely many MGs.
\qed \end{proof} Observe that complementation potentially constructs MG with increased degree. This next degree is bounded by the largest finite weight in the current graph minus one, but nevertheless, an increase of degree cannot be avoided. Therefore, classes of $\text{MG}^n$-definable sets are \emph{not} closed under complementation. \begin{example}
\label{ex:3}
The set $S= \{\nu\ |\ \nu(x) - \nu(y) \ge 5\}$ corresponds to the gap-formula $\varphi = (x-y\ge 5)$. Its MG $\{(x\step{5}y)\}$ is of degree $0$. However, its complement is characterized by the MG $\{(y\step{-4}x)\}$, which has degree $4$.
\end{example}
\newcommand{\sqsubseteq}{\sqsubseteq} It remains to show that we can compute ${\it Pre}^*(S)$ for MG-definable sets $S$. We recall the following partial ordering on monotonicity graphs and its properties \cite{Cer1994}. \begin{definition}
Let $M,N$ be MG over $({\it Var}, {\it Const})$. We say that
$M$ \emph{covers} $N$ (write $N\sqsubseteq M$) if for all
$x,y\in {\it Var}\cup {\it Const}$ it holds that $N(x,y)\le M(x,y)$. \end{definition} \begin{lemma}\ \label{lem:mg-order}
\begin{enumerate}
\item If $N\sqsubseteq M$ then $Sat(N)\supseteq Sat(M)$.
\item
For every $n\in\mathbb{N}$,
$\sqsubseteq$ is a well-order on the set
of MG over $({\it Var},{\it Const})$ with degree $\le n$.
\end{enumerate} \end{lemma} \begin{proof}
For the first claim, assume $\nu\in Sat(M)=Sat(|M|)$.
Then, for every $x,y\in{\it Var}\cup{\it Const}$, we have
$\nu(x)-\nu(y)\ge M(x,y)\ge N(x,y)$. So
$\nu\in Sat(|N|)=Sat(N)$.
The second claim follows from Dickson's Lemma if we interpret each
MG $M$ with degree $n$ as $|{\it Var}\cup{\it Const}|^2$-dimensional vector where
the component for the pair $(x,y)$ has value $n+M(x,y)$. \end{proof} Note that point 1 states that a $\sqsubseteq$-bigger MG is more restrictive and hence has a smaller denotation. Also notice that $\sqsubseteq$ is \emph{not} a well order on the set of all MG due the lack of a bound on finite, negative weights: for instance, the sequence $(M_n)_{n\in\mathbb{N}}$ of MG, where for every $n$, the graph $M_n$ has edges $x\step{n}y\step{-n}x$, is an infinite antichain.
\begin{lemma} \label{lem:ef} Let $S$ be a $\text{MG}^n$-definable set of valuations. Then ${\it Pre}^*(S)$ is $\text{MG}^n$-definable and a representation of ${\it Pre}^*(S)$
can be computed from a representation of $S$. \end{lemma} \begin{proof}
It suffices to show the claim for a set $S$
characterized by a single monotonicity graph $M_S$, because
${\it Pre}^*(S\cup S') = {\it Pre}^*(S)\cup {\it Pre}^*(S')$.
Assume that $M_S$ has degree $n$.
We proceed by exhaustively building a finite tree of MG, starting in $M_S$. For every node $N$ we
compute children $G\circ N$ for all of the finitely many transitional MG $G$ in the system.
Point 4) of Lemma~\ref{lem:mg-ops} guarantees that all intermediate representations have
degree $\le n$. By Lemma~\ref{lem:mg-order}, point 2, any branch eventually ends in a node that
covers a previous one and Lemma~\ref{lem:mg-order}, point 1 allows us to stop exploring such a
branch.
We conclude that ${\it Pre}^*(M)$ can be characterized by the finite union of all intermediate MG.
\qed \end{proof} Finally, we are ready to prove our main result. \begin{theorem} \label{thm:EF-dec}
\text{EF}\ model checking is decidable for Gap-order constraint systems.
Moreover, the set $\DEN{\psi}$ of valuations satisfying an \text{EF}-formula $\psi$ is effectively
gap definable. \end{theorem} \begin{proof}
We can evaluate a formula bottom up, representing the sets satisfying subformulae by finite sets
of MG. Atomic propositions are either ${\it true}$
or gap clauses and can thus be written directly as
MG. For composite formulae we use the properties that MG definable
sets are effectively closed under
intersection (Lemma~\ref{lem:mg-ops}) and negation (Lemma~\ref{lem:neg}),
and that we can compute
representations of $\Pre{a}{S}$ and ${\it Pre}^*(S)$ for MG-definable sets $S$ by
Lemmas~\ref{lem:mg-ops} and \ref{lem:ef}.
The key observation is that although negation (i.e., complementing)
may increase the degree of the intermediate
MG, this happens only finitely often in the bottom up evaluation of an \text{EF}\ formula.
Computing representations for
modalities $\langle a\rangle$ and $\OP{EF}$ does not increase the degree.
\qed \end{proof} \begin{remark}
Since steps in GCS are described by positive transitional gap constraints,
it is straightforward to extend
this positive result of Theorem~\ref{thm:EF-dec} to
model checking GCS w.r.t.~the slightly more general logic
$\text{EF}_\mathcal{C}$, in which the next-state and reachability
modalities
$\exnext{a}_\Con{C}$ and
$\OP{EF}_\Con{C}$ are subject to
transitional gap clauses $\Con{C}$. \end{remark}
The exact complexity of the EF model checking problem for GCS is still open. However, a \textsc{PSpace}\ lower bound already holds for reachability of the simpler model of {\em boolean programs}. Moreover, a rather crude Ackermannian upper complexity bound on EF model checking for GCS can be obtained by bounding ``bad sequences'' in our use of Dickson's Lemma. In the remainder of this section we will elaborate on these reductions.
\subsection{A \textsc{PSpace}\ lower complexity bound} Several equivalent notions of boolean programs are used in different application domains (see e.g., \cite{CS1999,BR2000}). Essentially, they consist of a finite control unit that manipulates finitely many boolean variables.
We define \emph{boolean programs} as finite-state machines with transitions of the form $s\step{g(\vec{x})/a}t$, where $s$ and $t$ are control-states, $a$ is an assignment $x = 0$ or $x=1$ for some variable $x$, and $g(\vec{x})$ is a boolean formula with free variables $\vec{x}$. The semantics of boolean programs is given by the binary step relation $\step{}$ over pairs of control-states and variable valuations. Let ${\it Var}=\{x_1,x_2,\dots,x_k\}$ be the set of variables in the system and let $\nu,\nu':{\it Var}\to\{0,1\}$ be two valuations. A transition $s\step{g(\vec{x})/y=b}t$ induces a step $s,\nu \step{} t,\nu'$ if 1) $\nu\models g$, 2) $\nu'(y) = b$ and 3) $\nu'(x) = \nu(x)$ for $x\in{\it Var}\setminus\{y\}$.
The \emph{state-reachability problem} for boolean programs asks, for two given control states $s$ and $t$, whether there exists a valuation $\nu$ and a finite number of steps from $s,\nu_0$ to $t,\nu$. Here, $\nu_0: x\mapsto 0$ assigns the value $0$ to every variable. We will show that this problem is \textsc{PSpace}\ hard, by reduction from the \textsc{PSpace}-complete satisfiability problem for \emph{quantified boolean formulae} (QBF). Notice that boolean programs can be directly simulated by gap-order constraint systems. The same lower bound thus holds for the reachability problem, and consequently also for the EF model checking problem for GCS.
\newcommand{\Qin}[1]{\mathit{eval}_{#1}} \newcommand{\Qout}[1]{\mathit{out}_{#1}} Let $Q_1x_1 Q_2x_2 \dots Q_kx_k \varphi$ be a QBF formula in prenex normal form. We construct a boolean program that contains control-states $\Qin{i}$ and $\Qout{i}$ as well as variables $x_i$ for each $1\le i \le k$. The program evaluates the formula top down: a subformula $\varphi_i = Q_ix_i, Q_{i+1}x_{i+1}\dots Q_kx_k \varphi$ is verified by a run from control-state $\Qin{i}$ to $\Qout{i}$. If the current valuation does not satisfy the subformula, the program deadlocks and $\Qout{i}$ is not reachable.
The quantifier-free subformula $\varphi$ is directly evaluated using a transition $\Qin{k+1}\step{\varphi/y_{k+1}=1} \Qout{k+1}$. Notice that a pair $\Qin{i},\nu$ is a deadlock unless $\nu\models\varphi$. For an existential quantifier $Q_i$, there are transitions \begin{equation*} \Qin{i}\step{/x_i=0} \Qin{i+1}, \quad \Qin{i}\step{/x_i=1} \Qin{i+1}, \quad \Qout{i+1}\step{} \Qout{i} \end{equation*} For a universal quantifier $Q_i$, there is an extra variable $y_i$ and transitions \begin{align*} &\Qin{i}\step{/x_i=0} \Qin{i+1}, &&\Qin{i}\step{y_{i}=1/x_i=1} \Qin{i+1},\\ &\Qout{i+1}\step{/y_{i}=1} \Qin{i}, &&\Qout{i+1}\step{y_{i}=1/y_{i}=0} \Qout{i}. \end{align*} Notice that the variable $y_i$ serves as a flag to indicate that the subformula $\varphi_i$ has been successfully verified for value $x_i=0$. Just observe that for any valuation $\nu$ with $\nu(y_i)=0$, there is a path from $\Qin{i},\nu$ to some $\Qout{i},\nu'$ iff $\nu_{[x_i=0]}\models\varphi_i$ and $\nu_{[x_i=1]}\models\varphi_i$ Moreover, the existence of such a path implies that $\nu'(y_i)=0$. An induction on $i$ shows that the given formula is indeed satisfiable if, and only if, there exists $\nu$ such that $\Qin{1},\nu_0\step{*}\Qout{1},\nu$.
\subsection{An Ackermann upper complexity bound} Due to our use of Dickson's Lemma in Lemma~\ref{lem:ef}, we can derive an Ackermannian upper bound for EF model checking using the approach of Schmitz et.al.~\cite{FFSS2011}. We show how to to bound the size of our representation of ${\it Pre}^*(S)$ in terms of fast-growing functions. This implies that the space required by the procedure of Theorem~\ref{thm:EF-dec} can be bounded by an Ackermannian function.
\newcommand{\F}[1]{F_{#1}} \newcommand{\FS}[1]{\mathbf{F}_{#1}} \newcommand{\FR}[1]{\mathcal{F}_{#1}}
The family of \emph{fast-growing functions} $F_n:\mathbb{N}\to\mathbb{N}$ is inductively defined as follows for all $x,k\in\mathbb{N}$. \begin{align*}
\F{0}(x) = x+1 \qquad\mbox{and}\qquad
\F{k+1}(x) = \F{k}^{x+1}(x). \end{align*} A variant of the Ackermann function is $\F{\omega}:\mathbb{N}\to\mathbb{N}$, defined as $\F{\omega}(x)=\F{x}(x)$.
Consider $d$-dimensional tuples of natural numbers with the pointwise ordering $\le$. A sequence $x_0 x_1\dots x_l \in (\mathbb{N}^d)^*$ of tuples is called \emph{good} if there exist indices $0\le i<j\le l$
such that $x_i\le x_j$ and \emph{bad} otherwise. I.e., a bad sequence is an antichain w.r.t.\ the ordering on the tuples. By Dickson's Lemma, every bad sequence is finite, but there exist bad sequences of arbitrary length, because there is no assumption on the increase in one dimension if another dimension decreases. The \emph{norm} of $x\in\mathbb{N}^d$ is $\norminf{x}=\max\{x(i)\;|\;0\le i\le d\}$. The sequence is \emph{$t$-controlled} by a function $f:\mathbb{N}\to\mathbb{N}$ if $\norminf{x_i} < f(i+t)$ for every index $0\le i\le l$. \newcommand{\BS}[3]{L_{#1,#2}(#3)} Let $\BS{d}{f}{t}$ denote the maximal length of a bad sequence in $\mathbb{N}^d$ that is $t$-controlled by $f$.
Schmitz et.al.~\cite{FFSS2011}, show how to bound such controlled bad sequences in terms of fast-growing functions. It follows from their work that for every $d\ge 1$ and $c,k,x\in\mathbb{N}$, \begin{equation} \label{BS-bound} \BS{d}{\F{k}^c}{t} \le \F{k+d-1}^{(c + d +2)^d}(t). \end{equation}
We are now ready to bound the size of computed representations of ${\it Pre}^{*}(S)$ for a given MG-definable sets $S$. Fix a GCS with variables ${\it Var}$, constants ${\it Const}$ and $\delta$-many transitional gap constraints. For the sake of readability we assume an unlabeled GCS; the bounds we provide directly apply for the labeled case as well. Recall that a satisfiable monotonicity graph $M$ has the property that $M(x,y)<\infty$ for all $x,y\in{\it Var}\cup{\it Const}$. We identify such a graph with the vector $v_M\in\mathbb{N}^d$ of dimension
$d=|({\it Var}\cup{\it Const})^2|$, where the component for the pair $(x,y)$ has value $0$ if $M(x,y)=-\infty$ and $n + M(x,y) +1$ otherwise. In particular, notice that $\norminf{M}$ is bounded by $n+1+\max\{M(x,y)\mid x,y\in{\it Var}\cup{\it Const}\}$.
Let $S$ be a MG-definable set of valuations represented by a single monotonicity graph and consider a branch of the tree constructed in the proof of Lemma~\ref{lem:ef}. It provides a bad sequence $M_0M_1\dots M_l$ where for each $0<i$, the graph $M_{i}$ is the result of composing its predecessor $M_{i-1}$ with one of the transitional monotonicity graphs $G$ of the system. Wlog., assume that all $M_i$ are satisfiable, because otherwise it (and with it all $M_j$ for $i\le j\le l$) represents the empty set and does not contribute to ${\it Pre^*}(S)$. By definition of compositions $G\circ M$ (see Definition~\ref{def:mg-ops}) we therefore get \newcommand{c}{c} $\norminf{M_{i}} \le \norminf{M_{i-1}} +c$, for every $0< i\le l$, where $c$ is the maximal constant in the system. Consequently, the branch is $\norminf{M_0}$-controlled by $f:x\mapsto xc$. Since $f$ is dominated by $\F{1}^c$, equation~\eqref{BS-bound} provides the bound \begin{equation} \label{eq:} l ~\le~ \BS{d}{f}{t} ~\le~ \BS{d}{\F{1}^c}{t} ~\le~ \F{d}^{(c + d +2)^d}(t) \end{equation} on the length of the branch, where $t=\norminf{M_0}$. If we instead let $t = \max\{\norminf{M_0}, (c + d +2)^d + \delta \}$, then we can bound $l$ by $\F{d}^{(c + d +2)^d}(t) \le \F{d}^{t+1}(t) = \F{d+1}(t)$. In particular, this means that the norm $\norminf{M_l}$ is bounded by $c \cdot \F{d+1}(t)\le \F{d+2}(t)$. Moreover, since $\F{2}^n(x) = x^n+x$, the total number of nodes in the tree is bounded by $\delta^l \le \F{2} \F{d+1}(t) \le \F{d+3}(t)$. We have shown the following lemma. \begin{lemma}
\label{lem:prestar-rep}
\def\F{d+2}(t){\F{d+2}(t)}
Let $S$ be a set of valuations represented by $m$ monotonicity graphs
and let $t\ge (c + d +2)^d + \delta$ be an upper bound on their norms.
Then ${\it Pre}^*(S)$ can be effectively represented by
no more than $m \cdot \F{d+3}(t)$ graphs with norm at most $\F{d+2}(t)$. \end{lemma}
The claim of the next lemma directly follows from Definitions~\ref{def:MGs}, \ref{def:mg-ops}, Lemma~\ref{lem:neg} as well as the definition of the vector $v_M$ representing the MG $M$. Notice that complementing a single MG $M$ results in at most $d$ graphs of degree $<\norminf{v_M}$. Each of them therefore corresponds to a vector with norm bounded by $2\norminf{v_M} + 1 = \F{1}(\norminf{v_M})$. \begin{lemma}
\label{lem:easy-reps}
Let $S,S'$ be sets of valuations, each
represented by a set of $m$ monotonicity graphs of
norm at most $t$.
Then,
\begin{enumerate}
\item $S\cup S'$ can be effectively represented by $m+m$ graphs
of norm at most $t$,
\item $S\cap S'$ can be effectively represented by $m\cdot m$
MG of norm at most $t$,
\item $\Pre{a}{S}$ can be effectively represented by $m$ graphs
of norm at most $t+c$, where $c$ is the maximal absolute value
of any constant in the system,
\item ${\it Val} \setminus S$ can be effectively represented by $d^m$ graphs
of norm at most $\F{1}(t)$.
\end{enumerate} \end{lemma} \begin{proposition}
Let $({\it Var},{\it Const},\mathit{Act},\Delta,\lambda)$ be a gap-order constraint system
and let $d=(|{\it Var}| +|{\it Const}|)^2$, $\delta=|\Delta|$ and $c=\max\{|x| : x\in{\it Const}\}$.
For every EF-formula $\varphi$ of nesting depth $k$, one can effectively
compute a representation of the set $\DEN{\varphi}$,
in space $\F{d+4}((c+d+2)^d+\delta + k)$. \end{proposition} \begin{proof}
Let $t=(c + d+2)^d+\delta)$.
We show by induction on the nesting depth $k$ of subformulae
that
$\DEN{\varphi}$ can be represented by at most
$\F{d+3}^k(t)$ monotonicity graphs with norms bounded by
$\F{d+2}^k(t)$.
For the base case, observe that atomic propositions are either $\varphi={\it true}$
or stated as single gap constraint $\varphi=\Con{C}$.
Either way, $\DEN{\varphi}$ can be expressed as single monotonicity graph
$M_\varphi$ with norm $\norminf{M_\varphi}\le c\le \F{d+2}^0((c +
d+2)^d+\delta)$.
For the induction step, we assume that
the claim is true for all formulae of height $i$ and consider
a formula $\varphi$ of height $i+1$. If the principal connector of $\varphi$
is $\land,\lor,\neg$ or $\exnext{a}$ for some action $a$,
then the claim follows by Lemma~\ref{lem:easy-reps}.
To see this, just notice that for all $m\in\mathbb{N}$,
$\F{d+2}(m)\ge \F{1}(m)= 2m+1$ and
$\F{d+3}(m)\ge \F{3}(m)\ge m^m$.
If $\varphi$ is of the form ${\it EF}\phi$, then,
by induction hypothesis,
$\DEN{\phi}$ can be effectively represented by
no more than $\F{d+3}^i(t)$ graphs with norm $\le \F{2}^i(t)$.
Lemma~\ref{lem:prestar-rep} thus implies that
$\DEN{\varphi}$ is representable by $\F{d+3}^{i+1}(t)$ graphs,
each with norm at most $\F{d+2}^{i+1}(t)$ as required.
The claim now follows from the observation that the total
space required for the above representation is
$d\cdot \log \F{d+2}^k(t) \cdot \F{d+3}^k(t) \le \F{d+4}(t+k)$. \end{proof}
\section{Applications}
\label{sec:app}
We consider labeled transition systems induced by GCS. In a weak semantics, one abstracts from non-observable actions modeled by a dedicated action $\tau\in \mathit{Act}$. The \emph{weak step} relation $\wstep{}$ is defined by \[ \begin{array}{lcl} \wstep{\tau} & = & \Step{\tau}{*}{} \\ \wstep{a} & = & \Step{\tau}{*}{}\cdot\step{a}\cdot\Step{\tau}{*}{}, \quad\mbox{for $a\neq\tau$} \end{array} \]
Bisimulation and weak bisimulation \cite{Par1981,Mil1989} are semantic equivalences in van Glabbeek's linear time -- branching time spectrum \cite{Gla2001}, which are used to compare the behavior of processes. Their standard co-inductive definition relative to a given LTS is as follows. \begin{definition}
A binary relation $R\subseteq V^2$ on the states of a labeled transition
system is a \emph{bisimulation} if $sRt$ implies that
\begin{enumerate}
\item for all $s\step{a}s'$ there is a $t'$ such that $t\step{a}t'$ and $s'Rt'$, and
\item for all $t\step{a}t'$ there is a $s'$ such that $s\step{a}s'$ and $s'Rt'$.
\end{enumerate}
Similarly, $R$ is a \emph{weak bisimulation} if in both conditions above, $\step{}$ is replaced by
$\wstep{}$.
(Weak) bisimulations are closed under union, so there exist
unique maximal bisimulation $\sim$ and weak bisimulation $\approx$ relations, which
are equivalences on $V$.
By the maximal (weak) bisimulation between two LTS with state sets $S$ and $T$
we mean the maximal (weak) bisimulation in their union projected into $(S\times T)\cup (T\times S)$. \end{definition} The \emph{Equivalence Checking Problem} is the following decision problem.
\begin{tabular}{ll}
{\sc Input:} &Given LTS $T_1=(V_1,\mathit{Act},\step{})$ and $T_2=(V_2,\mathit{Act},\step{})$,\\
&states $s\in V_1$ and $t\in V_2$ and an equivalence $R$.\\
{\sc Question:} &$s R t$? \end{tabular}
\noindent In particular, we are interested in checking strong and weak bisimulation between LTS induced by GCS and finite systems. Note that the decidability of weak bisimulation implies the decidability of the corresponding strong bisimulation because $\sim$ and $\approx$ coincide for LTS without $\tau$ labels.
Finite systems admit \emph{characteristic formulae} up to weak bisimulation in \text{EF}\ (see e.g.~\cite{KJ2006,JKM1998}). \begin{theorem}
\label{thm:char-formulae}
Let $T_1=(V_1,\mathit{Act},\step{})$ be an LTS with finite state set $V_1$
and $T_2=(V_2,\mathit{Act},\step{})$ be an arbitrary LTS.
For every state $s\in V_1$ one can construct an \text{EF}-formula $\psi_s$
such that $t\approx s \iff t\models \psi_s$ for all states $t\in V_2$. \end{theorem}
The following is a direct consequence of Theorems~\ref{thm:char-formulae} and \ref{thm:EF-dec}.
\begin{theorem} \label{thm:wbsim-ecp}
For every GCS $\mathcal{G}=(Var,Const,\mathit{Act},\Delta,\lambda)$
and every LTS $T=(V,\mathit{Act},\step{})$ with finite state set $V,$
the maximal weak bisimulation $\approx$
between $T_{\mathcal{G}}$ and $T$ is effectively gap definable. \end{theorem} \begin{proof}
By Theorem~\ref{thm:char-formulae} we can compute, for every state $s$ of $T,$
a characteristic formula $\psi_s$ that characterizes the set of
valuations $\{\nu\ |\ \nu\approx s\} = \DEN{\psi_s}$.
By Theorem~\ref{thm:EF-dec}, these sets are MG- and thus gap definable.
Since the class of gap definable sets is effectively closed under
finite union
and $\approx\ = \bigcup_{s\in V}\DEN{\psi_s}$, the result follows.
\qed \end{proof}
Considering that gap formulae are particular formulae of Presburger arithmetic, we know that gap definable sets have a decidable membership problem. Theorem~\ref{thm:wbsim-ecp} thus implies the decidability of equivalence checking between GCS processes and finite systems w.r.t.\ strong and weak bisimulation.
\section{Conclusion and Open Questions}
We have shown that model checking gap-order constraint systems with the logic \text{EG}\ is undecidable while the problem remains decidable for the logic \text{EF}. An immediate consequence of the latter result is the decidability of strong and weak bisimulation checking between GCS and finite systems.
The decidability of \text{EF}\ model checking is shown by using finite sets of monotonicity graphs or equivalently, gap formulae to represent intermediate results in a bottom-up evaluation. This works because the class of arbitrary gap definable sets is effectively closed under union and complements and for a gap definable set $S$ and a GCS $\mathcal{G}$, $\Pre{}{S}$ and ${\it Pre}^*(S)$ are effectively gap definable.
Our decidability result relies on a well-quasi-ordering argument to ensure termination of the fixpoint computation for ${\it Pre}^*(S)$, which does not yield any strong upper complexity bound. So far, there is only an Ackermannian upper bound and a \textsc{PSpace}\ lower bound.
Interesting open questions include determining the exact complexity of model checking GCS with respect to \text{EF}. We also plan to investigate the decidability and complexity of checking behavioral equivalences like strong and weak bisimulation between two GCS processes, as well as checking (weak) simulation preorder and trace inclusion.
\end{document}
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\begin{document}
\title[Optimal gradient estimates for the insulated conductivity problem]{Optimal gradient estimates for the insulated conductivity problem with dimensions\\ more than two}
\author[L.J. Ma]{Linjie Ma} \address[L.J. Ma]{Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, People's Republic of China.} \email{[email protected]}
\date{\today}
\maketitle
\begin{abstract}
In high-contrast composite materials, the electric (or stress) field may blow up in the narrow region between inclusions. The gradient of solutions depend on $\varepsilon$, the distance between the inclusions, where $\varepsilon$ approaches to $0$. By using the maximum principle techniques, we give another proof of the Dong-Li-Yang estimates \cite{DLY} for any convex inclusions of arbitrary shape with $n\geq 3$. This result solves the problem raised by \cite{W}, where the spherical inclusions with $n\geq 4$ is considered. Moreover, we also generalize the above results with flatter boundaries near touching points. \end{abstract}
\section{Introduction and main results} \subsection{Background} Let $D$ be a bounded open set in $\mathbb{R}^{n}$, $n\geq3$, containing two subdomains $D_{1}$ and $D_{2}$, with $\varepsilon$-apart, for a small positive constant $\varepsilon$. For a given appropriate function $g$, we consider the following conductivity problem with Dirichlet boundary data \begin{equation}\label{eq_ak}
\begin{cases}
-\nabla(a_{k}(x)\nabla u_{k,\varepsilon})=0,\quad&\mbox{in}~D,\\
u_{k,\varepsilon}=g,\quad&\mbox{on}~\partial D,
\end{cases} \end{equation} where \begin{equation*}
a_{k}(x)=
\begin{cases}
k\in[0,1)\cup(1,\infty],&\mbox{in}~D_{1}\cup{D}_{2},\\
1,&\mbox{in}~D_{0}:=D\setminus\overline{D_{1}\cup{D}_{2}}.
\end{cases} \end{equation*} In the context of electric conduction, the elliptic coefficients $a_k$ refer to conductivity, and the solution $u_{k,\varepsilon}$ represents voltage potential. From an engineering point of view, the most important quality is $\nabla u_{k,\varepsilon}$, representing the electric field. The above model arises from the study of composite material \cite{BAPK}, where Babu\v{s}ka etc. analyzed numerically that the high concentration of extreme electric field will occur in the narrow region between the adjacent inclusions or between inclusions and boundaries. Bonnetier and Vogelius \cite{BV} proved that $\nabla u_{k,\varepsilon}$ is bounded for a fixed $k$, which is far away from $0$ and $\infty$, and circular inclusions in two dimension. Later, Li and Vogelius\cite{LV} proved the boundedness of $\nabla u_{k,\varepsilon}$ for general second order elliptic equations of divergence form with piecewise H\"{o}lder coefficients and general shape of inclusions in any dimensions. In \cite{LN}, Li and Nirenberg extended the results in \cite{LV} to general second order elliptic systems of divergence form.
When $k$ equals to $\infty$ (perfect conductor) or $0$ (insulator), the gradient of solutions is much different. It was shown in \cite{BC,K1,M} that the gradient general become unbounded, as $\varepsilon\rightarrow 0$. Ammari et.al. in \cite{AKL,AKLLL} considered the perfect and insulate conductivity problem for the disk inclusions in dimension two, and gave the blow up rate $\varepsilon^{-1/2}$ in both cases. They also showed that the blow up rate is optimal. For the perfect conductivity problem in high dimensions, Yun extended the results to any bounded strictly convex smooth domains\cite{Y1,Y3}. Bao, Li and Yin in \cite{BLY1,BLY2} considered the higher dimensions and gave the optimal blow up rate, $\varepsilon^{-1/2}$ for $n=2$, $\varepsilon^{-1/2}\ln\varepsilon$ for $n=3$, $\varepsilon^{-1}$ for $n\geq4$. For further works, see e.g.\cite{L,LLY,LLBY,lx,BT2,Y1,Y2,BLL,BLY1,KLY2,LY,ABTV,ACKLY,ADKL,AKLLZ,BLL2,KLY,CL,CLX} and their references therein.
When $k$ goes to $0$, $u_{k,\varepsilon}$ converges to the solution of the following insulated conductive problem: \begin{equation}\label{ins}
\begin{cases}
\Delta u_\varepsilon=0\quad&\mbox{in}~D_0,\\
\frac{\partial u_\varepsilon}{\partial\nu}=0\quad&\mbox{on}~\partial D_1\cup\partial D_2,\\
u_\varepsilon=g\quad&\mbox{on}~\partial D.
\end{cases} \end{equation} where $\nu$ is the outward unit normal vector. \begin{figure}
\caption{The model for $n=2$.}
\end{figure} For the insulated conductivity problem, it was proved in \cite{BLY2} that the optimal blow up rate is $\varepsilon^{-1/2}$ in $\mathbb{R}^2$. Yun in \cite{Y3} considered two circle balls and gave the optimal blow rate $\varepsilon^{\frac{\sqrt{2}-2}{2}}$. Li and Yang in \cite{LYang,LYang2} improved the upper bound in dimension $n\geq 3$ to be of order $\varepsilon^{-1/2+\beta}$ for some $\beta>0$. Later, Weinkove in \cite{W} gave the blow-up rate $\gamma^*$ as the positive solution of the quadratic equation: \begin{equation}\label{W} (n-2)(\gamma^*)^2+(n^2-4n+5)\gamma^*-(n^2-5n+5)=0 \end{equation} for $n\geq 4$, which improved the result in \cite{LYang}. Several months later, Dong, Li and Yang considered the optimal gradient estimate in \cite{DLY} and gave the optimal blow up rate for $n\geq 3$: \begin{equation}\label{rate} \frac{-(n-1)+\sqrt{(n-1)^2+4(n-2)}}{2}. \end{equation}
Until now, for the insulated conductivity problem \eqref{ins} with any dimensions, the blow-up rate has been determined. But there are still some interesting questions to consider. As we know, Weinkove in \cite{W} used the maximum principle techniques to deal with the problem, which is completely different from \cite{DLY}. Whether this techniques can be used to give another proof of Dong-Li-Yang optimal estimates \eqref{rate} is an interesting question raised by Weinkove himself in \cite{W}.
In this paper, we try to consider this open problem. As we know, Weinkove in\cite{W} didn't give the blow-up rate for $n=3$ and only deal with the case for spherical inclusions rather than any convex inclusions of arbitrary shape. These two points are the main difficulties to overcome in this paper. \subsection{Our domain}
Before stating our main result, we firstly fix our domain. We use $x=(x',x_n)$ to denote a point in $\mathbb{R}^n$, $x'=(x_1,x_2,\ldots,x_{n-1})$, $n\geq3$. Let $D$ be a bounded open set in $\mathbb{R}^n$ that contains a pair of subdomain $D_1$ and $D_2$ with $2\varepsilon$ distance. \begin{figure}
\caption{The narrow region $\Omega_r$.}
\end{figure} Fix a constant $R_0<1$, independent of $\varepsilon$, such that the portions of $\partial D_j$ near the origin (which denoted by $\Gamma_{\pm}$) can be parameterized by $(x',h_1(x')+\varepsilon)$ and $(x',h_2(x')-\varepsilon)$, respectively. That is, \begin{align*} &\Gamma_{+}=\left\lbrace
x_n=h_1(x')+\varepsilon,\ |x'|<R_0\right\rbrace ,\\ &\Gamma_{-}=\left\lbrace
x_n=h_2(x')-\varepsilon,\ |x'|<R_0\right\rbrace, \end{align*} where $h_1$ and $h_2$ satisfy the following assumptions: \begin{equation}\label{eq1.2}
h_1(x')>h_2(x')\qquad\mbox{for}\ \ |x'|<R_0. \end{equation} Moreover, by the convexity assumptions on $\partial D_1$ and $\partial D_2$, after a rotation of the coordinates, if necessary, we assume that \begin{equation}\label{eq1.3+}
h_1(x')=\lambda_1|x'|^2+O(|x'|^{2+\alpha}),\quad
h_2(x')=-\lambda_2|x'|^2+O(|x'|^{2+\alpha})\quad\mbox{for}\ \ |x'|<R_0, \end{equation} where $\alpha\in(0,1)$, $\lambda_1$ and $\lambda_2$ are some positive constants depending on the curvature of $\partial D_1$ and $\partial D_2$, and \begin{align}\label{eq1.6}
\|h_1\|_{C^{2,\alpha}(B_{2R_0})}+\|h_2\|_{C^{2,\alpha}( B_{2R_0})}\leq \mu, \end{align} for some constant $\mu$. Here and throughout the paper, we use the notation $O(A)$ to denote a quantity that can be bounded by $CA$, where $C$ is some positive constant independent of $\varepsilon$. For $0<r\leq R_0$, define \begin{equation*}
\Omega_{r}:=\left\lbrace (x',x_n)\in \mathbb{R}^n\Big|\ h_2(x')-\varepsilon<x_n<h_1(x')+\varepsilon,\ |x'|<r\right\rbrace . \end{equation*} By standard elliptic estimates, the solution $u\in H^1(D_0)$ of \eqref{ins} satisfies \begin{equation*}
\|u\|_{C^1(D_0\backslash\Omega_{R_0/2})}\leq C. \end{equation*} We will focus on the following problem near the origin: \begin{equation}\label{ins1}
\begin{cases}
\Delta u=0,\quad&\mbox{in}~D_0,\\
\frac{\partial u}{\partial\nu}=0\quad&\mbox{on}~\Gamma_{+}\cup\Gamma_{-},\\
\|u\|_{L^\infty}\leq 1.
\end{cases} \end{equation}
The idea of this paper comes from \cite{W}, which using the maximum principle to deal with a specific quantity in the narrow region between the insulators: \begin{equation}\label{eq1.10}
\left[ (|x'|^2+\sigma)^{1-\gamma}+\varepsilon^{1-\gamma(1-\delta)}-A(bx_n^2+|x'|^4+\sigma)^{\gamma-1/2}\right] |\nabla u|^2. \end{equation}
$(|x'|^2+\sigma)^{1-\gamma}+\varepsilon^{1-\gamma(1-\delta)}$ is the main term, and $A(bx_n^2+|x'|^4+\sigma)^{\gamma-1/2}$ is the lower order term which is used to adjust the quantity in boundary and interior. $\sigma$ and $\delta$ are small positive constants. We revise \eqref{eq1.10} by \begin{align}\label{eq1.11}
\Bigg\{ (|x'|^2+\varepsilon)^{1-\gamma}-&\frac{b}{2^{\gamma}}\varepsilon^{1-\gamma}-A(|x'|^2+\varepsilon)^{2-\gamma}\cos\frac{x_n}{|x'|^2+\varepsilon}\nonumber\\
&+B(|x'|^2+\varepsilon)^{1-\gamma}\left( \frac{|x'|^2}{2}+\varepsilon\right) \cos\left(\frac{\pi}{2}\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{\frac{|x'|^2}{2}+\varepsilon} \right)
\Bigg\} |\nabla u|^2, \end{align}
Here, $(|x'|^2+\varepsilon)^{1-\gamma}-\frac{b}{2^{\gamma}}\varepsilon^{1-\gamma}\geq C(|x'|^2+\varepsilon)^{1-\gamma}$. Compared to the main term in \eqref{eq1.10} and \eqref{eq1.11}, they are equivalent. The lower order term $ -A(|x'|^2+\varepsilon)^{2-\gamma}\cos\frac{x_n}{|x'|^2+\varepsilon}+B(|x'|^2+\varepsilon)^{1-\gamma}\left( \frac{|x'|^2}{2}+\varepsilon\right) \cos\left(\frac{\pi}{2}\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{\frac{|x'|^2}{2}+\varepsilon} \right)$ plays a very important role in this quantity: $B(|x'|^2+\varepsilon)^{1-\gamma}\left( \frac{|x'|^2}{2}+\varepsilon\right) \cos\left(\frac{\pi}{2}\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{\frac{|x'|^2}{2}+\varepsilon} \right)$ is used to keep the normal derivative along the boundaries with good sign and $-A(|x'|^2+\varepsilon)^{2-\gamma}\cos\frac{x_n}{|x'|^2+\varepsilon}$ is used to make the second order derivative have a positive term, which is very useful to keep this quantity from blowing up in the interior of the narrow region. The constants $A,B,b$ are chosen to optimal the estimates.
Next, we give our main results. \subsection{Main results.}
\begin{theorem}\label{th-ins}
Let $D$, $D_1$, $D_0$ be defined as above and satisfy \eqref{eq1.2}-\eqref{eq1.6}. $g\in C^{1,\alpha}(\partial D)$. Assume that $u\in H^1(D)\cap C^1(\overline{D}_0)$ is a solution of system \eqref{ins}, then for $n\geq 3$, we have
\begin{align}\label{eq1.3}
\|\nabla u\|_{L^\infty(\Omega_{R/2})}\leq \frac{C \|g\|_{C^{1,\alpha}(\partial D)} }{(\varepsilon+|x'|^2)^{(1-\gamma)/2}},\quad 0<\gamma\leq\gamma^*,
\end{align} where \begin{equation}\label{index} \gamma^*=\gamma^*(n):=\frac{-(n-1)+\sqrt{(n-1)^2+4(n-2)}}{2}\in(0,1). \end{equation} \end{theorem}
\begin{remark} From \eqref{index}, when $n=3$, $\gamma^*=\sqrt{2}-1$, the blow up rate is $\frac{\sqrt{2}-2}{2}$, which is consistent with \cite{DLY} and \cite{Y3}. In table 1, we give the exact and approximate numerical values of blow up rate $-\frac{1-\gamma^*}{2}$ for $n=3,4,5,6,\infty$.
\begin{table}[htbp] \centering \setlength{\abovecaptionskip}{1cm}
\caption{The approximate numerical values of blow up rate.} \renewcommand\arraystretch{1.8}
\begin{tabular}{|c|c|c|c|} \hline \ \ \ \ \ \ \ $n$\ \ \ \ \ \ \ &\ \ \ \ \ \ \ $\gamma^*$\ \ \ \ \ \ \ &\ \ \ \ \ \ \ $-\frac{1-\gamma^*}{2}$\ \ \ \ \ \ \ &\ \ \ \ \ \ \ approx.\ \ \ \ \ \ \ \\ \hline 3 &$\sqrt{2}-1$ &$-\frac{2-\sqrt{2}}{2}$ &-0.2929\\ \hline 4 &$\frac{\sqrt{17}-3}{2}$ &$-\frac{5-\sqrt{17}}{4}$ &-0.2192\\ \hline 5 &$\sqrt{7}-2$ &$-\frac{3-\sqrt{7}}{2}$ &-0.1771\\ \hline 6 &$\frac{\sqrt{41}-5}{2}$ &$-\frac{7-\sqrt{41}}{4}$ &-0.1492\\ \hline $\infty$ &1 &0&0\\ \hline \end{tabular} \end{table}
Obviously, the blow up rate is monotonically increasing about $n$, that means the electric field concentration phenomenon will disappear as $n\rightarrow\infty$. \end{remark}
The above procedure can be applied to deal with the following generalized $m-$ convex inclusion cases. For simplicity, we assume that for $x\in\Omega_r$, \begin{align}\label{eq1.8}
h_1(x')=\lambda_1|x'|^m+O(|x'|^{m+\alpha}),\quad
h_2(x')=-\lambda_2|x'|^m+O(|x'|^{m+\alpha}),\quad m>2. \end{align}
\begin{theorem}\label{th-insm}
Let $D$, $D_1$, $D_0$ be defined as above and satisfy \eqref{eq1.2},\eqref{eq1.6} and \eqref{eq1.8}. $g\in C^{1,\alpha}(\partial D)$. Assume that $u\in H^1(D)\cap C^1(\overline{D}_0)$ is a solution of system \eqref{ins}, then for $n\geq 3$, we have
\begin{align}\label{eq1.4}
\|\nabla u\|_{L^\infty(\Omega_{R/2})}\leq \frac{C \|g\|_{C^{1,\alpha}(\partial D)} }{(\varepsilon+|x'|^m)^{(1-\gamma)/2}},\quad 0<\gamma\leq\gamma^*,
\end{align} where \begin{equation*}
\gamma^*=\gamma^*(n):=\frac{-(n-1)+\sqrt{(n-1)^2+4(n-2)}}{2}\in(0,1). \end{equation*} \end{theorem}
The proofs of Theorem \ref{th-ins} and Theorem \ref{th-insm} are given in section \ref{sect2}.
\section{Proof of Theorem \ref{th-ins} and Theorem \ref{th-insm} }\label{sect2}
Firstly, we have the following lemma, which is similar to Lemma 2.1 in \cite{W}. \begin{lemma}\label{lemma3.2} Under the assumption of \eqref{eq1.2}-\eqref{eq1.6}, at any point of $\Gamma_+$ and $\Gamma_-$, we have \begin{equation}\label{eq2.0}
\frac{\partial}{\partial\nu}(|\nabla u|^2)
\leq\frac{4\lambda_k}{\sqrt{1+4\lambda_k^2|x'|^2}}|\nabla u|^2, \end{equation} where $k=1$ on $\Gamma_+$, $k=2$ on $\Gamma_-$. \end{lemma} \begin{proof}
We only give the proof of the point on $\Gamma_+$, the proof on $\Gamma_-$ is similar. From $(2.3)$ in \cite{W}, one has
\begin{align}\label{eq2.2}
\frac{\partial}{\partial\nu}(|\nabla u|^2)
=&2\sum_{i,j=1}^{n-1}\left(\frac{{h_1}_{x_ix_j}}{\sqrt{1+\sum_{k=1}^{n-1}{h_1}_{x_k}^2}}-{h_1}_{x_i}{h_1}_{x_j} \right) u_{x_i}u_{x_j}+2u_{x_n}^2\nonumber\\
=&\frac{2}{\sqrt{1+\sum_{k=1}^{n-1}{h_1}_{x_k}^2}}\sum_{i,j=1}^{n-1}{h_1}_{x_ix_j} u_{x_i}u_{x_j}\quad on\ \ \Gamma_+,
\end{align} where we use the boundary data $\frac{\partial u}{\partial\nu}=0$, which is equivalent to \begin{equation} -\sum_{i=1}^{n-1}{h_1}_{x_i}u_{x_i}+u_{x_n}=0. \end{equation} Then by the assumptions, \eqref{eq2.0} holds. \end{proof}
\begin{proof}[Proof of Theorem \ref{th-ins}]
Without loss of general, we assume that $\lambda_1+\lambda_2=1$. We consider the quantity \begin{align*}
Q=F |\nabla u|^2\ \ \ \ \mbox{in}\ \ \Omega_{r}, \end{align*} where \begin{align*}
F:=(|x'|^2+\varepsilon)^{1-\gamma}-&\frac{b}{2^{\gamma}}\varepsilon^{1-\gamma}-A(|x'|^2+\varepsilon)^{2-\gamma}\cos\frac{x_n}{|x'|^2+\varepsilon}\nonumber\\
&+B(|x'|^2+\varepsilon)^{1-\gamma}\left( \frac{|x'|^2}{2}+\varepsilon\right) \cos\left(\frac{\pi}{2}\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{\frac{|x'|^2}{2}+\varepsilon} \right), \end{align*} $A,B,b$ are uniform positive constants satisfying \begin{equation}\label{eq2.6+}
A>\frac{4\pi\gamma\max\{\lambda_1,\lambda_2\}+54n-70-\left(42+54n-\frac{40}{n-1}\right) \gamma}{\cos1-\pi\sin1}, \end{equation} \begin{align}\label{eq2.6} \frac{2}{\pi}\Big( 4\gamma&\max\{\lambda_1,\lambda_2\}+A\sin1\Big)\nonumber\\ &<B<\frac{2}{\pi^2}\left[A\cos1+ \left(42+54n-\frac{40}{n-1}\right) \gamma-54n+70\right], \end{align} \begin{equation}\label{eq2.5}
2^\gamma(1-\gamma)<b\leq\frac{4}{5}. \end{equation}
By inequality $a^p+b^p\geq(a+b)^p\geq2^{p-1}(a^p+b^p)$ for $a,b>0$, $0<p<1$, we know that \begin{align}\label{eq2.7}
F\geq\frac{1}{2^{\gamma}}|x'|^{2(1-\gamma)}+\frac{1-b}{2^{\gamma}}\varepsilon^{1-\gamma}\geq \frac{1-b}{2^{\gamma}}(|x'|^2+\varepsilon)^{1-\gamma}, \end{align} where we use that \begin{align}\label{eq4.6}
-A&(|x'|^2+\varepsilon)^{2-\gamma}\cos\frac{x_n}{|x'|^2+\varepsilon}\nonumber\\
&+B(|x'|^2+\varepsilon)^{1-\gamma}\left( \frac{|x'|^2}{2}+\varepsilon\right) \cos\left(\frac{\pi}{2}\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{\frac{|x'|^2}{2}+\varepsilon} \right)
\ll (|x'|^2+\varepsilon)^{1-\gamma}. \end{align}
We assume that the quantity $Q$ achieves a maximum at $p$ in $\overline{\Omega}_r$. If $p$ is in $\partial\Omega_r\backslash(\Gamma_+\cup\Gamma_-)$, by \eqref{eq2.7}, we have \begin{equation}\label{eq2.4}
(|x'|^2+\varepsilon)^{1-\gamma}|\nabla u|^2
\leq C\|g\|_{C^{1,\alpha}(\partial D)}^2, \end{equation} thus \eqref{eq1.3} holds.
In the following, we will prove that the quantity $Q$ can only achieve its maximum on $\partial\Omega_r\backslash(\Gamma_+\cup\Gamma_-)$.
Firstly, we assume that $Q$ achieves its maximum at a point $p\in \Gamma_+$, then by Lemma \ref{lemma3.2}, we have \begin{align}\label{eq2.10}
0\leq\frac{\partial Q}{\partial\nu}
=&\frac{\partial F}{\partial\nu}|\nabla u|^2
+F\frac{\partial }{\partial\nu}(|\nabla u|^2)\nonumber\\
\leq&\left( \frac{\partial F}{\partial\nu}+\frac{4\lambda_1}{\sqrt{1+4\lambda_1^2|x'|^2}} F \right) |\nabla u|^2\ \ \mbox{on}\ \ \Gamma_+. \end{align}
Since $x_n=\lambda_1|x'|^2+\varepsilon+O(|x'|^{2+\alpha})$ on $\Gamma_+$ and the fact that \begin{equation*}
\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{\frac{|x'|^2}{2}+\varepsilon}=1 \quad \mbox{on}\ \Gamma_{+}, \end{equation*}
we have \begin{align*}
\frac{\partial F}{\partial\nu}\Big|_{\Gamma_+}
=&-\frac{1}{\sqrt{1+4\lambda_1^2|x'|^2}} \left( \sum_{i=1}^{n-1}2\lambda_1x_iF_{x_i}-F_{x_n}\right)\\
=&\frac{1}{\sqrt{1+4\lambda_1^2|x'|^2}}\Big\{-4\lambda_1(1-\gamma)(|x'|^2+\varepsilon)^{-\gamma}|x'|^2\nonumber\\
&+4A\lambda_1(2-\gamma)(|x'|^2+\varepsilon)^{1-\gamma}|x'|^2\cos\frac{x_n}{|x'|^2+\varepsilon}\nonumber\\
&+4A\lambda_1 (|x'|^2+\varepsilon)^{-\gamma}|x'|^2x_n\sin\frac{x_n}{|x'|^2+\varepsilon}\nonumber\\
&-B\pi\lambda_1\left( \lambda_1-\lambda_2+1\right) (|x'|^2+\varepsilon)^{1-\gamma}|x'|^2\nonumber\\
&+A(|x'|^2+\varepsilon)^{1-\gamma}\sin\frac{x_n}{|x'|^2+\varepsilon}-\frac{\pi}{2}B(|x'|^2+\varepsilon)^{1-\gamma}
+O(|x'|^{1-\gamma+\alpha})\Big\}\nonumber\\
\leq&\frac{1}{\sqrt{1+4\lambda_1^2|x'|^2}}\Big\{\left(-4\lambda_1(1-\gamma)+A\sin1-\frac{\pi}{2}B\right) (|x'|^2+\varepsilon)^{1-\gamma}\nonumber\\
&+4\lambda_1(1-\gamma)\varepsilon(|x'|^2+\varepsilon)^{-\gamma}
+4A\lambda_1(2-\gamma)(|x'|^2+\varepsilon)^{1-\gamma}|x'|^2\nonumber\\
&+4A\lambda_1 (|x'|^2+\varepsilon)^{-\gamma}|x'|^2x_n\sin1
-B\pi\lambda_1\left( \lambda_1-\lambda_2+1\right) (|x'|^2+\varepsilon)^{1-\gamma}|x'|^2\nonumber\\
&+O(|x'|^{1-\gamma+\alpha})\Big\}. \end{align*} Then \begin{align}\label{eq2.15} &\frac{\partial F}{\partial\nu}
+\frac{4\lambda_1}{\sqrt{1+4\lambda_1^2|x'|^2}}F \nonumber\\
\leq&\frac{1}{\sqrt{1+4\lambda_1^2|x'|^2}}\Big\{\left( 4\lambda_1\gamma+A\sin1-\frac{\pi}{2}B \right) (|x'|^2+\varepsilon)^{1-\gamma}\nonumber\\ &+\left[ 4(1-\gamma)-2^{2-\gamma}b\right]\lambda_1 \varepsilon^{1-\gamma}
-4A\lambda_1(|x'|^2+\varepsilon)^{2-\gamma}\cos1\nonumber\\
&+4A\lambda_1(2-\gamma)(|x'|^2+\varepsilon)^{1-\gamma}|x'|^2
+4A\lambda_1 (|x'|^2+\varepsilon)^{-\gamma}|x'|^2x_n\sin1\nonumber\\
&-B\pi\lambda_1\left( \lambda_1-\lambda_2+1\right) (|x'|^2+\varepsilon)^{1-\gamma}|x'|^2
+O(|x'|^{1-\gamma+\alpha})\Big\}\nonumber\\ <&0\ \ \mbox{on}\ \ \Gamma_+. \end{align} where for the last line, we used the inequalities \eqref{eq2.6}, \eqref{eq2.5} and the fact that \begin{align*}
&-4A\lambda_1(|x'|^2+\varepsilon)^{2-\gamma}\cos1
+4A\lambda_1(2-\gamma)(|x'|^2+\varepsilon)^{1-\gamma}|x'|^2 \nonumber\\
&+4A\lambda_1 (|x'|^2+\varepsilon)^{-\gamma}|x'|^2x_n\sin1
-B\pi\lambda_1\left( \lambda_1-\lambda_2+1\right) (|x'|^2+\varepsilon)^{1-\gamma}|x'|^2\nonumber\\
\ll&(|x'|^2+\varepsilon)^{1-\gamma}. \end{align*} Combining \eqref{eq2.10} and \eqref{eq2.15}, \begin{align*}
0\leq\frac{\partial Q}{\partial\nu}
<0\ \ \mbox{on}\ \ \Gamma_+, \end{align*} which is a contraction.
Similarly, we can also prove that the maximum cannot attained on $\Gamma_-$.
Next, we assume that the quantity $Q$ achieves a maximum at a point $p\in \Omega_r$, then we have \begin{align}\label{eq3.6}
0\geq\Delta Q
=&\Delta F |\nabla u|^2
+2\nabla F\cdot \nabla (|\nabla u|^2)+2F|\nabla\nabla u|^2. \end{align} In the following, we prove that \begin{equation}\label{eq4.9}
\Delta Q>0. \end{equation}
\textbf{Step 1. Estimates of $2\nabla F\cdot \nabla (|\nabla u|^2)$.}
For $2\partial_{x_n} F\cdot \partial_{x_n}(|\nabla u|^2)$, by Cauchy inequality, immediately we have \begin{align}\label{eq2.9}
2\partial_{x_n} F\cdot \partial_{x_n}(|\nabla u|^2)
=&4F_{x_n}\sum_{j=1}^nu_{x_j}u_{x_jx_n}\nonumber\\
\geq&-2\eta|\nabla u|^2-\frac{2}{\eta}|F_{x_n}|^2|\nabla\nabla u|^2, \end{align} where $\eta$ is some small positive constant which may differ from line to line, and which can be shrunk at the expense of shrinking $\varepsilon$ or $r$.
Since at maximum of $Q$, it holds that \begin{equation*}
0=Q_{x_i}=F_{x_i} |\nabla u|^2+F\partial_{x_i}(|\nabla u|^2), \end{equation*} for $i=1,\ldots,n-1$, we have that \begin{align*}
\partial_{x_i}\left( |\nabla u|^2\right) =-\frac{F_{x_i}|\nabla u|^2}{F}, \end{align*} which leads \begin{align*}
2\partial_{x_i} F\cdot \partial_{x_i}(|\nabla u|^2)
=-\frac{2}{F}F_{x_i}^2 |\nabla u|^2. \end{align*} Sum above for $i$ from $1$ to $n-1$, one has \begin{align}\label{eq3.11}
\sum_{i=1}^{n-1}2\partial_{x_i} F\cdot \partial_{x_i}(|\nabla u|^2)
=-\frac{\sum_{i=1}^{n-1}2F_{x_i}^2}{F} |\nabla u|^2. \end{align} Combining \eqref{eq3.11} and \eqref{eq2.9} together, we get \begin{align}\label{eq2.9+}
\mathcal{F}_1:=2\nabla F\cdot \nabla (|\nabla u|^2)
\geq-\left[\frac{\sum_{i=1}^{n-1}2F_{x_i}^2}{F}+2\eta \right] |\nabla u|^2
-\frac{2}{\eta}|F_{x_n}|^2|\nabla\nabla u|^2. \end{align}
On the other hand, we may make a change of coordinates so that $x_2=\cdots=x_{n-1}=0$ and $x_1\geq 0$ and hence \begin{equation*}
\nabla|x'|^2=(2|x'|,0,\ldots,0). \end{equation*} Next, we use the fact that at the maximum of $Q$ we have \begin{equation*}
0=Q_{x_1}=F_{x_1} |\nabla u|^2+F\partial_{x_1}(|\nabla u|^2), \end{equation*} that is \begin{equation*}
F_{x_1}=-\frac{F\partial_{x_1}(|\nabla u|^2) }{|\nabla u|^2}. \end{equation*} Then by Cauchy-Schwarz inequality, \begin{align*}
2\partial_{x_1} F\cdot \partial_{x_1}(|\nabla u|^2)
=-2F\frac{\left[ \partial_{x_1}(|\nabla u|^2)\right]^2 }{|\nabla u|^2}
=&-8F\frac{\left(\sum_{j=1}^nu_{x_j}u_{x_jx_1} \right) ^2}{|\nabla u|^2}\\
\geq&-8F\sum_{j=1}^nu_{x_jx_1}^2 . \end{align*} Using the fact that $u$ is harmonic, one has \begin{align*}
\sum_{j=1}^nu_{x_jx_1}^2
=&\frac{n-1}{n}u_{x_1x_1}^2+\frac{1}{n}u_{x_1x_1}^2
+\sum_{\substack{j=2}}^nu_{x_jx_1}^2\\
\leq&\frac{n-1}{n}u_{x_1x_1}^2
+\frac{1}{n}\left(\sum_{k=2}^n u_{x_kx_k}\right)^2
+\frac{1}{2}\sum_{\substack{i,j=1\\j\neq i}}^nu_{x_jx_i}^2\\
\leq&\frac{n-1}{n}u_{x_1x_1}^2
+\frac{n-1}{n}\sum_{k=2}^n u_{x_kx_k}^2
+\frac{n-1}{n}\sum_{\substack{i,j=1\\j\neq i}}^nu_{x_jx_i}^2\\
\leq&\frac{n-1}{n}\sum_{i,j=1}^nu_{x_ix_j}^2
=\frac{n-1}{n}|\nabla\nabla u|^2, \end{align*} we have \begin{align}\label{eq3.16}
2\partial_{x_1} F\cdot \partial_{x_1}(|\nabla u|^2)
\geq&-\frac{8F(n-1)}{n}|\nabla\nabla u|^2 . \end{align} From \eqref{eq3.16} and \eqref{eq2.9}, one has \begin{align}\label{eq3.16+}
\mathcal{F}_2:=2\nabla F\cdot \nabla (|\nabla u|^2)
=&2\partial_{x_1} F\cdot \partial_{x_1}(|\nabla u|^2)+2\partial_{x_n} F\cdot \partial_{x_n}(|\nabla u|^2)\nonumber\\
\geq&-2\eta|\nabla u|^2-\left[ \frac{8F(n-1)}{n}+\frac{2}{\eta}|F_{x_n}|^2\right] |\nabla\nabla u|^2. \end{align}
Combining \eqref{eq2.9+} and \eqref{eq3.16+}, for $0<\xi<1$, we can write \begin{align}\label{eq3.30}
2\nabla F\cdot \nabla (|\nabla u|^2)
=&\xi\mathcal{F}_1+(1-\xi)\mathcal{F}_2\nonumber\\
\geq&-\left[\frac{\sum_{i=1}^{n-1}2F_{x_i}^2}{F}\xi
+2\eta \right] |\nabla u|^2\nonumber\\
&-\left[ \frac{8F(n-1)}{n}(1-\xi)
+\frac{2}{\eta}|F_{x_n}|^2\right] |\nabla\nabla u|^2. \end{align} Substituting \eqref{eq3.30} into \eqref{eq3.6}, we have \begin{align}\label{eq2.28}
\Delta Q
\geq&\left[ \Delta F-\sum_{i=1}^{n-1}\frac{2F_{x_i}^2}{F}\xi-2\eta\right] |\nabla u|^2\nonumber\\
&+\left[ 2F-\frac{8F(n-1)}{n}(1-\xi)-\frac{2}{\eta}|F_{x_n}|^2\right] |\nabla\nabla u|^2. \end{align}
\textbf{Step 2: Estimates of $\left[ \Delta F-\sum_{i=1}^{n-1}\frac{2F_{x_i}^2}{F}\xi-2\eta\right] |\nabla u|^2$.}
By simple computation, \begin{align}
\Delta F
=&2(n-1)(1-\gamma)(|x'|^2+\varepsilon)^{-\gamma}
-4\gamma(1-\gamma)(|x'|^2+\varepsilon)^{-\gamma-1}|x'|^2 \nonumber\\
&+2A(|x'|^2+\varepsilon)^{-\gamma}\cos\frac{x_n}{|x'|^2+\varepsilon}\Big\{2(|x'|^2+\varepsilon)^{-2}|x'|^2x_n^2\nonumber\\
&-(n-1)(2-\gamma)(|x'|^2+\varepsilon)
-2(2-\gamma)(1-\gamma)|x'|^2\Big\}\nonumber\\
&+2A(|x'|^2+\varepsilon)^{-\gamma}\sin\frac{x_n}{|x'|^2+\varepsilon}\Big\{
6(|x'|^2+\varepsilon)^{-1}|x'|^2x_n-(n-1)x_n\Big\}
\nonumber\\
&+B(|x'|^2+\varepsilon)^{-\gamma}\sin\left(\frac{\pi}{2}\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{\frac{|x'|^2}{2}+\varepsilon} \right)\bigg(\lambda_1-\lambda_2+\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{\frac{|x'|^2}{2}+\varepsilon}
\bigg) \nonumber\\
&\cdot\bigg\{\frac{\pi}{2}(n-1)(|x'|^2+\varepsilon)+2\pi(1-\gamma)|x'|^2
\bigg\}\nonumber\\
&+B(|x'|^2+\varepsilon)^{-\gamma}\cos\left(\frac{\pi}{2}\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{\frac{|x'|^2}{2}+\varepsilon} \right)\bigg\{
[(2-\gamma)|x'|^2+(3-2\gamma)\varepsilon]\nonumber\\
&\cdot\left(n-1-2\gamma|x'|^2(|x'|^2+\varepsilon)^{-1}\right) +2(2-\gamma)|x'|^2\nonumber\\
&-\frac{\pi^2}{4}(|x'|^2+\varepsilon)|x'|^2(\frac{|x'|^2}{2}+\varepsilon)^{-1}\left(\lambda_1-\lambda_2+\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{ \frac{|x'|^2}{2}+\varepsilon} \right)^2
\bigg\}\nonumber\\
&+A(|x'|^2+\varepsilon)^{-\gamma}\cos\frac{x_n}{|x'|^2+\varepsilon}\nonumber\\
&-\frac{\pi^2}{4}B(|x'|^2+\varepsilon)^{1-\gamma}\left(\frac{|x'|^2}{2}+\varepsilon \right)^{-1}\cos\left(\frac{\pi}{2}\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{\frac{|x'|^2}{2}+\varepsilon} \right). \end{align} Since \begin{align*}
&2A(|x'|^2+\varepsilon)^{-\gamma}\cos\frac{x_n}{|x'|^2+\varepsilon}\Big\{2(|x'|^2+\varepsilon)^{-2}|x'|^2x_n^2\nonumber\\
&-(n-1)(2-\gamma)(|x'|^2+\varepsilon)
-2(2-\gamma)(1-\gamma)|x'|^2\Big\}\nonumber\\
&+2A(|x'|^2+\varepsilon)^{-\gamma}\sin\frac{x_n}{|x'|^2+\varepsilon}\Big\{
6(|x'|^2+\varepsilon)^{-1}|x'|^2x_n-(n-1)x_n\Big\}
\nonumber\\
&+B(|x'|^2+\varepsilon)^{-\gamma}\sin\left(\frac{\pi}{2}\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{\frac{|x'|^2}{2}+\varepsilon} \right)\bigg(\lambda_1-\lambda_2+\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{\frac{|x'|^2}{2}+\varepsilon}
\bigg) \nonumber\\
&\cdot\bigg\{\frac{\pi}{2}(n-1)(|x'|^2+\varepsilon)+2\pi(1-\gamma)|x'|^2
\bigg\}\nonumber\\
&+B(|x'|^2+\varepsilon)^{-\gamma}\cos\left(\frac{\pi}{2}\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{\frac{|x'|^2}{2}+\varepsilon} \right)\bigg\{
[(2-\gamma)|x'|^2+(3-2\gamma)\varepsilon]\nonumber\\
&\cdot\left(n-1-2\gamma|x'|^2(|x'|^2+\varepsilon)^{-1}\right) +2(2-\gamma)|x'|^2\nonumber\\
&-\frac{\pi^2}{4}(|x'|^2+\varepsilon)|x'|^2(\frac{|x'|^2}{2}+\varepsilon)^{-1}\left(\lambda_1-\lambda_2+\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{ \frac{|x'|^2}{2}+\varepsilon} \right)^2
\bigg\}\nonumber\\
\ll& (|x'|^2+\varepsilon)^{-\gamma}, \end{align*} one has \begin{align}\label{eq2.20}
\Delta F
\geq\left[2(n-1)(1-\gamma)-4\gamma(1-\gamma)
+A\cos1-\frac{\pi^2}{2}B-\eta\right] (|x'|^2+\varepsilon)^{-\gamma}. \end{align}
For $F_{x_i}^2$, we can write \begin{align}\label{eq2.21}
&F_{x_i}^2\nonumber\\
=&\Bigg\{
2(1-\gamma)(|x'|^2+\varepsilon)^{-\gamma}x_i
-2A(2-\gamma)(|x'|^2+\varepsilon)^{1-\gamma}x_i\cos\frac{x_n}{|x'|^2+\varepsilon}\nonumber\\
&-2A(|x'|^2+\varepsilon)^{-\gamma}x_ix_n\sin\frac{x_n}{|x'|^2+\varepsilon}\nonumber\\
&+B(|x'|^2+\varepsilon)^{-\gamma}x_i\left((2-\gamma ) |x'|^2+(3-2\gamma) \varepsilon \right) \cos\left(\frac{\pi}{2}\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{\frac{|x'|^2}{2}+\varepsilon} \right)\nonumber\\ &+\frac{B\pi}{2}(|x'|^2+\varepsilon)^{1-\gamma}x_i\left(\lambda_1-\lambda_2+\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{\frac{|x'|^2}{2}+\varepsilon}\right) \sin\left(\frac{\pi}{2}\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{\frac{|x'|^2}{2}+\varepsilon} \right)
\Bigg\}^2\nonumber\\
\leq& 4(1+\eta)(1-\gamma)^2(|x'|^2+\varepsilon)^{-2\gamma}x_i^2, \end{align} where we use that \begin{align*}
&-2A(2-\gamma)(|x'|^2+\varepsilon)^{1-\gamma}x_i\cos\frac{x_n}{|x'|^2+\varepsilon}\nonumber\\
&-2A(|x'|^2+\varepsilon)^{-\gamma}x_ix_n\sin\frac{x_n}{|x'|^2+\varepsilon}\nonumber\\
&+B(|x'|^2+\varepsilon)^{-\gamma}x_i\left((2-\gamma ) |x'|^2+(3-2\gamma) \varepsilon \right) \cos\left(\frac{\pi}{2}\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{\frac{|x'|^2}{2}+\varepsilon} \right)\nonumber\\ &+\frac{B\pi}{2}(|x'|^2+\varepsilon)^{1-\gamma}x_i\left(\lambda_1-\lambda_2+\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{\frac{|x'|^2}{2}+\varepsilon}\right) \sin\left(\frac{\pi}{2}\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{\frac{|x'|^2}{2}+\varepsilon} \right)\nonumber\\
\leq&2\eta (1-\gamma)(|x'|^2+\varepsilon)^{-\gamma}|x_i|. \end{align*} Hence, from \eqref{eq2.21} and \eqref{eq2.5}, we can write \begin{align}\label{eq4.18}
-\sum_{i=1}^{n-1}\frac{2F_{x_i}^2}{F}
\geq&-\frac{8(1+\eta)(1-\gamma)^2(|x'|^2+\varepsilon)^{-2\gamma}|x'|^2}{\frac{1-b}{2^\gamma}(|x'|^2+\varepsilon)^{1-\gamma}}\nonumber\\
\geq&-\frac{80(1+\eta)(1-\gamma)^2(|x'|^2+\varepsilon)^{-2\gamma}|x'|^2}{(|x'|^2+\varepsilon)^{1-\gamma}}. \end{align}
Combining \eqref{eq2.20} and \eqref{eq4.18}, one has \begin{align}\label{eq2.27} &\Delta F-\sum_{i=1}^{n-1}\frac{2F_{x_i}^2}{F}\xi-2\eta\nonumber\\ \geq& \Big[2(n-1)(1-\gamma)-4\gamma(1-\gamma)+A\cos1-\frac{\pi^2}{2}B
-80\xi_0(1-\gamma)^2 -3\eta\Big](|x'|^2+\varepsilon)^{-\gamma}\nonumber\\
:=&M(n,\gamma)(|x'|^2+\varepsilon)^{-\gamma}, \end{align} where $\xi_0=\xi(1+\eta)$ and \begin{align*} M(n,\gamma):=&2(n-1)(1-\gamma)-4\gamma(1-\gamma)+A\cos1-\frac{\pi^2}{2}B-80\xi_0(1-\gamma)^2-3\eta. \end{align*}
Define \begin{equation}\label{rho} \rho:=-\left[ \gamma^2+(n-1)\gamma-(n-2)\right]\geq 0, \end{equation} and choose \begin{equation}\label{xi}
\xi_0=1-\frac{n}{4(n-1)}+\eta, \end{equation} then $M$ can be written by \begin{align*} M(n,\gamma) =&\left(56-\frac{20}{n-1}\right)\rho +\left(42+54n-\frac{40}{n-1}\right) \gamma-54n+70+A\cos1\nonumber\\ &-\frac{\pi^2}{2}B-3\eta\nonumber\\ \geq&\left(42+54n-\frac{40}{n-1}\right) \gamma-54n+70+A\cos1-\frac{\pi^2}{2}B-3\eta >0. \end{align*} where in the last inequality we use \eqref{eq2.6}. Thus, \begin{align}\label{eq2.23}
\left[\Delta F-\sum_{i=1}^{n-1}\frac{2F_{x_i}^2}{F}\xi-2\eta\right] |\nabla u|^2>0. \end{align}
\textbf{Step 3: Estimates of $\left[ 2F-\frac{8F(n-1)}{n}(1-\xi)-\frac{2}{\eta}|F_{x_n}|^2\right] |\nabla\nabla u|^2$.}
Since \begin{align*}
&|F_{x_n}|^2\nonumber\\
=&\bigg[A(|x'|^2+\varepsilon)^{1-\gamma}\sin\frac{x_n}{|x'|^2+\varepsilon}-\frac{\pi}{2}B(|x'|^2+\varepsilon)^{1-\gamma}\sin\left(\frac{\pi}{2}\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^2}{\frac{|x'|^2}{2}+\varepsilon} \right) \bigg]^2\nonumber\\
\leq& \eta^2(|x'|^2+\varepsilon)^{1-\gamma}, \end{align*} we have \begin{equation*}
\frac{2}{\eta}|F_{x_n}|^2\leq \eta (|x'|^2+\varepsilon)^{1-\gamma}. \end{equation*} From \eqref{xi}, we know \begin{equation*} 1-\xi=\frac{n}{4(n-1)(1+\eta)}, \end{equation*} which leads \begin{align*}
2F-\frac{8F(n-1)}{n}(1-\xi)-\frac{2}{\eta}|F_{x_n}|^2
\geq2\eta F-\eta(|x'|^2+\varepsilon)^{1-\gamma}>0. \end{align*} Thus \begin{equation}\label{eq4.31}
\left[ 2F-\frac{8F(n-1)}{n}(1-\xi)-\frac{2}{\eta}|F_{x_n}|^2\right] |\nabla\nabla u|^2>0. \end{equation}
Combining \eqref{eq2.23}, \eqref{eq4.31} and \eqref{eq2.28}, we have \eqref{eq4.9}, which is contradictory with \eqref{eq3.6}.
Hence, we have ruled out the possibility that $Q$ obtains its maximum point at the boundary $\partial\Omega_r\backslash(\Gamma_+\cup\Gamma_-)$ Thus, increasing $r$ if necessary, we have \begin{align*}
Q\leq C\|g\|_{C^{1,\alpha}(\partial D)}^2, \end{align*} \eqref{eq1.3} follows.
\end{proof}
\begin{proof}[Proof of Theorem \ref{th-insm}]
Under the assumption of \eqref{eq1.2}, \eqref{eq1.6} and \eqref{eq1.8}, from Lemma \ref{lemma3.2}, we know that at any point of $\Gamma_+$ and $\Gamma_-$, we have
\begin{equation}\label{eq3.0}
\frac{\partial}{\partial\nu}(|\nabla u|^2)
\leq\frac{2m(m-1)\lambda_k|x'|^{m-2}}{\sqrt{1+m^2\lambda_k^2|x'|^m}} |\nabla u|^2,
\end{equation}
where $k=1$ on $\Gamma_+$, $k=2$ on $\Gamma_-$.
We consider the quantity
\begin{align*}
Q=F |\nabla u|^2\ \ \ \ \mbox{in}\ \ \Omega_{r},
\end{align*}
where \begin{align}
F=(|x'|^m+\varepsilon)&^{1-\gamma}-A(|x'|^m+\varepsilon)^{2-\gamma}|x'|^{m-2}\cos\frac{x_n}{|x'|^m+\varepsilon}\nonumber\\
&+B(|x'|^m+\varepsilon)^{1-\gamma}|x'|^{m-2}\left( \frac{|x'|^m}{2}+\varepsilon\right) \cos\left(\frac{\pi}{2}\frac{x_n-\frac{\lambda_1-\lambda_2}{2}|x'|^m}{\frac{|x'|^m}{2}+\varepsilon} \right) \end{align}
$A,B$ are uniform constants satisfying
\begin{equation*}
A>\frac{(m-2+m\gamma)m\pi\max\{\lambda_1,\lambda_2\}-m\left(\frac{mn}{2}-\frac{m}{n-1}-n+3 \right)\gamma
-m\left(m+n-3-\frac{mn}{2} \right) }{\cos1-\pi\sin1},
\end{equation*}
\begin{align*}
\frac{2}{\pi}\Big\{&(m-2+m\gamma)m\max\{\lambda_1,\lambda_2\}+A\sin1\Big\}\nonumber\\
&<B<\frac{2}{\pi^2}\Big\{m\left(\frac{mn}{2}-\frac{m}{n-1}-n+3 \right)\gamma
+m\left(m+n-3-\frac{mn}{2} \right) +A\cos1\Big\},
\end{align*}
Similar to the proof of Theorem \ref{th-ins}, we have the result. \end{proof}
\textbf{Acknowledgments.} The author would like to thank to Professor Haigang Li and Dr. Yu Chen for their suggestions and constant encouragement regarding this work. The author thanks anonymous referees for helpful suggestions which improve the exposition.
\def$'${$'$}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Dahlberg's bilinear estimate for solutions of divergence form complex elliptic equations}
\author[S. Hofmann]{Steve Hofmann} \address{Department of Mathematics, University of Missouri, Columbia, Missouri 65211, USA} \email{[email protected]} \thanks{The author was supported by the National Science Foundation}
\subjclass{42B20, 42B25, 35J25}
\maketitle \begin{abstract} We consider divergence form elliptic operators $L=-\operatorname{div} A(x)\nabla$, defined in $\mathbb{R}^{n+1}=\{(x,t)\in\mathbb{R}^{n}\times\mathbb{R}\},\, n \geq 2$, where the $L^{\infty}$ coefficient matrix $A$ is $(n+1)\times(n+1)$, uniformly elliptic, complex and $t$-independent. Using recently obtained results concerning the boundedness and invertibility of layer potentials associated to such operators, we show that if $Lu=0$ in $\mathbb{R}^{n+1}_+$, then for any vector-valued ${\bf v} \in W^{1,2}_{loc},$ we have the bilinear estimate
$$\left|\iint_{\mathbb{R}^{n+1}_+} \nabla u \cdot \overline{{\bf v}} dx dt \right|\leq C\sup_{t>0}
\|u(\cdot,t)\|_{L^2(\mathbb{R}^n)}\left( \||t \nabla {\bf v}\|| +
\|N_*{\bf v}\|_{L^2(\mathbb{R}^n)}\right),$$
where $\||F\|| \equiv \left(\iint_{\mathbb{R}^{n+1}_+} |F(x,t)|^2 t^{-1} dx dt\right)^{1/2},$ and where $N_*$ is the usual non-tangential maximal operator. The result is new even in the case of real symmetric coefficients, and generalizes the analogous result of Dahlberg for harmonic functions on Lipschitz graph domains. \end{abstract}
\section{Introduction \label{s1}}
In \cite{D}, B. Dahlberg considered the bilinear singular integral form \begin{equation} \int_\Omega \nabla u \cdot\overline{{\bf v}},\end{equation} where $u$ is harmonic in the domain $\Omega \equiv \{(x,t) \in \mathbb{R}^{n+1}: t > \varphi (x)\},$ with $\varphi $ Lipschitz, and where ${\bf v} \in W^{1,2}_{loc}$ is vector valued. He showed that the bilinear form (1.1) is bounded by the $L^2$ norm of the square function plus the non-tangential maximal function of $u$, times the same expression for ${\bf v}$. In the present note, we generalize Dahlberg's Theorem to variable coefficient divergence form elliptic operators. To be precise, let \begin{equation*} L=-\operatorname{div} A\nabla\equiv-\sum_{i,j=1}^{n+1}\frac{\partial}{\partial x_{i}}\left(A_{i,j} \,\frac{\partial}{\partial x_{j}}\right)\end{equation*} be defined in $\mathbb{R}^{n+1}=\{(x,t)\in\mathbb{R}^{n}\times\mathbb{R}\},\, n\geq 2,$ (we use the convention that $x_{n+1}=t$), where $A=A(x)$ is an $(n+1)\times(n+1)$ matrix of complex-valued $L^{\infty}$ coefficients, defined on $\mathbb{R}^{n}$ (i.e., independent of the $t$ variable), and satisfying the uniform ellipticity (accretivity) condition \begin{equation}
\label{eq1.1} \lambda|\xi|^{2}\leq\Re e\,\langle A(x)\xi,\xi\rangle, \, \,\,
\Vert A\Vert_{L^{\infty}(\mathbb{R}^{n})}\leq\Lambda, \end{equation}
for some $\lambda>0$, $\Lambda<\infty$, and for all $\xi\in\mathbb{C}^{n+1}$, $x\in\mathbb{R}^{n}$. Here, $\langle\cdot,\cdot\rangle$ denotes the usual hermitian inner product in $\mathbb{C}^{n+1}$, so that \begin{equation*} \langle A(x)\xi,\xi\rangle\equiv\sum_{i,j=1}^{n+1}A_{ij}(x)\xi_{j}\bar{\xi_{i}}\end{equation*}
In order to state our theorem, we first recall that the non-tangential maximal operator $N_{\ast}$ (and a variant $\widetilde{N}_*$) are defined as follows. Given $x_0\in\mathbb{R}^{n}$, let
$$\gamma(x_0)=\{(x,t)\in\mathbb{R}_{+}^{n+1}:|x_0-x|<t\}$$ denote the cone with vertex at $x_0$. Then for $U $ defined in $\mathbb{R}_{+}^{n+1}$,
\begin{equation*} N_{\ast} U(x_0) \equiv\sup_{(x,t)\in\gamma(x_0)}|U(x,t)|,\quad \widetilde{N}_{\ast} U(x_0) \equiv\sup_{(x,t)\in \gamma(x_0)}
\left(\fint\!\!\fint_{\substack{|x-y|<t\\ |t-s|<t/2}}|U(y,s)|^{2}dyds\right)^{\frac{1}{2}}.\end{equation*}
Our main result is the following: \begin{theorem} \label{t1.3} Suppose that $L$ is an operator of the type described above, with
\begin{equation}\label{eq1.small}\|A - A_0\|_\infty \leq \epsilon ,\end{equation} for some real, symmetric, $L^\infty$, elliptic, and $t$-independent matrix $A_0$. Suppose also that $Lu=0$, and that ${\bf v} \in W^{1,2}_{loc}(\mathbb{R}^{n+1}, \mathbb{C}^{n+1}).$ If $\epsilon\leq\epsilon_0$, with $\epsilon_0$ sufficiently small, depending only on dimension and ellipticity, then we have the bilinear estimate
$$\left|\iint_{\mathbb{R}^{n+1}_+} \nabla u \cdot \overline{{\bf v}} \, dx dt \,\right| \leq C \sup_{t>0}
\|u(\cdot,t)\|_{L^2(\mathbb{R}^n)}\left( \||t \nabla {\bf v}\|| +
\|N_*{\bf v}\|_{L^2(\mathbb{R}^n)}\right),$$
where $ C = C(n,\lambda,\Lambda)$ and $$\||F\|| \equiv \left(\iint_{\mathbb{R}^{n+1}_+} |F(x,t)|^2 t^{-1} dx dt\right)^{1/2}.$$ \end{theorem}
We remark that the result is new even in the case of real, symmetric coefficients. The analogous result was proved by Dahlberg for harmonic functions in Lipschitz graph domains, using a special change of variable found by Kenig and Stein, and independently by Maz'ya. Our theorem includes that of Dahlberg, as may be seen by pulling back under the mapping $(x,t) \to (x,\varphi (x) +t)$. Dahlberg's original method seems inapplicable to the variable coefficient case, unless the coefficients are differentiable and satisfy an appropriate sort of Carleson condition as in the work of Kenig and Pipher \cite{KP}. In the present setting, in lieu of the special change of variable, we use recently obtained results of \cite {AAAHK} concerning the boundedness and invertibility of layer potentials associated to variable coefficient $t$-independent operators.
The paper is organized as follows. In the next section, we recall some of the aforementioned results of \cite{AAAHK}. In Section \ref{s3}, we prove Theorem \ref{t1.3}, and in Section \ref{s4} we discuss the analogue of another result of \cite{D} concerning the domain of the infinitesimal generator of the Poisson semigroup for the equation $Lu=0$ in $\mathbb{R}_+^{n+1}$.
Let us now set some notation and terminology that we shall use in the sequel. We shall employ the standard convention that the generic constant $C$ is allowed to vary from one instance to the next, and may depend upon dimension and ellipticity. The symbol $\fint$ denotes the mean value, i.e.,
$\fint_E f \equiv |E|^{-1} \int_E f .$ We shall use the notation $$D_j\equiv\partial_{x_j}, \,\,\,1\leq j\leq n+1,$$ bearing in mind that $x_{n+1} = t,$ and we use $e_j,\, 1\leq j\leq n+1,$ to indicate the standard unit basis vector in the $x_j$ direction. The symbol $\nabla$ denotes the full $(n+1)$-dimensional
gradient, acting in both $x$ and $t$, and we use $\nabla_x$ or $\nabla_\|$ to indicate the $n$-dimensional gradient acting only in $x$. We use $ad\!j$ to denote the hermitian adjoint of an operator acting on functions defined on $\mathbb{R}^n$. We define the homogeneous Sobolev space $\dot{L}_{1}^{2}$ to be the completion of $C_0^{\infty}$ with respect to the seminorm
$\|\nabla F\|_2.$ As is well known, for $n\geq 2$, this space can be identified (modulo constants) with the space $I_1(L^2) \equiv \Delta^{-1/2}(L^2)$. We write $F \to f\,n.t.$ to mean that for $a.e.$
$x \in \mathbb{R}^n$, $F(y,t) \to f(x)$, as $(y,t) \to (x,0)$, with
$(y,t) \in \gamma(x).$
\section{Results for variable coefficient layer potentials \label{s2}}
We now recall the definitions of the layer potentials. We first note that by \eqref{eq1.small}, the stability result of \cite{A}, and the classical De Giorgi-Nash Theorem \cite{DeG,N}, solutions of $Lu=0$ are locally H\"older continuous. Let $\Gamma (x,t,y,s)$ denote the fundamental solution for $L$ (we refer the reader to \cite{HK} for the construction of, and estimates for, $\Gamma$ in the case of complex coefficients, assuming De Giorgi-Nash bounds). By $t$-independence, \begin{equation}\label{2.1a}\Gamma (x,t,y,s) \equiv \Gamma (x,t-s,y,0).\end{equation} We define the single and double layer potentials, respectively, in the usual way:
\begin{equation} \begin{split}\label{eq1.5}S_{t}f(x) & \equiv\int_{\mathbb{R}^{n}}\Gamma(x,t,y,0)\,f(y)\,dy, \,\,\, t\in \mathbb{R}\\ \mathcal{D}_{t}f(x) & \equiv\int_{\mathbb{R}^{n}}\overline{\partial_{\nu^*(y)} \Gamma^* (y,0,x,t)}\,f(y)\,dy,\,\,\, t \neq 0,\end{split} \end{equation}
where $\partial_{\nu^*}$ is the adjoint exterior
conormal derivative; i.e., if $A^{\ast}$ denotes the hermitian adjoint of $A$, then \begin{equation}\label{eq2.2a} \partial_{\nu^*(y)} \Gamma^* (y,0,x,t) =-\sum^{n+1}_{j=1}A_{n+1,j}^{\ast}(y)\frac{\partial \Gamma^*}{\partial y_{j}}(y,0,x,t)=-e_{n+1}\cdot A^{\ast}(y) \nabla_{y,s}\Gamma^*(y,s,x,t) \mid_{s=0}\end{equation}
(recall that $y_{n+1}=s$), where $e_{n+1}\equiv (0,...0,1)$ is the unit basis vector in the $t$ direction.
Here, $\Gamma^*$ is the fundamental solution for
$L^*$, the hermitian adjoint of $L$. Thus, $\Gamma^*$ is the conjugate transpose of $\Gamma$; i.e.,
$$\overline{\Gamma^*(y,s,x,t)} =\Gamma(x,t,y,s).$$ We also define (formally) the
boundary singular integral \begin{equation} \label{eq1.6}Kf(x) \equiv
``p.v."\int_{\mathbb{R}^{n}}\overline{\partial_{\nu^*} \Gamma^* (y,0,x,0)}\,f(y)\,dy. \end{equation}
(For the precise definiton of the latter operator in the
case of non-smooth coefficients, see \cite{AAAHK}, Section 4). In a departure from tradition impelled by the context of complex coefficients,
$K^*, S^*$ and $\mathcal{D}^*$ will denote the analogues of $K,S$ and $\mathcal{D}$ corresponding to $L^*$.
In order to prove our Theorem, we shall require some of
the main results of \cite{AAAHK}, which we summarize as follows: \begin{theorem} \cite{AAAHK}. \label{t2.6} Suppose that $L$ satisfies the hypotheses of Theorem 1.3. There exists a small constant $\epsilon_0= \epsilon_0(n,\lambda,\Lambda)$ such that if $\epsilon$ in \eqref{eq1.small} satisfies $\epsilon \leq \epsilon_0$, then the layer potential operators $\pm \frac{1}{2} I + K,\, \pm\frac{1}{2}I + K^*$ are isomorphisms on $L^2 (\mathbb{R}^n)$, with the implicit constants depending only upon dimension and ellipticity. In addition,
\begin{equation}\label{eq2.squarefunction}\sup_{t>0}\|\mathcal{D}_t f\|_2+
\||t \nabla \partial_t S_{\pm t} f\|| +\sup_{1\leq j \leq n}
\||t \partial_t S_{\pm t} D_jf\||\leq C \|f\|_2,\end{equation} and $\mathcal{D}_{\pm t}f \to (\pm\frac{1}{2}I + K)f$ $n.t.$ and in $L^2$, for $f\in L^2$. Moreover, the corresponding statements hold also for $\mathcal{D}_t^*, K^*$ and $S_t^*$. Finally, the solution to the Dirichlet problem \begin{equation} \begin{cases} Lu=0\text{ in }\mathbb{R}_{+}^{n+1}=\{(x,t)\in\mathbb{R}^{n}\times(0,\infty)\}\\ \lim_{t\to 0}u(\cdot,t)=f\text{ in } L^{2}(\mathbb{R}^{n}) \text{ and } n.t.\\
\sup_{t>0}\|u(\cdot,t)\|_{L^2(\mathbb{R}^n)}<\infty, \end{cases}\tag{D2}\label{D2}\end{equation} which exists by virtue of the aforementioned facts about layer potentials, is unique. \end{theorem}
We shall also require the following technical facts. \begin{lemma}\label{l2.9a} \cite{AAAHK} (Lemma 2.2) Suppose that $L$ satisfies the hypotheses of Theorem \ref{t1.3}, with $\epsilon \leq \epsilon_0$. Set $K_t(x,y) =t\partial_t^2\,\Gamma(x,t,y,0)$. Then \begin{equation}\label{eq2.10a}
|K_t(x,y)|\leq C \frac{t}{(t + |x-y|)^{n+1}}. \end{equation} \end{lemma} \begin{lemma}\label{l2.11a}\cite{AAAHK} (Lemma 2.8) Suppose that $L$ satisfies the hypotheses of Theorem \ref{t1.3}, with $\epsilon \leq \epsilon_0$. Then
$$\|t^2\nabla\partial_t^2S_{\pm t}^* f\|_{L^2(\mathbb{R}^n)} \leq C\|f\|_2.$$ \end{lemma} \begin{lemma}\label{l2.15}. Suppose that $\{ R_t\}_{t>0}$ is a family of operators defined by $$R_tf(x)\equiv \int_{\mathbb{R}^n}K_t(x,y) f(y) dy, $$ where the kernel $K_t$ satisfies \eqref{eq2.10a}. Suppose also that $R_t1=0$ for all $t\in \mathbb{R}$. Then for $h\in \dot{L}^2_1 (\mathbb{R}^n)$,
\begin{equation}\label{eq2.15a}\int_{\mathbb{R}^n}|R_th|^2\leq Ct^2\int_{\mathbb{R}^n}|\nabla_x h|^2.\end{equation} \end{lemma}
The proof of the last lemma is a standard exercise in the use of Poincar\'e's inequality. We omit the details, but see, e.g. \cite{AAAHK} (Lemma 3.5), for a more general result.
Finally, we shall use the following special case of the ``Fatou Theorem" of \cite{AAAHK}. \begin{lemma}\label{l2.12} \cite{AAAHK} (Corollary 4.41) Suppose that $L$ satisfies the hypotheses of Theorem \ref{t1.3}, with $\epsilon$ sufficiently small, and that $Lu=0$ in $\mathbb{R}^{n+1}.$ Suppose also that
\begin{equation*} \sup_{t>0} \|u(\cdot,t)\|_2 <\infty. \end{equation*} Then $u(\cdot,t)$ converges $n.t.$ and in $L^2$ as $t \to 0$. \end{lemma}
\section{Proof of Theorem \ref{t1.3} \label{s3}} The proof of the theorem will use the following
\begin{lemma}\label{l3.0}Suppose that $L$ satisfies the hypotheses of Theorem \ref{t1.3}, with $\epsilon$ sufficiently small. Suppose also that $Lu=0$ in $\mathbb{R}^{n+1}$, with $\sup_{t>0}\|u(\cdot,t)\|_2 < \infty.$ Then
$$\||t\nabla u\|| \leq C \sup_{t>0}\|u(\cdot,t)\|_2.$$ \end{lemma} \begin{proof} It is enough to show that for each fixed $\eta \in (0,10^{-10}), $ $$ \fint_\eta^{2\eta}\int_\delta^{1/\delta}\!\!\int_{\mathbb{R}^n}
|\nabla u|^2 t dx dt d\delta\leq C\sup_{t>0}\|u(\cdot,t)\|_2.$$ Integrating by parts in $t$ on the left side of the last inequality, we obtain \begin{equation}\label{eq3.0b}-\Re e \fint_\eta^{2\eta}\int_\delta^{1/\delta}\langle \partial_t \nabla u,\nabla u\rangle t^2 dt d\delta + \text { boundary},\end{equation} where the boundary terms are dominated by
\begin{equation}\label{eq3.0a}\sup_{r>0} \fint_r^{2r}\!\!\int_{\mathbb{R}^n}r^2|\nabla u(x,t)|^2dxdt \leq C\sup_{t>0}\|u(\cdot,t)\|^2_2,\end{equation} as desired, and in the last step we have split $\mathbb{R}^n$ into cubes of side length $\approx r$ and used Caccioppoli's inequality. By Cauchy's inequality, the main term in \eqref{eq3.0b} is no larger than $$\frac{\varepsilon}{2} \fint_\eta^{2\eta}\int_\delta^{1/\delta}\!\!\int_{\mathbb{R}^n}
|\nabla u|^2 t dx dt d\delta + \frac{1}{2\varepsilon}\fint_\eta^{2\eta} \int_\delta^{1/\delta}\!\!\int_{\mathbb{R}^n}
|\nabla \partial_t u|^2 t^3 dx dt d\delta \equiv I + II,$$ where $\varepsilon >0$ is at our disposal. Choosing $\varepsilon$ small, we may hide term $I$. Having fixed $\varepsilon$, and applying Caccioppoli's inequality in Whitney boxes, we obtain that
$$II \leq C \iint_{\mathbb{R}^{n+1}_+} |\partial_t u|^2 dx \,t dt.$$ By the Fatou Theorem of \cite{AAAHK}, Section 4, $u$ converges in $L^2(\mathbb{R}^n)$ to some
$f $, with $$\|f\|_2 \leq \sup_{t>0}\|u(\cdot,t)\|_2.$$ Thus, by Theorem \ref{t2.6}, $u(\cdot,t) = \mathcal{D}_t \left(-\frac{1}{2}I + K\right)^{-1}f.$ By \eqref{eq2.squarefunction}, the bijectivity of $\left(-\frac{1}{2}I+K\right)$ and the definition of $\mathcal{D}_t$, we obtain that
$$\iint_{\mathbb{R}^{n+1}_+} |\partial_t u|^2 dx \,t dt \leq C\|f\|_2\leq C\sup_{t>0}\|u(\cdot,t)\|_2.$$ \end{proof}
\begin{proof}[Proof of Theorem \ref{t1.3}]By the previous lemma, it is enough to establish the bound \begin{multline}\label{eq3.1} \sup_{0<\eta<10^{-10}}\fint_\eta^{2\eta}
\left|\int_\delta^{1/\delta}\!\!\int_{\mathbb{R}^n} \nabla u\cdot \overline{{\bf v}}\, dx dt \right|d\delta\\ \leq C
\left(\||t \nabla u\||+\sup_{t>0}
\|u(\cdot,t)\|_{L^2(\mathbb{R}^n)}\right)\left( \||t \nabla {\bf v}\|| +\sup_{t>0}
\|{\bf v}(\cdot,t)\|_{L^2(\mathbb{R}^n)}\right). \end{multline} We may suppose that the right hand side of \eqref{eq3.1} is finite, otherwise there is nothing to prove. On the left hand side of \eqref{eq3.1}, for each fixed $\eta$, we integrate by parts in $t$ to obtain the bound
\begin{equation}\label{eq3.2}\left|\fint_\eta^{2\eta}\! \!\int_\delta^{1/\delta}\! \!\int_{\mathbb{R}^n} \nabla
u\cdot \overline{\partial_t{\bf v}} \,dx\, tdt d\delta\right|
+\left|\fint_\eta^{2\eta}\! \!\int_\delta^{1/\delta}\! \!\int_{\mathbb{R}^n} \nabla
\partial_t u\cdot \overline{{\bf v}} \,dx\,t dtd\delta \right| +\text{ boundary},\end{equation} where the boundary terms are dominated by
\begin{multline*}C\left(\sup_{r>0} \fint_r^{2r}\!\!\int_{\mathbb{R}^n}r^2|\nabla u(x,t)|^2dxdt
\right)^{1/2}\left( \sup_{t>0}\|{\bf v}(\cdot,t)\|_2\right)\\\leq\,\, C\left( \sup_{t>0}\|u(\cdot,t)\|_2\right)
\left( \sup_{t>0}\|{\bf v}(\cdot,t)\|_2\right),\end{multline*} and we have used \eqref{eq3.0a} in the last step. Moreover, by Cauchy-Schwarz, the first term in \eqref{eq3.2} is no larger than
$\||t\nabla u\|| \,\, \||t\nabla {\bf v}\||.$
It therefore remains to treat the middle term in \eqref{eq3.2}. To this end, we write
$\nabla = \nabla_x + \partial_t e_{n+1},$ and ${\bf v} = \text{v}_\| + \text{v}_{n+1}e_{n+1},$ where $v_{n+1}\equiv {\bf v}\cdot e_{n+1}.$
Now, \begin{multline*}\left|\fint_\eta^{2\eta}\! \!\int_\delta^{1/\delta}\! \!\int_{\mathbb{R}^n} \nabla_x
\partial_t u\cdot \overline{\text{v}_\|} \,dx\,t dtd\delta \right| \\=\,\,
\left|\fint_\eta^{2\eta}\! \!\int_\delta^{1/\delta}\! \!\int_{\mathbb{R}^n}
\partial_t u\,\operatorname{div}_x \!\overline{\text{v}_\|} \,dx\,t dtd\delta \right|
\,\,\leq\,\, C\||t\nabla u\|| \,\, \||t\nabla {\bf v}\||,\end{multline*} as desired. Thus, it is enough to consider
\begin{multline}\label{eq3.3}\left|\fint_\eta^{2\eta}\! \!\int_\delta^{1/\delta}\! \!\int_{\mathbb{R}^n}
\partial_t^2 u\, \overline{\text{v}_{n+1}}dx\,t dtd\delta \right|\,\,\leq\,\,
\frac{1}{2}\left|\fint_\eta^{2\eta}\! \!\int_\delta^{1/\delta}\! \!\int_{\mathbb{R}^n}
\partial_t^3 u\, \overline{\text{v}_{n+1}}dx\,t^2 dtd\delta\right|\\ +\,\,\frac{1}{2}\left| \fint_\eta^{2\eta}\! \!\int_\delta^{1/\delta}\! \!\int_{\mathbb{R}^n}
\partial_t^2 u\, \overline{\partial_t\text{v}_{n+1}}dx\,t^2 dtd\delta\right| \,\,+\,\,|\mathcal{B}|\\\equiv |I| + |II| +
|\mathcal{B}|,\end{multline} where we have again integrated by parts in $t$, and $\mathcal{B}$ denotes boundary terms which satisfy
\begin{multline*}|\mathcal{B}| \leq C\left(\sup_{r>0} \fint_r^{2r}\!\!\int_{\mathbb{R}^n}r^4|\partial_t^2 u(x,t)|^2dxdt
\right)^{1/2}\left( \sup_{t>0}\|{\bf v}(\cdot,t)\|_2\right)\\\leq C\left( \sup_{t>0}\|u(\cdot,t)\|_2\right)
\left( \sup_{t>0}\|{\bf v}(\cdot,t)\|_2\right),\end{multline*} by a double application of Caccioppoli's inequality. Moreover,
$$|II| \leq C \,\||t^2 \partial_t^2u\|| \, \,\||t\partial_t {\bf v}\|| \leq C \,\||t \partial_t u\|| \, \,\||t\partial_t {\bf v}\||,$$ where we have used Caccioppoli in Whitney boxes to bound the first factor. Turning to the main term, we have that
\begin{equation}\label{eq3.4}|I| = C\left|\fint_\eta^{2\eta} \! \!\int_{\delta/2}^{1/(2\delta)}\! \!\int_{\mathbb{R}^n}
\partial_t^3 u(x,2t)\, \overline{\text{v}_{n+1}(x,2t)}\,dx\,t^2 dtd\delta \right|,\end{equation} where we have made the change of variable $t \to 2t$. For $t$ momentarily fixed, set $$g_t(x) \equiv \partial_t u(x,t).$$ By Theorem \ref{t2.6} (i.e., the result of \cite{AAAHK}), we have that $$\tilde{u}_t(\cdot,s) \equiv \mathcal{D}_s \left(-\frac{1}{2}I + K \right)^{-1} g_t$$ is the unique solution of (D2) with data $g_t$. Hence, by $t$-independence, $$\tilde{u}_t(\cdot,s) = \partial_tu(\cdot,t+s).$$ Setting $s=t$, we therefore obtain that $$(D_{n+1}^3 u)(\cdot,2t) =\left( \partial_t^2\mathcal{D}_t\right)\left(-\frac{1}{2}I + K \right)^{-1} g_t.$$ We observe that by \eqref{2.1a}, \eqref{eq1.5} and \eqref{eq2.2a}, \begin{equation*} ad\!j\left(\partial_t^2\mathcal{D}_{t}\right)= \partial_{\nu^*}\partial_t^2S^*_{-t}.\end{equation*} Consequently, \eqref{eq3.4} becomes
$$|I| = C\left|\fint_\eta^{2\eta}\! \!\int_{\delta/2}^{1/(2\delta)}\! \!\int_{\mathbb{R}^n} \left(-\frac{1}{2}I + K\right)^{-1}\partial_t u(\cdot,t)\,
\overline{\partial_{\nu^*}D_{n+1}^2S^*_{-t}\text{v}_{n+1}(\cdot,2t)}\,dx\,t^2 dtd\delta \right|,$$ so by Cauchy-Schwarz and Theorem \ref{t2.6}, it suffices to prove
\begin{equation}\label{eq3.5}\||t^2\nabla D_{n+1}^2 S^*_{-t}
{\bf v}(\cdot, 2t)\||\leq C\left( \||t\nabla {\bf v}\|| + \|N_*{\bf v}\|_2\right).\end{equation} The left hand side of \eqref{eq3.5} equals \begin{multline}\label{eq3.6}\left(\sum_{k=-\infty}^\infty \int_{2^k}^{2^{k+1}}
\!\!\int_{\mathbb{R}^n} |t^2\nabla D_{n+1}^2 S_{-t}^* {\bf v}(\cdot,2t)
|^2 \frac{dx\, dt}{t}\right)^{1/2}\\\leq\,\,\left(\sum_{k=-\infty}^\infty \int_{2^k}^{2^{k+1}}
\!\!\int_{\mathbb{R}^n} |t^2\nabla D_{n+1}^2 S_{-t}^*\left( {\bf v}(\cdot,2t)
-{\bf v}(\cdot,2t_k)\right)|^2 \frac{dx\, dt}{t}\right)^{1/2}\\ +\,\,\left(\sum_{k=-\infty}^\infty \int_{2^k}^{2^{k+1}}
\!\!\int_{\mathbb{R}^n} |t^2
\nabla D_{n+1}^2 S_{-t}^*{\bf v}(\cdot,2t_k)|^2 \frac{dx\, dt}{t}\right)^{1/2}\equiv \,\,III+IV\end{multline} where $t_k = 2^{k-1}$. We consider term $IV$ first. Dividing $\mathbb{R}^n$ into cubes of side length $2^k$, and using Caccioppoli's inequality, we deduce that \begin{multline*}IV\, \leq \,C \left(\sum_{k=-\infty}^\infty \int_{2^{k-1}}^{2^{k+2}}
\!\!\int_{\mathbb{R}^n} |t D_{n+1}^2 S_{-t}^*{\bf v}(\cdot,2t_k)|^2 \frac{dx\, dt}{t}\right)^{1/2}\\\leq\, C\left(\sum_{k=-\infty}^\infty \int_{2^{k-1}}^{2^{k+2}}
\!\!\int_{\mathbb{R}^n} |t D_{n+1}^2 S_{-t}^*\left( {\bf v}(\cdot,2t_k)
-{\bf v}(\cdot,2t)\right)|^2 \frac{dx\, dt}{t}\right)^{1/2}\\
+\,\,\||\left(tD_{n+1}^2S_{-t}^*1\right)\left(P_t {\bf v}(\cdot,2t)\right)\||\,\,+
\,\,||R_t {\bf v}(\cdot,2t)\||\,\,\equiv\,\, IV_1 + IV_2 + IV_3, \end{multline*} where $$R_t \equiv tD_{n+1}^2S_{-t}^*- \left(tD_{n+1}^2S_{-t}^*1\right)P_t,$$ and $P_t$ is a nice approximate identity with a smooth, compactly supported kernel. By Lemma \ref{l2.15},
$$IV_3 \leq \||t\nabla {\bf v}\||.$$ By \eqref{eq2.squarefunction}, Lemma \ref{l2.9a}, and a well known argument of Fefferman and Stein \cite{FS}, we have that
$|t \partial_t^2 S^*_{-t} 1|^2\frac{dx dt}{t}$ is a Carleson measure, whence
$$IV_2 \leq C \|N_* {\bf v}\|_2.$$ By Lemma \ref{l2.9a}, the operator $f\to tD_{n+1}^2 S_{-t}^* f$ is bounded on $L^2(\mathbb{R}^n)$, so that \begin{multline*}IV_1 \leq C\left(\sum_{k=-\infty}^\infty \int_{2^{k-1}}^{2^{k+2}}
\!\!\int_{\mathbb{R}^n}\left|\frac{1}{\sqrt{t}}\int_{2t_k}^{2t}\partial_s{\bf v}(x,s)ds\right|^2 dx\, dt\right)^{1/2} \\\leq \, C \left(\sum_{k=-\infty}^\infty
\iint_{\mathbb{R}^{n+1}}\left|\fint_{2t_k}^{2t}
1_{\{2^{k}\leq s \leq 2^{k+3}\}}\sqrt{s}\partial_s{\bf v}(x,s)ds\right|^2 dx\, dt\right)^{1/2}\leq C
\||t \partial_t {\bf v}\||, \end{multline*} where in the last step we have used the boundedness of the Hardy-Littlewood Maximal function.
Finally, we consider term $III$ in \eqref{eq3.6}. By Lemma \ref{l2.11a}, $III$ may be handled like $IV_1$ above. We omit the details. \end{proof}
\section{The domain of the generator of the Poisson semigroup \label{s4}} In this section we generalize to our setting a result of \cite{D} concerning the domain of the generator of the Poisson semigroup. We continue to suppose that the hypotheses of Theorem \ref{t1.3} hold. By Theorem \ref{t2.6}, if $\epsilon \leq \epsilon_0$ is sufficiently small, then the Dirichlet problem (D2) has a unique solution. Consequently, the solution operator $f\to\mathcal{P}(t) f \equiv u(\cdot,t),$ where $u$ solves (D2) with data $f$, satisfies
\begin{equation}\label{eq4.1}\sup_{t>0}\|\mathcal{P}(t)\|_{2\to2} \leq C,\quad
\lim_{t\to0}\|\mathcal{P}(t) f - f\|_2 =0, \end{equation} and \begin{equation}\label{eq4.2}\mathcal{P}(t+s) =\mathcal{P}(t)\mathcal{P}(s), \end{equation} where the last identity uses also $t$-independence of the coefficients. Standard semigroup theory therefore implies that the semigroup $\{\mathcal{P}(t)\}$ has a densely defined infinitesimal generator on $L^2(\mathbb{R}^n)$, which we denote by $\mathcal{A}$. We will show that the domain $D(\mathcal{A})$ of this generator is the Sobolev space $L^2_1 \equiv L^2 \cap\dot{L}^2_1$. More precisely, we have the following \begin{theorem}\label{t4.3}Suppose that $L$ is as above. Then $D(\mathcal{A}) = L^2_1,$ and
$$\|\mathcal{A}f\|_2 \approx \|\nabla_xf\|_2.$$ \end{theorem} We remark that this last theorem can be viewed as an extension of the Kato square root problem (\cite{CMcM},\cite{HMc}, \cite{AHLT},\cite{HLMc} and \cite{AHLMcT}) to the case that the coefficient matrix $A$ is a full $(n+1) \times (n+1)$ matrix. Indeed, the Kato problem corresponds to the case that the coefficient matrix has the special ``block" structure \begin{equation}
\left[\begin{array}{c|c}
& 0\\ B & \vdots\\
& 0\\ \hline 0\cdots0 & 1\end{array}\right]\label{eq4.4}\end{equation}
where $B=B(x)$ is a $n\times n$ matrix. In the latter case
the generator of the Poisson semigroup is $$-\sqrt{-\operatorname{div}_x B \nabla_x},$$
and the conclusion of Theorem \ref{t4.3} is the (now established) Kato conjecture.
We also note that by Theorem \ref{t2.6}, we have the representation \begin{equation}\label{eq4.represent} \mathcal{P}(t) = \mathcal{D}_t\left(-\frac{1}{2}I + K\right)^{-1}.\end{equation}
In order to prove the theorem we shall require the following result from
\cite{AAAHK}.
\begin{theorem}\cite{AAAHK}\label{t4.5}
Suppose that L satisfies the hypotheses of Theorem \ref{t1.3}.
There exists a small constant $\epsilon_0= \epsilon_0(n,\lambda,\Lambda)$ such
that if $\epsilon$ in \eqref{eq1.small} satisfies $\epsilon \leq \epsilon_0$,
then the single layer potential satisfies
\begin{equation}\label{eq4.6} \sup_{t \in \mathbb{R}}\|\nabla S_t \|_{2\to 2} \leq C,
\end{equation}
and $S_0\equiv S_t|_{t=0} : L^2(\mathbb{R}^n) \to \dot{L}^2_1(\mathbb{R}^n)$ is a bijection.
Moreover, there is a unique solution to the Regularity problem
\begin{equation} \tag{R2}\begin{cases} Lu=0\text{ in }\mathbb{R}_{+}^{n+1}\\ u(\cdot,t)\to f\in\dot{L}_{1}^{2}(\mathbb{R}^{n}) \, n.t.\\ \widetilde{N}_{\ast}(\nabla u)\in L^{2}(\mathbb{R}^{n}),\end{cases}\label{R2}\end{equation} which has the representation \begin{equation} \label{eq4.rep} u(\cdot,t) \equiv S_t \left(S_0^{-1} f\right),\end{equation}
and $\partial_t u(\cdot,t)$ converges $n.t.$ and in $L^2(\mathbb{R}^n)$ as $t\to 0$. Finally, \begin{equation}\label{eq4.7}\left(\nabla S_{t}\right)|_{t=\pm s} f \to \mp\frac{1}{2}\cdot\frac{f(x)}{A_{n+1,n+1}(x)}e_{n+1}+\mathcal{T}f \end{equation} weakly in $L^2(\mathbb{R}^n)$, where $\mathcal{T} : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n,\mathbb{C}^{n+1}) $
(see \cite{AAAHK}, Lemma 4.18 for a precise definition of $\mathcal{T}$). \end{theorem}
\begin{proof}[Proof of Theorem \ref{t4.3}]
The deep results underlying Theorem \ref{t4.3} are Theorems \ref{t2.6} and \ref{t4.5}, and we shall deduce the first as a straightforward corollary of the latter two. We observe that if $u$ solves (R2) with data $f \in L^2_1, $ then $$ \lim_{t\to 0} \partial_t u (\cdot,t) = \lim_{t\to 0} \partial_t\,\mathcal{P}(t) f \equiv \mathcal{A} f. $$ Thus, by \eqref{eq4.6}, \eqref{eq4.rep} and the bijectivity of $S_0$,
$$\|\mathcal{A}f\|_2 \leq C \|\nabla_xf\|_2.$$
The proof of the opposite inequality is only a bit harder, and we sketch the details briefly here.
We modify slightly the strategy of Verchota \cite{V}.
By the well known Rellich identity (see, e.g., \cite{K}), and the case $\epsilon = 0$ of
Theorem \ref{t4.5}, for $A_0$ real and symmetric
we have that
\begin{equation}\label{eq4.9}\|\nabla_x u_0(\cdot,t)\|_2 \approx
\|\partial_t u_0 (\cdot,t)\|_2,\end{equation}
uniformly in $t\geq 0$,
when $u_0(\cdot,t) \equiv S^0_t f,$ and $S^0_t$ is the single layer potential
asociated to $L_0 \equiv -\operatorname{div} A_0 \nabla.$ By Theorem \ref{t4.5} and analytic perturbation theory,
\begin{equation}\label{eq4.lip}\|\left(\nabla S_t^0 - \nabla S_t\right) f\|\leq C \|A_0-A\|_\infty \,
\|f\|_2.\end{equation} The latter estimate, combined with \eqref{eq4.9} yields, uniformly in $t\geq 0$,
\begin{equation*}C^{-1}\|\partial_t S_t f\|_2-C\epsilon_0\|f\|_2\leq
\|\nabla_x S_t f\|_2 \leq C\|\partial_t S_t f\|_2 + C\epsilon_0\|f\|_2.\end{equation*} Since the tangential derivatives $\nabla_x S_t f$ do not jump across the boundary, the latter bound, plus its analogue for the lower half space, and \eqref{eq4.7} imply
\begin{equation*}\left\|\frac{1}{2}(A_{n+1,n+1})^{-1}f+\mathcal{T}_{n+1}f \right\|_2\leq
\left\|\frac{1}{2}(A_{n+1,n+1})^{-1}f-\mathcal{T}_{n+1}f \right\|_2 + C\epsilon_0\|f\|_2,\end{equation*} where $\mathcal{T}_{n+1} \equiv \mathcal{T}\cdot e_{n+1}$. Thus , by the accretivity of $A_{n+1,n+1}$ we have
\begin{multline*}\|f\|_2 \leq \left\|\frac{1}{2}(A_{n+1,n+1})^{-1}f+\mathcal{T}_{n+1}f \right\|_2 +
\left\|\frac{1}{2}(A_{n+1,n+1})^{-1}f-\mathcal{T}_{n+1}f \right\|_2\\
\leq C \left\|\frac{1}{2}(A_{n+1,n+1})^{-1}f-\mathcal{T}_{n+1}f \right\|_2 + C\epsilon_0\|f\|_2.
\end{multline*}
For $\epsilon_0$ small enough, we may first hide the small term, and then obtain invertibility
on $L^2$ of
$-\frac{1}{2}(A_{n+1,n+1})^{-1} I +\mathcal{T}_{n+1}$ using \eqref{eq4.lip} and
the method of continuity as in \cite{AAAHK}. Now, given $f \in D(\mathcal{A})$, we set
$$\tilde{u}(\cdot,t) \equiv \mathcal{P}(t) f,\quad u(\cdot,t)\equiv S_t \left(-\frac{1}{2}(A_{n+1,n+1})^{-1}I +\mathcal{T}\right)^{-1} \mathcal{A}f,$$
so that $\partial_t \tilde{u}(\cdot,t),\,\partial_t u(\cdot,t) \to \mathcal{A}f \, n.t.$ and in $L^2$ as $t \to 0$. By uniqueness in (D2), $\partial_t \tilde{u} =\partial_t u$, hence
$\tilde{u} - u$ depends only on $x$, and therefore, since $L(\tilde{u}-u) = 0$, and $\tilde{u}(\cdot,t)
-u(\cdot,t)\in \dot{L}^2_1$, for each fixed $t>0$ (for $\tilde{u}$, this is a
consequence of the representation \eqref{eq4.represent}),
we deduce that $\tilde{u}-u = constant$. Thus, we have that $$\|\nabla_x f\|_2 \leq \sup_{t\geq 0}\|\nabla_x \tilde{u}(\cdot,t)\|_2
=\sup_{t\geq 0}\|\nabla_x u(\cdot,t)\|_2 \leq C\|\mathcal{A}f\|_2,$$ where in the last step we have used \eqref{eq4.6} and the bijectivity of $-\frac{1}{2}(A_{n+1,n+1})^{-1} I +\mathcal{T}_{n+1}$.
\end{proof}
\noindent {\bf Acknowledgements}. The author thanks Zongwei Shen for posing the question.
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\date{ {\small October 11, 2011}} \title{Convergence of the spectral measure of non normal matrices} \abstract{We discuss regularization by noise of the spectrum of large random non-Normal matrices. Under suitable conditions, we show that the regularization of a sequence of matrices that converges in $*$-moments to a regular element $a$, by the addition of a polynomially vanishing Gaussian Ginibre matrix, forces the empirical measure of eigenvalues to converge to the Brown measure of $a$.} \section{Introduction} Consider a sequence $A_N$ of $N\times N$ matrices, of uniformly bounded operator norm, and assume that
$A_N$ converges in $*$-moments toward an element $a$ in a $W^*$ probability space $({\cal A}, \|\cdot\|, *, \varphi)$, that is, for any non-commutative polynomial $P$, $$ \frac1N \mbox{\rm tr} P(A_N,A_N^*)\to_{N\to\infty} \varphi(P(a,a^*))\,.$$ We assume throughout that the tracial state $\varphi$ is faithful; this does not represent a loss of generality. If $A_N$ is a sequence of Hermitian matrices, this is enough in order to conclude that the empirical measure of eigenvalues of $A_N$, that is the measure $$ L_N^A:=\frac1N \sum_{i=1}^N \delta_{\lambda_i(A_N)},$$ where $\lambda_i(A_N), i=1\ldots N$ are the eigenvalues of $A_N$, converges weakly to a
limiting measure $\mu_a$,
the spectral measure of $a$,
supported on a compact subset of
$\mathbb{R}$. (See \cite[Corollary 5.2.16, Lemma 5.2.19]{AGZ} for this standard
result and further background.) Significantly, in the Hermitian case,
this convergence is stable under
small bounded perturbations: with $B_N=A_N+E_N$ and $\|E_N\|<\epsilon$,
any subsequential limit of $L_N^B$ will belong to
$B_L(\mu_a,\delta(\epsilon))$, with $\delta(\epsilon)\to_{\epsilon\to 0} 0$
and $B_L(\nu_a,r)$ is the ball (in say, the L\'{e}vy metric)
centered at $\nu_a$ and of radius $r$.
Both these statements fail when $A_n$ is not self adjoint. For a standard
example (described in \cite{Sniady}),
consider the nilpotent matrix $$ T_N=\left(\begin{array}{lllll}
0&1&0&\ldots&0\\
0&0&1&0&\ldots\\ \ldots&\ldots&\ldots&\ldots&\ldots\\ 0&\ldots&\ldots&0&1\\ 0&\ldots&\ldots&\ldots&0 \end{array} \right)\,.$$ Obviously, $L_N^T=\delta_0$, while a simple computation reveals that $T_N$ converges in $*$-moments to a Unitary Haar element of ${\cal A}$,
that is \begin{equation}
\label{eq-tn}
\frac1N \mbox{\rm tr}(T_N^{\alpha_1} (T_N^*)^{\beta_1} \ldots T_N^{\alpha_k} (T_N^*)^{\beta_k}) \to_{N\to\infty} \left\{\begin{array}{ll}
1, & \mbox{\rm if} \sum_{i=1}^k\alpha_i=\sum_{i=1}^k\beta_i,\\
0, & \mbox{\rm otherwise}. \end{array} \right. \end{equation} Further, adding to $T_N$ the matrix whose entries are all $0$ except for the bottom left, which is taken as $\epsilon$, changes the empirical measure of eigenvalues drastically - as we will see below, as $N$ increases, the empirical measure converges to the uniform measure on the unit circle in the complex plane.
Our goal in this note is to explore this phenomenun in the context of small random perturbations of matrices. We recall some notions. For $a\in {\cal A}$, the {\it Brown measure} $\nu_a$ on $\mathbb{C}$ is the measure satisfying
$$ \log \mbox{\rm det}(z-a)=\int \log |z-z'| d\nu_a(z'),\quad {\bf z}\in \mathbb{C},$$ where $\mbox{\rm det}$ is the Fuglede-Kadison determinant; we refer to \cite{brown,haageruplarsen} for definitions. We have in particular that $$ \log \mbox{\rm det}(z-a)=\int \log x d\nu_{a}^z(x)\,\quad z\in \mathbb{C}\,,$$
where $\nu_a^z$ denotes the spectral measure of the operator $|z-a|$. In the sense of distributions, we have $$ \nu_a=\frac{1}{2\pi} \Delta \log \mbox{\rm det}(z-a)\,.$$ That is, for smooth compactly supported function $\psi$ on $\mathbb{C}$, \begin{eqnarray*}
\int \psi(z) d\nu_a(z)&=&
\frac{1}{2\pi}\int dz \ \Delta \psi(z)
\int \log|z-z'| d\nu_a(z')\\
&=&
\frac{1}{2\pi}\int dz\ \Delta \psi(z)
\int \log x d\nu_a^z (x)\,. \end{eqnarray*} A crucial assumption in our analysis is the following. \begin{definition}[Regular elements]
An element $a\in {\cal A}$ is {\em regular} if
\begin{equation}
\label{eq-3}
\lim_{\epsilon\to 0}
\int_{\mathbb{C}}dz \Delta \psi(z) \int_0^\epsilon \log x d\nu_a^z(x) =0\,, \end{equation} for all smooth functions $\psi$ on $\mathbb{C}$ with compact support. \end{definition} Note that regularity is a property of $a$, not merely of its Brown measure $\nu_a$.
We next introduce the class of Gaussian perturbations we consider. \begin{definition}[Polynomially vanishing Gaussian matrices] A sequence of $N$-by-$N$ random Gaussian matrices is called {\em polynomially vanishing} if its entries $(G_N(i,j))$ are independent centered complex Gaussian variables, and there exist $\kappa>0$, $\kappa' \geq 1+\kappa$ so that
$$ N^{-\kappa'}\leq E|G_{ij}|^2 \leq N^{-1-\kappa}\,.$$
\end{definition} \begin{remark} \label{rem-1} As will be clear below, see the beginning of the proof of Lemma \ref{lem-5}, the Gaussian assumption only intervenes in obtaining a uniform lower bound on singular values of certain random matrices. As pointed out to us by R. Vershynin, this uniform estimate extends to other situations, most notably to the polynomial rescale of matrices whose entries are i.i.d. and possess a bounded density. We do not discuss such extensions here. \end{remark}
Our first result is a stability, with respect to polynomially vanishing Gaussian perturbations, of the convergence of spectral measures for non-normal matrices. Throughout, we denote by $\|M\|_{op}$ the operator norm of a matrix $M$. \begin{theorem}\label{theo1}
Assume that the uniformly bounded (in the operator norm)
sequence of $N$-by-$N$ matrices
$A_N$ converges in $*$-moments to a regular element $a$. Assume further that $L_N^A$ converges weakly to the Brown measure $\nu_a$.
Let $G_N$ be a sequence of polynomially vanishing Gaussian matrices, and set $B_N=A_N+G_N$.
Then, $L_N^{B}\to \nu_a$ weakly, in probability.
\end{theorem} Theorem \ref{theo1} puts rather stringent assumptions on the sequence $A_N$. In particular, its assumptions are not satisfied by the sequence of nilpotent matrices $T_N$ in \eqref{eq-tn}. Our second result corrects this defficiency, by showing that small Gaussian perturbations ``regularize'' matrices that are close to
matrices satisfying the assumptions of Theorem \ref{theo1}. \begin{theorem}\label{theo2} Let $A_N$, $E_N$ be a sequence of bounded (for the operator norm) $N$-by-$N$ matrices, so that
$A_N$ converges in $*$-moments to a regular element $a$. Assume that $\|E_N\|_{op}$ converges to zero polynomially fast in $N$, and that $L_N^{A+E}\to \nu_a$ weakly. Let $G_N$ be a sequence of polynomially vanishing Gaussian matrices, and set $B_N=A_N+G_N$. Then, $L_N^B\to \nu_a$ weakly, in probability.
\end{theorem} Theorem \ref{theo2} should be compared to earlier results of Sniady \cite{Sniady}, who used stochastic calculus to show that a perturbation by an asymptotically vanishing Ginibre Gaussian matrix regularizes arbitrary matrices. Compared with his results, we allow for more general Gaussian perturbations (both structurally and in terms of the variance) and also show that the Gaussian regularization can decay as fast as wished in the polynomial scale. On the other hand, we do impose a regularity property on the limit $a$ as well as on the sequence of matrices for which we assume that adding a polynomially small matrix is enough to obtain convergence to the Brown measure.
A corollary of our general results is the following. \begin{corollary}
\label{cor-tn}
Let $G_N$ be a sequence of polynomially vanishing Gaussian matrices and let
$T_N$ be as in \eqref{eq-tn}. Then $L_N^{T+G}$ converges weakly, in
probability,
toward the uniform measure on the unit circle in $\mathbb{C}$. \end{corollary}
\noindent In Figure~\ref{fig2}, we give a simulation of the setup in Corollary~\ref{cor-tn} for various $N$.
\begin{figure}
\caption{The eigenvalues of $T_N+N^{-3-1/2}G_N$, where $G_N$ is iid complex Gaussian with mean 0, variance 1 entries.
}
\label{fig2}
\end{figure}
We will now define class of matrices $T_{b,N}$ for which, if $b$ is chosen correctly, adding a polynomially vanishing Gaussian matrix $G_N$ is not sufficient to regularize $T_{b,N}+G_N$. Let $b$ be a positive integer, and define $T_{b,N}$ to be an $N$ by $N$ block diagonal matrix which each $b+1$ by $b+1$ block on the diagonal equal $T_{b+1}$ (as defined in \eqref{eq-tn}. If $b+1$ does not divide $N$ evenly, a block of zeros is inserted at bottom of the diagonal. Thus, every entry of $T_{b,N}$ is zero except for entries on the superdiagonal (the superdiagonal is the list of entries with coordinates $(i,i+1)$ for $1\le i\le N-1$), and the superdiagonal of $T_{b,N}$ is equal to $$(\underbrace{1,1,\dots,1}_b,0,\underbrace{1,1,\dots,1}_b,0,\dots,\underbrace{1,1,\dots,1}_b,\underbrace{0,0,\dots,0}_{\le b}).$$
Recall that the spectral radius of a matrix is the maximum absolute value of the eigenvalues. Also, we will use $\|A \|=\tr(A^*A)^{1/2}$ to denote the Hilbert-Schmidt norm.
\begin{proposition}\label{prop-lb}
Let $b= b(N)$ be a sequence of positive integers such that $b(N)\ge log N$ for all $N$, and let $T_{b,N}$ be as defined above. Let $R_N$ be an $N$ by $N$ matrix satisfying $\| R_N\| \le g(N)$, where for all $N$ we assume that $g(N) < \frac1{3 b \sqrt N}$. Then $$ \rho( T_{b,N} + R_N) \le (Ng(N))^{1/b} +o(1),$$ where $\rho(M)$ denotes the spectral radius of a matrix $M$, and $o(1)$ denotes a small quantity tending to zero as $N\to \infty$. \end{proposition}
Note that $T_{b,N}$ converges in $*$-moments to a Unitary Haar element of $\mathcal A$ (by a computation similar to \eqref{eq-tn}) if $b(N)/N$ goes to zero, which is a regular element. The Brown measure of the Unitary Haar element is uniform measure on the unit circle; thus, in the case where $(Ng(N))^{1/b}<1$, Proposition~\ref{prop-lb} shows that $T_{b,N}+R_N$ does not converge to the Brown measure for $T_{b,N}$.
\begin{corollary}\label{cor-lb} Let $R_N$ be an iid Gaussian matrix where each entry has mean zero and variance one.
Set $b=b(N)\ge \log N$ be a sequence of integers, and let $\gamma >5/2$ be a constant. Then, with probability tending to 1 as $N\to \infty$, we have $$\rho(T_{b,N} + \exp(-\gamma b) R_N) \le \exp\left(-\gamma + \frac{2\log N}{b}\right) + o(1),$$ where $\rho$ denotes the spectral radius and where $o(1)$ denotes a small quantity tending to zero as $N\to\infty$. Note in particular that the bound on the spectral radius is strictly less than $\exp(-1/2)< 1$ in the limit as $N\to\infty$, due to the assumptions on $\gamma$ and $b$. \end{corollary}
Corollary~\ref{cor-lb} follows from Proposition~\ref{prop-lb} by noting that, with probability tending to 1, all entries in $R_N$ are at most $C\log N$ in absolute value for some constant $C$, and then checking that the hypotheses of Proposition~\ref{prop-lb} are satisfied for $g(N)=\exp(-\gamma b) CN(\log N)^{1/4}$. There are two instances of Corollary~\ref{cor-lb} that are particularly interesting: when $b=N-1$, we see that a exponentially decaying Gaussian perturbation does not regularize $T_N=T_{N-1,N}$, and when $b=\log(N)$, we see that polynomially decaying Gaussian perturbation does not regularize $T_{\log N,N}$ (see Figure~\ref{fig3}).
\begin{figure}
\caption{The eigenvaules of $T_{\log N,N}+N^{-3-1/2}G_N$, where
$G_N$ is iid complex Gaussian with mean 0, variance 1 entries. The spectral radius is roughly $0.07$, and the bound from Corollary~\ref{cor-lb} is $\exp(-1)\approx 0.37$.}
\label{fig3}
\end{figure}
We will prove Proposition~\ref{prop-lb} in Section~\ref{sec-proppf}. The proof of our main results (Theorems~\ref{theo1} and \ref{theo2}) borrows from the methods of \cite{GKZ}. We introduce notation. For any $N$-by-$N$ matrix $C_N$, let
$$\widetilde C_N=\left( \begin{array}{cc} 0& C_N\cr C_N^*&0\cr\end{array} \right).$$
We denote by $G_{C}$ the Cauchy-Stieltjes transform of the spectral measure of the matrix $\widetilde C_N$, that is $$ G_C(z)=\frac1{2N}\tr(z-\widetilde C_N)^{-1}\,, \quad z\in \mathbb{C}_+\,.$$ The following estimate is immediate from the definition and the resolvent identity: \begin{equation}
\label{eq-comp}
|G_C(z)-G_D(z)|\leq \frac{ \|C-D\|_{op}}
{|\Im z|^2}\,. \end{equation}
\section{Proof of Theorem \ref{theo1}} We keep throughout the notation and assumptions of the theorem. The following is a crucial simple observation. \begin{proposition}\label{prop1} For all complex number $\xi$,
and all $z$ so that $\Im z\ge N^{-\delta}$ with $\delta<\kappa/4$,
$$\mbox{\bf E}|\Im G_{B_N+\xi}(z)|\le \mbox{\bf E}|\Im G_{A_N+\xi}(z)|+1$$ \end{proposition} \proof Noting that \begin{equation} \label{eq-doublestar}
\mbox{\bf E}\|B_N-A_N\|_{op}^k=\mbox{\bf E}\|G_N\|_{op}^k\leq C_k N^{-\kappa k/2}, \end{equation} the conclusion follows from \eqref{eq-comp} and H\"{o}lder's inequality. \qed
We continue with the proof of Theorem \ref{theo1}. Let $\nu_{A_N}^z$ denote the empirical measure of the eigenvalues of the matrix $\widetilde{A_N-z}$. We have that, for smooth test functions $\psi$,
$$ \int dz \Delta \psi(z) \int \log |x| d\nu_{A_N}^z(x)= \frac{1}{2\pi}\int \psi(z) dL_N^A(z)\,.$$ In particular, the convergence of $L_N^A$ toward $\nu_a$ implies that
$$ \mbox{\bf E}\int dz \Delta \psi(z) \int \log |x| d\nu_{A_N}^z(x) \to \int \psi(z)d\nu_a(z)= \int dz \Delta \psi(z) \int \log x d\nu_a^z(x)\,. $$ On the other hand, since $x\mapsto \log x$ is bounded continuous on compact subsets of $(0,\infty)$, it also holds that for any continuous bounded function $\zeta: \mathbb{R}_+\mapsto \mathbb{R}$ compactly supported in $(0,\infty)$, $$ \mbox{\bf E}\int dz \Delta \psi(z) \int \zeta(x)\log x d\nu_{A_N}^z(x) \to \int dz \Delta \psi(z)\int \zeta(x) \log x d\nu_a^z(x)\,.$$ Together with the fact that $a$ is regular and that $A_N$ is uniformly bounded, one concludes therefore that
$$\lim_{{\varepsilon}\downarrow 0}\lim_{N\rightarrow\infty}
\mbox{\bf E} \int \int_0^{\varepsilon} \log |x| d\nu_{A_N}^z(x)dz=0\,.$$
Our next goal is to show that the same applies to $B_N$. In the following, we let $\nu_{B_N}^z$ denote the empirical measure of the eigenvalues of $\widetilde{B_N-z}$. \begin{lemma}
\label{lem-5} $$\lim_{{\varepsilon}\downarrow 0}\lim_{N\rightarrow\infty}
\int \mbox{\bf E}[ \int_0^{\varepsilon} \log |x|^{-1} d\nu_{B_N}^z(x) ]dz=0$$ \end{lemma} Because
$\mbox{\bf E}\|B_N-A_N\|_{op}^k\to 0$ for any $k>0$, we have for any fixed smooth $w$ compactly supported in $(0,\infty)$ that
$$ \mbox{\bf E}|\int dz \Delta \psi(z) \int w(x)\log x d\nu_{A_N}^z(x)
- \int dz \Delta \psi(z) \int w(x)\log x d\nu_{B_N}^z(x)| \to_{N\to\infty} 0\,,$$ Theorem \ref{theo1} follows at once from Lemma \ref{lem-5}.
\noindent {\bf Proof of lemma \ref{lem-5}:} Note first that by \cite[Theorem 3.3]{SST} (or its generalization in \cite[Proposition 16]{GKZ} to the complex case), there exists a constant $C$ so that for any $z$, the smallest singular value $\sigma^z_N$ of $B_N+zI$ satisfies $$P(\sigma^z_N\le x)\le C\left( N^{\frac{1}{2}+\kappa'} x\right)^\beta$$ with $\beta=1$ or $2$ according whether we are in the real or the complex case. Therefore, for any $\zeta>0$, uniformly in $z$ \begin{eqnarray*}
\mbox{\bf E}[ \int_0^{N^{-\zeta}} \log |x|^{-1} d\nu_{B_N}^z(x) ] &\le & \mbox{\bf E}[\log(\sigma^z_N)^{-1} 1_{\sigma^z_N\le N^{-\zeta}}]\\ &=& C\left( N^{\frac{1}{2}+\kappa'-\zeta} \right)^\beta \log (N^\zeta) +\int_0^{N^{-\zeta}} \frac{1}{x} C\left( N^{\frac{1}{2}+\kappa'} x\right)^\beta dx\\ \end{eqnarray*} goes to zero as $N$ goes to infinity as soon as $\zeta>\frac{1}{2}+\kappa'$. We fix hereafter such a $\zeta$ and we may and shall restrict the integration from $N^{-\zeta}$ to ${\varepsilon}$. To compare the integral for the spectral measure of $ A_N$ and $B_N$, observe that for all probability measure $P$, with $P_\gamma$ the Cauchy law with parameter $\gamma$ \begin{equation}
\label{ez1}
P([a,b])\le P*P_\gamma([a-\eta,b+\eta])+P_\gamma([-\eta,\eta]^c) \le P*P_\gamma([a-\eta,b+\eta]) +\frac{\gamma}{\eta} \end{equation}
whereas for $b-a>\eta$ \begin{equation}
\label{ez2}
P([a,b])\ge P*P_\gamma([a+\eta,b-\eta])-\frac{\gamma}{\eta}
\,. \end{equation} Recall that \begin{equation}
\label{ez3}
P*P_\gamma([a,b])= \int_a^b |\Im G(x+i\gamma)| dx. \end{equation} Set $\gamma=N^{-\kappa/5}$, $\kappa''=\kappa/2$ and $\eta=N^{-\kappa''/5}$. We have, whenever $b-a\geq 4\eta$,
\begin{eqnarray*} \mbox{\bf E}\nu^z_{B_N}([a,b])&\le& \int_{a-\eta}^{b+\eta}
\mbox{\bf E}|\Im G_{B_n+z}(x+i\gamma)| dx + N^{-(\kappa-\kappa'')/5}\\ &\le & (b-a+2N^{-\kappa''/5} ) + \nu^z_{A_N}*P_{N^{-\kappa/5}}( [a-N^{-\kappa/10}, b+N^{-\kappa/10}])+N^{-\kappa/10}\\ &\le& (b-a+2N^{-\kappa/10}) + \nu^z_{A_N}( [(a-2N^{-\kappa/10})_+, (b+2N^{-\kappa/10})])+2N^{-\kappa/10}\,, \end{eqnarray*} where the first inequality is due to \eqref{ez1} and \eqref{ez3}, the second is due to Proposition \ref{prop1}, and the last uses \eqref{ez2} and \eqref{ez3}. Therefore, if $b-a=CN^{-\kappa/10}$ for some fixed $C$ larger than 4, we deduce that there exists a finite constant $C'$ which only depends on $C$ so that $$\mbox{\bf E}\nu^z_{B_N}([a,b])\le C'(b-a)+ \nu^z_{A_N}( [(a-2N^{-\kappa/10})_+, (b+2N^{-\kappa/10})])\,.$$ As a consequence, as we may assume without loss of generality that $\kappa'>\kappa/10$, \begin{eqnarray*}
&&\mbox{\bf E}[\int_{N^{-\zeta}}^{\varepsilon}\log |x|^{-1} d\nu_{B_N}^z(x)]\\ &\le&\sum_{k=0}^{[N^{\kappa/10}{\varepsilon}]} \log(N^{-\zeta} +2Ck N^{-\kappa/10})^{-1} \mbox{\bf E}[\nu_{B_N}^z]([N^{-\zeta} +2Ck N^{-\kappa/10}, N^{-\zeta}+2C(k+1) N^{-\kappa/10}])\,. \end{eqnarray*} We need to pay special attention to the first term that we bound by noticing that \begin{eqnarray*} &&\log(N^{-\zeta} )^{-1} \mbox{\bf E}[\nu_{B_N}^z([N^{-\zeta}, N^{-\zeta}+2C N^{-\kappa/10}])]\\ &\le& \frac{10\zeta}{\kappa} \log(N^{-\kappa/10} )^{-1} \mbox{\bf E}[\nu_{B_N}^z([0, 2(C+1) N^{-\kappa/10}])]\\ &\le& \frac{10\zeta}{\kappa} \log(N^{-\kappa/10} )^{-1} (2C' N^{-\kappa/10} + \nu_{A_N}^z([0,(C+2) N^{-\kappa/10}]))\\ &\le& \frac{20 C'\zeta}{\kappa} \log(N^{-\kappa/10} )^{-1} N^{-\kappa/10}
+C''\int_0^{2(C+2) N^{-\kappa/10}}\log |x|^{-1} d \nu_{A_N}^z(x)\\ \end{eqnarray*} For the other terms, we have \begin{eqnarray*} \sum_{k=1}^{[N^{\kappa/10}{\varepsilon}]} &&\log(N^{-\zeta} +2Ck N^{-\kappa/10})^{-1} \mbox{\bf E}[\nu_{B_N}^z]([N^{-\zeta} +2Ck N^{-\kappa/10}, N^{-\zeta}+2C(k+1) N^{-\kappa/10}])\\ &\le& 2C' \sum_{k=1}^{[N^{\kappa/10}{\varepsilon}]} \log(Ck N^{-\kappa/10})^{-1} CN^{-\kappa/10}\\ &&+\sum_{k=1}^{[N^{\kappa/10}{\varepsilon}]} \log(Ck N^{-\kappa/10})^{-1} \nu_{A_N}^z([2C(k-1) N^{-\kappa/10}, 2C(k+2) N^{-\kappa/10}])\,. \end{eqnarray*} Finally, we can sum up all these inequalities to find that there exists a finite constant $C'''$ so that
$$\mbox{\bf E}[\int_{N^{-\zeta}}^{\varepsilon}\log |x|^{-1} d\nu_{B_N}^z(x)]
\le C'''\int_{0}^{\varepsilon}\log |x|^{-1} d\nu_{A_N}^z(x)
+C''' \int_0^{\varepsilon} \log |x|^{-1} dx$$ and therefore goes to zero when $n$ and then ${\varepsilon}$ goes to zero. This proves the claim. \qed \section{Proof of Theorem \ref{theo2}.} From the assumptions, it is clear that $(A_N+E_N)$ converges in $*$-moments to the regular element $a$. By Theorem \ref{theo1}, it follows that $L_N^{A+E+G}$
converges (weakly, in probability) towards $\nu_a$.
We can now remove $E_N$. Indeed, by \eqref{eq-comp} and \eqref{eq-doublestar}, we have for any $\chi<\kappa'/2$
and all $\xi\in\mathbb C$
$$|G^N_{A+G +\xi}(z)-G^N_{A+G+E+\xi}(z)|\le \frac{N^{-\chi}}{\Im z^2} $$ and therefore for $\Im z\ge N^{-\chi/2}$,
$$|\Im G^N_{A+G +\xi}(z)|\le |\Im G^N_{A+G+E+\xi}(z)| +1.$$ Again by \cite[Theorem 3.3]{SST} (or its generalization in \cite[Proposition 16]{GKZ}) to the complex case), for any $z$, the smallest singular value $\sigma^z_N$ of $A_N+G_N+z$ satisfies $$P(\sigma^z_N\le x)\le C\left( N^{\frac{1}{2}+\kappa'} x\right)^\beta$$ with $\beta=1$ or $2$ according whether we are in the real or the complex case. We can now rerun the proof of Theorem \ref{theo1}, replacing $A_N$ by $A_N'=A_N+E_N+G_N$ and $B_N$ by $A_N'-E_N$. \qed
\section{Proof of Corollary \ref{cor-tn}} We apply Theorem \ref{theo2} with $A_N=T_N$, $E_N$ the $N$-by-$N$ matrix with $$E_N(i,j)=\{\begin{array}{ll} \delta_N=N^{-(1/2+\kappa')},& i=1,j=N\\ 0,& \mbox{\rm otherwise}\,, \end{array} $$ where $\kappa'>\kappa$. We check the assumptions of Theorem \ref{theo2}. We take $a$ to be a Unitary Haar element in ${\cal A}$, and recall that its Brown measure
$\nu_a$ is the uniform measure on $\{z\in\mathbb{C}: |z|=1\}$. We now check that $a$ is regular. Indeed, $\int x^k d\nu_a^z(x)=0$ if $k$ is odd by symmetry while for $k$ even, $$\int x^{k} d\nu_a^z(x)=\varphi([(z-a)(z-a)^*]^{k/2})=
\sum_{j=1}^{k/2} (|z|^2+1)^{k-j} \left(\begin{array}{l}
k\\2j \end{array}\right) \left(\begin{array}{l}
2j\\j \end{array}\right)\,,
$$
and one therefore verifies that for $k$ even,
$$\int x^kd\nu_a^z(x)=\frac1{2\pi}\int (|z|^2+1+2|z|\cos \theta)^{k/2} d\theta\,.$$ It follows that $$\int_0^{\varepsilon} \log x d\nu_a^z(x)=\frac1{4\pi}\int_0^{2\pi}
\log(|z|^2+1+2|z|\cos \theta){\bf 1}_{\{|z|^2+1+2|z|\cos\theta<{\varepsilon}\}}d\theta \to_{\epsilon\to 0} 0\,,$$ proving the required regularity.
Further, we claim that $L_N^{A+E}$ converges to $\nu_a$. Indeed the eigenvalues $\lambda$ of $A_N+E_N$ are such that there exists a non-vanishing vector $u$ so that $$u_N\delta_N=\lambda u_1,u_{i-1}=\lambda u_i\,,$$ that is $$\lambda^N=\delta_N.$$ In particular, all the $N$-roots of $\delta_N$ are (distinct) eigenvalues, that is the eigenvalues $\lambda_j^N$ of $A_N$ are
$$\lambda_j^N=|\delta_N|^{1/N} e^{2i\pi j/N},\quad 1\le j\le N\,.$$ Therefore,
for any bounded continuous $g$ function on $\mathbb{C}$, $$ \lim_{N\rightarrow\infty} \frac1N \sum_{i=1}^N g(\lambda_j^N)=\frac1{2\pi} \int g(\theta) d\theta\,,$$ as claimed. \qed
\section{Proof of Proposition~\ref{prop-lb}}
\label{sec-proppf}
In this section we will prove the following proposition:
\begin{proposition}\label{prop-lb2}
Let $b= b(N)$ be a sequence of positive integers, and let $T_{b,N}$ be as in Proposition~\ref{prop-lb}. Let $R_N$ be an $N$ by $N$ matrix satisfying $\| R_N\| \le g(N)$, where for all $N$ we assume that $g(N) < \frac1{3 b \sqrt N}$. Then $$ \rho( T_{b,N} + R_N) \le \left( O\left( \sqrt{Nb} \left(2 N^{1/4} g^{1/2}\right)^b\right)\right)^{1/(b+1)} + \left( b^2 Ng\right)^{1/(b+1)}\,.$$
\end{proposition}
Proposition~\ref{prop-lb} follows from Proposition~\ref{prop-lb2} by adding the assumption that $b(N) \ge log(N)$ and then simplifying the upper bound on the spectral radius.
\noindent {\bf Proof of Proposition~\ref{prop-lb2}:}
To bound the spectral radius, we will use the fact that $\rho( T_{b,N} + R_N) \le \left\| (T_{b,N} + R_N)^k \right\|^{1/k}$ for all integers $k \ge 1$. Our general plan will be to bound $\left\| (T_{b,N} + R_N)^k \right\|$ and then take a $k$-th root of the bound. We will take $k=b+1$, which allows us to take advantage of the fact that $T_{b,N}$ is $(b+1)$-step nilpotent. In particular, we make use of the fact that for any positive integer $a$, \begin{equation}
\label{eqn-tbna}
\| T_{b,N}^a \| = \begin{cases}
(b-a+1)^{1/2} \floor{\frac{N}{b+1}}^{1/2} & \mbox{ if } 1 \le a \le b\\
0 & \mbox{ if } b+1 \le a.
\end{cases} \end{equation}
We may write \begin{align} \norm{ (T_{b,N} + R_N)^{b+1} } &\le \sum_{\lambda \in \{0,1\}^{b+1}} \norm{ \prod_{i=1}^{b+1} T_{b,N}^{\lambda_i} R_N^{1-\lambda_i} }\nonumber\\ &= \sum_{\ell =0}^{b+1} \mathop{\sum_{\lambda \in \{0,1\}^{b+1}}}_{\lambda \mathrm{\ has\ } \ell \mathrm{\ ones}} \norm{ \prod_{i=1}^{b+1} T_{b,N}^{\lambda_i} R_N^{1-\lambda_i} }\nonumber \end{align} When $\ell$ is large, we will make use of the following lemma.
\begin{lemma}
\label{lem-secsum1} If $\lambda\in \{0,1\}^{k}$ has $\ell$ ones and $\ell \ge (k+1)/2$, then $$\norm{ \prod_{i=1}^{k} T_{b,N}^{\lambda_i} R_N^{1-\lambda_i} } \le \norm{ T_{b,N}^{\floor{\frac{\ell}{k-\ell+1}}}}^{k-\ell+1} \norm{R_N}^{k-\ell}. $$ \end{lemma}
We will prove Lemma~\ref{lem-secsum1} in Section~\ref{lempfs}.
Using Lemma~\ref{lem-secsum1} with $k=b+1$ along with the fact that $\norm{AB}\le \norm A \norm B$, we have \begin{align} \norm{ (T_{b,N} + R_N)^{b+1} } &\le \sum_{\ell =0}^{\floor{\frac{b+2}{2}}} \binom{b+1}{\ell} \norm{T_{b,N}}^\ell \norm{R_n}^{b-\ell+1} \nonumber\\ &\qquad
+ \sum_{\ell=\ceiling{\frac{b+2}{2}}}^{b+1} \binom{b+1}{\ell} \norm{ T_{b,N}^{\floor{\frac{\ell}{b-\ell+2}}}}^{b-\ell+2} \norm{R_N}^{b-\ell+1}.\nonumber\\ &\le \sum_{\ell =0}^{\floor{\frac{b+2}{2}}} \binom{b+1}{\ell} \norm{T_{b,N}}^\ell g^{b-\ell+1} \label{firstsum}\\ &\qquad
+ \sum_{\ell=\ceiling{\frac{b+2}{2}}}^{b+1} \binom{b+1}{\ell} \norm{ T_{b,N}^{\floor{\frac{\ell}{b-\ell+2}}}}^{b-\ell+2} g^{b-\ell+1},\label{secondsum} \end{align}
where the second inequality comes from the assumption $\|R_N\|\le g = g(N)$.
We will bound \eqref{firstsum} and \eqref{secondsum} separately. To bound \eqref{firstsum} note that \begin{align}
\sum_{\ell =0}^{\floor{\frac{b+2}{2}}} \binom{b+1}{\ell} \norm{T_{b,N}}^\ell g^{b-\ell+1} &\le \sum_{\ell =0}^{\floor{\frac{b+2}{2}}} \binom{b+1}{\ell} \left( (b+1)\floor{\frac{N}{b+1}}\right)^{\ell/2} g^{b-\ell+1} \nonumber\\ &\le \frac{b+4}{2} \binom{b+1}{\floor{(b+1)/2}} N^{(b+2)/4} g^{b/2} \nonumber\\ &=O\left(\sqrt{Nb}(2N^{1/4}g^{1/2})^b\right).\label{fsbd} \end{align}
Next, we turn to bounding \eqref{secondsum}. We will use the following lemma to show that the largest term in the sum \eqref{secondsum} comes from the $\ell=b$ term. Note that when $\ell=b+1$, the summand in \eqref{secondsum} is equal to zero by \eqref{eqn-tbna}.
\begin{lemma};
\label{lem-secsum}
If $\norm{ T_{b,N}^{\floor{\frac{\ell+1}{b-\ell+1}}}} > 0$ and $\ell \le b-1$ and $$ g \le \frac{2}{e^{3/2} N^{1/2}b}, $$ then $$ \binom{b+1}{\ell} \norm{ T_{b,N}^{\floor{\frac{\ell}{b-\ell+2}}}}^{b-\ell+2} g^{b-\ell+1} \le \binom{b+1}{\ell+1} \norm{ T_{b,N}^{\floor{\frac{\ell+1}{b-\ell+1}}}}^{b-\ell+1} g^{b-\ell}. $$ \end{lemma}
We will prove Lemma~\ref{lem-secsum} in Section~\ref{lempfs}.
Using Lemma~\ref{lem-secsum} we have \begin{align}
\sum_{\ell=\ceiling{\frac{b+2}{2}}}^{b+1} \binom{b+1}{\ell} \norm{ T_{b,N}^{\floor{\frac{\ell}{b-\ell+2}}}}^{b-\ell+2} g^{b-\ell+1} &\le \frac b2 (b+1) \norm{ T_{b,N}^{\floor{\frac{b}{2}}}}^{2} g^{1} \nonumber\\ &\le \frac b2 (b+1) (b-\floor{b/2} +1)\frac{N}{b+1} g \nonumber\\ &\le b^2 Ng. \label{ssbd} \end{align}
Combining \eqref{fsbd} and \eqref{ssbd} with \eqref{firstsum} and \eqref{secondsum}, we may use the fact that $(x+y)^{1/(b+1)} \le x^{1/(b+1)} +y^{1/(b+1)}$ for positive $x,y$ to complete the proof of Proposition~\ref{prop-lb2}. It remains to prove Lemma~\ref{lem-secsum1} and Lemma~\ref{lem-secsum}, which we do in Section~\ref{lempfs} below. \qed
\subsection{Proofs of Lemma~\ref{lem-secsum1} and Lemma~\ref{lem-secsum}}\label{lempfs}
\noindent {\bf Proof of Lemma~\ref{lem-secsum1}:} Using \eqref{eqn-tbna}, it is easy to show that \begin{equation}
\label{relation1}
\norm{T_{b,N}^a}\norm{T_{b,N}^c} < \norm{T_{b,N}^{a-1}}\norm{T_{b,N}^{c+1}} \mbox{ for integers } 3\le c+2 \le a \le b. \end{equation} It is also clear from \eqref{eqn-tbna} that \begin{equation}
\label{relation2} \norm{T_{b,N}^a} \le \norm{T_{b,N}^{a-1}} \mbox{ for all positive integers $a$}. \end{equation}
Let $\lambda \in \{0,1\}^{k}$ have $\ell$ ones. Then, using the assumption that $\ell \ge k-\ell +1$, we may write $$\prod_{i=1}^{k} T_{b,N}^{\lambda_i} R_N^{1-\lambda_i} = T_{b,N}^{a_1} R_N^{b_1} T_{b,N}^{a_2} R_N^{b_2} \cdotsT_{b,N}^{a_{k-\ell}} R_N^{b_{k-\ell}}T_{b,N}^{a_{k-\ell+1}},$$ where $a_i \ge 1$ for all $i$ and $b_i \ge 0$ for all $i$. Thus $$\norm{\prod_{i=1}^{k} T_{b,N}^{\lambda_i} R_N^{1-\lambda_i}} \le \norm{R_N}^{k-\ell} \prod_{i=1}^{k-\ell+1} \norm{T_{b,N}^{a_i}}.$$ Applying \eqref{relation1} repeatedly, we may assume that two of the $a_i$ differ by more than 1, all without changing the fact that $\sum_{i=1}^{k-\ell+1}a_i = \ell$. Thus, some of the $a_i$ are equal to $\floor{\frac{\ell}{k-\ell+1}}$ and some are equal to $\ceiling{\frac{\ell}{k-\ell+1}}$. Finally, applying \eqref{relation2}, we have that $$\prod_{i=1}^{k-\ell+1} \norm{T_{b,N}^{a_i}} \le \norm{T_{b,N}^{\floor{\frac{\ell}{k-\ell+1}}}}^{k-\ell+1}.$$ \qed
\noindent {\bf Proof of Lemma~\ref{lem-secsum}:} Using \eqref{eqn-tbna} and rearranging, it is sufficient to show that $$\frac{\ell+1}{b-\ell+1} \left(b -\floor{\frac{\ell}{b-\ell+2}} +1\right) ^{1/2} \floor{\frac{N}{b+1}}^{1/2} g \le \left( \frac{b-\floor{\frac{\ell+1}{b-\ell+1}}+1} {b-\floor{\frac{\ell}{b-\ell+2}}+1} \right)^{\frac{b-\ell+1}{2}} $$ Using a variety of manipulations, it is possible to show that \begin{eqnarray*} \left( \frac{b-\floor{\frac{\ell+1}{b-\ell+1}}+1} {b-\floor{\frac{\ell}{b-\ell+2}}+1} \right)^{\frac{b-\ell+1}{2}} &\ge& \exp\left( -\frac{(b-\ell+2)(b-\ell+1)} {(b+2)(b-\ell+2)-\ell} - \frac{b+2} {(b+2)(b-\ell+2)-\ell} \right)\\& \ge&\exp(-3/2). \end{eqnarray*} Thus, it is sufficient to have $$\frac b2 N^{1/2} g \le \exp(-3/2),$$ which is true by assumption. \qed
\end{document}
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{The Splitting of Primes in Division Fields of
Elliptic Curves }
\author{W.Duke \thanks{Supported by NSF grant DMS-98-01642, the Clay Mathematics Institute and the American Institute of Mathematics} and \'A. T\'oth \thanks{Supported by a Rackham grant} }
\date{} \maketitle
\begin{center} {\it Dedicated to the memory of Petr \~{C}i\v{z}ek} \end{center}
\begin{abstract} In this paper we will give a global description of the Frobenius for the division fields of an elliptic curve $E$ which is strictly analogous to the cyclotomic case. This is then applied to determine the splitting of a prime $p$ in subfields of such a division field. Such fields include a large class of non-solvable quintic extensions and our application provides an arithmetic counterpart to Klein's ``solution'' of quintic equations using elliptic functions. A central role is played by the discriminant of the ring of endomorphisms of the elliptic curve reduced modulo $p$. \end{abstract}
\section*{Introduction}
Given a Galois extension $L/K$ of number fields with Galois group $G$, a fundamental problem is to describe the (unramified) primes
${\mathfrak p}$ of $K$ whose Frobenius automorphisms lie in a given conjugacy class $C$ of $G$. In particular, all such primes have the same splitting type in a sub-extension of $L/K$. In general, all that is known is that the primes have density $|C|/|G|$ in the set of all primes, this being the Chebotarev theorem.
For $L/K$ an abelian extension Artin reciprocity describes such primes by means of their residues in generalized ideal classes of $K$. In the special case that $L$ is obtained explicitly by adjoining to $K$ the $q$-th division points of the unit circle we have that $G \subset GL(1,q)=(\mathbb{Z}/q\mathbb{Z})^*$ and the Frobenius of ${\mathfrak p}$ is determined by the norm $N({\mathfrak p})$ modulo $q$. If $K=\mathbb{Q}$ (cyclotomic fields) we have that $G = GL(1,q)$ and any abelian extension of $\mathbb{Q}$ occurs as a subfield of such an $L$ for a suitable $q$ (Kronecker-Weber). Here the Chebotarev theorem reduces to the prime number theorem in arithmetic progressions.
In a similar manner an elliptic curve $E$ over $K$ gives rise to its $q$-th division field $L_q$ by adjoining to $K$ all the coordinates of the $q$-torsion points. Now $L_q$ is a (generally non-abelian) Galois extension of $K$ with Galois group $G$, a subgroup of $GL(2,\mathbb{Z}/q\mathbb{Z})$ (see \cite{Serre}). In this paper we will give a global description of the Frobenius for the division fields of an elliptic curve $E$ which is strictly analogous to the cyclotomic case. This is then applied to determine the splitting of primes in fields contained in $L_q$ or, as we shall say, uniformized by $E$. As observed by Klein (see \cite{Klein}), such fields include a large class of non-solvable quintic extensions. Our aim in this application is to provide an arithmetic counterpart to Klein's ``solution'' of quintic equations using elliptic functions.
By using CM curves we may uniformize all abelian extensions of imaginary quadratic fields. A classical application here is the result of Gauss that $$ x^3-2 $$ factors completely modulo a prime $p>3$ if and only if $$p=x^2+27y^2$$ for integers $x$ and $y$ (see \cite{Cox}). One way to derive this is to determine the Frobenius class of $p$ in the field obtained by
adjoining to $\mathbb{Q}$ the x-coordinates of the 3-division points of the elliptic curve given by $$y^2=x^3-15x+22,$$ which has CM by the quadratic order of discriminant -12.
Analogous results for non-solvable quintics require non-CM curves. Consider the quintic $$f(x)=x^5+90x^3+3645x-6480$$ which has discriminant $(2)^{12}(3)^{16}(5)^5(7)^6$. Its splitting field has Galois group $S_5$ over $\mathbb{Q}$. It follows from the results of this paper that $f(x)$ factors completely modulo $p >7$ if and only if $$p=x^2-25 \Delta_p y^2$$ where $\Delta_p$ is the discriminant of the ring of endomorphisms of the elliptic curve $$ y^2=x(x-1)(x-3)$$ reduced mod $p$. The first two such primes are 1259 and 1951 for which $\Delta_{1259}=-31$ and $\Delta_{1951}=-51$ and where $$1259=(22)^2+25\cdot31\cdot1^2$$ and $$1951=(26)^2+25\cdot51\cdot1^2.$$ As may be checked, $$f(x) \equiv (x+734)(x+322)(x+26)(x+851)(x+585)\bmod 1259$$ and $$f(x) \equiv (x+1029)(x+1222)(x+839)(x+1771)(x+992) \bmod 1951.$$ In the non-CM case $\Delta_p$ is not determined by arithmetic progressions in $p$. A goal of this paper is to complement that of Shimura \cite{Shimura} by pointing out the role of $\Delta_p$ in such questions.
\noindent {\sc Acknowledgement:} We would like to thank N. Katz for his helpful comments.
\section*{Outline of results.}\label{}
Given an elliptic curve $E$ defined over a number field $K$ and a prime ideal ${\mathfrak p}$ in $\cal{O}_{K}$ of good reduction for $E$ we shall define an integral matrix $[{\mathfrak p}]$ of determinant $N({\mathfrak p})$ whose reduction modulo $q$ gives the action of the Frobenius for $L_q$, the $q$-th division field of $E$. Let $a_{\mathfrak p}$ be defined as usual by \begin{equation}\label{1} \#E_{\mathfrak p}(k) = N({\mathfrak p}) - a_{\mathfrak p} + 1 \end{equation} \noindent where $E_{{\mathfrak p}}$ is the reduction of $E$ at $\mathfrak p$ and is defined over $k$, the residue field of ${\mathfrak p}$ which satisfies $\#k=N({\mathfrak p})= p^n.$
Let $R$ be the ring of those endomorphisms of $E$ that are rational polynomial expressions of the Frobenius endomorphism $\phi_k$. If $\phi_p$ is multiplication by an integer $R=\mathbb{Z}$ and we define $\Delta_{\mathfrak p}=1$ and $b_{\mathfrak p}=0$. Otherwise the ring $R$ is the centralizer of the Frobenius endomorphism in the endomorphism ring of $E_{\mathfrak p}$ over $k$ and is an imaginary quadratic order whose discriminant we denote by $\Delta_{\mathfrak p}$. Then we shall see that $p$ does not divide the conductor $m$ of $\Delta_{\mathfrak p}$ and that there is a unique non-negative integer $b_{\mathfrak p}$ so that \begin{equation}\label{2} 4N({\mathfrak p}) \;=\; a_{\mathfrak p}^2 \;-\;\Delta _{\mathfrak p}^{ }\;b_{\mathfrak p}^2. \end{equation} We associate to ${\mathfrak p}$ the following integral matrix of determinant $N({\mathfrak p})$:
\begin{equation}\label{3} [{\mathfrak p}] = \begin{bmatrix} (a_{\mathfrak p} + b_{\mathfrak p} \delta_{\mathfrak p})/2 & b_{\mathfrak p} \\ b_{\mathfrak p}(\Delta_{\mathfrak p}- \delta_{\mathfrak p})/4 & (a_{\mathfrak p} - b_{\mathfrak p} \delta_{\mathfrak p} )/2 \end{bmatrix} \end{equation} \noindent where for a discriminant $\Delta$ we have $\delta = 0,1$ according to whether $\Delta \equiv 0,1\bmod 4$. We shall show that $[{\mathfrak p}]$ gives a global representation of the Frobenius class over ${\mathfrak p}$ for each $q$-th division field of $E$ by reducing it modulo $q$, provided $p$ is prime to $q$.
\noindent
\begin{theorem}\label{Thm1} Let $E$ be an elliptic curve defined over a number field $K$ and $q>1$ an integer. Let $L_q$ be the $q$-th division field of $E$ with Galois group $G$ over $K$. Let ${\mathfrak p}$ be a prime of good reduction for $E$ with $N(\mathfrak p)$ prime to $q$. Then ${\mathfrak p}$ is unramified in $L_q$ and the integral matrix $[{\mathfrak p}]$ defined in (\ref{3}), when reduced modulo $q$, represents the class of the Frobenius of ${\mathfrak p}$ in $G$. \end{theorem}
\noindent The proof we give of this uses the theory of canonical lifts of endomorphisms due originally to Deuring.
In analogy with the cyclotomic case, we have associated to each curve a sequence of prime power matrices, defined in terms of arithmetic data from the reduced elliptic curve which give the Frobenius in all of the division fields. Let $C$ be a conjugacy class of $G$ and let $\pi_E(X;q,C)$ be the number of primes ${\mathfrak p}$ of good reduction with $N({\mathfrak p})\leq X$ such that $[{\mathfrak p}] \equiv C_0~\bmod q$ for some $C_0 \in C.$ By the Chebotarev theorem \cite{Cheb} we derive the following strict analogue of the prime number theorem in progressions for the sequence $[{\mathfrak p}]:$
$$\pi_E(X;q,C) \sim \frac{|C|}{|G|} \pi_K(X)$$ as $X \rightarrow \infty$, where $\pi_K(X)$ counts all primes of $K$ with $N({\mathfrak p})\leq X.$
Of more interest for us here is the fact that the splitting type of
${\mathfrak p}$ in any field between $K$ and the $q$-th division
field $L_q$ is determined by $[{\mathfrak p}] \bmod q$. For example
we get immediately a criterion for complete splitting in the full
division field in terms of the invariants $a_{\mathfrak p}$ and
$b_{\mathfrak p}$ modulo $q$, provided $q$ is odd.
\begin{cor}\label{cor1} Let $E$ be an elliptic curve defined over a number field $K$ and $q>1$ an odd integer. Then ${\mathfrak p}$ a prime of good reduction for $E$ with $N(\mathfrak p)$ prime to $q$ splits completely in
$L_q$ if and only if $a_{\mathfrak p} \equiv 2 \bmod q $ and $b_{\mathfrak p} \equiv 0 \bmod q $. \end{cor}
For a discriminant $\Delta$ let $$Q_{\Delta}(x,y)= x^2 + \delta x y -((\Delta-\delta)/4)y^2$$ be the principal form where $\delta = 0,1$ according to whether $\Delta \equiv 0,1\bmod 4$. For ${\mathfrak p}$ a prime of good reduction for $E$ we get a representation \begin{equation}\label{4} N({\mathfrak p})= Q_{\Delta_{\mathfrak p}}(x,y) \end{equation} with integral $x,y$ upon using the change of variables \begin{equation}\label{5} x=(a_{\mathfrak p}-b_{\mathfrak p} \delta_{\mathfrak p})/2 \,\,\,\,\,\,\,\,\,\,\,\, y=b_{\mathfrak p} \end{equation} in (2). This representation is primitive if $\mathfrak p$ is ordinary. Let $L_q^+$ be the extension of $K$ obtained by adjoining only the Weber functions of the $q$-th division points, that is the $x$-coordinates unless $j(E)=0$ or $j(E)=1728$, in which case we must first cube or square the coordinates, respectively. By Theorem 1 we may determine which sufficiently large ordinary primes split completely in $L_q^+$ from any such primitive representation.
\begin{cor}\label{cor3} Let $E$ be an elliptic curve defined over a number field $K$ as above and $q \geq 1$ an integer. Then there is a constant $C_0$ depending only on $E$ and $q$ so that for every ordinary prime ${\mathfrak p}$ of $K$ with $N({\mathfrak p})>C_0$ we have that $\mathfrak p$ splits completely in $L_q^+$ if and only if $x \equiv \pm 1 \bmod q$ and $y \equiv 0 \bmod q$ in any primitive representation $$N({\mathfrak p})= Q_{\Delta_{\mathfrak p}}(x,y).$$ \end{cor}
If $E$ has CM by the ring of integers in an imaginary quadratic field of discriminant $\Delta$ then the splitting completely condition in $L_q^+$ becomes simply $$N({\mathfrak p})= Q_{\Delta}(x,y)$$ with integers $x \equiv \pm 1 \bmod q$ and $y \equiv 0 \bmod q$. Actually, suppose we take for $E$ the elliptic curve with lattice given by the ring of integers of an imaginary quadratic field $F$ of discriminant $\Delta$ and take $K=F(j(E))$, the Hilbert class field of $F$. It follows from Corollary 2 that a sufficiently large rational prime $p$ splits in $L_q^+$ iff $p=Q_{\Delta}(x,y)$ with integers $x \equiv \pm 1 \bmod q$ and $y \equiv 0 \bmod q$. This is a well known result of CM theory.
Another simple consequence in the CM case, this time of Corollary 1, is that the conditions $$\#E_{\mathfrak p}(k) \equiv 0 \bmod q^2 \,\, and \,\, N({\mathfrak p}) \equiv 1 \bmod q,$$ which are clearly necessary for ${\mathfrak p}$ of good reduction to split completely in $L_q$, are also sufficient, at least when $q$ is odd.
Our main application is to describe the primes which split completely in certain non-solvable quintic extensions $M/K$. Suppose $M$ is given by adjoining to $K$ a solution of a principal quintic over $K$: $$f(x)=x^5+ax^2+bx+c=0$$ and that the discriminant of $f$ is $D$. Suppose further that the Galois group of the normal closure $L$ of $M$ is $S_5$ and that $\sqrt{5D} \in K.$
\begin{theorem}\label{splitting-quintic} Let $M/K$ be a non-solvable quintic extension as above. There exists an elliptic curve $E$ defined over $K$ so that a prime ${\mathfrak p}$ of $K$ which has good reduction for $E$ and is prime to 5 splits completely in $M$ if and only if
$$ b_{\mathfrak p} \equiv 0 \bmod 5 $$ where $b_{\mathfrak p}$ is associated to the elliptic curve $E$. \end{theorem}
In general we have the following determination of the splitting type of $\mathfrak p$:
\begin{tabular}{|c||r|r|c|} \hline $\textrm{Splitting type of} \,\,\,\,\mathfrak p \,\,\textrm{in} \,\,\ M$ &{$\displaystyle{\left(\frac{a_{\mathfrak p}^2-4{N(\mathfrak p})}{5}\right)}$} & { $\displaystyle{\left(\frac{N(\mathfrak p)}{5}\right)}$}& \\ \hline \hline $(1)(2)^2$ & $1\ \ \ \ \ $ & $1\ \ \ $ & \\ \hline $(1)(4)$ & $1\ \ \ \ \ $ & $-1\ \ \ $ & \\ \hline $(1)^2(3)$ &$ -1\ \ \ \ \ $ & $1\ \ \ $ & \\ \hline $(1)^3(2)$ &$ -1\ \ \ \ \ $ & $-1\ \ \ $ & if $ 5\vert a_{\mathfrak p}$\\ \hline $(2)(3)$ & $-1\ \ \ \ \ $ & $-1\ \ \ $ & if $ 5\not\vert a_{\mathfrak p}$\\ \hline
$(5)$ & $0\ \ \ \ \ $ & \multicolumn{2}{|c|}{ if $ 5\not\vert b_{\mathfrak p}\ \ \ $} \\ \hline
$(1)^5$ & $0\ \ \ \ \ $ & \multicolumn{2}{|c|}{ if $ 5\vert b_{\mathfrak p}\ \ \ $} \\ \hline
\end{tabular}
\noindent Concerning the determination of $E$ from $f$, it is enough to find the $j$-invariant of $E$. Explicit computations are provided below. We remark that it is also possible to formulate a similar result for $A_5$ extensions of $K$ under otherwise identical assumptions. Furthermore, by allowing the elliptic curve to be defined over a quadratic or a biquadratic extension of $K$ one may uniformize all non-solvable quintic extensions.
It is also possible to explicitly uniformize certain degree 7 extensions whose normal closure have Galois group simple of order 168 by using the seventh division fields of elliptic curves (see \cite{Radford} and the references cited there.) By Theorem 1 one may similarly characterize the primes with a given splitting type in such extensions.
\section*{A global representation of the Frobenius}
In this section we will prove Theorem 1 and its corollaries using an approach that compares the action of the Frobenius on the prime-to $p$ division points with the action of the matrix (\ref{3}) on $\mathbb{Z}^2$.
Let $E$ be an elliptic curve defined over a number field $K$. Let $\mathfrak p$ be a prime ideal in $\cal{O}_{K}$ with residue field $k$ $E_{\mathfrak p}$, the reduction of $E$ mod $\mathfrak p$ (it is assumed that $E$ has good reduction at ${\mathfrak p}$). That ${\mathfrak p}$ is unramified in the field $L_q$ is well known, see e.g. \cite{Silverman} VII.\S4. Also note that there is nothing to prove when $\phi_{\mathfrak p} \in \mathbb{Z}$, so we will assume throughout that this is not the case. The idea of the proof is that modulo ${\mathfrak p}$ the curve $E$ can be replaced by a curve $\tilde{E}$ with complex multiplication so that the following diagram commutes:
\noindent \begin{equation}\label{diagram} \begin{CD} E[q] @>{red}>> E_{\mathfrak p}[q] @<{red}<< \tilde{E}[q] \\ @V{F_{\mathfrak P}}VV @V{\phi_{\mathfrak p}}VV @V{\tilde{\phi}_{\mathfrak p}}VV \\ E[q] @>{red}>> E_{\mathfrak p}[q] @<{red}<< \tilde{E}[q] \\ \end{CD} \end{equation}
\noindent where as usual $[q]$ stands for the $q$-division points on the curves in the algebraic closures of the appropriate fields.
We now explain this diagram in detail. To simplify matters we fix a Weierstrass equation for $E$ as in \cite{Silverman} III.\S1. Let $\overline{K}, \overline{k}$ be the algebraic closures of $K, k$. To specify the horizontal maps $red$ we choose an embedding of $\overline{K}$ into the algebraic closure $\overline{K_{\mathfrak p}}$ of $K_{\mathfrak p}$, the completion of $K$ at the valuation arising from ${\mathfrak p}$. We call the subgroup of torsion points whose orders are relatively prime to $p$ the $p'$-torsion. Then the $p'$-torsion points on $E(\overline{K})$ is mapped into the $p'$-torsion of $E(\overline{K_{\mathfrak p}})$ and this being defined over an unramified extension, reduction modulo a prime ${\mathfrak P}$ above ${\mathfrak p}$ maps this latter group into the $p'$-torsion of $E(\overline{k})$. Both of these maps are isomorphisms on $p'$ torsion. This is the map $red$ for reduction, though as explained above it depends on many choices. Note that after these choices are made there is a unique element $F_{\mathfrak p} \in Gal(K_{\mathfrak p}^{unram}/K_{\mathfrak p})$ that satisfies $F_{\mathfrak p}(t) \equiv t^{\#k} \bmod {\mathfrak P}$, for all $t\in K_{\mathfrak p}^{unram}$.
We are interested in the action of the Frobenius automorphism $\phi_{\mathfrak p} \in Gal(\overline{k}/k)$ on the $\overline{k}$-valued points. In terms of the Weierstrass equation for $E$, this action on the coordinates is simply $(x,y) \mapsto (x^{\#k},y^{\#k})$. By abuse of notation we also denote this action and the restriction of it to the $q$-division points by $\phi_{\mathfrak p}$.
Now the commutativity of the left half of the diagram is merely a restatement of the choices made above.
By Deuring's lifting theorem (\cite{Deuring}, \cite{Lang} p.184), there exists an elliptic curve $\tilde{E}$ defined over ${K_{\mathfrak p}}$ and an endomorphism $\tilde{\phi}_{\mathfrak p}$ of $\tilde{E}$ so that $\tilde{E}$ reduces to $E_{\mathfrak p}$ modulo ${\mathfrak p}{\cal O}_{\mathfrak p}$ and that $\tilde{\phi}_{\mathfrak p} \in End(\tilde{E}) $ reduces to $\phi_{\mathfrak p} \in End(E_{\mathfrak p})$. If $E$ is super-singular $\tilde{\phi}_{\mathfrak p}$ will be defined over a ramified extension. Reduction still makes sense since $\tilde{\phi}_{\mathfrak p}$ is an endomorphism and not a Galois automorphism.
This shows the commutativity of the right half of diagram (\ref{diagram}).
To prove our theorem we need to determine the endomorphism ring $S$ of $\tilde{E}$. Recall that the ring $R_{\mathfrak p}$ defined in the introduction is the centralizer of $\phi_{\mathfrak p}$ in the endomorphism ring of $E_{\mathfrak p}$ and is a quadratic order. We claim that $S$ is isomorphic to $R$. Since $R \subset S$, Deuring's reduction theorem implies equality if we can show that the conductor of $R$ is prime to $N({\mathfrak p})$, a fact that is trivial in the ordinary case and follows from \cite{Wat} in the super-singular case.
Let $\Delta_{\mathfrak p}$ be the discriminant of $R_{\mathfrak p}$. By choosing a complex square root of $\Delta_{\mathfrak p}$ we identify $R_{\mathfrak p}$ with a lattice in $\mathbb{C}$. After this identification $\phi_{\mathfrak p}$ corresponds to some complex number $\phi=(a_{\mathfrak p}+b_{\mathfrak p} \sqrt{\Delta_{\mathfrak p}})/2$. Clearly the lattice $R$ is preserved by multiplication by $\phi$ and leads to the integral matrix (3), where we may choose $b_{\mathfrak p} \geq 0$. Instead of $R$ one could in fact use any lattice whose endomorphism ring is $R$.
To finish the proof of Theorem 1, consider an embedding $\alpha$ of the algebraic closure of $K_{\mathfrak p}$ into $\mathbb{C}$. It allows us to view $\tilde{E}$ as an elliptic curve over the complex numbers, that we denote $E_{\alpha}$. Since $E_{\alpha}$ has complex multiplication by $R$ and $Gal(\mathbb{C}/\mathbb{Q})$ acts transitively on the set of elliptic curves with $R$ as its endomorphism ring, we may and will assume the $j(E_{\alpha})=j(R)$.
By choosing a non-trivial holomorphic differential $\omega$ on $\tilde{E}_\alpha$ appropriately the lattice of periods $\{ \int_\gamma \omega : \gamma \in H_1(\tilde{E}_\alpha, \mathbb{Z} \}=R$. Then the period mapping $ \Pi:\tilde{E}_\alpha \rightarrow \mathbb{C}/R$ is a biholomorphic isomorphism of complex analytic manifolds. The action of $\tilde{\phi}_{\mathfrak p}$ on $\tilde{E}$ defines an endomorphism of $\tilde{E}_\alpha$ and gives rise to a map $\phi_*$ on $R$. Since the Frobenius automorphism $\phi_{\mathfrak p}$ satisfies a quadratic equation
\begin{equation}\label{frobenius} \phi_{\mathfrak p}^2 - a_{\mathfrak p}\;\phi_{\mathfrak p} + N({\mathfrak p}) = 0. \end{equation}
\noindent $\phi_*$ can be identified with multiplication by one of the complex roots of this equation i.e. multiplication by $\phi: R \rightarrow R$ (viewed as complex numbers). Getting back to the $q$-division points we can again summarize the situation in the following diagram:
\begin{equation}\label{diagram2} \begin{CD} \tilde{E}_{\mathfrak p}[q] @>{\alpha}>> \tilde{E}_\alpha[q] @>q \times {\Pi}>> R /qR \\
@V{\tilde{\phi}_{\mathfrak p}}VV @V{\phi_\alpha}VV @VV{\phi_*}V \\
\tilde{E}_{\mathfrak p}[q] @>{\alpha}>> \tilde{E}_\alpha[q] @>q \times{\Pi}>> R/qR \\ \end{CD} \end{equation}
\noindent where $q \times{\Pi}$ is the period map followed by multiplication by $q$. This proves Theorem 1.
\noindent {\it Remark:} If $E$ is replaced by an Abelian variety $V$ then the unramifiedness of $\mathfrak{p}$ still holds \cite{Shimura-Taniyama} and the left square of diagram \ref{diagram} makes sense. If in addition $V$ has ordinary reduction at ${\mathfrak p}$ then the right square in diagram \ref{diagram} generalizes as shown by Deligne \cite{Del} (and therefore the whole proof works). However the general case leads to substantial difficulties \cite{Oort}.
\noindent Corollary 1 is an immediate consequence of Theorem 1.
\noindent
We now prove Corollary 2. Let $E$ be an elliptic curve defined over a number field $K$ as above and $q \geq 1$ an integer. Let ${\mathfrak p}$ be a prime of ordinary reduction for $E$. Given a primitive representation $$p^n= Q_{\Delta_{\mathfrak p}}(x,y)$$ we know that $x$ and $y$ are uniquely determined up to (proper or improper) automorphs of $Q_{\Delta_{\mathfrak p}}.$ If $-\Delta_{\mathfrak p}>4$ and $x \equiv \pm 1 \bmod q$ and $y \equiv 0 \bmod q$ then it follows that \begin{equation}\label{7} [{\mathfrak p}] \equiv \begin{bmatrix} x+ \delta y & y \\ y (\Delta_{\mathfrak p}- \delta_{\mathfrak p})/4 & x \end{bmatrix} \bmod q \end{equation} and hence that $\mathfrak p$ splits completely in $L_q^+$. If $j=j(E)$ is not 0 or 1728 then for $\mathfrak p$ with $N(\mathfrak p)$ sufficiently large we have that $-\Delta_{\mathfrak p}>4$. To see this write $j=\alpha/\beta$ for $\alpha,\beta \in \cal O_K$. We know that $j \equiv j(R_{\mathfrak p}) \bmod \mathfrak p$. If $j(R_{\mathfrak p})=0$ or 1728 then assuming that $j - j(R_{\mathfrak p}) \neq 0$ we have
$$N(\mathfrak p) \leq \max(|N(\alpha)|,|N(\alpha-1728\beta)|).$$ In case $j=0$ or $j=1728$ the altered definition of $L_q^+$ leads again to the result.
\noindent
Finally we prove the consequence of Corollary 1 mentioned below Corollary 2 above that, in the CM case, a prime of good reduction ${\mathfrak p}$ splits completely in $L_q$ if $a_{\mathfrak p} \equiv N({\mathfrak p})+1 \bmod q^2 $ and $N({\mathfrak p}) \equiv 1 \bmod q$, provided $q$ is odd. Since these conditions immediately imply that $a_{\mathfrak p} \equiv 2 \bmod{q},$ by Corollary 1 we only must show that $q \mid b_{\mathfrak p}$. By our assumption $$ a_p^2 \equiv (N({\mathfrak p})-1)^2 +4N({\mathfrak p}) \equiv 4N({\mathfrak p}) \bmod{q^2} $$ we get, using $$4N({\mathfrak p}) \;=\; a_{\mathfrak p}^2 \;-\;\Delta_{\mathfrak p} \;b_{\mathfrak p}^2,$$ that $$q^2 \mid \Delta_{\mathfrak p} b_{\mathfrak p}^2.$$ For a CM curve with fundamental $\Delta$ the only possible prime dividing the square part of $\Delta_{\mathfrak p}$ is 2. In fact, $\Delta_{\mathfrak p}= \Delta$ for ordinary $\mathfrak p$ and for super-singular $\mathfrak p$ $\Delta_{\mathfrak p}=-p$ or $-4p$ where $N(\mathfrak p)=p^n.$ Since $q$ is odd this implies that $q \mid b_{\mathfrak p}$.
\noindent
\section*{Quintics}
In the section we prove Theorem 2 and justify the general splitting criteria given after it as well as the example given in the introduction. Let $M$ be given by adjoining to $K$ a root of a principal quintic $$f(x)=x^5+ax^2+bx+c=0$$ defined over $K$. If the discriminant of $f$ is $5$ times a square then, by means of a Tschirnhausen transformation (\cite{Dickson} p.218.) we may assume that $M$ is determined by a Brioschi quintic $$ f_t(x)=x^5-10tx^3 + 45t^2x-t^2 $$ for some $t \in K$ with $t \neq 0, \frac{1}{1728}$ . It was shown by Kiepert \cite{Kiepert} already in 1879 (see \cite{King} for an exposition) that $M$ is contained in $L_5^+$ for any elliptic curve $E$ over $K$ with $j$-invariant $1728-t^{-1}$. Recall that $L_5^+$ is in this case obtained by adjoining to $K$ the $x$-coordinates of the 5 division points. One may take for instance the curve $E_t$ given by $$ E_t: y^2+xy = x^3+36tx+t. $$
If the splitting field of $f$ over $K$ is an $S_5$ extension then it must be the fixed field of the subgroup of scalars of $G$ since $PGL_2(\mathbb{F}_5) \simeq S_5$. Theorem 2 now follows easily from Theorem 1.
A calculation of conjugacy classes based on the identification of $S_5$ with $PGL_2(\mathbb{F}_5)$ leads to the determination of the splitting type of a prime $\mathfrak p$ of good reduction for $E_t$ which is prime to 5. Recall that $A \in GL_2(\mathbb{F}_5)$ is called regular if it has different eigenvalues. Clearly $A$ is regular if the discriminant of the characteristic equation $tr(A)^2-4\det(A)$ is non-zero. Given such $A$ its conjugacy class is determined by its trace and determinant. It is clear that the values of the following Legendre symbols $$ \sigma=\left(\frac{\det(A)}{5}\right)\ \ \ \text{ and } \ \ \ \rho=\left(\frac{tr(A)^2-4\det(A)}{5}\right) $$ \noindent are determined by the conjugacy class of $A$ in $PGL_2(\mathbb{F}_5)$. Now in case the characteristic polynomial of $A$ splits, that is $\rho=1$, the matrix $A$ is conjugate to a diagonal matrix in $GL_2(\mathbb{F}_5)$ and so the value of $\sigma$ already determines the cycle type of such matrices. When $\rho=-1$ one must take into account whether $tr(A)\equiv 0 \text{ or } \not \equiv 0 \bmod 5$. For $A$ non-regular $tr(A)^2 -4 \det(A)=0$ and one needs to know if $A$ is semi-simple or unipotent. This information cannot be extracted from the trace and determinant alone, but it is determined by the value of $b_{\mathfrak p}$. All that remains to be done is to identify each conjugacy classes with its cycle type.
The example in the introduction is obtained by taking $K=\mathbb{Q}$ and $t=\frac{-3^2}{2^85^2} $. Here we observe that since $E$ has four 2-torsion points over $\mathbb{Q}$ both $a_p$ and $b_p$ will be even for $p$ with good reduction. Thus the representation $$4p \;=\; a_{p}^2 \;-\;\Delta _{p}\;b_{p}^2$$ yields $$p \;=\; x^2 \;-\;\Delta _{p}\;y^2$$ and the condition for splitting completely is that $y \equiv 0 \bmod 5,$ since $x$ and $y$ are determined uniquely up to sign.
\section*{Some computational issues}
In this section we discuss some of the computational issues which arise when considering examples.
First, given a principal quintic (slightly modified from above) \begin{equation}\label{principal} f(x)=x^5+5ax^2+5bx+c=0 \end{equation} \noindent defined over $K$ with discriminant $D$ such that $\sqrt{5D} \in K$ we must determine $t$ so that the Brioschi quintic \begin{equation}\label{brioschi} f_t(x)=x^5-10tx^3 + 45t^2x-t^2. \end{equation} \noindent determines the same extension. This is done using a Tschirnhausen transformation and is described in detail in \cite{King}, p 103. (see also \cite{Dickson} p.218.) Here we will simply record the result in the case $a \neq 0.$
One determines $t, \lambda$ and $\mu$ in the map \begin{equation}\label{tschirn} x\mapsto \frac{\lambda+\mu x}{(x^2/t) -3} \end{equation} \noindent in order to transform the general principal quintic (\ref{principal}) to the Brioschi quintic (\ref{brioschi}).
An analysis using invariant polynomials for the icosahedral group acting on the Riemann sphere leads eventually to the quadratic equation for $\lambda$ given by $$ (a^4+abc-b^3)\lambda^2-(11a^3-ac^2+2b^2c)\lambda+64a^2b^2-27a^3c-bc^2=0.$$ The discriminant of this quadratic is $$5^{-5}a^2D$$ and so $\lambda \in K$. Choose either solution and let $$j=\frac{(a\lambda^2-3b\lambda-3c)^3}{a^2(\lambda ac-\lambda b^2 -bc)}.$$ Then, provided $j \neq 0, 1728$ we may take $$t=1/(1728-j)$$
in (\ref{brioschi}) and choose for the elliptic curve any curve with this $j$ invariant, say $$ E_t: y^2+xy = x^3+36tx+t $$ as above. Also, one may determine $\mu$ in (\ref{tschirn}) to be given by $$ \mu=\frac{j a^2-8{\lambda}^3a-72{\lambda}^2b-72\lambda c}{{\lambda}^2a+\lambda b +c}.$$ Note that the discriminant of $f_t$ is $$D_t=5^5t^8(1728t-1)^2$$ while that of $E_t$ is $$-t(1728t-1)^2.$$
Another issue is to compute the invariants $\Delta_p$ and $b_p$ in the rational case. This problem is mentioned in the Woods Hole seminar on Elliptic curves and formal groups by Lubin-Serre-Tate. An important study of $\Delta_p$ was made by Schoof in \cite{Schoof}.
The most straightforward way to determine $b_p$ and to find the order $R$ that appears in Deuring's theorem is to check all the possible singular invariants until we find one that is congruent to the given $j$-value modulo $p$. (Note that the discriminant of $R$ must divide $a_p^2-4p$.) We assume that our input is an elliptic curve $E$, given in Weierstrass equation, and $p$ is a prime number that does not divide the discriminant of $E$. After computing $a_p$ (a serious task by itself) we find $\Delta_p$ for an ordinary curve as follows; we first compute the square-free part $D$ of $a_p^2-4p$ and then create a vector whose values are all possible discriminants
$$\Delta=b^2 D|(a_p^2-4p).$$
\noindent For a possible conductor $\Delta$, we find the class group ${\cal C}(\Delta)$ of the proper ideal classes (using quadratic forms) and compute the integer \[ X_{\Delta}=\displaystyle{\prod_{\Lambda \in {\cal C}(\Delta)} \left(j(E)-j(\Lambda)\right)}. \]
\noindent Note that the canonical lift $\tilde{E}$ is distinguished by the fact that its endomorphism ring is $R$ and that $j(E) \equiv j(\tilde{E}) \bmod{P}$ for some prime $P$ dividing $p$. Therefore for any complex embedding $\alpha: \mathbb{Q}_p \rightarrow \mathbb{C},$ $$\alpha(j(\tilde{E})) \in \{j(\mathbb{C}/\Lambda): \Lambda \in {\cal C}_{\Delta_p}\},$$
where $\Delta_p$ is the actual discriminant of $R$. Also note that if $\Lambda' \in {\cal C}_{\Delta'}$ for $\Delta'\neq \Delta_p$, then the corresponding elliptic curve reduces to a curve whose endomorphism ring has discriminant $\Delta'$ for any place above $p$. Therefore $\Delta_p$ is uniquely characterized by the fact that $$X_{\Delta_p} \equiv 0 \bmod p.$$
Occasionally the computation of $X_{\Delta}$ involves complex numbers of rather large size. To make the algorithm efficient, one needs to determine the needed precision in advance. Assume that the lattices are given in the form $\mathbb{Z}+\mathbb{Z}\tau_i$, with $\tau_i$ in the upper half plane. Then the number of significant digits one must use is approximately
\[ \sum_{\tau_i}^{} \frac{\log(j(E)) + 2 \pi Im(\tau_i)}{\log(10)}. \]
\noindent
\end{document}
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\begin{document}
\title[Universal deformation rings of group representations]{Universal deformation rings of group representations, with an application of Brauer's generalized decomposition numbers}
\author{Frauke M. Bleher} \address{Department of Mathematics\\University of Iowa\\ Iowa City, IA 52242-1419} \email{[email protected]} \thanks{The author was supported in part by NSA Grant H98230-11-1-0131.} \subjclass[2010]{Primary 20C20; Secondary 20C15, 16G10} \keywords{Universal deformation rings; Brauer's generalized decomposition numbers; tame blocks; dihedral defect groups; semidihedral defect groups; generalized quaternion defect groups}
\begin{abstract} We give an introduction to the deformation theory of linear representations of profinite groups which Mazur initiated in the 1980's. We then consider the case of representations of finite groups. We show how Brauer's generalized decomposition numbers can be used in some cases to explicitly determine universal deformation rings. \end{abstract}
\maketitle
\section{Introduction} \label{s:intro}
Let $p$ be a prime number and let $(K,\mathcal{O},k)$ be a $p$-modular system, where $\mathcal{O}$ is a complete discrete valuation ring of characteristic 0 with maximal ideal $\mathfrak{m}_\mathcal{O}$, $K=\mathrm{Frac}(\mathcal{O})$ is its fraction field and $k=\mathcal{O}/\mathfrak{m}_\mathcal{O}$ is its residue field of characteristic $p$. Let $G$ be a group and let $V$ be a $kG$-module of finite $k$-dimension. It is a classical question to ask whether $V$ can be lifted to an $\mathcal{O}$-free $\mathcal{O}G$-module. Green showed in \cite{green} that this is possible if $\mathrm{Ext}^2_{kG}(V,V)=0$. However, this is only a sufficient criterion, as there are many cases when $\mathrm{Ext}^2_{kG}(V,V)\neq 0$ and $V$ can still be lifted to an $\mathcal{O}$-free $\mathcal{O}G$-module. Even if we know that there is such a lift of $V$ to $\mathcal{O}$, we may still not be able to characterize all possible lifts to $\mathcal{O}$. On the other hand, if $V$ cannot be lifted to $\mathcal{O}$, it is natural to ask if $V$ can be lifted to other complete local commutative rings with residue field $k$. This can be formulated as the following two natural questions: \begin{itemize} \item[(i)] How can all possible lifts of $V$ to $\mathcal{O}$ be described? \item[(ii)] Over which complete local commutative rings with residue field $k$ can $V$ be lifted? Is there one particular such complete local ring from which all these lifts arise? \end{itemize} To answer both of these questions, it is necessary to develop a systematic way to study isomorphism classes of lifts, also called deformations, of $kG$-modules $V$. In the 1980's, Mazur developed a deformation theory of representations of profinite Galois groups over finite fields which can be used to give answers to these questions in certain cases.
The goal of this paper is to give an introduction to this deformation theory, with an emphasis on the particular case of deformations of representations of finite groups.
In section \ref{s:maz}, we will first focus on Mazur's general deformation theory by studying the deformation functor, universal deformation rings, tangent spaces, obstructions, and deformation rings in number theory. In section \ref{s:udrfinite}, we will then concentrate on deformation rings and deformations of representations of finite groups. In section \ref{s:brauer}, we will describe how Brauer's generalized decomposition numbers can be used to explicitly determine the universal deformation rings of certain representations belonging to blocks of tame representation type.
We close this introduction by discussing a few classical examples which show that lifts occur naturally both in representation theory and number theory.
\begin{examples} \hspace*{1cm} \begin{enumerate} \item[(i)] Let $G$ be a finite group and consider permutation modules for $kG$.
Recall that direct summands of permutation modules are also called $p$-permutation modules or trivial source modules. Scott proved in \cite{scott} that $p$-permutation modules can be lifted to $\mathcal{O}$. Rickard used this result, for example, in \cite{splendid} to show that tilting complexes defining splendid equivalences can be lifted from $k$ to $\mathcal{O}$.
Another class of permutation modules is given by endo-permutation modules for $kG$, i.e. $kG$-modules $V$ for which $\mathrm{End}_k(V)$ is a permutation module. A special subclass is provided by the endo-trivial modules, for which $\mathrm{End}_k(V)$ is isomorphic as a $kG$-module to a direct sum of the trivial simple $kG$-module and a projective $kG$-module. Alperin proved in \cite{alperinendotrivial} that if $G$ is a $p$-group and $V$ is an endo-trivial $kG$-module, then $V$ can be lifted to an endo-trivial $\mathcal{O}G$-module.
\item[(ii)] Let $k=\mathbb{F}_p=\mathbb{Z}/p$ and let $\mathcal{O}=\mathbb{Z}_p$ be the ring of $p$-adic integers. Suppose $E$ is an elliptic curve over $\mathbb{Q}$ and define $G=G_\mathbb{Q}= \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$. Then $G$ acts on the torsion points $E[p^n]\cong \mathbb{Z}/p^n\times \mathbb{Z}/p^n$ for all $n\ge 1$. In particular, for $n=1$, we obtain a $kG$-module $V$ of $k$-dimension 2. Since the action of $G$ on $E[p^n]$ commutes with the multiplication by $p$ on $E[p^n]$, $G$ acts naturally on the Tate module $T_p(E)=\displaystyle\lim_{\longleftarrow}E[p^n] \cong \mathbb{Z}_p\times\mathbb{Z}_p$. Hence $T_p(E)$ defines a lift of $V$ over $\mathcal{O}=\mathbb{Z}_p$. \end{enumerate} \end{examples}
\section{Mazur's deformation theory} \label{s:maz}
In the 1980's, Mazur developed a deformation theory of Galois representations to systematically study $p$-adic lifts of representations of profinite Galois groups \cite{maz1}. His initial motivation seems to have come from Hida's work on ordinary $p$-adic modular forms (see for example Gouv\^{e}a's summary in \cite[Lecture 2]{gouvea2001}). Mazur's deformation theory became a fundamental tool in number theory when it was shown to play a crucial role in the proof of Fermat's Last Theorem, and more generally in the proof of the modularity conjecture for elliptic curves over $\mathbb{Q}$ (see \cite{wi,wita,bcdt}). For more information and background, we also refer the reader to the survey articles \cite{gouvea1991,gouvea2001,maz2} on Mazur's deformation theory and the collection of articles in \cite{CMS17,cornell} on the proof of Fermat's Last Theorem by Wiles and Taylor.
\subsection{The deformation functor} \label{ss:deffunctor}
Let $k$ be a perfect field of positive characteristic $p$, and let $W=W(k)$ be the ring of infinite Witt vectors over $k$. Recall that since $k$ is perfect, $W$ is the unique (up to isomorphism) complete discrete valuation ring of characteristic zero which is absolutely unramified, meaning that the valuation of $p$ is 1 so that $p$ is a generator of the maximal ideal $\mathfrak{m}_W$. Let $\hat{\mathcal{C}}$ denote the category whose objects are all complete local commutative Noetherian rings with residue field $k$ and whose morphisms are local homomorphisms of complete local Noetherian rings that induce the identity on $k$. Strictly speaking, each object of $\hat{\mathcal{C}}$ is a pair $(R,\pi_R)$ consisting of a complete local commutative Noetherian ring $R$ and a fixed reduction map $\pi_R:R\to k$ inducing an isomorphism $R/\mathfrak{m}_R\cong k$. Note that all rings $R$ in $\hat{\mathcal{C}}$ have a natural $W$-algebra structure, which means that the morphisms in $\hat{\mathcal{C}}$ can also be viewed as continuous $W$-algebra homomorphisms inducing the identity on the residue field $k$.
Let $G$ be a profinite group, and let $V$ be a finite dimensional vector space over $k$ with a continuous $k$-linear action of $G$ on $V$, which is given by a continuous group homomorphism from $G$ to the discrete group $\mathrm{Aut}_k(V)$. A \emph{lift} of $V$ over a ring $R$ in $\hat{\mathcal{C}}$ is defined to be a pair $(M,\phi)$ consisting of a finitely generated free $R$-module $M$ on which $G$ acts continuously together with a $G$-equivariant isomorphism $\phi: k \otimes_R M \to V$ of $k$-vector spaces. We say two lifts $(M,\phi)$ and $(M',\phi')$ of $V$ over $R$ are \emph{isomorphic} if there exists an $R$-linear $G$-equivariant isomorphism $f:M\to M'$ satisfying $\phi'\circ(k\otimes f)=\phi$. We define the set $\mathrm{Def}_G(V,R)$ of \emph{deformations} of $V$ over $R$ to be the set of isomorphism classes of lifts of $V$ over $R$. The \emph{deformation functor} $F_V:\hat{\mathcal{C}}\to \mathrm{Sets}$ is defined to be the covariant functor which sends a ring $R$ in $\hat{\mathcal{C}}$ to $\mathrm{Def}_G(V,R)$ and a morphism $\alpha:R\to R'$ in $\hat{\mathcal{C}}$ to the set map \begin{eqnarray*} F_V(\alpha):\qquad \mathrm{Def}_G(V,R) & \to & \mathrm{Def}_G(V,R')\\
\; [M,\phi] &\mapsto& [R'\otimes_{R,\alpha}M,\phi_\alpha] \end{eqnarray*} where $\phi_\alpha$ is the composition $k\otimes_{R'}(R'\otimes_{R,\alpha}M)\cong k\otimes_RM\xrightarrow{\phi} V$.
Sometimes it is useful to describe the deformation functor $F_V$ in terms of matrix groups. Choosing a $k$-basis of $V$, we can identify $V$ with $k^n$ where $n=\mathrm{dim}_k\,V$. The $G$-action on $V$ is then given by a continuous homomorphism $\overline{\rho}:G \to \mathrm{GL}_n(k)$, which is called a \emph{residual representation}. Let $R$ be a ring in $\hat{\mathcal{C}}$ and denote the reduction map $\mathrm{GL}_n(R)\to \mathrm{GL}_n(k)$ induced by $\pi_R:R\to k$ also by $\pi_R$. By a \emph{lift} of $\overline{\rho}$ over $R$ we mean a continuous homomorphism $\rho:G\to \mathrm{GL}_n(R)$ such that $\pi_R\circ\rho=\overline{\rho}$. Such a lift defines a $G$-action on $M=R^n$, and with the obvious isomorphism $\phi:k\otimes_RM\to V$ such a lift defines a deformation $[M,\phi]$ of $V$ over $R$. Two lifts $\rho,\rho':G\to \mathrm{GL}_n(R)$ of $\overline{\rho}$ over $R$ give rise to the same deformation if and only if they are \emph{strictly equivalent}, that is, if one can be brought into the other by conjugation by a matrix in the kernel of $\pi_R$. In this way, the choice of a basis of $V$ gives rise to an identification of $\mathrm{Def}_G(V,R)$ with the set $\mathrm{Def}_G(\overline{\rho},R)$ of strict equivalence classes of lifts of $\overline{\rho}$ over $R$. In particular, the deformation functor $F_{\overline{\rho}}:\hat{\mathcal{C}}\to\mathrm{Sets}$ associated to strict equivalence classes of lifts of $\overline{\rho}$ is naturally isomorphic to the deformation functor $F_V$. Note that if $\alpha:R\to R'$ is a morphism in $\hat{\mathcal{C}}$, then $F_{\overline{\rho}}(\alpha)$ is the set map $\mathrm{Def}_G(\overline{\rho},R) \to \mathrm{Def}_G(\overline{\rho},R')$ which sends $[\rho]$ to $[\alpha\circ\rho]$, where we denote the morphism $\mathrm{GL}_n(R)\to \mathrm{GL}_n(R')$ induced by $\alpha$ also by $\alpha$. In the following, we identify $F_V=F_{\overline{\rho}}$.
\subsection{Universal deformation rings} \label{ss:udr}
Assume the notation from the previous subsection. The functor $F_V$ is representable if there is a ring $R(G,V)$ in $\hat{\mathcal{C}}$ and a lift $(U(G,V),\phi_U)$ of $V$ over $R(G,V)$ such that for all $R$ in $\hat{\mathcal{C}}$ the map \begin{eqnarray*} f_R:\qquad \mathrm{Hom}_{\hat{\mathcal{C}}}(R(G,V),R) &\to& \mathrm{Def}_G(V,R)\\ \;\alpha &\mapsto& F_V(\alpha)([U(G,V),\phi_U]) \end{eqnarray*} is bijective. Put differently, this is the case if and only if $F_V$ is naturally isomorphic to the Hom functor $\mathrm{Hom}_{\hat{\mathcal{C}}}(R(G,V),-)$. If this is the case then $R(G,V)$ is called the \emph{universal deformation ring} of $V$ and $[U(G,V),\phi_U]$ is called the \emph{universal deformation} of $V$. The defining property determines $R(G,V)$ and $[U(G,V),\phi_U]$ uniquely up to a unique isomorphism.
A slightly weaker notion can be useful if the functor $F_V$ is not representable. The ring $k[\epsilon]$ of dual numbers with $\epsilon^2 = 0$ has a $W$-algebra structure such that the maximal ideal $\mathfrak{m}_W$ of $W$ annihilates $k[\epsilon]$. One says $R(G,V)$ is a \emph{versal deformation ring} of $V$ if the maps $f_R$ are surjective for all $R$, and bijective for $R=k[\epsilon]$. These conditions determine $R(G,V)$ uniquely up to isomorphism, but the isomorphism need not be unique.
\begin{theorem} {\rm (\cite[\S 1.2]{maz1}, \cite[Prop. 7.1]{desmitlenstra})} \label{thm:mazurudr} Suppose $G$ satisfies the following $p$-finiteness condition: \\[.2cm] $\mathbf{(\Phi_p)}$ For every open subgroup $J$ of finite index in $G$, there are only finitely many continuous homomorphisms from $J$ to $\mathbb{Z}/p$. \\[.2cm] Then every finite dimensional continuous representation $V$ of $G$ over $k$ has a versal deformation ring. If $\mathrm{End}_{kG}(V)=k$, then $V$ has a universal deformation ring. \end{theorem}
To prove the existence of versal deformation rings, Mazur verified Schlessinger's criteria \cite{Sch} of pro-representability of Artin functors for the deformation functor $F_V$. He also proved that $F_V$ is continuous, meaning that for all objects $R$ in $\hat{\mathcal{C}}$ we have $\displaystyle F_V(R)=\lim_{\longleftarrow} F_V(R/\mathfrak{m}_R^i)$. Note that Mazur assumed $k$ to be a finite field in \cite{maz1}. However, his proofs in \cite[\S 1.2]{maz1} go through in the more general case we are considering.
In the case when $\mathrm{End}_{kG}(V)=k$, de Smit and Lenstra took a different approach in \cite{desmitlenstra} which proceeds in three main steps: First they let $G$ be finite and considered the functor which assigns to each $R$ a certain set of homomorphisms $G\to\mathrm{GL}_n(R)$. They showed that this functor is representable by defining the corresponding universal ring by generators and relations. Taking projective limits, they obtained a similar result for profinite $G$. Finally, they concluded the construction by passing to a suitable closed subring, which is either generated by the traces of the elements of $G$ if $V$ is absolutely irreducible, or by a larger collection of elements as suggested by Faltings if $\mathrm{End}_{kG}(V)=k$.
In the case when $V$ is absolutely irreducible, another approach was given by Rouquier in \cite{rouquier}, using pseudo-characters and the results in \cite{nyssen}. The main point in this latter construction is that being a pseudo-character function has a universal solution, providing another description of the universal deformation ring in this case.
\subsection{Pseudocompact modules} \label{ss:pseudocompact}
In this subsection, we briefly describe another viewpoint of lifts and deformations using pseudocompact modules. This is useful, for example, when generalizing deformations of group representations to deformations of objects in derived categories (see for example \cite{bcderived1,bcderived2}). Pseudocompact rings, algebras and modules have been studied, for example, in \cite{ga1,ga2,brumer}, which also serve as references for the following statements.
Assume the notation from subsection \ref{ss:deffunctor}. For $R \in \mathrm{Ob}(\hat{\mathcal{C}})$, let $R[[G]]$ be the completed group algebra of the usual abstract group algebra $RG$ of $G$ over $R$, i.e. $R[[G]]$ is the projective limit of the usual group algebras $R[G/U]$ as $U$ ranges over the open normal subgroups of $G$ (where we put brackets around $G/U$ for better readability). Giving a finitely generated free $R$-module $M$ on which $G$ acts continuously is the same as giving a topological $R[[G]]$-module $M$ which is finitely generated and free as an $R$-module.
The completed group algebra $R[[G]]$ is a so-called \emph{pseudocompact ring}, i.e. it is a complete Hausdorff topological ring which admits a basis of open neighborhoods of $0$ consisting of two-sided ideals $J$ for which $R[[G]]/J$ is an Artinian ring. In particular, $R[[G]]$ is the projective limit of Artinian quotient rings having the discrete topology. Since $R$ is a commutative pseudocompact ring and $R[[G]]$ is an $R$-algebra and since the open neighborhood basis of $0$ can be chosen to consist of two-sided ideals $J$ for which $R[[G]]/J$ has finite length as $R$-module, $R[[G]]$ is moreover a \emph{pseudocompact $R$-algebra}. A complete Hausdorff topological $R[[G]]$-module $M$ is said to be a \emph{pseudocompact $R[[G]]$-module} if $M$ has a basis of open neighborhoods of $0$ consisting of submodules $N$ for which $M/N$ has finite length. It follows that an $R[[G]]$-module is pseudocompact if and only if it is the projective limit of $R[[G]]$-modules of finite length having the discrete topology. We denote the category of pseudocompact $R[[G]]$-modules by $\mathrm{PCMod}(R[[G]])$. Note that $\mathrm{PCMod}(R[[G]])$ is an abelian category with exact projective limits.
A pseudocompact $R[[G]]$-module $M$ is said to be \emph{topologically free} on a set $X=\{x_i\}_{i\in I}$ if $M$ is isomorphic to the product of a family $(R[[G]]_i)_{i\in I}$ where $R[[G]]_i=R[[G]]$ for all $i$. In particular, a topologically free pseudocompact $R[[G]]$-module on a finite set is the same as a finitely generated abstractly free $R[[G]]$-module.
As before, assume $V$ is a finite dimensional vector space over $k$ with a continuous $k$-linear action of $G$ on $V$, which is given by a continuous group homomorphism from $G$ to the discrete group $\mathrm{Aut}_k(V)$. Then $V$ is a pseudocompact $k[[G]]$-module. Moreover, any lift of $V$ over a ring $R$ in $\hat{\mathcal{C}}$ is given by a pseudocompact $R[[G]]$-module $M$ which is finitely generated and abstractly free as an $R$-module together with an isomorphism $\phi: k \otimes_R M \to V$ in $\mathrm{PCMod}(k[[G]])$. Note that in principle, we should use the completed tensor product $\hat{\otimes}_R-$ (see \cite[\S2]{brumer}) rather than the usual tensor product $\otimes_R$. However, since $k$ is finitely generated as a pseudocompact $R$-module, it follows that the functors $k\otimes_R -$ and $k\hat{\otimes}_R-$ are naturally isomorphic.
Every topologically free pseudocompact $R[[G]]$-module is a projective object in $\mathrm{PCMod}(R[[G]])$, and every pseudocompact $R[[G]]$-module is the quotient of a topologically free $R[[G]]$-module. Hence $\mathrm{PCMod}(R[[G]])$ has enough projective objects. If $M$ and $N$ are pseudocompact $R[[G]]$-modules, then we define the right derived functors $\mathrm{Ext}^n_{R[[G]]}(M,N)$ by using a projective resolution of $M$.
For all positive integers $n$, we have an isomorphism of $k$-vector spaces \begin{equation} \label{eq:cohomology} \mathrm{H}^n(G,\mathrm{Hom}_k(V,V))\cong \mathrm{Ext}^n_{k[[G]]}(V,V) \end{equation} where $\mathrm{H}^n$ is ``continuous'' cohomology. Since $V$ is finite dimensional and discrete, this isomorphism can be deduced by looking at direct limits over all open normal subgroups $U$ of $G$ which act trivially on $V$.
\subsection{Tangent space} \label{ss:tangent}
We continue to assume the notation from subsection \ref{ss:deffunctor}. Moreover, we assume that $G$ satisfies the $p$-finiteness condition $\mathbf{(\Phi_p)}$ from Theorem \ref{thm:mazurudr}. For simplicity, we also assume that $V$ has a universal deformation ring $R(G,V)$. In other words, the deformation functor $F_V$ is represented by $R(G,V)$ and thus naturally isomorphic to the Hom functor $\mathrm{Hom}_{\hat{\mathcal{C}}}(R(G,V),-)$.
The \emph{tangent space} of $F_V$ is defined as $$t_{F_V}=F_V(k[\epsilon])\cong \mathrm{Hom}_{\hat{\mathcal{C}}}(R(G,V),k[\epsilon])$$ where, as before, $k[\epsilon]$ is the ring of dual numbers with $\epsilon^2 = 0$. Since in $\hat{\mathcal{C}}$ there is only one morphism $R(G,V)\to k$, we obtain an isomorphism $$\mathrm{Hom}_{\hat{\mathcal{C}}}(R(G,V),k[\epsilon])\xrightarrow{\cong} \mathrm{Hom}_k\left( \frac{\mathfrak{m}_{R(G,V)}}{\mathfrak{m}_{R(G,V)}^2+pR(G,V)}\,,\,k\right)$$ of $k$-vector spaces. For $R$ in $\hat{\mathcal{C}}$, we call $$t_R^*=\frac{\mathfrak{m}_R}{\mathfrak{m}_R^2+pR}$$ the Zariski cotangent space of $R$. Hence we obtain an isomorphism of tangent spaces $$t_{F_V}\cong \mathrm{Hom}_k(t_{R(G,V)}^*,k)=t_{R(G,V)}.$$ The following result gives a connection of $t_{F_V}$ to the cohomology of $\mathrm{Hom}_k(V,V)$.
\begin{proposition} {\rm (\cite[\S1.2]{maz1}, \cite[\S 22]{maz2})} \label{prop:tangentspace} There is a natural isomorphism of $k$-vector spaces $$t_{F_V}\cong \mathrm{H}^1(G,\mathrm{Hom}_k(V,V))\cong \mathrm{Ext}^1_{k[[G]]}(V,V).$$ If $r=\mathrm{dim}_k\mathrm{Ext}^1_{k[[G]]}(V,V)$ then $R(G,V)$ is isomorphic to a quotient algebra of the power series algebra $W[[t_1,\ldots,t_r]]$ in $r$ commuting variables and $r$ is minimal with this property. \end{proposition}
As in the previous subsection, $\mathrm{Ext}^1_{k[[G]]}(V,V)$ means $\mathrm{Ext}^1$ in the category of pseudocompact $k[[G]]$-modules; see also Equation (\ref{eq:cohomology}). The main idea of the proof is to notice that if $(M,\phi)$ is a lift of $V$ over $k[\epsilon]$ then by restricting the scalars from $k[\epsilon]$ to $k$ we may view $M$ as a $k$-vector space of dimension $2\cdot\mathrm{dim}_kV$, with a $k$-linear continuous action of $G$. Identifying the $k[[G]]$-modules $\epsilon M$ and $M/\epsilon M$ with $V$ (using $\phi$), we then see $M$ as an extension of $V$ by $V$ in the category of pseudocompact $k[[G]]$-modules: $$\mathcal{E}:\quad 0\to V \xrightarrow{\iota} M \xrightarrow{\tau} V\to 0.$$ Sending the element of $t_{F_V}$ corresponding to the isomorphism class $[M,\phi]$ to the element of $\mathrm{Ext}^1_{k[[G]]}(V,V)$ corresponding to $\mathcal{E}$, we obtain a well-defined map $$s:t_{F_V}\to \mathrm{Ext}^1_{k[[G]]}(V,V)$$ which is a $k$-vector space homomorphism. The inverse map of $s$ is obtained by going backward: Given an extension $\mathcal{E}$ as above, we define a $k[\epsilon]$-structure on $M$ by letting $\epsilon$ act as the composition $\iota\circ \tau$, enabling us to view $M$ as a lift of $V$ over $k[\epsilon]$.
\subsection{Obstructions} \label{ss:obstruct}
We continue to assume the notation from subsection \ref{ss:deffunctor}, that $G$ satisfies $\mathbf{(\Phi_p)}$ and that $V$ has a universal deformation ring $R(G,V)$. Let $\overline{\rho}:G\to\mathrm{GL}_n(k)$ be a residual representation corresponding to $V$.
Suppose we have a surjective morphism $R_1\to R_0$ in $\hat{\mathcal{C}}$ and assume that the kernel $I$ satisfies $I\cdot \mathfrak{m}_{R_1}=0$, so $I$ has the structure of a $k$-vector space. Suppose we have a lift $\rho_0:G\to\mathrm{GL}_n(R_0)$ of $\overline{\rho}$ over $R_0$. We want to describe the obstruction to lifting $\rho_0$ to $R_1$.
Let $\gamma_1:G\to\mathrm{GL}_n(R_1)$ be a set-theoretic lift of $\rho_0$. Since $\gamma_1$ is a homomorphism modulo $I$, we obtain a 2-cocycle \begin{eqnarray*} c:\quad G\times G &\to& 1+\mathrm{Mat}_n(I)\\ (g_1,g_2) &\mapsto& \gamma_1(g_1g_2)\gamma_1(g_2)^{-1}\gamma_1(g_1)^{-1}. \end{eqnarray*} Identifying the multiplicative group $1+\mathrm{Mat}_n(I)\subset \mathrm{GL}_n(R_1)$ with the additive group $\mathrm{Mat}_n(I)^+$, we obtain the following isomorphisms: $$1+\mathrm{Mat}_n(I)\cong \mathrm{Mat}_n(I)^+\cong\mathrm{Mat}_n(k)\otimes_kI \cong \mathrm{Hom}_k(V,V)\otimes_kI.$$ Here the action of $G$ on $\mathrm{Mat}_n(k)$, respectively $\mathrm{Hom}_k(V,V)$, is given by the usual conjugation action; note that this is also called the \emph{adjoint representation} of $\overline{\rho}$ and denoted by $\mathrm{ad}(\overline{\rho})$. If we replace $\gamma_1$ by a different set-theoretic lift, this changes $c$ by a 2-coboundary. Therefore we obtain an element $$[c]\in\mathrm{H}^2(G,\mathrm{Hom}_k(V,V)\otimes_kI)\cong \mathrm{H}^2(G,\mathrm{Hom}_k(V,V))\otimes_kI$$ which gives the \emph{obstruction} to lifting $\rho_0$ to $R_1$. The class $[c]$ is sometimes called the \emph{obstruction class} of $\rho_0$ relative to the morphism $R_1\to R_0$.
\begin{proposition} {\rm (\cite[\S1.6]{maz1}, \cite[Thm. 2.4]{bockle})} \label{prop:obstructions} If $r=\mathrm{dim}_k\mathrm{Ext}^1_{k[[G]]}(V,V)$ and $s=\mathrm{dim}_k\mathrm{Ext}^2_{k[[G]]}(V,V)$ then $R(G,V)$ is isomorphic to a quotient algebra $W[[t_1,\ldots,t_r]]/J$ where $s$ is an upper bound on the minimal number of generators of $J$. \end{proposition}
The main idea of the proof is as follows: Let $\rho_u:G\to\mathrm{GL}_n(R(G,V))$ be a universal lift of $\overline{\rho}$. Let $\hat{R}=W[[t_1,\ldots,t_r]]$ and consider the sequence $$0\to J'=J/(J\mathfrak{m}_{\hat{R}})\to R'=\hat{R}/(J\mathfrak{m}_{\hat{R}}) \to R(G,V)\to 0.$$ Since $\mathfrak{m}_{R'}$ annihilates $J'$, we have an obstruction $[c]\in \mathrm{H}^2(G,\mathrm{Hom}_k(V,V))\otimes_kJ'$ to lifting $\rho_u$ to $R'$. This obstruction depends only on the strict equivalence class of $\rho_u$ and not on the chosen representation $\rho_u$. One then shows that the $k$-linear map \begin{eqnarray*} \mathrm{Hom}_k(J',k) &\to& \mathrm{H}^2(G,\mathrm{Hom}_k(V,V)) \cong \mathrm{Ext}^2_{k[[G]]}(V,V)\\ f &\mapsto& (1\otimes f)([c]) \end{eqnarray*} is injective, which implies the proposition.
\subsection{Deformation rings in number theory} \label{ss:explicitudr}
One of the main uses of deformation rings in number theory has been to establish a relationship between certain Galois representations and automorphic forms. More precisely, let $\overline{\rho}:G_\mathbb{Q}\to\mathrm{GL}_2(k)$ be an absolutely irreducible representation, where $k$ is a finite field of characteristic $p>2$ and $G_\mathbb{Q}$ is the absolute Galois group of $\mathbb{Q}$. Suppose that $\overline{\rho}$ is modular in the sense that it corresponds to a modular form (modulo $p$) which is an eigenfunction of Hecke operators. The idea is to prove that all reasonable lifts of $\overline{\rho}$ to $p$-adic representations are modular by establishing an isomorphism between a universal deformation ring, which parameterizes lifts of $\overline{\rho}$ with bounded ramification and satisfying appropriate deformation conditions, and a Hecke algebra, which parameterizes certain lifts of $\overline{\rho}$ which are modular of some fixed level.
Taylor and Wiles established such an isomorphism in \cite{wita}, which then led to the proof of Fermat's Last Theorem in \cite{wi}. The Taylor-Wiles method has been further refined by many people, such as Diamond \cite{diamond} and Fujiwara \cite{fujiwara}. In \cite{khare2003}, Khare gave an alternative approach for semistable $\overline{\rho}$ by establishing an isomorphism $R_Q\cong T_Q$, where $Q$ is a so-called auxiliary set of primes, $R_Q$ is the universal deformation ring for lifts of $\overline{\rho}$ minimally ramified away from $Q$ and satisfying appropriate deformation conditions and $T_Q$ is the analogous Hecke algebra.
Apart from the proof of Fermat's Last Theorem in \cite{wita,wi} and the proof of the general Shimura-Taniyama-Weil conjecture in \cite{bcdt}, deformation rings also played an important role in the proof of Serre's modularity conjecture \cite{serremod} by Khare-Wintenberger \cite{khawin} and Kisin \cite{kisin2}, which asserts that every absolutely irreducible representation $\overline{\rho}:G_\mathbb{Q}\to\mathrm{GL}_2(k)$ with odd determinant is modular.
Suppose $k$ is a finite field of characteristic $p>2$, $K$ is a number field, $S$ is a finite set of primes of $K$ containing the primes over $p$ and the infinite ones, and $\overline{\rho}:G_K\to\mathrm{GL}_2(k)$ is an absolutely irreducible representation unramified outside $S$. It is often desirable to have an explicit presentation of a universal deformation ring $R$, which parameterizes lifts of $\overline{\rho}$ satisfying certain deformation conditions, in terms of a power series algebra over $W=W(k)$ modulo an ideal given by a (minimal) number of generators. In the following, we describe several results of B\"ockle in this respect, some of which also played an important role in \cite{khawin}.
Since the relations occurring in the universal deformation ring $R$ often come from the obstructions of the associated local deformation problems $\overline{\rho}_{\mathfrak{p}}:G_{K_{\mathfrak{p}}}\to\mathrm{GL}_2(k)$ for $\mathfrak{p}\in S$, B\"ockle considered in \cite{bodemuskin} the problem of finding the universal deformation, or a smooth cover of it, in the local case where the relevant pro-$p$ group is an arbitrary Demu\v{s}kin group. He showed that the corresponding universal deformation ring is a complete intersection, flat over $W$, and with the (minimal) number of generators given by the $k$-dimension of $\mathrm{H}^2(G_{K_{\mathfrak{p}}},\mathrm{ad}_{\overline{\rho}})$. Moreover, he applied his local results to the global situation. For example, he gave conditions under which the universal deformation ring of an odd, absolutely irreducible representation $G_{\mathbb{Q}}\to \mathrm{GL}_2(k)$, unramified outside $S$, can be described explicitly, thus generalizing a result of Boston \cite{boston}.
In \cite{bockle,bopresent}, B\"ockle studied in more detail the connection between local and global deformation functors. In \cite{bockle}, he presented a rather general class of (global) deformation functors of $\overline{\rho}$ that satisfy local deformation conditions and investigated for those, under what conditions the global deformation functor is determined by the local deformation functors corresponding to primes $\mathfrak{p}\in S$. B\"ockle gave precise conditions under which the local functors are sufficient to describe the global functor. These conditions involve the vanishing of a second Shafarevich-Tate group and auxiliary primes as introduced by Taylor and Wiles in \cite{wita}. In \cite{bopresent}, B\"ockle provided generalizations and simplified proofs for some of the results in \cite{bockle}.
\section{Universal deformation rings of modules for finite groups} \label{s:udrfinite}
Assume the notation from subsection \ref{ss:deffunctor}. If $\mathrm{End}_{kG}(V)=k$, the construction of the universal deformation ring $R(G,V)$ by de Smit and Lenstra in \cite{desmitlenstra} shows that $R(G,V)$ is the inverse limit of the universal deformation rings $R(H,V)$ when $H$ ranges over all finite discrete quotients of $G$ through which the $G$-action on $V$ factors. Thus to answer questions about the ring structure of $R(G,V)$, it is natural to first consider the case when $G=H$ is finite.
For the remainder of this paper, we assume that $G$ is finite. The representation theory of $kG$ when $p$ divides $\# G$ is very beautiful but difficult. To avoid rationality questions and to simplify notation, we assume that $k$ is algebraically closed. More precisely, we make the following assumptions.
\begin{hypothesis} \label{hypo:finite} Let $k$ be an algebraically closed field of positive characteristic $p>0$, let $W=W(k)$ be the ring of infinite Witt vectors over $k$, let $G$ be a \textbf{finite} group, and let $V$ be a finitely generated $kG$-module. \end{hypothesis}
It follows as before that $V$ has a universal deformation ring if its endomorphism ring $\mathrm{End}_{kG}(V)$ is isomorphic to $k$. Note that $\mathrm{End}_{kG}(V)\cong \mathrm{H}^0(G,\mathrm{Hom}_k(V,V))$. When $G$ is finite, Tate cohomology groups often play an important role. Therefore, the question arises if we can use the 0-th Tate cohomology group, $$\hat{\mathrm{H}}^0(G,\mathrm{Hom}_k(V,V)) = \mathrm{End}_{kG}(V)/(s_G\cdot \mathrm{End}_k(V))$$ where $s_G=\sum_{g\in G} g$, to obtain a criterion for the existence of a universal deformation ring of $V$. Let $\mathrm{PEnd}_{kG}(V)$ denote the ideal of $\mathrm{End}_{kG}(V)$ consisting of all $kG$-module endomorphisms of $V$ factoring through a projective $kG$-module. Then $\mathrm{PEnd}_{kG}(V)$ is equal to $s_G\cdot \mathrm{End}_k(V)$, which implies that $\hat{\mathrm{H}}^0(G,\mathrm{Hom}_k(V,V)) = \mathrm{End}_{kG}(V)/\mathrm{PEnd}_{kG}(V)$. The quotient ring $\mathrm{End}_{kG}(V)/\mathrm{PEnd}_{kG}(V)$ is called the \emph{stable endomorphism ring of $V$} and is denoted by $\underline{\mathrm{End}}_{kG}(V)$. Note that in general $\mathrm{PEnd}_{kG}(V)\neq 0$, i.e. $\mathrm{End}_{kG}(V)$ properly surjects onto $\underline{\mathrm{End}}_{kG}(V)$. We have the following result:
\begin{proposition} {\rm (\cite[Prop. 2.1]{bc}, \cite[Rem. 2.1]{3quat})} \label{prop:blchin} Assume Hypothesis $\ref{hypo:finite}$ and that the stable endomorphism ring $\underline{\mathrm{End}}_{kG}(V)$ is isomorphic to $k$. Then $V$ has a universal deformation ring. Moreover, if $R$ is in $\hat{\mathcal{C}}$ and $(M,\phi)$ and $(M',\phi')$ are lifts of $~V$ over $R$ such that $M$ and $M'$ are isomorphic as $RG$-modules then $[M,\phi]=[M',\phi']$. \end{proposition}
The main point of the proof of this proposition is to show that if $(M,\phi)$ is a lift of $V$ over an Artinian object $R$ in $\hat{\mathcal{C}}$, then the ring homomorphism $R \to\underline{\mathrm{End}}_{RG}(M)$ coming from the action of $R$ on $M$ via scalar multiplication is surjective.
\begin{remark} \label{rem:nophi} The last statement of Proposition \ref{prop:blchin} means that the particular $kG$-module isomorphism $k\otimes_RM\xrightarrow{\phi} V$ from a lift $(M,\phi)$ is not important when $\underline{\mathrm{End}}_{kG}(V)\cong k$, which significantly simplifies computations.
Note that this is not true in general, as can be seen in the following example. Let $G=\langle \sigma\rangle$ be a cyclic group of order $p$ and let $V=k\oplus k$ with trivial $G$-action. Let $M$ be the $k[\epsilon]G$-module with $M=k[\epsilon]\oplus k[\epsilon]$ and $\sigma$ acting as $\left[\begin{array}{cc}1&\epsilon\\ 0&1\end{array}\right]$. Consider $\phi,\phi':k\otimes_{k[\epsilon]}M\to V$ with $\phi=\left[\begin{array}{cc}1&0\\ 0&1\end{array}\right]$ and $\phi'=\left[\begin{array}{cc}0&1\\1&0\end{array}\right]$. Then $(M,\phi)$ and $(M,\phi')$ are two non-isomorphic lifts of $V$ over $k[\epsilon]$. \end{remark}
The following result analyzes $R(G,V)$ further in the case when $\underline{\mathrm{End}}_{kG}(V)\cong k$. Here $\Omega$ denotes the syzygy functor or Heller operator, i.e. if $\pi:P_V\to V$ is a projective $k$G-module cover of $V$ then $\Omega(V)$ denotes the kernel of $\pi$ (see, for example, \cite[\S20]{alp}).
\begin{proposition} \label{prop:defhelp} {\rm (\cite[Cors. 2.5 and 2.8]{bc}).} Assume Hypothesis $\ref{hypo:finite}$ and that the stable endomorphism ring $\underline{\mathrm{End}}_{kG}(V)$ is isomorphic to $k$. \begin{enumerate} \item[(i)] Then $\underline{\mathrm{End}}_{kG}(\Omega(V))\cong k$, and $R(G,V)$ and $R(G,\Omega(V))$ are isomorphic. \item[(ii)] There is a non-projective indecomposable $kG$-module $V_0$ $($unique up to isomorphism$)$ such that $\underline{\mathrm{End}}_{kG}(V_0)\cong k$, $V$ is isomorphic to $V_0\oplus Q$ for some projective $kG$-module $Q$, and $R(G,V)$ and $R(G,V_0)$ are isomorphic. \end{enumerate} \end{proposition}
The main ideas of the proof are as follows: Since the deformation functor $F_V$ is continuous, most of the arguments can be carried out for the restriction of $F_V$ to the full subcategory $\mathcal{C}$ of $\hat{\mathcal{C}}$ of Artinian objects. For part (i), one shows that the syzygy functor $\Omega$ induces an isomorphism between the restrictions of the functors $F_V$ and $F_{\Omega(V)}$ to $\mathcal{C}$. For part (ii), one uses that the projective $kG$-module $Q$ can be lifted to a projective $RG$-module $Q_R$ for every $R$ in $\mathcal{C}$ to show that there is an isomorphism between the restrictions of the functors $F_V$ and $F_{V_0}$ to $\mathcal{C}$.
Recall that $kG$ can be written as a finite direct product of blocks $$kG=B_1\times \cdots \times B_r$$ where the blocks $B_1,\ldots,B_r$ are in one-to-one correspondence with the primitive central idempotents of $kG$. (For a good introduction to block theory, we refer the reader to \cite[Chap. IV]{alp}.) If $B$ is a block of $kG$, there is associated to it a conjugacy class of $p$-subgroups of $G$, called the defect groups of $B$. The defect groups measure how far $B$ is away from being a full matrix ring; they also determine the representation type of $B$. More precisely, $B$ has finite representation type if and only if its defect groups are cyclic; $B$ has infinite tame representation type if and only if $p=2$ and the defect groups of $B$ are dihedral, semi-dihedral or generalized quaternion; and $B$ has wild representation type in all other cases. (This result follows from \cite{hi,br,bd}. A description of this result together with an introduction to the representation type can also be found in \cite[Intro. and Sect. I.4]{erd}.)
Proposition \ref{prop:defhelp}(ii) says that if the stable endomorphism ring of $V$ is isomorphic to $k$ and we want to determine the universal deformation ring $R(G,V)$ then we may assume that $V$ is non-projective indecomposable. But then $V$ belongs to a unique block of $kG$, and we can use the theory of blocks, as introduced by Brauer and developed by many other authors, to determine the universal deformation ring $R(G,V)$.
\section{Brauer's generalized decomposition numbers and universal deformation rings} \label{s:brauer}
We continue to assume Hypothesis \ref{hypo:finite}. Our goal in this section is to show how Brauer's generalized decomposition numbers can be used in certain cases to determine the isomorphism type of the universal deformation ring $R(G,V)$. We first give a brief introduction to these generalized decomposition numbers.
\subsection{Brauer's generalized decomposition numbers} \label{ss:Brauergendec}
The usual decomposition numbers were introduced by Brauer and Nesbitt in \cite{brauernesbitt1937} (see also \cite{brauernesbitt}). They allow us to express the values of the ordinary irreducible characters of $G$ on $p$-regular elements of $G$, i.e. elements of order prime to $p$, by means of the absolutely irreducible $p$-modular characters of $G$. More precisely, if $\zeta_1, \zeta_2,\ldots$ are the ordinary irreducible characters of $G$ and $\varphi_1,\varphi_2,\ldots$ are the absolutely irreducible $p$-modular characters of $G$, then we have a formula \begin{equation} \label{eq:usualdec} \zeta_\mu(g) = \sum_\nu d_{\mu\nu}\varphi_\nu(g) \end{equation} provided $g$ is a $p$-regular element of $G$. The $d_{\mu\nu}$ are non-negative integers, called the \emph{decomposition numbers of $G$ for $p$}. As Brauer wrote in \cite[p. 192]{brauerordinaryandmodular}:
\smallbreak {``We may say that the group characters $\zeta_\mu$ of $G$ are built up by the modular characters $\varphi_\nu$, and it is possible to obtain a deeper insight into the nature of the ordinary group characters by the use of the modular characters and their properties. However, it is disturbing that we have to restrict ourselves to $p$-regular elements.''} \smallbreak
For this reason, Brauer introduced generalized decomposition numbers in \cite{brauerordinaryandmodular}. The value $\zeta_\mu(g)$ on an element $g\in G$ whose order is divisible by $p$ is then expressed by means of the absolutely irreducible $p$-modular characters of certain subgroups $C_i$ of $G$. The corresponding generalized decomposition numbers $d^i_{\mu\nu}$ are not necessarily rational integers, but they are algebraic integers in a cyclotomic field of $p$-power order roots of unity. More precisely, Brauer defined $d^i_{\mu\nu}$ as follows.
Suppose $\# G=p^a m'$ where $m'$ is relatively prime to $p$, and let $P$ be a fixed Sylow $p$-subgroup of $G$. Let $u_0 = 1, u_1, u_2, \ldots,u_h$ be a complete system of representatives of $G$-conjugacy classes of $p$-power order elements in $G$ with $u_i\in P$ for all $1\le i\le h$. Every conjugacy class of $G$ contains an element of the form
$u_i v$ where $i\in\{0,1,\ldots, h\}$ is uniquely determined by the class and $v$ is a $p$-regular element in the centralizer $C_G(u_i)$. For each $0\le i\le h$, let $v_{i,1},\ldots,v_{i,\ell_i}$ be a complete system of representatives of $C_G(u_i)$-conjugacy classes of $p$-regular elements in $C_G(u_i)$ with $v_{i,1}=1$. Then $\{u_iv_{i,j}\;|\; 0\le i\le h,1\le j\le \ell_i\}$ is a complete set of representatives of the conjugacy classes of $G$. Moreover, for each $0\le i\le h$, there are precisely $\ell_i$ absolutely irreducible $p$-modular characters of $C_G(u_i)$, which we denote by $\varphi_1^i,\ldots,\varphi_{\ell_i}^i$. As before, let $\zeta_1,\zeta_2,\ldots$ be the ordinary irreducible characters of $G$. Then \begin{equation} \label{eq:brauer1} \zeta_\mu(u_iv_{i,j})= \sum_{\nu=1}^{\ell_i} d_{\mu\nu}^i \,\varphi_\nu^i(v_{i,j}) \end{equation} for all $0\le i\le h$, $1\le j\le \ell_i$. The $d_{\mu\nu}^i$ are called the \emph{generalized decomposition numbers of $G$}. For $i = 0$, we have $u_0=1$ and $C_G(u_0)=G$, and the $d_{\mu\nu}^i$ coincide with the usual decomposition numbers $d_{\mu\nu}$ of $G$ in Equation (\ref{eq:usualdec}). In general, $d_{\mu\nu}^i$ is an algebraic integer in the field of the $p^{\alpha_i}$-th roots of unity where $p^{\alpha_i}$ is the order of $u_i$. In particular, $d_{\mu\nu}^i$ can be viewed to belong to $W[\omega_i]$ if $\omega_i$ is a primitive $p^{\alpha_i}$-th root of unity. In \cite[Sect. 6]{brauerdarst2}, Brauer moreover showed that if $\zeta_\mu$ belongs to the block $B$ of $kG$, then the generalized decomposition number $d_{\mu\nu}^i$ vanishes if $\varphi_\nu^i$ belongs to a block of $kC_G(u_i)$ whose Brauer correspondent in $G$ is not equal to $B$.
\subsection{Universal deformation rings of certain modules belonging to infinite tame blocks} \label{ss:udrtame}
We now focus on a certain class of modules belonging to blocks of infinite tame representation type for which Brauer's generalized decomposition numbers can be used to determine their universal deformation rings. This subsection is based on the paper \cite{brauerpaper}, and details can be found there. Recall from section \ref{s:udrfinite} that a block $B$ of $kG$ has infinite tame representation type if and only if $p=2$ and the defect groups of $B$ are either dihedral, or semi-dihedral, or generalized quaternion.
In \cite{brIV,brauer2,olsson}, Brauer and Olsson determined the generalized decomposition numbers for all the ordinary irreducible characters belonging to infinite tame blocks. Moreover, they proved that an infinite tame block has at most three isomorphism classes of simple modules. In \cite{erd}, Erdmann classified all infinite tame blocks up to Morita equivalence by providing a list of quivers and relations for their basic algebras.
We make the following assumptions.
\begin{hypothesis} \label{hyp:alltheway} Assume Hypothesis $\ref{hypo:finite}$. Additionally, assume that $p=2$, $V$ is indecomposable with $\underline{\mathrm{End}}_{kG}(V)\cong k$, and that $V$ belongs to a non-local block $B$ of $kG$ of infinite tame representation type with a defect group $D$ of order $2^n$. Let $F$ be the fraction field of $W$, and let $\overline{F}$ be a fixed algebraic closure of $F$. \end{hypothesis}
We want to concentrate on those $V$ for which Brauer's generalized decomposition numbers carry the most information. More precisely, we call a module $V$ as in Hypothesis \ref{hyp:alltheway} \emph{maximally ordinary} if the $2$-modular character of $V$ is the restriction to the $2$-regular conjugacy classes of an ordinary irreducible character $\chi$ such that for every $\sigma\in D$ of maximal $2$-power order, Brauer's generalized decomposition numbers corresponding to $\sigma$ and $\chi$ do not all lie in $\{0,\pm 1\}$. In other words, using the notation of Equation (\ref{eq:brauer1}), if $\chi=\zeta_\mu$ and $\sigma$ is conjugate in $G$ to $u_i$, then there exists an absolutely irreducible $2$-modular character $\varphi_\nu^i$ of $C_G(u_i)$ such that $d_{\mu\nu}^i \not\in\{0,\pm 1\}$.
By \cite{brauer2,olsson}, there are precisely $2^{n-2}-1$ ordinary irreducible characters of height 1 belonging to $B$ if $n\ge 4$. Moreover, they all define the same $2$-modular character when they are restricted to the $2$-regular conjugacy classes. If $n=3$, then there are either 1 or 3 ordinary irreducible characters of height 1 belonging to $B$, depending on whether $D$ is dihedral or quaternion. If $n=2$, then there are no ordinary irreducible characters of height 1 belonging to $B$. Recall that the height of an ordinary irreducible character $\chi$ belonging to $B$ is $b-a+n$, where $2^a$ (resp. $2^b$) is the maximal $2$-power dividing $\# G$ (resp. $\mathrm{deg}(\chi)$). Since $n$ is the defect of the block $B$, it follows that $b-a+n$ is a non-negative integer (see, for example, \cite[Sect. 56.E and Cor. (57.19)]{CR}).
Suppose $n\ge 4$. By \cite{brauer2,olsson}, exactly one of the $2^{n-2}-1$ ordinary irreducible characters of height 1 belonging to $B$ is realizable over $F$, i.e. it corresponds to an absolutely irreducible $FG$-module. Moreover, the remaining $2^{n-2}-2$ characters of height 1 are precisely the ordinary irreducible characters belonging to $B$ for which the generalized decomposition numbers corresponding to maximal $2$-power order elements in $D$ do not all lie in $\{0,\pm 1\}$. We have the following result:
\begin{theorem} {\rm (\cite[Thm. 1.1 and Cor. 6.2]{brauerpaper}).} \label{thm:supermain} Assume Hypothesis $\ref{hyp:alltheway}$. Then $V$ is maximally ordinary if and only if $n\ge 4$ and the $2$-modular character of $V$ is equal to the restriction to the $2$-regular conjugacy classes of an ordinary irreducible character of $G$ of height $1$. Suppose $V$ is maximally ordinary. There exists a monic polynomial $q_n(t)\in W[t]$ of degree $2^{n-2}-1$ which depends on $D$ but not on $V$ and which can be given explicitly such that either \begin{enumerate} \item[(i)] $R(G,V)/2R(G,V)\cong k[[t]]/(t^{2^{n-2}-1})$, in which case $R(G,V)$ is isomorphic to $W[[t]]/(q_n(t))$, or \item[(ii)] $R(G,V)/2R(G,V)\cong k[[t]]/(t^{2^{n-2}})$, in which case $R(G,V)$ is isomorphic to $W[[t]]/(t\,q_n(t),2\,q_n(t))$. \end{enumerate} In all cases, the ring $R(G,V)$ is isomorphic to a subquotient ring of $WD$, and it is a complete intersection if and only if we are in case $(i)$. \end{theorem}
A precise description of the maximally ordinary modules $V$ belonging to $B$ is given in \cite[Lemma 6.1 and Cor. 6.2]{brauerpaper}. A formula for the polynomials $q_n(t)$ can be found in \cite[Def. 5.3 and Rem. 5.4]{brauerpaper}.
We now discuss the main ideas of the proof of Theorem \ref{thm:supermain}. For details we refer the reader to \cite{brauerpaper}. The first statement of the theorem follows from the results in \cite{brIV,brauer2,olsson}. Suppose now that $n\ge 4$. As noted above, there are then precisely $2^{n-2}-1$ ordinary irreducible characters of height 1 belonging to $B$. Moreover, these characters fall into $n-2$ Galois orbits under the action of $\mathrm{Gal}(\overline{F}/F)$: $$\mathcal{F}_0,\mathcal{F}_1,\ldots,\mathcal{F}_{n-3}$$ where $\#\mathcal{F}_j=2^j$ for $0\le j\le n-3$. If $\xi$ is the ordinary character which is the sum of all the characters of height 1, then $\xi$ can be realized by an $FG$-module $$X=X_0\oplus X_1\oplus \cdots \oplus X_{n-3}$$ where each $X_j$ is a simple $FG$-module with Schur index 1 corresponding to the orbit $\mathcal{O}_j$.
The main steps to prove Theorem \ref{thm:supermain} are as follows: Suppose $V$ is maximally ordinary. First, we use Erdmann's description of the basic algebra of $B$ to show that $R(G,V)$ is a quotient algebra of $W[[t]]$ and that $V$ can be lifted to $k[[t]]/(t^{n-2}-1)$. Moreover, we show that this lift is given by an indecomposable $B$-module $\overline{U'}$ of $V$ such that $\mathrm{End}_{kG}(\overline{U'})\cong k[[t]]/(t^{n-2}-1)$. Next, we use the usual decomposition numbers, together with the description of the projective indecomposable $B$-modules and \cite[Prop. (23.7)]{CR} and \cite[Lemma 2.3.2]{3sim} to show that $\overline{U'}$ can be lifted to $W$. Moreover, we show that this lift is given by an indecomposable $WG$-module $U'$ which is free over $W$ with $F\otimes_WU'\cong X$. Then we use Brauer's generalized decomposition numbers to show that $\mathrm{End}_{WG}(U')\cong W[[t]]/(q_n(t))$ and that $U'$ is free as a module for $\mathrm{End}_{WG}(U')$. This then implies that $W[[t]]/(q_n(t))$ is a quotient ring of the universal deformation ring $R(G,V)$.
To complete the proof of Theorem \ref{thm:supermain}, we use again Erdmann's description of the basic algebra of $B$ to determine the universal mod 2 deformation ring $R(G,V)/2 R(G,V)$. It follows that the isomorphism type of $R(G,V)/2 R(G,V)$ depends on whether or not the stable Auslander-Reiten quiver $\Gamma_s(B)$ of $B$ contains 3-tubes. Note that if $D$ is dihedral then $\Gamma_s(B)$ always contains 3-tubes, whereas if $D$ is generalized quaternion then $\Gamma_s(B)$ never contains 3-tubes, and if $D$ is semi-dihedral then $\Gamma_s(B)$ may or may not contain 3-tubes. We show that $R(G,V)/2R(G,V)\cong k[[t]]/(t^{n-2}-1)$ if $\Gamma_s(B)$ contains no 3-tubes, and that $R(G,V)/2R(G,V)\cong k[[t]]/(t^{n-2})$ if $\Gamma_s(B)$ does contain 3-tubes. In the first case, the universal deformation ring of $V$ is $R(G,V)\cong W[[t]]/(q_n(t))$, whereas in the second case we use \cite[Lemma 2.3.3]{3sim} to show that $R(G,V)\cong W[[t]]/(t\,q_n(t),2\,q_n(t))$.
\end{document}
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\begin{document}
\title{Three-party quantum private comparison of equality based on genuinely maximally entangled six-qubit states} \author{Cai Zhang \and Zhiwei Sun \and Xiang Huang \and Dongyang Long}
\institute{C. Zhang \and X. Huang \and D. Long \at
School of Information Science and Technology, Sun Yat-sen University, Guangzhou, Guangdong, 510006, China \\
\email{[email protected] \\
Z. Sun \at
College of Information Engineering, Shenzhen University, Shenzhen, Guangdong, 518060, China
} }
\maketitle
\begin{abstract} We propose a new three-party quantum private comparison protocol using genuinely maximally entangled six-qubit states. In our protocol, three participants can determine whether their private information are equal or not without an external third party who helps compute the comparison result. At the same time the participants can preserve the privacy of their inputs, respectively. Our protocol does not need any unitary operations to encode information due to the excellent properties of genuinely maximally entangled six-qubit states. Additionally, the protocol uses one-step quantum transmission and it is congenitally free from Trojan horse attacks. We have also shown that our protocol is secure against outside and participant attacks in this paper. \keywords{Three-party quantum private comparison\and Genuinely maximally entangled six-qubit state \and Trojan horse attack}
\end{abstract}
\section{Introduction} \label{intro} The first quantum key distribution (QKD) protocol was proposed by Bennett and Brassard\cite{BB84} in 1984. After that, an increasing number of quantum cryptographic protocols such as quantum secret sharing (QSS) \cite{HBB99,KKI99,KB02,ZL07,JWGQG12,S12,long2012quantum}, quantum secure direct communication (QSDC) \cite{BF02,DLL03,GYW05,LWGZ08,LCJ12,sun2012quantum,sun2012quantum1}, quantum key agreement (QKA) \cite{zhou2004quantum,chong2010quantum,liu2013multiparty,sun2013improvements}, quantum summation \cite{H02,HN03,DCWZ07,CXYW10,zhang2014high} and quantum private comparison (QPC) \cite{yang2009efficient,chen2010efficient,tseng2012new} have been presented. Quantum private comparison, as a subfield of quantum cryptography, has attracted more and more researchers. The aim of QPCs is to compare the participants' private information without publicly revealing their respective private information.\par Since Yao \cite{yao1982protocols} presented a protocol for the millionaires' problem in which the participants try to determine which one is richer without revealing their actual wealth, the protocols of private comparison have widely been investigated. Boudot et al. \cite{boudot2001fair} proposed a protocol to decide whether two millionaires are equally rich or not. But Lo \cite{Lo97} showed that the equality function cannot be securely evaluated in a two-party scenario. Thus, some additional assumptions such as a semi-honest third party are required to achieve the goal of private comparison.\par Yang et al. \cite{yang2009efficient} proposed the first quantum private comparison protocol in which the entanglement of Einstein-Podolsky-Rosen (EPR) pairs and the one-way hash function are employed. Chen et al. \cite{chen2010efficient} presented an efficient protocol of equality using triplet states. Ref.\cite{tseng2012new} gave a more efficient quantum private comparison of equality protocol without the entanglement of EPR pairs. Liu et al. \cite{liu2011efficient,liu2012protocol,liu2012new,liu2012quantum} presented QPC protocols employing the triplet W states, $\chi$-type genuine four-particle entangled states and the Bell states. Huang et al. \cite{huang2013robust} designed a quantum private comparison of equality protocol with collective detection-noise channels. Liu et al. \cite{liu2013efficient} employed single photons and collective detection to devise an efficient quantum private comparison protocol. Zhang et al. \cite{zhang2013quantum} proposed a quantum private comparison protocol based on an quantum search algorithm. Li et al. \cite{li2014efficient} presented an efficient protocol for equal information comparison based on four-particle entangled W state and Bell entangled states swapping. All the above protocol works for two-party who wish to compare their private information. Recently, some multi-user quantum private comparison protocols were presented. Chang et al. \cite{chang2013multi} gave a multi-user private comparison protocol using GHZ class states. Liu et al. \cite{liu2013multi} presented a multi-party quantum private comparison protocol using $d$-dimensional basis states without entanglement swapping. \par All the above protocols include a third party who helps participants compute comparison results. Lin et al. \cite{lin2014quantum} used EPR pairs and a one-way hash function to design a quantum private comparison of equality protocol without a third party that works for the two-party scenario. If the number of participants exceeds 2, can one find a protocol for quantum private comparison without an external third party who helps compute the comparison result? We find such a protocol for three-party quantum private comparison. The protocol is based on genuinely maximally entangled six-qubit states (we name it BPB state). In our protocol, three participants can determine whether their private information are equal without the external third party, and meanwhile preserve the privacy of their inputs, respectively. Any unitary operations are not required to encode information due to the excellent properties of genuinely maximally entangled six-qubit states. Furthermore, the protocol employs one-step quantum transmission, hence it will not suffer Trojan horse attacks. Our protocol is also proven to be secure against various attacks including outside and participant attacks.\par The rest of this paper is organized as follows. In Section 2, we analyze the structure of the genuinely maximally BPB state and show the excellent properties which are useful for designing our protocol. In Section 3, we propose a protocol of quantum private comparison based on genuinely maximally BPB states. In Section 4, we analyze the correctness and the security of the presented protocol. Finally, we make a conclusion in Section 5.
\section{The genuinely maximally entangled six-qubit state}
Quantum entanglement, as a physical resource, plays a key role in many applications such as quantum teleportation \cite{bennett1993teleporting}, quantum dense coding \cite{bennett1992communication}, quantum key distribution \cite{ekert1991quantum}, quantum secret sharing \cite{cleve1999share} and quantum secure direct communication \cite{sun2012quantum,sun2012quantum1}. \par By using a numeric searching program, Borras et al. \cite{borras2007multiqubit} found the genuinely maximally BPB state, which is \begin{eqnarray}
\frac{1}{\sqrt{32}} [ (|000 000\rangle &+& |111 111\rangle + |000011\rangle + |111100\rangle \nonumber\\
&+& |000101\rangle + |111010\rangle + |000110\rangle + |111001\rangle \nonumber\\
&+& |001001\rangle + |110110\rangle + |001111\rangle + |110000\rangle \nonumber\\
&+& |010001\rangle + |101110\rangle + |010010\rangle + |101101\rangle \nonumber\\
&+& |011000\rangle + |100111\rangle + |011101\rangle + |100010\rangle) \nonumber\\
&-&( |010100\rangle + |101011\rangle + |010111\rangle + |101000\rangle \nonumber\\
&+& |011011\rangle + |100100\rangle + |001010\rangle + |110101\rangle \nonumber\\
&+& |001100\rangle + |110011\rangle + |011110\rangle + |100001\rangle)]_{123456}. \end{eqnarray}
We denote this six-qubit state by $\Psi_{6qb}$. From the above formula, we can see that $\Psi_{6qb}$ includes 32 terms, each of which has even $|0\rangle$ and equal coefficient.\par
To show the engtangled property of $\Psi_{6qb}$, we can rewrite it as
\begin{eqnarray}\label{first} \Psi_{6qb} =
\frac{1}{\sqrt{8}} [|000\rangle|\gamma_{1}^{1}\rangle &+& |001\rangle|\gamma_{2}^{1}\rangle + |010\rangle|\gamma_{3}^{1}\rangle + |011\rangle|\gamma_{4}^{1}\rangle \nonumber\\
&-& |100\rangle|\gamma_{5}^{1}\rangle - |101\rangle|\gamma_{6}^{1}\rangle + |110\rangle|\gamma_{7}^{1}\rangle \nonumber \\
&+& |111\rangle|\gamma_{8}^{1}\rangle
]_{123456}, \end{eqnarray}
where $\{|\gamma_{j}^{1}\rangle|j=1,2,\ldots,8\}$ forms an orthogonal basis on Hilbert space $C_{2} \otimes C_{2} \otimes C_{2}$ such that \begin{eqnarray}
|\gamma_{1}^{1}\rangle = \frac{1}{\sqrt{2}}(|0\rangle|\Phi^{+}\rangle + |1\rangle|\Psi^{+}\rangle), \
|\gamma_{2}^{1}\rangle = \frac{1}{\sqrt{2}}(|0\rangle|\Psi^{-}\rangle - |1\rangle|\Phi^{-}\rangle), \nonumber \\
|\gamma_{3}^{1}\rangle = \frac{1}{\sqrt{2}}(|0\rangle|\Psi^{+}\rangle - |1\rangle|\Phi^{+}\rangle), \
|\gamma_{4}^{1}\rangle = \frac{1}{\sqrt{2}}(|0\rangle|\Phi^{-}\rangle + |1\rangle|\Psi^{-}\rangle), \nonumber \\
|\gamma_{5}^{1}\rangle = \frac{1}{\sqrt{2}}(|0\rangle|\Psi^{-}\rangle + |1\rangle|\Phi^{-}\rangle), \
|\gamma_{6}^{1}\rangle = \frac{1}{\sqrt{2}}(|0\rangle|\Phi^{+}\rangle - |1\rangle|\Psi^{+}\rangle), \nonumber \\
|\gamma_{7}^{1}\rangle = \frac{1}{\sqrt{2}}(|0\rangle|\Phi^{-}\rangle - |1\rangle|\Psi^{-}\rangle), \
|\gamma_{8}^{1}\rangle = \frac{1}{\sqrt{2}}(|0\rangle|\Psi^{+}\rangle + |1\rangle|\Phi^{+}\rangle), \end{eqnarray}
and $|\Phi^{\pm}\rangle$ and $|\Psi^{\pm}\rangle$ are Bell states in the form of \begin{eqnarray}\label{BellStates}
|\Phi^{+}\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle) = \frac{1}{\sqrt{2}}(|++\rangle + |--\rangle), \label{firstBell} \\
|\Phi^{-}\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle) = \frac{1}{\sqrt{2}}(|+-\rangle + |-+\rangle),\\
|\Psi^{+}\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle) = \frac{1}{\sqrt{2}}(|++\rangle - |--\rangle),\\
|\Psi^{-}\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle) = \frac{1}{\sqrt{2}}(|-+\rangle - |+-\rangle). \label{lastBell} \end{eqnarray}
Here, $|+\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$ and $|-\rangle=\frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)$ are the eigenvectors of the Pauli operator $\sigma_{X}$.\par
We can also rewrite $\Psi_{6qb}$ as \begin{eqnarray}\label{second} \Psi_{6qb}=
\frac{1}{\sqrt{8}} [|+++\rangle|\gamma_{1}^{2}\rangle &+& |++-\rangle|\gamma_{2}^{2}\rangle + |+-+\rangle|\gamma_{3}^{2}\rangle + |+--\rangle|\gamma_{4}^{2}\rangle \nonumber\\
&+& |-++\rangle|\gamma_{5}^{2}\rangle + |-+-\rangle|\gamma_{6}^{2}\rangle + |--+\rangle|\gamma_{7}^{2}\rangle \nonumber\\
&+& |---\rangle|\gamma_{8}^{2}\rangle
]_{123456}, \end{eqnarray}
where $\{|\gamma_{j}^{2}\rangle|j=1,2,\ldots,8\}$ also forms an orthogonal basis on Hilber space $C_{2} \otimes C_{2} \otimes C_{2}$ such that \begin{eqnarray}
|\gamma_{1}^{2}\rangle = \frac{1}{\sqrt{2}}(|+\rangle|\Psi^{+}\rangle + |-\rangle|\Phi^{-}\rangle), \
|\gamma_{2}^{2}\rangle = \frac{1}{\sqrt{2}}(-|+\rangle|\Psi^{-}\rangle + |-\rangle|\Phi^{+}\rangle), \nonumber \\
|\gamma_{3}^{2}\rangle = \frac{1}{\sqrt{2}}(-|+\rangle|\Phi^{-}\rangle - |-\rangle|\Psi^{+}\rangle), \
|\gamma_{4}^{2}\rangle = \frac{1}{\sqrt{2}}(|+\rangle|\Phi^{+}\rangle - |-\rangle|\Psi^{-}\rangle), \nonumber \\
|\gamma_{5}^{2}\rangle = \frac{1}{\sqrt{2}}(|+\rangle|\Psi^{-}\rangle + |-\rangle|\Phi^{+}\rangle), \
|\gamma_{6}^{2}\rangle = \frac{1}{\sqrt{2}}(|+\rangle|\Psi^{+}\rangle - |-\rangle|\Phi^{-}\rangle), \nonumber \\
|\gamma_{7}^{2}\rangle = \frac{1}{\sqrt{2}}(|+\rangle|\Phi^{+}\rangle + |-\rangle|\Psi^{-}\rangle), \
|\gamma_{8}^{2}\rangle = \frac{1}{\sqrt{2}}(|+\rangle|\Phi^{-}\rangle - |-\rangle|\Psi^{+}\rangle), \end{eqnarray}
where $|\Phi^{\pm}\rangle$ and $|\Psi^{\pm}\rangle$ are Bell states as Eqs.(\ref{firstBell}-\ref{lastBell}) and $|+\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$ and $|-\rangle=\frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)$ are the eigenvectors of the Pauli operator $\sigma_{X}$.\par We will use Eq. (\ref{first}) and Eq. (\ref{second}) in our protocol to check whether the participant who distributes the state $\Psi_{6qb}$ is honest or not. \par Let us further investigate the properties of the state $\Psi_{6qb}$.
\begin{eqnarray}
\Psi_{6qb}&=&\frac{1}{2}(|\Phi^{+}\rangle_{12}|\Phi^{+}\rangle_{36}|\Phi^{+}\rangle_{45}
+|\Phi^{-}\rangle_{12}|\Psi^{-}\rangle_{36}|\Psi^{+}\rangle_{45} \nonumber\\
&+&|\Psi^{-}\rangle_{12}|\Psi^{+}\rangle_{36}|\Phi^{-}\rangle_{45}
+|\Psi^{+}\rangle_{12}|\Phi^{-}\rangle_{36}|\Psi^{-}\rangle_{45}) \label{twoSplit1}\\
&=&\frac{1}{2}(-|\Phi^{-}\rangle_{13}|\Phi^{-}\rangle_{24}|\Phi^{+}\rangle_{56}
+|\Phi^{+}\rangle_{13}|\Psi^{+}\rangle_{24}|\Psi^{+}\rangle_{56} \nonumber\\
&-&|\Psi^{+}\rangle_{13}|\Psi^{-}\rangle_{24}|\Phi^{-}\rangle_{56}
-|\Psi^{-}\rangle_{13}|\Phi^{+}\rangle_{24}|\Psi^{-}\rangle_{56}) \\
&=&\frac{1}{2}(|\Phi^{-}\rangle_{14}|\Phi^{+}\rangle_{26}|\Phi^{-}\rangle_{35}
+|\Phi^{+}\rangle_{14}|\Psi^{+}\rangle_{26}|\Psi^{+}\rangle_{35} \nonumber\\
&+&|\Psi^{-}\rangle_{14}|\Psi^{-}\rangle_{26}|\Phi^{+}\rangle_{35}
+|\Psi^{+}\rangle_{14}|\Phi^{-}\rangle_{26}|\Psi^{-}\rangle_{35}) \\
&=&\frac{1}{2}(|\Phi^{+}\rangle_{15}|\Phi^{+}\rangle_{23}|\Phi^{+}\rangle_{46}
+|\Phi^{-}\rangle_{15}|\Psi^{+}\rangle_{23}|\Psi^{-}\rangle_{46} \nonumber\\
&+&|\Psi^{+}\rangle_{15}|\Psi^{-}\rangle_{23}|\Phi^{-}\rangle_{46}
+|\Psi^{-}\rangle_{15}|\Phi^{-}\rangle_{23}|\Psi^{+}\rangle_{46}) \\
&=&\frac{1}{2}(|\Phi^{-}\rangle_{16}|\Phi^{+}\rangle_{25}|\Phi^{-}\rangle_{34}
+|\Phi^{+}\rangle_{16}|\Psi^{-}\rangle_{25}|\Psi^{-}\rangle_{34} \nonumber\\
&+&|\Psi^{+}\rangle_{16}|\Psi^{+}\rangle_{25}|\Phi^{+}\rangle_{34}
+|\Psi^{-}\rangle_{16}|\Phi^{-}\rangle_{25}|\Psi^{+}\rangle_{34}). \label{twoSplit5}
\end{eqnarray}
From the above Eqs.(\ref{twoSplit1}-\ref{twoSplit5}), it is obvious to see that the other four qubits will collapse to the tensor product of two pairs of EPR when any two qubits of $\Psi_{6qb}$ are measured with the Bell Basis $\{|\Phi^{+}\rangle,|\Phi^{-}\rangle,|\Psi^{+}\rangle,|\Psi^{-}\rangle\}$. However, these two-split forms of $\Psi_{6qb}$ are not suitable for our task. We should rewrite $\Psi_{6qb}$ as
\begin{eqnarray}\label{beUsed}
\Psi_{6qb}=\frac{1}{4}[
|\Phi^{+}\rangle_{12}(&|\Phi^{+}\rangle_{34}&|\Phi^{+}\rangle_{65}
+|\Phi^{-}\rangle_{34}|\Phi^{-}\rangle_{65} \nonumber\\
+&|\Psi^{+}\rangle_{34}&|\Psi^{+}\rangle_{65}
+|\Psi^{-}\rangle_{34}|\Psi^{-}\rangle_{65}) \nonumber\\
+|\Phi^{-}\rangle_{12}(-&|\Phi^{+}\rangle_{34}&|\Phi^{-}\rangle_{65}
+|\Phi^{-}\rangle_{34}|\Phi^{+}\rangle_{65} \nonumber\\
-&|\Psi^{+}\rangle_{34}&|\Psi^{-}\rangle_{65}
+|\Psi^{-}\rangle_{34}|\Psi^{+}\rangle_{65}) \nonumber\\
+|\Psi^{+}\rangle_{12}(&|\Phi^{+}\rangle_{34}&|\Psi^{+}\rangle_{65}
+|\Phi^{-}\rangle_{34}|\Psi^{-}\rangle_{65} \nonumber\\
-&|\Psi^{+}\rangle_{34}&|\Phi^{+}\rangle_{65}
-|\Psi^{-}\rangle_{34}|\Phi^{-}\rangle_{65}) \nonumber\\
+|\Psi^{-}\rangle_{12}(-&|\Phi^{+}\rangle_{34}&|\Psi^{-}\rangle_{65}
+|\Phi^{-}\rangle_{34}|\Psi^{+}\rangle_{65} \nonumber\\
+&|\Psi^{+}\rangle_{34}&|\Phi^{-}\rangle_{65}
-|\Psi^{-}\rangle_{34}|\Phi^{+}\rangle_{65})].
\end{eqnarray}
Let us agree on the following encoding: \begin{eqnarray}
&|\Phi^{+}\rangle& \rightarrow 00, \quad |\Phi^{-}\rangle \rightarrow 01, \nonumber \\
&|\Psi^{+}\rangle& \rightarrow 10, \quad |\Psi^{-}\rangle \rightarrow 11. \label{bEncoding} \end{eqnarray}
We denote the encoding of $x$ as $Encod(x)$ where $x \in \{ |\Phi^{+}\rangle, |\Phi^{-}\rangle, |\Psi^{+}\rangle, |\Psi^{-}\rangle \}$. For example, $Encod(|\Phi^{-}\rangle)=01$. We can let $Encod(x)=Encod(-x)$ because the measurement outcome of $-x$ will be $x$ with certainty if it is measured with Bell basis. Actually, we can say that $x$ and $-x$ are the same up to a global phase factor $-1$. \par After the above encoding Eq. (\ref{bEncoding}) , the Eq. (\ref{beUsed}) tells us that if we measure particles (1, 2) , particles (3, 4) and particles (6, 5) with Bell basis, respectively, then the responding measurement outcomes $R_{12}$, $R_{34}$ and $R_{65}$ satisfy the following equation: \begin{equation}\label{zero}
Encod(R_{12}) \oplus Encod(R_{34}) \oplus Encod(R_{65})=00. \end{equation} The Eqs.(\ref{beUsed}-\ref{zero}) allow us to design a three-party quantum comparison protocol in which three participants can determine whether their private information are equal or not without the help of an external third party and keep their inputs secret, respectively.
\section{The three-party quantum private comparison protocol} In our protocol, we assume that the classical and quantum channels are authenticated. Suppose that three participants $P_{1}$, $P_{2}$ and $P_{3}$ have private information (secret bit strings) $M_{1}$, $M_{2}$ and $M_{3}$, respectively. They wish to determine whether $M_{1}=M_{2}=M_{3}$ or not and preserve the privacy of their information, respectively. The length of secret bit string is $L$. We assume that the participant $P_{1}$ prepares the genuinely maximally BPB states $\Psi_{6qb}$, then the process of the multi-party quantum private comparison protocol can be described as follows.
\begin{enumerate}[{(S1)}]
\item $P_{1}$ first prepares $(\lceil \frac{L}{2} \rceil + \delta)$ ($\lceil \quad \rceil$ denotes the ceiling function) genuinely maximally BPB states $\Psi_{6qb}$. Then he picks up the particles (3, 4) (particles (6, 5)) from each $\Psi_{6qb}$ to form an ordered sequence $S_{34}$ ($S_{65}$). After that $P_{1}$ prepares $d$ decoy particles, each of which is in one of the quantum states $\{ |0\rangle, |1\rangle, |+\rangle, |-\rangle \}$. He then randomly inserts the $d$ decoy particles into the sequence $S_{34}$ ($S_{65}$) to form a new sequence $S_{34}^{*}$ ($S_{65}^{*}$). Note that anyone does not know the initial states and positions of the $d$ decoy particles except $P_{1}$. At last, $P_{1}$ transmits $S_{34}^{*}$ ($S_{65}^{*}$) to participant $P_{2}$ ($P_{3}$) and keeps the ordered sequence $S_{12}$ of particles (1, 2) from each $\Psi_{6qb}$ in his lab.
\item \label{outCheck}Confirming that participant $P_{2}$ ($P_{3}$) has received all the particles $S_{34}^{*}$ ($S_{65}^{*}$) sent by $P_{1}$. $P_{1}$ announces the positions and the bases of the decoy particles to $P_{2}$ ($P_{3}$). In the following, participant $P_{2}$ ($P_{3}$) measures the decoy particles with one of the two bases $\{|0\rangle, |1\rangle\}$ and $\{|+\rangle, |-\rangle\}$ according to $P_{1}$'s announced information. And then $P_{2}$ ($P_{3}$) publishes his measurement outcomes. Later, $P_{1}$ can determine the error rate according to the $d$ decoy particles' initial states. If the error rate exceeds the threshold, then this protocol will be aborted and repeat the step ($S1$). Otherwise, the protocol will go to the next step.
\item \label{step:colCheck}$P_{2}$ and $P_{3}$ collaborate to check whether $P_{1}$ distributes the intended particles to them. Namely, $P_{2}$ ($P_{3}$) should receive the ordered sequence $S_{34}$ ($S_{65}$) of particles (3, 4) (particles (6, 5)). First, $P_{2}$ ($P_{3}$) removes the decoy particles from $S_{34}^{*}$ ($S_{65}^{*}$) to get $S_{34}$ ($S_{65}$). They then randomly choose $\delta$ genuinely maximally BPB states $\Psi_{6qb}$ (we call them sample states) for checking and tell $P_{1}$ the positions of the sample states. After that they ask $P_{1}$ to measure the particles (1, 2) in each sample state with one of the two bases $\{|0\rangle, |1\rangle\}$ and $\{|+\rangle, |-\rangle\}$ randomly. If $P_{1}$ measures the particles (1, 2) with the basis $\{|0\rangle, |1\rangle\}$ ($\{|+\rangle, |-\rangle\}$), then $P_{2}$ measures the particle 3 with the basis $\{|0\rangle, |1\rangle\}$ ($\{|+\rangle, |-\rangle\}$), and $P_{2}$ and $P_{3}$ measure the particles (4, 5, 6) with the basis $\{|\gamma_{j}^{1}\rangle|j=1,2,\ldots,8\}$ ($\{|\gamma_{j}^{2}\rangle|j=1,2,\ldots,8\}$). Finally, $P_{2}$ and $P_{3}$ can determine the error rate of the correlation of their outcomes according to Eq. (\ref{first}) and Eq. (\ref{second}). If the error rate exceeds the threshold, then this protocol will be aborted and repeat the step ($S1$). Otherwise, the protocol will go to the next step.
\item \label{step:key} By removing the particles of the sample states, $P_{1}$ ($P_{2}$, $P_{3}$) measures particles (1, 2) (particles (3, 4) , particles (6, 5)) of the $ith$ $\Psi_{6qb}$ ($i=1, 2, \ldots, \lceil \frac{L}{2} \rceil$) with the Bell basis. According to their measurement outcomes and the encoding arrangement Eq. (\ref{bEncoding}), $P_{1}$ ($P_{2}$, $P_{3}$) will get the key $K_{1}$ ($K_{2}$, $K_{3}$) that will be kept secret. For example, the possible measurement outcomes of the $ith$ $\Psi_{6qb}$ may be $R_{12}^{i}=|\Psi^{-}\rangle$ ($R_{34}^{i}=|\Psi^{+}\rangle$, $R_{65}^{i}=|\Phi^{-}\rangle$), thus the $ith$ two bits of $K_{1}$ ($K_{2}$, $K_{3}$) is $11$ ($10$, $01$). After that $P_{1}$ ($P_{2}$, $P_{3}$) computes $C_{1}=M_{1}\oplus K_{1}$ ($C_{2}=M_{2}\oplus K_{2}$, $C_{3}=M_{3}\oplus K_{3}$) (Here, $\oplus$ denotes the addition module 2.).
\item \label{step:last}$P_{2}$ ($P_{3}$) sends $C_{2}$ ($C_{3}$) to $P_{1}$, $P_{1}$ then can determine whether $M_{2}=M_{3}$ or not. If $M_{2}\neq M_{3}$, $P_{1}$ announces the result and the protocol finishes. Otherwise, $P_{1}$ randomly computes $C_{13}=C_{1}\oplus C_{3}$ or $C_{12}=C_{1}\oplus C_{2}$. If $P_{1}$ computes $C_{13}=C_{1}\oplus C_{3}$ ($C_{12}=C_{1}\oplus C_{2}$), he then sends $C_{13}$ ($C_{12}$) to $P_{2}$ ($P_{3}$). Subsequently $P_{2}$ ($P_{3}$) can determine whether $M_{1}=M_{3}$ ($M_{1}=M_{2}$) depending on the key $K_{2}$ ($K_{3}$) and $C_{13}$ ($C_{12}$). Finally, they can determine whether $M_{1}=M_{2}=M_{3}$ or not and preserve the privacy of their information, respectively.
\end{enumerate}
\section{Analysis of the presented protocol} In this section, we will analyze the correctness and the security of our protocol.
\subsection{Correctness}
According to the Eq. (\ref{zero}), we can find in the step (S\ref{step:key}) that $K_{1}\oplus K_{2}\oplus K_{3}=0$. When $P_{1}$ receives $C_{2}$ and $C_{3}$ from $P_{2}$ and $P_{3}$, respectively, he could compute
\begin{eqnarray}\label{equ:P1forP23}
K_{1}\oplus C_{2}\oplus C_{3}&=&K_{1} \oplus K_{2}\oplus M_{2} \oplus K_{3}\oplus M_{3}\nonumber \\&=& K_{1} \oplus K_{2}\oplus K_{3} \oplus M_{2}\oplus M_{3}\nonumber \\
&=& M_{2} \oplus M_{3}.
\end{eqnarray}
Later, he can determine whether $M_{2} \oplus M_{3}=0$ depending on the Eq. (\ref{equ:P1forP23}). If he finds that $M_{2} \oplus M_{3}\neq 0$, he can simply announce that $M_{1}=M_{2}=M_{3}$ is false and the protocol finishes. Otherwise, the protocol will continue and $P_{2}$ ($P_{3}$) can also determine whether $M_{1}=M_{3}$ ($M_{1}=M_{2}$) or not in the step (S\ref{step:last}) using the similar method as that of $P_{1}$. Finally, the protocol could correctly determine $M_{1}=M_{2}=M_{3}$ or not.
\subsection{Security}
Compared with the quantum cryptography protocols such as quantum key distribution (QKD)\cite{BB84,LC99,ShorPreskill2000,He2011,FFBLSTW12}, quantum secret sharing (QSS)\cite{HBB99,KKI99,KB02,ZL07,JWGQG12,S12,long2012quantum} and quantum secure direct communication (QSDC)\cite{BF02,DLL03,GYW05,LWGZ08,LCJ12,sun2012quantum,sun2012quantum1}, the security analysis of multi-party quantum private comparison protocol is more complicated. Because the attacks from all participants have to be considered in multi-party quantum private comparison protocols. Outside eavesdroppers have the desire to get the participants' private inputs. In addition, some participants may do their utmost to derive other participants' private secret information. Therefore, multi-party quantum private comparison protocols must be secure against outside and participant attacks.\par
\subsubsection{Outside attacks}
Similar to the detection method for outside eavesdropping used in the BB84 QKD protocol \cite{BB84}, we employ the decoy particles to prevent the eavesdropping. It has been proven to be unconditionally secure by Ref. \cite{ShorPreskill2000}. Any outside eavesdropping will be detected in the step (S\ref{outCheck}), thus outside Eve's all kinds of attacks, such as the intercept-resend attack, the measurement-resend attack, the entanglement-measurement attack, are useless in our protocol. We take the intercept-resend attack as an example here: suppose that the initial decoy particle state is $|0\rangle$, and Eve randomly measures it with one of the two bases $\{|0\rangle, |1\rangle\}$ and $\{|+\rangle, |-\rangle\}$, and then she sends the fake particle prepared by herself according to the measurement outcomes to $P_{2}$ ($P_{3}$). Obviously, the probability of being detected during the step (S\ref{outCheck}) is $\frac{1}{4}$. When we use $d$ decoy particles for eavesdropping detection, the probability of being detected will be $ 1 - (\frac{3}{4})^{d} $. We can see that if $d$ is large enough, the probability of being detected will approach to 1. Therefore, Eve will be detected in the step (S\ref{outCheck}). \par
On the other hand, Eve may get the ciphertexts $C_{2}$, $C_{3}$, $C_{13}$ and $C_{12}$ that are one-time pad ciphertexts in the protocol. However, she cannot get the keys $K_{1}$, $K_{2}$ or $K_{3}$ that are kept secret by the participants. Thus, she fails to derive the participants' private inputs.\par
Trojan horse attack \cite{DLZZ05,GFKZR06,LDZ06}, such as the delay-photon Trojan horse attack and the invisible photon eavesdropping (IPE) Trojan horse attack, exists in two-way quantum communication protocols. Our protocol is congenitally free from these attacks because the presented protocol employs one-step quantum transmission.
\subsubsection{Participant attack: one of the participants wants to steal others' inputs}
First, we can assume that $P_{2}$ wishes to steal the input of $P_{1}$ ($P_{3}$) because the role of the participant $P_{3}$ is the same as the participant $P_{2}$. \par
\begin{table}[htb]
\caption{The relations of the participants' keys and their measurement outcomes.}
\label{tab:1}
{\begin{tabular}{llll}
\hline\noalign{
}
& $P_{1}$ & $P_{2}$ & $P_{3}$ \\
\noalign{
}\hline\noalign{
}
\tabincell{l}{Possible measurement\\ outcomes} & \tabincell{l}{ $|\Phi^{+}\rangle_{12}$ ($|\Phi^{-}\rangle_{12}$)} & \tabincell{l}{ $|\Phi^{+}\rangle_{34}$ ($|\Phi^{+}\rangle_{34}$)} & \tabincell{l}{ $|\Phi^{+}\rangle_{65}$ ($|\Phi^{-}\rangle_{65}$)}\\
Corresponding keys & $00$ ($01$) & $00$ ($00$) & $00$ ($01$)\\ \\
\tabincell{l}{Possible measurement\\ outcomes} & \tabincell{l}{ $|\Phi^{+}\rangle_{12}$ ($|\Phi^{-}\rangle_{12}$)} & \tabincell{l}{ $|\Phi^{-}\rangle_{34}$ ($|\Phi^{-}\rangle_{34}$)} & \tabincell{l}{ $|\Phi^{-}\rangle_{65}$ ($|\Phi^{+}\rangle_{65}$)}\\
Corresponding keys & $00$ ($01$) & $01$ ($01$) & $01$ ($00$)\\ \\
\tabincell{l}{Possible measurement\\ outcomes} & \tabincell{l}{ $|\Phi^{+}\rangle_{12}$ ($|\Phi^{-}\rangle_{12}$)} & \tabincell{l}{ $|\Psi^{+}\rangle_{34}$ ($|\Psi^{+}\rangle_{34}$)} & \tabincell{l}{ $|\Psi^{+}\rangle_{65}$ ($|\Psi^{-}\rangle_{65}$)}\\
Corresponding keys & $00$ ($01$) & $10$ ($10$) & $10$ ($11$)\\ \\
\tabincell{l}{Possible measurement\\ outcomes} & \tabincell{l}{ $|\Phi^{+}\rangle_{12}$ ($|\Phi^{-}\rangle_{12}$)} & \tabincell{l}{ $|\Psi^{-}\rangle_{34}$ ($|\Psi^{-}\rangle_{34}$)} & \tabincell{l}{ $|\Psi^{-}\rangle_{65}$ ($|\Psi^{+}\rangle_{65}$)}\\
Corresponding keys & $00$ ($01$) & $11$ ($11$) & $11$ ($10$)\\ \\
\tabincell{l}{Possible measurement\\ outcomes} & \tabincell{l}{ $|\Psi^{+}\rangle_{12}$ ($|\Psi^{-}\rangle_{12}$)} & \tabincell{l}{ $|\Phi^{+}\rangle_{34}$ ($|\Phi^{+}\rangle_{34}$)} & \tabincell{l}{ $|\Psi^{+}\rangle_{65}$ ($|\Psi^{-}\rangle_{65}$)}\\
Corresponding keys & $10$ ($11$) & $00$ ($00$) & $10$ ($11$)\\ \\
\tabincell{l}{Possible measurement\\ outcomes} & \tabincell{l}{ $|\Psi^{+}\rangle_{12}$ ($|\Psi^{-}\rangle_{12}$)} & \tabincell{l}{ $|\Phi^{-}\rangle_{34}$ ($|\Phi^{-}\rangle_{34}$)} & \tabincell{l}{ $|\Psi^{-}\rangle_{65}$ ($|\Psi^{+}\rangle_{65}$)}\\
Corresponding keys & $10$ ($11$) & $01$ ($01$) & $11$ ($10$)\\ \\
\tabincell{l}{Possible measurement\\ outcomes} & \tabincell{l}{ $|\Psi^{+}\rangle_{12}$ ($|\Psi^{-}\rangle_{12}$)} & \tabincell{l}{ $|\Psi^{+}\rangle_{34}$ ($|\Psi^{+}\rangle_{34}$)} & \tabincell{l}{ $|\Phi^{+}\rangle_{65}$ ($|\Phi^{-}\rangle_{65}$)}\\
Corresponding keys & $10$ ($11$) & $10$ ($10$) & $00$ ($01$)\\ \\
\tabincell{l}{Possible measurement\\ outcomes} & \tabincell{l}{ $|\Psi^{+}\rangle_{12}$ ($|\Psi^{-}\rangle_{12}$)} & \tabincell{l}{ $|\Psi^{-}\rangle_{34}$ ($|\Psi^{-}\rangle_{34}$)} & \tabincell{l}{ $|\Phi^{-}\rangle_{65}$ ($|\Phi^{+}\rangle_{65}$)}\\
Corresponding keys & $10$ ($11$) & $11$ ($11$) & $01$ ($00$)\\
\noalign{
}\hline
\end{tabular}} \end{table}
In order to do that, he should find out $P_{1}$'s key $K_{1}$ and $P_{3}$'s key $K_{3}$. However, he is unable to complete this task. From the Eq. (\ref{beUsed}) and the encoding arrangement Eq. (\ref{bEncoding}), he can just have $K_{1}\oplus K_{2}\oplus K_{3}=0$. $P_{2}$ is unable to exactly figure out the values of $K_{1}$ and $K_{3}$. In the step (S\ref{step:last}), $P_{2}$ may get $C_{13}$ ($C_{12}$) that's the ciphertext of $M_{1}\oplus M_{3}$ ($M_{1}\oplus M_{2}$) encrypted by $K_{2}$ ($K_{3}$). But he cannot obtain $C_{13}$ and $C_{12}$ at the same time in our protocol. If $P_{2}$ gets $C_{12}$, he could compute $C_{12}\oplus C_{2}=C_{1}=K_{1}\oplus M_{1}$. He could also get $C_{3}$ and $C_{13}$, computing $C_{3}\oplus C_{13}=K_{3}\oplus M_{3} \oplus K_{1} \oplus M_{1} \oplus K_{3} \oplus M_{3}=K_{1}\oplus M_{1}$ which is the ciphertext of $M_{1}$ encrypted by $K_{1}$. Obviously, he cannot have the $P_{1}$'s private input $M_{1}$ because he does not know the exact value of $K_{1}$. $P_{2}$ also fails to get $P_{3}$'s private input $M_{3}$ because he cannot get $P_{3}$'s key $K_{3}$. Therefore, $P_{2}$ couldn't have the private inputs of $P_{1}$ and $P_{3}$. We take the two bits keys as an example so as to see the relations of the participants' keys and their measurement outcomes. The details see Table \ref{tab:1}. From this table, we know that $P_{2}$ cannot definitely determine the keys of $P_{1}$ and $P_{3}$ depending on his own key.\par
Second, we will show that our protocol is still secure if $P_{1}$ wants to steal others' inputs. In the proposed protocol, the participant $P_{1}$ who prepares the state $\Psi_{6qb}$ is more powerful than $P_{2}$ and $P_{3}$. He may prepare some particular fake particles, sending them to $P_{2}$ and $P_{3}$, respectively. And then he can determine the keys of $P_{2}$ and $P_{3}$ with certainty. Hence he could get the private inputs of $P_{2}$ and $P_{3}$. For instance, $P_{1}$ can send particles in the ordered sequence in Bell state $|\Psi^{-}\rangle$ ($|\Psi^{+}\rangle$) to $P_{2}$ ($P_{3}$) if he wishes to decide some two bits of $P_{2}$'s ($P_{3}$'s) key to be 11 (10). He could also prepare the real state $\Psi_{6qb}$, but then sends the particular particles to $P_{2}$ ($P_{3}$) according to Eqs.(\ref{twoSplit1}-\ref{twoSplit5}). He can finally steal the key of $P_{2}$ ($P_{3}$) depending on the relation of their measurement outcomes, obtaining their private inputs. Unfortunately, these attacks will be detected in the step (S\ref{step:colCheck}) of our protocol if $\delta$ is large enough. The general attack of $P_{1}$ can be described by a unitary operation: $U_{A}$ performed on qubits, including the state $\Psi_{6qb}$ and the probe state initialized as $|0\rangle_{A}$ before $P_{1}$ sends the particles to $P_{2}$ and $P_{3}$. We can prove that the final state of $\Psi_{6qb}$ would not be entangled with $P_{1}$'s probe state, which implies $P_{1}$ cannot get any information about exact measurement outcomes of $P_{2}$ and $P_{3}$ through his probe if there is no error to occur. Thus he could not get the keys of $P_{2}$ and $P_{3}$.\par
The most general operation $P_{1}$ can do is to entangle the state $\Psi_{6qb}$ with the probe state initialized as $|0\rangle_{A}$, which can be written as
\begin{eqnarray}\label{smit}
U_{A}\Psi_{6qb}|0\rangle_{A}
=\frac{1}{4}[
|\Phi^{+}\rangle_{12}(&|\Phi^{+}\rangle_{34}&|\Phi^{+}\rangle_{65}|A_{1}\rangle
+|\Phi^{-}\rangle_{34}|\Phi^{-}\rangle_{65}|A_{2}\rangle \nonumber\\
+&|\Psi^{+}\rangle_{34}&|\Psi^{+}\rangle_{65}|A_{3}\rangle
+|\Psi^{-}\rangle_{34}|\Psi^{-}\rangle_{65}|A_{4}\rangle) \nonumber\\
+|\Phi^{-}\rangle_{12}(-&|\Phi^{+}\rangle_{34}&|\Phi^{-}\rangle_{65}|A_{5}\rangle|
+|\Phi^{-}\rangle_{34}|\Phi^{+}\rangle_{65}|A_{6}\rangle \nonumber\\
-&|\Psi^{+}\rangle_{34}&|\Psi^{-}\rangle_{65}|A_{7}\rangle
+|\Psi^{-}\rangle_{34}|\Psi^{+}\rangle_{65}|A_{8}\rangle) \nonumber\\
+|\Psi^{+}\rangle_{12}(&|\Phi^{+}\rangle_{34}&|\Psi^{+}\rangle_{65}|A_{9}\rangle
+|\Phi^{-}\rangle_{34}|\Psi^{-}\rangle_{65}|A_{10}\rangle \nonumber\\
-&|\Psi^{+}\rangle_{34}&|\Phi^{+}\rangle_{65}|A_{11}\rangle
-|\Psi^{-}\rangle_{34}|\Phi^{-}\rangle_{65}|A_{12}\rangle) \nonumber\\
+|\Psi^{-}\rangle_{12}(-&|\Phi^{+}\rangle_{34}&|\Psi^{-}\rangle_{65}|A_{13}\rangle
+|\Phi^{-}\rangle_{34}|\Psi^{+}\rangle_{65}|A_{14}\rangle \nonumber\\
+&|\Psi^{+}\rangle_{34}&|\Phi^{-}\rangle_{65}|A_{15}\rangle
-|\Psi^{-}\rangle_{34}|\Phi^{+}\rangle_{65}|A_{16}\rangle)],
\end{eqnarray}
where $|A_{i}\rangle$ ($i=1, 2,\ldots, 16$) are some unnormalized states in $P_{1}$'s probe space. We will prove that that the final state of $\Psi_{6qb}$ would not be entangled with $P_{1}$'s probe state if he can escape the detection in the step (S\ref{step:colCheck}) of our protocol. \par
On one hand, if $P_{1}$ measures the particles (1, 2) with the basis $\{|0\rangle, |1\rangle\}$, then $P_{2}$ measures the particle 3 with the basis $\{|0\rangle, |1\rangle\}$, and $P_{2}$ and $P_{3}$ measure the particles (4, 5, 6) with the basis $\{|\gamma_{j}^{1}\rangle|j=1,2,\ldots,8\}$. Then, the state in Eq. (\ref{smit}) can be rewritten as follows:
\begin{eqnarray}\label{01check}
\frac{1}{\sqrt{8}}\{|000\rangle[|\gamma_{1}^{1}\rangle(|A_{1}\rangle&+&|A_{3}\rangle+|A_{6}\rangle+|A_{8}\rangle)
+|\gamma_{4}^{1}\rangle(|A_{2}\rangle-|A_{4}\rangle-|A_{5}\rangle+|A_{7}\rangle)\nonumber\\
+|\gamma_{6}^{1}\rangle(|A_{1}\rangle&-&|A_{3}\rangle+|A_{6}\rangle-|A_{8}\rangle)
+|\gamma_{7}^{1}\rangle(|A_{2}\rangle+|A_{4}\rangle-|A_{5}\rangle-|A_{7}\rangle)] \nonumber\\
+|001\rangle[|\gamma_{2}^{1}\rangle(|A_{2}\rangle&+&|A_{4}\rangle+|A_{5}\rangle+|A_{7}\rangle)
+|\gamma_{3}^{1}\rangle(|A_{1}\rangle+|A_{3}\rangle-|A_{6}\rangle-|A_{8}\rangle)\nonumber\\
+|\gamma_{5}^{1}\rangle(|A_{4}\rangle&-&|A_{2}\rangle-|A_{5}\rangle+|A_{7}\rangle)
+|\gamma_{8}^{1}\rangle(|A_{1}\rangle+|A_{3}\rangle-|A_{6}\rangle-|A_{8}\rangle)] \nonumber\\
+|010\rangle[|\gamma_{3}^{1}\rangle(|A_{10}\rangle&+&|A_{12}\rangle+|A_{13}\rangle+|A_{15}\rangle)
+|\gamma_{2}^{1}\rangle(|A_{9}\rangle-|A_{11}\rangle-|A_{14}\rangle+|A_{16}\rangle)\nonumber\\
+|\gamma_{5}^{1}\rangle(|A_{9}\rangle&+&|A_{11}\rangle-|A_{14}\rangle-|A_{16}\rangle)
+|\gamma_{8}^{1}\rangle(|A_{10}\rangle-|A_{12}\rangle+|A_{13}\rangle-|A_{15}\rangle)] \nonumber\\
+|011\rangle[|\gamma_{4}^{1}\rangle(|A_{9}\rangle&+&|A_{11}\rangle+|A_{14}\rangle+|A_{16}\rangle)
+|\gamma_{1}^{1}\rangle(|A_{12}\rangle-|A_{10}\rangle+|A_{13}\rangle-|A_{15}\rangle)\nonumber\\
+|\gamma_{6}^{1}\rangle(|A_{10}\rangle&+&|A_{12}\rangle-|A_{13}\rangle-|A_{15}\rangle)
+|\gamma_{7}^{1}\rangle(|A_{11}\rangle-|A_{9}\rangle-|A_{14}\rangle+|A_{16}\rangle)] \nonumber\\
-|100\rangle[|\gamma_{5}^{1}\rangle(|A_{9}\rangle&+&|A_{11}\rangle+|A_{14}\rangle+|A_{16}\rangle)
+|\gamma_{2}^{1}\rangle(|A_{9}\rangle-|A_{11}\rangle+|A_{14}\rangle-|A_{16}\rangle)\nonumber\\
+|\gamma_{3}^{1}\rangle(|A_{10}\rangle&+&|A_{12}\rangle-|A_{13}\rangle-|A_{15}\rangle)
+|\gamma_{8}^{1}\rangle(|A_{10}\rangle-|A_{12}\rangle-|A_{13}\rangle+|A_{15}\rangle)] \nonumber\\
-|101\rangle[|\gamma_{6}^{1}\rangle(|A_{10}\rangle&+&|A_{12}\rangle+|A_{13}\rangle+|A_{15}\rangle)
+|\gamma_{1}^{1}\rangle(|A_{12}\rangle-|A_{10}\rangle-|A_{13}\rangle+|A_{15}\rangle)\nonumber\\
+|\gamma_{4}^{1}\rangle(|A_{9}\rangle&+&|A_{11}\rangle-|A_{14}\rangle-|A_{16}\rangle)
+|\gamma_{7}^{1}\rangle(|A_{11}\rangle-|A_{9}\rangle+|A_{14}\rangle-|A_{16}\rangle)] \nonumber\\
+|110\rangle[|\gamma_{7}^{1}\rangle(|A_{2}\rangle&+&|A_{4}\rangle+|A_{5}\rangle+|A_{7}\rangle)
+|\gamma_{1}^{1}\rangle(|A_{1}\rangle+|A_{3}\rangle-|A_{6}\rangle-|A_{8}\rangle)\nonumber\\
+|\gamma_{4}^{1}\rangle(|A_{2}\rangle&-&|A_{4}\rangle+|A_{5}\rangle-|A_{7}\rangle)
+|\gamma_{6}^{1}\rangle(|A_{1}\rangle-|A_{3}\rangle-|A_{6}\rangle+|A_{8}\rangle)] \nonumber\\
+|111\rangle[|\gamma_{8}^{1}\rangle(|A_{1}\rangle&+&|A_{3}\rangle+|A_{6}\rangle+|A_{8}\rangle)
+|\gamma_{2}^{1}\rangle(|A_{2}\rangle+|A_{4}\rangle-|A_{5}\rangle-|A_{7}\rangle)\nonumber\\
+|\gamma_{3}^{1}\rangle(|A_{3}\rangle&-&|A_{1}\rangle-|A_{6}\rangle+|A_{8}\rangle)
+|\gamma_{5}^{1}\rangle(|A_{4}\rangle-|A_{2}\rangle+|A_{5}\rangle-|A_{7}\rangle)]\}_{123456A}. \nonumber\\
\end{eqnarray}
According to Eq. (\ref{first}), if $P_{1}$ introduces no error, the following conditions should be satisfied:
\begin{eqnarray}\label{01should}
|A_{1}\rangle&=&|A_{3}\rangle=|A_{6}\rangle=|A_{8}\rangle, \nonumber\\
|A_{2}\rangle&=&|A_{4}\rangle=|A_{5}\rangle=|A_{7}\rangle, \nonumber\\
|A_{9}\rangle&=&|A_{11}\rangle=|A_{14}\rangle=|A_{16}\rangle, \nonumber\\
|A_{10}\rangle&=&|A_{12}\rangle=|A_{13}\rangle=|A_{5}\rangle.
\end{eqnarray}
On the other hand, if $P_{1}$ measures the particles (1, 2) with the basis $\{|+\rangle, |-\rangle\}$, then $P_{2}$ measures the particle 3 with the basis $\{|+\rangle, |-\rangle\}$, and $P_{2}$ and $P_{3}$ measure the particles (4, 5, 6) with the basis $\{|\gamma_{j}^{2}\rangle|j=1,2,\ldots,8\}$. Then, the state in Eq. (\ref{smit}) can be rewritten as follows:
\begin{eqnarray}\label{+-check}
\frac{1}{\sqrt{8}}\{|+++\rangle[|\gamma_{1}^{2}\rangle(|A_{2}\rangle&+&|A_{3}\rangle+|A_{13}\rangle+|A_{16}\rangle)
+|\gamma_{4}^{2}\rangle(|A_{1}\rangle-|A_{4}\rangle+|A_{14}\rangle-|A_{15}\rangle)\nonumber\\
+|\gamma_{6}^{2}\rangle(|A_{3}\rangle&-&|A_{2}\rangle+|A_{13}\rangle-|A_{16}\rangle)
+|\gamma_{7}^{2}\rangle(|A_{1}\rangle+|A_{4}\rangle-|A_{14}\rangle-|A_{15}\rangle)] \nonumber\\
+|++-\rangle[|\gamma_{2}^{2}\rangle(|A_{1}\rangle&+&|A_{4}\rangle+|A_{14}\rangle+|A_{15}\rangle)
+|\gamma_{3}^{2}\rangle(|A_{3}\rangle-|A_{2}\rangle-|A_{13}\rangle+|A_{16}\rangle)\nonumber\\
+|\gamma_{5}^{2}\rangle(|A_{1}\rangle&-&|A_{4}\rangle-|A_{14}\rangle+|A_{15}\rangle)
+|\gamma_{8}^{2}\rangle(|A_{2}\rangle+|A_{3}\rangle-|A_{13}\rangle-|A_{15}\rangle)] \nonumber\\
+|+-+\rangle[|\gamma_{3}^{2}\rangle(|A_{5}\rangle&+&|A_{8}\rangle+|A_{10}\rangle+|A_{11}\rangle)
+|\gamma_{2}^{2}\rangle(|A_{6}\rangle-|A_{7}\rangle+|A_{9}\rangle-|A_{12}\rangle)\nonumber\\
+|\gamma_{5}^{2}\rangle(|A_{6}\rangle&+&|A_{7}\rangle-|A_{9}\rangle-|A_{12}\rangle)
+|\gamma_{8}^{2}\rangle(|A_{8}\rangle-|A_{5}\rangle+|A_{10}\rangle-|A_{11}\rangle)] \nonumber\\
+|+--\rangle[|\gamma_{4}^{2}\rangle(|A_{6}\rangle&+&|A_{7}\rangle+|A_{9}\rangle+|A_{12}\rangle)
+|\gamma_{1}^{2}\rangle(|A_{8}\rangle-|A_{5}\rangle-|A_{10}\rangle+|A_{11}\rangle)\nonumber\\
+|\gamma_{6}^{2}\rangle(|A_{5}\rangle&+&|A_{8}\rangle-|A_{10}\rangle-|A_{11}\rangle)
+|\gamma_{7}^{2}\rangle(|A_{6}\rangle-|A_{7}\rangle-|A_{9}\rangle+|A_{12}\rangle)] \nonumber\\
+|-++\rangle[|\gamma_{5}^{2}\rangle(|A_{6}\rangle&+&|A_{7}\rangle+|A_{9}\rangle+|A_{12}\rangle)
+|\gamma_{2}^{2}\rangle(|A_{6}\rangle-|A_{7}\rangle-|A_{9}\rangle+|A_{12}\rangle)\nonumber\\
+|\gamma_{3}^{2}\rangle(|A_{5}\rangle&+&|A_{8}\rangle-|A_{10}\rangle-|A_{11}\rangle)
+|\gamma_{8}^{2}\rangle(|A_{8}\rangle-|A_{5}\rangle-|A_{10}\rangle+|A_{11}\rangle)] \nonumber\\
+|-+-\rangle[|\gamma_{6}^{2}\rangle(|A_{5}\rangle&+&|A_{8}\rangle+|A_{10}\rangle+|A_{11}\rangle)
+|\gamma_{1}^{2}\rangle(|A_{8}\rangle-|A_{5}\rangle+|A_{10}\rangle-|A_{11}\rangle)\nonumber\\
+|\gamma_{4}^{2}\rangle(|A_{6}\rangle&+&|A_{7}\rangle-|A_{9}\rangle-|A_{12}\rangle)
+|\gamma_{7}^{2}\rangle(|A_{6}\rangle-|A_{7}\rangle+|A_{9}\rangle-|A_{12}\rangle)] \nonumber\\
+|--+\rangle[|\gamma_{7}^{2}\rangle(|A_{1}\rangle&+&|A_{4}\rangle+|A_{14}\rangle+|A_{15}\rangle)
+|\gamma_{1}^{2}\rangle(|A_{2}\rangle+|A_{3}\rangle-|A_{13}\rangle-|A_{16}\rangle)\nonumber\\
+|\gamma_{4}^{2}\rangle(|A_{1}\rangle&-&|A_{4}\rangle-|A_{14}\rangle+|A_{15}\rangle)
+|\gamma_{6}^{2}\rangle(|A_{3}\rangle-|A_{2}\rangle+|A_{13}\rangle-|A_{16}\rangle)] \nonumber\\
+|---\rangle[|\gamma_{8}^{2}\rangle(|A_{2}\rangle&+&|A_{3}\rangle+|A_{13}\rangle+|A_{16}\rangle)
+|\gamma_{2}^{2}\rangle(|A_{1}\rangle+|A_{4}\rangle-|A_{14}\rangle-|A_{15}\rangle)\nonumber\\
+|\gamma_{3}^{2}\rangle(|A_{3}\rangle&-&|A_{2}\rangle+|A_{13}\rangle-|A_{16}\rangle)
+|\gamma_{5}^{2}\rangle(|A_{1}\rangle-|A_{4}\rangle+|A_{14}\rangle-|A_{15}\rangle)]\}_{123456A}. \nonumber\\
\end{eqnarray}
According to Eq. (\ref{second}), if $P_{1}$ introduces no error, the following conditions should be satisfied:
\begin{eqnarray}\label{+-should}
|A_{1}\rangle&=&|A_{4}\rangle=|A_{14}\rangle=|A_{15}\rangle, \nonumber\\
|A_{2}\rangle&=&|A_{3}\rangle=|A_{13}\rangle=|A_{16}\rangle, \nonumber\\
|A_{5}\rangle&=&|A_{8}\rangle=|A_{10}\rangle=|A_{11}\rangle, \nonumber\\
|A_{6}\rangle&=&|A_{7}\rangle=|A_{9}\rangle=|A_{12}\rangle.
\end{eqnarray}
We can derive from Eq.~(\ref{01should}) and Eq.~(\ref{+-should}) that $|A_{1}\rangle = |A_{2}\rangle = \ldots = |A_{16}\rangle$, which means that the state $\Psi_{6qb}$ and the prob state prepared by $P_{1}$ are entirely not entangled. Thus the subsequent measurement outcome of the prob state tells $P_{1}$ nothing. \par
$P_{1}$ may use the similar attack as $P_{2}$ to derive the inputs of $P_{2}$ and $P_{3}$ in accordance with the cipertexts $C_{2}$ and $C_{3}$ that are $M_{2}$ and $M_{3}$ encoded with $K_{2}$ and $K_{3}$, respectively. However, according to the Eq. (\ref{beUsed}) and the encoding arrangement Eq. (\ref{bEncoding}), he can have $K_{1}\oplus K_{2}\oplus K_{3}=0$ but the keys $K_{2}$ and $K_{3}$ and therefore is unable to steal $M_{2}$ and $M_{3}$ offered by $P_{2}$ and $P_{3}$, respectively. So the protocol remains secure against this attack.\par
In the step (S\ref{step:last}) of our protocol, $P_{1}$ first determines whether $M_{2}=M_{3}$ using his own key $K_{1}$, $P_{2}$ ($P_{3}$) then determines if $M_{1}=M_{3}$ ($M_{1}=M_{2}$) or not based on his key $K_{2}$ ($K_{3}$) and at last they can get the comparison result. In fact, the order in which one participant decides whether or not the other two participants' private inputs are equal is not important because their private inputs are encrypted by their keys that are kept secret.\par
Unfortunately, any two participants can collude with each other to derive the third one's key according to $K_{1} \oplus K_{2} \oplus K_{3}=0$, obtaining the corresponding private input. So it would be interesting to design multi-party quantum private comparison protocols that are still secure against such an attack.\par
\subsubsection{Security analysis over lossy and noise channel}
In the above analysis, the quantum channels are assumed to be under the ideal condition (i.e. noiseless and lossless). But quantum channels are usually lossy and noisy in the real world. In this section, we show that our protocol remains secure in lossy and noisy quantum channels. The eavesdropper, Eve, is assumed to be powerful enough to establish an ideal channel with any participant. We discuss the lossy and noisy quantum channels in case as follows.
\textbf{Case 1. Lossy quantum channel}
In such a quantum channel, Eve may intercepts particles sent from $P_{1}$ to $P_{2}$ and $P_{3}$. She then keeps some of them and transmits the other particles to $P_{2}$ and $P_{3}$ through an ideal channel. If the intercepted particles are not decoy particles, she is able to perform measurements on the related particles with Bell basis. The measurement results will lead to the leakage of the some key bits for $P_{1}$, she will finally get some information of $P_{1}$'s private input. Fortunately, our protocol remains secure against such a attack. In the Step 2 of our protocol, $P_{2}$ ($P_{3}$) informs $P_{1}$ which particles have been received and which are lost during the transmission. $P_{1}$ and $P_{2}$ ($P_{3}$) only employ the received particles to make a public discussion and finish equality comparison. The intercepted particles are useless and Eve will fail to extract any information about $P_{1}$'s key that is used to encrypt his private input.
\textbf{Case 2. Noisy quantum channel}
Eve can intercept the particles sent from $P_{1}$ to $P_{2}$ ($P_{3}$), performing intercept-resend attack or entangle-measure attack, forwarding these tampered particles to $P_{2}$ ($P_{3}$) through an idea channel established by herself. In this situation, Eve tries her best to cover up the tampering of particles as the noise existed on the quantum channel between $P_{1}$ and $P_{2}$ ($P_{3}$). We have learned that these attacks will be caught if the eavesdropper detection rate of our protocol is smaller than the quantum bit error rate of noise (QBER). In accordance with \cite{jennewein2000quantum,hughes2002practical,gobby2004quantum}, the QBER is roughly between $2\%$ and $8.9\%$ depending on the different channel situations (e.g., distance, etc.). Fortunately, the detection rate for decoy particle in our protocol is $25\%$ that is greater than the error rate of the quantum channel. Therefore, our protocol is also secure in the noisy quantum channel.
Up until now, we have completed the analysis of the correctness and the security of our protocol.
Note that if Lin et al.'s protocol \cite{lin2014quantum} is used for three participants' private comparison, it needs run twice in the worst case. But our protocol needs run only once even in the worst case. For Chang et al.'s protocol \cite{chang2013multi}, in the case of the number of the participants is 3, our protocol needs more $6\delta$ particles than their protocol do and our protocol will suffer collusion attack as mentioned in the previous security analysis. This is the disadvantages of our protocol! However, such disadvantages happen because of the lack of an external third party. The more $6\delta$ particles serves as participant attack detection, because the initial quantum states are prepared by $P_{1}$. We need to check if he is honest. If we include the external third party who is assumed to be semi-honest and prepares the initial quantum state, these defects will vanish. But it will require more various quantum states and different quantum operations. Chang et al.'s protocol may not be able to finish the private task if it lacks the semi-honest third party. In practice, resorting to an external third party for help in quantum private comparison may unexpectedly result in some private information leakage and this help is usually not free. In this sense, designing quantum private comparison protocols without an external third party is necessary.
\section{Conclusions}
We present a new three-party quantum private comparison protocol based on genuinely maximally entangled six-qubit states. Three participants can determine whether their private information are equal without the assistance of an external third party and in the meantime keep their inputs secret, respectively. The proposed protocol does not require any unitary operations to encode information for the sake of the excellent properties of genuinely maximally entangled six-qubit states. Because the proposed protocol utilizes one-step quantum transmission, it can be prevented from Trojan horse attacks. Finally, we also analyze the correctness and security of our protocol.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Binary classification with ambiguous training data
}
\author{Naoya Otani\affmark[1] \and
Yosuke Otsubo\affmark[1] \and Tetsuya Koike\affmark[1] \and Masashi Sugiyama\textsuperscript{2,3} }
\authorrunning{ Naoya Otani \and Yosuke Otsubo \and Tetsuya Koike \and Masashi Sugiyama}
\institute{Naoya Otani \at
Nikon Corporation, Research \& Development Division\\
471, Nagaodai-cho, Sakae-ku, Yokohama-city, Kanagawa, 244-8533, Japan.\\
Tel.: +81-45-853-8584\\
Fax: +81-45-853-8585\\
\email{[email protected]} \\ \\
\affaddr{\affmark[1] Nikon Corporation, Research \& Development Division}\\
\affaddr{\affmark[2] RIKEN Center for Advanced Intelligence Project}\\
\affaddr{\affmark[3] The University of Tokyo, Graduate School of Frontier Sciences } }
\date{Received: date / Accepted: date}
\maketitle
\begin{abstract} In supervised learning, we often face with \emph{ambiguous} (A) samples that are difficult to label even by domain experts. In this paper, we consider a binary classification problem in the presence of such A samples. This problem is substantially different from semi-supervised learning since unlabeled samples are not necessarily difficult samples. Also, it is different from 3-class classification with the positive (P), negative (N), and A classes since we do not want to classify test samples into the A class. Our proposed method extends binary classification with reject option, which trains a classifier and a rejector simultaneously using P and N samples based on the 0-1-$c$ loss with rejection cost $c$. More specifically, we propose to train a classifier and a rejector under the 0-1-$c$-$d$ loss using P, N, and A samples, where $d$ is the misclassification penalty for ambiguous samples. In our practical implementation, we use a convex upper bound of the 0-1-$c$-$d$ loss for computational tractability. Numerical experiments demonstrate that our method can successfully utilize the additional information brought by such A training data.
\keywords{Ambiguous samples \and Classification with reject option \and Binary classification}
\end{abstract}
\section{Introduction} Supervised learning has been successfully deployed in various real-world applications, such as medical diagnosis \citep{bar2015chest,wang2016deep,esteva2017dermatologist} and manufacturing systems \citep{park2016machine,ren2017generic}. However, when the amount of labeled data is limited, current supervised learning methods still do not work reliably \citep{pesapane2018artificial}.
To efficiently obtain labeled data, domain knowledge has been used in many application areas \citep{ren2017generic,cruciani2018automatic,konishi2019practical,bejnordi2017diagnostic}. However, as some studies have pointed out \citep{wagner2005physiological,li2016facial,shahriyar2018approach}, there are often \emph{ambiguous} samples that are substantially difficult to label even by domain experts.
The goal of this paper is to propose a novel classification method that can handle such ambiguous data. More specifically, we consider a binary classification problem where, in addition to positive (P) and negative (N) samples, ambiguous (A) samples are available for training a classifier.
Naively, we may consider employing 3-class classification methods for the P, N, and A classes. However, since we classify test samples only in the P or N class, not in the A class, naive 3-class methods cannot be directly used in our problem. Moreover, they cannot utilize the information that the A class exists between the P and N classes. Another related approach is classification with reject option \citep{bartlett2008classification,cortes2016learning}, where ambiguous test samples are not classified into the P or N classes, but rejected. However, classification methods with reject option do not consider ambiguous samples in the training phase and thus they cannot be employed in the current scenario.
Semi-supervised learning may also be related to the current problem, where unlabeled data is used for training a classifier in addition to P and N data \citep{odena2016semi,sakai2017semi}. In semi-supervised learning, unlabeled samples are P and N samples that have not yet been labeled and they are not necessarily difficult samples to be labeled. On the other hand, in our target problem of classification with ambiguous data, ambiguous data are typically distributed in the intersection of the P and N classes. Thus, since the problem setups are intrinsically different, merely using semi-supervised learning methods in the current problem may not be promising.
Classification with imperfect labeling \citep{cannings2020classification} allows incorrect labels in training data, so it can be useful to deal with a dataset where annotators forcibly give positive or negative labels to all samples. Open-set classification \citep{scheirer2012toward} detects samples classified into none of the training classes and classifies them into the ``unknown'' class, so we can apply it when we deal with a dataset where annotators skip labeling hard-to-label samples and we classify test samples into the P, N or A class. However, those two approaches are still different from our problem setting in terms of labels in input and output data. Table~\ref{tab} summerizes the problem setting of different approaches.
\begin{table} \centering \caption{Problem settings of related and our methods.} \begingroup \renewcommand{1.1}{1.1} \scriptsize \begin{tabular}{llll} \hline \\[-8pt] \begin{tabular}{l}Methods\end{tabular} & \begin{tabular}{l}Labels in\\training data\end{tabular} & \begin{tabular}{l}Labels predicted\\in test phase\end{tabular} & \begin{tabular}{l}Relationship\\among classes\end{tabular}\\ \hline \hline \begin{tabular}{l}Binary \\classification\end{tabular} & \begin{tabular}{ll}Positive\\Negative\end{tabular} & \begin{tabular}{ll}Positive\\Negative\end{tabular} & \begin{tabular}{l}None\end{tabular}\\ \hline \begin{tabular}{l}3-class \\classification\end{tabular} & \begin{tabular}{ll}Class 1\\Class 2\\Class 3\end{tabular} & \begin{tabular}{ll}Class 1\\Class 2\\Class 3\end{tabular}&\begin{tabular}{l}None\end{tabular} \\ \hline \begin{tabular}{l}Classification \\with reject \\option\end{tabular}& \begin{tabular}{ll}Positive\\Negative\end{tabular} & \begin{tabular}{ll}Positive\\Rejected\\Negative\end{tabular} &\begin{tabular}{l}Rejected samples \\are in P/N mixed\\ regions\end{tabular} \\ \hline \begin{tabular}{l}Semi-supervised\\learning\end{tabular}& \begin{tabular}{ll}Positive\\Unlabeled\\Negative\end{tabular} & \begin{tabular}{ll}Positive\\Negative\end{tabular} &\begin{tabular}{l}Unlabeled samples \\belong to P or N\end{tabular} \\ \hline \begin{tabular}{l}Classification \\ with imperfect \\labeling\end{tabular} & \begin{tabular}{ll}Positive\\Negative\end{tabular} & \begin{tabular}{ll}Positive\\Negative\end{tabular} &\begin{tabular}{l}Training data can\\ contain incorrect \\labels\end{tabular} \\ \hline \begin{tabular}{l}Open-set \\ classification\end{tabular} & \begin{tabular}{ll}Positive\\Negative\end{tabular} & \begin{tabular}{ll}Positive\\Negative\\Unknown\end{tabular} &\begin{tabular}{l}Test data can\\contain neigher\\positive nor\\negative samples\end{tabular} \\ \hline \begin{tabular}{l}Our proposal\end{tabular} & \begin{tabular}{ll}Positive\\Ambiguous\\Negative\end{tabular} & \begin{tabular}{ll}Positive\\Negative\end{tabular} &\begin{tabular}{l}Ambiguous \\samples are in \\P/N mixed regions\end{tabular} \\ \hline \end{tabular}\label{tab} \endgroup \end{table}
To effectively solve the problem of classification with ambiguous data, we propose to extend classification with reject option that trains a classifier and a rejector simultaneously using P and N samples based on the 0-1-$c$ loss with rejection cost $c$ \citep{cortes2016learning}. Our proposed method trains a classifier and a rejector under the 0-1-$c$-$d$ loss using P, N, and A samples, where $d$ is the misclassification penalty for ambiguous samples. Then, in the test phase, we use the trained classifier to assign P or N labels to test samples. However, in the same way as the 0-1-$c$ loss, directly performing optimization with the 0-1-$c$-$d$ loss is cumbersome due to its discrete nature. To cope with this problem, we introduce a convex upper bound of the 0-1-$c$-$d$ loss and use it in our practical implementation. Through experiments, we demonstrate that the proposed method can improve the test classification accuracy by utilizing A samples in the training phase. We also consider a simple heuristic that we randomly relabel ambiguous samples into the positive or negative class and apply classification with reject option. We show that the heuristic is essentially equivalent to a special case of the proposed method, and thus it can be an easy-to-implement alternative to the proposed method.
The rest of this paper is organized as follows.
We briefly review supervised learning in Section 2. Then, we define our problem setting and describe the details of our proposed method in Section 3. In Section 4, we experimentally evaluate its performance. Finally, in Section 5, we summarize our contributions and describe future works.
\section{Supervised classification} In this section, we first define the standard supervised classification problem and then review its standard solution.
\subsection{Formulation} Let $x \in \mathcal{X}$ be an input point and $y \in \mathcal{Y}=\{1,-1\}$ denote a binary label, which corresponds to the positive and negative classes, respectively. Suppose that we are given a set of positive and negative samples $\{(x_i, y_i)\}_{i=1}^N$ drawn independently from the probability distribution with density $p(x,y)$ defined on $\mathcal{X}\times\mathcal{Y}$. Let $h: \mathcal{X}\rightarrow \mathbb{R}$ denote a discriminant function, with which a class label is predicted for test input point $x$ as $\hat{y} = \mathop{\rm sign}\limits(h(x))$.
The goal is to train the discriminant function $h$ so that the expected misclassification rate is minimized. Let us define the 0-1 loss as \begin{equation} L_{01}(h,x,y) = 1_{yh(x)\leq0}, \label{0-1loss} \end{equation} where $1_A$ is the indicator function that takes 1 if statement $A$ is true and 0 otherwise. Then, we can express this problem as \begin{eqnarray*} h^* &=& \mathop{\rm argmin}\limits_{h} R(h), \\ R(h) &=& \mathbb{E}_{p(x,y)} \left[ L_{01}(h, x, y) \right] , \end{eqnarray*} where $h^*$ denotes the optimal discriminant function and $\mathbb{E}_{p(x,y)}$ denotes the expectation over $p(x,y)$. In practice, since we do not know the true density $p(x,y)$, we usually use the empirical distribution to approximate the expectation: \begin{equation} \hat{R}(h) = \frac{1}{N} \sum_{i=1}^{N} L_{01}(h, x_i, y_i). \label{empiricalrisk} \end{equation}
Based on Eqs.~(\ref{0-1loss}) and (\ref{empiricalrisk}), we can formulate various classification methods depending on loss functions \citep{book:Sugiyama:2015}. In the rest of this section, we introduce the support vector machine (SVM) \citep{book:Vapnik:1995}, which is one of the most basic algorithms of binary classification.
\subsection{Support vector machine (SVM)} Because optimization with $L_{01} (h,x,y)$ is computationally intractable, it is not practical to optimize the empirical risk $\hat{R}(h)$ directly. To overcome this problem, the hinge loss, an upper bound of $L_{01}(h,x,y)$ called the hinge loss, defined by \begin{equation} L_\mathrm{H}(h,x,y)=\max\left(1-yh(x),0\right), \label{L_H} \end{equation} has been introduced as its surrogate. Since the hinge loss is convex, optimization can be reduced to a convex program. Further introducing the L2 regularization, basis functions $\phi_1(x),\ldots,\phi_N(x)$, and slack variables $\xi=(\xi_1,\ldots,\xi_N)^\top$ with $^\top$ being the transpose, the following quadratic program can be obtained as a dual optimization problem:
\begin{eqnarray}
(\hat{w}, \hat{\xi}) &=& \mathop{\rm argmin}\limits_{(w, \xi)}\left[\frac{\lambda}{2} \| w \|^2 + \frac{1}{N} \sum_{i=1}^{N}\xi_i \right] \nonumber \\ &&\mathrm{s.t.} \left( \begin{array}{c} \xi_i \geq 1 - y_i h_i\\ \xi_i \geq 0 \end{array} \right) \mathrm{for}~i=1,\ldots,N, \end{eqnarray} where $w=(w_1,\ldots,w_N)^\top$ are the coefficients of the discriminant function, $\lambda>0$ is the L2 regularization parameter, and $h_i$ is the value of the discriminant function at sample point $x_i$ given by $h_i=\sum_{j=1}^{N}w_j\phi_{j}(x_i)$. The resulting discriminant function is given by $h(x; \hat{w}) = \sum_{j=1}^N \hat{w}_j \phi_j(x)$.
\section{Classification with ambiguous data}\label{sec:cad} In this section, we formulate our target problem called \emph{classification with ambiguous data} (CAD) and propose a new method for solving the CAD.
\subsection{Formulation} We consider three class labels, i.e,. positive, ambiguous, and negative: $y \in \mathcal{Y}_0=\{1,0,-1\}$. Suppose that we are given a set of positive, ambiguous, and negative samples $\{(x_i,y_i)\}_{i=1}^{N} $ drawn independently from the probability distribution with density $p_0(x,y)$ defined on $\mathcal{X}\times\mathcal{Y}_0$. Our goal is still to learn a discriminant function that classifies test samples into either the positive or negative class (not in the ambiguous class). Our key question in this scenario is if we can utilize the ambiguous training data to improve the classification accuracy of the discriminant function.
In this section, we develop a new method based on a method of \emph{classification with reject option} (CRO) \citep{cortes2016learning}. For this reason, before deriving the new method, we first review the CRO method.
\subsection{Classification with reject option by SVM (CRO-SVM)} \cite{cortes2016learning} introduced a rejection function $r:\mathcal{X}\rightarrow\mathbb{R}$ to identify the regions with high risk for misclassification, in addition to discriminating the positive and negative classes. When the rejection function takes a positive value, the corresponding sample is accepted and is classified into the positive or negative class by classifier $h$; otherwise, the sample is rejected and is not classified. When a sample is rejected, the rejection cost $c$ is incurred, which trades off the risk of misclassification and the cost of rejection. To realize this idea, the \emph{0-1-c loss} was introduced: \begin{equation} L_\mathrm{01c}(h,r,x,y) = 1_{yh(x)\leq0}1_{r(x)>0}+c1_{r(x)\leq0}. \label{L_0-1-c} \end{equation}
When $c=0$, all samples are rejected because the loss function does not incur any cost. On the other hand, when $c\geq0.5$, no samples are rejected because the expectation of the 0-1 loss is less than $0.5$; in that case, the 0-1-$c$ loss is reduced to the 0-1 loss. Therefore, effectively, we only consider $c$ such that $0<c<0.5$. The rejection function and the discriminant function are simultaneously learned from training data.
Similarly to the 0-1 loss, the 0-1-$c$ loss has discrete nature and thus its direct optimization is computationally intractable. To avoid the discontinuity, the following surrogate loss called the \emph{max-hinge (MH) loss} was introduced: \begin{equation} L_\mathrm{MH}(h,r,x,y) = \max\left(1+\frac{\alpha}{2}\left(r(x)-yh(x)\right),c\left(1-\beta r(x)\right),0\right), \end{equation} where $\alpha, \beta>0$ are the hyperparameters to control the shape of the surrogate loss.
In the same manner as the original SVM, introducing the L2 regularization, basis functions, and slack variables yields the following quadratic program: \begin{eqnarray}
(\hat{w}, \hat{u}, \hat{\xi}) &=& \mathop{\rm argmin}\limits_{(w, u, \xi)}\left[\frac{\lambda}{2} \| w \|^2 + \frac{\lambda'}{2} \| u\|^2 + \frac{1}{N} \sum_{i=1}^{N}\xi_i \right] \nonumber \\ &&\mathrm{s.t.} \left( \begin{array}{c} \xi_i \geq 1 + \frac{\alpha}{2}\left(r_i - y_i h_i \right) \\ \xi_i \geq c(1-\beta r_i) \\ \xi_i \geq 0 \end{array} \right) \mathrm{for}~i=1,\ldots,N, \label{h_i} \end{eqnarray} where $w=(w_1,\ldots,w_N)^\top$ are the coefficients of the discriminant function, $u=(u_1,\ldots,u_N)^\top$ are the coefficients of the rejection function, $\lambda,\lambda'>0$ are the L2 regularization parameters, and
$h_i$ and $r_i$ denote the values of the discriminant function and rejection function at sample point $x_i$ given by $h_i = \sum_{j=1}^{N}w_j \phi_j(x_i)$ and $r_i=\sum_{j=1}^{N}u_j \phi_j(x_i)$, respectively. The resulting discriminant function and rejection function are given by $h(x; \hat{w}) = \sum_{j=1}^N \hat{w}_j \phi_j(x)$ and $r(x; \hat{u}) = \sum_{j=1}^N \hat{u}_j \phi_j(x)$.
We refer to this method as CRO-SVM.
\subsection{Proposed method: classification with ambiguous data by SVM (CAD-SVM)} To handle ambiguous training data in the SVM formulation, we extend the 0-1-$c$ loss to the \emph{0-1-c-d loss} defined as \begin{equation} L_\mathrm{01cd}(h,r,x,y) = 1_{y^2=1}\left(1_{yh(x)\leq0}1_{r(x)>0}+c1_{r(x)\leq0}\right) + d 1_{y=0} 1_{r(x)>0}. \label{L_0-1-c-d} \end{equation} Table~\ref{fig_0-1-c_loss} and Table~\ref{fig_0-1-c-d_loss} compare the behavior of the 0-1-$c$ loss and the 0-1-$c$-$d$ loss. For positive and negative samples, the 0-1-$c$-$d$ loss behaves the same as the 0-1-$c$ loss. On the other hand, for ambiguous samples, the 0-1-$c$-$d$ loss incurs penalty $d$ when they are classified into the positive or negative class. Therefore, ambiguous samples tend to be classified into the ambiguous class if we employ the 0-1-$c$-$d$ loss. Compared to the CRO formulation, where a rejector cannot be learned explicitly from positive and negative samples, the CAD utilizes ambiguous samples to learn a rejector explicitly.
The above discussion may mislead us as if we are just solving a 3-class problem with the positive, ambiguous, and negative classes. However, we do not classify test samples into the ambiguous class, but only into the positive and negative classes. To solve the CAD problem, we utilize a binary discriminant function $h$ and a rejection function $r$, as in the CRO formulation reviewed above. More specifically, we train $h$ and $r$ under the 0-1-$c$-$d$ loss, and we only use $h$ in the test phase to classify test samples into the positive and negative classes. Thanks to the interplay between $h$ and $r$ in the 0-1-$c$-$d$ loss, we can utilize ambiguous data to train $h$ through $r$.
\begin{table} \centering \caption{The 0-1-$c$ loss function.} \scriptsize
\begin{tabular}{|l|c|c|c|} \hline \diagbox{Label $y$}{Judgement $(h,r)$} & \begin{tabular}{c}Positive\\$h>0$\\$r>0$\end{tabular} & \begin{tabular}{c}Rejected \\$ r\leq 0 $\end{tabular}& \begin{tabular}{c}Negative \\$ h \leq 0$ \\ $ r > 0 $\end{tabular} \\ \hline Positive $\quad y=1$ & 0 & $c$ & 1 \\ \hline Negative $\quad y=-1$ & 1 & $c$ & 0 \\ \hline \end{tabular}\label{fig_0-1-c_loss}
\vspace*{5mm} \caption{The 0-1-$c$-$d$ loss function.} \scriptsize
\begin{tabular}{|l|c|c|c|} \hline \diagbox{Label $y$}{Judgement $(h,r)$} & \begin{tabular}{c}Positive\\$h>0$\\$r>0$\end{tabular} & \begin{tabular}{c}Ambiguous \\$ r\leq 0 $\end{tabular}& \begin{tabular}{c}Negative \\$ h \leq 0$ \\ $ r > 0 $\end{tabular} \\ \hline Positive $\quad y=1$ & 0 & $c$ & 1 \\ \hline Ambiguous $\quad y=0$ & $d$ & 0 & $d$ \\ \hline Negative $\quad y=-1$ & 1 & $c$ & 0 \\ \hline \end{tabular}\label{fig_0-1-c-d_loss} \end{table}
Similarly to the 0-1-$c$ loss, we consider the following convex upper bound of the 0-1-$c$-$d$ loss called the \emph{max-hinge-ambiguous (MHA) loss} as a surrogate to avoid its discrete nature: \begin{eqnarray}
L_\mathrm{01cd}(h,r,x,y) &\leq& 1_{y^2=1} L_\mathrm{MH}(h,r,x,y) + d 1_{y=0} \max\left(1+\beta r(x), 0\right) \nonumber \\ &=& y^2 \max \left( 1 + \frac{\alpha}{2}\left(r(x) - yh(x)\right), c\left(1 - \beta r(x)\right), 0 \right) \nonumber \\ && + (1-y^2) \max \left(d\left(1+\beta r(x)\right), 0\right) \nonumber \\ &\leq& y^2 \max \left( 1 + \frac{\alpha}{2}\left(r(x) - yh(x)\right), \eta c\left(1 - \beta r(x)\right), 0 \right) \nonumber \\ && + (1-y^2) \max \left(\eta d\left(1+\beta r(x)\right), 0\right) \nonumber \\ &\equiv& L_\mathrm{MHA} (h,r,x,y), \end{eqnarray} where $\eta\geq 1$ is the hyperparameter to control the shape of the surrogate loss. See Figure~\ref{fig7} for its visualization.
\begin{figure}
\caption{The 0-1-$c$-$d$ loss $L_\mathrm{01cd}$ and its surrogate loss $L_\mathrm{MHA}$ for the penalty values $(c, d) = (0.2, 0.5)$.}
\label{fig7}
\end{figure}
Then, in the same way as the CRO-SVM, we have the following quadratic program:
\begin{eqnarray}
(\hat{w}, \hat{u}, \hat{\xi})&=& \mathop{\rm argmin}\limits_{(w,u,\xi)}\left[ \frac{\lambda}{2} \|w \|^2 + \frac{\lambda'}{2} \| u \|^2 + \frac{1}{N}\sum_{i=1}^{N}\xi_i \right] \nonumber \\ &&\mathrm{s.t.} \left( \begin{array}{c} \xi_i \geq y_i^2\left(1+\frac{\alpha}{2}(r_i - y_i h_i)\right) \\ \xi_i \geq y_i^2 \eta c(1-\beta r_i) \\ \xi_i \geq (1-y_i^2) \eta d(1+\beta r_i) \end{array} \right) \mathrm{for}~i=1,\ldots,N. \end{eqnarray} This is our proposed method called CAD-SVM.
The computational complexity of the CAD-SVM depends on implementation of the quadratic program. It naively costs $O(N^3)$, but if we use a fixed number of basis functions, the complexity reduces to $O(N)$.
The MHA loss depends on the choice of hyperparameters $(\alpha, \beta, \eta)$. To find good hyperparameter values, let us analyze the 0-1-$c$-$d$ loss first.
For each $x\in\mathcal{X}$, let $\pi_+(x) = p_0(y=1|x)$, $\pi_0(x) = p_0(y=0|x)$, and $\pi_-(x) = p_0(y=-1|x)$, where $\pi_+(x) +\pi_0(x) + \pi_-(x) = 1$. Then the following lemma shows how $c$ and $d$ are related to $\pi_+(x)$ and $\pi_-(x)$ for the optimal classifier and rejector (its proof is available in Appendix~\ref{sec:proof-lemma}): \begin{lemma} \label{lem1} For each $x\in\mathcal{X}$, let \begin{equation} \left(h_\mathrm{01cd}^*, r_\mathrm{01cd}^*\right)
= \mathop{\rm argmin}\limits_{(h,r)} \mathbb{E}_{p_0(y|x)} \left[ L_\mathrm{01cd}(h,r,x,y) \right]. \end{equation} Then \begin{equation} \left\{ \begin{array}{ll} \mathop{\rm sign}\limits(h^*_\mathrm{01cd})=1,~\mathop{\rm sign}\limits(r^*_\mathrm{01cd})=1 & \mathrm{if}~\pi_+\geq\frac{\displaystyle d+(1-c-d)\pi_-}{\displaystyle c+d}, \\ \mathop{\rm sign}\limits(h^*_\mathrm{01cd})=-1,~\mathop{\rm sign}\limits(r^*_\mathrm{01cd})=1 & \mathrm{if}~\pi_-\geq\frac{\displaystyle d+(1-c-d)\pi_+}{\displaystyle c+d}, \\ \mathop{\rm sign}\limits(r^*_\mathrm{01cd})=-1& \mathrm{otherwise}. \end{array} \right. \end{equation}
\end{lemma} Figure~\ref{fig1} illustrates the above results. This shows that when $\pi_+$ (or $\pi_-$) is large (i.e., imbalanced classification), the rejector accepts the sample and the classifier classifies that sample into the positive (or negative) class. On the other hand, when both $\pi_+$ and $\pi_-$ are not so large, the rejector rejects the sample.
\begin{figure}
\caption{Optimal solutions for the 0-1-$c$-$d$ loss.}
\label{fig1}
\end{figure}
Next, based on the above lemma, we have the following theorem for the MHA loss (its proof is given in Appendix~\ref{sec:proof-theorem}): \begin{theorem} \label{theo1} For each $x\in\mathcal{X}$, let \begin{equation} \left(h_\mathrm{MHA}^*, r_\mathrm{MHA}^*\right)
= \mathop{\rm argmin}\limits_{(h,r)} \mathbb{E}_{p_0(y|x)} \left[ L_\mathrm{MHA}(h,r,x,y) \right]. \end{equation} Then, for \begin{equation}
\alpha^* = 2(1-2c), \quad\beta^* = 1+2c, \quad\eta^*=\frac{2}{1+2c}, \label{theo1eq1} \end{equation} the signs of $(h^*_\mathrm{MHA}, r^*_\mathrm{MHA})$ match those of $(h^*_\mathrm{01cd},r^*_\mathrm{01cd})$. \end{theorem}
Based on the above theorem, we use Eq.~(\ref{theo1eq1}) as hyperparameter values in our experiments in the next section and demonstrate that they work well in practice. Nevertheless, given that the above theorem is valid only for the optimal solutions, we may cross-validate better hyperparameter values around Eq.~(\ref{theo1eq1}) to further improve the classification performance. Note that Eq.~(\ref{theo1eq1}) does not include $d$.
\section{Numerical experiments} \label{Num} In this section, we report experimental results.
\subsection{Datasets} For experiments, we use a toy dataset, a public dataset, and an in-house dataset.
\subsubsection{Toy dataset} To understand the behavior of our method and related methods, we created a toy classification problem and applied the methods to it. The problem contains three regions: the positive, negative, and mixed regions (see Figure~\ref{fig_MM}). The positive and negative regions are clearly separable, whereas the mixed region has no good discriminant function. For this problem, we want to clearly discriminate the positive and negative regions, with the influence of the mixed region avoided.
We also studied how the proportion of ambiguous samples influences the results by changing the proportion of ambiguous samples in the mixed region $r$. The number of all samples is 400, the total number of positive and negative samples in the separable regions is $200$, and the total number of positive and negative samples in the mixed region is $200 (1-r)$. Therefore, the expected maximum accuracy is $\{1 + (1-r) \times0.5\}/\{1 + (1-r)\}$.
\begin{figure}
\caption{Toy dataset. The lower-left and lower-right regions are negative and positive regions, while the upper region is the mixed region. The numbers indicate the numbers of samples in each region. For each region, samples are distributed uniformly.}
\label{fig_MM}
\end{figure}
\subsubsection{Public datasets (PD1, PD2, PD3)}
We processed a regression dataset, the Boston Housing Dataset \citep{harrison1978hedonic}, to convert it to a classification dataset with ambiguous data. The original dataset consists of 13 features $\xi_i\in\mathbb{R}^{13}$ associated with the average house prices $\zeta_i\in\mathbb{R}$ for 506 districts, where $1\leq i \leq 506$ denotes the sample number. We annotated all the samples to be positive, negative, and ambiguous according to the following procedure:
\begin{description} \item[PD1 (P/N/A separable):] Simply, the samples with $\zeta_i > 23$ were labeled as positive, the samples with $\zeta_i < 19$ were labeled as negative, and the other samples were labeled as ambiguous. The numbers of samples were 190, 173, and 143 for the positive, negative, and ambiguous classes, respectively. \item[PD2 (P/N/A mixed):] The samples with $\zeta_i > 23$ and the samples with $\zeta_i < 19$ were still labeled as positive and negative, respectively, but the remaining 143 samples were randomly labeled as positive, negative, or ambiguous.
\item[PD3 (separable and mixed):] We considered a hyperplane $v\cdot\xi=0$ in the feature space and divided all the samples into two parts, the mixed part $\{i|v\cdot\xi_i\geq0\}$ and the separable part $\{i|v\cdot\xi_i<0\}$. The coefficients of the hyperplane $v$ are selected so that the averages of $\zeta_i$ over the samples in both parts were approximately matched. For the mixed part, each sample is randomly labeled as positive, negative or ambiguous, whereas for the separable part, the samples with $\zeta_i > 21$ were labeled as positive and the others were labeled as negative. The numbers of samples in the separable region were 83, 0, and 87, and those in the mixed region were 114, 107, and 115 for the positive, negative, and ambiguous classes, respectively. \end{description}
Figure~\ref{figs_PCA} (a)--(c) and (e)--(g) show the 2-dimensional plots of the above datasets visualized by the principle component analysis (PCA) and the locality preserving projection (LPP) \citep{NIPS16:He+Niyogi:2004}. On the whole, ambiguous points are located between positive and negative points.
\subsubsection{In-house dataset from a cell culture process (ID)} As a real-world application, we prepared an in-house cell culture dataset. This dataset contained 124 fields of view (FOV). For each FOV, 2 images were acquired: one was in the middle, and the other was at the end of the culturing process. Each middle image was analyzed by the image processing software, CL-Quant \citep{alworth2010teachable}, and converted to 8 morphological features such as the average brightness and average area of cells.
Each final image was annotated by experts. If the cells in the image overall looked healthy/damaged, the image was labeled as positive/negative. However, some images contained both healthy and damaged cells, and they were labeled as ambiguous. The numbers of samples were 41, 59, and 24 for positive, negative, and ambiguous, respectively.
Our motivation was to predict the final state of each FOV annotated by the experts, using morphological features obtained in the middle of the culturing process. If we could predict it accurately, the culturing cost would be saved by aborting the culturing process where the cells would be damaged while keeping the healthy cells cultured. Therefore, we should focus on reducing misclassification between the positive and negative classes and we hope that information from ambiguous samples could be useful to improve the prediction accuracy. That was why test samples did not have the ambiguous label and were not classified into the ambiguous class in our scenario. Though ambiguous samples may occur in the test phase in the actual curturing process, we did not care which class they were classified into.
In the same manner as the PD1, PD2, and PD3 datasets, this dataset was also visualized by the PCA and the LPP in Figure~\ref{figs_PCA} (d) and (h), respectively. They show that ambiguous points are roughly located between positive and negative points.
\begin{figure}
\caption{Two-dimensional visualization of the PD1, PD2, PD3, and ID datasets by the PCA and the LPP. Red, blue, and green points correspond to positive, negative and ambiguous samples, respectively.}
\label{figs_PCA}
\end{figure}
\subsection{Experimental settings} Using the above datasets, we compared the classification performance of the SVM, the SVM-RL (random label), the LapSVM \citep{belkin2006manifold}, the two-step SVM, the CRO-SVM, the CRO-SVM-RL, and the CAD-SVM. The SVM-RL employs the SVM algorithm, but ambiguous samples are randomly relabeled as positive or negative, which effectively utilizes information of the ambiguous samples. The LapSVM is a semi-supervised learning method based on the SVM, which employs a regularization term defined by the graph Laplacian. Ambiguous samples are treated as unlabeled samples in the LapSVM. The two-step SVM is the method which learns the rejection function and the discriminant function sequentially---first the rejection function is learned to judges whether the sample is ambiguous or not, and then the discriminant function is learned only using the samples which are not rejected by the rejection function. When the rejection function is learned, class weights $c$ and $d$ are applied to non-ambiguous (i.e., positive and negative) and ambiguous samples, respectively. The CRO-SVM-RL is the CRO-SVM with random labels for ambiguous samples in the same manner as SVM-RL.
For each method, 500 test runs were performed by changing the training and test datasets which were randomly divided from the original dataset. The dividing ratio of the training and test dataset was 4:1 for the ID dataset and 1:2 for the other datasets. For each test run, 5-fold cross validation was performed to determine the parameters below. For validation and in the test phase, only positive and negative samples were applied to the discriminant function, thus we were able to evaluate the binary classification accuracy.
Note that our goal was to maximize the binary classification accuracy in the test phase, and that was equivalent to minimizing the expected 0-1 loss. Though we trained the models using various loss functions such as the hinge loss, the MH loss, and the MHA loss, the 0-1 loss was minimized by cross validation.
In total, we had 10 hyperparameters $(\lambda, \lambda', \sigma, \sigma', \tau, c, d, \alpha, \beta, \eta)$, where $(\lambda, \lambda')$ were the L2 regularization parameters, $\sigma$ was the width of the Gaussian radial basis function in the basis functions $\phi_i(x) = \exp\left( -\frac{\| x-x_i\|^2}{2\sigma^2}\right)$, $\sigma'$ was the hyperparameter of the weight matrix $W$ of the graph Laplacian as $W_{ij} = \exp\left( - \frac{\|x_i-x_j\|^2}{2\sigma'^2}\right)$ (only in the LapSVM), $\tau$ was the coefficient of the graph Laplacian regularization (only in the LapSVM), $(c, d)$ were the hyperparameters of the 0-1-$c$ loss (in the CRO-SVM and CRO-SVM-RL), the 0-1-$c$-$d$ loss (in the CAD-SVM) or the class weights (in the two-step SVM), and $(\alpha, \beta, \eta)$ were the hyperparameters of the MH and MHA loss functions (in the CRO-SVM, CRO-SVM-RL and CAD-SVM). We used 5-fold cross validation in terms of the classification accuracy to choose the hyperparameters from $(\lambda, \lambda')\in\{10^{-3}, 10^{-5}, 10^{-7}\}$, $(\sigma, \sigma') \in \{10^{0.5}, 10^{0.75}, 10^1\}$, $\tau\in\{10^{-1}, 10^{-2}, 10^{-3}\}$, $c \in \{0.03, 0.06, 0.20, 0.45\}$, and $d\in\{0.03, 0.06, 0.20, 0.50\}$. The other hyperparameters $(\alpha, \beta, \eta)$ were determined by Eq.~(\ref{theo1eq1}). Quadratic programming problems were solved using cvxopt \citep{cvxopt}.
\subsection{Results}
Figure~\ref{figs_MM} shows an example of the discriminant results for the toy dataset with $r=0.5$. The upper half of the domain was the mixed region of positive and negative samples and samples in that region were expected to be classified into the ambiguous class. On the other hand, the lower half of the domain was perfectly separable into the positive and negative regions and samples in those regions were expected to be discriminated accurately (see Figure~\ref{fig_MM}). The SVM made some misclassifications in the lower half of the domain, which were caused by the mixed region. The SVM-RL and LapSVM could not reduce the number of misclassifications, though they utilized information of ambiguous samples. This suggests that ambiguous samples should not be simply relabeled to positive or negative samples or should not be treated as unlabeled samples. The two-step SVM could not also reduce the number of misclassifications and it could not learn the ambiguous region. Thus, it would be disadvantageous to learn the ambiguous region by combining positive and negative classes. The CRO-SVM and CRO-SVM-RL successfully learned the ambiguous region but that did not lead to learning more accurate discriminant functions. The CAD-SVM also successfully learned the ambiguous region and then it was able to learn a more accurate discriminant function.
Overall, the CAD-SVM was shown to be the most appropriate method in this toy experiment.
\begin{figure}
\caption{An example of discriminant results for the toy dataset.}
\label{figs_MM}
\end{figure}
Tables~\ref{tab_toy2} shows the test accuracy of the toy dataset by changing the parameters $r$ which are the propotion of ambiguous samples in the mixed region. When the proportion of the ambiguous samples was small, the proposed method did not work effectively since the number of ambiguous samples was too small. On the other hand, when the proportion of the ambiguous samples was large, it also did not show better performance. This was because, under this condition, only small numbers of positive and negative samples existed in the mixed region. Therefore, methods which do not utilize ambiguous samples showed reasonably good enough performance. However, when the proportion of ambiguous samples was $r=0.5$, the CAD-SVM achieved a better score than the other methods.
\begin{table}[] \centering
\caption{Test accuracy for the toy dataset by changing the ratio $r$ of ambiguous samples in the mixed region. The boldface numbers show the best and equivalent results with 5\% t-test.} \scriptsize \begin{tabular}{lrrrrr} \hline $r$&0.1&0.3&0.5&0.7&0.9\\ \hline \hline SVM&$\bf{0.739}$&$0.767$&$0.814$&$0.853$&$0.931$\\ \hline SVM-RL&$\bf{0.739}$&$0.766$&$0.812$&$0.850$&$0.926$\\ \hline LapSVM&$\bf{0.739}$&$0.766$&$0.814$&$0.850$&$0.930$\\ \hline Two-step SVM&$\bf{0.737}$&$\bf{0.771}$&$0.816$&$0.854$&$\bf{0.935}$\\ \hline CRO-SVM&$0.736$&$0.767$&$\bf{0.820}$&$0.857$&$0.931$\\ \hline CRO-SVM-RL&$0.735$&$0.768$&$0.817$&$\bf{0.861}$&$0.926$\\ \hline CAD-SVM&$0.736$&$0.767$&$\bf{0.822}$&$0.857$&$0.932$\\ \hline
\end{tabular} \label{tab_toy2} \end{table}
Table~\ref{tab_res} summarizes the test accuracy of each method for the PD1, PD2, PD3, and ID datasets, respectively. For the PD1 dataset, the CAD-SVM was not superior to other methods since this condition was similar to the toy dataset with higher $r$. For the PD2 dataset, though the dataset had a mixed region, the CAD-SVM also did not show good performance. However, for the PD3 dataset, which had a mixed region and a separable region, the CAD-SVM achieved the best performance among the compared methods. It is suggested that our proposed method, the CAD-SVM, works effectively when the dataset has both of a mixed region and a separable boundary between the positive and negative classes. The CAD-SVM utilizes ambiguous samples to learn a rejector which rejects mixed regions, thus it would be able to focus on learning a separable region. The CRO-SVM and the CRO-SVM-RL also learn a rejector from positive and negative samples, but the CAD-SVM would be superior since the CAD-SVM can explicitly learn a rejector using ambiguous samples. For the ID dataset, the CAD-SVM performed better than the other method, except for the CRO-SVM-RL. Overall, ambiguous samples can improve the binary classification accuracy under some conditions, and the CAD-SVM is one of the solutions that can utilize such ambiguous samples.
\begin{table}[] \centering \caption{Test accuracy for the PD1, PD2, PD3, and ID datasets, where $\pm$ denotes the standard deviation. The boldface numbers show the best and equivalent results with 5\% t-test.} \scriptsize \begin{tabular}{lrrrr} \hline &PD1&PD2&PD3&ID\\ \hline \hline SVM&$\bf{0.924\pm0.019}$&$\bf{0.828\pm0.019}$&$0.635\pm0.029$&$0.803\pm0.089$\\ \hline SVM-RL&$0.918\pm0.021$&$\bf{0.827\pm0.020}$&$0.629\pm0.030$&$0.806\pm0.088$\\ \hline LapSVM&$0.921\pm0.020$&$0.824\pm0.021$&$0.629\pm0.031$&$0.802\pm0.088$\\ \hline Two-step SVM&$0.918\pm0.024$&$0.823\pm0.021$&$0.632\pm0.030$&$0.790\pm0.091$\\ \hline CRO-SVM&$\bf{0.922\pm0.020}$&$\bf{0.827\pm0.020}$&$0.635\pm0.029$&$\bf{0.812\pm0.090}$\\ \hline CRO-SVM-RL&$0.917\pm0.026$&$\bf{0.828\pm0.022}$&$0.632\pm0.029$&$\bf{0.820\pm0.086}$\\ \hline CAD-SVM&$0.921\pm0.020$&$\bf{0.828\pm0.019}$&$\bf{0.639\pm0.028}$&$\bf{0.818\pm0.089}$\\ \hline
\end{tabular} \label{tab_res} \end{table}
\subsection{Discussions}
Through all the experiments, the CRO-SVM-RL gave as good performance as the CAD-SVM. As detailed in Appendix~\ref{sec:proof-cad-svm-rl}, we can show the following relations between the CRO-SVM-RL and the CAD-SVM: \begin{enumerate} \item{The 0-1-$c$ loss function for the randomly labeled (RL) dataset, in which we randomly relabeled ambiguous samples as positive or negative, reduces to the 0-1-$c$-$d$ loss with $d=\frac{1}{2}-c$.} \item{For the RL dataset, the MH loss can be regarded as a surrogate loss of the 0-1-$c$-$d$ loss with $d=\frac{1}{2}-c$, and it is calibrated under the conditions of Eq.~(\ref{theo1eq1}).} \end{enumerate} Thus, though it is a simple heuristic, the CRO-SVM-RL is essentially equivalent to the CAD-SVM except for that the hyperparameter $d$ is fixed to $\frac{1}{2}-c$. In practice, the CRO-SVM-RL is easier to implement and thus it may be used as an alternative to the CAD-SVM. However, we note that the CRO-SVM-RL alone does not provide rich theoretical insights that we have shown in Section~\ref{sec:cad}.
Our goal for each experiment was minimizing the expected 0-1 loss in the test phase. Therefore, the SVM was naively an appropriate method since the hinge loss was calibrated to the 0-1 loss. Nevertheless, though the CRO-SVM-RL and the CAD-SVM minimized a surrogate of the 0-1-$c$-$d$ loss in the training phase, they were able to achieve the better accuracy than the SVM in the test phase. It is suggested that the providing a reject option and incorporating ambiguous training samples could work as a kind of regularization, but further studies will be needed to clarify its mathematical properties.
We note that ambiguous samples are intrinsically hard-to-label samples even by experts, so they usually contain little information for classifying positive and negative samples, and thus we cannot expect large improvement to the binary classification accuracy. Nevertheless, our proposed method achieved statistically significant improvements for some cases.
\section{Conclusion} In this study, we aimed to reduce the labeling cost and improve the classification accuracy by allowing labelers to give ``ambiguous'' labels for difficult samples. We extended a method of classification with reject option and proposed a novel classification method named the CAD-SVM that uses the 0-1-$c$-$d$ loss. We derived a surrogate loss for the 0-1-$c$-$d$ loss, which allowed us to convert the optimization problem into a convex quadratic program. We carried out numerical experiments and showed that ambiguous labels can be effectively used to improve the classification accuracy. We also showed that the CRO-SVM-RL, in which we randomly relabeled ambiguous samples to be positive or negative and applied classification with reject option, can be a practical alternative to the proposed method since it is essentially equivalent to the proposed method.
Though our proposed method was based on the SVM, it would be more useful if it can be applied to other models especially deep neural networks. However, further experimental studies will be needed to confirm if a naive application of the proposed MHA loss works well in practice. Indeed, it is known that changing models can cause other problems such as overfitting \citep{kiryo2017positive}. Moreover, for deep neural networks, though we usually use the softmax cross entropy as the loss function, even the 0-1-$c$ loss function has not been extended to the softmax cross entropy. So, analyzing the influence of changing loss functions is also an important issue to be further investigated.
In addition to the experimental analysis with more complex models, our future study will conduct theoretical analysis of the proposed method such as statistical consistency and the rate of convergence. Extending the proposed loss function to semi-supervised problems, imperfect labeling problems or multi-class problems is also a promising direction to be pursued.
\appendix \section{Proof of~Lemma \ref{lem1}}\label{sec:proof-lemma} We calculate the expectation value of $L_\mathrm{01cd}$ as follows: \begin{eqnarray}
&&\mathbb{E}_{y\sim p_0(y|x)}[L_\mathrm{01cd}(h,r,x,y)] \nonumber \\ &=&\pi_+ L_\mathrm{01cd}(h,r,x,+1) + \pi_0 L_\mathrm{01cd}(h,r,x,0) \nonumber + \pi_- L_\mathrm{01cd}(h,r,x,-1) \nonumber \\ &=&\left\{ \begin{array}{ll} d\pi_0 + \pi_- & \mathrm{if}~\mathop{\rm sign}\limits(h)=1, \mathop{\rm sign}\limits(r)=1,\\ d\pi_0 + \pi_+ & \mathrm{if}~\mathop{\rm sign}\limits(h)=-1, \mathop{\rm sign}\limits(r)=1, \\ c(\pi_+ + \pi_-) & \mathrm{otherwise}. \end{array} \right. \label{lem1eq1} \end{eqnarray} Then, the minimum of the expectation value is \begin{eqnarray}
&&\min_{(h,r)}\mathbb{E}_{y\sim p_0(y|x)}[L_\mathrm{01cd}(h,r,x,y)] \nonumber \\ &=& \min\left( d\pi_0 + \pi_- , d\pi_0 + \pi_+, c(\pi_+ + \pi_-) \right) \nonumber \\ &=& \left\{ \begin{array}{ll} d\pi_0+\pi_- & \mathrm{if}~\pi_+\geq\frac{ d+(1-c-d)\pi_-}{ c+d},\\ d\pi_0+\pi_+ & \mathrm{if}~\pi_-\geq\frac{ d+(1-c-d)\pi_+}{ c+d},\\ c(\pi_+ + \pi_-)& \mathrm{otherwise}. \end{array} \right. \label{lem1eq2} \end{eqnarray} From the comparison of Eqs.~(\ref{lem1eq1}) and (\ref{lem1eq2}), the optimal $(h,r)$ subject to $(\pi_+, \pi_-)$ are determined. \qed
\section{Proof of Theorem~\ref{theo1}}\label{sec:proof-theorem} At the condition of Eq. (\ref{theo1eq1}),
the expectation value of the loss is \begin{eqnarray}
&&\mathbb{E}_{y\sim p_0(y|x)}[L_\mathrm{MHA}(h,r,x,y)] \nonumber \\ &=&\pi_+ \max \left( 1 + (1-2c)\left(r(x) - h(x)\right), \frac{2c}{1+2c} - 2c r(x), 0 \right) \nonumber\\* &&+\pi_- \max \left( 1 + (1-2c)\left(r(x) + h(x)\right), \frac{2c}{1+2c} - 2c r(x), 0 \right) \nonumber\\* &&+\pi_0 \max \left(\frac{2d}{1+2c} + 2d r(x), 0\right) . \label{theo1eq2} \end{eqnarray} To find the minimum of the expectation value, we derive a linear programming problem considering $r(x), h(x)$ as independent variables.
As shown in Figure~\ref{theo1fig1}, we can calculate the boundary conditions subject to $(h, r)$ as, \begin{equation}
\mathbb{E}_{y\sim p_0(y|x)}[L_\mathrm{MHA}(h,r,x,y)] = \left\{ \begin{array}{ll} \frac{4}{1+2c}(d\pi_0+\pi_-) & \mathrm{if}~(h,r)=\left(\frac{ 2}{ 1-4c^2}, \frac{ 1}{ 1+2c}\right), \\ \frac{4}{1+2c}(d\pi_0+\pi_+) & \mathrm{if}~(h,r)=\left(-\frac{ 2}{ 1-4c^2}, \frac{ 1}{ 1+2c}\right), \\ \frac{4}{1+2c}c(\pi_++\pi_-) & \mathrm{if}~(h,r)=\left(0, -\frac{ 1}{ 1+2c}\right). \end{array} \right.\label{theo1eq3} \end{equation}
\begin{figure}
\caption{The boundary conditions in the linear programming problem of Eq.~(\ref{theo1eq2}).}
\label{theo1fig1}
\end{figure}
Thus, we determine the minimizers of $(h, r)$ subject to $(\pi_+, \pi_-)$.
\begin{eqnarray}
&(h^*_\mathrm{MHA}, r^*_\mathrm{MHA}) = \mathop{\rm argmin}\limits_{(h,r)} \mathbb{E}_{y\sim\mathrm{Pr_0}(y|x)}[L_\mathrm{MHA}(h,r,x,y)],&\\ & \left\{ \begin{array}{ll} h^*_\mathrm{MHA}=\frac{2}{1-4c^2}>0,\quad r^*_\mathrm{MHA}=\frac{ 1}{ 1+2c}>0 & \mathrm{if}~\pi_+\geq\frac{ d+(1-c-d)\pi_-}{ c+d},\\ h^*_\mathrm{MHA}=-\frac{2}{1-4c^2}<0,\quad r^*_\mathrm{MHA}=\frac{ 1}{ 1+2c}>0 & \mathrm{if}~\pi_-\geq\frac{ d+(1-c-d)\pi_+}{ c+d},\\ h^*_\mathrm{MHA}=0,\quad r^*_\mathrm{MHA}=-\frac{ 1}{ 1+2c}<0 & \mathrm{otherwise}. \end{array} \right.& \end{eqnarray} \qed
\section{Relation between the CRO-SVM-RL and the CAD-SVM}\label{sec:proof-cad-svm-rl} For a dataset which contains positive, negative and ambiguous samples, we define a RL (randomly labeled) dataset as a dataset in which ambiguous samples are randomly relabeled into the positive or negative classes. \begin{theorem} \label{theo2} The risk of the 0-1-$c$ loss function for the RL dataset is equal to the risk of the 0-1-$c$-$d$ loss with $d=\frac{1}{2}-c$ for the original dataset. \end{theorem}
{\it Proof}
Let a relabeled label be $z\in\{-1, 1\}$ which satisfies \begin{equation}
\mathrm{Pr}(z=1|x)=\pi_+(x)+{\scriptstyle\frac{1}{2}}\pi_0(x), \qquad \mathrm{Pr}(z=-1|x)=\pi_-(x)+{\scriptstyle\frac{1}{2}}\pi_0(x). \end{equation} Thus, we can calculate the risk of the 0-1-$c$ loss for the relabeled label $z$ as \begin{eqnarray}
&&\mathbb{E}_{z\sim \mathrm{Pr}(z|x)}[L_\mathrm{01c}(h,r,x,z)] \nonumber \\ &=&\left( \pi_+(x) + {\scriptstyle\frac{1}{2}}\pi_0(x) \right) (1_{h<0}1_{r\geq0} + c1_{r<0}) + \left( \pi_-(x) + {\scriptstyle\frac{1}{2}}\pi_0(x) \right) (1_{h>0}1_{r\geq0} + c1_{r<0}) \nonumber \\
&=&\pi_+(x) (1_{h<0}1_{r\geq0} + c1_{r<0}) + \pi_-(x)(1_{h>0}1_{r\geq0} + c1_{r<0}) + \pi_0(x) \left({\scriptstyle\frac{1}{2}}-c\right) 1_{r\geq0} + c \pi_0(x) \nonumber \\
&=&\mathbb{E}_{y\sim p_0(y|x)}\left[\left.L_\mathrm{01cd}\left(h,r,x,y\right)\right|_{d={\scriptstyle\frac{1}{2}}-c}\right] + \pi_0(x)c. \end{eqnarray} Therefore, the risk is equal to that of the 0-1-$c$-$d$ loss for the original label $y$. Note that the second term in the right-hand side is constant with respect to $h$ and $r$. \qed
\begin{theorem} \label{theo3} The MH loss for the RL dataset is convex and an upper bound of the 0-1-$c$-$d$ loss with $d=\frac{1}{2}-c$ for the original dataset. \end{theorem}
{\it Proof}
For the ambiguous label, we can calculate the expectation value of the MH loss function over the relabeling process as \begin{eqnarray}
&&\mathbb{E}_{z\sim \mathrm{Pr}(z|x, y=0)}[L_\mathrm{MH}(h,r,x,z)] \nonumber \\ &=&{\scriptstyle\frac{1}{2}}L_\mathrm{MH}(h,r,x,z=1)+{\scriptstyle\frac{1}{2}}L_\mathrm{MH}(h,r,x,z=-1) \nonumber \\ &=&{\scriptstyle\frac{1}{2}}\max\left(1+{\scriptstyle\frac{\alpha}{2}}\left(r(x)-h(x)\right),c\left(1-\beta r(x)\right),0\right) \nonumber \\ &&+{\scriptstyle\frac{1}{2}}\max\left(1+{\scriptstyle\frac{\alpha}{2}}\left(r(x)+h(x)\right),c\left(1-\beta r(x)\right),0\right) \nonumber \\ &\geq&L_\mathrm{01cd}(h, r, x, y=0). \end{eqnarray} Since $L_\mathrm{MH}(h,r,x,z=\pm1)$ is convex, the expectation is also convex. For positive and negative labels, it can be calculated in the same manner. \qed
\begin{theorem} \label{theo4} The expectation of the risk of the MH loss for the RL dataset is calibrated to the 0-1-$c$-$d$ loss with $d=\frac{1}{2}-c$ for the original dataset under the conditions of Eq.~(\ref{theo1eq1}). \end{theorem}
{\it Proof}
We minimize the expectation of the MH loss with respect to $(r, h)$ for the RL dataset as \begin{eqnarray}
&&\min_{(h,r)}\mathbb{E}_{z\sim \mathrm{Pr}(z|x)}[L_\mathrm{MH}(h,r,x,z)] \nonumber \\ &=&\min_{(h,r)} \left[ \pi_+(x) L_\mathrm{MH}(h,r,x,z=1) +\pi_-(x) L_\mathrm{MH}(h,r,x,z=-1) \vphantom{\scriptstyle\frac{1}{2}}\right. \nonumber \\ &&\hphantom{\min} \left. + \pi_0(x) {\scriptstyle\frac{1}{2}}\left(L_\mathrm{MH}(h,r,x,z=1)+L_\mathrm{MH}(h,r,x,z=-1)\right) \right]\nonumber \\ &=&\min_{(h,r)}\left[\frac{2}{1+2c}\times \left\{ \begin{array}{ll} 2\pi_+ + \pi_0 & (h=-\frac{2}{1-4c^2}, r=\frac{1}{1+2c}) \\ 2\pi_- + \pi_0 & (h=\frac{2}{1-4c^2}, r=\frac{1}{1+2c}) \\ 2c & (h=0, r=-\frac{1}{1+2c}) \end{array} \right.\right]. \end{eqnarray} Then, we calculate the minimizers as \begin{eqnarray}
&&\mathop{\rm argmin}\limits_{(h,r)}\mathbb{E}_{z\sim \mathrm{Pr}(z|x)}[L_\mathrm{MH}(h,r,x,z)] \nonumber \\ &=&\left\{ \begin{array}{ll} (-\frac{2}{1-4c^2}, \frac{1}{1+2c}) & \mathrm{if}~\pi_- \geq (1-2c) + \pi_+, \\ (\frac{2}{1-4c^2}, \frac{1}{1+2c}) & \mathrm{if}~\pi_+ \geq (1-2c) + \pi_-, \\ (0, -\frac{1}{1+2c}) & \mathrm{otherwise}. \end{array} \right. \end{eqnarray} The derived minimizers are consistent with Lemma~\ref{lem1} with $d=\frac{1}{2} - c $. \qed
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\author{Wenbin Guo\\ {\small Department of Mathematics, University of Science and Technology of China,}\\ {\small Hefei 230026, P. R. China}\\ {\small E-mail: [email protected]}\\ \\ Evgeny P. Vdovin\\ {\small Sobolev Institute of Mathematics and Novosibirsk State University,}\\ {\small Novosibirsk 630090, Russia}\\ {\small E-mail: [email protected]}}
\date{} \title{ space{-1cm}
\pagenumbering{arabic} \begin{abstract}
Denote by $\nu_p(G)$ the number of Sylow $p$-subgroups of $G$. It is not difficult to see that $\nu_p(H)\leqslant\nu_p(G)$ for $H\leqslant G$, however $\nu_p(H)$ does not divide $\nu_p(G)$ in general. In this paper we reduce the question whether $\nu_p(H)$ divides $\nu_p(G)$ for every $H\leqslant G$ to almost simple groups. This result substantially generalizes the previous result by G. Navarro and also provides an alternative proof for the Navarro theorem.
\noindent {\bf Key words:} Finite group, Number of Sylow subgroups, group of induced automorphism, $(rc)$-series.
\noindent {\bf MSC2010:} 20D20 \end{abstract}
\section*{Introduction}
Throughout this paper, all groups are finite. Let $G$ be a group and $p$ be a prime. Denote by $\nu_p(G)$ the number of Sylow $p$-subgroups of $G$. It is a trivial exercise to check that $\nu_p(H)\leqslant \nu_p(G)$ for every subgroup $H$ of $G$. However $\nu_p(H)$ does not necessarily divide $\nu_p(G)$ in general. For example, let $G=A_5$ and $H=A_4$, then $\nu_3(G)=10$ and $\nu_3(H)=4$. In 2003, G. Navarro in \cite{Navarro} proved that if $G$ is $p$-solvable then $\nu_p(H)$ divides $\nu_p(G)$ for every $H\leqslant G$.
We say that a group $G$ satisfies $\mathbf{DivSyl}(p)$ if $\nu_p(H)$ divides $\nu_p(G)$ for every $H\leqslant G$. Our goal is to prove that $G$ satisfies $\mathbf{DivSyl}(p)$ provided every nonabelian composition factor of $G$ satisfies $\mathbf{DivSyl}(p)$ (see Theorem~\ref{main} below). This result substantially generalizes the result by Navarro since in $p$-solvable groups all nonabelian composition factors evidently satisfy $\mathbf{DivSyl}(p)$. Our technique can also be applied to derive the Navarro theorem, so we provide an alternative proof for the Navarro theorem.
In order to formulate the main theorem, we need to recall some definitions.
Let $A,B,H$ be subgroups of $G$ such that $B\trianglelefteqslant A$. Define the {\em normalizer} $N_H(A/B)$ of $A/B$ in $H$ by $N_H(A)\cap N_H(B)$. If $x\in N_H(A/B)$, then $x$ induces an automorphism of $A/B$ acting by $Ba\mapsto B x^{-1}ax$. Thus there exists a homomorphism $N_H(A/B)\rightarrow \Aut(A/B)$. The image of the homomorphism is denoted by $\Aut_H(A/B)$ and is called the {\em group of $H$-induced automorphisms} on $A/B$, while the kernel of the homomorphism is denoted by $C_H(A/B)$. If $B=1$, then we simply write $\Aut_H(A)$. The groups of induced automorphisms were introduced by F. Gross in \cite{GrossExistence}, where the author says that this notion is taken from unpublished lectures by H. Wielandt. Clearly $C_H(A/B)=C_G(A/B)\cap H$, so $$\Aut_H(A/B)=N_H(A/B)/C_H(A/B)\simeq N_H(A/B)C_G(A/B)/C_G(A/B)\leqslant \Aut_G(A/B),$$ that is, $\Aut_H(A/B)$ can be embedded into $\Aut_G(A/B)$ in a natural way. Hence, without lose of generality, we may assume that $\Aut_H(A/B)$ is a subgroup of $\Aut_G(A/B)$.
A composition series is called an {\em $(rc)$-series}\footnote{This notion was introduced by V. A. Vedernikov in \cite{Ved}} if it is a refinement of a chief series.
\begin{Theo}\label{main} {\em (Main Theorem)} Let $$1=G_0<G_1<\ldots<G_n=G$$ be an $(rc)$-series of $G$. Assume that, for each nonabelian $G_i/G_{i-1}$ and for every $p$-subgroup $P$ of $\Aut_G(G_i/G_{i-1})$, the group $P(G_i/G_{i-1})$ satisfies {\em $\mathbf{DivSyl}(p)$}. Then $G$ satisfies $\mathbf{DivSyl}(p)$. \end{Theo}
At the end of the paper, we discuss possible way of improving this statement and also explain how Navarro's result can be derived from this theorem.
\section{Preliminaries}
We write $H\leqslant G$ and $H\trianglelefteqslant G$ if $H$ is a subgroup of $G$ and $H$ is a normal subgroup of $G$ respectively, $p$ denotes a prime number. A natural number $n$ is called a {\em $p$-number} (respectively a {\em $p'$-number}) if it is a power of $p$ (respectively if it is coprime to $p$). We denote by $n_p$ and $n_{p'}$ the $p$- and $p'$- part of the natural number $n$. A group is said to be a {\em $p$-group} (respectively a {\em $p'$-group}) if its order is a $p$-number (respectively a $p'$-number). A group $G$ is called {\em $p$-solvable} if it possesses a subnormal series with all sections either $p$- or $p'$-groups. If $\varphi:X\rightarrow Y$ is a map, then $\varphi(x)$ denotes the image of $x\in X$, while for $\sigma\in\Sym_k$ we write $i\sigma$ for the image of $i\in\{1,\ldots,k\}$ under $\sigma$. By $Syl_p(G)$ we denote the set of all Sylow $p$-subgroups of~$G$. All unexplained notations and definitions can be found in~\cite{Isaacs}.
\begin{Lemma}\label{nupfactor} Assume that $A\trianglelefteqslant G$. Then $\nu_p(G)=\nu_p(G/A)\cdot \nu_p(PA)$ for some (hence for every) $P\in Syl_p(G)$, and $\nu_p(PA)=\vert A:N_A(P)\vert$. \end{Lemma}
\begin{proof} Consider a natural homomorphism $$\overline{\phantom{G}}:G\rightarrow \overline{G}=G/A.$$ Clearly $\nu_p(PA)$ is the number of all Sylow $p$-subgroups $Q$ of $G$ such that image of $Q$ equals $\overline{P}$ since $\overline{P}=\overline{Q}$ if and only if $Q\leqslant PA$. By Sylow theorem, for every $Q\in Syl_p(G)$, there exists $x\in G$ such that $P^x=Q$, so \begin{equation*} (PA)^x=P^xA^x=P^xA=QA. \end{equation*} This shows that $\nu_p(PA)$ does not depend on the choice of $P$, and so it is the same for any Sylow $p$-subgroup of $G$. Thus we obtain $\nu_p(G)=\nu_p(G/A)\cdot \nu_p(PA)$. By Sylow theorem, \begin{equation*} \nu_p(PA)=\vert PA:N_{PA}(P)\vert =\vert A:N_A(P)\vert, \end{equation*} where the second identity follows from the Dedekind theorem. \end{proof}
\begin{Lemma}\label{extension} Assume that $A\trianglelefteqslant G$ and both $AP$ and $G/A$ satisfy $\mathbf{DivSyl}(p)$ for some (hence for every) $P\in Syl_p(G)$. Then $G$ satisfies $\mathbf{DivSyl}(p)$. \end{Lemma}
\begin{proof} Let $H\leqslant G$. Choose $Q\in Syl_p(G)$ so that $Q\cap H\in Syl_p(H)$. Since $PA$ and $QA$ are conjugate, $QA$ satisfies $\mathbf{DivSyl}(p)$. Now by Lemma \ref{nupfactor}, $\nu_p(G)=\nu_p(G/A)\cdot \nu_p(QA)$, and \begin{equation*} \nu_p(H)=\nu_p(H/(H\cap A))\cdot \nu_p((Q\cap H)(H\cap A))=\nu_p(HA/A)\cdot \nu_p((Q\cap H)(H\cap A)). \end{equation*} Since $HA/A\leqslant G/A$ and $(Q\cap H)(H\cap A)\leqslant QA$, the conditions of the lemma imply that $\nu_p(HA/A)$ divides $\nu_p(G/A)$ and $\nu_p((Q\cap H)(H\cap A))$ divides $\nu_p(QA)$. This implies that $\nu_p(H)$ divides~$\nu_p(G)$. \end{proof}
The following lemma is proven in \cite{Navarro}, but now it can be easily derived from the above two lemmas and we provide here an alternating proof of it.
\begin{Lemma}\label{psolvable} {\em \cite[Theorem~A]{Navarro}} Let $H$ be a subgroup of a $p$-solvable group $G$. Then $\nu_p(H)$ divides $\nu_p(G)$. \end{Lemma}
\begin{proof} Assume by contradiction that $G$ is a counterexample of minimal order, $H\leqslant G$ is chosen so that $\nu_p(H)$ does not divides $\nu_p(G)$ and $A$ is a minimal normal subgroup of $G$. Let $P\in Syl_p(G)$ with $P\cap H\leqslant Syl_p(H)$. By Lemma \ref{nupfactor}, we have $$\nu_p(G)=\nu_p(G/A)\cdot \nu_p(PA)$$ and $$\nu_p(H)=\nu_p(HA/A)\cdot \nu_p((H\cap P)(H\cap A)).$$ By induction, $\nu_p(HA/A)$ divides $\nu_p(G/A)$. Since $G$ is $p$-solvable, $A$ is either a $p$-group or a $p'$-group.
If $A$ is a $p$-group, then $\nu_p((H\cap P)(H\cap A))=1$, so $\nu_p(H)$ divides $\nu_p(G)$.
If $A$ is a $p'$-group, then $\vert A\vert$ and $\vert P\vert$ are corpime, in particular $P\cap A=1$. Since $P$ normalizes $A$, we have that $[N_A(P),P]\leqslant A\cap P=1$, so $N_A(P)=C_A(P)$. The known properties of coprime action (for example, see \cite[Excersize~3E.4]{Isaacs}) and Lemma 1 implies that $\nu_p((H\cap P)(H\cap A))=\vert (H\cap A):C_{H\cap A}(H\cap P)\vert$ divides $\vert A:C_A(H\cap P)\vert$, while $\vert A:C_A(H\cap P)\vert$ divides $\vert A:C_A(P)\vert=\nu_p(PA)$. Thus $\nu_p(H)$ divides $\nu_p(G)$, which contradicts the choice of $G$ and $H$. \end{proof}
Recall that a group $A$ is said to be an almost simple group if there exists a simple nonabelian group $S$ such that $S\leqslant A\leqslant \Aut(S).$ By Schreier conjecture $\Aut(S)/S$ is solvable for each simple group $S$.
\begin{Cor}\label{almsimple} Let $A$ be an almost simple group with simple socle $S$ and $P\in Syl_p(A)$. If $PS$ satisfies $\mathbf{DivSyl}(p)$, then $A$ satisfies $\mathbf{DivSyl}(p)$. \end{Cor}
\begin{proof} This follows from Lemmas \ref{extension} and \ref{psolvable} since $A/S$ is solvable. \end{proof}
\begin{Lemma}\label{inducedautomoprhisms} {\em \cite[Theorem~1]{VdoInd}} Let $$1=G_0<G_1<\ldots<G_n=G$$ be an $(rc)$-series of $G$ and denote the section $G_i/G_{i-1}$ by $S_i$. Let $$1=H_0<H_1<\ldots<H_n=G$$ be a composition series of $G$. Then there exist a permutation $\sigma\in \Sym_n$ such that $\Aut_G(H_i/H_{i-1})\leqslant \Aut_G(S_{i\sigma})$ for $i=1,\ldots,n$. Moreover, if the second series is an $(rc)$-series, then $\sigma$ can be chosen so that $\Aut_G(H_i/H_{i-1})\simeq\Aut_G(S_{i\sigma})$ for $i=1,\ldots,n$. \end{Lemma}
The following simple lemma plays technical role.
\begin{Lemma}\label{normpbypdashbyp} Let $G$ be a finite group, possessing a normal series $$1\lhd H\lhd K\lhd G$$ such that $H$ is a $p$-group, $K/H$ is a $p'$-group and $G/K$ is a $p$-group. Let $Q$ be a $p$-subgroup of $G$ containing $H$ and $P$ be a Sylow $p$-subgroup of $G$ containing $Q$. Consider a natural homomorphism $$\overline{\phantom{G}}:G\rightarrow \overline{G}=G/H.$$ Then $\overline{N_K(Q)}=N_{\overline{K}}(\overline{Q})=C_{\overline{K}}(\overline{Q})$ and $N_K(P)\leqslant N_K(Q)$. \end{Lemma}
\begin{proof} $\overline{N_K(Q)}\subseteq N_{\overline{K}}(\overline{Q})$ is evident. Consider $\bar{x}\in N_{\overline{K}}(\overline{Q})$. Then every inverse image $x$ of $\bar{x}$ normalizes the full inverse image of $\overline{Q}$. Since $H\leqslant Q$, the full inverse image of $\overline{Q}$ equals $Q$, so $x$ normalizes $Q$. Consequently, $N_{\overline{K}}(\overline{Q})\subseteq \overline{N_K(Q)}$. By the conditions of the lemma, $\overline{K}=O_{p'}(\overline{G})$, so $\overline{K}\trianglelefteqslant\overline{G}$ and $\vert \overline{K}\vert$ and $\vert \overline{Q}\vert$ are coprime. Thus $N_{\overline{K}}(\overline{Q})=C_{\overline{K}}(\overline{Q})$. Since $H\leqslant P$ and $P$ is a $p$-group the identity $\overline{N_K(P)}=N_{\overline{K}}(\overline{P})=C_{\overline{K}}(\overline{P})$ holds. Clearly $C_{\overline{K}}(\overline{P})\leqslant C_{\overline{K}}(\overline{Q}).$ It follows that $N_K(P)\leqslant N_K(Q)$. \end{proof}
In this paper we often consider subgroups of a permutation wreath product, so we fix notations for such groups. If $L$ is a group and $K$ is a subgroup of the symmetric group $\Sym_k$, then $L\wr K$ denotes the permutation wreath product. By definition $L\wr K$ is isomorphic to a semidirect product $(L_1\times\ldots\times L_k)\rtimes K$, where $L_1\simeq \ldots\simeq L_k\simeq L$ and $K$ permutes the $L_i$-s, that is arbitrary $\sigma\in K$ acts by \begin{equation*}\sigma:(g_1,\ldots,g_k)\mapsto (g_{1\sigma^{-1}},\ldots,g_{k\sigma^{-1}}).\end{equation*} We always denote $L_1\times\ldots\times L_k$ by $\mathbf{L}$.
Suppose that $G$ is a subgroup of $L\wr K$. Let $\rho:G\rightarrow \Sym_k$ be the homomorphism corresponding to the action of $G$ on $\{L_1,\ldots,L_k\}$ by conjugation. Each element $g$ of $G$ as an element from $L\wr K$ can be uniquely written as \begin{equation}\label{canonicalformwreath} g=(g_1,\ldots,g_k)\rho(g),\text{ where }g_i\in L_i,\text{ and }\rho(g)\in\Sym_k\text{ permutes the }g_i\text{-s}. \end{equation} Given $g=(g_1,\ldots,g_k)\rho(g)\in L\wr K$ define $\pi_i(g)$ to be equal to $g_i$. In particular, the restriction of $\pi_i$ on $\mathbf{L}$ is the natural projection on~$L_i$.
Now if $S$ is a normal subgroup of $L$ then we can construct a normal subgroup $\mathbf{S}=S_1\times\ldots\times S_k$ of $L\wr K$ such that $S_1\simeq\ldots\simeq S_k\simeq S$ and $(L\wr K)/\mathbf{S}\simeq (L/S)\wr K$. In order to construct $\mathbf{S}$ we need to take one representative $L_j$ from every $K$-orbit on $\{L_1,\ldots L_k\}$, the in each representative we need to take $S_j\trianglelefteqslant L_j$ such that $S_j\simeq S$ and $L_j/S_j\simeq L/S$, and finally define $\mathbf{S}$ to be equal to the normal closure of the subgroup generated by $S_j$-s. From the construction it is evident, that if $K$ is transitive, then $\mathbf{S}$ is uniquely defined by the choice of $S_1\leqslant L_1$.
\begin{Lemma}\label{inclusionwreathproduct} Assume that $T$ is a unique minimal normal subgroup of $G$. Assume also that $T$ is nonabelian, that is, $T=S_1\times\ldots\times S_k$, where $S_1,\ldots,S_k$ are isomorphic (moreover, $G$-conjugate) nonabelian simple groups. Let $\rho:G\rightarrow \Sym_k$ be the permutation representation corresponding to the action of $G$ on $\{S_1,\ldots,S_k\}$ by conjugation. Then there exists an injective homomorphism $$\varphi:G\rightarrow \Aut_G(S_1)\wr \rho(G)=\left(\Aut_G(S_1)\times\ldots\times \Aut_G(S_k)\right)\rtimes \rho(G).$$ Moreover, if $H\leqslant G$ and $HT=G$, then $\varphi$ can be chosen so that for every $H\leqslant X\leqslant G$ we have $$\varphi(X)\leqslant \Aut_X(S_1)\wr\rho(X)=\Aut_X(S_1)\wr \rho(G)=\left(\Aut_X(S_1)\times\ldots\times \Aut_X(S_k)\right)\rtimes \rho(X).$$ \end{Lemma}
\begin{proof} The existence of $\varphi$ is known (see, for example, \cite[Lemma~2]{VdoZen}). Hence, we only need to show that $\varphi(X)\leqslant \Aut_X(S_1)\wr \rho(X)$. Here we provide the construction of $\varphi$ and remain all technical details for the reader.
First notice that $\rho(G)$ is transitive since $T$ is a minimal normal subgroup. By the Fundamental counting principle (see \cite[Theorem~1.4]{Isaacs}), the action of $G$ on $\{S_1,\ldots,S_k\}$ is equivalent to the action of $G$ on right cosets of $N_G(S_1)$ by right multiplication. By \cite[Theorem~IV.1.4 a)]{Huppert}, there exists an injective homomorphism \begin{equation*}
\psi:G\rightarrow \left(N_G(S_1)\times\ldots\times N_G(S_k)\right)\rtimes\rho(G)=N_G(S_1)\wr\rho(G), \end{equation*} defined in the following way. Fix a right coset representatives $r_1,\ldots,r_k$ of $N_G(S_1)$ in $G$. For given $g\in G$, define the element $n_i(g)$ of $N_G(S_1)$ by \begin{equation*}
r_ig=n_i(g)r_{i\rho(g)}. \end{equation*} Then $\psi$ maps $g$ to $(n_1(g),\ldots,n_k(g))\rho(g)$.
Consider a normal subgroup $C_G(S_1)$ of $N_G(S_1)$. Since $\rho(G)$ is transitive, $C_G(S_1)\times\ldots\times C_G(S_k)$ is a normal subgroup. Consider a natural homomorphism \begin{equation*}
\theta:N_G(S_1)\wr\rho(G)\rightarrow \left(N_G(S_1)\wr\rho(G)\right)/\left(C_G(S_1)\times\ldots\times C_G(S_k)\right). \end{equation*} Since $T$ is the unique minimal normal subgroup of $G$ and $T$ is nonabelian, we have ${\psi(G)\cap Ker(\theta)=1}$, while $\theta(N_G(S_1)\wr\rho(G))=\Aut_G(S_1)\wr\rho(G)$ by definition.
Now assume that $H\leqslant G$ is chosen so that $HT=G$. Then the right coset representatives $r_1,\ldots,r_k$ of $N_G(S_1)$ can be chosen from $H$. Hence if $H\leqslant X\leqslant G$, then for every $x\in X$ and $i=1,\ldots,k$, we have $n_i(x)=r_ixr_{i\rho(x)}^{-1} \in X\cap N_G(S_1)=N_X(S_1)$. Therefore $\psi(X)\leqslant N_X(S_1)\wr\rho(X)$, and so $\varphi(X)=\theta\psi(X)\leqslant \Aut_X(S_1)\wr\rho(X)$. \end{proof}
\begin{Lemma}\label{numpwreath} Let $L$ be a group with a normal subgroup $S$ such that $L/S$ is a $p$-group. Let $K$ be a transitive $p$-subgroup of $\Sym_k$. Choose $S_1\leqslant L_1$ with $S_1\simeq S$ and $L_1/S_1\simeq L/S$, then $\mathbf{S}=S_1^{L\wr K}=S_1\times \ldots\times S_k$. Let $G$ be a subgroup of $L\wr K$ satisfying the following conditions: \begin{itemize} \item[{\em (a)}] $\mathbf{S}\leqslant G$; \item[{\em (b)}] $G\mathbf{L}=L\wr K$; \item[{\em (c)}] $\pi_i(N_G(S_i))=L_i$. \item[{\em (d)}] for some $Q\in Syl_p(G)$, if $P_i=\pi_i(N_Q(S_i))$, then
$Q\leqslant (P_1\times\ldots\times P_k)\rtimes K$. \end{itemize} Then the restriction of $\pi_i$ on $N_G(S_i)$ is a homomorphism and $\nu_p(G)=\vert S\vert_{p'}^{k-1}\cdot\nu_p(L)$. \end{Lemma}
\begin{proof} Since $G\mathbf{L}=L\wr K$, we have $\rho(G)=G\mathbf{L}/\mathbf{L}\simeq G/(G\cap \mathbf{L})\simeq K$, in particular, $G/(G\cap \mathbf{L})$ is a $p$-group. Hence $G=Q (G\cap \mathbf{L})$ and $L\wr K=Q\mathbf{L}$, in particular $\rho(Q)=K$. Now $g=(g_1,\ldots,g_k)\rho(g)\in N_{L\wr K}(S_i)$ if and only if $i\rho(g)=i$. For every $g,h\in L\wr K$ we have $\pi_i(g)\cdot \pi_i(h)=g_i\cdot h_i$. On the other hand, if we take $g,h\in N_{L\wr K}(S_i)$, then \begin{multline*}
\pi_i(gh)=\pi_i((g_1,\ldots,g_k)\rho(g)\cdot (h_1,\ldots,h_k)\rho(h))\\= \pi_i((g_1,\ldots,g_k)\cdot (h_1,\ldots,h_k)^{\rho(g)^{-1}}\cdot\rho(g) \rho(h))\\=
\pi_i((g_1,\ldots,g_k)\cdot (h_{1\rho(g)},\ldots,h_i,\ldots,h_{k\rho(g)})\cdot\rho(g) \rho(h))\\= \pi_i((g_1\cdot h_{1\rho(g)},\ldots, g_i\cdot h_i,\ldots g_k
\cdot h_{k\rho(g)})\cdot\rho(g)\rho(h)))=g_i\cdot h_i, \end{multline*} thus the restriction of $\pi_i$ on $N_G(S_i)$ is a homomorphism.
Since $L/S$ is a $p$-group and $K$ is a $p$-group, we have that $(L\wr K)/\mathbf{S}$ is a $p$-group. Hence $G/\mathbf{S}$ is a $p$-group and $Q\mathbf{S}=G$. Denote $\pi_i(Q\cap \mathbf{S})$ by $Q_i$. Clearly $Q\cap S_i\leqslant Q_i$. On the other hand, since $S_i$ is subnormal in $G$, $Q\cap S_i$ is a Sylow $p$-subgroup of $S_i$, so $Q\cap S_i=Q_i$. Since $\mathbf{S}$ normalizes $S_i$ and $G=Q\mathbf{S}$, we have $N_G(S_i)=N_Q(S_i)\mathbf{S}$, therefore $\pi_i(N_G(S_i))=P_iS_i=L_i$. But since \begin{multline*} \vert P_i\vert=\vert P_i/(P_i\cap S_i)\vert \cdot \vert P_i\cap S_i\vert=\\ \vert P_i/(P_i\cap S_i)\vert \cdot \vert \pi_i(N_Q(S_i))\cap S_i\vert= \vert P_i/(P_i\cap S_i)\vert \cdot \vert \pi_i(N_Q(S_i)\cap S_i)\vert=\\ \vert P_i/(P_i\cap S_i)\vert \cdot \vert \pi_i(Q\cap S_i)\vert = \vert L_i/S_i\vert\cdot \vert Q_i\vert=\vert L_i\vert_p, \end{multline*} we obtain $P_i\in Syl_p(L_i)$. Moreover, since $Q$ as a subgroup of $G$ permutes $S_i$-s, we have that $Q$ permutes $N_Q(S_i)$-s, and so $Q$ permutes $P_i$-s. This implies that $Q$ normalizes $P_1\times \ldots\times P_k$. Consequently, $\rho(Q)=K$ normalizes $P_1\times \ldots\times P_k$ and the subgroup $P=(P_1\times \ldots\times P_k)\rtimes K$ in condition (d) of the lemma is correctly defined. Since $P_i\in Syl_p(L_i)$ and $K$ is a $p$-group, we also have that $P\in Syl_p(L\wr S)$. Then by condition (d), $Q\leqslant P$, and so $Q=P\cap G$.
Since $G/S$ is a $p$-group, Lemma \ref{nupfactor} implies that \begin{equation*} \nu_p(G)=\vert \mathbf{S}:N_{\mathbf{S}}(Q)\vert \end{equation*} so \begin{equation*}
\nu_p(G)=\frac{\vert \mathbf{S}\vert_{p'}}{\vert N_{\mathbf{S}}(Q)\vert_{p'}}. \end{equation*}
Since $L_1=P_1S_1$, by Lemma \ref{nupfactor} again, we have \begin{equation}\label{step6identity} \nu_p(L_1)=\vert S_1:N_{S_1}(P_1)\vert=\frac{\vert S_1\vert_{p'}}{\vert N_{S_1}(P_1)\vert_{p'}}. \end{equation} If we can show that \begin{equation}\label{principalstep6}
\vert N_{\mathbf{S}}(Q)\vert_{p'}=\vert N_{S_1}(P_1)\vert_{p'}, \end{equation}
then \eqref{step6identity} and \eqref{principalstep6} would imply \begin{equation*} \nu_p(G)=\frac{\vert \mathbf{S}\vert_{p'}}{\vert N_{\mathbf{S}}(Q)\vert_{p'}}=\frac{\vert \mathbf{S}\vert_{p'}}{\vert S_1\vert_{p'}}\cdot \frac{\vert S_1\vert_{p'}}{\vert N_{S_1}(P_1)\vert_{p'}}=\vert S_1\vert_{p'}^{k-1}\cdot \nu_p(L_1) \end{equation*} and so the lemma follows.
Thus we remain to prove \eqref{principalstep6}.
Denote $N_{S_i}(P_i)$ by $X_i$ and let $\mathbf{X}=X_1\times\ldots\times X_k$. Clearly $Q_i=P_i\cap S_i\trianglelefteqslant X_i$, and since $Q_i$ is a Sylow $p$-subgroup of $S_i$, $Q_i$ is a (normal) Sylow $p$-subgroup of $X_i$. Therefore $Q_1\times\ldots\times Q_k=O_p(\mathbf{X})\in Syl_p(\mathbf{X})$ and $\mathbf{X}$ is an extension of a $p$-group by a $p'$-group. We show that $P$ normalizes $\mathbf{X}$ and that $N_{\mathbf{S}}(Q)$ lies in $\mathbf{X}$. By construction, $P$ permutes $P_i$-s, so $P$ permutes $X_i$-s, therefore $P$ normalizes $X_1\times\ldots\times X_k=\mathbf{X}$. In particular, $\rho(P)=\rho(Q)=K$ normalizes $\mathbf {X}$ and $\mathbf{X}K=X_1\wr K$.
In order to show that $N_{\mathbf{S}}(Q)\leqslant \mathbf{X}$, consider $x=(x_1,\ldots,x_k)\in N_{\mathbf{S}}(Q)$. Since $x$ normalizes $Q$ and $S_i$, we obtain that $x$ normalizes $N_Q(S_i)$ for $i=1,\ldots,k$. Take $y=(y_1,\ldots,y_k)\rho(y)\in N_Q(S_i)$. Then $i\rho(y)=i$. Recall that by condition (d) of the lemma $Q\leqslant P$, so $y_i\in P_i$ for $i=1,\ldots,k$. Since $x_j$ centralizes $y_t$ for $j\not=t$, we obtain \begin{multline*}
y^x=(y_1^{x_1},\ldots,y_k^{x_k})\cdot \rho(y)^x= (y_1^{x_1},\ldots,y_k^{x_k})\cdot (x_1^{-1},\ldots,x_k^{-1})\cdot
(x_1,\ldots,x_k)^{\rho(y)^{-1}}\cdot \rho(y)\\
=(y_1^{x_1},\ldots,y_k^{x_k})\cdot (x_1^{-1},\ldots,x_i^{-1},\ldots,x_k^{-1})\cdot (x_{1\rho(y)},\ldots,x_i,\ldots, x_{k\rho(y)})\cdot \rho(y)\\
= (y_1^{x_1}\cdot x_1^{-1}\cdot x_{1\rho(y)},\ldots,y_i^{x_i},\ldots y_k^{x_k}\cdot x_k^{-1}\cdot x_{k\rho(y)})\cdot \rho(y). \end{multline*} Thus $\pi_i(y^x)=y_i^{x_i}\in P_i$, i.e. $x_i$ lies in $X_i$ and so $N_{\mathbf{S}}(Q)$ lies in $\mathbf{X}$.
Consider $\mathbf{X}P\leqslant L\wr K$. Since $Q_1\times\ldots\times Q_k=O_p(\mathbf{X})\in Syl_p(\mathbf{X})$, the group $\mathbf{X}P$ has a normal series $$1\lhd Q_1\times\ldots\times Q_k=O_p(\mathbf{X})\trianglelefteqslant \mathbf{X}\trianglelefteqslant \mathbf{X}P.$$ By Lemma \ref{normpbypdashbyp}, $\overline{N_{\mathbf{X}}(Q)}=N_{\overline{\mathbf{X}}}(\overline{Q})=C_{\overline{\mathbf{X}}}(\overline{Q})$, where $$\overline{\phantom{G}}:P\mathbf{X}\rightarrow \overline{P\mathbf{X}}=P\mathbf{X}/(Q_1\times\ldots\times Q_k)$$ is a natural homomorphism.
Now consider the corresponding natural homomorphisms $$\overline{\phantom{G}}:(X_iP_i)\rightarrow \overline{X_iP_i}=(X_iP_i)/Q_i$$ (we simply restrict the above homomorphism on $X_iP_i$) for $i=1,\ldots,k$. Then each $x=(x_1,\ldots,x_k)\in \mathbf{X}$ is mapped to $\bar{x}=(\bar{x}_1,\ldots,\bar{x}_k)$ and $y=(y_1,\ldots,y_k)\rho(y)\in Q$ is mapped to $(\bar{y}_1,\ldots,\bar{y}_k)\rho(y)$. Since $X_i$ normalizes $P_i$, we obtain that $\overline{X}_i$ centralizes $\overline{P_i}$, i.e. $(\bar{x}_1,\ldots,\bar{x}_k)$ centralizes $(\bar{y}_1,\ldots,\bar{y}_k)$. So \begin{multline*} \bar{x}^{\bar{y}}=(\bar{x}_1,\ldots,\bar{x}_k)^{(\bar{y}_1,\ldots,\bar{y}_k)\rho(y)}=(\bar{x}_1^{\bar{y}_1},\ldots,\bar{x}_k^{\bar{y}_k})^{\rho(y)}=\\ (\bar{x}_1,\ldots,\bar{x}_k)^{\rho(y)}=(\bar{x}_{1\rho(y)^{-1}},\ldots,\bar{x}_{k\rho(y)^{-1}}). \end{multline*} Since $\rho(Q)=K\leqslant\Sym_k$ is transitive, we obtain that $$C_{\overline{\mathbf{X}}}(\overline{Q})=C_{\overline{\mathbf{X}}}(\rho(Q))=\{(\bar{x}_1,\ldots,\bar{x}_k)\mid \bar{x}_1=\ldots=\bar{x}_k\}.$$ Thus $$\vert N_{\mathbf{S}}(Q)\vert_{p'}=\vert \overline{N_{\mathbf{X}}(Q)}\vert=C_{\overline{X}}(\overline{Q})=\vert \overline{X_1}\vert = \vert X_1\vert_{p'}=\vert N_{S_1}(P_1)\vert_{p'}$$ and \eqref{principalstep6} holds. \end{proof}
\begin{Lemma}\label{numberindirectproduct} Assume $T$ is a normal subgroup of $G$ such that $T=G_1\times\ldots\times G_k$, and $G/T$ is a $p$-group. Assume also that $G$ acts on $\{G_1,\ldots,G_k\}$ by conjugation, i.e. for every $x\in G$ and $i=1,\ldots,k$ there exists $j\in\{1,\ldots,k\}$ with $G_i^x=G_j$. Denote by $\pi_i$ the natural projection $\pi_i:T\rightarrow G_i$. Let $H$ be a subgroup of $G$ such that for every $i=1,\ldots,k$ we have $\pi_i(H\cap T)$ equals either $G_i$ or $1$, that is, $H\cap T$ is a subdirect product of some of $G_i$-s. Then $\nu_p(H)$ divides $\nu_p(G)$. \end{Lemma}
\begin{proof} Suppose that this lemma is false and consider a counterexample $(G, H)$ with
$|G|$ minimal. Then $H\leqslant G$ is such that $H\cap T$ is a subdirect product of some of $G_i$-s and $\nu_p(H)$ does not divides $\nu_p(G)$.
Let $\Omega_1,\ldots,\Omega_s$ be $G$-orbits of $\{G_1,\ldots, G_k\}$ and $G_{i_j}$ be a fixed element of $\Omega_j$. In order to simplify notation, we assume $G_{i_1}=G_1$. Assume that $L_1$ is a minimal characteristic subgroup of $G_1$, then $L_1$ is a direct product of isomorphic simple groups, and either $L_1$ is elementary abelian or $L_1$ is a direct product of isomorphic nonabelian simple groups. For every $i=2,\ldots,s,$ fix the corresponding characteristic subgroup $L_{i_j}$ of $G_{i_j}$ so that $L_1\simeq L_{i_j}$ and $G_1/L_1\simeq G_{i_j}/L_{i_j}$. Denote by $L$ the normal closure of $\langle L_1,\ldots, L_{i_s}\rangle $ in $G$, that is, $L=\langle L_{i_j}^x\mid x\in G, j=1,\ldots,s\rangle$. Evidently $L$ is a direct product of some groups isomorphic to $L_1$. In particular, $L$ is abelian if and only if $L_1$ is abelian and $L$ is a direct product of isomorphic nonabelian simple groups if and only if $L_1$ is a direct product of isomorphic nonabelian simple groups. Since $G$ acts transitively by conjugation on each $G$-orbit, we obtain that $L=\pi_1(L)\times\ldots\times \pi_k(L)$, $\pi_1(L)\simeq\ldots\simeq\pi_k(L)$, and $G$ also acts by conjugation on $\{\pi_1(L),\ldots,\pi_k(L)\}$. Hence, if we consider the natural homomorphism $$\overline{\phantom{G}}:G\rightarrow \overline{G}=G/L,$$ then $\overline{T}=\overline{G}_1\times\ldots\times\overline{G}_k=(G_1/\pi_1(L))\times\ldots\times (G_k/\pi_k(L))$. Clearly $\overline{G}$ acts by conjugation on $\{\overline{G}_1,\ldots,\overline{G}_k\}$ and $\overline{G}/\overline{T}\simeq G/T$ is a $p$-group. This shows that $\overline{G}$ satisfies the conditions of the lemma. We also have \begin{equation*}
\pi_i(\overline{H}\cap \overline{T})=\pi_i(\overline{HL\cap T}), \end{equation*} so $\pi_i(\overline{H}\cap \overline{T})=\overline{G_i}$ if $\pi(H\cap T)=G_i$ and $\pi_i(\overline{H}\cap\overline{T})=1$ if $\pi_i(H\cap T)=1$. Thus $\overline{H}$ also satisfies conditions of the lemma. By induction, $\nu_p(\overline{H})$ divides $\nu_p(\overline{G})$.
Let $Q$ be a Sylow $p$-subgroup of $H$ and $P$ be a Sylow $p$-subgroup of $G$ containing $Q$. By Lemma \ref{nupfactor}, $\nu_p(H)=\nu_p(Q(H\cap L))\cdot \nu_p(H/(H\cap L))=\nu_p(Q(H\cap L))\cdot\nu_p(\overline{H})$ and $\nu_p(G)=\nu_p(PL)\cdot \nu_p(G/L)=\nu_p(PL)\cdot \nu_p(\overline{G})$. Thus we need to show that $\nu_p(Q(H\cap L))$ divides $\nu_p(PL)$ in order to obtain a contradiction with the choice of $(G, H)$.
Assume first that $L$ is abelian. Then $PL$ is solvable, hence $\nu_p(Q(H\cap L))$ divides $\nu_p(PL)$ by Lemma~\ref{psolvable}. Thus we may assume that $G_1$ has no characteristic abelian subgroups and so $G_1$ has no normal abelian subgroups.
Now assume that $L$ is nonabelian. As we note above, $L_1$ is a is a direct product of isomorphic nonabelian simple groups. So $L$ is also a direct product of isomorphic nonabelian simple groups, say $L=S_1\times\ldots\times S_m$. Then $G$ acts by conjugation on $\{S_1,\ldots,S_m\}$. Denote by $\psi_j:L\rightarrow S_j$ the natural projection. If $\pi_i(H\cap T)=1$, then clearly $\psi_j(H\cap L)=1$ for every $S_j\leqslant G_i$. Assume that $\pi_i(H\cap T)=G_i$. Since $H\cap L\trianglelefteqslant H$, we obtain that $\pi_i(H\cap L)$ is a normal subgroup of $\pi_i(H\cap T)=G_i$. Moreover, $\pi_i(H\cap L)$ is a normal subgroup of $\pi_i(L)=\prod_{S_j\leqslant L_i} S_j$, so $\pi_i(H\cap L)$ is a product of some of $S_j$-s. Therefore, $H\cap L$ is a subdirect product of $S_j$-s. This shows that $(PL, Q(H\cap L))$ satisfies conditions of the theorem. If $PL\not=G$, then by induction $\nu_p(Q(H\cap L))$ divides $\nu_p(PL)$. Thus we may assume that $PL=G$.
Assume that $L$ is not a minimal normal subgroup in $PL$. Then there exists $K\trianglelefteqslant PL$ such that $K< L$. Since $L=S_1\times\ldots\times S_m$ is a direct product of isomorphic simple groups, we obtain $K=S_{i_1}\times\ldots\times S_{i_t}$ for some $\{i_1,\ldots,i_t\}\subseteq \{1,\ldots,m\}$. Consider a natural homomorphism $\varphi:PL\rightarrow PL/K$. Clearly $(\varphi(PL), \varphi(Q(H\cap L)))$ satisfies conditions of the lemma, so by induction $\nu_p(\varphi(Q(H\cap L)))$ divides $\nu_p(\varphi(PL))$. It is also clear that $(PK, Q(H\cap K))$ also satisfies the conditions of the lemma, so by induction $\nu_p(Q(H\cap K))$ divides $\nu_p(PK)$. By Lemma \ref{nupfactor}, $\nu_p(Q(H\cap L))=\nu_p(\varphi(Q(H\cap L)))\cdot \nu_p(Q(H\cap K))$ and $\nu_p(PL)=\nu_p(\varphi(PL))\cdot \nu_p(PK)$, so $\nu_p(Q(H\cap L))$ divides $\nu_p(PL)$. Hence, we may assume that $L$ is a minimal normal subgroup of~$PL$.
Now we have that $G=PL$, where $P\in Syl_p(G)$, $L=S_1\times\ldots\times S_m$ is a minimal normal subgroup of $G$, and $S_1,\ldots,S_m$ are isomorphic (even $G$-conjugate) nonabelian simple groups. By Lemma \ref{inclusionwreathproduct}, there exists an embedding $\varphi:G\rightarrow \Aut_G(S_1)\wr \rho(G)$, where $\rho:G\rightarrow \Sym_m$ is the permutation representation corresponding to the action of $G$ on $\{S_1,\ldots,S_m\}$ by conjugation. Moreover, since $G=PL$, $\varphi$ can be chosen so that for every $P\leqslant X\leqslant G$ we have \begin{equation}\label{incl}\varphi(X)\leqslant \Aut_X(S_1)\wr \rho(X)=\Aut_X(S_1)\wr\rho(G).\end{equation} We show that $\varphi(G)$ as a subgroup of $\Aut_G(S_1)\wr\rho(G)$ satisfies conditions of Lemma~\ref{numpwreath}.
Since $S_1\times \ldots\times S_m\leqslant Ker\ \rho$, we obtain that $\rho(G)=\rho(P)$ is a transitive $p$-subgroup of $\Sym_m$. Now $S_2\times \ldots \times S_m$ centralizes $S_1$, so $\Aut_G(S_1)/S_1$ is a $p$-group. Thus $\Aut_G(S_1)$ with normal subgroup $S_1$ plays the role of group $L$ in the notations of Lemma \ref{numpwreath}, while $\rho(G)$ is a group $K$ in the notations of Lemma~\ref{numpwreath}.
Since $\varphi(S_1\times\ldots\times S_m)=S_1\times\ldots\times S_m$, we obtain that $S_1\times\ldots\times S_m\leqslant \varphi(G)$ and condition (a) of Lemma \ref{numpwreath} is satisfied. Clearly $\varphi(G)(\Aut_G(S_1)\times\ldots\times\Aut_G(S_m))=\Aut_G(S_1)\wr\rho(G)$, so condition (b) of Lemma \ref{numpwreath} is also satisfied. By definition, $\pi_i(N_G(S_i))=\Aut_G(S_i)$, thus condition (c) is true. Finally, by \eqref{incl} we have $\varphi(P)\leqslant (\Aut_P(S_1)\times\ldots\times\Aut_P(S_m))\rtimes \rho(P)$ and by definition $\Aut_P(S_i)=\pi_i(N_P(S_i))$, so condition (d) is also true. Thus \begin{equation}\label{nupG} \nu_p(G)=\vert S_1\vert_{p'}^{m-1}\cdot \nu_p(\Aut_G(S_1)). \end{equation} On the other hand, by Lemma \ref{nupfactor}, $\nu_p(PL)=\vert L:N_L(P)\vert$ and $\nu_p(Q(H\cap L))=\vert (H\cap L):N_{H\cap L}(Q)\vert$.
Assume that $H\cap L\not=L$. Since $L$ is a direct product of nonabelian simple groups and $H\cap L$ is a subdirect product of some simple factors of $L$, we obtain that $\vert H\cap L\vert =\vert S_1\vert^s$ for some $s<m$. Clearly $\vert (H\cap L):N_{H\cap L}(Q)\vert$ divides $\vert S_1\vert_{p'}^s$, since $\vert S_1\vert_{p'}^s$ is the index of a Sylow $p$-subgroup of $H\cap L$. Since $s\leqslant m-1$, \eqref{nupG} implies $\nu_p(H)$ divides $\nu_p(G)$.
Assume finally that $H\cap L=L$. Then from $Q\leqslant P$ we obtain $Q\cap L=P\cap L\in Syl_p(L)$. Hence the group $PN_L(P\cap L)$ has the following normal series:$$1\lhd (P\cap L)\lhd N_L(P\cap L)\lhd PN_L(P\cap L),$$ where $P\cap L$ is a $p$-group, $N_L(P\cap L)/(P\cap L)$ is a $p'$-group, and $PN_L(P\cap L)/N_L(P\cap L)\simeq P/(P\cap N_L(P\cap L))$ is a $p$-group. By Lemma \ref{normpbypdashbyp}, $N_{N_L(P\cap L)}(Q)\geqslant N_{N_L(P\cap L)}(P)$. But, clearly, $N_{H\cap L}(Q)=N_L(Q)$ normalizes $Q\cap L$, so lies in $N_L(P\cap L)$. Hence $$N_{H\cap L}(Q)=N_L(Q)=N_{N_L(P\cap L)}(Q)\geqslant N_{N_L(P\cap L)}(P)= N_L(P),$$ and so $\nu_p(Q(H\cap L))$ divides $\nu_p(PL)$. \end{proof}
\section{Proof of the main theorem}
Assume this theorem is false and let $G$ be a counterexample of minimal order. We proceed in a series of steps to get a contradiction.
{\slshape Step 1.} If $1\not=A\trianglelefteqslant G$, then $G/A$ satisfies $\mathbf{DivSyl}(p)$.
Indeed, by Lemma \ref{inducedautomoprhisms}, the group of induced automorphisms does not depend on the choice of an $(rc)$-series. Hence we may assume that $$1=G_0<G_1<\ldots<G_n=G$$ goes through $A$, i.e. $A=G_i$ for some $i$. Consider a natural homomorphism $\overline{\phantom{G}}:G\rightarrow \overline{G}=G/A$. Clearly $$\overline{1}=\overline{G}_i<\overline{G}_{i+1}<\ldots<\overline{G}_n=\overline{G}$$ is an $(rc)$-series of $\overline{G}$. By \cite[Lemma~1.2]{VdCart}, $\Aut_{\overline{G}}(\overline{G}_j/\overline{G}_{j-1})\simeq \Aut_G(G_j/G_{j-1})$ for $j=i+1,\ldots,n$, so $\overline{G}$ satisfies $\mathbf{DivSyl}(p)$ by induction.
{\slshape Step 2.} The solvable radical $S(G)$ (that is the largest solvable normal subgroup of $G$) of $G$ is trivial.
Otherwise $G/S(G)$ satisfies $\mathbf{DivSyl}(p)$ by Step 1. Moreover, for given $P\in Syl_p(G)$, the group $PS(G)$ satisfies $\mathbf{DivSyl}(p)$ by Lemma \ref{psolvable}. Hence $G$ satisfies $\mathbf{DivSyl}(p)$ by Lemma \ref{extension}, a contradiction.
{\slshape Step 3.} Let $T=S_1\times\ldots\times S_k$ be a minimal normal subgroup of $G$. Then $G/T$ is a $p$-group. Consequently, $T$ is the unique minimal normal subgroup of $G$, and $S_i$ is a nonabelian simple group for $i=1, \ldots, k.$
Indeed, let $P$ be a Sylow $p$-subgroup of $G$. If $G/T$ is not a $p$-group, then $PT$ is a proper subgroup of $G$. By Lemma \ref{inducedautomoprhisms}, we may assume that $$1=G_0<G_1<\ldots<G_n=G$$ goes through $T$, that is, $T=G_k$ for some $k\in \{1, \ldots, n-1\}.$ Denote the image of $P\cap N_G(G_i/G_{i-1})$ in $\Aut_G(G_i/G_{i-1})$ by $P_i$ for $i=1,\ldots,k$. By definition, $P_i(G_i/G_{i-1})=\Aut_{PT}(G_i/G_{i-1})\leqslant \Aut_G(G_i/G_{i-1})$, and the conditions of the theorem says that for every $p$-subgroup $Q$ of $P_i(G_i/G_{i-1})$ the group $Q(G_i/G_{i-1})$ satisfies $\mathbf{DivSyl}(p)$. Hence $PT$ satisfies the conditions of the theorem, so by induction, $PT$ satisfies $\mathbf{DivSyl}(p)$. Then Step 1 and Lemma \ref{extension} imply that $G$ satisfies $\mathbf{DivSyl}(p)$, a contradiction.
Therefore $G/T$ is a $p$-group and so $G=PT$ for some (hence for every) Sylow $p$-subgroup $P$ of $G$. Consequently, $G$ has the unique minimal normal subgroup $T=L_1\times\ldots\times L_k$, where all $L_i$-s are isomorphic nonabelian simple groups by Step 2.
{\slshape Step 4.} If $H$ is a subgroup of $G$ containing $T$, then $\nu_p(H)$ divides $\nu_p(G)$.
Assume that $T\leqslant H$. Let $Q$ be a Sylow $p$-subgroup of $H$. Since $G/T$ is a $p$-group, so is $H/T$, in particular, $H=QT$. Let $P\in Syl_p(G)$ be chosen so that $P\cap H=Q$. Then $P\cap T=Q\cap T\in Syl_p(T)$ is a normal subgroup of $P$ and of $Q$. Denote $N_T(P\cap T)$ by $K$. Since $P\cap T\in Syl_p(T)$, we have $P\cap T\in Syl_p(K)$, so $K/(P\cap T)$ is a $p'$-group, and $PK$ satisfies conditions of Lemma \ref{normpbypdashbyp}. Clearly both $N_T(Q)$ and $N_T(P)$ lie in $K$, so $N_T(Q)=N_K(Q)$ and $N_T(P)=N_K(P)$. By Lemma \ref{normpbypdashbyp}, $N_T(Q)=N_K(Q)\geqslant N_K(P)=N_T(P)$. Finally by Lemma 1, $\nu_p(H)=\nu_p(QT)=\vert T:N_T(Q)\vert$ and $\nu_p(G)=\nu_p(PT)=\vert T:N_T(P)\vert$. Thus $\nu_p(H)$ divides~$\nu_p(G).$
Denote $\Aut_G(S_i)$ by $L_i$. Since $T$ is a unique minimal normal subgroup and $T$ is nonabelian, by Lemma \ref{inclusionwreathproduct}, there exists an injective homomorphism $\varphi:G\rightarrow L_1\wr \rho(G)$, where $\rho:G\rightarrow \Sym_k$ is the permutation representation corresponding to the action of $G$ on $\{S_1,\ldots,S_k\}$ by conjugation. Moreover, since $S_i$ is a unique minimal normal subgroup of $L_i$ and $\rho(G)$ is transitive, we have $\mathbf{S}=S_1\times\ldots\times S_k$ is a unique minimal normal subgroup of $L\wr \rho(G)$. In particular, for every such inclusion $\varphi$ the identity $\varphi(T)=\mathbf{S}$ holds. Below we identify $G$ with its image $\varphi(G)\leqslant L_1\wr \rho(G)$ and $T$ with $\mathbf{S}$. For every $H\leqslant G$ denote $\pi_i(H\cap T)$ by $H_i$, where $\pi_i$ is defined before Lemma \ref{inclusionwreathproduct}. Clearly $H\cap T$ is a subdirect product of $H_1,\ldots,H_k$.
{\slshape Step 5.} If $H\leqslant G$ is chosen so that $HT=G$, then $\vert H_1\vert=\ldots=\vert H_k\vert$ and $H$ normalizes $H_1\times \ldots\times H_k$.
Since $T$ is a minimal normal subgroup of $G$ and $T$ is nonabelian, $G$ acts transitively by conjugation on $\{S_1,\ldots,S_k\}$. Since $HT=G$ and $T\leqslant Ker(\rho)$, we obtain $\rho(H)=\rho(G)$, in particular, for every $i=1,\ldots,k$, there exists $x_i\in H$ such that $S_1^{x_i}=S_i$.
Let $h\in H\cap T$. Then $h=h_1\cdot\ldots\cdot h_k$, where $h_i=\pi_i(h)\in L_i$. Since $H\cap T\trianglelefteqslant H$, it follows that $h^{x_i}\in H\cap T$ for $i=1,\ldots,k$. On the other hand, $h^{x_i}=h_1^{x_i}\cdot\ldots\cdot h_k^{x_i}$ and this identity with the fact that $x_i$ permutes $S_i$-s imply $\pi_i(h^{x_i})=h_1^{x_i}$. Whence $H_1^{x_i}\subseteq H_i$. The same arguments show that $H_i^{x_i^{-1}}\subseteq H_1$. Hence $H_1^{x_i}=H_i$ and $\vert H_1\vert=\vert H_i\vert$ for $i=1,\ldots,k$. These arguments also show that arbitrary $x\in H$ permutes $H_i$-s, so $x$ normalizes $H_1\times\ldots\times H_k$.
{\slshape Step 6.} Assume $H\leqslant G$ is chosen so that $HT=G$. If $H\cap T=H_1\times \ldots\times H_k$, then $\nu_p(H)=\nu_p(\Aut_H(S_1))\cdot\vert H_1\vert_{p'}^{k-1}$ divides $\nu_p(G)$. If $H\cap T\not =H_1\times \ldots\times H_k$, then $\nu_p(H)$ divides $\nu_p(\Aut_H(S_1))\cdot\vert H_1\vert_{p'}^{k-1}$, and so $\nu_p(H)$ also divides $\nu_p(G)$.
Let $Q$ be a Sylow $p$-subgroup of $H$ and $P\in Syl_p(G)$ is chosen so that $P\cap H=Q$. By Step 3, $G/T$ is a $p$-group. So $H=Q(H\cap T)$ and $G=QT$. By Lemma \ref{inclusionwreathproduct}, there exists an embedding $\varphi:G\rightarrow \Aut_G(S_1)\wr \rho(G)$ such that for every $Q\leqslant X\leqslant G$ we have $\varphi(X)\leqslant \Aut_X(S_1)\wr \rho(Q)$, so we choose the embedding $\varphi$ with this property. Clearly $G$ as a subgroup of $\Aut_G(S_1)\wr\rho(G)$ with Sylow $p$-subgroup $P\leqslant \Aut_P(S_1)\wr \rho(G)$ satisfies conditions of Lemma \ref{numpwreath}, so $\nu_p(G)=\vert S_1\vert_{p'}^{k-1}\cdot \nu_p(\Aut_G(S_1))$. If $H\cap T=H_1\times\ldots\times H_k$, then $H$ as a subgroup of $\Aut_H(S_1)\wr \rho(H)$ with Sylow $p$-subgroup $Q\leqslant \Aut_Q(S_1)\wr\rho(G)$ satisfies conditions of Lemma \ref{numpwreath}, so $\nu_p(H)=\vert H_1\vert_{p'}^{k-1}\cdot\nu_p(\Aut_H(S_1))$. Clearly $\vert \Aut_H(S_1)\vert$ divides $\vert\Aut_G(S_1)\vert$, while $\nu_p(\Aut_H(S_1))$ divides $\nu_p(\Aut_G(S_1))$ by the conditions of the theorem. Hence $\nu_p(H)$ divides $\nu_p(G)$.
If $H\cap T\not = H_1\times\ldots\times H_k$, then $H$ as a subgroup of $H(H_1\times\ldots\times H_k)$ satisfies conditions of Lemma \ref{numberindirectproduct}. So $\nu_p(H)$ divides $\nu_p(H(H_1\times\ldots\times H_k))$, and $\nu_p(H(H_1\times\ldots\times H_k))=\vert H_1\vert_{p'}^{k-1}\cdot\nu_p(\Aut_H(S_1))$ divides $\nu_p(G)$ in view of the previous paragraph. Hence $\nu_p(H)$ divides $\nu_p(G)$.
{\slshape Step 7.} If $H\leqslant G$ and $HT\neq G$, then $\nu_p(H)$ divides $\nu_p(G)$.
Since nonabelian composition factors of $HT$ and $G$ are the same, like in the proof of Step 3, we can show that $HT$ satisfies conditions of the theorem, so $HT$ satisfies $\mathbf{DivSyl}(p)$ by induction. Therefore $\nu_p(H)$ divides $\nu_p(HT)$. But by Step 4, $\nu_p(HT)$ divides $\nu_p(G)$. Consequently, $\nu_p(H)$ divides $\nu_p(G)$.
In view of Steps 4, 6 and 7, we obtain that in any case $\nu_p(H)$ divides $\nu_p(G)$, which contradicts the choice of $G$. This completes the proof.
\section{Remarks and questions}
In this section we discuss natural question arising after the main theorem. We now can prove the following
\begin{Prop}\label{psolvalmsimple} Let $G$ be an almost simple group with simple socle $S$ such that $p$ does not divides $\vert S\vert$. Then $G$ satisfies $\mathbf{DivSyl}(p)$. \end{Prop}
\begin{proof} By Corollary \ref{almsimple}, we only need to verify that $PS$ satisfies $\mathbf{DivSyl}(p)$ for $P\in Syl_p(G)$. Let $H$ be a subgroup of $PS$. We may assume that $P\cap H\in Syl_p(H)$. Since $p$ does not divides $\vert S\vert$, we have $PS=S\rtimes P$, and so $H=(H\cap S)\rtimes (H\cap P)$. The known properties of coprime action (see \cite[Excersize~3E.4]{Isaacs}, for example) implies that $\nu_p((H\cap S)\rtimes (H\cap P))=\vert (H\cap S):C_{H\cap S}(H\cap P)\vert$ divides $\vert S:C_S(H\cap P)\vert$, while $\vert S:C_S(H\cap P)\vert$ divides $\vert S:C_S(P)\vert=\nu_p(PS)$. Therefore $\nu_p(H)$ divides $\nu_p(PS)$. \end{proof}
From Proposition \ref{psolvalmsimple} we immediately get the following remark
\begin{Remark} The main theorem is the direct generalization of Navarro theorem since all nonabelian composition factors of a $p$-solvable group are $p'$-groups. \end{Remark}
\begin{Remark} In conditions of the main theorem, we assume that $P(G_i/G_{i-1})$ satisfies $\mathbf{DivSyl}(p)$ for every $p$-subgroup $P$ of $\Aut_G(G_i/G_{i-1})$. Corollary \ref{almsimple} implies that under this condition every group $L$ with $G_i/G_{i-1}\leqslant L\leqslant \Aut_G(G_i/G_{i-1})$ satisfies $\mathbf{DivSyl}(p)$. However the authors do not know any example of almost simple group $L$ with simple socle $S$ such that $L$ satisfies $\mathbf{DivSyl}(p)$, while $S$ does not. So it is natural to assert
\begin{Conj}\label{simpleDivSyl} Let $S$ be a simple group satisfying $\mathbf{DivSyl}(p)$. Then every $L$ such that $S\leqslant L\leqslant \Aut(S)$ satisfies $\mathbf{DivSyl}(p)$. \end{Conj} \end{Remark}
\begin{Remark}\label{examples} The of the normalizers of Sylow $2$-subgroups in finite simple groups (see \cite{Kond}) implies that in (infinitely) many almost simple groups the Sylow $2$-subgroups are self-normalizing, so these groups satisfy $\mathbf{DivSyl}(2)$. Also the classification of subgroups in $\SL_2(p^t)$ implies that $\PSL_2(p^t)$ satisfies $\mathbf{DivSyl}(p)$. Of course, there are over examples. We provide these examples just to show that there are may almost simple groups satisfying $\mathbf{DivSyl}(p)$ that are not $p'$-groups. \end{Remark}
In connection with Conjecture \ref{simpleDivSyl} and Remark \ref{examples}, the following question arising naturally:
\begin{Prob} How often almost simple groups satisfy $\mathbf{DivSyl}(p)$? \end{Prob}
\begin{Prob} What almost simple groups satisfy $\mathbf{DivSyl}(p)$ for every prime~$p$? \end{Prob}
\begin{Remark} Another possible questions arise: If we try to consider $\pi$-Hall subgroups instead of Sylow $p$-subgroups. In \cite{Turull} A. Turull prove that if $G$ is $\pi$-separable, then $\nu_\pi(H)$ divides $\nu_\pi(G)$ for every $H\leqslant G$. It is not hard to see that Lemmas \ref{nupfactor}--\ref{psolvable} hold if we consider $\pi$-separable groups and $\pi$-Hall subgroups, so an alternative proof of Turull theorem can be also obtained by the same arguments. But in order to extend Turull theorem, one needs to introduce appropriate class of finite groups so that each proper subgroup satisfies $E_\pi$ at least. We do not go into details in this case, we only mention that such possibility exists and could be considered later. \end{Remark}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title[On the Closing Lemma problem on the torus]{On the Closing Lemma problem \\ for vector fields of bounded type on the torus} \author{Simon Lloyd} \address{School of Mathematics, University of New South Wales, Sydney, Australia} \email{[email protected]}
\begin{abstract} We investigate the open Closing Lemma problem for vector fields on the $2$-dimensional torus. Under the assumption of bounded type rotation number, the $C^r$ Closing Lemma is verified for smooth vector fields that are area-preserving at all saddle points. Namely, given such a $C^r$ vector field $X$, $r\geq 4$, with a non-trivially recurrent point $p$, there exists a vector field $Y$ arbitrarily near to $X$ in the $C^r$ topology and obtained from $X$ by a twist perturbation, such that $p$ is a periodic point of $Y$.
The proof relies on a new result in $1$-dimensional dynamics on the non-existence of semi-wandering intervals of smooth maps of the circle. \end{abstract}
\maketitle
\section{Introduction}
The $C^r$ Closing Lemma is a longstanding open problem in topological dynamics and was listed by Smale in 2000 as one of the `problems for the next century' \cite{SmaleProblems}. Its importance arises in the fact that a positive solution would lead to many deep results in relation to genericity, stability and bifurcations. The classical $C^r$ Closing Lemma problem is stated as follows: \begin{CL} Let $M$ be a smooth compact manifold and let $r\geq 1$ be an integer. Given a $C^r$ vector field $X\in\mathfrak{X}^r(M)$ on $M$, a non-trivially recurrent point $p\in M$ and a neighbourhood $\mathcal{U}\subset \mathfrak{X}^r(M)$ of $X$, does there exist $Y\in \mathcal{U}$ with a periodic orbit passing through $p$? \end{CL}
The first partial results in this direction are due to Peixoto \cite{Peixoto62} for vector fields on orientable surfaces and to Anosov \cite{Anosov} for uniformly hyperbolic systems. For the $C^1$ topology, this question was answered fully and affirmatively by the celebrated paper of Pugh \cite{PughClosingLemma}. For $r\geq 2$, the problem is open.
There are partial results for $r\geq 2$, for vector fields on the $2$-dimensional torus $\mathbb{T}^2$, the simplest manifold which supports a non-trivial recurrent vector field. Gutierrez \cite{GutierrezUnboundedType} proved the $C^r$ Closing Lemma for a large class of vector fields on $\mathbb{T}^2$: the vector fields with `unbounded type' combinatorics. This result was sharpened by Carroll \cite{Carroll92}, and has also been extended to all closed orientable surfaces \cite{Gutierrezhighgenus,AMZ}. Gutierrez and Pires \cite{GutierrezPires} have also demonstrated $C^r$ closing for a class of vector fields on surfaces with an asymptotic contraction property. In each of these cases, the closing perturbation can take a simple form, a so-called `twist perturbation', supported on an annulus and transverse to the vector field.
In this article we obtain a positive result for complementary case of `bounded type' vector fields, although all current results for this case are negative. Gutierrez \cite{GutierrezCounterexample} constructs a vector field on $\mathbb{T}^2$, and shows that no local perturbation is a $C^2$ closing perturbation. Carroll showed that, unlike in the unbounded type case, twist perturbations cannot achieve $C^2$ closing for all bounded type vector fields on $\mathbb{T}^2$. The obstruction to closing is that all sufficiently small $C^2$ perturbations produce either another Cantor recurrent set, or else eliminate all one-dimensional recurrent orbits due to the intersection of basins of attractors and repellors.
We study vector fields on $\mathbb{T}^2$ with singularities in terms of their induced action on a transverse loop $\Sigma$. The maps of the circle that arise in this way are typically not diffeomorphisms, since if $\Sigma$ intersects the basin $B$ of an attractor (or repellor), then an interval of points in $B\cap\Sigma$ fails to return to the transverse loop in forward (or backward) time. These intervals create plateaus and discontinuities in the induced map. Thus we instead look at the larger class of order-preserving maps, which have no assumption of continuity, injectivity or surjectivity. These maps can possess dynamical features not exhibited by diffeomorphisms, such as `semi-wandering' intervals, whose iterates are disjoint intervals in forward or backward time only.
In Section \ref{semiwandering} we prove the following non-existence of wandering intervals result for order-preserving maps with bounded type rotation number.
\begin{theoremA} Let $f:S^1\to S^1$ be an order-preserving map of the circle with rotation number of bounded type. Suppose that $f$ is $C^1$ except possibly at a finite number of points, and that $\mathop{\mathrm{var}}_{\mathrm{supp}(\mathrm{D}f)} \log \mathrm{D}f$ is finite.
\noindent Then $f$ has either no forward wandering intervals, or else no backward wandering intervals. \end{theoremA}
In Section \ref{blackcells}, we discuss the features of phase portraits of vector fields on $\mathbb{T}^2$, in particular the grey cells that are characteristic of Denjoy vector fields, and the black cells of Cherry vector fields. We prove a structure theorem for vector fields with bounded type rotation number.
\begin{theoremB} Let $X\in\mathfrak{X}^r(\mathbb{T}^2)$ be a bounded type $C^r$ vector field on the torus, $r\geq 4$. Suppose there are at most finitely many singularities, all hyperbolic, and that the divergence is zero at each saddle point.
\noindent Then $X$ has no grey cells, and any black cells are co-directed. \end{theoremB}
There are related results on the non-existence of grey cells for vector fields that are repellor-free, and thus automatically have co-directed black cells. Repellor-free vector fields possess backward black cells but no grey cells if the divergence is non-positive at all saddles (see Martens \emph{et al} \cite{MvSdMM}) or else non-negative at all saddles (see Aranson \emph{et al} \cite{Zhuzhoma}). The existence of grey cells in the mixed case, with saddle points with divergence of both signs, is an open problem.
In Section \ref{closing}, we show that for a vector field with co-directed black cells, it is possible to choose a transverse loop such that the induced map extends to a continuous map of the circle, see Lemma \ref{lem:transversal}. In this case, any $1$-parameter family of $C^r$-twist perturbations contains arbitrarily $C^r$-small closing perturbations.
\begin{theoremC} Let $X\in\mathfrak{X}^r(\mathbb{T}^2)$, $r\geq 4$, be a bounded type vector field with finitely many singularities all hyperbolic, and suppose the divergence is zero at each saddle.
\noindent Then for any recurrent point $p$, there exists a loop $\Sigma$ through $p$ and transverse to $X$ such that any $1$-parameter family of $C^r$ twist perturbations along $\Sigma$ contains closing perturbations for $p$ that are arbitrarily $C^r$ near to $X$. \end{theoremC}
By Proposition \ref{prop:twist}, $C^r$-twist perturbations are sufficient for closing in any vector fields with co-directed black cells, but as Carroll's examples show \cite{Carroll92}, there are vector fields, with contra-directed black cells, for which $C^r$-twist perturbations are ineffective for closing recurrent orbits. Carroll's examples use \emph{degenerate} saddle singularities, as does the vector field of Gutierrez with poor $C^2$ closing properties \cite{GutierrezCounterexample}. It is an open problem whether there exist smooth vector fields with contra-directed black cells for which all singularities are hyperbolic.
\section{Semi-wandering intervals}\label{semiwandering}
Let $S^1$ be the unit circle, with orientation and metric $d$ induced by $\pi:\mathbb{R}\to S^1$, the canonical projection $\pi(x)= x\ (\mathrm{mod}\ 1)$. We say a function $f:S^1\to S^1$ is a \emph{(strictly) order-preserving map} if there exists a (strictly increasing) non-decreasing map $F:\mathbb{R}\to\mathbb{R}$ such that: \begin{itemize} \item $F(x+1)=F(x)+1$; \item $\pi\circ F = f\circ \pi$. \end{itemize} Note that order-preserving maps are not assumed to be continuous, injective nor surjective. However, because of their monotonicity, order-preserving maps are continuous except for at most countably many discontinuities of jump type, and are also locally injective, except for countably many intervals of constancy, called \emph{plateaus}.
Denjoy proved that a $C^1$ diffeomorphism with a non-trivially recurrent orbit and with derivative of bounded variation is \emph{minimal}: every orbit is dense \cite{Denjoy32}. This result is sharp: for arbitrarily small $\epsilon>0$, one can construct a $C^{2-\epsilon}$ diffeomorphism which is not minimal due to the presence of a \emph{wandering interval}, that is, an interval $I$ whose iterates $f^k(I)$, $k\in\mathbb{Z}$, are pairwise disjoint intervals not reduced to a point and the $\omega$-limit set of $I$ is not a periodic orbit. In such cases the minimal set is a Cantor set, and is said to be \emph{exceptional}.
Exceptional minimal sets can also occur for order-preserving maps due to the presence of ``semi-wandering intervals''. Given a map of the circle $f$, we say an interval $I\in\mathbb{T}$ is \emph{forward (backward) wandering} if $f^k(I)$ are disjoint intervals not reduced to a point, for integer $k\geq 0$ ($k\leq 0$) and the $\omega$-limit ($\alpha$-limit) of $I$ is not a periodic orbit. If an interval $I$ is either forward wandering or backward wandering but not both, then it is said to be \emph{semi-wandering}. In Theorem \ref{thm:A} we show that for sufficiently smooth maps with bounded type rotation number, forward wandering intervals and backward wandering intervals cannot both occur for the same map.
Let $R:S^1\to S^1$ denote the rigid rotation $R=R_\alpha:x\mapsto x+\alpha\ (\mathrm{mod}\ 1)$. Recall that for $\alpha$ rational, every orbit is periodic, and that for $\alpha$ irrational, every orbit is dense. In the latter case, information about the ordering of the iterates can be gained from the continued fraction expansion of $\alpha$. Let $[a_0,a_1,a_2,\ldots]$ denote the continued fraction expansion for $\alpha$, where $$ \alpha=a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\cdots}}}. $$ Let $(p_n/q_n)$ be the seqence of partial quotients of $\alpha$, obtained by truncating the sequence at the $n$th term. The pair of sequences $(p_n)$ and $(q_n)$ generated in this way each obey the same recursive relation $$ p_n=a_n p_{n-1}+p_{n-2}, \quad q_n=a_n q_{n-1}+q_{n-2} $$ but have different starting values $$ p_0=a_0,\ p_1=a_0 a_1+1,\quad q_0=1,\ q_1=a_1. $$
The sequence $(q_n)$ is called the sequence of \emph{closest return times}, since $$ d(R^{q_n}(x),x)=\min\{d(R^k(x),x):0<k\leq q_n\}, $$ for any $x\in S^1$. We call the points $R^{q_n}(x)$ the sequence of \emph{closest returns} to $x$.
Note that the sequences $R^{q_{2n}}(x)$ and $R^{q_{2n+1}}(x)$ each converge monotonically to $x$ and from opposite sides. Denote by $I_{2n}$ and $I_{2n+1}$ the decreasing sequences of closed intervals with endpoints $x$ and $R^{2n}(x)$ and $x$ and $R^{2n+1}(x)$ respectively.
The sequence $(a_n)$ from the continued fraction dictates the numbers of ``intermediate returns". Consider the interval $I_n\backslash I_{n+2}$, which has endpoints $R^{q_n}(x)$ and $R^{q_{n+2}}(x)$ from the sequence of closest returns. Any iterates $R^i(x)$, $q_n<i<q_{n+2}$, lying in the interval $I_n\backslash I_{n+2}$ are called \emph{intermediate return points}, and these occur at times $q_n+q_{n+1},q_n+2q_{n+1},\ldots,q_n+(a_n-1) q_{n+1}$: thus they are precisely $a_n-1$ in number.
For the proof of the Theorem \ref{thm:A}, we need to know how much certain collections of intervals overlap. Given a collection $\mathcal{X}=\{X_i\}_{i\in\mathcal{J}}$ of subsets of a space $X$, the \emph{intersection multiplicity} of $\mathcal{X}$ is defined to be $\textrm{IM}(\mathcal{X})=\max_{x\in X} \#\{i\in\mathcal{J}:x\in X_i\}$.
\begin{lem}\label{lem:Disjointness} Let $S_n(x)=\{x,R(x),\ldots,R^{q_n-1}(x)\}$, and let $T$ be an interval whose interior is contained in $S^1\backslash S_n(x)$. Then $$ \mathrm{IM}(\{T,R(T),\ldots,R^{q_n-1}(T)\}) \leq 2 (a_n+1). $$ \end{lem} \begin{proof} The collection of intervals \begin{eqnarray}\label{TS} \bigcup_{i=0}^{q_{n}-1} R^i(I_{n-1}) \cup \bigcup_{i=0}^{q_{n-1}-1} R^i(I_{n}) \end{eqnarray} cover the circle $\mathbb{T}$, and have pairwise disjoint interiors, see \cite{MS}.
It follows from (\ref{TS}) that the points \begin{eqnarray}\label{TSptn} \{x,R(x),\ldots,R^{q_{n-1}+q_{n-2}-1}(x)\} \end{eqnarray}
partition $S^1$ into intervals of length $|I_{n-1}|$ and $|I_{n-2}|$. Since $q_{n}\geq q_{n-1}+q_{n-2}$, the set $S_n(x)$ is a refinement of partition (\ref{TSptn}), and thus the distance between two adjacent points of $S_{n}(x)$ is also no greater than $|I_{n-2}|$.
From (\ref{TS}), for any \emph{open} interval $U$ of length $|I_{n-2}|$, the collection $\{ R^i(U) \}_{i=0}^{q_{n-1}-1}$ is pairwise disjoint.
Thus, allowing for the endpoints, $\textrm{IM}(\{ R^i(\overline{U}) \}_{i=0}^{q_{n-1}-1})\leq 2$. Since $q_n=a_n q_{n-1}+q_{n-2}< (a_n+1)q_{n-1}$ and since $|T|\leq |I_{n-2}|$, we have the required bound. \end{proof}
We use an assumption of boundedness of the variation of $\log\mathrm{D}f$ in order to control the distortion of iterates of $f$. Recall that for a function $g:U\to \mathbb{R}$, $U\subset S^1$, its \emph{variation} $\mathop{\mathrm{var}}_U(g)$ is defined to be $$
\mathop{\mathrm{var}}_U(g)=\sup_{x_0<\cdots<x_n}\left\{ \sum_{i=1}^n |g(x_i)-g(x_{i-1})| \right\}, $$ where the supremum is taken over all finite partitions $\{x_i\}_{i=0}^n\subset U$.
\begin{thm}\label{thm:A} Let $f:S^1\to S^1$ be an order-preserving map of the circle with rotation number of bounded type. Suppose that $f$ is $C^1$ except possibly at a finite number of points, and that $\mathop{\mathrm{var}}_{\mathrm{supp}(\mathrm{D}f)} \log \mathrm{D}f$ is finite.
\noindent Then $f$ has either no forward wandering intervals, or else no backward wandering intervals. \end{thm} \begin{proof} We assume, for a contradiction, that exist a forward wandering interval $I_+$ and a backward wandering interval $I_-$. Let the rotation number of $f$ be $\alpha=[0,a_1,a_2,\ldots]$, with $\liminf a_n =K$ say. Let $\mathcal{N}$ be the set of positive integers such that for all $n\in\mathcal{N}$, $a_n=K$. Note that the set $\mathcal{N}$ is infinite.
Since $\alpha$ is irrational $f$ is semiconjugate to the rigid rotation $R=R_\alpha$: that is, there exists a continuous surjective map $h:S^1\to S^1$ such that $h\circ f(x)=R\circ h(x)$ for all $x\in S^1$.
Since $R$ is minimal, $h$ maps $I_+$ and $I_-$ to single points. Let $z_+=h(I_+)$ and $z_-=h(I_-)$. We suppose that $I_+$ and $I_-$ are maximal, that is, they are not subsets of larger semi-wandering intervals: otherwise we consider $h^{-1}(z_+)$ and $h^{-1}(z_-)$ instead. By taking an iterate of $I_+$ if necessary, we may assume that $f^i(I_+)\cap f^{-j}(I_-)=\emptyset$ for every pair of integers $i,j>0$. Since $f$ is $C^1$ except at a finite number of points, we may assume $f^n|_{I_+}$ is $C^1$ for each $n\in\mathbb{N}$, otherwise we replace $I_+$ by some suitably high forward iterate. Similarly, we may assume $f^{-n}|_{I_-}$ is $C^1$ for each $n\in\mathbb{N}$.
Given $n\in\mathcal{N}$, let $q_n$ be the $n$th closest return time for $R$. Let $\mathcal{F}^+_n=\{f^i(I_+):i=0,\ldots,q_n-1\}$ and $\mathcal{F}^-_n=\{f^{-i}(I_-):i=0,\ldots,q_n-1\}$. Consider the shortest length interval from the collection $\mathcal{T}_n= \mathcal{F}^+_n \cup \mathcal{F}^-_n$. Either the shortest length interval in $\mathcal{T}_n$ occurs in the family $\mathcal{F}^+=\{f^i(I_+):i\geq 0\}$ or otherwise in the family $\mathcal{F}^-=\{f^{-i}(I_-):i\geq 0\}$. We denote by $S_+$ the subset of $\mathcal{N}$ non-negative integers for which an interval of $\mathcal{F}_+$ has shortest length among all intervals of $\mathcal{T}_n$, and define $S_-$ correspondingly. Thus $\mathcal{N}\subset S_+\cup S_-$, and so at least one of $S_+$ and $S_-$ is an infinite set.
Suppose then that $S_+$ is infinite: the case $S_-$ is similar. Fix $n\in S_+$, and suppose $k=k(n)\geq 0$ is such that the shortest length interval of $\mathcal{T}_n$ is the iterate $f^k(I_+)$. Note that by the disjointness of the forward iterates of the forward wandering domain $I_+$, we have that $k(n)\to \infty$ as $n\to\infty$ in $S_+$.
Let $f^{-j_1}(I_-)$ and $f^{-j_2}(I_-)$ be the closest images of $I_-$ on either side of $f^{k}(I_+)$ satisfying $0\leq j_1,j_2 < q_n$. Let $T$ be the shortest closed interval neighbourhood of $I_+$ containing $f^{-j_1-k}(I_-)$ and $f^{-j_2-k}(I_-)$.
By Lemma \ref{lem:Disjointness}, the intersection multiplicity of $T,f(T),\ldots,f^{q_n-1}(T)$ is at most $2a_n+2$, and hence the intersection multiplicity of $T,f(T),\ldots,f^k(T)$ is certainly less than or equal to $2K+2$.
The function $\log \mathrm{D}f$ is real-valued when restricted to the domain $U=\mathrm{supp} (\mathrm{D}f)$. Let $V=\mathop{\mathrm{var}}_U \log \mathrm{D}f$.
Consider $x,y\in \mathop{\mathrm{supp}}\mathrm{D}(f^k)\subset U \cap T$. Then \begin{eqnarray*} \log \frac{\mathrm{D}f^k(y)}{\mathrm{D}f^k(x)} & = & \log \frac{\mathrm{D}f(y)\cdot\mathrm{D}f(f(y))\cdots \mathrm{D}f(f^{k-1}(y))}{\mathrm{D}f(x)\cdot\mathrm{D}f(f(x))\cdots \mathrm{D}f(f^{k-1}(x))} \\ & = & \sum_{i=0}^{k-1} \log \mathrm{D}f(f^i(y)) - \log \mathrm{D}f(f^i(x)). \end{eqnarray*} Thus, since each point of the circle lies in at most $2(K+1)$ intervals of the collection $\bigcup_{i=0}^k f^i(T)$, we have $$ \sup_{x,y\in T\cap\mathop{\mathrm{supp}}\mathrm{D}(f^k)} \log \frac{\mathrm{D}f^k(y)}{\mathrm{D}f^k(x)} \leq 2(K+1)V. $$ Let $\beta:=e^{2(K+1)V}>0$.
By the Mean Value Theorem there exist points $y_i\in f^{-j_i-k}(I_-)\subset T$, $i=1,2$, and $x\in I_+\subset T$ such that \begin{eqnarray*}
\mathrm{D}f^{k}(x) & = & \frac{|f^k(I_+)|}{|I_+|}, \\
\mathrm{D}f^{k}(y_i) & = & \frac{|f^{-j_i}(I_-)|}{|f^{-j_i-k}(I_-)|}. \end{eqnarray*} Hence, for $i=1,2$, $$
\beta \geq \frac{|I_+|}{|f^k(I_+)|}\cdot\frac{|f^{j_i}(I_-)|}{|f^{-j_i-k}(I_-)|} \geq \frac{|I_+|}{|f^{-j_i-k}(I_-)|}, $$
where the final inequality follows from our choice of $k$ which ensures $|f^k(I_+)|\geq |f^{-j_i}(I_-)|$. So for $i=1,2$, we have $|f^{-j_i-k}(I_-)|\geq |I_+|/\beta$.
Hence for any $n\in S_+$ there exists $k(n)$, where $k(n)\to \infty$ as $n\to\infty$, and $m\geq k(n)$, such that $$
|f^{-m}(I_-)|\geq \frac{|I_+|}{\beta}. $$
Thus $\liminf_{n\to\infty} |f^{-n}(I_-)|\geq |I_+|/\beta$, which is impossible since $I_-$ is a backward wandering interval and thus has disjoint backward iterates. \end{proof}
\begin{rem}
In the proof of Theorem \ref{thm:A}, we do not know beforehand whether we will need to use the boundedness of the distortion of high iterates of $f$ or of $f^{-1}$. This bi-directional nature of the argument means that it cannot be generalised in the manner of Yoccoz \cite{Yoccoz} or Martens \emph{et al} \cite{MvSdMM} to the case where $f\sim \pm|x|^\alpha$, $\alpha\geq 1$ at discontinuities and the endpoints of plateaus, and so we must assume that $\log\mathrm{D}f$ is bounded on $\mathrm{supp}(\mathrm{D}f)$.
Thus in order to use Theorem \ref{thm:A} in the proof of Theorem \ref{thm:B}, we must restrict ourselves to vector fields where the divergence at each saddle point is zero. \end{rem}
\section{Phase portraits of vector fields on the torus}\label{blackcells}
Given a vector field $X\in\mathfrak{X}^r(\mathbb{T}^2)$, $r\geq 2$, we denote by $\Phi^t(x)$ the solution to the ordinary differential equation $\dot{z}=X(z)$ with initial value $z(0)=x\in\mathbb{T}^2$. The map $\Phi:\mathrm{R}\times \mathbb{T}^2\to\mathbb{T}^2$ is called the \emph{flow} associated to $X$. The \emph{forward (or backward) orbit} of a point $p$ is the set $\mathrm{o}^+(p)=\{\Phi^t(p):t\geq 0\}$ (or $\mathrm{o}^-(p)=\{\Phi^t(p):t\leq 0\}$ respectively); the \emph{orbit} of $p$ is the set $\mathrm{o}(p)=\mathrm{o}^+(p)\cup\mathrm{o}^-(p)$. A point $p$ (or its orbit $\mathrm{o}(p)$) is \emph{recurrent} if for any neighbourhood $U$ of $p$, $\mathrm{o}(p)\cap U\neq\emptyset$. Simple examples of recurrent orbits are \emph{singularities}, for which $\mathrm{o}(p)=\{p\}$, and \emph{closed orbits}, for which $\mathrm{o}(p)$ is homeomorphic to a circle. An orbit is said to be \emph{non-trivially recurrent} if it is recurrent but neither a singularity nor a closed orbit.
If $X\in\mathfrak{X}^r(\mathbb{T}^2)$ has a non-trivially recurrent orbit, and a transverse loop $\Sigma$, we denote by $\rho(X,\Sigma)$ the rotation number of the induced map $\Sigma\to\Sigma$. If we replace $\Sigma$ by a homologically distinct loop $\Sigma'$, then the new rotation number $\rho(X,\Sigma')$ is related to $\rho(X,\Sigma)$ by a linear fractional transformation. The \emph{rotation orbit} $$ \rho(X)=\left\{ \frac{a\rho(X,\Sigma)+b}{c\rho(X,\Sigma)+d}:a,b,c,d\in\mathbb{Z},\quad ad-bc=\pm1 \right\} $$ is independent of the choice of transverse loop $\Sigma$, and a topological invariant of the vector field $X$, see \cite{Zhuzhoma}. Since a fractional linear transformation acts as a right shift on the coefficients of a continued fraction, the upper and lower limits $\liminf a_n$ and $\limsup a_n$ of the continued fraction of an element $\alpha=[0,a_1,a_2,\ldots]\in\rho(X)$ of the rotation orbit are independent of the choice of $\alpha$. Thus we may describe a vector field as having \emph{(un)bounded type} without reference to any particular transverse loop.
We recall some terminology for describing the orbit structure of vector fields on the torus. Given a non-trivially recurrent vector field $X\in\mathfrak{X}^r(\mathbb{T}^2)$, the closure of any non-trivially recurrent point is called a \emph{quasiminimal set}. By Maier's estimate \cite{Maier}, the torus can support at most one quasiminimal set: thus if $p$ and $p'$ are non-trivially recurrent points, then $\overline{\mathrm{o}(p')}=\overline{\mathrm{o}(p)} = Q$. We say that an invariant set $A\neq Q$ is an \emph{attractor (or repellor)} if there exists a neighbourhood $U\supset A$ such that $\omega(x)=A$ (respectively $\alpha(x)=A$) for all $x\in U$; the maximal such $U$ is called the \emph{basin} of $A$.
A \emph{cell} is a maximal open path-connected invariant set $C$ such that the limit sets $\alpha(x)$ and $\omega(x)$ are independent of $x\in C$. A cell $C$ is a \emph{grey cell} if it lies in the complement of $Q$ and satisfies $\alpha(C)=\omega(C)=Q$. Denjoy \cite{Denjoy32} showed that for singularity-free $C^r$ vector fields, grey cells can occur if $r<2$, but if $X$ is $C^1$ with derivative of bounded variation, then $Q=\mathbb{T}^2$ and thus there are no grey cells.
In order to describe the phase portraits of vector fields with singularities, we make use of definitions introduced in the study of Cherry vector fields. Cherry \cite{Cherry38} constructed an analytic vector field on the torus with a non-trivially recurrent orbit that has two singularities -- a source and a saddle. The basin of the source is a cell which lies outside of the quasiminimal set: a cell disjoint from $Q$ that has a limit set other than $Q$ is called a \emph{black cell}. More specifically, we say a cell $C$ is called a \emph{forward black cell} if $\alpha(C)\neq \omega(C) = Q$, a \emph{backward black cell} if $\omega(C)\neq \alpha(C) = Q$, otherwise $\alpha(C)\neq Q$ and $\omega(C)\neq Q$ in which case we call $C$ a \emph{medial black cell}. In each case, the non-quasiminimal limit set, whether an attractor or repellor, is either a node or a limit cycle.
The phase portrait of a vector field with black cells may be obtained from a singularity-free flow on a torus by `blowing-up' a individual orbits (see \cite{Zhuzhoma}): that is, replacing one or more orbits with either of the phase portraits shown in Figure \ref{fig:BlackCells}.
\begin{figure}
\caption{Phase portraits of a forward black cell $C^+$ and a backward black cell $C^-$.}
\label{fig:BlackCells}
\end{figure}
Cherry constructed a non-trivially recurrent analytic vector field on the torus with a backward black cell, and asked whether such examples can have grey cells. Martens \emph{et al} \cite{MvSdMM} show that repellor-free smooth vector fields with non-positive divergence at all saddles have backward black cells but no grey cells. The techniques of Yoccoz \cite{Yoccoz} show that the same is true if the divergence at all saddles is non-negative, see \cite{Zhuzhoma}. It is an open problem whether grey cells can exist in repellor-free vector fields for which the quasiminimal set contains two saddle points with divergences of opposite signs. A vector field is said to have \emph{co-directed} black cells if there are no forward black cells or else no backward black cells; a vector field has \emph{contra-directed} black cells if it has a forward black cell and also a backward black cell.
The case of smooth non-trivially recurrent toral vector fields with both attractors and repellors is also little understood, and no explicit examples have yet been constructed where forward and backward black cells coexist. The following result gives a condition under which the black cells are \emph{co-directed}.
\begin{thm}\label{thm:B} Let $X\in\mathfrak{X}^r(\mathbb{T}^2)$ be a bounded type $C^r$ vector field on the torus, $r\geq 4$. Suppose there are at most finitely many singularities, all hyperbolic, and that the divergence is zero at each saddle point.
\noindent Then $X$ has no grey cells, and any black cells are co-directed. \end{thm} \begin{proof} We assume $X$ has singularities, otherwise there are no black cells, and by Denjoy's theorem \cite{Denjoy32} there are no grey cells.
Since $X$ has a non-trivially recurrent orbit, we may construct a $C^r$ loop $\Sigma$ transverse to $X$, see \cite{PalisDeMelo}. Let $f:\Sigma\to\Sigma$ be an order-preserving extension to $\Sigma$ of the induced map. As $X$ is $C^4$ and the divergence at each saddle points is zero, it follows that $\mathop{\mathrm{var}}_{\mathrm{supp}(\mathrm{D}f)}\log \mathrm{D}f$ is finite, see for example \cite{MvSdMM}.
Assume for a contradiction that there exist a forward black cell $C^+$ and a backward black cell $C^-$. Then $C^+\cap\Sigma$ contains an interval $I^+$ that is a forward wandering interval for $f$, and $C^-\cap\Sigma$ contains a backward wandering interval $I^-$ for $f$. As this contradicts Theorem \ref{thm:A} applied to $f$, it follows that $X$ has either no forward black cells or else no backward black cells.
Assume now that $X$ has a grey cell $C$. Then $C\cap\Sigma$ contains backward and forward wandering intervals $I^-$ and $I^+=f(I^-)$ for $f$. We again obtain a contradiction to Theorem \ref{thm:A}, and so $X$ has no grey cells. \end{proof}
\section{Closing Lemma results}\label{closing}
When studying vector fields on the torus in terms of the return map to a transverse loop, it is common to consider `twist perturbations'. Let $U$ be an annular neighbourhood of a transverse loop $\Sigma$ that is free of singularities. A \emph{family of $C^r$ twist perturbations} is a continuous family of $C^r$ vector fields $Y:[0,a_0]\to \mathfrak{X}^r(\mathbb{T}^2)$ such that $Y_0\equiv X$ and for $a>0$, $\mathrm{supp}(Y_a-X)= U$ and $Y_a|_{\mathrm{int}\ U}$ is transverse to $X|_{\mathrm{int}\ U}$.
We say $X$ admits \emph{$C^r$ closing by twist perturbations along $\Sigma$} if for any non-trivially recurrent point $p\in\Sigma$, any annular neighbourhood $U$ of $\Sigma$ and any family $\{Y_a\}_{0\leq a<a_0}$ of $C^r$ twist perturbations, there exists $(a_n)\downarrow 0$ and points $x_n\in\Sigma$, $(x_n)\to p$, such that $x_n$ lies on a closed orbit of $Y_{a_n}$.
As noted by Pugh \cite{PughClosingLemma}, in such a case we obtain an affirmative answer to the $C^r$ Closing Lemma problem as follows. Let $Z_n:\mathbb{T}^2\to \mathbb{T}^2$ denote the translation of the torus which takes $p$ to $x_n$. If $X$ admits $C^r$ closing by twist perturbations along a transverse loop $\Sigma$, then by setting $X_n(x):=Y_{a_n}(Z_n(x))$, we obtain a sequence of vector fields with closed orbits through $p$ and satisfying $\|X-X_n\|_{C^r}\to 0$.
The return map induced by a vector field on a transverse loop $\Sigma$ typically exhibits plateaus, caused points lying in the basin of an attractor whose forward orbits do not return to $\Sigma$, and also discontinuities caused by points in basins of repellors whose backward orbits do not return to $\Sigma$. We begin by proving that when black cells are co-directed, it is possible to find a transverse loop such that the induced map lacks one or the other of these features.
\begin{lem}\label{lem:transversal} Let $X\in\mathfrak{X}^r(\mathbb{T}^2)$, $r\geq 1$, be a vector field with finitely many singularities all hyperbolic, and suppose $X$ has a non-trivially recurrent point $p$. If there are no forward black cells, then there exists a transverse $C^r$ loop $\Sigma$ through $p$ such that the induced map $P:\Sigma\to\Sigma$ extends to a continuous map. \end{lem} \begin{proof} It is enough to show that we can construct a tranverse loop disjoint from all repellor basins. Since there are no backward black cells, each repellor basin consists of a disjoint union of medial black cells. Let $Q$ be the quasiminimal set and let $H$ denote the union of the medial black cells. Since $\overline{H}\cap Q$ consists only of saddle points and separatrices, we have that $\mathrm{o}(p)$ is disjoint from $\overline{H}$. Since $p$ lies in the open set $\mathbb{T}^2\backslash \overline{H}$, we may take a small open disc $U$ around $p$ disjoint from $\overline{H}$. Let $p_0,p_1$ be the first exit and first re-entry points of $\mathrm{o}^+(p)$ to $U$. Let $\gamma=\mathrm{o}^+(p_0)\cap\mathrm{o}^-(p_1)$ be the segment of orbit from $p_0$ to $p_1$, and let $L$ be an open tubular neighbourhood of $\gamma$ disjoint from $\overline{H}$. For $\epsilon$ sufficiently small, the orbit of $p$ under the vector field $X+\epsilon X^\perp$ exits $U$ at a point $q_0$ and re-enters $U$ at a point $q_1$ with $\ell=\mathrm{o}^+(p_0)\cap\mathrm{o}^-(p_1)$ contained in $L$. By connecting $q_1$ to $q_1$ with a $C^r$ path $\ell'\subset U$ through $p$ transverse to $X$, we obtain a $C^r$ loop $\Sigma=\ell\cup \ell'$ that is transverse to $X$ and disjoint from $\overline{H}$.
Let $P:\Sigma\to\Sigma$ be the induced return map. By construction, $\Sigma$ does not intersect the basins of any repellors. Thus $P$ extends to a continuous map $\Sigma\to\Sigma$. \end{proof}
\begin{lem}\label{lem:periodicpoints} Let $f_0:S^1\to S^1$ be a continuous map of the circle that is strictly increasing except for a finite number of plateaus, and let $p\in S^1$ be a non-trivially recurrent point whose orbit accumulates on $p$ from both sides. Suppose $f_{\epsilon}:=h_{\epsilon}\circ f_0$, where $h_\epsilon:S^1\to S^1$ is a continuous family of homeomorphisms, with lifts $H_\epsilon:\mathbb{R}\to \mathbb{R}$ such that $H_0=\mathrm{Id}$ and $\epsilon\mapsto H_\epsilon(x)$ is continuous, non-decreasing at each $x$ and strictly increasing at $p$.
Then there exists a sequence $s_n\downarrow 0$ such that for each $n\in\mathbb{N}$, $f_{s_n}$ has a periodic point $x_n\to p$ and moreover the orbit of $x_n$ is disjoint from the plateaus of $f_{s_n}$. \end{lem} \begin{proof} Since $p$ is non-trivially recurrent, $\rho(f_0)$ is irrational. Moreover, since $\epsilon\mapsto H_\epsilon(p)$ is strictly increasing, the mapping $\epsilon\mapsto \rho(f_\epsilon)$ is strictly increasing at $\epsilon=0$. As $f_0$ is continuous, the mapping $\epsilon\mapsto \rho(f_\epsilon)$ is continuous, and thus we can find a sequence $(s_n)\downarrow 0$ such that $\rho(f_{s_n})=p_n/q_n$.
If for arbitrarily large $n$, there is a periodic orbit through $p$ disjoint from all plateaus of $f_{s_n}$ we are done. Thus, shifting $(s_n)$ if necessary, we suppose this is not the case. Let $I_n$ denote the shortest interval containing $p$ and two distinct periodic points $x_n,y_n$ (of period $q_n$) of $f_{s_n}$ as its endpoints. Since $\epsilon\mapsto \rho(f_\epsilon)$ is locally constant at rational values, we may perturb each $s_n$ slightly, leaving $\rho(f_{s_n})$ unchanged, and ensure that the orbits of $x_n$ and $y_n$ do not intersect the endpointa of any plateau of $f_{s_n}$. It then follows from the Intermediate Value Theorem applied to $(f_{s_n})^{q_n}$ that at least one of the periodic endpoints, $x_n$ say, has an orbit disjoint from all plateaus of $f_{s_n}$.
It remains to show that $x_n\to p$. Assume for a contradiction that there exists an interval $J\ni p$, $J\subset I_n$ for arbitrarily large $n$. Since $I_n$ has period $q_n\to\infty$, it follows that $\{(f_{s_n})^k(J):0\leq k<q_n\}$ are pairwise disjoint intervals. Thus $p$ is contained in a wandering interval for $f_0$, which contradicts the assumption that $\mathrm{o}^+(p)$ accumulates on $p$ from both sides. \end{proof}
\begin{prop}\label{prop:twist} Let $X\in\mathfrak{X}^r(\mathbb{T}^2)$, $r\geq 1$, be a vector field with co-directed black cells and a non-trivially recurrent point $p$. Then there exists a loop $\Sigma$ along which $X$ admits $C^r$ closing by twist perturbations. \end{prop} \begin{proof} Since $X$ is has co-directed black cells, by Lemma \ref{lem:transversal} there exists a transverse loop $\Sigma$ through $p$ such that the induced map to $\Sigma$ extends to a continuous map $P:\Sigma\to\Sigma$. Let $U$ be a singularity-free annular neighbourhood of $\Sigma$ and let $\{Y_a\}_{0\leq a<a_0}$ be a family of $C^r$ twist perturbations. Reducing $a_0$ if necessary, we may assume $\Sigma$ is transverse to $Y_a$ for all $a\in [0,a_0)$.
Let $f_a:\Sigma\to\Sigma$ be the map induced by $Y_a$. Then the family $\{f_a\}_{0\leq a<a_0}$ satisfies the hypotheses of Lemma \ref{lem:periodicpoints}, and thus there exist sequences $x_n\to p$ and $(a_n)\downarrow 0$, where $x_n\in\Sigma$ is a periodic point of $f_{a_n}$. Moreover, since the orbit of $x_n$ does not intersect any plateaus of $f_{a_n}$, we have that the orbit of $x_n$ under $Y_{a_n}$ is a closed orbit. \end{proof}
\begin{thm}\label{thm:C} Let $X\in\mathfrak{X}^r(\mathbb{T}^2)$, $r\geq 4$, be a bounded type vector field with finitely many singularities all hyperbolic, and suppose the divergence is zero at each saddle.
\noindent Then for any recurrent point $p$, there exists a loop $\Sigma$ through $p$ and transverse to $X$ such that any $1$-parameter family of $C^r$ twist perturbations along $\Sigma$ contains closing perturbations for $p$ that are arbitrarily $C^r$ near to $X$. \end{thm} \begin{proof} The result follows directly from Theorem \ref{thm:B} and Proposition \ref{prop:twist}. \end{proof}
\begin{rem} In Theorems \ref{thm:B} and \ref{thm:C} we assume $X$ is $C^4$ because of the resonance of the eigenvalues of $X$ at the saddle points (see for example \cite{Stowe,MvSdMM}). Both theorems remain true when $X$ is $C^2$, so long as there are $C^2$ linearising coordinates for each saddle point. \end{rem}
\end{document}
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\begin{document}
\title{Sequential generation of entangled multi-qubit states}
\author{C. Sch\"on,$^{1}$ E. Solano,$^{1,2}$ F. Verstraete,$^{1,3}$ J. I. Cirac$^{1}$, and M. M. Wolf,$^{1}$}
\affiliation{$^{1}$Max-Planck-Institut f\"ur Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany \\ $^{2}$Secci\'on F\'{\i}sica, Departamento de Ciencias, Pontificia Universidad Cat\'olica del Per\'u, Apartado Postal 1761, Lima, Peru\\ $^{3}$Institute for Quantum Information, California Institute of Technology, CA 91125, USA}
\pacs{03.67.-a,42.50.-p,03.65.Ud} \date{\today}
\begin{abstract} We consider the deterministic generation of entangled multi-qubit states by the sequential coupling of an ancillary system to initially uncorrelated qubits. We characterize all achievable states in terms of classes of matrix product states and give a recipe for the generation on demand of any multi-qubit state. The proposed methods are suitable for any sequential generation-scheme, though we focus on streams of single photon time-bin qubits emitted by an atom coupled to an optical cavity. We show, in particular, how to generate familiar quantum information states such as W, GHZ, and cluster states, within such a framework. \end{abstract}
\maketitle
Entangled multi-qubit states are a valuable resource for the implementation of quantum computation and quantum communication protocols, like distributed quantum computing~\cite{DQC}, quantum cryptography~\cite{Ekert1991} or quantum secret sharing \cite{CGL99}. Using photonic degrees of freedom as qubits, say, polarization states or time-bins of energy eigenstates, has the advantage that photons propagate safely over long distances. Consequently, photonic devices are the most promising systems for quantum communication tasks. For this purpose, a lot of effort has been made in recent years to develop efficient and deterministic single photon sources~\cite{Law1997,Kuhn2002,McKeever2004, Lange2004,Deppe2004, Forchel2004}.
Photonic multi-qubit states can be generated by letting a source emit photonic qubits in a sequential manner \cite{Gheri2000}. If we do not initialize the source after each step, the created qubits will be in general entangled. Moreover, if we allow for specific operations inside the source before each photon emission, we will be able to create different multi-qubit states at the output. In fact, this is a particular instance of a general sequential generation scheme, where an ancillary system is coupled in turn to a number of initially uncorrelated qubits.
It is the purpose of this paper to provide a complete characterization of all multipartite quantum states achievable within a sequential generation scheme. It turns out that the classes of states attainable with increasing resources are exactly given by the hierarchy of so-called matrix-product states (MPS)~\cite{Zittartz,Fannes92}. These states typically appear in the theory of one-dimensional spin systems \cite{AKLT}, as they are the variational set over which Density Matrix Renormalization Group techniques are carried out~\cite{DMRG}. Thus, our analysis stresses the importance of MPS, since we show that they naturally appear in a completely different and relevant physical context. Moreover, particular instances of low-dimensional MPS, like cluster states~\cite{Briegel2001} or GHZ states~\cite{GHZ}, are a valuable resource in quantum information \cite{VerstCirQC}. Conversely, we will provide a recipe for the generation on demand of any multi-qubit state within a sequential generation scheme. Due to the general validity of these results, we will first state and prove them without referring to any particular physical system. This will be then applicable to all sequential setups, like streams of photonic qubits emitted either by a cavity QED (CQED) source~\cite{Law1997,Kuhn2002,McKeever2004, Lange2004} or by a quantum dot coupled to a microcavity~\cite{Deppe2004, Forchel2004}.
In the second part, we will focus on the physical implementation of these ideas within the realm of CQED. The role of the ancillary system will be performed by a $D$-level atom coupled to a single mode of an optical cavity. The sequentially generated qubits will be time-bin qubits $|0\rangle$ and $|1\rangle$, describing the absence and presence of a photon emitted from the cavity in a certain time interval (see Fig.\;1).
\vspace*{0.2cm}
\begin{figure}
\caption{A trapped D-level atom is coupled to a cavity qubit, determined by the energy eigenstates $| 0 \rangle$ and $| 1 \rangle$. After arbitrary bipartite source-qubit operations, photonic time-bins are sequentially and coherently emitted at the cavity output, creating a desired entangled multi-qubit stream.}
\label{figure}
\end{figure}
We will concentrate on setups where all intermediate operations are arbitrary unitaries and the ancilla decouples in the last step. The latter enables us to generate pure entangled states in a deterministic manner without the need of measurements. Let ${\cal H}_{\cal A}\simeq\mathbb{C}^D$ and ${\cal H}_{\cal B}\simeq\mathbb{C}^2$ be the Hilbert spaces characterizing a $D$-dimensional ancillary system and a single qubit (e.g. a time-bin qubit) respectively. In every step of the sequential generation of a multi-qubit state, we consider a unitary time evolution of the joint system
${\cal H}_{\cal A}\otimes{\cal H}_{\cal B}$. Assuming that each qubit is initially in the state $|0\rangle$ (i.e., the time-bin is empty), we disregard the qubit at the input and write the evolution in the form of an isometry $V:{\cal H}_{\cal A}\rightarrow{\cal H}_{\cal A}\otimes{\cal H}_{\cal B}$. Expressing the latter in terms of a basis
$V=\sum_{i,\alpha,\beta}V_{\alpha,\beta}^i |\alpha, i \rangle\langle\beta|$, the isometry condition reads $\sum_{i=0}^1 V^{i\dagger} V^i
=\mbox{$1 \hspace{-1.0mm} {\bf l}$}$, where each $V^i$ is a $D \times D$ matrix. We choose a basis where $\{ | \alpha \rangle , | \beta \rangle \}$ are any of the $D$ ancillary levels. If we now apply successively $n$, not necessarily identical, operations of this form to an initial state
$|\varphi_I\rangle\in{\cal H}_{\cal A}$, we obtain the state \begin{equation} |\Psi\rangle =
V_{[n]} \ldots V_{[2]}V_{[1]}|\varphi_I\rangle\;. \end{equation} Here, and in the following, indices in squared brackets represent the steps in the generation sequence. The $n$ generated qubits are in general entangled with the ancilla as well as among themselves. Assuming that in the last step the ancilla decouples from the system, such that $|\Psi\rangle=|\varphi_F\rangle\otimes|\psi\rangle$, we are left with the
$n$-qubit state \begin{equation}\label{MPSiso} |\psi\rangle=\sum_{i_1\ldots i_n=0}^1\langle\varphi_F|V_{[n]}^{i_n}\ldots V_{[1]}^{i_1}|\varphi_I\rangle\;|i_n,\ldots, i_1\rangle . \end{equation} States of this form are called matrix-product states (MPS)~\cite{Zittartz,Fannes92}, and play a crucial role in the theory of one-dimensional spin systems. Equation~(\ref{MPSiso})
shows that all sequentially generated multi-qubit states, arising from a $D$-dimensional ancillary system ${\cal H}_{\cal A}$, are instances of MPS with $D \times D$ matrices $V^i$ and open boundary conditions specified by $|\varphi_I\rangle$ and $|\varphi_F\rangle$. We will now prove that the converse is also true, i.e., that every MPS of the form \begin{equation}\label{MPS} |\tilde{\psi}\rangle = \langle\tilde{\varphi}_F|
\tilde{V}_{[n]}\ldots \tilde{V}_{[1]}|\tilde{\varphi}_I\rangle,\end{equation} with arbitrary maps
$\tilde{V}_{[k]}:{\cal H}_{\cal A}\rightarrow{\cal H}_{\cal A}\otimes{\cal H}_{\cal B}$, can be generated by isometries of the same dimension, and such that the ancillary system decouples in the last step. Since every state has a MPS representation, this is at the same time a general recipe for its sequential generation. The idea of the proof is an explicit construction of all involved isometries by subsequent application of singular value decompositions (SVD). We start by writing $\langle\tilde{\varphi}_F| \tilde{V}_{[n]}=V_{[n]}'M_{[n]}$, where the $2\times 2$ matrix $V_{[n]}'$ is the left unitary in the SVD and $M_{[n]}$ is the remaining part. The recipe for constructing the isometries is the induction \begin{equation} \big(M_{[k]}\otimes\mbox{$1 \hspace{-1.0mm} {\bf l}$}_2\big)\tilde{V}_{[k-1]} =V_{[k-1]}'M_{[k-1]}\label{induction},\end{equation} where the isometry $V_{[k-1]}'$ is constructed from the SVD of the left hand side, and $M_{[k-1]}$ is always chosen to be the remaining part. After
$n$ applications of Eq.~(\ref{induction}) in Eq.~(\ref{MPS}), from left to right, we set $|\varphi_I\rangle=M_{[1]}|\tilde{\varphi}_I\rangle$, producing \begin{equation} |\tilde{\psi}\rangle = V_{[n]}'\ldots V_{[1]}'|\varphi_I\rangle.\end{equation} Simple rank considerations show that $V_{[n-k]}'$ has dimension $2\min{[D,2^k]}\times \min{[D,2^{k+1}]}$. In particular, every $V_{[k]}'$ could be embedded into an isometry $V_{[k]}$ of dimension $2D \times D$. Physically, this just means we would have redundant ancillary levels that we need not to use. Finally, decoupling the ancilla in the last step is guaranteed by the fact that, after the application of $V_{[n-1]}$, merely two levels of ${\cal H}_{\cal A}$ are yet occupied, and can be mapped entirely onto the system ${\cal H}_{\cal B}$. This is precisely the action of $V_{[n]}$ through its embedded unitary $V_{[n]}'$.
This proves the equivalence of two sets of $n$-qubit states, which are described either as $D$-dimensional MPS with open boundary conditions, or as states that are generated sequentially and isometrically via a $D$-dimensional ancillary system which decouples in the last step. Motivated by current cavity QED setups, we will now provide a third equivalent characterization, namely, a set of multi-qubit states that are sequentially generated by a source consisting of a $2D$-level atom. In contrast to the first sequential scheme, the latter will not require arbitrary isometries.
Consider an atomic system with $D$ states $| a_i \rangle$ and $D$
states $| b_i \rangle$, so that
${\cal H}_{\cal A}={\cal H}_a\oplus{\cal H}_b\simeq\mathbb{C}^d\otimes\mathbb{C}^2$. That is, we will write $|\varphi\rangle|1\rangle$ for a superposition of $|
a_i \rangle$ states, whereas $|\varphi\rangle|0\rangle$ corresponds to a superposition of $| b_i \rangle$ states. Since the last qubit marks the atomic level, whether it belongs to the $| a_i \rangle$
or to the $| b_i \rangle$ subspace, we will refer to it as the tag-qubit and write ${\cal H}_{\cal A}={\cal H}_{{\cal A}'}\otimes{\cal H}_{\cal T}$. Now consider the atomic transitions from each $| a_i \rangle$ state to its respective $| b_i \rangle$ state accompanied by the generation of a photon in a certain time-bin. This is described by a unitary evolution, since now called ``D-standard map'', of the form \begin{eqnarray} T:\;|\varphi\rangle_{{\cal A}'}|1\rangle_{\cal T} |0\rangle_{\cal B} &\mapsto&
|\varphi\rangle_{{\cal A}'}|0\rangle_{\cal T} |1\rangle_{\cal B}\;, \nonumber
\\ |\varphi\rangle_{{\cal A}'}|0\rangle_{\cal T} |0\rangle_{\cal B} &\mapsto&
|\varphi\rangle_{{\cal A}'}|0\rangle_{\cal T} |0\rangle_{\cal B}\; . \end{eqnarray} Hence, $T$
effectively interchanges the tag-qubit with the time-bin qubit. If, additionally, arbitrary atomic unitaries $U_{{\cal A}}$ are allowed at any time, we can exploit the swap caused by $T$ in order to generate the operation \begin{equation} V|\varphi\rangle=\<0|_{\cal T} T\Big(U_{{\cal A}}\big(|\varphi\rangle_{{\cal A}'}|0\rangle_{\cal T}\big)|0\rangle_{\cal B}\Big) \label{IsofromT}\;,\end{equation} which is the most general isometry $V:{\cal H}_{{\cal A}'}\rightarrow {\cal H}_{{\cal A}'}\otimes{\cal H}_{\cal B}$. Therefore, the so generated $n$-qubit states include all possible states arising from subsequent applications of $2D \times D$-dimensional isometries. On the other hand, they are a subset of the MPS in Eq.~(\ref{MPS}) with arbitrary $2D \times D$-dimensional maps, assuming that the atom decouples at the end. Hence, these three sets are all equivalent.
Now, we show how these results can be applied in the realm of cavity QED, where an atom is trapped inside a high-$Q$ optical cavity, and we aim at generating multi-photon entangled states. A laser may excite the atom, producing subsequently a photon in the cavity mode, which, after some time, is emitted outside the cavity (Fig.\ 1). We consider two different scenarios, corresponding to the two families of states considered above. First, we may have fast and complete access to the atom-cavity system. In consequence, after the implementation of the desired isometry in each step, we should wait until the photon leaks out of the cavity before starting the next step. In this case, according to the analysis above, we will be able to produce arbitrary
$D$-dimensional MPS with $D$ equal to the number of involved atomic levels. Second, we may have a 2$D$--level atom ($D$ $| a_i
\rangle$ levels and $D$ $| b_i \rangle$ levels) and two kind of operations: (i) fast arbitrary operations which allow us to manipulate at will all atomic levels; (ii) an operation which maps each $| a_i \rangle$ state to its corresponding $| b_i \rangle$ state while creating a single cavity photon, allowing a taylored output. Here, we will also be able to produce arbitrary $D$-dimensional MPS.
In the following, we will illustrate the above statements with a specific example which is based on present cavity QED experiments
\cite{Kuhn2002,McKeever2004, Lange2004}. We consider a three--level atom coupled to a single cavity mode in the strong-coupling regime. An external laser field drives the transition from level $|a\rangle$ to the upper level $|u\rangle$ with coupling strength $\Omega_0$, and the cavity mode drives the transition between $|u\rangle$ and level $|b\rangle$ with coupling strength
$g$, in a typical $\Lambda$ configuration [see Fig.\ 2(a)]. We choose the detunings $\Delta$, with $|\Delta| \gg \{ g, \Omega_0
\}$, and assume that the cavity decay rate $\kappa$ is smaller than any other frequency in the problem, so that we can ignore cavity damping during the atom-cavity manipulations. By eliminating level $|u\rangle$, we remain with an effective $D=2$ atomic system plus the cavity mode. We will show how, by controlling the laser frequency and intensity, it is possible to generate arbitrary $2$-dimensional MPS. Note that, by allowing the manipulation of $D$ effective atomic levels, it is straightforward to extend these results to the generation of $D$-dimensional MPS.
According to the results presented above, we just have to show that we can implement any isometry $V:{\cal H}_{\cal A}\rightarrow{\cal H}_{\cal A}\otimes{\cal H}_{\cal B}$. In fact, we will show how it is possible to implement arbitrary operations on the atomic qubit and the $\sqrt{SWAP}$ operation on the atom-cavity system, which suffice to generate any isometry $V$ (since they give rise to a universal set of gates for the two qubit system~\cite{Nielsen2000}). The atomic qubit can be manipulated at will using a Raman laser system, as it is normally done with trapped ions~\cite{NIST,Innsbruck}. In order to implement the $\sqrt{SWAP}$, we notice that the atom-cavity coupling is described in terms of the Jaynes--Cummings model [see Fig.~2(b)], where the coupling constant $\Omega_0$ is controlled by the laser. Thus, application of laser pulses with the appropriate duration and phase \cite{Innsbruck,SelectiveCQED} will implement the unitary operation $U=e^{-iG}$, where generator
$G=(|a,0\rangle\langle b,1| + {\rm H.c.})\pi/4$, which corresponds to the desired $\sqrt{SWAP}$ operation. In order to generalize this scheme to an arbitrary $D$-level system, we notice that we can view the atom as a set of $M$ qubits (with $D\le 2^M$). Thus, if we are able to perform arbitrary atomic operations, together with the $\sqrt{SWAP}$ operation on two specific atomic levels as explained above, we can then implement a universal set of gates and, in consequence, any arbitrary isometry.
\begin{figure}\label{figure2}
\end{figure}
In the rest of the paper, we will use another setup which is closely related to current experiments \cite{Kuhn2002,McKeever2004, Lange2004} and optimizes our second method for MPS generation. In this frame, we will show how to generate familiar multi-qubit states like $W$~\cite{Wstates}, $GHZ$~\cite{GHZ}, and cluster states~\cite{Briegel2001}, which are all MPS with $D=2$ \cite{VerstCirQC}.
For the purpose, we consider an atom with three effective levels
$\{|a\rangle,|b_1\rangle,|b_2\rangle\}$ trapped inside an optical cavity. With the help of a laser beam, state $|a\rangle$ is mapped to the internal state $|b_1\rangle$, and a photon is generated, whereas the other states remain unchanged. This physical process is described by the map \begin{eqnarray}
M_{{\cal A}{\cal B}}: \; | a \rangle &\mapsto& | b_1 \rangle | 1 \rangle \; , \nonumber\\
| b_1 \rangle &\mapsto& | b_1 \rangle |0\rangle \; , \nonumber\\ | b_2 \rangle
&\mapsto& | b_2 \rangle |0\rangle \; , \end{eqnarray} and can be realized with the techniques used in~\cite{Kuhn2002,McKeever2004,Lange2004}. After the application of this process, an arbitrary operation is applied to the atom, which can be performed by using Raman lasers. The photonic states that are generated after several applications are those MPS where the isometries are given by $V_{[i]}=M_{{\cal A}{\cal B}}U_{\cal A}^{[i]}$, with ${i=1,\dots,n}$, $U_{\cal A}^{[i]}$ being arbitrary unitary atomic operators.
For example, to generate a W-type state of the form
\begin{eqnarray}
| \psi_{\rm W} \rangle
&=& e^{i\Phi_1}\sin\Theta_1 |0...01\rangle + \cos\Theta_1
e^{i\Phi_2}\sin\Theta_2 |0...010\rangle \nonumber\\&& +...+
\cos\Theta_1...\cos\Theta_{n-2} e^{i\Phi_{n-1}}\sin\Theta_{n-1}
|010...0\rangle\nonumber\\&& + \cos\Theta_1...\cos\Theta_{n-1}
|10...0\rangle,
\end{eqnarray} we choose the initial atomic state $|\varphi_I\rangle=| b_2 \rangle$ and operations $U_{\cal A}^{[i]} = U_{a {b_2}}^{b_1}(\Phi_i,\Theta_i)$, with $i=1,\dots,n-1$, where \begin{eqnarray} U_{kl}^m(\Phi_i,\Theta_i) &
\!\!\!\!\! = \!\!\!\! & \cos\Theta_i|k\rangle\<k| + \cos\Theta_i
|l\rangle\<l|+ e^{i\Phi_i} \! \sin\Theta_i|k\rangle\<l| \nonumber\\&& -
e^{-i\Phi_i}\sin\Theta_i|l\rangle\<k| + |m\rangle\<m| ,\end{eqnarray} and $\{ k,l,m \} = \{ a,b_1,b_2 \}$. To decouple the atom from the photon state, we choose the last atomic operation $U_{\cal A}^{[n]} = U_{a
{b_2}}^{b_1}(0,\pi/2)$ and, after the last map $M_{{\cal A}{\cal B}}$, the decoupled atom will be in state $|b_1\rangle$.
To produce a GHZ-type state in similar way, we choose
$|\varphi_I\rangle=|a\rangle$, $U_{\cal A}^{[1]}=U_{a {b_2}}^{b_1}(\Phi_1,\Theta_1)$, $U_{\cal A}^{[i]} = U_{a {b_1}}^{b_2}(0,\pi/2)$, with $i=2,\dots,n-1$, and $U_{\cal A}^{[n]} = U_{{b_1} {b_2}}^a(0,\pi/2)U_{a {b_1}}^{b_2}(0,\pi/2)$.
For generating cluster states, we choose $|\varphi_I\rangle=|b_2\rangle$, $U_{\cal A}^{[i]} = U_{a {b_2}}^{b_1}(\Phi_i,\Theta_i) U_{a {b_1}}^{b_2}(0,\pi/2)$, with $i=1,\dots,n-1$, and $U_{\cal A}^{[n]} = U_{a {b_1}}^{b_2}(\Phi_n,\Theta_n) U_{{b_1}{b_2}}^a(0,\pi/2) U_{a
{b_1}}^{b_2}(0,\pi/2)$, obtaining \begin{equation} |\psi \rangle =
\bigotimes_{i=1}^n \left(O_{i-1}^0 |0\rangle_i + O_{i-1}^1
|1\rangle_i\right) , \end{equation} where $O^0_{i-1} = \cos\Theta_i |0\rangle_{i-1}\<0|
-e^{-i\Phi_i}\sin\Theta_i |1\rangle_{i-1}\<1|$ and $O^1_{i-1}=
e^{i\Phi_i}\sin\Theta_i |0\rangle_{i-1}\<0| + \cos\Theta_i
|1\rangle_{i-1}\<1|$, with ${i=2,\dots,n-1}$. Operators $O_{i-1}^0$ and $O_{i-1}^1$ act on the nearest neighbor-qubit $i-1$ under the assumption $O_0^0 \equiv \cos\Theta_1$ and $O_0^1 \equiv e^{i\Phi_1}\sin\Theta_1$. If one chooses $\Phi_i=0$ and $\Theta_i=\pi/4$ this leads to the cluster states defined by \begin{equation}
|\psi_{\rm cl}\rangle = \frac{1}{2^{n/2}} \bigotimes_{i=1}^n
\left(\sigma^z_{i-1}|0\rangle_i + |1\rangle_i\right) \,\, , \,\, {\rm with} \,\,\, \sigma^z_0\equiv 1 . \end{equation}
The formalism presented here is also valid for other types of single-photon sources, in the context of cavity QED or quantum dots. For example, it could be extended to characterize the polarization-entangled multi-qubit photon states generated by an analogous cavity QED photon source~\cite{Gheri2000}. In fact, the presented ideas and proofs apply to any multi-qudit state with ${\cal H}_{{\cal B}}\simeq\mathbb{C}^d$ that is generated sequentially by a $D$-dimensional source.
In a wider scope, we have established a formalism describing a general sequential quantum factory, where the source is able to perform arbitrary unitary source-qudit operations before each qudit leaves. Apart from the multiphoton states, the present formalism applies also to many other physical scenarios: (a) to coherent microwave cavity QED experiments~\cite{Haroche}, where atoms sequentially cross a cavity, and thus the outcoming atoms end up in a MPS with the dimensions given by the effective number of states used in the cavity mode; (b) a light pulse crossing several atomic ensembles~\cite{Polzik2003}, which will be left in a matrix product Gaussian state \cite{Norbert}; (c) trapped ion experiments where each ion interacts sequentially with a collective mode of the motion~\cite{NIST,Innsbruck,CiracZoller}. Note also that one can include dissipation in the present formalism, by replacing MPS by matrix product density operators ~\cite{Fannes92,MPDO}. This description applies, for example, to the micromaser setup ~\cite{Walther2001} and other realistic scenarios.
C.S. wants to thank K. Hammerer for useful discussions. This work was supported by EU through RESQ project and the "Kompetenznetzwerk Quanteninformationsverarbeitung der Bayerischen Staatsregierung".
\end{document}
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Hopf algebroids, Hopf categories,\\ and their Galois theories}
\author{Clarisson Rizzie Canlubo\\ University of Copenhagen\\ \textit{[email protected]}}
\maketitle
\pagestyle{fancy} \headheight=17pt \textwidth=500pt \fancyhead{} \fancyhead[L]{\footnotesize{\leftmark}} \fancyhead[R]{\thepage} \renewcommand{1pt}{1pt} \renewcommand{1pt}{1pt} \fancyfoot{} \fancyfoot[R]{} \fancyfoot[L]{\scriptsize{Hopf algebroids, Hopf categories, and their Galois theories}} \fancyfoot[C]{}
\begin{abstract} Hopf algebroids are generalization of Hopf algebras over non-commutative base rings. It consists of a left- and a right-bialgebroid structure related by a map called the antipode. However, if the base ring of a Hopf algebroid is commutative one does not necessarily have a Hopf algebra. Meanwhile, a Hopf category is the categorification of a Hopf algebra. It consists of a category enriched over a braided monoidal category such that every hom-set carries a coalgebra structure together with an antipode functor. In this article, we will introduce the notion of a topological Hopf category$-$ a small category whose set of objects carries a topology and whose categorical structure maps are sufficiently continuous. The main result of this paper is to describe the relation between finitely-generated projective Hopf algebroids over commutative unital $C^{*}$-algebras and topological coupled Hopf categories of finite-type whose space of objects is compact and Hausdorff. We will accomplish this by using methods in algebraic geometry and spectral theory. Lastly, we will show that not only the two objects are tightly related, but so are their respective Galois theories.
\
\noindent \textit{Mathematics Subject Classification} (2010): 16T05, 14A20, 18F99, 18B40, 58B34
\
\noindent \textit{Keywords}: Hopf algebroid, Hopf category, Galois theory. \end{abstract}
\tableofcontents
\normalsize
\section{Introduction}\label{S1.0}
Hopf algebras are robust generalization of groups. Recently, many authors have studied much more general Hopf-like structures: weak Hopf algebras, Hopf monads, $\times_{R}$-Hopf algebras, compact quantum groups to name a few. In this article, we will mainly be interested with Hopf algebroids and Hopf categories. In the literature, there are plenty of inequivalent notions of a Hopf algebroid. For the exposition on these notions, see B\"ohm \cite{bohm}. Batista et al. \cite{bcv} introduced the notion of a Hopf category which is the natural categorification of a Hopf algebra. Motivated by a fundamental related to his PhD thesis, the author tries to describe the geometry of Hopf algebroids over $C(X)$. This geometric description necessitates a structure closely related to a Hopf category, but which has not appeared in the literature as far as the author's knowledge. We will define such structures in section (\ref{S3.0}).
We will recall in section (\ref{S2.0}) definitions and properties of Hopf algebroids. For completeness, we will also give a short exposition on the representation theoretic and Galois theoretic aspects of Hopf algebroids. Most of section (\ref{S2.0}) follows \cite{bohm} except for the definition of morphisms of Hopf algebroids and the definition of a coupled Hopf algebra. In section (\ref{S3.0}), we will define what topological Hopf categories are and we will also define coupled Hopf categories and their topological version. We will end that section with a formulation of Galois theory for Hopf categories and all the related variant we will introduce in that section.
One of the main result of this paper is theorem (\ref{T4.1}). It gives a bijective correspondence between finitely-generated projective Hopf algebroids over $C(X)$ and topological coupled Hopf categories of finite type. Using algebraic geometric and spectral theoretic methods, spanning the entirety of section (\ref{S4.0}), we will prove this result. The second main result is theorem (\ref{T5.1}), which states that, not only is there a bijection between Hopf algebroids and topological Hopf categories, their Galois theories also matched in a bijective manner.
Following David Hilbert's statement:
\begin{adjustwidth}{2cm}{2cm} \textit{"The art of doing mathematics consists in finding that special case which contains all the germs of generality."} \end{adjustwidth}
\noindent we will discuss a very important example in section (\ref{S3.2}) which completely illustrates the general situation.
\textbf{Acknowledgement.} I would like to thank my PhD supervisor Ryszard Nest for guiding me through my studies in non-commutative geometry and for the valuable discussions that help me write this article. I would also like to thank DSF Grant, UP Diliman and the support of the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).
\section{Hopf algebroids}\label{S2.0}
\subsection{Definitions}\label{S2.1}
There are several inequivalent notions of a Hopf algebroid. We will briefly present here the one defined in B\"ohm \cite{bohm}. An $R$-\textit{ring} is a monoid object in the category of $R$-bimodules. Explicitly, an $R$-ring is a triple $(A,\mu,\eta)$ where $A\otimes_{R}A\stackrel{\mu}{\longrightarrow}A$ and $R\stackrel{\eta}{\longrightarrow}A$ are $R$-bimodule maps satisfying the associativity and unit axioms similar for algebras over commutative rings. A morphism of $R$-rings is a monoid morphism in category of $R$-bimodules. It is important to note that there is a bijection between $R$-rings $(A,\mu,\eta)$ and $k$-algebra morphisms $R\stackrel{\eta}{\longrightarrow}A$. Similar to the case of algebras over commutative rings, we can define modules over $R$-rings. For an $R$-ring $(A,\mu,\eta)$, a \textit{right} (resp. \textit{left}) $(A,\mu,\eta)$-\textit{module} is an algebra for the monad $-\otimes_{R}A$ (resp. $A\otimes_{R}-$) on the category $\mathcal{M}_{R}$ (resp. ${}_{R}\mathcal{M}$) of right (resp. left) modules over $R$.
We can dualize all the objects we have defined in the previous paragraph. An $R$-\textit{coring} is a comonoid in the category of $R$-bimodules, i.e a triple $(C,\Delta,\epsilon)$ where $C\stackrel{\Delta}{\longrightarrow}C\otimes_{R}C$ and $C\stackrel{\epsilon}{\longrightarrow}R$ are $R$-bimodule maps satisfying the coassociativity and counit axioms dual to those axioms satisfied by the structure maps of an $R$-ring. A morphism of $R$-corings is a morphism of comonoids. Given an $R$-coring $(C,\Delta,\epsilon)$, similar to coalgebras over commutative rings, we define a \textit{right} (resp. \textit{left}) $(C,\Delta,\epsilon)$-\textit{comodule} as a coalgebra for the comonad $-\otimes_{R}C$ (resp. $C\otimes_{R}-$) on the category $\mathcal{M}_{R}$ (resp. ${}_{R}\mathcal{M}$).
\begin{defn}\label{D2.1} A \textit{right} (resp. \textit{left}) $R$-\textit{bialgebroid} $B$ is an $R\otimes_{k}R^{op}$-ring $(B,s,t)$ and an $R$-coring $(B,\Delta,\epsilon)$ satisfying:
\begin{enumerate} \item [(a)] $R\stackrel{s}{\longrightarrow}B$ and $R^{op}\stackrel{t}{\longrightarrow}B$ are $k$-algebra maps with commuting images defining the $R\otimes_{k}R^{op}$-ring structure on $B$ which is compatible to the $R$-bimodule structure as an $R$-coring thru the following relation: \[ r\cdot b \cdot r':=bs(r')t(r), \ \ (\text{resp.} \ r\cdot b \cdot r':=s(r)t(r')b,) \hspace{.15in} \forall r,r'\in R, b\in B. \]
\item [(b)] With the above $R$-bimodule structure on $B$ one can form $B\otimes_{R}B$. The coproduct $\Delta$ is required to corestrict to a $k$-algebra map to
\[ B\times_{R}B:=\left\{\sum\limits_{i}b_{i}\otimes_{R}b_{i}'\left|\sum\limits_{i}s(r)b_{i}\otimes_{R}b_{i}'=\sum\limits_{i}b_{i}\otimes_{R}t(r)b_{i}',\forall r\in R\right.\right\} \] \noindent respectively,
\[ B \prescript{}{R}{\times} \ B:=\left\{\sum\limits_{i}b_{i}\otimes_{R}b_{i}'\left|\sum\limits_{i}b_{i}t(r)\otimes_{R}b_{i}'=\sum\limits_{i}b_{i}\otimes_{R}b_{i}'s(r),\forall r\in R\right.\right\}. \]
\item [(c)] The counit $B\stackrel{\epsilon}{\longrightarrow}R$ extends the right (resp. left) regular $R$-module structure on $R$ to a right (resp. left) $(B,s)$-module. \end{enumerate}
\noindent A \textit{morphism} of $R$-bialgebroids is a morphism of $R\otimes R^{op}$-rings and $R$-corings. \end{defn}
\begin{rem}\label{R2.1} \begin{enumerate} \item[] \item[(1)] The $k$-algebra maps $s$ and $t$ define a $k$-algebra map $\eta=s\otimes_{k}t$. As we have noted, such $k$-algebra uniquely determines an $R\otimes_{k}R^{op}$-ring structure on $B$. The maps $s$ and $t$ are called the \textit{source} and \textit{target} maps, respectively.
\item[(2)] The $k$-submodule $B\times_{R}B$ (resp. $B\prescript{}{R}{\times} \ B$) of $B\otimes_{R}B$ is a $k$-algebra with factorwise multiplication. This is called the \textit{Takeuchi product}. The map $R\otimes_{k}R^{op}\longrightarrow B\times_{R}B$, $r\otimes_{k}r'\mapsto t(r')\otimes_{R}s(r)$ is easily seen to be a $k$-algebra morphism and hence, $B\times_{R}B$ is an $R\otimes_{k}R^{op}$-ring. The corestriction of $\Delta$ is an $R\otimes_{k}R^{op}$-bimodule map. Hence, $\Delta$ is an $R\otimes R^{op}$-ring map. The same is true for $B\prescript{}{R}{\times} \ B$.
\item[(3)] The source map $s$ is a k-algebra map and so it defines a unique $R$-ring structure on $B$. The right version of condition (c) explicitly means that $r\cdot b:=\epsilon(s(r)b)$, $\forall r\in R, b\in B $ defines a right $(B,s)$-action on $R$. \end{enumerate} \end{rem}
\begin{defn}\label{D2.2} Let $k$ be a commutative, associative unital ring and let $L$ and $R$ be associative $k$-algebras. A \textit{Hopf algebroid} $\mathcal{H}$ is a triple $\mathcal{H}=(H_{L},H_{R},S)$. $H_{L}$ and $H_{R}$ are bialgebroids having the same underlying $k$-algebra $H$. Specifically, $H_{L}$ is a left $L$-bialgebroid with $(H,s_{L},t_{L})$ and $(H,\Delta_{L},\epsilon_{L})$ as its underlying $L\otimes_{k}L^{op}$-ring and $L$-coring structures. Similarly, $H_{R}$ is a right $R$-bialgebroid with $(H,s_{R},t_{R})$ and $(H,\Delta_{R},\epsilon_{R})$ as its underlying $R\otimes_{k}R^{op}$-ring and $R$-coring structures. Let us denote by $\mu_{L}$ (resp. $\mu_{R}$) the multiplication on $(H,s_{L})$ (resp. $(H,s_{R})$). $S$ is a (bijective) $k$-module map $H\stackrel{S}{\longrightarrow}H$, called the \textit{antipode}. The compatibility conditions of these structures are as follows.
\begin{enumerate} \item[(a)] the sources $s_{R},s_{L}$, targets $t_{R},t_{L}$ and counits $\epsilon_{R},\epsilon_{L}$ satisfy
\[ s_{L}\circ \epsilon_{L}\circ t_{R}=t_{R}, \hspace{.15in} t_{L}\circ \epsilon_{L}\circ s_{R}=s_{R}, \hspace{.15in} s_{R}\circ \epsilon_{R}\circ t_{L}=t_{L}, \hspace{.15in} t_{R}\circ \epsilon_{R}\circ s_{L}=s_{L}, \]
\item[(b)] the left- and right-regular comodule structures commute, i.e.
\[ \xymatrix{ H \ar[d(1.7)]_-{\Delta_{L}} \ar[r(1.7)]^-{\Delta_{R}} & & H\tens{R} H \ar[d(1.6)]^-{\Delta_{L}\tens{R} id}\\ & & \\ H\tens{L} H \ar[r(1.6)]_-{id \tens{L}\Delta_{R}} & & H\tens{L} H\tens{R} H } \hspace{0.75in} \xymatrix{ H \ar[d(1.7)]_-{\Delta_{R}} \ar[r(1.7)]^-{\Delta_{L}} & & H\tens{L} H \ar[d(1.6)]^-{\Delta_{R}\tens{L} id}\\ & & \\ H\tens{R} H \ar[r(1.6)]_-{id \tens{R}\Delta_{L}} & & H\tens{R} H\tens{L} H } \]
\item[(c)] for all $l\in L, r\in R$ and for all $h\in H$ we have $S(t_{L}(l)ht_{R}(r))=s_{R}(r)S(h)s_{L}(l)$,
\item[(d)] $S$ is the convolution inverse of the identity map i.e., the following diagram commute
\[ \xymatrix{ & & H\tens{L} H \ar[rrrr]^-{S\tens{L} id} & & & & H\tens{L} H \ar[rrd]^-{\mu_{L}} & & \\ H \ar[rru]^-{\Delta_{L}} \ar[rrrr]^-{\epsilon_{R}} & & & & R \ar[rrrr]^-{s_{R}} & & & & H \\ H \ar[rrd]_-{\Delta_{R}} \ar[rrrr]_-{\epsilon_{L}} & & & & L \ar[rrrr]_-{s_{L}} & & & & H \\ & & H\tens{R} H \ar[rrrr]_-{id\tens{R} S} & & & & H\tens{R} H \ar[rru]_-{\mu_{R}} & & \\ } \]
\end{enumerate} \end{defn}
\begin{rem}\label{R2.2} \begin{enumerate} \item[] \item[(1)] In the constituent bialgebroids $H_{R}$ and $H_{L}$, the counits $\epsilon_{R}$ and $\epsilon_{L}$ extend the regular module structures on the base rings $R$ and $L$ to the $R$-ring $(H,s_{R})$ and to the $L$-ring $(H,s_{L})$, respectively. Equivalently, the counits extend the regular module structures on the base rings $R$ and $L$ to the $R^{op}$-ring $(H,t_{R})$ and to the $L^{op}$-ring $(H,t_{L})$. This particularly implies that the maps $s_{L}\circ \epsilon_{L}$, $t_{L}\circ \epsilon_{L}$, $s_{R}\circ \epsilon_{R}$ and $t_{R}\circ \epsilon_{R}$ are idempotents. This means that the images of $s_{R}$ and $t_{L}$ coincides in $H$. Same is true for the images of $s_{L}$ and $t_{R}$.
\item[(2)] Notice that for condition (b) to make sense, apart from being an $L$-bimodule map, $\Delta_{L}$ has to be an $R$-bimodule map. This is the case using remark (1). Similarly, $\Delta_{R}$ is an $L$-bimodule map.
\item[(3)] We can equip $H$ with two $(R,L)$-bimodule structures one using $t_{R}$ and $t_{L}$ and the other using $s_{R}$ and $s_{L}$. Condition (c) relates these two $(R,L)$-bimodules structures via the antipode $S$ which in turn makes the diagram in condition (d) defined.
\item[(4)] A most convenient way to summarize the property of the antipode of a Hopf algebra is to express it as the inverse of the identity map in the convolution algebra of endomorphisms of that Hopf algebra. For Hopf algebroids, the antipode is the inverse of the identity map in the appropriate category, called the \textit{convolution category} of $\mathcal{H}$. As before, $R$ and $L$ are $k$-algebras. Let $X$ and $Y$ be $k$-modules such that $X$ has an $R$-coring $(X, \Delta_{R}, \epsilon_{R})$ and an $L$-coring $(X,\Delta_{L},\epsilon_{L})$ structures and $Y$ has an $L\otimes_{k}R$-ring structure with multiplications $\mu_{R}:Y\otimes_{R}Y\longrightarrow Y$ and $\mu_{L}:Y\otimes_{L} Y\longrightarrow Y$. Define the convolution category $Conv(X,Y)$ to be the category with two objects labelled $R$ and $L$. For $I,J\in \left\{R,L\right\}$, a morphism $I\longrightarrow J$ is a $J$-$I$-bimodule map $X\longrightarrow Y$. For $I,J,K\in \left\{R,L\right\}$ and morphisms $J\stackrel{f}{\longrightarrow}I$ and $K\stackrel{g}{\longrightarrow}J$, we define the composition $f\ast g$ to be the following convolution
\[ f\ast g = \mu_{J} \circ (f\tens{J}g) \circ \Delta_{J}. \]
\noindent The antipode $S$ of a Hopf algebroid $\mathcal{H}$ is the inverse of the identity map $H\stackrel{id}{\longrightarrow}H$ viewed as an arrow in $Conv(H,H)$.
\item[(5)] Let us note that condition (c) in the definition of a bialgebroid implies that $\epsilon_{L}\circ s_{L}:L\longrightarrow L$ is the identity. Similarly, $\epsilon_{R}\circ s_{R}:R\longrightarrow R$ is also the identity. Using condition (a) in the definition of a Hopf algebroid, we see that the following compositions define pairs of inverse $k$-algebra maps.
\[ \xymatrix{ L \ar[rr]^-{\epsilon_{R}\circ s_{L}} & & R^{op} \ar[rr]^-{\epsilon_{L}\circ t_{R}} & & L} \hspace{.5in} \xymatrix{ R \ar[rr]^-{\epsilon_{L}\circ s_{R}} & & L^{op} \ar[rr]^-{\epsilon_{R}\circ t_{L}} & & R}\]
\noindent This is particular implies that $R$ and $L$ are anti-isomorphic $k$-algebras.
\item[(6)] Since there are two coproducts involved in a Hopf algebroid, namely $\Delta_{L}$ and $\Delta_{R}$, we will use different Sweedler notations for their corresponding components. We will write $\Delta_{L}(h)=h_{[1]}\otimes_{L}h_{[2]}$ and $\Delta_{R}(h)=h^{[1]}\otimes_{R}h^{[2]}$ for $h\in H$.
\item[(7)] With a fixed bijective antipode $S$, the constituent left- and right-bialgebroids of a Hopf algebroid determine each other, see for example the article \cite{bohm-szlachanyi}. In view of this and the fact that $L$ and $R$ are anti-isomorphic, in the sequel where we will be mainly interested with Hopf algebroids with bijective antipodes we will simply call $\mathcal{H}$ a Hopf algebroid \textit{over} $R$ instead of explicitly mentioning $L$.
\end{enumerate} \end{rem}
\begin{defn} \label{D2.3} Let $(H_{L},H_{R},S)$ and $(H_{L}^{'},H_{R}^{'},S^{'})$ be Hopf algebroids over $R$. An \textit{algebraic morphism} $(H_{L},H_{R},S)\longrightarrow(H_{L}^{'},H_{R}^{'},S^{'})$ of Hopf algebroids is a pair $(\varphi_{L},\varphi_{R})$ of a left-bialgebroid morphism $\varphi_{L}$ and a right-bialgebroid morphism $\varphi_{R}$ for which the following diagrams commute
\[ \xymatrix{ H_{L} \ar[dd]_-{\varphi_{L}} \ar[rr]^-{S} & & H_{R} \ar[dd]^-{\varphi_{R}}\\ & & \\ H_{L}^{'} \ar[rr]_-{S^{'}} & & H_{R}^{'} } \hspace{0.75in} \xymatrix{ H_{R} \ar[dd]_-{\varphi_{R}} \ar[rr]^-{S} & & H_{L} \ar[dd]^-{\varphi_{L}}\\ & & \\ H_{R}^{'} \ar[rr]_-{S^{'}} & & H_{L}^{'} } \]
\noindent and composition of such a pair is componentwise.
Let $R$ and $R^{'}$ be $k$-algebras and $(H_{L},H_{R},S)$ and $(K_{L^{'}},K_{R^{'}},S^{'})$ be Hopf algebroids over $R$ and $R^{'}$, respectively. In view of remark (\ref{R2.2}) (7) above, denote by $L=R^{op}$ and $L^{'}=(R^{'})^{op}$. A \textit{geometric morphism} $(H_{L},H_{R},S)\longrightarrow(K_{L^{'}},K_{R^{'}},S^{'})$ of Hopf algebroids is a pair $(f,\phi)$ of $k$-algerba maps $R\stackrel{f}{\longrightarrow}R^{'}$ and $H\stackrel{\phi}{\longrightarrow} K$, where $H,K$ denote the underlying $k$-algebra structures of the Hopf algebroids under consideration. These two maps satisfy the following compatibility conditions.
\begin{enumerate}
\item[(a)] $f$ and $\phi$ intertwines the source, target and counit maps of the left-bialgebroid structures of $\mathcal{H}$ and $\mathcal{K}$, i.e.
\[ \xymatrix{ H \ar[dd]_-{\phi} \ar[rr]^-{\epsilon^{H}_{L}} & & L \ar[dd]^-{f}\\ & & \\ K \ar[rr]_-{\epsilon^{K}_{L}} & & L^{'} } \hspace{0.15in} \xymatrix{ L \ar[dd]_-{f} \ar[rr]^-{t^{H}_{L}} & & H \ar[dd]^-{\phi}\\ & & \\ L^{'} \ar[rr]_-{t^{K}_{L}} & & K } \hspace{0.15in} \xymatrix{ L \ar[dd]_-{f} \ar[rr]^-{s^{H}_{L}} & & H \ar[dd]^-{\phi}\\ & & \\ L^{'} \ar[rr]_-{s^{K}_{L}} & & K. } \]
\noindent Same goes for the source, target and counit maps of the right-bialgebroid structures.
\item[(b)] In view of condition $(a)$, the $k$-bimodule map $\phi\otimes_{k}\phi$ defines $k$-bimodule maps
\[ \xymatrix{ H \prescript{}{L}{\otimes} \ H \ar[rr]^-{\phi\prescript{}{f}{\otimes} \ \phi} && K \prescript{}{L^{'}}{\otimes} \ K, } \hspace{.25in} \xymatrix{ H \otimes_{R} \ H \ar[rr]^-{\phi\ \otimes_{f} \phi} && K \otimes_{R^{'}} \ K. } \]
\noindent We then require that the following diagrams commute
\[ \xymatrix{ H \prescript{}{L}{\otimes} \ H \ar[rr]^-{\phi\prescript{}{f}{\otimes} \ \phi} \ar[dd]_-{\mu^{H}_{L}} && K \prescript{}{L^{'}}{\otimes} \ K \ar[dd]^-{\mu^{K}_{L}} \\ && \\ H \ar[rr]_-{\phi} && K } \hspace{.25in} \xymatrix{ H \otimes_{R} \ H \ar[rr]^-{\phi\ \otimes_{f} \phi} \ar[dd]_-{\mu^{H}_{R}} && K \otimes_{R^{'}} \ K \ar[dd]^-{\mu^{K}_{R}} \\ && \\ H \ar[rr]_-{\phi} && K} \]
\item[(c)] Also by of condition $(a)$, the $k$-bimodule maps $\phi\prescript{}{f}{\otimes} \ \phi$ and $\phi\ \otimes_{f} \phi$ of condition $(b)$ further define $k$-bimodule maps
\[ \xymatrix{ H \prescript{}{L}{\times} \ H \ar[rr]^-{\phi\prescript{}{f}{\times} \ \phi} && K \prescript{}{L^{'}}{\times} \ K, } \hspace{.25in} \xymatrix{ H \times_{R} \ H \ar[rr]^-{\phi\ \times_{f} \phi} && K \times_{R^{'}} \ K. } \]
\noindent We then require that the following diagrams commute.
\[ \xymatrix{ H \ar[rr]^-{\phi} \ar[dd]_-{\Delta^{H}_{L}} && K \ar[dd]^-{\Delta^{K}_{L}} \\ && \\ H \prescript{}{L}{\times} \ H \ar[rr]_-{\phi\prescript{}{f}{\times} \ \phi} && K \prescript{}{L^{'}}{\times} \ K } \hspace{.25in} \xymatrix{ H \ar[rr]^-{\phi} \ar[dd]_-{\Delta^{H}_{R}} && K \ar[dd]^-{\Delta^{K}_{R}} \\ && \\ H \times_{R} \ H \ar[rr]_-{\phi\ \times_{f} \phi} && K \times_{R^{'}} \ K } \]
\item[(d)] $\phi$ intertwines the antipodes of $\mathcal{H}$ and $\mathcal{K}$, i.e. $\phi\circ S_{H}=S_{K}\circ\phi$.
\end{enumerate} \end{defn}
\begin{rem}\label{R2.3} \begin{enumerate} \item[]
\item[(1)] For a $k$-algebra $R$, let us denote by $HALG^{alg}(R)$ the category whose objects are Hopf algebroids over $R$ and morphisms are algebraic morphisms. For a fixed $k$, let us denote by $HALG^{geom}(k)$ the category whose objects are Hopf algebroids over $k$-algebras and morphisms are geometric morphisms. The existence of these two naturally defined categories reflect the fact that Hopf algebroids are generalization of both Hopf algebras and groupoids.
\item[(2)] Equip $R^{e}$ with the Hopf algebroid structure defined in example 5 of the next section. Let $(H_{L},H_{R},S)$ be a Hopf algebroid over $R$. Then the unit maps $\eta_{L},\eta_{R}$ together with the identity map on $R$ define geometric morphisms $(id,\eta_{L}):R^{e}\longrightarrow \mathcal{H}$ and $(id,\eta_{R}):R^{e}\longrightarrow \mathcal{H}$. \end{enumerate} \end{rem}
\subsection{Examples}\label{S2.2}
\begin{exa}\label{E2.1} \textbf{Hopf algebras}. A Hopf algebra $H$ over the commutative unital ring $k$ gives an example of a Hopf algebroid. Here, we take $R=L=k$ as $k$-algebras, take $s_{L}=t_{L}=s_{R}=t_{R}=\eta$ to be the source and target maps, set $\epsilon_{L}=\epsilon_{R}=\epsilon$ to be the counits, and $\Delta_{L}=\Delta_{R}=\Delta$ to be the coproducts. \end{exa}
\begin{exa}\label{E2.2} \textbf{Coupled Hopf algebras}. It might be tempting to think that Hopf algebroids for which $R=L=k$ must be Hopf algebras. This is not entirely the case. We will give a general set of examples for which this is not true. Two Hopf algebra structures $H_{1}=(H,m,\eta,\Delta_{1},\epsilon_{1},S_{1})$ and $H_{2}=(H,m,\eta,\Delta_{2},\epsilon_{2},S_{2})$ over the same $k$-algebra $H$ are said to be \textit{coupled} if
\begin{enumerate} \item[(a)] there exists a $k$-module map $C:H\longrightarrow H$, called the \textit{coupling map} such that
\[ \xymatrix{ & & H\otimes H \ar[r(3)]^-{C\otimes id} & & & & H\otimes H \ar[rd(1.6)]^-{m} & & \\ & & & & & & & & \\ H \ar[ru(1.8)]^-{\Delta_{1}} \ar[rd(1.8)]_-{\Delta_{2}} \ar@<1ex>[rrrr]^-{\epsilon_{2}} \ar@<-1ex>[rrrr]_-{\epsilon_{1}} & & & & k \ar[rrrr]^-{\eta} & & & & H \\ & & & & & & & & \\ & & H\otimes H \ar[r(3)]_-{id\otimes C} & & & & H\otimes H \ar[ru(1.6)]_-{m} & & \\ } \]
\noindent commutes, and
\item[(b)] the coproducts $\Delta_{1}$ and $\Delta_{2}$ in $H$ commutes. \end{enumerate}
\noindent Coupled Hopf algebras give rise to Hopf algebroids over $k$. The left $k$-bialgebroid is the underlying bialgerba of $H_{1}$ while the right $k$-bialgebroid is the underlying bialgebra of $H_{2}$. The coupling map plays the role of the antipode.
Let us give examples of coupled Hopf algebras. Connes and Moscovici constructed \textit{twisted} antipodes in \cite{cm}. Let us show that such a twisted antipode is a coupling map for some coupled Hopf algebras. Let $H=(H,m,1,\Delta,\epsilon,S)$ be a Hopf algebra. Take $H_{1}=H$ as Hopf algebras. Let $\sigma:H\longrightarrow k$ be a character. Define $\Delta_{2}:H\longrightarrow H\otimes H$ by $h\mapsto h_{(1)}\otimes\sigma(S(h_{(2)}))h_{(3)}$. Take $\epsilon_{2}=\sigma$. Define $S_{2}:H\longrightarrow H$ by $h\mapsto \sigma(h_{(1)})S(h_{(2)})\sigma(h_{(3)})$. Note the Sweedler-legs of $h$ appearing in the definition of $S_{2}$ is the one provided by $\Delta$ and not by $\Delta_{2}$. Then, $H_{2}=(H,m,1,\Delta_{2},\epsilon_{2},S_{2})$ is a Hopf algebra coupled with $H_{1}$ by the coupling map $S^{\sigma}:H\longrightarrow H$ defined by $h\mapsto \sigma(h_{(1)})S(h_{(2)})$. \end{exa}
\begin{exa}\label{E2.3} \textbf{Groupoid algebras}. Given a small groupoid $\mathcal{G}$ with finitely many objects and a commutative unital ring $k$, we can construct what is called the groupoid algebra of $\mathcal{G}$ over $k$, denoted by $k\mathcal{G}$. For such a groupoid $\mathcal{G}$, let us denote by $\mathcal{G}^{(0)}$ its set of objects, $\mathcal{G}^{(1)}$ its set of morphisms, $s,t:\mathcal{G}^{(1)}\longrightarrow \mathcal{G}^{(0)}$ the source and target maps, $\iota:\mathcal{G}^{(0)}\longrightarrow \mathcal{G}^{(1)}$ the unit map, $\nu:\mathcal{G}^{(1)}\longrightarrow \mathcal{G}^{(1)}$ the inversion map, $\mathcal{G}^{(2)}=\mathcal{G}^{(1)}\prescript{}{t}\times_{s} \ \mathcal{G}^{(1)}$ the set of composable pairs of morphisms, and $m:\mathcal{G}^{(2)}\longrightarrow \mathcal{G}^{(1)}$ the partial composition. The groupoid algebra $k\mathcal{G}$ is the $k$-algebra generated by $\mathcal{G}^{(1)}$ subject to the relation
\[ ff^{'}= \begin{cases}
f\circ f^{'},& \text{if } f,f^{'} \ \text{are composable}\\
& \\
0, & \text{otherwise} \end{cases} \]
\noindent for $f,f^{'}\in \mathcal{G}^{(1)}$. The groupoid algebra $k\mathcal{G}$ is a Hopf algebroid as folows. The base algebras $R$ and $L$ are both equal to $k\mathcal{G}^{(0)}$ and the two bialgebroids $H_{R}$ and $H_{L}$ are isomorphic as bialgebroids with underlying $k$-module $k\mathcal{G}^{(1)}$. The partial groupoid composition $m$ dualizes and extends to a multiplication $m:k\mathcal{G}^{(1)}\otimes k\mathcal{G}^{(1)}\longrightarrow k\mathcal{G}^{(1)}$ which then factors through the canonical surjection $k\mathcal{G}^{(1)}\otimes k\mathcal{G}^{(1)}\longrightarrow k\mathcal{G}^{(1)}\otimes_{k\mathcal{G}^{(0)}} k\mathcal{G}^{(1)}$ to give the product $k\mathcal{G}^{(1)}\otimes_{k\mathcal{G}^{(0)}} k\mathcal{G}^{(1)}\longrightarrow k\mathcal{G}^{(1)}$. The source and target maps $s,t$ of the groupoid give the source and target maps $s,t:k\mathcal{G}^{(0)}\longrightarrow k\mathcal{G}^{(1)}$, respectively. The unit map gives the counit map $\epsilon:k\mathcal{G}^{(1)}\longrightarrow k\mathcal{G}^{(0)}$. Finally, the inversion map gives the antipode map $S:k\mathcal{G}^{(1)}\longrightarrow k\mathcal{G}^{(1)}$. Note that the underlying bimodule structures of the right and the left bialgerboid is related by the antipode map. \end{exa}
\begin{exa}\label{E2.4} \textbf{Weak Hopf algebras}. Another structure that generalize Hopf algebras, called weak Hopf algebras, also are Hopf algebroids. Explicitly, a weak Hopf algebra $H$ over a commutative unital ring $k$ is a unitary associative algebra together with $k$-linear maps $\Delta:H\longrightarrow H\otimes H$ (weak coproduct), $\epsilon:H\longrightarrow k$ (weak counit) and $S:H\longrightarrow H$ (weak antipode) satisfying the following axioms:
\begin{enumerate} \item[(i)] $\Delta$ is multiplicative, coassociative, and weak-unital, i.e. \[(\Delta(1)\otimes 1)(1\otimes \Delta(1))=\Delta^{(2)}(1)=(1\otimes\Delta(1))(\Delta(1)\otimes 1),\]
\item[(iii)] $\epsilon$ is counital, and weak-multiplicative, i.e. for any $x,y,z\in H$ \[ \epsilon(xy_{(1)})\epsilon(y_{(2)}z)=\epsilon(xyz)=\epsilon(xy_{(2)})\epsilon(y_{(1)}z),\]
\item[(v)] for any $h\in H$, $S(h_{(1)})h_{(2)}S(h_{(3)})=S(h)$ and \[ h_{(1)}S(h_{(2)})=\epsilon(1_{(1)}h)1_{(2)}, \hspace{.5in} S(h_{(1)})h_{(2)}=1_{(1)}\epsilon(h1_{(2)}) \] \end{enumerate}
Let us sketch a proof why a weak Hopf algebra $H$ is a Hopf algebroid. Consider the maps $p_{R}:H\longrightarrow H$, $h\mapsto 1_{(1)}\epsilon(h1_{(2)})$ and $p_{L}:H\longrightarrow H$, $h\mapsto \epsilon(1_{(1)}h)1_{(2)}$. By $k$-linearity and weak-multiplicativity of $\epsilon$, $p_{R}$ and $p_{L}$ are idempotents.
Multiplicativity and coassiociativity of $\Delta$ and counitality of $\epsilon$ implies that for any $h\in H$,
\[ h_{(1)}\otimes p_{L}(h_{(2)})=1_{(1)}h\otimes 1_{(2)} \hspace{.5in} p_{R}(h_{(1)})\otimes h_{(2)}=1_{(1)}\otimes h1_{(2)}.\]
\noindent Now, using these relations and coassiociativity of $\Delta$ we get
\[ 1_{(1)}1_{(1')}\otimes 1_{(2)} \otimes 1_{(2')} = 1_{(1')(1)}\otimes p_{L}(1_{(1')(2)})\otimes 1_{(2')}=1_{(1)}\otimes p_{L}(1_{(2)})\otimes 1_{(3)}\]
\[ 1_{(1)}\otimes 1_{(1')}\otimes 1_{(2)}1_{(2')} = 1_{(1)}\otimes p_{L}(1_{(2)(1)})\otimes 1_{(2)(2)}=1_{(1)(1)}\otimes p_{L}(1_{(1)(2)})\otimes 1_{(2)}\]
\noindent Thus, the first tensor factor of the left-hand side of the first equation above is in the image of $p_{R}$. Similarly, the last tensor factor of the left-hand side of the second equation above is in the image of $p_{L}$. Clearly, $p_{R}(1)=p_{L}(1)=1$. Hence, the images of $p_{R}$ and $p_{L}$ are unitary subalgebras of $H$. Denote these subalgebras by $R$ and $L$, respectively. By the weak-unitality of $\Delta$ we see that these subalgebras are commuting subalgebras of $H$.
Taking the source map $s$ as the inclusion $R\longrightarrow H$ and the target map as $t:R^{op}\longrightarrow H$, $r\mapsto\epsilon(r1_{(1)})1_{(2)}$ equips $H$ with an $R\otimes_{k}R^{op}$-ring structure. Taking $\epsilon_{R}=p_{R}$ and $\Delta_{R}$ as the composition
\[ \xymatrix{H \ar[rr]^-{\Delta} & & H\otimes_{k}H \ar@{->>}[rr] & & H\otimes_{R}H }\]
\noindent equips $H$ with an $R$-coring structure $(H,\Delta_{R},\epsilon_{R})$. The ring and coring structures just constructed gives $H$ a structure of right $R$-bialgebroid $H_{R}$.
Using $R^{op}$ in place of $R$ in the above construction, we get a left $R^{op}$-bialgebroid $H_{R^{op}}$. Together with the right $R$-bialgebroid constructed and the existing weak antipode $S$, we get a Hopf algebroid $(H_{R^{op}},H_{R},S)$. \end{exa}
\subsection{Representation of Hopf algebroids}\label{S2.3}
In this section, we will look at representations of Hopf algebroids. Towards the end of the section, we will look at the descent theoretic aspect of a special class of modules over Hopf algebroids, the so called relative Hopf modules. Let $\mathcal{H}=(H_{L},H_{R},S)$ be a Hopf algebroid with underlying $k$-module $H$. $H$ carries both a left $L$-module sctructure and a left $R$-module structure via the maps $s_{L}$ and $t_{R}$, respectively. A \textit{right} $\mathcal{H}$-\textit{comodule} $M$ is a right $L$-module and a right $R$-module together with a right $H_{R}$-coaction $\rho_{R}:M\longrightarrow M\otimes _{R}H$ and a right $H_{L}$-coaction $\rho_{L}:M\longrightarrow M\otimes_{L}H$ such that $\rho_{R}$ is an $H_{L}$-comodule map and $\rho_{L}$ is an $H_{R}$-comodule map.
For the coaction $\rho_{R}$, let us use the following Sweedler notation:
\[ \rho_{R}(m) = m^{[0]}\tens{R} m^{[1]} \]
\noindent and for the coaction $\rho_{L}$, let us use the following Sweedler notation:
\[ \rho_{L}(m) = m_{[0]} \tens{L} m_{[1]}. \]
\noindent With these notations, the conditions above explicitly means that for all $m\in M$, $l\in L$ and $r\in R$ we have
\[ (m\cdot l)^{[0]}\tens{R}(m\cdot l)^{[1]}=\rho_{R}(m\cdot l)=m^{[0]}\tens{R} t_{L}(l)m^{[1]} \]
\[ (m\cdot r)_{[0]}\tens{L}(m\cdot r)_{[1]}=\rho_{L}(m\cdot r)=m_{[0]}\tens{L} m_{[1]}s_{R}(r). \]
\noindent We further require that the two coactions satify the following commutative diagrams
\begin{equation}\label{eq2.1} \xymatrix{ M \ar[d(1.7)]_-{\rho_{R}} \ar[r(1.7)]^-{\rho_{L}} & & M\tens{L} H \ar[d(1.6)]^-{\rho_{R}\tens{L} id}\\ & & \\ M\tens{R} H \ar[r(1.6)]_-{id \tens{R}\Delta_{L}} & & M\tens{R} H\tens{L} H } \hspace{0.75in} \xymatrix{ M \ar[d(1.7)]_-{\rho_{L}} \ar[r(1.7)]^-{\rho_{R}} & & M\tens{R} H \ar[d(1.6)]^-{\rho_{L}\tens{R} id}\\ & & \\ M\tens{L} H \ar[r(1.6)]_-{id \tens{L}\Delta_{R}} & & M\tens{L} H\tens{R} H } \end{equation}
We will denote by $\mathcal{M}^{\mathcal{H}}$ the category of right $\mathcal{H}$-comodules. Symmetrically, we can define left $\mathcal{H}$-comodules and we denote the category of a such by $^{\mathcal{H}}\mathcal{M}$.
Comodules over Hopf algebroids are comodules over the constituent bialgebroids. Thus, one can speak of two different coinvariants, one for each bialgebroid. For a given right $\mathcal{H}$-comodule $M$, they are defined as follows:
\[ M^{co \ H_{R}} = \left\{m\in M\left| \ \rho_{R}(m)=m\tens{R} 1\right.\right\}, \]
\[ M^{co \ H_{L}} = \left\{m\in M\left| \ \rho_{L}(m)=m\tens{L} 1\right.\right\}. \]
\noindent In the general case, we have $M^{co \ H_{R}}\subseteq M^{co \ H_{L}}$. But in our case, where we assume $S$ is bijective these two spaces coincide. This will be important in the formulation of Galois theory for Hopf algebroids. To see that these coinvariants coincide, consider the following map
\[ \Phi_{M}:M\tens{R}H\longrightarrow M\tens{L}H \] \[ m\tens{R} h \mapsto \rho_{L}(m)\cdot S(h) \]
\noindent Here, $H$ acts on the right of $M\otimes_{L}H$ through the second factor. If $m\in M^{co \ H_{R}}$, then we have
\begin{eqnarray} \nonumber\label{} \rho_{L}(m)&=&\rho_{L}(m)\cdot S(h) = \Phi_{M}(m\tens{R}1) = \Phi_{M}(\rho_{R}(m))\\ \nonumber &=& \Phi_{M}(m^{[0]}\tens{R} m^{[1]}) = \rho_{L}(m^{[0]})\cdot S(m^{[1]})\\ \nonumber &=& (m^{[0]}_{[0]}\tens{L} m^{[0]}_{[1]})\cdot S(m^{[1]}) = m^{[0]}_{[0]}\tens{L} m^{[0]}_{[1]}S(m^{[1]})\\ \nonumber &=& m_{[0]}\tens{L} m^{[0]}_{[1]}S(m^{[1]}_{[1]})= m_{[0]}\tens{L} s_{L}(\epsilon_{L}(m_{[1]}))\\ \nonumber &=& m_{[0]}s_{L}(\epsilon_{L}(m_{[1]}))\tens{L} 1 = m\tens{L} 1\\ \nonumber \end{eqnarray}
\noindent This shows the inclusion $M^{co \ H_{R}}\subseteq M^{co \ H_{L}}$. To show the other inclusion, one can run the same computation but using the inverse of $\Phi_{M}$ which is the following map
\[ \Phi_{M}^{-1}:M\tens{L}H\longrightarrow M\tens{R}H \] \[ m\tens{L}h\mapsto S^{-1}(h)\cdot\rho_{R}(m). \]
\noindent In this case, we can simply write $M^{co \ \mathcal{H}}$ for $M^{co \ H_{R}}=M^{co \ H_{L}}$ and refer to it as the $\mathcal{H}$-coinvariants of $M$ instead of distinguishing the $H_{R}$- from the $H_{L}$-coinvariants, unless it is necessary to do so.
Let us now discuss monoid objects in $\mathcal{M}^{\mathcal{H}}$. They are called $\mathcal{H}$-comodule algebras. A right $\mathcal{H}$-\textit{comodule algebra} is an $R$-ring $(M,\mu,\eta)$ such that $M$ is a right $\mathcal{H}$-comodule and $\eta:R\longrightarrow M$ and $\mu:M\otimes_{R}M\longrightarrow M$ are $\mathcal{H}$-comodule maps. Using Sweedler notation for coactions, this explicitly means that for any $m,n\in M$ we have
\begin{equation}\label{eq2.2} (mn)^{[0]}\tens{R}(mn)^{[1]}=\rho_{R}(mn)=m^{[0]}n^{[0]}\tens{R} m^{[1]}n^{[1]}, \end{equation}
\begin{equation}\label{eq2.3} (mn)_{[0]}\tens{L}(mn)_{[1]}=\rho_{L}(mn)=m_{[0]}n_{[0]}\tens{L} m_{[1]}n_{[1]}, \end{equation}
\begin{equation}\label{eq2.4} 1_{M}^{[0]}\tens{R}1_{M}^{[1]}=\rho_{R}(1_{M})=1_{M}\tens{R} 1_{H}, \end{equation}
\begin{equation}\label{eq2.5} (1_{M})_{[0]}\tens{L}(1_{M})_{[1]}=\rho_{L}(1_{M})=1_{M}\tens{L} 1_{H}. \end{equation}
Let $\mathcal{H}=(H_{L},H_{R},S)$ be a Hopf algebroid with underlying $k$-module $H$. A $k$-algebra extension $A\subseteq B$ is said to be (\textit{right}) $H_{R}$-\textit{Galois} if $B$ is a right $H_{R}$-comodule algebra with $B^{co \ H_{R}}=A$ and the map
\[ \xymatrix{B\tens{A}B \ar[rr]^-{\mathfrak{gal}_{R}} & & B\tens{R}H } \] \[ a\tens{A}b\longmapsto ab^{[0]}\tens{R}b^{[1]}\]
\noindent is a bijection. The map $\mathfrak{gal}_{R}$ is called the Galois map associated to the bialgebroid extension $A\subseteq B$. Symmetrically, the extension $A\subseteq B$ is (\textit{right}) $H_{L}$-\textit{Galois} if $B$ is a right $H_{L}$-comodule algebra with $B^{co \ H_{L}}=A$ and the map
\[ \xymatrix{B\tens{A}B \ar[rr]^-{\mathfrak{gal}_{L}} & & B\tens{L}H } \] \[ a\tens{A}b\longmapsto a_{[0]}b\tens{L}a_{[1]}\]
\noindent is a bijection. We say that a $k$-algebra extension $A\subseteq B$ is $\mathcal{H}$-\textit{Galois} if it is both $H_{R}$-Galois and $H_{L}$-Galois. It is not known in general if the bijectivity of $\mathfrak{gal}_{R}$ and $\mathfrak{gal}_{L}$ are equivalent. However, if the antipode $S$ is bijective (which is part of our standing assumption) then $\mathfrak{gal}_{R}$ is bijective if and only if $\mathfrak{gal}_{L}$. To see this, note that $\mathfrak{gal}_{L}=\Phi_{B} \circ \mathfrak{gal}_{R}$ where $\Phi_{B}$ is the map defined in the previous section for $M=B$. Since $S$ is bijective, $\Phi_{B}$ is an isomorphism which gives the desired equivalence of bijectivity of $\mathfrak{gal}_{R}$ and $\mathfrak{gal}_{L}$. Thus, the extension $A\subseteq B$ is $\mathcal{H}$-Galois if it is a bialgebroid Galois extension for any of its constituent bialgebroids.
\section{Hopf categories} \label{S3.0}
\subsection{Definitions and properties}\label{S3.1}
Batista et al. \cite{bcv} introduced the notion of a Hopf category over an arbitrary strict braided monoidal $\mathcal{V}$. In this section, we will introduce its topological version. For this purpose, we specialize $\mathcal{V}$ as the category of complex vector spaces whose braiding is the usual flip of tensor factors. Also, we will assume that the underlying categories of such Hopf categories are small. We will be primarily interested with \textit{finite-type} $\mathcal{V}$-enriched categories, by which we mean the hom-sets are finite-dimensional vector spaces. Before giving the definition of a Hopf category, let is introduce some notation first. For two $\mathcal{V}$-enriched categories $\mathscr{A}$ and $\mathscr{B}$ with the same set of objects $X$, we define $\mathscr{A}\otimes_{X}\mathscr{B}$ to the the $\mathcal{V}$-enriched category with $X$ as the set of objects and for $x,y\in X$, the hom-set of arrows from $x$ to $y$ is the vector space
\begin{equation}\label{eq3.1} \left(\mathscr{A}\otimes_{X}\mathscr{B}\right)_{x,y}:=\mathscr{A}_{x,y} \otimes \mathscr{B}_{x,y}. \end{equation}
\noindent We call $\mathscr{A}\otimes_{X}\mathscr{B}$ the \textit{tensor product} of $\mathscr{A}$ and $\mathscr{B}$. With this $\otimes_{X}$, the category of $\mathcal{V}$-enriched categories over $X$ becomes a strict monoidal category whose monoidal unit, denoted by $\mathbb{1}^{X}$, is the category over $X$ such that for any $x,y\in X$ we have $\mathbb{1}^{X}_{x,y}=\mathbb{C}$.
\begin{defn}\label{D3.1} A \textit{Hopf category} $\mathscr{H}$ over $X$ is a $\mathcal{V}$-enriched category satisfying the following conditions.
\begin{enumerate}
\item[(a)] There are functors
\[ \xymatrix{\mathscr{H} \ar[rr]^-{\Delta} && \mathscr{H}\tens{X} \mathscr{H}}, \hspace{.5in} \xymatrix{\mathscr{H} \ar[rr]^-{\epsilon} && \mathbb{1}^{X}} \]
\noindent called the \textit{coproduct} and \textit{counit}, respectively, such that $\Delta$ is \textit{coassociative} and \textit{counital} with respect to $\epsilon$, i.e. the diagram of functors
\[ \xymatrix{ \mathscr{H} \ar[rr]^-{\Delta} \ar[dd]_-{\Delta} && \mathscr{H}\tens{X} \mathscr{H} \ar[dd]|-{id\otimes_{X}\Delta} \\ && \\ \mathscr{H}\tens{X} \mathscr{H} \ar[rr]_-{\Delta\otimes_{X} id} && \mathscr{H}\tens{X} \mathscr{H}\tens{X} \mathscr{H} \\ } \hspace{.25in} \xymatrix{ \mathscr{H} \ar@{=}[rr] \ar@{=}[dd] \ar[rrdd]^-{\Delta} && \mathscr{H}\tens{X} \mathbb{1}^{X} \\ && \\ \mathbb{1}^{X}\tens{X} \mathscr{H} && \mathscr{H}\tens{X} \mathscr{H} \ar[ll]^-{\epsilon\otimes_{X} id} \ar[uu]_-{is\otimes_{X}\epsilon} \\ }
\]
\noindent commute.
\item[(b)] There is a functor $S:\mathscr{H}\longrightarrow \mathscr{H}^{op}$, called the \textit{antipode}, satisfying
\[ \xymatrix{ && \mathscr{H}\tens{X} \mathscr{H} \ar[rr]^-{S\otimes_{X} id} & & \mathscr{H}^{op}\tens{X} \mathscr{H} \ar[rrd]^-{\circ} && \\ \mathscr{H} \ar[rru]^-{\Delta} \ar[rrd]_-{\Delta} \ar[rrr]^-{\epsilon} & & & \mathbb{1}_{X} \ar[rrr]^-{\eta} & & & \mathscr{H} \\ && \mathscr{H}\tens{X} \mathscr{H} \ar[rr]_-{id\otimes_{X} S} & & \mathscr{H}\tens{X} \mathscr{H}^{op} \ar[rru]_-{\circ} && \\} \]
\noindent Here, $\circ$ denotes the bifunctor induced by the categorical composition in $\mathscr{H}$ and $\eta$ is the functor that send $1\in\mathbb{1}^{X}_{x,y}$ to the identity element of $\mathscr{H}_{x,y}$. \end{enumerate} \end{defn}
\begin{rem}\label{R3.1} Functoriality of $\Delta$ and $\epsilon$ means that for any $x,y\in X$, we have linear maps
\[ \xymatrix{\mathscr{H}_{x,y} \ar[rr]^-{\Delta_{x,y}} && \mathscr{H}_{x,y}\otimes \mathscr{H}_{x,y}} \hspace{.5in} \xymatrix{\mathscr{H}_{x,y} \ar[rr]^-{\epsilon_{x,y}} && \mathbb{C}} \]
\noindent where $\Delta_{x,y}$ is coassociative and counital with respect to $\epsilon_{x,y}$ in the usual sense. This implies that $\mathscr{H}_{x,y}$ is a coalgebra. If we denote by $C{V}$ the category of coalgebras on $\mathcal{V}$, another way to package part $(a)$ of definition (\ref{D3.1}) is to say that $\mathscr{H}$ is enriched over $C(\mathcal{V})$. \end{rem}
For the main results of this paper, we will be mostly interested with the case $X$ is a topological space. In such a case, it makes sense to reflect \textit{continuity} on the functors $\Delta$, $\epsilon$ and $S$ along with the categorical structure maps. This calls for the following definition.
\begin{defn}\label{D3.2} Let $X$ be a topological space and let $\mathcal{O}_{X}$ be the sheaf of continuous complex-valued functions on $X$. A \textit{topological Hopf category} $\mathscr{H}$ over $X$ is a Hopf category together with a sheaf $H^{sh}$ over $X\times X$ (with the product topology) of $\mathcal{O}_{X}$-bimodules satisfying the following conditions. \begin{enumerate} \item[(a)] Denote by $\pi_{1},\pi_{2}:X\times X\longrightarrow X$ the projection onto the first and second factor, respectively. Over an open set $U\subseteq X\times X$, for any $\sigma\in H^{sh}(U)$, $f\in \mathcal{O}_{X}(\pi_{1}U)$ and $g\in \mathcal{O}_{X}(\pi_{2}U)$ we have
\[ \left(f\cdot\sigma\cdot g\right)(x,y)=f(x)\sigma(x,y)g(y) \]
\noindent for any $(x,y)\in U$.
\item[(b)] $\mathscr{H}_{x,y}$ is the fiber of $H^{sh}$ at $(x,y)\in X\times X$.
\item[(c)] $\circ$, $\eta$, $\Delta$, $\epsilon$ and $S$ are the induced maps on global sections of the following map of sheaves
\[ \xymatrix{H^{sh}\tens{\mathcal{O}_{X}} H^{sh} \ar[rr]^-{\circ^{sh}} && H^{sh} }, \hspace{.5in} \xymatrix{\mathcal{O}_{X} \ar[rr]^-{\eta^{sh}} && H^{sh} }, \]
\[ \xymatrix{H^{sh} \ar[rr]^-{\Delta^{sh}} && H^{sh}\tens{\mathcal{O}_{X}} H^{sh}}, \hspace{.5in} \xymatrix{H^{sh} \ar[rr]^-{\epsilon^{sh}} && \mathcal{O}_{X}}, \]
\[ \xymatrix{ H^{sh} \ar[rr]^-{S^{sh}} && \left(H^{sh}\right)^{op} } \]
\noindent respectively. Here, $\left(H^{sh}\right)^{op}$ is the pullback of the sheaf $H^{sh}$ along the map $X\times X\longrightarrow X\times X$ flipping the factors. \end{enumerate} \end{defn}
\begin{rem}\label{R3.2} The bimodule tensor product $\otimes_{\mathcal{O}_{X}}$ used in part $(c)$ for $\Delta^{sh}$ of definition (\ref{D3.2}) is the tensor product of the appropriately modified $\mathcal{O}_{X}$-bimodule $H^{sh}$, one in which we have
\[ f\cdot \left(\sigma \tens{\mathcal{O}_{X}} \tau\right)\cdot g = \left(\sigma\cdot g\right) \tens{\mathcal{O}_{X}} \left(f\cdot\tau\right) \]
\noindent for any $f,g\in \mathcal{O}_{X}$. For the bimodule tensor product $\otimes_{\mathcal{O}_{X}}$ used for $\circ^{sh}$ is the one with
\[ f\cdot \left(\sigma \tens{\mathcal{O}_{X}} \tau\right)\cdot g = \left(f\cdot\sigma\right) \tens{\mathcal{O}_{X}} \left(\tau\cdot g\right) \]
\noindent for any $f,g\in\mathcal{O}_{X}$. \end{rem}
The following, which will play an important role in our formulation of the main result, is the categorification of a coupled Hopf algebra.
\begin{defn}\label{D3.3} A \textit{coupled Hopf category} $\mathscr{H}$ is a $\mathcal{V}$-enriched category with two $C(\mathcal{V})$-enrichments, denoted by $\mathscr{H}_{L}$ and $\mathscr{H}_{R}$, with coproducts $\Delta^{L},\Delta^{R}$ and counits $\epsilon^{L},\epsilon^{R}$, respectively; and a functor $S:\mathscr{H}\longrightarrow \mathscr{H}^{op}$, called the \textit{coupling} functor, such that the following conditions are satisfied:
\begin{enumerate} \item[(a)] The following diagrams, indicating the \textit{coupling condition}, commute.
\[ \xymatrix{ & & \mathscr{H}\tens{X} \mathscr{H} \ar[rr]^-{S\otimes_{X} id} & & \mathscr{H}^{op}\tens{X} \mathscr{H} \ar[rrd]^-{\circ} & & \\ \mathscr{H} \ar[rru]^-{\Delta^{L}} \ar[rrr]^-{\epsilon^{R}} & & & \mathbb{1}^{X} \ar[rrr]^-{\eta} & & & \mathscr{H} \\ \mathscr{H} \ar[rrd]_-{\Delta^{R}} \ar[rrr]_-{\epsilon^{L}} & & & \mathbb{1}^{X} \ar[rrr]_-{\eta} & & & \mathscr{H} \\ & & \mathscr{H}\tens{X} \mathscr{H} \ar[rr]_-{id\otimes_{X} S} & & \mathscr{H}\tens{X} \mathscr{H}^{op} \ar[rru]_-{\circ} & & \\ }\]
\item[(b)] The coproducts $\Delta^{L}$ and $\Delta^{R}$ commute, i.e.
\[ \xymatrix{ \mathscr{H} \ar[dd]_-{\Delta^{L}} \ar[rr]^-{\Delta^{R}} & & \mathscr{H}\tens{X} \mathscr{H} \ar[dd]^-{\Delta^{L}\tens{X} id}\\ & & \\ \mathscr{H}\tens{X} \mathscr{H} \ar[rr]_-{id \tens{X}\Delta^{R}} & & \mathscr{H}\tens{X} \mathscr{H}\tens{X} \mathscr{H} } \hspace{.25in} \xymatrix{ \mathscr{H} \ar[dd]_-{\Delta^{R}} \ar[rr]^-{\Delta^{L}} & & \mathscr{H}\tens{X} \mathscr{H} \ar[dd]^-{\Delta^{R}\tens{X} id}\\ & & \\ \mathscr{H}\tens{X} \mathscr{H} \ar[rr]_-{id \tens{X}\Delta^{L}} & & \mathscr{H}\tens{X} \mathscr{H}\tens{X} \mathscr{H} } \]
\end{enumerate} \end{defn}
\begin{rem}\label{R3.3} \begin{enumerate} \item[] \item[(1)] Coupled Hopf categories are almost the categorification of coupled Hopf algebras. While the constituent bialgebras of a coupled Hopf algebra is a Hopf algebras in itself, the constituent categories $\mathscr{H}_{L}$ and $\mathscr{H}_{R}$ of a coupled Hopf category $\mathscr{H}$ need not be Hopf categories.
\item[(2)] Just like Hopf categories, we can also \textit{topologize} coupled Hopf categories. We can take definition (\ref{D3.2}): assert the existence of a sheaf $H^{sh}$ over $X\times X$ of $\mathcal{O}_{X}$-bimodules, take conditions $(a)$ and $(b)$ as they are, and replace condition $(c)$ by \begin{enumerate} \item[(c')] $\Delta^{L}$, $\Delta^{R}$, $\epsilon^{L}$, $\epsilon^{R}$ and $S$ are the induced maps on global sections of the following map of sheaves
\[ \xymatrix{H^{sh} \ar[rr]^-{(\Delta^{L})^{sh}} && H^{sh}\tens{\mathcal{O}_{X}} H^{sh}}, \hspace{.5in} \xymatrix{H^{sh} \ar[rr]^-{(\epsilon^{L})^{sh}} && \mathcal{O}_{X}}, \]
\[ \xymatrix{H^{sh} \ar[rr]^-{(\Delta^{R})^{sh}} && H^{sh}\tens{\mathcal{O}_{X}} H^{sh}}, \hspace{.5in} \xymatrix{H^{sh} \ar[rr]^-{(\epsilon^{R})^{sh}} && \mathcal{O}_{X}}, \]
\[ \xymatrix{ H^{sh} \ar[rr]^-{S^{sh}} && \left(H^{sh}\right)^{op} } \]
\noindent respectively, making the following diagram
\begin{flushleft} \xymatrix{ & H^{sh}(U)\tens{\mathcal{O}_{X}(U)} H^{sh}(U) \ar[rr]^-{S_{U}\tens{\mathcal{O}_{X}(U)} id} & & H^{sh}(U)^{op}\tens{\mathcal{O}_{X}(U)} H^{sh}(U) \ar[rd]^-{\mu_{U}} & \\ H^{sh}(U) \ar[ru]^-{\left(\Delta^{L}\right)^{sh}(U)} \ar[rr]^-{(\epsilon^{R})^{sh}(U)} & & \mathcal{O}_{X}(U) \ar[rr]^-{\eta_{U}} & & H^{sh}(\pi^{diag}_{2}U) \\ H^{sh}(U) \ar[rd]_-{(\Delta^{R})^{sh}(U)} \ar[rr]_-{(\epsilon^{L})^{sh}(U)} & & \mathcal{O}_{X}(U) \ar[rr]_-{\eta_{U}} & & H^{sh}(\pi^{diag}_{1}U) \\ & H^{sh}(U)\tens{\mathcal{O}_{X}(U)} H^{sh}(U) \ar[rr]_-{id\tens{\mathcal{O}_{X}(U)} S_{U}} & & H^{sh}(U)\tens{\mathcal{O}_{X}(U)} (H^{sh})(U)^{op} \ar[ru]_-{\mu_{U}} & \\} \end{flushleft}
\noindent commute for any $U\subseteq X\times X$. Here, $\mu_{U}$ and $\eta_{U}$ denote the maps induced by the composition and unit maps of $\mathscr{C}$. The maps $\pi_{1}^{diag}$ and $\pi_{2}^{diag}$ denote $X\times X\longrightarrow X\times X$, $(x,y)\mapsto(x,x)$ and $X\times X\longrightarrow X\times X$, $(x,y)\mapsto(y,y)$, respectively. \end{enumerate}
\end{enumerate} \end{rem}
\subsection{A good example of a Hopf category}\label{S3.2}
In this section, we will look at a very important example of a Hopf category. This example will also be an example of our main result. This is a special case of proposition 7.1 of \cite{bcv}. Consider a finite set $X$ whose elements are conveniently labelled as $1,2,...,n$. Equipped $X$ with the discrete topology. Consider the category $C$ whose set of objects is $X$ and define $C_{x,y}=\mathbb{C}$. The category $C$ is obviously a Hopf category. By proposition 7.1 of \cite{bcv}, $\mathcal{H}=\bigoplus_{x,y\in X}C_{x,y}$ is a weak Hopf algebra. Using the arguments in example 4 of section (\ref{S2.2}), $\mathcal{H}$ is a Hopf algebroid over $A=\mathbb{C}^{n}=\mathcal{O}_{X}(X)$.
The Hopf algebroid $\mathcal{H}$ has a more familiar form. It is isomorphic, as a Hopf algebroid, the algebra $M_{n}(\mathbb{C})$ over its diagonal $D_{n}=Diag_{n}(\mathbb{C})$. With the $D_{n}$-bimodule structure on $M_{n}(\mathbb{C})$ defined as
\[ P\cdot M\cdot Q := MPQ, \hspace{.5in} P,Q\in D_{n}, M\in M_{n}(\mathbb{C}), \]
\noindent the coproduct $\Delta_{R}$ and the counit $\epsilon_{R}$ are given as
\[ \Delta_{R}(E_{ij})=E_{ij}\otimes_{D_{n}}E_{ij}, \hspace{.35in} \epsilon_{R}(M)=\sum\limits_{i=1}^{n}E_{ii}\phi(ME_{ii}) \]
\noindent where $\phi$ is the linear functional defined by $\phi(E_{ij})=1$ for all $i,j\in X$. With the usual matrix multiplication and unit, $\Delta_{R}$ and $\epsilon_{R}$ constitutes a right $D_{n}$-bialgebroid structure on $M_{n}(\mathbb{C})$. For completeness, let us define the structure maps of the left $D_{n}$-bialgebroid structure of $M_{n}(\mathbb{C})$. Consider the $D_{n}$-bimodule structure on $M_{n}(\mathbb{C})$ defined as
\[ P\cdot M\cdot Q := PQM, \hspace{.5in} P,Q\in D_{n}, M\in M_{n}(\mathbb{C}). \]
\noindent The coproduct $\Delta_{L}$ and the counit $\epsilon_{L}$ are defined as
\[ \Delta_{L}(E_{ij})=E_{ij}\otimes_{D_{n}}E_{ij}, \hspace{.35in} \epsilon_{L}(M)=\sum\limits_{i=1}^{n}\phi(E_{ii}M)E_{ii} \]
\noindent where $\phi$ is the same linear functional used to defined $\epsilon_{R}$. The antipode $S$ of this Hopf algebroid is defined as $S(E_{ij})=E_{ji}$.
As a weak Hopf algebra, $\phi$ is the counit of $\mathcal{H}$. The coproducts $\Delta_{L}$ and $\Delta_{R}$ are the extension of the weak coproduct $\Delta$ to $M_{n}(\mathbb{C})\otimes_{D_{n}}M_{n}(\mathbb{C})$ relative to the $D_{n}$-bimodule structure used. As we will see in section (\ref{S4.0}), this is not a coincidence. This is in fact a special case of a more general result which we shall prove at the end of that section.
\subsection{Galois extensions of Hopf categories}\label{S3.3}
Formulation of Galois theory for Hopf category is straightforward. Recall that in the case of Hopf algebras, only the underlying bialgebra structure is relevant. In the coaction picture, the coalgebra is used to make sense of a coaction while the algebra structure is used to make sense of the Galois map. All these ingredients are already present in the case of a Hopf category. We will discuss the situation for topological Hopf categories. The case for Hopf categories follow almost immediately by dropping any manifestation of topology.
Before giving the definition of the categorical analogue of a comodule algebra, let us first discuss what a topological category is, at least for our purpose. A $\mathcal{V}$-enriched category $\mathscr{M}$ over a space $X$ is a \textit{topological category} if there is a sheaf $M^{sh}$ of $\mathcal{O}_{X}$-bimodules such that conditions $(a)$, $(b)$ and the relevant part of condition $(c)$ of definition (\ref{D3.2}) hold.
\begin{defn}\label{D3.4} Let $\mathscr{H}$ be a topological Hopf category with space of objects $X$, coproduct $\Delta$, counit $\epsilon$ and antipode $S$ with associated sheaf $H^{sh}$.
\begin{enumerate} \item[(1)] A topological category $\mathscr{M}$ over $X$ enriched over $\mathcal{V}$, with associated sheaf $M^{sh}$, is a \textit{right} $\mathscr{H}$-\textit{comodule} if there is a functor $\rho:\mathscr{M}\longrightarrow \mathscr{M}\otimes_{X}\mathscr{H}$ such that the following conditions hold.
\begin{enumerate} \item[(a)] $\rho$ is coassociative with respect to $\Delta$ and counital with respect to $\epsilon$, i.e. the diagrams of functors
\[ \xymatrix{ \mathscr{M} \ar[rr]^-{\rho} \ar[dd]_-{\rho} && \mathscr{M}\otimes_{X}\mathscr{H} \ar[dd]|-{id\otimes_{X}\Delta} \\ &&& \\ \mathscr{M}\otimes_{X}\mathscr{H} \ar[rr]_-{\rho\otimes_{X} id} && \mathscr{M}\otimes_{X}\mathscr{H}\otimes_{X}\mathscr{H} \\ } \hspace{-.5in} \xymatrix{ \mathscr{M} \ar@{=}[rr] \ar[rdd]_-{\rho} && \mathscr{M}\otimes_{X}\mathbb{1}^{X} \\ && \\
& \mathscr{M} \otimes_{X} \mathscr{H} \ar[ruu]|-{id\otimes_{X}\epsilon} & \\} \]
\noindent commute, and
\item[(b)] the functor $\rho$ is the map induced by the map of sheaves $M^{sh}\longrightarrow M^{sh}\otimes_{\mathcal{O}_{X}}H^{sh}$ where the tensor product is the same as the first one we described in remark (\ref{R3.2}). A \textit{left} $\mathscr{H}$-\textit{comodule} can be symmetrically defined. \end{enumerate}
\item[(2)] A \textit{morphism} $\mathscr{M}\stackrel{\phi}{\longrightarrow}\mathscr{N}$ of right $\mathscr{H}$-comodules is a functor that commutes with the right coactions, i.e. one which makes the following diagram commute
\[ \xymatrix{ \mathscr{M} \ar[rrr]^-{\rho^{M}} \ar[dd]_-{\phi} &&& \mathscr{M}\otimes_{X}\mathscr{H} \ar[dd]|-{\phi\otimes_{X}id} \\ &&& \\ \mathscr{N} \ar[rrr]_-{\rho^{N}} &&& \mathscr{N}\otimes_{X}\mathscr{H}. \\ } \]
\noindent Here, $\rho^{M}$ and $\rho^{N}$ are the coactions of $\mathscr{H}$ on $\mathscr{M}$ and $\mathscr{N}$, respectively.
\item[(3)] A right $\mathscr{H}$-comodule $\mathscr{M}$ is a \textit{right} $\mathscr{H}$-\textit{comodule-category} if in addition, the composition map $\mathscr{M}\otimes_{X}\mathscr{M}\stackrel{\circ}{\longrightarrow} \mathscr{M}$ is a map of right $\mathscr{H}$-comodules, where $\mathscr{M}\otimes_{X}\mathscr{M}$ is equipped with the diagonal coaction.
\item[(4)] The \textit{coinvariants} of a right $\mathscr{H}$-comodule-category $\mathscr{M}$ is the subcategory $\mathscr{M}^{co\ \mathscr{H}}$ whose space of objects is $X$ and whose hom-sets are defined as
\[ \left(\mathscr{M}^{co\ \mathscr{C}}\right)_{x,y}:=\left\{ \alpha\in \mathscr{M}_{x,y}|\rho(\alpha)=\alpha\otimes id_{y} \right\} \]
\noindent for any $x,y\in X$. \end{enumerate} \end{defn}
\begin{rem}\label{R3.4} A Hopf category is the categorification of a Hopf algebra with categorical composition corresponding to the algebra product. A right $\mathscr{H}$-comodule $\mathscr{M}$ is in particular a category, it already has a composition. This means that we only need to impose requirement $(3)$ in definition (\ref{D3.4}) to get a categorification of the notion of a comodule-algebra. In the classical set-up, one has to require the existence of a product and assert its compatibility with the comodule structures. \end{rem}
In the set-up of Hopf-Galois theory with respect to Hopf algebras, there is a well-understood notion for extensions of $k$-algebras $A\subseteq B$ to be $H$-Galois for a Hopf algebra $H$ even if $A\neq k$. This is because $B\otimes_{A}B$ makes sense as a $k$-module. All that is left to do is require $A=B^{co\ H}$ and that the map $B\otimes_{A}B\longrightarrow B\otimes H$, $a\otimes b\mapsto (a\otimes 1)\rho(b)$ is bijective. On the other hand, in the situation of a Hopf category $\mathscr{H}$ and extensions of comodule-categories $\mathscr{A}\subseteq\mathscr{M}$ with $\mathscr{A}=(\mathscr{M})^{co\ \mathscr{H}}$, we can only make sense of the product $\mathscr{M}\otimes_{\mathscr{A}}\mathscr{M}$ in the case $\mathscr{A}$ is the subcategory of $\mathscr{M}$ whose hom-sets $\mathscr{A}_{x,y}$ are all zero except when $x=y$, in which case $\mathscr{A}_{x,x}=\mathbb{C}$. In this case, we identify $\mathscr{M}\otimes_{\mathscr{A}}\mathscr{M}$ with $\mathscr{M}\otimes_{X}\mathscr{M}$. Let us call such a category the \textit{trivial linear category} over $X$, and denote by $I_{X}$. There might be a way to consider Galois extensions by Hopf categories in which the subcategory of coinvariants is strictly larger than $I_{X}$, but at present it is not clear to the author how to make sense of it. Fortunately, for our purpose of proving theorem (\ref{T5.1}) it is enough to have $I_{X}$ as the subcategory of coinvariants.
\begin{defn}\label{D3.5} A right $\mathscr{H}$-comodule-category $\mathscr{M}$ is a $\mathscr{H}$-\textit{Galois extension} of $I_{X}$ provided \begin{enumerate} \item[(a)] $\mathscr{M}^{co\ \mathscr{H}}=I_{X}$, and
\item[(b)] the functor
\[ \mathscr{M}\otimes_{X}\mathscr{M} \stackrel{\mathfrak{gal}}{\longrightarrow} \mathscr{M}\otimes_{X}\mathscr{H}, \] \[ \alpha \otimes \beta \mapsto \left(\alpha\circ\beta_{[0]}\right)\otimes \beta_{[1]}\]
\noindent called the \textit{Galois morphism}, is fully faithful. \end{enumerate} \end{defn}
\begin{rem}\label{R3.5} \begin{enumerate} \item[] \item[(1)] We are using Sweedler notation for the legs of the coaction
\[ \rho:\mathscr{M}\longrightarrow \mathscr{M}\otimes_{X}\mathscr{H}. \]
\noindent In other words, for any $x,y\in X$ and $\alpha\in\mathscr{M}_{x,y}$, we have $\rho(\alpha)=\alpha_{[0]}\otimes\alpha_{[1]}$, where $\alpha_{[0]}\in\mathscr{M}_{x,z}$ and $\alpha_{[1]}\in\mathscr{H}_{z,y}$ for some $z\in X$. This, in particular, tells us that the map $\mathfrak{gal}$ above make sense. \item[(2)] Galois extension by a coupled Hopf category $\mathscr{H}=(\mathscr{H}_{L},\mathscr{H}_{R},S)$ means simultaneous Galois extensions of the constituent $C(\mathcal{V})$-enriched categories $\mathscr{H}_{L}$ and $\mathscr{H}_{R}$. \end{enumerate} \end{rem}
\section{The category associated to a Hopf algebroid} \label{S4.0}
In this section, we will consider Hopf algebroids $\mathcal{H}$ over a commutative unital $C^{*}$-algebra $A$. We will restrict to the case where $\mathcal{H}$ is finitely-generated and projective as a left and a right $A$-module. With the underlying assumption that the antipode is bijective, by \cite{bohm-szlachanyi}, finitely-generated projectivity of any of the $A$-module structures of $\mathcal{H}$ coming from the source and target maps are all equivalent. Note that even though $A$ is commutative, its image under the source or the target map need not be central in $\mathcal{H}$. We will deal with this general situation and specialize in the case when we have centrality.
\subsection{Local eigenspace decomposition} \label{S4.1}
Let $\mathcal{H}=(H_{L},H_{R},S)$ be a Hopf algebroid over $A$, a commutative unital $C^{*}$-algebra. Assume that $H_{L}$ is finitely-generated and projective as a left- and a right-$A$-module via the source and target maps. With our standing assumption, $H_{R}$ has the same properties.
Let us first consider the left bialgebroid $H_{L}$. The Gelfand duality implies that $A\cong C(X)$ for some compact Hausdorff space $X$. The Serre-Swan theorem applied to the left $A$-module $H_{L}$ gives us a finite-rank vector bundle $E\stackrel{p}{\longrightarrow}X$ such that $H_{L}\cong\Gamma(X,E)$ as left modules, where the left $C(X)$-module structure on $\Gamma(X,E)$ is by pointwise multiplication, i.e. $(f\cdot\sigma)(x)=f(x)\sigma(x)$ for all $x\in X, f\in C(X)$ and $\sigma\in \Gamma(X,E)$. By the bimodule nature of $H_{L}$, the right $A$-module structure of $\Gamma(X,E)$ commutes with the left $A$-module which implies that we have a representation $C(X)\stackrel{\rho}{\longrightarrow} End(E)$ of $C(X)$ into the endomorphism bundle of $E\stackrel{p}{\longrightarrow}X$. Since $C(X)$ is abelian and $\rho$ is a $*$-morphism, $\rho(C(X))$ lands in a maximal abelian subalgebra $D(n)$ of $End(E)$.
Choose a finite collection of open sets $\left\{U_{i}|i=1,2,...,m\right\}$ that cover $X$ over which $E$ is trivializable. Choose a system of coordinates such that $E$ trivial over each $U_{i}$, i.e. $\left.E\right|_{U_{i}}\cong U_{i}\times V$, where $V$ is a finite-dimensional vector space.Choosing a basis $v_{1},v_{2},...,v_{n}\in V$ one has $End(\left.E\right|_{U_{i}})=C(U_{i},M_{n}(\mathbb{C}))$ where $n$ is the rank of $E$. Commutativity of $C(X)$ implies that up to unitaries $V_{i}\in U(n)$, we have
\[ \xymatrix{ C(X) \ar[rr]^-{\rho} && C(U_{i}, Diag(n))} \]
\noindent where $Diag(n)$ denotes the subalgebra of diagonal matrices on $M_{n}(\mathbb{C})$ and
\[ V_{i}\cdot C(U_{i}, Diag(n)) \cdot V_{i}^{*}=\left.D(n)\right|_{U_{i}}. \]
\noindent For each $i=1,2,...,m$, choosing a set of central orthogonal idempotents $\left\{e_{j}|j=1,...,n\right\}$ gives $n$ projections $p^{i}_{j}$ given by the following composition
\[ \xymatrix{ C(X) \ar[rr]^-{\rho_{i}} && C(U_{i}, Diag(n))\cong \bigoplus\limits_{k=1}^{n}C(U_{i}) \ar[rr]^-{proj_{j}} && C(U_{i}) } \]
\noindent These projections are in particular continuous $C^{*}$-morphisms. Hence, they give, for each $i=1,2,...,m$, (possibly non-distinct) $n$ continuous injective maps $U_{i}\stackrel{\varphi^{i}_{j}}{\longrightarrow}X$, $j=1,...,n$. Geometrically, the situation is depicted figure (\ref{fg4.1}).
\begin{figure}
\caption{Local eigenspace decomposition of $E$.}
\label{fg4.1}
\end{figure}
Let us describe the nature of the set $Z=\bigcup_{i,j}\varphi_{j}^{i}(U_{i})$ over the intersections $U_{\alpha}\cap U_{\beta}$. Over $U_{\alpha}\cap U_{\beta}\subseteq U_{\alpha}$ we get a unitary $V_{\alpha}$ which gives $n$ central orthogonal idempotents and up to ordering of such idempotents, one gets the sets $\varphi^{i}_{j}(U_{i})$. The union $\bigcup_{j}\varphi^{i}_{j}(U_{i})$ does not depend on the ordering of these idempotents. Thus, over $U_{\alpha}\cap U_{\beta}$ one gets unitaries $V_{\alpha}$ and $V_{\beta}$ which simultaneously diagonalize $\rho(C(X))$. Thus, we have
\[ \bigcup_{j}\varphi^{\alpha}_{j}(U_{\alpha}\cap U_{\beta}) = \bigcup_{j}\varphi^{\beta}_{j}(U_{\alpha}\cap U_{\beta}) \]
\noindent from which we get that
\[ \left(\bigcup_{j}\varphi^{\alpha}_{j}(U_{\alpha})\right)\cap\left(\bigcup_{j}\varphi^{\beta}_{j}(U_{\beta})\right)= \bigcup_{j}\varphi^{\alpha}_{j}(U_{\alpha}\cap U_{\beta}) \]
\noindent that is, the sets $\bigcup_{j}\varphi^{i}_{j}(U_{i})$ agree on the intersections.
A subset $T\subseteq X\times X$ is called \textit{transverse} if
\[ \left.proj_{1}\right|_{T}: X\times X\longrightarrow X, \hspace{.5in} \left.proj_{2}\right|_{T}: X\times X\longrightarrow X \]
\noindent are homeomorphisms, where $proj_{1}$ and $proj_{2}$ denotes the projection onto the first and second factor, respectively. In particular, $T$ is homeomorphic to $X$. Using the above argument, we have the following proposition.
\begin{prop}\label{P4.1} For every $i=1,2,...,m$, $j=1,2,...,n$ the set $\varphi_{j}^{i}(U_{i})$ extends to a transverse subset of $X\times X$ completely contained in $Z$. In particular, $Z$ is the union of $n$ (possibly overlapping) transverse subsets of $X\times X$. \end{prop}
\noindent This means that the curves in figure (\ref{fg4.1}) overlap.
\begin{rem}\label{R4.1} Another way to see why the closed subset $Z\subset X\times X$ is the union of transverse subsets of $X\times X$ is by the fact the we can run the construction of the sets $\varphi^{i}_{j}(U_{i})$ described in the beginning of this section in a symmetric fashion, one for each factor of $X\times X$. \end{rem}
The whole picture (\ref{fg4.1}) is a decomposition of $X\times X$ into $X\times U_{i}$, $i\in I$. The graphs of $\varphi^{i}_{j}$ are labelled accordingly. Note that each $f(x)\in End(E_{x})$, $f\in C(X)$ are diagonalizable since they commute with their adjoint $f(x)^{*}\in C(X)$. And since such operators commute with each other, the collection $\left\{f(x)\in End(E_{x})|f\in C(X)\right\}$ is simultaneously diagonalizable. Over a point $x\in U_{1}$, the fiber $E_{x}$ decomposes into joint eigenspaces of $\left\{f(x)\in End(E_{x})|f\in C(X)\right\}$. The dimension of these eigenspaces are determined by the number of intersections of the vertical dotted line through $x\in U_{1}$ with the graphs of $\varphi^{1}_{j}$. Using this eigenspace decomposition, we have the following proposition which describes geometrically the right $C(X)$-module structure of $H_{L}$.
\begin{prop}\label{P4.2} Given $\sigma\in \Gamma(X,E)$ and $f\in C(X)$ the section $\sigma\cdot f\in\Gamma(X,E)$ is given as
\begin{eqnarray} \left(\sigma\cdot f\right)(x)=\sum\limits_{j=1}^{n} f(\varphi^{i}_{j}(x))e_{j}\cdot\sigma(x) \label{eq4.1}. \end{eqnarray}
\noindent where $x\in U_{i}$ and $\sigma(x)=\sum\limits_{j=1}^{n} e_{j}\cdot\sigma(x)$. \end{prop}
\begin{rem}\label{R4.2} \begin{enumerate} \item[]
\item[(1)] In case $C(X)$ is central in $H_{L}$, the above picture reduce to $\left\{U_{i}|i\in I\right\}$ the trivial cover and $\varphi:X\longrightarrow X$ is the identity, i.e. the graph in the above picture is the diagonal of $X\times X$. The action defined by equation (\ref{eq4.1}) then reduces to pointwise multiplication which then coincides with the left $C(X)$-module structure of $H_{L}\cong\Gamma(X,E)$.
\item[(2)] One can understand the right action above as \textit{pointwise-eigenvalue-scaled} action. Compared to the central case, every $f\in C(X)$ acts on a $\sigma\in\Gamma(X,E)$ in a way that $f(x)$ acts diagonally on $\sigma(x)$, i.e. $E_{x}$ constitutes a single eigenspace for the operator $f(x)$ corresponding to the eigenvalue $f(x)\in\mathbb{C}$. In the noncentral case, the action is still pointwise. However, the operator $f(x)$ no longer has a single eigenspace. The eigenspaces are labelled by the points $\varphi^{i}_{j}(x)\in X$ where $x\in U_{i}$ and the eigenvalues of $f(x)$ are $f\left(\varphi^{i}_{j}(x)\right)$, $j=1,...,n$. \end{enumerate} \end{rem}
\begin{prop}\label{P4.3} As a $C(X)$-bimodule, $H_{L}\cong\Gamma(Z,\mathcal{E})$ where $\mathcal{E}$ is a sheaf of complex vector spaces over $X\times X$ supported on a closed subset $Z\subset X\times X$. The $C(X)$-bimodule structure on $\Gamma(Z,\mathcal{E})$ is defined as
\[ (f\cdot \sigma\cdot g)(x,y)=f(x)\sigma(x,y)g(y) \]
\noindent for $f,g\in C(X)$ and $\sigma\in \Gamma(Z,\mathcal{E})$. \end{prop}
The $C(X)\otimes C(X)^{op}$ is dense in $C(X\times X)$ thus we can extend the $C(X)\otimes C(X)^{op}$-module structure of $H_{L}$ to a $C(X\times X)$-module structure. Consider the annihilator of $H_{L}$,
\[ Ann(H_{L})=\left\{f\in C(X\times X)| f\cdot \sigma = 0, \text{\ for all \ } \sigma\in B_{L}\right\}. \]
\noindent Then, there is an open set $U\subset X\times X$ such that $Ann(H_{L})=C(U)$. Then $Z=(X\times X)-U$, the support of the bimodule $H_{L}$.
\begin{prop}\label{P4.4} The subset $Z\subseteq X\times X$ is completely determined by the $C(X)$-bimodule structure of $H_{L}$. Moreover, $Z$ is the support of $H_{L}\cong\Gamma(X\times X, \mathcal{E})$. \end{prop}
By proposition (\ref{P4.1}), $Z$ is the union of transverse subsets of $X\times X$ which is individually are unions of graphs of $\varphi^{i}_{j}$. Let
\[ E_{(x,y)}= \bigoplus\limits_{\varphi^{i}_{j}(x)=y}\left(E_{x}\right)_{\varphi^{i}_{j}(x)} \]
\noindent be the fiber of $\mathcal{E}$ over $(x,y)\in Z$, where $\left(E_{x}\right)_{\varphi^{i}_{j}(x)}$ denotes the eigensubspace of $E_{x}$ over the point $\varphi^{i}_{j}(x)$. This defines a sheaf of vector spaces on $X\times X$ supported on $Z$. A section of $\tau\in\Gamma(X,E)$ defines a section $\hat{\tau}\in \Gamma(Z,\mathcal{E})$ whose value at a point $(x,y)$ is
\[ \hat{\tau}(x,y)= \begin{cases}
proj_{ij}\tau(x) ,& \text{if } y=\varphi^{i}_{j}(x) \ \text{for some \ } i,j \\
& \\
0, & \text{otherwise,} \end{cases} \]
\noindent where $proj_{ij}$ denotes the projection $E_{x}\longrightarrow \left(E_{x}\right)_{\varphi^{i}_{j}(x)}$. Conversely, any section $\tau\in\Gamma(Z,\mathcal{E})$ defines a section $\check{\tau}\in\Gamma(X,E)$ by
\[ \check{\tau}(x)=\sum\limits_{y}\tau(x,y). \]
\noindent Now, given $h\in C(X)\otimes C(X)$ we have
\[ h(x,y)=\sum\limits_{k}f_{k}(x)g_{k}(y) \]
\noindent for some $f_{k},g_{k}\in C(X)$. For any $\sigma\in B\cong\Gamma(X,E)$ we have
\begin{eqnarray*} \left(h\cdot \hat{\sigma}\right)(x,y) &=&\sum\limits_{k}f_{k}(x)\hat{\sigma}(x,y)g_{k}(y) \\ &=& \sum\limits_{k}f_{k}(x)g_{k}(\varphi^{i}_{j}(x))e_{j}\left(\sigma(x)\right) \\ &=& proj_{ij}\left(\sum\limits_{k}f_{k}\cdot\sigma\cdot g_{k}\right)(x,y) \\ \end{eqnarray*}
\noindent which shows that $\phantom{A}^{\wedge}:\Gamma(X,E)\longrightarrow \Gamma(Z,\mathcal{F})$, $\tau\mapsto \hat{\tau}$ is a bimodule map whose inverse is the map $\phantom{A}^{\vee}:\Gamma(Z,\mathcal{F})\longrightarrow \Gamma(X,E)$, $\tau\mapsto\check{\tau}$.
Using proposition (\ref{P4.1}), we can relate the vector bundles the Serre-Swan theorem gives when applied to the left and right $C(X)$-module structure of $H_{L}$ as follows.
\begin{prop}\label{P4.5} Let $E_{1}\stackrel{p_{1}}{\longrightarrow}X$ and $E_{2}\stackrel{p_{2}}{\longrightarrow}X$ be the vector bundles given by the Serre-Swan theorem applied to the finitely-generated projective left and right $C(X)$-module $H_{L}$, respectively. Then $E_{1}$ and $E_{2}$ are the direct-images of the sheaf $\mathcal{E}$ along $\pi_{1}$ and $\pi_{2}$, respectively. \end{prop}
First, the direct-image of $\mathcal{E}$ along $\pi_{1}$ is easily seen to be a vector bundle and the space of sections $\Gamma(X,(\pi_{1})_{*}\mathcal{E})$ is easily seen to be isomorphic as left $C(X)$-modules to the left $C(X)$-module $\Gamma(Z,\mathcal{E})$. By proposition (\ref{P4.3}), $\Gamma(Z,\mathcal{E})\cong\Gamma(X,E_{1})$ as left $C(X)$-modules. Thus, by corollary 2.8 of \cite{b007} we see that $E_{1}$ and $(\pi_{1})_{*}\mathcal{E})$ are isomorphic as vector bundles. Similar argument works for $E_{2}$.
Let us say more about the nature of the eigenspaces $E_{(x,y)}$, $x,y\in X$ in relation to the subset $Z$.
\begin{prop}\label{P4.6} \begin{enumerate} \item[] \item[(i)] $E_{x}=\bigoplus\limits_{y\in X} E_{(x,y)}$
\item[(ii)] $dim\left( E_{(x,y)}\right)$ is the number of transverse subsets of $X\times X$ contained in $Z$ passing through $(x,y)$, with multiplicities.
\item[(iii)] $dim\left(\bigoplus\limits_{x\in X}E_{(x,y)}\right)=n$ for any $y\in X$. \end{enumerate} \end{prop}
\subsection{The geometry of \textit{C(X)}-ring structures}\label{S4.2}
The previous section describes the geometry of $H_{L}$ using its bimodule structure over $C(X)$. But $H_{L}$ has more structure than just being a bimodule. In particular, it is a $C(X)$-ring via the left source map $C(X)\stackrel{s_{L}}{\longrightarrow}H_{L}$. In this section, we will look at what this additional structure contributes to the geometry of $H_{L}$. We will keep the notations of the previous section.
The $C(X)$-ring structure on $H_{L}\cong\Gamma(X,\mathcal{E})$ via the source map $s_{L}$ consists of a pair of $C(X)$-bimodule maps
\[ \Gamma(Z,\mathcal{E})\tens{C(X)}\Gamma(Z,\mathcal{E}) \stackrel{\mu}{\longrightarrow} \Gamma(Z,\mathcal{E}) \]
\[ C(X)\stackrel{\eta}{\longrightarrow}\Gamma(Z,\mathcal{E}) \]
\noindent satisfying the associativity and unitality conditions. For brevity we will write $\eta=s_{L}$.
\begin{figure}
\caption{The geometry of the product and unit maps.}
\label{fg4.2}
\end{figure}
The unit map $\eta$ gives an element $1\in\Gamma(Z,\mathcal{E})$ satisfying $f\cdot 1=1\cdot f$ for all $f\in C(X)$. Since $X$ is Hausdorff, if $x\neq y$ then we can find an $f\in C(X)$ such that $f(x)=1$ and $f(y)=0$. Thus, for $x\neq y$ we have
\[ 1(x,y)=f(x)1(x,y)=(f\cdot 1)(x,y)=(1\cdot f)(x,y)=1(x,y)f(y)=0. \]
\noindent Thus, the source map $A\stackrel{s_{L}}{\longrightarrow} H_{L}$ is implemented by $C(X)\longrightarrow \Gamma(Z,\mathcal{E})$, $f\mapsto f\cdot 1$. This means that $s_{L}\circ f(x)=f(x)1(x,x)$ and choosing $f$ such that $f(x)\neq 0$ and $s_{L}\circ f(x)\neq 0$ we see that $1(x,x)\in E_{(x,x)}$ is a nonzero element. Thus, the diagonal $\slashed{\Delta}$ of $X\times X$ is in $Z$. See figure (\ref{fg4.3}).
\begin{figure}
\caption{Support $Z$ of the bimodule $B$.}
\label{fg4.3}
\end{figure}
Note that $\Gamma(Z,\mathcal{E})\tens{C(X)}\Gamma(Z,\mathcal{E})\cong\Gamma(Z,\mathcal{E}^{(2)})$ where $\mathcal{E}^{(2)}$ is the sheaf of vector spaces whose fiber at a point $(x,z)\in Z$ is the vector space
\[ \bigoplus\limits_{y\in X} \left(E_{(x,y)} \otimes E_{(y,z)}\right) \]
\noindent due to the balancing condition $\sigma\cdot f\otimes_{C(X)}\tau=\sigma\otimes_{C(X)}f\cdot\tau$ for $\sigma,\tau\in\Gamma(Z,\mathcal{E})$ and $f\in C(X)$. Notice that all but finitely many summands above are zero. Specifically, only those $y\in X$ for which $(x,y)$ and $(y,z)$ are both in $Z$ contribute nontrivially. Let us denote these $y\in X$ as $y_{1},y_{2},...,y_{n}$.
By proposition (\ref{P4.3}), $\Gamma(Z,\mathcal{E})\cong\Gamma(X,E)$ as $C(X)$-bimodules. Since $\Gamma(X,-)$ is a fully faithful functor by corollary 2.8 of \cite{b007}, we can convert the global ring structures $\mu$ and $\eta$ into something fiber-wise. In particular, the product map $\mu$ induces a map
\begin{eqnarray}\label{eq4.2} \xymatrix{ E_{(x,y_{1})}\otimes E_{(y_{1},z)} \oplus ... \oplus E_{(x,y_{n})}\otimes E_{(y_{n},z)} \ar[rr]^-{\mu_{*}} && E_{(x,z)} } \end{eqnarray}
\noindent illustrated in figure (\ref{fg4.2}). By the universal property of direct sums, there are maps
\[ \xymatrix{ E_{(x,y_{i})}\otimes E_{(y_{i},z)}\ar[rr]^-{\mu^{y_{i}}_{*}} && E_{(x,z)} }\]
\noindent one for each $y_{i}$. The collection of these maps satisfy a set of conditions which, though derivable from associativity, is complicated to write down. See (3) of the remark below for these conditions. However, for the maps $E_{(x,x)}\otimes E_{(x,x)}\stackrel{\mu^{x}_{*}}{\longrightarrow}E_{(x,x)}$ these conditions are precisely the associativity condition. Likewise, the map $\eta$ induces maps $\eta_{*}^{x,y}:\mathbb{C}\longrightarrow E_{(x,y)}$ which is nonzero when $x=y$ and zero otherwise. The map $\mu_{*}^{x}$ together with $\eta_{*}^{x}=\eta_{*}^{x,x}$ makes the vector space $E_{(x,x)}$ a unital algebra, whose dimension depend on the multiplicity of the associated eigenvalue. The following proposition is then immediate from these arguments.
\begin{prop}\label{P4.7} Let $A^{'}$ be the $C(X)$-sub-bimodule of $\Gamma(Z,\mathcal{E})$ supported on the diagonal $\slashed{\Delta}$. Then $A^{'}$ is an $A$-subring of $H_{L}$ where the multiplication is pointwise. Moreover, $A^{'}$ is the centralizer of $A$ in $H_{L}$. \end{prop}
\begin{rem}\label{R4.3} \begin{enumerate} \item[]
\item[(1)] Using abuse of notation, let us identify $A$ with its image in $H_{L}$. In case $A$ is central in $H_{L}$, the fibers of the vector bundle $E\longrightarrow X$ are algebras. These algebras correspond to $E_{(x,x)}$ together with the maps $E_{(x,x)}\otimes E_{(x,x)}\stackrel{\mu_{*}^{x}}{\longrightarrow} E_{(x,x)}$ and $\mathbb{C}\stackrel{\eta_{*}^{x}}{\longrightarrow} E_{(x,x)}$ since in the central case, $E_{(x,x)}=E_{x}$. Thus, $A^{'}=H_{L}$ in the central case which is not surprising at all knowing that $A^{'}$ is the centralizer of $A$.
\item[(2)] The maps $E_{(x,y_{i})}\otimes E_{(y_{1},z)}\stackrel{\mu^{i}_{*}}{\longrightarrow}E_{(x,z)}$ are only restricted by the associativity of $\mu$. Since $\Gamma(Z,\mathcal{E})\cong\Gamma(X,E)$ and $\Gamma(X,-)$ is known to be a fully faithful functor by corollary 2.8 of \cite{b007}, we have
\[ \xymatrix{ \bigoplus\limits_{y_{i},y_{j}} \left(E_{(x,y_{i})}\otimes E_{(y_{i},y_{j})} \otimes E_{(y_{j},z)}\right) \ar[rr]^-{\left(\bigoplus\limits_{i}\mu^{i}_{*}\right)\otimes id} \ar[dd]_-{id\otimes \left(\bigoplus\limits_{j}\mu^{j}_{*}\right)} && \bigoplus\limits_{y_{j}}\left(E_{(x,y_{j})} \otimes E_{(y_{j},z)}\right) \ar[dd]^-{\bigoplus\limits_{j}\mu^{j}_{*}} \\
&& \\ \bigoplus\limits_{y_{i}}\left(E_{(x,y_{i})} \otimes E_{(y_{i},z)}\right) \ar[rr]_-{\bigoplus\limits_{i}\mu^{i}_{*}} && E_{(x,z).} } \]
\noindent Universal property of direct sums gives us
\[ \xymatrix{ E_{(x,y_{i})}\otimes E_{(y_{i},y_{j})} \otimes E_{(y_{j},z)} \ar[rr]^-{\mu^{i}_{*}\otimes id} \ar[dd]_-{id\otimes \mu^{j}_{*}} && E_{(x,y_{j})} \otimes E_{(y_{j},z)} \ar[dd]^-{\mu^{j}_{*}} \\
&& \\ E_{(x,y_{i})} \otimes E_{(y_{i},z)} \ar[rr]_-{\mu^{i}_{*}} && E_{(x,z)}. } \]
\noindent This justifies the argument before proposition (\ref{P4.7}). We can also use this to say more about the fibers of $\mathcal{E}$ which we state in the next proposition.
\end{enumerate} \end{rem}
\begin{prop}\label{P4.8} $E_{(x,y)}$ is a left $E_{(x,x)}-E_{(y,y)}-$bimodule for every $x,y\in X$. \end{prop}
\begin{rem}\label{R4.4} Using remark (\ref{R4.3}) above, we can construct a small category $\mathscr{H}_{L}$ enriched over the category of complex vector spaces. The set of objects of $\mathscr{H}_{L}$ is $X$. For every $x,y\in X$, we define
\[ Hom(x,y):= \begin{cases}
E_{(x,y)} ,& \text{if } y=\varphi^{i}_{j}(x) \ \text{for some \ } i,j \\
& \\
\left\{0\right\}, & \text{otherwise.} \end{cases} \]
\noindent We will call $\mathscr{H}_{L}$ the \textit{associated category} of the left $A$-bialgebroid $H_{L}$. In the next section, we will see the additional properties of $\mathscr{H}_{L}$ coming from the $A$-coring structure of $H_{L}$. On a different note, let us give a complete geometric description of the $A$-ring structure of $H_{L}$. \end{rem}
\begin{prop}\label{P4.9} Denote by $a \ast_{i}b:=\mu_{*}^{y_{i}}(a,b)$, $a\in E_{(x,y_{i})}$ and $b\in E_{(y_{i},z)}$. The product of $\sigma,\tau\in\Gamma(Z,\mathcal{E})$ takes the form
\[ (\sigma\tau)(x,z)=\sum\limits_{i} \sigma(x,y_{i})\ast_{i}\tau(y_{i},z) \]
\noindent for all $(x,z)\in Z$. \end{prop}
This follows immediately from equation (\ref{eq4.2}). Notice the resemblance of this formula to the one for matrix multiplication. This should remind the reader of an example we discussed in section (\ref{S3.2}). One can view a $C(X)$-ring to be a "matrix" of vector spaces whose entries are indexed by $X\times X$ and what sits in entry $(x,y)$ is the vector space $E_{(x,y)}$. As we have defined after proposition (\ref{P4.4}), the vector space $E_{(x,y)}$ is the zero vector space if $(x,y)\notin Z$. For matrix algebras $M_{n}(\mathbb{C})$, $X$ would be an $n$-element set and the vector spaces $E_{(x,y)}$ would all be $\mathbb{C}$. There are a plethora of algebraic structures package into a bialgebroid let alone in a Hopf algebroid. Before we end this section, let us take a detour to describe the relationships among the structures of $H$: being a $\mathbb{C}$-algebra, the $A$-ring and the $A^{e}$-ring structures being a left-bialgebroid over $A=C(X)$.
For the purpose of this discussion, let us denote by $(\mu_{\mathbb{C}},\eta_{\mathbb{C}})$ the $\mathbb{C}$-algebra structure of $H$ and recall that $(\mu_{L},s_{L})$ and $(\mu_{A^{e}},\eta_{L})$ denote the relevant $A$-ring and $A^{e}$-ring structures of $H$, respectively. As we mentioned in section (\ref{S2.1}), for a $k$-algebra $R$, $R$-ring structures are in bijection with $k$-algebra maps $\eta:k\longrightarrow R$. Thus, the complex algebra structure of $H$ is uniquely determined by the unit map $\eta_{\mathbb{C}}:\mathbb{C}\longrightarrow H$. Similarly, the $A$-ring and the $A^{e}$-ring structures are determined by the $\mathbb{C}$-linear maps $s_{L}$ and $\eta_{L}$. These maps satisfy the following commutativity relations.
\[\xymatrix{\mathbb{C} \ar[rr]^-{\eta_{\mathbb{C}}} \ar[dd] && H\\ &&\\
A \ar[rr] \ar[rruu]|-{s_{L}} && A^{e} \ar[uu]_-{\eta_{L}}} \hspace{.5in} \xymatrix{H\otimes H \ar@{->>}[rr] \ar[rrdd]_-{\mu_{\mathbb{C}}} && H\tens{A}H \ar@{->>}[rr] \ar[dd]_-{\mu_{L}} && H\tens{A^{e}}H. \ar[lldd]^-{\mu_{A^{e}}} \\ &&&& \\ && H &&}\]
\noindent In terms of the local eigenspace decomposition, the map $\mu_{\mathbb{C}}$ induces maps
\[\xymatrix{ E_{(x,w)} \otimes E_{(z,y)} \ar[rr] && E_{(x,y)} } \]
\noindent while, by (\ref{eq4.2}), we have maps
\[ \xymatrix{ E_{(x,z)} \otimes E_{(z,y)} \ar[rr] && E_{(x,y)} }. \]
\noindent On the other hand, because the $C(X\times X)$-bimodule structure of $H$ is given as follows,
\[ (f\cdot\sigma)(x,y)=f(x,y)\sigma(x,y), (\sigma\cdot f)(x,y)=f(y,x)\sigma(x,y), \]
\noindent for any $f\in C(X\times X)$, $\sigma\in H$, and $x,y\in X$, the product $\mu_{A^{e}}$ induces maps
\[ \xymatrix{ E_{(x,z)}\otimes E_{(z,x)} \ar[rr] && E_{(x,x)} }. \]
Another way of seeing this is by noting that the product $\mu_{L}$ uses the tensor product $\otimes_{A}$ which kills products $\xymatrix{ E_{(x,w)}\otimes E_{(z,y)} \ar[rr] && E_{(x,y)} }$ for which $w\neq z$. Likewise, the tensor product $\otimes_{A^{e}}$ kills products $\xymatrix{ E_{(x,z)}\otimes E_{(z,y)} \ar[rr] && E_{(x,y)} }$ for which $x\neq y$.
\subsection{The geometry of \textit{C(X)}-coring structures}\label{S4.3}
In this section, using the techniques and results we have developed in sections (\ref{S4.1}) and (\ref{S4.2}) we will describe what the coring structure of $H_{L}$ contributes to the geometry of $\mathcal{E}$. We will keep the notations of the previous two sections.
The $C(X)$-bimodule structure of the underlying $A$-coring structure of $H_{L}$ is related to the $C(X)$-bimodule structure of the underlying $A$-ring via
\begin{equation}\label{eq4.3} (f\cdot \sigma\cdot g)(x,y)=f(x)g(x)\sigma(x,y) \end{equation}
\noindent for $\sigma\in \Gamma(Z,\mathcal{E})$, $f,g\in C(X)$, and $x,y\in X$. The left-hand side of equation (\ref{eq4.3}) concerns the bimodule structure one has for the underlying $A$-coring of $H_{L}$ while the right-hand side concerns its $A$-ring structure. This, in particular, implies that if we run the construction we have in section (\ref{S4.1}) for the bimodule structure of the $A$-coring of $H_{L}$, we will get the same sheaf $\mathcal{E}$ supported over the same closed subset $Z$.
The coproduct $\Delta_{L}$ of $H_{L}$, $H_{L}\stackrel{\Delta_{L}}{\longrightarrow}H_{L}\otimes_{A}H_{L}$, uses a different $A$-bimodule structure from the $A$-bimodule structure involved in the $A$-ring structure. Thus, $\otimes_{C(X)}$ means different from the $\otimes_{C(X)}$ we have in the product $\mu$. With this, let us denote by $\boxtimes_{A}$ this new tensor product. thus, we have
\begin{equation}\label{eq4.4} \xymatrix{\Gamma(Z,\mathcal{E}) \ar[rr]^-{\Delta_{L}} && \Gamma(Z,\mathcal{E})\boxtimes_{C(X)}\Gamma(Z,\mathcal{E}) }. \end{equation}
\noindent However, using the relation (\ref{eq4.3}) the codomain of $\Delta_{L}$ can be expressed as
\[ \Gamma(Z,\mathcal{E})\boxtimes_{C(X)}\Gamma(Z,\mathcal{E})\cong\Gamma(Z,\mathcal{E}^{\left\langle 2\right\rangle}), \]
\noindent where $\mathcal{E}^{\left\langle 2\right\rangle}$ is the sheaf of vector spaces whose fiber at $(x,z)\in X\times X$ is
\[ \bigoplus\limits_{y^{'},y^{''}\in X}\left(E_{(x,y^{'})}\otimes E_{(x,y^{''})}\right). \]
\noindent Using the same argument we used in the previous section, the map $\Delta_{L}$ induces a map $\left(\Delta_{L}\right)_{*}:\mathcal{E}\longrightarrow\mathcal{E}^{\left\langle 2\right\rangle}$ of sheaves over $Z$. Over point a $(x,y)\in Z$, we have a map
\begin{equation}\label{eq4.5} \xymatrix{ E_{(x,y)} \ar[rr]^-{\left(\Delta_{L}\right)_{*}^{(x,y)}} && \bigoplus\limits_{z',z''\in X}\left(E_{(x,z')}\otimes E_{(x,z'')}\right) } \end{equation}
\noindent Meanwhile, the counit $\epsilon_{L}:\Gamma(Z,\mathcal{E})\longrightarrow C(X)$ induces a map $\mathcal{E}\longrightarrow \mathcal{E}^{'}$ of sheaves over $Z$ and $\slashed{\Delta}$, respectively. Here, $Z\longrightarrow\slashed{\Delta}$ is the map $(x,y)\mapsto (x,x)$ for any $(x,y)\in Z$ and $\mathcal{E}^{'}$ is the subsheaf of $\mathcal{E}$ where the fiber of $\mathcal{E}^{'}$ at $(x,y)$ is $\left\{0\right\}$ unless $x=y$, to which the fiber is $\mathbb{C}$ viewed as the one-dimensional subalgebra of $E_{(x,x)}$ spanned by its unit $1(x,x)$. Hence, over a point $(x,y)\in Z$ we have $\left(\epsilon_{L}\right)_{*}^{(x,y)}:E_{(x,y)}\longrightarrow\mathbb{C}$.
Counitality of $\Delta_{L}$ with respect to $\epsilon_{L}$ implies that for fixed but arbitrary $x,y\in X$ we have
\begin{equation}\label{eq4.6} \xymatrix@R=2mm{ & & \bigoplus\limits_{z,z^{'}}\left(E_{(x,z)}\otimes E_{(x,z^{'})}\right) \ar[ddddd]^-{\bigoplus\limits_{z^{'}}id\otimes\left(\epsilon_{L}\right)_{*}^{(x,z^{'})}} \\ & & \\ & & \\ & & \\ & & \\ E_{(x,y)} \ar[rruuuuu]^-{\left(\Delta_{L}\right)_{*}^{(x,y)}} \ar@{=}[rr] & & \bigoplus\limits_{z}\left(E_{(x,z)}\otimes\mathbb{C}\right) \\
v \ar@{|->}[rr] && v\otimes 1 \\} \hspace{.15in} \xymatrix@R=2mm{ & & \bigoplus\limits_{z,z^{'}}\left(E_{(x,z)}\otimes E_{(x,z^{'})}\right) \ar[ddddd]^-{\bigoplus\limits_{z}\left(\epsilon_{L}\right)_{*}^{(x,z)}\otimes id} \\ & & \\ & & \\ & & \\ & & \\ E_{(x,y)} \ar[rruuuuu]^-{\left(\Delta_{L}\right)_{*}^{(x,y)}} \ar@{=}[rr] & & \bigoplus\limits_{z^{'}}\left(\mathbb{C}\otimes E_{(x,z^{'})}\right) \\
v \ar@{|->}[rr] && 1\otimes v \\} \end{equation}
\noindent The bottom isomorphisms imply that $\left(\epsilon_{L}\right)_{*}^{(x,z)}$ and $\left(\epsilon_{L}\right)_{*}^{(x,z^{'})}$ are nonzero maps for $z=y$ and $z^{'}=y$. Since $x$ and $y$ are arbitrary to start with, we have the following proposition.
\begin{prop}\label{P4.10} For any $(x,y)\in Z$, we have $\left(\epsilon_{L}\right)_{*}^{(x,y)}\neq 0$. \end{prop}
\noindent Another thing we can infer from the diagrams (\ref{eq4.6}), using the isomorphisms in the bottom and the fact that $y$ is among the $z$ and $z^{'}$ that appears as indices, is that the image of $\left(\Delta_{L}\right)_{*}^{(x,y)}$ is contained in
\[ \left(E_{(x,y)}\otimes E_{(x,y)}\right) \oplus \bigoplus\limits_{z,z^{'}} \left(ker\left(id\otimes\left(\epsilon_{L}\right)_{*}^{(x,z^{'})}\right)+ker\left(\left(\epsilon_{L}\right)_{*}^{(x,z)}\otimes id\right)\right) \]
\noindent We will show in the next section that more can be said. In fact, the image of $\left(\Delta_{L}\right)_{*}^{(x,y)}$ is completely contained in $E_{(x,y)}\otimes E_{(x,y)}$.
\subsection{Hopf algebroids over \textit{C(X)}}\label{S4.4}
In this section, we will complete our description of the geometry of the Hopf algebroid $\mathcal{H}$ over $C(X)$. In doing so, we will be able to illustrate the main point of this article. That to such a Hopf algebroid, one can associate a highly structured category.
So far, we have considered only the constituent left bialgebroid $H_{L}$ of $\mathcal{H}$. Running the arguments we have presented in sections (\ref{S4.1}) and (\ref{S4.2}) for $H_{R}$, we see that there is a sheaf of vector spaces $\mathcal{E}^{'}$ over $X\times X$ such that $H_{R}\cong\Gamma(X\times X,\mathcal{E}^{'})$. Let us denote by $Z^{'}$ the support of $H_{R}$ under the isomorphism $H_{R}\cong\Gamma(X\times X,\mathcal{E}^{'})$. The following proposition relates these two sheaves.
\begin{prop}\label{P4.11} For $H_{L}\cong\Gamma(X\times X,\mathcal{E})$ and $H_{R}\cong\Gamma(X\times X,\mathcal{E}^{'})$ as $C(X)$-bimodules as constructed in sections (\ref{S4.1}) and (\ref{S4.2}), where $\mathcal{E}$ and $\mathcal{E}^{'}$ are sheaves of vector spaces supported on $Z,Z^{'}\subseteq X\times X$, we have \begin{enumerate} \item[(1)] $Z=Z^{'}$. \item[(2)] $\mathcal{E}\cong\mathcal{E}^{'}$ as sheaves over $Z$. \end{enumerate} \end{prop}
\begin{prf} Condition (c) of the definition of a Hopf algebroid implies that the antipode $S$ of $\mathcal{H}$ flips the $C(X)$-bimodule structure used for the $C(X)$-ring structure of $H_{L}$ to that of the $C(X)$-bimodule structure used for the $C(X)$-ring structure of $H_{R}$. Likewise, $S$ flips the bimodule structures of the underlying $C(X)$-coring structures of $H_{L}$ and $H_{R}$. In particular, this tells us that $S$ induces a map $S_{*}:\mathcal{E}\longrightarrow\mathcal{E}^{'}$ which on fibers does $S_{*}(E_{(x,y)})=E_{(y,x)}$ for any $(x,y)\in Z$. Symmetrically, we also have a map denoted the same, $S_{*}:\mathcal{E}^{'}\longrightarrow\mathcal{E}$, which on fibers does $S_{*}(E_{(y,x)})=E_{(x,y)}$ for any $(y,x)\in Z^{'}$. This proves proposition (\ref{P4.11}). $\blacksquare$ \end{prf}
In view of proposition (\ref{P4.11}), we have $\Delta_{R}:\Gamma(Z,\mathcal{E})\longrightarrow\Gamma(Z,\mathcal{E})$. Similar to equation (\ref{eq4.5}), $\Delta_{R}$ induces maps
\begin{equation}\label{eq4.7} \xymatrix{ E_{(x,y)} \ar[rr]^-{\left(\Delta_{R}\right)_{*}^{(x,y)}} && \bigoplus\limits_{z',z''\in X}\left(E_{(z',y)}\otimes E_{(z'',y)}\right) } \end{equation}
\noindent for $(x,y)\in Z$. As we promised at the end of section (\ref{S4.3}), $\left(\Delta_{L}\right)_{*}^{(x,y)}$ maps $E_{(x,y)}$ into $E_{(x,y)}\otimes E_{(x,y)}$, for any $(x,y)\in Z$. same holds for $\left(\Delta_{R}\right)_{*}^{(x,y)}$. Let us summarize these statements into the following proposition.
\begin{prop}\label{P4.12} For every $(x,y)\in Z$, \begin{enumerate} \item[(1)] $E_{(x,y)}$ is a coalgebra with coproduct $\left(\Delta_{L}\right)_{*}^{(x,y)}$ and counit $\left(\epsilon_{L}\right)_{*}^{(x,y)}$, and \item[(2)] $E_{(x,y)}$ is a coalgebra with coproduct $\left(\Delta_{R}\right)_{*}^{(x,y)}$ and counit $\left(\epsilon_{R}\right)_{*}^{(x,y)}$. \end{enumerate} \end{prop}
\begin{prf} We will only prove part $(2)$. The proof for part $(1)$ is similar. The second commutation relation of $\Delta_{L}$ and $\Delta_{R}$ in part $(b)$ of the definition (\ref{D2.2}) gives the following diagram
\begin{equation}\label{eq4.8} \xymatrix{ E_{(x,y)} \ar[rrrr]^-{\left(\Delta_{L}\right)_{*}^{(x,y)}} \ar[ddd]_-{\left(\Delta_{R}\right)_{*}^{(x,y)}} &&&& \bigoplus\limits_{z^{'},z^{''}}\left(E_{(x,z^{'})}\otimes E_{(x,z^{''})}\right) \ar[dd]^-{\bigoplus\limits_{z^{'}}\left(\Delta_{R}\right)_{*}^{(x,z^{'})}\otimes id}\\ &&&& \\ &&&& \bigoplus\limits_{z^{'},z^{''}}\bigoplus\limits_{\alpha^{'},\alpha^{''}}\left(E_{(\alpha^{'},z^{'})}\otimes E_{(\alpha^{''},z^{'})}\otimes E_{(x,z^{''})}\right) \ar@{=}[d] \\ \bigoplus\limits_{\beta^{'},\beta^{''}}\left(E_{(\beta^{'},y)}\otimes E_{(\beta^{''},y)}\right) \ar[rrrr]_-{\bigoplus\limits_{\beta^{''}}id\otimes\left(\Delta_{L}\right)_{*}^{(\beta^{''},y)}} &&&& \bigoplus\limits_{\beta^{'},\beta^{''}}\bigoplus\limits_{\gamma^{'},\gamma^{''}}\left(E_{(\beta^{'},y)}\otimes E_{(\beta^{''},\gamma^{'})}\otimes E_{(\beta^{''},\gamma^{''})}\right)\\ } \end{equation}
\noindent for fixed but arbitrary $(x,y)\in Z$. In the composite $\left(\bigoplus\limits_{z^{'}}\left(\Delta_{R}\right)_{*}^{(x,z^{'})}\otimes id\right)\circ\left(\Delta_{L}\right)_{*}^{(x,y)}$, the third leg lands in $\bigoplus\limits_{z^{''}}E_{(x,z^{''})}$. On the other hand, the third leg of the composite $\left(\bigoplus\limits_{\beta^{''}}id\otimes\left(\Delta_{L}\right)_{*}^{(\beta^{''},y)}\right)\circ\left(\Delta_{R}\right)_{*}^{(x,y)}$ lands in $\bigoplus\limits_{\beta^{''},\gamma^{''}}E_{(\beta^{''},\gamma^{''})}$. This implies that for $\beta^{''}\neq x$, $E_{(\beta^{''},y)}\subseteq ker\ \left(\Delta_{L}\right)_{*}^{(\beta^{''},y)}$. From our last statement in section (\ref{S4.3}), $\left(\Delta_{L}\right)_{*}^{(\beta^{''},y)}\left(E_{(\beta^{''},y)}\right)$ is contained in
\[ \left(E_{(\beta^{''},y)}\otimes E_{(\beta^{''},y)} \right)\oplus\bigoplus\limits_{f^{'},f^{''}}\left(ker\left(id\otimes\left(\epsilon_{L}\right)_{*}^{(\beta^{''},f^{'})}\right)+ker\left(\left(\epsilon_{L}\right)_{*}^{(\beta^{''},f^{''})}\otimes id\right)\right). \]
\noindent Counitality of $\Delta_{L}$ with respect to $\epsilon_{L}$, implemented locally by diagram (\ref{eq4.6}), gives
\[ \xymatrix@R=.2cm{ E_{(\beta^{''},y)} \ar[r]^-{\cong} & \bigoplus\limits_{f^{'}}\left(id\otimes\left(\epsilon_{L}\right)_{*}^{(\beta^{''},f^{'})}\right)\left(\Delta_{L}\right)_{*}^{(\beta^{''},y)}(E_{(\beta^{''},y)}) \ar@{=}[r] & \left\{0\right\}. \\
v \ar@{|->}[r] & v\otimes 1 & \\} \]
\noindent By assumption, $E_{(\beta^{''},y)}$ are nontrivial. This is a contradiction unless the summands corresponding to $\beta^{''}\neq x$ of the direct sum in the lower left corner of diagram (\ref{eq4.8}) do not intersect the image of $\left(\Delta_{R}\right)_{*}^{(x,y)}$.
Using the first commutation relation in part $(b)$ of the definition (\ref{D2.2}), we get a diagram similar to diagram (\ref{eq4.8}). Inspecting that resulting diagram tells us that the image of $\left(\Delta_{R}\right)_{*}^{(x,y)}$ does not intersect those summands of the direct sum in the lower left corner of diagram (\ref{eq4.8}) corresponding to $\beta^{'}\neq x$. This shows that, indeed,
\[ \left(\Delta_{R}\right)_{*}^{(x,y)}: E_{(x,y)}\longrightarrow E_{(x,y)}\otimes E_{(x,y)}. \]
\noindent The coassociativity of $\left(\Delta_{R}\right)_{*}^{(x,y)}$ follows from coassociativity of $\Delta_{R}$ and its counitality with respect to $\left(\epsilon_{R}\right)_{*}^{(x,y)}$ follows from counitality of $\Delta_{R}$ with respect to $\epsilon_{R}$. This proves part $(2)$ of the above proposition. Exchanging the roles of $\Delta_{L}$ and $\Delta_{R}$ with minor modifications proves part $(1)$. $\blacksquare$ \end{prf}
Following the arguments in sections (\ref{S4.1}), (\ref{S4.2}) and (\ref{S4.3}) for $H_{R}$, we see that we can similarly associate a category $\mathscr{H}_{R}$ enriched over $\mathcal{V}$. Denoting by $C(\mathcal{V})$ by the category of coalgebras on $\mathcal{V}$, we have the following proposition.
\begin{prop}\label{P4.13} The categories $\mathscr{H}_{L}$ and $\mathscr{H}_{R}$ are enriched over $C(\mathcal{V})$. \end{prop}
These categories are strongly related. By proposition (\ref{P4.11}), we have the following corollary.
\begin{cor}\label{P4.14} The $C(\mathcal{V})$-enriched categories $\mathscr{H}_{L}$ and $\mathscr{H}_{R}$ have isomorphic underlying $\mathcal{V}$-enriched categories. \end{cor}
Note that the underlying $\mathcal{V}$-enriched category of $\mathscr{H}_{L}$ and $\mathscr{H}_{R}$ only depends on the $C(X)$-ring structures of $H_{L}$ and $H_{R}$, respectively. Another way to prove corollary (\ref{P4.14}) is to use the fact that $H_{L}$ and $H_{R}$ have the isomorphic $C(X)$-ring structures. To see why $H_{L}$ and $H_{R}$ have isomorphic $C(X)$-ring structures, note that the source map of $H_{L}$ is the target map of $H_{L}$ while the target map of $H_{L}$ is the source map of $H_{R}$. In the general definition of a Hopf algebroid, one can either use the source or the target map to select a particular ring structure to consider, see for example \cite{bohm}. Using the general fact that for a general $k$-algebra $R$, $R$-rings $(A,\mu,\eta)$ corresponds uniquely to $k$-algebra maps $\eta$, we see that $H_{L}$ and $H_{R}$ are isomorphic as $C(X)$-rings.
\begin{rem}\label{R4.5} Another way to see why $H_{L}$ and $H_{R}$ are isomorphic as $C(X)$-rings is the fact that general Hopf algebroids $\mathcal{H}$ with bijective antipode over a \textit{commutative} ring $K$ is a coupled $K$-Hopf algebra. \end{rem}
Unlike the ring structures, the $C(X)$-coring structures of $H_{L}$ and $H_{R}$ can vary wildly as illustrated by coupled Hopf algebras. This implies that the $C(\mathcal{V})$-enrichments $\mathscr{H}_{L}$ and $\mathscr{H}_{R}$ need not be isomorphic. However, they form a topological coupled Hopf category. The coupling functor is the one induced by the antipode $S$ of the Hopf algebroid $\mathcal{H}$. We formalize this in the following theorem.
\begin{thm} \label{T4.1} Given a finitely-generated projective Hopf algebroid $\mathcal{H}$ over $C(X)$ with bijective antipode, one can associate a topological coupled Hopf category $\mathscr{H}$ via the construction we presented in sections (\ref{S4.1}) and (\ref{S4.2}). Conversely, to any topological coupled Hopf category $\mathscr{H}$, the space of sections $\Gamma(X\times X,H)$ of the associated sheaf $H$ of $\mathscr{H}$ is a Hopf algebroid over $C(X)$. \end{thm}
The proof of the first statement is basically the breadth of section (\ref{S4.0}). For the second statement, one can consider the bimodule structures presented in proposition (\ref{P4.3}). The rest of the structures are given by the rest of the structure maps of $\mathscr{H}$. The above theorem is a generalization of the example in \cite{bcv} where they constructed out of a $k$-linear category with finitely many objects a weak Hopf algebra. The theorem not only recovers an inverse to the construction they presented but it also work for weak Hopf algebra as long as the subalgebra spanned by the left and the right units are commutative. The above theorem is the generalization of the example we discussed in section (\ref{S3.2}).
\subsection{The central case}\label{S4.5}
Although $C(X)$ is commutative, it may not be central in $\mathcal{H}$. Let us look at the special case when $C(X)$ is central in $\mathcal{H}$, by which we mean that the images of the source and target maps are central in the relevant $C(X)$-ring structure of $\mathcal{H}$. For simplicity, we will blur the disctinction between $C(X)$ at its images under the source maps of $\mathcal{H}$.
Let us consider first the constituent left bialgebroid $H_{L}$ of $\mathcal{H}$. By proposition (\ref{P4.7}), $H_{L}$ is supported along the diagonal $\slashed{\Delta}\subseteq X\times X$. This means that the sheaf $\mathcal{E}$ coincides with the vector bundle $E\longrightarrow X$. We can simply identify the diagonal $\slashed{\Delta}$ with $X$. With this, the multiplication $\mu_{L}$ in $H_{L}$ via the identification $H_{L}\cong\Gamma(X,E)$ is pointwise, i.e. the fibers of the vector bundle $E\longrightarrow X$ are (possibly nonisomorphic) unital complex alegrbas $\left(E_{x},\left(\mu_{L}\right)_{*}^{x},\left(s_{L}\right)_{*}^{x}\right)$, where $\left(\mu_{L}\right)_{*}^{x}$ and $\left(s_{L}\right)_{*}^{x}$ are the maps induced by $\mu_{L}$ and $s_{L}$ on the fiber $E_{x}$.
By proposition (\ref{P4.12}), the coproduct $\Delta_{L}$ and counit $\epsilon_{L}$ of $H_{L}$ also descends into a coproduct $\left(\Delta_{L}\right)_{*}^{x}$ and a counit $\left(\epsilon_{L}\right)_{*}^{x}$ for the fibers $E_{x}$, $x\in X$, making them coalgebras. Using condition $(b)$ in the definition of a bialgebroid, we see that $\left(\Delta_{L}\right)_{*}^{x}$ is multiplicative for any $x\in X$. Meanwhile, using condition $(c)$ of the definition of a bialgebroid we see that $\left(\epsilon_{L}\right)_{*}^{x}$ is multiplicative for any $x\in X$. This gives us the following proposition.
\begin{prop}\label{P4.15} If $C(X)$ is central in $H_{L}$, then for any $x\in X$,
\[ \left(E_{x},\left(\mu_{L}\right)_{*}^{x},\left(s_{L}\right)_{*}^{x},\left(\Delta_{L}\right)_{*}^{x},\left(\epsilon_{L}\right)_{*}^{x}\right) \]
\noindent is a bialgebra. Moreover, the bialgebroid $H_{L}$ is a bundle of bialgebras via $H_{L}\cong\Gamma(X,E)$. \end{prop}
Similar statement holds for the constituent right bialgebroid $H_{R}$. Since for very $x\in X$ the maps $(s_{L})_{*}^{x}$ and $(s_{R})_{*}^{x}$ induced by the source maps $s_{L}$ and $s_{R}$ are the same, the multiplications $(\mu_{L})_{*}^{x}$ and $(\mu_{R})_{*}^{x}$ coincide. Assuming mild nondegeneracy conditions for $(\Delta_{L})_{*}^{x}$ and $(\Delta_{R})_{*}^{x}$, we get the following proposition.
\begin{prop}\label{P4.16} Let $\mathcal{H}=(H_{L},H_{R},S)$ be a Hopf algebroid over $A=C(X)$ where $A$ is central in both $H_{L}$ and $H_{R}$. Denote by $H$ the underlying complex algebra of $\mathcal{H}$. Suppose that the maps
\[ \xymatrix@R=2mm{H\tens{A}H \ar[rr]^-{\mathfrak{gal}_{L}} && H\tens{A}H \\
a\tens{A}b \ar@{|->}[rr] && ab_{[1]}\tens{A}b_{[2]}\\}, \hspace{.5in} \xymatrix@R=2mm{H\tens{A}H \ar[rr]^-{\mathfrak{gal}_{R}} && H\tens{A}H \\
a\tens{A}b \ar@{|->}[rr] && ab^{[1]}\tens{A}b^{[2]}\\} \]
\noindent are bijections. Then \begin{enumerate} \item[(i)] $H$ is a coupled Hopf algebra with constituent Hopf algebras $H_{L}$ and $H_{R}$ and coupling map $S$. \item[(ii)] Each fiber $E_{x}$ is a Hopf algebra and $H_{L}\cong\Gamma(X,E)$ as Hopf algebras, where the structure maps of $\Gamma(X,E)$ are all pointwise. Same is true for $H_{R}$. \item[(iii)] $\mathcal{H}$ is a bundle of coupled Hopf algebras over $X$ such that the constituent Hopf algebras at a point $x\in X$ are the fiber Hopf algebras of $H_{L}$ and $H_{R}$. \end{enumerate} \end{prop}
\begin{prf} Centrality of $A$ in both $H_{L}$ and $H_{R}$ implies that $H_{L}$ and $H_{R}$ are in fact bialgebras over $A$ (not just bialgebroids). The nondegeneracy conditions assumed in the proposition implies that $H$ is a Galois extension for both bialgebras $H_{L}$ and $H_{R}$. By \cite{sch1}, the bialgebras $H_{L}$ are $H_{R}$ are in fact Hopf algebras, i.e. the identity maps $H_{L}\stackrel{id}{\longrightarrow}H_{L}$ and $H_{R}\stackrel{id}{\longrightarrow}H_{R}$ are invertible in the respective convolution algebras associated to the bialgebras $H_{L}$ and $H_{R}$. The rest of the conditions for $\mathcal{H}$ to be a Hopf algebroid imply that $H_{L}$ and $H_{R}$ are coupled Hopf algebras with coupling map $S$, the antipode of $\mathcal{H}$. This proves part $(i)$.
To prove part $(ii)$, we argue that the maps $\mathfrak{gal}_{L}$ and $\mathfrak{gal}_{R}$ are $A$-bimodule maps. Thus, there descend into fiberwise bijections. Using the same argument we did for part $(i)$, see that the fibers are coupled Hopf algebras. Part $(iii)$ readily follows from the proofs of parts $(i)$ and $(ii)$. $\blacksquare$ \end{prf}
\section{Correspondence of Galois extensions}\label{S5.0}
In this section, we will see that the correspondence between Hopf algebroids and coupled Hopf categories we established in theorem (\ref{T4.1}) persists to their corresponding Galois theories. To be precise, we will prove the following theorem.
\begin{thm} \label{T5.1} Let $\mathcal{H}=(H_{L},H_{R},S)$ be a Hopf algebroid over $A=C(X)$ for some compact Hausdorff space $X$. Let $\mathscr{H}$ be the corresponding topological coupled Hopf category of $\mathcal{H}$. Then $\mathcal{H}$-Galois extensions of $A$ corresponds bijectively to $\mathscr{H}$-Galois extensions of $\mathbb{1}^{X}$. \end{thm}
Before proving the above theorem, let us comment on what we mean by Galois extension by a (topological) coupled Hopf category $\mathscr{H}=(\mathscr{H}_{L},\mathscr{H}_{R},S)$. By this, we mean an inclusion of categories $\mathbb{1}^{X}\subseteq \mathscr{M}$ which is simultaneously $\mathscr{H}_{L}$-Galois and $\mathscr{H}_{R}$-Galois in the sense of section (\ref{S3.3}). Note that by definition (\ref{D3.3}), we are not requiring $\mathscr{H}_{L}$ and $\mathscr{H}_{R}$ to be Hopf categories (individually, they are only $C(\mathcal{V})$-enriched categories). In particular, they do not necessarily have antipodes. Fortunately, Galois extension in the sense described in section (\ref{S3.3}) does not really make use of the antipode.
\begin{prf} Let $B$ be a (left) $\mathcal{H}$-Galois extension of $A$. In particular, $B$ is an $A$-ring. Note that the arguments we used in sections (\ref{S4.1}) and (\ref{S4.2}) only use the $A$-ring structure of the Hopf algebroid $\mathcal{H}$. Using the same arguments, $B\cong\Gamma(X\times X, \mathcal{B})$ where $\mathcal{B}$ is a sheaf of vector spaces over $X\times X$. By the Galois condition, we see that $\mathcal{B}$ has the same support $Z\subseteq X\times X$ as the sheaf $\mathcal{E}$ we get from either $H_{L}$ or $H_{R}$. Similar to remark (\ref{R4.4}), we get a small category $\mathscr{B}$ over $X$ enriched over $\mathcal{V}$ whose associated sheaf is $\mathcal{B}$.
The (right) $H_{L}$-coaction $\rho_{L}:B\longrightarrow B\otimes_{A}H$ induces a map $\mathcal{B}\longrightarrow \mathcal{B}\prescript{}{X}{\times}_{X}\mathcal{E}$ of sheaves of $\mathcal{O}_{X}$-bimodules over $X\times X$. By definition, $B$ is a right $A$-module and a right $A^{op}$-module. Using this, the $A$-bimodule structure on $B$ is as follows:
\[ a\cdot b\cdot a^{'}=b(aa^{'}) \]
\noindent for any $a,a^{'}\in A$ and $b\in B$. Similar to (\ref{eq4.5}), the right $H_{L}$-coaction induces, for every $(x,y)\in Z$, linear maps
\begin{equation}\label{eq5.1} \xymatrix{ B_{(x,y)} \ar[rr]^-{(\rho_{L})_{*}^{(x,y)}} && \bigoplus\limits_{z^{'},z^{''}\in X} B_{(z^{'},y)}\otimes E_{(z^{''},y)}} \end{equation}
\noindent where $B_{(x,y)}$ the fiber of $\mathcal{B}$ at the point $(x,y)$. As before, $E_{(x,y)}$ denotes the fiber of $\mathcal{E}$ over $(x,y)$. Likewise, the right $H_{R}$-coaction $\rho_{R}$ induces linear maps
\begin{equation}\label{eq5.2} \xymatrix{ B_{(x,y)} \ar[rr]^-{(\rho_{R})_{*}^{(x,y)}} && \bigoplus\limits_{z^{'},z^{''}\in X} B_{(x,z^{'})}\otimes E_{(x,z^{''})}} \end{equation}
By diagram (\ref{eq2.1}), we have
\begin{equation}\label{eq5.3} \xymatrix{ B_{(x,y)} \ar[rrrr]^-{\left(\rho_{L}\right)_{*}^{(x,y)}} \ar[ddd]_-{\left(\rho_{R}\right)_{*}^{(x,y)}} &&&& \bigoplus\limits_{z^{'},z^{''}}\left(B_{(x,z^{'})}\otimes E_{(x,z^{''})}\right) \ar[dd]^-{\bigoplus\limits_{z^{'}}\left(\rho_{R}\right)_{*}^{(x,z^{'})}\otimes id}\\ &&&& \\ &&&& \bigoplus\limits_{z^{'},z^{''}}\bigoplus\limits_{\alpha^{'},\alpha^{''}}\left(B_{(\alpha^{'},z^{'})}\otimes E_{(\alpha^{''},z^{'})}\otimes E_{(x,z^{''})}\right) \ar@{=}[d] \\ \bigoplus\limits_{\beta^{'},\beta^{''}}\left(B_{(\beta^{'},y)}\otimes E_{(\beta^{''},y)}\right) \ar[rrrr]_-{\bigoplus\limits_{\beta^{''}}id\otimes\left(\Delta_{L}\right)_{*}^{(\beta^{''},y)}} &&&& \bigoplus\limits_{\beta^{'},\beta^{''}}\bigoplus\limits_{\gamma^{'},\gamma^{''}}\left(B_{(\beta^{'},y)}\otimes E_{(\beta^{''},\gamma^{'})}\otimes E_{(\beta^{''},\gamma^{''})}\right)\\ } \end{equation}
Meanwhile, counitality of the left coaction $\rho_{L}$ implies that
\[ \xymatrix@R=2mm{ & & \bigoplus\limits_{z^{'},z^{''}}\left(B_{(x,z^{'})}\otimes E_{(x,z^{''})}\right) \ar[ddddd]^-{\bigoplus\limits_{z^{''}}id\otimes\left(\epsilon_{L}\right)_{*}^{(z,y)}} \\ & & \\ & & \\ & & \\ & & \\ B_{(x,y)} \ar[rruuuuu]^-{\left(\rho_{L}\right)_{*}^{(x,y)}} \ar@{=}[rr] & & \bigoplus\limits_{z^{'}}\left(B_{(x,z^{'})}\otimes\mathbb{C}\right) \\
v \ar@{|->}[rr] && v\otimes 1 \\} \]
\noindent from which, using a similar argument we to the proof of proposition (\ref{P4.12})(1), gives
\[ \xymatrix{ B_{(x,y)} \ar[rr]^-{(\rho_{L})_{*}^{(x,y)}} && B_{(x,y)}\otimes E_{(x,y)}}. \]
\noindent Similarly, we have
\[ \xymatrix{ B_{(x,y)} \ar[rr]^-{(\rho_{R})_{*}^{(x,y)}} && B_{(x,y)}\otimes E_{(x,y)}}. \]
\noindent These tell us that $\mathscr{B}$ is a right $\mathscr{H}_{L}$- and a right $\mathscr{H}_{R}$-comodule. The composition $\circ$ in $\mathscr{B}$ is induced by the $A$-product on $B$. By equations (\ref{eq2.2}) to (\ref{eq2.5}), this composition $\circ$ is a map of right $\mathscr{H}_{L}$- and a right $\mathscr{H}_{R}$-modules. Thus, $\mathscr{B}$ is a right $\mathscr{H}_{L}$- and a right $\mathscr{H}_{R}$-comodule-category. It is not hard to see that the right coactions of $\mathscr{H}_{L}$ and $\mathscr{H}_{R}$ on $\mathscr{B}$ are both Galois whose subcategories of coinvariants are both the same as $I_{X}$. These imply that $\mathscr{B}$ is a Galois extension of $I_{X}$ by the topological coupled Hopf category $\mathscr{H}=(\mathscr{H}_{L},\mathscr{H}_{R},S)$.
The inverse of this correspondence is easily seen as the the one that associates to an $(\mathscr{H}_{L},\mathscr{H}_{R},S)$-Galois extension $I_{X}\subseteq \mathscr{B}$ the $(H_{L},H_{R},S)$-Galois extension $A\subseteq B$ where $H_{L},H_{R},B$ and $A$ are the space of global sections of the associated sheaves to $\mathscr{H}_{L},\mathscr{H}_{R},\mathscr{B}$ and $I_{X}$, respectively. The compatibility conditions in the categorical side precisely correspond to the analogous compatibility conditions in the algebraic side. $\blacksquare$ \end{prf}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Classification of elliptic and K3 fibrations birational to some $\mathbb{Q}$-Fano 3-folds} \author{Daniel Ryder\footnote{Email: \texttt{[email protected]}}} \date{August 2005} \maketitle
\begin{abstract} {\sloppy \noindent A complete classification is presented of elliptic and K3 fibrations birational to certain mildly singular complex Fano 3-folds. Detailed proofs are given for one example case, namely that of a general hypersurface $X$ of degree 30 in weighted $\mathbb{P}^4$ with weights 1,4,5,6,15; but our methods apply more generally. For constructing birational maps from $X$ to elliptic and K3 fibrations we use Kawamata blowups and Mori theory to compute anticanonical rings; to exclude other possible fibrations we make a close examination of the strictly canonical singularities of $\XnH$, where $\mathcal{H}$ is the linear system associated to the putative birational map and $n$ is its anticanonical degree. } \end{abstract}
\section{Introduction}
In \cite{CPR} Corti, Pukhlikov and Reid proved that a general quasismooth complex variety \linebreak $X = X_d \subset \mathbb{P}(1,a_1,\ldots,a_4)$ in one of the `famous~95 families' of $\mathbb{Q}$-Fano 3-fold weighted hypersurfaces is \emph{birationally rigid} --- that is, if $X$ is birational to some Mori fibre space $Y/S$ then in fact $Y \simeq X$. A related problem is to classify elliptic and K3 fibrations birational to general hypersurfaces in these families; it was Ivan Cheltsov \cite{Ch00} who first proved classification results of this kind for several birationally rigid smooth Fano varieties, including a general quartic 3-fold $X_4 \subset \mathbb{P}^4$ and a double cover of $\mathbb{P}^3$ branched in a general sextic surface, i.e., $X = X_6 \subset \mathbb{P}(1,1,1,1,3)$. In \cite{Ry02} the classification of elliptic and K3 fibrations birational to general members of the remaining 93 families was addressed, but completed only for family~5, $X_7 \subset \mathbb{P}(1,1,1,2,3)$. In the present paper we aim firstly to give concise proofs of some of the more generally applicable results of \cite{Ry02} and secondly to present a complete proof of the following theorem for family 75, which is the family referred to in the abstract. Furthermore, we state similar theorems for families 34, 88 and 90; these can be proved using essentially the same techniques.
\begin{thm} \label{thm:75main} Let $X = X_{30} \subset \mathbb{P}(1,4,5,6,15)_{x,y,z,t,u}$ be a general member of family~75 of the~95. \begin{itemize} \item[(a)] Suppose $\Phi \colon X \dashrightarrow Z/T$ is a birational map from $X$ to a K3 fibration $g \colon Z \to T$ (see~\ref{defns:fibr} below for our assumptions on K3 fibrations, and also on elliptic fibrations and Fano 3-folds). Then there exists an isomorphism $\mathbb{P}^1 \to T$ such that the diagram below commutes, where $\pi = (x^4,y) \colon X \dashrightarrow \mathbb{P}^1$. \[ \xymatrix@C=1.6cm{ X \ar@{-->}[r]^{\Phi} \ar@{-->}[d]_{\pi} & Z \ar[d]^g \\ \mathbb{P}^1 \ar[r]^{\simeq} & T \\ } \] \item[(b)] There does not exist an elliptic fibration birational to $X$. \item[(c)] If $\Phi \colon X \dashrightarrow Z$ is a birational map from $X$ to a Fano 3-fold $Z$ with canonical singularities then $\Phi$ is actually an isomorphism (so in particular $Z \simeq X$ has terminal singularities). \end{itemize} \end{thm}
Part (b) of this theorem was recently proved independently by Cheltsov and Park~\cite{CP05} using somewhat different methods. It is an interesting result because of its relevance to the question of whether $\mathbb{Q}$-rational points of $X$ are potentially dense: a birational elliptic fibration is one key geometric construction used to prove potential density (see~\cite{HT00}, \cite{Ha03} and \cite{HT01}). The proof presented here requires close examination of $\CSXnH$ (see~\ref{comments_on_excl_sec}), where $\mathcal{H}$ is the linear system associated to a putative birational map and $n$ is its anticanonical degree; in~\cite{CP05} more general methods are used. Our approach has the advantage that the other parts of Theorem~\ref{thm:75main} follow immediately from our complete classification of possible sets of strictly canonical centres.
The following results are analogous to Theorem~\ref{thm:75main}; they can be proved with the same techniques, though we do not include all the details here. From now on we abbreviate conclusions such as that in Theorem~\ref{thm:75main}(a) by stating that \emph{up to a birational twist of the base,} $g \circ \Phi = (x^4,y) \colon X \dashrightarrow \mathbb{P}^1$. The birational twist $\mathbb{P}^1 \to T$ of the base is an isomorphism in the above case because $T$ is a smooth curve. (We assume all fibrations are morphisms of normal varieties: see~\ref{defns:fibr}.)
\begin{thm} \label{thm:34main} Let $X = X_{18} \subset \mathbb{P}(1,1,2,6,9)_{x_0,x_1,y,z,t}$ be a general member of family~34 of the~95. \begin{itemize} \item[(a)] If $\Phi \colon X \dashrightarrow Z/T$ is a birational map from $X$ to a K3 fibration $g \colon Z \to T$ then, up to a birational twist of the base (see above for explanation), $g \circ \Phi = (x_0,x_1) \colon X \dashrightarrow \mathbb{P}^1$. \item[(b)] Suppose $\Phi \colon X \dashrightarrow Z/T$ is a birational map from $X$ to an elliptic fibration $g \colon Z \to T$. Then, up to a birational twist of the base, $g \circ \Phi = (x_0,x_1,y) \colon X \dashrightarrow \mathbb{P}(1,1,2)$. \item[(c)] If $\Phi \colon X \dashrightarrow Z$ is a birational map from $X$ to a Fano 3-fold $Z$ with canonical singularities then $\Phi$ is actually an isomorphism (so in particular $Z \simeq X$ has terminal singularities). \end{itemize} \end{thm}
\begin{thm} \label{thm:88main} Let $X = X_{42} \subset \mathbb{P}(1,1,6,14,21)_{x_0,x_1,y,z,t}$ be a general member of family~88 of the~95. Under assumptions corresponding to those in Theorem~\ref{thm:34main} we can conclude as follows. \begin{itemize} \item[(a)] Up to a birational twist of the base, $g \circ \Phi = (x_0,x_1) \colon X \dashrightarrow \mathbb{P}^1$. \item[(b)] Up to a birational twist of the base, $g \circ \Phi = (x_0,x_1,y) \colon X \dashrightarrow \mathbb{P}(1,1,6)$. \item[(c)] $\Phi$ is actually an isomorphism. \end{itemize} \end{thm}
\begin{thm} \label{thm:90main} Let $X = X_{42} \subset \mathbb{P}(1,3,4,14,21)_{x,y,z,t,u}$ be a general member of family~90 of the~95. Under assumptions corresponding to those in Theorem~\ref{thm:34main} we can conclude as follows. \begin{itemize} \item[(a)] Up to a birational twist of the base, $g \circ \Phi = (x,y) \colon X \dashrightarrow \mathbb{P}(1,3)$. \item[(b)] Up to a birational twist of the base, $g \circ \Phi = (x,y,z) \colon X \dashrightarrow \mathbb{P}(1,3,4)$. \item[(c)] $\Phi$ is actually an isomorphism. \end{itemize} \end{thm}
\subsection*{Outline of paper}
Our proof of Theorem~\ref{thm:75main} has essentially three parts; this division of the argument is closely modelled on the approach of Cheltsov to similar problems for smooth varieties --- see e.g.~\cite{Ch00}. In brief, the parts are: constructing the K3 fibration birational to our $X$ in family~75 (see~\S\ref{sec:constr}); proving a technical result, Theorem~\ref{thm:75aux}, using exclusion arguments (\S\ref{sec:excl}); and deriving Theorem~\ref{thm:75main} from Theorem~\ref{thm:75aux} (in~\S\ref{sec:class}) using an analogue of the Noether--Fano--Iskovskikh inequalities (\ref{nfi}) together with an adaptation of the framework of~\cite{Ch00}. We now make some comments on each of these three.
\begin{emp} In \S\ref{sec:constr} we show that the projection $(x^4,y) \colon X \dashrightarrow \mathbb{P}^1$ is indeed a K3 fibration, after resolution of indeterminacy. The construction is Mori-theoretic: we make a Kawamata blowup of $X$ and play out the two ray game. We also outline constructions of the elliptic fibrations in Theorems~\ref{thm:34main}(b),~\ref{thm:88main}(b) and~\ref{thm:90main}(b). \end{emp}
\begin{comments_on_excl_sec} \label{comments_on_excl_sec} First let us state the technical theorem mentioned above, which is proved in~\S\ref{sec:excl}. We need the following.
\begin{notn*} Let $X$ be a normal complex projective variety, $\mathcal{H}$ a mobile linear system on $X$ and $\alpha \in \mathbb{Q}_{\ge 0}$. We denote by $\operatorname{CS}(X,\alpha\mathcal{H})$ the set of centres on $X$ of valuations that are strictly canonical or worse for $K_X + \alpha\mathcal{H}$ --- that is, \[ \operatorname{CS}(X,\alpha\mathcal{H}) = \{\operatorname{Centre}_X(E) \mid a(E,X,\alpha\mathcal{H}) \le 0\}. \] Occasionally we also use $\operatorname{LCS}(X,\alpha\mathcal{H})$, which is defined similarly as \[ \operatorname{LCS}(X,\alpha\mathcal{H}) = \{\operatorname{Centre}_X(E) \mid a(E,X,\alpha\mathcal{H}) \le {-1}\}. \] \end{notn*}
\begin{thm} \label{thm:75aux} Let $X = X_{30} \subset \mathbb{P}(1,4,5,6,15)_{x,y,z,t,u}$ be a general member of family~75 of the~95. Suppose $\mathcal{H}$ is a mobile linear system of degree $n$ on $X$ with $\KXnH$ nonterminal. Then in fact $\KXnH$ is strictly canonical and $\CSXnH = \{Q_1,Q_2\}$, where $Q_1,Q_2 \sim \frac{1}{5}(1,4,1)_{x,y,t}$ are the two singularities of $X$ on the $zu$-stratum. \end{thm}
For an introduction to the relationship between this result and Theorem~\ref{thm:75main}, see~\ref{comments_on_final_sec} below. Our proof of~\ref{thm:75aux} in~\S\ref{sec:excl} is by exclusion arguments, as mentioned above --- e.g., we show in Theorem~\ref{thm:smpts} that no smooth point can be a centre on $X$ of a valuation strictly canonical for $\KXnH$; we refer to this as \emph{excluding\/} any smooth point. As well as \emph{absolute\/} exclusions such as this, there is an interesting \emph{conditional\/} exclusion result for singular points, Theorem~\ref{thm:Tmethodcond}. Some of the exclusion results of~\S\ref{sec:excl} are extensions of arguments in~\cite{CPR}, but Theorem~\ref{thm:Tmethodcond} is an example of strikingly new behaviour, and there are substantial differences in method also for exclusion of curves~(\S\ref{subsec:curves}). It should be noted that, though the main aim of~\S\ref{sec:excl} is to prove Theorem~\ref{thm:75aux}, several of the results obtained apply to many of the~95 families other than number~75 (in particular, numbers~34,~88 and~90) --- and the techniques of proof apply more generally still. \cite[App.\ A]{Ry02} contains a detailed list of results analogous to Theorem~\ref{thm:75aux} for most of the~95 families, including many for which only a conjectural birational classification of elliptic and K3 fibrations is currently known. In the case of family~34, for instance, we have the following.
\begin{thm} \label{thm:34aux} Let $X = X_{18} \subset \mathbb{P}(1,1,2,6,9)_{x_0,x_1,y,z,t}$ be a general member of family~34 of the~95. Suppose $\mathcal{H}$ is a mobile linear system of degree $n$ on $X$ with $\KXnH$ nonterminal. Then in fact $\KXnH$ is strictly canonical and $\CSXnH$ is either $\{C,P,Q_1,Q_2,Q_3\}$ or $\{P\}$. Here $C = \{x_0=x_1=0\}\cap X$ is irreducible by generality of $X$, and $P$, $Q_1$, $Q_2$ and $Q_3$ are the singularities of $X$; $P \sim \frac{1}{3}(1,1,2)_{x_0,x_1,y}$ lies on the $zt$-stratum and $Q_1,Q_2,Q_3 \sim \frac{1}{2}(1,1,1)_{x_0,x_1,t}$ lie on the $yz$-stratum. \end{thm}
\noindent Like Theorem~\ref{thm:34main}, this result is not proved in this paper, but the techniques for doing so, and for proving analogues for families~88 and~90, are essentially those we use below for Theorem~\ref{thm:75aux}. \end{comments_on_excl_sec}
\begin{comments_on_final_sec} \label{comments_on_final_sec} In a sense the relationship between Theorems~\ref{thm:75aux} and~\ref{thm:75main} is obvious: the K3 fibration given by $(x^4, y)$ has the singularities $Q_1$ and $Q_2$ of~\ref{thm:75aux} as centres, and no other set of centres is possible --- so if, say, we try to grow an elliptic fibration birational to $X$, we must start with an extremal extraction of $Q_1$ or $Q_2$, but then we find we have to extract the other $Q_i$ as well. It turns out that to make this rigorous we need some abstract machinery --- in particular, results concerning the log Kodaira dimension of $(X,(\frac{1}{n} + \varepsilon)\mathcal{H})$ for small $\varepsilon$. It should be noted that for many of the~95 families an argument such as the above does not apply directly, because sets of centres do not distinguish objects of interest: for example, \cite{Ry02} contains many examples of $\mathbb{Q}$-Fanos $X$ birational to, say, an elliptic fibration and also a Fano with canonical singularities, and with the two linear systems having the same $\operatorname{CS}$ on $X$.
To describe the approach of~\S\ref{sec:class} we consider a more abstract setup. Let $X$ be a Mori Fano variety (see~\ref{defns:Mfs} below), $Z$ a variety with canonical singularities and $\Phi \colon X \dashrightarrow Z$ a birational map. Assume furthermore that one of the following holds.
(a) $g \colon Z \to T$ is a $K$-trivial fibration (see~\ref{defns:fibr})
with $0 < \dim T < \dim Z$. Let $\mathcal{H}_Z = g^*|H|$ be the pullback of a very ample complete linear system of Cartier divisors on $T$ and $\mathcal{H}$ its birational transform on $X$.
(b) $Z$ is a Fano variety with canonical singularities
(see~\ref{defns:Mfs}) and $\Phi$ is not an isomorphism. Let $\mathcal{H}_Z = |H|$ be a very ample complete linear system of Cartier divisors on $Z$ and $\mathcal{H}$ its birational transform on $X$.
In either case (a) or (b), define $n \in \mathbb{Q}$ by $\KXnH \sim_\QQ 0$.
\begin{nfi}[{\cite{Ch00}}] \label{nfi} In either of the above situations, $\KXnH$ is nonterminal, that is, strictly canonical or worse. \end{nfi}
This is a standard result used in~\cite{Ch00} and~\cite{Is01}; but we give a proof of it in~\S\ref{sec:class}, under the assumptions~(a), because in this situation it follows from results we need anyway. Clearly~\ref{nfi} is motivation enough to address Theorem~\ref{thm:75aux} on the way towards proving Theorem~\ref{thm:75main}, but more work is needed to complete the proof of the latter. \S\ref{sec:class} contains the necessary arguments; two of the propositions are results of Cheltsov (though we give a proof of one of them), but in order to conclude we need a rather delicate argument that traces a log Kodaira dimension through a two ray game diagram. \end{comments_on_final_sec}
\subsection*{What is special about families 34, 75, 88 and 90?}
It is natural to ask why we are able to prove Theorems~\ref{thm:75main},~\ref{thm:34main},~\ref{thm:88main} and~\ref{thm:90main} for families~75,~34,~88 and~90, but are not able to prove similar theorems for other families out of the~95 with the same methods. There is no one answer to this, but the following are major factors. \begin{itemize} \item General members of families~34,~75,~88 and~90 are \emph{superrigid}, i.e., there are no nonautomorphic birational selfmaps. For nonsuperrigid families (the majority of the~95) one obtains much bigger anticanonical rings after Kawamata blowups of the singularities that are centres of involutions: see~\cite{CPR}. This makes impossible any direct generalisation of arguments such as that in our final proof of Theorem~\ref{thm:75main}(b). \item As already mentioned, it frequently occurs for other families out of the~95 that sets of centres on $X$ do not distinguish between different birational maps to elliptic or K3 fibrations or to Fanos with canonical singularities. \cite[Ch.\ 4]{Ry02} discusses this phenomenon in some detail using family~5, $X_7 \subset \mathbb{P}(1,1,1,2,3)$, as an extended example. Whilst the analogue of Theorem~\ref{thm:75main} is eventually proved for this family in \cite{Ry02}, the proof requires complicated exclusion arguments on blown up models of $X$, and there is no obvious way of avoiding these. See~\cite{CM04} for a situation that is in some ways analogous. \end{itemize}
The recent work of Cheltsov and Park~\cite{CP05} proves a number of results that complement those here: they show, for example, that a general member of any of the~95 families is birational to a K3 fibration, but do not classify the K3 fibrations so obtained; they also prove that a general member of family~$N$ is not birational to an elliptic fibration if and only if $N \in \{3,75,84,87,93\}$ and, for some 23 values of $N$ not in this set, they show that up to a birational twist of the base there is a unique birational elliptic fibration: it is obtained by projection $(x_0,x_1,x_2)$ onto the first three coordinates~\cite[4.13]{CP05}. They do not, however, prove full analogues of our Theorem~\ref{thm:75aux} for these families; this is one reason why they cannot classify birational K3 fibrations. Our methods do permit K3 fibrations to be classified, but only in cases where whenever we blow up canonical centres not excluded by general results we obtain small, manageable anticanonical rings. With our current technology this restricts us to families such as~34,~75,~88 and~90, where there are few birational maps to elliptic and K3 fibrations and no nontrivial birational maps to Fanos with canonical singularities.
\subsection*{Conventions and assumptions}
Our notations and terminology are mostly as in, for example,~\cite{KM}, but we list here some conventions that are nonstandard, together with assumptions that will hold throughout.
\begin{emp} All varieties considered are complex, and they are projective and normal unless otherwise stated. \end{emp}
\begin{emp} For details on the famous~95 families, see~\cite{Fl00} and~\cite{CPR}. In brief, $X_d \subset \mathbb{P}(1,a_1,\ldots,a_4)$ belongs to one of the families if (1) $X$ is \emph{quasismooth}, i.e., its singularities are all quotient singularities forced by the weighted $\mathbb{P}^4$; (2) the singularities are \emph{terminal} --- 3-fold terminal quotient singularities are necessarily of the form $\frac{1}{r}(1,a,r-a)$ with $r \ge 1$ and $(a,r) = 1$; and (3) $a_1 + \cdots + a_4 = d$, so by the adjunction formula ${-K_X} = \mathcal O_X(1)$ is very ample. Whenever $X$ is a member of one of the~95 families, we let $A = {-K_X} = \mathcal O_X(1)$ denote the positive generator of the class group; moreover, if $f \colon Y \to X$ is a birational morphism then $B$ denotes ${-K_Y}$. \end{emp}
\begin{emp} As in~\cite{CPR} and~\cite{Ry02}, we refer to the weighted blowup with weights $\frac{1}{r}(1, a, r-a)$ of a 3-fold terminal quotient singularity $\frac{1}{r}(1, a, r-a)$ with $r \ge 2$ and $(a, r) = 1$ as the \emph{Kawamata blowup\/} --- see~\cite{Ka96}, the main theorem of which is reproduced here as Theorem~\ref{thm:Ka}. \end{emp}
\begin{defns} \label{defns:Mfs} A \emph{Mori fibre space\/} $f \colon X \to S$ is a Mori extremal contraction of fibre type, that is, $\dim S < \dim X$. This means that $X$ and $S$ are projective varieties, $X$ has $\mathbb{Q}$-factorial, terminal singularities, $f_*\mathcal O_X = \mathcal O_S$, $\rho(X/S) = 1$ and ${-K_X}$ is $f$-ample. If $S = \{*\}$ is a point then $X$ is a \emph{Mori Fano variety}. We use the term \emph{Fano variety\/} more generally to refer to any normal, projective variety $X$ with $-K_X$ ample and $\rho(X) = 1$. \end{defns}
\begin{defns} \label{defns:fibr} Let $Z$ be a normal projective variety with canonical singularities. A \emph{fibration\/} is a morphism $g \colon Z \to T$ to another normal projective variety $T$ such that $\dim T < \dim Z$ and $g_*\mathcal O_Z = \mathcal O_T$. We say the fibration is \emph{$K$-trivial\/} if and only if $K_Z C = 0$ for every contracted curve~$C$. $g$ is an \emph{elliptic
fibration}, resp.\ a \emph{K3 fibration}, if and only if its general fibre is an elliptic curve, resp.\ a K3 surface. \end{defns}
\begin{emp} Usually when we write an equation explicitly or semi-explicitly in terms of coordinates we omit scalar coefficients of monomials; this is the `coefficient convention'. \end{emp}
\subsection*{Acknowledgments}
Most of the techniques in this paper, and some of the theorems, are from my PhD thesis,~\cite{Ry02}. I would like to thank my supervisor, Miles Reid, and also Gavin Brown, Alessio Corti and Hiromichi Takagi, for their help and generosity with their ideas; I would also like to thank Ivan Cheltsov for his helpful comments during the final stages of preparing this paper. My PhD studies were supported financially by the British EPSRC.
\section{Constructions} \label{sec:constr}
\subsection{K3 fibrations}
The following observation does not apply to family~75, our main object of study, but it does apply to families~34 and~88; in any case, it needs to be noted because it describes all `easy' K3 fibrations birational to members of the famous 95 --- cf.~Lemma~\ref{lem:75tworay} for the `hard' case.
\begin{prop} \label{prop:easyK3s} Let $X = X_d \subset \mathbb{P}(1,1,a_2,a_3,a_4)$ be general in one of the families with $a_1 = 1$ and $a_2 > 1$. Then a general fibre $S$ of $\pi = (x_0,x_1) \colon X \dashrightarrow \mathbb{P}^1$ is a quasismooth Du Val K3 surface and, setting $\mathcal P$ to be the pencil $\left<x_0,x_1 \right>$, we have \[ \operatorname{CS}(X,\mathcal P) = \{C,P_1,\ldots,P_r\}, \] where $C$ is the curve $\{x_0 = x_1 = 0\} \cap X$, which is irreducible by generality of $X$, and $P_1,\ldots,P_r$ are all the singularities of $X$. \end{prop}
\begin{proof}[\textsc{Proof}]
Because $S$ is a general element of $|\mathcal O_X(1)|$, it is certainly quasismooth. The adjunction formula for $S_d \subset \mathbb{P}(1,a_2,a_3,a_4) =: \mathbb{P}$ gives $K_S = 0$ and the cohomology long exact sequence from \[ 0 \to \mathcal I_{S,\mathbb{P}}=\mathcal O_{\mathbb{P}}(-d) \to \mathcal O_{\mathbb{P}} \to \mathcal O_S \to 0, \] with the standard cohomology results for weighted projective space, gives $h^1(S,\mathcal O_S) = 0$. Therefore $S$ is a quasismooth Du Val K3 surface.
Let $f \colon Y \to X$ be the blowup of the ideal sheaf $\mathcal I_{C,X}$ of $C$ and $E \subset Y$ the unique exceptional divisor of $f$ which dominates $C$. Then clearly $m_E(\mathcal P) = a_E(K_X) = 1$, so $C \in \operatorname{CS}(X,\mathcal P)$. The fact that $P_1,\ldots,P_r \in \operatorname{CS}(X,\mathcal P)$ is a consequence of Corollary~\ref{cor:Kalemma:2} of Kawamata's Lemma (below). Therefore \[ \operatorname{CS}(X,\mathcal P) \supset \{C,P_1,\ldots,P_r\}; \] the reverse inclusion follows from Theorem~\ref{thm:smpts}. \end{proof}
\begin{rk} If $X = X_d \subset \mathbb{P}(1,1,1,a_3,a_4)$ is general in a family with $a_1 = a_2 = 1$ then clearly a general element $S$ of any pencil $\mathcal P \subset
|\mathcal O_X(1)|$ is a Du Val K3 surface, provided one can prove it is quasismooth. This can be fiddly, the problem being that while $X$ is general and $S \in \mathcal P$ is general, $\mathcal P$ must be able to be
\emph{any\/} pencil inside $|\mathcal O_X(1)|$. \end{rk}
In contrast to the situation considered above, for families $X_d \subset \mathbb{P}(1,a_1,\ldots,a_4)$ with $a_1 > 1$ it is not immediately clear whether there exist K3 fibrations birational to $X$: to construct them, or at least to make sense of the construction, we need Mori theory. Here we consider only family~75, but the technique applies to many other families --- see~\cite{Ry02} --- and, in particular, to family~90, the subject of our Theorem~\ref{thm:90main}.
\begin{lemma} \label{lem:75tworay} Let $X=X_{30} \subset \mathbb{P}(1,4,5,6,15)_{x,y,z,t,u}$ be a general member of family~75 of the~95; for the notations $P,Q_1,Q_2,R_1,R_2$ for the singularities of $X$, see Theorem~\ref{thm:75aux}. Let $Q_i \in X$ be either $Q_1$ or $Q_2$ and $f \colon Y \to X$ the Kawamata blowup of $Q_i$. \begin{itemize} \item[(1)] Let $R \subset \operatorname{\overline{NE}} Y$ be the ray with $\operatorname{cont}_R = f$. Then the other ray $Q \subset \operatorname{\overline{NE}} Y$ is contractible and its contraction $g = \operatorname{cont}_Q \colon Y \to Z$ is antiflipping. \item[(2)] The antiflip $Y \dashrightarrow Y'$ of $g$ exists and $Y'$ has canonical singularities. \item[(3)] Let $Q' \subset \operatorname{\overline{NE}} Y'$ be the ray whose contraction is $g' \colon Y' \to Z$ and $R' \subset \operatorname{\overline{NE}} Y'$ the other ray. Then $R'$
is contractible and its contraction $f' \colon Y' \to \mathbb{P}^1$, which is in fact the anticanonical morphism $\varphi_{|{-4K_{Y'}}|} = \varphi_{|4B'|}$, is a K3 fibration --- that is, a general fibre $T'$ of $f'$ has Du Val singularities, $K_{T'} =0$ and $h^1(T',\mathcal O_{T'}) = 0$. \end{itemize} It follows that the total composite $X \dashrightarrow \mathbb{P}^1$ of the two ray game we have played, illustrated below, is $\pi = (x,y) \colon X \dashrightarrow \mathbb{P}(1,4) = \mathbb{P}^1$. \[ \xymatrix@C=0.4cm{
& Y \ar[dl]_f \ar[dr]^g & & Y' \ar[dl]_{g'} \ar[dr]^{f'} & \\
X & & Z & & \mathbb{P}^1 \\ } \] Therefore $R(Y,B) = R(Y',B') = k[x,y]$. \end{lemma}
\begin{proof}[\textsc{Proof of \ref{lem:75tworay}}] (1) The first part of the following argument --- showing that the curve $C$ defined below generates $Q \subset \operatorname{\overline{NE}} Y$ --- is one case of \cite[5.4.3]{CPR}; but in \cite{CPR} this point $Q_i$ is excluded as a maximal centre by the test class method, so there is no need for the two ray game to be played out.
By generality of $X$, the curve $\{x=y=0\} \cap X$ is irreducible. $f \colon Y \to X$ is the $\frac{1}{5}(1,4,1)_{x,y,t}$ weighted blowup of the $\frac{1}{5}(1,4,1)_{x,y,t}$ point
$Q_i$. Let $S \in \left|B\right|$ be the unique effective surface and
$T \in \left|4B\right|$ a general element, where as always $B = {-K_Y}$. One can check explicitly, by looking at the three affine pieces of $Y$ locally over $Q_i$, that $C := S \cap T$ is an irreducible curve inside $Y$; this uses the generality of $X$. We also know that \[ B^3 = A^3 - \frac{1}{ra(r-a)} = \frac{1}{60} - \frac{1}{20} < 0, \] so in particular $BC = 4B^3 < 0$. It follows that the ray $Q \subset \operatorname{\overline{NE}} Y$ is generated by $C$ --- indeed, suppose this is not the case; then $C$ is in the interior of the 2-dimensional cone $\operatorname{\overline{NE}} Y$, so we can pick an effective 1-cycle $\sum_{i = 0}^p \alpha_i C_i$ that lies strictly between $Q$ and the half-line generated by $C$. This 1-cycle is $B$-negative, because $R$ is $B$-positive (i.e., $K$-negative) and
$BC < 0$; but $\operatorname{Bs}\left|4B\right|$ is supported on $C$ and therefore one of the $C_i$, say $C_0$, is in fact $C$. The geometry of the cone now implies that, after we subtract off $\alpha_0 C_0$, $\sum_{i =
1}^p \alpha_i C_i$ is again strictly between $Q$ and the half-line generated by $C$; so we can repeat the argument to deduce that some other $C_i$ is $C$ --- and of course this contradicts our initial (implicit) assumption that the $C_i$ were distinct. This argument has also shown that $C$ is the only irreducible curve in the ray $Q$.
We show $Q$ is contractible with a general Mori-theoretic trick --- afterwards, clearly, $g = \operatorname{cont}_Q$ is antiflipping, because $C$ is the only contracted curve and $K_Y C = {-BC} > 0$. Firstly note that $S$ has canonical singularities and so in particular is klt. We apply Shokurov's inversion of adjunction (see \cite[5.50]{KM}) to deduce that the pair $(Y,S)$ is plt. (In fact $K_Y + S$ is Cartier so the log discrepancy of any valuation for $(Y,S)$ is an integer, and therefore $(Y,S)$ is canonical.) But plt is an open condition so for $\varepsilon \in \mathbb{Q}_{>0}$, $\varepsilon \ll 1$, the pair $(Y,S + \varepsilon T)$ is plt as well. Now \[ (K_Y + S + \varepsilon T)C = \varepsilon TC = \varepsilon B^3 < 0, \] so $Q$ is contractible by the (log) contraction theorem for $(Y,S + \varepsilon T)$.
(2) This follows immediately from Mori's result \cite[20.11]{FA}: $S$ and $T$ are effective divisors, $T \sim 4S$ and $T \cap S = C$ is precisely the exceptional set of $g$. Since $S \sim B = {-K_Y}$ the antiflip of $g$ is precisely its `opposite with respect to $S$', to use the language of \cite{FA}. This can be constructed as the normalisation of the closure of the image of \[ g \times \pi_Y \colon Y \dashrightarrow Z \times \mathbb{P}^1, \] where $\pi_Y = \pi \circ f \colon Y \dashrightarrow \mathbb{P}(1,4) = \mathbb{P}^1$
corresponds to the pencil $\left<4S,T\right> = \left|4B\right|$.
Alternatively, since we have already observed that the ray $Q$ is $(K_Y + S + \varepsilon T)$-negative, (2) follows from Shokurov's general result that log flips of lc pairs exist in dimension 3.
(3) From the construction of the antiflip, the transforms $S'$ and $T'$ of $S$ and $T$ on $Y'$ are disjoint, so
$\operatorname{Bs}\left|{-4K_{Y'}}\right| = \operatorname{Bs}\left|4B'\right| =
\emptyset$. Therefore $f' = \operatorname{cont}_{R'}$ exists and is (the Stein factorisation of) $\varphi_{\left|4B'\right|}$; $T'$ is a general fibre of $f'$. In the diagram below, $\nu \colon U \to g(T)$ is the normalisation of $g(T)$. \[ \xymatrix@C=0.1cm{
& Y & \supset & T \ar[dl]_f \ar@/_0.3cm/[ddrr]_g \ar[drr]^h & & & &
T' \ar[dll]_{h'} \ar@/^0.3cm/[ddll]^{g'} \ar[dr]^{f'} & \subset &
Y'& \\ X & \supset & T_X & & & U \ar[d]^{\nu} & & & \{*\} & \subset
& \mathbb{P}^1 \\ & & & & & g(T) & & & & & \\ } \]
Now $K_T = (K_Y + T)|_T = 3B|_T = 3C$, so $K_U = h_*(3C) = 0$. Furthermore, the Leray spectral sequence for $f\colon T\to T_X$, \[ 0 \to 0 = H^1\left(T_X,\mathcal O_{T_X}\right) \to
H^1\left(T,\mathcal O_T\right) \to H^0\left(T_X,R^1f_*\mathcal O_T\right) = 0 , \] shows that $h^1(T,\mathcal O_T) = 0$ (here $h^0(T_X,R^1f_*\mathcal O_T) = 0$ because the singularity $\frac{1}{5}(1,1)$ of $T_X$ at $Q_i$ is rational); and now the Leray spectral sequence for $h \colon T \to U$, \[ 0 \to H^1\left(U,\mathcal O_U\right) \to H^1\left(T,\mathcal O_T\right) = 0 , \] shows that $h^1(U,\mathcal O_U) = 0$. The only thing left to do is to show that $U$ has Du Val singularities --- it is then clear that $T'$ is a Du Val K3 surface because
\[ K_{T'} = \left(K_{Y'} + T'\right)|_{T'} = 3B'|_{T'} = 0 , \] so the minimal resolution $\widetilde{U} \to U$ of $U$ factors through $h' \colon T' \to U$.
To show $U$ has Du Val singularities we observe first that $T$ has two singularities: a $\frac{1}{5}(1,1)_{x,t}$ point over $Q_j \in T_X$, where $\{i,j\} = \{1,2\}$, and a $\frac{1}{3}(1,2)_{x,z}$ point over $P \in T_X$. The curve $C$ passes through both and is locally defined by $x = 0$ in a neighbourhood of each. Let $\xi \colon \widetilde{T} \to T$ be the minimal resolution of $T$, with exceptional curves $E_1$, $E_2$ lying over $P$ and $E_3$ lying over $Q_j$, where $E_1^2 = E_2^2 = {-2}$, $E_1 E_2 = 1$, $E_1 \widetilde{C} = 0$, $E_2 \widetilde{C} = 1$, $E_3^2 = {-5}$ and $E_3 \widetilde{C} = 1$ (here $\widetilde{C}$ is the birational transform of $C$ on $\widetilde{T}$). We can calculate that $(\widetilde{C})^2 < 0$ and $K_{\widetilde{T}}\widetilde{C} < 0$, from which it follows that $\widetilde{C}$ is a $-1$-curve. (To do this, first show that $C^2_T < 0$ and $K_T C < 0$; then express $\widetilde{C}$ as the pullback $\xi^*C$ minus nonnegative multiples of $E_1,E_2,E_3$; then use the fact that $(E_i E_j)_{i,j = 1}^2$ is negative definite, $K_{\widetilde{T}}(\xi^*C) = K_T C < 0$ and $K_{\widetilde{T}}$ is $\xi$-nef by minimality of $\xi$.)
Now the minimal resolution $\widetilde{U}$ of $U$ is obtained from $\widetilde{T}$ by running a minimal model program over $U$ --- so we start by contracting $\widetilde{C}$, which is exceptional over $U$, and then $E_2$, which has become a $-1$-curve, and finally $E_1$. $E_3$ is left as a $-2$-curve. This MMP can be summarised by \[ (2,2,1,5) \to (2,1,4) \to (1,3) \to (2). \] It follows that $U$ has a single $A_1$ Du Val singularity, as required. \end{proof}
\subsection{Elliptic fibrations}
Our main aim is to prove Theorem~\ref{thm:75main}, part of which is the statement that there is no elliptic fibration birational to a general $X$ in family~75. In this subsection, however, we digress briefly to discuss the construction of elliptic fibrations birational to hypersurfaces in other families out of the~95. In particular we are concerned with families~34,~88 and~90, the subjects of Theorems~\ref{thm:34main},~\ref{thm:88main} and~\ref{thm:90main}.
\begin{ex} Let $X = X_{18} \subset \mathbb{P}(1,1,2,6,9)_{x_0, x_1, y, z, t}$ be a general member of family~34 of the~95. We claim that \begin{itemize} \item[(a)] the projection $\pi = (x_0, x_1, y) \colon X \dashrightarrow \mathbb{P}(1,1,2)$ gives an elliptic fibration birational to $X$ after resolution of indeterminacy; and \item[(b)] the indeterminacy may be resolved as shown below, where $f \colon Y \to X$ is the Kawamata blowup of the unique singularity $P \sim \frac{1}{3}(1,1,2)_{x_0,x_1,y}$ on the $zt$-stratum, and $\pi_Y := \pi
\circ f$ is the anticanonical morphism $\varphi_{|2B|}$, $B = {-K_Y}$. \[ \xymatrix@C=1.6cm{ Y \ar[rd]^(.4){\pi_Y} \ar[d]_{f} & \\ X \ar@{-->}[r]^(.4){\pi} & \mathbb{P}(1,1,2) \\ } \] \end{itemize}
\begin{proof}[\textsc{Proof}] (a) This is easy to see: a general fibre of the rational map $\pi$ is a curve $E_{18} \subset \mathbb{P}(1,6,9)_{x,z,t}$ and, when we write down the Newton polygon of the defining equation of such a curve, there is a unique internal monomial (namely $x^3 z t$, the vertices being $t^2$, $x^{18}$ and~$z^3$). Consequently $E_{18}$ is birational to an elliptic curve, by standard toric geometry.
(b) The linear system $\mathcal L$ defining $\pi$ is $|2A| = \left< x_0^2, x_1^2, y \right>$ and one can calculate directly that its birational transform $\mathcal L_Y$ on $Y$ is free. Furthermore \[ \mathcal L_Y = f^*\mathcal L - \textstyle\frac{2}{3} E \sim 2B \] and $\mathcal L_Y$ is clearly a complete linear system. \end{proof} \end{ex}
As well as exhibiting one elliptic fibration birational to a general hypersurface in family~34 (actually, according to Theorem~\ref{thm:34main}, the only such), this example shows us one way to look for elliptic fibrations when we consider other families: namely, find a singular point $P$ with $B^3 = 0$ (for $B = {-K_Y}$, $f \colon Y \to X$ the Kawamata blowup of $P$), take the anticanonical morphism on $Y$ (if $B$ is eventually free) and see if it maps to a surface. It turns out that this method works for families~88 and~90, as well as~34, so as far as the present paper is concerned, we are done. There are, however, other ways in which elliptic fibrations occur birational to hypersurfaces in the~95 families. Here is a brief list; see~\cite{Ry02} for more details. \begin{itemize} \item Sometimes elliptic fibrations have more than one singular point of $X$ in $\CSXnH$; they can be constructed by blowing up all these points and taking the anticanonical morphism. \item {\sloppy It can also occur that $\CSXnH$ consists of only one singular point $P$, but \linebreak $\operatorname{CS}\left( Y, \frac{1}{n}\mathcal{H}_Y \right) \ne \emptyset$, where $Y = \operatorname{B}_P X$ is the Kawamata blowup; in this case further blowups of $Y$ are necessary before taking the anticanonical morphism.} \item Finally, there are examples of elliptic fibrations with a curve in $\CSXnH$ --- but only for families~1 and~2; see~\cite{Ry02}. \end{itemize}
\section{Exclusions, absolute and conditional} \label{sec:excl}
For an initial introduction to the contents of this section, see~\ref{comments_on_excl_sec}. We divide the material for proving Theorem~\ref{thm:75aux} into three subsections: in~\S\ref{subsec:curves} we show that all curves are excluded absolutely and in~\S\ref{subsec:smpts} we prove the corresponding result for smooth points; finally in~\S\ref{subsec:singpts} we deal with singular points. Much of the material in this section applies more widely than to family~75: for example, out of all the singular points on members of the~95 families, we show that those satisfying a certain condition turn out to be excluded absolutely, while those satisfying a different (but closely related) condition are excluded \emph{conditionally} --- that is, we prove that if they belong to $\CSXnH$ for some $\mathcal{H}$ then other centres must also exist in $\CSXnH$.
\subsection{Curves} \label{subsec:curves}
First we state the main curve exclusion theorem proved in \cite{Ry02}.
\begin{thm}[{\cite[Curves Theorem A]{Ry02}}] \label{thmA} Let $X = X_d \subset \mathbb{P} = \mathbb{P}(1,a_1,a_2,a_3,a_4)$ be a general hypersurface in one of the 95 families and $C \subset X$ a reduced, irreducible curve. Suppose $\mathcal{H}$ is a mobile linear system of degree $n$ on $X$ such that $K_X + \frac{1}{n}\mathcal{H}$ is strictly canonical and $C \in \CSXnH$. Then there exist two linearly independent forms $\ell,\ell'$ of degree $1$ in $(x_0,\ldots,x_4)$ such that \begin{equation} C \subset \{\ell = \ell' = 0\} \cap X. \label{CinPi} \end{equation} \end{thm}
We do not reproduce the proof in full here, but instead restrict ourselves to the case we need, namely $X_{30} \subset \mathbb{P}(1,4,5,6,15)$. This follows immediately from the following lemmas, the first of which is standard.
\begin{lemma} \label{lem:degC} Let $X$ be any hypersurface in one of the 95 families and $C \subset X$ a curve, reduced but possibly reducible. Suppose $\mathcal{H}$ is a mobile linear system of degree $n$ on $X$ such that $K_X + \frac{1}{n}\mathcal{H}$ is strictly canonical and each irreducible component of $C$ belongs to $\CSXnH$. Then $\deg C = AC \le A^3$. \end{lemma}
\begin{lemma} \label{curvesinwps1} Let $X = X_d \subset \mathbb{P} = \mathbb{P}(1,a_1,a_2,a_3,a_4)$ be a hypersurface in one of the families with $a_1 > 1$ and suppose that either \begin{itemize} \item[(a)] $d < a_1a_4$ or \item[(b)] $d < a_2a_4$ and the curve $\{x = y = 0 \}\cap X$ is irreducible (which holds for general $X$ in a family with $a_1 > 1$ by Bertini's theorem). \end{itemize} Then any curve $C \subset X$ that is not contracted by $\pi_4 \colon X \dashrightarrow \mathbb{P}(1,a_1,a_2,a_3)$ has $\deg C > A^3$. Consequently $C$ is excluded absolutely by Lemma \ref{lem:degC}. \end{lemma}
For the proofs of Lemmas~\ref{lem:degC} and~\ref{curvesinwps1}, see below. It is straightforward to check that they imply the following.
\begin{cor} \label{cor:75curves} Let $X = X_{30} \subset \mathbb{P}(1,4,5,6,15)_{x,y,z,t,u}$ be any (quasismooth) member of family~75. Then Theorem~\ref{thmA} holds for $X$, that is, no reduced, irreducible curve $C \subset X$ can belong to $\CSXnH$ for any mobile system $\mathcal{H}$ of degree $n$ with $\KXnH$ strictly canonical. \end{cor}
\begin{proof}[\textsc{Proof of Lemma \ref{lem:degC}}] Let $s$ be a natural number such that $sA$ is Cartier and very ample, and pick general members $H,H' \in \mathcal{H}$. Now by assumption \[ \operatorname{mult}_{C_i}(H) = \operatorname{mult}_{C_i}(H') = n \] for each irreducible component $C_i$ of $C$, so for a general $S \in
\left|sA \right|$ \[ A^3 sn^2 = SHH' \ge sn^2 AC = sn^2 \deg C, \] which proves $\deg C \le A^3$. \end{proof}
\begin{proof}[\textsc{Proof of Lemma \ref{curvesinwps1}}] We prove this lemma under the additional assumption that $(a_1,a_2) = 1$ --- which is the case for family 75 ($a_1 = 4$, $a_2 = 5$); if $(a_1,a_2) > 1$ then a little trick, described in \cite{Ry02}, is needed.
Suppose that $C \subset X$ has $\deg C \le A^3$ and is not contracted by $\pi_4$; let $C' \subset \mathbb{P}(1,a_1,a_2,a_3)$ be the set-theoretic image $\pi_4(C)$. Note that $\deg C' \le \deg C$ --- indeed, if $H$ denotes the hyperplane section of $\mathbb{P}(1,a_1,\ldots,a_4)$ and $H'$ that of $\mathbb{P}(1,a_1,a_2,a_3)$, we pick
$s \ge 1$ such that $\left|sH\right|$ and $\left|sH'\right|$ are very ample, and calculate that \begin{eqnarray*} s\deg C & = & (sH)C = \pi_4^*(sH')C \\
& = & sH'(\pi_4)_*C = srH'C' = sr\deg C' \ge s\deg C', \end{eqnarray*}
where $r \ge 1$ is the degree of the induced morphism $\pi_4|_C \colon C \to C'$. So in fact $\deg C$ is a multiple of $\deg C'$ by the positive integer $r$.
(The point of $\left|sH\right|$ being very ample is that we can move it away from $P_4$, where $\pi_4$ is undefined, and apply the projection formula to the morphism $\pi_4|_{\mathbb{P}(1,a_1,\ldots,a_4) \smallsetminus \{P_4\}}$.)
Now form the diagram below. \[ \xymatrix@C=0.1cm{
C \ar[d] & \subset & \mathbb{P}(1,a_1,a_2,a_3,a_4) \ar@{-->}[d]^{\pi_4}\\
C' \ar[d] & \subset & \mathbb{P}(1,a_1,a_2,a_3) \ar@{-->}[d]^{\pi_3}\\
\{*\} & \subset & \mathbb{P}(1,a_1,a_2)\\ } \] $C'$ is contracted by $\pi_3$
--- indeed, if its image were a curve $C''$ we would have \[ \deg C'' \le \deg C' \le \deg C \le A^3, \] but $A^3 = d/(a_1a_2a_3a_4) < 1/(a_1a_2)$, since $d < a_3a_4$ in either case (a) or (b), and on the other hand $1/(a_1a_2) \le \deg C''$ simply because $C'' \subset \mathbb{P}(1,a_1,a_2)$ --- contradiction.
Now by our extra assumption $(a_1,a_2) = 1$, the point $\{*\} \subset \mathbb{P}(1,a_1,a_2)$ is, up to coordinate change, one of \[ \{y = z = 0\}, \quad \{y^{a_2} + z^{a_1} = x = 0\}, \quad \{x = z = 0\} \quad \mbox{and} \quad \{x = y = 0\}, \] using the coefficient convention in $y^{a_2} + z^{a_1} = 0$. It follows that the curve $C' \subset \mathbb{P}(1,a_1,a_2,a_3)$ is defined by the same equations. In the first case, this means that $\deg C' = 1/a_3 > d/(a_1a_2a_3a_4) = A^3$, contradiction. In the second case $\deg C' = 1/a_3$ again, because $C' \simeq \{y^{a_2} + z^{a_1} = 0\} \subset \mathbb{P}(a_1,a_2,a_3)$ passes only through the singularity $(0,0,1)$, using $(a_1,a_2) = 1$ --- so we obtain a contradiction as in the first case. In the case $C' = \{x = z = 0\}$, we have $\deg C' = 1/(a_1a_3)$ and we easily obtain a contradiction from $a_2a_4 > d$. In the final case, $C' = \{x = y = 0\}$, if the assumptions in part (a) of the statement hold then we have \[ \deg C' = 1/(a_2a_3) > d/(a_1a_2a_3a_4) = A^3, \] contradiction; while if the assumptions in part (b) hold then \[ C = \{x = y = 0 \}\cap X \] (because the right hand side is irreducible), but \[ \deg\left(\{x = y = 0 \}\cap X \right) = a_1A^3 > A^3, \] since we also assumed $a_1 > 1$ --- contradiction. \end{proof}
Note that for the case $a_1 = 1, a_2 > 1$ there is the following analogue of Lemma~\ref{curvesinwps1} --- see~\cite[3.3]{Ry02} for the proof, which is similar to but much shorter than the one we have just seen.
\begin{lemma} \label{curvesinwps2} Let $X = X_d \subset \mathbb{P} = \mathbb{P}(1,1,a_2,a_3,a_4)$ be a hypersurface in one of the families with $a_1 = 1$ and $a_2 > 1$; suppose that $d < a_2a_4$. Then any curve $C \subset X$ that is not contracted by $\pi_4$ and that satisfies $\deg C \le A^3$ is contained in $\{x_0 = x_1 = 0\} \cap X$. \end{lemma}
\noindent When $a_1 = a_2 = 1$, however, the situation is different and more work is required to prove sufficiently strong results.
We note also for completeness that in the case of family~75 we have $P_4 = P_u \not\in X$ so the question of whether $C$ is contracted by $\pi_4 \colon X \dashrightarrow \mathbb{P}(1,a_1,a_2,a_3)$ never arises. For families that do contain curves contracted by $\pi_4$, \cite[3.5]{Ry02} shows that in almost all cases these curves are of degree greater than $A^3$, so they are excluded by Lemma~\ref{lem:degC}. The remaining few families to which this result does not apply are also dealt with in \cite{Ry02}.
\begin{crvs_fams34etc} We have dealt with curves inside a general $X$ in family~75. The arguments presented above, or variants of them, deal also with curves in general members of families~88 and~90, given the following observation: family~88 has $a_1 = 1$ and $a_2 > 1$, and consequently there is a birational K3 fibration obtained by the projection $(x_0, x_1) \colon X \dashrightarrow \mathbb{P}^1$. The corresponding pencil $\mathcal P = \left< x_0, x_1 \right>$ has $C \in \operatorname{CS}(X, \mathcal P)$, where $C$ is the curve $\{ x_0 = x_1 = 0 \} \cap X$. The important point is that, by taking $X$ general in its family, $C$ is \emph{irreducible}. If it were not, and we had \[ \{ x_0 = x_1 = 0 \} \cap X = C_0 \cup \cdots \cup C_r, \] we would have to prove a conditional exclusion result of the form `if $C_0 \in \CSXnH$ then all $C_i \in \CSXnH$'. This can be done --- see~\cite[3.12]{Ry02} --- but we avoid it using our generality assumption on $X$.
The case of family~34 is more problematic because Lemma~\ref{curvesinwps2} above does not apply. We omit the argument for curve exclusion for this family; it can be found in~\cite[\S3.4]{Ry02}. \end{crvs_fams34etc}
\subsection{Smooth points} \label{subsec:smpts}
In this subsection we present a proof of the following theorem.
\begin{smptsthm} \label{thm:smpts} Let $X = X_d \subset \mathbb{P} = \mathbb{P}(1,a_1,\ldots,a_4)$ be a general hypersurface in one of families $3,4,\ldots,95$ and $P \in X$ a smooth point. For any $n \in \mathbb{Z}_{\ge1}$ and any mobile linear system $\mathcal{H}$ of degree $n$ on $X$ we have $P \not\in \CSXnH$. \end{smptsthm}
The proof closely follows the argument used in \cite{CPR} to exclude smooth points as maximal centres of a pair $\XnH$. First we need to quote some theoretical results.
\begin{thm}[Shokurov's inversion of adjunction] \label{invadj} Let $P \in X$ be the germ of a smooth 3-fold and $\mathcal{H}$ a mobile linear system on $X$. Assume $n \in \mathbb{Z}_{\ge 1}$ is such that $P \in \operatorname{CS}\left(X,\frac{1}{n}\mathcal{H}\right)$. Then for any normal irreducible divisor $S$
containing $P$ such that $\mathcal{H}|_S$ is mobile we have $P \in
\operatorname{LCS}\left(S,\frac{1}{n}\mathcal{H}|_S\right)$. \end{thm}
For a readable account of the proof see \cite[5.50]{KM}. Note that under the given assumptions, but without assuming $\left.\mathcal{H}\right|_S$ is mobile, \cite[5.50]{KM} says
$K_S + \frac{1}{n}\mathcal{H}|_S$ is not klt near $P$, which does not preclude the centre on $S$ of the relevant valuation being a \emph{curve\/} containing $P$ rather than $P$
itself. This curve would of course be in $\operatorname{Bs}(\mathcal{H})$, so the problem is eliminated by assuming $\mathcal{H}|_S$ is mobile --- and it will be clear we may assume this in our application. With the extra assumption
$K_S + \frac{1}{n}\mathcal{H}|_S$ is not plt near $P$, as required.
\begin{thm}[Corti] \label{Cothm} Let $\left(P \in \Delta_1 + \Delta_2 \subset S \right) \simeq \left(0 \in \{xy = 0 \} \subset \mathbb C^2 \right)$ be the analytic germ of a normal crossing curve on a smooth surface; let $\mathcal L$ be a mobile linear system on $S$ and $L_1,L_2 \in \mathcal L$ general members. Suppose there exist $n \in \mathbb{Z}_{\ge 1}$ and $a_1,a_2 \in \mathbb{Q}_{\ge 0}$ such that \[ P \in \operatorname{LCS}\left(S, (1-a_1)\Delta_1 + (1-a_2)\Delta_2 + \textstyle\frac{1}{n}\mathcal L \right). \] Then \[ (L_1 \cdot L_2)_P \ge \left\{ \begin{array}{l@{\quad\mbox{if}\hspace{1.5ex}}l} 4a_1a_2n^2 & a_1 \le 1 \mbox{\hspace{1ex} or \hspace{1ex}} a_2 \le 1; \\ 4(a_1+a_2-1)n^2 & \mbox{both\hspace{1.5ex}} a_1,a_2 > 1. \end{array} \right. \] \end{thm} This is proved as in the original \cite{Co00}, but replacing `log canonical' by `purely log terminal' and strict inequalities by $\le$ or $\ge$ as appropriate. Now combining Theorems \ref{invadj} and \ref{Cothm} we obtain the following.
\begin{cor} \label{multthm} Let $P \in X$ be the germ of a smooth 3-fold and $\mathcal{H}$ a mobile linear system on $X$. Assuming as in Theorem \ref{invadj} that $P \in \CSXnH$ for some $n \in \mathbb{Z}_{\ge 1}$ we have \[ \operatorname{mult}_P(H \cdot H') \ge 4n^2 \] where $H,H' \in \mathcal{H}$ are general and $H \cdot H'$ is their intersection cycle. \end{cor}
Now we need to borrow an additional result from \cite{CPR}. First we recall the following definition.
\begin{defn}[cf. {\cite[5.2.4]{CPR}}] \label{defn:Gammaisol} Let $L$ be a Weil divisor class in a 3-fold $X$ and $\Gamma \subset X$ an irreducible curve or a closed point. We say that \emph{$L$ isolates $\Gamma$}, or is a \emph{$\Gamma$-isolating class\/}, if and only if there exists $s \in \mathbb{Z}_{\ge1}$ such that the linear system
$\mathcal L_{\Gamma}^s := \left|\mathcal I_{\Gamma}^s(sL) \right|$ satisfies \begin{itemize} \item $\Gamma \in \operatorname{Bs} \mathcal L_{\Gamma}^s$ is an isolated component (i.e., in some neighbourhood of $\Gamma$ the subscheme $\operatorname{Bs} \mathcal L_{\Gamma}^s$ is supported on $\Gamma$); and \item if $\Gamma$ is a curve, the generic point of $\Gamma$ appears with multiplicity 1 in $\operatorname{Bs} \mathcal L_{\Gamma}^s$. \end{itemize} \end{defn}
\begin{thm}[{\cite[5.3.1]{CPR}}] \label{Pisol} Let $X = X_d \subset \mathbb{P}(1,a_1,\ldots,a_4)$ be a general hypersurface in one of families $3,4,\ldots,95$ and $P \in X$ a smooth point. Then for some positive integer $l < 4/A^3$ the class $lA$ is $P$-isolating. \end{thm}
\begin{rk} \label{Pisolrk} \cite[5.3.1]{CPR} says $l \le 4/A^3$, but the statement there is for all the families except number 2 --- that is, including number 1 --- and a trivial check shows that in fact $l = 4/A^3$ only for number 1. \end{rk}
\begin{proof}[\textsc{Proof of Theorem \ref{thm:smpts}}]
We know that $P \in \operatorname{Bs}\left|\mathcal I_P^s(slA) \right|$ is an isolated component for some $l,s \in \mathbb{Z}_{\ge1}$ with $l < 4/A^3$. Take a general surface $S \in \left|\mathcal I_P^s(slA) \right|$ and general elements $H,H' \in \mathcal{H}$. If we assume that $P$ belongs to $\CSXnH$ then Corollary \ref{multthm} tells us that $\operatorname{mult}_P \left(H\cdot H' \right) \ge 4n^2$, so \[ {S \cdot H \cdot H'} \ge \left(S \cdot H \cdot H' \right)_P \ge 4sn^2. \] But we know \[ {S \cdot H \cdot H'} = sln^2A^3 < \frac{4}{A^3}sn^2A^3 = 4sn^2, \] contradiction. \end{proof}
\subsection{Singular points} \label{subsec:singpts}
The following two results are fundamental.
\begin{thm}[Kawamata, \cite{Ka96}] \label{thm:Ka} Let $P \in X \simeq \frac{1}{r}(1,a,r-a)$, with $r \ge 2$ and $(a,r) = 1$, be the germ of a 3-fold terminal quotient singularity, and \[ f \colon (E \subset Y) \to (\Gamma \subset X) \] a divisorial contraction such that $P \in \Gamma$ (so $Y$ has terminal singularities, $\operatorname{Exc} f = E$ is an irreducible divisor and $-K_Y$ is $f$-ample). Then $\Gamma = P$ and $f$ is isomorphic over $X$ to the $(1,a,r-a)$ weighted blowup of $P \in X$. \end{thm}
\begin{lemma}[Kawamata, \cite{Ka96}] \label{lem:Ka} Let $P \in X \simeq \frac{1}{r}(1,a,r-a)$ be as in Theorem \ref{thm:Ka} and $f \colon (E \subset Y) \to (P \in X)$ the $(1,a,r-a)$ weighted blowup; let $g \colon \widetilde{X} \to X$ be a resolution of singularities with exceptional divisors $\{E_i \}$. Fix an effective Weil divisor $H$ on $X$ and define $a_i = a_{E_i}\left(K_X\right)$ and $m_i = m_{E_i}(H)$ in the usual way via \begin{eqnarray}
K_{\widetilde{X}} & = & g^*K_X + \textstyle\sum a_iE_i, \nonumber \\
g^{-1}_*H & = & g^*H - \textstyle\sum m_iE_i; \nonumber \end{eqnarray} define $a_E$ and $m_E$ similarly using $f$. Then $m_i / a_i \le m_E / a_E$ for all $i$. \end{lemma}
In \cite{Ka96} Lemma \ref{lem:Ka} is used to prove Theorem \ref{thm:Ka}, but it is an interesting result in its own right; in particular, it has two corollaries that are of great importance for our problem.
\begin{cor} \label{cor:Kalemma:1} Let $P \in X \simeq \frac{1}{r}(1,a,r-a)$ and $f \colon (E \subset Y) \to (P \in X)$ the Kawamata blowup as in Lemma \ref{lem:Ka}. Suppose $\mathcal{H}$ is a mobile linear system on $X$ and $n \in \mathbb{Z}_{\ge1}$ is such that $P \in \CSXnH$. Then the valuation $v_E$ of $E$ is strictly canonical or worse for $\XnH$. \end{cor}
\begin{proof}[\textsc{Proof}] Let $g \colon \widetilde{X} \to X$ be any resolution of singularities with exceptional divisors $\{E_i\}$ and $H \in \mathcal{H}$ a general element. The assumption $P \in \CSXnH$ means that $n \le m_i/a_i$ for some $i$, so by Lemma \ref{lem:Ka} $n \le m_E / a_E$ as well. \end{proof}
This Corollary \ref{cor:Kalemma:1} tells us that we can exclude a singular point from any $\CSXnH$ simply by excluding the valuation $v_E$; it has other uses as well.
\begin{cor} \label{cor:Kalemma:2} Let $P \in X \simeq \frac{1}{r}(1,a,r-a)$ and $f \colon Y \to X$ be the Kawamata blowup of $P$ as in Lemma \ref{lem:Ka}. Suppose $C \subset X$ is a curve containing $P$, $\mathcal{H}$ is a mobile linear system on $X$ and $n \in \mathbb{Z}_{\ge1}$ is such that $C \in \CSXnH$. Then $P \in \CSXnH$ also. \end{cor}
\begin{proof}[\textsc{Proof}] Let $g \colon \widetilde{X} \to X$ be a resolution of singularities with exceptional divisors $\{E_i\}$ at least one of which has centre $C$ on $X$ and is strictly canonical or worse for $\XnH$. The rest of the proof is the same as that of Corollary~\ref{cor:Kalemma:1}. \end{proof}
\begin{absexclsingpts} Suppose $P$ is a singular point of a hypersurface $X$ in one of the~95 families and $P \in X$ is locally isomorphic to $\frac{1}{r}(1,a,r-a)$. Let $f \colon (E \subset Y) \to (P \in X)$ be the Kawamata blowup and suppose $B^3 < 0$ (where as always $B =
{-K_Y}$). We denote by $S$ the surface $f^{-1}_*\{x_0=0\}$, which is an element of $\left|B\right|$ and is irreducible, assuming $X$ is general. \end{absexclsingpts}
\begin{lemma}[{see~\cite[5.4.3]{CPR}}] \label{Texists}
If $B^3 < 0$ then there exist integers $b,c$ with $b > 0$ and $b/r \ge c \ge 0$ and a surface $T \in \left|bB + cE \right|$ such that \begin{itemize} \item[(a)] the scheme theoretic complete intersection $\Gamma = S \cap T$ is a reduced, irreducible curve and \item[(b)] $T\Gamma \le 0$. \end{itemize} \end{lemma}
\begin{thm} \label{thm:Tmethodabs} Suppose $B^3 < 0$ and the integer $c$ provided by Lemma \ref{Texists} is strictly positive. Then $P$ is excluded absolutely, that is, there does not exist a mobile linear system $\mathcal{H}$ of degree $n$ on $X$ such that $K_X + \frac{1}{n}\mathcal{H}$ is canonical and $P \in \CSXnH$. \end{thm}
For the proof we need only the following two lemmas.
\begin{lemma} \label{lem:test_class} Let $X$ be a Fano 3-fold hypersurface in one of the 95 families and $\mathcal{H}$ a mobile linear system of degree $n$ on $X$ such that $K_X + \frac{1}{n}\mathcal{H}$ is strictly canonical; suppose $\Gamma\subset X$ is an irreducible curve or a closed point satisfying $\Gamma\in\CSXnH$, and furthermore that there is a Mori extremal divisorial contraction \[ f\colon (E\subset Y) \to (\Gamma\subset X), \quad \operatorname{Centre}_X E = \Gamma, \] such that $E\in\VXnH$. Then $B^2 \in \operatorname{\overline{NE}} Y$. \end{lemma}
\begin{proof}[\textsc{Proof}] We know that \[ K_Y + \textstyle\frac{1}{n}\mathcal{H}_Y \sim_\QQ f^*\left(K_X + \textstyle\frac{1}{n}\mathcal{H}\right) \sim_\QQ 0 . \] It follows that $B \sim_\QQ \frac{1}{n}\mathcal{H}_Y$, and therefore $B^2 \in \operatorname{\overline{NE}} Y$, because $\mathcal{H}_Y$ is mobile. \end{proof}
\begin{lemma}[{see~\cite[5.4.6]{CPR}}] \label{NEbarY} If $B^3 < 0$, let $T$ and $\Gamma = S \cap T$ be as in the conclusion of Lemma \ref{Texists}. Write $R$ for the extremal ray of $\operatorname{\overline{NE}} Y$ contracted by $f \colon Y \to X$ and let $Q \subset \operatorname{\overline{NE}} Y$ be the other ray. Then $Q = \mathbb R_{\ge0}[\Gamma]$. \end{lemma}
\begin{proof}[\textsc{Proof of Theorem \ref{thm:Tmethodabs}}] Suppose $\mathcal{H}$ is a mobile linear system of degree $n$ on $X$ such that $K_X + \frac{1}{n}\mathcal{H}$ is canonical and $P \in \CSXnH$. Corollary~\ref{cor:Kalemma:1} of Kawamata's Lemma tells us that the Kawamata blowup $f \colon Y \to X$ of $P$ extracts a valuation $v_E$ (where $E = \operatorname{Exc} f$) which is strictly canonical for $\XnH$. The test class Lemma~\ref{lem:test_class} now implies that $B^2 \in \operatorname{\overline{NE}} Y$.
But the ray $Q \subset \operatorname{\overline{NE}} Y$ is generated by $(bB+cE)B$ for some $b,c > 0$, and certainly $EB \in \operatorname{\overline{NE}} Y$, so if $B^2 \in \operatorname{\overline{NE}} Y$ we have both $B^2,EB \in Q$ (by definition of `extremal'). It follows that $EB$ is numerically equivalent to $\alpha B^2$ for some positive $\alpha \in \mathbb{Q}$; but \[ EB \cdot B = E \Big(A-\frac{1}{r}E \Big)^2 = \frac{1}{r^2}E^3 = \frac{1}{a(r-a)} > 0, \] while $B^2 \cdot B = B^3 < 0$ by assumption --- contradiction. \end{proof}
\begin{cor} \label{cor:75singpts} Let $X = X_{30} \subset \mathbb{P}(1,4,5,6,15)_{x,y,z,t,u}$ be a general member of family~75 of the~95 and $\mathcal{H}$ a mobile linear system of degree $n$ on $X$ with $\KXnH$ strictly canonical. Then no singular point of $X$ other than the two $\frac{1}{5}(1,4,1)$ points in the $zu$-stratum can belong to $\CSXnH$. \end{cor}
\begin{proof}[\textsc{Proof}] Here is the complete list of singular points of $X$, together with the sign of $B^3$ and $bB + cE \sim T$ (see Lemma~\ref{Texists}) for each of them. \begin{tabbing} \hspace*{1cm} \=$P_y P_t \cap X = R_1, R_2 \sim \frac{1}{2}(1,1,1)_{x,z,u}$ space \= sample text \= \kill \>$P_y \sim \frac{1}{4}(1,1,3)_{x,z,u}$ \>$B^3 < 0$ \>$10B + E$ \\ \>$P_t P_u \cap X = P \sim \frac{1}{3}(1,1,2)_{x,y,z}$ \>$B^3 < 0$ \>$5B + E$ \\ \>$P_z P_u \cap X = Q_1, Q_2 \sim \frac{1}{5}(1,4,1)_{x,y,t}$ \>$B^3 < 0$ \>$4B$ \\ \>$P_y P_t \cap X = R_1, R_2 \sim \frac{1}{2}(1,1,1)_{x,z,u}$ \>$B^3 < 0$ \>$5B + 2E$ \end{tabbing} Clearly all these apart from $Q_1$ and $Q_2$ satisfy the hypotheses of Theorem~\ref{thm:Tmethodabs}, and consequently are excluded absolutely. \end{proof}
\begin{condexclsingpts} Assume $P \in X$ is a singular point of a hypersurface in one of the~95 families which is locally isomorphic to $\frac{1}{r}(1,a,r-a)$ with $B^3 < 0$. We keep the notations $T \sim bB + cE$, $S = f^{-1}_*\{x_0=0\}$ and $\Gamma = S \cap T$ of Lemma~\ref{Texists}; in the following paragraphs we consider the case where the integer $c$ provided by Lemma \ref{Texists} is zero. Out of all such singular points, the vast majority live in a family with $b = a_1 < a_2$, where $a_1,a_2$ are the weights of $x_1,x_2$. For such points we have the following result. \end{condexclsingpts}
\begin{thm} \label{thm:Tmethodcond} Let $P \in X$ be a singular point satisfying $B^3 < 0$, $T \sim bB$ and $b = a_1 < a_2$. Assume that the curve $C = \{x_0 = x_1 = 0\}\cap X$ is irreducible. Then $R(Y,B) = k[x_0,x_1]$. It follows that if $\mathcal{H}$ is a mobile linear system of degree $n$ on $X$ such that $\KXnH$ is canonical and $P \in \CSXnH$ then in fact $\CSXnH =
\operatorname{CS}\left(X,\frac{1}{b}f_*\left|bB\right|\right)$, where $f \colon Y \to X$ is the Kawamata blowup of $P$. \end{thm}
\begin{proof}[\textsc{Proof for family 75}]
We do not prove this theorem here for every case. As explained in~\cite{Ry02}, the first statement, namely $R(Y, B) = k[x_0, x_1]$, follows from two ray game calculations such as that in the proof of our Lemma~\ref{lem:75tworay}; we have already shown this for family~75. The second part of the theorem follows easily from the first: let $\mathcal{H}$ be mobile of degree $n$ on $X$ with $\KXnH$ canonical and $P \in \CSXnH$. Then $\mathcal{H}_Y \subset |nB| = \left<x_0^n,x_0^{n-b}x_1,\ldots,x_0^rx_1^q\right>$, where $n = qb +
r$, $0 \le r < b$ --- so $\mathcal{H} \subset f_*|nB| =
\left<x_0^n,\ldots,x_0^rx_1^q\right>$, while of course $f_*|bB| = \left<x_0^b,x_1\right>$, and therefore $\CSXnH =
\operatorname{CS}\left(X,\frac{1}{b}f_*|bB|\right)$. \end{proof}
We are now in a position to put together all the exclusion results obtained so far to prove Theorem~\ref{thm:75aux} for family~75.
\begin{proof}[\textsc{Proof of Theorem~\ref{thm:75aux}}] First, since $X$ is superrigid by~\cite[\S6]{CPR}, $\KXnH$ nonterminal implies $\KXnH$ strictly canonical; therefore $\CSXnH$ is nonempty. Corollary~\ref{cor:75curves} tells us that no curve belongs to $\CSXnH$, Theorem~\ref{thm:smpts} says that the same is true for smooth points and Corollary~\ref{cor:75singpts} shows the same for all singular points other than $Q_1, Q_2 \sim \frac{1}{5}(1,4,1)_{x,y,t}$. Therefore at least one $Q_i \in \CSXnH$; without loss of generality we may assume this holds for $Q_1$.
Now by Theorem~\ref{thm:Tmethodcond}
\[ \CSXnH = \operatorname{CS}\left( X, \textstyle\frac{1}{b} f_* |bB| \right) \] where $b = 4$ because $T \sim 4B$ (in the notation above) and $f \colon Y \to X$ is the Kawamata blowup of $Q_1$. But $R(Y, B) =
k[x_0, x_1] = k[x, y]$, so $f_* |bB|$ is just $\left< x^4, y \right>$; and both $x^4$ and $y$ have local vanishing order $4/5$ at $Q_2$, so $Q_2 \in \CSXnH$ also, as required. \end{proof}
\section{Birational classification of elliptic and K3 fibrations} \label{sec:class}
We have now proved Theorem~\ref{thm:75aux} for family~75; all that remains is to deduce the main Theorem~\ref{thm:75main} from it. For this we need the following theorem--definition and propositions.
\begin{thmdefn} \label{Kod} Let $X$ be a variety, normal and projective over $\mathbb C$ as always, and $\mathcal{H}$ a mobile linear system on $X$. Fix $\alpha\in\mathbb{Q}_{\ge0}$ and let $f \colon Y \to X$ be a birational morphism such that $K_Y + \alpha\mathcal{H}_Y$ is canonical. We define the \emph{log Kodaira dimension $\kappa(X,\alpha\mathcal{H})$} to be the $D$-dimension of $K_Y + \alpha\mathcal{H}_Y$, that is, \[ \kappa(X,\alpha\mathcal{H}) = D(K_Y + \alpha\mathcal{H}_Y) = \max
\{\dim(\operatorname{im}\varphi_{\left|m(K_Y + \alpha\mathcal{H}_Y)\right|}) \} , \]
taking the max over all $m\ge1$ such that $m(K_Y + \alpha\mathcal{H}_Y)$ is integral; if all the linear systems $\left|m(K_Y + \alpha\mathcal{H}_Y)\right|$ are empty, by definition \[\kappa(X,\alpha\mathcal{H}) = D(K_Y + \alpha\mathcal{H}_Y) = {-\infty}.\] Then $\kappa(X,\alpha\mathcal{H})$ is independent of the choice of $Y$ and is attained for a particular $Y$ using any sufficiently large $m$ such that $m(K_Y + \alpha\mathcal{H}_Y)$ is integral. \end{thmdefn}
\begin{proof}[\textsc{Proof}] This result is standard and is used in, e.g.,~\cite{Ch00} and~\cite{Is01}. The methods employed in \cite{Sh96} to show uniqueness of a log canonical model can be used to prove it; alternatively see \cite[Ch.\ 2]{FA}, particularly Theorem 2.22 and the Negativity Lemma 2.19, which is an essential ingredient. \end{proof}
\begin{emp} Now let $X$ be a Mori Fano variety and $\mathcal{H}$ a mobile linear system of degree $n$ on $X$, that is, $n \in \mathbb{Q}$ is such that $K_X + \frac{1}{n}\mathcal{H} \sim_\QQ 0$ --- of course $n\in\mathbb{Z}_{\ge 1}$ if $X$ is a hypersurface in one of the 95 families. \end{emp}
\begin{prop} \label{prop:varyep} Assume that $K_X + \frac{1}{n}\mathcal{H}$ is canonical. \begin{itemize} \item[(a)] Let $\varepsilon \in \mathbb{Q}$. Then \[ \kappa\left(X, \left(\textstyle \frac{1}{n} + \varepsilon \right)\mathcal{H} \right) = \left\{\begin{array}{l@{\quad\mbox{if}\hspace{1.5ex}}l} {- \infty} & \varepsilon < 0 \\ 0 & \varepsilon = 0 \\ d \ge 1 & \varepsilon > 0 \end{array} \right.\] \item[(b)] If $1 \le \kappa(X, (\frac{1}{n} + \varepsilon )\mathcal{H}) \le \dim X - 1$ for some $\varepsilon \in \mathbb{Q}_{>0}$ then $K_X + \frac{1}{n}\mathcal{H}$ is nonterminal, i.e., strictly canonical (so $K_X + (\frac{1}{n} + \varepsilon)\mathcal{H}$ is noncanonical). \end{itemize} \end{prop}
\begin{proof}[\textsc{Proof}] See the survey~\cite[III.2.3--2.4]{Is01}. \end{proof}
Recall that the NFI-type inequality~\ref{nfi} was stated under two alternative sets of assumptions: either \begin{itemize} \item[(a)] $X$ is a Mori Fano and $\Phi \colon X \dashrightarrow Z/T$ a birational map to the total space $Z$ of a $K$-trivial fibration $g \colon Z \to T$; or \item[(b)] $X$ is a Mori Fano and $\Phi \colon X \dashrightarrow Z$ a birational map to a Fano variety with canonical singularities. \end{itemize}
\begin{prop} \label{dimT} In situation (a) above, assume that $\KXnH$ is canonical. Then for any $\varepsilon \in \mathbb{Q}_{>0}$, $\kappa\XnepH = \dim T$. \end{prop}
\begin{proof}[\textsc{Proof}] Fix $\varepsilon\in\mathbb{Q}_{>0}$. $K_X + \frac{1}{n}\mathcal{H}$ is canonical so $\kappa(X,\frac{1}{n}\mathcal{H}) = 0$, and therefore $\kappa(Z,\frac{1}{n}\mathcal{H}_Z) = 0$ by the birational invariance of log Kodaira dimension (\ref{Kod}). But in fact $K_Z + \frac{1}{n}\mathcal{H}_Z$ is canonical (as is $K_Z + (\frac{1}{n} + \varepsilon)\mathcal{H}_Z$, because $K_Z$ is canonical and $\mathcal{H}_Z$ is free), so $\kappa(Z,\frac{1}{n}\mathcal{H}_Z)$ (and $\kappa(Z,(\frac{1}{n} + \varepsilon)\mathcal{H}_Z)$) can be computed on $Z$ as the ordinary $D$-dimension. Consequently for $m \gg 0$ and such that $m(K_Z + \frac{1}{n}\mathcal{H}_Z)$ is integral, it is in fact effective and fixed. Fix such an $m$ with the additional property that $m\varepsilon \in
\mathbb N$, so that $m(K_Z + (\frac{1}{n}+\varepsilon)\mathcal{H}_Z)$ is integral as well; let $F \in \left|m(K_Z + \frac{1}{n}\mathcal{H}_Z)\right|$ be the unique element. Now for any curve $C$ contracted by $g$, $FC = 0$, because by assumption $K_Z C = 0$ and $\mathcal{H}_Z = g^*\left|H\right|$. But $F$ is effective, so it must be a pullback $g^* F_T$ of some effective $F_T$ on $T$. Furthermore, for any $m' \in \mathbb N$, \[ H^0(T,m'F_T) = H^0(Z,g^*(m'F_T)) = H^0(Z,m'F), \] so $F_T$ is fixed, $D(F_T) = 0$ and it is easy to see that $D(F_T + m\varepsilon H) = \dim T$, because $H$ is ample and $F_T$ is effective. Now $g^*(F_T + m\varepsilon H) = F + m\varepsilon\mathcal{H}_Z$, so \begin{eqnarray*} \textstyle \kappa\left(X,\left(\frac{1}{n} + \varepsilon \right)\mathcal{H} \right) & = & \textstyle D\left(m\left(K_Z + \left(\frac{1}{n} + \varepsilon \right)\mathcal{H}_Z\right) \right) \\ & = & D(F+ m\varepsilon\mathcal{H}_Z) = \dim T \end{eqnarray*} as required. \end{proof}
\begin{proof}[\textsc{Proof of NFI-type inequality \ref{nfi} in situation} \emph{(a)}] We note that under the assumptions (a),~\ref{nfi} is an immediate consequence of Propositions~\ref{prop:varyep} and~\ref{dimT}. Under assumptions~(b), the proof of~\cite[4.2]{Co95} can be easily adapted to give an argument. Like Theorem--Definition~\ref{Kod}, this is a standard result used in~\cite{Ch00} and~\cite{Is01}, so we omit the details. \end{proof}
All that remains is to prove the main theorem for family~75. Arguments similar to the following also prove Theorems~\ref{thm:34main},~\ref{thm:88main} and~\ref{thm:90main}.
\begin{proof}[\textsc{Proof of Theorem \ref{thm:75main}}] It is simplest to prove part (b) first; we then indicate how the argument can be easily adapted to demonstrate (a) and (c) also.
{(b) \sloppy Suppose that $\Phi \colon X \dashrightarrow Z/T$ is a birational map from $X$ to an elliptic fibration \linebreak $g \colon Z \to T$. By the NFI-type inequality~\ref{nfi}, $\KXnH$ is nonterminal, where as usual the system \linebreak[4] $\mathcal{H} = \Phi^{-1}_*\mathcal{H}_Z =
\Phi^{-1}_*g^*|H|$ is the transform of a very ample complete system of Cartier divisors on $T$. By Theorem~\ref{thm:75aux}, $\KXnH$ is strictly canonical and $\CSXnH = \{Q_1,Q_2\}$. }
Let $Q$ be either $Q_1$ or $Q_2$. As in Lemma~\ref{lem:75tworay} we blow up $Q$ and play out the two ray game; in the notation of the lemma, \[ R(Y,B) = R(Y',B') = k[x,y], \] $f'$ is the anticanonical morphism of $Y'$ and the composite $\pi \colon X \dashrightarrow \mathbb{P}^1$ is $(x^4,y)$. Now because $Q \in \CSXnH$, we have that
\[ \mathcal{H}_Y \subset |{-nK_Y}| = |nB| = k[x,y]_n = k\left[x^n,x^{n-4}y,\ldots,x^{n \bmod 4}y^{\lfloor n/4\rfloor}\right]; \] the same is true of $Y'$ (since $Y$ and $Y'$ are isomorphic in codimension one) and therefore \linebreak $\mathcal{H}_{Y'} = (f')^*\mathcal{H}_{\mathbb{P}^1}$ is the pullback of a mobile system on $\mathbb{P}^1$. But any mobile system on $\mathbb{P}^1$ is free, so we can deduce (as in the proof of Proposition~\ref{dimT}) that for any $\varepsilon \in \mathbb{Q}$ with $0 < \varepsilon \ll 1$, \begin{equation} \label{kale1} \kappa\XnepH = \kappa\left(Y',\left(\textstyle\frac{1}{n} + \varepsilon\right)\mathcal{H}_{Y'}\right) = D(F + m\varepsilon\mathcal{H}_{\mathbb{P}^1}) \le 1 \end{equation} where $F$ is a fixed effective divisor on $\mathbb{P}^1$ (so in fact $F = 0$) and $m \in \mathbb{Z}_{>0}$.
But by Proposition~\ref{dimT} applied to $\Phi \colon X \dashrightarrow Z/T$ we have $\kappa\XnepH = \dim T = 2$, which contradicts (\ref{kale1}). This proves (b).
(a) We can follow the proof of (b) but in the end, rather than a contradiction, we deduce that the system $\mathcal{H} = \Phi^{-1}_*g^*|H|$ is actually a pullback $\pi^{-1}_*\mathcal{H}_{\mathbb{P}^1}$ via the map $\pi = (x^4,y) \colon X \dashrightarrow \mathbb{P}^1$. This induces an isomorphism $\mathbb{P}^1 \to T$ such that the specified diagram commutes.
(c) If we assume $\Phi$ is not an isomorphism, we can follow the argument for (b) to deduce that $\kappa\XnepH = 1$, which is obviously a contradiction since $\mathcal{H}_Z = |H|$ is very ample. (Note that the NFI-type inequality~\ref{nfi} requires us to assume $\Phi$ is not an isomorphism in the Fano case; for the elliptic and K3 cases this is of course not necessary.)
This completes the proof. \end{proof}
\small
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\date{}
\title{Algorithms for Testing Monomials in Multivariate Polynomials}
\author{Zhixiang Chen \hspace{5mm} Bin Fu \hspace{5mm} Yang Liu \hspace{5mm} Robert Schweller
\\ \\ Department of Computer Science\\
University of Texas-Pan American\\
Edinburg, TX 78539, USA\\ \{chen, binfu, yliu, schwellerr\}@cs.panam.edu\\\\
} \maketitle
\begin{abstract} This paper is our second step towards developing a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by an arithmetic circuit has some types of monomials in its sum-product expansion. The complexity aspects of this problem and its variants have been investigated in our first paper by Chen and Fu (2010), laying a foundation for further study. In this paper, we present two pairs of algorithms. First, we prove that there is a randomized $O^*(p^k)$ time algorithm for testing $p$-monomials in an $n$-variate polynomial of degree $k$ represented by an arithmetic circuit, while a deterministic $O^*(6.4^k + p^k)$ time algorithm is devised when the circuit is a formula, here $p$ is a given prime number. Second, we present a deterministic $O^*(2^k)$ time algorithm for testing multilinear monomials in $\Pi_m\Sigma_2\Pi_t\times \Pi_k\Pi_3$ polynomials, while a randomized $O^*(1.5^k)$ algorithm is given for these polynomials. The first algorithm extends the recent work by Koutis (2008) and Williams (2009) on testing multilinear monomials. Group algebra is exploited in the algorithm designs, in corporation with the randomized polynomial identity testing over a finite field by Agrawal and Biswas (2003), the deterministic noncommunicative polynomial identity testing by Raz and Shpilka (2005) and the perfect hashing functions by Chen {\em at el.} (2007). Finally, we prove that testing some special types of multilinear monomial is W[1]-hard, giving evidence that testing for specific monomials is not fixed-parameter tractable.
\end{abstract}
\section{Introduction} \subsection{Overview}
We begin with two examples to exhibit the motivation and necessity of the study about the monomial testing problem for multivariate polynomials. The first is about testing a $k$-path in any given undirected graph $G=(V,E)$ with $|V| = n$, and the second is about the satisfiability problem. Throughout this paper, polynomials refer to those with multiple variables.
For any fixed integer $c\ge 1$, for each vertex $v_i \in V$, define a polynomial $p_{k,i}$ as follows: \begin{eqnarray} p_{1,i} &=& x_i^c, \nonumber \\ p_{k+1,i} &=& x_i^c \left(\sum_{(v_i,v_j)\in E} p_{k,j}\right), \ k >1. \nonumber \end{eqnarray} We define a polynomial for $G$ as \begin{eqnarray} p(G, k) &=& \sum^{n}_{i=1} p_{k,i}. \nonumber \end{eqnarray} Obviously, $p(G,k)$ can be represented by an arithmetic circuit. It is easy to see that the graph $G$ has a $k$-path $v_{i_1}\cdots v_{i_k}$ iff $p(G, k)$ has a monomial $x_{i_1}^c\cdots x_{i_k}^c$ of degree $ck$ in its sum-product expansion. $G$ has a Hamiltonian path iff $p(G, n)$ has the monomial $x_1^c\cdots x_n^c$ of degree $cn$ in its sum-product expansion. One can also see that a path with some loop can be characterized by a monomial as well. Those observations show that testing monomials in polynomials is closely related to solving $k$-path, Hamiltonian path and other problems about graphs. When $c=1$, $x_{i_1}\cdots x_{i_k}$ is multilinear. The problem of testing multilinear monomials has recently been exploited by Koutis \cite{koutis08} and Williams \cite{williams09} to design innovative randomized parameterized algorithms for the $k$-path problem.
Now, consider any CNF formula $f= f_1 \wedge \cdots \wedge f_m$, a conjunction of $m$ clauses with each clause $f_i$ being a disjunction of some variables or negated ones. We may view conjunction as multiplication and disjunction as addition, so $f$ looks like a {\em "polynomial"}, denoted by $p(f)$. $p(f)$ has a much simpler $\Pi\Sigma$ representation, as will be defined in the next section, than general arithmetic circuits. Each {\em "monomial"} $\pi = \pi_1 \ldots \pi_m$ in the sum-product expansion of $p(f)$ has a literal $\pi_i$ from the clause $f_i$. Notice that a boolean variable $x \in Z_2$ has two properties of $x^2 = x$ and $x \bar{x} = 0$. If we could realize these properties for $p(f)$ without unfolding it into its sum-product, then $p(f)$ would be a {\em "real polynomial"} with two characteristics: (1) If $f$ is satisfiable then $p(f)$ has a multilinear monomial, and (2) if $f$ is not satisfiable then $p(f)$ is identical to zero. These would give us two approaches towards testing the satisfiability of $f$. The first is to test multilinear monomials in $p(f)$, while the second is to test the zero identity of $p(f)$. However, the task of realizing these two properties with some algebra to help transform $f$ into a needed polynomial $p(f)$ seems, if not impossible, not easy. Techniques like arithmetization in Shamir \cite{shamir92} may not be suitable in this situation. In many cases, we would like to move from $Z_2$ to some larger algebra so that we can enjoy more freedom to use techniques that may not be available when the domain is too constrained. The algebraic approach within $Z_2[Z^k_2]$ in Koutis \cite{koutis08} and Williams \cite{williams09} is one example along the above line. It was proved in Bshouty {\em et al.} \cite{bshouty95} that extensions of DNF formulas over $Z^n_2$ to $Z_N$-DNF formulas over the ring $Z^n_N$ are learnable by a randomized algorithm with equivalence queries, when $N$ is large enough. This is possible because a larger domain may allow more room to utilize randomization.
There has been a long history in theoretical computer science with heavy involvement of studies and applications of polynomials. Most notably, low degree polynomial testing/representing and polynomial identity testing have played invaluable roles in many major breakthroughs in complexity theory. For example, low degree polynomial testing is involved in the proof of the PCP Theorem, the cornerstone of the theory of computational hardness of approximation and the culmination of a long line of research on IP and PCP (see, Arora {\em at el.} \cite{arora98} and Feige {\em et al.} \cite{feige96}). Polynomial identity testing has been extensively studied due to its role in various aspects of theoretical computer science (see, for examples, Chen and Kao \cite{chen00}, Kabanets and Impagliazzo \cite{kabanets03}) and its applications in various fundamental results such as Shamir's IP=PSPACE \cite{shamir92} and the AKS Primality Testing \cite{aks04}. Low degree polynomial representing \cite{minsky-papert68} has been sought for so as to prove important results in circuit complexity, complexity class separation and subexponential time learning of boolean functions (see, for examples, Beigel \cite{beigel93}, Fu\cite{fu92}, and Klivans and Servedio \cite{klivans01}). These are just a few examples. A survey of the related literature is certainly beyond the scope of this paper.
The above two examples of the $k$-path testing and satisfiability problems, the rich literature about polynomial testing and many other observations have motivated us to develop a new theory of testing monomials in polynomials represented by arithmetic circuits or even simpler structures. The monomial testing problem is related to, and somehow complements with, the low degree testing and the identity testing of polynomials. We want to investigate various complexity aspects of the monomial testing problem and its variants with two folds of objectives. One is to understand how this problem relates to critical problems in complexity, and if so to what extent. The other is to exploit possibilities of applying algebraic properties of polynomials to the study of those critical problems. As a first step, Chen and Fu \cite{chen-fu10} have proved a series of results: The multilinear monomial testing problem for $\Pi\Sigma\Pi$ polynomials is NP-hard, even when each clause has at most three terms. The testing problem for $\Pi\Sigma$ polynomials is in P, and so is the testing for two-term $\Pi\Sigma\Pi$ polynomials. However, the testing for a product of one two-term $\Pi\Sigma\Pi$ polynomial and another $\Pi\Sigma$ polynomial is NP-hard. This type of polynomial products is, more or less, related to the polynomial factorization problem. We have also proved that testing $c$-monomials for two-term $\Pi\Sigma\Pi$ polynomials is NP-hard for any $c > 2$, but the same testing is in P for $\Pi\Sigma$ polynomials. Finally, two parameterized algorithms have been devised for three-term $\Pi\Sigma\Pi$ polynomials and products of two-term $\Pi\Sigma\Pi$ and $\Pi\Sigma$ polynomials. These results have laid a basis for further study about testing monomials.
\subsection{Contributions and Methods}
The major contributions of this paper are two pairs of algorithms. For the first pair, we prove that there is a randomized $O^*(p^k)$ time algorithm for testing $p$-monomials in an $n$-variate polynomial of degree $k$ represented by an arithmetic circuit, while a deterministic $O^*(6.4^k + p^k)$ time algorithm is devised when the circuit is a formula, here $p$ is a given prime number. The first algorithm extends two recent algorithms for testing multilinear monomials, the $O^*(2^{3k/2})$ algorithm by Koutis \cite{koutis08} and the $O(2^k)$ algorithm by Williams \cite{williams09}. Koutis \cite{koutis08} initiated the application of group algebra $Z_2[Z^k_2]$ to randomized testing of multilinear monomials in a polynomial. Williams \cite{williams09} incorporated the randomized Schwartz-Zippel polynomial identity testing with the group algebra $\mbox{GF}(2^{\ell})[Z^k_p]$ for some relatively small ${\ell}$ in comparison with $k$ to achieve the design of his algorithm. The success of applying group algebra to designing multilinear monomial testing algorithms is based on two simple but elegant properties found by Koutis, by which annihilating non-multilinear monomials is possible via replacements of variables by vectors in $Z^k_2$. When extending the group algebra from $Z_2[Z^k_p]$ to $Z_p[Z^k_p]$ for a given prime $p$ these two properties, as addressed in Section \ref{sec-2}, are fortunately no longer valid. To make the matter worse, the Schwartz-Zippel algorithm is not applicable to the larger algebra due to the lack of these two properties. Nevertheless, we find new characteristics about $Z_p[Z^k_p]$ and integrate these with a more powerful randomized polynomial identity testing algorithm by Agrawal and Biswas \cite{agrawal-biswas03} to accomplish the design of our algorithm. Our deterministic algorithm is obtained via derandomizing the two random processes involved in the first algorithm: deterministic selection of a set of linearly independent vectors for an unknown monomial to guarantee its survivability from vector replacements; and deterministic polynomial identity testing. The first part is realized with the perfect hashing functions by Chen {\em at el.} \cite{jianer-chen07}, while the second is carried out by the Raz and Shpilka \cite{raz05} algorithm for noncommunicative polynomials.
For the second pair of our algorithms, we present a deterministic $O^*(2^k)$ time algorithm for testing multilinear monomials in $\Pi_m\Sigma_2\Pi_t\times \Pi_k\Pi_3$ polynomials, while a randomized $O^*(1.5^k)$ algorithm is given for these polynomials. It has been proved in Chen and Fu \cite{chen-fu10} that testing multilinear monomials in $\Pi_m\Sigma_2\Pi_t$ or $\Pi_k\Pi_3$ polynomials is solvable in polynomial time. However, the problem becomes NP-hard for $\Pi_m\Sigma_2\Pi_t\times \Pi_k\Pi_3$ polynomials. Our two algorithms use the quadratic algorithm by Chen and Fu \cite{chen-fu10} for testing multilinear monomials in $\Pi_m\Sigma_2\Pi_t$ polynomials as the base case algorithm. Both new algorithms improve the $O^*(3^k)$ algorithm in \cite{chen-fu10}.
Finally, we prove that testing some special types of multilinear monomials is W[1]-hard, giving evidence that testing for specific monomials is not fixed-parameter tractable. One shall notice that difference between the general monomial testing and the specific monomial testing. The former asks for the existence of {\em "any one''} from a set of possibly many monomials that are needed. The latter asks for {\em "a specific one''} from the set.
\subsection{Organization} The rest of the paper is organized as follows. In Section 2, we introduce the necessary notations and definitions. In Section 3, we prove new properties about the group algebra $Z_p[Z^k_p]$ to help annihilate any monomials that are not $p$-monomials. These properties are then integrated with the randomized polynomial identity testing over a finite field to help design the randomize $p$-monomial testing algorithm. In Section 4, the two randomized processes involved in the randomized algorithm obtained in the previous section will be derandomized for polynomials represented by formulas. The success is based on combining deterministic construction of perfect hashing functions with deterministic noncommunicative polynomial identity testing. Section 5 first presents a deterministic parameterized algorithm for testing multilinear monomials in $\Pi_m\Sigma_2\Pi_t \times \Pi_k\Sigma_3$ polynomials, and then gives a more efficient randomized parameterized algorithm for the these polynomials. Finally, we show in Section 5 that testing some special type of multilinear monomials, called $k$-clique monomials, is W[1]-hard.
\section{Preliminaries} \subsection{Notations and Definitions}
For variables $x_1, \dots, x_n$, let ${\cal P} [x_1,\cdots,x_n]$ denote the communicative ring of all the $n$-variate polynomials with coefficients from a finite field ${\cal P}$. For $1\le i_1 < \cdots <i_k \le n$, $\pi =x_{i_1}^{j_1}\cdots x_{i_k}^{j_k}$ is called a monomial. The degree of $\pi$, denoted by $\mbox{deg}(\pi)$, is $\sum^k_{s=1}j_s$. $\pi$ is multilinear, if $j_1 = \cdots = j_k = 1$, i.e., $\pi$ is linear in all its variables $x_{i_1}, \dots, x_{j_k}$. For any given integer $c\ge 1$, $\pi$ is called a $c$-monomial, if $1\le j_1, \dots, j_k < c$.
An arithmetic circuit, or circuit for short, is a direct acyclic graph with $+$ gates of unbounded fan-in, $\times$ gates of fan-in two, and all terminals corresponding to variables. The size, denoted by $s(n)$, of a circuit with $n$ variables is the number of gates in it. A circuit is called a {\rm formula}, if the fan-out of every gate is at most one, i.e., its underlying direct acyclic graph is a tree.
By definition, any polynomial $F(x_1,\dots,x_n)$ can be expressed as a sum of a list of monomials, called the sum-product expansion. The degree of the polynomial is the largest degree of its monomials in the expansion. With this expression, it is trivial to see whether $F(x_1,\dots,x_n)$ has a multilinear monomial, or a monomial with any given pattern. Unfortunately, this expression is essentially problematic and infeasible to realize, because a polynomial may often have exponentially many monomials in its expansion.
In general, a polynomial $F(x_1,\dots,x_n)$ can be represented by a circuit or some even simpler structure as defined in the following. This type of representation is simple and compact and may have a substantially smaller size, say, polynomially in $n$, in comparison with the number of all monomials in the sum-product expansion. The challenge is how to test whether $F(x_1,\dots,x_n)$ has a multilinear monomial, or some other needed monomial, efficiently without unfolding it into its sum-product expansion?
Throughout this paper, the $O^*(\cdot)$ notation is used to suppress $\mbox{poly}(n,k)$ factors in time complexity bounds.
\begin{definition}\quad\nopagebreak Let $F(x_1,\dots,x_n)\in {\cal P}[x_1,\dots,x_n]$ be any given polynomial. Let $m, s, t\ge 1$ be integers. \begin{itemize} \item $F(x_1,\ldots, x_n)$ is said to be a $\Pi_m\Sigma_s\Pi_t$ polynomial, if $F(x_1,\dots,x_n)=\prod_{i=1}^t F_i$, $F_i = \sum_{j=1}^{r_i} X_{ij}$ and $1\le r_i \le s$, and $X_{ij}$ is a product of variables with $\mbox{deg}(X_{ij})\le t$. We call each $F_i$ a clause. Note that $X_{ij}$ is not a monomial in the sum-product expansion of $p(x_1,\dots,x_n)$ unless $m=1$. To differentiate this subtlety, we call $X_{ij}$ a term.
\item In particular, we say $F(x_1,\dots,x_n)$ is a $\Pi_{m}\Sigma_s$ polynomial, if it is a
$\Pi_m\Sigma_s\Pi_1$ polynomial. Here, each clause in $f$ is a linear addition
of single variables. In other word, each term has degree $1$.
\item $F(x_1,\dots,x_n)$ is called a $\Pi_m\Sigma_s\Pi_t \times \Pi_k\Sigma_{\ell}$ polynomial, if $F(x_1,\dots,x_n) = f_1 \cdot f_2$ such that $f_1$ is a $\Pi_m\Sigma_s\Pi_t$ polynomial and $f_2$ is a $\Pi_k\Sigma_{\ell}$ polynomial.
\end{itemize} \end{definition}
When no confusion arises from the context, we use $\Pi\Sigma\Pi$ and $\Pi\Sigma$ to stand for $\Pi_m\Sigma_s\Pi_t$ and $\Pi_m\Sigma_s$, respectively.
\subsection{The Group Algebra $F[Z_p^k$]} For any prime $p$ and integer $k \ge 2$, we consider the group $Z^k_p$ with the multiplication $\cdot$ defined as follows. For $k$-dimensional column vectors $\vec{x}, \vec{y} \in Z^k_p$ with $\vec{x} = (x_1, \ldots, x_k)^T$ and $\vec{y} = (y_1, \ldots, y_k)^T$, \begin{eqnarray} \vec{x} \cdot \vec{y} &=& (x_1+y_1 \pmod{p}, \ldots, x_k+y_k \pmod{p}). \end{eqnarray} $\vec{\bf 0}=(0, \ldots, 0)^T$ is the zero element in the group. For any field $F$, the group algebra $F[Z^k_p]$ is defined as follows. Every element $u \in F[Z^k_p]$ is a linear addition of the form \begin{eqnarray}\label{exp-2}
u &=& \sum_{\vec{x}\in Z^k_p, a_{\vec{x}}\in F} a_{\vec{x}} \vec{x}. \end{eqnarray} For any element \begin{eqnarray} v &=& \sum_{\vec{x}\in Z^k_p, b_{\vec{x}}\in F} b_{\vec{x}} \vec{x}, \nonumber \end{eqnarray} We define \begin{eqnarray}
u + v &=& \sum_{a_{\vec{x}, b_{\vec{x}}}\in F,\ \vec{x}\in Z^k_p} (a_{\vec{x}}+b_{\vec{x}}\pmod{p})
\vec{x}, \ \mbox{and} \\ u \cdot v &=& \sum_{a_{\vec{x}}, b_{\vec{y}}\in F, \mbox{ and } \vec{x}, \vec{y}\in Z^k_p} (a_{\vec{x}} b_{\vec{y}}\pmod{p}) (\vec{x}\cdot \vec{y}). \end{eqnarray} For any scalar $w \in F$, \begin{eqnarray}
w u &=& a \left(\sum_{\vec{x}\in Z^k_p, \ a_{\vec{x}}\in F} a_{\vec{x}} \vec{x}\right)
= \sum_{\vec{x}\in Z^k_p,\ a_{\vec{x}}\in F} (w a_{\vec{x}} \pmod{p})\vec{x}. \end{eqnarray} The zero element in $F[Z^k_p]$ is the one as represented in expression (\ref{exp-2}) with zero coefficients in $F$: \begin{eqnarray} {\bf 0} &=& \sum_{\vec{x}\in Z^k_p} 0 \vec{x} = 0\vec{\bf 0}. \end{eqnarray} The identity element in $F[Z^k_p]$ is \begin{eqnarray}
{\bf 1} &=& 1 \vec{\bf 0} = \vec{\bf 0}. \end{eqnarray}
For any vector $\vec{v} =(v_1, \ldots, v_k)^T \in Z_p^k$, for $i\ge 0$, let \begin{eqnarray} (\vec{v})^i &=& (i v_1 \pmod{p}, \ldots, i v_k\pmod{p})^T. \nonumber \end{eqnarray}
In particular, we have
\begin{eqnarray}
& & (\vec{v})^0 = (\vec{v})^p = \vec{\bf 0}. \nonumber \end{eqnarray}
When it is clear from the context, we will simply use $x y $ and $x+y$ to stand for $x y(\bmod{p})$ and $x+y \pmod{p}$, respectively.
\section{Randomized Testing of $p$-Monomials}\label{sec-2}
Group algebra $Z_2[Z^k_2]$ was first used by Koutis \cite{koutis08} and later by Williams \cite{williams09} to devise a randomized $O^*(2^k)$ time algorithm to test multilinear monomials in $n$-variate polynomials represented by arithmetic circuits. We shall extend $Z_2[Z^k_2]$ to $Z_p[Z^d_p]$ to test $p$-monomials for some $d>k$. Two key properties in $Z_2[Z^k_2]$, as first found by Koutis \cite{koutis08}, that are crucial to multilinear monomial testing are unfortunately no longer valid in $Z_p[Z^d_p]$. Instead, we establish new properties in Lemmas \ref{lem3} and \ref{lem4}. Also, the Schwartz-Zippel algorithm \cite{motwani95} for randomized polynomial identity testing adopted by Williams \cite{williams09} is not applicable to our case. Instead, we have to use a more advanced randomized polynomial identity testing algorithm, the Agrawal and Biswas algorithm \cite{agrawal-biswas03}.
Let $p$ be a prime number. Following conventional notations in linear algebra, for any vectors $\vec{v}_1, \ldots, \vec{v}_t \in Z^k_p$ with $k\ge 1$ and $t\ge 1$, let $\mbox{span}(\vec{v}_1, \ldots, \vec{v}_t)$ be the linear space spanned by these vectors. That is, \begin{eqnarray} \mbox{span}(\vec{v}_1, \ldots, \vec{v}_t) &=&
\{a_1\vec{v}_1+\cdots+a_t\vec{v}_t | a_1, \ldots, a_t \in Z_p\}. \nonumber \end{eqnarray}
We first give two simple properties about $\pmod{p}$ operation.
\begin{lemma}\label{lem1} For any $x, y \in Z_p$, we have $(x + y)^p = x^p + y^p \pmod{p}$. \end{lemma}
\begin{proof} $(x+y)^p = \sum^p_{i=0} (^p_i) x^{p-i} y^i
= x^p + y^p + \sum^{p-1}_{i=1} (^p_i)x^{p-i}y^{i}. $ Since $p$ is prime, $(^p_i)$ has a factor $p$, implying $(^p_i) = 0 \pmod{p}$, $1\le i\le p-1$. Hence, $(x+y)^p = x^p + y^p \pmod{p}$. \end{proof}
\begin{lemma}\label{lem2} For any $x, y \in Z_p$, we have $((p-1)x + y)^p = (p-1)x^p + y^p \pmod{p}$. \end{lemma}
\begin{proof} By Lemma \ref{lem1}, $((p-1)x + y)^p \equiv (p-1)^px^p + y^p \pmod{p}$. By Fermat's Little Theorem, $(p-1)^p = (p-1) \pmod{p}$. Thus, $((p-1)x + y)^p = (p-1)x^p + y^p \pmod{p}$. \end{proof}
The first crucial, though simple, property observed by Koustis \cite{koutis08} about testing multilinear monomials is that replacing any variable $x$ by $(\vec{v}+\vec{\bf 0})$ will annihilate $x^t$ for any $t\ge 2$, where $\vec{v}\in Z_2^k$ and $\vec{v}_0$ is the zero vector. This property is not valid in $Z_p[Z^d_p]$. However, we shall prove the following lemma that helps annihilate any monomials that are not $p$-monomials.
\begin{lemma}\label{lem3} Let $\vec{v}_0 \in Z_p^d$ be the zero vector and $\vec{v}_i \in Z_p^d$ be any vector. Then, we have \begin{eqnarray}\label{exp-lem3} ((p-1)\vec{v}_i + \vec{v}_0)^p &=& {\bf 0}, \end{eqnarray} i.e., the zero element in $Z_p[Z_p^d]$. \end{lemma}
\begin{proof} By Lemma \ref{lem2}, we have $((p-1)\vec{v}_i + \vec{v}_0)^p = (p-1)(\vec{v}_i)^p + (\vec{v}_0)^p \pmod{p}
= (p-1)\vec{v}_0 + \vec{v}_0 = p\vec{v}_0 \pmod{p} = {\bf 0}.$ \end{proof}
The second crucial property found by Koutis \cite{koutis08} has two parts: (a) Replacing variables $x_{i_j}$ in a multilinear monomial $x_{i_1}\cdots x_{i_k}$ with $(\vec{v}_{i_j}+\vec{v}_0)$ will annihilate the monomial, if the vectors $\vec{v}_{i_j}$ are linearly dependent in $Z_2^k$. (b) If these vectors are linearly independent, then the sum-product expansion of the monomial after the replacements will yield a sum of all $2^k$ vectors in $Z^k_2$. However, neither (a) nor (b) is in general true in $Z_p[Z^k_p]$. Fortunately, we have the following lemma, though not as {\em "structurally"} perfect as (b).
\begin{lemma}\label{lem4} Let $x_1^{m_1}\cdots x_{t}^{m_t}$ be any given $p$-monomial of degree $k$. If vectors $\vec{v}_1, \ldots, \vec{v}_t \in Z_p^d$ are linearly independent, then there are nonzero coefficients $c_i \in Z_p$ and distinct vector $\vec{u}_j \in Z_p^d$ such that \begin{eqnarray}\label{exp-lem4} ((p-1)\vec{v}_1 + \vec{v}_0)^{m_1} \cdots ((p-1)\vec{v}_t + \vec{v}_0)^{m_t} &=& c_1\vec{\bf 0} + \left( \sum^{(m_1+1)(m_2+1)\cdots (m_t+1)}_{i=2}\right) c_i \vec{u}_i, \end{eqnarray} where $c_1 = 1$. \end{lemma}
\begin{proof} \begin{eqnarray} & & ((p-1)\vec{v}_1 + \vec{v}_0)^{m_1} \cdots ((p-1)\vec{v}_t + \vec{v}_0)^{m_t} \nonumber \\ & & = \left(\sum^{m_1}_{i_1=0}(^{m_1}_{i_1})(p-1)^{i_1} (\vec{v}_1)^{i_1}\right) \left(\sum^{m_2}_{i_2=0}(^{m_2}_{i_2})(p-1)^{i_2} (\vec{v}_2)^{i_2}\right) \cdots \left(\sum^{m_t}_{i_t=0}(^{m_t}_{i_t})(p-1)^{i_t} (\vec{v}_t)^{i_t}\right) \nonumber \\ & & = \sum^{m_1}_{i_1=0} \sum^{m_2}_{i_2=0} \cdots \sum^{m_t}_{i_t=0} (^{m_1}_{i_1})(^{m_2}_{i_2}) \cdots (^{m_t}_{i_t}) (p-1)^{i_1+t_2\cdots+i_t} (\vec{v}_1)^{i_1} (\vec{v}_2)^{i_2} \cdots (\vec{v}_t)^{i_t} \end{eqnarray} As noted in the previous section, in the vector space $Z_p^d$, we have \begin{eqnarray}\label{exp-a} (\vec{v}_1)^{i_1} (\vec{v}_2)^{i_2} \cdots (\vec{v}_t)^{i_t} &=& i_1 \vec{v}_i + i_2 \vec{v}_2 + \cdots + t_t \vec{v}_t. \end{eqnarray} Since $\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_t$ are linearly independent, by expression (\ref{exp-a}) we have \begin{eqnarray} \label{exp-b} (\vec{v}_1)^{i_1} (\vec{v}_2)^{i_2} \cdots (\vec{v}_t)^{i_t} = \vec{\bf 0} &\mbox{iff} & i_1 =i_2 = \cdots = i_t = 0. \end{eqnarray} The linear independence of $\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_t$ implies that any non-empty subset of these vectors are also linearly independent. Similar to expression (\ref{exp-b}), this further implies that, for any $0\le j_i\le m_i$, $i=1, 2,\ldots, t$, \begin{eqnarray}\label{exp-c} (\vec{v}_1)^{i_1} (\vec{v}_2)^{i_2} \cdots (\vec{v}_t)^{i_t} = (\vec{v}_1)^{j_1} (\vec{v}_2)^{j_2} \cdots (\vec{v}_t)^{j_t} &\mbox{iff} & i_1 =j_1, i_2=j_2, \ldots, \mbox{ and } i_t = j_t. \end{eqnarray} Furthermore, since $p$ is prime and $m_i \in Z_p$, we have \begin{eqnarray}\label{exp-d} c(i_1,i_2,\ldots c_t) &= &(^{m_1}_{i_1})(^{m_2}_{i_2}) \cdots (^{m_t}_{i_t}) (p-1)^{i_1+t_2\cdots+i_t} \pmod{p} \nonumber \\
&\not=& 0 \pmod{p} \end{eqnarray} Combining expressions (\ref{exp-c}) and (\ref{exp-d}), we have \begin{eqnarray}\label{exp-e} && ((p-1)\vec{v}_1 + \vec{v}_0)^m_i \cdots ((p-1)\vec{v}_t + \vec{v}_0)^m_t \nonumber \\ && = 1 \vec{\bf 0} + \left(\sum_{0\le i_j\le m_j,\ 0\le j\le t,\ \mbox{\ and\ } i_1+i_2\cdots+i_t>0} c(i_1,i_2,\ldots,i_t) \cdot ((\vec{v}_1)^{i_1} (\vec{v}_2)^{i_2} \cdots (\vec{v}_t)^{i_t}) \right). \end{eqnarray} In the above expression (\ref{exp-e}), all the coefficients are nonzero, and all the $(m_1+1)(m_2+1) \cdot (m_t+1) \le p^k$ vectors are distinct. Hence, expression (\ref{exp-lem4}) is obtained. \end{proof}
{\bf Remark.} Lemma \ref{lem4} guarantees that replacing variables in a $p$-monomial by linearly independent vectors will prevent the monomial from being annihilated. Note that the total number of distinct vectors in expression \ref{exp-lem4} is at most $p^k$.
Lemmas \ref{lem3} and \ref{lem4} have laid a basis for designing randomized algorithms to test $p$-monomials. One additional help will be drawn from randomized polynomial identity testing over a finite field. We are ready to present the algorithm and show how to integrate group algebra with polynomial identity testing to aid our design. To simplify description, we assume, like in Koutis \cite{koutis08} and Williams \cite{williams09}, that the degree of $p$-monomials in a polynomial is at least $k$, provided that such monomials exist. Otherwise, we can simply multiply some new variables to the given polynomial to satisfy the requirement.
\begin{theorem}\label{thm-1} Let $p$ be a prime number. Let $F(x_1,x_2,\ldots,x_n)$ be an $n$-variate polynomial of degree $k$ represented by an arithmetic circuit $C$ of size $s(n)$. There is a randomized $O^*(p^k)$ time algorithm to test with high probability whether $F$ has a $p$-monomial of degree $k$ in its sum-product expansion. \end{theorem}
\begin{proof} Let $d = k+\log_p k + 1$, we consider the group algebra $Z_p[Z^d_p]$. As in Williams \cite{williams09}, we first expand the circuit $C$ to a new circuit $C'$ as follows. For each multiplication gate $g_i$, we attach a new gate $g'_i$ that multiplies the output of $g_i$ with a new variable $y_i$, and feed the output of $g'_i$ to the gate that reads the output of $g_i$. Assume that $C$ has $h$ multiplications gates. Then, $C'$ will have $h$ new multiplications gates corresponding to new variables $y_1, y_2, \ldots, y_h$. Let $F'(y_1, y_1, \ldots, y_h, x_1, x_2, \ldots, x_n)$ be he new polynomial represented by $C'$. The algorithm for testing whether $F$ has a $p$-monomial of degree $k$ is given in the following.
\begin{quote} Algorithm \mbox{RT-MLM} (\underline{R}andomized \underline{T}esting of \underline{M}ulti\underline {l}inear \underline{M}onomials): \begin{enumerate} \item Select uniform random vectors $\vec{v}_1,\ldots,\vec{v}_n \in Z^d_p-\{\vec{\bf 0}\}$. \item Replace each variable $x_i$ with $(\vec{v}_i + \vec{v}_0)$, $1\le i \le n$.
\item Use $C'$ to calculate \begin{eqnarray}\label{exp-thm-rt-mlm} F'(y_1,\ldots,y_h,(\vec{v}_1+\vec{v}_0),\ldots,(\vec{v}_n+\vec{v}_0)) & = & \sum_{j=1}^{2^d} f_j(y_1,\ldots,y_h) \cdot \vec{z}_j, \end{eqnarray} where each $f_j$ is a polynomial of degree $k$ over the finite field $Z_p$, and $\vec{z}_j$ with $1\le j\le 2^d$ are the $2^d$ distinct vectors in $Z^d_p$. \item Perform polynomial identity testing with the Agrawal and Biswas algorithm \cite{agrawal-biswas03} for every $f_{j}$ over $Z_p$. Return {\em "yes"} if one of them is not identical to zero, or {\em "no"} otherwise. \end{enumerate} \end{quote}
It follows from Lemma \ref{lem3} that all monomials that are not $p$-monomials in $F$ (and hence in $F'$) will become zero, when variables $x_i$ is replaced by $(\vec{v}_i + \vec{v}_0)$ at Step ii. We shall estimate that with high probability some $p$-monomials will survive from those replacements, i.e., will not become the zero element $\bf{0}$ in $Z_p[Z^d_p]$.
Consider any given $p$-monomial $\pi = x_{i_1}^{m_1}\cdots x_{i_t}^{m_t}$ of degree $k$ with $1\le m_i < p$ and $k= m_1 +\cdots+m_t$, $i=1,\ldots,t$. For any $1\le j \le t$, \begin{eqnarray} \mbox{Pr}\left[\vec{v}_j \in \mbox{span}(\vec{v}_{i_1},\ldots, \vec{v}_{i_{j-1}})\right] & = & \frac{p^{j-1}}{p^d}, \nonumber \end{eqnarray}
since $|\mbox{span}(\vec{v}_{i_1},\ldots, \vec{v}_{i_{j-1}})| =
p^{j-1}$ and $|Z_p^d| = p^d$. Hence, \begin{eqnarray}\label{prop} & & \mbox{Pr}\left[(\exists j \in \{1,\ldots, t\}) [ \vec{v}_{i_j} \in \mbox{span}(\vec{v}_{i_1},\ldots, \vec{v}_{i_{j-1}})]\right] \nonumber \\ & & = \mbox{Pr}\left[ [\vec{v}_1 = \vec{\bf 0}] \vee [ \vec{v}_{i_2} \in \mbox{span}(\vec{v}_{i_1})] \vee \cdots \vee [ \vec{v}_{i_t} \in \mbox{span}(\vec{v}_{i_1},\ldots, \vec{v}_{i_{t-1}})] \right] \nonumber \\ & & \le \mbox{Pr}[ \vec{v}_1 = \vec{\bf 0}] + \mbox{Pr}[\vec{v}_{i_2} \in \mbox{span}(\vec{v}_{i_1})] + \cdots + \mbox{Pr}[\vec{v}_{i_t} \in \mbox{span}(\vec{v}_{i_1},\ldots, \vec{v}_{i_{t-1}})] \nonumber \\ && = \frac{p^0}{p^d} + \frac{p^1}{p^d} + \cdots + \frac{p^{t-1}}{p^d} \le t \frac{p^{t-1}}{p^d} \nonumber \\ & & \le k \frac{p^{k-1}}{p^{k+\log_p k +1}} \le \frac{1}{p^2} \le \frac{1}{4}. \end{eqnarray} Because $\vec{v}_{i_1},\ldots,\vec{v}_{i_t}$ are linearly independent iff there is no $\vec{v}_{i_j} \in \mbox{span}(\vec{v}_{i_1},\ldots, \vec{v}_{i_{j-1}})$, by expression (\ref{prop}) the probability that $\vec{v}_{i_1},\ldots,\vec{v}_{i_t}$ are linearly independent is at least $\frac{3}{4}$. This implies, by Lemma \ref{lem4}, that the monomial $\pi$ will survive from the replacements at Step ii with probability at least $\frac{3}{4}$. Furthermore, by expression (\ref{exp-lem4}) in Lemma \ref{lem4}, \begin{eqnarray}\label{new-poly} ((p-1)\vec{v}_1 + \vec{v}_0)^{m_i} \cdots ((p-1)\vec{v}_t + \vec{v}_0)^{m_t} &=& \sum^{p^k}_{i=1}c(\pi)_i \vec{u}_i(\pi), \end{eqnarray} where $c(\pi)_i$ are coefficients in $Z_p$ such that $(m_1+1)(m_2+1)\cdots (m_t+1)$
of them are nonzero, and $\vec{u}_i(\pi)$ are distinct vectors in $Z^d_p$. Let $\psi(\pi)$ be the product of the new variables $y_j$ that are added with respect to the gates in $C$ such that those gates produce the monomial $\pi$. Then, $\psi(\pi)$ is a monomial that is generated by $C'$. Hence, at Step iii, by expression (\ref{new-poly}) $F'$ will have monomials respect to $\pi$ as given in the following expansion: \begin{eqnarray}\label{new-monomial} \phi(\pi) & = & \psi(\pi) \cdot ((p-1)\vec{v}_1 + \vec{v}_0)^{m_i} \cdots ((p-1)\vec{v}_t + \vec{v}_0)^{m_t} \nonumber \\
&=& \sum^{p^k}_{i=1}c(\pi)_i \cdot \psi(\pi) \cdot \vec{u}_i(\pi). \end{eqnarray} Let ${\cal S}$ be the set of all those $p$-monomials that survive from the variable replacements. Then, \begin{eqnarray}\label{new-F} &&F'(y_1,\ldots,y_h,(\vec{v}_1+\vec{v}_0),\ldots,(\vec{v}_n+\vec{v}_0)) = \sum_{\pi \in {\cal S}} \phi(\pi) \nonumber \\ & & = \sum_{\pi\in {\cal S}} \left(\sum^{p^k}_{i=1}c(\pi)_i \cdot \psi(\pi) \cdot \vec{u}_i(\pi)\right) \nonumber \\ & & = \sum_{j=1}^{2^d} \left(\sum_{\pi\in {\cal S} \mbox{ and } \vec{z}_j = \vec{u}_i(\pi)} c(\pi)_i \cdot \psi(\pi) \right) \cdot \vec{z}_j \end{eqnarray} Let \begin{eqnarray} f_j(y_1,\ldots,y_h) & = & \sum_{\pi\in {\cal S} \mbox{ and } \vec{z}_j = \vec{u}_i(\pi)} c(\pi)_i \cdot \psi(\pi), \nonumber \end{eqnarray} then the degree $k$ polynomial with respect to $\vec{z}_j$ is obtained for $F'$ in expression (\ref{exp-thm-rt-mlm}).
Recall that when constructing the circuit $C'$, each new gate is associated with a new variable. This means that for any two monomials $\pi'$ and $\pi''$ in $F$, we have $\psi(\pi')\not= \psi(\pi'')$. This implies that we cannot add $c(\pi') \cdot \psi(\pi')$ to $c(\pi'') \cdot \psi(\pi'')$ in $f_j$. Thus, the possibility of a {\rm "zero-sum"} of coefficients from different surviving monomials is completely avoided during the construction of $f_j$. Therefore, conditioned on that ${\cal S}$ is not empty, $F'$ must not be identical to zero, i.e., there exists at least one $f_j$ that is not identical to zero. At Step iv, we use the randomized algorithm by Agrawal and Biswas \cite{agrawal-biswas03} to test whether $f_j$ is identical to zero. It follows from Theorem 4.6 in Agrawal and Biswas \cite{agrawal-biswas03} that this testing can be done with probability at least $\frac{5}{6}$ in time polynomially in $s(n)$ and $\log q$. Since ${\cal S}$ is not empty with probability at least $\frac{3}{4}$, the probability of overall success of testing whether $F$ has a $p$-monomial is at least $\frac{5}{8}$.
Finally, we address the issues about how to calculate $F'$ and the time needed to do so. Naturally, every element in the group algebra $Z_p[Z^d_p]$ can be represented by a vector in $Z^{p^d}_p$. Adding two elements in $Z_p[Z^d_p]$ is equivalent to adding the two corresponding vectors in $Z_p^{p^d}$, and the latter can be done in $O(p^d \log p)$ time via component-wise sum. In addition, multiplying two elements in $Z_p[Z^d_p]$ is equivalent to multiplying the two corresponding vectors in $Z_p^{p^d}$, and the latter can be done in $O(dp^d\log^2 p)$ with the help of a similar Fast Fourier Transform style algorithm as in Williams \cite{williams09}. Calculating $F'$ consists of $s(n)$ arithmetic operations of either adding or multiplying two elements in $Z_p[Z^d_p]$ based on the circuit $C$ or $C'$. Hence, the total time needed is $O(s(n) d p^d log ^2 p)$. At Step iv, we run the Agrawal and Biswas \cite{agrawal-biswas03} algorithm to $F'$ to simultaneously testing whether there is one $f_j$ such that $f_j$ is not identical to zero. We choose a probability $\frac{5}{6}$, the by Theorem 4.6 in Agrawal and Biswas \cite{agrawal-biswas03}, this testing can be done in $O^*((s(n))^4 n^4 log^2 p)$ time, suppressing a $\mbox{poly}(\log s(n), \log n, \log \log p)$ factor. Recall that $d = k+log_p k +1$. The total time for the entire algorithm is $O^*(p^k)$. \end{proof}
\section{Derandomization}
In this section, we turn our attention to formulas instead of general arithmetic circuits and shall design a deterministic algorithm to test $p$-monomials for polynomials represented by a formula. Recall that the algorithm RT-MLM has only two randomized processes at Step i to select $n$ uniform random variables and at Step iv to test whether one $f_j$ from $F'$ is identical to zero over $Z_p$. In this section, we shall derandomize these two randomized processes respectively with the help of two advanced techniques of perfect hashing by Chen {\em at al.} \cite{jianer-chen07} and Naor {\em at el.} \cite{naor95} and noncommunicative multivariate polynomial identity testing by Raz and Shpilka \cite{raz05}.
Let $n$ and $k$ be two integers such that $1\le k\le n$. Let ${\cal A} =\{1, 2, \ldots, n\}$ and ${\cal K} = \{1, 2, \ldots, k\}$. A $k$-coloring of the set ${\cal A}$ is a function from ${\cal A}$ to ${\cal K}$. A collection ${\cal F}$ of $k$-colorings of ${\cal A}$ is a $(n,k)$-family of {\em perfect hashing functions} if for any subset $W$ of $k$ elements in ${\cal A}$, there is a $k$-coloring $h \in {\cal F}$ that is injective from $W$ to ${\cal K}$, i.e., for any $x, y \in W$, $h(x)$ and $h(y)$ are distinct elements in ${\cal K}$.
\begin{theorem}\label{thm-2} Let $p$ be a prime number. Let $F(x_1,x_2,\ldots,x_n)$ be an $n$-variate polynomial of degree $k$ represented by a formula $C$ of size $s(n)$. There is a deterministic $O(6.4^k + p^k)$ time algorithm to test whether $F$ has a $p$-monomial of degree $k$ in its sum-product expansion. \end{theorem}
\begin{proof} As in the proof of Theorem \ref{thm-1}, we consider the group algebra $Z_p[Z^k_p]$. Here, we do not need to expand the dimension $k$ to $d>k$. We also construct a new formula $C'$ from $C$ by adding new variable $y_i$ for each multiplication gate $g_i$ in the same way as what we did for Theorem \ref{thm-1}. Assume that $C$ has $h$ many multiplication gates, then $C'$ will have $h$ new multiplication gates corresponding to new variables $y_1, y_2, \ldots, y_h$. The algorithm for testing whether $F$ has a $p$-monomial of degree $k$ is given as follows.
\begin{quote} Algorithm \mbox{DT-MLM} (\underline{D}eterministic \underline{T}esting of \underline{M}ulti\underline{l}inear \underline{M}onomials): \begin{enumerate} \item Construct with the algorithm by Chen {\em at el.} \cite{jianer-chen07} an $(n,k)$-family of perfect hashing functions ${\cal H}$ of size $O(6.4^k\log^2 n)$. \item Select $k$ linearly independent vectors $\vec{v}_1,\ldots,\vec{v}_k \in Z^k_p$. (No randomization is needed at this step.) \item For each perfect hashing function $\tau \in{\cal H}$ do \begin{quote} a. For each variable $x_i$, replace it by $(\vec{v}_{\tau(i)} + \vec{v}_0)$.
b. Use $C'$ to calculate \begin{eqnarray}\label{exp-thm-dt-mlm} & &F'(y_1,\ldots,y_h,(\vec{v}_1+\vec{v}_0),\ldots,(\vec{v}_n+\vec{v}_0)) \nonumber \\ && = \sum_{j=1}^{2^k} f_j(y_1,y_2,\ldots,y_h) \cdot \vec{z}_j, \end{eqnarray} where each $f_j$ is a polynomial of degree $k$ over the finite field $Z_p$, and vectors $\vec{z}_j$ with $1\le j\le 2^k$ are the $2^k$ distinct vectors in $Z^k_p$.
c. Perform polynomial identity testing with the Raz and Shpilka algorithm \cite{raz05} for every $f_j$ over $Z_p$. Stop and return {\em "yes"} if one of them is not identical to zero. \end{quote} \item[iv.] If all perfect hashing functions in ${\cal H}$ have been tried without returning {\em "yes"}, then stop and output {\em "no"}. \end{enumerate} \end{quote}
By Chen {\em at el.}\cite{jianer-chen07}, Step i can be done in $O(6.4^k n \log^2 n)$ times. Step ii can be easily done in $O(k^2\log p)$ time.
It follows from Lemma \ref{lem3} that all those monomials that are not $p$-monomials in $F$, and hence in $F'$, will be annihilated, when variables $x_i$ are replaced by $(\vec{v}_i + \vec{v}_0)$ at Step iii.a.
Consider any given $p$-monomial $\pi = x_{i_1}^{m_1}\cdots x_{i_t}^{m_t}$ of degree $k$ with $1\le m_i < p$ and $k= m_1 +\cdots+m_t$, $i=1,\ldots,t$. Because of the nature of ${\cal H}$, there is at least one perfect hashing function $\tau$ in ${\cal H}$ such that $\tau(i_{j'}) \not= \tau(i_{j''})$ if $i_{j'} \not= i_{j''}$, $1\le j', j'' \le t \le k$. This means that $\vec{v}_{\tau(i_1)}, \ldots, \vec{v}_{\tau(i_t)}$ are distinct and hence linearly independent. By Lemma \ref{lem4}, $\pi$ will survive from the replacements at Step iii.a. Let ${\cal S}$ be the set of all surviving $p$-monomials. Following the same analysis as in the proof of Theorem \ref{thm-1}, we have $F'$ that is not identical to zero if ${\cal S}$ is not empty. That is, there is at least one $f_j$ that is not identical to zero, if ${\cal S}$ is not empty. Moreover, the time needed for calculating $F'$ is $O(kp^k\log^2 p)$.
We now consider imposing noncommunicativity on $C'$ as follows. Inputs to an arithmetic gate are ordered so that the formal expressions $y_{i_1}\cdot y_{i_2} \cdot \cdots \cdot y_{i_r}$ and $y_{j_1}\cdot y_{j_2} \cdot \cdots \cdot y_{j_l}$ are the same iff $r=l$ and $i_q =j_q$ for $q=1,\ldots,r$. Finally, we use the algorithm by Raz and Shpilka \cite{raz05} to test whether $f_j(y_1,\ldots,y_h)$ is identical to zero of not. This can be done in time polynomially in $s(n)$ and $n$, since $f_j$ is a non-communicative polynomial represented by a formula.
Combining the above analysis, the total time of the algorithm DT-MLM is $O(6.4^k n \log^2 n + k p^k (s(n) n)^{O(1)}\log^2 p) = O^*(6.4^k + p^k)$. \end{proof}
\section{$\Pi_m\Sigma_2\Pi_t \times \Pi_k\Sigma_3$ Polynomials}
It has been proved by Chen and Fu \cite{chen-fu10} that the problem of testing monomials in $\Pi_m\Sigma_s$ polynomials is solvable in $(ms\sqrt{m+s})$ time, and in $\Pi_m\Sigma_2\Pi_t$ polynomials is in $O((mt)^2)$ time. On the other hand, it has also been proved by in \cite{chen-fu10} that the problem for $\Pi_m\Sigma_3$ and $\Pi_m\Sigma_2\Pi_t \times \Pi_k\Sigma_3$ polynomials is respectively NP-complete. Moreover, a $O(tm^2 1.7751^m)$ time algorithm was obtained for $\Pi_m\Sigma_3\Pi_t$ polynomials, and so was a $O((mt)^2 3^k)$ algorithm obtained for $\Pi_m\Sigma_2\Pi_t \times \Pi_k\Sigma_3$ polynomials. In this section, we shall devise two parameterized algorithms, one deterministic and the other randomized, for testing multilinear monomials in $\Pi_m\Sigma_2\Pi_t \times \Pi_k\Sigma_3$ polynomials, improving the $O((mt)^2 3^k)$ upper bound in \cite{chen-fu10}.
\begin{theorem}\label{yang-thm-1} There is a deterministic algorithm of time $O(((mt+k)^2+k) 2^k)$ to test whether any $\Pi_m\Sigma_2\Pi_t\times\Pi_k\Sigma_3$ polynomial has a multilinear monomial in its sum-product expansion. \end{theorem}
\begin{proof} Let $F = F_1 \cdot F_2$ such that $F_1 = f_1\cdots f_m$ is a $\Pi_m\Sigma_2\Pi_t$ polynomial and $F_2=g_1\cdots g_k$ is a $\Pi_k\Sigma_3$ polynomial, where $f_i = (T_{i1}+T_{i2})$ and $g_j=(x_{j1}+x_{j2}+x_{j3})$, $1\le i\le m$, $1\le j\le k$.
Consider variable $x_{11}$ in the clause $g_1$. We devise a branch and bound process to divide the testing for $F$ into the testing for two new polynomials. We eliminate all $x_{11}$ in $g_j$ for $j=1, \ldots, k$. Let $g'_j$ be the clause resulted from $g_j$ after the eliminating process. Let $h_1 = F_1 \cdot g'_1$, $h_2 = F_1 \cdot x_{11}$, $q = g'_2 \ldots g'_k$. Note that exactly one of the three variable $x_{11}, x_{12}$ and $x_{13}$ in the clause $g_1$ must be selected to form a monomial (hence a multilinear monomial) for $F$ in the sum-product expansion of $F$. We have two cases concerning the selection of $x_{11}$:
(1) $x_{11}$ can not be selected to help form any multilinear monomial. In this case, $F$ has a multilinear monomial, iff $h_1 \cdot q$ has a multilinear monomial.
(2) $x_{11}$ can be selected to form a multilinear monomial. Thus, $F$ has a multilinear monomial, iff $h_2 \cdot q$ has a multilinear monomial.
In either case, the new polynomial is a product of two polynomials with the first being a $\Pi_{m+1}\Sigma_2\Pi_t$ polynomial and the second a $\Pi_k\Sigma_3$ polynomial. Furthermore, the second is the common $q$, which has one fewer clause than $F_2$.
Let $T(k)$ denote the time for testing multilinear monomials in $F$. Notice that the eliminating process for $x_{11}$ takes $O(k)$ time. Then, $T(k)$ is bounded as follows $$ T(k) \le 2T(k-1) + O(k) \le 2^k(T(0) + O(k)). $$ $T(0)$ is the time to test multilinear monomials in a $\Pi_{m+k}\Sigma_2\Pi_t$ polynomial with a size of $O(mt+k)$. By the algorithm in \cite{chen-fu10} for this type of polynomials, $T(0) = O((mt+k)^2)$. Therefore, $T(k) = O(((mt+k)^2 + k) 2^k)$. \end{proof}
We now show that the upper bound in the above theorem can be further improved via randomization.
\begin{theorem}\label{yang-thm-2} There is a $O((mt+k)^2 1.5^k))$ time randomized algorithm that finds a multilinear monomial for any $\Pi_m\Sigma_2\Pi_t\times\Pi_k\Sigma_3$ polynomial with probability at least $1-\frac{1}{e}$ if such monomials exist, or returns {\em "no''} otherwise. \end{theorem}
\begin{proof} Like in Theorem \ref{yang-thm-1}, let $F = F_1 \cdot F_2$ such that $F_1 = f_1\cdots f_m$ is a $\Pi_m\Sigma_2\Pi_t$ polynomial and $F_2=g_1\cdots g_k$ is a $\Pi_k\Sigma_3$ polynomial with $f_i = (T_{i1}+T_{i2})$ and $g_j=(x_{j1}+x_{j2}+x_{j3})$.
Assume that $F$ has a multilinear monomial $\pi$. Then, one of the three variables in $g_j$ must be included in $\pi$, $1\le j \le k$. We uniformly select two distinct variables $y_{j1}$ and $y_{j2}$ from $g_j$, then $g'_j = (y_{j1}+y_{j2})$ contains a desired variable for $\pi$ with a probability at least $2/3$. Let $$ F'=F_1\cdot (g'_1 \cdots g'_k), $$ then $F'$ has a multilinear monomial with a probability at least $(\frac{2}{3})^k$. On the other hand, if $F$ does not have any multilinear monomials in its sum-product expansion, then $F'$ must not have any multilinear monomials. Notice that $F'$ is a $\Pi_{m+k}\Sigma_2\Pi_t$ polynomial with a size of $O(mt+k)$. By the algorithm for this type of polynomials by Chen and Fu in \cite{chen-fu10}, one can find a multilinear monomial in $F'$ in time $O((mt+k)^2)$. In other words, the above randomized process will fail to find a multilinear monomial in $F$ with a probability of at most $1-\left(\frac{2}{3}\right)^k$ if such monomials exist, or return {\em "no''} otherwise.
Repeat the above randomized process $\left(\frac{3}{2}\right)^k$ many times. If $F$ has multilinear monomials, then these processes will fail to find one with a probability of at most $$ \left[1-\left(\frac{2}{3}\right)^k\right]^{\left(\frac{3}{2}\right)^k} < \frac{1}{e}. $$ Hence, the processes will find a multilinear monomial in $F$ with a probability of at least $1-\frac{1}{e}$. If $F$ does not have any multilinear monomial, then none of these repeated processes will find one in $F$. The total time of all the repeated processes is $O((mt+k)^2 1.5^k)$. \end{proof}
It is justified in \cite{chen-fu10} that the resemblance of $\Pi\Sigma\Pi$ and $\Pi\Sigma$ polynomials with SAT formulas is {\em "superficial"}. For example, The multilinear monomial testing problem for $\Pi_m\Sigma_3\Pi_1$ polynomials is in P, but 3SAT is NP-complete. As another example to show such superficial resemblance, one might consider to apply Sch\"{o}ning's algorithm for 3SAT \cite{schoning02} to the multilinear monomial testing problem. However, this is problematic. For the 3SAT problem, it is easy to find an unsatisfied 3-clause. On the other hand, for the multilinear monomial testing problem, we do not know which term in which clause leads to a confliction. Therefore, it is difficult to decide the change of the hamming distance between the current solution and any target solution. This difficulty constitutes a major barrier towards applying the Sch\"{o}ning's algorithm to monomial testing.
\section{W[1]-Hardness}
Although deterministic and randomized parameterized algorithms have been devised for testing monomials in previous three sections as well as in \cite{koutis08,williams09,chen-fu10}, yet we shall prove in this section that testing some special type of monomials in polynomials represented by arithmetic circuits is not fixed-parameter tractable, unless some unlikely collapse occurs in the fixed parameter complexity theory.
One shall notice that difference between the general monomial testing and the specific monomial testing. The former asks for the existence of {\em "any one''} from a set of possibly many monomials that are needed. The latter asks for {\em "a specific one''} from the set. For example, there may be $2^n -1$ multilinear monomials in the sum-product expansion of a $n$-variate polynomials. Testing for any one from these many monomials is certainly different from testing for a specific one, say, $x_1x_3x_7x_{11}$.
Downey and Fellows \cite{downey-fellows95} have established a hierarchy of parameterized complexity, named the W hierarchy, and proved that the $k$-Clique problem is W[1]-hard.
\begin{definition} Let $C = \{i_1, i_2, \ldots, i_k\}$ be a set of $k$ positive integers. A $k$-clique monomial with respect to $C$ is the multilinear monomial $\prod_{1\le j < \ell \le k}x_{i_j i_\ell}$ of degree $\frac{k(k-1)}{2}$. \end{definition}
\begin{theorem} It is W[1]-hard to test whether any given $n-$variate polynomial of degree $\frac{k(k-1)}{2}$ represented by an arithmetic circuit has a $k$-clique monomial in its sum-product expansion. \end{theorem}
\begin{proof}
We shall reduce the $k$-clique problem to the $k$-clique monomial testing problem. Let $G=(V,E)$ be an undirected graph and $k$ an integer parameter. $V=\{v_1, v_2, \ldots, v_m\}$ is the set of vertices. Each $(i, j) \in E$ represents the edge connecting vertices $v_i$ and $v_j$. For each edge $(i, j) \in E$, we define a variable $x_{ij}$. Let $n= |E|$. We construct a polynomial $f$ with $n$ variables. \begin{eqnarray} f(G, 1) & = & 1, \nonumber \\ f(G, 2) &=& \sum_{(i, j)\in E} x_{ij}, \nonumber\\ f(G, t+1) & = & \sum^{m}_{i=1} \left(\sum_{(i,j)\in E} x_{ij}\right)^t \cdot f(G, t) \nonumber \end{eqnarray}
As followed from the above definition, $f(G, k)$ has $n=|E|$ variables and its degree is $\frac{k(k-1)}{2}$. It is easy to see that $f(G, k)$ can be computed by an arithmetic circuit.
If $G$ has a $k$-clique $A=\{i_1,i_2\ldots,i_k\}$, then there are $\frac{k(k-1)}{2}$ edges connecting any two vertices in $A$. By definition, $f(G, k)$ has a term $(x_{i_1 i_2} + \cdots + x_{i_1 i_k} + \cdots + x_{i_{k-1}i_k})^{k-1} \cdot f(G, k-1).$ So, we can select $\pi_1 =x_{i_1 i_2} \cdots x_{i_1 i_k}$ from the first factor of this term. By simple induction, we can select a $(k-1)$-clique monomial of degree $\frac{(k-1)(k-2)}{2}$ with respect to $A - \{i_1\}$. Then, $\pi_1 \cdot \pi_2$ is a $k$-clique monomial with respect to $A$. On the other hand, it $f(G, k)$ has a $k$-clique monomial with respect to $A$, then by definition, $A$ is a $k$-clique for $G$. \end{proof}
\section*{Acknowledgments}
We thank Ioannis Koutis for helping us understand his group algebra in \cite{koutis08}. Bin Fu's research is supported by an NSF CAREER Award, 2009 April 1 to 2014 March 31.
\end{document}
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\begin{document}
\setlength\parindent{15pt} \setlength\parskip{6pt}
\title{Optimization of Triangular Networks with Spatial Constraints}
\author{ Valentin R.\ Koch\footnote{ Design and Creation Products (DCP), Autodesk, Inc. Email: \texttt{[email protected]}} ~and~ Hung M.\ Phan \footnote{ Department of Mathematical Sciences, Kennedy College of Sciences, University of Massachusetts Lowell, MA~01854, USA. E-mail: \texttt{hung\[email protected]}} }
\date{April 03, 2019}
\maketitle
\begin{abstract} A common representation of a three dimensional object in computer applications, such as graphics and design, is in the form of a triangular mesh. In many instances, individual or groups of triangles in such representation need to satisfy spatial constraints that are imposed either by observation from the real world, or by concrete design specifications of the object. As these problems tend to be of large scale, choosing a mathematical optimization approach can be particularly challenging. In this paper, we model various geometric constraints as convex sets in Euclidean spaces, and find the corresponding projections in closed forms. We also present an interesting idea to successfully maneuver around some important nonconvex constraints while still preserving the intrinsic nature of the original design problem. We then use these constructions in modern first-order splitting methods to find optimal solutions. \end{abstract}
\noindent{\small {\bf AMS Subject Classification:}
Primary
26B25,
65D17;
Secondary
49M27,
52A41,
90C25.}
\noindent{\bf Keywords:}
alignment constraint,
convex optimization,
Douglas--Rachford splitting,
maximum slope,
minimum slope,
oriented slope,
projection methods.
\section{Introduction and Motivation}
The notation in the paper is fairly standard and follows largely \cite{BC2017}. $\ensuremath{\mathbb R}$ denotes the set of real numbers. By ``$x:=y$", or equivalently ``$y=:x$", we mean that ``$x$ is defined by $y$". The {\em assignment operators} are denoted by ``$\longleftarrow $" and ``$\longrightarrow$". The angle between two vectors $\vec{n}$ and $\vec{q}$ is denoted by $\angle(\vec{n},\vec{q})$. The cross product of $\vec{n}_1$ and $\vec{n}_2$ in $\ensuremath{\mathbb R}^3$ is denoted by $\vec{n}_1\times\vec{n}_2$.
\subsection{Abstract Problem Formulation} \label{ss:abs_prob}
Throughout the paper, we assume that $n\in\{3,4,\ldots\}$ and $X=\ensuremath{\mathbb R}^n$ with standard inner product $\scal{\cdot}{\cdot}$ and induced norm $\|\cdot\|$. Assume that $G=(V_0,E_0)$ is a given {\em triangular mesh} on $\ensuremath{\mathbb R}^2$ where $V_0$ is the set of vertices and $E_0$ is the set of (undirected) edges, i.e., \begin{align*} V_0&:=\menge{p_i:=(p_{i1},p_{i2})\in\ensuremath{\mathbb R}^2} {i\in\{1,2,\ldots,n\}},\\ E_0&\subseteq\menge{ p_ip_j\equiv p_jp_i}{ p_i,p_j\in V_0}. \end{align*} From $G$, we can derive the set of triangles of the mesh as follows \begin{equation*} T_0:=\menge{\Delta=(p_ip_jp_k)}{\{p_ip_j,p_jp_k,p_kp_i\}\subseteq E_0}. \end{equation*} We first aim to \begin{subequations}\label{e:feas} \begin{equation} \text{find a vector}\quad z=(z_1,\ldots,z_n)\in X \end{equation} such that the points \begin{equation} \{P_i:=(p_{i1},p_{i2},z_i)\}_{i\in\{1,\ldots,n\}} \quad\text{satisfy a given set of constraints.} \end{equation} \end{subequations} Clearly, the points $\{P_i\}_{i\in\{1,\ldots,n\}}$ also form a corresponding triangular mesh $S$ in three dimensions. Therefore, if there is no confusion, we will also use $E_0,V_0,T_0$ to denote the sets of vertices, edges, and triangles of the three dimensional mesh.
Several types of constraints for triangular meshes are listed below: \begin{itemize} \item {\bf interval constraints:} For a given subset $I$ of $\{1,2,\ldots,n\}$, for all $i\in I$, the entries $z_i$ must lie in a given interval of $\ensuremath{\mathbb R}$. \item {\bf edge slope constraints:} For a given subset $E$ of the edges $E_0$, and for every edge $P_iP_j\in E$, the slope \begin{equation*} s_{ij}:=\frac{z_i-z_j}{ \sqrt{(p_{i1}-p_{j1})^2+(p_{i2}-p_{j2})^2}} \end{equation*} must lie in a given subset of $\ensuremath{\mathbb R}$.
\item {\bf edge alignment constraints:} For a given pair of edges $P_iP_j, P_mP_n \in E_0$, the edges must have the same slope $s_{ij} = s_{mn}$.
\item {\bf surface alignment constraints:} For a given pair of triangles $\Delta_i,\Delta_j \in T_0$ the \emph{normal vectors} $\vec{n}_{\Delta_i}$ and $\vec{n}_{\Delta_j}$ must be parallel.
\item {\bf surface orientation constraints:} For a given subset $T$ of the triangles $T_0$ and for each triangle $\Delta\in T$, there is a given set of vectors $Q_\Delta\subset\ensuremath{\mathbb R}^3$ such that for each $\vec{q}\in Q_\Delta$, the {\em angle} between the {\em normal vector} $\vec{n}_\Delta$ and $\vec{q}$ must lie in a given subset $\theta_{\vec{q}}$ of $\left[0,\pi\right]$. \end{itemize}
Suppose there are $J$ constraints imposed on a model. For $j\in\{1,2,\ldots,J\}$, we denote by $C_j$ the set of all points that satisfies the $j$-th constraint. Thus, \eqref{e:feas} can be written in the mathematical form \begin{equation}\label{e:feas2} \text{find a point}\quad x\in C:=\bigcap_{j\in\{1,2,\ldots,J\}}C_j, \end{equation} which we refer to as {\em feasibility problem}. Moreover, of the infinitude of possible solutions for \eqref{e:feas2}, one may be particularly interested in those that are {\em optimal} in some sense. For instance, it could be desirable to find a solution that minimizes the slope change between adjacent triangles, a solution that minimizes the volume between the initial triangles and the triangles in the final solution, or variants and combinations thereof. If more than one objective function is of interest, it is common to additively combine these functions, perhaps by scaling the functions based on their different levels of importance. In summary, our goal is to solve the problem \begin{equation}\label{e:fprob} \min\quad{F(z)} \quad\text{subject to}\quad z\in C:=\bigcap_{j\in\{1,2,\ldots,J\}}C_j, \end{equation} where $F$ may be a sum of (scaled) objective functions. The function $F$ is typically nonsmooth which prevents the use of standard optimization methods.
\subsection{Computer-Aided Design for Architecture and Civil Engineering Structures} \label{ss:civil}
The abstract problem formulation in Section~\ref{ss:abs_prob} has some concrete applications in computational surface generation of triangular meshes. In particular, in {Computer-Aided Design} (CAD), triangular meshes are widely used in various engineering disciplines. For example, in architecture and civil engineering, existing and finished ground surfaces are represented by triangulated irregular networks. Slopes are relevant in the context of drainage, in both, civil engineering (transportation structures), and architecture (roof designs), as well as in the context of surface alignments such as solar farms, embankment dams, and airport infrastructure layouts.
A concrete problem that arises in civil engineering design is the grading of a parking lot. Within a given area, the engineer has to define grading slopes such that the parking lot fits within existing structures, the drainage requirements on the lot are met, and safety and comfort is taken into account. Besides these requirements, the engineer would like to change the existing surface as little as possible, in order to save on earthwork costs.
\begin{figure}
\caption{Parking A}
\label{fig:parkingplanA}
\caption{Parking B}
\label{fig:parkingplanB}
\caption{Roundabout}
\label{fig:roundaboutplan}
\caption{Drainage schemes for various engineering structures.}
\label{f:parking}
\end{figure}
Consider Figure~\ref{f:parking}. The triangular mesh in Figure~\ref{fig:parkingplanA} represents the existing ground of a planned parking lot. The engineer would like the water to drain away from the building, and along the blue drain lines into the four corners. Red lines indicate where the triangle edges need to be aligned. In Figure~\ref{fig:parkingplanB}, the engineer would like to study a different scheme, where the drainage happens in parallel, on either side of the building. Lastly, Figure~\ref{fig:roundaboutplan} represents the triangular mesh of existing ground for a roundabout, where a minimum slope is required from the inner circle to the outer one, and water needs to drain from the road at the top along the outer circle to the roads on either side at the bottom.
\subsection{Objective and Outline of This Paper}
This paper aims to provide a framework for modeling practical design problems using geometric constraints in three dimensions. These problems can then be solved by iterative optimization methods. This involves the introduction and computation of projection/proximity operators of new constraints and objective functions. Once all required formulas are accomplished, they will be used in iterative optimization methods to obtain the solutions.
The paper is organized as follows. Section~\ref{s:overvw} contains an overview of iterative methods that will be employed. From Section~\ref{s:linconstr} to Section~\ref{s:prj_minsl}, we derive the projection operators of the above spatial constraints in closed forms. Particularly in Section~\ref{s:prj_minsl}, we present an interesting idea to {\em modify} certain nonconvex constraints into convex ones that retain the intrinsic nature of the original design problem. To the best of our knowledge, this is the {\em first} methodology to successfully maneuver around such nonconvexity. Utilizing these constructions, we include in Section~\ref{s:optim} some optimization problems of interest. Finally, we present the numerical experiment in Section~\ref{s:experiment} and some remarks in Section~\ref{s:conclus}.
\section{Methods Overview} \label{s:overvw}
\subsection{Projections onto Constraint Sets}
A constraint set $C\subseteq X$ is the collection of all feasible data points, i.e., points that satisfy some requirements. Suppose the given data point $z\in X$ is not feasible (i.e., $z\not\in C$), we aim to {\em modify} $z$ so that the newly obtained point $x$ is feasible (i.e., $x\in C$); and we would like to do it with {\em minimal} adjustment on $z$. This task can be achieved by using the projection onto $C$. Recall that the projection of $z$ onto $C$, denoted by $\ensuremath{\operatorname{P}}_Cz$, is the solution of the optimization problem \begin{equation*}
\ensuremath{\operatorname{P}}_C z =\ensuremath{\operatorname*{argmin}}_{x\in C}\|z-x\|=
\menge{x\in C}{\|z-x\|=\min_{y\in C} \|z-y\|}. \end{equation*} It is well known that if $C$ is nonempty, closed, and {\em convex}\footnote{$C$ is convex if for all $x,y\in C$ and $\lambda\in\left[0,1\right]$, we have $(1-\lambda)x+\lambda y\in C$}, then $\ensuremath{\operatorname{P}}_C z$ is singleton, see, for example, \cite[Theorem~2.10]{rusz2006}.
\subsection{Proximity Operators}
Suppose $f:X\to\ensuremath{\,\left(-\infty,+\infty\right]}$ is a proper convex lower semicontinuous function\footnote{See, e.g., \cite{rock70} and \cite{BC2017} for relevant materials in convex analysis} and $x$ is a given point in $X$. Then it is well known (see~\cite[Section~12.4]{BC2017}) that the function \begin{equation*}
X\to\ensuremath{\,\left(-\infty,+\infty\right]}:y\mapsto f(y)+\tfrac{1}{2}\|x-y\|^2 \end{equation*} has a unique minimizer, which we will denote by $\ensuremath{\operatorname{Prox}}_f(x)$. The induced operator $\ensuremath{\operatorname{Prox}}_f:X\to X$ is called the {\em proximal mapping} or {\em proximity operator} of $f$ (see~\cite{moreau65}). Note that if $f$ is the {\em indicator function} of a set $C$ (the indicator function $\iota_C$ is defined by $\iota_C(x)=0$ if $x\in C$ and $\iota_C(x)=+\infty$ otherwise), then $\ensuremath{\operatorname{Prox}}_f=\ensuremath{\operatorname{P}}_C$. Thus, proximity operators are generalizations of projections.
\subsection{Iterative Methods for Optimization Problems}
Iterative optimization methods are often used for solving \eqref{e:fprob}, which may require the computations of proximity and projection operators for the functions and constraint sets involved.
It turns out that all spatial constraints encountered in our settings are convex and closed. Hence, their projections always exist and are unique. Moreover, we also successfully obtain explicit formulas for these projections. In the coming sections, we will make the formulas as convenient as possible for software implementation. As we will make proximity operators available for several types of objective functions, any iterative optimization methods that utilize proximity operators, for example, \cite{BCH2013,bricenocombettes2011,comb-pesq-08,lion-mercer-79}, can be used to solve the corresponding problems. Let us describe one such method, the Douglas--Rachford (DR) splitting algorithm. The DR algorithm emerged from the field of differential equations \cite{doug-rach-56}, and was later made widely applicable in other areas thanks to the seminal work~\cite{lion-mercer-79}.
To formulate DR algorithm, we first use indicator functions to convert \eqref{e:fprob} into \begin{equation} \label{e:fprob2} \min\quad F(x)+\iota_{C_1}+\iota_{C_2}+\cdots+\iota_{C_J} \quad\text{subject to}\quad x\in X. \end{equation} So, it suffices to present DR for the following general optimization problem \begin{equation} \label{e:fprob2b} \min\quad \sum_{i=1}^m f_i(x) \quad\text{subject to}\quad x\in X, \end{equation} where each $f_i$ is a proper convex lower semicontinuous function on $X$. The DR operates in the product space $\ensuremath{{\mathbf{X}}}:=X^{m}$ with inner product $\scal{\ensuremath{\mathbf{x}}}{\ensuremath{\mathbf{y}}}:=\sum_{i=1}^m\scal{x_i}{y_i}$ for $\ensuremath{\mathbf{x}}=(x_i)_{i=1}^m$ and $\ensuremath{\mathbf{y}}=(y_i)_{i=1}^m$. Set the starting point $\ensuremath{\mathbf{x}}_0=(z,\ldots,z)\in\ensuremath{{\mathbf{X}}}$, where $z\in X$. Given $\ensuremath{\mathbf{x}}_k=(x_{k,1},\ldots,x_{k,m})\in\ensuremath{{\mathbf{X}}}$, we compute \begin{subequations}\label{e:dr} \begin{align} &\overline{x}_k :=\frac{1}{m}\sum_{i=1}^m x_{k,i},\\ \forall i\in \{1,\ldots,m\}:\quad &x_{k+1,i}:=x_{k,i}-\overline{x}_k+\ensuremath{\operatorname{Prox}}_{\gamma f_i}(2\overline{x}_{k}-x_{k,i}),\\ \text{then update}\quad &\ensuremath{\mathbf{x}}_{k+1}:=(x_{k+1,1},\ldots,x_{k+1,m}). \end{align} \end{subequations} Then the sequence $(\overline{x}_k)_{k\in\NN}$ converges to a solution of \eqref{e:fprob2b}. We note that $(\ensuremath{\mathbf{x}}_k)_{k\in\NN}$ and $(\overline{x}_k)_{k\in\NN}$ are referred to as {\em governing} and {\em monitored} sequences, respectively.
{It is worth mentioning that when all $f_i$'s are indicator functions of the constraints (thus, all proximity operators become projections), then \eqref{e:fprob2b} becomes a pure feasibility problem \begin{equation}\label{e:intersec_prob} \text{find a point}\quad x\in C:=\bigcap_{j\in\{1,2,\ldots,J\}}C_j. \end{equation} Therefore, all optimization methods that work for \eqref{e:fprob2b} can also be used for \eqref{e:intersec_prob}. Nevertheless, it is worth mentioning that there are many iterative projection methods that are specifically designed for feasibility problems. At the first glance, it might be tempting to solve such problem by finding the projection onto the intersection $C$ directly. However, this is often not possible due to the complexity of $C$. A workaround is to utilize the projection $\ensuremath{\operatorname{P}}_{C_j}$ onto each constraint, if the explicit formula is available. Then with an initial point, one can {\em iteratively execute} the projections $P_{C_j}$'s to derive a solution for \eqref{e:intersec_prob}. Let $z_0\in X$ be the initial data point. The following are two simplest instances among iterative projection methods (see \cite{BC2017,cegielskibook2012,cen_zen_par1997} and the references therein) \begin{itemize} \item {\bf cyclic projections:} Given $z_k$, we update $z_{k+1}:= T z_k$ where $T:=\ensuremath{\operatorname{P}}_{C_J}\ensuremath{\operatorname{P}}_{C_{J-1}}\cdots \ensuremath{\operatorname{P}}_{C_2}\ensuremath{\operatorname{P}}_{C_1}$. \item {\bf parallel projections:} Given $z_k$, we update $z_{k+1}:= T z_k$ where $T:=\frac{1}{J}\big(\ensuremath{\operatorname{P}}_{C_1}+\ensuremath{\operatorname{P}}_{C_2}+\cdots +\ensuremath{\operatorname{P}}_{C_J}\big)$. \end{itemize} In addition, when projection methods succeed (see~\cite{cen_chen_com_12} for some interesting examples), they have various attractive features: they are easy to understand, simple for implementation and maintenance, and sometime can be very fast. We refer the readers to \cite{BBsirev93,BK15,cegielskibook2012,cen_zen_par1997} for more comprehensive discussions on projection methods.}
\section{Projections onto Linear Constraints} \label{s:linconstr}
By linear constraints, we refer to any constraint on the triangular mesh that can be represented by a system of linear equalities and inequalities. Indeed, this class includes several important constraints in design problems. In this section, we will analyze those constraints and their projectors.
\subsection{Interval Constraint} \label{ss:prj_interp}
Similar to \cite[Section~2.2]{BK15}, we assume that $I$ is a subset of $\{1,2,\ldots,n\}$ and $(l_i)_{i\in I},(u_i)_{i\in I}\in\ensuremath{\mathbb R}^I$ are given. Define \begin{equation*} Y:=\menge{z=(z_1,z_2,\ldots,z_n)\in X}{\forall i\in I:\ l_i\leq z_i\leq u_i}. \end{equation*} Then one can readily check that $Y$ is closed and convex. The following explicit formula is for the projection onto $Y$, whose proof is straightforward, thus, omitted. \begin{align*} \ensuremath{\operatorname{P}}_Y&:X \to X: (z_1,z_2,\ldots,z_n)\mapsto (x_1,x_2,\ldots,x_n),\\ &\text{where}\ x_i=\begin{cases} \max\{l_i,\min\{u_i,z_i\}\},& i\in I,\\ z_i,& i\in\{1,2,\ldots,n\}\smallsetminus I. \end{cases} \end{align*}
\subsection{Edge Minimum Slope Constraint} Let $P_i=(p_{i1},p_{i2},z_i)$ and $P_j=(p_{j1},p_{j2},z_j)$. The designer may require that the (directional) slope from $P_j$ to $P_i$ must be no less than a threshold level $s_{ij}$. More specifically, since the value $p_{i1},p_{i2},p_{j1},p_{j2}$ are fixed, we can write this constraint as \begin{equation*} z_{i}-z_{j}\geq \alpha_{ij} :=s_{ij}\sqrt{(p_{i1}-p_{j1})^2+(p_{i2}-q_{j2})^2}, \end{equation*} which we will call the {\em edge minimum slope constraint}. This constraint is a linear inequality, thus, convex. The projection formula onto this type of constraint can be derived analogously to \cite[Section~2.3]{BK15}.
\subsection{Low Point Constraint} \begin{definition}[low point] A point $P_j=(p_{j1},p_{j2},z_j)$ on the mesh is called a low point if each point $P_i=(p_{i1},p_{i2},z_i)$ connected to $P_j$ satisfies the edge minimum slope constraint \begin{equation*} z_i-z_j\geq \alpha_{i} :=s_{i}\sqrt{(p_{i1}-p_{j1})^2+(p_{i2}-q_{j2})^2}, \end{equation*} where all $s_{i}\in\ensuremath{\mathbb R}$ are given. \end{definition}
We can treat low point constraint as a {\em single} constraint even though it is the intersection of finitely many edge slope constraints. The following result shows how to project onto this constraint. First, we need a simple lemma
\begin{lemma}\label{l:181227c} Let $a,b\in\ensuremath{\mathbb R}$ and $k\geq 1$. Assume that $ka\leq b+(k-1)\max\{a,b\}$. Then $a\leq b$. \end{lemma} \begin{proof} By assumption, we must have either $ka\leq b+(k-1)a$ or $ka\leq b+(k-1)b$, and any one of them readily implies $a\leq b$. \end{proof}
\begin{theorem}[projection onto a low point constraint] \label{t:lwpt} Let $\alpha_2, \ldots,\alpha_m\in\ensuremath{\mathbb R}$. Define the set \begin{equation*} E:=\menge{(x_1,\ldots,x_m)\in\ensuremath{\mathbb R}^m}{\forall i\in\{2,\ldots,m\}:\ x_i-x_1\geq\alpha_i}. \end{equation*} Let $z:=(z_1,\ldots,z_m)\in\ensuremath{\mathbb R}^m$. Let $\delta_1:=z_1$ and let $\delta_2\leq\delta_3\leq \cdots\leq \delta_m$ be the values $\{z_i-\alpha_i\}_{i\in\{2,\ldots,m\}}$ in nondecreasing order. Let $k$ be the largest number in $\{1,\ldots,m\}$ such that $\delta_k\leq(\delta_1+\cdots+\delta_k)/k$. Then the projection $x:=(x_1,x_2,\ldots,x_m):=\ensuremath{\operatorname{P}}_Ez$ is given by \begin{equation*} x_1=(\delta_1+\cdots+\delta_k)/k\quad \text{and}\quad \forall i\in\{2,\ldots,m\}:\ x_i=\max\{x_1,z_i-\alpha_i\}+\alpha_i. \end{equation*} \end{theorem} \noindent{\it Proof}. \ignorespaces First, $x$ is the solution of \begin{subequations}\label{e:161002a} \begin{align} \min\quad &(x_1-z_1)^2+\ldots+(x_m-z_m)^2\\ \text{s.t}\quad& x_1-x_i+\alpha_i\leq 0,\quad i\in\{2,\ldots,m\}. \end{align} \end{subequations} Let $y:=(y_1,\ldots,y_m)$ where $y_1=x_1$ and $y_i:=x_i-\alpha_i$ for $i\in\{2,\ldots,m\}$. Without relabeling, we may assume $\delta_i=z_i-\alpha_i$ for all $i\in\{2,\ldots,m\}$. Then \eqref{e:161002a} becomes \begin{subequations}\label{e:161002b} \begin{align} \min\quad &\varphi(y):=(y_1-\delta_1)^2+\ldots+(y_m-\delta_m)^2\\ \text{s.t}\quad& g_i(y):=y_1-y_i\leq 0,\quad i\in\{2,\ldots,m\}. \end{align} \end{subequations} To find the (unique) solution, we use Lagrange multipliers: there exist $\lambda_2,\ldots,\lambda_m$, such that \begin{subequations}\label{e:161002c} \begin{align} (1/2)\nabla \varphi(y)+\lambda_2\nabla g_2(y)+\ldots + \lambda_m\nabla g_m(y)&=0,\label{e:161002ca}\\ \forall i\in\{2,\ldots,m\}:\quad \lambda_ig_i(y) =0,\quad \lambda_i\geq 0,\quad g_i(y)&\leq 0.\label{e:161002cb} \end{align} \end{subequations} Then \eqref{e:161002ca} implies \begin{align*} (y_1-\delta_1)+\lambda_2+\cdots+\lambda_m&=0\\ \forall i\in\{2,\ldots,m\}:\quad (y_i-\delta_i)-\lambda_i &=0. \end{align*} So $y_1\leq \delta_1$ because $\lambda_2,\ldots,\lambda_m\geq 0$. By substitution, we get \begin{equation}\label{e:161002d} y_1+y_2+\ldots+y_m=\delta_1+\delta_2+\ldots +\delta_m. \end{equation} Also, \eqref{e:161002cb} reads as, for each $i\in\{2,\ldots,m\}$, \begin{equation*} 0=\lambda_i g_i(y)=(y_i-\delta_i)(y_1-y_i),\quad y_i-\delta_i\geq 0,\quad \text{and}\quad y_1-y_i\leq 0. \end{equation*} It follows that either $y_i=\delta_i\geq y_1$ or $y_1=y_i\geq\delta_i$, i.e., \begin{equation}\label{e:170107a} \forall i\in\{2,\ldots,m\}:\quad y_i=\max\{y_1,\delta_i\}. \end{equation} Substituting \eqref{e:170107a} into \eqref{e:161002d} yields \begin{equation}\label{e:161002e} \delta_1+\cdots +\delta_m= y_1+\max\{y_1,\delta_2\}+\cdots+\max\{y_1,\delta_m\}. \end{equation} So for all $j\in\{1,\ldots,m\}$, \eqref{e:161002e} implies \begin{equation}\label{e:181227a} \delta_1+\cdots +\delta_m\geq \underbrace{y_1+\cdots+y_1}_{\text{$j$ terms}}+\delta_{j+1}+\cdots+\delta_m \quad\Longrightarrow\quad y_1\leq \frac{\delta_1+\cdots+\delta_j}{j}. \end{equation} Next, let $k$ be the largest number in $\{1,\ldots,m\}$ that satisfies \begin{equation}\label{e:161002g} \delta_k\leq\frac{\delta_1+\cdots+\delta_k}{k}. \end{equation} The number $k$ is well defined since at least \eqref{e:161002g} is true for $k=1$. Now we claim that \begin{equation}\label{e:181227d} y_1=\frac{\delta_1+\cdots+\delta_k}{k} \end{equation} by considering two cases.
{\it Case~1:} $k=m$. Using \eqref{e:161002g}, \eqref{e:161002e}, and $\delta_2\leq\delta_3\leq\cdots\leq\delta_m$, we have \begin{align*} m\delta_m&\leq\delta_1+\cdots+\delta_m =y_1+\max\{y_1,\delta_2\}+\cdots+\max\{y_1,\delta_m\}\\ &\leq y_1+(m-1)\max\{y_1,\delta_m\}. \end{align*} Then Lemma~\ref{l:181227c} implies $\delta_m\leq y_1$. Using this in \eqref{e:161002e}, we derive $\delta_1+\cdots+\delta_m=my_1$ which is \eqref{e:181227d}.
{\it Case~2:} $k<m$. By the choice of $k$, we have $\frac{\delta_1+\cdots+\delta_{k+1}}{k+1}<\delta_{k+1}$, which implies \begin{equation*}\label{e:181227b} \frac{\delta_1+\cdots+\delta_{k}}{k}<\delta_{k+1}. \end{equation*} Combining with \eqref{e:181227a}, we conclude that $y_1<\delta_{k+1}$. Using $y_1<\delta_{k+1}$ and \eqref{e:161002g} in \eqref{e:161002e}, we obtain \begin{align*} k\delta_k+\delta_{k+1}+\cdots+\delta_m &\leq(\delta_1+\cdots +\delta_k)+\delta_{k+1}+\cdots+\delta_m\\ &= y_1+\max\{y_1,\delta_2\}+\cdots+\max\{y_1,\delta_m\}\\ &= y_1+\underbrace{\max\{y_1,\delta_2\}+\cdots+\max\{y_1,\delta_k\}}_{\text{$k-1$ terms}} +\delta_{k+1}+\cdots+\delta_m\\ &\leq y_1+(k-1)\max\{y_1,\delta_k\}+\delta_{k+1}+\cdots+\delta_m. \end{align*} This implies \begin{equation*} k\delta_k\leq y_1+(k-1)\max\{y_1,\delta_k\}. \end{equation*} Again, Lemma~\ref{l:181227c} implies $\delta_k\leq y_1$. Now using $\delta_k\leq y_1<\delta_{k+1}$ in \eqref{e:161002e}, we have \begin{equation*} \delta_1+\cdots +\delta_m=ky_1+\delta_{k+1}+\cdots+\delta_m, \quad\text{which implies \eqref{e:181227d}}. \end{equation*}
So, \eqref{e:181227d} is true. Finally, we compute $y_i$'s from \eqref{e:170107a} and \eqref{e:181227d}, then use them to derive $x_i$'s. \ensuremath{
\quad \square}
\subsection{Edge Alignment Constraint}
On the triangular mesh, the designer may want a {\em constant slope} on a particular path, in which case we say the path is ``aligned". Such a path is sometimes called a {\em feature line}. To formulate this constraint, suppose the feature line is given by adjacent points $A_1,A_2,\ldots,A_m$ on the triangular mesh where $A_i=(a_{i1},a_{i2},x_i)\in\ensuremath{\mathbb R}^3$. For $A_iA_{i+1}$, the length of its ``shadow'' on the $xy$-plane is the euclidean distance \begin{equation}\label{e:deltai}
\delta_i:=\|(a_{i+1,1},a_{i+1,2})-(a_{i,1},a_{i,2})\|=\sqrt{(a_{i+1,1}-a_{i,1})^2+(a_{i+1,2}-a_{i,2})^2}. \end{equation} Define also \begin{equation}\label{e:ti} t_1:=0\ ,\ t_2:=\delta_1\ ,\ t_3:=\delta_1+\delta_2\ ,\ \ldots\ ,\ t_m:=\delta_1+\cdots+\delta_{m-1}. \end{equation} Then the alignment constraint is written as \begin{equation}\label{e:algn} \forall i\in\{1,\ldots,\,m-2\}:\ \ (x_{i+1}-x_{i})/(t_{i+1}-t_i) = (x_{i+2}-x_{i+1})/(t_{i+2}-t_{i+1}). \end{equation}
\begin{theorem}[projection onto an edge alignment constraint] \label{t:algnRm} Suppose the points $(a_{i1},a_{i2})\in\ensuremath{\mathbb R}^2$ with ${i\in\{1,\ldots,m\}}$, forms a path in $\ensuremath{\mathbb R}^2$. Let $\delta_i$ and $t_i$ be given respectively by \eqref{e:deltai} and \eqref{e:ti}. Let $F$ be the set of points $(x_1,\ldots,x_m)\in\ensuremath{\mathbb R}^m$ such that \eqref{e:algn} is satisfied.
Let $z=(z_1,\ldots,z_m)\in\ensuremath{\mathbb R}^m$. Then the projection $\ensuremath{\operatorname{P}}_F z\in\ensuremath{\mathbb R}^m$ is given by \begin{equation*} \forall i\in\{1,\ldots,m\}:\ \left(\ensuremath{\operatorname{P}}_F z\right)_i=f(t_i)=\alpha+\beta t_i. \end{equation*} where $f(t)=\alpha + \beta t$ is the least squares line for the points $(t_i,z_i)\in\ensuremath{\mathbb R}^2$, ${i\in\{1,\ldots,m\}}$. \end{theorem} \noindent{\it Proof}. \ignorespaces First, $F$ is convex since all constraints in \eqref{e:algn} are linear. Next, we consider the points $(t_i,z_i)$ in $\ensuremath{\mathbb R}^2$. The problem is to find $(x_1,\ldots,x_m)$ such that the points $(t_i,x_i)$ are aligned and \begin{equation*}
\|x-z\|^2=\sum_{i=1}^m(x_i-z_i)^2 \quad\text{is minimized}. \end{equation*} This is the {\em least squares} problem for the points $(t_i,z_i)$. The proof is complete. \ensuremath{
\quad \square}
\subsection{Surface Alignment Constraint} \label{ss:prj_surf_algn}
The designer may want to patch several adjacent triangles on the mesh into a single polygon, in which case we say that these triangles are ``aligned". This is equivalent to requiring all vertices of the triangles to lie on the same plane. So we have the following result.
\begin{theorem}[projection onto a surface alignment constraint] Let $(a_{i1},a_{i2})$, $i\in\{1,\ldots,m\}$ be a collection of points in $\ensuremath{\mathbb R}^2$ that are not on the same line. Let $F$ be the set of all points $(x_1,\ldots,x_m)\in\ensuremath{\mathbb R}^{m}$ such that the points $\{(a_{i1},a_{i2},x_i)\}_{i\in\{1,\ldots,m\}}$ lie on the same plane in $\ensuremath{\mathbb R}^3$. Let $z=(z_1,\ldots,z_m)\in\ensuremath{\mathbb R}^{m}$. Then the projection $\ensuremath{\operatorname{P}}_Fz\in\ensuremath{\mathbb R}^m$ is given by \begin{equation*} \forall i\in\{1,\ldots,m\}:\quad \big(\ensuremath{\operatorname{P}}_F z\big)_i=f(a_{i1},a_{i2})=\alpha+\beta a_{i1}+\gamma a_{i2}, \end{equation*} where $f(t_1,t_2)=\alpha+\beta t_{1}+\gamma t_{2}$ is the least squares plane for the points $(a_{i1},a_{i2},z_i)\in\ensuremath{\mathbb R}^3$, ${i\in\{1,\ldots,m\}}$. \end{theorem} \noindent{\it Proof}. \ignorespaces Clearly, $F$ is a convex set. Let $x:=(x_1,\ldots,x_m)=\ensuremath{\operatorname{P}}_F z$, then $x$ minimizes \begin{equation*}
\|x-z\|^2=\sum_{i=1}^{m}|x_i-z_i|^2 \end{equation*} subject to the constraint that $(a_{i1},a_{i2},x_i)$, ${i\in\{1,\ldots,m\}}$, lie on the same plane in $\ensuremath{\mathbb R}^3$. Thus, it is equivalent to finding the least squares plane $f:\ensuremath{\mathbb R}^2\to\ensuremath{\mathbb R}$ for the points $(a_{i1},a_{i2},z_i)$, ${i\in\{1,\ldots,m\}}$, which has unique solution since these points are not on the same line. The conclusion follows. \ensuremath{
\quad \square}
\section{Projections onto General Surface Slope Constraints} \label{s:prj_surf}
Surface slope constraints are any requirement imposed on the {\em slope} of a triangle. In this section, we provide a general setup for projections onto such constraints.
\subsection{Normal Vector and Surface Slope Constraint} \label{ss:slConstr}
Let $(\ensuremath{{\mathbf{e}}}_1,\ensuremath{{\mathbf{e}}}_2,\ensuremath{{\mathbf{e}}}_3)$ be the standard basis of $\ensuremath{\mathbb R}^3$. Given three points $P_1=(p_{11},p_{12},h_1)$, $P_2=(p_{21},p_{22},h_2)$, and $P_3=(p_{31},p_{32},h_3)$ in $\ensuremath{\mathbb R}^3$, the normal vector to the plane $P_1P_2P_3$ is the cross product \begin{equation}\label{e:nt1t2} \vec{n} =\matrx{p_{11}-p_{31}\\ p_{12}-p_{32}\\ h_1-h_3} \times \matrx{p_{21}-p_{31}\\ p_{22}-p_{32}\\ h_2-h_3} =: \matrx{a_1\\ b_1\\ t_1} \times \matrx{a_2\\ b_2\\ t_2} =\matrx{b_1 t_2-b_2 t_1\\ a_2 t_1-a_1 t_2\\ a_1b_2-a_2b_1}, \end{equation} where the new variables $a_1,a_2,b_1,b_2,t_1,t_2$ are defined correspondingly, e.g., $a_1:=p_{11}-p_{31}$, etc. Also, we always assume that the ``shadows" of $P_1,P_2,P_3$ on $xy$-plane $(p_{11},p_{12})$, $(p_{21},p_{22})$, $(p_{31},p_{32})$ are not on the same line. Thus, $a_1b_2-a_2b_1\neq 0$.
So we can rescale and use \begin{equation*} \vec{n}=\left(\frac{b_1 t_2-b_2 t_1}{a_1b_2-a_2b_1}, -\frac{a_1 t_2-a_2 t_1}{a_1b_2-a_2b_1},1\right). \end{equation*} If we define \begin{equation}\label{e:t12uv} \matrx{t_1\\ t_2}=:\matrx{a_1 & b_1\\ a_2 & b_2}\matrx{u\\ v} \quad\iff\quad \matrx{u\\ v}:=\frac{1}{a_1b_2-a_2b_1} \matrx{b_2 & -b_1\\ -a_2 & a_1}\matrx{t_1\\ t_2}, \end{equation} then $\vec{n}=(-u,-v,1)$. Obviously, surface slope constraints depend solely on the normal vector $\vec{n}$. Thus, we can represent a surface slope constraint as \begin{equation*} g(u,v)\leq 0. \end{equation*} An important case of such constraints is the {\em surface orientation constraint} below.
\begin{definition}[surface orientation constraint] \label{d:surorc} Let $\Delta$ be a triangle with normal vector $\vec{n}=(-u,-v,1)\in\ensuremath{\mathbb R}^3$ as above. Let $\vec{q}=(q_1,q_2,q_3)\in\ensuremath{\mathbb R}^3\smallsetminus\{0\}$ be a unit vector and $\theta$ be an angle in $[0,\pi]$, the constraint \begin{equation*} \angle(\vec{n},\vec{q})\leq \theta, \quad\text{or equivalently,}\quad \cos\angle(\vec{n},\vec{q}) \geq \cos\theta, \end{equation*} is called the {\em surface orientation constraint} specified by the pair $(\vec{q},\theta)$. \end{definition} In Section~\ref{ss:prj_sl_general}, we develop the general framework for projection onto surface orientation constraint. Then in Sections~\ref{s:prj_maxsl} and \ref{s:prj_minsl} respectively, we will consider two special surface orientation constraints: the {\em surface maximum slope constraint} and the {\em surface oriented minimum slope constraint}.
\subsection{Projection onto a Surface Slope Constraint} \label{ss:prj_sl_general}
Consider a single triangle determined by three points $Q_1=(p_{11},p_{12},w_1)$, $Q_2=(p_{21},p_{22},w_2)$, and $Q_3=(p_{31},p_{32},w_3)$ in $\ensuremath{\mathbb R}^3$. Without loss of generality, we can assume \begin{equation*} w_1+w_2+w_3=0. \end{equation*} Projecting onto a slope constraint is to find the new heights $h_1,h_2,h_3$ that is a solution to the problem \begin{align*} \min_{(h_1,h_2,h_3)\in\ensuremath{\mathbb R}^3}\quad
&\|(h_1,h_2,h_3)-(w_1,w_2,w_3)\|^2\\ \text{subject to}\quad& \text{the triangle $P_1P_2P_3$ satisfies a given slope constraint,}\\ &\text{where $P_1=(p_{11},p_{12},h_1)$, $P_2=(p_{11},p_{12},h_2)$, and $P_3=(p_{11},p_{12},h_3)$.} \end{align*} Defining $a_i,b_i,t_i,u,v$ as in \eqref{e:nt1t2} and \eqref{e:t12uv}, we have $h_1=a_1u+b_1v+h_3$ and $h_2=a_2u+b_2v+h_3$, i.e., \begin{equation}\label{e:h123uv} \matrx{h_1\\ h_2\\ h_3}=\matrx{a_1 & b_1 & 1\\ a_2 & b_2 & 1\\ 0 & 0 & 1}\matrx{u\\ v\\ h_3}. \end{equation} Suppose also the slope constraint on the triangle $P_1P_2P_3$ is written as $g(u,v)\leq 0$. Then the projection problem is converted to \begin{subequations}\label{e:160716c} \begin{align} \min_{(u,v,h_3)\in\ensuremath{\mathbb R}^3} \quad& \phi(u,v,h_3):=
\left\|\matrx{a_1 & b_1 & 1\\ a_2 & b_2 & 1\\
0 & 0 & 1}\matrx{u\\ v\\ h_3}-\matrx{w_1\\ w_2\\ w_3}\right\|^2 \label{e:160716c-a}\\ \text{subject to}\quad& g(u,v)\leq 0.\label{e:160716c-b} \end{align} \end{subequations} Next, suppose further changing variables is necessary, for instance, $(u,v)$ is replaced by the new variables $(\wh{u},\wh{v})$ under a linear transformation \begin{equation*} \matrx{\wh{u}\\ \wh{v}} :=Q\matrx{u\\ v}, \quad\text{where $Q$ is an invertible matrix}. \end{equation*} Then \eqref{e:160716c} is equivalent to \begin{subequations}\label{e:160716h} \begin{align} \min_{(\wh{u},\wh{v},h_3)\in\ensuremath{\mathbb R}^3} \quad&
\left\|\matrx{\wh{a}_1 & \wh{b}_1 & 1\\ \wh{a}_2 & \wh{b}_2 & 1\\
0 & 0 & 1}\matrx{\wh{u}\\ \wh{v}\\ h_3}-\matrx{w_1\\ w_2\\ w_3}\right\|^2\\ \text{subject to}\quad& \wh{g}(\wh{u},\wh{v}):=g(u,v)\leq 0, \end{align} \end{subequations} where $\matrx{ \wh{a}_1 & \wh{b}_1\\ \wh{a}_2 & \wh{b}_2}:=\matrx{a_1 & b_1\\ a_2 & b_2}Q^{-1}$. That means we can treat \eqref{e:160716h} as \eqref{e:160716c} with {\em new} coefficients $\wh{a}_i,\wh{b}_i$ and constraint function $\wh{g}$.
Next, we will simplify the model \eqref{e:160716c} further. Since \eqref{e:160716c-b} does not involve $h_3$, we can convert problem~\eqref{e:160716c} into two variables $(u,v)$ as follows: first, set the derivative of $\phi$ with respect to $h_3$ to zero \begin{subequations} \begin{align*} \nabla_{h_3}\phi &=2\matrx{1 & 1 & 1}\left[\matrx{a_1 & b_1 & 1\\ a_2 & b_2 & 1\\ 0 & 0 & 1}\matrx{u\\ v\\ h_3}-\matrx{w_1\\ w_2\\ w_3}\right]\\ &=2\left[(a_1+a_2)u+(b_1+b_2)v-3h_3-(w_1+w_2+w_3)\right]=0. \end{align*} \end{subequations} Since $w_1+w_2+w_3=0$, we obtain \begin{equation}\label{e:h3uv} h_3=-(1/3)\big[(a_1+a_2)u+(b_1+b_2)v\big]. \end{equation} Substituting \eqref{e:h3uv} into \eqref{e:160716c-a}, we have \begin{subequations}\label{e:190102a} \begin{align}
\phi(u,v,h_3)&=\left\| \matrx{a_1 & a_2 & 1\\ a_2 & b_2 & 1\\ 0 & 0 & 1} \matrx{1 & 0\\ 0 & 1\\ -\frac{a_1+a_2}{3}& -\frac{b_1+b_2}{3}} \matrx{u\\ v}
-\matrx{w_1\\ w_2\\ -(w_1+w_2)}\right\|^2\\ &=:\frac{1}{9}\left[ \frac{1}{2}\matrx{u & v}\matrx{A & C\\ C & B}\matrx{u\\ v}-\matrx{w_a & w_b}\matrx{u\\ v}+Z\right] \end{align} \end{subequations} where $Z$ is some constant independent of $(u,v)$ and \begin{subequations}\label{e:ABCwab} \begin{align} A&:=2(a_1^2+a_2^2-a_1a_2)>0,\ B:=2(b_1^2+b_2^2-b_1b_2)>0, \label{e:ABCwab-a}\\ C&:=2a_1b_1+2a_2b_2-a_1b_2-a_2b_1,\quad\text{and} \label{e:ABCwab-b}\\ \matrx{w_a & w_b}&:= 3\matrx{w_1 & w_2}\matrx{a_1 & b_1\\ a_2 & b_2}. \end{align} \end{subequations} Thus, \eqref{e:160716c} is equivalent to the problem \begin{subequations}\label{e:160716b} \begin{align} \min_{(u,v)\in\ensuremath{\mathbb R}^2}\quad &\varphi(u,v):= \frac{1}{2}\matrx{u & v}\matrx{A & C\\ C& B}\matrx{u\\ v}-\matrx{w_a& w_b}\matrx{u\\ v} \label{e:160716b-a}\\ \text{subject to}\quad & g(u,v)\leq 0. \label{e:160716b-b} \end{align} \end{subequations} As long as we can find the solution $(u,v)$, we can obtain $(h_1,h_2,h_3)$ by using \eqref{e:h3uv} and \eqref{e:h123uv}. In the case new variables $\wh{u},\wh{v}$ are used, we will use the corresponding coefficients $(\wh{a}_i,\wh{b}_i,\wh{u},\wh{v})$ in place of $(a_i,b_i,u,v)$. Finally, we show that $\varphi(u,v)$ is strictly convex. \begin{lemma}\label{l:phi_scvx} Suppose $a_1,a_2,b_1,b_2\in\ensuremath{\mathbb R}$ such that $a_1b_2-a_2b_1\neq 0$. Define $A,B,C$ by \eqref{e:ABCwab-a}--\eqref{e:ABCwab-b}. Then $\matrx{A & C\\ C& B}\succ 0$. Consequently, the function $\varphi(u,v)$ in problem \eqref{e:160716b} is strictly convex. \end{lemma} \begin{proof} Note from \eqref{e:190102a} that $\matrx{A & C\\ C&B}=M^TM$ where $M:=\matrx{a_1 & a_2 & 1\\ a_2 & b_2 & 1\\ 0 & 0 & 1} \matrx{1 & 0\\ 0 & 1\\ -\frac{a_1+a_2}{3}& -\frac{b_1+b_2}{3}}$. Since $a_1b_2-a_2b_1\neq 0$, $M$ has full column rank, which implies $M^TM$ is positive definite. It follows that $\varphi$ is strictly convex. \end{proof}
\section{Projections onto Surface Maximum Slope Constraints} \label{s:prj_maxsl} Adopting the notation in Section~\ref{ss:slConstr}, we let $P_1P_2P_3$ be a triangle in $\ensuremath{\mathbb R}^3$ with normal vector $\vec{n}=(-u,-v,1)$. In certain cases, it is required that the angle between $\vec{n}$ and a given direction $\vec{q}$ must not exceed a given threshold. For example, suppose $P_1P_2P_3$ represents the desired ground, that cannot be too steep with respect to gravity, i.e., the slope of the plane $P_1P_2P_3$ must not exceed a threshold $s:=s_{\rm max}\in\ensuremath{\mathbb{R}_+}$. Then the angle between $\vec{n}$ and $\ensuremath{{\mathbf{e}}}_3:=(0,0,1)$ must satisfy $\angle(\vec{n},\ensuremath{{\mathbf{e}}}_3)\leq \tan^{-1}(s)$, which is equivalent to \begin{subequations}\label{e:maxsl-def2} \begin{align} \cos\angle(\vec{n},\ensuremath{{\mathbf{e}}}_3)
=\frac{\scal{\vec{n}}{\ensuremath{{\mathbf{e}}}_3}}{\|\vec{n}\|} \geq\cos(\tan^{-1}(s))=\frac{1}{\sqrt{1+s^2}}\\ \iff\quad \frac{1}{\sqrt{u^2+v^2+1}}\geq\frac{1}{\sqrt{1+s^2}} \quad\iff\quad u^2+v^2-s^2\leq 0. \end{align} \end{subequations}
\begin{definition}[surface maximum slope constraint] We call \eqref{e:maxsl-def2} the {\em surface maximum slope constraint} with maximum slope $s$. This is a special case of surface orientation constraint where $(\vec{q},\theta)=(\ensuremath{{\mathbf{e}}}_3,\tan^{-1}(s))$ (see~Section~\ref{ss:abs_prob}). \end{definition} Using the general model \eqref{e:160716b}, we convert the projection onto surface maximum slope constraint to \begin{subequations}\label{e:160716e} \begin{align} \min_{(u,v)\in\ensuremath{\mathbb R}^2}\quad &\varphi(u,v) =\frac{1}{2}\matrx{u & v}\matrx{A & C\\ C& B}\matrx{u\\ v}-\matrx{w_a& w_b}\matrx{u\\ v} \label{e:160716e-a}\\ \text{subject to}\quad & u^2+v^2-s^2\leq 0, \label{e:160716e-b} \end{align} \end{subequations} where $A,B,C,w_a,w_b$ are given by \eqref{e:ABCwab}. Rescaling by $(u,v,w_a,w_b) \longleftarrow \Big(\frac{u}{s},\frac{v}{s},\frac{w_a}{s},\frac{w_b}{s}\Big)$, we obtain an equivalent problem \begin{subequations}\label{e:160716d} \begin{align} \min_{(u,v)\in\ensuremath{\mathbb R}^2}\quad &\varphi(u,v)= \frac{1}{2}\matrx{u & v}\matrx{A & C\\ C& B}\matrx{u\\ v}-\matrx{w_a& w_b}\matrx{u\\ v} \label{e:160716d-a}\\ \text{subject to}\quad & g(u,v):=u^2+v^2-1\leq 0. \label{e:160716d-b} \end{align} \end{subequations} This problem can be solved by several ways including numerical methods. For example, \eqref{e:160716d} is a special case of trust region subproblem which can be solved by means of generalized eigenvalue problems \cite{adachi17}.
As this is a projection problem that is needed in iterative optimization methods, it is desirable to have a closed form solution. Thus, in the rest of this section, we will aim to find such solution via Lagrange multipliers, also known as Karush-Kuhn-Tucker (KKT) conditions, see \cite{karush1939,kuhntucker1950} or \cite[Theorem~11.5]{beck14}, and Ferrari's method for quartic equations \cite{irving2013}.
First, due to Lemma~\ref{l:phi_scvx}, \eqref{e:160716d} is the problem of minimizing a strictly convex quadratic function $\varphi(u,v)$ over a closed, bounded, convex set in $\ensuremath{\mathbb R}^2$. Thus, there exists a unique solution. To solve \eqref{e:160716d}, we start by finding the vertex $(u_0,v_0)$ of $\varphi(u,v)$, which is the unique solution of \begin{equation*} \nabla\varphi(u,v)= \matrx{A & C\\ C& B}\matrx{u\\ v}-\matrx{w_a\\ w_b}=0. \end{equation*} Now we check if the vertex $(u_0,v_0)$ is inside or outside the feasible region:
{\em Case~1:} $g(u_0,v_0)\leq 0$ (inside). Then $(u_0,v_0)$ is the solution of \eqref{e:160716d}.
{\it Case~2:} $g(u_0,v_0)>0$ (outside). Then $\nabla\varphi(u,v)\neq 0$ for all $g(u,v)\leq 0$. Observe that for each value $\eta\geq 0$, the level set $\varphi(u,v)=\eta$ is an ellipse in $\ensuremath{\mathbb R}^2$ whose center is the vertex $(u_0,v_0)$ (see Figure~\ref{fig:maxlvset}). \begin{figure}
\caption{Level sets of $\varphi(u,v)$ vs. feasible region.}
\label{fig:maxlvset}
\end{figure} Hence, the tangent point of the unit circle $g(u,v)=0$ and the {\em smallest} elliptical level set of $\varphi$ that intersects the circle (see Figure~\ref{fig:maxlvset}), denoted by $(u_1,v_1)$, is the unique solution of the minimization problem \eqref{e:160716d}; and any tangent point of the unit circle $g(u,v)=0$ and the {\em largest} elliptical level set of $\varphi$ that intersects the circle (see Figure~\ref{fig:maxlvset}), denoted by $(u_2,v_2)$, is a solution of the maximization problem \begin{equation*} \max_{(u,v)\in\ensuremath{\mathbb R}^2}\quad \varphi(u,v) \quad\text{subject to}\quad g(u,v)\leq 0. \end{equation*} Based on this observation, there exist Lagrange multipliers $\lambda_1\geq 0$ and $\lambda_2\leq 0$ such that $(u_1,v_1,\lambda_1)$ and $(u_2,v_2,\lambda_2)$ satisfy the Lagrange multipliers system \begin{subequations}\label{e:larg1} \begin{align} \nabla \varphi(u,v)+\tfrac{\lambda}{2}\nabla g(u,v)&=0,\label{e:larg1-a}\\ u^2+v^2-1&=0.\label{e:larg1-c} \end{align} \end{subequations} Note that $\nabla\varphi(u_1,v_1)\neq 0$ and $\nabla\varphi(u_2,v_2)\neq 0$, so we must have $\lambda_1>0$ and $\lambda_2<0$. Moreover, since the minimization problem has a unique solution, we therefore conclude that {\em the system \eqref{e:larg1} must possess a unique solution $(u_1,v_1,\lambda_1)$ with $\lambda_1>0$, and at least a solution $(u_2,v_2,\lambda_2)$ with $\lambda_2<0$}.
In summary, {\em Case~2} reduces to finding the unique solution $(u,v,\lambda)$ of \eqref{e:larg1} with $\lambda>0$. First, \eqref{e:larg1-a} yields \begin{equation}\label{e:uv2} \matrx{A+\lambda & C\\ C & B+\lambda}\matrx{u\\ v} =\matrx{w_a\\ w_b}. \end{equation} By Lemma~\ref{l:phi_scvx} and the assumption that $\lambda>0$, the matrix in \eqref{e:uv2} is positive definite. It follows that \eqref{e:uv2} has a unique solution \begin{subequations}\label{e:uvlm} \begin{align} u&=\frac{w_a(B+\lambda)-w_bC}{(A+\lambda)(B+\lambda)-C^2} =\frac{\lambda w_a +w_aB-w_bC}{\lambda^2+(A+B)\lambda+AB-C^2},\\ v&=\frac{w_b(A+\lambda)-w_aC}{(A+\lambda)(B+\lambda)-C^2} =\frac{\lambda w_b +w_bA-w_aC}{\lambda^2+(A+B)\lambda+AB-C^2}, \end{align} \end{subequations} Next, substituting $u$ and $v$ into \eqref{e:larg1-c} yields \begin{align*} [\lambda^2+(A+B)\lambda+AB-C^2]^2 &=[w_a\lambda+(w_aB-w_bC)]^2+[w_b\lambda+(w_bA-w_aC)]^2\\ &= (w_a^2+w_b^2)\lambda^2+2(w_a^2B+w_b^2A-2w_aw_bC)\lambda\\ &\quad +(w_aB-w_bC)^2+(w_bA-w_aC)^2. \end{align*} Defining the constants accordingly, we rewrite as \begin{equation*} (\lambda^2+R_1\lambda+R_2)^2=R_3\lambda^2+2R_4\lambda+R_5. \end{equation*}
One can simply solve this equation by the classic Ferrari's method for quartic equations \cite{cardano1968}. Nevertheless, since there is a unique positive $\lambda$, we will find its explicit formula following Ferrari's technique. We introduce a real variable $y$ \begin{subequations}\label{e:qrtlm} \begin{align} (\lambda^2+R_1\lambda+R_2+y)^2 &=R_3\lambda^2+2R_4\lambda+R_5 +2(\lambda^2+R_1\lambda+R_2)y+y^2\\ &=(R_3+2y)\lambda^2 +2\big(R_4+R_1y\big)\lambda +R_5+2R_2y+y^2. \end{align} \end{subequations} We choose $y$ such that the right hand side is a perfect square in $\lambda$. Thus, the right hand side must have zero discriminant \begin{align*} &[R_4+R_1y]^2-(R_3+2y)(R_5+2R_2y+y^2)=0 \label{e:ycb-a}\\ \iff\quad &-2y^3+[R_1^2-R_3-4R_2]y^2+[2R_1R_4-2R_2R_3-2R_5]y +R_4^2-R_3R_5=0. \end{align*} This is a cubic equation in $y$, so we use Cardano's method \cite{cardano1968} to find one real solution $y_0$. Then \eqref{e:qrtlm} becomes \begin{equation}\label{e:qrtlm2} \big(\lambda^2+R_1\lambda+R_2+y_0\big)^2= (R_3+2y_0)\bigg(\lambda +\frac{R_4+R_1y_0}{R_3+2y_0}\bigg)^2. \end{equation} As discussed above, this equation must have at least one positive and one negative solutions. Next, we solve for the (unique) positive $\lambda$:
{\em Case~2a:} $R_3+2y_0<0$. Then $\lambda=-(R_4+R_1y_0)/(R_3+2y_0)$ must be the unique solution, which is a contradiction. Thus, this case cannot happen.
{\em Case~2b:} $R_3+2y_0=0$. Then $y_0=-R_3/2$ and $\lambda^2+R_1\lambda+R_2+y_0=0$. So \begin{equation*} \lambda =\tfrac{1}{2} \Big(-R_1\pm\sqrt{R_1^2-4R_2+2R_3}\Big). \end{equation*} Since there is only one positive $\lambda$, we take $\lambda= \big(-R_1+\sqrt{R_1^2-4R_2+2R_3}\big)/2$.
{\em Case~2c:} $R_3+2y_0>0$. Define $r:=\sqrt{R_3+2y_0}$. Then \eqref{e:qrtlm2} becomes \begin{equation*} (\lambda^2+R_1\lambda+R_2+y_0)^2 =\Big(\lambda r +\tfrac{R_4+R_1y_0}{r}\Big)^2. \end{equation*} This leads to two quadratic equations in $\lambda$ \begin{equation}\label{e:160925a} \lambda^2+\left(R_1\pm r\right)\lambda +\big(R_2+y_0 \pm\tfrac{R_4+R_1y_0}{r}\big)=0. \end{equation} Now if $R_4+R_1y_0=0$, then \eqref{e:160925a} consists of two quadratic equations that have constant term $R_2+y_0$. Thus, it yield an even number (possibly none) of positive solutions. Therefore, we must have $R_4+R_1y_0\neq 0$. Moreover, we must take the equation with negative constant term. So we set $r\ \longleftarrow\ r\cdot\ensuremath{\operatorname{sgn}}(R_4+R_1y_0)$ and take only the equation \begin{equation*} \lambda^2+\left(R_1-r\right)\lambda +\big(R_2+y_0 -\tfrac{R_4+R_1y_0}{r}\big)=0. \end{equation*} It follows that the positive $\lambda$ is \begin{equation*} \lambda=\tfrac{1}{2}\Big(r-R_1 +\sqrt{(r-R_1)^2-4\big(R_2+y_0 -\tfrac{R_4+R_1y_0}{r}\big)}\,\Big). \end{equation*} Next, we obtain $u$ and $v$ from \eqref{e:uvlm} and then rescale variables $(u,v)\longleftarrow (su,sv)$. Finally, we obtain $(h_1,h_2,h_3)$ by using \eqref{e:h3uv} and \eqref{e:h123uv}.
\section{Projections onto Surface Oriented Minimum Slope Constraints} \label{s:prj_minsl}
\subsection{Motivation from a Nonconvex Constraint}
Let $P_1P_2P_3$ be a triangle with the normal vector $\vec{n}=(-u,-v,1)$ as in Section~\ref{ss:slConstr}. In some cases, it is required that the angle $\angle(\vec{n},\vec{q})\geq \alpha$ for some given vector $\vec{q}$ and number $\alpha$. This happens, for example, if $P_1P_2P_3$ must have a slope at least $s:=s_{\rm min}\in\ensuremath{\mathbb{R}_+}$. In this case, the angle between $\vec{n}$ and $\ensuremath{{\mathbf{e}}}_3=(0,0,1)$ satisfies \begin{subequations}\label{e:drain} \begin{align} &\angle{(\vec{n},\ensuremath{{\mathbf{e}}}_3)}\geq \tan^{-1}(s)\label{e:drain-a}\\ \Leftrightarrow\quad &\cos\angle(\vec{n},\ensuremath{{\mathbf{e}}}_3)
={\scal{\vec{n}}{\ensuremath{{\mathbf{e}}}_3}}/{\|\vec{n}\|} \leq\cos(\tan^{-1}(s))={1}/{\sqrt{1+s^2}}\\ \Leftrightarrow\quad &\frac{1}{\sqrt{u^2+v^2+1}}\leq\frac{1}{\sqrt{1+s^2}} \quad\Leftrightarrow\quad u^2+v^2-s^2\geq 0. \end{align} \end{subequations} We refer to \eqref{e:drain} as the {\em surface minimum slope constraint} with minimum slope $s$. This is clearly a {\em nonconvex} constraint in $(u,v)$. Despite the projection onto this constraint is still available, nonconvexity may prevent iterative methods from convergence. It is worth to mention that similar minimum slope constraints are also critical in road design problem \cite{BK15}, which is again nonconvex and thus, somewhat hinders the theoretical analysis. Therefore, we will next present a novel idea to modify this constraint in a way such that minimum slope is preserved.
\subsection{The Surface Oriented Minimum Slope Constraint}
Condition \eqref{e:drain-a} implies the angle between $\vec{n}$ and the $xy$-plane satisfies \begin{equation}\label{e:drain2} \angle (\vec{n},\text{$xy$-plane})\leq ({\pi}/{2})-\tan^{-1}(s). \end{equation} In some cases, it is reasonable to align the plane $P_1P_2P_3$ toward a given location/direction. For instance, in civil engineering drainage, the designer may want to direct the water to certain drains. Suppose we want $P_1P_2P_3$ inclined towards a unit direction $\vec{q}=(q_1,q_2,0)\in\ensuremath{\mathbb R}^3$. Then we can fulfill \eqref{e:drain2} by requiring $\angle{(\vec{n},\vec{q})}\leq \frac{\pi}{2}-\tan^{-1}(s)$, i.e., \begin{equation*} \cos\angle{(\vec{n},\vec{q})}
={\scal{\vec{n}}{\vec{q}}}/{\|\vec{n}\|}\geq \cos\big(\tfrac{\pi}{2}-\tan^{-1}(s)\big)={s}/{\sqrt{1+s^2}}, \end{equation*} Substituting $\vec{n}=(-u,-v,1)$ and $\vec{q}=(q_1,q_2,0)$, we obtain $\frac{-q_1u-q_2v}{\sqrt{u^2+v^2+1}}\geq \frac{s}{\sqrt{1+s^2}}$, which is equivalent to \begin{equation}\label{e:minsl} q_1u+q_2v<0,\quad u^2+v^2+1-\tfrac{1+s^2}{s^2}(q_1u+q_2v)^2\leq 0. \end{equation} Hence, we arrive at the following definition. \begin{definition}[surface oriented minimum slope constraint] \label{d:omsl} We call \eqref{e:minsl} the {\em surface oriented minimum slope constraint} specified by $(\vec{q},s)$, where $\vec{q}=(q_1,q_2,0)\in\ensuremath{\mathbb R}^3$ is a unit vector and $s\in\ensuremath{\mathbb{R}_{++}}$ is the minimum slope. \end{definition}
Definition~\ref{d:omsl} is a special case of the {\em surface orientation constraint} in Section~\ref{ss:abs_prob} where $(\vec{q},\theta)=(\vec{q},\tfrac{\pi}{2}-\tan^{-1}(s))$. It is worth mentioning that the constraint \eqref{e:minsl} not only guarantees surface minimum slope but also generates a {\em convex} feasible set.
\subsection{The Projection onto \eqref{e:minsl}}
By employing Section~\ref{ss:prj_sl_general}, the projection onto the surface oriented minimum slope constraint is given by the solution to the problem \begin{align*} \min_{(u,v)\in\ensuremath{\mathbb R}^2}\quad &(a_1u+b_1v+h_3-w_1)^2+(a_2u+b_2v+h_3-w_2)^2+(h_3-w_3)^2\\ \text{subject to}\quad &\eqref{e:minsl}. \end{align*} Again, we first simplify this problem. Define $Q:=\matrx{ q_1 & q_2\\ q_2 & -q_1}$ where $(q_1,q_2,0)$ is the unit direction vector that defines the constraint \eqref{e:minsl}. Then $Q^2=\ensuremath{\operatorname{Id}}$, which implies $Q=Q^T=Q^{-1}$. Next, we change variables \begin{equation*} \matrx{ \wh{u}\\ \wh{v}} := Q\matrx{u\\ v} =\matrx{q_1 u+q_2 v\\ q_2 u-q_1 v} \quad\Leftrightarrow\quad \matrx{u\\ v} =Q\matrx{\wh{u}\\ \wh{v}} =\matrx{q_1 \wh{u}+q_2 \wh{v}\\ q_2 \wh{u}-q_1 \wh{v}}. \end{equation*} Then the second inequality in \eqref{e:minsl} becomes \begin{equation*} (q_1\wh{u}+q_2\wh{v})^2+(q_2\wh{u}-q_1\wh{v})^2+1 -\Big(1+\frac{1}{s^2}\Big)\wh{u}^2 =\Big(-\frac{1}{s^2}\Big)\wh{u}^2+\wh{v}^2+1 \leq 0. \end{equation*} Thus, \eqref{e:minsl} becomes \begin{equation}\label{e:minsl2} \wh{u}<0,\quad \wh{v}^2-\frac{\wh{u}^2}{s^2}+1\leq 0. \end{equation} Following Section~\ref{ss:prj_sl_general}, we change coefficients (without relabeling) \begin{equation}\label{e:new_aibi} \matrx{a_1 & a_2\\ b_1 & b_2} \longleftarrow Q \matrx{a_1 & a_2\\ b_1 & b_2}, \end{equation} and obtain an equivalent problem \begin{equation}\label{e:160714a} \min_{(\wh{u},\wh{v})\in\ensuremath{\mathbb R}^2}\quad \frac{1}{2}\matrx{\wh{u} & \wh{v}} \matrx{A & C\\ C & B}\matrx{\wh{u}\\ \wh{v}} -\matrx{w_a & w_b}\matrx{\wh{u}\\ \wh{v}} \quad\text{subject to}\quad\eqref{e:minsl2}. \end{equation} where $A,B,C,w_a,w_b$ are defined by \eqref{e:ABCwab} with the new coefficients $a_1,a_2,b_1,b_2$ from \eqref{e:new_aibi}. Next, we change variables by $(u,v,A,B,w_b) \longleftarrow \Big(\frac{\wh{u}}{s},\wh{v},sA,\tfrac{B}{s},\frac{w_b}{s}\Big)$, then \eqref{e:160714a} is equivalent to \begin{subequations}\label{e:160614c}
\begin{align}
\min_{(u,v)\in\ensuremath{\mathbb R}^2}\
\quad&\varphi(u,v)=
\frac{1}{2}\matrx{u & v}
\matrx{A & C\\ C & B}\matrx{u\\ v}
-\matrx{w_a & w_b}\matrx{u\\ v}
\label{e:160614c-a}\\
\text{subject to}\quad&
g_1(u,v):=u<0,\quad
g_2(u,v):=v^2-u^2+1\leq0.\label{e:160614c-b}
\end{align} \end{subequations} Since all matrices in \eqref{e:new_aibi} are nonsingular, the new coefficients $a_1,a_2,b_1,b_2$ satisfy $a_1b_2-a_2b_1\neq 0$, which implies that $\varphi(u,v)$ is strictly convex by Lemma~\ref{l:phi_scvx}. The feasible set \eqref{e:160614c-b} is the left half of a hyperbola, thus, also a convex region (see Figure~\ref{fig:minlvset}). Therefore, \eqref{e:160614c} is the problem of minimizing a strictly convex quadratic function over a closed convex region, which must yield a unique solution.
Similar to Section~\ref{s:prj_maxsl}, we now will present a way to solve \eqref{e:160614c} by Lagrange multipliers. First, we find the vertex $(u_0,v_0)$ of $\varphi(u,v)$, which is the unique solution of \begin{equation*} \nabla\varphi(u,v)= \matrx{A & C\\ C& B}\matrx{u\\ v}-\matrx{w_a\\ w_b}=0. \end{equation*}
{\em Case~1:} $(u_0,v_0)$ is feasible, i.e., $u_0<0$ and $g_2(u_0,v_0)\leq0$. Then clearly $(u_0,v_0)$ is the unique solution of \eqref{e:160614c}.
{\em Case~2:} $(u_0,v_0)$ is not feasible. Then the unique solution $(u_1,v_1)$ of \eqref{e:160614c} is the tangent point of the left branch hyperbola curve $C:=\menge{(u,v)}{u<0,\ v^2-u^2+1=0}$ and the smallest level set of $\varphi(u,v)$ that intersects $C$, which is an ellipse centered at $(u_0,v_0)$. Figure~\ref{fig:minlvset} illustrates two possible scenarios. \begin{figure}
\caption{$g_2(u_0,v_0)>0$.}
\label{fig:minlv_a}
\caption{$g_2(u_0,v_0)\leq0$ and $u_0>0$.}
\label{fig:minlv_b}
\caption{Level sets of $\varphi(u,v)$ vs. feasible region.}
\label{fig:minlvset}
\end{figure} In both cases, we see that $g_1(u,v)=u<0$ is always an inactive constraint. Therefore, the Lagrange multipliers system reduces to \begin{subequations}\label{e:larg2} \begin{align} \nabla\varphi(u,v)+\tfrac{\lambda}{2} \nabla g_2(u,v) &=0,\label{e:larg2-ab}\\ g_2(u,v)=v^2-u^2+1&=0.\label{e:larg2-c} \end{align} \end{subequations} Utilizing Figure~\ref{fig:minlvset}, we conclude the following:
If $g_2(u_0,v_0)>0$ (see Figure~\ref{fig:minlv_a}), then \eqref{e:larg2} must have a unique solution $(u_1,v_1,\lambda_1)$ with $\lambda_1>0$ and $u_1<0$, and a unique solution $(u_2,v_2,\lambda_2)$ with $\lambda_2>0$ and $u_2>0$.
If $g_2(u_0,v_0)\leq 0$ and $g_1(u_0,v_0)=u_0>0$ (see Figure~\ref{fig:minlv_b}), then \eqref{e:larg2} must have a unique solution $(u_1,v_1,\lambda_1)$ with $\lambda_1>0$ and $u_1<0$. Also, \eqref{e:larg2} must have at least one solution $(u_2,v_2,\lambda_2)$ with $\lambda_2<0$.
In summary, {\em Case~2} reduces to finding the solution $(u_1,v_1,\lambda_1)$ of \eqref{e:larg2} where $\lambda_1>0$ and $u_1<0$. It then follows that $(u_1,v_1)$ is the unique solution of \eqref{e:160614c}. First, \eqref{e:larg2-ab} is \begin{equation}\label{e:uv3} \matrx{A-\lambda & C\\ C & B+\lambda}\matrx{u\\ v}=\matrx{w_a\\ w_b}. \end{equation} Define \begin{subequations}\label{e:DDuDv} \begin{align*} D&:=(A-\lambda)(B+\lambda)-C^2 =-\lambda^2+(A-B)\lambda+(AB-C^2),\\ D_u&:=w_a(B+\lambda)-w_bC =\lambda w_a + w_aB- w_b C,\\ D_v&:=w_b(A-\lambda)-w_a C =-\lambda w_b + (w_b A- w_aC). \end{align*} \end{subequations} Suppose $D\neq 0$, then $u=D_u/D$ and $v=D_v/D$. Substituting into \eqref{e:larg2-c} yields \begin{subequations} \begin{align*} \Big[\lambda^2+(B- A)\lambda+ (C^2-AB)\Big]^2 &=\big(\lambda w_a + w_aB- w_bC\big)^2 -\big[\lambda w_b + (w_aC- w_b A)\big]^2\\ &=\Big( w_a^2-w_b^2\Big)\lambda^2 +2\Big(w_a^2 B+ w_b^2 A -2 w_a w_b C\Big)\lambda\\ &\qquad+( w_a B- w_b C)^2 -( w_a C- w_b A)^2. \end{align*} \end{subequations} By defining the constants accordingly, we rewrite the above identity as \begin{equation*} (\lambda^2+ R_1\lambda + R_2)^2= R_3\lambda^2 + 2 R_4\lambda + R_5. \end{equation*} Again, one can analyze this equation analogously to Section~\ref{s:prj_maxsl}. However, complication arises since there are possibly more than one positive $\lambda$'s. Instead, we expand \begin{equation*} \lambda^4+2R_1\lambda^3+(R_1^2+2R_2-R_3)\lambda^2 +2(R_1R_2-R_4)\lambda+R_2^2-R_5=0, \end{equation*} and use the classic Ferrari's method for quartic equation. Next, for each $\lambda>0$, we solve \eqref{e:uv3} as follows.
{\em Case~2a:} $D\neq 0$. Then we simply compute $u={D_u}/{D}$ and $v={D_v}/{D}$.
{\em Case~2b:} $D=0$. If either $D_u\neq0$ or $D_v\neq0$, then \eqref{e:uv3} has no solution $(u,v)$. So we now consider the remaining case that $D_u=D_v=0$. Then from \eqref{e:uv3} and \eqref{e:larg2-c}, we have \begin{subequations} \begin{align*} Cu+(B+\lambda)v&= w_b,\\ v^2- u^2+1&=0. \end{align*} \end{subequations} Since $B>0$ and $\lambda>0$, the first equation implies $\displaystyle v={(w_b-Cu)}/{(B+\lambda)}$. So the second one becomes $( w_b-Cu)^2-(B+\lambda)^2 u^2+(B+\lambda)^2=0$, i.e., \begin{equation} \Big[C^2-(B+\lambda)^2\Big]u^2 -2w_bC u+ [w_b^2+(B+\lambda)^2]=0.\label{e:170316a} \end{equation} If the discriminant $(w_bC)^2 -(C^2-(B+\lambda)^2)( w_b^2+(B+\lambda)^2)\geq 0$, then we obtain two solutions $u$. Since there is a unique pair $(u,v)$ with $u<0$, the quadratic equation \eqref{e:170316a} must yield one positive and one negative solutions \begin{equation*} u= \frac{w_bC\pm\sqrt{ (w_bC)^2 -(C^2-(B+\lambda)^2)( w_b^2+(B+\lambda)^2)}} {C^2-(B+\lambda)^2}, \end{equation*} and we must also have $C^2-(B+\lambda)^2<0$. Therefore, the negative solution $u$ is \begin{equation*} u=\frac{w_bC+\sqrt{ (w_bC)^2 -(C^2-(B+\lambda)^2)( w_b^2+(B+\lambda)^2)}} {C^2-(B+\lambda)^2}. \end{equation*}
Next, among all $(u,v)$'s, choose the unique pair with $u<0$. Then rescale variables $u\longleftarrow su$. Finally, we obtain $(h_1,h_2,h_3)$ by \eqref{e:h3uv} and \eqref{e:h123uv}.
\section{Curvature Minimization} \label{s:optim}
In some design problems, the designer may wish to construct a surface that is as ``smooth" as possible. This problem is referred to as minimizing the {\em curvature} between adjacent triangles in the mesh. In this section, we will address this problem.
\noindent \begin{minipage}{.53\textwidth} \begin{figure}
\caption{Curvature between two triangles.}
\label{fg:ddn}
\end{figure} \end{minipage} \begin{minipage}{.44\textwidth} Given the points $P_i=(p_{i1},p_{i2},h_i)$, $i=1,2,3,4$, so that they form two adjacent triangles $\Delta_1=P_1P_2P_4$ and $\Delta_2=P_2P_3P_4$ (see Figure~\ref{fg:ddn}). \end{minipage}
Define \ $a_i:=p_{i1}-p_{41}$ and $b_i:=p_{i2}-p_{42}$, for $i=1,2,3$. Then the respective normal vectors are \begin{align*} \vec{n}_{1}&= \matrx{a_1\\ b_1\\ h_1-h_4}\times \matrx{a_2\\ b_2\\ h_2-h_4}= \matrx{-b_2h_1+b_1h_2+(b_2-b_1)h_4\\ a_2 h_1-a_1 h_2+(a_1-a_2)h_4\\ a_1b_2-a_2b_1},\\ \vec{n}_{2}&= \matrx{a_2\\ b_2\\ h_2-h_4}\times \matrx{a_3\\ b_3\\ h_3-h_4}= \matrx{-b_3h_2+b_2h_3+(b_3-b_2)h_4\\ a_3 h_2-a_2 h_3+(a_2-a_3)h_4\\ a_2b_3-a_3b_2}. \end{align*} We rescale and obtain \begin{align*} \vec{n}_{1}&= \bigg(\frac{-b_2h_1+b_1h_2+(b_2-b_1)h_4}{a_1b_2-a_2b_1}, \frac{a_2 h_1-a_1 h_2+(a_1-a_2)h_4}{a_1b_2-a_2b_1}, 1\bigg),\\ \vec{n}_{2}&= \Big(\frac{-b_3h_2+b_2h_3+(b_3-b_2)h_4}{a_2b_3-a_3b_2}, \frac{a_3 h_2-a_2 h_3+(a_2-a_3)h_4}{a_2b_3-a_3b_2}, 1\Big). \end{align*} The curvature can be represented by the difference between $\vec{n}_1$ and $\vec{n}_2$, i.e., \begin{equation*} \vec{\delta}_{12}:=\vec{\delta}_{\Delta_1\Delta_2}=\vec{n}_{1}-\vec{n}_{2} =:\matrx{\scal{u}{h}\\ \scal{v}{h}}, \end{equation*} where $h:=(h_1,h_2,h_3,h_4)$, $u:=(u_1,u_2,u_3,u_4)$, $v:=(v_1,v_2,v_3,v_4)$, \begin{subequations}\label{e:uv1234} \begin{alignat}{4} u_1=-b_2/(a_1b_2-a_2b_1)\quad, &&\quad u_2&=b_1/(a_1b_2-a_2b_1)+b_3/(a_2b_3-a_3b_2)\ , \\ u_3=-b_2/(a_2b_3-a_3b_2)\quad, &&\quad u_4&=-(u_1+u_2+u_3)\ ,\\ v_1=a_2/(a_1b_2-a_2b_1)\quad, &&\quad v_2&=-a_1/(a_1b_2-a_2b_1)-a_3/(a_2b_3-a_3b_2)\ , \\ v_3=a_2/(a_2b_3-a_3b_2)\quad, &&\quad v_4&=-(v_1+v_2+v_3). \end{alignat} \end{subequations} So for each pair $(\Delta_i,\Delta_j)$ of adjacent triangles, we find the corresponding vectors $h_{ij}=(h_1^{ij},h_2^{ij},h_3^{ij},h_4^{ij})\in\ensuremath{\mathbb R}^4$, $u_{ij},v_{ij}\in\ensuremath{\mathbb R}^4$, and compute the corresponding curvature $\vec{\delta}_{ij}=\left(\scal{u_{ij}}{h_{ij}},\scal{v_{ij}}{h_{ij}}\right)$. We then aim to minimize all the curvatures between adjacent triangles. Thus, we arrive at the objective \begin{equation*}
G_{1,*}(x):=\sum_{\text{all triangle pairs $(\Delta_i,\Delta_j)$}}\|\vec{\delta}_{ij}\|_* \end{equation*}
where $\|\cdot\|_*$ can be either $1$-norm or max-norm in $\ensuremath{\mathbb R}^2$. For $1$-norm, the objective is \begin{equation}\label{e:170109a}
G_{1,1}(x)=\sum_{\text{all triangle pairs $(\Delta_i,\Delta_j)$}} |\scal{u_{ij}}{h_{ij}}|+|\scal{v_{ij}}{h_{ij}}|. \end{equation} For max-norm, the objective is \begin{equation}\label{e:170109b}
G_{1,\infty}(x)=\sum_{\text{all triangle pairs $(\Delta_i,\Delta_j)$}} \max\{|\scal{u_{ij}}{h_{ij}}|,|\scal{v_{ij}}{h_{ij}}|\}. \end{equation}
\begin{remark}[simplified computations for symmetric cases] Suppose the two dimensional mesh satisfies the following symmetry: for every adjacent triangles $P_1P_2P_4$ and $P_2P_3P_4$, there exists $t\in\ensuremath{\mathbb R}$ such that \begin{equation*} \overrightarrow{P_4P_1}+\overrightarrow{P_4P_3}=t\overrightarrow{P_4P_2}. \end{equation*} Then it follows that $(a_1,b_1)+(a_3,b_3)=t(a_2,b_2)$. So we can deduce $a_1b_2-a_2b_1=a_2b_3-a_3b_2$. From \eqref{e:uv1234}, we have \begin{equation*} u=\frac{-b_2}{a_1b_2-a_2b_1}\big(1,-t,1,t-2\big) \ \ \text{and}\ \ v=\frac{a_2}{a_1b_2-a_2b_1}\big(1,-t,1,t-2\big), \end{equation*} i.e., $u$ and $v$ are parallel. This simplifies the computations for \eqref{e:170109a} and \eqref{e:170109b}. \end{remark}
Since our optimization methods require proximity operators, we will derive the necessary formulas. It is sometimes convenient to compute the proximity operator $\ensuremath{\operatorname{Prox}}_f$ via the proximity operator of its {\em Fenchel conjugate} $f^*$, which is defined by $f^*:X\to\ensuremath{\mathbb R}: x^*\mapsto\sup_{x\in X}\big(\scal{x^*}{x}-f(x)\big)$. Indeed, if $\gamma>0$, then (see, e.g., \cite[Theorem~14.3(ii)]{BC2017}) \begin{equation*} \forall x\in X:\quad x=\ensuremath{\operatorname{Prox}}_{\gamma f}(x)+\gamma\ensuremath{\operatorname{Prox}}_{\gamma^{-1}f^*}(\gamma^{-1}x). \end{equation*} We also recall a useful formula from \cite[Lemma~2.3]{BKP16}: if $f:X\to\ensuremath{\mathbb R}$ is convex and positively homogeneous, $\alpha>0$, $w\in X$, and $h:X\to\ensuremath{\mathbb R}:x\mapsto \alpha f(x-w)$, then \begin{equation*} \ensuremath{\operatorname{Prox}}_{h}(x)= w+\alpha\ensuremath{\operatorname{Prox}}_f\Big(\frac{x-w}{\alpha}\Big) =x-\alpha\ensuremath{\operatorname{Prox}}_{f^*}\Big(\frac{x-w}{\alpha}\Big). \end{equation*}
\begin{theorem}\label{t:proxmax} Let $\{u_i\}_{i\in I}$ be a system of finitely many vectors in $\ensuremath{\mathbb R}^n$, and \begin{equation*}
f:\ensuremath{\mathbb R}^n\to\ensuremath{\mathbb R}:x\to \max_{i\in I}\{|\scal{u_i}{x}|\}. \end{equation*} Then $f^*=\iota_{D}$ where $D:=\ensuremath{\operatorname{conv}}\bigcup_{i\in I}\{u_i,-u_i\}$. Consequently, $\ensuremath{\operatorname{Prox}}_{f}=\ensuremath{\operatorname{Id}} -\ensuremath{\operatorname{P}}_D$. \end{theorem} \noindent{\it Proof}. \ignorespaces Suppose $x^*\in D$, then we can express \begin{equation*} x^*=\sum_{i\in I}\lambda_i u_i \quad\text{where}\quad
\sum_{i\in I}|\lambda_i|\leq 1. \end{equation*} It follows that for all $x\in X$, \begin{equation*}
\scal{x^*}{x}-f(x)=\sum_{i\in I}\lambda_i \scal{u_i}{x}-\max_{i\in I}|\scal{u_i}{x}|
\leq\Big(\sum_{i\in I}\lambda_i-1\Big)\max_{i\in I}|\scal{u_i}{x}|\leq 0. \end{equation*} So, $\displaystyle f^*(x^*)=\sup_{x\in X}\big[\scal{x^*}{x}-f(x)\big]\leq0$. Notice that equality happens if we set $x=0$. Thus, $f^*(x^*)=0$.
Now suppose $x^*\not\in D$. Since $D$ is nonempty, closed, and convex, the classic separation theorem implies that there exists $x\in X$ such that \begin{equation*} \scal{x^*}{x}>\scal{u}{x}\quad\text{for all}\quad u\in D. \end{equation*} This leads to $\scal{x^*}{x}-f(x)>0$. Since $f$ is homogeneous, we have \begin{equation*} f^*(x^*)\geq \scal{x^*}{\lambda x}-f(\lambda x)\to +\infty \quad\text{as}\quad \lambda\to+\infty. \end{equation*} So $f^*(x^*)=+\infty$. Therefore, we can conclude that $f^*=\iota_D$. \ensuremath{
\quad \square}
Finally, Examples~\ref{ex:seg} and \ref{ex:parlgrm} provide the necessary formulas to compute the proximity operators of the objectives in \eqref{e:170109a} and \eqref{e:170109b}, respectively.
\begin{example} \label{ex:seg} Given $\alpha>0$, a vector $u\in\ensuremath{\mathbb R}^n\smallsetminus\{0\}$ and the function \begin{equation*}
f:\ensuremath{\mathbb R}^n\to\ensuremath{\mathbb R}:x\mapsto\alpha|\scal{u}{x}|. \end{equation*} By Theorem~\ref{t:proxmax}, the proximity operator of $f$ is \begin{equation*} \ensuremath{\operatorname{Prox}}_f=\ensuremath{\operatorname{Id}}-\ensuremath{\operatorname{P}}_D \ \text{where}\ D:=[-\alpha u,\alpha u]. \end{equation*} The explicit projection onto $D$ is given by (see \cite[Theorem~2.7]{BKP16}) \begin{equation*} \ensuremath{\operatorname{P}}_D x=
\min\Big\{1,\max\Big\{-1,\frac{\scal{\alpha u}{x}}{\|\alpha u\|^2}\Big\}\Big\}\alpha u
=\min\Big\{\alpha,\max\Big\{-\alpha,\frac{\scal{ u}{x}}{\|u\|^2}\Big\}\Big\}u. \end{equation*} \end{example}
\begin{example} \label{ex:parlgrm} Given two vectors $u,v\in\ensuremath{\mathbb R}^n$ and \begin{equation*}
f(x):\ensuremath{\mathbb R}^n\to\ensuremath{\mathbb R}:x\mapsto\max\{|\scal{u}{x}|,|\scal{v}{x}|\}. \end{equation*} By Theorem~\ref{t:proxmax}, the proximity operator of $f$ is \begin{equation*} \ensuremath{\operatorname{Prox}}_f=\ensuremath{\operatorname{Id}}-\ensuremath{\operatorname{P}}_D \ \text{where}\ D:=\ensuremath{\operatorname{conv}}\{u,v,-u,-v\}. \end{equation*} In general, $\ensuremath{\operatorname{Prox}}_D$ is the projection onto a parallelogram in $\ensuremath{\mathbb R}^n$. \end{example}
\section{Experiments} \label{s:experiment}
With the formulas for proximity and projection operators, we are ready to apply iterative methods to solve feasibility and optimization problems. In particular, we present an application to the civil engineer problem in Section~\ref{ss:civil} which is part of our motivation.
{\em Experiment setup:} In each of the three problems outlined in Figure~\ref{f:parking}, we aim to minimize the surface curvature using the objective function \eqref{e:170109a}, and subject to the requirements that: the maximum slope of all triangles does not exceed $4\%$; each triangle must incline toward its closest drain line (marked in blue) with minimum slope of $0.5\%$; and the triangle edges (marked in red) must be aligned. The DR algorithm \eqref{e:dr} will be used and it will stop when distance between two consecutive {\em governing} iterations are less than the tolerance $\varepsilon=0.001$ and the {\em monitored} iteration meets all constrained with (same) tolerance $\varepsilon$.
{\em Results:} In Figures~\ref{fig:parkingA}, \ref{fig:parkingB}, and \ref{fig:roundabout}, we show the solutions after various iterations of the DR algorithm. Triangles colored in red violate the design constraints (maximum and/or minimum slopes), whereas triangles in green satisfy the constraints. Below the surfaces are the contours, which become more regular with increasing iterations due to curvature minimization objective.
\begin{figure}
\caption{Starting conditions.}
\label{fig:pAstart}
\caption{At $k=20$}
\label{fig:pA10}
\caption{At $k=40$}
\label{fig:pA20}
\caption{Final solution.}
\label{fig:pAfinal}
\caption{Grading design for a parking lot with corner drainage.}
\label{fig:parkingA}
\end{figure}
\begin{figure}
\caption{Starting conditions.}
\label{fig:pBstart}
\caption{At $k=15$}
\label{fig:pB10}
\caption{At $k=100$}
\label{fig:pB20}
\caption{Final solution.}
\label{fig:pBfinal}
\caption{Grading design for a parking lot with side drainage.}
\label{fig:parkingB}
\end{figure}
\begin{figure}
\caption{Starting conditions.}
\label{fig:rstart}
\caption{At $k=10$}
\label{fig:r10}
\caption{At $k=20$}
\label{fig:r20}
\caption{Final solution.}
\label{fig:rfinal}
\caption{Grading design for a roundabout.}
\label{fig:roundabout}
\end{figure}
\section{Conclusion} \label{s:conclus} The manipulation of triangle meshes has many applications in computer graphics and computer-aided design. The paper presents a general framework for triangular design problems with spatial constraints. In particular, we model several important constraints and costs in suitable forms so that projection and proximity operators can be computed explicitly. With the help of iterative splitting methods, we are able to solve some complex design problems on these triangular meshes. Therefore, modeling constraints and their proximity operators, and using them in modern first-order optimization methods can be a successful approach to solve large-scale problems in industry and science.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Combining cumulative sum change-point detection tests for assessing the stationarity of univariate time series}
\begin{abstract} We derive tests of stationarity for univariate time series by combining change-point tests sensitive to changes in the contemporary distribution with tests sensitive to changes in the serial dependence. The proposed approach relies on a general procedure for combining dependent tests based on resampling. After proving the asymptotic validity of the combining procedure under the conjunction of null hypotheses and investigating its consistency, we study rank-based tests of stationarity by combining cumulative sum change-point tests based on the contemporary empirical distribution function and on the empirical autocopula at a given lag. Extensions based on tests solely focusing on second-order characteristics are proposed next. The finite-sample behaviors of all the derived statistical procedures for assessing stationarity are investigated in large-scale Monte Carlo experiments and illustrations on two real data sets are provided. Extensions to multivariate time series are briefly discussed as well.
\noindent {\it Keywords:} copula, dependent p-value combination, multiplier bootstrap, rank-based statistics, tests of stationarity.
\noindent {\it MSC 2010:} 62E20, 62G10, 62G09. \end{abstract}
\section{Introduction}
Testing the stationarity of a time series is of great importance prior to any modeling. Existing approaches assessing whether a time series is stationary could roughly be grouped into two main categories: procedures that mostly work in the frequency domain, and those that mostly work in the time domain. Among the tests in the former group, one finds for instance approaches testing the constancy of a spectral functional \citep[see, e.g.,][]{PriSub69,Pap10}, procedures comparing a time-varying spectral density estimate with its stationary approximation \citep[see, e.g.,][]{DetPreVet11,PreVetDet13,PucPre16} and approaches based on wavelets \citep[see, e.g.,][]{vonNeu00,Nas13,CarNas13,CarNas16}. As far as the second category of tests is concerned, one mostly finds approaches based on the autocovariance / autocorrelation function such as \cite{LeeHaNa03}, \cite{DwiSub11}, \cite{JinWanWan15} and \cite{DetWuZho15}. In particular, the works of \cite{LeeHaNa03} and \cite{DetWuZho15} also clearly pertain to the change-point detection literature \citep[see, e.g.,][for an overview]{CsoHor97,AueHor13}. The latter should not come as a surprise. Indeed, any test for change-point detection may be seen as a test of stationarity designed to be sensitive to a particular type of departure from stationarity.
To illustrate the latter point, let $X_1,X_2,\dots$ be a stretch from a univariate time series and consider the classical \emph{cumulative sum} (CUSUM) test ``for a change in the mean'' \cite[see, e.g.,][]{Pag54,Phi87}. The latter is usually regarded as a test of $$ H_0: X_1, X_2, \ldots \text{ have the same expectation} $$ but it only holds its level asymptotically if $X_1,X_2, \dots$ is a stretch from a time series whose autocovariances at all lags are constant \citep{Zho13}. Without the latter assumption, a small p-value can only be used to conclude that $X_1,X_2, \ldots$ is not a stretch from a second-order stationary time series. In other words, without the additional assumption of constant autocovariances, the classical CUSUM test ``for a change in the mean'' is merely a test of second-order stationarity that is particularly sensitive to a change in the expectation.
Obtaining a large p-value when carrying out the previously mentioned test should clearly not be interpreted as no evidence against second-order stationarity since a change in mean is only one possible departure from second-order stationarity. Following \cite{DetWuZho15}, complementing the previous test by tests for change-point detection particularly sensitive to changes in the variance and in the autocorrelation at some fixed lags may, in case of large p-values, comfort a practitioner in considering that $X_1,X_2,\dots$ might well be a stretch from a second-order stationary time series. The aim of this work is to adopt a similar perspective on assessing stationarity but without only restricting the analysis to second-order characteristics. In fact, all finite dimensional distributions induced by a time series could be potentially tested.
More formally, suppose we observe a stretch $X_1,\dots,X_{N}$ from a time series of univariate continuous random variables. For some $2\le h \le N$, set $n=N-h+1$ and let $\bm Y_1^{\scriptscriptstyle (h)},\dots,\bm Y_n^{\scriptscriptstyle (h)}$ be $h$-dimensional random vectors defined by \begin{equation} \label{eq:Yi} \bm Y_i^{(h)} = (X_i,\dots,X_{i+h-1}), \qquad i \in \{1,\dots,n\}. \end{equation} Note that the quantity $h$ is sometimes called the \emph{embedding dimension} and $h-1$ can be interpreted as the maximum lag under investigation. As an imperfect alternative, we shall focus on tests particularly sensitive to departures from the hypothesis \begin{equation} \label{eq:H0:Fh} H_0^{(h)}: \,\exists \, F^{(h)} \text{ such that } \bm Y_1^{(h)},\dots,\bm Y_n^{(h)} \text{ have the distribution function (d.f.) } F^{(h)}. \end{equation}
To derive such tests, a first natural approach would be to apply to the random vectors in~\eqref{eq:Yi} non-parametric CUSUM tests such as those based on differences of empirical d.f.s studied in~\cite{GomHor99}, \cite{Ino01} and \cite{HolKojQue13} (see also Section~\ref{sec:dftest} below), or on differences of empirical characteristic functions; see, e.g., \cite{HusMei06a} and \cite{HusMei06b}. However, preliminary numerical experiments (some of which are reported in Section~\ref{sec:MC}) revealed the low power of such an adaptation in the case of the empirical d.f.-based tests, especially when the non-stationarity of the underlying univariate time series is a consequence of changes in the serial dependence. These empirical conclusions, in line with those drawn in \cite{BucKojRohSeg14} in a related context, prompted us to consider the alternative approach consisting of assessing changes in the ``contemporary'' distribution (that is, of the $X_i$) separately from changes in the serial dependence.
Suppose that $H_0^{\scriptscriptstyle (h)}$ in~\eqref{eq:H0:Fh} holds and recall that $X_1, \dots, X_{n+h-1}$ is assumed to be a stretch from a time series of univariate continuous random variables. Then, the common d.f.\ of $\bm Y_i^{\scriptscriptstyle (h)}$ can be written \citep{Skl59} as $$ F^{(h)}(\bm x)=C^{(h)} \{ G(x_1),\dots,G(x_h) \}, \qquad \bm x \in \mathbb{R}^h, $$ where $C^{(h)}$ is the unique \emph{copula} (merely an $h$-dimensional d.f.\ with standard uniform margins) associated with $F^{(h)}$, and $G$ is the common marginal univariate d.f.\ of all the components of the $\bm Y_i^{\scriptscriptstyle (h)}$, $i \in \{1,\dots,n\}$. The copula $C^{(h)}$ controls the dependence between the components of the~$\bm Y_i^{\scriptscriptstyle (h)}$. Equivalently, it controls the \emph{serial dependence} up to lag $h-1$ in the time series, which is why it is sometimes called the lag $h-1$ \emph{serial copula} or \emph{autocopula} in the literature.
Notice further that, slightly abusing notation, the hypothesis $H_0^{\scriptscriptstyle (h)}$ in~\eqref{eq:H0:Fh} can be written as $H_0^{\scriptscriptstyle (1)} \cap H_{0,c}^{\scriptscriptstyle (h)}$, where \begin{equation} \label{eq:H0:1} H_0^{(1)}: \,\exists \, G \text{ such that } X_1,X_2, \dots \text{ have the d.f.\ } G, \end{equation} and \begin{equation} \label{eq:H0:Ch} H_{0,c}^{(h)}: \,\exists \, C^{(h)} \text{ such that } \bm Y_1^{(h)},\dots,\bm Y_n^{(h)} \text{ have the copula } C^{(h)}. \end{equation} In other words, $H_0^{\scriptscriptstyle (h)}$ in~\eqref{eq:H0:Fh} holds if all the $X_i$ have the same (contemporary) distribution and if all the $\bm Y_i^{\scriptscriptstyle (h)}$ have the same copula.
A sensible strategy for assessing whether $H_0^{\scriptscriptstyle (h)}$ in~\eqref{eq:H0:Fh} is plausible would thus naturally consist of combining two tests: a test particularly sensitive to departures from $H_0^{\scriptscriptstyle (1)}$ in~\eqref{eq:H0:1} and a test particularly sensitive to departures from $H_{0,c}^{\scriptscriptstyle (h)}$ in~\eqref{eq:H0:Ch}. For the former, as already mentioned, a natural candidate in the general context under consideration is the CUSUM test based on differences of empirical d.f.s studied in~\cite{GomHor99} and \cite{HolKojQue13}. We shall briefly revisit the latter approach in the setting of serially dependent observations. One of the main goals of this work is to derive a test that is particularly sensitive to departures from $H_{0,c}^{\scriptscriptstyle (h)}$ in~\eqref{eq:H0:Ch}, that is, to changes in the serial dependence. The idea is not new but seems to have been employed only with respect to second-order characteristics of a time series: see, e.g., \cite{LeeHaNa03} for tests on the autocovariance in a CUSUM setting, and \cite{DwiSub11} and \cite{JinWanWan15} for tests in a different setting. Specifically, one of the main contributions of this work is to propose a CUSUM test that is sensitive to departures from $H_{0,c}^{\scriptscriptstyle (h)}$. It will be based on a serial version of the so-called \emph{empirical copula} that we should naturally refer to as the \emph{empirical autocopula} hereafter.
Because the aforementioned test based on empirical d.f.s (particularly sensitive to departures from $H_0^{\scriptscriptstyle (1)}$ in~\eqref{eq:H0:1} by construction) and the test based on empirical autocopulas (designed to be sensitive to departures from $H_{0,c}^{\scriptscriptstyle (h)}$ in~\eqref{eq:H0:Ch}) rely on the same type of resampling, bootstrap replicates on the underlying statistics $S_{n,G}$ and $S_{n,C^{(h)}}$ can be generated jointly to reproduce, approximately, the distribution of $(S_{n,G},S_{n,C^{(h)}})$ under stationarity. Under such an assumption, another main contribution of this work, that may be of independent interest, is a general procedure for combining dependent bootstrap-based tests, relying on appropriate extensions of well-known p-value combination methods such as those of \cite{Fis33} or \cite{StoEtAl49}.
An interesting and desirable feature of the resulting global testing procedure is that it is rank-based. It is therefore expected to be quite robust in the presence of heavy-tailed observations. Still, in the case of Gaussian time series, some tests based on second-order characteristics might be more powerful. A natural competitor to our aforementioned global test could thus be obtained by combining tests particularly sensitive to changes in the expectation, variance and autocovariances up to lag $h-1$. Interestingly enough, CUSUM versions of such tests can be cast in the setting considered in \cite{BucKoj16b}: they can all be carried out using the same type of resampling and thus, as described in the previous paragraph, their (dependent) p-values can be combined, leading to a test that could be regarded as a test of second-order stationarity.
The paper is organized as follows. The proposed procedure for combining dependent boot\-strap-based tests is described in Section~\ref{sec:combine:tests}, conditions under which it is asymptotically valid under the conjunction of the component null hypotheses are stated and its consistency is theoretically investigated. The detailed description of the combined rank-based test involving empirical d.f.s and empirical autocopulas is given in Section~\ref{sec:rank}, along with theoretical results about its asymptotic validity under the null hypothesis of stationarity. The choice of the embedding dimension $h$ is discussed in Section~\ref{sec:hchoice}. The fourth section is devoted to related combined tests based on second-order characteristics: the corresponding testing procedures are provided and asymptotic validity results under the null are stated. Section~\ref{sec:MC} reports Monte Carlo experiments that are used to empirically study the previously described tests. Some illustrations on real-world data are presented in Section~\ref{sec:illus}. Finally, concluding remarks are provided in Section~\ref{sec:conc}, one of which, in particular, discusses multivariate extensions of the proposed tests.
Auxiliary results and all proofs are deferred to a sequence of appendices. Additional theoretical and simulation results are provided in a supplementary material. The studied tests are implemented in the package {\tt npcp} \citep{npcp} for the \textsf{R} statistical system \citep{Rsystem}. In the rest of the paper, the arrow~`$\leadsto$' denotes weak convergence in the sense of Definition~1.3.3 in \cite{vanWel96}, while the arrow~`$\scsp$' denotes convergence in probability. All convergences are for $n\to\infty$ if not mentioned otherwise. Finally, given a set $S$, $\ell^\infty(S)$ denotes the space of all bounded real-valued functions on $S$ equipped with the uniform metric.
\section{A general procedure to combine dependent tests based on resampling} \label{sec:combine:tests}
As argued in the introduction, to assess whether stationarity is likely to hold, it might be beneficial to combine several tests, each of which being designed to be sensitive to a particular form of non-stationarity. As the need for similar approaches may arise in other contexts than stationarity testing, in this section, we propose a very general strategy for combining tests based on resampling by relying on well-known p-value combination methods such as those of \cite{Fis33} or \cite{StoEtAl49}. Recall that, given $r$ p-values $p_1,\dots,p_r$ for right-tailed tests of corresponding null hypotheses $H_0^{\scriptscriptstyle (1)}, \dots, H_0^{\scriptscriptstyle (r)}$ with corresponding strictly positive weights $w_1,\dots,w_r$ that quantify the importance of each test in the combination, the latter method consists of computing, up to a rescaling term, the global statistic \begin{equation} \label{eq:stouffer} \psi_S(p_1,\dots,p_r) = \sum_{j=1}^r w_j \Phi^{-1}(1 - p_j), \end{equation} where $\Phi^{-1}$ is the quantile function of the standard normal. Large values provide evidence against the global null hypothesis $H_0 = H_0^{\scriptscriptstyle (1)} \cap \dots \cap H_0^{\scriptscriptstyle (r)}$. By analogy, the corresponding weighted version of the global statistic in Fisher's p-value combination method can be defined by \begin{equation} \label{eq:fisher} \psi_F(p_1,\dots,p_r) = -2 \sum_{j=1}^r w_j \log(p_j). \end{equation} If the p-values $p_1, \dots, p_r$ are independent and uniformly distributed on $(0,1)$, then it can be verified that $\psi_S(p_1,\dots,p_r)$ or $\psi_F(p_1,\dots,p_r)$ are pivotal, giving rise to simple exact global tests. If the component tests are dependent, however, the distributions of the previous statistics are not pivotal and computing the corresponding global p-values is not straightforward anymore.
Let $\bm X_n$ denote the available data (apart from measurability, no assumptions are made on $\bm X_n$, but it is instructive to think of $\bm X_n$ as an $n$-tuple of possibly multivariate observations which may be serially dependent) and let $T_{n,1}=T_{n,1}(\bm X_n),\dots,T_{n,r}=T_{n,r}(\bm X_n)$ be the statistics, each $\mathbb{R}$-valued, of the $r$ tests to be combined.
We assume furthermore that, for any $j \in \{1,\dots,r\}$, large values of $T_{n,j}$ provide evidence against the hypothesis $H_0^{\scriptscriptstyle (j)}$. As we continue, we let $\bm T_n = \bm T_n(\bm X_n)$ denote the $r$-dimensional random vector $(T_{n,1},\dots,T_{n,r}) = (T_{n,1}(\bm X_n),\dots,T_{n,r}(\bm X_n))$.
We suppose additionally that we have available a resampling mechanism which allows us to obtain a sample of $M$ bootstrap replicates $\bm T_n^{\scriptscriptstyle [i]} = \bm T_n^{\scriptscriptstyle [i]}(\bm X_n, \bm V_n^{\scriptscriptstyle [i]})$, $i \in \{1,\dots,M\}$, of $\bm T_n$ where $\bm V_n^{\scriptscriptstyle [1]}, \dots, \bm V_n^{\scriptscriptstyle [M]}$ are independent and identically distributed (i.i.d.) $\mathbb{R}^n$-valued random vectors representing the additional sources of randomness involved in the resampling mechanism and such that, for any $i \in \{1,\dots,M\}$, $T_{n,j}^{\scriptscriptstyle [i]}$ depends on the data $\bm X_n$ and $\bm V_n^{\scriptscriptstyle [i]}$, that is, $T_{n,j}^{\scriptscriptstyle [i]} = T_{n,j}^{\scriptscriptstyle [i]}(\bm X_n, \bm V_n^{\scriptscriptstyle [i]})$ for all $j \in \{1,\dots,r\}$. Note that the previous setup naturally implies that the components $T_{n,\scriptscriptstyle 1}^{\scriptscriptstyle [i]},\dots,T_{n,r}^{\scriptscriptstyle [i]}$ of $\bm T_n^{\scriptscriptstyle [i]}$ are bootstrap replicates of the components $T_{n,1},\dots,T_{n,r}$ of $\bm T_n$. The fact that all the components of $\bm T_n^{\scriptscriptstyle [i]}$ depend on the same additional source of randomness $\bm V_n^{\scriptscriptstyle [i]}$ makes it possible to expect that the bootstrap replicates $\bm T_n^{\scriptscriptstyle [i]}$, $i \in \{1,\dots,M\}$, be, approximately, i.i.d.\ copies of $\bm T_n$ under the global null hypothesis $H_0 = H_0^{\scriptscriptstyle (1)} \cap \dots \cap H_0^{\scriptscriptstyle (r)}$. For the individual test based on $T_{n,j}$, $j \in \{1,\dots,r\}$, an approximate p-value could then naturally be computed as \[
\frac1M \sum_{i=1}^M \mathbf{1}(T_{n,j}^{[i]} \ge T_{n,j}). \]
Let $\psi$ be a continuous function from $(0,1)^r$ to $\mathbb{R}$ that is decreasing in each of its $r$ arguments (such as $\psi_S$ or $\psi_F$ in~\eqref{eq:stouffer} and~\eqref{eq:fisher}, respectively). To compute an approximate $p$-value for the global statistic $\psi \{ p_{n,M}(T_{n,1}),\dots,p_{n,M}(T_{n,r}) \}$, we propose the following procedure: \begin{enumerate}
\item Let $\bm T_n^{\scriptscriptstyle [0]} = \bm T_{n}$.
\item Given a large integer $M$, compute the sample of $M$ bootstrap replicates $\bm T_n^{\scriptscriptstyle [1]},\dots,\bm T_n^{\scriptscriptstyle [M]}$ of~$\bm T_n^{\scriptscriptstyle [0]}$.
\item Then, for all $i \in \{0,1,\dots,M\}$ and $j \in \{1,\dots,r\}$, compute \begin{equation} \label{eq:pval_T} p_{n,M}(T_{n,j}^{[i]}) = \frac{1}{M+1} \bigg\{\frac{1}{2} + \sum_{k=1}^M \mathbf{1} \left( T_{n,j}^{[k]} \geq T_{n,j}^{[i]} \right) \bigg\}. \end{equation}
\item Next, for all $i \in \{0,1,\dots,M\}$, compute \begin{equation} \label{eq:WnMi} W_{n,M}^{[i]} = \psi \{ p_{n,M}(T_{n,1}^{[i]}),\dots,p_{n,M}(T_{n,r}^{[i]}) \}. \end{equation} \item The global statistic is $W_{n,M}^{\scriptscriptstyle [0]}$ and the corresponding approximate $p$-value is given by \begin{equation} \label{eq:pval_W} p_{n,M}(W_{n,M}^{[0]}) =\frac{1}{M} \sum_{k=1}^M \mathbf{1} \left( W_{n,M}^{[k]} \geq W_{n,M}^{[0]} \right). \end{equation} \end{enumerate}
Note that the quantities $p_{n,M}(T_{n,j}^{\scriptscriptstyle [i]})$, $j \in \{1,\dots,r\}$, in Step~3 can be regarded as approximate p-values for the ``statistic values'' $T_{n,j}^{\scriptscriptstyle [i]}$, $j \in \{1,\dots,r\}$. The offset by $1/2$ and the division by $M+1$ instead of $M$ in the formula is carried out to ensure that $p_{n,M}(T_{n,j}^{\scriptscriptstyle [i]})$ belongs to the interval $(0,1)$ so that Step~4 is well-defined.
The next result, proved in Appendix~\ref{app:proofs}, provides conditions under which the global test based on $W_{n,M}^{\scriptscriptstyle [0]}$ given by~\eqref{eq:WnMi} is asymptotically valid under the global null hypothesis $H_0 = H_0^{\scriptscriptstyle (1)} \cap \dots \cap H_0^{\scriptscriptstyle (r)}$ and the natural assumption that $M = M_n \to \infty$ as $n \to \infty$. Before proceeding, note that $W_{\scriptscriptstyle n,M_n}^{\scriptscriptstyle [0]}$ is a Monte Carlo approximation of the unobservable statistic \begin{equation} \label{eq:Wn} W_n = \psi\{\mathbb{P}(T_{n,1}^{\scriptscriptstyle [1]} \geq T_{n,1} \,|\, \bm X_n),\dots,\mathbb{P}(T_{n,r}^{\scriptscriptstyle [1]} \geq T_{n,r} \,|\, \bm X_n)\}. \end{equation}
\begin{prop} \label{prop:combined:general} Let $M=M_n\to\infty$ as $n\to\infty$. Assume that $H_0 = H_0^{\scriptscriptstyle (1)} \cap \dots \cap H_0^{\scriptscriptstyle (r)}$ holds, that $\bm T_n$ converges weakly to $\bm T = (T_1,\dots,T_r)$, where $\bm T$ has a continuous d.f., and that either \begin{equation} \label{eq:uncond:Tn} (\bm T_n, \bm T_n^{[1]}, \bm T_n^{[2]}) \leadsto (\bm T, \bm T^{[1]}, \bm T^{[2]}), \end{equation} where $\bm T^{[1]}$ and $\bm T^{[2]}$ are independent copies of $\bm T$, or \begin{equation} \label{eq:cond:Tn}
\sup_{\bm x \in \mathbb{R}^r} | \mathbb{P}(\bm T_n^{[1]} \leq \bm x \,|\, \bm X_n) - \mathbb{P}(\bm T_n \leq \bm x)| \p 0. \end{equation} Then, for any $N \in \mathbb{N}$, \begin{equation} \label{eq:uncondM} (W_{n,M_n}^{[0]}, W_{n,M_n}^{[1]},\dots, W_{n,M_n}^{[N]}) \leadsto (W,W^{[1]},\dots,W^{[N]}), \end{equation} where \begin{equation} \label{eq:W} W = \psi\{\bar F_{T_1}(T_1), \dots, \bar F_{T_r}(T_r) \} \end{equation} is the weak limit of $W_n$ in~\eqref{eq:Wn} with $\bar F_{T_j}(x) = \mathbb{P}(T_j \geq x)$, $x \in \mathbb{R}$, $j \in \{1,\dots,r\}$, and $W^{[1]},\dots,W^{[N]}$ are independent copies of $W$. Furthermore, if $\psi$ is chosen in such a way that the random variable $W$ has a continuous d.f., then \begin{align} \label{eq:condM}
&\sup_{x \in\mathbb{R}} | \mathbb{P}(W_{n,M_n}^{[1]} \le x \,|\, \bm X_n) - \mathbb{P}(W_n \le x) | \p 0, \\ \label{eq:MC}
&\sup_{x \in\mathbb{R}} \bigg| \frac{1}{M_n} \sum_{i=1}^{M_n} \mathbf{1}(W_{n,M_n}^{[i]} \le x) - \mathbb{P}(W_n \le x) \bigg| \p 0, \end{align} and, as a consequence, $p_{n,M_{n}}(W_{\scriptscriptstyle n,M_n}^{\scriptscriptstyle [0]}) \leadsto \text{Uniform}(0,1)$, where $p_{n,M_{n}}(W_{\scriptscriptstyle n,M_n}^{\scriptscriptstyle [0]})$ is defined by~\eqref{eq:pval_W}. \end{prop}
It is worthwhile mentioning that, by Lemma~2.2 of \cite{BucKoj18} and the assumption of continuity for the d.f.\ of $\bm T$, the statements~\eqref{eq:uncond:Tn} and~\eqref{eq:cond:Tn} are actually equivalent in the setting under consideration. Notice also that the resulting unconditional bootstrap consistency statement in~\eqref{eq:uncondM} does not require $W$ in~\eqref{eq:W} to have a continuous d.f. Proving the latter might actually be quite complicated as shall be illustrated in a particular case in Section~\ref{sec:comb:cop:df}.
We end this section by providing a result, proved in Appendix~\ref{app:proofs}, that states conditions under which the global test based on $W_{n,M}^{\scriptscriptstyle [0]}$ given by~\eqref{eq:WnMi} leads to the rejection of the global null hypothesis $H_0 = H_0^{\scriptscriptstyle (1)} \cap \dots \cap H_0^{\scriptscriptstyle (r)}$.
\begin{prop}
\label{prop:combined:alternative}
Let $M=M_n\to\infty$ as $n\to\infty$. Assume that
\begin{enumerate}[(i)]
\item the combining function $\psi$ is of the form
\[
\psi(p_1, \dots, p_r) = \sum_{j=1}^r w_j \varphi(p_j),
\]
where $\varphi$ is decreasing, non-negative and one-to-one from $(0,1)$ to $(0,\infty)$,
\item there exists $j_0 \in \{1,\dots,r\}$ such that the null hypothesis $H_0^{\scriptscriptstyle (j_0)}$ of $j_0$th test $T_{n,j_0}$ does not hold and $\mathbb{P}(T_{n,j_0}^{\scriptscriptstyle [1]} \ge T_{n,j_0})$ converges to zero,
\item for any $j \in \{1,\dots,r\}$, the sample of bootstrap replicates $T_{n,j}^{\scriptscriptstyle [1]}, \dots, T_{n,j}^{\scriptscriptstyle [M_n]}$ does not contain ties.
\end{enumerate}
Then, the approximate p-value $p_{n,M_n}(W_{\scriptscriptstyle n,M_n}^{\scriptscriptstyle [0]})$ of the global test converges to zero in probability, where $p_{n,M_n}(W_{\scriptscriptstyle n,M_n}^{\scriptscriptstyle [0]})$ is defined by~\eqref{eq:pval_W}. \end{prop}
Let us comment on the assumptions of the previous proposition. Assumption~$(i)$ is satisfied by the function $\psi_F$ defined by~\eqref{eq:fisher} but not by the function $\psi_S$ defined by~\eqref{eq:stouffer}. A result similar to Proposition~\ref{prop:combined:alternative}, which can be used to handle the function~$\psi_S$, is stated and proved in the supplementary material. Assumption~$(ii)$ can for instance be shown to hold under the hypothesis of one change in the contemporary d.f.\ of a time series when $T_{n,j_0}$ is a test statistic such as the one to be defined in Section~\ref{sec:dftest}, the observations are i.i.d., and the underlying resampling mechanism is a particular multiplier bootstrap. Specifically, in that case, one can rely on Theorem~3 of \cite{HolKojQue13} to show that, under the hypothesis of one change in the contemporary d.f., $T_{n,j_0}$ diverges to infinity in probability while $T_{n,j_0}^{\scriptscriptstyle [1]}$ is bounded in probability, implying that $T_{n,j_0}^{\scriptscriptstyle [1]} - T_{n,j_0}$ diverges to $-\infty$ in probability, and thus that $\mathbb{P}(T_{n,j_0}^{\scriptscriptstyle [1]} \ge T_{n,j_0})$ converges to zero. Finally, assumption~$(iii)$ appears empirically to be satisfied for most bootstrap-based tests for time series of continuous random variables.
\section{A rank-based combined test sensitive to departures from $H_{0}^{(h)}$} \label{sec:rank}
The aim of this section is to use the results of the previous section to derive a global test of stationarity by combining a test that is particularly sensitive to departures from $H_0^{\scriptscriptstyle (1)}$ in~\eqref{eq:H0:1} with a test that is particularly sensitive to departures from $H_{0,c}^{\scriptscriptstyle (h)}$ in~\eqref{eq:H0:Ch}. We start by describing the latter test and provide conditions under which it is asymptotically valid under stationarity. The available data, denoted generically by $\bm X_n$ in Section~\ref{sec:combine:tests}, take here, as in the introduction, the form of a stretch $X_1,\dots,X_{n+h-1}$ from a univariate time series, where $h$ is the chosen embedding dimension and where each $X_i$ is assumed to have a continuous d.f.
\subsection{A copula-based test sensitive to changes in the serial dependence}
The test that we consider has the potential of being sensitive to all types of changes in the serial dependence up to lag $h-1$. Under $H_0^{\scriptscriptstyle (h)}$ in~\eqref{eq:H0:Fh}, this serial dependence is completely characterized by the (auto)copula $C^{(h)}$ in~\eqref{eq:H0:Ch}. It is then natural to base the test on \emph{empirical (auto)copulas} \citep[see, e.g.,][]{Deh79,Deh81} calculated from portions of the data. For any $1 \leq k \leq l \leq n$, let \begin{equation} \label{eq:Cklh} C_{k:l}^{(h)}(\bm u) = \frac{1}{l-k+1} \sum_{i=k}^l \prod_{j=1}^h \mathbf{1}\{ G_{k:l}(X_{i+j-1}) \leq u_j \}, \qquad \bm u \in [0,1]^h, \end{equation} where \begin{equation} \label{eq:Gkl} G_{k:l}(x) = \frac{1}{l+h-k} \sum_{j=k}^{l+h-1} \mathbf{1}(X_j \leq x), \qquad x \in \mathbb{R}, \end{equation} with the convention that $C_{k:l}^{(h)} = 0$ if $k > l$. The quantity $C_{k:l}^{\scriptscriptstyle (h)}$ is a non-parametric estimator of $C^{(h)}$ based on $\bm Y_k^{\scriptscriptstyle (h)},\dots,\bm Y_l^{\scriptscriptstyle (h)}$ that, as already mentioned, we shall call the \emph{lag $h-1$ empirical autocopula}. The latter was for instance used in \cite{GenRem04} for testing serial independence. It can be verified that it is a straightforward transposition of one of the usual definitions of the empirical copula (when computed from a subsample) to the serial context under consideration.
\subsubsection{Test statistic}
The CUSUM statistic that we consider is \begin{equation} \label{eq:SnCh} S_{n,C^{(h)}} = \sup_{s \in [0,1]} \int_{[0,1]^h} \left\{ \mathbb{D}_{n,C^{(h)}}(s,\bm{u}) \right\}^2 \mathrm{d} C_{1:n}^{(h)} (\bm u) = \max_{1 \leq k \leq n-1} \int_{[0,1]^h} \left\{ \mathbb{D}_{n,C^{(h)}}(k/n,\bm{u}) \right\}^2 \mathrm{d} C_{1:n}^{(h)} (\bm u), \end{equation} where, as mentioned earlier, $\ip{.}$ is the floor function, \begin{equation} \label{eq:Dnh} \mathbb{D}_{n,C^{(h)}}(s,\bm{u}) = \sqrt{n} \lambda_n(0,s) \lambda_n(s,1) \left\{ C_{1:\ip{ns}}^{(h)}(\bm u) - C_{\ip{ns}+1:n}^{(h)} (\bm u) \right\}, \qquad (s, \bm u) \in [0,1]^{h+1}, \end{equation} and $\lambda_n(s,t) = (\ip{nt}-\ip{ns}) / n$, $(s,t) \in \Delta = \{ (s,t) \in [0,1]^2: s \le t\}$.
Under $H_0^{\scriptscriptstyle (h)}$ in~\eqref{eq:H0:Fh}, the difference between $C_{\scriptscriptstyle 1:k}^{\scriptscriptstyle (h)}$ and $C_{\scriptscriptstyle k+1:n}^{\scriptscriptstyle (h)}$ should be small for all $k \in \{1,\dots,n-1\}$, resulting in small values of $S_{n,C^{(h)}}$. At the opposite, large values of $S_{n,C^{(h)}}$ provide evidence of non-stationarity. The coefficient $\sqrt{n} \lambda_n(0,s) \lambda_n(s,1)$ in~\eqref{eq:Dnh} is the classical normalizing term in the CUSUM approach. It ensures that, under suitable conditions, $S_{n,C^{(h)}}$ converges in distribution under the null hypothesis of stationarity. Analogously to what was explained in the introduction, the test based on $S_{n,C^{(h)}}$ should in general not be used to reject $H_{\scriptscriptstyle 0,c}^{\scriptscriptstyle (h)}$ in~\eqref{eq:H0:Ch}: It is merely a test of stationarity that is particularly sensitive to a change in the lag $h-1$ autocopula.
\subsubsection{Limiting null distribution}
The limiting null distribution of $S_{n,C^{(h)}}$ turns out to be a corollary of a recent result by \cite{BucKoj16} and \cite{BucKojRohSeg14}. Under $H_0^{\scriptscriptstyle (h)}$ in~\eqref{eq:H0:Fh}, it can be verified that $\mathbb{D}_{n,C^{(h)}}$ in \eqref{eq:Dnh} can be written as \begin{equation} \label{eq:DnhH0}
\mathbb{D}_{n,C^{(h)}}(s,\bm{u}) = \lambda_n(s,1) \, \mathbb{C}_{n,C^{(h)}}(0,s,\bm{u}) - \lambda_n(0,s) \, \mathbb{C}_{n,C^{(h)}}(s,1,\bm{u}), \qquad (s,\bm{u}) \in [0,1]^{h+1}, \end{equation} where \begin{equation}
\label{eq:Cnh}
\mathbb{C}_{n,C^{(h)}}(s, t, \bm{u}) = \sqrt{n} \, \lambda_n(s, t) \, \{ C_{\ip{ns}+1: \ip{nt}}^{(h)}(\bm{u}) - C^{(h)}(\bm{u}) \}, \qquad (s, t, \bm{u}) \in \Delta \times [0, 1]^h. \end{equation} Hence, the null weak limit of the empirical process $\mathbb{D}_{n,C^{(h)}}$ follows from that of $\mathbb{C}_{n,C^{(h)}}$, which we shall call the \emph{sequential empirical autocopula process}.
The following usual condition on the partial derivatives of $C^{(h)}$ \citep[see][]{Seg12} is considered as we continue.
\begin{cond} \label{cond:pd} For any $j \in \{1,\dots,h\}$, the partial derivative $\dot C_{\scriptscriptstyle j}^{\scriptscriptstyle (h)} = \partial C^{\scriptscriptstyle (h)}/\partial u_j$ exists and is continuous on $V_{\scriptscriptstyle j}^{\scriptscriptstyle (h)} = \{ \bm{u} \in [0, 1]^h : u_j \in (0,1) \}$. \end{cond}
Condition~\ref{cond:pd} is nonrestrictive in the sense that it is necessary so that the candidate weak limit of $\mathbb{C}_{n,C^{(h)}}$ exists pointwise and has continuous sample paths. In the sequel, following \cite{BucVol13}, for any $j \in \{1,\dots,h\}$, we define $\dot C_{\scriptscriptstyle j}^{\scriptscriptstyle (h)}$ to be zero on the set $\{ \bm{u} \in [0, 1]^h : u_j \in \{0,1\} \}$. Also, as we continue, for any $j \in \{1,\dots,h\}$ and any $\bm{u} \in [0, 1]^h$, $\bm{u}^{(j)}$ will stand for the vector of $[0, 1]^h$ defined by $u^{\scriptscriptstyle (j)}_i = u_j$ if $i = j$ and 1 otherwise.
The null weak limit of $\mathbb{C}_{n,C^{(h)}}$ follows in turn from that of the sequential serial empirical process \begin{equation} \label{eq:seqep}
\mathbb{B}_{n,C^{(h)}}(s, t, \bm{u})
= \frac{1}{\sqrt{n}} \sum_{i=\ip{ns}+1}^{\ip{nt}} \bigg[ \prod_{j=1}^h \mathbf{1}\{ G(X_{i+j-1}) \leq u_j \} - C^{(h)}(\bm{u}) \bigg], \qquad (s, t,\bm{u}) \in \Delta \times [0, 1]^h, \end{equation} with the convention that $\mathbb{B}_{n,C^{(h)}}(s, t, \cdot) = 0$ if $\ip{nt} - \ip{ns} = 0$.
The following result, stating the weak limit of $\mathbb{C}_{n,C^{(h)}}$ and proved in Appendix~\ref{app:proofs}, is a consequence of the results of \cite{BucKoj16} and \cite{BucKojRohSeg14}. It considers $X_1,\dots,X_{n+h-1}$ as a stretch from a \emph{strongly mixing sequence}. For a sequence of random variables $(Z_i)_{i \in \mathbb{Z}}$, the $\sigma$-field generated by $(Z_i)_{a \leq i \leq b}$, $a, b \in \mathbb{Z} \cup \{-\infty,+\infty \}$, is denoted by $\mathcal{F}_a^b$. The strong mixing coefficients corresponding to the sequence $(Z_i)_{i \in \mathbb{Z}}$ are then defined by $\alpha_0^Z = 1/2$, \begin{equation} \label{eq:alpha}
\alpha_r^Z = \sup_{p \in \mathbb{Z}} \sup_{A \in \mathcal{F}_{-\infty}^p,B\in \mathcal{F}_{p+r}^{+\infty}} \big| \mathbb{P}(A \cap B) - \mathbb{P}(A) \mathbb{P}(B) \big|, \qquad r \in \mathbb{N}, \, r > 0. \end{equation} The sequence $(Z_i)_{i \in \mathbb{Z}}$ is said to be \emph{strongly mixing} if $\alpha_r^Z \to 0$ as $r \to \infty$.
\begin{prop} \label{prop:weak_Cnh_sm} Let $X_1,\dots,X_{n+h-1}$ be drawn from a strictly stationary sequence $(X_i)_{i \in \mathbb{Z}}$ of continuous random variables whose strong mixing coefficients satisfy $\alpha_r^X = O(r^{-a})$ for some $a > 1$ as $r \to \infty$. Then, provided Condition~\ref{cond:pd} holds, \begin{equation*}
\sup_{(s, t,\bm{u})\in \Delta \times [0, 1]^h} \Big| \mathbb{C}_{n,C^{(h)}}(s, t, \bm{u}) - \mathbb{B}_{n,C^{(h)}}(s, t, \bm{u}) + \sum_{j=1}^h \dot C_j^{(h)}(\bm{u}) \, \mathbb{B}_{n,C^{(h)}}(s, t, \bm{u}^{(j)}) \Big| \p 0. \end{equation*} Consequently, $\mathbb{C}_{n,C^{(h)}} \leadsto \mathbb{C}_{C^{(h)}}$ in $\ell^\infty(\Delta \times [0, 1]^h)$, where, for any $(s, t, \bm{u}) \in \Delta \times [0, 1]^h$, \begin{equation} \label{eq:CbCh} \mathbb{C}_{C^{(h)}}(s, t, \bm{u}) = \mathbb{B}_{C^{(h)}}(s, t, \bm{u}) - \sum_{j=1}^h \dot C_j^{(h)}(\bm{u}) \, \mathbb{B}_{C^{(h)}}(s, t, \bm{u}^{(j)}), \end{equation} and $\mathbb{B}_{C^{(h)}}$ in $\ell^\infty(\Delta \times [0, 1]^h)$, a tight centered Gaussian process, is the weak limit of $\mathbb{B}_{n,C^{(h)}}$ in~\eqref{eq:seqep}. \end{prop}
Since they are not necessary for the subsequent derivations, the expressions of the covariances of $\mathbb{B}_{C^{(h)}}$ and $\mathbb{C}_{C^{(h)}}$ are not provided. The latter can however be deduced from the above mentioned references.
The next result, proved in Appendix~\ref{app:proofs}, and partly a simple consequence of the previous proposition and the continuous mapping theorem, gives the limiting distribution of $S_{n,C^{(h)}}$ under the null hypothesis of stationarity.
\begin{prop} \label{prop:acs} Under the conditions of Proposition~\ref{prop:weak_Cnh_sm}, $\mathbb{D}_{n,C^{(h)}} \leadsto \mathbb{D}_{C^{(h)}}$ in $\ell^\infty([0,1]^{h+1})$, where, for any $(s,\bm{u}) \in [0,1]^{h+1}$, \begin{equation}
\label{eq:DCh}
\mathbb{D}_{C^{(h)}}(s,\bm{u}) = \mathbb{C}_{C^{(h)}}(0,s,\bm{u}) - s \, \mathbb{C}_{C^{(h)}}(0,1,\bm{u}), \end{equation} and $\mathbb{C}_{C^{(h)}}$ is defined by~\eqref{eq:CbCh}. As a consequence, we get \begin{equation} \label{eq:SCh}
S_{n,C^{(h)}} \leadsto S_{C^{(h)}} = \sup_{s \in [0,1]} \int_{[0, 1]^h} \{ \mathbb{D}_{C^{(h)}}(s,\bm{u}) \}^2 \, \mathrm{d} C^{(h)}(\bm{u}). \end{equation} Moreover, the distribution of $S_{C^{(h)}}$ is absolutely continuous with respect to the Lebesgue measure. \end{prop}
\subsubsection{Bootstrap and computation of approximate p-values}
The null weak limit of $S_{n,C^{(h)}}$ in~\eqref{eq:SCh} is unfortunately untractable. Starting from Proposition~\ref{prop:weak_Cnh_sm} and adapting the approach of \cite{BucKoj16} and \cite{BucKojRohSeg14}, we propose to base the computation of approximate p-values for $S_{n,C^{(h)}}$ on \emph{multiplier} resampling versions of $\mathbb{C}_{n,C^{(h)}}$ in~\eqref{eq:Cnh}. For any $m \in \mathbb{N}$ and any $(s,t,\bm{u}) \in \Delta \times [0,1]^h$, let \begin{equation} \label{eq:hatCbnm}
\hat{\mathbb{C}}_{n,C^{(h)}}^{[m]}(s, t, \bm{u}) = \hat{\mathbb{B}}_{n,C^{(h)}}^{[m]}(s, t, \bm{u}) - \sum_{j=1}^h \dot C_{j,1:n}^{(h)}(\bm{u}) \, \hat{\mathbb{B}}_{n,C^{(h)}}^{[m]}(s, t, \bm{u}^{(j)}), \end{equation} where $$ \dot{C}_{j,1:n}^{(h)}(\bm{u}) = \frac{C_{1:n}^{(h)}( \bm{u} + h \bm{e}_j ) - C_{1:n}^{(h)}( \bm{u} - h \bm{e}_j )}{\min(u_j+h,1) - \max(u_j-h, 0)} $$ with $\bm e_j$ the $j$-th unit vector and \begin{equation} \label{eq:hatBnm} \hat{\mathbb{B}}_{n,C^{(h)}}^{[m]}(s, t,\bm{u}) = \frac{1}{\sqrt{n}} \sum_{i=\ip{ns}+1}^{\ip{nt}} \xi_{i,n}^{[m]} \bigg[ \prod_{j=1}^h \mathbf{1}\{ G_{1:n}(X_{i+j-1}) \leq u_j \} - C_{1:n}^{(h)}(\bm{u}) \bigg], \end{equation} with $C_{1:n}^{\scriptscriptstyle (h)}$ and $G_{1:n}$ defined by~\eqref{eq:Cklh} and~\eqref{eq:Gkl}, respectively. The sequences of random variables $(\xi_{\scriptscriptstyle i,n}^{\scriptscriptstyle [m]})_{i \in \mathbb{Z}}$, $m \in \mathbb{N}$, appearing in the expressions of the processes $\hat{\mathbb{B}}_n^{\scriptscriptstyle (h),[m]}$ in~\eqref{eq:hatBnm}, $m \in \mathbb{N}$, are independent copies of what was called a \emph{dependent multiplier sequence} in \cite{BucKoj16}. Details on that definition, on how such a sequence can be generated and on how a respective block length parameter can be chosen adaptively are presented in Appendix~\ref{app:dep}.
Next, starting from~\eqref{eq:hatCbnm} and having~\eqref{eq:DnhH0} in mind, multiplier resampling versions of $\mathbb{D}_{n,C^{(h)}}$ are then naturally given, for any $m \in \mathbb{N}$ and $(s,\bm u) \in [0,1]^{h+1}$, by \begin{align*} \nonumber
\hat{\mathbb{D}}_{n,C^{(h)}}^{[m]}(s,\bm{u}) &= \lambda_n(s,1) \, \hat{\mathbb{C}}_{n,C^{(h)}}^{[m]}(0, s, \bm{u}) - \lambda_n(0,s) \, \hat{\mathbb{C}}_{n,C^{(h)}}^{[m]}(s, 1, \bm{u}) \\ &= \hat{\mathbb{C}}_{n,C^{(h)}}^{[m]}(0,s,\bm{u}) - \lambda_n(0,s) \, \hat{\mathbb{C}}_{n,C^{(h)}}^{[m]}(0, 1, \bm{u}). \end{align*} Corresponding multiplier resampling versions of the statistic $S_{n,C^{(h)}}$ in~\eqref{eq:SnCh} are finally \begin{equation} \label{eq:SnChm} \hat{S}_{n,C^{(h)}}^{[m]} = \sup_{s \in [0,1]} \int_{[0, 1]^h} \{ \hat{\mathbb{D}}_{n,C^{(h)}}^{[m]} (s, \bm{u}) \}^2 \, \mathrm{d} C_{1:n}^{(h)}(\bm{u}), \end{equation} which suggests computing an approximate p-value for $S_{n,C^{(h)}}$ as $M^{-1} \sum_{m=1}^M \mathbf{1} \big( \hat{S}_{n,C^{(h)}}^{[m]} \geq S_{n,C^{(h)}} \big)$ for some large integer $M$.
The following proposition establishes the asymptotic validity of the multiplier resampling scheme under the null hypothesis of stationarity. The proof is given in Appendix~\ref{app:proofs}.
\begin{prop} \label{prop:S_mult} Assume that $X_1,\dots,X_{n+h-1}$ are drawn from a strictly stationary sequence $(X_i)_{i \in \mathbb{Z}}$ of continuous random variables whose strong mixing coefficients satisfy $\alpha_r^X = O(r^{-a})$ as $r \to \infty$ for some $a > 3+3h/2$, and $(\xi_{\scriptscriptstyle i,n}^{\scriptscriptstyle [1]})_{i \in \mathbb{Z}},(\xi_{\scriptscriptstyle i,n}^{\scriptscriptstyle [2]})_{i \in \mathbb{Z}}, \dots$ are independent copies of a dependent multiplier sequence satisfying~($\mathcal{M} 1$)--($\mathcal{M} 3$) in Appendix~\ref{app:dep} with $\ell_n = O(n^{1/2 - \gamma})$ for some $0 < \gamma < 1/2$. Then, for any $M \in \mathbb{N}$, $$
\big(\mathbb{C}_{n,C^{(h)}}, \hat{\mathbb{C}}_{n,C^{(h)}}^{[1]}, \dots, \hat{\mathbb{C}}_{n,C^{(h)}}^{[M]} \big)
\leadsto
\big(\mathbb{C}_{C^{(h)}}, \mathbb{C}_{C^{(h)}}^{[1]}, \dots, \mathbb{C}_{C^{(h)}}^{[M]} \big) $$ in $\{\ell^\infty(\Delta \times [0, 1]^h)\}^{M+1}$, where $\mathbb{C}_{C^{(h)}}$ is defined by~\eqref{eq:CbCh}, and $\mathbb{C}_{C^{(h)}}^{\scriptscriptstyle [1]},\dots,\mathbb{C}_{C^{(h)}}^{\scriptscriptstyle [M]}$ are independent copies of $\mathbb{C}_{C^{(h)}}$. As a consequence, for any $M \in \mathbb{N}$, $$
\big( \mathbb{D}_{n,C^{(h)}}, \hat{\mathbb{D}}_{n,C^{(h)}}^{[1]}, \dots, \hat{\mathbb{D}}_{n,C^{(h)}}^{[M]} \big)
\leadsto
\big( \mathbb{D}_{C^{(h)}}, \mathbb{D}_{C^{(h)}}^{[1]}, \dots, \mathbb{D}_{C^{(h)}}^{[M]} \big) $$ in $\{ \ell^\infty([0,1]^{h+1}) \}^{M+1}$, where $\mathbb{D}_{C^{(h)}}$ is defined by~\eqref{eq:DCh} and $\mathbb{D}_{C^{(h)}}^{\scriptscriptstyle [1]}, \dots, \mathbb{D}_{C^{(h)}}^{\scriptscriptstyle [M]}$ are independent copies of $\mathbb{D}_{C^{(h)}}$. Finally, for any $M \in \mathbb{N}$, \begin{equation*}
\big( S_{n,C^{(h)}}, \hat{S}_{n,C^{(h)}}^{[1]}, \dots, \hat{S}_{n,C^{(h)}}^{[M]} \big) \leadsto
\big( S_{C^{(h)}},S_{C^{(h)}}^{[1]}, \dots, S_{C^{(h)}}^{[M]} \big), \end{equation*} where $S_{C^{(h)}}$ is defined by~\eqref{eq:SCh} and $S_{C^{(h)}}^{\scriptscriptstyle [1]},\dots,S_{C^{(h)}}^{\scriptscriptstyle [M]}$ are independent copies of $S_{C^{(h)}}$. \end{prop}
Notice that, by Lemma~2.2 of \cite{BucKoj18} and the continuity of the d.f.\ of $S_{C^{(h)}}$ (see Proposition~\ref{prop:acs} above), the last statement of Proposition~\ref{prop:S_mult} is equivalent to the following more classical formulation of bootstrap consistency: $$
\sup_{x \in \mathbb{R}} | \mathbb{P}(\hat S_{n,C^{(h)}}^{[1]} \leq x \,|\, \bm X_n) - \mathbb{P}(S_{n,C^{(h)}} \leq x)| \p 0. $$
Furthermore, Lemma~4.2 in \cite{BucKoj18} ensures that the test based on $S_{n,C^{(h)}}$ with approximate p-value $p_{n,M}(S_{n,C^{(h)}}) = M^{-1} \sum_{m=1}^M \mathbf{1} \big( \hat{S}_{n,C^{(h)}}^{\scriptscriptstyle [m]} \geq S_{n,C^{(h)}} \big)$ holds its level asymptotically under the null hypothesis of stationarity as $n$ and $M$ tend to the infinity. By Corollary~4.3 in the same reference, this implies that $p_{n,M_n}(S_{n,C^{(h)}}) \leadsto \text{Uniform}(0,1)$ when $n\to \infty$, for any sequence $M_n \to \infty$.
\subsection{A d.f.-based test sensitive to changes in the contemporary distribution} \label{sec:dftest}
We propose to combine the previous test with a test particularity sensitive to departures from $H_0^{\scriptscriptstyle (1)}$ in~\eqref{eq:H0:1}. As mentioned in the introduction, a natural candidate is the CUSUM test studied in~\cite{GomHor99} and extended in \cite{HolKojQue13}. For the sake of a simpler presentation, we proceed as if the only available observations were $X_1,\dots,X_n$, thereby ignoring the remaining $h-1$ ones. The test statistic can then be written as \begin{equation} \label{eq:Sng} S_{n,G} = \sup_{s \in [0,1]} \int_{\mathbb{R}} \left\{ \mathbb{E}_n(s,x) \right\}^2 \mathrm{d} G_{1:n}(x), \end{equation} where \begin{equation} \label{eq:En} \mathbb{E}_n(s,x) = \sqrt{n} \lambda_n(0,s) \lambda_n(s,1) \left\{ G_{1:\ip{ns}}(x) - G_{\ip{ns}+1:n} (x) \right\}, \qquad (s, x) \in [0,1] \times \mathbb{R}, \end{equation} and, for any $1 \leq k \leq l \leq n$, $G_{k:l}$ is defined as in~\eqref{eq:Gkl} but with $h=1$. As one can see, the test involves the comparison of the empirical d.f.\ of $X_1,\dots,X_k$ with the one of $X_{k+1},\dots,X_n$ for all $k \in \{1,\dots,n-1\}$. Under $H_0^{\scriptscriptstyle (1)}$ in~\eqref{eq:H0:1}, it can be verified that $\mathbb{E}_n$ in \eqref{eq:En} can be written as \begin{equation*}
\mathbb{E}_n(s,x) = \mathbb{G}_n(s,x) - \lambda_n(0,s) \, \mathbb{G}_n(1,x), \qquad (s, x) \in [0,1] \times \mathbb{R}, \end{equation*} where \begin{equation}
\label{eq:Gn}
\mathbb{G}_n(s, x) = \sqrt{n} \, \lambda_n(0, s) \, \{ G_{1: \ip{ns}}(x) - G(x) \}, \qquad (s, x) \in [0,1] \times \mathbb{R}. \end{equation}
The following result, proved in Appendix~\ref{app:proofs} and providing the null weak limit of $S_{n,G}$ in~\eqref{eq:Sng}, is partly an immediate consequence of Theorem~1 of \cite{Buc15} and of the continuous mapping theorem.
\begin{prop} \label{prop:weak_Gn_sm} Let $X_1,\dots,X_{n}$ be drawn from a strictly stationary sequence $(X_i)_{i \in \mathbb{Z}}$ of continuous random variables whose strong mixing coefficients satisfy $\alpha_r = O(r^{-a})$ for some $a > 1$, as $r \to \infty$. Then, $\mathbb{G}_n \leadsto \mathbb{G}$ in $\ell^\infty([0, 1] \times \mathbb{R})$, where $\mathbb{G}$ is a tight centered Gaussian process with covariance function $$ \mathrm{Cov}\{\mathbb{G}(s,x), \mathbb{G}(t, y) \} = \min(s,t) \sum_{k \in \mathbb{Z}} \mathrm{Cov}\{\mathbf{1}(X_0 \leq x) \mathbf{1}(X_k \leq y) \}. $$ Consequently, $\mathbb{E}_n \leadsto \mathbb{E}$ in $\ell^\infty([0, 1] \times \mathbb{R})$, where \begin{equation} \label{eq:E} \mathbb{E}(s,x) = \mathbb{G}(s,x) - s \mathbb{G}(1,x), \qquad (s, x) \in [0,1] \times \mathbb{R}, \end{equation} and $S_{n,G} \leadsto S_G$ with \begin{equation} \label{eq:SG} S_G = \sup_{s \in [0,1]} \int_{\mathbb{R}} \left\{ \mathbb{E}(s,x) \right\}^2 \mathrm{d} G(x). \end{equation} Moreover, the distribution of $S_G$ is absolutely continuous with respect to the Lebesgue measure. \end{prop}
Following \cite{GomHor99}, \cite{HolKojQue13} and \cite{BucKoj16}, we shall compute approximate p-values for $S_{n,G}$ using multiplier resampling versions of $\mathbb{G}_n$ in~\eqref{eq:Gn}. Let $(\xi_{i,n}^{\scriptscriptstyle [m]})_{i \in \mathbb{Z}}$, $m \in \mathbb{N}$, be independent copies of the same dependent multiplier sequence. For any $m \in \mathbb{N}$ and any $(s,x) \in [0,1] \times \mathbb{R}$, let \begin{align}
\nonumber
\hat{\mathbb{G}}_n^{[m]}(s, x) &= \frac{1}{\sqrt{n}} \sum_{i=1}^{\ip{ns}} \xi_{i,n}^{[m]} \left\{ \mathbf{1}(X_i \leq x) - G_{1:n}(x) \right\}, \\
\nonumber
\hat \mathbb{E}_n^{[m]}(s,x) &= \mathbb{G}_n^{[m]}(s,x) - \lambda_n(0,s) \, \mathbb{G}_n^{[m]}(1,x), \\
\label{eq:SnGm}
\hat{S}_{n,G}^{[m]} &= \sup_{s \in [0,1]} \int_{\mathbb{R}} \left\{ \hat \mathbb{E}_n^{[m]}(s,x) \right\}^2 \mathrm{d} G_{1:n}(x). \end{align} An approximate p-value for $S_{n,G}$ will then be computed as $p_{n,M}(S_{n,G}) = M^{-1} \sum_{m=1}^M \mathbf{1} \big( \hat{S}_{n,G}^{\scriptscriptstyle [m]} \geq S_{n,G} \big)$ for some large integer $M$. The asymptotic validity of this approach under the null hypothesis of stationarity can be shown as for the test based on $S_{n,C^{(h)}}$ presented in the previous section. The result is a direct consequence of Corollary~2.2 in \cite{BucKoj16}; see also Proposition~\ref{prop:combined1} in the next section. In particular, $p_{n,M_n}(S_{n,G}) \leadsto \text{Uniform}(0,1)$ when $n \to \infty$, for any sequence $M_n\to\infty$.
\subsection{Combining the two tests} \label{sec:comb:cop:df}
To combine the two tests, we use the general procedure described in Section~\ref{sec:combine:tests} with $r=2$, $T_{n,1} = S_{n,C^{(h)}}$ and $T_{n,2} = S_{n,G}$, for some suitable function $\psi:(0,1)^2 \to \mathbb{R}$ such as $\psi_S$ in~\eqref{eq:stouffer} or $\psi_F$ in~\eqref{eq:fisher}. To be able to apply Proposition~\ref{prop:combined:general}, we need to find conditions under which $\bm T_n = (T_{n,1}, T_{n,2})$ and its bootstrap replicates satisfy~\eqref{eq:uncond:Tn} or, equivalently,~\eqref{eq:cond:Tn}. A natural prerequisite is to compute the $M$ bootstrap replicates of $T_{n,1} = S_{n,C^{(h)}}$ and $T_{n,2} = S_{n,G}$ in~\eqref{eq:SnChm} and~\eqref{eq:SnGm}, respectively, using the same $M$ dependent multiplier sequences. Since a moving average approach is used to generate such sequences, it follows from~\eqref{eq:movave} that it is sufficient to impose that the same $M$ initial independent normal sequences be used for both tests. In practice, prior to using~\eqref{eq:movave} to generate the $M$ independent copies of the same dependent multiplier sequence, we estimate the key bandwidth parameter $\ell_n$ from $X_1,\dots,X_{n+h-1}$ using the approach proposed in \citet[Section 5.1]{BucKoj16}, briefly overviewed in Appendix~\ref{app:dep}.
The next result, proven in Appendix~\ref{app:proofs}, provides conditions under which~\eqref{eq:uncond:Tn} holds.
\begin{prop} \label{prop:combined1} Under the conditions of Proposition~\ref{prop:S_mult}, for any $M \in \mathbb{N}$, $$
\big((\mathbb{D}_{n,C^{(h)}},\mathbb{E}_n), (\hat{\mathbb{D}}_{n,C^{(h)}}^{[1]}, \hat{\mathbb{E}}_n^{[1]}), \dots, (\hat{\mathbb{D}}_{n,C^{(h)}}^{[M]}, \hat{\mathbb{E}}_n^{[M]}) \big)
\leadsto
\big((\mathbb{D}_{C^{(h)}}, \mathbb{E}), (\mathbb{D}_{C^{(h)}}^{[1]}, \mathbb{E}^{[1]}), \dots, (\mathbb{D}_{C^{(h)}}^{[M]}, \mathbb{E}^{[M]}) \big) $$ in $\{\ell^\infty([0, 1] \times \mathbb{R})\}^{2(M+1)}$, where $\mathbb{D}_{C^{(h)}}$ and $\mathbb{E}$ are defined by~\eqref{eq:DCh} and~\eqref{eq:E}, respectively, and $(\mathbb{D}_{C^{(h)}}^{\scriptscriptstyle [1]},\mathbb{E}^{[1]}),\dots,(\mathbb{D}_{C^{(h)}}^{\scriptscriptstyle [M]},\mathbb{E}^{[M]})$ are independent copies of $(\mathbb{D}_{C^{(h)}},\mathbb{E})$. Note that we do not specify the joint law of $(\mathbb{D}_{C^{(h)}},\mathbb{E})$; it will only be important that $(\hat{\mathbb{D}}_{n,C^{(h)}}^{\scriptscriptstyle [m]}, \hat{\mathbb{E}}_n^{\scriptscriptstyle [m]})$, $m \in \{1,\dots,M\}$, can be considered to have the same joint law as $(\mathbb{D}_{C^{(h)}},\mathbb{E})$ asymptotically. As a consequence, $$
\big( (S_{n,C^{(h)}},S_{n,G}),( \hat{S}_{n,C^{(h)}}^{[1]}, \hat{S}_{n,G}^{[1]}), \dots, (\hat{S}_{n,C^{(h)}}^{[M]}, \hat{S}_{n,G}^{[M]}) \big)
\leadsto
\big( (S_{C^{(h)}},S_G), (S_{C^{(h)}}^{[1]},S_G^{[1]}), \dots, (S_{C^{(h)}}^{[M]}, S_G^{[M]}) \big), $$ where $S_{C^{(h)}}$ and $S_G$ are defined by~\eqref{eq:SCh} and~\eqref{eq:SG}, respectively, and where the random vectors $(S_{C^{(h)}}^{\scriptscriptstyle [1]},S_G^{\scriptscriptstyle [1]}), \dots, (S_{C^{(h)}}^{\scriptscriptstyle [M]},S_G^{\scriptscriptstyle [M]})$ are independent copies of $(S_{C^{(h)}},S_G)$. \end{prop}
A consequence of the previous proposition is that the unconditional bootstrap consistency statement in~\eqref{eq:uncondM} holds under the conditions of Proposition~\ref{prop:S_mult}. To conclude that the conditional statements given in~\eqref{eq:condM} and~\eqref{eq:MC} hold has well, it is necessary to establish that $W$, given generically by~\eqref{eq:W}, has a continuous d.f. Proving the latter might actually be quite complicated: unlike $S_{C^{(h)}}$ in~\eqref{eq:SCh} and $S_G$ in~\eqref{eq:SG}, $W$ is not a convex function of some Gaussian process, whence the general results from \cite{DavLif84} and the references therein are not applicable. Proving the absolute continuity of the vector $(S_{C^{(h)}},S_G)$ could be a first step but the latter does not seem easy either: available results in the literature are mostly based on complicated conditions from Malliavin Calculus, see, e.g., Theorem~2.1.2 in \cite{Nua06}. For these reasons, we do not pursue such investigations any further in this paper. Nonetheless, we conjecture that $W$ will have a continuous d.f.\ in all except a few very pathological situations.
Under suitable conditions on alternative models, it can further be shown that at least one of the statistics $S_{n,G}$ or $S_{n,C^{(h)}}$ (for $h$ suitably chosen) diverges to infinity in probability at rate~$n$. For instance, for $S_{n,G}$, under the assumption of at most one change in the contemporary d.f.\ of the time series, the latter can be shown by adapting to the serially dependent case the arguments used in \citet[Proof of Theorem 3~(i)]{HolKojQue13}. Further details are omitted for the sake of brevity. As far as bootstrap replicates of $S_{n,G}$ or $S_{n,C^{(h)}}$ are concerned, based on our extensive simulation results, we conjecture that, for many alternative models, the bootstrap replicates are of lower order than $O_\mathbb{P}(n)$. As a consequence, assuming the aforementioned results, and when the combining function $\psi$ is $\psi_F$ in~\eqref{eq:fisher}, one can rely on Proposition~\ref{prop:combined:alternative} to show the consistency of the test based on~$W_{\scriptscriptstyle n,M_n}^{\scriptscriptstyle [0]}$ in~\eqref{eq:WnMi}.
\subsection{On the choice of the embedding dimension $h$} \label{sec:hchoice}
The methodology described in the previous sections depends on the embedding dimension~$h$. In this section, we will provide some intuition about the trade-off between the choice of small and large values of $h$. Based on the developed arguments, and on the large-scale simulation study in Section~\ref{sec:MC} and in the supplementary material, we will make a practical suggestion at the end of this section.
Let us start by considering arguments in favour of choosing a large value of $h$. For that purpose, note that stationarity is equivalent to $H_0^{\scriptscriptstyle (1)}$ in~\eqref{eq:H0:1} and $H_{\scriptscriptstyle 0,c}^{\scriptscriptstyle (h)}$ in~\eqref{eq:H0:Ch} for all $h\ge 2$, and that a test based on the embedding dimension $h$ can only detect alternatives for which $H_{\scriptscriptstyle 0,c}^{\scriptscriptstyle (h)}$ does not hold. Hence, since $H_{\scriptscriptstyle 0,c}^{\scriptscriptstyle (2)} \Leftarrow H_{\scriptscriptstyle 0,c}^{\scriptscriptstyle (3)} \Leftarrow \dots$, we would like to choose $h$ as large as possible to be consistent against as many alternatives as possible. Note that, at the same time, the potential gain in moving from $h$ to $h+1$ should decrease with $h$: first, the larger $h$, the less likely it seems that real-life phenomena satisfy $H_{\scriptscriptstyle 0,c}^{\scriptscriptstyle (h)}$ but not $H_{\scriptscriptstyle 0,c}^{\scriptscriptstyle (h+1)}$; second, from a model-engineering perspective, the larger the value of~$h$, the more difficult and artificial it becomes to construct sensible models that satisfy $H_{\scriptscriptstyle 0,c}^{\scriptscriptstyle (h)}$ but not $H_{\scriptscriptstyle 0,c}^{\scriptscriptstyle (h+1)}$. To illustrate the latter point, constructing such a model on the level of copulas would amount to finding (at least two) different $(h+1)$-dimensional copulas $C^{(h+1)}$ that have the same lower-dimensional (multivariate) margins. More formally and given the serial context under consideration, this would mean finding a model such that \[ C^{(h+1)}(1,\ldots,1,u_i,\ldots,u_{i+k-1},1,\ldots,1)=C^{(k)}(u_i,\ldots,u_{i+k-1}), \]
for all $k\in \{ 2,\ldots, h\}$, $i \in \{1,\ldots,h-k+2\}$, $u_i,\ldots,u_{i+k-1} \in [0,1]$, for some given $k$-dimensional copulas $C^{(k)}$. This problem is closely related to the so-called compatibility problem (\citealp{Nel06}, Section 3.5) and, to the best of our knowledge, has not yet a general solution. Some necessary conditions can be found in \citet[Theorem~4]{Rus85} for the case of copulas that are absolutely continuous with respect to the Lebesgue measure on the unit hypercube. As another example, consider as a starting point the autoregressive process $X_i=\beta X_{i-h} + \varepsilon_i$, where the noises $\varepsilon_i\sim \mathcal N(0,\tau^2)$ are i.i.d. and where $|\beta|<1$. The components of the vectors $\bm Y^{\scriptscriptstyle (h)}_i=(X_i, \dots,X_{i+h-1})$ are then i.i.d.\ $\mathcal N(0,\tau^2/(1-\beta^2))$. Hence, $C^{(h)}$ is the independence copula and $H_0^{\scriptscriptstyle (h)}$ in~\eqref{eq:H0:Fh} is met, while $H_{0,c}^{\scriptscriptstyle (h+1)}$ in~\eqref{eq:H0:Ch} would not be met should the parameters $\tau$ and $\beta$ change (smoothly or abruptly) in such a way that $\tau^2/(1-\beta^2)$ stays constant; a rather artificial example. More generally, one could argue that, the larger $h$, the more artificial instances of common time series models (such as ARMA- or GARCH-type models) for which $H_{\scriptscriptstyle 0,c}^{\scriptscriptstyle (h)}$ holds but not $H_{\scriptscriptstyle 0,c}^{\scriptscriptstyle (h+1)}$ seem to be.
The previous paragraph suggests to choose $h$ as large as possible, even if the marginal gain of an increase of $h$ becomes smaller for larger and larger $h$. At the opposite, there are also good reasons for choosing $h$ rather small. Indeed, for many sensible models, the power of the test based on $S_{n,C^{(h)}}$ {in~\eqref{eq:SnCh} is a decreasing function of $h$, at least from some small value onwards. This observation will for instance be one of the results of our simulation study in Section~\ref{sec:MC} (see, e.g., Figure~\ref{fig:h}), but it can also be supported by more theoretical arguments. Indeed, consider for instance the following simple alternative model: $X_1,X_2,\dots$ have the same d.f.\ $G$ and, for some $s^* \in (0,1)$, $\bm Y_i^{\scriptscriptstyle (h)}$, $i \in \{1,\dots,\ip{ns^*}-\ip{h/2}\}$, have copula $C_1^{\scriptscriptstyle (h)}$ and $\bm Y_i^{\scriptscriptstyle (h)}$, $i \in \{\ip{ns^*}+1+\ip{h/2},\dots,n\}$, have copula $C_2^{\scriptscriptstyle (h)} \ne C_1^{\scriptscriptstyle (h)}$. For simplicity, we do not specify the laws of the $\bm Y_i^{\scriptscriptstyle (h)}$ for $i\in \{\ip{ns^*}-\ip{h/2}+1,\ldots, \ip{ns^*}+\ip{h/2}\}$ (these observations induce negligible effects in the following reasoning), whence, asymptotically, we can do ``as if'' $\bm Y_i^{\scriptscriptstyle (h)}$, $i \in \{1,\dots,\ip{ns^*}\}$, have copula $C_1^{\scriptscriptstyle (h)}$ and $\bm Y_i^{\scriptscriptstyle (h)}$, $i \in \{\ip{ns^*}+1,\dots,n\}$, have copula $C_2^{\scriptscriptstyle (h)}$. Under this model and additional regularity conditions, we obtain that \[ n^{-1} S_{n,C^{(h)}} \leadsto \kappa_h \equiv \{ s^*(1-s^*) \}^2 \int_{[0,1]^h} \{ C_1^{(h)} (\bm u) - C_2^{(h)}(\bm u) \}^2 \, \mathrm{d} C_{s^*}^{(h)}(\bm u), \] where $C_{\scriptscriptstyle s^*}^{\scriptscriptstyle (h)} = s^* C_1^{\scriptscriptstyle (h)} + (1-s^*) C_2^{\scriptscriptstyle (h)}$. In other words, the dominating term in an asymptotic expansion of $S_{n,C^{(h)}}$ diverges to infinity at rate $n$, with scaling factor $\kappa_h$ depending on $h$. Since we conjecture that the bootstrap replicates of $S_{n,C^{(h)}}$ are of lower order than $O_\mathbb{P}(n)$ for any $h$, we further conjecture that the power curves of the test will be controlled to a large extent by the ``signal of non-stationarity'' $\kappa_h$. The impact of $h$ on this quantity is ambiguous, but, in many sensible models, it is decreasing in $h$ eventually, inducing a sort of ``curse of dimensionality''. This results in a smaller power of the corresponding test for larger $h$ and fixed sample size $n$, as will be empirically confirmed in several scenarios considered in the Monte Carlo experiments of Section~\ref{sec:MC} and in the supplementary material.
Additionally, several arguments lead us to assume that smaller values of~$h$ also yield a better approximation of the nominal level. From an empirical perspective, this will be confirmed for all the scenarios under stationarity in our Monte Carlo experiments. While we are not aware of any theoretical result for our quite general serially dependent setting (that would include the dependent multiplier bootstrap), some results are available for the i.i.d.\ or non-bootstrap case. For instance, \cite{CheCheKat13} provide bounds on the approximation error of i.i.d.\ sum statistics by an i.i.d.\ multiplier bootstrap; the bounds are increasing in the dimension~$h$. Moreover, the asymptotics of our test statistics relying on the asymptotics of empirical processes, we would be interested in a good approximation of empirical processes by their limiting counterparts. As shown in \cite{DedMerRio14} for the case of beta-mixing random variables, the approximation error by strong approximation techniques is again increasing in $h$.
Globally, the above arguments as well as the results of the simulation study in Section~\ref{sec:MC} below and in the supplementary material suggest that a rather small value of $h$, for instance in \{2,3,4\}, should be sufficient to test strong stationarity in many situations. Such a choice would provide relatively powerful tests for many interesting alternatives without strongly suffering from the curse of dimensionality. Depending on the ultimate interest, one might also consider choosing $h$ differently, e.g., as the ``forecast horizon''. Finally, a natural research direction would consist of developing data-driven procedures for choosing $h$, for instance following ideas developed in \cite{EscLob09} for testing serial correlation in a time series. However, such an analysis appears to be a research topic in itself and lies beyond the scope of the present paper.
\section{A combined test of second-order stationarity} \label{sec:sotests}
Starting from the general framework considered in \cite{BucKoj16b} and proceeding as in Section~\ref{sec:rank}, one can derive a combined test of second-order stationarity. Given the embedding dimension $h \geq 2$ and the available univariate observations $X_1,\dots,X_{n+h-1}$, let $\bm Z_i^{\scriptscriptstyle (q)}$, $i \in \{1,\dots,n\}$, be the random variables defined by \begin{equation} \label{eq:Zi} \bm Z_i^{(q)} = \left\{ \begin{array}{ll} X_i, & \mbox{if } q=1, \\ (X_i,X_{i+q-1}), & \mbox{if } q \in \{2,\dots,h\}. \end{array} \right. \end{equation} Let $\phi$ be a symmetric, measurable function on $\mathbb{R} \times \mathbb{R}$ or on $\mathbb{R}^2 \times \mathbb{R}^2$. Then, the \emph{$U$-statistic of order~2 with kernel $\phi$} obtained from the subsample $\bm Z_k^{\scriptscriptstyle (q)},\dots,\bm Z_l^{\scriptscriptstyle (q)}$, $1 \leq k < l \leq n$, is given by \begin{equation} \label{eq:Uklqphi} U_{k:l,q,\phi,} = \frac{1}{\binom{l-k+1}{2}} \sum_{k \leq i < j \leq l} \phi(\bm Z_i^{(q)},\bm Z_j^{(q)}). \end{equation} We focus on CUSUM tests for change-point detection based on the generic statistic \begin{equation} \label{eq:Snqphi}
S_{n,q,\phi} = \max_{2 \leq k \leq n-2} | \mathbb{U}_{n,q,\phi} (k/n) | = \sup_{s \in [0,1]} | \mathbb{U}_{n,q,\phi}(s) |, \end{equation} where \begin{equation*} \mathbb{U}_{n,q,\phi}(s) = \sqrt n \lambda_n(0,s) \lambda_n(s,1) ( U_{1:\ip{ns},q,\phi} - U_{\ip{ns}+1:n,q,\phi} ) \qquad \mbox{if } s \in [2/n,1-2/n], \end{equation*} and $\mathbb{U}_{n,q,\phi}(s)=0$ otherwise.
With the aim of assessing whether second-order stationarity is plausible, the following possibilities for $q \in \{1,\dots,h\}$ and the kernel $\phi$ are of interest: If $q=1$ and $\phi(z,z') = m(z, z') = z$, $z,z' \in \mathbb{R}$, the statistic $S_{n,q,\phi} = S_{n,1,m}$ is (asymptotically equivalent to) the classical CUSUM statistic that is particularly sensitive to changes in the expectation of $X_1,\dots,X_n$. Similarly, setting $q=1$ and $\phi(z,z') = v(z, z') = (z - z')^2 / 2$, $z,z' \in \mathbb{R}$, gives rise to the statistic $S_{n,1,v}$ particularly sensitive to changes in the variance of the observations. For $q \in \{2,\dots,h\}$, setting $\phi(\bm z, \bm z') = a(\bm z, \bm z') = (z_1 - z_1') (z_2 - z_2') / 2$, $\bm z, \bm z' \in \mathbb{R}^2$, results in the CUSUM statistic $S_{n,q,a}$ sensitive to changes in the autocovariance at lag $q-1$.
From \cite{BucKoj16b}, CUSUM tests based on $S_{n,1,m}$, $S_{n,1,v}$ and $S_{n,q,a}$, $q \in \{ 2, \dots, h\}$, sensitive to changes in the expectation, variance and autocovariances, respectively, can all be carried out using a resampling scheme based on dependent multiplier sequences. As a consequence, they can be combined by proceeding as in Sections~\ref{sec:combine:tests} and~\ref{sec:comb:cop:df}. Specifically, for the generic test based on $S_{n,q,\phi}$, let $(\xi_{i,n}^{\scriptscriptstyle [m]})_{i \in \mathbb{Z}}$, $m \in \mathbb{N}$, be independent copies of the same dependent multiplier sequence and, for any $m \in \mathbb{N}$ and $s \in [0,1]$, let \begin{equation*}
\hat \mathbb{U}_{n,q,\phi}^{[m]} (s) = \frac{2}{\sqrt{n}} \sum_{i=1}^{\ip{ns}} \xi_{i,n}^{(m)} \hat \phi_{1,1:n} (\bm Z_i^{(q)}) - \lambda_n(0,s) \times \frac{2}{\sqrt{n}} \sum_{i=1}^n \xi_{i,n}^{(m)} \hat \phi_{1,1:n}(\bm Z_i^{(q)}), \qquad \mbox{if } s \in [2/n,1-2/n], \end{equation*} and $\hat \mathbb{U}_{n,q,\phi}^{[m]} (s) = 0$ otherwise, where \begin{equation*}
\hat \phi_{1,1:n}(\bm Z_i^{(q)}) = \frac{1}{n-1} \sum_{j=1, j \neq i}^n \phi(\bm Z_i^{(q)}, \bm Z_j^{(q)}) - U_{1:n,q,\phi}, \qquad i \in \{1,\dots,n\}, \end{equation*} with $U_{1:n,q,\phi}$ defined by~\eqref{eq:Uklqphi}. Then, multiplier replications of $S_{n,q,\phi}$ are given by \begin{equation*}
\hat S_{n,q,\phi}^{[m]} = \max_{2 \leq k \leq n-2} | \hat \mathbb{U}_{n,q,\phi}^{[m]} (k/n) | = \sup_{s \in [0,1]} | \hat \mathbb{U}_{n,q,\phi}^{[m]}(s) |, \qquad m \in \mathbb{N}, \end{equation*} and an approximate p-value for $S_{n,q,\phi}$ can be computed as $p_{n,M}(S_{n,q,\phi}) = M^{-1} \sum_{m=1}^M \mathbf{1} \big( \hat S_{n,q,\phi}^{\scriptscriptstyle [m]} \geq S_{n,q,\phi} \big)$ for some large integer $M \in \mathbb{N}$.
To obtain a test of second-order stationarity, we use again the combining procedure of Section~\ref{sec:combine:tests}, this time, with $r = h + 1$, $T_{n,1} = S_{n,1,m}$, $T_{n,2} = S_{n,1,v}$ and $T_{n,q+1} = S_{n,q,a}$, $q \in \{2,\dots,h\}$, for some function $\psi:(0,1)^{h+1} \to \mathbb{R}$ decreasing in each of its arguments such as $\psi_S$ in~\eqref{eq:stouffer} or $\psi_F$ in~\eqref{eq:fisher}. As in Section~\ref{sec:comb:cop:df}, to compute bootstrap replicates of the components of $\bm T_n = (T_{n,1},\dots,T_{n,r})$, we use the same $M$ dependent multiplier sequences. Specifically, we first estimate $\ell_n$ from $X_1,\dots,X_{n}$ as explained in \citet[Section 2.4]{BucKoj16b} for $\phi=m$. Then, with the obtained value of $\ell_n$, we generate $M$ independent copies of the same dependent multiplier sequence using~\eqref{eq:movave} and compute the corresponding multiplier replicates $\hat S_{n,q,\phi}^{\scriptscriptstyle [1]},\dots,\hat S_{n,q,\phi}^{\scriptscriptstyle [M]}$ for $q=1$ and $\phi \in \{m,v\}$, and for $q \in \{ 2, \dots, h\}$ and $\phi = a$.
As in Section~\ref{sec:comb:cop:df}, to establish the asymptotic validity of the global test under stationarity using Proposition~\ref{prop:combined:general}, we need to establish conditions under which, using the notation of Section~\ref{sec:combine:tests}, $\bm T_n = (T_{n,1},\dots,T_{n,r})$ and its bootstrap replicates satisfy~\eqref{eq:uncond:Tn} or, equivalently,~\eqref{eq:cond:Tn}. The latter can be proved by starting from Proposition~2.5 in \cite{BucKoj16b} and by proceeding as in the proofs of the results stated in Section~\ref{sec:comb:cop:df}. For the sake of simplicity, the conditions in the following proposition require that $X_1,\dots,X_{n+h-1}$ is a stretch from an absolutely regular sequence. Indeed, assuming that $(X_i)_{i \in \mathbb{Z}}$ is only strongly mixing leads to significantly more complex statements. Recall that the absolute regularity coefficients corresponding to a sequence $(Z_i)_{i \in \mathbb{Z}}$ are defined by $$
\beta_r^Z = \sup_{p \in \mathbb{Z}} \mathrm{E} \sup_{A \in \mathcal{F}_{p+r}^\infty} | \mathbb{P}(A \,|\, \mathcal{F}_{-\infty}^p ) - \mathbb{P}(A) |, \qquad r \in \mathbb{N}, \, r > 0, $$ where $\mathcal{F}_a^b$ is defined above~\eqref{eq:alpha}. The sequence $(Z_i)_{i \in \mathbb{N}}$ is then said to be \emph{absolutely regular} if $\beta_r \to 0$ as $r \to \infty$. As $\alpha_r^Z \leq \beta_r^Z$, absolute regularity implies strong mixing.
\begin{prop}
Let $X_1,\dots,X_{n+h-1}$ be drawn from a strictly stationary sequence $(X_i)_{i \in \mathbb{Z}}$ such that $\mathrm{E}\{|X_1|^{2(4+\delta)}\} < \infty$ for some $\delta > 0$. Also, let $(\xi_{\scriptscriptstyle i,n}^{\scriptscriptstyle [1]})_{i \in \mathbb{Z}}$ and $(\xi_{\scriptscriptstyle i,n}^{\scriptscriptstyle [2]})_{i \in \mathbb{Z}}$ be independent copies of a dependent multiplier sequence satisfying~($\mathcal{M} 1$)--($\mathcal{M} 3$) in Appendix~\ref{app:dep} with $\ell_n = O(n^{1/2 - \gamma})$ for some $1/(6+2\delta) < \gamma < 1/2$. Then, if $\beta_r^X = O(r^{-b})$ for some $b > 2(4 + \delta)/\delta$ as $r \to \infty$,~\eqref{eq:uncond:Tn} or, equivalently,~\eqref{eq:cond:Tn}, hold.
\end{prop}
\section{Monte Carlo experiments} \label{sec:MC}
Extensive simulations were carried out in order to try to answer several fundamental questions (hereafter in bold) regarding the tests proposed in Sections~\ref{sec:rank} and~\ref{sec:sotests}. For the sake of readability, we only present a small subset of the performed Monte Carlo experiments in detail and refer the reader to the supplementary material for more results. Before formulating the questions, we introduce abbreviations for the components tests whose behavior we investigated: \vspace{-
amount} \begin{itemize}\parskip0pt \item d for the d.f.\ test based on $S_{n,G}$ in~\eqref{eq:Sng}, \item c for the empirical autocopula test at lag $h-1$ based on $S_{n,C^{(h)}}$ in~\eqref{eq:SnCh} (the value of $h$ will always be clear from the context), \item m for the sample mean test based on $S_{n,m}^{\scriptscriptstyle (1)}$ defined generically by~\eqref{eq:Snqphi}, \item v for the variance test based on $S_{n,v}^{\scriptscriptstyle (1)}$ defined generically by~\eqref{eq:Snqphi}, and \item a for the autocovariance test at lag $q-1$ based on $S_{n,a}^{\scriptscriptstyle (q)}$, $q \in \{2,\dots,h\}$, defined generically by~\eqref{eq:Snqphi} (the value of $q$ will always be clear from the context). \end{itemize} \vspace{-
amount}
\noindent With these conventions, the following abbreviations are used for the combined tests: \vspace{-
amount} \begin{itemize}\parskip0pt \item dc: equally weighted combination of the tests d and c for embedding dimension $h$ or, equivalently, lag $h-1$, \item va: combination of the test v with weight 1/2 and the autocovariance tests a for lags $q \in \{1,\dots,h-1\}$ with equal weights $1/\{2(h-1)\}$, \item mva: combination of the test m with weight 1/3, of the variance test v with weight 1/3 and the autocovariance tests a for lags $q \in \{1,\dots,h-1\}$ with equal weights $1/\{3(h-1)\}$, \item dcp: combination of the test d with weight 1/2 with pairwise bivariate empirical autocopula tests for lags $1,\dots,h-1$ with equal weights $1/\{2(h-1)\}$; in other words, the d.f.\ test based on $S_{n,G}$ in~\eqref{eq:Sng} is combined with $S_{n,C^{\scriptscriptstyle (2)}}$ in~\eqref{eq:SnCh} and $S_{n,\tilde C^{\scriptscriptstyle (3)}},\dots,S_{n,\tilde C^{(h)}}$, where the latter are the analogues of $S_{n, C^{\scriptscriptstyle (2)}}$ but for lags $2,\dots,h-1$ (that is, they are computed from~\eqref{eq:Zi} for $q \in \{3,\dots,h\}$). \end{itemize} \vspace{-
amount} The above choices for the weights are arbitrary and thus clearly debatable. An ``optimal'' strategy for the choice of the weights is beyond the scope of this work. For the function $\psi$ in Sections~\ref{sec:rank} and~\ref{sec:sotests}, we only consider $\psi_F$ in~\eqref{eq:fisher} as the use of $\psi_S$ in~\eqref{eq:stouffer} sometimes gave inflated levels.
Let us now state the fundamental questions concerning the studied tests that we attempted to answer empirically by means of a large number of Monte Carlo experiments.
\paragraph{Do the studied component and combined tests maintain their level?} As is explained in detail in the supplementary material, ten strictly stationarity models, including ARMA, GARCH and nonlinear autoregressive models with either normal or Student $t$ with 4 degrees of freedom innovations, were used to generate observations under the null hypothesis of stationarity. The rank-based tests of Section~\ref{sec:rank}, that is, d, c, dc and dcp, were never found to be too liberal, while some of the second-order tests of Section~\ref{sec:sotests}, namely, v, va and mva, were found to reject stationarity too often for a particular GARCH model mimicking S\&P500 daily log-returns.
\paragraph{How do the rank-based tests of Section~\ref{sec:rank} compare to the second-order tests of Section~\ref{sec:sotests} in terms of power?} As presented in detail in the supplementary material, to investigate the power of the tests, eight models connected to the literature on locally stationary processes were considered alongside with four models more in line with the change-point detection literature. All tests were found to have reasonable power for at least one (and, usually, several) of the alternatives under consideration. The combined rank-based tests proposed in Section~\ref{sec:rank}, that is, dc or dcp, were found, overall, to be more powerful than the combined second-order tests, namely, va or mva, even in situations involving changes in the second-order characteristics of the underlying time series.
\paragraph{How are the powers of the proposed component and combined tests related?} For the sake of illustration, we only focus on the component tests d and c, and the combined test dc, and consider three simple data generating models: \vspace{-
amount} \begin{itemize} \parskip0pt \item[D($\sigma$) -] ``Change in the contemporary distribution only'': The $n/2$ first observations are i.i.d.\ from the $N(0,\sigma^2)$ distribution and the $n/2$ last observations are i.i.d.\ from the $N(0,1)$ distribution.
\item[S($\beta$) -] ``Change in the serial dependence only'': The $n/2$ first observations are i.i.d.\ standard normal and the $n/2$ last observations are drawn from an AR(1) model with parameter $\beta$ and centered normal innovations with variance $(1-\beta^2)$. The contemporary distribution is thus constant and equal to the standard normal.
\item[DS($\sigma$, $\beta$) -] ``Change in the contemporary distribution and the serial dependence'': The $n/2$ first observations are i.i.d.\ from the $N(0,\sigma^2)$ distribution and the $n/2$ last observations are drawn from an AR(1) model with parameter $\beta$ and $N(0,1)$ innovations.
\end{itemize} At the 5\% significance level, the rejection percentages of the null hypothesis of stationarity computed from 1000 samples of size $n=128$ from model D($\sigma$), S($\beta$) or DS($\sigma$, $\beta$) for various values of $\sigma$ and $\beta$ are given in Table~\ref{relationship} for the tests d, c and dc for $h=2$. As one can see from the first four rows of the table, when one of the component tests has hardly any power, a ``dampening effect'' occurs for the combined test. However, when the two components tests tend to detect changes, most of the time, simultaneously, a ``reinforcement effect'' seems to occur for the combined test as can be seen from the last two rows of the table.
\begin{table}[t!] \centering \caption{Percentages of rejection of the null hypothesis of stationarity computed from 1000 samples of size $n=128$ from model D($\sigma$), S($\beta$) or DS($\sigma$, $\beta$) for various values of $\sigma$ and $\beta$. The meaning of the abbreviations d, c, dc is given in Section~\ref{sec:MC}.} \label{relationship} \begin{tabular}{lrrr}
\hline
\multicolumn{1}{c}{} & \multicolumn{3}{c}{$h=2$ or lag 1} \\ \cmidrule(lr){2-4} Model & d & c & dc \\ \hline D(2): `Small change in contemporary dist.\ only' & 33.6 & 2.2 & 16.4 \\
D(3): `Large change in contemporary dist.\ only' & 81.6 & 1.6 & 59.2 \\
S(0.3): `Small change in serial dep.\ only' & 6.4 & 19.6 & 16.6 \\
S(0.9): `Large change in serial dep.\ only' & 13.8 & 64.2 & 62.8 \\
DS(2, 0.4): `Small change in both' & 17.2 & 28.8 & 35.4 \\
DS(4, 0.7): `Large change in both' & 75.6 & 70.0 & 92.6 \\
\hline \end{tabular} \end{table}
\paragraph{Is the combined test dc truly more powerful than a simple multivariate extension of the test d designed to be directly sensitive to departures from $H_0^{(h)}$ in~\eqref{eq:H0:Fh}?} Note that to implement the latter test for a given embedding dimension $h$, it suffices to proceed as in Section~\ref{sec:dftest} but by using the $h$-dimensional empirical d.f.s of the $h$-dimensional random vectors $\bm Y_i^{\scriptscriptstyle (h)}$ in~\eqref{eq:Yi} instead of the one-dimensional empirical d.f.s generically given by~\eqref{eq:Gkl}. Let dh be the abbreviation of this test. To provide an empirical answer to the above question, we consider a similar setup as previously. The rejection percentages of the null hypothesis of stationarity computed from 1000 samples of size $n=128$ from model D($\sigma$), S($\beta$) or DS($\sigma$, $\beta$) for various values of $\sigma$ and $\beta$ are given in Table~\ref{joint} for the tests dc, dcp and dh for $h \in \{2,3\}$. As one can see, the test dh seems to have hardly any power when the non-stationarity is only due to a change in the serial dependence. Furthermore, even when the non-stationarity results from a change in the contemporary distribution, the test dh appears to be less powerful, overall, than the combined tests dc and dcp.
\begin{table}[t!] \centering \caption{Percentages of rejection of the null hypothesis of stationarity computed from 1000 samples of size $n=128$ from model D($\sigma$), S($\beta$) or DS($\sigma$, $\beta$) for various values of $\sigma$ and $\beta$. The meaning of the abbreviations dc, dcp and dh is given in Section~\ref{sec:MC}.} \label{joint} \begin{tabular}{lrrrrr}
\hline
\multicolumn{1}{c}{} & \multicolumn{2}{c}{$h=2$} & \multicolumn{3}{c}{$h=3$ or lag 2} \\ \cmidrule(lr){2-3} \cmidrule(lr){4-6} Model & dc & dh & dc & dcp & dh \\ \hline D(2): `Small change in contemporary dist.\ only' & 16.4 & 21.8 & 17.8 & 26.6 & 24.8 \\
D(3): `Large change in contemporary dist.\ only' & 59.2 & 52.4 & 58.8 & 73.0 & 44.0 \\
S(0.3): `Small change in serial dep.\ only' & 16.6 & 7.2 & 18.2 & 13.0 & 9.0 \\
S(0.9): `Large change in serial dep.\ only' & 62.8 & 15.6 & 63.0 & 65.0 & 16.0 \\
DS(2, 0.4): `Small change in both' & 35.4 & 20.6 & 42.2 & 34.8 & 30.0 \\
DS(4, 0.7): `Large change in both' & 92.6 & 67.6 & 92.4 & 91.6 & 71.6 \\
\hline \end{tabular} \end{table}
\paragraph{What is the influence of the choice of the embedding dimension $h$ on the empirical levels and the powers of the proposed tests?} The extensive simulations results available in the supplementary material indicate that, under the null hypothesis of stationarity, the tests c and dc tend, overall, to become more and more conservative as $h$ increases for fixed sample size~$n$. For fixed $h$, the empirical levels seem to get closer to the 5\% nominal level as $n$ increases, as expected theoretically. To convey some intuitions on the influence on $h$ on the empirical power under non-stationarity, we consider again the same setup as before and plot the rejection percentages of c and dc computed from 1000 samples of size $n=128$ from models D(2), D(3), S(0.3), S(0.9), DS(2,0.4) and DS(4,0.7) against the embedding dimension~$h$. As one can see from Figure~\ref{fig:h}, for the models under consideration, the empirical powers of the tests c and dc essentially decrease as $h$ increases. Additional simulations presented in the supplementary material and involving an AR(2) model instead of an AR(1) model for the serial dependence show that a similar pattern occurs from $h=3$ onwards in that case. Indeed, as discussed in Section~\ref{sec:hchoice}, for many models including those that were just mentioned, the power of the tests appears to be a decreasing function of $h$, at least from some small value of $h$ onwards.
\begin{figure}
\caption{ Rejection percentages of c and dc against the embedding dimension $h \in \{2,3,4,8\}$ computed from 1000 samples of size $n=128$ from models D(2), D(3), S(0.3), S(0.9), DS(2,0.4) and DS(4,0.7).}
\label{fig:h}
\end{figure}
\paragraph{How do the studied tests compare to existing competitors?}
As mentioned in the introduction, many tests of stationarity were proposed in the literature. Unfortunately, only a few of them seem to be implemented in statistical software. In the supplementary material, we report the results of Monte Carlo experiments investigating the finite-sample behavior of the tests of \cite{PriSub69}, \cite{Nas13} and \cite{CarNas13} that are implemented in the \textsf{R} packages \texttt{fractal} \citep{fractal}, \texttt{locits} \citep{locits} and \texttt{costat} \citep{costat}, respectively. Note that we did not consider the test of \cite{CarNas16} (implemented in the \textsf{R} package \texttt{BootWPTOS}) because we were not able to understand how to initialize the arguments of the corresponding \textsf{R} function. Under stationarity, unlike the rank-based tests d, c, dc and dcp, the three aforementioned tests were found to be too liberal for at least one of the considered models. Their behavior under the null turned out to be even more disappointing when heavy tailed innovations were used. In terms of empirical power, the results presented in the supplementary material allow in principle for a direct comparison with the results reported in \cite{CarNas13} and \cite{DetPreVet11}. Since the tests available in \textsf{R} considered in \cite{CarNas13} are far from maintaining their levels, a comparison in terms of power with these tests is clearly not meaningful. As far as the tests of \cite{DetPreVet11} are concerned, they appear, overall, to be more powerful for some of the considered models. It is however unknown whether they hold their levels when applied to stationary heavy-tailed observations as only Gaussian time series were considered in the simulations of \cite{DetPreVet11}.
\section{Illustrations} \label{sec:illus}
By construction, the tests based on the sample mean, variance and autocovariance proposed in Section~\ref{sec:sotests} are only sensitive to changes in the second-order characteristics of a time series. The results of the simulations reported in the previous section and in the supplementary material seem to indicate that the latter tests do not always maintain their level (for instance, in the presence of conditional heteroskedasticity) and that the rank-based tests proposed in Section~\ref{sec:rank} are more powerful, even in situations only involving changes in the second-order characteristics. Therefore, we recommend the use of the rank-based tests in general.
To illustrate their application, we consider two real datasets, both available in the \textsf{R} package \texttt{copula} \citep{copula}. The first one consists of daily log-returns of Intel, Microsoft and General Electric stocks for the period from 1996 to 2000. It was used in \citet[Chapter~5]{McNFreEmb05} to illustrate the fitting of elliptical copulas. The second dataset was initially considered in \cite{GreGenGen08} to illustrate the so-called \emph{copula--GARCH} approach \citep[see, e.g.,][]{CheFan06,Pat06}. It consists of bivariate daily log-returns computed from three years of daily prices of crude oil and natural gas for the period from July 2003 to July 2006.
Prior to applying the methodologies described in the aforementioned references, it is crucial to assess whether the available data can be regarded as stretches from stationary multivariate time series. As multivariate versions of the proposed tests would need to be thoroughly investigated first (see the discussion in the next section), as an imperfect alternative, we applied the studied univariate versions to each component time series. The results are reported in Table~\ref{illus}. For the sake of simplicity, we shall ignore the necessary adjustment of p-values or global significance level due to multiple testing.
\begin{table}[t!] \centering \caption{Approximate p-values (multiplied by 100) of the rank-based tests of stationarity proposed in Section~\ref{sec:rank} for embedding dimension $h \in \{2,3,4\}$ applied to the component times series of the trivariate log-return data considered in \citet[Chapter~5]{McNFreEmb05} and the bivariate log-return data considered in \cite{GreGenGen08}. The daily log-returns of the Intel, Microsoft and General Electric stocks are abbreviated by INTC, MSFT and GE, respectively. The meaning of the abbreviations d, c, dc and dcp is given in Section~\ref{sec:MC}. The columns c2 and c3 report the results for the bivariate analogues of the test based on $S_{n,C^{\scriptscriptstyle (2)}}$ defined by~\eqref{eq:SnCh} (which arise in the combined test dcp) for lags 2 and 3.} \label{illus} \begin{tabular}{lrrrrrrrrrrr}
\hline
\multicolumn{2}{c}{} & \multicolumn{2}{c}{$h=2$ or lag 1} & \multicolumn{4}{c}{$h=3$ or lag 2} & \multicolumn{4}{c}{$h=4$ or lag 3} \\ \cmidrule(lr){3-4} \cmidrule(lr){5-8} \cmidrule(lr){9-12} Variable & d & c & dc & c & dc & c2 & dcp & c & dc & c3 & dcp \\ \hline INTC & 0.0 & 2.0 & 0.0 & 4.8 & 0.0 & 32.5 & 0.0 & 7.9 & 0.0 & 30.2 & 0.0 \\
MSFT & 0.2 & 92.3 & 2.2 & 80.7 & 0.8 & 47.3 & 0.0 & 86.4 & 0.1 & 37.2 & 0.0 \\
GE & 0.1 & 62.1 & 0.7 & 15.9 & 0.1 & 67.2 & 0.0 & 22.4 & 0.6 & 16.7 & 0.1 \\
\hline \ oil & 89.6 & 22.1 & 52.5 & 55.3 & 84.0 & 46.5 & 67.8 & 89.0 & 97.2 & 5.6 & 49.0 \\
gas & 5.0 & 16.5 & 3.9 & 17.4 & 5.4 & 90.5 & 7.4 & 43.9 & 8.8 & 85.2 & 6.2 \\
\hline \end{tabular} \end{table}
As one can see from the results of the combined tests dc and dcp for embedding dimension $h \in \{2,3,4\}$, there is strong evidence against stationarity in the component series of the trivariate log-return data considered in \citet[Chapter~5]{McNFreEmb05}. For all three series, the very small p-values of the combined tests are a consequence of the very small p-value of the test d focusing on the contemporary distribution. For the Intel stock (line INTC), it is also a consequence of the small p-value of the test c for $h=2$. Although it is for instance very tempting to conclude that the non-stationarity in the log-returns of the Intel stock is due to $H_0^{\scriptscriptstyle (1)}$ in~\eqref{eq:H0:1} and $H_{0,c}^{\scriptscriptstyle (2)}$ in~\eqref{eq:H0:Ch} not being satisfied, such a reasoning is not formally valid without additional assumptions, as explained in the introduction. From the second horizontal block of Table~\ref{illus}, one can also conclude that there is no evidence against stationarity in the log-returns of the oil prices and only weak evidence against stationarity in the log-returns of the gas prices.
\section{Concluding remarks} \label{sec:conc}
Unlike some of their competitors that are implemented in various \textsf{R} packages, the rank-based tests of stationarity proposed in Section~\ref{sec:rank} were never observed to be too liberal for the rather typical sample sizes considered in this work. As discussed in Section~\ref{sec:hchoice}, and as empirically confirmed by the experiments of Section~\ref{sec:MC} and the supplementary material, the tests are nevertheless likely to become more conservative and less powerful as the embedding dimension $h$ is increased. The latter led us to make the rather general recommendation that they should be typically used with a small value of the embedding dimension $h$ such as 2, 3 or 4. It is however difficult to assess the breadth of that recommendation and it might be meaningful for the practitioner to consider the issue of the choice of $h$ in all its subtlety as attempted in the discussion of Section~\ref{sec:hchoice}.
While, unsurprisingly, the recommended tests seem to display good power for alternatives connected to the change-point detection literature, their power was not observed to be very high, overall, for the locally stationary alternatives considered in our Monte Carlo experiments. A promising approach to improve on the latter aspect would be to derive extensions of the tests allowing the comparison of blocks of observations in the spirit of \cite{HusSla01} and of \cite{KirMuh16}: once the time series is divided into moving blocks of equal length, the main idea is to compare successive pairs of blocks by means of a statistic based on a suitable extension of the process in~\eqref{eq:Dnh} (if the focus is on serial dependence) or in~\eqref{eq:En} (if the focus is on the contemporary distribution), and to finally aggregate the statistics for each pair of blocks.
Additional future research may consist of extending the proposed tests to multivariate time series. To fix ideas, let us focus on lag $h-1$ and consider a stretch $\bm X_i=(X_{i,1}, \dots, X_{i,d})$, $i \in \{1,\dots,n+h-1\}$ from a continuous $d$-dimensional time series. A straightforward extension of the approach considered in this work is first to define the $d\times h$-dimensional random vectors $\bm Y_i^{\scriptscriptstyle (h)} = (\bm X_i, \dots, \bm X_{i+h-1})$, $i \in \{1,\dots,n\}$. As argued in the introduction and in Section~\ref{sec:MC}, it will then be helpful in terms of finite sample power properties to split the hypothesis $H_0^{\scriptscriptstyle (h)}$ in~\eqref{eq:H0:Fh} into suitable sub-hypotheses. For $A\subset \{1, \dots, d\}$ and $B\subset \{0,\dots, h-1\}$, let \begin{align*} H_0^{(1)}(A): &\,\exists \, G^{A} \text{ such that } (X_{1,j})_{j\in A},\dots, (X_{n-h+1,j})_{j\in A} \text{ have d.f.\ } G^{A}, \\ H_{0,c}^{(h)}(A,B): &\,\exists \, C^{(h), A,B} \text{ such that } (X_{1+s,j})_{s\in B, j \in A},\dots,(X_{n+s,j})_{s\in B, j \in A} \text{ have copula } C^{(h), A,B}. \end{align*} Letting $\bar d = \{1, \dots, d\}$ and $\bar h=\{0, \dots, h-1\}$, Sklar's theorem suggests the decomposition $H_0^{\scriptscriptstyle (h)} = H_0^{\scriptscriptstyle (1)}(\{1\}) \cap \dots \cap H_0^{\scriptscriptstyle (1)}(\{d\}) \cap H_{0,c}^{\scriptscriptstyle (h)}(\bar d, \bar h)$. However, preliminary numerical experiments indicate that a straightforward extension of the approach proposed in Section~\ref{sec:comb:cop:df} to this combined hypothesis does not seem to be very powerful. The latter might be due to the curse of dimensionality identified in Section~\ref{sec:hchoice} and the fact that, under stationarity, the $d \times h$-dimensional copula $C^{\scriptscriptstyle (h), \bar d, \bar h}$ of the $\bm Y_i^{\scriptscriptstyle (h)}$ arising in the aforementioned decomposition does not solely control the serial dependence in the time series but also the cross-sectional dependence. As a consequence, alternative combination strategies would need to be investigated in the multivariate case. As an imperfect alternative, one might for instance consider the following hypothesis \begin{align*} \Big( \textstyle \bigcap_{j =1}^d H_0^{(1)}(\{j\}) \Big) \cap \Big( H_{0,c}^{(h)}(\bar d, \{0\})\Big) \cap \Big( \bigcap_{j =1}^d H_{0,c}^{(h)}(\{j\}, \bar h) \Big), \end{align*} a combined test of which would be sensible to any changes in the marginals, the contemporary dependence or the marginal serial dependence. One may easily include further hypotheses related to cross-sectional and cross-serial dependencies, like for instance $\bigcap_{i\ne j\in \bar d}H_{0,c}^{\scriptscriptstyle (h)}(\{i,j\}, \{0,1\})$. The amount of potential adaptations appears to be very large, whence a further investigation, in particular from a finite-sample point-of-view, is beyond the scope of this paper.
\appendix
\section{Dependent multiplier sequences} \label{app:dep}
A sequence of random variables $(\xi_{i,n})_{i \in \mathbb{Z}}$ is a \emph{dependent multiplier sequence} if the three following conditions are fulfilled: \begin{enumerate}[({$\mathcal{M}$}1)]
\item The sequence $(\xi_{i,n})_{i \in \mathbb{Z}}$ is independent of the available sample $X_1,\dots,X_{n+h-1}$ and strictly stationary with $\mathrm{E}(\xi_{0,n}) = 0$, $\mathrm{E}(\xi_{0,n}^2) = 1$ and $\sup_{n \geq 1} \mathrm{E}(|\xi_{0,n}|^\nu) < \infty$ for all $\nu \geq 1$. \item There exists a sequence $\ell_n \to \infty$ of strictly positive constants such that $\ell_n = o(n)$ and the sequence $(\xi_{i,n})_{i \in \mathbb{Z}}$ is $\ell_n$-dependent, i.e., $\xi_{i,n}$ is independent of $\xi_{i+p,n}$ for all $p > \ell_n$ and $i \in \mathbb{N}$.
\item There exists a function $\varphi:\mathbb{R} \to [0,1]$, symmetric around 0, continuous at $0$, satisfying $\varphi(0)=1$ and $\varphi(x)=0$ for all $|x| > 1$ such that $\mathrm{E}(\xi_{0,n} \xi_{p,n}) = \varphi(p/\ell_n)$ for all $p \in \mathbb{Z}$. \end{enumerate} Roughly speaking, such sequences extend to the serially dependent setting the multiplier sequences that appear in the \emph{multiplier central limit theorem} \citep[see, e.g.,][Theorem 10.1 and Corollary 10.3]{Kos08}. The latter result lies at the heart of the proof of the asymptotic validity of many types of bootstrap schemes for independent observations. In particular and as it shall become clearer below, the bandwidth parameter $\ell_n$ plays a role somehow similar to the block length in the block bootstrap of \cite{Kun89}.
Two ways of generating dependent multiplier sequences are discussed in \citet[Section 5.2]{BucKoj16}. Throughout this work, we use the so-called \emph{moving average approach} based on an initial independent and identically distributed (i.i.d.) standard normal sequence and Parzen's kernel $$
\kappa(x) = (1 - 6x^2 + 6|x|^3) \mathbf{1}(|x| \leq 1/2) + 2(1-|x|)^3 \mathbf{1}(1/2 < |x| \leq 1), \quad x \in \mathbb{R}. $$ Specifically, let $(b_n)$ be a sequence of integers such that $b_n \to \infty$, $b_n = o(n)$ and $b_n \geq 1$ for all $n \in \mathbb{N}$. Let $Z_1,\dots,Z_{n+2b_n-2}$ be i.i.d. $\mathcal{N}(0,1)$. Then, let $\ell_n=2b_n-1$ and, for any $j \in \{1,\dots,\ell_n\}$, let $w_{j,n} = \kappa\{(j-b_n)/b_n\}$ and $\tilde w_{j,n} = w_{j,n} ( \sum_{j'=1}^{\ell_n} w_{j',n}^2 )^{-1/2}$. Finally, for all $i \in \{1,\dots,n\}$, let \begin{equation} \label{eq:movave} \xi_{i,n} = \sum_{j=1}^{\ell_n} \tilde w_{j,n} Z_{j+i-1}. \end{equation} Then, as verified in \citet[Section 5.2]{BucKoj16}, the infinite size version of $\xi_{1,n},\dots,\xi_{n,n}$ satisfies Assumptions~($\mathcal{M} 1$)-($\mathcal{M} 3$), when $n$ is sufficiently large.
As can be expected, the bandwidth parameter $\ell_n$ (or, equivalently, $b_n$) will have a crucial influence on the finite-sample performance of the tests studied in this work. In practice, for the rank-based (resp.\ second-order) tests of Section~\ref{sec:rank} (resp.\ Section~\ref{sec:sotests}), we apply to the available univariate sequence $X_1,\dots,X_{n+h-1}$ the data-adaptive procedure proposed in \citet[Section 5.1]{BucKoj16} \citep[resp.][Section 2.4]{BucKoj16b}, which is based on the seminal work of \cite{PapPol01}, \cite{PolWhi04} and \cite{PatPolWhi09}, among others. Roughly speaking, the latter amounts to choosing $\ell_n$ as $K_n n^{1/5}$, which asymptotically minimizes a certain integrated mean squared error, for a constant $K_n$ that can be estimated from $X_1,\dots,X_{n+h-1}$.
Monte Carlo experiments studying the finite-sample behavior of the data-adaptive procedure of \citet[Section 5.1]{BucKoj16} for estimating the bandwidth parameter $b_n$ can be found in \citet[Section 6]{BucKoj16}. A small simulation showing how the automatically-chosen bandwidth parameter $b_n$ is affected by the strength of the serial dependence in an AR(1) model is presented in the supplementary material.
\section{Proofs} \label{app:proofs}
\begin{proof}[Proof of Proposition~\ref{prop:combined:general}] As we continue, we adopt the notation $\bar F^*_{T_j}(x) = \mathbb{P}(T_{n,j}^{\scriptscriptstyle [1]} \geq x \,|\, \bm X_n)$, $x \in \mathbb{R}$, $j \in \{1,\dots,r\}$. Note in passing that the functions $\bar F^*_{T_j}$ are random and that we can rewrite $W_n$ in~\eqref{eq:Wn} as $W_n = \psi\{ \bar F_{T_1}^*(T_{n,1}), \dots, \bar F_{T_r}^*(T_{n,r}) \}$. In addition, recall that $\bar F_{T_j}(x) = \mathbb{P}(T_j \geq x)$, $x \in \mathbb{R}$, $j \in \{1,\dots,r\}$. Combining either~\eqref{eq:uncond:Tn} or~\eqref{eq:cond:Tn} with Lemma~2.2 in \cite{BucKoj18} and Problem 23.1 in \cite{Van98}, we obtain that \begin{equation} \label{eq:aeSj}
\sup_{x \in \mathbb{R}} | \bar F_{T_j}^*(x) - \bar F_{T_j}(x)| \p 0, \quad j \in \{1,\dots,r\}. \end{equation} Furthermore, Lemma~2.2 in \cite{BucKoj18} implies that~\eqref{eq:aeSj} is equivalent to \begin{equation} \label{eq:aeSj2}
\sup_{x \in \mathbb{R}} \Big|\textstyle \frac{1}{M_n} \sum_{i=1}^{M_n} \mathbf{1}(T_{n,j}^{[i]} \geq x) - \bar F_{T_j}(x) \Big| \p 0, \quad j \in \{1,\dots,r\}. \end{equation} Again, from Lemma~2.2 in \cite{BucKoj18}, we also have that~\eqref{eq:uncond:Tn} or~\eqref{eq:cond:Tn} imply that $$ (\bm T_n, \bm T_n^{[1]}, \dots, \bm T_n^{[N]}) \leadsto (\bm T, \bm T^{[1]}, \dots, \bm T^{[N]}), $$ for all $N \in \mathbb{N}$, where $\bm T^{[1]},\dots,\bm T^{[N]}$ are independent copies of $\bm T$. Combining this last result with the continuous mapping theorem, we immediately obtain that, for any $N\in\mathbb{N}$, \begin{equation} \label{eq:jointFS} (\bar{\bm F}_T(\bm T_n), \bar{\bm F}_T(\bm T_n^{[1]}), \dots, \bar{\bm F}_T(\bm T_n^{[N]})) \leadsto (\bar{\bm F}_T(\bm T), \bar{\bm F}_T(\bm T^{[1]}), \dots, \bar{\bm F}_T(\bm T^{[N]})), \end{equation} where $\bar{\bm F}_T(\bm x) = (\bar F_{T_1}(x_1),\dots,\bar F_{T_r}(x_r))$, $\bm x \in \mathbb{R}^r$. Combining~\eqref{eq:jointFS} with~\eqref{eq:aeSj2}, the continuity of~$\psi$ and the continuous mapping theorem, we obtain that~\eqref{eq:uncondM} holds for all $N \in \mathbb{N}$.
From now on, assume that $W$ has a continuous d.f. As a straightforward consequence of~\eqref{eq:aeSj} and the continuous mapping theorem, the weak convergence in \eqref{eq:jointFS} implies that, for any $N\in\mathbb{N}$, \begin{equation*} (W_n,W_n^{[1]},\dots,W_n^{[N]}) \leadsto (W,W^{[1]},\dots,W^{[N]}), \end{equation*} where $W_n$ is defined by~\eqref{eq:Wn} and $W_n^{\scriptscriptstyle [i]} = \psi\{ \bar F_{T_1}^*(T_{n,1}^{\scriptscriptstyle [i]}), \dots, \bar F_{T_r}^*(T_{n,r}^{\scriptscriptstyle [i]}) \}, i \in \{1,\dots,N\}.$ The previous display has the following two consequences: first, by Problem 23.1 in \cite{Van98},\begin{equation} \label{eq:unif}
\sup_{x \in \mathbb{R}} | \mathbb{P}(W_n \leq x) - \mathbb{P}(W \leq x)| \p 0. \end{equation} Second, since $W_n^{\scriptscriptstyle [1]},\dots,W_n^{\scriptscriptstyle [N]}$ are identically distributed and independent conditionally on the data, by Lemma~2.2 in \cite{BucKoj18}, we have that \begin{equation} \label{eq:cond}
\sup_{x \in \mathbb{R}} | \mathbb{P}(W_n^{[1]} \leq x\,|\, \bm X_n) - \mathbb{P}(W_n \leq x)| \p 0. \end{equation}
Let us next prove~\eqref{eq:condM}. In view of~\eqref{eq:cond},
it suffices to show that \begin{equation} \label{eq:goal}
\sup_{x \in\mathbb{R}} | \mathbb{P}(W_{n,M_n}^{[1]} \le x \,|\, \bm X_n) - \mathbb{P}(W_n^{[1]} \leq x \,|\, \bm X_n) | \p 0. \end{equation} Using the fact that, for any $a,b,x\in\mathbb{R}$ and $\varepsilon>0$, \begin{equation} \label{eq:tool}
| \mathbf{1}(a \le x) - \mathbf{1}(b \le x) | \le \mathbf{1}(|x-a| \le \varepsilon) +\mathbf{1}(|a-b|> \varepsilon), \end{equation} we have that \begin{multline*}
\sup_{x \in\mathbb{R}} \big | \mathbb{P} (W_n^{[1]} \le x \,|\, \bm X_n) - \mathbb{P}(W_{n,M_n}^{[1]} \leq x \,|\, \bm X_n) \big | \le
\sup_{x \in\mathbb{R}} \mathbb{P}( |W_n^{[1]} - x | \le \varepsilon \,|\, \bm X_n ) \\
+ \mathbb{P}( | W_n^{[1]} - W_{n,M_n}^{[1]} | > \varepsilon \,|\, \bm X_n ). \end{multline*}
From~\eqref{eq:unif} and~\eqref{eq:cond}, $\sup_{x \in \mathbb{R}} \mathbb{P}( |W_n^{\scriptscriptstyle [1]} - x | \le \varepsilon\,|\, \bm X_n)$ converges in probability to $\sup_{x \in \mathbb{R}} \mathbb{P} ( |W - x | \le \varepsilon )$ which can be made arbitrary small by decreasing $\varepsilon$. From~\eqref{eq:aeSj},~\eqref{eq:aeSj2},~\eqref{eq:jointFS} and the continuous mapping theorem, we obtain that $W_n^{\scriptscriptstyle [1]} - W_{\scriptscriptstyle n,M_n}^{\scriptscriptstyle [1]} =o_\mathbb{P}(1)$, which implies that \begin{equation} \label{eq:tool2}
\mathbb{P}( | W_n^{[1]} - W_{n,M_n}^{[1]} | > \varepsilon ) = \mathrm{E} \{ \mathbb{P}( | W_n^{[1]} - W_{n,M_n}^{[1]} | > \varepsilon \,|\, \bm X_n ) \} \to 0, \end{equation}
and thus that $\mathbb{P}( | W_n^{\scriptscriptstyle [1]} - W_{\scriptscriptstyle n,M_n}^{\scriptscriptstyle [1]} | > \varepsilon \,|\, \bm X_n ) =o_\mathbb{P}(1)$. Hence,~\eqref{eq:goal} holds and thus so does~\eqref{eq:condM}.
Finally, let show that~\eqref{eq:MC} holds. Since $W_n^{\scriptscriptstyle [1]},\dots,W_n^{\scriptscriptstyle [N]}$ are identically distributed and independent conditionally on the data, by Lemma~2.2 in \cite{BucKoj18}, we have that~\eqref{eq:cond} implies \begin{equation} \label{eq:MCunobs}
\sup_{x \in \mathbb{R}} \bigg| \frac{1}{M_n} \sum_{i=1}^{M_n} \mathbf{1}(W_n^{[i]} \le x) - \mathbb{P}(W_n \leq x)\bigg| \p 0. \end{equation} Whence~\eqref{eq:MC} is proved if we show that \begin{equation} \label{eq:goal2}
\sup_{x \in\mathbb{R}} \bigg| \frac{1}{M_n} \sum_{i=1}^{M_n} \mathbf{1}(W_{n,M_n}^{[i]} \le x) - \frac{1}{M_n} \sum_{i=1}^{M_n} \mathbf{1}(W_n^{[i]} \le x) \bigg| \p 0. \end{equation} Using again~\eqref{eq:tool}, the term on the left of the previous display is smaller than $$
\sup_{x \in\mathbb{R}} \frac{1}{M_n} \sum_{i=1}^{M_n} \mathbf{1}( | W_n^{[i]} - x| \le \varepsilon) + \frac{1}{M_n} \sum_{i=1}^{M_n} \mathbf{1}( | W_n^{[i]} - W_{n,M_n}^{[i]} | \ge \varepsilon). $$
From~\eqref{eq:MCunobs} and~\eqref{eq:unif}, the first term converges in probability to $\sup_{x \in \mathbb{R}} \mathbb{P} ( |W - x | \le \varepsilon )$ which can be made arbitrary small by decreasing $\varepsilon$. The second term converges in probability to zero by Markov's inequality: for any $\lambda > 0$, \begin{align*}
\mathbb{P}\left\{ \frac{1}{M_n} \sum_{i=1}^{M_n} \mathbf{1}( | W_n^{[i]} - W_{n,M_n}^{[i]} | \ge \varepsilon) > \lambda \right\} &\leq
\lambda^{-1} \mathrm{E} \left\{ \frac{1}{M_n} \sum_{i=1}^{M_n} \mathbf{1}( | W_n^{[i]} - W_{n,M_n}^{[i]} | \ge \varepsilon) \right\} \\ &\leq
\lambda^{-1} \mathbb{P}( | W_n^{[1]} - W_{n,M_n}^{[1]} | \ge \varepsilon) \to 0 \end{align*} since the $W_n^{\scriptscriptstyle [i]} - W_{\scriptscriptstyle n,M_n}^{\scriptscriptstyle [i]}$ are identically distributed and by~\eqref{eq:tool2}. Therefore,~\eqref{eq:goal2} holds and, hence, so does~\eqref{eq:MC}. Note that, from the fact that $\bm T$ and $W$ have continuous d.f.s, we could have alternatively proved the analogue statement with `$\le$' replaced by `$<$'. As a consequence, we immediately obtain that $p_{n,M_n}(W_{\scriptscriptstyle n,M_n}^{\scriptscriptstyle [0]})$ has the same weak limit as $\bar F_{W_n}(W_{\scriptscriptstyle n,M_n}^{\scriptscriptstyle [0]})$, where $\bar F_{W_n}(w) = \mathbb{P}(W_n \ge w)$, $w \in \mathbb{R}$. By the analogue to~\eqref{eq:unif} with `$\le$' replaced by `$<$', the latter has the same asymptotic distribution as $\bar F_W(W_{\scriptscriptstyle n,M_n}^{\scriptscriptstyle [0]})$, where $\bar F_W(w) = \mathbb{P}(W \ge w)$, $w \in \mathbb{R}$. By the weak convergence $W_{\scriptscriptstyle n,M_n}^{\scriptscriptstyle [0]} \leadsto W$ following from~\eqref{eq:uncondM} and the continuous mapping theorem, $\bar F_W(W_{\scriptscriptstyle n,M_n}^{\scriptscriptstyle [0]})$ is asymptotically standard uniform. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:combined:alternative}] Notice first that assumption~$(ii)$ implies that the corresponding approximate p-value $p_{n,M_n}(T_{n,j_0}^{\scriptscriptstyle [0]})$ given by~\eqref{eq:pval_T} converges to zero in probability. Indeed, $$ \mathbb{E}\{ p_{n,M_n}(T_{n,j_0}^{[0]}) \} = \frac1{M_n+1} \left\{ \frac12 + \sum_{k=1}^{M_n} \mathbb{P}(T_{n,j_0}^{[k]} \ge T_{n,j_0}^{[0]}) \right\} = \mathbb{P}(T_{n,j_0}^{[1]} \ge T_{n,j_0}^{[0]}) + O(M_n^{-1}). $$ Next, a consequence of assumption~$(iii)$ is that, for any $j \in \{1, \dots, r\}$, \[ \big( p_{n,M_n}(T_{n,j}^{[1]}) , \dots, p_{n,M_n}(T_{n,j}^{[M_n]}) \big) \] is a permutation of the vector \[ \big( \tfrac{3}{2M_n + 2} , \dots, \tfrac{2M_n + 1}{2M_n+ 2} \big). \] It follows that, for any $x \in (0,1)$, \begin{equation}
\label{eq:emp:df} \frac{1}{M_n} \sum_{k=1}^{M_n} \mathbf{1} \big\{ p_{n,M_n}(T_{n,j}^{[k]}) \leq x \big\} = \frac{1}{M_n} \sum_{k=1}^{M_n} \mathbf{1} \big( \tfrac{2k + 1}{2M_n+ 2} \leq x \big) = x + O(M_n^{-1}), \end{equation} where $\ip{.}$ is the floor function. Then, let $\bar w = \max_{j \in \{1,\dots,r\}} w_j$. Starting from \eqref{eq:pval_W}, and relying on assumptions $(i)$ and $(iii)$, we successively obtain \begin{align*} p_{n,M_n}(W_{n,M_n}^{[0]}) &= \frac1{M_n} \sum_{k=1}^{M_n} \mathbf{1} \Big[ \sum_{j=1}^r w_j \varphi \big\{ p_{n,M_n}(T_{n,j}^{[k]}) \big\} \ge \sum_{j=1}^r w_j \varphi \big\{ p_{n,M_n}(T_{n,j}^{[0]}) \big\} \Big] \\ &\le \frac1{M_n} \sum_{k=1}^{M_n} \mathbf{1} \Big[ \bar w \sum_{j=1}^r \varphi \big\{ p_{n,M_n}(T_{n,j}^{[k]}) \big\} \ge w_{j_0} \varphi \big\{ p_{n,M_n}(T_{n,j_0}^{[0]}) \big\} \Big] \\ &\le \frac1{M_n} \sum_{k=1}^{M_n} \mathbf{1} \Big[ \exists\, j \in \{1, \dots, r \} : r \bar w \varphi \big\{ p_{n,M_n}(T_{n,j}^{[k]}) \big\} \ge w_{j_0} \varphi \big\{ p_{n,M_n}(T_{n,j_0}^{[0]}) \big\} \Big] \\ &\le \frac1{M_n} \sum_{k=1}^{M_n} \sum_{j=1}^r \mathbf{1} \Big[ \varphi \big\{ p_{n,M_n}(T_{n,j}^{[k]}) \big\} \ge \tfrac{w_{j_0}}{r \bar w} \varphi \big\{ p_{n,M_n}(T_{n,j_0}^{[0]}) \big\} \Big] \\
&= \frac{r}{M_n} \sum_{k=1}^{M_n} \mathbf{1} \Big[ \varphi \big( \tfrac{2k+1}{2M_n+ 2} \big) \ge \tfrac{w_{j_0}}{r \bar w} \varphi \big\{ p_{n,M_n}(T_{n,j_0}^{[0]}) \big\} \Big]
\\
&= \frac{r}{M_n} \sum_{k=1}^{M_n} \mathbf{1} \left( \tfrac{2k+1}{2M_n+ 2} \le \varphi^{-1} \Big[ \tfrac{w_{j_0}}{r \bar w} \varphi \big\{ p_{n,M_n}(T_{n,j_0}^{[0]}) \big\} \Big] \right) \\
&= r \varphi^{-1} \Big[ \tfrac{w_{j_0}}{r \bar w} \varphi \big\{ p_{n,M_n}(T_{n,j_0}^{[0]}) \big\} \Big] +O(M_n^{-1}) \p 0, \end{align*} where the last statement follows from~\eqref{eq:emp:df} and the fact that $p_{n,M_n}(T_{n,j_0}^{\scriptscriptstyle [0]}) \p 0$. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:weak_Cnh_sm}] The result is a consequence of Proposition~3.3 in \cite{BucKojRohSeg14} and the fact that the strong mixing coefficients of the sequence $(\bm Y_i^{\scriptscriptstyle (h)})_{i \in \mathbb{Z}}$ defined through~\eqref{eq:Yi} can be expressed from those of the sequence $(X_i)_{i \in \mathbb{Z}}$ as $\alpha_r^{\bm Y} = \alpha_{\scriptscriptstyle (r-h+1) \vee 0}^{X}, r \in \mathbb{N}$, where $\vee$ is the maximum operator. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:acs}] The assertions concerning weak convergence are simple consequences of the continuous mapping theorem and Proposition~\ref{prop:weak_Cnh_sm}. It remains to show that $\mathcal{L}(S_{C^{(h)}})$, the distribution of $S_{C^{(h)}}$, is absolutely continuous with respect to the Lebesgue measure. For that purpose, note that, with probability one, the sample paths of $\mathbb{D}_{C^{(h)}}$ are elements of $\mathcal C([0,1]\times [0,1]^{h})$, the space of continuous real-valued functions on $[0,1]\times [0,1]^{h}$. We may write $S_{C^{(h)}}= \{ f(\mathbb{D}_{C^{(h)}}) \}^2$, where \[ f:\mathcal C([0,1]^{h+1}) \to \mathbb{R}, \quad f(g) = \sup_{s \in [0,1]} \left\{ \int_{[0, 1]^h} g^2(s,\bm{u}) \, \mathrm{d} C^{(h)}(\bm{u}) \right\}^{1/2}, \]
and it is sufficient to show that $\mathcal{L} \{ f(\mathbb{D}_{C^{(h)}}) \}$ is absolutely continuous. Now, if $\mathcal C([0,1]^{h+1})$ is equipped with the supremum norm $\|\cdot\|_\infty$, then $f$ is continuous and convex. We may hence apply Theorem 7.1 in \cite{DavLif84}: $\mathcal{L} \{ f(\mathbb{D}_{C^{(h)}}) \}$ is concentrated on $[a_0,\infty)$ and absolutely continuous on $(a_0, \infty)$, where \[ a_0 = \inf\{ f(g) : g \text{ belongs to the support of } \mathcal{L} (\mathbb{D}_{C^{(h)}}) \}. \]
It hence remains to be shown that $\mathcal{L} \{ f(\mathbb{D}_{C^{(h)}}) \}$ has no atom at $a_0$. First of all, note that $a_0=0$. Indeed, by Lemma 1.2(e) in \cite{DerFehMatSch03}, we have $\mathbb{P}(\|\mathbb{D}_{C^{(h)}}\|_\infty\le \varepsilon) >0$ for any $\varepsilon>0$. Hence, for any $\varepsilon>0$, there exist functions $g$ in the support of the distribution of $\mathbb{D}_{C^{(h)}}$ such that $f(g)\le \varepsilon$, whence $a_0=0$ as asserted. Moreover, $f(\mathbb{D}_{C^{(h)}})=0$ holds if and only if $\mathbb D_{C^{(h)}}(s,\bm u) = 0$ for any $s\in[0,1]$ and any $\bm u$ in the support of the distribution induced by $C^{(h)}$ (by continuity of the sample paths). Then, choose an arbitrary point $\bm u^*$ in the latter support such that $\sigma^2 = \mathrm{Var}\{\mathbb C_{C^{(h)}}(0,1,\bm u^*)\}>0$. A straightforward calculation shows that $\mathbb C_{C^{(h)}}(0,1/2, \bm u^*)$ and $\mathbb C_{C^{(h)}}(1/2,1, \bm u^*)$ are uncorrelated and have the same variance $\tfrac12 \sigma^2$.
Hence, \begin{align*} \mathrm{Var} \{ \mathbb D_{C^{(h)}}(\tfrac12, \bm u^*) \} &= \mathrm{Var}\{ \tfrac12 \mathbb C_{C^{(h)}}(0,1/2, \bm u^*) - \tfrac12 \mathbb C_{C^{(h)}}(1/2,1 ,\bm u^*) \} \\ &= \tfrac14 \mathrm{Var}\{ \mathbb C_{C^{(h)}}(0,1/2, \bm u^*) \} + \tfrac14 \mathrm{Var} \{ \mathbb C_{C^{(h)}}(1/2,1, \bm u^*) \} = \tfrac14 \sigma^2 >0. \end{align*} As consequence, $ \mathbb{P}(f(\mathbb{D}_{C^{(h)}})=0) \le \mathbb{P}(\mathbb D_{C^{(h)}}(\tfrac12 ,\bm u^*) = 0 ) = 0, $ which finally implies that $\mathcal{L}(f(\mathbb{D}_{C^{(h)}}))$ and therefore $\mathcal{L}(S^{\scriptscriptstyle (h)})$ is absolutely continuous. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:S_mult}] The result is a consequence of Proposition~4.2 in \cite{BucKojRohSeg14} and the fact that the strong mixing coefficients of the sequence $(\bm Y_i^{\scriptscriptstyle (h)})_{i \in \mathbb{Z}}$ can be expressed as $\alpha_r^{\bm Y} = \alpha_{\scriptscriptstyle (r-h+1) \vee 0}^{X}, r \in \mathbb{N}$. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:weak_Gn_sm}] The assertions concerning weak convergence are simple consequences of Theorem~1 of \cite{Buc15} and of the continuous mapping theorem. Absolute continuity of $S_G$ can be shown along similar lines as for $S_{C^{(h)}}$ in Proposition~\ref{prop:acs}. \end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:combined1}] To prove the first claim, one first needs to show that the finite-dimensional distributions of $\big(\mathbb{D}_{n,C^{(h)}}, \hat{\mathbb{D}}_{\scriptscriptstyle n,C^{(h)}}^{\scriptscriptstyle [1]}, \dots, \hat{\mathbb{D}}_{\scriptscriptstyle n,C^{(h)}}^{\scriptscriptstyle [M]}, \mathbb{E}_n, \hat{\mathbb{E}}_n^{\scriptscriptstyle [1]}, \dots, \hat{\mathbb{E}}_n^{\scriptscriptstyle [M]} \big)$ converge weakly to those of $\big(\mathbb{D}_{C^{(h)}}, \mathbb{D}_{\scriptscriptstyle C^{(h)}}^{\scriptscriptstyle [1]}, \dots, \mathbb{D}_{\scriptscriptstyle C^{(h)}}^{\scriptscriptstyle [M]}, \mathbb{E}, \mathbb{E}^{[1]}, \dots, \mathbb{E}^{[M]} \big)$. The proof is a more notationally involved version of the proof of Lemma~A.1 in \cite{BucKoj16}. Joint asymptotic tightness follows from Proposition~\ref{prop:S_mult} as well as from the fact that, for any $m \in \mathbb{N}$, $\hat{\mathbb{E}}_n^{\scriptscriptstyle [m]} \leadsto \mathbb{E}^{[m]}$ in $\ell^\infty([0,1] \times \mathbb{R})$ as a consequence of Corollary~2.2 in \cite{BucKoj16} and the continuous mapping theorem. \end{proof}
\section*{Acknowledgments}
The authors would like to thank two anonymous Referees and a Co-Editor for their constructive and insightful comments on an earlier version of this manuscript. Axel Bücher gratefully acknowledges support by the Collaborative Research Center ``Statistical modeling of nonlinear dynamic processes'' (SFB 823) of the German Research Foundation. Parts of this paper were written when Axel Bücher was a postdoctoral researcher at Ruhr-Universität Bochum, Germany. Jean-David Fermanian's work was supported by the grant ``Investissements d’Avenir'' (ANR-11-IDEX0003/Labex Ecodec) of the French National Research Agency.
\thispagestyle{empty}
\begin{center} {\LARGE Supplementary material for \\ [2mm] ``Combining cumulative sum change-point detection tests for assessing the stationarity of univariate time series''}
{\large Axel B\"ucher\footnote{Heinrich-Heine-Universität D\"usseldorf, Mathematisches Institut, Universit\"atsstr.~1, 40225 D\"usseldorf, Germany. {E-mail:} \texttt{[email protected]}} \hspace{1cm} Jean-David Fermanian\footnote{CREST-ENSAE, J120, 3, avenue Pierre-Larousse, 92245 Malakoff cedex, France. {E-mail:} \texttt{[email protected]}} \hspace{1cm} Ivan Kojadinovic\footnote{CNRS / Universit\'e de Pau et des Pays de l'Adour, Laboratoire de math\'ematiques et applications -- IPRA, UMR 5142, B.P. 1155, 64013 Pau Cedex, France. {E-mail:} \texttt{[email protected]}}
\today
} \end{center}
\begin{abstract} After providing additional results stating conditions under which the procedure for combining dependent tests described in Section~\ref{sec:combine:tests} is consistent, we briefly illustrate the data-adaptive procedure used to estimate the bandwidth parameter arising in dependent multiplier sequences and report the results of additional Monte Carlo experiments investigating the finite-sample performance of the tests that were proposed in Sections~\ref{sec:rank} and~\ref{sec:sotests}. \end{abstract}
\section{Additional results on the consistency of the procedure for combining dependent tests}
The following result is an analogue of Proposition~\ref{prop:combined:alternative} that allows one to consider the function~$\psi_S$ in~\eqref{eq:stouffer} as combining function $\psi$ in the proposed global testing procedure.
\begin{prop}
Let $M=M_n\to\infty$ as $n\to\infty$. Assume that
\begin{enumerate}[(i)]
\item the combining function $\psi$ is of the form
\[
\psi(p_1, \dots, p_r) = \sum_{j=1}^r w_j \varphi(p_j),
\]
where $\varphi$ is decreasing, non-negative on $(0,1/2)$ and one-to-one from $(0,1)$ to $(-\infty,\infty)$,
\item the global statistic $W_{n,M_n}^{[0]}$ diverges to infinity in probability,
\item for any $j \in \{1,\dots,r\}$, the sample of bootstrap replicates $T_{n,j}^{\scriptscriptstyle [1]}, \dots, T_{n,j}^{\scriptscriptstyle [M_n]}$ does not contain ties.
\end{enumerate}
Then, the approximate p-value $p_{n,M_n}(W_{\scriptscriptstyle n,M_n}^{\scriptscriptstyle [0]})$ of the global test converges to zero in probability, where $p_{n,M_n}(W_{\scriptscriptstyle n,M_n}^{\scriptscriptstyle [0]})$ is defined by~\eqref{eq:pval_W}. \end{prop}
\begin{proof}
Let $\bar w = \max_{j \in \{1,\dots,r\}} w_j$ and let $\varphi^+ = \varphi \vee 0$, where $\vee$ is the maximum operator. Starting from \eqref{eq:pval_W}, and relying on assumptions $(i)$ and $(iii)$, we successively obtain \begin{align*} p_{n,M_n}(W_{n,M_n}^{[0]}) &= \frac1{M_n} \sum_{k=1}^{M_n} \mathbf{1} \Big[ \sum_{j=1}^r w_j \varphi \big\{ p_{n,M_n}(T_{n,j}^{[k]}) \big\} \ge W_{n,M_n}^{[0]} \big\} \Big] \\ &\le \frac1{M_n} \sum_{k=1}^{M_n} \mathbf{1} \Big[ \bar w \sum_{j=1}^r \varphi^+ \big\{ p_{n,M_n}(T_{n,j}^{[k]}) \big\} \ge W_{n,M_n}^{[0]} \Big] \\ &\le \frac1{M_n} \sum_{k=1}^{M_n} \mathbf{1} \Big[ \exists\, j \in \{1, \dots, r \} : r \bar w \varphi^+ \big\{ p_{n,M_n}(T_{n,j}^{[k]}) \big\} \ge W_{n,M_n}^{[0]} \Big] \\
&\le \frac1{M_n} \sum_{k=1}^{M_n} \sum_{j=1}^r \mathbf{1} \Big[ \varphi^+ \big\{ p_{n,M_n}(T_{n,j}^{[k]}) \big\} \ge (r \bar w)^{-1} W_{n,M_n}^{[0]} \Big] \\
&= \frac{r}{M_n} \sum_{k=1}^{M_n} \mathbf{1} \Big\{ \varphi^+ \big( \tfrac{2k+1}{2M_n+ 2} \big) \ge (r \bar w)^{-1} W_{n,M_n}^{[0]} \Big\}. \end{align*} Using assumption~$(i)$, we have that, for $x > 0$, \begin{align*}
\frac1{M_n} \sum_{k=1}^{M_n} \mathbf{1} \Big\{ \varphi^+ \big( \tfrac{2k+1}{2M_n+ 2} \big) \ge x \Big\} &= \frac1{M_n} \sum_{k=1}^{\ip{M_n/2}} \mathbf{1} \Big\{ \varphi \big( \tfrac{2k+1}{2M_n+ 2} \big) \ge x \Big\} + O(M_n^{-1}) \\
&= \frac1{M_n} \sum_{k=1}^{\ip{M_n/2}} \mathbf{1} \Big\{ \tfrac{2k+1}{2M_n+ 2} \le \varphi^{-1}(x) \Big\} + O(M_n^{-1}) \\
&= \varphi^{-1}(x)/2 + O(M_n^{-1}) \end{align*} since $\varphi^{-1}(x) \in (0,1/2)$. As a consequence, we obtain that $$ p_{n,M_n}(W_{n,M_n}^{[0]}) = r \varphi^{-1} \big\{ (r \bar w)^{-1} W_{n,M_n}^{[0]} \big\} /2 + O(M_n^{-1}) \p 0. $$ \end{proof}
The next results provides sufficient conditions so that assumption~$(ii)$ of the previous proposition is satisfied.
\begin{prop}
Let $M=M_n\to\infty$ as $n\to\infty$. Assume that
\begin{enumerate}[(i)]
\item the combining function $\psi$ is of the form
\[
\psi(p_1, \dots, p_r) = \sum_{j=1}^r w_j \varphi(p_j),
\]
where $\varphi$ is decreasing, non-negative on $(0,1/2)$ and one-to-one from $(0,1)$ to $(-\infty,\infty)$,
\item there exists $r_0 \in \{1,\dots,r-1\}$ such that
\begin{enumerate}[(a)]
\item $H_0^{(1)} \cap \dots \cap H_0^{(r_0)}$ holds, $(T_{n,1},\dots,T_{n,r_0})$ converges weakly to a random vector having a continuous d.f., and
$$
\sup_{\bm x \in \mathbb{R}^{r_0}} | \mathbb{P}(T_{n,1}^{[1]} \leq x_1, \dots, T_{n,r_0}^{[1]} \leq x_{r_0} \,|\, \bm X_n) - \mathbb{P}(T_{n,1} \leq x_1, \dots, T_{n,r_0} \leq x_{r_0})| \p 0,
$$
\item for any $j \in \{r_0+1,\dots,r\}$, $H_0^{(j)}$ does not hold and $\mathbb{P}(T_{n,j}^{\scriptscriptstyle [1]} \ge T_{n,j})$ converges to zero.
\end{enumerate}
\end{enumerate}
Then, $W_{n,M_n}^{\scriptscriptstyle [0]}$ diverges to infinity in probability. \end{prop}
\begin{proof}
As a consequence of assumption~$(ii)$~$(a)$ and Proposition~\ref{prop:combined:general}, we have that the random variable $\sum_{j=1}^{r_0} w_j \varphi\{p_{n,M_n}(T_{n,j}^{\scriptscriptstyle [0]})\}$ converges in distribution, and thus, that
\begin{equation}
\label{eq:bounded:proba}
\sum_{j=1}^{r_0} w_j \varphi\{p_{n,M_n}(T_{n,j}^{\scriptscriptstyle [0]})\} = O_\mathbb{P}(1).
\end{equation}
From assumption~$(ii)$~$(b)$, proceeding as in the proof of Proposition~\ref{prop:combined:alternative}, we immediately obtain that $p_{n,M_n}(T_{n,j}^{\scriptscriptstyle [0]})$ converges to zero in probability for all $j \in \{r_0+1,\dots,r\}$, which combined with assumption~$(i)$ and the continuous mapping theorem implies that
\begin{equation}
\label{eq:div:proba}
\sum_{j=r_0+1}^{r} w_j \varphi\{p_{n,M_n}(T_{n,j}^{\scriptscriptstyle [0]})\} \p \infty.
\end{equation}
The desired result follows from~\eqref{eq:bounded:proba} and~\eqref{eq:div:proba}. \end{proof}
\section{Data-adaptive bandwidth parameter of dependent multiplier sequences for an AR(1) model}
As explained in Appendix~\ref{app:dep}, the bandwidth parameter $\ell_n$ (or, equivalently, $b_n$) arising in the generation of dependent multiplier sequences through~\eqref{eq:movave} will have a crucial influence on the finite-sample performance of the tests studied in this work. In this section, we conduct a small simulation to illustrate the finite-sample properties of the data-adaptive procedure proposed in \citet[Section 5.1]{BucKoj16}. For $\beta \in \{0,0.1,\dots,0.9\}$, we generate 1000 samples of size $n=128$ from an AR(1) model with standard normal innovations and parameter~$\beta$. From each sample, we estimate the bandwidth parameter $b_n$. The mean and standard deviations of the 1000 estimates are represented in Figure~\ref{fig:b} against the value of $\beta$. As can be seen, the stronger the serial dependence, the larger $b_n$ is on average, suggesting that $b_n$ (or, equivalently, $\ell_n$) do indeed play a role similar to the block length in the block bootstrap of \cite{Kun89}.
\begin{figure}
\caption{ Mean and standard deviation of 1000 estimates of the bandwidth parameter $b_n$ arising in dependent multiplier sequences computed from samples of size $n=128$ from an AR(1) model with standard normal innovations and parameter $\beta$.}
\label{fig:b}
\end{figure}
\section{Additional Monte Carlo experiments}
After describing the main data generating models for studying the finite-sample properties of the proposed tests, we investigate in detail the behavior of some competitor tests available in \textsf{R}, study how well the proposed tests hold their level and finally assess their power under various alternatives, some belonging to the locally stationary process literature, others more in line with the change-point detection literature.
\subsection{Data generating processes}
The following ten strictly stationary models were used to generate observations under the null hypothesis of stationarity. Either standard normal or standardized Student $t$ with 4 degrees of freedom innovations were considered (standardization refers to the fact that the Student $t$ with 4 degrees of freedom distribution was rescaled to have variance one). The first seven models were considered in \cite{Nas13} (and are denoted by S1--S7 therein): \vspace{-
amount} \begin{enumerate}[{N}1 -][10]\setcounter{enumi}{0}\parskip0pt \item i.i.d.\ observations from the innovation distribution. \item AR(1) model with parameter $0.9$. \item AR(1) model with parameter $-0.9$. \item MA(1) model with parameter $0.8$. \item MA(1) model with parameter $-0.8$. \item ARMA(1, 0, 2) with the AR coefficient $-0.4$, and the MA coefficients $(-0.8, 0.4)$. \item AR(2) with parameters $1.385929$ and $-0.9604$. This process is stationary, but close to the ``unit root'': a ``rough'' stochastic process with spectral peak near $\pi/4$. \item GARCH(1,1) model with parameters $(\omega,\beta,\alpha)=(0.012,0.919,0.072)$. The latter values were estimated by \cite{JonPooRoc07} from SP500 daily log-returns. \item the exponential autoregressive model considered in \cite{AueTjo90} whose generating equation is \begin{equation*} X_{t} = \{ 0.8 - 1.1 \exp ( - 50 X_{t-1}^2 ) \} X_{t-1} + 0.1 \varepsilon_{t}. \end{equation*} \item the nonlinear autoregressive model used in \citet[Section 3.3]{PapPol01} whose generating equation is \begin{equation*} X_t = 0.6 \sin( X_{t-1} ) + \varepsilon_{t}. \end{equation*} \end{enumerate} \vspace{-
amount} For all these models, a burn-in period of 100 observations was used.
To simulate observations under the alternative hypothesis of non-stationarity, models connected to the literature on locally stationary processes were considered first. The first four are taken from \cite{DetPreVet11} and were used to generate univariate series $X_{1,n},\dots,X_{n,n}$ of length $n \in \{128,256,512\}$ by means of the following equations: \vspace{-
amount} \begin{enumerate}[{A}1 -][10]\setcounter{enumi}{0}\parskip0pt
\item $X_{t,n} = 1.1 \cos\{1.5 - \cos(4 \pi t / n)\} \varepsilon_{t-1} + \varepsilon_t$,
\item $X_{t,n} = 0.6 \sin(4 \pi t / n) X_{t-1,n} + \varepsilon_t$,
\item $X_{t,n} = (0.5 X_{t-1,n} + \varepsilon_t) \mathbf{1}(t \in \{1,\dots,n/4\} \cup \{3n/4+1,\dots,n\}) + (-0.5 X_{t-1,n} + \varepsilon_t) \mathbf{1}(t \in \{n/4+1,\dots,3n/4\})$,
\item $X_{t,n} = (-0.5 X_{t-1,n} + \varepsilon_t) \mathbf{1}(t \in \{1,\dots,n/2\} \cup \{n/2+n/64+1,\dots,n\}) + 4 \varepsilon_t \mathbf{1}(t \in \{n/2+1,\dots,n/2+n/64\})$, \end{enumerate} \vspace{-
amount}
\noindent where $\varepsilon_0,\varepsilon_1,\dots,\varepsilon_n$ are i.i.d.\ standard normal and with the convention that $X_{0,n} = 0$. The next four models under the alternative were considered in \cite{Nas13} (and are denoted P1--P4 therein), the last three ones being locally stationary wavelet (LSW) processes \citep[see, e.g., Equation~(1) in][]{Nas13}: \vspace{-
amount} \begin{enumerate}[{A}1 -][10]\setcounter{enumi}{4}\parskip0pt \item A time-varying AR model $X_t = \alpha_t X_{t-1} + \varepsilon_t$ with i.i.d.\ standard normal innovations and an AR parameter evolving linearly from 0.9 to $-0.9$ over the $n$ observations. \item A LSW process based on Haar wavelets with spectrum $S_j(z) = 0$ for $j > 1$ and $S_1 (z) = 1/4 - (z - 1/2 )^2$ for $z \in (0, 1)$. This process is a time-varying moving average process. \item A LSW process based on Haar wavelets with spectrum $S_j (z) = 0$ for $j > 2$, $S_1 (z)$ as for A6 and $S_2 (z) = S_1 (z + 1/2 )$ using periodic boundaries (for the construction of the spectrum only). \item A LSW process based on Haar wavelets with spectrum $S_j (z) = 0$ for $j = 2$ and $j > 4$. Moreover, $S_1 (z) = \exp\{-4(z - 1/2 )^2 \}$, $S_3 (z) = S_1(z - 1/4 )$ and $S_4 (z) = S_1(z + 1/4 )$, again assuming periodic boundaries. \end{enumerate} \vspace{-
amount}
\noindent Models A1--A8 considered thus far are connected to the literature on locally stationary processes. In a second set of experiments, we focused on models that are more in line with the change-point detection literature: \vspace{-
amount} \begin{enumerate}[{A}1 -][10]\setcounter{enumi}{8}\parskip0pt \item An AR(1) model with one break: the $n/2$ first observations are i.i.d.\ from the innovation distribution (standard normal or standardized $t_4$) and the $n/2$ last observations are from an AR(1) model with parameter $\beta \in \{-0.8,-0.4,0,0.4,0.8\}$. \item An AR(2) model with one break: the $n/2$ first observations are i.i.d.\ from the innovation distribution (standard normal or standardized $t_4$) and the $n/2$ last observations are from an AR(2) model with parameter $(0,\beta)$ with $\beta \in \{-0.8,-0.4,0,0.4,0.8\}$. \end{enumerate} \vspace{-
amount}
\noindent Note that both the contemporary distribution and the serial dependence are changing under these scenarios (unless $\beta=0$). Also note that there is no relationship between $X_t$ and $X_{t-1}$ for Model A10, that is, $H_{0,c}^{\scriptscriptstyle (2)}$ in~\eqref{eq:H0:Ch} is met with $C^{\scriptscriptstyle (2)}$ the bivariate independence copula.
Finally, we considered two simple models under the alternative for which the contemporary distribution remains unchanged: \vspace{-
amount} \begin{enumerate}[{A}1 -][10]\setcounter{enumi}{10}\parskip0pt \item An AR(1) model with a break affecting the innovation variance: the $n/2$ first observations are i.i.d.\ standard normal and the $n/2$ last observations are drawn from an AR(1) model with parameter $\beta \in \{0,0.4,0.8\}$ and centered normal innovations with variance $(1-\beta^2)$. The contemporary distribution is thus the standard normal. \item A max-autoregressive model with one break: the $n/2$ first observations are i.i.d.\ standard Fr\'echet, and the last $n/2$ observations follow the recursion \begin{equation*}
X_t = \max\{ \beta X_{t-1}, (1-\beta) Z_t\}, \end{equation*} where $\beta \in \{0,0.4,0.8\}$ and the $Z_t$ are i.i.d\ standard Fr\'echet. The contemporary distribution is standard Fr\'echet regardless of the choice of $\beta$, see, e.g., Example 10.3 in \cite{BeiGoeSegTeu04}. \end{enumerate} \vspace{-
amount}
\subsection{Some competitors to our tests} \label{sec:competitors}
As mentioned in the introduction, many tests of stationarity have been proposed in the literature. Unfortunately, only a few of them seem to have been implemented in statistical software. In this section, we focus on the tests of \cite{PriSub69}, \cite{Nas13} and \cite{CarNas13} that have been implemented in the \textsf{R} packages \texttt{fractal} \citep{fractal}, \texttt{locits} \citep{locits} and \texttt{costat} \citep{costat}, respectively. Note that we did not include the test of \cite{CarNas16} (implemented in the \textsf{R} package \texttt{BootWPTOS}) in our simulations because we were not able to understand how to initialize the arguments of the corresponding \textsf{R} function.
\begin{table}[t!] \centering \caption{Percentages of rejection of the null hypothesis of stationarity computed from 1000 samples of size $n \in \{128, 256, 512\}$ generated from Models N1--N10, first with $N(0,1)$ innovations and then with standardized $t_4$ innovations. The column PSR.T corresponds to the test of \cite{PriSub69} implemented in the \textsf{R} package \texttt{fractal}, the column hwtos2 corresponds to the test of \cite{Nas13} implemented in the \textsf{R} package \texttt{locits} and the column BTOS corresponds to the test of \cite{CarNas13} implemented in the \textsf{R} package \texttt{costat}. } \label{H0others} \begingroup\footnotesize \begin{tabular}{lrrrrrrr}
\hline
\multicolumn{2}{c}{} & \multicolumn{3}{c}{$N(0,1)$ innovations} & \multicolumn{3}{c}{Standardized $t_4$ innovations} \\ \cmidrule(lr){3-5} \cmidrule(lr){6-8} Model & $n$ & PSR.T & hwtos2 & BTOS & PSR.T & hwtos2 & BTOS \\ \hline N1 & 128 & 7.0 & 0.4 & 0.0 & 30.6 & 4.2 & 5.8 \\
& 256 & 5.7 & 3.3 & 0.0 & 47.3 & 15.1 & 8.5 \\
& 512 & 5.9 & 3.0 & 0.1 & 62.1 & 19.0 & 13.5 \\
N2 & 128 & 46.3 & 1.0 & 0.0 & 66.6 & 2.4 & 0.0 \\
& 256 & 22.0 & 3.6 & 0.0 & 59.5 & 12.5 & 0.0 \\
& 512 & 11.1 & 4.6 & 0.0 & 65.5 & 13.6 & 0.0 \\
N3 & 128 & 6.8 & 9.7 & 21.1 & 31.1 & 12.5 & 32.2 \\
& 256 & 6.7 & 15.8 & 35.8 & 46.0 & 22.8 & 41.4 \\
& 512 & 5.7 & 17.7 & 38.5 & 63.3 & 29.1 & 52.6 \\
N4 & 128 & 7.2 & 2.0 & 0.0 & 34.8 & 3.5 & 1.0 \\
& 256 & 7.4 & 4.7 & 0.0 & 46.8 & 18.8 & 2.4 \\
& 512 & 6.3 & 3.9 & 0.0 & 61.1 & 15.8 & 2.6 \\
N5 & 128 & 12.7 & 0.0 & 0.4 & 38.1 & 0.1 & 6.6 \\
& 256 & 9.1 & 0.0 & 0.5 & 48.3 & 7.4 & 11.6 \\
& 512 & 6.3 & 0.3 & 0.8 & 64.1 & 6.7 & 16.7 \\
N6 & 128 & 24.3 & 0.2 & 2.0 & 44.0 & 0.2 & 12.9 \\
& 256 & 8.6 & 0.7 & 2.4 & 46.8 & 6.5 & 16.6 \\
& 512 & 7.1 & 0.2 & 3.2 & 63.7 & 4.9 & 21.1 \\
N7 & 128 & 62.8 & 1.1 & 0.5 & 63.8 & 1.3 & 0.6 \\
& 256 & 63.1 & 7.1 & 5.0 & 73.5 & 9.1 & 8.3 \\
& 512 & 49.2 & 20.6 & 17.5 & 78.6 & 29.0 & 22.5 \\
N8 & 128 & 20.1 & 1.9 & 2.5 & 58.1 & 6.0 & 17.4 \\
& 256 & 38.5 & 8.9 & 4.1 & 81.0 & 30.1 & 33.8 \\
& 512 & 56.5 & 11.5 & 6.5 & 94.8 & 44.3 & 47.9 \\
N9 & 128 & 34.8 & 5.9 & 0.0 & 68.9 & 13.6 & 0.4 \\
& 256 & 25.8 & 16.6 & 0.2 & 71.3 & 34.8 & 1.6 \\
& 512 & 17.9 & 22.8 & 0.5 & 75.6 & 41.9 & 3.1 \\
N10 & 128 & 5.2 & 0.4 & 0.0 & 19.8 & 1.6 & 3.7 \\
& 256 & 4.2 & 2.0 & 0.0 & 36.1 & 12.2 & 5.3 \\
& 512 & 4.6 & 2.7 & 0.1 & 61.1 & 12.3 & 8.7 \\
\hline \end{tabular} \endgroup \end{table}
The rejection percentages of the three aforementioned tests were estimated for Models N1--N10 generating observations under the null. These tests were carried out at the 5\% significance level and the empirical levels were estimated from 1000 samples. As one can see in Table~\ref{H0others}, all these tests turn out to be too liberal in at least one scenario. Overall, the empirical levels are even higher when heavy tailed innovations are used instead of standard normal ones.
\subsection{Empirical levels of the proposed tests}
To estimate the levels of the proposed tests, we considered the same setting as in the previous section. We started with the component tests described in Sections~\ref{sec:rank} and~\ref{sec:sotests}, and then considered various combinations of them, based on the weighted version of Stouffer's method and a weighted version of Fisher's method. As explained previously, the former (resp.\ latter) consists of using $\psi_S$ in~\eqref{eq:stouffer} (resp.\ $\psi_F$ in~\eqref{eq:fisher}) as the function $\psi$ in Sections~\ref{sec:rank} and~\ref{sec:sotests}. As Stouffer's method sometimes gave inflated levels, for the sake of brevity, we only report the results for Fisher's method in this section.
As previously, all the tests were carried out at the 5\% significance level and the empirical levels were estimated from 1000 samples generated from Models N1--N10. The values 128, 256 and 512 were considered for the sample size $n$ and the embedding dimension $h$ was taken to be in the set $\{2,3,4,8\}$. To save space in the forthcoming tables providing the results, each component test is abbreviated by a single letter, as already explained in Section~\ref{sec:MC}.
\begin{sidewaystable}[t!] \centering \caption{Percentages of rejection of the null hypothesis of stationarity computed from 1000 samples of size $n \in \{128, 256, 512\}$ generated from Models N1--N5, first with $N(0,1)$ innovations and then with standardized $t_4$ innovations. The meaning of the abbreviations d, m, v, c, dc, dcp, a, va and mva is given in Section~\ref{sec:MC}.} \label{H0_1} \begingroup\footnotesize \begin{tabular}{llrrrrrrrrrrrrrrrrrrrrr}
\hline
\multicolumn{6}{c}{} & \multicolumn{5}{c}{$h=2$ or lag 1} & \multicolumn{5}{c}{$h=3$ or lag 2} & \multicolumn{5}{c}{$h=4$ or lag 3} & \multicolumn{2}{c}{$h=8$}\\ \cmidrule(lr){7-11} \cmidrule(lr){12-16} \cmidrule(lr){17-21} \cmidrule(lr){22-23} Model & Innov. & $n$ & d & m & v & c & dc & a & va & mva & c & dc & dcp & va & mva & c & dc & dcp & va & mva & c & dc \\ \hline N1 & N(0,1) & 128 & 4.0 & 4.7 & 3.8 & 3.0 & 3.9 & 3.0 & 3.5 & 3.6 & 4.6 & 4.1 & 4.2 & 3.9 & 4.1 & 3.2 & 3.4 & 5.7 & 4.1 & 4.5 & 0.0 & 0.7 \\
& & 256 & 4.9 & 5.3 & 4.1 & 4.4 & 4.1 & 4.5 & 4.8 & 4.6 & 5.5 & 5.2 & 5.9 & 4.4 & 5.1 & 4.2 & 4.7 & 5.9 & 4.4 & 4.9 & 0.0 & 0.5 \\
& & 512 & 5.8 & 4.7 & 4.6 & 4.6 & 5.0 & 4.3 & 5.5 & 4.4 & 6.2 & 6.6 & 5.0 & 5.4 & 4.3 & 6.6 & 6.3 & 5.2 & 4.9 & 5.0 & 0.2 & 2.3 \\
& Stand. $t_4$ & 128 & 4.5 & 3.4 & 1.6 & 4.2 & 5.0 & 5.2 & 3.7 & 3.3 & 4.9 & 5.3 & 5.7 & 4.7 & 3.0 & 3.4 & 4.0 & 5.9 & 3.7 & 2.4 & 0.0 & 0.7 \\
& & 256 & 5.1 & 3.8 & 1.4 & 4.2 & 5.9 & 4.2 & 3.6 & 4.0 & 5.9 & 5.5 & 5.4 & 3.0 & 2.6 & 5.3 & 5.4 & 5.4 & 2.8 & 3.0 & 0.0 & 0.4 \\
& & 512 & 4.7 & 4.5 & 2.6 & 5.8 & 5.7 & 4.9 & 3.8 & 4.0 & 8.6 & 6.7 & 4.5 & 4.0 & 3.1 & 7.1 & 6.0 & 5.9 & 3.2 & 3.1 & 0.3 & 1.2 \\
N2 & N(0,1) & 128 & 2.5 & 2.5 & 1.3 & 1.6 & 3.3 & 1.3 & 4.9 & 3.2 & 1.6 & 2.6 & 5.1 & 7.5 & 3.8 & 1.5 & 2.0 & 6.4 & 8.7 & 4.1 & 0.2 & 0.0 \\
& & 256 & 5.1 & 4.2 & 1.8 & 2.3 & 3.5 & 1.3 & 5.7 & 3.0 & 2.2 & 3.9 & 5.5 & 8.0 & 4.4 & 2.1 & 3.6 & 6.3 & 9.1 & 4.4 & 0.5 & 1.8 \\
& & 512 & 5.1 & 5.0 & 1.5 & 1.5 & 2.2 & 1.7 & 4.4 & 4.2 & 1.9 & 3.1 & 3.8 & 6.9 & 5.2 & 1.4 & 3.1 & 4.8 & 7.5 & 5.3 & 0.7 & 2.5 \\
& Stand. $t_4$ & 128 & 2.4 & 2.0 & 1.1 & 2.1 & 3.4 & 1.0 & 4.4 & 3.0 & 1.3 & 3.0 & 5.2 & 6.7 & 3.3 & 1.6 & 2.2 & 5.7 & 7.5 & 3.9 & 0.0 & 0.4 \\
& & 256 & 5.6 & 4.8 & 1.2 & 1.0 & 1.3 & 1.1 & 4.5 & 2.8 & 0.8 & 1.9 & 4.4 & 7.3 & 4.0 & 0.5 & 2.0 & 4.9 & 8.3 & 4.4 & 0.9 & 1.3 \\
& & 512 & 6.1 & 5.9 & 1.9 & 0.6 & 1.9 & 1.4 & 4.5 & 3.5 & 1.0 & 2.7 & 4.1 & 6.4 & 4.5 & 1.1 & 2.7 & 5.0 & 7.6 & 4.7 & 0.9 & 2.8 \\
N3 & N(0,1) & 128 & 0.5 & 0.0 & 0.5 & 2.2 & 3.2 & 0.5 & 2.8 & 0.7 & 0.7 & 0.9 & 6.5 & 3.3 & 0.7 & 0.0 & 0.0 & 6.3 & 4.1 & 1.0 & 0.0 & 0.0 \\
& & 256 & 1.4 & 0.0 & 0.5 & 2.4 & 4.1 & 0.6 & 2.3 & 0.7 & 0.1 & 1.0 & 6.7 & 3.6 & 1.0 & 0.0 & 0.0 & 6.6 & 4.0 & 1.2 & 0.0 & 0.0 \\
& & 512 & 2.7 & 0.3 & 1.4 & 2.2 & 4.8 & 1.1 & 4.2 & 1.6 & 0.5 & 2.0 & 7.7 & 5.9 & 1.9 & 0.2 & 0.5 & 7.9 & 6.9 & 2.1 & 0.0 & 0.0 \\
& Stand. $t_4$ & 128 & 0.4 & 0.1 & 0.4 & 2.0 & 3.7 & 0.1 & 1.8 & 0.4 & 0.1 & 0.3 & 5.0 & 2.5 & 0.3 & 0.0 & 0.0 & 5.4 & 2.6 & 0.2 & 0.0 & 0.0 \\
& & 256 & 2.2 & 0.1 & 0.8 & 2.7 & 5.3 & 0.4 & 2.8 & 0.6 & 0.4 & 2.0 & 7.9 & 4.4 & 0.8 & 0.0 & 0.3 & 8.5 & 5.4 & 0.8 & 0.0 & 0.0 \\
& & 512 & 2.8 & 0.7 & 1.0 & 2.9 & 6.3 & 0.6 & 3.4 & 1.8 & 0.3 & 2.0 & 7.7 & 5.2 & 2.0 & 0.1 & 0.7 & 8.7 & 5.2 & 2.2 & 0.0 & 0.2 \\
N4 & N(0,1) & 128 & 4.8 & 5.3 & 3.9 & 2.5 & 4.2 & 2.5 & 7.3 & 5.5 & 3.3 & 4.5 & 7.3 & 7.0 & 4.9 & 3.7 & 4.6 & 6.6 & 6.9 & 4.7 & 0.1 & 1.1 \\
& & 256 & 5.7 & 7.3 & 3.5 & 2.5 & 4.6 & 3.7 & 7.9 & 7.7 & 4.0 & 5.3 & 7.0 & 8.2 & 6.6 & 4.9 & 5.7 & 6.5 & 6.8 & 6.4 & 0.9 & 1.9 \\
& & 512 & 5.6 & 6.6 & 4.6 & 4.9 & 5.4 & 4.7 & 7.1 & 7.9 & 6.2 & 5.8 & 6.9 & 7.4 & 7.2 & 6.6 & 5.6 & 7.3 & 7.4 & 6.3 & 1.9 & 3.3 \\
& Stand. $t_4$ & 128 & 3.4 & 3.9 & 2.7 & 3.8 & 4.6 & 1.2 & 4.5 & 2.8 & 3.5 & 5.0 & 6.4 & 6.2 & 3.7 & 3.5 & 4.7 & 6.3 & 6.4 & 3.4 & 0.2 & 1.4 \\
& & 256 & 4.4 & 3.6 & 1.5 & 2.4 & 3.7 & 1.5 & 4.3 & 2.3 & 3.8 & 5.0 & 5.9 & 5.4 & 3.1 & 4.9 & 5.5 & 6.4 & 4.8 & 2.5 & 0.3 & 0.8 \\
& & 512 & 5.2 & 4.7 & 2.7 & 4.4 & 5.2 & 1.5 & 5.8 & 4.5 & 5.8 & 5.5 & 6.4 & 5.9 & 4.7 & 6.6 & 6.0 & 6.8 & 5.8 & 5.0 & 2.3 & 4.5 \\
N5 & N(0,1) & 128 & 1.3 & 0.0 & 1.1 & 3.2 & 2.0 & 1.3 & 3.4 & 0.6 & 2.2 & 1.8 & 2.1 & 3.1 & 0.5 & 0.3 & 0.2 & 2.2 & 2.7 & 0.2 & 0.0 & 0.0 \\
& & 256 & 2.0 & 0.0 & 1.3 & 3.5 & 3.4 & 1.3 & 3.5 & 0.5 & 2.6 & 2.7 & 3.6 & 3.6 & 0.1 & 0.8 & 1.3 & 3.2 & 3.2 & 0.0 & 0.0 & 0.2 \\
& & 512 & 3.2 & 0.1 & 2.0 & 4.7 & 4.4 & 3.1 & 5.9 & 1.6 & 3.6 & 3.7 & 5.0 & 4.9 & 1.3 & 1.7 & 2.6 & 4.6 & 4.9 & 0.9 & 0.0 & 0.2 \\
& Stand. $t_4$ & 128 & 1.4 & 0.0 & 1.1 & 3.8 & 3.0 & 1.6 & 4.3 & 0.4 & 2.5 & 1.7 & 3.3 & 3.9 & 0.2 & 0.5 & 0.6 & 3.3 & 3.2 & 0.1 & 0.0 & 0.0 \\
& & 256 & 2.4 & 0.0 & 0.7 & 3.9 & 2.9 & 1.5 & 3.4 & 0.3 & 2.6 & 2.1 & 4.2 & 3.2 & 0.1 & 0.8 & 0.8 & 2.9 & 2.6 & 0.0 & 0.0 & 0.2 \\
& & 512 & 3.9 & 0.0 & 1.1 & 4.0 & 4.0 & 1.3 & 4.0 & 0.6 & 3.6 & 4.3 & 5.1 & 3.2 & 0.3 & 1.5 & 1.8 & 5.1 & 2.9 & 0.1 & 0.0 & 0.7 \\
\hline \end{tabular} \endgroup \end{sidewaystable}
\begin{sidewaystable}[t!] \centering \caption{Continued from Table~\ref{H0_1}. Percentages of rejection of the null hypothesis of stationarity computed from 1000 samples of size $n \in \{128, 256, 512\}$ generated from Models N6--N10, first with $N(0,1)$ innovations and then with standardized $t_4$ innovations. The meaning of the abbreviations d, m, v, c, dc, dcp, a, va and mva is given in Section~\ref{sec:MC}.} \label{H0_2} \begingroup\footnotesize \begin{tabular}{llrrrrrrrrrrrrrrrrrrrrr}
\hline
\multicolumn{6}{c}{} & \multicolumn{5}{c}{$h=2$ or lag 1} & \multicolumn{5}{c}{$h=3$ or lag 2} & \multicolumn{5}{c}{$h=4$ or lag 3} & \multicolumn{2}{c}{$h=8$}\\ \cmidrule(lr){7-11} \cmidrule(lr){12-16} \cmidrule(lr){17-21} \cmidrule(lr){22-23} Model & Innov. & $n$ & d & m & v & c & dc & a & va & mva & c & dc & dcp & va & mva & c & dc & dcp & va & mva & c & dc \\ \hline N6 & N(0,1) & 128 & 2.5 & 0.1 & 1.1 & 3.4 & 3.7 & 1.2 & 4.1 & 1.0 & 0.2 & 0.9 & 5.0 & 5.8 & 1.2 & 0.0 & 0.0 & 5.0 & 5.5 & 1.1 & 0.0 & 0.0 \\
& & 256 & 4.1 & 1.1 & 2.9 & 2.7 & 5.5 & 2.4 & 7.9 & 3.1 & 1.3 & 3.3 & 7.2 & 8.6 & 3.9 & 0.0 & 0.5 & 6.8 & 7.8 & 3.2 & 0.0 & 0.2 \\
& & 512 & 5.5 & 1.6 & 3.8 & 4.2 & 5.8 & 4.3 & 10.3 & 6.5 & 1.5 & 4.0 & 7.1 & 12.1 & 7.1 & 0.2 & 1.1 & 8.2 & 11.8 & 5.8 & 0.0 & 0.8 \\
& Stand. $t_4$ & 128 & 1.8 & 0.1 & 1.1 & 2.6 & 3.0 & 1.2 & 4.0 & 0.9 & 0.1 & 0.9 & 4.5 & 4.4 & 1.2 & 0.0 & 0.2 & 4.9 & 3.8 & 0.8 & 0.0 & 0.0 \\
& & 256 & 4.1 & 0.7 & 2.2 & 2.7 & 4.3 & 1.5 & 5.1 & 2.4 & 1.1 & 2.3 & 5.9 & 7.1 & 2.3 & 0.2 & 0.9 & 6.2 & 6.2 & 1.7 & 0.0 & 0.5 \\
& & 512 & 4.6 & 1.5 & 2.5 & 4.7 & 5.5 & 2.2 & 6.9 & 3.2 & 1.1 & 2.5 & 6.9 & 8.4 & 3.4 & 0.0 & 0.6 & 7.1 & 7.5 & 3.1 & 0.0 & 0.5 \\
N7 & N(0,1) & 128 & 0.0 & 0.0 & 0.0 & 0.1 & 0.0 & 0.0 & 0.0 & 0.0 & 0.4 & 0.4 & 0.8 & 2.1 & 0.0 & 3.5 & 3.4 & 5.1 & 2.8 & 0.0 & 0.1 & 0.1 \\
& & 256 & 1.8 & 0.0 & 0.1 & 0.0 & 0.0 & 0.2 & 3.8 & 0.1 & 0.0 & 0.7 & 0.7 & 2.2 & 0.1 & 0.8 & 1.6 & 2.2 & 4.3 & 0.1 & 0.1 & 0.5 \\
& & 512 & 6.8 & 0.0 & 0.9 & 0.0 & 0.5 & 0.8 & 4.9 & 0.9 & 0.0 & 2.1 & 3.2 & 2.3 & 0.1 & 1.0 & 3.7 & 5.2 & 5.3 & 0.4 & 0.2 & 1.0 \\
& Stand. $t_4$ & 128 & 0.0 & 0.0 & 0.0 & 0.1 & 0.0 & 0.0 & 0.2 & 0.0 & 0.6 & 0.3 & 0.9 & 2.2 & 0.0 & 2.8 & 2.4 & 4.0 & 2.7 & 0.0 & 0.0 & 0.0 \\
& & 256 & 1.6 & 0.0 & 0.3 & 0.0 & 0.1 & 0.0 & 3.6 & 0.2 & 0.0 & 0.3 & 0.6 & 1.4 & 0.0 & 0.9 & 1.4 & 3.3 & 4.2 & 0.1 & 0.2 & 0.2 \\
& & 512 & 6.5 & 0.0 & 0.6 & 0.0 & 1.0 & 0.9 & 3.2 & 0.7 & 0.0 & 2.1 & 3.1 & 2.5 & 0.4 & 0.5 & 3.2 & 5.2 & 3.5 & 0.7 & 0.3 & 0.8 \\
N8 & N(0,1) & 128 & 5.9 & 5.0 & 23.5 & 4.4 & 6.0 & 4.3 & 17.7 & 11.5 & 4.9 & 5.5 & 6.3 & 22.4 & 11.6 & 3.0 & 4.2 & 7.0 & 23.4 & 12.3 & 0.0 & 0.7 \\
& & 256 & 6.3 & 5.7 & 31.9 & 5.8 & 6.6 & 4.0 & 26.2 & 19.1 & 5.6 & 6.4 & 6.7 & 30.8 & 24.1 & 4.3 & 5.6 & 6.9 & 31.9 & 23.2 & 0.0 & 1.2 \\
& & 512 & 7.1 & 5.5 & 37.2 & 5.7 & 6.4 & 4.1 & 30.6 & 25.4 & 7.9 & 8.8 & 7.7 & 33.0 & 27.7 & 8.0 & 8.5 & 7.4 & 35.5 & 28.5 & 0.5 & 2.4 \\
& Stand. $t_4$ & 128 & 6.0 & 3.6 & 13.0 & 4.0 & 4.4 & 4.3 & 11.2 & 8.0 & 4.6 & 5.8 & 4.9 & 16.5 & 7.5 & 3.2 & 4.9 & 6.0 & 15.6 & 7.9 & 0.0 & 0.7 \\
& & 256 & 6.8 & 4.3 & 19.5 & 3.4 & 5.7 & 2.8 & 14.4 & 10.4 & 6.2 & 7.1 & 6.6 & 20.2 & 12.3 & 5.2 & 6.1 & 7.7 & 21.5 & 12.3 & 0.0 & 1.5 \\
& & 512 & 8.2 & 4.4 & 23.6 & 4.1 & 6.3 & 3.1 & 19.6 & 15.2 & 6.0 & 8.0 & 7.3 & 21.3 & 15.9 & 6.2 & 7.9 & 8.3 & 24.4 & 16.0 & 0.2 & 2.2 \\
N9 & N(0,1) & 128 & 5.0 & 4.6 & 1.8 & 2.6 & 4.0 & 0.9 & 5.3 & 2.5 & 2.7 & 3.8 & 5.2 & 6.0 & 2.4 & 2.2 & 3.4 & 6.5 & 4.7 & 2.2 & 0.0 & 0.5 \\
& & 256 & 5.6 & 4.8 & 1.9 & 2.3 & 4.4 & 2.4 & 4.9 & 3.6 & 2.5 & 4.2 & 6.2 & 6.4 & 3.6 & 2.4 & 4.2 & 6.4 & 6.5 & 3.3 & 0.1 & 1.3 \\
& & 512 & 4.8 & 4.1 & 1.4 & 3.0 & 4.0 & 1.0 & 4.8 & 3.0 & 2.9 & 4.2 & 5.3 & 6.3 & 3.5 & 2.6 & 4.2 & 5.1 & 6.7 & 3.5 & 0.7 & 2.3 \\
& Stand. $t_4$ & 128 & 3.0 & 2.1 & 1.7 & 2.4 & 3.5 & 1.0 & 4.7 & 1.9 & 2.4 & 2.8 & 4.1 & 5.4 & 1.9 & 1.4 & 2.5 & 4.5 & 5.5 & 2.0 & 0.0 & 0.3 \\
& & 256 & 5.5 & 4.6 & 1.4 & 2.5 & 2.9 & 1.1 & 3.6 & 2.6 & 2.1 & 3.8 & 4.9 & 5.4 & 2.9 & 1.3 & 3.4 & 4.9 & 5.1 & 2.5 & 0.1 & 0.7 \\
& & 512 & 4.6 & 4.2 & 1.8 & 2.9 & 4.3 & 1.6 & 5.1 & 3.9 & 3.3 & 5.2 & 5.8 & 7.0 & 4.3 & 3.2 & 4.8 & 6.8 & 7.3 & 4.1 & 0.5 & 2.8 \\
N10 & N(0,1) & 128 & 5.9 & 5.9 & 3.9 & 3.7 & 5.2 & 4.0 & 5.2 & 4.5 & 3.8 & 5.8 & 6.5 & 5.7 & 4.7 & 3.1 & 5.9 & 7.1 & 5.1 & 4.4 & 0.2 & 0.8 \\
& & 256 & 6.6 & 6.8 & 3.3 & 3.5 & 5.1 & 3.0 & 5.1 & 4.7 & 3.3 & 5.9 & 6.8 & 6.2 & 5.2 & 3.9 & 5.3 & 7.5 & 5.3 & 5.2 & 0.6 & 1.6 \\
& & 512 & 7.4 & 6.7 & 3.8 & 4.5 & 6.6 & 4.0 & 5.4 & 6.8 & 5.0 & 7.2 & 7.8 & 6.1 & 6.7 & 5.4 & 6.6 & 8.6 & 6.1 & 6.3 & 3.1 & 3.6 \\
& Stand. $t_4$ & 128 & 7.0 & 7.8 & 1.6 & 4.0 & 6.1 & 4.8 & 4.6 & 6.1 & 4.4 & 5.9 & 7.7 & 5.5 & 5.4 & 4.0 & 5.9 & 9.4 & 4.1 & 5.5 & 0.1 & 1.7 \\
& & 256 & 6.9 & 5.3 & 2.0 & 3.6 & 6.1 & 3.9 & 3.8 & 4.1 & 4.0 & 5.9 & 7.9 & 2.7 & 4.1 & 4.8 & 6.6 & 8.2 & 3.0 & 4.5 & 0.4 & 2.3 \\
& & 512 & 6.7 & 5.8 & 2.2 & 3.4 & 5.6 & 3.6 & 3.3 & 4.0 & 5.0 & 6.1 & 6.4 & 3.7 & 4.3 & 5.3 & 6.1 & 7.2 & 3.0 & 4.4 & 4.2 & 4.0 \\
\hline \end{tabular} \endgroup \end{sidewaystable}
The empirical levels of the tests are reported in Tables~\ref{H0_1} and~\ref{H0_2} for $h \in \{2,3,4,8\}$ (for $h=8$, to save computing time, only the tests c and dc were carried out). As one can see, the rank-based tests d, c, dc and dcp of Section~\ref{sec:rank} never turned out, overall, to be too liberal (unlike their competitors considered in Section~\ref{sec:competitors} -- see Table~\ref{H0others}). Their analogues of Section~\ref{sec:sotests} focusing on second-order characteristics behave reasonably well except for Model N8. The latter is due to the fact that the test v is way too liberal for Model N8 as can be seen from Table~\ref{H0_2}, a probable consequence of the conditional heteroskedasticity of the model. For fixed $n$, as $h$ increases, the empirical autocopula test c (and thus the combined test dc) can be seen to be more and more conservative, as already mentioned in Section~\ref{sec:hchoice}. The latter clearly appears by considering the last vertical blocks of Tables~\ref{H0_1} and~\ref{H0_2} corresponding to $h=8$. Nonetheless, the rejection percentages therein hint at the fact that the empirical levels, as expected theoretically, should improve as $n$ increases further.
\subsection{Empirical powers}
The empirical powers of the proposed tests were estimated under Models A1--A12 from 1000 samples of size $n \in \{128,256,512\}$ for $h \in \{2,3,4,8\}$ (again, for $h=8$, only the tests c and dc were carried out). For Models A1--A8 (those that are connected to the literature on locally stationary processes), the rejection percentages of the null hypothesis of stationarity are reported in Table~\ref{H1}. As one can see, for $h \in \{2,3,4\}$, the rank-based combined tests of Section~\ref{sec:comb:cop:df} (dc and dcp) almost always seem to be more powerful than the combined tests of second-order stationarity that have been considered in Section~\ref{sec:sotests} (va and mva). Furthermore, the tests focusing on the contemporary distribution (d, m and v) hardly have any power overall, suggesting that the distribution of $X_t$ does not change (too) much for the models under consideration (note in passing the very disappointing behavior of the test m for Models A6--A8). The latter explains why the test c is more powerful than the combined tests dc and dcp, and why the test a is almost always more powerful than va and mva for $h=2$. Finally, note that, except for A8, the power of all the tests focusing on serial dependence decreases, overall, as $h$ increases (see also the discussion in Section~\ref{sec:hchoice}). At least for Models A1--A4, the latter is a consequence of the fact that the serial dependence is completely determined by the distribution of $(X_t,X_{t-1})$.
\begin{sidewaystable}[t!] \centering \caption{Percentages of rejection of the null hypothesis of stationarity computed from 1000 samples of size $n \in \{128, 256, 512\}$ generated from Models A1--A8. The meaning of the abbreviations d, m, v, c, dc, dcp, a, va and mva is given in Section~\ref{sec:MC}.} \label{H1} \begingroup\footnotesize \begin{tabular}{lrrrrrrrrrrrrrrrrrrrrr}
\hline
\multicolumn{5}{c}{} & \multicolumn{5}{c}{$h=2$ or lag 1} & \multicolumn{5}{c}{$h=3$ or lag 2} & \multicolumn{5}{c}{$h=4$ or lag 3} & \multicolumn{2}{c}{$h=8$} \\ \cmidrule(lr){6-10} \cmidrule(lr){11-15} \cmidrule(lr){16-20} \cmidrule(lr){21-22} Model & $n$ & d & m & v & c & dc & a & va & mva & c & dc & dcp & va & mva & c & dc & dcp & va & mva & c & dc \\ \hline A1 & 128 & 3.3 & 2.5 & 4.3 & 5.0 & 4.3 & 5.5 & 6.5 & 5.1 & 6.7 & 5.2 & 5.5 & 6.5 & 4.0 & 3.2 & 3.8 & 6.2 & 6.3 & 4.6 & 0.2 & 0.9 \\
& 256 & 5.5 & 5.0 & 5.2 & 11.4 & 10.5 & 12.9 & 12.9 & 10.4 & 13.2 & 11.0 & 9.1 & 11.8 & 8.8 & 6.9 & 6.7 & 9.1 & 10.9 & 8.5 & 0.2 & 1.2 \\
& 512 & 4.2 & 4.4 & 6.4 & 42.5 & 27.5 & 37.7 & 30.9 & 23.1 & 42.5 & 26.6 & 16.0 & 19.9 & 14.9 & 25.2 & 15.6 & 10.6 & 15.3 & 11.5 & 1.4 & 3.5 \\
A2 & 128 & 4.9 & 5.2 & 2.1 & 9.7 & 9.3 & 13.2 & 10.1 & 9.5 & 10.1 & 8.2 & 8.7 & 9.2 & 8.2 & 4.7 & 5.0 & 8.3 & 7.1 & 6.4 & 0.0 & 0.8 \\
& 256 & 6.1 & 6.3 & 3.9 & 36.1 & 26.2 & 34.9 & 26.2 & 22.2 & 31.9 & 23.5 & 15.0 & 16.9 & 14.6 & 17.2 & 14.3 & 12.1 & 14.4 & 11.4 & 0.9 & 2.1 \\
& 512 & 5.6 & 5.5 & 4.1 & 76.1 & 58.2 & 61.6 & 40.6 & 34.1 & 72.4 & 51.0 & 28.1 & 21.6 & 18.1 & 54.8 & 37.6 & 18.1 & 17.4 & 13.0 & 8.7 & 7.1 \\
A3 & 128 & 6.2 & 6.3 & 4.3 & 13.1 & 12.9 & 18.8 & 16.9 & 15.1 & 11.8 & 11.9 & 10.4 & 13.5 & 11.4 & 6.0 & 7.2 & 10.1 & 11.7 & 10.1 & 0.1 & 1.1 \\
& 256 & 5.8 & 5.4 & 4.9 & 42.1 & 30.6 & 40.7 & 28.3 & 23.6 & 36.1 & 25.2 & 17.6 & 18.7 & 14.3 & 19.5 & 16.1 & 13.8 & 15.1 & 11.0 & 1.4 & 2.3 \\
& 512 & 5.7 & 6.0 & 4.6 & 92.4 & 77.0 & 83.8 & 63.6 & 51.0 & 89.3 & 71.4 & 36.9 & 30.7 & 23.5 & 75.2 & 54.4 & 25.8 & 23.4 & 16.8 & 15.9 & 11.5 \\
A4 & 128 & 2.0 & 1.9 & 1.8 & 3.3 & 2.5 & 2.4 & 4.9 & 2.9 & 3.0 & 2.5 & 4.1 & 5.7 & 3.0 & 1.4 & 1.2 & 3.8 & 6.1 & 2.9 & 0.0 & 0.0 \\
& 256 & 4.0 & 2.0 & 1.7 & 3.4 & 3.4 & 2.2 & 5.1 & 2.9 & 3.3 & 3.2 & 4.3 & 6.1 & 3.0 & 1.6 & 1.9 & 4.8 & 5.4 & 2.5 & 0.0 & 0.2 \\
& 512 & 3.4 & 2.1 & 2.3 & 3.9 & 5.1 & 2.5 & 5.8 & 3.7 & 4.0 & 4.9 & 6.0 & 6.5 & 3.7 & 2.8 & 4.0 & 5.5 & 6.8 & 3.4 & 0.0 & 0.7 \\
A5 & 128 & 4.9 & 5.6 & 3.2 & 64.0 & 49.9 & 45.6 & 44.2 & 37.1 & 53.4 & 44.5 & 28.3 & 31.8 & 24.7 & 38.5 & 29.9 & 19.8 & 26.7 & 20.3 & 1.2 & 2.5 \\
& 256 & 5.6 & 5.4 & 4.0 & 84.5 & 75.5 & 60.4 & 54.4 & 43.6 & 77.6 & 68.2 & 46.2 & 40.9 & 29.9 & 63.6 & 52.1 & 35.8 & 36.6 & 25.6 & 12.1 & 9.5 \\
& 512 & 4.9 & 4.4 & 8.4 & 96.1 & 91.5 & 76.5 & 72.4 & 63.2 & 93.0 & 86.8 & 75.0 & 64.6 & 46.5 & 84.4 & 74.9 & 67.2 & 62.9 & 42.1 & 38.8 & 31.2 \\
A6 & 128 & 1.2 & 0.0 & 0.1 & 3.8 & 3.6 & 0.1 & 0.4 & 0.0 & 2.3 & 2.1 & 3.2 & 0.3 & 0.0 & 0.4 & 0.6 & 3.4 & 0.5 & 0.0 & 0.0 & 0.0 \\
& 256 & 7.1 & 0.0 & 0.3 & 4.6 & 8.2 & 0.3 & 1.4 & 0.0 & 3.9 & 6.9 & 10.4 & 1.6 & 0.0 & 0.7 & 2.7 & 10.5 & 1.9 & 0.0 & 0.0 & 0.4 \\
& 512 & 45.6 & 0.0 & 1.2 & 5.1 & 28.0 & 1.4 & 5.3 & 0.7 & 4.2 & 27.9 & 39.2 & 4.6 & 0.3 & 2.4 & 18.3 & 45.0 & 5.3 & 0.3 & 0.0 & 5.8 \\
A7 & 128 & 0.2 & 0.0 & 2.5 & 3.9 & 0.9 & 2.0 & 3.4 & 0.4 & 6.9 & 1.8 & 0.5 & 4.2 & 0.7 & 4.7 & 1.3 & 0.7 & 3.5 & 0.5 & 0.0 & 0.0 \\
& 256 & 0.8 & 0.0 & 1.4 & 6.5 & 1.8 & 2.2 & 2.3 & 0.2 & 6.4 & 1.8 & 1.8 & 2.8 & 0.3 & 4.9 & 1.8 & 2.2 & 2.7 & 0.2 & 0.0 & 0.1 \\
& 512 & 2.8 & 0.0 & 2.0 & 19.9 & 10.3 & 1.9 & 1.9 & 0.1 & 18.1 & 9.3 & 8.8 & 4.3 & 0.2 & 12.7 & 6.5 & 7.1 & 3.6 & 0.1 & 0.0 & 0.2 \\
A8 & 128 & 0.0 & 0.0 & 12.2 & 12.5 & 4.0 & 7.5 & 15.8 & 7.2 & 20.0 & 7.3 & 3.6 & 18.0 & 4.3 & 23.8 & 9.6 & 5.4 & 24.3 & 5.9 & 7.8 & 2.0 \\
& 256 & 0.0 & 0.0 & 12.1 & 30.4 & 11.4 & 11.3 & 19.9 & 6.4 & 39.3 & 21.7 & 16.5 & 25.6 & 6.2 & 45.3 & 27.8 & 21.3 & 34.9 & 10.4 & 42.4 & 20.7 \\
& 512 & 0.4 & 0.0 & 16.4 & 37.6 & 26.6 & 16.8 & 24.8 & 17.1 & 54.2 & 34.8 & 38.3 & 26.3 & 17.1 & 73.6 & 46.7 & 52.7 & 30.6 & 22.5 & 87.3 & 57.2 \\
\hline \end{tabular} \endgroup \end{sidewaystable}
The results of Table~\ref{H1} allow in principle for a direct comparison with the results reported in \cite{CarNas13} and \cite{DetPreVet11}. Since the tests available in \textsf{R} considered in \cite{CarNas13} and in Section~\ref{sec:competitors} are far from maintaining their levels, a comparison in terms of power with these tests is clearly not meaningful. As far as the tests of \cite{DetPreVet11} are concerned, they appear, overall, to be more powerful for Models A1--A4 (results for Models A5--A8 are not available in the latter reference). It is however unknown whether they hold their levels when applied to stationary heavy-tailed observations as only Gaussian time series were considered in the simulations of \cite{DetPreVet11}.
While Models A1--A8 considered thus far are connected to the literature on locally stationary processes, the remaining Models A9--A12 are more in line with the change-point literature. For the latter, all our tests (except m) turn out to display substantially more power. This should not come as a surprise given that the tests are based on the CUSUM approach and are hence designed to detect alternatives involving one single break.
Table~\ref{H1ar1} reports the empirical powers of the proposed tests for Model A9. Recall that both the contemporary distribution and the serial dependence is changing under this scenario (unless $\beta=0$). As one can see, even in this setting that should possibly be favorable to the tests focusing on second-order stationarity, the rank-based tests involving test c appear more powerful, overall, than those involving test a. Furthermore, with a few exceptions, the test c is always at least slightly more powerful than the combined test~dc. As expected given the data generating model and in line with the discussion of Section~\ref{sec:hchoice}, the increase of $h$ leads to a decrease in the power of c and dc. In addition, for $h \in \{3,4\}$, dcp is more powerful than dc, which can be explained by the fact that the serial dependence in the data generating model is solely of a bivariate nature.
\begin{sidewaystable}[t!] \centering \caption{Percentages of rejection of the null hypothesis of stationarity computed from 1000 samples of size $n \in \{128, 256, 512\}$ generated from Model A9 with $\beta \in \{-0.8, -0.4, 0, 0.4, 0.8\}$. The meaning of the abbreviations d, m, v, c, dc, dcp, a, va and mva is given in Section~\ref{sec:MC}.} \label{H1ar1} \begingroup\footnotesize \begin{tabular}{lrrrrrrrrrrrrrrrrrrrr}
\hline
\multicolumn{6}{c}{} & \multicolumn{5}{c}{$h=2$ or lag 1} & \multicolumn{5}{c}{$h=3$ or lag 2} & \multicolumn{5}{c}{$h=4$ or lag 3} \\ \cmidrule(lr){7-11} \cmidrule(lr){12-16} \cmidrule(lr){17-21} Innov. & $n$ & $\beta$ & d & m & v & c & dc & a & va & mva & c & dc & dcp & va & mva & c & dc & dcp & va & mva \\ \hline N(0,1) & 128 & -0.8 & 10.0 & 1.6 & 14.4 & 61.2 & 59.5 & 42.1 & 52.5 & 37.7 & 49.8 & 52.9 & 58.8 & 56.2 & 36.4 & 20.6 & 25.8 & 49.6 & 54.3 & 33.0 \\
& & -0.4 & 4.6 & 4.8 & 4.6 & 32.0 & 22.6 & 34.2 & 28.2 & 20.2 & 26.3 & 18.7 & 14.3 & 21.5 & 13.3 & 14.9 & 10.4 & 11.5 & 15.9 & 9.8 \\
& & 0.0 & 3.9 & 3.1 & 3.8 & 3.6 & 4.2 & 3.9 & 4.2 & 3.8 & 5.3 & 5.1 & 5.2 & 4.6 & 2.7 & 3.8 & 3.7 & 5.9 & 4.3 & 2.7 \\
& & 0.4 & 8.6 & 8.3 & 3.9 & 28.0 & 26.6 & 31.2 & 24.1 & 20.8 & 27.9 & 25.2 & 19.0 & 15.5 & 12.2 & 17.8 & 16.7 & 16.4 & 12.5 & 9.8 \\
& & 0.8 & 6.7 & 6.1 & 12.7 & 52.8 & 45.2 & 39.5 & 47.5 & 32.4 & 49.2 & 43.2 & 41.2 & 50.9 & 32.1 & 39.6 & 34.4 & 33.4 & 45.9 & 27.8 \\
& 256 & -0.8 & 40.9 & 2.5 & 46.1 & 95.5 & 97.0 & 74.2 & 81.8 & 69.5 & 93.5 & 95.0 & 96.6 & 83.1 & 70.7 & 69.4 & 84.8 & 95.2 & 82.8 & 67.6 \\
& & -0.4 & 5.7 & 4.6 & 10.2 & 66.4 & 52.8 & 69.0 & 60.3 & 48.6 & 57.3 & 44.7 & 34.9 & 45.0 & 33.0 & 35.4 & 28.2 & 23.3 & 32.1 & 22.3 \\
& & 0.0 & 3.7 & 4.0 & 4.6 & 4.4 & 3.3 & 4.7 & 4.1 & 4.1 & 6.2 & 5.1 & 4.6 & 4.9 & 4.8 & 5.3 & 4.2 & 4.8 & 5.1 & 4.8 \\
& & 0.4 & 6.8 & 6.6 & 6.6 & 63.6 & 54.9 & 69.8 & 57.3 & 48.4 & 60.8 & 51.7 & 37.6 & 39.7 & 28.7 & 50.6 & 42.3 & 25.1 & 27.3 & 19.5 \\
& & 0.8 & 10.0 & 6.9 & 37.6 & 89.4 & 85.0 & 74.7 & 81.2 & 68.2 & 86.9 & 82.2 & 80.8 & 84.0 & 67.0 & 80.5 & 76.5 & 72.1 & 82.0 & 63.3 \\
& 512 & -0.8 & 87.7 & 2.0 & 85.9 & 100.0 & 100.0 & 95.2 & 97.7 & 93.9 & 100.0 & 100.0 & 100.0 & 98.3 & 94.7 & 99.2 & 99.9 & 99.9 & 98.4 & 93.7 \\
& & -0.4 & 6.8 & 5.6 & 17.3 & 95.7 & 89.8 & 98.1 & 93.8 & 88.6 & 89.6 & 83.9 & 75.8 & 81.4 & 67.2 & 75.2 & 62.9 & 48.8 & 63.2 & 48.4 \\
& & 0.0 & 4.2 & 4.5 & 4.2 & 5.2 & 5.2 & 5.0 & 4.1 & 4.0 & 6.5 & 5.6 & 4.9 & 4.2 & 4.7 & 6.3 & 4.9 & 5.1 & 4.6 & 4.9 \\
& & 0.4 & 7.0 & 5.5 & 12.8 & 95.0 & 89.7 & 96.2 & 90.5 & 86.4 & 91.9 & 86.4 & 74.9 & 77.8 & 63.4 & 86.8 & 78.2 & 50.5 & 60.0 & 48.0 \\
& & 0.8 & 11.2 & 6.8 & 84.2 & 99.7 & 99.7 & 96.1 & 98.5 & 94.3 & 99.6 & 99.5 & 99.5 & 99.1 & 94.4 & 99.0 & 98.3 & 98.8 & 98.6 & 93.7 \\
St.\ $t_4$ & 128 & -0.8 & 15.8 & 1.0 & 7.7 & 61.5 & 67.1 & 34.4 & 37.6 & 23.5 & 51.5 & 59.9 & 67.7 & 42.5 & 23.5 & 21.8 & 35.1 & 60.3 & 41.0 & 22.1 \\
& & -0.4 & 4.4 & 2.5 & 2.6 & 38.3 & 27.8 & 25.9 & 21.0 & 11.2 & 33.5 & 23.5 & 16.4 & 13.2 & 6.1 & 15.9 & 11.7 & 12.0 & 10.6 & 4.3 \\
& & 0.0 & 5.1 & 3.6 & 1.2 & 3.2 & 4.6 & 4.4 & 4.1 & 2.8 & 5.8 & 3.9 & 5.3 & 3.9 & 1.9 & 3.9 & 3.2 & 6.1 & 3.5 & 1.7 \\
& & 0.4 & 9.4 & 7.0 & 2.1 & 37.3 & 32.0 & 26.1 & 19.8 & 14.2 & 32.0 & 29.1 & 23.8 & 12.5 & 8.9 & 22.3 & 21.2 & 21.0 & 10.6 & 6.7 \\
& & 0.8 & 9.4 & 7.4 & 7.9 & 56.3 & 52.9 & 33.5 & 37.2 & 27.3 & 50.2 & 48.0 & 47.2 & 39.4 & 25.2 & 39.2 & 38.9 & 39.7 & 36.8 & 21.1 \\
& 256 & -0.8 & 56.3 & 2.2 & 26.6 & 96.1 & 98.5 & 65.3 & 69.8 & 54.7 & 95.8 & 98.6 & 97.7 & 73.4 & 56.2 & 73.3 & 92.2 & 96.9 & 72.1 & 53.0 \\
& & -0.4 & 5.4 & 3.0 & 3.9 & 74.5 & 61.4 & 58.9 & 44.6 & 31.9 & 65.2 & 54.4 & 43.3 & 30.3 & 18.1 & 41.5 & 32.4 & 28.0 & 21.5 & 12.3 \\
& & 0.0 & 4.7 & 4.8 & 1.7 & 4.8 & 4.1 & 4.8 & 3.5 & 3.2 & 6.9 & 6.4 & 5.1 & 2.2 & 2.5 & 6.6 & 6.2 & 5.3 & 2.3 & 2.4 \\
& & 0.4 & 6.4 & 5.6 & 3.2 & 71.5 & 62.9 & 58.1 & 45.3 & 35.6 & 67.7 & 57.8 & 44.4 & 27.8 & 19.4 & 56.5 & 47.1 & 30.6 & 18.1 & 13.2 \\
& & 0.8 & 11.2 & 6.1 & 20.6 & 92.4 & 87.6 & 62.1 & 65.6 & 51.2 & 89.2 & 86.1 & 84.2 & 68.7 & 51.5 & 83.9 & 81.0 & 77.0 & 66.0 & 47.5 \\
& 512 & -0.8 & 96.1 & 2.8 & 58.4 & 100.0 & 100.0 & 88.4 & 92.3 & 83.9 & 100.0 & 100.0 & 100.0 & 93.9 & 84.6 & 99.9 & 100.0 & 100.0 & 93.8 & 84.0 \\
& & -0.4 & 8.8 & 4.7 & 5.1 & 98.6 & 96.2 & 89.3 & 81.5 & 69.7 & 95.3 & 91.3 & 85.2 & 65.7 & 46.4 & 84.6 & 74.8 & 60.0 & 46.3 & 27.3 \\
& & 0.0 & 4.9 & 3.3 & 1.6 & 5.1 & 5.6 & 4.3 & 3.9 & 5.1 & 6.7 & 6.2 & 5.4 & 3.9 & 3.4 & 7.2 & 6.2 & 6.0 & 2.4 & 2.3 \\
& & 0.4 & 7.0 & 5.4 & 4.9 & 98.1 & 95.8 & 87.8 & 79.3 & 69.7 & 97.2 & 92.8 & 81.4 & 62.6 & 46.1 & 93.1 & 85.4 & 59.5 & 40.8 & 28.1 \\
& & 0.8 & 14.9 & 5.7 & 57.1 & 99.9 & 100.0 & 87.5 & 89.7 & 82.5 & 99.7 & 99.6 & 99.9 & 91.7 & 83.1 & 99.7 & 99.4 & 99.5 & 91.9 & 82.2 \\
\hline \end{tabular} \endgroup \end{sidewaystable}
The rejection percentages for Model A10 are reported in Table~\ref{H1ar2}. As expected, the empirical powers of tests c and a are very low for $h=2$ since there is no relationship between $X_t$ and $X_{t-1}$. The tests focusing on the contemporary distribution are more powerful, in particular the test v. Consequently, the combined tests at lag 1 involving v do display some power. For $h \in \{3,4\}$, the two most powerful tests are dcp and va. The fact that dcp is more powerful than dc can again be explained by the bivariate nature of the serial dependence.
\begin{sidewaystable}[t!] \centering \caption{Percentages of rejection of the null hypothesis of stationarity computed from 1000 samples of size $n \in \{128, 256, 512\}$ generated from Model A10 with $\beta \in \{-0.8, -0.4, 0, 0.4, 0.8\}$. The meaning of the abbreviations d, m, v, c, dc, dcp, a, va and mva is given in Section~\ref{sec:MC}.} \label{H1ar2} \begingroup\footnotesize \begin{tabular}{lrrrrrrrrrrrrrrrrrrrr}
\hline
\multicolumn{6}{c}{} & \multicolumn{5}{c}{$h=2$ or lag 1} & \multicolumn{5}{c}{$h=3$ or lag 2} & \multicolumn{5}{c}{$h=4$ or lag 3} \\ \cmidrule(lr){7-11} \cmidrule(lr){12-16} \cmidrule(lr){17-21} Innov. & $n$ & $\beta$ & d & m & v & c & dc & a & va & mva & c & dc & dcp & va & mva & c & dc & dcp & va & mva \\ \hline N(0,1) & 128 & -0.8 & 2.6 & 0.4 & 11.6 & 1.5 & 3.7 & 2.8 & 10.2 & 4.4 & 16.1 & 21.9 & 26.7 & 35.4 & 15.3 & 11.3 & 16.2 & 20.0 & 29.9 & 11.0 \\
& & -0.4 & 2.4 & 2.5 & 7.0 & 3.7 & 2.6 & 2.3 & 4.2 & 2.6 & 9.3 & 5.6 & 9.3 & 19.4 & 9.8 & 8.7 & 4.9 & 6.2 & 15.5 & 7.2 \\
& & 0.0 & 3.9 & 3.1 & 3.8 & 3.6 & 4.2 & 3.9 & 4.2 & 3.8 & 5.3 & 5.1 & 5.2 & 4.6 & 2.7 & 3.8 & 3.7 & 5.9 & 4.3 & 2.7 \\
& & 0.4 & 8.8 & 11.3 & 4.3 & 5.2 & 9.1 & 6.2 & 6.2 & 7.8 & 9.0 & 11.3 & 21.0 & 19.0 & 15.8 & 7.4 & 10.1 & 17.3 & 14.8 & 12.7 \\
& & 0.8 & 7.3 & 8.6 & 11.3 & 1.5 & 5.6 & 4.1 & 14.1 & 13.4 & 2.5 & 5.6 & 25.2 & 38.7 & 27.4 & 2.8 & 5.0 & 22.7 & 36.7 & 23.9 \\
& 256 & -0.8 & 28.8 & 0.8 & 33.9 & 1.2 & 13.0 & 1.3 & 18.5 & 10.0 & 67.5 & 80.8 & 74.7 & 63.8 & 38.0 & 52.5 & 73.3 & 61.4 & 56.6 & 30.9 \\
& & -0.4 & 4.0 & 3.2 & 8.8 & 3.4 & 4.2 & 3.2 & 6.4 & 4.3 & 21.6 & 15.4 & 24.9 & 39.0 & 25.3 & 20.4 & 13.9 & 15.8 & 27.5 & 18.1 \\
& & 0.0 & 3.7 & 4.0 & 4.6 & 4.4 & 3.3 & 4.7 & 4.1 & 4.1 & 6.2 & 5.1 & 4.6 & 4.9 & 4.8 & 5.3 & 4.2 & 4.8 & 5.1 & 4.8 \\
& & 0.4 & 9.1 & 9.2 & 8.9 & 4.7 & 7.9 & 6.0 & 9.2 & 10.8 & 11.6 & 15.6 & 33.2 & 41.5 & 35.8 & 14.5 & 17.1 & 25.6 & 30.2 & 26.0 \\
& & 0.8 & 10.4 & 7.6 & 36.2 & 1.1 & 6.1 & 3.2 & 27.2 & 21.2 & 4.4 & 8.4 & 41.8 & 71.0 & 53.3 & 5.8 & 11.1 & 31.1 & 66.1 & 45.2 \\
& 512 & -0.8 & 81.3 & 1.5 & 78.2 & 2.9 & 56.0 & 1.9 & 53.0 & 35.4 & 99.5 & 99.8 & 98.7 & 90.7 & 75.7 & 97.4 & 99.5 & 97.0 & 88.1 & 69.3 \\
& & -0.4 & 4.1 & 3.6 & 17.4 & 3.7 & 3.0 & 4.4 & 10.9 & 7.7 & 47.8 & 35.3 & 58.9 & 74.0 & 51.7 & 49.0 & 35.3 & 32.7 & 53.4 & 35.1 \\
& & 0.0 & 4.2 & 4.5 & 4.2 & 5.2 & 5.2 & 5.0 & 4.1 & 4.0 & 6.5 & 5.6 & 4.9 & 4.2 & 4.7 & 6.3 & 4.9 & 5.1 & 4.6 & 4.9 \\
& & 0.4 & 8.0 & 7.1 & 15.8 & 4.2 & 7.8 & 5.3 & 11.7 & 11.8 & 23.7 & 20.7 & 62.5 & 72.2 & 57.6 & 32.4 & 25.9 & 36.0 & 51.9 & 40.0 \\
& & 0.8 & 17.1 & 8.1 & 77.7 & 1.4 & 8.3 & 3.8 & 60.4 & 46.3 & 16.2 & 27.7 & 83.2 & 93.8 & 84.2 & 27.8 & 40.3 & 58.5 & 91.9 & 78.3 \\
St.\ $t_4$ & 128 & -0.8 & 5.7 & 0.4 & 6.9 & 1.4 & 5.1 & 4.4 & 7.6 & 4.5 & 17.0 & 29.5 & 36.0 & 26.6 & 10.2 & 11.5 & 21.6 & 28.1 & 21.7 & 6.5 \\
& & -0.4 & 2.0 & 1.9 & 2.5 & 2.8 & 1.8 & 4.2 & 4.3 & 3.2 & 13.4 & 6.7 & 12.2 & 14.4 & 6.9 & 10.1 & 6.6 & 8.1 & 11.1 & 4.5 \\
& & 0.0 & 5.1 & 3.6 & 1.2 & 3.2 & 4.6 & 4.4 & 4.1 & 2.8 & 5.8 & 3.9 & 5.3 & 3.9 & 1.9 & 3.9 & 3.2 & 6.1 & 3.5 & 1.7 \\
& & 0.4 & 11.8 & 10.3 & 2.8 & 5.5 & 9.8 & 7.4 & 7.7 & 8.4 & 8.0 & 11.6 & 23.9 & 15.9 & 14.9 & 8.2 & 10.9 & 21.5 & 11.6 & 11.5 \\
& & 0.8 & 9.9 & 7.9 & 8.5 & 1.6 & 7.1 & 8.4 & 16.0 & 14.5 & 1.9 & 6.4 & 30.3 & 35.1 & 22.7 & 1.6 & 5.4 & 26.1 & 33.4 & 22.4 \\
& 256 & -0.8 & 44.7 & 0.2 & 19.7 & 0.6 & 20.1 & 3.2 & 12.5 & 5.9 & 69.4 & 88.5 & 83.0 & 46.8 & 24.9 & 55.7 & 82.3 & 74.3 & 40.4 & 18.4 \\
& & -0.4 & 3.0 & 1.9 & 4.3 & 3.9 & 3.0 & 2.9 & 4.5 & 2.8 & 23.3 & 16.0 & 28.8 & 27.8 & 15.1 & 22.9 & 14.9 & 16.3 & 17.8 & 7.9 \\
& & 0.0 & 4.7 & 4.8 & 1.7 & 4.8 & 4.1 & 4.8 & 3.5 & 3.2 & 6.9 & 6.4 & 5.1 & 2.2 & 2.5 & 6.6 & 6.2 & 5.3 & 2.3 & 2.4 \\
& & 0.4 & 10.0 & 9.1 & 2.9 & 4.4 & 9.1 & 5.6 & 6.5 & 7.8 & 11.0 & 15.3 & 41.3 & 28.0 & 21.0 & 14.8 & 17.6 & 28.9 & 18.2 & 15.7 \\
& & 0.8 & 13.7 & 7.9 & 20.3 & 1.3 & 6.7 & 5.1 & 18.5 & 14.3 & 3.0 & 8.3 & 50.1 & 53.4 & 39.2 & 4.7 & 11.7 & 35.8 & 49.0 & 32.5 \\
& 512 & -0.8 & 92.3 & 2.1 & 52.5 & 2.9 & 70.6 & 2.1 & 29.2 & 18.8 & 99.8 & 100.0 & 99.8 & 77.7 & 55.1 & 98.3 & 99.7 & 98.6 & 72.0 & 46.4 \\
& & -0.4 & 7.5 & 2.6 & 8.1 & 4.3 & 6.0 & 3.3 & 5.8 & 4.9 & 60.1 & 49.9 & 69.8 & 55.7 & 32.6 & 57.6 & 47.5 & 41.9 & 33.5 & 18.9 \\
& & 0.0 & 4.9 & 3.3 & 1.6 & 5.1 & 5.6 & 4.3 & 3.9 & 5.1 & 6.7 & 6.2 & 5.4 & 3.9 & 3.4 & 7.2 & 6.2 & 6.0 & 2.4 & 2.3 \\
& & 0.4 & 10.5 & 8.3 & 6.5 & 6.0 & 8.2 & 5.4 & 6.5 & 6.6 & 27.1 & 28.2 & 74.4 & 53.6 & 39.7 & 36.0 & 33.6 & 49.4 & 33.0 & 25.7 \\
& & 0.8 & 19.1 & 6.7 & 51.8 & 1.1 & 9.7 & 4.3 & 37.7 & 27.8 & 14.2 & 28.3 & 86.5 & 80.7 & 64.6 & 27.1 & 42.7 & 64.2 & 75.0 & 57.7 \\
\hline \end{tabular} \endgroup \end{sidewaystable}
Finally, the rejection percentages for Models A11 and A12 are given in Table~\ref{H1otherAR}. The columns c2 and c3 report the results for the bivariate analogues of the tests based on $S_{n,C^{\scriptscriptstyle (2)}}$ defined by~\eqref{eq:SnCh} for lags 2 and 3 (these tests arise in the combined test dcp). To save computing time, we did not include the tests of second-order stationarity as these were found less powerful, overall, in the previous experiments (for Models A12, moments do not exist, whence an application would not even be meaningful). Comparing the results for Model A11 with those of Table~\ref{H1ar1} for the same values of $h$ reveals, as expected, a higher power of the test c. In addition, the test c for lag 1 is more powerful than the test c2, which, in turn, is more powerful than the test c3, a consequence of the data generating models. Finally, the fact that the test d displays some power for $\beta = 0.8$ seems to be only a consequence of the sample sizes under consideration and the very strong serial dependence in the second half of the observations.
\begin{table}[t!] \centering \caption{Percentages of rejection of the null hypothesis of stationarity computed from 1000 samples of size $n \in \{128, 256, 512\}$ generated from Models A11 and A12 with $\beta \in \{0, 0.4, 0.8\}$. The meaning of the abbreviations d, c, dc, dcp is given in Section~\ref{sec:MC}. The columns c2 and c3 report the results for the bivariate analogues of the test based on $S_{n,C^{\scriptscriptstyle (2)}}$ defined by~\eqref{eq:SnCh} for lags 2 and 3 (these tests arise in the combined test dcp).} \label{H1otherAR} \begingroup\footnotesize \begin{tabular}{lrrrrrrrrrrrrr}
\hline
\multicolumn{4}{c}{} & \multicolumn{2}{c}{$h=2$ or lag 1} & \multicolumn{4}{c}{$h=3$ or lag 2} & \multicolumn{4}{c}{$h=4$ or lag 3} \\ \cmidrule(lr){5-6} \cmidrule(lr){7-10} \cmidrule(lr){11-14} Model & $n$ & $\beta$ & d & c & dc & c & dc & c2 & dcp & c & dc & c3 & dcp \\ \hline A11 & 128 & 0.0 & 4.3 & 4.9 & 4.5 & 4.8 & 3.8 & 5.2 & 5.1 & 4.1 & 3.2 & 6.4 & 5.1 \\
& & 0.4 & 8.3 & 69.3 & 60.7 & 60.6 & 53.3 & 8.4 & 41.8 & 48.6 & 40.4 & 7.1 & 30.3 \\
& & 0.8 & 22.1 & 91.9 & 90.4 & 86.4 & 84.6 & 80.0 & 91.2 & 80.8 & 79.5 & 55.5 & 88.6 \\
& 256 & 0.0 & 4.3 & 3.8 & 4.3 & 6.3 & 5.9 & 4.8 & 5.0 & 5.9 & 4.9 & 5.6 & 5.5 \\
& & 0.4 & 6.4 & 99.5 & 96.1 & 96.9 & 91.2 & 20.0 & 78.4 & 92.3 & 84.5 & 6.2 & 54.7 \\
& & 0.8 & 25.3 & 99.3 & 98.5 & 96.4 & 94.9 & 95.8 & 98.6 & 92.4 & 90.4 & 85.9 & 96.8 \\
& 512 & 0.0 & 4.6 & 6.6 & 5.9 & 7.0 & 6.5 & 4.2 & 5.2 & 5.6 & 6.2 & 5.0 & 5.0 \\
& & 0.4 & 6.4 & 100.0 & 100.0 & 99.9 & 100.0 & 56.7 & 99.2 & 99.8 & 99.6 & 10.7 & 83.6 \\
& & 0.8 & 22.1 & 100.0 & 100.0 & 100.0 & 100.0 & 100.0 & 100.0 & 99.9 & 99.7 & 99.7 & 100.0 \\
A12 & 128 & 0.0 & 5.6 & 3.7 & 4.7 & 4.9 & 4.9 & 4.4 & 5.9 & 3.3 & 4.1 & 6.3 & 7.4 \\
& & 0.4 & 6.5 & 30.8 & 24.9 & 31.3 & 25.5 & 6.8 & 18.7 & 23.5 & 20.1 & 6.5 & 14.3 \\
& & 0.8 & 10.0 & 69.9 & 60.0 & 67.6 & 61.1 & 32.8 & 55.8 & 62.0 & 57.8 & 15.3 & 48.1 \\
& 256 & 0.0 & 4.5 & 4.4 & 3.7 & 6.8 & 5.2 & 5.9 & 4.2 & 5.1 & 4.8 & 5.5 & 4.7 \\
& & 0.4 & 7.1 & 67.0 & 55.2 & 66.3 & 53.7 & 11.1 & 35.6 & 54.8 & 42.5 & 6.0 & 24.0 \\
& & 0.8 & 7.9 & 97.8 & 93.5 & 97.1 & 92.9 & 74.3 & 91.5 & 95.3 & 91.2 & 41.0 & 85.0 \\
& 512 & 0.0 & 4.8 & 4.7 & 4.6 & 8.6 & 7.5 & 4.7 & 4.9 & 7.4 & 6.9 & 4.4 & 4.6 \\
& & 0.4 & 6.5 & 96.0 & 90.8 & 94.3 & 88.5 & 22.4 & 75.5 & 89.4 & 83.1 & 8.2 & 51.6 \\
& & 0.8 & 7.2 & 100.0 & 99.7 & 100.0 & 99.6 & 99.3 & 99.6 & 99.9 & 99.2 & 89.2 & 99.6 \\
\hline \end{tabular} \endgroup \end{table}
\end{document}
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arXiv
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\begin{document}
\title{Vertex Algebras and the Equivariant Lie Algebroid Cohomology}
\begin{abstract} A vertex-algebraic analogue of the Lie algebroid complex is constructed, which generalizes the ``small" chiral de Rham complex on smooth manifolds. The notion of VSA-inductive sheaves is also introduced. This notion generalizes that of sheaves of vertex superalgebras. The complex mentioned above is constructed as a VSA-inductive sheaf. With this complex, the equivariant Lie algebroid cohomology is generalized to a vertex-algebraic analogue, which we call the chiral equivariant Lie algebroid cohomology. In fact, the notion of the equivariant Lie algebroid cohomology contains that of the equivariant Poisson cohomology. Thus the chiral equivariant Lie algebroid cohomology is also a vertex-algebraic generalization of the equivariant Poisson cohomology. A special kind of complex is introduced and its properties are studied in detail. With these properties, some isomorphisms of cohomologies are developed, which enables us to compute the chiral equivariant Lie algebroid cohomology in some cases. Poisson-Lie groups are considered as such a special case. \end{abstract}
\tableofcontents
\section{Introduction}\label{section: Introduction} The \textit{chiral de Rham complex} (\textit{CDR}) was introduced by Malikov-Schechtman-Vaintrob in \cite{MSV99}. It is a sheaf of vertex superalgebras containing the usual de Rham complex. Gorbounov-Malikov-Schechtman generalized this notion by introducing the sheaf of \textit{chiral differential operators}, and studied the sheaf in a series of papers \cite{GMS00,GMS03,GMS04}. Another construction of the CDR was done by means of formal loop spaces in \cite{KV04} by Kapranov-Vasserot. Moreover the CDR was studied in relation to elliptic genera and mirror symmetry in \cite{BL02,Bo01,BL00}. Recently Lian-Linshaw introduced a new equivariant cohomology theory in \cite{LL}, and studied in detail the CDR in the $C^\infty$-setting. The CDR was also investigated in terms of \textit{SUSY vertex algebras} in \cite{BZHS08,EHKZ13,Hel09,HK07,HZ10,HZ11}.
Let $G$ be a compact connected Lie group with complexified Lie algebra $\mathfrak{g}$ and let $M$ be a $G$-manifold. Then the algebra $\Omega(M)$ of differential forms on $M$ has a canonical $G$-action. Together with the Lie derivatives and the interior products, this action of $G$ makes $\Omega(M)$ a $G^*$-\textit{algebra}. It is well- known that the equivariant cohomology of $M$ is computed as the equivariant cohomology of the $G^*$-algebra $\Omega(M)$ (see \cite{GS99}). As a vertex-algebraic analogue of this equivariant cohomology theory, the \textit{chiral equivariant cohomology} was introduced by Lian-Linshaw in \cite{LL}. This cohomology was defined for $O(\mathfrak{sg})$-\textit{algebras}, a vertex-algebraic analogue of $G^*$-algebras. The key to the construction is the fact that the \textit{semi-infinite Weil complex} $\mathcal{W}(\mathfrak{g})$ introduced by Feigin-Frenkel in \cite{FF91} has an $O(\mathfrak{sg})$-algebra structure. This fact was proved by Lian-Linshaw together with the fact that the space $\mathcal{Q}(M)$ of global sections of the CDR of $M$ has an $O(\mathfrak{sg})$-algebra structure. Later in \cite{LLS1}, Lian-Linshaw-Song introduced the notion of $\mathfrak{sg}[t]$-\textit{modules} as an analogue of $\mathfrak{g}$-differential complexes, a complex with a compatible action of the Lie superalgebra $\mathfrak{sg}=\mathfrak{g}\ltimes_{\mathrm{ad}} \mathfrak{g}$. As a classical equivariant cohomology theory, the construction of the chiral equivariant cohomology was generalized for the case of $\mathfrak{sg}[t]$-\textit{modules}. Moreover in \cite{LLS1}, Lian-Linshaw-Song also introduced a ``small" CDR as a subcomplex of the CDR, and pointed out that the global section of the subcomplex is an $\mathfrak{sg}[t]$-module. The small CDR itself is trivial since its vertex superalgebra structure is commutative. However when one consider the corresponding chiral equivariant cohomology, the vertex superalgebra structure is very complicated in general. Such a vertex superalgebra were studied in \cite{LLS1,LLS2}.
An interesting example of $\mathfrak{g}$-differential complexes was considered in \cite{Gin99} by Ginzburg. This example comes from the space of multi-vector fields of a Poisson manifold with an action of $\mathfrak{g}$ and the corresponding equivariant cohomology is called the \textit{equivariant Poisson cohomology}. He also pointed out that one can define the same kind of equivariant cohomology for Lie algebroids.
In this paper, we construct $\mathfrak{sg}[t]$-modules from Lie algebroids with an action of $\mathfrak{g}$, generalizing the small CDR of Lian-Linshaw-Song. We then define the chiral equivariant Lie algebroid cohomology.
The CDR as well as the small version constructed by Lian-Linshaw on smooth manifolds is actually not a sheaf but a presheaf with a property, called a \textit{weak sheaf} by Lian-Linshaw-Song. For this reason, one has a little ambiguity in the choice of morphisms or the gluing properties. Therefore we introduce the notion of VSA-inductive sheaves, which generalize that of sheaves of vertex superalgebras, and formulate the morphisms and the gluing properties. We construct a VSA-inductive sheaf associated with a Lie algebroid and we obtain vertex-algebraic analogue of the Lie algebroid complex. Moreover we prove that the complex above has an $\mathfrak{sg}[t]$-module structure when the Lie algebroid has an action of $\mathfrak{g}$. This leads us to the definition of the chiral equivariant Lie algebroid cohomology. When the Lie algebroid is a tangent bundle, we recover the CDR of Lian-Linshaw-Song.
In the classical equivariant cohomology theory, an important role is played by special complexes called $W^*$-\textit{modules}. They have remarkable properties, which make it easy to compute their equivariant cohomologies (see \cite{GS99}). Motivated by this fact, we introduce the notion of chiral $W^*$-modules and prove that they have some properties analogous to those of $W^*$-modules. Moreover we prove that the complexes obtained from the VSA-inductive sheaves associated with a type of Lie algebroid containing the \textit{cotangent Lie algebroids} of Poisson-Lie groups have chiral $W^*$-module structures. We then compute their chiral equivariant Lie algebroid cohomologies by using the properties of chiral $W^*$-modules mentioned above.
The article is organized as follows: in Section 2 we recall some basics of vertex superalgebras and the chiral equivariant cohomology. In Section 3, we introduce the chiral $W^*$-modules and a chiral Cartan model for $\mathfrak{sg[t]}$-modules. We mainly consider this chiral Cartan model when $\mathfrak{g}$ is commutative. The result in this section will be used in the last part of Section 6. In Section 4, we introduce the notion of VSA-inductive sheaves. Then we establish some gluing properties. Moreover we construct VSA-inductive sheaves from presheaves of degree-weight-graded vertex superalgebras with some properties.
In Section 5, after recalling the notion of Lie algebroids and the Lie algebroid cohomology, we first construct an important VSA-inductive sheaf on $\mathbb{R}^m$ and its small version, which we denote respectively by $\Omega_\mathrm{ch}(\mathbb{R}^{m|r})$ and $\Omega^{\gamma c}_\mathrm{ch}(\mathbb{R}^{m|r})$. Using the gluing property proved in Section 4, we glue the small ones $\Omega^{\gamma c}_\mathrm{ch}(\mathbb{R}^{m|r})$ into the global VSA-inductive sheaf associated with an arbitrary vector bundle. Next for a Lie algebroid, we construct a differential on the VSA-inductive sheaf associated with the Lie algebroid, using the vertex operators of the bigger one $\Omega_\mathrm{ch}(\mathbb{R}^{m|r})$. Thus we obtain a vertex-algebraic analogue of the Lie algebroid complex. Moreover in Section 6, we equip this complex with an $\mathfrak{sg}[t]$-module structure, when the Lie algebroid has an action of a Lie algebra $\mathfrak{g}$. For this construction, we also need the vertex operators of the bigger VSA-inductive sheaf. Then we introduce the chiral equivariant Lie algebroid cohomology. In the last part of Section 6, we compute this cohomology for an important Lie algebroid called a \textit{transformation Lie algebroid}. In particular, we compute that cohomology for the cotangent Lie algebroids associated with \textit{Poisson-Lie groups}.
Throughout this paper, $\mathbb{K}$ is the field of real numbers $\mathbb{R}$ or that of complex numbers $\mathbb{C}$, and we will work over $\mathbb{K}$. We assume that a grading on a super vector space is compatible with the super vector space structure.
\section{Preliminaries}\label{section: Preliminaries} \subsection{Vertex Superalgebras}
We first recall basic definitions and facts concerning vertex superalgebras, which was introduced in \cite{Bor86}. We will follow the formalism and results in \cite{FBZ04,Kac01,LL04}.
A \textit{vertex superalgebra} is a quadruple $(V, \mathbf{1},T,Y)$ consisting of a super vector space $V,$ an even vector $\mathbf{1} \in V$, called the vacuum vector, an even linear operator $T : V \to V$, called the translation operator, and an even linear operation $ Y=Y(\, \cdot \, , z) : V \to (\mathrm{End} V)[[z^{\pm1}]], $ taking each $A \in V$ to a field on $V$, called the vertex operator, $$ Y(A,z)= \sum_{n \in \mathbb{Z}} A_{(n)} z^{-n-1}, $$ such that \begin{enumerate}[$\bullet$]
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\setlength{\itemindent}{0pt} \item (vacuum axiom)\\ $Y(\mathbf{1} ,z) = \mathrm{id}_V$;\\ $ Y(A,z)\mathbf{1} \in V[[z]], $ and $
Y(A,z)\mathbf{1} | _{z=0} = A $ \quad for any $A \in V$; \item (translation axiom)\\ $T \mathbf{1} = 0$;\\ $ [T,Y(A,z)] = \frac{\mathit{d}}{\mathit{d}z} Y(A,z) $ \quad for any $A \in V$; \item (locality axiom)\\ For any $A, B \in V,$ $Y(A,z)$ and $Y(B,z)$ are mutually local. \end{enumerate} A vertex superalgebra $(V, \mathbf{1}, T, Y)$ is said to be \textit{$\mathbb{Z}$-graded} when the super vector space $V$ is given a $\mathbb{Z}$-grading $V = \oplus_{n \in \mathbb{Z}} V[n]$ such that $\mathbf{1}$ is a vector of weight $0$, $T$ is a homogeneous linear operator of weight $1$, and if $A \in V[n]$, then the field $Y(A,z)$ is homogeneous of conformal dimension $n$. We refer to such a grading as a \textit{weight-grading} on the vertex superalgebra. Note that $\mathrm{Im}\,Y\subset(\mathrm{End}\,V)[[z^{\pm1}]]$ has a canonical structure of vertex superalgebra and that $Y: V\to \mathrm{Im}\,Y$ is an isomorphism of vertex superalgebras. We will often identify these two vertex superalgebras.
For a vertex superalgebra $(V, \mathbf{1}, T, Y)$, the data of $Y$ is equivalent to that of bilinear maps $$ (n): V\times V\to V, \quad (A, B)\mapsto A_{(n)}B. $$ Therefore vertex algebras are also written as $(V, \mathbf{1}, T, (n); n\in \mathbb{Z})$. Moreover the translation operator $T$ and the vertex operator $Y(A, z)$ are often denoted by $\partial$ and by $A(z)$, respectively. A purely even vertex superalgebra is called simply a \textit{vertex algebra}.
We use the following notation for the \textit{operator product expansion (OPE)} of the mutually local fields $A(z)$ and $B(z)$: $$ A(z)B(w)\sim \sum_{n\ge 0}C_n(w)(z-w)^{-n-1}, $$ where $C_n(w)$ are some fields. Note that the OPE formula gives the commutation relations among the coefficients of $A(z)$ and $B(w)$. (See \cite{FBZ04} and \cite{Kac01} for details.)
The following examples of vertex superalgebras will be used for the construction of some kinds of cohomology later.
\begin{example}[affine vertex superalgebras]\label{ex:affine va} Let $\mathfrak{g}$ be a Lie superalgebra with a supersymmetric invariant bilinear form $B$. Let $\Hat{\mathfrak{g}}=\mathfrak{g} [t^{\pm1}]\oplus \mathbb{K}K$ be the affine Lie algebra associated with $(\mathfrak{g}, B).$ Set $$ N(\mathfrak{g}, B):=U(\hat{\mathfrak{g}})\otimes_{U(\mathfrak{g}[t]\oplus \mathbb{K}K)}\mathbb{K}_1, $$ where $\mathbb{K}_1$ is the one-dimensional $\mathfrak{g}[t]\oplus \mathbb{K}K$-module on which $\mathfrak{g}[t]$ acts by zero and $K$ by $1.$ This $\Hat{\mathfrak{g}}$-module $N(\mathfrak{g}, B)$ has a $\mathbb{Z}_{\ge 0}$-graded vertex superalgebra structure called the \textit{affine vertex superalgebra} associated with $\mathfrak{g}$ and $B$. Note that the operator $a_{(n)}$ has weight $-n$, where $a_{(n)}$ stands for the operator on $N(\mathfrak{g}, B)$ corresponding to $at^n\in \hat{\mathfrak{g}}$. The Lie superalgebra $\mathfrak{g}$ can be seen as a subspace of $N(\mathfrak{g}, B)$ by the injection $\mathfrak{g}\to N(\mathfrak{g}, B),\ a\mapsto a_{(-1)}\mathbf{1}$. We denote by $O(\mathfrak{g}, B)$ the corresponding vertex superalgebra $\mathrm{Im}(Y)=Y(N(\mathfrak{g}, B))\subset (\mathrm{End}\,N(\mathfrak{g}, B))[[z^{\pm1}]]$. \end{example}
\begin{example}[$\beta \gamma$-systems]\label{ex:beta_gamma-systems} Let $V$ be a finite-dimensional vector space. Let $\mathfrak{h}(V)$ $=(V[t^{\pm1}]\oplus V^*[t^{\pm1}]dt)\oplus \mathbb{K}\mathbf{\tau}$ be the \textit{Heisenberg Lie algebra} associated with $V$. Set $$ \mathcal{S}(V):=U(\mathfrak{h}(V))\otimes_{U(V[t]\oplus V^*[t]dt\oplus \mathbb{K}\mathbf{\tau})}\mathbb{K}_1, $$ where $\mathbb{K}_1$ is the one-dimensional $(V[t]\oplus V^*[t]dt\oplus \mathbb{K}\mathbf{\mathbf{\tau}})$-module in which $V[t]\oplus V^*[t]dt$ acts by zero and $\mathbf{\tau}$ by $1.$ We denote by $\beta^{v}_n,\gamma^{\phi}_n$ the elements $v\otimes t^n, \phi \otimes t^{n-1}dt \in \mathfrak{h}(V)$, respectively. The $\mathfrak{h}(V)$-module $\mathcal{S}(V)$ has a $\mathbb{Z}_{\ge 0}$-graded vertex algebra structure called the $\beta \gamma$-\textit{system} associated with $V$. We sometimes denote $\beta^v_n$ and $\gamma^\phi_n$ by $\beta^v_{(n)}$ and $\gamma^\phi_{(n-1)}$, respectively. \end{example}
\begin{example}[$bc$-systems]\label{ex:bc-systems} Let $V$ be a finite-dimensional vector space. We regard $V[t^{\pm1}]\oplus V^*[t^{\pm1}]dt$ as an odd abelian Lie algebra. Consider the one-dimensional central extension $ \mathfrak{j}(V)=(V[t^{\pm1}]\oplus V^*[t^{\pm1}]dt)\oplus \mathbb{K}\mathbf{\tau} $ of that odd abelian Lie algebra with bracket \begin{multline*} [v_1\otimes f_1+\phi_1 \otimes g_1dt,v_2\otimes f_2+\phi_2 \otimes g_2dt] \\ =(\langle v_1,\phi_2\rangle\mathrm{Res}_{t=0}f_1g_2dt +\langle v_2,\phi_1\rangle\mathrm{Res}_{t=0}f_2g_1dt)\mathbf{\tau}. \end{multline*} Set $$ \mathcal{E}(V):=U(\mathfrak{j}(V))\otimes_{U(V[t]\oplus V^*[t]dt\oplus \mathbb{K}\mathbf{\tau})}\mathbb{K}_1, $$ where $\mathbb{K}_1$ is the one-dimensional $(V[t]\oplus V^*[t]dt\oplus \mathbb{K}\mathbf{\tau})$-module in which $V[t]\oplus V^*[t]dt$ acts by zero and $\mathbf{\tau}$ by $1.$ We denote by $b^{v}_n, c^{\phi}_n$ the elements $v\otimes t^n, \phi \otimes t^{n-1}dt,$ respectively. The $\mathfrak{j}(V)$-module $\mathcal{E}(V)$ has a $\mathbb{Z}_{\ge 0}$-graded vertex superalgebra structure called the $bc$-\textit{system} associated with $V$. We sometimes denote $b^v_n$ and $c^\phi_n$ by $b^v_{(n)}$ and $c^\phi_{(n-1)}$, respectively. \end{example}
\begin{example}[semi-infinite Weil algebras]\label{ex:semi-infinite_Weil_algebras} For a vector space $V$, the tensor product vertex superalgebra $$ \mathcal{W}(V):=\mathcal{E}(V)\otimes \mathcal{S}(V), $$ is called the \textit{semi-infinite Weil algebra} associated with $V$ (\cite{FF91}). \end{example}
We recall some graded structures on vertex superalgebras. A vertex superalgebra $V$ is \textit{degree-graded} if it is given a $\mathbb{Z}$-grading $V=\bigoplus_{p\in\mathbb{Z}}V^p$ such that $ A_{(n)}B\in V^{p+q} $ for all $A\in V^p, B\in V^q$, $n\in\mathbb{Z}$ and $\mathbf{1}\in V^0$. Recall that a $\mathbb{Z}$-grading $V=\bigoplus_{n\in \mathbb{Z}}V[n]$ on a vertex superalgebra $V$ is called a weight-grading if $ A_{(k)}B\in V[n+m-k-1] $ for all $A\in V[n]$, $B\in V[m]$ and $k\in \mathbb{Z}$, and $\mathbf{1}\in V[0]$. A vertex superalgebra $V$ is \textit{degree-weight-graded} if $V$ is both degree and weight-graded and the gradings are compatible, that is, $V=\bigoplus_{p, n\in \mathbb{Z}}V^p[n]$, where $V^p[n]=V^p\cap V[n]$.
\begin{example}\label{ex: another grading on N(g, 0)} We can define a degree-weight-grading on the affine vertex superalgeba $N(\mathfrak{g}, B)$ when $\mathfrak{g}$ has two compatible $\mathbb{Z}$-grading and the invariant bilinear form $B$ is $0$. We call the one grading on $\mathfrak{g}$ the weight-grading and the other the degree-grading. Then $N(\mathfrak{g}, 0)$ becomes a degree-weight-graded vertex superalgebra if we give the weight-grading by $ \mathrm{wt}\, a^1_{(n_1)}\dots a^r_{(n_r)}\mathbf{1} :=\sum_{i=1}^r(-n_i+\mathrm{wt}_\mathfrak{g}a^i), $ and the degree-grading by $ \mathrm{deg}\, a^1_{(n_1)}\dots a^r_{(n_r)}\mathbf{1} :=\sum_{i=1}^r\mathrm{deg}_\mathfrak{g}a^i, $ for degree-weight-homogeneous elements $a^1, \dots, a^r \in \mathfrak{g}$ and $n_1, \dots, n_r\in \mathbb{Z}_{<0}$. We call this grading the degree-weight-grading on the vertex superalgebra $N(\mathfrak{g}, 0)$ associated with the grading on $\mathfrak{g}$. \end{example}
In the sequel, we will always assume that an action of a degree-weight-graded vertex superalgebra on a degree-weight-graded super vector space is compatible with the gradings.
We give some lemmas used in Section \ref{section: Chiral Lie Algebroid Cohomology} and \ref{section: Chiral Equivariant Lie Algebroid Cohomology}. Recall the notion of vertex superalgebra derivation. A \textit{derivation} on a vertex superalgebra $V$ with parity $\bar{i}$ is an endomorphism $d$ on $V$ with parity $\bar{i}$ such that $ [d, Y(A, z)]=Y(d\cdot A, z) $ for any $A\in V$.
\begin{lemma}\label{lem: odd derivation is 0 if so on generators}
Let $V$ be a vertex superalgebra generated by a subset $S\subset V$. Let $D$ be an odd derivation on the vertex superalgebra $V$ such that $D^2|_S=0$. Then $D^2=0$ hold on $V$. \end{lemma} \begin{proof} Since the operator $[D,D]=2D^2$ is also a derivation, our assertion holds. \end{proof}
\begin{lemma}\label{lem: sufficient condition for B-linearilty} Let $f: V\to W$ be a morphism of vertex superalgebras. Let $N$ be a non-negative integer. Suppose $V$ is generated by a subset $S\subset V$. Let $(A_{(n)})_{0\le n \le N}$ and $(B_{(n)})_{0\le n \le N}$ be linear maps on $V$ and $W$, respectively. Suppose the following hold: \begin{gather}\label{eq: B-commutation relation} [A_{(n)},v_{(k)}]=\sum_{i\ge0}\binom{n}{i}(A_{(i)}v)_{(n+k-i)}, \\ \label{eq: B'-commutation relation} [B_{(n)},f(v)_{(k)}]=\sum_{i\ge0}\binom{n}{i}(B_{(i)}f(v))_{(n+k-i)}, \end{gather} for all $v\in S$, $0\le n\le N$, and $k\in\mathbb{Z}$ Then if $f\circ A_{(n)}=B_{(n)}\circ f$ on $S$ for all $0\le n\le N$, then $f\circ A_{(n)}=B_{(n)}\circ f$ holds on $V$ for any $0\le n\le N$. \end{lemma} \begin{proof} It suffices to show that $(f\circ A_{(n)})(v)=(B_{(n)}\circ f)(v)$ for $v\in V$ of the form $s^1_{(n_1)}\cdots s^r_{(n_r)}\mathbf{1}$ with $s^1, \dots, s^r\in S$ and $n_1, \dots, n_r\in\mathbb{Z}$. The assertion is proved by induction on $r$. Note that $A_{(n)}\mathbf{1}=0$ and $B_{(n)}\mathbf{1}=0$ are proved by induction on $n\in\mathbb{N}$ with \eqref{eq: B-commutation relation} and \eqref{eq: B'-commutation relation}. \end{proof}
\begin{lemma}\label{lem: generalized commutant} Let $V$ be a vertex superalgebra. Let $\mathcal{A}=(A_{(m)}^\lambda)_{m\ge0, \lambda\in\Lambda}$ be a family of $\mathbb{Z}/2\mathbb{Z}$-homogeneous linear maps on $V$ such that $$ [A_{(m)}^\lambda, v_{(k)}]=\sum_{i\ge0}\binom{m}{i}(A_{(i)}^\lambda v)_{(m+k-i)}, $$ for all $m\ge0$, $\lambda\in\Lambda$, $k\in\mathbb{Z}$ and $v\in V$.
Then $V^\mathcal{A}:=\{v\in V \bigm| A_{(m)}^\lambda v=0\ \textrm{for all}\ m\ge0\ \textrm{and}\ \lambda\in\Lambda\}$ is a subalgebra of $V$. \end{lemma} \begin{proof} The equality $A_{(m)}^\lambda \mathbf{1}=0$ is proved by induction on $n$. The subspace $V^\mathcal{A}$ is closed under the $n$-th product due to the assumption. \end{proof}
\begin{lemma}\label{lem: tensor commutant} Let $V$ and $W$ be vertex superalgebras. Let $\mathcal{A}=(A_{(m)}^\lambda)_{m\ge0, \lambda\in\Lambda}$ be a family of $\mathbb{Z}/2\mathbb{Z}$-homogeneous linear maps on $V$ and $\mathcal{B}=(B_{(m)}^\lambda)_{m\ge0, \lambda\in\Lambda}$ a family of $\mathbb{Z}/2\mathbb{Z}$-homogeneous linear maps on $W$ such that \begin{gather*} [A_{(m)}^\lambda, v_{(k)}]=\sum_{i\ge0}\binom{m}{i}(A_{(i)}^\lambda v)_{(m+k-i)}, \\ [B_{(m)}^\lambda, w_{(k)}]=\sum_{i\ge0}\binom{m}{i}(B_{(i)}^\lambda w)_{(m+k-i)}, \end{gather*} for all $m\ge0$, $\lambda\in\Lambda$, $k\in\mathbb{Z}$, $v\in V$ and $w\in W$. Then the family of $\mathbb{Z}/2\mathbb{Z}$-homogeneous linear maps on $V\otimes W$, $(A_{(m)}^\lambda\otimes \mathrm{id}+\mathrm{id}\otimes B_{(m)}^\lambda)_{m\ge0, \lambda\in\Lambda}$, satisfies the relation $$ [A_{(m)}^\lambda\otimes \mathrm{id}+\mathrm{id}\otimes B_{(m)}^\lambda, x_{(k)}]=\sum_{i\ge0}\binom{m}{i}\bigl((A_{(m)}^\lambda\otimes \mathrm{id}+\mathrm{id}\otimes B_{(m)}^\lambda) x\bigr)_{(m+k-i)}, $$ for any $m\ge0$, $\lambda\in\Lambda$, $k\in\mathbb{Z}$, $x\in V\otimes W$. \end{lemma} \begin{proof} The assertion is proved by
direct computations. \end{proof}
\subsection{Chiral Equivariant Cohomology}\label{subsection: Chiral Equivariant Cohomology}
We next recall the definition of the chiral equivariant cohomology. We refer the reader to \cite{LL,LLS1} and partly to \cite{FF91},
for more details.
Let $\mathfrak{g}$ be a Lie algebra. The Lie superalgebra $\mathfrak{sg}$ is defined by $$ \mathfrak{sg}:=\mathfrak{g}\ltimes\mathfrak{g}_{-1}, $$ where $\mathfrak{g}_{-1}$ is the adjoint representation of $\mathfrak{g}$.
Let $O(\mathfrak{sg}, 0)$ be the affine vertex superalgebra associated with the Lie superalgebra $\mathfrak{sg}$ with an invariant bilinear form $0$. The Lie superalgebra derivation $$ \mathfrak{sg} \to \mathfrak{sg}, \quad (\xi, \eta)\mapsto (\eta, 0), $$ induces a vertex superalgebra derivation $$ \mathbf{d}: O(\mathfrak{sg}, 0)\to O(\mathfrak{sg}, 0),\quad (\xi, \eta)(z)\mapsto (\eta, 0)(z). $$ This makes $O(\mathfrak{sg}):=(O(\mathfrak{sg},0), \mathbf{d})$ a differential degree-weight-graded vertex algebra, that is, a degree-weight-graded vertex superalgebra with a square-zero odd vertex superalgera derivation of degree $1$, where the gradings are given by $\mathrm{deg} ((\xi, 0)(z))=0, \mathrm{deg} ((0, \eta)(z))=-1$ and $\mathrm{wt} ((\xi, \eta)(z))=1$.
Recall the notion of $O(\mathfrak{sg})$-algebras from \cite{LL}. An $O(\mathfrak{sg})$-\textit{algebra} is a differential degree-weight-graded vertex superalgebra $(\mathcal{A}, d)$ equipped with a morphism of differential degree-weight-graded vertex superalgebras $\Phi_\mathcal{A}: O(\mathfrak{sg})\to (\mathcal{A}, d)$. Next we recall from \cite{LLS1} the notion of $\mathfrak{sg}[t]$-modules containing that of $O(\mathfrak{sg})$-algebras. Recall the Lie superalgebra $\mathfrak{sg}[t]=\mathfrak{sg}\otimes\mathbb{K}[t]$ has a differential $d$ defined by $$ d: \mathfrak{sg}[t] \to \mathfrak{sg}[t], \quad (\xi, \eta)t^n\mapsto (\eta, 0)t^n. $$ An $\mathfrak{sg}[t]$-\textit{module} is a degree-weight-graded complex $(\mathcal{A}, d_{\mathcal{A}})$ equipped with a Lie superalgebra morphism $$ \rho: \mathfrak{sg}[t]\to \mathrm{End} (\mathcal{A}), \quad (\xi, \eta)t^n\mapsto \rho((\xi, \eta)t^n)=L_{\xi, (n)}+\iota_{\eta, (n)}, $$ such that for all $x\in \mathfrak{sg}[t]$ we have \begin{enumerate}[$\bullet$]
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\setlength{\itemindent}{0pt} \item $\rho(dx)=[d_\mathcal{A},\rho(x)]$; \item $\rho(x)$ has degree $0$ whenever $x$ is even in $\mathfrak{sg}[t]$, and degree $-1$ whenever $x$ is odd, and has weight $-n$ if $x\in \mathfrak{sg}t^n$. \end{enumerate}
In this paper, we assume that the differential has odd parity and that the action of $\mathfrak{sg}[t]$ on $\mathcal{A}$ with the discrete topology is continuous, that is, for any $v\in\mathcal{A}$, $t^k\mathfrak{sg}[t]\cdot v=0$ for some sufficiently large $k\in\mathbb{N}$. Moreover we call an $\mathfrak{sg}[t]$-module a \textit{ differential} $\mathfrak{sg}[t]$-\textit{module}, emphasizing its differential.
For a differential $\mathfrak{sg}[t]$-module $(\mathcal{A}, d_\mathcal{A})$, we often write $L_{\xi, (n)}$ and $\iota_{\xi, (n)}$ as $L_{\xi, (n)}^\mathcal{A}$ and $\iota_{\xi, (n)}^\mathcal{A}$, respectively.
We will recall the notion of the chiral equivariant cohomology. Consider the semi-infinite Weil algebra $\mathcal{W}=\mathcal{W}(\mathfrak{g})$ associated with a finite-dimensional Lie algebra $\mathfrak{g}$ (see Example \ref{ex:semi-infinite_Weil_algebras}). Let $(\xi_i)_i$ be a basis of $\mathfrak{g}$ with the dual basis $(\xi^*_i)_i$ for $\mathfrak{g}^*$. Recall that the vertex superalgebra $\mathcal{W}(\mathfrak{g})$ is degree-weight-graded. The weight and degree-grading come from the diagonalizable operator ${\omega_{\mathcal{W}}}_{(1)}=(\omega_{\mathcal{E}}+\omega_{\mathcal{S}})_{(1)}$ and the operator ${j_{bc}}_{(0)}+2{j_{\beta\gamma}}_{(0)}$, respectively. Here $\omega_{\mathcal{S}}:=\sum_{i=1}^{\dim \mathfrak{g}}\beta^{x^i}_{-1}\partial\gamma^{x^{*}_i}_{0}\mathbf{1}$, $\omega_{\mathcal{E}}:=-\sum_{i=1}^{\dim \mathfrak{g}}b^{x^i}_{-1}\partial c^{x^{*}_i}_0\mathbf{1}$ and $j_{bc}:=-\sum_{i=1}^{\dim \mathfrak{g}}b^{\xi_i}_{-1}c^{\xi^*_i}_0\mathbf{1}$, $j_{\beta\gamma}:=\sum_{i=1}^{\dim \mathfrak{g}}\beta^{\xi_i}_{-1}\gamma^{\xi^*_i}_0\mathbf{1}$.
Set \begin{gather} D:=J+K, \\ \quad J:=-\sum_{i, j=1}^{\dim \mathfrak{g}}\beta^{[\xi_i,\xi_j]}_{-1}\gamma^{\xi^*_j}_0 c^{\xi^*_i}_0\mathbf{1}-\frac{1}{2}\sum_{i, j=1}^{\dim \mathfrak{g}}c^{\xi^*_i}_0c^{\xi^*_j}_0b^{[\xi_i,\xi_j]}_{-1}\mathbf{1}, \quad K:=\sum_{i=1}^{\dim \mathfrak{g}}\gamma^{\xi^*_i}_0b^{\xi_i}_{-1}\mathbf{1}. \end{gather} Then the operator $D_{(0)}$ is a differential on $\mathcal{W}$. The differential degree-weight-graded vertex superalgebra $(\mathcal{W}^\bullet, d_\mathcal{W})$ is called the \textit{semi-infinite Weil complex}, where $d_\mathcal{W}=D_{(0)}$ (\cite{FF91}).
Set \begin{gather} \Theta_\mathcal{W}^\xi:=\Theta_\mathcal{E}^\xi+\Theta_\mathcal{S}^\xi, \quad\\
\Theta_\mathcal{E}^\xi:=\sum_{i=1}^{\dim \mathfrak{g}}b^{[\xi,\xi_i]}_{-1}c^{\xi^*_i}_0\mathbf{1}, \quad \Theta_\mathcal{S}^\xi:=-\sum_{i=1}^{\dim \mathfrak{g}}\beta^{[\xi,\xi_i]}_{-1}\gamma^{\xi^*_i}_0\mathbf{1}, \end{gather} for $\xi\in \mathfrak{g}$. The following theorem is proved in \cite[theorem 5.11]{LL04}.
\begin{theorem}[Lian-Linshaw] The vertex superalgebra morphism $O(\mathfrak{sg})\to(\mathcal{W}(\mathfrak{g}), D_{(0)}),$ $(\xi, \eta)(z)$ $\mapsto\Theta_{\mathcal{W}}^\xi(z)+b^\eta(z)$ defines an $O(\mathfrak{sg})$-algebra structure on $\mathcal{W}(\mathfrak{g})$. \end{theorem}
We will use the following relations proved in \cite[Lemma 5.12]{LL}.
\begin{lemma}[Lian-Linshaw] Let $\eta, \xi$ be elements of $\mathfrak{g}$ and $\xi^*$ an element of $\mathfrak{g}^*$. Then the following hold: \begin{gather} \Theta_\mathcal{W}^\xi(z)c^{\xi^*}(w)\sim c^{\mathrm{ad}^*\xi\cdot\xi^*}(w)(z-w)^{-1}, \\ D_{(0)}c^{\xi^*}_0\mathbf{1}=-\frac{1}{2}\sum_{i=1}^{\dim{\mathfrak{g}}}c^{\mathrm{ad}^*\xi_i\cdot\xi^*}_0c^{\xi^*_i}_0\mathbf{1}+\gamma^{\xi^*}_0\mathbf{1}, \\ D_{(0)}\gamma^{\xi^*}_0\mathbf{1}=\sum_{i=1}^{\dim{\mathfrak{g}}}\gamma^{\mathrm{ad}^*\xi_i\cdot\xi^*}_0c^{\xi^*_i}_0\mathbf{1}, \end{gather} where the first formula stands for the OPE of the fields $\Theta_\mathcal{W}^\xi(z)$ and $c^{\xi^*}(z)$, and $\mathrm{ad}^*$ is the coadjoint action of $\mathfrak{g}$ on $\mathfrak{g}^*$. \end{lemma}
Recall the notion of the chiral horizontal, invariant and basic subspaces from \cite{LL,LLS1}. Let $(\mathcal{A}, d)$ be a differential $\mathfrak{sg}[t]$-module. The \textit{chiral horizontal, invariant} and \textit{basic subspaces} of $\mathcal{A}$ are respectively \begin{align*}
\mathcal{A}_{hor}&:=\bigl\{a\in \mathcal{A} \bigm| \iota_{\eta, (n)}a=0 \ \text{for all}\ \eta\in\mathfrak{g}, n\ge0 \bigr\}, \\
\mathcal{A}_{inv}&:=\bigl\{a\in \mathcal{A} \bigm| L_{\xi, (n)}a=0\ \text{for all}\ \xi\in\mathfrak{g}, n\ge0 \bigr\}, \ \text{and} \\ \mathcal{A}_{bas}&:=\mathcal{A}_{hor}\cap\mathcal{A}_{inv}. \end{align*} Note that if $(\mathcal{A}, d)$ is a differential $\mathfrak{sg}[t]$-module then the subspaces $\mathcal{A}_{hor}$ and $\mathcal{A}_{bas}$ are subcomplexes of $(\mathcal{A}, d)$.
We then recall the definitions of the chiral basic and equivariant cohomologies.
Let $G$ be a compact connected Lie group. Set $\mathfrak{g}=\mathrm{Lie}(G)^{\mathbb{K}}$. Let $(\mathcal{A}, d)$ be a differential $\mathfrak{sg}[t]$-module. Its \textit{chiral basic cohomology} $\mathbf{H}_{bas}{(\mathcal{A})}$ is the cohomology of the complex $(\mathcal{A}_{bas}, d|_{\mathcal{A}_{bas}})$. The \textit{chiral equivariant cohomology} $\mathbf{H}_{G}{(\mathcal{A})}$ of $(\mathcal{A}, d)$ is the chiral basic cohomology of the tensor product $ (\mathcal{W}(\mathfrak{g})\otimes \mathcal{A}, d_\mathcal{W}\otimes1+1\otimes d_\mathcal{A}). $
\section{Chiral $W^*$-Modules}\label{section: Chiral $W^*$-Modules}
\subsection{Definition of Chiral $W^*$-Modules and the Chiral Cartan Model} Let $\mathfrak{g}$ be a finite-dimensional Lie algebra. We denote by $\langle c, \gamma \rangle$ or $\mathcal{W}'$ the subalgebra of the semi-infinite Weil algebra $\mathcal{W}=\mathcal{W}(\mathfrak{g})$ generated by $c^{\xi^*}_0\mathbf{1}$, $\gamma^{\xi^*}_0\mathbf{1}$ with $\xi^*\in\mathfrak{g}^*$. Note that $\mathcal{W}'$ is preserved by the differential $d_\mathcal{W}$. Therefore we have a
subcomplex $(\mathcal{W}', d_{\mathcal{W}'})$, where $d_{\mathcal{W}'}:=d_{\mathcal{W}}|_{\mathcal{W}'}$. Note that $(\mathcal{W}', d_{\mathcal{W}'})$ is acyclic. This follows from the same argument as that for the proof of the acyclicity of $(\mathcal{W}, d_\mathcal{W})$ in \cite[Proposition 5]{Akm93}. (See Section \ref{subsection: Chiral Equivariant Cohomology} for the definition of the semi-infinite Weil complex $(\mathcal{W}(\mathfrak{g}), d_\mathcal{W})$.)
We denote by $\delta(z-w)_-$ the formal distribution $\sum_{n\ge 0}z^{-n-1}w^n$.
\begin{definition}\label{df: chiral W^*-modules} A \textbf{chiral} $W^*$\textbf{-module} (with respect to $\mathfrak{g}$) is a differential $\mathfrak{sg}[t]$-module $(\mathcal{A}, d_{\mathcal{A}})$ given a module structure over the vertex superalgebra $\langle c, \gamma \rangle$ $$ Y^\mathcal{A}(\ , z): \langle c, \gamma \rangle \to (\mathrm{End} \mathcal{A})[[z^{\pm1}]], $$ such that \begin{enumerate}[$(1)$\ ] \setlength{\topsep}{1pt}
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\setlength{\itemindent}{0pt} \item $[d_\mathcal{A}, Y^\mathcal{A}(x, z)]= Y^\mathcal{A}(d_{\mathcal{W}'}x, z)$, \quad for all $x \in \langle c, \gamma \rangle$, \item $[L_\xi^\mathcal{A}(z)_{-},Y^\mathcal{A}(c^{\xi^*}_0\mathbf{1}, w)]=Y^\mathcal{A}(c^{\mathrm{ad}^*\xi \cdot\xi^*}_0\mathbf{1}, w)\delta(z-w)_{-}$, for all elements $\xi$ of $\mathfrak{g}$ and all elements $\xi^*$ of $\mathfrak{g}^*$, \item $[\iota_\xi^\mathcal{A}(z)_{-},Y^\mathcal{A}(c^{\xi^*}_0\mathbf{1}, w)]=\langle {\xi^*}, \xi \rangle \delta(z-w)_{-}$, for all $\xi\in \mathfrak{g}$ and all ${\xi^*} \in \mathfrak{g}^*$, \end{enumerate} where $L_{\xi}^\mathcal{A}(z)_{-}:=\sum_{n\ge0}L_{\xi, (n)}^\mathcal{A}z^{-n-1}$ and $\iota_\xi^\mathcal{A}(z)_{-}:=\sum_{n\ge0}\iota_{\xi, (n)}^\mathcal{A}z^{-n-1}$ for an element $\xi$ of $\mathfrak{g}$. \end{definition}
For a $\langle c, \gamma \rangle$-module $(\mathcal{A}, Y^\mathcal{A})$, we often use the following notation: \begin{align*} Y^\mathcal{A}(c^{\xi^*}_0\mathbf{1}, z)&=c^{\xi^*, \mathcal{A}}(z)=\sum_{n\in\mathbb{Z}}c^{\xi^*, \mathcal{A}}_{(n)}z^{-n-1}, \\ Y^\mathcal{A}(\gamma^{\xi^*}_0\mathbf{1}, z)&=\gamma^{\xi^*, \mathcal{A}}(z)=\sum_{n\in\mathbb{Z}}\gamma^{\xi^*, \mathcal{A}}_{(n)}z^{-n-1}. \end{align*} for an element $\xi^*$ of $\mathfrak{g}^*$.
Note that the formula (2) in the preceding definition is equivalent to the commutation relations $[L_{\xi, (m)}^\mathcal{A},c_{(n)}^{\xi^*, \mathcal{A}}]=c_{(m+n)}^{\mathrm{ad}^*\xi\cdot\xi^*, \mathcal{A}}$ for all $m\in\mathbb{Z}_{\ge0}$ and $n\in\mathbb{Z}$. Similarly the formula (3) is equivalent to the relations $[\iota_{\xi, (m)}^\mathcal{A},c_{(n)}^{\xi^*, \mathcal{A}}]=\langle {\xi^*}, \xi \rangle\delta_{m+n, -1}$ for all $m\in\mathbb{Z}_{\ge0}$ and $ n\in\mathbb{Z}$.
Let $(\xi _i)_i$ be a basis of $\mathfrak{g}$ and $(\xi^*_i)_i$ the dual basis for $\mathfrak{g}^*$. Let $(\mathcal{A}, d_\mathcal{A}, Y^\mathcal{A})$ be a chiral $W^*$-module and $(\mathcal{B}, d_\mathcal{B})$ a differential $\mathfrak{sg}[t]$-module. Set $$ \Phi=\Phi_{\mathcal{A}, \mathcal{B}}:=\mathrm{exp}(\phi(0)_{\ge 0}) \in GL(\mathcal{A}\otimes\mathcal{B}), $$ where $\phi(0)_{\ge 0}:= \sum_{i=1}^{\dim \mathfrak{g}}\sum_{n\ge 0}c^{\xi^*_i, \mathcal{A}}_{(-n-1)}\otimes\iota^\mathcal{B}_{\xi_i, (n)}$. Set \begin{gather} C(\mathcal{A}; \mathcal{B}):=\Phi((\mathcal{A}\otimes\mathcal{B})_{bas}), \\
d=d_{\mathcal{A}, \mathcal{B}}:=\Phi\circ(d_\mathcal{A}\otimes1 +1 \otimes d_\mathcal{B})\circ\Phi^{-1}|_{C(\mathcal{A}; \mathcal{B})}. \end{gather} Note that $C(\mathcal{A}; \mathcal{B})$ is degree-weight-graded as a subspace of the degree-weight-graded super vector space $\mathcal{A}\otimes\mathcal{B}$ since $\Phi$ preserves the degree and weight-gradings. Then we have the following.
\begin{lemma}\label{lem: CHIRAL CARTAN MODEL} Let $(\mathcal{A}, d_\mathcal{A}, Y^\mathcal{A})$ be a chiral $W^*$-module and $(\mathcal{B}, d_\mathcal{B})$ a differential $\mathfrak{sg}[t]$-module. Then the map $\Phi=\Phi_{\mathcal{A}, \mathcal{B}}$ restricted to the chiral basic subspace is an isomorphism of degree-weight-graded complexes: $$
\Phi: ((\mathcal{A}\otimes\mathcal{B})_{bas}, (d_\mathcal{A}\otimes1 +1\otimes d_\mathcal{B})|_{(\mathcal{A}\otimes\mathcal{B})_{bas}})\to (C(\mathcal{A}; \mathcal{B}), d_{\mathcal{A}, \mathcal{B}}). $$ \end{lemma}
We call the complex $(C(\mathcal{A}; \mathcal{B}), d_{\mathcal{A}, \mathcal{B}})$ the \textbf{chiral Cartan model} for the differential $\mathfrak{sg}[t]$-module $(\mathcal{B}, d_\mathcal{B})$ with respect to the chiral $W^*$-module $(\mathcal{A}, d_\mathcal{A}, Y^\mathcal{A})$.
The following proposition is a variation of \cite[Theorem 4.6]{LL}.
\begin{proposition}\label{prop: chiral Cartan model for chiral W^*-modules} Let $(\mathcal{A}, d_\mathcal{A}, Y^\mathcal{A})$ be a chiral $W^*$-module and $(\mathcal{B}, d_\mathcal{B})$ a differential $\mathfrak{sg}[t]$-module. Then the following equalities hold in $\mathrm{End} (\mathcal{A}\otimes\mathcal{B})$: \begin{multline}\label{eq: the transformation of the differential} \Phi\circ(d_\mathcal{A}\otimes 1+ 1\otimes d_\mathcal{B})\circ \Phi^{-1}\\ =d_\mathcal{A}\otimes 1+ 1\otimes d_\mathcal{B} - \sum_{i=1}^{\dim \mathfrak{g}}\sum_{n\ge0}\gamma^{\xi^*_i, \mathcal{A}}_{(-n-1)}\otimes\iota^\mathcal{B}_{\xi_i, (n)}+\sum_{i=1}^{\dim \mathfrak{g}}\sum_{n\ge0}c^{\xi^*_i, \mathcal{A}}_{(-n-1)}\otimes L^\mathcal{B}_{\xi_i, (n)}\\ +\sum_{i, j=1}^{\dim \mathfrak{g}}\sum_{m,n\ge0}c_{(n)}^{\xi^*_i, \mathcal{A}}c_{(-n-m-2)}^{\xi^*_j, \mathcal{A}}\otimes\iota_{[\xi_i,\xi_j], (m)}^\mathcal{B}, \end{multline} and \begin{multline}\label{eq: the transformation of L} \Phi\circ(L^\mathcal{A}_{\xi, (n)}\otimes 1+1\otimes L^\mathcal{B}_{\xi, (n)})\circ\Phi^{-1} \\ \quad =L^\mathcal{A}_{\xi, (n)}\otimes 1+ 1\otimes L^\mathcal{B}_{\xi, (n)} +\sum_{i=1}^{\dim \mathfrak{g}}\sum_{0\le k<n}c^{\xi^*_i, \mathcal{A}}_{(n-k-1)}\otimes \iota^\mathcal{B}_{[\xi,\xi_i], (k)}, \end{multline} \begin{equation}\label{eq: the transformation of iota} \Phi \circ (\iota^\mathcal{A}_{\xi, (n)}\otimes 1 + 1\otimes \iota^\mathcal{B}_{\xi, (n)})\circ \Phi^{-1}=\iota^\mathcal{A}_{\xi, (n)}\otimes 1, \end{equation} for all $n \ge 0$, $\xi\in \mathfrak{g}$. \end{proposition} \begin{proof} The assertion is proved by direct computations of $\mathrm{ad}(\phi(0)_{\ge0})^l$, where $\mathrm{ad}(\phi(0)_{\ge0})=[\phi(0)_{\ge0}, \ \ ]$. Note that we have $\mathrm{ad}(\phi(0)_{\ge0})^3(d_\mathcal{A}\otimes 1+ 1\otimes d_\mathcal{B})=0$, $\mathrm{ad}(\phi(0)_{\ge0})^2 (L^\mathcal{A}_{\xi, (n)}\otimes 1+1\otimes L^\mathcal{B}_{\xi, (n)})=0$
and $(\mathrm{ad}(\phi(0)_{\ge0})^2(\iota^\mathcal{A}_{\xi, (n)}\otimes 1 + 1\otimes \iota^\mathcal{B}_{\xi, (n)})=0$. \end{proof}
By Proposition \ref{prop: chiral Cartan model for chiral W^*-modules},
we have $ C(\mathcal{A}; \mathcal{B})=(\mathcal{A}_{hor}\otimes\mathcal{B})^{\Phi\mathfrak{g}[t]\Phi^{-1}}, $ where the right-hand side stands for the invariant subspace under the modified action of $\mathfrak{g}[t]$: $$ \mathfrak{g}[t]\to \mathrm{End}(\mathcal{A}\otimes\mathcal{B})\stackrel{\mathrm{Ad}(\Phi)}{\longrightarrow} \mathrm{End}(\mathcal{A}\otimes\mathcal{B}). $$
\subsection{Commutative Cases} Consider the case when the Lie group $G=T$ is commutative. Set $\mathfrak{t}=\mathrm{Lie}(T)^\mathbb{K}$. The \textit{small chiral Cartan model} for $O(\mathfrak{st})$-algebras was introduced in \cite{LL}. By Lemma \ref{lem: CHIRAL CARTAN MODEL}, we can also define the small chiral Cartan model for any differential $\mathfrak{st}[t]$-module. Note that the action of $\mathfrak{t}[t]$ on $\mathcal{W}(\mathfrak{t})$ is trivial since $\mathfrak{t}$ is commutative.
Let $(\mathcal{A}, d_\mathcal{A})$ be a differential $\mathfrak{st}[t]$-module. Set $$ \mathcal{C}(\mathcal{A}):=\langle\gamma\rangle\otimes\mathcal{A}_{inv}\subset C(\mathcal{W}(\mathcal{\mathfrak{t}}); \mathcal{A}), $$ where we denote by $\langle \gamma \rangle$ the subalgebra of $\mathcal{W}'$ generated by $\gamma^{\xi^*}_0\mathbf{1}$ with $\xi^*\in\mathfrak{g}$.
Since $\mathfrak{t}$ is commutative, $d_\mathcal{W}|_{\langle \gamma \rangle}=0$ and $[\iota^{\mathcal{A}}_\xi(z)_-,L^{\mathcal{A}}_\eta(w)_- ]=0$ for any $\xi, \eta\in\mathfrak{t}$. Therefore it follows that $\mathcal{C}(\mathcal{A})$ is preserved by the differential $d_{\mathcal{W}(\mathfrak{t}), \mathcal{A}}$. We can prove the following lemma by the same argument as in \cite[Theorem 6.4]{LL}, where the case when $\mathcal{A}$ is an $O(\mathfrak{st})$-algebra is considered.
\begin{proposition}\label{prop: small Cartan model} The inclusion $$
(\mathcal{C}(\mathcal{A}), d_{\mathcal{W}(\mathfrak{t}), \mathcal{A}}|_{\mathcal{C}(\mathcal{A})})\to(C(\mathcal{W}(\mathcal{\mathfrak{t}}); \mathcal{A}), d_{\mathcal{W}(\mathfrak{t}), \mathcal{A}}), $$ is a quasi-isomorphism. \end{proposition}
We call $(\mathcal{C}(\mathcal{A}), d_{\mathcal{W}(\mathfrak{t}), \mathcal{A}}|_{\mathcal{C}(\mathcal{A})})$ the \textbf{small chiral Cartan model} for the differential $\mathfrak{st}[t]$-module $(\mathcal{A}, d_\mathcal{A})$.
The following lemma is proved by an argument similar to that in \cite[Lemma 2.6]{LLS1}, where the case when $\mathcal{A}$ is contained in an $O(\mathfrak{st})$-algebra is considered.
\begin{lemma}\label{lem: small Cartan model} $$ \Phi_{\mathcal{W}(\mathfrak{t}), \mathcal{A}}^{-1}(\mathcal{C}(\mathcal{A}))=(\langle c, \gamma\rangle\otimes\mathcal{A}_{inv})_{hor}. $$ \end{lemma}
Let $(\mathcal{A}, d_\mathcal{A}, Y^\mathcal{A})$ be a chiral $W^*$-module. Set $$ C'(\mathcal{A}, \mathcal{W}(\mathfrak{t})):=(\Phi_{\mathcal{A}, \mathcal{W}(\mathfrak{t})}\circ\tau\circ\Phi_{\mathcal{W}(\mathfrak{t}), \mathcal{A}}^{-1})(\mathcal{C}(\mathcal{A})) \subset C(\mathcal{A}, \mathcal{W}(\mathfrak{t})), $$ where $\tau: \mathcal{W}(\mathfrak{t})\otimes\mathcal{A}\to\mathcal{A}\otimes\mathcal{W}(\mathfrak{t})$ is the switching map. Then the following proposition follows from Proposition \ref{prop: small Cartan model}.
\begin{proposition}\label{prop: small Cartan model for chiral W^*-modules} There exists a canonical isomorphism $$ H(C'(\mathcal{A}; \mathcal{W}(\mathfrak{t})), d'_{\mathcal{A}, \mathcal{W}(\mathfrak{t})})\cong \mathbf{H}_{T}{\mathcal{(A)}}, $$
where $d'_{\mathcal{A}, \mathcal{W}(\mathfrak{t})}:=d_{\mathcal{A}, \mathcal{W}(\mathfrak{t})}|_{C'(\mathcal{A}, \mathcal{W}(\mathfrak{t}))}$. \end{proposition}
The following lemma leads us to the next important proposition.
\begin{lemma} $$ C'(\mathcal{A}, \mathcal{W}(\mathfrak{t}))=\mathcal{A}_{bas}\otimes\langle c, \gamma\rangle. $$ \end{lemma} \begin{proof} The operators $c^{\xi^*, \mathcal{A}}_{(n)}$ and $L_{\xi, (k)}^\mathcal{A}$ commute with each other since $\mathfrak{t}$ is commutative. Therefore the operators $c^{\xi^*, \mathcal{A}}_{(n)}$ preserve the subspace $\mathcal{A}^{\mathfrak{t}[t]}$. Thus we have $\Phi_{\mathcal{A}, \mathcal{W}(\mathfrak{t})}(\mathcal{A}^{\mathfrak{t}[t]}\otimes \langle c, \gamma\rangle)$ $\subset \mathcal{A}^{\mathfrak{t}[t]}\otimes \langle c, \gamma\rangle$. Therefore the subspace $$ \Phi_{\mathcal{A}, \mathcal{W}(\mathfrak{t})}\bigl(\big( \mathcal{A}^{\mathfrak{t}[t]}\otimes \langle c, \gamma\rangle\big)_{hor}\bigr)=\Bigl(\Phi_{\mathcal{A}, \mathcal{W}(\mathfrak{t})}\bigl(\mathcal{A}^{\mathfrak{t}[t]}\otimes \langle c, \gamma\rangle\bigr)\Bigr)^{\Phi_{\mathcal{A}, \mathcal{W}(\mathfrak{t})} \mathfrak{t}_{-1}[t] \Phi_{\mathcal{A}, \mathcal{W}(\mathfrak{t})}^{-1}}, $$ is contained in $\bigl(\mathcal{A}^{\mathfrak{t}[t]}\otimes\langle c, \gamma \rangle \bigr)^{\Phi_{\mathcal{A}, \mathcal{W}(\mathfrak{t})} \mathfrak{t}_{-1}[t] \Phi_{\mathcal{A}, \mathcal{W}(\mathfrak{t})}^{-1}}$. By the formula \eqref{eq: the transformation of iota}, we have \begin{equation}\label{eq: rewrite A_bas c gamma} \bigl(\mathcal{A}^{\mathfrak{t}[t]}\otimes\langle c, \gamma \rangle \bigr)^{\Phi_{\mathcal{A}, \mathcal{W}(\mathfrak{t})} \mathfrak{t}_{-1}[t] \Phi_{\mathcal{A}, \mathcal{W}(\mathfrak{t})}^{-1}}=\mathcal{A}_{bas}\otimes \langle c, \gamma \rangle. \end{equation} Thus we have \begin{equation}\label{eq: C'(A; W) is contained in A_bas c gamma} \Phi_{\mathcal{A}, \mathcal{W}(\mathfrak{t})}\bigl(\big( \mathcal{A}^{\mathfrak{t}[t]}\otimes \langle c, \gamma\rangle\big)_{hor}\bigr)\subset\mathcal{A}_{bas}\otimes \langle c, \gamma \rangle. \end{equation} On the other hand, we have \begin{align*} \Phi_{\mathcal{A}, \mathcal{W}(\mathfrak{t})}^{-1}(\mathcal{A}_{bas}\otimes \langle c, \gamma \rangle)&=\Phi_{\mathcal{A}, \mathcal{W}(\mathfrak{t})}^{-1}\Bigl(\bigl(\mathcal{A}^{\mathfrak{t}[t]}\otimes\langle c, \gamma \rangle \bigr)^{\Phi_{\mathcal{A}, \mathcal{W}(\mathfrak{t})} \mathfrak{t}_{-1}[t] \Phi_{\mathcal{A}, \mathcal{W}(\mathfrak{t})}^{-1}}\Bigr) \\ &=\Bigl(\Phi_{\mathcal{A}, \mathcal{W}(\mathfrak{t})}^{-1}\bigl(\mathcal{A}^{\mathfrak{t}[t]}\otimes\langle c, \gamma \rangle \bigr)\Bigr)^{\mathfrak{t}_{-1}[t]} \\ &\subset \bigl(\mathcal{A}^{\mathfrak{t}[t]}\otimes\langle c, \gamma \rangle \bigr)^{\mathfrak{t}_{-1}[t]}=\bigl(\mathcal{A}^{\mathfrak{t}[t]}\otimes\langle c, \gamma \rangle \bigr)_{hor}. \end{align*} The first equality follows from \eqref{eq: rewrite A_bas c gamma} and the inclusion follows from the formula $\Phi_{\mathcal{A}, \mathcal{W}(\mathfrak{t})}^{-1}=\mathrm{exp}\bigl(-\sum_{i=1}^{\dim \mathfrak{t}}\sum_{n\ge 0}c^{\xi^*_i, \mathcal{A}}_{(-n-1)}\otimes\iota^\mathcal{W}_{\xi_i, (n)}\bigr)$. Together with \eqref{eq: C'(A; W) is contained in A_bas c gamma}, we have $$ \Phi_{\mathcal{A}, \mathcal{W}(\mathfrak{t})}\bigl(\big( \mathcal{A}^{\mathfrak{t}[t]}\otimes \langle c, \gamma\rangle\big)_{hor}\bigr)=\mathcal{A}_{bas}\otimes \langle c, \gamma \rangle. $$ By Lemma \ref{lem: small Cartan model}, the left-hand side is equal to $C'(\mathcal{A}, \mathcal{W}(\mathfrak{t}))$. This completes the proof. \end{proof}
The following proposition is a chiral analogue of \cite[Theorem 4.3.1]{GS99}.
\begin{proposition} Let $(\mathcal{A}, d_\mathcal{A}, Y^\mathcal{A})$ be a chiral $W^*$-module. The inclusion \begin{equation}\label{eq: basic into basic otimes W'} (\mathcal{A}_{bas}, d_\mathcal{A})\to (C'(\mathcal{A}; \mathcal{W}(\mathfrak{t})), d'_{\mathcal{A}, \mathcal{W}(\mathfrak{t})}) , \quad a\mapsto a\otimes \mathbf{1}, \end{equation} is a quasi-isomorphism. \end{proposition} \begin{proof} Set \begin{gather*} d:=d'_{\mathcal{A}, \mathcal{W}(\mathfrak{t})}=d_1+d_2, \\ d_1:=1\otimes d_{\mathcal{W'}}=\sum_{i=1}^{\dim \mathfrak{t}}\sum_{n\ge 0}1\otimes\gamma^{\xi^*_i, \mathcal{W}}_{(-n-1)}b^{\xi_i, \mathcal{W}}_{(n)}, \\ d_2:=d_\mathcal{A}\otimes1 - \sum_{i=1}^{\dim \mathfrak{t}}\sum_{n\ge 0}\gamma^{\xi^*_i, \mathcal{A}}_{(-n-1)}\otimes b^{\xi_i, \mathcal{W}}_{(n)}. \end{gather*} Since $(\mathcal{W}', d_{\mathcal{W}'})$ is acyclic, we have \begin{equation}\label{eq: the cohomology of (A_bas otimes W', d_1)} H^i(\mathcal{A}_{bas}\otimes \langle c, \gamma \rangle, d_1)= \begin{cases} \mathcal{A}_{bas}\otimes\mathbb{K}\mathbf{1}, & \text{when}\ i=0,\\ 0, & \text{otherwise}. \end{cases} \end{equation} For $i, j \ge 0$, we set $ C^i:=\mathcal{A}_{bas}\otimes {\mathcal{W}'}^i, C_j:=\bigoplus_{0\le i \le j}C^i, $ and $C_{-1}:=0$. Note that $d_1$ has degree $1$ with respect to this grading $C'(\mathcal{A}; \mathcal{W}(t))=\bigoplus_{i\ge 0}C^i$ and $d_2$ preserves that filtration $C'(\mathcal{A}; \mathcal{W}(t))=\bigcup_{j\ge 0}C_j$. First we prove that for any $j\ge0$ if $\mu\in C_j$ and $d\mu=0$ then there exist an element $\nu\in C_{j-1}$ and $a\in \mathcal{A}_{bas}$ such that $\mu=d\nu+a\otimes \mathbf{1}$ and $d_\mathcal{A}a=0$. This implies that the map induced by the inclusion \eqref{eq: basic into basic otimes W'} on the cohomology is surjective. We use induction on $j$. Assume $j=0$. Let $\mu\in C_0$ with $d\mu=0$. Since $C_0=C^0=\mathcal{A}_{bas}\otimes\mathbb{K}\mathbf{1}$, we have an element $a\in\mathcal{A}_{bas}$ such that $\mu=a\otimes\mathbf{1}$. Considering the degree in the formula $0=d\mu=d_1\mu+d_2\mu$, we have $d_1\mu=0$. Therefore $d_2\mu=0$. From $d_2\mu=d_\mathcal{A}a\otimes\mathbf{1}$, we see $d_\mathcal{A}a=0$. Thus the proof for $j=0$ is completed. Next we assume $j>0$. Let $\mu\in C_j$ with $d\mu=0$. We can write $\mu$ as $ \mu=\mu_j+\mu_{j-1}+\dots+\mu_0 $ for some $\mu_i\in C^i$ with $i=1, \dots, j$. Since $d_1\mu_j$ is the component of $d\mu$ with the maximum degree, we have $d_1\mu_j=0$. By \eqref{eq: the cohomology of (A_bas otimes W', d_1)} and $j\neq0$, we have an element $\nu_{j-1}\in C^{j-1}$ such that $\mu_j=d_1\nu_{j-1}$. Therefore we have \begin{align*} \mu&=d_1\nu_{j-1}+ (\text{terms in}\ C_{j-1}) \\ &=(d\nu_{j-1}-d_2\nu_{j-1})+(\text{terms in}\ C_{j-1}) \\ &=d\nu_{j-1}+\mu', \end{align*} where $\mu'$ is some element of $C_{j-1}$. From $d\mu=0$, we have $d\mu'=0$. By the induction hypothesis, we have an element $\nu'\in C_{j-2}$ and $a'\in \mathcal{A}_{bas}$ such that $\mu'=d\nu'+a'\otimes\mathbf{1}$ and $d_\mathcal{A}a'=0$. Therefore we have $\mu=d\nu_{j-1}+\mu'=d(\nu_{j-1}+\nu')+a'\otimes\mathbf{1}$.
It remains to show that the map induced by \eqref{eq: basic into basic otimes W'} on the cohomology is injective. It suffices to show that if $a$ is an element of $\mathcal{A}_{bas}$ such that $d_\mathcal{A}a=0$ and $a\otimes\mathbf{1}=d\nu$ for some $\nu\in \mathcal{A}_{bas}\otimes\mathcal{W}'$ then there exists an element $b$ of $\mathcal{A}_{bas}$ such that $a=d_\mathcal{A}b$. Denote by $\mathcal{W}'^{(i, j)}$ the subspace of $\mathcal{W}'$ of degree $i$ and $j$ with respect to the operators ${j_{bc}}(0)_{\ge0}:=\sum_{i=1}^{\dim \mathfrak{t}}\sum_{n\ge 0}c^{\xi^*_i}_{(-n-1)}b^{\xi_i}_{(n)}$ and ${j_{\beta\gamma}}(0)_{\ge0}:=\sum_{i=1}^{\dim \mathfrak{t}}\sum_{n\ge 0}\gamma^{\xi^*_i}_{(-n-1)}\beta^{\xi_i}_{(n)}$, respectively. Note that $\mathcal{W}'{(0, 0)}=\mathbb{K}\mathbf{1}$ and $\mathcal{W}'=\bigoplus_{i, j \in \mathbb{N}}\mathcal{W}'^{(i, j)}$. Set $ D^{(i, j)}:= \mathcal{A}_{bas}\otimes \mathcal{W}'^{(i, j)}. $ Then we have the following homogeneous operators: \begin{equation}\label{eq: degree of d_1, d_A, d_3} \begin{aligned} d_1: D^{(i, j)}\to D^{(i-1, j+1)}, \\ d_\mathcal{A}\otimes1: D^{(i, j)}\to D^{(i, j)}, \\ d_3: D^{(i, j)}\to D^{(i-1, j)}, \end{aligned} \end{equation} where $d_3:=-\sum_{i=1}^{\dim \mathfrak{t}}\sum_{n\ge 0}\gamma^{\xi^*_i, \mathcal{A}}_{(-n-1)}\otimes b^{\xi_i, \mathcal{W}}_{(n)}$.
Let $a$ be an element of $\mathcal{A}_{bas}$ such that $d_\mathcal{A}a=0$ and $a\otimes \mathbf{1}=d\nu$, where $\nu$ is an element of $\mathcal{A}_{bas}\otimes\mathcal{W}'$. We can write $\nu$ as $ \nu=\sum_{i, j\ge0}\nu^{(i, j)}, $ where $\nu^{(i, j)}$ is an element of $D^{(i, j)}$ and the elements $\nu^{(i, j)}$ are $0$ for all but finitely many $(i, j)$. From $a\otimes\mathbf{1} =d\nu$ and \eqref{eq: degree of d_1, d_A, d_3}, we have \begin{equation}\label{eq: D^(0, 0) term} a\otimes\mathbf{1}=(d_\mathcal{A}\otimes1)\nu^{(0, 0)}+d_3\nu^{(1, 0)}, \end{equation} and \begin{equation}\label{eq: D^(i, j) term} 0=(d_\mathcal{A}\otimes1)\nu^{(i, j)}+d_3\nu^{(i+1, j)}+d_1\nu^{(i+1, j-1)}, \end{equation} for all $i, j\ge0$ with $i>0$ or $j>0$, where we set $\nu^{(i, -1)}:=0$. From \eqref{eq: D^(i, j) term} with $j=0$, we have \begin{equation}\label{eq: relations in D^(i, 0)} 0=(d_\mathcal{A}\otimes1)\nu^{(i, 0)}+d_3\nu^{(i+1, 0)}, \end{equation} for all $i>0$. Note that we have \begin{equation}\label{eq: rewrite d_3} d_3=\Bigl[d_\mathcal{A}\otimes1,\sum_{i=1}^{\dim \mathfrak{t}}\sum_{n\ge0}c^{\xi^*_i, \mathcal{A}}_{(-n-1)}\otimes b^{\xi_i, \mathcal{W}}_{(n)}\Bigr]. \end{equation} Indeed by the definition of chiral $W^*$-modules and the commutativity of $\mathfrak{t}$, we have $ \gamma^{\xi^*_i, \mathcal{A}}_{(-n-1)}=[d_\mathcal{A},c^{\xi^*_i, \mathcal{A}}_{(-n-1)}] $ for any $n\ge0$ and $i=1, \dots, \dim \mathfrak{t}$. The formula \eqref{eq: rewrite d_3} follows from this relation and the definition of $d_3$. We set $S:=\sum_{i=1}^{\dim \mathfrak{t}}\sum_{n\ge0}c^{\xi^*_i, \mathcal{A}}_{(-n-1)}\otimes b^{\xi_i, \mathcal{W}}_{(n)}$. Note that we have $[S, [d_\mathcal{A}\otimes1,S]]=0$. Then we have \begin{equation}\label{eq: d_3v^(1, 0) is exact} d_3\nu^{(1, 0)}=(d_\mathcal{A}\otimes1)\sum_{i=1}^N S^{[i]}\nu^{(i, 0)}-S^{[N]}(d_\mathcal{A}\otimes1)\nu^{(N, 0)}, \end{equation} for any $N>0$. This is proved by induction on $N$ with formulae \eqref{eq: relations in D^(i, 0)}, \eqref{eq: rewrite d_3} and $[S, [d_\mathcal{A}\otimes1,S]]=0$. We have a natural number $N(>0)$ such that $\nu^{(N, 0)}=0$. Therefore from the formulae \eqref{eq: D^(0, 0) term} and \eqref{eq: d_3v^(1, 0) is exact}, we have $ a\otimes\mathbf{1}=(d_\mathcal{A}\otimes1)\Bigl(\nu^{(0, 0)}+\sum_{i=1}^N S^{[i]}\nu^{(i, 0)}\Bigr). $ Notice that $S^{[l]}\nu^{(l, 0)}$ belongs to $D^{(0, 0)}$ since $S$ maps $D^{(i, j)}$ into $D^{(i-1, j)}$. Therefore we have $ \nu^{(0, 0)}+\sum_{i=1}^N S^{[i]}\nu^{(i, 0)}=b\otimes\mathbf{1}, $ for some $b\in \mathcal{A}_{bas}$. Thus we have $a=d_\mathcal{A}b$. This completes the proof. \end{proof}
By the preceding proposition and Proposition \ref{prop: small Cartan model for chiral W^*-modules}, we have the following theorem.
\begin{theorem}\label{thm: CHIRAL BASIC=CHIRAL EQUIVARIANT} Let $G$ be a compact connected Lie group with the Lie algebra $\mathfrak{g}=\mathrm{Lie}(G)^\mathbb{K}$. Let $(\mathcal{A}, d_\mathcal{A}, Y^\mathcal{A})$ be a chiral $W^*$-module with respect to $\mathfrak{g}$. Assume that $G$ is commutative. Then there exists a canonical isomorphism \begin{equation}\label{eq: ch basic = ch equiv} \mathbf{H}_{bas}{(\mathcal{A})}\cong\mathbf{H}_{G}{(\mathcal{A})}. \end{equation} \end{theorem}
\subsection{More on Chiral $W^*$-Modules} Let $\mathfrak{g}$ be a finite-dimensional Lie algebra. Let $(\xi_i)_i$ be a basis of $\mathfrak{g}$ and $(\xi^*_i)_i$ the dual basis for $\mathfrak{g}^*$.
Denote by $\langle c \rangle$ the subalgebra of $\mathcal{W}'$ generated by $c^{\xi^*}_0\mathbf{1}$ with $\xi^*\in\mathfrak{g}$.
\begin{lemma}\label{lem: sufficient condition for d-compatibility} Let $(A, d_\mathcal{A})$ be a differential $\mathfrak{sg}[t]$-module and $Y^\mathcal{A}$ a $\mathcal{W}'$-module structure on $\mathcal{A}$. Assume \begin{equation}\label{eq: d-compatibility for c} Y^\mathcal{A}(d_{\mathcal{W}'}c^{\xi^*}_0\mathbf{1}, z)=[d_\mathcal{A},c^{\xi^*, \mathcal{A}}(z)], \end{equation} for all $\xi^*\in\mathfrak{g}$. Then the following holds: \begin{equation}\label{eq: d-compatibility for x} Y^\mathcal{A}(d_{\mathcal{W}'}x, z)=[d_\mathcal{A},Y^\mathcal{A}(x, z)], \end{equation} for any $x\in \mathcal{W}'$. \end{lemma} \begin{proof} Let $S\subset \mathcal{W}'$ be a subset. Using the fact that $d_{{\mathcal{W}'}}$ commutes with the translation operator, we can check by induction that if \eqref{eq: d-compatibility for x} holds for all $x\in S$ then \eqref{eq: d-compatibility for x} holds for all $x\in \langle S \rangle$. Therefore it suffices to show that \eqref{eq: d-compatibility for x} holds for $x=\gamma^{\xi^*}_0\mathbf{1}$ with $\xi^*\in\mathfrak{g}^*$. Note that \eqref{eq: d-compatibility for x} holds for all $x\in \langle c \rangle$ by the assumption \eqref{eq: d-compatibility for c}. Let $\xi^*\in\mathfrak{g}^*$. From the formula \begin{equation}\label{eq: formula for d_W c} \gamma^{\xi^*}_0\mathbf{1}=d_{\mathcal{W}'}c^{\xi^*}_0\mathbf{1}+\frac{1}{2}\sum_{i=1}^{\dim \mathfrak{g}}c^{\mathrm{ad}^*\xi_i\cdot\xi^*}_0c^{\xi^*_i}_0\mathbf{1}, \end{equation} we have $$ Y^\mathcal{A}(d_{\mathcal{W}'}\gamma^{\xi^*}_0\mathbf{1}, z)=\frac{1}{2}\sum_{i=1}^{\dim \mathfrak{g}}Y^\mathcal{A}(d_{\mathcal{W}'}c^{\mathrm{ad}^*\xi_i\cdot\xi^*}_0c^{\xi^*_i}_0\mathbf{1}, z). $$ Since \eqref{eq: d-compatibility for x} holds for all $x\in \langle c \rangle$, the right-hand side equals $$ \Bigl[d_\mathcal{A},\frac{1}{2}\sum_{i=1}^{\dim \mathfrak{g}}Y^\mathcal{A}(c^{\mathrm{ad}^*\xi_i\cdot\xi^*}_0c^{\xi^*_i}_0\mathbf{1}, z)\Bigr]. $$ From \eqref{eq: d-compatibility for c} and \eqref{eq: formula for d_W c}, we can see this is equal to $[d_\mathcal{A},\gamma^{\xi^*, \mathcal{A}}(z)]$. \end{proof}
The following proposition is useful for checking that a differential $\mathfrak{sg}[t]$-module with a $\mathcal{W}'$-module structure is a chiral $W^*$-module.
\begin{proposition}\label{prop: sufficient condition for chiral W^*-modules} Let $(A, d_\mathcal{A})$ be a differential $\mathfrak{sg}[t]$-module and $Y^\mathcal{A}$ a $\mathcal{W}'$-module structure on $\mathcal{A}$. Assume the following: \begin{gather} \label{assump: [iota,gamma]=0} [\iota^\mathcal{A}_\xi(z)_{-}, \gamma^{\xi^*, \mathcal{A}}(w)]=0, \\ \label{assump: Y(dc)=[d,Y(c)]} Y^\mathcal{A}(d_{\mathcal{W}'}c^{\xi^*}_0\mathbf{1}, z)=[d_\mathcal{A},c^{\xi^*, \mathcal{A}}(z)], \\ \label{assump: [iota,c]=delta} [\iota^\mathcal{A}_\xi(z)_{-}, c^{\xi^*, \mathcal{A}}(w)]=\langle \xi^*, \xi \rangle \delta(z-w)_{-}, \end{gather} for all $\xi\in\mathfrak{g}$ and $\xi^*\in\mathfrak{g}^*$. Then the triple $(\mathcal{A}, d_\mathcal{A}, Y^\mathcal{A})$ is a chiral $W^*$-module. \end{proposition} \begin{proof} By Lemma \ref{lem: sufficient condition for d-compatibility}, it suffices to check $$ [L_\xi^\mathcal{A}(z)_{-},c^{\xi^*, \mathcal{A}}(w)]=c^{\mathrm{ad}^*\xi\cdot\xi^*, \mathcal{A}}(w)\delta(z-w)_{-}, $$ for all $\xi\in\mathfrak{g}$ and $\xi^*\in\mathfrak{g}^*$. Let $\xi\in\mathfrak{g}$ and $\xi^*\in\mathfrak{g}^*$. Applying $\mathrm{ad}(d_\mathcal{A})$ to the both sides of \eqref{assump: [iota,c]=delta}, we have $$ [d_\mathcal{A},[\iota^\mathcal{A}_\xi(z)_{-},c^{\xi^*, \mathcal{A}}(w)]]=0. $$ Therefore from \eqref{assump: Y(dc)=[d,Y(c)]} and $[d_\mathcal{A},\iota^\mathcal{A}_\xi(z)_{-}]=L_\xi^\mathcal{A}(z)_{-}$, we have $$ [L_\xi^\mathcal{A}(z)_{-},c^{\xi^*, \mathcal{A}}(w)]=[\iota_\xi^\mathcal{A}(z)_{-},Y^\mathcal{A}(d_{\mathcal{W}'}c^{\xi^*}_0\mathbf{1}, w)]. $$ By \eqref{assump: [iota,c]=delta}, \eqref{assump: [iota,gamma]=0} and the formula $d_{\mathcal{W}'}c^{\xi^*}_0\mathbf{1}=\gamma^{\xi^*}_0\mathbf{1}-1/2\sum_{i=1}^{\dim \mathfrak{g}}c^{\mathrm{ad}^*\xi_i\cdot\xi^*}_0c^{\xi^*_i}_0\mathbf{1}$, we can see the right-hand side equals $$ -\frac{1}{2}\sum_{i=1}^{\dim \mathfrak{g}}\langle \mathrm{ad}^* \xi_i \cdot\xi^*, \xi \rangle \delta(z-w)_{-}c^{\xi^*_i}(w)+ \frac{1}{2}\sum_{i=1}^{\dim \mathfrak{g}}c^{\mathrm{ad}^*\xi_i\cdot\xi^*}(w)\langle \xi^*_i, \xi\rangle \delta(z-w)_{-}. $$ This is just equal to $c^{\mathrm{ad}^*\xi\cdot\xi^*, \mathcal{A}}(w)\delta(z-w)_{-}$. \end{proof}
The following is useful when we equip a differential $\mathfrak{sg}[t]$-module with a chiral $W^*$-module structure.
\begin{proposition}\label{prop: construction of chiral W^*-modules} Let $(\mathcal{A}, d_\mathcal{A})$ be a differential $\mathfrak{sg}[t]$-module. Suppose given a module structure of the vertex superalgebra $\langle c \rangle$ on $\mathcal{A}$ $$ Y^\mathcal{A}_0: \langle c \rangle \to (\mathrm{End} \mathcal{A})[[z^{\pm1}]], $$ such that \begin{equation}\label{eq: [c,[d,c]]=0} [ c^{\xi^*, \mathcal{A}}(z),[d_\mathcal{A}, c^{\eta^*, \mathcal{A}}(w)]]=0, \end{equation} for all $\xi^*, \eta^*\in \mathfrak{g}^*$. Then there exists a unique $\langle c, \gamma \rangle$-module structure on $\mathcal{A}$, $$ Y^\mathcal{A}: \langle c, \gamma \rangle \to (\mathrm{End} \mathcal{A})[[z^{\pm1}]], $$
such that $Y^\mathcal{A}|_{\langle c \rangle}=Y^\mathcal{A}_0$ and $[d_\mathcal{A}, Y^\mathcal{A}(x, z)]=Y^\mathcal{A}(d_{\mathcal{W}'}x, z)$ for all $x\in \mathcal{W}'$. Moreover if the operation $Y^\mathcal{A}$ satisfies \begin{align} [\iota^\mathcal{A}_\xi(z)_{-}, c^{\xi^*, \mathcal{A}}(w)]&=\langle \xi^*, \xi \rangle \delta(z-w)_{-},\\ \label{eq: curvature is horizontal in prop: construction of chiral W^*-modules} [\iota^\mathcal{A}_\xi(z)_{-},\gamma^{\xi^*, \mathcal{A}}(w)]&=0, \end{align} for all $\xi \in \mathfrak{g}$ and $\xi^*\in\mathfrak{g}^*$, then the triple $(\mathcal{A}, d_\mathcal{A}, Y^\mathcal{A})$ is a chiral $W^*$-module. \end{proposition} \begin{proof} The uniqueness of the operation $Y^\mathcal{A}$ follows from the formula $d_{\mathcal{W}'} c^j_0\mathbf{1}=\gamma^j_0\mathbf{1}-1/2\sum_{i, k=1}^{\dim{g}}\Gamma_{i k}^j c^{i}_0c^{k}_0\mathbf{1}$, where $\Gamma_{i j}^k$ is the structure constants of the Lie algebra $\mathfrak{g}$, that is, $[\xi_i, \xi_j]=\sum_{k=1}^{\dim \mathfrak{g}}\Gamma_{i j}^k\xi_k$ for $i, j=1, \dots, \dim \mathfrak{g}$. We check the existence of such a $\mathcal{W}'$-module structure $Y^\mathcal{A}$. We set \begin{equation}\label{eq: df of gamma(z)} \gamma^{\xi^*_j, \mathcal{A}}(z):=[d_\mathcal{A}, c^{\xi^*_j, \mathcal{A}}(z)]+\frac{1}{2}\sum_{i, k=1}^{\dim \mathfrak{g}}\Gamma_{i k}^j \,{\baselineskip0pt\lineskip0.3pt\vcenter{\hbox{$\cdot$}\hbox{$\cdot$}}\,c^{\xi^*_i, \mathcal{A}}(z) c^{\xi^*_k, \mathcal{A}}(z)\,\vcenter{\hbox{$\cdot$}\hbox{$\cdot$}}}, \end{equation} for $j=1, \dots, \dim \mathfrak{g}$.
Note that these operators have even parity. We check $[c^{\xi^*_i, \mathcal{A}}(z), \gamma^{\xi^*_j, \mathcal{A}}(w)]=0$ and $[\gamma^{\xi^*_i, \mathcal{A}}(z),\gamma^{\xi^*_j, \mathcal{A}}(w)]=0$ for $i, j=1, \dots, \dim \mathfrak{g}$. This implies the existence of a $\mathcal{W}'$-module structure $Y^\mathcal{A}$ such that $Y^\mathcal{A}|_{\langle c \rangle}=Y^\mathcal{A}_0$. Applying $\mathrm{ad}( d_\mathcal{A})$ to the both sides of \eqref{eq: [c,[d,c]]=0}, we have \begin{equation}\label{eq: [[d,c],[d,c]]=0} \bigl[[d_\mathcal{A},c^{\xi^*, \mathcal{A}}(z)],[d_\mathcal{A},c^{\eta^*, \mathcal{A}}(w)]\bigr]=0, \end{equation} for all $\xi^*, \eta^*\in\mathfrak{g}^*$. We have
$ [\gamma^{\xi^*_j, \mathcal{A}}(z),\gamma^{\xi^*_{\Tilde{j}}, \mathcal{A}}(w)]=0 $
by \eqref{eq: [c,[d,c]]=0} and \eqref{eq: [[d,c],[d,c]]=0}. The formula $[\gamma^{\xi^*_i, \mathcal{A}}(z),\gamma^{\xi^*_j, \mathcal{A}}(w)]=0$ follows directly from \eqref{eq: [c,[d,c]]=0}. Thus we have a $\mathcal{W}'$-module structure $Y^\mathcal{A}$ such that $Y^\mathcal{A}|_{\langle c \rangle}=Y^\mathcal{A}_0$. It remains to check $[d_\mathcal{A},Y^\mathcal{A}(x, z)]=Y^\mathcal{A}(d_{\mathcal{W}'} x, z)$ for all $x\in \mathcal{W}'$. By the construction of $Y^\mathcal{A}$, this holds for $x=c^{\xi^*}_0\mathbf{1}$ with $\xi^*\in\mathfrak{g}^*$. Therefore by Lemma \ref{lem: sufficient condition for d-compatibility}, it holds for all $x\in\mathcal{W}'$. Thus we proved the existence part. The latter half of our assertion follows from Proposition \ref{prop: sufficient condition for chiral W^*-modules}. \end{proof}
\begin{remark} The condition \eqref{eq: curvature is horizontal in prop: construction of chiral W^*-modules} in Proposition \ref{prop: construction of chiral W^*-modules} can be replaced by the following condition: \begin{equation} [L^\mathcal{A}_\xi(z)_-, c^{\xi^*, \mathcal{A}}(w)]= c^{\mathrm{ad}^*\xi\cdot\xi^*}(w)\delta(z-w)_{-}, \end{equation} for all $\xi\in\mathfrak{g}$ and $\xi^*\in\mathfrak{g}^*$. \end{remark}
\section{VSA-Inductive Sheaves}\label{section: VSA-inductive Sheaves} In this section, we introduce the VSA-inductive sheaves. In the next section, we will construct a vertex-algebraic analogue of the Lie algebroid complex as a VSA-inductive sheaf.
\subsection{Ind-Objects} Let $\mathcal{C}$ be a category. Recall the category $\mathrm{Ind} (\mathcal{C})$ of ind-objects of $C$ introduced by Grothendieck (\cite{AGV72}). An \textit{inductive system} of $\mathcal{C}$ is a functor $$ X: A \to \mathcal{C}, \quad \mathrm{Ob}(A)\ni \alpha \mapsto X(\alpha)=X_\alpha \in \mathrm{Ob}(\mathcal{C}), $$ from a small filtered category $A$ to $\mathcal{C}$. An inductive system $X: A \to \mathcal{C}$ is also written as $(X_\alpha)_{\alpha\in A}$. An \textit{ind-object} associated to an inductive system $(X_\alpha)_{\alpha\in A}$ is a symbol $``\displaystyle\varinjlim_{\alpha\in A}" X_\alpha$. The objects of the category $\mathrm{Ind}(\mathcal{C})$ are the ind-objects of $\mathcal{C}$. We often express $``\displaystyle\varinjlim_{\alpha\in A}" X_\alpha$ by the corresponding functor $X$ like $X=``\displaystyle\varinjlim_{\alpha\in A}" X_\alpha$. The morphisms of $\mathrm{Ind}(\mathcal{C})$ are defined by $$ \mathrm{Hom}_{\mathrm{Ind}({\mathcal{C}})}(``\varinjlim_{\alpha\in A}" X_\alpha, ``\varinjlim_{\beta\in B}" Y_\beta):=\varprojlim_{\alpha\in A}\varinjlim_{\beta\in B}\mathrm{Hom}_{\mathcal{C}}(X_\alpha, Y_\beta), $$ where the limits in the right-hand side stand for those in the category of sets. For a morphisms of ind-objects $F: (X_\alpha)_{\alpha\in A}\to (Y_\beta)_{\beta\in B}$, $F$ is written as $ F=\bigl([F_\alpha^{j(\alpha)}]\bigr)_{\alpha\in A}, $ where $j: \mathrm {Ob}(A) \to \mathrm{Ob}(B)$ is a map and $[F_\alpha^{j(\alpha)}]$ is an equivalence class of a morphism $F_\alpha^{j(\alpha)}: X_\alpha \to Y_{j(\alpha)}$ in $\displaystyle\varinjlim_{\beta\in B} \mathrm{Hom}_{\mathcal{C}}(X_\alpha, Y_\beta)$. The composition is defined by $ F\circ G := \bigl([F_{j_G(\alpha)}^{j_F(j_G(\alpha))}\circ G_\alpha^{j_G(\alpha)}]\bigr)_{\alpha\in A} $ for morphisms of ind-objects $F=\bigl([F_\beta^{j_F(\beta)}]\bigr)_{\beta\in B}: (Y_\beta)_{\beta \in B}\to (Z_\gamma)_{\gamma\in \Gamma}$ and $G=\bigl([G_\alpha^{j_G(\alpha)}]\bigr)_{\alpha\in A}: (X_\alpha)_{\alpha\in A}\to (Y_\beta)_{\beta\in B}$. For a small filtered category $A$, we will use the notation $f_{\alpha' \alpha}$ to express a morphism $f\in \mathrm{Hom}_A(\alpha, \alpha')$, emphasizing the source and the target.
Let $\mathit{Presh}_X(\mathit{Vec}^{\mathrm{super}}_{\mathbb{K}})$ be the category of presheaves on a topological space $X$ of super vector spaces over $\mathbb{K}$. Consider the category of ind-objects of the category $\mathit{Presh}_X(\mathit{Vec}^{\mathrm{super}}_{\mathbb{K}})$, $\mathrm{Ind}(\mathit{Presh}_X(\mathit{Vec}^{\mathrm{super}}_{\mathbb{K}}))$. There exists a functor
\begin{equation}\label{eq: functor IndPresh to Presh} \underrightarrow{\mathrm{Lim}}\, : \mathrm{Ind}(\mathit{Presh}_X(\mathit{Vec}^{\mathrm{super}}_\mathbb{K}))\to \mathit{Presh}_X(\mathit{Vec}^{\mathrm{super}}_{\mathbb{K}}), \end{equation} sending an object $\mathcal{F}=``\displaystyle\varinjlim_{\alpha\in A}"\mathcal{F}_\alpha$ to the inductive limit presheaf $\underrightarrow{\mathrm{Lim}}\, \mathcal{F}:=\varinjlim_{\alpha\in A}\mathcal{F}_\alpha$, but not its sheafification, and sending a morphism $F=( [F_\alpha^{j(\alpha)}])_{\alpha\in A}: ``\displaystyle\varinjlim_{\alpha\in A}"\mathcal{F}_\alpha \to ``\displaystyle\varinjlim_{\beta\in B}"\mathcal{G}_\beta$ to the morphism of presheaves $\underrightarrow{\mathrm{Lim}}\, F$ defined by
\begin{equation}\label{eq: DEFINITION OF injlimF} \underrightarrow{\mathrm{Lim}}\, F(U): \varinjlim_{\alpha\in A}\mathcal{F}_\alpha(U) \to \varinjlim_{\beta \in B}\mathcal{G}_\beta(U), \quad [x_{\alpha}] \mapsto [F_{\alpha}^{j(\alpha)}x_{\alpha}], \end{equation} for each open subset $U \subset X$. Set \begin{multline}\label{df: bilinear morphisms of ind-objects} \mathrm{Bilin}_{\mathrm{Ind}(\mathit{Presh}_X(\mathit{Vec}^{\mathrm{super}}_\mathbb{K}))}\bigl(``\varinjlim_{\alpha\in A}"\mathcal{F}_\alpha, ``\varinjlim_{\beta\in B}"\mathcal{G}_\beta; ``\varinjlim_{\gamma\in\Gamma}"\mathcal{H}_\gamma\bigr) \\ :=\varprojlim_{(\alpha, \beta)\in A\times B} \Biggl(\varinjlim_{\gamma\in\Gamma}\mathrm{Bilin}_{\mathit{Presh}_X(\mathit{Vec}^{\mathrm{super}}_{\mathbb{K}})}(\mathcal{F}_\alpha, \mathcal{G}_\beta; \mathcal{H}_\gamma)\Biggr), \end{multline} for ind-objects $``\displaystyle\varinjlim_{\alpha\in A}"\mathcal{F}_\alpha, ``\varinjlim_{\beta\in B}"\mathcal{G}_\beta, ``\varinjlim_{\gamma\in\Gamma}"\mathcal{H}_\gamma$ of the category $\mathit{Presh}_X(\mathit{Vec}^{\mathrm{super}}_{\mathbb{K}})$, where $\mathrm{Bilin}_{\mathit{Presh}_X(\mathit{Vec}^{\mathrm{super}}_{\mathbb{K}})}(\mathcal{F}_\alpha, \mathcal{G}_\beta; \mathcal{H}_\gamma)$ is the set of all bilinear morphisms of presheaves from $\mathcal{F}_\alpha\times \mathcal{G}_\beta$ to $\mathcal{H}_\gamma$. We call an element $F$ of the set \eqref{df: bilinear morphisms of ind-objects} a \textit{bilinear morphism} of ind-objects and write it as $$ F: ``\varinjlim_{\alpha\in A}"\mathcal{F}_\alpha\times ``\varinjlim_{\beta\in B}"\mathcal{G}_\beta\to ``\varinjlim_{\gamma\in\Gamma}"\mathcal{H}_\gamma. $$ Each ind-object $``\displaystyle\varinjlim_{\gamma\in\Gamma}"\mathcal{H}_\gamma$ gives rise to a canonical contravariant functor \begin{align*} &\mathrm{Ind}(\mathit{Presh}_X(\mathit{Vec}^{\mathrm{super}}_{\mathbb{K}})\bigr)\times \mathrm{Ind}(\mathit{Presh}_X(\mathit{Vec}^{\mathrm{super}}_{\mathbb{K}}))\to \mathit{Set}, \\ &\Bigl(``\varinjlim_{\alpha\in A}"\mathcal{F}_\alpha, ``\varinjlim_{\beta\in B}"\mathcal{G}_\beta \bigr)\mapsto \mathrm{Bilin}_{\mathrm{Ind}(C)}\bigl(``\varinjlim_{\alpha\in A}"\mathcal{F}_\alpha, ``\varinjlim_{\beta\in B}"\mathcal{G}_\beta; ``\varinjlim_{\gamma\in\Gamma}"\mathcal{H}_\gamma\bigr), \end{align*} where $\mathit{Set}$ is the category of sets. Similarly, each pair of ind-objects $\Bigl(``\displaystyle\varinjlim_{\alpha\in A}"\mathcal{F}_\alpha,$ $ ``\displaystyle\varinjlim_{\beta\in B}"\mathcal{G}_\beta \bigr)$ induces a canonical covariant functor \begin{align*} \mathrm{Ind}(\mathit{Presh}_X(\mathit{Vec}^{\mathrm{super}}_{\mathbb{K}})) &\to \mathit{Set}, \\
``\varinjlim_{\gamma\in\Gamma}"\mathcal{H}_\gamma &\mapsto \mathrm{Bilin}_{\mathrm{Ind}(C)}\bigl(``\varinjlim_{\alpha\in A}"\mathcal{F}_\alpha, ``\varinjlim_{\beta\in B}"\mathcal{G}_\beta; ``\varinjlim_{\gamma\in\Gamma}"\mathcal{H}_\gamma\bigr). \end{align*} Moreover, a bilinear morphism of ind-objects $$ F=\bigl([F_{(\alpha, \gamma)}^{j(\alpha, \gamma)}]\bigr)_{(\alpha, \gamma)\in A\times B}: ``\varinjlim_{\alpha\in A}"\mathcal{F}_\alpha\times ``\varinjlim_{\beta\in B}"\mathcal{G}_\beta\to ``\varinjlim_{\gamma\in\Gamma}"\mathcal{H}_\gamma, $$ induces a bilinear morphism of presheaves $$ \underrightarrow{\mathrm{Lim}}\, F: \varinjlim_{\alpha\in A}\mathcal{F}_\alpha\times \varinjlim_{\beta\in B}\mathcal{G}_\beta\to \varinjlim_{\gamma\in\Gamma}\mathcal{H}_\gamma, $$ in the same way as in \eqref{eq: DEFINITION OF injlimF}.
Let $\mathit{Sh}_X(\mathit{Vec}^{\mathrm{super}}_{\mathbb{K}})$ be the full subcategory of $\mathit{Presh}_X(\mathit{Vec}^{\mathrm{super}}_{\mathbb{K}})$ consisting of sheaves on $X$ of super vector spaces over $\mathbb{K}$. Then the corresponding category of ind-objects $\mathrm{Ind}(\mathit{Sh}_X(\mathit{Vec}^{\mathrm{super}}_{\mathbb{K}}))$ is a full subcategory of $\mathrm{Ind}(\mathit{Presh}_X(\mathit{Vec}^{\mathrm{super}}_{\mathbb{K}}))$. Note that the category $\mathrm{Ind}(\mathit{Sh}_X(\mathit{Vec}^{\mathrm{super}}_{\mathbb{K}}))$ is bigger than the category of \textit{ind-sheaves} introduced by Kashiwara and Schapira (\cite{KS99}). The latter is the category of ind-objects of the category of sheaves with compact supports.
\subsection{Definition of VSA-Inductive Sheaves}
We denote by $\mathbb{K}_X$ the presheaf on a topological space $X$ of constant $\mathbb{K}$-valued functions. We regard $\mathbb{K}_X$ as an inductive system indexed by a set with one element and denote by $``\displaystyle\varinjlim"\mathbb{K}_X$ the corresponding ind-object.
\begin{definition} A \textbf{vertex superalgebra inductive sheaf (VSA-inductive sheaf)} on a topological space $X$ is a quadruple $\bigl( \mathcal{F}, \underline {\mathbf{1}}, \underline{T}, \underline{(n)}; n\in \mathbb{Z}\bigr)$ consisting of \begin{enumerate}[$\bullet$]
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\setlength{\itemindent}{5pt} \item an ind-object of $\mathit{Sh}_X(\mathit{Vec}^{\mathrm{super}}_{\mathbb{K}})$, $ \mathcal{F}=``\displaystyle\varinjlim_{\alpha\in A}"\mathcal{F}_\alpha, $ \item an even morphism of ind-objects $\underline{\mathbf{1}}: ``\varinjlim"\mathbb{K}_X\to \mathcal{F}$, \item an even morphism of ind-objects $\underline{T}: \mathcal{F}\to\mathcal{F}$, \item an even bilinear morphisms of ind-objects $\underline{(n)}: \mathcal{F}\times \mathcal{F} \to \mathcal{F}$, \end{enumerate} such that the map $\mathcal{F}(f)$ is even and injective for any morphism $f$ in $A$ and the quadruples $ \bigl(\varinjlim_{\alpha\in A}\mathcal{F}_\alpha(U), \mathbf{1}, T, (n); n\in\mathbb{Z}\bigr) $ are vertex superalgebras for all open subsets $U\subset X$, where $\mathbf{1}=\mathbf{1}(U):=\bigl(\displaystyle\underrightarrow{\mathrm{Lim}}\, {\underline{\mathbf{1}}}(U)\bigr)(1)$, $T=T(U):=\displaystyle\underrightarrow{\mathrm{Lim}}\, \underline{T}(U)$, $(n)=(n)(U):=\displaystyle\underrightarrow{\mathrm{Lim}}\, \underline{(n)}(U)$. \end{definition}
Let $\mathcal{V}_1=\bigl( \mathcal{F}_1, \underline {\mathbf{1}}_1, \underline{T}_1, \underline{(n)}_1; n\in \mathbb{Z}\bigr)$ and $\mathcal{V}_2=\bigl( \mathcal{F}_2, \underline {\mathbf{1}}_2, \underline{T}_2, \underline{(n)}_2; n\in \mathbb{Z}\bigr)$ be VSA-inductive sheaves on the same topological space $X$. We call a morphism of ind-objects $\Phi=([\Phi_\alpha^{j(\alpha)}])_{\alpha\in \mathrm{Ob}(A)}: \mathcal{F}_1\to \mathcal{F}_2$ a \textbf{base-preserving morphism} of VSA-inductive sheaves from $\mathcal{V}_1$ to $\mathcal{V}_2$ if $\Phi$ satisfies $\Phi\circ\underline{\mathbf{1}}_1=\underline{\mathbf{1}}_2$, $\Phi\circ\underline{T}_1=\underline{T}_2\circ\Phi$ and $\Phi\circ\underline{(n)}_1=\underline{(n)}_2\circ(\Phi\times\Phi)$ for all $n\in\mathbb{Z}$, where $\Phi\times\Phi$ is the morphism of ind-objects given by $\Phi\times\Phi:=\bigl([\Phi_\alpha^{j(\alpha)}\times\Phi_{\alpha'}^{j(\alpha')}]\bigr)_{(\alpha, \alpha')\in\mathrm{Ob}(A)\times\mathrm{Ob}(A)}$.
\begin{remark}\label{rem: VSA-inductive sheaves form a category} Let $X$ be a topological space. VSA-inductive sheaves on $X$ form a category with base-preserving morphisms of VSA-inductive sheaves. This category is a subcategory of $\mathrm{Ind}(\mathit{Sh}_{X}(\mathit{Vec}^{\mathrm{super}}_{\mathbb{K}}))$ and hence of $\mathrm{Ind}(\mathit{Presh}_{X}(\mathit{Vec}^{\mathrm{super}}_{\mathbb{K}}))$. \end{remark}
\begin{notation} Denote by $\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}_X$ the category of VSA-inductive sheaves on a topological space $X$ obtained in Remark \ref{rem: VSA-inductive sheaves form a category}. \end{notation}
\begin{lemma}\label{lem: PRESHEAF associated with A VSA-inductive sheaf} Let $\mathcal{V}=\bigl( \mathcal{F}, \underline {\mathbf{1}}, \underline{T}, \underline{(n)}; n\in \mathbb{Z}\bigr)$ be a VSA-inductive sheaf on a topological space $X$. Then the assignment $$ U\to \bigl(\underrightarrow{\mathrm{Lim}}\, \mathcal{F}(U), \mathbf{1}, T, (n); n\in\mathbb{Z}\bigr), $$ with restriction maps of the presheaf $\underrightarrow{\mathrm{Lim}}\, \mathcal{F}$ defines a presheaf on $X$ of vertex superalgebras. \end{lemma} \begin{proof} We must check the restriction maps are vertex superalgebra morphisms. We can see this since $\displaystyle\underrightarrow{\mathrm{Lim}}\, {\underline{\mathbf{1}}}$, $\displaystyle\underrightarrow{\mathrm{Lim}}\, \underline{T}$, $\displaystyle\underrightarrow{\mathrm{Lim}}\, \underline{(n)}$ are morphisms of presheaves. \end{proof}
Let $\mathcal{V}$ and $\mathcal{V}'$ be VSA-inductive sheaves. We write a morphism $\Phi$ as $\Phi: \mathcal{V}\to\mathcal{V}'$ even when $\Phi$ is not a morphism of VSA-inductive sheaves but simply a morphism of the underlying ind-objects of sheaves. When we say that $\Phi: \mathcal{V}\to\mathcal{V}'$ is a morphism of ind-objects, we mean $\Phi$ is a morphism between the underlying ind-objects of sheaves.
\begin{lemma} Let $\Phi: \mathcal{V}_1\to \mathcal{V}_2$ be a base-preserving morphism of VSA-inductive sheaves on the same topological space. Then the map $\underrightarrow{\mathrm{Lim}}\, \Phi: \underrightarrow{\mathrm{Lim}}\, {\mathcal{V}_1} \to \underrightarrow{\mathrm{Lim}}\, {\mathcal{V}_2}$ is a morphism of presheaves of vertex superalgebras. \end{lemma} \begin{proof} This follows directly from the definition of the morphisms. \end{proof}
\begin{remark} When we restrict the functor given in \eqref{eq: functor IndPresh to Presh} $$ \underrightarrow{\mathrm{Lim}}\, : \mathrm{Ind}(\mathit{Presh}_{X}(\mathit{Vec}^{\mathrm{super}}_{\mathbb{K}})) \to \mathit{Presh}_X(\mathit{Vec}_{\mathbb{K}}^{\mathrm{super}}), $$ to the subcategory $\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}_X$, we have a functor \begin{equation}\label{eq: functor from VSA-ISh} \mathit{VSA_{\mathbb{K}}}\textit{-IndSh}_X \to \mathit{Presh}_X(\mathit{VSA_{\mathbb{K}}}), \end{equation} where $\mathit{Presh}_X(\mathit{VSA_{\mathbb{K}}})$ is the category of presheaves on $X$ of vertex superalgebras over $\mathbb{K}$. \end{remark} We also denote by $\underrightarrow{\mathrm{Lim}}\, $ the functor \eqref{eq: functor from VSA-ISh}.
\begin{remark}\label{rem: LOCAL CALCULATION OF THE PRESHEAF associated with A Vsa IndSh} Let $\mathcal{V}=\bigl( \mathcal{F}, \underline {\mathbf{1}}, \underline{T}, \underline{(n)}; n\in \mathbb{Z}\bigr)$ be a VSA-inductive sheaf. Let $U\subset X$ be an open subset and $U=\bigcup_{\lambda\in\Lambda}U_{\lambda}$ an open covering. Then the map induced by restriction maps $$ \underrightarrow{\mathrm{Lim}}\, {\mathcal{V}}(U)\to \prod_{\lambda \in\Lambda}\underrightarrow{\mathrm{Lim}}\, {\mathcal{V}}(U_\lambda), $$ is injective since $\mathcal{F}_\alpha$ are sheaves, where $\mathcal{F}= ``\displaystyle\varinjlim_{\alpha\in A}"\mathcal{F}_\alpha$, and the map $\mathcal{F}(f)$ is injective for any morphism $f$ in $A$. \end{remark}
\begin{definition}\label{df: grading operator on a VSA-inductive sheaf} Let $\mathcal{V}=\bigl( \mathcal{F}=``\displaystyle\varinjlim_{\alpha\in A}"\mathcal{F}_{\alpha}, \underline {\mathbf{1}}, \underline{T}, \underline{(n)}; n\in \mathbb{Z}\bigr)$ be a VSA-inductive sheaf on a topological space $X$. \begin{enumerate}[(i)]
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\renewcommand{\upshape}{\upshape} \item A \textbf{Hamiltonian} or a \textbf{weight-grading operator} on $\mathcal{V}$ is an even morphism $\underline{H}: \mathcal{F}\to\mathcal{F}$ of ind-objects such that there exists a family $(H^\alpha_\alpha: \mathcal{F}_\alpha\to\mathcal{F}_\alpha)_{\alpha\in \mathrm{Ob}(A)}$ of even morphisms of sheaves satisfying the following conditions: \begin{enumerate}[$(1)$]
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\setlength{\itemindent}{0pt} \item $\underline{H}=([H_\alpha^\alpha])_{\alpha\in\mathrm{Ob}(A)}$. \item Each $H_\alpha^\alpha$ is a diagonalizable operator on $\mathcal{F}_\alpha^\alpha$, namely, the operator $H_\alpha^\alpha(U): \mathcal{F}_\alpha(U)\to\mathcal{F}_\alpha(U)$ is diagonalizable for any open subset $U\subset X$. \item For all $n\in \mathbb{Z}$, \begin{equation}\label{eq: Hamiltonian} \underline{(n)}\circ(\mathrm{id}\times\underline{H}+\underline{H}\times\mathrm{id})=(\underline{H}-(-n-1))\circ\underline{(n)}. \end{equation} \end{enumerate} \item A \textbf{degree-grading operator} on $\mathcal{V}$ is an even morphism $\underline{J}: \mathcal{F}\to\mathcal{F}$ of ind-objects such that there exists a family $(J^\alpha_\alpha: \mathcal{F}_\alpha\to\mathcal{F}_\alpha)_{\alpha\in \mathrm{Ob}(A)}$ of even morphisms of sheaves satisfying the following conditions: \begin{enumerate}[$(1)$]
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\setlength{\itemindent}{0pt} \item $\underline{J}=([J_\alpha^\alpha])_{\alpha\in\mathrm{Ob}(A)}$. \item Each $J_\alpha^\alpha$ is a diagonalizable operator on $\mathcal{F}_\alpha^\alpha$. \item For all $n\in \mathbb{Z}$, \begin{equation}\label{eq: degree-grading operator} \underline{(n)}\circ(\mathrm{id}\times\underline{J}+\underline{J}\times\mathrm{id})=\underline{J}\circ\underline{(n)}. \end{equation} \end{enumerate} \end{enumerate} \end{definition}
\begin{remark} In the above definition, the family $(H^\alpha_\alpha)_{\alpha\in\mathrm{Ob}(A)}$ and $(J^\alpha_\alpha)_{\alpha\in\mathrm{Ob}(A)}$ are unique by the relation $(1)$ and the injectivity of the morphisms in the inductive system $\mathcal{F}$. \end{remark}
\begin{definition} \begin{enumerate}[(i)]
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\renewcommand{\upshape}{\upshape} \item A $\mathbb{Z}$-\textbf{graded VSA-inductive sheaf} is a VSA-inductive sheaf given a Hamiltonian with only integral eigenvalues. \item A \textbf{degree-graded VSA-inductive sheaf} is a VSA-inductive sheaf given a degree-grading operator with only integral eigenvalues. \item A VSA-inductive sheaf $\mathcal{V}=\bigl( \mathcal{F}=``\displaystyle\varinjlim_{\alpha\in A}"\mathcal{F}_{\alpha}, \underline {\mathbf{1}}, \underline{T}, \underline{(n)}; n\in \mathbb{Z}\bigr)$ is said to be \textbf{degree-weight-graded} if $\mathcal{V}$ is given a Hamiltonian $\underline{H}=([H_\alpha^\alpha])_{\alpha\in\mathrm{Ob}(A)}$ and a degree-grading operator $\underline{J}=([J_\alpha^\alpha])_{\alpha\in\mathrm{Ob}(A)}$ such that for each $\alpha\in\mathrm{Ob}(A)$, $\mathcal{F}_\alpha$ $=\bigoplus_{n, l\in\mathbb{Z}} \mathcal{F}_\alpha^l[n]$, where $\mathcal{F}_\alpha^l[n]:=\mathcal{F}_\alpha[n]\cap\mathcal{F}_\alpha^l$. Here $\mathcal{F}_\alpha[n]$ and $\mathcal{F}_\alpha^l$ are the subsheaves $\mathrm{Ker}(H_\alpha^\alpha-n)$ and $\mathrm{Ker}(J_\alpha^\alpha-l)$, respectively. \end{enumerate} \end{definition}
\begin{lemma} If $(\mathcal{V}, \underline{H})$ is a $\mathbb{Z}$-graded VSA-inductive sheaf, then $(\varinjlim\mathcal{V}, \underrightarrow{\mathrm{Lim}}\, \underline{H})$ is a presheaf of $\mathbb{Z}$-graded vertex superalgebras. The same type of assertion holds in the degree-graded case and in the degree-weight-graded case. \end{lemma} \begin{proof} We will prove the assertion in the weight-graded case. The others are proved similarly. Let $\mathcal{V}=\bigl( \mathcal{F}=``\displaystyle\varinjlim_{\alpha\in A}"\mathcal{F}_{\alpha}, \underline {\mathbf{1}}, \underline{T}, \underline{(n)}; n\in \mathbb{Z}\bigr)$ be a $\mathbb{Z}$-graded VSA-inductive sheaf with a Hamiltonian $\underline{H}=([H_\alpha^\alpha])_{\alpha\in\mathrm{Ob}(A)}$. Let $f_{\alpha, \alpha'}: \alpha'\to\alpha$ be a morphism in $A$. Then the corresponding morphism $\mathcal{F}(f_{\alpha, \alpha'}): \mathcal{F}_{\alpha'}\to\mathcal{F}_\alpha$ preserves the grading. Indeed, from the injectivity of the morphisms in the inductive system $\mathcal{F}$ and the relation $H_\alpha^\alpha\circ\mathcal{F}(f_{\alpha, \alpha'})\sim H_{\alpha'}^{\alpha'}$ in $\varinjlim_{\beta\in A}\mathrm{Hom}(\mathcal{F}_{\alpha'}, \mathcal{F}_\beta)$, we have $H_\alpha^\alpha\circ\mathcal{F}(f_{\alpha, \alpha'})=\mathcal{F}(f_{\alpha, \alpha'})\circ H_{\alpha'}^{\alpha'}$. Thus we have $\varinjlim_{\alpha\in A}\mathcal{F}_\alpha$ $=\varinjlim_{\alpha\in A}(\bigoplus_{n\in\mathbb{Z}}\mathcal{F}_\alpha[n])$ $=\bigoplus_{n\in\mathbb{Z}}(\varinjlim_{\alpha\in A}\mathcal{F}_\alpha[n])$. Therefore $\underrightarrow{\mathrm{Lim}}\, \underline{H}$ is a diagonalizable operator with only integral eigenvalues on the presheaf $\underrightarrow{\mathrm{Lim}}\, \mathcal{F}=\varinjlim_{\alpha\in A}\mathcal{F}_\alpha$. From the relation \eqref{eq: Hamiltonian}, the operator $\underrightarrow{\mathrm{Lim}}\, \underline{H}$ is a Hamiltonian on $\underrightarrow{\mathrm{Lim}}\, \mathcal{V}$. \end{proof}
\begin{notation} Denote by $\textit{DegWt-}\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}_X$ the category of degree-weight-graded VSA-inductive sheaves on a topological space $X$. Its morphisms are morphisms of VSA-inductive sheaves on $X$ commuting with the Hamiltonians and the degree-grading operators. \end{notation}
\subsection{Gluing Inductive Sheaves} Let $X$ be a topological space and $A$ a small filtered category. We consider the subcategory of $\textit{DegWt-}\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}_X$ whose objects are degree-weight-graded VSA-inductive sheaves $\mathcal{V}=\bigl( \mathcal{F}, \underline{\mathbf{1}}, \underline{T}, \underline{(n)}; n\in \mathbb{Z}\bigr)$ such that $\mathcal{F}$ is a functor from $A$ and whose morphisms are morphisms $\Phi: \mathcal{F} \to \mathcal{G}$ of degree-weight-graded VSA-inductive sheaves such that there exist a family of morphisms of sheaves $(\Phi_\alpha^\alpha: \mathcal{F}_\alpha \to \mathcal{G}_\alpha)_{\alpha\in A}$ satisfying $F=\bigl([\Phi_\alpha^\alpha]\bigr)_{\alpha\in A}$. We denote this category by $\textit{DegWt-}\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}_X^A$. We will often call a morphism of this category a \textbf{strict morphism}. If two degree-weight-graded VSA-inductive sheaves are isomorphic via a strict isomorphism, by which we mean a isomorphism in $\textit{DegWt-}\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}_X^A$, then we say they are \textbf{strictly isomorphic}. We also call a morphism of ind-objects $\Phi: \mathcal{F} \to \mathcal{G}$, not necessarily a morphism of VSA-inductive sheaves, \textit{strict} if the same condition above is satisfied.
\begin{remark}\label{rem: restriction of VSA-inductive sheaves} Let $\mathcal{V}=\bigl(\mathcal{F}, \underline{\mathbf{1}}, \underline{T}, \underline{(n)}; n\in \mathbb{Z} \bigr)$ be a VSA-inductive sheaf on $X$. Let $U\subset X$ be an open subset. Consider the inductive system \begin{align*}
\mathcal{F}|_U: A &\to \mathit{Sh}_U(\mathit{Vec}^{\mathrm{super}}_{\mathbb{K}}), \\
\text{objects}:\quad \alpha &\mapsto \mathcal{F}_\alpha|_U, \\
\text{morphisms}:\quad f &\mapsto \mathcal{F}(f)|_U, \end{align*}
obtained by restricting $\mathcal{F}$ to $U$. The corresponding ind-object $``\displaystyle\varinjlim_{\alpha\in A}"(\mathcal{F}|_U)_\alpha$ is a VSA-inductive sheaf on $U$ with $\underline{\mathbf{1}}, \underline{T}, \underline{(n)}$ restricted to $U$. \end{remark} For a VSA-inductive sheaf $\mathcal{V}$ on
$X$ and an open subset $U$ of $X$, we denote by $\mathcal{V}|_U$ the VSA-inductive sheaf on $U$ given in Remark \ref{rem: restriction of VSA-inductive sheaves} and call it the \textbf{restriction} of the VSA-inductive sheaf $\mathcal{V}$.
Let us glue VSA-inductive sheaves.
Let $X=\bigcup_{\lambda \in \Lambda}U_\lambda$ be an open covering of $X$ and $(\mathcal{V}^\lambda)_{\lambda\in\Lambda}$ a family of degree-weight-graded VSA-inductive sheaves, where $\mathcal{V}^\lambda=\bigl(\mathcal{F}^\lambda, \underline{\mathbf{1}}^\lambda, \underline{T}^\lambda, \underline{(n)}^\lambda; n\in\mathbb{Z} \bigr)$ is an object of $\textit{DegWt-}\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}_{U_\lambda}^A$. Let $\underline{H}^\lambda$ and $\underline{J}^\lambda$ be the Hamiltonian and the degree-grading operator on $\mathcal{V}^\lambda$, respectively. Suppose given a family of strict isomorphisms of degree-weight-graded VSA-inductive sheaves $(\vartheta_{\lambda \mu}: \mathcal{V}^\mu|_{U_\mu \cap U_\lambda} \to \mathcal{V}^\lambda|_{U_\lambda \cap U_\mu})_{\lambda, \mu \in \Lambda}$ satisfying the following condition: \begin{align*}
(0)\quad \vartheta_{\lambda \lambda}=\mathrm{id},\quad \text{and}\quad (\vartheta_{\lambda \mu}|_{U_\lambda\cap U_\mu\cap U_\nu})\circ(\vartheta_{\mu \nu}|_{U_\mu\cap U_\nu\cap U_\lambda})=(\vartheta_{\lambda \nu}|_{U_\nu\cap U_\lambda\cap U_\mu}), \quad\\ \text{for all}\quad \lambda, \mu, \nu \in \Lambda.\quad\quad\quad\quad \end{align*}
We will often omit the subscripts such as $|_{U_\lambda\cap U_\mu\cap U_\nu}$ in the sequel.
In addition to the condition $(0)$, we assume the following conditions: \begin{enumerate}[$(1)$]
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\setlength{\itemindent}{0pt} \item For any $\alpha, \alpha' \in \mathrm{Ob}(A)$, we have $\mathrm{Hom}_A(\alpha, \alpha')\neq\emptyset$ or $\mathrm{Hom}_A(\alpha', \alpha)\neq\emptyset$, \item There exist a $\alpha_0\in \mathrm{Ob}(A)$ and sheaf morphisms $\underline{\mathbf{1}}^{\lambda, \alpha_0}: \mathbb{K}_X\to \mathcal{F}^\lambda_{\alpha_0}$ with $\lambda\in\Lambda$ such that $$ \underline{\mathbf{1}}^\lambda=[\underline{\mathbf{1}}^{\lambda, \alpha_0}], $$ for all $\lambda\in\Lambda$. (Note that the element $\alpha_0$ can be taken independently of $\lambda\in\Lambda$.) \item There exist a map $j_T: \mathrm{Ob}(A)\to \mathrm{Ob}(A)$ and sheaf morphisms $\underline{T}^{\lambda, j_T(\alpha)}_\alpha: \mathcal{F}^\lambda_\alpha \to \mathcal{F}^\lambda_{j_T(\alpha)}$ with $\alpha\in \mathrm{Ob}(A)$ and $\lambda \in \Lambda$ such that $$ \underline{T}^\lambda=\bigl([\underline{T}^{\lambda, j_T(\alpha)}_\alpha]\bigr)_{\alpha\in \mathrm{Ob}(A)}, $$ for all $\lambda \in \Lambda$. (Note that the map $j_T$ can be taken independently of $\lambda\in \Lambda$.) \item For each $n\in \mathbb{Z}$, there exist a map $j_{(n)}: \mathrm{Ob}(A)\times\mathrm{Ob}(A)\to\mathrm{Ob}(A)$ and bilinear sheaf morphisms $\underline{(n)}_{(\alpha, \alpha')}^{\lambda, j_{(n)}(\alpha, \alpha')}: \mathcal{F}^\lambda_\alpha\times \mathcal{F}^\lambda_{\alpha'}\to \mathcal{F}^\lambda_{j_{(n)}(\alpha, \alpha')}$ with $(\alpha, \alpha')\in \mathrm{Ob}(A)\times \mathrm{Ob}(A)$ and $\lambda\in\Lambda$ such that $$ \underline{(n)}^\lambda=\bigl([\underline{(n)}_{(\alpha, \alpha')}^{\lambda, j_{(n)}(\alpha, \alpha')}]\bigr)_{(\alpha, \alpha')\in \mathrm{Ob}(A)\times\mathrm{Ob}(A)}, $$ for all $\lambda\in\Lambda$. (Note that the map $j_{(n)}$ can be taken independently of $\lambda \in \Lambda$.)
\item The degree and weight-grading on each $\mathcal{F}_\alpha^\lambda$ are bounded from the above and the below uniformly with respect to $\lambda$. Moreover the weight-grading on $\mathcal{F}_\alpha^\lambda$ is bounded from the below uniformly with respect to $\alpha$ as well as $\lambda$. In other words, there exist an integer $N$ and natural numbers $N_\alpha, L_\alpha$ with $\alpha\in\mathrm{Ob}(A)$ such that $\mathcal{F}_\alpha^\lambda=\bigoplus_{N\le n \le N_\alpha}\bigoplus_{|l|\le L_\alpha}(\mathcal{F}_\alpha^\lambda)^l[n]$, where $(\mathcal{F}_\alpha^\lambda)^l[n]$ is the subsheaf of degree $l$ and weight $n$. \end{enumerate}
The uniqueness in the following proposition means that if $\bigl(\mathcal{V}, (\Phi^\lambda: \mathcal{V}|_{U_\lambda}\to\mathcal{V}^\lambda)_{\lambda\in\Lambda}\bigr)$ and $\bigl(\mathcal{V}', (\Phi'^\lambda: \mathcal{V}'|_{U_\lambda}\to\mathcal{V}^\lambda)_{\lambda\in\Lambda}\bigr)$ are the pairs as in the proposition below then there exists a strict isomorphism of degree-weight-graded VSA-inductive sheaves $F: \mathcal{V}\to \mathcal{V}'$ such that $\Phi'^\lambda\circ F|_{U_\lambda}=\Phi^\lambda$ for all $\lambda\in \Lambda$.
\begin{proposition}\label{prop: GLUING VsaIndShs} Under the above assumptions,
there exists an object $\mathcal{V}$ of $\textit{DegWt-}\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}_X^A$ and strict isomorphisms $(\Phi^\lambda: \mathcal{V}|_{U_\lambda}\to \mathcal{V}^\lambda)_{\lambda\in\Lambda}$ of degree-weight-graded VSA-inductive sheaves such that $(\Phi^\lambda|_{U_\lambda\cap U_\mu}) \circ (\Phi^\mu|_{U_\mu\cap U_\lambda})^{-1}=\vartheta_{\lambda \mu}$ for all $\lambda, \mu \in \Lambda$. Moreover such a pair is unique up to strict isomorphisms. \end{proposition} \begin{proof} First we see the existence part. Since $\vartheta_{\lambda \mu}$ is strict, we can write $\vartheta_{\lambda \mu}$ as $ \vartheta_{\lambda \mu}=([\vartheta_{\lambda\mu, \alpha}^\alpha])_{\alpha\in\mathrm{Ob}(A)}, $
where $\vartheta_{\lambda\mu, \alpha}^\alpha$ is a sheaf morphism from $\mathcal{F}^\mu_\alpha|_{U_\mu\cap U_\lambda}$ to $\mathcal{F}^\lambda_\alpha|_{U_\lambda\cap U_\mu}$. For each $\lambda\in \Lambda$ we have $ \mathrm{id}=\vartheta_{\lambda\lambda}=([\vartheta_{\lambda\lambda, \alpha}^\alpha])_{\alpha\in\mathrm{Ob}(A)} $ and therefore $ \mathrm{id}\sim\vartheta_{\lambda\lambda, \alpha}^\alpha $ for all $\alpha\in\mathrm{Ob}(A)$, where $\sim$ means that the two sheaf morphisms are equivalent in $\displaystyle\varinjlim_{\alpha'}\mathrm{Hom}(\mathcal{F}^\lambda_{\alpha}, \mathcal{F}^\lambda_{\alpha'})$. By the injectivity of the morphisms of the inductive system $\mathcal{F}^\lambda$, we have \begin{equation}\label{eq: id=theta_lambda-lambda} \mathrm{id}=\vartheta_{\lambda\lambda, \alpha}^\alpha. \end{equation} Similarly we have $ \vartheta_{\lambda\mu, \alpha}^\alpha\circ\vartheta_{\mu\nu, \alpha}^\alpha=\vartheta_{\lambda\nu, \alpha}^\alpha $ for all $\alpha\in\mathrm{Ob}(A)$ and $\lambda, \mu, \nu\in\Lambda$ by the assumption. Therefore for each $\alpha\in \mathrm{Ob}(A)$, we can glue the sheaves $(\mathcal{F}^\lambda_\alpha)_{\lambda\in\Lambda}$ with the sheaf morphisms $(\vartheta_{\lambda\mu, \alpha}^\alpha)_{\lambda, \mu\in\Lambda}$. Denote the resulting sheaf on $X$ by $\mathcal{F}_\alpha$. For each $f_{\alpha\alpha'}: \alpha'\to \alpha$, we will glue the morphisms $(\mathcal{F}^\lambda(f_{\alpha \alpha'}))_{\lambda\in\Lambda}$ to obtain a sheaf morphism $\mathcal{F}_{\alpha'}\to\mathcal{F}_\alpha$. By the definition of morphisms of ind-objects, we have $$ \vartheta_{\lambda\mu, \alpha}^\alpha\circ\mathcal{F}^\mu(f_{\alpha \alpha'})\sim\vartheta_{\lambda\mu, \alpha'}^{\alpha'}, $$
where $\sim$ means that the two sheaf morphisms are equivalent in the inductive limit $\displaystyle\varinjlim_{\alpha\in A}\mathrm{Hom}(\mathcal{F}^\mu_{\alpha'}|_{U_\mu\cap U_\lambda}, \mathcal{F}^\lambda_\alpha|_{U_\lambda\cap U_\mu})$. Therefore we have $\vartheta_{\lambda\mu, \alpha}^\alpha\circ\mathcal{F}^\mu(f_{\alpha \alpha'})=\mathcal{F}^\lambda(f_{\alpha\alpha'})\circ\vartheta_{\lambda\mu, \alpha'}^{\alpha'}$ by the injectivity of the morphisms in the inductive systems. Thus we can glue the morphisms $(\mathcal{F}^\lambda(f_{\alpha \alpha'}))_{\lambda\in\Lambda}$ to obtain the sheaf morphism $\mathcal{F}_{\alpha'}\to\mathcal{F}_\alpha$, which we denote by $\mathcal{F}(f_{\alpha \alpha'})$. Note that $\mathcal{F}(f_{\alpha \alpha'})$ is injective since $\mathcal{F}^\lambda(f_{\alpha \alpha'})$ are injective for all $\lambda\in\Lambda$. By the construction, the assignment \begin{align*} \mathcal{F}&: A\to \mathit{Sh}_X(Vec_\mathbb{K}^\mathrm{super}), \\ \text{objects}&: \alpha\mapsto \mathcal{F}_\alpha, \\ \text{morphisms}&: f_{\alpha \alpha'}\mapsto \mathcal{F}(f_{\alpha \alpha'}), \end{align*} defines a functor. Thus we have an inductive system $\mathcal{F}$ in $\mathit{Sh}_X(Vec_\mathbb{K}^{\mathrm{super}})$, therefore the corresponding ind-object $``\displaystyle\varinjlim_{\alpha\in A}"\mathcal{F}_\alpha$. By the assumption, for each $n\in \mathbb{Z}$, we have a map $j_{(n)}: \mathrm{Ob}(A)\times\mathrm{Ob}(A)\to\mathrm{Ob}(A)$ and bilinear sheaf morphisms $\underline{(n)}_{(\alpha, \alpha')}^{\lambda, j_{(n)}(\alpha, \alpha')}: \mathcal{F}^\lambda_\alpha\times \mathcal{F}^\lambda_{\alpha'}\to \mathcal{F}^\lambda_{j_{(n)}(\alpha, \alpha')}$ with $(\alpha, \alpha')\in \mathrm{Ob}(A)\times \mathrm{Ob}(A)$ and $\lambda\in\Lambda$ such that $$ \underline{(n)}^\lambda=\bigl([\underline{(n)}_{(\alpha, \alpha')}^{\lambda, j_{(n)}(\alpha, \alpha')}]\bigr)_{(\alpha, \alpha')\in \mathrm{Ob}(A)\times\mathrm{Ob}(A)}, $$ for all $\lambda\in\Lambda$. Fix $n\in\mathbb{Z}$. For each $(\alpha, \alpha')\in\mathrm{Ob}(A)\times\mathrm{Ob}(A)$, we will glue morphisms $\bigl(\underline{(n)}_{(\alpha, \alpha')}^{\lambda, j_{(n)}(\alpha, \alpha')}\bigr)_{\lambda\in\Lambda}$ to get a bilinear sheaf morphism $\mathcal{F}_\alpha\times\mathcal{F}_{\alpha'}\to\mathcal{F}_{j_{(n)}(\alpha, \alpha')}$. It suffices to check that the morphisms $\bigl(\underline{(n)}_{(\alpha, \alpha')}^{\lambda, j_{(n)}(\alpha, \alpha')}\bigr)_{\lambda\in\Lambda}$ commute with the gluing maps. Since each $\vartheta_{\lambda\mu}$ is a morphism of VSA-inductive sheaves, we have $$ \underline{(n)}^\lambda \circ (\vartheta_{\lambda\mu}\times\vartheta_{\lambda\mu})=\vartheta_{\lambda\mu}\circ\underline{(n)}^\mu, $$ and therefore $$ \underline{(n)}^{\lambda, j_{(n)}(\alpha, \alpha')}_{(\alpha, \alpha')}\circ(\vartheta_{\lambda\mu, \alpha}^\alpha\times \vartheta_{\lambda\mu, \alpha'}^{\alpha'})\sim \vartheta_{\lambda\mu, j_{(n)}(\alpha, \alpha')}^{j_{(n)}(\alpha, \alpha')}\circ\underline{(n)}^{\mu, j_{(n)}(\alpha, \alpha')}_{(\alpha, \alpha')}, $$ for all $(\alpha, \alpha')\in \mathrm{Ob}(A)\times\mathrm{Ob}(A)$. By the same argument for proving \eqref{eq: id=theta_lambda-lambda}, we have $$ \underline{(n)}^{\lambda, j_{(n)}(\alpha, \alpha')}_{(\alpha, \alpha')}\circ(\vartheta_{\lambda\mu, \alpha}^\alpha\times \vartheta_{\lambda\mu, \alpha'}^{\alpha'})= \vartheta_{\lambda\mu, j_{(n)}(\alpha, \alpha')}^{j_{(n)}(\alpha, \alpha')}\circ\underline{(n)}^{\mu, j_{(n)}(\alpha, \alpha')}_{(\alpha, \alpha')}, $$ for all $(\alpha, \alpha')\in \mathrm{Ob}(A)\times\mathrm{Ob}(A)$. Thus we can glue the morphisms $\underline{(n)}_{(\alpha, \alpha')}^{\lambda, j_{(n)}(\alpha, \alpha')}$ with ${\lambda\in\Lambda}$. We denote by $\underline{(n)}_{(\alpha, \alpha')}^{j_{(n)}(\alpha, \alpha')}$ the resulting bilinear morphism of sheaves. We claim that $\underline{(n)}:=\bigl([\underline{(n)}_{(\alpha, \alpha')}^{j_{(n)}(\alpha, \alpha')}]\bigr)_{(\alpha, \alpha')\in \mathrm{Ob}(A)\times\mathrm{Ob}(A)}$ is a bilinear morphism of ind-objects. We must check \begin{equation}\label{eq: (n) f*f sim (n)} \underline{(n)}_{(\alpha, \alpha')}^{j_{(n)}(\alpha, \alpha')}\circ(\mathcal{F}(f_{\alpha \Tilde{\alpha}})\times \mathcal{F}(f_{\alpha' \Tilde{\alpha}'})) \sim \underline{(n)}_{(\Tilde{\alpha}, \Tilde{\alpha}')}^{j_{(n)}(\Tilde{\alpha}, \Tilde{\alpha}')}, \end{equation} for any object $(\Tilde{\alpha}, \Tilde{\alpha}')\in \mathrm{Ob}(A)\times\mathrm{Ob}(A)$ and morphism $f_{\alpha \Tilde{\alpha}}\times f_{\alpha' \Tilde{\alpha}'}$. Let $(\Tilde{\alpha}, \Tilde{\alpha}')\in \mathrm{Ob}(A)\times\mathrm{Ob}(A)$ be an arbitrary object and $f_{\alpha \Tilde{\alpha}}\times f_{\alpha' \Tilde{\alpha}'}$ an morphism. For each $\lambda\in\Lambda$, we have $$ \underline{(n)}_{(\alpha, \alpha')}^{\lambda, j_{(n)}(\alpha, \alpha')}\circ(\mathcal{F}^\lambda(f_{\alpha \Tilde{\alpha}})\times \mathcal{F}^\lambda(f_{\alpha' \Tilde{\alpha}'})) \sim \underline{(n)}_{(\Tilde{\alpha}, \Tilde{\alpha}')}^{\lambda, j_{(n)}(\Tilde{\alpha}, \Tilde{\alpha}')}. $$ Therefore for each $\lambda\in\Lambda$, we have an object $\alpha''(\lambda)\in\mathrm{Ob}(A)$ and morphisms $f_{\alpha''(\lambda)\, j_{(n)}(\alpha, \alpha')}: j_{(n)}(\alpha, \alpha') \to \alpha''(\lambda)$, $f_{\alpha''(\lambda)\, j_{(n)}(\Tilde{\alpha}, \Tilde{\alpha}')}: j_{(n)}(\Tilde{\alpha}, \Tilde{\alpha}') \to \alpha''(\lambda)$ in $A$ such that \begin{multline}\label{eq: f (n) f*f = f (n) in lambda} \mathcal{F}^\lambda(f_{\alpha''(\lambda)\, j_{(n)}(\alpha, \alpha')})\circ\underline{(n)}_{(\alpha, \alpha')}^{\lambda, j_{(n)}(\alpha, \alpha')}\circ(\mathcal{F}^\lambda(f_{\alpha \Tilde{\alpha}})\times \mathcal{F}^\lambda(f_{\alpha' \Tilde{\alpha}'}))\\ = \mathcal{F}^\lambda(f_{\alpha''(\lambda)\, j_{(n)}(\Tilde{\alpha}, \Tilde{\alpha}')})\circ\underline{(n)}_{(\Tilde{\alpha}, \Tilde{\alpha}')}^{\lambda, j_{(n)}(\Tilde{\alpha}, \Tilde{\alpha}')}. \end{multline} By the assumption of this proposition, we have \begin{equation*} \mathrm{Hom}_A(j_{(n)}(\alpha, \alpha'), j_{(n)}(\Tilde{\alpha}, \Tilde{\alpha}'))\neq\emptyset\quad \text{or} \quad \mathrm{Hom}_A(j_{(n)}(\Tilde{\alpha}, \Tilde{\alpha}'), j_{(n)}(\alpha, \alpha'))\neq\emptyset. \end{equation*} When $\mathrm{Hom}_A(j_{(n)}(\alpha, \alpha'), j_{(n)}(\Tilde{\alpha}, \Tilde{\alpha}'))\neq\emptyset$, we have a morphism $f_{j_{(n)}(\alpha, \alpha')\, j_{(n)}(\Tilde{\alpha}, \Tilde{\alpha}')}$ in this set. The right-hand side of \eqref{eq: f (n) f*f = f (n) in lambda} is equivalent to $$ \mathcal{F}^\lambda(f_{\alpha''(\lambda)\, j_{(n)}(\alpha, \alpha')})\circ\mathcal{F}^\lambda(f_{j_{(n)}(\alpha, \alpha')\, j_{(n)}(\Tilde{\alpha}, \Tilde{\alpha}')})\circ\underline{(n)}_{(\Tilde{\alpha}, \Tilde{\alpha}')}^{\lambda, j_{(n)}(\Tilde{\alpha}, \Tilde{\alpha}')}, $$ in $\varinjlim_{\beta\in A}\mathrm{Bilin}_{\mathit{Presh}_{U_\lambda}(\mathit{Vec}^{\mathrm{super}}_{\mathbb{K}})}(\mathcal{F}_{\Tilde{\alpha}}^\lambda, \mathcal{F}_{\Tilde{\alpha}'}^\lambda; \mathcal{F}_\beta^\lambda)$. By the injectivity of the morphism of the inductive system $\mathcal{F}^\lambda$, we have $$ \underline{(n)}_{(\alpha, \alpha')}^{\lambda, j_{(n)}(\alpha, \alpha')}\circ(\mathcal{F}^\lambda(f_{\alpha \Tilde{\alpha}})\times \mathcal{F}^\lambda(f_{\alpha' \Tilde{\alpha}'})) =\mathcal{F}^\lambda(f_{j_{(n)}(\alpha, \alpha')\, j_{(n)}(\Tilde{\alpha}, \Tilde{\alpha}')})\circ\underline{(n)}_{(\Tilde{\alpha}, \Tilde{\alpha}')}^{\lambda, j_{(n)}(\Tilde{\alpha}, \Tilde{\alpha}')}. $$ We have this equality for any $\lambda\in\Lambda$. Note that $f_{j_{(n)}(\alpha, \alpha')\, j_{(n)}(\Tilde{\alpha}, \Tilde{\alpha}')}$ does not depend on $\lambda\in\Lambda$. Therefore we have the following relation for glued morphisms: $$ \underline{(n)}_{(\alpha, \alpha')}^{j_{(n)}(\alpha, \alpha')}\circ(\mathcal{F}(f_{\alpha \Tilde{\alpha}})\times \mathcal{F}(f_{\alpha' \Tilde{\alpha}'})) =\mathcal{F}(f_{j_{(n)}(\alpha, \alpha')\, j_{(n)}(\Tilde{\alpha}, \Tilde{\alpha}')})\circ\underline{(n)}_{(\Tilde{\alpha}, \Tilde{\alpha}')}^{j_{(n)}(\Tilde{\alpha}, \Tilde{\alpha}')}. $$ This means \eqref{eq: (n) f*f sim (n)}. When $\mathrm{Hom}_A(j_{(n)}(\Tilde{\alpha}, \Tilde{\alpha}'), j_{(n)}(\alpha, \alpha'))\neq\emptyset$, we can also obtain \eqref{eq: (n) f*f sim (n)} in a similar way. Thus we have a bilinear morphism of ind-objects $\underline{(n)}=\bigl([\underline{(n)}_{(\alpha, \alpha')}^{j_{(n)}(\alpha, \alpha')}]\bigr)_{(\alpha, \alpha')\in \mathrm{Ob}(A)\times\mathrm{Ob}(A)}: \mathcal{F}\times\mathcal{F}\to\mathcal{F}$.
In a similar way, we obtain morphisms of ind-objects $ \underline{T}=\bigl([\underline{T}_\alpha^{j_T(\alpha)}]\bigr)_{\alpha\in\mathrm{Ob}(A)}: \mathcal{F}\to\mathcal{F}, $ $\underline{\mathbf{1}}=[\underline{\mathbf{1}}^{\alpha_0}]: ``\displaystyle\varinjlim"\mathbb{K}_X\to\mathcal{F}, $ $\underline{H}=([H_\alpha^{\alpha}])_{\alpha\in\mathrm{Ob}(A)}: \mathcal{F}\to\mathcal{F}$ and $\underline{J}=([J_\alpha^{\alpha}])_{\alpha\in\mathrm{Ob}(A)}: \mathcal{F}\to\mathcal{F}$ from the morphisms $\underline{T}^\lambda=\bigl([\underline{T}_\alpha^{\lambda, j_T(\alpha)}]\bigr)_{\alpha\in\mathrm{Ob}(A)}$, $\underline{\mathbf{1}}^\lambda=[\underline{\mathbf{1}}^{\lambda, \alpha_0}]$, $\underline{H}^\lambda=([H_\alpha^{\lambda, \alpha}])_{\alpha\in\mathrm{Ob}(A)}$ and $\underline{J}^\lambda=([J_\alpha^{\lambda, \alpha}])_{\alpha\in\mathrm{Ob}(A)}$ with $\lambda\in\Lambda$, respectively. Since the gluing maps commutes with the Hamiltonians and the degree-grading operators, we can glue the sheaves $(\mathcal{F}_\alpha^\lambda)^l[n]$ with $\lambda\in\Lambda$. Denote the resulting sheaf on $X$ by $\mathcal{F}_\alpha^l[n]$. By the assumption (5), $(\mathcal{F}_\alpha)^l[n]=0$ for all but finitely many $l$ and $n$. Therefore the presheaf $\bigoplus_{n, l\in\mathbb{Z}}(\mathcal{F}_\alpha)^l[n]$ is a sheaf. Thus the sheaf $\mathcal{F}_\alpha$ is canonically isomorphic to the sheaf $\bigoplus_{n, l\in\mathbb{Z}}(\mathcal{F}_\alpha)^l[n]$ since they are locally isomorphic. This grading comes from the operators $H_\alpha^\alpha$ and $J_\alpha^\alpha$. In other words, the operators $H_\alpha^\alpha$ and $J_\alpha^\alpha$ are diagonalizable. The relation \eqref{eq: Hamiltonian} for $\underline{H}$ and the relation \eqref{eq: degree-grading operator} for $\underline{J}$ follow from the fact that the corresponding relations hold locally.
We must check the quadruple $\mathcal{V}:=(\mathcal{F}, \underline{\mathbf{1}}, \underline{T}, \underline{(n)}; n\in\mathbb{Z})$ is a VSA-inductive sheaf on $X$. Let $V$ be an arbitrary open subset of $X$. It suffices to show that the quadruple $(\varinjlim_{\alpha\in A}\mathcal{F}_\alpha(V), \mathbf{1}, T, (n); n\in\mathbb{Z})$ is a vertex superalgebra, where $\mathbf{1}:=\underrightarrow{\mathrm{Lim}}\, \underline{\mathbf{1}}(V)(1)$, $T:=\underrightarrow{\mathrm{Lim}}\, \underline{T}(V)$ and $(n):=\underrightarrow{\mathrm{Lim}}\, \underline{(n)}(V)$.
The map induced by the restriction maps and the isomorphisms $\mathcal{F}_\alpha|_{U_\lambda}\cong\mathcal{F}^\lambda_\alpha$, \begin{equation}\label{eq: inclusion into the product VSA} \varinjlim_{\alpha\in A}\mathcal{F}_\alpha(V)\to\prod_{\lambda\in\Lambda}\varinjlim_{\alpha\in A}\mathcal{F}_\alpha(V\cap U_\lambda)\cong\prod_{\lambda\in\Lambda}\varinjlim_{\alpha\in A}\mathcal{F}_\alpha^\lambda(V\cap U_\lambda), \end{equation} is injective since the morphisms $\mathcal{F}^\lambda(f_{\alpha\alpha'})$ are all injective. Via this map,
we regard $\varinjlim_{\alpha\in A}\mathcal{F}_\alpha(V)$ as a subspace of $\prod_{\lambda\in\Lambda}\varinjlim_{\alpha\in A}\mathcal{F}_\alpha^\lambda(V\cap U_\lambda)$. Then by the construction, $\varinjlim_{\alpha\in A}\mathcal{F}_\alpha(V)$ is preserved by $\prod_{\lambda\in\Lambda}T^\lambda$ and $\prod_{\lambda\in\Lambda}(n)^\lambda$ with $n\in\mathbb{Z}$, where $T^\lambda$ and $(n)^\lambda$ are the translation operator and the $n$-th product of $\varinjlim_{\alpha\in A}\mathcal{F}_\alpha^\lambda(V\cap U_\lambda)$, respectively. Moreover $(\mathbf{1}^\lambda)_{\lambda\in\Lambda}\in \varinjlim_{\alpha\in A}\mathcal{F}_\alpha(V)$, where $\mathbf{1}^\lambda$ is the vacuum vector of $\varinjlim_{\alpha\in A}\mathcal{F}_\alpha^\lambda(V\cap U_\lambda)$. Note that $T=(\prod_{\lambda\in\Lambda}T^\lambda)|_{\varinjlim_{\alpha\in A}\mathcal{F}_\alpha(V)}$, $(n)=(\prod_{\lambda\in\Lambda}(n)^\lambda)|_{\varinjlim_{\alpha\in A}\mathcal{F}_\alpha(V)}$ and $\mathbf{1}=(\mathbf{1}^\lambda)_{\lambda\in\Lambda}$. Moreover by the assumption (5), the weight-grading on $\varinjlim_{\alpha\in A}\mathcal{F}_\alpha(V)$ is bounded from the below. Therefore the formal distribution $\sum_{n\in\mathbb{Z}}A_{(n)}z^{-n-1}$ is a field for any $A\in\varinjlim_{\alpha\in A}\mathcal{F}_\alpha(V)$. Thus $(\varinjlim_{\alpha\in A}\mathcal{F}_\alpha(V), \mathbf{1}, T, (n); n\in \mathbb{Z})$ is a vertex superalgebra.
Therefore the quadruple $\mathcal{V}=(\mathcal{F}, \underline{\mathbf{1}}, \underline{T}, \underline{(n)}; n\in\mathbb{Z})$ with $\underline{H}$ and $\underline{J}$ is an object of the category $\textit{DegWt-}\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}_X^A$. It remains to construct a strict isomorphism $\mathcal{V}|_{U_\lambda}\cong\mathcal{V}^\lambda$ for each $\lambda\in \Lambda$. Let $\lambda\in\Lambda$. We set $$ \Phi^\lambda:=\bigl([\Phi^{\lambda, \alpha}_\alpha]\bigr)_{\alpha\in \mathrm{Ob}(A)}, $$
where $\Phi^{\lambda, \alpha}_\alpha$ is the usual sheaf isomorphism from $\mathcal{F}_\alpha|_{U_\lambda}$ to $\mathcal{F}_\alpha^\lambda$, which preserves the degree-weight-grading. This defines a morphism of ind-objects, or equivalently, $$ \Phi_\alpha^{\lambda, \alpha}\circ\mathcal{F}(f_{\alpha \alpha'})\sim\Phi_{\alpha'}^{\lambda, \alpha'}, $$ for any object $\alpha, \alpha'$ of $A$ and morphism $f_{\alpha \alpha'}: \alpha'\to \alpha$ in $A$. Indeed by the construction of $\mathcal{F}(f_{\alpha \alpha'})$, we have $$
\Phi_\alpha^{\lambda, \alpha}\circ (\mathcal{F}(f_{\alpha \alpha'})|_{U_\lambda})=\mathcal{F}^\lambda(f_{\alpha \alpha'})\circ\Phi_{\alpha'}^{\lambda, \alpha'}, $$ for each object $\alpha, \alpha'$ of $A$ and morphism $f_{\alpha \alpha'}: \alpha'\to \alpha$ in $A$.
The relation $(\Phi^\lambda|_{U_\lambda\cap U_\mu}) \circ (\Phi^\mu|_{U_\mu\cap U_\lambda})^{-1}=\vartheta_{\lambda \mu}$ holds since $(\Phi^{\lambda, \alpha}_\alpha|_{U_\lambda\cap U_\mu}) \circ (\Phi^{\mu, \alpha}_\alpha|_{U_\mu\cap U_\lambda})^{-1}=\vartheta_{\lambda \mu, \alpha}^\alpha$ for all $\alpha\in \mathrm{Ob}(A)$. Moreover $\Phi^\lambda$ is a strict isomorphism of VSA-inductive sheaves. In other words, $\Phi^\lambda$ commutes with $\underline{(n)}$, $\underline{T}$ and $\underline{\mathbf{1}}$ and in addition the strict inverse morphism $(\Phi^\lambda)^{-1}$ exists. This follows from the construction of the operators $\underline{(n)}$, $\underline{T}$, $\underline{\mathbf{1}}$ and from the definition of $\Phi^\lambda$. Moreover $\Phi^\lambda$ commutes with the Hamiltonians and the degree-grading operators since each $\Phi_\alpha^{\lambda, \alpha}$ does. Thus the existence part is proved. The uniqueness part is proved by the argument as in usual sheaf cases as well as the arguments used above with the fact that the morphisms $\mathcal{F}^\lambda(f_{\alpha'\alpha})$ are injective for all $f_{\alpha'\alpha}\in\mathrm{Hom}_A(\alpha, \alpha')$. \end{proof}
We can also glue morphisms. Let $X=\bigcup_{\lambda \in \Lambda}U_\lambda$ be as before. Let $\bigl((\mathcal{V}^\lambda)_{\lambda\in\Lambda},$ $ (\vartheta_{\lambda \mu})_{\lambda, \mu \in \Lambda}\bigr)$, $ \bigl((\mathcal{V}'^\lambda)_{\lambda\in\Lambda},$ $(\vartheta'_{\lambda \mu})_{\lambda, \mu \in \Lambda}\bigr)$ be families of degree-weight-graded VSA-inductive sheaves with strict isomorphisms as in Proposition \ref{prop: GLUING VsaIndShs}. In other words,
$\mathcal{V}^\lambda=\bigl(\mathcal{F}^\lambda, \underline{\mathbf{1}}^\lambda, \underline{T}^\lambda,$ $ \underline{(n)}^\lambda; n\in\mathbb{Z} \bigr)$ and $\mathcal{V}'^\lambda=\bigl(\mathcal{F}'^\lambda, \underline{\mathbf{1}'}^\lambda, \underline{T}'^\lambda, \underline{(n)}'^\lambda; n\in\mathbb{Z} \bigr)$ are objects of $\textit{DegWt-}\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}_{U_\lambda}^A$, and $\vartheta_{\lambda \mu}: \mathcal{F}^\mu|_{U_\mu \cap U_\lambda} \to \mathcal{F}^\lambda|_{U_\lambda \cap U_\mu}$ and $\vartheta'_{\lambda \mu}: \mathcal{F}'^\mu|_{U_\mu \cap U_\lambda} \to \mathcal{F}'^\lambda|_{U_\lambda \cap U_\mu}$ are strict isomorphisms of degree-weight-graded VSA-inductive sheaves on $U_\lambda \cap U_\mu$ such that the conditions $(0)$-$(5)$ hold. Let $\mathcal{V}$ with $(\Phi^\lambda)_{\lambda \in \Lambda}$ and $\mathcal{V}'$ with $(\Phi'^\lambda)_{\lambda\in\Lambda}$ be objects of $\textit{DegWt-}\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}_X^A$ with strict isomorphisms obtained by gluing $\bigl((\mathcal{V}^\lambda)_{\lambda\in\Lambda}, (\vartheta_{\lambda \mu})_{\lambda, \mu \in \Lambda}\bigr)$ and $\bigl((\mathcal{V}'^\lambda)_{\lambda\in\Lambda},$ $ (\vartheta'_{\lambda \mu})_{\lambda, \mu \in \Lambda}\bigr)$, respectively.
Suppose given a family of morphisms of ind-objects of sheaves $(F^\lambda: \mathcal{V}^\lambda \to \mathcal{V}'^\lambda)_{\lambda\in\Lambda}$ such that $\vartheta'_{\lambda \mu}\circ (F^\mu|_{U_\mu\cap U_\lambda}) = (F^\lambda|_{U_\lambda\cap U_\mu})\circ \vartheta_{\lambda \mu}$ for all $\lambda, \mu \in \Lambda$. We assume that there exist a map $j_F: \mathrm{Ob}(A)\to\mathrm{Ob}(A)$ and sheaf morphisms $F^{\lambda, j_F(\alpha)}_\alpha$ with $\alpha\in \mathrm{Ob}(A)$ and $\lambda\in \Lambda$ such that $ F^\lambda=\bigl( [F^{\lambda, j_F(\alpha)}_\alpha] \bigr)_{\alpha\in \mathrm{Ob}(A)} $ for all $\lambda \in \Lambda$.
\begin{proposition}\label{prop: GLUING MORPHISMS OF VsaIndSh} In the above situation, there exists a unique strict morphism $F: \mathcal{V}\to \mathcal{V}'$ of ind-objects of sheaves on $X$ such that
$\Phi'^\lambda\circ F|_{U_\lambda}=F^\lambda\circ \Phi^\lambda$ for any $\lambda\in\Lambda$. Moreover if the morphisms $F^\lambda$ are all morphisms of VSA-inductive sheaves, the resulting morphism of ind-objects $F$ is also a morphism of VSA-inductive sheaves. \end{proposition} \begin{proof} We can construct $F: \mathcal{V}\to \mathcal{V}'$, following the same argument as for the construction of $\underline{(n)}$ in Proposition \ref{prop: GLUING VsaIndShs}. The uniqueness part is proved by the same argument as in usual sheaf cases as well as the same arguments as in the proof of Proposition \ref{prop: GLUING VsaIndShs}. The latter half of this proposition is also checked by arguments similar to those above. \end{proof}
Consider three families of degree-weight-graded VSA-inductive sheaves with strict isomorphisms as in Proposition \ref{prop: GLUING VsaIndShs}, $\bigl((\mathcal{V}^\lambda)_{\lambda\in\Lambda}, (\vartheta_{\lambda \mu})_{\lambda, \mu \in \Lambda}\bigr)$, $\bigl((\mathcal{V}'^\lambda)_{\lambda\in\Lambda},$ $ (\vartheta'_{\lambda \mu})_{\lambda, \mu \in \Lambda}\bigr)$ and $\bigl((\mathcal{V}''^\lambda)_{\lambda\in\Lambda},$ $ (\vartheta''_{\lambda \mu})_{\lambda, \mu \in \Lambda}\bigr)$.
Suppose given families of morphisms of ind-objects of sheaves $(F^\lambda: \mathcal{V}^\lambda \to \mathcal{V}'^\lambda)_{\lambda\in\Lambda}$ and $(F'^\lambda: \mathcal{V}'^\lambda \to \mathcal{V}''^\lambda)_{\lambda\in\Lambda}$ as in Proposition \ref{prop: GLUING MORPHISMS OF VsaIndSh}. In other words, $\vartheta'_{\lambda \mu}\circ (F^\mu|_{U_\mu\cap U_\lambda}) = (F^\lambda|_{U_\lambda\cap U_\mu})\circ \vartheta_{\lambda \mu}$ and $\vartheta''_{\lambda \mu}\circ (F'^\mu|_{U_\mu\cap U_\lambda}) = (F'^\lambda|_{U_\lambda\cap U_\mu})\circ \vartheta'_{\lambda \mu}$ hold for all $\lambda, \mu \in \Lambda$, and moreover, there exist maps $j_F: \mathrm{Ob}(A)\to\mathrm{Ob}(A)$, $j_{F'}: \mathrm{Ob}(A)\to\mathrm{Ob}(A)$ and sheaf morphisms $F^{\lambda, j_F(\alpha)}_\alpha$, $F'^{\lambda, j_{F'}(\alpha)}_\alpha$ with $\alpha\in \mathrm{Ob}(A)$ and $\lambda\in \Lambda$ such that $ F^\lambda=\bigl( [F^{\lambda, j_F(\alpha)}_\alpha] \bigr)_{\alpha\in \mathrm{Ob}(A)} $ and $ F'^\lambda=\bigl( [F'^{\lambda, j_{F'}(\alpha)}_\alpha] \bigr)_{\alpha\in \mathrm{Ob}(A)} $ for all $\lambda \in \Lambda$. Let $F: \mathcal{V}\to \mathcal{V}'$, $F': \mathcal{V}'\to\mathcal{V}''$ and $G: \mathcal{V}\to\mathcal{V}''$ be the morphisms obtained by gluing $(F^\lambda)_{\lambda\in\Lambda}$, $(F'^\lambda)_{\lambda\in\Lambda}$ and $(F'^\lambda\circ F^\lambda)_{\lambda\in\Lambda}$, respectively.
\begin{proposition}\label{prop: FUNCTORIALITY OF GLUING} In the above situation, the composite $F'\circ F$ agrees with the morphism $G$. \end{proposition} \begin{proof} This proposition is a direct corollary of Proposition \ref{prop: GLUING MORPHISMS OF VsaIndSh}. \end{proof}
\begin{remark}\label{rem: Isomorphic as VSA-inductive sheaves if locally isomorphic} Two VSA-inductive sheaves are strictly isomorphic if they are strictly isomorphic locally via strict isomorphisms which coincide on the overlaps of their domains. \end{remark}
\begin{remark}\label{rem: morphisms coincides if so locally} Two morphisms of ind-objects between VSA-inductive sheaves coincide with each other if they coincide locally. \end{remark}
\subsection{From Presheaves to VSA-Inductive Sheaves}\label{subsection: From Presheaves to VSA-Inductive Sheaves} We construct VSA-inductive sheaves from presheaves of vertex superalgebras with some properties.
We denote by $\mathit{Presh}_X(\textit{DegWt-VSA}_\mathbb{K})_\mathrm{bdw, sh}$ the full subcategory of the category of presheaves on $X$ of degree-weight-graded vertex superalgebras over $\mathbb{K}$ whose objects are presheaves $\Tilde{\mathcal{V}}$ of degree-weight-graded vertex superalgebras on $X$ such that the weight-grading on $\Tilde{\mathcal{V}}(U)$ is bounded from the below uniformly with respect to open subsets $U\subset X$ and the subpresheaf $\Tilde{\mathcal{V}}[n]$ defined by the assignment $ U\mapsto \Tilde{\mathcal{V}}(U)[n] $ is a sheaf of super vector spaces for any $n \in\mathbb{Z}$.
Let $\Tilde{\mathcal{V}}$ be an object of $\mathit{Presh}_X(\textit{DegWt-VSA}_\mathbb{K})_\mathrm{bdw, sh}$. We set $$ \Tilde{\mathcal{V}}[\le N]:=\bigoplus_{n\le N}\Tilde{\mathcal{V}}[n]. $$ for $N\in \mathbb{N}$. These are sheaves by the assumptions. Consider the canonical inductive system of sheaves $(\Tilde{\mathcal{V}}[\le N])_{N\in \mathbb{N}}$. Then the corresponding ind-object $``\displaystyle\varinjlim_{n\in \mathbb{N}}"\Tilde{\mathcal{V}}[\le N]$ has a canonical VSA-inductive sheaf structure induced by the morphisms $\underline{\mathbf{1}}^0: \mathbb{K}_X\to \Tilde{\mathcal{V}}[\le 0]$ defined by $\mathbb{K}\to \Gamma(\Tilde{\mathcal{V}}[\le 0]), 1\mapsto \mathbf{1}$, $T_N: \Tilde{\mathcal{V}}[\le N]\to\Tilde{\mathcal{V}}[\le N+1]$, and $(n)_{N, M}: \Tilde{\mathcal{V}}[\le N]\times \Tilde{\mathcal{V}}[\le M]\to \Tilde{\mathcal{V}}[\le N+M-n-1]$, where $\mathbf{1}$, $T_N$, $(n)_{N, M}$ come from the vertex superalgebra structure on $\Tilde{\mathcal{V}}$. Note that the VSA-inductive sheaf $``\displaystyle\varinjlim_{n\in \mathbb{N}}"\Tilde{\mathcal{V}}[\le N]$ has a canonical degree-weight-graded structure. We refer to this degree-weight-graded VSA-inductive sheaf as the degree-weight-graded VSA-inductive sheaf associated with $\Tilde{\mathcal{V}}$.
\begin{lemma}\label{lem: MAKE VsaIndSh from PRESHEAVES OF Z-GRADED Vsa} There exists a canonical functor $$ \mathit{Presh}_X(\textit{DegWt-VSA}_\mathbb{K})_\mathrm{bdw, sh} \to \textit{DegWt-}\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}_X^\mathbb{N}, $$ sending an object $\Tilde{\mathcal{V}}$ of $\mathit{Presh}_X(\textit{DegWt-VSA}_\mathbb{K})_\mathrm{bdw, sh}$ to the degree-weight-graded VSA-inductive sheaf associated with $\Tilde{\mathcal{V}}$. \end{lemma} \begin{proof}
For a morphism $\Tilde{F}: \Tilde{\mathcal{V}}\to\Tilde{\mathcal{W}}$ in $\mathit{Presh}_X(\textit{DegWt-VSA}_\mathbb{K})_\mathrm{bdw, sh}$, we assign the morphism $F:=\Bigl([\Tilde{F}|_{\Tilde{\mathcal{V}}[\le N]}\bigr]\Bigr)_{N\ge 0}$ in $\textit{DegWt-}\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}_X^\mathbb{N}$. The functoriality follows from the definition directly. \end{proof}
\begin{remark} The composite of the functor $\underrightarrow{\mathrm{Lim}}\, $ given in \eqref{eq: functor from VSA-ISh} and the one given in Lemma \ref{lem: MAKE VsaIndSh from PRESHEAVES OF Z-GRADED Vsa} is the identity functor. \end{remark}
\begin{remark}\label{rem: homogeneous morphism of presheaves induces that of ind-objects}
Let $\Tilde{\mathcal{V}}, \Tilde{\mathcal{W}}$ be objects of the category $\mathit{Presh}_X(\textit{DegWt-VSA}_\mathbb{K})_\mathrm{bdw, sh}$ and $\mathcal{V}, \mathcal{W}$ the VSA-inductive sheaves associated with $\Tilde{\mathcal{V}}, \Tilde{\mathcal{W}}$, respectively. Let $\Tilde{F}: \Tilde{\mathcal{V}}\to\Tilde{\mathcal{W}}$ be a homogeneous linear morphism of degree $d$. In other words, $\Tilde{F}(U): \Tilde{\mathcal{V}}(U)\to \Tilde{\mathcal{W}}(U)$ is a homogeneous linear map of degree $d$ for any open subset $U\subset X$. Then $\Tilde{F}$ induces a morphism $F$ of ind-objects: $F:=\bigl([\Tilde{F}|_{\Tilde{\mathcal{V}}[\le N]}]\bigr)_{N\ge 0}: \mathcal{V}\to \mathcal{W}$. Moreover the corresponding morphism $\underrightarrow{\mathrm{Lim}}\, F$ of presheaves is nothing but the morphism $\Tilde{F}: \Tilde{\mathcal{V}}=\underrightarrow{\mathrm{Lim}}\, \mathcal{V}\to\Tilde{\mathcal{W}}=\underrightarrow{\mathrm{Lim}}\, \mathcal{W}$. \end{remark}
\begin{remark} The functor given in Lemma \ref{lem: MAKE VsaIndSh from PRESHEAVES OF Z-GRADED Vsa} commutes with the restriction. More precisely, if $U\subset X$ is an open subset and
$\mathcal{V}$ is the degree-weight-graded VSA-inductive sheaf associated with an object $\Tilde{\mathcal{V}}$ of $\mathit{Presh}_X(\textit{DegWt-VSA}_\mathbb{K})_\mathrm{bdw, sh}$ then we have $\mathcal{V}|_U=\mathcal{V}_U$, where $\mathcal{V}_U$ stands for the VSA-inductive sheaf associated with the presheaf $\Tilde{\mathcal{V}}|_U$. Here we denote by $\Tilde{\mathcal{V}}|_U$ the presheaf, not its sheafification, obtained by restricting $\Tilde{\mathcal{V}}$ to $U$. \end{remark}
\subsection{More on VSA-Inductive Sheaves} Let $\varphi: X\to Y$ be a continuous map between topological spaces. Consider the functor induced by the push-forward functor $\varphi_*$ of presheaves: \begin{align} \label{eq: push-forward functor of ind-objects} \varphi_*&: \mathrm{Ind}(\mathit{Presh}_X(Vec_\mathbb{K}^\mathrm{super}))\longrightarrow \mathrm{Ind}(\mathit{Presh}_Y(Vec_\mathbb{K}^\mathrm{super})), \\ \text{objects}&:\quad\quad\quad \mathcal{F}=``\varinjlim_{\alpha\in A}\mathcal{F}"\longmapsto \varphi_*\mathcal{F}:=``\varinjlim_{\alpha\in A}"\varphi_*\mathcal{F}, \\ \label{df: push-forward of morphism of ind-objects} \text{morphisms}&: F=\bigl([F_\alpha^{j(\alpha)}]\bigr)_{\alpha\in \mathrm{Ob}(A)}\longmapsto \varphi_*F:=\bigl([\varphi_*F_\alpha^{j(\alpha)}]\bigr)_{\alpha\in \mathrm{Ob}(A)}. \end{align} The push-forward of bilinear morphism of ind-objects is given in a way similar to that in \eqref{df: push-forward of morphism of ind-objects}.
\begin{lemma} Let $\varphi: X\to Y$ be a continuous map between topological spaces and $\mathcal{V}=\bigl( \mathcal{F}, \underline {\mathbf{1}}, \underline{T}, \underline{(n)}; n\in \mathbb{Z}\bigr)$ a VSA-inductive sheaf on $X$. The quadruple $\varphi_*\mathcal{V}:=\bigl( \varphi_*\mathcal{F}, \varphi_*\underline {\mathbf{1}}, \varphi_*\underline{T}, \varphi_*\underline{(n)}; n\in \mathbb{Z}\bigr)$ is a VSA-inductive sheaf on $Y$. Moreover if $F$ is a morphism in $\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}_X$ then $\varphi_*F$ is a morphism in $\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}_Y$. \end{lemma} \begin{proof} We can immediately see that $\varphi_*\mathcal{V}$ is a VSA-inductive sheaf. The latter half of this lemma follows from the functoriality of the push-forward functor of presheaves. \end{proof}
By the above lemma, we can restrict the functor \eqref{eq: push-forward functor of ind-objects} to obtain a functor $\varphi_*: \mathit{VSA_{\mathbb{K}}}\textit{-IndSh}_X\to\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}_Y$ sending an object $\mathcal{V}$ to $\varphi_*\mathcal{V}$ and a morphism $F$ to $\varphi_*F$. We call the VSA-inductive sheaf $\varphi_*\mathcal{V}$ the \textbf{push-forward} of $\mathcal{V}$.
\begin{remark} The push-forward functor commutes with the functor $\underrightarrow{\mathrm{Lim}}\, $ as well as the one given in Lemma \ref{lem: MAKE VsaIndSh from PRESHEAVES OF Z-GRADED Vsa}. \end{remark}
Let $\mathcal{V}_1=\bigl( \mathcal{F}_1, \underline {\mathbf{1}}_1, \underline{T}_1, \underline{(n)}_1; n\in \mathbb{Z}\bigr)$ and $\mathcal{V}_2=\bigl( \mathcal{F}_2, \underline {\mathbf{1}}_2, \underline{T}_2, \underline{(n)}_2;$ $n\in \mathbb{Z}\bigr)$ be VSA-inductive sheaves on topological spaces $X_1$ and $X_2$, respectively. A \textbf{morphism} of vertex superalgebra inductive sheaves from $\mathcal{V}_1$ to $\mathcal{V}_2$ is by definition a pair $(\varphi, \Phi)$ of a continuous map $\varphi: X_2 \to X_1$ and a base-preserving morphism $\Phi: \mathcal{V}_1 \to \varphi_*\mathcal{V}_2$ of VSA-inductive sheaves on $Y$.
\begin{remark}\label{rem: ALL VSA-inductive sheaves form a category} VSA-inductive sheaves form a category with morphisms defined above, whose composition is defined by $ (\varphi', \Phi')\circ(\varphi, \Phi):=(\varphi'\circ\varphi, \Phi'\circ\varphi'_*\Phi) $ for morphisms of VSA-inductive sheaves $(\varphi, \Phi): \mathcal{V}_1\to \mathcal{V}_2$ and $(\varphi', \Phi'): \mathcal{V}_2\to \mathcal{V}_3$. \end{remark}
\begin{notation} Denote by $\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}$ the category of VSA-inductive sheaves obtained in Remark \ref{rem: ALL VSA-inductive sheaves form a category}. \end{notation}
\begin{remark} If $\varphi: X\to Y$ is a continuous map and $(\mathcal{V},\underline{H} ,\underline{J})$ is a degree-weight-graded VSA-inductive sheaf on $X$, then $(\varphi_*\mathcal{V}, \varphi_*\underline{H}, \varphi_*\underline{J})$ is a degree-weight-graded VSA-inductive sheaf on $Y$. In a similar way above, the category of degree-weight-graded VSA-inductive sheaves is defined. \end{remark}
\begin{notation} Let us denote by $\textit{DegWt-}\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}$ the category of degree-weight-graded VSA-inductive sheaves. \end{notation}
Let $\mathcal{V}$ be a VSA-inductive sheaf and $\mathcal{F}=``\displaystyle\varinjlim_{\alpha\in A}"\mathcal{F}_{\alpha}$ the underlying ind-object. Then we have two canonical ind-objects of sheaves, $\mathcal{F}_{\bar{0}}:=``\displaystyle\varinjlim_{\alpha\in A}"(\mathcal{F}_\alpha)_{\bar{0}}$ and $\mathcal{F}_{\bar{1}}:=``\displaystyle\varinjlim_{\alpha\in A}"(\mathcal{F}_\alpha)_{\bar{1}}$. In addition, suppose that $\mathcal{V}$ is degree-graded. Then we have ind-objects of sheaves, $\mathcal{F}^l:=``\displaystyle\varinjlim_{\alpha\in A}"\mathcal{F}_\alpha^l$, where $\mathcal{F}_\alpha^l$ is the subsheaf of degree $l\in\mathbb{Z}$.
\begin{definition} Let $\mathcal{V}=\bigl( \mathcal{F}, \underline {\mathbf{1}}, \underline{T}, \underline{(n)}; n\in \mathbb{Z}\bigr)$ be a degree-weight-graded VSA-inductive sheaf. A \textbf{differential} on $\mathcal{V}$ is an odd morphism of ind-objects, $D: \mathcal{F}\to\mathcal{F}$ such that \begin{equation}\label{eq: deg 1 wt 0 condition of differential on VSA-inductive sheaf} [\underline{H}, D]=0, \quad [\underline{J}, D]=D, \end{equation} \begin{equation}\label{eq: square 0 condition of differential on VSA-inductive sheaf} D^2=0, \end{equation} \begin{equation}\label{eq: derivation condition of differential on VSA-inductive sheaf} D\circ\underline{(n)}-(-1)^{i}\underline{(n)}\circ(\mathrm{id}\times D)=\underline{(n)}\circ (D\times\mathrm{id}), \end{equation} on $\mathcal{F}_{\bar{i}}\times\mathcal{F}$ for all $n\in\mathbb{Z}$ and $i=0, 1$, and \begin{equation}\label{eq: degree derivation condition of differential on VSA-inductive sheaf} D\circ\underline{(n)}-(-1)^l\underline{(n)}\circ(\mathrm{id}\times D)=\underline{(n)}\circ (D\times\mathrm{id}), \end{equation} on $\mathcal{F}^l\times\mathcal{F}$ for all $l\in\mathbb{Z}$. Here $\underline{H}$ is the Hamiltonian of $\mathcal{V}$ and $\underline{J}$ is the degree-grading operator of $\mathcal{V}$. \end{definition}
By a \textbf{differential degree-weight-graded VSA-inductive sheaf}, we mean a degree-weight-graded VSA-inductive sheaf given a differential.
\begin{remark}\label{rem: presheaf of differential VSA-inductive sheaf} Let $(\mathcal{V}, D)$ be a differential degree-weight-graded VSA-inductive sheaf. Then $(\underrightarrow{\mathrm{Lim}}\, \mathcal{V},$ $\underrightarrow{\mathrm{Lim}}\, D)$ is a presheaf of differential degree-weight-graded vertex superalgebras. \end{remark}
\begin{notation} We denote by $\textit{Diff-}\textit{DegWt-}\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}_X$ the category of differential degree-weight-graded VSA-inductive sheaves on a topological space $X$, whose morphisms are morphisms of degree-weight-graded VSA-inductive sheaves on $X$ commuting with the differentials. \end{notation}
\begin{remark} Let $\varphi: X\to Y$ be a continuous map between topological spaces and $(\mathcal{V}, D)$ a differential degree-weight-graded VSA-inductive sheaf on $X$. Then the pair $(\varphi_*\mathcal{V}, \varphi_*D)$ is a differential degree-weight-graded VSA-inductive sheaf on $Y$. \end{remark}
\begin{notation} We denote by $\textit{Diff-}\textit{DegWt-}\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}$ the category of differential degree-weight-graded VSA-inductive sheaves, whose morphisms are morphisms of degree-weight-graded VSA-inductive sheaves which commute with the differentials. \end{notation}
\section{Chiral Lie Algebroid Cohomology}\label{section: Chiral Lie Algebroid Cohomology} In this section, we will construct VSA-inductive sheaves associated with vector bundles. For that purpose, we will construct VSA-inductive sheaves on an affine space. Then we will glue them to obtain a VSA-inductive sheaves on a manifold, using the facts proved in the preceding section.
By a manifold, we will mean a $C^\infty$-manifold. Let $M$ be a manifold. We simply denote by $TM$ the tangent bundle tensored by $\mathbb{K}$, $TM\otimes_{\mathbb{R}}\mathbb{K}$. We use a similar notation for the cotangent bundle $T^*M$, the sheaf of functions $C^\infty=C^\infty_M$ and the sheaf of vector fields $\mathscr{X}=\mathscr{X}_M$. We will mean a real or complex vector bundle simply by a vector bundle, following $\mathbb{K}$ is $\mathbb{R}$ or $\mathbb{C}$.
$TM\otimes_{\mathbb{R}}\mathbb{K}$.
\subsection{Lie Algebroids}\label{subsection: Lie Algebroids} In this subsection, we recall the notion of Lie algebroids. We refer the reader to \cite{DZ05,HM90,Mac05,Pra67,Vai97,Vai91} for more details.
Let $M$ be a manifold. A \textit{Lie algebroid} on $M$ is a vector bundle $A$ together with a vector bundle map $a: A \to TM$ over $M$, called the anchor map of $A$, and a $\mathbb{K}$-linear Lie bracket $[\,,]$ on $\Gamma(A)$ such that $ [X,fY]=f[X,Y]+a(X)(f)Y $ for all $X, Y\in \Gamma(A), f\in C^\infty(M).$ When $\mathbb{K}$ is $\mathbb{R}$, the corresponding Lie algebroids are called \textit{real Lie algebroids}. Similarly when $\mathbb{K}=\mathbb{C}$, the corresponding Lie algebroids are called \textit{complex Lie algebroids}.
\begin{example}[tangent bundles]\label{ex: tangent bundle Lie algebroid} The tangent bundle $TM$ of a manifold $M$ with bracket the Lie bracket of vector fields and with anchor the identity of $TM$ is a Lie algebroid on $M.$ \end{example}
\begin{example}[transformation Lie algebroids]\label{ex: transformation Lie algebroid} Let $ \rho: \mathfrak{g} \to \mathscr{X}(M) $ be an infinitesimal action of a Lie algebra $\mathfrak{g}$ on a manifold $M.$ Then there is a natural Lie algebroid structure on the trivial vector bundle $M \times \mathfrak{g}$ with the anchor map $ a_\rho(m, \xi):=\rho(\xi)(m) $ for $(m, \xi)\in M\times \mathfrak{g}$ and the bracket on $\Gamma(M\times \mathfrak{g})\cong C^\infty(M, \mathfrak{g})$ $$ [X,Y]_\rho:=[X,Y]_{\mathfrak{g}}+a_\rho(X)Y-a_\rho(Y)X, $$ for $X, Y\in C^\infty(M, \mathfrak{g}).$ This Lie algebroid $(M\times \mathfrak{g}, a_\rho, [\,,]_\rho)$ is called the \textit{transformation Lie algebroid} associated with $\rho$. \end{example}
\begin{example}[cotangent Lie algebroids]\label{ex: cotangent Lie algebroid} Let $(M, \Pi)$ be a Poisson manifold with the Poisson bivector field $\Pi.$ Consider the map $$ \Pi^\sharp: T^*M \to TM,\ \Pi^\sharp(\alpha)(\beta):=\Pi(\alpha, \beta)\ \text{for}\ \alpha, \beta \in \Gamma(T^*M). $$ Then, with $\Pi^\sharp$ as the anchor map, the cotangent bundle $T^*M$ becomes a Lie algebroid on $M$, where the Lie bracket on $\Gamma(T^*M)$ is given by $$ [\alpha,\beta]:=d(\Pi(\alpha, \beta))+i_{\Pi^\sharp\alpha}d\beta-i_{\Pi^\sharp\beta}d\alpha, $$ for $\alpha, \beta \in \Gamma(T^*M).$ The Lie algebroid $(T^*M, \Pi^\sharp, [\,,])$ is called the \textit{cotangent Lie algebroid} of $(M, \Pi).$ \end{example}
Let us recall the notion of the Lie algebroid representation. Let $(A, a, [\,,])$ be a Lie algebroid on a manifold $M$ and $E$ a vector bundle on $M$. An $A$-\textit{connection} on $E$ is a map $ \nabla: \Gamma(A) \times \Gamma(E) \to \Gamma(E) $ such that \begin{enumerate}[$\bullet$]
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\setlength{\itemindent}{10pt} \item $\nabla_{X+Y}s=\nabla_X s+\nabla_Y s,$ \item $\nabla_X(s+s')=\nabla_X s+\nabla_X s',$ \item $\nabla_{fX}s=f\nabla_X s,$ \item $\nabla_X(fs)=f\nabla_X s +a(X)(f)s,$ \end{enumerate} for all $X, Y\in \Gamma(A),\ s, s'\in \Gamma(E),$ and $f\in C^\infty(M).$ An $A$-connection $\nabla$ on $E$ is said to be \textit{flat} if $ \nabla_{[X,Y]}s=\nabla_X(\nabla_Y s)-\nabla_Y(\nabla_X s) $ for all $X, Y\in \Gamma(A)$ and $s\in \Gamma(E).$ A flat $A$-connection on $E$ is also called a \textit{representation} of $A$ on $E$.
\begin{example}[trivial representations]\label{ex: trivial representation of Lie algebroid} Let $A$ be a Lie algebroid on $M$ and $V$ a vector space. The \textit{trivial representation} of $A$ on $M \times V$ is given by $ \nabla_X f:=a(X)(f) $ for $X\in \Gamma(A)$ and $f: M \to V.$ \end{example}
Let us now recall the definition of the Lie algebroid cohomology.
For a Lie algebroid $(A, a, [\,,])$ on a manifold $M$ and a representation $(E, \nabla)$ of $A$ on a vector bundle $E$, consider a complex $\Omega^\bullet(A; E):=\Gamma((\wedge^\bullet A^*)\otimes E)$ and a differential $d_{\text{Lie}}^E: \Omega^\bullet(A; E) \to \Omega^{\bullet+1}(A; E)$ defined by \begin{multline*} (d_{\text{Lie}}^E \omega)(X_1, \dots, X_{n+1}) = \sum_{i=1}^{n+1}(-1)^{i+1}\nabla_{X_i}(\omega(X_1, \dots, \check{X_i}, \dots, X_{n+1})) \\ +\sum_{1\le i<j\le n+1}(-1)^{i+j}\omega([X_i,X_j], X_1, \dots, \check{X_i}, \dots, \check{X_j}, \dots, X_{n+1}), \end{multline*} for $\omega \in \Omega^n(A; E)$ and $X_1, \dots, X_{n+1}\in \Gamma(A).$ The cohomology space $H^\bullet(A; E)$ of the complex $(\Omega^\bullet(A; E), d_{\text{Lie}}^E)$ is called the \textit{Lie algebroid cohomology} with coefficients in $E$.
Any $X\in \Gamma(A)$ induces the \textit{Lie derivative} $L_X: \Omega^\bullet(A; E) \to \Omega^\bullet(A; E)$ and the \textit{interior product} $\iota_X: \Omega^\bullet(A; E) \to \Omega^{\bullet-1}(A; E):$ \begin{gather*} (L_X\omega)(X_1, \dots, X_n)=\nabla_X(\omega(X_1, \dots, X_n)) -\sum_{i=1}^{n}\omega(X_1, \dots, [X,X_i], \dots, X_n), \\ (\iota_X\omega)(X_1, \dots, X_{n-1})=\omega(X, X_1, \dots, X_{n-1}), \end{gather*} where $\omega \in \Omega^n(A; E)$ and $X_1, \dots, X_n \in \Gamma(A).$ The Lie derivatives are derivations of degree $0$ and the interior products are derivations of degree $-1$. They satisfy the Cartan relations: \begin{gather*} [d_{\text{Lie}}^E,\iota_X]=L_X, \\ [L_X,L_Y]=L_{[X,Y]}, \\ [L_X,\iota_Y]=\iota_{[X,Y]}, \\ [\iota_X,\iota_Y]=0, \end{gather*} for all $X, Y\in \Gamma(A).$
When $(E, \nabla)$ is the trivial representation on the trivial line bundle, we simply denote by $(\Omega^\bullet(A), d_{\text{Lie}})$ and $H^\bullet(A)$ the corresponding complex and cohomology, respectively.
\subsection{VSA-Inductive Sheaves on $\mathbb{R}^m$}
In this subsection, we define an important object of the category $\textit{DegWt-}\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}_{\mathbb{R}^m}^\mathbb{N}$, which we will denote by $\Omega_\mathrm{ch}(\mathbb{R}^{m|r})$.
To this end, we first construct an object, denoted by $\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})$, of the category $\mathit{Presh}_{\mathbb{R}^m}(\textit{DegWt-VSA}_\mathbb{K})_\mathrm{bdw, sh}$, following the argument in \cite{LL}.
Fix natural numbers $m$ and $r$. We consider the supermanifold $\mathbb{R}^{m|r}=\bigl(\mathbb{R}^m,$ $ C^\infty_{\mathbb{R}^m}\otimes_{\mathbb{K}} \bigwedge_{\mathbb{K}}(\theta^1, \dots, \theta^r)\bigr)$.
We denote the structure sheaf $C^\infty_{\mathbb{R}^m}\otimes_\mathbb{K} \bigwedge_\mathbb{K}(\theta^1, \dots, \theta^r)$ by $C^\infty_{\mathbb{R}^{m|r}}$. Let $U$ be an open subset of $\mathbb{R}^m$.
Consider the supercommutative superalgebra of functions on $U\subset \mathbb{R}^{m|r}$, $
C^{\infty}_{\mathbb{R}^{m|r}}(U)=C^\infty_{\mathbb{R}^m}(U)\otimes_{\mathbb{K}}\bigwedge\nolimits_{\mathbb{K}}(\theta^1, \dots, \theta^r), $ even derivations $\partial/\partial x^1, \dots, \partial/\partial x^m$ and odd derivations $\partial/\partial \theta^1, \dots, \partial/\partial \theta^r$ on it. Here $(x^1, \dots, x^m, \theta^1, \dots, \theta^r)$ is a standard supercoordinate on $U$. We consider the supercommutative Lie superalgebra $$
D^{m|r}(U):=\mathrm{Span}_\mathbb{K}\{ \partial/\partial x^i, \partial/\partial \theta^j | i=1, \dots, m,\ j=1, \dots, r\}, $$
acting on $C^{\infty}_{\mathbb{R}^{m|r}}(U)$ naturally, and set $$
\Lambda^{m|r}(U):= D^{m|r}(U) \ltimes C^{\infty}_{\mathbb{R}^{m|r}}(U). $$ We put $$
\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})(U):=N\bigl(\Lambda^{m|r}(U), 0\bigr)/\mathcal{I}^{m|r}(U), $$
where $\mathcal{I}^{m|r}(U)$ is the ideal of the affine vertex superalgebra $N(\Lambda^{m|r}(U), 0) \cong O(\Lambda^{m|r}(U), 0)$ (see Example \ref{ex:affine va}) generated by \begin{gather*} \frac{d}{dz}f(z)-\sum_{i=1}^m\,{\baselineskip0pt\lineskip0.3pt\vcenter{\hbox{$\cdot$}\hbox{$\cdot$}}\,{\frac{d}{dz}x^i(z) \frac{\partial f}{\partial x^i}(z)}\,\vcenter{\hbox{$\cdot$}\hbox{$\cdot$}}}\,-\sum_{j=1}^r\baselineskip0pt\lineskip0.3pt\vcenter{\hbox{$\cdot$}\hbox{$\cdot$}}\,\frac{d}{dz}\theta^j(z) \frac{\partial f}{\partial \theta^j}(z)\,\vcenter{\hbox{$\cdot$}\hbox{$\cdot$}}, \\ (fg)(z)-\baselineskip0pt\lineskip0.3pt\vcenter{\hbox{$\cdot$}\hbox{$\cdot$}}\,f(z)g(z)\,\vcenter{\hbox{$\cdot$}\hbox{$\cdot$}}, \ 1(z)-\mathrm{id}, \end{gather*}
where $f, g\in C^\infty_{\mathbb{R}^{m|r}}(U)\subset\Lambda^{m|r}(U)$. For $A\in N\bigl(\Lambda^{m|r}(U), 0\bigr)$, we denote by $\overline{A}$ the corresponding element in $\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})(U)$.
The Lie superalgebra $\Lambda^{m|r}(U)$ has compatible two gradings when we give $D^{m|r}(U)$, $C^{\infty}_{\mathbb{R}^{m|r}}(U)$ weight $0, -1$, respectively, and if we give $\partial/\partial x^i$, $\partial/\partial\theta^j$ and $C^\infty_{\mathbb{R}^m}(U)\otimes\bigwedge^l(\theta^1, \dots, \theta^r)$ degree $0$, $-1$ and $l$, respectively. This induces a degree-weight-grading on the vertex superalgebra $N\bigl(\Lambda^{m|r}(U), 0\bigr)$ (see Example \ref{ex: another grading on N(g, 0)}). Then the ideal $\mathcal{I}^{m|r}(U)$ is a homogeneous ideal. Therefore the vertex superalgebra $\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})(U)$ becomes a degree-weight-graded vertex superalgebra.
We set $$ \Gamma_{m}(U):=N\bigl(D_m(U)\ltimes C^\infty_{\mathbb{R}^m}(U), 0\bigr)/\mathcal{I}_m(U), $$
where $D_m(U)$ is the commutative Lie algebra $\mathrm{Span}_{\mathbb{K}}\{ \partial/\partial x^i | i=1, \dots, m\}$
and $\mathcal{I}_m(U)$ is the ideal of the affine vertex algebra $N\bigl(D_m(U)\ltimes C^\infty_{\mathbb{R}^m}(U), 0\bigr)$ generated by \begin{gather}\label{eq: relations for Gamma_m(U)} \frac{d}{dz}f(z)-\sum_{i=1}^m\baselineskip0pt\lineskip0.3pt\vcenter{\hbox{$\cdot$}\hbox{$\cdot$}}\,\frac{d}{dz}x^i(z)\frac{\partial f}{\partial x^i}(z) \,\vcenter{\hbox{$\cdot$}\hbox{$\cdot$}}\,, \\ (fg)(z)-\baselineskip0pt\lineskip0.3pt\vcenter{\hbox{$\cdot$}\hbox{$\cdot$}}\,f(z)g(z)\,\vcenter{\hbox{$\cdot$}\hbox{$\cdot$}}, \ 1(z)-\mathrm{id}, \end{gather}
where $f, g\in C^\infty_{\mathbb{R}^m}(U)$. Regard $D_m(U)\ltimes C^\infty_{\mathbb{R}^m}$ as a degree-weight-graded Lie subalgebra of $\Lambda^{m|r}(U)$. Then $\Gamma_m(U)$ becomes a degree-weight-graded vertex algebra (the degree-grading is trivial). For $A\in N\bigl(D_m(U)\ltimes C^\infty_{\mathbb{R}^m}(U), 0\bigr)$, we denote by $\overline{A}$ the corresponding element in $\Gamma_m(U)$.
\begin{remark}\label{rem: generators of Gamma_m(U)} $\Gamma_m(U)$ is spanned by the vectors of the form $$ \overline{\partial/\partial x^{i_1}}_{(n_1)}\dots\overline{\partial/\partial x^{i_k}}_{(n_k)}\overline{x^{i'_1}}_{(n'_1)}\dots\overline{x^{i'_{k'}}}_{(n'_{k'})}\overline{f}_{(-1)}\mathbf{1}, $$ with $n_1, \dots, n_k\le-1,\ n'_1, \dots, n'_{k'}\le-2,$ $\ i_1, \dots, i_k, i'_1, \dots, i'_{k'}\in \{ 1, \dots, m\}$ and $f\in C^\infty_{\mathbb{R}^m}(U)$. This follows from the relations \eqref{eq: relations for Gamma_m(U)} by induction. \end{remark}
We can rewrite the vertex superalgebra $\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})(U)$, using the vertex superalgebras $\Gamma_m(U)$ and $\mathcal{E}(W_r)$. Here $\mathcal{E}(W_r)$ is the $bc$-system associated with $W_r=\mathrm{Span}_{\mathbb{K}}\{ \theta^j |\ j=1, \dots, r\}$ regarded as an even vector space (see Example \ref{ex:bc-systems} for the definition of $bc$-systems and the notation used in the following proof). The $\mathbb{Z}$-graded vertex superalgebra $ \Gamma_m(U)\otimes\mathcal{E}(W_r)$ is degree-weight-graded when the degree-grading is given by the operator $j_{bc, (0)}$, where $j_{bc}=-\sum_{j=1}^r b^{\partial/\partial\theta^j}_{-1}c^{\theta^j}_{0}\mathbf{1}$.
\begin{lemma}\label{lem: omega_ch=beta-gamma_bc} There exists a canonical isomorphism of degree-weight-graded vertex superalgebras $$
\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})(U)\cong \Gamma_m(U)\otimes \mathcal{E}(W_r). $$ \end{lemma} \begin{proof}
The assertion follows by the same argument as in \cite{LL}, where the case when $m=r$ is considered. The canonical isomorphism is induced by the linear map $\alpha: \Lambda^{m|r}(U) \to \Gamma_m(U)\otimes \mathcal{E}(W_r)$ defined by \begin{align*} \alpha(\partial/\partial x^i)&:= \overline{\partial/\partial x^i}_{(-1)}\mathbf{1} \otimes \mathbf{1},\ i=1, \dots, m, \\ \alpha(\partial/\partial \theta^j)&:= \mathbf{1} \otimes b^{\partial/\partial\theta^{j}}_{-1}\mathbf{1},\ j=1, \dots, r, \\ \alpha(f\otimes \theta^{j_1}\dotsm\theta^{j_l})&:=\overline{f}_{(-1)}\mathbf{1} \otimes c^{\theta^{j_1}}_0\dotsm c^{\theta^{j_l}}_0\mathbf{1}, \\ &\text{for}\ f\otimes \theta^{j_1}\cdots\theta^{j_l}\in C^\infty_{\mathbb{R}^m}(U)\otimes \bigwedge(\theta^1, \dots, \theta^r), \end{align*} where we regard $(\partial/\partial\theta^{1}, \dots, \partial/\partial\theta^{r})$ as the basis dual to $(\theta^{1}, \dots, \theta^{r})$ for $W_r^*$. \end{proof}
We identify $\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})(U)$ with $\Gamma_m(U)\otimes \mathcal{E}(W_r)$ via this isomorphism.
We denote by $b^j_n$ and $c^j_n$ the element $b^{\partial/\partial\theta^{j}}_{n}=b^{\partial/\partial\theta^{j}}\otimes t^n$ and $c^{\theta^{j}}_n=\theta^{j}\otimes t^{n-1}dt$ in $\mathfrak{j}(W_r)$ (see Example \ref{ex:bc-systems} for the definition of the Lie superalgebra $\mathfrak{j}(W_r)$).
Consider the vector space $V_m(U):=\oplus_{i=1}^m\mathbb{K}x^i\subset C^\infty(U)$. We regard $(\partial/\partial x^1,$ $\dots,$ $\partial/\partial x^m)$ as the basis dual to $(x^1, \dots, x^m)$ and identify the Lie algebra $D_m(U)=\mathrm{Span}_\mathbb{K}\{ \partial/\partial x^1, \dots, \partial/\partial x^m \}$ as the dual vector space $V_m(U)^*$. Consider the Heisenberg Lie algebra associated with $D_m(U)$: $$ \mathfrak{h}_m(U):=\mathfrak{h}(D_m(U))=(D_m(U)[t^{\pm1}]\oplus V_m(U)[t^{\pm1}]dt)\oplus \mathbb{K}\mathbf{\tau}, $$ with commutation relations $ [\beta^{\partial/\partial x^i}_p,\gamma^{x^{i'}}_q]=\delta_{p, -q}\frac{\partial}{\partial x^i}(x^{i'})\mathbf{\tau}, $ where $\beta^{\partial/\partial x^i}_p$ and $\gamma^{x^{i'}}_q$ stand for, respectively, $\partial/\partial x^i\otimes t^p$ and $x^{i'}\otimes t^{q-1}dt$ as in Example \ref{ex:beta_gamma-systems}.
Consider the Heisenberg Lie algebra associated with the vector space $\mathbb{K}^m$: $$ \mathfrak{h}_m:=\mathfrak{h}(\mathbb{K}^m)=\mathbb{K}^m[t^{\pm1}]\oplus (\mathbb{K}^m)^*[t^{\pm1}]dt\oplus\mathbb{K}\mathbf{\tau}. $$ Let $(e_1, \dots, e_m)$ be the standard basis of $\mathbb{K}^m$ and $(\phi_1, \dots, \phi_m)$ the dual basis. We denote by $\beta^i_p$ and $\gamma^j_q$ the elements $e_1\otimes t^p$ and $\phi_i \otimes t^{q-1}dt$. We identify $\mathfrak{h}_m(U)$ with $\mathfrak{h}_m$ through the isomorphism induced by $ V_m(U)\to \mathbb{K}^m,\ x^i\mapsto e^i,\ i=1, \dots, m. $ We also denote $\beta^{\partial/\partial x^i}_p$ and $\gamma^{x^{i'}}_q$ by $\beta^i_p$ and $\gamma^{i'}_q$, respectively. We emphasize that the Lie algebra $\mathfrak{h}_m$ does not depend on the open subset $U$.
Let ${\mathfrak{h}_m(U)}_{\ge0}\subset \mathfrak{h}_m(U)$ be the Lie subalgebra generated by $\mathbf{\tau}$ and $\beta^i_p, \gamma^i_p$ with $i=1, \dots, m$ and $p, q \ge0$. We make $C^\infty_{\mathbb{R}^m}(U)$ a ${\mathfrak{h}_m(U)}_{\ge0}$-module by setting \begin{gather*} \beta^i_p\cdot f=\gamma^i_p\cdot f=0,\ p>0, \\ \beta^i_0\cdot f=\frac{\partial}{\partial x^i}(f),\ \gamma^i_0\cdot f=x^i f,\ \mathbf{\tau} \cdot f=f, \end{gather*} where $i=1, \dots, m$ and $f\in C^\infty_{\mathbb{R}^m}(U)$. As in \cite{LL}, we consider the $\mathfrak{h}_m(U)$-module $$ \Tilde{\Gamma}_m(U):=U(\mathfrak{h}_m(U))\otimes_{U({\mathfrak{h}_m(U)}_{\ge0})}C^\infty_{\mathbb{R}^m}(U). $$ Following \cite{LL}, we define the action of the commutative Lie algebra $C^\infty_{\mathbb{R}^m}(U)[t^{\pm1}]$ on $\Tilde{\Gamma}_m(U)\cong U(\mathfrak{h}_m(U)_{< 0})\otimes C^\infty_{\mathbb{R}^m}(U)$ as follows, where $\mathfrak{h}_m(U)_{<0}$ stands for the Lie subalgebra of $\mathfrak{h}_m(U)$ generated by $\beta^i_p, \gamma^i_p$ with $p<0$ and $i=1, \dots, m$. We define the action of $ft^k\in C^\infty_{\mathbb{R}^m}(U)[t^{\pm1}]$, denoted by $f_{(k)}$, using the PBW monomial basis of $U(\mathfrak{h}_m(U)_{< 0})$ for the basis $\beta^i_p, \gamma^i_p,\ p<0, i=1, \dots, m$ with an order such that $\beta^i_p>\gamma^{i'}_{q}$ for any $i, i'=1, \dots, m$ and $p, q<0$. First, for $ft^k\in C^\infty_{\mathbb{R}^m}(U)[t^{\pm1}]$ with $k\ge -1$ we set $$ f_{(k)}(1\otimes g):=\delta_{k, -1}(1\otimes fg), \quad g\in C^\infty_{\mathbb{R}^m}(U), $$ and for $ft^k\in C^\infty_{\mathbb{R}^m}(U)[t^{\pm1}]$ with $k < -1$ we inductively define $f_{(k)}$ on $\mathbb{K}1\otimes C^\infty_{\mathbb{R}^m}(U)$ by setting $$ f_{(k)}(1\otimes g):=\frac{1}{k+1}\sum_{i=1}^m\sum_{q<0}q\gamma^i_q\Bigl(\frac{\partial f}{\partial x^i}\Bigr)_{(k-q)}(1\otimes g), \quad g\in C^\infty_{\mathbb{R}^m}(U). $$ Next we define $f_{(k)}(P\otimes g)$ for $ft^k\in C^\infty_{\mathbb{R}^m}(U)[t^{\pm1}]$, a PBW monomial $P$ of positive length and $g\in C^\infty_{\mathbb{R}^m}(U)$. The PBW monomial $P$ is of the form $\gamma^i_p P'$ or $\beta^i_p P'$ with a PBW monomial $P'$ of length less than that of $P$. We set $$ f_{(k)}(P\otimes g):= \begin{cases} \gamma^i_pf_{(k)}(P'\otimes g), & \text{if}\ P=\gamma^i_p P', \\ \beta^i_pf_{(k)}(P'\otimes g)-\bigl(\frac{\partial f}{\partial x^i}\bigl)_{(k+p)}(P'\otimes g), & \text{if}\ P=\beta^i_p P'. \end{cases} $$ Thus we get operators $f_{(k)}$ on $\Tilde{\Gamma}_m(U)$.
We put $$ \gamma^i(z):=\sum_{n\in\mathbb{Z}}\gamma^i_n z^{-n}, \quad \beta^i(z):=\sum_{n\in\mathbb{Z}}\beta^i_n z^{-n-1}, $$ for $i=1, \dots, m$ and $$ f(z):=\sum_{k\in\mathbb{Z}}f_{(k)}z^{-k-1}, $$ for $f \in C^\infty_{\mathbb{R}^m}(U)$. Note that when $f=x^i$, we have $x^i(z)=\gamma^i(z)$, or equivalently, $x^i_{(k)}=\gamma^i_{k+1}$ for all $k\in \mathbb{Z}$. For $f\in C^\infty_{\mathbb{R}^m}(U)$, we also use the notation $$ f(z)=\sum_{n\in\mathbb{Z}}f_n z^{-n}, $$ or equivalently, $f_{(k)}=f_{k+1}$ for $k\in \mathbb{Z}$ and simply denote by $f$ the element $1\otimes f\in \Tilde{\Gamma}(U)$. Since $\beta^i(z)$ and $\gamma^i(z)$ come from the action of the Heisenberg Lie algebra $\mathfrak{h}_m(U)$, we have \begin{gather} \label{eq: [beta(z),beta(w)]=0} [\beta^i(z),\beta^{i'}(w)]=0, \\ [\gamma^i(z),\gamma^{i'}(w)]=0, \quad [\beta^i(z),\gamma^{i'}(w)]=\delta_{i, i'}\delta(z-w), \notag \end{gather} for $i, i'=1, \dots, m$. The following lemmas are proved in \cite[Section 2.5]{LL}.
\begin{lemma}[Lian-Linshaw] \label{lem: [beta(z),f(w)]=df/dx(w)delta(z-w)} The following hold: \begin{equation}\label{eq: [beta(z),f(w)]=df/dx(w)delta(z-w)} [\gamma^i(z),f(w)]=0, \quad [\beta^i(z),f(w)]=\frac{\partial f}{\partial x^i}(w)\delta(z-w). \end{equation} for all $f \in C^\infty_{\mathbb{R}^m}(U)$ and $i=1, \dots, m$. \end{lemma}
\begin{lemma}[Lian-Linshaw]\label{lem: relations for f(z)} The following hold: \begin{gather} \label{eq: [f(z),g(z)]=0} [f(z),g(w)]=0, \\ \label{eq: df(z)=df/dx(z)dx(z)} \frac{d}{dz}f(z)=\sum_{i=1}^m\frac{d}{dz}\gamma^i(z)\frac{\partial f}{\partial x^i}(z), \end{gather} for $f, g \in C^\infty_{\mathbb{R}^m}(U)$ and \begin{equation} \label{eq: 1(z)=1} 1(z)=\mathrm{id}. \end{equation} \end{lemma}
As a corollary of the above two lemmas, we have the following proposition, which corresponds to \cite[Corollary 2.21]{LL}.
\begin{proposition}\label{prop: VERTEX ALGEBRA STRUCTURE ON GAMMA-TILDE} $\Tilde{\Gamma}_m(U)$ has a unique vertex algebra structure such that \begin{enumerate}[$\bullet$]
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\setlength{\itemindent}{0pt} \item the vacuum vector is $\mathbf{1} :=1\otimes 1,$ \item the translation operator is $T:=\mathrm{Res}_{z=0}\sum_{i=1}^m\,{\baselineskip0pt\lineskip0.3pt\vcenter{\hbox{$\cdot$}\hbox{$\cdot$}}\,\beta^i(z)\partial_z\gamma^i(z)\,\vcenter{\hbox{$\cdot$}\hbox{$\cdot$}}}$, \item the vertex operators satisfy $Y(1\otimes f, z)=f(z)$ for $f\in C^\infty_{\mathbb{R}^m}(U)$ and $Y(\beta^i_{-1}\otimes1, z)=\beta^i(z)$ for $i=1, \dots, m.$ \end{enumerate} \end{proposition} \begin{proof} We will apply the existence theorem of Frenkel-Kac-Radul-Wang (see \cite[Proposition 3.1]{FKRW95}). The relations $[T,f(z)]=\partial_zf(z)$ and $[T,\beta^i(z)]$ $=\partial_z\beta^i(z)$ are checked by direct computations with \eqref{eq: [beta(z),f(w)]=df/dx(w)delta(z-w)}. By Lemmas \ref{lem: [beta(z),f(w)]=df/dx(w)delta(z-w)} and \ref{lem: relations for f(z)}, the other conditions in the existence theorem are satisfied. \end{proof}
By Proposition \ref{prop: VERTEX ALGEBRA STRUCTURE ON GAMMA-TILDE} and $f_{(-1)}g=fg$, we have \begin{equation}\label{eq: (fg)(z)=f(z)g(z)} (fg)(z)=f(z)g(z), \end{equation} for $f, g\in C^\infty_{\mathbb{R}^m}(U)$. By OPEs \eqref{eq: [beta(z),beta(w)]=0}, \eqref{eq: [f(z),g(z)]=0} and \eqref{eq: (fg)(z)=f(z)g(z)}, we have a vertex algebra morphism $ N\bigl(D_m(U)\ltimes C^\infty_{\mathbb{R}^m}(U), 0\bigr)\to \Tilde{\Gamma}_m(U) $ sending $f$ and $\partial/\partial x^i$ to $f=1\otimes f$ and $\beta^i_{-1}\mathbf{1}=\beta^i_{-1}\otimes1$, respectively. Moreover by the relations \eqref{eq: df(z)=df/dx(z)dx(z)}, \eqref{eq: 1(z)=1} and \eqref{eq: (fg)(z)=f(z)g(z)}, this morphism factors through the ideal $\mathcal{I}_m(U)$, and hence we have a morphism from $\Gamma_m(U)$. We can see this map is bijective, by taking into consideration the form of the basis of $\Tilde{\Gamma}_m(U)$ and Lemma \ref{rem: generators of Gamma_m(U)}. Thus we have the following.
\begin{proposition}\label{prop: GAMMA=GAMMA-TILDE} The linear map \begin{align*} D_m(U)\ltimes C^\infty_{\mathbb{R}^m}(U)&\to \Tilde{\Gamma}_m(U), \\
f &\to 1\otimes f, \\ \partial/\partial x^i &\to \beta^i_{-1}\otimes1, \end{align*} induces an isomorphism of vertex algebras $$ \Gamma_m(U) \xrightarrow{\cong} \Tilde{\Gamma}_m(U). $$ \end{proposition}
We identify $\Gamma_m(U)$ with $\Tilde{\Gamma}_m(U)$ through this isomorphism.
Let $\mathfrak{h}_{m, \ge0}$ and $\mathfrak{h}_{m, <0}$ be the Lie subalgebras of $\mathfrak{h}_m$ generated by $\beta^i_p, \gamma^i_p$ with $p\ge 0,\ i=1, \dots, m$, and $\beta^i_p, \gamma^i_p$ with $p<0,\ i=1, \dots, m$, respectively. By Poincar\'{e}-Birkhoff-Witt theorem, the decomposition $\mathfrak{h}_m=\mathfrak{h}_{m, <0}\oplus \mathfrak{h}_{m, \ge0}$ induces an isomorphism as vector spaces $$ \Tilde{\Gamma}_m(U)\cong U(\mathfrak{h}_{m, <0})\otimes C^\infty_{\mathbb{R}^m}(U). $$ Using the isomorphism in Lemma \ref{prop: GAMMA=GAMMA-TILDE}, we have an isomorphism
\begin{equation}\label{eq: basis of Gamma} \Gamma_m(U)\cong U(\mathfrak{h}_{m, <0})\otimes C^\infty_{\mathbb{R}^m}(U). \end{equation} Therefore, by Lemma \ref{lem: omega_ch=beta-gamma_bc} we have
\begin{equation}\label{eq: omega_ch=beta-gamma_bc-C_infty}
\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})(U)\cong U(\mathfrak{h}_{m, <0})\otimes C^\infty_{\mathbb{R}^m}(U)\otimes \mathcal{E}(W_r). \end{equation}
By restricting this isomorphism to the space of weight $n\in \mathbb{N}$, $\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})(U)[n]$, we get an isomorphism
\begin{equation}\label{eq: omega_ch=beta-gamma_bc-C_infty[n]}
\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})(U)[n]\cong (U(\mathfrak{h}_{m, <0})\otimes \mathcal{E}(W_r))[n]\otimes C^\infty_{\mathbb{R}^m}(U), \end{equation} where $(U(\mathfrak{h}_{m, <0})\otimes \mathcal{E}(W_r))[n]$ stands for the weight $n$ space of $U(\mathfrak{h}_{m, <0})\otimes \mathcal{E}(W_r)$ with respect to the grading induced by the $\mathbb{Z}_{\ge0}$-grading on $\mathcal{E}(W_r)$ and the grading on $U(\mathfrak{h}_{m, <0})$ given by $\mathrm{wt}\,\beta^i_p=\mathrm{wt}\,\gamma^i_p=-p$. Note that $(U(\mathfrak{h}_{m, <0})\otimes \mathcal{E}(W_r))[n]$ is finite-dimensional.
For two open subsets $V\subset U\subset \mathbb{R}^m$, we define the restriction map $
\mathrm{Res}^{m|r}_{V, U}: \underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})(U)\to \underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})(V) $ as follows.
The restriction map $C^\infty_{\mathbb{R}^{m|r}}(U)\to C^\infty_{\mathbb{R}^{m|r}}(V)$ induces a Lie superalgebra morphism $
\Lambda^{m|r}(U)\to\Lambda^{m|r}(V) $ and this morphism induces a morphism $
N(\Lambda^{m|r}(U), 0) \to N(\Lambda^{m|r}(V), 0) $ of vertex superalgebras.
Since this vertex superalgebra morphism is induced by an algebra morphism, namely, a morphism preserving the product and the unit, the generators of the ideal $\mathcal{I}^{m|r}(U)$ is mapped into $\mathcal{I}^{m|r}(V)$. Therefore we have a vertex superalgebra morphism
from $\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})(U)$ to $\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})(V)$. We define the restriction map $$
\mathrm{Res}^{m|r}_{V, U}: \underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})(U)\to \underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})(V), $$ as this vertex superalgebra morphism. Note that this restriction map preserves the degree-weight-grading.
The assignment $$
U\mapsto\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})(U), $$
and the maps $\mathrm{Res}^{m|r}_{V, U}$ define a presheaf of degree-weight-graded vertex superalgebras.
We denote by $\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})$ the presheaf. We claim that this presheaf is an object of the category $\mathit{Presh}_{\mathbb{R}^m}(\textit{DegWt-VSA}_\mathbb{K})_\mathrm{bdw, sh}$ (see Section \ref{subsection: From Presheaves to VSA-Inductive Sheaves} for the definition of this category). It suffices to check the presheaves $\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})[n]$ are sheaves for all $n\in \mathbb{N}$, where the presheaf $\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})[n]$ is defined by the assignment $$
U\mapsto \Gamma(U, \underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r}))[n], $$
and the maps $\mathrm{Res}^{m|r}_{V, U}\big|_{\Gamma(U, \underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r}))[n]}$.
The restriction map $
\mathrm{Res}^{m|r}_{V, U}\big|_{\Gamma(U, \underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r}))[n]} $ becomes $$ \mathrm{id}\otimes\mathrm{Res}_{V, U}\!\!: \!(U(\mathfrak{h}_{m, <0})\otimes \mathcal{E}(W_r))[n]\,\otimes C^\infty_{\mathbb{R}^m}(U)\!\to\! (U(\mathfrak{h}_{m, <0})\otimes \mathcal{E}(W_r))[n]\,\otimes C^\infty_{\mathbb{R}^m}(V), $$ through the isomorphism \eqref{eq: omega_ch=beta-gamma_bc-C_infty[n]}, where $\mathrm{Res}_{V, U}$ stands for the restriction map of $C^\infty_{\mathbb{R}^m}$. Therefore we have an isomorphism of presheaves $$
\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})[n] \cong (U(\mathfrak{h}_{m, <0})\otimes \mathcal{E}(W_r))[n]\,\otimes_\mathbb{K} C^\infty_{\mathbb{R}^m}. $$
Since $C^\infty_{\mathbb{R}^m}$ is a sheaf and $(U(\mathfrak{h}_{m, <0})\otimes \mathcal{E}(W_r))[n]$ is finite-dimensional, the presheaf $(U(\mathfrak{h}_{m, <0})\otimes \mathcal{E}(W_r))[n]\,\otimes C^\infty_{\mathbb{R}^m}$ is a sheaf of super vector spaces, and therefore $\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})[n]$ is also a sheaf. Thus $\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})$ is an object of the category $\mathit{Presh}_{\mathbb{R}^m}(\textit{DegWt-VSA}_\mathbb{K})_\mathrm{bdw, sh}$.
\begin{notation}
We denote by $\Omega_\mathrm{ch}(\mathbb{R}^{m|r})$ the VSA-inductive sheaf associated with the object $\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})$ of $\mathit{Presh}_{\mathbb{R}^m}(\textit{DegWt-VSA}_\mathbb{K})_\mathrm{bdw, sh}$. (See Lemma \ref{lem: MAKE VsaIndSh from PRESHEAVES OF Z-GRADED Vsa}.) \end{notation}
\begin{remark}\label{rem: SUPERFUNCTIONS INCLUDED IN OMEGA_CHIRAL} By the isomorphism \eqref{eq: omega_ch=beta-gamma_bc-C_infty[n]} with $n=0$, there exists an isomorphism, $$
\Gamma(U, \underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r}))[0]\cong C^\infty_{\mathbb{R}^m}(U)\otimes \bigwedge(c^1_0, \dots, c^r_0), $$ for any open subset $U\subset \mathbb{R}^m$. Therefore the following isomorphism of sheaves exists:
\begin{equation}\label{eq: WEIGHT ZERO SPACE IS CLASSICAL}
\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})[0] \cong C^\infty_{\mathbb{R}^m}\otimes\bigwedge(\theta^1, \dots, \theta^r). \end{equation} Here $\theta^j$ is identified with $c^j_0=c^{\theta^j}_0 $ for $j=1, \dots, r$. \end{remark}
We identify $\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})[0]$ with $C^\infty_{\mathbb{R}^m}\otimes\bigwedge(\theta^1, \dots, \theta^r)$ via the isomorphism \eqref{eq: WEIGHT ZERO SPACE IS CLASSICAL}.
\subsection{VSA-Inductive Sheaves for Vector Bundles} Let $E$ be a vector bundle of rank $r$ on a manifold $M$ of dimension $m$.
Let $\mathcal{U}=\bigl(U_\lambda \bigr)_{\lambda \in \Lambda}$ be an arbitrary family of
open subsets $U_\lambda$ in $M$ with a chart $\mathbf{x}_\lambda=(x_\lambda^1, \dots, x_\lambda^m)$ of $U_\lambda$, and a frame $\mathbf{e}_\lambda=(e^1_\lambda, \dots, e^r_\lambda)$ of $E|_{U_\lambda}$ such that $\bigl\{ (U_\lambda, \mathbf{x}_\lambda)\bigr\}_{\lambda \in \Lambda}$ is an altas on $M$ and is contained in the $C^\infty$-structure of $M$. We call such a family $\mathcal{U}$ a \textbf{framed covering} of $E$. Let $\bigl(f^E_{\lambda \mu}=(f^{E, j j'}_{\lambda \mu})_{1\le j, j'\le r}\bigr)_{\lambda, \mu\in\Lambda}$ be the transition functions of $E$ associated with $(\mathbf{e}_\lambda)_{\lambda\in\Lambda}$. In other words, we have $e^j_\mu=\sum_{j'=1}^r f^{E, j' j}_{\lambda \mu}e^{j'}_\lambda$. We denote by $\mathbf{c_\lambda}=(c_\lambda^1, \dots, c_\lambda^r)$ the frame dual to $\mathbf{e}_\lambda=(e^1_\lambda, \dots, e^ r_\lambda)$ for $E^*|_{U_\lambda}$.
We set $$
\Omega_\mathrm{ch}(E; \mathcal{U})_\lambda:=(\mathbf{x_\lambda}^{-1})_*(\Omega_\mathrm{ch}(\mathbb{R}^{m|r})|_{\mathbf{x_\lambda}(U_\lambda)}). $$
Note that $\Omega_\mathrm{ch}(E; \mathcal{U})_\lambda$ is nothing but the VSA-inductive sheaf associated with the object $(\mathbf{x_\lambda}^{-1})_*(\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})|_{\mathbf{x_\lambda}(U_\lambda)})$ of $\mathit{Presh}_{U_\lambda}(\textit{DegWt-VSA}_\mathbb{K})_\mathrm{bdw, sh}$, where $\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})|_{\mathbf{x_\lambda}(U_\lambda)}$ is the presheaf restricted to $\mathbf{x_\lambda}(U_\lambda)$ but not its sheafification. Notice that $\underrightarrow{\mathrm{Lim}}\,\Omega_\mathrm{ch}(E; \mathcal{U})_\lambda=(\mathbf{x_\lambda}^{-1})_*(\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})|_{\mathbf{x_\lambda}(U_\lambda)})$.
We consider the following two subpresheaves of $\underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})$:
\begin{align}
\underrightarrow{\Omega^{\gamma c}_{\mathrm{ch}}}(\mathbb{R}^{m|r})&: U \mapsto \Gamma(U, \underrightarrow{\Omega^{\gamma c}_{\mathrm{ch}}}(\mathbb{R}^{m|r})):=\bigl\langle C^\infty_{\mathbb{R}^m}(U)\bigr\rangle \otimes \langle c^1_0\mathbf{1}, \dots, c^r_0\mathbf{1}\rangle, \\
\underrightarrow{\Omega^{\gamma bc}_{\mathrm{ch}}}(\mathbb{R}^{m|r})&: U \mapsto \Gamma(U, \underrightarrow{\Omega^{\gamma bc}_{\mathrm{ch}}}(\mathbb{R}^{m|r})):=\bigl\langle C^\infty_{\mathbb{R}^m}(U)\bigr\rangle \otimes \mathcal{E}(W_r), \end{align}
where $\bigl\langle C^\infty_{\mathbb{R}^m}(U)\bigr\rangle \subset\Gamma_m(U)$ and $\langle c^1_0\mathbf{1}, \dots, c^r_0\mathbf{1}\rangle\subset\mathcal{E}(W_r)$ stand for the subalgebra generated by $C^\infty_{\mathbb{R}^m}(U)$ and $\{c^1_0\mathbf{1}, \dots, c^r_0\mathbf{1}\}$, respectively. The presheaves $\underrightarrow{\Omega^{\gamma c}_{\mathrm{ch}}}(\mathbb{R}^{m|r})[n]$ and $\underrightarrow{\Omega^{\gamma bc}_{\mathrm{ch}}}(\mathbb{R}^{m|r})[n]$ are sheaves for all $n\in\mathbb{N}$ since we have an isomorphism $\bigl\langle C^\infty_{\mathbb{R}^m}(U)\bigr\rangle\cong U\bigl(\langle\gamma^i_p\ |\ p<0, i=1, \dots, m\rangle\bigr) \otimes C^\infty_{\mathbb{R}^m}(U)$ from the isomorphism in Proposition \ref{prop: GAMMA=GAMMA-TILDE}, where $
U\bigl(\langle\gamma^i_p\ |\ p<0, i=1, \dots, m\rangle\bigr)$ is the universal enveloping algebra of the commutative Lie subalgebra of $\mathfrak{h}_m$ generated by $\gamma^i_p$ with $p<0, i=1, \dots, m$. Therefore the presheaves $\underrightarrow{\Omega^{\gamma c}_{\mathrm{ch}}}(\mathbb{R}^{m|r})$ and $\underrightarrow{\Omega^{\gamma bc}_{\mathrm{ch}}}(\mathbb{R}^{m|r})$ are objects of the category $\mathit{Presh}_{\mathbb{R}^m}(\textit{DegWt-VSA}_\mathbb{K})_\mathrm{bdw, sh}$. Thus we have VSA-inductive sheaves $\Omega^{\gamma c}_\mathrm{ch}(\mathbb{R}^{m|r})$ and $\Omega^{\gamma bc}_{\mathrm{ch}}(\mathbb{R}^{m|r})$, where $\Omega^{\gamma c}_\mathrm{ch}(\mathbb{R}^{m|r})$ and $\Omega^{\gamma bc}_{\mathrm{ch}}(\mathbb{R}^{m|r})$ are the VSA-inductive sheaves associated with $\underrightarrow{\Omega^{\gamma c}_{\mathrm{ch}}}(\mathbb{R}^{m|r})$ and $\underrightarrow{\Omega^{\gamma bc}_{\mathrm{ch}}}(\mathbb{R}^{m|r})$, respectively. Note that the vertex superalgebra $\Gamma(U, \underrightarrow{\mathrm{Lim}}\, \Omega^{\gamma c}_\mathrm{ch}(\mathbb{R}^{m|r}))=\Gamma(U, \underrightarrow{\Omega^{\gamma c}_{\mathrm{ch}}}(\mathbb{R}^{m|r}))$ is generated by the weight $0$ space for any open subset $U\subset \mathbb{R}^m$.
Set $$ \Gamma_{m}'(U):=N\bigl(C^\infty_{\mathbb{R}^m}(U), 0\bigr)/\mathcal{I}'_m(U), $$ where $\mathcal{I}'_m(U)$ is the ideal of the affine vertex algebra $N\bigl(C^\infty_{\mathbb{R}^m}(U), 0\bigr)$ generated by \begin{gather*} \frac{d}{dz}f(z)-\sum_{i=1}^m\,{\baselineskip0pt\lineskip0.3pt\vcenter{\hbox{$\cdot$}\hbox{$\cdot$}}\,\frac{d}{dz}x^i(z)\frac{\partial f}{\partial x^i}(z)\,\vcenter{\hbox{$\cdot$}\hbox{$\cdot$}}}, \\ (fg)(z)-\,{\baselineskip0pt\lineskip0.3pt\vcenter{\hbox{$\cdot$}\hbox{$\cdot$}}\,f(z)g(z)\,\vcenter{\hbox{$\cdot$}\hbox{$\cdot$}}}, \quad 1(z)-\mathrm{id}, \end{gather*} with $f, g\in C^\infty_{\mathbb{R}^m}(U)$.
\begin{remark} The canonical morphism, $ \Gamma_m'(U)\to\Gamma_m(U), $ induced by the inclusion of Lie algebras $C^\infty_{\mathbb{R}^m}(U)\to D_m(U)\ltimes C^\infty_{\mathbb{R}^m}(U)$ is injective. This follows from the form of the basis of $\Gamma_m(U)$. Therefore there exists an isomorphism of degree-weight-graded vertex algebras from $\Gamma_m'(U)$ to $\bigl\langle C^\infty_{\mathbb{R}^m}(U)\bigr\rangle \subset \Gamma_m(U)$. \end{remark}
We set \begin{align}
\Omega^{\gamma c}_\mathrm{ch}(E; \mathcal{U})_\lambda&:=(\mathbf{x_\lambda}^{-1})_*(\Omega^{\gamma c}_\mathrm{ch}(\mathbb{R}^{m|r})|_{\mathbf{x_\lambda}(U_\lambda)}), \\
\Omega^{\gamma bc}_{\mathrm{ch}}(E; \mathcal{U})_\lambda&:=(\mathbf{x_\lambda}^{-1})_*(\Omega^{\gamma bc}_{\mathrm{ch}}(\mathbb{R}^{m|r})|_{\mathbf{x_\lambda}(U_\lambda)}). \end{align} We will glue $\bigl(\Omega^{\gamma c}_\mathrm{ch}(E; \mathcal{U})_\lambda\bigr)_{\lambda\in\Lambda}$ and $\bigl(\Omega^{\gamma bc}_{\mathrm{ch}}(E; \mathcal{U})_\lambda\bigr)_{\lambda\in\Lambda}$.
Fix $U_\lambda$ with $\mathbf{x}_\lambda, \mathbf{e}_\lambda$ and $U_{\Tilde{\lambda}}$ with $\mathbf{x}_{\Tilde{\lambda}}, \mathbf{e}_{\Tilde{\lambda}}$ such that $U_\lambda\cap U_{\Tilde{\lambda}}\neq \emptyset$. Set $U:=\mathbf{x}_\lambda(U_\lambda\cap U_{\Tilde{\lambda}})$, $\Tilde{U}:=\mathbf{x}_{\Tilde{\lambda}}(U_{\Tilde{\lambda}}\cap U_\lambda)$ and
$\varphi=\varphi_{\Tilde{\lambda} \lambda}:= (\mathbf{x}_{\Tilde{\lambda}}|_{U_{\Tilde{\lambda}}\cap U_\lambda})\circ(\mathbf{x}_\lambda|_{U_\lambda\cap U_{\Tilde{\lambda}}})^{-1}: U\to \Tilde{U}$. We construct strict isomorphisms of VSA-inductive sheaves \begin{gather*}
\Omega^{\gamma c}_\mathrm{ch}(E; \mathcal{U})_{\Tilde{\lambda}}\big|_{U_{\Tilde{\lambda}}\cap U_\lambda} \to \Omega^{\gamma c}_\mathrm{ch}(E; \mathcal{U})_\lambda\big|_{U_\lambda\cap U_{\Tilde{\lambda}}}, \\
\Omega^{\gamma bc}_{\mathrm{ch}}(E; \mathcal{U})_{\Tilde{\lambda}}\big|_{U_{\Tilde{\lambda}}\cap U_\lambda} \to \Omega^{\gamma bc}_{\mathrm{ch}}(E; \mathcal{U})_\lambda\big|_{U_\lambda\cap U_{\Tilde{\lambda}}}, \end{gather*} motivated by gluing vector fields on the supermanifold $\Pi E=\bigr(M, \bigwedge(E^*)\bigr)$.
Let $V\subset U_\lambda\cap U_{\Tilde{\lambda}}$ be an open subset.
We denote by $\Tilde{\beta}^i_n$, $\Tilde{\gamma}^i_n$, $\Tilde{b}^j_n$ and $\Tilde{c}^j_n$ the operators $\beta^i_n$, $\gamma^i_n$, $b^j_n$ and $c^j_n$ on $\Gamma\Bigl(\mathbf{x}_{\Tilde{\lambda}}(V), \underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})\Bigr)$, respectively. We define elements of the vertex superalgebra
$\Gamma\Bigl(\mathbf{x}_\lambda(V), \underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})\Bigr)$ as follows:
\begin{gather} \label{df: phi^*f} \varphi^*\Tilde{f}:=\Tilde{f}, \\ \label{df: phi^*b} \varphi^*\Tilde{b}^j:=\sum_{1\le j'\le r}f^{E, j' j}_{\lambda \Tilde{\lambda}}b^{j'}_{-1}\mathbf{1}, \\ \label{df: phi^*c} \varphi^*\Tilde{c}^j:=\sum_{1\le j'\le r}f^{E, j j'}_{\Tilde{\lambda} \lambda}c^{j'}_0\mathbf{1}, \end{gather} for $\Tilde{f}\in C^\infty(V)$, $i=1, \dots, m$ and $j=1, \dots, r$. Here we use a usual notation for functions, identifying $C^\infty_{\mathbb{R}^m}\bigl(\mathbf{x}_\lambda(V)\bigr)$ and $C^\infty_{\mathbb{R}^m}\bigl(\mathbf{x}_{\Tilde{\lambda}}(V)\bigr)$ with $C^\infty(V)$ via $\mathbf{x}_\lambda$ and $\mathbf{x}_{\Tilde{\lambda}}$, respectively.
\begin{lemma}\label{lem: OPEs of OMEGA_CHIRAL} The following OPEs hold: \begin{gather} (\varphi^*\Tilde{f})(z)(\varphi^*\Tilde{g})(w)\sim 0, \\ (\varphi^*\Tilde{b}^j)(z)(\varphi^*\Tilde{b}^{j'})(w)\sim 0, \\ (\varphi^*\Tilde{c}^j)(z)(\varphi^*\Tilde{c}^{j'})(w)\sim 0, \\ (\varphi^*\Tilde{b}^j)(z)(\varphi^*\Tilde{c}^{j'})(w)\sim \frac{\delta_{j, j'}}{z-w}, \\ (\varphi^*\Tilde{f})(z)(\varphi^*\Tilde{b}^j)(w)\sim0, \\ (\varphi^*\Tilde{f})(z)(\varphi^*\Tilde{c}^j)(w)\sim0. \end{gather} \end{lemma} \begin{proof} These OPEs are checked by direct computations. \end{proof}
\begin{remark} When we consider the transformation rules of vector fields, it is natural to set $$ \varphi^*\Tilde{\beta}^i:=\sum_{1\le i'\le m}\beta^{i'}_{-1}\frac{\partial x^{i'}}{\partial \Tilde{x}^i}+\sum_{1\le j, k, l\le r}\frac{\partial f^{E, j k}_{\lambda \Tilde{\lambda}}}{\partial \Tilde{x}^i}f^{E, k l}_{\Tilde{\lambda} \lambda}c^l_0b^j_{-1}\mathbf{1}. $$ But the required relation $(\varphi^*\Tilde{\beta}^i)(z)(\varphi^*\Tilde{\beta}^{i'})(w) \sim 0$ does not hold in general. \end{remark}
By Lemma \ref{lem: OPEs of OMEGA_CHIRAL}, we have morphisms of degree-weight-graded vertex superalgebras \begin{align}
\underrightarrow{\vartheta'_{\lambda \Tilde{\lambda}}}(V): \Gamma\bigl(\mathbf{x}_{\Tilde{\lambda}}(V), \underrightarrow{\Omega^{\gamma c}_{\mathrm{ch}}}(\mathbb{R}^{m|r})\bigr) &\to \Gamma\bigl(\mathbf{x}_\lambda(V), \underrightarrow{\Omega^{\gamma c}_{\mathrm{ch}}}(\mathbb{R}^{m|r})\bigr), \\ \Tilde{f} &\mapsto \varphi^*\Tilde{f}, \notag \\ \Tilde{c}^j_0\mathbf{1} &\mapsto \varphi^*\Tilde{c}^j, \notag \\ \intertext{and,}
\underrightarrow{\vartheta''_{\lambda \Tilde{\lambda}}}(V): \Gamma\bigl(\mathbf{x}_{\Tilde{\lambda}}(V), \underrightarrow{\Omega^{\gamma bc}_{\mathrm{ch}}}(\mathbb{R}^{m|r})\bigr) &\to \Gamma\bigl(\mathbf{x}_\lambda(V), \underrightarrow{\Omega^{\gamma bc}_{\mathrm{ch}}}(\mathbb{R}^{m|r})\bigr), \\ \Tilde{f} &\mapsto \varphi^*\Tilde{f}, \notag \\ \Tilde{c}^j_0\mathbf{1} &\mapsto \varphi^*\Tilde{c}^j, \notag \\ \Tilde{b}^j_0\mathbf{1} &\mapsto \varphi^*\Tilde{b}^j. \notag \end{align} These morphisms $\Bigl(\underrightarrow{\vartheta'_{\lambda \Tilde{\lambda}}}(V)\Bigr)_V$ and $\Bigl(\underrightarrow{\vartheta''_{\lambda \Tilde{\lambda}}}(V)\Bigr)_V$ form morphisms of presheaves of degree-weight-graded vertex superalgebras $\underrightarrow{\vartheta'_{\lambda \Tilde{\lambda}}}$ and $\underrightarrow{\vartheta''_{\lambda \Tilde{\lambda}}}$, respectively. Therefore by Lemma \ref{lem: MAKE VsaIndSh from PRESHEAVES OF Z-GRADED Vsa} we have strict morphisms of VSA-inductive sheaves \begin{gather}
\vartheta'_{\lambda \Tilde{\lambda}}: \Omega^{\gamma c}_\mathrm{ch}(E; \mathcal{U})_{\Tilde{\lambda}}\big|_{U_{\Tilde{\lambda}}\cap U_\lambda} \to \Omega^{\gamma c}_\mathrm{ch}(E; \mathcal{U})_\lambda\big|_{ U_\lambda\cap U_{\Tilde{\lambda}}}, \\
\vartheta''_{\lambda \Tilde{\lambda}}: \Omega^{\gamma bc}_{\mathrm{ch}}(E; \mathcal{U})_{\Tilde{\lambda}}\big|_{U_{\Tilde{\lambda}}\cap U_\lambda} \to \Omega^{\gamma bc}_{\mathrm{ch}}(E; \mathcal{U})_\lambda\big|_{U_\lambda\cap U_{\Tilde{\lambda}}}. \end{gather} Thus we have families of strict morphisms $(\vartheta'_{\lambda \mu})_{\lambda, \mu \in \Lambda}$ and $(\vartheta''_{\lambda \mu})_{\lambda, \mu \in \Lambda}$. These morphisms satisfy the following.
\begin{lemma} The following hold: \begin{gather*} \vartheta'_{\lambda \lambda}= \mathrm{id}, \quad \vartheta'_{\lambda \mu}\circ\vartheta'_{\mu \nu}=\vartheta'_{\lambda \nu}, \\ \vartheta''_{\lambda \lambda}=\mathrm{id}, \quad \vartheta''_{\lambda \mu}\circ\vartheta''_{\mu \nu}=\vartheta''_{\lambda \nu}, \end{gather*} for all $\lambda, \mu, \nu\in \Lambda$. \end{lemma} \begin{proof} It suffices to check \begin{gather*} \underrightarrow{\vartheta'_{\lambda \lambda}}= \mathrm{id}, \quad \underrightarrow{\vartheta'_{\lambda \mu}}\circ\underrightarrow{\vartheta'_{\mu \nu}}=\underrightarrow{\vartheta'_{\lambda \nu}}, \\ \underrightarrow{\vartheta''_{\lambda \lambda}}=\mathrm{id}, \quad \underrightarrow{\vartheta''_{\lambda \mu}}\circ\underrightarrow{\vartheta''_{\mu \nu}}=\underrightarrow{\vartheta''_{\lambda \nu}}, \end{gather*} for all $\lambda, \mu, \nu\in \Lambda$. We can see that these relations hold on the generators from \eqref{df: phi^*f}, \eqref{df: phi^*b} and \eqref{df: phi^*c}. Thus we get the above relations. \end{proof}
By the construction, $\bigl((\Omega^{\gamma c}_\mathrm{ch}(E; \mathcal{U})_\lambda)_{\lambda\in\Lambda}, (\vartheta'_{\lambda \mu})_{\lambda, \mu \in \Lambda}\bigr)$ satisfies the assumptions (1)-(5) in Proposition \ref{prop: GLUING VsaIndShs}. Therefore by Proposition \ref{prop: GLUING VsaIndShs}, we can glue the degree-weight-graded VSA-inductive sheaves $\bigl((\Omega^{\gamma c}_\mathrm{ch}(E; \mathcal{U})_\lambda)_{\lambda\in\Lambda}, (\vartheta'_{\lambda \mu})_{\lambda, \mu \in \Lambda}\bigr)$ to a degree-weight-graded VSA-inductive sheaf on $M$. We denote by $\Omega^{\gamma c}_\mathrm{ch}(E; \mathcal{U})$ the resulting one. In the same way, we have a weight-degree-graded VSA-inductive sheaf on $M$, which we denote by $\Omega^{\gamma bc}_{\mathrm{ch}}(E; \mathcal{U})$, by gluing $\bigl((\Omega^{\gamma bc}_{\mathrm{ch}}(E; \mathcal{U})_\lambda)_{\lambda\in\Lambda},$ $(\vartheta''_{\lambda \mu})_{\lambda, \mu \in \Lambda}\bigr)$.
\begin{remark}\label{rem: we can take any framed covering} By Remark \ref{rem: Isomorphic as VSA-inductive sheaves if locally isomorphic}, the objects $\Omega^{\gamma c}_\mathrm{ch}(E; \mathcal{U})$ and $\Omega^{\gamma bc}_{\mathrm{ch}}(E; \mathcal{U})$ of the category $\textit{DegWt-}\mathit{VSA_{\mathbb{K}}}\textit{-IndSh}_M^\mathbb{N}$ \textit{do not} depend on the choice of framed coverings and are unique up to isomorphism. \end{remark} Therefore we simply denote by $\Omega^{\gamma c}_\mathrm{ch}(E)$ and $\Omega^{\gamma bc}_{\mathrm{ch}}(E)$ the VSA-inductive sheaves $\Omega^{\gamma c}_\mathrm{ch}(E; \mathcal{U})$ and $\Omega^{\gamma bc}_{\mathrm{ch}}(E; \mathcal{U})$, respectively.
\begin{remark} From the way of gluing and Remark \ref{eq: WEIGHT ZERO SPACE IS CLASSICAL}, the sheaf $\underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(E)}[0]$ is canonically isomorphic to the sheaf of local sections of $\bigwedge E^*$, denoted by $\underline{\bigwedge E^*}$. \end{remark}
\begin{remark} Let $E$ be the tangent bundle $TM$ of a manifold $M$. The presheaf associated with the VSA-inductive sheaf $\Omega^{\gamma c}_\mathrm{ch}(TM; \mathcal{U}^M)$ coincides with the small CDR for $M$ constructed in \cite{LL}, denoted by $\mathcal{Q}'_M$, where $\mathcal{U}_M=(U_\lambda)_{\lambda\in\Lambda}$ is the framed covering consisting of all open subsets $U_\lambda$ with a chart $\mathbf{x}_\lambda$ of $U_\lambda$ and the standard frame $(\partial/\partial x_\lambda^i)_{i=1, \dots, m}$. \end{remark}
\subsection{Chiral Lie Algebroid Complex} We consider the case when the vector bundle $E$ is a Lie algebroid. We construct a differential on $\Omega^{\gamma c}_\mathrm{ch}(E)$.
Let $(A, a, [ , ])$ be a Lie algebroid on a manifold $M$. We set $m:=\dim M$ and $r:=\mathrm{rank}\, A$. Let $\mathcal{U}=(U_\lambda)_{\lambda\in\Lambda}$ be a framed covering of $A$. As before we denote by $\mathbf{x}_\lambda$ and $\mathbf{e}_\lambda$ the chart and the frame associated with $U_\lambda$, respectively. First we define a differential on each $\Omega^{\gamma c}_\mathrm{ch}(A; \mathcal{U})_\lambda$. Then we will glue them.
Fix $\lambda \in \Lambda$. We write the anchor map and the bracket as $$ a(e_\lambda^j)=\sum_{\substack{1\le i\le m}}f^{\lambda, {i j}}\frac{\partial}{\partial x_\lambda^i}, $$ and $$ [e_\lambda^j,e_\lambda^k]=\sum_{1\le l \le r}\Gamma^{\lambda, j k}_l e_\lambda^l, $$ for $j, k=1, \dots, r$, where $f^{\lambda, {i j}}, \Gamma^{\lambda, j k}_l \in C^\infty(U_\lambda)$.
Let $V$ be an open subset of $U_\lambda$. Motivated by the differential $d_{\mathrm{Lie}}$ for the Lie algebroid cohomology, we define an odd element $Q^\lambda(V)$ of degree $1$ and weight $1$ in $\Gamma\Bigl(V, \underrightarrow{\mathrm{Lim}}\, {\Omega_\mathrm{ch}(A; \mathcal{U})_\lambda}\Bigr)=\Gamma\Bigl(\mathbf{x}_\lambda(V), \underrightarrow{\Omega_{\mathrm{ch}}}(\mathbb{R}^{m|r})\Bigr)$ by setting $$ Q^\lambda(V):=\sum_{\substack{1\le i \le m \\ 1\le j \le r}}\beta^i_{-1}f^{\lambda, {i j}} c^j_{0}\mathbf{1} -\frac{1}{2}\sum_{1\le j, k, l \le r}\Gamma^{\lambda, j k}_{l} c^j_0 c^k_0 b^l_{-1}\mathbf{1}. $$
Notice that the corresponding vertex operator $\underrightarrow{D^\lambda_{\mathrm{Lie}}}(V):=Q^\lambda(V)_{(0)}$ preserves the subspace $\Gamma\Bigl(V, \underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A; \mathcal{U})_\lambda}\Bigr)=\Gamma\Bigl( \mathbf{x}_\lambda(V), \underrightarrow{\Omega^{\gamma c}_{\mathrm{ch}}}(\mathbb{R}^{m|r}) \Bigr)$ and moreover coincides with the differential $d_{\mathrm{Lie}}$ on the weight $0$ space identified with $\Gamma(V, \bigwedge E^*)$.
\begin{lemma}\label{lem: D^2=0 PRESHEAF VERSION} $$ (\underrightarrow{D^\lambda_{\mathrm{Lie}}}(V))^2=0. $$ \end{lemma} \begin{proof} It suffices to check $ (\underrightarrow{D^\lambda_{\mathrm{Lie}}}(V))^2=0 $ on the weight $0$ subspace by Lemma \ref{lem: odd derivation is 0 if so on generators}. This follows from the fact that the differential $\underrightarrow{D^\lambda_{\mathrm{Lie}}}(V)$ on the weight $0$ subspace is nothing but the differential $d_{\mathrm{Lie}}$ for the Lie algebroid cohomology. \end{proof}
We can see that the linear maps $\underrightarrow{D^\lambda_{\mathrm{Lie}}}(V)$ with open subsets $V\subset U_\lambda$ define a linear endomorphism $\underrightarrow{D^\lambda_{\mathrm{Lie}}}$ of the presheaf $\underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A; \mathcal{U})_\lambda}$ by the definition of the restriction maps. Thus we have a differential $\underrightarrow{D^\lambda_{\mathrm{Lie}}}$ on $\underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A; \mathcal{U})_\lambda}$.
\begin{lemma}\label{lem: THE DIFFERENTIAL IS GLUED PRESHEAF VERSION} The following holds: $$
\underrightarrow{D^\lambda_{\mathrm{Lie}}}|_{U_\lambda\cap\mu}\circ\underrightarrow{\vartheta'_{\lambda \mu}}=\underrightarrow{\vartheta'_{\lambda \mu}}\circ \underrightarrow{D^\mu_{\mathrm{Lie}}}|_{U_\mu\cap U_\lambda}, $$ for any $\lambda, \mu \in \Lambda$. \end{lemma} \begin{proof} By Lemma \ref{lem: sufficient condition for B-linearilty} with $f=\underrightarrow{\vartheta'_{\lambda \mu}}$ and $N=0$, it suffices to check the relation on the weight $0$ subspace. This follows from the fact that $\underrightarrow{\vartheta'_{\lambda \mu}}$ and $\underrightarrow{D^\lambda_{\mathrm{Lie}}}$ coincide with the gluing map of the sheaf of sections $\underline{\bigwedge A^*}$ and the differential for the Lie algebroid cohomology, respectively. \end{proof}
The morphism $\underrightarrow{D^\lambda_{\mathrm{Lie}}}$ consists of homogeneous maps. By Remark \ref{rem: homogeneous morphism of presheaves induces that of ind-objects}, the operator $\underrightarrow{D^\lambda_{\mathrm{Lie}}}$ induces a strict morphism of ind-objects of sheaves $D^\lambda_{\mathrm{Lie}}\!: \Omega^{\gamma c}_\mathrm{ch}(A; \mathcal{U})_\lambda\!\to \Omega^{\gamma c}_\mathrm{ch}(A; \mathcal{U})_\lambda$.
\begin{lemma}\label{lem: THE DIFFERENTIAL IS GLUED} The following holds: $$
D^\lambda_{\mathrm{Lie}}|_{U_\lambda\cap\mu}\circ\vartheta'_{\lambda \mu}=\vartheta'_{\lambda \mu}\circ D^\mu_{\mathrm{Lie}}|_{U_\mu\cap U_\lambda}, $$ for any $\lambda, \mu \in \Lambda$. \end{lemma} \begin{proof} This relation follows from \ref{lem: THE DIFFERENTIAL IS GLUED PRESHEAF VERSION}. \end{proof}
By Lemma \ref{lem: THE DIFFERENTIAL IS GLUED}, we can glue the strict morphisms $(D^\lambda_{\mathrm{Lie}})_{\lambda\in\Lambda}$ into a morphism $D_{\mathrm{Lie}}$ on $\Omega^{\gamma c}_\mathrm{ch}(A)$. Note that by Remark \ref{rem: morphisms coincides if so locally}, the morphism $D_{\mathrm{Lie}}$ is independent of the choice of framed coverings.
\begin{lemma} The morphism $D_{\mathrm{Lie}}$ is a differential on $\Omega^{\gamma c}_\mathrm{ch}(A)$. \end{lemma} \begin{proof} By Remark \ref{rem: morphisms coincides if so locally}, it suffices to check the relations \eqref{eq: deg 1 wt 0 condition of differential on VSA-inductive sheaf}, \eqref{eq: square 0 condition of differential on VSA-inductive sheaf}, \eqref{eq: derivation condition of differential on VSA-inductive sheaf} and \eqref{eq: degree derivation condition of differential on VSA-inductive sheaf} locally. This follows from the fact that $\underrightarrow{D_{\mathrm{Lie}}^\lambda}$ is a differential. \end{proof}
Let $\Gamma\bigl(M, \underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A)}\bigr)$ be the space of all global sections of the presheaf $\underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A)}$ obtained by applying the functor $\underrightarrow{\mathrm{Lim}}\, $ to the degree-weight-graded VSA-inductive sheaf $\Omega^{\gamma c}_\mathrm{ch}(A)$. By the lemma above and Remark \ref{rem: presheaf of differential VSA-inductive sheaf}, we have the following.
\begin{theorem} The pair $\Bigl(\Gamma\bigl(M, \underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A)}\bigr), \underrightarrow{\mathrm{Lim}}\, {D_{\mathrm{Lie}}}(M)\Bigr)$ is a differential degree-weight-graded vertex superalgebra. \end{theorem}
The theorem above leads us to the following definition.
\begin{definition} Let $A$ be a Lie algebroid on a manifold $M$. Its \textbf{chiral Lie algebroid cohomology}, denoted by $H_{\mathrm{ch}}(A)$, is the cohomology of the complex $\Bigl(\Gamma\bigl(M, \underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A)}\bigr), \underrightarrow{\mathrm{Lim}}\, {D_{\mathrm{Lie}}}(M)\Bigr)$. \end{definition}
\begin{remark} From the construction, the chiral Lie algebroid cohomology $\!H_{\mathrm{ch}}(A)$ is a $\mathbb{Z}_{\ge 0}$-graded vertex superalgebra and the subspace of weight $0$, $H_{\mathrm{ch}}(A)[0]$, coincides with the classical Lie algebroid cohomology. \end{remark}
\section{Chiral Equivariant Lie Algebroid Cohomology}\label{section: Chiral Equivariant Lie Algebroid Cohomology}
\subsection{Definition of Chiral Equivariant Lie Algebroid Cohomology} Let $(A, a, [ , ])$ be a Lie algebroid on a manifold $M$. We will first equip $\Bigl(\Gamma\bigl(M, \underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A)}\bigr),$ $\underrightarrow{\mathrm{Lim}}\, {D_{\mathrm{Lie}}}(M)\Bigr)$ with a differential $\mathfrak{s}\Gamma(M, A)[t]$-module structure, where $\Gamma(M, A)$ is the Lie algebra of global sections of $A$.
Set $m:=\dim M$ and $r:=\mathrm{rank}\, A$. Let $\mathcal{U}=(U_\lambda)_{\lambda\in\Lambda}$ be a framed covering of $A$. As before we denote by $\mathbf{x}_\lambda$ and $\mathbf{e}_\lambda$ the chart and the frame associated with $U_\lambda$, respectively.
Let $X\in \Gamma(M, A)$ be a global section. Fix $\lambda\in\Lambda$. We can write $X$ on $U_\lambda$ as $X|_{U_\lambda}=\sum_{j=1}^r f^{\lambda, j} e_\lambda^j$, where $f^{\lambda, j}$ is a function on $U_\lambda$. We set $$
\iota_{X}(V):=\sum_{j=1}^r f^{\lambda, j} b^j_{-1}\mathbf{1} \in \Gamma\Bigl(\mathbf{x}_\lambda (V), \underrightarrow{\Omega^{\gamma bc}_{\mathrm{ch}}}(\mathbb{R}^{m|r})\Bigr)=\Gamma(V, \underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma bc}_{\mathrm{ch}}(A; \mathcal{U}}_{\lambda})), $$ for an open subset $V\subset U_\lambda$. For each $n\in\mathbb{Z}$, the corresponding vertex operators $\iota_X(V)_{(n)}$ with open subsets $V\subset U_\lambda$ form a morphism $\underrightarrow{\iota_{X, (n)}^\lambda}$ on the presheaf $\underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma bc}_{\mathrm{ch}}(A; \mathcal{U}}_{\lambda})$. Note that for $n\ge0$, the morphism $\underrightarrow{\iota_{X, (n)}^\lambda}$ preserves the subpresheaf $\underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A; \mathcal{U})_{\lambda}}$. Let $n\ge0$. Since the morphism $\underrightarrow{\iota_{X, (n)}^\lambda}$ is homogeneous of weight $-n$, it induces a strict morphism of ind-objects $\iota_{X, (n)}^\lambda: \Omega^{\gamma c}_\mathrm{ch}(A; \mathcal{U})_{\lambda}\to \Omega^{\gamma c}_\mathrm{ch}(A; \mathcal{U})_{\lambda}$ by Remark \ref{rem: homogeneous morphism of presheaves induces that of ind-objects}.
\begin{lemma} Let $n\in \mathbb{Z}_{\ge0}$. Then $$
\iota_{X, (n)}^\lambda|_{U_\lambda\cap\mu}\circ\vartheta'_{\lambda \mu}=\vartheta'_{\lambda \mu}\circ \iota_{X, (n)}^\mu|_{U_\mu\cap U_\lambda}, $$ hold for all $\lambda, \mu\in\Lambda$. \end{lemma} \begin{proof}
It suffices to show $\underrightarrow{\iota_{X, (n)}^\lambda}\big|_{U_\lambda\cap\mu}\circ\underrightarrow{\vartheta'_{\lambda \mu}}=\underrightarrow{\vartheta'_{\lambda \mu}}\circ \underrightarrow{\iota_{X, (n)}^\mu}\big|_{U_\mu\cap U_\lambda}$ for each $\lambda, \mu\in\Lambda$. By Lemma \ref{lem: sufficient condition for B-linearilty}, it suffices to check the relation on the weight $0$ subspace. This follows from the fact that the operator $\underrightarrow{\iota_{X, (0)}^\lambda}$ coincides with the inner product for $X$ on the weight $0$ subspace and the fact that the operator $\underrightarrow{\iota_{X, (n)}^\lambda}$ has weight $n$. \end{proof}
Thus for $n\ge0$, we have a morphism of ind-objects $\iota_{X, (n)}: \Omega^{\gamma c}_\mathrm{ch}(A)\to\Omega^{\gamma c}_\mathrm{ch}(A)$ by gluing the strict morphisms $(\iota_{X, (n)}^\lambda)_{\lambda\in\Lambda}$, which is independent of the choice of framed covering as in the case of $D_{\mathrm{Lie}}$. We set $$ L_{X, (n)}:= [D_{\mathrm{Lie}},\iota_{X, (n)}], $$ for $n \ge 0$.
\begin{lemma}\label{lem: CHIRAL CARTAN RELATIONS} Let $X, Y$ be global sections of $A$ and $n, k$ non-negative integers. Then the following relations hold: \begin{enumerate}[(i)]
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\renewcommand{\upshape}{\upshape} \item $[L_{X, (n)},\iota_{Y, (k)}]=\iota_{[X,Y], (n+k)}$, \item $[L_{X, (n)},L_{Y, (k)}]=L_{[X,Y], (n+k)}$, \item $[D_{\mathrm{Lie}},L_{X, (n)}+\iota_{Y, (k)}]=L_{Y, (k)}$. \end{enumerate} \end{lemma} \begin{proof} By Remark \ref{rem: morphisms coincides if so locally}, it suffices to check the relations locally.
Since the operators $D_{\mathrm{Lie}}^\lambda=D_{\mathrm{Lie}}|_{U_\lambda}$, $\iota_{Y, (k)}^\lambda=\iota_{Y, (k)}|_{U_\lambda}$ and $L_{X, (n)}^\lambda:=L_{X, (n)}|_{U_\lambda}$ come from morphisms of presheaves, it suffices to check the corresponding morphisms $\underrightarrow{D_{\mathrm{Lie}}^\lambda}=\underrightarrow{\mathrm{Lim}}\, D_{\mathrm{Lie}}^\lambda$, $\underrightarrow{\iota_{Y, (k)}^\lambda}=\underrightarrow{\mathrm{Lim}}\, \iota_{Y, (k)}^\lambda$ and $\underrightarrow{L_{X, (n)}^\lambda}:=\underrightarrow{\mathrm{Lim}}\, L_{X, (n)}^\lambda$ on $\underrightarrow{\mathrm{Lim}}\, \Omega^{\gamma c}_\mathrm{ch}(A; \mathcal{U})_\lambda$ satisfy the same relations. This is done by direct computations of OPEs. \end{proof}
\begin{theorem} Let $\mathfrak{g}$ be a Lie algebra. Suppose given a morphism of Lie algebras $ x^A: \mathfrak{g}\to \Gamma(M, A), \quad \xi\mapsto x^A_\xi. $ Then the assignment $$ \mathfrak{sg}[t]\ni(\xi, \eta)t^n\mapsto \underrightarrow{\mathrm{Lim}}\, {L_{x^A_\xi, (n)}}(M)+\underrightarrow{\mathrm{Lim}}\, {\iota_{x^A_\eta, (n)}}(M)\in\mathrm{End}\bigl(\Gamma\bigl(M, \underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A)}\bigr)\bigr), $$ defines a differential $\mathfrak{sg}[t]$-module structure on the complex $\Bigl(\Gamma\bigl(M, \underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A)}\bigr),$ $ \underrightarrow{\mathrm{Lim}}\, {D_{\mathrm{Lie}}}(M)\Bigr)$. \end{theorem} \begin{proof} By the above lemma, it suffices to check that the operator $\underrightarrow{\mathrm{Lim}}\, {\iota_{X, (n)}}(M)$ has degree $-1$ and weight $-n$ for any $n\in\mathbb{N}$ and $X\in\Gamma(M, A)$. Note that the continuity of the action of $\mathfrak{sg}[t]$ follows from the fact that the weight-grading on $\Gamma\bigl(M, \underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A)}\bigr)$ is bounded from the below. For that purpose, it suffices to show that $[\underline{J}, \iota_{X, (n)}]=-\iota_{X, (n)}$ and $[\underline{H}, \iota_{X, (n)}]=-n\iota_{X, (n)}$ for any $n\in\mathbb{N}$ and $X\in\Gamma(M, A)$, where $\underline{J}$ and $\underline{H}$ is the degree-grading operator and the Hamiltonian of $\Gamma\bigl(M, \underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A)}\bigr)$. These relations follow from the fact that the same relations hold locally. \end{proof}
This leads us to the following definitions. (See Section \ref{subsection: Chiral Equivariant Cohomology} for the definitions of differential $\mathfrak{sg}[t]$-modules, the chiral basic cohomology and the chiral equivariant cohomology)
\begin{definition} Let $G$ be a compact connected Lie group and $\mathfrak{g}$ the Lie algebra $\mathrm{Lie}(G)^\mathbb{K}$. Let $A$ be a Lie algabroid on a manifold $M$ with a Lia algebra morphism $\mathfrak{g}\to \Gamma(M, A)$. The \textbf{chiral basic Lie algebroid cohomology} of $A$, denoted by $H_{\mathrm{ch}, bas}(A)$, is the chiral basic cohomology of the differential $\mathfrak{sg}[t]$-module $\Bigl(\Gamma\bigl(M, \underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A)}\bigr), \underrightarrow{\mathrm{Lim}}\, {D_{\mathrm{Lie}}}(M)\Bigr)$. The \textbf{chiral equivariant Lie algebroid cohomology} of $A$, denoted by $H_{\mathrm{ch}, G}(A)$, is the chiral equivariant cohomology of the differential $\mathfrak{sg}[t]$-module $\Bigl(\Gamma\bigl(M, \underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A)}\bigr), \underrightarrow{\mathrm{Lim}}\, {D_{\mathrm{Lie}}}(M)\Bigr)$. \end{definition}
\begin{remark} The chiral basic Lie algebroid cohomology $H_{\mathrm{ch}, bas}(A)$ and the chiral equivariant Lie algebroid cohomology $H_{\mathrm{ch}, G}(A)$ are $\mathbb{Z}_{\ge0}$-graded vertex superalgebras. Indeed, $[\underrightarrow{\mathrm{Lim}}\, {L_{x^A_\xi, (n)}}(M), v_{(k)}]=\sum_{i\ge0}\binom{n}{i}(\underrightarrow{\mathrm{Lim}}\, {L_{x^A_\xi, (i)}}(M)v)_{(n+k-i)}$ and $[\underrightarrow{\mathrm{Lim}}\, {\iota_{x^A_\xi, (n)}}(M), v_{(k)}]=\sum_{i\ge0}\binom{n}{i}(\underrightarrow{\mathrm{Lim}}\, {\iota_{x^A_\xi, (i)}}(M)v)_{(n+k-i)}$ hold for any $n\ge0$, $k\in\mathbb{Z}$, $v\in\Gamma\bigl(M, \underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A)}\bigr)$ and $\xi\in\mathfrak{g}$ since the same relations hold locally. Then the assertion follows from Lemma \ref{lem: generalized commutant} and Lemma \ref{lem: tensor commutant} together with the fact that $\Gamma\bigl(M, \underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A)}\bigr)$ is $\mathbb{Z}_{\ge0}$-graded. Moreover $H_{\mathrm{ch}, bas}(A)[0]$ and $H_{\mathrm{ch}, G}(A)[0]$ coincide with the classical basic Lie algebroid cohomology and the classical equivariant Lie algebroid cohomology, respectively. This follows from the fact that $\Gamma\bigl(M, \underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A)}\bigr)$ is $\mathbb{Z}_{\ge0}$-graded and the fact that $\underrightarrow{\mathrm{Lim}}\, {L_{x^A_\xi, (0)}}(M)$ and $\underrightarrow{\mathrm{Lim}}\, {\iota_{x^A_\xi, (0)}}(M)$ coincides with the classical Lie derivative and the classical interior product, respectively. \end{remark}
\subsection{Transformation Lie Algebroid Cases} Let $M$ be a manifold with an infinitesimal action of a finite-dimensional Lie algebra $\mathfrak{g}$: $$ x^M: \mathfrak{g}\to \mathscr{X}(M), \quad \xi\mapsto x^M_\xi. $$ Let $A=M\times \mathfrak{g}$ be the corresponding transformation Lie algebroid (see Example \ref{ex: transformation Lie algebroid}). Let $(\xi_j)_j$ be a basis of $\mathfrak{g}$, $(\xi^*_j)_j$ the dual basis for $\mathfrak{g}^*$ and $(\Gamma^k_{i j})_{i, j, k}$ the structure constants, that is, constants satisfying $[\xi_i,\xi_j]=\sum_{k=1}^{\dim \mathfrak{g}}\Gamma^k_{i j}\xi_k$ for each $i, j=1, \dots, \dim\mathfrak{g}$.
Let $\mathcal{U}=(U_\lambda)_{\lambda\in\Lambda}$ be a framed covering of $A$ consisting of open subsets with the constant frame $(\xi_j)_j$ as the frame on them. We can use the VSA-inductive sheaf $\Omega^{\gamma c}_\mathrm{ch}(A; \mathcal{U})$ for computations of the cohomologies $H_{\mathrm{ch}}(A)$, $H_{\mathrm{ch}, bas}(A)$ and $H_{\mathrm{ch}, G}(A)$, since they do not depend on the choice of framed coverings. By the construction of $\Omega^{\gamma c}_\mathrm{ch}(A; \mathcal{U})$, we see $\underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A; \mathcal{U})}=\underrightarrow{\mathrm{Lim}}\, {C^{\infty, \gamma c}_{\mathrm{ch}, M}}\otimes\langle c \rangle$, where $C^{\infty, \gamma c}_{\mathrm{ch}, M}:=\Omega^{\gamma c}_\mathrm{ch}(M\times \{0\})$, and $\langle c \rangle$ is the subalgebra of $\mathcal{E}(\mathfrak{g})$ generated by $c^{\xi^*}_0\mathbf{1}$ with $\xi^*\in\mathfrak{g}^*$. Therefore by the construction, the differential for the corresponding chiral Lie algebroid cohomology $H_{\mathrm{ch}}(A)$ is nothing but the differential for the continuous Lie algebra cohomology with coefficients in the $\mathfrak{g}[t]$-module $\underrightarrow{\mathrm{Lim}}\, {C^{\infty, \gamma c}_{\mathrm{ch}, M}}(M)$. The action of the element $\xi_j t^n\in\mathfrak{g}[t]$ on $\underrightarrow{\mathrm{Lim}}\, {C^{\infty, \gamma c}_{\mathrm{ch}, M}}(M)$ is given by the operator $$ \sum_{k\ge0}\sum_{i=1}^{\dim M}f^{i j}_{-k-n} \beta^i_k, $$ on each space $\underrightarrow{\mathrm{Lim}}\, {C^{\infty, \gamma c}_{\mathrm{ch}, M}}(U_\lambda)$ of local sections,
where the vector field $x^M_{\xi_j}|_{U_\lambda}$ is written as $\sum_{i=1}^{\dim M}f^{i j} \partial/\partial x^i$ with $f^{i j}\in C^\infty(U_\lambda)$. Thus we have $$ H_{\mathrm{ch}}(A)=H\bigl(\mathfrak{g}[t]; \underrightarrow{\mathrm{Lim}}\, {C^{\infty, \gamma c}_{\mathrm{ch}, M}}(M)\bigr). $$
Consider the Lie algebra morphism \begin{equation}\label{eq: canonical Lie algebra morphism of transformation Lie algebroids} \mathfrak{g}\to \Gamma(M, A)=C^\infty(M)\otimes \mathfrak{g}, \quad \xi \mapsto 1\otimes \xi. \end{equation} We will compute the corresponding chiral equivariant cohomology of $\mathcal{A}:=\Gamma(M, \underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A; \mathcal{U})})$, namely, $H_{\mathrm{ch}, G}(A)$. Notice that \begin{equation}\label{eq: iota for transformation Lie algebroids} \iota^\mathcal{A}_{\xi, (n)}=b^{\xi}_n, \end{equation} for all $\xi\in\mathfrak{g}$ and $n\ge0$. Then the chiral basic cohomology $H_{\mathrm{ch}, bas}(A)$ of $\mathcal{A}=\Gamma(M, \underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A; \mathcal{U})})$ is as follows:
\begin{equation}\label{eq: chiral basic cohomology of transformation Lie algebroid} H_{\mathrm{ch}, bas}^i(A)= \begin{cases} \underrightarrow{\mathrm{Lim}}\, {C^{\infty, \gamma c}_{\mathrm{ch}, M}}(M)^{\mathfrak{g}[t]}, & \text{when $i=0$,} \\ 0, & \text{otherwise.} \end{cases} \end{equation} Indeed, from \eqref{eq: iota for transformation Lie algebroids} we have $\bigl(\Gamma(M, \underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A; \mathcal{U})})\bigr)_{hor}=\underrightarrow{\mathrm{Lim}}\, {C^{\infty, \gamma c}_{\mathrm{ch}, M}}(M)$ and therefore $$ \bigl(\Gamma(M, \underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A; \mathcal{U})})\bigr)_{bas}=\underrightarrow{\mathrm{Lim}}\, {C^{\infty, \gamma c}_{\mathrm{ch}, M}}(M)^{\mathfrak{g}[t]}. $$
Equip $\mathcal{A}=\Gamma(M, \underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A; \mathcal{U})})=\underrightarrow{\mathrm{Lim}}\, {C^{\infty, \gamma c}_{\mathrm{ch}, M}}(M)\otimes\langle c \rangle$ with the $\langle c \rangle$-module structure given by the left multiplication. We claim that this $\langle c \rangle$-module structure induces a chiral $W^*$-module structure. Proposition \ref{prop: construction of chiral W^*-modules} will be applied. Let $d_\mathcal{A}$ be the differential for $\mathcal{A}$, that is, that for the Lie algebra cohomology $H\bigl(\mathfrak{g}[t], \underrightarrow{\mathrm{Lim}}\, {C^{\infty, \gamma c}_{\mathrm{ch}, M}}(M)\bigr)$. From the definition of $d_\mathcal{A}$ and the commutation relations in $\mathcal{E}(\mathfrak{g})$, we have \begin{align} [d_\mathcal{A},c^{\xi^*_l, \mathcal{A}}(z)]&=\Bigl[-\frac{1}{2}\sum_{i, j , k =1}^{\dim \mathfrak{g}}\sum_{\substack{s, t\le0,\\ \notag s+t+u=0}}\Gamma_{i j}^kc^{\xi^*_i}_sc^{\xi^*_j}_tb^{\xi_k}_u,\ c^{\xi^*_l, \mathcal{A}}(z)\Bigr] \\ \notag &=-\frac{1}{2}\sum_{i, j , k =1}^{\dim \mathfrak{g}}\sum_{\substack{s, t\le0,\\ s+t+u=0}}\Gamma_{i j}^kc^{\xi^*_i}_sc^{\xi^*_j}_t\langle \xi^*_l, \xi_k\rangle z^u \\ \notag &=-\frac{1}{2}\sum_{i, j=1}^{\dim \mathfrak{g}}\sum_{\substack{s, t\le0,\\ s+t+u=0}}\Gamma_{i j}^lc^{\xi^*_i}_sc^{\xi^*_j}_t z^u \\ &=-\frac{1}{2}\sum_{i, j=1}^{\dim \mathfrak{g}}\Gamma_{i j}^lc^{\xi^*_i, \mathcal{A}}(z)c^{\xi^*_j, \mathcal{A}}(z).\label{eq: the formula of [d_A,c] for translation Lie algebroid} \end{align} Therefore we have $[c^{\xi^*, \mathcal{A}}(z),[d_\mathcal{A}, c^{\eta^*, \mathcal{A}}(w)]]=0$ for all $\xi^*, \eta^*\in\mathfrak{g}$. By Lemma \ref{prop: construction of chiral W^*-modules}, we obtain a $\langle c, \gamma \rangle$-module structure $Y^\mathcal{A}$ on $\mathcal{A}$ by extending the above $\langle c \rangle$-module structure. From the definition of $\gamma^{\xi^*_l, \mathcal{A}}(z)$ (see the proof of Proposition \ref{prop: construction of chiral W^*-modules}) and \eqref{eq: the formula of [d_A,c] for translation Lie algebroid}, we have $$ \gamma^{\xi^*_l, \mathcal{A}}(z)=[d_\mathcal{A},c^{\xi^*_l, \mathcal{A}}(z)]+\frac{1}{2}\sum_{i, j=1}^{\dim \mathfrak{g}}\Gamma_{i j}^lc^{\xi^*_i, \mathcal{A}}(z)c^{\xi^*_j, \mathcal{A}}(z)=0, $$ for all $l=1, \dots, \dim \mathfrak{g}$. Therefore $[\iota_\xi^\mathcal{A}(z)_-,\gamma^{\xi^*, \mathcal{A}}(x)]=0$ for all $\xi\in\mathfrak{g}$ and $\xi^*\in\mathfrak{g}^*$. Recall that $\iota^\mathcal{A}_{\xi, (n)}=b^{\xi}_n$ for all $\xi\in\mathfrak{g}$ and $n\ge0$. From this, we have $[\iota_\xi^\mathcal{A}(z)_-,c^{\xi^*, \mathcal{A}}(w)]=\langle\xi^*, \xi\rangle\delta(z-w)_-$ for all $\xi\in\mathfrak{g}$ and $\xi^*\in\mathfrak{g}^*$. Thus we can apply Proposition \ref{prop: construction of chiral W^*-modules} and we see that the triple $(\mathcal{A}, d_\mathcal{A}, Y^\mathcal{A})$ is a chiral $W^*$-module. We have proved the following.
\begin{theorem} For a transformation Lie algebroid $A=M\times\mathfrak{g}$ with the Lie algebra morphism \eqref{eq: canonical Lie algebra morphism of transformation Lie algebroids}, the differential $\mathfrak{sg}[t]$-module $\Gamma(M, \underrightarrow{\mathrm{Lim}}\, {\Omega^{\gamma c}_\mathrm{ch}(A)})$ has a canonical structure of a chiral $W^*$-module. \end{theorem}
Therefore by Theorem \ref{thm: CHIRAL BASIC=CHIRAL EQUIVARIANT} and \eqref{eq: chiral basic cohomology of transformation Lie algebroid}, we have the following.
\begin{corollary}\label{prop: chiral equivariant Lie algebroid cohomology for tramsf. Lie algebroids when comm}
Let $G$ be a compact connected Lie group, $\mathfrak{g}$ the Lie algebra $\mathrm{Lie}(G)^\mathbb{K}$ and $A=M\times\mathfrak{g}$ a transformation Lie algebroid. Assume that $G$ is commutative. Then the following holds: \begin{equation} H_{\mathrm{ch}, G}^{i}(A)= \begin{cases} \underrightarrow{\mathrm{Lim}}\, {C^{\infty, \gamma c}_{\mathrm{ch}, M}}(M)^{\mathfrak{g}[t]}, & \text{when $i=0$,} \\ 0, & \text{otherwise.} \end{cases} \end{equation} \end{corollary}
We consider a special case. Let $(G, \Pi)$ be a compact connected Poisson-Lie group with the Lie algebra $\mathfrak{g}$ and $(G^*, \Pi^*)$ the dual Poisson-Lie group of $G$. Recall the Lie algebra morphism $$ \mathfrak{g}\to\Gamma(G^*, T^*G^*), \quad \xi\mapsto \xi^l, $$ where we denote by $\xi^l$ the left invariant $1$-form on $G^*$ whose value at $e$ is $\xi\in\mathfrak{g}$. Consider the corresponding chiral equivariant cohomology. By \cite[Proposition 5.25]{Lu90}, we have an isomorphism of Lie algebroids $$ T^*G^*\cong G^*\times \mathfrak{g}, $$ using the left invariant one-forms on $G^*$. Here we equip the trivial bundle $G^*\times \mathfrak{g}$ with the transformation Lie algebroid structure defined by the infinitesimal left dressing action. The following is a chiral analogue of \cite[Corollary 4.20]{Gin99}.
\begin{proposition} In the above setting,
assume that $(G, \Pi)$ is commutative. Then the following holds: \begin{equation} H_{\mathrm{ch}, G}^{i}(T^*G^*)= \begin{cases} \underrightarrow{\mathrm{Lim}}\, {C^{\infty, \gamma c}_{\mathrm{ch}, G^*}}(G^*), & \text{when $i=0$,} \\ 0, & \text{otherwise.} \end{cases} \end{equation} \end{proposition} \begin{proof} The infinitesimal left dressing action is trivial since $T$ is commutative. Therefore our assertion follows from Corollary \ref{prop: chiral equivariant Lie algebroid cohomology for tramsf. Lie algebroids when comm}. \end{proof}
\section*{Acknowledgments}
The author wishes to express his sincere gratitude to his advisor Professor Atsushi Matsuo for helpful advice and continuous encouragement during the course of this work. He is also thankful to Professor Hiroshi Yamauchi (Tokyo Women's Christian University) for advice and encouragement.
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\begin{document}
\title[Dispersion-managed NLS]{Averaging for the dispersion-managed NLS}
\author{Luccas Campos} \address{IMECC, State University of Campinas (UNICAMP), Campinas, SP, Brazil} \email{[email protected]} \author{Jason Murphy} \address{Missouri University of Science \& Technology} \email{[email protected]} \author{Tim Van Hoose} \address{Missouri University of Science \& Technology} \email{[email protected]}
\begin{abstract} We establish global-in-time averaging for the $L^2$-critical dispersion-managed nonlinear Schr\"odinger equation in the fast dispersion management regime. In particular, in the case of nonzero average dispersion, we establish averaging with any subcritical data, while in the case of a strictly positive dispersion map, we obtain averaging for data in $L^2$. \end{abstract}
\maketitle
\section{Introduction}
We consider dispersion-managed nonlinear Schr\"odinger equations (DMNLS) of the form \begin{equation}\label{DMNLS}
i\partial_t u + \gamma(t)\Delta u = \mu |u|^p u,\quad (t,x)\in\mathbb{R}\times\mathbb{R}^d, \end{equation} where the dispersion map $\gamma:\mathbb{R}\to\mathbb{R}$ is a $1$-periodic function. This model arises in nonlinear fiber optics, particularly when $\gamma$ is taken to be piecewise constant (see, for example, \cite{Agrawal, Kurtzke}). Equations of the form \eqref{DMNLS} (along with some variants) have been the topic of some recent mathematical interest; we refer the reader to \cite{AntonelliSautSparber, CHL, CL, EHL, Fanelli, GT1, GT2, GMO, HL, HL2, MV, MVH, PZ, ZGJT} for a representative sample of results. Throughout this note, we will focus on the case of the cubic NLS in two space dimensions, which is of particular interest in the optics setting. In the case of constant dispersion, this leads to an $L^2$-critical problem (that is, the scaling symmetry that preserves the class of solutions also preserves the $L^2$-norm), although (as we will discuss below) much of our analysis can be adapted to more general choices of $(d,p)$.
Our interest in this note is the phenomenon of `averaging' in the so-called \emph{fast dispersion management regime}. In particular, for each $\varepsilon>0$, we consider the equation \begin{equation}\label{DMNLSeps} \begin{cases}
i\partial_t u + \gamma(\tfrac{t}{\varepsilon})\Delta u = |u|^2 u, \\
u|_{t=0}=\varphi. \end{cases} \end{equation} Our aim to prove that in the limit $\varepsilon\to 0$, the solutions to \eqref{DMNLSeps} converge in a suitable sense to the solution to the equation \begin{equation}\label{DMNLS0} \begin{cases}
i\partial_t u + \langle \gamma\rangle\Delta u = |u|^2 u, \\
u|_{t=0}=\varphi, \end{cases} \end{equation} where $\langle\gamma\rangle$ is the average dispersion, defined by \[ \langle\gamma\rangle=\int_0^1\gamma(t)\,dt. \]
A related problem (known as \emph{strong dispersion management}) instead considers maps of the form $\tfrac{1}{\varepsilon}\gamma(\tfrac{t}{\varepsilon})$, where one expects convergence to a non-local effective model (see e.g. \cite{CL, Lushnikov, GT1, EHL, HL, PZ, ZGJT}); this problem is related to the study of so-called \emph{dispersion managed solitons}. See also \cite{dBD, DT} for the case of averaging with random dispersion maps.
In the fast dispersion management regime, the averaging problem has been previously studied in works including \cite{AntonelliSautSparber, BK1, BK2, YKT}. In particular, \cite{AntonelliSautSparber} proved that for a piecewise-constant dispersion map $\gamma$ and $\varphi\in H^2$, one can obtain averaging in the $L_t^\infty H_x^2$-norm on any finite subinterval of the maximal lifespan of the solution to \eqref{DMNLS0}. In their case, the map $\gamma$ itself must be bounded away from zero, although the case $\langle\gamma\rangle=0$ is permitted. The high regularity on $\varphi$ is imposed in order to utilize the embedding $H^2\hookrightarrow L^\infty$ in dimensions $d\leq 3$.
Our main goals in this work are twofold: (i) to reduce the regularity required on the initial condition, and (ii) to obtain global-in-time averaging results, when possible. Our analysis also permits a slightly more general class of dispersion maps than the piecewise-constant case, although $\gamma$ itself must remain bounded away from zero (and the piecewise-constant case is probably the most important example, anyway). In particular, we will prove that for suitable dispersion maps with nonzero average dispersion and $\varphi \in H^s$ for some $s>0$, we can obtain averaging in $S:=L_t^\infty L_x^2\cap L_{t,x}^4$ on any interval on which the solution to the averaged equation exists and obeys finite $S$-norms. As the $2d$ cubic equation is $L^2$-critical, we therefore obtain averaging for any scaling-subcritical data. In fact, under stronger assumptions on $\gamma$ (namely, strict positivity), we can obtain the same averaging result for merely $L^2$ initial data, yielding a critical result.
For the case of zero average dispersion, we offer no improvement over the result of \cite{AntonelliSautSparber}. Indeed, our arguments rely heavily on the use of Strichartz estimates (both for the averaged equation and the $\varepsilon$-dependent equations), which break down in this setting. In general, it seems difficult to treat the case of zero average dispersion without working with bounded solutions.
To state our main results precisely, we first introduce the notion of an admissible dispersion map, as defined in \cite{MVH}:
\begin{definition}\label{D:admissible} We call $\gamma:\mathbb{R}\to\mathbb{R}$ \emph{admissible} if: \begin{itemize} \item $\gamma(t+1)=\gamma(t)$ for all $t\in\mathbb{R}$. \item $\gamma$ and $\tfrac{1}{\gamma}$ belong to $L_t^\infty(\mathbb{R})$. \item $\gamma$ has at most finitely many discontinuities in $[0,1]$.
\end{itemize} \end{definition} In particular, given admissible $\gamma$ with non-zero average dispersion, \cite{MVH} established global-in-time Strichartz estimates for the underlying dispersion-managed Schr\"odinger equation. Our Theorem~\ref{T}(i) below relies fundamentally on these estimates; in particular, this ultimately explains why we can work with (nearly) critical data.
Next, we recall the main result of \cite{Dodson}, which states that for $\varphi\in L^2$ and $\langle\gamma\rangle>0$, the solution to \eqref{DMNLS0} with $u|_{t=0}\in L^2(\mathbb{R}^2)$ is global in time, scatters in $L^2$, and satisfies $u\in S(\mathbb{R}),$ where \begin{equation}\label{def:S}
\|u\|_{S(I)}:=\|u\|_{L_t^\infty L_x^2(I\times\mathbb{R}^2)}+\|u\|_{L_{t,x}^4(I\times\mathbb{R}^2)}. \end{equation}
Our main result is as follows:
\begin{theorem}\label{T} Let $\gamma$ be an admissible dispersion map satisfying either \begin{itemize} \item[(i)] $\int_0^1\gamma(t)\,dt>0$, or \item[(ii)] $\gamma(t)>0$ for all $t\in\mathbb{R}$. \end{itemize}
In case (i), let $s>0$ and $\varphi \in H^s(\mathbb{R}^2)$. In case (ii), let $\varphi\in L^2(\mathbb{R}^2)$.
Let $u:\mathbb{R}\times\mathbb{R}^2\to\mathbb{C}$ be the solution to \eqref{DMNLS0} with $u|_{t=0}=\varphi$. For all $\varepsilon>0$ sufficiently small, the solution $u^\varepsilon$ to \eqref{DMNLSeps} with $u^\varepsilon|_{t=0}=\varphi$ is global in time, with $u^\varepsilon\in S(\mathbb{R})$ and \begin{equation}\label{convergence!}
\lim_{\varepsilon\to 0}\|u^\varepsilon-u\|_{S(\mathbb{R})}= 0. \end{equation} \end{theorem}
We have arranged matters so that the underlying averaged equation is the defocusing model, in which case all solutions enjoy global space-time bounds. More generally, the analysis will show that we can obtain averaging on any interval on which the solution to the averaged equation has finite space-time bounds. Thus, for example, if $\langle\gamma\rangle<0$, we would obtain global-in-time averaging for masses below a suitable ground state threshold. We have stated Theorem~\ref{T} in this way to emphasize the fact that our result \emph{allows} for global-in-time averaging, when it is in fact possible.
Similarly, we have chosen to prove our result only for the $L^2$-critical problem. This makes it clear that in case (i) we are only slightly off from the critical result, while (ii) obtains the critical result in a restricted setting. More generally, our analysis should carry over to settings in which one can obtain a reasonable well-posedness and stability theory using traditional Strichartz spaces. Notably, however, this would exclude the important $1d$ cubic model.
In the rest of the introduction, we briefly give the ideas behind the proof of Theorem~\ref{T}.
In case (i), the proof builds upon the strategy of \cite{AntonelliSautSparber}. In particular, we run a continuity argument based on examining the difference of the Duhamel formulas for the solutions $u^\varepsilon$ and $u$. As in \cite{AntonelliSautSparber}, the key is to exploit the fact that the underlying linear propagator for \eqref{DMNLSeps} should converge to that of \eqref{DMNLS0}. In contrast to \cite{AntonelliSautSparber}, we use a quantitative form of this convergence, which costs us slightly in regularity and ultimately explains why Theorem~\ref{T}(i) is an inherently subcritical result. With this bound in place, we can then interpolate with the Strichartz estimates of \cite{MVH} (which, by a change of variables, can be shown to hold uniformly in $\varepsilon$). With the full range of Strichartz estimates at our disposal, we can run a fairly standard continuity argument to establish the desired convergence (globally in time). In fact, in this setting we obtain a quantitative bound of the form \[
\|u^\varepsilon-u\|_{S(\mathbb{R})} \lesssim \varepsilon^c \] for some small $c=c(s)>0$, where $H^s$ is the regularity of the initial condition.
In case (ii), we combine ideas from \cite{Fanelli, GMO}, which studied NLS with time-dependent dispersion, and \cite{Ntekoume}, which studied homogenization for NLS with a spatially inhomogeneous nonlinearity. In particular, in the case of a strictly positive dispersion map, we can utilize a change of variables as in \cite{Fanelli, GMO} to convert \eqref{DMNLSeps} into an equation with constant dispersion, but with a time-dependent factor in the nonlinearity. We then follow the approach of \cite{Ntekoume} to obtain homogenization for this equation. The basic idea is to show that the solution to the averaged equation defines an approximate solution to the $\varepsilon$-dependent equation. That this is possible relies on the fact that the solution itself obeys good space-time bounds, while the difference in the nonlinear factors should vanish as $\varepsilon\to 0$. To witness this vanishing, however, we first need to perform an integration by parts in time, which introduces a few minor technical complications. In the end, however, we can obtain the the desired homogenization result. We then `undo' the change of variables in order to reach the desired averaging result in Theorem~\ref{T}(ii).
The rest of this paper is organized as follows: Section~\ref{S:preliminary} collects some preliminary lemmas. We then prove Theorem~\ref{T}(i) in Section~\ref{S:i} and Theorem~\ref{T}(ii) in Section~\ref{S:ii}.
\section{Preliminaries}\label{S:preliminary} We write $A\lesssim B$ to denote $A\leq CB$ for some $C>0$. We denote dependence of implicit constants on parameters via subscripts. For example, $A\lesssim_\gamma B$ means $A\leq CB$ for some $C=C(\gamma)$.
We write $\langle\nabla\rangle^s$ to denote the Fourier multiplier operator $(1-\Delta)^{s/2}$. We employ the standard Littlewood--Paley frequency projectors, denoted $P_{\leq N}$, $P_N$, $P_{>N}$, and so on. We will also make use of the standard Bernstein inequalities, e.g. \begin{align*}
\|P_{>N} f\|_{L^p} &\lesssim N^{-s} \||\nabla|^s f\|_{L^p}, \\
\| |\nabla|^s P_{\leq N}f\|_{L^p} &\lesssim N^s \|f\|_{L^p}, \\
\| P_{\leq N} f\|_{L^q} &\lesssim N^{\frac{d}{r}-\frac{d}{q}}\|P_{\leq N} f\|_{L^r}\quad(r\leq q), \end{align*} and so on.
We will use the notation $\text{\O}(X)$ to denote a finite linear combination of terms that resemble $X$ up to frequency projection and/or complex conjugation. In particular, we will write \[
|u|^2 u = |u_{\leq N}|^2 u_{\leq N} + \text{\O}(u_{>N} u^2). \]
We will need to make use of the Strichartz estimates for the dispersion-managed equation established in \cite{MVH}. To state these precisely in two dimensions, recall that we call $(q,r)$ an \emph{admissible pair} if $2<q,r\leq\infty$ and $\tfrac{2}{q}+\tfrac{2}{r}=1$. (In higher dimensions, the endpoint $q=2$ is allowed.) Theorem~2 in \cite{MVH} showed that for admissible dispersion maps $\gamma$ (in the sense of Definition~\ref{D:admissible}) with $\langle \gamma\rangle\neq 0$, we have \begin{equation}\label{MVH-Strichartz}
\bigl\| e^{i\Gamma(\cdot,s)\Delta}\|_{L^2\to L_t^q L_x^r} \lesssim_\gamma 1 \end{equation} uniformly in $s\in\mathbb{R}$, where $\Gamma(t,s):=\int_s^t\gamma(\tau)\,d\tau$. Combining this with the method of $TT^*$ and the Christ--Kiselev lemma, this yields the full range of Strichartz estimates (minus the $L_t^2$ endpoint, in the higher-dimensional case).
For $\varepsilon>0$, we now define \begin{equation}\label{gamma-eps} \gamma_\varepsilon(t)=\gamma(\tfrac{t}{\varepsilon}) \qtq{and} \Gamma_{\varepsilon}(t,t_0)=\int_{t_0}^t \gamma_\varepsilon(\tau)\,d\tau. \end{equation}
We will see below that \eqref{MVH-Strichartz} holds for $\Gamma_\varepsilon$ as well, with implicit constant independent of $\varepsilon$ (cf. \eqref{uniform-Strichartz}).
The following elementary estimate plays a key role in much of what follows. Indeed, it explains exactly why one should expect convergence to the averaged equation.
\begin{lemma}\label{L:crucial-estimate} Let $\gamma$ be an admissible dispersion map, with $\langle\gamma\rangle=\int_0^1\gamma(t)\,dt$. For any $t,t_0\in\mathbb{R}$, \begin{equation}\label{crucial-estimate}
\bigl| \Gamma_\varepsilon(t,t_0)-\langle\gamma\rangle(t-t_0)\bigr| \leq 4\varepsilon\bigl\{\|\gamma\|_{L^\infty}+|\langle\gamma\rangle|\bigr\}. \end{equation} \end{lemma}
\begin{proof} We change variables to write \begin{align*} \Gamma_\varepsilon(t,t_0) & = \varepsilon\int_{\frac{t_0}{\varepsilon}}^{\frac{t}{\varepsilon}}\gamma(\tau)\,d\tau \\ &= \varepsilon\biggl[\int_{\frac{t_0}{\varepsilon}}^{\lceil \frac{t_0}{\varepsilon}\rceil} \gamma(\tau)\,d\tau + \int_{\lfloor \frac{t}{\varepsilon}\rfloor}^{\frac{t}{\varepsilon}} \gamma(\tau)\,d\tau\biggr] + \varepsilon\int_{\lceil \frac{t_0}{\varepsilon}\rceil}^{\lfloor\frac{t}{\varepsilon}\rfloor}\gamma(\tau)\,d\tau, \end{align*} and we similarly decompose \[ \langle\gamma\rangle(t-t_0)=\varepsilon\langle\gamma\rangle\bigl[ \tfrac{t}{\varepsilon}-\lfloor\tfrac{t}{\varepsilon}\rfloor - (\tfrac{t_0}{\varepsilon} - \lceil\tfrac{t_0}{\varepsilon}\rceil)\bigr] +\varepsilon\langle\gamma\rangle\bigl(\lfloor \tfrac{t}{\varepsilon}\rfloor - \lceil\tfrac{t_0}{\varepsilon}\rceil\bigr). \]
Using the fact that \[ \int_{\lceil \frac{t_0}{\varepsilon}\rceil}^{\lfloor\frac{t}{\varepsilon}\rfloor}\gamma(\tau)\,d\tau = \langle\gamma\rangle\bigl(\lfloor \tfrac{t}{\varepsilon}\rfloor - \lceil\tfrac{t_0}{\varepsilon}\rceil\bigr), \] we obtain \eqref{crucial-estimate}. \end{proof}
Finally, we record a stability result for the class of equations of the form \begin{equation}\label{E:NLSmu}
i\partial_t u + \Delta u = \mu(t)|u|^2 u,\quad \mu\in L_t^\infty(\mathbb{R}). \end{equation}
A similar result was proven in \cite{Ntekoume}, with a spatially-dependent coefficient rather than a time-dependent coefficient. As the proof is essentially identical, we will omit it and instead refer the reader to \cite[Theorem~3.4]{Ntekoume}.
\begin{proposition}\label{L:stability} Let $\mu \in L^\infty(\mathbb{R})$ and $I=[0,T]$. Suppose $\tilde u$ satisfies \begin{equation} \begin{cases}
i\partial_t\tilde{u} + \Delta \tilde{u} = \mu(t)|\tilde u|^2\tilde u + e \\
\tilde u|_{t=0}=\tilde u_0\in L^2. \end{cases} \end{equation}
Assume that \[
\|\tilde{u}\|_{L_t^\infty L_x^2(I \times \mathbb{R}^2)} \leq M,\quad\|u_0 - \tilde{u}_0\|_{L_x^2} \leq M',\qtq{and}
\|\tilde{u}\|_{L_{t,x}^4(I \times \mathbb{R}^2)} \leq L \]
for some $M, M', L >0$ and $u_0 \in L^2(\mathbb{R}^2)$. There exists $\eta_1=\eta_1(M,M',L,\|\mu\|_{L^\infty})$ sufficiently small so that if \begin{align}
\|e^{it\Delta}(u_0 - \widetilde{u}_0)\|_{L_{t,x}^4(I \times \mathbb{R}^2)} \leq \eta \qtq{and}
\left\| \int_0^t e^{i(t-s)\Delta}e(s) \,ds \right\|_{L_{t,x}^4} \leq \eta, \end{align}
then there exists a unique solution $u$ to \eqref{E:NLSmu} with $u|_{t=0} = u_0$ obeying \begin{align}
\|u - \widetilde{u}\|_{L_{t,x}^4} \lesssim \eta,\quad\|u - \widetilde{u}\|_{S(I)} \lesssim M',\qtq{and}
\|u\|_{S(I)} &\lesssim 1. \end{align} \end{proposition}
This result will be used in Section~\ref{S:ii}, where the coefficients $\mu$ will depend on the parameter $\varepsilon$. Importantly, however, the $L^\infty$-norm of these coefficients will \emph{not} depend on $\varepsilon$, and hence the parameter $\eta_1$ appearing in the statement of Proposition~\ref{L:stability} will be uniform in $\varepsilon$.
\section{Proof of Theorem~\ref{T}(i)}\label{S:i}
Throughout this section, we fix an admissible dispersion map $\gamma$ satisfying \[ \langle\gamma\rangle = \int_0^1 \gamma(t)\,dt>0. \]
Given $\varepsilon>0$, we define $\gamma_\varepsilon$ and $\Gamma_\varepsilon$ as in \eqref{gamma-eps}.
\begin{proposition}[Convergence of propagators]\label{P:CoP} There exists $C_\gamma>0$ such that for any $t_0\in\mathbb{R}$ and any $\theta\in[0,1]$, \begin{equation}\label{E:CoP}
\|e^{i\Gamma_\varepsilon(\cdot,t_0)\Delta}-e^{i\langle\gamma\rangle(\cdot-t_0)\Delta}\|_{\dot H_x^\theta\to L_t^\infty L_x^2} \leq C_\gamma\varepsilon^{\frac{\theta}{2}}. \end{equation} \end{proposition}
\begin{proof} This estimate is essentially contained in \cite{AntonelliSautSparber}. We fix $t_0,t\in\mathbb{R}$, $\theta\in[0,1]$, and and $\varphi \in \dot H^\theta$. Using \eqref{crucial-estimate}, we estimate \begin{align*}
\|& (e^{i\Gamma_\varepsilon(t,t_0)\Delta}-e^{i\langle\gamma\rangle(t-t_0)\Delta})\varphi\|_{L^2}^2 \\
& = \int \bigl| (e^{-i[\Gamma_\varepsilon(t,t_0)-\langle\gamma\rangle(t-t_0)]|\xi|^2}-1)\hat\varphi(\xi)\bigr|^2\,d\xi \\
& \leq \int |\Gamma_\varepsilon(t,t_0)-\langle \gamma\rangle(t-t_0)|^{\theta} \ \bigl| |\xi|^{\theta}\hat \varphi(\xi)|^2\,d\xi \lesssim_\gamma \varepsilon^{\theta}\|\varphi\|_{\dot H^\theta}^2. \end{align*} \end{proof}
\begin{corollary}\label{C:CoP} Let $\varepsilon>0$ and $\theta\in[0,1]$. \begin{itemize} \item[(i)] For any Schr\"odinger admissible pair $(q,r)$ and any $t_0\in\mathbb{R}$, \begin{equation}\label{interp-1}
\| e^{i\Gamma_\varepsilon(\cdot,t_0)\Delta}-e^{i\langle\gamma\rangle(\cdot-t_0)\Delta}\|_{H_x^{\theta}\to L_t^q L_x^r} \lesssim_\gamma \varepsilon^{(1-\frac{2}{q})\frac\theta2}. \end{equation} \item[(ii)] For any admissible pairs $(q,r)$, $(\tilde q,\tilde r)$ we have \begin{equation}\label{interp-2}
\biggl\| \int_{t_0}^t [e^{i\Gamma_\varepsilon(t,s)\Delta}-e^{i\langle\gamma\rangle(t-s)\Delta}]F(s)\,ds \biggr\|_{L_t^q L_x^r} \lesssim_\gamma \varepsilon^{(2-\frac{2}{q}-\frac{2}{\tilde q})\frac\theta2} \|\langle\nabla\rangle^{2\theta}F\|_{L_t^{\tilde q'}L_x^{\tilde r'}}. \end{equation} \end{itemize} \end{corollary}
\begin{proof} Fix an admissible pair $(q,r)$, $t_0\in\mathbb{R}$, and $\varepsilon>0$, and define \[ u(t,x)=e^{i\Gamma_{\varepsilon}(t,t_0)\Delta}\varphi. \] It follows that \[ v(t,x):=u(\varepsilon t,\sqrt{\varepsilon}x) \qtq{solves}\begin{cases} i\partial_t v + \gamma(t)\Delta v = 0, \\ v(\tfrac{t_0}{\varepsilon},x) = \varphi(\sqrt{\varepsilon}x), \end{cases} \] i.e. \[ v(t,x) = e^{i\Gamma_1(t,\varepsilon^{-1}t_0)\Delta}[\varphi(\sqrt{\varepsilon}x )]. \]
Thus, by \cite[Theorem~2]{MVH}, we have \begin{equation}\label{str-with-eps}
\|u(\varepsilon t,\sqrt{\varepsilon} x)\|_{L_t^q L_x^r} \lesssim_\gamma \|\varphi(\sqrt{\varepsilon}x)\|_{L^2}, \end{equation} where the implicit constant is independent of $\varepsilon$. As $(q,r)$ is an admissible pair, a change of variables reduces the estimate \eqref{str-with-eps} to \[
\|u(t,x)\|_{L_t^q L_x^r} \lesssim_\gamma \|\varphi\|_{L^2}. \]
It follows that \begin{equation}\label{uniform-Strichartz}
\|e^{i\Gamma_\varepsilon(t,t_0)\Delta}\|_{L^2\to L_t^q L_x^r}\lesssim_\gamma 1 \qtq{uniformly in}\varepsilon>0. \end{equation}
As we have the same bounds for $e^{i\langle\gamma\rangle(\cdot-t_0)\Delta}$, we can now obtain \eqref{interp-1} by the triangle inequality and interpolation with \eqref{E:CoP}.
We now deduce (ii) from (i) using the method of ${TT}^*$ and the Christ--Kiselev lemma \cite{CK}. \end{proof}
\begin{remark} We can obtain these estimates in other dimensions, as well, as long as we avoid the $L_t^2$ endpoint in (ii). \end{remark}
We are now in a position to prove Theorem~\ref{T}(i).
\begin{proof}[Proof of Theorem~\ref{T}(i)] We let $u$ be the solution to \eqref{DMNLS0} with $u|_{t=0}=\varphi$ and $u^\varepsilon$ the solution to \eqref{DMNLSeps} with $u^\varepsilon|_{t=0}=\varphi$.
By the result of \cite{Dodson} and persistence of regularity for \eqref{DMNLS0}, we have that $u$ exists globally in time and satisfies \[
\|\langle\nabla \rangle^s u\|_{S(\mathbb{R})}\leq M <\infty, \] where $S(\cdot)$ is as in \eqref{def:S}.
By the standard local well-posedness theory for \eqref{DMNLSeps} (see e.g. \cite{AntonelliSautSparber} or \cite{MVH}), we have that $u^\varepsilon$ exists at least locally in time; furthermore, as long as $u^\varepsilon$ remains bounded in the $S$-norm, we can continue the solution. In fact, the time of dependence for $u^\varepsilon$ is independent of $\varepsilon$, which can be deduced from the fact that the implicit constants in the Strichartz estimates for \eqref{DMNLSeps} are independent of $\varepsilon$ (cf. \eqref{uniform-Strichartz}).
Thus, in what follows, it will suffice to assume that the solution $u^\varepsilon$ exists and then establish \emph{a priori} bounds on the difference between $u$ and $u^\varepsilon$. Without loss of generality, we estimate the difference for times $t\in[0,\infty)$. In general, the implicit constants below may depend on $\gamma$, but will be independent of $\varepsilon$. We will write $C_\gamma>0$ to explicitly indicate various implicit constants appearing in the Strichartz estimates below.
Writing $F(z)=|z|^2 z$, we first use the Duhamel formula to obtain \begin{align} u^\varepsilon(t) - u(t) & = e^{i\Gamma_\varepsilon(t,t_0)\Delta}[u^\varepsilon(t_0)-u(t_0)] \label{dif-data1} \\ & \quad + [e^{i\Gamma_\varepsilon(t,t_0)\Delta}-e^{i\langle\gamma\rangle(t-t_0)\Delta}]u(t_0) \label{dif-data2} \\ &\quad + \int_{t_0}^t e^{i\Gamma_\varepsilon(t,s)\Delta}[F(u^\varepsilon(s))-F(u(s))]\,ds \label{dif-non}\\ & \quad + \int_{t_0}^t [e^{i\Gamma_\varepsilon(t,s)\Delta}-e^{i\langle\gamma\rangle(t-s)\Delta}]F(u(s))\,ds \label{dif-prop} \end{align} for any $t,t_0\in\mathbb{R}$. Thus, for any time interval $I\ni t_0$, we can use Corollary~\ref{C:CoP}, Strichartz (see \eqref{uniform-Strichartz}), and the fractional product rule to obtain the \emph{a priori} estimate \begin{align*}
\|&u^\varepsilon-u\|_{S} \\
& \lesssim \|u^\varepsilon(t_0)-u(t_0)\|_{L^2} + \varepsilon^c \|u\|_{L_t^\infty H_x^{\frac{s}{2}}} + \| F(u^\varepsilon)-F(u)\|_{L_{t,x}^{\frac43}} + \varepsilon^c \|\langle\nabla\rangle^{s}F(u)\|_{L_{t,x}^{\frac43}} \\
& \lesssim \|u^\varepsilon(t_0)-u(t_0)\|_{L^2} + M\varepsilon^c + \bigl\{ \|u^\varepsilon\|_{L_{t,x}^4}^2+\|u\|_{L_{t,x}^4}^2\bigr\}\|u^\varepsilon-u\|_{L_{t,x}^4} + \\
& \quad + \varepsilon^c \| u\|_{L_{t,x}^4}^2 \|\langle \nabla\rangle^{s}u\|_{L_{t,x}^4} \\
& \lesssim \|u^\varepsilon(t_0)-u(t_0)\|_{L^2} + M\varepsilon^c + \|u\|_{L_{t,x}^4}^2 \|u^\varepsilon-u\|_{L_{t,x}^4} + \|u^\varepsilon-u\|_{L_{t,x}^4}^3 \\
&\quad + M \varepsilon^c \|u\|_{L_{t,x}^4}^2 \\
& \lesssim \|u^\varepsilon(t_0)-u(t_0)\|_{L^2} + M\varepsilon^c + \|u\|_{L_{t,x}^4}^2 \|u^\varepsilon-u\|_{S} + \|u^\varepsilon-u\|_{S}^3 + M\varepsilon^c \|u\|_{L_{t,x}^4}^2 \end{align*} for some $c=c(s)>0$, where all space-time norms are over $I\times\mathbb{R}^2$.
We now let $\eta>0$ be a small parameter to be determined below (ultimately depending only on $\gamma$), and split $[0,\infty)$ into $J=J(M,\gamma)$ intervals $I_j=[t_j,t_{j+1}]$ such that \begin{equation}\label{franged}
\|u\|_{L_{t,x}^4(I_j\times\mathbb{R}^2)}<\eta \qtq{for all}j\geq 0. \end{equation}
We will prove by induction that for $\varepsilon=\varepsilon(M,\gamma,s)$ sufficiently small, \begin{equation}\label{induction}
\|u^\varepsilon - u\|_{S(I_j)} \leq A_j \varepsilon^c \qtq{for all}j\geq 0, \end{equation} where \[ A_0:=8C_\gamma M\qtq{and} A_{j}:=4C_\gamma[A_{j-1}+2M]\qtq{for} j\geq 1. \]
Note that as $J=J(M,\gamma)$ is finite, we have that $A_j \lesssim_{\gamma,M} 1$ uniformly in $j$.
For $j=0$, we note that $u^\varepsilon(t_0)=u(t_0)=\varphi$, so that using \eqref{franged}, the \emph{a priori} estimate on an interval $I=[0,t]\subset I_0$ reduces to \begin{align*}
\|u^\varepsilon - u\|_{S(I)} \lesssim 2M\varepsilon^c + \eta^2\|u^\varepsilon-u\|_{S(I)} + \|u^\varepsilon-u\|_{S(I)}^3. \end{align*}
Choosing $\eta=\eta(\gamma)$ small enough, this implies \[
\|u^{\varepsilon} - u\|_{S(I)} \leq 4C_\gamma M\varepsilon^c + C_\gamma\| u^\varepsilon-u\|_{S(I)}^3. \]
As this holds for any $I=[0,t]\subset I_0$, a continuity argument then implies (for $\varepsilon=\varepsilon(\gamma,M,s)$ small enough) \[
\|u^{\varepsilon}-u\|_{S(I_0)}\leq 8C_\gamma M \varepsilon^c, \] which is \eqref{induction} for $j=0$.
Now suppose that that \eqref{induction} holds up to level $j-1$. Then, in particular, we have \[
\|u^\varepsilon(t_j)-u(t_j)\|_{L^2} \leq A_{j-1} \varepsilon^c. \]
Applying again the \emph{a priori} estimate above, we find that for any $I=[t_j,t]\subset I_j$, \[
\|u^\varepsilon-u\|_{S(I)} \lesssim A_{j-1}\varepsilon^c + 2M\varepsilon^c + \eta^2\|u^\varepsilon-u\|_{S(I)} + \|u^\varepsilon-u\|_{S(I)}^3 \] which (for $\eta$ small, as before) yields \[
\|u^\varepsilon-u\|_{S(I)} \leq 2C_\gamma[A_{j-1}+2M]\varepsilon^c + C_\gamma\|u^\varepsilon-u\|_{S(I)}^3 \]
As long as $\varepsilon=\varepsilon(M,\gamma,s)$ is small enough (depending on $M$, $C_\gamma$, and $\sup_j A_j$), this yields \[
\|u^{\varepsilon}-u\|_{S(I_j)}\leq 4C_\gamma[A_{j-1}+2M]\varepsilon^c=A_j\varepsilon^c, \] which is \eqref{induction} at level $j$. This completes the proof of \eqref{induction}.
Adding the estimates \eqref{induction} and recalling that $J=J(M,\gamma)$ is finite, we derive \[
\|u^\varepsilon-u\|_{S([0,\infty))} \lesssim_{\gamma,M} \varepsilon^c, \] which yields the result. \end{proof}
\section{Proof of Theorem~\ref{T}(ii)} \label{S:ii}
Throughout this section we assume that $\gamma$ is an admissible dispersion map satisfying $\gamma(t)>0$ for all $t\in\mathbb{R}$. In fact, by the definition of admissibility, this guarantees that $\gamma$ is bounded away from zero.
We define $\gamma_\varepsilon(t)$ and $\Gamma_\varepsilon(t,t_0)$ as in \eqref{gamma-eps}. To simplify notation, we will abbreviate $\Gamma_\varepsilon(t,0)$ by $\Gamma_\varepsilon(t)$. Note that $\gamma_1$ corresponds to the original dispersion map $\gamma$; accordingly, we will denote the function $\Gamma_1$ by $\Gamma$.
The strategy in this section will be to utilize a change of variables as in \cite{Fanelli, GMO} to convert \eqref{DMNLSeps} into an equation with constant dispersion but time-dependent nonlinearity (see \eqref{NLSeps} below). We then adapt the arguments of \cite{Ntekoume}, which considered the homogenization problem for $2d$ cubic NLS with spatially dependent nonlinearity, to obtain a homogenization result for \eqref{NLSeps}. Changing back to the original equation, we can complete the proof of Theorem~\ref{T}(ii).
As $\gamma_\varepsilon$ is bounded away from zero, the function $\Gamma_\varepsilon$ is invertible. We denote $c_\varepsilon=\Gamma_\varepsilon^{-1}$ and observe that \begin{equation}\label{gece} \Gamma_\varepsilon(c_\varepsilon(t)) \equiv t \implies \gamma_\varepsilon(c_\varepsilon(t))c_\varepsilon'(t) \equiv 1. \end{equation}
In particular, one sees that if $u^\varepsilon$ solves \eqref{DMNLSeps}, then \[ w^\varepsilon(t,x):=u^\varepsilon(c_\varepsilon(t),x) \] solves the equation \begin{equation}\label{NLSeps}
i\partial_t w + \Delta w = c_\varepsilon'(t)|w|^2 w. \end{equation}
In the case of constant dispersion map $\gamma(t)\equiv\langle\gamma\rangle$, the same computations lead instead to the time-independent equation \begin{equation}\label{NLS0}
i\partial_t w + \Delta w = \tfrac{1}{\langle\gamma\rangle} |w|^2 w. \end{equation}
Note that the result of \cite{Dodson} implies that solutions to \eqref{NLS0} with $w|_{t=0}\in L^2$ are global and scatter in $L^2$, with $w\in S(\mathbb{R})$.
We will prove the following homogenization result for \eqref{NLSeps}.
\begin{proposition}\label{P:w} Let $\varphi\in L^2$, and let $w\in S(\mathbb{R})$ denote the solution to \eqref{NLS0} with $w|_{t=0}=\varphi$. For all $\varepsilon>0$ sufficiently small, the solution $w^\varepsilon$ to \eqref{NLSeps} with $w^\varepsilon|_{t=0}=\varphi$ is global in time, with $w^\varepsilon\in S(\mathbb{R})$ and \[
\lim_{\varepsilon\to 0}\|w^\varepsilon-w\|_{S(\mathbb{R})}=0. \] \end{proposition}
\begin{proof} We follow the strategy in \cite{Ntekoume} and prove that $w$ is an approximate solution to \eqref{NLS0}. In particular, given $\eta>0$ small, we need to show that for all $\varepsilon>0$ sufficiently small, we have \begin{equation}\label{NTS}
\biggl\| \int_0^t e^{i(t-s)\Delta}[c_\varepsilon'(s)-\tfrac{1}{\langle\gamma\rangle}]F(w(s))\,ds \biggr\|_{L_{t,x}^4(\mathbb{R}\times\mathbb{R}^2)} \lesssim \eta, \end{equation}
where we denote $F(z)=|z|^2z$ and allow the implicit constants to depend on $\gamma$ and $\|w\|_{S(\mathbb{R})}$. We will show below that $c_\varepsilon(s)-\tfrac{s}{\langle\gamma\rangle}=\mathcal{O}(\varepsilon)$ uniformly in $s$, a fact we can exploit after an integration by parts in time. However, this produces derivatives acting on $F(w)$, which we can control only if we first restrict to low frequencies. Thus, we will proceed by decomposing \begin{equation}\label{decompose} F(w) = [F(w)-F(w_{\leq N})] + F(w_{\leq N})= \text{O}(w_{>N}w^2) + F(w_{\leq N}) \end{equation} for some $N\gg 1$. We will then show that the contribution of high frequencies can be made as small as we wish, and only use integration by parts on the low frequency piece.
We begin with the contribution of the $\O(w_{>N}w^2)$ term. The key will be to prove the following: \begin{equation}\label{high-freqs-small}
\|w_{>N}\|_{L_{t,x}^4(\mathbb{R}\times\mathbb{R}^2)} \lesssim \eta\qtq{for}N\qtq{sufficiently large.} \end{equation}
To see this, first choose $N_0>0$ large enough that \[
\|P_{>N_0}\varphi\|_{L^2} <\eta. \]
We then let $v$ denote the solution to \eqref{NLS0} with initial data $P_{\leq N_0}\varphi$. In particular, by the scattering theory and persistence of regularity for \eqref{NLS0}, we have that $v$ is global in time and satisfies \[
\|\langle\nabla\rangle v\|_{S(\mathbb{R})}\lesssim N_0. \]
Moreover, as \[
\|w|_{t=0}-v|_{t=0}\|_{L^2}=\|P_{>N_0}\varphi\|_{L^2}<\eta, \] the stability theory for \eqref{NLS0} implies that \[
\|w-v\|_{L_{t,x}^4} \lesssim \eta \] provided $\eta$ is sufficiently small. Thus, by Bernstein's inequalities, \begin{align*}
\|w_{>N}\|_{L_{t,x}^4} & \lesssim \|w-v\|_{L_{t,x}^4} + N^{-1}\|\nabla v\|_{L_{t,x}^4} \lesssim \eta + N^{-1}N_0, \end{align*} which yields \eqref{high-freqs-small} for $N\geq N_0\eta^{-1}$.
Using \eqref{high-freqs-small}, Strichartz, and \eqref{gece} (which implies $c_\varepsilon' \in L_t^\infty$, with bound independent of $\varepsilon$), we can now estimate \begin{equation}\label{high-freq-part} \begin{aligned}
\biggl\| \int_0^t e^{i(t-s)\Delta}[c_\varepsilon'(s)&-\tfrac{1}{\langle\gamma\rangle}]\text{\O}(w_{>N}w^2)(s)\,ds \biggr\|_{L_{t,x}^4(\mathbb{R}\times\mathbb{R}^2)} \\
& \lesssim\|c_\varepsilon'-\tfrac{1}{\langle\gamma\rangle}\|_{L^\infty} \|w_{>N}\|_{L_{t,x}^4} \|w\|_{L_{t,x}^4}^2 \lesssim \eta, \end{aligned} \end{equation} which is acceptable.
We turn to the contribution of the $F(w_{\leq N})$ term in \eqref{decompose}. We will prove \begin{equation}\label{low-freq-part}
\biggl\| \int_0^t e^{i(t-s)\Delta}\tfrac{d}{ds}[c_\varepsilon(s)-\tfrac{s}{\langle \gamma\rangle}]\cdot F(w_{\leq N}(s))\,ds\biggr\|_{L_{t,x}^4(\mathbb{R}\times\mathbb{R}^2)}\lesssim N^{2}\varepsilon. \end{equation} Integrating by parts with respect to time, we first write \begin{align} \int_0^t & e^{i(t-s)\Delta}\tfrac{d}{ds}[c_\varepsilon(s)-\tfrac{s}{\langle \gamma\rangle}] \cdot F(w_{\leq N}(s))\,ds \nonumber \\ & = [c_\varepsilon(t)-\tfrac{t}{\langle\gamma\rangle}] F(w_{\leq N}(t)) \label{boundary-term} \\ & \quad -i\int_0^t e^{i(t-s)\Delta}[c_\varepsilon(s)-\tfrac{s}{\langle\gamma\rangle}](i\partial_s + \Delta)F(w_{\leq N}(s))\,ds \label{derivative-term}, \end{align} where we have used the fact that $c_\varepsilon(0)=0$. Thus it suffices to estimate the contribution of \eqref{boundary-term} and \eqref{derivative-term}. For each of these terms, the key will be to use the following bound: \begin{equation}\label{eps-gain}
\|c_\varepsilon(t)-\tfrac{t}{\langle\gamma\rangle}\|_{L_t^\infty} \lesssim_\gamma \varepsilon, \end{equation} which we may prove as follows: by Lemma~\ref{L:crucial-estimate} and the definition of $c_\varepsilon$, we have \begin{align*}
|c_\varepsilon(\Gamma_\varepsilon(t)) - \tfrac{1}{\langle\gamma\rangle}\Gamma_\varepsilon(t)| & = \tfrac{1}{\langle\gamma\rangle}|\langle \gamma\rangle t - \Gamma_\varepsilon(t)| \lesssim_\gamma \varepsilon \end{align*} uniformly in $t\in\mathbb{R}$. As $\Gamma:\mathbb{R}\to\mathbb{R}$ is surjective, the estimate \eqref{eps-gain} follows.
We turn to the estimate of \eqref{boundary-term} and \eqref{derivative-term}. For \eqref{boundary-term}, we use Bernstein's inequality (with the fact that $F(w_{\leq N})$ is still localized to frequencies $\lesssim N$) to estimate \begin{align*}
\|[c_\varepsilon(t)-\tfrac{t}{\langle\gamma\rangle}] F(w_{\leq N})\|_{L_{t,x}^4} & \lesssim \|c_\varepsilon(t)-\tfrac{t}{\langle\gamma\rangle}\|_{L_t^\infty} \| F(w_{\leq N})\|_{L_{t,x}^4} \\
& \lesssim \varepsilon N^{\frac32} \| F(w_{\leq N})\|_{L_t^4 L_x^1} \\ &\lesssim \varepsilon N^{\frac32} \|w\|_{L_t^\infty L_x^2}\|w_{\leq N}\|_{L_t^\infty L_x^4} \|w\|_{L_{t,x}^4} \\
& \lesssim \varepsilon N^{2}\|w\|_{L_t^\infty L_x^2}^2 \|w\|_{L_{t,x}^4} \lesssim \varepsilon N^{2}, \end{align*} which is acceptable.
For \eqref{derivative-term}, we apply Strichartz and \eqref{eps-gain} and find that it will suffice to prove \[
\|\partial_s F(w_{\leq N})\|_{L_{t,x}^\frac{4}{3}} + \| \Delta F(w_{\leq N})\|_{L_{t,x}^{\frac{4}{3}}} \lesssim N^2. \]
For the second term, we use the frequency localization and H\"older's inequality to obtain \[
\|\Delta F(w_{\leq N})\|_{L_{t,x}^{\frac43}} \lesssim N^2\|w\|_{L_{t,x}^4}^3 \lesssim N^2, \] which is acceptable. For the first term, we use the fact that $w_{\leq N}$ solves \[
i\partial_t w_{\leq N} = - \Delta w_{\leq N} + \tfrac{1}{\langle \gamma\rangle} P_{\leq N}(|w|^2 w) \] and Bernstein and H\"older to estimate as follows: \begin{align*}
\|\partial_s F(w_{\leq N})\|_{L_{t,x}^{\frac43}} & \lesssim \| w_{\leq N}^2 \Delta w_{\leq N}\|_{L_{t,x}^{\frac43}} + \|w_{\leq N}^2 P_{\leq N}(|w|^2 w)\|_{L_{t,x}^{\frac43}} \\
& \lesssim N^2\|w\|_{L_{t,x}^4}^3 + \|w_{\leq N}\|_{L_{t,x}^\infty}^2\|P_{\leq N}(|w|^2 w)\|_{L_{t,x}^{\frac43}} \\
& \lesssim N^2+ N^2\|w\|_{L_t^\infty L_x^2}^2\|w\|_{L_{t,x}^4}^3 \lesssim N^2, \end{align*} which is acceptable. This completes the proof of \eqref{low-freq-part}.
Combining \eqref{high-freq-part} and \eqref{low-freq-part}, we obtain \[
\biggl\| \int_0^t e^{i(t-s)\Delta}[c_\varepsilon'(s)-\tfrac{1}{\langle\gamma\rangle}] F(w(s))\,ds \biggr\|_{L_{t,x}^4} \lesssim \eta + N^2\varepsilon. \] Choosing $\varepsilon$ sufficiently small, we therefore obtain \eqref{NTS}.
With \eqref{NTS} in place, we can now apply the stability result (Proposition~\ref{L:stability}) to obtain that \[
\|w^\varepsilon-w\|_{S(\mathbb{R})} \lesssim \eta \] for all $\varepsilon$ sufficiently small. As $\eta>0$ was arbitrary, this completes the proof of Proposition~\ref{P:w}. \end{proof}
It remains to show that Proposition~\ref{P:w} implies the desired averaging result appearing in Theorem~\ref{T}(ii).
\begin{proof}[Proof of Theorem~\ref{T}(ii)] We define $u$ and $u^\varepsilon$ as in the statement of Theorem~\ref{T}(ii), and define the corresponding solutions $w$ and $w^\varepsilon$ to \eqref{NLS0} and \eqref{NLSeps} as in Proposition~\ref{P:w}. In particular, Proposition~\ref{P:w} guarantees that the $u^\varepsilon$ are global in time for small enough $\varepsilon>0$ and satisfy \begin{equation}\label{converge0}
\lim_{\varepsilon\to 0} \|u^\varepsilon(c_\varepsilon(t),\cdot) - u(\tfrac{t}{\langle\gamma\rangle},\cdot) \|_{S(\mathbb{R})} = 0. \end{equation}
We will prove that \begin{equation}\label{converge1}
\lim_{\varepsilon\to 0}\|u(c_\varepsilon(t),\cdot)-u(\tfrac{t}{\langle\gamma\rangle},\cdot)\|_{S(\mathbb{R})} = 0, \end{equation} so that by the triangle inequality we obtain \[
\lim_{\varepsilon\to 0} \|u^\varepsilon(c_\varepsilon(t),\cdot)-u(c_\varepsilon(t),\cdot)\|_{S(\mathbb{R})}=0. \]
As $c_\varepsilon:\mathbb{R}\to\mathbb{R}$ is invertible and \eqref{gece} guarantees that $\tfrac{1}{c_\varepsilon'}\in L_t^\infty$ (uniformly in $\varepsilon$), a change of variables then implies the desired convergence \eqref{convergence!}.
It therefore remains to establish \eqref{converge1}. We let $\eta>0$ and consider the $L_t^\infty L_x^2$- and $L_{t,x}^4$-norms separately.
First, for the $L_{t,x}^4$-norm, we choose $\psi\in C_c^\infty(\mathbb{R}\times\mathbb{R}^2)$ so that \[
\|u-\psi\|_{L_{t,x}^4(\mathbb{R}\times\mathbb{R}^2)}<\eta. \]
Then, using a change of variables, the fundamental theorem of calculus, and \eqref{eps-gain}, we have \begin{align*}
\|u(c_\varepsilon(t))-u(\tfrac{t}{\langle \gamma\rangle})\|_{L_{t,x}^4} & \lesssim \|\psi(c_\varepsilon(t))-\psi(\tfrac{t}{\langle\gamma\rangle})\|_{L_{t,x}^4}+\eta \\
& \lesssim \|c_\varepsilon(t)-\tfrac{t}{\langle\gamma\rangle}\|_{L_t^\infty} \|\partial_t\psi\|_{L_{t,x}^\infty}\|1\|_{L_{t,x}^4(\text{supp}(\psi))}+\eta \\ & \lesssim C(\psi)\varepsilon + \eta. \end{align*}
Next, for the $L_t^\infty L_x^2$-norm, we first argue as we did for \eqref{high-freqs-small} above to find $N=N(\eta)$ sufficiently large that \[
\|u_{>N}\|_{L_{t,x}^4}\lesssim \eta. \]
Writing $I_\varepsilon(t)$ for the interval between $c_\varepsilon(t)$ and $\tfrac{t}{\langle \gamma\rangle}$, we use the Duhamel formula, \eqref{decompose}, Strichartz estimates, Bernstein estimates, and \eqref{eps-gain} to obtain \begin{align*}
\|u(c_\varepsilon(t))-u(\tfrac{t}{\langle\gamma\rangle})\|_{L_x^2} & = \tfrac{1}{\langle\gamma\rangle}\biggl\| \int_{I_\varepsilon(t)} e^{-is\Delta}[\text{\O}(u_{>N}u^2)+F(u_{\leq N})]\,ds\biggr\|_{L_x^2} \\
& \lesssim \|u_{>N}\|_{L_{t,x}^4}\|u\|_{L_{t,x}^4}^2 + \|F(u_{\leq N})\|_{L_{t,x}^{\frac43}(I_\varepsilon(t))} \\
& \lesssim \eta + |I_\varepsilon(t)|^{\frac34} \|u_{\leq N}\|_{L_t^\infty L_x^4}^3 \\
& \lesssim \eta + \varepsilon^{\frac34} N^{\frac32}\|u\|_{L_t^\infty L_x^2}^3 \lesssim \eta + \varepsilon^{\frac34}N^{\frac32} \end{align*} uniformly in $t$.
Combining the estimates above, we find \[
\|u(c_\varepsilon(t),\cdot)-u(\tfrac{t}{\langle\gamma\rangle},\cdot)\|_{S(\mathbb{R})} \lesssim \eta + C(\psi)\varepsilon + \varepsilon^{\frac34}N^{\frac32}. \]
Choosing $\varepsilon$ sufficiently small, we deduce \[
\|u(c_\varepsilon(t),\cdot)-u(\tfrac{t}{\langle\gamma\rangle},\cdot)\|_{S(\mathbb{R})} \lesssim \eta. \]
As $\eta>0$ was arbitrary, we obtain \eqref{converge0}, as desired. \end{proof}
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\maketitle
\begin{abstract} We show that a certain linear representation of the singular braid monoid $SB_3$ is faithful. Furthermore we will give a second - group theoretically motivated - solution to the word problem in $SB_3$. \end{abstract}
\section {Preliminaries}
While knots are important for our daily life - ties, shoelaces etc. - singular knots are not. However, since the theory of Vassiliev invariants started (see e.g. \cite{BL}) mathematicians became more and more interested in singular knotted objects, i.e. objects having a finite number of transversal self intersections. One of these objects are singular braids which form the singular braid monoid $SB_n$, i.e. the monoid generated by the standard generators $\s_1, \dots, \s_{n-1}$ of the braid group $B_n$ plus the additional singular generators $\t_1, \dots, \t_{n-1}$.
A presentation for the singular braid monoid in terms of these generators, build up from the usual choice of a presentation for the braid group is not hard to deduce (cf. \cite{Birman2}).
The theory of Vassiliev invariants suggests a homomorphism from $SB_n$ into the integral group ring $\Z B_n$ of the braid group. This homomorphism is given by mapping a singular generator $\t_i$ to $\s_i - \s_i^{-1}$ and $\s_i$ to itself (cf. \cite{Birman2}). A famous conjecture of Joan Birman (\cite {Birman2}, \cite{FRZ}) asserts that this homomorphism is injective.
As it appeared to be very useful for knot theorist to have a complete understanding of the braid group $B_3$ (see e.g. \cite{BM}) the purpose of this paper is to give some analogous results for the braid monoid $SB_3$.
By results of Fenn et al. \cite{FKR} the monoid $SB_n$ embeds in a group $SG_n$. Using this result we will prove that an extension to $SB_n$ of the Burau representation for $B_n$ that was defined in \cite{Gemein1} is faithful for $n=3$. Since the Burau representation of $B_n$ is known to be unfaithful for $n \geq 6$ \cite{LP} this result cannot carry over at least for $n \geq 6$. So we will give a second solution to the word problem for $SB_3$ by using the group theoretical structure of $SG_3$.
It is worth mentioning that the faithfulness of the singular Burau representation yields an algorithm for the solution of the word problem for $SB_3$ that has a time complexity which is linear in the length of the word.
This paper was written during a visit of the first author at the Columbia University. He would like to thank Columbia for the warm hospitality and especially Joan Birman for many discussions and fruitful comments on an earlier version of this text.
The second author would like to thank Wilhelm Singhof for many useful suggestions and remarks.
Furthermore both authors are very grateful to E. Mail for her invaluable help.
\section{A new presentation for the singular braid monoid} We recall the well-known presentation for the monoid of singular braids on $n$ strands:
\begin{proposition}[Baez \cite {Baez}, Birman \cite{Birman2}] The monoid $SB_n$ is generated by the elements $$\s_i^{\pm 1}, \, i=1, \dots, n-1, \, \t_i, \, i=1, \dots, n-1$$ (see Figure \ref{generators}) satisfying the following relations: \begin{eqnarray} \s_i \s_i^{-1}&= & 1 \mbox{ for all } i\\ \s_i \s_{i+1} \s_i = \s_{i+1} \s_i \s_{i+1} &\mbox{and}& \s_i \s_j = \s_j \s_i \quad \mbox{ for } \, \, j>i+ 1\\ \t_{i+1} \s_i \s_{i+1} &=& \s_i \s_{i+1} \t_i\\ \s_{i+1} \s_i \t_{i+1} &=& \t_i \s_{i+1} \s_i\\ \s_i \t_i &=& \t_i \s_i\\ \s_i \t_j &=& \t_j \s_i \quad \mbox{ for } \,\, j>i+ 1\\ \s_j \t_i &=& \t_i \s_j \quad \mbox{ for } \,\, j>i+ 1\\ \t_i \t_j &=& \t_j \t_i \quad \mbox{ for } \,\, j>i+ 1. \end{eqnarray} \end{proposition}
We use this proposition to derive a new presentation for the singular braid monoid which is more suitable for our purposes:
\begin{figure}
\caption{ $\sigma_i, \sigma_i^{-1}$ and $\tau_i$}
\label{generators}
\end{figure}
\begin{proposition} The monoid $SB_n$ is generated by the elements $$\s_i^{\pm 1}, \, i=1, \dots, n-1, \, \mbox{ and } \t$$ satisfying the following relations: \begin{eqnarray} \s_i \s_i^{-1}&= &1 \mbox{ for all } \, i \label{Braid relation 1}\\ \s_i \s_{i+1} \s_i = \s_{i+1} \s_i \s_{i+1} &\mbox{and}& \s_i \s_j = \s_j \s_i \quad \mbox{ for } \, \, j>i+ 1 \label{Braid relation 3}\\ (\s_2 \s_1)^3 \t &=& \t (\s_2 \s_1)^3\\ \s_i \t &=& \t \s_i \quad \mbox{ for } \, i \neq 2 \label{commutator}\\ \s_2 \s_3 \s_1 \s_2 \t \s_2 \s_3 \s_1 \s_2 \t &=& \t \s_2 \s_3 \s_1 \s_2 \t \s_2 \s_3 \s_1 \s_2 \mbox{ for } n > 3. \label{t1t3} \end{eqnarray} \end{proposition}
\begin{proof} We will see that the new relations are included in the old ones. So we will concentrate ourselves to show that all other relations can be derived from our relations.
First we note that we have \begin{eqnarray} \label{t_i Definition} \t_{i+1} = \s_i \s_{i+1} \t_i \s_{i+1}^{-1} \s_i ^{-1}. \end{eqnarray}
\begin{enumerate} \item {\bf Relations of the form $\s_i \t_i = \t_i \s_i$}
We will show that these relations can be deduced from $\s_1 \t_1 = \t_1 \s_1$, Relation (\ref{t_i Definition}) and the relations in the braid group (\ref {Braid relation 1})- (\ref {Braid relation 3}). We will proceed by induction on $i$: \begin{eqnarray*} \s_i \t_i &=& \s_i \s_{i-1} \s_i \t_{i-1} \s_i^{-1} \s_{i-1}^{-1}\\ &=& \s_{i-1} \s_i \s_{i-1} \t_{i-1} \s_i^{-1} \s_{i-1}^{-1}\\ &=& \s_{i-1} \s_i \t_{i-1} \s_{i-1} \s_i^{-1} \s_{i-1}^{-1}\\ &=& \s_{i-1} \s_i \t_{i-1} \s_i^{-1} \s_{i-1}^{-1} \s_i\\ &=& \t_i \s_i \end{eqnarray*}
\item {\bf Relations of the form $\s_i \t_j = \t_j \s_i, \, \vert i-j \vert >1$}
We will show that all relations of this form can be derived from the relations $\s_i \t_1 = \t_1 \s_i, \quad i\neq 2$ as well as (\ref{t_i Definition}) and the braid group relations (\ref {Braid relation 1}) - (\ref {Braid relation 3}).
{\bf Case 1: } If $1 < j <i-1$ then we have \begin{eqnarray*} \s_i \t_j &=& \s_i \s_{j-1} \s_{j} \t_{j-1} \s_j^{-1} \s_{j-1}^{-1}\\ &=& \s_{j-1} \s_{j} \t_{j-1} \s_j^{-1} \s_{j-1}^{-1} \s_i\\ &=& \t_j \s_i \end{eqnarray*} where we use (\ref{t_i Definition}), (\ref {Braid relation 3}) and induction on $j$.
{\bf Case 2: } In the braid group with $w:=\s_{i+1} \s_{i+2} \s_i \s_{i+1}$ it holds: \begin{eqnarray} \s_i w&=& w \s_{i+2} \label{computation 1}. \end{eqnarray}
Thus, for $j = i+2$: \begin{eqnarray*} \s_i \t_{i+2} &=& \s_i w \t_i w^{-1}\\ &=& w \s_{i+2} \t_i w^{-1}\\ &=& w \t_i \s_{i+2} w^{-1}\\ &=& w \t_i w^{-1} \s_i\\ &=& \t_{i+2} \s_i \end{eqnarray*} where we use (\ref{t_i Definition}), (\ref{computation 1}) and Case $1$.
{\bf Case 3: } If $j \geq i +3$ then we can proceed as in Case 1 to show that $$\s_i \t_j = \t_j \s_i, \quad i>1+j$$ follows from our relations and Case $2$.
\item {\bf Relations of the form $\t_i \t_j = \t_j \t_i, \, i<j, j-i>1$}
Our aim is to reduce these relations to $\t_1 \t_3 = \t_3 \t_1$ with the help of (\ref{t_i Definition}) and the relation $\s_i \t_j= \t_j \s_i, \,$ for $\vert j-i \vert \neq 1$. We use induction on the pair $(i,j)$ that is ordered lexicographically.
If $i>1$ then \begin{eqnarray*} \t_i \t_j &=& \s_{i-1} \s_{i} \t_{i-1} \s_{i}^{-1} \s_{i-1}^{-1} \t_j\\ &=& \t_j \s_{i-1} \s_{i} \t_{i-1} \s_{i}^{-1} \s_{i-1}^{-1}\\ &=& \t_j \t_i. \end{eqnarray*}
In the same way one can deduce $\t_1 \t_j = \t_j \t_1$ for $j>3$.
\item {\bf Relations of the form $\s_{i+1} \s_i \t_{i+1} = \t_i \s_{i+1} \s_i$}
We will show that these relations follow from Relation (\ref {t_i Definition}) together with the relations in the braid group and the initial relation: \begin{eqnarray} \s_2 \s_1 \t_2 = \t_1 \s_2 \s_1 \label{Induction relation} \end{eqnarray}
We will need the following two relations in the braid group which can be easily tested: \begin{eqnarray} \s_i^{-1} \s_{i+1}^{-1} \s_{i-1}^{-1} \s_i^{-1}&=& \s_{i-1}^{-1} \s_i^{-1} \s_{i+1}^{-1} \s_{i-1}^{-1} \s_i^{-1} \s_{i+1}\\ \mbox{and} \qquad \s_{i-1}^{-1} \s_i^{-1} \s_{i+1}^{-1} \s_{i-1} \s_i \s_{i+1}&=& \s_i \s_{i+1} \s_{i-1}^{-1} \s_i^{-1} \end{eqnarray}
\begin{eqnarray*} \s_i^{-1} \s_{i+1}^{-1} \t_i \s_{i+1} \s_i &=& \s_i^{-1} \s_{i+1}^{-1} \s_{i-1}^{-1} \s_i^{-1} \t_{i-1} \s_i \s_{i-1} \s_{i+1} \s_i\\ &=& \s_{i-1}^{-1} \s_i^{-1} \s_{i+1}^{-1} \s_{i-1}^{-1} \s_i^{-1} \s_{i+1} \t_{i-1} \s_{i+1}^{-1} \s_i \s_{i-1} \s_{i+1} \s_i \s_{i-1} \\ &=& \s_{i-1}^{-1} \s_i^{-1} \s_{i+1}^{-1} \s_{i-1}^{-1} \s_i^{-1} \t_{i-1} \s_i \s_{i-1} \s_{i+1} \s_i \s_{i-1}\\ &=& \s_{i-1}^{-1} \s_i^{-1} \s_{i+1}^{-1} \s_{i-1} \s_i \t_{i-1} \s_i^{-1} \s_{i-1}^{-1} \s_{i+1} \s_i \s_{i-1}\\ &=& \s_{i-1}^{-1} \s_i^{-1} \s_{i+1}^{-1} \s_{i-1} \s_i \s_{i+1} \t_{i-1} \s_{i+1}^{-1} \s_i^{-1} \s_{i-1}^{-1} \s_{i+1} \s_i \s_{i-1}\\ &=&\s_i \s_{i+1} \s_{i-1}^{-1} \s_i^{-1} \t_{i-1} \s_i \s_{i-1} \s_{i+1}^{-1} \s_i^{-1}\\ &=&\t_{i+1} \end{eqnarray*} \end{enumerate} Now we are left with the braid group relations, Relation (\ref{commutator}) as well as Relation (\ref {t_i Definition}) and \begin{eqnarray} \t_2 &=& \s_1^{-1} \s_2^{-1} \t_1 \s_2 \s_1 \label{left1}\\ \t_1 \t_3 &=& \t_3 \t_1 \label{left2}. \end{eqnarray}
Relation (\ref{left2}) is equivalent to Relation (\ref{t1t3}).
Relation (\ref{left1}) is equivalent to: \begin{eqnarray*} \s_2 \s_1 \s_1 \s_2 \t_1 &=& \t_1 \s_2 \s_1 \s_1 \s_2 \\ \Longleftrightarrow \s_2 \s_1 \s_1 \s_2 \s_1^{2} \t_1 \s_1^{-2} &=& \t_1 \s_2 \s_1 \s_1 \s_2 \\ \Longleftrightarrow (\s_2 \s_1)^3 \t_1 &=& \t_1 (\s_2 \s_1)^3. \end{eqnarray*}
Finally, we can skip the Relations (\ref {t_i Definition}) because the $\t_j, \, j>1$, only occur in these relations and we set $\tau:=\tau_1$. \end {proof}
\begin{remark} As the braid group $B_n$ admits a presentation with two generators $\s_1$ and $A:=\s_1 \cdots \s_{n-1}$ for every $n$ (cf. Artin's initial paper \cite{Artin}) we can rewrite the presentation for the singular braid monoid in terms of three generators $\s_1, A$ and $\t_1$. We will omit the details here. \end{remark}
\begin{corollary} \label{Corollary_SB_3} The monoid $SB_3$ is generated by the elements $\se^{\pm 1}, \sz^{\pm 1}, \t_1$ satisfying the following relations: \begin{enumerate} \item $\se \se^{-1} = \sz \sz^{-1}= 1$. \item $\se \sz \se = \sz \se \sz$ \item $\t_1 (\s_2 \s_1)^3 = (\s_2 \s_1)^3 \t_1$ \item $\se \t_1 = \t_1 \se$ \end{enumerate} \end{corollary}
The following theorem of Fenn, Keyman and Rourke \cite {FKR} will make our arguments on $SB_n$ much easier:
\begin{theorem} \label{embedding} Let $SG_n$ be the group given by the monoid presentation of $SB_n$ considered as a group presentation. Then the natural homomorphism of $SB_n$ into $SG_n$ is an embedding. \end{theorem}
\section{A faithful representation of $SB_3$}
One can find a representation of the singular braid groups which is an extension of the famous Burau representation of the braid groups itself (cf. \cite{Gemein1}). For $SB_3$ this representation looks like:
\begin{proposition} The map $\beta_s$ given by $$ \se \mapsto \mat {-t} 1 0 1, \, \sz \mapsto \mat 1 0 t {-t}, \, \t_1 \mapsto \mat {1 - y -t y} y 0 1$$
yields a representation of the singular braid monoid into a matrix ring: $$\beta_s: B_3 \rightarrow M_2(\Z [ t, t^{-1}, y]).$$ \end{proposition}
We will show that this representation is faithful. For this purpose we need the following easy consequence (cf. \cite {Fine}) of a theorem of P.M. Cohn \cite{Cohn}:
\begin{theorem} \label{Cohn} Let $d \neq 1,2,3,7,11$ be square-free, i.e. $d$ is not divisible by the square of an integer, and $\omega:=\sqrt{-d}$ if $d \equiv 1$ or $2$ modulo $4$ or $\omega:= (1+\sqrt{-d})/2$ for $d \equiv 3$ modulo $4$ and let $\O_d= \Z + \omega \Z$ be the imaginary quadratic integers in $\Q[\omega]$. Furthermore, let $A, B$ and $C$ be the following elements in $PSL_2(\O_d)$: $$A:= \mat 1 1 0 1, \, B:= \mat 0 {-1} 1 0 \, \mbox{ and } \, C:= \mat 1 {\omega} 0 1.$$
The subgroup $PE_2(\O_d)$ of $PSL_2(\O_d)$ generated by all matrices of the forms $$ \mat 1 x 0 1 \, \mbox{ and } \, \mat 1 0 y 1 $$ for $x$ and $y$ in $\O_d$, has the presentation:
$$PE_2(\O_d) = \pr {A,B,C} {B^2=(A B)^3=[A,C]=1}.$$
With $\Sigma_1:=A, T_1:=C$ and $$\Sigma_2:=(A B A)^{-1}=\mat 1 0 {-1} 1$$ we get - after some easy transformations - the presentation: $$PE_2(\O_d) =\pr {\Sigma_1, \Sigma_2, T_1} {(\Sigma_1 \Sigma_2 \Sigma_1)^2= (\Sigma_1 \Sigma_2)^3=1, \, \Sigma_1 T_1 = T_1 \Sigma_1}.$$ \end{theorem}
Now we have all the tools to show that \begin {theorem} The singular Burau representation $\beta_s: SB_3 \longrightarrow M_2(\Z[t,t^{-1},y])$ is faithful. \end{theorem} \begin{proof} The arguments are essentially the same as in the proof of Magnus and Peluso (cf. \cite{Birman3}) of the faithfulness of the Burau representation for 3-string braids. While in this proof the well-known presentation of the group $PSL_2(\Z)$ is used we will make use of the presentation of $PE_2(\Z[\omega])$ of Theorem \ref{Cohn} for a suitable ring of integers $\Z[\omega]$.
For reasons of convenience we choose $d=5$, so $\omega:=\sqrt{-5}$.
We will denote the extension of $\beta_s$ to $SG_3$ also by $\beta_s$. So $$\beta_s: SG_3 \rightarrow M_2(\Z[t,t^{-1}, y, 1/(1-y-t y)]).$$ The image of $\beta_s$ in this matrix group is - of course - a quotient of $SG_3$. Furthermore by setting $t:=-1$ and $y:=\omega$ it naturally maps onto $PE_2(\Z[\omega])$.
Thus we have a homomorphism from $SG_3$ onto $PE_2(\Z[\omega])$ given by $\s_1 \mapsto \Sigma_1, \s_2 \mapsto \Sigma_2$ and $\tau_1 \mapsto T_1$ where $\Sigma_1, \Sigma_2$ and $T_1$ are as in Theorem \ref{Cohn}. Moreover by comparing the presentations of $SG_3$ and $PE_2(\Z[\omega])$ we see that the kernel of this homomorphism is the normal closure of the element $(\s_1 \s_2 \s_1)^2$, which is - as is easy to see - a central element in $SG_3$. Hence, the kernel is cyclic.
Thus, the image of $SG_3$ under $\beta_s$ is isomorphic to $SG_3$ modulo a power of $\beta_s((\s_1 \s_2 \s_1)^2)$. Since $(\s_1 \s_2 \s_1)^2$ is mapped by $\beta_s$ to the element $$\mat {t^3} 0 0 {t^3}$$ of infinite order, we see that $SG_3$ must be isomorphic to its image under $\beta_s$.
Because of the Embedding Theorem \ref{embedding} the theorem follows. \end{proof}
\subsection{The Birman-Conjecture for $SB_3$}
Let $\eta: SB_n \longrightarrow \Z B_n$ be the Birman homomorphism which maps $\t_1 \mapsto \s_1 - \s_1^{-1}$ and $\s_j$ to itself. In \cite{Gemein2} the follwing result is proved:
\begin{theorem} \label{Gemein-Injectivity} Let $b$ and $b'$ be two braids in $SB_n$ with $\eta(b) = \eta(b')$. Then we have $\beta_s(b) = \beta_s(b').$ \end{theorem}
Because the singular Burau representation is faithful for $n=3$ this means that the Birman homomorphism is also faithful, i.e. the Birman conjecture is valid for $n=3$. We learned that this result was already obtained by Antal J\'arai \cite {Jarai}.
It might be interesting, however, for the reader to see how the linearity of $SB_3$ and the injectivity of $\eta$ relate. Therefore we will give a sketchy and informal proof of Theorem \ref{Gemein-Injectivity}. For full details the reader is referred to \cite{Gemein2}.
For notational reasons we will consider the special case $SB_3$. However, the argument holds also for higher $n$. Since the determinant of the Burau matrix of a given braid with $m$ singularities equals $t^k \cdot (1-y-ty)^m$ for some $k \in \Z$, two braids with a different number of singularities cannot map to the same matrix under $\beta_s$. The same holds for the Birman homomorphism. Therefore we may restrict ourselves to braids with a fixed number $m$ of singularities. The set of all braids with exactly $m$ singularities will be denoted by $SB_3^{(m)}$ in the sequel. \par The main idea of the proof is to imitate the Birman homomorphism on the level of Burau matrices. If we deal with braids having exactly one singularity, this can be done easily: Substituting $y$ by 1 corresponds to a right-handed resolution of the singularity, substituting $y$ by $x^{-1}$ corresponds to a left-handed resolution of the singularity. This induces a well defined homomorphism from the matrix ring $M_2(\Z[t,t^{-1},y])$ into the group ring $\Z[M_2(\Z[t,t^{-1}])]$ imitating the Birman homomorphism. \par If the number of singularities is greater than one, we cannot proceed in the same way. The two indicated substitutions would correspond to a right-handed (resp. left-handed) resolution of all the singularities. Unfortunately, we also have to consider cases where some singularities are resolved in a right-handed way while others are resolved in a left-handed way. \par Therefore we change our point of view slightly: First, we number the singularities of our braid $b$ from 1 to $m$. Afterwards we assign to the $j$-th singularity of the braid the matrix \[ \mat {1-y_j-ty_j} {y_j}
0 1 \] rather than the usual matrix \[ \mat {1-y-ty} y 0 1 . \]
In this way we define a modification of the Burau matrix of $b$. Its entries take values in the polynomial ring $\Z[t,t^{-1},y_1, \dots , y_m]$. Of course, the numbering of the singularities was somewhat arbitrary. Therefore we shall regard this modified matrices only up to permutation of the indices of the $y_i$. (To be more precise, we let the symmetric group $\Sigma_m$ act on $M_2(\Z[t,t^{-1},y_1, \dots , y_m])$ and consider the orbits. By abuse of notation we shall denote the set of these orbits also with $M_2(\Z[t,t^{-1},y_1, \dots , y_m])$.) It is obvious that we obtain the Burau representation of $b$ out of its modified Burau matrix by the projection $p$ which sends all the $y_i$ to $y$. \par We have introduced the modified Burau matrix in order to compute the (regular) Burau matrices of all possible resolutions. In fact, define a matrix resolution $$r: M_2(\Z[t,t^{-1},y_1, \dots , y_m]) \rightarrow M_2(\Z[t,t^{-1}])$$ as a projection where, in addition, any $y_i$ is mapped either to $1$ or to $t^{-1}$. The index $\mu(r)$ is defined to be the number of $y_i$ which are sent to $t^{-1}$. Clearly, a given resolution is not well defined on our orbits. However, taking formal sums over all possible resolutions gives a well defined map $\rho$. So, if $M$ is a modified Burau matrix, then $$\rho(M) = \bigoplus_{r} (-1)^{\mu(r)} \cdot r(M).$$ Note that the sum in the formula is a formal sum in the group ring $\Z[M_2(\Z[t,t^{-1}])].$ \par Easy calculations show that the application $\rho$ corresponds to the Birman homomorphism $\eta$ on the level of matrices. In fact, we get the following commutative diagram:
\unitlength1cm \begin{picture}(13,6) \put(3.2,0){$\Z[B_3]$} \put(3.3,3){$SB_3^{(m)}$} \put(8.5,0){$\Z[M_2(t,t^{-1})]$} \put(8,3){$M_2(t,t^{-1},y_1,\dots,y_m)$} \put(8.5,5){$M_2(t,t^{-1},y)$} \put(3.6,2.5){\vector(0,-1){2}} \put(9.3,2.5){\vector(0,-1){2}} \put(4.6,0.1){\vector(1,0){3.5}} \put(4.5,3.1){\vector(1,0){3.2}} \put(9.3,3.6){\vector(0,1){1}} \put(4.3,3.6){\vector(3,1){4}} \put(3.9,1.5){$\eta$} \put(9.6,1.5){$\rho$} \put(9.6,4){$p$} \put(6,0.3){$\Z[\beta]$} \put(6,2.6){$\tilde \beta$} \put(5.9,4.5){$\beta_s$} \end{picture}
\par
Here $\tilde \beta$ denotes the application which maps a braid to its corresponding modified Burau matrix. With $\Z[\beta]$ we denote the extension of the usual (regular) Burau homomorphism to the group rings. \par We now claim: \begin{description} \item[Claim 1] Let $M,M' \in M_2(x^{\pm 1}, y_1, \dots , y_m)$ be two elements in the image of $\tilde \beta$ with $\rho(M) = \rho (M')$. Then we have $p(M) = p(M')$. \end{description} Let us assume that the claim is true. then the proof of Theorem 3.4 becomes easy diagram chasing: \par Let $b,b'$ be elements of $SB_3^{(m)}$ and suppose that $\eta(b) = \eta(b')$. It follows that $(\Z[\beta] \circ \eta)(b) = (\Z[\beta] \circ \eta)(b')$ and by commutativity of the diagram that $(\rho \circ \tilde \beta) (b) = (\rho \circ \tilde \beta) (b')$. Using Claim 1 we get $(p \circ \tilde \beta) (b) = (p \circ \tilde \beta) (b')$ and - again by commutativity of the diagram - $\beta (b) = \beta (b')$. \par Thus, we only have to show that Claim 1 holds. This is the most technical part of the proof. In fact, we have to figure out in how far the matrices of the formal sum $\rho(M)$ determine the matrix $M$. This leads to one system of linear equations for each entry of the matrix, which may be solved after having observed the following two facts: \begin{enumerate} \item If $M$ is in the image of $\tilde \beta$, then each $y_i$ cannot appear in the matrix with powers greater than $1$. \item We may use the determinant of the matrices in our formal sum in order to compute the index of the resolution which has produced them. This fact is important when solving the equations. \end{enumerate} With these two observations and some tedious computations, we derive that two matrices $M$ and $M'$ are mapped to the same formal sum under $\rho$ if their entries differ by permutations of the indices of the $y_i$. Hence, they vanish under the projection $p$.
\section{A second solution to the word problem in $SB_3$}
To give a second solution to the word problem for $SB_3$, i.e. the problem whether two elements in $SB_3$ are equivalent, we will need Britton's Lemma that can be applied to the group $SG_3$ \begin{lemma}[Britton \cite{Britton}] Let $H= \pr {S} {R}$ be a presentation of the group $H$ with a set of generators $S$ and relations $R$ in these generators.
Furthermore let $G$ be a HNN-extension of $H$ of the following form: $$ G = \pr {S,t} {R, t^{-1} X_i t = X_i, i\in I}$$ for some index set $I$, where $X_i$ are words over $S$.
Let $W$ be a word in the generators of $G$ which involves $t$.
If $W=1$ in $G$ then $W$ contains a subword $t^{-1} C t$ or $t C t^{-1}$ where $C$ is a word in $S$, and $C$, regarded as an element of the group $H$, belongs to the subgroup $X$ of $H$ generated by the $X_i$. \end{lemma}
We will rather solve the word problem for $SG_3$ than for $SB_3$. By Corollary \ref{Corollary_SB_3} $SG_3$ has a presentation as in Britton's Lemma. So to solve the word problem in $SG_3$ we have to decide whether a given word in the generators of $B_3$ is element of the subgroup $H_3$ generated by the elements with which $\t_1$ commutes: $\s_1$ and $(\s_2 \s_1)^3$.
This decision problem, called membership problem, would not be hard to solve with the help of the Burau representation of $B_3$. However we promised to give a puristic proof which gives more hope for a generalization to braids and singular braids with more than three strands.
So we will choose the approach of Xu \cite{Xu} for the word problem for $B_3$ - which was generalized most recently to arbitrary $B_n$ by Birman, Ko and Lee \cite{BKL} - to solve the membership problem for the subgroup $H_3$.
We briefly recall this approach using the notation of Birman, Ko and Lee. The first step is to rewrite the presentation of $B_3$ in terms of the new generators: $\aa:=\s_1, \ab:=\s_2$ and $\ac:=\s_2 \s_1 \s_2^{-1}$.
So we get a new presentation $$ B_3 = \pr {\aa, \ab, \ac}{\ab \aa = \ac \ab = \aa \ac}. $$
Using the element $\delta := \ab \aa$ one can show now that every element of $B_3$ can be brought into a unique normal form $\delta ^k P$ for some $k$ with $P$ a positive word, i.e. only positive exponents occurs, in the generators $\aa, \ab$ and $\ac$, such that none of the subwords $\ab \aa$, $\ac \ab$ or $\aa \ac$ appear in $P$.
\begin{lemma} \label{membership_problem} The membership problem for the subgroup $H_3$ of $B_3$ generated by the elements $\s_1$ and $(\s_2 \s_1)^3$ can be solved. \end{lemma}
\begin{proof} First of all we see that $H_3$ is abelian and $(\s_2 \s_1)^3=\delta^3$. Therefore if we want to bring a word into the normal form we only have to look for $\s_1^k$ for $k \in \Z$. If $k$ is not negative, then the normal form for an element $(\s_2 \s_1)^{3l} \s_1^k$ is simply $\delta ^{3l} \aa ^k$.
If $k\leq 0$ then the following identities are easy to see: \begin{eqnarray*} (\s_2 \s_1)^{3l} \s_1^{3k} &=& \delta^{3l+3k} (\ac \aa \ab)^{-k}\\ (\s_2 \s_1)^{3l} \s_1^{3k-1} &=& \delta^{3l+3k-1} \ab (\ac \aa \ab)^{-k}\\ (\s_2 \s_1)^{3l} \s_1^{3k-2} &=& \delta^{3l+3k-2} \aa \ab (\ac \aa \ab)^{-k}. \end{eqnarray*}
Therefore for every word $w$ in the braid group $B_3$ we can bring it into its unique normal form and compare this form with the normal forms for the elements in $H_3$. Hence, the membership problem for $H_3$ is solvable. \end{proof}
Thus we have proved: \begin{theorem} Given two words $w_1 = b_1 \t_1 b_2 \t_1 \cdots \t_1 b_m$ and $w_2= c_1 \t_1 c_2 \t_1 \cdots \t_1 c_l$ in the generators $\s_1, \s_2$ and $\t_1$ in $SB_3$, where the $b_j$ and $c_j$ are words in $B_3$.
Then $w_1$ and $w_2$ are equal in $SB_3$ if and only if $b_m c_l^{-1}$ is in the subgroup $H_3$ of $B_3$ that is generated by $\s_1$ and $(\s_2 \s_1)^3$ and $b_1 \t_1 b_2 \t_1 \cdots b_{m-1} b_m$ and $c_1 \t_1 c_2 \t_1 \cdots c_{l-1} c_l$ are equal in $SB_3$.
This gives a solution to the word problem in $SB_3$ because the membership problem for $H$ is solvable by Lemma \ref{membership_problem} and the word problem in $B_3$ is solvable - as mentioned above - as well. \end{theorem}
\subsection{An algebraic proof of the embedding theorem for $SB_3$ into $SG_3$}
Actually - as a Corollary of our approach - one can get a purely algebraic proof of the Embedding Theorem \ref{embedding} of \cite{FKR} for the special case $n=3$:
\begin{corollary} $SB_3$ embeds into $SG_3$. \end{corollary}
\begin{proof} We have to show that if two elements $w_1$ and $w_2$ in $SB_3$ are different then their images in $SG_3$ are different. By the HNN-structure of $SG_3$ the subgroup $B_3$ embeds in it. Furthermore two elements $w_1$ and $w_2$ that map to the same element in $SG_3$ must have the same number of singular points.
Now let $w_1 = b_1 \t_1 b_2 \t_1 \cdots \t_1 b_m$ and $w_2= c_1 \t_1 c_2 \t_1 \cdots \t_1 c_m$, $b_j$ and $c_i \in B_3$, be two different elements of $SB_3$ that map to the same element in $SG_3$, by slight abuse of notation also denoted by the same word. We assume that $w_1$ and $w_2$ are minimal examples with respect to the number of singular points. Since $w_1 \beta \neq w_2 \beta \iff w_1 \neq w_2$ for an element $\beta \in B_3$ we further may assume that $c_m = 1$. Then by Britton's Lemma $b_m$ must lie in the subgroup $H_3$ defined above. Since all the elements of $H_3$ commute with $\t_1$ we have $\t_1 b_m = b_m \t_1$ both in $SB_3$ and $SG_3$.
So $w_1$ is equal within $SB_3$ to $w_1 = b_1 \t_1 b_2 \t_1 \cdots b_{m-1} b_m \t_1$. Now consider the two word $w_1' = b_1 \t_1 b_2 \t_2 \cdots b_{m-1} b_m$ and $w_2'= c_1 \t_1 \cdots c_{m-1}$ in $SB_3$. These two words represent different elements in $SB_3$ - otherwise we would have $w_1 = w_1' \t_1 = w_2' \t_1 = w_2$ - but map to the same element in $SG_3$. This contradicts our assumption. \end{proof}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
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\begin{document}
\title{A note on the Huq-commutativity of normal monomorphisms} \author{James Richard Andrew Gray and Tamar Janelidze-Gray} \maketitle \abstract{ We give an alternative criteria for when a pair of Bourn-normal monomorphisms Huq-commute in a unital category. We use this to prove that in a unital category, in which a morphism is a monomorphism if and only if its kernel is zero morphism, a pair of Bourn-normal monomorphisms with the same codomain Huq-commute as soon as they have trivial pullback. As corollaries we show that several facts known only in the protomodular context are in fact true in more general contexts. } \section{Introduction} It is well known and easy to prove that if $K$ and $L$ are normal subgroups of a group $G$ and $K\cap L = 0$, then each element in $K$ commutes with each element of $L$. This fact has several known generalizations to categories.
An immediate generalization is obtained in the context where there is a suitable notion of a commutator $[-,-]$ defined for normal subobjects, (which is commutative and) satisfying the property that if $K,L$ are normal subobjects then $[K,L]\leq K$. In this context, if $K$ and $L$ are normal then $[K,L]\leq K\wedge L$. Therefore if $K$ and $L$ are trivial it immediately follows that $K\wedge L$ is trivial, which implies that $K$ and $L$ commute. This is the case for the Huq commutator in a normal unital category.
An alternative generalization was obtained by D. Bourn (Theorem 11 \cite{BOURN:2000a}) in the context of pointed protomodular category \cite{BOURN:1991} (also introduced by D. Bourn): he proved that if the meet of $k$ and $l$ is $0$, and if $k$ and $l$ are Bourn-normal with the same codomain, then $k$ and $l$ Huq-commute \cite{HUQ:1968}. Recall that in a pointed finitely complete category, a Bourn-normal monomorphism is essentially the zero class of an internal equivalence relation.
We show (Corollary \ref{corollary:_meet_trivial_commutes}) that this latter fact is true in the wider context of a unital category \cite{BORCEUX_BOURN:2004} (introduced by F. Borceux and D. Bourn) satisfying Condition \ref{zero_det}, which simply requires a morphism to be a monomorphism as soon as it's kernel is zero. This context is sufficiently wide so that it includes every normal unital category which implies that the former result also becomes a special case. In doing so we produce an alternative criteria (Theorem \ref{theorem:char_of_huq_commutes_for_bourn_normal}) for when a pair of Bourn-normal monomorphisms commute in a unital category, which closely resembles Proposition 2.6.13 of \cite{BORCEUX_BOURN:2004}.
We briefly study Condition \ref{zero_det}, and in particular: (i) we explain that it is a special case of a known condition (see Remark \ref{remark:zero_dat_known}) and that it together with regularity is easily equivalent to normality; (ii) we give examples of categories satisfying it as well as our other conditions (some of which are not normal categories); (iii) we characterize it in terms of the fibration of points (Proposition \ref{characterization_of_zero_det}). Using in part this characterization, we show that in a pointed Mal'tsev category \cite{CARBONI_LAMBECK_PEDICCHIO:1991} satisfying Condition \ref{zero_det}, the join of Bourn-normal monomorphisms, with the same codomain and trivial meet, exists and is Bourn-normal. In addition, we show that the characterization of abelian objects, via the normality of their diagonal in the product, lifts from pointed protomodular categories to strongly unital categories \cite{BORCEUX_BOURN:2004} satisfying Condition \ref{zero_det}.
\section{Preliminaries} In this section we recall the necessary definitions and preliminary facts, and introduce the notation we will use.
For a pointed category $\mathbb{C}$ we write $0$ for the zero object as well as for each zero morphism between each pair of objects. For objects $X$ and $Y$ we will often write $\pi_1 :X\times Y \to X$ and $\pi_2 :X\times Y\to Y$ for the first and second product projections (when they exists), and for morphisms $f:W\to X$ and $g:W\to Y$ we will write $\langle f,g\rangle:W\to X\times Y$ for the unique morphism with $\pi_1\langle f,g\rangle = f$ and $\pi_2\langle f,g\rangle=g$. Recall that a category $\mathbb{C}$ is unital if $\mathbb{C}$ it is pointed, finitely complete, and for each pair of objects $X$ and $Y$ the unique morphisms $\langle 1,0\rangle: X\to X\times Y$ and $\langle 0,1\rangle : Y\to X\times Y$ are jointly strongly epimorphic.
A pair of morphisms $f:X\to A$ and $g:Y\to A$ in a unital category $\mathbb{C}$ are said to Huq-commute, if there exists a unique morphism $\varphi : X\times Y\to A$ making the diagram \begin{equation}\label{} \vcenter{ \xymatrix{ X \ar[r]^-{\langle 1,0\rangle} \ar[dr]_{f} & X\times Y \ar[d]^{\varphi} & Y \ar[l]_-{\langle 0,1\rangle} \ar[dl]^{g} \\ & A & } } \end{equation} commute. The morphism $\varphi$ is called the cooperator of $f$ and $g$. A morphism $f:X\to A$ is called a central monomorphism if it is a monomorphism and it Huq-commutes with $1_A$.
We will also need the following lemmas (see e.g \cite{BORCEUX_BOURN:2004} and the references there): \begin{lemma} \label{lemma:mono_cancels} For $u:X'\to X$, $v:Y'\to Y$, $f:X\to A$ and $g:Y\to A$ morphisms in $\mathbb{C}$ and $m:A\to B$ and monomorphism. \begin{enumerate}[(i)] \item The morphisms $f$ and $g$ Huq-commute if and only if the morphisms $g$ and $f$ Huq-commute; \item The morphisms $mf$ and $mg$ Huq-commute if and only if the morphisms $f$ and $g$ Huq-commute; \item If the morphisms $f$ and $g$ Huq-commute, then so do the morphisms $fu$ and $gv$. \end{enumerate} \end{lemma} \begin{lemma} \label{lemma:product_decomposes} For $f:X\to A$, $g:Y\to A$, $f':X'\to A'$ and $g':Y'\to A'$ in $\mathbb{C}$, the morphisms $f\times f'$ and $g\times g'$ Huq-commute if and only if both the morphisms $f$ and $g$, and the morphisms $f'$ and $g'$ Huq-commute. \end{lemma} \begin{lemma} \label{lemma:product_decomposes_2} For $f:X\to A$, $g:Y\to A$, $f':X\to A'$ and $g':Y\to A'$ in $\mathbb{C}$, the morphisms $\langle f, f'\rangle$ and $\langle g, g'\rangle$ Huq-commute if and only if $f$ and $g$ Huq-commute, and $f'$ and $g'$ Huq-commute. \end{lemma}
\section{The results} Throughout this section we assume that $\mathbb{C}$ is a unital category. Let $k:X\to A$ and $l:Y\to A$ be monomorphisms, let $r_1,r_2:R\to A$ and $s_1,s_2:S\to A$ be equivalence relations and let $\kappa$ and $\lambda$ be morphisms such that the diagrams \begin{equation}\label{Def k and l}\vcenter{ \xymatrix{ X \ar[r]^{\kappa} \ar[d]_{k} & R \ar[d]^{\langle r_1,r_2\rangle} & Y \ar[r]^{\lambda} \ar[d]_{l} & S \ar[d]^{\langle s_1,s_2\rangle} \\ A \ar[r]_-{\langle 1,0\rangle} & A\times A & A \ar[r]_-{\langle 0,1\rangle} & A\times A } } \end{equation} are pullbacks. Note that in pointed context this amounts to saying $k$ and $l$ are Bourn-normal. In particular, this includes the case when $k$ and $l$ are the kernels of some morphisms $f$ and $g$: in this case, $r_1,r_2$ and $s_1,s_2$ can be constructed as the kernel pairs of $f$ and $g$ respectively, and $\kappa$ and $\lambda$ are the unique morphisms with $r_1 \kappa = k$, $r_2\kappa=0$, $s_1\lambda = 0$, and $s_2\lambda=0$.
We will need the relation $R\sqz S$, which is a pointed counter part to $R\sq S$ introduced by A. Carboni, M.C. Pedicchio and N. Pirovano in \cite{CARBONI_PEDICCHIO_PIRAVANO:1992}. In the context of pointed sets has elements \[
\{(x,a,y)\in X\times A\times Y\,|\,(k(x),a)\in S \text{ and } (a,l(y))\in R\}. \] Note that an element $(x,a,y)$ in $R\sqz S$ can, after identifying $k(x)$ and $x$, and $l(y)$ and $y$, be displayed as follows \[ \xymatrix@R=5ex@C=5ex{ x \ar[r]^{R} \ar[d]_{S} & 0 \ar[d]^{S} \\ a \ar[r]_{R} & y. } \] Categorically this relation can be built via the pullbacks \begin{equation}\vcenter{\label{Def Composite of relations} \xymatrix{ & & P \ar[dl]_{p_1} \ar[dr]^{p_2} & & \\ & S\ar[dl]_{s_1} \ar[dr]^{s_2} & & R \ar[dl]_{r_1} \ar[dr]^{r_2} & \\ A & & A & & A }} \end{equation} \begin{equation}\label{Def theta}\vcenter{ \xymatrix{ R\sqz S \ar[r]^{\theta} \ar[d]_{\psi} & X\times Y \ar[d]^{k\times l} \\ P \ar[r]_-{\langle s_1p_1,r_2p_2\rangle} & A\times A }} \end{equation} or directly as the limit of the outer arrows of what is easily seen to be a limiting cone \begin{equation} \label{limit_defining_box_zero} \vcenter{ \xymatrix@C=6ex@R=7ex{ A & X \ar[l]_{k} & \\ S \ar[u]^{s_1} \ar[d]_{s_2} & R\sqz S \ar[u]^{\pi_1\theta} \ar[l]_{p_1\psi} \ar[r]^{\pi_2\theta} \ar[d]^{p_2\psi} & Y \ar[d]^{l} \\ A & R \ar[l]_{r_1} \ar[r]^{r_2} & A. } } \end{equation} Let $\alpha:X\to R\sqz S$ and $\beta: Y\to R\sqz S$ be the unique cone morphisms induced by the cones \begin{equation}\label{Def alpha and beta}\vcenter{ \xymatrix@C=9ex@R=7ex{ A & X \ar[l]_{k} & \\ S \ar[u]^{s_1} \ar[d]_{s_2} & X \ar[u]^{1_X} \ar[l]_{e_S k} \ar[r]^{0} \ar[d]^{\kappa} & Y \ar[d]^{l} \\ A & R \ar[l]_{r_1} \ar[r]^{r_2} & A } \xymatrix@C=9ex@R=7ex{ A & X \ar[l]_{k} & \\ S \ar[u]^{s_1} \ar[d]_{s_2} & Y \ar[u]^{0} \ar[l]_{\kappa} \ar[r]^{1_Y} \ar[d]^{e_R l} & Y \ar[d]^{l} \\ A & R \ar[l]_{r_1} \ar[r]^{r_2} & A. } } \end{equation}
Note that, in particular, it follows that $\alpha$ and $\beta$ are morphisms making the two triangles in the diagram \begin{equation*} \label{theta_def} \vcenter{ \xymatrix{
& R\sqz S \ar[d]^{\theta}\\ X \ar[r]_-{\langle 1,0\rangle} \ar[ru]^-{\alpha} & X\times Y & Y \ar[l]^-{\langle 0,1\rangle} \ar[lu]_{\beta}, } } \end{equation*} where $\theta$ is defined as in Diagram \ref{Def theta}, are commutative. Since $\mathbb{C}$ is a unital catgory, this means (see e.g. Theorem 1.2.12 of \cite{BORCEUX_BOURN:2004}): \begin{proposition} \label{proposition:theta_strong} The morphism $\theta$ in \eqref{theta_def} is a strong epimorphism. \qed \end{proposition} Using in part the previous fact, we are now ready to state and prove our alternative criteria for when a pair of Bourn-normal monomorphisms commute. \begin{theorem} \label{theorem:char_of_huq_commutes_for_bourn_normal} The following conditions are equivalent: \begin{enumerate}[(a)] \item $k:X\to A$ and $l:Y\to A$, as defined in \eqref{Def k and l}, Huq-commute; \item $\alpha:X\to R\sqz S$ and $\beta:Y\to R\sqz S$, as defined in \eqref{Def alpha and beta}, Huq-commute; \item $\theta :R\sqz S\to X\times Y$ is a split epimorphism of cospans with domain $(R\sqz S,\alpha,\beta)$ and $(X\times Y,\langle 1,0\rangle,\langle 0,1\rangle)$. \end{enumerate} \end{theorem} \begin{proof} Let $m : R\sqz S \to X\times A\times Y$ be the morphism defined by $m=\langle \pi_1\theta,s_2p_1\psi,\pi_2 \theta \rangle$. An easy calculation shows that $m$ is a monomorphism. Noting that $m\alpha = \langle 1,k,0\rangle$ and $m\beta = \langle 0,l,1\rangle$, it follows from Lemma \ref{lemma:mono_cancels} that $\alpha$ and $\beta$ Huq-commute if and only if $\langle 1,k,0\rangle$ and $\langle 0,l,1\rangle$ Huq-commute. However, by Lemma \ref{lemma:product_decomposes_2} this latter condition is equivalent to requiring $k$ and $l$ to Huq-commute. This proves $(a)\Leftrightarrow (b)$. To prove that $(b)\Rightarrow (c)$ we note that (b) is equivalent to requiring that there is a morphism $\sigma : X\times Y \to R\sqz S$ making the upper part of the diagram \begin{equation} \label{diag:char_of_huq_commutes_for_bourn_normal} \vcenter{ \xymatrix{
& X\times Y \ar[d]^{\sigma} & \\ X \ar[ur]^{\langle 1,0\rangle} \ar[r]^-{\alpha} \ar[dr]_{\langle 1,0\rangle} & R\sqz S \ar[d]^{\theta} & Y \ar[ul]_-{\langle 0,1\rangle} \ar[l]_-{\beta} \ar[dl]^-{\langle 0,1\rangle} \\ & X\times Y & } } \end{equation} commute. However, since $\langle 1,0\rangle$ and $\langle 0,1\rangle$ are jointly epimorphic any such morphism must satisfy $\theta \sigma = 1_{X\times Y}$ and so (c) holds. The converse is immediate, since (c) implies that there is a morphism $\sigma$ making the upper part of \eqref{diag:char_of_huq_commutes_for_bourn_normal} commute, and as mentioned (b) is equivalent to the existence of such a morphism.
\end{proof} \begin{lemma} \label{lemma:kernel_of_theta} The objects $\textnormal{Ker}(\theta)$ and $X\times_A Y$, where $X\times_A Y$ is the pullback of $k:X\to A$ and $l:Y\to A$, are isomorphic. \end{lemma} \begin{proof} Note that since \eqref{Def theta} is a pullback, it follows that $\textnormal{Ker}(\theta)\cong \textnormal{Ker}(\langle s_1p_1,r_2p_2\rangle)$. Now consider the diagram \begin{equation} \label{diag:proof_kernel_of_theta} \vcenter{ \xymatrix@!@C=-0.75ex@R=0ex{ & & & \textnormal{Ker}(\theta) \ar[dl]_{v} \ar[dr]^{u}
\ar@{}[dd]|{(*)} \\ & & \textnormal{Ker}(s_1p_1) \ar[dl]_{j} \ar[dr]_(0.6){\textnormal{ker}(s_1p_1)} & & \textnormal{Ker}(r_2p_2) \ar[dl]^(0.6){\textnormal{ker}(r_2p_2)} \ar[dr]^{i} & & \\ & Y \ar[dl] \ar[dr]^{\lambda} & & P \ar[dl]_{p_1} \ar[dr]^{p_2} & & X \ar[dl]_{\kappa} \ar[dr] & \\ 0 \ar[dr] & & S\ar[dl]_{s_1} \ar[dr]^{s_2} & & R \ar[dl]_{r_1} \ar[dr]^{r_2} & & 0 \ar[dl] \\ & A & & A & & A & }} \end{equation} consisting of the diagram \eqref{Def Composite of relations} and in which: \begin{description} \item{-} $i$ and $j$ are the unique morphism such that $\lambda j =p_1 \textnormal{ker}(s_1p_1)$ and $\kappa i = p_2 \textnormal{ker}(r_2p_2)$; \item{-} $u$ and $v$ are the unique morphisms making $(*)$ in the diagram above, commute. \end{description} Since each diamond in \eqref{diag:proof_kernel_of_theta} is a pullback and $r_1\kappa = k$ and $s_2\lambda =l$, it follows that the diagram \[ \xymatrix{ \textnormal{Ker}(\theta) \ar[d]_{iu} \ar[r]^{jv} & Y \ar[d]^{l} \\ X \ar[r]_{k} & A } \] is a also a pullback, and therefore, $\textnormal{Ker}(\theta)\cong X\times_A Y$ as desired.
\end{proof} Let $\mathbb{X}$ be a pointed category. Consider the condition: \begin{condition}\label{zero_det} A morphism $f:A\to B$ in $\mathbb{X}$ is a monomorphism if and only if the kernel of $f$ is $0$. \end{condition} \begin{remark}\label{remark:zero_dat_known} Note that a pointed category $\mathbb{X}$ satisfies Condition \ref{zero_det} if and only if each reflexive relation in $\mathbb{X}$ satisfies what was called Condition ($*\pi_0)$ in \cite{GRAN_JANELIDZE:2014}, with respect to the ideal of zero morphisms. \end{remark} Recall that a regular category \cite{BARR:1971} is normal \cite{JANELIDZE_Z:2010} if and only if every regular epimorphism is a normal epimorphism. The following proposition follows from Corollary 2.3 of \cite{GRAN_JANELIDZE:2014}, however we give a direct proof in order to avoid introducing notation and terminology that would not otherwise be needed in this paper. \begin{proposition} A regular category $\mathbb{X}$ with cokernels is normal if and only if it satisfies Condition \ref{zero_det}. \end{proposition} \begin{proof} It is immediate that a normal category satisfies Condition \ref{zero_det}. It remains to prove the converse. Suppose $f:A\to B$ is a regular epimorphism and consider the diagram \[ \xymatrix{ \textnormal{Ker}(f) \ar[r]^{\textnormal{ker}(f)} \ar@{-->}[d]_{u} & A \ar[r]^{f} \ar@{-->}[d]^{q} & B \\ \textnormal{Ker}(r) \ar[r]_{\textnormal{ker}(r)} & Q \ar[ur]_{r} } \] in which $q$ is cokernel of $\ker(f)$, $r$ is the unique morphism with $rq=f$, and $u$ the unique morphism with $\textnormal{ker}(r)u=q\textnormal{ker}(f)$. Since the left hand square is a pullback it follows that $u$ is a regular epimorphism. Since $\textnormal{ker}(r)u= q\textnormal{ker}(f)=0$, it follows that $\textnormal{ker}(r)=0$, and therefore $r$ is monomorphism. Since $r$ is also a regular epimorphism, the latter implies that $r$ is an isomorphism. \end{proof} Recall that for a category $\mathbb{X}$ and an object $B$ is $\mathbb{X}$, the category $\mathbf{Pt}_{\mathbb{X}}(B)$ of points, in the sense of D. Bourn, has objects triples $(A,\alpha,\beta)$, where $A$ is an object in $\mathbb{X}$, and $\alpha : A\to B$ and $\beta : B\to A$ are morphisms in $\mathbb{X}$ such that $\alpha\beta=1_B$. A morphism $f$ from $(A,\alpha,\beta)$ to $(A',\alpha',\beta')$ in $\mathbf{Pt}_{\mathbb{X}}(B)$ is a morphism $f:A\to A'$, such that $\alpha'f=\alpha$ and $f\beta=\beta'$. Furthermore, a morphism $p:E\to B$ in $\mathbb{X}$ determines a pullback functor $p^*:\mathbf{Pt}_{\mathbb{X}}(B)\to \mathbf{Pt}_{\mathbb{X}}(E)$ which sends $(A,\alpha,\beta)$ in $\mathbf{Pt}_{\mathbb{X}}(B)$ to $(E\times_B A,\pi_1,\langle 1,\beta p\rangle)$ in $\mathbf{Pt}_{\mathbb{X}}(E)$, with objects and morphism defined as in the following commutative diagram \[ \xymatrix{ E \ar@/^2ex/[drr]^{\beta p} \ar@/_2ex/[ddr]_{1_E}
\ar[dr]|{\langle 1,\beta p\rangle} & & \\ & E\times_B A \ar[r]^{\pi_2} \ar[d]_{\pi_1}
\ar@{}[dr]|{\boxed{1}} & A \ar[d]^{\alpha} \\ & E \ar[r]_{p} &B } \] in which $\boxed{1}$ is a pullback. When $\mathbb{X}$ is a pointed category, pullback functors along morphisms of the form $0\to B$ are essentially the same as kernel functors $\text{Ker}_{B}:\mathbf{Pt}_{\mathbb{X}}(B)\to \mathbb{X}$. \begin{proposition}\label{characterization_of_zero_det} For a pointed finitely complete category $\mathbb{X}$ the following are equivalent: \begin{enumerate}[(a)] \item The category $\mathbb{X}$ satisfies Condition \ref{zero_det}; \item For each object $B$ in $\mathbb{X}$ the functor $\textnormal{Ker}_B$ reflects terminal objects; \item For each object $B$ in $\mathbb{X}$ the functor $\textnormal{Ker}_B$ reflects monomorphisms; \item For each object $B$ in $\mathbb{X}$ the category $\mathbf{Pt}_{\mathbb{X}}(B)$ satisfies Condition \ref{zero_det}; \item For each morphism $p:E\to B$ in $\mathbb{X}$ the functor $p^*:\mathbf{Pt}_{\mathbb{X}}(B)\to \mathbf{Pt}_{\mathbb{X}}(E)$ reflects terminal objects; \item For each morphism $p:E\to B$ in $\mathbb{X}$ the functor $p^*:\mathbf{Pt}_{\mathbb{X}}(B)\to \mathbf{Pt}_{\mathbb{X}}(E)$ reflects monomorphisms. \end{enumerate} \end{proposition} \begin{proof} For a morphism $f:A\to B$ in $\mathbb{X}$, note that: \begin{enumerate}[(i)] \item $f:A\to B$ is a monomorphism if and only if in the pullback diagram \[ \xymatrix{ A\times_B A \ar[d]_{\pi_1} \ar[r]^-{\pi_2} &A \ar[d]^{f} \\ A \ar[r]_{f} & B } \]
$\pi_1$ is a isomorphism. \item The morphism $\pi_1$ is an isomorphism whenever $(A\times_B A,\pi_1,\langle 1,1\rangle)$ is a terminal object in $\mathbf{Pt}_{\mathbb{X}}(A)$; \item The kernel of $f$ is isomorphic to the kernel of $\pi_1$. \end{enumerate}
Combining these observations we see that (a)$\Leftrightarrow$(b). For any functor $F$ between pointed categories which preserves terminal objects,
since morphisms into the terminal object are necessarily split epimorphisms, one easily shows that if $F$ reflects monomorphisms, then it reflects terminal objects. Therefore (f)$\Rightarrow$(e) and (c)$\Rightarrow$(b). Recalling that if a composite of functors $FG$ reflects some property and $F$ preserves it, then $G$ reflects it, and noting that kernel functors certainly preserve terminal objects, one easily sees that (b)$\Rightarrow$(e) (just note that for each morphism $p:E\to B$ the functor $\textnormal{Ker}_E\circ p^*$ is isomorphic to $\textnormal{Ker}_B$). Since each pullback functor between points along a morphism in a category of points of $\mathbb{X}$ is up to isomorphism a pullback functor between points for $\mathbb{X}$ it follows that (e)$\Rightarrow$(d). For a functor $F$ between pointed finitely complete categories satisfying Condition 3.4, preserving limits and reflecting terminal objects, if $F(f)$ is a monomorphism then $F(\text{Ker}(f))\cong \text{Ker}(F(f)) \cong 0$ and hence $\text{Ker}(f)\cong 0$ which forces $f$ to be a monomorphism. This proves (e)$\Rightarrow$(f) since we already know that (e)$\Rightarrow$(d). The proof is completed by noting that trivially (f)$\Rightarrow$(c) and (d)$\Rightarrow$(a). \end{proof} \begin{proposition} Let $\mathcal{V}$ be a (quasi)-variety of universal algebras considered as a category, and let $\mathbb{X}$ be a category with finite limits. If $\mathcal{V}$ satisfies Condition \ref{zero_det}, then $\mathcal{V}(\mathbb{X})$ satiesfies Condition \ref{zero_det}. \end{proposition} \begin{proof} Since the Yoneda embedding $Y : \mathbb{X} \to \mathbf{Set}^{\mathbb{X}^{\textnormal{op}}}$ preserves and reflects limits and $\mathcal{V}(\mathbf{Set}^{\mathbb{X}^{\textnormal{op}}})=\mathcal{V}^{\mathbb{X}^{\textnormal{op}}}$, taking internal $\mathcal{V}$ algebras we obtain a functor $\tilde Y : \mathcal{V}(\mathbb{X}) \to \mathcal{V}^{\mathbb{X}^{\textnormal{op}}}$ which preserves and reflects limits. The claim now follows by noting that Condition \ref{zero_det} lifts to functor categories. \end{proof} \begin{example} Recall that an implication algebra is a triple $(X,\cdot,1)$ where $X$ is a set, $\cdot$ is a binary operation and $1$ is constant satisfying the axioms: $(xy)x=x$, $(xy)y=(yx)x$, $x(yz)=y(xz)$, $11=1$. H.\ P.\ Gumm and A.\ Ursini showed in \cite{GUMM_URSINI:1984} that the variety of implication algebras form an ideal determined variety of universal algebras which is not congruence permutable. This means that the category of implication algebras is ideal determined but not Mal'tsev \cite{JANELIDZE_MARKI_THOLEN_URSINI:2010}. Since the two element boolean algebra $2=(2,\to,1)$ forms an implication algebra and $\{(0,1),(1,0),(1,1)\}$ is a sub-algebra of $2\times 2$, we see that it is not a unital category. However adding an independent binary operation $*$ satisfying $x*1=1*x=x$ will produce a unital ideal determined category, and hence a strongly unital normal category. We leave as open problems whether this latter variety is Mal'tsev or not and if their exists a normal strongly unital variety which is not Mal'tsev. On the other hand the previous proposition tells us that internal such algebras in a category with finite limits always produce a category which is strongly unital and satisfies Condtion \ref{zero_det}.
\end{example} \begin{example} It is easy to show that the quasi-variety $\mathcal{V}$ of universal algebras, with terms $p(x,y)$ and $s(x,y)$ satisfying $p(x,0)=p(0,x)=x$, $s(x,0)=x$, $s(x,x)=0$, and $s(x,y)=0 \Rightarrow x=y$, is a normal strongly unital category. In fact, it turns out that this quasi-variety is almost exact (i.e every regular epimorphism is an effective descent morphism) and is not Mal'tsev. As before, by the previous proposition, we obtain that internal such algebras in a finitely complete catgory will produce strongly unital categories satisfying Condition \ref{zero_det}. In particular, if the base category is the product of the category of sets with the quasi-variety $\mathcal{W}$ of abelian groups satisfying $4x=0 \Rightarrow 2x=0$, then resulting category will on the one hand not be Mal'tsev since $\mathcal{V}$ is not, and on the other hand not be regular (and hence not normal) since $\mathcal{V}(\mathcal{W})=\mathcal{W}$ which is not regular. \end{example} \begin{corollary} \label{corollary:_meet_trivial_commutes} Let $k$ and $l$ be Bourn-normal monomorphisms in a unital category $\mathbb{C}$ satisfying Condition \ref{zero_det}. If $k$ and $l$ have trivial pullback, then $k$ and $l$ commute. \end{corollary} \begin{proof} If $k$ and $l$ have trivial pullback, then by Lemma \ref{lemma:kernel_of_theta} the morphism $\theta$ has trivial kernel and hence is a monomorphism. Moreover, since by Proposition \ref{proposition:theta_strong} the morphism $\theta$ is strong epimorphism it foolws that it is an isomorphism. The claim now follows from Theorem \ref{theorem:char_of_huq_commutes_for_bourn_normal} . \end{proof} \begin{lemma} \label{lemma:kernel_of_coperator} Let $k:X\to A$ and $l:Y\to A$ be monomorphisms in a strongly unital category $\mathbb{C}$ which commute, and let $\varphi : X\times Y \to A$ be their cooperator. If $\langle u,v \rangle : W\to X\times Y$ is the kernel of $\varphi$, then $u$ and $v$ are central monomorphisms and $ku=k(-v)$. \end{lemma} \begin{proof} Since each of the squares in the following diagram \[ \xymatrix{ 0 \ar[r] \ar[d] & W \ar[d]_{\langle u,v\rangle} & 0 \ar[l] \ar[d] \\ X \ar[r]_{\langle 1,0\rangle} & X\times Y & Y \ar[l]^{\langle 0,1\rangle} } \] are pullbacks, we see via Lemma \ref{lemma:product_decomposes}, that $u$ and $v$ are central. To complete the proof just note that $\langle u,0\rangle = \langle u,v\rangle+\langle 0,-v\rangle$ and therefore \begin{align*} ku&=\varphi \langle u,0\rangle\\ &= \varphi (\langle u,v\rangle + \langle 0,-v\rangle)\\ &= \varphi \langle u,v\rangle + \varphi \langle 0,-v\rangle\\ &= \varphi \langle 0,-v\rangle\\ &=l(-v). \end{align*} \end{proof} \begin{proposition} \label{prop:copoerator_join} Let $k$ and $l$ be Bourn-normal monomorphisms in a strongly unital category $\mathbb{C}$ satisfying Condition \ref{zero_det}. If $k$ and $l$ have trivial pullback, then $k$ and $l$ commute and their cooperator $\varphi : X\times Y\to A$ is a monomorphism, which is also their join. \end{proposition} \begin{proof} By Corollary \ref{corollary:_meet_trivial_commutes} we know that $k$ and $l$ commute. It remains to show that their cooperator is a monomorphism and is their join. The first point follows from Condition \ref{zero_det} since Lemma \ref{lemma:kernel_of_coperator} implies the kernel of $\varphi$ is zero. The final point follows immediately from the fact that $\langle 1,0\rangle$ and $\langle 0,1\rangle$ are jointly strongly epimorphic. \end{proof} Recall that in the Mal'tsev context an equivalence relation $r_1,r_2:R\to A$ is essentially the same thing as a monomorphism \[ \xymatrix{ R \ar@<0.5ex>[dr]^{r_1} \ar[rr]^-{\langle r_1,r_2\rangle} & & A\times A \ar@<0.5ex>[dl]^{\pi_1} \\ & A \ar@<0.5ex>[ul]^{e} \ar@<0.5ex>[ur]^{\langle 1,1\rangle} & } \] in the category $\mathbf{Pt}(A)$. Moreover such a monomorphism $\langle r_1,r_2 \rangle$ is necessarily Bourn-normal. To see why, consider the pullback diagram \[ \xymatrix{ R\times_A R \ar[d]_{p_1} \ar[r]^{p_2} & R \ar[d]^{r_1} \\ R \ar[r]_{r_1} & A. } \] It follows that $\langle r_1p_1,r_2p_1\rangle,\langle r_1p_1,r_2p_2\rangle: (R\times_A R,r_1p_1,\langle e,e\rangle) \to (A\times A,\pi_1,\langle 1,1\rangle)$ (where $e$ is the splitting of $r_1$ and $r_2$) is an equivalence relation and the diagrams \[ \xymatrix{ A\times (A\times A) \ar[d]_{1\times \pi_1} \ar[r]^-{1\times \pi_2} & A\times A\ar[d]^{\pi_1}\\ A\times A\ar[r]_{\pi_1} & A } \xymatrix@C=10ex{ R \ar[d]_{\langle r_1,r_2\rangle} \ar[r]^-{\langle 1,r_1 e\rangle} & R\times_A R \ar[d]^{\langle r_1\pi_1,\langle r_2p_1,r_2p_2\rangle\rangle} \\ A\times A \ar[r]_-{1\times \langle p_2,p_1\rangle} & A\times (A\times A) } \] are pullbacks. \begin{theorem} Let $k$ and $l$ be Bourn-normal monomorphisms in a Mal'tsev category $\mathbb{C}$ satisfying Condition \ref{zero_det}. If $k$ and $l$ have trivial pullback, then $k$ and $l$ commute and their cooperator $\varphi : X\times Y\to A$ is a Bourn-normal monomorphism which is also their join. \end{theorem} \begin{proof} By Proposition \ref{characterization_of_zero_det} we see that $X\wedge Y=0$ implies $R\wedge S=0$ when considered as subobjects of $(A\times A,\pi_1,\langle 1,1\rangle)$ in the category of points over $A$. It now follows from Proposition \ref{prop:copoerator_join} that $R\times S$ (in $\mathbf{Pt}(A)$) is a subobject of $(A\times A,\pi_1,\langle 1,1\rangle)$, and hence is an equivalence relation with zero class the cooperator of $k$ and $l$. \end{proof} \begin{corollary} Let $\mathbb{C}$ be strongly unital category satisfying Condition \ref{zero_det}. For an object $X$ in $\mathbb{C}$ the following conditions are equivalent: \begin{enumerate}[(a)] \item $X$ is abelian; \item $\langle 1,1\rangle : X \to X\times X$ is a normal monomorphism; \item $\langle 1,1\rangle : X\to X\times X$ is a Bourn-normal monomorphisn. \end{enumerate} \end{corollary} \begin{proof} The implications (a) $\Rightarrow$ (b)$\Rightarrow$ (c) are immediate. It is therefore sufficient to show that (a) follows from (c). Suppose $X$ is object in $\mathbb{C}$ and $\langle 1,1\rangle$ is Bourn-normal. Since $\langle 1,0\rangle$ and $\langle 1,1\rangle$ have trivial pullback, it follows that they commute and hence we obtain a morphismi $\psi$ making the diagram \[ \xymatrix{ X \ar[r]^-{\langle 1,0\rangle} \ar@{=}[d] & X\times X \ar[d]^{\psi} & X \ar[l]_-{\langle 0,1\rangle} \ar@{=}[d] \\ X \ar[r]^-{\langle 1,0\rangle} & X\times X & X \ar[l]_-{\langle 1,1\rangle} } \] commute. The claim now follows from Corollary 1.8.20 of \cite{BORCEUX_BOURN:2004}, since $\pi_1\psi$ is a cooperator for $1_X$ and $1_X$. \end{proof} \begin{remark} Given that abelianess is a property in a subtractive category, and abelianization is obtained by forming the cokernel of the diagonal in a regular subtractive category (provided the cokernel exists) \cite{BOURN_JANELIDZE_Z:2016}, one expects that the above corollary is true in a wider context. \end{remark}
\end{document}
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\begin{document}
\title{On the chromatic number of random regular hypergraphs} \author{Patrick Bennett\thanks{Research supported in part by Simons Foundation Grant \#426894.}\\Department of Mathematics,\\ Western Michigan University\\ Kalamazoo MI 49008\and Alan Frieze\thanks{Research supported in part by NSF Grant DMS1661063 }\\ Department of Mathematical Sciences,\\ Carnegie Mellon University,\\ Pittsburgh PA 15213. } \date{}
\maketitle
\begin{abstract}
We estimate the likely values of the chromatic and independence numbers of the random $r$-uniform $d$-regular hypergraph on $n$ vertices for fixed $r$, large fixed $d$, and $n \rightarrow \infty$. \end{abstract}
\section{Introduction} The study of the chromatic number of random graphs has a long history. It begins with the work of Bollob\'as and Erd\H{o}s \cite{BE} and Grimmett and McDiarmid \cite{GM} who determined $\chi(G_{n,p})$, $p$ constant to within a factor 2, w.h.p.~Matula \cite{M} reduced this to a factor of 3/2. Then we have the discovery of martingale concentration inequalities by Shamir and Spencer \cite{SS} leading to the breakthrough by Bollob\'as \cite{B} who determined $\chi(G_{n,p})$ asymptotically for $p$ constant.
The case of $p\to 0$ proved a little more tricky, but {\L}uczak \cite{L1} using ideas from Frieze \cite{F} and \cite{M} determined $\chi(G_{n,p}),p=c/n$ asymptotically for large $c$. {\L}uczak \cite{L2} showed that w.h.p.~$\chi(G_{n,p}),p=c/n$ took one of two values. It was then that the surprising power of the second moment method was unleashed by Achlioptas and Naor \cite{AN}. Since then there has been much work tightening our estimates for the $k$-colorability threshold, $k\ge 3$ constant. See for example Coja-Oghlan \cite{C}.
Random regular graphs of low degree were studied algorithmically by several authors e.g. Achlioptas and Molloy \cite{AM} and by Shi and Wormald \cite{SW}. Frieze and {\L}uczak \cite{FL92} introduced a way of using our knowledge of $\chi(G_{n,p}),p=c/n$ to tackle $\chi(G_{n,r})$ where $G_{n,r}$ denotes a random $r$-regular graph and where $p=r/n$. Subsequently Achlioptas and Moore \cite{AM} showed via the second moment method that w.h.p.~$\chi(G_{n,r})$ was one of 3 values. This was tightened basically to one value by Coja-Oghlan, Efthymiou and Hetterich \cite{CE}.
For random hypergraphs, Krivelevich and Sudakov \cite{KS98} established the asymptotic chromatic number for $\chi({\mathcal H}_r(n,p)$ for $\binom{n-1}{r-1}p$ sufficiently large. Here ${\mathcal H}_r(n,p)$ is the binomial $r$-uniform hypergraph where each of the $\binom{n}{r}$ possible edges is included with probability $p$. There are several possibilities of a proper coloring of the vertices of a hypergraph. Here we concentrate on the case where a vertex coloring is proper if no edge contains vertices of all the same color. Dyer, Frieze and Greehill \cite{DFG} and Ayre, Coja-Oghlan and Greehill \cite{ACG} established showed that w.h.p.~$\chi({\mathcal H}_r(n,p)$ took one or two values. When it comes to what ew denote by $\chi({\mathcal H}_r(n,d)$, a random $d$-regular, $r$-uniform hypergraph, we are not aware of any results at all. In this paper we extend the approach of \cite{FL92} to this case: \begin{theorem}\label{thm:main} For all fixed $r$ and $\varepsilon} \def\f{\phi} \def\F{{\Phi}} \def\vp{\varphi} \def\g{\gamma>0$ there exists $d_0=d_0(r, \varepsilon} \def\f{\phi} \def\F{{\Phi}} \def\vp{\varphi} \def\g{\gamma)$ such that for any fixed $d \ge d_0$ we have that w.h.p. \begin{equation}
\abrac{ \frac{ \chi(\mc{H}_r(n, d)) - \rbrac{\frac{(r-1)d}{r \log d}}^{\frac{1}{r-1}}}{\rbrac{\frac{(r-1)d}{r \log d}}^{\frac{1}{r-1}}}} \le \varepsilon} \def\f{\phi} \def\F{{\Phi}} \def\vp{\varphi} \def\g{\gamma, \qquad \abrac{ \frac{ \alpha} \def\b{\beta} \def\d{\delta} \def\D{\Delta(\mc{H}_r(n, d)) - \rbrac{\frac{r \log d}{(r-1)d}}^{\frac{1}{r-1}}n}{\rbrac{\frac{r \log d}{(r-1)d}}^{\frac{1}{r-1}}n}} \le \varepsilon} \def\f{\phi} \def\F{{\Phi}} \def\vp{\varphi} \def\g{\gamma \end{equation} Here $\alpha} \def\b{\beta} \def\d{\delta} \def\D{\Delta$ refers to the independence number of a hypergraph. \end{theorem}
\section{Preliminaries}
\subsection{Tools} We will be using the following forms of Chernoff's bound (see, e.g., \cite{FK}).
\begin{lemma}[Chernoff bound] Let $X\sim \textrm{Bin}(n,p)$. Then for all $0<\lambda<np$ \begin{equation}\label{Chernoff}
\mathbb{P}(|X-np| \ge \lambda ) \le 2\exp\rbrac{-\frac{ \lambda^2}{3np}}. \end{equation} \end{lemma}
\begin{lemma}[McDiarmid's inequality]
Let $X=f(\Vec{Z})$ where $\Vec{Z}= (Z_1, \ldots Z_t)$ and the $Z_i$ are independent random variables. Assume the function $f$ has the property that whenever $\vec{z}, \vec{w}$ differ in only one coordinate we have $|f(\vec{z}) - f(\vec{w})| \le c$. Then for all $\lambda>0$ we have \begin{equation}\label{Mcdiarmid}
\mathbb{P}(|X-\mathbb{E}[X]| \ge \lambda ) \le 2\exp\rbrac{-\frac{ \lambda^2}{2c^2 t}}. \end{equation} \end{lemma}
Bal and the first author \cite{BB21} showed the following.
\begin{theorem}[Claim 4.2 in \cite{BB21}]\label{thm:BB} Fix $r \ge 3$, $d \ge 2$, and $0 < c < \frac{r-1}{r}.$ Let $z_2$ be the unique positive number such that \begin{equation}\label{p1} \frac{z_2 \sbrac{\rbrac{z_2 + 1}^{r-1} - z_2^{r-1}}}{\rbrac{z_2 + 1}^r - z_2^r} = c \end{equation} and let \beq{p2}{ z_1 = \frac{d}{r\sbrac{\rbrac{z_2 + 1}^r - z_2^r}}. } Let $h(x) = x \log x$. If it is the case that \begin{equation}\label{eqn:BB}
h\rbrac{\frac{d}{r}} + h(d c) + h(d(1-c)) - h(c) -h(1-c) - h(d) - \frac{d}{r} \log z_1 - dc \log z_2 <0 \end{equation}
then w.h.p.~$\alpha(\mc{H}_r(n, d))< cn$. \end{theorem}
Krivelevich and Sudakov \cite{KS98} proved the following.
\begin{theorem}[Theorem 5.1 in \cite{KS98}]\label{thm:KS} For all fixed $r$ and $\varepsilon} \def\f{\phi} \def\F{{\Phi}} \def\vp{\varphi} \def\g{\gamma>0$ there exists $d_0=d_0(r, \varepsilon} \def\f{\phi} \def\F{{\Phi}} \def\vp{\varphi} \def\g{\gamma)$ such that whenever $D=D(p):=\binom{n-1}{r-1}p \ge d_0$ we have that \begin{equation*}
\abrac{ \frac{ \chi(\mc{H}_r(n, p)) - \rbrac{\frac{(r-1)D}{r \log D}}^{\frac{1}{r-1}}}{\rbrac{\frac{(r-1)D}{r \log D}}^{\frac{1}{r-1}}}} \le \varepsilon} \def\f{\phi} \def\F{{\Phi}} \def\vp{\varphi} \def\g{\gamma, \qquad \qquad \abrac{ \frac{ \alpha} \def\b{\beta} \def\d{\delta} \def\D{\Delta(\mc{H}_r(n, p)) - \rbrac{\frac{r \log D}{(r-1)D}}^{\frac{1}{r-1}}n}{\rbrac{\frac{r \log D}{(r-1)D}}^{\frac{1}{r-1}}n}} \le \varepsilon} \def\f{\phi} \def\F{{\Phi}} \def\vp{\varphi} \def\g{\gamma \end{equation*} with probability at least $1-o(1/n)$. \end{theorem}
\section{Proof}
In this section we prove Theorem \ref{thm:main}. First we give an overview. We show in Subsection \ref{sec:UBalpha} that the upper bound on $\alpha} \def\b{\beta} \def\d{\delta} \def\D{\Delta$ follows from Theorem \ref{thm:BB} and some straightforward calculations. Then the lower bound on $\chi$ follows as well. Thus we will be done once we prove the upper bound on $\chi$ (since that proves the lower bound on $\alpha} \def\b{\beta} \def\d{\delta} \def\D{\Delta$). This will be in Subsection \ref{sec:UBchi}. For that we follow the methods of Frieze and \L uczak \cite{FL92}.
We will assume $r \ge 3$ since Frieze and \L uczak \cite{FL92} covered the graph case. We will use standard asymptotic notation, and we will use big-O notation to suppress any constants depending on $r$ but not $d$. Thus, for example we will write $r=O(1)$ and $d^{-1} = O(1)$ but not $d=O(1)$. This is convenient for us because even though our theorem is for fixed $d$, it requires $d$ to be sufficiently large.
\subsection{Upper bound on the independence number}\label{sec:UBalpha}
We will apply Theorem \ref{thm:BB} to show an upper bound on $\alpha(\mc{H}_r(n, d)).$ Fix $\varepsilon} \def\f{\phi} \def\F{{\Phi}} \def\vp{\varphi} \def\g{\gamma, r$ (but not $d$) and let $c=c(d):= (1+\varepsilon} \def\f{\phi} \def\F{{\Phi}} \def\vp{\varphi} \def\g{\gamma)\rbrac{\frac{r \log d}{(r-1)d}}^{\frac{1}{r-1}}$. Let $z_2$ be as defined in \eqref{p1} and $z_1$ be as defined in \eqref{p2}. We see that \begin{lemma} \[
z_2 = \frac{c}{1-c} +O\rbrac{c^ {r}} \] \end{lemma} \begin{proof} After some algebra, we re-write \eqref{p1} as \[ z_2-\frac{z_2^{r}}{(1+z_2)^{r-1}}=\frac{c}{1-c}. \] and the claim follows. \end{proof} Now we check \eqref{eqn:BB}. \begin{align}
& h\rbrac{\frac{d}{r}} + h(d c) + h(d(1-c)) - h(c) -h(1-c) - h(d) - \frac{d}{r} \log z_1 - dc \log z_2 \nonumber\\
=& \frac{d}{r} \log \rbrac{\frac{d}{r}} + dc \log(d c) + d(1-c) \log (d(1-c)) - c \log c -(1-c) \log (1-c) \nonumber\\
& \qquad - d\log d - \frac{d}{r} \log z_1 - dc \log z_2\nonumber\\
=& dc \log \sbrac{\frac{c}{(1-c)z_2}}+ \frac dr \log \sbrac{(z_2 + 1)^r-z_2^r} + d \log (1-c) - c \log c - (1-c) \log(1-c). \label{eqn:uppersimp} \end{align} Now note that the first term of \eqref{eqn:uppersimp} is \[ dc \log \sbrac{\frac{c}{(1-c)z_2}} = dc \log \sbrac{\frac{c}{(1-c)\rbrac{\frac{c}{1-c} + O\rbrac{c^{r+1}}}}} = dc \log \sbrac{\frac{1}{1 + O\rbrac{c^{r}}}} = O\rbrac{dc^{r+1}}. \] The second term of \eqref{eqn:uppersimp} is \begin{align*}
\frac dr \log \sbrac{(z_2 + 1)^r-z_2^r} &= \frac dr \log \sbrac{\rbrac{\frac{1}{1-c} + O\rbrac{c^{r+1}}}^r-\rbrac{\frac{c}{1-c} + O\rbrac{c^{r+1}}}^r}\\
&=\frac dr \log \sbrac{\rbrac{\frac{1}{1-c}}^r \rbrac{1-c^r + O\rbrac{c^{r+1}}}}\\
&=\frac dr \log \rbrac{\frac{1}{1-c}}^r + \frac dr \log \rbrac{1-c^r + O\rbrac{c^{r+1}}}\\
&= -d \log (1-c) - \frac dr c^r + O\rbrac{dc^{r+1}}. \end{align*} The last term of \eqref{eqn:uppersimp} is \[ (1-c) \log(1-c) = O(c). \] Therefore \eqref{eqn:uppersimp} becomes \begin{align*}
&-\frac dr c^r - c \log c +O\rbrac{c+ dc^{r+1}} \\
=& -c\sbrac{\frac dr c^{r-1} + \log c}+O\rbrac{c+ dc^{r+1}}\\
=& -c\sbrac{\frac dr (1+\varepsilon} \def\f{\phi} \def\F{{\Phi}} \def\vp{\varphi} \def\g{\gamma)^{r-1} \frac{r \log d}{(r-1)d} + \log \rbrac{(1+\varepsilon} \def\f{\phi} \def\F{{\Phi}} \def\vp{\varphi} \def\g{\gamma)\rbrac{\frac{r \log d}{(r-1)d}}^{\frac{1}{r-1}}}}+O\rbrac{c+ dc^{r+1}}\\
=& -c\sbrac{ (1+\varepsilon} \def\f{\phi} \def\F{{\Phi}} \def\vp{\varphi} \def\g{\gamma)^{r-1} \frac{ \log d}{r-1} -\frac{ \log d}{r-1} +O(\log \log d)}+O\rbrac{c+ dc^{r+1}}\\
=& -\Omega\rbrac{c \log d}. \end{align*} It follows from Theorem \ref{thm:BB} that w.h.p. \beq{uppalpha}{ \alpha(\mc{H}_r(n, d))\leq(1+\varepsilon} \def\f{\phi} \def\F{{\Phi}} \def\vp{\varphi} \def\g{\gamma)\rbrac{\frac{r \log d}{(r-1)d}}^{\frac{1}{r-1}}. } \subsection{Upper bound on the chromatic number}\label{sec:UBchi}
Our proof of the upper bound uses the method of Frieze and \L uczak \cite{FL92}. We will generate $\mc{H}_r(n, d)$ in a somewhat complicated way. The way we generate it will allow us to use known results on $\mc{H}_r(n, p)$ due to Krivelevich and Sudakov \cite{KS98}.
Set
\begin{equation}
m:=\rbrac{\frac {d- d^{1/2} \log d}{r} } n.
\end{equation}
Let $\mc{H}^*_r(n, m)$ be an $r$-uniform multi-hypergraph with $m$ edges, where each multi-edge consists of $r$ independent uniformly random vertices chosen with replacement. We will generate $\mc{H}^*_r(n, m)$ as follows. We have $n$ sets (``buckets'' ) $V_1, \ldots V_n$ and a set of $rm$ points $P:=\{p_1, \ldots p_{rm}\}$. We put each point $p_i$ into a uniform random bucket $V_{\f(i)}$ independently. We let $\mc{R}=\{R_1, \ldots, R_m\}$ be a uniform random partition of $P$ into sets of size $r$. Of course, the idea here is that the buckets $V_i$ represent vertices and the parts of the partition $\mc{R}$ represent edges. Thus $R_i$ defines a hyper-edge $\set{\f(j):j\in R_i}$ for $i=1,2,\ldots,m$. We denote the hypergraph defined by $\mc{R}$ by $H_{\mc{R}}$.
Note that since $r \ge 3$ the expected number of pairs of multi-edges in $\mc{H}^*_r(n, m)$ is at most
\[
\binom nr \binom m2 \rbrac{\frac{1}{ \binom nr}}^2 = O\rbrac{\frac{m^2}{n^r}} = O(n^{-1}).
\]
Thus, w.h.p.~there are no multi-edges. Now the expected number of ``loops'' (edges containing the same vertex twice) is at most
\[
n m \binom r2 \rbrac{\frac 1n}^2 = O(1).
\]
Thus w.h.p.~there are at most $\log n$ loops. We now remove all multi-edges and loops, and say that $M$ is the (random) number of edges remaining, where $m - \log n \le M \le m$. The remaining hypergraph is distributed as $\mc{H}(n, M)$, the random hypergraph with $M$ edges chosen uniformly at random without replacement. Next we estimate the chromatic number of $\mc{H}_r(n, M)$.
\begin{claim}\label{clm:hnm}
W.h.p.~we have
\begin{equation}
\abrac{ \frac{ \chi(\mc{H}_r(n, M)) - \rbrac{\frac{(r-1)d}{r \log d}}^{\frac{1}{r-1}}}{\rbrac{\frac{(r-1)d}{r \log d}}^{\frac{1}{r-1}}}} \le \frac \e2 , \qquad \qquad \abrac{ \frac{ \alpha} \def\b{\beta} \def\d{\delta} \def\D{\Delta(\mc{H}_r(n, M)) - \rbrac{\frac{r \log D}{(r-1)D}}^{\frac{1}{r-1}}n}{\rbrac{\frac{r \log D}{(r-1)D}}^{\frac{1}{r-1}}n}} \le \frac \e2 \nonumber. \end{equation}
\end{claim}
\begin{proof}
We will use Theorem \ref{thm:KS} together with a standard argument for comparing $\mc{H}_r(n, p)$ with $\mc{H}_r(n, m)$. Set $p:=m / \binom nr$ and apply Theorem \ref{thm:KS} with $\varepsilon} \def\f{\phi} \def\F{{\Phi}} \def\vp{\varphi} \def\g{\gamma$ replaced with $\varepsilon} \def\f{\phi} \def\F{{\Phi}} \def\vp{\varphi} \def\g{\gamma/4$ so we get \begin{equation}\label{eqn:clm1D}
\abrac{ \frac{ \chi(\mc{H}_r(n, p)) - \rbrac{\frac{(r-1)D}{r \log D}}^{\frac{1}{r-1}}}{\rbrac{\frac{(r-1)D}{r \log D}}^{\frac{1}{r-1}}}} \le \frac\e4 \end{equation} with probability at least $1-o(1/n)$. Note that here \[ D = \binom{n-1}{r-1}p = \binom{n-1}{r-1}m / \binom nr = rm/n = d-d^{1/2} \log d. \] Now since $d, D$ can be chosen to be arbitrarily large and $d=D+O(D^{1/2} \log D)$ we can replace $D$ with $d$ in \eqref{eqn:clm1D} without changing the left hand side by more than $\varepsilon} \def\f{\phi} \def\F{{\Phi}} \def\vp{\varphi} \def\g{\gamma / 4$ to obtain \begin{equation}\label{eqn:clm1d}
\abrac{ \frac{ \chi(\mc{H}_r(n, p)) - \rbrac{\frac{(r-1)d}{r \log d}}^{\frac{1}{r-1}}}{\rbrac{\frac{(r-1)d}{r \log d}}^{\frac{1}{r-1}}}} \le \frac \e2 \end{equation} with probability at least $1-o(1/n)$. But now note that with probability $\Omega(n^{-1/2})$ the number of edges in $\mc{H}_r(n, p)$ is precisely $M$. Thus we have that
\begin{equation}
\abrac{ \frac{ \chi(\mc{H}_r(n, M)) - \rbrac{\frac{(r-1)d}{r \log d}}^{\frac{1}{r-1}}}{\rbrac{\frac{(r-1)d}{r \log d}}^{\frac{1}{r-1}}}} \le \frac \e2 \nonumber \end{equation} with probability at least $1-o(n^{-1/2})$. This proves the first inequality, and the second one follows similarly.
\end{proof} Now we will start to transform $\mc{H}_r(n, m)$ to the random regular hypergraph $\mc{H}_r(n, d)$. This transformation will involve first removing some edges from vertices of degree larger than $d$, and then adding some edges to vertices of degree less than $d$. We define the \tbf{rank} of a point $p_i \in V_j$, to be the number of points $p_{i'} \in V_j$ such that $i'\le i$. We form a new set of points $P' \subseteq P$ and a partition $\mc{R}'$ of $P'$ as follows. For any $R_k \in \mc{R}$ containing a point with rank more than $d$, we delete $R_k$ from $\mc{R}$ and delete all points of $R_k$ from $P$. Note that each bucket contains at most $r$ points of $P'$. Note also that $\mc{R}' $ is a uniform random partition of $P'$. We let $\mc{H}_{\mc{R}'}$ be the natural hypergraph associated with $\mc{R}'$.
Now we would like to put some more points into the buckets until each bucket has exactly $d$ points, arriving at some set of points $P'' \supseteq P'$. We would also like a uniform partition $\mc{R}''$ of $P''$ into sets of size $r$, and we would like $\mc{R}''$ to have many of the same parts as $\mc{R}'$. We will accomplish this by constructing a sequence $P'_1:=P'\subseteq P'_2 \subseteq \ldots \subseteq P'_\ell=:P''$ of point sets and a sequence $\mc{R}'_1:=\mc{R}', \mc{R}'_2, \ldots, \mc{R}'_\ell=:\mc{R}''$ where $\mc{R}'_j$ is a uniform random partition of $P'_j$.
We construct $P'_{j+1}, \mc{R}'_{j+1}$ from $P'_{j}, \mc{R}'_{j}$ as follows. Suppose $|\mc{R}'_j|=a$ (in other words $\mc{R}'_j$ has $a$ parts), so $|P'_j| = ra$. $P'_{j+1}$ will simply be $P'_{j}$ plus $r$ {\bf new} points. Now we will choose a random value $K \in \{1, \ldots, r\}$ using the distribution $\mathbb{P}[K=k] = q_k(a)$, where $q_k(a)$ is defined as follows. \begin{definition} Consider a random partition of $ra+r$ points into $a+1$ parts of size $r$, and fix some set $Q$ of $r$ points. Then for $1 \le k \le r$, the number $q_k(a)$ is defined to be the probability that $Q$ meets exactly $k$ parts of the partition. \end{definition} We will then remove a uniform random set of $K-1$ parts from $\mc{R}'_{j}$, leaving $Kr$ points in $P'_{j+1}$ which are not in any remaining part of $\mc{R}'_{j}$. We partition those points into $K$ parts of size $r$ such that each part contains at least one new point (each such partition being equally likely), arriving at our partition $\mc{R}'_{j+1}$.
We claim that $\mc{R}'_{j+1}$ is a uniform random partition of $P'_{j+1}$ into parts of size $r$. Indeed, first consider the $r$ new points that are in $P'_{j+1}$ which were not in $P'_{j}$. The probability that a uniform random partition of $P'_{j+1}$ would have exactly $k$ parts containing at least one new point is $q_k$. So we can generate such a random partition as follows: first choose a random value $K$ with $\mathbb{P}[K=k]=q_k$; next we choose a uniform random set of $(K-1)r$ points from $P'_j$; next we choose a partition of the set of points consisting of $P'_{j+1} \setminus P'_{j}$ together with the points from $P'_{j}$ we chose in the last step, where the partition we choose is uniformly random from among all partitions such that each part contains at least one point of $P'_{j+1} \setminus P'_{j}$; finally, we choose a uniform partition of the rest of the points. In our case this partition of the rest of the points comprises the current partition of the ``unused'' $(a-K+1)r$ points. At the end of this process we have that $\mc{H}_{\mc{R}''}$ is distributed as $\mc{H}_r(n,d)$. \subsubsection{Bounding the number of low degree vertices in $\mc{H}_{\mc{R'}}$} We define some sets of buckets. We show that w.h.p.~there are few small buckets i.e few vertices of low degree in the hypregraph $H_{\mc{R'}}$. Let $S_0$ be the buckets with at most $d-3d^{1/2}\log d$ points of $P'$, and let $S_1$ be the buckets with at most $d-2d^{1/2}\log d$ points of $P$. Let $S_2$ be the set of buckets that, when we remove points from $P'$ to get $P$, have at least $d^{1/2}\log d$ points removed. Then $S_0 \subseteq S_1 \cup S_2$. Our goal is to bound the probability that $S_0$ is too large.
Fix a bucket $V_j$ and let $X \sim \textrm{Bin}\rbrac{rm, \frac{1}{n}}$ be the number of points of $P$ in $V_j$. Then the probability that $V_j$ is in $S_1$ satisfies \mults{ \mathbb{P}[V_j \in S_1] = \mathbb{P}\sbrac{X \le d-2d^{1/2}\log d} = \mathbb{P}\sbrac{X -\frac{rm}{n} \le -d^{1/2}\log d } \\ \le \exp\sbrac{- \frac{d \log^2 d}{3(d-d^{1/2}\log d)}}= \exp\sbrac{-\Omega\rbrac{ \log^2 d}}, }
where for our inequality we have used the Chernoff bound (Lemma \ref{Chernoff}). Therefore $\mathbb{E}[|S_0|] \le \exp\sbrac{-\Omega\rbrac{ \log^2 d}} n$. Now we argue that $|S_1|$ is concentrated using McDiarmid's inequality (Lemma \ref{Mcdiarmid}). For our application we let $X=|S_1|$ which is a function (say $f$) of the vector $(Z_1, \ldots Z_{rm})$ where $Z_i$ tells us which bucket the $i^{th}$ point of $P$ went into. Moving a point from one bucket to another can only change $|S_1|$ by at most 1 so we use $c=1$. Thus we get the bound \begin{equation}\label{Az}
\mathbb{P}(|X-\mathbb{E}[X]| \ge n^{2/3} ) \le 2\exp\rbrac{-\frac{ n^{4/3}}{2 rm}} = o(1). \end{equation}
Now we handle $S_2$. For $1 \le j \le n$ let $Y_j$ be the number of parts $R_k \in \mc{R}$ such that $R_k$ contains a point in the bucket $V_j$ as well as a point in some bucket $V_{j'}$ where $|V_{j'}| > d$. Note that if $V_j \in S_2$ then $Y_j \ge d^{1/2} \log d.$ We view $R_k$ as a set of $r$ points, say $\{q_1, \ldots, q_r\}$ each going into a uniform random bucket. Say $q_i$ goes to bucket $V_{j_i}$. The probability that $R_k$ is counted by $Y_j$ is at most \begin{align*}
& r\mathbb{P}[j_1=j \mbox{ and } |V_{j_1}|>d ] + r(r-1) \mathbb{P}[j_1=j \mbox{ and } |V_{j_2}|>d ] \\
& = \frac{r}{n} \mathbb{P}[|V_{j_1}|>d \big| j_1=j]+ \frac{r(r-1)}{n} \mathbb{P}[|V_{j_2}|>d \big| j_1=j]\\
& \le \frac{r^2}{n} \mathbb{P}[|V_{j_1}|>d \big| j_1=j]\\
& \le \frac{r^2}{n} \mathbb{P}[\textrm{Bin}(rm-1, 1/n) \ge d] = \frac{r^2}{n}\exp\sbrac{-\Omega\rbrac{ \log^2 d}}.\\ \end{align*} Thus we have \[ \mathbb{E}[Y_j] = m \cdot \frac{r^2}{n}\exp\sbrac{-\Omega\rbrac{ \log^2 d}} \le rd\exp\sbrac{-\Omega\rbrac{ \log^2 d}} = rd^{1/2}\exp\sbrac{-\Omega\rbrac{ \log^2 d}} \] and so Markov's inequality gives us \[ \mathbb{P}\sbrac{Y_j \ge d^{1/2} \log d} \le \frac{rd\exp\sbrac{-\Omega\rbrac{ \log^2 d}}}{d^{1/2} \log d} = \exp\sbrac{-\Omega\rbrac{ \log^2 d}} \]
and so $\mathbb{E}[|S_2|] = n \exp\sbrac{-\Omega\rbrac{ \log^2 d}}.$ We use McDiarmid's inequality once more, this time with $X=|S_2|$. A change in choice of bucket changes $|S_2|$ by at most one and so \eqref{Az} continues to hold. Thus \[
|S_0|=n \exp\sbrac{-\Omega\rbrac{ \log^2 d}}.\quad \text{w.h.p.} \] \subsubsection{A property of independent subsets of $\mc{H}_r(n,m)$}\label{322} Fix $1 \le j \le r-1$. Set \[ a:= \brac{1+\frac \e2}\rbrac{\frac{r \log d}{(r-1)d}}^{\frac{1}{r-1}}, \qquad \k_j:= \frac {10d}r \binom rj a^j,\qquad p:=\frac{d(r-1)!}{n^{r-1}}. \] The expected number of independent sets $A$ in $\mc{H}_r(n, p)$ of size at most $an$ such that there are $\k_j n$ edges each having $j$ vertices in $A$ is at most \begin{align*}
& \sum_{s=1}^{an}\binom{n}{s}(1-p)^{\binom{s}{r}} \binom{\binom{s}{j}\binom{n}{r-j}}{\k_j n }p^{\k_j n}\\
& \le\sum_{s=1}^{an} \exp\cbrac{ s \log \rbrac{\frac{en}{s}} - \binom sr p + \k_j n \log \rbrac{\frac{e \frac{(an)^j}{j!} \frac{n^{r-j}}{(r-j)!}p}{\k_j n}}}\\ &=\sum_{s=1}^{an} \exp\cbrac{ s \log \rbrac{\frac{en}{s}} - \binom sr p + \k_j n\log\bfrac{ea^j}{10}}\\
& \le an \cdot \exp\cbrac{ \sbrac{ \log \rbrac{\frac ea} - \frac {10d}r \binom rj a^{j-1} \log \rbrac{\frac{10}{e}} }an }\\
&=o(1/n) \end{align*} where the last line follows since as $d \rightarrow \infty$ we have \[ \log \rbrac{\frac ea} \sim \frac{1}{r-1} \log d \] and \[
\frac {10d}r \binom rj a^{j-1} \log \rbrac{\frac{10}{e}} = \Omega\rbrac{d^{\frac{r-j}{j-1}} \log^{-\frac{j-1}{r-1}} d} \gg \log d. \] Thus with probability $1-o(1/n)$, $\mc{H}_r(n, p)$ has a coloring using $(1+\varepsilon} \def\f{\phi} \def\F{{\Phi}} \def\vp{\varphi} \def\g{\gamma/2)\rbrac{\frac{(r-1)d}{r \log d}}^{\frac{1}{r-1}}$ colors such that for each color class $A$ and for each $1 \le j \le r-1$ there are at most $\k_j n$ edges with $j$ vertices in $A$. The hypergraph $\mc{H}_r(n, m)$, $m=\binom{n}rp$ will have this property w.h.p.. \subsubsection{Transforming $\mc{H}_{\mc{R}'}$ into $\mc{H}_r(n,d)$} Now we will complete the transformation to the random regular hypergraph $\mc{H}_r(n, d)$. We are open to the possibility that doing so will render our coloring no longer proper, since this process will involve changing some edges which might then be contained in a color class. We will keep track of how many such ``bad" edges there are and then repair our coloring at the end.
We have to add at most $(3d^{1/2} \log d+ d \exp\sbrac{-\Omega\rbrac{ \log^2 d}})n < (4d^{1/2} \log d)n$ points, which takes at most as many steps. For each color class $A$ of $\mc{H}_{\mc{R}'}$ define $X_{A, j}=X_{A, j}(i)$ to be the number of edges with $j$ vertices in $A$ at step $i$. We have already established that $X_{A, j}( 0) \le \k_j n$. This follows from Section \ref{322} and the fact that we have removed edges from $\mc{H}(n,m)$ to obtain $\mc{H}_{\mc{R}'}$. Let $\mc{E}_i$ be the event that at step $i$ we have that for each color class $A$ and for each $1 \le j \le r-1$ we have $X_{A, j}( i) \le 2\k_j n$. Then, assuming $\mc{E}_i$ holds, the probability that $X_{A, j}$ increases at step $i$ is at most \[
\sum_{\substack{1\leq k \leq r,\;\; j_\ell \ge 1 \\ j_1 + \cdots + j_k = j}} \prod_{1\le \ell \le k} \frac{2\k_{j_\ell} n}{nd/r} = \sum_{\substack{1\leq k \leq r,\;\; j_\ell \ge 1 \\ j_1 + \cdots + j_k = j}} \prod_{1\le \ell \le k} 20 \binom r{j_k} a^{j_k} \le \sum_{\substack{1\leq k \leq r,\;\; j_\ell \ge 1 \\ j_1 + \cdots + j_k = j}} 20^r 2^{r^2} a^j \le 40^r 2^{r^2} a^j. \] Also, the largest possible increase in $X_{A, j}$ in one step is $r$. Thus, the final value of $X_{A, j}$ after at most $(4d^{1/2} \log d)n$ steps is stochastically dominated by $\k_j n + r Y$ where $Y \sim \textrm{Bin}\big((4d^{1/2} \log d)n, 40^r 2^{r^2} a^j \big)$. An easy application of the Chernoff bound tells us \begin{equation}\label{eqn:probbound}
\mathbb{P}\rbrac{Y > 2\mathbb{E}[Y]} \le \exp(-\Omega(n)). \end{equation} Note that here \[ \frac{2\mathbb{E}[Y]}{\k_j n} = \frac{8d^{1/2} \log d \cdot 40^r 2^{r^2} a^j n }{10d \binom rj a^j n/r} = O(d^{-1/2} \log d) < 1 \] for sufficiently large $d$. Thus, using \eqref{eqn:probbound} and the union bound over all color classes $A$, we have w.h.p.~the final value of $X_{A, j}$ is at most $\k_j n + 2\mathbb{E}[Y] \le 2 \k_j n$ for all $1 \le j \le r-1$.
Now we address ``bad'' edges, i.e. edges contained in a color class. Assuming $\mc{E}_i$ holds, the expected number of new edges contained in any color class at step $i$ is at most $ r (40)^r 2^{r^2+2r} a^r = O\rbrac{\rbrac{\frac{\log d}{d}}^\frac{r}{r-1}}$ (because it would have to be one of the colors of one of the vertices we are adding points to). Thus the expected number of bad edges created in $(4d^{1/2} \log d)n$ steps is stochastically dominated by $Z \sim r\cdot \textrm{Bin}\big((4d^{1/2} \log d)n, O\rbrac{\rbrac{\frac{\log d}{d}}^\frac{r}{r-1}} \big)$. Another easy application of Chernoff shows that w.h.p.~$Z \le 2\mathbb{E}[Z] =O(d^{-1/2} n)$.
We repair the coloring as follows. First we uncolor one vertex from each bad edge, and let the set of uncolored vertices be $U$ where $|U|=u = O\rbrac{d^{-1/2} n}$. Let \[ \d := \frac \e2 \rbrac{\frac{(r-1)d}{r \log d}}^{\frac{1}{r-1}}. \]
We claim that for every $S \subseteq U,|S|=s$, the hypergraph induced on $S$ has at most $\d s / r$ edges. This will complete our proof since it implies that the minimum degree is at most $\d$ and so $U$ can be recolored using a fresh set of $\d$ colors, yielding a coloring of $\mc{H}_r(n, d)$ using at most \[ \chi(\mc{H}_r(n, M)) + \d \le \rbrac{1+\frac \e2}\rbrac{\frac{(r-1)d}{r \log d}}^{\frac{1}{r-1}} + \frac \varepsilon} \def\f{\phi} \def\F{{\Phi}} \def\vp{\varphi} \def\g{\gamma 2\rbrac{\frac{(r-1)d}{r \log d}}^{\frac{1}{r-1}} = \rbrac{1+ \varepsilon} \def\f{\phi} \def\F{{\Phi}} \def\vp{\varphi} \def\g{\gamma}\rbrac{\frac{(r-1)d}{r \log d}}^{\frac{1}{r-1}} \] colors.
The expected number of sets $S$ with more than $\d s / r$ edges is at most \begin{align}
& \sum_{1 \le s \le u} \binom{n}{s} \binom{\binom{ds}{r}}{\d s / r} \frac{1}{\binom{dn}{r} \binom{dn-r}{r} \ldots \binom{dn-\d s+r}{r}}\nonumber\\
& \le \sum_{1 \le s \le u} \rbrac{\frac{ne}{s}}^s \rbrac{\frac{ (dse/r)^r e}{\d s / r}}^{\d s / r} \frac{(r!)^{\d s / r}}{(dn-\d s)^{\d s}}\nonumber\\
& \le \sum_{1 \le s \le u} \sbrac{\frac{ne}{s} \rbrac{\frac{dse}{(dn-\d s)r}}^\d \rbrac{\frac{er \cdot r!}{\d s}}^{\d/r}}^s.\label{eqn:S1} \end{align} Now for $1 \le s \le \sqrt{n}$ the term in \eqref{eqn:S1} is at most \[ \sbrac{O(n) \cdot \rbrac{O(n^{-1/2})}^\d \cdot O(1)}^s = o(1/n) \] since $\d$ can be made arbitrarily large by choosing $d$ large. Meanwhile for $\sqrt{n} \le s \le u$ we have that the term in \eqref{eqn:S1} is at most \[ \sbrac{O(n^{1/2}) \cdot O(1) \cdot \rbrac{O(n^{-1/2})}^{\d/r}}^s = o(1/n). \] Now since \eqref{eqn:S1} has $O(n)$ terms the whole sum is $o(1)$ and we are done. This completes the proof of Theorem \ref{thm:main}. \section{Summary} We have asymptotically computed the chromatic number of random $r$-uniform, $d$-regular hypergraphs when proper colorings mean that no edge is mono-chromatic. It would seem likely that the approach we took would extend to other definitions of proper coloring. We have not attempted to use second moment calculations to further narrow our estimates. These would seem to be two natural lines of further research.
\end{document}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Minimal disturbance measurement for coherent states is non-Gaussian} \author{Ladislav Mi\v{s}ta Jr.} \affiliation{Department of Optics, Palack\' y University, 17. listopadu 50, 772~07 Olomouc, Czech Republic} \date{\today}
\begin{abstract} In standard coherent state teleportation with shared two-mode squeezed vacuum (TMSV) state there is a trade-off between the teleportation fidelity and the fidelity of estimation of the teleported state from results of the Bell measurement. Within the class of Gaussian operations this trade-off is optimal, i.e. there is not a Gaussian operation which would give for a given output fidelity a larger estimation fidelity. We show that this trade-off can be improved by up to $2.77\%$ if we use a suitable non-Gaussian operation. This operation can be implemented by the standard teleportation protocol in which the shared TMSV state is replaced with a suitable non-Gaussian entangled state. We also demonstrate that this operation can be used to enhance the transmission fidelity of a certain noisy channel. \end{abstract} \pacs{03.67.-a}
\maketitle
\section{Introduction} In quantum mechanics there is not an operation which would give some information on an unknown quantum state without disturbing the state. This property of quantum mechanics is closely related to the no-cloning theorem \cite{Wootters_82} which forbids to perfectly duplicate an unknown quantum state. A question that can be risen in this context is which operation allowed by quantum mechanics approximates best this non-existing operation, i.e. which operation introduces for a given information gain the least possible disturbance. Naturally, this operation, conventionally denoted as minimal disturbance measurement (MDM), will in general depend on the set of input states, their a-priori distribution, and on the figures of merit used to quantify the information gain and the state disturbance. A convenient approach to the problem on finding the MDM was developed in \cite{Banaszek_01a}. In this approach the classical information gained on the input state from a quantum operation is converted into the estimate of the input state and the information gain is then quantified by the average estimation fidelity $\bar{G}$, i.e. the fidelity $G$ between the estimate and the input state averaged over the distribution of the input states. On the other hand, the disturbance introduced by the operation into the input state is quantified by the average output fidelity $\bar{F}$, i.e. the fidelity $F$ between the input state and the state at the output of the operation averaged over the distribution of the input states. According to the laws of quantum mechanics for a given set of input states there exists a specific optimal trade-off between the fidelities $\bar{G}$ and $\bar{F}$ which cannot be overcome by any quantum operation. In terms of the fidelities $\bar{G}$ and $\bar{F}$ the MDM then can be defined as a quantum operation which saturates this optimal trade-off. First optimal fidelity trade-offs and the corresponding MDMs were derived in the context of finite-dimensional quantum systems and observables with discrete spectra. To be more specific, the MDMs were found analytically for a completely unknown \cite{Banaszek_01a} and partially known \cite{Mista_05} $d$-level particle and numerically for $N$ identical copies of a completely unknown $2$-level particle (qubit) \cite{Banaszek_01b}. In addition, the MDMs for a completely unknown as well as partially known qubit were demonstrated experimentally using a single-photon polarization qubit \cite{Sciarrino_05}. Besides being of fundamental interest MDM can be applied to increase the transmission fidelity of a certain lossy channel \cite{Ricci_05}.
Only recently the concept of MDM was also extended into the realm of systems with infinite-dimensional Hilbert spaces and observables with continuous spectrum-continuous variables (CVs). The attention has been paid to MDMs on Gaussian states, i.e. states represented by Gaussian Wigner function, realized by covariant Gaussian operations, i.e. operations preserving Gaussian states which are invariant under displacement transformations. These operations are advantageous since for coherent input states they posses state-independent output fidelity $F$ and estimation fidelity $G$ which can be conveniently used for characterization of state disturbance and information gain. Within the class of such operations optimal trade-off between the two fidelities as well as the corresponding MDM for the set of all coherent states with uniform a-priori distribution were derived in \cite{Andersen_05,Fiurasek_05} and realized experimentally in \cite{Andersen_05}. In addition, also this covariant Gaussian MDM was shown to be capable to increase transmission fidelity of some noisy channels \cite{Andersen_05}.
In this paper we address a natural question of whether the covariant Gaussian MDM for uniformly distributed coherent states can be improved. We answer this question in the affirmative. We show that the fidelity trade-off corresponding to this measurement can be increased by up to $2.77\%$ if we use a suitable non-Gaussian operation thus showing that MDM for coherent states is non-Gaussian. This MDM can be implemented using the standard continuous-variable (CV) teleportation protocol \cite{Vaidman_94,Braunstein_98} in which the participants share an appropriate non-Gaussian entangled state. Further, we demonstrate that our non-Gaussian MDM gives a higher transmission fidelity of a certain noisy channel in comparison with that achieved in \cite{Andersen_05}. As a by-product we also derive for the set of all coherent states with uniform a-priori distribution a lower bound for the optimal fidelity trade-off for any covariant quantum operation. The present paper is inspired by the recent result that fidelity of quantum cloning of coherent states can be increased by using a non-Gaussian entangled state \cite{Cerf_04}.
The paper is organized as follows. Section~\ref{sec_1} deals with optimal Gaussian fidelity trade-off and corresponding MDM for uniformly distributed coherent states. In Section~\ref{sec_2} we derive for this set of states a lower bound on optimal fidelity trade-off for any covariant quantum operation. Section~\ref{sec_3} is dedicated to implementation of a quantum operation saturating this bound and its application. Section~\ref{sec_4} contains conclusions.
\section{Gaussian minimal disturbance measurement}\label{sec_1}
The Gaussian MDM can be realized by at least three ways \cite{Andersen_05} encompassing the asymmetric cloning followed by a joint measurement, a linear-optical scheme with a feed forward or by the standard CV teleportation protocol proposed by Braunstein and Kimble (BK) \cite{Braunstein_98} and demonstrated experimentally in \cite{Furusawa_98}. With respect to what follows it is convenient to start with the implementation of the Gaussian MDM via BK teleportation protocol. Here we use an optical notation in which CV systems are realized by single modes of an optical field and the role of CVs is played by quadratures $x_{j}$ and $p_{k}$ ($[x_{j},p_{k}]=i\delta_{jk}$) of these modes.
In the BK protocol an unknown coherent state $|\alpha\rangle_{\rm in}$ of a mode ``in'' is teleported by sender Alice ($A$) to receiver Bob ($B$). In each run of the protocol, the state is chosen randomly with uniform distribution from the set of all coherent states. Initially, Alice and Bob share a two-mode squeezed vacuum (TMSV) state which is a Gaussian entangled state of two optical modes $A$ and $B$ described in the Fock basis as
\begin{equation}\label{TMSV}
|TMSV\rangle_{AB}=\sum_{n=0}^{\infty}\tilde c_{n}|n,n\rangle_{AB}, \quad \tilde c_{n}=\sqrt{1-\lambda^2}\lambda^{n}, \end{equation}
where $\lambda=\tanh r$ ($r$ is the squeezing parameter). Then, Alice performs the so called Bell measurement that consists of superimposing of modes ``in'' and $A$ on a balanced beam splitter and subsequent detection of the quadrature variables $x_{1}=(x_{\rm in}-x_{A})/\sqrt{2}$ and $p_{2}=(p_{\rm in}+p_{A})/\sqrt{2}$ at its outputs. She obtains classical results of the measurement $\bar x_{1}$ and $\bar p_{2}$ and sends them via classical channel to Bob who displaces his part of the shared state as $x_{B}\rightarrow x_{\rm out}=x_{B}+\sqrt{2}\bar x_{1}$ and $p_{B}\rightarrow p_{\rm out}=p_{B}+\sqrt{2}\bar p_{2}$. As a result, Bob's output quadratures read as $x_{\rm out}=x_{\rm in}-\sqrt{2}e^{-r}x_{B}^{(0)}$ and $p_{\rm out}=p_{\rm in}+\sqrt{2}e^{-r}p_{A}^{(0)}$ \cite{Loock_00a}, where $x_{i}^{(0)}$ and $p_{j}^{(0)}$ stand for initial vacuum quadratures. Since the squeezing $r$ is always finite in practice Bob has only an approximate replica $\rho_{\rm out}(\alpha)$ of the input state. Moreover, for the same reason Alice gains some information on the input state from results of the Bell measurement that can be converted into a classical estimate $\rho_{\rm est}(\alpha)$ of the input state by displacing a vacuum mode $E$ as $x_{E}^{(0)}\rightarrow x_{\rm est}=x_{E}^{(0)}+\sqrt{2}\bar x_{\rm 1}$ and $p_{E}^{(0)}\rightarrow p_{\rm est}=p_{E}^{(0)}+\sqrt{2}\bar p_{\rm 2}$. Hence she obtains $x_{\rm est}=x_{\rm in}+x_{E}^{(0)}-(e^{r}x_{A}^{(0)}+e^{-r}x_{B}^{(0)})/\sqrt{2}$ and $p_{\rm est}=p_{\rm in}+p_{E}^{(0)}+(e^{-r}p_{A}^{(0)}+e^{r}p_{B}^{(0)})/\sqrt{2}$. Quantifying now the resemblance of the states $\rho_{\rm out}(\alpha)$ and $\rho_{\rm est}(\alpha)$
to the input state $|\alpha\rangle$ by the output fidelity $F$ and the estimation fidelity $G$
\begin{equation}\label{fidelities}
F=\langle\alpha|\rho_{\rm out}(\alpha)|\alpha\rangle,\quad
G=\langle\alpha|\rho_{\rm est}(\alpha)|\alpha\rangle, \end{equation}
one finds using the latter formulas that
\begin{equation}\label{FGBK} F_{\rm BK}=\frac{1}{1+e^{-2r}},\quad G_{\rm BK}=\frac{1}{1+\cosh^{2}(r)}. \end{equation}
Expressing $\cosh^{2}(r)$ in terms of $F_{\rm BK}$ using the first formula and inserting this into the formula for $G_{\rm BK}$ we finally arrive at the following trade-off between the fidelities $F_{\rm BK}$ and $G_{\rm BK}$:
\begin{equation}\label{trade-off} G_{\rm BK}=\frac{1}{1+\frac{1}{4F_{\rm BK}(1-F_{\rm BK})}}. \end{equation}
The obtained trade-off is depicted by dashed curve in Fig.~\ref{fig1}. It is obvious from the figure that as one would expect the obtained fidelities exhibit complementary behavior, i.e. the larger is the estimation fidelity the smaller is the output fidelity and vice versa. Interestingly, it was shown in \cite{Andersen_05,Fiurasek_05} that if one restricts only to the covariant Gaussian operations, then the trade-off (\ref{trade-off}) is optimal. It means in other words that the standard BK teleportation protocol with shared TMSV state realizes (within the class of all covariant Gaussian operations) MDM for coherent states. In the following sections we demonstrate that the optimal Gaussian trade-off (\ref{trade-off}) can be improved by a suitable covariant non-Gaussian operation.
\begin{figure}
\caption{Trade-off between the otput fidelity $F$ and the estimation fidelity $G$ for the BK teleportation scheme with the shared optimized non-Gaussian state (solid curve), TMSV state (\ref{TMSV}) (dashed curve) and TMSV state de-gaussificated by local single photon subtraction from each mode (\ref{subtracted}) (dotted-dashed curve). See text for details.}
\label{fig1}
\end{figure}
\section{Optimal fidelity trade-off for coherent states}\label{sec_2}
We start by a suitable mathematical formulation of the task on finding the optimal fidelity trade-off for CVs. For this purpose we use a general method developed in \cite{Mista_05}. We restrict our attention to
coherent input states $|\alpha\rangle=D(\alpha)|0\rangle$, where $\alpha$ lies in the complex plane $\mathbb{C}$, which form the orbit of the Weyl-Heisenberg group.
Here $|0\rangle$ is the vacuum state and the displacement operators $D(\alpha)=\mbox{exp}(\alpha a^{\dag}-\alpha^{\ast}a)$, $\alpha\in\mathbb{C}$, where $a$ and $a^{\dag}$ are the standard annihilation and creation operators satisfying boson commutation rule $[a,a^{\dag}]=\openone$, comprise the irreducible unitary representation of this group. We also assume that the a-priori distribution of the input states coincides with the invariant measure on the group $d^{2}\alpha/\pi=d(\mbox{Re}\alpha)d(\mbox{Im}\alpha)/\pi$.
The standard BK teleportation protocol can be formally viewed as a trace-preserving quantum operation which is covariant, i.e. invariant under displacement transformations, and which can give as a measurement outcome any complex number $\beta$ ($\beta\equiv\bar x_{1}+i\bar p_{2}$ in the case of teleportation). Therefore, we will seek for the optimal fidelity trade-off on this set of quantum operations. To each outcome $\beta$ of such an operation we can assign a trace-decreasing completely positive (CP) map that can be represented by the following positive-semidefinite operator on the tensor product ${\mathcal H}_{\rm in}\otimes{\mathcal H}_{\rm out}$ of the input and output Hilbert spaces $\mathcal{H}_{\rm in}$ and $\mathcal{H}_{\rm out}$ \cite{Jamiolkowski_72}
\begin{equation}\label{covariant} \chi(\beta)=[D_{\rm in}(\beta^{\ast}) \otimes D_{\rm out}(\beta)] \chi_{0} [D_{\rm in}^{\dag}(\beta^{\ast}) \otimes D_{\rm out}^{\dag}(\beta)], \end{equation}
where $\chi_{0}$ is a positive-semidefinite operator. For the measurement outcome
$\beta$ the estimated state is $|\beta\rangle$ whereas the output state reads
\begin{equation}\label{out}
\rho(\beta|\alpha)=\mathrm{Tr}_{\rm in}[\chi(\beta)
(|\alpha\rangle_{\rm in}\langle\alpha|)^{\rm T}\otimes \openone_{\rm out}], \label{rhoout} \end{equation}
where $\openone_{\rm out}$ is the identity operator on $\mathcal{H}_{\rm out}$. Since the map $\chi(\beta)$ is trace-decreasing the output state (\ref{rhoout}) is not normalized to unity
and its norm $P(\beta|\alpha)=\mathrm{Tr}_{\rm out}[\rho(\beta|\alpha)]$ is equal to the probability density of this outcome. The entire operation should be trace-preserving which imposes the following constraint:
\begin{equation}\label{constraint} \frac{1}{\pi}\int_{\mathbb{C}}\mathrm{Tr}_{\rm out}[\chi(\beta)]d^{2}\beta =\openone_{\rm in}, \end{equation}
where $\int_{\mathbb{C}}$ denotes integration over the whole complex plane
and $\openone_{\rm in}$ is the identity operator on $\mathcal{H}_{\rm in}$. For the input state $|\alpha\rangle$ the studied operation produces on average the output state and the estimated state in the form
\begin{eqnarray}\label{rho}
\rho_{\rm out}(\alpha)&=&\frac{1}{\pi}\int_{\mathbb{C}}\rho(\beta|\alpha)d^{2}\beta,\nonumber\\
\rho_{\rm est}(\alpha)&=&\frac{1}{\pi}\int_{\mathbb{C}} P(\beta|\alpha)|\beta\rangle\langle\beta|d^{2}\beta. \end{eqnarray}
Making use of Eqs.~(\ref{covariant}), (\ref{out}), (\ref{rho}) and the formula
$D(-\beta)|\alpha\rangle=\mbox{exp}[i\mbox{Im}(\alpha\beta^{\ast})]|\alpha-\beta\rangle$ one finally finds the fidelities (\ref{fidelities}) to be
\begin{equation} F= \mathrm{Tr}[\chi_{0}R_F], \qquad G= \mathrm{Tr}[\chi_{0}R_G], \label{FGchi} \end{equation}
where $R_{F}$ and $R_{G}$ are the positive-semidefinite Gaussian operators defined as
\begin{eqnarray}\label{R}
R_F&=&\frac{1}{\pi}\int_{\mathbb{C}}|\gamma^{\ast}\rangle_{\rm in}\langle\gamma^{\ast}|\otimes
|\gamma\rangle_{\rm out}\langle\gamma|d^{2}\gamma, \nonumber\\
R_G&=&\frac{1}{\pi}\int_{\mathbb{C}}e^{-|\gamma|^{2}}|\gamma\rangle_{\rm in}\langle\gamma| d^{2}\gamma\otimes\openone_{\rm out}. \end{eqnarray}
The optimal trade-off between the fidelities $F$ and $G$ can be found by finding the maximum of the weighted sum
\begin{equation}\label{transmission} \mathcal{F}(p)=pF+(1-p)G \end{equation}
of these two fidelities \cite{Fiurasek_03}, where the parameter $p\in[0,1]$ controls the ratio between the information gained from the input state and the disturbance of this state. We can write $\mathcal{F}(p)=\mathrm{Tr}[\chi_0 R(p)]$, where
\begin{equation}\label{Rp} R(p)=p R_F+(1-p)R_G. \end{equation}
Making use of the inequality $R(p)\leq\lambda_{\rm max}(p)(\openone_{\rm in}\otimes\openone_{\rm out})$, where $\lambda_{\rm max}(p)$ is the maximum eigenvalue of $R(p)$ and taking into account the condition $\mathrm{Tr}[\chi_{0}]=1$ which we obtain from the constraint (\ref{constraint}) using the formula \cite{D'Ariano_04}
\begin{equation}\label{Schur} \frac{1}{\pi}\int_{\mathbb{C}}D(\alpha)XD^{\dag}(\alpha)d^{2}\alpha=\mathrm{Tr}[X]\openone \end{equation}
following from Schur's lemma, one finds $\mathcal F(p)$ to be upper bounded as $\mathcal{F}(p)\leq\lambda_{\rm max}(p)$. Now, if we find a normalized eigenvector
$|\chi_{\rm max}(p)\rangle$ of the operator $R(p)$ corresponding to the maximum eigenvalue $\lambda_{\rm max}(p)$, then the map (\ref{covariant}) generated from
any such $\chi_{0,{\rm max}}=|\chi_{\rm max}(p)\rangle\langle\chi_{\rm max}(p)|$ satisfies the trace-preservation condition (\ref{constraint}) as follows from Eq.~(\ref{Schur}). Consequently, the map $\chi_{0,{\rm max}}$ is the optimal one that saturates optimal trade-off between $F$ and $G$.
The finding of the optimal fidelity trade-off thus boils down to the diagonalization of the operator $R(p)$ which acts on the direct product of two infinite-dimensional spaces. For this purpose it is convenient to express the operators $R_F$ and $R_G$ in the form:
\begin{eqnarray}\label{explicitR}
R_F&=&\sum_{K=0}^{\infty}\frac{{K!}}{2^{K+1}}\sum_{n,m=0}^{K}\frac{|n\rangle_{\rm in}\langle K-m|
\otimes|m\rangle_{\rm out}\langle K-n|}{\sqrt{{n!}{(K-m)!}{m!}{(K-n)!}}}, \nonumber \\
R_G&=&\sum_{n=0}^{\infty}\frac{1}{2^{n+1}}|n\rangle_{\rm in}\langle n|\otimes\openone_{\rm out}, \end{eqnarray}
which can be calculated from Eq.~(\ref{R}) using the formulas
$|\gamma\rangle=e^{-|\gamma|^{2}/2}\sum_{n=0}^{\infty}(\gamma^{n}/\sqrt{n!})|n\rangle$
and $\int_{\mathbb{C}}e^{-s|\gamma|^{2}}\gamma^{n}{\gamma^{\ast}}^{m}d^{2}\gamma= (\pi{n!}/s^{n+1})\delta_{nm}$ (Re$s>0$). The present eigenvalue problem can be simplified, if we notice that both the operators (\ref{explicitR}) and therefore also the operator $R(p)$ commute with the operator of the photon number difference $N_{-}=n_{\rm in}-n_{\rm out}$, where $n_{i}=a_{i}^{\dag}a_{i}$, $i={\rm in},{\rm out}$. Consequently, the total Hilbert space splits into the direct sum $\displaystyle {\mathcal H}_{\rm in}\otimes{\mathcal H}_{\rm out}= \mathop{\oplus}_{N=-\infty}^{+\infty}{\mathcal H}^{(N)}$ of the characteristic subspaces ${\mathcal H}^{(N)}$ of the operator $N_{-}$ corresponding to the eigenvalues $N=-\infty,\ldots,+\infty$. The infinite-dimensional subspaces ${\mathcal H}^{(+L)}$ (${\mathcal H}^{(-L)}$), $L=0,1,\ldots$ are spanned by the basis vectors
$\{|n+L,n\rangle_{\rm in,out}, n=0,1,\ldots\}$ ($\{|n,n+L\rangle_{\rm in,out}, n=0,1,\ldots\}$). Hence, it remains to diagonalize the operator $R(p)$ in the subspaces ${\mathcal H}^{(\pm L)}$ where it is represented by infinite-dimensional matrices $R^{(\pm L)}(p)=pR_{F}^{(\pm L)}+(1-p)R_{G}^{(\pm L)}$, where
\begin{eqnarray}\label{RL} \left(R_F^{(\pm L)}\right)_{nm}&=&\frac{\sqrt{\left({n+m+L \atop n}\right)\left({n+m+L \atop m}\right)}} {2^{L+n+m+1}}, \nonumber \\ \left(R_G^{(+L)}\right)_{nm}&=&\frac{\delta_{nm}}{2^{n+L+1}},\quad \left(R_G^{(-L)}\right)_{nm}= \frac{\delta_{nm}}{2^{n+1}},\nonumber\\ \end{eqnarray}
where $n,m=0,1,\ldots$.
We have accomplished this task numerically and the obtained fidelity trade-off is depicted by the solid curve in Fig.~\ref{fig1}. The figure clearly demonstrates that this trade-off beats the optimal Gaussian trade-off (\ref{trade-off}) (dashed curve). In order to see the degree of improvement better we have plotted by a solid curve in Fig.~\ref{fig2} the dependence of difference $\Delta G=G-G_{\rm BK}$ between the estimation fidelity $G$ in the improved trade-off and the estimation fidelity $G_{\rm BK}$ in the optimal Gaussian trade-off on the output fidelity $F$. Numerical analysis reveals that, for instance, $\Delta G\approx 1.04\%$ is achieved for $F\approx 0.794$ and the maximum improvement of $\Delta G\approx 2.77\%$ is attained for $F=F_{\rm max}\approx 0.963$.
It should be stressed that we have in fact found a lower bound on the optimal trade-off because we have approximated each original infinite-dimensional matrix $R^{(\pm L)}(p)$ by its $\mathcal{N}$-dimensional submatrix $R_{\mathcal{N}}^{(+L)}(p)$ ($R_{\mathcal{N}}^{(-L)}(p)$) on the $\mathcal{N}$-dimensional subspace ${\mathcal H}_{\mathcal{N}}^{(+L)}$ (${\mathcal H}_{\mathcal{N}}^{(-L)}$) spanned by the basis vectors
$\{|n+L,n\rangle_{\rm in,out}, n=0,1,\ldots,\mathcal{N}-1\}$
($\{|n,n+L\rangle_{\rm in,out}, n=0,1,\ldots,\mathcal{N}-1\}$). This follows from the inequality
\begin{equation}\label{inequality}
\langle\psi_{\mathcal{N}}|R_{\mathcal{N}}^{(\pm L)}(p)|\psi_{\mathcal{N}}\rangle
=\langle\psi_{\mathcal{N}}|R(p)|\psi_{\mathcal{N}}\rangle\leq\lambda_{\rm max}(p), \end{equation}
which holds for any $|\psi_{\mathcal{N}}\rangle\in{\mathcal H}_{\mathcal{N}}^{(\pm L)}$. Therefore, because we only calculated the eigenvector corresponding to maximum eigenvalue of matrices $R_{\mathcal{N}}^{(\pm L)}(p)$ (here we took $\mathcal{N}=500$ and $L\leq 30$), the optimal trade-off can be slightly larger than that given by the solid curve in Fig.~\ref{fig1}.
\begin{figure}
\caption{Difference $\Delta G=G-G_{\rm BK}$ between the estimation fidelity $G$ in the improved fidelity trade-off and the estimation fidelity $G_{\rm BK}$ in the optimal Gaussian trade-off versus the output fidelity $F$ for $\mathcal{N}=4$ (dotted curve), $\mathcal{N}=12$ (dashed curve), $\mathcal{N}=50$ (dotted-dashed curve), $\mathcal{N}=500$ (solid curve). See text for details.}
\label{fig2}
\end{figure}
\section{Implementation}\label{sec_3}
The above analysis shows that the optimal eigenvector $|\chi_{\rm max}(p)\rangle$ lies in one of the subspaces ${\mathcal H}^{(\pm L)}$. We have a strong numerical evidence that it lies in the subspace corresponding to $L=0$, i.e. it has the structure
\begin{equation}\label{chimax}
|\chi_{\rm max}(p)\rangle=\sum_{n=0}^{\infty}c_{n}|n,n\rangle_{AB}, \end{equation}
and, in addition, the probability amplitudes $c_{n}$ comprising the dominant eigenvector of the matrix $R^{(0)}(p)$ are nonnegative. The latter statement follows immediately from positivity of elements of the matrix $R(p)$. The optimal CP map (\ref{covariant}) corresponding to the vector (\ref{chimax}) can be implemented by the BK teleportation scheme in which the TMSV state (\ref{TMSV}) is replaced by this vector. This can be shown if we describe the BK teleportation by the transfer operator method \cite{Hofmann_01}. In this formalism, the action of the BK teleporter
with shared entangled state $\sum_{n=0}^{\infty}c_{n}|n,n\rangle_{AB}$ is described by the set of
transfer operators $\{T(\beta)=D(\beta)T(0)D^{\dag}(\beta)\}_{\beta\in\mathbb{C}}$, where $T(0)=\sum_{n=0}^{\infty}c_{n}|n\rangle\langle n|$. If the input state is $|\alpha\rangle$ and the Bell measurement gives the outcome $\beta=\bar x_{1}+i\bar p_{2}$
the output state reads as $|\psi_{\rm out}(\beta|\alpha)\rangle=T(\beta)|\alpha\rangle$. The state $|\chi_{0}\rangle$ forming the positive-semidefinite operator $\chi_{0}$ describing the considered teleportation protocol then can be calculated by acting with the operator $T(0)$ on one part of maximally entangled state
$\sum_{n=0}^{\infty}|n,n\rangle_{AB}$ \cite{Jamiolkowski_72}. This finally gives
the state $|\chi_{0}\rangle=T_{B}(0)\sum_{n=0}^{\infty}|n,n\rangle_{AB}$, which coincides exactly with the state (\ref{chimax}).
Likewise, we can implement the quantum operation which saturates the fidelity trade-off depicted by the solid curve in Fig.~\ref{fig1}. For this purpose, we need to prepare the entangled state
\begin{equation}\label{suboptimal}
\sum_{n=0}^{\mathcal{N}-1}c_{n}|n,n\rangle_{AB}, \end{equation}
where nonnegative probability amplitudes $c_{n}$ form the dominant eigenvector of the matrix $R_{\mathcal{N}}^{(0)}(p)$. Apparently, the improvement $\Delta G$ which can be achieved when using the state (\ref{suboptimal}) will vary with the dimension $\mathcal{N}$ of the truncated space ${\mathcal H}_{\mathcal{N}}^{(0)}$. This dependence is depicted in Fig.~\ref{fig2}. We see from the figure that the maximum improvement increases and moves towards larger values of $F$ as $\mathcal{N}$ grows. It is also seen from the figure that in order to achieve $\Delta G\approx 1\%$ one needs at least $\mathcal{N}\geq 12$. For values of $\mathcal{N}$ where the improvement $\Delta G$ achieves at least a few tenths of percent one can calculate the probability amplitudes $c_{n}$ of the state (\ref{suboptimal}) only numerically. In order to demonstrate the difference between the state (\ref{suboptimal}) and the optimal Gaussian state (\ref{TMSV}) we display in Fig.~{\ref{fig3}} the difference $\Delta c_{n}=c_{n}-\tilde c_{n}$ of the Schmidt coefficient $c_{n}$ of the state (\ref{suboptimal}) with $\mathcal{N}=500$ and the Schmidt coefficient $\tilde c_{n}$ of the TMSV state (\ref{TMSV}) for $F=F_{\rm max}\approx 0.963$ versus the photon number. The state (\ref{suboptimal}) can be prepared, at least in principle, using the probabilistic scheme for preparation of an arbitrary two-mode state with finite Fock state expansion based on linear optics \cite{Kok_02}.
The specific feature of the optimal state (\ref{chimax}) is that it possesses perfect correlations in photon number as the TMSV state (\ref{TMSV}). However, there is a sharp difference between the two states, because in contrast with the latter state the former one is non-Gaussian. To show this assume on the contrary that the state (\ref{chimax}) is a Gaussian state of two modes $A$ and $B$. Such a state is completely characterized by the first moments
$\langle\xi_k\rangle=\langle\chi_{\rm max}(p)|\xi_k|\chi_{\rm max}(p)\rangle$, where $\xi=(x_{A},p_{A},x_{B},p_{B})^{\rm T}$, and by the variance matrix $V$ with elements $V_{kl}=\langle\{\Delta \xi_{k},\Delta\xi_{l}\}\rangle$, where $\Delta\xi_{k}=\xi_{k}-\langle\xi_{k}\rangle$ and $\{A,B\}\equiv(1/2)(AB+BA)$. As $\langle\xi\rangle=0$ for the state (\ref{chimax}), it is completely described just by the variance matrix which reads as
\begin{eqnarray}\label{standard} V=\left(\begin{array}{cccc} a & 0 & c & 0 \\ 0 & a & 0 & -c \\ c & 0 & a & 0 \\ 0 & -c & 0 & a\\ \end{array}\right), \end{eqnarray}
where $a=\sum_{n=0}^{\infty}nc_{n}^{2}+1/2$ and $c=\sum_{n=0}^{\infty}(n+1)c_{n}c_{n+1}\geq0$. Taking into account the purity of the state, which imposes the constraint $\sqrt{\mbox{det}V}=a^{2}-c^{2}=1/4$ \cite{Trifonov_97} we see that such a state would be a TMSV state (\ref{TMSV}) with $\lambda=\sqrt{(a-1/2)/(a+1/2)}$ which does not beat the trade-off of the BK scheme and thus we arrive to a contradiction. Therefore, the state (\ref{chimax}) is inevitably non-Gaussian. Thus we have found a non-Gaussian operation which possesses a better trade-off between output and estimation fidelities than any covariant Gaussian operation which implies that MDM for a completely unknown coherent state is non-Gaussian.
\begin{figure}
\caption{Dependence of the difference $\Delta c_{n}=c_{n}-\tilde c_{n}$ of the Schmidt coefficient $c_{n}$ of the entangled state (\ref{suboptimal}) with $\mathcal{N}=500$ and the Schmidt coefficient $\tilde c_{n}$ of the TMSV state (\ref{TMSV}) for $F=F_{\rm max}\approx 0.963$ on the photon number.}
\label{fig3}
\end{figure}
It can be interesting to compare the fidelity trade-off derived by us with the trade-off that would be obtained when teleporting with the state produced by local single photon subtraction from each mode of a TMSV state. Originally investigated in the context of increase of teleportation fidelity via local operations and classical communication \cite{Opatrny_00,Cochrane_02,Olivares_03} the state was also shown to be suitable for loophole-free Bell test based on homodyne detection \cite{Nha_04,Garcia-Patron_04,Olivares_04,Garcia-Patron_05}. The reason for studying the state here is twofold. First, the state is a non-Gaussian state of the form \cite{Cochrane_02,Garcia-Patron_05}
\begin{equation}\label{subtracted}
\sqrt{\frac{(1-T^{2}\lambda^{2})^{3}}{1+T^{2}\lambda^{2}}}\sum_{n=0}^{\infty}(n+1)(T\lambda)^{n}|n,n\rangle_{AB}, \end{equation}
where $T$ is a transmittance of an unbalanced beam splitter used for photon subtraction and therefore it possesses perfect correlations in photon number as the optimal state (\ref{chimax}). Second, the subtraction of a single photon was already demonstrated experimentally for a single-mode squeezed vacuum state \cite{Wenger_04}. Making use of the formulas
\begin{eqnarray*}
F=\sum_{m,n=0}^{\infty}{m+n \choose n}\frac{c_m^{\ast}c_{n}}{2^{m+n+1}},\quad G=\sum_{n=0}^{\infty}\frac{|c_{n}|^{2}}{2^{n+1}} \end{eqnarray*}
for the teleportation and estimation fidelities in the BK teleportation with the shared state (\ref{chimax}) we can find using Eq.~(\ref{subtracted}) the teleportation fidelity to be \cite{Olivares_03}
\begin{eqnarray*}\label{Fsubtracted} F_{\rm s}=\frac{(1+T\lambda)^{3}(2-2T\lambda+T^{2}\lambda^{2})}{4(1+T^{2}\lambda^{2})}, \end{eqnarray*}
while the estimation fidelity reads as
\begin{eqnarray*}\label{Fsubtracted} G_{\rm s}=2\left(\frac{2+T^{2}\lambda^{2}}{1+T^{2}\lambda^{2}}\right)\left(\frac{1-T^{2}\lambda^{2}}{2-T^{2}\lambda^{2}}\right)^{3}. \end{eqnarray*}
The trade-off between the fidelities $F_{\rm s}$ and $G_{\rm s}$ is depicted by the dotted-dashed curve in Fig.~\ref{fig1}. The figure clearly reveals that the trade-off is even worse than the optimal Gaussian trade-off (\ref{trade-off}). Thus while the single photon subtraction can be a useful method for distillation of the CV entanglement and test of Bell inequalities it is not suitable for preparation of a non-Gaussian entangled state which would improve fidelity trade-off in teleportation of coherent states.
\begin{figure}
\caption{Dependence of the difference $\Delta F=\mathcal{F}-\mathcal{F}_{\rm BK}$ of the transmission fidelity $\mathcal{F}=pF+(1-p)G$ for the non-Gaussian operation and the transmission fidelity $\mathcal{F}_{\rm BK}=pF_{\rm BK}+(1-p)G_{\rm BK}$ maximized with respect to the squeezing parameter $r$ on the transmission probability $p$. See text for details.}
\label{fig4}
\end{figure}
The non-Gaussian operation realized by the BK teleportation protocol with shared non-Gaussian state (\ref{suboptimal}) can be applied to enhance the transmission fidelity of a certain noisy channel. The channel in question transmits perfectly with probability $p$ the input coherent state while with probability $1-p$ the state is completely absorbed by the channel. For the set of all coherent states with uniform a-priori distribution the channel possesses the average transmission fidelity equal to $\mathcal{F}_{\rm av}(p)=p$. In \cite{Andersen_05} it was demonstrated that for $0<p<4/5$ the transmission fidelity can be improved by using the Gaussian MDM in front of the channel while for $p\geq4/5$ it is better to entirely use the channel. Using the BK teleportation protocol to realize the MDM the improved scheme works as follows. Instead of sending directly the input coherent state through the channel one sends through it one part of the TMSV state (\ref{TMSV}). In the next step the other part of thus obtained state is used for teleportation of the input coherent state. The transmission fidelity for this scheme is given by the formula $\mathcal{F}_{\rm BK}(p)=pF_{\rm BK}+(1-p)G_{\rm BK}$, where the fidelities $F_{\rm BK}$ and $G_{\rm BK}$ are given in Eq.~(\ref{FGBK}). By maximizing the transmission fidelity $\mathcal{F}_{\rm BK}(p)$ with respect to the squeezing parameter $r$ we can reach optimal performance of the scheme when in the interval $0<p<4/5$ $\mathcal{F}_{\rm BK}(p)>\mathcal{F}_{\rm av}(p)$ \cite{Andersen_05}. Interestingly, the transmission fidelity of the channel can be further improved provided that we use in the BK teleportation the non-Gaussian entangled state (\ref{suboptimal}) ($\mathcal{N}=500$) as a quantum channel. This scheme must be inevitably optimal since within the class of all covariant operations it is designed in such a way that it maximizes the quantity (\ref{transmission}) which is in fact the transmission fidelity of the considered channel. The dependence of the improvement $\Delta F(p)=\mathcal{F}(p)-\mathcal{F}_{\rm BK}(p)$ on the probability $p$ for our scheme is depicted in Fig.~\ref{fig4}. The figure reveals that for $0<p\lesssim0.85$ the scheme really allows to slightly improve the transmission fidelity the maximum improvement of $\Delta F\approx 0.81\%$ being achieved for $p=0.67$. In the region of $p\gtrsim0.85$ $\Delta F$ attains negative values which is a numerical artefact caused by the truncation of the infinite-dimensional matrix $R^{(0)}(p)$ to the finite-dimensional matrix $R_{\mathcal{N}}^{(0)}(p)$. Therefore, in order to achieve $\Delta F>0$ also for some $p\gtrsim0.85$ we would need to use in teleportation the state (\ref{suboptimal}) with $\mathcal{N}>500$. If this is not possible then for $p\gtrsim0.85$ it is better to send directly the input coherent state through the channel rather than to use our non-Gaussian operation. Thus we have illustrated also practical utility of the studied non-Gaussian operation for increase of the transmission fidelity of a specific quantum channel.
\section{Conclusions}\label{sec_4}
In conclusion, we have shown that there exists a covariant non-Gaussian quantum operation which gives for a completely unknown coherent state a better trade-off between the output fidelity and the estimation fidelity than any covariant Gaussian operation. This means that the covariant MDM for a completely unknown coherent state is non-Gaussian. The non-Gaussian operation can be implemented by the standard BK teleportation protocol with a suitable non-Gaussian entangled state as a quantum channel and can be utilized to enhance the transmission fidelity of a certain channel. As a by-product we also derived a lower bound for the optimal fidelity trade-off for a completely unknown coherent state within the class all covariant quantum operations. Our result thus clearly illustrates that one can extract more information on an unknown coherent state while preserving the degree of disturbance introduced into it by this procedure by using a suitable non-Gaussian operation.
\acknowledgments
I would like to thank Jarom\'{\i}r Fiur\'a\v{s}ek, Radim Filip, Ulrik Andersen, Miroslav Gavenda, and Radek \v{C}elechovsk\'y for valuable discussions. The research has been supported by the research project: ``Measurement and Information in Optics,'' No. MSM 6198959213 and by the COVAQIAL (FP6-511004) and SECOQC (IST-2002-506813) projects of the sixth framework program of EU.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Squeezing concentration for Gaussian states with unknown parameter} \author{Radim Filip, \footnote{email:[email protected], tel:+420-68-5631572, fax:+420-68-5224246} and Ladislav Mi\v sta Jr.} \affiliation{Department of Optics, Research Center for Optics,\\ Palack\' y University,\\ 17. listopadu 50, 772~07 Olomouc, \\ Czech Republic} \date{\today}
\begin{abstract} A continuous-variable analog of the Deutsch's distillation protocol locally operating with two copies of the same Gaussian state is suggested. Irrespectively of the impossibility of Gaussian state distillation, we reveal that this protocol is able to perform a squeezing concentration of the Gaussian states with {\em unknown} displacement. Since this operation cannot be implemented using only single copy of the state, it is a new application of the distillation protocol which utilizes two copies of the same Gaussian state. \end{abstract}
\pacs{03.67.-a}
\maketitle
\section{Introduction}
The exploitation of entangled states of quantum systems makes new quantum information protocols possible such as quantum teleportation \cite{Bennett93}, entanglement swapping \cite{Pan98} and quantum cryptography \cite{Ekert91}. The common feature of all these protocols is that they require to transmit the entangled state from the common source to the distant partners. In practice, the transmission is always accompanied by losses that contaminate the shared state and thus reduce the degree of entanglement. Since the efficiency of the protocols is considerably dependent on the degree of entanglement shared by the partners, the question that naturally arises is whether the partners can eliminate the influence of losses and to enhance the entanglement employing only local operations and classical communication (LOCC). Such the protocols, commonly called distillation protocols, leading to the concentration of entanglement contained in several copies of partially entangled two-qubit states into a smaller number of singlet states has been proposed \cite{Bennett96a,Bennett96b} and also experimentally realized \cite{Pan01}. Further, efficient distillation procedure converting any two-qubit state with fidelity relative to maximally entangled state $F>1/2$ has been suggested in \cite{Deutsch96}. Also entanglement swapping based distillation of partially entangled pure states of two qubits has been proposed in the literature \cite{Bose99}. The higher-dimensional generalizations of the qubit distillation protocols have been given in \cite{Horodecki99a} and \cite{Alber01}.
With increase of the interest in quantum information processing with continuous variables \cite{Braunstein}, \cite{Furusawa01} the natural need for distillation of continuous-variable (CV) entangled states has arisen. The existing proposals of CV distillation schemes allow either to distill the bipartite Gaussian states at the expense of non-Gaussian operations such as photon number measurement \cite{Duan00} or they are capable to distill the bipartite non-Gaussian states \cite{Parker00}. Unfortunately, any of these protocols is not satisfactory with respect to the current experiment, in which only Gaussian operations with Gaussian states are well managed. Recent results which apply to the "purely" Gaussian distillation protocols have led to the conclusion that no such a protocol exists. Namely, it was shown by means of the Gaussian completely positive (CP) maps technique that any trace-decreasing LOCC Gaussian CP map acting on known Gaussian state can be replaced by trace-preserving LOCC Gaussian map CP \cite{Fiurasek02}. From that it follows particularly, that a single copy of two-mode bipartite entangled Gaussian state cannot be distilled to a more entangled state using only LOCC Gaussian operations. Further it was proved, that the bipartite symmetric two-mode Gaussian state cannot be distilled to more entangled state with the aid of another identical copy of the state using only local Gaussian operations \cite{Eisert02}. The general proof that Gaussian states cannot be distilled by local Gaussian operations and classical communication was given in \cite{Giedke02}. Having these facts in mind the question that naturally arises is whether there exists some Gaussian operation on bipartite entangled Gaussian state that two distant observers employing only local Gaussian operations wish to carry out that can be performed only at the expense of utilization of more than single copy of the state. This is the subject of the present article. The paper is organized as follows. In Sec. II we investigate the continuous-variable analog of the Deutsch's qubit distillation protocol. The example of Gaussian operation requiring 'collective' local operations on two copies of entangled Gaussian state is given in Sec. III. Finally, Sec. IV contains conclusion.
\section{CV analog of Deutsch's protocol}
\begin{figure}
\caption{Two-mode squeezing concentration scheme: S -- source pair of the modes, T -- target pair of the modes, HOM -- homodyne detection, D -- coordinate unitary displacement.}
\end{figure}
The properties of the CV analog of the Deutsch's protocol \cite{Deutsch96}, which will be described below, are best studied on the example of the displaced two-mode squeezed vacuum state
\begin{eqnarray}\label{nopa}
|x_{0},\sigma_{+},\sigma_{-}\rangle=\frac{1}{\sqrt{\pi\sqrt{\sigma_{+}\sigma_{-}}}} \int\!\!\!\int_{-\infty}^{\infty}dxdy\times\nonumber\\ \exp\left[-\frac{(x-y-x_{0})^{2}}{4\sigma_{-}} -\frac{(x+y-x_{0})^{2}}{4\sigma_{+}}
\right]|x\rangle_{A}|y\rangle_{B}, \end{eqnarray}
where $\sigma_{+}=\langle[\Delta(X_{A}+X_{B})]^{2}\rangle= \mbox{e}^{2r_{1}}$, $\sigma_{-}=\langle[\Delta(X_{A}-X_{B})]^{2}\rangle= \mbox{e}^{-2r_{2}}$ $r_1$, $r_2$ are squeezing parameters; $x_{i}$ and $p_{j}$, $i,j=A,B$, $[X_{i},P_{j}]=i\delta_{ij}$ are standard position and momentum quadrature operators of modes $A$ and $B$, respectively; $x_{0}$ is an {\em unknown} coherent displacement of quadrature $X_{A}$. Note, that as $\sigma_{-}$ decreases, $\sigma_{+}$ increases, and this state approaches the well-known Einstein-Podolsky-Rosen state. The state (\ref{nopa}) can be prepared by mixing of momentum-squeezed vacuum state with squeezing $r_1$ and position-squeezed vacuum state with squeezing $r_2$ at a balanced beamsplitter \cite{Furusawa01}, followed by unitary displacement operation $U=\mbox{exp}(-ix_{0}P_{A})$ on mode $A$.
Let us assume that two identical copies, conventionally called source (S) and target (T), of the state (\ref{nopa}) where $x_{0}=0$ are distributed between two observers, Alice (A) and Bob (B), as is depicted in Fig.~1. The natural extension of the Deutsch's qubit distillation protocol \cite{Deutsch96} to CV then consists of (i) local QND measurement of source position quadrature (CV analog of XOR-gate)
\begin{equation}
|x\rangle_{S}|x'\rangle_{T} \rightarrow
|x\rangle_{S}|x'-x\rangle_{T}, \end{equation}
performed both by Alice and by Bob, followed by (ii) local homodyne measurement of the target quadratures $X_{{T}_{A}}$ and $X_{{T}_{B}}$. Alice and Bob then can communicate their measurement outcomes via classical channel and, as in the original protocol, discard both the pairs if the outcomes do not coincide. A straightforward calculation reveals, however, that irrespectively of the measurement outcomes all the output states of the protocol can be brought into the single state that is independent on the measurement outcomes, by applying two local unitary displacement transformations
\begin{eqnarray}\label{displ} x\rightarrow x+{\cal X}/2,\hspace{0.3cm} y\rightarrow y+{\cal Y}/2 \end{eqnarray}
on Alice's and Bob's side, if the measurement outcomes obtained by them are ${\cal X}$ and ${\cal Y}$, respectively. Thus the selection of the subensemble of the output states based on the results of target measurements is not useful and the protocol becomes {\em deterministic}. This property of the CV analog of the Deutsch's scheme is in sharp contrast with the original protocol for qubits. It has been shown recently \cite{Fiurasek02}, that this behaviour is characteristic to all Gaussian distillation protocols for apriori {\em known} input Gaussian states, i.e. that any such 'purely' Gaussian distillation protocol would be deterministic.
After application of this CV protocol and displacements (\ref{displ}), one finds the output state of the form (\ref{nopa}) with the variances changed according to the rule $\tilde{\sigma}_{\pm}=\frac{\sigma_{\pm}}{2}$. Irrespectively of the increase of the position correlations, the marginal purity $P=2\sqrt{\sigma_{+}\sigma_{-}}/(\sigma_{+}+\sigma_{-})$ of outgoing pure state remains unchanged and the entanglement in the state (\ref{nopa}) is not enhanced. In order to decrease the marginal purity $P$ and consequently to obtain more entangled state, we would need different behaviour in these variances, for example, $\sigma_{-}$ decreases and $\sigma_{+}$ remains constant at a time.
Surprisingly, our CV procedure can be substituted by the {\em unconditional} scheme employing only {\em single copy} of the source state. It consists of two single-mode squeezers, performing the following squeezing transformations ${\tilde X_{A}}=\frac{1}{\sqrt{2}}X_{A}$ and ${\tilde X_{B}}=\frac{1} {\sqrt{2}} X_{B}$ on Alice's and Bob's side. This procedure is capable of transformation of nonsymmetrically entangled states with $\sigma_{+}\sigma_{-}>1$ to symmetrical ones, having $\sigma_{+}\sigma_{-}=1$, and vice versa, as has been previously discussed in \cite{Bowen01}. Thus, due to the equivalence between aforementioned two-copy and single-copy schemes, one can have doubt about the usefulness of our CV procedure based on operations on two copies of the input state. The question that naturally arises in this context is, whether the utilization of two copies of a two-mode Gaussian state enables us to perform operation, which cannot be equivalently carried out only on single copy of the state. In the following Section, we demonstrate that our CV analog of the Deutsch's protocol can be useful in this respect, particularly for squeezing concentration of the Gaussian states (\ref{nopa}) with an {\em unknown} displacement $x_{0}$.
\section{Squeezing concentration}
\begin{figure}
\caption{Squeezing concentration procedure: the noise ellipse is squeezed without a change of {\em unknown} mean value $x_{0}$.}
\end{figure}
We start from a simple example illustrating the usefulness of the protocol discussed in the previous Section in manipulating with an {\em unknown} Gaussian state. First, we analyse a local part of the protocol, for example, on the Alice's side. Let us consider the following single-mode pure Gaussian state
\begin{equation}\label{single}
|x_{0},\sigma_{X}\rangle=(2\pi\sigma)^{-1/4}\int_{-\infty}^{\infty}\exp
\left[-\frac{(x-x_{0})^{2}}{4\sigma_{X}}\right]|x\rangle dx \end{equation}
where $x_{0}=\langle X\rangle$ is an {\em unknown} parameter representing a coherent signal and $\sigma_{X}=\langle (\Delta X)^{2}\rangle= \mbox{e}^{-2r}/2$, $r$ is the squeezing parameter. This state can be prepared by displacing the squeezed vacuum state by the value $x_{0}$. The displacement is often used to encode information into the quantum state, for example, in the CV dense coding \cite{Braunstein00}. Since the transmission channel introduces extra noise into the state (\ref{single}), it is desirable to reduce the noise affecting the transmitted information before subsequent processing. It can be achieved by means of appropriate squeezing in the position quadrature. Employing only single copy of the state (\ref{single}), we can simply squeeze the position fluctuations in the single-mode squeezer, however, in this case also the value of {\em unknown} parameter $x_{0}$ is changed. The reason is that single-mode squeezer reduces both the mean value $\langle X\rangle$ and the variance $\sigma=\langle (\Delta X)^{2}\rangle$ of the position quadrature. If we do not know the displacement $x_{0}$, then we cannot restore the mean value to the original one without errors. Because the information is encoded in the mean value $\langle X\rangle$, its preservation during squeezing is important. On the other hand, knowing the parameter $x_{0}$, we can restore the initial mean value by a suitable displacement.
However, if we consider two copies of the state (\ref{single}), we are able to squeeze the position fluctuations without changing {\em unknown} mean value $\langle X\rangle=x_{0}$, as is depicted in Fig.~2. It can be achieved by above discussed protocol, employing, for instance, only Alice's part of the scheme outlined in Fig.~1. The procedure produces single copy of the state (\ref{single}) with new reduced variance $\tilde{\sigma}=\frac{\sigma}{2}$ and without changing the mean value $\langle X\rangle$. Since it cannot be achieved employing only single copy of the state (\ref{single}), this example illustrates a new application of the CV analog of the distillation protocol in manipulation with {\em unknown} Gaussian states.
In general, this protocol can be looked at as a state transformation of the source Wigner function $W_{S}(x,p)$ with the help of target Wigner function $W_{T}(x,p)$. After QND measurement, position measurement on the target mode and subsequent displacement, we arrive at the output source Wigner function \begin{eqnarray}\label{wigner1} \tilde{W}_{S}(x,p)&=&\int\!\!\!\int_{-\infty}^{\infty}W_{S}(x-{\cal X}/2,p')\times\nonumber\\ & & W_{T}(x+{\cal X}/2,p-p')dp'd{\cal X}. \end{eqnarray} If we are interested in the position and momentum distributions of the outgoing source mode separately, we can integrate the Wigner function (\ref{wigner1}) over momentum $p$ and position $x$, respectively and obtain the following marginal distributions \begin{eqnarray}\label{marg} \tilde{p}_{S}(x)&=&\int_{-\infty}^{\infty}p_{S}(x-{\cal X}/2)p_{T}(x+{\cal X}/2)d{\cal X},\nonumber\\ \tilde{p}_{S}(p)&=&\int_{-\infty}^{\infty}p_{S}(p')p_{T}(p-p')dp', \end{eqnarray} where $p_{S}$ and $p_{T}$ are input source and target marginal distributions, respectively. Assuming both the source and target state to be in the same Gaussian state with marginal distributions \begin{eqnarray} p_{S}(x)=p_{T}(x)&=&\frac{1}{\sqrt{2\pi\langle(\Delta X_{S})^{2}\rangle}} \exp\left[-\frac{(x-x_{0})^{2}}{2\langle(\Delta X_{S})^{2}\rangle}\right],\nonumber\\ p_{S}(p)=p_{T}(p)&=&\frac{1}{\sqrt{2\pi\langle(\Delta P_{S})^{2}\rangle}} \exp\left[-\frac{p^{2}}{2\langle(\Delta P_{S})^{2}\rangle}\right],\nonumber\\ \end{eqnarray} substituting them into the formulas (\ref{marg}) and performing the integrations, we obtain that the {\em unknown} mean value $\langle{\tilde X}_{S}\rangle=\langle X_{S}\rangle=x_{0}$ is preserved, whereas the position fluctuations are squeezed to half of initial value \begin{equation} \langle(\Delta\tilde{X}_{S})^{2}\rangle= \frac{\langle(\Delta X_{S})^{2}\rangle}{2},\hspace{0.3cm} \langle(\Delta\tilde{P}_{S})^{2}\rangle= 2\langle(\Delta P_{S})^{2}\rangle \end{equation} around this mean value. Consequently, due to principle of complementarity, the momentum fluctuations are enhanced. Employing $N$ identical copies of the same Gaussian state, we are able to squeeze the position fluctuations to $1/N$ of initial value, without changing the mean value of the position quadrature. Naturally, similar procedure can be constructed for the squeezing of momentum fluctuations. Note, that a discrete-variable analogue of this procedure has been discussed as the qubit purification protocol \cite{Bowmeester00}.
Let us apply now the idea of this procedure to the two-mode case, considering two copies of the state (\ref{nopa}), for simplicity. If the parameter $x_{0}$ is known, then we can utilize the Bowen's ``concentrating'' procedure and manipulate with two-mode correlations on single copy \cite{Bowen01}. If, however, the value $x_{0}$ is a priori not known, then we are not able to coherently manipulate with the variance $\langle[\Delta(X_{A}-X_{B})]^{2}\rangle$, without changing the unknown parameter $x_{0}$. On the other hand, employing entire two-mode setup depicted in Fig.~1, we are able to squeeze coherently the fluctuations ${\tilde\sigma}_{+}=\sigma_{+}/2$, ${\tilde\sigma}_{-}=\sigma_{-}/2$ again without changing the {\em unknown} parameter $x_{0}$. In terms of Wigner functions, our concentrating procedure can be expressed, in analogy with the single-mode case (\ref{wigner1}), by the formula \begin{widetext} \begin{eqnarray}\label{wigner2} \tilde{W}_{S}(x_{A},p_{A},x_{B},p_{B})&=& \int\!\!\!\int\!\!\!\int\!\!\! \int_{-\infty}^{\infty} W_{S}(x_{A}-{\cal X}/2,p'_{A},x_{B}-{\cal Y}/2,p'_{B})\times\nonumber\\ & &W_{T}(x_{A}+{\cal X}/2,p_{A}-p'_{A},x_{B}+{\cal Y}/2,p_{B}-p'_{B})dp'_{A}dp'_{B}d{\cal X}d{\cal Y}. \end{eqnarray} \end{widetext} If we are interested separately in position correlations and momentum correlations between Alice and Bob, we can integrate the function (\ref{wigner2}) over momentum variables $p_{A}$, $p_{B}$ to obtain joint position distribution \begin{eqnarray} \tilde{p}_{S}(x_{A},x_{B})=\int\!\!\!\int_{-\infty}^{\infty}p_{S}(x_{A}-{\cal X}/2,x_{B}-{\cal Y}/2)\times\nonumber\\ p_{T}(x_{A}+{\cal X}/2,x_{B}+{\cal Y}/2)d{\cal X}d{\cal{Y}} \end{eqnarray} or we can integrate over position variables $x_{A}$, $x_{B}$ to obtain joint momentum distribution \begin{eqnarray} \tilde{p}_{S}(p_{A},p_{B}) &=&\int\!\!\!\int_{-\infty}^{\infty}p_{S}(p'_{A},p'_{B})\times\nonumber\\ & &p_{T}(p_{A}-p'_{A},p_{B}-p'_{B})dp'_{A}dp'_{B}. \end{eqnarray} Let us assume, that the source and target have the same joint probability distributions of the form \begin{eqnarray}\label{twomode} p({\bf x})&=&\frac{1}{2\pi\sqrt{\det {\bf V_{X}}}} \exp\left[-({\bf x}-{\bf x}_{0})^{T}{(2{\bf V}_{X})}^{-1}({\bf x}-{\bf x}_{0})\right],\nonumber\\ p({\bf p})&=&\frac{1}{2\pi\sqrt{\det {\bf V_{P}}}} \exp\left[-{\bf p}^{T}{(2{\bf V}_{P})}^{-1}{\bf p}\right], \end{eqnarray} where ${\bf x}=(x_{A},x_{B})$, ${\bf x}_{0}=(x_{0},x_{0})$, ${\bf p}=(p_{A},p_{B})$, $T$ denotes operation of transposition and where \begin{equation} ({\bf V}_{X})_{ij}=\langle \Delta X_{i}\Delta X_{j}\rangle, \hspace{0.3cm} ({\bf V}_{P})_{ij}=\langle \Delta P_{i}\Delta P_{j}\rangle \end{equation} are the elements of position and momentum variance matrices, where $\Delta X_{i}=X_{i}-\langle X_{i}\rangle$, $\Delta P_{j}=P_{j}-\langle P_{j}\rangle$, $i,j=A,B$. The simple calculation then yields the output probability distributions of the same form as in Eq.~(\ref{twomode}), however, with the transformed variance matrices ${\tilde{\bf V}}_{X}={\bf V}_{X}/2$ and ${\tilde{\bf V}}_{P}=2{\bf V}_{P}$. Thus, the correlations in positions increase as follows \begin{equation} \langle[\Delta(\tilde{X}_{A}-\tilde{X}_{B})^{2}]\rangle= \frac{\langle[\Delta({X}_{A}-{X}_{B})^{2}]\rangle}{2}, \end{equation} whereas the mean values $\langle X_{A}\rangle$ and $\langle X_{B}\rangle$ are preserved. On the other hand, the momentum anticorrelations decrease \begin{equation} \langle[\Delta(\tilde{P}_{A}+\tilde{P}_{B})]^{2}\rangle= 2\langle[\Delta({P}_{A}+{P}_{B})]^{2}\rangle, \end{equation} and thus the entanglement and total entropy of the source state are preserved. This example illustrates that although the CV analog of the Deutsch's distillation protocol is not useful for enhancement of entanglement, it can be useful when one wishes to locally concentrate the two-mode squeezing in a partially {\em unknown} Gaussian state.
We have analysed the CV analog of specific distillation protocol for two copies of pure two-mode Gaussian state. We have demonstrated that it is only able to manipulate with squeezing, while the entanglement is preserved. Irrespectively to impossibility of entanglement increasing, it can be useful from another point of view. Employing two-copies of a two-mode Gaussian state displaced in the position by an {\em unknown} value, we can utilize this local-operation protocol to enhance the particular correlations, without changing the {\em unknown} position displacement. Because this procedure cannot be implemented if we have only single copy of this state, we have revealed a new application of the CV analog of the discrete-variable distillation protocol.
\noindent {\bf Acknowledgments} The authors would like to thank J. Fiur\' a\v{s}ek for fruitful discussions. The work was supported by the project LN00A015 and CEZ:J14/98 of the Ministry of Education of Czech Republic and by the EU grant under QIPC project IST-1999-13071 (QUICOV).
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title[Subsonic-Sonic Limit to M-D Steady Euler Equations] {Subsonic-Sonic Limit of Approximate Solutions to Multidimensional Steady Euler Equations}
\author{Gui-Qiang Chen} \address{G.-Q. Chen, School of Mathematical Sciences, Fudan University\\ Shanghai 200433, China; Mathematical Institute, University of Oxford\\ Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK; Academy of Mathematics and Systems Science\\ Academia Sinica, Beijing, 100190, P. R. China} \email{[email protected]}
\author{Fei-Min Huang} \address{F.-M. Huang, Institute of Applied Mathematics\\
Academy of Mathematics and Systems Science\\ Academia Sinica, Beijing, 100190, P. R. China} \email{[email protected]}
\author{Tian-Yi Wang} \address{T.-Y. Wang, Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan, Hubei, 430070, P. R. China; Academy of Mathematics and Systems Science, Academia Sinica, Beijing, 100190, P. R. China; Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK} \email{[email protected]; [email protected]} \date{\today}
\begin{abstract} A compactness framework is established for approximate solutions to subsonic-sonic flows governed by the steady full Euler equations for compressible fluids in arbitrary dimension. The existing compactness frameworks for the two-dimensional irrotational case do not directly apply for the steady full Euler equations in higher dimensions. The new compactness framework we develop applies for both non-homentropic and rotational flows. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass balance and the vorticity, along with the Bernoulli law and the entropy relation, through a more delicate analysis on the phase space. As direct applications, we establish two existence theorems for multidimensional subsonic-sonic full Euler flows through infinitely long nozzles. \end{abstract}
\keywords{Multidimensional, subsonic-sonic limit, steady flow, full Euler equations, homentropic, rotation, compactness framework, strong convergence, exact solutions, approximate solutions} \subjclass[2010]{ 35Q31; 35M30; 35L65; 76N10; 76G25; 35B40; 35D30 }
\maketitle
\section{Introduction} The full Euler equations for steady compressible flows in $\R^n$ read \begin{eqnarray}\label{1.5} \begin{cases} \mbox{div}\,(\rho u)=0,\\ \mbox{div}\,(\rho u\otimes u+ p I)=0,\\ \mbox{div}\, (\rho u E+ u p)=0, \end{cases} \end{eqnarray} where $x=(x_1, \cdots, x_n)\in \R^n, n\ge 2$, $u=(u_1,\cdots,u_n)\in \R^n$ is the fluid velocity, and $$
q:=|u|=\Big(\sum_{i=1}^n u_i^2\Big)^{1/2} $$ is the speed, while $\rho$, $p$, and $E$ represent the density, pressure, and total energy, respectively, and $I$ is the $n\times n$ unit matrix. The nonnegative quantities $\rho$, $q$, $p$, and $E$ are not independent.
For ideal polytropic gas, $$ E=\frac{q^2}{2}+\frac{p}{(\gamma-1)\rho} $$ with adiabatic exponent $\gamma>1$. In this case, the Bernoulli law is written as \begin{equation}\label{1.6} \frac{q^2}{2}+ h(\rho, p)=B, \end{equation} where $h(\rho, p)=\frac{\gamma p}{(\gamma-1)\rho}$ is the enthalpy, and $B$ is a Bernoulli function determined by appropriate additional conditions (such as boundary conditions and/or asymptotic conditions at infinity). The sound speed of the flow is \begin{equation}\label{1.7} c=\sqrt{\frac{\gamma p}{\rho}}, \end{equation} and the Mach number is defined as \begin{equation}\label{1.8} M=\frac{q}{c}. \end{equation} Then, for a fixed Bernoulli function $B$, there is a critical speed $q_{\rm cr}=\sqrt{2\frac{\gamma-1}{\gamma+1}B}$ such that, when $q\le q_{\rm cr}$, the flow is subsonic-sonic (that is, $M \le 1$); otherwise, it is supersonic (that is, $M>1$).
When the flow is homentropic, the pressure is a function of the density, and $(\ref{1.5})$ then reduces to \begin{eqnarray}\label{1.1} \begin{cases} \mbox{div}(\rho u)=0,\\[2mm] \mbox{div}(\rho u\otimes u+pI)=0. \end{cases} \end{eqnarray} As usual, we require $$ p'(\rho)>0, \quad 2p'(\rho) + \rho p''(\rho) > 0 \qquad \mbox{for $\rho>0$}, $$ which include the $\gamma$-law flows with $p=\kappa \rho^\gamma$, $\gamma>1$ and $\kappa>0$, and the isothermal flows with $p=\kappa \rho$; see \cite{Courant-Friedrichs}. In this case, the Bernoulli law has the same formula as $(\ref{1.6})$, while the enthalpy $h(p(\rho),\rho)=h(\rho)$ with $h'(\rho)=\frac{p'(\rho)}{\rho}$. The sound speed of the flow is $c=\sqrt{p'(\rho)}$, and the Mach number is defined as $M=\frac{q}{c}$. Then, for a fixed Bernoulli function $B$, the condition $2p'(\rho) + \rho p''(\rho) > 0$ for $\rho>0$ implies the existence of a unique critical speed $q_{\rm cr}=q_{\rm cr}(B)$ so that the flow can be classified by subsonic-sonic or supersonic by $q\leq q_{\rm cr}$ or $q>q_{\rm cr}$, respectively.
It is well known that the steady Euler equations for compressible fluids are of composite-mixed type, which is determined by the Mach number $M$. That is, the system can be reduced to a system such that two of the equations are elliptic-hyperbolic mixed: elliptic when $M < 1$ and hyperbolic when $M > 1$, while the other $n$ equations are hyperbolic. For the homentropic case, there are two mixed characteristics and $n-1$ hyperbolic characteristics.
During the 1950s, the effort on system \eqref{1.1} was focused mainly on the irrotational case, namely when $u$ is constrained to satisfy the additional equation $\mbox{curl} \,u=0$. Since the equations of uniform subsonic flow possess ellipticity, solutions have better regularity than those corresponding to transonic or supersonic flow. The airfoil problem for irrotational, two-dimensional subsonic flow was solved; {\it cf.} Frankl-Keldysh \cite{Frankl}, Shiffman \cite{Shiffman2}, Bers \cite{Bers2}, and Finn-Gilbarg \cite{Finn1}. The first result for three-dimensional subsonic flow past an obstacle was given by Finn and Gilbarg \cite{Gilbarg1} under some restrictions on the Mach number. Dong \cite{Dong1} and Dong-Ou \cite{Dong2} extended the results to the maximum Mach number $M < 1$ for the arbitrarily dimensional case. Also see Du-Xin-Yan \cite{Xin3} for the construction of a smooth uniform subsonic flow in an infinitely long nozzle in $\R^n, n\geq 2$.
For the rotational case, the global existence of homentropic subsonic flow through two-dimensional infinitely long nozzles was proved in Xie-Xin \cite{Xin4}. The result was also extended to the two-dimensional periodic nozzles in Chen-Xie \cite{CX} and to axisymmetric nozzles in Du-Duan \cite{DD}. For the full Euler flow, the first result was given by Chen-Deng-Xiang \cite{ChenDX} for two-dimensional infinitely long nozzles. Bae \cite{Bea} showed the stability of contact discontinuities for subsonic full Euler flow in the two-dimensional, infinitely long nozzles. Duan-Luo \cite{Duan-Luo} recently considered the axisymmetric nozzle problem for the smooth subsonic flow.
On the other hand, few results are currently known for the cases of subsonic-sonic flow and transonic flow, since the uniform ellipticity is lost and shocks may be present. That is, smooth solutions may not exist. Instead, one must consider weak solutions. Morawetz \cite{Morawetz2,Morawetz3} introduced an approach via compensated compactness to analyze irrotational steady flow of the Euler equations. Indeed, Morawetz established a compactness framework under the assumption that the solutions are free of stagnation points and cavitation points. Morawetz's result has been improved by Chen-Slemrod-Wang \cite{Chen8}, who have shown that the approximate solutions away from cavitation are constructed by a viscous perturbation.
The compactness framework for subsonic-sonic irrotational flow allowing for stagnation in two dimensions was due to Chen-Dafermos-Slemrod-Wang \cite{Chen6} by combining the mass conservation, momentum, and irrotational equations. The key observation in \cite{Chen6} is that the two-dimensional steady flow can be regarded as a one-dimensional unsteady system of conservation laws, that is, one of the spatial variables can be regarded as the time variable, so that the div-curl lemma can be applied to the two momentum equations. In fact, the momentum equations are first employed in \cite{Chen6} to reduce the support of the corresponding Young measure to two points, and then the irrotational equation and the mass equation are used to reduce the Young measure to a Dirac measure. Using a similar idea, Xie-Xin \cite{Xin1} investigated the subsonic-sonic limit of the two-dimensional irrotational, infinitely long nozzle problem. Later, in \cite{Xin2}, they extended the result to the three-dimensional axisymmetric flow through an axisymmetric nozzle. The compactness framework in the multidimensional irrotational case was established in Huang-Wang-Wang \cite{Huang-Wang-Wang}.
The compactness framework established for irrotational flow no longer applies directly for the steady full Euler equations in $\R^n$ with $n\geq 2$. When $n\ge 3$, the equations cannot be reduced to a one-dimensional system of conservation laws. More importantly, the div-curl lemma is no longer valid for the momentum equations, due to the presence of linear characteristics. One of our main observations is that it is still possible to achieve the same compactness result, {\it i.e.}, to reduce the Young measure to a Dirac measure, by using only natural weak estimates for the mass balance and the vorticity, along with the Bernoulli law and entropy relation, through a more delicate analysis on the phase space. In particular, the Bernoulli function and entropy function play a key role in our proof. We then establish a compactness framework for approximate solutions for steady full Euler flows in arbitrary dimension.
The rest of this paper is organized as follows. In Section 2, we establish the compactness framework for subsonic-sonic approximate solutions to subsonic-sonic flows governed by the steady full Euler equations for compressible fluids in $\R^n$ with $n\geq2$, as well as by the steady homentropic Euler equations with weaker conditions. In Sections 3--4, we give two direct applications of the compactness framework to establish the existence of subsonic-sonic full Euler flow through infinitely long nozzles in $\R^n$ with $n\geq 2$.
\section{Compactness Framework for Approximate Steady Full Euler Flows}
In this section, we establish the compensated compactness framework for approximate solutions of the steady full Euler equations in $\R^n$ with $n\ge 2$ with the form: \begin{eqnarray}\label{3.1} \begin{cases} \mbox{div}(\rho^\varepsilon u^\varepsilon)=e_1(\varepsilon),\\[1mm] \mbox{div}(\rho^\varepsilon u^\varepsilon\otimes u^\varepsilon+ p^\varepsilon I)=e_2(\varepsilon),\\[1mm] \mbox{div} (\rho^\varepsilon u^\varepsilon E^\varepsilon+ u^\varepsilon p^\varepsilon)=e_3(\varepsilon), \end{cases} \end{eqnarray} where $e_1(\varepsilon)$, $e_2(\varepsilon)=(e_{21}(\varepsilon), \cdots, e_{2n}(\varepsilon))^\top$, and $e_3(\varepsilon)$ are sequences of functions depending on the parameter $\varepsilon$.
Let a sequence of functions $\rho^\varepsilon(x)$, $u^\varepsilon(x)=(u^\varepsilon_1, \cdots, u^\varepsilon_n)(x)$, and $p^\varepsilon(x)$ be defined on an open subset $\Omega\subset \mathbb{R}^n$ such that the following qualities: \begin{eqnarray}
&&q^\varepsilon := |u^\varepsilon|=\sqrt{\sum_{i=1}^n (u_i^\varepsilon)^2}, \quad c^\varepsilon := \sqrt{\frac{\gamma p^\varepsilon}{\rho^\varepsilon}}, \quad M^\varepsilon := \frac{q^\varepsilon}{c^\varepsilon},\label{2.2a}\\[3mm] &&B^\varepsilon := \frac{(q^\varepsilon)^2}{2}+\frac{\gamma p^\varepsilon}{(\gamma-1)\rho^\varepsilon}, \qquad S^\varepsilon := \frac{\gamma p^\varepsilon}{(\gamma-1)(\rho^\varepsilon)^\gamma} \label{2.2b} \end{eqnarray} can be well defined and satisfy the following conditions:
(A.1). $M^\varepsilon\leq 1$ {\it a.e.} in $\Omega$;
(A.2). $S^\varepsilon$ and $B^\varepsilon$ are uniformly bounded and, for any compact set $K$, there exists a uniform constant $c(K)$ such that $\inf\limits_{x\in K} S^\varepsilon(x)\ge c(K)>0$. Moreover, $(S^\varepsilon, B^\varepsilon)\to (\overline{S}, \overline{B})$ {\it a.e.} in $\Omega$;
(A.3). $\mbox{curl}\ u^\varepsilon$ and $e_1(\varepsilon)$ are in a compact set in $W_{loc}^{-1, p}$ for some $1<p \le 2$.
\noindent Then we have
\begin{theorem}[Compensated compactness framework for the full Euler case] \label{thm3.1} Let a sequence of functions $\rho^\varepsilon(x)$, $u^\varepsilon(x)=(u^\varepsilon_1, \cdots, u^\varepsilon_n)(x)$, and $p^\varepsilon(x)$ satisfy conditions {\rm (A.1)}--{\rm (A.3)}. Then there exists a subsequence (still labeled) $(\rho^\varepsilon, u^\varepsilon, p^\varepsilon)(x)$ such that $$ \rho^\varepsilon(x)\to \rho(x), \quad u^\varepsilon(x)\to (u_1, \cdots, u_n)(x), \quad p^\varepsilon(x)\to p(x) \qquad \mbox{a.e. in $x \in \Omega$ as $\varepsilon\rightarrow 0$}, $$ and $$ M(x):=\frac{q(x)}{c(x)}\leq 1 \qquad \mbox{a.e. $x\in \Omega$}. $$ \end{theorem}
\noindent\textbf{Proof}. We divide the proof into three steps.
{\it Step 1. The strong convergence of $(\rho^\varepsilon, p^\varepsilon)$ follows from the strong convergence of $q^\varepsilon=|u^\varepsilon|$}.
We employ \eqref{2.2b} to obtain \begin{equation}\label{3.3f} \frac{(q^\varepsilon)^2}{2}+S^\varepsilon(\rho^\varepsilon)^{\gamma-1}=B^\varepsilon. \end{equation}
From these, the three variables $\rho^\varepsilon$, $p^\varepsilon$, and $q^\varepsilon$ are determined by one of them. In other words, the pressure $p^\varepsilon$ and density $\rho^\varepsilon$ can be regarded as functions of $q^\varepsilon$ through $(B^\varepsilon, S^\varepsilon)$ with \begin{eqnarray} &&\rho^\varepsilon=\rho(q^\varepsilon; B^\varepsilon, S^\varepsilon) =\left(\frac{2B^\varepsilon- (q^\varepsilon)^2}{2S^\varepsilon}\right)^{\frac{1}{\gamma-1}},\label{3.3a1}\\[2mm] &&p^\varepsilon=p(q^\varepsilon; B^\varepsilon, S^\varepsilon) =\frac{\gamma-1}{2\gamma} \frac{\left(2B^\varepsilon-(q^\varepsilon)^2\right)^{\frac{\gamma}{\gamma-1}}}{(2S^\varepsilon)^{\frac{1}{\gamma-1}}}. \label{3.3a2} \end{eqnarray} Since $(B^\varepsilon, S^\varepsilon)$ strongly converge to $(\overline{B}, \overline{S})$ {\it a.e.}, the strong convergence of the density $\rho^\varepsilon$ and pressure $p^\varepsilon$ becomes a nature consequence of the strong convergence of the speed $q^\varepsilon$.
{\it Step 2. The $H^{-1}_{loc}$--compactness}. From the uniform boundedness of the Bernoulli function $B^\varepsilon$ and the subsonic-sonic condition $M^\varepsilon\le1$, it is easy to see $$ q^\varepsilon\le \sup\limits_{\varepsilon}\sqrt{2\frac{\gamma-1}{\gamma+1}B^\varepsilon}, $$ which implies the uniform boundedness of the velocity $u^\varepsilon(x)=(u^\varepsilon_1, \cdots, u^\varepsilon_n)(x)$. This shows that $$ (\mbox{curl}\,u^\varepsilon)_{ij}=\partial_{x_i} u^\varepsilon_j - \partial_{x_j} u^\varepsilon_i \qquad\mbox{is bounded in $W^{-1, \infty}$}. $$ On the other hand, $\mbox{curl}\, u^\varepsilon$ is compact in $W^{-1, p}$ for some $1< p \le 2$. By the interpolation theorem, \begin{equation}\label{2.1a} \mbox{curl}\, u^\varepsilon \qquad \mbox{is compact in} \,\,\, H^{-1}_{loc}. \end{equation} For a fixed compact set $K$, the density $\rho^\varepsilon$ can be controlled by $\sup\limits_{\varepsilon}\Big(\sup\limits_{x\in K} B^\varepsilon/\inf\limits_{x\in K}S^\varepsilon\Big)^{\frac{1}{\gamma-1}}$. Similarly, we have \begin{equation}\label{2.1b} \mbox{div}(\rho^\varepsilon u^\varepsilon) \qquad \mbox{is compact in} \,\,\, H^{-1}_{loc}. \end{equation}
{\it Step 3. The strong convergence of $u^\varepsilon(x)$, which also leads to the strong convergence of $(\rho^\varepsilon, p^\varepsilon)(x)$ from Step 1}. By the div-curl lemma of Murat \cite{Murat} and Tartar \cite{Tartar}, the Young measure representation theorem for a uniformly bounded sequence of functions ({\it cf.} Tartar \cite{Tartar}; also Ball \cite{Ball}), and \eqref{2.1a}--\eqref{2.1b}, we have the following commutation identity: \begin{eqnarray}\label{3.3} \sum\limits_{i=1}^n\langle \nu( u), \,\rho(q) u_i\rangle \langle \nu( u), u_i\rangle&=&\langle \nu(u), \,\sum\limits_{i=1}^n\rho(q) u_i u_i\rangle. \end{eqnarray} Here and hereafter, for simplicity of notation, we have used that $\nu(u):=\nu_{x}(\rho(q; \overline{B}(x), \overline{S}(x)), u)$ denotes the associated Young measure (a probability measure) for the sequence $u^\varepsilon(x)$, and $\rho(q):=\rho(q; \overline{B}(x), \overline{S}(x))$ for the limits $\overline{B}(x)$ and $\overline{S}(x)$ of $B^\varepsilon(x)$ and $S^\varepsilon(x)$ respectively.
Then the main point in this step for the compensated compactness framework is to prove that $\nu$ is in fact a Dirac measure, which in turn implies the compactness of the sequence $u^\varepsilon(x)$.
Combining both sides of $(\ref{3.3})$ together, we have \begin{equation} \langle \nu(u^{(1)})\otimes\nu(u^{(2)}), \,\sum\limits_{i=1}^n\rho(q^{(1)}) u^{(1)}_i (u_i^{(1)}-u_i^{(2)})\rangle=0. \end{equation} Exchanging indices $(1)$ and $(2)$, we obtain the following symmetric commutation identity: \begin{equation}\label{3.4} \langle \nu(u^{(1)})\otimes\nu(u^{(2)}), I(u^{(1)}, u^{(2)})\rangle=0, \end{equation} where \begin{equation}\label{3.5} I(u^{(1)},u^{(2)}) =\sum_{i=1}^n(\rho(q^{(1)}) u_i^{(1)}-\rho(q^{(2)})u_i^{(2)})(u_i^{(1)}-u_i^{(2)}). \end{equation}
\noindent Then it remains to prove the strong convergence of the velocity $u^\varepsilon$ from the above identity.
Notice that \begin{eqnarray}\label{3.6}
I(u^{(1)}, u^{(2)})&=&\sum_{i=1}^n\big(\rho(q^{(1)}) u_i^{(1)}-\rho(q^{(2)})u_i^{(2)}\big)\big(u_i^{(1)}-u_i^{(2)}\big)\nonumber\\
&=&\sum_{i=1}^n \big(\rho(q^{(1)}) |u_i^{(1)}|^2-\rho(q^{(1)}) u_i^{(1)}u_i^{(2)}-\rho(q^{(2)}) u_i^{(2)}u_i^{(1)}-\rho(q^{(2)}) |u_i^{(2)}|^2\big) \nonumber\\ &=&\rho(q^{(1)})\big((q^{(1)})^2-\sum\limits_{i=1}^n u_i^{(1)}u_i^{(2)}\big)+\rho(q^{(2)})\big((q^{(2)})^2-\sum\limits_{i=1}^n u_i^{(1)}u_i^{(2)}\big). \end{eqnarray} The Cauchy inequality implies \begin{eqnarray}\label{3.7} I(u^{(1)}, u^{(2)})&\geq &\rho(q^{(1)})\big((q^{(1)})^2-q^{(1)} q^{(2)}\big)+\rho(q^{(2)})\big((q^{(2)})^2-q^{(1)}q^{(2)}\big)\nonumber\\ &=&\big(q^{(1)}-q^{(2)}\big)\big(\rho(q^{(1)})q^{(1)}-\rho(q^{(2)})q^{(2)}\big)\nonumber\\ &=&\big(q^{(1)}-q^{(2)}\big)^2 \frac{d(\rho q)}{dq}(\tilde{q}), \end{eqnarray} where $\tilde{q}$ lies between $q^{(1)}$ and $q^{(2)}$ by the mean-value theorem.
Taking derivative with respect to $q$ on $(\ref{3.3f})$, we obtain $$ \frac{d \rho}{d q}=-\frac{q}{(\gamma-1)\rho^{\gamma-2}\overline{S}}=-\frac{\rho q}{c^2}. $$ Then $$ \frac{d(\rho q)}{d q}=\rho(1-M^2). $$ For subsonic-sonic flows, {\it i.e.}, $q^{(1)}, q^{(2)}\leq q_{\rm cr}(\overline{B})$, we have $$ M^2(\tilde{q})\le 1. $$ On the other hand, $$ \rho(\tilde{q})\geq \left(\frac{2\overline{B}}{(\gamma+1)\overline{S}}\right)^{\frac{1}{\gamma-1}}\geq 0. $$ Notice that $M^2(\tilde{q})=1$ if and only if $q^{(1)}=q^{(2)}=q_{\rm cr}(\overline{B})$, while $\rho(\tilde{q})=0$ if and only if $\overline{B}=0$ and $q^{(1)}=q^{(2)}=q_{\rm cr}(\overline{B})$. Then \begin{equation}\label{3.8} I(u^{(1)}, u^{(2)})\ge\big(q^{(1)}-q^{(2)}\big)^2 \, \rho(\tilde{q})\big(1-M^2(\tilde{q})\big)\geq 0. \end{equation} With $(\ref{3.4})$, it implies that $$ q^{(1)}=q^{(2)}, $$ which also deduces $$ \rho^{(1)}=\rho^{(2)}, \qquad p^{(1)}=p^{(2)}. $$
Again, using $(\ref{3.4})$ and $(\ref{3.5})$, we further obtain \begin{eqnarray}\label{3.10} 0&=&\langle \nu(u^{(1)})\otimes\nu(u^{(2)}), I(u^{(1)}, u^{(2)})\rangle \nonumber\\ &=&\langle \nu(u^{(1)})\otimes\nu(u^{(2)}), \rho(q^{(1)})\sum\limits_{i=1}^n\big(u_i^{(1)}-u_i^{(2)}\big)^2\rangle, \end{eqnarray} which immediately implies $$ u^{(1)}=u^{(2)}, $$ {\it i.e.}, $\nu(u)$ concentrates on a single point. If this would not be the case, we could suppose to have two different points $\acute{u}$ and $\grave{u}$ in the support of $\nu$. Then $(\acute{u}, \acute{u})$, $(\acute{u}, \grave{u})$, $(\grave{u}, \acute{u})$, and $(\grave{u}, \grave{u})$ would be in the support of $\nu\otimes\nu$, which contradicts with $u^{(1)}=u^{(2)}$. Therefore, the Young measure $\nu$ is a Dirac measure, which implies the strong convergence of $(\rho^\varepsilon, u^\varepsilon, p^\varepsilon)$. This completes the proof.
For the homentropic case, the entropy function $S$ is constant. Then the pressure $p$ may be regarded as a function of the density $\rho$ in $C^1\left(\mathbb{R}^+\cup\{0\}\right)\cap C^2\left(\mathbb{R}^+\right)$, which satisfies $p'(\rho)>0$ and $2p'(\rho) + \rho p''(\rho) > 0$ for $\rho>0$.
Without loss of generality, we set the enthalpy $h(\rho)$ as $$ h(\rho):=\int_{1}^{\rho}\frac{p'(s)}{s}ds, $$ so that $h(1)=0$. It is noticeable that, in this case, we do not have the property that the Bernoulli function is greater or equal to zero. To replace the nonnegative property, we introduce a lower bound $$ B_{\rm min}=\lim_{\rho\to 0^+}f(\rho), $$ while $$ f(\rho):=\frac{p'(\rho)}{2}+h(\rho) $$ belongs to $C^1(\mathbb{R}^{+}\cup\{0\})$. Since $2p'(\rho) + \rho p''(\rho)> 0$ for $\rho>0$, $f(\rho)$ is an increase function in $\rho>0$. Then $f(\rho)>B_{\rm min}$ for $\rho>0$ and $B_{\rm min}$ is lower bound so that $B_{\rm min}<f(1)=\frac{p'(1)}{2}$.
In the homentropic case, conditions (A.1)--(A.3) can be reformulated as:
(B.1). $M^\varepsilon\leq 1$ {\it a.e.} in $\Omega$;
(B.2). $B^\varepsilon= \frac{(q^\varepsilon)^2}{2}+\int_{1}^{\rho^\varepsilon}\frac{p'(s)}{s}ds$ are uniformly bounded and $B^\varepsilon \ge B_{\rm min}$;
(B.3). $\mbox{curl}\ u^\varepsilon$, $\frac{e_1(\varepsilon)}{\rho^\varepsilon}$, and $\frac{e_2(\varepsilon)}{\rho^\varepsilon}$ are uniformly bounded measures.
Similar to Theorem $\ref{thm3.1}$, we have
\begin{theorem}[Compensated compactness framework for the homentropic case]\label{thm2.1} Let a sequence of functions $\rho^\varepsilon(x)$ and $u^\varepsilon(x)=(u^\varepsilon_1, \cdots, u^\varepsilon_n)(x)$ satisfy conditions {\rm (B.1)--(B.3)}. Then there exists a subsequence (still labeled) $(\rho^\varepsilon, u^\varepsilon)(x)$ such that $$ \rho^\varepsilon(x)\rightarrow \rho(x), \quad u^\varepsilon(x)\rightarrow u(x)=(u_1, \cdots, u_n)(x)
\qquad \mbox{a.e. $x\in\Omega\, \,$ as $\varepsilon\rightarrow 0$} $$ and $$ M(x)\leq 1 \qquad \mbox{a.e. $x\in \Omega$.} $$ \end{theorem}
\noindent\textbf{Proof}. First, since $p'(\rho)>0$ for $\rho>0$, we can employ the implicit function theorem to conclude $\rho(q; B)=h^{-1}(B-\frac{q^2}{2})$. Then we can regard $\rho^\varepsilon$ as a function of $q^\varepsilon$ and $B^\varepsilon$, that is, $\rho^\varepsilon=\rho(q^\varepsilon; B^\varepsilon)$. As a consequence, the sequence $\rho^\varepsilon$ is nonnegative and uniformly bounded. Conditions (B.1)--(B.2) indicate directly that the speed sequence $q^\varepsilon$ is uniformly bounded.
Differentiating the Bernoulli functions $B^\varepsilon$ with respect to $x_i$ yields \begin{equation}\label{2.2} \begin{array}{ll} \partial_{x_i} B^\varepsilon &=\sum\limits_{j=1}^n u_j^\varepsilon\, \partial_{x_i} u_j^\varepsilon+\frac{p'(\rho^\varepsilon)}{\rho^\varepsilon}\partial_{x_i}\rho^\varepsilon\\ & = \sum\limits_{j=1}^n u_j^\varepsilon\,\partial_{x_i}\, u_j^\varepsilon
+ \frac{\partial_{x_i} p(\rho^\varepsilon)}{\rho^\varepsilon}\\ & = \sum\limits_{j=1}^n u_j^\varepsilon\, \partial_{x_i} u_j^\varepsilon + \frac{e_{2i}(\varepsilon)-\sum\limits_{j=1}^n\rho^\varepsilon u_j^\varepsilon\, \partial_{x_j} u_i^\varepsilon-u_i^\varepsilon e_1(\varepsilon)}{\rho^\varepsilon}\\ & = \sum\limits_{j=1}^n u_j^\varepsilon (\partial_{x_i} u_j^\varepsilon-\partial_{x_j} u_i^\varepsilon)+ \frac{e_{2i}(\varepsilon)-u_i^\varepsilon e_1(\varepsilon)}{\rho^\varepsilon}\\ & = \sum\limits_{j=1}^n u_j^\varepsilon\, \omega^\varepsilon_{ij}+ \frac{e_{2i}(\varepsilon)-u_i^\varepsilon e_1(\varepsilon)}{\rho^\varepsilon}. \end{array} \end{equation}
From the boundedness of $q^\varepsilon$ and condition (B.3), we conclude that $$ \nabla B^\varepsilon \qquad\mbox{is uniformly bounded measures}. $$ Since $B^\varepsilon$ is uniformly bounded, the total-variation norm of $B^\varepsilon$ is uniformly bounded, which implies that $B^\varepsilon(x)$ converges to $\overline{B}(x)$ in $L^1_{loc}$, as $\varepsilon$ tends to $0$, and $\overline{B}(x)\ge B_{\rm min}$ {\it a.e.} in $\Omega$.
From the definition, $(\mbox{curl}\, u^\varepsilon)_{ij}=\omega_{ij}^\varepsilon=\partial_{x_i} u_j^\varepsilon - \partial_{x_j} u_i^\varepsilon$ is bounded in $W^{-1, \infty}$. On the other hand, $\mbox{curl}\, u^\varepsilon$ is a uniformly bounded measure sequence, which implies that $\mbox{curl}\, u^\varepsilon$ is compact in $W^{-1, q}$ for each $1\le p <\frac{n}{n-1}$. From the interpolation theorem, $$ \mbox{curl}\, u^\varepsilon \qquad \mbox{is compact in}~ H^{-1}_{loc}. $$ Similarly, we have $$ \mbox{div}(\rho^\varepsilon u^\varepsilon) \qquad \mbox{is compact in}~ H^{-1}_{loc}. $$
Next, we will discuss the strong convergence in two cases.
{\it Case 1: $\overline{B}>B_{\rm min}$}. From the monotonicity of $f(\rho)$, we have $$ \rho_{\rm cr}>0, $$ which implies $f(\rho_{\rm cr})=\overline{B}$ and, on the support of $\nu$, $$ \rho(q)\in[\rho_{\rm cr}, h^{-1}(\overline{B})]. $$ Following similar argument for Theorem $\ref{thm3.1}$, we obtain the following commutation identity: \begin{equation}\label{2.41} \langle \nu(u^{(1)})\otimes\nu(u^{(2)}), \, I( u^{(1)}, u^{(2)})\rangle=0, \end{equation} and \begin{equation}\label{2.51} I(u^{(1)}, u^{(2)}) =\sum_{i=1}^n\big(\rho(q^{(1)}) u_i^{(1)}-\rho(q^{(2)})u_i^{(2)}\big)\big(u_i^{(1)}-u_i^{(2)}\big). \end{equation}
By the same calculation as in the previous argument, we have \begin{equation}\label{2.7} I(u^{(1)}, u^{(2)})\geq\big(q^{(1)}-q^{(2)}\big)^2 \frac{d(\rho q)}{dq}(\tilde{q}), \end{equation} where $\tilde{q}$ lies between $q^{(1)}$ and $q^{(2)}$.
From the definition of $\rho(q)$, we obtain that $\frac{d \rho}{d q}=-\frac{\rho q}{c^2}$, which implies $\frac{d(\rho q)}{dq}=\rho(1-M^2)$. From subsonic-sonic flows, {\it i.e.}, $q^{(1)}, q^{(2)}\leq q_{\rm cr}$, we have $$ \rho(\tilde{q})\big(1-M^2(\tilde{q})\big)\geq 0. $$ Notice that $M^2(\tilde{q})=1$ if and only if $q^{(1)}=q^{(2)}=q_{\rm cr}$. Then \begin{equation}\label{2.8} I(u^{(1)}, u^{(2)})\ge\left(q^{(1)}-q^{(2)}\right)^2 \, \rho(\tilde{q})\big(1-M^2(\tilde{q})\big)\geq 0. \end{equation} Thus, from $(\ref{2.41})$, we obtain $$ q^{(1)}=q^{(2)}. $$
Again using $(\ref{2.41})$ and $(\ref{2.51})$, we have \begin{eqnarray}\label{3.10a} 0&=&\langle \nu(u^{(1)})\otimes\nu(u^{(2)}), \, I(u^{(1)}, u^{(2)})\rangle \nonumber\\ &=&\langle \nu(u^{(1)})\otimes\nu(u^{(2)}), \, \rho(q^{(1)})\sum\limits_{i=1}^n\big(u_i^{(1)}-u_i^{(2)}\big)^2\rangle, \end{eqnarray} which immediately implies $u^{(1)}=u^{(2)}$, {\it i.e.}, the Young measure $\nu$ is a Dirac measure. Thus, we conclude the strong convergence of $(\rho^\varepsilon, u^\varepsilon)(x)$ to $(\rho, u)(x)$ with $M(x)\le 1$ a.e $x\in \Omega$.
{\it Case 2. $\overline{B}=B_{\rm min}$}. Considering the boundedness of $\overline{B}$, we can see $\lim\limits_{\rho\rightarrow0^+}\int_{1}^{\rho}\frac{p'(s)}{s}ds$ is finite, which implies $p'(0)=0$. On the other hand, the monotonicity of $f(\rho)$ shows that $\rho(q)=0$ for any $q$ belonging to the support of the Young measure $\nu$. It follows from the subsonic-sonic condition that $0\le q^2\le p'(0)=0$. Then the strong convergence of $(\rho^\varepsilon, u^\varepsilon)$ is shown in this case.
\begin{remark}\label{rem3} The main difference between Theorems $\ref{thm3.1}$ and $\ref{thm2.1}$ is that the homentropic case has the Bernoulli-vortex relation $(\ref{2.2})$, while the full Euler case has \begin{equation}\label{2.17} \partial_{x_i} B^\varepsilon= \sum\limits_{j=1}^n u^\varepsilon_j\, \omega^\varepsilon_{ij} +\frac{(\rho^\varepsilon)^{\gamma-1}}{\gamma}\partial_{x_i} S^\varepsilon +\frac{e_{2i}(\varepsilon)-u_i^\varepsilon e_1(\varepsilon)}{\rho^\varepsilon} \end{equation} so that the gradient of $B^\varepsilon$ cannot be achieved only by the vorticity $\omega^\varepsilon$. \end{remark}
\begin{remark} The main theorem in Huang-Wang-Wang \cite{Huang-Wang-Wang} is included in Theorem $\ref{thm2.1}$. Condition {\rm (B.1)} is the same as the one in \cite{Huang-Wang-Wang}, while $B^\varepsilon$ is assumed to be constant in \cite{Huang-Wang-Wang} which clearly satisfies condition {\rm (B.2)}. As a consequence, condition {\rm (B.3)} could be regarded as the $H^{-1}_{loc}$--compactness of $\mbox{curl}\, u^\varepsilon$. The irrotational condition in \cite{Huang-Wang-Wang} is removed. Thus, Theorem $\ref{thm2.1}$ includes more physical consideration. From $(\ref{2.2})$, the irrotational condition implies that the Bernoulli function is a constant in the flow field. \end{remark}
\begin{remark}\label{rem4} Consider any function $Q(\rho, u, p)=(Q_1,\cdots,Q_n)(\rho, u, p)$ satisfying \begin{equation}\label{3.12} \mbox{\rm div}\,(Q(\rho^\varepsilon, u^\varepsilon, p^\varepsilon))=o_Q(\varepsilon), \end{equation} where $o_Q(\varepsilon)\rightarrow 0$ in the distributional sense as $\varepsilon\rightarrow 0$. We can see from the strong convergence of $(\rho^\varepsilon, u^\varepsilon, p^\varepsilon)$ ensured by Theorem $\ref{thm3.1}$ that $\mbox{\rm div} (Q(\rho, u, p))=0$ holds in the distributional sense. Thus, if \begin{equation} \mbox{\rm div} \, (\rho^\varepsilon u^\varepsilon\otimes u^\varepsilon+p^\varepsilon I)=e_2(\varepsilon)\rightarrow 0 \qquad \mbox{in the sense of distributions}, \end{equation} the weak solution also satisfies the momentum equations in $(\ref{1.5})_2$ and the energy equation $(\ref{1.5})_3$ in the distributional sense. The statement is also valid for Theorem $\ref{thm2.1}$. \end{remark}
Then, as corollaries, we conclude the following theorems.
\begin{theorem}[Convergence of approximate solutions for the full Euler flow]\label{thm3.2} Let $\rho^\varepsilon(x)$, $u^\varepsilon(x)=(u^\varepsilon_1, \cdots, u^\varepsilon_n)(x)$, and $p^\varepsilon(x)$ be a sequence of approximate solutions satisfying {\rm (A.1)}--{\rm (A.3)} and $e_j(\varepsilon)\rightarrow 0, j=1,2,3$, in the distributional sense as $\varepsilon\rightarrow 0$. Then there exists a subsequence (still labeled) $(\rho^\varepsilon, u^\varepsilon, p^\varepsilon)(x)$ that converges a.e. as $\varepsilon\rightarrow 0$ to a weak solution $(\rho, u, p)$ to the Euler equations of $(\ref{1.5})$, which satisfies $M(x) \le 1$, {\it a.e.} $x\in \Omega$. \end{theorem}
\begin{theorem}[Convergence of approximate solutions for the homentropic Euler flow]\label{thm2.2} Let $\rho^\varepsilon(x)$ and $u^\varepsilon(x)=(u^\varepsilon_1, \cdots, u^\varepsilon_n)(x)$ be a sequence of approximate solutions satisfying {\rm (B.1)}--{\rm (B.3)} and $e_j(\varepsilon)\rightarrow 0, j=1,2$, in the distributional sense as $\varepsilon\rightarrow 0$. Then there exists a subsequence (still labeled) $(\rho^\varepsilon, u^\varepsilon)(x)$ that converges {\it a.e.} as $\varepsilon\rightarrow 0$ to a weak solution $(\rho, u)$ to the Euler equations of $(\ref{1.1})$, which satisfies $M(x) \le 1$ {\it a.e.} $x\in \Omega$. \end{theorem}
There are various ways to construct approximate solutions by either numerical methods or analytical methods such as vanishing viscosity methods. As direct applications, we show two examples in Sections 3--4 to apply the compactness framework built above in establishing existence theorems for multidimensional subsonic-sonic full Euler flows through infinitely long nozzles.
\section{Subsonic-Sonic Limit for Two-Dimensional Steady Full Euler Flows \\ in an Infinitely Long Nozzle}
In this section, as a direct application of the compactness framework established in Theorem \ref{thm3.1}, we obtain the subsonic-sonic limit of steady subsonic full Euler flows in a two-dimensional, infinitely long nozzle.
The infinitely long nozzle is defined as \begin{equation*} \Omega=\{(x_1,x_2)\, :\, f_1(x_1)<x_2<f_2(x_1), \,-\infty<x_1<\infty\}, \end{equation*} with the nozzle walls $\partial\Omega:=W_1\cup W_2$, where \begin{equation*} W_i=\{(x_1,x_2)\, :\, x_2=f_i(x_1)\in C^{2,\alpha}, ~-\infty<x_1<\infty\}, \qquad i=1,2. \end{equation*}
Suppose that $W_1$ and $W_2$ satisfy \begin{align}\label{cdx-3} &f_2(x_1)>f_1(x_1) ~ \qquad \mbox{for} ~x_1\in(-\infty, \infty),\nonumber\\ &f_1(x_1)\rightarrow 0, \quad f_2(x_1)\rightarrow 1 \qquad \mbox{as} ~x_1\rightarrow -\infty, \nonumber\\ &f_1(x_1)\rightarrow a, \quad f_2(x_1)\rightarrow b>a
\qquad \mbox{as} ~x_1\rightarrow \infty, \end{align} and there exists $\alpha>0$ such that \begin{equation}\label{cdx-4}
\|f_i\|_{C^{2,\alpha}(\mathbb{R})}\leq C, \qquad i=1,2, \end{equation} for some positive constant $C$. It follows that $\Omega$ satisfies the uniform exterior sphere condition with some uniform radius $r>0$. See Fig \ref{Fig1}.
\begin{figure}
\caption{Two-Dimensional Infinitely Long Nozzle}
\label{Fig1}
\end{figure}
Suppose that the nozzle has impermeable solid walls so that the flow satisfies the slip boundary condition: \begin{equation}\label{cdx-5} u \cdot\nu =0 \qquad \mbox{on} ~\partial \Omega, \end{equation} where $u=(u_1, u_2)\in \R^2$ is the velocity and $\nu=(\nu_1, \nu_2)$ is the unit outward normal to the nozzle wall. In the flow without vacuum, it can be written as \begin{equation}\label{cdx-sc} (\rho u) \cdot\nu =0 \qquad \mbox{on} ~\partial \Omega. \end{equation}
It follows from $(\ref{1.5})_1$ and $(\ref{cdx-sc})$ that \begin{equation}\label{cdx-6} \int_s\, (\rho u) \cdot l \, ds\equiv m \end{equation} holds for some constant $m$, which is the mass flux, where $s$ is any curve transversal to the $x_1$--direction, and $l$ is the normal of $s$ in the positive $x_1$--axis direction.
We assume that the upstream entropy function is given, {\it i.e.}, \begin{equation}\label{cdx-8} \frac{\gamma p}{(\gamma-1)\rho^\gamma}\longrightarrow S_-(x_2) \qquad \mbox{as} ~x_1\rightarrow -\infty, \end{equation} and the upstream Bernoulli function is given, {\it i.e.}, \begin{equation}\label{cdx-11} \frac{q^2}{2}+\frac{\gamma p}{(\gamma-1)\rho}\longrightarrow B_-(x_2) \qquad\mbox{as} ~x_1\rightarrow -\infty, \end{equation} where $B(x_2)$ is a function defined on $[0,1]$.
$\mathbf{Problem~1}(m)$: Solve the full Euler system $(\ref{1.5})$ with the boundary condition $(\ref{cdx-sc})$, the mass flux condition $(\ref{cdx-6})$, and the asymptotic conditions $(\ref{cdx-8})$--$(\ref{cdx-11})$.
Set \begin{equation*} \underline{S}=\inf_{x_2\in[0,1]}S_-(x_2), \qquad \underline{B}=\inf_{x_2\in[0,1]}B_-(x_2). \end{equation*}
For this problem, the following theorem has been established in Chen-Deng-Xiang \cite{ChenDX}.
\begin{theorem}\label{thm5.1} Let the nozzle walls $\partial \Omega$ satisfy $(\ref{cdx-3})$--$(\ref{cdx-4})$, and let $\underline{S}>0$ and $\underline{B}>0$. Then there exists
$\delta_0>0$ such that, if $\|(S_--\underline{S},B_--\underline{B})\|_{C^{1,1}([0,1])}\leq \delta$ for $0<\delta\leq\delta_0$, $(S_-B_-^{-\gamma})'(0)\geq 0$, and $(S_-B_-^{-\gamma})'(1)\leq 0$, there exists $\hat{m}\geq 2\delta_0^{{1}/{8}}$ such that, for any $m\in(\delta^{{1}/{4}},\hat{m})$, there exists a global solution ({\it i.e.} a full Euler flow) $(\rho, u, p) \in C^{1,\alpha}(\overline{\Omega})$ of $\mathbf{Problem~1}(m)$ such that the following hold:
{\rm (i)} Subsonicity and positivity of the horizontal velocity: The flow is uniformly subsonic with positive horizontal velocity in the whole nozzle, {\it i.e.}, \begin{equation}\label{cdx3} \sup_{\overline{\Omega}}(q^2-c^2)<0, \quad u_1>0 \qquad \mbox{in} ~\overline{\Omega}; \end{equation}
{\rm (ii)} The flow satisfies the following asymptotic behavior in the far fields: As $x_1\rightarrow -\infty$, \begin{equation}\label{cdx4} p\rightarrow p_->0, \qquad u_1\rightarrow u_-(x_2)>0, \qquad (u_2,\rho)\rightarrow (0,\rho_-(x_2;p_-)), \end{equation} \begin{equation}\label{cdx5} \nabla p\rightarrow 0, \qquad \nabla u_1\rightarrow (0, u_-'(x_2)), \qquad \nabla u_2\rightarrow 0, \qquad \nabla\rho\rightarrow (0,\rho_-'(x_2;p_-)) \end{equation} uniformly for $x_2\in K_1\Subset(0,1)$, where $ \rho_-(x_2;p_-)=\big(\frac{\gamma p_-}{(\gamma-1)S_-(x_2)}\big)^{{1}/{\gamma}}, $ the constant $p_-$ and function $u_-(x_2)$ can be determined by $m$, $S_-(x_2)$, and $B_-(x_2)$ uniquely;
{\rm (iii)} Uniqueness: The full Euler flow of $\mathbf{Problem~1} (m)$ satisfying $(\ref{cdx3})$ and the asymptotic behavior $(\ref{cdx4})$--$(\ref{cdx5})$ is unique.
{\rm (iv)} Critical mass flux: $\hat{m}$ is the upper critical mass flux for the existence of subsonic flow in the following sense: Either $\sup\limits_{\overline{\Omega}}(q^2-c^2)\rightarrow 0$ as $m\rightarrow \hat{m}$, or there is no $\sigma>0$ such that, for all $m\in(\hat{m},\hat{m}+\sigma)$, there are full Euler flows of $\mathbf{Problem~1} (m)$ satisfying $(\ref{cdx3})$, the asymptotic behavior $(\ref{cdx4})$--$(\ref{cdx5})$, and $\sup\limits_{m\in(\hat{m},\hat{m}+\sigma)} \sup\limits_{\overline{\Omega}}(c^2-q^2)>0$. \end{theorem}
We note that Theorem \ref{thm5.1} does not apply to the critical flows, that is, those flows for which $m=\hat{m}$ must be sonic at some point. Now we can employ Theorem $\ref{thm3.1}$ to establish a more general result.
\begin{theorem}[Subsonic-sonic limit of two-dimensional full Euler flows]\label{thm5.2} Let $\delta^{{1}/{4}}<m^{\varepsilon}<\hat{m}$ be a sequence of mass fluxes, and let $(\rho^\varepsilon, u^\varepsilon, p^\varepsilon)(x)$ be the corresponding sequence of solutions to $\mathbf{Problem~1} (m^\varepsilon)$. Then, as $m^{\varepsilon}\rightarrow \hat{m}$, the solution sequence possesses a subsequence (still denoted by) $(\rho^\varepsilon, u^{\varepsilon}, p^\varepsilon)(x)$ that converges strongly {\it a.e.} in $\Omega$ to a vector function $(\rho, u, p)(x)$ which is a weak solution of $\mathbf{Problem~1} (\hat{m})$. Furthermore, the limit solution $(\rho, u, p)(x)$ also satisfies $(\ref{1.5})$ in the distributional sense and the boundary conditions $(\ref{cdx-sc})$ as the normal trace of the divergence-measure field $\rho u$ on the boundary in the sense of Chen-Frid {\rm \cite{Chen7}}. \end{theorem}
\noindent{\bf Proof}. We divide the proof into three steps.
1. We first need to show that $(\rho^\varepsilon, u^\varepsilon, p^\varepsilon)(x)$ satisfy condition (A.1)--(A.3). For $B^\varepsilon$ and $S^\varepsilon$, we have \begin{eqnarray}\label{3.2.1} \begin{cases} \partial_{x_1}(\rho^\varepsilon u_1^\varepsilon)+\partial_{x_2}(\rho^\varepsilon u_2^\varepsilon)=0,\\[2mm] \partial_{x_1}(\rho^\varepsilon u_1^\varepsilon B^\varepsilon)+\partial_{x_2}(\rho^\varepsilon u_2^\varepsilon B^\varepsilon)=0,\\[2mm] \partial_{x_1}(\rho^\varepsilon u_1^\varepsilon S^\varepsilon)+\partial_{x_2}(\rho^\varepsilon u_2^\varepsilon S^\varepsilon)=0. \end{cases} \end{eqnarray}
From $(\ref{3.2.1})_1$, we introduce the following stream function $\psi^\varepsilon$: \begin{eqnarray} \begin{cases} \partial_{x_1}\psi^\varepsilon=-\rho^\varepsilon u_2^\varepsilon,\\[2mm] \partial_{x_2}\psi^\varepsilon=\rho^\varepsilon u_1^\varepsilon, \end{cases} \end{eqnarray} which means that $\psi^\varepsilon$ is constant along the streamlines.
From the far-field behavior of the Euler flows, we define $$ \psi^\varepsilon_-(x_2):=\lim\limits_{x_1\rightarrow-\infty}\psi^\varepsilon(x_1, x_2). $$ Since both the upstream Bernoulli and entropy functions are given, $B^\varepsilon$ and $S^\varepsilon$ have the following expression: $$ B^\varepsilon(x)=B_{-}((\psi^\varepsilon_{-})^{-1}(\psi^\varepsilon(x))), \qquad S^\varepsilon(x)=S_{-}((\psi^\varepsilon_{-})^{-1}(\psi^\varepsilon(x))), $$ where $(\psi^\varepsilon_{-})^{-1}\psi^\varepsilon(x)$ is a function from $\Omega$ to $[0,1]$. For fixed $x_1$, it can be regarded as a backward characteristic map with $$ \frac{\partial ((\psi^\varepsilon_{-})^{-1}\psi^\varepsilon)}{\partial x_2} =\frac{\rho^\varepsilon u_1^\varepsilon}{\rho^\varepsilon_-u^\varepsilon_-}>0. $$ The boundedness and positivity of $\rho^\varepsilon_-u_-^\varepsilon$ and $\rho^\varepsilon u_1^\varepsilon$ show that the map is not degenerate. Thus, we have \begin{eqnarray} \begin{cases} \partial_{x_1}B^\varepsilon(x)=-B'_{-}((\psi^\varepsilon_{-})^{-1}(\psi^\varepsilon(x)))\frac{\rho^\varepsilon u^\varepsilon_2}{\rho^\varepsilon_-u^\varepsilon_{-}},\\[2mm] \partial_{x_2}B^\varepsilon(x)=B'_{-}((\psi^\varepsilon_{-})^{-1}(\psi^\varepsilon(x)))\frac{\rho^\varepsilon u^\varepsilon_1}{\rho^\varepsilon_-u^\varepsilon_-}. \end{cases} \end{eqnarray} Then $B^\varepsilon$ is uniformly bounded in $BV$, which implies its strong convergence. The similar argument can lead to the strong convergence of $S^\varepsilon$.
2. For the corresponding vorticity sequence $\omega^\varepsilon$, (\ref{2.17}) can be written as \begin{eqnarray} \begin{cases} \partial_{x_1} B^\varepsilon= u_2^\varepsilon \omega^\varepsilon+\frac{1}{\gamma}(\rho^\varepsilon)^{\gamma-1}\partial_{x_1} S^\varepsilon,\\[2mm] \partial_{x_2} B^\varepsilon= - u_1^\varepsilon \omega^\varepsilon+\frac{1}{\gamma}(\rho^\varepsilon)^{\gamma-1}\partial_{x_2} S^\varepsilon. \end{cases} \end{eqnarray} By direct calculation, we have \begin{eqnarray} \omega^\varepsilon&=&\frac{1}{(q^\varepsilon)^2} \Big(u_2^\varepsilon\big(\partial_{x_1} B^\varepsilon-\frac{1}{\gamma}(\rho^\varepsilon)^{\gamma-1} \partial_{x_1} S^\varepsilon\big) -u_1^\varepsilon\big(\partial_{x_2} B^\varepsilon-\frac{1}{\gamma}(\rho^\varepsilon)^{\gamma-1}\partial_{x_2} S^\varepsilon\big)\Big)\nonumber\\ &=&\frac{1}{\rho_-^\varepsilon u_-^\varepsilon}\Big(\rho^\varepsilon B_-'-\frac{1}{\gamma}(\rho^\varepsilon)^\gamma S_-'\Big), \end{eqnarray} which implies that $\omega^\varepsilon$ as a measure sequence is uniformly bounded, which is compact in $H^{-1}_{loc}$.
Then Theorem $\ref{thm3.1}$ immediately implies that the solution sequence has a subsequence (still denoted by) $(\rho^\varepsilon, u^{\varepsilon}, p^\varepsilon)(x)$ that converges {\it a.e.} in $\Omega$ to a vector function $(\rho, u, p)(x)$.
Since $(\ref{1.5})$ holds for the sequence of subsonic solutions $(\rho^\varepsilon, u^{\varepsilon}, p^\varepsilon)(x)$, it is straightforward to see that $(\rho, u, p)$ also satisfies $(\ref{1.5})$ in the distributional sense.
3. The boundary condition is satisfied in the sense of Chen-Frid \cite{Chen7}, which implies \begin{equation} \int_{\partial \Omega}\phi(w)(\rho u)(w)\cdot \nu (w)\, d\mathcal{H}^{1}(w)
= \int_{\Omega} (\rho u)(x) \cdot \nabla \phi(x)\, dx + \langle \mbox{div}(\rho u)|_{\Omega}, \phi\rangle \end{equation}
for $\psi\in\mathbf{C}^1_0$. From above, we can see that $\langle \mbox{div}(\rho u)|_{\Omega}, \phi\rangle=0$. Also, \begin{equation} \int_{\Omega}(\rho u)(x) \cdot \nabla \phi(x)\, dx =\lim\limits_{\varepsilon \rightarrow 0}\int_{\Omega}(\rho^\varepsilon u^\varepsilon)(x) \cdot \nabla \phi(x)\, dx=0. \end{equation} Then we have \begin{equation} \int_{\partial \Omega}\phi(w)(\rho u)(w)\cdot\nu(w)\, d\mathcal{H}^{1}(w)=0, \end{equation} that is, $(\rho u)\cdot \nu = 0$ on $\partial \Omega$ in $\mathcal{D}'$.
This completes the proof.
\section{Subsonic-Sonic Limit for the Full Euler Flows \\ in an Infinitely Long Axisymmetric Nozzle}
We consider flows though an infinitely long axisymmetric nozzle given by \begin{equation*} \Omega=\{(x_1,x_2,x_3)\in\mathbb{R}^3\, : \, 0\leq \sqrt{x_2^2+x_3^2}<f(x_1),~ -\infty<x_1< \infty\}, \end{equation*} where $f(x_1)$ satisfies \begin{align}\label{dd-2} &f(x_1)\rightarrow 1 \qquad \mbox{as} ~x_1\rightarrow-\infty,\nonumber\\ &f(x_1)\rightarrow r_0 \qquad \mbox{as} ~x_1\rightarrow \infty,\nonumber\\
&\|f\|_{C^{2,\alpha}(\mathbb{R})}\leq C \qquad \mbox{for} ~\mbox{some} ~\alpha>0 ~ \mbox{and} ~C>0, \\ & \inf_{x_1\in\mathbb{R}}f(x_1)=b>0. \end{align} See Fig. \ref{Fig3}.
\begin{figure}
\caption{Infinitely Long Axisymmetric Nozzle}
\label{Fig3}
\end{figure}
The boundary condition is set as follows: Since the nozzle wall is solid, the flow satisfies the slip boundary condition: \begin{equation}\label{dL-3} u\cdot\nu=0 \qquad \mbox{on} ~\partial \Omega, \end{equation} where $u=(u_1,u_2,u_3)$, and $\nu=(\nu_1, \nu_2, \nu_3)$ is the unit outward normal to the nozzle wall. In the flow without vacuum, it can be written as \begin{equation}\label{dL-sc} (\rho u) \cdot\nu =0 \qquad \mbox{on} ~\partial \Omega. \end{equation} The continuity equation in $(\ref{1.5})_1$ and the boundary condition $(\ref{dL-sc})$ imply that the mass flux \begin{equation}\label{dd-4} \int_\Sigma \, (\rho u)\cdot l\, ds\equiv m_0 \end{equation} remains for some positive constant $m_0$, where $\Sigma$ is any surface transversal to the $x_1$--axis direction, and $l$ is the normal of $\Sigma$ in the positive $x_1$--axis direction.
In Duan-Luo \cite{Duan-Luo}, the axisymmetric flows without swirl are considered for the fluid density $\rho=\rho(x_1,r)$, the velocity $$ u=(u_1, u_2, u_3)=(U(x_1,r), V(x_1, r)\frac{x_2}{r}, V(x_1, r)\frac{x_3}{r}), $$ and the pressure $p=p(x_1, r)$ in the cylindrical coordinates, where $u_1$, $u_2$, $u_3$ are the axial velocity, radial velocity, and swirl velocity, respectively, and $r=\sqrt{x_2^2+x_3^2}$. Then, instead of $(\ref{1.5})$, we have \begin{equation}\label{dd-6}\left\{ \begin{split} &\partial_{x_1}(r\rho U)+\partial_r(r\rho V)=0,\\ &\partial_{x_1}(r\rho U^2)+\partial_r(r\rho UV)_r+r\partial_{x_1}p=0,\\ &\partial_{x_1}(r\rho UV)+\partial_r(r\rho V^2)_r+r\partial_rp=0,\\ &\partial_{x_1}(r \rho U (E+\frac{p}{\rho}))+\partial_r(r \rho V (E+\frac{p}{\rho}))=0. \end{split} \right.\end{equation} Rewrite the axisymmetric nozzle as \begin{equation*} \Omega=\{(x_1,r)\, :\, 0\leq r<f(x_1), ~-\infty<x_1<\infty\} \end{equation*} with the boundary of the nozzle: \begin{align*} \partial \Omega=\{(x_1,r)\, :\, r=f(x), ~-\infty<x_1<\infty\}. \end{align*}
The boundary condition $(\ref{dL-3})$ becomes \begin{equation}\label{dd-7} (U,V,0)\cdot \tilde{\nu}=0 \qquad \mbox{on} ~\partial \Omega, \end{equation} where $\tilde{\nu}$ is the unit outer normal of the nozzle walls in the cylindrical coordinates. The mass flux condition $(\ref{dd-4})$ can be rewritten in the cylindrical coordinates as \begin{equation}\label{dd-9} \int_\Sigma (r\rho U, r \rho V,0)\cdot {l}\, dS\equiv m:=\frac{m_0}{2\pi}, \end{equation} where $\Sigma$ is any curve transversal to the $x_1$-axis direction, and ${l}$ is the unit normal of $\Sigma$.
The quantities $B=h(\rho, p)+\frac{U^2+V^2}{2}$ and $S=\frac{\gamma p}{(\gamma-1) \rho^\gamma}$ are both constants along each streamline. For the full Euler flows in the axisymmetric nozzle, we assume that the upstream Bernoulli and entropy functions are given, that is, \begin{equation}\label{dd-13} h(\rho,p)+\frac{U^2+V^2}{2}\longrightarrow B_-(r) \qquad \mbox{as} ~x_1\rightarrow -\infty, \end{equation} \begin{equation}\label{dd-13s} \frac{\gamma p}{(\gamma-1) \rho^\gamma}\longrightarrow S_-(r) \qquad \mbox{as} ~x_1\rightarrow -\infty, \end{equation} where $B_-(r)$ and $S_-(r)$ are smooth functions defined on $[0, 1]$.
Set \begin{equation} \underline{B}=\inf\limits_{r\in[0,1]} B_-(r),
\quad \sigma_1=\|B'_-\|_{C^{0,1}([0,1])}, \end{equation} \begin{equation} \underline{S}=\inf\limits_{r\in[0,1]} S_-(r),
\quad \sigma_2=\|S'_-\|_{C^{0,1}([0,1])}. \end{equation} We denote the above problem as $\mathbf{Problem~2} (m)$. It is shown in \cite{Duan-Luo} that
\begin{theorem}\label{thm4.5} Suppose that the nozzle satisfies $(\ref{dd-2})$. Let the upstream Bernoulli function $B_-(r)$ and entropy function $S_-(r)$ satisfy $\underline{B}>0$, $B_{-}'(r)\in C^{1,1}([0,1])$, $B_{-}'(0)=0$, $B_{-}'(r)\geq 0$ on $r\in [0,1]$; and $\underline{S}>0$, $S_{-}'(r)\in C^{1,1}([0,1])$, $S_{-}'(0)=0$, $S_{-}'(r)\leq 0$ on $r\in [0,1]$. Then
\begin{enumerate} \item[(i)] There exists $\delta_0>0$ such that, if $\delta:=\max\{\sigma_1, \sigma_2\}\leq\delta_0$, then there is $\hat{m}\leq 2\delta_0^{{1}/{8}}$ so that, for any $m\in (\delta^{{1}/{4}},\hat{m})$, there exists a global $C^1$--solution ({\it i.e.} a full Euler flow) $(\rho, U, V, p)\in C^1(\overline{\Omega})$ through the nozzle with mass flux condition $(\ref{dd-9})$ and the upstream asymptotic condition $(\ref{dd-13})$. Moreover, the flow is uniformly subsonic, and the axial velocity is always positive, {\it i.e.}, \begin{equation}\label{dd2} \sup_{\overline{\Omega}}(U^2+V^2-c^2)<0 \quad \mbox{and} \quad U>0 \qquad \mbox{in} ~\overline{\Omega}. \end{equation}
\item[(ii)] The subsonic flow satisfies the following properties: As $x_1\rightarrow-\infty$, \begin{align}\label{dd4} &\rho\rightarrow\rho_->0, \qquad \nabla\rho\rightarrow 0, \qquad p\rightarrow \frac{\gamma-1}{\gamma}S_-(r)\rho_-^\gamma, \qquad \nabla p\rightarrow (0, \frac{\gamma-1}{\gamma}S_-'(r)\rho_-^\gamma), \nonumber\\ &(U, V)\rightarrow (U_-(r), 0), \qquad \nabla U\rightarrow(0,U_-'(r)), \qquad \nabla V\rightarrow 0 \end{align} uniformly for $r\in K_1\Subset(0,1)$, where $\rho_-$ is a positive constant, and $\rho_-$ and $U_-(r)$ can be determined by $m$, $B_-(r)$, and $S_-(r)$ uniquely.
\item[(iii)] There exists at most one smooth axisymmetric subsonic flow through the nozzle which satisfies $\eqref{dd2}$ and the properties in {\rm (ii)}.
\item[(iv)] There exists a critical mass flux $\hat{m}$ such that, for any $m\in(\delta^{{1}/{4}}, \hat{m})$, there exists a unique axisymmetric subsonic flow through the nozzle with the mass flux condition $(\ref{dd-9})$ and the asymptotic behavior $(\ref{dd4})$. Moreover, $\hat{m}$ is the upper critical mass flux for the existence of subsonic flow in the following sense: Either $\sup\limits_{\overline{\Omega}}(U^2+V^2-c^2)\rightarrow 0$ as $m\rightarrow \hat{m}$, or there is no $\sigma>0$ such that, for all $m\in(\hat{m} ,\hat{m} +\sigma)$, there is an Euler flow with the mass flux $m$ through the nozzle which satisfies the upstream asymptotic condition $(\ref{dd-13})$--$(\ref{dd-13s})$, the downstream asymptotic behavior $(\ref{dd4})$, and $ \sup\limits_{m\in(\hat{m},\hat{m}+\sigma)}\sup\limits_{\overline{\Omega}} (c^2-(U^2+V^2))>0. $ \end{enumerate} \end{theorem}
As above, we have the subsonic-sonic limit theorem for this case.
\begin{theorem}[Subsonic-sonic limit of three-dimensional Euler flows through an axisymmetric nozzle]\label{thm4.3} Let $\delta^{{1}/{4}}<m^{\varepsilon}<\hat{m}$ be a sequence of mass fluxes, and let $\rho^\varepsilon$, $u^\varepsilon=(u^\varepsilon_1, u^\varepsilon_2, u^\varepsilon_3)$, and $p^\varepsilon$ be the corresponding solutions to $\mathbf{Problem~2}~(m^\varepsilon)$. Then, as $m^{\varepsilon}\rightarrow \hat{m}$, the solution sequence possesses a subsequence (still denoted by) $(\rho^\varepsilon, u^{\varepsilon}, p^\varepsilon)$ that converges strongly {\it a.e.} in $\Omega$ to a vector function $(\rho, u, p)$ with $u=(u_1, u_2, u_3)$ which is a weak solution of $\mathbf{Problem~2} (\hat{m})$. Furthermore, the limit solution $(\rho, u, p)$ also satisfies $(\ref{1.5})$ in the distributional sense and the boundary conditions $(\ref{cdx-sc})$ as the normal trace of the divergence-measure field $(\rho u_1,\rho u_2, \rho u_3)$ on the boundary in the sense of Chen-Frid {\rm \cite{Chen7}}. \end{theorem}
\noindent{\bf Proof}. First, we need to show that $(\rho^\varepsilon, u^\varepsilon, p^\varepsilon)$ satisfy condition $(A.1)$--$(A.3)$ in $\Omega$.
For the approximate solutions, $B^\varepsilon$ and $S^\varepsilon$ satisfy \begin{eqnarray} && \partial_{x_1}(r\rho^\varepsilon U^\varepsilon B^\varepsilon)+\partial_r (r\rho^\varepsilon V^\varepsilon B^\varepsilon) = 0,\\[2mm] && \partial_{x_1}(r\rho^\varepsilon U^\varepsilon S^\varepsilon)+\partial_r (r\rho^\varepsilon V^\varepsilon S^\varepsilon) = 0. \end{eqnarray} From $\partial_{x_1}(r\rho^\varepsilon U^\varepsilon)+\partial_r (r\rho^\varepsilon V^\varepsilon)=0$, we introduce $\psi^\varepsilon$ as \begin{eqnarray} \begin{cases} \partial_{x_1}\psi^\varepsilon=-r \rho^\varepsilon V^\varepsilon,\\[2mm] \partial_{r}\psi^\varepsilon=r \rho^\varepsilon U^\varepsilon. \end{cases} \end{eqnarray} From the far-field behavior of the Euler flows, we define $\psi^\varepsilon_-(r):=\lim\limits_{x_1\rightarrow-\infty}\psi^\varepsilon(x_1, r)$. Similar to the argument in Theorem \ref{thm5.2}, $(\psi^\varepsilon_{-})^{-1}(\psi^\varepsilon)$ are nondegenerate maps. A direct calculation yields \begin{eqnarray*} && B^\varepsilon(x_1, x_2, x_3)=B_{-}((\psi^\varepsilon_{-})^{-1}(\psi^\varepsilon(x_1, \sqrt{x_2^2+x_3^2}))),\\[2mm] && S^\varepsilon(x_1, x_2, x_3)=S_{-}((\psi^\varepsilon_{-})^{-1}(\psi^\varepsilon(x_1, \sqrt{x_2^2+x_3^2}))). \end{eqnarray*} Therefore, we have \begin{eqnarray} \begin{cases} \partial_{x_1}B^\varepsilon=-r\rho^\varepsilon V^\varepsilon\frac{B'_{-}}{(\psi^\varepsilon_-)'},\\[2mm] \partial_{x_2}B^\varepsilon= x_2\rho^\varepsilon U^\varepsilon\frac{B'_{-}}{(\psi^\varepsilon_-)'},\\[2mm] \partial_{x_3}B^\varepsilon=x_3\rho^\varepsilon U^\varepsilon\frac{B'_{-}}{(\psi^\varepsilon_-)'}. \end{cases} \end{eqnarray} Notice that \begin{equation} \frac{B'_-}{(\psi^\varepsilon_-)'}((\psi^\varepsilon_-)^{-1}(\psi^\varepsilon)) =\frac{B'_-((\psi^\varepsilon_-)^{-1}(\psi^\varepsilon))} {(\psi^\varepsilon_-)^{-1}(\psi^\varepsilon)\rho^\varepsilon_-U^\varepsilon_-((\psi^\varepsilon_-)^{-1}(\psi^\varepsilon))}. \end{equation} Since $B'_-(0)=0$ and $B_-\in C^{1,1}$, we conclude that $\frac{B'_-(s)}{s}$ is bounded. Then the sequence $B^\varepsilon$ is uniformly bounded in $BV$, which implies its strong convergence. The similar argument can lead to the strong convergence of $S^\varepsilon$.
On the other hand, the vorticity $\omega^\varepsilon$ has the following expression: \begin{eqnarray} \begin{cases} \omega_{1,2}^\varepsilon=\partial_{x_1}u_2^\varepsilon-\partial_{x_2}u_1^\varepsilon
=\frac{x_2}{r}(\partial_{x_1}V^\varepsilon-\partial_{r}U^\varepsilon),\\[2mm] \omega_{2,3}^\varepsilon=\partial_{x_2}u_3^\varepsilon-\partial_{x_3}u_2^\varepsilon=0,\\[2mm] \omega_{3,1}^\varepsilon=\partial_{x_3}u_1^\varepsilon-\partial_{x_1}u_3^\varepsilon
=-\frac{x_3}{r}(\partial_{x_1}V^\varepsilon-\partial_{r}U^\varepsilon). \end{cases} \end{eqnarray} A direct computation gives \begin{equation} \partial_{x_1}V^\varepsilon-\partial_{r}U^\varepsilon =\frac{r}{(\psi^\varepsilon_-)'}\big(\rho^\varepsilon B_-'-\frac{(\rho^\varepsilon)^\gamma S_-'}{\gamma}\big), \end{equation} which implies that $\omega^\varepsilon$ is uniformly bounded in the bounded measure space.
Since $(\ref{1.5})$ holds for the sequence of subsonic solutions $(\rho^\varepsilon, u^{\varepsilon}, p^\varepsilon)(x)$, it is straightforward to see from Theorem $\ref{thm3.1}$ that there exists a subsequence (still denoted by) $(\rho^\varepsilon,u^\varepsilon,p^\varepsilon)$ which converges to a vector function $(\rho,u,p)$ {\it a.e.} in $\Omega$ satisfying $(\ref{1.5})$ in the distributional sense.
The boundary condition is satisfied in the sense of Chen-Frid \cite{Chen7}, which implies \begin{equation} \int_{\partial \Omega}\phi(w) (\rho u)(w)\cdot \nu (w)\, d\mathcal{H}^{1}(w)
= \int_{\Omega} (\rho u)(x) \cdot \nabla \phi(x)\, dx + \langle \mbox{div}(\rho u)|_{\Omega}, \phi\rangle \end{equation} for $\psi\in\mathbf{C}^1_0$. From above, we can see
$\langle \mbox{div}(\rho u)|_{\Omega}, \phi\rangle=0$. Furthermore, we have \begin{equation} \int_{\Omega}(\rho u)(x) \cdot \nabla \phi(x)\, dx =\lim\limits_{\varepsilon \rightarrow 0} \int_{\Omega}(\rho^\varepsilon u^\varepsilon)(x)\cdot \nabla \phi(x)\, dx=0, \end{equation} which yields \begin{equation} \int_{\partial \Omega}\phi(w)(\rho u)(w)\cdot\nu(w) \, d\mathcal{H}^{1}(w)=0, \end{equation} that is, $(\rho u)\cdot \nu = 0$ on $\partial\Omega$ in $\mathcal{D}'$.
This completes the proof.
\begin{remark} In the homentropic case, the subsonic results of \cite{CX,DD,Xin4} can be also extended to the subsonic-sonic limit by using Theorem $\ref{thm2.1}$. \end{remark}
\noindent {\bf Acknowledgments:} The research of Gui-Qiang Chen was supported in part by the UK EPSRC Science and Innovation Award to the Oxford Centre for Nonlinear PDE (EP/E035027/1), the NSFC under a joint project Grant 10728101, and the Royal Society--Wolfson Research Merit Award (UK). The research of Fei-Min Huang was supported in part by NSFC Grant No. 10825102 for distinguished youth scholars, and the National Basic Research Program of China (973 Program) under Grant No. 2011CB808002. The research of Tian-Yi Wang was supported in part by the China Scholarship Council No. 201204910256 as an exchange graduate student at the University of Oxford, the UK EPSRC Science and Innovation Award to the Oxford Centre for Nonlinear PDE (EP/E035027/1), and the NSF of China under Grant 11371064.
\end{document}
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arXiv
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\begin{document}
\title{Topological transformation monoids} \author{Z.\ Mesyan, J.\ D.\ Mitchell, and Y.\ P\'eresse}
\maketitle
\begin{abstract}
We investigate semigroup topologies on the full transformation monoid $\Omega^{\Omega}$ of an infinite set $\Omega$. We show that the standard pointwise topology is the weakest Hausdorff semigroup topology on $\Omega^{\Omega}$, show that this topology is the unique Hausdorff semigroup topology on $\Omega^{\Omega}$ that induces the pointwise topology on the group $\mathrm{Sym}(\Omega)$ of all permutations of $\Omega$, and construct $|\Omega|$ distinct Hausdorff semigroup topologies on $\Omega^{\Omega}$. In the case where $\Omega$ is countable, we prove that the pointwise topology is the only Polish semigroup topology on $\Omega^{\Omega}$. We also show that every separable semigroup topology on $\Omega^{\Omega}$ is perfect, describe the compact sets in an arbitrary Hausdorff semigroup topology on $\Omega^{\Omega}$, and show that there are no locally compact perfect Hausdorff semigroup topologies on $\Omega^{\Omega}$ when $|\Omega|$ has uncountable cofinality.
\noindent \emph{Keywords:} transformation monoid, topological semigroup, semitopological semigroup, pointwise topology, Polish space
\noindent \emph{2010 MSC numbers:} 08A35, 20M20, 54H15 \end{abstract}
\section{Introduction}
Recall that a \emph{topological group} is a group $G$ together with a topology on $G$ that makes the multiplication and inversion operations continuous. We shall refer to topologies of this sort as \emph{group topologies}. The discrete and trivial topologies are group topologies on every group, but the question of finding interesting group topologies has received a great deal of attention in the literature. We begin with a brief overview of this literature, to motivate our work in this paper.
Markov~\cite{Markoff1944aa} sparked significant interest in the subject when he asked whether there exist infinite groups with no nondiscrete Hausdorff group topologies. This question was answered in the affirmative by Shelah~\cite{Shelah1980aa}, assuming the continuum hypothesis, and, in ZFC, by Olshanskii~\cite{Olshanskij1980aa} in the same year. There are even infinite groups where every quotient of every subgroup has no nondiscrete Hausdorff group topologies~\cite{Klyachko2012aa}.
A substantial portion of the literature on topological groups has focused on \emph{Polish} groups, which arise in many areas of mathematics, particularly in descriptive set theory. These are topological groups where the topology is \emph{Polish}, that is, completely metrizable and separable. Compared to the situation explored by Markov, it is easy to find examples both of Polish groups and of topological groups that are not Polish. Specifically, since by the Cantor-Bendixson Theorem~\cite[Theorem 6.4]{Kechris}, every Polish space has cardinality either at most $\aleph_0$ or equal to $2^{\aleph_0}$, any topological group with cardinality greater than $2^{\aleph_0}$ is not Polish. Moreover, there is no nondiscrete Polish group topology on any free group. (This follows from the result of Dudley~\cite{Dudley1961aa} that every homomorphism from a complete metric group to a free group, with the discrete topology, is continuous.) On the other end of the spectrum are Polish groups with infinitely many non-homeomorphic Polish group topologies. Perhaps the simplest such example is that of the additive group $\mathbb{R}$ of real numbers, which is a Polish group with respect to the standard topology. As vector spaces over the field $\mathbb{Q}$ of the rational numbers, $\mathbb{R}$ and $\mathbb{R}^n$ are isomorphic for every positive integer $n$, and so $\mathbb{R}$ and $\mathbb{R}^n$ are isomorphic as additive groups also. But $\mathbb{R}^n$ is homeomorphic to $\mathbb{R}^m$ if and only if $m = n$, and so there are infinitely many non-homeomorphic Polish group topologies on $\mathbb{R}$. In contrast, Shelah~\cite{Shelah1984aa} showed that it is consistent with the Zermelo-Fraenkel axioms of set theory, without the axiom of choice, that every Polish group has a unique Polish group topology. Of course, in the above example, the axiom of choice was needed to show that $\mathbb{R}$ and $\mathbb{R}^n$ are isomorphic.
One of the most extensively studied topological groups is the group $\mathrm{Sym}(\Omega)$ of all permutations of a set $\Omega$, which has a natural group topology, known as the \emph{pointwise} topology. This topology is Polish in the case where $\Omega$ is countable. Gaughan~\cite{Gaughan} showed that every Hausdorff group topology on $\mathrm{Sym}(\Omega)$ contains the pointwise topology, and that there is no nondiscrete locally compact Hausdorff group topology on $\mathrm{Sym}(\Omega)$. The latter answered problem 96 in the Scottish Book~\cite{Mauldin2015aa}, posed by Ulam. Kechris and Rosendal~\cite{KR} showed that any homomorphism from $\mathrm{Sym}(\Omega)$ into a separable group is continuous, which together with Gaughan's result about the pointwise topology implies that there is a unique Polish group topology on $\mathrm{Sym}(\Omega)$, when $\Omega$ is countable. Moreover, Rosendal~\cite{Rosendal2005aa} extended Gaughan's local compactness result to show that there are not even any non-trivial homomorphisms from $\mathrm{Sym}(\Omega)$ into a locally compact Polish group. Gaughan's result regarding the pointwise topology was also recently extended, by Banakh, Guran, and Protasov~\cite{BGP}, to any subgroup of $\mathrm{Sym}(\Omega)$ containing the permutations with finite support.
There are many further examples of groups with a unique Polish group topology, such as the group of isometries of Minkowski spacetime (the usual framework for special relativity)~\cite{Kallman2010aa}; see also \cite{Gartside2008aa, Kallman1986aa}. Furthermore, there is a wealth of examples of infinite groups with no Polish group topologies~\cite{Cohen2016aa, Mann2017aa}. Additional references include \cite{Chang2017aa, Dixon, Hart2014aa}.
\emph{Topological semigroups}, that is, semigroups with topologies that make multiplication continuous, have received somewhat less attention in the literature than topological groups. However, some notable recent papers on the topic include~\cite{Banakh2014aa, Bodirsky2017aa, Bodirsky2017-2}.
In this paper we focus on the natural semigroup analogue of $\mathrm{Sym}(\Omega)$, namely the semigroup $\Omega^{\Omega}$ of all functions from $\Omega$ to $\Omega$, which also has a natural \emph{pointwise} topology. Our goal is to explore the pointwise topology on $\Omega^{\Omega}$ in detail, along with semigroup topologies on $\Omega^{\Omega}$ in general. In Section~\ref{basics-section} we give a brief review of the basics of topological semigroups, along with some methods for constructing topologies on semigroups. We then show in Theorem~\ref{point-finer} that the pointwise topology is the weakest $T_1$ (and hence also Hausdorff) semigroup topology on $\Omega^{\Omega}$. More generally, the same holds for the topology induced by the pointwise topology on any subsemigroup of $\Omega^{\Omega}$ that contains all the elements of ranks $1$ and $2$. This is analogous to the result of Gaughan regarding the pointwise topology on $\mathrm{Sym}(\Omega)$ mentioned above.
Using Theorem~\ref{point-finer} we show in Proposition~\ref{group-top} that the pointwise topology is the unique $T_1$ semigroup topology on $\Omega^{\Omega}$ that induces the pointwise topology on $\mathrm{Sym}(\Omega)$. From this we conclude in Theorem~\ref{polish-thrm} that, analogously to a result about $\mathrm{Sym}(\Omega)$ stated above, the pointwise topology is the only Polish semigroup topology on $\Omega^{\Omega}$, when $\Omega$ is countably infinite. We also show in Theorem~\ref{many-isolated} that if $\Omega$ is infinite, then there are either no isolated points or $2^{\operatorname{cf}(|\Omega|)}$ isolated points in any semigroup topology on $\Omega^{\Omega}$, where $\operatorname{cf}(|\Omega|)$ denotes the cofinality of $|\Omega|$. In particular, every separable semigroup topology on $\Omega^{\Omega}$ is perfect (i.e., has no isolated points), when $\Omega$ is infinite (Corollary~\ref{separable-perfect}).
In Section~\ref{compact-section} we describe the compact sets in an arbitrary $T_1$ semigroup topology on $\Omega^{\Omega}$ (Proposition~\ref{compact-prop}). We then show that there are no locally compact perfect $T_1$ semigroup topologies on $\Omega^{\Omega}$ when $|\Omega|$ has uncountable cofinality (Theorem~\ref{loc-comp-perf}). This is a partial analogue of Gaughan's local compactness result mentioned above.
Along the way we give various examples to illustrate our results. For instance, we construct $|\Omega|$ distinct Hausdorff semigroup topologies on $\Omega^{\Omega}$, for $\Omega$ infinite (Proposition~\ref{top-chain-prop}). We also construct a natural perfect Hausdorff semigroup topology on $\Omega^{\Omega}$ for $\Omega$ of regular cardinality, which is separable when $\Omega$ is countable, and has a very different flavour from the pointwise topology (Proposition~\ref{open-prod-top}). The paper concludes with an open question.
\section{Topological semigroups} \label{basics-section}
A \emph{topological semigroup} is a semigroup $S$ together with a topology on $S$, such that the semigroup multiplication, viewed as a function $\mathfrak{m} : S\times S \to S$, is continuous; where the topology on $S\times S$ is the corresponding product topology. A semigroup is \emph{semitopological} if for every $s\in S$ the maps $\mathfrak{r}_s:S\rightarrow S$ and $\mathfrak{l}_s:S\rightarrow S$ induced by right, respectively left, multiplication by $s$ are continuous with respect to the relevant topology. It is a standard and easily verified fact that every topological semigroup is semitopological.
Next we give several simple but useful methods for constructing topologies on semigroups.
\begin{lem} \label{hom-lemma} Let $f : S_1 \to S_2$ be a homomorphism of semigroups. Suppose that $S_2$ is a topological semigroup with respect to a topology $\, \mathcal{T}_2$, and let $\, \mathcal{T}_1$ be the least topology on $S_1$ such that $f$ is continuous. Then $\, \mathcal{T}_1 = \set{(A)f^{-1}}{A \in \mathcal{T}_2}$, and $S_1$ is a topological semigroup with respect to $\, \mathcal{T}_1$. \end{lem}
\begin{proof} Noting that $(\bigcup_{i \in I} A_i)f^{-1} = \bigcup_{i \in I} (A_i)f^{-1}$ and $(\bigcap_{i \in I} A_i)f^{-1} = \bigcap_{i \in I} (A_i)f^{-1}$ for any collection $\set{A_i}{i \in I}$ of subsets of $S_2$, it follows that $\set{(A)f^{-1}}{A \in \mathcal{T}_2}$ is a topology on $S_1$. Clearly, this topology is contained in $\mathcal{T}_1$, and $f$ is continuous with respect to it. Therefore $\mathcal{T}_1 = \set{(A)f^{-1}}{A \in \mathcal{T}_2}$.
Next, let $\mathfrak{m}_i : S_i \times S_i \to S_i$ denote the multiplication map on $S_i$, for $i \in \{1, 2\}$, and define $f_c : S_1 \times S_1 \to S_2 \times S_2$ by $(x,y)f_c = ((x)f,(y)f)$ for all $x,y \in S_1$. Then, viewing $S_1 \times S_1$ and $S_2 \times S_2$ as topological spaces in the product topologies induced by $\mathcal{T}_1$ and $\mathcal{T}_2$, respectively, $f_c$ is continuous, since both of its coordinate functions, namely $f : S_1 \to S_2$, are continuous. (See, e.g.,~\cite[Exercise 18.10]{Munkres}.)
To show that $\mathfrak{m}_1$ is continuous, let $A \in \mathcal{T}_1$, and let $A'\in \mathcal{T}_2$ be such that $A = (A')f^{-1}$. Then $$(x,y) \in (A)\mathfrak{m}_1^{-1} \Leftrightarrow xy \in (A')f^{-1} \Leftrightarrow (x)f(y)f \in A'$$ $$\Leftrightarrow ((x)f,(y)f)\in (A')\mathfrak{m}_2^{-1} \Leftrightarrow (x,y)\in (A')\mathfrak{m}_2^{-1}f_c^{-1}$$ for all $x,y \in S_1$, and hence $(A)\mathfrak{m}_1^{-1} = (A')\mathfrak{m}_2^{-1}f_c^{-1}$. Since the composite $f_c \mathfrak{m}_2 : S_1 \times S_1 \to S_2$ of continuous functions is continuous, and $A' \in \mathcal{T}_2$, it follows that $(A)\mathfrak{m}_1^{-1}$ is open in $S_1\times S_1$. Hence $\mathfrak{m}_1$ is continuous, and therefore $S_1$ is a topological semigroup with respect to $\mathcal{T}_1$. \end{proof}
\begin{lem} \label{semi-top-lemma} Let $S$ be a semigroup. \begin{enumerate} \item[$(1)$] Given an ideal $I$ of $S$, let $\, \mathcal{T}_1 = \set{A}{A \subseteq S\setminus I}\cup\{S\}$ and $\, \mathcal{T}_2 = \mathcal{T}_1 \cup \set{A \cup I}{A \in \mathcal{T}_1}$. Then $S$ is a topological semigroup with respect to $\, \mathcal{T}_1$ and $\, \mathcal{T}_2$.
\item[$(2)$] Let $\, \mathcal{T}_1$ and $\, \mathcal{T}_2$ be topologies on $S$, and let $\, \mathcal{T}_3$ be the topology generated by $\, \mathcal{T}_1 \cup \mathcal{T}_2$. If $S$ is a topological semigroup with respect to $\, \mathcal{T}_1$ and $\, \mathcal{T}_2$, then the same holds for $\, \mathcal{T}_3$. \end{enumerate} \end{lem}
\begin{proof} Let $\mathfrak{m} : S \times S \to S$ denote the multiplication map on $S$.
(1) It is easy to see that $\mathcal{T}_1$ is closed under unions and (finite) intersections, and is hence a topology on $S$. Since $I$ is an ideal, $(A) \mathfrak{m}^{-1} \subseteq (S\setminus I) \times (S\setminus I)$, and hence $(A) \mathfrak{m}^{-1}$ is open in the product topology on $S \times S$ induced by $\mathcal{T}_1$, for any $A \in \mathcal{T}_1 \setminus \{S\}$. It follows that $S$ is a topological semigroup with respect to $\mathcal{T}_1$.
Next, let $f: S \to S/I$ be the natural homomorphism. Then the elements of $\mathcal{T}_2$ are precisely the preimages under $f$ of the subsets open in the discrete topology on $S/I$. Thus $S$ is a topological semigroup with respect to $\mathcal{T}_2$, by Lemma~\ref{hom-lemma}.
(2) Let $A \in \mathcal{T}_3$. Then $A = \bigcup_i (B_i \cap C_i)$ for some $B_i \in \mathcal{T}_1$ and $C_i \in \mathcal{T}_2$. Thus $$(A)\mathfrak{m}^{-1} = \bigcup_i ((B_i \cap C_i)\mathfrak{m}^{-1}) = \bigcup_i ((B_i)\mathfrak{m}^{-1} \cap (C_i)\mathfrak{m}^{-1}).$$ Since $S$ is a topological semigroup with respect to $\mathcal{T}_1$ and $\mathcal{T}_2$, for each $i$ we can write $(B_i)\mathfrak{m}^{-1} = \bigcup_j (B_{ij} \times B'_{ij})$ and $(C_i)\mathfrak{m}^{-1} = \bigcup_l (C_{il} \times C'_{il})$, for some $B_{ij}, B'_{ij} \in \mathcal{T}_1$ and $C_{il}, C'_{il} \in \mathcal{T}_2$. Thus $$(A)\mathfrak{m}^{-1} = \bigcup_i \bigcup_j \bigcup_l ((B_{ij} \times B'_{ij}) \cap (C_{il} \times C'_{il})) = \bigcup_i \bigcup_j \bigcup_l ((B_{ij} \cap C_{il}) \times (B'_{ij} \cap C'_{il})),$$ since it is easy to see that $$(X_1\times Y_1) \cap (X_2 \times Y_2) = (X_1 \cap X_2) \times (Y_1\cap Y_2)$$ for any sets $X_1$, $X_2$, $Y_1$, $Y_2$. Hence $(A)\mathfrak{m}^{-1}$ is open in the product topology on $S\times S$ induced by $\mathcal{T}_3$. It follows that $S$ is a topological semigroup with respect to $\mathcal{T}_3$. \end{proof}
The two obvious topologies on a semigroup $S$ can be viewed as extremal cases of the constructions in Lemma~\ref{semi-top-lemma}(1). Specifically, if $I=S$, then $\mathcal{T}_1 = \mathcal{T}_2$ is the trivial topology, while if $I = \emptyset$, then $\mathcal{T}_1 = \mathcal{T}_2$ is the discrete topology. As we shall see in Proposition~\ref{top-chain-prop}, the two topologies $\mathcal{T}_1$ and $\mathcal{T}_2$ can also be distinct from each other.
Recall that a topological space $X$ is $T_1$ if for any two distinct points $x, y \in X$ there is an open neighbourhood of $x$ that does not contain $y$, and $X$ is $T_2$, or \emph{Hausdorff}, if for any two distinct points $x,y \in X$ there are open neighbourhoods $U$ and $V$ of $x$ and $y$, respectively, such that $U \cap V = \emptyset$.
We conclude this section with an observation that, while interesting, will not be used in the rest of the paper.
\begin{prop}\label{prop-congruence} Let $S$ be a topological semigroup with respect to a topology $\, \mathcal{T}$, define $$A_x = \bigcap\set{U}{U\in \mathcal{T} \text{ and } x \in U}$$ for every $x\in S$, and let $$\rho_S = \set{(x, y)}{A_x = A_y} \subseteq S\times S.$$ Then the following hold. \begin{enumerate} \item[$(1)$] The relation $\rho_S$ is a congruence on $S$. \item[$(2)$] If $\, \mathcal{T}$ is $T_1$, then $\rho_S = \set{(x, x)}{x \in S}$. \item[$(3)$] The topology $\, \mathcal{T}$ is contained in the least topology on $S$ with respect to which the natural homomorphism $S \to S/\rho_S$ is continuous, where $S/\rho_S$ is endowed with the discrete topology. \end{enumerate} \end{prop}
\begin{proof} (1) It is clear that $\rho_S$ is an equivalence relation on $S$, and so it suffices to show that $(xs, ys), (sx, sy)\in \rho_S$ for all $s\in S$ and $(x,y) \in \rho_S$. Thus suppose that $(x,y) \in \rho_S$, and let $U$ is an open neighbourhood of $xs$, for some $s \in S$. By the continuity of multiplication, there exist open neighbourhoods $V$ and $W$ of $x$ and $s$, respectively, such that $VW \subseteq U$. But $A_x = A_y$, and so, in particular, $y \in V$. Thus $ys\in VW \subseteq U$, which shows that $ys \in A_{xs}$. By symmetry, also $xs \in A_{ys}$, giving $(xs, ys)\in \rho_S$. The proof that $(sx, sy)\in \rho_S$ is dual.
(2) If $\mathcal{T}$ is $T_1$, then clearly $A_x = \{x\}$ for all $x \in S$, which implies that $\rho_S = \set{(x, x)}{x \in S}$.
(3) Let $f: S \to S/\rho_S$ be the natural homomorphism defined by $(s)f = s/\rho_S$, and let $$\mathcal{P}= \set{(V)f^{-1}}{V\subseteq S/\rho_S}$$ be the least topology such that $f$ is continuous (by Lemma~\ref{hom-lemma}). It is straightforward to verify that $U\in \mathcal{P}$ if and only if $U$ is a union of $\rho_S$-classes. Hence to prove that $\mathcal{T} \subseteq \mathcal{P}$, it suffices to show that if $x\in V\in \mathcal{T}$, then $y\in V$ for all $y\in S$ such that $(x, y)\in \rho_S$. But if $(x, y)\in \rho_S$, then every open neighbourhood of $x$ is also an open neighbourhood of $y$, and hence $y\in V$, as required. \end{proof}
\section{The pointwise topology}
Given a set $\Omega$, we denote by $\Omega^{\Omega}$ the \emph{full transformation monoid} of $\Omega$, consisting of all functions from $\Omega$ to $\Omega$, under composition. The \emph{pointwise} (or \emph{function}, or \emph{finite}) \emph{topology} on $\Omega^{\Omega}$ has a base of open sets of the following form: $$[\sigma : \tau] = \set{f \in \Omega^{\Omega}}{(\sigma_i)f = \tau_i \text{ for all } 0 \leq i\leq n},$$ where $\sigma=(\sigma_0, \sigma_1, \dots, \sigma_n)$ and $\tau=(\tau_0, \tau_1, \dots, \tau_n)$ are sequences of elements of $\Omega$, and $n \in \mathbb{N}$ (the set of the natural numbers). It is straightforward to see that this coincides with the product topology on $\Omega^{\Omega} = \prod_\Omega \Omega$, where each component set $\Omega$ is endowed with the discrete topology. As a product of discrete spaces, this space is Hausdorff. It is well-known and easy to see that $\Omega^{\Omega}$ is a topological semigroup with respect to the pointwise topology. If $\Omega$ is finite, then $\Omega^{\Omega}$ is discrete in this topology. Finally, it is a standard fact that if $\Omega$ is countable, then the pointwise topology on $\Omega^{\Omega}$ is Polish, i.e., separable and completely metrisable (see, e.g.,~\cite[Section 3.A, Example 3]{Kechris}).
We are now ready to prove an analogue of a result of Gaughan~\cite[Theorem 1]{Gaughan} for infinite symmetric groups, which shows that the pointwise topology is the weakest Hausdorff semigroup topology on $\Omega^{\Omega}$.
Recall that the \emph{rank} of a function $f \in \Omega^{\Omega}$ is the cardinality of the image $(\Omega)f$ of $f$.
\begin{theorem} \label{point-finer} Let $\, \Omega$ be a set, let $S$ be a subsemigroup of $\, \Omega^{\Omega}$ that contains all the transformations of rank at most $\, 2$, and let $\, \mathcal{T}$ be a topology on $S$ with respect to which $S$ is a semitopological semigroup. Then the following are equivalent. \begin{enumerate} \item[$(1)$] $\, \mathcal{T}$ is Hausdorff. \item[$(2)$] $\, \mathcal{T}$ is $T_1$. \item[$(3)$] Every set of the form $\, [\sigma : \tau] \cap S$ is open in $\, \mathcal{T}$. \emph{(}I.e., $\, \mathcal{T}$ contains the topology induced on $S$ by the pointwise topology on $\, \Omega^{\Omega}$.\emph{)} \item[$(4)$] Every set of the form $\, [\sigma : \tau] \cap S$ is closed in $\, \mathcal{T}$. \end{enumerate} \end{theorem}
\begin{proof} We may assume that $\Omega$ is infinite, since each of the conditions (1)--(4) is equivalent to $\mathcal{T}$ being discrete in the case where $\Omega$ is finite. We note, however, that our arguments below only require that $\Omega$ contains at least $2$ elements. We shall prove that $(1) \Rightarrow (2) \Rightarrow (3) \Rightarrow (1)$ and $(3) \Rightarrow (4) \Rightarrow (2)$.
$(1) \Rightarrow (2)$ This is a tautology.
$(2) \Rightarrow (3)$ Assuming that $\mathcal{T}$ is $T_1$, we start by showing that for any $\alpha,\beta \in \Omega$, the set $$F_{\alpha,\beta}=\set{f\in S}{(\alpha)f\not=\beta}$$ is closed in $\mathcal{T}$. Let $\gamma \in \Omega \setminus\{\alpha\}$, let $f\in S$ be the constant function with image $\alpha$, and define $g\in S$ by $$(\delta)g = \left\{ \begin{array}{ll} \gamma & \text{if } \, \delta=\beta\\ \alpha & \text{if } \, \delta \neq \beta \end{array}\right..$$ Since $\mathcal{T}$ is $T_1$, the singleton $\{f\}$ is closed, and since $S$ is semitopological with respect to $\mathcal{T}$, the composite $\mathfrak{l}_f\circ \mathfrak{r}_g: S \rightarrow S$ of multiplication maps is continuous. Hence $$(f)(\mathfrak{l}_f\circ \mathfrak{r}_g)^{-1}=\set{h\in S}{fhg=f}=F_{\alpha,\beta}$$ must be closed in $\mathcal{T}$.
Let $\sigma=(\sigma_0, \sigma_1, \dots, \sigma_n)$ and $\tau=(\tau_0, \tau_1, \dots, \tau_n)$ be arbitrary sequences of elements of $\Omega$, of the same finite length. Then $$S \setminus ([\sigma : \tau]\cap S) = \bigcup_{i=0}^n F_{\sigma_i,\tau_i},$$ is closed, being a finite union of closed sets. Thus $[\sigma : \tau] \cap S$ is open in $\mathcal{T}$.
$(3) \Rightarrow (1)$ Since the pointwise topology on $\Omega^{\Omega}$ is Hausdorff, so is the topology it induces on $S$, and hence so is any topology on $S$ that contains this induced topology.
$(3) \Rightarrow (4)$ Let $\sigma=(\sigma_0, \sigma_1, \dots, \sigma_n)$ and $\tau=(\tau_0, \tau_1, \dots, \tau_n)$ be arbitrary finite sequences of elements of $\Omega$, of length $n+1$. Then $$S \setminus ([\sigma : \tau]\cap S) = \bigcup_{\phi \neq \tau} ([\sigma : \phi] \cap S),$$ where $\phi=(\phi_0, \phi_1, \dots, \phi_n)$ runs over all sequences of elements of $\Omega$ of length $n+1$, distinct from $\tau$. Thus, if each $[\sigma : \phi] \cap S$ is open in $\mathcal{T}$, then so is $S \setminus ([\sigma : \tau]\cap S)$. In this case $[\sigma : \tau] \cap S$ must be closed in $\mathcal{T}$.
$(4) \Rightarrow (2)$ Let $f \in S$ be any function. Then $$\{f\} = \bigcap_{\alpha \in \Omega} \{g \in S : (\alpha)g = (\alpha)f\} = \bigcap_{\alpha \in \Omega} ([(\alpha) : ((\alpha)f)]\cap S).$$ Thus, if (4) holds, then $\{f\}$ is closed in $\mathcal{T}$. Since $f \in S$ was arbitrary, it follows that $\mathcal{T}$ is $T_1$. \end{proof}
We shall show in Proposition~\ref{top-chain-prop} that $\Omega^{\Omega}$ has many Hausdorff semigroup topologies different from the pointwise one.
Next we give an example of a semitopological subsemigroup $S$ of $\Omega^{\Omega}$ with a topology that is $T_1$ but not Hausdorff, to show the necessity of the hypothesis on $S$ in Theorem~\ref{point-finer}.
Given a set $X$ we denote by $|X|$ the cardinality of $X$.
\begin{eg}
Let $\Omega$ be an infinite set, let $S$ be an infinite subsemigroup of $\Omega^{\Omega}$ consisting of bijections (so $S$ is a subsemigroup of $\mathrm{Sym}(\Omega)$), and let $\mathcal{T}$ be the cofinite topology on $S$. That is, $\mathcal{T}$ consists of precisely $\emptyset$ and the cofinite subsets of $S$ (i.e., $X \subseteq S$ such that $|S\setminus X| < \aleph_0$). Clearly, $\mathcal{T}$ is $T_1$. (In fact, the cofinite topology is the weakest $T_1$ topology on any set.) On the other hand, since $S$ is infinite, the intersection of any two nonempty elements of $\mathcal{T}$ is infinite, and hence $\mathcal{T}$ is not Hausdorff. We shall show that $S$ is semitopological with respect to $\mathcal{T}$.
Let $U \in \mathcal{T} \setminus \{S\}$ be a nonempty open set, write $S\setminus U = \{g_0, \dots, g_n\}$ for some $n \in \mathbb{N}$, and let $f \in S$. Since $f$ is a bijection, and hence invertible both on the left and the right (in $\mathrm{Sym}(\Omega)$), for each $i \in \{0, \dots, n\}$ there can be at most one $h \in S$ such that $fh = g_i$, and at most one $h \in S$ such that $hf = g_i$. Hence $$(U)\mathfrak{l}_f^{-1} = \set{h\in S}{fh \in U} = \set{h \in S}{fh \notin \{g_0, \dots, g_n\}},$$ and therefore $|S \setminus (U)\mathfrak{l}_f^{-1}| \leq n+1$. Similarly, $|S \setminus (U)\mathfrak{r}_f^{-1}| \leq n+1$. It follows that $(U)\mathfrak{l}_f^{-1}, (U)\mathfrak{r}_f^{-1} \in \mathcal{T}$ for all $U \in \mathcal{T}$, and hence $S$ is semitopological with respect to $\mathcal{T}$.
\qedsymbol \end{eg}
Given a set $\Omega$, the \emph{pointwise topology} on the group $\mathrm{Sym}(\Omega)$ of all permutations of $\Omega$ is the subspace topology induced on $\mathrm{Sym}(\Omega)$ by the pointwise topology on $\Omega^{\Omega}$. It is well-known and easy to see that $\mathrm{Sym}(\Omega)$ is a topological group with respect to the pointwise topology. (That is, $\mathrm{Sym}(\Omega)$ is a topological semigroup with respect to this topology, and the inversion map $\mathrm{Sym}(\Omega) \to \mathrm{Sym}(\Omega)$ is continuous.) Moreover, as with $\Omega^{\Omega}$, if $\Omega$ is countable, then the pointwise topology on $\mathrm{Sym}(\Omega)$ is Polish (see, e.g.,~\cite[Section 9.A, Example 7]{Kechris}).
It turns out that the pointwise topology on $\Omega^{\Omega}$ is the only $T_1$ semigroup topology that induces the pointwise topology on $\mathrm{Sym}(\Omega)$.
\begin{prop} \label{group-top} Let $\, \Omega$ be a set, and let $\, \mathcal{P}$ denote the pointwise topology on $\, \Omega^{\Omega}$. Suppose that $\, \Omega^{\Omega}$ is a semitopological semigroup with respect to a $T_1$ topology $\, \mathcal{T}$, and that $\, \mathcal{T}$ induces a topology on $\, \mathrm{Sym}(\Omega)$ that is contained in the pointwise topology. Then $\, \mathcal{T} = \mathcal{P}$. \end{prop}
\begin{proof} We may assume that $\Omega$ is infinite, since otherwise the only $T_1$ topology on $\Omega^{\Omega}$ is the discrete topology, which coincides with $\mathcal{P}$ in this case.
Write $\Omega = \bigcup_{\alpha \in \Omega} \Sigma_{\alpha}$, where the union is disjoint, and $|\Sigma_{\alpha}| = |\Omega|$ for each $\alpha \in \Omega$. Let $g_1 \in \Omega^{\Omega}$ be an injective function such that $|\Omega \setminus (\Omega)g_1| = |\Omega|$, and let $g_2 \in \Omega^{\Omega}$ be the function that takes each $\Sigma_{\alpha}$ to $\alpha$. Then it is easy to show that $g_1\mathrm{Sym}(\Omega) g_2 = \Omega^{\Omega}$ (see the proof of~\cite[Theorem 12]{ZM} for the details).
Let $U \in \mathcal{T}$ be a nonempty open set, and let $f \in U$. Also let $h \in \mathrm{Sym}(\Omega)$ be such that $g_1hg_2 = f$. Then $h \in (U)(\mathfrak{l}_{g_1} \circ \mathfrak{r}_{g_2})^{-1} \in \mathcal{T}$, and hence there is an open neighbourhood $V \subseteq \mathrm{Sym}(\Omega) \cap (U)(\mathfrak{l}_{g_1} \circ \mathfrak{r}_{g_2})^{-1}$ of $h$ in the topology induced on $\mathrm{Sym}(\Omega)$ such that $$V \supseteq \set{g \in \mathrm{Sym}(\Omega)}{(\beta_0)g=(\beta_0)h, \dots, (\beta_n)g=(\beta_n)h},$$ for some distinct $\beta_0, \dots, \beta_n \in \Omega$. Let $\Xi = (\{\beta_0, \dots, \beta_n\})g_1^{-1}$, and let $$W = \set{g \in \Omega^{\Omega}}{(\alpha)g = (\alpha)g_1hg_2 \text{ for all } \alpha \in \Xi}.$$ Then for each $g \in W$, there exists $p \in V$ such that $(\alpha)g_1p \in \Sigma_{(\alpha)g}$ for all $\alpha \in \Omega$, and hence $g = g_1pg_2 \in g_1Vg_2$. Thus $f \in W \subseteq g_1V g_2 \subseteq U$. Since $f \in U$ was arbitrary and $W \in \mathcal{P}$, this implies that $U \in \mathcal{P}$, and hence $\mathcal{T} \subseteq \mathcal{P}$. Finally, since $\mathcal{T}$ was assumed to be $T_1$, we conclude that $\mathcal{T} = \mathcal{P}$, by Theorem~\ref{point-finer}. \end{proof}
Recall that a topological space is \emph{separable} if it contains a countable dense subset. Kechris and Rosendal showed in~\cite[Theorem 6.26]{KR} that the pointwise topology is the only nontrivial separable group topology on $\mathrm{Sym}(\Omega)$, when $\Omega$ is countably infinite. In particular, the pointwise topology is the only Polish group topology on $\Omega$ in this situation, since any Polish topology must be Hausdorff, and hence nontrivial. Using Proposition~\ref{group-top}, we can prove the analogous statement for $\Omega^{\Omega}$.
\begin{theorem} \label{polish-thrm} Let $\Omega$ be a countably infinite set. Then the pointwise topology is the only Polish topology on $\, \Omega^{\Omega}$ with respect to which it is a semitopological semigroup. \end{theorem}
\begin{proof} Let $\mathcal{P}$ denote the pointwise topology on $\Omega^{\Omega}$, and let $\mathcal{T}$ be any Polish topology on $\Omega^{\Omega}$ with respect to which it is a semitopological semigroup. Then, in particular, $\mathcal{T}$ is Hausdorff and so $\mathcal{T}\supseteq \mathcal{P}$, by Theorem \ref{point-finer}.
It is a standard fact (see, e.g.,~\cite[Theorem 3.11]{Kechris}) that a subspace of a Polish space is Polish if and only if it is $G_{\delta}$ (i.e., a countable intersection of open sets). Since $\mathrm{Sym}(\Omega)$ is Polish in the topology induced by $\mathcal{P}$, it follows that $\mathrm{Sym}(\Omega)$ is a $G_{\delta}$ subspace of $\Omega^{\Omega}$ with respect to $\mathcal{P}$. From the fact that $\mathcal{T} \supseteq \mathcal{P}$, we conclude that $\mathrm{Sym}(\Omega)$ is also a $G_{\delta}$ subspace of $\Omega^{\Omega}$ with respect to $\mathcal{T}$, and hence $\mathrm{Sym}(\Omega)$ is a Polish space in the topology induced by $\mathcal{T}$. Since $\Omega^{\Omega}$ is a semitopological semigroup with respect to $\mathcal{T}$, it is easy to see that the same holds for $\mathrm{Sym}(\Omega)$ with respect to the topology induced by $\mathcal{T}$, and therefore it is a Polish semigroup in this topology.
According to a result of Montgomery~\cite[Theorem 2]{Montgomery}, if a group is a semitopological semigroup with respect to a Polish topology, then it is a topological group with respect to that topology. Hence $\mathrm{Sym}(\Omega)$ is a Polish group in the topology induced by $\mathcal{T}$, and therefore this topology must be the pointwise topology on $\mathrm{Sym}(\Omega)$, by~\cite[Theorem 6.26]{KR}. Finally, Proposition~\ref{group-top} implies that $\mathcal{T}=\mathcal{P}$. \end{proof}
We note that~\cite[Theorem 6.26]{KR} and Proposition~\ref{group-top} imply a stronger statement than what is used in the final paragraph of the above proof. Specifically, the pointwise topology is the only $T_1$ semitopological semigroup topology on $\Omega^{\Omega}$ that induces a separable group topology on $\mathrm{Sym}(\Omega)$, when $\Omega$ is countably infinite. To complement this observation and Theorem~\ref{polish-thrm}, we record the following fact.
\begin{cor} \label{separable-cor} Let $\, \Omega$ be an uncountable set. Then $\, \Omega^{\Omega}$ does not admit a separable $T_1$ topology with respect to which it is a semitopological semigroup. \end{cor}
\begin{proof}
Suppose that $\Omega^{\Omega}$ is a semitopological semigroup with respect to a $T_1$ topology $\mathcal{T}$, and that $X \subseteq \Omega^{\Omega}$ is countable and dense in $\mathcal{T}$. By Theorem~\ref{point-finer}, $\mathcal{T}$ contains the pointwise topology, and hence each subset of $\Omega^{\Omega}$ of the form $[\sigma : \tau]$ contains an element of $X$. Letting $\alpha \in \Omega$ be any element, we can find some $\beta \in \Omega \setminus (\alpha)X$, since $|(\alpha)X| \leq \aleph_0$. Hence $[(\alpha) : (\beta)] \cap X = \emptyset$, producing a contradiction. \end{proof}
\section{Topologies with isolated points}
It is well-known (see~\cite[Lemma 1]{Malcev}) that the proper ideals of $\Omega^{\Omega}$ are precisely the subsemigroups of the form $$I_\lambda = \set{f\in \Omega^{\Omega}}{|(\Omega) f|<\lambda},$$ where $\lambda$ is a cardinal satisfying $1 \leq \lambda \leq |\Omega|$. This fact allows us to construct $|\Omega|$ distinct Hausdorff semigroup topologies on $\Omega^{\Omega}$ (containing the pointwise topology). These topologies, along with all the others considered in this section, have \emph{isolated} points, i.e., points $x$ such that $\{x\}$ is open.
\begin{prop} \label{top-chain-prop}
Let $\, \Omega$ be an infinite set, let $\, \mathcal{T}_P$ denote the pointwise topology on $\, \Omega^{\Omega}$, and for each nonempty proper ideal $I_\lambda$ of $\, \Omega^{\Omega}$, let $$\, \mathcal{T}_{I_\lambda^1} = \set{A}{A \subseteq \Omega^{\Omega}\setminus I_\lambda}\cup\{\Omega^{\Omega}\}$$ and $$\, \mathcal{T}_{I_\lambda^2} = \mathcal{T}_{I_\lambda^1} \cup \set{A \cup I_\lambda}{A \in \mathcal{T}_{I_\lambda^1}}.$$ Also let $\, \mathcal{T}_{P \cup I_\lambda^i}$ be the topology on $\, \Omega^{\Omega}$ generated by $\, \mathcal{T}_P$ and $\, \mathcal{T}_{I_\lambda^i}$ $\, (i \in \{1,2\}, \ 1 < \lambda \leq |\Omega|)$. Then the following diagram consists of Hausdorff topologies with respect to which $\, \Omega^{\Omega}$ is a topological semigroup: $$\begin{array}{ccccccccccc} \mathcal{T}_P & \subset & \cdots & \subset & \mathcal{T}_{P \cup I_\lambda^2} & \subset & \cdots & \subset & \mathcal{T}_{P \cup I_3^2} & \subset & \mathcal{T}_{P \cup I_2^2}\\
\verteq &&&& \cup &&&& \cup && \cup \\ \mathcal{T}_P & \subset & \cdots & \subset & \mathcal{T}_{P \cup I_\lambda^1} & \subset & \cdots & \subset & \mathcal{T}_{P \cup I_3^1} & \subset & \mathcal{T}_{P \cup I_2^1} \\
\end{array}$$ $(1 < \lambda \leq |\Omega|)$. Moreover, $\mathcal{T}_{P \cup I_2^2}$ is the discrete topology. \end{prop}
\begin{proof} By Lemma~\ref{semi-top-lemma}(1), $\Omega^{\Omega}$ is a topological semigroup with respect to each $\mathcal{T}_{I_\lambda^i}$, and hence, by Lemma~\ref{semi-top-lemma}(2), the same holds for each $\mathcal{T}_{P \cup I_\lambda^i}$. Moreover, the topologies $\mathcal{T}_{P \cup I_\lambda^i}$ are Hausdorff, since they contain the pointwise topology $\mathcal{T}_P$.
It is easy to see that for any $1 < \lambda \leq |\Omega|$ the topology $\mathcal{T}_{P \cup I_\lambda^1}$ consists of all sets of the form $A\cup B$, where $A \in \mathcal{T}_P$ and $B \subseteq \Omega^{\Omega}\setminus I_\lambda$, while the topology $\mathcal{T}_{P \cup I_\lambda^2}$ consists of all sets of the form $(A\cap I_\lambda) \cup B$, where $A \in \mathcal{T}_P$ and $B \subseteq \Omega^{\Omega}\setminus I_\lambda$. Thus, $\mathcal{T}_{P \cup I_\lambda^1} \subset \mathcal{T}_{P \cup I_\lambda^2}$ for each $1 < \lambda \leq |\Omega|$. Moreover, given two cardinals $1 < \lambda < \kappa \leq |\Omega|$ and $i \in \{1,2\}$, $I_\lambda \subseteq I_\kappa$ and $$(A\cap I_\kappa) \cup B = (A\cap I_\lambda) \cup (B \cup (A \cap (I_\kappa \setminus I_\lambda)))$$ imply that $\mathcal{T}_{P \cup I_\kappa^i} \subseteq \mathcal{T}_{P \cup I_\lambda^i}$. Noting that any $f \in I_{\kappa} \setminus I_{\lambda}$ is isolated in $\mathcal{T}_{P \cup I_\lambda^i}$, to conclude that $\mathcal{T}_{P \cup I_\kappa^i} \subset \mathcal{T}_{P \cup I_\lambda^i}$ it suffices to show that $f \in I_{\kappa} \setminus I_{\lambda}$ is not isolated in $\mathcal{T}_{P \cup I_\kappa^i}$. But this follows immediately from the fact that $A \cap I_\kappa$ is either empty or infinite, for any $A \in \mathcal{T}_P$ and $\kappa \geq 3$, and hence any open set in $\mathcal{T}_{P \cup I_\kappa^i}$ containing $f$ is infinite.
Finally, $\mathcal{T}_{P \cup I_2^2}$ is the discrete topology, since $I_2 \cap [(\alpha) : (\alpha)] \in \mathcal{T}_{P \cup I_2^2}$ is the singleton set consisting of the constant function with value $\alpha$, for any $\alpha \in \Omega$, and all non-constant elements of $\Omega^{\Omega}$ are isolated in $\mathcal{T}_{P \cup I_2^2}$, by definition. \end{proof}
In addition to describing the ideals of $\Omega^{\Omega}$, Mal'cev~\cite{Malcev} classified all the congruences $\rho$ on this semigroup. Thus, one can obtain additional Hausdorff semigroup topologies on $\Omega^{\Omega}$ by putting the discrete topology on $\Omega^{\Omega}/\rho$, and considering the topology generated by the pointwise topology together with the one induced on $\Omega^{\Omega}$ via Lemma~\ref{hom-lemma} (with $f : \Omega^{\Omega} \to \Omega^{\Omega}/\rho$ taken to be the natural projection). However, the topologies obtained this way typically, though not always, coincide with ones described in the previous proposition. Since non-Rees congruences on $\Omega^{\Omega}$ tend to be complicated to describe, we shall not discuss the resulting topologies in further detail here.
All the topologies on $\Omega^{\Omega}$ constructed in Proposition~\ref{top-chain-prop} have $2^{|\Omega|}$ isolated points, since for example, all the surjective elements of $\Omega^{\Omega}$ are isolated in these topologies. Our next goal is to show that any semigroup topology on $\Omega^{\Omega}$ with isolated points also must have ``many" of them. We begin with a couple of lemmas.
\begin{lem}\label{lem-dual-kernels} Let $\, \Omega$ be a set, and suppose that $\, \Omega^{\Omega}$ is a semitopological semigroup with respect to some topology. Also suppose that $f, g\in \Omega^{\Omega}$ have the property that there exist $b_0, \ldots, b_n \in \Omega^{\Omega}$ and injective $a_0, \ldots, a_n\in \Omega^{\Omega}$, such that $$\, (\Omega) b_0 \cup \cdots \cup (\Omega) b_n = \Omega$$ and $$b_0ga_0 = \cdots = b_nga_n = f.$$ If $f$ is isolated, then so is $g$. \end{lem}
\begin{proof} Suppose that $i\in \{0, \ldots, n\}$ and that $h\in \Omega^{\Omega}$ is such that $b_iha_i = f$. Then $b_iha_i = b_iga_i$, and since $a_i$ is injective, $b_ih = b_ig$. In other words, $h$ and $g$ agree on $(\Omega)b_i$. Hence if $b_iha_i = f$ for all $i \in \{0, \ldots, n\}$, then $h = g$, since $(\Omega) b_0 \cup \cdots \cup (\Omega) b_n = \Omega$. Thus $$\bigcap_{i=0}^{n}(f) (\mathfrak{l}_{b_i} \circ \mathfrak{r}_{a_i})^{-1} = \{g\},$$ and therefore if $f$ is isolated, then so is $g$. \end{proof}
A \emph{kernel class} of an element $f\in \Omega^{\Omega}$ is a nonempty set of the form $$(\beta)f^{-1}=\set{\alpha\in \Omega}{(\alpha)f=\beta}.$$ The \emph{kernel} $\ker(f)$ of $f \in \Omega^{\Omega}$ is the collection of the kernel classes of $f$. Finally, the \emph{kernel class type} of $f$ is the collection $\set{f_\kappa}{1\leq \kappa \leq |\Omega|}$, where $f_\kappa$ is the cardinality of the set of kernel classes of $f$ of size $\kappa$.
\begin{lem}\label{lem-kernels}
Let $\, \Omega$ be a nonempty set, and suppose that $\, \Omega^{\Omega}$ is a semitopological semigroup with respect to some topology. Also let $f,g \in \Omega^{\Omega}$ be such that there is a bijection $p$ from the set of kernel classes of $g$ to the set of kernel classes of $f$, with the property that $\, |(\Sigma)p| \geq |\Sigma|$ for each kernel class $\, \Sigma$ of $g$. If $f$ is isolated, then so is $g$.
In particular, if an element of $\, \Omega^{\Omega}$ is isolated, then so is every other element of $\, \Omega^{\Omega}$ with the same kernel class type. \end{lem}
\begin{proof}
We may assume that $|\Omega| \geq 2$, since otherwise $f=g$.
Since the kernel classes of $f$ and $g$ are in one-to-one correspondence, $|(\Omega)f| = |(\Omega)g|$, and hence we can write $(\Omega) f=\set{\beta_{\iota}}{\iota \in \Gamma}$ and $(\Omega) g= \set{\gamma_{\iota}}{\iota \in \Gamma}$, for some index set $\Gamma$. Moreover, by the hypothesis on $p$, we may choose the $\beta_{\iota}$ and $\gamma_{\iota}$ such that $|(\beta_{\iota})f^{-1}| \geq |(\gamma_{\iota})g^{-1}|$ for all $\iota \in \Gamma$. Let $b\in \Omega^{\Omega}$ be any surjection such that $\left((\beta_{\iota})f^{-1}\right)b = (\gamma_{\iota})g^{-1}$ for all $\iota \in \Gamma$. Also, let $a_0, a_1 \in \Omega^{\Omega}$ be any elements that take $\gamma_{\iota}$ to $\beta_{\iota}$ for all $\iota \in \Gamma$, and that have constant value $\delta_0, \delta_1 \in \Omega$, respectively, on $\Omega \setminus (\Omega) g$, where $\delta_0 \neq \delta_1$. Then $bga_0=bga_1=f$. We shall show that $g$ is the only element of $\Omega^{\Omega}$ with this property.
Suppose that $bha_0=f$ for some $h\in \Omega^{\Omega}$. Then $(\alpha)bh=(\alpha)bg$ for any $\alpha\in \Omega$ satisfying $(\alpha)f\not = \delta_0$. Similarly, if $bha_1=f$ and $(\alpha)f \not = \delta_1$, then $(\alpha)bh=(\alpha)bg$. Thus if both $bha_0=f$ and $bha_1=f$, then $bh=bg$, and hence $h=g$, since $b$ is surjective and therefore left-invertible. Thus, we have shown that $$(f) (\mathfrak{l}_{b} \circ \mathfrak{r}_{a_0})^{-1} \cap (f) (\mathfrak{l}_{b} \circ \mathfrak{r}_{a_1})^{-1} = \{g\},$$ and therefore if $f$ is isolated, then so is $g$.
The final claim is immediate. \end{proof}
The \emph{cofinality} $\operatorname{cf}(\kappa)$ of a cardinal $\kappa$ is the least cardinal $\lambda$ such that $\kappa$ is the union of $\lambda$ cardinals, each smaller than $\kappa$. A cardinal $\kappa$ is \emph{regular} if $\operatorname{cf}(\kappa) = \kappa$. It is a standard and easily verified fact that $\operatorname{cf}(\kappa) \leq \kappa$ for every cardinal $\kappa$. Let us also recall that for an infinite cardinal $\kappa$, the cofinality $\operatorname{cf}(\kappa)$ is necessarily infinite. (Otherwise it would be the case that $\kappa = \lambda_0 + \dots + \lambda_n$ for some $n \in \mathbb{N}$ and infinite cardinals $\lambda_i < \kappa$. But $\sum_{i=0}^n \lambda_i = \max\{\lambda_0, \dots, \lambda_n\},$ since the $\lambda_i$ are infinite, contradicting $\lambda_i < \kappa$.)
We are now ready for the main result of this section.
\begin{theorem} \label{many-isolated}
Let $\, \Omega$ be an infinite set, let $\kappa = \operatorname{cf}(|\Omega|)$, and suppose that $\, \Omega^{\Omega}$ is a semitopological semigroup with respect to some topology. Then there are either no isolated points or at least $\, 2^{\kappa}$ isolated points in $\, \Omega^{\Omega}$.
In particular, if $\, |\Omega|$ is regular, then there are either no isolated points or $\, 2^{|\Omega|}$ isolated points in $\, \Omega^{\Omega}$. \end{theorem}
\begin{proof}
Suppose that $f\in \Omega^{\Omega}$ is isolated. If $f$ has at least $\kappa$ distinct kernel classes, then the set of those $g \in \Omega^{\Omega}$ with the same kernel class type as $f$ has cardinality at least $2^{\kappa}$. (Since $\kappa$ is infinite, one can obtain $2^{\kappa}$ such $g$ by permuting $\kappa$ of the elements in the image of $f$.) Hence, there are at least $2^{\kappa}$ isolated points in $\Omega^{\Omega}$, by Lemma~\ref{lem-kernels}. Let us therefore assume that $f$ has strictly fewer than $\kappa$ distinct kernel classes, and so, in particular, $\Omega \setminus (\Omega)f \neq \emptyset$, since $\kappa \leq |\Omega|$.
It cannot be the case that every kernel class of $f$ has cardinality strictly less than $|\Omega|$, since then the union of these (fewer than $\kappa$) kernel classes would have cardinality strictly less than $|\Omega|$, by the definition of ``cofinality". This would contradict the fact that the union of the kernel classes of any element of $\Omega^{\Omega}$ is $\Omega$. Therefore, there must be a kernel class $\Sigma$ of $f$ such that $|\Sigma| = |\Omega|$. Let $(\Sigma)f = \beta$, and partition $\Sigma$ as $\Sigma = \Lambda_0 \cup \Lambda_1$, such that $\Lambda_0$ and $\Lambda_1$ are nonempty. Let $g\in \Omega^{\Omega}$ be defined by $$(\alpha)g = \left\{ \begin{array}{cl} (\alpha)f & \text{if } \, \alpha \in \Omega\setminus \Sigma\\ \beta & \text{if } \, \alpha \in \Lambda_0\\ \gamma & \text{if } \, \alpha \in \Lambda_1 \end{array}\right.,$$
where $\gamma$ is any value in $\Omega \setminus (\Omega) f$ (which exists by assumption). Let $b_0, b_1\in \Omega^{\Omega}$ be any functions that act as the identity on $\Omega\setminus \Sigma$, and where $(\Sigma)b_0 = \Lambda_0$ and $(\Sigma)b_1 = \Lambda_1$. Also let $a_0 \in \Omega^{\Omega}$ be the identity function, and let $a_1 \in \Omega^{\Omega}$ be the transposition interchanging $\beta$ and $\gamma$. Then $$(\Omega) b_0 \cup(\Omega) b_1 = \Lambda_0 \cup \Lambda_1 \cup (\Omega\setminus \Sigma) = \Omega,$$ and $f = b_0ga_0 = b_1ga_1$. Hence $g$ is isolated, by Lemma~\ref{lem-dual-kernels}. Since there are $2^{|\Sigma|} = 2^{|\Omega|}$ ways to partition $\Sigma$ into $\Lambda_0$ and $\Lambda_1$, resulting in $2^{|\Omega|} \geq 2^{\kappa}$ functions $g$, the desired conclusion follows.
For the final claim, we recall the fact that $2^{\lambda} = \mu^{\lambda}$ for any infinite cardinal $\lambda$ and any cardinal $\mu$ satisfying $2 \leq \mu \leq \lambda$ (see, e.g.,~\cite[Lemma 5.6]{Jech}). Thus $2^{|\Omega|} = |\Omega|^{|\Omega|} = |\Omega^{\Omega}|$, since $\Omega$ is assumed to be infinite. Therefore, if $|\Omega|$ is regular and there are isolated points in $\Omega^{\Omega}$, then there are precisely $2^{\operatorname{cf}(|\Omega|)} = 2^{|\Omega|}$ of them. \end{proof}
Recall that a topology is \emph{perfect} if it has no isolated points.
\begin{cor} \label{separable-perfect} Let $\, \Omega$ be an infinite set. Then every separable topology on $\, \Omega^{\Omega}$, with respect to which $\, \Omega^{\Omega}$ is a semitopological semigroup, is perfect. \end{cor}
\begin{proof}
Suppose that $\Omega^{\Omega}$ is a semitopological semigroup with respect to a separable topology $\mathcal{T}$. By Theorem~\ref{many-isolated}, if $\mathcal{T}$ has isolated points, then it must have at least $2^{\operatorname{cf}(|\Omega|)} \geq 2^{\aleph_0}$ of them, since as noted above, $\operatorname{cf}(|\Omega|) \geq |\Omega| \geq \aleph_0$. But since $\mathcal{T}$ is separable, it can have at most countably many isolated points, and therefore $\mathcal{T}$ must be perfect. \end{proof}
We conclude this section with another result about isolated points. It sheds additional light on the topologies constructed in Proposition~\ref{top-chain-prop} and strengthens Lemma~\ref{lem-kernels}.
\begin{prop} Let $\, \Omega$ be an infinite set, and suppose that $\, \Omega^{\Omega}$ is a semitopological semigroup with respect to some topology. Let $f,g \in \Omega^{\Omega}$ be such that $f$ is isolated. If either of the following conditions holds, then $g$ is also isolated. \begin{enumerate}
\item[$(1)$] $|(\Omega)f| \leq |(\Omega)g| < \aleph_0$.
\item[$(2)$] $|(\Omega)f| = |(\Omega)g| \geq \aleph_0$, and there is an injection $p$ from the set $K_g$ of kernel classes of $g$ to the set $K_f$ of kernel classes of $f$, with the property that $\, |(\Sigma)p| \geq |\Sigma|$ for each $\Sigma \in K_g$. \end{enumerate} \end{prop}
\begin{proof}
Suppose that (1) holds. Let $\Xi_0, \dots, \Xi_{n}$ be the kernel classes of $f$, where we may assume that $|\Xi_0| = |\Omega|$, and let $\Upsilon_0, \dots, \Upsilon_m$ be the kernel classes of $g$. Also, write $(\Xi_i)f = \alpha_i$ and $(\Upsilon_j)g = \beta_j$ for all $i \in \{0, \dots, n\}$ and $j \in \{0, \dots, m\}$. For each $j \in \{0, \dots, m\}$ let $a_j \in \Omega^{\Omega}$ be an injection such that $(\beta_j)a_j = \alpha_0$ and $$\{\alpha_0, \dots, \alpha_n\} \subseteq (\{\beta_0, \dots, \beta_m\})a_j.$$ Such functions exist, since $n \leq m$. Also for each $j \in \{0, \dots, m\}$ let $b_j \in \Omega^{\Omega}$ be such that $(\Xi_0)b_j = \Upsilon_j$, and $(\Xi_i)b_j \subseteq \Upsilon_l$, where $\beta_l = (\alpha_i)a_j^{-1}$, for each $i \in \{1, \dots, n\}$. Such functions exist, since $|\Xi_0| = |\Omega|$. Then $b_jga_j = f$ for each $j \in \{0, \dots, m\}$, and $$\Omega \supseteq \bigcup_{i=0}^m(\Omega)b_j \supseteq \bigcup_{j=0}^m \Upsilon_j = \Omega,$$ which implies that $$(\Omega) b_0 \cup \cdots \cup (\Omega) b_m = \Omega.$$ Therefore $g$ is isolated, by Lemma~\ref{lem-dual-kernels}.
Now suppose that (2) holds. Upon replacing $f$ with another function having the same kernel, we may assume, by Lemma~\ref{lem-kernels}, that $|\Omega \setminus (\Omega)f| = |\Omega|$. Since the image of $g$ is infinite, we can write $$K_g = \set{\Xi_{\iota}}{\iota \in \Gamma} \cup \set{\Upsilon_{\zeta}}{\zeta \in \Delta},$$ where the union is disjoint, and $|\Gamma| = |\Delta| = |(\Omega)g|$. Let $b_0 \in \Omega^{\Omega}$ be such that $((\Xi_{\iota})p)b_0 = \Xi_{\iota}$ for each $\iota \in \Gamma$, and $b_0$ takes $K_f \setminus (\set{\Xi_{\iota}}{\iota \in \Gamma})p$ injectively into $\set{\Upsilon_{\zeta}}{\zeta \in \Delta}$. (That is, $b_0$ maps all the points in an element of $K_f \setminus (\set{\Xi_{\iota}}{\iota \in \Gamma})p$ to some $\Upsilon_{\zeta}$.) Such a transformation exists, since $|(\Xi_{\iota})p| \geq |\Xi_{\iota}|$ for each $\Xi_{\iota}$, and $|\Delta| = |(\Omega)f|$. Similarly, let $b_1 \in \Omega^{\Omega}$ be such that $((\Upsilon_{\zeta})p)b_1 = \Upsilon_{\zeta}$ for each $\zeta \in \Delta$, and $b_1$ takes $K_f \setminus (\set{\Upsilon_{\zeta}}{\zeta \in \Delta})p$ injectively into $\set{\Xi_{\iota}}{\iota \in \Gamma}$. Then $$\Omega \supseteq (\Omega) b_0 \cup (\Omega) b_1 \supseteq \bigcup_{\iota \in \Gamma} \Xi_{\iota} \cup \bigcup_{j\in \Delta} \Upsilon_{\zeta} = \Omega,$$ giving $(\Omega)b_0 \cup (\Omega)b_1 = \Omega$. Moreover, $$\ker(b_0g) = \ker(b_1g) = \ker(f),$$ and since $|\Omega \setminus (\Omega)f| = |\Omega|$, there exist injective $a_0, a_1\in \Omega^{\Omega}$ such that $b_0ga_0 = b_1ga_1 = f$. Hence, by Lemma~\ref{lem-dual-kernels}, $g$ is isolated. \end{proof}
\section{Topologies obtained by restricting images}
Given a set $\Omega$ and collection $\set{\Sigma_{\alpha}}{\alpha \in \Omega}$ of subsets of $\Omega$, we identify $\prod_{\alpha \in \Omega} \Sigma_{\alpha}$ with $$\set{f \in \Omega^{\Omega}}{(\alpha)f \in \Sigma_{\alpha} \text{ for all } \alpha \in \Omega}.$$ Next we construct a perfect Hausdorff topology on $\Omega^{\Omega}$, different from the pointwise topology, by declaring certain sets of the form $\prod_{\alpha \in \Omega} \Sigma_{\alpha}$ open.
\begin{prop} \label{open-prod-top}
Let $\, \Omega$ be an infinite set, let $$\mathcal{B} = \bigg\{\prod_{\alpha \in \Omega} \Sigma_{\alpha} : |\set{\alpha \in \Omega}{\beta \notin \Sigma_{\alpha}}| <|\Omega| \text{ for all } \beta \in \Omega\bigg\},$$ and let $\, \mathcal{T} = \langle \mathcal{B} \rangle$ denote the topology on $\, \Omega^{\Omega}$ generated by $\mathcal{B}$. Then the following hold. \begin{enumerate} \item[$(1)$] $\mathcal{B}$ is closed under finite intersections, and in particular is a base for $\, \mathcal{T}$.
\item[$(2)$] Any base for $\, \mathcal{T}$ must have strictly more than $\, |\Omega|$ elements. \item[$(3)$] $\mathcal{T}$ strictly contains the pointwise topology. In particular, $\, \mathcal{T}$ is Hausdorff. \item[$(4)$] $\mathcal{T}$ is perfect.
\item[$(5)$] There is a subset of $\, \Omega^{\Omega}$ of cardinality $\, | \sum_{\kappa < |\Omega|} |\Omega|^{\kappa} |$ that is dense in $\, \mathcal{T}$. In particular, if $\, \Omega$ is countable, then $\, \mathcal{T}$ is separable. \item[$(6)$] No nonempty element of $\mathcal{B}$ is contained in a compact subset of $\, \Omega^{\Omega}$.
\item[$(7)$] If $\, |\Omega|$ is regular, then $\, \Omega^{\Omega}$ is a topological semigroup with respect to $\, \mathcal{T}$. \end{enumerate} \end{prop}
\begin{proof}
(1) Let $n \in \mathbb{N}$, and let $\prod_{\alpha \in \Omega} \Sigma_{i, \alpha} \in \mathcal{B}$ for $i \in \{0, \dots, n\}$. Then $$\bigcap_{i=0}^n \bigg(\prod_{\alpha \in \Omega} \Sigma_{i, \alpha}\bigg) = \prod_{\alpha \in \Omega} \bigg(\bigcap_{i=0}^n \Sigma_{i, \alpha}\bigg).$$ Now $$\bigg|\bigg\{\alpha \in \Omega : \beta \notin \bigcap_{i=0}^n\Sigma_{i, \alpha}\bigg\}\bigg| \leq \sum_{i=0}^n |\set{\alpha \in \Omega}{\beta \notin \Sigma_{i, \alpha}}| < \sum_{i=0}^n |\Omega| = |\Omega|$$ for any $\beta \in \Omega$. Hence $\bigcap_{i=0}^n (\prod_{\alpha \in \Omega} \Sigma_{i, \alpha}) \in \mathcal{B}$, showing that $\mathcal{B}$ is closed under finite intersections. Since the elements of $\mathcal{B}$ cover $\Omega^{\Omega}$ (e.g., because $\Omega^{\Omega} \in \mathcal{B}$), it follows that $\mathcal{B}$ is a base for $\mathcal{T} = \langle \mathcal{B} \rangle$.
(2) Let $\set{X_{\alpha}}{\alpha \in \Omega} \subseteq \mathcal{T}$ be a collection of nonempty open subsets of $\Omega^{\Omega}$. We shall construct sets $\Sigma_{\alpha} \subseteq \Omega$ such that $\prod_{\alpha \in \Omega}\Sigma_{\alpha} \in \mathcal{B}$ and $X_{\beta} \not\subseteq \prod_{\alpha \in \Omega}\Sigma_{\alpha}$ for all $\beta \in \Omega$, which implies that $\set{X_{\alpha}}{\alpha \in \Omega}$ cannot be a base for $\mathcal{T}$, and hence that any base for $\mathcal{T}$ must have strictly more than $|\Omega|$ elements.
Without loss of generality, we may assume that $\Omega$ is a cardinal. Since, by (1), $\mathcal{B}$ is a base for $\mathcal{T}$, for each $\beta \in \Omega$ we can find a nonempty $\prod_{\alpha \in \Omega}\Sigma_{\alpha, \beta} \in \mathcal{B}$ such that $\prod_{\alpha \in \Omega}\Sigma_{\alpha, \beta} \subseteq X_{\beta}$. Next we shall construct recursively a function $f : \Omega \to \Omega$. Assuming that $(\alpha)f$ is defined for all $\alpha < \beta \in \Omega$, let $(\beta)f$ be such that $(\beta)f \neq (\alpha)f$ for all $\alpha < \beta$, and such that $\beta \in \Sigma_{(\beta)f, \beta}$. Such $(\beta)f \in \Omega$ exists, since $$|\set{(\alpha)f}{\alpha < \beta} \cup \set{\alpha \in \Omega}{\beta \notin \Sigma_{\alpha, \beta}}| < \Omega.$$ Now for each $\alpha \in \Omega$ let $$\Sigma_{\alpha} = \left\{ \begin{array}{cl} \Omega \setminus \{\beta\} & \text{if } \, \alpha = (\beta)f\\ \Omega & \text{if } \, \alpha \notin (\Omega)f \end{array}\right..$$ This is well-defined, since $f$ is injective, and clearly $\prod_{\alpha \in \Omega}\Sigma_{\alpha} \in \mathcal{B}$. Moreover, $\prod_{\alpha \in \Omega}\Sigma_{\alpha, \beta} \not\subseteq \prod_{\alpha \in \Omega}\Sigma_{\alpha}$ for all $\beta \in \Omega$, since $\beta \in \Sigma_{(\beta)f, \beta} \setminus \Sigma_{(\beta)f}$. Hence $X_{\beta} \not\subseteq \prod_{\alpha \in \Omega}\Sigma_{\alpha}$ for all $\beta \in \Omega$, as desired.
(3) Let $\sigma=(\sigma_0, \sigma_1, \dots, \sigma_n)$ and $\tau=(\tau_0, \tau_1, \dots, \tau_n)$ be two finite sequences of elements of $\Omega$ of the same length, and let $$\Sigma_{\alpha} = \left\{ \begin{array}{cl} \{\tau_i\} & \text{if } \, \alpha = \sigma_i \text{ for some } i \in \{0, \dots, n\} \\ \Omega & \text{if } \, \alpha \in \Omega \setminus \{\sigma_0, \dots, \sigma_n\} \end{array}\right..$$
Then $[\sigma : \tau] = \prod_{\alpha \in \Omega} \Sigma_{\alpha} \in \mathcal{T}$, showing that $\mathcal{T}$ contains the pointwise topology, and is hence Hausdorff. Since the sets of the form $[\sigma : \tau]$ constitute a base for the pointwise topology of cardinality $|\Omega|$, we conclude from (2) that $\mathcal{T}$ strictly contains the pointwise topology.
(4) If $\prod_{\alpha \in \Omega} \Sigma_{\alpha} \in \mathcal{B}$, then for any $\beta, \gamma \in \Omega$ there must be infinitely many $\alpha \in \Omega$ such that $\beta, \gamma \in \Sigma_{\alpha}$. Thus each nonempty element of $\mathcal{B}$ is (uncountably) infinite, and hence by (1), $\mathcal{T}$ has a base consisting of infinite sets. In particular, no point can be isolated in $\mathcal{T}$.
(5) Fix $\alpha \in \Omega$. Given $\Gamma \subseteq \Omega$, where $0 < |\Gamma| < |\Omega|$, let $X_{\Gamma} \subseteq \Omega^{\Omega}$ consist of all $f \in \Omega^{\Omega}$ such $(\Omega \setminus \Gamma)f = \alpha$. Then letting $X$ be the union of the $X_{\Gamma}$, we see that $X$ is dense in $\mathcal{T}$. Moreover $$|X| = \sum_{\Gamma} |X_{\Gamma}| = \sum_{\Gamma} |\Omega^{\Gamma}| = \sum_{\kappa < |\Omega|} |\Omega|^{\kappa}.$$ In the case where $\Omega$ is countable, $|X| = \aleph_0$, which implies that $\mathcal{T}$ is separable.
(6) Let $\prod_{\alpha \in \Omega}\Sigma_{\alpha} \in \mathcal{B}$ be nonempty, and suppose that $\prod_{\alpha \in \Omega}\Sigma_{\alpha} \subseteq X$ for some compact $X \subseteq \Omega^{\Omega}$. Then, by the definition of $\mathcal{B}$, we can find an infinite $\Gamma \subseteq \Omega$ and distinct $\gamma_{\beta} \in \Omega$, such that $\gamma_{\beta} \in \Sigma_{\beta}$ and $|\Sigma_{\beta}| \geq 2$ for all $\beta \in \Gamma$. For each $\beta \in \Gamma$ let $$\Xi_{\alpha, \beta} = \left\{ \begin{array}{cl} \{\gamma_{\beta}\} & \text{if } \, \alpha = \beta \\ \Omega & \text{if } \, \alpha \neq \beta \end{array}\right.,$$ and let $$\Delta_{\beta} = \left\{ \begin{array}{cl} \Omega \setminus \{\gamma_{\beta}\} & \text{if } \, \beta \in \Gamma \\ \Omega & \text{if } \, \beta \notin \Gamma \end{array}\right..$$ Also let $$Y = \bigg\{\prod_{\alpha \in \Omega} \Delta_{\alpha}\bigg\} \cup \bigg\{\prod_{\alpha \in \Omega} \Xi_{\alpha, \beta} : \beta \in \Gamma\bigg\}.$$ Then $\prod_{\alpha \in \Omega} \Xi_{\alpha, \beta} \in \mathcal{B}$ for each $\beta \in \Gamma$, and $\prod_{\alpha \in \Omega} \Delta_{\alpha} \in \mathcal{B}$. Moreover, $Y$ is an open cover of $\Omega^{\Omega}$, and hence also of $X$, since for each $f \in \Omega^{\Omega} \setminus \prod_{\alpha \in \Omega} \Delta_{\alpha}$ there is some $\beta \in \Gamma$ such that $(\beta)f = \gamma_{\beta}$, and hence $f \in \prod_{\alpha \in \Omega} \Xi_{\alpha, \beta}$. However, it is easy to see that $\prod_{\alpha \in \Omega}\Sigma_{\alpha}$ is not contained in the union of any finite collection of elements of $Y$, and hence neither is $X$, contradicting $X$ being compact. Therefore $\prod_{\alpha \in \Omega}\Sigma_{\alpha}$ is not contained in any compact subset of $\Omega^{\Omega}$.
(7) Without loss of generality we may assume that $\Omega$ is a (regular) cardinal, and so in particular, it is well-ordered. Let $\prod_{\alpha \in \Omega}\Sigma_{\alpha} \in \mathcal{B}$ and $f,g \in \Omega^{\Omega}$ be such that $fg \in \prod_{\alpha \in \Omega}\Sigma_{\alpha}$. We shall construct $\prod_{\alpha \in \Omega}\Gamma_{\alpha}, \prod_{\alpha \in \Omega}\Delta_{\alpha} \in \mathcal{B}$ such that $f \in \prod_{\alpha \in \Omega}\Gamma_{\alpha}$, $g \in \prod_{\alpha \in \Omega}\Delta_{\alpha}$, and \begin{equation*} \tag{$\dagger$} \bigg(\prod_{\alpha \in \Omega}\Gamma_{\alpha}\bigg)\bigg(\prod_{\alpha \in \Omega}\Delta_{\alpha}\bigg) \subseteq \prod_{\alpha \in \Omega}\Sigma_{\alpha}. \end{equation*} The existence of such sets implies that the multiplication map on $\Omega^{\Omega}$ is continuous with respect to $\mathcal{T}$, and hence that $\Omega^{\Omega}$ is a topological semigroup.
For each $\alpha \in \Omega$ let $$\Gamma_{\alpha} = \{(\alpha)f\} \cup \set{\beta \in \Omega}{(\beta)g \in \Sigma_{\alpha} \text{ and } \beta < \alpha}$$ and $$\Delta_{\alpha} = \bigcap_{\set{\beta \in \Omega}{\alpha \in \Gamma_{\beta}}} \Sigma_{\beta}.$$ Then clearly $f \in \prod_{\alpha \in \Omega}\Gamma_{\alpha}$. If $\beta \notin \Gamma_{\alpha}$ for some $\alpha, \beta \in \Omega$, then either $\alpha \leq \beta$ or $(\beta)g \notin \Sigma_{\alpha}$. Since $$|\set{\alpha \in \Omega}{\alpha \leq \beta} \cup \set{\alpha \in \Omega}{(\beta)g \notin \Sigma_{\alpha}}| <\Omega$$ for all $\beta \in \Omega$, we see that $|\set{\alpha \in \Omega}{\beta \notin \Gamma_{\alpha}}| <\Omega$ for all $\beta \in \Omega$, and hence $\prod_{\alpha \in \Omega}\Gamma_{\alpha} \in \mathcal{B}$.
To show that $g \in \prod_{\alpha \in \Omega}\Delta_{\alpha}$, let $\alpha \in \Omega$, and let $\beta \in \Omega$ be such that $\alpha \in \Gamma_{\beta}$. Then either $\alpha = (\beta)f$, or $(\alpha)g \in \Sigma_{\beta}$. In both cases, $(\alpha)g \in \Sigma_{\beta}$, and hence $(\alpha)g \in \Delta_{\alpha}$, as desired.
To prove that $\prod_{\alpha \in \Omega}\Delta_{\alpha} \in \mathcal{B}$ we note that for any $\beta \in \Omega$, $$|\set{\alpha \in \Omega}{\beta \notin \Delta_{\alpha}}| = |\set{\alpha \in \Omega}{\exists \gamma \in \Omega \ (\alpha \in \Gamma_{\gamma} \land \beta \notin \Sigma_{\gamma})}| = \Big|\bigcup_{\set{\gamma \in \Omega}{\beta \notin \Sigma_{\gamma}}} \Gamma_{\gamma}\Big| < \Omega,$$ since $|\set{\gamma \in \Omega}{\beta \notin \Sigma_{\gamma}}| < \Omega$, $|\Gamma_{\gamma}| < \Omega$ for each $\gamma \in \Omega$, and $\Omega$ is regular.
Finally, let $\gamma \in \Omega$, $f' \in \prod_{\alpha \in \Omega}\Gamma_{\alpha}$, and $g' \in \prod_{\alpha \in \Omega}\Delta_{\alpha}$. Then $(\gamma)f' \in \Gamma_{\gamma}$, and so $$(\gamma)f'g' \in \Delta_{(\gamma)f'} = \bigcap_{\set{\beta \in \Omega}{(\gamma)f' \in \Gamma_{\beta}}} \Sigma_{\beta} \subseteq \Sigma_{\gamma}.$$ Hence $f'g' \in \prod_{\alpha \in \Omega}\Sigma_{\alpha}$, which shows that $(\dagger)$ holds. \end{proof}
The next result suggests that making sets of the form $\prod_{\alpha \in \Omega} \Sigma_{\alpha}$ open, other than those in the above construction, tends to result in semigroup topologies on $\Omega^{\Omega}$ that are not perfect. More specifically, only non-perfect semigroup topologies on $\Omega^{\Omega}$ have open sets $\prod_{\alpha \in \Omega} \Sigma_{\alpha}$ not of the above form that contain constant functions. In particular, no perfect semigroup topology on $\Omega^{\Omega}$ can have an open set of the form $\prod_{\alpha \in \Omega} \Sigma$, where $\Sigma$ is a proper subset of $\Omega$.
\begin{prop} \label{iso-prop} Let $\, \Omega$ be an infinite set, and suppose that $\, \Omega^{\Omega}$ is a semitopological semigroup with respect to some topology. \begin{enumerate}
\item[$(1)$] Suppose that there exist $\beta \in \Omega$, a constant function $f \in \Omega^{\Omega}$, and an open neighbourhood $U$ of $f$, such that $U \subseteq \prod_{\alpha \in \Omega} \Sigma_{\alpha}$ and $\, |\set{\alpha \in \Omega}{\beta \notin \Sigma_{\alpha}}| = |\Omega|$. Then $f$ is isolated.
\item[$(2)$] Let $\, \set{\Sigma_{\alpha}}{\alpha \in \Omega}$ be a collection of disjoint nonempty subsets of $\, \Omega$, and suppose that $U \subseteq \prod_{\alpha \in \Omega} \Sigma_{\alpha}$ is open. Then every element of $U$ is isolated. \end{enumerate} \end{prop}
\begin{proof} (1) Write $(\Omega)f = \gamma$, set $\Gamma = \set{\alpha \in \Omega}{\beta \in \Sigma_{\alpha}}$, and let $g \in \Omega^{\Omega}$ be defined by $$(\alpha)g = \left\{ \begin{array}{cl} \beta & \text{if } \, \alpha \in \Omega\setminus \{\gamma\}\\ \gamma & \text{if } \, \alpha = \gamma
\end{array}\right..$$ Then $$(U)\mathfrak{r}_g^{-1} \subseteq \bigg(\prod_{\alpha \in \Omega} \Sigma_{\alpha}\bigg)\mathfrak{r}_g^{-1} \subseteq \set{y \in \Omega^{\Omega}}{(\alpha)y = \gamma \text{ for all } \alpha \in \Omega \setminus \Gamma},$$ and $f \in (U)\mathfrak{r}_g^{-1}$. If $\Gamma = \emptyset$, then this implies that $(U)\mathfrak{r}_g^{-1} = \{f\}$, and this set is open, since $U$ is. Hence we may assume that $\Gamma \neq \emptyset$. Since $|\Omega \setminus \Gamma| = |\Omega|$, there exists an $h \in \Omega^{\Omega}$ such that $(\Omega \setminus \Gamma)h = \Gamma$. Then $$(U)\mathfrak{r}_g^{-1} \circ \mathfrak{l}_h^{-1} \subseteq \bigg(\prod_{\alpha \in \Omega} \Sigma_{\alpha}\bigg)\mathfrak{r}_g^{-1} \circ \mathfrak{l}_h^{-1} \subseteq \set{y \in \Omega^{\Omega}}{(\alpha)y = \gamma \text{ for all } \alpha \in \Gamma},$$ and $f \in (U)\mathfrak{r}_g^{-1} \circ \mathfrak{l}_h^{-1}$. Since $U$ is open, it follows that $(U)\mathfrak{r}_g^{-1} \cap ((U)\mathfrak{r}_g^{-1} \circ \mathfrak{l}_h^{-1}) = \{f\}$ is open.
(2) We may assume that $U\neq \emptyset$, since otherwise there is nothing to prove. Let $f \in U$ be arbitrary. For each $\alpha \in \Omega$ we wish to construct $\Xi_{\alpha} \subseteq \Omega$, such that the following properties are satisfied: \begin{enumerate}
\item[$($a$)$] $|\Sigma_{\alpha}|\leq |\Xi_{\alpha}|$ for all $\alpha \in \Omega$, \item[$($b$)$] $\Sigma_{\alpha} \cap \Xi_{\alpha} = \{(\alpha)f\}$ for all $\alpha \in \Omega$, \item[$($c$)$] $\Xi_{\alpha} \cap \Xi_{\beta} = \emptyset$ for all distinct $\alpha, \beta \in \Omega$, \item[$($d$)$] $\bigcup_{\alpha \in \Omega} \Xi_{\alpha} = \Omega$. \end{enumerate}
First, partition $\Omega$ as $\Omega = \bigcup_{\alpha \in \Omega} \Lambda_{\alpha}$, where $|\Lambda_{\alpha}| = |\Omega|$ for each $\alpha \in \Omega$. Then $|(\bigcup_{\beta \in \Lambda_{\alpha}} \Sigma_{\beta})\setminus \Sigma_{\alpha}| = |\Omega|$ for each $\alpha \in \Omega$, since the $\Sigma_{\alpha}$ are disjoint and $|\Lambda_{\alpha}| = |\Omega|$. Hence for each $\alpha \in \Omega$ we can construct $\Xi_{\alpha}$ that satisfies (a) and (b) by choosing $|\Sigma_{\alpha}|$ elements from $(\bigcup_{\beta \in \Lambda_{\alpha}} \Sigma_{\beta})\setminus \Sigma_{\alpha}$, along with $(\alpha)f$. Since the $\Lambda_{\alpha}$ are disjoint, the sets so constructed also satisfy (c). Now add any remaining elements of $\Omega$ to the $\Xi_{\alpha}$ in any way that preserves $\Sigma_{\alpha} \cap \Xi_{\alpha} = \{(\alpha)f\}$ for all $\alpha \in \Omega$. The resulting sets $\Xi_{\alpha}$ will then satisfy (a), (b), (c), and (d).
Since the $\Xi_{\alpha}$ satisfy (a), (b), and (c), we can find a $g\in \Omega^{\Omega}$ such that $(\Xi_{\alpha})g = \Sigma_{\alpha}$ and $(\alpha)fg = (\alpha)f$ for all $\alpha \in \Omega$. If $h\in \Omega^{\Omega}$ is such that $hg \in U$, then $(\alpha)hg \in \Sigma_{\alpha}$, and hence $(\alpha)h\in (\Sigma_{\alpha})g^ {-1} = \Xi_{\alpha}$, for all $\alpha \in \Omega$ (since the $\Sigma_{\alpha}$ are disjoint and $\bigcup_{\alpha \in \Omega} \Xi_{\alpha} = \Omega$). Thus $(U)\mathfrak{r}_g^{-1} \subseteq \prod_{\alpha \in \Omega}\Xi_{\alpha}$, and therefore $U \cap (U)\mathfrak{r}_g^{-1} = \{f\}$, in light of condition (b). Since $U$ is open, it follows that $f$ is isolated. \end{proof}
\section{Compactness} \label{compact-section}
Our next goal is to describe the compact sets in an arbitrary $T_1$ semigroup topology on $\Omega^{\Omega}$. We begin with a characterisation of the compact sets in the pointwise topology on $\Omega^{\Omega}$. If $\Omega$ is countable, in which case the pointwise topology on $\Omega^{\Omega}$ is Polish, and hence metrisable, this characterisation can be obtained from the standard fact that a subset of a metric space is compact if and only if it is complete and totally bounded. Proving the fact in question for arbitrary $\Omega$ is also easy, but we provide the details for the convenience of the reader.
Given a topological space $X$, a subset $U$ of $X$ is \emph{nowhere dense} if $X \setminus \overline{U}$ is dense in $X$, where $\overline{U}$ denotes the closure of $U$.
\begin{lem}\label{compact-lem} Let $\, \Omega$ be a set, let $X \subseteq \Omega^{\Omega}$, and let $\, \mathcal{T}$ denote the pointwise topology on $\, \Omega^{\Omega}$. Then the following hold. \begin{enumerate}
\item[$(1)$] $X$ is compact in $\, \mathcal{T}$ if and only if $X$ is closed in $\, \mathcal{T}$ and $\, |(\alpha)X| < \aleph_0$ for all $\alpha \in \Omega$. \item[$(2)$] If $\, \Omega$ is infinite and $X$ is compact in $\, \mathcal{T}$, then $X$ is nowhere dense. \end{enumerate} \end{lem}
\begin{proof} (1) We may assume that $\Omega$ is nonempty, since otherwise every subset of $\Omega^{\Omega}$ is both compact and closed in $\mathcal{T}$.
Suppose that $X$ is compact. It is a standard fact that in a Hausdorff space every compact subset is closed (see, e.g.,~\cite[Theorem 26.3]{Munkres}), and hence $X$ is closed in $\mathcal{T}$. Now, let $\alpha \in \Omega$. Then $$X = \bigcup_{\beta \in \Omega} ([(\alpha) : (\beta)] \cap X).$$ Since $X$ compact in $\mathcal{T}$, $$X = \bigcup_{i=0}^n ([(\alpha) : (\beta_i)] \cap X)$$ for some $\beta_0, \dots, \beta_n \in \Omega$ and $n \in \mathbb{N}$. Hence $(\alpha)X \subseteq \{\beta_0, \dots, \beta_n\}$, giving $|(\alpha)X| < \aleph_0$.
Conversely, suppose that $X$ is closed in $\mathcal{T}$ and $|(\alpha)X| < \aleph_0$ for all $\alpha \in \Omega$. For each $\alpha \in \Omega$ let $\Sigma_{\alpha} = (\alpha)X$, and let $Y = \prod_{\alpha \in \Omega} \Sigma_{\alpha} \subseteq \Omega^{\Omega}$ consist of all functions $f \in \Omega^{\Omega}$ such that $(\alpha)f \in \Sigma_{\alpha}$. Then $X \subseteq Y$. Since every closed subset of a compact set is compact (see, e.g.,~\cite[Theorem 26.2]{Munkres}), it suffices to show that $Y$ is compact. Now the subspace topology on $Y$ induced by $\mathcal{T}$ is precisely the product topology on $\prod_{\alpha \in \Omega} \Sigma_{\alpha}$ resulting from endowing each $\Sigma_{\alpha}$ with the discrete topology. Since each $\Sigma_{\alpha}$ is finite, and hence compact, $Y$ is a product of compact sets. Therefore $Y$ compact, by Tychonoff's theorem (see, e.g.,~\cite[Theorem 37.3]{Munkres}), as desired.
(2) If $X$ is compact, then by (1), for each $\alpha \in \Omega$ the set $(\alpha)X = \Sigma_{\alpha}$ is finite. As before we have $X \subseteq Y = \prod_{\alpha \in \Omega} \Sigma_{\alpha}$. Now let $\sigma$ and $\tau$ be any two sequences of elements of $\Omega$ of the same finite length. Since $\Omega$ is infinite, we can find some $\alpha \in \Omega \setminus \sigma$, and some $f \in [\sigma : \tau]$ such that $(\alpha)f \notin \Sigma_{\alpha}$. Then $f \in \Omega^{\Omega}\setminus Y \subseteq \Omega^{\Omega}\setminus X$. Since sets of the form $[\sigma : \tau]$ constitute a base for $\mathcal{T}$, this implies that $\Omega^{\Omega} \setminus \overline{X} = \Omega^{\Omega} \setminus X$ is dense in $\Omega^{\Omega}$, and hence $X$ is nowhere dense. \end{proof}
\begin{prop} \label{compact-prop}
Let $\, \Omega$ be a set, and suppose that $\, \Omega^{\Omega}$ is a semitopological semigroup with respect to some $T_1$ topology. If $X \subseteq \Omega^{\Omega}$ is compact, then $X$ is closed and $|(\alpha)X| < \aleph_0$ for all $\alpha \in \Omega$. \end{prop}
\begin{proof} By Theorem~\ref{point-finer}, if $\mathcal{T}$ is a $T_1$ topology with respect to which $\Omega^{\Omega}$ is a semitopological semigroup, then $\mathcal{T}$ contains the pointwise topology. Hence a set compact in such a topology $\mathcal{T}$ is also compact in the pointwise topology. The desired conclusion now follows from Lemma~\ref{compact-lem}(1). \end{proof}
In general, it is not the case that all subsets of $\Omega^{\Omega}$ (for $\Omega$ infinite) that are closed and satisfy $|(\alpha)X| < \aleph_0$ for all $\alpha \in \Omega$ are compact in a $T_1$ semigroup topology. For example, consider the discrete topology, where only finite sets are compact.
Recall that a topological space is \emph{locally compact} if every element has an open neighbourhood that is contained in some compact set. The remainder of this section is devoted to showing that locally compact semigroup topologies on $\Omega^{\Omega}$ are generally not well-behaved.
\begin{cor} \label{loc-com-base}
Let $\, \Omega$ be an infinite set, and suppose that $\, \Omega^{\Omega}$ is a semitopological semigroup with respect to some locally compact $T_1$ topology $\, \mathcal{T}$. Then any base for $\, \mathcal{T}$ must have strictly more than $\, |\Omega|$ elements. \end{cor}
\begin{proof}
Suppose that there is a base $\mathcal{B}$ for $\mathcal{T}$ such that $|\mathcal{B}| \leq |\Omega|$. Since $\mathcal{T}$ is locally compact, for every $f \in \Omega^{\Omega}$ there exists some $U_f \in \mathcal{B}$ and a compact set $X_f \subseteq \Omega^{\Omega}$ such that $f \in U_f \subseteq X_f$. Hence $\bigcup_{f \in \Omega^{\Omega}} X_f = \Omega^{\Omega}$. Since $|\mathcal{B}| \leq |\Omega|$, we can find some subset $\set{X_{\alpha}}{\alpha \in \Omega}$ of $\set{X_f}{f \in \Omega^{\Omega}}$, such that $\bigcup_{\alpha \in \Omega} X_{\alpha} = \Omega^{\Omega}$. In view of each $X_{\alpha}$ being compact, it follows from Proposition~\ref{compact-prop} that $(\alpha)X_{\alpha}$ is finite for each $\alpha \in \Omega$. But since $\Omega$ is infinite, we can find some $f \in \Omega^{\Omega}$ such that $(\alpha)f \notin (\alpha)X_{\alpha}$ for all $\alpha \in \Omega$, which contradicts $\bigcup_{\alpha \in \Omega} X_{\alpha} = \Omega^{\Omega}$. Thus any base for $\, \mathcal{T}$ must have $>|\Omega|$ elements. \end{proof}
We observe that while the topology on $\Omega^{\Omega}$ constructed in Proposition~\ref{open-prod-top} satisfies the conclusion of the previous result, it is not locally compact, since Proposition~\ref{open-prod-top}(6) implies that no element of $\Omega^{\Omega}$ has an open neighbourhood that is contained in a compact set.
We are now ready to give a partial analogue of the result~\cite[Section 4]{Gaughan} of Gaughan that there is no nondiscrete locally compact Hausdorff group topology on $\mathrm{Sym}(\Omega)$.
\begin{theorem} \label{loc-comp-perf}
Let $\, \Omega$ be an infinite set such that $\, |\Omega|$ has uncountable cofinality, and suppose that $\, \Omega^{\Omega}$ is a semitopological semigroup with respect to a locally compact topology $\, \mathcal{T}$. Then $\, \mathcal{T}$ is either not perfect or not $T_1$. \end{theorem}
\begin{proof}
Seeking a contradiction, suppose that $\mathcal{T}$ is perfect and $T_1$, and let $f \in \Omega^{\Omega}$ be a constant function. Since $\mathcal{T}$ is locally compact, there must be a compact set $X$ and an open neighbourhood $U$ of $f$ such that $U \subseteq X$. Letting $\Sigma_{\alpha} = (\alpha)X$ for each $\alpha \in \Omega$, we have $U \subseteq \prod_{\alpha \in \Omega} \Sigma_{\alpha}$. Since $\mathcal{T}$ is perfect, by Proposition~\ref{iso-prop}(1), it must be the case that $|\set{\alpha \in \Omega}{\beta \notin \Sigma_{\alpha}}| < |\Omega|$ for each $\beta \in \Omega$. Then letting $\Gamma \subseteq \Omega$ be a countably infinite set, $$|\set{\alpha \in \Omega}{\Gamma \not\subseteq \Sigma_{\alpha}}| \leq \Big|\bigcup_{\beta \in \Gamma}\set{\alpha \in \Omega}{\beta \notin \Sigma_{\alpha}}\Big| < |\Omega|,$$ since $|\Omega|$ is assumed to have uncountable cofinality. Hence $\Gamma \subseteq \Sigma_{\alpha}$ for some $\alpha \in \Omega$, making $\Sigma_{\alpha}$ infinite. But since $\mathcal{T}$ is $T_1$, by Proposition~\ref{compact-prop}, $|\Sigma_{\alpha}| < \aleph_0$ for all $\alpha \in \Omega$, producing the desired contradiction. \end{proof}
In light of Theorem~\ref{loc-comp-perf} we ask the following.
\begin{question} Does there exist an infinite set $\, \Omega$ and a perfect locally compact $T_1$ (or Hausdorff) topology on $\, \Omega^{\Omega}$ with respect to which it is a topological semigroup? \end{question}
\noindent Z.\ Mesyan, Department of Mathematics, University of Colorado, Colorado Springs, CO, 80918, USA
\noindent \emph{Email:} \href{mailto:[email protected]}{[email protected]}
\noindent J.\ D.\ Mitchell, Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, Scotland
\noindent \emph{Email:} \href{mailto:[email protected]}{[email protected]}
\noindent Y.\ P\'eresse, University of Hertfordshire, Hatfield, Hertfordshire, AL10 9AB, UK
\noindent \emph{Email:} \href{mailto:[email protected]}{[email protected]}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title[Regularity Properties of Monotone Measure-preserving Maps]{Regularity Properties of Monotone Measure-preserving Maps}
\author[A.\ Figalli]{Alessio Figalli} \address{ETH Z\"{u}rich, Department of Mathematics, R\"{a}mistrasse 101, Z\"{u}rich 8092, Switzerland} \email{[email protected]} \author[Y.\ Jhaveri]{Yash Jhaveri} \address{Rutgers University\,--\,Newark, Department of Mathematics \& Computer Science, Smith Hall, 101 Warren Street, Newark, New Jersey 07102, USA} \email{[email protected]}
\begin{abstract} In this note, we extend the regularity theory for monotone measure-preserving maps, also known as optimal transports for the quadratic cost optimal transport problem, to the case when the support of the target measure is an arbitrary convex domain and, on the low-regularity end, between domains carrying certain invariant measures.\\
\begin{center} {\it In honor of David Jerison for his 70th birthday.} \end{center} \end{abstract} \maketitle
\section{Introduction}
Given a pair of atom-free Borel probability measures $\mu$ and $\nu$ on $\mathbb{R}$, the monotone rearrangement theorem asserts that the function $y = y(x)$ defined implicitly by \[ \int_{-\infty}^x \, \mathrm{d} \mu = \int_{-\infty}^{y(x)} \, \mathrm{d} \nu \] is measure-preserving, i.e., \[ \mu(y^{-1}(E)) = \nu(E)\quad \text{for any Borel set} \quad E \subset \mathbb{R}. \] In particular, $y$ is unique $\mu$ almost everywhere and can be made to be monotone, or, equivalently, the derivative of a convex function.
When $\mathbb{R}$ is replaced by $\mathbb{R}^n$, however, before Brenier's discovery (see \cite{B}), a proper generalization of the monotone rearrangement theorem was myth; and only after the work of McCann, in \cite{M}, was the myth made real. Precisely, he proved that if $\mu$ vanishes on every Lipschitz $(n-1)$-dimensional surface\footnote{\,McCann actually assumes that $\mu$ vanishes on all (Borel) sets of Hausdorff dimension $n-1$. This guarantees that the set of non-differentiability points of a convex function are $\mu$-negligible. However, this assumption can be weakened. Since convex functions are differentiable outside of a countable union of Lipschitz hypersurfaces (see \cite{Z}), McCann's theorem holds assuming that $\mu$ vanishes on every Lipschitz $(n-1)$-dimensional surface.}, then a convex potential $u : \mathbb{R}^n \to \mathbb{R} \cup \{ + \infty \}$ exists whose gradient map $\nabla u = \nabla u(x)$ is unique $\mu$ almost everywhere and pushes $\mu$ forward to $\nu$, i.e., \[ \mu((\nabla u) ^{-1}(E)) = \nu(E) \quad \text{for any Borel set} \quad E \subset \mathbb{R}^n. \] (Brenier's theorem guaranteed the same conclusion as McCann's theorem, but under some restrictive technical conditions on $\mu$ and $\nu$.)
The first general regularity result on these Brenier--McCann maps was proved by Caffarelli in \cite{C2}: provided that $\mu$ and $\nu$ are absolutely continuous with respect to $n$-dimensional Lebesgue measure $\mathscr{L}^n$, their respective densities $f$ and $g$ vanish outside of and are bounded away from zero and infinity on open bounded sets $X$ and $Y$ respectively, and $Y$ is convex, he showed that $u$ is strictly convex in $X$ (see, e.g., the proof of \cite[Theorem 4.6.2]{F}). This result opened the door to the development of a regularity theory for mappings with convex potentials based on the regularity theory for strictly convex solutions to the Monge--Amp\`{e}re equation (see, e.g., \cite{C0}); indeed, \[ (\nabla u)_{\#} f = g \text{ is formally, at least, equivalent to } \det D^2 u = \frac{f}{g(\nabla u)}. \]
Unfortunately, Caffarelli's boundedness assumptions on the domains $X$ and $Y$ are restrictive, since many probability densities, especially those found in applications, are supported on all of $\mathbb{R}^n$: Gaussian densities, for example. Motivated by this, in \cite{FCE}, Cordero-Erausquin and Figalli showed that, in several situations of interest, one can ensure the regularity of monotone measure-preserving maps even if the measures under consideration have unbounded supports. However, missing from their collection is the situation where $Y$ is an {\it arbitrary} convex domain. Lifting this restriction is a main goal of this paper.
\subsection{Results} Our main theorem is an extension of Caffarelli's theorem, on the strict convexity of $u$, in two ways. First, we allow $X$ and $Y$ to be unbounded. Second, we permit $X$ and $Y$ to carrying certain invariant measures that we call {\it locally doubling measures} (qualitatively, our notion replaces balls in the classical notion of a doubling measure with ellipsoids, in order to account for the affine invariance of our setting).
\begin{definition} A nonnegative measure $\lambda$ is {\it locally doubling (on ellipsoids)} if the following holds: for every ball $B$, there is a constant $C \geq 1$ such that \[ \lambda(\mathcal{E}) \leq C \lambda(\tfrac{1}{2}\mathcal{E}) \] for all ellipsoids $\mathcal{E} \subset B$ with center (of mass) in $\spt(\lambda)$. Here $\frac{1}{2}\mathcal{E}$ is the dilation of $\mathcal{E}$ with respect to its center by $1/2$. \end{definition} This notion of doubling was introduced by Jhaveri and Savin in \cite{JS}.\footnote{\,This family of measures is strictly larger than the family of measures locally comparable to Lebesgue measure on their supports. (See \cite{JS} for examples of locally doubling measures not comparable to Lebesgue measure on their supports.)} That said, the first consideration of measures with a ``doubling like'' property in the world of solutions to Monge--Amp\`{e}re equations can be traced back to the work of Jerison \cite{J} and then Caffarelli \cite{C1}. In particular, in \cite{C1}, Caffarelli showed that Alexandrov solutions to \[ \det D^2 v = \rho, \] where the measure $\rho$ is doubling on a specific collection of convex sets called sections\footnote{\,These are sets of the form $\{ v \leq \ell \}$ for any affine function $\ell$.}, share the same geometric properties as Alexandrov solutions to Monge--Amp\`{e}re equations with right-hand sides comparable to Lebesgue measure (see \cite{C0}).
We now state our main theorem.
\begin{theorem} \label{thm: strict convexity} Let $\mu$ and $\nu$ be two locally doubling probability measures on $\mathbb{R}^n$ that vanish on Lipschitz $(n-1)$-dimensional surfaces and are concentrated on two open sets $X$ and $Y$ respectively, and suppose that $Y$ is convex. Then any convex potential $u$ associated to the Brenier--McCann map pushing $\mu$ forward to $\nu$ is strictly convex in $X$. \end{theorem}
\begin{remark}
It is well-known that $Y$ needs to be convex. When $Y$ is not convex, $u$ can fail to be strictly convex and $\nabla u$ can behave rather poorly. If we consider Pogorelov's counterexample to the strict convexity of solutions to the Monge--Amp\`{e}re equation in three dimensions, we see that $Y$ needs to be convex in order to guarantee the strict convexity of potentials of Brenier--McCann maps. In particular, let $u(x',x_3) = |x'|^{4/3}(1+x_3^2)$, let $Q_r = \{ |x_i| < r/2 \text{ for } i = 1,2,3 \}$ be the cube with side length $r > 0$ and centered at the origin in $\mathbb{R}^3$, and let $Y = \nabla u(Q_r)$. If $r \ll 1$, then $f = \det D^2 u$ is analytic and positive in $Q_r$, and $\nabla u$ is the Brenier--McCann map pushing forward $\mu = (f/\|f\|_{L^1(Q_r)}) \mathscr{L}^3 \mres Q_r$ to $\nu = (1/\mathscr{L}^3(Y)) \mathscr{L}^3 \mres Y$. The set $Y$ is open but not convex, and $u = 0$ along $\{ x' = 0 \}$. Moreover, as demonstrated in, for instance, \cite{C2} and \cite{Jh}, $\nabla u$ easily fails to be continuous when $Y$ is not convex. \end{remark}
\begin{remark} While the target measure $\nu$ need not vanishes on all Lipschitz $(n - 1)$-dimensional surfaces in order to invoke McCann's theorem (which asks this only of the source measure $\mu$), it must in order to ensure our main theorem holds. If $\mu = \mathscr{L}^2 \mres Q_1$ is the uniform measure on $Q_1$ the unit cube centered at the origin in $\mathbb{R}^2$ and $\nu = \mathscr{H}^1 \mres Q_1 \cap \{ x_2 = 0 \}$ is the $1$-dimensional Hausdorff measure restricted to the central horizontal axis of $Q_1$, then the Brenier--McCann map pushing $\mu$ forward to $\nu$ is the projection map $(x_1,x_2) \mapsto x_1$. Up to a constant, this map's convex potential is $\frac{1}{2}x_1^2$, which is not strictly convex. With respect our proof of Theorem~\ref{thm: strict convexity}, asking this of both $\mu$ and $\nu$ guarantees the validity of the mass balance formula in Lemma~\ref{lem: Brenier soln} (see also Remark~\ref{rmk:balance}), our main tool. \end{remark}
\begin{remark}
A simple case to which Theorem~\ref{thm: strict convexity} applies, but the corresponding results in \cite{C2} and \cite{FCE} do not, is when $\mu = g \mathscr{L}^3 \mres \{|x_1| < 1 \} \times \mathbb{R}^2$ and $\nu = g \mathscr{L}^3 \mres \mathbb{R}^2 \times \{|x_3| < 1 \}$, and $g$ is the standard Gaussian density on $\mathbb{R}^3$ appropriately normalized to make $\mu$ and $\nu$ probability measures. \end{remark}
With our main theorem in hand, our second and third theorems further extend the known regularity theory for monotone measure-preserving maps, completing the story started by Cordero-Erausquin and Figalli in \cite{FCE} on monotone transports between unbounded domains.
\begin{theorem} \label{thm: C1} Let $\mu$ and $\nu$ be two locally doubling probability measures on $\mathbb{R}^n$ that vanish on Lipschitz $(n-1)$-dimensional surfaces and are concentrated on two open sets $X$ and $Y$ respectively, and suppose that $Y$ is convex. Then the Brenier--McCann map $\nabla u$ pushing $\mu$ forward to $\nu$ is a homeomorphism from $X$ onto a full measure subset of $Y$. Moreover, for every $A \Subset X$, a constant $\alpha > 0$ exists such that $\nabla u \in C^{0,\alpha}(A)$. Furthermore, $\nabla u(X) = Y$ whenever $X$ is convex. \end{theorem}
\begin{theorem} \label{thm: higher reg} Let $f$ and $g$ be two functions on $\mathbb{R}^n$ that define locally doubling probability measures concentrated on two open sets $X$ and $Y$ respectively, and suppose that $Y$ is convex. Assume that $f$ and $g$ are bounded away from zero and infinity on compact subsets of $X$ and $Y$ respectively. Then for every $E \Subset X$, a constant $\epsilon > 0$ exists such that any convex potential $u$ associated to the Brenier--McCann map pushing $f$ forward to $g$ is $W^{2,1+\epsilon}(E)$. Also, $\nabla u$ is locally a $C^{k+1,\beta}$-diffeomorphism from $X$ onto its image provided $f$ and $g$ are locally $C^{k,\beta}$ in $X$ and $Y$ respectively. \end{theorem}
\begin{remark} We note that the proof of the Theorem~\ref{thm: higher reg}, given the strict convexity of $u$ (provided by Theorem~\ref{thm: strict convexity}), is classical. Indeed, it suffices to localize classical regularity results for the Monge--Amp\`ere equation. We refer the reader to \cite[Section 4.6.1]{F} for more details. \end{remark}
\subsection{Structure} This remainder of this paper is structured as follows.
In Section~\ref{sec: strict convexity}, we prove Theorem~\ref{thm: strict convexity}. Our proof is self-contained apart from some facts in convex analysis; we provide explicit references to these used but unproved facts. We remark that our proof is inspired by the proof of the Alexandrov maximum principle in \cite{JS} (and, of course, Caffarelli's original proof of the strict convexity of potential functions of optimal transports/solutions to Monge--Amp\`{e}re equations). If the reader is familiar with \cite{FCE} or \cite{C2}, then they might consider directing their attention to Case 2. Case 2b is completely novel. Case 2a illustrates our argument in the setting of \cite{C2}, which builds on the work of \cite{C0} and is the foundation for Case 2b.
Section~\ref{sec: C1} is dedicated to the proof of Theorem~\ref{thm: C1}. Our proof here is similarly self-contained (and an adaptation of Caffarelli's argument of the same result in \cite{C0}, but, of course, using the line of reasoning developed to prove Theorem~\ref{thm: strict convexity}). The H\"{o}lder regularity of $\nabla u$ is a consequence of appropriately localizing the arguments of \cite{JS}.
\section{Proof of Theorem~\ref{thm: strict convexity}} \label{sec: strict convexity}
Before we begin, it will be convenient to replace the potential $u$ by the following lower-semicontinuous extension of $u$ outside of $X$:\footnote{\,Here $\partial u(z)$ is called the subdifferential of $u$ at $z$ and is defined as follows: \[ \partial u(z) := \{ p \in \mathbb{R}^n : u(x) \geq u(z) + p \cdot (x-z) \text{ for all $x \in X$} \}. \] Moreover, for a set $E \subset \mathbb{R}^n$, we define $\partial u(E) := \cup_{z \in E} \partial u(z)$. } \[ \underline{u}(x) := \sup_{\substack{z \in X \\ p \in \partial u(z)}} \{ u(z) + p \cdot (x-z) \}. \]
Observe that $\underline{u}|_X = u|_X$. For notational simplicity, we shall not distinguish $\underline{u}$ from $u$; so when we write $u$ in what follows, we mean $\underline{u}$.
We shall denote the domain of $u$ by $\dom(u)$, namely, $\dom(u) := \{u<+\infty\}$. Note that $\dom(u)$ is convex. We recall that convex functions are locally Lipschitz inside their domain (see, e.g., \cite[Appendix A.4]{F}). Furthermore, we shall denote the convex hull of a set $A$ by $\conv(A)$.
Let $\ell$ define a supporting plane to the graph of $u$ at a point in $X$. Precisely, \[ \ell(x)=u(z) + p \cdot (x-z)\quad \text{for some}\quad (z,p) \in X \times \mathbb{R}^n \] and $\ell \leq u$. Note that \[ \Sigma := \{ u = \ell \} = \{ u \leq \ell \} \] is closed, as $u$ is lower-semicontinuous, \[ \Sigma, X \subset \dom(u), \] and, because $Y$ is convex, \begin{equation} \label{eqn: subdiff props} \partial u(\mathbb{R}^n) \subset \overline{\partial u(X)} \subset \overline{Y}\quad \text{and}\quad \mathscr{L}^n(Y \setminus \partial u(X)) = 0. \end{equation} (A proof of \eqref{eqn: subdiff props} can be found in \cite{FCE}.)
Recall that an exposed point $\hat{x}$ of $\Sigma\subset \mathbb{R}^n$ is one for which there exists a hyperplane $\Pi\subset \mathbb{R}^n$ tangent to $\Sigma$ at $\hat{x}$ such that $\Pi\cap \Sigma=\{\hat{x}\}$. Also, remember that optimal/monotone transports balance mass, in the following way.
\begin{lemma}[Mass Balance Formula] \label{lem: Brenier soln} Let $u : \mathbb{R}^n \to \mathbb{R} \cup \{ + \infty \}$ be convex and such that $(\nabla u)_\# \mu = \nu$ where $\mu$ and $\nu$ are two Borel measures that vanish on all Lipschitz $(n-1)$-dimensional surfaces. Then for all Borel sets $E \subset \mathbb{R}^n$, \[ \mu(E) = \nu (\partial u(E) ). \] \end{lemma}
\begin{remark} \label{rmk:balance} The mass balance formula was originally proved for measures that are absolutely continuous with respect to Lebesgue measure (see, e.g., \cite[Lemma 4.6]{V}). However, with respect to absolute continuity, the proof only relies on the measures in question not giving mass to the set of non-differentiable points of a convex function. As observed in the introduction, such points are contained in a countable union of Lipschitz $(n-1)$-dimensional surfaces. So the set of non-differentiable points of a convex function is negligible both for $\mu$ and $\nu$ under our assumption. \end{remark}
Finally, recall that if a nonnegative measure is locally doubling (on ellipsoids), then it is locally doubling on all bounded convex domains (see \cite[Corollary 2.5]{JS}). After this preliminary discussion, we can now prove our main theorem.
\begin{proof}[Proof of Theorem~\ref{thm: strict convexity}] Proving that $u$ is strictly convex in $X$ corresponds to proving that for any supporting plane $\ell$ to (the graph of) $u$ at a point in $X$, the set $\Sigma=\{u=\ell\}$ is a singleton. Assuming that $\Sigma$ is not a singleton, we will show that $\Sigma$ both has and does not have exposed points, which cannot be; thus, $\Sigma$ is a singleton, as desired.
\subsection*{Case 1}{\bf $\Sigma$ has no exposed points.} If $\Sigma$ has no exposed points, then $\Sigma \supset \mathbb{R}\mathbf{e}$ for some unit vector $\mathbf{e}$. In turn, $\partial u(\mathbb{R}^n) \subset \mathbf{e}^\perp$ (see, e.g., \cite[Lemma A.25]{F}). But this is impossible given \eqref{eqn: subdiff props}: \[ 0 < \mathscr{L}^n(\partial u(X) \cap Y) \leq \mathscr{L}^n(\partial u(\mathbb{R}^n) \cap Y) \leq \mathscr{L}^n(\mathbf{e}^\perp \cap Y) = 0. \]
\subsection*{Case 2}{\bf $\Sigma$ has an exposed point $\hat{x}$ in $\overline{X}$.} Up to a translation and a rotation, we can assume that \[ \hat{x} = 0 \in \overline{X},\quad \Sigma \subset \{ x_1 \leq 0 \},\quad \text{and}\quad \Sigma \cap \{ x_1 = 0 \} = \{ 0 \}. \] Since $X$ is open and $\Sigma\cap X$ is nonempty by construction, there is a point $x_{int}\in \Sigma \cap X$ and a ball centered at this point completely contained in $X$. Thus, up to a shearing transformation $x \mapsto x - \eta x_1$ with $\eta \cdot \mathbf{e}_1 = 0$, and a dilation, we may assume that \[ x_{int} = - \mathbf{e}_1 \quad \text{and}\quad B_d(- \mathbf{e}_1 ) \Subset X \] for some $d > 0$. Finally, up to subtracting $\ell$ from $u$, we can assume that \[ \ell \equiv 0. \]
\subsection*{Case 2a: $0 \in \interior(\dom(u)) \cap \overline{X}$.} As our exposed point $0$ and all of the points in $\overline{B_d(-\mathbf{e}_1)}$ belong to $\interior(\dom(u))$, which is convex (and, by definition, open), the convex hull of the union of $\overline{B_d(-\mathbf{e}_1)}$ and $\{0\}$ is contained in $\interior(\dom(u))$. So there exists an open, bounded set $U \Subset \interior (\dom (u))$ containing $\conv (\overline{B_d(-\mathbf{e}_1)} \cup \{0\} )$. Moreover, we know that \[ \partial u(U) \subset \conv(\overline{\nabla u(U)}) =: \Upsilon \subset B_R \cap \overline{Y} \] for some $R > 0$ (see, e.g., \cite[Lemma A.22]{F}). Let $u^\ast$ be the Legendre transform of $u$, namely \begin{equation} \label{eq:u ast} u^\ast(q):=\sup_{x \in \mathbb{R}^n}\{q\cdot x-u(x)\} \end{equation} and define \[ \Omega := \partial u^\ast(\Upsilon) \supset U. \] Recalling that $\partial u$ and $\partial u^\ast$ are inverses of each other (see, e.g., \cite[Section A.4.2]{F}), we deduce that $(\nabla u)_{\#} \rho = \gamma$ where \[ \rho := \mu \mres \Omega\quad \text{and} \quad \gamma := \nu \mres \Upsilon. \] In particular, if we let $\phi : \mathbb{R}^n \to \mathbb{R} \cup \{ + \infty \}$ be defined by \[ \phi(x) := \sup_{\substack{z \in \Omega \\ p \in \partial u(z)}} \{ u(z) + p \cdot (x-z) \}, \] then, by construction, $\phi$ and $u$ agree on $\Omega$, $\phi$ is (globally) Lipschitz, \[ \partial \phi(\mathbb{R}^n) = \partial \phi (\Omega) = \Upsilon, \] and $0 \in \Omega$ is an exposed point for $\{ \phi = 0 \} = \{ \phi \leq 0 \}$.
Now, let $ \phi_\epsilon(x) := \phi(x) - \epsilon(x_1 + 1)$ and define \[ S_0 := \{ \phi = 0 \} \cap \{ x_1 \geq -1 \} \quad \text{and}\quad S_\epsilon := \{ \phi_\epsilon \leq 0 \}. \] Also, let $\gamma_\epsilon$ be defined by \[ \gamma_\epsilon := (\mathrm{Id} - \epsilon \mathbf{e}_1)_{\#} \gamma . \] Notice that, by construction, $S_0$ is compact, $S_0\subset\{x_1\leq 0\}$, $0,-\mathbf{e}_1 \in S_0$, and $S_\epsilon \to S_0$ in the Hausdorff sense as $\epsilon \to 0$; in particular, there exists $D>0$ such that \[ S_\epsilon \subset B_D\quad \text{for all}\quad \epsilon \ll 1. \] Also, if $a_\epsilon > 0$ is such that $\Pi_\epsilon := \{ x_1 = a_\epsilon \}$ is a supporting plane to $S_\epsilon$, we see that \[ a_\epsilon \to 0 \quad\text{as}\quad \epsilon \to 0 \] and \[
\epsilon = |\phi_\epsilon(0)| \leq \max_{S_\epsilon} |\phi_\epsilon| \leq (1 + a_\epsilon) \epsilon. \]
Let $A_\epsilon$ be the John transformation (affine map) that normalizes $S_\epsilon$ (see, e.g., \cite{Gu}): \[ A_\epsilon x := L_\epsilon(x - x_\epsilon), \] where $x_\epsilon$ is the center of mass of $S_\epsilon$ and $L_\epsilon:\mathbb{R}^n\to\mathbb{R}^n$ is a symmetric and positive definite linear transformation. Set \[ \tilde{\phi}_\epsilon(x) := \frac{\phi_\epsilon(A_\epsilon^{-1}x)}{\epsilon} \quad\text{and}\quad \tilde{S}_\epsilon := A_\epsilon(S_\epsilon). \] Then \[ B_1 \subset \tilde{S}_\epsilon \subset B_{n^{3/2}} \] and \[
1 = |\tilde{\phi}_\epsilon(\tilde{0}_\epsilon)| \leq \max_{\tilde{S}_\epsilon} |\tilde{\phi}_\epsilon| \leq 1 + a_\epsilon \quad \text{with}\quad \tilde{0}_\epsilon := A_\epsilon(0). \] Recall that affine transformations preserve the ratio of the distances between parallel planes; therefore, letting $\Pi_{-1} := \{ x_1 = -1 \}$, $\Pi_0 := \{ x_1 = 0 \}$, and $\tilde{\Pi}_i := A_\epsilon(\Pi_i)$ for $i = -1, 0, \epsilon$, we have that \[ \frac{\dist(\tilde{\Pi}_0,\tilde{\Pi}_\epsilon)}{\dist(\tilde{\Pi}_{-1},\tilde{\Pi}_\epsilon)} = \frac{\dist(\Pi_0,\Pi_\epsilon)}{\dist(\Pi_{-1},\Pi_\epsilon)} = \frac{a_\epsilon}{1+a_\epsilon}. \] In turn, \[ \dist(\tilde{0}_\epsilon, \partial \tilde{S}_\epsilon) \leq \dist(\tilde{\Pi}_0,\tilde{\Pi}_\epsilon) \leq \dist(\tilde{\Pi}_{-1},\tilde{\Pi}_\epsilon)\frac{a_\epsilon}{1+a_\epsilon} \leq \diam(\tilde{S}_\epsilon)a_\epsilon \leq 2n^{3/2}a_\epsilon, \] and considering the cone generated by $\partial \tilde{S}_\epsilon$ over $(\tilde{0}_\epsilon,\tilde{\phi}_\epsilon(\tilde{0}_\epsilon))$, we find that \[ K_\epsilon := \conv \left( B_{r_n} \cup \left\{ \tfrac{r_n}{a_\epsilon}\mathbf{e}_1 \right\} \right) \subset \partial \tilde{\phi}_\epsilon(\tilde{S}_\epsilon) \quad \text{with}\quad r_n := \frac{1}{2n^{3/2}}. \] (For more details on this inclusion, see, e.g., \cite[Theorem 2.8]{F}.) So if we let \[ \tilde{\rho}_\epsilon := (A_\epsilon)_{\#} \rho \quad \text{and}\quad \tilde{\gamma}_\epsilon := (\epsilon^{-1}L_\epsilon^{-1})_{\#}\gamma_\epsilon, \] then $(\nabla \tilde{\phi}_\epsilon)_\# \tilde{\rho}_\epsilon = \tilde{\gamma}_\epsilon$, and, by the mass balance formula, \begin{equation} \label{eqn: mb1} \tilde{\gamma}_\epsilon (K_\epsilon) \leq \tilde{\gamma}_\epsilon\big(\partial \tilde{\phi}_\epsilon(\tilde{S}_\epsilon)\big) = \tilde{\rho}_\epsilon (\tilde{S}_\epsilon)\leq \tilde{\rho}_\epsilon(B_{n^{3/2}}). \end{equation} On the other hand, since $S_\epsilon \subset B_D$ and $\tilde S_\epsilon \supset B_1$ for $\epsilon \ll 1$, we see that \[
|A_\epsilon(w) - A_\epsilon(z)| \geq \frac{1}{D}|w - z| \quad \text{for all}\quad w,z \in \mathbb{R}^n. \] In turn, for $\epsilon \ll 1$, \[ \tilde{\Omega}_\epsilon := A_\epsilon(\Omega) \supset A_\epsilon(B_d(-\mathbf{e}_1)) \supset B_{\frac{d}{D}}(A_\epsilon(-\mathbf{e}_1)). \] Therefore, if we define \[ \tilde{S}_{\epsilon,d} := \left\{ \dist(\,\cdot\,,\partial \tilde{S}_\epsilon) \geq \tfrac{d}{2D} \right\}, \] then there exists a dimensional constant $C_n > 0$ and a point $\tilde{z}_d$ such that \[ A_\epsilon(B_d(-\mathbf{e}_1)) \cap \tilde{S}_{\epsilon,d} \supset B_{\frac{d}{C_nD}}(\tilde{z}_d). \] Also, by, for example, \cite[Corollary A.23]{F}, \[ \partial \tilde{\phi}_\epsilon (\tilde{S}_{\epsilon,d}) \subset B_{\frac{6D}{d}} . \] Thus, for all $\epsilon \ll 1$, \begin{equation} \label{eqn: mb2} \begin{split} \tilde{\rho}_\epsilon (B_{n^{3/2}}) \leq \tilde{\mu}_\epsilon (B_{2n^{3/2}}(\tilde{z}_d))
\leq C_\mu^k \tilde{\mu}_\epsilon\left(B_{\frac{d}{C_n D}}(\tilde{z}_d)\right) = C_\mu^k \tilde{\rho}_\epsilon\left(B_{\frac{d}{C_n D}}(\tilde{z}_d)\right), \end{split} \end{equation} where $\tilde{\mu}_\epsilon := (A_\epsilon)_{\#}\mu$, the number $k \in \mathbb{N}$ is such that $2n^{3/2} \leq 2^k \frac{d}{C_nD}$, and $C_\mu$ is the doubling constant for $\mu$ in $B_{4Dn^{3/2}}$. (The last equality holds since $\tilde{\mu}_\epsilon$ and $\tilde{\rho}_\epsilon$ agree on $\tilde{\Omega}_\epsilon$.) Moreover, using the mass balance formula again, we deduce that \begin{equation} \label{eqn: mb3} \begin{split} \tilde{\rho}_\epsilon\left(B_{\frac{d}{C_n D}}(\tilde{z}_d)\right) &\leq \tilde{\rho}_\epsilon\left(A_\epsilon(B_d(-\mathbf{e}_1)) \cap \tilde{S}_{\epsilon,d}\right) \\ &= \tilde{\gamma}_\epsilon\left(\partial \tilde{\phi}_\epsilon (A_\epsilon(B_d(-\mathbf{e}_1)) \cap \tilde{S}_{\epsilon,d})\right) \leq \tilde{\gamma}_\epsilon\big(B_{\frac{6D}{d}}\big) \end{split} \end{equation} for all $\epsilon \ll 1$. Consequently, combining the three chains of inequalities \eqref{eqn: mb1}, \eqref{eqn: mb2}, and \eqref{eqn: mb3}, we have that \begin{equation} \label{eqn: mb4} \tilde{\gamma}_\epsilon(K_\epsilon) \leq C_\mu^k\tilde{\gamma}_\epsilon\big(B_{\frac{6D}{d}}\big). \end{equation} Now, let $t_m \mathbf{e}_1 \in K_\epsilon$ for $m = 1, \dots, M$ be a sequence of points chosen\footnote{\,A possible way to construct such a sequence is to choose $t_m=5^m$. To ensure that $t_m \mathbf{e}_1 \in K_\epsilon$ for any $m=1,\ldots,M$, one needs $M \leq \frac{\log r_n - \log a_\epsilon}{\log 5}$.} so that \[ \frac{1}{2}K_m \subset K_m \setminus K_{m-1} \quad \text{with}\quad K_m := \conv ( B_{r_n} \cup \{t_m\mathbf{e}_1\} ) \quad\text{and}\quad K_0 := B_{r_n}. \] By construction, $\{ \frac{1}{2}K_m \}_{m = 1}^M$ is a disjoint family, and \begin{equation} \label{eq:M} M = M(a_\epsilon) \to \infty\quad \text{as}\quad a_\epsilon \to 0. \end{equation} Hence, since $\epsilon L_\epsilon(K_\epsilon) \subset \Upsilon$, we find that \[ M\tilde{\gamma}_\epsilon(B_{r_n}) \leq \sum_{m=1}^M \tilde{\gamma}_\epsilon(K_m) \leq C_\gamma \sum_{m=1}^M \tilde{\gamma}_\epsilon(\tfrac{1}{2} K_m) \leq C_\gamma \tilde{\gamma}_\epsilon(K_\epsilon), \] with $C_\gamma$ denoting the doubling constant for $\gamma$ in $B_{2R}$, which is the same as the doubling constant for $\nu$ in $B_{2R}$; since $\Upsilon$ is convex, $\gamma$ inherits its doubling property from $\nu$. All in all, considering the above chain of inequalities and \eqref{eqn: mb4}, and denoting by $j\in \mathbb N$ the smallest number such that $\frac{6D}{d} \leq 2^j r_n$, we see that \[ 0 < M\tilde{\gamma}_\epsilon(B_{r_n}) \leq C_\mu^kC_\gamma \tilde{\gamma}_\epsilon\big(B_{\frac{6D}{d}}\big) \leq C_\mu^kC_\gamma^{j+1}\tilde{\gamma}_\epsilon(B_{r_n}), \] or, equivalently, \[ M \leq C_\mu^kC_\gamma^{j+1}. \] But this is impossible for small $\epsilon$, concluding the proof.
\subsection*{Case 2b: $0 \in \partial (\dom(u)) \cap \overline{X}$.} In this subcase, let $u_\epsilon(x) := u(x) - \epsilon(x_1 + 1)$ and define \[ S_0 := \Sigma \cap \{ x_1 \geq -1 \} \quad \text{and} \quad S_\epsilon := \{ u_\epsilon \leq 0 \}. \] Like before, for all $\epsilon \ll 1$, \[ S_\epsilon \subset B_D \] for some $D > 0$. Here, however, as $0 \in \partial (\dom(u))$, we have that \[ \partial \tilde{u}_\epsilon(\tilde{S}_\epsilon) \supset \conv (B_{r_n} \cup \mathbb{R}^+\mathbf{e}_1 ) \quad \text{with}\quad r_n := \frac{1}{2n^{3/2}}. \] The function $\tilde{u}_\epsilon$ is defined in an analogous fashion to how $\tilde{\phi}_\epsilon$ was defined in Case 2a (but replacing $\phi$ by $u$) and, again, $\tilde{S}_\epsilon := A_\epsilon(S_\epsilon)$ with $A_\epsilon$ denoting the John map associated to $S_\epsilon$ whose linear part is $L_\epsilon$. In turn, arguing as we did in Case 2a, where again $k \in \mathbb{N}$ is such that $2n^{3/2} \leq 2^k \frac{d}{C_nD}$ and $C_\mu$ is the doubling constant for $\mu$ in $B_{4Dn^{3/2}}$, but in the original variables, we deduce that \[ \nu_\epsilon\big(\epsilon L_\epsilon (K_\epsilon)\big) \leq C_\mu^k \nu_\epsilon\big(\epsilon L_\epsilon (B_{\frac{6D}{d}} )\big) \] for all $\epsilon \ll 1$ (cf. \eqref{eqn: mb4}). Here, instead, $\nu_\epsilon := (\mathrm{Id} - \epsilon \mathbf{e}_1)_{\#} \nu$ and \[ K_\epsilon := \conv \left( B_{r_n} \cup \left\{ \tfrac{r_n}{\epsilon}\mathbf{e}_1 \right\} \right). \] Now notice that \[
|L_\epsilon(\mathbf{e}_1)| = |A_\epsilon(0) - A_\epsilon(-\mathbf{e}_1)| \leq 2n^{3/2}. \] Moreover, we claim there exists an $N \gg 2n^{3/2} > 0$ such that \[
\|\epsilon L_\epsilon\| \leq N \quad \text{for all}\quad \epsilon \ll 1. \]
Indeed, if not, then we can find a sequence of points $z_\epsilon \in S_\epsilon$ and slopes $p_\epsilon \in \partial u_\epsilon(z_\epsilon) \cap \Span (S_0)^\perp$ such that $|p_\epsilon| \to \infty$. In particular, in the limit, we find a point $z_0 \in S_0$ such that $\partial u(z_0) \cap \Span (S_0)^\perp$ contains a sequence of slopes $\{p_j\}_{j \in \mathbb{N}}$ with $|p_j| = j$. But as $p_j \in \Span (S_0)^\perp$, we see that $p_j \cdot (x - z_0) = p_j \cdot x = p_j \cdot (x - z)$ for any $z \in S_0$. Hence, $p_j \in \partial u(z)$ for all $z \in S_0$ and $j \in \mathbb{N}$. However, this is impossible; $S_0 \cap \interior(\dom(u))$ is nonempty, and on this set, $u$ is locally Lipschitz, proving the claim.
Therefore, \[ \epsilon L_\epsilon(K_\epsilon) \subset B_N \quad \text{and}\quad \epsilon L_\epsilon \big(B_{\frac{6D}{d}}\big) \subset B_{\frac{6DN}{d}}. \] And so, arguing exactly like we did in Case 2a, we find that \[ M \leq C_\mu^kC_\nu^{j+1}, \] where $C_\nu$ is the doubling constant for $\nu$ in $B_{6DN/d}$ and $M = M(\epsilon) \to \infty$ as $\epsilon \to 0$ is the analogous count for this case's $K_\epsilon$ (cf. \eqref{eq:M}). But, again, this is impossible.
\subsection*{Case 3}{\bf $\Sigma$ has an exposed point $\hat{x}$ in $\mathbb{R}^n \setminus \overline{X}$.} In this case, up to a translation, a dilation, a rotation, and subtracting $\ell$ from $u$, we can assume that \[ \hat{x} = 0,\quad \Sigma \subset \{ x_1 \leq 0 \},\quad \ell \equiv 0,\quad \text{and}\quad S_0 := \Sigma \cap \{ x_1 \geq - 1 \} \Subset \mathbb{R}^n \setminus \overline{X}. \] Like before, let $u_\epsilon(x) := u(x) - \epsilon(x_1 + 1)$ and define \[ S_\epsilon := \{ u_\epsilon \leq 0 \} \quad \text{and}\quad \nu_\epsilon := (\mathrm{Id} - \epsilon \mathbf{e}_1)_{\#} \nu. \] Again, $S_\epsilon \to S_0$ as $\epsilon \to 0$, so \[ \diam(S_\epsilon) \leq 2\diam(S_0) \quad\text{and}\quad S_\epsilon \Subset \mathbb{R}^n \setminus \overline{X} \] for all $\epsilon \ll 1$. For these small positive $\epsilon$, then, \[ 0 = \mu(S_\epsilon) = \nu_\epsilon(\partial u_\epsilon(S_\epsilon)), \] where the second equality follows from the mass balance formula. (Recall that $\mu$ vanishes on $\mathbb{R}^n \setminus \overline{X}$.) Moreover, as $Y$ is convex, \[ \partial u_\epsilon(S_\epsilon) \subset \spt (\nu_\epsilon) = \overline{Y} - \epsilon\mathbf{e}_1 \] (cf. \eqref{eqn: subdiff props}). Thus, any open subset of $\partial u_\epsilon(S_\epsilon)$ must be in the interior of the support of $\nu_\epsilon$. In turn, considering the cone generated by $\partial S_\epsilon$ over $(0,u_\epsilon(0))$, for $0 < \epsilon \ll 1$, we find that \[
0 = \nu_\epsilon(\partial u_\epsilon(S_\epsilon)) \geq \nu_\epsilon(B_{r_\epsilon})> 0 \quad\text{with}\quad r_\epsilon := \frac{|u_\epsilon(0)|}{2\diam(S_0)}. \] (Again, for more details on this inclusion, see, e.g., \cite[Theorem 2.8]{F}.) This is a contradiction and concludes the proof. \end{proof}
\section{Proof of Theorem~\ref{thm: C1}} \label{sec: C1} Again, we replace $u$ by its lower-semicontinuous extension outside of $X$, exactly as we did at the beginning of Section~\ref{sec: strict convexity}. We split the proof in three parts.
\subsection*{Part 1: $u$ is continuously differentiable inside $X$.} We follow the argument used to prove \cite[Corollary 1]{C0}. Assume for the sake of a contradiction that the result is false. Up to a translation, let $0 \in X$ be a point at which $u$ has two distinct supporting planes. After a rotation, dilation, and subtracting off an affine function from $u$, we may assume that \[ u(x) \geq \max \{ x_1,0 \},\quad u(0) = 0, \quad\text{and}\quad\frac{u(-s\mathbf{e}_1)}{s} \to 0 \quad\text{as}\quad s \to 0. \] Now consider the function $u_\sigma$ defined by \[ u_\sigma(x) := u(x) - \tau( x_1 + 2\sigma) \quad\text{with}\quad \tau := \frac{u(-\sigma\mathbf{e}_1 )}{\sigma}. \] Note that $\tau \to 0$ as $\sigma \to 0$. If \[ S_\sigma := \{ u_\sigma \leq 0 \}, \] then, by the strict convexity of $u$ provided by Theorem~\ref{thm: strict convexity}, we see that \[ S_\sigma \subset B_D \Subset X \cap \interior (\dom(u)) \] for some $D > 0$ and for all $\sigma \ll 1$; also, for these small positive $\sigma$, \[ \partial u_\sigma (S_\sigma) \subset B_R \cap Y \] for some $R > 0$. Moreover, if $\Pi_{-a} := \{ x_1 = -a \}$ and $\Pi_{b} := \{ x_1 = b \}$ denote the two parallel planes that tangentially sandwich $S_\sigma$, we see that \[ a > \sigma\quad \text{and}\quad b < \frac{2\tau \sigma}{1 - \tau}, \] provided $\sigma > 0$ is small enough to guarantee that $\tau < 1$. Furthermore, \[
\max_{S_\sigma} |u_\sigma| = |u_\sigma(0)| = 2\tau \sigma, \] and \[ \frac{\dist(\Pi_b,\Pi_0)}{\dist(\Pi_{-a},\Pi_b)} = \frac{b}{a+b} \leq \frac{b}{a} \leq \frac{2\tau}{1 - \tau} \to 0 \quad\text{as}\quad \sigma \to 0, \] where $\Pi_0 := \{ x_1 = 0\}$.
Now, set \[ \tilde{u}_\sigma(x) := \frac{u_\sigma(A_\sigma^{-1}x)}{2\tau\sigma} \quad \text{and}\quad \tilde{S}_\sigma : = A_\sigma(S_\sigma), \] where $A_\sigma$ is the John map that normalizes $S_\sigma$. Arguing as we did in the proof of Theorem~\ref{thm: strict convexity}, we find the same contradiction as we did in Case 2 when $\tau$ sufficiently small; the only difference is that we consider a slightly different chain of inequalities: \[ \begin{split} \tilde{\nu}_\sigma(K_\sigma) \leq \tilde{\nu}_\sigma\big(\partial \tilde{u}_\sigma(\tilde{S}_\sigma)\big) &= \tilde{\mu}_\sigma(\tilde{S}_\sigma) \\ &\leq C_\mu \tilde{\mu}_\sigma(\tfrac{1}{2}\tilde{S}_\sigma) = C_\mu\tilde{\nu}_\sigma\big(\partial \tilde{u}_\sigma(\tfrac{1}{2}\tilde{S}_\sigma)\big) \leq C_\mu\tilde{\nu}_\sigma(B_{\frac{1}{r_n}}) \end{split} \] (cf. \eqref{eqn: mb4}) where $\tilde{\mu}_\sigma$ and $\tilde{\nu}_\sigma$ are defined so that $(\nabla \tilde{u}_\sigma)_{\#} \tilde{\mu}_\sigma = \tilde{\nu}_\sigma$ and \[ K_\sigma := \conv\left(B_{r_n} \cup \left\{ \tfrac{r_n+n(1-\tau)}{2\tau}\mathbf{e}_1\right\}\right) \quad \text{with}\quad r_n := \frac{1}{2n^{3/2}}. \] This proves that $u$ is differentiable.
By \cite[Lemma A.24]{F}, for example, we know that differentiable convex functions are continuously differentiable. So we conclude that $u$ is continuously differentiable in $X$.
\subsection*{Part 2: $\nabla u(X)=Y$ when $X$ is convex.} Because $\nabla u$ is continuous in $X$, its image $Y':=\nabla u(X)$ is an open set of full $\nu$-measure contained inside $Y$. Also, as the assumptions on $\mu$ and $\nu$ are symmetric, the optimal transport map $\nabla v$ from $\nu$ to $\mu$ is continuous, and $X':=\nabla v(Y)$ is an open set of full $\mu$-measure contained inside $X$. Hence, recalling that $\nabla u$ and $\nabla v$ are inverses of each other (see, e.g., \cite[Corollary 2.5.13]{FG}), we conclude that $X'=X$ and $Y'=Y$, as desired.
\subsection*{Part 3: $\nabla u$ is locally H\"older continuous inside $X$.} Thanks to the strict convexity and $C^1$ regularity of $u$, we can localize the arguments of the proof of \cite[Theorem 1.1]{JS} to obtain the local H\"older continuity of $u$ inside $X$.
More precisely, if $u^\ast$ denotes the Legendre transform of $u$ (see \eqref{eq:u ast}), as in \cite{JS}, one can show that $u^\ast$ satisfies a weak form of Alexandrov's Maximum principle (see \cite[Lemma 3.2]{JS}), from which one deduces the engulfing property for the sections of $u^\ast$ (see \cite[Lemma 3.3]{JS}). Iteratively applying this engulfing property, one obtains a polynomial strict convexity bound for $u^\ast$. This bound implies the local H\"older continuity of $u$ inside $X$ (see \cite[Proof of Theorem 1.1]{JS}). We leave the details of this adaptation to the interested reader.
\noindent {\bf Conflict of Interest:} Authors state no conflict of interest.
\\ \noindent {\bf Funding Information:} AF acknowledges the support of the ERC grant No. 721675 ``Regularity and Stability in Partial Differential Equations (RSPDE)'' and of the Lagrange Mathematics and Computation Research Center. YJ was supported in part by NSF grant DMS-1954363.
\end{document}
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\begin{document}
\begin{Titul} {\large \bf DUNKL HARMONIC ANALYSIS AND\\ FUNDAMENTAL SETS OF CONTINUOUS FUNCTIONS\\[0.2em] ON THE UNIT SPHERE }\\[3ex] {{\bf Roman~A.~Veprintsev} \\[5ex]} \end{Titul}
\begin{Anot} {\bf Abstract.} We establish a necessary and sufficient condition on a continuous function on $[-1,1]$ under which the family of functions on the unit sphere $\mathbb{S}^{d-1}$ constructed in the described manner is fundamental in $C(\mathbb{S}^{d-1})$. In our construction of functions and proof of the result, we essentially use Dunkl harmonic analysis.
{\bf Key words and phrases:} fundamental set, continuous function, unit sphere, Dunkl intertwining operator, $\kappa$-spherical harmonics
{\bf MSC 2010:} 42B35, 42C05, 42C10 \end{Anot}
\section{Introduction and preliminaries}
We need some elements of the general Dunkl theory (see \cite{dunkl_xu_book_orthogonal_polynomials:2014,dunkl_article_integral_kernels:1991,dunkl_article_reflection_groups:1988,dai_xu_book_approximation:2013,dunkl_article_operators:1989}); for a background on reflection groups and root systems the reader is referred to \cite{Humphreys_book_reflection:1990,dunkl_xu_book_orthogonal_polynomials:2014}.
Let $\mathbb{R}^d$ denote $d$-dimensional Euclidean space. For $x\in\mathbb{R}^d$, we write $x=(x_1,\dots,x_d)$. The inner product of $x,y\in\mathbb{R}^d$ is denoted by $\langle x,y\rangle=\sum\limits_{i=1}^d x_iy_i$, and the norm of $x$ is denoted by $\|x\|=\sqrt{\langle x,x\rangle}$.
The unit sphere $\mathbb{S}^{d-1}$, $d\geq 2$, and the unit ball $\mathbb{B}^d$ of $\mathbb{R}^d$ are defined by \begin{equation*}
\mathbb{S}^{d-1}=\{x\colon\, \|x\|=1\}\quad\text{and}\quad \mathbb{B}^d=\{x\colon\, \|x\|\leq 1\}. \end{equation*}
For a nonzero vector $v\in\mathbb{R}^d$, define the reflection $\sigma_v$ by \begin{equation*}
\sigma_v(x)=x-2\frac{\langle x,v\rangle}{\|v\|^2}\,v,\quad x\in\mathbb{R}^d. \end{equation*} Each reflection $\sigma_v$ is contained in the orthogonal group $O(\mathbb{R}^d)$.
We give some basic definitions and notions which will be important.
\begin{definition} Let $R\subset\mathbb{R}^d\setminus\{0\}$ be a finite set. Then $R$ is called a root system if
$(1)$ $R\cap \mathbb{R}v=\{\pm v\}$ for all $v\in R$;
$(2)$ $\sigma_v(R)=R$ for all $v\in R$.
The subgroup $G=G(R)\subset O(\mathbb{R}^d)$ which is generated by the reflections $\{\sigma_v\colon\, v\in R\}$ is called the reflection group associated with $R$. \end{definition}
For any root system $R$ in $\mathbb{R}^d$, the reflection group $G=G(R)$ is finite. The set of reflections contained in $G(R)$ is exactly $\{\sigma_v\colon\, v\in R\}$.
Each root system can be written as a disjoint union $R=R_+\cup -R_+$, where $R_+$ and $-R_+$ are separated by a hyperplane through the origin. Such a set $R_+$ is called a positive subsystem. Its choice is not unique.
\begin{definition} A nonnegative function $\kappa$ on a root system $R$ is called a multiplicity function on $R$ if it is $G$-invariant, i.e. $\kappa(v)=\kappa(g(v))$ for all $v\in R$, $g\in G$. \end{definition}
\begin{definition} The Dunkl operators are defined by \begin{equation*} \mathcal{D}_if(x)=\frac{\partial f(x)}{\partial x_i}+\sum\limits_{v\in R_+} \kappa(v)\frac{f(x)-f(\sigma_v(x))}{\langle x,v\rangle} \langle v,e_i\rangle,\quad 1\leq i\leq d, \end{equation*} where $e_1,\dots,e_d$ are the standard unit vectors of $\mathbb{R}^d$. \end{definition}
The above definition does not depend on the special choice of $R_+$, thanks to the $G$-invariance of $\kappa$. In case $\kappa=0$, the Dunkl operators reduce to the corresponding partial derivatives.
Suppose $\Pi^d$ is the space of all polynomials in $d$ variables with complex coefficients, $\mathcal{P}_n^d$ $\bigl(n\in \mathbb{N}_0=\{0,1,2,\dots\}\bigr)$ is the subspace of homogeneous polynomials of degree $n$ in $d$ variables.
According to \cite{dunkl_article_integral_kernels:1991}, there exists a unique linear isomorphism $V_\kappa$ of $\Pi^d$ such that \begin{equation*} V_\kappa(\mathcal{P}_n^d)=\mathcal{P}_n^d,\,\,\, n\in\mathbb{N}_0,\quad V_\kappa1=1,\quad\text{and}\quad\mathcal{D}_iV_\kappa=V_\kappa\frac{\partial}{\partial x_i},\quad 1\leq i\leq d. \end{equation*} This operator is called the Dunkl intertwining operator. If $\kappa=0$, $V_\kappa$ becomes the identity operator.
M.~R\"{o}sler has proved in \cite{rosler_article_positivity:1999} that for each $x\in\mathbb{R}^d$ there exists a unique probability measure $\mu_x^\kappa$ on the Borel $\sigma$-algebra of $\mathbb{R}^d$ with support in $\{y\colon\,\|y\|\leq\|x\|\}$, such that for all polynomials $p$ on $\mathbb{R}^d$ we have \begin{equation}\label{rosler_measure} V_\kappa p(x)=\int\nolimits_{\mathbb{R}^d} p(y)\,d\mu_x^\kappa(y). \end{equation}
For $X=[-1,1]$ or $X=\mathbb{S}^{d-1}$, we denote by $C(X)$ the space of continuous complex-valued functions on $X$.
\begin{definition} We define the Dunkl truncated intertwining operator at an arbitrary point $\xi\in\mathbb{S}^{d-1}$ \begin{equation*} V_\kappa(\xi)\colon C([-1,1])\to C(\mathbb{S}^{d-1}) \end{equation*} by \begin{equation*} V_\kappa(\xi;g,x)=\int\nolimits_{\mathbb{B}^d} g(\langle \xi,\zeta\rangle)\, d\mu_x^\kappa(\zeta),\quad x\in\mathbb{S}^{d-1},\quad g\in C[-1,1], \end{equation*} where $\mu_x^\kappa$ is the measure given in \eqref{rosler_measure}. \end{definition}
The operator $V_\kappa(\xi)$ is well defined. Indeed, there exists a sequence $\{p_n\}$ of polynomials in $\Pi^1$ such that \begin{equation*}
\sup\limits_{t\in[-1,1]} \, |g(t)-p_n(t)|\to 0,\quad n\to\infty. \end{equation*} Note that $p_n(\langle \xi,\cdot\rangle)\in\Pi^d$. Thus, for all $x\in\mathbb{B}^d$, \begin{equation*} \begin{split}
\bigl|V_\kappa&(\xi;g,x)-V_\kappa\bigl[p_n(\langle \xi,\cdot\rangle)\bigr](x)\bigr|\leq \int\nolimits_{\mathbb{B}^d} \bigl|g(\langle\xi,\zeta\rangle)-p_n(\langle\xi,\zeta\rangle)\bigr| \, d\mu_x^\kappa(\zeta)\\
&\leq \sup\limits_{\zeta\in\mathbb{B}^d} \, \bigl|g(\langle\xi,\zeta\rangle)-p_n(\langle\xi,\zeta\rangle)\bigr|=\sup\limits_{t\in[-1,1]} \, |g(t)-p_n(t)|\to 0,\quad n\to\infty. \end{split} \end{equation*} As $V_\kappa\bigl[p_n(\langle\xi,\cdot\rangle)\bigr]$ is continuous on $\mathbb{R}^d$, then we deduce that $V_\kappa(\xi;g)$ is continuous on the unit ball $\mathbb{B}^d$.
In the present paper, we establish a necessary and sufficient condition on a function $g\in C([-1,1])$ under which the family of functions $\{V_\kappa(x;g)\colon x\in\mathbb{S}^{d-1}\}$ is fundamental in $C(\mathbb{S}^{d-1})$. This result generalizes Theorem 2 in \cite{sun_cheney_article_fundamental_sets:1997}. We state and prove the main result in section~\ref{section_with_main_result}.
Recall that a set $\mathcal{F}$ in a Banach space $\mathcal{E}$ is said to be fundamental if the linear span of $\mathcal{F}$ is dense in $\mathcal{E}$.
\section{Fundamentality in $C(\mathbb{S}^{d-1})$}
Suppose that the unit sphere $\mathbb{S}^{d-1}$ is equipped with a positive Borel measure and notions of orthogonality for functions on $\mathbb{S}^{d-1}$ are defined in terms of this Borel measure.
Let there be given an orthogonal sequence $\{U_n\}_{n=1}^\infty$ of finite-dimensional subspaces in $C(\mathbb{S}^{d-1})$. It is assumed that $\bigcup\limits_{n=1}^\infty U_n$ is fundamental in the space $C(\mathbb{S}^{d-1})$. For each $n$, let $\{u_{nj}\colon 1\leq j\leq\dim U_n\}$ denote a real-valued orthonormal basis of $U_n$.
Assume that we possess a summability method, given by an infinite matrix $A\!=\!\!\bigl[A_{nm}\bigr]_{n,m=1}^\infty$ with complex entries that has these properties:
\begin{itemize} \item[(i)] each row of $A$ has only finitely many nonzero elements;
\item[(ii)] $\lim\limits_{n\to\infty} \, A_{nm}$ exists for each $m=1,2,\ldots$;
\item[(iii)] the sequence of functions $f_n(x,y)=\sum\nolimits_{m} \, A_{nm}\,\sum\nolimits_{j} \, u_{mj}(x)u_{mj}(y)$, where $x,y\in\mathbb{S}^{d-1},$ converges (as $n\to\infty$) uniformly in $x$ and $y$ to a limit function $f(x,y)$. \end{itemize}
\begin{teoen}\label{main_general_theorem} Suppose that the hypotheses given above are satisfied, and let $f$ be as in {\upshape (iii)}. In order that the set of functions $\{x\mapsto f(x,y)\colon\, y\in\mathbb{S}^{d-1}\}$ be fundamental in $C(\mathbb{S}^{d-1})$ it is necessary and sufficient that $\lim\limits_{n\to\infty} \, A_{nm}$ be nonzero for all $m$. \end{teoen}
This theorem was proved in \cite[Section~2, Theorem~1]{sun_cheney_article_fundamental_sets:1997} in a more general setting.
\section{Some facts of Dunkl harmonic analysis on the unit sphere}\label{section_with_Dunkl_harmonic_analysis}
The Dunkl Laplacian is defined by \begin{equation*} \Delta_\kappa=\mathcal{D}_1^2+\dots+\mathcal{D}_d^2. \end{equation*} The Dunkl Laplacian plays the role of the ordinary Laplacian. In the special case $\kappa=0$, $\Delta_\kappa$ reduces to the ordinary Laplacian.
A $\kappa$-harmonic polynomial $P$ of degree $n\in\mathbb{N}_0$ is a homogeneous polynomial $P\in \mathcal{P}_n^d$ such that $\Delta_\kappa P=0$. The $\kappa$-spherical harmonics of degree $n$ are the restriction of $\kappa$-harmonics of degree $n$ to the unit sphere $\mathbb{S}^{d-1}$. Let $\mathcal{A}_n^d(\kappa)$ be the space of $\kappa$-spherical harmonics of degree $n$ and let $N(n,d)$ be the dimension of $\mathcal{A}_n^d(\kappa)$.
The weighted inner product of $f,h\in C(\mathbb{S}^{d-1})$ is denoted by \begin{equation*} \langle f,h\rangle_\kappa=\frac{1}{\sigma_d^\kappa}\int\nolimits_{\mathbb{S}^{d-1}} f(x)\overline{h(x)}\, w_\kappa(x)d\omega(x), \end{equation*} where $d\omega$ is the Lebesgue measure on $\mathbb{S}^{d-1}$, $w_\kappa$ is the weight function, invariant under the reflection group $G$, defined by \begin{equation*}
w_\kappa(x)=\prod\limits_{v\in R_+} |\langle v,x\rangle|^{2\kappa(v)},\quad x\in\mathbb{S}^{d-1}, \end{equation*} and $\sigma_d^\kappa$ is the constant chosen such that $\langle 1,1\rangle_{\kappa}=1$.
Note that if $\kappa=0$, then $w_\kappa=1$.
In the rest of this section, we assume that $\kappa\not=0$ if $d=2$.
The following properties hold:
\begin{itemize} \item[(I)] $\bigcup\limits_{n=0}^\infty \mathcal{A}_n^d(\kappa)$ is fundamental in $C(\mathbb{S}^{d-1})$.
\item[(II)] For any real-valued orthonormal basis $\{S_{n,j}^{\kappa}\colon\, 1\leq j\leq N(n,d)\}$ of $\mathcal{A}_n^d(\kappa)$,
\begin{equation*}
V_\kappa(x;C_n^{\lambda_\kappa},y)=\frac{\lambda_\kappa}{n+\lambda_\kappa}\, \sum\limits_{j=1}^{N(n,d)} S_{n,j}^{\kappa}(x)S_{n,j}^{\kappa}(y),\quad x,y\in\mathbb{S}^{d-1},
\end{equation*}
where $C_n^{\lambda_\kappa}(\cdot)$ denotes the Gegenbauer polynomial, $\lambda_\kappa$ is a positive constant defined by
\begin{equation}\label{lambda_kappa}
\lambda_\kappa=\sum\limits_{v\in R_+} \kappa(v)+\frac{d-2}{2}
\end{equation} (see, for example, \cite[Section~7.2]{dai_xu_book_approximation:2013}). \item[(III)] If $n\not=m$, then $\mathcal{A}_{n}^{d}(\kappa)\perp\mathcal{A}_m^d(\kappa)$, i.e. $\langle P,Q\rangle_\kappa=0$ for $P\in\mathcal{A}_{n}^{d}(\kappa)$ and $Q\in \mathcal{A}_m^d(\kappa)$ \cite[Theorem~1.6]{dunkl_article_reflection_groups:1988}. \end{itemize}
Property (I) follows from the Weierstrass approximation theorem: if $f$ is continuous on $\mathbb{S}^{d-1}$, then it can be uniformly approximated by polynomials restricted to $\mathbb{S}^{d-1}$. According to \cite[Theorem~1.7]{dunkl_article_reflection_groups:1988}, these restrictions belong to the linear span of $\bigcup\limits_{n=0}^\infty \mathcal{A}_n^d(\kappa)$.
\section{Main result and its proof}\label{section_with_main_result}
We can now establish the main result of the paper.
\begin{teoen} Fix $d\geq 2$. Suppose $R$ is a fixed root system in $\mathbb{R}^d$, $\kappa$ is a multiplicity function on $R$. Assume that $\kappa\not=0$ if $d=2$. Let $g\in C([-1,1])$. In order that the family of functions $\{V_\kappa(x;g)\colon\, x\in\mathbb{S}^{d-1}\}$ be fundamental in $C(\mathbb{S}^{d-1})$ it is necessary and sufficient that \begin{equation*} \int\nolimits_{-1}^{1} g(t) C_n^{\lambda_\kappa}(t) \, (1-t^2)^{\lambda_\kappa-1/2}dt\not=0,\quad n=0,1,2,\ldots, \end{equation*} where the constant $\lambda_\kappa$ is defined in~\eqref{lambda_kappa}. \end{teoen}
\proofen For $\lambda>0$, let \begin{equation*}c_{\lambda}=\Bigr(\int\nolimits_{-1}^1 (1-t^2)^{\lambda-1/2}\,dt\Bigr)^{-1}\end{equation*} be the normalizing constant. Then the Gegenbauer expansion of $g$ takes the form \begin{equation}\label{Gegenbauer_expansion} g(t)\sim\sum\limits_{n=0}^\infty b_n\frac{n+\lambda}{\lambda} C_{n}^{\lambda}(t)\quad\text{with}\quad b_n=\frac{c_{\lambda}}{C_n^{\lambda}(1)} \int\nolimits_{-1}^1 g(t) C_n^{\lambda}(t) \, (1-t^2)^{\lambda-1/2} dt, \end{equation} since $c_\lambda\int\nolimits_{-1}^1 \bigl(C_n^\lambda(t)\bigr)^2 \,(1-t^2)^{\lambda-1/2}\,dt=C_n^\lambda(1)\lambda/(n+\lambda)$.
For $\delta>0$, the Ces\`{a}ro $(C,\delta)$ means of the above series are \begin{equation}\label{Cesaro_means} S_n^\delta g(t)=\frac{1}{A_n^\delta}\sum\limits_{m=0}^n A_{n-m}^\delta b_m\frac{m+\lambda}{\lambda}C_m^\lambda(t),\quad A_n^\delta=\binom{n+\delta}n. \end{equation} Note that $A_{n-m}^\delta/A_n^\delta\to1$ as $n\to\infty$ for each $m$.
If $\delta>\lambda$, then it follows from \cite[Theorem~1.3]{chanillo_muckenhoupt_estimates:1993} that the sequence $\{S_n^\delta g\}$ converges uniformly on $[-1,1]$ to the function $g$.
Let $\lambda=\lambda_\kappa$ and $\delta>\lambda$. Then the sequence $\{V_\kappa(x;S_n^\delta g,y)\}$ converges uniformly in $x,y\hm\in\mathbb{S}^{d-1}$ to $V_\kappa(x;g,y)$. Indeed, \begin{equation*}\begin{split}
\sup\limits_{x,y\in\mathbb{S}^{d-1}} \, &\bigl|V_\kappa(x;g,y)-V_\kappa(x;S_n^\delta g,y)\Bigr|\\&=\sup\limits_{x,y\in\mathbb{S}^{d-1}} \, \Bigl|\int\nolimits_{\mathbb{B}^d} g(\langle x,\xi\rangle)\, d\mu_y^\kappa(\xi)-\int\nolimits_{\mathbb{B}^d} S_n^\delta g(\langle x,\xi\rangle) \, d\mu_y^\kappa(\xi)\Bigr|\\
&\leq\sup\limits_{x,\xi\in\mathbb{S}^{d-1}} \, \bigl|g(\langle x,\xi\rangle)-S_n^\delta g(\langle x,\xi\rangle)\bigr|\\
&=\sup\limits_{t\in[-1,1]} \, \bigl|g(t)-S_n^\delta g(t)\bigr|\to0,\quad n\to\infty. \end{split} \end{equation*} Using Property (II) of the $\kappa$-spherical harmonics (see Section~\ref{section_with_Dunkl_harmonic_analysis}) and \eqref{Cesaro_means}, we get \begin{equation*} V_\kappa(x;g,y)=\lim\limits_{n\to\infty} \, \sum\limits_{m=0}^n \frac{A_{n-m}^\delta}{A_n^\delta} \, b_m \, \sum\limits_{j=1}^{N(m,d)} S_{m,j}^\kappa(x)S_{m,j}^\kappa(y). \end{equation*}
We can apply Theorem~\ref{main_general_theorem} with $U_n=\mathcal{A}_n^d(\kappa)$, $u_{nj}=S_{n,j}^\kappa$, and $A_{nm}=(A_{n-m}^\delta b_m)/A_n^\delta$ for $m\leq n$, $A_{nm}=0$ for $m>n$, to conclude that the condition \begin{equation*} \lim\limits_{n\to\infty} \, \Bigl(\frac{A_{n-m}^\delta}{A_n^\delta} \, b_m\Bigr)\not=0\quad \text{for each $m$} \end{equation*} is the necessary and sufficient condition for fundamentality. Obviously, this condition reduces to $b_m\not=0$ for all $m$. By \eqref{Gegenbauer_expansion}, the latter is equivalent to the integral condition described in the theorem.
$\square$
The approach used in the proof of this theorem is as in \cite[Theorem~2]{sun_cheney_article_fundamental_sets:1997}.
\begin{Biblioen}
\bibitem{chanillo_muckenhoupt_estimates:1993}S. Chanillo and B. Muckenhoupt, Weak type estimates for Ces\`{a}ro sums of Jacobi polynomial series, \emph{Mem. Am. Math. Soc.} \textbf{102}:487 (1993), 1--90.
\bibitem{dai_xu_book_approximation:2013}F. Dai and Y. Xu, \emph{Approximation theory and harmonic analysis on spheres and balls}, Springer, Berlin--New York, 2013.
\bibitem{dunkl_article_reflection_groups:1988}C. F. Dunkl, Reflection groups and orthogonal polynomials on the sphere, \emph{Math. Z.} \textbf{197} (1988), 33--60.
\bibitem{dunkl_article_operators:1989}C. F. Dunkl, Differential-difference operators associated to reflection groups, \emph{Trans. Amer. Math. Soc.} \textbf{311}:1 1989, 167--183.
\bibitem{dunkl_article_integral_kernels:1991}C. F. Dunkl, Integral kernels with reflection group invariance, \emph{Can. J. Math.} \textbf{43}:6 (1991), 1213--1227.
\bibitem{dunkl_xu_book_orthogonal_polynomials:2014}C. F. Dunkl and Y. Xu, \emph{Orthogonal polynomials of several variables}, 2nd ed., Cambridge Univ. Press, 2014.
\bibitem{Humphreys_book_reflection:1990}J. E. Humphreys, \emph{Reflection groups and Coxeter groups}, Cambridge Univ. Press, 1990.
\bibitem{rosler_article_positivity:1999}M. R\"{o}sler, Positivity of Dunkl's intertwining operator, \emph{Duke Math. J.} \textbf{98}:3 (1999), 445--463.
\bibitem{sun_cheney_article_fundamental_sets:1997}Xingping Sun and E. W. Cheney, Fundamental sets of continuous functions on spheres, \emph{Constr. Approx.} \textbf{13}:2 (1997), 245--250.
\end{Biblioen}
\noindent \textsc{Department of Applied Mathematics and Computer Science, Tula State University, Tula, Russia }
\noindent \textit{E-mail address}: \textbf{[email protected]}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Non-Markovianity over ensemble averages in quantum complex networks}
\maketitle
\begin{abstract} We consider bosonic quantum complex networks as structured finite environments for a quantum harmonic oscillator and investigate the interplay between the network structure and its spectral density, excitation transport properties and non-Markovianity. After a review of the formalism used, we demonstrate how even small changes to the network structure can have a large impact on the transport of excitations. We then consider the non-Markovianity over ensemble averages of several different types of random networks of identical oscillators and uniform coupling strength. Our results show that increasing the number of interactions in the network tends to suppress the average non-Markovianity. This suggests that tree networks are the random networks optimizing this quantity. \end{abstract}
\section{Introduction} \setcounter{equation}{0}
Understanding the dynamics of open quantum systems is important in several fields of physics and chemistry including problematics dealing, e.g., with quantum to classical transition and decoherence with its harmful effects for quantum information processing and communication. In general, formulating or deriving a suitable equation of motion for the density matrix plöt for the open system is often a daunting task. Perhaps the most celebrated and most used theoretical result in this context is the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation \cite{GKS76,L76}
\begin{equation} \dfrac{d \rho_s(t)}{dt}=-i[H_s,\rho_s(t)]+\sum_k \gamma_k \left( C_k\rho_s(t) C_k^\dagger-\dfrac{1}{2} \left \{ C_k^\dagger C_k,\rho_s(t) \right \} \right), \end{equation}
\noindent with the associated completely positive and trace preserving dynamical map with semigroup property. Above, $H_s$ is the open system Hamiltonian, $\gamma_k$ are positive constant rates, and $C_k$ are the jump operators with $k$ indexing the different decoherence channels. Indeed, this master equation and the corresponding publications had recently 40th anniversary celebrations in the Symposium on Mathematical Physics in Toru{\'n} in June 2016.
GKSL master equation $(1.1)$ describes Markovian memoryless open system dynamics and during the last 10-15 years there has been an increasing amount of research activities in understanding memory effects and quantifying non-Markovianity for open systems beyond the semigroup property \cite{Rivas14,Vega17,Breuer16}. A pair of complementary approaches here include a description based on quantifying the information flow between the open system and its environment \cite{Breuer09} or the characterization of dynamical maps in terms of their divisibility properties \cite{Rivas14,Ch14} while a large number of other ways to characterize non-Markovianity also exist, see e.g. \cite{Wolf08,Lu10,Luo12,Lorenzo13,Bylicka14}. Most of the research so far has focused on non-Markovianity using discrete variable open systems as examples while in the current work we are interested in the memory effects in a continuous variable (CV) open system with controlled environmental structure.
Indeed, here we consider structured finite environments modeled by bosonic quantum complex networks. While this and other kinds of quantum complex networks have recieved increasing attention in recent years in the context of perfect state transfer \cite{Christandl04,Yung05}, quantum random walks \cite{Kempe10,Faccin13}, efficient entanglement distribution \cite{Cirac97,Acin07,Cuquet09} and the unification of classical and quantum network theory \cite{Bianconi15,Biamonte17}, here the focus is on the interplay between the network structure and the reduced dynamics of an open quantum system attached to it. To this end, we investigate the impact of the structure on the network spectral density, excitation transport properties and non-Markovianity of the reduced dynamics.
The paper is organized as follows. Section 2 concerns the network itself. Here we present the microscopic model and briefly discuss the connection between the network Hamiltonian and certain matrix representations of abstract graphs in classical graph theory. The dynamics of the network is given in terms of a symplectic matrix acting on the vector of operators at initial time. In Section 3, we describe how complex quantum networks can be treated in the framework of the theory of open quantum systems as tunable structured environments. We demonstrate how small changes in the network structure can have a large impact on its excitation transport properties. In Section 4, we consider the non-Markovianity of the reduced dynamics using a recently introduced witness based on non-monotonicity of the evolution of Gaussian interferometric power. Finally, conclusions are drawn in Section 5.
\section{Bosonic quantum complex networks} \setcounter{equation}{0}
\subsection{The Hamiltonian}
We set $\hbar=1$ and work with position and momentum operators defined as $q=(a^\dagger+a)/\sqrt{2\omega}$ and $p=(a^\dagger-a)i\sqrt{\omega/2}$, satisfying the commutation relation $[q,p]=i$. We consider networks of $N$ unit mass quantum harmonic oscillators coupled by springlike couplings. The general form of a Hamiltonian for such networks is
\begin{equation} H_E=\dfrac{\mathbf{p}^T\mathbf{p}}{2}+\mathbf{q}^T\mathbf{Aq}, \end{equation}
\noindent where we have introduced the vectors of position and momentum operators $\mathbf{q}^T=\{q_1,...,q_N\}$ and $\mathbf{p}^T=\{p_1,...,p_N\}$, and where $\mathbf{A}$ is the matrix containing the coupling terms and frequencies. It has elements $\mathbf{A}_{ij}=\delta_{ij}\tilde{\omega}_i^2/2-(1-\delta_{ij})g_{ij}/2$, where $g_{ij}$ is the strength of the springlike coupling $g_{ij}(q_i-q_j)^2/2$ between the position operators of oscillators $i$ and $j$, and $\tilde{\omega}_i^2=\omega_i^2+\sum_j g_{ij}$ is the effective frequency of oscillator $i$ resulting from absorbing the quadratic parts of the coupling terms into the free Hamiltonians of the oscillators.
The matrix $\mathbf{A}$, which completely determines the network Hamiltonian, can be related to some of the typical matrix representations of weighted graphs, i.e. abstract networks of nodes connected by weighted edges. By weighted, we mean that a magnitude is assigned to each connection. This can be used to establish a link between the properties of the network and results from graph theory. A paradigmatic example is the adjacency matrix $\mathbf{V}$ having elements $\mathbf{V}_{ij}=w_{ij}$, where $w_{ij}$ is the weigth of the connection between nodes $i$ and $j$; a weigth of 0 corresponds to the nodes being disconnected. Another matrix that arises very naturally is the Laplace matrix $\mathbf{L}$, related to the adjacency matrix as $\mathbf{L}=\mathbf{D}-\mathbf{V}$, where $\mathbf{D}$ is diagonal with elements $\mathbf{D}_{ii}=\sum_j w_{ij}$. In terms of them, matrix $\mathbf{A}$ can be written as $\mathbf{A}=\mathbf{\Delta}_{\tilde{\omega}}^2/2-\mathbf{V}/2$ or as $\mathbf{A}=\mathbf{\Delta}_{\omega}^2/2+\mathbf{L}/2$, where $\mathbf{\Delta}_{\tilde{\omega}}$ and $\mathbf{\Delta}_{\omega}$ are diagonal matrices of the effective and bare frequencies of the network oscillators, respectively, and weights are given by the coupling strengths. The graph aspect of this and other kinds of quantum networks have been very recently used to, e.g., develop a local probe for the connectivity and coupling strength of a quantum complex network by using results of spectral graph theory \cite{Nokkala17a}, and constructing Bell-type inequalities for quantum communication networks by mapping the task to a matching problem of an equivalent unweighted bipartite graph \cite{Luo17}.
The Hamiltonian $(2.1)$ is a special case of the quadratic Hamiltonian $H=\mathbf{x}^T\mathbf{Mx}$, where the vector $\mathbf{x}$ contains both the position and momentum operators and $\mathbf{M}$ is a $2N \times 2N$ matrix such that $H$ is Hermitian. It can be shown \cite{Ticochinsky79} that quadratic Hamiltonians can be diagonalized to arrive at an equivalent eigenmode picture of uncoupled oscillators provided that $\mathbf{M}$ is positive definite. Since $H$ is Hermitian, this is equivalent with the positivity of the eigenvalues of $\mathbf{M}$. In the case at hand, $H_E$ may be diagonalized with an orthogonal matrix $\mathbf{K}$ such that $\mathbf{K}^T\mathbf{AK}=\mathbf{\Delta}$, where the diagonal matrix $\mathbf{\Delta}$ holds the eigenvalues of $\mathbf{A}$. By defining new operators
\begin{equation} \begin{cases} \mathbf{Q}=\mathbf{K}^T\mathbf{q} \\ \mathbf{P}=\mathbf{K}^T\mathbf{p}, \\ \end{cases} \end{equation}
\noindent the diagonal form of $H_E$ reads
\begin{equation} H_E=\dfrac{\mathbf{P}^T\mathbf{P}}{2}+\mathbf{Q}^T\mathbf{\Delta}\mathbf{Q}, \end{equation}
\noindent which is the Hamiltonian of $N$ decoupled oscillators with frequencies $\Omega_i=\sqrt{2\mathbf{\Delta}_{ii}}$.
\subsection{The dynamics of the network}
A bosonic quantum complex network is also an interesting system to study in its own right. Below, we review the mathematical tools useful for the task, adopting the definitions for a commutator and anti-commutator between two operator-valued vectors used in \cite{Menicucci11}. While we will be later concerned with networks initially in the thermal state, we will also briefly discuss the case of an initial Gaussian state without displacement. For a more detailed review of Gaussian formalism in phase space, see \cite{Ferraro05}. What is presented here is straightforward to apply to the case where interactions with external oscillators is considered, and we will do so in Section $3$.
Let $\mathbf{x}$ be a vector containing the position and momentum operators of the network oscillators, and define the commutator between two operator valued vectors as $[\mathbf{x}_1,\mathbf{x}_2^T]=\mathbf{x}_1\mathbf{x}_2^T-(\mathbf{x}_2\mathbf{x}_1^T)^T$. Now canonical commutation relations give rise to a symplectic form $\mathbf{J}$, determined by $[\mathbf{x},\mathbf{x}^T]=i\mathbf{J}$. Let $\mathbf{x}'=\mathbf{S}\mathbf{x}$, where $\mathbf{S}$ is a $2N \times 2N$ matrix of real numbers. In order for $\mathbf{S}$ to be a canonical transformation of $\mathbf{x}$, the commutation relations must be preserved. This requirement gives $i\mathbf{J}=[\mathbf{x}',\mathbf{x}'^T]=[\mathbf{S}\mathbf{x},(\mathbf{S}\mathbf{x})^T]=\mathbf{S}[\mathbf{x},\mathbf{x}^T]\mathbf{S}^T=i\mathbf{S}\mathbf{J}\mathbf{S}^T$, implying that $\mathbf{SJS}^T=\mathbf{J}$. Such a matrix is called symplectic with respect to symplectic form $\mathbf{J}$. Symplectic matrices form the symplectic group $Sp(2N,\mathbb{R})$ with respect to matrix multiplication, which can be used to define a symplectic representation of the Gaussian unitary group, meaning that (up to an overall phase factor) the two groups are bijective.
We fix $\mathbf{x}^T=\{\mathbf{q}^T,\mathbf{p}^T\}=\{q_1,...,q_N,p_1,...,p_N\}$ throughout the rest of the present work. Then the symplectic form becomes $\mathbf{J}=\left(\begin{smallmatrix} 0 & \mathbf{I}_N \\ -\mathbf{I}_N & 0 \end{smallmatrix}\right)$, where $\mathbf{I}_N$ is the $N\times N$ identity matrix. By defining the vector of eigenmode operators to be $\mathbf{X}^T=\{\mathbf{Q}^T,\mathbf{P}^T\}=\{Q_1,...,Q_N,P_1,...,P_N\}$, we can express the transformation that diagonalizes the network Hamiltonian as $\mathbf{X}=\left(\begin{smallmatrix} \mathbf{K}^T & 0 \\ 0 & \mathbf{K}^T \end{smallmatrix}\right)\mathbf{x}$; a direct calculation shows that the matrix diagonalizing the Hamiltonian is both symplectic and orthogonal.
In the eigenmode picture, the equations of motion are those of noninteracting oscillators. By defining the auxiliary diagonal matrices with elements $\mathbf{D}_{\cos ii}^\Omega=\cos(\Omega_i t)$, $\mathbf{D}_{\sin ii}^\Omega=\sin(\Omega_i t)$ and $\mathbf{\Delta}_{\Omega ii}=\Omega_i$, we can express them as
\begin{equation} \begin{pmatrix} \mathbf{Q}(t) \\ \mathbf{P}(t) \end{pmatrix} = \begin{pmatrix}
\mathbf{D}_{\cos}^\Omega & \mathbf{\Delta}_{\Omega}^{-1}\mathbf{D}_{\sin}^\Omega \\
-\mathbf{\Delta}_{\Omega}\mathbf{D}_{\sin}^\Omega & \mathbf{D}_{\cos}^\Omega \end{pmatrix} \begin{pmatrix} \mathbf{Q}(0) \\ \mathbf{P}(0) \end{pmatrix}, \end{equation}
\noindent where the block matrix acting on the vectors is again symplectic. To recover the dynamics of the network oscillators, we may use Eq. $(2.2)$ to express $\mathbf{x}(t)$ in terms of either $\mathbf{X}(0)$ as
\begin{equation} \begin{pmatrix} \mathbf{q}(t) \\ \mathbf{p}(t) \end{pmatrix} = \begin{pmatrix}
\mathbf{K}\mathbf{D}_{\cos}^\Omega & \mathbf{K}\mathbf{\Delta}_{\Omega}^{-1}\mathbf{D}_{\sin}^\Omega \\
-\mathbf{K}\mathbf{\Delta}_{\Omega}\mathbf{D}_{\sin}^\Omega & \mathbf{K}\mathbf{D}_{\cos}^\Omega \end{pmatrix} \begin{pmatrix} \mathbf{Q}(0) \\ \mathbf{P}(0) \end{pmatrix}, \end{equation}
\noindent or in terms of $\mathbf{x}(0)$ as
\begin{equation} \begin{pmatrix} \mathbf{q}(t) \\ \mathbf{p}(t) \end{pmatrix} = \begin{pmatrix}
\mathbf{K}\mathbf{D}_{\cos}^\Omega\mathbf{K}^T & \mathbf{K}\mathbf{\Delta}_{\Omega}^{-1}\mathbf{D}_{\sin}^\Omega\mathbf{K}^T \\
-\mathbf{K}\mathbf{\Delta}_{\Omega}\mathbf{D}_{\sin}^\Omega\mathbf{K}^T & \mathbf{K}\mathbf{D}_{\cos}^\Omega\mathbf{K}^T \end{pmatrix} \begin{pmatrix} \mathbf{q}(0) \\ \mathbf{p}(0) \end{pmatrix}. \end{equation}
\noindent Notice that the group properties of symplectic matrices quarantees that in both cases the block matrix remains symplectic.
If we now restrict our attention to Gaussian states with zero mean, we may define the covariance matrix of the initial state as
\begin{equation} \mathrm{cov}(\mathbf{x}(0))=\tfrac{1}{2}\langle[\mathbf{x}(0),\mathbf{x}^T(0) ]_+\rangle, \end{equation}
\noindent where the anti-commutator is defined as $[\mathbf{x}_1,\mathbf{x}_2 ]_+=\mathbf{x}_1\mathbf{x}_2^T+(\mathbf{x}_2\mathbf{x}_1^T)^T$. If $\mathbf{x}(t)=\mathbf{S}\mathbf{x}(0)$, then the covariance matrix at time $t$ becomes
\begin{equation} \begin{aligned} \mathrm{cov}(\mathbf{x}(t)) & =\mathrm{cov}(\mathbf{S}\mathbf{x}(0))=\tfrac{1}{2}\langle[(\mathbf{S}\mathbf{x}(0),(\mathbf{S}\mathbf{x}(0))^T ]_+\rangle \\ & =\tfrac{1}{2}\mathbf{S}\langle[\mathbf{x}(0),\mathbf{x}(0))^T ]_+\rangle\mathbf{S}^T=\mathbf{S}\mathrm{cov}(\mathbf{x}(0))\mathbf{S}^T. \end{aligned} \end{equation}
In the present case of symplectic matrices appearing in Eqs. $(2.5)$ and $(2.6)$, the choice depends on the basis where the initial covariance matrix is defined. A particular subtlety concerns an initial thermal state for the network, where either choice might seem natural. Here, assuming the usual thermal expectation values for non-interacting oscillators in the real oscillator basis, i.e. a diagonal $\mathrm{cov}(\mathbf{x}(0))$, corresponds to the case where the interactions are suddenly switched on at $t=0+$. As here the state is not the stationary state with respect to the Hamiltonian $(2.1)$, one will see the excitations of each network oscillator evolve with time. On the other hand, if one assumes the covariance matrix to be diagonal in the eigenmode basis instead, the excitations will be frozen. In this work we are using the latter approach as it is quite natural to assume an initial stationary state for the environment of an open quantum system.
While here the correlation structure in the state of the network is not studied, it is of great interest in the emerging field of continuous-varibale quantum information processing and in particular in the study of so-called cluster states \cite{Menicucci06,Zhang06}, which are multi-mode correlated states used as a resource in measurement-based quantum computing. In this context, it is typically the state, rather than the Hamiltonian, that is represented with a graph. It has been shown that specific quadratic Hamiltonians have cluster states as their ground state, which can then be adiabatically prepared by cooling a set of non-interacting modes to zero temperature and then switching on the interactions \cite{Aolita11}.
Finally, we mention the complementary viewpoint of open quantum networks, where the network is considered as the open system interacting with an environment of infinite size. The dynamics can then be described with a master equation for the network density matrix. Collective phenomena, such as synchronization, can occur in a network relaxing towards a steady state \cite{Manzano13}.
\subsection{Experimental aspect}
To implement an oscillator network, the basic requirements to meet are a static topology, harmonic potential and quantum regime for the oscillators. To match the form of the Hamiltonian $(2.1)$, the couplings between the oscillator position operators should be springlike, and any other interactions between them should either be eliminated or minimized.
More challenging requirements include the scalability to many nodes and the ability to implement also long-range couplings in order to have a nontrivial topology. The biggest difficulties are related to the implementation of generic networks: essentially a platform reconfigurable to a desired static topology would be needed, i.e. independent control and tunability over all couplings would be necessary.
A possible way to implement a simple oscillator network is to use vibrational modes of trapped ions. In this way, it is possible to implement simple oscillator chains that interact in a harmonic way via Coulomb force in single or segmented traps \cite{ions1,ions2}. The main limitations are related to scalability and independent control of couplings. In particular, if the couplings are mediated by Coulomb force, then they cannot be controlled in an independent way, which limits the networks that can be realized in this way. Proposals for scalable arrays of trapped ions have been made \cite{ionsproposal}.
One can also consider cold atoms trapped in optical lattices. They offer a scalable platform to simulate different many-body systems, in particular the Bose-Hubbard Hamiltonian, which describes interacting bosons in a lattice. While the Hamiltonian is different, it still shares some similarities with that of an oscillator network. The parameters of the Hamiltonian can be tuned, but it cannot be used to implement an arbitrary topology.
An array of coupled micro- or nanomechanical resonators acting as phonon traps is a natural candidate for an experimental realization. The setup has good scalability, as experimental implementations of arrays of up to 400 resonators have been reported \cite{resonatorarray}. In the case of mechanically coupled devices, independent control of the coupling strengths might not be possible, however a proposal of a fully reconfigurable resonator array based on optical couplings has been made \cite{reconfigurablearray}. Other challenges include the suppression of intrinsic nonlinearities of the devices, as well as cooling them to reduce thermal noise. First steps in this direction have been taken, as coherent phonon manipulation has been reported in a system of two resonators with a tunable mechanical coupling \cite{coherentphonons}.
Perhaps the most promising alternative is the very recently proposed optical implementation of the dynamics given by the Hamiltonian $(2.1)$, based on a simultaneous downconversion of the components of an optical frequency comb from a femtosecond laser followed by pulse shaping and mode-selective measurements \cite{Nokkala17b}. By mapping the Hamiltonian to quadrature operators of the optical field modes and determining the so called Bloch-Messiah decomposition of either the symplectic matrix $(2.5)$ or $(2.6)$, one will find the pulse shape and measurement basis necessary to implement it. In particular, since the network structure is mapped into the parameters of the platform, changing the network does not require a change in the optical setup. The result is a deterministic and highly reconfigurable implementation of quantum complex networks with in principle arbitrary structure. In practice, producing the required pump shape to a sufficiently good accuracy will require further theoretical and experimental work before the proposal can be tested.
\section{Quantum networks as structured environments} \setcounter{equation}{0}
\subsection{Attaching external oscillators}
We consider as the open quantum system a single additional quantum harmonic oscillator interacting with one of the network oscillators. While this is sufficient to our present purposes, what follows is straightforward to extend to the case of multiple external oscillators or interactions with multiple network nodes. Moreover, we will fix the states of the open system and the network to be a Gaussian state and a thermal state of temperature $T$, respectively, assume factorizing initial conditions and work in such units that the Boltzmann constant $k_B=1$.
The open system Hamiltonian is $H_S=(p_S^2+\omega_S^2q_S^2)/2$, and the form of the interaction Hamiltonian reads $H_I=-kq_Sq_i$, or equivalently, $H_I=-kq_S\sum_j^N \mathbf{K}_{ij}Q_j$ in the basis of eigenmodes, where $k$ is the coupling strength between the open system and the network. The total Hamiltonian is now $H=H_S+H_E+H_I$. By including the operators of the open system as the final elements of the vectors of operators, we may express it analogously to Hamiltonian $(2.1)$ as
\begin{equation}
H=\dfrac{\{\mathbf{P},p_S\}^T\{\mathbf{P},p_S\}}{2}+\{\mathbf{Q},q_S\}^T\mathbf{B}\{\mathbf{Q},q_S\}, \end{equation}
\noindent where the matrix $\mathbf{B}$ has diagonal elements $\mathbf{B}_{ii}=\Omega_i^2/2$ for $i<N+1$ and $\mathbf{B}_{N+1,N+1}=\omega_S^2/2$, while $\mathbf{B}_{N+1,i}=\mathbf{B}_{i,N+1}=-k\mathbf{K}_{li}/2$ for $i<N+1$; here the index $l$ is the index of the network oscillator directly interacting with the open system. We may diagonalize the matrix $\mathbf{B}$ as $\mathbf{O}^T\mathbf{B}\mathbf{O}=\mathbf{F}$ where $\mathbf{O}$ is orthogonal and $\mathbf{F}$ diagonal with elements $F_{ii}=f_i^2/2$, where $f_i$ will be the frequencies of the modes in the fully diagonal picture. If we define the new operators as
\begin{equation} \begin{cases} \bm{\mathcal{Q}}=\mathbf{O}^T \{\mathbf{Q},q_S\}\\ \bm{\mathcal{P}}=\mathbf{O}^T \{\mathbf{P},p_S\}, \\ \end{cases} \end{equation}
\noindent the total Hamiltonian reads
\begin{equation} H_E=\dfrac{\bm{\mathcal{P}}^T\bm{\mathcal{P}}}{2}+\bm{\mathcal{Q}}^T\mathbf{F}\bm{\mathcal{Q}}. \end{equation}
We are now in position to write down the symplectic matrix giving the dynamics of the total Hamiltonian. By following the steps leading from Hamiltonian $(2.3)$ to Eq. $(2.6)$, we arrive at
\begin{equation} \begin{pmatrix} \mathbf{Q}(t) \\ q(t)\\ \mathbf{P}(t) \\ p(t)\\ \end{pmatrix} = \begin{pmatrix}
\mathbf{O}\mathbf{D}_{\cos}\mathbf{O}^T & \mathbf{O}\mathbf{\Delta}_{f}^{-1}\mathbf{D}_{\sin}\mathbf{O}^T \\
-\mathbf{O}\mathbf{\Delta}_{f}\mathbf{D}_{\sin}\mathbf{O}^T & \mathbf{O}\mathbf{D}_{\cos}\mathbf{O}^T \end{pmatrix} \begin{pmatrix} \mathbf{Q}(0) \\ q(0)\\ \mathbf{P}(0) \\ p(0)\\ \end{pmatrix}, \end{equation}
\noindent where we have introduced the diagonal matrices $\mathbf{D}_{\cos ii}=\cos(f_i t)$, $\mathbf{D}_{\sin ii}=\sin(f_i t)$ and $\mathbf{\Delta}_{f ii}=f_i$.
As we will consider an initial thermal state for the network, throughout the rest of the present work we will consider as the initial basis the one on the R.H.S. of the equation above, where the initial covariance matrix of the network is diagonal with elements $\langle Q_i(0)^2 \rangle=(n_i+1/2)/\Omega_i$ and $\langle P_i(0)^2 \rangle=(n_i+1/2)\Omega_i$, where $n_i=(\exp(\Omega_i/T)-1)^{-1}$.
If we are interested in the dynamics of the operators in the network basis, we may use Eq. $(2.2)$ and define the symplectic and orthogonal $N+1 \times N+1$ matrix $\tilde{\mathbf{K}}$ with elements $\tilde{\mathbf{K}}_{N+1,i}=\mathbf{K}_{i,N+1}=0$ for $i<N+1$, $\tilde{\mathbf{K}}_{N+1,N+1}=1$, and $\tilde{\mathbf{K}}_{ij}=\mathbf{K}_{ij}$ otherwise. Now
\begin{equation} \begin{pmatrix} \mathbf{q}(t) \\ q(t)\\ \mathbf{p}(t) \\ p(t)\\ \end{pmatrix} = \begin{pmatrix}
\tilde{\mathbf{K}}\mathbf{O}\mathbf{D}_{\cos}\mathbf{O}^T & \tilde{\mathbf{K}}\mathbf{O}\mathbf{\Delta}_{f}^{-1}\mathbf{D}_{\sin}\mathbf{O}^T \\
-\tilde{\mathbf{K}}\mathbf{O}\mathbf{\Delta}_{f}\mathbf{D}_{\sin}\mathbf{O}^T & \tilde{\mathbf{K}}\mathbf{O}\mathbf{D}_{\cos}\mathbf{O}^T \end{pmatrix} \begin{pmatrix} \mathbf{Q}(0) \\ q(0)\\ \mathbf{P}(0) \\ p(0)\\ \end{pmatrix}. \end{equation}
\noindent The dynamics can be readily determined from the initial covariance matrix of the total system, as outlined in Eq. $(2.8)$, where the result will be the covariance matrix at time $t$ in the basis of either the eigenmodes or the network oscillators, depending on which symplectic matrix is used. If the open system has displacement, also the evolution of its first moments needs to be considered to determine the evolution of its state.
While we are concerned with the dynamics of the open system as well as the network oscillators, we mention here the possibility to treat the network in the framework of Gaussian channels. For a general Gaussian state and for $\mathbf{x}(0)=\left(\begin{smallmatrix}q(0) \\p(0) \end{smallmatrix}\right)$, the elements of the covariance matrix of a single mode system are $\mathrm{cov}(\mathbf{x}(0))_{ij}=\langle \mathbf{x}(0)_i \mathbf{x}(0)_j+\mathbf{x}(0)_j \mathbf{x}(0)_i \rangle/2-\langle \mathbf{x}(0)_i \rangle\langle \mathbf{x}(0)_j\rangle$. For any Gaussian channel taking the covariance matrix to time $t$, the transformation can be written as
\begin{equation} \mathrm{cov}(\mathbf{x}(t))=\mathbf{C}(t)\mathrm{cov}(\mathbf{x}(0))\mathbf{C}(t)^T+\mathbf{L}(t), \end{equation}
\noindent where $\mathbf{C}(t)$ and $\mathbf{L}(t)$ are real matrices and $\mathbf{L}(t)$ is symmetric. In terms of the elements of the symplectic matrix $\mathbf{S}$ of Eq. $(3.4)$, we may find the elements using Eq. $(2.8)$ to be
\begin{equation} \mathbf{C}(t)=\begin{pmatrix}\mathbf{S}_{N+1,N+1}&\mathbf{S}_{N+1,2N+2}\\\mathbf{S}_{2N+2,N+1}&\mathbf{S}_{2N+2,2N+2}\end{pmatrix}, \end{equation}
\noindent and
\begin{equation} \mathbf{L}(t)=\sum_i \langle \mathbf{X}_i(0)^2 \rangle \begin{pmatrix} \mathbf{S}_{N+1,i}^2 & \mathbf{S}_{N+1,i}\mathbf{S}_{2N+2,i}\\ \mathbf{S}_{N+1,i}\mathbf{S}_{2N+2,i} & \mathbf{S}_{2N+2,i}^2 \end{pmatrix}, \end{equation}
\noindent where the sum is taken to $2N+1$ excluding $N+1$, such that $\mathbf{L}(t)$ is independent of the initial expectation values of the open system. The matrices $\mathbf{C}(t)$ and $\mathbf{L}(t)$ now completely characterize the channel, allowing, e.g. to make comparisons with channels defined by a master equation or to construct intermediate channels taking the system from time $t>0$ to $s>t$ and checking if the resulting channel is completely positive or not, as is done in a recently introduced measure of non-Markovianity for Gaussian channels \cite{Torre15}. The difficulty in implementing this measure in the present case is neither in the construction of the intermediate map nor checking its complete positivity, but rather in the fact that it considers the limit $s \rightarrow t$, and it is not clear how to take such a limit in the case of numerical, rather than analytical, matrices $\mathbf{C}(t)$ and $\mathbf{L}(t)$.
\subsection{The spectral density}
One of the central concepts in the theory of open quantum systems is the spectral density of environmental couplings $J(\omega)$, which encodes the relevant information in the environment and interaction Hamiltonians into a single function of frequency. The reduced dynamics of the open system can then be determined once the initial state of the total system as well as the system Hamiltonian are fixed \cite{Weiss}. In particular, a heat bath is completely characterized by its spectral density and temperature. The definition of the spectral density, in terms of the environment eigenfrequencies $\Omega_i$ and coupling strengths to eigenmodes $g_i$, reads
\begin{equation} J(\omega)=\dfrac{\pi}{2}\sum_i\dfrac{g_i^2}{\Omega_i}\delta(\omega-\Omega_i), \end{equation}
\noindent where $\delta$ is the Dirac's delta function. The definition is rarely used in practice, since in the case of an infinite heat bath with a continuum of frequencies the spectral density becomes a continuous function, and phenomenological spectral densities are defined instead.
In the case of finite environments it is convenient to use the relation between $J(\omega)$ and the damping kernel $\gamma(t)$, the latter appearing in the generalized quantum Langevin equations giving the dynamics for the open system operators \cite{Weiss}. It is defined as
\begin{equation} \gamma(t)=\sum_i\dfrac{g_i^2}{\Omega_i^2}\cos(\Omega_i t), \end{equation}
\noindent and the relation is given by
\begin{equation} J(\omega)=\omega\int_0^\infty\gamma(t)\cos(\omega t) dt, \end{equation}
If the environment is finite, both Eq. $(3.9)$ and Eq. $(3.11)$ will result in delta spikes. However, by replacing the upper limit of integration by a finite time $t_{max}$, the intermediate form of the spectral density can be considered instead. If a quantum network defined by a Hamiltonian of the form $(2.1)$ is sufficiently symmetric, the reduced dynamics will have a regime where the system interacts with a continuum of frequencies as if the environment was infinite. This is evident from the damping kernel having a very small value during this transient, until finite size effects cause a revival of oscillations. The duration of this continuous regime of reduced dynamics depends on the structure and size of the finite environment.
In the present case of quantum complex networks, the coupling strengths to eigenmodes $g_i$ are determined by the interaction Hamiltonian $H_I$ and the matrix $\mathbf{K}$ diagonalizing the Hamiltonian $(2.1)$ as $g_i=-k \mathbf{K}_{li}$, where $l$ is the index of the network oscillator directly interacting with the system and $k$ the interaction strength in the network basis. In Fig. \ref{figJY}, we show two examples of damping kernels and spectral densities for quantum networks. The symmetric network is a chain with nearest and next nearest couplings. Additionally, the chain is made homogeneous by setting the effective frequencies of the ends of the chain equal with the rest. The spectral density is continuous for the used value of $t_{max}$. If the interaction time is sufficiently short, it would not be possible to tell from the reduced dynamics of an open quantum system coupled to the network alone that the environment is in fact finite. In contrast, the disorder in the other network results in a highly structured spectral density that does not have a continuous regime.
\begin{figure*}
\caption{ (Color online) A comparison of the spectral densities and damping kernels for a symmetric and a disordered network. The black dots are probed values of $J(\omega)$ extraced from the reduced dynamics of an open quantum system interacting with the networks. The symmetric network is a chain of $N=100$ oscillators with nearest and next nearest neighbor couplings with magnitudes of $g_1=0.1$ and $g_2=0.02$, respectively, while the disordered network is a random network of $N=30$ oscillators with a constant coupling strength $g=0.05$. For both, the bare frequency of the oscillators was $\omega_0=0.25$, the system-network interaction strength was $k=0.01$ and the states of the system and the network where a thermal state of $T=1$ and vacuum, respectively. The system is coupled to the first oscillator in the chain and to a random oscillator of the disordered network. }
\label{figJY}
\end{figure*}
In general, it may be asked whether $J(\omega)$ of a quantum complex network can be deduced from the reduced dynamics of the system. It can be shown \cite{Nokkala16} that, provided the coupling to the network $k$ is weak and the network is in a thermal state, the system excitation number is well approximated by the expression $\langle n(t) \rangle=\exp(-\Gamma t)\langle n(0) \rangle+n(\omega_S)(1-\exp(-\Gamma t)$, where $\Gamma=J(\omega_S)/\omega_S$ and $n(\omega_S)=(\exp(\omega_S/T)-1)^{-1}$, or the thermal average boson number at system frequency $\omega_S$. The value of the spectral density at system frequency is then approximated by
\begin{equation} J(\omega_S)=\dfrac{\omega_S}{t}\ln\left( \dfrac{\Delta n(0)}{\Delta n(t)} \right), \end{equation}
\noindent where $\Delta n(t)=n(\omega_S)-\langle n(t) \rangle$. If $T$ is known, the local value of the spectral density can be determined by performing measurements on the system only. This is demonstrated in Fig. \ref{figJY}, where the dots are probed values of the spectral density with each circle corresponding to one value of the system frequency. By keeping the interaction time fixed to the used value of $t_{max}$, it can be seen that even for networks with disorder, the probed values follow the shape of $J(\omega)$.
It is also worth mentioning that the machinery introduced so far can be used to approximate an infinite heat bath, determined by its spectral density, with a finite one. Together with its temperature, the finite bath is completely characterized by the coupling strengths $g_i$ and frequencies $\Omega_i$. While there is considrebale freedom in choosing $\Omega_i$, they should cover the non-vanishing parts of $J(\omega)$ and there should be enough of them to push the finite size effects to interaction times longer than what is being considered. Next, the couplings are determined from the spectral density as follows. From Eq. $(3.9)$, it can be seen that $\int_0^\infty\frac{2}{\pi}J(\omega)\omega d\omega=\sum_i g_i^2$. Approximating the integral on the left hand side with, e.g., a Riemann sum, and identifying the terms on both sides then gives $g_i^2=\frac{2}{\pi}J(\Omega_i)\Omega_i\Delta\Omega_i$, where $\Delta\Omega_i=|\Omega_i-\Omega_{i+1}|$ is the sampling interval. The range of interaction times where the approximation is valid can be checked by comparing the damping kernels calculated for the finite bath from Eq. $(3.10)$ and for the infinite bath from the inversion of Eq. $(3.11)$, namely, $\gamma(t)=\frac{2}{\pi}\int_0^\infty\frac{J(\omega)}{\omega}\cos(\omega t)d\omega$. The two will be similar up to the point where finite size effects manifest. This can be of advantage when considering early or intermediate dynamics in the case of a strong coupling, since the dynamics given by Eqs. $(3.4)$ or $(3.5)$ is exact.
\subsection{Engineering aspect and excitation transport}
Reservoir engineering aims to modify the properties of the environment of an open quantum system, typically to protect non-classicality of the system or to increase the efficiency of some task. In the present case, the environment is a quantum network determined by the matrix $\mathbf{A}$. To assess its properties as an environment, it is convenient to consider the effect of the structure on the spectral density $J(\omega)$, which can be returned to the effect of the structure on the eigenfrequencies $\Omega_i$ and coupling strengths to eigenmodes $g_i$. By changing the structure by, e.g., adding or removing links, one can try to effectively decouple the system from the network by finding a configuration where $J(\omega_S)$ has a small value, or alternatively to look for structures with increases transport efficiency.
In fact, assuming that the system can be freely coupled to any single node in the network, a single network can produce as many spectral densities as it has nodes. This is because a coupling to a single node corresponds to a set of coupling strengths $g_i$, which are in turn directly proportional to a row of the matrix $\mathbf{K}$ diagonalizing the network. On the other hand, the set of eigenfrequencies $\Omega_i$ are completely determined by the eigenvalues of the matrix $\mathbf{A}$ and as such are independent of where in the network the system is coupled.
Even small changes to the network structure can have a large impact on both the network spectral density and excitation transport properties. Generally speaking, when the reduced dynamics has a continuous regime, the flow of energy is steady provided that the system is resonant with the network. Furthermore, excitations can freely be exchanged between different nodes in the network. On the other hand, when the degree of disorder in the network is high, the excitations typically become locked to a subset of the network nodes and cannot spread effectively. We present examples of this in Fig. \ref{figtransport}, where the same symmetric network is considered as in Fig. \ref{figJY}. Rewiring randomly only a single coupling changes the path taken by the majority of excitations. Also shown is the excitation dynamics in a random network.
\begin{figure*}
\caption{ (Color online) Examples of excitation transport in quantum networks. In all examples, the interaction time is shown on the horizontal axis while the vertical axis corresponds to the index of the network oscillator. The color bar shows the difference between initial excitations and excitations at time $t$. On the left, the network is the symmetric network of Fig. \ref{figJY}. Excitations propagate freely along the chain. In the middle, a single randomly chosen link in the symmetric network has been rewired, changing the transport properties. On the right, evolution of excitations in the network oscillators of a random network of $N=100$ oscillators with bare frequency $\omega_0=0.25$ and coupling strengths $g=0.05$ is shown. Excitations become locked to a subset of network oscillators. }
\label{figtransport}
\end{figure*}
While transport is inefficient in most random networks, a search can be carried out for exceptions, and indeed it can be shown that when sampling the distribution of random networks, some rare cases have vastly superior transport properties robust against ambient dephasing \cite{Scholak11}. One may also ask whether there is any connection between the excitation transport properties and non-Markovianity. While in the spin-boson model non-Markovianity and the back-flow of excitations can behave similarly with respect to the environment parameters \cite{Guarnieri16a}, there does not seem to be such a connection in the case of a continuous variable system \cite{Guarnieri16b}. Furthermore, even in the spin-boson model, information and excitation backflows can occur without the other \cite{Schmidt16}.
\section{Non-Markovianity in complex quantum networks} \setcounter{equation}{0}
\subsection{Generalities}
The dynamics of an open quantum system can significantly deviate from the memoryless Markovian case when the interaction between the open system and the environment is strong, or if the environment is structured. Previous investigations \cite{Vasile14} of harmonic chains with nearest neighbor couplings, having a Hamiltonian of the form $(2.1)$, show that the strongest memory effects occur when the system frequency is located near the edges of the spectral density. A $J(\omega)$ with a single band will then have two regimes of system frequency where memory effects are strong while one with band-gaps will have more. In this work, the Breuer-Laine-Piilo \cite{BLP} and Rivas-Huelga-Plenio \cite{RHP} measures were used.
To the best of our knowledge, however, there have been no studies of non-Markovianity attempting to connect it to the structure of a complex network. While it is the case that any spectral density of an oscillator network with non-regular structure can be replicated with an oscillator chain with nearest neighbor couplings \cite{linearmappingA,linearmappingB,linearmappingC}, it is nevertheless of interest to ask whether the amount of non-Markovianity could be tied to the statistical properties of complex networks by comparing the average non-Markovianity over many realizations, and whether adding more structure typically increases the non-Markovianity or not.
To this end, we considered three types of random networks presented in Figure \ref{figgraphs}. For all three cases, we fixed the size of the network to be $N=30$ and assumed that the network is connected, i.e. any node can be reached from any other by following the links. The Erd\H{o}s-R\'{e}nyi network $G(N,p)$ \cite{ER} is constructed from the completely connected network of $N$ nodes by independently selecting each link to be part of the final network with a probability $p$. The Barab\'{a}si-Albert network $G(N,l)$ \cite{BA} is constructed from a connected network of $3$ nodes and repeatedly adding a new node with $l$ links, connecting it randomly to existing nodes but favoring nodes which already have a high number of links, until the size $N$ is reached. Setting $l=1$ is an important special case, as the resulting network is a tree, i.e. it has the smallest possible number of links that a connected network of size $N$ can have. Finally, a Watts-Strogatz network $G(N,p,n)$ \cite{WS} is constructed starting from a circular network where all nodes are connected to $n$-th nearest neighbors, and then rewiring each link with the probability $p$. In this work, we fixed $n=2$.
\begin{figure*}
\caption{ (Color online) Schematics for the used networks. Each column corresponds to a network type while the rows correspond to different parameter values. When the connection probability for the Erd\H{o}s-R\'{e}nyi network is increased the number of links grows, but links are chosen randomly. This is at variance with the Barab\'{a}si-Albert network where nodes with a higher number of links are preferred when introducing new links, resulting in highly connected nodes when the connectivity parameter grows. Watts-Strogatz networks are constructed from a cycle graph by rewiring each link with a given probability. As this rewiring probability grows, the average distance between the nodes decreases, but the total number of links remains constant. }
\label{figgraphs}
\end{figure*}
\subsection{Non-Markovianity quantified by the non-monotonicity of Gaussian interferometric power}
The key concept used in several witnesses and measures of non-Markovianity is to track the dynamics of a quantity that can be shown to behave differently under Markovian and non-Markovian evolutions. In this work, we consider a recently introduced measure and a witness based on the non-monotonicity of Gaussian interferometric power under non-divisible dynamical maps \cite{Souza15}.
Gaussian inteferometric power $\mathcal{Q}$ quantifies the worst-case precision achievable in black-box phase estimation using a bipartite Gaussian probe composed of modes $A$ and $B$. It is also a measure of discord-type correlations between the two modes, as it vanishes for product states. For quantifying non-Markovianity, it is enough to consider the case where mode $A$ is subjected to a local Gaussian channel while mode $B$ remains unchanged. Then the expression for the Gaussian interferometric power $\mathcal{Q}$ has a closed form in terms of the symplectic invariants of the two-mode covariance matrix $\sigma_{AB}$ \cite{Adesso14}.
For Markovian channels, $\mathcal{Q}$ is a monotonically non-increasing function of time, implying that $\dfrac{d}{dt}\mathcal{Q}(\sigma_{AB})\leq 0$. Any period of time where this does not hold is then a sign of non-Markovianity. Once the initial covariance matrix $\sigma_{AB}$ has been fixed, the degree of non-Markovianity of the reduced dynamics can then be quantified as
\begin{equation}
\mathcal{N}_{GIP}=\frac{1}{2}\int_0^\infty (|\mathcal{D}(t)|+\mathcal{D}(t))dt, \end{equation}
\noindent where $\mathcal{D}(t)=\dfrac{d}{dt}\mathcal{Q}(\sigma_{AB})$. While the related measure is defined with a maximization over all initial states for the bi-partite system, Eq. $(4.1)$ provides a lower bound for this measure. Since there is strong numerical evidence that squeezed thermal states are particularly suited for witnessing non-Markovianity of this type \cite{Souza15}, we fix the initial state of the two-mode system to be a squeezed thermal state with two-mode squeezing parameter $r=\frac{1}{2}\cosh^{-1}(5/2)$ and initial thermal excitations $n_A=n_B=1/2$ for both modes. The network initial state is taken to be the vacuum.
Besides disorder in the network structure, additional sources of non-Markovianity include finite size effects that become stronger as the interaction time is increased, and memory effects at the boundaries of the spectral density. To better assess the non-Markovianity arising from the structure, we will restrict the interaction time to an intermediate value of $t=50$ and fix the frequency of the system to be the $15th$ eigenfrequency of the networks to ensure that it is resonant.
\begin{figure*}
\caption{ (Color online) A comparison of the non-Markovianity for three different types of quantum complex networks. The columns correspond to the type and the rows to interaction strength between the network and the system. The size of each network is fixed to $N=30$ while a parameter controlling the structure of the network is varied. The parameters are connection probability $p$, connectivity parameter $l$ and rewiring probability $p$ for Erd\H{o}s-R\'{e}nyi, Barab\'{a}si-Albert and Watts-Strogatz networks, respectively. Refer to main text for details. Results are averaged over 1000 realizations for each parameter value. }
\label{figGIP}
\end{figure*}
The results are shown in Fig. \ref{figGIP}. For all considered cases, changing the interaction strength affects the magnitude but not the behaviour of non-Markovianity against the network parameter. For Erd\H{o}s-R\'{e}nyi and Barab\'{a}si-Albert networks, the number of couplings between network oscillators grows with the parameter, reducing the amount of non-Markovianity. On the other hand, the number of couplings in the network is constant for the Watts-Strogatz network. The results suggest that when the system is resonant with the network, non-Markovianity is highest for networks with a small amount of random couplings. For all considered coupling strengths, the highest non-Markovianity is achieved when the network is a tree. If the network is highly symmetric, as is the case with Watts-Strogatz networks with a low rewiring probability, the amount of non-Markovianity in the resonant case is very small. Non-Markovianity is increased by introducing disorder into the network through rewiring of the couplings.
Besides the results we present here, we also checked that increasing the network temperature decreases the non-Markovianity. Furthermore, for comparison we determined the non-Markovianity in the simple case of a homogeneous chain with nearest-neighbor couplings only and found that even at the edges of the spectral density, where memory effects are strongest, $\mathcal{N}_{GIP}$ has a similar value than Erd\H{o}s-R\'{e}nyi and Barab\'{a}si-Albert networks have in the resonant case.
\section{Conlusions and outlook} \setcounter{equation}{0}
In this work, we have studied bosonic quantum complex networks in the framework of open quantum systems. After briefly investigating the effect of the network stucture on the spectral density and transport of excitations, we focused on the non-Markovianity in the reduced dynamics of an open quantum system interacting with the network.
We considered non-Markovianity over ensemble averages of different types of random networks of identical oscillators and constant coupling strength between the network oscillators. Previous work shows that strong memory effects can occur in symmetric networks at the edges of the spectral density and near band gaps. Here we have shown that increasing the disorder of the network can lead to a high degree of non-Markovianity also when the system is resonant with the network, however increasing the number of interactions between network oscillators appears to suppress it, suggesting that trees optimize the ensemble averaged non-Markovianity.
While here we considered only the lower bound of a single non-Markovianity measure, it would be interesting to extend the investigations to other measures such as the measure introduced by Torre, Roga and Illuminati \cite{Torre15}. We expect that a systematic study could perhaps link some of the graph invariants, such as the mean distance between nodes, to non-Markovianity and other non-classical properties of the quantum networks, such as the ability to generate or transport entanglement. Such a link could pave way to structural control of non-classical properties of quantum complex networks. Indeed, in the case of quantum walks on classical complex networks, it can be shown that the quantumness of the walk is a function of both the initial state and specific graph invariants. Furthermore, for a deeper understanding of quantum networks the introduction of purely quantum graph invariants without a classical counterpart would be needed.
\section*{Acknowledgments} The authors acknowledge financial support from the Horizon 2020 EU collaborative projects QuProCS (Grant Agreenement No. 641277). J. N. acknowledges the Wihuri foundation for financing his graduate studies.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Arbitrarily large Morita Frobenius numbers}
\begin{abstract}
We construct blocks of finite groups with arbitrarily large Morita Frobenius numbers, an invariant which determines the size of the minimal field of definition of the associated basic algebra. This answers a question of Benson and Kessar. This also improves upon a result of the second author where arbitrarily large $\mathcal O$-Morita Frobenius numbers are constructed. \end{abstract}
\section{Introduction}
Let $\ell$ be a prime and $k$ an algebraically closed field of characteristic $\ell$. For a finite-dimensional $k$-algebra $A$ we define the $n^{\operatorname{th}}$ Frobenius twist of $A$, denoted $A^{(\ell^n)}$, as follows: as a set, and indeed as a ring, $A^{(\ell^n)}$ is equal to $A$, but for $\lambda \in k$ and $a\in A^{(\ell^n)}$ we set $\lambda\cdot a = \lambda^{\ell^{-n}} a$ (the multiplication on the right hand side being that of $A$). That is, if we think of a $k$-algebra as a ring with a distinguished embedding $k\hookrightarrow Z(A)$, then that embedding is precomposed with the $n^{\operatorname{th}}$ power of the Frobenius automorphism to obtain the $k$-algebra structure of $A^{(\ell^n)}$. The result of this construction is clearly isomorphic to $A$ as a ring, but not necessarily as a $k$-algebra. This leads to the following notion, first defined by Kessar~\cite{KessarDon}. \begin{defi}
The \emph{Morita Frobenius number} of $A$, denoted $\operatorname{mf}(A)$, is the smallest $n\in \mathbb N$ such that $A$ is Morita equivalent to $A^{(\ell^n)}$ as a $k$-algebra. \end{defi} As an alternative characterisation, Kessar~\cite{KessarDon} showed that, for a basic algebra $A$, $\operatorname{mf}(A)$ is the smallest $n\in\mathbb N$ such that $A\cong k\otimes_{\mathbb F_{\ell^n}} A_0$ for some $\mathbb F_{\ell^n}$-algebra $A_0$. For fixed $n\in\mathbb N$ there are only finitely many possibilities for $A_0$ in any given dimension, which is why Morita Frobenius numbers are being used to approach Donovan's famous finiteness conjecture (more on this further below).
In the present paper we are interested in the Morita Frobenius numbers of blocks of finite groups $G$. For a block $B$ of $kG$, the Frobenius twist $B^{(\ell^n)}$ is isomorphic as a $k$-algebra to $\sigma^n(B)$, where $\sigma$ is the ring automorphism \begin{equation}\label{algn:ring_auto} \sigma:\ kG \longrightarrow kG,\ \sum_{g\in G}\alpha_g\phantom{\hspace{0.8pt}} g\mapsto\sum_{g\in G}\alpha_g^{\ell}\phantom{\hspace{0.8pt}} g. \end{equation} We can therefore think of a Frobenius twist of a block simply as another block of the same group algebra, Galois conjugate to the original one. And while there is no bound on the number of Galois conjugates of a block, Benson~and~Kessar~\cite{BenKesIneq} observed that Morita Frobenius numbers of blocks tend to be very small, with no known example exceeding Morita Frobenius number two. This prompted them to ask the following. \begin{question}[{Benson-Kessar, \cite[Question 6.2]{BenKesIneq}}]
Is there a universal bound on the Morita Frobenius numbers of $\ell$-blocks of finite groups? \end{question} This question, to which we give a negative answer in the present article, has gained much interest in recent years. In~\cite[Examples 5.1, 5.2]{BenKesIneq} Benson and Kessar constructed blocks with Morita Frobenius number two, the first discovered to be greater than one. The relevant blocks all have a normal, abelian defect group and abelian $\ell'$ inertial quotient with a unique isomorphism class of simple modules. It was also proved that amongst such blocks the Morita Frobenius numbers cannot exceed two~\cite[Remark 3.3]{BenKesIneq}. In work of Benson, Kessar and Linckelmann~\cite[Theorem 1.1]{BenKesLinNorm} the bound of two was extended to blocks that don't necessarily have a unique isomorphism class of simple modules. One can also define Morita Frobenius numbers over a complete discrete valuation ring $\mathcal O$ of characteristic zero with residue field $k$, and in \cite{BenKesLinNorm} it was also shown that the aforementioned bound of two applies equally to the Morita Frobenius numbers of the corresponding blocks defined over $\mathcal O$. Finally, Farrell~\cite[Theorem 1.1]{FarrellMFno} and Farrell and Kessar~\cite[Theorem 1.1]{FarrellKessarRational} proved that the Morita Frobenius number of any block of a finite quasi-simple group is at most four (both over $k$ and over $\mathcal O$).
Our main result (see Theorem~\ref{thm:main}) is the $k$-analogue of~\cite[Theorem 3.6]{LiveseyOLarge}, where the corresponding result is proved for blocks defined over $\mathcal O$. Note that the result in the current paper takes significantly longer to prove as Weiss' criterion to detect $\ell$-permutation modules does not hold over $k$.
\begin{thm*}
For any $n\in\mathbb N$ there exists an $\ell$-block $B$ of $kG$, for some finite group $G$, such that $\operatorname{mf}(B)=n$. \end{thm*} Hence the questions \cite[Questions 6.2, 6.3]{BenKesIneq} (the second of which is the one mentioned above) both have negative answers.
The blocks realising arbitrarily large Frobenius numbers have elementary abelian defect groups and metabelian $\ell'$ inertial quotients, and turn out to be Morita equivalent to a \emph{twisted tensor product} of the algebra $k[D\rtimes P]$ with itself, where $D$ is an elementary abelian $\ell$-group and $P$ is a cyclic $\ell'$-group. It should be mentioned that while the blocks realising Morita Frobenius number two in \cite{BenKesIneq} are described as \emph{quantum complete intersections}, such algebras can also be realised as iterated twisted tensor products of group algebras of cyclic groups.
To close, let us quickly explain how the initial motivation to consider Morita Frobenius numbers came from their link with Donovan's conjecture. \begin{conjecture}[Donovan]
Let $D$ be a finite $\ell$-group. Then, amongst all finite groups $G$ and blocks $B$ of $k G$ with defect group isomorphic to $D$, there are only finitely many Morita equivalence classes. \end{conjecture} If true, Donovan's conjecture would imply that Morita Frobenius numbers are bounded in terms of a function of the isomorphism class of the defect group.
\begin{conjecture}[{see \cite[Question 6.1]{BenKesIneq}}]\label{con:bdmf}
Let $D$ be a finite $\ell$-group. Then, amongst all finite groups $G$ and blocks $B$ of $k G$ with defect group isomorphic to $D$, the Morita Frobenius number $\operatorname{mf}(B)$ is bounded. \end{conjecture} While we cannot contribute much to this, we should point out that the defect of our blocks realising Morita Frobenius number $n\in \mathbb N$ grows exponentially in $n$. Hence our result does not contradict Conjecture~\ref{con:bdmf} and it would in fact be consistent with a logarithmic bound of Morita Frobenius numbers in terms of the rank of $D$.
In~\cite[Theorem 1.4]{KessarDon} Kessar proved that Donovan's conjecture is equivalent to Conjecture~\ref{con:bdmf} together with the so-called Weak Donovan conjecture, which further highlights the importance of understanding and bounding Morita Frobenius numbers. \begin{conjecture}[Weak Donovan]
Let $D$ be a finite $\ell$-group. Then there exists a constant $c(D)\in\mathbb N$ such that if $G$ is a finite group and $B$ is a block of $k G$ with defect group isomorphic to $D$, then the entries of the Cartan matrix of $B$ are bounded by $c(D)$. \end{conjecture}
The article is organised as follows. In $\S$\ref{sec setup} we introduce the block $B(\theta)$, which becomes the focus of study for the remainder of the paper. We describe the simple $B(\theta)$-modules in $\S$\ref{sec simples} and in $\S$\ref{sec subalg} we study a certain subalgebra of $B(\theta)$ more closely. We introduce $B(\theta)_0$, a $k$-algebra Morita equivalent to $B(\theta)$, in $\S$\ref{sec basic alg} and prove our main theorem in $\S$\ref{sec Mor equ}.
\subsection*{Notation} For an $\ell'$-group $G$, $e_\chi\in kG$ will denote the primitive central idempotent corresponding to $\chi\in\operatorname{Irr}(G)$ and $1_G\in\operatorname{Irr}(G)$ will signify the trivial character. We will often use the fact that $\operatorname{IBr}(G)=\operatorname{Irr}(G)$ for such a group $G$.
For an arbitrary group $G$, a normal subgroup $N\lhd G$ and $\chi\in\operatorname{Irr}(N)$, we set $\operatorname{Irr}(G|\chi)$ to be the set of characters of $G$ appearing as a non-zero constituent in $\chi\uparrow_N^G$.
For $x\in kN$ and $g\in G$, we denote by $x^g=g^{-1}xg\in kN$. Similarly, for $\chi\in\operatorname{Irr}(N)$, we signify by $\chi^g$ the character of $N$ given by $\chi^g(h)=\chi(h^{g^{-1}})$, for all $h\in N$. Note this definition ensures that $e_\chi^g=e_{\chi^g}$. If $\chi,\chi'\in \operatorname{Irr}(N)$ such that $\chi'=\chi^g$, for some $g\in G$, we write $\chi\sim_G \chi'$.
\section{Setup}\label{sec setup}
In this section we will define the groups and blocks which we later show realise arbitrarily large Morita Frobenius numbers over $k$. All notation introduced in this section will be used throughout the paper. We start by setting,
for $i\in \{1,2\}$,
\begin{equation}
P_i=(\mathbb F_p,+)=C_p
\end{equation}
for some prime $p\neq \ell$, and
\begin{equation}
D_i=\prod_{P_i} C_\ell \bigg/ \left\langle (x,\ldots,x) \ | \ x \in C_\ell \right\rangle\cong C^{p-1}_{\ell}.
\end{equation}
We set $d_i^x=(1,\ldots,1,d,1,\ldots,1)\in D_i$, where $d$ is a fixed generated of $C_\ell$, and the position of $d$ is the direct factor of $\prod C_\ell$ labeled by $x\in P_i$. In particular, all $d_i^x$ taken together generate $D_i$. The group
$P_i$ acts on $D_i$ by permuting the direct factors, i.e. by setting $(d_i^x)^y=d_i^{xy}$ for $x,y\in P_i$. Hence we can form the algebra
\begin{equation}
A_i=k[D_i\rtimes P_i].
\end{equation}
Let $L=\langle g_0 \rangle \subseteq \mathbb F_p^\times \cong C_{p-1}$ be an $\ell'$-subgroup of order $r>1$. Set
\begin{equation}
H=\langle g_1,g_2,g_z : g_1^r=g_2^r=g_z^r=1, [g_1,g_z]=[g_2,g_z]=1, [g_1,g_2]=g_z\rangle
\end{equation}
where we adopt the convention that $[g,h]=ghg^{-1}h^{-1}$,
and define the subgroups
\begin{equation}
L_1=\langle g_1\rangle,\
L_2=\langle g_2\rangle\textrm{ and }Z =\langle g_z\rangle.
\end{equation}
We have $L_1\cong L_2\cong Z\cong L\cong C_r$.
Note we have an action of $\mathbb F_p^\times$ (and hence $L$) on each $P_i$ given by multiplication.
We can now define an action of $H$ on $(D_1\rtimes P_1)\times (D_2\rtimes P_2)$, with kernel $Z$, in the following way. If $\{i,j\}=\{1,2\}$, then $L_i$ acts on $D_i\rtimes P_i$ by setting
\begin{equation}
(d_i^xy)^w=d_i^{(x^w)}y^w\quad \textrm{for $x,y\in P_i$ and $w\in L_i$,}
\end{equation}
and setting the action of $L_i$ on $D_j\rtimes P_j$ to be trivial.
\begin{defi}[The group $G$, and the block $B(\theta)$]
Define
\begin{equation}
G=((D_1\rtimes P_1)\times (D_2\rtimes P_2))\rtimes H.
\end{equation}
and the following subgroups of $G$
\begin{equation}
D=D_1\times D_2,\quad\textrm{and}\quad E=(P_1\times P_2)\rtimes H.
\end{equation}
Let $\theta\in \operatorname{Irr}(Z)$ be a faithful character, and let $e_{\theta}$ be the associated central-primitive idempotent. Define a block $B(\theta)=kGe_\theta$.
\end{defi}
\begin{defi}\label{def h chi}
Let $\theta\in \operatorname{Irr}(Z)$ be faithful, as before, and
let $\{i,j\}=\{1,2\}$.
For each $\chi \in \operatorname{Irr}(L_i)$ define an element $h^\theta_{\chi,i}\in L_j$ such that
\begin{equation}\label{eqn chi theta}
\chi(-)=\theta([h^\theta_{\chi,i},-])
\end{equation}
and
\begin{equation}\label{eqn hchi multiplication}
h^\theta_{\chi,i}h^\theta_{\eta,i}=h^\theta_{\chi\eta,i} \quad\textrm{ for all $\chi,\eta \in \operatorname{Irr}(L_i)$.}
\end{equation}
We will often refer to $h^\theta_{\chi,i}$ as $h_{\chi,i}$ where the choice of $\theta$ is clear from the context.
\end{defi}
Note that in the foregoing definition, the existence of an $h^\theta_{\chi,i}$ satisfying \eqref{eqn chi theta} is guaranteed by \cite[Lemma 4.1]{HolKesQuant} and the uniqueness of such an $h^\theta_{\chi,i}$ in $L_j$ follows by the fact that $C_H(L_i)=Z\times L_i$.
In order to see that \eqref{eqn hchi multiplication} holds we note that, since $[H,H]\subseteq Z \subseteq Z(H)$, we have
\begin{equation}
[h^\theta_{\chi,i}, g]\phantom{\hspace{0.8pt}} [h^\theta_{\eta,i}, g]=h^\theta_{\chi,i} g (h^\theta_{\chi,i})^{-1}g^{-1} [h^\theta_{\eta,i}, g]=h^\theta_{\chi,i} [h^\theta_{\eta,i}, g] g (h^\theta_{\chi,i})^{-1}g^{-1}=[h^\theta_{\chi,i} h^\theta_{\eta,i}, g]
\end{equation}
for all $g\in L_i$.
Effectively, the above just fixes an isomorphism
between $\operatorname{Irr}(L_i) \cong \operatorname{Hom}(C_r, k^\times)\cong C_r$ and $L_j \cong C_r$.
\section{Simple modules and Brauer characters}\label{sec simples}
From now on, unless we are explicitly considering $B(\theta)$ and $B(\theta')$ for two $\theta,\theta'\in\operatorname{Irr}(Z)$, we denote $B(\theta)$ simply by $B$. Since $D$ acts trivially on every simple $B$-module, we can and do identify $\operatorname{IBr}(B)$ with $\operatorname{Irr}(E|\theta)$. In what follows, by an abuse of notation, we often use $1$ to denote $1_{P_i}$, for $i=1,2$. We define the following elements of $\operatorname{Irr}(E|\theta)$,
\begin{equation}\label{algn:lab_char}
\begin{split}
(1,1)&=(\theta\otimes 1_{P_1\times (P_2\rtimes L_2)})\uparrow_{Z\times P_1\times(P_2\rtimes L_2)}^E=(\theta\otimes 1_{(P_1\rtimes L_1)\times P_2})\uparrow_{Z\times(P_1\rtimes L_1)\times P_2}^E,\\
(\phi,1)&=(\theta\otimes \phi\otimes 1_{P_2\rtimes L_2})\uparrow_{Z\times P_1\times (P_2\rtimes L_2)}^E,\\
(1,\psi)&=(\theta\otimes 1_{P_1\rtimes L_1}\otimes \psi)\uparrow_{Z\times (P_1\rtimes L_1)\times P_2}^E,\\
(\phi,\psi)&=(\theta\otimes \phi\otimes \psi)\uparrow_{Z\times P_1\times P_2}^E,
\end{split}
\end{equation}
for all $\phi\in\operatorname{Irr}(P_1)\setminus\{1\}$ and $\psi\in\operatorname{Irr}(P_2)\setminus\{1\}$. Note that, since $C_{L_1}(L_2)=C_{L_2}(L_1)=\{1\}$ and $\theta$ is faithful, ${\operatorname{Stab}}_H(\theta\otimes 1_{L_i})=Z\times L_i$, for all $i=1,2$. Also, as any non-trivial $\phi\in\operatorname{Irr}(P_i)$ is faithful, ${\operatorname{Stab}}_{L_i}(\phi)=\{1\}$, for all $i=1,2$. It follows that all the characters in \eqref{algn:lab_char} are indeed irreducible.
\begin{lemma}\label{lem:simples}
$\operatorname{Irr}(E|\theta)=\{(\phi,\psi)|\phi\in\operatorname{Irr}(P_1),\ \psi\in \operatorname{Irr}(P_2)\}$ and $(\phi,\psi)=(\phi',\psi')$ if and only if either $\phi=\phi'$ and $\psi=\psi'$,
or $\phi,\psi,\phi',\psi'\neq 1$ and $\phi\sim_{L_1}\phi'$, $\psi\sim_{L_2}\psi'$.
Moreover,
\begin{align*}
\deg(1,1)=\deg(\phi,1)=\deg(1,\psi)=r, \quad \deg(\phi,\psi)=r^2,
\end{align*}
for all $\phi\in\operatorname{Irr}(P_1)\setminus\{1\}$ and $\psi\in\operatorname{Irr}(P_2)\setminus\{1\}$.
\end{lemma}
\begin{proof}
We claim that
\begin{align*}
\{(1,1)\}&=\operatorname{Irr}(E|\theta\otimes 1_{P_1}\otimes 1_{P_2}),\\
\{(\phi,1)\}_{\phi\in\operatorname{Irr}(P_1)\setminus\{1\}}&=\bigcup_{\mu\in\operatorname{Irr}(P_1)\setminus\{1\}}\operatorname{Irr}(E|\theta\otimes\mu\otimes 1_{P_2}),\\
\{(1,\psi)\}_{\psi\in\operatorname{Irr}(P_2)\setminus\{1\}}&=\bigcup_{\nu\in\operatorname{Irr}(P_2)\setminus\{1\}}\operatorname{Irr}(E|\theta\otimes 1_{P_1}\otimes\nu),\\
\{(\phi,\psi)\}_{\phi\in\operatorname{Irr}(P_1)\setminus\{1\}, \psi\in\operatorname{Irr}(P_2)\setminus\{1\}}&=\bigcup_{\substack{\mu\in\operatorname{Irr}(P_1)\setminus\{1\}\\ \nu\in\operatorname{Irr}(P_2)\setminus\{1\}}} \operatorname{Irr}(E|\theta\otimes \mu\otimes\nu).
\end{align*}
These equalities can all be readily checked. The main point is that, by the comments preceding the lemma, $L_1$ acts regularly on $\operatorname{Irr}(Z\times L_2|\theta)$ and $L_2$ on $\operatorname{Irr}(Z\times L_1|\theta)$. These facts are needed to prove the first three equalities. The fourth is more straightforward. The fact that there are no duplicates, other than the desired ones, is again a consequence of the regularity of these actions.
It is a simple task to verify the degrees.
\end{proof}
\begin{prop}\label{prop isoms}
\begin{enumerate}
\item Let $g_i\in\operatorname{Aut}(P_i) \cong \mathbb F_p^\times$ for $i\in\{1,2\}$. The following automorphism of $G$ induces an automorphism of $B(\theta)$,
\begin{equation}
\begin{array}{lll} d_i^xy&\mapsto d_i^{(x^{g_i})}\phantom{\hspace{0.8pt}} y^{g_i},&\text{ for all }i\in\{1,2\}\text{ and }x,y\in P_i,\\
h&\mapsto h,&\text{ for all }h\in H.
\end{array}
\end{equation}
Furthermore, the corresponding permutation of $\operatorname{IBr}(B(\theta))$ is given by ${(\phi,\psi)\mapsto(\phi^{g_1},\psi^{g_2}})$, for all $\phi\in\operatorname{Irr}(P_1), \psi\in\operatorname{Irr}(P_2)$.
\item The following automorphism of $G$ induces an isomorphism $B(\theta)\xrightarrow{\sim} B(\theta^{-1})$,
\begin{equation}
\begin{array}{lll}
(x_1,x_2)&\mapsto(x_2,x_1),&\text{ for all }(x_1,x_2)\in (D_1\rtimes P_1)\times (D_2\rtimes P_2),\\
z&\mapsto z^{-1},&\text{ for all }z\in Z,\\
g_i&\mapsto g_j,&\text{ (for $\{i,j\}=\{1,2\}$)},
\end{array}
\end{equation}
where $g_1$ and $g_2$ are the generators for $L_1$ and $L_2$ defined in \S\ref{sec setup}.
Also, for the topmost assignment recall that $P_1$ and $P_2$, as well as $D_1$ and $D_2$, are defined as two copies of the same group, i.e. we may identify them.
Furthermore, the corresponding bijection $\operatorname{IBr}(B(\theta))\longrightarrow\operatorname{IBr}(B(\theta^{-1}))$ is given by $(\phi,\psi)\mapsto(\psi,\phi)$, where we identify $\operatorname{Irr}(P_1)$ and $\operatorname{Irr}(P_2)$.
\end{enumerate} \end{prop}
\begin{proof}
This is all straightforward to check. \end{proof}
\section{Generators and relations for $A_i=k[D_i\rtimes P_i]$}\label{sec subalg}
Let us now give a description of the (isomorphic) algebras $A_i$ for $i\in\{1,2\}$ in terms of quiver and relations. This description will be used implicitly throughout the remainder of the paper. For the sake of readability, we will use the same notation for the generators of $A_1$ and $A_2$.
\begin{defi}\label{defi e s}
Set
\begin{equation}
s_{\phi} = \sum_{g\in P_i} \phi(g^{-1}) \phantom{\hspace{0.8pt}} d_i^g \in k[D_i] \subset A_i \quad \textrm{for $\phi \in \operatorname{Irr}(P_i)\setminus \{1\}$},
\end{equation}
as well as
\begin{equation}
s_{\psi,\phi} = e_\psi \phantom{\hspace{0.8pt}} s_{\phi} \in A_i\quad \textrm{for $\psi \in \operatorname{Irr}(P_i)$ and $\phi \in \operatorname{Irr}(P_i)\setminus \{1\}$}.
\end{equation}
\end{defi}
Note that $\operatorname{Irr}(P_i)\setminus \{1\}$ equals the set of constituents of the (multiplicity-free) $k[P_i]$-module $k\otimes_{\mathbb F_\ell}D_i$ and hence, by \cite[Proposition 5.2]{EiseleSL2}, of $J(k[D_i])/J^2(k[D_i])$.
\begin{prop}[{see \cite[Proposition 5.3]{EiseleSL2}}]
\begin{enumerate}
\item
The $e_{\psi}$ for $\psi\in\operatorname{Irr}(P_i)$ form a full set of primitive idempotents in $A_i$.
\item The $s_{\psi,\phi}$ map to a basis of $J(A_i)/J^2(A_i)$ and $e_{\psi}s_{\psi,\phi}=s_{\psi,\phi}e_{\psi\phantom{\hspace{0.8pt}} \phi}$. That is, the $s_{\psi,\phi}$ correspond to arrows in the quiver of $A_i$.
\item
The relations between the arrows are generated by
\begin{equation}
s_{\psi,\phi}\phantom{\hspace{0.8pt}} s_{\psi\phantom{\hspace{0.8pt}} \phi,\zeta}=s_{\psi,\zeta}\phantom{\hspace{0.8pt}} s_{\psi\phantom{\hspace{0.8pt}} \zeta, \phi}\quad \textrm{for $\psi \in \operatorname{Irr}(P_i)$ and $\phi,\zeta \in \operatorname{Irr}(P_i)\setminus \{1\}$}
\end{equation}
and
\begin{equation}\label{eqn l power relation}
s_{\psi,\phi} \phantom{\hspace{0.8pt}} s_{\psi\phantom{\hspace{0.8pt}} \phi,\phi}\phantom{\hspace{0.8pt}} \cdots\phantom{\hspace{0.8pt}} s_{\psi\phantom{\hspace{0.8pt}} \phi^{\ell-1},\phi}=0\quad \textrm{for $\psi \in \operatorname{Irr}(P_i)$ and $\phi \in \operatorname{Irr}(P_i)\setminus \{1\}$}.
\end{equation}
\end{enumerate}
\end{prop}
A basis of $J(A_i)$ is given by elements of the form
\begin{equation}
s_{\psi,\boldsymbol{\phi}}=s_{\psi,\phi_1}s_{\psi\phi_1,\phi_2}\cdots s_{\psi\phi_1\cdots \phi_{m-1},\phi_m} \textrm{ where $\psi \in \operatorname{Irr}(P_i)$, $\boldsymbol{\phi}=(\phi_1,\ldots,\phi_m) \in \bigcup_{m=1}^\infty(\operatorname{Irr}(P_i)\setminus \{1\})^m$}. \ \end{equation}
To be more precise, we get a basis when we let $\boldsymbol{\phi}$ range over a transversal of
\begin{equation}
\bigcup_{m=1}^\infty(\operatorname{Irr}(P_i)\setminus \{1\})^m\big/\operatorname{Sym}_m,
\end{equation}
where, in addition, $(\phi_1,\ldots,\phi_m)$ must not involve any element of $\operatorname{Irr}(P_i)\setminus \{1\}$ more than $\ell-1$ times. Let $\mathcal I$ denote the set of all possible values for $\boldsymbol{\phi}$ which give rise to non-zero $s_{\psi, \boldsymbol \phi}$'s, and let $\mathcal I/\sim$ denote equivalence classes of $\boldsymbol \phi$'s that give rise to the same $s_{\psi, \boldsymbol \phi}$ (all of this is independent of $\psi$). We have $|\mathcal I/\sim|=\ell^{p-1}-1$, as $\mathcal I/\sim$ is naturally in bijection with maps $\operatorname{Irr}(P_1)\setminus \{1\}\longrightarrow\{0,1,\ldots,\ell-1\}$ which are not identically zero.
\section{The algebra $B(\theta)_0$}\label{sec basic alg}
We now need to distinguish between the two sets of generators for $A_1$ and $A_2$ introduced earlier. We will always do this implicitly though, and keep the notation from the previous section. Also, $\theta\in\operatorname{Irr}(Z)$ will denote a fixed faithful character in this section.
\begin{defi} We define the $k$-algebra $B(\theta)_0=C_{B(\theta)}(kHe_\theta)$.
\end{defi}
As with $B(\theta)$, we will usually denote $B(\theta)_0$ simply by $B_0$. Note that, by Lemma \ref{lem:simples},
\begin{equation}
|\operatorname{Irr}(H|\theta)|=|\operatorname{Irr}(E|\theta\otimes 1_{P_1}\otimes 1_{P_2})|=|\{(1,1)\}|=1
\end{equation}
and so $kHe_{\theta}\cong M_r(k)$. In particular,
\begin{equation}
B \cong B_0\otimes_k kHe_\theta\cong B_0\otimes_k M_r(k),
\end{equation}
where the first isomorphism is given by multiplication. Naturally this shows that $B$ and $B_0$ are Morita equivalent. Moreover, the dimensions of the simple $B_0$-modules are equal to the dimensions of the corresponding simple $B$-modules divided by $r$.
Therefore, since by Lemma~\ref{lem:simples} $\deg(\phi,\psi)=r^2$, for any $\phi\neq 1$ and $\psi\neq 1$, $B_0$ is not basic, but it is sufficiently small for our purposes.
The structure of the algebra $B_0$ described in Definition~\ref{defi twisted tensor} and Proposition~\ref{prop emb ai}~(1)--(4) below is also known as a \emph{twisted tensor product} of $A_1$ and $A_2$, a notion originally introduced in \cite{TwistedTensorOrig} (see also \cite{TwistedTensorModern} for the special type of twisted tensor product that appears in our context).
\begin{defi}\label{defi twisted tensor}
\begin{enumerate}
\item Define a linear map
\begin{equation}
\pi = \pi_\theta:\ B_0 \longrightarrow A_1\otimes_k A_2
\end{equation}
as the restriction of the linear map $kG\phantom{\hspace{0.8pt}} e_\theta\longrightarrow k[(D_1\rtimes P_1)\times (D_2\rtimes P_2)]\cong A_1\otimes_k A_2$ which sends $n\phantom{\hspace{0.8pt}} h\phantom{\hspace{0.8pt}} e_{\theta}$ to $n$ for any $n\in (D_1\rtimes P_1)\times (D_2\rtimes P_2)$ and $h\in L_1\cdot L_2\subset H$ (note that $L_1\cdot L_2$ is not a group).
\item For $i\in \{1,2\}$ let
\begin{equation}
A_i = \bigoplus_{\chi \in \operatorname{Irr} (L_i)} A_i^\chi
\end{equation}
be the decomposition of $A_i$ as an $L_i$-module into isotypical components, i.e. $a^g=\chi(g)\phantom{\hspace{0.8pt}} a$ whenever $a\in A_i^\chi$ and $g\in L_i$. We refer to the elements of
any one of the spaces $A_i^\chi$ as \emph{homogeneous}.
\item For $i\in \{1,2\}$ define the linear map $\iota_i=\iota_{i,\theta}$ as follows:
\begin{equation}
\iota_i:\ A_i \longrightarrow B_0,\ a \mapsto a\phantom{\hspace{0.8pt}} h_{\chi,i}^{-1} \phantom{\hspace{0.8pt}} e_\theta \quad \textrm{for all $a\in A_i^\chi$ and $\chi\in \operatorname{Irr}(L_i)$.}
\end{equation}
\end{enumerate}
\end{defi}
\begin{remark}
We will often use without further mention that $e_1 \in A_i^1$ for $i\in\{1,2\}$. Analogous statements are not true for the other idempotents.
\end{remark}
The next proposition summarises the properties of the maps $\pi$, $\iota_1$ and $\iota_2$, which relate the structure of $B_0$ to that of $A_1\otimes_k A_2$, which we understand completely by \S\ref{sec subalg}. The $\iota_i$ turn out to be actual algebra homomorphisms. The map $\pi$ induces a bijection between $B_0$ and $A_1\otimes_k A_2$. And while $\pi$ is not an algebra isomorphism, it nevertheless shares some of the properties of an algebra isomorphism (e.g. point~\eqref{point radical} of the proposition below would be obvious if $\pi$ were a isomorphism).
\begin{prop}\label{prop emb ai}
\begin{enumerate}
\item The map $\iota_i:\ A_i \hookrightarrow B_0$
is a $k$-algebra homomorphism for $i\in\{1,2\}$.
\item \label{point commutation} For $\chi \in \operatorname{Irr}(L_1)$ and $\eta\in\operatorname{Irr}(L_2)$ we have
\begin{equation}\label{eqn commutation new}
\iota_1(a) \phantom{\hspace{0.8pt}} \iota_2(b) = \theta([h_{\eta,2}, h_{\chi,1}])\phantom{\hspace{0.8pt}} \iota_2(b) \phantom{\hspace{0.8pt}} \iota_1(a) \quad \textrm{for all $a\in A_1^\chi, b\in A_2^\eta$}.
\end{equation}
\item The map $\pi:\ B_0\longrightarrow A_1\otimes_k A_2$ is bijective.
\item\label{point formula pi}
For all $a\in A_1$ and $b\in A_2$ we have
\begin{equation}\label{eqn pi prod homogeneous}
\pi(\iota_1(a)\phantom{\hspace{0.8pt}} \iota_2(b)) = a\otimes b.
\end{equation}
\item \label{point radical} For all $i\geqslant 1$ we have
\begin{equation}
\pi(J^i(B_0))=J^i(A_1\otimes_k A_2)=\sum_{j=1}^i J^j(A_1)\otimes_k J^{i-j}(A_2).
\end{equation}
\item\label{point ideals} Let $a_1,a_2\in A_1$ and $b_1,b_2 \in A_2$ be homogeneous elements. Then
\begin{equation}
\pi( \iota_1(a_1)\iota_2(b_1) \phantom{\hspace{0.8pt}} B_0 \phantom{\hspace{0.8pt}} \iota_1(a_2)\iota_2(b_2)) = (a_1\otimes b_1) \phantom{\hspace{0.8pt}} (A_1\otimes_k A_2) \phantom{\hspace{0.8pt}} (a_2\otimes b_2).
\end{equation}
\end{enumerate}
\end{prop}
\begin{proof}
\begin{enumerate}
\item For $a\in A_i^\chi$ and $b\in A_i^\eta$, we have
\begin{equation}
\iota_i(a)\phantom{\hspace{0.8pt}}\iota_i(b)= a\phantom{\hspace{0.8pt}} h_{\chi,i}^{-1} \phantom{\hspace{0.8pt}} e_{\theta} \phantom{\hspace{0.8pt}} b \phantom{\hspace{0.8pt}} h_{\eta,i}^{-1} \phantom{\hspace{0.8pt}} e_\theta = ab\phantom{\hspace{0.8pt}} h_{\chi,i}^{-1}\phantom{\hspace{0.8pt}} h_{\eta,i}^{-1} \phantom{\hspace{0.8pt}} e_\theta = ab\phantom{\hspace{0.8pt}} h_{\chi\eta,i}^{-1} \phantom{\hspace{0.8pt}} e_\theta = \iota_i(ab),
\end{equation}
using that $h_{\chi,i} h_{\eta,i}=h_{\chi\eta,i}$, as we saw earlier. The above shows that $\iota_i$ is a $k$-algebra homomorphism, since the various $A_i^\chi$ span $A_i$.
\item We have
\begin{equation}
\begin{array}{rcl} \iota_1(a)\phantom{\hspace{0.8pt}}\iota_2(b)&=& a\phantom{\hspace{0.8pt}} h_{\chi,1}^{-1} \phantom{\hspace{0.8pt}} e_{\theta} \phantom{\hspace{0.8pt}} b \phantom{\hspace{0.8pt}} h_{\eta,2}^{-1} \phantom{\hspace{0.8pt}} e_\theta = a\phantom{\hspace{0.8pt}} b^{h_{\chi,1}}\phantom{\hspace{0.8pt}} h_{\chi,1}^{-1}\phantom{\hspace{0.8pt}} h_{\eta,2}^{-1} \phantom{\hspace{0.8pt}} e_\theta\\ &=& \eta({h_{\chi,1}})\phantom{\hspace{0.8pt}} b\phantom{\hspace{0.8pt}} a\phantom{\hspace{0.8pt}} h_{\eta,2}^{-1}\phantom{\hspace{0.8pt}} h_{\chi,1}^{-1}\phantom{\hspace{0.8pt}} \theta([h_{\eta,2},h_{\chi,1}^{-1}]) \phantom{\hspace{0.8pt}} e_\theta= b\phantom{\hspace{0.8pt}} a\phantom{\hspace{0.8pt}} h_{\eta,2}^{-1}\phantom{\hspace{0.8pt}} h_{\chi,1}^{-1} \phantom{\hspace{0.8pt}} e_\theta\\ &=& \chi(h_{\eta,2}^{-1})\phantom{\hspace{0.8pt}} b\phantom{\hspace{0.8pt}} h_{\eta,2}^{-1} \phantom{\hspace{0.8pt}} a \phantom{\hspace{0.8pt}} h_{\chi,1}^{-1} \phantom{\hspace{0.8pt}} e_\theta = \theta([h_{\eta,2}, h_{\chi,1}])\phantom{\hspace{0.8pt}} \iota_2(b)\phantom{\hspace{0.8pt}} \iota_1(a). \end{array} \end{equation}
\item Surjectivity of $\pi$ will follow immediately from point \eqref{point formula pi} below. For injectivity we compare dimensions. The image of $\pi$ has dimension $\dim(A_1)\phantom{\hspace{0.8pt}} \dim(A_2)=|D_1|^2\phantom{\hspace{0.8pt}} |P_1|^2$. On the other hand ${\dim(C_{B(\theta)}(H))\phantom{\hspace{0.8pt}} \dim(kHe_\theta)= \dim(C_{B(\theta)}(H))\phantom{\hspace{0.8pt}} r^2=\dim(B)=|D_1|^2|P_1|^2r^2}$, which shows that $\dim(B_0)=\dim(C_{B(\theta)}(H))= |D_1|^2|P_1|^2$, which is the same as the dimension of the image of $\pi$.
\item It suffices to check formula~\eqref{eqn pi prod homogeneous} for $a$ and $b$ homogeneous. So assume $a\in A_1^\chi$ and $b\in A_2^\eta$. As in the proof of \eqref{point commutation} we have
$
\iota_1(a)\phantom{\hspace{0.8pt}}\iota_2(b)
= a\phantom{\hspace{0.8pt}} b\phantom{\hspace{0.8pt}} h_{\eta,2}^{-1} \phantom{\hspace{0.8pt}} h_{\chi,1}^{-1}\phantom{\hspace{0.8pt}} e_\theta,
$
Since, slightly counter-intuitively, $h_{\chi,1}\in L_2$ and $h_{\eta,2}\in L_1$, we have that $h_{\eta,2}^{-1} \phantom{\hspace{0.8pt}} h_{\chi,1}^{-1}\in L_1\phantom{\hspace{0.8pt}} L_2$.
Therefore, by definition, $\pi$ maps the above element to $a\otimes b$.
\item Recall that $\pi$ was defined as the restriction of a linear map $\hat \pi:\ kG\phantom{\hspace{0.8pt}} e_\theta\longrightarrow k[(D_1\rtimes P_1)\times (D_2\rtimes P_2)]\cong A_1\otimes_k A_2$ which is a homomorphism of left $k[D_1\times D_2]$-modules. In particular, $\hat \pi$ will map $J^i(kG\phantom{\hspace{0.8pt}} e_\theta)$ onto $J^i(A_1\otimes A_2)$ for all $i\geqslant 0$.
This uses that $D_1\times D_2$ is a normal Sylow $\ell$-subgroup of $G$, and therefore $J(kG)=J(k[D_1\times D_2])\phantom{\hspace{0.8pt}} kG$ and an analogous expression for $J(A_1\otimes A_2)$.
Now $J^i(kGe_\theta) = J^i(B_0)\phantom{\hspace{0.8pt}} kHe_\theta$, and by the definition of $\hat \pi$ we have
$\hat \pi(J^i(B_0)\phantom{\hspace{0.8pt}} kHe_\theta) = \pi(J^i(B_0))$, which proves the claim.
\item It suffices to check that
\begin{equation} \pi( \iota_1(a_1)\iota_2(b_1) \phantom{\hspace{0.8pt}} (\iota_1(A_1^\chi)\iota_2(A_2^\eta)) \phantom{\hspace{0.8pt}} \iota_1(a_2)\iota_2(b_2)) = (a_1\otimes b_1) \phantom{\hspace{0.8pt}} (A_1^\chi\otimes_k A_2^\eta )\phantom{\hspace{0.8pt}} (a_2\otimes b_2)
\end{equation}
for all $\chi\in \operatorname{Irr}(L_1)$ and $\eta\in\operatorname{Irr}(L_2)$.
However, by formula~\eqref{eqn commutation new}, the relevant factors in the argument of $\pi$ above commute up to a non-zero scalar (which does not affect the image). Hence the left hand side of the above is equal to
$
\pi(\iota_1(a_1A_1^\chi a_2)\phantom{\hspace{0.8pt}} \iota_2(b_1A_2^\eta b_2)),
$
which equals the right hand side of the above by point~\eqref{point formula pi}.
\qedhere
\end{enumerate}
\end{proof}
\begin{prop}[Explicit formula for $\iota_i$]\label{prop formula iota}
For $\{i,j\}=\{1,2\}$ we have
\begin{equation}\label{eqn sjjsjjssj}
\iota_i(a) = \sum_{g\in L_i} (a\phantom{\hspace{0.8pt}} e_{1_{L_j}} \phantom{\hspace{0.8pt}} e_\theta)^g\quad \textrm{for all $a\in A_i=k[D_i\rtimes P_i]$.}
\end{equation}
\end{prop}
\begin{proof}
Assume without loss of generality that $i=1$ and $j=2$.
It suffices to prove this for $a\in A_1^\chi$ for a fixed $\chi\in \operatorname{Irr}(L_1)$. Note the idempotent $e_{1_{L_2}}$ can also be written as $r^{-1}\phantom{\hspace{0.8pt}} \sum_{\eta\in \operatorname{Irr}(L_1)} h_{\eta,1}$. Now by character orthogonality (also using $h_{\eta,1} =[h_{\eta,1}, g^{-1}] \phantom{\hspace{0.8pt}} h_{\eta,1}^g$),
\begin{equation}
\pi\left(\sum_{g\in L_1} a^g\phantom{\hspace{0.8pt}} h_{\eta,1}^g \phantom{\hspace{0.8pt}} e_{\theta}\right)= \pi\left(\sum_{g\in L_1} \chi(g)\eta(g)\phantom{\hspace{0.8pt}} a \phantom{\hspace{0.8pt}} h_{\eta,1} \phantom{\hspace{0.8pt}} e_\theta\right) = \left\{\begin{array}{ll} 0 & \textrm{if $\chi\neq \eta^{-1}$}\\ r\phantom{\hspace{0.8pt}} a\otimes 1 & \textrm{if $\chi = \eta^{-1}$} \end{array},\right.
\end{equation}
for all $\eta\in\operatorname{Irr}(L_1)$. As $\pi(\iota_1(a))= a\otimes 1$, summing over all such $\eta$ gives that $\pi$ applied to both sides of \eqref{eqn sjjsjjssj} holds true. Since $\pi$ is bijective, the result follows.
\end{proof}
By the above, the algebra $B_0=B(\theta)_0$ can be thought of as being graded by $\operatorname{Irr}(P_1)\times \operatorname{Irr}(P_2)$, and the character $\theta$ determines how the homogeneous components commute through equation~\eqref{eqn commutation new}. We will ultimately recover $\theta$ from $B_0$ by showing that certain subspaces of the homogeneous components are preserved under isomorphisms modulo $J^3(B_0)$. However, we will proceed in a more elementary way, and for that we will need idempotents and certain arrows explicitly.
\begin{defi} \label{defi eps s}
\begin{enumerate}
\item For $\phi \in \operatorname{Irr}(P_1),\psi\in\operatorname{Irr}(P_2)$ set
\begin{equation}
\varepsilon_{(\phi,1)}=\iota_1(e_{\phi})\iota_2(e_1)\quad\textrm{and}\quad
\varepsilon_{(1,\psi)}=\iota_1(e_1)\iota_2(e_\psi).
\end{equation}
For $\phi\in\operatorname{Irr}(P_1)\setminus\{1\}, \psi\in \operatorname{Irr}(P_2)\setminus\{1\}$
set
\begin{equation}
\varepsilon_{(\phi,\psi)} =\iota_1(e_{[\phi]}) \iota_2(e_{[\psi]}),\quad\textrm{where}\ e_{[\psi]}= \sum_{g\in L_i} e_{\psi^g}\in A_i^1.
\end{equation}
\item For $\psi,\phi\in \operatorname{Irr}(P_1)$, $\xi,\zeta\in \operatorname{Irr}(P_2)$ such that $\phi\neq 1$, $\zeta\neq 1$ set
\begin{equation}
S_{\psi,\phi} = \iota_1(s_{\psi,\phi})\iota_2(e_1) \quad \textrm{ and }\quad T_{\xi,\zeta} = \iota_1(e_1)\iota_2(s_{\xi,\zeta}).
\end{equation}
\item For $1\neq \phi\in \operatorname{Irr}(P_1)$, $1\neq \zeta\in \operatorname{Irr}(P_2)$, $\chi\in \operatorname{Irr}(L_1)$ and $\eta\in \operatorname{Irr}(L_2)$ set
\begin{equation}
\tilde S_\phi^\chi = \sum_{g\in L_1} \chi(g^{-1}) S_{1,\phi^g}S_{\phi^g, (\phi^{-1})^g}\quad\textrm{and}\quad
\tilde T_\zeta^\eta = \sum_{g\in L_2} \eta(g^{-1}) T_{1,\zeta^g}T_{\zeta^g, (\zeta^{-1})^g}.
\end{equation}
\end{enumerate} \end{defi}
\begin{remark}\label{remark iota hom}
Note that by Proposition~\ref{prop emb ai}~\eqref{point commutation}, for $\{i,j\}=\{1,2\}$, the element $\iota_j(e_1)$, and more generally every element of $\iota_j(A_j^1)$, commutes with every element of $\iota_i(A_i)$.
It follows that the $\varepsilon_{(\psi,\phi)}$ are idempotents, and
\begin{equation}\label{eqn s arrows}
S_{\psi,\phi} = \varepsilon_{(\psi,1)}\phantom{\hspace{0.8pt}} S_{\psi,\phi}\phantom{\hspace{0.8pt}} \varepsilon_{(\psi\phi,1)} \quad\textrm{and}\quad
T_{\xi,\zeta} = \varepsilon_{(\xi,1)}\phantom{\hspace{0.8pt}} T_{\xi,\zeta}\phantom{\hspace{0.8pt}} \varepsilon_{(\xi\zeta,1)}.
\end{equation}
Moreover, it follows that
\begin{equation}
S_{\psi_1,\phi_1}\cdots S_{\psi_m,\phi_m} = \iota_1(s_{\psi_1,\phi_1}\cdots s_{\psi_m,\phi_m})\iota_2(e_1)\quad\textrm{ for any $m> 0$,}
\end{equation}
\begin{equation}
S_{\psi_1,\phi_1}\cdots S_{\psi_m,\phi_m} T_{\xi_1,\zeta_1}\cdots T_{\xi_n,\zeta_n} =
\iota_1(s_{\psi_1,\phi_1}\cdots s_{\psi_m,\phi_m}e_1)\iota_2(e_1s_{\xi_1,\zeta_1}\cdots s_{\xi_n,\zeta_n})\quad\textrm{for any $m,n> 0$,}
\end{equation}
for any admissible choice of $\psi_i$, $\phi_i$, $\xi_i$ and $\zeta_i$. This will be useful since the image of the right hand side under $\pi$ can readily be determined using Proposition~\ref{prop emb ai}~\eqref{point formula pi}. \end{remark}
\begin{prop}\label{prop idempotents}
If $(\psi,\phi)$ and $(\psi',\phi')$ label two different Brauer characters of $B$ (using the notation of \S\ref{sec simples}), then $\varepsilon_{(\phi,\psi)}$ and $\varepsilon_{(\phi',\psi')}$ are distinct orthogonal idempotents. The idempotent $\varepsilon_{(\phi,\psi)}$ annihilates all simple $B_0$-modules except for the one corresponding to the character $(\phi,\psi)$. In particular, the idempotents $\varepsilon_{(\psi,1)}$ and $\varepsilon_{(1,\phi)}$ are primitive in $B_0$ for any $\psi\in\operatorname{Irr}(P_1), \phi\in\operatorname{Irr}(P_2)$. \end{prop} \begin{proof}
Let $\phi\in\operatorname{Irr}(P_1)$ and $\psi\in\operatorname{Irr}(P_2)$. Using Proposition~\ref{prop formula iota} and the fact that $\iota_2(e_1)=e_{1_{P_2}}$ (immediately from the definition~of~$\iota_2$) we get
\begin{equation}
\varepsilon_{(\phi,1)} = \iota_1(e_\phi) \phantom{\hspace{0.8pt}} \iota_2(e_1)=
\sum_{g\in L_1} (e_\phi \phantom{\hspace{0.8pt}} e_{1_{L_2}} \phantom{\hspace{0.8pt}} e_{\theta})^g \phantom{\hspace{0.8pt}} e_{1_{P_2}} = \sum_{g\in L_1} (e_\phi \phantom{\hspace{0.8pt}} e_{1_{P_2\rtimes L_2}} \phantom{\hspace{0.8pt}} e_{\theta})^g.
\end{equation}
The idempotent on the right hand side is clearly the central-primitive idempotent in $kE$ which belongs to the induced module given in equation~\eqref{algn:lab_char}. That is, $\varepsilon_{(\phi,1)}$ is the idempotent attached to the character $(\phi,1)$. By the same argument $\varepsilon_{(1,\psi)}$ belongs to the character $(1,\psi)$. If $\phi\neq 1$, $\psi\neq 1$ then, using only the definition of $\iota_i$ this time,
\begin{equation}
\varepsilon_{(\phi,\psi)}=\iota_1(e_{[\phi]})\phantom{\hspace{0.8pt}} \iota_2(e_{[\psi]}) = \sum_{g\in L_1}e_{\phi^g}\phantom{\hspace{0.8pt}} e_\theta \phantom{\hspace{0.8pt}} \sum_{h\in L_2} e_{\psi^h} \phantom{\hspace{0.8pt}} e_\theta = \sum_{g\in L_1} \sum_{h\in L_2} (e_{\phi} \phantom{\hspace{0.8pt}} e_{\psi}\phantom{\hspace{0.8pt}} e_\theta)^{gh},
\end{equation}
which is clearly the central-primitive idempotent in $kE$ which belongs to the character $(\phi,\psi)$.
The distinctness and orthogonality of the various $\varepsilon_{(\phi,\psi)}$'s follows immediately from the fact that these are central-primitive idempotents belonging to distinct simple modules (albeit in the algebra $kE$). As each simple $kE$-module gives rise to a simple $kG$-module, whose restriction to $B_0$ is a direct sum of copies of a single simple module (since $B_0$ and $B$ are naturally Morita equivalent), the claim regarding the action of these idempotents on the simple $B_0$-modules follows as well. It is also clear that the $\varepsilon_{(\phi,1)}$ and $\varepsilon_{(1,\psi)}$ are primitive, as the corresponding simple $B_0$-modules are one-dimensional. \end{proof}
\begin{lemma}\label{lemma arrow properties}
\begin{enumerate}
\item\label{point ext} For all $\psi,\phi\in\operatorname{Irr}(P_1)$ and $\xi,\zeta\in\operatorname{Irr}(P_2 )$ with $\phi\neq 1$ and $\zeta\neq 1$,
\begin{eqnarray}
\varepsilon_{(\psi,1)}(J(B_0)/J^2(B_0))\varepsilon_{(\psi\phi,1)}&=&\langle S_{\psi,\phi}\rangle_k+J^2(B_0),\\
\varepsilon_{(1,\xi)}(J(B_0)/J^2(B_0))\varepsilon_{(1,\xi\zeta)}&=&\langle T_{\xi,\zeta}\rangle_k+J^2(B_0),
\end{eqnarray}
and all of these spaces are one-dimensional. That is, the $S_{\psi,\phi}$ and $T_{\xi,\zeta}$ correspond to arrows in the quiver of $B_0$. Moreover,
\begin{equation}
\varepsilon_{(\psi,1)}(J(B_0)/J^2(B_0))\varepsilon_{(\psi,1)} =0
\quad \textrm{and}\quad
\varepsilon_{(1,\xi)}(J(B_0)/J^2(B_0))\varepsilon_{(1,\xi)}=0.
\end{equation}
\item\label{point all psi equal} Let $\phi_1,\dots,\phi_\ell\in\operatorname{Irr}(P_1)\setminus\{1\}$ such that $\phi_1\phantom{\hspace{0.8pt}}\ldots\phantom{\hspace{0.8pt}}\phi_\ell=1$. Then
\begin{equation}
S_{1,\phi_1}S_{\phi_1,\phi_2}\cdots S_{\prod_{i=1}^{\ell-1}\phi_i,\phi_\ell}\in J^{\ell+1}(B_0)
\end{equation}
if and only if $\phi_1=\dots=\phi_\ell$. An analogous statement holds for the $T_{\xi,\zeta}$'s.
\item\label{point lin indep} The sets
\begin{equation}
\left\{S_{1,\phi}S_{\phi,\phi^{-1}}\right\}_{\phi\in\operatorname{Irr}(P_1)\setminus\{1\}} \quad\textrm{and}\quad \left\{S_{1,\phi}S_{\phi,\phi^{-1}}T_{1,\zeta}T_{\zeta,\zeta^{-1}}\right\}_{\phi\in\operatorname{Irr}(P_1)\setminus\{1\}, \zeta\in\operatorname{Irr}(P_2)\setminus\{1\}}
\end{equation}
are linearly independent modulo $J^3(B_0)$ and $J^5(B_0)$, respectively. An analogous statement to the first one holds for the $T_{\xi,\zeta}$'s.
\item\label{point schi comm} For any $1\neq \phi\in \operatorname{Irr}(P_1)$, $1\neq \zeta\in \operatorname{Irr}(P_2)$, $\chi\in \operatorname{Irr}(L_1)$ and $\eta\in \operatorname{Irr}(L_2)$
\begin{equation}
\tilde S^\chi_{\phi} \tilde T^\eta_{\zeta} = \theta([h_{\eta,2}, h_{\chi,1}]) \phantom{\hspace{0.8pt}} \tilde T^\eta_{\zeta} \tilde S^\chi_{\phi}.
\end{equation}
\end{enumerate} \end{lemma} \begin{proof}
\begin{enumerate}
\item
First of all note that ``$\supseteq$'' is clear by equation~\eqref{eqn s arrows}. By Proposition~\ref{prop emb ai}~\eqref{point formula pi} it follows that $\pi(S_{\psi,\phi})=s_{\psi,\phi}\otimes e_1$, and this element is not contained in $J^2(A_1\otimes_k A_2)$. Hence $S_{\psi,\phi}\not\in J^2(B_0)$ by Proposition~\ref{prop emb ai}~\eqref{point radical}. That is, the spaces on the right hand side are all one-dimensional.
By definition we have
\begin{equation}
\sum_{\psi\in\operatorname{Irr}(P_1)} \varepsilon_{(\psi,1)} = \iota_2(e_1).
\end{equation}
Since the other inclusion is already known, to prove ``$\subseteq$'' it will suffice to show that $\iota_2(e_1) (J(B_0)/J^2(B_0)) \iota_2(e_1)$ (which contains all of the $\varepsilon_{(\psi,1)}(J(B_0)/J^2(B_0))\varepsilon_{(\psi\phi,1)}$)) is spanned by elements of the form $S_{\psi,\phi} +J^2(B_0)$. Since $\pi$ is bijective we may as well consider the images under $\pi$. Using Proposition~\ref{prop emb ai}~\eqref{point radical}~and~\eqref{point ideals}, as well as $\pi(\iota_2(e_1))=1\otimes e_1$, we have
\begin{equation}
\begin{array}{rl}
&\pi( \iota_2(e_1)(J(B_0)/J^2(B_0))\iota_2(e_1))\\
=&\left[J(A_1)/J^2(A_1)\otimes_k e_1(A_2/J(A_2))e_1\right] \oplus \left[A_1/J(A_1)\otimes_k e_1(J(A_2)/J^2(A_2))e_1\right],
\end{array}
\end{equation}
which is spanned by elements of the form $s_{\psi,\phi}\otimes e_1 + J^2(A_1\otimes_k A_2)=\pi(S_{\psi,\phi} +J^2(B_0))$, since the second bracket is zero. This proves the first claim.
The second claim follows from the fact that, just like the other spaces we considered, $\varepsilon_{(\phi,1)} (J(B_0)/J^2(B_0)) \varepsilon_{(\phi,1)}$ is contained in $\iota_2(e_1) (J(B_0)/J^2(B_0)) \iota_2(e_1)$. We saw that the latter is spanned by the $S_{\mu,\nu}+J^2(B_0)$. But $\varepsilon_{(\phi,1)} S_{\mu,\nu}\varepsilon_{(\phi,1)}=0$ for all choices of $\mu,\nu$.
\item By Proposition~\ref{prop emb ai}~\eqref{point formula pi} (and Remark~\ref{remark iota hom}) we have
\begin{equation}
\pi(S_{1,\phi_1}S_{\phi_1,\phi_2}\cdots S_{\prod_{i=1}^{\ell-1}\phi_i,\phi_\ell}) = s_{1,\phi_1}s_{\phi_1,\phi_2}\cdots s_{\prod_{i=1}^{\ell-1}\phi_i,\phi_\ell}\otimes e_1.
\end{equation}
By our knowledge of the basis of $A_1$ the right hand side is contained in $J^{\ell + 1}(A_1\otimes_k A_2)$ if and only if $\phi_1=\ldots=\phi_\ell$. Now the assertion follows by Proposition~\ref{prop emb ai}~\eqref{point radical}.
\item By Proposition~\ref{prop emb ai}~\eqref{point formula pi} (and Remark~\ref{remark iota hom}) we have
\begin{equation}
\pi(S_{1,\phi}S_{\phi,\phi^{-1}}) = s_{1,\phi}s_{\phi,\phi^{-1}}\otimes e_1,\quad
\pi(S_{1,\phi}S_{\phi,\phi^{-1}}T_{1,\zeta}T_{\zeta,\zeta^{-1}}) = s_{1,\phi}s_{\phi,\phi^{-1}}\otimes s_{1,\zeta}s_{\psi,\zeta^{-1}},
\end{equation}
and by our knowledge of the bases of $A_1$ and $A_2$ these elements are linearly independent modulo $J^3(A_1\otimes_k A_2)$ and $J^5(A_1\otimes_k A_2)$, respectively. Our claim follows using Proposition~\ref{prop emb ai}~\eqref{point radical}.
\item By Remark~\ref{remark iota hom} we see that
\begin{equation}
\tilde S^\chi_{\phi} = \iota_1\left(\sum_{g\in L_1} \chi(g^{-1}) s_{1,\phi^g} s_{\phi^g,(\phi^{-1})^g} \right)\iota_2(e_1) \in \iota_1(e_1A_1^\chi e_1)\iota_2(e_1),
\end{equation}
that is, $\tilde S^\chi_{\phi}=\iota_1(\tilde s^\chi_{\phi})\iota_2(e_1)$ for some $\tilde s^\chi_{\phi} \in e_1A_1^\chi e_1$. Analogously we have $\tilde T^\eta_{\zeta}=\iota_1(e_1)\iota_2(\tilde t^\eta_{\zeta})$ for some $\tilde t^\eta_{\zeta} \in e_1A_2^\eta e_1$. Hence
\begin{equation}
\tilde S^\chi_{\phi} \tilde T^\eta_{\zeta}=\iota_1(\tilde s^\chi_{\phi})\iota_2(\tilde t^\eta_{\zeta})\quad \textrm{and}\quad \tilde T^\eta_{\zeta}\tilde S^\chi_{\phi}=\iota_2(\tilde t^\eta_{\zeta})\iota_1(\tilde s^\chi_{\phi}).
\end{equation}
The statement now follows directly from Proposition~\ref{prop emb ai}~\eqref{point commutation}.\qedhere
\end{enumerate} \end{proof}
\section{Morita equivalences}\label{sec Mor equ}
From now on let $\theta, \theta'\in \operatorname{Irr}(Z)$ be two faithful characters. We keep all other notation from the previous section. The equivalent in $B_0'=B(\theta')_0$ of the various elements given in Definition~\ref{defi eps s} will be denoted with a prime, e.g. $\varepsilon_{(\phi,\psi)}'$, $S_{\psi,\phi}'$, $(\tilde S^\chi _{\phi})'$, and so on. We will show under which conditions $B_0$ and $B_0'$ are Morita equivalent. This will immediately enable us to prove the main theorem of this paper.
\begin{prop}\label{prop e11 is special}
\begin{enumerate}
\item\label{point dim preserved} A Morita equivalence between $B_0$ and $B'_0$ preserves the dimensions of simple modules. In particular, any such Morita equivalence is afforded by an isomorphism.
\item An isomorphism $\tau:\ B_0\longrightarrow B'_0$ can be modified by an inner automorphism such that
\begin{equation}
\left\{\sum_{\phi \in \operatorname{Irr}(P_1)} \tau(\varepsilon_{(\phi,1)}), \sum_{\psi \in \operatorname{Irr}(P_2)} \tau(\varepsilon_{(1,\psi)})\right\}=\left\{\sum_{\phi \in \operatorname{Irr}(P_1)} \varepsilon'_{(\phi,1)}, \sum_{\psi \in \operatorname{Irr}(P_2)} \varepsilon'_{(1,\psi)}\right\}.
\end{equation}
\end{enumerate} \end{prop}
\begin{proof}
We will prove this by finding distinguishing Morita invariant properties of the simple modules in $B_0$ and the attached idempotents. By definition we have $\pi(\varepsilon_{(\phi,\psi)})=e_{[\phi]}\otimes e_{[\psi]}$ if $\phi\neq 1$ and $\psi\neq 1$. Using Proposition~\ref{prop emb ai}~\eqref{point radical}~and~\eqref{point ideals}, we get
\begin{equation}
\begin{array}{rcll}
\pi(\varepsilon_{(\phi,\psi)}\phantom{\hspace{0.8pt}} J(B_0)/J^2(B_0)\phantom{\hspace{0.8pt}} \varepsilon_{(\phi,\psi)})&=&&e_{[\phi]}(J(A_1)/J^2(A_1))e_{[\phi]} \otimes_k e_{[\psi]}(A_2/J(A_2))e_{[\psi]} \\
&&\oplus& e_{[\phi]}(A_1/J(A_1))e_{[\phi]} \otimes_k e_{[\psi]}(J(A_2)/J^2(A_2))e_{[\psi]},
\end{array}
\end{equation}
which is clearly non-zero (e.g. $s_{\phi, \phi^{-1}\phi^g}\otimes e_{[\psi]}$ gives a non-trivial on the right hand side for any $1\neq g\in L_1$). In particular the simple modules belonging to the characters of the form $(\phi,\psi)$ all have non-trivial self-extensions. This implies the first assertion since by Lemma~\ref{lemma arrow properties}~\eqref{point ext} the other simples do not have non-trivial self-extensions.
From Lemma~\ref{lemma arrow properties}~\eqref{point ext} we already know that the ${\operatorname{Ext}}^1$ between two simple modules labeled by $(\phi,1)$ and $(\phi',1)$ is one-dimensional if $\phi \neq \phi'\in\operatorname{Irr}(P_1)$. The analogous statement holds for the simples labeled by $(1,\zeta)$ and $(1,\zeta')$. It therefore suffices to show that there are no non-trivial extensions between the simples labeled $(\phi,1)$ and $(1,\zeta)$, where $1\neq \phi \in \operatorname{Irr}(P_1)$ and $1\neq \zeta\in\operatorname{Irr}(P_2)$. The sum of the $\varepsilon_{(\phi,1)}$ for $\phi\neq 1$ is equal to
$f_1=\iota_1(1-e_1)\iota_2(e_1)$, and, analogously, the sum of the $\varepsilon_{(1,\zeta)}$ for $\zeta\neq 1$ is equal to $f_2=\iota_1(e_1)\iota_2(1-e_1)$. Note that $f_1$ and $f_2$ are suitable for application of Proposition~\ref{prop emb ai}~\eqref{point ideals}, and we have $\pi(f_1)=(1-e_1)\otimes e_1$ and $\pi(f_2)=e_1\otimes(1-e_1)$. Hence
\begin{equation}
\begin{split}
\pi(\varepsilon_{(\phi,1)}J(B_0)\varepsilon_{(1,\zeta)})
\subseteq ((1-e_1)\otimes e_1) J(A_1\otimes_k A_2)(e_1\otimes (1-e_1))\\
= (1-e_1)J(A_1)e_1\otimes_k e_1 J(A_2)(1-e_1)\subseteq J^2(A_1\otimes_k A_2).
\end{split}
\end{equation}
It follows that $\varepsilon_{(\phi,1)}(J(B_0)/J^2(B_0))\varepsilon_{(1,\zeta)}=0$, that is, there are no non-trivial extensions between the corresponding simple modules.
\end{proof}
\begin{remark}\label{remark complete set}
From Proposition~\ref{prop idempotents} and Proposition~\ref{prop e11 is special}~(2) it follows that
a $\tau$ as in Proposition~\ref{prop e11 is special}~(2) will, up to an inner automorphism, satisfy either
\begin{equation}
\tau(\varepsilon_{(\phi,1)})=\varepsilon'_{(\sigma(\phi),1)},\quad \textrm{or} \quad \tau(\varepsilon_{(\phi,1)})=\varepsilon'_{(1,\sigma(\phi))},
\end{equation}
for a bijective map $\sigma$ from $\operatorname{Irr}(P_1)$ to either $\operatorname{Irr}(P_1)$ or $\operatorname{Irr}(P_2)$.
The analogous statement holds for the $\varepsilon_{(1,\zeta)}$, and thus, in particular, $\tau(\varepsilon_{(1,1)})=\varepsilon'_{(1,1)}$.
\end{remark}
\begin{prop}\label{prop automorphisms rigid}
Let $\tau:\ B_0\longrightarrow B_0'$ be an isomorphism
such that
\begin{equation}
\tau(\varepsilon_{(\phi,1)}) = \varepsilon'_{(\sigma(\phi),1)}\quad\textrm{ for a bijective map $\sigma:\ \operatorname{Irr}(P_1)\longrightarrow \operatorname{Irr}(P_1)$},
\end{equation}
and $\sigma(1)=1$. Then $\sigma$ is a group automorphism.
\end{prop}
\begin{proof}
All we need to show is that $\sigma(\zeta^i)=\sigma(\zeta)^i$ for all $i\geqslant 0$ for some arbitrary generator $\zeta\in \operatorname{Irr}(P_1)$. For $i< 2$ this is clear.
By way of induction we may assume that $\sigma(\zeta^j)=\sigma(\zeta)^j$ for all $j< i$, and $i\geqslant 2$.
Now
for any $\psi\in \operatorname{Irr}(P_1)$ and $1\neq \phi \in \operatorname{Irr}(P_1)$ we can write (using Lemma~\ref{lemma arrow properties}~\eqref{point ext})
\begin{equation}
\tau(S_{\psi,\phi})+J^2(B_0')= c_{\psi,\phi}\phantom{\hspace{0.8pt}} S'_{\sigma(\psi), \sigma(\psi)^{-1}\sigma(\psi\phi)}+ J^2(B_0')\quad\textrm{for some $c_{\psi,\phi}\in k^\times$.}
\end{equation}
By applying $\tau$ to the element from Lemma~\ref{lemma arrow properties}~\eqref{point all psi equal} it follows that
\begin{equation}\label{eqn rrksks}
S'_{\sigma(\psi), \sigma(\psi)^{-1}\sigma(\psi\phi)}S'_{\sigma(\psi\phi), \sigma(\psi\phi)^{-1}\sigma(\psi\phi^2)} \cdots S'_{\sigma(\psi\phi^{\ell-1}), \sigma(\psi\phi^{\ell-1})^{-1}\sigma(\psi\phi^\ell)} \in J^{\ell+1}(B_0'),
\end{equation}
again for any $\psi\in \operatorname{Irr}(P_1)$ and $1\neq \phi \in \operatorname{Irr}(P_1)$. By Lemma~\ref{lemma arrow properties}~\eqref{point all psi equal} all second indices occurring in \eqref{eqn rrksks}, that is, all $\sigma(\psi\phi^q)^{-1}\sigma(\psi\phi^{q+1})$ for $0\leqslant q <\ell$, must be equal for the element to be contained in $J^{\ell+1}(B_0')$. In particular, if we specialise $\psi=\zeta^{i-2}$, $\phi=\zeta$ and look at $q=0$ and $q=1$, we get
\begin{equation}
\sigma(\zeta^{i-2})^{-1}\sigma(\zeta^{i-1})=\sigma(\zeta^{i-1})^{-1}\sigma(\zeta^{i}).
\end{equation}
The left hand side is equal to $\sigma(\zeta)$ by the induction hypothesis, which implies that
$\sigma(\zeta^i)=\sigma(\zeta^{i-1})\phantom{\hspace{0.8pt}} \sigma(\zeta)=\sigma(\zeta)^i$, which completes the induction step.
\end{proof}
Of course the analogue of the above statement with $(\phi,1)$ swapped for $(1,\zeta)$ holds as well.
\begin{prop}\label{prop non isom}
The block $B({\theta})$ is Morita equivalent to $B({\theta'})$ if and only if $\theta'=\theta^{\pm 1}$. \end{prop}
\begin{proof}
We first note that if $\theta'=\theta^{\pm 1}$, then, by Lemma \ref{prop isoms}~(2), $B$ is Morita equivalent to $B'$.
Conversely, suppose $B$ is Morita equivalent to $B'$. Of course this implies that $B_0$ and $B'_0$ are Morita equivalent. Moreover, by Proposition \ref{prop e11 is special}~\eqref{point dim preserved}, any such Morita equivalence must preserve the dimensions of the simple modules and so we may assume that it is induced by a $k$-algebra isomorphism $\tau:B_0\to B'_0$.
By Remark~\ref{remark complete set} we may assume, after pre-composing with an inner automorphism and the isomorphism from Lemma~\ref{prop isoms}~(2) (in which case we replace $\theta$ by $\theta^{-1}$), that
$\tau(\varepsilon_{(\phi,1)}) = \varepsilon'_{(\sigma_1(\phi),1)}$ and $\tau(\varepsilon_{(1,\zeta)}) = \varepsilon'_{(1,\sigma_2(\zeta))}$ for maps $\sigma_i:\ \operatorname{Irr}(P_i)\longrightarrow \operatorname{Irr}(P_i)$. Furthermore, by Proposition~\ref{prop automorphisms rigid} we may assume that the $\sigma_i$ are group automorphisms of $\operatorname{Irr}(P_i)$. Certainly every group automorphism of $\operatorname{Irr}(P_i)$ is induced by one of $P_i$ and so, possibly after pre-composing $\tau$ with an automorphism as in Lemma \ref{prop isoms}~(1), we may assume that both $\sigma_i$ are the identity. That is, we may assume
\begin{equation}\label{eqn idempot preserved}
\tau(\varepsilon_{(\phi,1)})=\varepsilon'_{(\phi,1)},\quad \tau(\varepsilon_{(1,\zeta)})=\varepsilon'_{(1,\zeta)} \quad \textrm{for any $\phi\in\operatorname{Irr}(P_1)$ and $\zeta\in\operatorname{Irr}(P_2)$}.
\end{equation}
From now on we fix a $\phi\in \operatorname{Irr}(P_1)\setminus\{1\}$ and $\zeta\in \operatorname{Irr}(P_2)\setminus\{1\}$ and define the spaces
\begin{align*}
\boldsymbol{S}_\phi&=\langle \tilde{S}_\phi^\chi|\chi\in\operatorname{Irr}(L_1)\rangle_k=\langle S_{1,\phi^h}S_{\phi^h,(\phi^h)^{-1}}|h\in L_1\rangle_k,\\
\boldsymbol{T}_\zeta&=\langle \tilde{T}_\zeta^\eta|\eta\in\operatorname{Irr}(L_2)\rangle_k=\langle T_{1,\zeta^h}T_{\zeta^h,(\zeta^h)^{-1}}|h\in L_2\rangle_k.
\end{align*}
The first assertion of Lemma~\ref{lemma arrow properties}~\eqref{point lin indep} gives that $\boldsymbol{S}_\phi$ and $\boldsymbol{T}_\zeta$ both have dimension $r$ modulo $J^3(B_0)$, with bases $(\tilde{S}_\phi^\chi+J^3(B_0))_{\chi\in\operatorname{Irr}(L_1)}$ and $(\tilde{T}_\zeta^\eta+J^3(B_0))_{\eta\in\operatorname{Irr}(L_2)}$ respectively. From the second assertion of Lemma~\ref{lemma arrow properties}~\eqref{point lin indep} and Lemma~\ref{lemma arrow properties}~\eqref{point schi comm}, we get that
\begin{equation}\label{algn comm S}
\begin{array}{rl}
\{x\in\boldsymbol{S}_\phi+J^3(B_0)|&xy-yx\in J^5(B_0)\text{ for all }y\in \boldsymbol{T}_\zeta+J^3(B_0)\}\\
=&\langle \tilde{S}_\phi^{1}\rangle_k+J^3(B_0)
\end{array}
\end{equation}
and similarly
\begin{equation}\label{algn comm T}
\begin{array}{rl}
\{x\in\boldsymbol{T}_\zeta+J^3(B_0)|&xy-yx\in J^5(B_0)\text{ for all }y\in \boldsymbol{S}_\phi+J^3(B_0)\}\\
=&\langle \tilde{T}_\zeta^{1}\rangle_k+J^3(B_0).
\end{array}
\end{equation}
Of course, the analogous assertions for the algebra $B'_0$ hold as well.
Equation \eqref{eqn idempot preserved} combined with Lemma~\ref{lemma arrow properties}~\eqref{point ext} implies that
\begin{align}\label{algn:ST mod J2}
\begin{split}
\tau(\langle S_{\mu,\nu}\rangle_k)+J^2(B_0)&=\langle S'_{\mu,\nu}\rangle_k+J^2(B'_0),\\
\tau(\langle T_{\gamma,\delta}\rangle_k)+J^2(B_0)&=\langle T'_{\gamma,\delta}\rangle_k+J^2(B'_0),
\end{split}
\end{align}
for all $\mu,\nu\in\operatorname{Irr}(P_1)$ and $\gamma,\delta\in\operatorname{Irr}(P_2)$, $\nu\neq 1$ and $\delta \neq 1$. Therefore, by Lemma~\ref{lemma arrow properties}~\eqref{point lin indep},
\begin{align*}
\tau(S_{1,\phi^g}S_{\phi^g,(\phi^g)^{-1}})+J^3(B_0)&=u_{g}\phantom{\hspace{0.8pt}} S'_{1,\phi^g}S'_{\phi^g,(\phi^g)^{-1}}+J^3(B'_0),\\
\tau(T_{1,\zeta^h}T_{\zeta^h,(\zeta^h)^{-1}})+J^3(B_0)&=v_{h}\phantom{\hspace{0.8pt}} T'_{1,\zeta^h}T'_{\zeta^h,(\zeta^h)^{-1}}+J^3(B'_0),
\end{align*}
for all $g\in L_1$, $h\in L_2$ and uniquely determined $u_{g},v_{h}\in k^\times$. Furthermore, \eqref{algn comm S} and \eqref{algn comm T} give
\begin{align*}
\tau(\langle \tilde S_\phi^{1}\rangle_k)+J^3(B_0)=\langle (\tilde S_\phi^{1})'\rangle_k+J^3(B'_0),\\
\tau(\langle \tilde T_\zeta^{1}\rangle_k)+J^3(B_0)=\langle (\tilde T_\zeta^{1})'\rangle_k+J^3(B'_0).
\end{align*}
Therefore, all the $u_{g}$'s are equal, say $u$. Similarly we set $v$ to be the common value of the $v_{h}$'s. In particular,
\begin{align*}
\tau(\tilde{S}_\phi^\chi)+J^3(B_0)&=u\phantom{\hspace{0.8pt}} (\tilde{S}_\phi^\chi)'+J^3(B'_0),\\
\tau(\tilde{T}_\zeta^\eta)+J^3(B_0)&=v\phantom{\hspace{0.8pt}} (\tilde{T}_\zeta^\eta)'+J^3(B'_0),
\end{align*}
for all $\chi\in\operatorname{Irr}(L_1)$ and $\eta\in\operatorname{Irr}(L_2)$. From Lemma~\ref{lemma arrow properties}~\eqref{point schi comm} we get the identities
\begin{equation}
\tilde S^\chi_{\phi} \tilde T^\eta_{\zeta} = \theta([h^\theta_{\eta,2}, h^\theta_{\chi,1}]) \phantom{\hspace{0.8pt}} \tilde T^\eta_{\zeta} \tilde S^\chi_{\phi}\quad\textrm{and}\quad(\tilde S^\chi_{\phi})' (\tilde T^\eta_{\zeta})' = \theta'([h^{\theta'}_{\eta,2}, h^{\theta'}_{\chi,1}]) \phantom{\hspace{0.8pt}} (\tilde T^\eta_{\zeta})' (\tilde S^\chi_{\phi})'
\end{equation}
Using the second assertion of Lemma~\ref{lemma arrow properties}~\eqref{point lin indep} we can apply $\tau$ to the first and compare to the second modulo $J^5(B_0')$. That gives
\begin{equation}
\eta(h^\theta_{\chi,1})=\theta([h^\theta_{\eta,2}, h^\theta_{\chi,1}])=\theta'([h^{\theta'}_{\eta,2} h^{\theta'}_{\chi,1}])=\eta(h_{\chi,1}^{\theta'}),
\end{equation}
for all $\chi\in\operatorname{Irr}(L_1)$ and $\eta\in\operatorname{Irr}(L_2)$. Therefore, $h^\theta_{\chi,1}=h^{\theta'}_{\chi,1}$, for all $\chi\in\operatorname{Irr}(L_1)$. Finally, since $[L_1,L_2]=Z$, it follows from Definition~\ref{def h chi} that $\theta=\theta'$. \end{proof}
\begin{thm}\label{thm:main}
Let $\ell$ be a prime and $n\in\mathbb N$. Then there exists an $\ell$-block $B$ of $kG$, for a finite group $G$, such that $\operatorname{mf}(B)=n$.
\end{thm}
\begin{proof}
Of course, we are setting to $G$ and $B$ to be as in the rest of the article. We just need an appropriate choice of $p$ and $r$ and $\theta$.
Set $r=\ell^n+1$. By the Dirichlet prime number theorem, we can set $p$ to be a prime congruent to $1$ modulo $\ell$ and modulo $r$. Set $\theta$ to be any faithful, linear character of $Z$. Note that $B(\theta)^{(\ell^m)}=B(\theta^{\ell^m})$, for all $m\in\mathbb{N}$, and so, by Proposition~\ref{prop non isom}, $\operatorname{mf}(B)$ is the smallest $m\in\mathbb{N}$, such that $\theta^{\ell^m}=\theta^{\pm1}$ or equivalently that $r|(\ell^m\pm 1)$. It is now clear that $\operatorname{mf}(B)=n$.
\end{proof}
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Practical quantum access network over a 10 Gbit/s Ethernet passive optical network}
\author{Bi-Xiao Wang,\authormark{1,2,6} Shi-Biao Tang,\authormark{3,6} Yingqiu Mao,\authormark{1,2} Wenhua Xu,\authormark{4} Ming Cheng,\authormark{5} Jun Zhang,\authormark{1,2} Teng-Yun Chen\authormark{1,2,7} and Jian-Wei Pan\authormark{1,2,8}} \address{\authormark{1}Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China\\ \authormark{2}CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China\\ \authormark{3}QuantumCTek Co., Ltd, Hefei 230088, China\\ \authormark{4}China Telecom Corporation Ltd. Shanghai Branch, Shanghai 200120, China\\ \authormark{5}Research Institute of China Telecom Co., Ltd, Shanghai 200122, China}
\email{\authormark{6}These authors contributed equally to this work.} \email{\authormark{7}[email protected]} \email{\authormark{8}[email protected]}
\begin{abstract} Quantum key distribution (QKD) provides an information-theoretically secure method to share keys between legitimate users. To achieve large-scale deployment of QKD, it should be easily scalable and cost-effective. The infrastructure construction of quantum access network (QAN) expands network capacity and the integration between QKD and classical optical communications reduces the cost of channel. Here, we present a practical downstream QAN over a 10 Gbit/s Ethernet passive optical network (10G-EPON), which can support up to 64 users. In the full coexistence scheme using the single feeder fiber structure, the co-propagation of QAN and 10G-EPON signals with 9 dB attenuation is achieved over 21 km fiber, and the secure key rate for each of 16 users reaches 1.5 kbps. In the partial coexistence scheme using the dual feeder fiber structure, the combination of QAN and full-power 10G-EPON signals is achieved over 11 km with a network capacity of 64-user. The practical QAN over the 10G-EPON in our work implements an important step towards the achievement of large-scale QKD infrastructure. \end{abstract}
\section{Introduction} Quantum key distribution (QKD) provides an effective solution to realize the remote secure communication by the laws of quantum mechanics\cite{Bennett1984,Gisin2002}. So far, remarkable progresses on QKD technology have been achieved, and various large-scale deployments of QKD have been reported\cite{Zhang:18}. Quantum access network (QAN) with point-to-multipoint connections is a practical approach to provide secure keys for multi-users and to extend the scale of QKD\cite{Townsend1997}. Meanwhile, the coexistence of QKD and classical optical communications in existing fiber infrastructures is a viable method to reduce the cost of the quantum channel and to improve the scalability of the QKD network\cite{Xu2020}.
The classical access network is an important type of telecommunications network which connects end-users to network infrastructure. One of the most typical forms of classical access networks is the passive optical network (PON), where the downstream signal from the optical line terminal (OLT) at central office is broadcasted to all users by a power splitter, and the upstream signal from only one optical network unit (ONU) at user node is transmitted to OLT at each time slot\cite{Keiser2003}. Similar structure can also be used to build a QAN, which has two configurations, i.e., downstream (QKD receiver at user nodes) \cite{Townsend1997} and upstream (QKD transmitter at user nodes)\cite{Froehlich2015}. Therefore, realizing QAN over a PON provides secure keys for the "last mile" and greatly reduces the cost of optical fiber resources.
The main challenge in the coexistence of QAN and PON is the spontaneous Raman scattering (SRS) noise generated by PON data signals\cite{Wang2015}. The effects of SRS noise are different for the upstream and downstream configurations. Due to backwards SRS, the upstream configuration suffers from higher noise\cite{Choi:10}. In addition, the implementation of the upstream QAN is more complicated, where one QKD receiver is allocated to all QKD transmitters by the time division multiplexing technology. One method is that different QKD transmitters alternately emit signal pulses to share the detector bandwidth. Yet, as the number of users increases, the difficulty in accurately assigning time slots will increase, and the secure key rate of each user will decrease significantly\cite{Froehlich2013}. The other method is that only one QKD transmitter occupies the detector at each time. Yet, after switching between different QKD transmitter, the QKD system link needs to be re-established, which increases the overall session time. The advantage of the upstream QAN is cost-effective, since the QKD receiver with single-photon detectors is relatively expensive. Compared with the upstream configuration, the downstream configuration has less SRS noise, and the secure key generation of each user is not influenced by other users, since each user holds SPDs independently from other users.
Many efforts have been taken to achieve the coexistence of QKD and PON. In the time-assignment scheme, the time-division multiplexing technique has been presented to reduce SRS noise, but it requires modification of the classical optical communication system\cite{Choi_2011}. In addition, other methods such as the post-processing scheme\cite{MartinezMateo2014} and the bypass structure scheme\cite{Sun2018} have been proposed to improve the signal-to-noise ratio of QKD. However, these schemes cannot realize operation for multi-user QANs. Recent work has demonstrated that the space-division multiplexing fiber provides additional isolation between quantum and classical signals, but such type of fiber is still far from practical applications\cite{Cai2020}.
In this paper, we demonstrate a practical QAN, as well as its coexistence with a 10 Gbit/s Ethernet passive optical network (10G-EPON). Considering the fiber resources and power budget of the realistic 10G-EPON network, different coexistence schemes are proposed and experimentally implemented. In the full coexistence scheme based on the single feeder fiber structure, the co-propagation distance reaches 21 km with a secure key rate of 1.5 kbps for each of 16 users, when an additional attenuation of 9 dB is added to the 10G-EPON signal. Further, using the dual feeder fiber structure\cite{Choi:10,Froehlich2015,Vokic2020}, two partial coexistence schemes, i.e., dual feeder fiber coexistence and dual power splitter coexistence, are realized. In the dual feeder fiber coexistence, QAN integrated with full-power 10G-EPON signals supports up to 64 users with a secure transmission distance of 11 km. In the dual power splitter coexistence, the secure key rate of each user is improved and flexibly distributed by an independent quantum power splitter of QAN.
\section{Experiment} We set up the coexisting access network based on the structure of the PON. In our experiment, a commercial 10G-EPON instrument (FiberHome AN5516-01) is used including one OLT, one optical distribution network (ODN), and three ONUs. For the QAN, QKD transmitter is placed with the OLT to form the central office, and each ONU is paired with a QKD receiver to form the user node. Details on the 10G-EPON pulses and QKD pulses are listed in Table 1. The ODN consists of three parts, i.e., the feeder fiber, the power splitter, and the drop fiber\cite{Tanaka2010}. The feeder fiber connects the central office with the power splitter, and the drop fiber connects the power splitter with the user nodes. In field environments, there are usually one or two feeder fiber links, while the number of drop fibers depends on the number of users. Typically, the distance of the drop fiber is shorter than that of the feeder fiber in order to save fiber resources. In this experiment, the distance of each drop fiber is fixed at 1 km. The single-mode fiber is based on ITU-T G.652.D and the fiber losses (Att) are also listed in Table 1. We note that the losses are higher than typical laboratory values to mimic field environments. In the 10G-EPON system, there is only one OLT laser. The 10G-OLT and 1G-OLT signals are integrated in an XFP transceiver (10 gigabit small form-factor pluggable), the average power of the overall OLT signal is 7.2 dBm. The number of ONU lasers equals to the number of users, which is three in this work. 10G-ONU, 1G-ONU, and 1G-ONU is assigned to User-1, User-2, and User-3, with peak powers of 5.7 dBm, 2.0 dBm, and 3.4 dBm, respectively. During the operation of quantum access network and the measurement of Raman noise, all four lasers are turned on.
\begin{table}[htbp] \renewcommand\arraystretch{2.38} \centering \caption{Description for the 10G-EPON and QKD pulses.}
\begin{tabular}{c|c|c|c|c|c} \hline
& \makecell{Wavelength range \\ (nm)} & \makecell{Center wavelength\\ (nm)}&\makecell{Data rate\\ (Gbps)}&Direction&\makecell{Att\\ (dB/km)}\\ \hline 1G-OLT& 1480$\sim$1500&1490&1.25&downstream&\multirow{2}*{0.31}\\
10G-OLT& 1575$\sim$1580&1577&10.3125&downstream&\\ \hline 10G-ONU&1260$\sim$1280&1270&10.3125&upstream&0.57\\
1G-ONU&1260$\sim$1360&1310&1.25&upstream&0.48\\
1G-ONU&1260$\sim$1360&1310&1.25&upstream&0.48\\ \hline QKD-Sig&-&1550.12&-&downstream&0.35\\
QKD-Syn&-&1569.59&-&downstream&0.34\\ \hline \end{tabular}
\label{tab:shape-functions} \end{table}
\begin{figure}
\caption{Schematic layout of three coexistience schemes. (a) Full coexistence. (b) Dual feeder fiber scheme. (c) Dual power splitter scheme. WDM MUX/DEMUX: wavelength division multiplexer/demultiplexer. DFB: distributed feedback laser, SI: Sagnac interferometer, VOA: variable optical attenuator, BS: beam splitter, PBS: polarization beam splitter, EPC: electric polarization controller, SPD: single-photon detector, PIN: PIN photodetector. User-1, User-2, and User-3 are real users. The remaining users represent the virtual users.}
\end{figure}
The downstream QAN based on polarization-encoding decoy-state BB84 protocol\cite{Hwang2003,Wang2005,Lo2005}, includes one QKD transmitter and three QKD receivers, as illustrated in Fig. 1a. In the QKD transmitter, one distributed feedback laser is used to generate quantum optical pulses (Sig) with a repetition rate of 625 MHz\cite{Wang:20}. Each photon emitted by the QKD transmitter randomly chooses one output path of power splitter and is distributed to one of the QKD receivers. The different intensities and polarization states are implemented by two Sagnac interferometers in sequence\cite{Shen2013}. After attenuation, the average photon numbers of signal, decoy state, and vacuum state are 0.4, 0.1, and 0, with an emission ratio of 6:1:1, respectively. Also, the synchronized clock pulses (Syn) from QKD transmitter is broadcasted to all QKD receivers with a repetition rate of 100 kHz. In each QKD receiver, four InGaAs/InP SPDs are applied to measure the Sig pulses\cite{Liang2012,Zhang2015}. Each SPD operates at a gate frequency of 1.25 GHz, with a detection efficiency of $\sim$15\%, and a dark count rate per gate of $\sim$2$\times$$10^{-7}$.
For classical reconciliation, QKD transmitter is equipped with three modules, and each one corresponds to a receiver module at the three QKD receivers. The different classical reconciliation modules in QKD transmitter do not exchange data with each other. In particular, the data traffic generated in the classical reconciliation stage is fed into the OLT and ONU to be exchanged through the network as 10G-EPON pulses.
The calibration of the QAN includes delay scanning, synchronization calibration, and polarization feedback\cite{Chen2021}, and the specific optical signals emitted by the QKD transmitter is broadcasted to each QKD receiver in each procedure, thus the calibration is performed between QKD transmitter and each QKD receiver simultaneously. Key generation will only begin after all three links are successfully calibrated.
The initial authentication of the public discussion channel between QKD transmitter and each QKD receiver is realized through the pre-stored keys, then periodically refreshed using the generated secure keys\cite{Mao:18}. In the post-processing, the Winnow algorithm\cite{Buttler2003} is used for error correction and the method of Toeplitz matrix\cite{Krawczyk1995} is applied for privacy amplification. Once the quantum bit error rate (QBER) exceeds the preset threshold of 4\%, QKD is aborted and the calibration of the QAN will restart.
\begin{figure}
\caption{The layout of WDM modules. (a) WDM MUX. (b) WDM DEMUX. (c) WDM Filter. FWDM: filter wavelength division multiplexer, passband 1510$\sim$1580 nm, reflectband 1260$\sim$1500 nm. CWDM: coarse wavelength division multiplexer, center wavelength 1578 nm. DWDM: 100 GHz dense wavelength division multiplexer, center wavelength 1550.12 nm (C34) and 1569.59 nm (C20). FBG: fiber Bragg grating, full width at half maximum 0.08 nm (10 GHz) and 0.16 nm (20 GHz).}
\end{figure}
In the single feeder fiber structure, the full coexistence of the QAN and 10G-EPON over the total ODN is achieved, as depicted in Fig. 1a. The coexistence is realized using a WDM MUX and DEMUX module, as illustrated in Fig. 2a and 2b. In the WDM MUX module, the isolation of 1550.12 nm from 1G-OLT signal is 23 dB by FWDM-1, and then the band-pass filter of 1G-OLT signal is realized by FWDM-2. Using the CWDM-1, the band-pass filter of 10G-OLT signal and the band-stop filter of 1550.12 nm component in 10G-OLT signal are achieved. In the WDM DEMUX module, both the isolation of ONU signal and 1G-OLT signal from 1550.12 nm are 107 dB, and the isolation of 10G-OLT signal from 1550.12 nm is 71 dB. The WDM modules provide sufficient cross-talk isolation, and the linear crosstalk of 10G-EPON signals is eliminated effectively.
The insertion losses of OLT, 10G-ONU, 1G-ONU and quantum signals in WDM MUX/DEMUX are 0.9/1.0 dB, 1.0/0.5 dB, 0.7/0.5 dB, and 0.8/3.4 dB, respectively. The network capacity of the coexistence network corresponds to the splitting ratio of power splitter, i.e., 1:4, 1:8, 1:16, 1:32, and 1:64, where their average insertion losses are 7.4 dB, 10.5 dB, 13.6 dB, 17.1 dB, and 20.2 dB, respectively.
Further, we investigate the dual feeder fiber structure. The second feeder fiber is assigned as quantum channel, which is generally a spare fiber for the PON. Two possible partial coexistence schemes, i.e., dual feeder fiber scheme and dual power splitter scheme, are depicted in Fig. 1b and 1c. In the partial coexistence schemes, the linear crosstalk of the OLT laser and the nonlinear SRS noise caused by the OLT signal in the feeder fiber are filtered with a WDM Filter module, before the quantum and classical signals are combined by the power splitter, effectively lowering the noise photons, as shown in Fig. 2c. In flexible network scenarios where the users can freely choose to be connected or not to the QAN, the dual power splitter scheme can be applied, where the quantum signals travel through a separate quantum power splitter from the classical signals.
\section{Results and discussion}
The SRS noise photons in the proposed coexistence network schemes are mainly generated by the OLT signal. The SRS noise caused by the ONU signal can be negligible, since the wavelength interval between ONU and quantum signals is more than 190 nm apart\cite{Wang2017}. The OLT signal generates SRS noise photons in both the feeder fiber and drop fiber, which is described as\cite{Chapuran_2009,Sun2018}, \begin{equation} S_F=P\frac{ \beta}{\alpha_{q}-\alpha_{c}} (e^{-\alpha_{c} L_F}-e^{-\alpha_{q} L_F})\frac{1}{N}e^{-\alpha_{q}L_D}, \end{equation}
\begin{equation} S_D=\frac{P}{N} e^{-\alpha_{c}L_F}\frac{\beta}{\alpha_{q}-\alpha_{c}} (e^{-\alpha_{c} L_D}-e^{-\alpha_{q} L_D}), \end{equation} where $\alpha_{c}$ and $\alpha_{q}$ correspond to the path attenuation coefficient of OLT and quantum signal, $L_F$ and $L_D$ are the distance of feeder fiber and drop fiber, $P$ is the launch power of the OLT signal, $N$ is the splitting ratio of power splitter, and $\beta$ is the SRS coefficient.
In the single feeder fiber structure, i.e., full coexistence, the total SRS noise detectable by QKD receiver is the sum of $S_F$ and $S_D$. The SRS noise in the full coexistence scheme is affected by three factors, i.e., the splitting ratio of power splitter, the distance of feeder fiber, and the launch power of the OLT signal.
\begin{figure}
\caption{Measured and simulated results for full coexistence scheme. (a) Secure key rate of each user under different network capacity. The distance of feeder fiber is fixed at 5 km. An attenuation of 5 dB is added to the OLT signal. Inset: secure key rate of each user verses time in a 64-user network. (b) User-1 secure key rate as a function of feeder fiber distance under different attenuation of the OLT signal. The splitting ratio is fixed at 1:16. The red solid, blue dash-dotted, and green dashed line represents the simulation under the 3 dB, 6 dB, and 9 dB attenuation of the OLT signal, respectively. Red circles, blue squares, and green triangles denote the experimental data under the 3 dB, 6 dB, and 9 dB attenuation of the OLT signal, respectively.}
\end{figure}
First, the splitting ratio of power splitter is adjusted, while the distance of feeder fiber and the launch power of the OLT signal are fixed. The results are shown in Fig. 3a. In addition, a key rate stability test of the QAN when the network capacity is 64 users is shown in the inset of Fig. 3a. The total transmission loss from QKD transmitter to QKD receiver is $\sim$26.5 dB, and an average key rate of 2.1 kbps per user is obtained over 9 hours.
Second, to investigate the maximum secure transmission distance of the QAN under different launch powers of the OLT signal, the splitting ratio of power splitter is fixed. The measured and simulated secure key rates of each user under different feeder fiber distance and the launch power of the OLT signal are plotted in Fig. 3b. The simulation is performed with methods from Ref.\cite{Ma2005}, and the secure key rate, $R$, is given by \begin{equation} R=q\lbrace Q_1[1-H_2(e_1)]-fQ_{\mu}H_2(E_{\mu})+Q_0\rbrace, \end{equation} where $q$ is the basis-sift factor, which is 1/2 in this experiment, $f$ is the correction efficiency of error correction, which is 1.2$\sim$1.5, $H_2(x)$ is the binary entropy function, $Q_\mu$ and $E_\mu$ are the overall gain and QBER of the signal state, $Q_1$ and $e_1$ are the gain and QBER of the single-photon of signal states, $Q_0$ is the background gain, which comes from the sum of dark and Raman noise ($S_F$ and $S_D$) counts in the full coexistence scheme.
We note that without additional attenuation to the OLT signal, the distance of feeder fiber only reaches 1 km. As the distance increases, the launch power of the OLT signal should be decreased. In order to generate secure keys over a typical distance for PON as long as 20 km of feeder fiber, the OLT signal must be attenuated by 9 dB. Nevertheless, the additional attenuation on the classical signals may affect the performance of the 10G-EPON, which requires its pulse power to be within the power budget.
\begin{table}[h!]
\centering \caption{Comparison of SRS noise photon count rate (kcps) in the full and partial coexistence schemes. } \begin{tabular}{ccc} \hline Feeder fiber distance & Full coexistence & Partial coexistence \\ \hline 5 km & 16.3 & 2.9 \\ 20 km & 18.1 & 1.0\\ \hline \end{tabular}
\label{tab:shape-functions} \end{table}
Further, we investigate the partial coexistence scheme. In such scheme, $S_F$ is eliminated before the multiplexing of the quantum and OLT signals, therefore, the SRS noise is only contributed by $S_D$. The measured SRS noise generated by the full-power OLT signal in the 16-user network of the full and partial coexistence schemes are listed in Table 2, from which one can see the SRS noise photons are drastically lowered with the partial coexistence scheme. Therefore, the QAN can be combined with full-power 10G-EPON over long distance ODN in the partial coexistence scheme.
\begin{figure}
\caption{Results for the partial coexistence schemes. Black triangles and red squares denote the experimental data of secure key rate and QBER. No attenuation is added to the OLT signal. (a) User-1 secure key rate and QBER under different network capacity in dual feeder fiber scheme. For the 4-user, 8-user, 16-user, and 32-user networks, the distance of feeder fiber is 20 km. In the 64-user network, the distance of feeder fiber is 10 km. (b) User-1 secure key rate and QBER under different configurations in the dual power splitter scheme. In each configuration, the first and the second are the network capacities of classical and quantum network, respectively. The distance of the feeder fiber is 20 km.}
\end{figure}
Figure 4a shows the performance of secure key rate and QBER in the dual feeder fiber scheme. When the total distance of ODN is fixed at 21 km, the maximum network capacity is 32 users. To support network capacity of 64 users, the maximum secure transmission distance of ODN is 11 km.
In the dual power splitter scheme, the network capacities of 10G-EPON and QAN are independent. The performance of the QAN under four network capacity configurations is shown in Fig. 4b. The secure key rate is increased by using a power splitter with lower splitting ratio for the quantum signals, see the configuration between $\lbrace$32C,32Q$\rbrace$ and $\lbrace$32C,16Q$\rbrace$, and $\lbrace$16C,16Q$\rbrace$ and $\lbrace$16C,8Q$\rbrace$. Compared with the $\lbrace$32C,16Q$\rbrace$ configuration, the splitting ratio of the classical power splitter is lower in the $\lbrace$16C,16Q$\rbrace$ configuration, and thus, the launch power of the OLT signal and the Raman noise generated by OLT signal in the drop fiber is higher, and the secure key rate is lower in the $\lbrace$16C,16Q$\rbrace$ configuration. Furthermore, the dual power splitter coexistence scheme has an additional benefit that it is highly flexible. The splitting ratio of power splitter can be imbalance, which is customized according to the user demand in this scheme, under the premise that the user with the minimum splitting ratio can also obtain the secure key.
\section{Conclusion} In summary, we have demonstrated a practical downstream QAN over a 10G-EPON with the maximum network capacity of 64-user. Based on the optical fiber topology and power budget of 10G-EPON, different coexistence schemes, i.e., full coexistence, dual feeder fiber coexistence, and dual power splitter coexistence, are proposed, which are suitable for different field network environments. The full coexistence scheme based on the single feeder fiber structure is suitable for scenarios where the 10G-EPON has sufficient power budget. When the 10G-EPON signal is attenuated by 9 dB, QAN supports 16 users with a secure key rate of 1.5 kbps per user and the secure transmission distance reaches 21 km. In the partial coexistence schemes, the SRS noise from feeder fiber is effectively eliminated by the dual feeder fiber structure, and QAN is integrated with full-power 10G-EPON signals. The secure key rate of each user can be customized according to the user demand in the dual power splitter coexistence scheme. Our work provides a practical and flexible method to implement the coexistence of QAN and 10G-EPON, which could be favourable for the deployment of QKD network infrastructure.
\begin{backmatter} \bmsection{Funding} China State Railway Group Co., Ltd. Scientific and Technological Research Project (K2019G062); National Natural Science Foundation of China (61875182); Anhui Initiative in Quantum Information Technologies.
\bmsection{Disclosures} The authors declare no conflicts of interest.
\bmsection{Data availability} Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request. \end{backmatter}
\end{document}
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\begin{document}
\title[Regularity of radicals and arithmetic degrees]{Upper bounds for regularity of radicals of ideals and arithmetic degrees}
\author{Yihui Liang} \address{Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN47907, USA} \curraddr{} \email{[email protected]} \thanks{The author was partially supported by the Ross-Lynn Research Scholar Fund from Purdue University.}
\subjclass[2020]{Primary 13D02, 13H15, 13P10}
\date{}
\commby{}
\keywords{Castelnuovo-Mumford regularity, radical, arithmetic degrees}
\begin{abstract} Let $S$ be a polynomial ring in $n$ variables over a field. Let $I$ be a homogeneous ideal in $S$ generated by forms of degree at most $d$ with $\text{dim}(S/I)=r$. In the first part of this paper, we show how to derive from a result of Hoa an upper bound for the regularity of $\sqrt{I}$. More specifically we show that $\text{reg}(\sqrt{I})\leq d^{(n-1)2^{r-1}}$. In the second part, we show that the $r$-th arithmetic degree of $I$ is bounded above by $2\cdot d^{2^{n-r-1}}$. This is done by proving upper bounds for arithmetic degrees of strongly stable ideals and ideals of Borel type. \end{abstract}
\maketitle
\section{Introduction} Let $S=\mathbb{K}[x_1,\dots,x_n]$ be a standard graded polynomial ring over a field. \linebreak Castelnuovo-Mumford regularity is an important homological invariant that measures the complexity of performing computations with finitely generated graded $S$-modules, for instance calculating graded free resolutions and the associated invariants. In \cite{Ravirad}, Ravi was the first to investigate the relationship between regularity of ideals and regularity of their radicals. Recall that the radical of an ideal $I$, denoted by $\sqrt{I}$, is defined as the set of all elements $x$ such that some power of $x$ lies in $I$. More specifically, Ravi showed that $\text{reg}(\sqrt{I})\leq \text{reg}(I)$ when $I$ is a monomial ideal, or when $R/\sqrt{I}$ is a Buchsbaum module, or in some cases when $\sqrt{I}$ defines a curve in $\mathbb{P}^3$. In addition, he raised the following question: Is it always true that $\text{reg}(\sqrt{I})\leq \text{reg}(I)$ for any homogeneous ideal $I$? This question is answered negatively by Chardin and D'Cruz \cite{chardincruz}, as they provided a family of ideals $I_{m,n}$ such that $\text{reg}(I_{m,n})=m+n+2$ but $\text{reg}(\sqrt{I})=mn+2$. It is natural to ask the next question: Does there exist a bound of $\text{reg}(\sqrt{I})$ in terms of $\text{reg}(I)$? To our best knowledge, so far there has not been any answer to the above question.
In the first part of the paper, we explain how to specialize an existing formula of Hoa \cite{hoareg} to obtain the following bound (see Corollary \ref{regraddegreebd}). Recall that the generating degree is bounded above by the regularity so one may replace $d$ by $\text{reg}(I)$ in the inequalities below.
\begin{theorem*}
Let $I$ be a homogeneous ideal in $S$ of dimension $r\geq 2$ and generated by forms of degree at most $d$, then
\[
\text{reg}(\sqrt{I})\leq \left(\frac{d^{n-r}(d^{n-r}-1)}{2}+d^{n-1}\right)^{2^{r-2}} \leq d^{(n-1)2^{r-1}}.
\] \end{theorem*}
In the second part of the paper, we focus on proving upper bounds for arithmetic degrees of homogeneous ideals in terms of their generating degree (see $\S$4). Arithmetic degrees were introduced by Bayer and Mumford in \cite{computealggeo}, and they arise as refinements of multiplicities which serve as an important complexity measure. For this reason, it is desirable to find good bounds for this invariant. In the literature, there are two classical results of \cite{sturmfels} and \cite{hoa1998castelnuovo} which give bounds on arithmetic degrees for arbitrary monomial ideals in terms of their generating degrees. Since the arithmetic degrees of ideals are bounded above by the arithmetic degrees of their initial ideals, in particular the generic initial ideals, it is helpful to consider strongly stable ideals and ideals of Borel type. For strongly stable ideals, bounds which depends on the primary decompositions of the ideals were given in \cite{doi:10.1080/00927872.2020.1726940}. Some bounds were also proved for square-free strongly stable ideals (see \cite{squarefree1} and \cite{squarefree2}).
In Section 4.1, we first prove upper bounds for arithmetic degrees of strongly stable ideals and ideals of Borel type (see Corollary \ref{stronglytstablebd} and \ref{weaklystablebd}). Then in Section 4.2, we derive from our result on ideals of Borel type the following general bounds for homogeneous ideals (see Theorem \ref{arithdeghomideal}):
\begin{theorem*}
Let $I$ be a homogeneous ideal in $S=\mathbb{K}[x_1,\dots,x_n]$ generated by forms of degree at most $d$. For all $r=0,\dots,n-1$, the $r$-th arithmetic degree of $I$ is bounded above by
\[
\text{arith-deg}_r(I)\leq 2\cdot d^{2^{n-r-1}}.
\] \end{theorem*} Currently all the existing bounds for arithmetic degrees of arbitrary homogeneous ideals depend on their regularities, for example in \cite{computealggeo} Bayer and Mumford showed that $\text{arith-deg}_r(I)\leq \text{reg}(I)^{n-r}$. With their formula, to get a bound in terms of the generating degree, one needs to combine it with a doubly exponential bound of regularity (for instance \cite[Corollary 2.13]{caviglia2021linearly}) and the inequality becomes $\text{arith-deg}_r(I)\leq d^{(n-r)2^{n-r-1}}$. Notice that our bound is a significant improvement as we eliminate the factor of $n-r$ in the exponent.
\section{Preliminaries}
In this section, we provide definitions and some basic facts about multiplicity, geometric degree, arithmetic degree, Castelnuovo–Mumford regularity, strongly stable ideal, and ideal of Borel type. For a more thorough introduction to these subjects, see \cite{eisenbook}, \cite{computealggeo}, and \cite{herzog}.
\subsection{Castelnuovo-Mumford regularity, multiplicity, geometric degree, and arithmetic degree}
Let $\mathbb{K}$ be an arbitrary field and $S$ be the polynomial ring $\mathbb{K}[x_1,\dots,x_n]$. All the invariants that will be introduced in this section do not change if we extend our field $\mathbb{K}$ to $\mathbb{K}(y)$ where $y$ is a new indeterminate, thus we may assume our field is infinite when needed.
Let $M$ be a finitely generated graded module over $S$. Let $\beta_{ij}(M):=$\linebreak$\text{dim}_{\mathbb{K}}(\text{Tor}^S_i(M,\mathbb{K})_j)$ be the graded Betti numbers of $M$. The \textit{Castelnuovo–Mumford regularity} of $M$ is defined as $\text{reg}(M)=\text{max}\{j : \beta_{i,i+j}(M)\neq 0 \text{ for some }i \}$.
For a graded module $M$, the \textit{multiplicity} of $M$, denoted by $e(M)$, is the normalized leading coefficient of the Hilbert polynomial of $M$. Let $I$ be a homogeneous ideal and $\prec$ be any monomial order, the initial ideal of $I$ with respect to $\prec$ is denoted by $\text{in}_\prec(I)$. Since the Hilbert function of $S/I$ agrees with the Hilbert function of $S/\text{in}_\prec(I)$, the multiplicities of $S/I$ and $S/\text{in}_\prec(I)$ agree as well.
For a homogeneous ideal $I$ in $S$, let $\text{Ass}(S/I)$ denote the set of associated primes of $S/I$ and $\text{Min}(S/I)$ denote the set of minimal primes of $S/I$. For a prime ideal $P$, the length-multiplicity of $P$ with respect to $I$, denoted by $\text{mult}_I(P)$, is defined as the length of the largest submodule of $(S/I)_P$ with finite length.
Let us recall the associativity formula for multiplicity: \[ e(S/I)=\sum_{P\in \text{Min}(S/I),\text{ dim}(S/P)=\text{dim}(S/I)}\text{mult}_I(P)\cdot e(S/P). \] Notice that for a minimal prime $P$, $(S/I)_P$ has finite length.
The \textit{$r$-th geometric degree} of $I$ is defined as: \begin{equation*}
\text{geom-deg}_r(I)=\sum_{P\in \text{Min}(S/I), \text{ dim}(S/P)=r}\text{mult}_I(P)\cdot e(S/P). \end{equation*} The \textit{geometric degree} of $I$, denoted by $\text{geom-deg}(I)$, is the sum of $\text{geom-deg}_r(I)$ for all $r$.
The \textit{$r$-th arithmetic degree} of $I$ is defined as: \begin{equation*}
\text{arith-deg}_r(I)=\sum_{P\in \text{Ass}(S/I), \text{ dim}(S/P)=r}\text{mult}_I(P)\cdot e(S/P). \end{equation*} The \textit{arithmetic degree} of $I$, denoted by $\text{arith-deg}(I)$, is the sum of $\text{arith-deg}_r(I)$ for all $r$. It is clear from the definitions that $e(S/I)\leq \text{geom-deg}(I)\leq \text{arith-deg}(I)$.
We present in Corollary \ref{geodegcoro} an upper bound for the geometric degree of $I$, which will be used in Section 3. This bound can be deduced from the theorem below (see \cite[Theorem 4.3]{sturmfels}). A bound for the geometric degree can also be obtained using \cite[Proposition 3.5]{computealggeo} which says the $i$-th geometric degree is bounded above by $d^{n-i}$ for all $i$. From this result one gets $\text{geom-deg}(I)= \sum_{i=1}^{r} \text{geom-deg}_i(I)\leq \sum_{i=1}^{r} d^{n-i}$, where $r$ is the dimension of $S/I$. Notice that Corollary \ref{geodegcoro} is a slight improvement of the above bound. \begin{theorem} [Sturmfels, Trung, Vogel] \label{geodeg}
Let $J\subset I$ be homogeneous ideals in $S$ such that $(S/J)_P$ is Cohen-Macaulay for every minimal prime $P$ of $I$. Let $f_1,\dots,f_m$ be forms in $I$ with degrees $d_1\geq d_2\geq \cdots \geq d_m$ such that $I=J+(f_1,\dots,f_m)$. Then
\[
\text{geom-deg}(I)\leq d_1d_2\cdots d_t\cdot \text{geom-deg}(J),
\]
where $t:=\text{max}\{\text{ht}(P/J) : P\in\text{Min}(I) \}$. \end{theorem}
\begin{corollary} \label{geodegcoro}
Let $I$ be a homogeneous ideal in $S$ of dimension $r\neq 0$ generated by forms of degree at most $d$, then the geometric degree of $I$ is bounded above by:
\[
\text{geom-deg}(I)\leq d^{n-1}.
\] \end{corollary}
\begin{proof}
Without loss of generality, we may assume the field is infinite.
Let $f_1,\dots,f_s$ be homogeneous polynomials that minimally generate $I$ with $\text{deg}(f_i)\leq d$. We may assume $f_1,\dots,f_h$ form a regular sequence where $h=n-r$ is the height of $I$, and $f_{h+1},\dots,f_s$ are relabeled so that $\text{deg}(f_{h+1})\geq \cdots \geq \text{deg}(f_{s})$. Let $J=(f_1,\dots,f_h)$, so we have $I=J+(f_{h+1},\dots,f_s)$. By assumption the homogeneous maximal ideal cannot be a minimal prime of $I$, therefore we get $t:=\text{max}\{\text{ht}(P/J):P \in\text{Min}(I)\}\leq n-h-1$. Notice that the geometric degree of $J$ is equal to its multiplicity which is equal to $\prod_{i=1}^{h}\text{deg}(f_i)$. Hence we get $\text{geom-deg}(I)\leq \prod_{i=h+1}^{t+h}\text{deg}(f_i) \cdot \text{geom-deg}(J)\leq d^{n-h-1} \cdot \prod_{i=1}^{h}\text{deg}(f_i) \leq d^{n-h-1}\cdot d^{h}=d^{n-1}$.
\end{proof}
\subsection{Strongly-stable ideals and ideals of Borel type}
Recall that a monomial ideal $I$ is an ideal of \textit{Borel type} (also called weakly stable or an ideal of nested type) if for every monomial $u\in I$, if $l$ is the maximum integer such that $x_i^l|u$, then for every $j\leq i$ there exists some $t\geq 0$ such that $x_j^tu/x_i^l\in I$. Another equivalent definition says $I$ is an ideal of Borel type if every associated prime of $I$ is of the form $(x_1,\dots,x_i)$ for some $i$.
A monomial ideal $I$ is \textit{strongly stable} if for every monomial $u\in I$, for any $i$ such that $x_i|u$ and $j\leq i$, we have $x_ju/x_i\in I$.
Given a set of monomials $U=\{u_1,\dots,u_s\}$, one can consider the smallest strongly stable ideal $I$ that contains $U$. These $u_i$'s are sometimes called Borel generators of $I$ (see \cite{FRANCISCO2011522} for more information about Borel generators).
Let $I$ be a homogeneous ideal and $\prec$ be any monomial order such that $x_n\prec x_{n-1}\prec \cdots \prec x_1$. If $\mathbb{K}$ is infinite, then there exists a nonempty Zariski open set $U\subset \text{GL}_n(\mathbb{K})$ such that $\text{in}_\prec(\alpha I)=\text{in}_\prec(\alpha^\prime I)$ for all $\alpha,\alpha^\prime \in U$. The \textit{generic initial ideal} of $I$ with respect to $\prec$ is defined as $\text{gin}_\prec(I):=\text{in}_\prec(\alpha I)$ for any $\alpha \in U$. It is well-known that $\text{gin}_\prec(I)$ is an ideal of Borel type. Moreover if $\text{char}(\mathbb{K})=0$, $\text{gin}_\prec(I)$ is a strongly stable ideal.
\section{Regularity of radical of ideal}
Let $I$ be a homogeneous ideal in $S$ generated by forms of degree at most $d$. In this section we use a theorem of Hoa in \cite{hoareg} to derive an upper bound for the regularity of radical of $I$. The theorem of Hoa is given below:
\begin{theorem} [Hoa] \label{hoathm}
Assume $S/J$ is a reduced ring of dimension $r\geq 2$ and multiplicity $e(S/J)$, then
\[
\text{reg}(J)\leq\left(\frac{e(S/J)(e(S/J)-1)}{2}+\text{arith-deg}(J)\right)^{2^{r-2}}.
\] \end{theorem}
We obtain the following upper bound of $\text{reg}(\sqrt{I})$ in terms of multiplicity and geometric degree of $I$ by applying the above inequality to $S/\sqrt{I}$.
\begin{corollary} \label{regradgeombd}
Let $I$ be a homogeneous ideal in $S$ of dimension $r\geq 2$, then
\[
\text{reg}(\sqrt{I})\leq \left(\frac{e(S/I)(e(S/I)-1)}{2}+\text{geom-deg}(I)\right)^{2^{r-2}}.
\] \end{corollary}
\begin{proof}
Since $\sqrt{I}$ is the intersection of all minimal primes of $I$, we get that $\text{Ass}(S/\sqrt{I})$\linebreak$=\text{Min}(S/\sqrt{I})=\text{Min}(S/I)$. This implies that $\text{arith-deg}(\sqrt{I})=\text{geom-deg}(\sqrt{I})$. Also we have $\text{dim}(S/\sqrt{I})=\text{dim}(S/I)=r$. For any minimal prime $P$ of $I$, notice that $(S/\sqrt{I})_P=S_P/PS_P$, so the length-multiplicity $\text{mult}_{\sqrt{I}}(P)=l((S/\sqrt{I})_P)=1$ which is clearly bounded above by $\text{mult}_{I}(P)$. It follows that $e(S/\sqrt{I})\leq e(S/I)$ and $\text{geom-deg}(\sqrt{I})\leq \text{geom-deg}(I)$. We get the desired bound by combining Theorem \ref{hoathm} with the above inequalities. \end{proof}
By applying Corollary \ref{geodegcoro}, we obtain the following upper bound for $\text{reg}(\sqrt{I})$ in terms of the generating degree of $I$. Note that the $0$-dimensional case is trivial since $\sqrt{I}$ is the homogeneous maximal ideal. \begin{corollary} \label{regraddegreebd}
Let $I$ be a homogeneous ideal in $S$ of dimension $r\neq 0$ and generated by forms of degree at most $d$, then
\[
\text{reg}(\sqrt{I})\leq \left.
\begin{cases}
d^{n-1}, & \text{if } r=1\\
\left(\frac{d^{n-r}(d^{n-r}-1)}{2}+d^{n-1}\right)^{2^{r-2}}, & \text{if } r\geq 2\\
\end{cases} \right\} \leq d^{(n-1)2^{r-1}}.
\]
\end{corollary}
\begin{proof}
First notice that $e(S/I)=\text{geom-deg}_r(I)$ by the associativity formula. The $r$-th geometric degree can be bounded by $\text{geom-deg}_r(I) \leq d^{n-r}$ according to \cite[Proposition 3.5]{computealggeo}.
If $r=1$, then $S/\sqrt{I}$ is a one-dimensional reduced ring, therefore it is Cohen-Macaulay. Then apply \cite[Theorem 1.2]{rossi2005castelnuovo} to get $\text{reg}(\sqrt{I})\leq e(S/\sqrt{I})$. By the proof in the previous theorem we have $e(S/\sqrt{I})\leq e(S/I)\leq d^{n-1}$.
Now assume $r\geq2$. Combining Corollary \ref{regradgeombd} with the inequality $e(S/I) \leq d^{n-r}$ and Corollary \ref{geodegcoro}, we have:
\begin{equation*}
\begin{split}
\text{reg}(\sqrt{I})
&\leq \left(\frac{e(S/I)(e(S/I)-1)}{2}+\text{geom-deg}(I)\right)^{2^{r-2}}\\
&\leq \left(\frac{d^{n-r}(d^{n-r}-1)}{2}+d^{n-1}\right)^{2^{r-2}}\\
&\leq d^{(n-1)2^{r-1}}.
\end{split}
\end{equation*} \end{proof}
\section{Upper bounds for arithmetic degrees} In this section, our goal is to obtain upper bounds for the arithmetic degrees of any homogeneous ideal $I$. By \cite[Theorem 2.3]{sturmfels}, $\text{arith-deg}_r(I)\leq \text{arith-deg}_r(\text{in}_\prec(I))$ \linebreak $ \text{ for all }r=0,1,\dots,n$ and for any monomial order $\prec$, in particular the arithmetic degrees of $I$ can be bounded above by the arithmetic degrees of its generic initial ideal. Therefore the problem reduces to the case where $I$ is an ideal of Borel-type, and when $\text{char}(\mathbb{K})=0$ we may further assume $I$ is strongly stable. \subsection{Arithmetic degrees of strongly-stable ideals and ideals of Borel type} Let us first consider the case where $I$ is simply a monomial ideal. Notice that every associated prime of $I$ is generated by a subset of the variables so it has the form $P_Z=(x_i : x_i\in \{x_1,\dots,x_n\}\setminus Z)$, and it follows that $e(S/P_Z)=1$. Therefore to compute the arithmetic degrees of $I$, it suffices to compute the length-multiplicities mult$_I(P_Z)$, which in fact have a combinatorial description due to Sturmfels, Trung, and Vogel (see \cite[\S 3]{sturmfels}). To see this we need to define the notion of standard pairs first.
\begin{definition} [Sturmfels, Trung, Vogel] \label{stdp}
Let $u$ be a monomial in S, let $Z$ be a subset of the variables $\{x_1,\dots,x_n\}$. A pair $(u,Z)$ is called a \textit{standard pair} with respect to the monomial ideal $I$ if the following conditions hold:
\begin{enumerate}
\item $Z\cap $ supp($u$)=$\emptyset$,
\item $u\mathbb{K}[Z]\cap I=\{0\}$,
\item if $(u^\prime,Z^\prime)\neq (u,Z)$ is another pair such that $Z^\prime\cap $ supp($u^\prime$)=$\emptyset$ and $u^\prime\mathbb{K}[Z^\prime]\cap I=\{0\}$, then $u\mathbb{K}[Z]\not\subset u^\prime\mathbb{K}[Z^\prime]$.
\end{enumerate}
Let std$_r(I)$ denote the number of standard pairs of the form $(\cdot,Z)$ such that $|Z|=r$. \end{definition}
The following lemma from \cite[Lemma 3.3]{sturmfels} allows us to transform the problem of computing the length-multiplicities into counting the number of standard pairs. \begin{lemma} [Sturmfels, Trung, Vogel]
For any subset $Z\subseteq \{x_1,\dots,x_n\}$, \linebreak $\text{mult}_I(P_Z)$ equals the number of standard pairs of the form $(\cdot,Z)$. In particular $\text{arith-deg}_r(I)=\text{std}_r(I)$ for all $r=0,\dots,n$. \end{lemma}
Now let us assume $I$ is an ideal of Borel type, so any associated prime of $I$ must have the form $P=(x_1,\dots,x_i)$ for some $i$. By the above lemma, computing the $r$-th arithmetic degree of $I$ is equivalent to counting the number of standard pairs of $I$ that have the form $(x_1^{c_1}\cdots x_{n-r}^{c_{n-r}},\{x_{n-r+1},\dots,x_n\})$. The following lemma gives us upper bounds on the number of such standard pairs.
Let $G(I)$ be the set of minimal monomial generators of $I$. For a monomial $u$, let Md$_i(u)=\text{max}\{c:x_i^c \text{ divides }u\}$. Denote $\text{Md}_i(I)=\text{max}\{\text{Md}_i(u): u\in G(I)\}$. \begin{lemma}
Let $I$ be an ideal of Borel type. For any $j=1,\dots,n$, \linebreak if $(x_1^{c_1}\cdots x_j^{c_j},\{x_{j+1},\dots,x_n\})$ is a standard pair with respect to $I$, then $0\leq c_i\leq \text{Md}_i(I)-1$ for all $i=1,\dots,j$. In particular $\text{std}_r(I)\leq \text{Md}_1(I)\text{Md}_2(I)\cdots \text{Md}_{n-r}(I)$ for all $r=0,\dots,n-1$. \end{lemma}
\begin{proof}
Assume for contradiction that $c_i\geq \text{Md}_i(I)$ for some $i=1,\dots,j$, we claim that either $x_1^{c_1}\cdots x_i^{c_i}\in I$ or $x_1^{c_1}\cdots x_{i-1}^{c_{i-1}}\mathbb{K}[x_{i},\dots,x_n]\cap I=\{0\}$. Notice that if $x_1^{c_1}\cdots x_i^{c_i}\in I$, then $x_1^{c_1}\cdots x_j^{c_j}\in I$, so the first case violates conditon (2) in the definition of standard pairs. The second case violates condition (3) since $x_1^{c_1}\cdots x_{j}^{c_{j}}\mathbb{K}[x_{j+1},\dots,x_n]\subset x_1^{c_1}\cdots x_{i-1}^{c_{i-1}}\mathbb{K}[x_{i},\dots,x_n]$. To prove the claim, assume there exists a monomial $x_1^{c_1}\cdots x_{i-1}^{c_{i-1}} x_i^{p_i}\cdots x_n^{p_n}\in$ $x_1^{c_1}\cdots x_{i-1}^{c_{i-1}}\mathbb{K}[x_{i},\dots,x_n]\cap I$. Since $I$ is an ideal of Borel type, there exists a monomial of the form $x_1^{c_1}\cdots x_{i-1}^{c_{i-1}}x_i^{q_i}\in I$. Let $x_1^{c_1^\prime}\cdots x_{i-1}^{c_{i-1}^\prime}x_{i}^{q_{i}^\prime}$ be the monomial in the minimal generating set that divides $x_1^{c_1}\cdots x_{i-1}^{c_{i-1}}x_{i}^{q_{i}}$. Then we have $c_1^\prime \leq c_1,\dots,c_{i-1}^\prime \leq c_{i-1},$ and $q_i^\prime \leq \text{Md}_i(I)\leq c_i$. It follows that $x_1^{c_1^\prime}\cdots x_{i-1}^{c_{i-1}^\prime}x_{i}^{q_{i}^\prime}$ divides $x_1^{c_1}\cdots x_i^{c_i}$, therefore $x_1^{c_1}\cdots x_i^{c_i}\in I$.
\end{proof}
\begin{corollary} \label{weaklystablebd}
Let $I$ be an ideal of Borel type in $S$ generated by monomials of degree at most $D$. Let $I_{[i]}$ denote the image of $I$ in $S/(x_{i+1},\dots,x_n)\cong \mathbb{K}[x_1,\dots,x_i]$ and $D(I_{[i]})$ denote the generating degree of $I_{[i]}$. Then $\text{arith-deg}_r(I)\leq \prod_{i=1}^{n-r} D(I_{[i]})$ for all $r=0,\dots,n-1$. \end{corollary}
\begin{proof}
Since $I$ is an ideal of Borel type, by \cite[Lemma 1.5]{cavigliasbarra} we have $\text{Md}_{i}(I_{[i]})=\text{Md}_i(I)$ for all $i=1,\dots,n$. Also it is clear that $\text{Md}_{i}(I_{[i]})\leq D(I_{[i]})$. The rest follows from the previous lemma. \end{proof}
Now we further assume that $I$ is a strongly stable. Given a set of monomials $U=\{u_1,\dots,u_s\}$, we can obtain the smallest strongly stable ideal containing $U$ by adjoining to $U$ all the monomials that can be obtained by swapping variables as in the strongly stable definition and letting this new set be the generating set. Let us first consider the simplest case where $I$ is the smallest strongly stable ideal containing a single monomial (sometimes called principal Borel ideal), in this case we can list all possible standard pairs of $I$.
\begin{lemma} \label{strongstd}
Let $I$ be the smallest strongly stable ideal containing a monomial $u=x_1^{l_1}\cdots x_n^{l_n}$. For all $j=1,\dots,n$, we have $x_1^{c_1}\cdots x_j^{c_j}\mathbb{K}[x_{j+1},\dots,x_n]\cap I=\{0\}$ if and only if $c_1\leq l_1-1$, or $c_1+c_2\leq l_1+l_2-1$, or $\cdots$, or $c_1+\cdots+c_j\leq l_1+\cdots+l_j-1$. \end{lemma}
\begin{proof}
Since $I$ is strongly stable and contains $u=x_1^{l_1}\cdots x_n^{l_n}$, it also contains all monomials of the form $x_1^{a_1}\cdots x_j^{a_j}x_{j+1}^{l_{j+1}}\cdots x_n^{l_n}$ for all $a_1,\dots,a_j\geq 0$ such that $a_1\geq l_1,a_1+a_2\geq l_1+l_2,\dots,$ and $a_1+\cdots+a_j\geq l_1+\cdots+l_j$. Notice that $x_1^{c_1}\cdots x_j^{c_j}\mathbb{K}[x_{j+1},\dots,x_n]\cap I=\{0\}$ if and only if for all such $a_1,\dots,a_j$ as above, either $c_1<a_1$, or $c_2<a_2$, or $\dots$, or $c_j<a_j$. The latter statement is equivalent to $c_1\leq l_1-1$, or $c_1+c_2\leq l_1+l_2-1$, or $\cdots$, or $c_1+\cdots+c_j\leq l_1+\cdots+l_j-1$. \end{proof}
\begin{corollary}
Let $I$ be the smallest strongly stable ideal containing a monomial $u=x_1^{l_1}\cdots x_n^{l_n}$. Then the set of all standard pairs of $I$ is equal to \linebreak $\bigcup_{j=1,\dots,n}\{(x_1^{c_1}\cdots x_j^{c_j},\{x_{j+1},\dots,x_n\}):c_1+\cdots+c_i\geq l_1+\cdots+l_i \text{ for all } i=1,\dots,j-1 \text{ and } c_1+\cdots+c_j\leq l_1+\cdots+l_j-1 \}$. \end{corollary}
\begin{proof}
Notice that if $(x_1^{c_1}\cdots x_j^{c_j},\{x_{j+1},\dots,x_n\})$ is a standard pair, then by condition (3) in Definition \ref{stdp}, we must have $x_1^{c_1}\cdots x_i^{c_i}\mathbb{K}[x_{i+1},\dots,x_n]\cap I\neq \{0\}$ for all $1\leq i<j$. The rest follows from the previous lemma. \end{proof}
In general for an arbitrary strongly stable ideal $I$, there exists a unique minimal set of monomials in $I$ such that $I$ is the smallest strongly stable ideal containing it. Such a set can often be much smaller than a minimal generating set. With information of this set, we can find upper bounds of the arithmetic degrees of $I$.
\begin{corollary} \label{stronglytstablebd}
Let $I$ be the smallest strongly stable ideal containing a set of monomials $\{u_i=x_1^{l_{i_1}}\cdots x_n^{l_{i_n}}: i=1,\dots,s\}$. Then for all $r=0,\dots,n-1,
$
\[
\text{arith-deg}_r(I)= \text{std}_r(I)\leq \binom{L_{n-r}+n-r-1}{n-r}
\]
where $L_{n-r}=\text{max}\{l_{i_1}+\cdots+l_{i_{n-r}}: i=1,\dots,s\}$.
\end{corollary}
\begin{proof}
\sloppy For all $i=1,\dots,s$, let $I_i$ be the smallest strongly stable ideal containing $u_i$. Then $x_1^{c_1}\cdots x_j^{c_j}\mathbb{K}[x_{j+1},\dots,x_n]\cap I=\{0\}$ if and only if $x_1^{c_1}\cdots x_j^{c_j}\mathbb{K}[x_{j+1},\dots,x_n]\cap I_i=\{0\}$ for all $i$. Let $(x_1^{c_1}\cdots x_{n-r}^{c_{n-r}}, \{x_{n-r+1},\dots,x_n\})$ be a standard pair of $I$, then by Definition \ref{stdp} and the above equivalence we have that $x_1^{c_1}\cdots x_{n-r}^{c_{n-r}}\mathbb{K}[x_{n-r+1},\dots,x_n]\cap I_i=\{0\}$ for all $i$ and $x_1^{c_1}\cdots x_{n-r-1}^{c_{n-r-1}}\mathbb{K}[x_{n-r},\dots,x_n]\cap I\neq \{0\}$. Assume for contradiction that $c_1+\cdots+c_{n-r}\geq l_{i_1}+\cdots+l_{i_{n-r}}$ for all $i$, then by Lemma \ref{strongstd}, this implies that $x_1^{c_1}\cdots x_{n-r-1}^{c_{n-r-1}}\mathbb{K}[x_{n-r},\dots,x_n]\cap I_i=\{0\}$ for all $i$ and hence $x_1^{c_1}\cdots x_{n-r-1}^{c_{n-r-1}}\mathbb{K}[x_{n-r},\dots,x_n]\cap I=\{0\}$, which is a contradiction. Thus for some $i=1,\dots,s$, we get $c_1+\cdots+c_{n-r}\leq l_{i_1}+\cdots+l_{i_{n-r}}-1$. In particular every standard pair $(x_1^{c_1}\cdots x_{n-r}^{c_{n-r}}, \{x_{n-r+1},\dots,x_n\})$ satisfies $c_1+\cdots+c_{n-r}\leq L_{n-r}-1$. Counting the number of all possible $c_1,\dots,c_{n-r}\geq 0$ gives us the desired upper bound. \end{proof}
\subsection{Upper bounds for arithmetic degrees of homogeneous ideals}
In this subsection, we derive bounds for arithmetic degrees of homogeneous ideals in terms of their generating degrees by passing to their generic initial ideals. Since the generic initial ideals are ideals of Borel type, we can apply our result from the previous subsection to obtain the following bound. \begin{theorem} \label{arithdeghomideal}
Let $I$ be a homogeneous ideal in $S=\mathbb{K}[x_1,\dots,x_n]$ generated by forms of degree at most $d$. For all $r=0,\dots,n-1$, the $r$-th arithmetic degree of $I$ is bounded above by
\[
\text{arith-deg}_r(I)\leq 2\cdot d^{2^{n-r-1}}.
\] \end{theorem}
\begin{proof}
Let $J=\text{gin}_{\text{revlex}}(I)$ be the generic initial ideal of $I$ with respect to the degree reverse lexicographical order. Then by \cite[Theorem 2.3]{sturmfels}, we have $\text{arith-deg}_r(I)\leq \text{arith-deg}_r(J)$. For any $i=1,\dots,n-r$, let $I_{\langle i \rangle}$ denote the image of $I$ in \linebreak $S/(l_{n},\dots,l_{i+1})\cong \mathbb{K}[x_1,\dots,x_i]$ where $l_{n},\dots,l_{i+1}$ are general linear forms. Notice that $D(J_{[i]})\leq \text{reg}(J_{[i]})= \text{reg}(I_{\langle i \rangle})$ for all $i$ by \cite[Remark 2.12]{zerogin}. When $i\geq 3$, we have $ \text{reg}(I_{\langle i \rangle})\leq d^{2^{i-2}}$ by the regularity bound given in \cite[Corollary 2.13]{caviglia2021linearly}. When $i=1,2$, we have $\text{reg}(I_{\langle 1 \rangle})\leq d$ and $\text{reg}(I_{\langle 2 \rangle})\leq2d-1$ by \cite[\S 2]{cavigliasbarra}. Combining the above inequalities with Corollary \ref{weaklystablebd}, we get that
\[
\begin{split}
\text{arith-deg}_r(I)
\leq \text{arith-deg}_r(J)
&\leq \prod_{i=1}^{n-r} D(J_{[i]})\\
&\leq d\cdot (2d) \cdot \prod_{i=3}^{n-r} d^{2^{i-2}}\\
&= 2\cdot d^{2^{n-r-1}}.\\
\end{split}
\] \end{proof}
{}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Weak and strong error analysis for mean-field rank-based particle approximations of one dimensional viscous scalar conservation laws} \author{O. Bencheikh and B. Jourdain\thanks{Cermics, \'Ecole des Ponts, INRIA, Marne-la-Vall\'ee, France. E-mails : [email protected], [email protected]. The authors would like to acknowledge financial support from Universit\'e Mohammed VI Polytechnique.}} \maketitle
\begin{abstract} In this paper, we analyse the rate of convergence of a system of $N$ interacting particles with mean-field rank-based interaction in the drift coefficient and constant diffusion coefficient. We first adapt arguments by Kolli and Shkolnikhov \cite{ShkoKol} to check trajectorial propagation of chaos with optimal rate $N^{-1/2}$ to the associated stochastic differential equations nonlinear in the sense of McKean. We next relax the assumptions needed by Bossy \cite{MBoss} to check the convergence in $L^1\left({\mathbb R}\right)$ with rate ${\mathcal O}\left(\frac{1}{\sqrt N} + h\right)$ of the empirical cumulative distribution function of the Euler discretization with step $h$ of the particle system to the solution of a one dimensional viscous scalar conservation law. Last, we prove that the bias of this stochastic particle method behaves as ${\mathcal O}\left(\frac{1}{N} + h\right)$. We provide numerical results which confirm our theoretical estimates. \end{abstract}
\section{Introduction}
The order of weak convergence in terms of the number $N$ of particles for the approximation of diffusions nonlinear in the sense of McKean solving \begin{equation}
X_t=X_0+\int_0^t\varsigma(s,X_s,\mu_s)\,dW_s+\int_0^t\vartheta(s,X_s,\mu_s)\,ds\mbox{ with $\mu_s$ denoting the probability distribution of $X_s$},\label{edsnlg} \end{equation} by the systems of $N$ interacting particles \begin{equation}
\check X^{i,N}_t=\check X^i_0+\int_0^t\varsigma(s,\check X^{i,N}_s,\check \mu^N_s)\,dW^i_s+\int_0^t\vartheta(s,\check X^{i,N}_s,\check \mu^N_s)\,ds,\;i\in\{1,\hdots,N\}\mbox{ with }\check\mu^N_s=\frac{1}{N}\sum_{i=1}^N\delta_{\check X^{i,N}_s},\label{syspart} \end{equation}
has been recently investigated in several papers \cite{Mischler,BenJou,Chassa,ChaudruFrikha}. Here $\left(W_t\right)_{t\ge 0}$ is a $d$-dimensional Brownian motion independent of the initial ${\mathbb R}^n$-valued random vector $X_0$, $\left(W^i,\check X^i_0\right)_{i\ge 1}$ are i.i.d. copies of $\left(W,X_0\right)$, $\varsigma:[0,T]\times{\mathbb R}^n\times{\mathcal P}({\mathbb R}^n)\rightarrow {\mathbb R}^{n\times d}$ and $\vartheta:[0,T]\times{\mathbb R}^n\times{\mathcal P}({\mathbb R}^n)\rightarrow {\mathbb R}^{n}$ with ${\mathcal P}({\mathbb R}^n)$ denoting the space of Borel probability distributions on ${\mathbb R}^n$. Typically, under some regularity assumptions, the weak error (or bias) $\displaystyle {\mathbb E}\left[\varphi\left(\check X^{1,N}_T\right)-\varphi\left(X_T\right)\right]={\mathbb E}\left[\int_{{\mathbb R}^n}\varphi(x)\check\mu^N_T(dx)\right]-\int_{{\mathbb R}^n}\varphi(x)\mu_T(dx)$ for test functions $\varphi:{\mathbb R}^n\to{\mathbb R}$ is of order $N^{-1}$. On the other hand, it is well known since \cite{Szn91} that the strong error $\max\limits_{1\le i\le N}{\mathbb E}\left[\sup_{t\in[0,T]}\left|\check X^{i,N}_t-X^i_t\right|\right]$ where $(X^i_t)_{t\in[0,T]}$ solves \eqref{edsnlg} with $(X_0,W)$ replaced by $(X^i_0,W^i)$ is of order $N^{-1/2}$. From a numerical perspective, this implies that simulating $N$ independent copies of the system with $N$ particles leads to a bias and a statistical error both of order $N^{-1}$ which is also the order of the global error resulting from one single simulation of the system with $N^2$ particles. When the computation time of the interaction is quadratic, then the cost of these $N$ copies is of order $N^3$ compared to the order $N^4$ of the computation cost of the system with $N^2$ particles.
In Theorem 6.1 \cite{Mischler}, Mischler, Mouhot and Wennberg prove that for $\varsigma$ uniformly elliptic and not depending on the time and measure arguments, $\displaystyle \sup_{t\in[0,T]}\left|{\mathbb E}\left[\varphi\left(\check X^{1,N}_t\right)\right]-\int_{{\mathbb R}^n}\varphi(x)\mu_t(dx)\right|\le \frac{C}{N}$ when $\varphi$ is Lispchitz and has some Sobolev regularity and $\vartheta(t,x,\mu)=Ax+\int U(x-y)\mu(dy)$ for some constant matrix $A$ and some function $U$ with Sobolev regularity. In \cite{BenJou}, we consider the case of interaction through moments:
$\left(\begin{array}{c}\varsigma \\\vartheta \end{array}\right)(s,x,\mu)=\left(\begin{array}{c}\sigma\\b \end{array}\right)\left(s,\displaystyle \int_{{\mathbb R}^n}\alpha(x)\mu(dx),x\right)$. When $\alpha:{\mathbb R}^n\to{\mathbb R}^p$, $\sigma:[0,T]\times{\mathbb R}^p\times {\mathbb R}^n\to{\mathbb R}^{n\times d}$, $b:[0,T]\times{\mathbb R}^p\times {\mathbb R}^n\to{\mathbb R}^{n}$ and $\varphi:{\mathbb R}^n\to{\mathbb R}$ are twice continuously differentiable with bounded derivatives and Lipschitz second order derivatives and $\sigma\sigma^*$ is globally Lipschitz, we obtain: $$\exists \, C<\infty,\;\forall h\in[0,T],\;\forall N\in{\mathbb N}^*,\quad \sup_{t\in[0,T]}\left|{\mathbb E}\left[\varphi\left(\check X^{1,N,h}_t\right)\right]-\int_{{\mathbb R}^n}\varphi(x)\mu_t(dx)\right|\le C\left(\frac{1}{N}+h\right)$$ where $\check X^{i,N,0}_t$ denotes the particle system \eqref{syspart} and $\check X^{i,N,h}_t$ its Euler discretization with step $h$ when $h>0$.
When $n=d$, in Theorem 2.17 \cite{Chassa}, Chassagneux, Szpruch and Tse prove the expansion of the bias $$\displaystyle {\mathbb E}\left[\Phi\left(\check \mu^N_T\right)\right]-\Phi\left(\mu_T\right)=\sum_{j=1}^{k-1}\frac{C_j}{N^j}+{\mathcal O}\left(\frac{1}{N^k}\right),$$ for time-homogeneous coefficients $\varsigma$ and $\vartheta$, $(2k+1)$-times differentiable with respect to both the spatial coordinates and the probability measure argument (for the notion of lifted differentiability introduced by Lions in his lectures at the Coll\`ege de France \cite{cardaliaguet}) with $\varsigma$ bounded and $X_0$ admitting a finite moment of order $(2k+1)$. They assume the same regularity on the test function $\Phi$ on the space of probability measures on ${\mathbb R}^d$ which is possibly nonlinear: $\Phi(\mu)$ is not necessarily of the form $\int_{{\mathbb R}^d}\varphi(x)\mu(dx)$. In Theorem 3.6 \cite{ChaudruFrikha}, under uniform ellipticity, Chaudru de Raynal and Frikha prove $\left|{\mathbb E}\left[\Phi\left(\check \mu^N_T\right)\right]-\Phi\left(\mu_T\right)\right|\le \frac{C}{N}$ when $\Phi$ has bounded and H\"older continuous first and second order linear functional derivatives and $\varsigma\varsigma^*$ and $\vartheta$ are bounded and globally H\"older continuous with respect to the spatial variables and have bounded and H\"older continuous first and second order linear functional derivatives with respect to the measure argument. Notice that the existence of a linear functional derivative requires less regularity than the Fr\'echet differentiability of the lift since the lifted derivative is the gradient of the linear functional derivative with respect to the spatial variables.\\
Our aim in the present paper is to check that the ${\mathcal O}\left(\frac{1}{N}+h\right)$ behaviour of the weak error for the Euler discretization with step $h$ of the system with $N$ particles generalizes to a stochastic differential equation with an even discontinuous drift coefficient. This SDE is one-dimensional ($n=d=1$) and has a constant diffusion coefficient $\varsigma(s,x,\mu)=\sigma$ for $\sigma>0$. The drift coefficient writes $\vartheta(s,x,\mu)=\lambda(\mu((-\infty,x]))$ where ${\mathbb R}\times{\mathcal P}({\mathbb R})\ni(x,\mu)\mapsto \mu\left((-\infty,x]\right)$ is not even continuous and $\lambda$ is the derivative of a $C^1$ function $\Lambda:[0,1] \to {\mathbb R}$: \begin{equation}\label{edsnl}
\left\{
\begin{aligned}
\displaystyle &X_t = X_0 + \sigma W_t + \int_0^t \lambda\left(F(s,X_s)\right)\,ds,\quad t\in[0,T]\\
&F(s,x) = \mathbb{P}\left(X_s \le x\right),\;\forall (s,x)\in[0,T]\times{\mathbb R}.\\
\end{aligned}
\right. \end{equation} We denote by $m$ the probability distribution of $X_0$ and by $F_0$ its cumulative distribution function. According to Section \ref{secmainres} in the paper \cite{BossTal} specialized to the case $\Lambda(u) = u^2/2$ and Proposition 1.2 and Theorem 2.1 \cite{JDP} for a general function $\Lambda$, weak existence and uniqueness hold for the SDE \eqref{edsnl}. By \cite{Veret}, it actually admits a unique strong solution. For $t>0$, by the Girsanov theorem, the law $\mu_t$ of $X_t$ admits a density $p(t,x)$ with respect to the Lebesgue measure (see Lemma \ref{ctyF} below). The function $p(t,x)$ is a weak solution to the Fokker-Planck equation $ \partial_t p(t,x) + \partial_x \left(\lambda(F(t,x)) p(t,x)\right) = \frac{\sigma^2}{2} \partial_{xx}p(t,x)$. By integration with respect to the spatial variable $x$, we deduce that $F(t,x)$ is a weak solution to the following viscous conservation law: \begin{equation}\label{vpde}
\left\{
\begin{aligned}
&\partial_t F(t,x) + \partial_x\Big(\Lambda(F(t,x))\Big) = \frac{\sigma^2}{2}\partial_{xx} F(t,x),\\
&F_0(x) = m \left((-\infty,x]\right).\\
\end{aligned}
\right.
\end{equation}
The corresponding particle dynamics is \begin{align}\label{PSDEbis}
\displaystyle \breve X^{i,N}_t = X^{i}_0 + \sigma W^{i}_t + \int_0^t \lambda\left(\frac 1 N\sum \limits_{j=1}^{N} \mathbf{1}_{ \left\{ \breve X^{j,N}_s \leq \breve X^{i,N}_s \right\} } \right)\,ds, \quad 1\le i\le N,\;t\in[0,T]. \end{align} As for the initial positions $\left(X^{i}_0\right)_{i \ge 1}$, we will consider both cases of the random initialization ($\left(X^i_0\right)_{i\ge 1}$ i.i.d. according to $m$) and an optimal deterministic initialization which will be made precise in Section \ref{secmainres}.
In fact, for $1\le i\le N$, the coefficient $\lambda(i/N)$ is close to \begin{align}\label{driftPart}
\lambda^{N}(i) = N \left( \Lambda\left(\frac{i}{N}\right) - \Lambda\left(\frac{i-1}{N}\right) \right) \end{align} so that the dynamics is close (see Corollary \ref{corec2systpart} for a precise statement) to the one introduced in \cite{JMal} : \begin{align}\label{PSDE}
\displaystyle X^{i,N}_t = X^{i}_0 + \sigma W^{i}_t + \int_0^t \lambda^N\left(\sum \limits_{j=1}^{N} \mathbf{1}_{ \left\{ X^{j,N}_s \leq X^{i,N}_s \right\} } \right)\,ds, \quad 1\le i\le N,\;t\in[0,T]. \end{align} We denote by $\mu^{N}_t = \frac{1}{N}\sum \limits_{i=1}^N \delta_{X^{i,N}_t}$ the empirical measure and by $F^N(t,x)=\frac 1N\sum\limits_{i=1}^N{\mathbf 1}_{\{X^{i,N}_t\le x\}}$ the empirical cumulative distribution function at time $t$ of this second particle system. Both dynamics are called rank-based models since the drift coefficient only depends on the rank of the $i$-th particle in the system. We call them mean-field rank-based since the interaction between the particles is also of mean-field type. The ability of rank-based models to reproduce stylized empirical properties observed on stock markets \cite{Fern}, has motivated their mathematical study \cite{BanFerKar}. By the Girsanov theorem, the stochastic differential equations \eqref{PSDEbis} and \eqref{PSDE} admit a unique weak solution and, according to \cite{Veret}, they actually admit a unique strong solution. Under concavity of $\Lambda$, Jourdain and Malrieu \cite{JMal} prove the propagation of chaos with optimal rate $N^{-1/2}$ and study the long-time behaviour of the particle system \eqref{PSDE} and its mean-field limit \eqref{edsnl}. For the particle system \eqref{PSDEbis}, this study is extended by Jourdain and Reygner \cite{jrProp} when the diffusion coefficient is no longer constant but also of mean-field rank-based type. For this more general model and without the concavity assumption, Kolli and Shkolnikhov \cite{ShkoKol} recently proved the propagation of chaos with optimal rate $N^{-1/2}$ and convergence of the associated fluctuations when the initial probability measure $m$ admits a bounded density with respect to the Lebesgue measure. We choose to focus on the modified dynamics \eqref{PSDE} because when $y^1<y^2<\hdots<y^N$, then the distribution derivative of $x\mapsto \Lambda\left(\frac 1N\sum\limits_{i=1}^N{\mathbf 1}_{\{y^i\le x\}}\right)$ is $x\mapsto \frac 1N\sum\limits_{i=1}^N\lambda^N(i){\mathbf 1}_{\{y^i\le x\}}$ and not (when $\Lambda$ is not affine) $x\mapsto \frac 1N\sum\limits_{i=1}^N\lambda(i/N){\mathbf 1}_{\{y^i\le x\}}$. For this reason, it is more closely connected to the PDE \eqref{vpde}. As our error analysis is based on a comparison of the mild formulation of the PDE \eqref{vpde} and the perturbed mild formulation satisfied by empirical cumulative distribution function of the Euler discretization of the particle system, we concentrate on \eqref{PSDE}, for which no extra error term appears in this perturbed version but we will also explain how our results extend to \eqref{PSDEbis}. This approach, also called integral representation technique, was used in \cite{TangTsai} to study certain system of particles and its limit. Let us also introduce the Euler discretization with time-step $h \in (0,T]$ of \eqref{PSDE} : \begin{align}\label{edsnh}
\displaystyle X^{i,N,h}_t &= X^i_0+ \sigma W^{i}_t + \int_0^t \lambda^N\left(\sum \limits_{j=1}^{N} \mathbf{1}_{ \left\{ X^{j,N,h}_{\tau^{h}_{s}} \le X^{i,N,h}_{\tau^{h}_{s}} \right\}} \right)\,ds,\;1\le i\le N,\;t\in[0,T] \mbox{ where }\tau^h_s=\lfloor s/h\rfloor h. \end{align} The empirical cumulative distribution function of $\mu^{N,h}_t = \frac{1}{N}\sum \limits_{i=1}^N \delta_{X^{i,N,h}_t}$ is $ F^{N,h}(t,x) := \frac{1}{N} \sum \limits_{j=1}^N \mathbf{1}_{\left\{X^{j,N,h}_t \le x \right\}} $. It is natural and convenient to consider that $\tau^0_s=s$ and $\left(X^{i,N,0}_t\right)_{t\in[0,T],1\le i\le N}=\left(X^{i,N}_t\right)_{t\in[0,T],1\le i\le N}$. Using these notations, we then have by convention that $F^{N,0}(t,x) = F^{N}(t,x)$. Moreover, we will refer to the empirical cumulative distribution function $F^{N,h}_0$ at initialization by $\hat F^N_0$ when choosing positions that are i.i.d. according to $m$ and by $\tilde F^N_0$ when choosing optimal deterministic initial positions. Finally let us define $\left(\breve X^{i,N,h}_t\right)_{t\in[0,T],1\le i\le N}$ like $\left(\breve X^{i,N}_t\right)_{t\in[0,T],1\le i\le N}$ by replacing $\lambda^N(k)$ by $\lambda(k/N)$ in \eqref{edsnh} and set $\breve\mu^{N,h}_t=\frac 1N \sum\limits_{i=1}^N\delta_{X^{i,N,h}_t}$ and $\breve F^{N,h}(t,x) := \frac{1}{N} \sum \limits_{j=1}^N \mathbf{1}_{\left\{\breve X^{j,N,h}_t \le x \right\}} $.
The paper is organized as follows. In Section $2$, we state our results. Taking advantage of the constant diffusion coefficient, we adapt the arguments in \cite{ShkoKol} to obtain propagation of chaos with optimal rate $N^{-1/2}$ for the particle systems \eqref{PSDE} and \eqref{PSDEbis} without any assumption on the initial probability measure $m$. Then we state that the strong rate of convergence of $\mu^{N,h}_t$ to $\mu_t$ for the Wasserstein distance with index one (or equivalently of $F^{N,h}(t,.)$ to $F(t,.)$ for the $L^1$ norm) is ${\mathcal O}\left(\frac{1}{\sqrt{N}}+h\right)$, a result already obtained long ago by Bossy \cite{MBoss} under more regularity assumptions on the initial probability measure $m$ and the function $\Lambda$. Our main result is that the weak rate of convergence is ${\mathcal O}(\frac{1}{N}+h)$. In Section $3$, we introduce the reordered particle system and establish the mild formulation of the PDE \eqref{vpde} satisfied by $F(t,x)$ and the perturbed version satisfied by $F^{N,h}(t,x)$. Section $4$ is dedicated to the proofs of the results in Section $2$. We finally provide numerical experiments in Section $5$ to illustrate our results. Beforehand, we introduce some additional notation.
\subsection*{Notation:} \begin{itemize}
\item We denote by $L_{\Lambda} = \sup_{u \in [0,1]} \left| \lambda(u)\right|$ the Lipchitz constant of $\Lambda$. When $\lambda$ is also assumed to be Lipschitz continuous, we denote similarly its Lipschitz constant by $L_\lambda$.
\item For $ 1 \le p < \infty$, we denote by $L^p({\mathbb R})$ the space of measurable real valued functions which are $L^p$-integrable for the Lebesgue measure i.e. $f\in L^p$ if $\displaystyle \left\Vert f\right\Vert_{L^p} = \left( \int_{{\mathbb R}} |f(x)|^p \; dx \right)^{\frac{1}{p}}<\infty$.
\item The space $L^{\infty}({\mathbb R})$ refers to the space of almost everywhere bounded measurable real valued functions endowed with the norm $\left\Vert f\right\Vert_{L^{\infty}} = \inf\{ C \ge 0: |f(x)|\le C \text{ for almost every } x \in {\mathbb R} \} $. \item The $*$ in $f*g$ stands for the standard convolution operator. \item The set of natural numbers starting at one is denoted by ${\mathbb N}^*$; and the interval notation $\llbracket a,b\rrbracket$ for $a,b\in{\mathbb N}^*$ with $a\le b$ stands for the set of natural numbers between $a$ and $b$. \item For $x,y\in{\mathbb R}$, we set $x\wedge y=\min(x,y)$, $x\vee y=\max(x,y)$ and denote the positive part of $y$ by $y^{+} = \max(y,0)$. \item We denote by $\Gamma$ the Gamma function defined by $\displaystyle \Gamma(x) = \int_{0}^{+ \infty}y^{x-1}\exp(-y)\,dy$ for $x\in (0,+\infty)$. \item To simplify the notations, when a function $g$ defined on $[0,T] \times {\mathbb R}$ and $x \in {\mathbb R}$, we may use sometimes the notation $g_0(x) := g(0,x)$. \end{itemize}
\section{Main results}\label{secmainres}
We will state the propagation of chaos result with optimal rate $N^{-1/2}$ before giving the main results concerning the convergence of the empirical cumulative distribution function $F^{N,h}$ of the Euler discretization with time-step $h$ of the system with $N$ interacting particles towards its limit $F$. We will make an intensive use of the interpretation of the $L^1$-norm of their difference as the Wasserstein distance with index $1$ between $\mu^{N,h}_t=\frac 1N\sum\limits_{i=1}^n\delta_{X^{i,N,h}_t}$ and the law $\mu_t$ of $X_t$.
The Wasserstein distance of index $\rho\ge 1$ between two probability measures $\mu$ and $\nu$ on ${\mathbb R}^d$ is defined by
$$\mathcal{W}_\rho^\rho(\mu, \nu)= \inf \left\{ {\mathbb E}\left[|X-Y|^\rho\right]; \text{Law}(X) = \mu, \text{Law}(Y) = \nu \right\}.$$ In dimension $d=1$, the Hoeffding-Fr\'echet or comonotone coupling given by the inverse transform sampling is optimal: \begin{align}\label{Wasserstein}
\displaystyle \mathcal{W}_\rho^\rho(\mu, \nu) = \int_0^1 \left|F^{-1}_{\mu}(u) - F^{-1}_{\nu}(u) \right|^\rho\,du \end{align}
where ${\mathbb R}\ni x\mapsto F_{\eta}(x) = \eta\left(\left(-\infty;x\right] \right)$ and $(0,1)\ni u\mapsto F^{-1}_{\eta}(u) = \inf\left\{ x \in {\mathbb R}: F_{\eta}(x) \geq u\right\}$ respectively denote the cumulative distribution function and the quantile function of a probability measure $\eta$ on ${\mathbb R}$. Since $\int_0^1 \left|F^{-1}_{\mu}(u) - F^{-1}_{\nu}(u)\right|\,du= \int_{{\mathbb R}}\left|F_{\mu}(x) - F_{\nu}(x) \right|\,dx$, the $\mathcal{W}_1$ distance between two probability measures $\mu$ and $\nu$ on the real line is equal to the $L^1$-norm of the difference between the cumulative distribution functions of $\mu$ and $\nu$:
\begin{equation}\label{w1fdr}
\mathcal{W}_1(\mu,\nu)= \int_{{\mathbb R}}\left|F_{\mu}(x) - F_{\nu}(x) \right|\,dx. \end{equation}
We will also take advantage of the dual formulation of the $\mathcal{W}_1$ distance which holds whatever $d\in{\mathbb N}^*$: \begin{align}
\mathcal{W}_1(\mu,\nu)= \sup_{\varphi \in \mathcal{L}} \left(\int_{{\mathbb R}^d}\varphi(x)\mu(dx)-\int_{{\mathbb R}^d}\varphi(x)\nu(dx) \right)\label{dualwass} \end{align} where $\mathcal{L}$ denotes the set of all $1$-Lipschitz function $\varphi:{\mathbb R}^d \to {\mathbb R}$.
\subsection{Propagation of chaos}
Kolli and Shkolnikov \cite{ShkoKol} prove a quantitative propagation of chaos result at optimal rate $N^{-1/2}$ and convergence of the associated fluctuations for the particle system without time-discretization in the much more general and difficult case when the diffusion coefficient is also mean-field rank-based. Taking advantage of the constant diffusion coefficient, we are going to relax their assumptions on $\lambda$ and $m$ to prove the following result. \begin{thm}\label{thmpropchaos}
Let the initial positions $X^i_0$ be i.i.d. according to $m$ and $(X^i_t)_{t\ge 0}$ denote the solution to the stochastic differential equation nonlinear in the sense of McKean \eqref{edsnl} starting from $X^i_0$ and driven by $(W^i_t)_{t\ge 0}$. If $\lambda$ is Lispchitz continuous, then
$$\forall \rho,T>0,\;\exists \, C<\infty,\;\forall N\in{\mathbb N}^*,\;\max_{1\le i\le N}{\mathbb E}\left[\sup_{t\in[0,T]}\left|X^i_t-X^{i,N}_t\right|^\rho+\sup_{t\in[0,T]}\left|X^i_t-\breve X^{i,N}_t\right|^\rho\right]\le C N^{-\rho/2}.$$
\end{thm}
The estimation ${\mathbb E}\left[\sup_{t\in[0,T]}\left|X^i_t-\breve X^{i,N}_t\right|^\rho\right]\le C N^{-\rho/2}$ follows from Theorem 1.6 \cite{ShkoKol} when $\lambda$ is differentiable with an H\"older continuous derivative and $m$ has a bounded density w.r.t. the Lebesgue measure and a finite moment of order $2+\varepsilon$ for some $\varepsilon>0$. An immediate consequence of Theorem \ref{thmpropchaos} is to quantify the proximity of the two particles dynamics \eqref{PSDEbis} and \eqref{PSDE}. \begin{cor}\label{corec2systpart}
Assume that the initial positions $X^i_0$ are i.i.d. according to $m$ and that $\lambda$ is Lispchitz continuous. Then:
$$\forall \rho,T>0,\;\exists C<\infty,\;\forall N\in{\mathbb N}^*,\;\max_{1\le i\le N}{\mathbb E}\left[\sup_{t\in[0,T]}\left|X^{i,N}_t-\breve X^{i,N}_t\right|^\rho\right]\le C N^{-\rho/2}.$$ \end{cor} \begin{remark} By Theorem 2.1 in \cite{Jabir}, propagation of chaos also holds at optimal rate $N^{-1/2}$ in total variation distance. \end{remark}
\subsection{Initialization error}
In addition to the random initialization of the particles which permits to obtain propagation of chaos, we will also consider deterministic initial positions. \begin{itemize} \item When choosing a random initialization, we denote by $\hat F^N_0(x)=\frac{1}{N}\sum \limits_{i=1}^N \mathbf{1}_{\left\{X^i_0 \le x \right\}}$ and $\hat \mu^{N}_0 = \frac{1}{N}\sum \limits_{i=1}^N \delta_{X^{i}_0}$ the empirical cumulative distribution function and the empirical measure of the $N$ first random variables in the sequence $(X^i_0)_{i\ge 1}$ i.i.d. according to $m$. \item When choosing a deterministic initialization, we seek to construct a family $x^{N}_{1}\le x_2^N\le \hdots\le x^N_N$ of initial positions minimizing the $L^1$ norm of the difference between the piecewise constant function $\tilde F^N_0(x) = \frac{1}{N}\sum \limits_{i=1}^N \mathbf{1}_{\left\{x^N_{i} \le x \right\}} $ and $F_0$. According to \eqref{Wasserstein},
$$\int_{\mathbb R}\left|\tilde F^N_0(x)-F_0(x)\right|\,dx=\sum \limits_{i=1}^N \int_{\frac{i-1}{N}}^{\frac{i}{N}} \left|x^N_{i} - F^{-1}_0(u) \right|\,du.$$
Since, as remarked in \cite{jrdcds}, for $i\in\llbracket1,N\rrbracket$, $\displaystyle y\mapsto N\int_{\frac{i-1}{N}}^{\frac{i}{N}} \left|y -F^{-1}_0(u) \right|\,du$ is minimal for $y$ equal to the median $F_0^{-1}\left(\frac{2i-1}{2N}\right)$ of the image of the uniform law on $\left[\frac{i-1}{N},\frac i N \right]$ by $F_0^{-1}$, we choose $X^i_0=x_i^N=F_0^{-1}\left(\frac{2i-1}{2N}\right)$. We denote by $\tilde \mu^{N}_0 = \frac{1}{N}\sum \limits_{i=1}^N \delta_{F_0^{-1}\left(\frac{2i-1}{2N}\right)}$ the associated empirical measure. \end{itemize}
The next proposition, discusses assumptions under which the $L^1$-norm of the difference between $F_0$ and $\hat F_0^N$ or $\tilde F_0^N$ is of order $N^{-1/2}$. \begin{prop}\label{propvitsqn}
We denote for simplicity $\displaystyle \int_{{\mathbb R}}|x|^{2+}m(dx)<\infty$ the existence of $\varepsilon>0$ such that $\displaystyle \int_{{\mathbb R}}|x|^{2+\varepsilon}m(dx)<\infty$ and $\displaystyle \int_{{\mathbb R}}|x|^{2-}m(dx)<\infty$ the fact that $\displaystyle \int_{{\mathbb R}}|x|^{2-\varepsilon}m(dx)<\infty$ for each $\varepsilon\in (0,2]$. We have the following results concerning the ${\mathcal O}(N^{-1/2})$ behaviour of the errors: \begin{equation}
\begin{array}{ccccc}
& & \displaystyle{\sup_{N\ge 1} }\sqrt{N}{\mathbb E} \left[\mathcal{W}_1\left(\hat \mu^N_0,m \right) \right]<\infty& &\\
& & \Updownarrow & &\\
\displaystyle \int_{{\mathbb R}}|x|^{2+}m(dx)<\infty & \Rightarrow &\displaystyle \int_{{\mathbb R}}\sqrt{F_0(x)(1-F_0(x))}\,dx<\infty & \Rightarrow &\displaystyle \int_{{\mathbb R}}|x|^{2}m(dx)<\infty
\end{array} \label{stiniid} \end{equation} \begin{equation}
\begin{array}{ccccc}
\displaystyle \int_{{\mathbb R}}|x|^{2}m(dx)<\infty &\Rightarrow & \displaystyle{\sup_{x\ge 0}}\,x^2\left(F_0(-x)+1-F_0(x)\right)<\infty&\Rightarrow &\displaystyle \int_{{\mathbb R}}|x|^{2-}m(dx)<\infty\\
& &\Updownarrow & & \\
& & \displaystyle{\sup_{N \ge 1} }\sqrt{N} \, \mathcal{W}_1 \left(\tilde \mu^N_0 ,m\right)<\infty& .
\end{array}\label{stindet} \end{equation} \end{prop} For the statements \eqref{stiniid} which concern the random initialization, we refer to \cite{bobkovledoux} Section 3.1. The statements \eqref{stindet} concerning the optimal deterministic initialization are proved in Section \ref{proofinitdet}.
Concerning the weak error, since the empirical cumulative distribution function of i.i.d. samples is unbiased, ${\mathbb E}\left[\hat F_0^N(x)\right]=F_0(x)$ for all $N\ge 1$ and $x\in{\mathbb R}$ and then \begin{equation}
\int_{\mathbb R} \left|{\mathbb E}\left[\hat F^N_0(x)\right]-F_0(x)\right|\,dx=0.
\label{iidsansbiais} \end{equation} As for the deterministic initialization, we have that: \begin{align}\label{w1opt}
\displaystyle \int_{\mathbb R} \left|\tilde F_0^N(x)-F_0(x)\right|\,dx = &\int_{-\infty}^{F_0^{-1}\left(\frac{1}{2N}\right)} F_0(x)\,dx + \sum_{i=1}^{N-1} \int_{F_0^{-1}\left(\frac{2i-1}{2N}\right)}^{F_0^{-1}\left(\frac{2i+1}{2N}\right)}\left|F_0(x)- \frac{i}{N} \right|\,dx +\int_{F_0^{-1}\left(\frac{2N-1}{2N}\right)}^{+\infty}\left(1-F_0(x)\right)\,dx \end{align} where the integrand is not greater than $1/(2N)$. When $m([c,d])=1$ with $-\infty<c\le d<\infty$, the integrand vanishes outside the interval $[c,d]$. One then easily deduces the next proposition proved in \cite{jrdcds} by using the alternative formulation: \begin{equation}
\displaystyle \int_{\mathbb R} \left|\tilde F_0^N(x)-F_0(x)\right|\,dx = \int_0^1\left|\left(\tilde F_0^N\right)^{-1}(u)-F_0^{-1}(u)\right|\,du = \sum_{i=1}^N \int_{\frac{2i-1}{2N}}^{\frac{i}{N}}\left(F_0^{-1}(u)-F_0^{-1}\left(u-\frac{1}{2N}\right)\right)\,du.\label{alterw1} \end{equation} \begin{prop}\label{detL1init}
When $m$ is compactly supported i.e. there exists $ -\infty<c\le d<\infty$ such that $m([c,d])=1$, then $$\mathcal{W}_1 \left(\tilde \mu^N_0 ,m\right) \le \frac{d-c}{2N}.$$ \end{prop}
\subsection{Strong and weak ${\cal W}_1$-errors} We recall that $\lambda$ is the derivative of the $C^1$ function $\Lambda:[0,1]\to{\mathbb R}$.
Let us state our estimation of the strong error which is proved in Section \ref{secproofthm1}.
\begin{thm}\label{thm1}
Assume that for some $\rho >1$, $\displaystyle \int_{{\mathbb R}}|x|^{\rho}m(dx) < \infty$ and assume either that the initial positions are optimal deterministic or the initial positions are i.i.d. according to $m$. Then
$$\exists C< \infty, \forall N \in {\mathbb N}^{*}, \quad \sup_{t \leq T}{\mathbb E}\left[ \mathcal{W}_1\left(\mu^{N,0}_t,\mu_t \right)\right] \leq C \left({\mathbb E}\left[ \mathcal{W}_1\left(\mu^{N}_0,m\right)\right] + \frac{1}{\sqrt{N}}\right). $$
Moreover, if $\lambda$ is Lipschitz continuous then:
$$\exists C< \infty, \forall N \in {\mathbb N}^{*}, \forall h \in (0,T], \quad \sup_{t \leq T}{\mathbb E}\left[ \mathcal{W}_1\left(\mu^{N,h}_t,\mu_t \right)\right] \le C \left({\mathbb E}\left[ \mathcal{W}_1\left(\mu^{N}_0,m \right)\right] + \frac{1}{\sqrt{N}} + h\right). $$ \end{thm}
Combining the theorem with Proposition \ref{propvitsqn}, we have the following corollary: \begin{cor}\label{cor1} Assume that the initial positions are \begin{itemize} \item either i.i.d. according to $m$ and $\displaystyle \int_{{\mathbb R}}\sqrt{F_0(x)(1 - F_0(x))}\,dx < \infty$,
\item or optimal deterministic and $\displaystyle \sup_{x \ge 1} x\int_{x}^{+\infty} \left(F_0(-y) + 1 - F_0(y) \right)\,dy<\infty$. \end{itemize} Then:
$$\exists C< \infty, \forall N \in {\mathbb N}^{*}, \quad \sup_{t \leq T}{\mathbb E}\left[ \mathcal{W}_1\left(\mu^{N,0}_t,\mu_t \right)\right] \le \frac{C}{\sqrt{N}}. $$
Moreover, if $\lambda$ is Lipschitz continuous then:
$$\exists C< \infty, \forall N \in {\mathbb N}^{*}, \forall h \in (0,T], \quad \sup_{t \leq T}{\mathbb E}\left[ \mathcal{W}_1\left(\mu^{N,h}_t,\mu_t \right)\right] \le C\left(\frac{1}{\sqrt{N}} + h\right). $$ \end{cor}
Let us now state our main result, proved in Section \ref{secproofthm2}, concerning the weak error : the $L^1$-weak error between the empirical cumulative distribution function $F^{N,h}$ of the Euler discretization with time-step $h$ of the system with $N$ interacting particles and its limit $F$ is $\mathcal O\left(\frac{1}{N} +h \right)$. We denote by ${\mathbb E}\left[\mu^{N,h}_t\right]$ the probability measure on ${\mathbb R}$ defined by
$$\int_{\mathbb R}\varphi(x) {\mathbb E}\left[\mu^{N,h}_t\right](dx)= {\mathbb E}\left[\int_{\mathbb R}\varphi(x)\mu^{N,h}_t(dx)\right]={\mathbb E} \left[\frac{1}{N}\sum \limits_{i=1}^N \varphi\left( X^{i,N,h}_t\right) \right]$$for each $\varphi:{\mathbb R}\to{\mathbb R}$ measurable and bounded. The cumulative distribution function of ${\mathbb E}\left[\mu^{N,h}_t\right]$ is equal to ${\mathbb E}\left[F^{N,h}(t,x)\right]$ and $\displaystyle \mathcal{W}_1 \left({\mathbb E}\left[\mu^{N,h}_t\right],\mu_t\right) = \int_{\mathbb R} \left|{\mathbb E}\left[F^{N,h}(t,x)\right]-F(t,x)\right|\,dx$. \begin{thm}\label{thmBias} Assume that $\lambda$ is Lipschitz continuous and the initial positions are \begin{itemize} \item either i.i.d. according to $m$ and $\displaystyle \int_{{\mathbb R}}\sqrt{F_0(x)(1 - F_0(x))}\,dx < \infty$,
\item or optimal deterministic and $\displaystyle \sup_{x \ge 1} x\int_{x}^{+\infty} \left(F_0(-y) + 1 - F_0(y) \right)\,dy<\infty$. \end{itemize} Then:
$$\exists C_b< \infty, \forall N \in {\mathbb N}^{*}, \forall h \in [0,T],\quad \sup_{t \leq T}\mathcal{W}_1 \left({\mathbb E}\left[\mu^{N,h}_t\right],\mu_t\right) \le C_b \left(\mathcal{W}_1 \left({\mathbb E}\left[\mu^{N}_0\right],m\right) + \left(\frac{1}{N} + h \right)\right). $$ \end{thm}
Combining the theorem with \eqref{iidsansbiais} and Proposition \ref{detL1init}, we obtain the following corollary: \begin{cor}\label{corBiais}
Assume that $\lambda$ is Lipschitz continuous and the initial positions are\begin{itemize}
\item either i.i.d. according to $m$ and $\displaystyle \int_{\mathbb R}\sqrt{F_0(x)(1-F_0(x))}\,dx<\infty$,
\item or optimal deterministic with $m$ compactly supported. \end{itemize}Then:
$$\exists C_b< \infty, \forall N \in {\mathbb N}^{*}, \forall h \in [0,T],\quad \sup_{t \leq T}\mathcal{W}_1 \left({\mathbb E}\left[\mu^{N,h}_t\right],\mu_t\right) \le C_b\left(\frac{1}{N} + h \right). $$ \end{cor}
Using the dual formulation \eqref{dualwass} of the Wasserstein distance, we deduce that if $\varphi:{\mathbb R}\to{\mathbb R}$ is Lipschitz continuous with constant ${\rm Lip}(\varphi)$ then $$\forall N \in {\mathbb N}^{*}, \forall t,h \in [0,T],\quad \left| {\mathbb E} \left[\frac{1}{N}\sum \limits_{i=1}^N \varphi\left( X^{i,N,h}_t\right) \right] - {\mathbb E} \left[\varphi(X_t)\right]\right| \le C_b{\rm Lip}(\varphi) \left(\frac{1}{N} + h\right).$$
\begin{remark}\label{remcheck}
For the dynamics \eqref{PSDEbis} with initial positions deterministic and given by $x^N_{i} = F^{-1}_0\left(\frac{i}{N}\right)$ when $i=1,\ldots,N-1$ and $x^N_N = F^{-1}_0\left( 1 - \frac{1}{2N}\right)$, Bossy \cite{MBoss} proved an estimation also dealing with the supremum of the expected error between $\breve F^{N,h}(t,x)$ and $F(t,x)$ similar to the last statement in Corollary \ref{cor1}: $$\exists C< \infty, \forall N \in {\mathbb N}^{*}, \forall h \in (0,T],\;\sup_{t \leq T}{\mathbb E}\left[ \mathcal{W}_1\left(\breve \mu^{N,h}_t,\mu_t \right)\right]+ \sup_{(t,x) \in [0,T]\times {\mathbb R}}{\mathbb E}\left[ \left|\breve F^{N,h}(t,x) - F(t,x)\right| \right]\le C\left(\frac{1}{\sqrt{N}} + h\right).$$
She assumes additional regularity on the coefficient $\Lambda$, namely that $\Lambda$ is $C^3$, and on the initial measure $m$, namely that $F_0$ is $C^2$ bounded with bounded first and second order derivatives in $x$ and that $\exists \;M, \beta>0, \; \alpha \geq 0 $ such that $|\partial_x F_0(x)| \leq \alpha \exp\left(-\beta x^2/2 \right)$ when $|x| >M$. Her proof is based on the regularity of the backward Kolmogorov PDE associated with the generator of the diffusion \eqref{edsnl}. By contrast, our approach is based on a comparison of the mild formulation of the forward in time PDE \eqref{vpde} satisfied by $F(t,x)$ and the pertubed mild formulation satisfied by $F^{N,h}(t,x)$.
In fact, all the above results hold with $\mu^{N,h}_t$ replaced by $\breve \mu^{N,h}_t$. For those concerning $\displaystyle \sup_{t \leq T}{\mathbb E}\left[ \mathcal{W}_1\left(\mu^{N}_t,\mu_t \right)\right]$, we just need to add the assumption that $\lambda$ is H\"older continuous with exponent $1/2$ to ensure that $\sup\limits_{1\le i\le N}\sqrt{N}\left|\lambda^N(i)-\lambda(i/N)\right|$ $<\infty$. Notice that under Lipschitz continuity of $\lambda$, we even get $\sup\limits_{1\le i\le N}N\left|\lambda^N(i)-\lambda(i/N)\right|<\infty$. See Remark \ref{mildEDSbis} below, where we outline how to adapt the proofs. \end{remark}
\section{Dynamics of the reordered particle system and mild formulations} The reordering of mean-field rank-based particle systems without time discretization has been first introduced in \cite{JSPA} and has proved to be a very useful tool in the study of the limit $N\to\infty$ with vanishing viscosity (the parameter $\sigma$ depends on $N$ and tends to $0$ as $N\to\infty$) \cite{JAppPro,JMelWoy} (the latter when the driving Brownian motions are replaced by symmetric $\alpha$-stable L\'evy processes with $\alpha>1$), the long time behaviour of both the particle system and its mean-field limit \cite{jrProp} and the small noise limit $\sigma\to 0$ of the particle system \cite{jrSmall}. Before deriving the dynamics of the reordering of the Euler discretization \eqref{edsnh}, let us check the existence of the density $p(t,x)$ of $X_t$ for $t>0$, which guarantees that, in the sense of distributions, $\partial_x \Lambda(F(t,x)) = \lambda(F(t,x))p(t,x)$ so that $F(t,x)$ is a weak solution of the viscous scalar conservation law \eqref{vpde}. Since the drift coefficient in \eqref{edsnl} is bounded, the Girsanov theorem easily implies the next statement. \begin{lem}\label{ctyF}
For $t>0$, $X_t$ admits a density $p(t,x)$ with respect to the Lebesgue measure. \end{lem}
Let for each $t\ge 0$, $\eta_t$ be a permutation of $\{1,\hdots,N\}$ such that $X^{\eta_t(1),N,h}_t\le X^{\eta_t(2),N,h}_t\le\hdots\le X^{\eta_t(N),N,h}_t$ and $\left(Y^{i,N,h}_t=X^{\eta_t(i),N,h}_t\right)_{i \in \llbracket1,N\rrbracket}$ denote the increasing reordering also called order statistics of $\left(X^{i,N,h}_t\right)_{i \in \llbracket1,N\rrbracket}$. Even if the empirical measures of the reordered and original positions do not coincide in general at the level of sample-paths, one has, for each $t\ge 0$, $ \frac{1}{N}\sum\limits_{i=1}^N\delta_{X^{i,N,h}_t}=\frac{1}{N}\sum\limits_{i=1}^N\delta_{Y^{i,N,h}_t}$ and therefore $\displaystyle F^{N,h}(t,x)=\frac{1}{N}\sum_{i=1}^N\mathbf{1}_{\left\{Y^{i,N,h}_t \le x \right\}}$. By the Girsanov Theorem, reasoning like in the proof of Lemma \ref{ctyF}, we show that for $t>0$, the vector $\left(X^{1,N,h}_t, X^{2,N,h}_t, \dots, X^{N,N,h}_t \right)$ admits a density with respect to the Lebesgue measure on ${\mathbb R}^N$ and therefore \begin{align}\label{exchanG}
\forall t>0, a.s., \text{ the original (resp. reordered) particles have distinct positions.} \end{align} The collisions are possible along time but have zero probability to occur at a fixed time $t>0$ (see e.g. \cite{Sarantsev}). We are going to check that the function $F(t,x)$ solves a mild formulation of the PDE \eqref{vpde} and $F^{N,h}(t,x)$ solves a perturbed version of this mild formulation. To do so, it is convenient to obtain the dynamics of the reordered positions $Y^{i,N,h}_t$. Let $\overline{\tau}^h_s = \lceil s/h \rceil h $ denote the discretization time right after $s$. We recall that $\tau^h_s=\lfloor s/h\rfloor h$ denotes the discretization time right before $s$. We set $t_k = kh$ for $k \in {\mathbb N}$. For $s \in [t_k,t_{k+1})$, $\tau^h_s = t_k$ and, for $s \in (t_k, t_{k+1}]$, $\overline{\tau}^h_s = t_{k+1}$. For $t>0$, let $\eta_t^{-1}$ denote the inverse of the permutation $\eta_t$. By \eqref{exchanG}, $a.s.$, for each $k\in{\mathbb N}^*$, the positions $\left(X^{i,N,h}_{t_k}\right)_{1\le i\le N}$ are distinct and for $t$ in the time-interval $[t_k,t_{k+1})$, $X^{i,N,h}_t$ evolves with the drift coefficient $\lambda^N\left(\eta_{t_k}^{-1}(i)\right)=\lambda^N\left(\eta_{\tau^h_t}^{-1}(i)\right)$. To obtain the same expression of the drift coefficient on the first time interval $[0,t_1)$ we will use from now on the convention \begin{equation}
\eta_0^{-1}(i)=\sum_{j=1}^{N}{\mathbf 1}_{\left\{X^{j}_0\le X^{i}_0 \right\}} \quad \mbox{ for }1\le i\le N.\label{convinvinit} \end{equation} With this convention, which is consistant with the usual definition of the inverse of a permutation only if the initial positions are distinct, we have $$dX^{i,N,h}_t=\sigma dW^i_t+\lambda^N\left(\eta_{\tau^h_t}^{-1}(i)\right)\,dt,\;1\le i\le N.$$ By the Girsanov theorem, we may define a new probability measure equivalent to the original one on each finite time horizon under which the processes $\left(X^{i,N,h}_t-X^i_0=W^i_t+\int_0^t\lambda^N\left(\eta_{\tau_s}^{-1}(i)\right)\,ds\right)_{t\ge 0,1\le i\le N}$ are independent Brownian motions. Applying Lemma 3.7 \cite{Szn91}, which states that under this probability measure, the reordered positions evolve as a $N$-dimensional Brownian motion normally reflected at the boundary of the simplex (also called Weyl chamber), we deduce that \begin{equation*}
dY^{i,N,h}_t = \sum \limits_{j=1}^{N} \mathbf{1}_{\left \{ Y^{i,N,h}_t = X^{j,N,h}_t \right \}}\left(\sigma dW^{j}_t+\lambda^N\left(\eta_{\tau^h_t}^{-1}(j)\right)\,dt\right)+ \left(\gamma^{i}_t - \gamma^{i+1}_t \right)\,d|K|_t,\;1\le i\le N
\end{equation*}
where the local-time process $K$ with coordinates $K^i_t=\int_0^t \left( \gamma^{i}_s - \gamma^{i+1}_s\right)\,d|K|_s$ is an ${\mathbb R}^N$-valued continuous process with finite variation $|K|$ such that: \begin{align}\label{procVarFin}
d|K|_t \; \text{a.e.} \;, \gamma^{1}_t = \gamma^{N+1}_t = 0\mbox{ and for }2\le i\le N,\; \gamma^{i}_t \geq 0 \; \text{and} \; \gamma^{i}_t\left(Y^{i,N,h}_t - Y^{i-1,N,h}_t \right) = 0. \end{align} Defining a $N$-dimensional Brownian motion $\left(\beta^1,\hdots,\beta^N\right)$ by $\beta^{i}_t = \sum \limits_{j=1}^{N} \displaystyle \int_0^t \mathbf{1}_{\left \{ Y^{i,N,h}_s = X^{j,N,h}_s \right \}}\,dW^{j}_s$ and using the definition of $\eta_t$ and \eqref{exchanG}, we have \begin{equation}\label{dynamicY}
dY^{i,N,h}_t = \sigma d\beta^{i}_t+ \lambda^N\left( \eta^{-1}_{\tau^h_t}\left(\eta_t(i)\right) \right)\,dt + \left(\gamma^{i}_t - \gamma^{i+1}_t \right)\,d|K|_t,\;1\le i\le N. \end{equation} Denoting by $G_t(x) = \exp(-\frac{x^2}{2\sigma^2t}) \Big/ \sqrt{2\pi\sigma^2t} $ the probability density function of the normal law $\mathcal{N}(0,\sigma^2t)$, we are now ready to state the mild formulation of the PDE \eqref{vpde} satisfied by $F(t,x)$ and the perturbed version satisfied by $F^{N,h}(t,x)$.
\begin{prop}\label{propspde} For each $t\ge 0$ and each $h \in [0,T]$ , we have $dx$ a.e.: \begin{align}
\displaystyle F(t,x) =& \quad G_t*F_0(x) - \int_0^t \partial_xG_{t-s}*\Lambda(F(s,.))(x)\,ds,\label{milf}\\
\text{a.s.} \quad F^{N,h}&(t,x) = \quad G_t*F^{N,h}_0(x) - \int_0^t \partial_xG_{t-s}*\Lambda(F^{N,h}(s,.))(x)\,ds - \frac{\sigma}{N}\sum \limits_{i=1}^N \int_0^t G_{t-s}(X^{i,N,h}_s -x)\,dW^{i}_s \notag \\
&+ \frac{1}{N}\sum \limits_{i=1}^N \int_0^t G_{t-s}(Y^{i,N,h}_s -x) \left[ \lambda^N\left(i\right) - \lambda^N\left(\eta_{\tau^h_s}^{-1}(\eta_s(i))\right) \right]\,ds.\label{milfnh} \end{align} \end{prop} \begin{remark} When $h=0$, one should notice that the fourth term in $F^{N,h}(t,x)$ is null so that: $$\displaystyle F^{N,0}(t,x) = G_t*F^{N,0}_0(x) - \int_0^t \partial_xG_{t-s}*\Lambda(F^{N,0}(s,.))(x)\,ds - \frac{\sigma}{N}\sum \limits_{i=1}^N \int_0^t G_{t-s}(X^{i,N,0}_s -x)\,dW^{i}_s.$$ \end{remark} \begin{remark}\label{mildEDSbis}
Let similarly for each $t\ge 0$, $\breve \eta_t$ be a permutation of $\{1,\hdots,N\}$ such that $\breve X^{\breve \eta_t(1),N,h}_t\le \breve X^{\breve \eta_t(2),N,h}_t\le\hdots\le \breve X^{\breve \eta_t(N),N,h}_t$ and $\left(\breve Y^{i,N,h}_t=\breve X^{\breve \eta_t(i),N,h}_t\right)_{i \in \llbracket1,N\rrbracket}$. Let also $\breve \eta_t^{-1}$ denote the inverse of the permutation $\breve \eta_t$ for $t>0$ and $\breve\eta_0^{-1}=\eta_0^{-1}$. Reasoning like in the proof of Proposition \ref{propspde}, we may derive the perturbed mild equation satisfied by the associated empirical cumulative distribution function $\breve F^{N,h}(t,x)$: \begin{align*} \breve F^{N,h}(t,x) = &\quad G_t*F^{N,h}_0(x) - \int_0^t \partial_xG_{t-s}*\Lambda(\breve F^{N,h}(s,.))(x)\,ds - \frac{\sigma}{N}\sum \limits_{i=1}^N \int_0^t G_{t-s}(\breve X^{i,N,h}_s -x)\,dW^{i}_s \notag \\
&+ \frac{1}{N}\sum \limits_{i=1}^N \int_0^t G_{t-s}(\breve Y^{i,N,h}_s -x) \left[ \lambda^N\left(i\right) - \lambda\left(\breve \eta_{\tau^h_s}^{-1}(\breve \eta_s(i))/N\right) \right]\,ds. \end{align*}
Using the estimation $$\int_{\mathbb R}\left|\frac{1}{N}\sum \limits_{i=1}^N\int_0^t G_{t-s}\left(\breve Y^{i,N,h}_s -x\right) \left[ \lambda^N\left(\breve \eta_{\tau^h_s}^{-1}(\breve \eta_s(i))\right)-\lambda\left(\breve \eta_{\tau^h_s}^{-1}(\breve \eta_s(i))/N\right) \right]\,ds\right|\,dx\le t\max_{1\le j\le N}|\lambda^N(j)-\lambda(j/N)|$$ of the additional error term in comparison with \eqref{milfnh}, we may adapt all proofs to check the statements at the end of Remark \ref{remcheck}. \end{remark} \begin{proof}
Let $t>0$, $f$ be a $C^1$ and compactly supported function on ${\mathbb R}$ and $\displaystyle \varphi(s,x)=\int_{\mathbb R} {\mathbf 1}_{\{x\le y\}}G_{t-s}*f(y)\,dy$ be the convolution of $G_{t-s}$ with $\displaystyle x\mapsto\int_x^{+\infty}f(y)\,dy$ for $(s,x)\in[0,t)\times{\mathbb R}$ and $\displaystyle \varphi(t,x)=\int_{\mathbb R} {\mathbf 1}_{\{x\le y\}}f(y)\,dy$. The function $\varphi(s,x)$ is continuously differentiable w.r.t. to $s$ and twice continuously differentiable w.r.t. to $x$ on $[0,t]\times{\mathbb R}$ and solves
\begin{equation}\label{edpvarphi}
\partial_s \varphi(s,x)+\frac{\sigma^2}{2}\partial_{xx}\varphi(s,x)=0\mbox{ for }(s,x)\in[0,t]\times{\mathbb R}.
\end{equation}
Computing $\varphi(t,X_t)$ where $(X_s)_{s\ge 0}$ solves \eqref{edsnl} and using \eqref{edpvarphi}, we obtain that:
$$\varphi(t,X_t)=\varphi(0,X_0)-\sigma\int_0^tG_{t-s}*f(X_s)\,dW_s-\int_0^t\lambda(F(s,X_s))G_{t-s}*f(X_s)\,ds.$$
Since, on $[0,t)\times{\mathbb R}$, $G_{t-s}*f(x)$ is bounded by the supremum of $|f|$, the expectation of the stochastic integral is zero. By Fubini's theorem and since $G_t$ is even, the expectations of $\varphi(t,X_t)$ and $\varphi(0,X_0)$ are respectively equal to $\displaystyle \int_{\mathbb R} \int_{\mathbb R}{\mathbf 1}_{\{x\le y\}}\mu_t(dx)f(y)\,dy=\int_{\mathbb R} F(t,y)f(y)\,dy$ and $\displaystyle \int_{\mathbb R} F_0(y)G_t*f(y)\,dy=\int_{\mathbb R} G_t*F_0(y)f(y)\,dy$. Using Fubini's theorem, the equality $\displaystyle G_{t-s}*f(x)=-\int_{\mathbb R}{\mathbf 1}_{\{x\le y\}}\partial_yG_{t-s}*f(y)\,dy$, the fact that, by the chain rule for continuous functions with finite variation, $\displaystyle \int_{\mathbb R}{\mathbf 1}_{\{x\le y\}}\lambda(F(s,x))p(s,x)\,dx=\Lambda(F(s,y))-\Lambda(0)$, the equality $\displaystyle \int_{\mathbb R}\partial_yG_{t-s}*f(y)\,dy=0$ and the oddness of $\partial_y G_{t-s}$, we obtain that the expectation of the last term on the right-hand side is equal to \begin{align*}
\displaystyle \int_0^t\int_{\mathbb R}\int_{\mathbb R}{\mathbf 1}_{\{x\le y\}}\lambda(F(s,x))p(s,x)\,dx\,\partial_yG_{t-s}*f(y)\,dy\,ds
&=\int_0^t\int_{\mathbb R}(\Lambda(F(s,y))-\Lambda(0))\partial_yG_{t-s}*f(y)\,dy\,ds\\
&=-\int_0^t\int_{\mathbb R}\partial_yG_{t-s}*\Lambda(F(s,.))(y)f(y)\,dy\,ds. \end{align*} Exchanging the time and space integrals by Fubini's theorem, we deduce that $$\int_{\mathbb R} F(t,x)f(x)dx=\int_{\mathbb R} G_t*F_0(x)f(x)\,dx-\int_{\mathbb R} f(x)\int_0^t\partial_x G_{t-s}*\Lambda(F(s,.))(x)\,ds\,dx.$$
Since $f$ is arbitrary, we conclude that $F$ satisfies the mild formulation \eqref{milf}.
Let us now establish that $F^{N,h}$ satisfies a perturbed version of this equation given by \eqref{milfnh}.
By computing $\varphi(t,Y^{i,N,h}_t)$ by It\^o's formula, using \eqref{edpvarphi} and summing over $i\in\{1,\hdots,N\}$, we obtain
\begin{align}\label{itoy}
\int_{{\mathbb R}}\sum_{i=1}^N{\mathbf 1}_{\{Y^{i,N,h}_t\le y\}}f(y)\,dy=&\int_{{\mathbb R}}\sum_{i=1}^N{\mathbf 1}_{\{Y^{i,N,h}_0\le y\}}G_t*f(y)\,dy-\sum_{i=1}^N\int_0^tG_{t-s}*f(Y^{i,N,h}_s)\left(\sigma\,d\beta^i_s+\lambda^N\left( \eta^{-1}_{\tau^h_s}(\eta_s(i) \right)\,ds\right)\\
&+\sum_{i=1}^N\int_0^t\partial_x\varphi(s,Y^{i,N,h}_s)(\gamma^i_s-\gamma^{i+1}_s)\,d|K|_s.\notag
\end{align}
By summation by parts and \eqref{procVarFin},
$$\sum_{i=1}^N\int_0^t\partial_x\varphi(s,Y^{i,N,h}_s)(\gamma^i_s-\gamma^{i+1}_s)\,d|K|_s =\sum_{i=2}^N\int_0^t(\partial_x\varphi(s,Y^{i,N,h}_s)-\partial_x\varphi(s,Y^{i-1,N,h}_s))\gamma^i_s\,d|K|_s=0.$$
Since the empirical cumulative distribution functions of the original and the reordered systems at time $t$ coincide and the function $G_t$ is even, the left-hand side and the first term in the right-hand side are respectively equal to $\displaystyle N\int_{\mathbb R} F^{N,h}(t,y)f(y)\,dy$ and $ \displaystyle N\int_{\mathbb R} G_t*F^{N,h}(0,y)f(y)\,dy$. The definition of the Brownian motion $\beta$ and \eqref {exchanG} imply that
$$\sum_{i=1}^N\int_0^tG_{t-s}*f(Y^{i,N,h}_s)\,d\beta^i_s=\sum_{j=1}^N\int_0^tG_{t-s}*f(X^{j,N,h}_s)\,dW^j_s.$$
Dividing \eqref{itoy} by $N$, we deduce that
\begin{align*}
\displaystyle \int_{\mathbb R} F^{N,h}(t,y)f(y)\,dy=&\int_{\mathbb R} G_t*F^{N,h}(0,y)f(y)\,dy-\frac{\sigma}{N}\sum_{i=1}^N\int_0^tG_{t-s}*f\left(X^{i,N,h}_s\right)\,dW^i_s\\
&-\frac{1}{N}\sum_{i=1}^N\int_0^tG_{t-s}*f(Y^{i,N,h}_s)\lambda^N\left( \eta^{-1}_{\tau^h_s}(\eta_s(i) \right)\,ds.
\end{align*}
We are going to add and substract \begin{align*}
\displaystyle \frac 1 N\sum_{i=1}^NG_{t-s}*f(Y^{i,N,h}_s)\lambda^N(i)&=-\int_{{\mathbb R}}\partial_yG_{t-s}*f(y)\sum_{i=1}^N{\mathbf 1}_{\{Y^{i,N,h}_s\le y\}}\left(\Lambda(i/N)-\Lambda((i-1)/N)\right)\,dy\\
&=-\int_{\mathbb R}\partial_yG_{t-s}*f(y)\left(\Lambda\left(F^{N,h}(s,y)\right)-\Lambda(0)\right)\,dy=\int_{\mathbb R} f(y)\partial_yG_{t-s}*\Lambda\left(F^{N,h}(s,.)\right)(y)\,dy. \end{align*}
On the other hand, since $f$ is square integrable and with the use of Young's inequality for the product and the estimate \eqref{GSquareEsp} from Lemma \ref{EstimHeatEq}, we have that:
\begin{align*}
\displaystyle \int_{{\mathbb R}} \left\{\int_0^t \left| G_{t-s}(X^{i,N,h}_s -x)f(x)\right|^2\,ds \right\}^{1/2}dx = &\int_{{\mathbb R}} |f(x)|\left\{\int_0^t| G_{t-s}(X^{i,N,h}_s -x)|^2\,ds \right\}^{1/2}\,dx \\
&\leq \frac{1}{2}\int_{{\mathbb R}} f^2(x)\,dx + \frac{1}{2}\int_{{\mathbb R}} \int_0^t G^2_{t-s}(X^{i,N,h}_s -x)\,ds\,dx \\
&= \frac{1}{2}\int_{{\mathbb R}} f^2(x)\,dx + \frac{1}{2\sigma} \sqrt{\frac{t}{\pi}} < \infty.
\end{align*}
For that reason, we can use a stochastic Fubini theorem stated by Veraar \cite{VERAA} and recalled in Lemma \ref{StochFub} to deduce that $\displaystyle \frac{\sigma}{N}\sum_{i=1}^N\int_0^tG_{t-s}*f\left(X^{i,N,h}_s\right)\,dW^i_s=\frac{\sigma}{N}\sum \limits_{i=1}^N \int_{{\mathbb R}}f(x)\left\{ \int_0^t G_{t-s}\left(X^{i,N,h}_s -x\right)\,dW^{i}_s\right\}\,dx$. Therefore \begin{align*}
\int_{\mathbb R} F^{N,h}(t,x)f(x)\,dx=&\int_{\mathbb R} G_t*F^{N,h}(0,x)f(x)\,dx-\frac{\sigma}{N}\sum \limits_{i=1}^N \int_{{\mathbb R}}f(x)\left\{ \int_0^t G_{t-s}(X^{i,N,h}_s -x)\,dW^{i}_s\right\}\,dx\\
&-\int_0^t\int_{\mathbb R} f(x)\partial_xG_{t-s}*\Lambda\left(F^{N,h}(s,.)\right)(x)\,dx\,ds\\
&+\frac{1}{N}\sum_{i=1}^N\int_0^t\int_{\mathbb R} G_{t-s}\left(Y^{i,N,h}_s-x\right)f(x)\,dx\left\{\lambda^N(i)-\lambda^N\left( \eta^{-1}_{\tau^h_s}(\eta_s(i) \right)\right\}\,ds. \end{align*} Since $f$ is bounded and $\Lambda$ is bounded on the interval $[0,1]$, using \eqref{FirstDerivG}, we check that we can apply Fubini's theorem to interchange the space and time integrals in the two last terms of the right-hand side. Since $f$ is arbitrary, we conclude that \eqref{milfnh} holds a.s. $dx$ a.e.. \end{proof}
\section{Proofs of the results in Section 2} \subsection{Quantitative propagation of chaos result} The proof of Theorem \ref{thmpropchaos} relies on the following Lemma which estimates for $t>0$ the $L^\infty$-norm of the density $p(t,x)$ of $X_t$ solution to \eqref{edsnl} which is guaranteed to exist according to Lemma \ref{ctyF}. \begin{lem}\label{estidens}
$$\forall \; T\in(0,+\infty),\exists \; C_{\infty,T}<\infty,\forall \; t\in(0,T],\quad \|p(t,.)\|_{L^\infty}\le C_{\infty,T} t^{-1/2}.$$ \end{lem} \begin{proof}
Reasoning like at the beginning of the proof of Proposition \ref{propspde} but with the function $\varphi(s,x)={\mathbf 1}_{[0,t[}(s)G_{t-s}*f(x)+{\mathbf 1}_{\{s=t\}}f(x)$ (in place of its spatial antiderivative) for $f$ $C^2$ and compactly supported on ${\mathbb R}$, we easily check that $p(t,x)$ satisfies the mild formulation :
\begin{equation}\label{mildp}
\displaystyle \forall t>0,\;dx\;a.e.,\quad p(t,x)=G_t*m(x)-\int_0^t\partial_x G_{t-s}*\left(\lambda(F(s,.))p(s,.)\right)(x)\,ds.
\end{equation}
Since for $t>0$, $\|G_t*m\|_{L^\infty}\le \|G_t\|_{L^\infty}=(2\pi \sigma^2t)^{-1/2}$, it is enough to check that the estimation holds for the time integral in the mild formulation. By Jensen's inequality, Minkowski's inequality, Young's inequality then then \eqref{GPrimSqu} and \eqref{GSquare}, \eqref{mildp} implies that, for $t>0$,
\begin{align*}
\displaystyle \|p(t,.)\|_{L^2}&\le \|G_t\|_{L^2}+\int_0^t\|\partial_x G_{t-s}\|_{L^2}\|\lambda(F(s,.))p(s,.)\|_{L^1}\,ds\\
&\le \frac{1}{\sqrt{2\sigma} (\pi t)^{1/4}}+L_\Lambda \int_0^t\frac{ds}{2\sigma^{3/2}\pi^{1/4}(t-s)^{3/4}}=\frac{1}{\sqrt{2\sigma} (\pi t)^{1/4}}+\frac{2L_\Lambda t^{1/4}}{\sigma^{3/2}\pi^{1/4}}
\end{align*} where the right-hand side is not greater than $\left(\frac{1}{\sqrt{2\sigma} (\pi)^{1/4}}+\frac{2L_\Lambda T^{1/2}}{\sigma^{3/2}\pi^{1/4}}\right) t^{-1/4}$ for $t\in(0,T]$.
With the boundedness of $\lambda$ and Young's inequality for convolutions, we deduce that for $t\in(0,T]$,
\begin{align*}
\|p(t,.)\|_{L^\infty}&\le \|G_t\|_{L^\infty}+L_\Lambda\int_0^t\|\partial_x G_{t-s}\|_{L^2}\|\lambda(F(s,.))p(s,.)\|_{L^2}\,ds\\
&\le (2\pi \sigma^2t)^{-1/2}+\frac{L_\Lambda}{2\sigma^{3/2}\pi^{1/4}}\left(\frac{1}{\sqrt{2\sigma} (\pi)^{1/4}}+\frac{2L_\Lambda T^{1/2}}{\sigma^{3/2}\pi^{1/4}}\right)\int_0^t\frac{ds}{(t-s)^{3/4}s^{1/4}}.
\end{align*}
Since $\displaystyle \int_0^t\frac{ds}{(t-s)^{3/4}s^{1/4}}ds=\int_0^1\frac{du}{(1-u)^{3/4}u^{1/4}}\le \int_0^1\frac{du}{(1-u)^{3/4}u^{1/4}}T^{1/2}t^{-1/2}$, we easily conclude. \end{proof} We are now ready to prove Theorem \ref{thmpropchaos} by adapting the proof of Theorem 1.6 \cite{ShkoKol}. Since, by Jensen's inequality, the conclusion with $\rho=1$ implies the conclusion with $\rho\in (0,1)$, we suppose without loss of generality that $\rho\ge 1$. Lemma \ref{estidens} implies the following estimation of the Lipschitz constant of $x\mapsto \lambda(F(t,x))$: \begin{equation}
\label{liplamft}\forall t\in(0,T],\;L_{\lambda(F(t,.))}\le C_{\infty,T}L_\lambda t^{-1/2}. \end{equation} We deduce that for a finite constant $C$ which may change from line to line and depends on $T$ but not on $N$: \begin{align*}
\displaystyle &\sup_{s\in[0,t]}\left|X^i_s-\breve X^{i,N}_s\right|^\rho\le \left(\int_0^t\left|\lambda\left(F\left(u,X^i_u\right)\right)-\lambda\left(\breve F^N\left(u,\breve X^{i,N}_u\right)\right)\right|\,du\right)^\rho \\
&= \left(\int_0^t u^{-\frac{\rho-1}{2\rho}} \times u^{\frac{\rho-1}{2\rho}}\left|\lambda\left(F\left(u,X^i_u\right)\right)-\lambda\left(\breve F^N\left(u,\breve X^{i,N}_u\right)\right)\right|\,du\right)^\rho\\
&\le \left(\int_0^t u^{-1/2}\,du\right)^{\rho-1} \int_0^tu^{(\rho-1)/2}\left(\left|\lambda\left(F\left(u,X^i_u\right)\right)-\lambda\left(F\left(u,\breve X^{i,N}_u\right)\right)\right|+\left|\lambda\left(F\left(u,\breve X^{i,N}_u\right)\right)-\lambda\left(\breve F^N\left(u,\breve X^{i,N}_u\right)\right)\right|\right)^\rho\,du\\
&\le C\int_0^t \left(u^{-1/2}\left|X^i_u-\breve X^{i,N}_u\right|^\rho +u^{(\rho-1)/2}\left|\lambda\left(F\left(u,\breve X^{i,N}_u\right)\right)-\lambda\left(\breve F^N\left(u,\breve X^{i,N}_u\right)\right)\right|^\rho \right)\,du, \end{align*} where we used H\"older's inequality for the second inequality. Using exchangeability of $(\breve X^{1,N},\hdots,\breve X^{N,N})$, denoting by $\breve Y^{1,N}_u\le \breve Y^{2,N}_u\le \hdots\le \breve Y^{N,N}_u$ (resp. $Y^1_u\le Y^2_u\le \hdots\le Y^N_u$) the increasing reordering of $\left(\breve X^{1,N}_u,\hdots,\breve X^{N,N}_u\right)$ (resp. $\left(\breve X^{1}_u,\hdots,\breve X^{N}_u\right)$) and using that \eqref{exchanG} and its proof generalizes to the particle system \eqref{PSDEbis} then \eqref{liplamft}, we obtain that \begin{align*}
{\mathbb E}&\left[\left|\lambda\left(F\left(u,\breve X^{i,N}_u\right)\right)-\lambda\left(\breve F^N\left(u,\breve X^{i,N}_u\right)\right)\right|^\rho\right]={\mathbb E}\left[\frac{1}{N}\sum_{j=1}^N\left|\lambda\left(F\left(u,\breve X^{j,N}_u\right)\right)-\lambda\left(\breve F^N\left(u,\breve X^{j,N}_u\right)\right)\right|^\rho\right]\\
&={\mathbb E}\left[\frac{1}{N}\sum_{j=1}^N\left|\lambda\left(F\left(u,\breve Y^{j,N}_u\right)\right)-\lambda\left(\breve F^N\left(u,\breve Y^{j,N}_u\right)\right)\right|^\rho\right]\\
&={\mathbb E}\left[\frac{1}{N}\sum_{j=1}^N\left|\lambda(F(u,\breve Y^{j,N}_u))-\lambda(F(u,Y^{j}_u))+\lambda(F(u,Y^{j}_u))-\lambda\left(\frac{j}{N}\right)\right|^\rho\right]\\
&\le C \left(u^{-\rho/2}{\mathbb E}\left[\frac{1}{N}\sum_{j=1}^N\left|\breve Y^{j,N}_u-Y^j_u\right|^\rho\right]+{\mathbb E}\left[\frac{1}{N}\sum_{j=1}^N\left|F(u,Y^{j}_u)-\frac{j}{N}\right|^\rho\right]\right). \end{align*}
Since $F(u,Y^1_u)\le F(u,Y^2_u)\le \hdots\le F(u,Y^N_u)$ is the increasing reordering of the random variables $(F(u,X^i_u))_{1\le i\le N}$ which are i.i.d. according to the uniform law on $[0,1]$, according to the proof of Theorem 1.6 \cite{ShkoKol}, the second expectation in the right-hand side is bounded from above by $CN^{-\rho/2}$. On the other hand, by \eqref{Wasserstein}, $\frac{1}{N}\sum \limits_{j=1}^N\left|\breve Y^{j,N}_u-Y^j_u\right|^\rho={\cal W}_\rho^\rho\left(\frac 1N\sum \limits_{i=1}^N\delta_{\breve Y^i_u},\frac 1N\sum \limits_{i=1}^N\delta_{X^i_u}\right)\le \frac{1}{N}\sum \limits_{j=1}^N\left|\breve X^{j,N}_u-X^j_u\right|^\rho$. We deduce that for all $t\in[0,T]$,
$$\displaystyle {\mathbb E}\left[\sup_{s\in[0,t]}\left|X^i_s-\breve X^{i,N}_s\right|^\rho\right]\le C\left(N^{-\rho/2}+\int_0^t u^{-1/2}{\mathbb E}\left[\left|X^i_u-\breve X^{i,N}_u\right|^\rho\right]\,du\right).$$
Performing the change of variable $v=\sqrt{u}$ in the integral and setting $f(t)={\mathbb E}\left[\sup_{s\in[0,t^2]}|X^i_s-\breve X^{i,N}_s|^\rho\right]$, we deduce that $\forall t\in\left[0,\sqrt{T}\right]$, $f(t)\le C\left(N^{-\rho/2}+\int_0^t f(v)dv\right)$. Since, by boundedness of $\lambda$, the function $f$ is locally bounded, we conclude using Gr\"onwall's lemma that ${\mathbb E}\left[\sup_{s\in[0,T]}|X^i_s-\breve X^{i,N}_s|^\rho\right]\le CN^{-\rho/2}$.
Remarking that $\displaystyle \left|X^{i,N}_t-X^i_0-\sigma W^i_t-\int_0^t\lambda\left(F^{N}\left(s,X^{i,N}_s\right)\right)\,ds\right|\le\frac{L_\lambda t}{2N}$, we may adapt the arguments to deal with the particle system \eqref{PSDE}.
\subsection{Rate of convergence of the strong $L^1$-error}\label{secproofthm1}
To prove Theorem \ref{thm1}, we need the following lemmas. \begin{lem}\label{XandY}
$$\forall 0 \le s \le t \le T,\;\forall \rho \ge 1, h \in [0,T],\; N \in {\mathbb N}^{*} \quad \sum \limits_{j=1}^N \left| Y^{j,N,h}_t - Y^{j,N,h}_s \right|^{\rho} \leq \sum \limits_{j=1}^N \left| X^{j,N,h}_t - X^{j,N,h}_s \right|^{\rho}.$$ \end{lem} \begin{proof}
By \eqref{Wasserstein}, we have $\displaystyle \mathcal{W}_{\rho}^{\rho}\left( \mu^{N,h}_s,\mu^{N,h}_t\right)=\frac{1}{N}\sum \limits_{j=1}^N \left|Y^{j,N,h}_t - Y^{j,N,h}_s \right|^{\rho}$. Since $\displaystyle \frac{1}{N} \sum \limits_{j=1}^N \delta_{\left(X^{j,N,h}_s,X^{j,N,h}_t\right)}$ is a coupling measure on ${\mathbb R}^2$ with first marginal $\mu^{N,h}_s$ and second marginal $\mu^{N,h}_t$, we conclude that $$\displaystyle \frac{1}{N}\sum \limits_{j=1}^N \left|X^{j,N,h}_t - X^{j,N,h}_s \right|^{\rho} \ge \mathcal{W}_{\rho}^{\rho}\left( \mu^{N,h}_s,\mu^{N,h}_t\right) = \frac{1}{N}\sum \limits_{j=1}^N \left|Y^{j,N,h}_t - Y^{j,N,h}_s \right|^{\rho}.$$
\end{proof}
This second lemma ensures the local integrability of $t \mapsto {\mathbb E}\left[ \mathcal{W}_1\left(\mu^{N,h}_t,\mu_t \right)\right]$. \begin{lem}\label{integrF} $$\forall t,h \in [0,T],\forall N \in {\mathbb N}^{*}, \quad \displaystyle {\mathbb E}\left[\mathcal{W}_1\left(\mu^{N,h}_t,\mu_t \right)\right] \leq 2 \, \sigma \, \sqrt{\frac{2 t}{\pi}} + 2 \, L_{\Lambda}t + {\mathbb E}\left[\mathcal{W}_1(\mu^{N,h}_0, m)\right]. $$
If $\displaystyle \int_{{\mathbb R}} |x|m(dx) < \infty$, then for each $N\in{\mathbb N}^*$ and each $h\in[0,T]$, $t\mapsto {\mathbb E}[{\cal W}_1(\mu^{N,h}_t,\mu_t)]$ is locally integrable on ${\mathbb R}_+$. \end{lem}
\begin{proof}
Using the triangle inequality, we have: \begin{align}\label{inegTriang}
{\mathbb E}\left[\mathcal{W}_1(\mu^{N,h}_t, \mu_t)\right] \leq {\mathbb E}\left[\mathcal{W}_1(\mu^{N,h}_t, \mu^{N,h}_0)\right] + {\mathbb E}\left[\mathcal{W}_1(\mu^{N,h}_0, m)\right] + \mathcal{W}_1(m, \mu_t). \end{align} Since \begin{equation}\label{estimW1}
\left\{
\begin{aligned}
& \mathcal{W}_1(m,\mu_t) \leq {\mathbb E}\left[\left| X_t - X_0 \right|\right] \leq {\mathbb E} \left[ \left| \sigma W_t\right|\right] + {\mathbb E}\left[\left| \displaystyle \int_0^t \lambda(F(s,X_s))\,ds \right|\right] = \sigma \, \sqrt{\frac{2 t}{\pi}} + L_{\Lambda}t, \\
&{\mathbb E}\left[\mathcal{W}_1(\mu^{N,h}_t, \mu^{N,h}_0)\right] \le \frac{1}{N} \sum \limits_{i=1}^N {\mathbb E}\left[\left|\sigma W^{i}_t + \displaystyle \int_0^t \lambda^N\left( \mathbf{1}_{\left\{ X^{j,N,h}_s \leq X^{i,N,h}_s \right\}} \right)\,ds \right| \right] \leq \sigma \,\sqrt{\frac{2 t}{\pi}} + L_{\Lambda}t.\\
\end{aligned}
\right. \end{equation}
then by injecting \eqref{estimW1} in \eqref{inegTriang}, we obtain the upper-bound of ${\mathbb E}\left[ \mathcal{W}_1\left(\mu^{N,h}_t,\mu_t \right)\right]$. Since for $i\in\{1,\hdots,N\}$, $x^N_{i} = F^{-1}_0 \left(\frac{2i-1}{2N} \right)$ minimizes $\displaystyle {\mathbb R}\ni y\mapsto \int_{\frac{i-1}{N}}^{\frac{i}{N}}\left|y - F^{-1}_0(u) \right|\,du$, we have that: \begin{align}
\displaystyle \mathcal{W}_1\left(\tilde \mu_0^N,m\right) =\sum_{i=1}^N \int_{\frac{i-1}{N}}^{\frac{i}{N}}\left|x^N_{i} - F^{-1}_0(u) \right|\,du &\le \sum_{i=1}^N \int_{\frac{i-1}{N}}^{\frac{i}{N}}\left| F^{-1}_0(u) \right|\,du = \int_{{\mathbb R}}|x|m(dx).\label{majow1det} \end{align} For the random initialization, by Theorem 3.5 \cite{bobkovledoux}, \begin{align*}
{\mathbb E}\left[ \mathcal{W}_1\left(\hat\mu^N_0,m\right)\right]\le 2\int_{\mathbb R}\mathbf{1}_{\{F_0(x)(1-F_0(x))\le \frac{1}{4N}\}}F_0(x)(1-F_0(x))dx+\frac{1}{\sqrt{N}}\int_{\mathbb R} \mathbf{1}_{\{F_0(x)(1-F_0(x))> \frac{1}{4N}\}}\sqrt{F_0(x)(1-F_0(x))}dx.\end{align*} Since $F_0(x)(1-F_0(x))> \frac{1}{4N}$ implies $\frac{1}{\sqrt{N}}<2\sqrt{F_0(x)(1-F_0(x))}$, we deduce that
$${\mathbb E}\left[ \mathcal{W}_1\left(\hat\mu^N_0,m\right)\right] \le 2\int_{\mathbb R} F_0(x)(1-F_0(x))dx\le 2\left(\int_{-\infty}^0F_0(x)dx+\int^{+\infty}_0(1-F_0(x))dx\right)=2\int_{\mathbb R} |x|m(dx).$$ Hence for both the random and the optimal deterministic initializations, the finiteness of the first order moment of $m$ implies the finiteness of ${\mathbb E}\left[\mathcal{W}_1(\mu^{N,h}_0, m)\right]$ and, with the first statement, the local integrability of $t \mapsto {\mathbb E}\left[ \mathcal{W}_1\left(\mu^{N,h}_t,\mu_t \right)\right]$. \end{proof}
The third lemma gives a control of the moments of order $\rho\ge 1$ of $X^{i,N,h}_t$, $\forall i \in \llbracket1,N \rrbracket$. \begin{lem}\label{controlM}
If $\displaystyle \int_{{\mathbb R}} |x|^{\rho}m(dx) < \infty$ for some $\rho\ge 1$, then $\forall N \in {\mathbb N}^{*}, \forall h \in [0,T]$, $$\sup_{t \leq T}{\mathbb E}\left[\frac{1}{N}\sum_{i=1}^N\left|X^{i,N,h}_t\right|^{\rho} \right] \leq M := 3^{\rho-1}\left( 2\int_{{\mathbb R}}|x|^{\rho}m(dx)+ \frac{1}{\sqrt \pi}\Gamma\left(\frac{\rho +1}{2} \right)(2\sigma^2T)^{\rho/2}+ (L_{\Lambda}T)^{\rho} \right). $$ \end{lem} \begin{proof}
We have $${\mathbb E}\left[\left|X^{i,N,h}_t\right|^{\rho} \right] \le 3^{\rho-1}\left( {\mathbb E}\left[|X^{i}_0|^{\rho}\right] + \sigma^\rho {\mathbb E}\left[|W^{i}_t|^{\rho}\right] + {\mathbb E}\left[\left|\int_0^t \lambda^N\left(\sum \limits_{j=1}^{N} \mathbf{1}_{ \{ X^{j,N,h}_{\tau^h_s} \le X^{i,N,h}_{\tau^h_s} \}}\right)\,ds\right|^{\rho}\right] \right).$$
Since ${\mathbb E}\left[|W^{i}_t|^{\rho}\right]^{1/\rho}=\frac{1}{\sqrt \pi}\Gamma\left(\frac{\rho +1}{2} \right)(2t)^{\rho/2}$, one easily concludes when the initial conditions are i.i.d. according to $m$. When they are optimal deterministic, we sum over $i\in\{1,\hdots,N\}$, divide by $N$ and use the estimation
$\frac{1}{N}\sum_{i=1}^N\left|F_0^{-1}\left(\frac{2i-1}{2N}\right)\right|^\rho\le 2\int_{\mathbb R}|x|^\rho m(dx)$ that we now prove.
Since
\begin{align*}
\displaystyle &\forall i\in\{1,\hdots,N-1\},\quad F_0^{-1}\left(\frac{2i-1}{2N}\right)\ge 0\Rightarrow \frac{\left|F_0^{-1}\left(\frac{2i-1}{2N}\right)\right|^\rho}{N}\le \int_{\frac{2i-1}{2N}}^{\frac{2i+1}{2N}}\left|F_0^{-1}(u)\right|^\rho\,du,\\
&\forall i\in\{2,\hdots,N\},\quad F_0^{-1}\left(\frac{2i-1}{2N}\right)\le 0\Rightarrow \frac{\left|F_0^{-1}\left(\frac{2i-1}{2N}\right)\right|^\rho}{N} \le \int_{\frac{2i-3}{2N}}^{\frac{2i-1}{2N}}\left|F_0^{-1}(u)\right|^\rho\,du,
\end{align*}
one has
\begin{align*}
\displaystyle \frac{1}{N}\sum_{i=1}^N\left|F_0^{-1}\left(\frac{2i-1}{2N}\right)\right|^\rho &\le \mathbf{1}_{\left\{F_0^{-1}\left(\frac{1}{2N}\right)\le 0 \right\}} \frac{\left|F_0^{-1}\left(\frac{1}{2N}\right)\right|^\rho}{N}+\int_{\frac{1}{2N}}^{\frac{2N-1}{2N}}\left|F_0^{-1}(u)\right|^\rho\,du + \mathbf{1}_{\left\{F_0^{-1}\left(\frac{2N-1}{2N}\right)\ge 0 \right \}} \frac{\left|F_0^{-1}\left(\frac{2N-1}{2N}\right)\right|^\rho}{N}\\
&\le \int_0^{\frac{1}{2N}}\left|F_0^{-1}(u)\right|^\rho\,du +\int_{0}^{1}\left|F_0^{-1}(u)\right|^\rho\,du+\int^1_{\frac{2N-1}{2N}}\left|F_0^{-1}(u)\right|^\rho\,du\\
&\le 2\int_0^1|F_0^{-1}(u)|^\rho du=2\int_{\mathbb R}|x|^\rho m(dx).\end{align*}\end{proof}
\subsubsection{Proof of Theorem \ref{thm1}} Defining for all $t,h \in [0,T], N \in {\mathbb N}^{*}$, \begin{align} \displaystyle &R^{N,h}(t,x) = - \frac{\sigma}{N}\sum \limits_{i=1}^N \int_0^t G_{t-s}(X^{i,N,h}_s -x)\,dW^{i}_s,\label{defrnh} \\ &E^{N,h}(t,x) = \frac{1}{N}\sum \limits_{i=1}^N \int_0^t G_{t-s}(Y^{i,N,h}_s-x) \left[ \lambda^{N}\left(i\right) - \lambda^{N}\left(\eta_{\tau^h_s}^{-1}(\eta_s(i))\right) \right]\,ds, \end{align}
we deduce from Proposition \ref{propspde} that: \begin{align}\label{eqFh}
\displaystyle F^{N,h}(t,x) - F(t,x) =& \quad G_t*\left(F^{N,h}_0 - F_0\right)(x) - \int_0^t \partial_x G_{t-s} * \left(\Lambda(F^{N,h}(s,.)) - \Lambda(F(s,.))\right)(x)\,ds \\
&+ R^{N,h}(t,x) + E^{N,h}(t,x). \notag \end{align} Using the triangle inequality and taking expectations, we deduce that: \begin{align*}
\displaystyle {\mathbb E}\left[\left\| F^{N,h}(t,.) - F(t,.) \right\|_{L^1} \right] &\le {\mathbb E}\left[\left\| G_t \ast \left(F^{N,h}_0 - F_0\right) \right\|_{L^1} \right]
+ {\mathbb E}\left[\left\| R^{N,h}(t,.) \right\|_{L^1} \right] + {\mathbb E}\left[\left\| E^{N,h}(t,.) \right\|_{L^1} \right] \\
&+ {\mathbb E}\left[\left\|\int_0^t \partial_x G_{t-s} * \left(\Lambda(F^{N,h}(s,.)) - \Lambda(F(s,.))\right)\,ds \right\|_{L^1} \right]. \end{align*} Using the estimate \eqref{FirstDerivG} from Lemma \ref{EstimHeatEq} and setting $\displaystyle A = \frac{L_{\Lambda}}{\sigma} \sqrt{\frac{2}{\pi}}$, we obtain: \begin{align*}
\displaystyle {\mathbb E}\left[\left\|\int_0^t \partial_x G_{t-s} * \left(\Lambda(F^{N,h}(s,.)) - \Lambda(F(s,.))\right)\,ds \right\|_{L^1} \right] &\leq \int_0^t \left\| \partial_x G_{t-s}\right\|_{L^1} L_{\Lambda}{\mathbb E}\left[\left\Vert F^{N,h}(s,.) - F(s,.) \right\Vert_{L^1}\right]\,ds \\
&= A\int_0^t \frac{1}{\sqrt{t-s}}{\mathbb E}\left[\left\Vert F^{N,h}(s,.) - F(s,.) \right\Vert_{L^1}\right]\,ds. \end{align*} Therefore, \begin{align}\label{controlConvoB}
\displaystyle {\mathbb E}\left[\left\| F^{N,h}(t,.) - F(t,.) \right\|_{L^1} \right] \le {\mathbb E}\left[\left\| F^{N,h}_0 - F_0 \right\|_{L^1} \right] &+ {\mathbb E}\left[\left\| R^{N,h}(t,.) \right\|_{L^1} \right] + {\mathbb E}\left[\left\| E^{N,h}(t,.) \right\|_{L^1} \right] \\
&+ A \int_0^t \frac{1}{\sqrt{t-s}}{\mathbb E}\left[\left\Vert F^{N,h}(s,.) - F(s,.) \right\Vert_{L^1}\right]\,ds.\notag \end{align}
The next lemma states that the random variable $R^{N,h}(t,x)$ is centered and provides an upper-bound for ${\mathbb E}\left[\left\| R^{N,h}(t,.)\right\|_{L^1} \right]$.
\begin{lem}\label{controlRN} We have $\forall N \in {\mathbb N}^{*}, \forall h,t \in [0,T]$, $ \left\| {\mathbb E}\left[R^{N,h}(t,.)\right]\right\|_{L^1} = 0 $. Moreover, if for some $\rho >1$, $\displaystyle \int_{{\mathbb R}} |x|^{\rho}m(dx) < \infty$, then:
$$ \exists R< \infty,\;\forall N \in {\mathbb N}^{*},\;\forall h \in [0,T],\quad \sup_{t \leq T}{\mathbb E}\left[\left\| R^{N,h}(t,.) \right\|_{L^1} \right] \leq \frac{R}{\sqrt{N}}.$$ \end{lem}
\begin{proof}
We have that $\displaystyle \int_{{\mathbb R}}{\mathbb E}\left[\int_0^t G^2_{t-s}(X^{i,N,h}_s - x)\,ds\right]\,dx = {\mathbb E}\left[\int_{{\mathbb R}}\int_0^t G^2_{t-s}(X^{i,N,h}_s - x)\,ds\,dx \right] \le \frac{1}{\sigma} \sqrt{\frac{t}{\pi}} < \infty $ according to the estimate \eqref{GSquareEsp} from Lemma \ref{EstimHeatEq}. Therefore, $ {\mathbb E}\left[R^{N,h}(t,x) \right] = 0 \quad \text{dx a.e.}$. Moreover, denoting $\displaystyle I_{\rho} = \int_{{\mathbb R}} \frac{dx}{1 + |x|^{\rho}}$ and using the It\^o isometry for the first equality then Young's inequality for the second inequality and last the estimate \eqref{GSquareEsp} from Lemma \ref{EstimHeatEq}, we obtain:
\begin{align*} \displaystyle {\mathbb E}\left[ \left\Vert R^{N,h}(t,.)\right\Vert_{L^1} \right] &\le \int_{{\mathbb R}} {\mathbb E}^{1/2}\left[ \left( \frac{\sigma}{N}\sum \limits_{i=1}^N \int_0^t G_{t-s}(X^{i,N,h}_s -x)\,dW^{i}_s\right)^2\right]\,dx \\ &=\frac{\sigma}{\sqrt{N}} \int_{{\mathbb R}} {\mathbb E}^{1/2} \left[ \frac{1}{N}\sum \limits_{i=1}^N \int_0^t G^2_{t-s}(X^{i,N,h}_s -x)\,ds\right]\,dx \\
&= \frac{\sigma}{\sqrt{N}} \int_{{\mathbb R}} {\mathbb E}^{1/2} \left[ \frac{1}{N}\sum \limits_{i=1}^N \int_0^t G^2_{t-s}(X^{i,N,h}_s -x)\,ds (1 + |x|^{\rho}) \right] \frac{dx}{\sqrt{1+|x|^{\rho}}} \\
&\le \frac{\sigma}{2\sqrt{N}} \int_{{\mathbb R}} \left(\frac{1}{1 + |x|^{\rho}} + {\mathbb E}\left[\frac{1}{N}\sum \limits_{i=1}^N \int_0^t G^2_{t-s}(X^{i,N,h}_s -x)\,ds (1 + |x|^{\rho}) \right]\right)\,dx \\
&= \frac{\sigma I_{\rho}}{2\sqrt{N}} + \frac{\sigma}{2\sqrt{N}}{\mathbb E}\left[\frac{1}{N}\sum \limits_{i=1}^N\int_0^t\left(\int_{{\mathbb R}}G^2_{t-s}(X^{i,N,h}_s -x)\,dx + \int_{{\mathbb R}} |x|^{\rho} G^2_{t-s}(X^{i,N,h}_s -x)\,dx \right)\,ds \right] \\ &= \frac{\sigma I_{\rho}}{2\sqrt{N}} + \frac{\sigma}{2\sqrt{N}} {\mathbb E}\left[ \frac{1}{N}\sum
\limits_{i=1}^N \frac{1}{\sigma} \, \sqrt{\frac{t}{\pi}} + \int_0^t \int_{{\mathbb R}} \frac{|X^{i,N,h}_s - y|^{\rho}}{2\sigma \, \sqrt{\pi (t-s)}}G_{(t-s)/2}(y)\,dy\,ds\right] \\ &\le \frac{\sigma I_{\rho}}{2\,\sqrt{N}} +\frac{1}{2}\sqrt{\frac{t}{N\pi}} + \frac{2^{\rho -1}\sigma}{2\sqrt{N}}
{\mathbb E}\left[\frac{1}{N}\sum \limits_{i=1}^N \int_0^t \frac{|X^{i,N,h}_s|^{\rho}}{2\sigma \, \sqrt{\pi(t-s)}}\,ds + \frac{\sigma^{\rho-1}}{2 \pi} \Gamma\left(\frac{\rho +1}{2} \right) \int_0^t (t-s)^{(\rho-1)/2} \,ds \right]\\ &= \frac{1}{2\sqrt{N}}\left(
\sigma I_{\rho} + \sqrt{\frac{t}{\pi}} \left(1 + \frac{2^{\rho -1}\sigma^{\rho}}{\rho +1}\Gamma\left(\frac{\rho +1}{2}\right)\sqrt{\frac{t^{\rho}}{\pi}}\right) + \frac{2^{\rho -1}}{\sqrt{\pi}}\int_0^t {\mathbb E}\left[ \frac{1}{N}\sum \limits_{i=1}^N |X^{i,N,h}_s|^{\rho} \right] \frac{ds}{2\sqrt{(t-s)}}\right). \end{align*} With the use of Lemma \ref{controlM}, we conclude by setting $\displaystyle R = \frac{1}{2} \left(\sigma I_{\rho}+ \sqrt{\frac{T}{\pi}}\left(1 +2^{\rho -1}M +\frac{2^{\rho -1}\sigma^{\rho}}{\rho +1}\Gamma\left(\frac{\rho +1}{2}\right)\sqrt{\frac{T^{\rho}}{\pi}}\right)\right)$. \end{proof} Therefore, Inequality \eqref{controlConvoB} becomes: \begin{align}\label{controlConvo}
\displaystyle {\mathbb E}\left[\left\| F^{N,h}(t,.) - F(t,.) \right\|_{L^1} \right] \le {\mathbb E}\left[\left\| F^{N,h}_0 - F_0 \right\|_{L^1} \right] &+\frac{R}{\sqrt{N}} +{\mathbb E}\left[\left\|E^{N,h}(t,.)\right\|_{L^1}\right] \\ &+A \int_0^t \frac{1}{\sqrt{t-s}}{\mathbb E}\left[\left\Vert F^{N,h}(s,.) - F(s,.) \right\Vert_{L^1}\right]\,ds.\notag \end{align} \textbullet $\;$ One should notice that for $h=0$, $E^{N,0}(t,x)=0 \quad \forall t \in [0,T], N \in {\mathbb N}^{*}, x \in {\mathbb R}$. Therefore, to control the term ${\mathbb E}\left[ \mathcal{W}_1\left(\mu^{N,0}_t,\mu_t \right)\right]$, we iterate Inequality \eqref{controlConvo} and obtain: \begin{align*}
\displaystyle {\mathbb E}\left[\left\Vert F^{N,0}(t,.) - F(t,.) \right\Vert_{L^{1}}\right] \leq \left(2A \sqrt{t} +1\right) &\left( {\mathbb E}\left[\left\| F^{N,h}_0 - F_0 \right\|_{L^1} \right] + \frac{R}{\sqrt N} \right) \\
&+ A^2 \displaystyle \int_0^t {\mathbb E}\left[\left\Vert F^{N,0}(r,.) - F(r,.) \right\Vert_{L^1}\right] \int_r^t \frac{ds}{\sqrt{t-s}\, \sqrt{s-r}}\,dr. \end{align*} Since $\displaystyle \int_r^t \frac{ds}{\sqrt{t-s}\,\sqrt{s-r}} = \pi$ and with the use of Lemma \ref{integrF}, we can apply Gr\"onwall's lemma to deduce that\\ $\displaystyle \forall N \in {\mathbb N}^{*}, \sup_{t \leq T}{\mathbb E}\left[ \mathcal{W}_1\left(\mu^{N,0}_t,\mu_t \right)\right] \le C\left({\mathbb E}\left[ \mathcal{W}_1\left(\mu^{N}_0,m \right)\right] + \frac{1}{\sqrt{N}} \right)$ where $ C = \max(1,R)\left(2A \sqrt{T} +1 \right)\exp\left( A^2 \pi T\right)$. This concludes the proof of Theorem \ref{thm1} when $h=0$.\\ \\
\textbullet $\;$ When $h>0$, we need to estimate ${\mathbb E}\left[\left\| E^{N,h}(t,.) \right\|_{L^1} \right], h \in (0,T]$. \begin{prop}\label{ControlFinalE}
We assume that for some $\rho >1$, $\displaystyle \int_{{\mathbb R}} |x|^{\rho}m(dx) < \infty$ and that the function $\lambda$ is Lipschitz continuous. Then $\exists Z <\infty, \forall N \in {\mathbb N}^{*}, \forall h \in (0,T],\forall t\in[0,T],$
$$\displaystyle {\mathbb E} \left[ \left\| E^{N,h}(t,.)\right\|_{L^1} \right] \leq Z \left( \frac{1}{\sqrt{N}} + h + \sqrt{h}{\mathbb E}\left[\left\|F^{N,h}(t,.) - F(t,.) \right\|_{L^1} \right] + \int_{0}^{t} \frac{1}{2\sqrt{t-s}}{\mathbb E}\left[\left\|F(s,.) - F^{N,h}(s,.) \right\|_{L^1}\right]\,ds \right).$$ \end{prop} Proposition \ref{ControlFinalE} will be proved in Section \ref{cetSec}. \\ \\ From Equation\eqref{controlConvo} and Proposition \ref{ControlFinalE}, we have that: \begin{align*}
\displaystyle &\left(1 - Z \sqrt{h} \right){\mathbb E}\left[\left\Vert F^{N,h}(t,.) - F(t,.) \right\Vert_{L^{1}} \right] \\
&\le {\mathbb E}\left[\left\Vert F^{N,h}_0 - F_0 \right\Vert_{L^{1}} \right]+ \frac{Z+R}{\sqrt{N}} + Z h + \left(A+\frac{Z}{2}\right) \int_0^t \frac{1}{\sqrt{t-s}}{\mathbb E}\left[\left\Vert F^{N,h}(s,.) - F(s,.) \right\Vert_{L^1}\right]\,ds. \end{align*} Hence, if we denote $\displaystyle J = 2\left(Z + R \right)$ and $K = 2A + Z$ then: \begin{align}\label{EquaIciiii}
\displaystyle &2\left(1 - Z \sqrt{h} \right) {\mathbb E}\left[\left\Vert F^{N,h}(t,.) - F(t,.) \right\Vert_{L^{1}} \right] \notag\\
&\le 2{\mathbb E}\left[\left\Vert F^{N,h}_0 - F_0 \right\Vert_{L^{1}} \right] + J \left(\frac{1}{\sqrt{N}} + h \right) + \int_0^t \frac{K}{\sqrt{t-s}}{\mathbb E}\left[\left\Vert F^{N,h}(s,.) - F(s,.) \right\Vert_{L^1}\right]\,ds. \end{align} \begin{itemize}
\item When $h \leq \frac{1}{4Z^2}$, Equation \eqref{EquaIciiii} implies: \begin{align*}
&{\mathbb E}\left[\left\|F^{N,h}(t,.) - F(t,.) \right\|_{L^1}\right] \\
&\le 2{\mathbb E}\left[\left\Vert F^{N,h}_0 - F_0 \right\Vert_{L^{1}} \right]+ J\left(\frac{1}{\sqrt{N}} +h \right) + \int_0^t \frac{K}{ \sqrt{t-s}}{\mathbb E}\left[ \left\Vert F^{N,h}(s,.) - F^N(s,.) \right\Vert_{L^1}\right]\,ds. \end{align*} We iterate this inequality to obtain: \begin{align*}
&{\mathbb E}\left[\left\|F^{N,h}(t,.) - F(t,.) \right\|_{L^1}\right] \\
&\le \left(1 + 2 K \sqrt{t} \right) \left(2{\mathbb E}\left[\left\Vert F^{N,h}_0 - F_0 \right\Vert_{L^{1}} \right] + J\left(\frac{1}{\sqrt{N}}+ h \right) \right) + K^2 \pi \int_0^t {\mathbb E}\left[ \left\Vert F^{N,h}(r,.) - F^N(r,.) \right\Vert_{L^1}\right]\,dr. \end{align*} With the use of Lemma \ref{integrF}, we can apply Gr\"onwall's Lemma and deduce that:
$$\forall t \in [0,T], \quad {\mathbb E}\left[\left\|F^{N,h}(t,.) - F(t,.) \right\|_{L^1}\right] \le \left(1 + 2 K \sqrt{t} \right)\exp\left(K^2\pi t \right)\left(2{\mathbb E}\left[\left\Vert F^{N,h}_0 - F_0 \right\Vert_{L^{1}} \right] + J\left(\frac{1}{\sqrt{N}}+ h \right) \right). $$
\item When $h>\frac{1}{4Z^2}$, by Lemma \ref{integrF} and \eqref{w1fdr}, \begin{align*}
\displaystyle {\mathbb E}\left[\left\|F^{N,h}(t,.) - F(t,.) \right\|_{L^1}\right] &\le 2 \sigma \sqrt{\frac{2t}{\pi}} + 2 L_{\Lambda} t + {\mathbb E}\left[\left\Vert F^{N,h}_0 - F_0 \right\Vert_{L^{1}} \right]\\
&\le 4Z^2h\left(2\sigma \sqrt{\frac{2t}{\pi}} + 2 L_{\Lambda} t\right) + {\mathbb E}\left[\left\Vert F^{N,h}_0 - F_0 \right\Vert_{L^{1}} \right]. \end{align*} \end{itemize} We choose $C = \max\left(\max(2,J)\left(1 + 2 K \sqrt{T} \right)\exp\left(K^2\pi T \right), 4Z^2\left(2\sigma \sqrt{\frac{2T}{\pi}} + 2 L_{\Lambda} T\right)\right) $ and conclude that: $$\forall N \in {\mathbb N}^{*},\forall h \in (0,T],\forall t \in [0,T], \quad \sup_{t \leq T}{\mathbb E}\left[ \mathcal{W}_1\left(\mu^{N,h}_t,\mu_t \right)\right] \le C\left({\mathbb E}\left[ \mathcal{W}_1\left(\mu^{N}_0,m \right)\right] + \left(\frac{1}{\sqrt{N}} + h \right)\right). $$
Let us now prove Proposition \ref{ControlFinalE} in the following section.
\subsubsection{Proof of Proposition \ref{ControlFinalE}}\label{cetSec}
We recall the expression of $E^{N,h}(t,x)$:
$$ \displaystyle E^{N,h}(t,x) = \frac{1}{N}\sum \limits_{i=1}^N \int_0^t G_{t-s}(Y^{i,N,h}_s-x) \left[ \lambda^{N}\left(i\right) - \lambda^{N}\left(\eta_{\tau^h_s}^{-1}(\eta_s(i))\right) \right]\,ds. $$ We do not know how to estimate the difference of values of $\lambda^N$ between the brackets. For $s>0$, we are going to take advantage of the permutation $\eta_s^{-1}\circ\eta_{\tau^h_s}$ (because of the convention \eqref{convinvinit}, this is not necessarily a permutation for $s=0$ and $\eta_{\tau^h_s}^{-1}\circ\eta_{\tau^h_s}$ is equal to the identity permutation for $s\ge h$ but not necessarily for $s\in[0,h)$) to change indices and obtain the same value multiplied by a difference of values of the smooth function $G_{t-s}$. Using this permutation for the first equality then that $Y^{\eta_s^{-1}(j),N,h}_s=X^{j,N,h}_s$ for $s>0$ and $1\le j\le N$ for the second one, we obtain that \begin{align}
E^{N,h}(t,x) &= \frac{1}{N}\sum \limits_{i=1}^N \int_0^t G_{t-s}(Y^{i,N,h}_s-x)\lambda^{N}\left(i\right)-G_{t-s}\left(Y^{\eta_s^{-1}(\eta_{\tau^h_s}(i)),N,h}_s-x\right)\lambda^{N}\left(\eta_{\tau^h_s}^{-1}(\eta_{\tau^h_s}(i))\right)\,ds\notag\\
&= \frac{1}{N}\sum \limits_{i=1}^N \int_0^t \left\{\left(G_{t-s}\left(Y^{i,N,h}_s -x\right)- G_{t-s}\left(X^{\eta_{\tau^h_s}(i),N,h}_s -x \right) \right)\right\}\lambda^N(i)\,ds,\notag\\
&+\frac{1}{N}\sum \limits_{i=1}^N \int_{0}^{t\wedge h}G_{t-s}(X^{\eta_0(i),N,h}_s-x)\left(\lambda^N(i)-\lambda^N\left(\eta_0^{-1}(\eta_0(i))\right)\right)\,ds\label{decompENh}.
\end{align}
Substracting $G_{t-\tau^h_s}\left(Y^{i,N,h}_{\tau^h_s}-x\right) - G_{t-\tau^h_s} \left(X^{\eta_{\tau^h_s}(i),N,h}_{\tau^h_s}-x\right) =0$ in the brace in the first term of the right-hand side makes apparent that this term is not too large since $\tau^h_s$ is close to $s$. Computing $G_{t-s}\left(Y^{i,N,h}_s -x\right)-G_{t-\tau^h_s}\left(Y^{i,N,h}_{\tau^h_s} -x\right)$ and $G_{t-s}\left(X^{\eta_{\tau^h_s}(i),N,h}_s -x \right) - G_{t-{\tau^h_s}}\left(X^{\eta_{\tau^h_s}(i),N,h}_{\tau^h_s} -x \right)$ by It\^o's formula, we obtain the following new expression of $E^{N,h}(t,x)$: \begin{lem}\label{ControlE} The process $\tilde \beta=\left(\tilde \beta^1,\hdots,\tilde \beta^N \right)$ where $\displaystyle \tilde \beta^i_t=\sum_{j=1}^N\int_0^t\mathbf{1}_{\left\{\eta_{\tau^h_s}(i)=j\right\}}dW^j_s$ is a $N$-dimensional Brownian motion and we can express $E^{N,h}(t,x)$ as $E^{N,h}(t,x)=\sum \limits_{p=0}^5 e^{N,h}_p(t,x)$ where: \begin{itemize}
\item $ \displaystyle e^{N,h}_0(t,x) = \frac{1}{N}\sum \limits_{i=1}^N \int_{0}^{t\wedge h}G_{t-s}\left(X^{\eta_0(i),N,h}_s-x\right)\left(\lambda^N(i)-\lambda^N\left(\eta_0^{-1}(\eta_0(i))\right)\right)\,ds $,
\item $\displaystyle e^{N,h}_1(t,x) = \frac{1}{N}\sum \limits_{i=2}^N \int_{0}^{t}(t \wedge \overline{\tau}^h_s -s)\left( \lambda^N(i) - \lambda^N(i-1) \right)\partial_x G_{t-s}\left(Y^{i,N,h}_s -x \right)\gamma^{i}_s\,d|K|_s$,
\item $\displaystyle e^{N,h}_2(t,x) = \frac{\sigma}{N} \sum \limits_{i=1}^N \int_{0}^{t}(t \wedge \overline{\tau}^h_s -s)\lambda^N(i) \partial_x G_{t-s}\left(Y^{i,N,h}_s -x \right)\,d\beta^{i}_s$,
\item $\displaystyle e^{N,h}_3(t,x) = - \frac{\sigma}{N}\sum \limits_{i=1}^N \int_{0}^{t}(t \wedge \overline{\tau}^h_s -s)\lambda^N(i) \partial_x G_{t-s}\left(X^{\eta_{\tau^h_s}(i),N,h}_s -x \right)\,d\widetilde{\beta}^i_s$,
\item $\displaystyle e^{N,h}_4(t,x) = \frac{1}{N} \sum \limits_{i=1}^N \int_{0}^{t}(t \wedge \overline{\tau}^h_s -s)\lambda^N(i) \lambda^N\left( \eta^{-1}_{\tau^h_s}(\eta_s(i))\right)\partial_x G_{t-s}\left(Y^{i,N,h}_s -x \right)\,ds$,
\item $ \displaystyle e^{N,h}_5(t,x) = - \frac{1}{N}\sum \limits_{i=1}^N \int_{0}^{t}(t \wedge \overline{\tau}^h_s -s) \lambda^N(i)\lambda^N\left( \eta^{-1}_{\tau^h_s}\left(\eta_{\tau^h_s}(i)\right)\right)\partial_x G_{t-s}\left(X^{\eta_{\tau^h_s}(i),N,h}_s -x \right)\,ds $. \end{itemize} \end{lem} Notice that in the definition of $e^{N,h}_5(t,x)$, $\lambda^N\left( \eta^{-1}_{\tau^h_s}(\eta_{\tau^h_s}(i))\right)=\lambda^N(i)$ for $s\ge h$, but because of the convention \eqref{convinvinit}, this equality does not necessarily hold for $s\in[0,h)$.
\begin{proof}
For $1\le i,k\le N$ and $t\ge 0$, one has
$$\langle \tilde \beta^i,\tilde \beta^k\rangle_t=\sum_{j=1}^N\int_0^t\mathbf{1}_{\left\{\eta_{\tau^h_s}(i)=j \right\}}\mathbf{1}_{\left\{\eta_{\tau^h_s}(k)=j \right\}}\,ds=\int_0^t\mathbf{1}_{\left\{\eta_{\tau^h_s}(i)=\eta_{\tau^h_s}(k)\right\}}\,ds=\mathbf{1}_{\{i=k\}}t,$$
since $\eta_{\tau^h_s}$ is a permutation for each $s\ge 0$ . One deduces that $\tilde \beta$ is a Brownian motion by applying L\'evy's characterization.
By \eqref{decompENh} and the equality $G_{t-\tau^h_s}\left(Y^{i,N,h}_{\tau^h_s}-x\right) - G_{t-\tau^h_s} \left(X^{\eta_{\tau^h_s}(i),N,h}_{\tau^h_s}-x\right) =0$, it is enough to check that
\begin{align*}
\frac{1}{N}\sum \limits_{i=1}^N \int_0^t &\Bigg\{\left(G_{t-s}\left(Y^{i,N,h}_s -x\right)-G_{t-\tau^h_s}\left(Y^{i,N,h}_{\tau^h_s} -x\right) \right) \\
&\phantom{\frac{1}{N}\sum \limits_{i=1}^N \int_0^t\{( - } - \left(G_{t-s}\left(X^{\eta_{\tau^h_s}(i),N,h}_s -x \right) - G_{t-{\tau^h_s}}\left(X^{\eta_{\tau^h_s}(i),N,h}_{\tau^h_s} -x \right) \right)\Bigg\} \lambda^N(i)\,ds=\sum_{p=1}^5e^{N,h}_p(t,x).
\end{align*} We are going to compute the two differences in the right-hand side by applying It\^{o}'s formula. To do so, let us recall the dynamics of $X^{\eta_{\tau^h_s}(i),N,h}_u$ for $u\in[\tau^h_s,\bar\tau^h_s)$ : $$dX^{\eta_{\tau^h_s}(i),N,h}_u = \sigma\,d\tilde \beta^{i}_u + \lambda^N \left(\eta_{\tau^h_s}^{-1}\left(\eta_{\tau^h_s}(i)\right)\right)\,du= \sigma\,d\tilde \beta^{i}_u + \lambda^N\left(\eta_{\tau^h_u}^{-1}\left(\eta_{\tau^h_u}(i)\right)\right)\,du.$$ We then have: \begin{align*}
\displaystyle &G_{t-s}\left(X^{\eta_{\tau^h_s}(i),N,h}_s -x \right) = G_{t-\tau^h_s}\left(X^{\eta_{\tau^h_s}(i),N,h}_{\tau^h_s} -x \right) + \sigma \int_{\tau^h_s}^s \partial_x G_{t-u}\left(X^{\eta_{\tau^h_s}(i),N,h}_u -x \right)\,d\tilde \beta^{i}_u \\
&+ \int_{\tau^h_s}^s \left(\partial_u G_{t-u}\left(X^{\eta_{\tau^h_s}(i),N,h}_u -x \right) + \frac{\sigma^2}{2}\partial_{xx}G_{t-u}\left(X^{\eta_{\tau^h_s}(i),N,h}_u -x \right) + \lambda^N\left(\eta_{\tau^h_u}^{-1}\left(\eta_{\tau^h_u}(i)\right)\right)\partial_x G_{t-u}\left(X^{\eta_{\tau^h_s}(i),N,h}_u -x \right)\right)\,du. \end{align*} Since $\partial_u G_{t-u} = - \partial_t G_{t-u}$, by the heat equation \eqref{heateq} from Lemma \ref{EstimHeatEq}, we have: \begin{align}\label{eqXG}
\displaystyle \int_{0}^t &\left( G_{t-s}\left(X^{\eta_{\tau^h_s}(i),N,h}_s -x \right)- G_{t-\tau^h_s}\left(X^{\eta_{\tau^h_s}(i),N,h}_{\tau^h_s} -x \right) \right)\,ds \\
&= \sigma \int_{0}^t \int_{\tau^h_s}^s \partial_x G_{t-u}\left(X^{\eta_{\tau^h_s}(i),N,h}_u -x \right)\,d\tilde \beta^{i}_u\,ds + \int_{0}^t \int_{\tau^h_s}^s \lambda^N \left(\eta_{\tau^h_u}^{-1}\left(\eta_{\tau^h_u}(i)\right)\right)\partial_x G_{t-u}\left(X^{\eta_{\tau^h_s}(i),N,h}_u -x \right)\,du\,ds. \notag \end{align} Let us suppose that $t>0$ and treat each term of the right-hand side of the above equation. For $x \in {\mathbb R}$, The function $u \mapsto \partial_xG_{t-u}\left(X^{\eta_{\tau^h_s}(i),N,h}_u -x \right)$ is continuous on $[0,t)$. Since $\left( X^{1,N,h}_t, \dots,X^{N,N,h}_t \right) $ admits a density, as stated after the proof of Lemma \ref{ctyF}, $\mathbb{P}\left(X^{\eta_{\tau^h_s}(i),N,h}_t = x \right) \le \sum\limits_{j=1}^N \mathbb{P}\left(X^{j,N,h}_t = x \right) = 0$ a.s.. Therefore, a.s. the previous function has a vanishing limit as $u\to t$ and is therefore bounded on the interval $[0,t]$. We can then apply Fubini's theorem to obtain: \begin{align*}
\displaystyle \int_{0}^t \int_{\tau^h_s}^s\lambda^N\left(\eta_{\tau^h_u}^{-1}\left(\eta_{\tau^h_u}(i)\right)\right)& \partial_x G_{t-u}\left(X^{\eta_{\tau^h_s}(i),N,h}_u -x \right)\,du\,ds \\&=\int_{0}^{t}\left(t \wedge \overline{\tau}^h_u -u \right) \lambda^N \left(\eta_{\tau^h_u}^{-1}\left(\eta_{\tau^h_u}(i)\right)\right)\partial_x G_{t-u}\left(X^{\eta_{\tau^h_u}(i),N,h}_u -x \right)\,du \quad a.s.. \end{align*} Secondly, with the use of Young's inequality and the same arguments of density of $\left( X^{1,N,h}_t, \dots,X^{N,N,h}_t \right) $, we get: \begin{align*}
\displaystyle &\int_0^t \left( \int_{\tau^h_s}^s \left|\partial_x G_{t-u}\left(X^{\eta_{\tau^h_s}(i),N,h}_u -x \right) \right|^2\,du\right)^{1/2}\,ds \\
&= \int_0^t \left( \int_{\tau^h_s}^s \frac{\left(X^{\eta_{\tau^h_s}(i),N,h}_u -x \right)^2}{2 \sigma^5 \sqrt{\pi} (t-u)^{5/2}} G_{(t-u)/2}\left(X^{\eta_{\tau^h_s}(i),N,h}_u -x \right) \,du\right)^{1/2}\,ds \\
&\leq \frac{t}{2} + \frac{1}{2}\int_0^t \int_{\tau^h_s}^s \frac{\left(X^{\eta_{\tau^h_s}(i),N,h}_u -x \right)^2}{2 \sigma^5 \sqrt{\pi} (t-u)^{5/2}} G_{(t-u)/2}\left(X^{\eta_{\tau^h_s}(i),N,h}_u -x \right)\,du\,ds < \infty \quad a.s.. \end{align*} Therefore, we can apply the stochastic Fubini Lemma \ref{StochFub} and obtain: \begin{align*}
\displaystyle \int_{0}^t \int_{\tau^h_s}^s \partial_x G_{t-u}\left(X^{\eta_{\tau^h_s}(i),N,h}_u -x \right)\,d\tilde \beta^{i}_u\,ds &= \int_{0}^{t}(t \wedge \overline{\tau}^h_s -u)\partial_x G_{t-u}\left(X^{\eta_{\tau^h_u}(i),N,h}_s -x \right)\,d\tilde \beta^{i}_u. \end{align*} Equation \eqref{eqXG} becomes: \begin{align}\label{eqXGnew}
\displaystyle& \int_{0}^t \left( G_{t-s}\left(X^{\eta_{\tau^h_s}(i),N,h}_s -x \right)- G_{t-\tau^h_s}\left(X^{\eta_{\tau^h_s}(i),N,h}_{\tau^h_s} -x \right) \right)\,ds \\
&= \sigma \int_{0}^{t}\left(t \wedge \overline{\tau}^h_s -s\right)\partial_x G_{t-s}\left(X^{\eta_{\tau^h_s}(i),N,h}_s -x \right)\,d\tilde \beta^{i}_s + \int_{0}^{t}\left(t \wedge \overline{\tau}^h_s -s\right)\lambda^N\left(\eta_{\tau^h_s}^{-1}\left(\eta_{\tau^h_s}(i)\right)\right)\partial_x G_{t-s}\left(X^{\eta_{\tau^h_s}(i),N,h}_s -x \right)\,ds . \notag \end{align}
Now, let us apply It\^{o}'s formula to $G_{t-s}\left(Y^{i,N,h}_s -x\right)$. Using once again the property \eqref{heateq} of the kernel $G_t(x)$ from Lemma \ref{EstimHeatEq} and the dynamics of $Y^{i,N,h}_s$ given by \eqref{dynamicY}, we have: \begin{align*}
\displaystyle G_{t-s}&\left(Y^{i,N,h}_s -x\right) - G_{t-\tau^h_s}\left(Y^{i,N,h}_{\tau^h_s} -x\right) = \sigma \int_{\tau^h_s}^s \partial_x G_{t-u}\left(Y^{i,N,h}_u -x \right)\,d\beta^{i}_u \\
&+ \int_{\tau^h_s}^s \partial_x G_{t-u}\left(Y^{i,N,h}_u -x \right) \lambda^N\left( \eta^{-1}_{\tau^h_u}(\eta_u(i) \right)\,du + \int_{\tau^h_s}^s \partial_x G_{t-u}\left(Y^{i,N,h}_u -x \right)\left(\gamma^{i}_u - \gamma^{i+1}_u \right)\,d|K|_u. \end{align*}
We use the same reasoning as for $X^{\eta_{\tau^h_s}(i),N,h}_s$ to treat the integrals from $0$ to $t$ of the first two terms : \begin{align*}
\displaystyle \int_{0}^t \int_{\tau^h_s}^s \partial_x G_{t-u}\left(Y^{i,N,h}_u -x \right)\,d\beta^{i}_u\,ds &= \int_{0}^{t}(t \wedge \overline{\tau}^h_s -s) \partial_x G_{t-s}\left(Y^{i,N,h}_s -x \right)\,d\beta^{i}_s, \\
\int_{0}^t \int_{\tau^h_s}^s \partial_x G_{t-u}\left(Y^{i,N,h}_u -x \right) \lambda^N\left( \eta^{-1}_{\tau^h_u}(\eta_u(i) \right)\,du\,ds &= \int_{0}^{t}(t \wedge \overline{\tau}^h_s -s) \partial_x G_{t-s}\left(Y^{i,N,h}_s -x \right)\lambda^N\left( \eta^{-1}_{\tau^h_s}(\eta_s(i) \right)\,ds. \end{align*}
As for the last term $ \displaystyle \int_{0}^t \int_{\tau^h_s}^s \partial_x G_{t-u}\left(Y^{i,N,h}_u -x \right)\left(\gamma^{i}_u - \gamma^{i+1}_u \right)\,d|K|_u\,ds$, we sum over $i \in \llbracket1,N\rrbracket$ after multiplying by $\lambda^N(i)$ then we apply Fubini's theorem. Using the property \eqref{procVarFin}, we finally obtain: \begin{align*}
\displaystyle \frac{1}{N}\sum \limits_{i=1}^N \lambda^N(i)&\int_{0}^t \int_{\tau^h_s}^s \partial_x G_{t-u}\left(Y^{i,N,h}_u -x \right)\left(\gamma^{i}_u - \gamma^{i+1}_u \right)\,d|K|_u\,ds \\
&= \frac{1}{N} \sum \limits_{i=2}^N \int_{0}^{t}(t \wedge \overline{\tau}^h_s -s) \left( \lambda^N(i)\partial_x G_{t-s}\left(Y^{i,N,h}_s -x \right) - \lambda^N(i-1)\partial_x G_{t-s}\left(Y^{i-1,N,h}_s -x \right) \right)\gamma^{i}_s\,d|K|_s \\
&= \frac{1}{N} \sum \limits_{i=2}^N \int_{0}^{t}(t \wedge \overline{\tau}^h_s -s) \left( \lambda^N(i) - \lambda^N(i-1) \right)\partial_x G_{t-s}\left(Y^{i,N,h}_s -x \right)\gamma^{i}_s\,d|K|_s. \end{align*} Therefore, \begin{align*}
\displaystyle &\frac{1}{N} \sum \limits_{i=1}^N \int_{0}^t \lambda^N(i) \left(G_{t-s}\left(Y^{i,N,h}_s -x\right) - G_{t-\tau^h_s}\left(Y^{i,N,h}_{\tau^h_s} -x\right) \right)\,ds \\
&= \frac{\sigma}{N} \sum \limits_{i=1}^N \lambda^N(i) \left\{\int_{0}^{t}(t \wedge \overline{\tau}^h_s -s) \partial_x G_{t-s}\left(Y^{i,N,h}_s -x \right)\,d\beta^{i}_s + \int_{0}^{t}(t \wedge \overline{\tau}^h_s -s) \lambda^N\left( \eta^{-1}_{\tau^h_s}(\eta_s(i) \right)\partial_x G_{t-s}\left(Y^{i,N,h}_s -x \right)\,ds \right\} \notag \\
&+ \frac{1}{N} \sum \limits_{i=2}^N\int_{0}^{t}(t \wedge \overline{\tau}^h_s -s) \left( \lambda^N(i) - \lambda^N(i-1) \right)\partial_x G_{t-s}\left(Y^{i,N,h}_s -x \right)\gamma^{i}_s\,d|K|_s. \notag \end{align*} We conclude by combining this equality and the sum over $i \in \llbracket1,N\rrbracket$ of \eqref{eqXGnew} multiplied by $\lambda^N(i)/N$. \end{proof}
Now that we got rid of the difference of $\lambda^N$ in the term $E^{N,h}(t,x)$, we can control the mean of the $L^1$-norm of this term. We present a succession of lemmas that will estimate each ${\mathbb E}\left[ \left\Vert e_p^{N,h}(t,.)\right\Vert_{L^1} \right]$ for $p \in \llbracket0,5\rrbracket$. \\
Since $G_{t-s}$ is a probability density, $\displaystyle {\mathbb E} \left[ \left\| e_{0}^{N,h}(t,.)\right\|_{L^1} \right] \le \frac{1}{N} \sum \limits_{i=1}^N \int_{0}^{t\wedge h}\left| \lambda^N(i) - \lambda^N\left( \eta^{-1}_{0}(\eta_{0}(i))\right) \right| \,ds $. Therefore, we obtain the following result concerning the term $e^{N,h}_0(t,x)$: \begin{lem}\label{ControlE0}
$$ \forall N \in {\mathbb N}^{*}, \forall h \in (0,T], \quad \sup_{t \le T} \left\| {\mathbb E}\left[ e^{N,h}_{0}(t,.)\right]\right\|_{L^1} \le \sup_{t \le T}{\mathbb E} \left[ \left\| e^{N,h}_{0}(t,.)\right\|_{L^1} \right] \le 2L_{\Lambda}h.$$\\ \end{lem}
We remark that the terms $e_4^{N,h}(t,x)$ and $e_5^{N,h}(t,x)$ are of the same nature. \begin{lem}\label{ControlDetE}
For $r \in \{4,5\}$: $$\exists C_{4,5}<\infty, \forall N \in {\mathbb N}^{*}, \forall h \in (0,T], \quad \sup_{t \le T} \left\| {\mathbb E}\left[ e^{N,h}_{r}(t,.)\right]\right\|_{L^1} \le \sup_{t \le T}{\mathbb E} \left[ \left\| e^{N,h}_{r}(t,.)\right\|_{L^1} \right] \le C_{4,5}\, h.$$ \end{lem} \begin{proof} Let us treat the term $e^{N,h}_5(t,x)$. \\ We have, using the estimate \eqref{FirstDerivG} from Lemma \ref{EstimHeatEq} for the second inequality, then the opposite monotonicities of the functions $s \mapsto \frac{1}{\sqrt{t-s}} $ and $s \mapsto (t \wedge t_{k+1} -s) $ on the time interval $[t\wedge t_k,t\wedge t_{k+1}]$ for the third inequality that: \begin{align*}
\displaystyle \left\| {\mathbb E}\left[ e^{N,h}_{5}(t,.)\right]\right\|_{L^1} &\le {\mathbb E} \left[ \left\| e_{5}^{N,h}(t,.)\right\|_{L^1} \right] \leq \frac{1}{N} \sum \limits_{i=1}^N \int_{0}^{t}(t \wedge \overline{\tau}^h_s -s) \left| \lambda^N(i) \right| \left| \lambda^N\left( \eta^{-1}_{\tau^h_s}(\eta_{\tau^h_s}(i))\right) \right| \left\Vert \partial_xG_{t-s} \right\Vert_{L^1}\,ds \\
&\le \frac{L_{\Lambda}^2}{\sigma}\sqrt{\frac{2}{\pi}}\sum_{k\in{\mathbb N}:t_k<t}\int_{t_k}^{t\wedge t_{k+1}}\frac{(t \wedge t_{k+1}-s)}{\sqrt{t-s}}\,ds \\&\le \frac{L_{\Lambda}^2}{\sigma}\sqrt{\frac{2}{\pi}} \left\{ \sum_{k\in{\mathbb N}:t_k<t}\frac{1}{t\wedge t_{k+1}-t_k}\int_{t_k}^{t\wedge t_{k+1}}(t \wedge t_{k+1}-s)ds\int_{t_k}^{t\wedge t_{k+1}}\frac{ds}{\sqrt{t-s}} \right\} \\
&\le \frac{L_{\Lambda}^2}{\sigma}\sqrt{\frac{2}{\pi}} \left\{ \frac{h}{2}\sum_{k\in{\mathbb N}:t_k<t}\int_{t_k}^{t\wedge t_{k+1}}\frac{ds}{\sqrt{t-s}} \right\}=\frac{L_{\Lambda}^2}{\sigma}\sqrt{\frac{2t}{\pi}}\,h. \end{align*} The term $e^{N,h}_4(t,x)$ can be estimated in the same way and the conclusion holds with $\displaystyle C_{4,5} = \frac{L_{\Lambda}^2}{\sigma}\sqrt{\frac{2T}{\pi}}$. \end{proof} We remark that the terms $e_2^{N,h}(t,x)$ and $e_3^{N,h}(t,x)$ are of the same nature as well. \begin{lem}\label{ControlStochE}
For $r \in \{2,3\}$, $\forall N \in {\mathbb N}^{*}, \forall h \in (0,T]$, $\left\| {\mathbb E}\left[ e^{N,h}_{r}(t,.)\right]\right\|_{L^1} = 0$. Moroever, if $\displaystyle \int_{{\mathbb R}} |x|^{\rho}m(dx) < \infty$ for some $\rho >1$, then:
$$\exists C_{2,3}<\infty, \quad \sup_{t \le T}{\mathbb E} \left[ \left\| e^{N,h}_{r}(t,.)\right\|_{L^1} \right] \leq \frac{C_{2,3}}{\sqrt{N}}.$$ \end{lem} \begin{proof} Let us treat the term $e_{3}^{N,h}(t,x)$. \\ Using the estimate \eqref{firstDerivSquar} from Lemma \ref{EstimHeatEq}, we obtain that \begin{align*}
\int_{{\mathbb R}} {\mathbb E}\left[ \int_0^t(t \wedge \overline{\tau}^h_s -s)^2 (\lambda^N(i))^2\left(\partial_xG_{t-s}\right)^2\left(X^{\eta_{\tau^h_s}(i),N,h}_s-x \right)\,ds \right]\,dx\le \int_0^t\frac{(t-s)^2L_\Lambda^2}{4\sigma^3(t-s)^{3/2}\sqrt{\pi}}ds<\infty. \end{align*} Therefore, $ {\mathbb E}\left[e^{N,h}_3(t,x) \right] = 0 $ dx a.e.. Moreover, using the estimate \eqref{GPrimSqu} from Lemma \ref{EstimHeatEq}, the It\^o isometry for the first equality, Young's inequality for the second inequality and last Fubini's theorem, we obtain: \begin{align*}
\displaystyle &{\mathbb E} \left[ \left\| e_{3}^{N,h}(t,.)\right\|_{L^1} \right] \leq \int_{{\mathbb R}} {\mathbb E}^{1/2}\left[ \left(\frac{\sigma}{N}\sum \limits_{i=1}^N \int_{0}^{t}(t \wedge \overline{\tau}^h_s -s) \lambda^N(i)\partial_x G_{t-s}\left(X^{\eta_{\tau^h_s}(i),N,h}_s -x \right)\,d\tilde \beta^{i}_s \right)^2 \right]\,dx \\
&= \frac{\sigma}{\sqrt{N}} \int_{{\mathbb R}} {\mathbb E}^{1/2}\left[ \frac{1}{N}\sum \limits_{i=1}^N \int_{0}^{t}(t \wedge \overline{\tau}^h_s -s)^2 \left|\lambda^N(i)\right|^2 \left(\partial_x G_{t-s}\right)^2\left(X^{\eta_{\tau^h_s}(i),N,h}_s -x \right)\,ds\right]\,dx \\
&= \frac{\sigma}{\sqrt{N}} \int_{{\mathbb R}} {\mathbb E}^{1/2}\left[ \frac{1}{N}\sum \limits_{i=1}^N \int_{0}^{t}(t \wedge \overline{\tau}^h_s -s)^2 \left|\lambda^N(i)\right|^2 \left(\partial_x G_{t-s}\right)^2\left(X^{\eta_{\tau^h_s}(i),N,h}_s -x \right)\,ds(1+|x|^{\rho}) \right]\frac{dx}{\sqrt{1+|x|^{\rho}}} \\
&\leq \frac{\sigma }{2\sqrt{N}} \int_{{\mathbb R}} \left( \frac{1}{1+|x|^{\rho}} + L_{\Lambda}^2{\mathbb E}\left[ \frac{1}{N}\sum \limits_{i=1}^N \int_{0}^{t}(t \wedge \overline{\tau}^h_s -s)^2 \left(\partial_x G_{t-s}\right)^2\left(X^{\eta_{\tau^h_s}(i),N,h}_s -x \right)\,ds(1+|x|^{\rho}) \right]\right)\,dx \\
&= \frac{\sigma I_{\rho} }{2\sqrt{N}} + \frac{L_{\Lambda}^2}{4 \sigma^4\sqrt{N \pi}}{\mathbb E}\Bigg[ \frac{1}{N}\sum \limits_{i=1}^N \int_{0}^{t} \frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{5/2}}\bigg\{ \int_{{\mathbb R}} \left(X^{\eta_{\tau^h_s}(i),N,h}_s -x \right)^2G_{(t-s)/2}\left(X^{\eta_{\tau^h_s}(i),N,h}_s -x \right)\,dx \\
&\phantom{\frac{\sigma \pi }{2\sqrt{N}} + \frac{L_{\Lambda}^2}{4 \sigma^4\sqrt{N \pi}} {\mathbb E}\Bigg[\frac{1}{N}\sum \limits_{i=1}^N \int_{0}^{t} \frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{5/2}} \bigg\{ \;}+ \int_{{\mathbb R}} |x|^{\rho}\left(X^{\eta_{\tau^h_s}(i),N,h}_s -x \right)^2G_{(t-s)/2}\left(X^{\eta_{\tau^h_s}(i),N,h}_s -x \right)\,dx\bigg\}\,ds\Bigg] \\
&= \frac{\sigma I_{\rho} }{2\sqrt{N}} + \frac{L_{\Lambda}^2}{4 \sigma^4\sqrt{N \pi}}{\mathbb E}\left[ \frac{1}{N}\sum \limits_{i=1}^N \int_{0}^{t} \frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{
5/2}}\left\{ \int_{{\mathbb R}} y^2G_{(t-s)/2}(y)\,dy + \int_{{\mathbb R}} \left|X^{\eta_{\tau^h_s}(i),N,h}_s -y \right|^{\rho}y^2G_{(t-s)/2}(y)\,dy \right\}\,ds\right] \\
&\le \frac{\sigma I_{\rho} }{2\sqrt{N}} + \frac{L_{\Lambda}^2}{4 \sigma^4\sqrt{N \pi}}{\mathbb E}\Bigg[ \frac{1}{N}\sum \limits_{i=1}^N \int_{0}^{t} \frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{5/2}}\Bigg\{ \left(1 + 2^{\rho -1} \left|X^{\eta_{\tau^h_s}(i),N,h}_s \right|^{\rho}\right) \frac{\sigma^2(t-s)}{2} \\
&\phantom{\frac{\sigma I_{\rho} }{2\sqrt{N}} + \frac{L_{\Lambda}^2}{4 \sigma^4\sqrt{N \pi}}{\mathbb E}\Bigg[ \frac{1}{N}\sum \limits_{i=1}^N \int_{0}^{t} \frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{5/2}}\Bigg\{ \left(1 + 2^{\rho -1} \left|X^{\eta_{\tau^h_s}(i),N,h}_s \right|^{\rho}\right) \Bigg\{\;} + 2^{\rho -1} \frac{\left(\sigma \sqrt{t-s} \right)^{2+\rho}}{\sqrt \pi}\Gamma\left(\frac{\rho +3}{2} \right) \Bigg\}\,ds\Bigg] \\
&= \frac{1}{2\sqrt{N}}\Bigg( \sigma I_{\rho} + \frac{L_{\Lambda}^2}{4 \sigma^2 \sqrt{\pi}} \Bigg\{ \int_0^t \frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{3/2}}\,ds + 2^{\rho-1}\int_0^t \frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{3/2}} {\mathbb E}\left[\frac{1}{N}\sum \limits_{i=1}^N \left|X^{\eta_{\tau^h_s}(i),N,h}_s\right|^{\rho} \right]\,ds \\
&\phantom{\frac{1}{2\sqrt{N}}\Bigg( \sigma I_{\rho} + \frac{L_{\Lambda}^2}{4 \sigma^2 \sqrt{\pi}} \Bigg\{ \;}+ \frac{2^{\rho -2}\sigma^{\rho}}{\sqrt \pi}\Gamma\left(\frac{\rho + 3}{2} \right) \int_0^t \frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{\frac{3-\rho}{2}}}\,ds \Bigg\} \Bigg). \end{align*} With the use of Lemma \ref{controlM}, we obtain: \begin{align*}
\displaystyle {\mathbb E} \left[ \left\| e_{3}^{N,h}(t,.)\right\|_{L^1} \right] &\le \frac{1}{2\sqrt{N}}\Bigg(\sigma I_{\rho} + \frac{L_{\Lambda}^2}{4 \sigma^2 \sqrt{\pi}} \Bigg\{\left(1+2^{\rho-1}M \right) \int_0^t \frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{3/2}}\,ds \\ &\phantom{\frac{1}{2\sqrt{N}}\Bigg(\sigma I_{\rho} + \frac{L_{\Lambda}^2}{4 \sigma^2 \sqrt{\pi}} \Bigg\{\left(1+2^{\rho-1}M \right) \Bigg\{\;} + \frac{2^{\rho -2}\sigma^{\rho}}{\sqrt \pi}\Gamma\left(\frac{\rho + 3}{2} \right) \int_0^t \frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{\frac{3-\rho}{2}}}\,ds \Bigg\} \Bigg). \end{align*} When $t \ge h$, $\displaystyle\int_0^t \frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{3/2}}\,ds \leq h^2\int_0^{t-h}\frac{ds}{(t-s)^{3/2}} + \int_{t-h}^t \sqrt{t-s}\,ds \leq \frac{8}{3}h^{3/2}$ and $\displaystyle \int_0^t \frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{\frac{3-\rho}{2}}}\,ds \le \frac{2}{\rho-1}h^2 t^{\frac{\rho-1}{2}} + \frac{2}{\rho +3}h^{\frac{\rho +3}{2}}\le \frac{2(\rho+2)}{(\rho-1)(\rho+3)}h^{\frac{\rho +3}{2}}$. When $t \le h$, $\displaystyle\int_0^t \frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{3/2}}\,ds = \int_0^t \sqrt{t-s}\,ds = \frac{2}{3}t^{3/2} \leq \frac{2}{3}h^{3/2}$ and $\displaystyle \int_0^t \frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{\frac{3-\rho}{2}}}\,ds \le \frac{2}{\rho +3}h^{\frac{\rho +3}{2}}$. The term $e^{N,h}_2(t,x)$ can be estimated in the same way and the conclusion holds with $$\displaystyle C_{2,3} = \frac{1}{2}\left( \sigma I_{\rho} + \frac{L_{\Lambda}^2 T^{3/2}}{4 \sigma^2 \sqrt{\pi}} \left( \frac{8}{3}\left(1+2^{\rho-1}M \right) + \frac{2^{\rho -1}\sigma^{\rho}}{\sqrt \pi}\times\frac{2(\rho+2)}{(\rho-1)(\rho+3)}\Gamma\left(\frac{\rho + 3}{2} \right) T^{\frac{\rho -1}{2}}\right) \right).$$ \end{proof}
Now, we finally treat the term $e^{N,h}_1(t,x)$ in the lemma below: \begin{lem}\label{ControlGam}
If $\lambda$ is Lipschitz continuous with constant $L_{\lambda}$ then $\exists C_{1}<\infty, \forall N \in {\mathbb N}^{*}, \forall h \in (0,T], \forall t \in [0,T]$,
$$\displaystyle {\mathbb E} \left[ \left\| e^{N,h}_{1}(t,.)\right\|_{L^1} \right] \leq C_1\left(h + \sqrt{h}{\mathbb E} \left[ \left\| F^{N,h}(t,.) - F(t,.) \right\|_{L^1} \right] + {\mathbf 1}_{\{t\ge h\}}\int_0^{t-h} \frac{h}{2(t-s)^{3/2}}{\mathbb E} \left[ \left\| F^{N,h}(s,.) - F(s,.) \right\|_{L^1} \right]\,ds \right).$$ \end{lem} The proof of this assertion relies on the following results. \begin{lem}\label{AccroissF}
We have: $$\exists Q < \infty, \forall 0 \le s \le t \le T, \quad \left\| F(t,.) - F(s,.)\right\|_{L^1} \leq Q \left(\sqrt{t}-\sqrt{s}\right) - L_{\Lambda}(t-s)\ln(t-s). $$ \end{lem} \begin{proof} Let $ 0 \le s \le t \le T$. We recall that: $$\displaystyle F(t,x) - F(s,x) = \left(G_t - G_s \right)* F_0(x) - \int_0^t \partial_xG_{t-u} * \Lambda\left(F(u,.) \right)(x)\,du + \int_0^s \partial_xG_{s-u} * \Lambda\left(F(u,.) \right)(x)\,du.$$ Using Equality \eqref{heateq} and the estimates \eqref{FirstDerivG} and \eqref{SecondDerivG} from Lemma \ref{EstimHeatEq}, as well as the fact that $(t-s) \leq 2\sqrt{T}\left(\sqrt{t} - \sqrt{s} \right)$ and $t\ln t-s\ln s=\int_s^t(1+\ln x)dx\le \int_s^txdx=\frac{t+s}{2}(t-s)\le T\times 2\sqrt{T}(\sqrt{t}-\sqrt{s})$, we obtain: \begin{align*}
\displaystyle &\left\|F(t,.) - F(s,.) \right\|_{L^1} \\
&\le \left\|\left(G_t - G_s \right)* F_0 \right\|_{L^1}+ \left\|\int_0^s\left(G_{t-u}-G_{s-u}\right)*\partial_x\Lambda\left(F(u,.)\right)\,du \right\|_{L^1} + \left\|\int_s^t G_{t-u}*\partial_x\Lambda\left(F(u,.)\right)\,du\right\|_{L^1} \\
&\le \left\|\int_s^t \left(\partial_u G_u*F_0\right)\,du \right\|_{L^1} + \left\|\int_0^s \int_{s-u}^{t-u} \partial_r G_r* \partial_x\Lambda\left(F(u,.)\right)\,dr\,du\right\|_{L^1} + \int_s^t \left\| G_{t-u}*\left(\lambda\left(F(u,.) \right)p(u,.)\right)\right\|_{L^1}\,du \\
&\le \frac{\sigma^2}{2}\int_s^t \left\| \left(\partial_{x}G_u*m\right)\right\|_{L^1} \,du + \frac{\sigma^2}{2}\int_0^s \int_{s-u}^{t-u} \left\| \partial_{xx}G_r* \left(\lambda\left(F(u,.)\right)p(u,.)\right) \right\|_{L^1} \,dr\,du + L_{\Lambda}(t-s) \\
&\le \frac{\sigma^2}{2}\int_s^t\sqrt{\frac{2}{\pi \sigma^2 u}}\,du + L_{\Lambda}\int_0^s \ln\left(\frac{t-u}{s-u}\right)\,du + L_{\Lambda}(t-s) \\
&= \sigma \sqrt{\frac{2}{\pi}} \left(\sqrt{t}-\sqrt{s} \right) + L_{\Lambda}\left( t\ln(t) - s\ln(s) - (t-s)\ln(t-s)\right) + L_{\Lambda}(t-s)\\
&\leq \left(\sigma \sqrt{\frac{2}{\pi}}+ 2L_{\Lambda} \sqrt{T}\left(1+T\right) \right)\left(\sqrt{t}-\sqrt{s}\right) - L_{\Lambda} (t-s)\ln(t-s). \end{align*} The conclusion holds with $Q=\sigma \sqrt{2 / \pi}+ 2 L_{\Lambda}\sqrt{T}\left(1+T\right)$. \end{proof}
The next lemma provides two different estimations of the term $\displaystyle {\mathbb E}\left[ \int_{s}^{t} \gamma^{i}_u d|K|_u \right]$. They are both useful to prove Lemma \ref{ControlGam}. \begin{lem}\label{estimGam} $\forall N \in {\mathbb N}^{*}, \forall i \in \llbracket2,N\rrbracket, \forall h \in (0,T], \forall 0 \le s \le t \le T$, \begin{align}\label{estimGam1}
&\displaystyle {\mathbb E}\left[\left(\int_{s}^{t}\gamma^{i}_u\,d|K|_u\right)^2 \right] \leq 9N^2 \left(\sigma^2 + L_{\Lambda}^2 T\right)(t-s), \end{align} \begin{align}\label{estimGam2}
&\mbox{and }\displaystyle {\mathbb E}\left[ \int_{s}^{t}\gamma^{i}_u\,d|K|_u \right] \leq N \left({\mathbb E}\left[ \left\|F^{N,h}(t,.) - F^{N,h}(s,.) \right\|_{L^1} \right] + L_{\Lambda} (t-s)\right). \end{align} \end{lem} \begin{proof}
Let $2 \le i \le N$. Since $\gamma^{N+1}_u = 0$, we have $\displaystyle \int_{s}^{t} \gamma^{i}_u\,d|K|_u = \int_{s}^{t} \sum \limits_{j=i}^{N} \left(\gamma^{j}_u - \gamma^{j+1}_u \right)\,d|K|_u$ and with the dynamics \eqref{dynamicY} of $Y^{j,N,h}$, we deduce that \begin{align*}
\displaystyle \int_{s}^{t} \gamma^{i}_u\,d|K|_u &= \int_{s}^{t} \sum \limits_{j=i}^{N} \left(\gamma^{j}_u - \gamma^{j+1}_u \right)\,d|K|_u = \sum \limits_{j=i}^{N}\left\{ \left( Y^{j,N,h}_t - Y^{j,N,h}_s \right) - \sigma\left( \beta^{j}_t - \beta^{j}_s \right) - \int_s^t \lambda^N \left( \sigma^{-1}_{\tau^h_u}(\eta_{u}(j))\right)\,du \right\}. \end{align*}
Let us start by proving the estimation of $\displaystyle {\mathbb E}\left[\left( \int_{s}^{t}\gamma^{i}_u\,d|K|_u \right)^2\right]$. With the use of Jensen's inequality and Lemma \ref{XandY} for $\rho = 2$, we obtain: \begin{align*}
\displaystyle {\mathbb E}\left[\left(\int_{s}^{t}\gamma^{i}_u\,d|K|_u\right)^2 \right] &\leq 3 N \left( \sum \limits_{j=1}^{N} {\mathbb E}\left[\left| Y^{j,N,h}_t - Y^{j,N,h}_s \right|^2\right] + \sum \limits_{j=1}^{N}\left(\int_{s}^{t} \left| \lambda^N\left( \sigma^{-1}_{\tau^h_u}(\eta_{u}(j))\right)\right|\,du \right)^2 + \sigma^2\sum \limits_{j=1}^{N} {\mathbb E}\left[ \left|\beta^{j}_t - \beta^{j}_s \right|^2\right] \right) \\
&\le 3N \left( \sum \limits_{j=1}^{N}{\mathbb E}\left[\left| X^{j,N,h}_t - X^{j,N,h}_s \right|^2\right] + N L_{\Lambda}^2 (t-s)^2 + N \sigma^2(t-s) \right)\\
&\le 3N \left( \sum \limits_{j=1}^{N}2{\mathbb E}\left[\sigma^2|W^j_t-W^j_s|^2+L_{\Lambda}^2 (t-s)^2
\right] + N L_{\Lambda}^2 (t-s)^2 + N \sigma^2(t-s) \right)\\&\leq 9N^2\left(\sigma^2 + L_{\Lambda}^2 T \right)(t-s). \end{align*}
Notice that because of the latter contribution of ${\mathbb E}\left[|W^j_t-W^j_s|^2\right]$, it was not useful to take advantage of the independence of the Brownian motions $\beta^j$ which ensures ${\mathbb E}\left[ \left|\sum \limits_{j=i}^{N}(\beta^{j}_t - \beta^{j}_s) \right|^2\right]=(N+1-i)(t-s)$. Let us now prove the second estimation of $\displaystyle {\mathbb E}\left[ \int_{s}^{t}\gamma^{i}_u\,d|K|_u \right]$. To do so, we use that, according to \eqref{Wasserstein} and \eqref{w1fdr}, $\displaystyle \frac{1}{N} \sum \limits_{i=1}^N \left|Y^{i,N,h}_t - Y^{i,N,h}_s \right| = \mathcal{W}_1\left(\mu^{N,h}_t, \mu^{N,h}_s\right) = \int_{{\mathbb R}}\left|F^{N,h}(t,x) - F^{N,h}(s,x)\right|\,dx = \left\|F^{N,h}(t,.) - F^{N,h}(s,.)\right\|_{L^1}$ to obtain: \begin{align*}
\displaystyle {\mathbb E}\left[\int_{s}^{t}\gamma^{i}_u\,d|K|_u \right] \leq& \sum \limits_{j=1}^{N} {\mathbb E}\left[\left| Y^{j,N,h}_t - Y^{j,N,h}_s \right|\right] + \sum \limits_{j=1}^{N} \int_{s}^{t} \left| \lambda^N\left( \sigma^{-1}_{\tau^h_u}(\eta_{u}(j)) \right)\right|\,du \\
\leq& N {\mathbb E}\left[ \left\|F^{N,h}(t,.) - F^{N,h}(s,.) \right\|_{L^1} \right] + N L_{\Lambda} (t-s). \end{align*} \end{proof}
Let us now prove Lemma \ref{ControlGam}. \begin{proof}
We recall that $\displaystyle e^{N,h}_1(t,x) = \frac{1}{N}\sum \limits_{i=2}^N \int_{0}^{t}(t \wedge \overline{\tau}^h_s -s)\left( \lambda^N(i) - \lambda^N(i-1) \right)\partial_x G_{t-s}\left(Y^{i,N,h}_s -x \right)\gamma^{i}_s\,d|K|_s$. For $i \in \llbracket 2,N \rrbracket$, we have $\displaystyle \left|\lambda^N(i)-\lambda^N(i-1) \right| = \left| N\int_{\frac{i-1}{N}}^{\frac{i}{N}} \left( \lambda(u) - \lambda\left(u-\frac{1}{N}\right)\right)\,du\right| \le \frac{L_{\lambda}}{N}$. Using the estimate \eqref{FirstDerivG} from Lemma \ref{EstimHeatEq} and the property \eqref{procVarFin}, we have: \begin{align}\label{here}
\displaystyle {\mathbb E}\left[\left\|e_{1}^{N,h}(t,.)\right\|_{L^1} \right] &\leq \frac{L_{\lambda}}{\sigma N}\sqrt{\frac{2}{\pi}}\left\{\frac{1}{N}\sum_{i=2}^N {\mathbb E}\left[ \int_{0}^{t} \frac{(t \wedge \overline{\tau}^h_s -s)}{\sqrt{t-s}}\gamma^{i}_s\,d|K|_s \right]\right\}. \end{align} \begin{itemize}
\item For $t \le h$, since $\frac{(t \wedge \overline{\tau}^h_s -s)}{\sqrt{t-s}} = \sqrt{t-s} \le \sqrt h$, we deduce from \eqref{estimGam1} that $\displaystyle {\mathbb E}\left[\left\|e_{1}^{N,h}(t,.)\right\|_{L^1} \right] \leq \frac{3L_{\lambda}}{\sigma}\sqrt{\frac{2(\sigma^2 + L_{\Lambda}^2T)}{\pi}}\,h$.
\item For $t \ge h$, we decompose the right-hand side of inequality \eqref{here} onto the sub-intervals $[0,t-h]$ and $[t-h,t]$ for a better control. Therefore,
\begin{align*}
\displaystyle {\mathbb E}\left[\left\|e_{1}^{N,h}(t,.)\right\|_{L^1} \right] &\le \frac{L_{\lambda}h}{\sigma N^2}\sqrt{\frac{2}{\pi}} \sum_{i=2}^N \left({\mathbb E}\left[ \int_{0}^{t-h} \frac{1}{\sqrt{t-s}}\gamma^{i}_s\,d|K|_s \right] + \frac{1}{\sqrt{h}}{\mathbb E}\left[ \int_{t-h}^{t}\gamma^{i}_s\,d|K|_s \right]\right).
\end{align*}
As for the first term of the right-hand side of the above inequality, we introduce $\displaystyle A_s = - \int_s^t \gamma_u^{i}d|K|_u$ and apply Fubini's theorem to obtain:
\begin{align*}
\displaystyle \int_0^{t-h}A_s \frac{ds}{2(t-s)^{3/2}} &= \int_0^{t-h} \left(A_0 + \int_0^sdA_r \right)\frac{ds}{2(t-s)^{3/2}} = A_0\left( \frac{1}{\sqrt h} - \frac{1}{\sqrt t}\right) + \int_0^{t-h}\int_0^s\frac{dA_r}{2(t-s)^{3/2}}\,ds \\
&= A_0\left( \frac{1}{\sqrt h} - \frac{1}{\sqrt t}\right) + \int_0^{t-h}\int_r^{t-h}\frac{ds}{2(t-s)^{3/2}}\,dA_r \\
&= A_0\left( \frac{1}{\sqrt h} - \frac{1}{\sqrt t}\right) + \int_0^{t-h} \left( \frac{1}{\sqrt h} - \frac{1}{\sqrt{t-r}}\right)\,dA_r \\
&= -\frac{1}{\sqrt t}A_0 + \frac{1}{\sqrt h}A_{t-h} - \int_0^{t-h}\frac{1}{\sqrt{t-r}}\,dA_r.
\end{align*}
Consequently, we obtain that:
\begin{align*}
\displaystyle {\mathbb E}\left[ \int_{0}^{t-h} \frac{1}{\sqrt{t-s}}\gamma^{i}_s\,d|K|_s \right] + \frac{1}{\sqrt{h}} {\mathbb E}\left[ \int_{t-h}^t\gamma_u^{i}\,d|K|_u\right] = \frac{1}{\sqrt{t}}{\mathbb E}\left[\int_0^t\gamma_u^{i}\,d|K|_u\right] + {\mathbb E}\left[\int_{0}^{t-h} \frac{1}{2(t-s)^{3/2}}\int_{s}^{t}\gamma^{i}_u\,d|K|_u\,ds \right].
\end{align*}
We shall use the estimate \eqref{estimGam1} and the estimate \eqref{estimGam2} from Lemma \ref{estimGam} for respectively the first term and the second term of the right-hand side of the following inequality:
\begin{align*}
\displaystyle &{\mathbb E}\left[\left\|e_{1}^{N,h}(t,.)\right\|_{L^1} \right] \\
&\le \frac{L_{\lambda}h}{\sigma N^2}\sqrt{\frac{2}{\pi}}\sum_{i=2}^N \left\{ \frac{1}{\sqrt{t}} {\mathbb E}\left[\int_0^t\gamma_u^{i}\,d|K|_u\right] +\int_{0}^{t-h} \frac{1}{2(t-s)^{3/2}}{\mathbb E}\left[ \int_{s}^{t}\gamma^{i}_u\,d|K|_u\right]\,ds \right\} \\
&\le \frac{L_{\lambda}h}{\sigma N^2}\sqrt{\frac{2}{\pi}}\sum_{i=2}^N \Bigg\{ 3N \sqrt{ \left(\sigma^2 + L_{\Lambda}^2 T\right)} + N \int_{0}^{t-h} \frac{L_{\Lambda}}{2\sqrt{t-s}}\,ds + N\int_{0}^{t-h} \frac{1}{2(t-s)^{3/2}}{\mathbb E}\left[ \left\|F^{N,h}(t,.) - F^{N,h}(s,.) \right\|_{L^1} \right] \,ds \Bigg\} \\
&= \frac{L_{\lambda}}{\sigma }\sqrt{\frac{2}{\pi}}\left(3\sqrt{\left(\sigma^2 + L_{\Lambda}^2 T\right)} + L_{\Lambda}\left(\sqrt{t}-\sqrt{h} \right) \right)h + \frac{L_{\lambda}}{\sigma}\sqrt{\frac{2}{\pi}}\int_{0}^{t-h} \frac{h}{2(t-s)^{3/2}}{\mathbb E}\left[ \left\|F^{N,h}(t,.) - F^{N,h}(s,.) \right\|_{L^1} \right]\,ds \\
&\le \frac{L_{\lambda}}{\sigma}\sqrt{\frac{2}{\pi}}\left( 3\sqrt{\left(\sigma^2 + L_{\Lambda}^2 T\right)} + L_{\Lambda}\sqrt{T}\right)h + \frac{L_{\lambda}}{\sigma}\sqrt{\frac{2}{\pi}}\int_{0}^{t-h} \frac{h}{2(t-s)^{3/2}}{\mathbb E}\left[ \left\|F^{N,h}(t,.) - F^{N,h}(s,.) \right\|_{L^1} \right]\,ds.
\end{align*}
Since
\begin{align*}
\displaystyle {\mathbb E}\left[ \left\|F^{N,h}(t,.) - F^{N,h}(s,.) \right\|_{L^1} \right] \leq {\mathbb E}\left[ \left\|F^{N,h}(t,.) - F(t,.) \right\|_{L^1} \right]&+ \left\|F(t,.) - F(s,.) \right\|_{L^1} + {\mathbb E}\left[ \left\|F(s,.) - F^{N,h}(s,.) \right\|_{L^1} \right],
\end{align*}
using Lemma \ref{AccroissF}, we obtain:
\begin{align*}
\displaystyle \int_{0}^{t-h} \frac{h}{2(t-s)^{3/2}}&{\mathbb E}\left[ \left\|F^{N,h}(t,.) - F^{N,h}(s,.) \right\|_{L^1} \right]\,ds \le \sqrt{h}{\mathbb E}\left[ \left\|F^{N,h}(t,.) - F(t,.) \right\|_{L^1} \right] \\
&+ h\int_{0}^{t-h} \frac{{\mathbb E}\left[ \left\|F(s,.) - F^{N,h}(s,.) \right\|_{L^1}\right]}{2(t-s)^{3/2}}\,ds + h\int_{0}^{t} \left(Q\frac{\sqrt{t}-\sqrt{s} }{2(t-s)^{3/2}} - \frac{L_{\Lambda}}{2} \frac{\ln(t-s)}{ \sqrt{t-s}}\right)\,ds.
\end{align*}
To treat the last term of the right-hand side of the above inequality, we will use the fact that $\underset{x>0}{\sup}\left\{\sqrt{x}\left(2 - \ln(x)\right)\right\} = 2$.
\begin{align*} \displaystyle \int_{0}^{t} \left(Q\frac{\sqrt{t}-\sqrt{s} }{2(t-s)^{3/2}} - \frac{L_{\Lambda}}{2} \frac{\ln(t-s)}{ \sqrt{t-s}}\right)\,ds &= Q \int_0^1 \frac{1 - \sqrt{x}}{2(1-x)^{3/2}}\,dx - L_{\Lambda}\left(\sqrt{t}\ln(t) - 2\sqrt{t} \right) \\ &= Q \left(\left[(1-x)^{-1/2}(1-\sqrt x)\right]_0^1 + \int_0^1 \frac{dx}{2 \sqrt{x} \sqrt{1-x}} \right) + L_{\Lambda} \sqrt{t}\left( 2 - \ln(t)\right) \\ &=Q\left( \frac{\pi}{2}-1\right) + L_{\Lambda} \sqrt{t}\left( 2 - \ln(t)\right) \\
&\le Q\left( \frac{\pi}{2}-1\right) + 2L_{\Lambda}.
\end{align*} Therefore, \begin{align*}
\displaystyle {\mathbb E}\left[\left\|e_{1}^{N,h}(t,.)\right\|_{L^1} \right] \leq& C_1 \left( h + \sqrt{h}{\mathbb E}\left[\left\|F^{N,h}(t,.) - F(t,.) \right\|_{L^1} \right] + \int_{0}^{t-h} \frac{h}{2(t-s)^{3/2}}{\mathbb E}\left[\left\|F(s,.) - F^{N,h}(s,.) \right\|_{L^1}\right]\,ds \right) \end{align*} where $\displaystyle C_1 = \frac{L_{\lambda}}{\sigma}\sqrt{\frac{2}{\pi}} \left[1 \vee \left( 3\sqrt{\left(\sigma^2 + L_{\Lambda}^2 T\right)} + L_{\Lambda}(2 +\sqrt{T}) + Q\left( \frac{\pi}{2}-1\right)\right)\right]$. \end{itemize} \end{proof}
Using Lemmas \ref{ControlE}, \ref{ControlE0}, \ref{ControlDetE}, \ref{ControlStochE} and \ref{ControlGam} and the fact that for $s\in[0,t-h]$, $\frac{h}{2(t-s)^{3/2}}\le \frac{1}{2 \sqrt{t-s}} $, we conclude the proof of Proposition \ref{ControlFinalE} for the choice $Z = 2 \max\left(L_{\Lambda}+C_1/2+C_{4,5}, C_{2,3} \right)$.
\begin{remark}
In Lemma \ref{estimGam}, we provide two estimations of $\displaystyle {\mathbb E}\left[\int_s^t \gamma^{i}_u d|K|_u\right]$. If we only use the first estimation \eqref{estimGam1} in the proof of Lemma \ref{ControlGam}, we obtain, using a decomposition that we will detail in Section \ref{SectBiais}, a rough estimation of ${\mathbb E}\left[\left\|e_{1}^{N,h}(t,.)\right\|_{L^1} \right]$ where we lose a $\ln(h)$ factor. \end{remark}
\subsection{Estimation of the bias}\label{secproofthm2}
We recall Equation \eqref{eqFh}: \begin{align*}
&F^{N,h}(t,x) - F(t,x) \\
&= \quad G_t*\left(F^{N,h}_0 - F_0\right)(x) - \displaystyle \int_0^t \partial_x G_{t-s} * \left(\Lambda(F^{N,h}(s,.)) - \Lambda(F(s,.))\right)(x)\,ds + R^{N,h}(t,x) + E^{N,h}(t,x), \end{align*} and we shall use the expression of $E^{N,h}(t,x)$ proved in Lemma \ref{ControlE}. The next lemma provides an upper-bound of $\left\Vert {\mathbb E}\left[E^{N,h}(t,.) \right] \right\Vert_{L^1}$. \begin{lem}\label{normL1ofE} Assume that $\lambda$ is Lipschitz continuous and the initial positions are \begin{itemize} \item either i.i.d. according to $m$ and $\displaystyle \int_{{\mathbb R}}\sqrt{F_0(x)(1 - F_0(x))}\,dx < \infty$,
\item or optimal deterministic and $\displaystyle \sup_{x \ge 1} x\int_{x}^{+\infty} \left(F_0(-y) + 1 - F_0(y) \right)\,dy<\infty$. \end{itemize}Then $$\exists Z_b <\infty, \forall N \in {\mathbb N}^{*}, \forall h \in (0,T], \quad \sup_{t\le T}\left\Vert {\mathbb E}\left[E^{N,h}(t,.) \right] \right\Vert_{L^1} \leq Z_b \left(\sqrt{\frac h N} + h\right).$$ \end{lem} \begin{proof} To estimate $\left\Vert {\mathbb E}\left[E^{N,h}(t,.) \right] \right\Vert_{L^1}$, we estimate each $\left\Vert {\mathbb E}\left[e^{N,h}_p(t,.) \right] \right\Vert_{L^1}, p \in \llbracket0,5\rrbracket$. From Lemmas \ref{ControlE0}, \ref{ControlDetE} and \ref{ControlStochE} we have $\left\Vert {\mathbb E}\left[E^{N,h}(t,.) \right] \right\Vert_{L^1} \le \left\Vert {\mathbb E}\left[e_1^{N,h}(t,.) \right] \right\Vert_{L^1} + 2L_{\Lambda}h+ 2C_{4,5}h$. By Lemma \ref{ControlGam} and Corollary \ref{cor1}, we have: \begin{align*} \displaystyle &\left\Vert {\mathbb E}\left[e_1^{N,h}(t,.) \right] \right\Vert_{L^1}\le {\mathbb E}\left[\left\Vert e_1^{N,h}(t,.) \right\Vert_{L^1}\right] \\
&\le C_1h + C_1\sqrt{h}{\mathbb E}\left[ \left\|F^{N,h}(t,.) - F(t,.) \right\|_{L^1} \right] + C_1{\mathbf 1}_{\{t\ge h\}}\int_{0}^{t-h} \frac{h}{2(t-s)^{3/2}}{\mathbb E}\left[ \left\|F^{N,h}(s,.) - F(s,.) \right\|_{L^1} \right]\,ds\\ &\le C_1h + C_1\sqrt{h} C \left( \frac{1}{\sqrt N} + h\right) + C_1{\mathbf 1}_{\{t\ge h\}}\int_{0}^{t-h} \frac{h}{2(t-s)^{3/2}}C\left( \frac{1}{\sqrt N} + h\right)\,ds \\ &\le C_1h+C_1C h^{3/2}+ C_1C \sqrt{\frac h N} + C_1C\left( \frac{1}{\sqrt N} + h\right)\sqrt h\\ &\le \left(C_1 + 2C_1C \sqrt T \right)h +2C_1C\sqrt{\frac h N}. \end{align*} The conclusion holds with $Z_b = \max(2L_{\Lambda} + 2C_{4,5} + C_1 + 2C_1C \sqrt T, 2C_1C)$. \end{proof}
The proof of Theorem \ref{thmBias} relies on the following Proposition that we will prove in Section \ref{SectBiais}. \begin{prop}\label{normL2}
Assume that $\displaystyle \int_{{\mathbb R}}|x|m(dx) < \infty$ and $\lambda$ is Lipschitz continuous. Then:
$$\exists M_b< \infty, \forall N \in {\mathbb N}^{*}, \forall h \in (0,T],\quad \sup_{t \leq T}{\mathbb E}\left[\left\Vert F^{N,h}(t,.) - F(t,.) \right\Vert^{2}_{L^2}\right] \le M_b\left(\frac{1}{N} + h \right). $$ \end{prop}
\subsubsection{Proof of Theorem \ref{thmBias}}
Taking the expectation of Equation \eqref{eqFh} and using Lemma \ref{controlRN}, we obtain that $dx$ a.e.: \begin{align*}
\displaystyle &{\mathbb E}\left[ F^{N,h}(t,x)\right] - F(t,x) \\
&= G_t * {\mathbb E}\left[F^N_0(x) - F_0(x) \right] - \int_0^t \partial_xG_{t-s}*{\mathbb E}\left[ \left(\Lambda(F^{N,h}(s,.)) - \Lambda(F(s,.))\right)(x) \right]\,ds + {\mathbb E}\left[E^{N,h}(t,x) \right].
\end{align*} Besides, using Taylor-Young's inequality, we have that: \begin{align*}
\left| \Lambda\left(F^{N,h}(s,.)\right) - \Lambda(F(s,.)) - \lambda(F(s,.)) \left[F^{N,h}(s,.) - F(s,.) \right] \right| \le \frac{L_{\lambda}}{2}\left| \left( F^{N,h}(s,.) - F(s,.) \right)^2 \right| \end{align*} which implies: \begin{align*}
\left\Vert{\mathbb E}\left[ \Lambda(F^{N,h}(s,.))\right] - \Lambda(F(s,.))\right\Vert_{L^1} \le \left\Vert \lambda(F(s,.)) \right\Vert_{L^{\infty}} & \left\Vert {\mathbb E}\left[F^{N,h}(s,.)\right] - F(s,.) \right\Vert_{L^1} + \frac{L_{\lambda}}{2}{\mathbb E}\left[\left\Vert F^{N,h}(s,.) - F(s,.) \right\Vert^{2}_{L^2}\right]. \end{align*} Therefore, using the fact that $G_t$ is a probability density and the estimate \eqref{FirstDerivG} from Lemma \ref{EstimHeatEq}, we obtain: \begin{align*}
\displaystyle \left\Vert {\mathbb E}\left[ F^{N,h}(t,.)\right] - F(t,.) \right\Vert_{L^1} \le \left\Vert {\mathbb E}\left[ F^{N}_0 \right] - F_0 \right\Vert_{L^1} + &\sqrt{\frac{2}{\pi \sigma^2}}\int_0^t \frac{1}{\sqrt{t-s}} \Bigg\{ L_{\Lambda} \left\Vert {\mathbb E}\left[ F^{N,h}(s,.)\right] - F(s,.) \right\Vert_{L^1} \\
&+ \frac{L_{\lambda}}{2}{\mathbb E}\left[\left\Vert F^{N,h}(s,.) - F(s,.) \right\Vert^{2}_{L^2}\right] \Bigg\}\,ds + \left\Vert{\mathbb E}\left[E^{N,h}(t,x) \right]\right\Vert_{L^1} . \end{align*}
Using Lemma \ref{normL1ofE} and Proposition \ref{normL2} then Young's inequality, we deduce that: \begin{align*}
\displaystyle &\left\Vert {\mathbb E}\left[ F^{N,h}(t,.)\right] - F(t,.) \right\Vert_{L^1} \\
&\le \left\Vert {\mathbb E}\left[ F^{N}_0 \right] - F_0 \right\Vert_{L^1} + \sqrt{\frac{2}{\pi \sigma^2}}\int_0^t \frac{1}{\sqrt{t-s}} \Bigg\{L_{\Lambda} \left\Vert {\mathbb E}\left[ F^{N,h}(s,.)\right] - F(s,.) \right\Vert_{L^1} +\frac{L_{\lambda}}{2} M_b \left(\frac 1 N + h \right)\Bigg\}\,ds + Z_b \left(\sqrt{\frac h N} + h\right) \\
&\le \left\Vert {\mathbb E}\left[ F^{N}_0 \right] - F_0 \right\Vert_{L^1} + \left( \frac{1+\sqrt 2}{2}Z_b + \frac{L_{\lambda}M_b}{\sigma} \sqrt{\frac{2t}{\pi}} \right) \left(\frac{1}{N} + h \right) + \frac{L_{\Lambda}}{\sigma} \sqrt{\frac{2}{\pi}}\int_0^t \frac{1}{\sqrt{t-s}} \ \left\Vert {\mathbb E}\left[ F^{N,h}(s,.) \right] - F(s,.) \right\Vert_{L^1}\,ds. \end{align*} We iterate this inequality to obtain:
\begin{align*} \displaystyle \left\Vert {\mathbb E}\left[ F^{N,h}(t,.) - F(t,.) \right]\right\Vert_{L^1} &\le \left( \frac{1+\sqrt2}{2} Z_b + \frac{L_{\lambda}M_b + (1 + \sqrt 2)L_{\Lambda}Z_b}{\sigma}\sqrt{\frac{2t}{\pi}} + \frac{4 L_{\lambda}L_{\Lambda}M_bt}{\sigma^2} \right)\left(\frac{1}{N} +h \right) \\ &+ \left(1 + \frac{L_{\Lambda}}{\sigma}\sqrt{\frac{8t}{\pi}} \right) \left\Vert {\mathbb E}\left[ F^{N}_0 \right] - F_0 \right\Vert_{L^1} + \frac{2L_{\Lambda}^2}{\sigma^2} \int_0^t \left\Vert {\mathbb E}\left[ F^{N,h}(r,.) - F(r,.) \right]\right\Vert_{L^1}\,dr. \end{align*} By Lemma \ref{integrF}, the application $t \mapsto \left\Vert {\mathbb E}\left[ F^{N,h}(t,.)\right] - F(t,.) \right\Vert_{L^1}$ is locally integrable $\forall h \in [0,T], N \in {\mathbb N}^{*}$. Therefore, we can apply Gr\"onwall's lemma and choosing $$ C_b = \max\left(1 + \frac{L_{\Lambda}}{\sigma}\sqrt{\frac{8T}{\pi}}, \frac{1+\sqrt2}{2} Z_b + \frac{L_{\lambda}M_b + (1 + \sqrt 2)L_{\Lambda}Z_b}{\sigma}\sqrt{\frac{2T}{\pi}} + \frac{4L_{\lambda}L_{\Lambda}M_bT}{\sigma^2} \right) \exp\left(\frac{2 L_{\Lambda}^2}{\sigma^2}T \right)$$ concludes the proof of the theorem.
\subsubsection{Proof of Proposition \ref{normL2}}\label{SectBiais} For all $t,h \in [0,T], N \in {\mathbb N}^{*}$, we use Jensen's inequality upon Equation \eqref{eqFh} and obtain: \begin{align}\label{eqFhSqu}
\displaystyle {\mathbb E}\left[\left\Vert F^{N,h}(t,.) - F(t,.) \right\Vert^{2}_{L^2}\right] \le 4 &\int_{{\mathbb R}} {\mathbb E}\left[G_t*\left(F^{N,h}_0 - F_0\right)^2(x)\right]\,dx + 4 \int_{{\mathbb R}}{\mathbb E}\left[R^{N,h}(t,x)^2\right]\,dx + 4 \int_{{\mathbb R}}{\mathbb E}\left[E^{N,h}(t,x)^2\right]\,dx \\
&+ 4\int_{{\mathbb R}}{\mathbb E}\left[\left( \int_0^t \partial_xG_{t-s} * \left(\Lambda(F^{N,h}(s,.)) - \Lambda(F(s,.))\right)(x)\,ds \right)^2 \right]\,dx. \notag \end{align}
On the one hand, we have using the definition \eqref{defrnh} of $R^{N,h}(t,x)$, It\^o's isometry and the estimate \eqref{GSquareEsp} from Lemma \ref{EstimHeatEq} that: $$\displaystyle \int_{{\mathbb R}}{\mathbb E}\left[R^{N,h}(t,x)^2\right]\,dx = \frac{\sigma^2}{N^2} \sum \limits_{i=1}^N \int_0^t \mathbb{E} \left[ \int_{\mathbb{R}} G^2_{t-s}\left(X^{i,N,h}_s -x\right)dx \right] ds = \frac{\sigma}{N}\sqrt{\frac{t}{\pi}}.$$ On the other hand, using Minkowski's, Young's and Cauchy-Schwarz's inequalities in addition to the estimate \eqref{FirstDerivG} from Lemma \ref{EstimHeatEq}, we get: \begin{align*} \displaystyle &\int_{{\mathbb R}}{\mathbb E}\left[\left(\int_0^t \partial_xG_{t-s}*\left(\Lambda\left(F^{N,h}(s,.)\right) - \Lambda(F(s,.))\right)(x)\,ds \right)^2 \right]\,dx ={\mathbb E}\left[\left\Vert \int_0^t \partial_xG_{t-s}*\left(\Lambda(F^{N,h}(s,.)) - \Lambda(F(s,.))\right)\,ds \right\Vert_{L^2}^2 \right] \\ &\le {\mathbb E}\left[\left( \int_0^t \left\Vert \partial_xG_{t-s} \right\Vert_{L^1}\left\Vert \Lambda(F^{N,h}(s,.)) - \Lambda(F(s,.)) \right\Vert_{L^2}\,ds\right)^2\right] \\ &\le {\mathbb E}\left[ \left(\int_0^t \sqrt{\frac{2 L_{\Lambda}^2}{\pi \sigma^2(t-s)}}\left\Vert F^{N,h}(s,.) - F(s,.) \right\Vert_{L^2}\,ds \right)^2 \right]\\
&\le \frac{2L_{\Lambda}^2}{\pi \sigma^2}\int_0^t \frac{du}{\sqrt{t-u}} \int_0^t \frac{1}{\sqrt{t-s}} {\mathbb E}\left[\left\Vert F^{N,h}(s,.) - F(s,.) \right\Vert^2_{L^2} \right]\,ds \\
&= \frac{4 L_{\Lambda}^2 \sqrt{t}}{\pi \sigma^2}\int_0^t \frac{1}{\sqrt{t-s}} {\mathbb E}\left[\left\Vert F^{N,h}(s,.) - F(s,.) \right\Vert^2_{L^2} \right]\,ds. \end{align*}
Therefore, Inequality \eqref{eqFhSqu} becomes: \begin{align*}
\displaystyle {\mathbb E}\left[\left\Vert F^{N,h}(t,.) - F(t,.) \right\Vert^{2}_{L^2}\right] \le &4\int_{{\mathbb R}}{\mathbb E}\left[\left(F^{N,h}_0 - F_0\right)^2(x)\right]\,dx + \frac{4 \sigma}{N} \sqrt{\frac t \pi} + 4 \int_{{\mathbb R}}{\mathbb E}\left[E^{N,h}(t,x)^2\right]\,dx \\
&+ \frac{16L_{\Lambda}^2 \sqrt{t}}{\pi \sigma^2}\int_0^t \frac{1}{\sqrt{t-s}} {\mathbb E}\left[\left\Vert F^{N,h}(s,.) - F(s,.) \right\Vert^2_{L^2} \right]\,ds . \notag \end{align*}
As for the intialization term, since $|\tilde F_0^N-F_0|$ is not greater than $1/(2N)$, using \eqref{majow1det} for the second inequality, we obtain that
$$ \left\|\tilde F^N_0-F_0 \right\|_{L^2}^2\le \frac{\left\|\tilde F^N_0-F_0 \right\|_{L^1}}{2N}=\frac{ \mathcal{W}_1\left(\tilde \mu_0^N,m\right)}{2N}\le\frac{1}{2N}\displaystyle \int_{{\mathbb R}}|x|m(dx).$$ On the other hand, since $N\hat F^N_0(x)$ is a binomial random variable with parameter $(N,F_0(x))$, \begin{align*}
N\int_{{\mathbb R}}{\mathbb E}\left[\left(\hat F^{N}_0 - F_0\right)^2(x)\right]\,dx=\int_{{\mathbb R}}F_0(x)(1-F_0(x))dx\le \int_{-\infty}^0F_0(x)dx+\int_0^{+\infty}(1-F_0(x))dx=\int_{\mathbb R}|x|m(dx). \end{align*}
Therefore we have $\displaystyle \int_{{\mathbb R}}{\mathbb E}\left[\left(F^{N,h}_0 - F_0\right)^2(x)\right]\,dx\le\frac{1}{N}\int_{\mathbb R}|x|m(dx)$ for both the random and the deterministic optimal initializations. With Lemma \ref{estimQuad} below which provides an estimation of the term $\displaystyle \int_{{\mathbb R}} {\mathbb E}\left[E^{N,h}(t,x)^2 \right]\,dx$, we deduce that: \begin{align*}
\displaystyle {\mathbb E}\left[\left\Vert F^{N,h}(t,.) - F(t,.) \right\Vert^{2}_{L^2}\right] &\le \frac{4}{N}\int_{\mathbb R}|x|m(dx) + \frac{4\sigma}{N}\sqrt{\frac t \pi} + 4Q_bh + \frac{16 L_{\Lambda}^2 \sqrt{t}}{\pi \sigma^2}\int_0^t \frac{1}{\sqrt{t-s}} {\mathbb E}\left[\left\Vert F^{N,h}(s,.) - F(s,.) \right\Vert^2_{L^2} \right]\,ds. \end{align*} Iterating the previous inequality, we obtain: \begin{align*} \displaystyle &{\mathbb E}\left[\left\Vert F^{N,h}(t,.) - F(t,.) \right\Vert^{2}_{L^2}\right] \\
&\le 4\left(1 + \frac{32L_{\Lambda}^2t}{\pi \sigma^2} \right) \left(\frac{1}{N}\int_{\mathbb R}|x|m(dx) +\frac{\sigma}{N}\sqrt{\frac{t}{\pi}}+ Q_bh \right) + \frac{256 L_{\Lambda}^4 t}{\pi \sigma^4}\int_0^t {\mathbb E}\left[\left\Vert F^{N,h}(r,.) - F(r,.) \right\Vert^2_{L^2} \right]\,dr. \end{align*}
By Lemma \ref{integrF} and since $\left|F^{N,h}(t,.) - F(t,.)\right| \le 1$, the function $t \mapsto {\mathbb E}\left[\left\Vert F^{N,h}(t,.) - F(t,.) \right\Vert^{2}_{L^2} \right]$ is locally integrable for all $h \in [0,T],\;N \in {\mathbb N}^{*}$. We use Gr\"onwall's lemma once again and conclude for the choice $$ M_b = 4 \max\left\{Q_b\left(1 + \frac{32L_{\Lambda}^2T}{\pi \sigma^2} \right),\left(1 + \frac{32L_{\Lambda}^2t}{\pi \sigma^2} \right)\left(\int_{{\mathbb R}}|x|m(dx) + \sigma\sqrt{\frac{T}{\pi}} \right) \right\}\exp\left(\frac{256 L_{\Lambda}^4T^2}{\pi \sigma^4} \right).$$ \begin{lem}\label{estimQuad} Assume that $\lambda$ is Lipschitz continuous. Then $$\exists Q_b, \forall h \in (0,T], \forall t \in [0,T], \quad \displaystyle \int_{{\mathbb R}} {\mathbb E}\left[E^{N,h}(t,x)^2 \right]\,dx \le Q_b \; h .$$ \end{lem} \begin{proof} We have that $\displaystyle \int_{{\mathbb R}} {\mathbb E}\left[E^{N,h}(t,x)^2 \right]\,dx \leq 6 \sum \limits_{p=0}^5 \int_{{\mathbb R}} {\mathbb E}\left[e^{N,h}_p(t,x)^2 \right]\,dx $. For this reason, we shall estimate, in what follows, each $\displaystyle \int_{{\mathbb R}} {\mathbb E}\left[e^{N,h}_p(t,x)^2 \right]\,dx, p \in \llbracket0,5\rrbracket$. \\ On the one hand, for $r \in \left\{ 2,3\right\}$, we have using It\^o's isometry and the estimate \eqref{GPrimSqu} from Lemma \ref{EstimHeatEq} that $\forall h \in (0,T]$: \begin{align*}
\displaystyle \int_{{\mathbb R}} {\mathbb E}\left[e^{N,h}_r(t,x)^2 \right]\,dx &= \frac{\sigma^2}{N^2} \sum \limits_{i=1}^N \int_0^t (t \wedge \overline{\tau}^h_s -s)^2 \left(\lambda^N(i)\right)^2 {\mathbb E} \left[\int_{{\mathbb R}} \left(\partial_x G_{t-s}\left( Y^{i,N,h}_s -x\right)\right)^2\,dx\right]\,ds \\
&\le \frac{L_{\Lambda}^2 }{4 \sigma \sqrt{\pi} N} \int_0^t \frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{3/2}} \,ds \le \frac{2L_{\Lambda}^2 }{3 \sigma \sqrt{\pi}} \frac{h^{3/2}}{N}, \end{align*} where the last inequality has already been derived at the end of the proof of Lemma \ref{ControlStochE}. The Cauchy-Schwarz inequality then a similar reasoning implies that, for $r \in \{4,5\}$, $\displaystyle \int_{{\mathbb R}} {\mathbb E}\left[e^{N,h}_r(t,x)^2 \right]\,dx \le \frac{2L_{\Lambda}^4T }{3 \sigma^3 \sqrt{\pi}}h^{3/2}$. As for the term $e^{N,h}_0$, we have using the estimate \eqref{GSquare} that $\displaystyle \int_{{\mathbb R}} {\mathbb E}\left[e^{N,h}_0(t,x)^2 \right]\,dx \le \frac{4L_{\Lambda}^2}{\sigma\sqrt{\pi}}h^{3/2}$.\\
On the other hand, using Cauchy-Schwarz's inequality twice, \eqref{GPrimSqu} and the estimation \eqref{estimGam1}, we obtain:
\begin{align*}
\displaystyle &\int_{{\mathbb R}} {\mathbb E}\left[e^{N,h}_1(t,x)^2 \right]\,dx = {\mathbb E} \left[\int_{{\mathbb R}} \left(\frac{1}{N}\sum \limits_{i=2}^N \int_{0}^{t}(t \wedge \overline{\tau}^h_s -s)\left( \lambda^N(i) - \lambda^N(i-1) \right)\partial_x G_{t-s}\left(Y^{i,N,h}_s -x \right)\gamma^{i}_s\,d|K|_s \right)^2\,dx \right] \\
&\leq {\mathbb E} \left[ \int_{{\mathbb R}}\frac{1}{N}\sum \limits_{i=2}^N\left(\int_{0}^{t}(t \wedge \overline{\tau}^h_s -s)^2\left(\lambda^N(i) - \lambda^N(i-1)\right)^2\left(\partial_x G_{t-s}\right)^2\left(Y^{i,N,h}_s -x \right)\gamma^{i}_s\,d|K|_s\right)\left(\int_0^t \gamma^{i}_r\,d|K|_r\right)\,dx \right] \\
&\leq \frac{L_{\lambda}^2}{4 \sigma^3 \sqrt\pi }{\mathbb E} \left[\frac{1}{N^3} \sum \limits_{i=2}^N \left(\int_0^t \frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{3/2}}\gamma^{i}_s\,d|K|_s\right)\left(\int_0^t \gamma^{i}_r\,d|K|_r\right) \right] \\
&\leq \frac{L_{\lambda}^2}{4 \sigma^3 \sqrt\pi} \frac{1}{N^3} \sum \limits_{i=2}^N {\mathbb E}^{1/2}\left[\left(\int_0^t \frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{3/2}}\gamma^{i}_s\,d|K|_s\right)^2 \right]{\mathbb E}^{1/2}\left[\left(\int_0^t \gamma^{i}_r\,d|K|_r\right)^2 \right]\\
&\le \frac{3L_{\lambda}^2}{4\sigma^3} \sqrt{\frac{t\left(\sigma^2 + L_{\Lambda}^2T\right)}{\pi}} \frac{1}{N^2} \sum \limits_{i=2}^N {\mathbb E}^{1/2}\left[\left(\int_0^t \frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{3/2}}\gamma^{i}_s\,d|K|_s\right)^2 \right]. \end{align*}
We denote by $m=\lceil\log_2\left(t/h \right) \rceil$ and rewrite the integral the following way:
$$\displaystyle \int_{0}^t\frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{3/2}}\gamma^{i}_s\,d|K|_s = \sum \limits_{k=0}^{m -1} \int_{t-t/2^{k}}^{t-t/2^{k+1}}\frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{3/2}}\gamma^{i}_s\,d|K|_s + \int_{t-t/2^{m}}^{t}\frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{3/2}}\gamma^{i}_s\,d|K|_s.$$ Therefore,
\begin{align*}
\displaystyle {\mathbb E}^{1/2}\left[\left(\int_{0}^t \frac{(t \wedge \overline{\tau}^h_s -s)^2}{(t-s)^{3/2}}\gamma^{i}_s\,d|K|_s\right)^2 \right] &\le \sum \limits_{k=0}^{m -1} \frac{h^2}{\left(\frac{t}{2^{k+1}}\right)^{3/2}} {\mathbb E}^{1/2}\left[\left(\int_{t-t/2^{k}}^{t-t/2^{k+1}} \gamma^{i}_s\,d|K|_s\right)^2 \right] + \sqrt{h} {\mathbb E}^{1/2}\left[\left(\int_{t-t/2^{m}}^{t} \gamma^{i}_s\,d|K|_s\right)^2 \right] \\
&\le 3N \sqrt{\left(\sigma^2 + L_{\Lambda}^2 T\right)} \left(h^2 \sum \limits_{k=0}^{m -1}\left(\frac{t}{2^{k+1}} \right)^{-1} + h \right) \\
&= 3N \sqrt{\left(\sigma^2 + L_{\Lambda}^2 T\right)} \left(\frac{2h^2}{t}(2^m -1) + h\right) \\
&\le 15Nh \sqrt{\left(\sigma^2 + L_{\Lambda}^2 T\right)}.
\end{align*}
We then have $\displaystyle \int_{{\mathbb R}} {\mathbb E}\left[e^{N,h}_1(t,x)^2 \right]\,dx \le \frac{45 L_{\lambda}^2\left(\sigma^2 + L_{\Lambda}^2 T\right)}{4 \sigma^3} \sqrt{\frac T \pi} \; h$. \\
The conclusion holds for the choice $\displaystyle Q_b =\frac{1}{\sigma }\sqrt{\frac T \pi} \left\{\frac{16L_{\Lambda}^2}{3}\left(1 + \frac{L_{\Lambda}^2 T}{4 \sigma^2} \right) + \frac{45 L_{\lambda}^2}{4}\left(1 + \frac{L_{\Lambda}^2 T}{\sigma^2}\right) \right\}$. \end{proof}
\subsection{Particle initialization : proof of \eqref{stindet}}\label{proofinitdet}
Since $\displaystyle \int_{{\mathbb R}}y^2m(dy)=2\left(-\int_{-\infty}^0 yF_0(y)\,dy+\int_0^{+\infty}y(1-F_0(y))\,dy\right)$, one has
\begin{align*}
\displaystyle \forall x>0,\quad \int_{-\infty}^{-x}F_0(y)\,dy+\int_x^{+\infty}(1-F_0(y))\,dy \le \frac{1}{2x}\int_{{\mathbb R}}y^2m(dy).
\end{align*}
Using the monotonicity of $F_0$ for the first inequality then the inequality $F_0(y)+(1-F_0(-y))=m((-\infty,-y]\cup (y,+\infty))\le 1$ valid for $y\ge 0$, we deduce that for all $x>0$,
\begin{align}
F_0(-x)+(1-F_0(x))\le \frac{2}{x}\int_{x/2}^x (F_0(-y)+(1-F_0(y))dy&\le \frac{2}{x}\int_{x/2}^{+\infty} (F_0(-y)+(1-F_0(y))dy\label{minointqueue}\\&\le\frac{2}{x^2}\int_{{\mathbb R}}y^2m(dy).\notag
\end{align}
Hence $\int_{{\mathbb R}}y^2m(dy)<\infty\Rightarrow \sup_{x\ge 0}x^2(F_0(-x)+(1-F_0(x)))<\infty$.
Let us now suppose that $\sup_{x\ge 0}x^2(F_0(-x)+(1-F_0(x)))=C<\infty$ and deduce that $\displaystyle \int_{{\mathbb R}}|x|^{2-\varepsilon}m(dx)<\infty$ for all $\varepsilon\in (0,1]$ (and therefore all $\varepsilon\in (0,2]$) and that $\sup_{N\ge 1}\sqrt{N}{\cal W}_1(\tilde\mu^N_0,m)<\infty$. We have \begin{equation}
\forall x>0,\;\int_x^{+\infty}(F_0(-y)+(1-F_0(y)))dy\le \int_x^{+\infty}\frac{C}{y^2}dy=\frac{C}{x}.\label{majointintsurv} \end{equation}
For $\varepsilon=1$, we have $$ \displaystyle \int_{{\mathbb R}}|x|m(dx)=\int_0^{+\infty}(F_0(-y)+(1-F_0(y)))\,dy \le 1+\int_1^{+\infty}(F_0(-y)+(1-F_0(y)))\,dy \le 1+C.$$
Let now $\varepsilon\in (0,1)$. Writing $\frac{x^{2-\varepsilon}}{(2-\varepsilon)(1-\varepsilon)}=\frac{1}{1-\varepsilon}\int_0^x y^{1-\varepsilon}dy=\int_0^x\int_0^yz^{-\varepsilon}dzdy$ and using Fubini's theorem, we obtain that
\begin{align*}
\displaystyle \frac{1}{(2-\varepsilon)(1-\varepsilon)}&\int_0^{+\infty} x^{2-\varepsilon}m(dx)= \int_0^{+\infty}z^{-\varepsilon} \int_z^{+\infty}(1-F_0(y))\,dy\,dz\\
&\le \int_0^{1}z^{-\varepsilon} dz\int_0^{+\infty}(1-F_0(y))dy+\int_1^{+\infty}z^{-\varepsilon}\int_z^{+\infty}(1-F_0(y))\,dy\,dz.
\end{align*}
Combining this inequality with the symmetric one then using the above estimation of $\displaystyle \int_0^{+\infty}(F_0(-y)+(1-F_0(y)))\,dy$ and \eqref{majointintsurv}, we conclude that
\begin{align*}
\displaystyle \frac{1}{2-\varepsilon}\int_{\mathbb R} |x|^{2-\varepsilon}m(dx) &\le \int_0^{+\infty}\left(F_0(-y)+(1-F_0(y))\right)\,dy+(1-\varepsilon)C\int_1^{+\infty}z^{-1-\varepsilon}\,dz \le 1+\frac{C}{\varepsilon}.
\end{align*} Let $N\ge 1$. Since $1-F_0(x) \le 1/(2N)$ for $x\ge F_0^{-1}(1-1/(2N))$ and $F_0^{-1}(1-1/(2N))\ge F_0^{-1}(1/2)$, we have \begin{align*}
\int_{F_0^{-1}\left(\frac{2N-1}{2N}\right)}^{+\infty}(1-F_0(x))\,dx &\le \int_{F_0^{-1}\left(\frac{2N-1}{2N}\right) \wedge \sqrt{N}}^{\sqrt{N}}(1-F_0(x))\,dx + \int_{\sqrt{N}}^{+\infty}(1-F_0(x))\,dx\\&\le\frac{(\sqrt{N}-F_0^{-1}(1/2))^+}{2N}+\int_{\sqrt{N}}^{+\infty}(1-F_0(x))\,dx.\end{align*} Dealing with $\int^{F_0^{-1}\left(\frac{1}{2N}\right)}_{-\infty}F_0(x)\,dx$ in a symmetric way and using \eqref{majointintsurv}, we deduce that \begin{align}
\int^{F_0^{-1}\left(\frac{1}{2N}\right)}_{-\infty}F_0(x)\,dx+\int_{F_0^{-1}\left(\frac{2N-1}{2N}\right)}^{+\infty}(1-F_0(x))\,dx\le \frac{\sqrt{N}-F_0^{-1}(1/2)\vee\left(-\sqrt{N}\right)}{2N}+\frac{C}{\sqrt{N}}.\label{majointqueuesbis} \end{align}
Either $F_0^{-1}(1-1/(2N))\le 0$ or $\frac{1}{2N}\le 1-F_0(F_0^{-1}(1-1/(2N))-)\le \frac{C}{(F_0^{-1}(1-1/(2N)))^2}$ so that $F_0^{-1}(1-1/(2N))\le \sqrt{2CN}$. By a symmetric reasoning, $F_0^{-1}(1/(2N))\ge -\sqrt{2CN}$.
Since $|\tilde F_0^N-F_0|$ is not greater than $1/(2N)$, we deduce that $\int_{F_0^{-1}\left(\frac{1}{2N}\right)}^{F_0^{-1}\left(\frac{2N-1}{2N}\right)} \left|\tilde F_0^N(x)-F_0(x)\right|\,dx\le \sqrt{\frac{2C}{N}}$. With \eqref{w1fdr} and \eqref{majointqueuesbis}, we conclude that \begin{align}
\displaystyle {\cal W}_1(\tilde\mu^N_0,m) \le \sqrt{\frac{2C}{N}}+\frac{\sqrt{N}+|F_0^{-1}(1/2)|\vee \sqrt{N}}{2N}+\frac{C}{\sqrt{N}}. \end{align} Let us now suppose that $\sup_{N\ge 1}{\cal W}_1(\tilde\mu^N_0,m)<\infty$. Then $\tilde C:=\sup_{N\ge 1}\sqrt{N}\int^{F_0^{-1}\left(\frac{1}{2N}\right)}_{-\infty}F_0(x)\,dx<\infty$ and for $N\ge 1$, since $F_0(x)\ge\frac{1}{4N}$ when $x\ge F_0^{-1}\left(\frac{1}{4N}\right)$, \begin{align*}
\frac{F_0^{-1}\left(\frac{1}{2N}\right)-F_0^{-1}\left(\frac{1}{4N}\right)}{4N}\le \int^{F_0^{-1}\left(\frac{1}{2N}\right)}_{F_0^{-1}\left(\frac{1}{4N}\right)}F_0(x)\,dx\le \frac{\tilde C}{\sqrt{N}}. \end{align*} For $k\in{\mathbb N}^*$, we deduce that $F_0^{-1}\left(2^{-(k+1)}\right)-F_0^{-1}\left(2^{-k}\right)\ge -\tilde C 2^{\frac{k+3}{2}}$, and after summation that $$F_0^{-1}\left(2^{-k}\right)\ge F_0^{-1}\left(1/2\right)-\frac{4\tilde C}{\sqrt{2}-1}(2^{\frac{k-1}{2}}-1).$$ With the monotonicity of $F_0^{-1}$, we deduce that $$\forall u\in(0,1/2],\;F_0^{-1}(u)\ge F_0^{-1}(1/2)-\frac{4\tilde C}{\sqrt{2}-1}(u^{-1/2}-1),$$ an therefore that $\inf_{u\in(0,1/2]}(\sqrt{u}F_0^{-1}(u))>-\infty$. With the inequality $F_0^{-1}(F_0(x))\le x$ valid for $x\in{\mathbb R}$, this implies that $\inf_{x\in{\mathbb R}:F_0(x)\le 1/2}(x\sqrt{F_0(x)})>-\infty$ and therefore that $\sup_{x\ge 0}x^2F_0(-x)<\infty$. With a symmetric reasoning, we conclude that $\sup_{x\ge 0}x^2(1-F_0(x))<\infty$.
\begin{remark}
In fact, we have proved that
\begin{align*}
\sup_{x\ge 0}x^2\left(F_0(-x)+1-F_0(x)\right)<\infty&\Leftrightarrow\sup_{x\ge 0}x\int_x^{+\infty}\left(F_0(-y)+1-F_0(y)\right)dy<\infty\\&\Leftrightarrow\sup_{N\ge 1}\sqrt{N}{\cal W}_1(\tilde\mu^N_0,m)<\infty\Leftrightarrow\sup_{u\in(0,1/2]}\sqrt{u}\left(F_0^{-1}(1-u)-F_0^{-1}(u)\right)<\infty.
\end{align*}
Let us assume that if $\displaystyle \limsup_{x\to\infty}x\int_x^{+\infty}(F_0(-y)+1-F_0(y))dy>0$, which by \eqref{minointqueue} is implied by $\displaystyle \limsup_{x\to\infty}x^2(F_0(-x)+1-F_0(x))>0$ and, by monotonicity of the integral implies that $\limsup_{N\to\infty}x_N\int_{x_N}^{+\infty}(F_0(-y)+1-F_0(y))dy>0$ along any sequence $(x_N)_{N\in{\mathbb N}}$ of positive numbers increasing to $+\infty$ and such that $\limsup_{N\to\infty}\frac{x_{N+1}}{x_N}<\infty$. Then, under the equivalent conditions,
$$\hat{C}:=\sup_{N\ge 1}\frac{F_0^{-1}(1-\frac{1}{2N})}{\sqrt{N}}\vee\sup_{N\ge 1}\frac{-F_0^{-1}(\frac{1}{2N})}{\sqrt{N}}\in(0,+\infty)$$ and
$$\sqrt{N}{\cal W}_1(\tilde\mu^N_0,m)\ge \sqrt{N}\left(\int_{-\infty}^{F_0^{-1}(\frac{1}{2N})}F_0(y)dy+\int^{+\infty}_{F_0^{-1}(1-\frac{1}{2N})}(1-F_0(y))dy\right)\ge \sqrt{N}\int_{\hat{C}\sqrt{N}}^{+\infty}(F_0(-y)+1-F_0(y))dy$$
so that $\limsup_{N\to\infty}\sqrt{N}{\cal W}_1(\tilde\mu^N_0,m)>0$.
\end{remark}
\section{Numerical experiments for the Burgers equation}
In order to confirm our theoretical estimates for the strong and the weak $L^1$-error between $F^{N,h}$ and its limit $F$, we consider, for the choice $\Lambda(u) = -(1-u)^2/2$ and the initial condition $F(0,x) = \mathbf{1}_{\{x \ge 0 \}}$, the following equation: $$\left\{
\begin{aligned}
&\partial_tF(t,x) + \partial_x F(t,x) \Big(1 - F(t,x)\Big) = \frac{\sigma^2}{2}\partial_{xx}F(t,x),\\
&F(0,x) = \mathbf{1}_{\{x \ge 0 \}}.
\end{aligned} \right.$$ We can notice that the function $\left(1-F(t,.)\right)$ is solution of the Burgers equation that was also used in \cite{BossTal}. The Cole-Hopf transformation yields the following closed-form expression of $F$: \begin{align*} \displaystyle F(t,x) = 1 - \frac{\mathcal{N}\left(\frac{t-x}{\sigma \sqrt t}\right)}{\mathcal{N}\left(\frac{t-x}{\sigma \sqrt t}\right) + \exp\left(\frac{2x-t}{2\sigma^2}\right)\mathcal{N}\left(\frac{x}{\sigma \sqrt t}\right)}, \end{align*} where $\displaystyle \mathcal{N}\left(x\right) = \int_{-\infty}^x \frac{\exp(y^2/2)}{\sqrt{2\pi}}\,dy$.
The drift coefficient of the $i^{th}$ particle in the increasing order is then equal to $\lambda^{N}(i) = 1 -\frac{2i -1}{2N} $ and the Euler discretization with step $h \in (0,T]$ of the particle system is: \begin{align*}
\displaystyle dX^{i,N,h}_t &= \sigma dW^{i}_t + \left(1 + \frac{1}{2N} - \frac{1}{N} \sum \limits_{j=1}^{N} \mathbf{1}_{ \left\{ X^{j,N,h}_{\tau^{h}_{t}} \leq X^{i,N,h}_{\tau^{h}_{t}} \right\}} \right)\,dt,\;1\le i\le N,\;t\in[0,T]. \end{align*} As $F_0$ is the cumulative function of the Dirac mass centered at zero, we place the $N$ particles at zero for their initialisation.\\
We seek to observe the dependence of the strong $L^1$-error ${\mathbb E}\left[ \mathcal{W}_1\left(\mu^{N,0}_T,\mu_T \right)\right]$ and the weak $L^1$-error $\mathcal{W}_1 \left({\mathbb E}\left[\mu^{N,h}_T\right],\mu_T\right)$ at time $T$ on the number $N$ of particles and on the time step $h$. We recall \eqref{Wasserstein} and \eqref{w1fdr} where the Wasserstein distance between a probability measure $\nu$ and $\mu_T$ can be expressed either using the quantile functions or the cumulative distribution functions: \begin{align*}
\mathcal{W}_1\left(\nu,\mu_T \right)&= \int_0^1 \left|F^{-1}_{\nu}(u) - F^{-1}_T(u) \right|\,du \\
&= \int_{{\mathbb R}}\left|F_{\nu}(x) - F(T,x)\right|\,dx. \end{align*} We choose to use the second expression because we have an explicit formula for $F(T,.)$ unlike the inverse $F^{-1}_T(.)$ (which can still be numerically estimated but this is costly and induces additional numerical error). \\
When $\nu$ is an empirical measure of the form $\frac{1}{N}\sum\limits_{i=1}^N \delta_{x^{i}}$, we choose to approximate the $\mathcal{W}_1$ distance not using a grid but in the following way. For $(y^{i})_{1 \le i \le N}$ denoting the increasing reordering of $(x^{i})_{1 \le i \le N}$, we have: $\mathcal{W}_1\left( \frac{1}{N}\sum\limits_{i=1}^N \delta_{x^{i}}, \mu_T \right)\simeq\Psi \left( y^1,y^2,\dots, y^N \right) $ where $$\displaystyle \Psi \left( y^1,y^2,\dots, y^N \right) = \sum \limits_{i=1}^{N-1}\frac{1}{2}\left( y^{i+1}-y^{i}\right)\left(\left|F(T,y^{i+1}) - \frac{i}{N}\right| + \left|F(T,y^{i}) - \frac{i}{N} \right|\right).$$
Therefore, for the strong $L^1$-error, $\left(Y^{i,N,h}_{t,r}\right)_{i \in \llbracket1,N\rrbracket}$ being the increasing reordering of the particles positions $\left(X^{i,N,h}_{t,r}\right)_{i\in \llbracket1,N\rrbracket}$, $t>0$ in the $r^{th}$ out of $R$ independent Monte-Carlo runs, we obtain the following approximation: \begin{align*} {\mathbb E}\left[ \mathcal{W}_1\left(\mu^{N,h}_T,\mu_T\right)\right] &\simeq \frac{1}{R} \sum \limits_{r =1}^R \Psi\left(Y^{1,N,h}_{T,r}, \dots, Y^{N,N,h}_{T,r}\right). \end{align*} We also define the precision of this estimation as half the width of the $95 \%$ confidence interval of the empirical error i.e. $\text{Precision} = 1.96 \times \sqrt{\text{Variance}/R}$ where Variance denotes the empirical variance over the runs of the empirical error over the particles. \\
Concerning the weak $L^1$-error, we approximate ${\mathbb E}\left[ \mu^{N,h}_T\right]$ by $\frac{1}{R \times N} \sum\limits_{r=1}^{R} \sum\limits_{i=1}^{N} \delta_{X^{i,N,h}_{T,r}}$. But as $R \times N$ will be as big as $10^8$ in our simulations, rather than using the previous grid free approximation, we use the grid $\left(F^{-1}_T\left(\frac{k}{K} \right)\right)_{1 \le k \le K-1}$ ($K$ will be chosen equal to $5000$) to compute the $\mathcal{W}_1$ distance. For $k \in \llbracket 0,K-1\rrbracket$ and $x \in \left[F^{-1}_T\left(\frac{k}{K} \right), F^{-1}_T\left(\frac{k+1}{K} \right) \right]$, we make the following approximation $F(T,x) \simeq \frac{2k+1}{2K}$. We also define the function $\varphi$ as: \begin{align*}
\varphi \left(u_0, u_1, \dots, u_{K-1} \right) = &\sum \limits_{k=1}^{K-2} \left|u_k - \frac{2k+1}{2K}\right| \left(F^{-1}_T\left(\frac{k+1}{K} \right)- F^{-1}_T\left(\frac{k}{K} \right) \right) + 2\left|u_0-\frac{1}{2K} \right|\left(F^{-1}_T\left(\frac{1}{K} \right)- F^{-1}_T\left(\frac{1}{2K} \right) \right) \\
&+ 2\left|u_{K-1} -\left(1 - \frac{1}{2K}\right) \right|\left(F^{-1}_T\left(1 -\frac{1}{2K} \right)- F^{-1}_T\left(1 - \frac{1}{K} \right) \right). \end{align*} Therefore, we can approach the weak $L^1$-error by $\mathcal{W}_1 \left({\mathbb E}\left[\mu^{N,h}_T\right],\mu_T\right) \simeq \varphi \left(\left(\frac{1}{R}\sum\limits_{r=1}^R F^{N,h}_r\left(T,F^{-1}_T\left(\frac{2k+1}{2K} \right)\right) \right)_{0 \le k \le K-1}\right)$. We divide the $R$ runs into $B$ batches of $M = R/B$ independent simulations in order to estimate the associated precision. Indeed, we estimate the empirical variance over the batches while estimating the weak error for each independent simulation over the batches. And by the delta method, we may expect, denoting $E={\mathbb E}\left[ \left(F^{N,h}\left(T,F^{-1}_T\left(\frac{2k+1}{2K} \right) \right)\right)_{0 \le k \le K-1}\right]$, that
\begin{align*} &\sqrt{\rho}\left[ \varphi \left(\left(\frac{1}{\rho}\sum\limits_{r=1}^\rho F^{N,h}_r\left(T,F^{-1}_T\left(\frac{2k+1}{2K} \right)\right) \right)_{0 \le k \le K-1}\right)- \varphi\left(E\right) \right] \\
&\phantom{\sqrt{\rho}\left[ \varphi \left(\left(\frac{1}{\rho}\sum\right.\right.\right.}
\overset{\mathcal L}{\longrightarrow} \mathcal{N}\Bigg(0,\nabla\varphi^{T}\left(E\right)\text{Cov}\left[ \left(F^{N,h}\left(T,F^{-1}_T\left(\frac{2k+1}{2K} \right) \right)\right)_{0 \le k \le K-1}\right]\nabla\varphi\left(E\right) \Bigg) \end{align*} when $\rho \to +\infty$. Applying this result with $\rho = M$ and $\rho = R$, one expects that $1/B$ times the empirical variance of $\left(\varphi \left(\left(\frac{1}{M}\sum\limits_{r=(b-1)M+1}^{bM} F^{N,h}_r\left(T,F^{-1}_T\left(\frac{2k+1}{2K} \right)\right) \right)_{0 \le k \le K-1}\right) \right)_{1 \le b \le B}$ over the batches provides an estimator of the variance of $\varphi \left(\left(\frac{1}{R}\sum\limits_{r=1}^R F^{N,h}_r\left(T,F^{-1}_T\left(\frac{2k+1}{2K} \right)\right) \right)_{0 \le k \le K-1}\right)$. So the precision is computed as $1.96$ times the square root of this estimator. \\
For both of the errors, we fix the time horizon $T=1$ and the diffusion coefficient $\sigma^2 = 0.2$.
\subsection{Strong $L^1$-error behaviour} We present numerical estimates of ${\mathbb E}\left[ \mathcal{W}_1\left(\mu^{N,h}_T,\mu_T\right)\right]$, computed as described above. \\
$\blacktriangleright$ \textbf{Dependence on $N$}:\\
We fix the time-step $h=0.002$ small enough in order to observe the effect of the number $N$ of particles on the error. The simulation is done with $R=100$ Monte-Carlo runs. We obtain the following results for the estimation of the error and the associated precision:\\
\begin{center}
\begin{tabular}{ |c|c|c|c| }
\hline
\multicolumn{4}{|c|}{Evolution of the strong $L^1$-error w.r.t. $N$} \\ \hline
Number of particles $N$ & Estimation & Precision & Ratio of decrease \\ \hline
$250$ & $0.03312361$ & $0.00290442$ & $\times$ \\ \hline
$1000$ & $0.01598253$ & $0.00133181$ & $2.07$ \\ \hline
$4000$ & $0.00841976$ & $0.00077491$ & $1.90$ \\ \hline
$16000$ & $0.00358799$ & $0.00028319$ & $2.35$ \\ \hline
$64000$ & $0.00193416$ & $0.00016111$ & $1.86$ \\
\hline
\end{tabular} \end{center}
We observe that the ratio of successive estimations $\frac{\text{Estimation}(N/4)}{\text{Estimation}(N)}$ is around $2$ when we multiply $N$ by $4$, which means that the strong $L^1$-error is roughly proportional to $N^{-1/2}$.
\begin{remark} This strong error was computed for the choice $\sigma^2 = 0.2$. When choosing the larger variance $\sigma^2 = 20$ with the same time-step $h = 0.002$, the error is approximately multiplied by $10$ and the $\mathcal{O}\left(N^{-1}\right)$ behaviour is still observed. For the smaller variance $\sigma^2 = 0.002$, we need to choose the smaller time-step $h = 0.001$ in order to recover the $\mathcal{O}\left(N^{-1}\right)$ behaviour in the number of particles since the estimated error is very small.
\end{remark}
$\blacktriangleright$ \textbf{Dependence on $h$}:\\
We apply the same strategy to study the dependence of the error on $h$ by choosing a large number $N = 500000$ of particles. The following table presents numerical estimates of the $L^1$-norm of the error and its associated precision for $R=100$ runs.\\
\begin{center}
\begin{tabular}{ |c|c|c|c|}
\hline
\multicolumn{4}{|c|}{Evolution of the strong $L^1$-error w.r.t. $h$} \\ \hline
Time-step $h$ & Estimation & Precision & Ratio of decrease \\ \hline
$1/2$ & $0.07963922$ & $5.8279 \times 10^{-5}$ & $\times$ \\ \hline
$1/4$ & $0.03550774$ & $5.9151 \times 10^{-5}$ & $2.24$ \\ \hline
$1/8$ & $0.01682159$ & $5.9598 \times 10^{-5}$ & $2.11$ \\ \hline
$1/16$ & $0.00817936$ & $6.5275 \times 10^{-5}$ & $2.06$ \\ \hline
$1/32$ & $0.00409004$ & $5.6482 \times 10^{-5}$ & $2.00$ \\ \hline
$1/64$ & $0.00206127$ & $5.3539 \times 10^{-5}$ & $1.98$ \\ \hline
$1/128$ & $0.00116977$ & $4.7468 \times 10^{-5}$ & $1.76$ \\ \hline
$1/256$ & $0.00084118$ & $5.9673 \times 10^{-5}$ & $1.39$ \\
\hline
\end{tabular} \end{center}
We observe that when the time step $h$ is divided by $2$, the ratio of decrease $\frac{\text{Estimation}(h)}{\text{Estimation}(h/2)}$ is approximately equal to $2$.
\begin{remark} This strong error was computed for the choice $\sigma^2 = 0.2$. When choosing the larger variance $\sigma^2 = 20$ with the same number of particles $N=500000$, the error is of the same order for the $3$ first time-steps but deteriorates afterwards and becomes larger compared to errors obtained for $\sigma^2 = 0.2$. The ratio also deteriorates quickly and tends to be constant. In order to only observe the effect of the time-step on the weak error, we need to increase the number of particles when $\sigma$ is large. For the smaller variance $\sigma^2 = 0.002$, the error is smaller than the ones obtained for greater $\sigma^2$. The $\mathcal{O}\left(h\right)$-behaviour is recovered when the time-steps are small.
\end{remark}
\subsection{Weak $L^1$-error behaviour}
We present numerical estimates of $\mathcal{W}_1 \left({\mathbb E}\left[\mu^{N,h}_T\right],\mu_T\right)$, computed as described above.\\
$\blacktriangleright$ \textbf{Dependence on $N$}:\\
We fix the time-step $h=0.002$ small enough once again to observe the effect of the number $N$ of particles on the weak error. The estimation is done with $B=100$ batches of $M = 200$ independent simulations for a total of $R = 20000$ Monte-Carlo runs and $K=5000$. The results are shown in the following table:
\begin{center}
\begin{tabular}{ |c|c|c|c| }
\hline
\multicolumn{4}{|c|}{Evolution of the weak $L^1$-error w.r.t. $N$} \\ \hline
Number of particles $N$ & Estimation & Precision & Ratio of decrease \\ \hline
$100$ & $0.01018160$ & $5.6947 \times 10^{-4}$ & $\times$ \\ \hline
$200$ & $0.00483151$ & $3.8455 \times 10^{-4}$ & $2.11$ \\ \hline
$400$ & $0.00248807$ & $2.0485 \times 10^{-4}$ & $1.94$ \\ \hline
$800$ & $0.00136491$ & $1.4707 \times 10^{-4}$ & $1.82$ \\ \hline
$1600$ & $0.00077723$ & $1.0822 \times 10^{-4}$ & $1.76$ \\ \hline
$3200$ & $0.00038285$ & $4.9747 \times 10^{-5}$ & $2.03$ \\
\hline
\end{tabular} \end{center}
We observe that multiplying the number of particles by $2$ implies a division of the error estimation by approximately $2$ which proves that the weak $L^1$-error is roughly proportional to $N^{-1}$.
\begin{remark} This weak error was computed for the choice $\sigma^2 = 0.2$. When choosing the larger variance $\sigma^2 = 20$ with the time-step $h = 0.002$, the $\mathcal{O}\left(N^{-1}\right)$ behaviour is still observed. When choosing the smaller variance $\sigma^2 = 0.05$ and the smaller time-step $h = 0.001$, the $\mathcal{O}\left(N^{-1}\right)$ behaviour is recovered. The weak error estimates only vary very slightly when varying $\sigma$. However, for smaller values of $\sigma$, the computation of the quantile function becomes difficult.
\end{remark}
$\blacktriangleright$ \textbf{Dependence on $h$}:\\
Once again, we do the same to study the dependence of the weak error on $h$ by choosing a large number $N=100000$ of particles, $B=20$ batchs of $M=50$ independent simulations for a total of $R = 1000$ Monte-Carlo runs and $K=5000$. \\
\begin{center}
\begin{tabular}{ |c|c|c|c|}
\hline
\multicolumn{4}{|c|}{Evolution of the weak $L^1$-error w.r.t. $h$} \\ \hline
Time-step $h$ & Estimation & Precision & Ratio of decrease \\ \hline
$1/2$ & $0.07954397$ & $4.7356 \times 10^{-5}$ & $\times$ \\ \hline
$1/4$ & $0.03546112$ & $4.7932 \times 10^{-5}$ & $2.24$ \\ \hline
$1/8$ & $0.01681185$ & $4.0437 \times 10^{-5}$ & $2.11$ \\ \hline
$1/16$ & $0.00816986$ & $4.1616 \times 10^{-5}$ & $2.06$ \\ \hline
$1/32$ & $0.00407191$ & $3.9306 \times 10^{-6}$ & $2.01$ \\ \hline
$1/64$ & $0.00199744$ & $3.0719 \times 10^{-5}$ & $2.04$ \\ \hline
$1/128$ & $0.00096767$ & $5.6043 \times 10^{-5}$ & $2.06$ \\ \hline
$1/256$ & $0.00048294$ & $3.6172 \times 10^{-5}$ & $2.00$ \\
\hline
\end{tabular} \end{center}
We observe that dividing the time step $h$ by $2$ implies a ratio of decrease $\frac{\text{Estimation}(h)}{\text{Estimation}(h/2)}$ greater or equal to $2$ which shows an $L^1$-weak error roughly proportional to $h$.
\begin{remark} This weak error was computed for the choice $\sigma^2 = 0.2$. When choosing the smaller variance $\sigma^2 = 0.05$ with the same number of particles $N=100000$, the $\mathcal{O}\left(h\right)$ behaviour is recovered. When choosing the larger variance $\sigma^2 = 20$, the $\mathcal{O}\left(h\right)$ behaviour is observed for the first time-steps until $1/32$. However, for smaller time-steps, the error becomes larger, compared to the values obtained for smaller $\sigma$, and the ratio tends to be constant. In order to only observe the effect of the time-step on the weak error, we need to increase the number of particles when $\sigma$ is large.
\end{remark}
\begin{appendix} \section{Appendix} The first lemma gives a condition under which we can interchange a Lebesgue and a stochastic integral. It is called the stochastic Fubini theorem and is a consequence of Theorem 2.2 proved by Veraar in \cite{VERAA}. \begin{lem}\label{StochFub}
Let $V:[0,T] \times {\mathbb R} \times \Omega \to {\mathbb R}$ be a progressively measurable function. If $\displaystyle \int_{{\mathbb R}}\left( \int_0^T |V(t,x)|^2dt \right)^{1/2}dx < \infty$ almost surely then one has:
$$ \forall t \in [0,T], \quad \text{a.s.}, \quad \displaystyle \int_{{\mathbb R}} \left(\int_0^t V(s,x)\,dW_s \right)\,dx = \int_0^t \left(\int_{{\mathbb R}} V(s,x)\,dx \right)\,dW_s. $$ \end{lem}
For $t>0$, let $G_t$ denote the probability density function of the normal law $\mathcal{N}(0,\sigma^2t)$: $$G_t(x) = \exp\left(-\frac{x^2}{2\sigma^2t}\right) \Big/ \sqrt[]{2\pi\sigma^2t} .$$ The following lemma provides a set of estimates that are very useful: \begin{lem}\label{EstimHeatEq} The function $G_t(x)$ solves the heat equation: \begin{align}\label{heateq}
\partial_t G_t(x) - \frac{\sigma^2}{2} \partial_{xx} G_t(x) = 0, \quad (t,x) \in [0,+\infty)\times {\mathbb R}. \end{align} We can express the square of the first order spatial derivative as: \begin{align}\label{firstDerivSquar}
\left(\partial_x G_t\right)^2(x) = \frac{x^2}{2 \sigma^5 t^{5/2} \sqrt{\pi}}G_{t/2}(x), \end{align} and deduce the $L^1$-norm of $(\partial_xG_t)^2$: \begin{align}\label{GPrimSqu}
\displaystyle \|(\partial_xG_t)^2\|_{L^1}=\|\partial_x G_t\|_{L^2}^2 = \frac{1}{4\sigma^3 t^{3/2} \sqrt{\pi}}. \end{align} Moreover, we have estimates of the $L^1$-norm of the spatial derivatives of $G$: \begin{align}\label{FirstDerivG}
\left\Vert \partial_xG_{t} \right\Vert_{L^1} = \sqrt{\frac{2}{\pi\sigma^2 t}}, \end{align} \begin{align}\label{SecondDerivG}
\left\Vert \partial_{xx}G_{t} \right\Vert_{L^1} \leq \frac{2}{\sigma^2 t}. \end{align} We may also compute the $L^1$-norm of $G^2_t$: \begin{align}\label{GSquare}
\displaystyle \|G_t^2\|_{L^1}=\|G_t\|_{L^2}^2 = \frac{1}{ 2 \sigma \, \sqrt{\pi t}}, \end{align} which implies that for every measurable function $ \; y:[0,T] \to {\mathbb R}$, \begin{align}\label{GSquareEsp}
\displaystyle \int_{{\mathbb R}} \int_{0}^{t} G^2_{t-s}\left(y(s)-x\right)\,ds\,dx = \frac{1}{\sigma} \; \sqrt[]{\frac{t}{\pi}}. \end{align} \end{lem} \begin{proof} The second estimate is obtained by rewriting $\partial_{xx}G_{t}(x)$ as $\partial_{xx}G_t(x) = - \frac{1}{\sigma^2t}G_t(x) + \frac{1}{\sigma^2t} \left(-x \partial_xG_t(x) \right)$. We apply an integration by parts for the second term and obtain: $\displaystyle \int_{{\mathbb R}}\partial_{xx}G_t(x)dx \leq \frac{2}{\sigma^2 t}\left\Vert G_{t} \right\Vert_{L^1} = \frac{2}{\sigma^2 t}$. \\
As for the values of $\left\Vert G_{t}^2 \right\Vert_{L^1}$ and $\displaystyle \int_{{\mathbb R}} \int_{0}^{t} G^2_{t-s}\left(y(s)-x\right)\,ds\,dx $, we use the fact that $G^2_{t}(x) = G_{t/2}(x) / 2 \sigma \, \sqrt{\pi t}$. \end{proof} \end{appendix}
\pagenumbering{gobble}
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\begin{abstract}
We provide a general contractibility criterion for subsets of Riemannian metrics on the disc. For instance, this result applies to the space of metrics that have positive Gauss curvature and make the boundary circle convex (or geodesic). The same conclusion is not known in any dimension $n\geq 3$, and (by analogy with the closed case) is actually expected to be false for many values of $n\geq 4$.
\end{abstract}
\maketitle
\begin{spacing}{1.04}
\section{Introduction} \label{sec:intro}
Let $M$ be a smooth, compact manifold of dimension $n\geq 2$, without boundary.
Two important questions in Riemannian Geometry are whether $M$ supports any metric of positive scalar curvature and, if that is the case, whether one can characterize the homotopy type of the subset $\mathfrak{R}^{+}(M)\subset \mathfrak{R}(M)$ consisting of all such metrics. In spite of many remarkable advances, these questions are still far from being fully resolved, except for the case of surfaces ($n=2$) and of 3-manifolds ($n=3$), the latter being the object of the very recent work by Bamler-Kleiner \cite{BK19}.
When $n\geq 3$, it is well-known that \emph{any} manifold $M$ as above can be endowed with metrics of negative scalar curvature (cf. \cite{KW75a, KW75b} for a more precise characterization), while pioneering works by Schoen-Yau \cite{SY79a, SY79b,SY82} and Gromov-Lawson \cite{GL80a,GL83} imply that, at the opposite end of the spectrum, manifolds that do not admit any metric of positive scalar curvature exist in abundance. The simplest such example is provided by the three-dimensional torus $S^1\times S^1\times S^1$, which is in fact a significant result as it implies the positive mass theorem for three-dimensional asymptotically flat manifolds (this is the compactification approach followed in \cite{SY17}). A complete description of those three-manifolds for which $\mathfrak{R}^{+}(M)\neq\emptyset$ follows, when $n=3$, by Perelman's papers on the Ricci flow with surgery \cite{Per02, Per03a, Per03b}, while for $n\geq 5$ and $M$ simply connected our knowledge relies on the combination of outstanding work by Gromov-Lawson \cite{GL80b} and Stolz \cite{Sto92}: $\mathfrak{R}^{+}(M)\neq\emptyset$ if and only if either $M$ does not admit any spin structure or, in case of spin manifolds, if $\alpha(M)=0$. However, for $n\geq 4$, we are far from a complete understanding of the topology of $\mathfrak{R}^{+}(M)$ (for partial, but often striking achievements see e.g. \cite{Hit74, Car88, KS93, BG96, Rub01, BHSW10, HSS14, BER17} among others). These works provide simple criteria (of dimensional character, and/or on the topology of $M$) implying that $\mathfrak{R}^{+}(M)$ is often \emph{not even path-connected}, which contrasts with the main theorem in \cite{Mar12} for $S^3$ (and, more generally, with the results in \cite{BK19}).
For closed surfaces, the Gauss-Bonnet theorem gives at once that $\mathfrak{R}^{+}(M)\neq\emptyset$ if and only if $\chi(M)>0$, that is if $M \cong S^2$ or $\chi\cong \mathbb{R}\mathbb{P}^2$. The topology of this space of metrics was first studied by Weyl \cite{Wey15} to the scope of proving the existence of isometric embeddings of positively curved spheres as convex bodies in $\mathbb{R}^3$ (a theorem later obtained by Nirenberg \cite{Nir53} in one of his very first papers). In particular, Weyl was able to prove that $\mathfrak{R}^{+}(S^2)$ is path-connected. To the best of our knowledge, the full characterization of the homotopy type of $\mathfrak{R}^{+}(S^2)$ and $\mathfrak{R}^{+}(\mathbb{R}\mathbb{P}^2)$, namely the theorem that these spaces are actually \emph{contractible}, was first sketched by Rosenberg-Stolz in their beautiful survey \cite{RS01}.
If we now let $M$ be a smooth, compact manifold of dimension $n\geq 2$, with (smooth) boundary, a remarkable theorem by Gromov implies that $M$ \emph{always} supports metrics of positive scalar (in fact: sectional) curvature \cite{Gr69}. Hence, \emph{boundary conditions} must be introduced for the problem not to be trivial. This leaves room for different sorts of choices, and we will focus on some of the most natural ones, namely those defined by pointwise conditions given in terms of the mean curvature of the boundary.
For instance, one can consider the space $\mathfrak{M}^{+}(M)$ that consists of those Riemannian metrics on $M$ that are smooth up to the boundary, and such that $(M,g)$ is a (strictly) mean-convex domain of positive scalar curvature.
Once again, one wonders under what conditions on $M$ it is the case that $\mathfrak{M}^{+}(M)\neq\emptyset$ and what is the topology of this space of metrics. Yet, some of the techniques that come into play in the study of closed manifolds do not have a straightforward extension to the case when $\partial M\neq\emptyset$, with the net result that comparatively little is known. Besides the analysis of the $n=3$ case, which is the object of the recent article \cite{CL19} by the first author and C. Li, we mention the work by Walsh \cite{Wal14} (although the focus there is on \emph{collar} boundary conditions) and Botvinnik-Kazaras \cite{BK18}. The main purpose of this note is to prove a general theorem for the (two-dimensional) closed disc $\overline{D}$, which easily provides various significant geometric applications, including a full answer to the question about the homotopy type of
$\mathfrak{M}^{+}(M)\neq\emptyset$ in the special case above.
In order to state our main result, let us first introduce some notation. Given $g\in\mathfrak{R}(M)$, thus in the space of Riemannian metrics on $M$ \emph{that are smooth up to the boundary}, denote by $[g]$ the pointwise conformal class of $g$; that is, $g_1 \in [g]$ if $g_1 = e^{2u} g$ for some smooth function $u$ on $M$. Let
\[
\mathfrak{C}(M) = \{ [ g ] : g \in \mathfrak{R}(M)\},
\]
which we always assume endowed with the so-called \emph{smooth topology} inherited (as a quotient) from $\mathfrak{R}(M)$; denote by $\pi: \mathfrak{R}(M)\to \mathfrak{C}(M)$ the associated projection map.
\begin{thm}\label{thm:Main}
Let $\mathfrak{M}^+(\overline{D})\subset \mathfrak{R}(\overline{D})$ be a non-empty space of Riemannian metrics satisfying the following two properties:
\begin{enumerate}
\item {for every $g\in \mathfrak{R}(\overline{D})$ the intersection $\mathfrak{M}^+(\overline{D}) \cap \pi^{-1}([g])$ is convex in the sense that if $g_i = e^{2u_i} g \in \mathfrak{M}^+(\overline{D}) \cap \pi^{-1} ([g])$, $i = 1, 2$, then
\[
e^{ 2 ( t_1 u_1 + t_2 u_2 )} g \in \mathfrak{M}^+ (\overline{D}) \cap \pi^{-1} [g] \ \text{for all} \ t_1, t_2\geq 0 \ \text{such that} \ t_1+t_2=1;
\]
}
\item {$\mathfrak{M}^+(\overline{D})$ is invariant under diffeomorphisms, i.e.
\[
g\in \mathfrak{M}^+(\overline{D}) \ \text{if and only if} \ \phi^{\ast}g\in\mathfrak{M}^+(\overline{D}) \ \text{for all} \ \phi\in \textup{Diff}(M).
\]
}
\end{enumerate}
Then $\mathfrak{M}^+(\overline{D})$ is contractible.
\end{thm}
The two assumptions above are often easy to verify: the second one is always satisfied when $\mathfrak{M}^+(\overline{D})$ is given by \emph{curvature} conditions (which are our main concern), while the first one holds (for instance) for linear inequalities (possibly equalities) involving the Gauss curvature of the disc in question, and the geodesic curvature of its boundary.
Indeed, we have the well-known equations
\[
e^{2u} K_{e^{2u} g} = K_g - \Delta_g u, \ \ \ \ e^{u}k_{e^{2u} g}=k_g+\nu(u)
\]
where $\nu$ is the outward-pointing unit normal along the boundary.
Thus, as a special case, we can derive what follows:
\begin{cor}\label{cor:R+D}
Let $\mathfrak{M}^+(\overline{D})$ denote one of the following spaces of Riemannian metrics:
\begin{itemize}
\item{$I:= \left\{g\in \mathfrak{R}(\overline{D}) \ : \ K_g\geq 0\right\}, \ II:= \left\{g\in \mathfrak{R}(\overline{D}) \ : \ K_g>0\right\}, \ III:= \left\{g\in \mathfrak{R}(\overline{D}) \ : \ K_g= 0\right\}$;}
\item{$I_{\partial}:= \left\{g\in \mathfrak{R}(\overline{D}) \ : \ k_g\geq 0\right\}, \ II_{\partial}:= \left\{g\in \mathfrak{R}(\overline{D}) \ : \ k_g> 0\right\}, \ III_{\partial}:= \left\{g\in \mathfrak{R}(\overline{D}) \ : \ k_g=0\right\}$.}
\end{itemize}
Then $\mathfrak{M}^+(\overline{D})$ is contractible; furthermore any space $\mathfrak{M}^+(\overline{D})$ obtained by intersecting one among $I, II, III$ with one among $I_{\partial}, II_{\partial}, III_{\partial}$ is either empty or contractible.
\end{cor}
The arguments we present turn out to be rather direct, and conceptually transparent. In addition, as we will see, they are somewhat more elementary than in the case of $S^2$ (cf. Remark \ref{rem:Contract}). Note that when $n\geq 4$ the analogy with the closed case (as briefly summarized above) suggests that even $\pi_0 (\mathfrak{M}^{+}(M))$ should be non-trivial in many cases of interest. It is reasonable to expect some sort of progress on these matters in the years to come.
\
\noindent \textit{Acknowledgements:} The first author would like to thank Alexander Kupers for various clarifications; the second author would like to thank Simon Donaldson for bringing Ahlfors' work on the Beltrami equation to his attention. The authors would also like to thank the anonymous referee for providing very helpful remarks that improved the presentation of this paper.
A. C. is partly supported by the National Science Foundation (through grant DMS 1638352) and by the Giorgio and Elena Petronio Fellowship Fund. D. W. is partly supported by the National Science Foundation (through grant DMS 1611745) and the Simonyi Endowment of the Institute for Advanced Study.
This project was developed at the Institute for Advanced Study during the special year \emph{Variational Methods in Geometry}: both authors would like to acknowledge the support of the IAS and the excellent working conditions.
\section{Proof of Theorem \ref{thm:Main}}\label{sec:proofs}
Let $\textup{Diff}(M)$ be the group of diffeomorphisms of $M$; when $M$ is a manifold with boundary we agree that diffeomorphisms are proper in the sense that they map $\partial M$ onto $\partial M$. When $M$ is oriented (as we assume), we denote by $\textup{Diff}^+(M)$ the orientation-preserving subgroup of $\textup{Diff}(M)$. In either case, we also tacitly employ the (strong) smooth topology on these spaces of maps. Notice that in case $\partial M\neq\emptyset$ we do \emph{not} require diffeomorphisms to restrict to the identity map along the boundary.
If $M$ is a compact surface and we fix three distinct points $x_1, x_2, x_3$ on $M$, then we let
\[
\textup{Diff}_{\bullet}^+(M) = \{ \phi \in \textup{Diff}^+(M): \phi (x_i) = x_i, \ i = 1, 2, 3\}.
\]
Specifically, for $M=\overline{D}$ we agree to choose
\[
x_1 = 1, \ x_2 = i, \ x_3 = -1.
\]
As usual, we denote by $C^{\infty}(M)$ be the set of (real-valued) smooth functions on $M$. The notation $\mathds{1}_X$ stands for the identity map on a set $X$.
Let $z = x + i y$ be the standard complex coordinate on $\mathbb{C}$, and let
\[ g_0= |dz|^2 := (d z \otimes d\bar{z} + d\bar{z} \otimes dz)/2= dx \otimes dx + dy \otimes dy,
\] the standard Euclidean metric on the plane.
More generally, we adopt the convention that $|\vartheta|^2:= \text{Re}(\vartheta\otimes\overline{\vartheta})=(\vartheta\otimes\overline{\vartheta}+\overline{\vartheta}\otimes\vartheta)/2$ for any complex valued 1-form $\vartheta$.
We will identify $D$ with
the unit disc $\mathbb{D}\subset\mathbb{C}$ (as smooth manifolds) and, correspondingly, we will write $C^{\infty}(\overline{D},D)\subset C^{\infty}(\overline{D},\mathbb{C})$ to denote the set of complex-valued maps defined on $\overline{D}$ whose image lies in $\left\{z\in\mathbb{C} \ : \ |z|<1 \right\}$ and that are smooth up to the boundary.
\
We shall employ the uniformization theorem in the following version:
\begin{thm} \label{th:Ufz-S2}
For any $g \in \mathfrak{R}(\overline{D})$, there exists a map $\phi \in \textup{Diff}^+(\overline{D})$ and a function $u \in C^{\infty}(\overline{D})$ such that
\[
g = \phi^* (e^{2u} g_0).
\]
\end{thm}
To properly interpret the result above, and for later scopes, it is actually convenient to briefly recall the proof, recast in terms of Beltrami's equation.
Following e.g. \cite{EE69, ES70} it is easy to construct a bijection from $\mathfrak{C}(\overline{D})$ to the space $C^{\infty}(\overline{D},D)$.
To define it, given $[g] \in \mathfrak{C}(\overline{D})$ with
\[
g = g_{11} dx\otimes dx + g_{12} (dx \otimes dy + dy \otimes dx) + g_{22} dy \otimes dy,
\]
observe that
\begin{align*}
\ \ \ 4 g &= (g_{11} - g_{22} - 2 i g_{12} ) dz \otimes dz + (g_{11} - g_{22} + 2 i g_{12}) d \bar{z} \otimes d \bar{z}
+ (g_{11} + g_{22}) (dz \otimes d\bar{z} + d \bar{z} \otimes dz) \\
\ \ \ &= \rho | dz + \mu d \bar{z}|^2,
\end{align*}
provided we set
\[
\rho = (g_{11}+g_{22}) + 2\sqrt{g_{11} g_{22} - g_{12}^2}, \ \
\mu = \frac{(g_{11} - g_{22}) + 2 i g_{12}}{(g_{11} + g_{22}) + 2 \sqrt{g_{11} g_{22} - g_{12}^2}}.
\]
Note that
\[
|\mu|^2 = \frac{(g_{11} + g_{22}) - 2 \sqrt{g_{11} g_{22} - g_{12}^2}}{(g_{11} + g_{22}) + 2 \sqrt{g_{11} g_{22} - g_{12}^2}} < 1,
\]
hence the map $[g] \mapsto \mu$ is well-defined.
Conversely, given $\mu \in C^{\infty}(\overline{D}, D)$, let
$
g = |dz + \mu d\bar{z}|^2.
$
Then, the formula
\begin{align*}
g & = |1 + \mu|^2 dx \otimes dx + |1 - \mu|^2 d y \otimes dy + i (\bar{\mu} - \mu) (dx \otimes dy + dy \otimes dx)
\end{align*}
defines a Riemannian metric on $\overline{D}$, since $|1 \pm \mu|^2 > 0$ and
\[
|1 + \mu|^2 |1 - \mu|^2 - ( i (\bar{\mu}-\mu))^2 = (1 - |\mu|^2)^2 > 0.
\]
It is readily checked that the map $\mu \mapsto [g]$ is the inverse of the previous one. Furthermore, both maps in question are continuous with respect to the smooth topologies, so that one actually obtains a homeomorphism $\mathfrak{C}(\overline{D})$ to the space $C^{\infty}(\overline{D},D)$. As a result, we can draw the following important conclusion.
\begin{prop}\label{pro:ContractibleD}
The space $\mathfrak{C}(\overline{D})$ is contractible.
\end{prop}
\begin{proof}
The assertion follows at once, by virtue of the homeomorphism above, from the (strict) convexity of $C^{\infty}(\overline{D}, D)$.
\end{proof}
\begin{rmk}\label{rem:Contract}
\emph{(a)} Proposition~\ref{pro:ContractibleD} might actually be regarded as a special case of a more general fact, as we discuss in Appendix \ref{app:ConfClass} below for the sake of completeness (cf. \cite[p. 213]{Don11} and, for compact oriented manifolds, see Theorem 2.2 in \cite{FT84} applied for $s=\infty$ based on Remark 2.5 therein).\newline \emph{(b)} In proving the contractibility of $\mathfrak{C}(S^2)$, which is crucial for the contractibility of $\mathfrak{R}^{+}(S^2)$, Rosenberg-Stolz invoke the celebrated theorem by Smale \cite{Sma59} on the space of diffeomorphisms of the sphere.\newline \emph{(c)} Let us explicitly remark that the sole contractibility of $\mathfrak{C}(M)$ does not suffice, in general, for the conclusion of Theorem \ref{thm:Main} to hold, as it is shown for instance by the case of $\mathfrak{R}^+(S^8)$, because of \cite{Hit74}. \end{rmk}
Let $\mathbb{H} = \{z \in \mathbb{C}: \textup{Im} (z) > 0\}$ and $h: \mathbb{H} \to \mathbb{D}$ be the Cayley transform $h(z) = -(z-i) / (z+i)$. Then $h$ is biholomorphic from $\mathbb{H}$ to $\mathbb{D}$, extends smoothly to a map $\overline{\mathbb{H}}:=\left\{\text{Im}(z)\geq 0\right\}\cup\left\{\infty \right\}\to \overline{\mathbb{D}}$ and defines (by restriction) a smooth map from the boundary $\partial \mathbb{H}$ onto the unit circle $\partial \mathbb{D}$. Therefore, given $g\in\mathfrak{R}(\overline{D})$ the conformal class $[g]$ uniquely determines a smooth function $\mu$ on $\overline{\mathbb{H}}$ with $\sup |\mu| \le c < 1$.
\
At that stage, it follows from Ahlfors-Bers~\cite{AB60} that there is a unique map $w$ which is a diffeomorphism from $\overline{\mathbb{H}}$ onto itself, leaves $0$, $1$, $\infty$ fixed, and satisfies the Beltrami equation $w_{\bar{z}} = \mu w_z$. This implies that $w \circ h^{-1}$ is the isothermal coordinate map for the metric $g$; hence, the map $\phi = h \circ w \circ h^{-1}$ is a biholomorphic map from $(D, g)$ onto $(D,g_0)$. Theorem \ref{th:Ufz-S2} for the closed disc follows at once. So, for $[g]\in\mathfrak{C}(\overline{D})$ we shall set $\Phi([g])=\phi$;
note that $\Phi([g])\in \textup{Diff}_{\bullet}^+(\overline{D})$.
\begin{prop}\label{pro:homeo} The map \[ \Phi: \mathfrak{C}(\overline{D})\to \textup{Diff}^+_{\bullet}(\overline{D}) \] is a homeomorphism. \end{prop}
\begin{proof}
The continuity of the map $\Phi$ follows from the continuity of the three maps \[ \begin{tikzcd} \mathfrak{C}(\overline{D}) \arrow{r} & C^{\infty}(\overline{D}, \mathbb{C}) \arrow{r} & \textup{Diff}^+_{\bullet} (\overline{\mathbb{H}}) \arrow{r} & \textup{Diff}^+_{\bullet} (\overline{D}) \\ \textup{[}g\textup{]} \arrow[mapsto]{r} & \mu \arrow[mapsto]{r} & w \arrow[mapsto]{r} & \phi:=h \circ w \circ h^{-1} \end{tikzcd} \]
the first and third claim being obvious, the second being proven in \cite[2B]{ES70}.
We then assert that the map in question is bijective, and that its inverse is the (patently continuous) map $\Psi: \textup{Diff}^+_{\bullet}(\overline{D}) \to \mathfrak{C}(\overline{D})$ defined by $\phi \mapsto [\phi^*g_0]$. The fact that $\Psi\circ \Phi=\mathds{1}_{\mathfrak{C}(\overline{D})}$ descends from the biholomorphicity of $\phi=\Phi([g])$ as a map from $(D,g)$ to $(D,g_0)$. Instead, the fact that \[ \Phi\circ \Psi=\mathds{1}_{\textup{Diff}^+_{\bullet}(\overline{D})} \] follows from observing that if $\tilde{\phi}^*(g_0)$ is (pointwise) conformal to $\phi^*(g_0)$, in the sense that $\tilde{\phi}^*(g_0) = e^{2u} \phi^*(g_0)$ for some $u \in C^{\infty}(\overline{D})$, then $\tilde{\phi} = \phi$. To see this, it is sufficient to show \begin{equation} \label{eq:tphi} \tilde{\phi} \circ \phi^{-1} \in \textup{Aut}(\overline{D}), \end{equation} since $\tilde{\phi} \circ \phi^{-1}$ fixes three points (see for example \cite{Lan99}). Here $\textup{Aut}(\overline{D})$ denotes the set of biholomorphic automorphisms of the unit disc in $\mathbb{C}$.
In order to verify \eqref{eq:tphi}, let $\eta \in \textup{Diff}^+(\overline{D})$ satisfy $ \eta^* g_0 = e^{2w} g_0$ for some $w \in C^{\infty}(\overline{D})$: we claim that any such map $\eta$ is holomorphic with respect to the coordinate $z$ on $\mathbb{C}$, i.e., \begin{equation} \label{eq:dphi=0} \frac{\partial \eta}{\partial \bar{z}} = 0. \end{equation} (It then follows from the holomorphic version of the inverse function theorem (cf. e.g. Theorem 9.6 in \cite{Gri89}) that $\eta^{-1}$ is also holomorphic, hence $\eta \in \textup{Aut}(\overline{D})$). To check that \eqref{eq:dphi=0} holds, first note that \[ \eta^* g_0 = \frac{\partial \eta}{\partial z} \frac{\partial \overline{\eta}}{\partial z} dz \otimes dz + \frac{\partial \eta}{\partial \bar{z}} \frac{\partial \overline{\eta}}{\partial \bar{z}} d\bar{z} \otimes d\bar{z}
+ \Big(\Big| \frac{\partial \eta}{\partial \bar{z}} \Big|^2 + \Big|\frac{\partial \eta}{\partial z} \Big|^2\Big) |dz|^2, \] so that, by our assumption, we have \[ \frac{\partial \eta}{\partial z} \frac{\partial \overline{\eta}}{\partial z} = 0, \quad \frac{\partial \eta}{\partial \bar{z}} \frac{\partial \overline{\eta}}{\partial \bar{z}} = 0. \] On the other hand, since $\eta$ preserves the orientation, \begin{align*} 0 < \textup{Jacobian} (\eta) = \frac{\partial (\eta, \overline{\eta}\,)}{\partial (z, \bar{z})}
= \Big| \frac{\partial \eta}{\partial z} \Big|^2 - \Big| \frac{\partial \eta}{\partial \bar{z}} \Big|^2, \end{align*} thus $\partial \eta/\partial z$ will not vanish at any point. Thereby, it follows that
$|\partial \eta/\partial \bar{z}|^2 = 0$, as we had to prove. \end{proof}
Theorem~\ref{thm:Main} follows immediately from Proposition~\ref{pro:ContractibleD} and the following Lemma~\ref{le:R+drC}.
\begin{lem} \label{le:R+drC}
Let $\mathfrak{M}^{+}(\overline{D})$ be as in the statement of Theorem~\ref{thm:Main}. Then it is homotopy equivalent to $\mathfrak{C}(\overline{D})$.
\end{lem}
\begin{proof}
Let $g_+\in \mathfrak{M}^{+}(\overline{D})$ (the set is assumed to be non-empty). By virtue of Theorem \ref{th:Ufz-S2}, we can write
$g_+=\phi_+^*(e^{2u_0}g_0)$ for some $\phi_+\in \text{Diff}^+(\overline{D})$, hence by our assumption (2) on the set $\mathfrak{M}^{+}(\overline{D})$ it must be that $g_D:=e^{2u_0}g_0\in\mathfrak{M}^{+}(\overline{D})$ as well.
Let $\pi: \mathfrak{M}^{+}(\overline{D}) \to \mathfrak{C}(\overline{D})$ denote the restriction of the projection map, i.e. $\pi(g)=[g]$.
We further define the map $\sigma: \mathfrak{C}(\overline{D}) \to \mathfrak{M}^{+}(\overline{D})$ by factoring through $\textup{Diff}^+_{\bullet}(\overline{D})$ as shown in the diagram
\[
\begin{tikzcd}
\mathfrak{C}(\overline{D}) \arrow{r}{\sigma} \arrow{dr}{\Phi} & \mathfrak{M}^{+}(\overline{D}) \\
& \textup{Diff}^+_{\bullet}(\overline{D}) \arrow{u},
\end{tikzcd}
\]
where $\Phi$ is the map defined above.
Indeed, given $[g] \in \mathfrak{C}(\overline{D})$, there exists a unique map $\Phi([g]) = \phi \in \textup{Diff}^+_{\bullet}(\overline{D})$ such that
\[
[g] = [\phi^*g_0] = [ \Phi([g])^* g_0].
\]
Then we define
\[
\sigma ([g]) = \phi^*g_D = \Phi([g])^* g_D,
\]
where it should be noted that $\sigma ([g]) \in \mathfrak{M}^{+}(\overline{D})$ by diffeomorphism invariance (assumption (2) in Theorem \ref{thm:Main}).
Clearly $\pi$ is continuous; the map $\sigma$ is also continuous, as $\Phi$ is, by virtue of Proposition \ref{pro:homeo}.
\
Note that $\pi \circ \sigma = \mathds{1}_{\mathfrak{C}}$, for indeed
$
\pi \circ \sigma ([g]) = \pi \big((\Phi([g])^* g_D\big) = [ \Phi([g])^*g_D]=[ \Phi([g])^*g_0] = [g].
$
Next, we will show that $\sigma \circ \pi \simeq \mathds{1}_{\mathfrak{M}^{+}(\overline{D})}$, where $\simeq$ stands for the homotopy relation. Define a map $H : [0, 1] \times \mathfrak{M}^{+}(\overline{D}) \to \mathfrak{M}^+(\overline{D})$ as follows. For each metric $g$ in $\mathfrak{M}^{+}(\overline{D})$,
we can write $g = e^{2u} \Phi([g])^*g_D$ for some $u \in C^{\infty}(\overline{D})$ which is uniquely determined by $g$, and continuously depending on $g$. We then set
\[
H(t, g) = e^{2(1-t)u} \Phi([g])^* g_D.
\]
Then $H$ is well-defined, by assumption (1) on the set $\mathfrak{M}^{+}(\overline{D})$, and continuous. Note that
\begin{align*}
H(0, g) & = e^{2u} \Phi([g])^*g_D = g, \\
H(1, g) & = \Phi([g])^*g_D = \sigma \circ \pi (g).
\end{align*}
This proves that $\sigma \circ \pi \simeq \mathds{1}_{\mathfrak{M}^{+}(\overline{D})}$. Hence, $\mathfrak{M}^{+}(\overline{D})$ is homotopy equivalent to $\mathfrak{C}(\overline{D})$.
\end{proof}
\begin{rmk}The conclusion of Theorem \ref{thm:Main} continues to hold if assumption (1) is relaxed to only require \emph{one} fiber to be \emph{star-shaped}. More precisely, let $\mathfrak{M}^+(\overline{D})$ satisfy condition (2) in that statement, together with the following assumption: there exists a metric $\bar{g} \in \mathfrak{M}^+(\overline{D})$ such that $e^{2tu} \bar{g} \in \mathfrak{M}^+(\overline{D}) \cap \pi^{-1}([\overline{g}])$ for all $0 \le t \le 1$, whenever $g_1 = e^{2u} \bar{g} \in \mathfrak{M}^+(\overline{D}) \cap \pi^{-1}([\bar{g}])$. Then, modifying the proof of \ref{le:R+drC} above by letting $g_+$ be $\bar{g}$ one can still conclude that $\mathfrak{M}^+(\overline{D})$ is contractible.
\end{rmk}
\appendix
\section{Contractibility of the space of conformal classes} \label{app:ConfClass}
\begin{lem} \label{le:CM-c}
Let $n\geq 2$ and let $M$ be an $n$-dimensional oriented manifold (possibly with non-empty boundary). Then, $\mathfrak{C}(M)$ is contractible.
\end{lem}
\begin{proof}
Fixed any $g_0\in\mathfrak{R}(M)$, define a continuous map $H: [0, 1] \times \mathfrak{R}(M) \to \mathfrak{C}(M)$ by
\[
H(t, g) = \pi \bigg( (1 - t) \bigg(\frac{\mu(g_0)}{\mu (g)}\bigg)^{2/n} g + t g_0 \bigg).
\]
Here $\mu(g)$ (resp. $\mu(g_0)$) stands for the volume element of $g$ (resp. $g_0$); in terms of local oriented coordinates $(x^1,\ldots, x^n)$ centered at an interior point of $M$
we have
\[
\mu(g) = \sqrt{ \det (g_{ij})} \, dx^1 \wedge \cdots \wedge dx^n.
\]
Note that the ratio
\[
\frac{\mu(g_0)}{\mu(g)} := \frac{\sqrt{\det (g_{0, ij})}}{\sqrt{\det (g_{ij})}}
\]
is a globally-defined, smooth positive function on $M$; furthermore, if $M$ has non-empty boundary, then $\mu (g_0) /\mu(g)$ extends continuously up to $\partial M$.
We claim that $H$ descends to a continuous map from $[0, 1] \times \mathfrak{C}(M)$ to $\mathfrak{C}(M)$. Indeed, if $\tilde{g} = e^{2u} g$ then
\[
\det (\tilde{g}_{ij}) = e^{2n u} \det (g_{ij}); \quad \textup{thus}, \quad \frac{\mu(\tilde{g})}{\mu(g_0)} = e^{nu} \frac{\mu(g)}{\mu(g_0)}.
\]
It follows that
\[
\bigg(\frac{\mu(g_0)}{\mu(\tilde{g})}\bigg)^{2/n} \tilde{g} = \bigg(\frac{\mu(g_0)}{\mu(g)}\bigg)^{2/n} g,
\]
hence $H(t, \tilde{g}) = H(t, g)$. The continuity of $H$ on $\mathfrak{C}(M)$ follows immediately from the definition of quotient topology; since $H(0, [g]) = [g]$ and $H(1, [g]) = [g_0]$ we conclude that $\mathfrak{C}(M)$ is homotopy equivalent to $[g_0]$.
\end{proof}
\end{spacing}
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Isomorphism Problems of Noncommutative Deformations of Type $D$ Kleinian singularities}{}{}
\begin{abstract} We construct all possible noncommutative deformations of a Kleinian singularity ${\mathbb C}^2/\Gamma$ of type $D_n$ in terms of generators and relations, and solve the problem of when two deformations are isomorphic. We prove that all isomorphisms arise naturally from the action of the normalizer $N_{\mathop{\rm SL}\nolimits(2)}(\Gamma)$ on ${\mathbb C}/\Gamma$. We deduce that the moduli space of isomorphism classes of noncommutative deformations in type $D_n$ is isomorphic to a vector space of dimension $n$. \end{abstract}
\section{Introduction}
Let $V$ be a complex vector space of dimension 2 and let $\Gamma$ be a finite subgroup of $\mathop{\rm SL}\nolimits(V)$. Such subgroups are classified: up to conjugacy, they are in one-to-one correspondence with the simply-laced Dynkin diagrams $A_n (n\geq 1),D_n (n\geq 4),E_6,E_7,E_8$. Let $\Delta$ be the Dynkin diagram of $\Gamma$. The quotient $V/ \Gamma$, which has coordinate ring ${\mathbb C}[V]^\Gamma$ and embeds as a hypersurface in ${\mathbb A}^3$ is a {\it Kleinian singularity} or {\it rational double point} of type $\Delta$. The Dynkin diagram $\Delta$ arises as the exceptional configuration of the minimal resolution of the singularity $V/\Gamma$ (see \cite[\S 6]{slodowy}), or as the type of the McKay graph of $\Gamma$, which is isomorphic to the extended Dynkin diagram $\hat\Delta$ (\cite{mckay}).
It follows from the identification of $V/\Gamma$ with a hypersurface in ${\mathbb C}^3$ that there is a Poisson bracket on ${\mathbb C}[V]^\Gamma$, and an associated Poisson structure on the polynomial ring ${\mathbb C}[X,Y,Z]$. In \cite{cbh}, Crawley-Boevey and Holland constructed a family of (in general non-commutative) deformations ${\cal O}_\lambda$ of (the Poisson bracket on) ${\mathbb C}[V]^\Gamma$, parametrised by $\lambda\in Z({\mathbb C}\Gamma)$. This generalized work of Hodges \cite{hodges} and Smith \cite{smith} who constructed deformations of, respectively, a Kleinian singularity of type $A$ and the corresponding Poisson structure on ${\mathbb C}[X,Y,Z]$. It is perhaps a little surprising that noone has attempted to describe the possible deformations of the non-type $A$ singularities in terms of generators and relations. In Sect. 1 we carry this out for type $D$. We show that construct all noncommutative deformations of a Kleinian singularity of type $D_n$, parametrised by a pair $(Q,\gamma)$ where $Q(t)$ is a monic polynomial of degree $(n-1)$ and $\gamma\in{\mathbb C}$. We denote the corresponding associative algebras by $D(Q,\gamma)$ (Def. \ref{DQgamma}). We also classify the noncommutative deformations of the corresponding Poisson structure on ${\mathbb C}[X,Y,Z]$, which are parametrised by $\gamma\in{\mathbb C}$ and a polynomial $P$ with leading term $(n-1)t^{n-2}$. We denote the associative algebras thus produced by $H(P,\gamma)$. (One could perform this process for the exceptional types. The calculations are rather detailed, but not impossible.)
Let ${\mathfrak g}$ be a complex simple Lie algebra with Dynkin diagram $\Delta$ and let $E$ be a subregular nilpotent element of ${\mathfrak g}$. Choose an $\mathfrak{sl}(2)$-triple $\{ H,E,F\}\subset{\mathfrak g}$ containing $E$. It was proved by Brieskorn \cite{brieskorn} that the intersection of the {\it Slodowy slice} $E+{\mathfrak z}_{\mathfrak g}(F)$ with the nilpotent cone of ${\mathfrak g}$ is isomorphic to $V/ \Gamma$. (This was generalized to the case of non-simply-laced $\Delta$ by Slodowy \cite{slodowy}.) In fact, one can carry out the same process for an arbitrary nilpotent orbit. In \cite{premslice}, Premet proved that all singularities constructed in this way have natural non-commutative deformations (see also \cite{gg}). It was recently proved (by Arakawa \cite{arakawa} for regular $E$, by De Sole-Kac et al for general $E$ \cite[Appendix]{desole-kac}) that Premet's deformations are isomorphic to the {\it finite (quantum) $W$-algebras} of mathematical physics, constructed via quantum Hamiltonian reduction and the BRST cohomology (see de Boer and Tjin \cite{deboertjin}).
The associative algebras constructed by Hodges and Smith have straightforward presentations in terms of generators and relations: if $a$ and $P$ are polynomials then let $A(a)$ (resp. $R(P)$) be the algebra with generators $e,f,h$ (resp. $H,A,B$) and relations $he-eh=e,hf-fh=-f,ef=a(h-1),fe=a(h)$ (resp. $HA-AH=A,HB-BH=-B,AB-BA=P(H)$). If $P(t)=a(t-1)-a(t)$, then clearly $A(a)$ is a quotient of $R(P)$. The algebras of Smith are sometimes called {\it generalized Weyl algebras} because of their similarity to the first Weyl algebra. The problem of when two algebras $A(a_1),A(a_2)$ are isomorphic was solved by Bavula and Jordan in \cite{bav-jord}: the isomorphisms are precisely the `obvious' ones. Namely, $A(a_1)\cong A(a_2)$ if and only if $a_1(t)=\eta a_2(\tau\pm t)$ for some $\eta\in{\mathbb C}^\times,\tau\in{\mathbb C}$. The analagous isomorphism theorem for the generalized Weyl algebras then follows. In Sect. 2 we tackle the isomorphism problem for the algebras $D(Q,\gamma)$. If $n>4$ then (assuming $Q$ monic) only the isomorphisms $D(Q,\gamma)\cong D(Q,-\gamma)$ occur (Thm. \ref{geq4}). It follows that the moduli space of isomorphism classes of deformations of (the Poisson bracket on) a Kleinian singularity of type $D_n$ is isomorphic to affine $n$-space, generalizing the result for type $A$ (Cor. \ref{moduli}). In Sect. 3 we carry out specific calculations for the case $n=4$, where the situation is rather more interesting. Here there are six sets of isomorphisms, corresponding to the six elements of $N_{\mathop{\rm SL}\nolimits(V)}(\Gamma)/\Gamma\cong S_3$ (Thm. \ref{main}). The moduli space of isomorphism classes is isomorphic to a vector space of dimension 4 (Cor. \ref{moduli4}). We also apply our results on $D(Q,\gamma)$ to solve the problem of when two algebras $H(P,\gamma),H(\tilde{P},\tilde\gamma)$ are isomorphic (Thm. \ref{Hn4} and Thm. \ref{H4}).
Our methods share a certain similarity with those of Bavula and Jordan \cite{bav-jord}, who adapted Dixmier's approach to solving the isomorphism problem for the case $\deg a=1,2$ (\cite{dixmier,dixmier2}). In particular, we construct a filtration of $D(Q,\gamma)$ by the additive monoid ${\mathbb Z}_{\geq 0}\times{\mathbb Z}_{\geq 0}$ (which compares to Bavula and Jordan's family of filtrations of $A(a)$ \cite[Thm. 3.14]{bav-jord}) and exploit a peculiar property of one of the generators for $D(Q,\gamma)$ to analyse its possible images in the corresponding graded algebra (Lemma \ref{reducetov}). On the other hand, at this point we diverge sharply from the path of \cite{bav-jord}, since elimination of the remaining cases requires an in-depth study of certain expressions involving commutators. However, one advantage of this analysis is the explicit construction of the `non-trivial' isomorphisms in type $D_4$ (Definition \ref{isos}).
Unfortunately, we have as yet been unable to determine whether every algebra $D(Q,\gamma)$ is isomorphic to some ${\cal O}_\lambda$. Given our description of the isomorphisms as essentially arising from the normalizer of $\Gamma$ in $\mathop{\rm SL}\nolimits(V)$ (which can be identified with the group of graph automorphisms of a root system of type $D_n$) we expect this to be the case.
{\it Notation.} We denote by $[x,y]$ for the commutator product $xy-yx$. If $m$ and $j$ are positive integers, then $[m/j]$ will denote the integer part of $m/j$.
{\it Acknowledgement.} Much of the research for this paper was carried out while at Aarhus University under a fellowship from the ``Lie GRITS" European Union research network. I am grateful both to Lie GRITS and to my Danish hosts for this opportunity. The article was completed at the \'Ecole Polytechnique F\'ed\'erale de Lausanne. I would like to thank Alexander Premet for alerting me to the problem and for several helpful conversations. I would also like to express my appreciation of the advice of Jens Carsten Jantzen and Iain Gordon.
\section{Generators and Relations}
Let $V$ be a complex vector space of dimension 2. Identify $\mathop{\rm SL}\nolimits(V)$ with $\mathop{\rm SL}\nolimits(2)$ by choice of a basis for $V$, and let $x,y$ be the corresponding coordinate functions on $V$. Up to conjugacy, there is a unique {\it binary dihedral group} $\Gamma\subset\mathop{\rm SL}\nolimits(V)$ of order $4(n-1)$ for each $n\geq 3$. Following \cite{slodowy}, we choose the following generators for $\Gamma$: $$\sigma=\left(\begin{array}{cc} \zeta & 0 \\ 0 & \zeta^{-1} \end{array}\right)\;\mbox{and}\;\tau=\left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right),\;\;\mbox{where }\zeta=e^{\pi i/(n-1)}.$$
The quotient $V/ \Gamma$ is a Kleinian singularity or rational double point of type $D_{n+1}$. It is easy to see that the coordinate ring ${\mathbb C}[V]^\Gamma$ is generated by $x^2y^2,(x^{2(n-1)}+y^{2(n-1)})$ and $xy(x^{2(n-1)}-y^{2(n-1)})$, hence is isomorphic to ${\mathbb C}[X,Y,Z]/(X^n+XY^2+Z^2)$.
Recall that a Poisson algebra is a commutative algebra $B$ endowed with a {\it Poisson bracket} $\{ .\, ,.\}$ satisfying:
(i) $(B,\{ .\, ,.\})$ is a Lie algebra,
(ii) $\{b,.\}$ and $\{.\, ,b\}$ are derivations of $B$ for each $b\in B$.
Any polynomial $\phi\in{\mathbb C}[X,Y,Z]$ induces a Poisson algebra structure on ${\mathbb C}[X,Y,Z]$, which we denote $\{ .\, ,.\}_\phi$, such that: $$\{X,Y\}_\phi=\partial\phi/ \partial Z, \{X,Z\}_\phi=-\partial\phi/ \partial Y, \{Y,Z\}_\phi=\partial\phi/ \partial X.$$
Moreover, since $(\phi)\subset{\mathbb C}[X,Y,Z]$ is a Poisson ideal, there is an induced Poisson bracket on the quotient ${\mathbb C}[X,Y,Z]/(\phi)$. In the case $\phi= X^n+XY^2+Z^2$, we will first construct (all possible) non-commutative deformations of the Poisson bracket on ${\mathbb C}[X,Y,Z]$. We denote the algebras thus produced by $H(P,\gamma)$, parametrised by a polynomial $P$ of the form $nt^{n-1}+\ldots$ and a scalar $\gamma$. The $H(P,\gamma)$ are the type $D$ analogues of the generalized Weyl algebras constructed by Smith \cite{smith}. In Lemma \ref{centreisp} we show that the centre of $H(P,\gamma)$ is a polynomial ring ${\mathbb C}[\Omega]$ on one generator, and provide a precise description of $\Omega$. The various factor algebras $H(P,\gamma)/(\Omega-c)$ thus determine all possible non-commutative deformations of the Kleinian singularity ${\mathbb C}[X,Y,Z]/(\phi)$ of type $D_{n+1}$.
For our purposes, a non-commutative deformation of a Poisson algebra $(A_0,\{ .\, ,.\})$ is an associative ${\mathbb C}[[t]]$-algebra ${\cal A}$, free as a ${\mathbb C}[[t]]$-module, such that:
(a) There is an isomorphism $\pi:{\cal A}/t{\cal A}\longrightarrow A_0$ of associative (commutative) algebras,
(b) For any $x,y\in {\cal A}$, $\pi(xy-yx+t{\cal A})=\{\pi(x+t{\cal A}),\pi(y+t{\cal A})\}$.
Note that freeness implies that any lift of a basis of $A_0$ is a (${\mathbb C}[[t]]$-)basis of ${\cal A}$. But it follows that if $A_0$ is a Kleinian singularity of type $D_{n+1}$ (resp. the corresponding Poisson algebra on ${\mathbb C}[X,Y,Z]$) then any deformation ${\cal A}$ of $A_0$ possesses a set $U,V,W$ of generators such that $\{ U^iV^jW^\epsilon \,:\, i,j\geq 0,\epsilon\in\{ 0,1\}\,\}$ (resp. $\{ U^iV^jW^k\,:\, i,j,k\geq 0\}$) is a ${\mathbb C}[[t]]$-basis for ${\cal A}$. Moreover, for any $\alpha,\beta\in{\mathbb C}^\times$ the quotients ${\cal A}/ (t-\alpha){\cal A}$ and ${\cal A}/ (t-\beta){\cal A}$ are naturally isomorphic, by appropriate scaling of the images of $U,V,W$. Hence we can (and will) abuse terminology and refer to the quotient $A={\cal A}/(t-1){\cal A}$ as the deformation of $A_0$. In less formal language, a noncommutative deformation of $A_0$ is a filtered associative algebra $A$ satisfying the appropriate condition as above on a generating set, such that $\mathop{\rm gr}\nolimits A=A_0$ and $\mathop{\rm gr}\nolimits [x,y]=\{ \mathop{\rm gr}\nolimits x,\mathop{\rm gr}\nolimits y\}$.
For the moment we wish to determine all noncommutative deformations of the Poisson algebra $({\mathbb C}[X,Y,Z],\{ .\, ,.\}_\phi)$. Hence suppose $A$ has generators $U,V,W$ such that $U^iV^jW^k$ is a basis. We require that $\mathop{\rm gr}\nolimits U=X$, $\mathop{\rm gr}\nolimits V=Y$, and $\mathop{\rm gr}\nolimits W=Z$: hence the filtration on $A$ satisfies $U\in A_4\setminus A_2,V\in A_{2n-2}\setminus A_{2n-4}$, and $W\in A_{2n}\setminus A_{2n-2}$. Moreover, with respect to this filtration $[U,V]=2W+\mbox{lower terms}$, $[U,W]=-2UV+\mbox{lower terms}$, and $[V,W]=V^2+nU^{n-1}+\mbox{lower terms}$. We wish to find the possible expressions for these commutators satisfying the Jacobi identity. But we may clearly replace $U$ (resp. $V,W$) by an equivalent element modulo the scalars (resp. $A_{2n-4},A_{2n-2}$).
Hence, after substituting for $W$, we assume that $[U,V]=2W$. Now $[U,W]=-2UV+\alpha W+\beta V+p(U)$ for some polynomial $p\in{\mathbb C}[t]$ of degree $\leq (n+1)/2$. Substituting $(U-\beta/2)$ for $U$, we may assume that $\beta=0$. Let $p=tq+\gamma$ for $q\in{\mathbb C}[t],\gamma\in{\mathbb C}$. Replacing $V$ by $(V-q(U)/2)$, we may assume that $[U,W]=-2UV+\alpha W+\gamma$. Finally, there exist polynomials $P,m_1,m_2\in{\mathbb C}[t]$ with $m_1$ of degree $\leq (n-3)/2$, $m_2$ of degree $\leq (n-2)/2$ and $P$ with leading term $nt^{n-1}$ such that $[V,W]=V^2+P(U)+m_1(U)W+m_2(U)V$. But the Jacobi identity now requires that $[U,[V,W]]=[V,[U,W]]$, hence that $m_1=m_2=0$ and $\alpha=2$.
\begin{definition} Let $P(t)$ be a polynomial of degree $(n-1)$ and let $\gamma\in{\mathbb C}$. The algebra $H(P,\gamma)$ has generators $U,V,W$ and relations $[U,V]=2W$, $[U,W]=-2UV+2W+\gamma$ and $[V,W]=V^2+P(U)$. \end{definition}
This definition does not require $P$ to have leading term $nt^{n-1}$; but by scaling generators $(U,V,W)\mapsto (U,\alpha V,\alpha W)$ we can easily see that $H(P,\gamma)$ is isomorphic to $H(\alpha^2 P,\alpha\gamma)$. It follows immediately from the above discussion:
\begin{lemma} Let $A$ be a noncommutative deformation of $({\mathbb C}[X,Y,Z],\{.\, ,.\}_\phi)$, where $\phi=X^n+XY^2+Z^2$. Then $A$ is isomorphic (as a filtered algebra) to $H(P,\gamma)$ for some polynomial $P(t)=nt^{n-1}+\ldots$ and $\gamma\in{\mathbb C}$. \end{lemma}
We now describe the centre of $H(P,\gamma)$. To do this we need a little preparation. By definition $[U,V]=2W$ and $[U,W]=-2UV+2W+\gamma$. It follows that there exist polynomials $\alpha_n,\beta_n\in{\mathbb C}[t]$ such that $[U^n,V]=\alpha_n(U)[U,V]+\beta_n(U)[U,W]$. Indeed, then $$\begin{array}{rl} [U^{n+1},V]= & U^n[U,V]+(\alpha_n(U)[U,V]+\beta_n(U)[U,W])U \\ = & (U^n+\alpha_n(U)+2U\beta_n(U))[U,V]+((U-2)\beta_n(U)-2\alpha_n(U))[U,W]\end{array}$$ Hence $\alpha_{n+1}=t^n+t\alpha_n+2t\beta_n$ and $\beta_{n+1}=(t-2)\beta_n-2\alpha_n$. To solve these difference equations, let $\iota:{\mathbb C}[t]\hookrightarrow{\mathbb C}[s]$ be the algebra embedding which sends $t$ to $-s(s+1)$. For $f\in{\mathbb C}[t]$ (temporarily) denote by $\overline{f}$ the image $\iota(f)$. Let $\rho_n=\overline{\alpha_n}-s\overline{\beta_n}$ and let $\mu_n=\overline{\alpha_n}+(s+1)\overline{\beta_n}$. A straightforward calculation shows that $\rho_{n+1}=(-s(s+1))^n-s(s-1)\rho_n$ and $\mu_{n+1}=(-s(s+1))^n-(s+1)(s+2)\mu_n$. But $\rho_1=\mu_1=1$, hence $\rho_n=((-s(s-1))^n-(-s(s+1))^n)/2s$ and $\mu_n=((-s(s+1))^n-(-(s+1)(s+2))^n)/2(s+1)$. It would be straightforward to write down explicit expressions for $\alpha_n$ and $\beta_n$, but this will suffice for our purposes. Let $\rho,\mu:{\mathbb C}[s]\rightarrow{\mathbb C}[s]$ be the linear maps given by $\rho(p)=(p(-s)-p(s))/2s$ and $\mu(p)=(p(-(s+1))-p(s+1))/2(s+1)$. We note that for any $f\in{\mathbb C}[t]$ there exist unique polynomials $\alpha(f),\beta(f)\in{\mathbb C}[t]$ such that $\overline{\alpha(f)}-s\overline{\beta(f)}=\rho(\overline{f})$. Indeed, by the above discussion there exist unique $\alpha(f),\beta(f)$ such that $\overline{\alpha(f)}-s\overline{\beta(f)}=\rho(\overline{f})$ and $\overline{\alpha(f)}+(s+1)\overline{\beta(f)}=\mu(\overline{f})$. But $-s(s+1)$ is stable under the algebra endomorphism of ${\mathbb C}[s]$ which sends $s$ to $-(s+1)$, hence the first condition implies the second. Hence we introduce the linear maps $\alpha,\beta:{\mathbb C}[t]\rightarrow{\mathbb C}[t]$ such that $\overline{\alpha(f)}-s\overline{\beta(f)}=\rho(\overline{f})$ for all $f\in{\mathbb C}[t]$.
\begin{lemma}\label{ucomms} (a) $[f(U),V]=\alpha(f)(U)[U,V]+\beta(f)(U)[U,W]$.
(b) $[f(U),W]=-U\beta(f)(U)[U,V]+(\alpha(f)+\beta(f))(U)[U,W]$. \end{lemma}
\begin{proof} The first assertion is an immediate consequence of the discussion in the paragraph above. For (b), we note that $2[U,W]=[U,[U,V]]=2\alpha(f)(U)[U,W]-2U\beta(f)(U)[U,V]+2\beta(f)(U)[U,W]$. \end{proof}
\begin{lemma}\label{centreisp} Let $Q$ be a monic polynomial, unique up to addition of scalars, such that $Q(-s(s-1))-Q(-s(s+1))=(s-1)P(-s(s-1))+(s+1)P(-s(s+1))$ and let $\Omega=Q(U)+UV^2+W^2-2WV-\gamma V\in H=H(P,\gamma)$. Then $Z(H)={\mathbb C}[\Omega]$. \end{lemma}
\begin{proof} Let $h$ be an element of $H$ of the form $Q(U)+UV^2+W^2+\alpha WV+\beta V^2+p_1(U)W+p_2(U)V$, where $Q\in{\mathbb C}[t]$ is monic of degree $n$ and $p_1,p_2\in{\mathbb C}[t]$ are polynomials of degrees $\leq (n-1)/2$ and $\leq n/2$ respectively. To find the possible $Q,p_1,p_2$ such that $h\in Z(H)$ we have only to find the conditions under which $[z,U]=[z,V]=0$ (since then $[z,[U,V]]=0$, hence $z\in Z(H)$). It is easy to see that $$[U,UV^2+W^2]=2UWV-2WUVv+4W^2+2\gamma W=[U,2WV+\gamma V]$$ It follows that $[U,h]=0$ if (and only if, though this is unnecessary) $h=Q(U)+UV^2+W^2-2WV-\gamma V$, for some monic polynomial $Q$ of degree $n$. Assume $h$ is of this form. To determine when $[h,V]=0$ we apply Lemma \ref{ucomms}. By a straightforward calculation, $$\begin{array}{rl}[UV^2+W^2-2WV,V]= & -[V,[V,W]]-2P(U)W+[P(U),W] \\ = & [P(U),V]+[P(U),W]-P(U)[U,V]\end{array}$$ But therefore $h\in Z(H)$ if and only if $[Q(U)+P(U),V]+[P(U),W]=P(U)[U,V]$. By Lemma \ref{ucomms}, $\alpha(Q)=P+t\beta(P)-\alpha(P)$ and $\beta(Q)=-(\alpha +2\beta)(P)$. It follows that $\overline{\alpha(Q)}-s\overline{\beta(Q)}=\overline{P}+(s-1)(\alpha-s\beta)(\overline{P})$. Hence $Q$ is the unique polynomial modulo addition of scalars such that $\overline{Q}(-s)-\overline{Q}(s)=(s-1)\overline{P}(-s)+(s+1)\overline{P}(s)$.
This proves that $\Omega=Q(U)+UV^2+W^2-2WV-\gamma V\in Z(H)$. Let $B$ be the Poisson algebra $({\mathbb C}[X,Y,Z],\{.\, ,.\}_\phi)$. It is well-known (and easy to check) that the Casimir elements $\mathop{\rm Cas}\nolimits B=\{ f\in B\,:\, \{f,g\}=0\,\forall g\in B\}={\mathbb C}[\phi]$. It is easy to see that if $h\in Z(H)$ then $\mathop{\rm gr}\nolimits h\in\mathop{\rm Cas}\nolimits B$. Suppose therefore that $h\in Z(H)$, but $h\not\in {\mathbb C}[\Omega]$. We may assume that the degree of $h$ is minimal subject to this condition. Then $\mathop{\rm gr}\nolimits h\in Cas B$, hence $\mathop{\rm gr}\nolimits h=\xi\phi^i$ for some $i$ and some $\xi\in{\mathbb C}^\times$. But now $h-\xi\Omega^i\in Z(H)$ has degree strictly less than $h$, which contradicts our original assumption. \end{proof}
We note that the condition on $Q,P$ is equivalent to the condition: \begin{eqnarray}\label{QP} Q(-s(s+1))+(s+1)P(-s(s+1)) \mbox{ is an even polynomial in $s$}\end{eqnarray} Moreover, for each monic polynomial $Q(t)$ there is a unique $P(t)$ satisfying (\ref{QP}), necessarily with leading term $nt^{n-1}$ (where $n$ is the degree of $Q$).
\begin{definition}\label{DQgamma} Let $Q(t)$ be a polynomial of degree $n$ and let $\gamma\in{\mathbb C}$. We define $D(Q,\gamma)$ to be the associative algebra with generators $u,v,w$ and relations: $$[u,v]=2w,\;\;[u,w]=-2uv+2w+\gamma,\;\;[v,w]=v^2+P(u)\;\;\mbox{and}\;\;Q(u)+uv^2+w^2-2wv-\gamma v=0$$
where $P(t)$ is the unique polynomial of degree $(n-1)$ such that $$Q(-s(s-1))-Q(-s(s+1))=(s-1)P(-s(s-1))+(s+1)P(-s(s+1)).$$ \end{definition}
In common with the convention for type $A$, we have not assumed that $Q$ is monic in the above definition. But the change of generators $(u,v,w)\mapsto (u,\xi v,\xi w)$ gives a natural isomorphism $D(Q,\gamma)\cong D(\xi^2 Q,\xi\gamma)$. Hence any such algebra $D(Q,\gamma)$ is isomorphic to some $D(Q_0,\gamma_0)$ with $Q_0$ monic.
\section{The Isomorphism Problem}
Recall that if $A$ is any ${\mathbb Z}$-filtered algebra, then there is a uniquely defined degree function on non-zero elements of $A$: $\deg x=\mathop{\rm min}\nolimits_{x\in A_i}i$. Fix a monic polynomial $Q(t)$ of degree $n\geq 3$ and $\gamma\in{\mathbb C}$, and let $A=D(Q,\gamma)$. Let $P(t)$ be the unique polynomial such that $Q(-s(s+1))+(s+1)P(-s(s+1))$ is even in $s$. By construction $A$ is a ${\mathbb Z}$-filtered algebra such that $u$ has degree 4, $v$ has degree $2 n-2$ and $w$ has degree $2n$. Specifically, $\{ u^iv^jw^\epsilon\,:\, i,j\in{\mathbb Z}\geq 0,\epsilon\in\{ 0,1\}\}$ is a basis for $A$ and $\deg \sum_{i,j,\epsilon}a_{ij\epsilon}u^iv^jw^\epsilon=\max_{a_{ij\epsilon}\neq 0}(4i+(2n-2)j+2n\epsilon)$.
However, for any $N>n$ we can also define a filtration on $A$ with degree function $\deg\sum_{i,j,\epsilon}a_{ij\epsilon}u^iv^jw^\epsilon=\max_{a_{ij\epsilon}\neq 0}(4i+(2N-2)j+2N\epsilon)$. To see this we have only to check that if $x,y\in A$ then $\deg xy\leq \deg x+\deg y$. Hence suppose $x=\sum a_{ij\epsilon}u^iv^jw^\epsilon$ and $y=\sum b_{ij\epsilon}u^iv^jw^\epsilon$. Then $4i+(2N-2)j+2N\epsilon\leq\deg x$ for all $a_{ij\epsilon}\neq 0$, and $4k+(2N-2)l+2N\eta\leq\deg y$ for all $b_{kl\eta}\neq 0$. It follows that $4(i+k)+(2N-2)(j+l)+2N(\epsilon+\eta)\leq\deg x+\deg y$ for all $a_{ij\epsilon}b_{kl\eta}\neq 0$. But $xy=\sum a_{ij\epsilon}b_{kl\eta}(u^{i+k}v^{j+l}w^{\epsilon+\eta}-u^{i+k}v^j[v^{l},w^\epsilon]w^\eta-u^i[u^k,v^jw^\epsilon]v^lw^\eta)$. Hence by induction on $\deg x,\deg y$ we have only to show that the commutator relations $[u,v]=2w$, $[u,w]=-2uv+2w+\gamma$, $[v,w]=v^2+P(u)$ and the substitution $w^2=-Q(u)-uv^2-2vw+2v^2+2P(u)+\gamma v$ are of non-positive degree, that is, the terms on the right are of equal or lower degree than each of the terms on the left. This is easily checked. It will be extremely useful to us to consider the `limit as $N$ tends to infinity' of these filtrations. Hence consider the additive monoid of pairs $(a,b)$ of non-negative integers, with the lex ordering $(a,b)>(a',b')$ if and only if $a>a'$ or $a=a'$ and $b>b'$. Let $A_0^{(0)}={\mathbb C}\subset A$, let $A_a^{(b)}$ be the subspace of $A$ spanned by all monomials of the form $u^iv^jw^\epsilon$ with $(j+\epsilon,2i+\epsilon)\leq (a,b)$ and let $A_a^{(\infty)}=\cup_{b\geq 0} A_a^{(b)}$. (We assume that $\epsilon\in\{ 0,1\}$, although this isn't strictly necessary.) It is straightforward to check that the commutation relations $[u,v]=2w$, $[u,w]=-2uv+2w+\gamma$ and $[v,w]=v^2+P(u)$ satisfy $\deg [x,y]=\deg x+\deg y-(0,1)$ and that the equality $w^2=-uv^2-Q(u)+2wv+\gamma v$ replaces $w^2$ be a term of equal degree $(2,2)$ (congruent to $-uv^2$ modulo $A_2^{(1)}$). It follows by the argument above that $A=\cup_{a,b\geq 0} A_a^{(b)}$ is a well-defined filtration of $A$, which we call the {\it limit filtration}. It is easy to see that the corresponding graded algebra is isomorphic to ${\mathbb C}[X,Y,Z]/(XY^2+Z^2)$, where $X$ has degree $(0,2)$, $Y$ has degree $(1,0)$ and $Z$ has degree $(1,1)$.
Until further notice we fix the limit filtration on $A$. It turns out to be significantly easier for us to calculate using the monomials $u^iwv^{j-1}$ rather than $u^iv^{j-1}w$. (This does not effect our definition of the filtration since $wv=vw+$terms of lower degree.) Hence we express elements of $A$ in terms of the basis $\{ u^iw^\epsilon v^j\,:\, i,j\geq 0,\epsilon=0,1\}$. Since any subset of the ordered set ${\mathbb Z}_{\geq 0}\times {\mathbb Z}_{\geq 0}$ has a minimal element, there is a well-defined degree function on non-zero elements of $A$. It is easy to see moreover that $u^iw^\epsilon v^j$ has degree $(a,2b)$ if and only if $i=b$, $j=a$ and $\epsilon=0$, and has degree $(a,2b+1)$ if and only if $a>0$, $\epsilon=1$, $i=b$ and $j=a-1$. Hence each summand in the grading of $\mathop{\rm gr}\nolimits A$ is of dimension 1. We will write $x=\xi u^iw^\epsilon v^j+$lower terms to mean that $x$ is congruent to $\xi u^iw^\epsilon v^j$ modulo $\cup_{(a,b)<(j+\epsilon,2i+\epsilon)}A_a^{(b)}$ (implicitly assuming $\xi\neq 0$). We refer to $\xi u^iw^\epsilon v^j$ as the `leading term' in $x$.
Note that $[u,v^m]=2mwv^{m-1}+$lower terms and $[u,wv^{m-1}]=-2muv^m+$lower terms, thus the cosets of $(\mathop{\rm ad}\nolimits u)^j(v^m)$, $j\geq 0$ form a basis for $A_m^{(\infty)}/A_{m-1}^{(\infty)}$. Define polynomials $F_m\in{\mathbb C}[S,T],\; m\in{\mathbb Z}_{\geq 0}$ by: $F_0=S$ and $F_m=(S^2-2m^2 S+m^2(m^2-1) + 4m^2T)$ for $m\geq 1$. Clearly $\mathop{\rm ad}\nolimits u$ and left multiplication by $u$, denoted $l_u$, commute.
\begin{lemma}\label{Fprod} Let $x\in A$. Then there exists $m$ such that $\prod_{i=0}^m F_i(\mathop{\rm ad}\nolimits u,l_u)(x)=0$. \end{lemma}
\begin{proof} Since $\mathop{\rm ad}\nolimits u$ and $l_u$ preserve each of the subspaces $A_m^{(\infty)}$, it is enough to show that $F_m(\mathop{\rm ad}\nolimits u,l_u)(x)\in A_{m-1}^{(\infty)}$ for any $x\in A_m^{(\infty)}$. But $A_m^{(\infty)}/A_{m-1}^{(\infty)}$ is spanned by (the cosets of) $(\mathop{\rm ad}\nolimits u)^i(v^m)$, $i\in{\mathbb Z}_{\geq 0}$, hence it will suffice to show that $F_m(\mathop{\rm ad}\nolimits u,l_u)(v^m)\in A_{m-1}^{(\infty)}$.
Clearly $$\begin{array}{rl}[u,v^m]= & 2wv^{m-1}+2vwv^{m-2}+\ldots+2v^{m-1}w \\ = & 2mwv^{m-1}+\sum_{j=1}^{m-1}[v^j,w]v^{m-1-j}\end{array}$$ But since $[v,w]\equiv v^2\;(\mathop{\rm mod}\nolimits A_0^{(\infty)})$, we deduce that $[v^j,w]v^{m-1-j}\equiv jv^m\;(\mathop{\rm mod}\nolimits A_{m-2}^{(\infty)})$. It follows that $[u,v^m]\equiv 2mwv^{m-1}+m(m-1)v^m\;(\mathop{\rm mod}\nolimits A_{m-2}^{(\infty)})$. Now $[u,wv^{m-1}]=-2uv^m+2wv^{m-1}+\gamma v^{m-1}+w[u,v^{m-1}]$. Thus $[u,wv^{m-1}]\equiv -2uv^m+((m-1)(m-2)+2)wv^{m-1}+\gamma v^{m-1}+2(m-1)w^2v^{m-2}\;(\mathop{\rm mod}\nolimits A_{m-2}^{(\infty)})$. But $w^2\equiv -uv^2+2wv+\gamma v\;(\mathop{\rm mod}\nolimits A_0^{(\infty)})$ by the defining relations for $A$. Hence $[u,wv^{m-1}]\equiv -2muv^m+m(m+1)wv^{m-1}+(2m-1)\gamma v^{m-1}\;(\mathop{\rm mod}\nolimits A_{m-2}^{(\infty)})$.
This proves the statement about $v^m$: in fact we have shown that $F_m(\mathop{\rm ad}\nolimits u,l_u)(v^m)\equiv 2m(2m-1)\gamma v^{m-1}\;(\mathop{\rm mod}\nolimits A_{m-2}^{(\infty)})$. \end{proof}
\begin{lemma}\label{prodform} Let $P(S,T)=\prod_{i=0}^m F_i(S,T)$. If $P$ is written in the form $\sum_{i=0}^{2m+1} a_i(T)S^i$, then $a_{2m+1}=1$ and $\deg a_{2m+1-i}\leq i/2$. \end{lemma}
\begin{proof} Let ${\cal S}$ be the set of all polynomials in ${\mathbb C}[S,T]$ of the form $\sum_{i=0}^N a_i(T)S^i$, where $a_N\neq 0$ and $\deg a_{N-i}\leq i/2$. The product of any two elements of ${\cal S}$ is also in ${\cal S}$, since the coefficient of $S^i$ in $(\sum a_j(T)S^j)(\sum b_l(T)S^l)$ is $\sum_{j+l=i}a_j(T)b_l(T)$. But clearly $F_i\in{\cal S}$ for all $i$, hence $P\in{\cal S}$. \end{proof}
Denote by $\mathop{\rm gr}\nolimits_{lim} A$ the graded algebra of $A$ corresponding to the limit filtration, identified with ${\mathbb C}[X,Y,Z]/(XY^2+Z^2)$. The Poisson bracket on $\mathop{\rm gr}\nolimits_{lim}A$ satisfies $\{ X,Y\}=2Z$, $\{ X,Z\}=-2XY$, $\{ Y,Z\}=Y^2$.
\begin{lemma}\label{monomials} Let $x$ be a monomial in $X,Y,Z$. Then unless $x=Y^b$ or $x=X^b$, there exists some monomial $y$ in $X,Y,Z$ such that $\{ x,.\}^M(y)\neq 0$ for all $M\geq 0$. \end{lemma}
\begin{proof} Since $Z^2=-XY^2$, we have only to prove the lemma in the case where $x=X^aY^bZ^\epsilon$ with $\epsilon\in\{ 0,1\}$. Suppose $x'=x^r$ for some $r\geq 2$. Then $\{ x',.\}(y)=rx^{r-1}\{ x,y\}$. Hence $\{ x',.\}^M(y)=r^Mx^{M(r-1)}\{ x,.\}^M(y)$. It follows that we need only prove the lemma in the case where $x$ cannot be expressed as a power of any other monomial. Note that if $(i,j)$ is the degree of $x$ in $\mathop{\rm gr}\nolimits_{lim} A$ then this holds if and only if $i$ and $j$ are coprime.
Suppose first of all that $x=X^aY^b$ such that $b$ and $2a$ are coprime (and $ab\neq 0$). By calculation $\{ X^aY^b,X^cY^d\}=2(ad-bc)X^{a+c-1}Y^{b+d-1}Z$. Moreover, $\{ X^aY^b,X^cY^{d-1}Z\}=(b(2c+1)-2ad))X^{a+c}Y^{b+d}$. Hence by our condition on $x$, $\{ x,y\}=0$ if and only if $y=x^r$ for some $r$. We claim that $\{ x,.\}^{2i+1}(X^cY^d)=0$ if and only if $(c,d)=(ka+j,kb)$ for some $k\geq 0$, $0\leq j\leq i$ and $\{ x,.\}^{2i}(X^cY^{d-1}Z)=0$ if and only if $(c,d)=(ka+j,kb)$ for some $k\geq 0$, $0\leq j\leq (i-1)$. This is true for $i=0$ by the above calculations. Hence suppose we know our claim to be true for $(i-1)$. Then $\{ x,.\}^{2i}(X^cY^{d-1}Z)=0$ if and only if $(b(2c+1)-2ad)\{ x,.\}^{2i-1}(X^{a+c}Y^{b+d})=0$. By the induction hypothesis, this is true if and only if $(c,d)=(ka+j,kb)$ for some $k\geq 0$ and some $j$, $0\leq j\leq (i-1)$. This proves the induction step for $X^cY^{d-1}Z$. But now $\{ x,.\}^{2i+1}(X^cY^d)=0$ if and only if $2(ad-bc)\{ x,.\}^{2i}(X^{a+c-1}Y^{b+d-1}Z)=0$. It follows from the above step that $\{ x,.\}^{2i+1}(X^cY^d)=0$ if and only if $(c,d)=k(a,b)$ or $(c-1,d)=(ka+j,kb)$ for some $j$, $0\leq j\leq i-1$. This proves our claim. Thus $\{ x,.\}^M(y)=0$ for some $M$ if and only if $y=X^{ka+c}Y^{kd}$ or $y=X^{kc+a}Y^{kd-1}Z$ for some $k\in{\mathbb N}$. Therefore (for example) $\{ x,.\}^M(Y)\neq 0$ for all $M\geq 0$.
Suppose now that $x=X^aY^{b-1} Z$ where $b$ and $(2a+1)$ are coprime. By the above $\{ x,X^cY^d\}=(2bc-d(2a+1))X^{a+c}Y^{b+d}$. It follows that $\{ x,X^cY^{d-1}Z\}=(b(2c+1)-d(2a+1))X^{a+c}Y^{b+d-1}Z$. Thus once more $\{ x,y\}=0$ if and only if $y=x^r$ for some $r$. If $a$ and $b$ are not both zero, then it follows that $\{ x,.\}^M(Z)$ is a non-zero multiple of $X^{Ma}Y^{Mb+M}Z$ for each $M\geq 0$. On the other hand, if $x=Z$ then $\{ x,.\}^M(Y)$ is a non-zero multiple of $Y^{M+1}$ for each $M\geq 1$. This completes the proof. \end{proof}
\begin{rk} We note that it follows from the proof of Lemma \ref{monomials}, if $x$ and $y$ are monomials of coprime degrees $(m,i)$ and $(m',i')$, then $\{ x,y\}=\pm (mi'-im') z$, where $z$ is a monomial of degree $(m+m',i+i'-1)$. \end{rk}
\begin{lemma}\label{reducetov} Suppose $f$ is an element of $A$ satisfying the condition that for any $a\in A$ there exists $m$ such that $\prod_{i=0}^m F_i(\mathop{\rm ad}\nolimits f,l_f)(a)=0$. Then either $f\in{\mathbb C}[u]$ or there exist $r$ and $\xi\neq 0$ such that $f=\xi v^r+$lower terms. \end{lemma}
\begin{proof} Let $f$ be such an element, let $a\in A$ and suppose $\prod_0^m F_i(\mathop{\rm ad}\nolimits f,l_f)(a)=0$. Recall by Lemma \ref{prodform} that $P(S,T)=\prod_{i=0}^m F_i(S,T)$ is of the form $S^{2m+1}+a_{2m}S^{2m}+a_{2m-1}(T)S^{2m-1}+\ldots +a_0(T)$, where $\deg a_{2m+1-i}\leq i/2$. Suppose $\mathop{\rm gr}\nolimits f=x$ is not of the form $\xi X^i$ or $\xi Y^i$. Then by Lemma \ref{monomials} there exists $y\in\mathop{\rm gr}\nolimits_{lim} A$ such that $\{x,.\}^M(y)\neq 0$ for all $M\geq 0$. Let $a\in A$ be such that $\mathop{\rm gr}\nolimits a=y$. Then it is easy to see that $\deg(\mathop{\rm ad}\nolimits f)^m(y)=\deg \{x,.\}^m(y)$. Let $\deg\mathop{\rm gr}\nolimits f=(r,s)$ with $r>0$. Then it follows that $\deg (\mathop{\rm ad}\nolimits f)^{2m+1}(x)=((2m+1)r+c,(2m+1)(s-1)+d)$. But each remaining term in the equation for $F_m(\mathop{\rm ad}\nolimits f,l_f)(x)$ is of strictly smaller degree. Hence $P(\mathop{\rm ad}\nolimits f,l_f)(a)\neq 0$, which contradicts the assumption on $f$. \end{proof}
Our approach here is similar to that of \cite{bav-jord} in that we exploit the Poisson structure on $\mathop{\rm gr}\nolimits_{lim} A$ to pin down the possible images of the minimal degree element $u\in A$. However, $u$ is not {\it strictly semisimple} in the sense of \cite[3.3]{bav-jord}. To determine all possible isomorphisms $D(Q_2,\gamma_2)\rightarrow D(Q_1,\gamma_1)$ we carry out a case-by-case study of the possible images of the standard generators for $D(Q_2,\gamma_2)$.
Hence let $Q_2$ (resp. $Q_1$) be monic of degree $N\geq 3$ (resp. $n\geq 3$) and let $f,g,h$ be the respective images of the standard generators for $D(Q_2,\gamma_2)$ in $D(Q_1,\gamma_1)$. Assume until further notice that $f=\xi v^r+$ lower terms. Recall from the proof of Lemma \ref{Fprod} that $F_1(\mathop{\rm ad}\nolimits f,l_f)(g)=2\gamma_2$. Thus $F_1(\mathop{\rm ad}\nolimits f,l_f)(2h)=F_1(\mathop{\rm ad}\nolimits f,l_f)([f,g])=[f,F_1(\mathop{\rm ad}\nolimits f,l_f)(g)]=0$. It follows that either $g=\xi' wv^{s-1}+$lower terms, or $g=\xi' v^s+$lower terms, for some $\xi'\neq 0$ and $s$. Similarly, either $h=\xi'' wv^{t-1}+$lower terms, or $h=\xi'' v^t+$lower terms, for some $\xi''\neq 0$ and $t$. By considering the equalities $[f,g]=2h$, $[f,h]=-2fg+2h+\gamma_2$ and $Q_2(f)+fg^2+h^2=2hg+\gamma_2 g$, we obtain the following exclusive list of possibilities:
(i) $g=\xi' wv^{(N/2-1)r-1}+$lower terms, $h=\xi'' v^{Nr/2}+$lower terms, where $\xi''^2+\xi^N=0$,
(ii) $g=\xi' v^{(N-1)r/2}+$lower terms, $h=\xi'' wv^{(N-1)r/2-1}+$lower terms, where $\xi'^2+\xi^{N-1}=0$,
(iii) $g=\xi' v^{(N-1)r/2}+$lower terms, $h=\xi'' v^t+$lower terms, where $(N-1)r/2<t<Nr/2$ and $\xi'^2+\xi^{N-1}=0$,
(iv) $g=\xi' v^s+$lower terms, $h=\xi'' v^{Nr/2}+$lower terms, where $(N/2-1)r<s<(N-1)r/2$ and $\xi''^2+\xi^N=0$,
(v) $g=\xi' v^s+$lower terms, $h=\xi'' v^{s+r/2}+$lower terms, where $s>(N-1)r/2$ and $\xi\xi'^2+\xi''^2=0$,
(vi) $g=\xi' v^{(N-1)r/2}+$lower terms, $h=\xi'' v^{Nr/2}+$lower terms, where $\xi^N+\xi\xi'^2+\xi''^2=0$.
To deal with these cases, we examine in detail the monomials in $f,g,h$ of highest degree in the expression for $\prod_{i=0}^{m-1} F_i(\mathop{\rm ad}\nolimits f,l_f)(g^m)$, and similarly for $hg^{m-1}$. We will show that the degree of any expression in $f,g,h$ is too high to be equal to $u$ unless $N=3$, where the only possible case is (ii) with $r=1$.
From now on, all monomials in $f,g,h$ will be of the form $f^ih^\epsilon g^j$ with $\epsilon\in\{ 0,1\}$. For each monomial $x$ in $f,g,h$ and for each non-negative integer $r$, let $J_r(x)$ denote the (finite-dimensional) subspace of $A$ generated by all monomials $f^ih^\epsilon g^j$ with $j+\epsilon\leq r$ and $\deg f^ih^\epsilon g^j<\deg x$.
\begin{lemma}\label{commprep} Let $f,g,h$ be as in one of the cases (i)-(vi) above, and let $m\geq 2$.
(a) $[f,g^m]=2mhg^{m-1}+m(m-1)g^m+m(m-1)Nf^{N-1}g^{m-2}+a$, for some $a\in J_{m-2}(f^{N-1}g^{m-2})$,
(b) $[f,hg^{m-1}]=-2mfg^m-2(m-1)f^Ng^{m-2}+m(m+1)hg^{m-1}+(2m-1)\gamma' g^{m-1}+(m-1)(m-2)Nf^{N-1}hg^{m-3} +a'$, where $a'\in J_{m-2}(f^{N-1}hg^{m-3})$. \end{lemma}
\begin{proof} Clearly $$[f,g^m]= 2\sum_{j=0}^{m-1} g^jhg^{m-1-j} = 2mhg^{m-1}+\sum_{j=0}^{m-1}[g^j,h]g^{m-1-j}$$ and $[g^j,h]=\sum_{l=0}^{j-1}g^l[g,h]g^{j-1-l}$. Moreover, $g^l[g,h]g^{j-1-l}=g^{j+1}+Ng^lf^{N-1}g^{j-1-l}=g^{j+1}+Nf^{N-1}g^{j-1}+b_l$ for some $b_l\in J_{j-1}(f^{N-1}g^{j-1})$. It follows that $[g^j,h]=jg^{j+1}+jNf^{N-1}g^{j-1}+b$ for some $b\in J_{j-1}(f^{N-1}g^{j-1})$. But then clearly $bg^{m-1-j}\in J_{m-2}(f^{N-1}g^{m-2})$. We deduce that $[f,g^m]\equiv 2mhg^{m-1}+m(m-1)g^m+m(m-1)Nf^{N-1}g^{m-2}\; (\mathop{\rm mod}\nolimits J_{m-2}(f^N g^{m-2}))$.
For (b), $[f,hg^{m-1}]=[f,h]g^{m-1}+h[f,g^{m-1}]$. By the definition of $D(Q_2,\gamma_2)$, $[f,h]g^{m-1}=-2fg^m+2hg^{m-1}+\gamma_2 g^{m-1}$. Moreover, $$h[f,g^{m-1}]=2(m-1)h^2g^{m-2}+(m-1)(m-2)hg^{m-1}+(m-1)(m-2)Nf^{N-1}hg^{m-3}+b'$$for some $b'\in hJ_{m-3}(f^{N-1}g^{m-3})\subseteq J_{m-2}(f^{N-1}hg^{m-3})$. The result now follows from the equality $h^2=-Q_2(f)-fg^2+2hg+\gamma_2 g$. \end{proof}
\begin{corollary}\label{Fiform} (a) If $m\geq 2$ then there exists $a\in J_m(fg^m)+J_{m-2}(f^Ng^{m-2})$ such that $$F_i(\mathop{\rm ad}\nolimits f,l_f)(g^m)=-4(m^2-i^2)fg^m-4m(m-1)f^Ng^{m-2}+a$$
(b) If $m\geq 3$ then there exists $a'\in J_m(fhg^{m-1})+J_{m-2}(f^Nhg^{m-3})$ such that $$F_i(\mathop{\rm ad}\nolimits f,l_f)(hg^{m-1})=-4(m^2-i^2)fhg^{m-1}-4(m-1)(m-2)f^Nhg^{m-3}+a'$$ \end{corollary}
\begin{proof} By Lemma \ref{commprep}: $$[f,[f,g^m]]=[f,2mhg^{m-1}+m(m-1)g^m+m(m-1)Nf^{N-1}g^{m-1}+a_0]$$ where $a_0\in J_{m-2}(f^Ng^{m-2})$. But then clearly $[f,g^m]\in J_m(fg^m)$ and $[f,f^{N-1}g^{m-2}],[f,a_0]\in J_{m-2}(f^Ng^{m-2})$. Applying Lemma \ref{commprep} again, we see that $[f,hg^{m-1}]= -2mfg^m-2(m-1)f^N g^{m-2}+a_1$ for some $a_1\in J_m(fg^m)+J_{m-2}(f^N g^{m-2})$. Hence the result for $F_i(\mathop{\rm ad}\nolimits f,l_f)(g^m)$ follows.
Similarly, Lemma \ref{commprep} implies that \begin{eqnarray}\nonumber\lefteqn{[f,[f,hg^{m-1}]]=[f,-2mfg^m-2(m-1)f^Ng^{m-2}} \\ \nonumber & +m(m+1)hg^{m-1}+(2m+1)\gamma' g^{m-1}+(m-1)(m-2)Nf^{N-1}hg^{m-3}+a_2]\end{eqnarray} where $a_2\in J_{m-2}(f^{N-1}hg^{m-3})$. But it is immediate that $[f,hg^{m-1}],[f,g^{m-1}]\in J_m(fhg^{m-1})$ and $[f,f^{N-1}hg^{m-3}],[f,a_2]\in J_{m-2}(f^N hg^{m-3})$. Hence the result for $F_i(\mathop{\rm ad}\nolimits f,l_f)(hg^{m-1})$ follows by Lemma \ref{commprep}(a). \end{proof}
For ease of notation, let $P_i=\prod_{j=0}^i F_j(\mathop{\rm ad}\nolimits f,l_f)$ for the rest of this section. Corollary \ref{Fiform} allows us to describe the monomials in $f,g,h$ which are of highest degree in the expression for $P_i(g^m),P_i(hg^{m-1})$. We begin with cases (i) and (iv) listed after Lemma \ref{reducetov}. Here we use the notation $x=\chi f^ih^\epsilon g^j+$lower terms to mean that $x=\chi f^ih^\epsilon g^j+a$, where $a$ is a sum of monomials in $f,g,h$ each of lower degree than $f^ih^\epsilon g^j$.
\begin{lemma}\label{caseiandiv} Suppose $h=\xi'' v^{Nr/2}+$lower terms and either $g=\xi' wv^{(N/2-1)r-1}+$lower terms (case (i)) or $g=\xi' v^s+$lower terms, where $(N/2-1)r<s<(N-1)r/2$ (case (iv)). Then for any $m\geq 1$:
(a) $P_i(g^{2m})=\left\{\begin{array}{ll} \chi_{i} f^{iN}hg^{2m-2i-1}+\mbox{lower terms} & \mbox{if $0\leq i<m$,} \\ \chi_{i} f^{i+(m-1)(N-1)}hg+\mbox{lower terms} & \mbox{if $m\leq i<2m$.} \end{array}\right.$
(b) $P_i(g^{2m-1})=\left\{\begin{array}{ll} \chi_{i} f^{iN}hg^{2m-2i-2}+\mbox{lower terms} & \mbox{if $0\leq i<m$,} \\ \chi_{i} f^{i+(m-1)(N-1)}h+\mbox{lower terms} & \mbox{if $m\leq i<2m-1$.}\end{array}\right.$
(c) $P_i(hg^{2m-1})=\left\{\begin{array}{ll} \eta_{i} f^{(i+1)N}g^{2m-(2i+2)}+\mbox{lower terms} & \mbox{if $0\leq i<m$,} \\ \eta_{i} f^{i+1+m(N-1)}+\mbox{lower terms} & \mbox{if $m\leq i<2m$.}\end{array}\right.$
(d) $P_i(hg^{2m-2})=\left\{\begin{array}{ll} \eta_{i} f^{(i+1)N}g^{2m-(2i+3)}+\mbox{lower terms} & \mbox{if $0\leq i<m-1$,} \\ \eta_{i} f^{i+1+(m-1)(N-1)}g+\mbox{lower terms} & \mbox{if $m-1\leq i<2m-1$.}\end{array}\right.$
Here $\chi_{i}$ (resp. $\eta_{i}$) is a real number of sign $(-1)^i$ (resp. $(-1)^{i+1}$). \end{lemma}
\begin{proof} Our proof is by induction on $m$ and $i$. Since $P_0=\mathop{\rm ad}\nolimits f$, the lemma is true for $i=0$ by Lemma \ref{commprep} and the fact that $\deg f^N>\deg fg^2$. For $m=1$, this proves (b) and (d). By a direct calculation, $P_1(g^2)=-48fhg+$lower terms and $P_1(hg)=24f^{N+1}+$lower terms. Hence (a) and (c) are also true for $m=1$. We assume therefore that $m\geq 2$.
By Cor. \ref{Fiform}, $P_1(g^l)=[f,-4(l^2-1)fg^l-4l(l-1)f^Ng^{l-2}+a]$ for some $a\in J_l(fg^l)+J_{l-2}(f^Ng^{l-2})$. Let $\delta$ be equal to $(r,-1)$ in case (i), and equal to $(Nr/2-s,0)$ in case (iv). Then $\deg [f,g^l]=\deg g^l+\delta$ for any $l\geq 1$. Moreover, if $x$ is any monomial in $f,g,h$ and $\sum a_{ij\epsilon} f^ih^\epsilon g^j$ is the unique expression for $[f,x]$ in terms of monomials in $f,g,h$ then it follows from Lemma \ref{commprep} that each non-zero term $a_{ij\epsilon}f^ih^\epsilon g^j$ has degree less than or equal to $\deg x+\delta$. But therefore $[f,a]$ and $[f,fg^l]$ are both of degree less than $f^Nhg^{l-3}$. Hence $P_1(g^l)=-8l(l-1)(l-2)f^Nhg^{l-3}+$lower terms for any $l\geq 3$. This proves (a) and (b) for $i=1$. A direct calculation establishes that $P_1(hg^2)=144f^{N+1}g+$lower terms. Hence (d) is true for $m=2$ and $i=1$. We therefore consider $P_1(hg^{l-1})$ for $l\geq 4$. By Cor. \ref{Fiform}, $F_1(\mathop{\rm ad}\nolimits f,l_f)(hg^{l-1})=-4(l^2-i^2)fhg^{l-1}-4(l-1)(l-2)f^Nhg^{l-3}+a'$, where $a'\in J_l(fhg^{l-1})+J_{l-2}(f^Nhg^{l-3})$. The highest degree term here is $f^Nhg^{l-3}$. Moreover, $[f,f^Nhg^{l-3}]=-2(l-3)f^{2N}g^{l-4}+$lower terms, and $\deg f^{2N}g^{l-4}=\deg f^Nhg^{l-3}+\delta$. By the remarks above, $P_1(hg^{l-1})=8(l-1)(l-2)(l-3)f^{2N}g^{l-4}+$lower terms, which confirms (c) and (d) for $i=1$.
An equivalent statement for the Lemma can be formulated in terms of degrees (and leading coefficients) of the $P_i(g^l),P_i(hg^{l-1})$. Specifically: $$\deg P_i(g^{2m})-\deg g^{2m}=\left\{\begin{array}{ll} (2i+1)\delta & \mbox{if $i<m$,} \\ (2m-1)\delta+(i-m+1)(r,0) & \mbox{if $m\leq i<2m$.}\end{array}\right.$$ $$\deg P_i(g^{2m-1})-\deg g^{2m-1}=\left\{\begin{array}{ll} (2i+1)\delta & \mbox{if $i<m$,} \\ (2m-1)\delta+(i-m+1)(r,0) & \mbox{if $m\leq i<2m-1$.}\end{array}\right.$$ $$\deg P_i(hg^{2m-1})-\deg hg^{2m-1}=\left\{\begin{array}{ll} (2i+1)\delta & \mbox{if $i<m$,} \\ (2m-1)\delta+(i-m+1)(r,0) & \mbox{if $m\leq i<2m$.}\end{array}\right.$$ $$\deg P_i(hg^{2m-2})-\deg hg^{2m-2}=\left\{\begin{array}{ll} (2i+1)\delta & \mbox{if $i<m-1$,} \\ (2m-3)\delta+(i-m+2)(r,0) & \mbox{if $m-1\leq i<2m-1$.}\end{array}\right.$$
(We retain of course the assumption on the signs of the leading coefficients $\chi_i,\eta_i$.)
Assume therefore that $i\geq 2$ and that (a)-(d) are known to be true for all pairs $(m',i')$ with $m'<m$ or $m'=m$ and $i'<i$. By Cor. \ref{Fiform}, $F_i(\mathop{\rm ad}\nolimits f,l_f)(g^{2m})=-4(4m^2-i^2)fg^{2m}-8m(2m-1)f^Ng^{2m-2}+a$ for some $a\in J_{2m}(fg^{2m})+J_{2m-2}(f^Ng^{2m-2})$. But let $a=a_1+a_2$, where $a_1\in J_{2m}(fg^{2m})$ and $a_2\in J_{2m-2}(f^Ng^{2m-2})$. By the induction hypothesis and the remarks above, $\deg P_{i-1}(a_1)-\deg a_1\leq \deg P_{i-1}(fg^{2m})-\deg fg^{2m}$ and $\deg P_{i-1}(a_2)-\deg a_2\leq \deg P_{i-1}(f^Ng^{2m-2})-\deg f^Ng^{2m-2}$. Hence $P_i(g^{2m})=-4(4m^2-i^2)P_{i-1}(fg^{2m})-8m(2m-1)P_{i-1}(f^Ng^{2m-2})+$lower terms. If $i<m$, then by the induction hypothesis $P_{i-1}(fg^{2m})=\chi_{i-1}f^{N(i-1)+1}hg^{2m-2i+2}+$lower terms and $P_{i-1}(f^Ng^{2m-2})=\chi_{i-1}'f^{iN}hg^{2m-2i-1}+$lower terms, where $\chi_{i-1}$ and $\chi'_{i-1}$ are both of sign $(-1)^{i-1}$. It follows that $P_i(g^{2m})=-8m(2m-1)\chi_{i-1}f^{iN}hg^{2m-2i-1}+$lower terms, which proves the induction step for (a) in the case $i<m$. If $2m-1>i\geq m$, then by the induction hypothesis $P_{i-1}(fg^{2m})=\chi_{i-1}f^{i+(m-1)(N-1)}hg+$lower terms and $P_{i-1}(f^Ng^{2m-2})=\chi_{i-1}'f^{i+(m-1)(N-1)}hg+$lower terms, where $\chi_{i-1}$ and $\chi'_{i-1}$ are of sign $(-1)^{i-1}$. This proves the induction step in this case. Finally, $P_{2m-2}(f^Ng^{2m-2})=P_{2m-2}(a_2)=0$, hence $P_{2m-1}(g^{2m})=-4(4m-1)P_{2m-2}(fg^{2m})+$lower terms. But $P_{2m-2}(g^{2m})=\chi_{2m-2}f^{2m-1+(m-1)(N-1)}hg+$lower terms, where $\chi_{2m-2}$ is positive. It follows that $P_{2m-1}(g^{2m})=-4(4m-1)\chi_{2m-2}f^{m+(m-1)N}hg+$lower terms.
This proves the induction step for (a). The arguments for (b) and (c) are identical. For (d) we need to be slightly careful, for if $x$ is a monomial in $J_{2m-1}(fhg^{2m-2})$ then it is not necessarily true that $\deg P_{i-1}(x)-\deg x\leq \deg P_{i-1}(fhg^{2m-2})-\deg fhg^{2m-2}$ (and similarly for $J_{2m-3}(f^Nhg^{2m-4})$). In fact, one can see easily from the description of degrees above that if $x$ is a monomial in $J_{2m-1}(fhg^{2m-2})$ then $\deg P_{i-1}(x)-\deg x>\deg P_{i-1}(fhg^{2m-2})-\deg fhg^{2m-2}$ if and only if $i=m$ and $x=g^{2m-1}$ or $x=fg^{2m-1}$. However, in this case we still have that $\deg P_{i-1}(x)\leq \deg P_{i-1}(fhg^{2m-2})$. Similarly, if $x\in J_{2m-3}(f^Nhg^{2m-4})$ then $\deg P_{i-1}(x)\leq \deg P_{i-1}(f^Nhg^{2m-4})$. It follows that $P_i(hg^{2m-2})=-4(4m^2-i^2)P_{i-1}(fhg^{2m-2})-4(m-1)(m-2)P_{i-1}(f^Nhg^{2m-4})+$lower terms. The rest of the argument now proceeds as above. \end{proof}
\begin{corollary}\label{iiv} Suppose $g,h$ are as in Lemma \ref{caseiandiv}. Then there is no possible expression for $u$ in terms of $f,g,h$. \end{corollary}
\begin{proof} Suppose there exists such an expression $u=\sum_{i,j\geq 0,\epsilon\in\{ 0,1\} } a_{ij\epsilon} f^ih^\epsilon g^j$, and let $m=\max_{\{ a_{ij\epsilon}\neq 0\}}(j+\epsilon)$. Clearly $\deg g^m<\deg hg^{m-1}<\deg fg^m$. Moreover, $\deg P_{m-1}(g^m)<\deg P_{m-1}(hg^{m-1})<\deg P_{m-1}(fg^m)$ by Lemma \ref{caseiandiv}. Applying $P_{m-1}$ to both sides of the equation, we have the equality $P_{m-1}(u)=\sum_{j+\epsilon=m}a_{ij\epsilon}P_{m-1}(f^ih^\epsilon g^j)$. Thus $\deg P_{m-1}(u)\geq \deg P_{m-1}(g^m)$. To show that there can be no such expression for $u$, it will therefore suffice to show that $\deg P_{m-1}(u)<\deg P_{m-1}(g^m)$. If $m=1$, then $\deg P_{m-1}(u)=(r,1)<\deg P_{m-1}(g)=(Nr/2,0)$. Suppose therefore that $m\geq 2$, hence $\deg P_{m-1}(u)<((2m-1)r,0)$.
If $m$ is even, then by Lemma \ref{caseiandiv}, $P_{m-1}(g^m)=\chi_{m-1}f^{(m/2-1)(N+1)+1}hg+$lower terms, hence $\deg P_{m-1}(g^m)=(m(N+1)r/2,1)$ in case (i) and $\deg P_{m-1}(g^m)=(m(N+1)r/2+(s-(N/2-1)r),0)>(m(N+1)r/2,0)$ in case (iv). But $N+1\geq 4$, hence $\deg P_{m-1}(g^m)>((2m-1)r,0)$ in both cases. Similarly, if $m$ is odd then $P_{m-1}(g^m)=\chi_{m-1}f^{(m-1)(N+1)/2}h+$lower terms, hence $\deg P_{m-1}(g^m)=((m(N+1)-1)r/2,0)>((2m-1)r,0)$. This completes the proof. \end{proof}
Next we deal with case (v). This case is fairly straightforward, since the highest degree term in the expression for $h^2$ is $fg^2$. Once more we write $x=\chi f^ih^\epsilon g^j+$lower terms to mean $x=\chi f^ih^\epsilon g^j+a$, where $a$ is a sum of terms $a_{kl\epsilon}f^kh^\epsilon g^l$, each of degree strictly less than that of $f^ih^\epsilon g^j$.
\begin{lemma}\label{casev} Suppose $g=\xi' v^s+$lower terms and $h=\xi'' v^{s+r/2}+$lower terms, where $s>(N-1)r/2$ (hence $r$ is even). Then for any $0\leq i<m$, $P_i(g^m)=\chi_{i}f^ihg^{m-1}+$lower terms, where $\chi_{i}$ is a real number of sign $(-1)^i$ and $P_i(hg^{m-1})=\eta_{i}f^{i+1}g^m+$lower terms, where $\eta_{i}$ is a real number of sign $(-1)^{i+1}$. \end{lemma}
\begin{proof} We apply a similar argument to that in the proof of Lemma \ref{caseiandiv}. The statement of the Lemma for $i=0$ follows immediately from Lemma \ref{commprep}. Hence assume $i\geq 1$, and that the Lemma is known to be true for all pairs $(m',i')$ with $i'<m'$ and either $m'<m$ or $m'=m$ and $i'<i$. Note that the induction hypothesis implies that $\deg P_{i'}(g^{m'})-\deg g^{m'}=(2i'+1)(r/2,0)$ for any such pair $(m',i')$, and similarly for $hg^{m-1}$. It follows that if $a\in J_m(fg^m)$ (resp. $a'\in J_m(fhg^{m-1})$) then $\deg P_{i-1}(a)\leq \deg f^ihg^{m-1}$ (resp. $\deg P_{i-1}(a')\leq\deg f^{i+1}g^m$). By Lemma \ref{Fiform}, $F_i(\mathop{\rm ad}\nolimits f,l_f)(g^m)=-4(m^2-i^2)fg^m+a$, where $a\in J_m(fg^m)$. Moreover, $P_{i-1}(fg^m)=\chi_{i-1}f^ihg^{m-1}+$lower terms, where $\chi_{i-1}$ is a real number of sign $(-1)^{i-1}$. Hence $P_i(g^m)=P_{i-1}(-4(m^2-i^2)fg^m+a)=-4(m^2-i^2)\chi_{i-1}f^ihg^{m-1}+$lower terms. This proves the induction step for $P_i(g^m)$. Similarly, $F_i(\mathop{\rm ad}\nolimits f,l_f)(hg^{m-1})=-4(m^2-i^2)fhg^{m-1}+a'$, where $a'\in J_m(fhg^{m-1})$. But $P_{i-1}(fhg^{m-1})=\eta_{i-1}f^{i+1}g^m+$lower terms, where $\eta_{i-1}$ is real of sign $(-1)^i$. Since $P_{i-1}(a')\in J_m(f^{i+1}g^m)$, we deduce that $P_i(a')=-4(m^2-i^2)f^{i+1}g^m+$lower terms. This completes the proof. \end{proof}
\begin{corollary}\label{v} Suppose $g$ and $h$ are as in Lemma \ref{casev}. Then there is no possible expression for $u$ in terms of $f,g,h$. \end{corollary}
\begin{proof} Suppose there exists an expression $u=\sum a_{ij\epsilon} f^ih^\epsilon g^j$, and as in the proof of Lemma \ref{caseiandiv}, let $m=\max_{\{ a_{ij\epsilon}\neq 0\} }(j+\epsilon)$. Applying $P_{m-1}$ to both sides, we have an equality $P_{m-1}(u)=\sum_{j+\epsilon=m}a_{ij\epsilon}f^i P_{m-1}(h^\epsilon g^j)$. Moreover, it is immediate from Lemma \ref{casev} that $\deg P_{m-1}(g^m)<\deg P_{m-1}(hg^{m-1})<\deg fP_{m-1}(g^m)$, hence $\deg P_{m-1}(u)\geq \deg P_{m-1}(g^m)$. Hence to prove the lemma we have only to prove that $\deg P_{m-1}(u)<\deg P_{m-1}(g^m)$. Clearly $[u,v^r]=2r wv^{r-1}+$lower terms, hence $\deg P_0(u)=(r,1)$. But $P_0(g)=2h$ is of degree $(s+r/2,0)> (Nr/2,0)>(r,1)$. On the other hand, if $m\geq 2$ then $\deg P_{m-1}(u)< ((2m-1)r,0)$. Furthermore, $\deg P_{m-1}(g^m)=(ms+(m+1/2)r,0)>((m(N+1)-1)r/2,0)$. Since $N\geq 3$, $\deg P_{m-1}(g^m)> \deg P_{m-1}(u)$. This completes the proof. \end{proof}
We have therefore eliminated cases (i), (iv) and (v) listed after Lemma \ref{reducetov}. Roughly speaking, the highest degree monomial in the expression for $h^2$ ($f^N$ in cases (i) and (iv), $fg^2$ in case (v)) contributes the highest degree monomial in the expression for $F_i(\mathop{\rm ad}\nolimits f,l_f)(g^m)$, and thus eventually in the expression for $P_i(g^m)$ (and similarly $P_i(hg^{m-1})$). For the remaining cases, we must replace $h^2$ by terms of possibly higher degree, since we wish to find the expressions for $P_{m-1}(g^m)$ and $P_{m-1}(hg^{m-1})$ in terms of the monomials $f^ih^\epsilon g^j$ with $\epsilon\in\{ 0,1\}$. In these circumstances $f^N$ and $fg^2$ are now of equal degree, hence our final expression for the leading term of $P_{m-1}(g^m)$ will contain a number of monomials in $f,g,h$ of equal degree. Here we write $x=\sum \chi_j f^{i+j(N-1)}hg^{m-2j-1}+$lower terms (resp. $x=\sum \eta_j f^{i+1+j(N-1)}g^{m-2j}+$lower terms) to mean that $x=\sum \chi_j f^{i+j(N-1)}hg^{m-2j-1}+a$ (resp. $x=\sum \eta_j f^{i+1+j(N-1)}g^{m-2j}+a$), where $a$ is a sum of monomials in $f,g,h$, each of degree less than that of $f^ihg^{m-1}$ (resp. $f^{i+1}g^m$).
\begin{lemma}\label{othercases} Suppose $g=\xi' v^{(N-1)r/2}+$lower terms and either $h=\xi'' wv^{(N-1)r/2-1}+$lower terms (case (ii)) or $h=\xi' v^t+$lower terms, where $(N-1)r/2<t\leq Nr/2$ (cases (iii) and (vi)). Then for any $i<m$:
(a) Let $l=\mathop{\rm min}\nolimits\{i,[(m-1)/2]\}$. Then $P_i(g^m)=\sum_{j=0}^{l} \chi_{j} f^{i+j(N-1)}hg^{m-2j-1}+$lower terms, where the $\chi_{j}$ are real numbers of sign $(-1)^i$.
(b) Let $l=\mathop{\rm min}\nolimits\{i+1,[m/2]\}$. Then $P_i(hg^{m-1})=\sum_{j=0}^{l} \eta_{j} f^{i+1+j(N-1)}g^{m-2j}+$lower terms, where the $\eta_{j}$ are real numbers of sign $(-1)^{i+1}$. \end{lemma}
\begin{proof} We follow a similar argument to the proofs of Lemmas \ref{caseiandiv} and \ref{casev}. If $i=0$, then (a) and (b) are direct consequences of Lemma \ref{commprep}. Assume therefore that $m>i\geq 1$ and that the Lemma is known to be true for all pairs $(m',i')$ with $i'<m'$ and either $m'<m$ or $m'=m$ and $i'<i$. By a direct calculation, $P_1(\mathop{\rm ad}\nolimits f,l_f)(g^2)=-48fhg+$lower terms and $P_1(hg)=48f^2g^2+24f^{N+1}+$lower terms. Hence assume $m\geq 3$. Let $\delta=\deg h-\deg g\leq (r/2,0)$. By Cor. \ref{Fiform}, $F_i(\mathop{\rm ad}\nolimits f,l_f)(g^m)=-4(m^2-i^2)fg^m-4m(m-1)f^Ng^{m-2}+a$ and $F_i(\mathop{\rm ad}\nolimits f,l_f)(hg^{m-1})=-4(m^2-i^2)fhg^{m-1}-4(m-1)(m-2)f^Nhg^{m-3}+a'$, where $a\in J_m(fg^m)$ and $a'\in J_m(f^Nhg^{m-1})$. We note that if $i-1<m'\leq m$ then by the induction hypothesis $P_{i-1}(g^{m'})$ is a sum of monomials in $f,g,h$, each of degree less than or equal to $\delta+(ir,0)+\deg g^m$. On the other hand, $P_{i-1}(hg^{m'-1})$ is a sum of monomials of degree less than or equal to $((i+1)r,0)-\delta+\deg hg^{m'-1}$. But therefore $P_{i-1}(a')\in J_m(f^{i+1}g^m)$ for any $a'\in J_m(hg^{m-1})$. It follows by the induction hypothesis that $$\begin{array}{rl}P_i(hg^{m-1})=& P_{i-1}(-4(m^2-i^2)fhg^{m-1}-4(m-1)(m-2)f^Nhg^{m-3}+a') \\ = & -4(m^2-i^2)P_{i-1}(fhg^{m-1})-4(m-1)(m-2)P_{m-1}(f^Nhg^{m-3})+b'\end{array}$$ where $b'\in J_m(f^{i+1}g^m)$. This proves the induction step for (b).
For (a) we have to be careful, for it is not in general true that if $x$ is a monomial in $f,g,h$ then $\deg P_{i-1}(x)-\deg x\leq \deg P_{i-1}(g^m)-\deg g^m$. In fact we can see that this is true if and only if $x=f^{i'}g^{m'}$ for some $i',m'$. On the other hand, if $x\in J_m(fg^m)$ is of the form $f^{i'}hg^{m'-1}$ then $\deg x\leq\deg g^m-(r,0)+\delta$. By our statement above on the degrees of $P_{i-1}(g^{m'})$, $P_{i-1}(hg^{m'-1})$ we nevertheless have that $P_{m-1}(x)\in J_m(f^ihg^{m-1})$, hence that $P_{m-1}(a)\in J_m(f^ihg^{m-1})$. The argument now proceed exactly as above. \end{proof}
Suppose therefore that $g,h$ are as in Lemma \ref{othercases}. Then $P_{m-1}(g^m)=\chi_{0}f^{m-1}hg^{m-1}+\chi_{1}f^{m+N-2}hg^{m-3}+\ldots+a$, where $a\in J_m(f^{m-1}hg^{m-1})$ and the $\chi_{j}$ are real numbers of the same sign. Since $\deg g^2=\deg f^{N-1}$, the monomials $f^{m-1+j(N-1)}hg^{m-2j-1}$ are of equal degree. We ask therefore whether it is possible that the highest degree terms of these monomials (expressed in terms of $u,v$, and $w$) cancel out. Specifically, this holds if and only if $\chi_{0}\xi'^{(m-1)}+\chi_{1}\xi^{N-1}\xi'^{(m-3)}+\chi_{2}\xi^{2(N-1)}\xi'^{(m-5)}+\ldots=0$, which is in turn true if and only if $\chi_{0}+\chi_{1}\mu+\chi_{2}\mu^2+\ldots=0$, where $\mu=\xi^{N-1}/(\xi')^{2}$. In case (ii) or (iii), $\mu=-1$. In case (vi) $\mu\neq -1$, since $\xi^N+\xi\xi'^2+\xi''^2=0$.
\begin{lemma}\label{coeffs} (a) $P_{m-1}(g^m)=\chi_0 f^{m-1}hg^{m-1}+\chi_1 f^{N+m-2}hg^{m-3}+\ldots+a$, where $a\in J_m(f^{m-1}hg^{m-1})$ and $\chi_i=\chi_0\cdot\left(\begin{array}{c} m-i-1 \\ i\end{array}\right)/4^i$ for $0\leq i\leq [(m-1)/2]$.
(b) $P_{m-1}(hg^{m-1})=\eta_0 f^mg^m+\eta_1 f^{m+N-1}g^{m-2}+\ldots+a'$, where $a'\in J_m(f^mg^m)$ and $\eta_i=\eta_0\cdot(\left(\begin{array}{c} m-i \\ i\end{array}\right)+\left(\begin{array}{c} m-i-1\\ i-1 \end{array}\right) )/4^i$ for $0\leq i\leq [m/2]$. \end{lemma}
\begin{proof} The fact that $P_{m-1}(g^m)$ has the above form for some constants $\chi_0,\chi_1,\ldots$ follows immediately from Lemma \ref{othercases}. Moreover, $F_m(\mathop{\rm ad}\nolimits f,l_f)(P_{m-1}(g^m))=0$. Let $\omega_i=(m-2i+1)(m-2i)\chi_{i-1}+((m-2i)^2-m^2)\chi_i$ for $1\leq i\leq [(m-1)/2]$. By application of (the argument in the proof of) Cor. \ref{Fiform}, $$F_m(\mathop{\rm ad}\nolimits f,l_f)(P_{m-1}(g^m))=-4\sum_1^{[(m-1)/2]}\omega_if^{m-1+i(N-1)}hg^{m-2i-1}+F_m(\mathop{\rm ad}\nolimits f,l_f)(a)$$ But by the observation in the proof of Lemma \ref{othercases}, $F_{m}(\mathop{\rm ad}\nolimits f,l_f)(a)\in J_m(f^{N+m-1}hg^{m-3})$. Hence each of the coefficients $(m-2i)(m-2i+1)\chi_{i-1}-4i(m-i)\chi_i$ is equal to zero. We deduce that $$\frac{\chi_i}{\chi_0}=\frac{(m-1)!}{(m-2i-1)!}\cdot\frac{1}{i!}\cdot\frac{(m-i-1)!}{(m-1)!}\cdot \frac{1}{4^i}=\left(\begin{array}{c} m-i-1 \\ i\end{array}\right)/4^i$$
Similarly, the existence of some constants $\eta_i$ and an expression for $P_{m-1}(hg^{m-1})$ as in (b) follows immediately from Lemma \ref{othercases}. We apply the same argument as above. Thus let $\omega'_i= (m-2i+2)(m-2i+1)\eta_{i-1}-4i(m-i)\eta_i$ for $1\leq i\leq [m/2]$. Then $$F_m(\mathop{\rm ad}\nolimits f,l_f)(P_{m-1}(hg^{m-1}))=-4\sum_0^{[m/2]} \omega_i'f^{m+1+i(N-1)}g^{m-2i}+F_m(\mathop{\rm ad}\nolimits f,l_f)(a')=0$$ By Lemma \ref{othercases}, $F_m(\mathop{\rm ad}\nolimits f,l_f)(a')\in J_m(f^{m+N}g^{m-2})$. Hence each of the coefficients $(m-2i+2)(m-2i+1)\eta_{i-1}-4i(m-i)\eta_i$ is equal to zero. We conclude that $$\frac{\eta_i}{\eta_0}=\frac{m!}{(m-2i)!}\cdot\frac{(m-i-1)!}{i!(m-1)!}\cdot\frac{1}{4^i}=\frac{(m-i-1)!}{i!(m-2i)!}\cdot\frac{m}{4^i}$$ from which (b) follows. \end{proof}
We thus introduce the polynomials $$p_m(t)=\sum_0^{[(m-1)/2]}\left(\begin{matrix} m-i-1 \\ i \end{matrix}\right)(t/4)^i$$ and $$q_m(t)=\sum_0^{[m/2]}(\left(\begin{matrix} m-i \\ i \end{matrix}\right)+\left(\begin{matrix} m-i-1 \\ i-1 \end{matrix}\right) )(t/4)^i.$$
\begin{lemma}\label{binomials} (a) $p_{m+1}(t)=p_m(t)+tp_{m-1}(t)/4$ and $q_{m+1}(t)=q_m(t)+tq_{m-1}(t)$.
(b) $q_m(t)=p_{m}(t)+tp_{m-1}(t)/2$.
(c) $p_m(-1)=m/2^{m-1}$ and $q_m(-1)=1/2^{m-1}$. \end{lemma} \begin{proof} Part (a) follows immediately from the fact that $\left(\begin{matrix} m-i-1 \\ i \end{matrix}\right)-\left(\begin{matrix} m-i-2 \\ iÊ\end{matrix}\right)=\left(\begin{matrix} m-i-2 \\ i-1 \end{matrix}\right)$ and similarly for $\left(\begin{matrix} m-i \\ i \end{matrix}\right)$, $\left(\begin{matrix} m-i-1 \\ i-1 \end{matrix}\right)$. For part (b), we see by a simple re-indexing exercise that $q_m(t)=p_{m+1}(t)+tp_{m-1}(t)/4$, hence that the equality is true by application of (a). If $\alpha_m=p_m(-1)$ and $\beta_m=q_m(-1)$ then it follows that $\alpha_{m+1}=\alpha_m-\alpha_{m-1}/4$ and $\beta_{m+1}=\beta_m-\beta_{m-1}/4$. The general solution to this difference equation is $(Am+B)/2^m$. But $\alpha_2=1,\alpha_3=3/4$. Hence $\alpha_m=m/2^m$. Similarly, $\beta_2=1/2$ and $\beta_3=1/4$, hence $\beta_m=1/2^{m-1}$. \end{proof}
\begin{corollary}\label{notzero} If $h=\xi'' wv^{(N-1)r/2-1}+$lower terms (case (ii)) or $h=\xi'' v^t+$lower terms, where $(N-1)r/2<t<Nr/2$ (case (iii)) then $\deg P_{m-1}(g^m)=\deg f^{m-1}hg^{m-1}$ and $\deg P_{m-1}(hg^{m-1})=\deg f^mg^m$. \end{corollary}
\begin{proof} We remarked after Lemma \ref{othercases} that the degree of $P_{m-1}(g^m)$ (resp. $P_{m-1}(hg^{m-1})$) is lower than that of $f^{m-1}hg^{m-1}$ (resp. $f^mg^m$) if and only if $\chi_0+\chi_1\mu+\ldots+\chi_{[(m-1)/2]}\mu^{[(m-1)/2]}=0$ (resp. $\eta_0+\eta_1\mu+\ldots+\eta_{[m/2]}\mu^{[m/2]}=0$). But here $\mu=-1$, hence by Lemmas \ref{coeffs} and \ref{binomials} $\chi_0+\chi_1\mu+\ldots+\chi_{[(m-1)/2]}\mu^{[(m-1)/2]}\neq 0$ and $\eta_0+\eta_1\mu+\ldots+\eta_{[m/2]}\mu^{[m/2]}\neq 0$. \end{proof}
\begin{corollary}\label{notzero2} If $\mu\neq -1$ then there exists no $m$ such that $p_m(\mu)$ and $q_m(\mu)$ are both zero. Hence if $h=\xi'' v^{Nr/2}+$lower terms (case (vi)) then either $\deg P_{m-1}(g^m)=\deg f^{m-1}hg^{m-1}$ or $\deg P_{m-1}(hg^{m-1})=\deg f^mg^m$. \end{corollary}
\begin{proof} Suppose there exists $m$ such that $p_m(\mu)=p_{m+1}(\mu)=0$. Let $m'$ be minimal such. Then by Lemma \ref{binomials}(a) $p_{m'-1}(\mu)=0$, which contradicts the minimality of $m'$. Hence there exists no $m$ such that $p_m(\mu)=p_{m+1}(\mu)=0$. But if $p_m(\mu)=q_m(\mu)=0$ then $p_{m-1}(\mu)=0$ by Lemma \ref{binomials}(b). \end{proof}
\begin{lemma}\label{potential} Let $a\in J_{m+1}(f^ihg^m)$ and $a'\in J_m(f^ig^m)$. If $h=\xi'' v^{Nr/2}+$lower terms (case (vi)) then $[f,a]\in J_{m+1}(f^{i+1}g^{m+1})$ and $[f,a']\in J_m(f^ihg^{m-1})$. \end{lemma}
\begin{proof} In case (vi), we have $[f,g]=2h$ and $\deg h-\deg g=(r/2,0)$. Moreover, $[f,h]=-2fg+$lower terms, and $\deg fg-\deg h=(r/2,0)$. It follows that $\deg [f,y]-\deg y\leq (r/2,0)$ for any monomial $y$ in $f,g,h$. Hence $[f,a]\in J_{m+1}(f^{i+1}g^{m+1})$ for any $a\in J_{m+1}(f^ihg^m)$ and $[f,a']\in J_m(f^ihg^{m-1})$ for any $a'\in J_m(f^ig^m)$. This completes the proof. \end{proof}
Note that Lemma \ref{potential} is not true in cases (ii) and (iii). (Hence the proof of Lemma \ref{Ngeq4} really requires different arguments for the cases $\mu=-1$, $\mu\neq -1$.)
\begin{lemma}\label{Ngeq4} Suppose $f,g,h$ are as in Lemma \ref{othercases}.
(a) If $N\geq 4$ then there is no possible expression for $u$ in terms of $f,g,h$.
(b) If $N=3$ then there exists no possible expression for $u$ in terms of $f,g,h$ unless $f=\xi v+$lower terms, $g=\xi' v+$lower terms, $h=\xi'' w+$lower terms. Moreover, in this case any expression for $u$ must be of the form $c_1 g+c_2 f+c_3$, where $c_1,c_2\in {\mathbb C}^\times$ and $c_3\in{\mathbb C}$. \end{lemma}
\begin{proof} Suppose there is such an expression $u=\sum_{ij\epsilon}a_{ij\epsilon}f^ih^\epsilon g^j$ (with the sum taken over all $i,j\geq 0$, $\epsilon\in\{ 0,1\}$). Let $m=\max_{a_{ij\epsilon}\neq 0}(j+\epsilon)$. Assume first of all that $m>1$: we will show that such an expression is impossible for all $N$. Suppose first of all that $a_{im0}f^ig^m$ is the term of highest degree in the expression for $u$ among those of the form $f^jg^m$, $f^jhg^{m-1}$. By Lemma \ref{othercases}, $P_{m-1}(u)=a_{im0}f^iP_{m-1}(g^m)+a$, where $a\in J_m(f^{i+m-1}hg^{m-1})$. If $\mu=-1$ (cases (ii) and (iii)) then by Cor. \ref{notzero}, $\deg P_{m-1}(g^m)=\deg f^{m-1}hg^{m-1}$. But $\deg f^{m-1}hg^{m-1}\geq ((m(N+1)/2-1)r,0)\geq ((2m-1)r,0)>\deg P_{m-1}(u)$, hence such an equality is impossible. On the other hand, if $\mu\neq -1$ (case (vi)) then either $\deg P_{m-1}(g^m)=\deg f^{m-1}hg^{m-1}$ or $\deg P_{m-1}(hg^{m-1})=\deg f^mg^m$. If $\deg P_{m-1}(g^m)=\deg f^{m-1}hg^{m-1}$, then the argument above provides a contradiction. If not, then we consider the equality $[f,u]=2ma_{im0}f^ihg^{m-1}+a'$, where $a'\in J_m(f^ihg^{m-1})$. By Lemmas \ref{potential} and \ref{othercases}, $P_{m-1}([f,u])=2ma_{im0}f^iP_{m-1}(hg^{m-1})+a$, where $a\in J_m(f^mg^m)$. But now $\deg P_{m-1}([f,u])<(2mr,0)$ and $\deg f^iP_{m-1}(hg^{m-1})\geq \deg f^mg^m=(m(N+1)r/2,0)$. Hence there is no such expression for $u$.
Assume therefore that $a_{i(m-1)1}f^ihg^{m-1}$ is the highest degree term in the expression for $u$ among those of the form $f^jg^m$, $f^jhg^{m-1}$. Once more we apply $P_{m-1}$: $P_{m-1}(u)=a_{i(m-1)1}f^iP_{m-1}(hg^{m-1})+a$, where $a\in J_m(f^{i+m}g^m)$. If $\mu=-1$ (cases (ii) and (iii)) then $\deg P_{m-1}(u)<((2m-1)r,0)$ and $\deg f^iP_{m-1}(hg^{m-1})\geq (m(N+1)r/2,0)$, hence such an equality is impossible. If $\mu\neq -1$, then either $\deg P_{m-1}(hg^{m-1})=(m(N+1)r/2,0)$ or $\deg P_{m-1}([f,hg^{m-1}])=((m(N+1)+1)r/2,0)$. In the first case, the argument for $\mu=-1$ shows that equality of degrees is impossible. In the second case, by Lemma \ref{potential} there is an equality $[f,u]=-2ma_{i(m-1)1}f^{i+1}g^m+a'$, where $a'\in J_m(f^{i+1}g^m)$. Then $\deg P_{m-1}([f,u])<(2mr,0)$. But $\deg f^iP_{m-1}(f^{i+1}g^m)>(m(N+1)r/2,0)$. This proves our claim.
Suppose therefore that $m=1$. Hence $u=m_1(f)g+m_2(f)h+m_3(f)$ for some polynomials $m_1(t),m_2(t),m_3(t)$. Applying $(\mathop{\rm ad}\nolimits f)$, we have an equality $[f,u]=m_1(f)[f,g]+m_2(f)[f,h]$. But $\deg h=\deg [f,g]<\deg [f,h]<\deg f[f,g]$ and $\deg [f,u]=(r,1)$, hence such an equality is only possible if $h=\xi'' w+$lower terms, that is to say, if $N=3$, $r=1$ and we are in case (ii). Thus (b) follows. \end{proof}
Lemma \ref{Ngeq4} essentially completes our determination of the isomorphisms $D(Q_2,\gamma_2)\rightarrow D(Q_1,\gamma_1)$ in the case $N\geq 4$ (or $n\geq 4$). The following straightforward lemma is the final step.
\begin{lemma}\label{outer} Let $Q_1$ and $Q_2$ be monic polynomials of degree $\geq 3$ and $\gamma_1,\gamma_2\in{\mathbb C}$. Suppose $\phi:D(Q_2,\gamma_2)\rightarrow D(Q_1,\gamma_1)$ is an isomorphism, and let $f,g,h$ be the images in $D(Q_1,\gamma_1)$ of the standard generators for $D(Q_2,\gamma_2)$. If $f\in {\mathbb C}[u]$, then $f=u$, $Q_1=Q_2$ and either:
(i) $\gamma_1=\gamma_2$ and $g=v,h=w$ (the trivial isomorphism) or,
(ii) $\gamma_1=-\gamma_2$ and $g=-v$, $h=-w$. \end{lemma}
\begin{proof} Since the cosets of $(\mathop{\rm ad}\nolimits u)^j(v^m)$, $j\geq 0$ form a basis for $A_m^{(\infty)}/A_{m-1}^{(\infty)}$, the centralizer of $u$ in $D(Q_1,\gamma_1)$ is ${\mathbb C}[u]$. But therefore the centralizer of $f$ in $D(Q_1,\gamma_1)$ is ${\mathbb C}[f]$. It follows that $f=au+b$ for some $a\in{\mathbb C}^\times$, $b\in{\mathbb C}$. By (the proof of) Lemma \ref{Fprod}, $F_1(\mathop{\rm ad}\nolimits f,l_f)(h)=0$. Suppose $g=\xi' u^iv^j+$lower terms (resp. $g=\xi' u^iwv^{j-1}+$lower terms). Then $h=\xi'' u^{i}wv^{j-1}+$lower terms (resp. $h=\xi'' u^{i+1}v^{j}+$lower terms). Thus $F_j(\mathop{\rm ad}\nolimits u,l_u)(h)\in A_{j-1}^{(\infty)}$ and if $P\in {\mathbb C}[S,T]$ is any polynomial of the form $S^2+c_1S+c_2+dT$ which is not equal to $F_j(\mathop{\rm ad}\nolimits u,l_u)$, then $P(\mathop{\rm ad}\nolimits u,l_u)(h)\not\in A_{j-1}^{(\infty)}$. (This is clear since $h,[u,h],uh$ are linearly independent over $A_{j-1}^{(\infty)}$.) Hence we must have $F_1(\mathop{\rm ad}\nolimits f,l_f)=a^2 F_j(\mathop{\rm ad}\nolimits u,l_u)$. It follows that $a=1/j^2$ and $b=1/4-1/4j^2$. But by the same argument for the inverse isomorphism $\phi^{-1}:D(Q_1,\gamma_1)\rightarrow D(Q_2,\gamma_2)$, we must have $u=f/k^2+(1-1/k^2)$ for some $k\geq 1$. It follows that $k=j=1$. Hence $f=u$.
Now $g=\xi' u^iv+$lower terms or $g=\xi' u^iw+$lower terms for some $i\geq 0$ and some $\xi'\in{\mathbb C}^\times$ by Lemma \ref{Fprod}. Applying the same argument to $\phi^{-1}$, there is an equality $v=q_1(u) g+q_2(u)h+q_3(u)$ for some polynomials $q_1(t),q_2(t),q_3(t)$. But if $g=\xi' u^iv+$lower terms (resp. $g=\xi' u^iw+$lower terms) then $h=\xi' u^iw+$lower terms (resp. $h=-\xi' u^{i+1}v+$lower terms). In other words, the leading terms of $q_1(u)g$ and $q_2(u)h$ are of different degrees. Hence $g=\xi' v+p(u)$ for some polynomial $p(t)$. Moreover, $F_1(\mathop{\rm ad}\nolimits u,l_u)(g)=2\xi'\gamma_1+4up(u)=2\gamma_2$, hence $p(t)=0$. Thus $g=\xi' v$ and $h=\xi' w$. But now $$Q_1(u)+uv^2+w^2-2wv-\gamma_1 v=Q_2(u)+(\xi')^2uv^2+(\xi')^2w^2-2(\xi')^2wv-\xi'\gamma_2v=0$$ It follows that $\xi'^2Q_1(u)-\xi'^2\gamma_1 v=Q_2(u)-\xi'\gamma_2 v$, hence that $\xi'=\pm 1$. This leaves only one non-trivial possibility: that $g=-v$ and $h=-w$. But one can clearly define such an isomorphism $D(Q_1,-\gamma_1)\rightarrow D(Q_1,\gamma_1)$. This completes the proof of the Lemma. \end{proof}
We have therefore solved the isomorphism problem in type $D_{n+1}$, $n\geq 4$.
\begin{definition} Let $Q(t)$ be a monic polynomial of degree $n\geq 4$, let $\gamma\in{\mathbb C}$ and let $u,v,w$ be the standard generators for $D(Q,\gamma)$. Then we denote by $\Theta$ the isomorphism $D(Q,\gamma)\rightarrow D(Q,-\gamma)$ which maps $u\mapsto u'$, $v\mapsto -v'$, $w\mapsto -w'$ where $u',v',w'$ are the standard generators for $D(Q,-\gamma)$. \end{definition}
\begin{rk} This definition of $\Theta$ should perhaps refer to the defining parameters $Q,\gamma$ in its definition. However, $\Theta$ can be thought of as the action of the non-identity element of the normalizer $N_{\mathop{\rm SL}\nolimits(V)}(\Gamma)$ on the space of noncommutative deformations of $V/\Gamma$. \end{rk}
\begin{theorem}\label{geq4} Let $Q(t)$ be a monic polynomial of degree $n\geq 4$ and let $\gamma\in{\mathbb C}$.
(a) If $\tilde{Q}$ is monic of degree greater than or equal to 3 and $\tilde{\gamma}\in{\mathbb C}$, then $D(Q,\gamma)$ is isomorphic to $D(\tilde{Q},\tilde{\gamma})$ if and only if $\tilde{Q}=Q$ and $\tilde\gamma=\pm\gamma$.
(b) The automorphism group of $D(Q,\gamma)$ is trivial unless $\gamma=0$, in which case the automorphism group is cyclic of order 2, generated by $\Theta$. \end{theorem}
\begin{proof} This follows from Lemma \ref{outer}, Lemma \ref{Ngeq4}, Cor. \ref{v} and Cor. \ref{iiv}. \end{proof}
\begin{corollary}\label{moduli} The moduli space of isomorphism classes of noncommutative deformations of a Kleinian singularity of type $D_n$, $n\geq 5$ is isomorphic to a vector space of dimension $n$. \end{corollary}
\begin{proof} The vector space $V$ of monic polynomials of degree $(n-1)$ is isomorphic to ${\mathbb C}^{n-1}$. Hence we map the isomorphism class of $D(Q,\gamma)$ to $(Q,\gamma^2)\in V\oplus{\mathbb C}\cong{\mathbb C}^n$. \end{proof}
We apply this to determine when to of the algebras $H(P,\gamma)$ ($P(t)$ has leading term $nt^{n-1}$, $n\geq 4$) are isomorphic.
\begin{theorem}\label{Hn4} Let $P(t)$ be a polynomial with leading term $nt^{n-1}$ ($n\geq 4$), $\tilde{P}(t)$ a polynomial with leading term $Nt^{N-1}$ ($N\geq 3$) and let $\gamma,\tilde\gamma\in{\mathbb C}$. Then $H(P,\gamma)\cong H(\tilde{P},\tilde\gamma)$ if and only if $P=\tilde{P}$ and $\gamma=\pm\tilde\gamma$. \end{theorem}
\begin{proof} Suppose there exists some isomorphism $\phi:H(P,\gamma)\rightarrow H(\tilde{P},\tilde\gamma)$. Let $Q(t)$ (resp. $\tilde{Q}(t)$) be the unique monic polynomial with zero constant term such that $Q(-s(s+1))+(s+1)P(-s(s+1))$ (resp. $\tilde{Q}(-s(s+1))+(s+1)\tilde{P}(-s(s+1))$) is an even polynomial in $s$. Let $\Omega=Q(U)+UV^2+W^2-2WV-\gamma V$ (resp. $\tilde\Omega=\tilde{Q}(\tilde{U})+\tilde{U}\tilde{V}^2+\tilde{W}^2-2\tilde{W}\tilde{V}-\tilde\gamma\tilde{V}$), where $U,V,W$ (resp. $\tilde{U},\tilde{V},\tilde{W}$) are the standard generators for $H(P,\gamma)$ (resp. $H(\tilde{P},\tilde\gamma)$). By Lemma \ref{centreisp}, $Z(H(P,\gamma))={\mathbb C}[\Omega]$ and $Z(H(\tilde{P},\tilde\gamma))={\mathbb C}[\tilde\Omega]$. But therefore $\phi(\Omega)=a\tilde\Omega+c$ for some $a\in{\mathbb C}^\times$, $c\in{\mathbb C}$. It follows that $\phi$ induces an isomorphism $H(P,\gamma)/(\Omega)\rightarrow H(\tilde{P},\tilde\gamma)/(\tilde\Omega-c/a)$. But $H(P,\gamma)/(\Omega)\cong D(Q,\gamma)$ and $H(\tilde{P},\tilde\gamma)/(\tilde\Omega-c/a)\cong D(\tilde{Q}-c/a,\tilde\gamma)$. It follows that $\tilde\gamma=\pm\gamma$ and $\tilde{Q}=Q+c/a$, hence $\tilde{P}=P$. \end{proof}
\section{Isomorphisms in type $D_4$}
Thm. \ref{geq4} solves the problem of determining all isomorphisms $D(Q_2,\gamma_2)\cong D(Q_1,\gamma_1)$ where the degree of $Q_2$ is greater than or equal to 4. Hence we have only to deal with the case $N=3$. On considering the inverse isomorphism, we see that $n=3$ as well. Furthermore, if $\phi:D(Q_2,\gamma_2)\rightarrow D(Q_1,\gamma_1)$ is not of the form described in Lemma \ref{outer} then by Lemma \ref{Ngeq4}, $f=\xi v+p(u)$, $g=\xi' v+q(u)$, $h=\xi'' w+$lower terms and $u=c_1 g+c_2f+c_3$ for some polynomials $p(t),q(t)$ and some $c_1,c_2\in{\mathbb C}^\times,c_3\in{\mathbb C}$. Replacing $\phi$ by $\phi^{-1}$, we may assume that $p$ is linear. (We will see that this holds for both $\phi$ and $\phi^{-1}$.) Since $Q_2(f)+fg^2+h^2-2hg-\gamma_2 g=0$, we must have $\xi'=\pm i\xi$. After composing with an isomorphism of the form given in Lemma \ref{outer}, if necessary, we may assume furthermore that $\xi'=i\xi$. Now $2\xi'' w=2h=[f,g]=i\xi [p(u),v]-\xi[q(u),v]$. It follows that $q$ is also linear. Hence assume $p(t)=at+b$ and $q(t)=ct+d$. Then $\xi''=\xi(ia-c)$. Moreover, $$\begin{array}{rl}[f,h]= & \xi^2(ia-c)[v,w]+\xi a(ia-c)[u,w] \\ = & \xi^2 (ia-c)(v^2+P_1(u))+a(ia-c)\xi(-2uv+2w+\gamma_1)\end{array}$$ where $P_1(t)$ is the unique polynomial such that $Q_1(-s(s+1))+(s+1)P_1(-s(s+1))$ is even in $s$. On the other hand, by assumption $[f,h]=-2fg+2h+\gamma_2$ and $fg=i\xi^2 v^2+((ai+c)u+(bi+d))\xi v-2ia\xi w+(au+b)(cu+d)$. We deduce that $a=-1/2$, $c=3i/2$, $d=-bi$ and $-4i\xi^2 P_1(u)+2i\xi\gamma_1=i(u-2b)(3u-2b)+2\gamma_2$. Suppose $P_1(t)=3t^2+X_1u+Y_1$. Then it follows that $\xi^2=-1/4$, $8b=-X_1$ and $\gamma_2=i(Y_1/2-X_1^2/32)+i\xi\gamma_1$. We choose $\xi=i/2$. (The case $\xi=-i/2$ will then arise as the inverse of the isomorphism we construct below, composed with the non-trivial isomorphism from Lemma \ref{outer}.) Assume therefore that: $$f=iv/2-u/2-X_1/8,\; g=-v/2+3iu/2+iX_1/8,\; h=w$$ We wish to determine for which values of $Q_2,\gamma_2$ there exists an isomorphism $\phi$ mapping the standard generators for $D(Q_2,\gamma_2)$ onto $f,g,h$. By the calculation above, we must have $\gamma_2=i(Y_1/2-X_1^2/32)-\gamma_1/2$. We note the following description of the coefficients of the polynomial $P(t)$ in terms of those of $Q(t)$.
\begin{lemma}\label{constants} Suppose $Q(t)=t^3+At^2+Bt+C$ and $P(t)=3t^2+Xt+Y$. Then $Q(-s(s+1))+(s+1)P(-s(s+1))$ is an even polynomial in $s$ if and only if $X=2A+8$ and $Y=2A+B+8$. \end{lemma}
\begin{proof} We have $$Q(-s(s+1))=-s^6-3s^5+(A-3)s^4+(2A-1)s^3+(A-B)s^2-Bs+C\mbox{, and}$$ $$(s+1)P(-s(s+1))=3s^5+9s^4+(9-X)s^3+(3-2X)s^2+(Y-X)s+Y$$ The Lemma follows on comparing odd powers of $s$. \end{proof}
To proceed, we therefore calculate $Q_2(f)+g^2f+h^2+2gh-\gamma_2 g$, assuming $Q_2(f)=f^3+A_2 f^2+B_2f+C_2$. By a straightforward calculation \begin{equation}\label{fsquared}f^2=-v^2/4-(u+X_1/4)iv/2+iw/2+(u+X_1/4)^2/4\end{equation} and \begin{equation}\label{gsquared}g^2=v^2/4-(3u+X_1/4)iv/2+3iw/2-(3u+X_1/4)^2/4\end{equation} Adding (\ref{fsquared}) and (\ref{gsquared}), we obtain $f^2+g^2=-2(u+X_1/8)iv+2iw-2u(u+X_1/8)$. Multiplying on the right by $f$, we obtain $f^3+g^2f=(u+X_1/8)v^2-wv+((X_1/4-2)u+X_1^2/32)iv-(3u+(X_1/2-2))iw+u(u+X_1/8)(u+X_1/4)+i\gamma_1$. Hence $f^3+g^2f+h^2=wv+(X_1/8)v^2+((X_1/4-2)u+ X_1^2/32-i\gamma_1)iv-(3u+(X_1/2-2))iw+u(u+X_1/8)(u+X_1/4)-Q_1(u)+i\gamma_1$. Moreover, $X_1/8=1+A_1/4$. We deduce that \begin{eqnarray}\nonumber\lefteqn{f^3+g^2f+h^2+2gh=A_1v^2/4+(A_1u/2+2(1+A_1/4)^2-i\gamma_1)iv} \\ \label{long}& & -A_1iw/2+u(u+1+A_1/4)(u+2+A_1/2)-Q_1(u)-P_1(u)+i\gamma_1\end{eqnarray} It follows that $A_2=A_1$. Multiplying (1) by $A_1$ and substituting for $X_1$, we have: \begin{eqnarray}\nonumber\lefteqn{A_1f^2=-A_1v^2/4-(A_1u/2+A_1(1+A_1/4))iv} \\ \label{A1f2} & & +A_1iw/2+A_1u^2/4+A_1(1+A_1/4)u+A_1(1+A_1/4)^2\end{eqnarray} We notice moreover that $\gamma_2=i(B_1/2+2(1-A_1^2/16)+i\gamma_1/2)$. It follows that \begin{eqnarray}\label{gamma2g}-\gamma_2 g=(B_1/4+1-A_1^2/16+i\gamma_1/4)iv+(B_1/4+1-A_1^2/16+i\gamma_1/4)(3u+2+A_1/2)\end{eqnarray} Let $B_2=6(A_1^2/16-1)+3i\gamma_1/2-B_1/2$. Taking the sum of (\ref{long}), (\ref{A1f2}) and (\ref{gamma2g}), we see that \begin{eqnarray}\nonumber\lefteqn{f^3+A_1f^2+g^2f+h^2+2gh-\gamma_2 g=-B_2 iv/2} \\ \label{b1}& \mbox{} & +B_2u/2+B_2(1+A_1/4)+(B_1-i\gamma_1+4(1-A_1^2/16))A_1/4-C_1\end{eqnarray} Adding $B_2 f$ to (\ref{b1}), we deduce that $Q_2(t)=t^3+A_1 t^2+B_2 t+C_2$ where $C_2=C_1-A_1(B_1/4-i\gamma_1/4+1-A_1^2/16)$.
\begin{lemma} Let $f=iv/2-u/2-(1+A_1/4)$, $g=-v/2+3iu/2+i(1+A_1/4)$ and $h=w$. Then $f^3+A_1f^2+B_2 f+C_2+fg^2+h^2-2hg-\gamma_2 g=0$, where $$B_2=6(A_1^2/16-1)+3i\gamma_1/2-B_1/2,$$ $$C_2=C_1-A_1(B_1/4-i\gamma_1/4+1-A_1^2/16),$$ $$\gamma_2=iB_1/2-2i(A_1^2/16-1)-\gamma_1/2.$$ Moreover, $[f,g]=2h$, $[f,h]=-2fg+2h+\gamma_2$ and $[g,h]=g^2+3f^2+(2A_1+8)f+(2A_1+B_2+8)$. \end{lemma}
\begin{proof} Let $Q_2(t)=t^3+A_1t^2+B_2t+C_2$. By construction, $[f,g]=2h$ and $[f,h]=-2fg+2h+\gamma_2$. Moreover, by the discussion above, $Q_2(f)+g^2 f+h^2+2gh-\gamma_2 g=0$. But it follows from the commutator relation $[f,g]=2h$ that $g^2f+2gh=fg^2-2hg$. Hence we have only to show that $[g,h]=g^2+3f^2+(2A+8)f+2A+B+8$. We deduce from the Jacobi identity $[f,[g,h]]=[[f,g],h]+[g,[f,h]]$ and the known commutator relations for $f$ that $[f,[g,h]]=2(gh+hg)=[f,g^2]$. Hence $[g,h]=g^2+z$ for some $z\in Z_{D(Q_1,\gamma_1)}(f)=Z(f)$. We claim that $Z(f)={\mathbb C}[f]$. Indeed, let $x$ be an element of $Z(f)\setminus{\mathbb C}[f]$ of minimal degree. By considering $\{ \mathop{\rm gr}\nolimits f,\mathop{\rm gr}\nolimits x\}$ we see that $x=\chi v^j+$lower terms for some $\chi\in{\mathbb C}^\times$ and $j$. But now $x-\chi(-2if)^j$ is an element of $Z(f)\setminus{\mathbb C}[f]$ of lower degree than $x$, which provides a contradiction. It follows that $z=p(f)$ for some polynomial $p(t)$. Thus there is a homomorphism $H(p,\gamma_2)\rightarrow D(Q_1,\gamma_1)$ which sends the standard generators $U,V,W$ for $H(p,\gamma)$ to $f,g,h$. The equality $Q_2(f)+fg^2+h^2-2hg-\gamma_2 g=0$ now implies that $p_2(t)=3t^2+(2A_1+8)t+2A_1+B_2+8$. \end{proof}
\begin{definition}\label{isos} For a monic polynomial $Q$ of degree $3$ and $\gamma\in{\mathbb C}$, let $\Theta$ be the isomorphism $D(Q,\gamma)\rightarrow D(Q,-\gamma)$ given by $u\mapsto u',v\mapsto -v',w\mapsto -w'$, where $u',v',w'$ are the standard generators of $D(Q,-\gamma)$.
Let $\tilde{B}=6(A^2/16-1)+3i\gamma/2-B/2$, $\tilde{C}=C-A(B/4-i\gamma/4+1-A^2/16)$, $\tilde\gamma=i(2(1-A^2/16)+B/2+i\gamma/2)$ and $\tilde{Q}(t)=t^3+At^2+\tilde{B}t+\tilde{C}$. Let $\Psi$ be the isomorphism $D(Q,\gamma)\rightarrow D(\tilde{Q},\tilde\gamma)$ given by $$u\mapsto i\tilde{v}/2-\tilde{u}/2-(1+A/4),\; v\mapsto -\tilde{v}/2+3i\tilde{u}/2+i(1+A/4),\; w\mapsto \tilde{w}$$ where $\tilde{u},\tilde{v},\tilde{w}$ are the standard generators of $D(\tilde{Q},\tilde\gamma)$. \end{definition}
\begin{rk} As in the case $n\geq 4$, the isomorphisms $\Theta,\Psi$ depend on the choice of $Q,\gamma$ and therefore should perhaps refer to these defining parameters in their definition. However, one can think of $\Theta$ and $\Psi$ as representatives of elements of $N_{\mathop{\rm SL}\nolimits(V)}(\Gamma)/\Gamma$ acting as transformations of the space of noncommutative deformations of $V/\Gamma$. Since any element of $N_{\mathop{\rm SL}\nolimits(V)}$ preserves the invariant ring ${\mathbb C}[V]^\Gamma$, there is a natural action of $N_{\mathop{\rm SL}\nolimits(V)}(\Gamma)/\Gamma$ on $V/\Gamma$. Our construction above and Thm. \ref{main} below therefore say that each such element of $N_{\mathop{\rm SL}\nolimits(V)}/\Gamma$ has an induced action on the space of noncommutative deformations of $V/\Gamma$, and this induced action produces all possible isomorphisms between points $(Q,\gamma)$. \end{rk}
\begin{lemma} (a) $\Psi^3$ is the identity map on each $D(Q,\gamma)$.
(b) $\Theta\circ\Psi\circ\Theta^{-1}=\Psi^2$. \end{lemma}
\begin{proof} Consider $\Psi:D(Q,\gamma)\rightarrow D(\tilde{Q},\tilde\gamma)$ and $\Psi:D(\tilde{Q},\tilde\gamma)\rightarrow D(\hat{Q},\hat\gamma)$. Let $u,v,w$ (resp, $\tilde{u},\tilde{v},\tilde{w}$, $\hat{u},\hat{v},\hat{w}$) be the standard generators for $D(Q,\gamma)$ (resp. $D(\tilde{Q},\tilde\gamma)$, $D(\hat{Q},\hat\gamma)$). Then by calculation $\Psi^2:u\mapsto -i\hat{v}/2-\hat{u}/2-1-A/4$, $v\mapsto -\hat{v}/2-3i\hat{u}/2-i(1+A/4)$ and $w\mapsto\hat{w}$. This proves (b). On considering the composition of $\Psi^2:D(Q,\gamma)\rightarrow D(\hat{Q},\hat\gamma)$ with $\Psi:D(\hat{Q},\hat\gamma)\rightarrow D(\overline{Q},\overline\gamma)$, we see that $\Psi^3:u\mapsto \overline{u},v\mapsto\overline{v},w\mapsto\overline{w}$. But therefore $\overline{Q}=Q$ and $\overline\gamma=\gamma$. \end{proof}
We therefore define the isomorphism $\Psi^{-1}=\Psi^2$: for $A,B,C,\gamma\in{\mathbb C}$ let $\hat{B}=6(A^2/16-1)-3i\gamma/2-B/2$, $\hat{C}=C-A(B/4+i\gamma/4+1-A^2/16)$, $\hat{\gamma}=i(2(A^2/16-1)-B/2)-\gamma/2$ and let $\hat{Q}(t)=t^3+At^2+\hat{B}t+\hat{C}$. Then there exists an isomorphism $\Psi^{-1}: D(Q,\gamma)\rightarrow D(\hat{Q},\hat\gamma)$ given by $u\mapsto -i\hat{v}/2-\hat{u}/2-(1+A/4)$, $v\mapsto -\hat{v}/2-3i\hat{u}/2-i(1+A/4)$, $w\mapsto\hat{w}$, where $\hat{u},\hat{v},\hat{w}$ are the standard generators for $D(\hat{Q},\hat\gamma)$.
Hence, we have completed our task.
\begin{theorem}\label{main} Let $Q$ be a monic polynomial of degree $3$ and let $\gamma\in{\mathbb C}$.
(a) There are exactly six isomorphims from $D(Q,\gamma)$ to algebras $D(\overline{Q},\overline\gamma)$, namely $\mathop{\rm Id}\nolimits_{D(Q,\gamma)}$ and $\Psi,\Psi^{-1},\Theta,(\Theta\circ\Psi),(\Theta\circ\Psi^{-1})$.
(b) If $\gamma=0$ and $B=4(A^2/16-1)$ then $\mathop{\rm Aut}\nolimits D(Q,\gamma)$ is isomorphic to the symmetric group $S_3$, and is generated by $\Psi$ and $\Theta$.
(c) $\mathop{\rm Aut}\nolimits D(Q,\gamma)$ is of order 2 if exactly one of $B-4(A^2/16-1)-i\gamma$, $B-4(A^2/16-1)+i\gamma$ and $\gamma$ is zero. If $B=4(A^2/16-1)+i\gamma$ (resp. $B=4(A^2/16-1)-i\gamma$) and $\gamma\neq 0$ then $\mathop{\rm Aut}\nolimits D(Q,\gamma)$ is generated by $\Theta\circ\Psi$ (resp. $\Theta\circ\Psi^{-1}$). If $\gamma=0$ but $B\neq 4(A^2/16-1)$ then $\mathop{\rm Aut}\nolimits D(Q,\gamma)$ is generated by $\Theta$.
(d) If $\gamma\neq 0$ and $B\neq 4(A^2/16-1)\pm i\gamma$ then there are no non-trivial automorphisms of $D(Q,\gamma)$. \end{theorem}
\begin{proof} This follows from Lemma \ref{reducetov}, Cor. \ref{iiv}, Cor. \ref{v}, Lemma \ref{Ngeq4}, Lemma \ref{outer} and the discussion above. \end{proof}
We therefore think of the symmetric group $S_3$ as acting on the space of noncommutative deformations via the representatives $\Theta,\Psi$. However, it is not difficult to describe the invariants with respect to this action. Let $\sigma=\left(\begin{array}{ccc} 1 & 2 & 3 \end{array}\right)$ and $\tau=\left(\begin{array}{cc} 1 & 2 \end{array}\right)$ be generators for $S_3$.
\begin{lemma}\label{invs} Let $S_3$ act as algebra automorphisms of ${\mathbb C}[A,B,C,\gamma]$ with the action of $\sigma$ (resp. $\tau$) given by that of $\Psi$ (resp. $\Theta$) on $Q(t)=t^3+At^2+Bt+C$ and $\gamma$. Let $x_1=B-4(A^2/16-1)-\gamma\sqrt{3},x_2=B-4(A^2/16-1)+\gamma\sqrt{3},x_3=6C-AB$ and $x_4=A$.
Then ${\mathbb C}[A,B,C,\gamma]={\mathbb C}[x_1,x_2,x_3,x_4]$ and $\sigma(x_1)=e^{2\pi i/3} x_1,\sigma(x_2)=e^{-2\pi i/3}x_2$, $\tau(x_1)=x_2,\tau(x_2)=x_1$. The action of $S_3$ on $x_3$ and $x_4$ is trivial. \end{lemma}
\begin{proof} The fact that ${\mathbb C}[A,B,C,\gamma]={\mathbb C}[x_1,x_2,x_3,x_4]$ is clear, since $x_2-x_1=2\gamma\sqrt3$, $x_1+x_2+4(x_4^2-1)=2B$ and $x_3+AB=6C$. The action of $\sigma$ and $\tau$ on $x_1,x_2,x_3,x_4$ follows immediately from definition \ref{isos}. \end{proof}
\begin{corollary}\label{moduli4} The moduli space of isomorphism classes of noncommutative deformations of a Kleinian singularity of type $D_4$ is isomorphic to a vector space of dimension 4. \end{corollary}
\begin{proof} By Lemma \ref{invs} the ring of invariants ${\mathbb C}[A,B,C,\gamma]^{S_3}$ is generated by $x_1^3+x_2^3,x_1x_2,x_3$ and $x_4$. But hence the map $D(Q,\gamma)\mapsto ((B-4(A^2/16-1))((B-4(A^2/16-1))^2+9\gamma^2),(B-4(A^2/16-1))^2-3\gamma^2,6C-AB,A)$ induces a bijective map on isomorphism classes of deformations. \end{proof}
Finally, we can now solve the problem of when two algebras $H(P,\gamma),H(\tilde{P},\tilde\gamma)$ are isomorphic.
\begin{theorem}\label{H4} Let $P(t)=3t^2+Xt+Y$, $\tilde{P}(t)=3t^2+\tilde{X}t+\tilde{Y}$ and $\gamma,\tilde\gamma\in{\mathbb C}$. Then $H(P,\gamma)\cong H(\tilde{P},\tilde\gamma)$ if and only if $X=\tilde{X}$ and either
(i) $\tilde{Y}=3(X+X^2/32+i\gamma/2)-Y/2$ and $\pm\tilde\gamma=i(Y/2-X-X^2/32)-\gamma/2$, or
(ii) $\tilde{Y}=3(X+X^2/32-i\gamma/2)-Y/2$ and $\pm\tilde\gamma=-i(Y/2-X-X^2/32)-\gamma/2$, or
(iii) $\tilde{Y}=Y$ and $\tilde\gamma=\pm\gamma$. \end{theorem}
\begin{proof} Let $\phi:H(P,\gamma)\rightarrow H(\tilde{P},\tilde\gamma)$ be an isomorphism. Let $Q(t)$ (resp. $\tilde{Q}(t)$) be the unique monic polynomial with zero constant term such that $Q(-s(s+1))+(s+1)P(-s(s+1))$ (resp. $\tilde{Q}(-s(s+1))+(s+1)\tilde{P}(-s(s+1))$) is even in $s$ and let $\Omega=Q(u)+uv^2+w^2-2wv-\gamma v$, $\tilde\Omega=\tilde{Q}(\tilde{u})+\tilde{u}\tilde{v}^2+\tilde{w}^2-2\tilde{w}\tilde{v}-\tilde\gamma \tilde{v}$. By Lemma \ref{centreisp}, $Z(H(P,\gamma))={\mathbb C}[\Omega]$ and $Z(H(\tilde{P},\tilde\gamma))={\mathbb C}[\tilde\Omega]$. It follows that $\phi(\Omega)=a\tilde\Omega+c$ for some $a\in{\mathbb C}^\times$, $c\in{\mathbb C}$. Hence $\phi$ induces an isomorphism $H(P,\gamma)/(\Omega)\rightarrow H(\tilde{P},\tilde\gamma)/(\tilde\Omega-c/a)$. But $H(P,\gamma)/(\Omega)\cong D(Q,\gamma)$ and $H(\tilde{P},\tilde\gamma)/(\tilde\Omega-c/a)\cong D(\tilde{Q}-c/a,\tilde\gamma)$. The theorem now follows from Thm. \ref{main} and the fact that $Q(t)=t^3+(X/2-4)t^2+(Y-X)t$, $\tilde{Q}(t)=t^3+(\tilde{X}/2-4)t^2+(\tilde{Y}-\tilde{X})t$. \end{proof}
\end{document}
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\begin{document}
\title[Sensitivity of iterated function systems]{Sensitivity of iterated function systems} \author[Ghane]{F. H. Ghane} \address{\centerline{Department of Mathematics, Ferdowsi University of Mashhad} \centerline{Mashhad, Iran.} } \email{[email protected]} \email{f\_h\[email protected]} \author[Rezaali]{E. Rezaali} \address{\centerline{Department of Mathematics, Ferdowsi University of Mashhad} \centerline{Mashhad, Iran.}} \email{[email protected]} \author[Sarizadeh]{A. Sarizadeh$^{*}$} \address{\centerline{Department of Mathematics, School of Science, Ilam University } \centerline{P. O. Box: 69315-516, Ilam, Iran.} } \email{[email protected]} \email{[email protected]}
\thanks{$^*$Corresponding author}
\subjclass[2010] {37B05; 37B10; 54H20; 58F03.} \keywords{iterated function systems, minimality, transitivity, weak topologically exact, equicontinuity, sensitivity.} \begin{abstract} The present work is concerned with the eqiucontinuity and sensitivity of iterated function systems (IFSs). Here, we consider more general case of IFSs, i.e. the IFSs generated by a family of relations. We generalize the concepts of transitivity, sensitivity and equicontinuity to these kinds of systems. This note investigates the relationships between these concepts.
Then, several sufficient conditions for sensitivity of IFSs are presented. We introduce the notion of weak topologically exact for IFSs generated by a family of relations. It is proved that non-minimal weak topologically exact IFSs are sensitive. That yields to different examples of non-minimal sensitive systems which are not an $M$-system. Moreover, some interesting examples are given which provide some facts about the sensitive property of IFSs. \end{abstract} \maketitle \thispagestyle{empty}
\section{Introduction} In deterministic dynamical systems, chaos concludes all random phenomena without any stochastic factors. On the study of chaos, the concept of sensitivity is a key ingredient. In fact, sensitivity characterizes the unpredictability of chaotic phenomena, and is one of the essential conditions in various definitions of chaos. Therefore, the study on sensitivity has attracted a lot of attention from many researchers
(e.g. \cite{ABC, AG, ak,bbcds, GW,HS,km,ls}). In 1971, Ruelle introduced the first precise definition for sensitivity \cite{RT}. Then a formulation of sensitivity was given by Guckenheimer on the study of interval maps, \cite{G}. In 1986, Devaney \cite{De} proposed the widely accepted definition of chaos (topological transitivity, dense periodic points and sensitivity), and emphasized the significance of sensitivity in describing dynamical systems. Afterwards, Li-Yorke sensitivity \cite{ak}, $n$-sensitivity \cite{Xe}, and collective sensitivity \cite{WW} were successively proposed, and each of these concepts is used to describe the complexity of dynamical systems. For continuous self-maps of compact metric spaces, Moothathu \cite{Mo} initiated a preliminary study of stronger forms of sensitivity formulated in terms of large subsets of $\mathbb{N}$. Mainly he considered syndetic sensitivity and cofinite sensitivity and proved that any syndetically transitive, non-minimal map is syndetically sensitive. Then, Liu, Liao and Wang \cite{LLW} introduced some other versions of sensitivities, including thick sensitivity and thickly syndetical sensitivity. After that Wang, Yin and Yan \cite{WYY} extended some of these results to semigroup actions. They presented some sufficient conditions for dynamical systems of semigroup actions to have these sensitivities.
Although the sensitivity is widely understood as the central idea in Devaney chaos, but it is implied by transitivity and density of periodic points \cite{bbcds}. This result was generalized in \cite{AAB}, changing density of periodic points to density of minimal points. In fact the authors proved that if a dynamical system $(X,f)$ is topologically transitive and has no equicontinuity point, then it is sensitive. In particular a minimal system is either equicontinuous or sensitive.
In \cite{GW}, Glasner and Weiss (see also Akin, Auslander and Berg \cite{AAB}) established a stronger result (for compact metric systems): they proved that if $(X,f)$ is a non-minimal $M$-system then $(X,f)$ is sensitive. For a compact metric space $X$, we recall that an $M$-system means that the set of almost periodic points is dense in $X$ (the Bronstein condition) and, in addition, the system is topologically transitive. These results were generalized for groups in \cite{G1}.
In \cite{km}, Kontorovich and Megrelishvili generalized these results for a wide class of topological semigroup actions including one-parameter semigroup actions on Polish spaces and $M$-systems.
Notice that the dynamical characterizations of $M$-systems have received special attention, see e.g. \cite{G1, G2, HU}.
Recently, Iglesias and Portela \cite{IP} proved that if a semigroup is almost open then it is sensitive or there exists a residual set of equicontinuity points. As a consequence they obtained a sufficient condition for sensitivity that generalizes that given in Kontorovich and Megrelishvili \cite{km}.
In this literature, Auslander-Yorke dichotomy theorem \cite{au} states that for a single transitive map, either the system is sensitive, or the set of equicontinuity points is exactly the same as the set of transitive points. It holds in the non-invertible case as well. On the other hand, any weak mixing system is sensitive and there are weak mixing systems which are not $M$-systems. Akin \cite{A} provided an example of a mixing homeomorphism on the torus with a fixed point as the unique minimal set. Furthermore, Akin and Kolyada in \cite{ak} introduced the notion of Li-Yorke sensitivity. They proved that every weak mixing system $(X, f)$, where $X$ is a compact metric space and $f$ a continuous map of $X$ is Li-Yorke sensitive and hence it is sensitive. An example of Li-Yorke sensitive system without weak mixing factors was given in \cite{Ci} (see also \cite{C2}). In \cite{M1}, Mel$\acute{\text{i}}$chov$\acute{\text{a}}$ proved that every minimal system with a weak mixing factor, is Li-Yorke sensitive.
The objective of this paper is to discuss the sensitivity of iterated function systems. An iterated function system, or IFS, is simply a finite collection of continuous self-maps of a topological space $X$. Then we can consider the semigroup generated by these transformations. IFSs provide a method for both generating and characterizing fractal images whenever the continuous maps are contraction. Iterated function system was firstly introduced and then popularized by Hutchinson \cite{H} and Barnsley \cite{B}. They have a wide variety of applications in many branches of nonlinear dynamics; for instance in the image processing theory \cite{E} and in the theory of stochastic growth models \cite{F}. Here, we discuss a general case of IFSs, i.e. we consider the IFSs which are generated by a family of continuous relations. We generalize some dynamical properties to continuous relations including transitivity, equicontinuity and sensitivity. Each IFS generated by a finite number of continuous relations $f_i:X \to X$, $i=1, \ldots, k$, can be considered as a continuous relation $F=\bigcup_{i=1}^k f_i$. Then, the investigation of the relationship between these notions for IFSs can be done by using the relation $F$.
The concept of transitivity could trace back to Birkhoff \cite{Br}. After that many articles dealt with such a topic. Transitivity is a widely accepted feature of chaos. It is often required in definitions of chaos as one of several ingredients. In the present work, we discuss some aspects of transitivity including transitivity, weak topological exactness and topological exactness in iterated function systems (IFSs) and investigate their relations with the notion of sensitivity. One of our main results states that non-minimal weak topologically exact IFSs are sensitive. As is well known, unlike ordinary dynamical systems, the minimality of an iterated function system $\IFS(X;\mathcal{F})$ with more than one generator is not equivalent to that of its inverse $\IFS(X;\mathcal{F}^{-1})$. To see such systems, we refer to \cite{BG}. This property of IFSs yields to existence of different examples of non-minimal sensitive IFSs which are not $M$-system. Additionally, we give an example of a sensitive minimal IFS that is not equicontinuous. \subsection{The paper is organized as follows}
In Section 2, we recall some standard definitions about relations, continuous relations and iterated function systems generated by a family of relations. We generalize some dynamical properties including transitivity, equicontinuity and sensitivity to continuous relations. Then, we investigate their relationships. We introduce the notion of weak topologically exact property for IFSs. It is proved that each IFS generated by a finite family of relations on a compact metric space so that their inverses are continuous relations is backward minimal iff it is weak topologically exact. We study the sensitivity and equicontinuity properties for IFSs in Section 3. We show that every non-minimal topologically exact IFS generated by a finite family of maps on a compact metric space can not be equicontinuous. Furthermore, we prove that each non-minimal topologically exact IFS generated by a finite family of open maps on a compact metric space is sensitive. In Section 4, we give several examples of IFSs. The first example shows that backward minimality can not be followed by minimality but the second example is a non-minimal weak topologically exact IFS which is backward minimal. This example provide a non-minimal sensitive systems which are not $M$-system. The last example is a non-minimal topologically exact system which is not an $M$-system.
\section{Dynamical Systems of Relations}
A dynamical system in the present paper is a triple $(\Gamma,X,\varphi)$, where $\Gamma$ is a semigroup, $X$ is a set and $\varphi : \Gamma\times X\to X$ is a relation. Sometimes we write the dynamical system as a pair $(\Gamma,X)$. It can be said that is the most general and common definition of dynamical systems.
We remark that a \emph{relation} $f : X \to Y$ is a subset of $X \times Y$ with $f(x) =\{ y : (x, y) \in f\}$, for $x\in X$. Following \cite{A}, the image of $A \subset X$ under a relation $f : X \to Y$ is given by $f(A) =\bigcup_{x \in A}f(x)$. We assume that $X$ and $Y$ are compact metric spaces. We define $f^{-1}= \{(y,x): (x,y) \in f\}.$ Thus, $f$ is a map when $f(x)$ is a singleton for all $x \in X$. We say that $f$ is \emph{surjective} when $f(X) = Y$ and $f^{-1}(Y ) = X$. For $f : X \to Y$ and $g : Y \to Z$ the \emph{composition} $g \circ f : X \to Z$ is the projection to $X \times Z$ of $(f \times Z) \cap (X \times g) \subset X \times Y \times Z$. Inductively, for $f : X \to X$, $f^{n+1}: = f \circ f^n$ with $f^0:=id$ the identity map.
Let us now equip $X,Y$ with two topologies and assume that they are compact. A relation $f : X \to Y$ is \emph{closed} when it is a closed subset of $X \times Y$ or, equivalently by compactness, when $f^{-1}(B)$ is closed for all closed set $B \subseteq Y$. That is, $f$ is \emph{upper semi-continuous}.
A relation $f : X \to Y$ is \emph{lower semi-continuous} when $f^{-1}(B)$ is open for all open set $B \subseteq Y$. $f$ is said to be \emph{continuous} when it is closed and lower semi-continuous.
That is $f$ is both upper and lower semi-continuous. By compactness the composition of closed relations is closed. A relation $f$ is called \emph{bi-continuous} or \emph{open} if $f$ and $f^{-1}$ are continuous. The composition of continuous (or bi-continuous) relations is continuous (resp. bi-continuous). The finite union of closed, continuous or bi-continuous relations is satisfies the corresponding property.
Let us remark two popular kinds of the dynamical systems: cascades and iterated function systems. Consider the semigroup generated by a relation $f : X \to X$ and denote it by $\Gamma$. Take $\varphi : \Gamma\times X\to X$ so that $\varphi(f^n,x)=\{y\in X: (x,y)\in f^n\}$, for every $n\in \mathbb{N}$. So, a pair $(\Gamma,X)$ is a dynamical system. If $\Gamma = \{f^n\}_{n\in\mathbb{N}}$ and $f : X \to X$ is a relation, then the classical dynamical system $(\Gamma,X)$ is called a \emph{cascade} and we use the standard notation: $(X, f)$.
Now, let $\mathcal{F}$ be a family of relations defined on a set $X$. We denote by $\mathcal{F}^+$ the semigroup generated by these relations. Take $\varphi : \Gamma\times X\to X$ so that $\varphi(h,x)=\{y\in X: (x,y)\in h\}$, where $\Gamma=\mathcal{F}^+$. In this case, the dynamical system $(\mathcal{F}^+,X)$ is called an \emph{iterated function system} (IFS) associated to $\mathcal{F}$. We use the usual notations: $\IFS(X;\mathcal{F})$ or $\IFS(\mathcal{F})$. Roughly speaking, an iterated function system (IFS) can be thought of as a collection of relations which can be applied successively in any order.
For the $\IFS(X;\mathcal{F})$ and $x \in X$, the \emph{forward orbit} of $x$ is defined by $$\mathcal{O}_\mathcal{F}^+(x)= \bigcup_{h\in \mathcal{F}^+}h(x).$$ Since $f:X\to Y$ is a relation, so is $f^{-1}:Y\to X$. Analogously, one can define the \emph{backward orbit} of $x$ by $$\mathcal{O}_\mathcal{F}^-(x)= \bigcup_{h\in \mathcal{F}^+}h^{-1}(x),$$ where $h^{-1}(x)=\{y:(y,x)\in h\}$. \begin{definition}\label{def1111}\cite{BG,km} Let $\IFS(\mathcal{F})$ be an iterated function system generated by a finite or infinite family of relations, $\mathcal{F}$, on a metric space $X$. \begin{enumerate}[label=(\roman*),ref=\roman*]
\item\label{it:0} $\IFS(\mathcal{F})$ is called \emph{symmetric},
if for each $f \in \mathcal{F}$ it holds that $f^{-1} \in \mathcal{F}$.
\item\label{it:1} $\IFS(\mathcal{F})$ is called \emph{ transitive}, if for any two non-empty open sets $U$ and
$V$ in $X$, there exists $h \in \mathcal{F}^+$ such that $h(U) \cap V \neq \emptyset$.
\item\label{it:2} A point $x$ is called a \emph{transitive point} if $\overline{\mathcal{O}_{\mathcal{F}}^+(x)}=X$,
where $\overline{A}$ is the closure of a subset $A$ of $X$. Let $\text{Trans}(\mathcal{F})$
denote the set of all transitive points.
\item\label{it:3} $\IFS(\mathcal{F})$ is called \emph{forward minimal}
if for every $x\in X$, $\text{Trans}(\mathcal{F})=X$.
\item\label{it:4} $\IFS(\mathcal{F})$ is called
\emph{backward minimal}, if $\IFS(\mathcal{F}^{-1})$ is minimal where $\mathcal{F}^{-1}=\{f^{-1}: f\in \mathcal{F}\}$, i.e. $\overline{\mathcal{O}_{\mathcal{F}}^-(x)}=X$ for every $x\in X$.
\item\label{it:4} A subset $A$ of $X$ is called \emph{forward invariant} for $\IFS(\mathcal{F})$
(or $\mathcal{F}$) if $f(A)\subset A$ for every $f\in \mathcal{F}$. \end{enumerate} \end{definition} \begin{remark}\label{rem1-4}The following statements can be immediately deduced from the above discussions.
\begin{itemize}
\item\label{re:01} A subset $A$ is forward invariant for a relation $f:X\to X$ if and only if its complement is
forward invariant for $f^{-1}$.
\item\label{re:02} A continuous map is a continuous relation.
\item\label{re:04} If $\IFS(X,\mathcal{F})$ is transitive, so is $\IFS(X,\mathcal{F}^{-1})$, because for any two nonempty open sets $U$ and $V$, $f^n(U)\cap V \neq \emptyset$ iff $U \cap f^{-n}(V)\neq \emptyset$.
\item\label{re:05} For an iterated function system $\IFS(X,\mathcal{F})$, if $\text{Trans}(\mathcal{F})$ is dense then $\IFS(\mathcal{F})$ is transitive.
When each $f$ in $\mathcal{F}$ is a continuous relation the converse is true.
\item\label{re:06} If $\IFS(X,\mathcal{F})$ is transitive then every non-empty open and $\mathcal{F}^{-1}$-invariant set $B$ is dense.
When each $f$ in $\mathcal{F}$ is a continuous relation the converse is true.
\item\label{re:07} If $\IFS(X,\mathcal{F})$ is transitive and each $f$ in $\mathcal{F}$ is a continuous relation then $\text{Trans}(\mathcal{F})$ is a dense $G_\delta$ set whenever $X$ is compact metric.
\item\label{re:08} The minimality of $\IFS(X;\mathcal{F})$ is equivalent to the following: every forward invariant non-empty closed subset
for $\IFS(\mathcal{F})$ is the whole space $X$.
\end{itemize} \end{remark} In what follows, we introduce the concept of weak topological exactness which has a key role in our results and is weaker than backward minimality. \begin{definition}Let $\mathcal{F}$ be a finite or infinite family of relations on a topological space $X$. \begin{enumerate}
\item We say that $\IFS(X;\mathcal{F})$ is weak topologically exact if for every open set $U$, there exists a finite sequence
$(T_i)_{i}^N$ in $\mathcal{F}^+$ so that $\bigcup_{i=1}^N T_i(U)$ is dense in $X$.
\item \cite{BFM2018}We say that $\IFS(X;\mathcal{F})$ is topologically exact if for every open set $U$, there exists a sequence
$(T_i)_{i\in\mathbb{N}}$ in $\mathcal{F}^+$ so that $\bigcup_{i} T_i(U)=X$. \end{enumerate} \end{definition}
The following proposition emphasizes that backward minimality is equivalent to weak topologically exactness, in the relations theme. \begin{proposition}\label{stransi} Let $\mathcal{F}$ be a family of relations on a compact normal space $X$ so that for every $f\in\mathcal{F}$, $f^{-1}$ is a continuous relation. If $\IFS(\mathcal{F})$ is backward minimal and $U$ is a nonempty open set then there exists $T_1,\dots,T_N\in \mathcal{F}^+$ such that $\bigcup_{n=1}^N T_{n}(U)=X$. Conversely, if for every nonempty open set $U$ there exists $T_1,\dots,T_N$ in $\mathcal{F}^+$ such that $\bigcup_{i=1}^{N} T_{i}(U)$ is dense in $X$ then $\IFS(\mathcal{F})$ is backward minimal. \end{proposition} \begin{proof} Assume that $\IFS(\mathcal{F})$ is backward minimal and $U$ is a nonempty open set. Since $\IFS(\mathcal{F}^{-1})$ is minimal, hence for each $x \in X$ there exist $T \in \mathcal{F}^+$ and $y \in T^{-1}(x)$ so that $y \in U$ and therefore $x \in T(U)$. By this fact and since for every $T\in \mathcal{F}^+$, the relation $T^{-1}$ is continuous and hence $T$ is open, so $\{T(U) : T \in \mathcal{F}^+\}$ is an open cover and therefore, by compactness of $X$, one has a finite sub-cover, i.e. there exist $T_i \in \mathcal{F}^+$, $i=1, \ldots, \ell$, such that $X=\bigcup_{i=1}^\ell T_i(U)$.
Conversely, assume $\IFS(\mathcal{F})$ is not backward minimal. Then there exists a closed, non-empty proper subset $A$ of $X$ so that $A$ is invariant for $\IFS(X;\mathcal{F}^{-1})$. Let $U$ be a non-empty open set whose closure $B$ is disjoint from $A$. Since for every $f\in\mathcal{F}$, $f^{-1}$ is a continuous relation, the set $\bigcup_{i=1}^\ell T_i(U)$ is an open subset of $X$ and a subset of the closed set $\bigcup_{i=1}^\ell T_i(B)$, for every finite sequence $(T_i )_{i=1}^\ell$ in $\mathcal{F}^+$. On the other hand, $\bigcup_{i=1}^\ell T_i(B)$ is disjoint from $A$ and so $\bigcup_{i=1}^\ell T_i(U)$ is not dense. \end{proof} \begin{remark} Notice that the above argument works for IFSs generated by a family of relations. For the IFSs generated that are generated by maps, the arguments and results may be slightly different. For example, consider the expanding map $f:\mathbb{S}^1\to \mathbb{S}^1;\ x\mapsto 2x\ (\text{mod} 1)$. This map is topologically exact and weak topologically exact but the relation $f^{-1}:\mathbb{S}^1\to \mathbb{S}^1$ is not a map and so we can not say about the backward minimality of $(\mathbb{S}^1,f)$ as a map. Thus, it can not be considered as a counterexample for Proposition \ref{stransi}, in some sense. \end{remark} \begin{proposition} Let $\mathcal{F}$ be a family of relations on a Lindel\"{o}f space $X$ so that for every $f\in\mathcal{F}$, $f^{-1}$ is a continuous relation. If $\IFS(\mathcal{F})$ is backward minimal then it is topologically exact. \end{proposition} \begin{proof} For given a nonempty open set $U$, let $A= X \setminus \bigcup_{T \in \mathcal{F}^+}T(U)$. By Remark \ref{rem1-4}, $A$ is a backward invariant closed subset of $X$. Also, it is clear $A \neq X$ (since $U \neq \emptyset$). If the iterated function system $\IFS(\mathcal{F})$ is backward minimal, then we have that $A=\emptyset$ and so, $X=\bigcup_{T \in \mathcal{F}^+}T(U) $. Since $X$ is a Lindel\"{o}f space, there exists a sequence $(T_i)_{i\in\mathbb{N}}$ in $\mathcal{F}$ such that $\bigcup_{i\in\mathbb{N}} T_{i}(U)=X$ and the proof is completed. \end{proof}
In Section \ref{exam}, we give an example of non-minimal weak topologically exact IFS generated by maps which is backward minimal. This example is important from another point of view. In fact, it provides a non-minimal sensitive system which is not an $M$-system, see \cite{km}.
\section{Equicontinuity and Sensitivity for iterated function systems} In this section, we give several sufficient conditions for sensitivity of IFSs. To this end, we introduce the concept of sensitivity for IFSs which is a generalized version of the existing definition for cascades.
Let $\mathcal{F}=\{f_i: X \to X:\ i=1,\dots,k\}$ be a finite family of relations defined on a compact metric space $(X,d)$. Symbolic dynamic is a way to represent the elements of $\mathcal{F}^+$. Indeed, consider the product space $\Sigma^+_k = \{1,\dots, k \}^\mathbb{N}$. For any sequence $\omega=(\omega_1\omega_2\dots\omega_n\dots)\in \Sigma_k^+$,
take $f^{0}_{\omega}:=Id$ and $$
f^n_{\omega}(x)=f^{n}_{\omega_{1}\dots\omega_{n}}(x)=f_{\omega_n}\circ f_\omega^{n-1}(x); \
\ \forall \ n\in \mathbb{N}. $$ Obviously, $f^n_{\omega}=f^{n}_{\omega_{1}\dots\omega_{n}}=f_{\omega_n}\circ f_\omega^{n-1}\in \mathcal{F}^+$, for every $ n\in \mathbb{N}$.
Define the metric \begin{equation}\label{metric} d_{\mathcal{F}}(x_1,x_2):=\sup_{\omega,n}d_H(f_\omega^n(x_1),f_\omega^n(x_2)), \end{equation} where $d_H$ is the Hausdorff distance, or Hausdorff metric. Clearly, $d_{\mathcal{F}} \geq d_H$. Notice that if the generators are maps, then $d_H=d$. \begin{definition} Let $\mathcal{F}$ be a finite family of relations on a compact metric space $X$. \begin{enumerate}
\item A point $x$ is an equicontinuity point when the identity map from $(X; d)$ to $(X; d_{\mathcal{F}})$ is continuous at $x$.
We denote the set of all equicontinuity points of $\IFS(\mathcal{F})$ by $\text{Eq}(X)$.
\item An $\IFS(\mathcal{F})$ is equicontinuous when every
point is an equicontinuity point and so $d_{\mathcal{F}}$ is a metric equivalent to $d$ on $X$.
\item For $\varepsilon > 0$, define $\text{Eq}_\varepsilon$ to be the union of
the open subsets with $d_{\mathcal{F}}$ diameter less than $\varepsilon$.
\item A point $x \in X$ is sensitivity point if it is not an equicontinuity point.
\item An $\IFS(\mathcal{F})$ is sensitive when there exists $\varepsilon > 0$ such that every nonempty open subset has
$d_{\mathcal{F}}$ diameter at least $\varepsilon$, i.e. $\text{Eq}_\varepsilon=\emptyset$. \end{enumerate}
\end{definition} \begin{remark}\label{rem21} Notice that \begin{itemize}
\item an $\IFS(\mathcal{F})$ is equicontinuous if and only if $\text{Eq}(X)=X$;
\item $Eq(X)=\bigcap_{\varepsilon>0}Eq_\varepsilon$. \end{itemize} \end{remark} Hereafter, let $\mathcal{F}=\{f_i: X \to X:\ i=1,\dots,k\}$ be a finite family of continuous maps defined on a metric space $(X,d)$, unless otherwise stated. Let $X=\bigcup_{i=1}^kf_i(X)$. Consider the multifunction $$ F:X\to \mathcal{P}(X);\ x\mapsto\bigcup_{i=1}^kf_i(x),$$ where $\mathcal{P}(X)$ is the set of all nonempty subsets of $X$. The image of a nonempty $B\in \mathcal{P}(X)$ under $F$ is $F(B) :=\bigcup_{b\in B} F(b)$. By this, we can consider $(X,F)$ as a dynamical system. Obviously, $F$ is a continuous relation. Also, $F$ is onto which means that for every $y\in X$ there exists $x\in X$ so that $y\in F(x)$ (or $\bigcup_{x\in X} F(x)=X$).
Notice that some topological properties of $(X,F)$ and $\IFS(\mathcal{F})$ are the same. For instance: \begin{itemize}
\item $A$ is an invariant set for $\IFS(\mathcal{F})$ if and only if it is an invariant set for $F$;
\item $(X,F)$ is transitive (resp. minimal) if and only if $\IFS(\mathcal{F})$ is transitive (resp. minimal);
\item $\text{Trans}(\mathcal{F})$=$\text{Trans}(F)$. \end{itemize} So, by similar arguments to \cite{AAB}, one can obtain some of its results for a cascade system $(X,F)$ and consequently, they hold for $\IFS(X;\mathcal{F})$. \begin{proposition}\label{openmap} Let $\mathcal{F}$ be a finite family of continuous maps defined on a metric space $(X,d)$ with $X=\bigcup_{f\in\mathcal{F}}f(X)$ and $F$ be as mentioned above. \begin{enumerate}
\item If $F$ is an open relation then for every $\varepsilon>0$, $\text{Eq}_\varepsilon(X) $ is open and $F$-invariant.
\item If $(X,F)$ is transitive with open relation $F$ so that $Eq(X)\neq \emptyset$ then $Eq(X)$ is a dense $G_\delta$ set whenever $X$ is a complete metric space.
\item If $(X,F)$ is transitive then $\text{Eq}(X)\subseteq\text{Trans}(F)$.
\item If $F$ is an open relation and $\text{Eq}(X)\neq\emptyset$ then $\text{Trans}(F^{-1})\subseteq \text{Eq}(X)$. \end{enumerate} \end{proposition} \begin{proof} \begin{enumerate}
\item That is an immediate consequence of the definition.
\item Indeed, $Eq(X)=\bigcap_{k\in \mathbb{N}}Eq_{1/k}$ and $(X,F^{-1})$ is transitive.
Since $Eq_{1/k}(X)$ is open and $(F^{-1})^{-1}$-invariant, the set $Eq_{1/k}$ must be open and dense.
\item Suppose that $x\in \text{Eq}(X)$ and $y$ is an arbitrary point of $X$. For every $\varepsilon>0$, there exists $\delta<\varepsilon$
so that $d_\mathcal{F}$-diameter $U=B_d(x,\delta)$ less than $\varepsilon$.
Since $(X,F)$ is transitive, there exists $T\in \mathcal{F}^+$ so that $T(U)\bigcap B(y,\varepsilon/2)\neq\emptyset$.
Therefore, $\mathcal{O}^+_\mathcal{F}(x)\bigcap B(y,\varepsilon)\neq\emptyset$.
\item Let $x$ be an arbitrary point in $\text{Trans}(F^{-1})$ and $\varepsilon>0$ be given.
Since $\text{Eq}_\varepsilon(X)$ is open and $x\in \text{Trans}(F^{-1})$, one can have $F^{-n}(x)\bigcap\text{Eq}_\varepsilon(X)\neq\emptyset$.
Take $y\in F^{-n}(x)\bigcap\text{Eq}_\varepsilon(X)$.
When $\text{Eq}_\varepsilon(X)$ is $F$-invariant, $x\in F^{n}(y)\bigcap\text{Eq}_\varepsilon(X)$ and so $x\in \text{Eq}_\varepsilon(X)$, for
every $\varepsilon>0$. \end{enumerate} \end{proof} Summing up what in the previous corollary, we get the next result. \begin{corollary}\label{corcor} If $(X,F)$ transitive with open relation $F$ and $\text{Eq}(X)\neq\emptyset$ then $$ \text{Trans}(F^{-1})\subseteq \text{Eq}(X)\subseteq\text{Trans}(F). $$ \end{corollary} \begin{proposition}\label{review} Suppose each $f_i:X\to X$ is a continuous open surjective map, for $i=1,\dots,k$, and $(X,F)$ is not sensitive. If $(X,F^{-1})$ is minimal, then each $f_i$ is a homeomorphism and $(X,F)$ is equicontinuous and minimal. \end{proposition} \begin{proof} Since $(X,F)$ is not sensitive, $\text{Eq}_\varepsilon(X)\neq \emptyset$, for all $\varepsilon>0$. By Proposition \ref{openmap}, each $\text{Eq}_\varepsilon(X)$ is $F$-invariant. So, minimality of $(X,F^{-1})$ and Corollary \ref{corcor} imply that $\text{Eq}_\varepsilon(X)=X$, for every $\varepsilon>0$. Hence, $(X,F)$ is equicontinuous.
To prove the minimality of $(X,F)$, we redundant one of our assumptions. In fact, the openness of $F$ is not a necessary condition in the following results. Minimality can be followed from the following result. \begin{sublemma}\label{22} If $(X,F)$ is transitive and equicontinuous then it is minimal. \end{sublemma} \begin{proof} Suppose that $A$ is a nonempty closed $F$-invariant subset of $X$. Clearly, for every $i=1,\dots,k$, $f_i(A)\subset A$. For arbitrary $\varepsilon>0$, the set $A_\varepsilon=\{x : d_{\mathcal{F}} (x,A) < \varepsilon\}$ is open and $F$-invariant. By Remark \ref{rem1-4}, $F^{-1}$ is transitive. Thus, for arbitrary $\varepsilon>0$, $A_\varepsilon$ is dense and hence $\overline{A_\varepsilon}=X$, for arbitrary $\varepsilon>0$. It follow that $A=X$ and so $F$ is minimal. \end{proof} At the end, notice that each $f_i$ is equicontinuous when $(X,F)$ is equicontinuous. The so-called Hippopotamus Hide Theorem \cite{A0} implies that each $f_i$ is an isometry and so is a homeomorphism. \end{proof} \begin{corollary} Under the assumption of Proposition \ref{review}, $(X,F^{-1})$ is equicontinuous. \end{corollary}
\begin{corollary} Let $\mathcal{F}$ be a finite family of maps on a space $X$. If $\IFS(\mathcal{F})$ is weak topologically exact but not minimal then $(X,F)$ can not be equicontinuous. \end{corollary} The following theorem says more than the mentioned corollary and it insures that weak topologically exact systems with non-minimality assumption are sensitive. Also, in the following theorem, we do not need where our generators are continuous. \begin{theorem}\label{nonmi} Let $\mathcal{F}=\{f_1,\dots,f_k\}$ be a finite family of maps on a compact space $X$ so that $f^{-1}_i$ is continuous relations, $i=1,\dots,k$. If $\IFS(\mathcal{F})$ is weak topologically exact but not minimal then $\IFS(\mathcal{F})$ is sensitive. \end{theorem} \begin{proof} Let $y\in X$ with $\overline{\mathcal{O}^+_{\mathcal{F}}(y)}\neq X$ and $z\in X \setminus \overline{\mathcal{O}^+_{\mathcal{F}}(y)}$. Take $V=B(z,\delta)$ where $\delta=\frac{1}{4}d(z,\overline{\mathcal{O}^+_{\mathcal{F}}(y)}).$ Let $x \in X$, and let $U\subset X$ be an arbitrary neighborhood of $x$. Since $\IFS(\mathcal{F})$ is weak topological exact, there exist $T_1,\dots,T_\ell$ in $\mathcal{F}^+$ so that the following holds:
$(H_1) \ \ X\subseteq \overline{\bigcup_{i=1}^\ell T_i(U)}.$\\ Furthermore, for every $i\in \{1,\dots,\ell\}$, there exist $T^{(i)}_1,\dots,T^{(i)}_{\ell_i}$ so that
$(H_2) \ \ X\subseteq \overline{\bigcup_{j=1}^{\ell_i} T^{(i)}_j(T_i(U))}; \ \forall \ 1\leq i\leq\ell.$\\
To verify the condition $(H_2)$ notice that $T_i^{-1}$ is a continuous relation,
hence $T_i(U)$ is a nonempty open subset of $X$. Therefore, $(H_2)$ follows by $(H_1)$.
Take $t:= \ell+\max \Xi$, where $\Xi=\{|T^{(i)}_j|: 1\leq i\leq\ell\ \& \ 1\leq j\leq \ell_i\}$
and $|T|$ is the length of $T$ (we say that the length of $T$ is equal to $n$ and write $|T|=n$ if $T$ is the combination of $n$ elements of the generating set $\mathcal{F}$). Choose a neighborhood $W$ around $y$ such that $\text{diam}(f_\rho^i(W))<\delta$, for every $\rho\in \Sigma^+_k$ and for every $i=0, 1, \dots, t$. Moreover, by the choice of $\delta$ we may assume that $d(f_\rho^i(W),V)\geq 2\delta$, for $i=0, 1, \ldots,t$ and for each $\rho\in\Sigma^+_k$.
The condition $(H_1)$ ensures that for some $1\leq s\leq \ell$, $T_{s}(U)\bigcap W\neq \emptyset$. On the other hand, according to $(H_2)$, there exists $1\leq j_0\leq \ell_s$ so that $T^{(s)}_{j_0}(T_s(U))\bigcap V\neq \emptyset$. Also, one has that $T^{(s)}_{j_0}(T_s(U))\bigcap T^{(s)}_{j_0}(W)\neq \emptyset$. By the choice of $W$, we have $d(T^{(s)}_{j_0}(W),V)\geq 2\delta$. Thus, $\text{diam}(T^{(s)}_{j_0}(T_s(U)))>\delta$. Since $x$ and $U$ are arbitrary and $\delta$ does not depend on $x$, the proof is complete. \end{proof} Now, consider a Barnsley $\IFS(X;\mathcal{F})$ i.e. $f_i$ is a contraction of the metric $d$ on $X$, for each $i=1,\dots,k$, where $(X,d)$ is a compact metric space. We remark that $f$ is contractive if there exists $\lambda<1$ such that $d(f(x_1),f(x_2))\leq\lambda d(x_1,x_2)$. Also, let $X=\bigcup_{i=1}^kf_i(X)$. In this case, $d_\mathcal{F}=d$ and so $\IFS(\mathcal{F})$ is equicontinuous. The induced map on the space of nonempty closed subsets of $X$ equipped with Hausdorff metric is contraction and so by the Banach fixed point theorem it has a unique fixed point. Since we have assumed that $F$ is surjective, the unique fixed point is $X$. It follows that $(X,F)$ is minimal and so $(X, F^{-1})$ is transitive but it need not be minimal. By Theorem \ref{nonmi}, $(X, F^{-1})$ is sensitive. In this way, a natural question arises for $\IFS(\mathcal{F})$ generated by homeomorphisms where $\text{Eq}(X)$ is a dense $G_\delta$-subset: under which assumptions $\IFS(X;\mathcal{F}^{-1})$ is sensitive?
We finalize this section with another sufficient condition for sensitivity. Let $f$ be a map on a metric space $(X,d)$ and $x \in X$ be a fixed point of $f$. Then the stable set $W^s(x)$ and unstable set $W^u(x)$ are defined, respectively, by $$W^s(x)=\{y \in X: f^n(y)\to x, \ \textnormal{as} \ n \to +\infty\},$$ $$W^u(x)=\{y \in X: f^n(y)\to x, \ \textnormal{as} \ n \to -\infty\}.$$ \begin{definition} We say that $x$ is an attracting (or repelling) fixed point of $f$ if the stable set $W^s(x)$ (or unstable set $W^u(x)$) contains an open neighborhood $B$ of $x$. Then $B$ is called local basin of attraction (or repulsion) of $x$. \end{definition} \begin{theorem}\label{pro1} Let $\mathcal{F}$ be a finite family of open continuous maps. If $\IFS(\mathcal{F})$ is a weak topologically exact iterated function system and the associated semigroup $\mathcal{F}^+$ has a map with a repelling fixed point then $\IFS(\mathcal{F})$ is sensitive on $X$. \end{theorem} \begin{proof} Suppose that the associated semigroup $\mathcal{F}^+$ has a map $h$ with a repelling fixed point $q$ and the local repulsion basin $B$. Suppose that $\textnormal{IFS}(\mathcal{F})$ is not sensitive. Then for each $n \in \mathbb{N}$, there is a non-empty open subset $U_n$ of $X$ such that the following holds: \begin{equation}\label{e11} diam(f_\omega^i(U_n))< \frac{1}{n}; \ \ \forall\omega \in \Sigma_k^+ \ \forall i \in \mathbb{N}. \end{equation} Take $n$ large enough, then for each $y, z \in B$ with $y\neq z$, there exists $m \in \mathbb{N}$ sufficiently large so that \begin{equation}\label{e12} d(h^m(y),h^m(z)) \geq \frac{1}{n}. \end{equation} Let us fix the integer $n \in \mathbb{N}$ for which (\ref{e12}) holds. Since $\textnormal{IFS}(\mathcal{F})$ is weak topologically exact and hence by Proposition \ref{stransi} it is backward minimal, there is $T \in \mathcal{F}^+$ and $q^{\prime}\in T^{-1}(q)$ so that $q^{\prime} \in U_n$. By continuity there is $\delta > 0$ such that $B_\delta(q) \subset B$ and there exists $B^{\prime} \subset T^{-1}(B_\delta(q))$, with $q^{\prime} \in B^{\prime}$ and $T(B^{\prime})=B_\delta(q)$, so that $B^{\prime}\subset U_n$ which implies that $B_\delta(q) \subset T(U_n)$.\\
Let us take $y,z \in B_\delta(q)$, $y^{\prime}\in T^{-1}(y)\cap B^{\prime}$ and $z^{\prime}\in T^{-1}(z)\cap B^{\prime}$. Then $y^{\prime},z^{\prime} \in B^{\prime} \subset U_n$ and hence by (\ref{e12})
$$d(h^m \circ T(y^{\prime}),h^m \circ T(z^{\prime})) = d(h^m(y),h^m(z)) \geq \frac{1}{n}$$ which contradicts (\ref{e11}). \end{proof}
\section{Examples}\label{exam} The first example belong to the folklore. \begin{example} Let us equipped the product space $\Sigma^+_2=\{1,2\}^\mathbb{N}$ to the metric $d(\eta,\omega)=\inf\{2^{1-n}:\ \eta_i=\omega_i, \forall i<n\}$. For $\theta=1,2$, let $(f_\theta(\eta))_1=\theta$ and $(f_\theta(\eta))_i=\eta_{i-1}$. Take $\mathcal{F}=\{f_1,f_2\}$. The inverse $F^{-1}$ on $\Sigma^+_2=\{1,2\}^\mathbb{N}$ is shift map given by $(F^{-1}(\eta))_i=(\sigma(\eta))=\eta_{i+1}$ Thus, the periodic points of $F^{-1}$ is dense and $F^{-1}$ is sensitive. However, the $\IFS(\mathcal{F})$ is minimal and equicontinuous. \end{example} It is a well known fact that for ordinary dynamical systems, the minimality of a map $f$ is equivalent to that of $f^{-1}$. Nevertheless this is not the case for iterated function systems as Kleptsyn and Nalskii pointed at ~\cite[pg.~271]{KN04}.
However, they omitted to include any example of forward but not backward minimal IFS. Recently, the authors in \cite{BG}, illustrated an example of forward but not backward minimal IFS which we expose below. That lead to an example of backward minimal IFS which is not forward minimal. So, it is sensitive. Since the previous examples of non-minimal sensitive systems (see, \cite{AAB,GW}) are $M$-system, this example maybe more valuable. Indeed, our example is non-minimal sensitive system which is not an $M$-system. Recall that $(S,X)$ is an \emph{$M$-system} if the set of almost periodic points\footnote{ A point $x$ is called almost periodic if the subsystem $\overline{\mathcal{O}^+_\mathcal{F}(x)}$ is minimal and compact.} is dense in $X$ (the Bronstein condition) and, in addition, the system is transitive. \begin{example} Consider a symmetric $\IFS(\mathcal{F})$ generated by homeomorphisms of the circle. Then, one of the following possibilities holds ~\cite{Navas,Gh01}: \begin{enumerate}[itemsep=0.1cm] \item there is a point $x$ so that $\text{Card}(\mathcal{O}^+_\mathcal{F}(x))<\infty$; \item $\IFS(\mathcal{F})$ is minimal; or \item there is a unique invariant minimal Cantor set $K$ for $\IFS(\mathcal{F})$, that is, $$ g(K)=K \ \ \text{for all $g\in \mathcal{F}$} \quad \text{and} \quad K=\overline{\mathcal{O}^+_\mathcal{F}(x)} \ \
\text{for all $x\in K$}. $$. \end{enumerate}
The Cantor set $K$ in the above third conclusion is usually called \emph{exceptional minimal set}. By~\cite[Exer.~2.1.5]{Navas}, one can ensure that there exists a symmetric semigroup $\mathcal{F}^+$ generated by homeomorphisms $f_1,\dots,f_k$ of $\mathbb{S}^1$ which admits an exceptional minimal set $K$ such that the orbit of every point of $\mathbb{S}^1\setminus K$ is dense in $\mathbb{S}^1$. So, the closed invariant subsets of $\mathbb{S}^1$ for $\IFS(\mathcal{F})$ are $\emptyset$, $K$ and $\mathbb{S}^1$.
Now, consider any homeomorphism $h$ of $\mathbb{S}^1$ such that $h(K)$ strictly contains $K$. Then the IFS generated by $f_1^{-1},\ldots,f_n^{-1},h^{-1}$ is backward minimal but not forward minimal. Therefore, the IFS generated by $f_1^{-1},\ldots,f_n^{-1},h^{-1}$ is weak topological exact and sensitive. Also, one can easily check that the IFS generated by $f_1^{-1},\ldots,f_n^{-1},h^{-1}$ is not an $M$-system. \end{example} As a consequence of the previous example we get the next result. \begin{corollary} There exists a weak topologically exact non minimal IFS which is sensitive but it is not an $M$-system. \end{corollary}
Finally, we provide another example of non-minimal topologically exact system which is not an $M$-system. \begin{example} Suppose $f$ is a north-south pole homeomorphism on the circle. By this we mean that the nonwandering set of $f$, $\Omega(f)$, consists of one fixed source, $q$, one fixed sink, $p$. Then $$\Omega(f)=\{p\}\cup\{q\}.$$ The salient feature of the north-south pole homeomorphism is that for every $x\in \mathbb{S}^1\setminus\Omega(f)$, $f^n(x)\to p$ and $f^{-n}(x)\to q$ as $n\to+\infty$. It is observed that $\mathbb{S}^1\setminus\{p,q\}$ composed of exactly two connected pieces $U_1$ and $U_2$. Now, consider continuous maps $h$, $h_1$ and $h_2$ of $\mathbb{S}^1$ such that \begin{enumerate}
\item $h(x)=p$, for every $x\in \mathbb{S}^1$,
\item $h_1(p)=p=h_2(p)$,
\item there exist connected closed set $I$ of the circle so that $q\in I$ and $p$, $q$ are not boundary points for $I$,
\item $\IFS(I,h_1|_I,h_2|_I)$ is minimal,
\item there exist $a\in I\cap U_1$ and $b\in I\cap U_2$ so that $f(a),f(b)\in I$. \end{enumerate} Take $\mathcal{F}=\{f,f^{-1},h,h_1,h_2\}$. One can check $$
\overline{\mathcal{O}^+_\mathcal{F}(x)}=\mathbb{S}^1;\ \ \ \ \forall \ x\in \mathbb{S}^1\setminus \{p\}. $$ Also, $g(p)=p$, for every $g\in \mathcal{F}^+$. Therefore, by Definition \ref{def1111}, the $\IFS(\mathbb{S}^1;\mathcal{F})$ is neither forward minimal nor backward minimal. However, there are two important facts: \begin{enumerate}
\item $\IFS(\mathbb{S}^1;\mathcal{F})$ is topologically exact,
\item $\IFS(\mathbb{S}^1;\mathcal{F})$ is not an $M$-system. \end{enumerate}
Indeed, $x$ is not almost periodic point when $x\neq p$ i.e. $\overline{\mathcal{O}^+_\mathcal{F}(x)}$ is not minimal for $x\neq p$. \end{example}
\section*{Acknowledgments} We would like to thank the anonymous reviewer whose comments and remarks improved the results and presentation of the paper.
\end{document}
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\begin{document}
\title{Parameterized Fine-Grained Reductions}
\begin{abstract} During recent years the field of fine-grained complexity has bloomed to produce a plethora of results, with both applied and theoretical impact on the computer science community. The cornerstone of the framework is the notion of fine-grained reductions, which correlate the exact complexities of problems such that improvements in their running times or hardness results are carried over. We provide a parameterized viewpoint of these reductions (PFGR) in order to further analyze the structure of improvable problems and set the foundations of a unified methodology for extending algorithmic results. In this context, we define a class of problems (FPI) that admit fixed-parameter improvements on their running time. As an application of this framework we present a truly sub-quadratic fixed-parameter algorithm for the orthogonal vectors problem. Finally, we provide a circuit characterization for FPI to further solidify the notion of improvement. \end{abstract}
\section{Introduction} Fine-Grained Complexity deals with the exact complexity of problems, and establishes a web of refined reductions, that preserve exact solving times. While many of the key ideas come from well-known frameworks (NP-completeness program, parameterized algorithms and complexity, etc.), this significant new perspective has emerged only recently \cite{Wil15}.
The main question posed by this new field is ``\textit{given a problem known to be solvable in $t(n)$ time, is there an $\varepsilon>0$ such that it can be solved in $t^{1-\varepsilon}(n)$?}''. In the case such a result exists, we can connect this improvement to algorithmic advances among different problems. In the case it does not, we can establish conditional lower bounds based on this hardness, as is usually done with popular conjectures \cite{AWY18,WW18}. Several such conjectures are used, such as the Orthogonal Vectors Conjecture (OVC), the Strong Exponential Time Hypothesis (SETH), the APSP conjecture etc. It has been shown that OVC is implied by SETH \cite{Wil05}, and the variants and consequences of both conjectures have been extensively studied.
A main difference between the fine-grained approach and classical complexity theory is that all NP-complete problems form an equivalence class modulo polynomial-time reductions. On the contrary, fine-grained reductions produce a much more complex web. Many problems stem from SETH/OVC, others from the 3-SUM conjecture (especially Computational Geometry problems \cite{GAJ012}), and very few equivalence classes are known (a significant exception is the equivalence class for APSP \cite{AGW15}, \cite{WW18}). These observations raise questions concerning the structural complexity of fine-grained reducibility, as has traditionally been the case in other fields of complexity theory: Conditional irreducibility results, the morphology of the equivalence classes formed, fine-grained completeness notions, consequences of partitioning problems into classes etc., propose a fine-grained structural complexity program.
On the other hand, the parameterized point of view is dominant in theoretical computer science during the last decades. ETH and SETH were introduced in that context, and used widely to establish conditional lower bounds (SETH-hardness). Additionally, Fixed Parameter Tractability (FPT)\footnote{A problem is called Fixed Parameter Tractable, if there is a parameterization $k$ such that the problem can be solved in time $f(k)\cdot poly(n)$, for a computable function $f$.}, gave a multi-variable view of complexity theory, as well as the means to \emph{concentrate the hardness} of problems to a certain parameter, instead of the input size: many problems have significant improvements on their complexity, if one restricts them to instances having fixed $k$, where $k$ is a parameter of the aforementioned problems. This can be viewed as an indication of the structural importance of $k$ in each problem.
Similar techniques can be used to differentiate versions of problems, as has been seen recently in the case of fine-grained conjectures (e.g. problems on sparse graphs \cite{GIKW17}).
\subsection{Motivation}
The conditional bounds shown by fine-grained reductions stem from relating improvements between the conjectured best running times of two problems. This has resulted in an effort to classify problems either through equivalence or hardness via a minimal element (OV-hardness) \cite{CW19}.
Additionally, most known fine-grained reductions inherently relate the problems in more than trivial ways, also mapping specific parameters of each problem to one another. This could indicate relations between the problems' property of concentrating hardness to certain parameters.
On the other hand, while parameterized complexity has traditionally been concerned with breaching the gap between polynomial and exponential running times, there has recently been interest in fixed-parameter improvements amongst polynomial time problems (sometimes referred to as FPT in P \cite{Wil15,GMN15}).
The ability of fine-grained complexity to express correlations between problems of various running times, comes with some inherent theoretical obstacles. Specifically, the new viewpoint of a problem's "hardness" is associated with the capability to improve its running time. This results in a "counter-intuitive" notion of hard problems, as they frequently correspond to (in classical terms) easy ones. Moreover, the foundation on which this framework is based, allows the computational resources of a fine-grained reduction to change depending on the participant problems. This produces vagueness regarding what would be considered a complexity class compatible with such reductions.
Our concern is to surpass the inherent difficulties of the field towards constructive methods that produce generalizable results, as well as to contribute to the effort of establishing structural foundations for the framework. This could result in furthering our understanding of what constitutes difficulty in computation, as well as to structurally define improvability.
\subsection{Our Results} We introduce Parameterized Fine-Grained Reductions (PFGR), a parameterized approach to fine-grained reducibility that is consistent with known fine-grained reductions and offers (a) tools to study structural correlations between problems, and (b) an extension to the suite of results that are obtained through the reductions. This provides a multi-variate approach of analyzing fine-grained reductions. Additionally, we give evidence that these reductions connect structural properties of the problems, such as the aforementioned concentration of hardness.
We define a class of problems (FPI) that admit parameterized improvements on their respective conjectured best running time algorithms. To this end, we treat improvements in the same way as fine-grained complexity (i.e. excluding polylogarithmic improvements in the running time). This gives us the expressive power to correlate structural properties of problems that belong in different complexity classes.
We prove that this class is closed under the aforementioned parameterized fine-grained reductions, which can be used as a tool to produce non-trivial parameterized algorithms (via the reduction process). We present such an application in the case of the reduction from OV to Diameter, in which we use a fixed parameter (with respect to treewidth) algorithm for Diameter to produce a new sub-quadratic fixed parameter algorithm for OV running in time $O\left( d^2(n+d)\log^d(n+d)\right)$ where $d$ is the dimension of the input vectors.
Finally, we use notions from parameterized circuit complexity to analyze membership in this class and introduce a circuit characterization, similar to the one used in the definition of the W-Hierarchy in parameterized complexity \cite{DT11}.
\subsection{Related Work}
The fine-grained reductions literature has quickly grown over the recent years (see \cite{Wil15} for a survey). The basis for reductions have been some conjectures that are widely considered to be true. Namely, the 3SUM, APSP, HITTING SET \& SETH conjectures, which are associated with the respective plausibility for improvement of each problem.
A large portion of known reductions stem from the Orthogonal Vectors problem, which is known to be SETH-hard \cite{Wil05}, thus OV plays a central role in the structure of the reductions web. Different versions of the OV conjecture were studied, usually parameterized by the dimension \cite{ABDN18}.
The logical expressibility of problems similar to OV were studied \cite{GIKW17}, research that created a notion of hardness for problems in first-order logic, and introduced various equivalence classes (\cite{GIKW17,GI19,CW19}) concerning different versions of the OV problem, with significant applications to other fields (such as Computational Geometry). Additionally, the OV conjecture was studied in restricted models, such as branching programs \cite{KW19}.
Structural implications of fine-grained irreducibility and hypotheses were studied, culminated to new conjectures, like NSETH \cite{CGIMPS16}. New implications of refuting the aforementioned hypotheses on Circuit Lower Bounds were discovered \cite{JMV15,AHWW16,AB18}, and the refutation of SETH would imply state-of-the-art lower bounds on non-uniform circuits. Recently, fine-grained hypotheses were connected to long-standing questions in complexity theory, such as Derandomization of Complexity Classes \cite{CIS18}.
The parameterized analysis of algorithms, one of the most active areas in theoretical and applied computer science, has been frequently used to provide tools in fine-grained complexity. As such, new conjectures were formed about the solvability of polynomial-time problems in terms of parameters \cite{AWW15}.
In a notable case of similar work \cite{BK18}, the authors analyze the multivariate (parameterized) complexity of the longest common subsequence problem (LCS), taking into account all of the commonly discussed parameters on the problem. As a result, they produce a general conditional lower bound accounting for the conjunction of different parameterized algorithms for LCS: Unless SETH fails, the optimal running time for LCS is $\left( n+min\{d,\delta \Delta,\delta m\}\right)^{1\pm o(1)} $ where $d,\delta,\Delta,m$ are the aforementioned parameters. Note that our work is in a different direction to this result. Instead of separately reducing SETH to each different parameterized case, we give the means to show correlations between parameters in such reductions, i.e. with this framework one can analyze a single reduction to show multiple dependencies between parameterized improvements for each problem. While this automatically produces several conditional bounds among parameterizations of problems, it proves most useful in the opposite direction, namely to transfer improvements between problems and thus derive new parameterized algorithms.
\section{Preliminaries} We denote with $[n]$, for $n\in\mathbb{N}$, the set $\{1,\ldots,n\}$.
\begin{definition}[Generalized Parameterized Languages] Let $L \subseteq \Sigma^*$, and $k_1,\dots,k_\ell$ parameterization functions, $k_i:\Sigma^* \to \mathbb{N}$, $1\le i \le \ell$. Let $\langle L,k_1,\dots, k_{\ell}\rangle$ denote the corresponding parameterized language. \end{definition} For simplicity, we will use $\langle L \rangle$ to abbreviate $\langle L,k_1,\dots, k_{\ell}\rangle$, and $I_L$ to denote an input instance for $\langle L \rangle$.
Note here the divergence from the classical definition, that associates each problem with only one parameter \cite{DT11}. We prefer the generalized version that allows us to describe simultaneously several structural measures of the problem, such as number of variables, number of nodes and more complex ones. For each one of those parameters we assume that there exists an index $j \in \{1, \ldots \ell \}$ such that $k_j$ corresponds to the mapping of the input instance $I_L$ to this specific parameter. In this way we can not only isolate and analyze different characteristics of structures but also treat each of these measures individually.
\begin{definition}[OV]
Define the Orthogonal Vectors problem ($OV$) as follows: Given two sets $A,B \subseteq \{0,1\}^d$, with $|A|=|B|=n$, are there two vectors $a\in A$, $b\in B$, such that $a\cdot b=\sum_{i=1}^d a[i]\cdot b[i] = 0$? \end{definition}
\begin{definition}[$3/2$-Approx-Diameter] Given a graph $G=(V,E)$, approximate its diameter, i.e. the quantity $\max_{u,v \in V}d(u,v)$, within a factor $3/2$. \end{definition}
We will also define the notion of treewidth as we will later present a result that utilizes it as a graph parameter.
\begin{definition}[Treewidth] A \textbf{tree decomposition} of a graph $G = (V, E)$ is a tree, $T$, with nodes $X_1,X_2 \ldots X_n$ (called bags), where each $X_i$ is a subset of $V$, satisfying the following properties: \begin{itemize}
\item The union of all sets $X_i$ equals $V$. That is, each graph vertex is contained in at least one tree node.
\item The tree nodes containing vertex v, form a connected subtree of T.
\item For every edge $(v, w)$ in the graph, there is a subset $X_i$ that contains both $v$ and $w$. \end{itemize} The \textbf{width} of a tree decomposition is the size of its largest set $X_i$ minus one. The \textbf{treewidth} $tw(G)$ of a graph $G$ is the minimum width among all possible tree decompositions of $G$. \end{definition}
For more information on parameterized complexity, and treewidth the reader is referred to \cite{DT11, DF13}.
We will utilize the following notions from circuit complexity theory (for more details the reader is referred to Ch. 6 of \cite{AB09}).
\begin{definition}[Circuit Complexity]
The circuit-size complexity of a Boolean function ${\displaystyle f} $ is the minimal size (number of gates) of any circuit computing ${\displaystyle f}$. The circuit-depth complexity of a Boolean function ${\displaystyle f}$ is the minimal depth of any circuit computing ${\displaystyle f}$. \end{definition}
We will also use the following definition of Fine-Grained reductions from \cite{WW18}. \begin{definition}[Fine-Grained Reduction] \label{FGR}
Let $a(n), b(n)$ be nondecreasing functions of $n$.
Problem $A$ is $(a,b)$-reducible to problem $B$ (denoted $A \le_{FG} B$), if for all $\varepsilon>0$ there exists a $\delta >0$, and an algorithm $F$ solving $A$ with oracle access to $B$ such that $F$ runs in at most $d\cdot a^{1-\delta}(n)$ time, making at most $k(n)$ oracle queries adaptively (i.e. the $j^{th}$ instance $B_{j}$ is a function of $\{B_i,a_i\}_{1\le i<j}$). The sizes $|B_{i}|=n_i$ for any choice of oracle answers $a_i$, obey the inequality: $$ \sum_{i=1}^{k(n)}b^{1-\varepsilon}(n_i) \le d \cdot a^{1-\delta}(n)$$ \end{definition}
\section{Parameterized Fine-Grained Reductions}
In this section we define Parameterized Fine-Grained Reductions (PFGR), along with some examples of applications, and show their relation to fine-grained reductions.
\begin{definition}[PFGR] Given problems $A$ and $B$ with $a(n)$, $b(n)$ their respective conjectured best running times: We say $\langle A,k_1,\dots, k_{i_A} \rangle \le_{PFG} \langle B,\lambda_1,\dots, \lambda_{i_B} \rangle$ if there exists and algorithm R such that \begin{enumerate} \item For every $\varepsilon>0$ there exists a $\delta>0$ such that R runs in $a^{1-\delta}(n)$ time on inputs $I_A$ of length $n$ by making $q$ query calls to $\langle B \rangle$ with query lengths $n_1,\dots,n_q$, and $\sum_{i=1}^q b^{1-\varepsilon}(n_i) \le c\cdot a^{1-\delta}(n)$, for some constant $c>0$, and R accepts iff $I_A \in \langle A \rangle$.
\item For every query $q_j ,j=1, \ldots, q$, there exists a computable function $g_j:\mathbb{N}^{i_A} \to \mathbb{N}^{i_B }$ defined as $g_j(k_1, \ldots, k_{i_A})= [g_{j,1},g_{j,2},\ldots, g_{j,i_B}]$ such that for every $ \lambda_i \in \langle \lambda_1,\dots, \lambda_{i_B} \rangle$\\$ \lambda_i \le g_{j,i}(k_1, \ldots, k_{i_A})$. \end{enumerate} \end{definition}
\begin{remark} The number of calls is specific to the type of reduction used. In the case of adaptive queries, the number of potential calls could exceed $q$ exponentially (in the worst case). A reasonable objection could arise here since the mappings $g_j$ are not assumed to have any time restriction. However, $g_j$ are not implemented by the reduction algorithm, and are merely correlating functions of the parameters. As such, they do not affect the running time of the reduction. \end{remark}
Additionally, one would suspect that this definition is a limitation on the original fine-grained framework and hence is only satisfied by some of the known reductions. The main problem is that most of the known reductions refer to non parameterized problems. This however can be easily surpassed by our formalization, as we can view these as projections of PFGR (see section \ref{equiv}):
Given a problem $P$ and an input instance $I_P$, the parameterized version of the problem can be produced by extending the input with the computable function that defines each parameter over it. We can now redefine any reductions it took place in, simply replacing the problem with its parameterized version. This essentially provides us with all of the possible parameterizations a problem can have, and uses them as a whole in order to preserve structural characteristics.
While analyzing reductions, some notable cases occur: Firstly, the case of only one query call, as observed in the majority of known fine-grained reductions. Secondly, the case where even though many query calls are made, the constructions of the input instance to $\langle B\rangle$ maintain uniform mappings of the parameters, i.e. $g_j=g_{j'}, \forall j,j' \in [q]$. Lastly, the case where the value of each parameter of problem $\langle B\rangle$ is only related to a single parameter of $\langle A\rangle$, i.e. $g_{j,i}:\mathbb{N} \to \mathbb{N}$. \footnote{this case is especially useful in transferring parameterized improvements, as we will see in section \ref{closure}.}
We provide some examples to further clarify our definition and notation: \begin{example} Consider the well-studied reduction $CNF\mbox{-}SAT \le_{2^n,n^2}k\mbox{-}OV$, presented in \cite{Wil05}. It is apparent that through one call, the number of clauses $m$ of the SAT instance corresponds to the dimension $d$ of the OV vectors instance, as well as that the number of variables $n$ is mapped to the number of vectors $N$ via the mapping: $g(n,m)=\langle 2^{n/k},m \rangle=\langle N,d\rangle=I_{OV}$. This means that the input instance to OV will contain $N=2^{n/k}$ vectors of dimension $d=m$. \end{example} This procedure is summarized in the first row of the following table, as well as other indicative reductions, in the same context. For a more detailed analysis of each reduction see the full version.
\begin{table}[h!] \[
\begin{array}{|l|l|c|c|} \hline \mbox{\textbf{Reduction}} & \mbox{\textbf{Mapping}} & \mbox{\textbf{calls}} & \mbox{\textbf{Ref.}} \\ \hline I_{SAT} \le I_{OV} & g(n,m): N = 2^{n/k}, d = m & 1 & \cite{Wil05} \\ \hline I_{APSP} \le I_{\text{MPProd}} & g(\text{Nodes}) : (n \times n , n \times n) & \lceil \log n \rceil & \cite{WW18} \\ \hline I_{\text{MPProd}}\le I_{APSP} & g(n_1 \times n_2,n_2 \times n_3): (\text{Nodes} = n_1+n_2+n_3) & 1 & \cite{WW18}\\ \hline I_{\text{MPProd}}\le I_{\text{NegTr}} & g( n_1 \times n_2, n_2 \times n_3): (\text{Nodes} = n_1+n_2+n_3) & \log n &\cite{WW18}\\ \hline I_{\text{NegTr}}\le I_{\text{Radius}} & g(\text{Nodes} = n,\text{Weights} = n^c): (4n,3n^c) & 1 & \cite{AGW15} \\ \hline I_{APNT} \le I_{\text{NegTr}} & g(\text{Nodes} = n) : (\text{Nodes'} = \sqrt[3]{n}) & n^2+\frac{n^3}{\sqrt[3]{n}} & \cite{WW18}\\ \hline \end{array} \] \caption{Some reductions using our notation (\textit{where APNT abbreviates the All-Pairs Negative Triangle problem, NegTr the Negative Triangle, and MPProd the Min-Plus product problem.})}\label{table1} \end{table}
\paragraph*{Consistency with fine grained complexity} \label{equiv} While our framework encapsulates many natural structural properties, there are problems that are fine-grained reducible to each other and either do not have an obvious correlation between their structures, or have connections that are not apparent. We will show here that our definition for Parameterized Fine-Grained Reduction is consistent with those cases, as the set of problems that are reducible to each other via fine grained reductions (denoted as $S_1$) and the respective set for Parameterized Fine Grained Reductions (denoted as $S_2$) are equivalent.
\begin{theorem} Let $S_1:=\{(A,B): A\le_{FG}B \}$ and $S_2:=\{(A,B): A\le_{PFG}B\}.$\\ Then $S_1=S_2$. \end{theorem}
\begin{itemize}
\subparagraph*{}
\item \textbf{{$S_2 \subseteq S_1$}}\\ Firstly, given problems $A$ and $B$ with $a(n)$, $b(n)$ their respective conjectured best running times, if $\langle A \rangle \le_{PFG} \langle B \rangle $, we can simply ignore the parameters involved in the reduction and treat it as a fine-grained reduction between A and B, as the time restrictions enforced in both definitions are identical ($a^{1-\delta}(n)$ bound for the reduction time, and $\sum_{i=1}^{q} b^{1-\varepsilon}(n_i)$ for the calls to problem $B$).
\subparagraph*{} \item \textbf{{$S_1 \subseteq S_2$}} \begin{lemma} Given problems $A$ and $B$ with $a(n)$, $b(n)$ their respective conjectured best running times, if $A\le_{FGR}B$, for every $\lambda_i$ in a given parameterization $\langle B \rangle $, for each query call $q_j$ made in the reduction, there exists a computable function $g_{j,i}:\mathbb{N}^{i_A}\longrightarrow\mathbb{N}$ such that $\lambda_i \le g_{j,i}(k_1,\ldots,k_{i_A})$, and as such $\langle A \rangle \le_{PFG} \langle B \rangle$. \end{lemma}
\begin{proof} We remind the reader here that for $\lambda_i$ to be considered a parameter of a problem $A$, it has to be the output of a computable function on the input of $A$. Hence, every parameter of $B$ has a computable function $f_i(I_B)$ associated with it. Now, since the reduction producing the instances of $B$ is $a^{1-\delta}$-time computable, it can trivially be viewed as a computable function $F$ having as domain field the inputs to problem $A$, and range the input instances of $B$ it produces. Having these, we can simply take the composition of $F$ and each $f_i$ to produce computable functions $f'_i=F\circ f_i$ that produce the aforementioned parameters of $B$. Hence, these parameters can be viewed both as parameters of $B$ and parameters of $A$. For these reasons, the fine-grained reduction can be viewed as $\langle A,k_1,\dots\rangle \le_{PFG} \langle B,\lambda_1,\dots \rangle$ for $k_i=\lambda_i$ (ergo having the identity function as $g_{j,i}$). \end{proof}
\end{itemize}
\section{Fixed Parameter Improvable Problems (FPI)}
In this section we define a class of problems that admit parameterized improvements on their conjectured best running times, prove that this class is closed under PFGR, as well as produce new parameterized improvements as an application of this closure.
\begin{definition}[FPI]
Let $A$ be a problem with conjectured best running time $a(n)$.
Then, $\langle A \rangle $ has the FPI property with respect to a set of parameters $K=(k_1,k_2, \ldots, k_x) \subseteq \langle A \rangle$ \footnote{in this context, $K\subseteq \langle A \rangle$ denotes a set of parameterization functions over the input of $A$.} (denoted
FPI$(A,K)$) if there exists an algorithm solving $\langle A \rangle $ in $O\left( a^{1-\varepsilon}(n)\cdot f(k_1,k_2, \ldots, k_x)\right)$ time, for some $\varepsilon>0$ and a computable function $f$.
\end{definition}
\begin{remark}
For simplicity, in the case of a single parameter, we denote as $FPI(A,k)$ the property $FPI(A,\{k\})$.
\end{remark}
\begin{theorem}\label{NPFPI} For every NP-hard problem $A$ that admits an FPT algorithm w.r.t.\ a parameter $k$, we have that $FPI(A,k)$, unless $P=NP$.
\end{theorem}
\begin{proof} Since all NP-hard problems are conjectured to demand exponential running time, any FPT algorithm that solves them in $O\left( n^c\cdot f(k)\right)$ time for some parameter $k$ can be viewed as an improvement to $O(a^{1-\varepsilon}(n)\cdot f(k))$; the actual improvement in the conjectured running time $a(n)$ is in fact much greater than $a^{\varepsilon}(n)$ . \end{proof}
\begin{corollary} The following problems, parameterized with the respective parameter are FPI: \begin{itemize}
\item $\langle$ Vertex Cover, Solution size $\rangle$
\item $\langle$ SAT, Number of clauses $\rangle$
\item $\langle$ k-knapsack,k $\rangle$ \end{itemize} \end{corollary}
\begin{definition}[Minimum Necessary Set] Let $\langle A,k_1,\dots, k_{i_A} \rangle$, $\langle B,\lambda_1,\dots, \lambda_{i_B} \rangle$ be parameterized problems such that $\langle A,k_1,\dots, k_{i_A} \rangle \le_{PFG} \langle B,\lambda_1,\dots, \lambda_{i_B} \rangle$. We define as $K_\Lambda$ to be the minimum necessary set needed to bound the set of parameters $\Lambda$ of $\langle B \rangle$ with respect to g, ergo the parameter set ${k_1,\ldots,k_x}$ of problem A for which $\exists \lambda_i \in \Lambda, \forall j \in[q]$ such that $g_{j,i}(k_1,\dots, k_{A}) \le h(k_1,\ldots,k_x),~ k_i \in K_\Lambda$ for some computable function $h$. \end{definition}
\begin{theorem}[Closure under PFGR]\label{closure} Let $\langle A,k_1,\dots, k_{i_A} \rangle$, $\langle B,\lambda_1,\dots, \lambda_{i_B} \rangle$ be parameterized problems. If $\langle A,k_1,\dots, k_{i_A} \rangle \le_{PFG} \langle B,\lambda_1,\dots, \lambda_{i_B}\rangle$ and $FPI(B,\Lambda)$, then $FPI(A,K_\Lambda)$, where $K_\Lambda$ is the minimum necessary set of $\Lambda$ w.r.t. $g$. \end{theorem}
\begin{proof} It suffices to prove that there exists an algorithm for $A$ running in $O(a^{1-\varepsilon}(n)\cdot f(k_1,\ldots,k_x))$ for $k_i,~i\in [x]$, parameters of $\langle A \rangle$.
Since $\langle A \rangle \le_{PFG} \langle B \rangle$, then for all $\varepsilon>0$ there exists a $\delta>0$ such that $I_A$ is evaluated by an $a^{1-\delta}(n)$ algorithm using $B$ as an oracle, and $\sum_{i=1}^{q} b^{1-\varepsilon}(n_i) \le a^{1-\delta}(n)$.
Also, $FPI(B,\Lambda)$, so there is an algorithm computing $B$ in $O(b^{1-\varepsilon'}(n)\cdot f(\lambda_1,\ldots,\lambda_{|\Lambda|}))$. We can use this algorithm to resolve the oracle calls in time $\sum_{i=1}^{q} b^{1-\varepsilon'}(n_i)f(\lambda_1,\ldots,\lambda_{|\Lambda|})$.
We can use $K_\Lambda$ to describe the running time for B utilizing function h. As such, the total running time of $A$ is:
$$a^{1-\delta}(n)+\sum_{i=1}^{q} b^{1-\varepsilon'}(n)f(\lambda_1,\ldots,\lambda_{|\Lambda|})
= a^{1-\delta}(n)+\sum_{i=1}^{q} b^{1-\varepsilon'}(n)f(h(k_1,\ldots,k_{|K_\Lambda|})) $$
$$\le a^{1-\delta}(n) \cdot f'(k_1,\ldots,k_{|K_\Lambda|}) $$ Hence, $FPI(A,K_\Lambda)$. \end{proof}
The above result essentially means that any parametric improvement can be carried through a valid PFGR.
\subsection{A subquadratic fixed-parameter algorithm for OV}
We will now provide an analysis of a known reduction from $OV$ to $3/2$-approx-diameter \cite{RW13} using the PFGR framework. Specifically, we show that since $3/2$-approx-diameter admits fixed parameter improvements \cite{AWW15} on the treewidth parameter (hence is in FPI), this can be used to provide a fixed-parameter improvement on the OV problem.
\begin{theorem}\label{ovtodiam} $\langle OV,dimension\rangle$ is PFG-reducible to $\langle 3/2$-approx-diameter$,treewidth\rangle $. \end{theorem}
\begin{proof}
Note that this reduction is implemented using only one call, and we analyze only one parameter. As such, we will simplify the notation of $g_{j,i}$ to $g$.
We begin with the construction given in \cite{RW13}: Given an $OV$ instance with sets $A$, $B$ as input, we create a graph as follows: for every $a \in A$ create a node $a \in G$, for every $b\in B$ create a node $b\in G$, and for every $i \in [d]$ create a node $c_i\in G$, as well as two nodes $x,y$. For every $a\in A$ and $i\in [d]$, if $a[i]=1$ we add the edge $(a,c_i)$. Similarly, for every $b\in B$ and $i\in [d]$, if $b[i]=1$, we add the edge $(b,c_i)$. Also, we add the edges $(x,a)$ for every $a\in A$, $(x,c_i)$ for every $i\in[d]$, $(y,b)$ for every $b\in B$, $(y,c_i)$ for every $i\in [d]$, and $(x,y)$.
It suffices to find which parameter is connected to treewidth via the reduction: As seen in \ref{figureTW}-(a), the graph produced by the reduction has a very specific structure. That is, all nodes of group $A$ are linked exclusively with nodes of group $C$ and with node $x$. Similarly, for group $B$ we have connections to group $C$ and node $y$. Therefore, to produce a tree decomposition of $G$ we can leave the nodes of group $A$ unrelated with those of group $B$.
Now, the specific connections of nodes from the groups $A$, $B$ to the nodes of group $C$ can vary, depending on the form of the OV instance. We can however give an upper bound to the treewidth of $G$ as shown in \ref{figureTW}-(b) by copying the whole group $C$ in all of the decomposition's bags. One can check that each component induced by a label is connected, and that all edges of $G$ are covered by the given bags. This decomposition is of maximum bag size $d+2$ and hence of width $d+1$, where $d$ is the size of group $C$.
As follows from the definition, $d+1$ is an upper bound for the treewidth of $G$. However, another tree decomposition with smaller width could exist. In order to prove that the treewidth of $G$ is exactly $d+1$, we must show that there exist instances of OV that produce graphs through this reduction, corresponding to treewidth $d+1$. Depending on the vectors containing a 1 coordinate in the suitable position, we could end up with a graph containing as a minor a complete $K\{|A|=|B|,|C|\}$ bipartite graph. Since complete bipartite graphs $K_{m,n}$ have treewidth exactly $\min \{m,n\}$, we can deduce that all graphs $G$ produced by this reduction will have treewidth in the worst case $d+1$ (as seen in chapter 10 of \cite{DF13}).
\begin{figure}
\caption{Graph produced by the reduction}
\caption{Tree Decomposition}
\caption{PFG reduction from $OV$ to $3/2$-approx-diameter.}
\label{figureTW}
\end{figure}
Since $d$ is exactly the dimension of the OV instance producing the graph, we can see that there exists a function $g$ that maps the dimension of the OV instance to the treewidth of the $3/2$- approx-dimension.
Therefore, the mapping is $g(\langle OV,d\rangle) = \langle 3/2$-approx-diameter $,treewidth-1\rangle$ \end{proof}
It was shown in \cite{AWW15} that $3/2$-approx-diameter parameterized by treewidth has a subquadratic algorithm. Hence, FPI($3/2$-approx-diameter,treewidth). Now, as follows by theorem \ref{closure} we should expect that FPI(OV,k) where $g(k)=treewidth$. Equivalently, we would expect a parametric improvement to the running time of OV for the instances that are related to the ones of the 3/2-approx-Diameter problem of bounded treewidth.
By theorem \ref{ovtodiam} we have that since $3/2$-approx-diameter has a parameterized improvement for fixed treewidth, OV has a subquadratic algorithm for fixed dimension of the vectors.
We will construct a subquadratic fixed-parameter algorithm, via the process described above.
Specifically, for the reduction time: The graph constructed contained $2n+d+2$ nodes and $2nd+2d+2n$ edges, which can be constructed in $O(nd)$ time from the OV instance.
The resulting graph has $O(n+d)$ nodes, and the decision problem of $3/2$-approx-diameter for this graph also gives an answer for the decision problem of the OV instance.
The parameterized complexity of the algorithm solving diameter is $k^2 n \log^{k-1}n$, where $k$ denotes the treewidth of the graph \cite{AWW15}.
Ergo, since $k=d+1$ (via our reduction) we can use the above to obtain an algorithm for OV running in time: $$(d+1)^2(n+d)\log^{d+1-1}(n+d)=O\left( d^2(n+d)\log^d(n+d)\right) $$
\begin{remark} Through our reduction, this result can be carried out to all problems PFG-reducible to OV, such as SAT or Hitting Set. See the full version for the respective analyses of these reductions. \end{remark}
\begin{remark}
As we have seen, if problem $B$ admits parameterized improvements on parameters $\Lambda$, then through the reduction this can be translated to improvements on problem $A$ and parameters $K_\Lambda$ such that $\lambda_i \le g(k_1,...,k_{|K_\Lambda|})$, for $i\in [|\Lambda|]$. However, whether we can locate which parameters constitute $K_\Lambda$ or not depends on the invertibility of $g$. In the case $g$ is not invertible one can only show the existence of such an algorithm, but not necessarily construct it. Nevertheless, the FPI property still holds through our definition, because we can abuse the notation to interpret each $\lambda_i$ as a parameter of A, as it is a byproduct of the reduction which is \emph{an ($a(n)$-time) computable function on the input of $A$}. \end{remark}
\section{Circuit Characterization of FPI}
We provide a characterization for FPI using circuit complexity. Specifically, it is known that any circuit of size $S(n)$ can be simulated by an algorithm with complexity $O(S(n))$, thus if one can design a circuit with size smaller than the conjectured complexity of the problem, then this can be translated into a faster algorithm.
As such, having a circuit of size $S(n)$, if we can fix any number of parameters $x$ such that the circuit can be seen as having $S'(n)\leq a(n)^{1-\varepsilon}f(k_1,\ldots,k_x)$ size, we can use this circuit to produce a truly sub-$a(n)$ algorithm for our problem.
Nevertheless, the smallest circuit solving the problem may differ from the one produced via a simulation of an algorithm\footnote{which is the only universal way to produce a circuit from an arbitrary algorithm.}. This means that an improvement in the size complexity of the circuit may not be enough to be translated into a more effective algorithm via an inverse simulation. In that case, for the improvement in the size of the circuit to be translated to a faster algorithm, it is necessary to exceed this difference. From now on, when referring to a circuit solving a problem, the reader should consider the one produced by the simulation procedure.
As shown in \cite{Fur82}, we can simulate any algorithm running in time $a(n)$ by a circuit of size $S(n) = a(n)log(a(n))$. Thus, we can use this as an upper bound on the overall size complexity of the circuit produced, to show that an improvement in the size of the circuit $S^{1-\varepsilon}(n)f(k_1,\ldots,k_x)$ (for some $x$) is always sufficient.
\begin{theorem}
Let $A$ be a problem with $a(n)$ conjectured best running time. Then, $FPI(A,K)$ if and only if for the uniform circuit family $\{C_n\}$ of size $S(n)$ computing $A$, for each $n\in\mathbb{N}$, $C_n$ has size $S'(n)=S^{1-\varepsilon}(n)\cdot f(k_1,\ldots,k_{|K|})$, for a computable function $f$. \end{theorem}
\begin{proof}
``$\Rightarrow$'':
\begin{eqnarray*}
S'(n) &=& S^{1-\varepsilon}(n)f(k_1,\ldots,k_{|K|})\\
&=& a^{1-\varepsilon}(n) ( \log a(n))^{1-\varepsilon} \le a^{1-\varepsilon}(n) a^{\delta}(n)f(k_1,\ldots,k_{|K|})\text{, for any }\delta>0. \end{eqnarray*}
If we choose $0<\delta<\varepsilon$, then $a^{1-\varepsilon+\delta}(n)f(k_1,\ldots,k_{|K|})=a^{1-\varepsilon'}(n)f(k_1,\ldots,k_{|K|})$ for $\varepsilon'=\varepsilon-\delta$, which is an FPI improvement on the running time of the algorithm, since we can simulate the circuit of size $S'(n)$ in linear time.
``$\Leftarrow$'':
if there is an algorithm and a parameter set $K$ for which the running time is $a^{1-\varepsilon}(n)f(k_1,\ldots,k_{|K|})$, then we can simulate it with a circuit of size: \begin{align*}
S'(n) &= a^{1-\varepsilon}(n)f(k_1,\ldots,k_{|K|})\log\left( a^{1-\varepsilon}(n) f(k_1,\ldots,k_{|K|})\right)\\ &=
a^{1-\varepsilon}(n)\left(f(k_1,\ldots,k_{|K|})\log a^{1-\epsilon}(n)+f(k_1,\ldots,k_{|K|})\log(f(k_1,\ldots,k_{|K|}))\right) \\&= a^{1-\varepsilon}(n) \log a^{1-\varepsilon}(n)f'(k_1,\ldots,k_{|K|}) \leq a^{1-\varepsilon}(n) a^\delta(n) f'(k_1,\ldots,k_{|K|})\text{, for any }\delta>0. \end{align*} \begin{remark} In the scope of parameterized complexity, we can transform the addition in the second line into multiplication, since it is equivalent, as seen in \cite{DF13}. \end{remark}
If we choose $0<\delta<\varepsilon$, then $S'(n)=a^{1-\varepsilon+\delta}(n)f'(k_1,\ldots,k_{|K|})=a^{1-\varepsilon'}(n)f'(k_1,\ldots,k_{|K|})\le S^{1-\varepsilon'}(n)f'(k_1,\ldots,k_{|K|})$, for $\varepsilon'=\varepsilon-\delta$. \end{proof}
\section{Conclusion} In this work we have introduced a framework for fine-grained reductions that can capture a deeper connection between the problems involved, namely, a correlation among their parameters. We have shown that this framework captures the essence of the fine-grained approach without restricting the results. As a byproduct of our analysis, we defined and studied the structure of improvable problems, and the implications of fine-grained reductions on such problems. Finally, we produced a fixed parameter improvement in the running time of the OV problem by utilizing its parametric correlation to the $3/2$-approx-diameter problem.
A notable discussion in this field, is whether or not this framework can be used to define a complexity class, since FPI as a property has some unusual features. Specifically, the inherent meaning of "hardness" that arises, results in the absence of maximal elements (at least currently) in the partial ordering defined by parameterized fine-grained reductions. Additionally, because of the conjectured nature of our notion of improvements, the property $FPI(A,K)$ is directly related to previous work on each problem. It is possible that a parameterized algorithm may be proven sub-optimal in the case a problem's conjectured best running time is updated, resulting in disproving said property. As such, if problems having this property are considered a class, inclusion in this class could be negated after the fact, which is inconsistent with traditional complexity classes.
Using this framework, one can follow the direction of Theorem \ref{ovtodiam} to produce parameterized improvements via the transitivity of PFGR. This analysis can be done for each reduction in the fine-grained reduction web, producing a wide variety of improved algorithms on many interesting problems.
A natural question to consider is the relation between our work and traditional parameterized approach. As seen in Theorem \ref{NPFPI}, it remains an open problem to find the exact relation between FPI and FPT, that is, to formally characterize the problems in FPT that are not FPI. Additionally, one could potentially utilize the plethora of results available through the framework on parameter tractable or harder problems. All of these results may be translated to our terminology given the appropriate assumptions.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Analogues of Weyl's Formula for Reduced Enveloping Algebras}
\author{J.E. Humphreys \\ Dept. of Mathematics \& Statistics, University of Massachusetts, Amherst, MA 01003 \\E-mail: [email protected] }
\maketitle
\begin{abstract} In this note we study simple modules for a reduced enveloping algebra $U_\chi(\frg)$ in the critical case when $\chi \in \mathfrak{g}^*$ is ``nilpotent''. Some dimension formulas computed by Jantzen suggest modified versions of Weyl's dimension formula, based on certain reflecting hyperplanes for the affine Weyl group which might be associated to Kazhdan--Lusztig cells. \end{abstract}
\section{Introduction}
In the last decade or so there has been significant progress in understanding the non-restricted representations of the Lie algebra of a reductive group over a field of prime characteristic. Friedlander and Parshall extended the earlier foundations laid by Kac and Weisfeiler, while Premet proved the Kac--Weisfeiler conjecture on the minimum $p$-power dividing dimensions. More recently the work of Jantzen has reinforced ideas of Lusztig which arise in the framework of affine Hecke algebras and Springer fibers in the flag variety.
In spite of the progress made, serious obstacles remain to a definitive treatment of the representations. Here we attempt to interpret Jantzen's explicit dimension calculations in terms of analogues of Weyl's classical formula, imitating Kazhdan--Lusztig theory for the case of restricted representations.
\section{Reduced enveloping algebras}\label{sec.red}
First we recall briefly some essential background and notation, referring for details to the survey \cite{hu98} and the lectures by Jantzen \cite{ja98}, whose notation we mainly follow. For Lusztig's perspective on these questions, see \cite{lu01}.
\subsection{} Let $G$ be a simply connected, semisimple algebraic group over an algebraically closed field of characteristic $p>0$, with Lie algebra $\mathfrak{g}$. Following work of Kac and Weisfeiler, the simple modules for the universal enveloping algebra $U(\mathfrak{g})$ partition into modules for quotients $U_\chi(\frg)$ of $U(\mathfrak{g})$ (reduced enveloping algebras) associated with linear functionals $\chi \in \mathfrak{g}^*$. All $\chi$ in a coadjoint $G$-orbit yield isomorphic algebras. If $\chi=0$, $U_\chi(\frg)$ is the restricted enveloping algebra, whose representations include those derived from representations of $G$.
It has been known since early work of Jacobson and Zassenhaus that the maximum possible dimension of a simple module for $\mathfrak{g}$ or $U(\mathfrak{g})$ is $p^N$ ($N=$ number of positive roots). The Steinberg module in the restricted case is an example where this dimension is achieved.
\subsection{} The ``nilpotent'' $\chi$ (including $\chi=0$) play the main role. These correspond to nilpotent elements of $\mathfrak{g}$ when $\mathfrak{g}$ can be identified in a $G$-equivariant way with $\mathfrak{g}^*$, and form finitely many $G$-orbits (corresponding naturally to the characteristic 0 orbits when $p$ is good). Probably the most important question about the representation theory of $U_\chi(\frg)$ is this:
\emph{Question. How does the geometry of the $G$-orbit $G \chi$ influence the category of $U_\chi(\frg)$-modules?}
The orbit geometry involves a number of important ideas which have played a major role in characteristic 0 representation theory: Springer's resolution of the nilpotent variety, the flag variety and Springer fibers, affine Weyl groups and Hecke algebras, Kazhdan--Lusztig theory. It seems clear from recent work of Lusztig that many of these same ideas should recur in prime characteristic. In particular, the affine Weyl group $W_p$ relative to $p$ (defined in terms of the Langlands dual of $G$) has for a long time been known to play a major role in organizing the representation theory of $G$.
\subsection{} The category of $U_\chi(\frg)$-modules can be enriched by adding a natural action of the centralizer group $C_G(\chi)$. When this group contains at least a 1-dimensional torus $T_0$, Jantzen is able to obtain graded versions of the Lie algebra actions and exploit translation functors much as in the restricted case.
The best-behaved case occurs when $\chi$ has \emph{standard Levi form} in the sense of Friedlander--Parshall \cite{fp90}: for some choice of Borel subalgebra $\mathfrak{b}$, we have $\chi(\mathfrak{b})=0$ while $\chi$ vanishes on all negative root vectors $x_{-\alpha}$ except for a set $I$ of simple roots $\alpha$. (This always happens in type $A$.) Then the simple $U_\chi(\frg)$-modules are parametrized uniformly by linked weights $w \cdot \lambda$ with $w$ running over coset representatives for the subgroup $W_I$ of the Weyl group $W$ generated by corresponding reflections.
In general the parametrization by weights is much less well understood. It may depend in part on the choice of a Borel subalgebra on which $\chi$ vanishes: such a Borel subalgebra lies on one or more irreducible components of the Springer fiber. It is also possible that the component group $C_G(\chi)/C_G(\chi)^\circ$ and its characters will play a significant role, as they do in Springer theory. Recent work of Brown and Gordon \cite{bg01} confirms, at any rate, that the blocks of $U_\chi(\frg)$ (when $\chi$ is nilpotent) are in natural bijection with linkage classes of restricted weights. Here we consider only the most generic situation, involving simple modules in blocks parametrized by $p$-regular weights. This requires $p \geq h$ (the Coxeter number).
\subsection{} So far the most striking general fact about $U_\chi(\frg)$-modules is the theorem of Premet \cite{pr95}, valid for arbitrary $\chi$ (under mild restrictions on $\mathfrak{g}$ and $p$):
\emph{If $d$ is half the dimension of the coadjoint orbit $G \chi$, then the dimension of every $U_\chi(\frg)$-module is divisible by $p^d$.}
This had been conjectured much earlier by Kac and Weisfeiler. In particular, when $\chi$ is regular, all simple modules have the maximum possible dimension $p^N$ ($N=$ number of positive roots). Premet's theorem suggests a natural question:
\emph{With $d$ as above, does there always exist a simple $U_\chi(\frg)$-module $L_\chi(\lambda)$ of the smallest possible dimension $p^d$?}
The answer is yes in the cases investigated so far, but for no obvious conceptual reason unless $\chi$ lies in a Richardson orbit (permitting an easy construction by parabolic induction from a trivial module). Our proposed interpretation of dimension formulas stems partly from trying to understand this question better.
\section{The restricted case}{\label{sec.res}}
We recall briefly the standard framework \cite{ja87} for the study of simple $G$-modules, which include all simple $U_\chi(\frg)$-modules when $\chi=0$.
For each dominant weight $\lambda$ there is a Weyl module $V(\lambda)$, whose formal character and dimension are given by Weyl's formulas. In particular, \[\dim V(\lambda) = \frac{\prod_{\alpha>0} \langle \lambda+\rho,\alpha^\vee \rangle} {\prod_{\alpha>0} \langle \rho,\alpha^\vee \rangle}.\] Each Weyl module has a unique simple quotient $L(\lambda)$. Those for which $\lambda$ is \emph{restricted} (the coordinates of $\lambda$ relative to fundamental weights lying between 0 and $p-1$) are precisely the $p^r$ simple $U_0(\mathfrak{g})$-modules, where $r$ is the rank. Knowing just these modules would allow one to recover all $L(\lambda)$ as twisted tensor products, by Steinberg's Tensor Product Theorem \cite{st63}. But so far the broader study of Weyl modules for $G$ has yielded the most concrete results.
Knowledge of the formal characters and dimensions of all $L(\lambda)$ is equivalent to knowledge of the composition factor multiplicities of all $V(\lambda)$. When $p<h$ (the Coxeter number), there is no specific program for finding these multiplicities, but for $p \geq h$ the answer is expected to be given by Lusztig's conjecture. (This is known to be true for ``sufficiently large'' $p$, from the work of Andersen--Jantzen--Soergel \cite{ajs94}.)
Lusztig's approach depends on the fact that composition factors of $V(\lambda)$ must have highest weights linked to $\lambda$ under the standard dot action of the affine Weyl group $W_p$ relative to $p$. Write dominant weights as $w \cdot \lambda$, where $\lambda$ lies in the lowest alcove of the dominant Weyl chamber. One can in principle express the character of $L(w \cdot \lambda)$ as an alternating sum (with multiplicities) of the known Weyl characters for various weights $w' \cdot \lambda \leq w \cdot \lambda$. The multiplicities are in turn predicted to be the values of Kazhdan--Lusztig polynomials for pairs in $W_p$ related to $(w',w)$, after evaluation at 1. This procedure is inherently recursive and even in low ranks cannot usually be expected to produce simple closed formulas for characters or dimensions of simple modules.
Note how use of the lowest dominant alcove as a starting point locates in a natural way the unique weight $\lambda=0$ for which $L(\lambda)$ has the smallest possible dimension $p^0=1$. This weight is as close as possible to all hyperplanes bounding the alcove below, i.e., minimizes the numerator of Weyl's formula. But cancellation by the denominator is needed to produce 1.
\section{Special cases}{\label{sec.spec}}
At the opposite extreme from $\chi=0$, in the case where the coadjoint orbit of $\chi$ is \emph{regular} (with $d=N$), one has $\dim L_\chi(\lambda) = p^N$ for all $\lambda$. Much less is known between these extremes.
In a series of recent papers, Jantzen has studied a number of special cases when $\chi$ is nilpotent and of small codimension in the nilpotent variety. He obtains explicit dimension formulas for simple modules, as well as many details about projective modules, Ext groups, etc. Unlike the case $\chi=0$, it is feasible here to work out closed formulas for dimensions.
\begin{itemize}
\item[(a)] Type $B_2$, with $\chi$ in the minimal (nonzero) nilpotent orbit was first treated by \emph{ad hoc} methods in \cite{ja97} and then more systematically in \cite{ja00}. We take a closer look at this in the following section.
\item[(b)] The case when $\chi$ lies in the subregular orbit ($d=N-1$) was initially treated in \cite{ja99a} for the two simple types $A_n, B_n$ where $\chi$ can be chosen in standard Levi form. A more comprehensive treatment was then given in \cite{ja99b}. The results are more complete for simply-laced types. When there are two root lengths, the number of simple modules in a typical block is less certain (leading to uncertainty about some of the dimensions), but everything is expected to agree with Lusztig's predictions.
\item[(c)] Unpublished work by Jantzen (assisted by B. Jessen for type $G_2$) deals with a number of other cases, including the nilpotent orbits of $G_2$ for which $d=3,4$ (while $N=6$) and the ``middle'' orbit of $A_3$ (with $d=4, N=6$). He also works out families of examples involving standard Levi form: $C_n (n \geq 3)$ with $I$ of type $C_{n-1}$ and $D_n (n \geq 4)$ with $I$ of type $D_{n-1}$. In each case $d=N-2$. The results are somewhat less complete in types $G_2$ ($d=3$) and $C_n$, just as in the subregular case.
\end{itemize}
It is a striking fact that, in all of these cases, the dimension formulas for simple modules have the same quotient format as Weyl's formula. There is a constant denominator, together with a numerator written as the product of $N$ factors: $p$ repeated $d$ times (in accordance with Premet's Theorem), as well as $N-d$ other factors. Each of these factors involves an affine expression in the coordinates of a $\rho$-shifted weight based in one reference alcove. One or more weights will minimize the numerator, giving a dimension equal to $p^d$ after dividing by the denominator.
The main drawback to these formulas is that there is a separate one for each simple module (or small family of simple modules) in a typical block. Moreover, there is no obvious way to predict the formulas in advance, apart from the occurrence of $p^d$.
\section{Example: $B_2$}{\label{sec.B2}}
To explain more concretely our approach to dimension formulas, we look at type $B_2$ (say $p \geq 5$). Denote the simple roots by $\alpha_1$ (long) and $\alpha_2$ (short), with corresponding fundamental weights $\varpi_1$ and $\varpi_2$.
\subsection{} Consider the case when $\chi$ lies in the minimal nilpotent orbit (with $N=4, d=2$). Here $\chi$ has standard Levi form, relative to the subset $I=\{\alpha_1\}$. There is a one-dimensional torus $T_0$ in $C_G(\chi)$ which acts naturally on $U_\chi(\frg)$-modules. A generic block has four simple modules $L_\chi(\lambda)$, each labelled by two ``highest'' weights $\lambda$ linked by the subgroup of $W$ generated by the simple reflection $s_1$. The dimensions of the $L_\chi(\lambda)$ were first worked out by Jantzen in \cite{ja97}; he later developed a streamlined version based on the systematic use of translation functors in \cite[\S 5]{ja00}. As required by Premet's Theorem, $p^2$ divides all dimensions.
To parametrize the simple modules by weights, Jantzen starts in the conventional lowest alcove of the dominant Weyl chamber, fixing a $p$-regular weight $\lambda$. In order to simplify formulas, he builds in the $\rho$-shift by writing $\lambda+\rho = r\varpi_1 + s\varpi_2 = (r,s)$. Thus $r,s>0$ while $2r+s<p$. The dimensions of simple modules corresponding to linked weights in the four restricted alcoves are: \[\frac{s(p-2r-s)}{2} p^2, \; \frac{2pr}{2} p^2, \; \frac{(2p-s)(p-2r-s)}{2} p^2 \; \frac{s(p+2r+s)}{2} p^2. \; \] Notice that there are two choices of $(r,s)$ which yield a simple module of smallest possible dimension $p^2$: $((p-3)/2,1)$ and $((p-3)/2,2)$. These weights (which parametrize a single module) lie in the second restricted alcove, which suggests that we might instead view that alcove as ``lowest'' and use a weight there to rewrite Jantzen's formulas. The alcove in question is labelled $A$ in Figure 1.
\begin{figure}
\caption{Some alcoves for type $B_2$}
\label{fig.one}
\end{figure}
The four dimensions above correspond to linked weights in the respective alcoves $A,B,C,D$. The simple module of dimension $p^2$ corresponds to two weights in the lower left corner of alcove $A$, as close to both vertical and horizontal walls as possible. Moreover, the dimension formula $s(p-2r-s)p^2/2$ for alcove $A$ corresponds in a transparent way to the defining equations $s=0$ and $2r+s=p$ of these two hyperplanes (if we retain Jantzen's standard coordinates). The two special weights minimize the numerator in this dimension formula, giving $2p^2$; the denominator is then needed to cancel the 2. In this way, we can begin to imitate the interpretation of Weyl's formula in Section \ref{sec.res}.
\subsection{} To develop further the analogy with the restricted case, we have to rewrite the dimension formula for alcove $A$ in terms of new ($\rho$-shifted) coordinates $(r,s)$ of a weight in this alcove. Set \[\delta(r,s) :=s(2r+s-p)p^2/2.\] Now the idea is to apply this formula to the ($\rho$-shifted) coordinates $(r,s)$ of an arbitrary weight, in the spirit of Weyl's formula. In particular the formula yields 0 when applied to a weight in the indicated orthogonal hyperplanes bounding alcove $A$ below.
Write briefly $\delta_A, \delta_B, \dots$ for the formal dimensions obtained by applying $\delta$ to linked weights in alcoves $A, B, \dots$ Denoting the corresponding simple $U_\chi(\frg)$-modules by $L_A, L_B, \dots,$ we find the following pattern: \begin{eqnarray*} \dim L_A &=& \delta_A \\ \dim L_B &=& \delta_B -\delta_A \\ \dim L_C &=& \delta_C - \delta_B + \delta_A \\ \dim L_D &=& \delta_D - \delta_B + \delta_A \end{eqnarray*} This in turn raises the question of the possible existence of modules $V_\chi(\lambda)$ (analogous to Weyl modules in the case $\chi=0$) having dimensions given by the function $\delta$. These should exist for weights lying in an appropriate collection of alcoves (here infinite) and should be modules for $U_\chi(\frg)$ as well as for $C_G(\chi)$.
The alternating sum formulas above are certainly suggestive of a general pattern, though we usually must expect (as for $\chi=0$) coefficients of absolute value $>1$ coming from Kazhdan--Lusztig theory. In our example, alcove $E$ should carry the simple module $L_A$, but the most likely alternating sum formula will produce a multiple of its dimension such as $3\delta_A$. This suggests associating to a weight in alcove $E$ a $U_\chi(\frg)$-module together with a nontrivial representation of an $SL_2$-type subgroup of $C_G(\chi)$ (whose trivial representation would occur for alcove $A$). Such a pairing, somewhat analogous to the Springer Correspondence, would be compatible with Lusztig's cell conjectures \cite[\S 10]{lu89}.
In any case, the main thrust of our formulation is the derivation of diverse-looking dimension formulas from a single formula based on a special choice of affine hyperplanes. This much can be conjectured in general, but the explanation for such regularity remains speculative.
\section{Kazhdan--Lusztig cells and hyperplanes}{\label{sec.cell}}
\subsection{} How can one identify suitable affine hyperplanes which might support a version of Weyl's dimension formula for arbitrary nilpotent $\chi$, in the spirit of the above discussion of the minimal nilpotent orbit for type $B_2$? An answer is suggested by Lusztig's bijection \cite{lu89} between nilpotent orbits (in good characteristic) and two-sided cells in the affine Weyl group (for the Langlands dual of $G$). As shown by Lusztig and Xi \cite{lx88}, each two-sided cell in turn meets the dominant Weyl chamber in a ``canonical'' left cell. In characteristic $p$ we identify the affine Weyl group in question with $W_p$, allowing us to view the cells as unions of $p$-alcoves.
For example, the minimal nilpotent orbit of $B_2$ corresponds to the canonical left cell whose lower portion (beginning with alcoves $A,B, \dots$) is the strip along one wall pictured in Figure 2. \begin{figure}
\caption{Lower part of a canonical left cell for type $B_2$}
\label{fig.two}
\end{figure} To rewrite the previous discussion of dimensions in terms of weights lying in these translated alcoves, we just have to redefine the function $\delta$ by \[\delta(r,s):=(s-p)(2r+s-2p)p^2/2.\] This leads to the same dimension formulas as in Sec. \ref{sec.B2}.
Empirical study of Jantzen's formulas in a variety of cases shows a strong correlation with the hyperplanes bounding below the canonical left cell for $\chi$ (but with hyperplanes bounding the dominant region omitted). This is the rationale for our reformulation of his $B_2$ results above.
In the $B_2$ example, there are two orthogonal hyperplanes, corresponding to an $A_1 \times A_1$ root system. In other cases studied one gets more complicated root systems, taking in each case the natural hyperplanes corresponding to the associated positive roots as a framework for the basic dimension formula. Of course, when $\chi=0$ we are just reverting to Weyl's formula in this way. Such hyperplane systems must come from various proper subsets of the extended Dynkin diagram.
\subsection{} This type of interpretation agrees well with the location of weights which yield $\dim L_\chi(\lambda) = p^d$, as in the $B_2$ example. However, this small example is oversimplified in some respects. There are several complicating factors in the attempt to correlate $U_\chi(\frg)$-modules with cells:
\begin{itemize}
\item[(a)] It is not easy to describe geometrically the lower boundary behavior of canonical left cells, though the work of Shi \cite{sh86} in type $A$ (and further work by him and his associates in other cases) provides a lot of combinatorial data. It is clear that one cannot in general expect to find a unique lowest alcove in a canonical left cell. For example, the minimal orbit for type $A_3$ (where $N=6$ and $d=3$) yields a cell with two symmetrically placed configurations of lower hyperplanes of $A_2$ type. Here one expects three factors corresponding to positive roots of $A_2$ in the conjectural dimension formulas.
\item[(b)] One cannot always point to an obvious hyperplane configuration of the right size. An extreme case to keep in mind is the minimal nilpotent orbit of $E_8$, where $N=120$ and $d=29$. The 91 expected factors in a dimension formula might well arise from a combination of the 28 positive roots in an $A_7$ root system and another 63 positive roots in an $E_7$ root system (both found in the extended Dynkin diagram). It is unclear how to predict such patterns in general, though they may be related to a duality for nilpotent orbits studied by Sommers. Note that for type $A$ one has a simple version of duality (based on transpose partitions) which might suggest a natural choice of hyperplanes.
\item[(c)] When the component group $C_G(\chi)/C_G(\chi)^\circ$ is nontrivial, it may permute a number of nonisomorphic simple modules having the same dimension \cite{ja99b}. This already shows up in subregular cases for $B_2$ or $G_2$, in a way that looks consistent with Lusztig's conjectures in \cite[\S 10]{lu89}: each intersection of a left cell with its inverse should correspond to an orbit of the component group in the set of simple modules belonging to a typical block of $U_\chi(\frg)$.
\end{itemize}
\subsection{} How can one construct modules $V_\chi(\lambda)$ for $U_\chi(\frg)$ and $C_G(\chi)$ which carry dimension formulas of the shape we have described? One approach has been initiated by Mirkovi\'c and Rumynin \cite{mr01}, but many technical problems remain. The natural starting point is the Springer fiber associated with $\chi$, whose dimension is $N-d$.
Ultimately all of this should connect naturally with the ideas of Lusztig \cite{lu97,lu98,lu99a,lu99b,lu01} involving Springer fibers, equivariant $K$-theory, affine Hecke algebras, cells, etc. Significant progress has recently been made by Bezrukavnikov, Mirkovi\'c, and Rumynin \cite{bmr}.
\noindent {\footnotesize \emph{Acknowledgments.} Conversations and correspondence with Jens Carsten Jantzen have been extremely useful in formulating the ideas here, though he should not be held responsible for my speculative suggestions. I am also grateful to Roman Bezrukavnikov, Paul Gunnells, Ivan Mirkovi\'c, Jian Yi Shi, and Eric Sommers for useful consultations.}
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\begin{document}
\title{The Fourier-Mukai transform of a universal family of stable vector bundles}
\author{Fabian Reede} \address{Institut f\"ur Algebraische Geometrie, Leibniz Universit\"at Hannover, Welfengarten 1, 30167 Hannover, Germany} \email{[email protected]}
\subjclass[2010]{14J60,14F05}
\begin{abstract} In this note we prove that the Fourier-Mukai transform $\Phi_{\mathcal{U}}$ of the universal family of the moduli space $\mathcal{M}_{\PP^2}(4,1,3)$ is not fully faithful. \end{abstract}
\maketitle
\section*{Introduction} To every smooth projective variety $X$ one can associate its bounded derived category of coherent sheaves $\Db(X)$. The derived category contains a lot of geometric information about $X$. In some cases one can even recover $X$ from $\Db(X)$ but there are also examples of different varieties with equivalent derived categories, see \cite{huy2} for an introduction.
To compare the derived categories of two smooth projective varieties $X$ and $Y$, one needs to study functors between them. As it turns out, most of the interesting functors are Fourier-Mukai transforms $\Phi_{\mathcal{F}}: \Db(X) \rightarrow \Db(Y)$ for some object $\mathcal{F}\in \Db(X\times Y)$.
In this note we are interested in fully faithful Fourier-Mukai transforms because they give a semi-orthogonal decomposition of the derived category $\Db(Y)$ in smaller admissible subcategories. For example Krug and Sosna prove in \cite{krug3} that the Fourier-Mukai transform $\Phi_{\mathcal{I}_{\mathcal{Z}}}: \Db(S) \rightarrow \Db(S^{[n]})$ induced by the universal ideal sheaf $\mathcal{I}_{\mathcal{Z}}$ of the Hilbert scheme $S^{[n]}$ is fully faithful for a surface $S$ with $p_g=q=0$, hence $\Db(S)$ is an admissible subcategory in $\Db(S^{[n]})$. This result was generalized for the Hilbert square $X^{[2]}$ to smooth projective varieties $X$ with exceptional structure sheaf and arbitrary dimension $\dim(X)\geq 2$, see \cite{bel}.
Another example of this behaviour is given by the moduli space $\mathcal{M}_C(2,L)$ of stable rank two vector bundles with fixed determinant $L$ of degree one on a smooth projective curve $C$ of genus $g\geq 2$. This moduli space is fine and thus there is a universal family $\mathcal{U}$ on $C\times \mathcal{M}_C(2,L)$. By work of Narasimhan, see \cite{nara1} and \cite{nara2}, as well as Fonarev and Kuznetsov, see \cite{fon}, it is known that the Fourier-Mukai transform $\Phi_{\mathcal{U}}: \Db(C)\rightarrow \Db(\mathcal{M}_C(2,L))$ is fully faithful. Thus $\Db(C)$ is an admissible subcategory of $\Db(\mathcal{M}_C(2,L))$. This also solves the so-called Fano visitor problem for smooth projective curves of genus $g\geq 2$. This result was generalized in \cite{bel2} to the higher rank case $\mathcal{M}_C(r,L)$ for a line bundle $L$ of degree $d$ such that $\text{gcd}(r,d)=1$ and curves of genus $g\geq g_0$ for some $g_0\in \mathbb{N}$.
In light of these examples one can ask if the Fourier-Mukai transform of the universal family $\mathcal{U}$ on a moduli space $\mathcal{M}_{\PP^2}(r,c_1,c_2)$ of stable sheaves on $\PP^2$ is also fully faithful. Our main result is, that this is not always the case. We prove: \begin{thm-non}
The Fourier-Mukai transform
\begin{equation*}
\Phi_{\mathcal{U}}: \Db(\PP^2) \rightarrow \Db(\mathcal{M}_{\PP^2}(4,1,3))
\end{equation*}
induced by the universal family $\mathcal{U}$ of the moduli space $\mathcal{M}_{\PP^2}(4,1,3)$ is not fully faithful. \end{thm-non} The structure of this note is as follows: in section \ref{sect1} we recall some facts about the moduli space we are interested in. We construct an explicit family of stable sheaves for this moduli space in section \ref{sect2}. The computation of some cohomology groups for the family of stable sheaves can be found in section \ref{sect3}. In the final section \ref{sect4} we prove the main result.
Everything in this note is defined over the field of complex numbers $\mathbb{C}$. The projective plane $\PP^2$ is polarized by $H=\OO_{\PP^2}(1)$, thus $\mu$-stability means $\mu_{H}$-stability. A cohomology group written in lowercase characters simply denotes its dimension as a $\mathbb{C}$-vector space.
\section{The moduli space}\label{sect1} We begin by studying the moduli space $\mathcal{M}_{\PP^2}(4,1,3)$ of $S$-equivalence classes of $\mu$-semistable torsion-free sheaves $E$ on the projective plane $\PP^2$ with the following numerical data: \begin{equation*} \rk(E)=4,\,\,\,c_1(E)=1\,\,\, c_2(E)=3. \end{equation*} (Since the first Chern class is just an integer multiple of the polarization $H$, we simply identify it with this number.)
By this choice of rank $r$ and Chern classes $c_1$ resp. $c_2$ we get: \begin{lem} The moduli space $\mathcal{M}_{\PP^2}(4,1,3)$ is fine and there are no proper semistable sheaves. \end{lem} \begin{proof} We have \begin{equation*}
\text{gcd}\left(r,c_1.H, \frac{1}{2}c_1.(c_1-K_{\PP^2})-c_2 \right)=\text{gcd}(4,1,-1)=1. \end{equation*} The result now follows from \cite[Corollary 4.6.7]{huy} and \cite[Remark 4.6.8]{huy}. \end{proof} \begin{rem} This lemma shows that the moduli space $\mathcal{M}_{\PP^2}(4,1,3)$ has a universal family, that is a sheaf $\mathcal{U}$ on $\PP^2\times\mathcal{M}_{\PP^2}(4,1,3)$ flat over $\mathcal{M}_{\PP^2}(4,1,3)$ such that for every $E$ with $\left[E\right]\in \mathcal{M}_{\PP^2}(4,1,3)$ there is an isomorphism $ \mathcal{U}_{\left[E\right]}\cong E$, where $\mathcal{U}_{\left[E \right]}$ denotes the restriction of $\mathcal{U}$ to the fiber over $\left[E \right]$. \end{rem}
The following properties of the moduli space are probably well known: \begin{lem}\label{prop} The moduli space $\mathcal{M}_{\PP^2}(4,1,3)$ is a smooth projective variety of dimension six. Furthermore all sheaves $E$ classified by this moduli space are locally free. \end{lem} \begin{proof} The space $\mathcal{M}_{\PP^2}(4,1,3)$ is projective by construction. Since every sheaf $E$ is stable, we get by Serre duality \begin{equation*}
\Ext^2(E,E)\cong \Hom(E,E(-3))^{\vee} = 0 \end{equation*} hence $\mathcal{M}_{\PP^2}(4,1,3)$ is smooth and by \cite{elli} it is also irreducible. We also recall \begin{equation*}
\dim(\mathcal{M}_{\PP^2}(r,c_1,c_2))=\Delta-(r^2-1)\chi(\PP^2,\OO_{\PP^2}) \end{equation*} where $\Delta=2rc_2-(r-1)c_1^2$ is the discriminant. So $\dim(\mathcal{M}_{\PP^2}(4,1,3))=6$.
The double dual of a $\mu$-stable torsion-free sheaf $E$ is still $\mu$-stable and defines a smooth point in $\mathcal{M}_{\PP^2}(4,1,3-\ell)$ with $\ell=\text{length}(E^{\vee\vee}/E)$. If $E$ were not locally free we would have $\ell\geq 1$ and $\mathcal{M}_{\PP^2}(4,1,3-\ell)$ would have negative dimension, which is not possible. \end{proof} \begin{rem}\label{locf} Using Lemma \ref{prop} together with \cite[Lemma 2.1.7.]{huy} shows that the universal family $\mathcal{U}$ is itself locally free on $\PP^2\times\mathcal{M}_{\PP^2}(4,1,3)$. This implies that the sheaves $\mathcal{U}_p$, the restriction the fiber over $p\in \PP^2$, are also locally free on the moduli space. \end{rem}
The sheaves classified by $\mathcal{M}_{\PP^2}(4,1,3)$ can be described more explicitly: \begin{lem}\label{exseq1}
Let $E$ be a locally free sheaf on $\PP^2$ with $\left[E \right]\in \mathcal{M}_{\PP^2}(4,1,3)$, then there is a length three subscheme $Z\subset \PP^2$ and an exact sequence \begin{equation*}
\begin{tikzcd}
0 \arrow[r] & \OO_{\PP^2}^{\oplus 3} \arrow[r] & E \arrow[r] & I_Z(1) \arrow[r] & 0.
\end{tikzcd} \end{equation*} \end{lem}
\begin{proof} Hirzebruch-Riemann-Roch shows $\chi(\PP^2,E)=3$. The stability of $E$ implies that we have $h^2(\PP^2,E)=0$ and thus $h^0(\PP^2,E)\geq 3$.
Choose a 3-dimensional subspace $U\subset H^0(\PP^2,E)$, then by \cite[Lemma 1.5]{naka} the natural evaluation map $\varphi: U\otimes \OO_{\PP^2} \rightarrow E$ is injective with torsion-free quotient $Q=\cok(\varphi)$. We get the exact sequence \begin{equation*}
\begin{tikzcd}
0 \arrow[r] & U\otimes\OO_{\PP^2} \arrow[r,"\varphi"] & E \arrow[r] & Q \arrow[r] & 0.
\end{tikzcd} \end{equation*} By comparing Chern classes we see that we must have $Q\cong I_Z(1)$ for a length three subscheme $Z\subset \PP^2$, that is $Z\in\HP$. This gives the desired exact sequence. \end{proof}
Lemma \ref{exseq1} shows that there is a close connection between $\mathcal{M}_{\PP^2}(4,1,3)$ and $\HP$. This connection will become clearer in the next sections.
\section{Construction of a family}\label{sect2} In this section we want to construct a $\HP$-family of $\mu$-stable locally free sheaves such that every member of this family is classified by $\mathcal{M}_{\PP^2}(4,1,3)$. The construction is based on a construction of Mukai, see \cite[Section 3]{muk}
The starting point of our construction is the observation that \begin{equation}\label{obs}
\ext^1(I_Z(1),\OO_{\PP^2})= h^1(\PP^2,I_Z(-2)) =3 \end{equation} for every $Z\in \HP$.
We define $V:=\Ext^1(I_Z(1),\OO_{\PP^2})$ and observe the isomorphism \begin{equation*} \Ext^1(I_Z(1),V^{\vee}\otimes\OO_{\PP^2})\cong\Ext^1(I_Z(1),\OO_{\PP^2})\otimes V^{\vee}\cong\Hom(V,V). \end{equation*} Hence there is a distinguished extension class $e\in \Ext^1(I_Z(1),V^{\vee}\otimes \OO_{\PP^2})$ corresponding to $\text{id}_V\in \Hom(V,V)$, giving rise to: \begin{equation}\label{exseq2}
\begin{tikzcd}
0 \arrow{r} & V^{\vee}\otimes\OO_{\PP^2} \arrow{r} & E_Z \arrow{r} & I_Z(1) \arrow{r} & 0.
\end{tikzcd} \end{equation}
\begin{rem}\label{hom}
The sheaf $E_Z$ is called the universal extension of $I_Z(1)$ by $\OO_{\PP^2}$. By construction we have $\Hom(E_Z,\OO_{\PP^2})=0$. \end{rem}
We want to study some of the properties of the sheaf $E_Z$. For example we have: \begin{lem} The sheaf $E_Z$ is a locally free sheaf on $\PP^2$. \end{lem}
\begin{proof} Tensor the exact sequence \eqref{exseq2} with $\omega_{\PP^2}$: \begin{equation*}
\begin{tikzcd}
0 \arrow{r} & V^{\vee}\otimes\omega_{\PP^2} \arrow{r} & E_Z\otimes \omega_{\PP^2} \arrow{r} & I_Z(-2) \arrow{r} & 0
\end{tikzcd}. \end{equation*} Now for every subscheme $Z'\subsetneq Z$ of length $0\leq d <3$ we have \begin{equation*}
h^1(\PP^2,I_{Z'}(-2))<h^1(\PP^2,I_{Z}(-2)), \end{equation*} which by \cite[Lemma 1.2.]{tyu} implies that $E_Z\otimes \omega_{\PP^2}$ is locally free, hence so is $E_Z$. \end{proof}
We also have the following result concerning the stability of $E_Z$: \begin{lem}
The locally free sheaf $E_Z$ is $\mu$-stable. \end{lem} \begin{proof} This follows from a more general result, see \cite[Lemma 1.4.]{naka}. But in this situation we can also give a direct proof:
Let $F$ be a torsion free quotient of $E_Z$ with $1\leq \rk(F)\leq 3$, then there is the following commutative diagram:
\begin{equation*}
\begin{tikzcd}
0 \arrow{r} & \mathcal{O}_{\mathbb{P}^2}^{\oplus 3} \arrow{r}\arrow{d} & E_{Z} \arrow{r}\arrow{d} & I_{Z}(1) \arrow{r}\arrow{d} & 0\\
0 \arrow{r} & F_0 \arrow{r} & F \arrow{r} & F_1 \arrow{r} & 0
\end{tikzcd} \end{equation*} with $F_0=\Ima(\mathcal{O}_{\mathbb{P}^2}^{\oplus 3}\hookrightarrow E_Z \rightarrow F)$. Thus all vertical arrows are surjective. Since $F_0$ is a quotient of a free sheaf we have $c_1(F_0).H\geq 0$. Furthermore $\rk(F_1)\in \left\lbrace0,1 \right\rbrace$ as $F_1$ is a quotient of a torsion free sheaf of rank 1. We distinguish two cases.
\underline{Case $\rk(F_1)=1$:}
In this case $F_1\cong I_Z(1)$ and hence $c_1(F).H=(c_1(F_0)+c_1(F_1)).H \geq 1$. This implies \begin{equation*}
\mu(F)=\frac{c_1(F).H}{\rk(F)}\geq \frac{1}{3} > \frac{1}{4} =\mu(E_Z). \end{equation*}
\underline{Case $\rk(F_1)=0$:}
In this case we have $c_1(F_1).H\geq 0$ as $F_1$ is a torsion sheaf. The only critical case is $c_1(F_0)=c_1(F_1)=0$, since otherwise $c_1(F)=d\geq 1$ and thus $\mu(F)\geq \frac{1}{3} > \frac{1}{4} =\mu(E_Z)$.
So assume $c_1(F_0)=c_1(F_1)=0$. Then $F_0$ is trivial itself, see for example \cite[p. 302]{laz}, and $F_1$ is supported in finitely many points. This implies \begin{equation*} \Hom(F,\OO_{\PP^2})\cong \mathbb{C}^{\rk(F_0)}. \end{equation*} On the other hand $\Hom(F,\OO_{\PP^2})\hookrightarrow \Hom(E_Z,\OO_{\PP^2})=0$ by Remark \ref{hom}. This shows $\rk(F_0)=0$ and hence $\rk(F)=0$. So for $\rk(F)\geq 1$ the case $c_1(F_0)=c_1(F_1)=0$ cannot occur and $E_Z$ is stable. \end{proof} The last two lemmas show: \begin{cor}\label{cor}
For every $Z\in \HP$ the sheaf $E_Z$ defines a point $\left[E_Z \right]\in \mathcal{M}_{\PP^2}(4,1,3)$. \end{cor}
We want to put the $\mu$-stable locally free sheaves $E_Z$ in a family classified by $\HP$. To do this we need the following maps: \begin{equation*}
\begin{tikzcd}[column sep=small]
\mathcal{Z}\arrow[hookrightarrow]{r} & \PP^2\times\HP \arrow[rightarrow]{dl}[swap]{p} \arrow[rightarrow]{dr}{q} & \\
\PP^2 && \HP
\end{tikzcd} \end{equation*} where $\mathcal{Z}$ is the universal family of length 3 subschemes.
\begin{rem} Recall that for any coherent sheaf $F$ on $\mathbb{P}^2$ there is the associated coherent \emph{tautological sheaf} $F^{[3]}$ on $\HP$ defined by
\begin{equation*}
F^{[3]}:=q_{*}\left(p^{*}F\otimes \OO_{\mathcal{Z}} \right).
\end{equation*} If $F$ is locally free of rank $r$ then $F^{[3]}$ is locally free of rank $3r$. \end{rem} To construct the family of stable sheaves, we first put the $\Ext^1(I_Z(1),\OO_{\PP^2})$ for $Z\in \HP$ in a family: \begin{lem}\label{rext}
The first relative $\Ext$-sheaf $\mathcal{V}:=\Rex(\mathcal{I}_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(1),\OO_{\PP^2\times \HP})$ is a locally free sheaf of rank three on $\HP$. It commutes with base change and there is an isomorphism
\begin{equation}\label{reliso}
\Rex(\mathcal{I}_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(1),\OO_{\PP^2\times \HP})^{\vee} \cong \OO_{\PP^2}(-2)^{[3]}.
\end{equation} \end{lem}
\begin{proof} The morphism $q$ is proper and flat and the map \begin{equation*}
\phi: \HP \rightarrow \mathbb{N},\,\,\, Z \mapsto \ext^1(I_Z(1),\OO_{\PP^2}) \end{equation*} is constant due to \eqref{obs}. So by \cite[Satz 3.]{schu} the first relative $\Ext$-sheaf is locally free of rank three on $\HP$ and commutes with base change, that is for every $Z\in \HP$ we have \begin{equation*}
\Rex(\mathcal{I}_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(1),\OO_{\PP^2\times \HP})\otimes k(Z) \cong \Ext^1(I_Z(1),\OO_{\PP^2}). \end{equation*}
Using relative Serre duality, see \cite[Corollary(24)]{klei}, gives an isomorphism
\begin{align*}
\Rex(\mathcal{I}_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(1),\OO_{\PP^2\times \HP})&\cong \Rex(\mathcal{I}_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(-2),\omega_{q})\\
&\cong \mathcal{H}om(R^1q_{*}(\mathcal{I}_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(-2)),\OO_{\HP}).
\end{align*} The exact sequence
\begin{equation*}
\begin{tikzcd}
0 \arrow{r} & \mathcal{I}_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(-2) \arrow{r} & p^{*}\OO_{\PP^2}(-2) \arrow{r} & \OO_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(-2) \arrow{r} & 0
\end{tikzcd}
\end{equation*} and standard cohomology and base change results, see \cite[II.5.]{mum}, show that there is an isomorphism \begin{equation*}
q_{*}(\OO_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(-2))\cong R^1q_{*}(\mathcal{I}_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(-2)). \end{equation*} We see that $R^1q_{*}(\mathcal{I}_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(-2))\cong \OO_{\PP^2}(-2)^{[3]}$ is locally free of rank three and thus we get the desired isomorphism \eqref{reliso}. \end{proof}
As the main result of this section we can now construct the desired family:
\begin{thm}\label{fam1} There is a locally free $\HP$-family $\mathcal{E}$ of $\mu$-stable locally free sheaves, given by the exact sequence \begin{equation*}
\begin{tikzcd}
0 \arrow{r} & q^{*}\mathcal{V}^{\vee} \arrow{r} & \mathcal{E} \arrow{r} & \mathcal{I}_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(1) \arrow{r} & 0,
\end{tikzcd} \end{equation*} i.e. for every $Z\in \HP$ the restriction to the fiber over $Z$ defines a point $\left[\mathcal{E}_Z \right]\in \mathcal{M}_{\PP^2}(4,1,3)$. \end{thm}
\begin{proof} For every $Z\in \HP$ we have $\Hom(I_Z(1),\OO_{\PP^2})=0$, so \begin{equation*}
\mathcal{E}xt^0_q(\mathcal{I}_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(1),\OO_{\PP^2\times\HP})=q_{*}\mathcal{H}om(\mathcal{I}_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(1),\OO_{\PP^2\times\HP})=0. \end{equation*} Using this fact and the projection formula for relative $\Ext$-sheaves \cite[Lemma 4.1.]{lan}, the five term exact sequence of the spectral sequence \begin{equation*}
H^i(\HP,\mathcal{E}xt^j_q(\mathcal{I}_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(1),q^{*}\mathcal{V}^{\vee})) \Rightarrow \Ext^{i+j}(\mathcal{I}_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(1),q^{*}\mathcal{V}^{\vee}) \end{equation*} reduces to an isomorphism \begin{align*}
\Ext^1(\mathcal{I}_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(1),q^{*}\mathcal{V}^{\vee})&\cong H^0(\HP,\mathcal{E}xt^1_q(\mathcal{I}_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(1),q^{*}\mathcal{V}^{\vee}))\\
&\cong H^0(\HP,\mathcal{E}xt^1_q(\mathcal{I}_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(1),\OO_{\PP^2\times\HP})\otimes \mathcal{V}^{\vee})\\
&\cong \Hom(\mathcal{V},\mathcal{V}). \end{align*} The identity $\id_{\mathcal{V}}$ gives rise to an extension on $\PP^2\times \HP$: \begin{equation}\label{fam}
\begin{tikzcd}
0 \arrow{r} & q^{*}\mathcal{V}^{\vee} \arrow{r} & \mathcal{E} \arrow{r} & \mathcal{I}_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(1) \arrow{r} & 0
\end{tikzcd} \end{equation} with $\mathcal{E}$ flat over $\HP$, since both other terms are. Restricting to the fiber over a point $Z\in \HP$ defines by flatness of $\mathcal{I}_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(1)$ a map \begin{equation*}
\Ext^1(\mathcal{I}_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(1),q^{*}\mathcal{V}^{\vee}) \rightarrow \Ext^1(I_Z(1),V^{\vee}\otimes\OO_{\PP^2}). \end{equation*} By \cite[Lemma 2.1.]{lan} the extension defined by $\id_{\mathcal{V}}$ restricts to the extension given by $\id_V$ on the fiber over $Z\in\HP$. Thus the pullback of \eqref{fam} to the fiber over $Z\in\HP$ is exactly the exact sequence \eqref{exseq2}, hence it defines a locally free sheaf classified by $\mathcal{M}_{\PP^2}(4,1,3)$. Using \cite[Lemma 2.1.7.]{huy} again, we see that $\mathcal{E}$ is itself locally free. \end{proof}
By the universal property of $\mathcal{M}_{\PP^2}(4,1,3)$ the family $\mathcal{E}$ comes with a classifying morphism \begin{equation*} f_{\mathcal{E}}: \HP \rightarrow \mathcal{M}_{\PP^2}(4,1,3),\,\, Z \mapsto \left[\mathcal{E}_Z \right]. \end{equation*} Furthermore there is $L\in \Pic(\HP)$ and an isomorphism \begin{equation}\label{univ}
(\id_{\PP^2}\times f_{\mathcal{E}})^{*}\mathcal{U}\otimes q^{*}L\cong \mathcal{E}. \end{equation}
We need to study some properties of the morphism $f_{\mathcal{E}}$. For this we need: \begin{lem}\label{iso}
Assume $Z\in \HP$ is not collinear. If there is an isomorphism $\alpha: E_{Z'} \cong E_{Z}$ for some $Z'\in \HP$, then $Z=Z'$. \end{lem} \begin{proof}
We look at the following diagram:
\begin{equation*}
\begin{tikzcd}
0 \arrow{r} & \mathcal{O}_{\mathbb{P}^2}^{\oplus 3} \arrow{r}{\iota} & E_{Z'} \arrow{r}\arrow{d}{\alpha}[swap]{\cong} & I_{Z'}(1) \arrow{r} & 0\\
0 \arrow{r} & \mathcal{O}_{\mathbb{P}^2}^{\oplus 3} \arrow{r} & E_{Z} \arrow{r}{q} & I_{Z}(1) \arrow{r} & 0.
\end{tikzcd}
\end{equation*}
Since $Z$ is not collinear the composition $\beta:=q\circ\alpha\circ\iota$ is zero. Consequently the free submodule of $E_{Z'}$ maps injectively to the free submodule of $E_Z$, which then must be an isomorphism, so we get in fact the following diagram:
\begin{equation*}
\begin{tikzcd}
0 \arrow{r} & \mathcal{O}_{\mathbb{P}^2}^{\oplus 3} \arrow{r}{}\arrow{d}[swap]{\cong} & E_{Z'} \arrow{r}\arrow{d}{\alpha}[swap]{\cong} & I_{Z'}(1) \arrow{r}\arrow{d}[swap]{\cong} & 0\\
0 \arrow{r} & \mathcal{O}_{\mathbb{P}^2}^{\oplus 3} \arrow{r} & E_{Z} \arrow{r}{} & I_{Z}(1) \arrow{r} & 0.
\end{tikzcd}
\end{equation*}
Therefore there is an induced isomorphism $I_{Z'}(1)\cong I_Z(1)$ and so $Z=Z'$. \end{proof} Thus the non-collinear subschemes in $\HP$ define sheaves $E_Z$ with exactly three global sections. It makes sense to study the Brill-Noether-locus $S$ in $ \mathcal{M}_{\PP^2}(4,1,3)$: \begin{equation*}
S:=\left\lbrace \left[E \right]\in \mathcal{M}_{\PP^2}(4,1,3)\,|\, h^0(\PP^2,E)=4 \right\rbrace. \end{equation*} \begin{rem}
We can write down the inverse $g$ to $f_{\mathcal{E}}$ on the complement of $S$:
\begin{equation*}
g: \mathcal{M}_{\PP^2}(4,1,3)\setminus S \rightarrow \HP,\,\,\, E \mapsto \supp(Q^{\vee\vee}/Q)
\end{equation*} where $Q$ is the cokernel of the (in this case) canonical evaluation map from Lemma \ref{exseq1}. \end{rem} By \cite[Corollary, p.14, lines 3-5]{tyu} we get for the $E_Z$ with collinear subschemes $Z$: \begin{lem}\label{niso}
Assume $Z, Z'\in \HP$ are collinear with $Z\neq Z'$ such that there is a line $\ell\subset \PP^2$ containing both $Z$ and $Z'$, then $E_Z= E_{Z'}$. \end{lem} \begin{rem} This shows that for a sheaf $E_Z$ with a collinear subscheme $Z\subset \PP^2$ we have \begin{equation*}
f_{\mathcal{E}}^{-1}(\left[ E_Z\right] )=\ell^{[3]}\cong \PP^3, \end{equation*} where $\ell\subset \PP^2$ is the line containing $Z$. \end{rem} The last two lemmas suggest that $f_{\mathcal{E}}$ is the blow up of $S$ in $\mathcal{M}_{\PP^2}(4,1,3)$. This is indeed the case since by \cite[5.29, Example 5.3.]{yosh} and we have: \begin{lem}\label{bir}
The Brill-Noether-locus $S$ in $\mathcal{M}_{\PP^2}(4,1,3)$ is isomorphic to $\PP^2$ and there is an isomorphism $\HP \cong \Bl_S\mathcal{M}_{\PP^2}(4,1,3)$ such that $f_{\mathcal{E}}$ can be identified with the blow up of $S$ in $\mathcal{M}_{\PP^2}(4,1,3)$. \end{lem}
\begin{rem} This description goes back to Drezet who proved this in terms of Kronecker modules in \cite[Th\'eor\`eme 4.]{dre}. \end{rem}
\begin{cor}\label{coh} For every locally free sheaf $F$ on $\mathcal{M}_{\PP^2}(4,1,3)$ we have isomorphisms \begin{equation*} H^i(\mathcal{M}_{\PP^2}(4,1,3),F)\cong H^i(\HP, f_{\mathcal{E}}^{*}F). \end{equation*} \end{cor} \begin{proof} Since $f_{\mathcal{E}}$ is birational by Lemma \ref{bir}, the result follows from $R^i\left(f_\mathcal{E} \right) _{*}\OO_{\HP}=0$ for $i\geq 1$, the projection formula and the Leray spectral sequence. \end{proof}
\section{Computations}\label{sect3} We want to understand the family $\mathcal{E}$ as a $\PP^2$-family, that is we want to understand the sheaves $\mathcal{E}_p$ on $\HP$ for $p\in \PP^2$. For this we first note that $\mathcal{Z}$ is not just flat over $\HP$ but also over $\PP^2$, see \cite[Theorem 2.1.]{krug1}, so restricting the exact sequence \begin{equation*}
\begin{tikzcd}
0 \arrow{r} & \mathcal{I}_{\mathcal{Z}} \arrow{r} & \OO_{\PP^2\times\HP} \arrow{r} & \OO_{\mathcal{Z}}\arrow{r} & 0
\end{tikzcd} \end{equation*} to the fiber over point $p\in \PP^2$ gives the exact sequence \begin{equation}\label{ide}
\begin{tikzcd}
0 \arrow{r} & I_{S_p} \arrow{r} & \OO_{\HP} \arrow{r} & \OO_{S_p} \arrow{r} & 0,
\end{tikzcd} \end{equation}
where $S_p:=\left\lbrace Z\in \HP\,|\, p\in \supp(Z) \right\rbrace$ is a codimension two subscheme in $\HP$.
Since $\mathcal{Z}$ is flat over $\PP^2$ we see, using \cite[Examples 5.4 vi)]{huy2}, that $\OO_{S_p}=k(p)^{[3]}$ is the tautological sheaf on $\HP$ associated to the skyscraper sheaf $k(p)$ of the point $p\in \PP^2$. This implies we can use \cite[Theorem 3.17.,Remark 3.20.]{krug3} to find the following cohomology groups: \begin{equation}\label{krug1}
\ext^i(\OO_{\HP},\OO_{\PP^2}(-2)^{[3]})=0\,\,\text{for all $i$ and}\,\, \ext^i(\OO_{S_p},\OO_{\PP^2}(-2)^{[3]})=\begin{cases} 1 & i=2\\ 0 & i\neq 2\end{cases} \end{equation} as well as \begin{equation}\label{krug2}
\ext^i(\OO_{\PP^2}(-2)^{[3]},\OO_{\HP})=\begin{cases} 6 & i=0\\ 0 & i\geq 1\end{cases}\,\,\text{and}\,\, \ext^i(\OO_{\PP^2}(-2)^{[3]},\OO_{S_p})=\begin{cases} 7 & i=0\\ 0 & i\geq 1.\end{cases} \end{equation}
Using these results we can prove: \begin{lem}\label{comp}
For $p\in \PP^2$ we have
\begin{equation*}
\ext^i(I_{S_p},\OO_{\PP^2}(-2)^{[3]})=\begin{cases} 1 & i=1\\ 0 & i\neq 1\end{cases}\,\,\text{and}\,\,\ext^i(\OO_{\PP^2}(-2)^{[3]},I_{S_p})=\begin{cases} \geq 1 & i=1\\ 0 & i\geq 2.\end{cases}
\end{equation*} \end{lem} \begin{proof} Applying $\Hom(-,\OO_{\PP^2}(-2)^{[3]})$ to \eqref{ide} we have $\Ext^6(I_{S_p},\OO_{\PP^2}(-2)^{[3]})=0$ as well as isomorphisms: \begin{equation*}
\Ext^i(I_{S_p},\OO_{\PP^2}(-2)^{[3]})\cong \Ext^{i+1}(\OO_{S_p},\OO_{\PP^2}(-2)^{[3]})\,\,\,\text{for $1\leq i\leq 5$}, \end{equation*} since $\Ext^i(\OO_{\HP},\OO_{\PP^2}(-2)^{[3]})=0$ for all $i$ by \eqref{krug1}. We also find $\Hom(I_{S_p},\OO_{\PP^2}(-2)^{[3]})=0$ by further using $\Ext^i(\OO_{S_p},\OO_{\PP^2}(-2)^{[3]})=0$ for $i=0,1$. This proves the first claim.
For the second claim we apply $\Hom(\OO_{\PP^2}(-2)^{[3]},-)$ to \eqref{ide}. As $\Ext^1(\OO_{\PP^2}(-2)^{[3]},\OO_{\HP})=0$ by \eqref{krug2} the first part of the long exact sequence gives \begin{equation*}
\begin{tikzcd}
0 \arrow{r} & \Hom(\OO_{\PP^2}(-2)^{[3]},I_{S_p}) \arrow{r} & \mathbb{C}^6 \arrow{r} & \mathbb{C}^7 \arrow{r} & \Ext^1(\OO_{\PP^2}(-2)^{[3]},I_{S_p}) \arrow{r} & 0
\end{tikzcd} \end{equation*} which shows that $\ext^1(\OO_{\PP^2}(-2)^{[3]},I_{S_p})\geq 1$. We also get isomorphisms \begin{equation*}
\Ext^i(\OO_{\PP^2}(-2)^{[3]},I_{S_p})\cong \Ext^{i-1}(\OO_{\PP^2}(-2)^{[3]},\OO_{S_p})\,\,\,\text{ for $2\leq i \leq 6$}. \end{equation*} Again using \eqref{krug2} proves the second claim. \end{proof}
To study the locally free sheaves $\mathcal{E}_p$ on $\HP$ we note that the exact sequence \begin{equation*}
\begin{tikzcd}
0 \arrow{r} & q^{*}\mathcal{V}^{\vee} \arrow{r} & \mathcal{E} \arrow{r} & \mathcal{I}_{\mathcal{Z}}\otimes p^{*}\OO_{\PP^2}(1) \arrow{r} & 0
\end{tikzcd} \end{equation*} restricts to the fiber over $p\in \PP^2$ as \begin{equation}\label{ep}
\begin{tikzcd}
0 \arrow{r} & \OO_{\PP^2}(-2)^{[3]} \arrow{r} & \mathcal{E}_p \arrow{r} & I_{S_p} \arrow{r} & 0
\end{tikzcd} \end{equation} by using flatness of $\mathcal{I}_{\mathcal{Z}}$ over $\PP^2$ and Lemma \ref{rext}. We can now prove:
\begin{thm}\label{ext1}
Let $\mathcal{E}$ be the $\HP$-family of $\mu$-stable locally free sheaves, then for any pair of closed points $p,q\in \PP^2$ with $p\neq q$ we have
\begin{equation*}
\ext^1(\mathcal{E}_p,\mathcal{E}_q)\geq 1.
\end{equation*} \end{thm}
\begin{proof} By \cite[Theorem 1.2]{krug2} the Fourier-Mukai transform \begin{equation*} \Phi_{\mathcal{I}_{\mathcal{Z}}}: \Db(\PP^2) \rightarrow \Db(\HP) \end{equation*} is fully faithful, that is for $p,q\in \PP^2$ with $p\neq q$ we have by flatness of $\mathcal{I}_{\mathcal{Z}}$ over $\PP^2$: \begin{equation*}
\Ext^i(I_{S_p},I_{S_q})\cong \Ext^i(k(p),k(q))= 0\,\,\,\text{for $0\leq i\leq 6$}. \end{equation*} So applying $\Hom(I_{S_q},-)$ with $q\neq p$ to \eqref{ep} gives isomorphisms \begin{equation*}
\Ext^i(I_{S_q},\mathcal{E}_p)\cong \Ext^i(I_{S_q},\OO_{\PP^2}(-2)^{[3]})\,\,\,\text{for $0\leq i\leq 6$}. \end{equation*} If we apply $\Hom(\OO_{\PP^2}(-2)^{[3]},-)$ and use \cite[Theorem 3.17.]{krug3} again to see \begin{equation*}
\ext^i(\OO_{\PP^2}(-2)^{[3]},\OO_{\PP^2}(-2)^{[3]})=\begin{cases}
1 & i=0 \\ 0 & i\geq 1
\end{cases} \end{equation*} we get an exact sequence \begin{equation*}
\begin{tikzcd}
0 \arrow{r} & \mathbb{C} \arrow{r} & \Hom(\OO_{\PP^2}(-2)^{[3]},\mathcal{E}_p)
\arrow{r} & \Hom(\OO_{\PP^2}(-2)^{[3]},I_{S_p})\arrow{r} & 0
\end{tikzcd} \end{equation*} and isomorphisms \begin{equation*}
\Ext^i(\OO_{\PP^2}(-2)^{[3]},\mathcal{E}_p)\cong \Ext^i(\OO_{\PP^2}(-2)^{[3]},I_{S_p})\,\,\,\text{for $1\leq i \leq 6$}. \end{equation*}
Finally applying $\Hom(-,\mathcal{E}_q)$ with $q\neq p$ to \eqref{ep} we get the following relevant part of the induced long exact sequence: \begin{equation*}
\begin{tikzcd}
\arrow{r} & \Ext^1(\mathcal{E}_p,\mathcal{E}_q) \arrow{r} & \Ext^1(\OO_{\PP^2}(-2)^{[3]},\mathcal{E}_q)
\arrow{r} & \Ext^2(I_{S_p},\mathcal{E}_q)\arrow{r} & {}
\end{tikzcd} \end{equation*} With the previous results this sequence gets: \begin{equation*}
\begin{tikzcd}
\arrow{r} &\Ext^1(\mathcal{E}_p,\mathcal{E}_q) \arrow{r} & \Ext^1(\OO_{\PP^2}(-2)^{[3]},I_{S_q}) \arrow{r} & \Ext^2(I_{S_p},\OO_{\PP^2}(-2)^{[3]}) \arrow{r} & {}
\end{tikzcd} \end{equation*} Using Lemma \ref{comp} we have $\Ext^2(I_{S_p},\OO_{\PP^2}(-2)^{[3]})=0$ and thus \begin{equation*}
\ext^1(\mathcal{E}_p,\mathcal{E}_q)\geq \ext^1(\OO_{\PP^2}(-2)^{[3]},I_{S_q})\geq 1.\qedhere \end{equation*} \end{proof}
\section{Non-full faithfulness of the universal family}\label{sect4} We want to study the full faithfulness of the Fourier-Mukai transform \begin{equation*}
\Phi_{\mathcal{U}}: \Db(\PP^2) \rightarrow \Db(\mathcal{M}_{\PP^2}(4,1,3)) \end{equation*} induced by the universal family $\mathcal{U}$ of the moduli space $\mathcal{M}_{\PP^2}(4,1,3)$.
We will use the following corollary of the Bondal-Orlov criterion for full faithfulness: \begin{lem}\cite[Corollary 7.5]{huy2}\label{orlo}
Let $X$ and $Y$ be two smooth projective varieties and $\mathcal{P}$ a coherent sheaf on $X\times Y$, flat over $X$. Then the Fourier-Mukai transform
\begin{equation*}
\Phi_{\mathcal{P}}: \Db(X) \rightarrow \Db(Y)
\end{equation*} is fully faithful if and only if the following two conditions are satisfied \begin{enumerate}[i)]
\item For any closed point $x\in X$ one has $\Ext^i(\mathcal{P}_x,\mathcal{P}_x)=\begin{cases}
\mathbb{C} & i=0 \\ 0 & i>\dim(X)
\end{cases}$
\item For any pair of closed points $x,y\in X$ with $x\neq y$ one has $\Ext^i(\mathcal{P}_x,\mathcal{P}_y)=0$ for all i. \end{enumerate} \end{lem}
To apply this lemma to $\mathcal{P}=\mathcal{U}$, the universal family of $\mathcal{M}_{\PP^2}(4,1,3)$, we need to be able to compute $\Ext^i(\mathcal{U}_p,\mathcal{U}_q)$. The following lemma reduces this problem to computing $\Ext^i(\mathcal{E}_p,\mathcal{E}_q)$: \begin{lem}\label{exts}
Let $\mathcal{U}$ be the universal family of $\mathcal{M}_{\PP^2}(4,1,3)$ and $\mathcal{E}$ be the $\HP$-family, then for any two points $p,q\in \PP^2$ there are the following isomorphisms for all $i$:
\begin{equation*}
\Ext^i(\mathcal{U}_p,\mathcal{U}_q)\cong \Ext^i(\mathcal{E}_p,\mathcal{E}_q).
\end{equation*} \end{lem} \begin{proof} We have the following chain of isomorphisms: \begin{align*}
\Ext^i(\mathcal{U}_p,\mathcal{U}_q) &\cong H^i(M,\mathcal{H}om(\mathcal{U}_p,\mathcal{U}_q))\\
&\cong H^i(\HP,f_{\mathcal{E}}^{*}\mathcal{H}om(\mathcal{U}_p,\mathcal{U}_q))\\
&\cong H^i(\HP,\mathcal{H}om(f_{\mathcal{E}}^{*}\mathcal{U}_p,f_{\mathcal{E}}^{*}\mathcal{U}_q))\\
&\cong \Ext^i(f_{\mathcal{E}}^{*}\mathcal{U}_p,f_{\mathcal{E}}^{*}\mathcal{U}_q)\\
&\cong \Ext^i(f_{\mathcal{E}}^{*}\mathcal{U}_p\otimes L,f_{\mathcal{E}}^{*}\mathcal{U}_q\otimes L)\\
&\cong \Ext^i(\mathcal{E}_p,\mathcal{E}_q) \end{align*} Here the first and third isomorphism use the locally freeness of $\mathcal{U}_p$, see Remark \ref{locf}. The second isomorphism is Corollary \ref{coh}. The fourth isomorphism uses locally freeness of $f_{\mathcal{E}}^{*}\mathcal{U}_p$, while the sixth isomorphism follows from restricting \eqref{univ} to the fiber over $p\in \PP^2$. \end{proof} We can now prove the main theorem of this note: \begin{thm}
The Fourier-Mukai transform
\begin{equation*}
\Phi_{\mathcal{U}}: \Db(\PP^2) \rightarrow \Db(\mathcal{M}_{\PP^2}(4,1,3))
\end{equation*}
induced by the universal family $\mathcal{U}$ of the moduli space $\mathcal{M}_{\PP^2}(4,1,3)$ is not fully faithful. \end{thm} \begin{proof} For the Fourier-Mukai transform $\Phi_{\mathcal{U}}$ to be fully faithful one needs \begin{equation}\label{van}
\Ext^i(\mathcal{U}_p,\mathcal{U}_q)=0 \end{equation} for any pair of points $p,q\in \PP^2$ with $p\neq q$ and any $i$ according to Lemma \ref{orlo}.
Lemma \ref{exts} shows that \eqref{van} is equivalent to \begin{equation*}
\Ext^i(\mathcal{E}_p,\mathcal{E}_q)=0. \end{equation*} But we have $\Ext^1(\mathcal{E}_p,\mathcal{E}_q)\neq 0$ by Lemma \ref{ext1}, so $\Phi_{\mathcal{U}}$ cannot be fully faithful. \end{proof}
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\begin{abstract} An equiangular hyperbolic Coxeter polyhedron is a hyperbolic polyhedron where all dihedral angles are equal to $\pi/n$ for some fixed $n\in\mathbb{Z}$, $n\geq 2.$ It is a consequence of Andreev's theorem that either $n=3$ and the polyhedron has all ideal vertices or that $n=2$. Volume estimates are given for all equiangular hyperbolic Coxeter polyhedra. \end{abstract}
\title{Volume estimates for equiangular hyperbolic Coxeter polyhedra}
\section{Introduction}
An orientable $3$--orbifold, $\mathcal{Q}$, is determined by an underlying $3$--manifold, $X_{\mathcal{Q}}$, and a trivalent graph, $\Sigma_{\mathcal{Q}}$, labeled by integers. If $\mathcal{Q}$ carries a hyperbolic structure then it is unique by Mostow rigidity, so the hyperbolic volume of $\mathcal{Q}$ is an invariant of $\mathcal{Q}$. Therefore, for hyperbolic orbifolds with a fixed underlying manifold, the volume is a function of the labeled graph $\Sigma.$ In this paper, methods for estimating the volume of orbifolds of a restricted type in terms of this labeled graph will be described.
The orbifolds studied in this paper are quotients of $\mathbb{H}^3$ by reflection groups generated by reflections in hyperbolic Coxeter polyhedra. A Coxeter polyhedron is one where each dihedral angle is of the form $\pi/n$ for some $n\in \mathbb{Z}$, $n\geq 2$. Given a hyperbolic Coxeter polyhedron $\mathcal{P}$, consider the group generated by reflections through the geodesic planes determined by its faces, $\Gamma(\mathcal{P})$. Then $\Gamma(\mathcal{P})$ is a Kleinian group which acts on $\mathbb{H}^3$ with fundamental domain $\mathcal{P}$. The quotient, $\mathcal{O}=\mathbb{H}^3/ \Gamma(\mathcal{P})$, is a non--orientable orbifold with singular locus $\mathcal{P}^{(2)}$, the $2$--skeleton of $\mathcal{P}$. One may think of obtaining $\mathcal{O}$ by ``mirroring" the faces of $\mathcal{P}$. It is a consequence of Andreev's theorem that any equiangular hyperbolic Coxeter polyhedron has either all dihedral angles equal to $\pi/3$ and is ideal or has all dihedral angles equal to $\pi/2$ \cite{andreev1, andreev2}. This paper gives two-sided, combinatorial volume estimates for all equiangular hyperbolic Coxeter polyhedra. In this paper, only polyhedra with finite volume will be considered.
Lackenby gave volume estimates for hyperbolic alternating link complements in \cite{lackenby} in terms of the twist number of the link. His work was part of what led to this investigation of how geometric data arises from associated combinatorial data. Some of the techniques used in this paper follow methods used by Lackenby. The lower bound given by Lackenby was improved by Agol, Storm, and Thurston in \cite{agolhaken}. In his thesis \cite{inoue}, Inoue has identified the two smallest-volume, compact, right-angled hyperbolic polyhedra. He also gave a method to order such polyhedra based on a decomposition into L\"{o}bell polyhedra, provided that the volume of any given right-angled polyhedron can be calculated exactly.
The results of this paper can be used to list all equiangular hyperbolic polyhedra with volume not exceeding some fixed value. A sample application of this is to classify all arithmetic Kleinian maximal reflection groups. Agol has shown in \cite{agolreflect} that the number of such groups is finite up to conjugacy. Given a maximal reflection group $\Gamma$ generated by reflections in a polyhedron $\mathcal{P}$, he gives an upper bound, independent of $\Gamma$, for the volume of $\mathcal{P}/\Theta$ where $\Theta$ is the group of symmetries of $\mathcal{P}$ which are not reflections. One could therefore attempt to classify such groups using the results of this paper by writing down a list of all polyhedra of sufficiently small volume and checking arithmeticity for those for which the quotient by additional symmetries has small enough volume.
\begin{acknowledgments} The author would like to thank his thesis advisor, Ian Agol, for his excellent guidance and the referee for many valuable comments. The author was partially supported by NSF grant DMS-0504975. \end{acknowledgments}
\section{Summary of results}
The results of the paper are outlined in this section. Theorems \ref{sharp2}, \ref{pi2finite}, and \ref{pi2general} concern the volumes of right-angled hyperbolic polyhedra. Theorem \ref{sharp3} concerns hyperbolic polyhedra with all angles $\pi/3$. Before stating any results, some terminology will be introduced.
An \textit{abstract polyhedron} is a cell complex on $S^2$ which can be realized by a convex Euclidean polyhedron. A theorem of Steinitz says that realizability as a convex Euclidean polyhedron is equivalent to the $1$--skeleton of the cell complex being $3$--connected \cite{steinitz}. A graph is $3$--connected if the removal of any $2$ vertices along with their incident edges leaves the complement connected.
A \textit{labeling} of an abstract polyhedron $P$ is a map $$\Theta : \mbox{Edges}(P) \to (0,\pi/2].$$ For an abstract polyhedron, $P$, and a labeling, $\Theta$, the pair $(P,\Theta)$ is a \textit{labeled abstract polyhedron.} A labeled abstract polyhedron is said to be \textit{realizable as a hyperbolic polyhedron} if there exists a hyperbolic polyhedron, $\mathcal{P}$, such that there is a label-preserving graph isomorphism between $\mathcal{P}^{(1)}$ with edges labeled by dihedral angles and $P$ with edges labeled by $\Theta$. A \textit{defining plane} for a hyperbolic polyhedron $\mathcal{P}$ is a hyperbolic plane $\Pi$ such that $\Pi \cap \mathcal{P}$ is a face of $\mathcal{P}$. A labeling $\Theta$ which is constantly equal to $\pi/n$ is \textit{$\pi/n$--equiangular}. Suppose $G$ is a graph and $G^*$ is its dual graph. A \textit{$k$--circuit} is a simple closed curve composed of $k$ edges in $G^*$. A \textit{prismatic $k$--circuit} is a $k$--circuit $\gamma$ so that no two edges of $G$ which correspond to edges traversed by $\gamma$ share a vertex. The following theorem is a special case of Andreev's Theorem, which gives necessary and sufficient conditions for a labeled abstract polyhedron to be realizable as a hyperbolic polyhedron.
\begin{theorem}[Andreev's theorem for $\pi/2$--equiangular polyhedra] A $\pi/2$--equiangular labeled abstract polyhedron $(P,\Theta)$ is realizable as a hyperbolic polyhedron, $\mathcal{P}$, if and only if the following conditions hold: \begin{enumerate} \item $P$ has at least $6$ faces. \item $P$ each vertex has degree $3$ or degree $4$. \item For any triple of faces of $P$, $(F_i,\, F_j,\, F_k),$ such that $F_i \cap F_j$ and $F_j \cap F_k$ are edges of $P$ with distinct endpoints, $F_i \cap F_k = \emptyset.$ \item $P^*$ has no prismatic $4$ circuits. \end{enumerate} Furthermore, each degree $3$ vertex in $P$ corresponds to a finite vertex in $\mathcal{P}$, each degree $4$ vertex in $P$ corresponds to an ideal vertex in $\mathcal{P}$, and the realization is unique up to isometry. \end{theorem}
The first result gives two-sided volume estimates for ideal, $\pi/2$--equiangular hyperbolic polyhedra.
\begin{theorem} \label{theorempi2} \label{sharp2} If $\mathcal{P}$ is an ideal $\pi/2$--equiangular polyhedron with $N$ vertices, then $$ (N-2) \cdot \frac{V_8}{4} \leq \mbox{\rm{Vol}}(\mathcal{P})\leq (N-4) \cdot \frac{V_8}{2}, $$ where $V_8$ is the volume of a regular ideal hyperbolic octahedron. Both inequalities are equality when $\mathcal{P}$ is the regular ideal hyperbolic octahedron.
There is a sequence of ideal $\pi/2$--equiangular polyhedra $\mathcal{P}_i$ with $N_i$ vertices such that $\mbox{\rm{Vol}}(\mathcal{P}_i)/N_{i}$ approaches $V_8/2$ as $i$ goes to infinity. \end{theorem}
The constant $V_8$ is the volume of a regular ideal hyperbolic octahedron. In terms of the Lobachevsky function,
$$\Lambda(\theta)=-\int_0^{\theta} \log |2\sin\,t|\, dt,$$
$V_8=8\Lambda(\pi/4).$ This volume can also be expressed in terms of Catalan's constant, $K$, as $V_8 = 4K$, where $$K=\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^2}.$$ The value of $V_8$ to five decimal places is $3.66386.$
The proof is spread throughout the paper. The lower bound will be shown in Section ~\ref{S:lower} to be a consequence of the stronger Theorem~\ref{pi2lower}, which depends also on information about the number of faces. The upper bound in Theorem~\ref{theorempi2} will be proved in Section ~\ref{S:upper} and will be shown to be asymptotically sharp in Section ~\ref{S:seq}.
Using similar techniques, the following theorem giving volume estimates for compact $\pi/2$--equiangular polyhedra will be proved:
\begin{theorem} \label{pi2finite} If $\mathcal{P}$ is a compact $\pi/2$--equiangular hyperbolic polyhedron with $N$ vertices, then $$ (N-8) \cdot \frac{V_8}{32} \leq \mbox{\rm{Vol}}(\mathcal{P}) < (N-10)\cdot \frac{5V_3}{8},$$ where $V_3$ is the volume of a regular ideal hyperbolic tetrahedron. There is a sequence of compact polyhedra, $\mathcal{P}_i$, with $N_i$ vertices such that $\mbox{\rm{Vol}}(\mathcal{P}_i)/N_i$ approaches $5V_3/8$ as $i$ goes to infinity. \end{theorem}
In terms of the Lobachevsky function, $V_3=2 \Lambda(\pi/6).$ To five decimal places, $V_3$ is $1.01494$.
Combining the methods of Theorems~\ref{theorempi2} and \ref{pi2finite}, estimates will be given for $\pi/2$--equiangular polyhedra with both finite and ideal vertices:
\begin{theorem} \label{pi2general} If $\mathcal{P}$ is a $\pi/2$--equiangular hyperbolic polyhedron with $N_{\infty}$ ideal vertices and $N_F$ finite vertices, then $$\frac{4N_{\infty}+N_F-8}{32} \cdot V_8 \leq \mbox{\rm{Vol}}(\mathcal{P}) < (\mbox{\rm{N}}_{\infty}-1)\cdot \frac{V_8}{2} +
N_F\cdot \frac{5 V_3}{8} .$$ \end{theorem}
The proofs of the lower bounds in Theorems~\ref{pi2finite} and \ref{pi2general} appear in Section \ref{S:lower}. The upper bounds will be proved in Section \ref{S:upper}.
Two-sided volume estimates for $\pi/3$--equiangular polyhedra are also given. First, the special case of Andreev's Theorem for $\pi/3$--equiangular polyhedra is stated.
\begin{theorem}[Andreev's theorem for $\pi/3$--equiangular polyhedra]\label{and3} A $\pi/3$--equiangular abstract polyhedron, $(P, \Theta)$, is realizable as a hyperbolic polyhedron, $\mathcal{P}$ if each vertex of $P$ has degree $3$ and $P^*$ has no prismatic $3$--circuits. Furthermore, each vertex of $\mathcal{P}$ is ideal and $\mathcal{P}$ is unique up to isometry. \end{theorem}
The next theorem gives analogous results to that of Theorem~\ref{theorempi2} for ideal $\pi/3$--equiangular polyhedra.
\begin{theorem} \label{sharp3} \label{theorempi3} If $\mathcal{P}$ is an ideal $\pi/3$--equiangular polyhedron with $N>4$ vertices, then $$N\cdot\frac{V_3}{3} \leq \mbox{\rm{Vol}}(\mathcal{P}) \leq (3N-14) \cdot \frac{V_3}{2},$$
where $V_3$ is the volume of a regular ideal hyperbolic tetrahedron. The upper bound is sharp for the regular ideal hyperbolic cube. There is a sequence of ideal $\pi/3$--equiangular polyhedra $\mathcal{P}_i$ with $N_i$ vertices such that $\mbox{\rm{Vol}}(\mathcal{P}_i)/N_{i}$ approaches $3V_3 / 2$ as $N_i$ increases to infinity. \end{theorem}
The lower bound in this theorem will be proved in Section \ref{S:lower3} by packing horoballs around the vertices. The upper bound will be proved in Section \ref{S:upper} and will be shown to be asymptotically sharp in Section \ref{S:seq}, mirroring the proofs in the $\pi/2$ case. In a personal communication, Rivin has indicated how to improve the lower bound to $N \cdot \frac{3 V_3}{8}$. His argument will be briefly described at the end of Section \ref{S:lower3}.
\section{Lower volume bound for ideal $\pi/2$--equiangular polyhedra}\label{S:lower} The key result used in proving the lower volume bound in Theorems~\ref{theorempi2} and \ref{pi2lower} is a theorem of Miyamoto which says that the volume of a complete hyperbolic $3$--manifold with totally geodesic boundary is greater than or equal to a constant multiple of the area of the boundary \cite{miyamoto}:
\begin{theorem}[Miyamoto] \label{miyamoto} If $\mathcal{O}$ is a complete hyperbolic $3$--orbifold with non-empty totally geodesic boundary, then
$$\mbox{\rm{Vol}}(\mathcal{O})\geq \mbox{\rm{Area}}(\partial \mathcal{O}) \cdot \frac{V_8}{4 \pi}$$ with equality only if $M$ can be decomposed into regular ideal hyperbolic octahedra. \end{theorem}
Miyamoto actually only stated this theorem for manifolds. The orbifold version follows immediately, however. Given a complete hyperbolic $3$--orbifold, $\mathcal{O}$, with non-empty totally geodesic boundary, Selberg's lemma implies the existence of an integer $m$ such that an $m$--fold cover of $\mathcal{O}$ is a manifold $M$ \cite{selberg}. Then since $M$ is a finite cover, $\mbox{\rm{Vol}} (M)=m \cdot \mbox{\rm{Vol}} (\mathcal{O})$ and $\mbox{\rm{Area}} (\partial M) = m \cdot \mbox{\rm{Area}} (\partial \mathcal{O})$.
Theorem~\ref{theorempi2} is a consequence of the following stronger theorem which also takes into account information about the faces of the polyhedron. Andreev's theorem for $\pi/2$--equiangular polyhedra implies that the $1$--skeleton of an ideal $\pi/2$--equiangular polyhedron is a four-valent graph on $S^2$. The faces, therefore, can be partitioned into a collection of black faces, $\mathcal{B},$ and a collection of white faces, $\mathcal{W}$, so that no two faces of the same color share an edge. Denote by $|\mathcal{B}|$ and
$|\mathcal{W}|$ the number of black and white faces, respectively.
\begin{theorem}\label{pi2lower}
Suppose $\mathcal{P}$ is an ideal $\pi/2$--equiangular polyhedron with $N$ vertices. If $\mathcal{P}$ is $2$--colored with $|\mathcal{B}|\geq|\mathcal{W}|$, then
$$ (N-|\mathcal{W}|) \cdot \frac{V_8}{2} \leq \mbox{\rm{Vol}}{(\mathcal{P})}.$$ This inequality is sharp for an infinite family of polyhedra obtained by gluing together regular ideal hyperbolic octahedra. \end{theorem}
\begin{proof} Given an ideal $\pi/2$--equiangular polyhedron $\mathcal{P}$, consider the orbifold $\mathbb{H}^3 / \Gamma(\mathcal{P})$, where $\Gamma(\mathcal{P})$ is the reflection group generated by $\mathcal{P}$, as described in the introduction. Denote the generators of $\Gamma(\mathcal{P})$ by the same symbol denoting the face of $\mathcal{P}$ through which it is a reflection. If $A$ is a face of $\mathcal{P}$, let $\Gamma_{A}(\mathcal{P})$ be the group obtained from $\Gamma(\mathcal{P})$ by removing the generator $A$ and all relations involving $A$. This may be thought of as ``un-mirroring" the face $A$. Then $CC(\mathbb{H}^3/\Gamma_{A}(\mathcal{P})),$ the convex core of $\mathbb{H}^3/\Gamma_{A}(\mathcal{P})$, is an orbifold with totally geodesic boundary and the same volume as $\mathcal{P}$. Note that the face $A$ is a suborbifold of $\mathbb{H}^3 / \Gamma(\mathcal{P})$ because all dihedral angles of $\mathcal{P}$ are $\pi/2$. In general, a face of a polyhedron meeting a dihedral angle not equal to $\pi/2$ will not be a suborbifold, and removing the generators and relations corresponding to that face from the reflection group will not give a totally geodesic boundary because the preimage of that face in $\mathbb{H}^3$ will not be a collection of disjoint geodesic planes.
To get the best lower bound on volume from Theorem~\ref{miyamoto}, the boundary should be chosen to have the greatest possible area. Given a collection of faces $\mathcal{A} =\bigcup_{i=1} ^M A_i$, denote the group obtained by removing all generators and relations involving the $A_i$ by $\Gamma_{\mathcal{A}} (\mathcal{P})$. If no two faces in such a collection $\mathcal{A}$ share an edge, the orbifold $CC(\mathbb{H}^3 / \Gamma_{\mathcal{A}} (\mathcal{P}))$ has totally geodesic boundary.
As described before the statement of the theorem, $2$--color the faces of $\mathcal{P}$
black and white so that no two faces of the same color share an edge. Suppose that coloring is chosen so that the number of black faces, $|\mathcal{B}|$, is at least the number of white faces, $|\mathcal{W}|$. This choice will ensure that the sum of the areas of the faces in $\mathcal{W}$ is at least the sum of the areas of the black faces, as will be seen in Lemma~\ref{areaw}. Then, $CC(\mathbb{H}^3 / \Gamma_{\mathcal{W}}(\mathcal{P}))$ is an orbifold with totally geodesic boundary consisting of the $W_i$. Theorem~\ref{miyamoto} applied to this orbifold gives $$\mbox{\rm{Vol}}(\mathcal{P}) \geq \mbox{\rm{Area}}(\mathcal{W}) \cdot \rho_3 (0).$$ The following lemma gives the area of $\mathcal{W}$ and completes the proof of the lower bound.
\begin{lemma}\label{areaw} With $\mathcal{W}$ as above,
$$\mbox{\rm{Area}}(\mathcal{W})=2\pi(N-|\mathcal{W}|).$$ \end{lemma} \begin{proof} Consider $D\mathcal{W}$, the double of $\mathcal{W}$ along its boundary. Recall that the orbifold Euler characteristic of a $2$--orbifold $Q$ is $$\chi(Q)=\chi(X_Q) - \sum_{i} (1-1/m_i),$$ where $X_Q$ is the underlying topological space and $Q$ has cone points of orders $m_i$ \cite{CHK}. The faces in $\mathcal{W}$ meet each of the $N$ vertices and each of the $E$ edges of $\mathcal{P}$. The orbifold $D\mathcal{W}$ is a union over all faces of $\mathcal{W}$ of doubled ideal polygons. Each of these doubled ideal polygons is a $2$--sphere with a cone point of order $\infty$ for each vertex of the polygon.
Each vertex of $\mathcal{P}$ contributes a cone point to two of these doubled polygons. Therefore
$$\chi(D\mathcal{W})= |\mathcal{W}| \cdot \chi(S^2) - 2N=2(|\mathcal{W}|-N).$$ The punctured surface $D\mathcal{W}$ is hyperbolic, being a union of hyperbolic $2$--orbifolds. Therefore, the Gauss--Bonnet theorem for orbifolds implies that
$$\mbox{\rm{Area}}(D\mathcal{W})=-2\pi \chi(D\mathcal{W})=4\pi (N-|\mathcal{W}|),$$ so that
$\mbox{\rm{Area}}(\mathcal{W})=2\pi(N-|\mathcal{W}|)$. \end{proof}
The polyhedra which realize the lower bound as claimed in Theorem~\ref{pi2lower} are constructed by gluing together octahedra. Consider a regular ideal hyperbolic octahedron with faces colored white and black, so that no two faces of the same color share an edge. For a single octahedron, $N=6$ and
$|\mathcal{W}|=4$, so the lower bound is equal to the volume. To obtain an infinite number of polyhedra which satisfy the claim, glue a finite collection of $2$--colored regular ideal hyperbolic octahedra together, only gluing black faces to black faces. Each successive gluing results in a polyhedron with $3$ more vertices and $1$ more white face. Therefore by induction, for each example constructed in this fashion, the lower inequality in Theorem~\ref{pi2lower} will be equality. See figure~\ref{figure:gluing}. Note that gluing octahedra in a different pattern than described yields examples which do not satisfy the claim. This completes the proof of Theorem~\ref{pi2lower}. \end{proof}
\begin{figure}
\caption{An example of the gluing.}
\label{figure:gluing}
\end{figure}
The argument giving the lower bound in Theorem~\ref{theorempi2} as a consequence of Theorem~\ref{pi2lower} is similar to the proof of Theorem $5$ in
\cite{lackenby}. The idea is to average the estimate coming from the black faces with the estimate coming from the white faces. Consider a sort of dual polyhedron, $\mathcal{G}$, to $\mathcal{P}$. The vertices of $\mathcal{G}$ are the white faces in the specified coloring of $\mathcal{P}$. For any two faces of $\mathcal{P}$ which share a vertex, the corresponding two vertices in $\mathcal{G}$ are connected by an edge. The $2$--skeleton is homeomorphic to $S^2$, so the Euler characteristic of $\mathcal{G}^{(2)}$ is $2$. The number of vertices, edges and faces of $\mathcal{G}^{(2)}$ respectively are $|\mathcal{W}|$, $N$, and
$|\mathcal{B}|$, where $N$ is the number of vertices of $\mathcal{P}$. Hence $|\mathcal{B}|= 2-|\mathcal{W}| +N$. An application of Lemma~\ref{areaw} yields $\mbox{\rm{Area}}(\mathcal{B})=2\pi
(N-|\mathcal{B}|)=2\pi(|\mathcal{W}|-2)$. Hence
$$\mbox{\rm{Vol}}(\mathcal{P}) \geq (|\mathcal{W}|-2) \cdot \frac{V_8}{2}.$$ Therefore, combining this inequality with the inequality from Theorem~\ref{pi2lower}, $$\mbox{\rm{Vol}}(\mathcal{P}) \geq (N-2) \cdot \frac{V_8}{4},$$ proving the lower bound of Theorem~\ref{theorempi2}.
The lower bound in Theorem~\ref{pi2general}, where $\mathcal{P}$ is a $\pi/2$--equiangular polyhedron with vertices which are either finite or ideal, is proved similarly. \begin{proposition} Suppose $\mathcal{P}$ is a $\pi/2$--equiangular hyperbolic polyhedron with $N_{\infty}$ ideal vertices, $N_F$
finite vertices, and $|\mathcal{F}|$ faces. Then
$$\frac{8N_{\infty}+3N_F-4|\mathcal{F}|}{32} \cdot V_8 \leq \mbox{\rm{Vol}}(\mathcal{P}).$$ \end{proposition} \begin{proof} By the four color theorem, a $4$--coloring of the faces of $\mathcal{P}$ may be found \cite{appelhaken,gonthier}. One of the collections of faces of the same color, say $\mathcal{B}$, has area at least $\mbox{\rm{Area}}(\partial\mathcal{P})/4$, where $\mbox{\rm{Area}}(\partial \mathcal{P})$ should be interpreted as the sum of the areas of all the faces of $\mathcal{P}$. The area of a hyperbolic $k$--gon with interior angles summing to $S$ is $(k-2)\pi - S$. The sum of the interior angles of a face of $\mathcal{P}$ is $n_F \cdot \pi/2,$ where $n_F$ is the number of finite vertices of the face. Hence the area of a single face is $$\pi \left(n_{\infty}+\frac{n_F}{2}-2\right),$$ where $n_{\infty}$ is the number of ideal vertices of the face. Summing over all faces and using the fact that each finite vertex is a vertex of three faces and each ideal vertex is a vertex of four faces gives
$$\mbox{\rm{Area}}(\partial \mathcal{P}) = \pi \cdot \frac{8N_{\infty}+3N_F -4 |\mathcal{F}|}{2}.$$ Then since $\mbox{\rm{Area}}(\mathcal{B}) \geq \mbox{\rm{Area}}(\partial \mathcal{P})/4$,
applying Miyamoto's theorem to $\mathcal{B}$ finishes the proof of the proposition. \end{proof}
The lower bound in Theorem~\ref{pi2finite} follows by setting $N_{\infty}=0$.
\section{Lower volume bound for ideal $\pi/3$--equiangular polyhedra}\label{S:lower3}
In this section the lower bound given in Theorem~\ref{theorempi3} will be proved:
\begin{proposition}\label{pi3prop} If $\mathcal{P}$ is an ideal $\pi/3$--equiangular polyhedron with $N>4$ vertices, then $$\mbox{\rm{Vol}}(\mathcal{P}) > N \cdot \frac{V_3}{3} .$$ \end{proposition} Before proving this proposition, two lemmas about ideal $\pi/3$--equiangular polyhedra with more than $4$ vertices are needed.
Consider a $\pi/3$--equiangular polyhedron, $\mathcal{P},$ in the upper half-space model for $\mathbb{H}^3$ with one vertex placed at the point at infinity. The link of each vertex is Euclidean, so must be an equilateral Euclidean triangle since all dihedral angles are $\pi/3$. Thus the image of $\mathcal{P}$ under the orthogonal projection to the bounding plane of $\mathbb{H}^3$ is an equilateral triangle. This triangle will be referred to as the {\it base triangle}. The three vertices adjacent to the vertex at infinity will be called {\it corner vertices}.
The following is a corollary of Andreev's theorem for $\pi/3$--equiangular polyhedra.
\begin{corollary}\label{cor4edges} If $\mathcal{P}$ is a $\pi/3$--equiangular polyhedron which has more than $4$ vertices, then each face of $\mathcal{P}$ has at least $4$ edges. \end{corollary}
\begin{proof} Suppose for contradiction that $\mathcal{P}$ has a triangular face, $\Delta_1$. Andreev's theorem for $\pi/3$--equiangular polyhedra (Theorem~\ref{and3}) says that the dual graph of $\mathcal{P}^{(1)}$ has no prismatic $3$--circuits, so at least two of the edges emanating from $\Delta_1$ share a vertex. Hence $\mathcal{P}$ contains two adjacent triangular faces, $\Delta_1$ and $\Delta_2$. Let $v_1$ be the vertex of $\Delta_1$ which is not contained in $\Delta_2$ and $v_2$ the vertex of $\Delta_2$ which is not contained in $\Delta_1$. Let $e_1$ and $e_2$ be the edges emanating from $v_1$ and $v_2$ respectively which are not contained in $\Delta_1$ or $\Delta_2$. The edges $e_1$ and $e_2$ are both contained in two common faces. See figure~\ref{figure:andedge}. Therefore by convexity, $e_1$ and $e_2$ must actually be the same edge, which contradicts the fact that $\mathcal{P}$ has more than $4$ vertices. \end{proof}
\begin{figure}
\caption{Note that $e_1$ and $e_2$ are both part of the ``front" face and the ``back" face.}
\label{figure:andedge}
\end{figure}
\begin{lemma}\label{cornersymmetry} Suppose that $\mathcal{P}$ is an ideal $\pi/3$--equiangular polyhedron with $N>4$ vertices. If coordinates for the upper half-space model of $\mathbb{H}^3$ are chosen so that a vertex of $\mathcal{P}$ is at the point at infinity, then the Euclidean distance from a corner vertex to each of the adjacent vertices in the base triangle are equal. \end{lemma}
\begin{proof} The fact that all dihedral angles are equal to $\pi/3$ implies that the arrangement of defining planes for the corner vertex is left invariant under reflection through a geodesic plane through infinity bisecting the angle between the two vertical planes defining the vertex. This proves the lemma. \end{proof}
The following lemma shows that if $\mathcal{P}$ is a non-obtuse polyhedron, then intersections of faces of $\mathcal{P}$ correspond to intersections of the defining planes of $\mathcal{P}$.
\begin{lemma}\label{orthgeod} If $\mathcal{P}$ is a non-obtuse hyperbolic polyhedron, then the closures of two faces $F_1$ and $F_2$ of $\mathcal{P}$ intersect if and only if $\overline{\Pi}_1$ and $\overline{\Pi}_2$ intersect in $\overline{\mathbb{H}}^3$ where $\Pi_i$ is the defining plane for $F_i$. \end{lemma}
\begin{proof} Sufficiency is clear.
For necessity, the contrapositive will be proved. Suppose that $F_1$ and $F_2$ are two faces of $\mathcal{P}$ such that their closures do not intersect. A geodesic orthogonal to both $F_1$ and $F_2$ will be constructed. This geodesic is also orthogonal to both $\Pi_1$ and $\Pi_2$, and such a geodesic exists only if the closures of the $\Pi_i$ are disjoint.
Choose any $x_0 \in F_1$ and $y_0 \in F_2$ and let $\gamma_0$ be the geodesic between them. The set, $$K(\gamma_0) = \{(x,y) \in F_1 \times F_2 \, \mid \, d(x,y) \leq l(\gamma_0) \},$$ is a closed subset of $F_1 \times F_2$. There exists open subsets $N_i$ of $F_i$ containing all of the ideal vertices of $F_i$ such that for any $z_1 \in N_1$ and $z_2 \in N_2$, $d(z_1,z_2) > l(\gamma_0).$ Hence $K(\gamma_0)$ is also a bounded subset of $F_1 \times F_2$, therefore compact.
It follows from compactness of $K(\gamma_0)$ that $$d_{min} = \mbox{\rm{min}} \{ d(x,y)\, \mid \, (x,y) \in K(\gamma_0) \}$$ is achieved for some $(x,y) \in K(\gamma_0).$ The geodesic segment, $\gamma$, between $x$ and $y$ must be orthogonal to both $F_1$ and $F_2$. If not, suppose $\gamma$ is not orthogonal to $F_1$. Since $\mathcal{P}$ is non-obtuse, the orthogonal projection of $y$ to $\Pi_1$ is contained in $F_1$. By the hyperbolic Pythagorean theorem, the geodesic between $y$ and its projection has length less than that of $\gamma$. This contradicts the construction of $\gamma$. The argument is identical if $\gamma$ is not orthogonal to $F_2$. \end{proof}
There is actually a simpler argument for the previous lemma in the case that $\mathcal{P}$ is a Coxeter polyhedron. The development of $\mathcal{P}$ into $\mathbb{H}^3$ gives a tessellation of $\mathbb{H}^3$ by copies of $\mathcal{P}$, so any intersection of defining planes must correspond to an edge of $\mathcal{P}$.
\begin{corollary}\label{3lem} Suppose $\mathcal{P}$ is an ideal $\pi/3$--equiangular polyhedron with $N>4$ vertices. Choose coordinates for the upper half-space model of $\mathbb{H}^3$ so that a vertex, $v_0,$ of $\mathcal{P}$ is at the point at infinity. Then if the distance in Lemma \ref{cornersymmetry} from the corner vertex, $u$, to the two adjacent vertices in the base triangle is $r$ and the edge length of the base triangle is $a$, then $0<r<\frac{3a}{4}$. \end{corollary}
\begin{proof} Suppose that the three defining planes which contain $v_0$ are $\Pi_1$, $\Pi_2$ and $\Pi_3$ and that the three defining planes containing $u$ are $\Pi_1$, $\Pi_2$ and $\Pi_4$. If $\frac{3a}{4} \leq r < a,$ then $\Pi_3$ intersects $\Pi_4$ with interior dihedral angle less than $\pi/3$. By Lemma~\ref{orthgeod}, the corresponding faces, $F_3$ and $F_4$, also have intersecting closures, and the interior dihedral angle between $F_3$ and $F_4$ will be less than $\pi/3$. If $r=a$, then $\mathcal{P}$ would be a tetrahedron and for $r > a$, $\mathcal{P}$ would have finite vertices at the points $\Pi_1 \cap \Pi_3 \cap \Pi_4$ and $\Pi_2 \cap \Pi_3 \cap \Pi_4$. \end{proof}
In what follows, the intersection with $\mathcal{P}$ of a closed horoball centered at a vertex $u$ of $\mathcal{P}$ which intersects only faces and edges containing $u$ will be called a {\it vertex neighborhood}. The next lemma is the main observation which leads to the lower volume bound. This lemma follows the approach of Adams in \cite{adams}.
\begin{lemma}\label{bump} Let $\mathcal{P}$ be an ideal $\pi/3$--equiangular polyhedron with more than $4$ vertices. Suppose two vertex neighborhoods of equal volume intersect with disjoint interiors. Then the volume of each of the vertex neighborhoods is at least $\frac{\sqrt{3}}{6}$. \end{lemma} \begin{proof} Let $B_1$ and $B_2$ about vertices $u_1$ and $u_2$, respectively, be vertex neighborhoods which intersect with disjoint interiors. Choose coordinates for the upper half-space model of $\mathbb{H}^3$ so that $u_1$ is the point at infinity and $B_1$ intersects $B_2$ at Euclidean height $1$ above the bounding plane. Let $\Gamma(\mathcal{P})$ be the reflection group generated by $\mathcal{P}$ and $\mathcal{G}_{\infty}$ the subgroup fixing the point at infinity: $$\mathcal{G}_{\infty} = \{ \gamma \in \Gamma(\mathcal{P})\, \mid \, \gamma(\{\infty\})=\{\infty\}\}.$$
Let $H_1=\mathcal{G}_{\infty} \cdot B_1$ be the horoball centered at infinity covering $B_1$ and let $H_2$ be one of the height $1$ horoballs contained in $\Gamma(\mathcal{P}) \cdot B_2$. The projection of $\mathcal{P}$ to the bounding plane of $\mathbb{H}^3$ is an equilateral triangle and the orbit of this triangle under the action of $\mathcal{G}_{\infty}$ tiles the plane. Let $\Delta$ be a triangle in this tiling containing the point of $\overline{\mathbb{H}^3}$ about which $H_2$ is centered.
The collection of height $1$ horoballs covering $B_2$ is equal to $\mathcal{G}_{\infty} \cdot H_2$ and, for each pair $g\neq h \in \mathcal{G}_{\infty},$ either $g H_2 \cap h H_2$ is empty, a single point, or $g H_2 = h H_2$. The proof breaks up into three cases. Either $u_2 \in \text{int}\,{\Delta}$, $u_2$ is contained in the interior of an edge of $\Delta$, or $u_2$ is a vertex of $\Delta$.
If $u_2 \in \text{int}\,{\Delta},$ then the projection of $H_2$ to the bounding plane must be a closed disk contained in $\Delta$. The minimum possible value of $\mbox{\rm{Vol}}(B_1)=\mbox{\rm{Vol}}(B_2)$ occurs when the projection of $H_2$ to the bounding plane is inscribed in $\Delta$ and $\Delta$ has edge length $\sqrt{3}$, as shown in figure~\ref{baryball}. Hence the area of $\Delta$ is $\frac{3 \sqrt{3}}{4}$ and $$\mbox{\rm{Vol}}(B_i)=\frac{3 \sqrt{3}}{4}\int_{1}^{\infty} \frac{dz}{z^3} = \frac{3\sqrt{3}}{8}.$$
\begin{figure}
\caption{Projection of $H_2$ inscribed in $\Delta$ }
\label{baryball}
\end{figure}
If $u_2$ is contained in the interior of an edge of $\Delta$, then the minimum possible value of the vertex neighborhood volume occurs when $\Delta$ has edge length $\frac{2\sqrt{3}}{3}$ and $u_2$ is at the midpoint of an edge of $\Delta$, as in figure~\ref{edgeball}. Calculating as above, $$\mbox{\rm{Vol}}(B_i)=\frac{\sqrt{3}}{6}.$$
\begin{figure}
\caption{Projection of $H_2$ for case $u_2$ contained in the interior of an edge of $\Delta$ }
\label{edgeball}
\end{figure}
Now suppose that $u_2$ is a vertex of $\Delta$. If the edge length of $\Delta$ is $a$, then $$\mbox{\rm{Vol}}(B_1)=\frac{a^2 \sqrt{3}}{4}\int_{1}^{\infty} \frac{dz}{z^3}=\frac{a^2 \sqrt{3}}{8}.$$ If $r$ is as in Corollary~\ref{3lem}, then $$\mbox{\rm{Vol}}(B_2)=\frac{\sqrt{3}}{8 r^2}.$$ By Corollary~\ref{3lem}, $0<r<\frac{3a}{4}$, so $$\mbox{\rm{Vol}}(B_2)> \frac{2 \sqrt{3}}{9a^2}.$$ Equating this lower bound with $\mbox{\rm{Vol}}(B_1)$ yields $a= \frac{2\sqrt{3}}{3}$. Therefore it may concluded that $$\mbox{\rm{Vol}}(B_i)>\frac{\sqrt{3}}{6}.$$ \end{proof}
To complete the proof of Theorem~\ref{pi3prop}, start with disjoint, equal volume vertex neighborhoods at each vertex. Expand the vertex neighborhoods so that the volumes remain equal at all time until two of the vertex neighborhoods intersect with disjoint interior intersection. Lemma~\ref{bump} then says that there is a vertex neighborhood at each vertex of volume at least $\sqrt{3}/6$. B\"or\"oczky and Florian in \cite{boro} show that the maximal density of a horoball packing in $\mathbb{H}^3$ is ${\sqrt{3}}/(2V_3)$. Applying this result gives $$\mbox{\rm{Vol}}(\mathcal{P}) > N\cdot \frac{V_3}{3}.$$
In a personal communication, Rivin has indicated how to improve the lower bound to $N\cdot \frac{3V_3}{8}.$ The idea of the argument is that for any given vertex $v$ in a $\pi/3$-equiangular polyhedron $\mathcal{P}$, $v$ along with the three vertices of $\mathcal{P}$ with which $v$ shares an edge are the vertices of a regular ideal hyperbolic tetrahedron contained in $\mathcal{P}$. A collection of such tetrahedra with disjoint interiors may be constructed by taking any independent set of vertices of $\mathcal{P}$. By a result of Heckman and Thomas, a trivalent graph with $N$ vertices contains an independent set of cardinality at least $\frac{3N}{8}$ \cite{indset}.
\section{The Upper Volume Bounds}\label{S:upper}
In this section, the upper volume bounds in Theorems~\ref{theorempi2}, \ref{pi2finite}, \ref{pi2general}, and \ref{theorempi3} will be proved using
arguments inspired by an argument of Agol and D. Thurston for an upper bound on the volume of an alternating link complement \cite{lackenby}. First, a decomposition of an arbitrary non-obtuse hyperbolic polyhedron into tetrahedra will be described. In each case, the volume contributed by the tetrahedra meeting at each vertex will be analyzed to obtain the volume bounds.
Let $\mathcal{P}$ be a non-obtuse hyperbolic polyhedron and $v_0$ a vertex of $\mathcal{P}$. For each face, $A_i$, not containing $v_0$, let $\gamma_i$ be the unique geodesic orthogonal to $A_i$ which passes through or limits to $v_0$, where $v_0$ is a finite or ideal vertex respectively. Define the nearest point projection, $u_i$, of $v_0$ to $A_i$ to be the intersection of $\gamma_i$ with $A_i$. The projection $u_i$ will lie on the interior of $A_i$ unless $A_i$ meets one of the faces containing $v_0$ orthogonally, in which case, $u_i$ will lie in the interior of an edge of $A_i$ or will coincide with a vertex of $A_i$ if $A_i$ meets two faces containing $v_0$ orthogonally. Cyclically label the vertices of $A_i$ by $v_{i,j}$ where $j\in \{ 1,\, 2, \dots \mbox{\rm{deg}}(A_i)\}$ is taken modulo $\mbox{\rm{deg}}(A_i)$. Let $w_{i,j}$ be the nearest point projection of $u_i$ onto the edge of $A_i$ with endpoints $v_{i,j}$ and $v_{i,j+1}$, where the nearest point projection is defined as above. Each face of a non-obtuse polyhedron is a non-obtuse polygon, so the nearest point projection of any point in $A_i$ to an edge $A_i$ actually lies in $A_i$. See figure~\ref{figure:tri}.
\begin{figure}
\caption{The figure on the left shows the case where $u_i$ is in the interior of a face. The figure on the right is the case where $u_i$ is in the interior of an edge.}
\label{figure:tri}
\end{figure}
Define $\Delta(i,j)$ to be the tetrahedron with vertices $v_0$, $u_i$, $w_{i,j}$, and $v_{i,j}$ and $\Delta'(i,j)$ to be the tetrahedron with vertices $v_0$, $u_i$, $w_{i,j}$, and $v_{i,j+1}$. In the case where $u_i$ coincides with $w_{i,j}$, both $\Delta(i,j)$ and $\Delta'(i,j)$ will be degenerate tetrahedra. Let $\mathcal{I}$ be the set of $(i,j)$ such that $\Delta(i,j)$ and $\Delta'(i,j)$ are nondegenerate. For each $(i,j)\neq (i',j') \in \mathcal{I}$, $\text{Int}(\Delta(i,j))\cap \text{Int}(\Delta(i',j'))=\emptyset$ and $\text{Int}(\Delta'(i,j))\cap \text{Int}(\Delta'(i',j'))=\emptyset.$ Also, the interior of each $\Delta$ is disjoint from the interior of each $\Delta'$. Then $$\mathcal{P}= \bigcup_{(i,j) \in \mathcal{I}} \left( \Delta(i,j) \cup \Delta'(i,j) \right).$$
This decomposition of $\mathcal{P}$ into tetrahedra will be analyzed to prove each of the upper bounds in Theorems~\ref{theorempi2}, \ref{pi2finite}, \ref{pi2general}, and \ref{theorempi3}. The following technical lemma is needed. It follows directly from the fact that the Lobachevsky function is concave down on the interval $[0,\pi/2]$.
\begin{lemma}\label{maxlem} Suppose $\overrightarrow{\alpha}=(\alpha_1, \dots, \alpha_M)$ where $\alpha_i\in[0,\pi/2]$. Let $$f(\overrightarrow{\alpha})=\frac{1}{2}\sum_{i=1}^{M} \Lambda(\pi/2-\alpha_i)$$ and $g(\overrightarrow{\alpha})=\alpha_1+ \cdots + \alpha_M$. Then the maximum value of $f(\overrightarrow{\alpha})$ subject to the constraint $g(\overrightarrow{\alpha})=C$ for some constant $C \in [0, M\pi/2]$ occurs for $\overrightarrow{\alpha}=(C/M, \dots, C/M)$. \end{lemma}
The next proposition gives the upper bound in Theorem~\ref{theorempi2}. \begin{proposition}\label{upper2} If $\mathcal{P}$ is an ideal $\pi/2$--equiangular polyhedron with $N$ vertices, then $$\mbox{\rm{Vol}}(\mathcal{P}) \leq (N-4) \cdot \frac{V_8}{2},$$ where $V_8$ is the volume of the regular ideal hyperbolic octahedron. Equality is achieved when $\mathcal{P}$ is the regular ideal hyperbolic octahedron. \end{proposition}
\begin{proof} Decompose $\mathcal{P}$ as above. Suppose that $v=v_{i,j}$ is a vertex of $\mathcal{P}$ which is not contained in a face containing $v_0$. Then $v$ is contained in exactly eight tetrahedra of the decomposition, say $T_1, \dots, T_8$. Suppose that $T_l$ coincides with $\Delta(m,n)$ in the decomposition. Then $T_l$ is a tetrahedron with $2$ ideal vertices, $v_0$ and $v$, and two finite vertices, $u_m$ and $w_{m,n}$. The dihedral angles along the edges between $v$ and $u_m$, between $u_m$ and $w_{m,n}$, and between $w_{m,n}$ and $v_0$ are all $\pi/2$. Suppose that the dihedral angle along the edge between $v$ and $v_0$ is $\alpha_l$. Then the dihedral angles along the remaining two edges are $\pi/2-\alpha_l$. See figure~\ref{figure:tet}.
\begin{figure}
\caption{One of the $T_l$}
\label{figure:tet}
\end{figure}
The volume of $T_l$ is given by $\Lambda(\pi/2-\alpha_l)/2$, where the Lobachevsky function, $\Lambda,$ is defined as $$\Lambda(\theta)=-\int_0
^{\theta} \log |2 \sin(t) | \, dt.$$ Therefore, the volume contributed by the tetrahedra adjacent to the vertex $v$ is a function of $\overrightarrow{\alpha}=(\alpha_1, \alpha_2, \dots, \alpha_8)$: $$f(\overrightarrow{\alpha})= \frac{1}{2} \sum_{l=1}^{8} \Lambda(\pi/2-\alpha_l).$$
The $\alpha_l$ must sum to $2\pi$. The maximum of $f(\overrightarrow{\alpha})$, subject to this constraint, occurs when $\overrightarrow{\alpha}=(\pi/4, \pi/4, \dots, \pi/4)$, by Lemma~\ref{maxlem}. Gluing $16$ copies of $T_l$ with $\alpha_l=\pi/4$ together appropriately yields a regular ideal hyperbolic octahedron. Hence $f(\pi/4, \pi/4, \dots, \pi/4) = V_{8}/2.$
Only two tetrahedra in the decomposition meet each of the four vertices which share an edge with $v_0$. A similar analysis as above shows that the volume contributed by the tetrahedra at each of these four vertices is no more than $V_{8}/8$.
Therefore, accounting for the vertex, $v_0$, at infinity and the fact that only $V_8 /8$ is contributed by each of the tetrahedra at the vertices adjacent to $v_0$, $$\mbox{\rm{Vol}}(\mathcal{P}) \leq (N-1)\cdot \frac{V_8}{2} - 4 \cdot 3\frac{V_8}{8}=(N-4) \cdot \frac{V_8}{2}.$$ Equality is clearly achieved when $\mathcal{P}$ is the regular ideal hyperbolic octahedron. \end{proof}
The proof of the upper bound in Theorem~\ref{theorempi3} is similar to the previous argument.
\begin{proposition}\label{upper3} If $\mathcal{P}$ is an ideal $\pi/3$--equiangular polyhedron with $N$ vertices, then $$\mbox{\rm{Vol}}(\mathcal{P}) \leq (3N-14) \cdot \frac{V_3}{2},$$ where $V_3$ is the volume of the regular ideal hyperbolic tetrahedron. Equality is achieved when $\mathcal{P}$ is the regular ideal hyperbolic cube. \end{proposition}
\begin{proof} Decompose $\mathcal{P}$ as described at the beginning of this section. Each vertex of $\mathcal{P}$ which is not contained in a face containing $v_0$ is a vertex of exactly six tetrahedra of the decomposition. Lemma~\ref{maxlem} implies that the sum of the volumes of the six tetrahedra around such a vertex is no more than $3V_3/2$.
If $v_1$ is one of the three vertices adjacent to $v_0$, then $v$ is a vertex of two tetrahedra, $T_1$ and $T_2$, say. The sum of the volumes of $T_1$ and $T_2$ is at most $V_3/3$ when $\alpha_1=\alpha_2=\pi/6$, again by Lemma~\ref{maxlem}.
By Corollary~\ref{cor4edges} each face containing $v_0$ has degree at least $4$. If $v_2$ is a vertex of such a face which does not share an edge with $v_0$, then $v_2$ is a vertex of four tetrahedra of the decomposition of $\mathcal{P}$. See figure~\ref{figure:edgevert}.
\begin{figure}
\caption{A view of $v_2$ as seen from $v_0$. The solid lines are edges of $\mathcal{P}$, and the dashed lines are edges of tetrahedra which are not also edges of $\mathcal{P}$.}
\label{figure:edgevert}
\end{figure}
The link of $v_2$ intersected with each of $T_i$, $i = 3,4$ is a Euclidean triangle with angles $\pi/2,$ $\pi/3$ and $\alpha_i$. Hence $\alpha_3 = \alpha_4 = \pi/6$. Using Lemma~\ref{maxlem} and the fact that $\alpha_5+\alpha_6=2\pi/3$, the sum of the volumes of these four tetrahedra is seen to have a maximum value of $5V_3/6$.
The upper bound is computed by assuming that the volume contributed by the tetrahedra containing each vertex other than $v_0$ is $3V_3/2$ and subtracting the excess for each of the three vertices which share an edge with $v_0$ and for the three vertices described in the previous paragraph: $$\mbox{\rm{Vol}}(\mathcal{P}) \leq \left( (N-1)\frac{3}{2} - 3\left(\frac{7}{6} \right) - 3\left(\frac{2}{3}\right)\right) \cdot V_3=\left(\frac{3N-14}{3}\right)\cdot \frac{3V_3}{2}.$$
The regular ideal hyperbolic cube has $N=8$ and volume $5V_3$. \end{proof}
The proofs of Theorems~\ref{pi2finite} and \ref{pi2general} require different methods than the previous two theorems because Lemma~\ref{maxlem} does not apply. The volume of the tetrahedra into which $\mathcal{P}$ is decomposed is given by the sum of three Lobachevsky functions, so the simple Lagrange multiplier analysis fails. The next lemma will play the role of Lemma~\ref{maxlem} in what follows.
\begin{lemma}\label{cube} The regular ideal hyperbolic cube has largest volume among all ideal polyhedra with the same combinatorial type. \end{lemma}
\begin{proof} Any ideal polyhedron, $\mathcal{Q}$, with the combinatorial type of the cube can be decomposed into five ideal tetrahedra as follows: Let $v_1, v_2, v_3, v_4$ be a collection of vertices of $\mathcal{Q}$ so that no two share an edge. The five ideal tetrahedra consist of the tetrahedron with vertices $v_1, v_2, v_3,$ and $v_4$, and the four tetrahedra with vertices consisting of $v_i$ along with the three adjacent vertices, for $i=1,2,3,4$.
Then since the regular ideal tetrahedron is the ideal tetrahedron of maximal volume, $\mbox{\rm{Vol}}(\mathcal{Q})\leq 5V_3$. The regular ideal hyperbolic cube is decomposed into five copies of the regular ideal tetrahedron when the above decomposition is applied, so has volume $5V_3$. \end{proof}
The next proposition proves the upper bound in Theorem~\ref{pi2finite}. \begin{proposition}\label{upper2finite} If $\mathcal{P}$ is a $\pi/2$--equiangular compact hyperbolic polyhedron with $N$ vertices, then $$\mbox{\rm{Vol}}(\mathcal{P}) < (N-10)\cdot \frac{5V_3}{8}.$$ \end{proposition}
\begin{proof} Decompose $\mathcal{P}$ into tetrahedra as described at the beginning of this section for some choice of $v_0$. The volume of $\mathcal{P}$ will be bounded above by considering tetrahedra with one ideal vertex. The reason for using tetrahedra with an ideal vertex to estimate the volume of a compact polyhedron is that $$\max_{v\in \text{Vert}(\mathcal{P})} d(v_0,v)$$ can be made arbitrarily large by choosing polyhedra $\mathcal{P}$ with a large enough number of vertices.
Suppose that $v=v_{i,j}$ is a vertex of $\mathcal{P}$ which is not contained in a face containing $v_0$. The vertex $v$ is contained in six tetrahedra of the decomposition. Consider $S$, the union of the six triangular faces of these tetrahedra which are contained in the faces of the polyhedron which contain
$v$. Let $\hat{v}$ be the point at infinity determined by the geodesic ray emanating from $v$ and passing through $v_0$. Define $\mathcal{T}$ to be the cone of $S$ to $\hat{v}$.
The cone, $\mathcal{T},$ is an octant of an ideal cube, $\mathcal{Q}$. By Lemma~\ref{cube}, $\mbox{\rm{Vol}}(\mathcal{Q}) \leq 5V_3.$ Then since $\mbox{\rm{Vol}}(\mathcal{Q}) = 8 \mbox{\rm{Vol}}(\mathcal{T}),$ $\mbox{\rm{Vol}}(\mathcal{T}) \leq \frac{5V_3}{8}.$
By Andreev's theorem, each face of $\mathcal{P}$ must be of degree at least $5$. Hence,
the three faces of $\mathcal{P}$ containing $v_0$ contain at least $10$ distinct vertices of $\mathcal{P}$, so there are at most $N-10$ vertices that do not share a face with $v_0$. Therefore the volume of $\mathcal{P}$ satisfies $$\mbox{\rm{Vol}}(\mathcal{P}) < (N-10)\cdot \frac{5V_3}{8}.$$ \end{proof}
Proposition~\ref{upper2general} combines the techniques of Proposition~\ref{upper2} and Proposition~\ref{upper2finite} and gives the upper bound in Theorem~\ref{pi2general}.
\begin{proposition}\label{upper2general} If $\mathcal{P}$ is a $\pi/2$--equiangular hyperbolic polyhedron with $N_{\infty}\geq 1$ ideal vertices and $N_F$ finite vertices, then $$\mbox{\rm{Vol}}(\mathcal{P}) < (\mbox{\rm{N}}_{\infty}-1)\cdot \frac{V_8}{2} +
N_F\cdot \frac{5 V_3}{8} .$$ \end{proposition}
\begin{proof} Assign to one of the ideal vertices the role of $v_0$ in the decomposition described at the beginning of this section. Then each ideal vertex of $\mathcal{P}$ which is not contained in a face of $\mathcal{P}$ containing $v_0$ will be a vertex of eight tetrahedra in the decomposition. These tetrahedra contribute no more than $V_8/2$ to the volume of $\mathcal{P}$, by Proposition~\ref{upper2}. Each finite vertex which is not contained in a face containing $v_0$ is a vertex of six tetrahedra. The volume contributed by these is no more than $5V_3/8$ by Lemma~\ref{cube} and the proof of Proposition~\ref{upper2finite}. Putting this all together completes the proof. \end{proof}
\section{Sequences of polyhedra realizing the upper bound estimates} \label{S:seq}
In this section, it is proved that the upper bounds in Theorems~\ref{theorempi2}, in \ref{pi2finite}, and in \ref{theorempi3} are asymptotically sharp. Results will first be established about the convergence of sequences of circle patterns in the plane and about the convergence of volumes of polyhedra which correspond to these circle patterns.
Define a {\it disk pattern} to be a collection of closed round disks in the plane such that no disk is the Hausdorff limit of a sequence of distinct disks and so that the boundary of any disk is not contained in the union of two other disks. Define the angle between two disks to be the angle between a clockwise tangent vector to the boundary of one disk at an intersection point of their boundaries and a counterclockwise tangent vector to the boundary of the other disk at the same point. Suppose that $D$ is a disk pattern such that for any two intersecting disks, the angle between them is in the interval $[0,\pi/2]$. Define $G(D)$ to be the graph with a vertex for each disk and an edge between any two vertices whose corresponding disks have non-empty interior intersection. The graph $G(D)$ inherits an embedding in the plane from the disk pattern. Identify $G(D)$ with its embedding. A face of $G(D)$ is a component of the complement of $G(D)$. Label the edges of $G(D)$ with the angles between the intersecting disks. The graph $G(D)$ along with its edge labels will be referred to as the {\it labeled $1$--skeleton for the disk pattern $D$}. A disk pattern $D$ is said to be {\it rigid} if $G(D)$ has only triangular and quadrilateral faces and each quadrilateral face has the property that the four corresponding disks of the disk pattern intersect in exactly one point. See \cite{he} for more details on disk patterns.
Consider the path metric on $G(D)$ obtained by giving each edge of $G(D)$ length $1$. Given a disk $d$ in a disk pattern $D$, the set of disks corresponding to the ball of radius $n$ in $G(D)$ centered at the vertex corresponding to $d$ will be referred to as {\it $n$ generations of the pattern about $d$}. Given disk patterns $D$ and $D'$ and disks $d\in D$ and $d' \in D'$, then $(D,d)$ and $(D',d')$ {\it agree to generation $n$} if there is a label preserving graph isomorphism between the balls of radius $n$ centered at the vertices corresponding to $d$ and $d'$. The following proposition is a slight generalization of the Hexagonal Packing Lemma in \cite{rodinsullivan}.
\begin{proposition}\label{hrs} Let $c_{\infty}$ be a disk in an infinite rigid disk pattern $D_{\infty}$. For each positive integer $n$, let $D_n$ be a rigid finite disk pattern containing a disk $c_n$ so that $(D_{\infty},c_{\infty})$ and $(D_n,c_n)$ agree to generation $n$. Then there exists a sequence $s_n$ decreasing to $0$ such that the ratios of the radii of any two disks adjacent to $c_n$ differ from $1$ by less than $s_n$. \end{proposition}
\begin{proof} With lemma 7.1 from \cite{he} playing the role of the ring lemma in \cite{rodinsullivan}, the proof runs exactly the same. The length--area lemma generalizes to this case with no change and any reference to the uniqueness of the hexagonal packing in the plane should be replaced with Rigidity Theorem 1.1 from \cite{he}. \end{proof}
A {\it simply connected disk pattern} is a disk pattern so that the union of the disks is simply connected. Disk patterns arising from finite volume hyperbolic polyhedra will all be simply connected, so all disk patterns will be implicitly assumed to be simply connected. If for a simply connected disk pattern $D$, all labels on $G(D)$ are in the interval $(0,\pi/2]$, Andreev's theorem implies that each face of $G(D)$ will be a triangle or quadrilateral. An {\it ideal disk pattern}, $D$, is one where the labels of $G(D)$ are in the interval $(0,\pi/2]$ and the labels around each triangle or quadrilateral in $G(D)$ sum to $\pi$ or $2\pi$ respectively. Ideal disk patterns correspond to ideal polyhedra. A {non-ideal disk pattern}, $D$, is one where $G(D)$ has only triangular faces and the sum of the labels around each face is greater than $\pi$. These disk patterns correspond to compact polyhedra.
Ideal disk patterns and their associated polyhedra will be dealt with first. The analysis for non-ideal disk patterns is slightly different and will be deferred until after the proofs of the remaining claims in Theorems~\ref{theorempi2} and \ref{theorempi3}. The upper half-space model for $\mathbb{H}^3$ will be used here. For each disk $d$ in the circle pattern, let $S(d)$ be the geodesic hyperbolic plane in $\mathbb{H}^3$ bounded by the boundary of $d$.
Suppose $c$ is a disk in $D$ which intersects $l$ neighboring disks, $d_{1}, \dots d_{l}$. In the case of an ideal disk pattern, the intersection of $S(c)$ with each of the $S(d_{i})$ is a hyperbolic geodesic. These $l$ geodesics bound an ideal polygon, $p(c) \subset \mathbb{H}^3$. If necessary, choose coordinates so that the point at infinity is not contained in $c$. Cone $p(c)$ to the point at infinity and denote the ideal polyhedron thus obtained by $C(p(c))$.
\begin{lemma}\label{vc} Suppose that $D_n$ and $D_{\infty}$ are simply connected, ideal, rigid, disk patterns such that $(D_n,c_n)$ and $(D_{\infty},c_{\infty})$ satisfy Proposition~\ref{hrs}. Then
$$\lim_{n\to \infty} \mbox{\rm{Vol}}(C(p(c_n)))=\mbox{\rm{Vol}}(C(p(c_{\infty})))$$
where $C(p(c_{\infty}))$ is the cone on the polygon determined by the disk $c_{\infty}$. Moreover, there exists a bounded sequence $0\leq \epsilon_n \leq K <\infty$ converging to zero such that $|
\mbox{\rm{Vol}}(C(p(c_n)))-\mbox{\rm{Vol}}(C(p(c_{\infty}))) | \leq \epsilon_n$. \end{lemma} \begin{proof} Suppose the dihedral angle between $S(c_n)$ and the vertical face which is a cone on the intersection of $S(c_n)$ and $S(d_{i,n})$ is $\alpha_{i}^{n}$ and that the corresponding dihedral angles in $C(p(c_{\infty}))$ are $\alpha_i^{\infty}$. Then by chapter 7 of \cite{thurstonnotes}, $$\mbox{\rm{Vol}}(C(p(c_n)))=\sum_{i=1}^{l} \Lambda(\alpha_i^{n}),$$ where $p(c_n)$ has degree $l$. For each $i$, $\alpha_i^{n}$ converges to $\alpha_i ^{\infty}$ because $\alpha _i^{n}$ is a continuous function of the angle between $c_n$ and $d_{i,n}$ and the radii of the two disks, which converge to the radii of the corresponding disks in the infinite packing by Proposition~\ref{hrs}. The function $\Lambda$ is continuous, so convergence of the $\alpha$'s implies the first statement of the lemma. The second statement is a consequence of the first and the fact that $\mbox{\rm{Vol}}(C(p(c_n)))$ is finite for all $n$ including $\infty.$ \end{proof}
The remaining claims in Theorems \ref{theorempi2} and \ref{theorempi3} can now be proved. First the following proposition is proved:
\begin{proposition}\label{asy2} There exists a sequence of ideal $\pi/2$--equiangular polyhedra $\mathcal{P}_i$ with $N_i$ vertices such that $$\lim_{i\to \infty} \frac {\mbox{\rm{Vol}}(\mathcal{P}_i)}{N_i} = \frac{V_8}{2}.$$ \end{proposition}
\begin{proof} Let $D_{\infty}$ be the infinite disk pattern defined as $$D_{\infty}=\bigcup d_{(p,q)},$$ where the union ranges over all $(p,q)\in \mathbb{Z}^2$ such that both $p$ and $q$ are even or both $p$ and $q$ are odd, and $d_{(p,q)}$ is the disk of radius $1$ centered at the point $(p,q)$. Consider the ideal hyperbolic polyhedron with infinitely many vertices, $\mathcal{P}_{\infty}$, corresponding to $D_{\infty}$. This polyhedron has all dihedral angles equal to $\pi/2$. Applying the decomposition into tetrahedra described in the proof of Proposition~\ref{upper2}, it is seen that the sum of the volumes of the tetrahedra meeting each vertex is exactly $\frac{V_8}{2}$. A sequence of polyhedra, $\mathcal{P}_{2k},$ which have volume-to-vertex ratio converging to that of $\mathcal{P}_{\infty}$ will be constructed.
For each even natural number $2k$, $k\geq3$, consider the set of lines in the plane $L_{2k}=\{(x,y) \in \mathbb{R}^2 \mid \, y=0,\, y=2k \text{, or }y=\pm x + z,\, z \in \mathbb{Z}\}$. Now let $\mathcal{P}_{2k}$ be the hyperbolic polyhedron with $1$--skeleton given by $$\mathcal{P}_{2k}^{(1)}=\{(x,y) \in L_{2k} \mid \, 0\leq y \leq 2k\}/\{(x,y)\sim(x+2k, y)\}$$ and all right angles. See figure~\ref{figure:p6} for an illustration of $\mathcal{P}_6$. The existence of such a hyperbolic polyhedron is guaranteed by Andreev's theorem. Equivalently, there is a simply connected rigid disk pattern, $D_{2k}$, in the plane with each disk corresponding to a face and right angles between disks which correspond to intersecting faces. The vertices and faces of $\mathcal{P}_{2k}$ will be referred to in terms of the $(x,y)$ coordinates of the corresponding vertices and faces of $L_{2k}$.
The polyhedra $\mathcal{P}_{2k}$ will prove the proposition. The volume of $\mathcal{P}_{2k}$ is expressed as the sum of volumes of cones on faces and Lemma~\ref{vc} is used to analyze the limiting volume-to-vertex ratio.
Choose coordinates for the upper half-space model of $\mathbb{H}^3$ so that the vertex $(0,0)$ of $\mathcal{P}_{2k}$ is located at infinity. Then the volume of $\mathcal{P}_{2k}$ may be written $$\mbox{\rm{Vol}}(\mathcal{P}_{2k})=\sum_{d} \mbox{\rm{Vol}}(C(p(d))),$$ where the sum is taken over all faces $d$ which do not meet the vertex $(0,0)$. Using Lemma~\ref{vc}, the volume of each $C(p(d))$ can be estimated in terms of the number of generations of disks surrounding $d$ which agree with $D_{\infty}$.
Fix a disk $d_{\infty} \in D_{\infty}.$ For each $m\in \mathbb{Z}$, $m\geq 0$, define $F_m$ to be the set of disks $d\in D_{2k}$ for which $(D_{2k},d)$ and $(D_{\infty}, d_{\infty})$ agree to generation $m$, but do not agree to generation $m+1$. The $2k$ faces of $\mathcal{P}_{2k}$ centered at the points $(i+1/2, k)$, $0\leq i \leq 2k-1$ as well as the $4k$ faces which share an edge with them except for the face centered at $(0,k-1/2)$ make up $F_{k-1}$. Thus
$$|F_{k-1}|=6k-1.$$ The set $F_{k-2}$ consists of the face centered at $(0,k-1/2)$ along with the faces centered at the $8k-1$ points with coordinates $(i + 1/2 , k \pm 1)$ and $(i, k \pm 3/2)$ for $0 \leq i \leq 2k-1$, excluding the face centered at $(0,k-3/2)$. In general, for $2 \leq l \leq k-1$, $F_{k-l}$ consists of the face centered at $(0, k-(2(l-1)-1)/2)$ along with the faces centered at the $8k-1$ points with coordinates $(i+1/2, k\pm (l-1))$ and $(i, k \pm (2l-1)/2)$ for $0\leq i \leq 2k-1$, excluding the face centered at $(0, k-(2l-1)/2)$. Hence for $2\leq l\leq k-1,$
$$|F_{k-l}| =8k.$$ See figure~\ref{figure:p6} for an example.
\begin{figure}
\caption{Identify the two vertical sides to obtain $\mathcal{P}_6$. A face labeled by an integer $n$ has $n$ generations of disks about it. Unlabeled faces have $0$ generations about them.}
\label{figure:p6}
\end{figure}
The polyhedron $\mathcal{P}_{2k}$ has $8k^2+2k+2$ faces. In the previous paragraph, it was found that
$$\left|\bigcup_{l=1}^{k-1} F_l\right|=8k^2-10k-1.$$ The remaining $12k+3$ faces consist of the following: $4$ vertical faces which do not contribute to the volume, one $2k$--gon, $4k-2$ triangular faces, and $8k$ rectangular faces in $F_0$. The maximum value of the Lobachevsky function, $\Lambda(\theta)$, is attained for $\theta=\pi/6$ \cite{thurstonnotes}. Hence, the formula for the volume of a cone on an ideal polygon given in the proof of Lemma~\ref{vc} implies that the volume of the cone on the $2k$--gon is less than or equal to $2k \Lambda(\pi/6)$. Similarly, each of the remaining triangular and rectangular faces have volume less than or equal to $4 \Lambda(\pi/6)$. This implies that the leftover faces have cone volume summing to a value $L\leq 14k\Lambda(\pi/6)$.
The volume of the cone to infinity of any face in $D_{\infty}$ is $4 \Lambda(\pi/4)=V_8 /2$. By Lemma~\ref{vc}, there exists a real-valued, positive function, $\delta_m$, on $F_m$ such that for each face $f \in F_m$, $0 \leq \delta_m(f) \leq \epsilon_m$ and $\mbox{\rm{Vol}}(C(f))=V_8/2 \pm \delta_m(f)$. Therefore, the volume of $\mathcal{P}_{2k}$ can written as:
$$\mbox{\rm{Vol}}(\mathcal{P}_{2k})=\sum_{l=1}^{k-1} \sum_{f\in F_{l}} \left(\frac{V_8}{2} \pm \delta_{l}(f)\right)+L. $$ Using the analysis of the $F_m$ from above, expand the sums and collect terms to get
$$\mbox{\rm{Vol}}(\mathcal{P}_{2k})= (8k^2-10k-1)\frac{V_8}{2} + \sum_{l=1}^{k-1} \sum_{f\in F_{l}}(\pm \delta_{l} (f)) +L .$$
The polyhedron $\mathcal{P}_{2k}$ has $N_{2k}=8k^2+2k$ vertices. Therefore $$\lim_{k\to\infty} \frac{8k^2-10k-1}{N_{2k}}\frac{V_8}{2}=\frac{V_8}{2}.$$
It remains to show that the ratio of the last two summands to the number of vertices converges to zero. Set $\overline{\delta}_{l}=\max_{f\in F_l} \delta_l(f)$. Then
$$\lim_{k \to \infty}\frac{\left|\sum_{l=1}^{k-1} \sum_{f\in F_l} (\pm
\delta_l(f)) \right|}{N_{2k}}\leq \lim_{k \to \infty} \left( \frac{(6k-1)\overline{\delta}_{k-1}}{N_{2k}}+
\frac{8k\sum_{l=1}^{k-2} \overline{\delta_l}}{N_{2k}}\right)=0$$ because $\overline{\delta}_l \to 0$ as $l \to \infty$. Also since $L<14k \Lambda(\pi/6)$, $$\lim_{k\to\infty}\frac{L}{N_{2k}}=0.$$
Therefore, $$\lim_{k \to \infty} \frac{\mbox{\rm{Vol}}(\mathcal{P}_{2k})}{N_{2k}}= \frac{V_8}{2}.$$ \end{proof}
The argument to prove Proposition~\ref{asy2} easily adapts to prove the following:
\begin{proposition}\label{asy3} There exists a sequence of ideal $\pi/3$--equiangular polyhedra $\mathcal{P}_i$ with $N_i$ vertices such that $$\lim_{i\to \infty} \frac {\mbox{\rm{Vol}}(\mathcal{P}_i)}{N_i} = \frac{3 V_3}{2}.$$ \end{proposition} \begin{proof} Consider the regular hexagon $H$ in the plane formed by the vertices $(0,0)$, $(1,0)$, $(3/2, \sqrt{3}/2)$, $(1,\sqrt{3})$, $(0,\sqrt{3})$, and $(-1/2,\sqrt{3}/2)$. Let $G$ be the lattice of translations generated by $\{(x,y)\mapsto (x+3,y) \}$ and $\{(x,y) \mapsto (x+3/2,y+ \frac{\sqrt{3}}{2})$. Now define $T$ to be the orbit of the hexagon $H$ under the action of $G$ on the plane. This orbit is a tiling of the plane by regular hexagons. As in the previous construction, define $$L_{2k}=\{(x,y)\in \mathbb{R}^2 \mid \, y=0, \, y=2k\sqrt{3} \text{, or } (x,y) \text{ lies on a vertex or edge of }T\}.$$
Let $\mathcal{Q}_{2k}$ be the hyperbolic polyhedron with $1$--skeleton $$\mathcal{Q}_{2k}^{(1)}=\{(x,y)\in L_{2k} \mid \, 0\leq y \leq 2k\sqrt{3}\}/\{(x,y) \sim (x+3k,y)\}$$
and all dihedral angles $\pi/3$. This is a polyhedron with $4k^2+k+2$ faces and $N_{2k}= 8k^2 +2k$ vertices. Let $D_{2k}$ be the associated simply connected rigid disk pattern and $D_{\infty}$ to be the infinite circle pattern with $G(D_{\infty})$ equal to a tiling of the plane by equilateral triangles with each edge labeled $\pi/3$.
\begin{figure}
\caption{Identify the left and right sides to get $\mathcal{Q}_6$. The labeling is as in the previous figure.}
\label{figure:hex}
\end{figure}
As in the proof of Proposition~\ref{asy2}, choose coordinates so that the vertex at $(0,0)$ is at infinity. Recall that for a fixed choice of $d_{\infty} \in D_{\infty}$, $F_m$ is defined to be the set of disks $d \in D_{2k}$ for which $(D_{2k},d)$ and $(D_{\infty},d_{\infty})$ agree to generation $m$, but do not agree to generation $m+1$. The set $F_{k-1}$ consists of $3k-1$ faces, while the remaining $F_{k-l}$ for $2 \leq l \leq k-1$ consist of $4k$ faces. Again, the faces not contained in $F_{k-l}$ for $1 \leq l \leq k-1$ have cone volume summing to a value $L$ bounded above by a constant multiple of $k$ where the bound is independent of $k$. The volume of the cone to infinity of any face in the regular hexagonal circle pattern is $6 \Lambda(\pi/6)=3V_3$. There exists a function, $\delta_{l}$, with the same properties as above. The volume of $\mathcal{Q}_{2k}$ is $$\mbox{\rm{Vol}}(\mathcal{Q}_{2k})= \sum_{l=1}^{k-1} \sum_{f\in F_{l}} \left(3V_3 \pm \delta_{l}(f)\right)+ L.$$ The argument finishes exactly as the all right-angled case to give $$\lim_{k \to \infty} \frac{\mbox{\rm{Vol}}(\mathcal{Q}_{2k})}{N_{2k}}=\frac{3V_3}{2}.$$ \end{proof}
To finish the proof of Theorem~\ref{pi2finite}, a minor modification to Lemma~\ref{vc} is made.
Recall that for a disk pattern $D$, $G(D)$ is the graph with a vertex for each disk and an edge connecting two vertices which have corresponding disks with non-empty interior intersection. Suppose that $D$ is a non-ideal disk pattern and that $c$ is a disk which intersects $l$ neighboring disks, $d_{1}, \dots d_{l}.$ The intersection of $S(c)$ with each of the $S(d_{i})$ is a finite length geodesic segment. The union of the $l$ geodesic segments along with the disk bounded by them in $S(c)$ is a polygon $p(c)$. Let $x_0\in \mathbb{H}^3$ be a point which is not contained in $S(c)$. Denote by $C(p(c),x_0)$ the cone of $p(c)$ to the point $x_0$. The cone to the point at infinity in the upper half-space model of $\mathbb{H}^3$ will be denoted by $C(p(c),\{\infty\})$.
Let $c$ be a disk in a simply connected, non-ideal, finite disk pattern, $D$, with associated polyhedron $\mathcal{P}$. Suppose $c_{max}$ realizes the quantity $$\max_{c'} d_{G(D)} ( c,c').$$ Define a \textit{cut point for $c$} to be any vertex of $p(c_{max})$.
\begin{lemma}\label{vcc} Suppose that $D_n$ and $D_{\infty}$ are simply connected, non-ideal, rigid, disk patterns, such that $(D_n,c_n)$ and $(D_{\infty},c_{\infty})$ satisfy Proposition~\ref{hrs} and $D_{\infty}$ fills the entire plane. Suppose also that $x_n$ is a cut point for $c_n$ in $D_n$. Then
$$\lim_{n\to \infty} \mbox{\rm{Vol}}(C(p(c_n),x_n))=\mbox{\rm{Vol}}(C(p(c_{\infty}),\{\infty\})).$$ Moreover, there exists a bounded sequence $0\leq \epsilon_n \leq K <\infty$ converging to zero such that
$| \mbox{\rm{Vol}}(C(p(c_n),x_n))-\mbox{\rm{Vol}}(C(p(c_{\infty}),\{\infty\})) | \leq \epsilon_n$ \end{lemma}
\begin{proof} Note that for any choice of points $y_n \in S(c_n)$, $d(x_n , y_n) \to \infty$ as $n \to \infty$. Also, since $D_{\infty}$ fills the entire plane, the distance measured in $G(D_n)$ from $c_n$ to a disk $c'_n$ such that $S(c'_n)$ contains $x_n$ goes to infinity. Hence as $n$ goes to infinity, $x_n$ approaches the point at infinity, so the compact cone $C(p(c_n), x_n)$ approaches the infinite cone $C(p(c_{\infty}), \{\infty\})$. Therefore it suffices to show that the volume of the compact cones approaches that of the infinite cone.
A \textit{$3$--dimensional hyperbolic orthoscheme} is a hyperbolic tetrahedron with a sequence of three edges $v_0v_1$, $v_1v_2$, and $v_2v_3$ such that $v_0v_1 \bot v_1v_2 \bot v_2v_3$. See figure~\ref{figure:comortho}. Suppose that the degree of $p(c_n)$ is $a_n$. The cone, $C(p(c_n),x_n)$, can be decomposed into $2a_n$ orthoschemes by the procedure described at the beginning of section~\ref{S:upper}. \begin{figure}
\caption{A compact orthoscheme, $T(\alpha_n, \beta_n, \gamma_n)$. The unlabeled edges have dihedral angle $\pi/2$}
\label{figure:comortho}
\end{figure}
The volume of one of the compact orthoschemes, $T(\alpha_n, \beta_n, \gamma_n)$ determined by angles $\alpha_n,$ $\beta_n$, and $\gamma_n,$ as shown in figure~\ref{figure:comortho}, is given by \begin{align*} \mbox{\rm{Vol}}(T(\alpha_n, \beta_n, \gamma_n) = \frac{1}{4} &\bigg( \Lambda(\alpha_n +\delta_n) - \Lambda(\alpha_n-\delta_n)+\Lambda(\gamma_n+\delta_n) - \Lambda(\gamma_n-\delta_n) \\ &- \Lambda \left(\frac{\pi}{2} -\beta_n + \delta_n \right) +\Lambda \left(\frac{\pi}{2} -\beta_n - \delta_n \right) + 2 \Lambda \left(\frac{\pi}{2}-\delta_n \right) \bigg) , \end{align*} where $$0 \leq \delta_n = \arctan{\frac{\sqrt{-\Delta_n}}{\cos{\alpha_n} \cos{\gamma_n}}} < \frac{\pi}{2},$$ and $$\Delta_n= \sin^2{\alpha_n} \sin^2{\gamma_n} - \cos^2{\beta_n}.$$ This is due to Lobachevsky. See, for example, page 125 of \cite{geomii}.
Similarly, the cone $C(p(c_{\infty}), \{\infty\})$ can be decomposed into orthoschemes of the form $T(\alpha_{\infty}, \pi/2-\alpha_{\infty}, \gamma_{\infty})$ with one ideal vertex . The volume of this orthoscheme is given by $$\mbox{\rm{Vol}}(T(\alpha_{\infty}, \pi/2 - \alpha_{\infty}, \gamma_{\infty}) = \frac{1}{4} \left( \Lambda(\alpha_{\infty} + \gamma_{\infty}) + \Lambda( \alpha_{\infty} - \gamma_{\infty}) + 2 \Lambda( \pi/2 - \alpha_{\infty}) \right).$$
As $n \to \infty$, $\Delta_n \to -\sin^2{\alpha_n} \cos^2{\gamma_n}$, so $\delta_n \to \alpha_n$. Therefore the sequence of volumes of the compact orthoschemes converges to that of the orthoscheme with one ideal vertex. Summing over all orthoschemes in the decomposition proves the lemma. \end{proof}
The next proposition will complete the proof of Theorem~\ref{pi2finite}.
\begin{proposition}\label{asycom2} There exists a sequence of compact $\pi/2$--equiangular polyhedra $\mathcal{P}_i$ with $N_i$ vertices such that $$\lim_{i\to \infty} \frac {\mbox{\rm{Vol}}(\mathcal{P}_i)}{N_i} = \frac{5 V_3}{8}.$$ \end{proposition}
\begin{proof} Define $L'_{2k}$ to be $L_{2k}$ as in the proof of Proposition~\ref{asy3} along with the tripods as shown in figure~\ref{figure:hexmod}. The tripods must be added to remove the degree $4$ faces. \begin{figure}
\caption{Identify the left and right sides to get $\mathcal{R}_6$. The labeling is as in the previous figures.}
\label{figure:hexmod}
\end{figure}
Let $\mathcal{R}_{2k}$ be the polyhedron with $1$--skeleton $$\mathcal{R}_{2k}^{(1)}=\{(x,y)\in L'_{2k} \mid \, 0\leq y \leq 2k\sqrt{3}\}/\{(x,y) \sim (x+3k,y)\}$$ and all dihedral angles equal to $\pi/2$. For each $k>2$, this can be realized as a compact hyperbolic polyhedron by Andreev's theorem. The rest of the proof of this proposition mirrors the proof of Proposition~\ref{asy3} exactly, using Lemma~\ref{vcc} in place of Lemma~\ref{vc}. \end{proof}
\end{document}
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\begin{abstract}
A ``$k$-rule" is a sequence $\vec A=((A_n,B_n):n<\omega)$ of pairwise disjoint sets $B_n$, each of cardinality $\le k$ and subsets $A_n\subseteq B_n$. A subset $X\subseteq \omega$ (a ``real'') follows a rule $\vec A$ if for infinitely many $n\in \omega$, $X\cap B_n=A_n$.
There are obvious cardinal invariants resulting from this definition: the least number of reals needed to follow all $k$-rules, $\mathfrak s_k$, and the least number of $k$-rules without a real following all of them, $\mathfrak r_k$.
Call $\vec A$ a {\em bounded} rule if $\vec A$ is a $k$-rule for some $k$. Let $\mathfrak r_\infty$ be the least cardinality of a set of bounded rules with no real following all rules in the set.
We prove the following: $\mathfrak r_\infty\ge\max(\operatorname{cov}(\mathbb K),\operatorname{cov}(\mathbb L))$ and $\mathfrak r=\mathfrak r_1\ge \mathfrak r_2=\mathfrak r_k$ for all $k\ge 2$. However, in the Laver model, $\mathfrak r_2<\mathfrak b=\mathfrak r_1$.
An application of $\mathfrak r_\infty$ is in Section 3: we show that below $\mathfrak r_\infty$ one can find proper extensions of dense independent families which preserve a pre-assigned group of automorphisms. The original motivation for discovering rules was an attempt to construct a maximal homogeneous family over $\omega$. The consistency of such a family is still open.
\end{abstract}
\title{Rules and Reals}
\section*{Introduction}
In the present paper we present new cardinal invariants which resulted from investigations of homogeneous families. These numbers have intrinsic interest (in fact we regard it as surprising that those numbers have not been discovered earlier).
In Section 1 we discuss cardinal invariants related to ``$k$-rules.'' A {\em $k$-rule}
is a sequence $\vec A=((A_n,B_n):n<\omega)$ of pairwise disjoint sets $B_n$, each of cardinality $\le k$, and subsets $A_n\subseteq B_n$. A subset $X\subseteq \omega$ (a ``real'') {\em follows} a rule $\vec A$ if for infinitely many $n\in \omega$, $X\cap B_n=A_n$.
A rule $\vec A$ is {\em bounded} if it is a $k$-rule for some $k\in \omega$.
The obvious cardinal invariants related to rules are the following: the least number of reals needed to follow all $k$-rules, $\mathfrak s_k$, and the least number of $k$-rules with no real following all of them, $\mathfrak r_k$. Let $\mathfrak r_\infty$ be the least number of bounded rules with no real following all of them.
We compare the $\mathfrak r_k$s and $\mathfrak r_\infty$ among themselves and to well known cardinal invariants: covering of category, covering of Lebesgue measure, $\mathfrak r$, $\mathfrak b$, $\mathfrak d$ and the evasion numbers $\mathfrak e_k$ which were studied by Blass and Brendle. We prove: \begin{itemize} \itm a
$\max(cov(\mathbb K), cov(\mathbb L)) \le \mathfrak r_\infty$; \itm b $\mathfrak r=\mathfrak r_1\ge \mathfrak r_2=\mathfrak r_k$ for all $k\ge 2$; \itm c
$\mathfrak s_2\le\mathfrak e_2$; \itm d
$\mathfrak r_\infty\le \min(\mathfrak r_2,\mathfrak d)$. \end{itemize}
In Section 2 we prove the consistency of $\mathfrak r_2<\mathfrak b$.
In Section 3 we show that below $\mathfrak r_\infty$ one can properly extend an independent family of subsets of $\omega$ preserving a prescribed group of automorphisms. This is the relevance of $\mathfrak r_\infty$ to the behavior of homogeneous families under inclusion, which was the original motivation for the discovery of rules.
\section{Rules}
\begin{definition} \begin{itemize} \itm 1 A {\em rule} is a sequence $\vec A =
( A_n, B_n: n \in \omega)$, where the sets
$B_n$ are disjoint and finite, and for all $n$, $A_n \subseteq
B_n\subseteq\omega$. \itm 2 We say that $X\in [\omega]^\omega $ {\em follows} the rule $\vec A$ if there are
infinitely many $n$ with $X \cap B_n = A_n$; otherwise $X$ is
said to {\em avoid} $\vec A$. \itm 3 For $k \in \omega$ we say that $\vec A$ is a $k$-rule if all sets
$B_n$ have size $\le k$. We say that $\vec A$ is a
{\em bounded} rule if $\vec A$ is a $k$-rule for some
$k$. \item 4 More generally, for any function
$f:\omega \to \omega$ we say that $\vec A$ is an $f$-rule if
for all $n$, $|B_n| \le f(n)$.
We say that $f$ is a ``slow'' function if
$$ \sum_{n=0}^\infty 2^{-f(n)} = \infty,$$ and we say that $\vec A$ is a slow rule if it is an $f$-rule for some slow $f$. \end{itemize} \end{definition}
\begin{definition} \begin{enumerate} \itm 1 For $k \in \omega $ let $\mathfrak r_k:=
\min \{ |\mathfrak R|: $ there is no $X$ which follows all $k$-rules from $\mathfrak R$ $\}$.
(Similarly $\mathfrak r_f$, when $f:\omega \to \omega$.) \itm 2 Dually, let $\mathfrak s_k:=
\min \{ |\mathfrak S|: $ every $k$-rule is followed by some $X\in \mathfrak S$ $\}$. \itm 3 We let $\mathfrak r_\infty = \min
\{ |\mathfrak R|: $ there is no $X$ which follows all bounded rules from $\mathfrak R$\nobreak$\}$. \end{enumerate} \end{definition}
We remark that $2^k$ trivially bounds the least cardinality of a set of $k$-rules with the property that every real follows some rule in the set.
Recall that the ``splitting'' number $\mathfrak s$ and the ``reaping'' number $\mathfrak r$ are defined as follows:
\begin{definition} If $s, X \in [\omega]^\omega $, then we say that $s$ ``splits'' $X$ if $s$ divides $X$ into two infinite parts, i.e., $s\cap X$ and $(\omega-s)\cap X$ are both infinite.
\begin{itemize}
\itm 1 $\mathfrak s:= \min \{|\mathfrak S|: \mathfrak S \subseteq [\omega]^\omega $,
every $X\in [\omega]^\omega $ is split by some
$s\in \mathfrak S$ $\}$
\itm 2 $\mathfrak r:= \min \{|\mathfrak R|$: $\mathfrak R \subseteq [\omega]^\omega $, there is no
$X\in [\omega]^\omega $ which
splits all $r\in \mathfrak R$ $\}$ \end{itemize} \end{definition}
\begin{Fact} $\cdots \le \mathfrak r_3 \le \mathfrak r_2 \le \mathfrak r_1 = \mathfrak r$,
and $\mathfrak s \le \mathfrak s_2 \le \mathfrak s_3 \le \cdots$. However, $\mathfrak s_1 = 2$, witnessed by $\mathfrak S =\{\emptyset, \omega \}$. \end{Fact}
\begin{Theorem} \begin{enumerate} \itm a Let $( N,{\in})$ be a model of ZFC* (a large enough fragment of ZFC). If a real $X$ follows all rules from $N$, then $X$ is Cohen over
$N$. (Conversely, a Cohen real over $N$ follows all rules from $N$.) \itm b
If $X$ is random over $N$, then $X$ follows all slow
rules from $N$ (so in particular, all bounded rules).
\itm c $\max(cov(\mathbb K), cov(\mathbb L)) \le \mathfrak r_\infty$. ($\operatorname{cov}(\mathbb K)$ is the smallest number of first category sets needed to cover the real line. $\operatorname{cov}(\mathbb L )$ is defined similarly using measure zero sets.) \end{enumerate} \end{Theorem}
\begin{proof}
(a): Assume that $X \subseteq \omega$ follows all rules from $N$. We claim that $\chi_X$, the characteristic function of $X$, is a Cohen real over $N$, that is, the set $\{ \chi_X\mathord{\restriction} n: n \in \omega\}$ is generic for the forcing notion $\pre2^{<\omega}$.
To verify this claim, consider any nowhere dense tree
$T \subseteq \pre 2^{<\omega}$ in $N$. We have to check that $\chi_X$ is not a branch of $T$.
Using the fact that $T$ is nowhere dense (and $T$ is in $N$) we can by induction (in $N$!)\ find sequences $( n_i: i < \omega)$ and $( \eta_i : i < \omega)$ such that for all $i< \omega$ we have: \begin{enumerate} \item $n_i < n_{i+1} $, $\eta_i \in \pre 2^{[n_i, n_{i+1})} $ \item For all $\nu \in \pre 2^{n_i}$, $\nu \cup \eta_i \notin T$. \end{enumerate}
Now let $B_i:= [n_i, n_{i+1})$, $A_i = \{ k: \eta_i(k)=1 \}$. Our assumption tells us
that $X$ follows the rule $( A_i, B_i: i \in \omega)$. So for some $i$ we have $X \cap B_i = A_i$, and hence $\chi_X \supseteq \eta_i$. Hence $\chi_X$ is not a branch of $T$.
This concludes the proof of (a).
The converse to (a) is obvious.
(b) is also easy: Let $X_n:= \{ X: X \cap B_n \not= A_n \}$. For $n\not=m$, the sets $X_n$ and $X_m$ are independent (in the probabilistic sense), and $\mu(X_n) = 1-2^{-f(n)}$, where $\mu$ is the Lebesgue measure on ${\Cal P} (\omega) \simeq \pre 2^\omega $. Hence $\mu(\bigcap_{n>m} X_n) = \prod_{n>m} (1-2^{-f(n)}) = 0$.
(c) follows from (a) and (b).
\end{proof}
\begin{Theorem}[Shelah]\label{rkr2} For $k\ge 2$, $\mathfrak r_k = \mathfrak r_2$ (and similarly, $\mathfrak s_2 = \mathfrak s_k$). \end{Theorem}
\begin{proof} We will show that $\mathfrak r_k=\mathfrak r_{k+1}$: Let $N_0$ be sufficiently closed (say, a model of ZFC*, but closed under some recursive functions is
sufficient) of size $<\mathfrak r_{k}$; we have to show that there is a real that follows all $k+1$-rules from $N_0$.
We define a sequence $(N_i, C_i: i \le k)$ such that $N_i \cup \{ C_i \} \subseteq N_{i+1}$, each $N_i$ is sufficiently
closed and of the same cardinality as $N_0$, and $C_i$ follows
all $k$-rules from $N_i$.
Now let $C$ be the ``average'' of the $C_i$:
$m \in C$ iff $m$ is in ``most'' of the $C_i$'s, or formally:
$$ C:= \{ m \in \omega: | \{i\le k: m \in C_i \} | > (k+1)/2 \}$$
Now we check that $C$ indeed follows all $k+1$-rules from $N_0$.
Let $(A_n, B_n: n \in \omega)$ be a $k+1$-rule in $N_0$. For $0\le i \le k$ we let $(A^i_n, B^i_n: n \in \omega)$ be the $k$-rule obtained by removing the each $i$th element of $B_n$. That is, letting $\{b_n^0, \ldots, b_n^k\}$ be the increasing enumeration of $B_n$ we let $B_n^i:= B_n \setminus \{b^i_n\}$, $A_n^i:= A_n \cap B_n^i$.
Let $E_0:= \omega$. For $0\le i \le k$ let $$E_{i+1}:= \{ n \in E_i:
B^{i}_n \cap C_i = A^{i}_n\}, $$ i.e., $E_{i+1}$ is the set of indices on which $C_i$ follows
the rule $(A_n^i, B_n^i: n \in E_i)$. Note that $E_i \in N_i$ and $C_{i} \in N_{i+1}$. By the choice of $C_i$ we know that each $E_{i+1}$ is infinite.
We conclude the proof by showing that for $n \in E_{k+1}$ we have $A_n = B_n \cap C$. Let $n \in E_{k+1}$ (so also $n \in E_i$ for all $i \le k$), and $m \in B_n$. Say $m = b_n^j$. Then for $i \not= j$ we have $m \in B_n^i$, so $m \in A_n^i \Leftrightarrow m \in C_i$.
Hence the cardinality of the set $ \{ i \le k: m \in C_i \}$ is either in $\{0,1\} $ (iff $m\notin A_n$) or in $\{ k, k+1\}$. In any case we
get $m \in C $ iff $m \in A_n$.
So $A_n = B_n \cap C$.
\end{proof}
\begin{theorem} $\mathfrak r_\infty \ge \min(\mathfrak r_2, \mathfrak d)$. In particular, if $\mathfrak r_2 \le \mathfrak d$ then $\mathfrak r_\infty = \mathfrak r_2$. \end{theorem} \begin{proof} Let $N$ be a model of ZFC* of cardinality $< \min(\mathfrak r_2, \mathfrak d)$. We will find a real $X$ following all bounded rules from $N$.
Define sequences $(N_i:i<\omega)$, $(X_i:i<\omega)$ satisfying the following conditions: \begin{enumerate} \item $N_0 = N$. \item $N_i$ is a model of ZFC*, $N_{i-1} \cup\{X_i\}
\subseteq N_i $.
\item $|N_i| = |N_0|$. \item $X_i$ follows all $i$-rules (and hence also all $j$-rules for $j\le i$) from $N_{i-1}$. \end{enumerate}
Let $N_\omega$ be a model of size $|N_0|$ containing
$(N_i:i<\omega)$ and $(X_i:i<\omega)$. Since $|N_\omega | < \mathfrak d$ we can find a strictly increasing function $f$ that is not dominated by any function from $N_ \omega $.
Define $X \subseteq \omega $ by requiring $X \cap ( f(i-1), f(i)] = X_i \cap (f(i-1), f(i)]$. We claim that $X$ follows all bounded rules from $N$.
To complete the proof, consider an arbitrary $k$-rule $(A_n,B_n:n \in \omega )$ from $N$. We may assume $\min \bigcup_n B_n > f(k)$. We define sequences $(E_i:k \le i < \omega ) $
satisfying the following conditions for all $i \ge k$. \begin{enumerate} \item $\forall n \in E_i$ $B_n \cap X_i = A_n$. \item $E_i \in N_i$. \item $E_{i+1} \subseteq E_i$. \end{enumerate} We can carry out this construction, because $(A_n, B_n: n \in E_i)$ is a rule in $N_i$, so we just choose $E_{i+1}$ to witness that $X_{i+1}$ follows this rule.
Now let $n_i:= \min E_i$. Clearly the function $i \mapsto n_i$ is in $N_\omega $. So we can find infinitely many $j$ such that $f(j) > \max B_{n_j} $.
We claim that for each such $j$,
$X \cap B_{n_j} = A_{n_j}$. For all $i \in [k,j]$
we have $n_j \in E_i$, so $X_i \cap B_j = A_j$. Note that $B_j \subseteq [f(k), f(j))$, so we also have $X \cap B_j=A_j$.
\end{proof}
\begin{Problem} Is $\mathfrak r_\infty < \mathfrak r_2$ consistent? \end{Problem}
We remark that in the random real model we have $\mathfrak r_2=\operatorname{cov}(\mathbb L) = \mathfrak c = \mathfrak r_\infty$, $\mathfrak d=\aleph_1$. So one cannot hope to prove $\mathfrak r_2\le \mathfrak d$.
We now consider the invariant that is dual to $\mathfrak r_k$, and we compare it with the well-known ``evasion'' number.
\begin{definition} $(\pi,D)$ is a $k$-predictor, if $D$ is an infinite subset of $ \omega $, $\pi = (\pi_n:n \in D)$, $\pi_n $ a function from $\pre k^n$ to $k$.
We say that $f\in \pre k^\omega$ evades $(\pi, D)$ if there are infinitely many $\ell\in D$ such that $f(\ell) \not= \pi_\ell(f\mathord{\restriction} \ell)$.
$$ \mathfrak e_k := \min \{ |N|: \forall \pi\, \exists f\in N:
\hbox {$f$ evades $\pi$} \}$$ \end{definition}
Brendle in \cite{evpr} investigated these and other cardinal invariants and showed that all $\mathfrak e_k$ are equal to each other.
The following construction connects rules with predictors.
\begin{definition} Let $R = (A_n, B_n: n \in \omega)$ be a 2-rule. Define a $2$-predictor $(\pi_R, D_R)$ as follows: \begin{enumerate} \item $D_R = \{\max B_n: n \in \omega \}$
\item If $\ell=\max B_n$, and $|A_n|=1$, then $\pi_\ell(f)=f(\min B_n)$ for all $f \in \pre 2^\ell$. Otherwise, $\pi_\ell(f)=1-f(\min B_n)$. \end{enumerate} \end{definition}
\begin{lemma} Let $X \subseteq \omega$. If $\chi_X$
evades $\pi_R$, then either $X$ or $\omega \setminus X$ follows $R$. \end{lemma} \begin{proof} Let $\ell_n:= \max B_n$, $i_n = \min(B_n)$ for all $n$.
$X$ evades $\pi_R$, so there are infinitely many $n$ such that $X(\ell_n) \not= \pi_{\ell_n}(X\mathord{\restriction} \ell_n)$.
\subsubsection*{Case 1:} There are infinitely
many such $n$ where in addition $|A_n| = 1$.
So for each such $n$, $X(\ell_n) \not= \pi_{\ell_n}(X\mathord{\restriction} \ell_n) = X(i_n)$. So $X(\ell_n) \not= X(i_n)$, so $X \cap B_n$ must be either $A_n$ or $B_n\setminus A_n$. One of the two alternatives holds infinitely often. Hence, either there are infinitely many $n$ such that $X\cap B_n = A_n$, or there are infinitely many $n$ such
that $(\omega \setminus X) \cap B_n = A_n$.
\subsubsection*{Case 2:} There are infinitely many such $n$ with
$X(\ell_n) \not= \pi_{\ell_n}(X\mathord{\restriction} \ell_n)$, where in addition $|A_n| = 2$, i.e., $A_n = B_n$. So for each such $n$, $X(\ell_n) \not= \pi_{\ell_n}(X\mathord{\restriction} \ell_n) = 1-X(i_n)$. So $X(\ell_n) = X(i_n)$, so $X \cap B_n$ must be either $B_n$ or $\emptyset$. One of the two alternatives holds infinitely often. So again we either get infinitely many $n$ such that $X\cap B_n = A_n$, or infinitely many $n$ such
that $(\omega \setminus X) \cap B_n = A_n$.
\subsubsection*{Case 3:} For infinitely many $n$ as above we have $A_n=\emptyset$. Similar to the above.
\end{proof}
\begin{corollary} $\mathfrak s_2 \le \mathfrak e_2$ \end{corollary}
\begin{proof} Let $N$ be a model (of set theory) witnessing $\mathfrak e_2$, i.e., for every 2-predictor $\pi$ there is a function $f\in N$ evading $\pi$.
Let $R$ be any 2-rule. There is $X\in N$ evading $\pi_R$, so either $X$ or $\omega \setminus X$ (both in $N$) follows $R$. \end{proof}
\begin{Remark} $\mathfrak s \le \mathfrak e_2$ is known. Brendle showed that $\mathfrak s<\mathfrak s_2$ is consistent (unpublished). \end{Remark}
\section{Consistency of $\mathfrak r_2<\mathfrak r$}
We show here in contrast to theorem \ref{rkr2} that $\mathfrak r$ is not provably equal to $\mathfrak r_2$. Moreover, whereas $\mathfrak b\le \mathfrak r$ is provable in ZFC (see \cite{Vaughan} for a collection of results on cardinal invariants), we show that $\mathfrak r_2 < \mathfrak b$ is consistent with ZFC.
The following definition is standard:
\begin{Definition} \begin{enumerate} \item $S$ is a {\em slalom} iff $\operatorname{dom}(S)= \omega$ and for all
$n\in \omega$, $S(n)$ is a finite set
of size $n$. \item If $f$ is a function with $\operatorname{dom}(f) = \omega $, $S$ a
slalom, then we say that $S$ {\em captures} $f$ iff
$\forall^\infty n \,\, f(n) \in S(n)$. \item Let $M \subseteq N$ be sets (typically: models of ZFC*). We say that $N$ has the {\em Laver property} over $M$ iff: \begin{quote} For every function $H\in \pre \omega^\omega \cap M$, for every function $f\in \pre \omega^\omega \cap N$ satisfying $f \le H$ there is a slalom $S\in M$ that captures $f$. \end{quote} \item A forcing notion $P$ has the Laver property iff $\Vdash_P $ ``$V^P$ has the Laver property over $V$.'' \end{enumerate} \end{Definition}
Before we formulate the main lemma, we need the following easy claim:
\begin{Claim}
Let $k> 2^n$. If $X \subseteq \pre 2^k$,
$|X| = n$ then there are $i<j$ in $k$
such that for all $f\in X$, $f(i)=f(j)$. \end{Claim}
\begin{proof} For $i<j$, $f\in X$, define an equivalence relation $\sim_f$ by: $i \sim_f j \iff f(i) = f(j)$. Let $i \sim j$ iff $i \sim_f j$ for all $f$ in $X$. Since each $\sim_f$ has at most $2$ equivalence classes, $\sim$ has at most $2^n$ classes, so there are $i \not= j$, $i \sim j$. \end{proof}
\begin{Lemma}\label{lavermain} Assume that $(N,{\in} )$ is a model of ZFC*, and that $V$ has the Laver property over $N$.
Then every real avoids some $2$-rule from $N$. \end{Lemma}
\begin{proof}
Let $a_0 = 0$, $a_{n+1} = a_n + 2^n+1$. The sequence $(a_n:n\in \omega )$ is in $N$.
For any $X\in {\Cal P}(\omega)$, we will find a rule in $N$ which $X$ does not follow.
Let $\chi_X\in \pre 2^\omega $ be the characteristic function of $X$. Define $X^* := (\chi_X \mathord{\restriction} [a_n, a_{n+1}): n \in \omega)$. Note that there are only $2^{2^n+1}$ many possibilities for $\chi_X\mathord{\restriction} [a_n, a_{n+1} )$.
Since $V$ has the Laver property over $N$ there is a sequence $\vec S = (S_n: n\in \omega)\in N$, $S_n \subseteq \pre 2^{[a_n,a_{n+1} )}$,
$|S_n| \le n$, and for all $n>0$, $\chi_X\mathord{\restriction} [a_n, a_{n+1}) \in S(n)$. By the above claim we can find $i_n< j_n$ in $[a_n, a_{n+1}) $ such that for all $z\in S(n)$, $z(i_n)=z(j_n)$. Since the sequence $\vec S$ is in $N$, we can find such a sequence $( i_n, j_n:n<\omega)$ in $N$.
Define a 2-rule $( A_n, B_n: n \in \omega)\in N$ by $A_n=\{i_n\}$, $B_n = \{i_n, j_n \}$. Since $i_n\in X $ iff $j_n \in X$, $X $ does not follow this rule.
\end{proof}
\begin{Lemma}\label{laverfact} \begin{enumerate} \itm a Let $\bar P = (P_i, Q_i: i< \omega_2 )$ be a countable support
iteration of proper forcing notions such that for
each $i$ we have $\Vdash_i$ ``$Q_i$ has the Laver
property.'' Then $P_{\omega_2}$, the countable support
limit of $\bar P$, also has the Laver property. \itm b Laver forcing is proper and has the Laver property. \itm c Laver forcing adds a real that dominates all reals
from the ground model. \end{enumerate} \end{Lemma}
\begin{proof} These facts are
well known and (at least for the case where each $Q_i$
is Laver forcing) appear implicitly or explicitly in Laver's paper \cite{L1}. \end{proof}
\begin{Conclusion} Let $P_{\omega_2} $ be the limit of a countable support iteration of Laver forcing over a model $V_0$ of GCH. Then $\Vdash_{P_{\omega_2}} \mathfrak b=\mathfrak r={\omega_2} $ and $\mathfrak r_2={\omega_1}$. \end{Conclusion} \begin{proof} Let $V_{{\omega_2}} = V^{P_{\omega_2}}$. $V_{\omega_2} \models \mathfrak b={\omega_2} $ is well known. (Let $f_i$ be the real added by the $i$th Laver forcing; then $( f_i:i<{\omega_2})$ is a strictly increasing and cofinal sequence in $\pre \omega^\omega$.
By \ref{laverfact}, $V_{{\omega_2}}$ has the Laver property over $V_0$. Hence, by \ref{lavermain}, every real avoids some rule from $V_0$. So $\mathfrak r_2 \le |\pre \omega^\omega \cap V_0| = \aleph_1 $. \end{proof}
\section{Application to independent families}
A family ${\mathcal F}\subseteq\Cal P(\omega)$ of subsets of $\omega$ is {\em independent} iff it generates a free boolean algebra in $\Cal P(\omega)/\text{fin}$. Equivalently, for any two disjoint finite subsets of ${\mathcal F}$, the intersection of all members in the first set with all complements of members in the second set is infinite.
The following is an example of an independent family of size continuum over a countable set: $\{A_r:r\in \Bbb R\}$ where $A_r=\{p\in \Bbb Z[X]: p(r)>0\}$.
A family ${\mathcal F}\subseteq \Cal P(\omega)$ is {\em dense} iff for any two finite disjoint subsets of $\omega$ there are infinitely many members of ${\mathcal F}$ that contain the first set and are disjoint to the second.
An interesting (proper) subclass of the class of dense independent families over $\omega$ is the class of homogeneous families, which was introduced in \cite{GGK}. Its study was continued in \cite{KjSh:499}.
While every dense independent family is contained in a maximal dense independent family, this is not obvious (and perhaps false) for homogeneous families. The existence, even the consistency, of a maximal homogeneous family over $\omega$ is still open. In particular, an increasing union of homogeneous families need not be homogeneous.
In the study of extendibility of homogeneous families, the following notion is fundamental: Let $G\subseteq \operatorname{Aut} {\mathcal F}$. We define $( {\mathcal F},G)\le ( {\mathcal F}',G')$ iff ${\mathcal F}\subseteq {\mathcal F}'$, $G\subseteq G'\subseteq \operatorname{Aut} {\mathcal F}'$. The usefulness of $\le$ is that unions of suitable $\le$ chains {\em are} homogeneous (see \cite{KjSh:499} for a detailed account of direct limits in the category of homogeneous families).
We show now that below $\mathfrak r_\infty$ one can get proper $\le$-extensions of independent families. This was our original motivation for discovering $\mathfrak r_\infty$.
\begin{theorem}\label{motivate} Suppose $G \subseteq \operatorname{Aut} {\mathcal F}$, ${\mathcal F}\subseteq\Cal P(\omega)$ is dense independent and
$|{\mathcal F}|+|G|<\mathfrak r_\infty $. Then there exists ${\mathcal F}'\supsetneqq {\mathcal F}$ such that $( {\mathcal F},G) \le ( {\mathcal F}',G)$ \end{theorem}
\begin{proof} Suppose that $G\subseteq \operatorname{Aut} {\mathcal F}$, ${\mathcal F}$ is dense independent and
$|G|+|{\mathcal F}|<r_\infty$. We shall find a real $X\subseteq \omega$ such that $X\notin {\mathcal F}$ and ${\mathcal F}\cup G[X]$ is independent, where $G[X]$ is the orbit of $X$ under $G$. This will suffice, since clearly $G\subseteq \operatorname{Aut}( {\mathcal F}\cup G[X])$ for any real $X$.
It is a priori unclear why such $X$ should exist. If for example there is some $\sigma\in G$ with finite support, then for no $X\subseteq \omega$ is even the orbit $G[X]$ itself independent. However, the following lemma takes care of this. Let $\operatorname{supp}({\sigma} ) = \{ n\in \omega : {\sigma}(n)\not=n\}$ for a permutation ${\sigma} \in \operatorname{Sym} \omega $.
\begin{lemma}\label{support} Suppose that ${\mathcal F}$ is dense independent and $\sigma\in \operatorname{Aut} {\mathcal F}$ is not the identity. Then there are distinct sets $C_n\in {\mathcal F}$ such that for all $n$, $C_{2n}-C_{2n+1}\subseteq \operatorname{supp} \sigma$. \end{lemma}
\begin{Remark}In particular, the supports of non-identity automorphisms have the finite intersection property and hence generate a filter.
This is the ``strong Mekler condition'' for $\operatorname{Aut} {\mathcal F}$ (see \cite{truss}). \end{Remark}
\begin{proof} Fix $k^*\in \omega$ for which $\sigma(k^*)\not=k^*$. Find $C_{2n}$, $C_{2n+1}$ by induction on $n$. Suppose $C_m$ is chosen for $m< 2n$. By density, there are infinitely many $C\in {\mathcal F}$ for which $k^*\in C$, $ {\sigma} (k^*)\notin C$. Choose some such $C$ so that neither $C$ nor $ {\sigma} [C]$ are among $\{C_m:m< 2n\}$. Let $C_{2n}=C$ and $C_{2n+1}= {\sigma} [C]$. Since $k^*\in C_{2n}-C_{2n+1}$, those sets are indeed distinct.
We claim that $C_{2n}-C_{2n+1} \subseteq \operatorname{supp}( {\sigma} )$. Indeed, for any $k\in C_{2n}-C_{2n+1}$ we have $ {\sigma} (k)\in C_{2n+1}$ but $k \notin C_{2n+1}$, so $k \not= {\sigma} (k)$. \end{proof} We now continue the proof of theorem \ref{motivate}. Let $M$ be a transitive model of ZFC* of cardinality $< \mathfrak r_\infty$ such that ${\mathcal F},G\in M$ and $G\subseteq M, {\mathcal F}\subseteq M$. Let $X$ be a real that follows all bounded rules from $M$. Clearly, $X\notin M$, and therefore $X\notin {\mathcal F}$.
We need to show that every boolean combination over ${\mathcal F}\cup G[X]$ is infinite. Suppose that $$D:= A\cap \sigma_0[X] \cap\cdots\cap \sigma_{n-1}[X]\cap (\omega-\sigma_{n}[X])\cap\cdots \cap(\omega-\sigma_{m-1}[X])$$ is a boolean combination over ${\mathcal F}\cup G[X]$, where $\sigma_i\in G$ for $i<m$, and $A$ is some boolean combination over ${\mathcal F}$. Clearly, $A\in M$.
Set $N= \binom{m}{2}$ and let $( \tau_i:i<N)$ be a list of all $\sigma_k\circ \sigma^{-1}_\ell$ for $k<\ell<m$. By induction find a sequence $(C_0, \ldots, C_{2n+1})$ of $2N+2$ many distinct sets such that no $C_k$ participates in $D$ and such that $C_{2k}-C_{2k+1}\subseteq \operatorname{supp} \tau_k$. $C_{2k}$ and $C_{2k+1}$ are constructed in the $k$-th step by using
lemma \ref{support}.
Since all the $C_k$ are distinct, the intersection $E=A\cap \bigcap_{k<N}C_{2k}-\bigcup_{k<N}C_{2k+1}$ is infinite. Clearly $E$ belongs to $M$. Define by induction an $m$-rule $( A_n,B_n:n<\omega)$ as follows: suppose $( A_k,B_k:k\le n)$ are defined. Find a point $j_n\in E$ such that $B= \{\sigma_k^{-1}(j_n): k<m\}$ is disjoint from $\bigcup_{\ell\le n} B_\ell$ and $j_n \notin \bigcup_{\ell\le n} B_\ell$. Let $B_{n+1}$ be $B$ and let $A_{n+1}=\{\sigma_\ell^{-1}(j_n): \ell< n\}$.
The rule we defined obviously belongs to $M$. Since $X$ satisfies all bounded rules from $M$, there are infinitely many $n$ for which $X\cap B_n=A_n$. For each such $n$, $X_n \in D$. \end{proof}
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\rm \baselineskip=0.33333in \begin{quote} \raggedleft TAUP 2513-98 \end{quote} \vglue 0.4in \begin{center}{\bf The Role of Inert Objects\\ in Quantum Mechanical Phase}
\end{center} \begin{center}E. Comay \end{center}
\begin{center} School of Physics and Astronomy \\ Raymond and Beverly Sackler Faculty \\ of Exact Sciences \\ Tel Aviv University \\ Tel Aviv 69978 \\ Israel \end{center} \vglue 0.5in \noindent Email: [email protected] \vglue 0.5in \noindent PACS No: 03.65.Bz
\vglue 0.5in \noindent Abstract:
Quantum mechanical foundations of the polarized neutron phase shift experiment are discussed. The fact that the neutron retains its ground state throughout the experiment is shown to be crucial for the phase shift obtained.
Experimental data of the interaction of the neutron's magnetic moment with a magnetized material have two aspects: they provide information on the structure of specific materials and help us understand physical properties of the neutron. Thus, neutron scattering off a magnetized material was used for deciding that the neutron's magnetic moment is analogous to that of a current loop[1-3]. A new aspect of this issue is discussed in this work.
Very sensitive results of neutron dynamics are derived from an analysis of its interference pattern. Consider the nonrelativistic electromagnetic Lagrangian[4] augmented by the interaction of the neutron's intrinsic magnetic moment {\boldmath $\mu$} with the external magnetic field {\boldmath $B$} [5] \begin{equation} L = \frac{1}{2}mv^2 + e \mbox{\boldmath $v\cdot A$} - eV + \mbox{\boldmath $\mu \cdot B$}. \label{eq:LAGQ} \end{equation} In this work expressions are written in units where $c = 1$. Obviously, in the case of a neutron, the electric charge $e$ vanishes and we are left with the first and the last terms of $(\!\!~\ref{eq:LAGQ})$. Hence, the action[4] derived from $(\!\!~\ref{eq:LAGQ})$ depends on the external magnetic field, and one obtains \begin{equation} dS = (\frac{1}{2}mv^2 + \mbox{\boldmath $\mu \cdot B$}) dt. \label{eq:DS} \end{equation} This expression shows how the magnetic field affects the action, and thereby the neutron's phase and its interference pattern. Recent experiments utilize this relation between interference pattern and the Lagrangian $(\!\!~\ref{eq:LAGQ})$[6,7] for understanding the dynamics of the system. The present work analyses results of [7] and shows why quantum mechanical properties of the neutron distinguish it from an ordinary classical current loop.
In the experiment reported in [7], polarized neutrons drift through an external time dependent magnetic field. The neutron's velocity and spin as well as the external magnetic field are parallel to each other. At the neutron's location, the magnetic field is practically uniform. Hence, neither force nor torque are exerted on the neutron. The experiment confirms that, in spite of these facts and due to the action $(\!\!~\ref{eq:DS})$, the neutron is affected by the magnetic field which induces a phase shift that modifies the neutron's interference pattern.
Evidently, the neutron behaves in the experiment mentioned above like an inert object. Indeed, as a quantum mechanical system, the neutron is not excited to higher baryonic states whose energy is several hundreds Mev above the ground state, by its very small interaction with the external magnetic field. It is shown in this work that this property of the neutron is crucial for the magnetically dependent phase shift of $(\!\!~\ref{eq:DS})$.
Let us examine a hypothetical experiment which is similar to the one described in [7]. All parts of this experiment are like those of [7], except the neutron which is replaced by a "classical neutron" whose structure is described below.
Consider a ring made of an insulating material. This material takes the shape of a thin circular pipe containing a positively charge fluid. The insulating material is charged uniformly with negative charge so that the device looks like an electrically neutral object. The charge to mass ratio of the fluid is very small. This fluid rotates frictionlessly in a clockwise direction. Let $a$ denote the ring's radius and $I$ is the electric current associated with the fluid's motion. The magnetic field of the device is like that of a tiny magnetic dipole[8] \begin{equation} \mbox{\boldmath $\mu $} = \pi a^2I\mbox{\boldmath $k$} \label{eq:MURING} \end{equation} where {\boldmath $k$} denotes a unit vector in the $z$-direction. This hypothetical experiment is carried out in conditions where the nonrelativistic limit holds. The rather small ring is a macroscopic object and the motion of the charged fluid can be treated classically.
The system's Lagrangian is the ordinary Lagrangian of an electromagnetic system[4], namely $(\!\!~\ref{eq:LAGQ})$ without its last term \begin{equation} L = \frac{1}{2}mv^2 + e \mbox{\boldmath $v\cdot A$} - eV. \label{eq:LAGC} \end{equation} Here the electric potential $V$ vanishes. Hence, for evaluating $(\!\!~\ref{eq:LAGC})$ we have to find the values of the mechanical part and that of the second one which is associated with the magnetic interaction. The mechanical part of the Lagrangian describing the present experiment consists of two terms. One term pertains to the motion of the device in the $z$-direction and the second one is associated with the rotation of the fluid. Hence, the required Lagrangian is \begin{equation} L = \frac{1}{2}Mv_z^2 + \frac{1}{2}m_fv_\perp ^2 +
\int \mbox{\boldmath $j\cdot A$}d^3r \label{eq:LAGC1} \end{equation} where $M$ and $m_f$ denote the mass of the entire device and of the rotating fluid, respectively and $v_\perp$ is the fluid's velocity in the $(x,y)$ plane. The last term of $(\!\!~\ref{eq:LAGC1})$ is the continuum analog of the single particle expression $e${\boldmath $v\cdot A$} (see [8], p. 596).
Let us evaluate the last term of $(\!\!~\ref{eq:LAGC1})$. Introducing the electric current $I$, one applies Stokes theorem, the spatial uniformity of the magnetic field {\boldmath $B$} and relation $(\!\!~\ref{eq:MURING})$ and finds \begin{eqnarray} \int \mbox{\boldmath $j\cdot A$}d^3r & = & I\oint_L \mbox{\boldmath $A\cdot $}d\mbox{\boldmath $l$} \nonumber \\ & = & I\int_S (curl \mbox{\boldmath $A)\cdot $}d\mbox{\boldmath $s$} \nonumber \\ & = & \pi a^2 IB \nonumber \\ & = & \mbox{\boldmath $\mu\cdot B$}. \label{eq:JA} \end{eqnarray} Here the integral subscript $L$ denotes the closed path along the ring and $S$ is the ring's area. It follows that in $(\!\!~\ref{eq:LAGC1})$, the term {\boldmath $j\cdot A$} of the classical device discussed here is analogous to the last term of $(\!\!~\ref{eq:LAGQ})$.
Unlike the neutron case, where its internal structure does not change while the external magnetic field is turned on, the kinetic energy of the rotating fluid depends on the external magnetic field. Since the charge to mass ratio of the rotating fluid is very small, the current $I$ is regarded in the following calculation as a constant. The power transmitted to the rotating charged fluid is \begin{eqnarray} P & = & I\oint_L \mbox{\boldmath $E\cdot $}d\mbox{\boldmath $l$} \nonumber \\ & = & I\int_S (curl \mbox{\boldmath $E)\cdot $}d\mbox{\boldmath $s$} \nonumber \\ & = & -I\pi a^2 \frac {\partial B}{\partial t} \label{eq:POWER} \end{eqnarray} where the spatial uniformity of {\boldmath $B$} is used. Applying the notation of $(\!\!~\ref{eq:LAGC1})$, let $v_\perp (T_0)$ denote the velocity of the charged fluid at $T_0$ before the external magnetic field is turned on. It follows that, at an instant $t$, this part of the kinetic energy of the fluid is \begin{eqnarray} \frac {1}{2} mv_\perp ^2(t) & = & \frac {1}{2} mv_\perp ^2(T_0) - I\pi a^2 \int \frac {\partial B}{\partial t} dt \nonumber \\ & = & \frac {1}{2} mv_\perp ^2(T_0) - I\pi a^2B. \nonumber \\ & = & \frac {1}{2} mv_\perp ^2(T_0) - \mbox{\boldmath $\mu\cdot B$}. \label{eq:DK} \end{eqnarray} Substituting $(\!\!~\ref{eq:JA})$ and $(\!\!~\ref{eq:DK})$ into $(\!\!~\ref{eq:LAGC1})$, one finds that in the classical analog of the neutron experiment, the value of the Lagrangian \begin{equation} L(t) = \frac{1}{2}Mv_z^2 + \frac{1}{2}m_fv_\perp ^2(T_0) =L(T_0) \label{eq:LAGCT} \end{equation} is independent of the external magnetic field. The same is true for the corresponding action.
The results of this work emphasize the quantum mechanical meaning of the polarized neutron interference experiment[7]. As stated in [7], the neutron is free of classical effects like force and torque. In spite of this, its quantum properties vary due to the external magnetic field. The discussion carried out above points out another quantum mechanical aspect of the experiment. The neutron's spin and its associated magnetic moment are properties of a quantum mechanical system whose state may vary only in quantum leaps. The excited baryonic states of the neutron are very high, so that in experiment [7], the neutron is always in its ground state and behaves as an inert object. As such, there exists no analog to the variation of the self energy of the rotating fluid $(\!\!~\ref{eq:DK})$. For this reason, the Lagrangian of the neutron experiment [7] and the corresponding action {\em depend} on the external magnetic field whereas in the analogous classical experiment discussed above they are {\em independent} of it. It can be concluded that a comparison of the two experiments, namely the actual experiment reported in [7] and the hypothetical one which uses a classical current loop, demonstrates another quantum mechanical aspect of the neutron experiment [7].
References: \begin{itemize} \item[{[1]}] C. G. Shull, E. O. Wollan and W. A. Strauser, Phys. Rev. {\bf 81} (1951) 483. \item[{[2]}] F. Mezei, Physica {\bf 137B-C} (1986) 295. \item[{[2]}] F. Mezei, Physica {\bf 151B-C} (1988) 74. \item[{[4]}] L. D. Landau and E. M. Lifshitz, {\em The Classical Theory of Fields} (Pergamon, Oxford, 1975). p. 46. \item[{[5]}] J. M. Blatt and V. F. Weisslopf, {\em Theoretical Nuclear Physics} (Wiley, New York, 1952) p. 32. \item[{[6]}] J. Summhammer et al., Phys. Rev. Lett. {\bf 75} (1995) 3206. \item[{[7]}] W. -T Lee, O. Motrunich, B. E. Allman and S. A. Werner, Phys. Rev. Lett. {\bf 80} (1998) 3165. \item[{[8]}] J. D. Jackson, {\em Classical Electrodynamics,} second edition (John Wiley, New York, 1975). pp. 178, 179.
\end{itemize}
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Uniformly closed sublattices of finite codimension}
\author{Eugene Bilokopytov} \email{[email protected]}
\author{Vladimir G. Troitsky} \email{[email protected]}
\address{Department of Mathematical and Statistical Sciences,
University of Alberta, Edmonton, AB, T6G\,2G1, Canada.}
\thanks{The second author was supported by an NSERC grant.} \keywords{vector lattice, Banach lattice, sublattice, uniform convergence, continuous functions} \subjclass[2010]{Primary: 46A40. Secondary: 46B42,46E05}
\date{\today}
\begin{abstract}
The paper investigates uniformly closed subspaces, sublattices, and
ideals of finite codimension in Archimedean vector lattices. It is
shown that every uniformly closed subspace (or sublattice) of finite
codimension may be written as an intersection of uniformly closed
subspaces (respectively, sublattices) of codimension one. Every
uniformly closed sublattice of codimension $n$ contains a uniformly
closed ideal of codimension at most $2n$. If the vector lattice is
uniformly complete then every ideal of finite codimension is
uniformly closed. Results of the paper extend (and are
motivated by) results of~\cite{Abramovich:90a,Abramovich:90b}, as
well as Kakutani's characterization of closed sublattices of $C(K)$
spaces. \end{abstract}
\maketitle
\section{Introduction and preliminaries}
This paper merges two lines of investigation. The first line goes back to the celebrated Krein-Kakutani Theorem \cite{Krein:40,Kakutani:41} that every Archimedean vector lattice with a strong unit can be represented as a norm dense sublattice of $C(K)$ for some compact Hausdorff space. Furthermore, Kakutani in~\cite{Kakutani:41} completely characterized closed sublattices of $C(K)$ spaces.
The second line was initiated by Abramovich and Lipecki in~\cite{Abramovich:90a,Abramovich:90b}, where they studied sublattices and ideals of finite codimension in vector and Banach lattices. They proved, in particular, that every vector lattice has sublattices of all finite codimensions and that every finite codimensional ideal in a Banach lattice is closed.
We start our paper by revisiting Kakutani's characterization of closed sublattices of $C(K)$ spaces. We provide an easier proof of the characterization; we also extend it from $C(K)$ to $C(\Omega)$, where $\Omega$ need not be compact. We characterize closed sublattices of Banach sequence spaces.
In the rest of the paper, we investigate uniformly closed subspaces, sublattices, and ideals of finite codimension in an arbitrary Archimedean vector lattice $X$. By ``uniformly closed'', we mean ``closed with respect to relative uniform convergence''. While~\cite{Abramovich:90a,Abramovich:90b} use algebraic techniques based on prime ideals, our approach is to use Krein-Kakutani Representation theorem to reduce the problems to the case of $C(\Omega)$ spaces, and then use Kakutani's characterization of closed sublattices there. In a certain sense, it is the same approach because one can use the technique of prime ideals to prove Krein-Kakutani Representation Theorem. However, using the theorem makes proofs easier and more transparent. We extend several of the results of~\cite{Abramovich:90a,Abramovich:90b} about topologically closed sublattices in Banach and F-lattices to uniformly closed sublattices in vector lattices.
For a linear functional $\varphi$ on $X$, we relate properties of $\varphi$ to those of the one codimensional subspace $\ker\varphi$. Namely, we show that $\varphi$ is positive or negative iff $\ker\varphi$ is full, $\varphi$ is order bounded iff $\ker\varphi$ is uniformly closed, and $\varphi$ is a difference of two lattice homomorphisms iff $\ker\varphi$ is a uniformly closed sublattice. We show that every uniformly closed subspace (or sublattice) of finite codimension may be written as an intersection of uniformly closed subspaces (respectively, sublattices) of codimension one. We show that every uniformly closed sublattice of codimension $n$ contains a uniformly closed ideal of codimension at most $2n$. We prove that if $X$ is uniformly complete then every ideal of finite codimension is uniformly closed.
Throughout the paper, $X$ stands for an Archimedean vector lattice. For background on vector lattices, we refer the reader to~\cite{Aliprantis:06,Luxemburg:71,Meyer-Nieberg:91}.
We will now provide a brief overview of standard linear algebra facts on subspaces of finite codimension that will be used throughout the paper. Let $E$ be a vector space over $\mathbb R$. By a functional on $E$ we always mean a linear functional. We write $E'$ for the linear dual of $E$. Given a subspace $F$ of $E$, we say that $F$ is of codimension $n$ in $E$ if $\dim E/F=n$. Equivalently, $F\cap G=\{0\}$ and $F+G=E$ for some subspace $G$ with $\dim G=n$. Equivalently, there exists linear independent functionals $\varphi_1,\dots,\varphi_n$ in $E'$ such that $F=\bigcap_{i=1}^n\ker\varphi_i=\{\varphi_1,\dots,\varphi_n\}_\perp$. It follows that if $\codim F=n$ and $G\subseteq E$ is another subspace then the codimension of $F\cap G$ in $G$ is at most $n$. If $G\subseteq F$ then the codimension of $F$ in $E$ equals the codimension of $F/G$ in $E/G$. If $F$ is a closed subspace of a normed space $E$ then the functionals $\varphi_1,\dots,\varphi_n$ above may be chosen in the topological dual $E^*$. In this case, $\codim F=\dim F^\perp$, where $F^\perp$ is the annihilator of $F$ in $E^*$.
\begin{lemma}\label{codim-compl}
Let $F$ be a closed subspace of codimension $n$ in a normed space
$E$. Then the closure $\overline{F}$ of $F$ in the completion
$\overline{E}$ of $E$ has codimension $n$. \end{lemma}
\begin{proof}
The annihilator of $F$ in $E^*$ has dimension $n$. The canonical
isometric isomorphism between $E^*$ and $\overline{E}^*$ that sends
every functional in $E^*$ to its unique extension in
$\overline{E}^*$ maps this annihilator to the annihilator of $F$ in
$\overline{E}^*$; hence the latter also has dimension $n$. It
follows that $\codim(F^\perp)_\perp=n$, where both operations
are performed with respect to the dual pair
$(\overline{E},\overline{E}^*)$. Finally, recall that
$(F^\perp)_\perp=\overline{F}$. \end{proof}
\section{Closed sublattices of $C(\Omega)$} \label{CK-cl-sublat}
In this section, we revisit Kakutani's characterization of closed sublattices of $C(K)$ spaces from ~\cite{Kakutani:41}. Throughout this section, $\Omega$ stands for a completely regular Hausdorff topological space (also known as a Tychonoff space), which is exactly the class of Hausdorff spaces where the conclusion of Urysohn's lemma holds. Recall that locally compact Hausdorff spaces and normal spaces are completely regular.
We equip the space $C(\Omega)$ of all real valued continuous functions on $\Omega$ with the compact-open topology, i.e., the topology of uniform convergence on compact sets. In the special case of a compact space $K$, this topology agrees with the supremum norm topology on $C(K)$. Note that for every $t\in\Omega$, the point evaluation functional $\delta_t(f)=f(t)$ is a continuous positive linear functional on $C(\Omega)$, i.e., $\delta_f\in C(\Omega)^*_+$. We will characterize closed sublattices of $C(\Omega)$, a well as describe the closure of a sublattice of $C(\Omega)$.
We start with some motivation and examples. It can be easily verified that the set of all functions $f$ in $C[0,1]$ satisfying $f(1)=2f(0)$ forms a closed sublattice. More generally, fix $s\ne t$ in $\Omega$ and a non-negative scalar $\alpha$, and put $Y=\bigl\{f\in C(\Omega)\::\: f(s)=\alpha f(t)\bigr\}$; then $Y$ is a closed sublattice of $C(\Omega)$. Note that $Y=\ker\mu$ where $\mu\in C(\Omega)^*$ is given by $\mu=\delta_s-\alpha\delta_t$. In case $\alpha=0$, we just have $\mu=\delta_s$; then $Y$ is actually an ideal and $t$ is irrelevant.
In the preceding paragraph, $Y$ was determined by a single constraint $f(s)=\alpha f(t)$. We may generalize this construction to an arbitrary family of constraints as follows. Consider a collection of triples $(s,t,\alpha)$, where $s$ and $t$ are two distinct points in $\Omega$ and $\alpha\geqslant 0$, and let $Y$ be the set of all functions $f$ in $C(\Omega)$ such that $f(s)=\alpha f(t)$ for every triple $(s,t,\alpha)$ in the collection. There is an alternative way to describe $Y$: let $\mathcal M$ be a family of linear functionals on $C(\Omega)$ of the form $\mu=\delta_s-\alpha\delta_t$, where the triple $(s,t,\alpha)$ is in the collection; then $Y=\bigcap_{\mu\in\mathcal M}\ker\mu=\mathcal M_\perp$. Again, it is easy to see that $Y$ is a closed sublattice of $C(\Omega)$.
It is proved in Theorem~3 in~\cite{Kakutani:41} that for a compact space $K$, every closed sublattice $Y$ of $C(K)$ is of this form. That is \begin{math}
Y=\mathcal M_\perp, \end{math} where $\mathcal M$ consists of all the functionals of the form $\delta_s-\alpha\delta_t$ with $s,t\in K$ and $\alpha\geqslant 0$ that vanish on $Y$. That is, \begin{displaymath}
Y=\Bigl(\bigl\{\delta_s-\alpha\delta_t\::\: s,t\in K,\ \alpha\geqslant 0\bigr\}
\cap Y^\perp\Bigr)_\perp. \end{displaymath} We are going to present a short proof of this and other characterizations of closed sublattices $C(\Omega)$ where $\Omega$ is not assumed to be compact.
\begin{theorem}\label{CK-sublat-closure}
Let $Y$ be a sublattice of $C(\Omega)$. Each of
the following sets equals $\overline{Y}$.
\begin{eqnarray*}
Z_1&=&\Bigl\{f\in C(\Omega)\::\: \forall\mbox{ finite }F\subseteq\Omega\
\exists g\in Y,\mbox{ $f$ and $g$ agree on }F\Bigr\},\\
Z_2&=&\Bigl\{f\in C(\Omega)\::\: \forall s,t\in\Omega\
\exists g\in Y,\ f(s)=g(s)\mbox{ and }f(t)=g(t)\Bigr\},\\
Z_3&=&\Bigl(\bigl\{\delta_s-\gamma\delta_t \::\: s,t\in\Omega,\
\gamma\geqslant 0\bigr\} \cap Y^\perp\Bigr)_\perp,\\
Z_4&=&\Bigl(\Span\bigl\{\delta_s\::\: s\in\Omega\bigr\}
\cap Y^\perp\Bigr)_\perp,\\
Z_5&=&\mbox{the intersection of all closed sublattices of } C(\Omega)\\
&&\qquad\mbox{of codimension at most 1 that contain $Y$}.
\end{eqnarray*} \end{theorem}
\begin{proof}
To show that $\overline{Y}\subseteq Z_1$, let $f\in\overline{Y}$ and
$F=\{t_1,\dots,t_n\}$ in $\Omega$. Define
$T\colon C(\Omega)\to\mathbb R^n$ via
$Tg=\bigl(g(t_1),\dots,g(t_n)\bigr)$. Since $T$ is continuous,
we have $TY\subseteq T\overline{Y}\subseteq\overline{TY}=TY$, hence
$T\overline{Y}=TY$. It follows that there exists $g\in Y$ with
$Tf=Tg$. Thus, $f\in Z_1$ and, therefore,
$\overline{Y}\subseteq Z_1$.
We trivially have $Z_1\subseteq Z_2$. We will now show that
$Z_2\subseteq\overline{Y}$. Let $f\in Z_2$, let $\varepsilon>0$ and
$K\subseteq\Omega$ be compact. It suffices to find $g\in Y$ such
that $\bigabs{f(t)-g(t)}<\varepsilon$ for all $t\in K$.
For every
$s$ and $t$ in $K$ find $g_{s,t}\in Y$ so that $g_{s,t}(s)=f(s)$ and
$g_{s,t}(t)=f(t)$. Put
\begin{eqnarray*}
U_{s,t} & = &
\bigl\{w\in\Omega\::\: g_{s,t}(w)<f(w)+\varepsilon\bigr\},
\mbox{ and }\\
V_{s,t} & = &
\bigl\{w\in\Omega\::\: g_{s,t}(w)>f(w)-\varepsilon\bigr\}.
\end{eqnarray*}
Clearly, each of these two sets is open and contains both $s$ and
$t$. It follows that for each $t\in K$ the collection of sets
$\{U_{s,t}\::\: s\in K\}$ is an open cover of $K$. Hence, there is a
finite subcover, $K=\bigcup_{i=1}^nU_{s_i,t}$. Put
$g_t=\bigwedge_{i=1}^ng_{s_i,t}$. Then $g_t\in Y$, $g_t(t)=f(t)$,
and $g_t\leqslant f+\varepsilon\mathbbold{1}$ on $K$. Put $V_t=\bigcap_{i=1}^nV_{s_i,t}$.
Then $V_t$ is open, $t\in V_t$, and $g_t(w)\geqslant f(w)-\varepsilon$ for
all $w\in V_t$. Again, the collection $\{V_t\::\: t\in K\}$ is an
open cover of $K$, hence there is a finite subcover
$K=\bigcup_{j=1}^mV_{t_j}$. Put $g=\bigvee_{j=1}^mg_{t_j}$, then
$g\in Y$ and $f-\varepsilon\mathbbold{1}\leqslant g\leqslant f+\varepsilon\mathbbold{1}$ on $K$.
It follows that $Z_2\subseteq\overline{Y}$, so that
$Z_1=Z_2=\overline{Y}$.
We show next that $Z_3\subseteq Z_2$. Suppose that $f\in Z_3$; fix
$s\ne t$ in $\Omega$. We need to find $h\in Y$ such that $f(s)=h(s)$
and $f(t)=h(t)$. Put $T\colon Y\to\mathbb R^2$ via
$Th=\bigl(h(s),h(t)\bigr)$. If $\Range T$ is all of $\mathbb R^2$
then we can find $h\in Y$ such that
$\bigl(h(s),h(t)\bigr)=Th=\bigl(f(s),f(t)\bigr)$. If
$\Range T=\{0\}$ then both $\delta_s$ and $\delta_t$ vanish on $Y$; it
follows that they vanish on $f$ because $f\in Z_3$, and we can take
$h=0$. Finally, suppose that $T$ has rank 1. Then, possibly after
interchanging $s$ and $t$, we find $\alpha\in\mathbb R$ such that
$g(s)=\alpha g(t)$ for all $g\in Y$. It also follows that
$g_0(t)\ne 0$ for some $g_0\in Y$. Since $\abs{g_0}\in Y$, we have
$\abs{g_0(s)}=\alpha\abs{g_0(t)}$, which yields $\alpha\geqslant 0$. Let
$\mu:=\delta_s-\alpha\delta_t$. Since $f\in Z_3$, we have
$f\in\ker\mu$. We take $h$ to be a scalar multiple of $g_0$ so
that $h(t)=f(t)$. It now follows that
$h(s)=\alpha h(t)=\alpha f(t)=f(s)$. Hence, $f\in Z_2$. We now
conclude from
$\overline{Y}\subseteq Z_3\subseteq Z_2=\overline{Y}$
that $Z_3=\overline{Y}$.
It is easy to see that each $Z_4$ and $Z_5$ contains $\overline{Y}$
and is contained in $Z_3$. Since we already know that
$Z_3=\overline{Y}$, this completes the proof. \end{proof}
\begin{corollary}\label{CK-sublat-closed}
Closed sublattices of $C(\Omega)$ are the sets of the form $\mathcal M_\perp$,
where $\mathcal M$ is a collection of functionals of the form
$\delta_s-\alpha\delta_t$ where $s,t\in\Omega$ ans
$\alpha\geqslant 0$. \end{corollary}
\begin{corollary}
Every closed sublattice of $C(\Omega)$ of codimension one is either
of the form $\bigl\{f\in C(\Omega)\::\: f(s)=0\bigr\}$ for some
$s\in \Omega$ or of the form
$\bigl\{f\in C(\Omega)\::\: f(s)=\alpha f(t)\bigr\}$ for some
$s\ne t$ in $\Omega$ and $\alpha>0$. \end{corollary}
\begin{remark}\label{equiv-classes}
Let $Y$ be a sublattice of $C(\Omega)$. By
Theorem~\ref{CK-sublat-closure}, $\overline{Y}=\mathcal M_\perp$
where
\begin{math}
\mathcal M=\bigl\{\delta_s-\alpha\delta_t \::\: s,t\in\Omega,\
\alpha\geqslant 0\bigr\} \cap Y^\perp.
\end{math}
Let $\ker Y=\bigl\{t\in\Omega\::\:\delta_t\in\mathcal M\bigr\}$. That
is, $t\in\ker Y$ iff $f(t)=0$ for all $f\in Y$. It follows from
$\ker Y=\bigcap_{f\in Y}\ker f$ that $\ker Y$ is closed. For any
$s,t\in\Omega\setminus\ker Y$, we set $s\sim t$ if there exists
$\alpha\ne 0$ such that $f(s)=\alpha f(t)$ for all $f\in Y$. Note
that $\alpha$ has to be positive. Indeed, since $s,t\notin\ker Y$,
there exist $f,g\in Y$ with $f(s)\ne 0$ and $g(t)\ne 0$. Put
$h=\abs{f}\vee\abs{g}$; then $h\in Y$. It follows from
$h(s)=\alpha h(t)$ and $h(s),h(t)>0$ that $\alpha>0$. It also
follows that $\alpha$ is uniquely determined by $s$ and $t$. One can
easily verify that $s\not\sim t$ iff there exists $f\in Y_+$ such
that $f(s)=1$ and $f(t)=0$. This yields that
$s\sim t$ is equivalent to the following condition: $f(s)=0$ iff
$f(t)=0$ for all $f\in Y$.
It is clear that $s\sim t$ iff $\delta_s$ and $\delta_t$ are
proportional on $Y$ and, therefore, on $\overline{Y}$. This
is an equivalence relation on $\Omega\setminus\ker Y$; hence yields
a partition of $\Omega\setminus\ker Y$ into equivalence classes
$\{A_\gamma\}$. We will call these sets the \term{clans} of $Y$.
Of course, of primary interests are the clans
consisting of more than one point.
Let $A_\gamma$ be a clan. Fix $e_\gamma\in Y_+$ which does not
vanish at some point (and, therefore, at every point) of
$A_\gamma$. Furthermore, let $f\in Y$ be arbitrary and fix $t\in
A_\gamma$. For every $s\in A_\gamma$, it follows from $s\sim t$ that
$\frac{f(s)}{e_\gamma(s)}=\frac{f(t)}{e_\gamma(t)}$. Therefore,
every function in $Y$ is proportional to $e_\gamma$ on
$A_\gamma$. It now follows that for $f\in C(\Omega)$, we have
$f\in\overline{Y}$ iff $f$ vanishes on $\ker Y$ and is proportional
to $e_\gamma$ on $A_\gamma$ for every $\gamma$. \end{remark}
\begin{corollary}\label{CK-sublat-proport}
Let $Y$ be a sublattice of $C(\Omega)$. Then there exist disjoint
subsets $A_0$ and $(A_\gamma)_{\gamma\in\Gamma}$ and functions
$(e_\gamma)_{\gamma\in\Gamma}$ in $Y$ such that $f\in \overline{Y}$
iff $f$ vanishes on $A_0$ and is proportional to $e_\gamma$ on
$A_\gamma$ for every $\gamma\in\Gamma$. \end{corollary}
It is easy to see that the converse is also true: every set of this form is a closed sublattice of $C(\Omega)$.
Now consider the special case when $Y$ is an ideal in $C(\Omega)$. It is easy to see that if $\delta_s-\alpha\delta_t$ vanishes on $Y$ for some $s\ne t$ and $\alpha>0$ then $\delta_s$ and $\delta_t$ vanish on $Y$, so that $s,t\in\ker Y$ in the notation of Remark~\ref{equiv-classes}. It is clear than that all the equivalence classes in $\Omega\setminus\ker Y$ are singletons and $\mathcal M$ consists of evaluation functionals only.
\begin{corollary}\label{closed-ideal}
Every closed ideal in $C(\Omega)$ is of the form
\begin{math}
J_F=\bigl\{f\in C(\Omega)\::\: f_{|F}=0\bigr\}
\end{math}
for some closed set $F$. More generally,
if $J$ is an ideal of $C(\Omega)$ then $\overline{J}=J_F$ where
$F=\ker J$. \end{corollary}
\begin{corollary}\label{CK-sublat-ideal}
Every closed sublattice $Y$ of $C(\Omega)$ of codimension $n$ contains a
closed ideal $J$ of $C(\Omega)$ of codimension at most $2n$. \end{corollary}
\begin{proof}
By Theorem~\ref{CK-sublat-closure}, $Y=\mathcal
M_\perp=(\Span\mathcal M)_\perp$ where
\begin{math}
\mathcal M=\bigl\{\delta_s-\gamma\delta_t \::\: s,t\in\Omega,\
\gamma\geqslant 0\bigr\} \cap Y^\perp.
\end{math}
It follows that $Y=\bigcap_{i=1}^n\mu_i$ for some linearly
independent $\mu_1,\dots,\mu_n\in\mathcal M$, so each $\mu_i$ is the
form $\delta_s$ or $\delta_s-\alpha\delta_t$ for some $s,t\in\Omega$
and $\alpha\geqslant 0$. Put $J$ to be the intersection of the kernels of
all these $\delta_s$'s and $\delta_t$'s. \end{proof}
\begin{example}\label{one-clan}
Let $Y$ be the one-dimensional sublattice of $C[0,1]$ consisting of
all scalar multiples of $f(t)=t$. Then $\ker Y=\{0\}$ and the only
clan of $A$ is $(0,1]$. \end{example}
\begin{example}\label{CK-sublat-ex}
Let $Y$ be the closed sublattice of $C[0,10]$ consisting of all
the functions $f$ satisfying the following constraints:
\begin{displaymath}
f(0)=f(1)=0,\ f(2)=f(3),\ f(4)=\tfrac45f(5),\
f(6)=\tfrac17f(7)=\tfrac18f(8)=\tfrac19f(9).
\end{displaymath}
Then $\ker Y=\{0,1\}$ and the
non-singleton clans of $Y$ are $A_1=\{2,3\}$, $A_2=\{4,5\}$, and
$A_3=\{6,7,8,9\}$. We can take $e_1$ to be any continuous function
which takes values 1 at 2 and 3 but vanishes at 0, 1, 4, 5, 6, 7, 8,
and 9; $e_2$ to be any continuous function on satisfying
$e_2(4)=\frac45e_2(5)$ and vanishing at 0, 1, 2, 3, 6, 7, 8, 9; and
$e_3$ be any continuous function satisfying
$e_3(6)=\tfrac17e_3(7)=\tfrac18e_3(8)=\tfrac19e_3(9)$ and vanishing
at 0, 1, 2, 3, 4, and 5. Furthermore $Y$ has codimension 7, and
$Y=\bigcap_{i=1}^7\ker\mu_i$ where $\mu_1=\delta_0$,
$\mu_2=\delta_1$, $\mu_3=\delta_2-\delta_3$,
$\mu_4=\delta_4-2\delta_5$, $\mu_5=\delta_6-\frac17\delta_7$,
$\mu_6=\delta_6-\frac18\delta_8$, and
$\mu_7=\delta_6-\frac19\delta_9$. Put
$J=\bigl\{f\in C(K)\::\: f(i)=0,\ i=0,\dots,9\bigr\}$. Then $J$ is an
ideal in $C[0,10]$ of codimension 10 and $J\subseteq Y$. \end{example}
\begin{example}
Modify the preceding example as follows: let $Y$ be the (non-closed)
subspace of $C[0,10]$ consisting of all piece-wise affine functions
$f$ satisfying the same constraints and, in addition, affine on
$[4,5]$. Then $\ker Y=\{0,1\}$ and the non-singleton clans of $Y$
are $A_1=\{2,3\}$, $A_2=[4,5]$, and $A_3=\{6,7,8,9\}$. We can take
$e_1$ and $e_3$ as in the previous example; for $e_2$ we take a
function such that $e_2(t)=t$ on $[4,5]$ and vanishes at 0, 1, 2, 3,
6, 7, 8, and 9; we additionally require now that $e_1$, $e_2$, and
$e_3$ are piece-wise affine. The set of constraints will now also
include $f(t)=\frac{t}{4}f(4)$ for all $t\in(4,5]$ (that is,
$\mu_t=\delta_t-\frac{t}{4}\delta_4$), or an equivalent collection
of functionals. The closure $\overline{Y}$ will now consists of all
continuous functions satisfying the same constraints, and it will be
of infinite codimension. \end{example}
Again, let $Y$ be a sublattice of $C(\Omega)$. It is clear that $\ker Y$ is closed. Example~\ref{one-clan} shows that clans need not be closed. However, we have the following:
\begin{proposition}
Let $Y$ be a sublattice of $C(\Omega)$.
Clans of $Y$ are closed in $\Omega\setminus\ker Y$. \end{proposition}
\begin{proof}
Fix a clan $A_\gamma$; let $e_\gamma$ be as in
Remark~\ref{equiv-classes}. We know that for every $f\in Y$ there
exists $\lambda_f\geqslant 0$ such that $f$ agrees with
$\lambda_fe_\gamma$ on $A_\gamma$. Let $F$ be the set of all $t$
such that $f(t)=\lambda_fe_\gamma(t)$ for all $f\in Y$.
Clearly, $F$ is closed and contains $A_\gamma\cup\ker Y$. It
suffices to show that $F\subseteq A_\gamma\cup\ker Y$ as this would
imply $A_\gamma=F\cap(\Omega\setminus\ker Y)$. Let
$s\not\in A_\gamma\cup\ker Y$ but $s\in F$. Then there exists
$t\in A_\gamma$ such that $t\not\sim s$. It follows that there
exists $f\in Y$ such that $f(s)\ne 0$ but $f(t)=0$. But the latter
implies that $\lambda_f=0$. Since $s\in F$, it follows that
$f(s)=0$, which is a contradiction. \end{proof}
There are well known characterizations of bands and projection bands in $C(\Omega)$ spaces; see, e.g., Corollary~2.1.10 in~\cite{Meyer-Nieberg:91} for the case when $\Omega$ is compact and Corollary~4.5 in~\cite{Kandic:19} for the non-compact case.
\section{Closed sublattices of Banach sequence spaces}
By a Banach sequence space we mean a Banach lattice whose order is given by a Schauder basis. It is clear that the basis has to be 1-unconditional. One can view a Banach function space as a sublattice of $\mathbb R^{\mathbb N}$, which may also be identified with $C(\mathbb N)$, where $\mathbb N$ is equipped with the discrete topology. When talking about a sequence of vectors in a Banach sequence space, it is sometimes convenient to use subscripts and superscripts. For example, $x^{(n)}_i$ stands for the $i$-th coordinate of the $n$-th term of the sequence $(x^{(n)})$. It can be easily verified that every Banach sequence space is order continuous. The following theorem extends Theorem~5.2 of~\cite{Radjavi:08}. We first provide a direct proof of it; we will later show that it may be easily deduced from a more general statement.
\begin{theorem}\label{sublat-discr}
Let $X$ be a Banach sequence space. Closed sublattices of $X$ are
exactly the closed spans of (finite or infinite) disjoint positive
sequences. \end{theorem}
\begin{proof}
Let $Y$ be a closed sublattice of $X$. As in
Remark~\ref{equiv-classes}, we consider
$\ker Y=\{i\in\mathbb N\::\:\forall y\in Y\ i_i=0\}$; for
$i,j\notin\ker Y$, we write $i\sim j$ if there exists $\alpha>0$
such that $y_i=\alpha y_j$ for every $y\in Y$. This is an equivalence
relation on $\mathbb N\setminus\ker Y$; the equivalence classes are
the clans of $Y$. Again, as in Remark~\ref{equiv-classes}, if
$i\not\sim j$ then there exists $y\in Y_+$ such that $y_i=1$ and
$y_j=0$.
Fix a clan $A$ and any $n\in A$. Let $P_A$ be the natural basis
projection onto $A$, that is, $P_Ax$ is the sequence that agrees
with $x$ on $A$ and vanishes outside of $A$. It is easy to see that
$P_Ax$ and $P_Ay$ are proportional for any $x,y\in Y$, hence
$P_A(Y)$ is one-dimensional. Enumerating
$\mathbb N\setminus(\ker Y\cup A)$, we produce a sequence
$(y^{(m)})$ in $Y_+$ such that $y^{(m)}_n=1$ for all $m$, and for
every $j\notin A$ there exists $m$ with $y^{(m)}_j=0$. Replacing
$y^{(m)}$ with $y^{(1)}\wedge\dots \wedge y^{(m)}$, we may assume
that $y^{(m)}\downarrow$. Since $X$ is order complete and
$y^{(m)}\geqslant 0$, we have $y^{(m)}\downarrow x$ for some $x\in
X_+$. Since $X$ is order continuous, $y^{(m)}$ converges to $x$ in
norm, hence $x\in Y$. Also, it follows from $y^{(m)}\downarrow x$
that $x_n=1$ and $x_i=0$ for all $i\notin A$. Clearly, $x_k\ne 0$
for all $k\in A$. This yields that the support of $x$ is $A$. It
follows that $P_A(Y)=\Span\{x\}$.
Let $A_1,A_2,\dots$ be the clans of $Y$. For
each $n$, fix $x^{(n)}\in Y_+$ such that $P_{A_n}(Y)=\Span
x^{(n)}$. Clearly, $(x^{(n)})$ is a disjoint sequence in $Y$. Take
any $z\in Y$, it suffices to show that
$z\in\overline{\rm span}\ x^{(n)}$. WLOG, $z\geqslant 0$ as, otherwise, we
can consider $z^+$ and $z^-$. For each $m$, put
$z^{(m)}=P_{A_1}z+\dots+P_{A_m}z$. Then
$z^{(m)}\in\Span\{x^{(n)}\}$. It follows from $z^{(m)}\uparrow z$
that that $z^{(m)}$ converge to $z$ in norm; hence
$z\in\overline{\rm span}\ x^{(n)}$. \end{proof}
The following is an easy corollary Theorem~\ref{sublat-discr}; it may also be deduced from Proposition~\ref{CK-sublat-proport} by taking $K=\{1,\dots,n\}$.
\begin{corollary}\label{sublat-Rn}
A subspace of $\mathbb R^n$ is a sublattice iff it can be written as
a span of disjoint positive vectors. \end{corollary}
Recall that a net $(x_\alpha)$ in a vector lattice $X$ \term{uo-converges} to $x$ if $\abs{x_\alpha-x}\wedge u$ converges to zero in order for every $u\geqslant 0$. A sublattice is order closed iff it is uo-closed. Furthermore, let $(x_\alpha)$ be a net in a regular sublattice $Y$ of $X$, then $x_\alpha\xrightarrow{\mathrm{uo}} 0$ in $X$ iff $x_\alpha\xrightarrow{\mathrm{uo}} 0$ in $Y$. We refer the reader to~\cite{Gao:17} for a review of order convergence, uo-convergence, and regular sublattices.
\begin{proposition}\label{ocont-COmega}
Let $X$ be an order continuous Banach lattice, continuously embedded
as a sublattice into $C(\Omega)$ for some locally compact $\Omega$;
let $Y$ be a sublattice of $X$. If $Y$ is norm closed in $X$
then $Y$ is closed in $X$ with respect to the compact-open topology
of $C(\Omega)$ restricted to $X$. It follows that
$Y=\overline{Y}^{C(\Omega)}\cap X$. \end{proposition}
\begin{proof}
We claim that $X$ is regular in $C(\Omega)$. Indeed, suppose that
$x_\alpha\downarrow 0$ in $X$. It follows that $x_\alpha\to 0$ in
norm of $X$ and, therefore, in the compact-open topology of
$C(\Omega)$. Since $(x_\alpha)$ is monotone, we have
$x_\alpha\downarrow 0$ in $C(\Omega)$ by Theorem~2.21c
of~\cite{Aliprantis:03}. This proves the claim.
Since $Y$ is norm closed in $X$ and $X$ is order continuous, $Y$ is
order closed in $X$, and, therefore, uo-closed in $X$. It follows
easily from Theorem~3.2 in~\cite{Bilokopytov:22} that compact-open
convergence on $C(\Omega)$ is stronger than uo-convergence in
$C(\Omega)$. The same holds true for their restrictions to $X$
because compact-open convergence is topological and $X$ is
regular. It follows that $Y$ is closed in the restriction of the
compact-open topology to $X$. \end{proof}
We can now easily deduce Theorem~\ref{sublat-discr} from Proposition~\ref{ocont-COmega}:
\begin{proof}[Alternative proof of Theorem~\ref{sublat-discr}]
Being a sequence space, $X$ may be viewed as an ideal in
$C(\mathbb N)$.
By Proposition~\ref{ocont-COmega}, for every closed
sublattice $Y$ of $X$, we have $Y=\overline{Y}\cap X$ , where
$\overline{Y}$ is the closure of $Y$ in $C(\mathbb N)$, hence is a
closed sublattice of $C(\mathbb N)$. We now apply
Corollary~\ref{CK-sublat-proport} to $\overline{Y}$; let $A_0$,
$A_\gamma$ and $e_\gamma$ be as in the corollary. Note that
$e_\gamma\cdot\mathbbold{1}_{A_\gamma}$ is in $\overline{Y}$ by
Corollary~\ref{CK-sublat-proport}. Since $X$ is an ideal in
$C(\mathbb N)$, we have $e_\gamma\cdot\mathbbold{1}_{A_\gamma}$ is in $X$ and,
therefore, in $Y$ because $Y=\overline{Y}\cap X$.
Hence, replacing each $e_\gamma$
with $e_\gamma\cdot\mathbbold{1}_{A_\gamma}$, we may assume that the
collection $(e_\gamma)$ is disjoint and, therefore, countable.
It follows from Corollary~\ref{CK-sublat-proport} that $Y$ is
the closed linear span of this collection. \end{proof}
\section{Uniformly closed subspaces of finite codimension}
Throughout the rest of the paper, $X$ stands for an Archimedean vector lattice. For $e\in X_+$, we write $I_e$ for the principal ideal of $e$. For $x\in X$, we put \begin{displaymath}
\norm{x}_e=\inf\bigl\{\lambda>0\::\:\abs{x}\leqslant\lambda e\bigr\}. \end{displaymath} This defines a lattice norm on $I_e$. By Kakutani-Krein representation theorem, $\bigl(I_e,\norm{\cdot}_e\bigr)$ is lattice isometric to a dense sublattice of $C(K)$ for some compact Hausdorff space $K$.
For a net $(x_\alpha)$ in $X$, we say that it converges \term{relatively uniformly} or just \term{uniformly} to $x\in X$ and write $x_\alpha\xrightarrow{\mathrm{u}} x$ if $\norm{x_\alpha-x}_e\to 0$ for some $e\in X_+$; this implies, in particular, that $x_\alpha-x\in I_e$ for all sufficiently large $\alpha$. A subset $A$ of $X$ is said to be \term{uniformly closed} if it is closed with respect to uniform convergence. Equivalently, $A\cap I_e$ is closed in $\bigl(I_e,\norm{\cdot}_e\bigr)$ for every $e\in X_+$. Actually, it suffices to verify this condition for every $e$ in a majorizing sublattice of $X$ because $e\leqslant u$ implies $\norm{\cdot}_u\leqslant\norm{\cdot}_e$. In a Banach lattice, every uniformly convergent sequence is norm convergent and every norm convergent sequence has a uniformly convergent subsequence (see, e.g., Lemma~1.1 of~\cite{Taylor:20}). It follows that a subset of a Banach lattice is uniformly closed iff it is norm closed.
Recall that if $J$ is an ideal in $X$ then $X/J$ is a vector lattice and the quotient map is a lattice homomorphism; $X/J$ is Archimedean iff $J$ is uniformly closed by Theorem~60.2 in~\cite{Luxemburg:71}.
A vector lattice is \term{uniformly
complete} if for every $e\in X_+$ the ideal $I_e$ is complete in $\norm{\cdot}_e$ and, as a consequence, $\bigl(I_e,\norm{\cdot}_e\bigr)$ is lattice isometric to $C(K)$ for some compact Hausdorff space $K$.
If $K$ is a compact Hausdorff space then (relative) uniform convergence in $C(K)$ agrees with compact-open convergence, which is, in this case, the same as the norm convergence, i.e., the convergence in $\norm{\cdot}_{\mathbbold{1}}$ norm. This is no longer true when $\Omega$ is just a Tychonoff space, which creates an unfortunate clash of terminologies. In this paper, uniform convergence in $C(\Omega)$ will always be interpreted as the relative uniform convergence rather than convergence in $\norm{\cdot}_{\mathbbold{1}}$. It is easy to see that uniform convergence in $C(\Omega)$ implies convergence in compact-open topology, so that sets that are closed in compact-open topology are uniformly closed.
The converse is false. Indeed, identify $\mathbb R^{\mathbb N}$ with $C(\mathbb N)$ and consider the double sequence $x_{n,m}=(0,\dots,0,n,n,\dots)$, with $m$ zeros at the head. It is easy to see that $(x_{n,m})$ converges to zero pointwise, hence in the compact-open topology. However, every tail of this net fails to be order bounded, hence it cannot converge uniformly.
Consider the special case when $\Omega$ is locally compact and $\sigma$-compact, i.e., can be written as a union of sequence of compact sets. We claim that, in this case, uniformly closed sets are compact-open closed. Indeed, it suffices to show that every net that converges in compact-open topology contains a sequence that converges uniformly. Since compact-open topology on $C(\Omega)$ is complete and metrizable by, e.g.,~\cite[pp.~62-64]{Beckenstein:77}, the proof of this fact is similar to the classical proof that every norm convergent sequence in a Banach lattice has a uniformly convergent subsequence; see, e.g., \cite[Lemma~1.1]{Taylor:20}
It was shown in \cite[Theorem~5.1]{Taylor:20} that an operator between vector lattices is order bounded iff it is \term{uniformly continuous}, i.e., maps uniformly convergent nets to uniformly convergent nets.
\begin{proposition}\label{obdd-ker-u-closed}
A linear functional $\varphi$ on $X$ is order bounded iff $\ker\varphi$ is
uniformly closed. \end{proposition}
\begin{proof}
If $\varphi$ is order bounded, it is uniformly continuous; it
follows that $\ker\varphi$ is uniformly closed. Conversely, suppose
that $\ker\varphi$ is uniformly closed. We need to show that
$\varphi$ is uniformly continuous. It suffices to show that the
restriction $\varphi_{|I_e}$ is $\norm{\cdot}_e$-continuous for
every $e\in X_+$, which is equivalent to $\ker\varphi_{|I_e}$ being
closed in $\bigl(I_e,\norm{\cdot}_e\bigr)$; the latter is
straightforward. \end{proof}
Since every subspace of codimension one is the kernel of a linear functional, we get the following:
\begin{corollary}\label{u-subsp-codim-1}
Uniformly closed subspaces of $X$ of codimension 1 are exactly the
kernels of order bounded functionals. \end{corollary}
\begin{example}
Let $X=c_{00}$, the space of all eventually zero sequences. Every
principal ideal in $c_{00}$ is finite-dimensional. It follows that
every subspace of $c_{00}$ is uniformly closed and, therefore, every linear
functional on $c_{00}$ is order bounded. \end{example}
\begin{lemma}
Let $Y$ and $Z$ be subspaces of $X$ such that $Y$ is uniformly
closed and $Z$ is finite dimensional. Then $Y+Z$ is uniformly
closed. \end{lemma}
\begin{proof}
Suppose that $x_\alpha\xrightarrow{\mathrm{u}} x$ for some $(x_\alpha)$ in $Y+Z$. Find
$e\in X_+$ with $\norm{x_\alpha-x}_e\to 0$. Since $Z$ is
finite-dimensional, we can find $u\geqslant e$ such that $Z\subseteq
I_u$. Then $(Y+Z)\cap I_u=(Y\cap I_u)+Z$. Since $Y\cap I_u$ is
closed in $I_u$, so is $(Y+Z)\cap I_u$. It follows from
$\norm{x_\alpha-x}_u\to 0$ that $x\in Y+Z$. \end{proof}
\begin{corollary}\label{nested}
Let $Y$ be a uniformly closed subspace of $X$ of finite
codimension. If $Y\subseteq Z\subseteq X$ for some subspace $Z$ then
$Z$ is uniformly closed. \end{corollary}
\begin{corollary}
Every uniformly closed subspace of $X$ of finite codimension is an
intersection of uniformly closed subspaces of codimention~1. \end{corollary}
\begin{question}
Under what conditions on $X$ can we say that \emph{every} uniformly
closed subspace is the intersection of uniformly closed subspaces of
codimension~1? This is not true in general as $X^\sim$ may be
trivial, and then $X$ has no uniformly closed subspaces of
codimension~1. However, this is true when $X$ is a Banach lattice
because in this case uniformly closed and norm closed sets agree. \end{question}
\begin{question}
Is every uniformly closed subspace the kernel of an
order bounded operator? \end{question}
In Proposition~\ref{obdd-ker-u-closed}, we describe the kernels of order bounded functionals. The following result characterizes kernels of positive functionals. Recall that a subspace $Y$ of $X$ is \term{full} if $x,y\in Y$ and $x\leqslant y$ imply $[x,y]\subseteq Y$. Clearly, every ideal is a full subspace. The converse is false: the straight line $y=-x$ in $\mathbb R^2$ is full but not an ideal.
\begin{lemma}
A subspace $Y$ is full iff the set
\begin{math}
\bigl\{x\in X\::\:\abs{x}\in Y\bigr\}
\end{math}
is an ideal. \end{lemma}
\begin{proof}
Suppose that $Y$ is full; we will show that the set
$J:=\bigl\{x\in X\::\:\abs{x}\in Y\bigr\}$ is an ideal. It is clear
that $J$ is closed under scalar multiplication. To show that it is
closed under addition, let $x,y\in J$, then $\abs{x}+\abs{y}\in Y$;
it now follows from $0\leqslant\abs{x+y}\leqslant\abs{x}+\abs{y}$ that
$\abs{x+y}\in Y$ and, therefore, $x+y\in J$. Finally, suppose that
$y\in J$ and $\abs{x}\leqslant\abs{y}$; it follows from
$0\leqslant\abs{x}\leqslant\abs{y}$ that $\abs{x}\in Y$ and, therefore, $x\in
J$.
Conversely, suppose that $J$ is an ideal. Suppose $x\in[y_1,y_2]$
where $y_1,y_2\in Y$. It follows from $0\leqslant x-y_1\leqslant y_2-y_1$ and
$y_2-y_1\in J$ that $x-y_1\in J$, hence $x-y_1\in Y$ and, therefore,
$x\in Y$. \end{proof}
\begin{proposition}\label{ker-full}
A one-codimensional subspace of $X$ is full iff it is the kernel of
a positive functional. \end{proposition}
\begin{proof}
Let $\varphi$ be a positive functional on $X$,
$y_1,y_2\in\ker\varphi$, and $y_1\leqslant x\leqslant y_2$. Then
$0=\varphi(y_1)\leqslant\varphi(x)\leqslant\varphi(y_2)=0$ implies
$x\in\ker\varphi$.
For the converse, suppose that $\ker\varphi$ is full but $\varphi$
is neither positive nor negative. Then there exist $x,y\in X_+$ such
that $\varphi(x)>0$ and $\varphi(y)<0$. Take $\alpha,\beta>0$
such that $\alpha+\beta=1$ and
$0=\alpha\varphi(x)+\beta\varphi(y)=\varphi\bigl(\alpha x+\beta
y)$. It now follows from $0\leqslant\alpha x\leqslant\alpha x+\beta
y\in\ker\varphi$ that $\alpha x\in\ker\varphi$, which is a contradiction. \end{proof}
The following two results are known; we include them for completeness.
\begin{proposition}\label{ker-ideal}
A subspace $J$ of co-dimension 1 is an ideal iff $J$ is the kernel
of a real-valued lattice homomorphism. \end{proposition}
\begin{proof}
If $J$ is an ideal then $J$ is the kernel of the quotient map $X\to
X/J$. The converse is trivial. \end{proof}
For a positive functional $\varphi$, we write $N_\varphi$ for its \term{null ideal}: $N_\varphi=\bigl\{x\in X\::\:\varphi\bigl(\abs{x}\bigr)=0\bigr\}$.
\begin{proposition}
An ideal $J$ in $X$ is a null ideal of some positive functional iff
$X/J$ admits a strictly positive functional. \end{proposition}
\section{Uniformly closed sublattices of finite codimension}
The following result is a part of Theorem~3 in~\cite{Abramovich:90a}. As it is important for our exposition, we include a short proof of it.
\begin{proposition}[\cite{Abramovich:90a}]\label{sublats-exist}
For every $n\in\mathbb N$ with $n\leqslant\dim X$, $X$ admits a sublattice
of codimension $n$. \end{proposition}
\begin{proof}
Clearly, it suffices to prove the statement for $n=1$. By
Theorem~33.4 in~\cite{Luxemburg:71} $X$ admits a proper ideal
$J$ which is prime, i.e., $x\wedge y\in J$ implies $x\in J$ or
$y\in J$. Let $Y$ be any subspace of $X$ of codimension 1 containing
$J$. If $x\in Y$ then $x^+$ or $x^-$ is in $J$, hence both $x^+$ and
$x^-$ are in $Y$. Therefore, $Y$ is a sublattice. \end{proof}
\begin{lemma}
Let $Y$ be a sublattice of $\mathbb R^n$ of codimension~$m$. Then
$Y$ contains at least $n-2m$ and at most $n-m$ of the standard unit
vectors. \end{lemma}
\begin{proof}
The lower estimate follows from Corollary~\ref{sublat-Rn}; the upper
estimate is obvious. \end{proof}
The following result was proved in \cite{Abramovich:90b} in the special case of sublattices of codimension 1.
\begin{proposition}\label{sublat-disj-2m}
Let $Y$ be a sublattice of $X$ of codimension $m$. Then any disjoint
set that does not meet $Y$ consists of at most $2m$ vectors. \end{proposition}
\begin{proof}
Suppose that $x_1,\dots,x_n$ are disjoint vectors in $X\setminus Y$
for some $n>2m$. Assume first that $x_1,\dots,x_n>0$. Let
$Z=\Span\{x_1,\dots,x_n\}$. Clearly, there exists a lattice
isomorphism $T$ from $Z$ onto $\mathbb R^n$ such that $Tx_i=e_i$ as
$i=1,\dots,n$. Note that $Y\cap Z$ is a sublattice of $Z$ of
codimension at most $m$. It follows from the preceding lemma that
$Y\cap Z$ contains some (at least $n-2m$) of the $x_i$'s, which
contradicts the assumptions. This proves the theorem for positive
vectors.
Suppose now that $x_1,\dots,x_n$ are arbitrary disjoint vectors in
$X\setminus Y$ for some $n>2m$. For each $i$, either $x_i^+$ or
$x_i^-$ (or both) is not in $Y$ (in particular, it is
non-zero). This yields a collection of $n$ disjoint positive vectors
in $X\setminus Y$, which is impossible by the first part of the
proof. \end{proof}
We now proceed to uniformly closed sublattices. We proved in Corollary~\ref{u-subsp-codim-1} that uniformly closed subspaces of $X$ of codimension 1 are the kernels of order bounded functionals. Also, we observed that every closed sublattice $Y$ of $C(\Omega)$ of codimension 1 is the kernel of $\delta_t-\alpha\delta_s$ for some $s\ne t$ in $\Omega$ and $\alpha\geqslant 0$; hence, $Y$ is the kernel of the difference of two lattice homomorphisms in $C(\Omega)^*$. In Theorem~2b of~\cite{Abramovich:90b}, the same result was established for vector latices of simple functions. We are now going to extend this to general vector lattices.
Recall that a positive operator $T$ between vector lattices is a lattice homomorphism iff $x\wedge y=0$ implies $Tx\wedge Ty=0$. We write $X^h$ for the set of all real-valued lattice homomorphisms on $X$. Clearly, $X^h$ is a subset (but not a subspace) of $X^\sim$. Recall that the elements of $X^h$ are atoms of $X^\sim$, see, e.g., \cite[Theorem~1.85]{Aliprantis:03}; in particular, any two elements of $X^h$ are either disjoint or proportional. It follows that $\varphi\in X^\sim$ is a difference of two lattice homomorphisms, iff $\varphi^+$ and $\varphi^-$ are lattice homomorphisms.
\begin{proposition}\label{sublat-codim-1}
Uniformly closed sublattices of $X$ of codimension 1 are exactly the
kernels of differences of two lattice homomorphisms in $X'$. \end{proposition}
\begin{proof}
Suppose that $Y=\ker(\varphi-\psi)$ where $\varphi$ and $\psi$ are
lattice homomorphisms. Being the set on which $\varphi$ and $\psi$
agree, $Y$ is a sublattice; it is uniformly closed by
Corollary~\ref{u-subsp-codim-1}.
Suppose that $Y$ is a uniformly closed sublattice of $X$ of
codimension 1. By Corollary~\ref{u-subsp-codim-1}, $Y=\ker\varphi$
for some $\varphi\in X^\sim$. It suffices to show that $\varphi^+$
and $\varphi^-$ are lattice homomorphisms. Let $x\wedge y=0$; we
need to show that $\varphi^+(x)\wedge\varphi^+(y)=0$ or,
equivalently, that either $\varphi^+(x)=0$ or
$\varphi^+(y)=0$. Suppose not, then by Riesz-Kantorovich Formulae,
there exist $u\in[0,x]$ and $v\in[0,y]$ such that $\varphi(u)$ and
$\varphi(v)$ are greater than zero. Then
$\varphi(u-\lambda v)=\varphi(u)-\lambda\varphi(v)=0$ for some
$\lambda>0$, hence $u-\lambda v\in\ker\varphi$. Since $u\perp v$ and
$\ker\varphi$ is a sublattice, it follows that
$u+\lambda v=\abs{u-\lambda v}\in\ker\varphi$; which is impossible.
The proof that $\varphi^-$ is a lattice homomorphism is similar. \end{proof}
\begin{example}
By Proposition~\ref{sublats-exist}, $L_p[0,1]$
($1\leqslant p<\infty$) contains a sublattice of codimension 1. It is not
uniformly closed as $L_p[0,1]$ admits no real-valued lattice
homomorphisms. \end{example}
\begin{question}
Let $\varphi$ be a functional on $X$.
Propositions~\ref{obdd-ker-u-closed}, \ref{ker-full},
\ref{ker-ideal}, and~\ref{sublat-codim-1} relate various order
properties of $\varphi$ with those of $\ker\varphi$. However, one
property is clearly missing from this list: it would be interesting
to characterize when $\ker\varphi$ is a sublattice
(not necessarily uniformly closed).
Lemma~2 in~\cite{Abramovich:90b} or Proposition~\ref{sublat-disj-2m}
in the current paper yield the following necessary condition: for
every three disjoint vectors, $\varphi$ must vanish at at least one
of them. On the other hand, Lemma~1b (and its proof)
in~\cite{Abramovich:90a} yields a sufficient condition: for every
two disjoint vectors, $\varphi$ must vanish at at least one of them
(i.e., $\varphi$ is disjointness preserving). \end{question}
We will now show in Theorem~\ref{closures} that much of Theorem~\ref{CK-sublat-closure} can be generalized from $C(\Omega)$ to general vector lattices, replacing evaluation functionals with lattice homomorphisms. In fact, most of Theorem~\ref{CK-sublat-closure} may be deduced from Theorem~\ref{closures}. We start with some linear algebra lemmas.
For a subset $A$ of a vector space $E$ and $n\in\mathbb N$, we write \begin{displaymath}
\Span_nA=\Bigl\{\alpha_1x_1+\dots+\alpha_nx_n\::\:
x_1,\dots,x_n\in A,\ \alpha_1,\dots,\alpha_n\in\mathbb R\bigr\}. \end{displaymath}
\begin{lemma}\label{span-n}
Let $F$ be a subspace of a vector space $E$, $A\subseteq E'$, and
$n\in\mathbb N$. For $x\in E$,
$x\in\bigl(\Span_nA\cap F^\perp\bigr)_\perp$ iff for every
$\varphi_1,\dots,\varphi_n\in A$ there exist $y\in F$ with
$\varphi_i(x)=\varphi_i(y)$ as $i=1,\dots,n$. \end{lemma}
Before proving the lemma let us first demystify its statement. The set $\Span_nA\cap F^\perp$ consists of the elements of $\Span_n A$ which vanish on $F$, and so the first set is the maximal subspace on which these functionals vanish. Meanwhile, the other set consists of the vectors that allow $n$-nod interpolation by the elements of $F$ with respect to $A$.
\begin{proof}
Denote the sets in question by $H$ and $G$. Suppose that $x\in G$ and
$\psi\in \Span_nA\cap F^\perp$. Then
$\psi=\alpha_1\varphi_1+\dots+\alpha_n\varphi_n$ for some
$\varphi_1,\dots,\varphi_n\in A$ and
$\alpha_1,\dots,\alpha_n\in\mathbb R$. Since $x\in G$, there exists
$y\in F$ such that $\varphi_i(x)=\varphi_i(y)$ as $i=1,\dots,n$. It
follows that
\begin{math}
\psi(x)=\psi(y)=0
\end{math}
because $\psi\in F^\perp$. This yields $x\in\bigl(\Span_nA\cap
F^\perp\bigr)_\perp$. Hence $G\subseteq H$.
Suppose $x\notin G$. Then there exist
$\varphi_1,\dots,\varphi_n\in A$ such that $Tx\notin TF$, where
$T\colon E\to\mathbb R^n$, $(Tx)_i=\varphi_i(x)$. It follows that
there is a functional on $\mathbb R^n$ that separates $Tx$ from
$TF$. That is, there exist $\bar a=(\alpha_1,\dots,\alpha_n)$ such
that $0=\langle\bar a,Ty\rangle=\sum_{i=1}^n\alpha_i\varphi_i(y)$
for all $y\in F$ but $\langle\bar a,Tx\rangle\ne 0$. Put
$\psi=\sum_{i=1}^n\alpha_i\varphi_i$; then $\psi$ vanishes on $F$
but $\psi(x)\ne 0$, so $x\notin H$. \end{proof}
\begin{corollary}\label{span-perp}
In the setting of Lemma~\ref{span-n}, $x\in\bigl(\Span A\cap
F^\perp\bigr)_\perp$ iff for every $n$ and every
$\varphi_1,\dots,\varphi_n\in A$ there exists $y\in F$ such that
$\varphi_i(x)=\varphi_i(y)$ as $i=1,\dots,n$. \end{corollary}
\begin{theorem}\label{closures}
Let $Y$ be a sublattice of $X$. The following sets are equal:
\begin{eqnarray*}
Y_1&=&\Bigl\{z\in X\::\: \forall\mbox{ finite }F\subseteq X^h
\ \exists y\in Y,\mbox{ $y$ and $z$ agree on $F$}\Bigr\};\\
Y_2&=&\Bigl\{z\in X\::\: \forall h_1,h_1\in X^h\
\exists y\in Y,\ h_1(z)=h_1(y)\mbox{ and }h_2(z)=h_2(y)\Bigr\};\\
Y_3&=&\Bigl(\bigl(\Span X^h\bigr)\cap Y^\perp\Bigr)_\perp;\\
Y_4&=&\Bigl(\bigl(\Span_2 X^h\bigr)\cap Y^\perp\Bigr)_\perp;\\
Y_5&=&\Bigl(\bigl\{h_1-h_2 \::\: h_1,h_2\in X^h\}
\cap Y^\perp\Bigr)_\perp;\\
Y_6&=&\mbox{the intersection of all uniformly closed sublattices of }X\\
&&\qquad\mbox{of codimension at most 1 that contain $Y$}.
\end{eqnarray*} \end{theorem}
\begin{proof}
It is clear that $Y_3\subseteq Y_4\subseteq Y_5$. Also, $Y_1=Y_3$
and $Y_2=Y_4$ by Corollary~\ref{span-perp} and Lemma~\ref{span-n},
respectively. By Proposition~\ref{sublat-codim-1}, $Y_5=Y_6$. We
will now show that $Y_5\subseteq Y_3$; this will complete the proof.
Suppose $z\in Y_5$. This means that if $h_1$ and $h_2$ in
$X^h$ agree on $Y$ then they also agree at $z$. In
particular, if $h\in X^h$ vanishes on $Y$ then
$h(z)=0$. Suppose now that
$f\in\bigl(\Span X^h\bigr)\cap Y^\perp$; we need to show that
$f(z)=0$. Let $f=\alpha_1h_1+\dots+\alpha_nh_n$, where
$h_1,\dots,h_n\in X^h$. For every $i=1,\dots,n$, if $h_i$
vanishes on $Y$ then $h_i(z)=0$, so we may assume that no $h_i$'s
vanishes on $Y$.
The restriction of every $h_i$ to $Y$ is a lattice homomorphism,
hence an atom in $Y^\sim$. Since any two atoms are either disjoint
or proportional and non-proportional atoms are linearly independent,
we may split the sum $\alpha_1h_1+\dots+\alpha_nh_n$ into groups
corresponding to non-proportional atoms in $Y^\sim$; then each group
vanishes on $Y$. Hence, without loss of generality, we may assume
that we have only one group, i.e., that ${h_i}_{|Y}$'s are all
proportional to some lattice homomorphism $g$ on $Y$, i.e.,
${h_i}_{|Y}=\beta_ig$. Since ${h_i}_{|Y}\ne 0$, we have $g\ne 0$ and
$\beta_i\ne 0$ for all $i$.
It follows from
\begin{math}
0=f_{|Y}=\sum_{i=1}^n\alpha_i{h_i}_{|Y}
=\Bigl(\sum_{i=1}^n\alpha_i\beta_i\Bigr)g
\end{math}
that $\sum_{i=1}^n\alpha_i\beta_i=0$. Also, for any $i\ne j$, the
restrictions of $\frac{1}{\beta_i}h_i$ and $\frac{1}{\beta_j}h_j$ to
$Y$ both equal $g$, hence are equal to each other. Since $z\in Y_5$
it follows that
$\frac{1}{\beta_i}h_i(z)=\frac{1}{\beta_j}h_j(z)$. This means that
$\frac{1}{\beta_i}h_i(z)$ does not depend on $i$; denote it
$\lambda$. It follows that
\begin{math}
f(z)=\sum_{i=1}^n\alpha_ih_i(z)=\sum_{i=1}^n\alpha_i\beta_i\lambda=0.
\end{math} \end{proof}
Unlike Theorem~\ref{CK-sublat-closure}, the sets in Theorem~\ref{closures} no longer equal the uniform closure of $Y$. It would be interesting to characterize vector lattices (in particular, Banach lattices) in which every uniformly closed sublattice $Y$ can be written as an intersection of uniformly closed sublattices of codimension one. In this case, the six sets in Theorem~\ref{closures} equal $\overline{Y}$. We will show in Theorem~\ref{codim-inters} that this is the case when $Y$ is of finite codimension. On the other hand, this fails when $X$ is a non-atomic order continuous Banach lattice as in this case $X$ has no closed sublattices of finite codimension (see Corollary~\ref{nonatom-ocont}), hence $Y_6=X$, even if $Y$ is a proper uniformly closed sublattice (even an ideal) of $X$.
\begin{question}
Suppose that $X^h$ separates the points of $X$. Does this imply that
every uniformly closed sublattice can be written as an intersection
of uniformly closed sublattices of codimension one? \end{question}
We will now extend Corollary~\ref{CK-sublat-ideal} to general vector latices.
\begin{lemma}\label{no-ideals}
Suppose that $X$ has a uniformly closed sublattice of
codimension $n$ such that it contains no ideals of $X$. Then $\dim
X\leqslant 2n$. \end{lemma}
\begin{proof}
Let $Y$ be such a sublattice. Suppose that $\dim X>2n$. Then we can
find linearly independent $x_1,\dots,x_{2n+1}$ in $X$. Put
$e=\bigvee_{i=1}^{2n+1}\abs{x_i}$. Then $x_i\in I_e$ as
$i=1,\dots,2n+1$, hence $\dim I_e>2n$. Clearly, $Y\cap I_e$ is a
sublattice of codimension at most $n$ in $I_e$, closed with respect
to $\norm{\cdot}_e$.
By Krein-Kakutani's Representation Theorem, we may identify $I_e$
with a dense sublattice of $C(K)$. Let $Z$ be the closure of
$Y\cap I_e$ in $C(K)$. By Lemma~\ref{codim-compl}, $Z$ is a closed
sublattice of $C(K)$ of codimension at most $n$. By
Theorem~\ref{CK-sublat-closure},
$Z=\bigcap_{i=1}^n\ker(\delta_{s_{2i-1}}-\alpha_i\delta_{s_{2i}})$
for some $s_1,\dots,s_{2n}$ in $K$ and $\alpha_i\geqslant 0$. Put
$J=\bigcap_{i=1}^{2n}\ker\delta_{s_i}$. Then $J$ is an ideal of
$C(K)$ contained in $Z$. It follows that $J\cap I_e$ is an ideal in
$I_e$, hence in $X$, contained in $Y$. By assumption,
$J\cap I_e=\{0\}$.
Let $t\notin\{s_1,\dots,s_{2n}\}$. Clearly, there exists $f\in C(K)$
such that $f$ vanishes on $\{s_1,\dots,s_{2n}\}$ but not at
$t$. Since $I_e$ is a dense sublattice of $C(K)$, we could use $Z_1$ in
Theorem~\ref{CK-sublat-closure} to find a function $g\in I_e$
which vanishes on $\{s_1,\dots,s_{2n}\}$ but not at $t$; this would
contradict $J\cap I_e=\{0\}$. It follows that such a $t$ does not
exist, i.e., $K=\{s_1,\dots,s_{2n}\}$. This yields $\dim
I_e\leqslant\dim C(K)\leqslant 2n$, which is a contradiction. \end{proof}
\begin{proposition}\label{codim-lat-ideal}
Every uniformly closed sublattice of $X$ of codimension $n$ contains
a uniformly closed ideal of $X$ of codimension at most $2n$. \end{proposition}
\begin{proof}
Let $Y$ be a uniformly closed sublattice of $X$ of codimension
$n$. Let $J$ be the union of all the ideals of $X$ contained in
$Y$. It is clear that $J$ is an ideal of $X$. Since the uniform
closure of an ideal is again an ideal by Theorem~63.1
in~\cite{Luxemburg:71}, it follows that $J$ is uniformly closed in
$X$. Therefore, the quotient $X/J$ is Archimedean by Theorem~60.2
in~\cite{Luxemburg:71}. Let $Q\colon X\to X/J$ be the quotient map.
Clearly, $Q(Y)$ is a sublattice of codimension $n$ in $X/J$.
It follows from Corollary~63.4 in~\cite{Luxemburg:71} (or can be
easily verified directly) that $QY$ is uniformly closed in
$X/J$. It follows from the maximality of $J$ in $Y$ that $Q(Y)$
contains no proper non-trivial ideals: if $H$ were such an ideal
that $Q^{-1}(H)$ would be an ideal of $X$ satisfying $J\subsetneq
Q^{-1}(H)\subsetneq Y$, which would be a contradiction.
It follows from Lemma~\ref{no-ideals} that $\dim(X/J)\leqslant 2n$ and,
therefore, $J$ has codimension at most $2n$ in $X$. \end{proof}
\begin{remark}
The preceding proposition has no infinite-codimensional analogue:
the (infinite codimensional) closed sublattice of $C[-1,1]$
consisting of all even functions contains no ideals. \end{remark}
\begin{remark}
The assumption that the sublattice is uniformly closed in
Proposition~\ref{codim-lat-ideal} cannot be dropped. Indeed, by
Proposition~\ref{sublats-exist}, $L_p[0,1]$ contains a sublattice of
codimension one. But $L_p[0,1]$ has no ideals of codimension 2 or of
any finite codimension; see Corollary~\ref{no-ideals-nonat}. \end{remark}
\begin{question}
Is the uniform closedness assumption in Lemma~\ref{no-ideals}
necessary? \end{question}
\begin{theorem}\label{codim-inters}
Every uniformly closed sublattice of $X$ of codimension $n$ may be
written as the intersection of $n$ uniformly closed sublattices of
codimension 1. \end{theorem}
\begin{proof}
Let $Y$ be a uniformly closed sublattice of $X$ of codimension
$n$. By Proposition~\ref{codim-lat-ideal}, $Y$ contains a uniformly
closed ideal $J$ of $X$ of codimension $m\leqslant 2n$. Let
$Q\colon X\to X/J$ be the quotient map. Clearly, $X/J$ is lattice
isomorphic to $\mathbb R^m$ and $Q(Y)$ is a sublattice of $X/J$. It
follows from Corollary~\ref{sublat-Rn} (or from
Theorem~\ref{CK-sublat-closure}) that $Q(Y)$ may be written as the
intersection of sublattices of codimension 1. Taking their
preimages, we get the required representation for $Y$. \end{proof}
Combining the theorem with Proposition~\ref{sublat-codim-1}, we get:
\begin{corollary}
Every uniformly closed sublattice of codimension $n$ in $X$ may be
written in the form $\bigcap_{i=1}^n\ker(\varphi_{2i-1}-\varphi_{2i})$, where
$\varphi_1,\dots,\varphi_{2n}$ are lattice homomorphisms in $X^\sim$. \end{corollary}
Note that some of the $\varphi_i$'s in the corollary may coincide or equal zero. Note also that taking $J=\bigcap_{i=1}^{2n}\ker\varphi_i$ yields a uniformly closed ideal of codimension at most $2n$ as per Proposition~\ref{codim-lat-ideal}.
Let $S$ and $T$ be lattice homomorphisms from $X$ to an Archimedean vector lattice $Z$ and put $Y=\ker(S-T)$. It follows from $Y=\{x\in X\::\: Sx=Tx\}$ that $Y$ is a sublattice; it is uniformly closed because $S-T$ is order bounded and, therefore, uniformly continuous.
\begin{corollary}
Uniformly closed sublattices of codimension at most $n$ in $X$ are
exactly the kernels of differences of two lattice homomorphisms from
$X$ to $\mathbb R^n$. \end{corollary}
\begin{proof}
Let $Y$ be a uniformly closed sublattice of codimension at most $n$
in $X$. Then $Y=\bigcap_{i=1}^n\ker(\varphi_i-\psi_i)$, where
$\varphi_i$ and $\psi_i$ are lattice homomorphisms in $X^\sim$ as
$i=1,\dots,n$. Define $S,T\colon X\to\mathbb R^n$ via
\begin{displaymath}
Sx=\bigl(\varphi_1(x),\dots,\varphi_n(x)\bigr)
\quad\mbox{and}\quad
Tx=\bigl(\psi_1(x),\dots,\psi_n(x)\bigr).
\end{displaymath}
It is easy to see that $S$ and $T$ are lattice homomorphisms and
$Y=\ker(S-T)$. The converse is trivial. \end{proof}
\begin{question}
Is every uniformly closed sublattice
the kernel of the difference of two lattice homomorphisms? \end{question}
We would like to mention two results that imply that many important classes of vector lattices lack uniformly closed sublattices of finite codimension. Recall that a sublattice $Y$ of $X$ is \term{order dense} if for every $0<x\in X$ there exists $y\in Y$ such that $0<y\leqslant x$.
\begin{proposition}[\cite{Wojtowicz:98}]
If $X$ is non-atomic then every finite codimensional sublattice is
order dense. \end{proposition}
\begin{proof}
Assume that $Y$ is a finite codimensional sublattice but is not
order dense. Then there exists $0<x\in X$ such that
$I_x\cap Y=\{0\}$. It follows that $\dim I_x<\infty$. Being a finite
dimensional vector lattice, $I_x$ and, therefore, $X$ admits an
atom; a contradiction. \end{proof}
Note that there exist proper finite codimensional uniformly closed sublattices that are order dense. For example, consider the sublattice of $C[0,1]$ consisting of all functions that vanish at $0$. However, no proper sublattice is both order closed and order dense. Recall that if $Y$ is an order dense sublattice then $x=\sup[0,x]\cap Y$ for every $x\in X_+$. It follows that an order dense sublattice of an order continuous Banach lattice has to be norm dense. We can now easily deduce the following result of \cite{Abramovich:90a}:
\begin{corollary}[\cite{Abramovich:90a}]\label{nonatom-ocont}
Let $X$ be a non-atomic order continuous Banach lattice. Then $X$
has no proper closed sublattices of finite codimension. \end{corollary}
We would like to mention an important corollary of the preceding result:
\begin{corollary}[\cite{Abramovich:90a,Wojtowicz:98}]
If $X$ is a non-atomic order continuous Banach lattice then $X^*$
is non-atomic. \end{corollary}
\begin{proof}
If $\varphi\in X^*$ is an atom, it is a lattice homomorphism, hence
$\ker\varphi$ is a closed ideal of codimension 1 in $X$. \end{proof}
\section{Finite codimensional ideals}
It was observed in II.5.3, Corollary~3 of~\cite{Schaefer:74} that every ideal of co-dimension 1 in a Banach lattice is closed. In~\cite{Abramovich:90a}, this result was extended to finite-codimensional ideals of F-lattices. We are now going to extend it to uniformly closed ideals.
\begin{proposition}
Every ideal of codimension 1 is uniformly closed. \end{proposition}
\begin{proof}
The quotient map may be viewed as a lattice homomorphism from $X$ to
$\mathbb R$. Now apply Corollary~\ref{u-subsp-codim-1}. \end{proof}
The following result and example are motivated by~\cite{Abramovich:90a}.
\begin{proposition}\label{idea-fcodim-ucl}
If $X$ is uniformly complete then every finite codimensional ideal
in $X$ is uniformly closed. \end{proposition}
\begin{proof}
Let $J$ be an ideal of finite codimension $n$.
Proof is by induction on $n$. For $n=1$, this is the preceding
proposition. Suppose $n>1$. We can then find an intermediate ideal
$H$ such that $J\subsetneq H\subsetneq X$: one can take the
preimage under the quotient map of any proper non-trivial ideal in
$X/J$. By induction hypothesis, $J$ is uniformly closed in $H$ and
$H$ is uniformly closed in $X$. Since uniform convergence in $H$
agrees with that inherited from $X$ by Proposition~2.12
in~\cite{Taylor:20} and the remark following it, we conclude that
$J$ is uniformly closed in $X$. \end{proof}
It is observed in Lemma~3 of \cite{Abramovich:90a} that for every ideal $J$ in $X$ of codimension $n$ there exists a chain of ideals $J=J_1\subsetneq J_2\subsetneq\dots\subsetneq J_n=X$.
\begin{example}
\emph{An ideal of codimension 2 that is not uniformly closed.} Let
$X$ be the vector lattice of all piece-wise affine functions on
$[0,1]$ and
\begin{math}
J=\bigl\{f\in X\::\:\exists r>0\ f\text{ vanishes on }[0,r]\bigr\}.
\end{math}
Clearly, $J$ is an ideal; its codimension equals 2 as
$X=J+\Span\{\mathbbold{1},f_0\}$, where $f_0(t)=t$. It can be easily verified
that $f_0$ is in the uniform closure of $J$, hence $J$ is
not uniformly closed.
Let $Y=\bigl\{f\in X\::\: f(0)=0\bigr\}$. Then
$J\subsetneq Y\subsetneq X$ is a chain of ideals, hence $J$ is
uniformly closed in $Y$ and $Y$ is uniformly closed in $X$, yet
$J$ is not uniformly closed in $X$. \end{example}
Since a subset of a Banach lattice is closed iff it is uniformly closed, Proposition~\ref{idea-fcodim-ucl} and Corollary~\ref{nonatom-ocont} easily yield the following two results from~\cite{Abramovich:90a}:
\begin{corollary}[\cite{Abramovich:90a}]
Every finite-codimensional ideal in a Banach lattice is closed. \end{corollary}
\begin{corollary}[\cite{Abramovich:90a}]\label{no-ideals-nonat}
A non-atomic order continuous Banach lattice admits no proper ideals
of finite codimension. \end{corollary}
Let $J$ be a closed ideal of codimension $n$ in $C(\Omega)$. By Theorem~\ref{CK-sublat-closure} and Corollary~\ref{closed-ideal}, there exist distinct $n$ points $t_1,\dots,t_n$ such that $f\in J$ iff $f$ vanishes at these points. That is, $J=\bigcap_{i=1}^n\ker\delta_{t_i}$. Note that $\delta_{t_i}$ is an ideal of codimension 1. This is generalized in the following proposition that we borrow from \cite{Abramovich:90a,Abramovich:90b}; we include it for the sake of completeness.
\begin{proposition}
Let $J$ be a uniformly closed ideal of $X$ of codimension $n$. There
exist unique (up to a permutation) ideals $H_1,\dots,H_n$ of
codimension $1$ such that $J=\bigcap_{i=1}^nH_i$. \end{proposition}
\begin{proof}
Consider the quotient map $Q\colon X\to X/J$.
Since $J$ is uniformly closed, $X/J$ is an Archimedean vector
lattice of dimension $n$, hence is lattice isomorphic to
$\mathbb R^n$. Therefore, there are exactly $n$ distinct ideals
$E_1,\dots,E_n$ of codimension $1$ in $X/J$ and
$\bigcap_{i=1}^nE_i=\{0\}$. Now take $H_i=Q^{-1}(E_i)$ as
$i=1,\dots,n$.
To prove uniqueness, let $H$ be an ideal of codimension 1 in $X$ such
that $J\subseteq H$. It is easy to see that $Q(H)$ is an ideal of
codimension 1 in $X/J$, hence $H=Q^{-1}(E_i)$ for some $i$. \end{proof}
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{ Quantum Computing with Very Noisy Devices }
\author{E. Knill \\[-6pt] \normalsize \emph{
Mathematical and Computational Sciences Division,}\\[-8pt] \normalsize
\emph{National Institute of Standards and Technology, Boulder, CO 80305} }
\date{} \maketitle
\textbf{ In theory, quantum computers can efficiently simulate quantum physics, factor large numbers and estimate integrals, thus solving otherwise intractable computational problems. In practice, quantum computers must operate with noisy devices called ``gates'' that tend to destroy the fragile quantum states needed for computation. The goal of fault-tolerant quantum computing is to compute accurately even when gates have a high probability of error each time they are used. Here we give evidence that accurate quantum computing is possible with error probabilities above $3\,\%$ per gate, which is significantly higher than what was previously thought possible. However, the resources required for computing at such high error probabilities are excessive. Fortunately, they decrease rapidly with decreasing error probabilities. If we had quantum resources comparable to the considerable resources available in today's digital computers, we could implement non-trivial quantum computations at error probabilities as high as $1\,\%$ per gate. }
\ignore{ Remaining terminological issues: * Two architectures: Postselected and error-correcting $C_4/C_6$, clarify terminology and use it? * ``postselected threshold''? }
Research in quantum computing is motivated by the great increase in computational power offered by quantum computers~\cite{shor:qc1995a,feynman:qc1982a,abrams:qc1999b}. There is a large and still growing number of experimental efforts whose ultimate goal is to demonstrate scalable quantum computing. Scalable quantum computing requires that arbitrarily large computations can be efficiently implemented with little error in the output. Criteria that need to be satisfied by devices used for scalable quantum computing have been specified by DiVincenzo~\cite{divincenzo:qc2000a}. One of the criteria is that the level of noise affecting the physical gates is sufficiently low. The type of noise affecting the gates in a given implementation is called the ``error model''. A scheme for scalable quantum computing in the presence of noise is called a ``fault-tolerant architecture''. In view of the low-noise criterion, studies of scalable quantum computing involve constructing fault-tolerant architectures and providing answers to questions such as the following: Q1: Is scalable quantum computing possible for error model ${\cal E}$? Q2: Can fault-tolerant architecture ${\cal A}$ be used for scalable quantum computing with error model ${\cal E}$? Q3: What resources are required to implement quantum computation ${\cal C}$ using fault-tolerant architecture ${\cal A}$ with error model ${\cal E}$?
To obtain broadly applicable results, fault-tolerant architectures are constructed for generic error models. Here, the error model is parametrized by an error probability per gate (or simply error per gate, EPG), where the errors are unbiased and independent. The fundamental theorem of scalable quantum computing is the threshold theorem and answers question Q1 as follows: If the EPG is smaller than a threshold, then scalable quantum computing is possible~\cite{preskill:qc1998a,kitaev:qc1997a,aharonov:qc1996a,knill:qc1998a}. Thresholds depend on additional assumptions on the error model and device capabilities. Estimated thresholds vary from below $10^{-6}$ [\citeonline{preskill:qc1998a,kitaev:qc1997a,aharonov:qc1996a,knill:qc1998a}] to $3\times 10^{-3}$ [\citeonline{steane:qc2002a}], with $10^{-4}$ [\citeonline{gottesman:qc1997b}] often quoted as the EPG to be achieved in experimental quantum computing.
Many experimental proposals for quantum computing claim to achieve EPGs below $10^{-4}$ in theory. However, in the few cases where experiments with two quantum bits (qubits) have been performed, the EPGs currently achieved are much higher, $3\times 10^{-2}$ or more in ion traps~\cite{leibfried:qc2003a,roos:qc2004a} and liquid-state NMR~\cite{knill:qc1999a,childs:qc2001b}. The first goal of our work is to give evidence that scalable quantum computing is possible at EPGs above $3\times 10^{-2}$. While this is encouraging, the fault-tolerant architecture that achieves this is extremely impractical because of large resource requirements. To reduce the resource requirements, lower EPGs are required. The second goal of our work is to give a fault-tolerant architecture (called the ``$C_4/C_6$ architecture'') well suited to EPGs between $10^{-4}$ and $10^{-2}$ and to determine its resource requirements, which we compare to the state of the art in scalable quantum computing as exemplified by the work of Steane~\cite{steane:qc2002a}.
Fault-tolerant architectures realize low-error qubits and gates by encoding them with error-correcting codes. A standard technique for amplifying error reduction is concatenation. Suppose we have a scheme that, starting with qubits and gates at one EPG, produces encoded qubits and gates that have a lower EPG. Provided the error model for encoded gates is sufficiently well behaved, we can then apply the same scheme to the encoded qubits and gates to obtain a next level of encoded qubits and gates with much lower EPGs. Thus, a concatenated fault-tolerant architecture involves a hierarchy of repeatedly encoded qubits and gates. The hierarchy is described in terms of levels of encoding, with the physical qubits and gates being at level $0$. The top level is used for implementing quantum computations and its qubits, gates, EPGs, etc.~are referred to as being ``logical''. Typically, the EPGs decrease superexponentially with number of levels, provided that the physical EPG is below the threshold for the architecture in question.
The $C_4/C_6$ architecture differs from previous ones in five significant ways. First, we use the simplest possible error-detecting codes, thus avoiding the complexity of even the smallest error-correcting codes. Error correction is added naturally by concatenation. Second, error correction is performed in one step and combined with logical gates by means of error-correcting teleportation. This minimizes the number of gates contributing to errors before they are corrected. Third, the fault-tolerant architecture is based on a minimal set of operations with only one unitary gate, the controlled-NOT. Although this set does not suffice for universal quantum computing, it is possible to bootstrap other gates. Fourth, verification of the needed ancillary states (logical Bell states) largely avoids the traditional syndrome-based schemes. Instead, we use hierarchical teleportations. Fifth, the highest thresholds are obtained by introducing the model of postselected computing with its own thresholds, which may be higher than those for standard quantum computing. Our fault tolerant implementation of postselected computing has the property that it can be used to prepare states sufficient for (standard) scalable quantum computing.
\phead{Basics.} For an introduction to quantum information, computing and error correction, see~[\citeonline{nielsen:qc2001a}]. The unit of quantum information is the qubit whose states are superpositions $\alpha\ket{0}+\beta\ket{1}$. Qubits are acted on by the Pauli operators $X=\sigma_x$ (bit flip), $Z=\sigma_z$ (sign flip) and $Y=\sigma_y=i\sigma_x\sigma_z$. The identity operator is $I$. One-qubit gates include preparation of $\ket{0}$ and $\ket{+}=(\ket{0}+\ket{1})/\sqrt{2}$, $Z$-measurement (distinguishing between $\ket{0}$ and $\ket{1}$), $X$-measurement (distinguishing between $\ket{+}$ and $\ket{-}=(\ket{0}-\ket{1})/\sqrt{2}$), and the Hadamard gate (HAD, $\alpha\ket{0}+\beta\ket{1}\mapsto\alpha\ket{+}+\beta\ket{-}$). We use one unitary two-qubit gate, the controlled-NOT (CNOT), which maps $\ket{00}\mapsto \ket{00}$, $\ket{01}\mapsto\ket{01}$, $\ket{10}\mapsto\ket{11}$, and $\ket{11}\mapsto\ket{10}$. This set of gates is a subset of the so-called Clifford gates, which are insufficient for universal quantum computing~\cite{gottesman:qc1997b}. Our minimal gate set $\cG_{\textnormal{\tiny min}}$ consists of $\ket{0}$ and $\ket{+}$ preparation, $Z$ and $X$ measurement and CNOT. Universality may be achieved with the addition of other one-qubit preparations or measurements, as explained below. The physical gates mentioned are treated as being implemented in one ``step''; the actual implementation may be more complex.
The $C_4/C_6$ architecture is based on two error-detecting stabilizer codes, $C_4$, a four-qubit code, and $C_6$, a three qubit-pair code, both encoding a qubit pair. A stabilizer code is a common eigenspace of a set of commuting products of Pauli operators (the ``check operators''). Such products are denoted by strings of $X$, $Y$, $Z$ and $I$. For example, $XIZ$ is a Pauli product for three qubits with $X$ acting on the first and $Z$ on the last. The shortest error-detecting code $C_4$ for qubits encodes two qubits in four and has check operators $XXXX$ and $ZZZZ$. The encoded qubits (labeled $L$ and $S$) are defined by encoded operators $X_L=XXII$, $Z_L=ZIZI$, $X_S=IXIX$ and $Z_S=IIZZ$. We use this code as a first level of encoding and call the encoded qubits ``level $1$'' qubits. Level $1$ qubits come in pairs, each encoded in a ``block'' of four physical qubits. The second code $C_6$ is constructed as a code on three qubit pairs able to detect any error acting on one pair. It encodes a qubit pair and has check operators $XIIXXX$, $XXXIIX$, $ZIIZZZ$ and $ZZZIIZ$ acting on three consecutive qubit pairs. A choice for encoded operators for $C_6$ is $X_L = IXXIII, Z_L = IIZZIZ, X_S = XIXXII, Z_S = IIIZZI$. This code is used for the second and higher levels of encoding. For example, the second level is obtained by using three level one pairs to obtain a level two pair. A level $l$ qubit pair requires a block of $4\times 3^{l-1}$ physical qubits. The block structure is depicted in Fig.~\ref{fig:c4/c6_struct}.
The concatenation of error-detecting codes allows for a flexible use of error detection and correction. Given a joint eigenstate of the check operators, its list of eigenvalues is called the ``syndrome''. The level $l$ encoding has check operators that can be derived from the check and encoded operators of $C_4$ and $C_6$. Ideally, the state of a level $l$ qubit exists in the subspace with syndrome $\mathbf{0}$ (all eigenvalues are $+1$). In the presence of errors this is typically not the case, so the state is defined only with respect to a current ``Pauli frame'' and an implicit recovery scheme. The Pauli frame is defined by a Pauli product that restores the error-free state of the block to the syndrome $\mathbf{0}$ subspace. The implicit recovery scheme determines the Pauli products needed to coherently map states with other syndromes to the syndrome of the error-free state. Defining the level $l$ state in this way makes it possible to avoid explicitly applying Pauli products for correction or teleportation compensation~\cite{steane:qc2002a}. Error-detection and correction are based on measurements that retroactively determine the syndrome of the state (the current syndrome has already been affected by further errors). An error is detected when the syndrome differs from that expected according to the Pauli frame. In ``postselected'' quantum computing, the state is then rejected and the computation restarted. In standard quantum computing, the syndrome information must be used to update the Pauli frame. With the $C_4/C_6$ architecture, it is possible to do so at level 2 and above by the following method: First check the level 1 $C_4$ syndromes of each block of four qubits. For each block where an error is detected, mark the encoded level 1 qubit pair as having an error. Proceed to level 2 and check the (encoded) $C_6$ syndrome for each block of three level 1 pairs. If exactly one of the level 1 pairs has an error, use the $C_6$ syndrome to correct it. This works because error-detecting codes can always correct an error at a known location. If not, mark the encoded level 2 pair as having an error unless none of the three level 1 pairs have an error and the $C_6$ syndrome is as expected according to the Pauli frame. Continue in this fashion through all higher levels. For optimizing state preparation, we can replace the error-correction step by error-detection at the top few levels depending on context as explained below.
\phead{Error model and assumptions.} All error models can be described by inserting errors (which act as quantum operations) after gates or before measurements. We could model correlations between the errors by extending the errors' quantum operations to a common external environment. However, here we assume that errors are independent. We further assume that a gate's errors consist of applications of Pauli products with probabilities determined by the gate. Ideally, we would obtain a threshold that does not depend on the details of the probability distributions of Pauli products. This is too difficult with available techniques, so a \emph{depolarizing} model is assumed for each gate: $\ket{0}$ ($\ket{+}$) state preparation erroneously produces $\ket{1}$ ($\ket{-}$) with probability $e_p$. A binary (e.g. $Z$ or $X$) measurement results in the wrong outcome with probability $e_m$. CNOT is followed by one of the $15$ possible non-identity Pauli products, each with probability $e_c/15$. HAD is modified by one of the Pauli operators, each with probability $e_h/3$. We further simplify by setting $e_c=\gamma$, $e_m=e_p= 4\gamma/15$, $e_h=4\gamma/5$. This choice is justified as follows: $4\gamma/5$ is the one-qubit marginal probability of error for the CNOT, and it is reasonable to expect that one-qubit gates have error below this. In fact, one-qubit gates have much lower error than CNOTs in experimental systems such as ion-traps and liquid-state NMR. As for preparation errors, if they are much larger than $4\gamma/15$, then it is possible to purify prepared states using a CNOT. For example, prepare $\ket{0}$ twice, apply a CNOT from the first to the second and $Z$ measure the second. Try again if the measurement outcome indicates $\ket{1}$, otherwise use the first state. The probability of error is given by $4\gamma/15 + O(\gamma^2)$, assuming that CNOT error is as above and measurement and preparation errors are proportional to $\gamma$. This also works for $\ket{+}$ preparation. \ignore{ The only order 1 (in $\gamma$) way to accept the wrong state involves having CNOT errors of the form
$(X|Y)(X|Y)$, which gives four possibilities. } To improve $Z$ measurement, it is necessary to introduce an ancilla in $\ket{0}$, apply a CNOT from the qubit to be measured to the ancilla, and measure both qubits, accepting the answer only if the measurements agree. The error probability conditional on acceptance is again $4\gamma/15+O(\gamma^2)$. Detected error is much more readily managed than undetected error~\cite{knill:qc2003b}. In our architecture, the primary role of state preparation implies that the conditional error is typically more relevant. To improve measurement without possibility of rejection requires an additional ancilla~\cite{divincenzo:qc2000a} and CNOT with majority decoding of the three measurement outcomes. However, the error probability is now $4\gamma/5 + O(\gamma^2)$. \ignore{ With ancillas numbered $2$ and $3$, use CNOT($1\rightarrow 2)$ followed by CNOT($1\rightarrow 3)$. If the state starts out as $\ket{0}$, The following outputs of weight $>1$ have probability $O(\gamma)$: $101$ ($8\gamma/15$), $111$ ($4\gamma/15$). }
The error model used here is idealized and does not match the error behavior of physical qubits and gates. There are three notable differences. First, real errors include coherent rotations, but any error can still be expressed as a linear combination of Pauli products. The syndrome measurements serve to ``collapse'' the linear combinations, so that these errors can be managed. The main problem with such errors is that consecutive errors can add coherently rather than probabilistically, resulting in more rapid error propagation. In principle, this problem can be eliminated by frequently applying known but random Pauli products, thus modulating the Pauli frame and reducing the likelihood of coherent addition. This also has the beneficial effect of decoupling~\cite{kern:qc2004a} weakly interacting environments. The second difference is that real errors on gates nearby in time and space have correlations. These correlations are expected to decay rapidly with distance, and their effect can be alleviated by known coding techniques such as block interleaving~\cite{steane:qc2002a}. The third difference is that in many cases, qubits are defined by a subspace of a quantum system from which amplitude can leak. An example of this problem is photon loss in optical quantum computing. Leakage errors, particularly if undetected, can be problematic. One advantage of error-correcting teleportation is that leakage is automatically controlled at each step.
The error model does not specify ``memory'' error or the amount of time used by a measurement~\cite{steane:qc2002a}. We assume that gates other than measurements take the same amount of time; that is, the error parameter should represent the total error including any delays for faster gates to equalize gate times. For the $C_4/C_6$ architecture, memory is an issue only when waiting for measurement outcomes that determine whether prepared states are good, or that are needed after teleportation, particularly when implementing non-Clifford gates~\cite{gottesman:qc1999a,zhou:qc2000b}. If the architecture is used for postselected computing, we can compute ``optimistically'', anticipating but not waiting for measurement outcomes. The output of the computation is accepted only if all measurement outcomes are as anticipated. Consider standard quantum computing with maximum parallelism. In teleportation, Bell measurements determine correction gates that need to be applied. If the correction gates are simple Pauli products, they can be absorbed into the Pauli frame and do not need to be known immediately. For teleportations used to implement non-Clifford gates, a non-Pauli compensation may be required. In this case, we must wait for the measurement outcomes. To avoid accumulating memory errors, we can maintain the logical state by repeatedly applying error-correcting teleportations with delays whose memory error is equivalent to that of physical one-qubit gates. The logical errors for these steps are comparable to logical one-qubit gates, which are small by design. The measurement outcomes for the teleportations determine only Pauli-frame updates and need not be known immediately. In state preparation, using additional resources, we may continue computing optimistically without waiting for measurement outcomes until the state is ready to be used for a logical gate. At this point it is necessary to wait for measurement outcomes to make sure the prepared state has no detected uncorrectable error. This adds at most the memory error incurred during a measurement time to each qubit. To account for this we assume that gate errors are set high enough to include this memory error.
Two additional assumptions are used in analyzing the $C_4/C_6$ architecture: The first is that there is no error and no speed constraint on classical computations required to interpret measurement outcomes and control future gates. The second is that two-qubit gates can be applied to any pair of qubits without delay or additional error. This assumption is unrealistic, but the effect on the threshold is due primarily to relatively short-range CNOTs acting within the ancillas needed for maintaining one or two blocks. This may be accounted for by use of a higher effective EPG~\cite{steane:qc2002a,svore:qc2004a}.
\phead{Clifford gates for $C_4$ and $C_6$.} The codes $C_4$, $C_6$ and their concatenations have the property that encoded CNOTs, HADs and measurements that act in parallel on both encoded qubits in a pair can be implemented ``transversally'' with physical qubit relabeling. For example, to apply an encoded CNOT between two encoded qubit pairs, it suffices to apply physical CNOTs transversally, that is between corresponding physical qubits in the encoding blocks. HAD requires in addition permuting the physical qubits in a block, which can be done by relabeling without physical manipulations or error, see the supplementary information (S.I. Sect.~\ref{sect:networks}). The transversal implementation ensures that errors from the physical gates apply independently to each physical qubit in a block so that they can be managed by error detection or correction. We use two methods for encoded state preparation. The first yields low-error encoded $\ket{0}$ and $\ket{+}$ states and follows from the Bell state-preparation scheme needed for error correction (S.I. Sect.~\ref{sect:networks}). The second uses teleportation to ``inject'' physical states into encoded qubits. The resulting encoded state has error but can be purified.
\phead{Error-correcting teleportation.} To correct (more accurately, to keep track of) errors, we use error-correcting teleportation, which generalizes gate teleportation~\cite{gottesman:qc1999a}. It involves preparing two blocks, each encoding a logical qubit pair so that the first pair is uniformly entangled with the second. Two such blocks form a ``logical Bell pair'', and its logical state is the ``logical Bell state''. Suppose that a logical Bell pair's error is as if each physical qubit were subject to independent error of order $\gamma$. A block used for logical computation can then be error-corrected by applying Bell measurements transversally between corresponding qubits in the computational block and the first block of the logical Bell pair. This is the first step of conventional quantum teleportation~\cite{bennett:qc1993b} and results in the transfer of the logical state to the second block of the logical Bell pair, up to a known change in the Pauli frame. The Bell measurement outcomes reveal the syndromes of the products of identical check operators on the two blocks. Provided that the combined errors from the two measured blocks are within the limits of what the codes used can handle, they can be determined to update the Pauli frame (S.I. Sect.~\ref{sect:teleportation}). Compared to the syndrome extraction methods of Steane~\cite{steane:qc2002a}, error-correcting teleportation involves only one step instead of at least two but requires preparing more complex states.
\phead{Logical Bell state preparation.} State preparation networks are detailed in S.I Sect.~\ref{sect:networks}. It is necessary to prepare logical Bell states so that any errors introduced are similar to independent physical one-qubit errors. We prepare such states by constructing encoded Bell states at each level, using them as a resource for constructing Bell states at the next level. An encoded Bell state can be obtained by preparing and verifying encoded $\ket{++}\ket{00}$ in two encoded qubit pairs and applying an encoded CNOT from the first to the second. The encoded CNOT is applied transversally but can introduce correlations between the first and second block. To limit these correlations to the current level, the subblocks are teleported using lower-level Bell states with error detection or correction depending on context. The remaining correlations do not appear to significantly affect logical errors but can be reduced by purification~\cite{duer:qc2003a} or by entanglement swapping~\cite{zukowski:qc1993a} with two encoded Bell states. A key observation for preparing encoded $\ket{00}$ and $\ket{++}$ is that for both $C_4$ and $C_6$, they are close to cat states such as $(\ket{0\ldots 0}+\ket{1\ldots 1})/\sqrt{2}$. In the case of $C_4$, encoded $\ket{00}$ and $\ket{++}$ are cat states on four qubits. For $C_6$, they are parallel three-qubit cat states on three qubit pairs modified by internal CNOTs on two of the pairs. To prepare verified cat states we use a minimal variant of the methods of Shor~\cite{shor:qc1996a} starting with Bell states. For the concatenations used here, the internal CNOTs can be implemented by relabeling the physical qubits.
\phead{Low error logical $\cG_{\textnormal{\tiny min}}$ gates.} The first step in establishing fault tolerance of the $C_4/C_6$ architecture is to implement logical $\cG_{\textnormal{\tiny min}}$ gates with low EPGs. For the purpose of establishing high thresholds, we first consider postselected $\cG_{\textnormal{\tiny min}}$ computing. Postselected computing is like standard quantum computing except that when a gate is applied, the gate may fail. If it fails, this is known. The probability of success must be non-zero. There may be gate errors conditional on success, but fault-tolerant postselected computing requires that such errors are small. Purely postselected computing has little computational power, but we can use it to prepare states needed to enable scalable quantum computing. A fault-tolerant architecture for postselected computing can be implemented by use of the $C_4/C_6$ architecture without error correction, aborting the computation whenever an error is detected. We have used two methods to determine threshold values for $\gamma$ below which fault-tolerant postselected $\cG_{\textnormal{\tiny min}}$ computing is possible. The first involves a computer-assisted heuristic analysis of the conditional errors in prepared encoded Bell pairs. The analysis is described for a $C_4$ architecture in~[\citeonline{knill:qc2004b}]. It requires that encoded Bell pairs are purified to ensure that errors are approximately independent between each Bell pair's two blocks. We obtained exact conditional errors for level $1$ encoded Bell pairs and then heuristically bounded them from above with an error model that is independent between the two blocks. This independence implies that the error model for gates at the next level also satisfies strict independence, so the process can be repeated at each level to bound the conditional logical errors. With this analysis, thresholds of above $\gamma=0.03$ were obtained. The second method involves direct simulation of the error behavior of postselected encoded CNOTs with error-detecting teleportation at up to two levels of encoding and physical EPGs of $.01 \leq \gamma \leq .0375$. The simulation method is outlined in S.I. Sect.~\ref{sect:simulation}. The resulting conditional logical errors are shown in Fig.~\ref{fig:condEPG} and suggest a threshold of above $\gamma = 0.06$ by extrapolation. At $\gamma=0.03$, the logical preparation and measurement errors were found to be consistent with being below the threshold.
Scalable $\cG_{\textnormal{\tiny min}}$ computing with the $C_4/C_6$ architecture requires lower EPGs and the use of error correction to increase the probability of success to near $1$. To optimize the resource requirements needed to achieve a given logical EPG, the last level at which error correction is used is $dl$ levels below the relevant top level, where $dl$ depends on context and $\gamma$. At higher levels, errors are only detected. For simplicity and to enable extrapolation by modeling, we examined a fixed strategy with $dl=1$ in all state-preparation contexts and $dl=0$ (maximum error correction) in the context of logical computation. The relevant top level in a state preparation context is the level of a block measurement or error-correcting teleportation of a subblock, not the logical level of the state that is eventually prepared. Each logical gate now has a probability of detected but uncorrectable error, and a probability of logical error conditional on not having detected an error. Fig.~\ref{fig:ecEPG} shows both error probabilities up to level $4$ for a logical CNOT with error-correcting teleportation and EPGs $\gamma \leq 0.01$. The data indicate that the threshold for this architecture is above $0.01$. The logical preparation and measurement errors were found to be comparatively low.
In designing and analyzing fault-tolerant architectures, particularly those based on concatenation, care must be taken to ensure that logical errors do not have correlations that lead to larger than expected errors when gates are composed. Such effects can be missed when inferring thresholds from analysis or simulation of just one level of concatenation. An additional complication is that the $C_4/C_6$ architecture's level $l+1$ gates are not implemented solely in terms of level $l$ gates. We therefore simulated the architecture at the highest levels possible. To verify that logical errors are sufficiently uncorrelated, we simulated sequential teleportation and checked the incremental error behavior of each step as shown in Fig.~\ref{fig:ecChain}.
\phead{Universal computation.} To complete the $\cG_{\textnormal{\tiny min}}$ gate set so that we can implement arbitrary quantum computations, it suffices to add HAD and preparation of the state $\ket{\pi/8}=\cos(\pi/8)\ket{0}+\sin(\pi/8)\ket{1}$~\cite{knill:qc1997a,knill:qc2004a}. We treat the qubits in a logical qubit pair identically and ignore one of them for the purpose of computation. See S.I. Sect.~\ref{sect:universal} for how to take full advantage of both qubits. The logical HAD is implemented similarly to the logical CNOT and uses one error-correcting teleportation. Its logical errors are are less than those of the logical CNOT. To prepare logical $\ket{\pi/8}$ in both qubits of a logical qubit pair, we obtain a logical Bell pair, decode the first block of the Bell pair into two physical qubits and make measurements to project the physical qubits' states onto $\ket{\pi/8}$ or the orthogonal state. If an orthogonal state is obtained, we adjust the Pauli frame by $Y$'s accordingly. Because of the entanglement between the physical qubits and the logical ones, this prepares the desired logical state, albeit with error. This procedure is called ``state injection''. To decode the first block of the Bell pair, we first decode the $C_4$ subblocks and continue by decoding six-qubit subblocks of $C_6$. Syndrome information is obtained in each step and can be used for error detection or correction. The error in decoding is expected to be dominated by the last decoding steps. Consequently, the error in the injected state should be bounded as the number of levels increase, which we verified by simulation to the extent possible. To remove errors from the injected states, logical purification can be used~\cite{bravyi:qc2004a,knill:qc2004a} and is effective if the error of the injected state is less than $0.141$~\cite{bravyi:qc2004a}. The purification method can be implemented fault tolerantly to ensure that the purified logical $\ket{\pi/8}$ states have errors similar to those of logical CNOTs (S.I. Sect.~\ref{sect:resources}). To simplify the implementation of quantum computations, other states can be prepared similarly.
Consider the threshold for postselected universal quantum computing. The logical HAD and injection errors at $\gamma=0.03$ and level $2$ are shown in Fig.~\ref{fig:condEPG}. The injection error is well below the maximum allowed and is not expected to increase substantially for higher levels. The injection error should scale approximately linearly with EPG, so the extrapolated threshold of $\gamma \geq 0.06$ may apply to universal postselected quantum computing.
The injection and purification method for preparing states needed to complete the gate set works with the error-correcting $C_4/C_6$ architecture. Consider state injection at $\gamma=0.01$. The context for injection is state preparation, which determines the combination of error-correction and detection as discussed above. The conditional logical error after state injection was determined to be $\expdata{8.6\aerrb{0.6}{0.5}\times 10^{-3}}{cliff/finaldata/injeec_l3_be.tex[0.01,:]}$ at level 3 and $\expdata{1.1\aerrb{0.1}{0.1}\times 10^{-2}}{cliff/finaldata/injeec_l4_be.tex[0.01,:]}$ at level 4, comparable to $\gamma$ and sufficiently low for $\ket{\pi/8}$ purification. As a result, the $C_4/C_6$ architecture enables scalable quantum computing at EPGs above $0.01$. To obtain higher thresholds, we use fault-tolerant postselected computing to prepare states in a code that can handle higher EPGs than $C_4/C_6$ concatenated codes can. The states are chosen so that we can implement a universal set of gates by error-correcting teleportation. Suppose that arbitrarily low logical EPGs are achievable with the $C_4/C_6$ architecture for universal postselected computing. To compute scalably, we choose a sufficiently high level $l$ for the $C_4/C_6$ architecture and a very good error-correcting quantum code $C_e$. The first step is to prepare the desired $C_e$-encoded states using level $l$ encoded qubits, in essence concatenating $C_e$ with level $l$ of the $C_4/C_6$ architecture. The second step is to decode each block of the $C_4/C_6$ architecture to physical qubits to obtain unconcatenated $C_e$-logical states. Once these states are successfully prepared, they can be used to implement each logical gate by error-correcting teleportation. Simulations show that the postselected decoding introduces an error $\lesssim\gamma$ for each decoded qubit (Fig.~\ref{fig:condEPG}). There is no postselection in error-correcting teleportation with $C_e$, and it is sensitive to decoding error in two blocks ($\approx 2\gamma$) as well as the error of the CNOT ($\approx\gamma$) and the two physical measurements ($\approx 8\gamma/15$) required for the Bell measurement. Hence, the effective error per qubit that needs to be corrected is $\approx 3.53\gamma$. The maximum error probability per qubit correctable by known codes $C_e$ is $\approx 0.19$~[\citeonline{divincenzo:qc1998c}]. Provided that $3.53\gamma \lesssim 0.19$ and $\gamma$ is below the postselected threshold for the $C_4/C_6$ architecture, the error in the state preparation before decoding together with the logical error in error-correcting teleportation can be made smaller than $10^{-3}$ (S.I. Sect.~\ref{sect:resources}). The $C_e$ architecture can therefore be concatenated with the error-correcting $C_4/C_6$ architecture to arbitrarily reduce the logical EPG. In view of the postselected threshold indicated by Fig.~\ref{fig:condEPG}, scalable quantum computing is possible at $\gamma = 0.03$ and perhaps up to $\gamma \approx 0.05$. Although the postselection overheads are extreme, this method is theoretically efficient.
\phead{Resources} The resource requirements for the error-correcting $C_4/C_6$ architecture can be mapped out as a function of $\gamma$ for different sizes of computations. Since we do not have analytical expressions for the resources for logical Bell state preparation or for the logical errors as a function of $\gamma$ and, with our current capabilities, we are not able to determine them in enough detail by simulation, we use naive models to approximate the needed expressions. The resources required are related to the number of physical CNOTs used, which dominates the number of state state preparations and measurements. HADs are used only for universality at the logical level. The number of physical CNOTs used in a logical Bell state preparation is modeled by functions of the form $C/(1-\gamma)^k$, which would be correct on average if the state-preparation network had $C$ gates of which $k$ failed independently with probability $\gamma$, and the network were repeatedly applied until none of the $k$ gates fail. $C$ and $k$ depend on the level of concatenation. The logical error probabilities are modeled at level $l\geq 1$ by $p_d(l) = d(l)\gamma^{f(l+1)}$ (detected error) and $p_c(l) = c(l)\gamma^{f(l+2)}$ (conditional logical error), where $f(0)=0,f(1) = 1, f(l+1) = f(l)+f(l-1)$ is the Fibonacci sequence. These expressions are asymptotically correct as $\gamma\rightarrow 0$. We verified that they model the desired values well and determined the constants at the lower levels by simulation (S.I. Sect.~\ref{sect:resources}). At high levels, the constants were estimated by extrapolating their level-dependent behavior. Using these expressions, we determined the level of concatenation that requires the fewest resources to implement a computation of a given size. The resulting resource graph is shown in Fig.~\ref{fig:resource_graph}.
Since interesting quantum computations use many non-Clifford gates, it is necessary to estimate the average resources required for preparing states such as the $\ket{\pi/8}$ state. One instance of this state suffices for implementing a $45{}^{\circ}$ $Y$ rotation. Two are required for a phase-variant of a Toffoli gate. Consider $\gamma=0.01$. At level $4$ of the $C_4/C_6$ architecture, one purification stage requires $\approx\expdata{370}{$\ket{\pi/8}$ state preparation overhead determined in S.I.}$ logical CNOTs (S.I. Sect.~\ref{sect:resources}). It is likely that this overhead can be significantly improved, but it must be accounted for when using the graphs of Fig.~\ref{fig:resource_graph} as discussed in the caption. It is possible to implement a computation with $100$ logical qubits and up to $1000$ $\ket{\pi/8}$-preparations using $\expdata{1.23\times 10^{14}}{see S.I.}$ physical CNOTs (S.I. Sect.~\ref{sect:resources}). This takes into account the probability that the computation fails with a detected error. The conditional probability of obtaining an incorrect output is $\approx\expdata{0.02}{see S.I.}$. Such a computation is non-trivial in the sense that its output is not efficiently predictable using known classical algorithms. The resource requirements are large but would be reasonable in the context of classical computing: Central processing units have $10^8$ or more transistors operating at rates faster than $10^9$ bit operations per second~\cite{intel.com:qc2004a}.
We compare our resource requirements to those of Steane's architecture based on an example at $\gamma=10^{-4}$ detailed in~[\citeonline{steane:qc2002a}]. Steane's architecture is based on non-concatenated block codes, which are expected to be more efficient at such low EPGs~\cite{preskill:qc1998a}. Steane's example has an effective logical error per qubit of $\approx 7\times 10^{-12}$ using $\approx 420$ physical CNOTs per qubit per gate. Our architecture achieves detected errors of $\expdata{5.5\times 10^{-9}}{see S.I.}$ (level $3$) or $\expdata{6\times 10^{-14}}{see S.I.}$ (level $4$) using respectively $\approx\expdata{2100}{see S.I.}$ or $\expdata{2.6\times 10^4}{see S.I.}$ physical CNOTs per qubit (S.I. Sect.~\ref{sect:resources}). The conditional logical errors are much smaller. The $C_4/C_6$ architecture's resource requirements are still within two orders of magnitude of Steane's at $\gamma=10^{-4}$. The $C_4/C_6$ architecture has the advantage of simplicity, of yielding more reliable answers conditional on having no detected errors, and of operating at higher EPGs.
\phead{Discussion.} How high must EPGs be so that it is not possible to scalably quantum compute? It is known that if unbiased one-qubit EPGs exceed $.5$, then we can simulate the effect of gates classically~\cite{bennett:qc1999c,harrow:qc2003a}. Furthermore, if one-qubit EPGs exceed $0.25$, then we cannot realize a quantum computation ``faithfully'', that is by encoding the computation's qubits with quantum codes~\cite{bennett:qc1996a,bruss:qc1998a}. This is because the quantum channel capacity vanishes at a depolarizing error probability above $0.25$. Faithful techniques are likely to require at least three sequential gates before an error can be eliminated (in our case these are preparation gates whose errors remain in the logical Bell pairs, a CNOT and a measurement for teleportation). Thus one would not expect to obtain thresholds above $\sim 0.09$ using faithful methods. This is not far from the extrapolated $0.05$ evidenced by our work. Note that the thresholds obtained here are similar to those for quantum communication~\cite{briegel:qc1999a}.
An important use of studies of fault-tolerant architectures is to provide guidelines for EPGs that should be achieved to meet the low-error criterion for scalability. Such guidelines should depend on the details of the relevant error models and constraints on two-qubit gates. Nevertheless, the value of $\gamma=10^{-4}$ has often been cited as the EPG to be achieved. With architectures such as Steane's~\cite{steane:qc2002a,reichardt:qc2004a} and the one introduced here, resource requirements at $\gamma=10^{-3}$ are now comparable to what they were for $\gamma=10^{-4}$~[\citeonline{steane:qc1997a}] at the time this value was starting to be cited.
\ignore{ E.g. Steane~\cite{steane:qc1997a} has overheads of order $10^5$ for error rates below $3*10^{-5}$. More specifically: $O(10^9)$ Toffoli gates, Steane took that as an algorithm requiring $2{\times}10^{12}$ cnots ($KQ = 4 10^{12}$ on our terms.) Preskills three-fold concatenated $[[7,1,3]]$ code at $\gamma=10^{-6}$ would require $10^17$ gates (multiply by two if these are mostly cnots, as I think they are). The highest error-tolerating code seems in this paper appears to be a $[[87,1,15]]$ code which at $\gamma=1.9{\times}10^{-5}$ requires $3.9\times 10^16$ gates. This gives overheads at these error rates of perhaps a few times $10^{4}$ on our terms. The question is, what would they be at $\gamma=10^{-4}$? The $[[7,1,3]]$ architecture would require at least one more level of concatenation, one would guess another order of magnitude or two in overhead (assuming it was even doable at the time). Ok, I should be able to get the behavior from Steane's equations (9) and (11). I think he must have used a threshold of $10^{-4}$ for the $[[7,1,3]]$ architecture, though I don't see it stated explicitly. Taking the more optimistic value of $10^{-3}$ in Eqs. (10) and (11), I get: $N/K \approx 26$, $L\approx 2$, $T/Q\approx 7*480*2150/\eta$, where $\eta$ is the ``average number of computational steps per correction of the whole computer'', which he ``safely'' chooses as $\eta=w$. This is an optimization, do more computational steps (i.e. logical gates) before correcting. $w$ is the average weight in a row of the classical parity check matrix for a CSS code. $w$ is said to be $d+1$, where $d$ is the minimum distance. So $9$ in my example. This makes $T/Q\approx 8*10^{5}$, and this is optimistic, assuming a speculative threshold for this architecture that may not have applied to the known ways of implementing it at the time. To look at the other code, he writes $m=(n-1)/2$, $n$ is the length of the code, $m$ occurs in the formulas. I am looking at eq. (9) for the $[[87,1,15]]$ code. $T/Q \approx (86*16+435)*2*r*2150/16 \approx 3.9*10^6$, where is the dependence on $\gamma$? He assumes that on average, $r=(d+1)/2$, it is the number of syndromes needed. The dependence on $\gamma$ only shows up in the logical error probability, which presumably is not good enough at a $5$ times higher $\gamma$ than given in the tables (probably decreases by at least $5^7$). It looks like somewhere between $10^{-7}$ and $10^{-8}$ from Fig. 4 (per correction, per block). For comparison, my resources at $\gamma=10^{-3}$ are $5.3\times 10^{5}$ using level $5$ for $KQ = 10^10$ and $3\times 10^6$ with level $5$ at $KQ = 10^12$. }
Several open problems arise from the work presented here. Can the high thresholds evidenced by our simulations be mathematically proven? Are thresholds for postselected computing strictly higher than thresholds for scalable standard quantum computing? Recent work by Reichardt~\cite{reichardt:qc2004a} shows that Steane's architecture can be made more efficient by the judicious use of error detection, improving Steane's threshold estimates to around $10^{-2}$. How do the available fault-tolerant architectures compare for EPGs between $10^{-3}$ and $10^{-2}$? It would be helpful to significantly improve the resource requirements of fault-tolerant architectures, particularly at high EPGs.
This work is a contribution of NIST, an agency of the U.S. government, and is not subject to U.S. copyright. Correspondence and requests for materials should be sent to E. Knill ([email protected]).
\pagebreak
\begin{herefig} \label{fig:c4/c6_struct} \begin{picture}(-1.4,3.5)(0,-3.3) \nputgr{0,0}{t}{scale=.5}{c4c6l=3} \nputbox{0,-1}{b}{$\overbrace{\rule{4.2in}{0pt}}$} \nputgr{0,-1}{t}{scale=.5}{c4c6l=2} \nputbox{-1.75,-1.8}{b}{$\overbrace{\rule{1.5in}{0pt}}$} \nputbox{0,-1.8}{b}{$\overbrace{\rule{1.5in}{0pt}}$} \nputbox{1.75,-1.8}{b}{$\overbrace{\rule{1.5in}{0pt}}$} \nputgr{0,-1.8}{t}{scale=.5}{c4c6l=1}
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$\left.\begin{array}{@{}l@{}} \textsf{\small Level $1$ encoded} \\[-3pt] \textsf{\small qubit pairs:}
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\nputbox{-4,-1.3}{l}{ $\left.\begin{array}{@{}l@{}} \textsf{\small Level $2$} \\[-3pt] \textsf{\small syndrome bits:}
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$\left.\begin{array}{@{}l@{}} \textsf{\small Level $2$ encoded} \\[-3pt] \textsf{\small qubit pairs:}
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\nputbox{-2,-.6}{l}{ $\left.\begin{array}{@{}l@{}} \textsf{\small Level $3$} \\[-3pt] \textsf{\small syndrome bits:}
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$\left.\begin{array}{@{}l@{}} \textsf{\small Level $3$ encoded} \\[-3pt] \textsf{\small qubit pairs:}
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\end{picture} \ifnature{\pagebreak}{} \herefigcap{Block structure of $C_4/C_6$ concatenated codes. The bottom line shows $9$ blocks of four physical qubits. Each block encodes a level $1$ qubit pair with $C_4$. The encoded qubit pairs are shown in the line above. Formally, each such pair is associated with two syndrome bits, shown below the encoded pair in a lighter shade, which are accessible by syndrome measurements or decoding for the purpose of error detection and correction. The next level groups three level $1$ qubit pairs into a block, encoding a level $2$ qubit pair with $C_6$ that is associated with $4$ syndrome bits. A level $2$ block consists of a total of $12$ physical qubits. Three level $2$ qubit pairs are used to form a level $3$ qubit pair, again with $C_6$ and associated with $4$ syndrome bits. The total number of physical qubits in a level $3$ block is $36$. } \end{herefig}
\pagebreak
\begin{herefig} \label{fig:condEPG} \begin{picture}(0,3.7)(0,-3.5) \nputbox{.17,-1.1}{tl}{\color{grey}\rule{2pt}{.25in}}
\nputgr{0,0}{t}{height=3in}{2teledet_le} \nputbox{-3,0}{tr}{\rotatebox{90}{\textsf{\small Logical CNOT conditional error probability}}} \nputbox{3,-3.0}{tr}{\textsf{\small Physical CNOT error probability $\gamma$.}}
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\textsf{\small
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\multicolumn{3}{@{}l@{}}{Level $2$ errors at $\gamma=0.03$.}\\[-3pt]
&CNOT : & $\expdata{1.5\aerrb{0.7}{0.5}\times 10^{-3}}{cliff/finaldata/2teledet_l2_be.dat[0.03,:]} $\\[-2pt]
&preparation : & $\expdata{2.1\aerrb{0.3}{0.2}\times 10^{-4}}{cliff/finaldata/prepedet_l2_be.dat[0.03,:]} $\\[-2pt]
&measurement : & $\expdata{4.1\aerrb{1.4}{1.1}\times 10^{-5}}{cliff/finaldata/measedet_l2_be.dat[0.03,:]} $\\[-2pt]
&HAD : & $\expdata{1.8\aerrb{0.4}{0.3}\times 10^{-3}}{cliff/finaldata/1teledet_l2_be.dat[0.03,:]}$\\[-2pt]
&decoding : & $\expdata{2.2\aerrb{0.2}{0.2}\times 10^{-2}}{cliff/finaldata/decedet_l2_be.dat[0.03,:]}$\\[-2pt]
&injection : & $\expdata{3.1\aerrb{0.2}{0.2}\times 10^{-2}}{cliff/finaldata/injedet_l2_be.dat[0.03,:]}$\\[-2pt]
\end{tabular}
}
}
} \end{picture} \ifnature{\pagebreak}{} \herefigcap{Conditional logical errors with postselection. The plot shows logical CNOT errors conditional on not detecting any errors as a function of EPG parameter $\gamma$ at levels $0$, $1$ and $2$. The logical CNOT is implemented with transversal physical CNOTs and two error-detecting teleportations, where the output state is accepted only if no errors are detected in the teleportations. The data show the incremental error attributable to the logical CNOT in the context of a longer computation, as explained in S.I. Sect.~\ref{sect:simulation}. The error bars are $68\,\%$ confidence intervals. The solid lines are obtained by least-squares interpolation followed by gradient-descent likelihood maximization. Extrapolations are shown with dashed lines and suggest that logical EPG improvements with increasing levels are possible above $\expdata{\gamma=0.06}{Determined by conservative visual inspection of the graph.}$. The error in the slope ($\expdata{4.32}{cliff/finaldata/2teledet_l2_be.ipol[1,2]}$) of the level $2$ line is estimated as $\expdata{0.52}{sqrt(cliff/finaldata/2teledet_l2_be.ipol[1,4])}$ by resampling. The smallest number of undetectable errors at level $2$ is $4$, which should be the slope as $\gamma$ goes to $0$. At high $\gamma$, the curves are expected to level off~\cite{steane:qc2002a}. Other operations' errors for $\gamma=0.03$ and level $2$ are shown in the inset table. Ratios between the preparation or measurement and CNOT errors are smaller than those assumed for the physical error model. The logical HAD error is expected to be between $0.5$ and $0.8$ of the logical CNOT error, which could not be confirmed because of the large error bars. The decoding error is the incremental error introduced by decoding a block into two physical qubits. The injection error is the error in a logical state that we prepare by decoding one block of a logical Bell pair and measuring the decoded qubits. The measurement error per qubit is assumed to be the same as that of $X$- and $Z$-measurements. Decoding and injection errors were found to decrease from level 1 (decoding error $\expdata{4.4\aerrb{0.4}{0.4}\times 10^{-2}}{cliff/finaldata/decedet_l1_be.tex[0.03,:]}$, injection error $\expdata{5.5\aerrb{0.5}{0.4}\times 10^{-2}}{cliff/finaldata/injedet_l1_be.tex[0.03,:]}$ ) to level 2.} \end{herefig}
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\begin{herefig} \label{fig:ecEPG}\label{fig:decEPG} \begin{picture}(0,4.2)(0,-4) \nputgr{-1.8,0}{t}{height=3in}{2teleec_le} \nputbox{-3.35,0}{tr}{\rotatebox{90}{\textsf{\small Logical CNOT conditional error probability}}} \nputbox{-2.8,-.2}{c}{\Large\textbf{a.}} \nputbox{-1.22,-2.5}{tl}{\rotatebox{9}{\scalebox{1.8}{$\downarrow$}{}}}
\nputbox{.25,0}{tr}{\rotatebox{90}{\textsf{\small Logical CNOT detected error probability}}} \nputgr{1.8,0}{t}{height=3in}{2teleec_de} \nputbox{3.4,-3.0}{tr}{\textsf{\small Physical CNOT error probability $\gamma$.}} \nputbox{.8,-.2}{c}{\Large\textbf{b.}} \end{picture} \ifnature{\pagebreak}{} \herefigcap{Conditional and detected logical errors with error correction. The plot shows incremental detected and conditional logical errors for a logical CNOT as a function of EPG parameter $\gamma$ up to level $4$. Error bars and lines are as described in the caption of Fig.~\ref{fig:condEPG}. The data is obtained as described in S.I. Sect.~\ref{sect:simulation}. The combination of error correction and detection is as required for the error-correcting $C_4/C_6$ architecture. Plot \textbf{a.} shows the logical CNOT's error conditional on not detecting an uncorrectable error. Plot \textbf{b.} shows the probability of detecting an uncorrectable error. At $\gamma=0.01$, the detected errors are are $\expdata{2.4\aerrb{0.0}{0.0}\times 10^{-2}}{cliff/finaldata/2teleec_l3_de.tex[0.01,:]}$ (level 3) and $\expdata{2.4\aerrb{1.0}{0.7}\times 10^{-3}}{cliff/finaldata/2teleec_l4_de.tex[0.01,:]}$ (level 4). The conditional errors are $\expdata{6.4\aerrb{0.6}{0.6}\times 10^{-4}}{cliff/finaldata/2teleec_l3_be.tex[0.01,:]}$ (level 3) and $\expdata{0.0\aerrb{4.4}{0.0}\times 10^{-4}}{cliff/finaldata/2teleec_l4_be.tex[0.01,:]}$ (level 4). For comparison, the preparation errors at levels 3,4 were found to be $\expdata{2.1\aerrb{0.3}{0.3}\times10^{-4}}{cliff/finaldata/prepec_l3_de.tex[0.01,:]}$, $\expdata{0.0\aerrb{1.0}{0.0}\times10^{-4}}{cliff/finaldata/prepec_l4_de.tex[0.01,:]}$ (detected error) and $\expdata{3.3\aerrb{7.5}{2.7}\times10^{-6}}{cliff/finaldata/prepec_l3_be.tex[0.01,:]}$, $\expdata{0.0\aerrb{1.0}{0.0}\times10^{-4}}{cliff/finaldata/prepec_l4_be.tex[0.01,:]}$ (conditional error). The measurement errors are $\expdata{4.7\aerrb{0.4}{0.4}\times10^{-4}}{cliff/finaldata/measec_l3_de.tex[0.01,:]}$, $\expdata{5.6\aerrb{12.8}{4.6}\times10^{-5}}{cliff/finaldata/measec_l4_de.tex[0.01,:]}$ (detected error) and $\expdata{3.3\aerrb{7.4}{2.7}\times10^{-6}}{cliff/finaldata/measec_l3_be.tex[0.01,:]}$, $\expdata{0.0\aerrb{1.0}{0.0}\times10^{-4}}{cliff/finaldata/measec_l4_be.tex[0.01,:]}$ (conditional error). Finally, the HAD errors at level 3 are $\expdata{1.3\aerrb{0.0}{0.0}\times 10^{-2}}{cliff/finaldata/1teleec_l3_de.tex[0.01,:]}$ (detected error) and $\expdata{3.5\aerrb{0.6}{0.5}\times 10^{-4}}{cliff/finaldata/1teleec_l3_be.tex[0.01,:]}$ (conditional error). } \end{herefig}
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\begin{herefig} \label{fig:ecChain} \begin{picture}(0,4.2)(0,-4) \nputgr{-1.8,0}{t}{height=3in}{chaincor_le} \nputbox{-3.35,0}{tr}{\rotatebox{90}{\textsf{\small Logical conditional error probability}}}
\nputbox{.25,0}{tr}{\rotatebox{90}{\textsf{\small Logical detected error probability}}} \nputbox{-2.8,-.2}{c}{\Large\textbf{a.}}
\nputgr{1.8,0}{t}{height=3in}{chaincor_de} \nputbox{3.4,-3.0}{tr}{\textsf{\small Number of steps.}} \nputbox{.8,-.2}{c}{\Large\textbf{b.}} \end{picture} \ifnature{\pagebreak}{} \herefigcap{Error-compounding behavior with and without error-correcting teleportation. Incremental conditional (\textbf{a.}) and detected (\textbf{b.}) error probabilities are shown for each step of a sequence of 30 steps of applying the one-qubit error associated with HAD to each physical qubit and teleporting or not teleporting the logical qubit pair's block. Error bars are $68\,\%$ confidence intervals. Level $3$ of the error-correcting architecture is used. The first step is omitted since it is biased by the error-free reference-state preparation as discussed in S.I. Sect.~\ref{sect:simulation}. The horizontal gray lines show the average incremental error if teleportation is used. Note that for the first four steps, the incremental conditional error is smaller if no teleportation is used. This may be exploited when optimizing networks, provided one takes account of the resulting spreading of otherwise localized error events~\cite{zalka:qc1996a,steane:qc2002a}. } \end{herefig}
\pagebreak
\begin{herefig} \label{fig:resource_graph} \begin{picture}(0,3.7)(0,-3.5)
\nputgr{0,0}{t}{height=3in}{opcntsmod_pl} \expdata{}{Output generated by matlab code in cliff/finaldata, see cliff/finaldata/catalogue.txt. Requires file prepared in octave code below} \nputbox{-3,0}{tr}{\rotatebox{90}{\textsf{\small
Resources per qubit and gate}}} \nputbox{3,-3.0}{tr}{\textsf{\small Physical CNOT error probability $\gamma$.}} \nputbox{-2.2,-1.6}{c}{\textsf{\small $10^{34}$}} \nputbox{-2,-1.8}{c}{\textsf{\small $10^{21}$}} \nputbox{-1.9,-1.98}{c}{\textsf{\small $10^{13}$}} \nputbox{-1.4,-2.14}{c}{\textsf{\small $10^{8}$}} \nputbox{-1.73,-2.45}{c}{\textsf{\small $10^{5}$}} \nputbox{.86,-2.4}{c}{\textsf{\small $10^{3}$}}
\nputbox{2.82,-1.3}{c}{\circle{.17}} \nputbox{2.73,-1.3}{c}{\textsf{\small 4}} \nputbox{1.87,-1.75}{c}{\circle{.17}} \nputbox{1.78,-1.75}{c}{\textsf{\small 4}} \nputbox{.54,-1.85}{c}{\circle{.17}} \nputbox{.45,-1.85}{c}{\textsf{\small 4}} \nputbox{-1.61,-1.85}{c}{\circle{.17}} \nputbox{-1.7,-1.85}{c}{\textsf{\small 4}} \nputbox{-1.69,-1.63}{c}{\circle{.17}} \nputbox{-1.78,-1.63}{c}{\textsf{\small 5}} \nputbox{-1.86,-1.43}{c}{\circle{.17}} \nputbox{-1.95,-1.43}{c}{\textsf{\small 6}}
\end{picture} \ifnature{\pagebreak}{} \herefigcap{ Resources per qubit and gate for different computation sizes. The size $KQ$ of a computation is the product of the number of gates (including ``memory'' gates) and the average number of qubits per gate. Each curve is labeled by the computation size and shows the number $\textsl{pcnot}$ of physical CNOTs required per qubit and gate to implement a computation of size $KQ$ with the $C_4/C_6$ architecture. The curves are based on naive models of resource usage in state preparation and of the logical errors (S.I. Sect.~\ref{sect:resources}). The circled numbers are at the point above which the indicated or a higher level must be used. The curves are most reliable for levels $<4$. To obtain the total computational resources, multiply $\textsl{pcnot}\times KQ$ by twice the average number of logical CNOTs needed for implementing a gate of the computation. It is assumed that these logical CNOTs are involved in state preparation required for universality but do not contribute to the error (S.I. Sect.~\ref{sect:resources}). The ``scale-up'' (number of physical qubits per logical qubit) depends on parallelism and level $l$ of concatenation. With maximum parallelism, the scale-up is of the same order as $\textsl{pcnot}$. For a completely sequential algorithm such as could be used if there is no memory error, this can be reduced to $3^{l-1}2$. With some memory error and logical gate parallelism, $\approx (1+2*(l-1))3^{l-1}2$ is more realistic (S.I. Sect.~\ref{sect:resources}). The steps in the curve arise from increasing the number of levels. The first step is to level $2$, and each subsequent step increments the level by $1$. The steps are smoothed because we can exploit error-detection to avoid using the next level. Improvements of only one to two orders of magnitude are obtained by reducing $\gamma$ from $0.001$ to $0.0001$, compared to at least five orders by reducing $\gamma$ from $0.01$ to $0.001$.} \end{herefig}
\pagebreak
\appendix
\begin{center} \huge\textbf{ Supplementary Information} \end{center}
\section{Explanation of Error-Correcting Teleportation} \label{sect:tec}\label{sect:teleportation}
For the basic theory of stabilizer codes, see [\citeonline{nielsen:qc2001a}]. Let $Q$ be the $l\times 2n$ binary check matrix with entries defining a stabilizer code on $n$ qubits for encoding $k=n-l$ qubits with good error-detecting or -correcting properties. The check matrix is obtained from an independent set of check operators $P_1, \ldots, P_l$ by placing a binary representation of $P_k$ into row $k$. The binary representation of $P$ is obtained by replacing the Pauli operator symbols according to $I\mapsto 00$, $X\mapsto 10$, $Z\mapsto 01$ and $Y\mapsto 11$. For example, the check operator $XIY$ is represented by the row vector $[100011]$. Commas are omitted in binary vectors, and square brackets are used to distinguish them from binary strings. The syndrome of a joint eigenstate of the check operators is denoted by a binary column vector, with $0$ ($1$) in the $k$'th position denoting a $1$ ($-1$, respectively) eigenvalue of $P_k$. The projection operator onto the eigenspace with syndrome $\mathbf{x}$ is denoted by $\Pi(Q,\mathbf{x})$. If a Pauli product $P$ with binary representation $\mathbf{g}$ is applied to a state with syndrome $\mathbf{x}$, the new syndrome $\mathbf{x}'$ is given by $\mathbf{x}' = \mathbf{x}+Q\mathbb{S}\mathbf{g}^T$. Arithmetic with binary vectors and matrices is modulo 2 and $\mathbb{S}$ is the $2n\times 2n$ block-diagonal matrix with blocks $\left(\begin{array}{cc}0&1\\1&0\end{array}\right)$.
Consider an $n$ qubit ``input'' block carrying $l$ qubits encoded in the stabilizer code for $Q$, where the block has been affected by errors. An effective way of detecting or correcting errors is to teleport each of the $n$ qubits of the input block using two blocks of $n$ qubits that form an ``encoded Bell pair''. That is, both blocks have syndrome $\mathbf{0}$ with respect to $Q$ and corresponding qubits encoded in the two blocks are in the state $(\ket{00}+\ket{11})/\sqrt{2}$. The state of the two blocks is defined by the following preparation procedure: Start with $n$ pairs of qubits in the standard Bell state $(\ket{00}+\ket{11})/\sqrt{2}$. The two blocks are formed from the first and second members of each pair, respectively. Use a $Q$-syndrome measurement on the $n$ second members of each pair to project them into one of the joint eigenspaces of $Q$. Finally, apply identical Pauli matrices to both members of pairs in such a way as to reset the syndromes to $\mathbf{0}$. To teleport, apply the usual protocol to corresponding qubits in the three blocks. In the absence of errors, this copies the encoded input state to the second block of the encoded Bell pair. We show that errors are revealed by parities of the teleportation measurement outcomes.
The standard quantum teleportation protocol begins with an arbitrary state $\kets{\psi}{1}$ in qubit $\mathsf{1}$ and the Bell state $(\kets{00}{23}+\kets{11}{23})/\sqrt{2}$ in qubits $\mathsf{2},\mathsf{3}$. The global initial state can be viewed as $\ket{\psi}$ encoded in the stabilizer code generated by $IXX$ and $IZZ$, whose check matrix has rows $\mathbf{b}_1=[001010]$ and $\mathbf{b}_2=[000101]$. Let $\slb{B}{23}$ be the check matrix whose rows are the $\mathbf{b}_i$. The stabilizer consists of the system-labeled Pauli products $I,\slb{\sigma_x}{2}\slb{\sigma_x}{3},\slb{\sigma_y}{2}\slb{\sigma_y}{3}$ and $\slb{\sigma_z}{2}\slb{\sigma_z}{3}$. To teleport, one makes a Bell-basis measurement on the first two qubits. This is equivalent to making a $\slb{B}{12}$-syndrome measurement, where $\slb{B}{12}$ has as rows $[101000]=[XXI]$ and $[010100]=[ZZI]$. This is identical to $\slb{B}{23}$ with qubits $\mathsf{2},\mathsf{3}$ exchanged for qubits $\mathsf{1},\mathsf{2}$. Depending on the syndrome $\mathbf{e}$ that results from the measurement, one applies correcting Pauli matrices to qubit $\mathsf{3}$ to restore $\ket{\psi}$ in qubit $\mathsf{3}$.
Consider the teleportation of $n$ qubits in a block as described above. The protocol is such that the $2n$ binary measurement outcomes linearly (with respect to computation modulo $2$) determine the Pauli product correction to be applied to the second block of the encoded Bell pair. Let $\mathbf{g}$ be the binary representation of the Pauli product correction. The syndrome of the input block constrains $\mathbf{g}$ as shown in Fig.~\ref{fig:teleec}. The principle is as explained in~[\citeonline{gottesman:qc1999a}] for unitary gates, but generalized to measurements. In this case, a stabilizer projection on the destination qubits before teleportation is equivalent to a projection after teleportation, where the syndrome associated with the projection is modified by the correction Pauli product used at the end of teleportation. The expression $Q\mathbb{S} \mathbf{g}^T$ must match the syndrome of the input block. Consequently, the syndrome of the input block can be deduced from $\mathbf{g}$, a function of the teleportation Bell measurement. Errors can be detected or corrected accordingly.
It is necessary to consider the effects of errors in the prepared encoded Bell pair. Errors on the second block propagate forward and must be handled by future teleportations. Because of the Bell measurement, errors on the first block have an effect equivalent to the same errors on the input block. Thus, using the inferred syndrome for detection or correction of errors deals with errors in both blocks, as long as their combination is within the capabilities of the code.
Error correction or detection by teleportation handles leakage errors in the same way as other errors. If a qubit ``leaked'', the outcome of its Bell measurement becomes undetermined. The Bell measurement can be filled in arbitrarily, because for the purpose of interpreting the syndrome, the effect is the same as if a Pauli error occurred depending on how the measurement result is filled in.
Note that as usual, none of the Pauli corrections actually have to be implemented explicitly. One can just update the Pauli frame as needed.
\pagebreak \begin{herefig} \label{fig:teleec} \begin{picture}(0,5.8)(0,-5.9) \nputgr{-3,-1}{l}{height=2.4in}{teleec_a} \nputbox{-3.3,.1}{lt}{$n$ input qubits} \nputbox{0.1,-.97}{lt}{$n$ output qubits} \nputbox{-2.3,-1.04}{c}{\rotatebox{-90}{Bell}} \nputbox{-1.58,-1.31}{c}{\small\rotatebox{-90}{$\Pi(Q,0)$}} \nputbox{-.82,-.5}{c}{\rotatebox{-90}{Bell}} \nputbox{-.21,-1.27}{c}{$P(\mathbf{g})$}
\nputbox{-1.7,-3}{r}{\scalebox{2.5}{$\Leftrightarrow$}} \nputgr{-1.5,-3}{l}{height=2.4in}{teleec_b} \nputbox{-.8,-3}{c}{\rotatebox{-90}{Bell}} \nputbox{-.05,-2.45}{c}{\rotatebox{-90}{Bell}} \nputbox{.55,-3.26}{c}{$P(\mathbf{g})$} \nputbox{1.25,-3.26}{c}{\small\rotatebox{-90}{$\Pi(Q,Q\mathbb{S}\mathbf{g}^T)$}}
\nputbox{-.2,-5}{r}{\scalebox{2.5}{$\Leftrightarrow$}} \nputgr{0,-5}{l}{height=2.4in}{teleec_c} \nputbox{2.75,-5.28}{c}{\small\rotatebox{-90}{$\Pi(Q,Q\mathbb{S}\mathbf{g}^T)$}} \end{picture} \nopagebreak \herefigcap{Teleporting with an encoded entangled state is equivalent to a syndrome measurement. The gray lines are the time lines of blocks of $n$ qubits. The boxes denote various operations. The Bell-state preparation on corresponding pairs of qubits in two blocks is depicted with a box angled to the right and labeled ``Bell''. The state used for teleportation in the top diagram is obtained after Bell-state preparation by projecting one of the blocks with $\Pi(Q,0)$ (the actual preparation procedure is different but has the same output). Projection operators are shown with boxes angled both ways with the operator written in the box. Bell measurement of corresponding pairs of qubits in two blocks is depicted with a box angled to the left and labeled ``Bell''. A Bell measurement on qubits $1$ and $2$ is implemented by applying a CNOT from qubit $1$ to $2$, performing an $X$-measurement on qubit $1$ and a $Z$-measurement on qubit $2$. The top diagram is the actual network implemented. The other two are logically equivalent. The Bell measurement outcome $\mathbf{g}$ is correlated with the effective projection in the bottom diagram. If the input state has a particular syndrome, then only $\mathbf{g}$ for which the projection is onto the subspace with this syndrome have non-zero probababilities.} \end{herefig} \pagebreak
\section{Networks for $C_4$ and $C_6$ State Preparation and Gates} \label{sect:networks}
\begin{herefig} \label{fig:elements} \begin{picture}(0,4.4)(0,-4.2)
\nputgr{-2,-.3}{r}{scale=.7}{blockprepZ}
\nputbox{-1.9,-.3}{l}{
\begin{tabular}{@{}l}
Encoded $\ket{0}$\\[-4pt] preparation.
\end{tabular}
}
\nputgr{1.8,-.3}{r}{scale=.7}{blockprepX}
\nputbox{1.9,-.3}{l}{
\begin{tabular}{@{}l}
Encoded $\ket{+}$\\[-4pt] preparation.
\end{tabular}
}
\nputgr{-2,-.9}{r}{scale=.7}{blockmeasZ}
\nputbox{-1.9,-.9}{l}{
\begin{tabular}{@{}l}
Transversal $Z$\\[-4pt] measurement.
\end{tabular}
}
\nputgr{1.8,-.9}{r}{scale=.7}{blockmeasX}
\nputbox{1.9,-.9}{l}{
\begin{tabular}{@{}l}
Transversal $X$\\[-4pt] measurement.
\end{tabular}
}
\nputgr{-2,-1.8}{r}{scale=.7}{trnsvlcnot}
\nputbox{-1.9,-1.8}{l}{
\begin{tabular}{@{}l}
Transversal CNOT.
\end{tabular}
}
\nputgr{1.8,-1.53}{r}{scale=.7}{blockhad}
\nputbox{1.9,-1.53}{l}{
\begin{tabular}{@{}l}
Encoded HAD.
\end{tabular}
}
\nputgr{1.8,-2.07}{r}{scale=.7}{trnsvlhad}
\nputbox{1.9,-2.07}{l}{
\begin{tabular}{@{}l}
Transversal HAD.
\end{tabular}
}
\nputgr{-2,-2.8}{r}{scale=.7}{blockmu}
\nputbox{-1.9,-2.8}{l}{
\begin{tabular}{@{}l}
$*u$ on a block\\[-4pt]
encoding two qubits
\end{tabular}
}
\nputgr{1.8,-2.8}{r}{scale=.7}{blockmusq}
\nputbox{1.9,-2.8}{l}{
\begin{tabular}{@{}l}
$*u^2$ on a block\\[-4pt]
encoding two qubits
\end{tabular}
}
\nputgr{-2,-3.6}{r}{scale=.7}{block4tel}
\nputbox{-1.9,-3.6}{l}{
\begin{tabular}{@{}l}
$C_4$ subblock teleportation,\\[-4pt]
involves four lower-\\[-4pt]
level blocks.
\end{tabular}
}
\nputgr{1.8,-3.6}{r}{scale=.7}{block3tel}
\nputbox{1.9,-3.6}{l}{
\begin{tabular}{@{}l}
$C_6$ subblock teleportation,\\[-4pt]
involves three lower\\[-4pt]
level blocks
\end{tabular}
}
\end{picture} \nopagebreak \herefigcap{Network elements. The elements shown represent networks
acting on blocks of qubits. Blocks (shown by thick gray lines) may
consist of only one physical qubit, so the elements can also
represent physical gates. Elements with ``fringes'' are transversal
gates: The indicated gate is applied to each physical qubit, or to
corresponding physical qubits in the input blocks. The measurements
have classical output indicated with a black line. Because they are
transversal, the output contains as many bits as there are physical
qubits. Because the codes used here are CSS codes, the check
operators and the encoded Pauli operators contain only one type
of non-identity Pauli operator. The output bits therefore contain
both error-check information and the encoded-measurement
answers. The $*u$ and $*u^2$ elements are defined as shown for
physical qubit pairs. The notation comes from a polynomial
construction of $C_6$ as a code on three quaternary qudits using the
four-element field $GF(4)$. The symbol $u$ denotes a third root of unity
over $GF(2)$. The gates transform Pauli operators by
multiplication with $u$ or $u^2$ in a $GF(4)$ labeling of these operators.} \end{herefig} \pagebreak
\begin{herefig} \label{fig:C_4_reductions:_preps} \begin{picture}(0,6.4)(0,-6.4)
\nputgr{-1.3,-1.6}{r}{scale=.7}{blockprepZ}
\nputbox{-.9,-1.6}{c}{\scalebox{2.5}{$\Leftrightarrow$}}
\nputgr{-.5,-1.6}{l}{scale=.5}{reducebprepZ}
\nputgr{-1.3,-4.8}{r}{scale=.7}{blockprepX}
\nputbox{-.9,-4.8}{c}{\scalebox{2.5}{$\Leftrightarrow$}}
\nputgr{-.5,-4.8}{l}{scale=.5}{reducebprepX} \end{picture} \pagebreak \herefigcap{Encoded state preparations in terms of lower-level elements for $C_4$. The lower-level blocks (``subblocks'') can be either physical or encoded single qubits and are represented by the merging lines in the networks on the right. In the $C_4/C_6$ architecture, $C_4$ is used only at level $1$, so the subblocks are always physical qubits. In this case, the output block contains an encoded qubit pair with each qubit in the pair in the state indicated by the preparation gate on the left. The physical states prepared are four-qubit cat states ($(\ket{0000}+\ket{1111})/\sqrt{2}$ in the case of the top network). If no error occurred, the four measurements in each network on the right have total parity $0$. For any single error in the state preparation network, if this error results in an error in the output state that is not equivalent to a single physical qubit error, then the parity is $1$, so this event can be detected. Thus, if the total measurement parity is $1$, the output state is rejected. This ensures that errors occurring with linear probability in the EPGs introduce no undetectable errors. Note that the networks on the right begin with Bell-state preparations. The teleportation steps are not implemented on physical qubits but are included for generality. The encoded $Z$- and $X$-preparations shown assume that the next step is a transversal CNOT followed by subblock teleportations. Otherwise it may be necessary to teleport subblocks immediately to avoid error propagation. } \end{herefig} \pagebreak
\begin{herefig} \label{fig:C_6_reductions:_preps} \begin{picture}(0,5.8)(0,-5.5)
\nputgr{-1.3,-1.3}{r}{scale=.7}{blockprepZ}
\nputbox{-.9,-1.3}{c}{\scalebox{2.5}{$\Leftrightarrow$}}
\nputgr{-.5,-1.3}{l}{scale=.5}{reduceb6prepZ}
\nputgr{-1.3,-4.0}{r}{scale=.7}{blockprepX}
\nputbox{-.9,-4.0}{c}{\scalebox{2.5}{$\Leftrightarrow$}}
\nputgr{-.5,-4.0}{l}{scale=.5}{reduceb6prepX} \end{picture} \nopagebreak \herefigcap{Encoded state preparations in terms of lower-level elements for $C_6$. The lower-level blocks (``subblocks'') contain encoded qubit pairs. The beginning of the network prepares two parallel three-qubit cat states $(\ket{000}+\ket{111})/\sqrt{2}$ in the top network) on corresponding members of the encoded qubit pairs. The encoded measurements in the cat-state preparation satisfy the parity constraint described in the caption of Fig.~\ref{fig:C_4_reductions:_preps} for each of the three corresponding qubits in the encoded qubit pairs. Because the measurements are implemented transversally, they also provide lower-level syndrome information that can be used for error detection or correction. Again, the networks begin with Bell pair preparations and the teleportations are only implemented on encoded qubit pairs. The last elements rotate the parallel cat states into $C_6$, so that the encoded qubit pair has both qubits in the desired state. Because the first level encoding uses $C_4$, they can be implemented as simple permutations, which can be accomplished by logical relabeling without delay or error, see Fig.~\ref{fig:C_6_reductions:_mults}. As in Fig.~\ref{fig:C_4_reductions:_preps}, the encoded $Z$- and $X$-preparations shown assume that the next step is a transversal CNOT followed by subblock teleportations. Otherwise it may be necessary to teleport subblocks immediately to avoid error propagation. } \end{herefig} \pagebreak
\begin{herefig} \label{fig:C_6_reductions:_mults} \begin{picture}(0,3.2)(0,-3)
\nputgr{-1.3,-1.2}{r}{scale=.7}{blockmu}
\nputbox{-.9,-1.2}{c}{\scalebox{2.5}{$\Leftrightarrow$}}
\nputgr{-.5,-1.2}{l}{scale=.5}{reduceb6mu4}
\nputgr{-1.3,-2.5}{r}{scale=.7}{blockmusq}
\nputbox{-.9,-2.5}{c}{\scalebox{2.5}{$\Leftrightarrow$}}
\nputgr{-.5,-2.5}{l}{scale=.5}{reduceb6musq4} \end{picture} \\ \begin{picture}(0,3.8)(0,-3.6)
\nputgr{-1.3,-1.2}{r}{scale=.7}{blockmu}
\nputbox{-.9,-1.2}{c}{\scalebox{2.5}{$\Leftrightarrow$}}
\nputgr{-.5,-1.2}{l}{scale=.5}{reduceb6mu}
\nputgr{-1.3,-2.8}{r}{scale=.7}{blockmusq}
\nputbox{-.9,-2.8}{c}{\scalebox{2.5}{$\Leftrightarrow$}}
\nputgr{-.5,-2.8}{l}{scale=.5}{reduceb6musq} \end{picture} \nopagebreak \herefigcap{Implementation of $*u$ and $*u^2$ on blocks of $C_4$ (top two) and $C_6$ (bottom two). } \end{herefig} \pagebreak
\begin{herefig} \label{fig:C_4/C_6_reductions:_tele} \begin{picture}(0,7.2)(0,-7)
\nputgr{-1.8,-2.0}{r}{scale=.7}{block4tel}
\nputbox{-1.4,-2.0}{c}{\scalebox{2.5}{$\Leftrightarrow$}}
\nputgr{-1,-1.6}{l}{scale=.5}{reducebtele}
\nputgr{-1.8,-6.15}{r}{scale=.7}{block3tel}
\nputbox{-1.4,-6.15}{c}{\scalebox{2.5}{$\Leftrightarrow$}}
\nputgr{-1,-5.3}{l}{scale=.5}{reduceb6tele} \end{picture} \nopagebreak \herefigcap{Implementation of subblock teleportation. If the blocks are physical qubits or pairs of physical qubits, the teleportation elements on the left are not implemented. Otherwise the networks shown on the right are used on the lower-level blocks. The top network is for $C_4$ and the bottom for $C_6$. Note that the networks look like traditional teleportation of each subblock. However, if the subblocks are not physical, the encoded Bell states used imply that the teleportations are error-detecting/correcting for the subblocks.} \end{herefig} \pagebreak
\begin{herefig} \label{fig:C_4/C_6_reductions:_had} \begin{picture}(0,4.2)(0,-4)
\nputgr{-1,-1.0}{r}{scale=.7}{blockhad}
\nputbox{0,-1.0}{c}{\scalebox{2.5}{$\Leftrightarrow$}}
\nputgr{1,-1}{l}{scale=.5}{reduceb4had}
\nputgr{-1,-3}{r}{scale=.7}{blockhad}
\nputbox{0,-3}{c}{\scalebox{2.5}{$\Leftrightarrow$}}
\nputgr{1,-3}{l}{scale=.5}{reduceb6had} \end{picture} \nopagebreak \herefigcap{Implementation of encoded HADs for $C_4$ and $C_6$. The top network is for $C_4$ and is transversal except for an interchange of the middle two qubits. The bottom is for $C_6$ and is transversal. Using the HAD and CNOT implementations, it is also possible to implement the encoded conditional sign flip transversally up to a physical qubit permutation implementable by relabeling. } \end{herefig} \pagebreak
As can be seen, all preparation networks are based ultimately on Bell state preparation followed by full or half Bell measurements. As shown, the networks use teleportation fastidiously. It may be possible to delay teleportation in some cases, but this was not confirmed by simulation. For postselected computing, there is no need to wait for measurement outcomes before proceeding to the next steps. However, this delays the rejection of states found later to be faulty, which incurs a large resource cost if the probability of detecting an error is high. For standard quantum computing, this resource cost can be avoided by delaying further processing and incurring some memory error instead. If error correction is used, at higher levels the probability of unrecoverable error decreases rapidly so one can again proceed optimistically, before measurement answers are known.
\section{Decoding $C_4$ and $C_6$.} \label{sect:decoding}
There are two reasons to explicitly decode logical states encoded by concatenating $C_4$ and $C_6$. First, at the highest EPGs, to implement a standard quantum computation with the postselected $C_4/C_6$ architecture requires preparing $C_4/C_6$ encoded states that are themselves states encoded in a code $C_e$ with very good error-correction capabilities. Once such a state is prepared, the $C_4/C_6$ concatenation hierarchy is decoded to obtain a physical block encoding a state in $C_e$. Second, to implement arbitrary quantum computations requires preparing special encoded states that are not reachable using $\cG_{\textnormal{\tiny min}}$ and HAD gates alone. These encoded states need not be error-free initially, since they can be purified using low-error logical $\cG_{\textnormal{\tiny min}}$ and HAD gates. A way to prepare these states with error that is bounded independently of the number of levels is to prepare a logical Bell state in two blocks, decode the first block into two physical qubits, and make a measurement of the physical qubits to project them into the desired state. (Alternatively, but with more error, the measurement can be replaced by a teleportation of the desired state prepared in another pair of physical qubits.) The entanglement between the physical qubits and the logical ones in the second block ensures that the state is injected into the logical qubits.
A good method for decoding the $C_4$/$C_6$ concatenation hierarchy is to decode ``bottom up''. That is, in the first step, the blocks of four physical qubits encoding qubit pairs in $C_4$ at the lowest level of the hierarchy are decoded. Syndrome information becomes available in a pair of ancillas for each block of $C_4$ and can be used for error detection. In subsequent steps, six physical qubits encoding qubit pairs in $C_6$ are similarly decoded. Error information obtained in previous decoding steps can be combined with new syndrome information for error detection or correction. The $C_4$ and $C_6$ decoding networks are shown in Fig.~\ref{fig:c4,c6_decoding}.
\pagebreak
\begin{herefig} \label{fig:c4,c6_decoding} \begin{picture}(0,3.6)(0,-3.2) \nputbox{-2.1,.3}{t}{ Decoding $C_4$:} \nputgr{-2.1,-1.3}{c}{scale=.65}{decodec4} \nputbox{1.7,.3}{t}{ Decoding $C_6$:} \nputgr{1.7,-1.3}{c}{scale=.65}{decodec6} \end{picture} \nopagebreak \herefigcap{Decoding $C_4$ (left) and $C_6$ (right). The gates are shown in plain form to indicate that they are physical, not encoded. The measurements reveal the syndrome and can be used for error detection. Error correction can be used if (a) the incoming qubits were decoded in an earlier step and (b) exactly one of them (for $C_4$) or one pair of them ($(1,2)$, $(3,4)$ or ($5,6$) for $C_6$) was detected to have an error. The $C_6$ decoding can be simplified if the first level of the full concatenation hierarchy uses $C_4$: The first step is a $*u^2$ operation on the first and last pair and can be implemented by relabeling before the level $1$ blocks of $C_4$ are decoded. Even if the $C_6$ decoding is implemented with maximum parallelism and without waiting for measurement outcomes, it has an initial memory delay on qubits $5$ and $6$ that was not taken into consideration in the simulations. } \end{herefig} \pagebreak
\section{Universal Computing} \label{sect:universal}
Universal computing with logical qubits encoded with $C_4/C_6$ can be accomplished by use of the logical $\cG_{\textnormal{\tiny min}}$ gates, HADs and $\ket{\pi/8}$-state preparation. However, since these operations do not distinguish between the two logical qubits encoded in one block, computations are implemented on only one of the two logical qubits in each block. Because the other one experiences the same evolution, the computation's output is obtained twice each time it is run. It is desirable to be able to address the two logical qubits in a block separately and have the ability to apply a CNOT from one to the other. One operation that is already available is the $*u$ gate and its inverse, which acts on a logical qubit pair as a swap followed by a CNOT. As with all stabilizer codes, it is also possible to apply arbitrary combinations of logical Pauli matrices by applying suitable products of physical Pauli matrices or by making a Pauli frame change. Fig.~\ref{fig:*u_univ} shows how to use the gates mentioned to implement a set of gates that is sufficiently rich to address individual qubits in a pair.
\begin{herefig} \label{fig:*u_univ} \begin{picture}(-.2,4.2)(0,-4)
\nputgr{-1.5,-1.0}{r}{scale=.7}{isselective_op}
\nputbox{-1,-1.0}{c}{\scalebox{2.5}{$\Leftrightarrow$}}
\nputgr{-.3,-1}{l}{scale=.7}{selective_op}
\nputgr{-1.5,-3.0}{r}{scale=.7}{iszz_op}
\nputbox{-1,-3.0}{c}{\scalebox{2.5}{$\Leftrightarrow$}}
\nputgr{-.3,-3}{l}{scale=.7}{zz_op} \end{picture} \nopagebreak \herefigcap{Implementing selective gates on qubit pairs. The networks are shown for physical qubits, but can be used with logical qubits by making the appropriate substitutions. The implementations on the right use only gates that do not distinguish the qubits in a pair, $*u$, $*u^2$ and Pauli products. The top network shows how to implement any gate $U$ selectively on one qubit in a pair. The implementation uses a selective Pauli operator and non-selective controlled-$U$ gates. The bottom network shows how to implement a type of controlled phase gate between the two logical qubits in a pair. It uses a $*u$ operation, a selective $90{}^{\circ}$ $z$-rotation (which can be implemented using the top network) and a $*u^2$ operation. The CNOT (without swap) between the qubits in a pair can be implemented in terms of the controlled phase gate shown and selective one-qubit gates. } \end{herefig}
The networks shown in Fig.~\ref{fig:*u_univ} do not result in particularly efficient ways of implementing gates on individual logical qubits. An alternative is to inject and purify states needed for one-qubit teleportation of the desired gates using the techniques given in~[\citeonline{zhou:qc2000b}]. An example of such a state is $\ket{0}\ket{+}$. Note that $\ket{0}\ket{+}$ is much more readily purified than $\ket{\pi/8}$. For example, to purify $\ket{0}\ket{+}$ one can apply the method suggested in the main text to reduce the preparation error. In the encoded setting, this requires a measurement of $Z$ and $X$ of the qubits in a pair, which cannot be done by a transversal encoded measurement. Instead, a third instance of $\ket{0}\ket{+}$ is introduced and involved in a transversal Bell measurement with the block of the qubits to be measured. As in error-correcting teleportation, the desired information can be extracted from parities of the Bell measurements. At the same time, syndrome information that can be used for error detection and correction is obtained.
\section{Simulation of Error Behavior} \label{sect:simulation}
To simulate the error behavior of fault-tolerant methods based on stabilizer codes, we use the result that computation with Clifford gates and feed-forward from $Z$- and $X$-measurements can be efficiently simulated~\cite{gottesman:qc1997b}. The Clifford gates include $Z$- and $X$-state preparations and measurements, HAD, CNOT and $90{}^{\circ}$ $Z$-rotations. Networks using these gates always result in stabilizer states, which are eigenstates of maximal sets of commuting check operators (the check matrix). Simulation requires tracking a complete independent set of such check operators and the syndrome (which gives the state's eigenvalues with respect to the check operators). Check operators can be represented by binary vectors (see Sect.~\ref{sect:tec}). To simplify the computations required for updating the check matrix and syndrome after applying gates, we maintain it in ``graph-state normal form''~\cite{schlingemann:qc2001b,grassl:qc2002a}. In this form, each qubit has an associated ``commuting'' operator, which is either $X$ or $Z$, and there is exactly one check operator acting on the qubit with an operator different from $I$ or the commuting operator. In addition to the check matrix and the ideal syndrome, we maintain the Pauli products representing the current effect of errors (the ``error vector'') and the Pauli frame. The error vector is known only to the simulation, not to the user implementing a computation. The error vector and Pauli frame are updated with each operation. For efficiency, blocks that have not yet interacted are associated with separate check matrices and ``merged'' when needed. Also, since it is necessary to accumulate as much statistics as possible, an array of error vectors and corresponding Pauli frames is used to represent multiple simultaneous preparation attempts without duplicating check matrices. For rapid prototyping purposes and fast array processing, we used Octave to implement the simulator. For simulating measurements and errors, a random-number generator is needed. We used the standard random-number generator provided with Octave. Because this implies that there are implicit correlations in the errors for the large-scale simulations undertaken here, the results obtained do not constitute full statistical proof. However, no artifacts not explainable by statistics were observed. In particular, in the few cases where an analytic expression for the data were available, the simulated data was as expected. This was checked for conditional error probabilities in postselected computing using concatenation with $C_4$ as discussed in~[\citeonline{knill:qc2004b}] for up to two levels (data not shown).
The simulations are used to determine the error behavior of various logical gates. For the data shown in Fig.~\ref{fig:condEPG},~\ref{fig:ecEPG} and~\ref{fig:ecChain}, we used the reference entanglement method~\cite{schumacher:qc1996a} for determining logical CNOT error probabilities. This involves applying the logical CNOT and error-detecting or -correcting teleportations to the first members of two error-free logical Bell pairs and then comparing the logical state to what would have been obtained if the logical CNOT had no error. The comparison is implemented by applying error-free CNOTs to disentangle the Bell pairs and making error-free logical $X$- or $Z$-measurements with error detection or correction depending on the context. The procedure was modified by (1) applying only the CNOT's physical error model associated with the transversal implementation and (2) applying the error model and error-detecting teleportation twice and determining the incremental error introduced the second time. (1) simplifies the verification without affecting the error probabilities. (2) is required so as to determine the effective error introduced in the middle of a computation, because the error-free Bell pairs have no initial error, contrary to what would be expected later. Using the second of two steps suffices because of the isolating properties of teleportation, which was verified by taking some data for more steps as shown in Fig.~\ref{fig:ecChain}. For detected error probabilities, the incremental error is determined as the fraction of trials in which an uncorrectable error was detected during the teleportations or in the verifying measurements. For conditional error probabilities, the incremental error is the fraction of trials with no detected uncorrectable error for which the logical measurement outcomes are incorrect but there was no undetected logical error in the preceding steps.
\section{Scalable Quantum Computing via Bootstrapping with Postselection} \label{sect:scalable}
The fault-tolerant architecture based on a good quantum error-correcting code $C_e$ using the $C_4/C_6$ architecture with postselection for state preparation is described in the main text. We claimed that if $3.53\gamma \lesssim 0.19$ and $\gamma$ is below the threshold for fault-tolerant postselected computing with the $C_4/C_6$ architecture, then the logical errors for the $C_e$ architecture can be made to be below $10^{-3}$, which is below the threshold for known fault-tolerant architectures. The estimate assumes that the decoding error per decoded qubit is $\approx\gamma$, in which case $\approx 3.53\gamma$ is the effective error per qubit that determines whether the error-correcting teleportation successfully corrects. With this assumption, the claim is proven as follows. Choose $\epsilon$ such that $3.53\gamma \lesssim 0.19-\epsilon$. Choose $C_e$ such that if a logical qubit is encoded in $C_e$ without error and each physical qubit is independently subjected to an error with probability $0.19-\epsilon$, then the logical state can be recovered with error at most $10^{-3}/4$. Such codes exist~\cite{divincenzo:qc1998c} although their length $n$ grows as $\epsilon$ goes to zero. Algorithms for encoding needed states in one to four blocks of $C_e$ require at most $c_1 n^2$ gates for some constant $c_1$ [\citeonline{cleve:qc1996b}]. Choose the level of the postselected $C_4/C_6$ architecture so that the logical gate error is well below $10^{-3}/(2 c_1 n^2)$. Then, before they are decoded, the postselected prepared states have logical error at most $10^{-3}/2$, since they required fewer than $c_1 n^2$ logical gates of the $C_4/C_6$ architecture. This error persists as a $C_e$-logical error after the $C_e$-error-correcting teleportation that uses this state after it is decoded. It adds to the logical error introduced by failure to error-correct in teleportation. However, because at most two error-correcting teleportations are involved, the total logical error is below $10^{-3}$. Note that if $\gamma$ is given and strictly below the threshold, then the resources required to achieve $C_e$-logical EPGs below $10^{-3}$ are determined. Because the fault-tolerant architectures that can be used with EPGs of $10^{-3}$ are known to be theoretically efficient, the combined architecture starting with $C_e$ is also theoretically efficient. The problem is that as $\gamma$ approaches the upper limit, the minimum length of the code $C_e$ grows, and as a result the probability of successfully preparing the required states by postselection goes down dramatically, making the combined architecture highly impractical.
\section{Resource Usage} \label{sect:resources}
The simulations keep track of the number of operations of different types that are applied in the course of implementing a quantum network. The resources required depend on whether the networks are implemented with maximum parallelism or sequentially: If they are implemented sequentially, one can take advantage of the ability to abort some computations early, but such implementations require quantum memory of sufficiently low error. Here we consider only the case of maximum parallelism. At the core of the fault-tolerant architecture is Bell pair preparation. One can analyze the resources required to construct a level $l+1$ Bell pair in terms of the number of level $l$ Bell pairs consumed. As a first step, consider the case of zero EPG. In this case no error is ever detected and all networks succeed on the first try. We count only the number of physical qubit state preparations, $p(l,\gamma)$, and the number of physical CNOTs, $c(l,\gamma)$. The number of physical qubit measurements is less than the number of qubit state preparations. A level $0$ (physical) Bell pair requires $p(0,0)=2$ qubit state preparations and $c(0,0)=1$ CNOT. A level $1$ Bell pair requires four level $0$ Bell pairs and four CNOTs for preparing and verifying the initial state of each of the two blocks to be used. Combining the blocks requires another four CNOTs. Thus $p(1,0) = 8p(0,0)$ and $c(1,0) = 8c(0,0)+12$. For $l\geq 1$, a level $l+1$ Bell pair requires three level $l$ Bell pairs and three level $l$ encoded CNOTs for preparing the initial state in each block. Combining the blocks to form the level $l+1$ Bell pair requires $3^{l-1}4\times 3$ physical CNOTs. To remove lower level errors, each of the six level $l$ subblocks is teleported. Each teleportation uses one level $l$ Bell pair and CNOT. Thus $p(l+1,0) = 12 p(l,0)$ and $c(l+1,0) = 12 c(l,0)+3^l 20$.
\ignore{
function p = preps(l);
if (l == 0);
p = 2;
elseif (l == 1);
p = preps(l-1)*8;
else;
p = preps(l-1)*12;
endif; endfunction; function c = CNOTs(l);
if (l == 0);
c = 1;
elseif (l == 1);
c = CNOTs(0)*8+12;
else;
c = CNOTs(l-1)*12 + 3^(l-1) *20;
endif; endfunction;
function [f, m] = fleads(x);
m = floor(log10(x));
f = x*10^(-m); endfunction; for l = (0:6);
p = preps(l); c = CNOTs(l);
[fp, ep] = fleads(p);
[fc, ec] = fleads(c);
if (p > 10^4);
printf('$
l,
fp, ep,
fc, ec);
else;
printf('$
l, p, c);
endif; endfor; }
\begin{heretab} \label{tab:zero_error_resources} \heretabcap{ Table of resources used for logical Bell pair preparation at EPG $\gamma=0$. \\[6pt]}
\begin{tabular}{c|cc} \hline Level & Preparations & CNOTs \\ \hline \expdata{}{Table inserted from octave output above} $0$ & $2$ & $1$ \\ $1$ & $16$ & $20$ \\ $2$ & $192$ & $300$ \\ $3$ & $2304$ & $3780$ \\ $4$ & $2.765{\times} 10^{4}$ & $4.590{\times} 10^{4}$ \\ $5$ & $3.318{\times} 10^{5}$ & $5.524{\times} 10^{5}$ \\ $6$ & $3.981{\times} 10^{6}$ & $6.634{\times} 10^{6}$ \\ \hline \end{tabular} \end{heretab}
With maximum parallelism, the average resource requirements increase by factors inversely related to the probability of success at various points in the preparation process. Preparing an encoded Bell state involves two sequential steps that may fail. The first verifies the initial states of each block before they are combined with CNOTs. Let the probability of successful verification of a block at level $l$ be given by $v(l,\gamma)$. The second involves teleportation of each subblock after the two blocks are combined. Let the overall probability of success of the teleportations be $t(l,\gamma)$. Note that both of these probabilities of success are with respect to the combination of error correction and detection used in state preparation, which differs from the full error correction used in logical computation. The above resource formulas are modified as follows: $p(0,\gamma) = 2$, $c(0,\gamma) = 1$, $p(1,\gamma) = 8p(0,\gamma)/v(1,\gamma)$, $c(1,\gamma) = 8c(0,\gamma)/v(1,\gamma)+12$, $p(l+1,\gamma) = (6p(l,\gamma)/v(l+1,\gamma) + 6p(l,\gamma))/t(l+1,\gamma)$, $c(l+1,\gamma) = ((6c(l,\gamma)+3^l 12)/v(l+1,\gamma)+ 6c(l,\gamma)+ 3^l 8)/t(l+1,\gamma)$. These formulas were obtained under the assumption that the verification of the two blocks proceeds independently with many simultaneous attempts, where the successful ones are then combined. This requires waiting for measurement outcomes and any associated memory error must be accounted for in $\gamma$. The subblock teleportations are not independent because of the immediately preceding transversal CNOT, which introduces correlated errors. Tables~\ref{tab:gamma_resources1},~\ref{tab:gamma_resources2},~\ref{tab:gamma_resources3} show the success probabilities up to level 5 for $\gamma=0.01, 0.001, 0.0001$ together with the resources estimated according to these recursive formulas and the resources determined by the simulation after averaging over the number of attempts made. The simulation is expected to show higher resource requirements because it involves some loss when combining unequal numbers of independently prepared blocks, as would be expected to occur in a real implementation. This was not taken into account in deriving the formulas.
\ignore{ function p = prepsf(l,v,t);
if (l == 0);
p = 2;
elseif (l == 1);
p = prepsf(l-1,v,t)*8/v(1);
else;
p = (6*prepsf(l-1,v,t)/v(l)+6*prepsf(l-1,v,t))/t(l);
endif; endfunction; function c = CNOTsf(l,v,t);
if (l == 0);
c = 1;
elseif (l == 1);
c = CNOTsf(0,v,t)*8/v(1)+12;
else;
c = ((CNOTsf(l-1,v,t)*6+3^(l-1)*12)/v(l)+6*CNOTsf(l-1,v,t)+3^(l-1)*8)/t(l);
endif; endfunction;
function [f, m] = fleads(x);
m = floor(log10(x));
f = x*10^(-m); endfunction; v2 = t2 = v3 = t3 = v4 = t4 = ones(5,1); for k = (1:5);
s2{k} = load('-ascii', \
sprintf('cliff/data/ana6cnts_s1.000_l
v2(k) = (s2{k}.vx(2)+s2{k}.vz(2))/2;
if (k == 1);
t2(k) = 1;
else;
if (struct_contains(s2{k},'pr'));
t2(k) = s2{k}.pr;
else;
t2(k) = s2{k}.p(2);
endif;
endif;
s3{k} = load('-ascii', \
sprintf('cliff/data/ana6cnts_s0.100_l
v3(k) = (s3{k}.vx(2)+s3{k}.vz(2))/2;
if (k == 1);
t3(k) = 1;
else;
if (struct_contains(s3{k},'pr'));
t3(k) = s3{k}.pr;
else;
t3(k) = s3{k}.p(2);
endif;
endif;
s4{k} = load('-ascii', \
sprintf('cliff/data/ana6cnts_s0.010_l
v4(k) = (s4{k}.vx(2)+s4{k}.vz(2))/2;
if (k == 1);
t4(k) = 1;
else;
if (struct_contains(s4{k},'pr'));
t4(k) = s4{k}.pr;
else;
t4(k) = s4{k}.p(2);
endif;
endif; endfor;
function tab = restab(v,t,s);
p = prepsf(1,v,t); c = CNOTsf(1,v,t);
tab = '';
tab = sprintf('
tab, v(1), p, s{1}.preps(2), c, s{1}.CNOTs(2), s{1}.tries);
for l = (2:5);
p = prepsf(l,v,t); c = CNOTsf(l,v,t);
if (p > 10^4);
[fp, mp] = fleads(p); [fc, mc] = fleads(c);
[tfp, tmp] = fleads(s{l}.preps(2));
[tfc, tmc] = fleads(s{l}.CNOTs(2));
tab = sprintf('
tab, l, v(l), t(l), fp, mp, tfp, tmp, fc, mc, tfc, tmc, s{l}.tries);
else;
tab = sprintf('
tab, l, v(l), t(l), p, s{l}.preps(2), c, s{l}.CNOTs(2), s{l}.tries);
endif;
endfor; endfunction;
tab2 = restab(v2,t2,s2) tab3 = restab(v3,t3,s3) tab4 = restab(v4,t4,s4) }
\begin{heretab} \label{tab:gamma_resources1} \heretabcap{ Table of success probabilities and resources used for
Bell state preparation at EPG
$\gamma = 0.01$. \\[6pt]
\begin{minipage}{\textwidth}The values $v(l,0.01)$, $t(l,0.01)$
and the numbers in the ``preparations'' and ``CNOTs''
columns are obtained by simulation using the number of
successful Bell pair preparations shown in the
``\# Bell pairs'' column. Because only two successful preparations
were used at level 5, the level 5 data have significant noise.
\end{minipage}\\[6pt]}
\begin{tabular}{@{}c|ccccccc@{}} \hline Level & $v(l,0.01)$ & $t(l,0.01)$ & $p(l,0.01)$ & Preparations & $c(l,0.01)$ & CNOTs & \# Bell pairs\\\hline $0$ & NA & NA & $2$ & $2$ & $1$ & $1$ & NA \\ \expdata{}{Table inserted from octave output tab2 above} $1$ & $0.940$ & NA & $17.01$ & $17.01$ & $20.51$ & $21.01$ & $10345$\\
$2$ & $0.722$ & $0.247$ & $984.6$ & $1022.2$ & $1485.5$ & $1542.7$ & $2279$\\
$3$ & $0.602$ & $0.100$ & $1.58{\times} 10^{5}$ & $1.60{\times} 10^{5}$ & $2.40{\times} 10^{5}$ & $2.45{\times} 10^{5}$ & $409$\\
$4$ & $0.885$ & $0.205$ & $9.84{\times} 10^{6}$ & $9.63{\times} 10^{6}$ & $1.50{\times} 10^{7}$ & $1.48{\times} 10^{7}$ & $70$\\
$5$ & $0.900$ & $0.500$ & $2.49{\times} 10^{8}$ & $3.08{\times} 10^{8}$ & $3.80{\times} 10^{8}$ & $4.72{\times} 10^{8}$ & $2$\\ \hline \end{tabular} \end{heretab}
\begin{heretab} \label{tab:gamma_resources2} \heretabcap{ Table of success probabilities and resources used for Bell
state preparation at EPG
$\gamma = 0.001$. \\[6pt] }
\begin{tabular}{@{}c|ccccccc@{}} \hline Level & $v(l,0.001)$ & $t(l,0.001)$ & $p(l,0.001)$ & Preparations & $c(l,0.001)$ & CNOTs & \# Bell pairs\\\hline $0$ & NA & NA & $2$ & $2$ & $1$ & $1$ & NA \\ \expdata{}{Table inserted from octave output tab3 above} $1$ & $0.994$ & NA & $16.10$ & $16.11$ & $20.05$ & $20.11$ & $10927$\\
$2$ & $0.970$ & $0.870$ & $225.6$ & $226.6$ & $351.2$ & $352.8$ & $2014$\\
$3$ & $0.957$ & $0.815$ & $3395.6$ & $3434.9$ & $5513.4$ & $5576.1$ & $401$\\
$4$ & $1.000$ & $0.970$ & $4.20{\times} 10^{4}$ & $4.67{\times} 10^{4}$ & $6.88{\times} 10^{4}$ & $7.61{\times} 10^{4}$ & $64$\\
$5$ & $1.000$ & $1.000$ & $5.04{\times} 10^{5}$ & $5.61{\times} 10^{5}$ & $8.27{\times} 10^{5}$ & $9.17{\times} 10^{5}$ & $2$\\ \hline \end{tabular} \end{heretab}
\begin{heretab} \label{tab:gamma_resources3} \heretabcap{ Table of success probabilities and resources used for
Bell state preparation at EPG
$\gamma = 0.0001$. \\[6pt] }
\begin{tabular}{@{}c|c@{\hspace*{3pt}}c@{\hspace*{3pt}}c@{\hspace*{3pt}}c@{\hspace*{3pt}}c@{\hspace*{3pt}}c@{\hspace*{3pt}}c@{}} \hline Level & $v(l,0.0001)$ & $t(l,0.0001)$ & $p(l,0.0001)$ & Preparations & $c(l,0.0001)$ & CNOTs & \# Bell pairs\\\hline $0$ & NA & NA & $2$ & $2$ & $1$ & $1$ & NA \\ \expdata{}{Table inserted from octave output tab4 above} $1$ & $0.999$ & NA & $16.01$ & $16.02$ & $20.01$ & $20.02$ & $10987$\\
$2$ & $0.995$ & $0.984$ & $195.7$ & $204.9$ & $305.6$ & $317.2$ & $2155$\\
$3$ & $0.994$ & $0.982$ & $2398.6$ & $2556.0$ & $3929.9$ & $4158.1$ & $429$\\
$4$ & $1.000$ & $1.000$ & $2.88{\times} 10^{4}$ & $3.13{\times} 10^{4}$ & $4.77{\times} 10^{4}$ & $5.16{\times} 10^{4}$ & $66$\\
$5$ & $1.000$ & $1.000$ & $3.45{\times} 10^{5}$ & $3.92{\times} 10^{5}$ & $5.74{\times} 10^{5}$ & $6.50{\times} 10^{5}$ & $2$\\ \hline \end{tabular} \end{heretab}
Resources for implementing logical gates transversally are dominated by those required for logical Bell state preparation. For example, the logical CNOT includes error-correcting teleportation and therefore requires two logical Bell states and three transversal CNOTs. The number of physical CNOTs in a transversal CNOT grows by a factor of $3$ for each level after the first, whereas the number of physical CNOTs required for logical Bell state preparation grows by a factor greater than $12$. This justifies focusing attention on the resources required for logical Bell state preparation. The biggest resource overhead is incurred when implementing non-Clifford gates such as $\ket{\pi/8}$-preparation (see below) or Toffoli gates. Note that two $\ket{\pi/8}$ states are needed to implement a Toffoli gate up to a reversible phase in the logical basis, which is all that is required for most uses of Toffoli gates. We have not attempted to optimize $\ket{\pi/8}$-preparation. Furthermore, it is possible that gates such as the Toffoli gate can be implemented more efficiently using other states, for example, using Steane's adaptation~\cite{steane:qc1999a} of Shor's method~\cite{shor:qc1996a}.
For completeness and to obtain an upper bound on the requirements for a minimal non-trivial quantum algorithm at $\gamma=0.01$, we outline one method for preparing good logical $\ket{\pi/8}$ states, discuss why the error when using these states is expected to be similar to that of one logical CNOT and estimate the average number of logical CNOTs required. A straightforward method for preparing a noisy logical $\ket{\pi/8}$-state is to prepare a logical Bell state, decode the first block and make a measurement in the basis $\ket{\pi/8},\ket{5\pi/8} = -\sin(\pi/8)\ket{0}+\cos(\pi/8)\ket{1}$ of each of the two decoded, now physical qubits. Note that $\ket{5\pi/8}$ differs from $\ket{\pi/8}$ by a $Y$ operator, so any measurement outcome is acceptable and can be accounted for by a change in Pauli frame if necessary. The simulations indicate that if the measurement has the same error probability as a $Z$- or $X$-measurement, then the error $\epsilon_{\pi/8}$ in the logical prepared state is near the EPG parameter $\gamma$. To reduce the noise in the logical $\ket{\pi/8}$ states, one can purify them. The simplest purification method known so far involves using $15$ prepared $\ket{\pi/8}$ states. One is encoded into the $[[7,1,3]]$ code~\cite{steane:qc1995a} (a code that encodes $1$ qubit in $7$ qubits with minimum distance $3$, which implies that it can correct any $(3-1)/2=1$ qubit error or detect any $(3-1)=2$ qubit errors). The other $2\times 7$ $\ket{\pi/8}=14$ states are used to implement a conditional logical HAD from an ancilla to realize an encoded HAD measurement. Note that $\ket{\pi/8}$ is the $+1$ eigenstate of HAD. In the last step, the $[[7,1,3]]$ code is decoded. If the measurement outcomes are as would be expected if no error had occurred, the state is accepted and has much reduced conditional error. The method is equivalent to Bravyi and Kitaev's scheme~\cite{bravyi:qc2004a} (Reichardt, private communication) and can be analyzed using their formulas. With no error in the $\cG_{\textnormal{\tiny min}}$ and HADs used to implement the procedure, the probability of error in successfully purified gates is $\epsilon^{(1)}_{\pi/8} = 35\epsilon_{\pi/8}^3+O(\epsilon_{\pi/8}^4)$. The probability of success is $p^{(1)}_{\pi/8} = 1-35\epsilon_{\pi/8}-O(\epsilon_{\pi/8}^2)$. Using the exact formulas, for $\epsilon_{\pi/8} = 0.01, 0.001, 0.0001$ we obtain $\epsilon^{(1)}_{\pi/8} = 3.6\times 10^{-5}, 3.5\times 10^{-8}, 3.5\times 10^{-11}$ and $p^{(1)}_{\pi/8} = 0.860, 0.985, 0.999$. The purification can be iterated, but for the example considered below, the dominant errors are from the logical $\cG_{\textnormal{\tiny min}}$ and HAD gates, so we use only one purification stage.
\ignore{
function e1 = eps8(e);
e1 = (1-15*(1-2*e)^7+15*(1-2*e)^8-(1-2*e)^15)/ \
(2*(1+15*(1-2*e)^8)); endfunction; function p1 = p8(e);
p1 = (1+15*(1-2*e)^8)/16; endfunction;
}
Consider the effect of logical gate error on the error in the purified $\ket{\pi/8}$. We conjecture that by using state injection with Steane's fault-tolerant methods for preparing states, the additional error on the purified $\ket{\pi/8}$ state is dominated by a decoding error of the order of the logical CNOT error. Specifically, one can encode one noisy logical $\ket{\pi/8}$ by teleportation into the $[[7,1,3]]$ code using a Bell state correlating a logical qubit and a $[[7,1,3]]$-encoded qubit. (Strictly speaking, our architecture requires the use of logical qubit pairs associated with blocks of the $C_4/C_6$ codes, but we treat each qubit in a pair identically.) This Bell state has minimum distance $4$, so that any combination of Pauli errors on up to three qubits results in an orthogonal state. It can therefore be well verified using Steane's methods. The error in the state teleported into the $[[7,1,3]]$ code is due to the initially prepared $\ket{\pi/8}$ state, initial error in the logical qubit of the Bell state used, and the CNOT and measurements needed for the teleportation Bell measurement. Because we are operating with logical qubits of the $C_4/C_6$ architecture, all but the first of these errors are comparatively small, assuming that the $C_4/C_6$ encoding level is chosen so as to significantly decrease CNOT errors. The errors have two effects. One is to modify the encoded state, which can be subsumed by considering this as additional error in the initial $\ket{\pi/8}$ state. The other is to perturb the syndrome of the encoded state. If two or fewer errors occurred, this can be detected in the decoding stage. The encoded state is verified using the controlled-HADs implemented with the other $14$ noisy logical $\ket{\pi/8}$ states. Each of these controlled-HADs involves at most five CNOTs~\cite{knill:qc2004a}. The error is dominated by that in the $\ket{\pi/8}$ states used. Additional error due to the logical CNOT either has a smaller effect, to be detected in decoding, or results in the wrong outcome in the encoded HAD measurement. The latter event could cause unintentional acceptance of the final state, but only if additional error occurred elsewhere. At the end of the procedure, the $[[7,1,3]]$-encoded qubit is decoded and the syndrome verified. One can decode directly or by reverse teleportation through the same type of Bell state used for the initial teleportation, verifying the syndrome in the teleportation process. The latter method may be more robust. In all cases, the effect of additional errors are either suppressed by the fault-tolerant methods used to encode and decode the $[[7,1,3]]$ code, or can be subsumed as a relatively small amount of additional error in the initial $\ket{\pi/8}$ states due to at most five logical CNOTs. Based on experience with $C_4/C_6$ codes, it is likely that the additional error from encoding and decoding is of the order of that of a logical CNOT, whereas the effective additional $\ket{\pi/8}$ error should be sufficiently small (because of significant decrease in CNOT errors at the level chosen) to have little effect on the error in the purified $\ket{\pi/8}$.
We estimate the number of logical CNOTs needed for the $\ket{\pi/8}$ purification process. The Bell state needed for injection into the $[[7,1,3]]$ code can be prepared from a logical Bell state by encoding one of the two blocks into the $[[7,1,3]]$ code. $11$ CNOTs suffice for encoding. The resulting state can be verified using Steane's methods. There are eight syndromes each of weight $4$ to check, each requires an ancilla preparation with five CNOTs and four CNOTs for the syndrome check. If memory is an issue, it may be necessary to add error-correcting teleportations not associated with a gate. We do not consider this here but note that this may add another four logical Bell states per syndrome check to the resources required. If the robust decoding scheme is used, two of the injection Bell states are required overall. The verification process using controlled-HADs requires about $5\times 7$ CNOTs. This gives a total of $201$ CNOTs, but does not take into account the probability of failure in the various checks. We can estimate this probability as $1-(1-p)^{201}$, where $p$ is the probability of detected error in a logical CNOT. For the relevant parameters, $p$ is below $\expdata{0.003}{detected error probability at $\gamma=0.01$ and level $4$}$. Taking the average number of trials required due to logical gate failure as $1/(1-p)^{201}$, we can upper bound the average number of CNOTs required as $\expdata{370}{computed here}$.
An obvious optimization of the $\ket{\pi/8}$-purification method in the context of the fault tolerant $C_4/C_6$ architecture is to concatenate with the $[[7,1,3]]$ code as a last level, lifting all logical states accordingly, but injecting $\ket{\pi/8}$ states to the last $C_4/C_6$ level as before for $\ket{\pi/8}$ purification purposes. This avoids having to decode the purified $\ket{\pi/8}$ states while achieving significantly lower error probabilities. Within the $C_4/C_6$ scheme, if more than one purification stage is required, it may be worthwhile injecting and purifying states at intermediate levels before injecting and purifying at the top.
As an example, consider $\gamma=0.01$, aiming for implementing a non-trivial quantum computation. The smallest non-trivial quantum computation must be one involving more qubits than can be directly simulated on existing classical computers. $100$ qubits is a safe number for this property. Such a quantum computation should also apply sufficiently many gates for a classical simulation with current computers not to be able to predict the output of the quantum computation by taking advantage of restrictions on the reachable states. If the number of gates applied involves sufficiently many parallel steps of non-Clifford gates involving all qubits, this is expected to be the case. Short of having an explicit example of a computation whose output is unknown and not believed to be accessible to classical computers, we assume that $10$ steps involving parallel CNOTs, HADs and $\ket{\pi/8}$-preparations suffice\footnote{Finding a computation with as few as $100$ qubits and fewer than $10^4$ gates with a definite and convincing answer of interest independent of quantum information theory would be very helpful and could be a boon for quantum information processing. For comparison, all fully worked out computations of this sort seem to require that the number of gates greatly exceeds $10^9$.}. We therefore take $1000$ as a minimal number of gates in a non-trivial quantum algorithm. Note that with EPGs of $0.01$ it is not possible to combine this many physical gates and still expect that a computation's output can be discerned. If more than $68$ physical gates at this EPG are applied, the probability that the output is correct cannot be guaranteed to be strictly greater than $0.5$. Although the output of a computation with such few gates may already be difficult to simulate with current classical computers, it is conceivably possible to do so.
Consider level $4$ of our scheme at $\gamma=0.01$. The detected error probability of a logical CNOT is $p_d=\expdata{2.4\aerrb{1.0}{0.7}\times 10^{-3}}{cliff/finaldata/2teleec_l4_de.tex[0.01,:]}$. The conditional probability of a logical error is much lower and estimated as $p_c=\expdata{2.3\times 10^{-5}}{see below}$ (see below). The logical $\ket{\pi/8}$-purification method ensures that similar error probabilities apply to uses of these states in the algorithm. The probability that there is a detected failure in $1000$ gates is $1-(1-p_d)^{1000} \approx \expdata{0.91}{computed here}$. Thus, the algorithm needs to be applied $\expdata{11.1}{computed here}$ times on average before a successful answer is obtained. Once an answer is obtained, its error probability is $1-(1-p_c)^{1000} \approx \expdata{0.02}{computed here}$. The average number of physical CNOTs required for obtaining an answer can now be estimated as \begin{equation} \expdata{1.23\times 10^{14}}{computed here} =
\begin{array}[t]{lclcl}
\underbrace{1000}_{
\textrm{\begin{tabular}{c}logical gates\\ in computation\end{tabular}}
} &\times&
\underbrace{370}_{
\textrm{\begin{tabular}{c} logical CNOTs per\\ purified $\ket{\pi/8}$-prep\end{tabular}}
} &\times&
\underbrace{1/0.09}_{
\textrm{\begin{tabular}{c}prob.${}^{-1}$~of \\ overall success\end{tabular}}
} \\[.7in] &\times &
\underbrace{2\times 1.5\times 10^7}_{
\textrm{\begin{tabular}{c}physical CNOTs \\ per logical CNOT\end{tabular}}
}
\end{array} \end{equation} Although this number of physical CNOTs vastly exceeds current capabilities, it may be compared to typical resources available today for classical computation. For example, today's central processing units in desktop computers have more than $\expdata{10^8}{http://computer.howstuffworks.com/microprocessor2.htm}$ transistors and operate at rates above $10^9$ bit operations per second~\cite{intel.com:qc2004a}.
Resource requirements for implementing a given computation decrease significantly with $\gamma$. Simulation is too inefficient for resolving the dependence of resource requirements on $\gamma$, particularly when error probabilities are extremely small. We therefore obtain and verify simple models for resources and errors as a function of $\gamma$ and level of concatenation. Ideally, we would like to obtain analytic expressions, however this is difficult to do, particularly since our schemes are not strictly concatenated, and the combination of error-detection and correction behaves differently depending on the level. Nevertheless, it is possible to derive functional forms for Bell state preparation resources and logical CNOT error behavior that are asymptotically valid as $\gamma$ goes to $0$.
We model the number of physical CNOTs required for preparing logical Bell states at level $l$ as $\textrm{rbell}(l,\gamma) = P(l)/(1-\gamma)^{k(l)}$. This is a naive model based on assuming that the resources are determined by applying a network with $P(l)$ physical CNOT gates, $k(l)$ of which fail independently with probability $\gamma$ each, and the network is repeatedly applied until no failure is detected. Perhaps surprisingly, this model matches the simulations well in the range shown in Fig.~\ref{fig:res_naive}.
\begin{herefig} \label{fig:res_naive} \begin{picture}(0,3.7)(0,-3.5)
\nputgr{0,0}{t}{height=3in}{opcnts_pl} \nputbox{-3,0}{tr}{\rotatebox{90}{\textsf{\small Number of physical CNOTs per logical Bell pair}}} \nputbox{3,-3.0}{tr}{\textsf{\small Physical CNOT error probability $\gamma$.}}
\end{picture} \herefigcap{Graphs of resources for logical Bell pair preparation. The points are obtained by simulation and counting the number of physical CNOTs used in preparing a number of logical Bell pairs at different values of $\gamma$. The error bars are not statistically rigorous. They are standard deviations computed from the number of prepared Bell pairs and assuming the naive model described in the text. The curves are least-squares fits to the data with functions of the form $P(l)/(1-\gamma)^{k(l)}$. } \end{herefig}
To understand the error behavior of the $C_4/C_6$ architecture, suppose more generally that we have a fault-tolerant scheme ${\cal A}$ for implementing an encoded gate, which results in a detected, uncorrectable error with probability $p_d$, or an undetected logical error with probability $p_c$, conditional on not having detected an error. Suppose that this is concatenated with a one-error detecting (minimum distance $2$) code $C$ and used in a scheme similar to the ones used here. $C$ can correct any error at a known location. If the implementation of $C$-encoded gates is fault tolerant and includes error-correcting teleportation or another method for determing the $C$-syndrome, any one error detected by ${\cal A}$ can be corrected with no resulting encoded error. The event that an error is detected but not correctable during implementation of a $C$-encoded gate therefore requires at least one undetected error or at least two detected errors. The conditional event that an undetected error occurs requires that the ${\cal A}$ gates used have one detected and one undetected error or two or more undetected errors. To lowest order, the detected and conditional error probabilities for $C$-encoded gates are therefore of the form $p'_d = D_c p_c + D_d p_d^2$ and $p'_c = L_c p_c^2 + C_d p_d C_c $ In our case, $p_d$ and $p_c$ depend on one parameter $\gamma$. After level $1$, the order in $\gamma$ of $p_c$ is always between $p_d$ and $p_d^2$, so the expressions can be simplified to $p'_d = D p_c$, $p'_c = C p_d p_c$, to lowest order in $\gamma$. Let $p_d(l)$ and $p_c(l)$ be the detected and logical error probabilities at level $l$ in the $C_4/C_6$ architecture. Using the above, we can write $p_d(l+1) = D(l)p_c(l)$ and $p_c(l+1)=C(l)p_d(l)p_c(l)$. At level $1$, $p_d(1) = d(1)\gamma$ and $p_c(1) = c(1)\gamma^2$, to lowest order in $\gamma$. Expanding the recursion at higher levels, we obtain $p_d(2) = d(2)\gamma^2, p_c(2) = c(2)\gamma^3$, $p_d(3) = d(3)\gamma^3, p_c(3) = c(3)\gamma^5$. The Fibonacci sequence $f(0) = 0, f(1) = 1, f(n+1) = f(n)+f(n-1)$ emerges as the relevant exponent so that $p_d(l) = d(l)\gamma^{f(l+1)}, p_c(l)= c(l)\gamma^{f(l+2)}$. As is typical of concatenation schemes, the exponent grows exponentially.
In view of the previous paragraph, we examine the data shown in Fig.~\ref{fig:ecEPG} to determine $C(l), D(l)$ for $l=1,2$ and $c(l), d(l)$ for $l=1,2,3$. The results are shown in Table~\ref{tab:errcnsts}. We computed the values of $c(l)$ and $d(l)$ by fitting the model curves to the error probabilities obtained by simulation. The points at $\gamma = 0.01$ were omitted for levels $1$, $2$, and $3$ to reduce the chance of introducing optimistic biases by the curves' leveling off at higher $\gamma$, although this effect has not been observed. We obtained the fits by starting with a least-squares fit of the log-log plots and then using a fastest-descent method to optimize the likelihood. We computed standard deviations by resampling the data according to the fitted curve and repeating the fitting process. The fitted curves are shown with the data in Fig.~\ref{fig:ecfibfits}. $C(l)$ and $D(l)$ were computed from $c(l)$, $c(l+1)$, $d(l)$ and $d(l+1)$ by solving the equations. Their uncertainty intervals are found by linear error analysis.
\begin{heretab} \label{tab:errcnsts} \heretabcap{ Table of $d(l)$, $c(l)$, $D(l)$ and $C(l)$ with uncertainty intevals based on standard deviations.
\\[6pt] }
$\begin{array}{@{}c|cccc@{}} \hline
\textrm{Level} & d(l) & c(l) & D(l) & C(l) \\\hline \expdata{}{data produced by octave code following this table.} 1 & 37.0\serrb{0.1} & 35.2\serrb{1.5} & 29.94\serrb{1.13} & 3.43\serrb{0.01} \\
2 & 1.06\serrb{0.01}{\times}10^{3} & 4.47\serrb{0.18}{\times}10^{3} & 4.87\serrb{0.14} & 1.69\serrb{0.14} \\
3 & 2.18\serrb{0.02}{\times}10^{4} & 7.95\serrb{1.01}{\times}10^{6} & 3.01\serrb{0.70} & \textrm{NA} \\
4 & 2.39\serrb{0.86}{\times}10^{7} & \textrm{NA} & \textrm{NA} & \textrm{NA} \\ \hline \end{array}$ \end{heretab} \ignore{
stat1d = load('cliff/finaldata/2teleec_to0.9pwr_l1_de.ipol'); stat1c = load('cliff/finaldata/2teleec_to0.9pwr_l1_be.ipol'); stat2d = load('cliff/finaldata/2teleec_to0.9pwr_l2_de.ipol'); stat2c = load('cliff/finaldata/2teleec_to0.9pwr_l2_be.ipol'); stat3d = load('cliff/finaldata/2teleec_to0.9pwr_l3_de.ipol'); stat3c = load('cliff/finaldata/2teleec_to0.9pwr_l3_be.ipol'); stat4d = load('cliff/finaldata/2teleec_topwr_l4_de.ipol');
function [f,df, m] = flead(x,dx);
m = floor(log10(x));
f = x*10^(-m);
df = dx*10^(-m); endfunction;
function [D,C] = concrate(d0, c0, d1, c1);
vD = d1(1)/c0(1);
vC = c1(1)/(d0(1)*c0(1));
dD = abs((-d1(1)/(c0(1)^2))*c0(2) + \
(1/c0(1))*d1(2));
dC = abs((-c1(1)/(d0(1)^2*c0(1)))*d0(2) + \
(-c1(1)/(d0(1)*c0(1)^2))*c0(2) + \
(1/(d0(1)*c0(1)))*c1(2));
D = [vD, dD]; C = [vC, dC];
endfunction;
tab = ''; sd1 = stat1d(3); dsd1 = sqrt(stat1d(6)); md1 = 0; sc1 = stat1c(3); dsc1 = sqrt(stat1c(6)); mc1 = 0; [sd2, dsd2, md2] = flead(stat2d(3), sqrt(stat2d(6))); [sc2, dsc2, mc2] = flead(stat2c(3), sqrt(stat2c(6))); [D1, C1] = concrate([sd1,dsd1]*10^md1, [sc1, dsc1]*10^mc1, \
[sd2, dsd2]*10^md2, [sc2, dsc2]*10^mc2); [sd3, dsd3, md3] = flead(stat3d(3), sqrt(stat3d(6))); [sc3, dsc3, mc3] = flead(stat3c(3), sqrt(stat3c(6))); [D2, C2] = concrate([sd2,dsd2]*10^md2, [sc2, dsc2]*10^mc2, \
[sd3, dsd3]*10^md3, [sc3, dsc3]*10^mc3); [sd4, dsd4, md4] = flead(stat4d(3), sqrt(stat4d(6))); [D3, C3] = concrate([sd3,dsd3]*10^md3, [sc3, dsc3]*10^mc3, \
[sd4, dsd4]*10^md4, [1, 0]);
tab = sprintf('
tab, 1, sd1, dsd1, sc1, dsc1, \
D1(1), D1(2), C1(1), C1(2) \
); tab = sprintf('
tab, 2, sd2, dsd2, md2, sc2, dsc2, mc2, \
D2(1), D2(2), C2(1), C2(2) \
);
tab = sprintf('
tab, 3, sd3, dsd3, md3, sc3, dsc3, mc3, \
D3(1), D3(2) \
);
tab = sprintf('
tab, 4, sd4, dsd4, md4 \
);
tab
global D; D = [sd3*10^md3, D2(1)*ones(1,1000)]; global C; C = [sc3*10^mc3, C2(1)*ones(1,1000)];
function [pd, pc] = pes(l, g);
global D;
global C;
if (l < 3); return; endif;
if (l == 3);
pd = D(1)*g^3;
pc = C(1)*g^5;
else;
[pdp, pcp] = pes(l-1, g);
pd = D(l-2)*pcp;
pc = C(l-2)*pdp*pcp;
endif; endfunction;
[pd, pc] = pes(30,0.028066); [pd, pc] = pes(30,0.028067);
probfid = fopen('cliff/finaldata/probs_pl.ipol','w');
fprintf(probfid, '
fprintf(probfid, '
fprintf(probfid, '
trnsD = D2(1);
trnsC = C2(1);
dl = sd2*10^md2;
cl = sc2*10^mc2;
for l = (3:10);
cln = dl*cl*trnsC;
dln = cl*trnsD;
dl = dln; cl = cln;
fprintf(probfid, '
endfor; fclose(probfid);
}
\pagebreak \begin{herefig} \label{fig:ecfibfits} \begin{picture}(0,7.2)(0,-7) \nputgr{0,0}{t}{height=3in}{2teleec_to09pwr_le} \nputbox{-3.0,0}{tr}{\rotatebox{90}{\textsf{\small Logical CNOT conditional error probability}}}
\nputbox{-3.0,-3.3}{tr}{\rotatebox{90}{\textsf{\small Logical CNOT detected error probability}}} \nputgr{0,-3.3}{t}{height=3in}{2teleec_to09pwr_de} \nputbox{3.0,-6.4}{tr}{\textsf{\small Physical CNOT error probability $\gamma$.}} \end{picture} \herefigcap{Fits to the error data for the logical CNOT. The model assumed is $p_d(l) = d(l)\gamma^{f(l+1)}, p_c(l)= c(l)\gamma^{f(l+2)}$, where $f$ is the Fibonacci sequence. The constants $d(l)$ and $c(l)$ are obtained by a maximum-likelihood method from the data points in the range of the solid lines. The gray dashed lines are extrapolations.} \end{herefig} \pagebreak
The constants $D(l), C(l)$ are significantly reduced for going from level $2$ to level $3$ compared to going from level $1$ to $2$. Level $2$ is the first stage of using $C_6$ and the first where error correction can be used. One may conjecture that the level $2$ to level $3$ behavior persists or improves at higher levels, as is the case for $D(3)$ compared to $D(2)$. For the purposes of modeling errors we use this conjecture to recursively obtain $d(l+1)$ and $c(l+1)$ with $D(2)$ and $C(2)$ in place of $D(l)$ and $C(l)$ for $l>2$. It is an interesting exercise to use the recursion implied by the $D(l)$ and $C(l)$ to obtain a threshold. The threshold thus obtained is conjectural, because the approximations made are not strictly valid, particularly at high $\gamma$, and because of the extrapolation of $D(l)$ and $C(l)$. By implementing the recursion numerically, we obtained a threshold of ${\approx}\expdata{0.028}{octave above}$ for this architecture, which does not seem unreasonable in view of the data shown in Fig.~\ref{fig:ecEPG}. Of course, the resource overheads diverge as any such threshold is approached from below.
We return to the question of resource requirements for implementing gates at $\gamma < 0.01$. As $\gamma$ decreases, the physical resources required per logical CNOT are reduced in two ways. First, the state preparation success probabilities at a given level of concatenation increase, see Tables~\ref{tab:gamma_resources1},~\ref{tab:gamma_resources2} and~\ref{tab:gamma_resources3} and Fig.~\ref{fig:res_naive}. This increase is particularly notable near the upper limit for $\gamma$. Second, fewer levels of concatenation suffice for achieving sufficiently low logical errors. Consider implementing a computation ${\cal C}$ with the product of the number of logical gates and average number of qubits per gate given by $KQ$. For computations that are not maximally parallel, this quantity should include memory delays in the gate count. To simplify the resource estimates, logical errors and physical gate counts are given in terms of ``effective'' error and physical gate counts per (logical) qubit and gate. For example, consider the logical cnot in the $C_4/C_6$ architecture. It acts on two logical qubit pairs, so its effective error per qubit is $1/4$ of its total error. Similarly, its effective physical gate count per qubit is $1/4$ of the total gate count. With this simplification, we can estimate the total error and number of physical gates for implementing the computation ${\cal C}$ by multiplying $KQ$ by the the appropriate effective quantity and a nontransversal-gate state preparation overhead. In making these estimates, we assume that (1) each of the logical gates needed by ${\cal C}$ can be implemented with effective error similar to that of the logical CNOT, (2) the implementation can take advantage of both logical qubits in the logical qubit pairs and (3) overhead for addressing individual logical qubits in the pairs is accounted for in the nontransversal-gate state preparation overhead. The assumptions require that the nontransversal-gate state preparations have the property that logical gates used in the preparations do not contribute additional error, as is the case for the $\ket{\pi/8}$ state preparation described above. The reason for not including the nontransversal-gate state preparation overhead in the effective quantities per qubit and gate is that this overhead can be optimized independent of the architecture and depends on the choice of elementary nontransversal gates. It is expected to add one to two orders of magnitude to the total implementation resources.
We estimate the optimal effective number $\textrm{pcnot}(KQ,\gamma)$ of physical CNOTs per qubit and gate as a function of the size $KQ$ of ${\cal C}$ and the EPG parameter $\gamma$. As noted above, other physical resources such as state preparation and measurement are comparable. We optimize $\textrm{pcnot}(KQ,\gamma)$ by choosing the level $l$ of the $C_4/C_6$ architecture and use it to repeatedly implement ${\cal C}$ until no uncorrectable error is detected in the logical gates. At this point the output of ${\cal C}$ must be correct with probability at least $2/3$. The value of $2/3$ is chosen to be strictly between $1/2$ and $1$ but otherwise not crucial. At the minimizing level, $\textrm{pcnot}(KQ,\gamma)$ is computed as the product of the average number of times ${\cal C}$ must be implemented until no error is detected and $1/2$ of the number of physical CNOTs, $\textrm{rbell}(l,\gamma)$, needed to prepare a logical Bell state (neglecting the relatively small additional number of physical cnots needed for transversal gates and for using the Bell state in an error-correcting teleportation). The factor of $1/2$ accounts for having two qubits in each block of the $C_4/C_6$ concatenated codes. The probability of success of a single instance of ${\cal C}$ can be estimated as $(1-p_d(l,\gamma)/4)^{KQ}$, which is approximately correct for our accounting using effective errors per qubit and gate, provided that $p_d(l,\gamma)$ is small. On average, ${\cal C}$ must be tried $1/(1-p_d(l,\gamma)/4)^{KQ}$ times to successfully obtain the output. The conditional probability of a successful output's being correct is $(1-p_c(l,\gamma)/4)^{KQ}$. Thus, given $KQ$, the optimal $\textrm{pcnot}(KQ,\gamma)$ is obtained as the minimum over $l$ of ${1 \over 2}\textrm{rbell}(l,\gamma)/(1-p_d(l,\gamma)/4)^{KQ}$ subject to $(1-p_c(l,\gamma)/4)^{KQ}\geq 2/3$. Curves for $\textrm{pcnot}(KQ,\gamma)$ for various $KQ$ as a function of $\gamma$ are plotted in Fig.~\ref{fig:resource_graph}.
The quantity $\textrm{pcnot}(KQ,\gamma)$ gives the overall ``work'' overhead for implementing a computation using the $C_4/C_6$ architecture, but does not differentiate between parallel and sequential resources or indicate the number of physical qubits needed per logical qubit (``scale-up''). The $C_4/C_6$ architecture does not determine these resources uniquely, as they depend on how the trade-off between parallelism and requirements for memory is resolved. In the case of maximum parallelism, the scale-up is close to $\textrm{pcnot}(KQ,\gamma)$. If minimum parallelism is used, this can be reduced to a small multiple of the minimum scale-up associated with the $C_4/C_6$ concatenated code at the level $l$ that is used. This minimum scale-up is given by $3^{l-1}2$ (taking into account that there are two qubits per block of $3^{l-1}4$ qubits). If there is no memory error at all, the additional overhead per block can be minimized by operating on only one block at a time. Otherwise, for each block, two additional blocks are needed in error-correcting teleportation. Logical Bell state preparation requires an additional overhead depending on the degree of parallelism required. If the subblock teleportations in the preparation are done in parallel, and taking into accounting lower level Bell state preparations, two more blocks or equivalent are needed for each level other than the first. This means that $1+2(l-1)$ blocks are needed per computational block. The $\ket{\pi/8}$-state preparation has additional overhead. Depending on how it is implemented it may require up to $14$ blocks with their own overhead of $1+2(l-1)$ or more blocks each. The contribution of $\ket{\pi/8}$-state preparation can be minimized by implementing the logical part of the computation sequentially but using memory steps to remove the effects of memory error as needed. Based on these estimates, the scale-up for low but not minimum parallelism is $\approx 3^{l-1}2(1+2(l-1))$. At levels $2$, $3$, $4$, this evaluates to $18$, $90$, $378$, respectively.
The error-correcting $C_4/C_6$ architecture is relatively simple and designed to work well at high EPGs. However, there is a minimum resource cost (of order $10^3$ per gate and qubit) to use it since error-correction kicks in only at level $2$. As a result, at low EPGs, architectures such as Steane's~\cite{steane:qc2002a} based on more efficient codes with little or no concatenation are more efficient and have more flexibility in achieving the desired logical error probabilities. This effect can be quantified by comparing the $C_4/C_6$ architecture to that of Steane using the illustrative example at $\gamma\approx 10^{-4}$ worked out in~[\citeonline{steane:qc2002a}]. Steane's error model differs from ours in that preparation, measurement and one-qubit gates all have error probability $\gamma$. In our analysis, preparation and measurement errors are $4\gamma/15$, which we justified with a purification scheme. This scheme could also be used in the context of Steane's error model. We compare the two architectures based on the resources per logical qubit of one logical step such as a CNOT, for which the $C_4/C_6$ architecture does not require one-qubit gates other than preparation and measurement. Steane's error model also includes memory error ($\gamma/100$ per step) and accounts for measurement times in excess of gate times ($25$ times the gate time). In our model and in the maximally parallel setting, this would require an additional error of $\gamma/4$ per qubit at the end of state preparation to delay for measurement outcomes that determine whether the state is good or not. The comparison is also complicated by Steane's method deferring some error correction to later steps (we do not account for the implicit overhead in this) and by our method having both detected and conditional logical error, with the latter typically being much lower (we use only the detected error for comparison).
Steane's example is based on a $[[127,43,13]]$ code, which encodes $43$ logical qubits. Full error correction of a block requires about $1.8\times 10^4$ physical CNOTs on average and has a probability of logical error (called ``crash probability'' in~[\citeonline{steane:qc2002a}]) of $\approx 3\times 10^{-10}$. This translates to $\approx 420$ physical CNOTs per qubit and gate and an effective error of $\approx 7\times 10^{-12}$ per logical qubit. The $C_4/C_6$ architecture at level 3 uses $\expdata{4158.1}{from table~\ref{tab:gamma_resources3}}$ physical CNOTs for an error-correcting teleportation. Including $36$ physical gates for a transversal operation, this gives $\approx \expdata{2100}{a little more than the previous number /2}$ physical CNOTs per qubit and gate. The detected error probability for a logical CNOT was estimated above as $\expdata{2.2\times 10^{-8}}{C*.0001^3 with C from cliff/finaldata/2teleec_to0.9pwr_l3.ipol[3]}$, which translates to $\approx \expdata{5.5 \times 10^{-9}}{previous number /4}$ effective error per qubit and gate. To meet the effective error probability achieved by Steane requires another level of encoding. At level $4$, the $C_4/C_6$ architecture uses $\expdata{2.6{\times}10^4}{from table~\ref{tab:gamma_resources3}, /2}$ physical CNOTs per qubit and gate and with a detected error probability of $\expdata{6\times 10^{-14}}{C*.0001^5/4 with C from cliff/finaldata/2teleec_to0.9pwr_l4_de.ipol[3]}$ per qubit and gate. One can also compare the scale-up for the two architectures at $\gamma=10^{-4}$: Steane's example has a scale-up of between $10$ and $20$ compared to from $378$ to over $2000$ for the $C_4/C_6$ architecture at level $3$, depending on parallelism. As expected, Steane's architecture requires fewer resources at low EPGs. It is however notable that the $C_4/C_6$ architecture requires only two orders of magnitude more resources at EPGs as low as $\gamma=10^{-4}$. The $C_4/C_6$ architecture has the advantage of simplicity and of yielding more reliable answers, conditional on having no detected errors.
\ignore{ Steane's ``illustrative example'' parameters from \cite{steane:qc2002a}. 43 logical qubits. n_{rep} = 2.5: number of ancilla blocks maintained per block per X or Z. w = 47: Maximum row/column weight of latin square. N_A = 1802: Total weight. Number of CNOT/csgn in verification. N_{GV} = 3689: Total number of gates in the G and V networks. N_{h} = 8893: Number of memory steps in the G and V networks. r = 5: r' = 4: r'' = 3: t_R = 143: recovery time steps. N_A/w = 38: gates per time step.
Consider one X or Z recovery step (need one of each per block): alpha = 0.74: Fraction of prepared ancillas passing verification on first try. beta = 0.8: Fraction of syndromes that are zero on first extraction try. (1-beta)*(r-1): Additional syndrome extractions.
Recovery gate count: 2*(N_{GV}/0.74)(0.8+0.2*5) = 17946
\bar p = 3 10^{-10} : Apparently the total probability of error for both recovery steps for one block, but not clear in the example.
This is ignoring the problem of trying to catch errors at later recoveries (delayed error recovery).
Regarding Steane's resources in \cite{steane:qc1999a}: Focuses on ``scale-up'' rather than overall work requirement. Scale-up of 22 for KQ \approx 10^16.
From \cite{steane:qc1997a}, apparently improving \cite{preskill:qc1998a}: KQ = 2*10^10 (*2 for this being two qubit gates, includes a factor
of 10 for Toffoli gate state preps) T(total number of CNOTs or equivalent, but includes the Toffoli overhead) = 4*10^16 at \gamma=3.2*10^{-5}. Hence, for comparison, the overhead is about 2*10^5. Generally, \cite{steane:qc1997a} comparable overheads are of the order of 10^5.
}
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Proposed all-versus-nothing violation of local realism in the Kitaev spin-lattice model} \author{Ming-Guang Hu} \author{Dong-Ling Deng} \author{Jing-Ling Chen}
\email{[email protected]} \affiliation{Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071, People's Republic of China}
\date{\today}
\begin{abstract} We investigate the nonlocal property of the fractional statistics in Kitaev¡¯s toric code model. To this end, we construct the Greenberger-Horne-Zeilinger paradox which builds a direct conflict between the statistics and local realism. It turns out that the fractional statistics in the model is purely a quantum effect and independent of any classical theory.We also discuss a feasible experimental scheme using anyonic interferometry to test this contradiction. \end{abstract}
\pacs{03.65.Ud, 03.67.Mn, 75.10.Jm, 42.50.Dv}
\maketitle
Quantum theory can predict results which are never achievable from the local realism (LR)~\cite{A.Einstein}. By definition, LR consists of two constraints of realism and locality: Any observable has a predetermined value, regardless of whether it is measured or not, and the choice of which observable to measure on one party of a multipartite system does not affect the results of the other parties. These constraints lead not only to the well-known Bell inequalities~\cite{J.S.Bell} which put bounds on the correlations and are violated statistically by certain quantum states, but also to the so-called Greenberger-Horne-Zeilinger (GHZ) paradox~\cite{GHZ-TH} which derives directly the inconsistent values of the correlations from LR and quantum mechanics. Such paradox is tested by the nonstatistical measurements, yielding succinctly an all-versus-nothing proof between LR and quantum mechanics.
In this work we investigate LR in the context of Kitaev's toric code spin-lattice model, which is an exactly solvable model and is crucial for fault-tolerant topological quantum computation (TQC)~\cite{2003Kitaev,2006Kitaev}. This model has the merits: Its degenerate ground states yield a topologically protected subspace that provides robustness against noise and quasilocal perturbations, arousing much interest in condensed matter and quantum optical physics to realize and control it \cite{2008Jiang,2006Micheli,2003Duan}; Excitations of the ground states known as anyons possess a class of fractional statistics \cite{1982Wilczek} intervening between the bosonic and fermionic statistics, that is, the quantum state of anyons can acquire an unusual phase factor when one anyon is exchanged with another one, in contrast to usual values $+1$ for bosons and $-1$ for fermions.
Since anyons are at the heart of TQC \cite{2007Nayak}, it is natural and important to ask whether the anyonic statistics, though defined in quantum mechanics can be described by LR. With this aim, we construct the GHZ paradox by using the string operators that are used in the model to move anyons on the lattice. According to this paradox, the results derived from anyonic statistics will contradict irreconcilably that derived from LR. In this way we conclude straightforwardly that the fractional statistics in Kitaev's model is purely a quantum effect and independent of any classical theory. In experiment, the GHZ paradox was tested only by using multi-photon systems \cite{2000Pan,1999Bouwmeester}. The model discussed here also provides a potential platform to test the GHZ paradox in the future and this is discussed briefly at the end.
The Kitaev's toric code spin-lattice model \cite{2003Kitaev} is introduced as follows. Considering a $k\times k$ square lattice on a torus ${\rm T}^2$ (see Fig. \ref{fig1}), one spin or qubit is attached to each edge of the lattice. Thus there are $2k^2$ qubits. For each vertex $\mathcal{V}$ and each face $\mathcal{F}$, consider operators of the following forms: \begin{equation*} A_\mathcal{V}=\prod_{j\in \mathcal{V}}\sigma_j^x,\quad B_\mathcal{F}=\prod_{j\in \mathcal{F}}\sigma_j^z, \end{equation*} where the $\sigma_j^X$ denotes the Pauli matrix with $X=x,y,z$ and it acts on the $j$-th qubit of a vertex $\mathcal{V}$ or face $\mathcal{F}$. These four-body interacting operators commute with each other because a vertex $\mathcal{V}$ and a boundary $\mathcal{F}$ consist of either $0$ or $2$ common qubits. From their definitions, we know that operators $A_\mathcal{V}$ and $B_\mathcal{F}$ have eigenvalues $\pm1$. Summing them together, it constructs the model Hamiltonian as \begin{equation*} H_0=-\sum_{\mathcal{V}\in {\rm T}^2}A_\mathcal{V}-\sum_{\mathcal{F}\in {\rm T}^2}B_\mathcal{F}, \label{eq-Hami} \end{equation*}
of which the ground states $|{\rm g}\rangle$ satisfy
$A_\mathcal{V}|{\rm g}\rangle=|{\rm g}\rangle$,
$B_{\mathcal{F}}|{\rm g}\rangle=|{\rm g}\rangle$ for all $\mathcal{V},\mathcal{F}$. Due to the topological property of torus, the ground states are four-fold degenerate and construct a four-dimensional Hilbert subspace, one basis of which can be explicitly written as \begin{equation}
|{\rm g_0}\rangle=\mathcal{J}\prod_{\mathcal{V}\in {\rm T}^2}(1+A_\mathcal{V})|0\rangle^{\otimes 2k^2}, \end{equation} with a normalization constant $\mathcal{J}$, while the remaining three can be given after we introduce the concept of string operators \cite{2005Hamma,2003Kitaev}.
\begin{figure}
\caption{(Color online) An illustration of the Kitaev spin-lattice model: Each black dot on the edge of the lattice represents a spin-$1/2$ particle or qubit; The interactions of the Hamiltonian $H_0$ are along edges that bound a face $\mathcal{F}$, and edges that meet at a vertex $\mathcal{V}$. The string $P_{x,z}$ indicate paths of products of $\sigma^{x,z}$ operators that are logical operators on the qubits.}
\label{fig1}
\end{figure}
Here we describe anyons as the quasiparticle excitations of the spin-lattice system with $H_0$. There are two types of anyons:
$z$-particles living on the vertices and $x$-particles living on the faces of the lattice. These anyons are created in pairs (of the same type) by string operators: $|\psi^z(P_z)\rangle=S_{P_z}^z|{\rm g}\rangle$ and $|\psi^x(P_x)\rangle=S_{P_x}^x|{\rm g}\rangle$ and they live at the end of strings, where \begin{equation}\label{eq-string} S^z_{P_z}=\prod_{r\in P_z}\sigma_r^z,\quad S^x_{P_x}=\prod_{r\in P_x}\sigma_r^x, \end{equation} are string operators associated with string $P_z$ on the lattice and string $P_x$ on the dual lattice, respectively (Fig. \ref{fig1}). Note that two anyons of the same type would annihilate each other when they meet and this is so called fusion rule. Then we can see that $A_\mathcal{V}$ and $B_\mathcal{F}$ are just two closed string operators. Especially, there are four nonequivalent classes of closed strings that are not contractible, e.g., \{$\mathcal{C}_{x1}$, $\mathcal{C}_{z1}$, $\mathcal{C}_{x2}$, $\mathcal{C}_{z2}$\} in Fig. \ref{fig1}. The corresponding string operators \{$S^x_{\mathcal{C}_{x1}}$, $S^z_{\mathcal{C}_{z1}}$, $S^x_{\mathcal{C}_{x2}}$, $S^z_{\mathcal{C}_{z2}}$\} have the same commutation relations as \{$\sigma_1^x$, $\sigma_1^z$, $\sigma_2^x$, $\sigma_2^z$\} and all of them commute with $A_\mathcal{V}$, $B_\mathcal{F}$ and thus $H_0$, which consequently give out the remaining three bases of the ground state subspace through
\{$S^x_{\mathcal{C}_{x1}}|{\rm g}_0\rangle$,
$S^x_{\mathcal{C}_{x2}}|{\rm g}_0\rangle$,
$S^x_{\mathcal{C}_{x1}}S^x_{\mathcal{C}_{x2}}|{\rm g}_0\rangle$\} \cite{2005Hamma,2003Kitaev}. In addition, if one utilizes string operators to move an $x$- (or $z$-) particle around a $z$- (or $x$-)particle one loop, a global phase factor $-1$ would be picked up in front of the initial wavefunction. This is the unusual statistical property of abelian anyons.
Now, we present our idea for the construction of GHZ paradox by using the string operators. We define four composite string operations denoted by $D_{1,2,3,4}$ by virtue of string operators (Fig. \ref{fig2}). For $D_1$, it can be written as
$D_1=S_{L_z^{651}}^zS_{L_x}^xS_{L_z^2}^z$. When it acts on a ground state $|{\rm g}\rangle$, anyons will be created, moved and annihilated as follows: First, a pair of $z$-particles are created by the string operator $S_{L_z^2}^z=\sigma_2^z$ with $L_z^j$ denoting the edge where the $j$-th qubit crosses the loop $L_z$ as shown in Fig. \ref{fig2}(a); Then a pair of $x$-particles are created, moved and annihilated along the loop $L_x$ by the string operator $S_{L_x}^x=\sigma_1^x\sigma_4^x\sigma_3^x\sigma_2^x$; At last, the string operator $S_{L_z^{651}}^z=\sigma_6^z\sigma_5^z\sigma_1^z$ moves one of the $z$-particles to meet and hence annihilate another one. As a result, we can see from Fig. \ref{fig2}(a) that the loops $L_x$ and $L_z$ construct a link. According to the anyonic statistics, a global phase factor $-1$ is picked up in front of the ground state, i.e.,
$D_1|{\rm g}\rangle=-|{\rm g}\rangle$. Likewise, as for the operations $D_2=S_{L_z^{872}}^zS_{L_x}^xS_{L_z^3}^z$, and $D_3=S^z_{L_z^{87651}}S_{L_x}^xS^z_{L_z^{3}}$, they have the same interpretations as $D_1$ but with different loops $L_{x,z}$. The links constructed by $L_{x,z}$ for $D_{2,3}$ are shown in Fig.
\ref{fig2}(b) and \ref{fig2}(c), respectively and due to the anyonic statistics, we have $D_{2,3}|{\rm g}\rangle=-|{\rm g}\rangle$. As for the operation $D_4=S_{Lx}^x$, it has a straightforward correspondence with $A_\mathcal{V}$ operator and its loop $L_x$ constructs a simple unknot shown in Fig. \ref{fig2}(d). Therefore when acting it on the ground state, no change would appear, i.e.,
$D_4|{\rm g}\rangle=|{\rm g}\rangle$.
Writing the above statements of composite string operations into more convenient forms, we have \begin{subequations} \begin{eqnarray}
D_1|{\rm g}\rangle&=&\sigma_6^z\sigma_5^z\sigma_4^x\sigma_3^x\sigma_2^y\sigma_1^y|{\rm g}\rangle=-|{\rm g}\rangle\label{D1},\\
D_2|{\rm g}\rangle&=&\sigma_8^z\sigma_7^z\sigma_4^x\sigma_3^y\sigma_2^y\sigma_1^x|{\rm g}\rangle=-|{\rm g}\rangle\label{D2},\\
D_3|{\rm g}\rangle&=&\sigma_8^z\sigma_7^z\sigma_6^z\sigma_5^z\sigma_4^x\sigma_3^y\sigma_2^x\sigma_1^y|{\rm g}\rangle=-|{\rm g}\rangle\label{D3},\\
D_4|{\rm g}\rangle&=&\sigma_4^x\sigma_3^x\sigma_2^x\sigma_1^x|{\rm g}\rangle=|{\rm g}\rangle\label{D4}, \end{eqnarray} \end{subequations} in which the algebraic relation $\sigma_j^z\sigma_j^x=i\sigma_j^y$ has been used. In the viewpoint of measurements, suppose there are eight observers and each of them has access to one spin (See Fig. \ref{fig2}) in the model. On the $i$-th spin, the corresponding observer measures the observable $\sigma_i^X$ without disturbing other spins and the measurement result is denoted by $m^X_i$. Since these results must satisfy the same functional relations satisfied by the corresponding operator, then from Eqs. (3), we can predict that, if all the operators in Eqs. (3) are measured, their results must satisfy \begin{subequations} \begin{eqnarray} m^z_6m^z_5m^x_4m^x_3m^y_2m^y_1&=&-1,\label{d1}\\ m^z_8m^z_7m^x_4m^y_3m^y_2m^x_1&=&-1,\label{d2}\\ m^z_8m^z_7m^z_6m^z_5m^x_4m^y_3m^x_2m^y_1&=&-1,\label{d3}\\ m^x_4m^x_3m^x_2m^x_1&=&+1.\label{d4} \end{eqnarray} \end{subequations} After obtaining this, we next reveal how it produces the contradiction according to LR.
Noting that Eqs.~(3) contain only local operators, the operators in each equation thereby commute and can all simultaneously have their eigenvalues.
Thus, from LR we can associate an element of reality to each of the eigenvalues in Eqs.~(4). For instance, the observers on particles ($2, 3,4,5,6$) measure, without disturbing each other, the observables ($\sigma_6^z,\sigma_5^z,\sigma_4^x,\sigma_3^x,\sigma_2^y$), respectively and if the multiplier of their results is $1$ (or $-1$), then from Eq.~(\ref{d1}) they can predict with certainty that the result of measuring $\sigma_1^y$ will be $-1$ (or $1$). That is, they can predict with certainty the value of quantity $\sigma_1^y$ by measuring other particles without disturbing particle $1$, and therefore an element of reality can be associated to the physical quantity $\sigma_1^y$. Analogously, we can associate elements of reality to all the physical quantities in Eqs.~(3). Then we can suppose that this result was somehow predetermined and initially hidden in the original state of the system. Such predictions with certainty would lead us to assign values $+1$ or $-1$ to all the observables in Eqs.~(3). However, such assignment cannot be consistent with rules of quantum mechanics because if we multiply Eqs.~(\ref{d1})-(\ref{d3}) together, it will lead to, $m^x_4m^x_3m^x_2m^x_1=-1$, which directly contradicts Eq.~(\ref{d4}). Therefore, we conclude that the four predictions of quantum mechanics given by Eqs.~(3) cannot be reproduced by LR. This completes the construction of the GHZ paradox in the context of the Kitaev spin-lattice model.
\begin{figure}\label{fig2}
\end{figure}
Further, the above GHZ paradox applies to more general situations. We can enlarge the loops $L_{x,z}$ in Fig. \ref{fig2} to generalize these $D_{1,2,3,4}$ operations. For each set of $D_{1,2,3,4}$ when acting on a ground state, it can admit the GHZ paradox only if they satisfy all of the following requirements: (i) the loop $L_x$ for all of them should be the same, no matter how large area they enclose; (ii) there is a loop $L_z$ for each of $D_{1,2,3}$ that should construct a link when combined with $L_x$; (iii) when we merge the $L_z$ of $D_1$ with the $L_z$ of $D_2$ together with overlapping edges vanishing, the resultant loop should be the same as the $L_z$ of $D_3$. In this case, the string operators of anyons give us a simple yet effective approach to look for various sets of $D_{1,2,3,4}$ operators to construct the GHZ paradox.
To sum up, it turns out that the GHZ paradox is very common in the Kitaev's toric code model. The all-versus-nothing violation of LR above well shows the anyonic statistics in the model as a pure quantum effect. In a way, it also indicates that the anyonic statistics may be at the conflictive regime between LR and quantum mechanics, which still needs an investigation in the future.
At the end, let us discuss briefly a feasible experimental implementation of the above consideration. The Kitaev spin-lattice model could be realized through dynamic laser manipulation of trapped atoms \cite{2003Duan} or molecules \cite{2006Micheli} in an optical lattice. In addition, an approach of anyonic interferometry in atomic systems was as well suggested recently by Jiang et al. \cite{2008Jiang} to measure topological degeneracy and anyonic statistics, enabling the measurement of the statistical phase associated with arbitrary braiding paths. By using this approach, it suffices to implement the operations $D_{1,2,3,4}$ in Eqs. (3) and to detect the sign. We briefly introduce this in the following.
Consider a spin lattice of trapped atoms or molecules inside an optical cavity (as shown in Fig. 2a of Ref. \cite{2008Jiang}), which provides a model Hamiltonian $H_0$ and on which the spins are called memory qubits. Except for the memory qubits, an additional ancilla spin is needed to probe the sign change before ground states and hence is called the probe qubit. To achieve controlled-string operations, an optical cavity associated with the quantum nondemolition interaction between the common cavity mode and selected spins is used to implement, e.g., a $z$-type string operation \begin{equation}
\Lambda[S_\mathcal{C}^z]=|1\rangle_{\rm A}\langle 1|\otimes S_\mathcal{C}^z+|0\rangle_{\rm A}\langle0|\otimes {\rm I}, \end{equation}
where the probe qubit is spanned by $\{|0\rangle_{\rm A},|1\rangle_{\rm A}\}$. It means: If the ancilla spin is in state
$|0\rangle_{\rm A}$, no operation is applied to the memory qubits; If the ancilla spin is in state $|1\rangle_{\rm A}$, the operation $S_\mathcal{C}^z$ is applied to the topological memory. For our spin-lattice system with $H_0$, we prepare its initial state
$|\Psi_{\rm initial}\rangle$ to be a ground state $|{\rm g}\rangle$
and $D_j|\Psi_{\rm initial}\rangle=\pm|\Psi_{\rm initial}\rangle$. Here the sign in front of the gound state $|{\rm g}\rangle$ is what we need to observe. The following interference experiment can be used to measure the sign. First, we prepare the probe qubit in a superposition state $(|0\rangle_A+|1\rangle_A)/\sqrt{2}$. We then use controlled-string operations to achieve interference of the following two possible evolutions: If the probe qubit is in state
$|0\rangle_{\rm A}$, no operation is applied to the memory qubits; If the probe qubit is in state $|1\rangle_{\rm A}$, the operation $D_j$ is applied to the topological memory, which picks up the extra phase factor $e^{i\theta_{j}}$ we want to measure. After the controlled-string operations, the probe qubit will be in state
$(|0\rangle_{\rm A}+e^{i\theta_j}|1\rangle_{\rm A})/\sqrt{2}$. Finally, we project the probe qubit to the basis of
$|\xi_\pm\rangle\equiv(|0\rangle_{\rm A}\pm e^{i\phi}|1\rangle_{\rm A})/\sqrt{2}$ with $\phi\in[0,2\pi)$, and measure the operator
$\sigma_\phi\equiv|\xi_+\rangle\langle\xi_+|-|\xi_-\rangle\langle\xi_-|$. The measurement of $\langle \sigma_\phi\rangle$ versus $\phi$ should have fringes with perfect contrast and a maximum shifted by $\phi=\theta_j$ for $\langle\sigma_\phi\rangle=\cos(\phi-\theta_j)$. In other words, for sigma operations $D_j$ ($j=1,2,3$), the $\phi$ shifts of the maximal $\langle\sigma_\phi\rangle$ will differ from those for $D_4$ by $\pi$.
In summary, we have shown the GHZ paradox in the context of Kitaev's toric code spin-lattice model by using the anyonic string operations. It shows that the anyonic statistics in the model cannot be described by LR but be a purely quantum effect. In return, the Kitaev model provides a potential platform for testing the GHZ paradox or LR in the future. A feasible experimental consideration by using the anyonic interferometry is discussed to test such a contradiction at the end. It is worth noting that the measurement employed in the above experimental scheme is non-destructive and can be repeated without disturbing the ground state. It is a predominant advantage of using Kitaev's model compared with the experimental tests by using multi-photon systems. Also recent experiments demonstrated string operations on small networks of interacting NMR qubits~\cite{2007Du} and non-interacting optical qubits~\cite{2007Pachos,2007Lu}, on the basis of which it is possible to realize our construction in advance on a small-scale qubit system.
Besides, the ground states of the Kitaev model belong to graph states, which is crucial in quantum information application. Bell inequalities have been shown to discuss LR for graph states \cite{2005Guhne}. Our construction here actually also contributes to the subject.
M. G. H. is indebted to J. Du for enlightenment at USTC. This work was supported in part by NSF of China (Grants No. 10575053 and No. 10605013), Program for New Century Excellent Talents in University, the Project-sponsored by SRF for ROCS, SEM, and LuiHui Center for Applied Mathematics through the joint project of Nankai and Tianjin Universities.
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\begin{abstract}
{It is proved that only a finite number of cohomological classes of a closed orientable irreducible three-dimensional Riemannian manifold can be represented by the Euler class of a tangent distribution to a smooth transversely oriented foliation of codimension one whose leaves have the modulus of the mean curvature bounded from above by a fixed constant. }
\end{abstract}
\maketitle \section{Introduction}
Let $ (M, g) $ be a closed oriented three-dimensional Riemannian manifold and let $ \mathcal F $ be a transversely oriented smooth foliation of codimension one on $ M $. A foliation is always assumed to be smooth, i.e. belonging to the class $ C ^{\infty} $. Recall that a foliation $\mathcal F $ is {\it taut} if its leaves are minimal submanifolds of $ M $ for some Riemannian metric on $ M $. D. Sullivan \cite{Sul} gave a description of taut foliations, namely, he proved that a foliation is taut if and only if each leaf of $\mathcal F$ is intersected by a transversal closed curve, which is equivalent to $\mathcal F$ does not contain generalized Reeb components (see bellow).
We previously proved the following result \cite {B1}.
\begin{thm} \label{result1}
Let $ V_0> 0, i_0> 0, K_0 \geq 0 $ be fixed constants, and $ M $ be a closed oriented three-dimensional Riemannian manifold with the following properties:
\begin{enumerate}
\item the volume $Vol(M)\leq V_0$;
\item the sectional curvature $K$ of $M$ satisfies the inequality $K \leq K_0$;
\item $\min \{inj(M),\frac{\pi}{2\sqrt{K_0}}\} \geq i_0 $, where $ inj (M) $ is the injectivity radius of of $ M $.
\end{enumerate}
Let us set
\[ H_0 = \begin{cases}
\min \{{\frac {2\sqrt{3}i_0^2}{V_0},\sqrt[3]{\frac{2\sqrt{3}}{V_0}}}\}, & \text{if $K_0 = 0$,}\\
\min \{ {\frac {2\sqrt{3}i_0^2}{V_0}, x_0 } \}, & \text{if $K_0 >0$,}
\end{cases} \]
where $x_0$ is the root of the equation
$$ \frac{1}{ K_0 }\arccot^2\frac{x}{\sqrt{K_0}} - \frac{V_0}{2\sqrt{3}}x=0. $$
Then any smooth transversely oriented foliation $ \mathcal F $ of codimension one on $ M $, such that the modulus of the mean curvature of its leaves satisfies the inequality $ | H | <H_0 $,
should be taut, in particular, minimal for some Riemannian metric on $ M $.
\end{thm}
\begin{rem}
A contact analog of Theorem \ref{result1} can be found in \cite{EKM}. \end{rem}
Recall that 3-manifold $M$ is called {\it irreducible} if each an embedded sphere is bounded a ball in $M$. In particular, $\pi_2(M)=0$ (see \cite{H}). Note that if $M$ admits a taut foliation, then $M$ is irreducible \cite{Nov}.
W. Thurston has proved \cite{Th} (see also \cite{ET}) that for each closed embedded orientable surface $M^2\subset M$ which is different from $S^2$ the Euler class $e(\mathcal F)$ of the tangent distribution to a transversely oriented taut foliation $ \mathcal F $ on $M$ satisfies the following estimate:
\begin{equation}\label{Chi}
|e(\mathcal F)[M^2]|\leq - \chi (M^2). \end{equation}
Since any integer homology class $H_2(M)$ can be represented by a closed oriented surface (see subsection \ref{3.1.}) the inequality above gives bounds for the possible values of the cohomology class $e(\cal F)$ on the generators of $H_2(M)$ and therefore the number of cohomological classes $ H^2 (M) $ realized as Euler classes $e(\mathcal F)$ of the tangent distribution to $ \mathcal F $ is finite.
The purpose of this work is to generalize the inequality \eqref{Chi} to the class of bounded mean curvature foliations. We prove the following theorem in this paper.
\begin{thm}[Main theorem]\label{main}
Let $M$ be a smooth closed three-dimensional orientable irreducibe Riemannian manifold. Then, for any fixed constant $H_0>0$, there are only finitely many cohomological classes of the group $H^2(M)$ that can be realized by the Euler class of a two-dimensional transversely oriented foliation $\mathcal F $ whose leaves have the modulus of mean curvature bounded above by the constant $H_0$. \end{thm}
\begin{rem}\label{cond}
Unlike Theorem \ref{result1} the Riemannian metric on $M$ in Main theorem is supposed to be fixed and all constants appearing in Theorem \ref{result1} are determined by the given Riemannian metric:
\begin{itemize}
\item the volume $Vol(M)$;
\item the injectivity radius $inj(M)$;
\item the smallest non-negative constant $K_0$ that bounds from above the sectional curvature of $M$;
\item $i_0=\min \{inj(M),\frac{\pi}{2\sqrt{K_0}}\}$
\end{itemize} \end{rem}
\begin{rem}
The foliation $\mathcal F$ does not contain a sphere as a leaf since in this case Reeb's stability theorem yields $M\simeq S^2\times S^1$, which contradicts the irreducibility of $M$. \end{rem}
\section{Geometry of folations} \label{sec:1} \subsection{Generalized Reeb components}
A subset of the manifold $ M $ with a given foliation $\mathcal F$ on it is called {\it a saturated set} if it consists of leaves of the folation $ \mathcal F $. A saturated set $ A $ of a three-dimensional compact orientable manifold $ M $ with a given transversely orientable foliation $ \mathcal F $ of codimension one is called a {\it generalized Reeb component} if $ A $ is a connected three-dimensional manifold with a boundary and any transversal to $ \mathcal F $ vector field restricted to the boundary of $ A $ is directed either everywhere inwards or everywhere outwards of the Reeb component $ A $. It is clear that $ \partial {A} $ consists of a finite set of compact leaves of the foliation $ \mathcal F $. In particular, the Reeb component $ R $ (see \cite {T}) is a generalized Reeb component. It is not difficult to show that $ \partial A $ is a family of tori (see \cite {Good}).
The next result is due to G. Reeb.
\begin{thm}(Reeb)\label{R}
Let $ (M, g) $ be a closed oriented three-dimensional Riemannian manifold and $ \mathcal F $ be a smooth transversely oriented foliation of codimension one on $ M $. Then
$$ d \chi = 2H \mu, $$
where $ \chi $ is the volume form of the foliation $ \mathcal F $, and $ \mu $ is the volume form on $ M $. \end{thm} \begin{proof} See, for example, \cite[Lemma 1]{B1}. \end{proof}
\begin{prop}\label{th1}
Let $ (M, g) $ be a closed oriented three-dimensional Riemannian manifold with a given transversely oriented smooth foliation $ \mathcal F $ of codimension one. Suppose that $ \mathcal F $ contains a generalized Reeb component $ A $ and the modulus of the mean curvature of the foliation $ \mathcal F $ is bounded above by $ | H | \leq H_0 $. Then
$
Area (\partial A) \leq 2H_0 Vol(A).
$ \end{prop} \begin{proof}
$$
0<Area (\partial A)= |\int_{\partial A}\chi|= |\int_Ad\chi| \overset{(Reeb)}=|2\int_{A} H\mu | \leq 2\int_{A} H_0\mu = 2H_0 Vol(A).
$$
It follows
$
Area (\partial A) \leq 2H_0 Vol(A).
$ \end{proof}
\begin{cor}
The generalized Reeb component $ A $ is an obstruction to the foliation $ \mathcal F $ being taut. \end{cor} \begin{rem}
The converse is also true, if the foliation is not taut, then it contains a generalized Reeb component (see \cite{Good}).
\end{rem}
\subsection{Geometry of a two-dimensional torus}
The Lowner theorem below gives an upper bound on the length of the shortest closed geodesic in a Riemannian two-dimensional torus.
\begin{thm}\label{sys1} (Lowner)\cite{Pu}
Let $T^2$ be a two-dimensional torus with an arbitrary Riemannian metric on it.
Denote by $sys$ (abbreviated from {\it systole}) the length of the shortest closed noncontractible geodesic on $T^2$. Then
\begin{equation}\label{sys2}
sys^2 \leq \frac {2} {\sqrt{3}} Area (T^2).
\end{equation} \end{thm}
\begin{prop}\label{prop1}
Let $T^2$ be a Riemannian torus for which $$sys \geq C_0 , \ Area (T^2)\leq S_0.$$ Then $T^2$ contains a pair of noncontractible closed curves that a nonhomologous to each other and have a length not exceeding some constant $C(C_0,S_0)$.
\end{prop}
\begin{proof}
By the uniformization theorem, the Riemannian metric $g$ is conformally flat, i.e., $g=\lambda^2 g_0$ for some flat metric $g_0$ and positive function $\lambda$ on $T^2$. Using the homothety we can choose both the function $\lambda$ and the flat metric $g_0$ so that $sys := sys (T^2,g) = sys (T^2,g_0) $. Let us represent the torus $(T^2,g_0)$ as a parallelogram $ABCD$ on the Euclidean plane with identified opposite sides: $AB = CD$ and $BC = AD$, where $|AD| = sys $. Here $|\ \ |$ denotes the Euclidean norm. Denote by $h$ the height of the parallelogram dropped to the side $AD$. Let $\gamma_s(t)$ be a family of parallelogram segments parallel to $AD$ which are naturally parametrized in the flat metric, where $s$ is the natural parameter on $h$\footnote{For simplicity, we denote the length of the height and the height itself by the same letter.}.
Taking into account that $\int_{\gamma_s}\lambda dt$ is the length of a closed curve $\gamma_s(t)$ nonhomotopic to zero on the torus $(T^2,g)$, and $\int_ {(T^2 ,g_0)} \lambda^2$ is nothing but $Area(T^2,g)$, we have:
\begin{equation}\label{eq1}
\int_{(T^2,g_0)} \lambda \overset {Fubini}= \int_0^h\int_{\gamma_s}\lambda \geq \min_s|\gamma_s|_g\cdot h\geq sys\cdot h =Area(T^2,g_0)
\end{equation}
Gelder's inequality gives the following estimate:
\begin{equation}\label{eq2}
\int_{(T^2,g_0)} \lambda \leq (\int_ {(T^2,g_0)} \lambda^2)^{\frac{1}{2}} \cdot (\int_ {(T^2,g_0)} 1)^{\frac{1}{2}}= Area(T^2,g)^{\frac{1}{2}}\cdot(Area(T^2,g_0))^{\frac{1}{2}}
\end{equation}
From \eqref {sys2} , \eqref{eq1} and \eqref{eq2} it follows:
\begin{equation}\label{eq3}
Area(T^2,g_0)\leq Area(T^2,g);
\end{equation}
\begin{equation}\label{eq4}
\frac{\sqrt{3}}{2}sys\leq h\leq \frac {Area(T^2,g)}{sys} ;
\end{equation}
\begin{equation}\label{eq5}
\min_s|\gamma_s|_g\cdot h\leq Area(T^2,g).
\end{equation}
Thus we get the following estimate:
\begin{equation}\label{eq6}
\min_s|\gamma_s|_g \leq \frac{2S_0}{\sqrt{3}C_0}.
\end{equation}
Without limiting generality, we can assume that the parallelogram $ABCD$ is obtained from a rectangle $A'B'C'D'$ with sides of length $|A'D'|=|AD|$ and $|A'B'|=h$ by the linear transformation (see fig. \ref{ris5})
$$ P:\Bbb R^2 \to \Bbb R^2 $$
$$ P= \begin{pmatrix}
1 & t \\
0 & 1
\end{pmatrix}, \text {where $t\geq 0$.}
$$
Then
\begin{align}
\overrightarrow {AB} &= \begin{pmatrix}
1 & t \\
0 & 1
\end{pmatrix}
\begin{pmatrix}
0 \\
h
\end{pmatrix}
=
\begin{pmatrix}
ht \\
h
\end{pmatrix}.
\end{align}
Suppose
\begin{equation}\label{n}
n \leq \frac{ht}{sys} < n+1
\end{equation}
for some $n\in \mathbb N$.
Denote by $F$ the point on the side $AB$ dividing $AB$ in a ratio of $1:n$.
\begin{align}
\overrightarrow {DF}=
&\begin{pmatrix}
1 & t \\
0 & 1
\end{pmatrix}
\begin{pmatrix}
-sys \\
\frac{h}{n}
\end{pmatrix}
=
\begin{pmatrix}
-sys + \frac{ht}{n} \\
\frac{h}{n}
\end{pmatrix}
\end{align}
\begin{figure}
\caption{Geometry of $T^2$}
\label{ris5}
\end{figure}
Considering \eqref{sys2} and \eqref{eq3}, we see that the circle on the torus $(T^2,g_0)$,
corresponding to direction $ \overrightarrow {DF} $ has the length
$$n | {DF}| < \sqrt{sys^2 + h^2}\leq \sqrt{\frac{2}{\sqrt{3}}S_0 + \frac{S_0^2}{C^2_0}}$$
In the case \begin{equation}\label{0}
0 \leq \frac{ht}{sys} < 1,
\end{equation}
we have exactly the same estimate for $|AB|$:
$$ |AB| < \sqrt{sys^2 + h^2}\leq \sqrt{\frac{2}{\sqrt{3}}S_0 + \frac{S_0^2}{C^2_0}}.$$
Cutting the torus $(T^2,g_0)$ along the systolic geodesic and the geodesic corresponding to one of the directions $ \overrightarrow {DF} $ (in the case \eqref {n}) or $ \overrightarrow {AB} $ (in the case \eqref{0}) we get a parallelogram, one of side of which still corresponds to a flat systolic geodesic, and the second has a length not exceeding $\sqrt{\frac{2}{\sqrt{3}}S_0 + \frac{S_0^2}{C^2_0}}.$ Without loss of generality, denote the resulting parallelogram by the same letters $ABCD$, where the side $AD$ corresponds to the systolic geodesic. Note that the height $h$ remains the same, since the base and area of the parallelogram have not changed.
Denote by $\hat h$ the height dropped to the side $AB$, and by $\hat\gamma_s(t)$ the family of parallelogram segments parallel to $AB$, naturally parametrized in the flat metric, where $s$ is a natural parameter in the flat metric on $\hat h$.
Considering \eqref{eq2} - \eqref{eq3}, we get:
\begin{equation}\label{eq7}
\min_s|\hat \gamma_s|_g\cdot \hat h\leq \int_0^{\hat h}\int_{\hat \gamma_s}\lambda = \int_{(T^2,g_0)} \lambda \leq Area(T^2,g).
\end{equation}
On the other hand,
\begin{equation}\label{eq8}
|AB|\cdot \hat h = Area (T^2,g_0)\geq \frac{\sqrt{3}}{2} sys^2,
\end{equation}
where
\begin{equation}\label{eq9}
\hat h \geq \frac { \frac{\sqrt{3}}{2} C_0^2}{\sqrt{\frac{2}{\sqrt{3}}S_0 + \frac{S_0^2}{ C^2_0}}}.
\end{equation}
And finally, considering \eqref{eq7} , we get the estimate:
\begin{equation}\label{eq10}
\min_s|\hat \gamma_s|_g \leq S_0\frac {\sqrt{\frac{2}{\sqrt{3}}S_0 + \frac{S_0^2}{C^2_0}}}{\frac{ \sqrt{3}}{2} C_0^2}.
\end{equation}
Since the circles corresponding to segments $AB$ and $AD$ are not homologous, setting
\begin{equation}\label{C}
C =\max \{ \frac{2S_0}{\sqrt{3}C_0}, S_0\frac {\sqrt{\frac{2}{\sqrt{3}}S_0 + \frac{S_0^2}{C^2_0}}}{\frac{ \sqrt{3}}{2} C_0^2}\}
\end{equation}
we see that estimates \eqref{eq6} and \eqref{eq10} yield the desired result. \end{proof}
\section{Some quantitative estimates} \subsection{Estimating the number of Reeb components}
Recall the following comparison theorem for the normal curvatures.
\begin{thm}\cite [\S 22.3.2.]{BZ}\label {BZ}
Let $p\in M$ and $\ \beta:[0,r]\to M$ be a radial geodesic of the ball $B(p,r)$ of radius $r$ centered at $p$. Let $\beta(r)$ be a point not conjugate with $p$ along $\beta$. Let the radius $r$ be such that there are no conjugate points in the space of constant curvature $K_0$ within the radius of length $r$. Then if at each point $\beta (t)$ the sectional curvatures $K$ of the manifold $M$ do not exceed $K_0$, then the normal curvature $k^S_n$ of the sphere $S(p,r)$ at the point $\beta(r)$ with respect to the normal $-\beta'$ is not less than the normal curvature $k_n^0$ of the sphere of radius $r$ in the space of constant curvature $K_0$. \end{thm}
By Theorem \ref{BZ} all normal curvatures of the sphere $S(r)\subset M$ of radius $r$ are positive, provided that $r<i_0$ and the normal to the sphere $S(r)$ is directed inside the ball $B(r)$ which it bounds\footnote {Note that the sphere indeed bounds the ball, since by definition $r< inj (M)$.}. We will call such a normal {\it inward}.
A hypersurface $S\subset M$ of the Riemannian manifold $M$ is called {\it supporting} to the subset $A\subset M^n$ at the point $p\in \partial A\cap S$ with respect to the normal $n_p \perp T_pS$, if $S$ cuts some spherical neighborhood $B_p$ of the point $p$ into two components, and $A\cap B_p$ is contained in that component to which the normal $n_p$ is directed . We will call the sphere $S(r)\subset M$ ($r< i_0$) the {\it supporting sphere} to the set $A\subset M$ at the point $q\in A\cap S(r)$ if it is the supporting sphere to $A $ at the point $q$ with respect to the inward normal.
It is easy to prove the following lemma.
\begin{lem}(\cite{B1})\label{l1}
Assume that the surface $F\subset M$ is tangent to the sphere $S(r_0)$ ($r_0< i_0$) at the point $q$ and the sphere $S(r_0)$ is the supporting sphere to $F$ at the point $q $. Then $k^S_n(v) \leq k_n^F(v)\ \forall v\in T_qS(r_0)$, where $k^S_n(v)$ and $k^F_n(v)$ denote corresponding normal curvatures of $S(r_0)$ and $F$ at the point $q$ in the direction $v$. \end{lem}
As a consequence of Lemma \ref{l1} and Theorem \ref{BZ} we obtain the following inequalities: \begin{equation}\label{ineq1} 0< H_r^0\leq H_r(q) \leq H (q), \end{equation} where $H^0_r $ and $H_r$ are the mean curvatures of the spheres $S(r)$ bounding the ball of radius $r, \ r<i_0,$ in the space of constant curvature $K_0$ and the manifold $M$ respectively, and $H$ is the mean curvature of the surface $F$ .
\begin{thm}\label{rc}
Let $M$ satisfy the conditions of Theorem \ref{main} and $\mathcal F$ be a codimension one transversely oriented foliation on $M$ whose leaves have a modulus of mean curvature bounded above by the constant $H_0$. Then:
\begin{enumerate}
\item The boundary torus of a Reeb component $R$ of the foliation $\mathcal F$ satisfies the condition of Proposition \ref{prop1} for some constants $C_0(H_0, Vol(M),K_0, i_0)$ and $S_0(H_0, Vol(M))$.
\item The number of Reeb components of the foliation $\mathcal F$ is finite and does not exceed $ \frac{4 H_0Vol(M)} {\sqrt{3}C^2_0}$.
\end{enumerate} \end{thm} \begin{proof}
Suppose $T^2 = \partial R$ has a systole $sys$, satisfying the inequality
$
\frac{sys}{2}< i_0 .
$
Let $l_{sys}$ be a systolic geodesic of $T^2$. Then there is a disk $D$ with $\partial D =l_{sys},$ in general position with respect to $\mathcal F$ which is inside of a ball $B(r)$, where $r\in (\frac{sys}{2}, i_0 )$. As shown by Novikov \cite{Nov} there is a vanishing cycle which belongs to $T'^2\cap D\subset B(r)$, where $T'^2\in\mathcal F $ is a torus bounding a Reeb component $R'$ \footnote{The case $R=R'$ is not excluded.}.
\begin{rem}\label{reg_r}
Applying Sard's theorem one can show that the set of $r\in (\frac{sys}{2}, i_0 )$ for which the torus $T'^2$ and the sphere $S(r):=\partial B(r)$ are in general position ($S(r)\pitchfork T'^2$ or $S(r)\cap T'^2 =\emptyset$) has a full measure in the interval $(\frac{sys}{2}, i_0 )$.
\end{rem}
In \cite[Proposition 2]{B1} we have proved that if $S(r)\cap R\not =\emptyset$ and $S(r)\pitchfork T'^2$, then $S(r)$ will be the supporting sphere with respect to the inward normal at the tangent point $q$ for some inner leaf of the Reeb component $ R'$.
In the case of $S(r)\cap R' =\emptyset$ we will achieve the tangency of the sphere $S(r)$ and $T'^2$ by decreasing the radius $r$, and the sphere $S(r)$ will become supporting for the torus $T'^2$.
From \eqref{ineq1} it follows:
\[ H_r^0 = \left\{\begin{array}{lr}
\sqrt{K_0} \cot (r\sqrt{K_0}) , &\text {if } K_0>0 \\
\frac{1}{r}, &\text {if } K_0=0 \end{array} \right\} \leq H_0.
\]
Considering Remark \ref{reg_r} we conclude that, in the general case, $sys $ must satisfy the inequality:
\[ sys\geq C_0:= \left\{\begin{array}{lr}
2 \min \{{i_0}, \frac{1}{ \sqrt{K_0}} \arccot\frac{H_0}{\sqrt{K_0}}\} , &\text {if } K_0>0 \\
2 \min \{i_0, \frac{1}{H_0}\}, &\text {if } K_0=0 \end{array} \right\}.
\]
Let us put $S_0 = 2H_0Vol(M).$
From Theorem \ref{sys1} and Proposition \ref{th1} we have
\begin{equation}\label{ineq2}
\frac{\sqrt{3}}{2}C_0^2\leq Area (\partial R) \leq 2H_0 Vol(R)\leq 2H_0 Vol(M)=S_0,
\end{equation}
which proves item $1$ of the theorem.
From \eqref{ineq2} it follows that
$
Vol(R)\geq \frac{\sqrt{3}C^2_0}{4H_0}.
$
Since the interiors of Reeb components do not intersect, the number of Reeb components does not exceed $ \frac{4 H_0Vol(M)} {\sqrt{3}C^2_0}$.
\end{proof}
\subsection{Estimates associated with generators in the second homology group}
Recall that \begin{equation}\label{S^1} H^1(M)\cong [M,S^1] \end{equation} and each cohomological class $a\in H^1(M)$ can be obtained as an image of the generator $[S^1]^*\in H^1(S^1)\cong \Z$ under the homomorphism $f^*:H^1 (S^1)\to H^1(M)$ induced by the mapping $f: M\to S^1$ uniquely defined up to homotopy. If $f$ is a circle-valued Morse function, then the preimage $f^{-1}(s)$ of a regular value $s\in S^1$ is a smooth not necessarily connected oriented submanifold $M^2\subset M$, which we identify with the image of the corresponding embedding $i: M^2 \hookrightarrow M$. Since $M^2$ is oriented, the pair $(M^2, i)$ can be understood as a singular cycle if we fix some triangulation on $M^2$. The singular homology class $[M^2,i]:=i_*[M^2]\in H_2(M)$ corresponding to the cycle $(M^2, i)$ is Poincar\'e dual to the cohomology class $a$, where $[M^2]$ denotes a fundamental class of $M^2$. We will understand the genus $g(M^2)$ of $M^2$ as the sum of genera of its connected components.
\begin{rem}\label{r2}
Recall that in the $C^2$-topology the $S^1$-valued Morse functions on a closed manifold $M$ are dense in the space of all $C^2$-smooth functions $f:M\to S^1$ representing a fixed $c\in H^1(M)$. Note that by Sard's theorem the set of regular values of $f$ is everywhere dense in $S^1$, and since $M$ is compact it is also an open set in $S^1$. \end{rem}
Let us choose Morse functions $f_i: M \to S^1, \ j=1,\dots, k,$ such that the homotopy classes $[f_i]$ represent the generators of $H^1(M)\cong \Z^k$ according to isomorphism \eqref{S^1}. \footnote{Since $M$ is orientable the group $H^1(M)$ is torsion free. }
Since $M$ is compact the Morse functions $f_i$ have only a finite number of singular values. Thus we can define the integer $g_{\max}$ as follows: $$ g_{\max}:= {\max }_{i\in \{1,\dots, k\}} \max_{x\in reg (f_i) }{ g(f_i^{-1}(x)) }, $$ where $reg(f_i)$ denotes the set of regular values of $f_i$.
It follows from Proposition \ref{prop1} and Theorem \ref{rc} that every torus $T^2_j $ bounding the Reeb component of ${ R}_j$ contains a simple closed smooth curve $\gamma_j$ which is non-homologous to zero in ${ R}_j$ and has a length bounded by the constant $C$ (see \eqref{C}). For convenience, let us introduce the following notations: $$ {\bf \gamma}:=\bigsqcup_j \gamma_j;\ {\bf T}:=\bigsqcup_j T^2_j;\ {\bf R}:=\bigsqcup_j {R}_j. $$
Using Theorem \ref {rc} we obtain the following upper bound on the length of ${\bf \gamma}$: \begin{equation}\label{length} l({\bf \gamma})\leq C_{{\bf \gamma}}:=C\cdot \frac{4H_0Vol(M)} {\sqrt{3}C^2_0}. \end{equation}
Since $M$ is compact then for each $i$ the mapping $f_i: M \to S^1$ to the unit circle $S^1$ is $c_i $-contracting for some positive constant $c_i$. It means that $|df_i(v)|< c_i|v|$. Let us identify the circle $S^1$ with the round circle $S^1(r)\subset E^2$ of radius $r_0:=\min\{ 1,1 /c_i; \ i=1,\dots,k\},$ by the radial homothety and induce the Riemannian metric on $S^1$ from $S^1(r_0)\subset E^2$. Thus each mapping $f_i: M \to S^1$ would be a contraction mapping in the sense that $|df_i(v)|\leq |v|$ for all $x\in M, \ v \in T_x(M)$, $i\in\{1,\dots,k\}$.
\begin{defn} Denote by $L_0$ the length $l(S^1(r_0)) = 2\pi r_0 $. \end{defn}
\begin{lem}\label{l2}
For each Morse function $f_i, \ i\in \{1,\dots, k\}$ the restriction $f_i|_{{\bf \gamma}}:{\bf \gamma}\to S^1$ has a regular value
$x_i\in S^1$ such that
\begin{enumerate}
\item [(1)]$card(f_i|_{{\bf \gamma}})^{-1}(x_i) \leq \frac{C_{{\bf \gamma}}}{L_0}$;
\item [(2)]The value $x_i$ is also a regular value for each of the functions $f_i:M \to S^1$ and $f_i|_{\bf T}: {\bf T}\to S^1$.
\end{enumerate}
\end{lem}
\begin{proof}
Assume that the statement of item $(1)$ is not true and
\begin{equation}{\label{card}}
card(f_i|_{{\bf \gamma}})^{-1}(x) > \frac{C_{{\bf \gamma}}}{L_0}
\end{equation}
for each regular value $x\in S^1$ of the function $f_i|_{{\bf \gamma}}$.
Since ${\bf \gamma} $ is compact, it follows from Remark \ref{r2} that the set of regular values $reg(f_i|_{{\bf \gamma}})$ of the function $f_i|_{{\bf \gamma}}$ is an open and everywhere dense set in $S^1$. Recall that nonempty open sets in $S^1$ are either all $S^1$ or a finite or countable union of open intervals in $S^1$.
Suppose $reg(f_i|_{{\bf \gamma}})$ does not coincide with $S^1$ or $\emptyset$. Then
$$
reg(f_i|_{{\bf \gamma}})=\bigsqcup_{\omega}J_{\omega}, \ \omega\in \Omega,
$$
where $\Omega$ is either a finite or a countable indexing set, and $J_{\omega}$ is an open interval in $S^1$ for each $\omega\in \Omega$.
Let $\it m$ denote the standard Lebesgue measure on a curve, which is defined by the length measure on the curve. The additivity of $\it m$ implies :
\begin{equation}\label{l}
l({\bf \gamma}) = \it m({\bf \gamma})\geq \it m(\sum_{\omega}(f_i|_{{\bf \gamma}})^{-1}(J_{\omega}))=\sum_{\ \omega}\it m((f_i|_{{\bf \gamma}})^{-1}(J_{\omega})).
\end{equation}
Since the mapping $f_i|_{{\bf \gamma}}:{\bf \gamma} \to S^1$ is a covering map on each preimage $(f_i|_{{\bf \gamma}})^{-1}(J_{\omega})$, then by assumption \eqref{card} the cardinality of this covering is greater than
$ \frac{C_{{\bf \gamma}}}{L_0}$. Since $f_i:M\to S^1$ is a contraction, we deduce:
$$
\sum_{\omega}\it m((f_i|_{{\bf \gamma}})^{-1}(J_{\omega}))> \frac{C_{{\bf \gamma}}}{L_0} \sum_{\omega} \it m(J_{\omega}) = \frac{C_{{\bf \gamma}}}{L_0}\it m(S^1) = \frac{C_{{\bf \gamma}}}{L_0}L_0=C_ {{\bf \gamma}},
$$
which together with \eqref{l} contradicts to \eqref{length}.
In the case
$ reg(f_i|_{{\bf \gamma}})=S^1$ we get the covering $f_i|_{{\bf \gamma}}:{\bf \gamma} \to S^1$ whose cardinality, by assumption \eqref{card}, is greater than $ \frac{C_{{\bf \gamma}}}{L_0}$, which for similar reasons leads to a contradiction with \eqref{length}:
$$
l({\bf \gamma}) = \it m({\bf \gamma})> \frac{C_{{\bf \gamma}}}{L_0}\it m(S^1) = C_{{\bf \gamma}}.
$$
Thus item $(1)$ is proven.
To prove item $(2)$ we note that for sufficiently close regular values $x,y\in S^1$ we have $card(f_i|_{{\bf \gamma}})^{-1}(x) = card( f_i|_{{\bf \gamma}})^{-1}(y)$. This follows from the fact that on the inverse image of a small neighborhood of a regular value the mapping $f_i|_{{\bf \gamma}}: {\bf \gamma} \to S^1 $ is a covering map.
Now the proof immediately follows from Remark \ref {r2}.
\end{proof}
Denote by $M^2_i$ the preimage $f_i^{-1}(x_i)$. Let $M^2\in\{M^2_i,\ i=1,\dots, k\}$. Clearly, if $x\in M^2 \cap {\bf \gamma} $, then $ {\bf \gamma} \pitchfork M^2 $ at the point $x$. Note that if $M^2\cap {\bf T }\not =\emptyset$, then by the construction $ M^2 \pitchfork {\bf T }$.
\begin{defn}\label{circ}
Denote by ${\mathcal C}$ the disjoint finite family (possibly empty) of circles $\{C_j\}$ such that $M^2\cap {\bf T } =\bigsqcup_j C_j$. \end{defn}
\begin{cor} \label{fincirc}
The number of those circles of the family ${\mathcal C}$, that represent the nontrivial kernel $\ker( {\bf i}_*: H_1({\bf T })\to H_1(\bf R))$ does not exceed $\frac{C_{{\bf \gamma}}}{L_0}$, where ${\bf i}_*$ is a homomorphism induced by the embedding ${\bf i}: {\bf T} \hookrightarrow {\bf R}$. \end{cor}
\begin{proof}
This is a direct consequence of Lemma \ref{l2} and the fact that ${\bf \gamma}$ necessarily intersects each of the circles in the family $\mathcal C $, which represent the non-trivial kernel $\ker ( {\bf i}_*: H_1({\bf T })\to H_1(\bf R ))$.
\end{proof}
\section{General position and Euler class}
\subsection {General position immersions} The following theorem describes general position immersions of a surface into the foliated manifold $M$.
\begin{thm}\cite [{Theorem 7.1.10}]{CC1}\label{CC1} Let $M$ be a 3-manifold with a smooth foliation $\mathcal F$ on it. Then for any $C^q$-mapping $f:N^2\to \text {int} \ (M)$, where $N^2$ is acompact surface such that $f|_{\partial N^2} \pitchfork \mathcal F$ (in the case $\partial N^2\not =\emptyset$), and for any $\delta> 0$ there exists a $\delta$-close to $f$ $C^q$ -immersion $p: N^2\to \text {int} \ (M)$ in $C^q(N^2,M)$ -topology, $q\geq 2$, such that:
\begin{enumerate}
\item [I.]The induced foliation $\mathcal F':=p^{-1}(\mathcal F\cap p(N^2))$ has only Morse singularities, each of which is either an elliptic singular point (center), in the neighborhood of which the foliation looks like a family of concentric circles, or a hyperbolic singular point (saddle point), in the neighborhood of which the foliation looks like a family of curves $x^2-y^2=t, \ t\in (-\varepsilon, +\varepsilon)$, including the singular point $(0,0)$;
\item [II.]There is at most one hyperbolic point on one leaf;
\item [III.] $p|_{\partial N^2} \pitchfork \mathcal F$.
\end{enumerate}
An immersion $p$ will be called a general position immersion. \end{thm}
\begin{prop}\label{prop2}
Let $M^2\in \{M^2_i\ i=1,\dots, k\}$ and $i:M^2 \hookrightarrow M$ be the corresponding embedding. Then there is a general position embedding $i': M^2 \hookrightarrow M$ with the image $M'^2:= i'(M^2)$ satisfying the following properties:
\begin{enumerate}
\item $M'^2\simeq M^2$. In particular, the genus of $M'^2$ does not exceed $g_{\max}$;
\item If $M'^2\cap {\bf T }\not =\emptyset $ then $M'^2\pitchfork {\bf T }$ and the intersection $M'^2\cap {\bf T}$ is a family of circles ${\mathcal C'}$.
\item The number of those circles of the family ${\mathcal C'}$ that represent the nontrivial kernel $\ker( {\bf i}_*: H_1({\bf T })\to H_1(\bf R ))$ does not exceed $\frac{C_{{\bf \gamma}}}{L_0}$;
\item $[M^2,i'] =[M^2,i]\in H_2(M)$.
\end{enumerate} \end{prop}
\begin{proof}
Let us consider a tubular neighborhood $W\subset M$ of the submanifold $M^2$ such that $W\cap {\bf T}$ consist of disjoint tubular neighborhoods $\{W_j\simeq C_j\times \R\}$ in $\bf T$ of the finite family of circles ${\mathcal C}=\{C_j\}$ defined in Definition \ref{circ}. Since $M^2$ and $M$ are orientable, $W$ is diffeomorphic to the trivial normal bundle $\nu M^2$ over $M^2$. We can identify $W$ with the direct product $ M^2\times \R$, where $M^2$ corresponds to the zero section $M^2\simeq M ^2\times 0 \overset{i_W}\hookrightarrow M^2\times \R \simeq W$. Let us identify the pair $(W,\bigsqcup_jW_j)$ with the pair of linear bundles $(\nu M^2, \nu M^2|_{\sqcup_jC_j})$.
Let $p:W\to M^2$ be a projection along the fibers of $W$. Recall that the identity component $Diff_0 ^2 (M^2, M^2)$ of $C^2$ - diffeomorphisms $Diff ^2 (M^2, M^2)$ is open in $C^2(M^2, M^2)$ (see \cite {Hi}) and it's preimage under the continuous mapping $C^2(M^2, W)\overset {p_*}\to C^2 (M^2, M^2)$, which is defined by $p_*(f) = p\circ f$, is an open neighborhood $V_1$ of the zero section $i_W:M^2 \to W$ (see \cite{PM}). Clearly, $V_1$ consists of some family of embeddings $M^2 \to W$ transversal to the fibers of $W$.
Since ${\bf T}\cap W$ is a closed subset of $W$, the subset of $C^2(M^2,W)$ transversal to ${\bf T}\cap W$ is open in $C^2(M^2,W)$ - topology (see \cite {PM}). Denote it by $V_2$.
Let $i'_W:M^2 \to W$ satisfy the condition $I$ and $II$ of Theorem \ref{CC1} and $i_W'\in V_1\cap V_2$.
Let us put $i':=i^W\circ i_W'$, where $i^W: W\hookrightarrow M$ is a natural embedding. Denote by $M'^2$ the image $i'(M^2)\subset M$. From the properties of $V_1$ and $V_2$ it follows that each fiber of $W$ transversely intersects the embedded submanifold $M'^2$ exactly at one point and the parts $1$ and $2$ immediately follow.
Since the fibers of the bundle $W_j$ are the fibers of $W$, then $M'^2 \pitchfork W_j$ and $M'^2\cap W_j$ is a circle $C'_j$ transversal to the fibers of $W_j$ for each $j$ and therefore $C'_j$ is homotopic to $C_j$ in $W_j$. If the circles $C_j$ and $C'_j$ are equipped by the corresponding orientation, then $[C_j] = [C_j']\in H_1(\bf T)$. Setting ${\cal C'}=\bigsqcup_jC'_j$ and taking into account Corollary \ref{fincirc}, we complete the proof of part $3$.
Since an arbitrary diffeomorphism belonging to $Diff ^2_0(M^2,M^2)$ induces the identity isomorphism of $H_2(M^2)$ and the embeddings $i$ and $i'$, up to such a diffeomorphism differ in deformation along the fibers $W$, part $4$ is proved.
\end{proof} \subsection {Euler class of foliations}\label{3.1.}
Let $p: N^2\to (M,\mathcal F)$ be a general position immersion of a closed oriented surface $N^2$. The induced foliation $\mathcal F' =p^{-1}(\mathcal F\cap p(N^2) )$ on $N^2$ can be oriented outside the singular points. To verify this let us take a normal vector field $n$ to the foliation $\mathcal F$ and for all $x=p(z)\in p(N^2)$ consider the orthogonal projection $n'(x)$ of the normal $n(x)$ to $\mathcal F$ onto the tangent plane $p_*(T_z (N^2))$, which in the case where $z$ is not a singular point uniquely determines the unit tangent vector $e'$ to the leaf $\mathcal L'_z\in \mathcal F', \ z\in \mathcal L'_z$, such that the frame $\{e',p_*^{-1}\frac {n'}{| n'|}\}$ defines a positive orientation of $T_z(N^2)$. Now we can define a smooth vector field $X$ on $N^2$ tangent to $\mathcal F'$ whose zeros correspond to the singular points of $\mathcal F'$ putting \begin{equation}\label{X}
X=|n'|e'. \end{equation} \begin{rem}\label{perp}
It is easy to define a vector field $X^{\perp}$ orthogonal to $\mathcal F'$ with respect to the Riemannian metric on $M'^2$. The vector field $X^{\perp}$ has the same singular points as $X$ and the integral curves of $X^{\perp}$ define a foliation $\mathcal F'^{\perp}$ orthogonal to $\mathcal F'$ on $N^2$. \end{rem}
Recall that a separatrix coming out of a singular point and returning to it, together with the singular point (a saddle) is called {\it a separatrix loop}. The general position conditions imply that the saddle singular point of $\mathcal F'$ belongs to at most two separatrix loops. A closed curve consisting of separatrix loops will be called {\it a graph}. A graph is called {\it a full graph} if it consists of two separatrix loops. A closed orbit $X$ will also be called {\it a closed orbit} of $\mathcal F'$.
As W. Thurston showed \cite{Th}, to calculate the value of the Euler class $e(T\mathcal F)$ of the foliation $\mathcal F$ on the class $[p,N^2]$, it suffices to calculate the total index of singular points of the vector field $X$ on $N^2$ taking into account the orientation of $p_*(T_q(N^2))$ at singular points. Since $M$ is oriented we can uniquely choose a unit normal vector $m\in T_{p(q)} M$ to the plane $ p_*(T_q(N^2), \ q\in N^2 $, which defines the orientation of $ p_*(T_q(N^2))$ coming from the orientation of $T_q(N^2)$.
We say that a singular point $q \in N^2$ is of {\it negative} type if $m(p(q))=-n(p(q))$. In the case when $m(p(q))=n(p(q))$ the type of the singular point is called {\it positive}.
We denote by $I_N$ the sum of indices of negative type singular points of the vector field $X$ and by $I_P$ the sum of indices of positive type singular points. Then, as W. Thurston showed, the value of the Euler class $e(T\mathcal F)$ on the homology class $[N^2,p]$ is calculated as follows: \begin{equation}\label{index} e(T\mathcal F)([N^2,p]) = e (p^*(T\mathcal F))([N^2]) =I_P -I_N. \end{equation}
Recall that the Poincaré-Hopf theorem states that
\begin{equation}\label{PH} \chi(N^2) = I_P + I_N. \end{equation}
\section { Surgeries and vanishing cycles}
\subsection{Surgeries on general position immersed surfaces }\label {3.3}
\begin{defn}
Let $p: N^2\to (M,\mathcal F)$ be a general position immersion of a closed oriented surface $N^2$ and $\mathcal F' =p^{-1}(\mathcal F\cap p(N^2) )$ be induced foliation on $N^2$. A closed curve in $N^2$ that is a closed orbit or a graph of $\mathcal F'$ is said to be {\it essential} if its $p$-image is not homotopic to zero in the leaf of the foliation $\mathcal F$ containing this curve. Otherwise, we call the curve {\it inessential}. \end{defn}
Let $i': M^2\hookrightarrow M$ be a general position embedding from Proposition \ref{prop2} and $l_1\in M^2$ be an inessential closed orbit of $\mathcal F'=i'^{-1}(\mathcal F\cap i'(M^2))$ such that $0\not =[l_1]\in \pi_1(M^2,y_1), \ y_1\in l_1$. Since $l_1$ is inessential, due to Jordan-Sch\"onflies theorem, $i'(l_1)$ bounds a disk in the leaf $ L\in\mathcal F$ containing $i'(l_1)$. Moreover, there is a {\it good neighborhood} $U_{l_1}\simeq l_1\times (-\varepsilon,\varepsilon)$ in $ M^2$, i.e., a neighborhood fibered by the inessential closed orbits $ l_1\times t, \ t\in (-\varepsilon,\varepsilon)$.
Let us choose a nonzero value ${\varepsilon_1}\in (0,\varepsilon)$ and produce a surgery on $M^2$ cutting out $U_{1} \simeq l_1\times (-{\varepsilon_1},{\varepsilon_1})\subset l\times (-\varepsilon,\varepsilon)\simeq U_{l_1}$ and gluing in the disks ${\mathcal D}_{1}\bigsqcup {\mathcal D}_{-1}$. Denote by $M^2_1$ the obtained manifold. Then we find next inessential closed orbit $l_2\subset M^2_1$ (if such exists) with the good collar $U_{l_2}\simeq l_2\times (-\varepsilon,\varepsilon)$ such that $0\not =[l_2]\in \pi_1(M_1^2,y_2), \ y_2\in l_2$. Choosing a nonzero value $\varepsilon_2 \in (0,\varepsilon)$ we make a surgery cutting out $U_{2} \simeq l_2\times (-{\varepsilon_2},{\varepsilon_2})\subset l_2\times (-\varepsilon,\varepsilon)\simeq U_{l_2}$ and gluing in the disks ${\mathcal D}_{2}\bigsqcup {\mathcal D}_{-2}$. We obtain the new manifold $M^2_2$. Then we select the next curve $l_3\subset M^2_2$ with the same properties, and so on. At the $i$-th step we get a manifold $M^2_i$.
Let $\{{\mathcal D}_{\pm i}\}, \ i\in \{1,\dots,\rho\},$ be a family of the disjoint disks surgically pasted instead of the cut out annuli $U_i\simeq l_i\times (-\varepsilon_i, \varepsilon_i)\subset l_i\times (-\varepsilon,\varepsilon)$, where $l_i\subset M^2_{i-1}$ is an inessential closed orbit such that $0\not =[l_i]\in \pi_1(M_{i-1}^2,y_i), \ y_i\in l_i$. Denote $l_{\pm i} = \partial {\mathcal D}_{\pm i}$. Let us endow $M^2_{\rho}$ with the structure of an differentiable oriented manifold, complementing the differentiable structures and corresponding orientations of disks $\bigsqcup^{\rho}_{i=1}{\mathcal D}_{\pm i}$ and $M^2\setminus \bigsqcup^{\rho}_{i=1}U_i $ with a differentiable structure and an agreed orientation of a tubular neighborhood of the boundary $\partial (M^2\setminus \bigsqcup^{\rho}_{i=1}U_i) $ (aee \cite{Hi}).
Let us extend $i'|_{M_{\rho}^2\setminus \text {int} \ {\bigsqcup_i {\mathcal D}}_{\pm i}}=i'|_{M^2\setminus \bigsqcup_i U_i}$ to all of $M_{\rho}^2$ by embeddings $h_{\pm i}: {\mathcal D}_{\pm i}\to M$, such that $h_{\pm i}({ {\mathcal D}}_{\pm i}) = D_{\pm i}$, where $D_{\pm i}\subset L_{\pm i}\in \mathcal F, \ i\in \{1,\dots,\rho\},$ are disks in the corresponding leaves of $\mathcal F$ such that $i'(l_{\pm i}) =\partial D_{\pm i}$.
Let us consider arbitrary small disjoin foliated neighborhoods $U_{\pm i}$ of $D_{\pm i}$. Applying an isotopy to $h_{\pm i}$ that is supported in $\mathcal D_{\pm i}$ and has a values in $U_{\pm i}$, which pushes out $D_{\pm i}$ into the side that $i'(U_i)$ belongs to, we can obtain a smooth general position immersion $i'_{\rho}: M_{\rho}^2 \to M$ which is a continuation of $i'|_{M_{\rho}^2\setminus \text {int} \ {\bigsqcup_i {\mathcal D}}_{\pm i}}$ such that the induced foliation ${i'_{\rho}}^{-1}(\mathcal F\cap i'_{\rho}({\mathcal D}_{\pm i}))$ on each ${ \mathcal D}_{{\pm i}}$ consists of inessential closed orbits surrounding a center and the immersion $i'_{\rho}$ is still transversal to $\bf T$.
\begin{lem}
$[M^2_{\rho},i'_{\rho}] =[M^2,i'] \in H_2(M)$. \end{lem} \begin{proof}
The singular cycles $(M^2,i')$ and $(M_{\rho}^2,i'_{\rho})$ differ by the sum of spherical cycles $\bigsqcup_i (S_i^2, g_i), \ i\in \{1,\dots,\rho\},$ where $S_i^2$ is identified with an annulus $A_i \cong \bar U_i$ to which two disks $ \mathcal D_{\pm i}$ are glued along the common boundary. Put $g_i|_{A_i} = i'$ and $g_i|_{ \mathcal D_{\pm i}} =i'_{\rho}$. From irreducibility of $M$ it follows that $g_i$ can be extended to a mapping of the ball $G_i: D_i^3\to M$. Taking into account the orientation coming from $M^2$ and $M_{\rho}^2$, on the level of singular chains we have $\partial (D_i^3, G_i) = (S_i^2, g_i)$, which implies the result. \end{proof}
\begin{rem}\label{tor}
To estimate the number of necessary surgeries we note that if an inessential closed orbit $l_k$ belongs to the toric component $T^2 \subset M^2_{k-1}$ and represents a nontrivial element of $ \pi_1(T^2)$, then the result of a surgery of $T^2$ along $l_k$ is a sphere $S^2$ and the singular cycles $(T^2,i'_{k-1})$ and $ (S^2,i'_{k})$ are homologous. But $M$ is supposed to be irreducible and therefore $(S^2,i'_{k})$ and $(T^2,i'_{k-1})$ are homologous to zero. Thus such surgeries are superfluous, and we will ignore them. \end{rem}
Let $\rho$ be the maximal number of surgeries on $M^2$ that do not result in a sphere. Since the genus of $g(M^2)\leq g_{\max}$ it is easy to show that \begin{equation}\label{gmax} \rho \leq g_{\max} - 1. \end{equation}
\begin{defn}\label{N2}
Denote by $N^2$ the union of those connected components of $M^2_{\rho}$, which represent nontrivial singular cycles by $i'_{\rho}$. In particular, $N^2$ does not contain spherical components. Denote by $p$ the restriction $i'_{\rho}|_{N^2}$.
As usual, let $\mathcal F'$ denote the induced foliation $p^{-1}(\mathcal F\cap p(N^2))$. \end{defn}
\begin{rem}\label{constr}
By the construction, taking into account Remark \ref{tor} and the Jordan-Sch\"onflies theorem, each inessential closed orbit of $\mathcal F'$ must bound a disk in $N^2$. \end{rem}
Let everywhere below $N^2,\mathcal F'$ and $p$ satisfy Definition \ref{N2}.
\subsection{Maximal vanishing cycles}
We slightly modify the concept of a vanishing cycle introduced by S.P. Novikov \cite{Nov} (see also \cite[Chapter 9]{CC2}). \begin{defn}\label{PO}
A closed curve ${\mathcal O}\subset N^2$ is called a {\it vanishing cycle} if $\mathcal O$ is either an essential closed orbit ({\it a regular vanishing cycle}) or a graph of the foliation $\mathcal F'$ and satisfies the following properties::
\begin{enumerate}
\item $\mathcal O$ is the boundary of a set $\mathcal P\subset N^2$, which is homeomorphic to either a disk or a pinched annulus (see fig. \ref{ris2});
\item There exists a {\it good collar} $V \subset \mathcal P$ of the vanishing cycle $\mathcal O$, which is the image of the continuous mapping $ i_V:S^1\times [0,1)\to \mathcal P$ with the following properties:
\begin{enumerate}
\item ${\mathcal O}= i_V(S^1\times 0)$;
\item $\forall t\in (0,1)$ the curve $ i_V(S^1\times t)$ is
an inessential closed orbit of $\mathcal F'$;
\item $i_V|_{S^1\times t}, \ t\in [0,1),$ is an embedding except when $t=0$ and ${\mathcal O}$ is a full graph. In the last case, the mapping $i_V|_{S^1\times 0}$ is one-to-one everywhere except for two points, which are mapped to the singular point of $\mathcal O$.
\item $i_V|_{x\times [0,1)}, \ x\in S^1,$ is an embedding transversal to $\mathcal F'$.
\end{enumerate}
\end{enumerate} \end{defn}
\begin{figure}
\caption{Pinched annulus $\mathcal P$.}
\label{ris2}
\end{figure}
\begin{rem}\label{inessent}
If $\mathcal O$ is an essential curve, then
due to Novikov \cite{Nov}, $p({\mathcal O})\in {T^2}$, where $T^2$ is the boundary torus of a Reeb component $ R$ and $[p({\mathcal O})]:= p_*[ i_V|_{S^1\times0}] \in \ker( i_*: \pi_1(T^2)\to \pi_1({R}))$. Since by the construction the immersion $p$ is transverse to $\bf T$, then $\mathcal O$ must be a regular vanishing cycle. Therefore, if $\mathcal O$ is a graph, then $\mathcal O$ must be inessential. In particular, if $\mathcal O$ is a full graph consisting of two separatrix loops ${\mathcal O}_1$ and $ {\mathcal O}_2$, then ${\mathcal O}$ can be of two types:
\begin{enumerate}
\item [A)] Both ${\mathcal O}_1$ and ${\mathcal O}_2 $ are inessential.
\item [B)] Both ${\mathcal O}_1$ and ${\mathcal O}_2$ are essential and $[p({\mathcal O}_1)] = - [p({\mathcal O}_2)] \in \pi_1 (\mathcal L)$, where $\mathcal L\in \mathcal F$ is a leaf containing $p(\mathcal O)$. It is easy to see that in this case there exists a lifting $\widetilde{p(\mathcal O)}$ of $p(\mathcal O)$ to the universal covering $\widetilde{\mathcal{L}}$ of the leaf $\mathcal L$ which is homeomorphic to the circle and by the Jordan-Sch\"onflies theorem, bounds a disk in $\widetilde{\mathcal{L}}$, whose image with respect to the covering mapping $\widetilde{\mathcal{L}}\to {\mathcal{L}}$ is a pinched annulus in $\mathcal L $ bounded by $p(\mathcal O)$\footnote{Note that by the construction of $\mathcal F'$ the restriction of $p: N^2 \to M$ to the arbitrary orbit of $\mathcal F'$ is an embedding.}.
\end{enumerate}
\end{rem}
\begin{rem} \label {PO1}
$\mathcal P$ is uniquely defined by $\mathcal O$ because an ambiguity can only arise when $\mathcal O$ is a closed orbit and the connected component of $N^2$ containing $\mathcal O$ is a sphere which is impossible. In this case, we will understand by $\mathcal P(\mathcal O)$ the set $\mathcal P$ from Definition \ref{PO} that is bounded by the vanishing cycle $\mathcal O$.
\end{rem}
The following lemma is well known (see \cite{Nov}, \cite{CC2}). For the convenience of the reader, we present a brief proof of this lemma. \begin{lem}\label{ll1}
Let ${B} \subset N^2$ be a disk of $N^2$ bounded by an inessential closed orbit of $\mathcal F'$. Then $B \subset \mathcal P(\mathcal O)$ for some vanishing cycle $\mathcal O$. \end{lem} \begin{proof} Due to the Reeb stability theorem (see \cite{T}), each inessential closed orbit $l_0$ of $\mathcal F'$ has a { good neighborhood} homeomorphic to $(-\varepsilon,\varepsilon) \times l_0$, where $l_s = s\times l_0$ is an inessential closed orbit of $\mathcal F'$. Let $U=\bigcup _{ t }B_t, \ t\in [0,\infty)$, be the union of disk neighborhoods of $B$ consisting of inessential closed orbits $l_{t } = \partial B_t$ of $\mathcal F'$, where $B_0=B$. Clearly, the family of disks $\{B_t\}$ is linearly ordered by the inclusion: $B_{t_1}\subset B_{t_2}$ if $t_1< t_2$.
Note that $\partial \bar U$ cannot be a center since $ N^2$ does not contain a connected component homeomorphic to $S^2$. Note also that
$\partial \bar U$ consists of orbits of $\mathcal F'$ which are not inessential closed orbits because such a closed orbits have good neighborhoods and cannot belong to $\partial \bar U$.
Note that $\partial \bar U$ is a saturated set, i.e. it consists of leaves of $\mathcal F'$ (see \cite{M}. If the closure $\partial \bar U$ contains a regular leaf $r \in \mathcal F'$ to which accumulate another leaves of $\partial \bar U$, then there exists a small transversal $\tau$ to $\mathcal F'$ through $r$ which contains the interval $J$ connecting two points $ a\in l_{t_1}, \ b\in l_{t_2},$ between which there are points of $\partial \bar U$. But it is impossible because $\mathcal F'^{\perp}$ is not degenerated on $l_t$ and $l_t$ separates $N^2$, therefore if $\tau $ leaves $B_t$, it will never come back.
We conclude that $\partial \bar U$ consists of at most a finite union ${\bf O}: =\bigsqcup_i \mathcal O_i $ of closed orbits or graphs of $\mathcal F'$. It remains to show that ${\bf O}$ is connected and is a vanishing cycle. Denote by ${\bf U}:=\bigsqcup_i U_i$ a disjoint union of tube neighborhoods $U_i$ of ${ O_i}$. Clearly, $\bar U \setminus \bf U$ is compact and is contained inside of $B_{t_0}$ for some $t_0\in (0,\infty)$. Since $U\setminus B_{t_0}$ is connected we immediately conclude that ${\bf O}$ is connected. From the orientibility of $N^2$ it follows that ${\bf O}$ divides $\bf U$ into connected components, closure of each of which in $N^2$ has a nonempty boundary consisting of orbits of ${\bf O}$. Since ${ U}\setminus B_{t_0}$ is connected it can only belong to one of these connected components and thus, by the definition, ${\bf O}$ is a vanishing cycle. \end{proof}
\begin{defn}
A vanishing cycle ${\mathcal O}_{\max}\subset N^2$ is called {\it maximal} if $\mathcal P({\mathcal O}_{\max})\subset \mathcal P(\mathcal O)$ implies ${\mathcal O}_{\max} = \mathcal O$. \end{defn}
\begin{lem}\label{ll2}
\
\begin{enumerate}
\item [(1)] For an arbitrary vanishing cycle $\mathcal O$ we have $\mathcal P({\mathcal O})\subset \mathcal P_{\max}$ for some $\mathcal P_{\max}:=\mathcal P({\mathcal O}_{\max})$.
\item [(2)] Let $\mathcal P_{\max}=\mathcal P({\mathcal O_{\max}})$ and $\mathcal P'_{\max}=\mathcal P({\mathcal O'_{\max}})$, where $O_{\max}$ and $O'_{\max}$ are maximal vanishing cycles. Then either $\mathcal P_{\max}=\mathcal P'_{\max}$ or $\mathcal P_{\max}\cap \mathcal P'_{\max}=\emptyset$. In particular, $\mathcal P_{\max}$ in part $(1)$ is unique.
\end{enumerate} \end{lem} \begin{proof}
Part $(1) $ immediately follows from the finiteness of both the set of singular points and the number of regular vanishing cycles of $\mathcal F'$.
To prove part $(2)$ it is enough to suppose that ${\mathcal O}_{\max}$ and $ {\mathcal O'}_{\max}$ are different. Otherwise we obtain a contradiction since $N^2$ does not contain $S^2$ as a connected component. One of the following takes place for ${\mathcal O}_{\max} \cap {\mathcal O'}_{\max}$:
\begin{enumerate}
\item [i.]$\emptyset$;
\item [ii.] a saddle point;
\item [iii.] a separatrix loop.
\end{enumerate}
In the cases $ii$ or $iii$ at least one of $\mathcal P_{\max}$ or $\mathcal P'_{\max}$ must be a disk and ${\mathcal O}_{\max}\cup {\mathcal O}'_{\max}$ is a full graph with a saddle point $s$. By Remark \ref{inessent}, $\mathcal O$ and $\mathcal O'$ are inessential and therefore by the Reeb stability theorem there exists an external good collar $V$ of $\mathcal P_{\max}\cup \mathcal P'_{\max}$ \footnote{Note that $\mathcal P_{\max}\cup \mathcal P'_{\max}$ is homeomorphic to either a disk or a bouquet of two disks.}. Let $l\subset V $ be an inessential closed orbit.
Clearly, $l$ bounds a disk $B$ containing $\mathcal P_{\max}\cup \mathcal P'_{\max}$. Applying Lemma \ref{ll1} we find a vanishing cycle ${\mathcal O}$ such that $\mathcal P_{\max}\cup \mathcal P'_{\max}\subset \mathcal P({\mathcal O})$ which contradicts the maximality of both ${\mathcal O}_{\max}$ and ${\mathcal O}'_{\max}$.
Let us consider case $i$.
Let us suppose that there exists $a\in \text {int} \ \mathcal P \cap \text {int} \ \mathcal P'$. Since $\mathcal P$ and $\mathcal P'$ are connected, $\mathcal P'_{\max}\not \subset \mathcal P_{\max}$ and $\mathcal P_{\max}\not \subset \mathcal P'_{\max}$, we have ${\mathcal O}_{\max}\subset \mathcal P'_{\max}$ and ${\mathcal O'}_{\max}\subset \mathcal P_{\max}$. Let $l\subset \mathcal P_{\max}$ be an inessential closed orbit of a good collar of $\mathcal O_{\max}$, which bounds a disk $B$ inside of $\mathcal P_{\max}$ such that $a\cup {\mathcal O'}_{\max}\subset \text {int} \ B$. Since $\mathcal P'_{\max}\not \subset \mathcal P_{\max}$ and $\mathcal P'$ is connected, we conclude that $l\subset \text {int} \ \mathcal P'_{\max}$ and by the Jordan-Sch\"onflies theorem $l$ bounds a disk $B'\subset \text {int} \ \mathcal P'_{\max}$. On the other hand, $l$ bounds $B\subset \mathcal P_{\max}$. Since $ {\mathcal O'}_{\max}\subset \text {int} \ B$ we conclude that $B\not = B'$ which implies that $\mathcal P \cup \mathcal P' \simeq S^2$. But this contradicts to the fact that $N^2$ does not contain a connected component homeomorphic to the sphere. We conclude that $ \text {int} \ \mathcal P \cap \text {int} \ \mathcal P'=\emptyset$ which implies the result.
\end{proof}
\begin{cor}\label {cmax}
Each center of $\mathcal F'$ belongs to the unique $\mathcal P_{\max}=\mathcal P({\mathcal O}_{\max})$. \end{cor}
\begin{proof} A center $\mathcal F'$ has a punctured neighborhood consisting of inessential closed orbits and the result immediately follows from Lemmas \ref{ll1} and \ref {ll2}. \end{proof}
\begin{lem}\label {ll4}
\begin{enumerate}
Let $\mathcal P_{\max} = \mathcal P({\mathcal O_{\max}})\subset N^2$ be a pinched annulus. Then the separatrix loops of $\mathcal O_{\max}$ are essential and their $p$ - images bound a pinched annulus in the leaf $\mathcal L\in \mathcal F$ containing $p(\mathcal O_{\max})$.
\end{enumerate} \end{lem}
\begin{proof}
According to Remark \ref{inessent} it is enough to show that there is no maximal vanishing cycle ${\mathcal O_{\max}}$ consisting of inessential separatrix loops.
Suppose that the separatrix loops of $\mathcal O_{\max}$ are inessential, then they have good exterior collars with respect to the pinched annulus $\mathcal P_{\max}$. By Remark \ref{constr} each closed orbit of this collar must bound a disk in $N^2$. Since there are no connected components of $N^2$ homeomorphic to $ S^2$, one of such disks contains $\mathcal O_{\max}$. Applying Lemma \ref {ll1} we conclude that ${{\mathcal O}_{\max} }\subset \text {int} \ \mathcal P(\mathcal O)$. The contradiction follows from Lemma \ref{ll2}.
\end{proof}
\section {Proof of Main Theorem}
Let $\{\mathcal P^k_{\max}=\mathcal P({\mathcal O}^k_{\max}), \ k\in \bf K\}$ be a family of disks and pinched annuli in $N^2$ bounded by maximal vanishing cycles of $\mathcal F'$, where $\bf K$ denotes a finite indexing set. Let $\{V_k \subset \mathcal P^k_{\max}, \ k\in \bf K\}$ denote good collars of ${\mathcal O}^k_{\max}$ and $\{l_k\subset V_k, k\in \bf K\}$ are fixed inessential closed orbits of $\mathcal F'$ inside of the good collars. Let us suppose that $V_k$ is small enough to $p|_{V_k}$ be an embedding. By Remark \ref{constr} and the definition, each $l_k$ bounds a disk $B_k$ in $N^2$, and $p (l_k)$ bounds a disk $D_k\subset L_k\in \mathcal F$ in the corresponding leaf $L_k$ of the foliation $\mathcal F$. Now we will do the same as in subsection \ref{3.3}. Let us redefine the mapping $p|_{ B_k}$ by the embedding $h_k: B_k \to M$ such that $h_k|_{l_k}= p|_{l_k}$ and $h_k(B_k) =D_k$.
Let us consider arbitrary small foliated neighborhoods $U_{k}$ of $D_{k}$. Applying an isotopy to $h_{k}$ that is supported in $B_{k}$ and has a values in $U_{k}$, which pushes out $D_{k}$ to the side away from $p(V_k)\cap U_{k}$, we can obtain a smooth general position immersion $p': N^2 \to M$ which is a continuation of $p|_{N^2\setminus \text {int} \ {\bigsqcup_k {B}}_{k}}$ such that the induced foliation ${p'}^{-1}(\mathcal F\cap p'({B}_{k}))$ on each ${B}_{{k}}$ consists of inessential closed orbits surrounding a center.
\begin{lem}
$[N^2,p] = [N^2,p'] \in H_2(M).$ \end{lem}
\begin{proof}
For each $k\in \bf K$ let $S^2_k:= ( B^1_k \bigsqcup B^2_k)/ (\partial B^1_k \sim \partial B^2_k) \simeq S^2$ be two copies of $ B_k$ with naturally identified boundaries.
Let us define a spheroid $g_k: S^2_k\to M$, where $g_k|_{ B^1_k}=p|_{ B^1_k}$ and $g_k|_{ B^2_k}=p'_{ B^2_k}$. Since $M$ is irreducible, $g_k$ can be extended to a mapping of the ball: $\Phi_k :D^3_k\to M $ such that $S^2_k=\partial D^3_k$. Taking into account the orientations of $ B^i_k,\ i=1,2,$ coming from the orientation of $ B_k$, on the level of singular chains we obtain $\partial (D^3_k,\Phi_k) =(S_k^2, g_k)$. It means that $(N^2,p) -( N^2,p')=\partial ( \bigsqcup_k (D^3_k,\Phi_k))$ which implies the result.
\end{proof}
Let us denote $\mathcal F'':= p'^{-1}(\mathcal F\cap p'(N^2))$. Let $\{\mathcal P^k_{\max}=\mathcal P({\mathcal O}^k_{\max}), \ k\in \bf K'\subset \bf K\}$ be a family of disks or pinched annuli, such that each ${\mathcal O}^k_{\max}$ is a graph. Let $(\mathcal P_{\max},{\mathcal O}_{\max}, V, l, L, D, B, U, h) \in \{(\mathcal P^k_{\max},{\mathcal O}^k_{\max}, V_k, l_k, L_k, D_k, B_k, U_k, h_k), \ k\in \bf K'\}$. From Remark \ref{inessent} and Lemma \ref{ll4} it follows that $p'({\mathcal O} _{\max})$ also bounds respectively a disk or a pinched annulus $D_{\max}$ in the leaf $L\in \mathcal F$ containing $p'({\mathcal O} _{\max})$.
Let us suppose that $D_{\max}$ is a pinched annulus, then $D_{\max}\subset A \subset L$ , where $A\simeq S^1\times (0,1)$ be an annular neighborhood of $D_{\max}$ in the leaf $L$ and $D_{\max}$ is a deformation retract of $A$. Since the collar $V$ of ${\mathcal O}_{\max}$ can be taken arbitrarily small, we can assume that the normal collar $N\simeq A\times [0,1)$ of $A=A\times 0$ contains $p'(V)$ and the foliation $\mathcal F\cap {N}$ is transversal to the interval fibers $\{*\times [0,1)\}$. The embedding $S^1\times 1/2\hookrightarrow S^1\times (0,1)\simeq A$ induces the embedding $S^1\times [0,1) \hookrightarrow A\times [0,1)\simeq N$ transversal to $\mathcal F\cap {N}$. Clearly, the foliation $\mathcal F\cap N$ is obtained from the foliation $ \mathcal F \cap (S^1\times [0,1))$ by multiplying by the interval $(0,1)$. Since $\mathcal F \cap (S^1\times [0,1))$ consists of intervals or circles representing a generator of $\pi_1(S^1\times [0,1))\cong \Z$, the foliation $\mathcal F\cap {N}$ consists of either annuli which are deformation retract of $N$ or contractible leaves. Since $p'(l)$ is free homotopic to $p'({\mathcal O}_{\max})$ (see Definition \ref{PO}), which represents a zero element in $\pi_1(A)$, the path $p'(l)$ is contracted inside of the the leaf $L\cap N$ of $\mathcal F\cap {N}$ containing $p'(l)$. By the Jordan-Sch\"onflies theorem $p'(l)$ bounds a disc in $L\cap N$. Since there is no leaves of $\mathcal F$ homeomorphic to sphere, this disc should be coincide with the disk $D$.
In the case ${ D}_{\max}$ is a disk, we denote by $A$ an open disk in $L$ containing $D_{\max}$. Then by Reeb stability theorem the induced foliation $\mathcal F\cap N$ of the normal collar $N\simeq A\times [0,1)$ containing $p'(V)$ is homeomorphic to the product foliation $\{A\times *, \ *\in[0,1)\}$, i.e. is a foliation by disks and by the Jordan-Sch\"onflies theorem $p'(l)$ will also bound the disc $D$ in $L\cap N$.
Since $U$ can be taken by an arbitrarily small neighborhood of $D$ we can assume that $p'(B)\subset N$. Let us denote $B_{\max}:=p'(B\cup V)$.
By construction, in the case ${ D}_{\max}$ is a pinched annulus, ${D}_{\max}\cup { B}_{\max}$ bounds a ball $Q^3$ with two points identified which we call a pinched ball. Using the same reasoning as for the disk $D$ we can show that the foliation $\mathcal F\cap Q^3= \{D_t, \ t\in [0,1]\}$, is a foliation by disks excepting the cases $t=0$, where $D_0\simeq \mathcal P_{\max}$, and $t=1,$ where $D_1 = p'(c)$. The foliation $\mathcal F\cap Q^3$ can be realized in $\R^3$ as it is shown on Fig. \ref{ris11}, where it is transverse to the vertical direction.
If ${ D}_{\max}$ is homeomorphic to a disk, then ${D}_{\max}\cup { B}_{\max}$ bounds a ball $B^3$ and we can realize the foliation $\mathcal F\cap B^3$ in $\R^3$ looks like a level set of a height function, where $\mathcal F\cap B^3$ is a foliation by disks which degenerate to a point $ {p'(c)} $ (see Fig. \ref{ris1}).
\begin{figure}
\caption{The ball $B^3$}
\label{ris1}
\end{figure}
In both cases, taking into account the behavior of the foliation in a neighborhood of a saddle point, one can see that when ${ \mathcal O}_{\max}$ is a graph with a saddle singular point $s$ then the types of the singular points $s$ and $c$ coincide (see Fig. \ref{ris11}). Taking into account part $2$ of Lemma \ref{ll2}, we conclude that when calculating the Euler class the pair of singular points $s$ and $c$ can be excluded since their total index in the sum \eqref{index} is equal to zero.
\begin{figure}
\caption{ The pinched ball $Q^3$}
\label{ris11}
\end{figure}
Note, that the surgeries made in Section \ref{3.3} do not generate new (i.e. not coming from $(M^2, \mathcal F')$) essential closed orbits of $(N^2,\mathcal F')$. Moreover, the surgeries increase the Euler characteristic. Taking into account Proposition \ref{prop2}, Lemma \ref{ll2} and Corollary \ref{cmax}, we conclude that the number of centers of $\mathcal F''$ which are not excluded above, i.e., centers corresponding to maximal regular vanishing cycles (see Remark \ref{inessent}), does not exceed $\frac{C_{{\bf \gamma}}}{L_0}$. Since $g(N^2)\leq g_{\max}$, using the formulas \eqref{index} and \eqref {PH}, considering the singularities excluded above, we get the following estimate: \begin{equation}\label{euler}
||e(T\mathcal F)([p',N^2])| \leq 2g_{\max} -2 + 2\frac{C_{{\bf \gamma}}}{L_0} . \end{equation}
Since the singular cycle $(N^2,p')$ can represent an arbitrary generator of the singular homology group $H_2(M)$ dual to $[f_i]\in H^1(M), \ i\in \{1,\dots,k\}$, the Main Theorem is proved.
I want to thank Vlad Yaskin, Sergej Maksimenko for their help and useful comments. I am also grateful to professor A. Borysenko for his attention to the work.
\end{document}
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arXiv
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\begin{document}
\begin{abstract} The aim of the paper is to characterize (pre)compactness in the spaces of Lipschitz/H\"older continuous mappings, which act from an arbitrary (not necessarily compact) metric space to a normed space. To this end some extensions and generalizations of already existing compactness criteria for the spaces of bounded and continuous mappings with values in normed spaces needed to be established. Those auxiliary results, which are interesting on their own since they use a concept of equicontinuity not seen in the literature, are based on an abstract compactness criterion related to the recently introduced notion of an equinormed set. \end{abstract}
\maketitle
\section{Introduction}
The importance of the spaces of Lipschitz and H\"older continuous functions in mathematics and other sciences cannot be overestimated. This introduction is too short even to list all the theories where such functions play a key role, let aside describe them in detail. Therefore, we will limit ourselves to mentioning only those instances which are related to compactness, i.e., the main object of our study. One of the oldest examples of an application of H\"older continuous functions dates back to 1930s and to Leray and Schauder, who used them in the theory of partial differential equations. (For more details, we refer the reader to the well-known monographs \cites{evans, gilbarg_trudinger}.) When talking about applications of the spaces of H\"older continuous functions one should not forget singular operators appearing in the fractional calculus. It turns out that such spaces are natural domains for the Riemann\--Liouville integral operator of fractional order or one of its most notable representatives, that is, the Abel operator (for more details, see~\cite{GV} and~\cite{samko}*{Chapters 3.1 and 3.2}). The other noteworthy situation where Lipschitz spaces emerge naturally is the Kantorovich–Rubinstein metric defined in the space of Borel measures on a compact metric space. These metrics are used to define the weak sequential convergence of measures and are applied in the treatment of the Monge--Kantorovich mass transport problem. (The introductory discussion on the subject, from the perspective of the spaces of Lipschitz continuous functions, may be found in~\cite{cobzas_book}*{Section~8.4}.)
Because of their applications -- in the previous paragraph we barely touched this topic -- Lipschitz and H\"older continuous functions constantly enjoy great interest among researchers. Each year numerous scientific articles and chapters are devoted just to studying their properties. Even whole books on Lipschitz and H\"older functions are written. Among the recent ones let us mention two monographs: one by Weaver, whose second (extended) edition was published in 2018 (see~\cite{weaver}), and the other by Cobza{\c{s}} \emph{et al.} published in 2019 (see~\cite{cobzas_book}). In those monographs one may find plenty of profound ideas, perspectives and applications related to the spaces of Lipschitz and H\"older continuous functions. What they lack, however, is a (strong) compactness criterion in such spaces. No matter how hard we tried, we could not find one in the literature, although, as it turned out, in the case of real-valued functions it had stayed hidden in a plain sight for more than half a century. In a paper of Cobza{\c{s}} from 2001 it is even stated: ``\emph{\textup[\ldots\textup] apparently there is no compactness criterion in spaces of H\"older functions, and some criteria given in the literature turned to be false}'' (see~\cite{cobzas2001}*{p.~9}).
Over the last years several attempts to find a characterization of compactness in the spaces of Lipschitz and/or H\"older continuous mappings have been made. For example, in~\cite{cobzas2001} Cobza{\c{s}} proved a precompactness criterion in the space consisting of those vector-valued Lipschitz maps that, roughly speaking, are restrictions of continuously differentiable function. In~\cite{banas_nalepa} the authors gave a sufficient condition for relative compactness of subsets of the space of H\"older continuous real-valued functions; this result was later repeated in~\cite{banas_nalepa2}. It should be underlined here that this condition, which we will refer to as \emph{uniform local flatness}, is far from being necessary. Many natural Lipschitz and H\"older continuous functions (like, for example, $f \colon [0,1] \to \mathbb R$ given by $f(x)=x$) simply do not satisfy it. It turns out, however, that uniform local flatness can be used to characterize precompactness in the so-called \emph{little Lipschitz \textup(H\"older\textup) spaces}. (For the appropriate definitions and the compactness criterion in little Lipschitz spaces see Section~\ref{sec:55} below.) This was already known to Johnson in the real-valued setting (see~\cite{johnson}*{Theorem~3.2}) and to Garc\'{\i}a-Lirola \emph{et al.} in a slightly different context of little Lipschitz functions that are continuous with respect to a topology which needs not to come from the metric of the underlying space (see~\cite{GPR}*{Lemma~2.7}). Finally, based on the ideas of Bana\'s and Nalepa, in~\cite{Saiedinezhad} Saiedinezhad used the uniform local flatness to provide a sufficient condition for relative compactness of a non-empty subset of the space $C^{k,\alpha}(X,\mathbb R)$. (Let us recall that $C^{k,\alpha}(X,\mathbb R)$ consists of real-valued multivariate functions that are $k$-times continuously differentiable and whose partial derivatives of order $k$ are H\"older continuous with exponent $\alpha$; here $X$ is a compact subset of $\mathbb R^n$.) Clearly, this result cannot be necessary. However, Saiedinezhad used it to introduce and study a measure of non-compactness in the space $C^{k,\alpha}(X,\mathbb R)$.
When thinking about possible approaches to proving a compactness criterion in a given normed space, two obvious methods come to mind: direct and indirect one. The former one is self-explanatory. The latter one often consists of two steps: finding another -- simpler -- space (linearly) isomorphic with our given space and proving a compactness criterion in this simpler space. It turns out that for $\Lip_\varphi(X,E)$ as the simpler space we can take the product of $E$ and the space of bounded (or bounded and continuous) functions. (Here, and in the sequel, $\Lip_\varphi(X,E)$ stands for the space of H\"older continuous mappings defined on a metric space $(X,d)$ and with values in a normed space $E$.) The isomorphism can be constructed using the so-called de Leeuw’s map. There is only one ``little'' detail that needs to be taken care of. To a given H\"older continuous function $f \colon X \to E$ the de Leeuw’s map assigns the bounded continuous mapping $\Phi(f)$, which for distinct $x,y \in X$ is defined by the formula $\Phi(f)(x,y)=(f(x)-f(y))/\varphi(d(x,y))$; here $\varphi$ is the so-called comparison function. (More details on comparison functions and de Leeuw’s map can be found in Section~\ref{sec:compact_in_lip} below and in the references given therein.) So, as we can see, the function $\Phi(f)$ is not defined on the whole product $X \times X$, but on its proper subset $(X\times X) \setminus \dset{(x,x)}{x \in X}$, which may not be compact even if $X$ is. And the compactness criteria for $B(Y,E)$ or $C(Y,E)$, in general, are not easy to come by, especially if $E$ is infinite-dimensional and $Y$ is an arbitrary non-empty set or a topological space. Here, and in the rest of this paper, by $B(Y,E)$ we denote the space of bounded functions defined on the non-empty set $Y$ with values in the normed space $E$, and by $C(Y,E)$ -- its subspace of continuous functions. Of course, in the case of $C(Y,E)$ we assume that $Y$ is at least a topological space. Note also that we do not require $Y$ to be bounded or compact.
One example of such a criterion for real-valued functions can be found in the celebrated monograph of Dunford and Schwarz \cite{DS}. Using an approach of Phillips (see~\cite{phillips} as well as~\cite{eveson}) it is shown that a non-empty and bounded subset $A$ of $B(Y,\mathbb R)$ or $C(Y,\mathbb R)$ is relatively compact if and only if for every $\varepsilon>0$ there is a finite cover $U_1,\ldots,U_N$ of $Y$ and points $t_i \in U_i$ such that $\sup_{f \in A}\sup_{x \in U_i}\abs{f(x)-f(t_i)}\leq \varepsilon$ for $i=1,\ldots,N$ (cf.~\cite{DS}*{Theorem~IV.5.6, p.~260 and Theorem~IV.6.5, p.~266}); here $Y$ is either an arbitrary (non-empty) set in the case of $B(Y,\mathbb R)$, or a topological space in the case of $C(Y,\mathbb R)$. (The condition appearing above is equivalent to the condition (DS) discussed in the present article. Because of that, we will refer to it as the (DS) condition as well.)
In our study we have been interested in a compactness criterion for the space of H\"older continuous mappings taking values in an arbitrary normed space $E$. (The space $E$ may be over the field of either real or complex numbers.) What may come as a surprise (having in mind, for example, the Arzel\`a--Ascoli theorem) is that the mere replacement of absolute values with norms in the results of Dunford and Schwarz simply does not work. It is not difficult to provide examples of compact (or even finite) subsets of the spaces $B(Y, E)$ or $C(Y,E)$, with $Y$ being a metric space, which do not satisfy the condition (DS) – for more details and examples see Section~\ref{sec:discussion} below. Because of that, we had to find a completely new (not based on Phillip's results) approach to establishing compactness criteria in $B(Y,E)$ and/or $C(Y,E)$. We also needed a completely new condition equivalent to precompactness in those spaces, which is more general than the condition (DS). (Note that we do not assume that $E$ is complete, hence we speak of precompactness rather than relative compactness.) Our method of proof is based on a version of a very recent abstract compactness criterion proved in~\cite{GKM}. This result, roughly speaking, allows to translate precompactness to a certain inequality involving the norm of the considered space. We should also mention that the ideas of Ambrosetti (see~\cite{Ambrosetti}*{Section~2} and~\cite{BBK}*{Lemma~1.2.8}) were of much help in our study. It is worth underlining here that our approach is more elementary than the one presented in the book of Dunford and Schwarz~\cite{DS} (in a sense that it does not use advanced concepts from functional analysis or topology). However, it allows not only to prove precompactness criteria in $B(Y, E)$, $C(Y,E)$ and various spaces of Lipschitz maps in the general case when $E$ is an arbitrary normed space, but also to draw some additional conclusion about the regularity of sets the $U_i$ covering the space $Y$. (It turns out that those sets might be chosen either closed or open, if the function space in question consists of continuous maps.) We refer the reader to Section~\ref{sec:discussion} below, where we provide a detailed discussion of the interplay between the dimension of the target space $E$ and various conditions related to precompactness, which are studied in the paper.
As already mentioned, using the results for $B(Y, E)$ and $C(Y,E)$, in the second part of the paper we prove precompactness criteria for various spaces of Lipschitz continuous functions defined on an arbitrary metric space. In general, we do not have to assume that the domain $X$ is either bounded or compact. It is also important to notice that in the case of unbounded domains the space $\Lip_\varphi(X,E)$ and its linear subspace $\BLip_\varphi(X,E)$, consisting of bounded maps, do not coincide. Further, the default norm on $\BLip_\varphi(X,E)$ is stronger than the norm inherited from $\Lip_\varphi(X,E)$. Nevertheless, we are able to formulate and discuss (pre)compactness conditions for both spaces $\Lip_\varphi(X,E)$ and $\BLip_\varphi(X,E)$. We even study precompactness in another important space of Lipschitz continuous mappings, that is, in $\Lip_0(X,E)$. And, we do not stop there. We analyse the main condition appearing in those results, and show that what really matters for compact domains is a certain uniform behaviour of functions for arguments which are close to each other. This allows us to formulate an equivalent -- localized -- version of the compactness criterion in $\Lip_\varphi(X,E)$, which we later use to obtain a compactness criterion in the little Lipschitz space $\lip_\varphi(X,E)$.
\section{Preliminaries} The main goal of this section is to introduce the notation and conventions used throughout the paper and to recall some basic facts. We will use default symbols to denote the classical spaces and norms. Let $(E, \norm{\cdot}_E)$ be a normed space over the field of either real or complex numbers. By $B(X,E)$ we will denote the normed space of all bounded $E$-valued maps defined on a non-empty set $X$, endowed with the supremum norm $\norm{f}_\infty:=\sup_{x \in X}\norm{f(x)}_E$. Assuming that $X$ is a topological space, by $C(X,E)$ we will denote the normed subspace of $B(X,E)$ consisting of all continuous maps $f \colon X \to E$. Given a set $A\subseteq B(X,E)$ and a point $x \in X$, let $A(x):=\dset{f(x)}{f \in A}$; we will call such a set a \emph{section} of $A$ at $x$. We also set $A(X):=\bigcup_{x \in X}A(x)$. The open and closed balls in a metric space $X$ with center at $x\in X$ and radius $r>0$ will be denoted by $B_X(x,r)$ and ${\overline{B}}_X(x,r)$, respectively. Finally, if $A, B$ are two non-empty subsets of a linear space, then $A+B:=\dset{a+b}{a\in A,\ b\in B}$ and $A-B:=\dset{a-b}{a\in A,\ b\in B}$. For functions $f,g \colon X \to E$, mapping a set $X$ into a normed space $E$, the difference $f-g$ will be always defined pointwise.
\subsection{Precompactness and relative compactness}
The definition of precompactness is well-known. However, it is sometimes mistaken with relative compactness. So, to avoid any ambiguity, let us state the definition explicitly. A metric space $X$ is \emph{precompact}, if its completion is compact. Equivalently, $X$ is precompact, if every sequence in $X$ contains a Cauchy subsequence. In the sequel, we will also use the fact that the classes of precompact and totally bounded metric spaces coincide. Let us recall that a metric space $X$ is \emph{totally bounded}, if for each $\varepsilon>0$ there is a finite collection of points $x_1,\ldots,x_n \in X$ such that $X= \bigcup_{i=1}^n{\overline{B}}_X(x_i,\varepsilon)$; such a collection is often called an \emph{$\varepsilon$-net}. A subset $A$ of a metric space $X$ is \emph{precompact}, if it is precompact as a metric space itself (with the metric inherited from $X$).
A non-empty subset $A$ of a metric space $X$ is called \emph{relatively compact in $X$}, if the closure of $A$ with respect to $X$ is compact. Equivalently, $A$ is relatively compact in $X$, if each sequence in $A$ contains a subsequence convergent to an element in $X$.
In complete metric spaces the above notions connected with compactness coincide. For more information concerning precompact and relatively compact metric spaces see~\cite{BBK}*{Section~1.1} or~\cite{MV}*{Chapter~4}.
\subsection{Abstract compactness criterion} The proofs of the main results of the paper will be based on a modification of a new abstract compactness criterion that was presented in~\cite{GKM}.
Let $(E,\norm{\cdot}_E)$ be a normed space endowed with a family $\bigl\{\norm{\cdot}_i\bigr\}_{i \in I}$ of semi-norms on $E$. We assume that this family satisfies the following two conditions: \begin{enumerate}[label=\textbf{\textup{(A\arabic*)}}]
\item\label{i} $\norm{x}_E=\sup_{i \in I}\norm{x}_i$ for $x \in E$,
\item\label{ii} for every $i,j \in I$ there is an index $k \in I$ such that $\norm{x}_i \leq \norm{x}_k$ and $\norm{x}_j \leq \norm{x}_k$ for $x \in E$; in other words, the family $\bigl\{\norm{\cdot}_i\bigr\}_{i \in I}$ forms a directed set. \end{enumerate} A non-empty subset $A$ of $E$ is called \emph{equinormed} (\emph{with respect to the family $\bigl\{\norm{\cdot}_i\bigr\}_{i \in I}$}), if for every $\varepsilon>0$ there exists $k \in I$ such that $\norm{x}_E\leq \varepsilon + \norm{x}_k$ for all $x \in A$.
Now, we are in position to state the afore-mentioned abstract compactness criterion. Although this is a slight modification of~\cite{GKM}*{Theorem~19}, we will skip the proof completely, as it is identical to the original one.
\begin{theorem}\label{thm:compactness_ver2} Let $(E,\norm{\cdot}_E)$ be a normed space equipped with a family $\bigl\{\norm{\cdot}_i\bigr\}_{i \in I}$ of semi-norms satisfying conditions~\ref{i} and~\ref{ii}. A non-empty subset $A$ of $E$ is precompact if and only if \begin{enumerate}[label=\textup{(\roman*)}]
\item\label{it:compactness_ver2_i} the set $A-A$ is equinormed, and
\item\label{it:compactness_ver2_ii} for each sequence $(x_n)_{n \in \mathbb N}$ of elements of $A$ there is a subsequence $(x_{n_k})_{k \in \mathbb N}$ which is Cauchy with respect to each semi-norm $\norm{\cdot}_i$. \end{enumerate} \end{theorem}
\begin{remark}\label{rem:boundedness} In Theorem~\ref{thm:compactness_ver2} we need not require the set $A$ to be bounded, as its boundedness follows easily from other assumptions. It is clear that if $A$ is precompact, it is also bounded. On the other hand, suppose that the set $A$ satisfies the conditions~\ref{it:compactness_ver2_i} and~\ref{it:compactness_ver2_ii} above, but is not bounded. Then, there is a sequence $(y_n)_{n \in \mathbb N}$ in $A$ such that $\norm{y_n}_E \to +\infty$ as $n \to +\infty$. Let $(y_{n_k})_{k \in \mathbb N}$ be its subsequence which is Cauchy with respect to each semi-norm $\norm{\cdot}_i$. In particular, $(y_{n_k})_{k \in \mathbb N}$ is bounded in each semi-norm, that is, $M_i:=\sup_{k \in \mathbb N}\norm{y_{n_k}}_i<+\infty$ for every $i \in I$. As the set $A-A$ is equinormed, there is an index $j \in I$ such that $\norm{\xi-\eta}_E \leq 1+\norm{\xi-\eta}_j$ for all $\xi,\eta \in A$. And then we have $\norm{y_{n_k}-y_{n_1}}_E \leq 1+ \norm{y_{n_k}-y_{n_1}}_j \leq 1+2M_j$ for $k \in \mathbb N$. This means that $\sup_{k \in \mathbb N}\norm{y_{n_k}}_E<+\infty$, which is impossible. \end{remark}
\section{Compactness in $B(X,E)$}
In Section~\ref{sec:compactness_B_X_E} we provide a full characterization of precompact subsets of the space of bounded mappings with values in a normed space. The proof of this result will not be a mere rewriting of the argument used in the case of real-valued functions. (Here, talking about the real-valued setting, we mainly think about~\cite{DS}*{Theorem~6, p.~260}.) A completely new approach is needed, partly because in the vector-valued case the condition imposed on a non-empty set $A\subseteq B(X,E)$ that guarantees its precompactness is different from the ``classical'' condition considered in~\cite{DS} (cf. condition (DS) below). Our proof will be based on Theorem~\ref{thm:compactness_ver2}. We will therefore begin Section~\ref{sec:compactness_B_X_E} by defining an appropriate family of semi-norms on $B(X,E)$. Next, we will introduce a new condition (called the condition (B)) that is the main ingredient of our precompactness criterion. Finally, we will state and prove the criterion itself.
In Section~\ref{sec:discussion_of_B} we discuss the condition (B) little bit more. We will provide examples illustrating that it cannot be easily simplified. We will also show that in the case of real-valued functions it agrees with the condition presented in~\cite{DS}*{Theorem~6, p.~260}.
\subsection{Compactness criterion in $B(X,E)$}\label{sec:compactness_B_X_E} Let us fix a non-empty set $X$ and a normed space $(E,\norm{\cdot}_E)$. Furthermore, let $\mathcal F$ be the family of all non-empty and finite subsets of $X$. For any $Y \in \mathcal F$ set \begin{equation}\label{eq:norms} \norm{f}_Y:=\sup_{x \in Y}\norm{f(x)}_E. \end{equation} It is clear that the family of semi-norms $\set[\big]{\norm{\cdot}_Y}_{Y \in \mathcal F}$ satisfies the conditions~\ref{i} and \ref{ii}.
Let us introduce a condition which plays a key role in our considerations.
\begin{definition} A non-empty set $A$ in $B(X,E)$ is said to satisfy the condition (B), if for every $\varepsilon>0$ there is a finite cover $U_1,\ldots,U_N$ of $X$ such that for every $i \in \{1,\ldots,N\}$ and every pair $x, y \in U_i$ for all $f \in A$ we have $\abs[\big]{\norm{f(x)}_E - \norm{f(y)}_E} \leq \varepsilon$. \end{definition}
Now, we are ready to state the precompactness criterion in $B(X,E)$.
\begin{theorem}\label{thm:compactness_BXE} A non-empty subset $A$ of $B(X,E)$ is precompact if and only if the set $A-A$ satisfies the condition \textup{(B)} and for every $x \in X$ the sections $A(x)$ are precompact. \end{theorem}
\begin{remark}\label{rem:relatively_compact} If $E$ is a Banach space, then the space $B(X,E)$ is clearly complete. Therefore, the classes of its relatively compact and precompact subsets coincide. Hence, in such a case Theorem~\ref{thm:compactness_BXE} provides necessary and sufficient conditions for relative compactness of a non-empty subset of $B(X,E)$. \end{remark}
Theorem~\ref{thm:compactness_BXE} is a consequence of the following three lemmas and Theorem~\ref{thm:compactness_ver2}.
\begin{lemma}\label{lem:5} Let $A$ be a non-empty subset of $B(X,E)$ such that the algebraic difference $A-A$ satisfies the condition \textup{(B)}. Moreover, assume that the sections $A(x)$ are precompact for every $x \in X$. Then, each sequence $(f_n)_{n \in \mathbb N}$ of elements of the set $A$ contains a subsequence $(f_{n_k})_{k\in \mathbb N}$ which is Cauchy with respect to each semi-norm $\norm{\cdot}_Y$ given by~\eqref{eq:norms}. \end{lemma}
\begin{proof} For each $i \in \mathbb N$ choose finitely many sets $U_1^i,\ldots,U_{N_i}^i$ satisfying the condition (B) with $\varepsilon=\frac{1}{i}$. In each set $U_j^i$ let us also fix a point $t_j^i$ and define $T:=\dset{t_j^i \in X}{\text{$i \in \mathbb N$ and $j=1,\ldots,N_i$}}$.
Now, let $(f_n)_{n \in \mathbb N}$ be a given sequence of elements of $A$. By the classical diagonal argument and the fact that all the sections $A(x)$ are precompact, we may select a subsequence $(f_{n_k})_{k \in \mathbb N}$ which is Cauchy at every point $t \in T$. We will show that this subsequence is Cauchy at every $x \in X$. Take an arbitrary $x \in X$, fix $\varepsilon>0$ and choose $i \in \mathbb N$ such that $\frac{1}{i}\leq \frac{1}{2}\varepsilon$. As the sets $U_1^i,\ldots,U_{N_i}^i$ cover $X$, the point $x$ must belong to at least one of them; denote this set by $U_j^i$. The sequence $(f_{n_k}(t_j^i))_{k \in \mathbb N}$ is Cauchy in $E$, so there is an index $K$ such that $\norm{f_{n_k}(t_j^i) - f_{n_l}(t_j^i)}_E \leq \frac{1}{2}\varepsilon$ for all $k,l \geq K$. Then, in view of the condition (B), for all such indices $k,l$ we have $\norm{f_{n_k}(x)-f_{n_l}(x)}_E=\norm{(f_{n_k}-f_{n_l})(x)}_E - \norm{(f_{n_k}-f_{n_l})(t_j^i)}_E + \norm{f_{n_k}(t_j^i) - f_{n_l}(t_j^i)}_E \leq \varepsilon$. This proves that the sequence $(f_{n_k}(x))_{k \in \mathbb N}$ is Cauchy. \end{proof}
\begin{lemma} Let $A$ be a non-empty subset of $B(X,E)$ such that $A-A$ satisfies the condition \textup{(B)}. Then, the algebraic difference $A-A$ is equinormed \textup(with respect to the family of semi-norms $\{\norm{\cdot}_Y\}_{Y \in \mathcal F}$ defined by~\eqref{eq:norms}\textup). \end{lemma}
\begin{proof} Similar to what we did in the proof of Lemma~\ref{lem:5} for each $i \in \mathbb N$ we choose finitely many sets $U_1^i,\ldots,U_{N_i}^i$ satisfying the condition (B) with $\varepsilon=\frac{1}{i}$. In each set $U_j^i$ we fix a point $t_j^i$. And, finally, we define $T:=\dset{t_j^i \in X}{\text{$i \in \mathbb N$ and $j=1,\ldots,N_i$}}$.
Now, let us fix $\varepsilon>0$ and let $i \in \mathbb N$ be such that $\frac{1}{i}\leq \frac{1}{2}\varepsilon$. Moreover, let $Y:=\{t_1^i,\ldots,t_{N_i}^i\}$. If $f,g \in A$, then there exists a point $x_\ast \in X$ such that $\norm{f-g}_{\infty} \leq \frac{1}{2}\varepsilon + \norm{(f-g)(x_\ast)}_E$. But $x_\ast$ belongs to one of the sets $U_1^i,\ldots,U_{N_i}^i$, say $U_j^i$, and so \begin{align*}
\norm{f-g}_{\infty} &\leq \tfrac{1}{2}\varepsilon + \norm{(f-g)(x_\ast)}_E\\
& = \tfrac{1}{2}\varepsilon + \norm{(f-g)(x_\ast)}_E - \norm{(f-g)(t_j^i)}_E + \norm{(f-g)(t_j^i)}_E\\
& \leq \varepsilon + \norm{f-g}_Y.
\end{align*} Therefore, the set $A-A$ is equinormed. \end{proof}
\begin{lemma}\label{lem:7} Let $A$ be a non-empty subset of $B(X,E)$ such that $A-A$ is equinormed \textup(with respect to the family of semi-norms $\{\norm{\cdot}_Y\}_{Y \in \mathcal F}$ defined by~\eqref{eq:norms}\textup). Moreover, assume that each sequence $(f_n)_{n \in \mathbb N}$ of elements of the set $A$ contains a subsequence $(f_{n_k})_{k\in \mathbb N}$ which is Cauchy with respect to each semi-norm $\norm{\cdot}_Y$. Then, $A-A$ satisfies the condition \textup{(B)} and the sections $A(x)$ for $x \in X$ are precompact. \end{lemma}
\begin{proof} Note that for any $x\in E$ the sequence $(f_n)_{n \in \mathbb N}$ in $B(X,E)$ is Cauchy with respect to the semi-norm induced by the singleton $Y:=\{x\}$ if and only if the sequence $(f_n(x))_{n \in \mathbb N}$ is Cauchy in $E$. Thus, we can easily conclude that each section $A(x)$, where $x\in X$, is precompact. So, we only need to show that the algebraic difference $A-A$ satisfies the condition (B).
Let $\varepsilon>0$ be fixed. Because the algebraic difference $A-A$ is equinormed there is a finite set $Y:=\{t_1,\ldots,t_m\} \subseteq X$ such that $\norm{f-g}_\infty \leq \frac{1}{16}\varepsilon + \norm{f-g}_Y$ for any $f,g \in A$. From the first part of the proof we know that each section $A(t_i)$, where $i=1,\ldots,m$, is precompact. Therefore, precompact is also the union $\bigcup_{i=1}^m A(t_i)$. Hence, there exists a finite collection of points $e_1,\ldots,e_n \in E$ such that $\bigcup_{i=1}^m A(t_i) \subseteq \bigcup_{i=1}^n B_E(e_i,\frac{1}{32}\varepsilon)$. By $\Psi$ denote the set of all functions $\psi\colon \{1,\ldots,m\}\to \{1,\ldots,n\}$, and for each $\psi \in \Psi$ let $A_\psi := \dset[\big]{f\in A}{\text{$f(t_i) \in B_E(e_{\psi(i)},\tfrac{1}{32}\varepsilon)$ for every $i=1,\ldots,m$}}$. Moreover, by $\Omega$ denote the set of all those $\psi\in\Psi$ for which $A_\psi\neq\emptyset$. It is obvious that for each $f\in A$ there exists $\psi_f\in \Omega$ such that $f\in A_{\psi_f}$. Let, additionally, for each $\psi\in\Omega$ a function $h_\psi\in A_\psi$ be fixed. As the set $\Omega$ is finite, there are only finitely many functions $h_\psi$. For simplicity, let us denote them as $h_1,\ldots,h_N$. Furthermore, let us define the function $G \colon X \to \mathbb R^{N^2}$ by the formula \begin{align*} & G(x):=\bigl(\norm{(h_1-h_1)(x)}_E, \norm{(h_1-h_2)(x)}_E, \ldots, \norm{(h_1-h_N)(x)}_E, \norm{(h_2-h_1)(x)}_E,\\ &\hspace{4.5cm} \ldots, \norm{(h_2-h_N)(x)}_E, \ldots, \norm{(h_N-h_1)(x)}_E,\ldots, \norm{(h_N-h_N)(x)}_E\bigr). \end{align*} In view of Remark~\ref{rem:boundedness} the set $A$ is bounded in $B(X,E)$. Let $M:=\sup_{f \in A}\norm{f}_{\infty}$. Then, $G(X)\subseteq [0,2M]^{N^2}$. Let $J_1,\ldots,J_k$ be a family of non-empty intervals of $\mathbb R$ covering $[0,2M]$ of diameters at most $\frac{1}{2}\varepsilon$. Moreover, let \begin{align*} & U_\lambda:=G^{-1}\bigl(J_{\lambda_{1,1}}\times J_{\lambda_{1,2}} \times \cdots \times J_{\lambda_{1,N}}\times J_{\lambda_{2,1}} \times \cdots \times J_{\lambda_{2,N}} \times \cdots \times J_{\lambda_{N,1}} \times \cdots \times J_{\lambda_{N,N}}\bigr), \end{align*} where $\lambda:=(\lambda_{1,1}, \lambda_{1,2},\ldots, \lambda_{1,N}, \lambda_{2,1},\ldots, \lambda_{2,N}, \ldots, \lambda_{N,1}, \ldots, \lambda_{N,N}) \in \{1,\ldots,k\}^{N^2}$. The sets $U_\lambda$ form a finite cover of $X$. Let $\Lambda$ be the set of all those indices $\lambda$ for which $U_\lambda \neq \emptyset$.
We will show that for the sets $U_\lambda$, where $\lambda \in \Lambda$, the condition (B) holds. Let us fix $\lambda \in \Lambda$, mappings $f,g \in A$ and points $x,y \in U_\lambda$. Furthermore, let $h_f$ and $h_g$ be those maps among $h_1,\ldots,h_N$ for which we have $h_f \in A_{\psi_f}$ and $h_g \in A_{\psi_g}$. (Note that $h_f$ and $h_g$ may coincide.) Further, instead of writing $\lambda_{p,q}$ with the appropriate numbers $p,q \in \{1,\ldots,N\}$ corresponding to $h_f$ and $h_g$, we will simply write $\lambda_{f,g}$. By the definition of the sets $A_\psi$, we have $f(t_i), h_f(t_i) \in B_E(e_{{\psi_f}(i)},\frac{1}{32}\varepsilon)$ and $g(t_i), h_g(t_i) \in B_E(e_{{\psi_g}(i)},\frac{1}{32}\varepsilon)$ for $i=1,\ldots,m$. This implies that $\norm{f-h_f}_Y\leq \frac{1}{16}\varepsilon$ and $\norm{g-h_g}_Y\leq \frac{1}{16}\varepsilon$. Consequently, $\norm{f-h_f}_\infty\leq \frac{1}{16}\varepsilon + \norm{f-h_f}_Y \leq \frac{1}{8}\varepsilon$ and $\norm{g-h_g}_\infty \leq \frac{1}{8}\varepsilon$. Likewise, by the definition of the sets $U_\lambda$, we have $\norm{(h_f-h_g)(x)}_E, \norm{(h_f-h_g)(y)}_E \in J_{\lambda_{f,g}}$. And so, $\abs[\big]{\norm{(h_f-h_g)(x)}_E - \norm{(h_f-h_g)(y)}_E}\leq \frac{1}{2}\varepsilon$. Note that \[
\norm{(f-g)(x)}_E \leq \norm{(f-h_f)(x)}_E + \norm{(h_f-h_g)(x)}_E + \norm{(h_g-g)(x)}_E \] and \[
\norm{(h_f-h_g)(y)}_E \leq \norm{(h_f-f)(y)}_E + \norm{(f-g)(y)}_E + \norm{(g-h_g)(y)}_E. \] Thus, \begin{align*} & \norm{(f-g)(x)}_E - \norm{(f-g)(y)}_E\\ & \quad \leq \norm{(f-h_f)(x)}_E + \norm{(h_f-h_g)(x)}_E + \norm{(h_g-g)(x)}_E\\ &\qquad + \norm{(h_f-f)(y)}_E - \norm{(h_f-h_g)(y)}_E + \norm{(g-h_g)(y)}_E\\ & \quad \leq 2\norm{f-h_f}_\infty + 2\norm{g-h_g}_\infty + \norm{(h_f-h_g)(x)}_E - \norm{(h_f-h_g)(y)}_E\\ & \quad \leq 2\cdot \tfrac{1}{8}\varepsilon + 2\cdot \tfrac{1}{8}\varepsilon + \tfrac{1}{2}\varepsilon = \varepsilon. \end{align*} Interchanging the roles of $x,y$, we get $\norm{(f-g)(y)}_E - \norm{(f-g)(x)}_E \leq\varepsilon$. This shows that the set $A-A$ satisfies the condition (B). \end{proof}
\subsection{Discussion of the condition (B)}\label{sec:discussion_of_B} \label{sec:discussion}
In this section we are going to study the condition (B) little bit more. There are some questions that can (and should) be raised. One of the most natural ones is whether in the statement of Theorem~\ref{thm:compactness_BXE} the phrase ``the set $A-A$ satisfies the condition (B)'' can be replaced by ``the set $A$ satisfies the condition (B)''. It may come as a surprise that the answer is negative. Moreover, it cannot be done in any infinite-dimensional normed space $E$.
\begin{proposition}\label{prop:B:zero:one} Let $X$ be the set consisting of all zero-one sequences with only finitely many non-zero terms. Moreover, let $E$ be an infinite-dimensional normed space. Then, there exists a non-empty set $A\subseteq B(X,E)$ with the following properties\textup: \begin{enumerate}[label=\textup{(\alph*)}]
\item $A$ satisfies the condition \textup{(B)},
\item $A-A$ does not satisfy the condition \textup{(B)},
\item for each $x \in X$ the section $A(x)$ is compact,
\item $A(X)$ is not precompact,
\item $A$ is not precompact. \end{enumerate} \end{proposition}
\begin{proof} By the well-known Riesz lemma (see e.g.~\cite{kreyszig}*{Theorem~2.5-4}), there exists a sequence $(e_n)_{n \geq 0}$ of unit vectors in $E$ such that $\norm{e_i-e_j}_E \geq \frac{1}{2}$ for all distinct $i,j\in \mathbb N\cup\{0\}$. For each $n \in \mathbb N$ let us define a bounded mapping $f_n \colon X \to E$ by \begin{equation} f_n(x) = \begin{cases}
e_0 & \text{if $\xi_n = 0$,}\\
e_n & \text{if $\xi_n = 1$;} \end{cases} \end{equation} here $x:=(\xi_n)_{n \in \mathbb N}$. Let also $A:=\dset{f_n}{n\in \mathbb N}$.
Note that the set $A$ satisfies the condition (B) with the cover of $X$ consisting of a single set $U_1:=X$, because for every $x,y\in X$ and $n \in \mathbb N$ we have $\norm{f_n(x)}_E = \norm{f_n(y)}_E = 1$. However, the algebraic difference $A-A$ does not satisfy condition (B). To see this let $U_1,\ldots,U_N$ be any finite cover of $X$ and let $k$ be a positive integer such that $2^{k-1} > N$. Since there are exactly $2^{k-1}$ different zero-one sequences of length $k-1$, there exists at least one set $U_i$ which contains at least two distinct elements $x=(\xi_n)_{n \in \mathbb N}$ and $y=(\eta_n)_{n \in \mathbb N}$ such that $\xi_n = \eta_n = 0$ for $n\geq k$. As $x\neq y$, an index $l<k$ exists such that $\xi_l\neq \eta_l$. Of course, we may assume that $\xi_l=1$ and $\eta_l=0$. So, $f_k(x) = f_k(y) = f_l(y)= e_0$ and $f_l(x)=e_l$. Therefore, we have \[ \abs[\big]{\norm{(f_k-f_l)(x)}_E - \norm{(f_k-f_l)(y)}_E} = \norm{e_0-e_l}_E \geq \tfrac{1}{2}. \] This shows that the condition (B) for the set $A-A$ is not satisfied.
Now, let us fix $x=(\xi_n)_{n \in \mathbb N} \in X$ and let $m \in \mathbb N$ be such that $\xi_n=0$ for $n \geq m$. Then, $A(x)\subseteq \{e_0,e_1,\ldots,e_{m-1}\}$, meaning that the section $A(x)$ is compact. On the other hand, $A(X)=\dset{e_n}{n \in \mathbb N\cup\{0\}}$. This implies that $A(X)$ is not precompact, as the sequence $(e_n)_{n \in \mathbb N}$ does not contain a Cauchy subsequence.
The fact that the set $A$ is not precompact follows from Theorem~\ref{thm:compactness_BXE}. \end{proof}
When dealing with compactness criteria for real-valued functions, a one-dimensional version of the following condition often can be encountered: \begin{enumerate}
\item[(DS)] for every $\varepsilon>0$ there exist finitely many non-empty subsets $U_1,\ldots,U_N$ of $X$ whose union is $X$ such that for every $i \in \{1,\ldots,N\}$ and every pair $x, y \in U_i$ for all $f \in A$ we have $\norm{f(x) - f(y)}_E \leq \varepsilon$ \end{enumerate} (cf.~\cite{alexiewicz1969analiza}*{pp. 370–371} or~\cite{DS}*{Section IV.4}). Clearly, it is stronger than the condition (B). Thus, another natural question concerning Theorem~\ref{thm:compactness_BXE} arises. Does this result still hold, if in its statement we replace the condition (B) with (DS)? Or, maybe we should impose the condition (DS) on the set $A$ rather than on its algebraic difference $A-A$. The answers to both these questions are negative, as shown by the following example.
\begin{example}\label{ex:B:comp} Let $X$ be the unit sphere of an infinite-dimensional normed space $E$. Consider the mappings $f,g \colon X \to E$ given by $f(x)=x$ and $g(x)=2x$. Moreover, let $A:=\{f,g\}$. It is easy to see that both the sets $A$ and $A-A$ satisfy the condition (B) with $U_1:=X$, as $\norm{f(x)}_E- \norm{f(y)}_E=0$, $\norm{g(x)}_E-\norm{g(y)}_E=0$ and $\norm{(f-g)(x)}_E-\norm{(f-g)(y)}_E=0$ for any $x,y \in X$. However, if either $A$ or $A-A$ satisfied the condition (DS), then for every $\varepsilon>0$ it would be possible to cover the unit sphere of an infinite-dimensional normed space with finitely many sets of diameter not greater than $\varepsilon$, which is absurd. Thus, neither the set $A$ nor the algebraic difference $A-A$ satisfy the condition (DS). Note also that for each $x \in X$ the section $A(x)$ is precompact (it is even compact), but the set $A(X)$ is not. \end{example}
Let us stop for a moment and look at the above example once more. The construction of the set $A$ with appropriate properties was possible, because the normed space $E$ was infinite-dimensional. In consequence, the set $A(X)$ was not precompact. However, if we assume that the subset $A(X)$ of $E$ is precompact, the situation changes significantly.
\begin{proposition}\label{prop:DS_equivalent_B} For a non-empty subset $A$ of $B(X,E)$ the following conditions are equivalent\textup: \begin{enumerate}[label=\textup{(\roman*)}]
\item\label{it:DS_B_i} $A$ satisfies the condition \textup{(DS)} and for every $x \in X$ the section $A(x)$ is precompact,
\item\label{it:DS_B_ii} $A-A$ satisfies the condition \textup{(B)} and the set $A(X)$ is precompact. \end{enumerate} \end{proposition}
\begin{proof} We begin with the proof of the implication~$\ref{it:DS_B_i}\Rightarrow\ref{it:DS_B_ii}$. Note that if the set $A$ satisfies the condition (DS), then so does $A-A$. And, moreover, for any $x,y\in X$ and $f,g \in A$ we have $\abs[\big]{\norm{(f-g)(x)}_E-\norm{(f-g)(y)}_E}\leq \norm{(f-g)(x)-(f-g)(y)}_E$. Now, we need only to show that the set $A(X)$ is precompact. Fix $\varepsilon>0$. By the condition (DS) the set $X$ can be covered with a finite collection of non-empty sets $U_1,\ldots,U_n$ such that $\sup_{f \in A}\norm{f(\xi)-f(\eta)}_E \leq \frac{1}{2}\varepsilon$ for every $i\in\{1,\ldots,n\}$ and every pair $\xi,\eta \in U_i$. In each set $U_i$ let us fix a point $t_i$. Then, for any $x \in U_i$ and $f \in A$ we have $f(x)=f(t_i)+f(x)-f(t_i) \in \{f(t_i)\}+{\overline{B}}_E(0,\frac{1}{2}\varepsilon)$. Thus, $A(U_i):=\bigcup_{x\in U_i}A(x) \subseteq A(t_i) + {\overline{B}}_E(0,\frac{1}{2}\varepsilon)$. But $A(t_i)$ is totally bounded by assumption, and so it can be covered with a finite family of balls ${\overline{B}}_E(y_1^i,\frac{1}{2}\varepsilon)$, \ldots, ${\overline{B}}_E(y_{m_i}^i,\frac{1}{2}\varepsilon)$, where $y_j^i \in A(t_i)$. This means that $A(X)\subseteq \bigcup_{i=1}^n A(U_i) \subseteq \bigcup_{i=1}^n \bigcup_{j=1}^{m_i} {\overline{B}}_E(y_j^i,\varepsilon)$, and shows that $A(X)$ is totally bounded/precompact.
Now, we prove the implication~$\ref{it:DS_B_ii}\Rightarrow\ref{it:DS_B_i}$. The reasoning is similar to the one we used to establish Lemma~\ref{lem:7}. Let us assume that the algebraic difference $A-A$ satisfies the condition (B). Then, given $\varepsilon>0$ there exists a finite cover $W_1,\ldots,W_n$ of $X$ such that for every $i \in \{1,\ldots,n\}$ and all $x,y \in W_i$ and $f,g \in A$ we have $\abs[\big]{\norm{(f-g)(x)}_E - \norm{(f-g)(y)}_E} \leq \frac{1}{8}\varepsilon$. For each $i \in \{1,\ldots,n\}$ let us also fix $w_i\in W_i$. Furthermore, let $e_1,\ldots,e_m \in E$ be a finite $\frac{1}{8}\varepsilon$-net for $A(X)$.
By $\Psi$ denote the set of all functions $\psi\colon \{1,\ldots,n\}\to \{1,\ldots,m\}$. For each $\psi \in \Psi$ let $A_\psi: = \dset{f\in A}{\text{$f(w_i) \in {\overline{B}}_E(e_{\psi(i)},\tfrac{1}{8}\varepsilon)$ for every $i=1,\ldots,n$}}$.
Moreover, by $\Omega$ denote the set of all those $\psi\in\Psi$ for which $A_\psi\neq\emptyset$.
For each $\psi\in\Omega$ let us chose a function $g_\psi\in A_\psi$. Thus, we obtain a finite family $g_1,\ldots,g_k$ of functions, where $k$ is the cardinality of $\Omega$.
It is obvious that for each $f\in A$ there exist $\psi_f\in \Omega$ and a corresponding function $g_l \in \{g_1,\ldots,g_k\}$ such that $f,g_l\in A_{\psi_f}$.
Define the mapping $G \colon X \to E^k$ by the formula $G(x)=(g_1(x),\ldots,g_k(x))$, and let $V_\lambda:=G^{-1}(\prod_{i=1}^k {\overline{B}}_E(e_{\lambda_i},\frac{1}{8}\varepsilon))$, where $\lambda:=(\lambda_1,\ldots,\lambda_k) \in \{1,\ldots,m\}^k$. Set $U_{i,\lambda}:=W_i \cap V_\lambda$. By $I$ let us denote the set of those pairs $(i,\lambda)$, where $i \in \{1,\ldots,n\}$ and $\lambda \in \{1,\ldots,m\}^k$, for which the sets $U_{i,\lambda}$ are non-empty. Clearly, the finite family $\{U_{i,\lambda}\}_{(i,\lambda) \in I}$ is a covering of $X$.
Let us now fix an index $(i,\lambda) \in I$, take any points $x,y\in U_{i,\lambda}$ and any function $f \in A$, together with the corresponding function $g_l \in \{g_1,\ldots,g_k\}$, such that $f,g_l\in A_{\psi_f}$. Then, by condition (B) and the fact that $U_{i,\lambda}\subseteq W_i$, we can see that \[ \norm{(f-g_l)(x)}_E\leq \norm{(f-g_l)(w_i)}_E + \tfrac{1}{8}\varepsilon. \] Similarly, \[ \norm{(f-g_l)(y)}_E\leq \norm{(f-g_l)(w_i)}_E + \tfrac{1}{8}\varepsilon. \] As $f,g_l \in A_{\psi_f}$, we have $\norm{(f-g_l)(w_i)}_E \leq \frac{1}{4}\varepsilon$. Using the fact that the set $U_{i,\lambda}$ is included in $V_\lambda$, we obtain $\norm{g_l(x)-g_l(y)}_E \leq \frac{1}{4}\varepsilon$. Therefore, $\norm{f(x)-f(y)}_E \leq \norm{(f-g_l)(x)}_E + \norm{g_l(x)-g_l(y)}_E + \norm{(f-g_l)(y)}_E \leq \tfrac{3}{8}\varepsilon + \tfrac{1}{4}\varepsilon + \tfrac{3}{8}\varepsilon = \varepsilon$.
Since a subset of a precompact set is also precompact, and $A(x)\subseteq A(X)$ for every $x \in X$, this ends the proof. \end{proof}
As bounded sets in finite-dimensional Banach spaces are precompact, from Theorem~\ref{thm:compactness_BXE} and Proposition~\ref{prop:DS_equivalent_B} (cf. also Remark~\ref{rem:relatively_compact}) we immediately get the following result. For $n=1$ and $\mathbb K:=\mathbb R$ it can be found (in a slightly different but equivalent form) on page~260 in~\cite{DS}.
\begin{theorem}\label{thm:DS} Let $n \in \mathbb N$ and $\mathbb K \in \{\mathbb R, \mathbb C\}$. A non-empty set $A \subseteq B(X,\mathbb K^n)$ is relatively compact if and only if it is bounded and satisfies the condition \textup{(DS)}. \end{theorem}
\section{Compactness in $C(X,E)$} \label{sec:compactness_in_CXE}
In this section, as corollaries to our compactness results established previously, we will state compactness criteria for the space of bounded and continuous maps. The most interesting is the first one, as it is probably not (widely) known. The other two (in the exact or similar version) are already known in the literature (see, e.g.,~\cite{alexiewicz1969analiza}*{pp.~370--371}, \cite{BBK}*{Corollary~1.2.9}, \cite{DS}*{Section~IV.6} or~\cite{kreyszig}*{Theorem~8.7-4}). We have decided to state them, because of two reasons. Firstly, we will need them when dealing with compactness in Lipschitz spaces. And, secondly, because, in contrast to the approach presented in~\cite{DS}, our method allows to say more about the regularity of the sets $U_1,\ldots,U_n$ appearing in the conditions (B) and (DS) (see Remarks~\ref{rem:open_closed} and~\ref{rem:open_closed2}).
As an application of Theorem~\ref{thm:compactness_BXE} to a closed subspace of $B(X,E)$ consisting of continuous mappings we get the following result.
\begin{theorem}\label{thm:compactness_in_CXE_1} Let $X$ be a topological space. Then, a non-empty subset $A$ of $C(X,E)$ is precompact if and only if the algebraic difference $A-A$ satisfies the condition \textup{(B)} and for every $x \in X$ the sections $A(x)$ are precompact. \end{theorem}
\begin{remark}\label{rem:open_closed} A closer look at the proof of Theorem~\ref{thm:compactness_BXE} allows us to say more about the regularity of the sets $U_1,\ldots,U_N$ appearing in the condition (B) in Theorem~\ref{thm:compactness_in_CXE_1}. If $A\subseteq C(X,E)$, then the function $G$ defined in the proof of Lemma~\ref{lem:7} is continuous. Thus, taking the open/closed covering $J_1,\ldots,J_k$ of the set $[0,2M]^{N^2}$, we may conclude that all the set $U_1,\ldots,U_N$ are open (closed) in $X$. \end{remark}
We already know that in some situations both conditions (B) and (DS) can be used to characterize precompact subsets of $B(X,E)$. Now, we prove a result similar to Proposition~\ref{prop:DS_equivalent_B} in the case when $A$ is a non-empty subset of the space $C(X,E)$.
\begin{corollary}\label{rem:DS_B_3} Let $X$ be a compact Hausdorff topological space. Moreover, assume that $A$ is a non-empty subset of $C(X,E)$ such that each section $A(x)$, where $x \in X$, is precompact. Then, $A$ satisfies the condition \textup{(DS)} if and only if $A-A$ satisfies the condition \textup{(B)}. \end{corollary}
\begin{proof} In view of Proposition~\ref{prop:DS_equivalent_B} it suffices to show that if the algebraic difference $A-A$ satisfies the condition (B), then the set $A(X)$ is precompact. So let us fix $\varepsilon>0$. Then, by Theorem~\ref{thm:compactness_in_CXE_1} the set $A$ is precompact, and hence it has a finite $\varepsilon$-net $g_1,\ldots,g_n \in A$. As the functions $g_i$ are continuous and the topological space $X$ is compact, the images $g_i(X)$ as well as their union $\bigcup_{i=1}^n g_i(X)$ are compact. For every $f \in A$ and $x \in X$ we thus have $f(x)=f(x)-g_j(x) + g_j(x) \in {\overline{B}}_E(0,\varepsilon)+g_j(X)$, where the function $g_j$ is an element of the $\varepsilon$-net $g_1,\ldots,g_n$ chosen so that $\norm{f-g_j}_\infty \leq \varepsilon$. Therefore, $A(X)\subseteq {\overline{B}}_E(0,\varepsilon)+\bigcup_{i=1}^n g_i(X)$. This proves that the set $A(X)$ is precompact (cf. the first part of the proof of Proposition~\ref{prop:DS_equivalent_B}). \end{proof}
\begin{remark} It is worth underlining that using a similar approach to the one we used in the proof of Proposition~\ref{prop:DS_equivalent_B}, we can avoid applying Theorem~\ref{thm:compactness_in_CXE_1} in the proof of Corollary~\ref{rem:DS_B_3}. We decided, however, to proceed along this less elegant route to simplify the reasoning and not utilizing the same arguments over and over again. \end{remark}
Combining Proposition~\ref{prop:DS_equivalent_B}, Theorem~\ref{thm:compactness_in_CXE_1} and Corollary~\ref{rem:DS_B_3}, yields yet another one compactness criterion in $C(X,E)$.
\begin{theorem}\label{thm:compactness_in_CXE_2} Let $n \in \mathbb N$ and $\mathbb K \in \{\mathbb R, \mathbb C\}$. Moreover, assume that either \begin{enumerate}[label=\textup{(\alph*)}]
\item\label{it:CXE_2_i} $X$ is an arbitrary topological space and $E=\mathbb K^n$, or
\item $X$ is a compact Hausdorff topological space and $E$ is an arbitrary normed space. \end{enumerate} Then, a non-empty subset $A$ of $C(X,E)$ is precompact if and only if it satisfies the condition \textup{(DS)} and the sections $A(x)$ for $x \in X$ are precompact. \end{theorem}
\begin{remark}\label{rem:17} Let $n \in \mathbb N$ and $\mathbb K \in \{\mathbb R, \mathbb C\}$. Note that when $E=\mathbb K^n$, the set $A(x)$ for $x\in X$ is precompact if and only if it is bounded. Therefore, in this case, in Theorem~\ref{thm:compactness_in_CXE_2} instead of the phrase ``the sections $A(x)$ for $x \in X$ are precompact'' we can equivalently write ``the sections $A(x)$ for $x \in X$ are bounded'', or even ``$A$ is a bounded subset of $C(X,\mathbb K^n)$''. However, when the normed space $E$ is infinite-dimensional, the precompactness of the sections $A(x)$ cannot be replaced with their boundedness, or the boundedness of the whole set $A$. To see this take a separable infinite-dimensional Banach space $E$ and set $X:={\overline{B}}_{E^\ast}(0,1)$, where $E^\ast$ is the dual of $E$. From classical facts from functional analysis we know that $X$ is a metrizable compact Hausdorff topological space when endowed with the weak$^\ast$ topology (see~\cite{Conway}*{Theorem~V.3.1 and Theorem~V.5.1}). Given a point $y \in X$ consider the constant mapping $g_y \colon X \to (E^\ast,\norm{\cdot}_{E^\ast})$ given by $g_y(x)=y$ for $x \in X$. Also, let $A:=\dset{g_y}{y \in X}$. It is straightforward to check that the set $A$ satisfies the condition (DS) (with a family consisting of a single open set $U_1:=X$) and is bounded in $C(X,E^{\ast})$. However, $A(x)={\overline{B}}_{E^\ast}(0,1)$ for any $x \in X$. So, no section is precompact in the norm of $E^\ast$. \end{remark}
\begin{remark}\label{rem:open_closed2} A remark similar to Remark~\ref{rem:open_closed} is also true in the case of Theorem~\ref{thm:compactness_in_CXE_2}. In other words, for a set $A\subseteq C(X,E)$, in the condition (DS) we may require all the sets $U_1,\ldots,U_N$ to be either open or closed in $X$. \end{remark}
As a corollary to Theorem~\ref{thm:compactness_in_CXE_2} we obtain the classical Arz\`ela--Ascoli compactness criterion. It uses the notion of an equicontinuous family of functions. Although, this notion is very well-known, just for completeness, let us recall it. A non-empty set $A \subseteq C(X,E)$ is \emph{equicontinuous}, if for every $\varepsilon>0$ there exists $\delta>0$ such that for every $x,y \in X$ with $d(x,y)\leq \delta$ and every $f \in A$ we have $\norm{f(x)-f(y)}_E \leq \varepsilon$.
\begin{theorem}\label{thm:compactness_in_CXE_3} Let $(X,d)$ be a compact metric space. Then, a non-empty subset $A$ of $C(X,E)$ is precompact if and only if it is equicontinuous and for every $x \in X$ the sections $A(x)$ are precompact. \end{theorem}
\begin{proof} Assume that $A\subseteq C(X,E)$ is equicontinous. Let us fix $\varepsilon>0$ and choose $\delta>0$ according to the definition of equicontinuity. Since the family $\bigl\{ B_X(x,\frac{1}{2}\delta)\bigr\}_{x \in X}$ is an open cover of the compact metric space $X$, there exist $x_1,\ldots,x_N \in X$ such that $X=\bigcup_{i=1}^N B_X(x_i,\frac{1}{2}\delta)$. Let $U_i:=B_X(x_i,\frac{1}{2}\delta)$ for $i=1,\ldots,N$. It is now clear that if $x,y \in U_i$, then $d(x,y)\leq \delta$, and hence $\sup_{f \in A}\norm{f(x)-f(y)}_E \leq \varepsilon$. This means that $A$ satisfies the condition (DS). As the sections $A(x)$ for $x \in X$ are precompact, it suffices to apply Theorem~\ref{thm:compactness_in_CXE_2} to conclude that $A$ is precompact.
Let us move to the second part of the proof. This time we assume that $A$ is precompact. Then, in view of Theorem~\ref{thm:compactness_in_CXE_2} and Remark~\ref{rem:open_closed2}, it satisfies the condition (DS) with open sets. Also, for each $x \in X$ the set $A(x)$ is precompact. Our aim is to show that $A$ is equicontinuous. So, let us fix $\varepsilon>0$, and let $U_1,\ldots,U_N$ be the open cover of $X$ appearing in the condition (DS). Given a set $U_i$ and $\xi_i \in U_i$, let $r_{\xi_i}>0$ be such that $B_X(\xi_i,r_{\xi_i})\subseteq U_i$. Then, the family $\bigcup_{i=1}^N \bigl\{B_X(\xi_i,\frac{1}{2}r_{\xi_i})\bigr\}_{\xi_i \in U_i}$ is an open cover of the compact metric space $X$. Hence, it contains a finite subcover. Let us denote the elements of this subcover by $B_X(x_1,\frac{1}{2}r_1),\ldots,B_X(x_m,\frac{1}{2}r_m)$. Moreover, let $\delta:=\frac{1}{2}\min_{1\leq k \leq m} r_k$. Now, choose arbitrary $x,y \in X$ such that $d(x,y)\leq \delta$, and fix $f \in A$. Then, $x \in B_X(x_j,\frac{1}{2}r_j)\subseteq U_l$ for some $j \in \{1,\ldots,m\}$ and $l \in \{1,\ldots,N\}$. Consequently, $y \in B_X(x_j,r_j) \subseteq U_l$. Now, it is enough to use the condition (DS) to conclude that $\norm{f(x)-f(y)}_E \leq \varepsilon$. In other words, $A$ is equicontinuous. \end{proof}
\section{Compactness in Lipschitz spaces} \label{sec:compact_in_lip}
The goal of this section is to provide a full characterization of precompact subsets of both the Lipschitz and little Lipschitz spaces. In the first part of this section we will work with functions defined on an arbitrary metric space. In the second one, we will move to the case of compact domains.
\begin{notation} Throughout this section for a given metric space $(X,d)$ we will write $\tilde{X}$ for the metric space $(X\times X) \setminus \dset{(x,x)}{x \in X}$. We endow $\tilde{X}$ (and any other subset of $X\times X$) with the maximum metric $d_\infty$ inherited from $X \times X$. For completeness, let us add that for $(x_1,x_2),(y_1,y_2) \in X\times X$ the \emph{maximum metric} is defined by the formula $d_\infty((x_1,x_2),(y_1,y_2)) = \max\{ d(x_1,y_1), d(x_2,y_2) \}$. \end{notation}
\subsection{Comparison function} A key ingredient when defining classes of Lipschitz continuous mappings is the so-called \emph{comparison function}, that is, a non-zero function $\varphi\colon [0,+\infty)\to[0,+\infty)$ that is right-continuous at $0$, concave and such that $\varphi(0)=0$. Typical examples of such functions are the power functions $\varphi(t)=t^\alpha$ for $\alpha \in (0,1]$. They are especially important, as they give rise to the classical classes of Lipschitz/H\"older continuous mappings (for the appropriate definitions see the next subsection). Another example of a comparison function is $\varphi(t)=\ln(1+t)$. It may be interesting, because near zero it behaves like the identity function, but when $t$ tends to infinity, it increases slower than any of the power functions defined above. In the sequel we will use several basic properties of such functions; we gathered them in the following lemma. We skip its proof, because it is straightforward and the arguments can be found scattered around various books on real functions or the internet (especially, on the Mathematics Stack Exchange forum -- see, for example, question 2757353).
\begin{lemma}\label{lem:comparision_function} Let $\varphi\colon [0,+\infty)\to[0,+\infty)$ be a comparison function. Then, \begin{enumerate}[label=\textup{(\alph*)}]
\item\label{lem:comparision_function_a} $\varphi$ is continuous on $[0,+\infty)$,
\item\label{lem:comparision_function_b} $\varphi(t)/t$ is non-increasing on $(0,+\infty)$,
\item\label{lem:comparision_function_c} $\varphi$ is sub-additive, that is, $\varphi(t+s)\leq \varphi(t)+\varphi(s)$ for $t,s\in [0,+\infty)$,
\item\label{lem:comparision_function_d} $\varphi$ is non-decreasing on $[0,+\infty)$,
\item\label{lem:comparision_function_e} $\varphi(t)>0$ for every $t \in (0,+\infty)$,
\item\label{lem:comparision_function_f} the limit $\lim_{t \to 0^+} \varphi(t)/t$ exists, and is either a positive number or $+\infty$. \end{enumerate} \end{lemma}
A word of caution is in order here. The notion of a comparison function is ambiguous and its definition varies throughout the literature. Different classes of comparison functions are defined by means of various combinations of the properties~\ref{lem:comparision_function_a}--\ref{lem:comparision_function_f} and many others. (See~\cite{AKORR}*{Section~2} for a detailed discussion of several types of comparison functions.) Our assumption of concavity may thus seem too strong. And, in fact, it is. However, we decided to use it because of two reasons. It makes all the unnecessary technicalities disappear. And, more importantly, the comparison functions corresponding to the Lipschitz/H\"older classes are concave. In other words, our results cover the most interesting cases.
\subsection{Lipschitz spaces} \label{sec:52} Fix a comparison function $\varphi$. Moreover, let $(X,d)$ be a metric space and let $(E,\norm{\cdot}_E)$ be a normed space. (Note that we \emph{do not} assume that $X$ is bounded or compact.) By $\Lip_\varphi(X,E)$ we denote the class of all $\varphi$-Lipschitz continuous mappings, that is, mappings $f \colon X \to E$ which satisfy the condition \[
\abs{f}_{\varphi}:=\sup_{\substack{x,y \in X\\x\neq y}}\frac{\norm{f(x)-f(y)}_E}{\varphi(d(x,y))}<+\infty. \] Now, let us fix a point $x_*$ in $X$; such a point is called a \emph{base} (or \emph{distinguished}) \emph{point} of the space $X$. It can be easily checked that $\Lip_\varphi(X,E)$ is a normed space, when endowed with the norm $\norm{f}_\varphi:=\norm{f(x_*)}_E+\abs{f}_{\varphi}$. It should be also evident that for different selections of base points the corresponding norms are equivalent. Furthermore, if $E$ is a Banach space, then the space $\Lip_\varphi(X,E)$ is complete.
If $\varphi(t)=t^\alpha$ for $\alpha \in (0,1)$, the space $\Lip_\varphi(X,E)$ coincides with the space of H\"older continuous maps with exponent $\alpha$, and is usually denoted by $\Lip_\alpha(X,E)$. Similarly, if $\varphi(t)=t$, then $\Lip_\varphi(X,E)$ is just the space of Lipschitz mappings $\Lip(X,E)$.
At the space $\Lip_\varphi(X,E)$ we may also look from a slightly different perspective. With the help of Lemma~\ref{lem:comparision_function} it is easy to see that if $\varphi$ is a comparison function and $d$ is a metric on $X$, so is $d_\varphi:=\varphi \circ d$. Therefore, if by $Y$ we denote the metric space $(X,d_\varphi)$, then the space $\Lip(Y,E)$ is well-defined. Moreover, in such a case, $\Lip_{\varphi}(X,E)$ and $\Lip(Y,E)$ are linearly isomorphic. (If we chose the same base point in $X$ and $Y$, then they are even isometrically isomorphic.) So, in the sequel, instead of working in spaces $\Lip_{\varphi}(X,E)$, we could focus on the space of Lipschitz continuous mappings entirely. However, we will not do that. The reason behind our decision is practical. Often, in applications, the more explicit the statements of the results one applies are, the better. But, generally, it is just a matter of taste, which approach to choose.
It is worth noting that because we do not require the space $X$ to have finite diameter, $\Lip_{\varphi}(X,E)$ may contain unbounded maps. In the literature, the space $\BLip_\varphi(X,E)$ of bounded $\varphi$-Lipschitz continuous mappings is also considered (cf.~\cite{cobzas_book}*{Chapter~8} or~\cite{weaver}*{Chapter~2}). It is endowed with the norm $\vnorm{f}_\varphi := \norm{f}_\infty + \abs{f}_\varphi$. For arbitrary domains, $\BLip_\varphi(X,E)$ is just a linear subspace of $\Lip_\varphi(X,E)$. Moreover, in general, the norm $\vnorm{\cdot}_\varphi$ on $\BLip_\varphi(X,E)$ is strictly stronger than $\norm{\cdot}_\varphi$ inherited from $\Lip_\varphi(X,E)$. However, if $X$ has finite diameter (or, in particular, is compact), then $\BLip_\varphi(X,E)$ is linearly isomorphic to $\Lip_\varphi(X,E)$.
Another space of Lipschitz continuous mappings, which plays a prominent role, is the space $\Lip_0(X,E)$. It consists of those Lipschitz maps that vanish at the base point, and is endowed with the norm $f \mapsto \abs{f}_1$; here by $\abs{\cdot}_1$ we denote the semi-norm $\abs{\cdot}_\varphi$ corresponding to the comparison function $\varphi(t)=t$. For an arbitrary domain $X$, $\Lip_0(X,E)$ is a closed subspace of $\Lip(X,E)$, with the quotient $\Lip(X,E)/\Lip_0(X,E)$ isometrically isomorphic to $E$. What may come as a surprise, however, is the fact that $\Lip_0(X,E)$ spaces are very closely related to spaces $\BLip(X,E)$. It turns out that for $E=\mathbb R$ every space of bounded Lipschitz continuous functions, as Weaver puts it on page~42 of his monograph~\cite{weaver}, ``\emph{effectively is, in every important sense, a Lip$_0$ space}.''
The main takeaway from the above discussion is that the (pre)compactness criteria we are going to prove are quite general. Not only can they be applied to the space $\Lip_{\varphi}(X,E)$ and $\Lip_0(X,E)$ with arbitrary metric space $X$, but also to spaces $\BLip_\varphi(X,E)$.
For more information on various classes of Lipschitz continuous functions and their properties, we refer the Reader to~\cites{cobzas_book, weaver}.
\subsection{De Leeuw's map}\label{sec:deLeeuw} When dealing with Lipschitz continuous mappings, the de Leeuw's map $\Phi$ comes in handy. To each $f \in \Lip_\varphi(X,E)$, it assigns the mapping $\Phi(f) \colon \tilde X \to E$ given by \[ \Phi(f)(x,y) = \frac{f(x)-f(y)}{\varphi(d(x,y))}. \] The mapping $\Phi(f)$ is clearly continuous and bounded on $\tilde X$. Moreover, $\Phi(f)=\Phi(g)$ if and only if the difference $f-g$ is a constant function. More on the de Leeuw's map can be found in~\cite{cobzas_book}*{Section 8.3.1} and \cite{weaver}*{Section 2.4}.
\subsection{Compactness criteria -- preparatory part} Before we study (pre)compactness in spaces of Lipschitz continuous mappings, for a non-empty set $A\subseteq L_\varphi(X,E)$ let us introduce the following two conditions: \begin{enumerate}
\item[(LDS)] for every $\varepsilon>0$ there is a finite cover $U_1, \ldots, U_N$ of $\tilde X$ such that for any $i\in \{1,\ldots, N\}$ we have
\[
\sup_{(x,y), (\xi,\eta) \in U_i} \sup_{f \in A} \norm[\bigg]{\frac{f(x) - f(y)}{\varphi(d(x,y))} - \frac{f(\xi) - f(\eta)}{\varphi(d(\xi,\eta))}}_E\leq \varepsilon, \]
\item[(L)] for every $\varepsilon>0$ there is a finite cover $U_1,\ldots,U_N$ of $\tilde X$ such that for any $i \in \{1,\ldots,N\}$ we have
\[
\sup_{(x,y), (\xi,\eta) \in U_i} \sup_{f \in A}\abs[\Bigg]{\frac{\norm{f(x) - f(y)}_E}{\varphi(d(x,y))} - \frac{\norm{f(\xi) - f(\eta)}_E}{\varphi(d(\xi,\eta))}}\leq \varepsilon.
\] \end{enumerate}
\begin{remark}\label{rem:open_closed_in_L_LDS} In the above conditions we may also require all the sets $U_1,\ldots,U_N$ to be either open or closed in $\tilde{X}$ (cf.~Remarks~\ref{rem:open_closed} and~\ref{rem:open_closed2}). \end{remark}
\subsection{Compactness criteria -- arbitrary domains} As the title suggests, in this section we will be interested in Lipschitz continuous maps that are defined on an arbitrary metric space. We begin with the space $\Lip_\varphi(X,E)$.
\begin{theorem}\label{thm:compact_in_Lip} A non-empty subset $A$ of $\Lip_\varphi(X,E)$ is precompact if and only if
the set $A-A$ satisfies the condition (L) and the sections $A(x)$ for every $x \in X$ are precompact. \end{theorem}
\begin{proof} Let $x_\ast$ be the base point of the metric space $X$. Consider the linear operator $T\colon \Lip_\varphi(X,E)\to E\times C(\tilde{X},E)$ given by $T(f)=(f(x_\ast), \Phi(f))$; here $\Phi$ is the de Leeuw's map. It is easy to check that $T$ is an isometric embedding of $\Lip_\varphi(X,E)$ into $E\times C(\tilde{X},E)$, when the target space is endowed with the norm $\norm{(e,g)}:=\norm{e}_E + \norm{g}_\infty$ for $(e,g) \in E\times C(\tilde{X},E)$. Note that the condition (L) for $A-A$ states that the set $\Phi(A)-\Phi(A) \subseteq C(\tilde X,E)$ satisfies the condition (B). Furthermore, precompactness of each section $A(x)$ for $x \in X$ is equivalent with the fact that the sets $A(x_\ast)$ and $\Phi(A)(x,y):=\dset{\Phi(f)(x,y)}{ f\in A}$ for $(x,y) \in \tilde X$ are precompact.
Now, let us assume that the non-empty subset $A$ of $\Lip_\varphi(X,E)$ is precompact. Then, precompact are also the sets $\pi_1(T(A))$ and $\pi_2(T(A))$; here $\pi_1$ and $\pi_2$ are the projections onto the first and second factor of $E\times C(\tilde{X},E)$, respectively. Note that $\pi_1(T(A))=A(x_\ast)$ and $\pi_2(T(A))=\Phi(A)$. So, by Theorem~\ref{thm:compactness_in_CXE_1} and our preliminary observations, we infer that the condition (L) is satisfied for $A-A$. Also, $A(x)$ is precompact for $x \in X$.
On the other hand, if $A-A$ satisfies the condition (L) and the sections $A(x)$ are precompact for every $x \in X$, then by~Theorem~\ref{thm:compactness_in_CXE_1} the sets $A(x_\ast) \subseteq E$ and $\Phi(A) \subseteq C(\tilde X,E)$ are precompact. Hence, precompact is also their product $A(x_\ast) \times \Phi(A)$. As precompactness is hereditary, this implies the precompactness of $T(A) \subseteq A(x_\ast) \times \Phi(A)$. To end the proof it suffices to note that $A=T^{-1}(T(A))$. \end{proof}
In the finite-dimensional case, using Theorem~\ref{thm:compactness_in_CXE_2} instead of Theorem~\ref{thm:compactness_in_CXE_1} and reasoning as in the proof of the above result, we get the following criterion (cf. also Remark~\ref{rem:17}).
\begin{theorem}\label{thm:compact_in_Lip2} Let $n \in \mathbb N$ and $\mathbb K \in \{\mathbb R, \mathbb C\}$. A non-empty subset $A$ of $\Lip_\varphi(X,\mathbb K^n)$ is precompact if and only if it is bounded and satisfies the condition (LDS). \end{theorem}
\begin{remark}\label{rem:Lip_0} It is worth underlining that Theorems~~\ref{thm:compact_in_Lip} and~\ref{thm:compact_in_Lip2} can be also applied to $\Lip_0(X,E)$ with any metric space $X$. This follows from the fact that $\Lip_0(X,E)$ is a closed subspace of $\Lip(X,E)$. \end{remark}
From Remark~\ref{rem:open_closed_in_L_LDS} we know that in the conditions (LDS) and (L) we may require all the sets $U_1,\ldots,U_N$ to be open in $\tilde{X}$. It would be tempting to replace those sets with open balls in Theorems~\ref{thm:compact_in_Lip} and~\ref{thm:compact_in_Lip2}. Unfortunately, this is impossible, even when $E$ is one-dimensional, as the following example shows.
\begin{example}\label{ex:balls_open_sets} Set $X:=[0,2]$ and consider the function $f \colon X \to \mathbb R$ given by $f(x)=1-\abs{x-1}$. It clearly belongs to $\Lip(X,\mathbb R)$. Now, take any finite family of open balls in $\tilde X$ that covers $\tilde X$. Let $B_{\tilde X}((\xi,\zeta),r)$ be a ball belonging to this family that contains infinitely many points of the form $(1-\frac{1}{n},1)$, where $n \in \mathbb N$. Then, $\abs{\xi-1+\frac{1}{n}}<r$ and $\abs{\zeta-1}<r$ for infinitely many $n \in \mathbb N$. Using continuity of the absolute value, we infer that for all but finitely many $m \in \mathbb N$ we have $\abs{\zeta-1+\frac{1}{m}}<r$. This implies that the open ball $B_{\tilde X}((\xi,\zeta),r)$ contains (at least) two points $(1-\frac{1}{n_1},1-\frac{1}{m_1})$ and $(1-\frac{1}{n_2},1-\frac{1}{m_2})$ with $n_1<m_1$ and $n_2>m_2$. And then \[
\abs[\Bigg]{\frac{f\bigl(1-\frac{1}{n_1}\bigr) - f\bigl(1-\frac{1}{m_1}\bigr)}{\abs[\big]{\frac{1}{m_1}-\frac{1}{n_1}}} - \frac{f\bigl(1-\frac{1}{n_2}\bigr) - f\bigl(1-\frac{1}{m_2}\bigr)}{\abs[\big]{\frac{1}{m_2}-\frac{1}{n_2}}}}=2. \] This shows that replacing the open sets $U_1,\ldots,U_N$ with open balls in the condition (LDS) in the statement of Theorem~\ref{thm:compact_in_Lip2} is not possible, even if $E=\mathbb R$ and the set $A$ is a singleton.
A similar example works also in the case of the condition (L) and Theorem~\ref{thm:compact_in_Lip}. It suffices to take the set $A:=\{f,g\}$ with the function $f$ defined above and $g(x)=0$ for $x \in X$ as well as the points $(1-\frac{1}{n},1)$ and $(1-\frac{1}{n}, 1+\frac{1}{n})$. (Note that for a sufficiently large $n \in \mathbb N$ those points must lie in the same open ball belonging to a given family of open balls covering $\tilde X$.) The details are left to the reader. \end{example}
We end this section with a brief discussion of precompactness criteria in $\BLip_\varphi(X,E)$. As the default norm $\vnorm{\cdot}_\varphi$ in this space is stronger than the norm $\norm{\cdot}_\varphi$ inherited from $\Lip_\varphi(X,E)$, we will need an additional condition.
\begin{theorem}\label{thm:compact_in_BLip} A non-empty subset $A$ of $\BLip_\varphi(X,E)$ is precompact if and only if
the set $A-A$ satisfies both the conditions (L) and (B), and the sections $A(x)$ are precompact for every $x \in X$. \end{theorem}
\begin{proof}[Sketch of the proof] Consider the linear operator $T\colon \BLip_\varphi(X,E)\to C(X,E)\times C(\tilde{X},E)$ given by $T(f)=(f, \Phi(f))$; here $\Phi$ is the de Leeuw's map. When we endow the target space with the norm $\norm{(g,h)}:=\norm{g}_\infty + \norm{h}_\infty$, then it is clear that $T$ is an isometric embedding of $\Lip_\varphi(X,E)$ into $C(X,E)\times C(\tilde{X},E)$. The rest of the proof is analogous to the proof of Theorem~\ref{thm:compact_in_Lip} with obvious changes. \end{proof}
In a similar vein to Theorem~\ref{thm:compact_in_Lip2} we get the following compactness criterion for $\BLip_\varphi(X,\mathbb K^n)$. Once again we skip the proof, because at this point it should be evident.
\begin{theorem}\label{thm:compact_in_BLip2} Let $n \in \mathbb N$ and $\mathbb K \in \{\mathbb R, \mathbb C\}$. A non-empty subset $A$ of $\BLip_\varphi(X,\mathbb K^n)$ is precompact if and only if it is bounded and satisfies the conditions (DS) and (LDS). \end{theorem}
Finally, we will show that in Theorems~\ref{thm:compact_in_BLip} and~\ref{thm:compact_in_BLip2} the additional conditions on the sets $A-A$ and $A$, respectively, are essential.
\begin{example} Let $X:=\mathbb R$ and choose $x_\ast=0$ as the base point. For each $n \in \mathbb N$, let $f_n \colon X \to \mathbb R$ be equal to $f_n(x)=1-\abs{\frac{1}{n}x-1}$ on $[0,2n]$, and zero elsewhere. Also, let $A:=\dset{f_n}{n \in \mathbb N}\cup \{f_0\}$, where $f_0$ denotes the zero function. If by $\norm{\cdot}_1$ we denote the norm in $\Lip(X,\mathbb R)$, then $\norm{f_n}_1 \to 0$ as $n \to +\infty$. Thus, $A$ is precompact as a subset of $\Lip(X,\mathbb R)$, that is, with respect to the norm (metric) inherited from $\Lip(X,\mathbb R)$. In particular, the sets $A-A$ and $A$ satisfy the conditions (L) and (LDS), respectively (see Theorems~\ref{thm:compact_in_Lip} and~\ref{thm:compact_in_Lip2}). Furthermore, $A(x)\subseteq [0,1]$ for $x \in X$, which means that all the sections $A(x)$ are precompact.
Now, our aim is to show that the sets $A-A$ and $A$ do not satisfy the conditions (B) and (DS), respectively. Because (DS) is stronger than (B) and is preserved when passing from a set to its algebraic difference, we need to focus on the condition (B) only. Let $U_1,\ldots,U_N$ be an arbitrary finite cover of $X$. Then, there is a set $U_i$ that contains infinitely many positive integers. Among those numbers, we can always find (at least) two of the form $n$ and $n+k$ for some integer $k \geq n$. And so, $\abs[\big]{\abs{(f_n-f_0)(n)}-\abs{(f_n-f_0)(n+k)}}=1$. Thus, the set $A-A$ does not satisfy the condition (B).
Finally, we show that although $A$ is bounded in $\BLip(X,E)$, it is not precompact in the norm of this space $\vnorm{\cdot}_1$; here $\BLip(X,E)$ denotes the subset of $\Lip(X,\mathbb R)$ consisting of bounded functions. Let $(f_{n_k})_{k \in \mathbb N}$ be an arbitrary subsequence of $(f_n)_{n\in \mathbb N}$. For a fixed $N \in \mathbb N$ choose $l \in \mathbb N$ so that $n_l \geq 2n_N$. Then, $\vnorm{f_{n_l}-f_{n_N}}_1 \geq \abs{f_{n_l}(n_l)-f_{n_N}(n_l)}=1$. Thus, $(f_n)_{n\in \mathbb N}$ does not admit a Cauchy subsequence with respect to $\vnorm{\cdot}_1$. Hence, $A$ is not precompact as a subset of $\BLip(X,\mathbb R)$. \end{example}
\subsection{Compactness criteria -- compact domains}
This time we assume that the metric space $X$ is compact. In such a case the spaces $\Lip_\varphi(X,E)$ and $\BLip_\varphi(X,E)$ coincide as sets. Moreover, their norms $\norm{\cdot}_\varphi$ and $\vnorm{\cdot}_\varphi$ are equivalent. Hence, we can pass from one to the other without any concern.
Our main goal in this section will be to show that for compact domains all that really matters is the behaviour of the difference quotients appearing in the condition (L) near the diagonal of $X\times X$. First, however, we will prove two technical lemmas.
\begin{lemma}\label{lem:introGH} Let $X$ be a compact metric space. Moreover, assume that $A$ is a non-empty and bounded subset of $\Lip_\varphi(X,E)$ such that the sections $A(x)$ are precompact for $x \in X$. For a fixed $\delta>0$ let $\tilde X_\delta:=(X\times X)\setminus \bigcup_{x\in X}(B_X(x,\delta)\times B_X(x,\delta))$. Also, let $\tilde{A}_\delta$ consist of all the mappings $h$ which are the restrictions of $\Phi(f)$ to $\tilde X_\delta$, where $f \in A$ and $\Phi$ is the de Leeuw's map. Then, for any $\varepsilon>0$ there is a finite open cover of $\tilde X_\delta$ such that $\norm{h(x,y)-h(\xi,\eta)}_E\leq \varepsilon$ for any $h\in \tilde{A}_\delta$ and any $(x,y),(\xi,\eta)$ contained in the same element of the cover of $\tilde X_\delta$. \end{lemma}
\begin{proof} Clearly, $\tilde X_\delta$ is a compact metric space in the maximum metric $d_\infty$ inherited from $X\times X$, and $\tilde A_\delta \subseteq C(\tilde X_\delta,E)$. Furthermore, observe that precompactness of $A(x)$ for $x\in X$ implies the precompactness of $\tilde A_\delta (x,y):=\dset{h(x,y)}{h \in \tilde A_\delta}$ for $(x,y) \in \tilde X_\delta$. Thus, by Corollary~\ref{rem:DS_B_3} we only need to show that the algebraic difference $\tilde A_\delta - \tilde A_\delta$ satisfies the condition (B) with open sets (cf.~Remarks~\ref{rem:open_closed} and~\ref{rem:open_closed2}).
By assumption, the set $A$ is bounded in $\Lip_\varphi(X,E)$. So, there is a positive constant $M$ such that $\vnorm{f}_\varphi=\norm{f}_\infty+\abs{f}_\varphi \leq M$ for $f\in A$. We may also assume that $M\geq \sup\dset{\varphi(d(x,y))}{x,y\in X}$. Because the diagonal $\dset{(x,x)}{x\in X}$ is compact and disjoint from the compact set $\tilde X_\delta$, there is a~positive constant $m$ such that $m\leq \inf\dset{\varphi(d(x,y))}{(x,y)\in \tilde X_\delta}$.
Now, fix an arbitrary $\varepsilon>0$. Recall that the comparison function is continuous on the non-negative half-axis $[0,+\infty)$ and $\varphi(0)=0$. Hence, there exists $r>0$ such that $\varphi(r) \leq \frac{m^2}{8M^2}\varepsilon$ and $\abs{\varphi(t)-\varphi(s)}\leq \frac{m^2}{8M}\varepsilon$ for any $t,s \in [0,\diam X]$ with $\abs{t-s}\leq 2r$. As the family $U_1,\ldots,U_N$ let us choose any finite covering of $\tilde X_\delta$ with open balls in $\tilde X_\delta$ of radius $\frac{1}{2}r$. Now, take any points $(x,y),(\xi,\eta) \in \tilde X_\delta$ belonging to the same member of the covering and any $h_1,h_2 \in \tilde A_\delta$. Then, $h_1=\Phi(f)|_{\tilde X_\delta}$ and $h_2=\Phi(g)|_{\tilde X_\delta}$ for some $f,g \in A$. Setting for simplicity $d_\varphi:=\varphi \circ d$, we obtain \begin{align*} &\abs[\Big]{\norm[\big]{(h_1-h_2)(x,y)}_E - \norm[\big]{(h_1-h_2)(\xi,\eta)}_E}\\ &\ \ = \abs[\Bigg]{\frac{\norm[\big]{(f-g)(x)-(f-g)(y)}_E}{d_\varphi(x,y)}-\frac{\norm[\big]{(f-g)(\xi)-(f-g)(\eta)}_E}{d_\varphi(\xi,\eta)}}\\ &\ \ \leq m^{-2} \abs[\Big]{\, d_\varphi(\xi,\eta)\cdot\norm[\big]{(f-g)(x)-(f-g)(y)}_E - d_\varphi(x,y)\cdot \norm[\big]{(f-g)(\xi)-(f-g)(\eta)}_E\,}\\ &\ \ \leq m^{-2}d_\varphi(\xi,\eta)\cdot \norm[\big]{(f-g)(x)-(f-g)(\xi) + (f-g)(\eta)-(f-g)(y)}_E\\ &\qquad + m^{-2}\abs[\big]{d_\varphi(x,y)-d_\varphi(\xi,\eta)}\cdot \norm[\big]{(f-g)(\xi)-(f-g)(\eta)}_E\\ &\ \ \leq m^{-2}d_\varphi(\xi,\eta)\cdot \norm[\big]{(f-g)(x)-(f-g)(\xi)}_E + m^{-2}d_\varphi(\xi,\eta)\cdot \norm[\big]{(f-g)(\eta)-(f-g)(y)}_E\\ &\qquad + m^{-2}\abs[\big]{d_\varphi(x,y)-d_\varphi(\xi,\eta)}\cdot \norm[\big]{(f-g)(\xi)}_E + m^{-2}\abs[\big]{d_\varphi(x,y)-d_\varphi(\xi,\eta)}\cdot \norm[\big]{(f-g)(\eta)}_E\\ &\ \ \leq 2m^{-2}M d_\varphi(\xi,\eta)\bigl[d_\varphi(x,\xi) + d_\varphi(\eta,y)\bigr] + 4m^{-2}M\abs[\big]{d_\varphi(x,y)-d_\varphi(\xi,\eta)}\\ &\ \ \leq 2m^{-2}M^2\bigl[\varphi(d(x,\xi)) + \varphi(d(\eta,y))\bigr] + 4m^{-2}M\abs[\big]{\varphi(d(x,y))-\varphi(d(\xi,\eta))}. \end{align*} Recall that $\tilde X_\delta$ is endowed with the maximum metric $d_\infty$. Hence, $\max\{d(x,\xi),d(\eta,y)\}\break=d_\infty((x,y),(\xi,\eta)) \leq r$ and $\abs{d(x,y)-d(\xi,\eta)}\leq d(x,\xi)+d(y,\eta)\leq 2r$. Thus, we obtain \begin{align*}
\abs[\Big]{\norm[\big]{(h_1-h_2)(x,y)}_E - \norm[\big]{(h_1-h_2)(\xi,\eta)}_E} \leq \tfrac{1}{2}\varepsilon + \tfrac{1}{2}\varepsilon =\varepsilon. \end{align*} Consequently, the algebraic difference $\tilde A_\delta - \tilde A_\delta$ satisfies the condition (B). \end{proof}
\begin{lemma}\label{lem:tube} Let $X$ be a compact metric space. Moreover, assume that for some $n \in \mathbb N$ the diagonal of $X\times X$ is covered by a finite collection of open balls ${B}_X(x_i,\frac{1}{n})\times {B}_X(x_i,\frac{1}{n})$, where $i=1,\ldots, m$. Then, there is a number $\delta>0$ such that $\bigcup_{x\in X}({B}_X(x,\delta)\times {B}_X(x,\delta)) \subseteq \bigcup_{i=1}^m({B}_X(x_i,\frac{1}{n})\times{B}_X(x_i,\frac{1}{n}))$. \end{lemma}
\begin{proof} Let $n \in \mathbb N$ such that $\dset{(x,x)}{x \in X} \subseteq \bigcup_{i=1}^m ({B}_X(x_i,\frac{1}{n})\times {B}_X(x_i,\frac{1}{n}))$ be fixed. Suppose that the claim is not true, that is, for any $q\in \mathbb{N}$ there is a point $(\xi^q,\eta^q)\in {B}_X(\zeta^q,\frac{1}{q})\times {B}_X(\zeta^q,\frac{1}{q})$ such that $(\xi^q,\eta^q) \notin \bigcup_{i=1}^m({B}_X(x_i,\frac{1}{n})\times {B}_X(x_i,\frac{1}{n}))$. By compactness of $X$ we may assume that the sequence of centers $(\zeta^q)_{q \in \mathbb N}$ converges to a point $z\in X$. As the set $\bigcup_{i=1}^m (B_X(x_i,\frac{1}{n})\times B_X(x_i,\frac{1}{n}))$ is open in $X\times X$, there is some $\delta>0$ such that $B_X(z,\delta)\times B_X(z,\delta)\subseteq \bigcup_{i=1}^m({B}_X(x_i,\frac{1}{n})\times {B}_X(x_i,\frac{1}{n}))$. But then, for all but finitely many $q \in \mathbb N$ we have ${B}_X(\zeta^q,\frac{1}{q})\times {B}_X(\zeta^q,\frac{1}{q})\subseteq B_X(z,\delta)\times B_X(z,\delta)$, which is impossible. \end{proof}
We saw in the proof of Lemma~\ref{lem:introGH} that compactness of $X$ was essential. Without it we would not have been able to apply Corollary~\ref{rem:DS_B_3}, or find the constants $m$ and $M$. On the other hand, in Lemma~\ref{lem:tube} this requirement seems a little bit artificial. Therefore, it is interesting to ask whether Lemma~\ref{lem:tube} would still hold, if the domain $X$ was allowed to be non-compact. Note that this question makes sense only for bounded metric spaces. Because, otherwise, the diagonal of $X\times X$ cannot be covered by a finite family of (open) balls. The following example shows that, if we replace ``compactness'' with ``boundedness'' in the statement of Lemma~\ref{lem:tube}, the result will fail.
\begin{example} Set $X:=\mathbb Z \setminus\{0\}$. Now, we define a metric on $X$. To simplify the formulas, we will write $(\pm k, \pm l)$ if both the numbers $k,l$ have the same sign, and $(\pm k, \mp l), (\mp k, \pm l)$ otherwise. Let \begin{align*} d(\pm n,\pm n)&:=0 & & \hspace*{-2cm} \text{for $n\in \mathbb N$,}\\ d(\pm 1,\mp 1)&:=\tfrac{1}{2}& &\\ d(\pm 1,\pm n)&:=d(\pm n,\pm 1):=\tfrac{1}{2}-\tfrac{1}{2n}& & \hspace*{-2cm} \text{for $n\geq 2$,}\\ d(\pm 1,\mp n)&:=d(\mp n,\pm 1):=\tfrac{1}{2}& &\hspace*{-2cm} \text{for $n\geq 2$,}\\ d(\pm n,\mp m)&:=\tfrac{1}{2n}+\tfrac{1}{2m} & &\hspace*{-2cm} \text{for $n, m\geq 2$,}\\ d(\pm n,\pm m)&:=\tfrac{1}{2n}+\tfrac{1}{2m} & &\hspace*{-2cm} \text{for $n,m\geq 2$ with $n\neq m$.} \end{align*} The proof that $d$ is indeed a metric on $X$ is quite straightforward, but tedious. So, we skip it.
Observe now that the diagonal of $X\times X$ is covered by $(B_X(1,\frac{1}{2})\times B_X(1,\frac{1}{2})) \cup (B_X(-1,\frac{1}{2}) \times B_X(-1,\frac{1}{2}))$. Further, given any fixed $\delta>0$ let $m,n \in \mathbb N$ be so that $d(m,-n)<\delta$. Then, clearly $(m,-n) \in B_X(m,\delta)\times B_X(m,\delta)$. But $(m,-n) \notin (B_X(1,\frac{1}{2})\times B_X(1,\frac{1}{2})) \cup (B_X(-1,\frac{1}{2}) \times B_X(-1,\frac{1}{2}))$. This shows that $\bigcup_{k \in X} (B_X(k,\delta)\times B_X(k,\delta)) \not\subseteq (B_X(1,\frac{1}{2})\times B_X(1,\frac{1}{2})) \cup (B_X(-1,\frac{1}{2}) \times B_X(-1,\frac{1}{2}))$ for any $\delta>0$. \end{example}
Before proving the main result of this section, for a non-empty set $A\subseteq \Lip_\varphi(X,E)$ let us introduce yet another condition: \begin{enumerate}
\item[($\Lambda$)] for every $\varepsilon>0$ and $n \in \mathbb N$ there is a radius $\delta>0$ and a finite number of open subsets $U_1,\ldots,U_N$ of $\tilde X$ with $\bigcup_{x\in X} (B_X(x,\delta)\times B_X(x,\delta))\cap\tilde X\subseteq \bigcup_{i=1}^N U_i\subseteq \bigcup_{x\in X} (B_X(x,\frac{1}{n})\times B_X(x,\frac{1}{n}))\cap\tilde X$ such that for any $f \in A$ and $i\in \{1,\ldots,N\}$ we have \[ \sup_{(x,y)\in U_i} \frac{\norm{f(x)-f(y)}_E}{\varphi(d(x,y))}-\inf_{(x,y)\in U_i}\frac{\norm{f(x)-f(y)}_E}{\varphi(d(x,y))}\leq \varepsilon. \] \end{enumerate} The above condition is a localized versions of (L) in a sense that instead of considering a covering of the whole $\tilde{X}$, we focus our attention on the diagonal of $X\times X$. Note also that here we use a slightly different formulation of the inequality appearing in (L). This change is sanctioned by the fact that for a bounded real function $g$ defined on a non-empty set $Y$ we have $\sup_{x,y \in Y}\abs{g(x)-g(y)}=\sup_{x \in Y} g(x) - \inf_{y \in Y}g(y)$ (see~\cite{Lojasiewicz}*{p.~4}).
And now the main result follows.
\begin{proposition}\label{lem:GH} Let $X$ be a compact metric space. Moreover, assume that $A$ is a non-empty subset of $\Lip_\varphi(X,E)$ such that the sections $A(x)$ are precompact for $x \in X$. Then, the following conditions are equivalent\textup: \begin{enumerate}[label=\textup{(\roman*)}]
\item\label{it:L} $A-A$ satisfies (L) with open sets $U_1,\ldots,U_N$,
\item\label{it:Lambda} $A-A$ satisfies ($\Lambda$). \end{enumerate} \end{proposition}
\begin{proof} $\ref{it:L}\Rightarrow\ref{it:Lambda}$ Fix any $\varepsilon>0$ and $n \in \mathbb N$. Then, there exists an open cover $U_1,\ldots,U_N$ of $\tilde X$ such that for any $i \in\{1,\ldots,N\}$ and any $f,g \in A$ we have \[ \sup_{(x,y)\in U_i}\frac{\norm{(f-g)(x) - (f-g)(y)}_E}{\varphi(d(x,y))} -\inf_{(x,y) \in U_i}\frac{\norm{(f-g)(x) - (f-g)(y)}_E}{\varphi(d(x,y))}\leq \varepsilon \] (see the observation before Proposition~\ref{lem:GH}). As the set $\tilde X$ is open in $X\times X$, so are the members of the collection $U_1,\ldots,U_N$. Since the diagonal $\dset{(x,x)}{x\in X}$ is a compact subset of the metric space $X\times X$, there is a finite number of points $x_1,\ldots,x_k \in X$ such that $\dset{(x,x)}{x\in X}\subseteq \bigcup_{j=1}^k (B_X(x_j,{\frac{1}{n}})\times B_X(x_j,{\frac{1}{n}}))$. For each $i\in \{1,\ldots,N\}$ and $j\in \{1,\ldots,k\}$ set $V_{i,j}:=(B_X(x_j,{\frac{1}{n}})\times B_X(x_j,{\frac{1}{n}}))\cap U_i$. By Lemma~\ref{lem:tube} it is clear that the family consisting of all the non-empty sets $V_{i,j}$ satisfies all the requirements of the condition ($\Lambda$) for $A-A$.
$\ref{it:Lambda}\Rightarrow\ref{it:L}$ We will divide this part into two steps. We begin with showing that, under the assumption of precompactness of the sections $A(x)$, the condition~\ref{it:Lambda} implies that the set $A$ is bounded in $\Lip_\varphi(X,E)$. Let $\delta>0$ and $U_1,\ldots, U_N$ be chosen as in the condition ($\Lambda$) for $A-A$ and $\varepsilon=n=1$.
Suppose that $\sup_{f\in A}\sup_{x\in X}\norm{f(x)}_E=+\infty$. Then, there exist two sequences $(x_m)_{m\in \mathbb{N}}$ in $X$ and $(f_m)_{m\in \mathbb{N}}$ in $A$ for which $\lim_{m\to\infty}\norm{f_m(x_m)}_E=+\infty$. Since the metric space $X$ is compact we may assume that $\lim_{m\to\infty}x_m=y\in X$. If $x_{m_l}=y$ for infinitely many indices $l \in \mathbb N$, we would get $\lim_{l\to\infty}\norm{f_{m_l}(y)}_E=+\infty$. This, in turn, would mean that $A(y)$ is not precompact, which is absurd. Therefore, we may assume that $x_m \neq y$ for all $m \in \mathbb N$. Hence, $(y,x_m) \in (B_X(y,\delta) \times B_X(y,\delta))\cap \tilde X \subseteq \bigcup_{j=1}^N U_j$ for all but finitely many $m \in \mathbb N$. As there are only $N$ sets $U_j$, this implies that there is an index $i \in \{1,\ldots,N\}$ such that $(y,x_m)\in U_i$ for infinitely many $m$'s. Passing to a subsequence yet another time, we may state that $(y,x_m)\in U_i$ for all $m \in \mathbb N$. Now, pick a function $f \in A$ and a point $(\xi,\eta) \in U_i$. Note that, because of the compactness of the metric space $X$ and the continuity of the functions $\varphi$ and $f$, there is a constant $R>0$ with $\varphi(d(x_m,y))\leq R$ and $\norm{f(x_m)}_E\leq R$ for $m \in \mathbb N$. Observe also that, since the sections $A(x)$ are precompact, (adjusting the constant $R$ if necessary) we have $\norm{(f_m-f)(z)}_E\leq R$ for $m \in \mathbb N$, where $z \in \{\xi,\eta,y\}$. Thus, for all $m \in \mathbb N$ we get \begin{align*} &\norm{f_m(x_m)}_E\\
&\ \ \leq \norm{f_m(x_m)-f(x_m)}_E+\norm{f(x_m)}_E\\
&\ \ \leq \norm{(f_m-f)(x_m)-(f_m-f)(y)}_E+\norm{(f_m-f)(y)}_E+\norm{f(x_m)}_E\\
&\ \ = \varphi(d(x_m,y))\cdot \frac{\norm{(f_m-f)(x_m)-(f_m-f)(y)}_E}{\varphi(d(x_m,y))}+\norm{(f_m-f)(y)}_E+\norm{f(x_m)}_E\\
&\ \ \leq \varphi(d(x_m,y)) \cdot \sup_{(p,q)\in U_i}\frac{\norm{(f_m-f)(p)-(f_m-f)(q)}_E}{\varphi(d(p,q))}+\norm{(f_m-f)(y)}_E+\norm{f(x_m)}_E\\
&\ \ \leq \varphi(d(x_m,y))\biggl(1+\frac{\norm{(f_m-f)(\xi)-(f_m-f)(\eta)}_E}{\varphi(d(\xi,\eta))}\biggr)+\norm{(f_m-f)(y)}_E+\norm{f(x_m)}_E\\ &\ \ \leq 3R + 2R^2/\varphi(d(\xi,\eta))<+\infty. \end{align*} This leads to a contradiction. Hence, $M:=\sup_{f\in A}\norm{f}_\infty<+\infty$.
Using a similar reasoning to the above one, we will show that the set $A$ is also bounded in the Lipschitz semi-norm $\abs{\cdot}_\varphi$. (Note that we will use the same letters as in the former part, but with possibly different meanings.) Suppose that \[
\sup_{f\in A}\sup_{(x,y)\in \tilde X} \frac{\norm{f(x)-f(y)}_E}{\varphi(d(x,y))}=+\infty. \] Hence, there are sequences $(x_m, y_m)_{m\in \mathbb{N}}$ in $\tilde X$ and $(f_m)_{m\in \mathbb{N}}$ in $A$ for which $\lim_{m\to\infty} \norm{f_m(x_m)-f_m(y_m)}_E/\varphi(d(x_m,y_m))=+\infty$. Because the metric space $X \times X$ is compact, we may assume that $\lim_{m \to \infty}(x_m,y_m)=(a,b) \in X\times X$. If $a\neq b$, then for every $m \in \mathbb N$ we would have \[
\frac{\norm{f_m(x_m)-f_m(y_m)}_E}{\varphi(d(x_m,y_m))} \leq \frac{2M}{\varphi(d(x_m,y_m))}. \] Consequently, \[
\lim_{m\to\infty} \frac{\norm{f_m(x_m)-f_m(y_m)}_E}{\varphi(d(x_m,y_m))}\leq \frac{2M}{\varphi(d(a,b))}<+\infty, \] which is absurd. Thus, $a=b$. This means that $(x_m,y_m) \in (B_X(a,\delta)\times B_X(a,\delta))\cap \tilde{X} \subseteq \bigcup_{j=1}^N U_j$ for all but finitely many $m \in \mathbb N$. Therefore, there is an index $i \in \{1,\ldots,N\}$ such that $(x_m,y_m) \in U_i$ for infinitely many $m$'s. Passing to a subsequence if necessary, we may actually claim that the above relation holds for all $m \in \mathbb N$. If we fix $f \in A$ and $(\xi,\eta)\in U_i$, then in view of the condition ($\Lambda$) for all $m \in \mathbb N$ we have \begin{align*}
\frac{\norm{f_m(x_m)-f_m(y_m)}_E}{\varphi(d(x_m,y_m))}& \leq \frac{\norm{(f_m-f)(x_m)-(f_m-f)(y_m)}_E}{\varphi(d(x_m,y_m))}+\frac{\norm{f(x_m)-f(y_m)}_E}{\varphi(d(x_m,y_m))}\\
& \leq \sup_{(p,q)\in U_i}\frac{\norm{(f_m-f)(p)-(f_m-f)(q)}_E}{\varphi(d(p,q))}+\frac{\norm{f(x_m)-f(y_m)}_E}{\varphi(d(x_m,y_m))}\\
& \leq 1+\frac{\norm{(f_m-f)(\xi)-(f_m-f)(\eta)}_E}{\varphi(d(\xi,\eta))}+\frac{\norm{f(x_m)-f(y_m)}_E}{\varphi(d(x_m,y_m))}\\
& \leq 1 + 4M/\varphi(d(\xi,\eta)) + \abs{f}_{\varphi}<+\infty. \end{align*} This is impossible. Therefore, $\sup_{f\in A} \abs{f}_{\varphi}<+\infty$. Consequently, the set $A$ is bounded in $\Lip_\varphi(X,E)$.
Now, we focus on the second -- main -- step of this part of the proof. (Once again we ``reset'' our notation.) Fix $\varepsilon>0$ and $n \in \mathbb N$. Then, by the condition~\ref{it:Lambda}, there is a radius $\delta>0$ and an open cover $U_1,\ldots,U_k$ of $\tilde X$ with $\bigcup_{x\in X} (B_X(x,\delta)\times B_X(x,\delta))\cap\tilde X\subseteq \bigcup_{i=1}^k U_i\subseteq \bigcup_{x\in X} (B_X(x,\frac{1}{n})\times B_X(x,\frac{1}{n}))\cap\tilde X$ such that for any $f,g\in A$ and $i\in \{1,\ldots,k\}$ we have \[ \sup_{(x,y)\in U_i} \frac{\norm{(f-g)(x)-(f-g)(y)}_E}{\varphi(d(x,y))}-\inf_{(x,y)\in U_i}\frac{\norm{(f-g)(x)-(f-g)(y)}_E}{\varphi(d(x,y))}\leq \varepsilon. \] From the previous step we know that the set $A$ is bounded in $\Lip_{\varphi}(X,E)$. Hence, we can apply Lemma~\ref{lem:introGH} with $\frac{1}{2}\varepsilon$ and $\frac{1}{4}\delta$ to find a finite open cover $W_{k+1},\ldots, W_N$ of $\tilde X_{\frac{1}{4}\delta}$ such that \[
\sup_{f \in A} \norm[\Bigg]{\frac{f(x)-f(y)}{\varphi(d(x,y))} - \frac{f(\xi)-f(\eta)}{\varphi(d(\xi,\eta))}}_E \leq \frac{1}{2}\varepsilon \] for any $(x,y),(\xi,\eta)$ belonging to the same member of the collection $W_{k+1},\ldots, W_N$. For every $i=k+1,\ldots,N$ let $U_i:=W_i\setminus \dset{(x,y) \in X\times X}{\inf_{z \in X}\max\{d(x,z),d(y,z)\}\leq \frac{1}{2}\delta}$. We may clearly assume that all the sets $U_i$ are non-empty; otherwise we just remove the empty ones from our family.
We claim that the sets $U_i$ for $i \in \{k+1,\ldots,N\}$ are open in $X \times X$, and hence in $\tilde X$. So, let us fix a set $U_j$ with $j \in \{k+1,\ldots,N\}$ and take any point $(a,b) \in U_j$. Since $(a,b) \in W_j$, and the set $W_j$ is open in $\tilde X_{\frac{1}{4}\delta}$, there is a radius $r>0$ such that $(B_X(a,r) \times B_X(b,r)) \cap \tilde X_{\frac{1}{4}\delta}\subseteq W_j$. Set $\rho:=\inf_{z \in X}\max\{d(a,z),d(b,z)\}$ and $R:=\min\{r,\frac{1}{4}\delta,\rho-\frac{1}{2}\delta\}$. Note that by definition we have $\rho>\frac{1}{2}\delta$. So, the radius $R$ is well-defined. We will show now that the open ball $B_{X}(a,R)\times B_X(b,R)$ in $X\times X$ is included in $U_j$. Clearly, it is included in $B_X(a,r) \times B_X(b,r)$. Furthermore, $B_{X}(a,R)\times B_X(b,R) \subseteq \tilde X_{\frac{1}{4}\delta}$. Otherwise, there would exist points $u,p,q \in X$ such that $d(a,p)<R$, $d(b,q)<R$, $d(p,u)<\frac{1}{4}\delta$ and $d(q,u)<\frac{1}{4}\delta$. This, in turn, would imply that $d(u,a)\leq d(a,p)+d(p,u)<\frac{1}{2}\delta$. And, similarly, $d(u,b)<\frac{1}{2}\delta$. But this is impossible, because $\rho= \inf_{z \in X}\max\{d(a,z),d(b,z)\}>\frac{1}{2}\delta$. Therefore, $B_{X}(a,R)\times B_X(b,R) \subseteq (B_X(a,r) \times B_X(b,r))\cap \tilde X_{\frac{1}{4}\delta} \subseteq W_j$.
It remains to show that the sets $\dset{(x,y) \in X\times X}{\inf_{z \in X}\max\{d(x,z),d(y,z)\}\leq \frac{1}{2}\delta}$ and $B_{X}(a,R)\times B_X(b,R)$ are disjoint. Suppose, on the contrary, that there is a point $(\alpha,\beta) \in B_{X}(a,R)\times B_X(b,R)$ such that $\inf_{z \in X}\max\{d(\alpha,z),d(\beta,z)\}\leq \frac{1}{2}\delta$. As the function $z \mapsto \max\{d(\alpha,z),d(\beta,z)\}$ is continuous and the metric space $X$ is compact, there is $\zeta\in X$ such that $\inf_{z \in X}\max\{d(\alpha,z),d(\beta,z)\}=\max\{d(\alpha,\zeta),d(\beta,\zeta)\}$. Now, we have \begin{align*} \rho&=\inf_{z \in X}\max\{d(a,z),d(b,z)\}\\ &\leq \max\{d(a,\zeta),d(b,\zeta)\}\\ &\leq \max\{d(a,\alpha),d(b,\beta)\} + \max\{d(\alpha,\zeta),d(\beta,\zeta)\}\\ &< \rho-\tfrac{1}{2}\delta + \tfrac{1}{2}\delta = \rho. \end{align*} This is absurd. Thus, the sets $\dset{(x,y) \in X\times X}{\inf_{z \in X}\max\{d(x,z),d(y,z)\}\leq \frac{1}{2}\delta}$ and $B_{X}(a,R)\times B_X(b,R)$ are disjoint. Consequently, $B_{X}(a,R)\times B_X(b,R) \subseteq U_j$. Hence, the sets $U_{k+1},\ldots,U_N$ are open in $\tilde{X}$.
The rest of the proof is straightforward. It is not difficult to see that the family of sets $U_1,\ldots, U_k, U_{k+1},\ldots, U_N$ satisfies all the requirements of the condition~\ref{it:L}. \end{proof}
From Remark~\ref{rem:open_closed_in_L_LDS}, Theorem~\ref{thm:compact_in_Lip} and Proposition~\ref{lem:GH} we immediately get the following second compactness criterion in the space of Lipschitz continuous functions $\Lip_\varphi(X,E)$.
\begin{theorem}\label{thm:compact_in_Lip'} Let $X$ be a compact metric space. A non-empty subset $A$ of $\Lip_\varphi(X,E)$ is precompact if and only if the set $A-A$ satisfies the condition ($\Lambda$) and the sections $A(x)$ for $x \in X$ are precompact. \end{theorem}
\begin{remark} Theorem~\ref{thm:compact_in_Lip'} provides also a compactness criterion in the space $\Lip_0(X,E)$ with $X$ being a compact metric space (cf.~Remark~\ref{rem:Lip_0}). \end{remark}
In Corollary~\ref{rem:DS_B_3} we saw that in the case of $C(X,E)$, when the domain $X$ is compact, we can replace the assumption ``$A-A$ satisfies the condition (B)'' with ``$A$ satisfies the condition (DS)''. The conditions (L) and (LDS), we consider in this section, can be viewed as versions of (B) and (DS) for $\Phi(A)$, respectively. Therefore, a natural question is whether in Theorem~\ref{thm:compact_in_Lip'} instead of the conditions ($\Lambda$) or (L) we can use (LDS)? The answer is negative, as shown by the following simple example.
\begin{example} Consider the Banach space $l^\infty$ of all bounded real sequences, endowed with the supremum norm $\norm{\cdot}_\infty$. For each $n \in \mathbb N$ let $x_n:=(0,\ldots,0,\frac{1}{n},0,\ldots)$, where the only non-zero element is at the $n$-th place. Moreover, let $x_0$ be the zero sequence. Set $X:=\dset{x_n}{n \in \mathbb N \cup\{0\}}$. Because $\norm{x_n}_\infty \to 0$ when $n \to +\infty$, the metric space $X$ is compact (in the metric inherited from $l^\infty$). Now, for $a \in [0,1]$ define $f_a \colon X \to l^\infty$ by $f_a(x)=ax$. And, consider $A:=\dset{f_a}{a\in [0,1]}$. Clearly, $A\subseteq \Lip(X,l^\infty)$.
Observe that $A(x_0)$ is a singleton, and $A(x_n)$ for every $n \in \mathbb N$ is homeomorphic with the interval $[0,1]$. This means that each section $A(x)$ for $x \in X$ is (pre)compact. Further, note that for any point $(x,y) \in \tilde{X}$ and any $a,b \in [0,1]$ we have \[
\frac{\norm{(f_a-f_b)(x)-(f_a-f_b)(y)}_{\infty}}{\norm{x-y}_{\infty}}=\abs{a-b}. \] Therefore, the set $A-A$ satisfies the condition (L) with $U_1:=\tilde{X}$. By Proposition~\ref{lem:GH}, it also satisfies the condition ($\Lambda$).
Finally, we claim that the set $A$ does not satisfy the condition (LDS). Let $W_1,\ldots,W_N$ be any cover of $\tilde X$. As there are infinitely many points $(x_n,x_0)$, there is an element $W_i$ of the cover that contains (at least) two of them, say $(x_n,x_0)$ and $(x_m,x_0)$. Then, \[
\norm[\Bigg]{\frac{f_1(x_n)-f_1(x_0)}{\norm{x_n-x_0}_\infty} - \frac{f_1(x_m)-f_1(x_0)}{\norm{x_m-x_0}_\infty}}_{\infty} = \norm[\Bigg]{\frac{x_n}{\norm{x_n}_\infty} - \frac{x_m}{\norm{x_m}_\infty}}_{\infty}=1. \] This proves our claim.
As a closing remark, note that the above example works only because the target space $l^\infty$ is infinite-dimensional. If its dimension were finite, it might not be possible to find a set $W_i$ containing two different points $(x_n,x_0)$ and $(x_m,x_0)$. \end{example}
\subsection{Compactness criterion in $\lip_\varphi(X,E)$} \label{sec:55}
In the theory of Lipschitz spaces special attention is paid to the subclass $\lip_\varphi(X,E)$ of $\Lip_\varphi(X,E)$ called the \emph{little Lipschitz space}. It, along with its various modifications and versions, appears naturally, when the problem of the second predual of $\Lip_\varphi(X,E)$ is investigated. Although it is possible to define little Lipschitz space with $X$ being non-compact (Weaver does it in Section~4.2 of his book~\cite{weaver}), it requires much more technicalities. Since $\lip_\varphi(X,E)$ is not the main object of our study, and the precompactness criterion for this space heavily relies on the results from the previous section, in the sequel we will assume that the metric space $X$ is compact.
Let us now introduce the little Lipschitz space properly. Fix a comparison function $\varphi$. Also, let $(X, d)$ be a compact metric space and let $(E,\norm{\cdot}_E)$ be a normed space. The class $\lip_\varphi(X,E)$ consists of all functions from $\Lip_\varphi(X,E)$ that satisfy the following condition \begin{equation*}\label{eq:little_lipschitz_def}
\lim_{\delta\to 0^+} \sup\dset[\Bigg]{\frac{\norm{f(x)-f(y)}_E}{\varphi(d(x,y))}}{\text{$x,y \in X$ with $0<d(x,y) \leq \delta$}} = 0. \end{equation*} Equivalently, a map $f \in \Lip_{\varphi}(X,E)$ belongs to $\lip_\varphi(X,E)$, if for every $\varepsilon>0$ there exists $\delta>0$ such that $\norm{f(x)-f(y)}_E\leq \varepsilon \varphi(d(x,y))$ for all $x,y \in X$ with $d(x,y) \leq \delta$. Mappings satisfying the above equivalent conditions are called \emph{locally flat}.
The space $\lip_\varphi(X,E)$ is endowed with one of the equivalent norms $\norm{\cdot}_\varphi$ or $\vnorm{\cdot}_\varphi$ inherited from $\Lip_\varphi(X,E)$. Further, it is a Banach space, when $E$ is complete. It is easy to check that $\lip_\varphi(X,E)$ is a closed subspace of $\Lip_\varphi(X,E)$. However, in general, $\lip_\varphi(X,E)\neq \Lip_\varphi(X,E)$; to see this it suffices to set $\varphi(t)=t$, $X:=[0,1]$, $E:=\mathbb R$ and take the function $f(x)=x$. Actually, it turns out that in this setting the little Lipschitz space consists only of the constant functions (see~\cite{weaver}*{Example~4.8}). What is surprising is that this phenomenon holds more generally. If $X$ is a compact connected Riemannian manifold and $\varphi(t)=t$, then $\lip_\varphi(X,\mathbb R)$ is one-dimensional and consists of constant functions only (see~\cite{weaver}*{Example~4.9}).
On the other hand, for any comparison function $\varphi$ such that $\lim_{t \to 0^+} \varphi(t)/t=+\infty$, the class $\Lip(X,\mathbb R)$ is dense in $\lip_\varphi(X, \mathbb R)$ (see~\cite{hanin}*{Proposition~4} and cf.~\cite{weaver2}*{Corollary~1.5}).
Now, we move to characterizing precompact subsets of the little Lipschitz space $\lip_\varphi(X,E)$. This time there is no need to distinguish between $E$ being finite- or infinite-dimensional. There is a very simple explanation of this phenomenon. Namely, if $f \in \lip_{\varphi}(X,E)$, then the mapping $\Phi(f)$, where $\Phi$ is the de Leeuw's map, can be extended continuously from $\tilde X$ over the whole (compact) product $X \times X$ simply by putting $\Phi(f)(x,x)=0$ for $x \in X$.
\begin{theorem}\label{th:little:lip} Let $X$ be a compact metric space. A non-empty subset $A$ of $\lip_\varphi(X,E)$ is precompact if and only if \begin{enumerate}[label=\textup{(\roman*)}]
\item the sections $A(x)$ are precompact for every $x \in X$, and
\item the set $A$ is \emph{uniformly locally flat}, that is, for every $\varepsilon>0$ there exists $\delta>0$ such that for any $x,y \in X$ with $d(x,y)\leq \delta$ we have $\norm{f(x)-f(y)}_E \leq \varepsilon \varphi(d(x,y))$ for every $f\in A$. \end{enumerate} \end{theorem}
\begin{proof} Fix $\varepsilon>0$. Since $A$ is a uniformly locally flat set of functions, there exists $\eta>0$ such that $\norm{f(x)-f(y)}_E\leq \frac{1}{2}\varepsilon \varphi(d(x,y))$ for all $f \in A$ and all $x,y \in X$ with $d(x,y)\leq \eta$. (To make it easier to refer to the previous results, we have changed the notation a little bit here.) Furthermore, let us fix $n \in \mathbb N$, and choose $m \in \mathbb N$ so that $\frac{1}{m}\leq \min\{\frac{1}{2}\eta,\frac{1}{n}\}$. Now, pick a finite family of points $x_1,\ldots,x_N \in X$ such that the diagonal of $X\times X$ is covered by the union of the balls $B_X(x_i,\frac{1}{m})\times B_X(x_i,\frac{1}{m})$, where $i\in \{1,\ldots,N\}$.
Define $U_i:=(B_X(x_i,\frac{1}{m})\times B_X(x_i,\frac{1}{m}))\cap \tilde X$ for $i\in \{1,\ldots,N\}$. Then, clearly $\bigcup_{i=1}^N U_i \subseteq \bigcup_{x\in X}(B_X(x,\frac{1}{n})\times B_X(x,\frac{1}{n}))\cap \tilde X$. Moreover, by Lemma~\ref{lem:tube} (applied with $m$) there is a radius $\delta>0$ such that $\bigcup_{x\in X}(B_X(x,\delta)\times B_X(x,\delta))\cap \tilde X \subseteq \bigcup_{i=1}^N U_i$. If we fix $f,g \in A$ and $i\in \{1,\ldots,N\}$, then by the uniform local flatness of $A$ we obtain \begin{align*} & \sup_{(x,y)\in U_i} \frac{\norm{(f-g)(x)-(f-g)(y)}_E}{\varphi(d(x,y))}-\inf_{(x,y)\in U_i}\frac{\norm{(f-g)(x)-(f-g)(y)}_E}{\varphi(d(x,y))}\\ &\qquad \leq \sup_{(x,y)\in U_i} \frac{\norm{(f-g)(x)-(f-g)(y)}_E}{\varphi(d(x,y))} \leq \varepsilon. \end{align*} Thus, by Theorem~\ref{thm:compact_in_Lip'} we can conclude that $A$ is precompact in $\Lip_\varphi(X,E)$, and consequently in $\lip_\varphi(X,E)$.
To prove the opposite implication, let us assume that $A$ is a precompact subset of $\lip_\varphi(X,E)$ and $f_1,\ldots,f_m\in A$ is its finite $\frac{1}{2}\varepsilon$-net. Choose $\delta>0$ so that $\max_{1\leq i\leq m}\norm{f_i(x)-f_i(y)}_E\leq \frac{1}{2}\varepsilon\varphi(d(x,y))$ for $x,y \in X$ with $d(x,y)\leq \delta$. For any $f\in A$ there is some $j\in \{1,\ldots,m\}$ for which we have $\norm{f-f_j}_\varphi \leq \frac{1}{2}\varepsilon$. Hence, \begin{align*} \norm{f(x)-f(y)}_E &\leq \norm{(f-f_j)(x)-(f-f_j)(y)}_E+\norm{f_j(x)-f_j(y)}_E\\ & \leq \tfrac{1}{2}\varepsilon\varphi(d(x,y))+ \tfrac{1}{2}\varepsilon\varphi(d(x,y))=\varepsilon\varphi(d(x,y)), \end{align*} whenever $x,y \in X$ are such that $d(x,y)\leq\delta$. This shows that the family $A$ is uniformly locally flat. To end the proof it suffices to observe that the Lipschitz norm $\norm{\cdot}_\varphi$ is stronger than the supremum norm $\norm{\cdot}_\infty$, and hence precompactness of $A$ implies precompactness of each section $A(x)$, where $x \in X$. \end{proof}
\begin{remark} Theorem~\ref{th:little:lip} can be also applied to the space $\lip_0(X,E)$, which consists of those functions from $\Lip_0(X,E)$ that are locally flat. \end{remark}
\begin{remark} Theorem~\ref{th:little:lip} in the case when $E:=\mathbb R$ and $\varphi(t)=t$ was proven by Johnson using a different approach than ours (see~\cite{johnson}*{Theorem~3.2}). A similar result was also obtained by Garc\'{\i}a-Lirola \emph{et al.}, who, motivated by some considerations of duality, studied compactness in a certain little Lipschitz space (see~\cite{GPR}*{Lemma~2.7}). To be more specific, they worked in a subspace of $\lip_0(X,\mathbb R)$ consisting of those functions that are also continuous with respect to a topology $\tau$ on $X$. Furthermore, they assumed that $X$ is compact with respect to $\tau$ and that the metric $d$ of $X$ is $\tau$-lower semi-continuous. \end{remark}
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\bib{weaver}{book}{ author = {Weaver, N.}, title = {Lipschitz Algebras}, publisher = {World Scientific Publishing Co. Pte. Ltd}, address={Singapore}, year = {2018}, }
\end{biblist} \end{bibdiv}
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\begin{document}
\title{Boundary crossing identities for diffusions having the time inversion property}
\author{ L. Alili} \address{Department of Statistics, The University of Warwick, Coventry CV4 7AL, United Kingdoms} \email{[email protected]}
\author{P. Patie} \address{Department of Mathematical Statistics and Actuarial Science, University of Bern, Alpeneggstrasse 22, CH-3012 Bern, Switzerland} \email{[email protected]}
\begin{abstract} We review and study a one-parameter family of functional transformations, denoted by $(S^{(\beta)})_{\beta\in \mathbb{R}}$, which, in the case $\beta<0$, provides a path realization of bridges associated to the family of diffusion processes enjoying the time inversion property. This family includes the Brownian motion, Bessel processes with a positive dimension and their conservative $h$-transforms. By means of these transformations, we derive an explicit and simple expression which relates the law of the boundary crossing times for these diffusions over a given function $f$ to those over the image of $f$ by the mapping $S^{(\beta)}$, for some fixed $\beta\in \mathbb{R}$. We give some new examples of boundary crossing problems for the Brownian motion and the family of Bessel processes. We also provide, in the Brownian case, an interpretation of the results obtained by the standard method of images and establish connections between the exact asymptotics for large time of the densities corresponding to various curves of each family. \end{abstract}
\keywords{Self-similar diffusions; Brownian motion; Bessel processes; Time inversion property;
Boundary crossing problem} \subjclass[2000]{60J60, 60J65, 60G18, 60G40}
\maketitle
\section{Introduction} \noindent Let $X:=(X_t, t\geq 0)$ be a $E$-valued, where $E= \mathbb{R}$ or $\mathbb{R}_+=[0,\infty)$, conservative, in the sense that it has an infinite life-time, $2$-self-similar homogenous diffusion enjoying the time inversion property in the sense of Shiga and Watanabe \cite{Shiga-Watanabe-73}. We denote by $(\mathbb{P}_x)_{x\in E}$ the family of probability measures of $X$ which act on $\mathcal{C}(\mathbb{R}_+,E)$, the space of continuous functions from $\mathbb{R}_{+}$ into $E$, such that $\mathbb{P}_x(X_0=x)=1$. The standard notation $\mathcal{F}_t$ is used for the $\sigma$-algebra generated by the process $X$ up to time $t$ and we write simply $\mathcal{F}=\mathcal{F}_{\infty}$. The above assumptions mean that, for any $x\in E$ and $c>0$, \begin{equation*} \textrm{the law of }(c^{-1}X_{c^2t},t\geq0;\mathbb{P}_{cx}) \textrm{ is } \mathbb{P}_x \qquad (\textrm{2-self-similarity}) \end{equation*} and the process \begin{equation} \label{eq:ti} i(X):=(tX_{1/t}, t>0; \mathbb{P}_x) \qquad(\textrm{time inversion}) \end{equation} is a homogeneous conservative diffusion. It is well known that the class of processes considered therein consists of Brownian motion and Bessel processes of positive dimensions, see Watanabe \cite{Watanabe-75}. Next, let $f\in \mathcal{C}(\mathbb{R}_+,E)$, such that $f(0) \neq X_0$ and set $$T^{f} = \inf \left\{s > 0; \:X_s = f(s) \right\}$$ with the usual convention that $\inf\{ \emptyset \}=+\infty$. The study of the distribution of the stopping time $T^{f}$ is known as the boundary crossing or first passage problem. Unfortunately, the explicit determination of these distributions is only attainable for a few specific functions. In this paper, we suggest a new method which allows to relate, in a simple and explicit manner, the law of $T^{f}$ with the ones of the family of stopping times $(T^{f^{(\beta)}})_{\beta \in \mathbb{R}}$ with $f^{(\beta)}:=S^{(\beta)}(f)$ and \begin{eqnarray} \label{definition-transformation} S^{(\beta)}: \quad \mathcal{C}(\mathbb{R}_+, E) & \rightarrow & \mathcal{C}([0,\zeta^{(\beta)}), E) \\ \nonumber f &\rightarrow& \left(1+\beta .\right)f\left(\frac{.}{1+\beta .}\right) \end{eqnarray}
where \[\zeta^{(\beta)}= \begin{cases} 1/|\beta|, \quad & \textrm{if } \beta <0, \\ +\infty, & \textrm{otherwise}.
\end{cases}\]
The results extend to $h$-transforms of the prescribed processes leading to conservative diffusions.
In the case $\beta<0$, our methodology has a simple description. Indeed, as a consequence of the time inversion property, Pitman and Yor \cite{Pitman-Yor-81} showed that the law of the process $(S^{(\beta)}(X)_t, 0\leq t< \zeta^{(\beta)})$ corresponds to the law of a bridge associated to $X$. Thus, one may relate, by means of a deterministic time change, the law of $T^{f}$ with the first crossing time of the curve $f^{(\beta)}$ by the bridge $(S^{(\beta)}(X)_t, 0\leq t< \zeta^{(\beta)})$. Finally, our identities are then obtained by using the construction of the law of bridges of Markov processes as a Doob $h$-transform of their laws, see Fitzsimmons et al.~\cite{Fitzsimmons-Pitman-Yor-93}. We also show that similar devices can be implemented in the case $\beta>0$.
The motivations for investigating the boundary crossing problem are of both practical and theoretical importance. Indeed, on the one hand, such studies were originally motivated by their connections to sequential analysis, see e.g.~Robbins and Siegmund \cite{Robbins-Siegmund-70} and Smirnov-Kolmogorov test, see e.g.~Lerche \cite{Lerche-86}. On the other hand, this problem has found many applications in several fields of sciences, such as mathematical physics \cite{Zambrini-Lescot-05}, neurology \cite{Lanski-Sacerdote-01}, epidemiology \cite{Martin-Lof-98} and mathematical finance \cite{Bielecki-Jeanblan-Rut-04} and \cite{Patie-th-04}.
Amongst general results, in the Brownian setting, Strassen \cite{Strassen-67} proved
that if $f$ is continuously differentiable then the distribution of $T^{f}$ is absolutely continuous with a continuous probability density function. Moreover, for elementary curves, the density is known explicitly for linear, square root and parabolic functions. For these curves, specific techniques proved to be efficient and we refer to Table \ref{table} in Section 2 for a description of these cases. Besides, the distribution is known for some concave curves solving implicit equations obtained by the celebrated standard method of images, see Lerche \cite{Lerche-86}. For some recent investigations, we refer to P\"otzelberger and Wang \cite{Potzelberger-Wang-01} and Borovkov and Novikov \cite{Borovkov-Novikov--05} for numerical approximations of the density, Peskir \cite{Peskir-02-2}, \cite{Peskir-02} for the study of the small time behavior of the density and Kendall et al.~ \cite{Kendall-Martin-Robert-04b} for statistical applications.
The remaining part of the paper is organized as follows. In the next Section we recall some recent results regarding diffusions which enjoy the time inversion property. Section \ref{Mainresults} is devoted to the statement of our main results and their proofs. Section \ref{Brownian-motion} is concerned with a detailed study of the Brownian motion case. In particular, we present several new explicit examples of the boundary crossing problem. We also show how our results translate and agree with the standard method of images. Finally, in Section \ref{Section-Application-Bessel}, we treat the example of Bessel processes and characterize the distribution of hitting times of straight lines. We also mention that, in the Brownian motion case, the boundary crossing identity \eqref{switching-identity}, stated below, has been published without proofs in the note \cite{Alili-Patie-cras-05}.
\section{Preliminaries}
Let us recall that $E$ is either $\mathbb{R}^+$ or $\mathbb{R}$ and
$X:=(X_t, t\geq 0)$ is a $2$-self-similar conservative homogenous diffusion enjoying the time inversion property. In recent papers, Gallardo and Yor \cite{Gallardo-Yor-05} and Lawi \cite{Lawi-08} characterized the class of self-similar Markov processes (which might have c\`adl\`ag paths) satisfying the time inversion property in terms of their semi-groups when the latter are assumed to be absolutely continuous and twice differentiable. More precisely, they showed that if we write $\mathbb{P}_x(X_t\in dy)=p_t(x,y)dy$ the transitions densities have the following form
\begin{equation*} p_t(x,y) = \frac{c}{\sqrt{t}} \Phi\left(\frac{xy}{t}\right) \left(\frac{y}{\sqrt{t}}\right)^{2\nu+1} e^{-\frac{x^2+y^2}{2t}}, \quad t>0,\: x,y\in E, \end{equation*} where the reals $\nu$ and $c$ are related to the function $\Phi:E\rightarrow \mathbb{R}_+$ as follows. \begin{enumerate} \item If $E=\mathbb{R}$, then $X$ is a Brownian motion, $\Phi(y)=e^{y}$ and necessarily $\nu=-1/2$ and $c=1/\sqrt{2\pi}$. \item If $E=\mathbb{R}^+$ then $X$ is Bessel process of dimension $\delta$ for some $\delta>0$, $c=1$, $\nu=\frac{\delta}{2}-1$ and $\Phi(y)=y^{-\nu}I_{\nu}(y)$ where $I_{\nu}$ is the modified Bessel function of the first kind of index $\nu$ which admits the following power series representation, see e.g.~\cite[Section 5.2]{Lebedev-72}, \begin{equation*} I_{\nu}(z)=\left(\frac{z}{2}\right)^{\nu}\sum_{n=0}^{\infty}\frac{(z/2)^{2n}}{n!\Gamma(\nu+n+1)},\:
|z|<\infty,\: |\textrm{ arg}(z)|<\pi. \end{equation*} Moreover, it is well known that, in this case, the state $0$ is \begin{itemize} \item[-] an entrance boundary if $\delta\geq2$, \item[-] a reflecting boundary if $0<\delta<2$. \end{itemize} \end{enumerate} Next, recalling that, for every fixed $y\in E$, the mapping $x\mapsto \Phi(yx)$ is $\frac{y^2}{2}$-excessive, see e.g.~\cite{Borodin-Salminen-02}, we can define a new family of probability measures as follows \begin{equation}\label{h-transform}
\hbox{d}\mathbb{P}_x^{(y)}{}_{|{\mathcal{F}_t}}=\frac{\Phi(y X_t)}{\Phi(y x)}e^{-\frac{1}{2}{y}^2 t} \hbox{d}\mathbb{P}_x{}_{|{\mathcal{F}_t}}, \quad t>0. \end{equation} Note, from the definition of $\Phi$, that $\mathbb{P}_x^{(0)}=\mathbb{P}_x$. Moreover, for any $y \in E$, $(\mathbb{P}_{x}^{(y)})_{x\in E}$ is precisely the family of probability measures of the process $i(X)$ when $X$ starts at $y$. We also recall that $i(X)$ is conservative and satisfies the time inversion property \eqref{eq:ti}, see \cite{Gallardo-Yor-05}. It follows that $i(X)$ has an absolutely continuous semi-group with densities given, for any $t>0$ and $x,a \in E$, by \begin{eqnarray} \label{eq:dens_h_transf} p^{(y)}_t(x,a)&=&\frac{1}{\sqrt{t}}\frac{\Phi(ya)}{\Phi(y x)}\Phi\left(\frac{ax}{t}\right)\left(\frac{a}{\sqrt{t}} \right)^{2\nu+1}e^{-\frac{1}{2}\left( ty^2+(x^2+a^2)/t \right)}. \end{eqnarray} Note, in particular, the following: \begin{enumerate} \item If $E=\mathbb{R}$ then $i(X)$ is a Brownian motion with drift $y$. \item If $E=\mathbb{R}^+$ then $i(X)$ is the so-called Bessel process in the wide sense introduced by Watanabe \cite{Watanabe-75}. \end{enumerate} Now, from the definition of the family of mappings $(S^{(\beta)})_{\beta \geq 0}$, see \eqref{definition-transformation}, we observe that for any $f\in \mathcal{C}(\mathbb{R}^+,\mathbb{R})$ and $\alpha, \beta \in\mathbb{R}$, we have $S^{(0)}(f) = f$ and $ S^{(\alpha)}\circ S^{(\beta)}=S^{(\alpha+\beta)}$ and so
$(S^{(\beta)})_{\beta \geq 0}$ is a semi-group on $\mathcal{C}(\mathbb{R}^+,\mathbb{R})$. Moreover, the family of mappings $(S^{(\beta)})_{\beta<0}$ naturally appears in the process of
construction of bridges associated to $X$. To explain the connection, for any fixed $T>0$
and $x\in E$, let us denote by $(\mathbb{P}^{(y)}_{x, z,T}, z\in E )$ a regular version of the family of conditional probability distributions $\left(\mathbb{P}^{(y)}_x(. |X_T = z), z \in E\right)$. That is, for some $z\in E$, $\mathbb{P}^{(y)}_{x,z,T}$ is the law of the bridge of length $T$ associated to $X$, between $x$ and $z$. This, according to Fitzsimmons and al.~\cite{Fitzsimmons-Pitman-Yor-93}, can be obtained from $\mathbb{P}^{(y)}_{x}$ as the following Doob $h$-transform
\begin{equation}\label{h-bridges}
\mathbb{P}^{(y)}_{x,z,T}=\frac{p^{(y)}_{T-s}(X_s, z)}{p^{(y)}_{T}(x,z)} \mathbb{P}^{(y)}_{x} \quad \textrm{on } \mathcal{F}_s,\: s<T.
\end{equation} Fitzsimmons \cite{Fitzsimmons-98} observed that these laws remain invariant under changes of probability measures of type \eqref{h-transform}. Finally, we recall that Pitman and Yor \cite[Theorem 5.8]{Pitman-Yor-81}, see also \cite[Theorem 1]{Gallardo-Yor-05}, showed, by means of the time inversion property that, for any $x$, $z\in E$, the processes \begin{equation} \label{eq:ibs} \{X_u, 0\leq u<T; \mathbb{P}_{x,Tz,T}^{(y)}\} \textrm{ and } \{S^{(-1/T)}(X)_u, 0\leq u<T; \mathbb{P}_x^{(z)} \} \end{equation} have the same law. Note that, in the Brownian case, (\ref{eq:ibs}) can also be seen from the unique decomposition, as a semi-martingale, in its own filtration, of the Brownian bridge of length $T=-\frac{1}{\beta}>0$, between $0$ and $0$. Indeed, for any
$0\leq t<T$, we can write \begin{equation*}
S^{(-1/T)}(X)_t=(T-t)\int_0^{t} \frac{d\tilde{X}_s}{T- s}, \quad X_t= T\int_0^{\frac{Tt}{T - t}} \frac{d\tilde{X}_s}{T- s},\quad t<T, \end{equation*} where $\tilde{X}$ is a Brownian motion on $[0, \zeta^{(-\beta)}]$ with respect to the filtration generated by $ (S^{(-\frac{1}{T})}(X)_t, 0\leq t<T)$. Thus, we have \begin{equation*} S^{(-1/T)}(X)_t = \tilde{X}_t - \int_0^t \frac{S^{(-1/T)}(X)_s}{T-s}ds,\quad t < T. \end{equation*} which coincides with the canonical decomposition of the standard Brownian bridge.
\section{Main results and proofs}\label{Mainresults}
\noindent We keep the notation and setting of the previous section and assume, throughout this section, that $\beta$ is some fixed real. We proceed by pointing out that the mapping $S^{(\beta)}$ can be defined similarly on the space of probability measures. For instance, in the absolutely continuous case, we associate to $\mu(dt)=h(t)dt$ the image $S^{(\beta)}(\mu)(dt)= h^{(\beta)}(t) dt$ where we recall that $h^{(\beta)}(t):= S^{(\beta)}(h)(t)$. We are now ready to state our main result.
\begin{theorem} \label{thm} Let $f\in \mathcal{C}(\mathbb{R}_+, E)$. Then, for any $x$, $y\in E$ such that $f(0)\neq x$, and $t<\zeta^{(\beta)}$, we have \begin{eqnarray*} \mathbb{P}^{(y)}_x(T^{f^{(\beta)}}\in dt)&=& (1+\beta t)^{\nu-2} \frac{\Phi(yf^{(\beta)}(t))}{\Phi(yf^{(\beta)}(t)/(1+\beta t))}e^{-\frac{\beta}{2}\frac{y^2t^2}{1+\beta t}}e^{-\frac{\beta}{2}{\frac{{f^{(\beta)}}^2(t)}{1+\beta
t}}+\frac{\beta}{2}x^2} \\ &\times & S^{(\beta)}\left(\mathbb{P}^{(y)}_x( T^{f} \in dt)\right).\nonumber \end{eqnarray*} \noindent The particular case $y=0$ yields \begin{equation}\label{switching-identity} \mathbb{P}_x\left(T^{f^{(\beta)}}\in dt\right)=(1+\beta t)^{\nu-2}e^{-\frac{\beta}{2}{\frac{{f^{(\beta)}}^2(t)}{1+\beta
t}+\frac{\beta}{2}x^2}} \times S^{(\beta)}\left(\mathbb{P}_x\left( T^{f} \in dt\right)\right). \end{equation} \end{theorem} Theorem \ref{thm} simplifies when the focus is on straight lines. Indeed, if we consider a constant function $f\equiv a$ where $a\neq 0$ then, with $\beta=b/a$, for some $b\in \mathbb{R}$, we have $f^{(b/a)}(t)=a + bt,\: t< \zeta^{(b/a)}$.
Note that if $b<0$ then the support of $T^{a+b\cdot }$ is $(0,-b/a)$. The previous result reads as follows. \begin{cor} \label{thm-straight-lines} For any fixed $a,b,x \in E$ such that $a \neq x$ and all $t<\zeta^{(b/a)}$, we have \begin{eqnarray*} \mathbb{P}^{(y)}_x\left(T^{a+b.}\in dt\right)&=&(1+b t/a)^{\nu-2} \frac{\Phi(y(a+ b t))}{\Phi(ya)}\exp{-\frac{b}{2}\left(a+bt+\frac{t^2y^2}{a+bt}- \frac{x^2}{a}\right)}\\ &\times& S^{(b/a)}\left(\mathbb{P}^{(y)}_x( T^{a} \in dt)\right).\nonumber \end{eqnarray*}
In particular, when $y=0$, we obtain \begin{eqnarray*} \mathbb{P}_x\left(T^{a+b.}\in dt\right)&=&(1+b t/a)^{\nu-2} \exp{-\frac{b}{2}(a+bt- x^2/a)}\\ &\times& S^{(b/a)}\left(\mathbb{P}_x( T^{a} \in dt)\right). \end{eqnarray*} \end{cor} Before proving Theorem \ref{thm} we need to prepare two intermediate results. To this end, let us denote by $H^{(\beta,f)}$ the first time when $S^{(\beta)}(X)$ crosses $f$ i.e. \begin{equation*} H^{(\beta, f)} = \inf \left\{0<s < \zeta^{(\beta)}; \:S^{(\beta)}(X)_s =f (s) \right\}. \end{equation*} The aim of the next result is to relate the stopping times $H^{(\beta,f)}$ and $T^{f^{(\beta)}}$. \begin{lemma} \label{doob} The identities \begin{equation*} H^{(-\beta,f)}= \frac{T^{f^{(\beta)}}}{1+\beta T^{f^{(\beta)}}} \quad \textrm{ and } \quad T^{f^{(\beta)}} = \frac{H^{(-\beta,f)}}{1-\beta H^{(-\beta,f)}} \end{equation*} hold almost surely. In particular, we have $\left\{H^{(-\beta,f)} <\zeta^{(-\beta)}\right\}=\left\{T^{f^{(\beta)}}<\zeta^{(\beta)} \right\}$. \end{lemma} \begin{proof} From the definition of $S^{(\beta)}(X)$ and by using a deterministic time change, we get \begin{eqnarray*} H^{(-\beta, f)} &=& \inf \left\{ 0< s < \zeta^{(-\beta)}; \: X_{\frac{s}{1-\beta s}} = \frac{f(s)}{1-\beta s}\right\} \\ &=& \inf \left\{ 0< s < \zeta^{(-\beta)}; \: X_{\frac{s}{1-\beta s}} = S^{(\beta)}(f)\left( \frac{s}{1-\beta s}\right)\right\}\\ &=&\frac{T^{f^{(\beta)}}}{1+\beta T^{f^{(\beta)}}}. \end{eqnarray*} \noindent The second identity is obtained in a similar way. The last statement follows then by observing that, for any $\beta \in \mathbb{R}$, we have $\zeta^{(\beta)}=\frac{\zeta^{(-\beta)}}{1-\beta \zeta^{(-\beta)}}$ in the limiting sense. \end{proof}
\noindent The image by the mapping $S^{(\beta)}$ of any homogeneous Markov process is clearly a non-homogeneous Markov process. However, the time inversion property allows to connect the law of $X$ and that of $S^{(\beta)}(X)$ via a simple time-space $h$-transform. To be more precise, we state the following result. \begin{lemma}\label{abs-diff} Let $x, y \in E$ and $\beta \in \mathbb{R}$. Under $\mathbb{P}_{x}^{(y)} $, the process $X^{(\beta)}=S^{(\beta)}(X)$, defined on $[0,\zeta^{(\beta)})$, is a non-homogeneous strong Markov process. Its law, denoted by $\mathbb{Q}_{x}^{y, \beta} $, is absolutely continuous with respect to $\mathbb{P}^{(y)}_x$ with Radon-Nikodym derivative $M_{x}^{y, \beta}(t,X_t)$, $t<\zeta^{(\beta)}$, given by \begin{equation*} M_x^{y,\beta}(t,X_t)=(1+\beta t)^{-\nu-1}\frac{\Phi(y X_t/(1+\beta t))}{\Phi(y X_t)}e^{\frac{\beta}{2} \frac{y^2 t^2}{1+\beta t}+\frac{\beta}{2}\frac{X_t^2}{1+\beta
t}-\frac{\beta}{2}x^2}. \end{equation*} \noindent In particular, for $y=0$, we have \begin{equation*} M_x^{0,\beta}(t,X_t)=(1+\beta t)^{-\nu-1}e^{\frac{\beta}{2}\frac{X_t^2}{1+\beta
t}-\frac{\beta}{2}x^2}. \end{equation*}
\end{lemma} \begin{proof} We start with the case $\beta<0$. Using successively the identities \eqref{eq:ibs} and \eqref{h-bridges}, we obtain, for any bounded measurable functional $F$ and $t<\zeta^{(\beta)}$, that \begin{eqnarray*} \mathbb{E}_x^{(y)}[F(S^{(\beta)}(X)_s, s\leq t)]&=&\mathbb{E}_x^{(y)}[F(X_s, s\leq t)\mid X_{-1/\beta}=-y/\beta] \\ &=&\mathbb{E}_x^{(y)}\left[\frac{p^{(y)}_{-1/\beta - t} (X_t,-y/\beta)}{p^{(y)}_{-1/\beta} (x,-y/\beta)}F(X_s, s\leq t)\right]\\ &=&\mathbb{E}^{(y)}\left[M_x^{y,\beta}(t,X_t)F(X_s, s\leq t)\right] \end{eqnarray*} where the last line follows from \eqref{eq:dens_h_transf}. Using the fact that, for any $ c,d \in E$, $\frac{\Phi(cy)}{\Phi(dy)}\rightarrow 1$ as $y\rightarrow 0$, we get the stated result for $y=0$.
Next, we treat the case $\beta>0$ and assume, first, that $X$ is a Bessel process of dimension $\delta>0$ and $y=0$. Using the It\^o formula, we see that the process $(M_x^{0,\beta}(t,X_t), t\geq 0)$ is a $\mathbb{P}_x$-local martingale. We now show that it is a true martingale. Recall that the squared Bessel process $X^2$ is the unique solution to the stochastic differential equation (for short sde) \begin{equation}\label{squared-Bessel} dX_t^2=2 X_tdB_t +\delta dt,\quad X_0^2=x^2, \end{equation} where $B$ is a Brownian motion, see \cite{Revuz-Yor-99}. Furthermore, we need to introduce the processes $Y$ and $\tilde{Y}$ which are defined, for a fixed $t\geq0$, by \[Y_t =(1+\beta t)\tilde{Y}_t=S^{(\beta)}(X)_t .\] It follows from (\ref{squared-Bessel}) and by performing a deterministic time change, that $\tilde{Y}$ solves the sde \[ d\tilde{Y}_t^2= \frac{\delta }{(1+\beta t)^2 }dt+\frac{2\tilde{Y}_t}{1+\beta t}d\gamma_t \] where $(\gamma_t, t\geq0)$ is a Brownian motion with respect to the filtration generated by the process $\tilde{Y}$. Similarly, $Y^2$ satisfies \begin{eqnarray}\label{intermediate-equation} dY_t^2&=&d\left((1+\beta t)^2\tilde{Y}_t^2 \right) \\\nonumber
&=& 2Y_t \left(d\gamma_t +\frac{\beta}{1+\beta t} Y_tdt\right) + \delta dt. \end{eqnarray} Now, from \eqref{squared-Bessel} and by Girsanov theorem, the law of the unique solution to (\ref{intermediate-equation}) is obtained from the law of $X^2$ by a change of probability measure using the local martingale $(M_{x}^{0, \beta}(t,X_t), t\geq0)$. The proof of the claim, in the case $y=0$, is completed by invoking the conservativeness property of $Y^2$, which implies that $(M_{x}^{0, \beta}(t,X_t), t\geq0)$ is a martingale. Finally, to recover the case $y\neq 0$, we use (\ref{h-transform}) which gives \begin{eqnarray*}
\hbox{d}\mathbb{Q}_x^{y,\beta}|_{\mathcal{F}_t}&=&\frac{\Phi(y X_t /(1+\beta t))}{\Phi(y x)}e^{-\frac{1}{2}
\frac{ty^2}{1+\beta t}}\hbox{d}\mathbb{Q}_x^{0,\beta}|_{\mathcal{F}_t}\\ &=& \frac{\Phi(y X_t/(1+\beta t))}{\Phi(y x)}e^{-\frac{1}{2} \frac{ty^2}{1+\beta t}} M_{x}^{0,
\beta}(t,X_t)\hbox{d}\mathbb{P}_x|_{\mathcal{F}_t}\\ &=& \frac{\Phi(y X_t/(1+\beta t))}{\Phi(y X_t)}e^{\frac{\beta}{2} \frac{y^2 t^2}{1+\beta t}} M_{x}^{0,
\beta}(t,X_t)\hbox{d}\mathbb{P}_x^{(y)}|_{\mathcal{F}_t}\\
&=&M_{x}^{y,\beta}(t,X_t)\hbox{d}\mathbb{P}_x^{(y)}|_{\mathcal{F}_t}.
\end{eqnarray*} This completes the proof in the Bessel case. The same arguments work for the Brownian case and this completes the proof. \end{proof}
\noindent We shall now return back to the proof of Theorem \ref{thm}. Lemmas \ref{doob} and \ref{abs-diff}, when combined with the optional stopping theorem, allow us to write, for any $\lambda>0$, \begin{eqnarray*} \mathbb{E}^{(y)}_x\left[e^{-\lambda T ^{ f^{(\beta)}}}\ind{T^{f^{(\beta)}}<\zeta^{(\beta)}} \right] &=&\mathbb{E}^{(y)}_x\left[ e^{-\lambda\frac{H ^{(-\beta,f)}}{1-\beta H ^{(-\beta,f)}}}\ind{H ^{(-\beta,f)}<\zeta^{(-\beta)}} \right]\\ &=&\mathbb{E}^{(y)}_x\left[e^{-\lambda\frac{T^{f}}{1-\beta T^{f}}}M_{x}^{y,-\beta}(T^{f},X_{ T^{f}}) \ind{T^{f}<\zeta^{(-\beta)}} \right]\\ &=&\int_0^{\zeta^{(-\beta)}} e^{-\lambda\frac{t}{1-\beta t}}M_{x}^{y,-\beta}(t,f(t))\mathbb{P}^{(y)}_x\left(T^{f}\in dt \right)\\ &=&\int_0^{\zeta^{(\beta)}} \frac{e^{-\lambda u}}{(1+\beta u)^3} M_{x}^{y,-\beta}\left(u/(1+\beta u), f(u/(1+\beta u)\right)\\ &\times& S^{(\beta)}\left(\mathbb{P}^{(y)}_x( T^{f} \in du)\right) \end{eqnarray*} where we have performed the change of variables $t/(1-\beta t)=u$. Simplifying the above expression and by the injectivity of the Laplace transform, we get our first assertion. The second statement follows by letting $y\rightarrow 0$.
\section{Brownian motion}\label{Brownian-motion} \noindent In this paragraph, we take $\mathbb{E}=\mathbb{R}$. Thus, $X$ is a Brownian motion. Thanks to the homogeneity property, in this case, it is clearly enough to consider the case $X_0=0$ and we simply write $\mathbb{P}$ for $\mathbb{P}_0$. To simplify the discussion, except in Subsection \ref{asymptotics-Section}, we assume that $f\in \mathcal{C}^{1}(\mathbb{R}_+,E)$, with $f(0)\neq 0$, which implies that the studied distributions are absolutely continuous with respect to the Lebesgue measure with continuous densities, see \cite{Strassen-67}. We write then $\mathbb{P}\left(T^{f}\in dt\right)=p^{f}(t)\: dt$ and read from Theorem \ref{thm} the identity \begin{equation}\label{identity-brownian} p^{f^{(\beta)}}(t)=\frac{1}{(1+\beta t)^{3/2}}e^{ -\frac{1}{2} \frac{\beta}{ 1+\beta t}f^{(\beta)}(t)^2}p^{f}\left(\frac{t}{1+\beta t}\right),\: t<\zeta^{(\beta)}. \end{equation} Observe, when $\beta>0$, the asymptotic behavior \begin{equation}\label{asymptotic-0} p^{f^{(\beta)}}(t)\sim (\beta t)^{-3/2}e^{-\frac{1}{2}\frac{\beta}{1+\beta t}{f^{(\beta)}}^2(t)}p^{f}(1/\beta) \quad \textrm{ as } t\rightarrow \infty, \end{equation} where $h(t)\sim g(t)$ as $t\rightarrow \zeta$ means that $h(t)/g(t)\rightarrow 1$ as $t\rightarrow \zeta$, for some $\zeta\in [0,\infty]$. It seems natural to examine the images of some curves by the mapping $S^{(\beta)}$. We gathered in Table \ref{table} the images of the most studied curves. \begin{table}[h] \begin{center}
\begin{tabular} {|c|c|c|c|} \hline\label{tab:cur}
& $f$ & $f^{(\beta)}$& $\textrm{References}$ \\ \hline \hline (1)& $ a + bt$ &$ a + (b+a\beta)t$ & \cite{Bachelier-41} \\
(2) & $\sqrt{1+2 b t}$ & $\sqrt{(1+\beta t)(1+(\beta + 2 b)t)}$& \cite{Breiman-67}, \cite{Novikov-71} \\
(3) & $ a+bt^2, \:ab>0 $ & $a(1+\beta t)+ bt^2/{(1+\beta t)}$& \cite{Groeneboom-89}, \cite{Salminen-88}\\ (4) & $ \frac{a}{2} -\frac{t}{a}\ln\left(\frac{b + \sqrt{b^2 + 4 b_1 e^{-\frac{a^2}{t}}}}{2} \right) $& $\frac{a ( 1+ \beta t)}{2} -\frac{t}{a}\ln\left(\frac{b + \sqrt{b^2 + 4 \hat{b}_1 e^{-\frac{a^2}{t}}}}{2}\right)$
& \cite{Daniels-69} \\ \hline \end{tabular} \end{center} \caption{Image by $S^{(\beta)}$ of $f$ and the corresponding references where the distribution of $T^{f}$ is studied with $a,b \in \mathbb{R}$, $b_1>-b^2/4$ and $\hat{b}=b_1e^{-a^2\beta}$. } \label{table} \end{table} The fact that $S^{(\beta)}$ preserves straight lines is well known, see for instance \cite{Pitman-Yor-81}. More generally, we observe that if $g_{\beta, \alpha}(.)=(1+\beta .)^{\alpha}$, for some reals $\alpha$ and $\beta$, then we have $g^{(-\beta)}_{\beta, \alpha}=g_{-\beta, 1-\alpha}$. In particular, if we take $f\equiv a$ and $\beta=b/a$, for some reals $a$ and $b$, then $f^{(b/a)}(t)=a+bt$, $t<\zeta^{(b/a)}$. An immediate application of Corollary \ref{thm-straight-lines}, combined with \begin{equation*}
\mathbb{P}(T^{ a}\in dt)=\frac{|a|}{\sqrt{2\pi t^3}}e^{-\frac{a^2}{2t}}dt,\: t>0, \end{equation*}
yields then the following well known Bachelier-L\'evy formula \begin{equation*}
\mathbb{P}\left(T^{a+b\cdot}\in dt \right)=\frac{|a|}{\sqrt{2 \pi t^3}}e^{-b a -\frac{b^2}{2}t-\frac{a^2}{2t}}\: dt. \end{equation*}
\subsection{Some new examples}\label{examples}
We now consider the boundary crossing problem associated to two families of curves consisting of the square root of second order polynomials and the reciprocal of affine functions.
\subsubsection{} First, we consider the distribution of the stopping time \begin{equation*} T^{(\lambda_1, \lambda_2)}_a=\inf\left\{ s> 0;\: X_s=a\sqrt{\left(1+\lambda_1 s\right)\left(1+\lambda_2 s \right)}\right\} \end{equation*} where $a$ and $\lambda_1<\lambda_2$ are some fixed reals. We do not consider the case $\lambda_1=\lambda_2$ which can be studied in a elementary way, with the extra cost of making use of the strong Markov property when $\lambda_1<0$. First, consider the case $\lambda_2=0$ and, to simplify the notation, set $\lambda_1=\lambda$ and $T_a^{(\lambda,0)}=T_a^{(\lambda)}$. This is the setting of the classical example studied by Breiman in \cite{Breiman-67}. $T_a^{(\lambda)}$ is related to the hitting time of a constant level by an Ornstein-Uhlenbeck process and we refer to Alili et al.~\cite{Alili-Patie-Pedersen-05} for a recent survey on this topic. That is, with \begin{equation*} U_t=e^{-\lambda t/2}\int_0^te^{\lambda s/2}dX_s, \quad t\geq 0, \end{equation*} and $H_a=\inf\{ s> 0; \: U_s=a\}$, we have \begin{equation} \label{eq:iht} T^{(\lambda)}_a \stackrel{(d)}{=}\lambda^{-1}\left( e^{\lambda H_a}-1 \right) \end{equation} where $\stackrel{(d)}{=}$ stands for the identity in distribution. By symmetry, it is enough to consider the case where $a$ is positive. We proceed by recalling that the distribution of $H_a$ is given, see for instance \cite{Alili-Patie-Pedersen-05}, by \begin{eqnarray*}
{\mathbb P}\left(H_a\in dt \right) = -\frac{1}{2} \lambda e^{-\lambda a^2/4}
\sum_{n=1}^{\infty} \frac{D_{\nu_{n,- a\sqrt{\lambda}}}(0)}
{D^{(\nu)}_{ \nu_{n,- a\sqrt{\lambda}}}(- a \sqrt{\lambda})} \: e^{-\lambda \nu_{n, -a\sqrt{\lambda}}
t/2},\quad t>0, \end{eqnarray*} where we used the notation $D^{(\nu)}_{\nu_{n,b}}(b) =
\frac{\partial D_{\nu }(b)}{\partial\nu}|_{\nu= \nu_{n,b}}$ and $(\nu_{j,b})_{j\geq 0}$ stands for the ordered sequence of the positive zeros of the parabolic function $\nu \rightarrow D_{\nu}(b)$. By means of the identity \eqref{eq:iht}, we get that \begin{equation*}
{\mathbb P}\left(T^{(\lambda)}_a\in dt \right)=\frac{1}{1+\lambda t}{\mathbb P}\left(H_a\in d\cdot \right){\big | }_{\cdot= \frac{1}{\lambda}\log\left(1+\lambda t \right)}dt,\quad t>0. \end{equation*} Next, we assume that $\lambda_1<\lambda_2$. Thus, the support of $T^{(\lambda_1, \lambda_2)}_a$ is $[0,\zeta^{(\lambda_1)})$ if $\lambda_1$ is positive and is the positive real line otherwise. We have \begin{equation*} S^{(\lambda_1)}\left(\sqrt{1+(\lambda_2-\lambda_1)\cdot} \right)=\sqrt{\left(1+\lambda_2 \cdot\right)\left(1+\lambda_1 \cdot \right)}. \end{equation*} We are now ready to use Theorem \ref{thm} and write \begin{equation*} \mathbb{P}\left(T^{(\lambda_1, \lambda_2)}_a\in dt\right)=\frac{1}{(1+\lambda_1 t)^{5/2}}e^{ -\frac{1}{2} \lambda_1( 1+\lambda_2 t)} S^{(\lambda_1)}\left(\mathbb{P}\left( T^{\left(\lambda_2-\lambda_1\right)}_a \in dt\right)\right),\quad t<\zeta^{(\lambda_1)}. \end{equation*}
\subsubsection{} We are now interested in computing the distribution of the stopping time $T^{h^{(\beta)}}$ defined by \begin{equation*} T^{h^{(\beta)}}=\inf\left\{ 0<s<\zeta^{(\beta)} ;\: X_s=\frac{1}{1+\beta s}\right\} \end{equation*} where $\beta$ is some real. To this end, we recall that Groeneboom \cite{Groeneboom-89} has computed the density of $T^{\tilde{h}}$ with $\tilde{h}(t)=1+\beta^2t^2$ as follows \begin{equation*} \mathbb{P}(T^{\tilde{h}} \in dt) = 2(\beta^2 c)^{2} e^{-\frac{2}{3}\beta^4 t^3} \sum_{k=0}^{\infty}\frac{Ai\left(z_k+2c\beta^2\right)}{Ai'(z_k)}e^{-z_k t}\:dt, \: t>0, \end{equation*} where $(z_k)_{k \geq 0}$ is the decreasing sequence of negative zeros of the Airy function, see e.g.~\cite{Lebedev-72}, and we have set $c=(2\beta^2)^{-\frac{1}{3}}$. Next, by means of the Cameron-Martin formula, we obtain, with $h(t)=(1-\beta t)^2=\tilde{h}(t)-2\beta t$, \begin{equation*} \mathbb{P}(T^{{h}} \in dt) = 2(\beta^2 c)^{2} e^{2\beta-2\beta^2 t \left(1+\frac{2}{3}\beta^2 t^2-\beta t\right)} \sum_{k=0}^{\infty}\frac{Ai\left(z_k+2c\beta^2\right)}{Ai'(z_k)}e^{-z_k t}\:dt,\: t>0. \end{equation*} Finally, since $h^{(\beta)}= S^{(\beta)}(h)$, we obtain, by applying Theorem \ref{thm}, that \begin{equation*} \mathbb{P}(T^{{h^{(\beta)}}} \in dt) = \frac{2(\beta^2 c)^{2} e^{2\beta}}{(1+\beta t)^{3/2}}e^{-2\beta^2 t \left(1+\frac{1}{4\beta}+\beta t +\frac{2}{3}\beta^2 t^2\right)} \sum_{k=0}^{\infty}\frac{Ai\left(z_k+2c\beta^2\right)}{Ai'(z_k)}e^{-z_k \frac{t}{1+\beta t}}\:dt, \quad t<\zeta^{(\beta)}. \end{equation*} We complete the example by stating the following asymptotic result \begin{equation*} \mathbb{P}(T^{{h^{(\beta)}}} \in dt) \sim \left(2(\beta^2 c)^{2} e^{2\beta}\sum_{k=0}^{\infty}\frac{Ai\left(z_k+2c\beta^2\right)}{Ai'(z_k)}e^{- \frac{z_k}{\beta}}\right) e^{-\frac{4}{3}\beta^4 t^3} \:dt\: \textrm{ as } t\rightarrow \infty \end{equation*} which holds provided that $\beta>0$.
\subsection{Interpretation of the mapping $S^{\beta}$ via the method of images}\label{images}
We aim to describe the impact of our methodology to the so-called standard method of images. To this end, let us assume that $X_0= 0$ and set \[h(t,x)dx=\mathbb{P}\left( T^{f}>t, X_t \in dx\right).\] We need to impose some conditions on the the curve $f$ in order to be able to apply the standard method of images. That is, we assume that
\begin{itemize}
\item[-] $f$ is infinitely often continuously differentiable, \item[-] $f(t)/t$ is monotone decreasing, \item[-] $f$ is concave. \end{itemize} Note that these properties are also satisfied by $f^{(\beta)}$ when $\beta >0$. The function $h$ is characterized as being the unique solution to the heat equation, see Lerche \cite[Chap.~I.1]{Lerche-86}, \begin{equation*} \frac{\partial h}{\partial t}=\frac{1}{2}\frac{\partial^2 h}{\partial x^2} \quad \textrm{on } \{(t,x)\in \mathbb{R}^+\times \mathbb{R};\:x\leq f(t) \}, \end{equation*}
with the boundary conditions \begin{equation*} h\left(t, f(t)\right)=0,\quad h(0, .)=\delta_0(.)\quad \hbox{on}\quad ]-\infty, f(0^+)] \end{equation*} where $\delta_0$ stands for the Dirac function at $0$ and $f(0^+)=\lim_{t\searrow 0}f(t)$. The standard method of images assumes that $h$ is known, whilst $f$ is unknown, and is given by \begin{equation}\label{space-time} h(t,x)=\frac{1}{\sqrt{2\pi t}}\left( e^{-\frac{x^2}{2t}}-\int_{0}^{\infty}e^{-\frac{(x-s)^2}{2t}} F(ds)\right) \end{equation} where $F(ds)$ is some positive $\sigma$-finite measure satisfying $\int_{0}^{\infty}e^{-\frac{\epsilon s^2}{2}} F(ds)<\infty$, for all $\epsilon>0$. In \cite{Lerche-86}, it is shown that if $f$ is the unique root of the equation $h(t,x)=0$ in the unknown $x$ for a fixed $t\geq 0$, then we have \begin{equation}\label{hitting-density-brut} \mathbb{P}\left(T^{f}\in dt \right)=\frac{dt}{\sqrt{2\pi t^3}}\int_{0}^{\infty}se^{-\frac{(s-f(t))^2}{2t}}F(ds),\quad t>0. \end{equation} Note that the implicit equation $h(t, x)=0$ may be written as \begin{equation} \label{impl} \int_{0}^{\infty}e^{-\frac{s^2}{2t}+s\frac{x}{t}}F(ds)=1. \end{equation}
With $F(ds)$ replaced by $F(ds)e^{-\mu s}$, $\mu>0$, the unique solution to \eqref{impl} is $f(t)+\mu t$ for a fixed positive real $t$ and this is easily checked by the Cameron-Martin formula. In the same spirit, the identity (\ref{identity-brownian}) has the following interpretation.
\begin{proposition}\label{images-interpretation} If $\beta>0$ then we have the following representation \begin{equation*}\mathbb{P}\left(T^{f^{(\beta)}}\in dt \right)=\frac{dt}{\sqrt{2\pi t^3}}\int_{0}^{\infty}se^{-\frac{(s-f^{(\beta)}(t))^2}{2t}}e^{-\beta \frac{s^2}{2}}F(ds), \quad t>0. \end{equation*} Furthermore, the $\sigma$-finite measure corresponding to the curve $f^{(\beta)}$ is $F^{(\beta)}(ds):=e^{-\beta s^2/2}F(ds)$ and \begin{equation*} \mathbb{P}\left( T^{f^{(\beta)}}>t, X_t\in dx\right)=\frac{dx}{\sqrt{2\pi t}}\left( e^{-\frac{x^2}{2t}}-\int_{0}^{\infty}e^{-\frac{(x-s)^2}{2t}} e^{-\beta^2 \frac{s}{2}}F(ds)\right). \end{equation*}
\end{proposition}
\begin{proof} Noting, from (\ref{hitting-density-brut}), that \begin{eqnarray*} \mathbb{P}\left(T^{f^{(\beta)}}\in dt \right)&=& \frac{dt}{\sqrt{2\pi t^3}}\int_{0}^{\infty}se^{-\frac{(s-f^{(\beta)}(t))^2}{2t}}F^{(\beta)}(ds) \end{eqnarray*} and by using (\ref{identity-brownian}) we obtain that \begin{eqnarray*} \frac{1}{\sqrt{2\pi t^3}}\int_{0}^{\infty}se^{-\frac{(s-f^{(\beta)}(t))^2}{2t}}F^{(\beta)}(ds) &=&\frac{1}{(1+\beta t)^{3/2}}e^{ -\frac{1}{2} \frac{\beta}{ 1+\beta t}f^{(\beta)}(t)^2}p^{f}\left(\frac{t}{1+\beta t}\right)\\ &=&\frac{1}{\sqrt{2\pi t^3}}e^{ -\frac{1}{2} \frac{\beta}{ 1+\beta t}f^{(\beta)}(t)^2}\int_{0}^{\infty}se^{-\frac{(s-f(t/(1+\beta t)))^2}{2t/(1+\beta t)}}F(ds)\\ &=&\frac{1}{\sqrt{2\pi t^3}}\int_{0}^{\infty}se^{-\frac{(s-f^{(\beta)}(t))^2}{2t}}e^{-\beta \frac{s^2}{2}}F(ds). \end{eqnarray*} This proves the first assertion and suggests, but doest not prove, that the $\sigma$-finite measure corresponding to the curve $f^{(\beta)}$ is $F^{(\beta)}(ds)$. Next, with \begin{equation*} h^{(\beta)}(t,x)=\frac{1}{\sqrt{2\pi t}}\left(e^{-\frac{x^2}{2t}}-\int_{0}^{\infty}e^{-\frac{(x-s)^2}{2t}} e^{-\beta^2 s/2}F(ds)\right), \end{equation*} we aim to solve $h^{(\beta)}(t,x)=0$ for each fixed $t>0$. Now, setting $x=f(t)$ and replacing $x$ and $t$, respectively, by $f(t)$ and $t/(1+\beta t)$ in (\ref{impl}), we find that $\int_{0}^{\infty}e^{-\frac{s^2}{2t}+s\frac{f^{(\beta)}(t)}{t}}e^{-\beta s^2/2}F(ds)=1$. It follows that $f^{(\beta)}(t)$ solves the implicit equation $h^{(\beta)}(t,x)=0$, $t>0$, which finishes the proof. \end{proof}
\subsection{Large asymptotics for the density of $T^{f^{(\beta)}}$} \label{asymptotics-Section} Following Anderson and Pitt \cite{Anderson-Pitt-97}, we consider the asymptotic of the density of the
distribution of $T^{f^{(\beta)}}$ as time gets close to $\zeta^{(\beta)}$. To start with, we exceptionally assume in this subsection that $f\in \mathcal{C}^1((0,\infty), \mathbb{R}_+)$. So the distribution of $T^{f}$ has a continuous density with respect to the Lebesgue measure on $(0,\infty)$. Assume that
the distribution of $T^{f}$ is defective. That is $r=\mathbb{P}(T^{f}<\infty)<1$ and, in this case, we say that $f$ is transient. By the classical Kolmogorov-Erd\"os-Petrovski theorem, see \cite{Erdos-42}, we know that if $t^{-1/2}f(t)$ is increasing for sufficiently large $t$, then $f$ is transient if and only if the integral test \begin{equation}\label{integral-test} \int_1^{\infty}t^{-3/2}f(t)e^{-f^2(t)/2t}\: dt<\infty \end{equation} holds. Clearly, if $\beta>0$ and $0<\beta f(1/\beta)<\infty$ then $f^{(\beta)}$ does not satisfy (\ref{integral-test}) but the general formula (\ref{asymptotic-0}) holds. Now, we consider the more interesting case $\beta<0$ and examine the asymptotic of the density of the distribution of $T^{f^{(\beta)}}$ as $t\rightarrow -1/\beta$. We need to work under the following three conditions borrowed from \cite{Anderson-Pitt-97}: \begin{itemize} \item[-] $f$ is increasing, concave, twice differentiable on $(0,\infty)$ and of regular variations at $\infty$ with index $\alpha\in [1/2,1)$,
\item[-]
$f(t)/\sqrt{t}$ is monotonic increasing at $\infty$, and $f(t)/t$
is convex and decreases to $0$ for sufficiently large $t$,
\item[-] There exist positive constants $c<1$ and $c'$ such
that we have the inequalities
$tf'(t)<cf(t)$ and $|t^2 f''(t)|\leq c' f(t)$ for a sufficiently large enough $t$. \end{itemize}
The behavior at $\infty$ imposed in the above conditions is granted in the examples where $f$ behaves like $$ f(t)=Ct^a(\log t)^b(\log\log t)^c (\log\log\log t)^d$$
with $1/2\leq a<1$, for large $t$. It is clear that these conditions are not always preserved by the family $S^{(\beta)}$. Equipped with this, we are now ready to state the following result. \begin{proposition}\label{asymptotics} Assume that $f$ is transient and satisfies the above conditions. Then,
for any $\beta<0$, we have \begin{equation*}
p^{f^{(\beta)}}(t) \sim \frac{1}{\sqrt{2\pi |\beta|^{3}}}\left(1-r \right)\tilde{f}(\beta, t) \quad \textrm{as } t\rightarrow \zeta^{(\beta)} \end{equation*} where \begin{equation*} \tilde{f}(\beta, t)=f\left(\frac{t}{1+\beta t}\right)- \frac{t}{1+\beta t}f'\left(\frac{t}{1+\beta t}\right). \end{equation*}
\end{proposition} \begin{proof} We read from Theorem 1 that \begin{equation} \label{eq:as_d} p^{f}(t) \sim \left(1-r \right)\frac{f(t)-tf'(t)}{\sqrt{2\pi}t^{3/2}}e^{-f^2(t)/2t} \quad \textrm{as } t\rightarrow \infty. \end{equation}
Moreover, as $t\rightarrow \infty$, $f(t)/t \downarrow 0$ and hence \begin{equation*} e^{-\frac{\beta}{2(1+\beta t)}{f^{(\beta)}}^2(t)-\frac{1+\beta t}{2t}f^2(t/(1+\beta t))}=e^{-{f^{(\beta)}}^2(t)/2t}\sim 1. \end{equation*} The proof is then completed by combining \eqref{eq:as_d} with (\ref{identity-brownian}). \end{proof}
\begin{remark}
In the defective case, the distribution of the last crossing time $\tilde{T}^{f}=\sup\{s>0; X_s= f(s) \}$ is shown to have an atom at 0 and its asymptotic as $t\rightarrow 0$ is determined in \cite{Strassen-67}. Similar questions can be treated for $\tilde{T}^{f^{(\beta)}}=\sup\{s>0; X_s= f^{(\beta)}(s) \}$ using this method. \end{remark}
\section{Bessel processes and straight lines}\label{Section-Application-Bessel} \noindent We investigate here the case when $X$ is a Bessel process of dimension $\delta>0$ and we refer to Revuz and Yor \cite[Chap. XI]{Revuz-Yor-99} for a concise treatment of these processes. For $\delta \geq 2$, and $x>0$, the Bessel process of dimension $\delta$ is the unique solution to \begin{equation*}
X_t = B_t +x+\frac{\delta -1}{2}\int_0^t \frac{ds}{ X_s},\quad t> 0, \end{equation*} where $B$ is a standard Brownian motion. For $0<\delta<2$, $X$ is defined as the square root of the unique non-negative solution of \eqref{squared-Bessel}. We recall that $0$ is an entrance boundary when $\delta\geq2$ and a reflecting boundary otherwise. We denote by $\mathbb{P}_{x}^{\nu}$ (resp.~$\mathbb{E}_x^{\nu}$) the law (resp.~the expectation operator) of a Bessel process of index $\nu=\delta/2-1$, starting at $x>0$. The semi-group of $X^2$ is characterized by \begin{eqnarray} \label{eq:sg} \mathbb{E}_x^{\nu}\left[ e^{- \lambda X^2_t} \right] &=& (1+2\lambda t)^{-\delta/2} e^{- \frac{ \lambda x^2}{1+2\lambda t}}, \quad \lambda,t\geq 0. \end{eqnarray} The densities of the semi-group of $X$, with respect to the Lebesgue measure, are \begin{eqnarray*} p^{\nu}_t(x,y) &=& \frac{y}{t}\left(\frac{y}{x}\right)^{\nu} e^{-\frac{x^2+y^2}{2t}} I_{\nu}\left( \frac{xy}{t}\right), \quad t, x,y >0. \end{eqnarray*}
For a given $f\in \mathcal{C}(\mathbb{R}_+,E)$ let us keep the notation used in the introduction and Section \ref{Mainresults}. Observe, that if $y=0$ then
Theorem \ref{thm} reads \begin{equation*} \mathbb{P}_x^{\nu}\left(T^{f^{(\beta)}}\in dt\right)={(1+\beta t)^{\nu -2}}e^{ -\frac{\beta}{2} \frac{f^{(\beta)}(t)^2}{1+\beta t}+\frac{\beta}{2}x^2}S^{(\beta)}\left(\mathbb{P}_x^{\nu}( T^{f} \in dt)\right) \end{equation*} for all $t<\zeta^{(\beta)}$. We end up our discussion by computing the distribution of the hitting time by $X$ of a straight line $a +b\cdot$ where $a>0$ and $b$ is a real. We keep the notation used in Corollary \ref{thm-straight-lines} and the reader is reminded about observations preceding its statement. To the best of our knowledge, the problem of the determination of the distribution of $T^{a+b.}$, which was raised in \cite{Pitman-Yor-81} for the case $a\neq 0$, remained open. Recall that the law of $T^{a}$, which corresponds to $b=0$, is characterized by \begin{eqnarray}\label{eq:laplace-bessel} \mathbb{E}_x^{\nu}\left[ e^{- \frac{\lambda^2}{2} T^{a}} \right] =
\left\{ \begin{array}{lll}
& \frac{x^{-\nu}I_{\nu}(x\lambda)}{a^{-\nu}I_{\nu}(a\lambda)}, & x \leq a, \\
& \\
& \frac{x^{-\nu}K_{\nu}(x\lambda)}{a^{-\nu}K_{\nu}(a\lambda)}, & x \geq a, \\ \end{array} \right. \end{eqnarray} for any $\lambda \geq 0$, where $K_{\nu}$ is the modified Bessel functions of the second kind of index $\nu$, see for instance Borodin and Salminen \cite{Borodin-Salminen-02}. Observe that when $x> a$ the distribution of $T^a$ is defective and $\mathbb{P}^{\nu}_x(T^{a}<\infty)=(a/x)^{2\nu}$. It is also well known that if $x<a$ then we have \begin{eqnarray}\label{eq:density-bessel} \mathbb{P}_x^{\nu}(T^{a} \in dt) &=&
\sum_{k=1}^{\infty} \frac{x^{-\nu}j_{\nu,k}J_{\nu}(j_{\nu,k}\frac{x}{a})}{a^{2-\nu}J_{\nu+1}(j_{\nu,k})} e^{-j^2_{\nu,k} t/2a^2} \: dt,\quad t>0, \end{eqnarray} where $(j_{\nu,k})_{k\geq1}$ is the ordered sequence of positive zeros of the Bessel function of the first kind $J_{\nu}(.)$, see \cite{Borodin-Salminen-02}. Furthermore, if $a=0$ and $b>0$ then $T^{b\cdot}$ and 1/$\sup\{s>0; X_s=b\}$ have the distribution which was determined in \cite{Pitman-Yor-81} by making use of the time inversion property. Now, we are ready to state the following result. \begin{theorem} For $0\leq x<a$ and $b\in \mathbb{R}$, we have for any $t<\zeta^{(b/a)}$ \begin{eqnarray*} \mathbb{P}_x^{\nu}(T^{a+b\cdot} \in dt) &=& \frac{e^{\frac{b}{2a}(a^2-x^2)+\frac{b^2}{2}t}}{(1+\frac{b}{a} t)^{\nu+2}} \sum_{k=1}^{\infty} \frac{x^{-\nu}j_{\nu,k}J_{\nu}(j_{\nu,k}\frac{x}{a})}{a^{2-\nu}J_{\nu+1}(j_{\nu,k})} e^{-j^2_{\nu,k}\frac{t}{2a(a+bt)}} \: dt. \end{eqnarray*} For any $x\geq 0$ and $a, b >0$, we have \begin{equation*} \mathbb{E}_x^{\nu}\left[ e^{- \frac{\lambda^2}{2} T^{a-b\cdot}} \right] =
\left\{ \begin{array}{lll}
& e^{\frac{b}{2a}(a^2-x^2)} \frac{x^{-\nu}}{a^{-\nu}} \int_0^{\infty} \frac{I_{\nu}(\sqrt{2}xu)}{I_{\nu}(\sqrt{2}au)} p^{\nu}_{b/2a}(\lambda_{b},u)\: du , & x \leq a, \\
& e^{\frac{b}{2a}(a^2-x^2)} \frac{x^{-\nu}}{a^{-\nu}} \int_0^{\infty} \frac{K_{\nu}(\sqrt{2}xu)}{K_{\nu}
(\sqrt{2}au)} p^{\nu}_{b/2a}(\lambda_{b},u)\: du, & x \geq a, \end{array} \right. \end{equation*} where $\lambda_{b}= \sqrt{(\lambda^2 + b^2)/2}$, $\lambda\in \mathbb{R}$. \end{theorem}
\begin{proof} The first statement results from a combination of Corollary \ref{thm-straight-lines} and relation (\ref{eq:density-bessel}). Next, using Lemmae \ref{doob} and \ref{abs-diff}, with
$\beta=-b/a$, we can write \begin{eqnarray*} & &\mathbb{E}_x^{\nu}\left[ e^{- \frac{\lambda^2}{2} T^{a-b\cdot}}; T^{a-b\cdot}<\frac{a}{b} \right] \\&=& \mathbb{E}_x^{\nu}\left[
e^{- \frac{\lambda^2}{2} \frac{H^{(b/a, a)} }{1+\frac{b}{a} H^{(b/a,a)}}}; H^{(b/a,a)}<\infty\right] \\ & = & \mathbb{E}_x^{\nu}\left[
e^{- \frac{\lambda^2}{2} \frac{T^{a}}{1+\frac{b}{a} T^{a}}} \left(1+\frac{b}{a}
T^{a}\right)^{-\delta/2}e^{ \frac{b}{2a} \left(\frac{a^2}{1+\frac{b}{a} T^{a}}-x^2\right)}\right] \\ & = & e^{\frac{b}{2a}(a^2-x^2)} \mathbb{E}_x^{\nu}\left[
e^{-\frac{\lambda_{b}^2 T^{a}}{1+\frac{b}{a} T^{a}}} \left(1+\frac{b}{a} T^{a}\right)^{-\delta/2}\right] \\ & = & e^{\frac{b}{2a}(a^2-x^2)} \int_0^{\infty} p^{\nu}_{b/2a}(\lambda_{b},u) \mathbb{E}_x^{\nu}\left[ e^{- u^2 T^{a}} \right]du \end{eqnarray*} where the last line follows from \eqref{eq:sg}. It remains to use (\ref{eq:laplace-bessel}) to conclude. \end{proof}
\begin{remark} The process $(Y_t:=X_t+bt,t\geq 0)$ is a non-homogeneous Markov process and solves the sde \begin{equation*} Y_t=B_t+\frac{\delta-1}{2}\int_0^t\frac{ds}{Y_s-bs}+bt,\quad t\geq 0. \end{equation*} This is to be distinguished from a Bessel process with a "naive" drift $b$ introduced in \cite{Yor-84} and defined as a solution to \begin{equation*} Z_t=B_t+\frac{\delta-1}{2}\int_0^t\frac{ds}{Z_s}+bt,\quad t\geq 0. \end{equation*} \end{remark}
\begin{remark} The Bessel process of dimension $\delta=1$ is, in fact, the reflected Brownian motion. It follows that the associated first hitting times can be interpreted as double barrier hitting times. That is, with $f$ as above, the time when a Brownian motion $B$ hits one of the curves $x\pm f(\cdot)$, i.e. $\inf\{s>0; \: B_s=x\pm f(s)\}$. \end{remark}
\section{Concluding remarks and some comments} \subsection{} It is plain that the results of Theorem \ref{thm} can be readily extended to any $h$-transform of the process $X$, but we need to take care of the life-times of the involved processes in the Bessel case. In the Brownian setting, one gets similar results for the process $X^{\epsilon}_t=X_t+y \epsilon t,\: t\geq0$, where $\epsilon$ is an independent symmetric Bernoulli random variable taking values in $\{-1,1\}$. Observe that $i(X^{\epsilon})$ is a strong Markov process. However, because the latter starts at the random point $ y \epsilon$, $X^{(\epsilon)}$ does not satisfy the time inversion property \eqref{eq:ti}.
\subsection{}
Assuming that $X$ is a $2$-self-similar strong Markov process with c\`adl\`ag paths (i.e.~with possible jumps) enjoying the time inversion property, then Theorem 5.8 in \cite{Pitman-Yor-81} or Theorem 1 in \cite{Gallardo-Yor-05} allow to extend Lemma \ref{doob} and Lemma \ref{abs-diff} and an analogue of Theorem \ref{thm} can be stated. However, we did not succeed to construct examples of $\mathbb{R}_+$-valued processes enjoying the time inversion property other than Bessel processes in the usual or wide sense. This is the reason why the setting is restricted to the continuous one. We refer to \cite{Patie-08a} where the second author characterizes, through its Mellin transform, the law of the first passage time above the square root boundary for spectrally negative positive $2$-self-similar Markov processes.
\subsection{}
In \cite{Durbin-85}, Durbin considered the studied boundary crossing problem for continuous gaussian processes and showed that, in the absolute-continuous case, the problem reduces to the computation of a conditional expectation. In particular, for the Brownian motion, if $f$ is continuously differentiable and $f(0)\neq 0$ then \[ \mathbb{P} \left(T^{f}\in dt \right)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{f^2(t)}{2t}}h(t) dt\] where \begin{equation*}
h(t)=\lim_{s\nearrow t}\frac{1}{t-s}\mathbb{E}\left[B_s-f(s); T^{f}>s \big{|}B_t=f(t) \right],\quad t>0. \end{equation*}
There seems to be no known way to compute the function $h$. It is shown in \cite{Durbin-88} that this method is in agreement with the standard method of images. With the obvious notation, it is clearly possible to relate $h^{(\beta)}$ to $h$.
\subsection{}
We learned from Kendall \cite{Kendall-04} an intuitive interpretation of Durbin's expression involving the family of local times of $X$ at $f$ denoted by $ (L_{t}^{X=f}, t\geq 0)$. Indeed, observe that Durbin's formula can be rewritten as ${\mathbb{P}} \left(T^{f}\in dt \right)=h(t){\mathbb E}[dL_t^{X=f} ]$, $t\geq 0$. The latter when integrated over $[0,a]$, yields ${\mathbb P}\left(T^{f}<a \right)={\mathbb E}\left[\int_0^{a}h(s)dL_s^{X=f} \right]$. Such a decomposition is not unique and many can be constructed from the above one. A natural non trivial one to consider is motivated by the following observation found in \cite{Kendall-04}. If $g:\mathbb{R}^+\times \mathbb{R}^+\rightarrow \mathbb{R}^+$ solves
${\mathbb E}\left[\int_t^{a}g(s,a)dL_s^{X=f}\big|X_t=f(t) \right]=1$, for $0< t<a$,
then clearly ${\mathbb E}\left[\int_0^{a}g(s,a)dL_s^{X=f} \right]={\mathbb P}\left(T^{f}<a \right)$. However, it is not clear how to express $g$ in
terms of $h$. W. Kendall told the first author that the above interpretation could be interesting for further investigations for the studied boundary crossing problem.
\subsection{} In the Brownian case, it is interesting to analyze the impact of our identities on some integral equations satisfied by the studied densities. However, some of them lead to
obvious facts explainable by change of variables. As an example, we observe that if $f$ is positive and does not vanish then $X$, when started at $x>f(0)$, must hit $f$ before reaching $0$. The strong Markov property then gives \begin{equation*} {x\over{\sqrt{2\pi t^3}}}e^{-{x^2\over {2t}}}=\int_0^t \mathbb{P}_x(T^{f} \in dr) {f(r)\over {\sqrt{2\pi {(t-r)^3}}}}e^{-{f(r)^2\over {2(t-r)}}},\:\:\:\: t>0. \end{equation*} This is easily shown to be in accordance with the result stated in Theorem \ref{thm}. For the above and other classical integral equations we refer to \cite{Durbin-85},
\cite{Ferebee-83}, \cite{Lerche-86} and also to
\cite{Peskir-02-2} for some more recent ones.
\subsection{}
The technics used in the example treated in Subsection \ref{examples}, for Brownian motion, can be applied
to Bessel processes and
square root curves. The required results, for that end, can be found in Delong \cite{Delong-81} or in Yor \cite{Yor-84}. This law is connected via a deterministic time change to the one of the first passage time to a fixed level by the radial norm of a $\delta$-dimensional Ornstein-Uhlenbeck process, where $\delta$ is some positive integer. Due to the
stationarity property, the law of the first passage times can be expressed as infinite convolutions
of mixture of exponential distributions, see Kent \cite{Kent-80}.
{\bf Acknowledgment:} We are grateful to F.~Delbaen, W.S.~Kendall, D.~Talay and M.~Yor for inspiring and fruitful discussions on the topic. The first author would like to thank the University of Bern for their kind invitation for a visit during which a part of this work was carried out. We thank two anonymous referees for their careful reading and useful comments that helped to improve the presentation of the paper.
\end{document}
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{The Geometry of Hida Families \Rmnum{2}: $\Lambda$-adic $(\varphi,\Gamma)$-modules and $\Lambda$-adic Hodge Theory}
\author{Bryden Cais} \address{University of Arizona, Tucson} \curraddr{Department of Mathematics, 617 N. Santa Rita Ave., Tucson AZ. 85721} \email{[email protected]}
\thanks{
During the writing of this paper, the author was partially supported by an NSA Young Investigator grant
(H98230-12-1-0238) and an NSF RTG (DMS-0838218).
}
\dedicatory{To Haruzo Hida, on the occasion of his $60^{\text{th}}$ birthday.}
\subjclass[2010]{Primary: 11F33 Secondary: 11F67, 11G18, 11R23} \keywords{Hida families, integral $p$-adic Hodge theory, de~Rham cohomology, crystalline cohomology.} \date{\today}
\begin{abstract}
We construct the $\Lambda$-adic crystalline and Dieudonn\'e analogues of Hida's
ordinary $\Lambda$-adic \'etale cohomology,
and employ integral $p$-adic Hodge theory to prove $\Lambda$-adic
comparison isomorphisms between these cohomologies and
the $\Lambda$-adic de Rham cohomology studied in \cite{CaisHida1}
as well as Hida's $\Lambda$-adic \'etale cohomology.
As applications of our work, we provide a ``cohomological" construction of the
family of $(\varphi,\Gamma)$-modules attached to Hida's ordinary $\Lambda$-adic \'etale
cohomology by \cite{Dee}, and we give a new and purely geometric
proof of Hida's finitenes and control theorems. We also prove suitable
$\Lambda$-adic duality theorems for each of the cohomologies we construct.
\end{abstract}
\maketitle
\section{Introduction}\label{intro}
\subsection{Motivation}
In a series of groundbreaking papers \cite{HidaGalois} and \cite{HidaIwasawa}, Hida constructed $p$-adic analytic families of $p$-ordinary Galois representations interpolating the Galois representations attached to $p$-ordinary cuspidal Hecke eigenforms in integer weights $k\ge 2$ by Deligne \cite{DeligneFormes}, \cite{CarayolReps}. At the heart of Hida's construction is the $p$-adic \'etale cohomology $H^1_{\et}:=\varprojlim_r H^1_{\et}(X_1(Np^r)_{\Qbar},\Z_p)$ of the tower of modular curves over $\Q$, which is naturally a module for the ``big" $p$-adic Hecke algebra $\H^*:=\varprojlim_r \H_r^*$, which is itself an algebra over the completed group ring $\Lambda:=\Z_p[\![\Delta_1]\!]\simeq \Z_p[\![T]\!]$ via the diamond operators $\Delta_r:=1+p^r\Z_p$. Writing $e^*\in \H^*$ for the idempotent attached to the (adjoint) Atkin operator $U_p^*$, Hida proves (via explicit computations in group cohomology) that the ordinary part $e^*H^1_{\et}$ of $H^1_{\et}$ is
finite and free as a module over $\Lambda$, and that the resulting Galois representation \begin{equation*}
\xymatrix{
{\rho: \scrG_{\Q}} \ar[r] & {\Aut_{\Lambda}(e^*H^1_{\et}) \simeq \GL_m(\Z_p[\![T]\!])}
} \end{equation*} $p$-adically interpolates the representations attached to ordinary cuspidal eigenforms.
By analyzing the geometry of the tower of modular curves, Mazur and Wiles \cite{MW-Hida} showed that both the inertial invariants and convariants of the the local (at $p$) representation $\rho_p$ are free of the finite same rank over $\Lambda$, and hence that the ordinary filtration of the Galois representations attached to ordinary cuspidal eigenforms interpolates in Hida's $p$-adic family.
As an application, they gave examples of cuspforms $f$ and primes $p$ for which the specialization of the associated Hida family of Galois representations to weight $k=1$ is not Hodge--Tate, and so does not arise from a weight one cuspform via the construction of Deligne-Serre \cite{DeligneSerre}. Shortly thereafter, Tilouine \cite{Tilouine} clarified the geometric underpinnings of \cite{HidaGalois} and \cite{MW-Hida}.
In \cite{OhtaEichler}, \cite{Ohta1} and \cite{Ohta2}, Ohta initiated the study of the $p$-adic Hodge theory of Hida's ordinary $\Lambda$-adic (local) Galois representation $\rho_p$. Using the invariant differentials on the tower of $p$-divisible groups over $R_{\infty}:=\Z_p[\mu_{p^{\infty}}]$ attached to the ``good quotient" modular abelian varieties introduced in \cite{MW-Iwasawa} and studied in \cite{MW-Hida} and \cite{Tilouine}, Ohta constructs a certain $\Lambda_{R_{\infty}}:= R_{\infty}[\![\Delta_1]\!]$-module $e^*H^1_{\Hodge}$, which is the Hodge cohomology analogue of $e^*H^1_{\et}$. Via an integral version of the Hodge--Tate comparison isomorphism \cite{Tate} for ordinary $p$-divisible groups, Ohta establishes a $\Lambda$-adic Hodge-Tate comparison isomorphism relating $e^*H^1_{\Hodge}$ and the semisimplification of the ``semilinear representation" $\rho_{p}\wh{\otimes} \O_{\C_p}$.
Using Hida's results, Ohta shows that $e^*H^1_{\Hodge}$ is free of finite rank over $\Lambda_{R_{\infty}}$ and specializes to finite level exactly as one expects. As applications of his theory, Ohta provides a construction of two-variable $p$-adic $L$-functions attached to families of ordinary cuspforms differing from that of Kitagawa \cite{Kitagawa}, and, in a subsequent paper \cite{Ohta2}, provides a new and streamlined proof of the theorem of Mazur--Wiles \cite{MW-Iwasawa} (Iwasawa's Main Conjecture for $\Q$; see also \cite{WilesTotallyReal}). We remark that Ohta's $\Lambda$-adic Hodge-Tate isomorphism is a crucial ingredient in the forthcoming proof of Sharifi's conjectures \cite{SharifiConj}, \cite{SharifiEisenstein} due to Fukaya and Kato \cite{FukayaKato}.
In \cite{CaisHida1}, we continued the trajectory begun by Ohta by constructing the de Rham analogue of $e^*H^1_{\et}$. Using the canonical integral structures in de Rham cohomology studied in \cite{CaisDualizing} and certain Katz-Mazur \cite{KM} integral models $\X_r$ of $X_1(Np^r)$ over $R_r:=\Z_p[\mu_{p^r}]$, for each $r> 0$ we constructed a canonical
short exact sequences of free $R_r$-modules
\begin{equation}
\xymatrix{
0\ar[r] & {H^0(\X_r, \omega_{\X_r/R_r})} \ar[r] & {H^1(\X_r/R_r)} \ar[r] & {H^1(\X_r,\O_{\X_r})}
\ar[r] & 0
}\label{finiteleveldRseq} \end{equation}
whose scalar extension to $K_r:=\Frac(R_r)$ recovers the Hodge filtration of the de Rham cohomology of $X_1(Np^r)$ over $K_r$. Extending scalars to $R_{\infty}$, taking projective limits, and passing to ordinary parts gives a sequence of modules over $\Lambda_{R_{\infty}}$ with semilinear $\Gamma:=\Gal(K_{\infty}/K_0)$-action and commuting linear $\H^*$-action \begin{equation}
\xymatrix{
0\ar[r] & {e^*H^0(\omega)} \ar[r] & {e^*H^1_{\dR}} \ar[r] & {e^*H^1(\O)} \ar[r] & 0
}.\label{orddRseq}
\end{equation} The main result of \cite{CaisHida1} is that (\ref{orddRseq}) is the correct de Rham analogue of Hida's ordinary $\Lambda$-adic \'etale cohomology and Ohta's ordinary $\Lambda$-adic Hodge cohomology (see \cite[Theorem 3.2.3]{CaisHida1}):
\begin{theorem}\label{dRMain}
Let $d=\sum_{k=3}^{p+1} d_k$ for $d_k:=\dim_{\F_p} S_k(\Upgamma_1(N);\F_p)^{\ord}$
the $\F_p$-dimension of the space of mod $p$ weight $k$ ordinary cuspforms for $\Upgamma_1(N)$.
Then $(\ref{orddRseq})$ is a short exact sequence of free $\Lambda_{R_{\infty}}$-modules
of ranks $d$, $2d$, and $d$, respectively.
Applying $\otimes_{\Lambda_{R_{\infty}}} R_{\infty}[\Delta_1/\Delta_r]$ to $(\ref{orddRseq})$
recovers the ordinary part of the scalar extension of $(\ref{finiteleveldRseq})$ to $R_{\infty}$. \end{theorem}
The natural cup-product auto-duality of $(\ref{finiteleveldRseq})$ over $R_r':=R_r[\mu_N]$ induces a canonical $\Lambda_{R_{\infty}'}$-linear and $\H^*$-equivariant auto-duality of (\ref{orddRseq}) which intertwines the dual semilinear action of $\Gamma \times \Gal(K_0'/K_0)\simeq \Gal(K_{\infty}'/K_0)$ with a certain $\H^*$-valued twist of its standard action; see \cite[Proposition 3.2.4]{CaisHida1} for the precise statement.
We moreover proved that, as one would expect, the $\Lambda_{R_{\infty}}$-module $e^*H^0(\omega)$ is canonically isomorphic to the module $eS(N,\Lambda_{R_{\infty}})$ of ordinary $\Lambda_{R_{\infty}}$-adic cusp forms of tame level $N$; see \cite[Corollary 3.3.5]{CaisHida1}.
\subsection{Results}\label{resultsintro}
In this paper, we cmplete our study of the geometry and $\Lambda$-adic Hodge theory of Hida families begun in \cite{CaisHida1} by constructing the crystalline counterpart to Hida's ordinary $\Lambda$-adic \'etale cohomology, Ohta's $\Lambda$-adic Hodge cohomology, and our $\Lambda$-adic de Rham cohomology. Via a careful study of the geometry of modular curves and abelian varieties and comparison isomorphisms in integral $p$-adic cohomology, we prove the appropriate control and finiteness theorems, and a suitable $\Lambda$-adic version of every integral comparison isomorphism one could hope for. In particular, we are able to recover the entire family of $p$-adic Galois representations $\rho_{p}$ (and not just its semisimplification) from our $\Lambda$-adic crystalline cohomology. A remarkable byproduct of our work is a {\em cohomological} construction of
the family of \'etale $(\varphi,\Gamma)$-modules attached to $e^*H^1_{\et}$ by Dee \cite{Dee}.
As an application of our theory, we give a new and purely geometric proof of Hida's freeness and control theorems for $e^*H^1_{\et}$.
In order to survey our main results more precisely, we introduce some notation. Throughout this paper, we fix a prime $p>2$ and a positive integer $N$ with $Np > 4$. Fix an algebraic closure $\Qbar_p$ of $\Q_p$ as well as a $p$-power compatible sequence $\{\varepsilon^{(r)}\}_{r\ge 0}$ of primitive $p^r$-th roots of unity in $\Qbar_p$. As above, we set $K_r:=\Q_p(\mu_{p^r})$ and $K_r':=K_r(\mu_N)$, and we write $R_r$ and $R_r'$ for the rings of integers in $K_r$ and $K_r'$, respectively. Denote by $\scrG_{\Q_p}:=\Gal(\Qbar_p/\Q_p)$ the absolute Galois group and by $\scrH$ the kernel of the $p$-adic cyclotomic character $\chi: \scrG_{\Q_p}\rightarrow \Z_p^{\times}$.
Using that $K_0'/\Q_p$ is unramified, we canonically identify $\Gamma=\scrG_{\Q_p}/\scrH$ with $\Gal(K_{\infty}'/K_0')$. We will denote by $\langle u\rangle$ (respectively $\langle v\rangle_N)$ the diamond operator\footnote{Note that $\langle u^{-1}\rangle=\langle u\rangle^*$ and $\langle v^{-1}\rangle_N = \langle v\rangle_N^*$, where $\langle\cdot\rangle^*$ and $\langle \cdot\rangle_N^*$ are the adjoint diamond operators; see \cite[\S2.2]{CaisHida1}. } in $\H^*$ attached to $u^{-1}\in \Z_p^{\times}$ (respectively $v^{-1}\in (\Z/N\Z)^{\times}$) and write $\Delta_r$ for the image of the restriction of $\langle\cdot\rangle :\Z_p^{\times}\hookrightarrow \H^*$ to $1+p^r\Z_p\subseteq \Z_p^{\times}$. For convenience, we put $\Delta:=\Delta_1$, and for any ring $A$ we write $\Lambda_{A}:=\varprojlim_r A[\Delta/\Delta_r]$ for the completed group ring on $\Delta$ over $A$; if $\varphi$ is an endomorphism of $A$, we again write $\varphi$ for the induced endomorphism of $\Lambda_A$ that acts as the identity on $\Delta$. For any ring homomorphism $A\rightarrow B$, we will write $(\cdot)_B:=(\cdot)\otimes_A B$ and $(\cdot)_B^{\vee}:=\Hom_B((\cdot)\otimes_A B , B)$ for these functors from $A$-modules to $B$-modules.\footnote{This convention is unfortunately somewhat at odds with our notation $\Lambda_A$, which (as an $A$-module) is in general neither the tensor product $\Lambda\otimes_{\Z_p} A$ nor (unless $A$ is a complete $\Z_p$-algebra) the completed tensor product $\Lambda \wh{\otimes}_{\Z_p} A$; we hope that this small abuse causes no confusion.}
If $G$ is any group of automorphisms of $A$ and $M$ is an $A$-module with a semilinear action of $G$, for any ``crossed" homomorphism\footnote{That is, $\psi(\sigma\tau) = \psi(\sigma)\cdot\sigma\psi(\tau)$ for all $\sigma,\tau\in\Gamma$,} $\psi:G\rightarrow A^{\times}$ we will write $M(\psi)$ for the $A$-module $M$ with ``twisted" semilinear $G$-action given by $g\cdot m:=\psi(g)g m$. Finally, we denote by $X_r:=X_1(Np^r)$ the usual modular curve over $\Q$ classifying (generalized) elliptic curves with a $[\mu_{Np^r}]$-structure, and by $J_r:=J_1(Np^r)$ its Jacobian.
We analyze the tower of $p$-divisible groups attached to the ``good quotient" modular abelian varieties introduced by Mazur-Wiles \cite{MW-Iwasawa}. To avoid technical complications with logarithmic $p$-divisible groups, following \cite{MW-Hida} and \cite{OhtaEichler}, we will henceforth remove the trivial tame character by working with the sub-idempotent ${e^*}'$ of $e^*$ corresponding to projection to the part where $\mu_{p-1}\subseteq \Z_p^{\times}$ acts {\em non}-trivially via the diamond operators. As is well-known (e.g. \cite[\S9]{HidaGalois} and \cite[Chapter 3, \S2]{MW-Iwasawa}), the $p$-divisible group $G_r:={e^*}'J_r[p^{\infty}]$ over $\Q$ extends to a $p$-divisible group $\G_r$ over $R_r$, and we write $\o{\G}_r:=\G_r\times_{R_r} \F_p$ for its special fiber. Denoting by $\D(\cdot)$ the contravariant Dieudonn\'e module functor on $p$-divisible groups over $\F_p$, we form the projective limits \begin{equation}
\D_{\infty}^{\star}:=\varprojlim_r \D(\o{\G}_r^{\star})\quad\text{for}\quad \star\in \{\et,\mult,\Null\},
\label{DlimitsDef} \end{equation} taken along the mappings induced by $\o{\G}_r\rightarrow \o{\G}_{r+1}$. Each of these is naturally a $\Lambda$-module equipped with linear (!) Frobenius $F$ and Verscheibung $V$ morphisms satisfying $FV=VF=p$, as well as a linear action of $\H^*$ and a ``geometric inertia" action of $\Gamma$, which reflects the fact that the generic fiber of $\G_r$ descends to $\Q_p$. The $\Lambda$-modules (\ref{DlimitsDef}) have the expected structure (see Theorem \ref{MainDieudonne}):
\begin{theorem}\label{DieudonneMainThm}
There is a canonical split short exact sequence of finite and free $\Lambda$-modules
\begin{equation}
\xymatrix{
0 \ar[r] & {\D^{\et}_{\infty}} \ar[r] & {\D_{\infty}} \ar[r] & {\D_{\infty}^{\mult}} \ar[r] & 0
}\label{Dieudonneseq}
\end{equation}
with linear $\H^*$ and $\Gamma$-actions.
As a $\Lambda$-module, $\D_{\infty}$ is free of rank $2d'$, while $\D_{\infty}^{\et}$
and $\D_{\infty}^{\mult}$ are free of rank $d'$, where $d':=\sum_{k=3}^p \dim_{\F_p} S_k(\Upgamma_1(N);\F_p)^{\ord}$.
For $\star\in \{\mult,\et,\Null\}$, there are canonical isomorphisms
\begin{equation*}
\D_{\infty}^{\star}\tens_{\Lambda} \Z_p[\Delta/\Delta_r] \simeq \D(\o{\G}_r^{\star})
\end{equation*}
which are compatible with the extra structures.
Via the canonical splitting of $(\ref{Dieudonneseq})$, $\D_{\infty}^{\star}$ for $\star=\et$
$($respetively $\star=\mult$$)$ is
identified with the maximal subpace of $\D_{\infty}$ on which $F$ $($respectively $V$$)$ acts
invertibly .
The Hecke operator $U_p^*\in \H^*$ acts as $F$ on $\D_{\infty}^{\et}$ and as $\langle p\rangle_N V$ on
$\D_{\infty}^{\mult}$,
while $\Gamma$ acts trivially on $\D_{\infty}^{\et}$ and via $\langle \chi(\cdot)\rangle^{-1}$
on $\D_{\infty}^{\mult}$. \end{theorem}
The short exact sequence (\ref{Dieudonneseq}) is very nearly $\Lambda$-adically auto-dual (see Proposition \ref{DieudonneDuality}):
\begin{theorem}\label{DDuality}
There is a canonical $\H^*$-equivariant isomorphism of exact sequences of $\Lambda_{R_0'}$-modules
\begin{equation*}
\xymatrix{
0 \ar[r] & {\D_{\infty}^{\et}(\langle \chi \rangle\langle a\rangle_N)_{\Lambda_{R_0'}}} \ar[r]\ar[d]^-{\simeq} &
{\D_{\infty}(\langle \chi \rangle\langle a\rangle_N)_{\Lambda_{R_0'}}}\ar[r]\ar[d]^-{\simeq} &
{\D_{\infty}^{\mult}(\langle \chi \rangle\langle a\rangle_N)_{\Lambda_{R_0'}}}\ar[r]\ar[d]^-{\simeq} & 0 \\
0\ar[r] & {(\D_{\infty}^{\mult})^{\vee}_{\Lambda_{R_0'}}} \ar[r] &
{(\D_{\infty})^{\vee}_{\Lambda_{R_0'}}} \ar[r] &
{(\D_{\infty}^{\et})^{\vee}_{\Lambda_{R_0'}}}\ar[r] & 0
}
\end{equation*}
that is $\Gamma\times \Gal(K_0'/K_0)$-equivariant,
and intertwines $F$
$($respectively $V$$)$ on the top row with $V^{\vee}$
$($respectively $F^{\vee}$$)$ on the bottom.\footnote{Here, $F^{\vee}$ (respectively $V^{\vee}$)
is the map taking a linear functional $f$ to $\varphi^{-1}\circ f\circ F$
(respectively $\varphi\circ f\circ V$), where $\varphi$
is the Frobenius automorphism of $R_0'=\Z_p[\mu_N]$.} \end{theorem}
In \cite{MW-Hida}, Mazur and Wiles relate the ordinary-filtration of ${e^*}'H^1_{\et}$ to the \'etale cohomology of the Igusa tower studies in \cite{MW-Analogies}. We can likewise interpret the slope filtraton (\ref{Dieudonneseq}) in terms of the crystalline cohomology of the Igusa tower as follows. For each $r$, let $I_r^{\infty}$ and $I_r^0$ be the two ``good" irreducible components of $\X_r\times_{R_r}\F_r$ (see the discussion preceding Proposition \ref{GisOrdinary}), each of which is isomorphic to the Igusa curve $\Ig(p^r)$ of tame level $N$ and $p$-level $p^r$. For $\star\in \{0,\infty\}$ we form the projective limit \begin{equation*}
H^1_{\cris}(I^{\star}):=\varprojlim_{r} H^1_{\cris}(I_r^{\star}/\Z_p); \end{equation*} with respect to the trace mappings on crystalline cohmology induced by the canonical degeneracy maps on Igusa curves. Then $H^1_{\cris}(I^{\star})$ is naturally a $\Lambda$-module with linear Frobenius $F$ and Verscheibung $V$ endomorphisms, and we write $H^1_{\cris}(I^{\star})^{V_{\ord}}$ (respecytively $H^1_{\cris}(I^{\star})^{F_{\ord}}$) for the maximal $V$- (respectively $F$-) stable submodule on which $V$ (respectively $F$) acts invertibly. Letting $f'$ be the idempotent of $\Lambda$ corresponding to projection to the part where $\mu_{p-1}\subseteq \Z_p^{\times}$ acts nontrivially via the diamond operators, we prove (see Theorem \ref{DieudonneCrystalIgusa}):
\begin{theorem}
There is a canonical isomorphism of $\Lambda$-modules, compatible with $F$ and $V,$
\begin{equation}
\D_{\infty} =\D_{\infty}^{\mult}\oplus \D_{\infty}^{\et}\simeq
f'H^1_{\cris}(I^{0})^{V_{\ord}} \oplus
f'H^1_{\cris}(I^{\infty})^{F_{\ord}}.\label{crisIgusa}
\end{equation}
which preserves the direct sum decompositions of source and target.
This isomorphism is Hecke and $\Gamma$-equivariant, with $U_p^*$ and $\Gamma$
acting as $\langle p\rangle_N V\oplus F$ and
$ \langle \chi(\cdot)\rangle^{-1}\oplus \id$, respectively,
on each direct sum. \end{theorem}
We note that our ``Dieudonn\'e module" analogue (\ref{crisIgusa}) is a significant sharpening of its \'etale counterpart \cite[\S4]{MW-Hida}, which is formulated only up to isogeny (i.e. after inverting $p$). From $\D_{\infty}$, we can recover the $\Lambda$-adic Hodge filtration (\ref{orddRseq}), so the latter is canonically split (see Theorem \ref{dRtoDieudonneInfty}):
\begin{theorem}\label{dRtoDieudonne}
There is a canonical $\Gamma$ and $\H^*$-equivariant isomorphism of
exact sequences
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] & {{e^*}'H^0(\omega)} \ar[r]\ar[d]^-{\simeq} &
{{e^*}'H^1_{\dR}} \ar[r]\ar[d]^-{\simeq} & {{e^*}'H^1(\O)} \ar[r]\ar[d]^-{\simeq} & 0 \\
0 \ar[r] & {\D_{\infty}^{\mult}\tens_{\Lambda} \Lambda_{R_{\infty}}} \ar[r] &
{\D_{\infty}\tens_{\Lambda} \Lambda_{R_{\infty}}} \ar[r] &
{\D_{\infty}^{\et}\tens_{\Lambda} \Lambda_{R_{\infty}}} \ar[r] & 0
}\label{dRcriscomparison}
\end{gathered}
\end{equation}
where the mappings on bottom row are the canonical inclusion and projection morphisms
corresponding to the direct sum decomposition $\D_{\infty}=\D_{\infty}^{\mult}\oplus \D_{\infty}^{\et}$.
In particular, the Hodge filtration exact sequence $(\ref{orddRseq})$ is canonically
split, and admits a canonical descent to $\Lambda$. \end{theorem}
We remark that under the identification (\ref{dRcriscomparison}),
the Hodge filtration (\ref{orddRseq}) and slope filtration (\ref{Dieudonneseq})
correspond, but in the opposite directions. As a consequence of Theorem
\ref{dRtoDieudonne}, we deduce (see Corollary \ref{MFIgusaDieudonne} and
Remark \ref{MFIgusaCrystal}):
\begin{corollary} \label{OhtaCor}
There is a canonical isomorphism of finite free $\Lambda$ $($respectively $\Lambda_{R_0'}$$)$-modules
\begin{equation*}
{e}'S(N,\Lambda) \simeq \D_{\infty}^{\mult}
\qquad\text{respectively}\qquad
e'\H\tens_{\Lambda} \Lambda_{R_0'} \simeq \D_{\infty}^{\et}(\langle a\rangle_N)\tens_{\Lambda}{\Lambda_{R_0'}}
\end{equation*}
that intertwines $T\in \H:=\varprojlim \H_r$ with $T^*\in \H^*$, where we let
$U_p^*$ act as $\langle p\rangle_N V$ on $\D_{\infty}^{\mult}$ and as $F$ on $\D_{\infty}^{\et}$.
The second of these isomorphisms is in addition $\Gal(K_0'/K_0)$-equivariant. \end{corollary}
We are also able to recover the semisimplification of ${e^*}'H^1_{\et}$ from $\D_{\infty}$. Writing $\I\subseteq \scrG_{\Q_p}$ for the inertia subgroup at $p$, for any $\Z_p[\scrG_{\Q_p}]$-module $M$, we denote by $M^{\I}$ (respectively $M_{\I}:=M/M^{\I}$) the sub (respectively quotient) module of invariants (respectively covariants) under $\I$.
\begin{theorem}\label{FiltrationRecover}
There are canonical isomorphisms of $\Lambda_{W(\o{\F}_p)}$-modules
with linear $\H^*$-action and semilinear actions of $F$, $V$, and $\scrG_{\Q_p}$
\begin{subequations}
\begin{equation}
\D_{\infty}^{\et} \tens_{\Lambda} \Lambda_{W(\o{\F}_p)}
\simeq ({e^*}'H^1_{\et})^{\I}\tens_{\Lambda} \Lambda_{W(\o{\F}_p)}
\label{inertialinvariants}
\end{equation}
and
\begin{equation}
\D_{\infty}^{\mult}(-1) \tens_{\Lambda} \Lambda_{W(\o{\F}_p)}
\simeq ({e^*}'H^1_{\et})_{\I}\tens_{\Lambda} \Lambda_{W(\o{\F}_p)}.
\label{inertialcovariants}
\end{equation}
\end{subequations}
Writing $\sigma$ for the
Frobenius automorphism of $W(\o{\F}_p)$,
the isomorphism $(\ref{inertialinvariants})$ intertwines $F\otimes \sigma$ with $\id\otimes\sigma$
and $\id\otimes g$ with $g\otimes g$ for $g\in \scrG_{\Q_p}$, whereas $(\ref{inertialcovariants})$
intertwines $V\otimes \sigma^{-1}$ with $\id\otimes\sigma^{-1}$ and $g\otimes g$ with $g\otimes g$,
where $g\in\scrG_{\Q_p}$ acts on the Tate twist
$\D_{\infty}^{\mult}(-1):=\D_{\infty}^{\mult}\otimes_{\Z_p}\Z_p(-1)$ as
$\langle \chi(g)^{-1}\rangle \otimes \chi(g)^{-1}$. \end{theorem}
Theorem \ref{FiltrationRecover} gives the following ``explicit" description of the semisimplification of ${e^*}'H^1_{\et}$:
\begin{corollary}
For any $T\in (e^*\H^{*})^{\times}$, let
$\lambda(T):\scrG_{\Q_p}\rightarrow e^*\H^{*}$ be the
unique continuous $($for the $p$-adic topology on $e^*\H^{*}$$)$
unramified character whose value on $($any lift of$)$
$\mathrm{Frob}_p$ is $T$.
Then $\scrG_{\Q_p}$ acts on $({e^*}'H^1_{\et})^{\I}$ through the
character $\lambda({U_p^*}^{-1})$ and on $({e^*}'H^1_{\et})_{\I}$
through $\chi^{-1} \cdot \langle \chi^{-1}\rangle \lambda(\langle p\rangle_N^{-1}U_p^*)$. \end{corollary}
We remark that Corollary \ref{OhtaCor} and Theorem \ref{FiltrationRecover} combined give a refinement of the main result of \cite{OhtaEichler}. We are furthermore able to recover the main theorem of \cite{MW-Hida} (that the ordinary filtration of ${e^*}'H^1_{\et}$ interpolates $p$-adic analytically):
\begin{corollary}\label{MWmainThmCor}
Let $d'$ be the integer of Theorem $\ref{DieudonneMainThm}$. Then each of
$({e^*}'H^1_{\et})^{\I}$ and $({e^*}'H^1_{\et})_{\I}$ is a free
$\Lambda$-module of rank $d'$, and for each $r\ge 1$ there are canonical
$\H^*$ and $\scrG_{\Q_p}$-equivariant isomorphisms of $\Z_p[\Delta/\Delta_r]$-modules
\begin{subequations}
\begin{equation}
({e^*}'H^1_{\et})^{\I} \tens_{\Lambda} \Z_p[\Delta/\Delta_r] \simeq
{e^*}'H^1_{\et}({X_r}_{\Qbar_p},\Z_p)^{\I}\label{HidaResultSub}
\end{equation}
\begin{equation}
({e^*}'H^1_{\et})_{\I} \tens_{\Lambda} \Z_p[\Delta/\Delta_r] \simeq
{e^*}'H^1_{\et}({X_r}_{\Qbar_p},\Z_p)_{\I}
\label{HidaResultQuo}
\end{equation}
\end{subequations} \end{corollary}
To recover the full $\Lambda$-adic local Galois representation ${e^*}'H^1_{\et}$, rather than just its semisimplification, it is necessary to work with the full Dieudonn\'e {\em crystal} of $\G_r$ over $R_r$. Following Faltings \cite{Faltings} and Breuil (e.g. \cite{Breuil}), this is equivalent to studying the evaluation of the Dieudonn\'e crystal of $\G_r\times_{R_r} R_r/pR_r$ on the ``universal" divided power thickening $S_r\twoheadrightarrow R_r/pR_r$, where $S_r$ is the $p$-adically completed PD-hull of the surjection $\Z_p[\![u_r]\!]\twoheadrightarrow R_r$ sending $u_r$ to $\varepsilon^{(r)}-1$. As the rings $S_r$ are too unwieldly to directly construct a good crystalline analogue of Hida's ordinary \'etale cohomology, we must functorially descend the ``filtered $S_r$-module" attached to $\G_r$ to the much simpler ring $\s_r:=\Z_p[\![u_r]\!]$. While such a descent is provided (in rather different ways) by the work of Breuil--Kisin and Berger--Wach, neither of these frameworks is suitable for our application: it is essential for us that the formation of this descent to $\s_r$ commute with base change as one moves up the cyclotomic tower, and it is not at all clear that this holds for Breuil--Kisin modules or for the Wach modules of Berger. Instead, we use the theory of \cite{CaisLau}, which works with frames and windows \`a la Lau and Zink to provide the desired functorial descent to a ``$(\varphi,\Gamma)$-module" $\m_r(\G_r)$ over $\s_r$. We view $\s_r$ as a $\Z_p$-subalgebra of $\s_{r+1}$ via the map sending $u_r$ to $\varphi(u_{r+1}):=(1+u_{r+1})^p -1$, and we write $\s_{\infty}:=\varinjlim \s_r$ for the rising union\footnote{As explained in Remark \ref{Slimits}, the $p$-adic completion of $\s_{\infty}$ is actually a very nice ring: it is canonically and Frobenius equivariantly isomorphic to $W(\F_p[\![u_0]\!]^{\perf})$, for $\F_p[\![u_0]\!]^{\perf}$ the perfect closure of the $\F_p$-algebra $\F_p[\![u_0]\!]$. } of the $\s_r$, equiped with its Frobenius {\em automorphism} $\varphi$ and commuting action of $\Gamma$ determined by $\gamma u_r:=(1+u_r)^{\chi(\gamma)} - 1$. We then form the projective limits \begin{equation*}
\m_{\infty}^{\star}:=\varprojlim (\m_r(\G_r^{\star})\tens_{\s_r} \s_{\infty})\quad\text{for}\quad \star\in\{\et,\mult,\Null\} \end{equation*} taken along the mappings induced by $\G_{r}\times_{R_r} R_{r+1}\rightarrow \G_{r+1}$ via the functoriality of $\m_r(\cdot)$ and its compatibility with base change. These are $\Lambda_{\s_{\infty}}$-modules equipped with a semilinear action of $\Gamma$, a linear and commuting action of $\H^*$, and a $\varphi$ (respectively $\varphi^{-1}$) semilinear endomorphism $F$ (respectively $V$) satisfying $FV=\omega$ and $VF = \varphi^{-1}(\omega)$, for $\omega:=\varphi(u_1)/u_1 = u_0/\varphi^{-1}(u_0)\in \s_{\infty}$, and they provide our crystalline analogue of Hida's ordinary \'etale cohomology (see Theorem \ref{MainThmCrystal}):
\begin{theorem}
There is a canonical short exact sequence of finite free $\Lambda_{\s_{\infty}}$-modules
with linear $\H^*$-action, semilinear $\Gamma$-action, and semilinear
endomorphisms $F$, $V$ satisfying $FV=\omega$, $VF=\varphi^{-1}(\omega)$
\begin{equation}
\xymatrix{
0 \ar[r] & {\m_{\infty}^{\et}} \ar[r] & {\m_{\infty}} \ar[r] & {\m_{\infty}^{\mult}} \ar[r] & 0
}.\label{CrystallineAnalogue}
\end{equation}
Each of $\m_{\infty}^{\star}$ for $\star\in \{\et,\mult\}$ is free of rank $d'$ over
$\Lambda_{\s_{\infty}}$, while $\m_{\infty}$ is free of rank $2d'$, where
$d'$ is as in Theorem $\ref{DieudonneMainThm}$. Extending scalars on $(\ref{CrystallineAnalogue})$
along the canonical surjection
$\Lambda_{\s_{\infty}}\twoheadrightarrow \s_{\infty}[\Delta/\Delta_r]$ yields the short exact
sequence
\begin{equation*}
\xymatrix{
0 \ar[r] & {\m_r(\G_r^{\et})\tens_{\s_r} \s_{\infty}} \ar[r] &
{\m_r(\G_r)\tens_{\s_r} \s_{\infty}} \ar[r] &
{\m_r(\G_r^{\mult})\tens_{\s_r} \s_{\infty}} \ar[r] &
0
}
\end{equation*}
compatibly with $\H^*$, $\Gamma$, $F$ and $V$. The Frobenius endomorphism $F$
commutes with $\H^*$ and $\Gamma$, whereas the Verscheibung $V$ commutes with $\H^*$ and satisfies
$V\gamma = \varphi^{-1}(\omega/\gamma\omega)\cdot \gamma V$ for all $\gamma\in \Gamma$. \end{theorem}
Again, in the spirit of Theorem \ref{DDuality} and \cite[Proposition 3.2.4]{CaisHida1}, there is a corresponding ``autoduality" result for $\m_{\infty}$ (see Theorem \ref{CrystalDuality}). To state it, we must work over $\s_{\infty}':=\varinjlim_r \Z_p[\mu_N][\![u_r]\!]$, with the inductive limit taken along the $\Z_p$-algebra maps sending $u_r$ to $\varphi(u_{r+1})$.
\begin{theorem}
Let $\mu:\Gamma\rightarrow \Lambda_{\s_{\infty}}^{\times}$ be the crossed homomorphism
given by $\mu(\gamma):=\frac{u_1}{\gamma u_1}\chi(\gamma) \langle \chi(\gamma)\rangle$.
There is a canonical $\H^*$ and $\Gal(K_{\infty}'/K_0)$-compatible isomorphism of
exact sequences
\begin{equation*}
\begin{gathered}
\xymatrix{
0\ar[r] & {\m_{\infty}^{\et}(\mu \langle a\rangle_N)_{\Lambda_{\s_{\infty}'}}} \ar[r]\ar[d]_-{\simeq} &
{\m_{\infty}(\mu \langle a\rangle_N)_{\Lambda_{\s_{\infty}'}}} \ar[r]\ar[d]_-{\simeq} &
{\m_{\infty}^{\mult}(\mu \langle a\rangle_N)_{\Lambda_{\s_{\infty}'}}} \ar[r]\ar[d]_-{\simeq} & 0\\
0\ar[r] & {(\m_{\infty}^{\mult})_{\Lambda_{\s_{\infty}'}}^{\vee}} \ar[r] &
{(\m_{\infty})_{\Lambda_{\s_{\infty}'}}^{\vee}} \ar[r] &
{(\m_{\infty}^{\et})_{\Lambda_{\s_{\infty}'}}^{\vee}} \ar[r] & 0
}
\end{gathered}
\end{equation*}
intertwining $F$ and $V$ on the top row with
$V^{\vee}$ and $F^{\vee}$, respectively, on the bottom. The action
of $\Gal(K_{\infty}'/K_0)$ on the bottom row is the standard one
$\gamma\cdot f:=\gamma f\gamma^{-1}$ on linear duals. \end{theorem}
The $\Lambda_{\s_{\infty}}$-modules $\m_{\infty}^{\et}$ and $\m_{\infty}^{\mult}$ have a particularly simple structure (see Theorem \ref{etmultdescent}):
\begin{theorem}
There are canonical $\H^*$, $\Gamma$, $F$ and $V$-equivariant isomorphisms
of $\Lambda_{\s_{\infty}}$-modules
\begin{subequations}
\begin{equation}
\m_{\infty}^{\et} \simeq \D_{\infty}^{\et}\tens_{\Lambda} \Lambda_{\s_{\infty}},
\end{equation}
intertwining $F$ and $V$ with
$F\otimes \varphi$ and $F^{-1}\otimes \varphi^{-1}(\omega)\cdot \varphi^{-1}$, respectively,
and $\gamma\in \Gamma$
with $\gamma\otimes\gamma$, and
\begin{equation}
\m_{\infty}^{\mult}\simeq \D_{\infty}^{\mult}\tens_{\Lambda} \Lambda_{\s_{\infty}},
\end{equation}
intertwing $F$ and $V$ with $V^{-1} \otimes \omega \cdot\varphi$
and $V\otimes\varphi^{-1}$, respectively,
and $\gamma$ with $\gamma\otimes \chi(\gamma)^{-1} \gamma u_1/u_1$.
In particular, $F$ $($respectively $V$$)$
acts invertibly on $\m_{\infty}^{\et}$ $($respectively $\m_{\infty}^{\mult}$$)$. \end{subequations} \end{theorem}
From $\m_{\infty}$, we can recover $\D_{\infty}$ and ${e^*}'H^1_{\dR}$, with their additional structures (see Theorem \ref{SRecovery}): \begin{theorem}\label{MinftySpecialize}
Viewing $\Lambda$ as a $\Lambda_{\s_{\infty}}$-algebra via the map induced by $u_r\mapsto 0$,
there is a canonical isomorphism of short exact sequences of finite free $\Lambda$-modules
\begin{equation*}
\xymatrix{
0 \ar[r] & {\m_{\infty}^{\et}\tens_{\Lambda_{\s_{\infty}}} \Lambda}\ar[d]_-{\simeq} \ar[r] &
{\m_{\infty}\tens_{\Lambda_{\s_{\infty}}} \Lambda}\ar[r] \ar[d]_-{\simeq}&
{\m_{\infty}^{\mult}\tens_{\Lambda_{\s_{\infty}}} \Lambda} \ar[r]\ar[d]_-{\simeq} & 0\\
0 \ar[r] & {\D_{\infty}^{\et}} \ar[r] & {\D_{\infty}} \ar[r] &
{\D_{\infty}^{\mult}} \ar[r] & 0
}
\end{equation*}
which is $\Gamma$ and $\H^*$-equivariant and carries $F\otimes 1$ to $F$
and $V\otimes 1$ to $V$.
Viewing $\Lambda_{R_{\infty}}$ as a $\Lambda_{\s_{\infty}}$-algebra via the map
$u_r\mapsto (\varepsilon^{(r)})^p - 1$, there is a canonical
isomorphism of short exact sequences of
$\Lambda_{R_{\infty}}$-modules
\begin{equation*}
\xymatrix{
0 \ar[r] & {\m_{\infty}^{\et}\tens_{\Lambda_{\s_{\infty}}} \Lambda_{R_{\infty}}}
\ar[d]_-{\simeq} \ar[r] &
{\m_{\infty}\tens_{\Lambda_{\s_{\infty}}} \Lambda_{R_{\infty}}}\ar[r] \ar[d]_-{\simeq}&
{\m_{\infty}^{\mult}\tens_{\Lambda_{\s_{\infty}}} \Lambda_{R_{\infty}}} \ar[r]\ar[d]_-{\simeq} & 0\\
0 \ar[r] & {{e^*}'H^1(\O)} \ar[r]_{i} &
{{e^*}'H^1_{\dR}} \ar[r]_-{j} & {{e^*}'H^0(\omega)} \ar[r] & 0
}
\end{equation*}
that is $\Gamma$ and $\H^*$-equivariant,
where $i$ and $j$ the splittings
given by Theorem $\ref{dRtoDieudonne}$. \end{theorem}
To recover Hida's ordinary \'etale cohomology from $\m_{\infty}$, we introduce the ``period" ring of Fontaine\footnote{Though we use the notation introduced by Berger and Colmez.} $\wt{\e}^+:=\varprojlim \O_{\C_p}/(p)$, with the projective limit taken along the $p$-power mapping; this is a perfect valuation ring of characteristic $p$ equipped with a canonical action of $\scrG_{\Q_p}$ via ``coordinates". We write $\wt{\e}$ for the fraction field of $\wt{\e}^+$ and $\wt{\a}:=W(\wt{\e})$ for its ring of Witt vectors, equipped with its canonical Frobenius automorphism $\varphi$ and $\scrG_{\Q_p}$-action induced by Witt functoriality. Our fixed choice of $p$-power compatible sequence $\{\varepsilon^{(r)}\}$ determines an element $\u{\varepsilon}:=(\varepsilon^{(r)}\bmod p)_{r\ge 0}$ of $\wt{\e}^+$, and we $\Z_p$-linearly embed $\s_{\infty}$ in $\wt{\a}$ via $u_r\mapsto \varphi^{-r}([\u{\varepsilon}]-1)$ where $[\cdot]$ is the Teichm\"uller section. This embedding is $\varphi$ and $\scrG_{\Q_p}$-compatible, with $\scrG_{\Q_p}$ acting on $\s_{\infty}$ through the quotient $\scrG_{\Q_p}\twoheadrightarrow \Gamma$.
\begin{theorem} \label{RecoverEtale}
Twisting the structure map $\s_{\infty}\rightarrow \wt{\a}$ by the Frobenius automorphism $\varphi$,
there is a canonical isomorphism of short exact sequences of $\Lambda_{\wt{\a}}$-modules
with $\H^*$-action
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] & {\m_{\infty}^{\et}\tens_{\Lambda_{\s_{\infty}},\varphi} \Lambda_{\wt{\a}}}
\ar[d]_-{\simeq} \ar[r] &
{\m_{\infty}\tens_{\Lambda_{\s_{\infty}},\varphi} \Lambda_{\wt{\a}}}\ar[r] \ar[d]_-{\simeq}&
{\m_{\infty}^{\mult}\tens_{\Lambda_{\s_{\infty}},\varphi} \Lambda_{\wt{\a}}} \ar[r]\ar[d]_-{\simeq} & 0\\
0 \ar[r] & {({e^*}'H^1_{\et})^{\I}\tens_{\Lambda} \Lambda_{\wt{\a}}} \ar[r] &
{{e^*}'H^1_{\et}\tens_{\Lambda} \Lambda_{\wt{\a}}} \ar[r] &
({e^*}'H^1_{\et})_{\I}\tens_{\Lambda} \Lambda_{\wt{\a}}\ar[r] & 0
}\label{FinalComparisonIsom}
\end{gathered}
\end{equation}
that is $\scrG_{\Q_p}$-equivariant for the ``diagonal" action of $\scrG_{\Q_p}$
$($with $\scrG_{\Q_p}$ acting on $\m_{\infty}$ through $\Gamma$$)$
and intertwines
$F\otimes \varphi$ with $\id\otimes\varphi$ and $V\otimes\varphi^{-1}$ with $\id\otimes \omega\cdot\varphi^{-1}$.
In particular, there is a canonical isomorphism of $\Lambda$-modules, compatible
with the actions of $\H^*$ and $\scrG_{\Q_p}$,
\begin{equation}
{e^*}'H^1_{\et} \simeq \left( \m_{\infty}\tens_{\Lambda_{\s_{\infty}},\varphi}
\Lambda_{\wt{\a}}\right )^{F\otimes\varphi = 1}.\label{RecoverEtaleIsom}
\end{equation} \end{theorem}
Theorem \ref{RecoverEtale} allows us to give a new proof of Hida's finiteness and control theorems for ${e^*}'H^1_{\et}$:
\begin{corollary}[Hida]\label{HidasThm}
Let $d'$ be as in Theorem $\ref{DieudonneMainThm}$. Then
${e^*}'H^1_{\et}$ is free $\Lambda$-module of rank $2d'$.
For each $r\ge 1$ there is a canonical isomorphism of $\Z_p[\Delta/\Delta_r]$-modules
with linear $\H^*$ and $\scrG_{\Q_p}$-actions
\begin{equation*}
{e^*}'H^1_{\et} \tens_{\Lambda} \Z_p[\Delta/\Delta_r] \simeq {e^*}'H^1_{\et}({X_r}_{\Qbar_p},\Z_p)
\end{equation*}
which is moreover compatible with the isomorphisms $(\ref{HidaResultSub})$ and $(\ref{HidaResultQuo})$
in the evident manner. \end{corollary}
We also deduce a new proof of the following duality result \cite[Theorem 4.3.1]{OhtaEichler} ({\em cf.} \cite[\S6]{MW-Hida}):
\begin{corollary}[Ohta]\label{OhtaDuality}
Let $\nu:\scrG_{\Q_p}\rightarrow \H^*$ be the
character $\nu:=\chi\langle \chi\rangle \lambda(\langle p\rangle_N)$.
There is a canonical $\H^*$ and $\scrG_{\Q_p}$-equivariant isomorphism
of short exact sequences of $\Lambda$-modules
\begin{equation*}
\xymatrix{
0 \ar[r] & {({e^*}'H^1_{\et})^{\I}(\nu)}
\ar[d]^-{\simeq} \ar[r] &
{{e^*}'H^1_{\et}(\nu)}\ar[d]^-{\simeq} \ar[r] &
{({e^*}'H^1_{\et})_{\I}(\nu)}
\ar[d]^-{\simeq}\ar[r] & 0 \\
0 \ar[r] & {\Hom_{\Lambda}(({e^*}'H^1_{\et})_{\I},\Lambda)} \ar[r] &
{\Hom_{\Lambda}({e^*}'H^1_{\et},\Lambda)} \ar[r] &
{\Hom_{\Lambda}(({e^*}'H^1_{\et})^{\I},\Lambda)}\ar[r] & 0
}
\end{equation*} \end{corollary}
The $\Lambda$-adic splitting of the ordinary filtration of $e^*H^1_{\et}$ was considered by Ghate and Vatsal \cite{GhateVatsal}, who prove (under certain technical hypotheses of ``deformation-theoretic nature") that if the $\Lambda$-adic family $\scrF$ associated to a cuspidal eigenform $f$ is primitive and $p$-distinguished, then the associated $\Lambda$-adic local Galois representation $\rho_{\scrF,p}$ is split if and only if some arithmetic specialization of $\scrF$ has CM \cite[Theorem 13]{GhateVatsal}. We interpret the $\Lambda$-adic splitting of the ordinary filtration as follows:
\begin{theorem}\label{SplittingCriterion}
The short exact sequence $(\ref{CrystallineAnalogue})$ admits
a $\Lambda_{\s_{\infty}}$-linear splitting which is compatible with $F$, $V$,
and $\Gamma$ if and only if the ordinary filtration of ${e^*}'H^1_{\et}$
admits a $\Lambda$-linear spitting which is compatible with the action of $\scrG_{\Q_p}$. \end{theorem}
\subsection{Overview of the article}\label{Overview}
Section \ref{Prelim} is preliminary: we first review in \S\ref{DDR}--\ref{Universal} some background material on Dieudonn\'e modules and crystals, as well as the integral $p$-adic cohomology theories of \cite{CaisDualizing} and \cite{CaisNeron}. In \S\ref{PhiGammaCrystals}, we summarize the theory developed in \cite{CaisLau}, which uses Dieudonn\'e crystals of $p$-divisible groups to provide a ``cohomological" construction of the $(\varphi,\Gamma)$-modules attached to potentially Barsotti--Tate representations. We then specialize these results to the case of ordinary $p$-divisible groups in \S\ref{pDivOrdSection}; it is precisely this theory which allows us to construct our crystalline analogue of Hida's ordinary $\Lambda$-adic \'etale cohomology.
Section \ref{results} constitutes the main body of this paper, and the reader who is content to refer back to \S\ref{DDR}--\ref{pDivOrdSection} as needed should skip directly there.
In section \ref{BTfamily}, we study the tower of $p$-divisible groups whose cohomology allows us to construct our $\Lambda$-adic Dieudonn\'e and crystalline analogues of Hida's \'etale cohomlogy in \S\ref{OrdDieuSection} and \S\ref{OrdSigmaSection}, respectively. We establish $\Lambda$-adic comparison isomorphisms between each of these cohomologies using the integral comparison isomorphisms of \cite{CaisNeron} and \cite{CaisLau}, recalled in \S\ref{Universal} and \S\ref{PhiGammaCrystals}--\ref{pDivOrdSection}, respectively. This enables us to give a new proof of Hida's freeness and control theorems and of Ohta's duality theorem in \S\ref{OrdSigmaSection}. A key technical ingredient in our proofs is the commutative algebra formalism developed in \cite[\S3.1]{CaisHida1} for dealing with projective limits of cohomology and establishing appropriate ``freeness and control" theorems by reduction to characteristic $p$.
As remarked in \S\ref{resultsintro}, and following \cite{OhtaEichler} and \cite{MW-Hida}, our construction of the $\Lambda$-adic Dieudonn\'e and crystalline counterparts to Hida's \'etale cohomology excludes the trivial eigenspace for the action of $\mu_{p-1}\subseteq \Z_p^{\times}$ so as to avoid technical complications with logarithmic $p$-divisible groups. In \cite{Ohta2}, Ohta uses the ``fixed part" (in the sense of Grothendieck \cite[2.2.3]{GroModeles}) of N\'eron models with semiabelian reduction to extend his results on $\Lambda$-adic Hodge cohomology to allow trivial tame nebentype character. We are confident that by using Kato's logarithmic Dieudonn\'e theory \cite{KatoDegen} one can appropriately generalize our results in \S\ref{OrdDieuSection} and \S\ref{OrdSigmaSection} to include the missing eigenspace for the action of $\mu_{p-1}$.
\subsection{Notation}\label{Notation}
If $\varphi:A\rightarrow B$ is any map of rings, we will often write $M_B:=M\otimes_{A} B$ for the $B$-module induced from an $A$-module $M$ by extension of scalars. When we wish to specify $\varphi$, we will write $M\otimes_{A,\varphi} B$. Likewise, if $\varphi:T'\rightarrow T$ is any morphism of schemes, for any $T$-scheme $X$ we denote by $X_{T'}$ the base change of $X$ along $\varphi$. If $f:X\rightarrow Y$ is any morphism of $T$-schemes, we will write $f_{T'}: X_{T'}\rightarrow Y_{T'}$ for the morphism of $T'$-schemes obtained from $f$ by base change along $\varphi$. When $T=\Spec(R)$ and $T'=\Spec(R')$ are affine, we abuse notation and write $X_{R'}$ or $X\times_{R} R'$ for $X_{T'}$. We frequently work with schemes over a discrete valuation ring $R$, and will write $\X,\Y,\ldots$ for schemes over $\Spec(R)$, reserving $X,Y,\ldots$ (respectively $\o{\X},\o{\Y},\ldots$) for their generic (respectively special) fibers. As this article is a continuation of \cite{CaisHida1}, we will freely use the notation and conventions therein.
\section{Dieudonn\'e Crystals and Dieudonn\'{e} Modules}\label{Prelim}
This section is devoted to recalling the geometric background we will need in our constructions. Much (though not all) of this material is contained in \cite{CaisDualizing}, \cite{CaisNeron}, and \cite{CaisLau}.
\subsection{Dieudonn\'e modules and {d}e~Rham cohomology}\label{DDR}
Let $k$ be a perfect field of characteristic $p$ and $X$ a smooth and proper curve over $k$. We begin by recalling the relation between the de Rham cohomology of $X$ over $k$ and the Dieudonn\'e module of the $p$-divisible group of the Jacobian of $X$.
Let us write $H(X/k)$ for the three-term ``Hodge filtration" exact sequence \begin{equation*}
\xymatrix{
0\ar[r] & {H^0(X,\Omega^1_{X/k})} \ar[r] & {H^1_{\dR}(X/k)} \ar[r] & {H^1(X,\O_X)}\ar[r] & 0.
} \end{equation*} Pullback by the absolute Frobenius gives an endomorphism of $F:H(X/k)\rightarrow H(X/k)$ that is semilinear over the $p$-power Frobenius automorphism $\varphi$ of $k$. Under the canonical cup-product autoduality of $H(X/k)$, we obtain $\varphi^{-1}$-semilinear endomorphism \begin{equation}
\xymatrix{
{V:={F}_*: H^1_{\dR}(X/k)} \ar[r] & {H^1_{\dR}(X/k)}
}\label{CartierOndR} \end{equation} whose restriction to $H^0(X,\Omega^1_{X/k})$ coincides with the Cartier operator \cite[\S2.3]{CaisHida1}. Let $A$ be the ``Dieudonn\'e ring", {\em i.e.}~the (noncommutative if $k\neq \F_p$) ring $A:=W(k)[F,V]$, where
$F$, $V$ satisfy $FV=VF=p$, $F\alpha=\varphi(\alpha)F$, and $V\alpha=\varphi^{-1}(\alpha)V$ for all $\alpha\in W(k)$. We view $H^1_{\dR}(X/k)$ as a left $A$-module in the obvious way. By Fitting's Lemma \cite[Lemma 2.3.3]{CaisHida1}, for $f=F$ or $V$, any finite left $A$-module $M$ admits a canonical direct sum decomposition \begin{equation}
M=M^{f_{\ord}}\oplus M^{f_{\nil}}
\label{ordnotation} \end{equation} where $M^{f_{\ord}}$ (respectively $M^{f_{\nil}}$) is the maximal $A$-submodule of $M$ on which $f$ is bijective (respectively $p$-adically topologically nilpotent).
\begin{proposition}[Oda]\label{OdaDieudonne}
Let $J:=\Pic^0_{X/k}$ be the Jacobian of $X$ over $k$ and $G:=J[p^{\infty}]$
its $p$-divisible group. Denote by $\D(\cdot)$ the contravariant
Dieudonn\'e crystal $($see $(\ref{DieudonneDef})$ below$)$, so the Dieudonn\'e module $\D(G)_W$ is naturally a left $A$-module, finite and
free over $W:=W(k)$.
\begin{enumerate}
\item \label{OdaIsom} There are canonical isomorphisms of left $A$-modules
\begin{equation*}
H^1_{\dR}(X/k)\simeq \D(J[p])_{k}\simeq \D(G)_k.
\end{equation*}
\item For any finite morphism $\rho:Y\rightarrow X$ of smooth and proper curves
over $k$, the identification of $(\ref{OdaIsom})$ intertwines $\rho_*$
with $\D(\Pic^0(\rho))$ and $\rho^*$ with $\D(\Alb(\rho))$.\label{OdaIsomFunctoriality}
\item \label{GetaleGmult} Let $G=G^{\et}\times G^{\mult}\times G^{\loc}$
be the canonical direct product decomposition of $G$ into its maximal \'etale,
multiplicative, and local-local subgroups.
Via the identification of $(\ref{OdaIsom})$, the canonical mappings
in the exact sequence $H(X/k)$ induce natural isomorphisms of left $A$-modules
\begin{equation*}
H^0(X,\Omega^1_{X/k})^{V_{\ord}} \simeq \D(G^{\mult})_k
\quad\text{and}\quad
H^1(X,\O_X)^{F_{\ord}} \simeq \D(G^{\et})_k
\end{equation*}
\item The isomorphisms of $(\ref{GetaleGmult})$ are dual to each other, using the
perfect duality on cohomology induced by the cup-product pairing \cite[Remark 2.3.3]{CaisHida1}
and the identification $\D(G)_k^t\simeq \D(G)_k$
resulting from the compatibility of $\D(\cdot)_k$ with duality and
the autoduality of $J$.\label{BBMDuality}
\end{enumerate} \end{proposition}
\begin{proof}
Using the characterizing properties of the Cartier operator
defined by Oda \cite[Definition 5.5]{Oda} and the explicit
description of the autoduality of $H^1_{\dR}(X/k)$ in terms
of cup-product and residues, one checks that the endomorphism
of $H^1_{\dR}(X/k)$ in \cite[Definition 5.6]{Oda} is adjoint
to $F^*$, and therefore coincides with the endomorphism
$V:={F}_*$ in (\ref{CartierOndR}); {\em cf.}
the proof of \cite[Proposition 9]{SerreTopology}.
We recall that one has a canonical isomorphism
\begin{equation}
H^1_{\dR}(X/k)\simeq H^1_{\dR}(J/k)\label{dRIdenJac}
\end{equation}
which is compatible with Hodge filtrations and duality (using the canonical
principal polarization to identify $J$ with its dual) and which, for
any finite morphism of smooth curves $\rho:Y\rightarrow X$ over $k$,
intertwines $\rho_*$ with $\Pic^0(\rho)^*$ and $\rho^*$ with $\Alb(\rho)^*$; see
\cite[Proposition 5.4]{CaisNeron}, noting that the proof given there works over any field $k$,
and {\em cf.}~Proposition \ref{intcompare}. It follows from these compatibilities
and the fact that the Cartier operator as defined in \cite[Definition 5.5]{Oda} is functorial
that the identification (\ref{dRIdenJac}) is moreover an isomorphism of left $A$-modules,
with the $A$-structure on $H^1_{\dR}(J/k)$ defined as in \cite[Definition 5.8]{Oda}.
Now by \cite[Corollary 5.11]{Oda} and \cite[Theorem 4.2.14]{BBM},
for any abelian variety $B$ over $k$,
there is a canonical isomorphism of left $A$-modules
\begin{equation}
H^1_{\dR}(B/k)\simeq \D(B)_k\label{AbVarDieuMod}
\end{equation}
Using the definition of this isomorphism in Proposition 4.2 and Theorem 5.10 of \cite{Oda},
it is straightforward (albeit tedious\footnote{Alternately,
one could appeal to
\cite{MM}, specifically to Chapter I, 4.1.7, 4.2.1, 3.2.3, 2.6.7
and to Chapter II, \S13 and \S15 (see especially Chapter II, 13.4 and 1.6).
See also \S2.5 and \S4 of \cite{BBM}.
})
to check that for any homomorphism $h:B'\rightarrow B$
of abelian varieties over $k$, the identification (\ref{AbVarDieuMod}) intertwines $h^*$ and $\D(h)$.
Combining (\ref{dRIdenJac}) and (\ref{AbVarDieuMod})
yields (\ref{OdaIsom}) and (\ref{OdaIsomFunctoriality}).
Now since $V={F}_*$ (respectively $F=F^*$) is the zero endomorphism of $H^1(X,\O_X)$
(respectively $H^0(X,\O_X)$), the canonical mapping
\begin{equation*}
\xymatrix{
{H^0(X,\Omega^1_{X/k})} \ar@{^{(}->}[r] & {H^1_{\dR}(X/k)\simeq \D(G)_k}
}
\quad\text{respectively}\quad
\xymatrix{
{\D(G)_k\simeq H^1_{\dR}(X/k)} \ar@{->>}[r] & {H^1(X,\O_X)}
}
\end{equation*}
induces an isomorphism on $V$-ordinary (respectively $F$-ordinary) subspaces.
On the other hand, by Dieudonn\'e theory one knows that
for {\em any} $p$-divisible group $H$, the semilinear endomorphism $V$
(respectively $F$) of $\D(H)_W$
is bijective if and only if $H$ is of multiplicative type (respectively \'etale).
The (functorial) decomposition $G=G^{\et}\times G^{\mult}\times G^{\loc}$
yields a natural isomorphism of left $A$-modules
\begin{equation*}
\D(G)_W\simeq \D(G^{\et})_W\oplus \D(G^{\mult})_W\oplus \D(G^{\loc})_W,
\end{equation*}
and it follows that the natural maps $\D(G^{\mult})_W\rightarrow \D(G)_W$,
$\D(G)_W\rightarrow \D(G^{\et})_W$ induce isomorphisms
\begin{equation}
\D(G^{\mult})_W \simeq \D(G)^{V_{\ord}}_W\quad\text{and}\quad
\D(G)^{F_{\ord}}_W\simeq \D(G^{\et})_W,\label{VordMultFordEt}
\end{equation}
respectively, which gives (\ref{GetaleGmult}). Finally, (\ref{BBMDuality})
follows from Proposition 5.3.13 and the proof of Theorem 5.1.8 in \cite{BBM},
using Proposition 2.5.8 of {\em op.~cit.}~and the compatibility of the isomorphism
(\ref{dRIdenJac}) with duality (for which see
\cite[Theorem 5.1]{colemanduality} and {\em cf.} \cite[Lemma 5.5]{CaisNeron}). \end{proof}
\subsection{Universal vectorial extensions}\label{Universal}
We now study the mixed characteristic analogue of the situation considered in \S\ref{DDR}. Fix a discrete valuation ring $R$ with field of fractions $K$ of characteristic zero and perfect residue field $k$ of characteristic $p$. Recall \cite[\S2.1]{CaisHida1} that by a {\em curve} over $S:=\Spec R$ we mean a flat finitly presented local complete intersection $f:X\rightarrow S$ of relative dimension one with geometrically reduced fibers. Let $f:X\rightarrow S$ be a normal and proper curve over $S$ with smooth and geometrically connected generic fiber $X_K$, and write
$\omega_{X/S}$ for the relative dualizing sheaf of $f$. The hypercohomology $H^i(X/R)$ of the two-term complex $\O_X\rightarrow \omega_{X/S}$ provides a canonical integral structure on the algebraic de Rham cohomology of the generic fiber $X_K$:
\begin{proposition}[{\cite[2.1.11]{CaisHida1}}]\label{HodgeIntEx}
Let $f:X\rightarrow S$ be a normal curve that is proper over $S=\Spec(R)$.
There is a canonical short exact sequence of finite free $R$-modules,
which we denote $H(X/R)$,
\begin{equation*}
\xymatrix{
0\ar[r] & {H^0(X,\omega_{X/S})} \ar[r] & {H^1(X/R)} \ar[r] & {H^1(X,\O_X)} \ar[r] & 0
}
\end{equation*}
that recovers the Hodge filtration of $H^1_{\dR}(X_K/K)$ after
extending scalars to $K$ and is canonically $R$-linearly self-dual
via the cup-product pairing on $H^1_{\dR}(X_K/K)$.
The exact sequence $H(X/R)$ is functorial in finite morphisms $\rho:Y\rightarrow X$
of normal and proper $S$-curves via pullback $\rho^*$ and trace $\rho_*$; these
morphisms recover the usual pullback and trace mappings on Hodge filtrations after extending scalars
to $K$ and are adjoint with respect to the cup-product autoduality of $H(X/R)$. \end{proposition}
There is an alternate description of the short exact sequence $H(X/R)$ of Proposition \ref{HodgeIntEx} in terms of Lie algebras and N\'eron models of Jacobians that will allow us to relate this cohomology to Dieudonn\'e modules. To explain this description and its connection with crystals, we first recall some facts from \cite{MM} and \cite{CaisNeron}.
Fix a base scheme $T$, and let $G$ be an fppf sheaf of abelian groups over $T$. A {\em vectorial extension} of $G$ is a short exact sequence (of fppf sheaves of abelian groups) \begin{equation}
\xymatrix{
0 \ar[r] & {V} \ar[r] & {E} \ar[r] & {G} \ar[r] & 0.
}\label{extension} \end{equation} with $V$ a vector group (i.e. an fppf abelian sheaf which is locally represented by a product of $\Ga$'s). Assuming that $\Hom(G,V)=0$ for all vector groups $V$, we say that a vectorial extension (\ref{extension}) is {\em universal} if, for any vector group $V'$ over $T$, the pushout map $\Hom_T(V,V')\rightarrow \Ext^1_T(G,V')$ is an isomorphism. When a universal vectorial extension of $G$ exists, it is unique up to canonical isomorphism and covariantly functorial in morphisms $G'\rightarrow G$ with $G'$ admitting a universal extension.
\begin{theorem}\label{UniExtCompat}
Let $T$ be an arbitrary base scheme.
\begin{enumerate}
\item If $A$ is an abelian scheme over $T$, then a universal vectorial
extension $\E(A)$ of $A$ exists, with $V=\omega_{\Dual{A}}$,
and is compatible with arbitrary base change on $T$.
\label{UniExtCompat1}
\item If $p$ is locally nilpotent on $T$ and $G$ is a $p$-divisible group over
$T$, then a universal vectorial extension $\E(G)$ of $G$ extsis, with $V=\omega_{\Dual{G}}$,
and is compatible with arbitrary base change on $T$.\label{UniExtCompat2}
\item If $p$ is locally nilpotent on $T$ and $A$ is an abelian scheme over $T$ with
associated $p$-divisible group $G:=A[p^{\infty}]$, then the canonical map of fppf sheaves
$G\rightarrow A$ extends to a natural map
\begin{equation*}
\xymatrix{
0 \ar[r] & {\omega_{\Dual{G}}} \ar[r]\ar[d] & {\E(G)} \ar[r]\ar[d] & {G}\ar[d] \ar[r] & 0\\
0 \ar[r] & {\omega_{\Dual{A}}} \ar[r] & {\E(A)} \ar[r] & {A} \ar[r] & 0
}
\end{equation*}
which induces an isomorphism of the corresponding short exact sequences of Lie algebras.
\label{UniExtCompat3}
\end{enumerate} \end{theorem}
\begin{proof}
For the proofs of (\ref{UniExtCompat1}) and (\ref{UniExtCompat2}), see
\cite[\Rmnum{1}, \S1.8 and \S1.9]{MM}. To prove (\ref{UniExtCompat3}), note that
pulling back the universal vectorial extension of $A$ along $G\rightarrow A$
gives a vectorial extension $\E'$ of $G$ by $\omega_{\Dual{A}}$. By universality, there then exists
a unique map $\psi:\omega_{\Dual{G}}\rightarrow \omega_{\Dual{A}}$ with the property
that the pushout of $\E(G)$ along $\psi$ is $\E'$, and this gives the map on universal extensions.
That the induced map on Lie algebras is an isomorphism follows from \cite[\Rmnum{2}, \S 13]{MM}. \end{proof}
For our applications, we will need a generalization of the universal extension of an abelian scheme to the setting of N\'eron models; in order to describe this generalization, we first recall the explicit description of the universal extension of an abelian scheme in terms of rigidified extensions.
For any commutative $T$-group scheme $F$, a {\em rigidified extension of $F$ by $\Gm$ over $T$} is a pair $(E,\sigma)$ consisting of an extension (of fppf abelian sheaves) \begin{equation}
\xymatrix{
0 \ar[r] & {\Gm} \ar[r] & {E} \ar[r] & {F} \ar[r] & 0
}\label{ExtRigDef} \end{equation} and a splitting $\sigma: \Inf^1(F)\rightarrow E$ of the pullback of (\ref{ExtRigDef}) along the canonical closed immersion $\Inf^1(F)\rightarrow F$. Two rigidified extensions $(E,\sigma)$ and $(E',\sigma')$ are equivalent if there is a group homomorphism $E\rightarrow E'$ carrying $\sigma$ to $\sigma'$ and inducing the identity on $\Gm$ and on $F$. The set $\Extrig_T(F,\Gm)$ of equivalence classes of rigidified extensions over $T$ is naturally a group via Baer sum of rigidified extensions\cite[\Rmnum{1}, \S2.1]{MM}, so the functor on $T$-schemes $T'\rightsquigarrow \Extrig_{T'}(F_{T'},\Gm)$ is naturally a group functor that is contravariant in $F$ via pullback (fibered product). We write $\scrExtrig_T(F,\Gm)$ for the fppf sheaf of abelian groups associated to this functor.
\begin{proposition}[Mazur-Messing]\label{MMrep}
Let $A$ be an abelian scheme over an arbitrary base scheme $T$.
The fppf sheaf $\scrExtrig_T(A,\Gm)$ is represented by a smooth and separated $T$-group scheme,
and there is a canonical short exact sequence of smooth group schemes over $T$
\begin{equation}
\xymatrix{
0\ar[r] & {\omega_A} \ar[r] & {\scrExtrig_T(A,\Gm)} \ar[r] & {\Dual{A}} \ar[r] & 0
}.\label{univextabelian}
\end{equation}
Furthermore, $(\ref{univextabelian})$ is naturally isomorphic to the universal extension of $\Dual{A}$ by a vector
group. \end{proposition}
\begin{proof}
See \cite{MM}, $\Rmnum{1}, \S2.6$ and Proposition 2.6.7. \end{proof}
In the case that $T=\Spec R$ for $R$ a discrete valuation ring of mixed characteristic $(0,p)$ with fraction field $K$, we have the following genaralization of Proposition \ref{MMrep}:
\begin{proposition}
Let $A$ be an abelian variety over $K$, with dual abelian variety $\Dual{A}$, and
write $\A$ and $\Dual{\A}$ for the N\'eron models of $A$ and $\Dual{A}$ over $T=\Spec(R)$.
Then the fppf abelian sheaf $\scrExtrig_T(\A,\Gm)$ on the category of smooth $T$-schemes
is represented by a smooth and separated $T$-group scheme. Moreover, there
is a canonical short exact sequence of smooth group schemes over $T$
\begin{equation}
\xymatrix{
0\ar[r] & {\omega_{\A}} \ar[r] & {\scrExtrig_T(\A,\Gm)} \ar[r] & {\Dual{\A}^0} \ar[r] & 0
}\label{NeronCanExt}
\end{equation}
which is contravariantly functorial in $A$ via homomorphisms of abelian varieties over $K$.
The formation of $(\ref{NeronCanExt})$ is compatible with smooth base change on $T$; in particular,
the generic fiber of $(\ref{NeronCanExt})$ is the universal extension of $\Dual{A}$ by a vector group. \end{proposition}
\begin{proof}
Since $R$ is of mixed characteristic $(0,p)$ with perfect residue field,
this follows from Proposition 2.6 and the discussion following Remark 2.9 in \cite{CaisNeron}. \end{proof}
In the particular case that $A$ is the Jacobian of a smooth, proper and geometrically connected curve $X$ over $K$ which is the generic fiber of a normal proper curve $\X$ over $R$, we can relate the exact sequence of Lie algebras attached to (\ref{NeronCanExt}) to the exact sequence $H(X/R)$ of Proposition \ref{HodgeIntEx}:
\begin{proposition} \label{intcompare}
Let $\X$ be a proper relative curve over $T=\Spec(R)$ with smooth generic fiber $X$ over $K$.
Write $J:=\Pic^0_{X/K}$ for the Jacobian of $X$ and $\Dual{J}$ for its dual,
and let $\J$, $\Dual{\J}$ be the corresponding N\'eron models over $R$.
There is a canonical homomorphism of exact sequences of finite free $R$-modules
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] & {\Lie\omega_{\J}} \ar[r]\ar[d] & {\Lie\scrExtrig_T(\J,\Gm)} \ar[r]\ar[d]
& {\Lie \Dual{\J}^0} \ar[r]\ar[d] & 0\\
0 \ar[r] & {H^0(\X,\omega_{\X/T})} \ar[r] & {H^1(\X/R)} \ar[r] & {H^1(\X,\O_{\X})}
\ar[r] & 0
}
\end{gathered}\label{IntegralComparisonMap}
\end{equation}
that is an isomorphism when $\X$ has rational singularities.\footnote{Recall that $\X$ is
said to have {\em rational singularities} if it admits a resolution of singularities
$\rho:\X'\rightarrow \X$ with the natural map $R^1\rho_*\O_{{\X'}}=0$. Trivially, any
regular $\X$ has rational singularities.}
For any finite morphism $\rho:\Y \rightarrow \X$ of $S$-curves satisfying the above hypotheses,
the map $(\ref{IntegralComparisonMap})$ intertwines $\rho_*$
$($respectively $\rho^*$$)$ on the bottom row with $\Pic(\rho)^*$
$($respectively $\Alb(\rho)^*$$)$ on the top. \end{proposition}
\begin{proof}
See Theorem 1.2 and (the proof of) Corollary 5.6 in \cite{CaisNeron}. \end{proof}
\begin{remark}\label{canonicalproperty}
Let $X$ be a smooth and geometrically connected curve over $K$
admitting a normal proper model $\X$ over $R$ that is a curve
having rational singularities.
It follows from Proposition \ref{intcompare}
and the N\'eron mapping property
that $H(\X/R)$ is a {\em canonical integral structure}
on the Hodge filtration of $H^1_{\dR}(X/K)$: it is
independent of the choice of proper model $\X$ that is normal with rational singularities,
and
is contravariantly (respectively covariantly) functorial by pullback
(respectively trace) in finite morphisms $\rho:Y\rightarrow X$
of proper smooth curves over $K$ which admit models over $R$ satisfying these hypotheses.
These facts can be proved in greater generality by appealing to resolution of singularities
for excellent surfaces and the flattening techniques of Raynaud--Gruson \cite{RayGrus};
see \cite[Theorem 5.11]{CaisDualizing} for details. \end{remark}
Finally, we will need to relate the universal extension of a $p$-divisible group as in Theorem \ref{UniExtCompat} (\ref{UniExtCompat2}) to its Dieudonn\'e crystal. In order to explain how this goes, we begin by recalling some basic facts from crystalline Dieudonn\'e theory, as discussed in \cite{BBM}.
Fix a perfect field $k$ and set $\Sigma:=\Spec(W(k))$, considered as a PD-scheme via the canonical divided powers on the ideal $pW(k)$. Let $T$ be a $\Sigma$-scheme on which $p$ is locally nilpotent (so $T$ is naturally a PD-scheme over $\Sigma$), and denote by $\Cris(T/\Sigma)$ the big crystalline site of $T$ over $\Sigma$, endowed with the {\em fppf} topology (see \cite[\S 2.2]{BBM1}). If $\scrF$ is a sheaf on $\Cris(T/\Sigma)$ and $T'$ is any PD-thickening of $T$, we write $\scrF_{T'}$ for the associated {\em fppf} sheaf on $T'$. As usual, we denote by $i_{T/\Sigma}:T_{fppf}\rightarrow (T/\Sigma)_{\Cris}$ the canonical morphism of topoi, and we abbreviate $\underline{G}:={i_{T/\Sigma}}_{*}G$ for any fppf sheaf $G$ on $T$.
Let $G$ be a $p$-divisible group over $T$, considered as an fppf abelian sheaf on $T$. As in \cite{BBM}, we define the (contravariant) {\em Dieudonn\'e crystal of $G$ over $T$} to be \begin{equation}
\D(G) := \scrExt^1_{T/\Sigma}(\underline{G},\O_{T/\Sigma}).\label{DieudonneDef} \end{equation} It is a locally free crystal in $\O_{T/\Sigma}$-modules, which is contravariantly functorial in $G$ and of formation compatible with base change along PD-morphisms $T'\rightarrow T$ of $\Sigma$-schemes thanks to 2.3.6.2 and Proposition 2.4.5 $(\rmnum{2})$ of \cite{BBM}. If $T'=\Spec(A)$ is affine, we will simply write $\D(G)_A$ for the finite locally free $A$-module associated to $\D(G)_{T'}$.
The structure sheaf $\O_{T/\Sigma}$ is canonically an extension of $\u{\mathbf{G}}_a$ by the PD-ideal $\J_{T/\Sigma}\subseteq \O_{T/\Sigma}$, and by applying $\scrHom_{T/\Sigma}(\underline{G},\cdot)$ to this extension one obtains (see Propositions 3.3.2 and 3.3.4 as well as Corollaire 3.3.5 of \cite{BBM}) a short exact sequence (the {\em Hodge filtration}) \begin{equation}
\xymatrix{
0\ar[r] & {\scrExt^1_{T/\Sigma}(\underline{G},\J_{T/\Sigma})}\ar[r] &
{\D(G)}\ar[r] &
{\scrExt^1_{T/\Sigma}(\underline{G},\u{\mathbf{G}}_a)}\ar[r] & 0
}\label{HodgeFilCrys} \end{equation} that is contravariantly functorial in $G$ and of formation compatible with base change along PD-morphisms $T'\rightarrow T$ of $\Sigma$-schemes. The following ``geometric" description of the value of (\ref{HodgeFilCrys}) on a PD-thickening of the base will be essential for our purposes:
\begin{proposition}\label{BTgroupUnivExt}
Let $G$ be a fixed $p$-divisible group over $T$ and let $T'$ be any
$\Sigma$-PD thickening of $T$. If $G'$ is any lifting of $G$ to a $p$-divisible
group on $T'$, then there is a natural isomorphism
\begin{equation*}
\xymatrix{
0 \ar[r] & {\omega_{G'}} \ar[r]\ar[d]^-{\simeq} & {\scrLie(\E(\Dual{G'}))} \ar[r]\ar[d]^-{\simeq} &
{\scrLie (\Dual{G'})}\ar[r]\ar[d]^-{\simeq} & 0\\
0\ar[r] & {\scrExt^1_{T/\Sigma}(\underline{G},\J_{T/\Sigma})_{T'}}\ar[r] & {\D(G)_{T'}}\ar[r] &
{\scrExt^1_{T/\Sigma}(\underline{G},\underline{\mathbf{G}}_a)_{T'}}\ar[r] & 0
}
\end{equation*}
that is moreover compatible with base change in the evident manner. \end{proposition}
\begin{proof}
See \cite[Corollaire 3.3.5]{BBM} and \cite[\Rmnum{2}, Corollary 7.13]{MM}. \end{proof}
\begin{remark}\label{MessingRem}
In his thesis \cite{Messing}, Messing showed that the Lie algebra of the universal extension
of $\Dual{G}$ is ``crystalline in nature" and used this as the {\em definition}\footnote{Noting
that it suffices to define the crystal $\D(G)$ on $\Sigma$-PD thickenings $T'$
of $T$ to which $G$ admits a lift.} of $\D(G)$. (See chapter $\Rmnum{4}$, \S2.5 of \cite{Messing} and especially 2.5.2). Although we
prefer the more intrinsic description (\ref{DieudonneDef}) of
\cite{MM} and \cite{BBM}, it is ultimately Messing's original
definition that will be important for us. \end{remark}
\subsection{Dieudonn\'e crystals and \texorpdfstring{$(\varphi,\Gamma)$}{(phi,Gamma)}-modules}\label{PhiGammaCrystals}
In this section, we summarize the main results of \cite{CaisLau}, which provides a classification of $p$-divisible groups by certain semi-linear algebra structures. These structures---which arise naturally via the Dieudonn\'e crystal functor--- are cyclotomic analogues of Breuil and Kisin modules, and are closely related to Wach modules.\footnote{See \cite{CaisLau} for the precise relationship.}
Fix a perfect field $k$ of characteristic $p$. Write $W:=W(k)$ for the Witt vectors of $k$ and $K$ for its fraction field, and denote by $\varphi$ the unique automorphism of $W(k)$ lifting the $p$-power map on $k$. Fix an algebraic closure $\overline{K}$ of $K$, as well as a compatible sequence $\{\varepsilon^{(r)}\}_{r\ge 1}$ of primitive $p$-power roots of unity in $\o{K}$, and set $\scrG_K:=\Gal(\o{K}/K)$. For $r\ge 0$, we put $K_r:=K(\mu_{p^r})$ and $R_r:=W[\mu_{p^r}]$, and we set $\Gamma_r:=\Gal(K_{\infty}/K_r)$, and $\Gamma:=\Gamma_0$.
Let $\s_r:=W[\![u_r]\!]$ be the power series ring in one variable $u_r$ over $W$, viewed as a topological ring via the $(p,u_r)$-adic topology. We equip $\s_r$ with the unique continuous action of $\Gamma$ and extension of $\varphi$ determined by \begin{align}
&\gamma u_r := (1+u_r)^{\chi(\gamma)} -1\quad \text{for $\gamma\in \Gamma$} && \text{and} &&
\varphi(u_r) := (1+u_r)^p -1.\label{gamphiact} \end{align} We denote by $\O_{\E_r}:=\widehat{\s_r[\frac{1}{u_r}]}$ the $p$-adic completion of the localization ${\s_r}_{(p)}$, which is a complete discrete valuation ring with uniformizer $p$ and residue field $k(\!(u_r)\!)$. One checks that the actions of $\varphi$ and $\Gamma$ on $\s_r$ uniquely extend to $\O_{\E_r}$.
For $r>0$, we write $\theta: \s_r\twoheadrightarrow R_r$ for the continuous and $\Gamma$-equivariant $W$-algebra surjection sending $u_r$ to $\varepsilon^{(r)}-1$, whose kernel is the principal ideal generated by the Eisenstein polynomial $E_r:=\varphi^r(u_r)/\varphi^{r-1}(u_r)$, and we denote by $\tau:\s_r\twoheadrightarrow W$ the continuous and $\varphi$-equivariant surjection of $W$-algebras determined by $\tau(u_r)=0$. We lift the canonical inclusion $R_r\hookrightarrow R_{r+1}$ to a $\Gamma$- and $\varphi$-equivariant $W$-algebra injection ${\s_r} \hookrightarrow {\s_{r+1}}$ determined by $u_r\mapsto \varphi(u_{r+1})$; this map uniquely extends to a continuous injection $\O_{\E_r}\hookrightarrow \O_{\E_{r+1}}$, compatibly with $\varphi$ and $\Gamma$. We will frequently identify $\s_r$ (respectively $\O_{\E_r}$) with its image in $\s_{r+1}$ (respectively $\O_{\E_{r+1}}$), which coincides with the image of $\varphi$ on $\s_{r+1}$ (respectively $\O_{\E_{r+1}})$. Under this convention, we have $E_{r}(u_r) = E_1(u_1) = u_0/u_1$ for all $r>0$, so we will simply write $\omega:=E_r(u_r)$ for this common element of $\s_r$ for $r>0$.
\begin{definition}
We write $\BT_{\s_r}^{\varphi}$ for the category of {\em Barsotti-Tate modules over $\s_r$},
{\em i.e.} the category whose objects are pairs $(\m,\varphi_{\m})$ where
\begin{itemize}
\item $\m$ is a free $\s_r$-module of finite rank.
\item $\varphi_{\m}:\m\rightarrow \m$ is a $\varphi$-semilinear
map whose linearization has cokernel killed by $\omega$,
\end{itemize}
and whose morphisms are $\varphi$-equivariant $\s_r$-module homomorphisms.
We write $\BT_{\s_r}^{\varphi,\Gamma}$ for the subcategory of $\BT_{\s_r}^{\varphi}$
consisting of objects $(\m,\varphi_{\m})$ which admit a semilinear $\Gamma$-action
(in the category $\BT_{\s_r}^{\varphi}$) with the property that $\Gamma_r$ acts trivially on
$\m/u_r\m$. Morphisms in $\BT_{\s_r}^{\varphi,\Gamma}$ are $\varphi$ and $\Gamma$-equivariant
morphisms of $\s_r$-modules.
We often abuse notation by writing $\m$ for the pair
$(\m,\varphi_{\m})$ and $\varphi$ for $\varphi_{\m}$. \end{definition}
If $(\m,\varphi_{\m})$ is any object of $\BT_{\s_r}^{\varphi,\Gamma}$, then $1\otimes\varphi_{\m}:\varphi^*\m\rightarrow \m$ is injective with cokernel killed by $\omega$, so there is a unique $\s_r$-linear homomorphism $\psi_{\m}:\m\rightarrow \varphi^*\m$ with the property that the composition of $1\otimes\varphi_{\m}$ and $\psi_{\m}$ (in either order) is multiplication by $\omega$. Clearly, $\varphi_{\m}$ and $\psi_{\m}$ determine eachother. We warn the reader that the action of $\Gamma$ does {\em not} commute with $\psi_{\m}$: instead, for any $\gamma\in \Gamma$, one has \begin{equation}
(\gamma\otimes\gamma)\circ\psi_{\m}=(\gamma\omega/\omega)\cdot\psi_{\m}\circ\gamma.
\label{psiGammarel} \end{equation}
\begin{definition}\label{DualBTDef}
Let $\m$ be an object of $\BT_{\s_r}^{\varphi,\Gamma}$. The {\em dual
of $\m$} is the object $(\m^{t},\varphi_{\m^{t}})$ of $\BT_{\s_r}^{\varphi,\Gamma}$
whose underlying $\s_r$-module is $\m^{t}:=\Hom_{\s_r}(\m,\s_r)$, equipped with
the $\varphi$-semilinear endomorphism
\begin{equation*}
\xymatrix@C=32pt{
{\varphi_{\m^{t}}: \m^{t}} \ar[r]^-{1\otimes \id_{\m^{t}}} & {\varphi^*\m^{t} \simeq (\varphi^*\m)^{t}}
\ar[r]^-{\psi_{\m}^{t}} & {\m^{t}}
}
\end{equation*}
and the commuting\footnote{As one checks
using the intertwining relation (\ref{psiGammarel}).} action of $\Gamma$ given for $\gamma\in \Gamma$ by
\begin{equation*}
(\gamma f)(m) := \chi(\gamma)^{-1}\varphi^{r-1}(\gamma u_r/u_r)\cdot\gamma (f(\gamma^{-1} m )).
\end{equation*} \end{definition}
There is a natural notion of base change for Barsotti--Tate modules. Let $k'/k$ be an algebraic extension (so $k'$ is automatically perfect), and write $W':=W(k')$, $R_r':=W'[\mu_{p^r}]$, $\s_r':=W'[\![u_r]\!]$, and so on. The canonical inclusion $W\hookrightarrow W'$ extends to a $\varphi$ and $\Gamma$-compatible $W$-algebra injection $\iota_r:\s_r\hookrightarrow \s_{r+1}'$, and extension of scalars along $\iota_r$ yields a canonical canonical base change functor ${\iota_r}_*: \BT_{\s_r}^{\varphi,\Gamma}\rightarrow \BT_{\s_{r+1}}^{\varphi,\Gamma}$ which one checks is compatible with duality.
Let us write $\pdiv_{R_r}^{\Gamma}$ for the subcategory of $p$-divisible groups over $R_r$ consisting of those objects and morphisms which descend (necessarily uniquely) to $K=K_0$ on generic fibers. By Tate's Theorem, this is of course equivalent to the full subcategory of $p$-divisible groups over $K_0$ which have good reduction over $K_r$. Note that for any algebraic extension $k'/k$, base change along the inclusion $\iota_r:R_r\hookrightarrow R_{r+1}'$ gives a covariant functor ${\iota_r}_*:\pdiv_{R_r}^{\Gamma}\rightarrow \pdiv_{R_{r+1}'}^{\Gamma}$.
The main result of \cite{CaisLau} is the following:
\begin{theorem}\label{CaisLauMain}
For each $r>0$, there is a contravariant functor
$\m_r:\pdiv_{R_r}^{\Gamma}\rightarrow \BT_{\s_r}^{\varphi,\Gamma}$ such that:
\begin{enumerate}
\item The functor $\m_r$ is an exact antiequivalence of categories, compatible with duality.
\label{exequiv}
\item The functor $\m_r$ is of formation compatible with base change:
for any algebraic extension $k'/k$, there is a natural isomorphism
of composite functors ${\iota_r}_*\circ \m_r \simeq \m_{r+1}\circ {\iota_{r}}_*$ on $\pdiv_{R_r}^{\Gamma}$.
\label{BaseChangeIsom}
\item For $G\in \pdiv_{R_r}^{\Gamma}$, put $\o{G}:=G\times_{R_r} k$ and $G_0:=G\times_{R_r} R_r/pR_r$.
\begin{enumerate}
\item There is a functorial and $\Gamma$-equivariant isomorphism of $W$-modules
\begin{equation*}
\m_r(G)\tens_{\s_r,\varphi\circ \tau} W \simeq \D(\o{G})_W,
\end{equation*}
carrying
$\varphi_{\m}\otimes \varphi$ to $F:\D(\o{G})_W\rightarrow \D(\o{G})_W$
and $\psi_{\m}\otimes 1$ to $V\otimes 1: \D(\o{G})_W \rightarrow \varphi^*\D(\o{G})_W$.
\label{EvaluationONW}
\item There is a functorial and $\Gamma$-equivariant isomorphism of $R_r$-modules
\begin{equation*}
\m_r(G)\tens_{\s_r,\theta\circ\varphi} R_{r} \simeq \D(G_0)_{R_r}.
\end{equation*}\label{EvaluationONR}
\end{enumerate}
\end{enumerate} \end{theorem}
We wish to explain how to functorially recover the $\scrG_K$-representation afforded by the $p$-adic Tate module $T_pG_K$ from $\m_r(G)$. In order to do so, we must first recall the necessary period rings; for a more detailed synopsis of these rings and their properties, we refer the reader to \cite[\S6--\S8]{Colmez}.
As usual, we put\footnote{Here we use the notation introduced by Berger and Colmez; in Fontaine's original notation, this ring is denoted $\R$.} $$\wt{\e}^+:=\varprojlim_{x\mapsto x^p} \O_{\c_K}/(p),$$ equipped with its canonical $\scrG_K$-action via ``coordinates" and $p$-power Frobenius map $\varphi$. This is a perfect ({\em i.e.} $\varphi$ is an automorphism) valuation ring of charteristic $p$ with residue field $\overline{k}$ and fraction field $\wt{\e}:=\Frac(\wt{\e}^+)$ that is algebraically closed. We view $\wt{\e}$ as a topological field via its valuation topology, with respect to which it is complete. Our fixed choice of $p$-power compatible sequence $\{\varepsilon^{(r)}\}_{r\ge 0}$ induces an element $\u{\varepsilon}:=(\varepsilon^{(r)}\bmod p)_{r\ge 0}$ of $\wt{\e}^+$ and we set $\e_{K}:=k(\!(\u{\varepsilon} - 1)\!)$, viewed as a topological\footnote{The valuation $v_{\e}$ on $\wt{\e}$ induces the usual discrete valuation on $\e_{K,r}$, with the unusual normalization $1/p^{r-1}(p-1)$.} subring of $\wt{\e}$; note that this is a $\varphi$- and $\scrG_K$-stable subfield of $\wt{\e}$ that is independent of our choice of $\u{\varepsilon}$. We write $\e:=\e_K^{\sep}$ for the separable closure of $\e_K$ in the algebraically closed field $\wt{\e}$. The natural $\scrG_K$-action on $\wt{\e}$ induces a canonical identification $\Gal(\e/\e_{K}) = \scrH:=\ker(\chi)\subseteq \scrG_K$, so $\e^{\scrH}=\e_{K}$. If $E$ is any subring of $\wt{\e}$, we write $E^+:=E\cap \wt{\e}^+$ for the intersection (taken inside $\wt{\e}$).
We now construct Cohen rings for each of the above subrings of $\wt{\e}$. To begin with, we put \begin{equation*}
\wt{\a}^+:=W(\wt{\e}^+),\qquad\text{and}\qquad \wt{\a}:=W(\wt{\e}); \end{equation*} each of these rings is equipped with a canonical Frobenius automorphism $\varphi$ and action of $\scrG_K$ via Witt functoriality. Set-theoretically identifying $W(\wt{\e})$ with $\prod_{m=0}^{\infty} \wt{\e}$ in the usual way, we endow each factor with its valuation topology and give $\wt{\a}$ the product topology.\footnote{This is what is called the {\em weak topology} on $\wt{\a}$. If each factor of $\wt{\e}$ is instead given the discrete topology, then the product topology on $\wt{\a}=W(\wt{\e})$ is the familiar $p$-adic topology, called the {\em strong} topology.} The $\scrG_K$ action on $\wt{\a}$ is then continuous and the canonical $\scrG_K$-equivariant $W$-algebra surjection $\theta:\wt{\a}^+\rightarrow \O_{\c_K}$ is continuous when $\O_{\c_K}$ is given its usual $p$-adic topology. For each $r\ge 0$, there is a unique continuous $W$-algebra map $j_r:\O_{\E_r}\hookrightarrow \wt{\a}$ determined by $j_r(u_r):=\varphi^{-r}([\u{\varepsilon}] - 1)$. These maps are moreover $\varphi$ and $\scrG_K$-equivariant, with $\scrG_K$ acting on $\O_{\E_r}$ through the quotient $\scrG_K\twoheadrightarrow \Gamma$, and compatible with change in $r$. We define $\a_{K,r}:=\im(j_r:\O_{\E_r}\rightarrow \wt{\a}),$ which is naturally a $\varphi$ and $\scrG_K$-stable subring of $\wt{\a}$ that is independent of our choice of $\u{\varepsilon}$. We again omit the subscript when $r=0$. Note that $\a_{K,r}=\varphi^{-r}(\a_K)$ inside $\wt{\a}$, and that $\a_{K,r}$ is a discrete valuation ring with uniformizer $p$ and residue field $\varphi^{-r}(\e_K)$ that is purely inseparable over $\e_K$. We define $\a_{K,\infty}:=\bigcup_{r\ge 0} \a_{K,r}$ and write $\wt{\a}_K$ (respectively $\wh{\a}_K$) for the closure of $\a_{K,\infty}$ in $\wt{\a}$ with respect to the weak (respectively strong) topology.
Let $\a_{K,r}^{\sh}$ be the strict Henselization of $\a_{K,r}$ with respect to the separable closure of its residue field inside $\wt{\e}$. Since $\wt{\a}$ is strictly Henselian, there is a unique local morphism $\a_{K,r}^{\sh}\rightarrow \wt{\a}$ recovering the given inclusion on residue fields, and we henceforth view $\a_{K,r}^{\sh}$ as a subring of $\wt{\a}$. We denote by $\a_r$ the topological closure of $\a_{K,r}^{\sh}$ inside $\wt{\a}$ with respect to the strong topology, which is a $\varphi$ and $\scrG_K$-stable subring of $\wt{\a}$, and we note that $\a_r = \varphi^{-r}(\a)$ and $\a_r^{\scrH}= \a_{K,r}$ inside $\wt{\a}$. We note also that the canonical map $\Z_p\hookrightarrow \wt{\a}^{\varphi=1}$ is an isomorphism, from which it immediately follows that the same is true if we replace $\wt{\a}$ by any of its subrings constructed above. If $A$ is any subring of $\wt{\a}$, we define $A^+:=A\cap \wt{\a}^+$, with the intersection taken inside $\wt{\a}$.
\begin{remark}\label{Slimits}
We will identify $\s_r$ and $\O_{\E_r}$ with their respective images
$\a_{K,r}^+$ and $\a_{K,r}$ in $\wt{\a}$ under $j_r$.
Writing $\s_{\infty}:=\varinjlim \s_r$
and $\O_{\E_{\infty}}:=\varinjlim \s_r$, we likewise
identify $\s_{\infty}$ with $\a_{K,\infty}^+$ and $\O_{\E_{\infty}}$
with $\a_{K,\infty}$.
Denoting by $\wh{\s}_{\infty}$ (respectively $\wt{\s}_{\infty}$) the $p$-adic (respectively
$(p,u_0)$-adic) completions, one has
\begin{equation*}
\wh{\s}_{\infty} = \wh{\a}_K^+ = W(\e_K^{\rad,+})\quad\text{and}\quad
\wt{\s}_{\infty} = \wt{\a}_K^+ = W(\wt{\e}_K^{+}),
\end{equation*}
for $\e_K^{\rad}:=\cup_{r\ge 0} \varphi^{-r}(\e_K)$ the radiciel ($=$perfect) closure of
$\e_K$ in $\wt{\e}$ and $\wt{\e}_K$ its topological completion.
Via these identifications, $\omega :=u_0/u_1\in \a_{K,1}^+$ is
a principal generator
of $\ker(\theta:\wt{\a}^+\twoheadrightarrow \O_{\C_K})$. \end{remark}
We can now explain the functorial relation between $\m_r(G)$ and $T_pG_K$:
\begin{theorem}\label{comparison}
Let $G\in \pdiv_{R_r}^{\Gamma}$, and write $H^1_{\et}(G_K):=(T_pG_K)^{\vee}$
for the $\Z_p$-linear dual of $T_pG_K$.
There is a canonical mapping
of finite free $\a_r^+$-modules with semilinear Frobenius and $\scrG_K$-actions
\begin{equation}
\xymatrix{
{\m_r(G)\tens_{\s_r,\varphi} \a_r^+} \ar[r] & {H^1_{\et}(G_K)\otimes_{\Z_p} \a_r^+}
}
\end{equation}
that is injective with cokernel killed by $u_1$.
Here, $\varphi$ acts as $\varphi_{\m_r(G)}\otimes \varphi$ on source
and as $1\otimes\varphi$ on target, while $\scrG_K$ acts diagonally
on source and target through the quotient $\scrG_K\twoheadrightarrow \Gamma$
on $\m_r(G)$.
In particular, there is a natural $\varphi$ and $\scrG_K$-equivariant
isomorphism
\begin{equation}
{\m_r(G)\tens_{\s_r,\varphi} \a_r} \simeq {H^1_{\et}(G_K)\otimes_{\Z_p} \a_r}.
\label{comparisonb}
\end{equation}
These mappings are compatible with duality and with change in $r$ in the obvious manner. \end{theorem}
\begin{corollary}\label{GaloisComparison}
For $G\in \pdiv_{R_r}^{\Gamma}$, there are functorial isomorphisms of $\Z_p[\scrG_K]$-modules
\begin{subequations}
\begin{align}
T_pG_K &\simeq \Hom_{\s_r,\varphi}(\m_r(G),\a_r^+)\\
H^1_{\et}(G_K) &\simeq (\m_r(G) \tens_{\s_r,\varphi} \a_r)^{\varphi_{\m_r(G)}\otimes \varphi=1}.
\label{FontaineModule}
\end{align}
\end{subequations}
which are compatible with duality and change in $r$. In the first isomorphism,
we view $\a_r^+$ as a $\s_r$-algebra via the composite of the usual structure map with
$\varphi$. \end{corollary}
\begin{remark}
By definition, the map $\varphi^r$ on $\O_{\E_r}$ is injective with image $\O_{\E}:=\O_{\E_0}$,
and so induces a $\varphi$-semilinear isomorphism of $W$-algebras
$\xymatrix@C=15pt{{\varphi^{r}:\O_{\E_r}} \ar[r]^-{\simeq}&{\O_{\E}} }$.
It follows from (\ref{FontaineModule}) of Corollary \ref{GaloisComparison} and Fontaine's theory of
$(\varphi,\Gamma)$-modules
over $\O_{\E}$ that $\m_r(G)\otimes_{\s_r,\varphi^r} \O_{\E}$ {\em is} the $(\varphi,\Gamma)$-module
functorially associated to the $\Z_p[\scrG_K]$-module $H^1_{\et}(G_K)$. \end{remark}
For the remainder of this section, we recall the construction of the functor $\m_r$, both because we shall need to reference it in what follows, and because we feel it is enlightening. For details, including the proofs of Theorems \ref{CaisLauMain}--\ref{comparison} and Corollary \ref{GaloisComparison}, we refer the reader to \cite{CaisLau}.
Fix $G\in \pdiv_{R_r}^{\Gamma}$ and set $G_0:=G\times_{R_r}{R_r/pR_r}.$ The $\s_r$-module $\m_r(G)$ is a functorial descent of the evaluation of the Dieudonn\'e crystal $\D(G_0)$ on a certain ``universal" PD-thickening of $R_r/pR_r$, which we now describe. Let $S_r$ be the $p$-adic completion of the PD-envelope of $\s_r$ with respect to the ideal $\ker\theta$, viewed as a (separated and complete) topological ring via the $p$-adic topology. We give $S_r$ its PD-filtration: for $q\in \Z$ the ideal $\Fil^q S_r$ is the topological closure of the ideal generated by the divided powers $\{\alpha^{[n]}\}$ for $\alpha\in \ker\theta$ and $n\ge q$. By construction, the map $\theta:\s_r\twoheadrightarrow R_r$ uniquely extends to a continuous surjection of $\s_r$-algebras $S_r\twoheadrightarrow R_r$ (which we again denote by $\theta$) whose kernel $\Fil^1 S_r$ is equipped with topologically PD-nilpotent\footnote{Here we use our assumption that $p>2$.} divided powers. Similarly, the continuous $W$-algebra map $\tau:\s_r\twoheadrightarrow W$ determined by $\tau(u_r)=0$ uniquely extends to a continuous, PD-compatible $W$-algebra surjection $\tau:S_r\twoheadrightarrow W$ whose kernel we denote by $I:=\ker(\tau)$. One shows that there is a unique continuous endomorphism $\varphi$ of $S_r$ extending $\varphi$ on $\s_r$, and that $\varphi(\Fil^1 S_r)\subseteq pS_r$; in particular, we may define $\varphi_1: \Fil^1 S_r\rightarrow S_r$ by $\varphi_1:=\varphi/p$, which is a $\varphi$-semilinear homomorphism of $S_r$-modules. Note that $v_r:=\varphi_1(E_r)$ is a unit of $S_r$, so the image of $\varphi_1$ generates $S_r$ as an $S_r$-module.
Since the action of $\Gamma$ on $\s_r$ preserves $\ker\theta$, it follows from the universal mapping property of divided power envelopes and $p$-adic continuity considerations that this action uniquely extends to a continuous and $\varphi$-equivariant action of $\Gamma$ on $S_r$ which is compatible with the PD-structure and the filtration. Similarly, the transition map $\s_r\hookrightarrow \s_{r+1}$ uniquely extends to a continuous $\s_r$-algebra homomorphism $S_r\rightarrow S_{r+1}$ which is moreover compatible with filtrations (because $E_r(u_r)=E_{r+1}(u_{r+1})$ under our identifications), and for nonnegative integers $s < r$ we view $S_r$ as an $S_s$-algebra via these maps.
Put $\lambda := \log(1+u_0)/{u_0},$ where $\log(1+X):\Fil^1 S_r\rightarrow S_r$ is the usual (convergent for the $p$-adic topology) power series and $u_0:=\varphi^r(u_r)\in S_r$.
One checks that $\lambda$ admits the convergent product expansion $\lambda=\prod_{i\ge 0} \varphi^i(v_r)$, so $\lambda\in S_r^{\times}$ and
\begin{equation}
\frac{\lambda}{\varphi(\lambda)} = \varphi(E_r)/p= v_r\qquad\text{and}\qquad
\frac{\lambda}{\gamma\lambda} = \chi(\gamma)^{-1}\varphi^r(\gamma u_r/u_r) \quad\text{for}\
\gamma\in \Gamma.\label{lambdaTransformation}
\end{equation}
\begin{definition}
Let $\BT_{S_r}^{\varphi}$ be the category of triples $(\scrM,\Fil^1\scrM, \varphi_{\scrM,1})$ where
\begin{itemize}
\item $\scrM$ is a finite free $S_r$-module and $\Fil^1\scrM\subseteq \scrM$ is an $S_r$-submodule.
\item $\Fil^1\scrM$ contains $(\Fil^1 S_r)\scrM$ and the quotient $\scrM/\Fil^1\scrM$ is a free
$S_r/\Fil^1S_r=R_r$-module.
\item $\varphi_{\scrM,1}:\Fil^1\scrM_r\rightarrow \scrM$ is a $\varphi$-semilinear map whose image
generates $\scrM$ as an $S_r$-module.
\end{itemize}
Morphisms in $\BT_{S_r}^{\varphi}$ are $S_r$-module homomorphisms which are compatible with the
extra structures. As per our convention, we will often write $\scrM$ for a triple
$(\scrM,\Fil^1\scrM,\varphi_{\scrM,1})$, and $\varphi_1$ for $\varphi_{\scrM,1}$ when it can cause
no confusion. We denote by $\BT_{S_r}^{\varphi,\Gamma}$ the subcategory of $\BT_{S_r}^{\varphi}$
consisting of objects $\scrM$ that are equipped
with a semilinear action of $\Gamma$ which preserves $\Fil^1\scrM$, commutes with $\varphi_{\scrM,1}$,
and whose restriction to $\Gamma_r$ is trivial on $\scrM/I\scrM$; morphisms in $\BT_{S_r}^{\varphi,\Gamma}$
are $\Gamma$-equivariant morphisms in $\BT_{S_r}^{\varphi}$. \end{definition}
The kernel of the surjection $S_r/p^nS_r\twoheadrightarrow R_r/pR_r$ is the image of the ideal $\Fil^1 S_r + pS_r$, which by construction is equipped topologically PD-nilpotent divided powers. We may therefore define \begin{equation}
\scrM_r(G)=\D(G_0)_{S_r}:=\varprojlim_n \D(G_0)_{S_r/p^nS_r}, \end{equation} which is a finite free $S_r$-module that depends contravariantly functorially on $G_0$. We promote $\scrM_r(G)$ to an object of $\BT_{S_r}^{\varphi,\Gamma}$ as follows. As the quotient map $S_r\twoheadrightarrow R_r$ induces a PD-morphism of PD-theckenings of $R_r/pR_r$, there is a natural isomorphism of free $R_r$-modules \begin{equation}
\scrM_r(G)\otimes_{S_r} R_r \simeq \D(G_0)_{R_r}.\label{surjR} \end{equation} By Proposition \ref{BTgroupUnivExt}, there is a canonical ``Hodge" filtration $\omega_G \subseteq \D(G_0)_{R_r}$, which reflects the fact that $G$ is a $p$-divisible group over $R_r$ lifting $G_0$, and we define $\Fil^1\scrM_r(G)$ to be the preimage of $\omega_G$ under the composite of the isomorphism (\ref{surjR}) with the natural surjection $\scrM_r(G)\twoheadrightarrow \scrM_r(G)\otimes_{S_r} R_r$; note that this depends on $G$ and not just on $G_0$. The Dieudonn\'e crystal is compatible with base change (see, {\em e.g.} \cite[2.4]{BBM1}), so the relative Frobenius $F_{G_0}:G_0\rightarrow G_0^{(p)}$ induces an canonical morphism of $S_r$-modules \begin{equation*}
\xymatrix{
{\varphi^*(\D(G_0)_{S_r}) \simeq \D(G_0^{(p)})_{S_r}} \ar[r]^-{\D(F_{G_0})} & {\D(G_0)_{S_r}}
}, \end{equation*} which we may view as a $\varphi$-semilinear map $\varphi_{\scrM_r(G)}:\scrM_r(G)\rightarrow \scrM_r(G)$. As the relative Frobenius map $\omega_{G_0^{(p)}}\rightarrow \omega_{G_0}$ is zero, it follows that the restriction of $\varphi_{\scrM_r(G)}$ to $\Fil^1 \scrM_r(G)$ has image contained in $p\scrM_r(G)$, so we may define $\varphi_{\scrM_r(G),1}:=\varphi_{\scrM_r(G)}/p$, and one proves as in \cite[Lemma A.2]{KisinFCrystal} that the image of $\varphi_{\scrM_r(G),1}$ generates $\scrM_r(G)$ as an $S_r$-module.
It remains to equip $\scrM_r(G)$ with a canonical semilinear action of $\Gamma$.
Let us write $G_{K_r}$ for the generic fiber of $G$ and $G_{K}$ for its unique descent to $K=K_0$. The existence of this descent is reflected by the existence of a commutative diagram with cartesian square \begin{equation} \begin{gathered}
\xymatrix{
{G_{K}\fiber_K K_r} \ar@/^/[rrd]^-{1\times \gamma} \ar@/_/[ddr] \ar@{.>}[dr]|-{\gamma} & & \\
&{\big(G_{K}\fiber_K K_r\big)_{\gamma}} \ar[r]_-{\pr_1} \ar[d]^-{\pr_2}\ar@{} [dr] |{\square} &
{G_{K}\fiber_K K_r} \ar[d]\\
&{\Spec(K_r)} \ar[r]_-{\gamma} &{\Spec(K_r)}
} \end{gathered} \label{GammaAction} \end{equation} for each $\gamma\in \Gamma$, compatibly with change in $\gamma$; here, the subscript of $\gamma$ denotes base change along the map of schemes induced by $\gamma$. Since $G$ has generic fiber $G_{K_r}=G_K\times_K K_r$, Tate's Theorem ensures that the dotted arrow above uniquely extends to an isomorphism of $p$-divisible groups over $R_r$ \begin{equation}
\xymatrix{
{G}\ar[r]^-{\gamma} & {G_{\gamma}}
},\label{TateExt} \end{equation} compatibly with change in $\gamma$.
By assumption, the action of $\Gamma$ on $S_r$ commutes with the divided powers on $\Fil^1 S_r$ and induces the given action on the quotient $S_r\twoheadrightarrow R_r$; in other words, $\Gamma$ acts by automorphisms on the object $(\Spec(R_r/pR_r)\hookrightarrow \Spec(S_r/p^nS_r))$ of $\Cris((R_r/pR_r)/W)$. Again using the compatibility of $\D(G_0)$ with base change, we therefore see that each $\gamma\in \Gamma$ gives an $S_r$-linear map \begin{equation*}
\xymatrix{
{\gamma^*(\D(G_0)_{S_r}) \simeq \D((G_0)_{\gamma})_{S_r}} \ar[r] & {D(G_0)_{S_r}}
} \end{equation*} and hence an $S_r$-semilinear (over $\gamma$) endomorphism $\gamma$ of $\scrM_r(G)$. One easily checks that the resulting action of $\Gamma$ on $\scrM_r(G)$ commutes with $\varphi_{\scrM,1}$ and preserves $\Fil^1\scrM_r(G)$. By the compatibility of $\D(G_0)$ with base change and the obvious fact that the $W$-algebra surjection $\tau:S_r\twoheadrightarrow W$ is a PD-morphism over the canonical surjection $R_r/pR_r\twoheadrightarrow k$, there is a natural isomorphism \begin{equation}
\scrM_r(G)\otimes_{S_r} W \simeq \D(\o{G})_W. \end{equation} It follows easily from this and the diagram (\ref{GammaAction}) that the action of $\Gamma_r$ on $\scrM_r(G)/I\scrM_r(G)$ is trivial.
To define $\m_r(G)$, we functorially descend the $S_r$-module $\scrM_r(G)$ along the structure morphism $\alpha_r:\s_r\rightarrow S_r$. More precisely, for $\m\in \BT_{\s_r}^{\varphi,\Gamma}$, we define ${\alpha_r}_*(\m):=(M,\Fil^1M,\Phi_1)\in \BT_{S_r}^{\varphi,\Gamma}$ via:
\begin{equation}
\begin{gathered}
M:=\m\tens_{\s_r,\varphi} S_r\qquad\text{with diagonal $\Gamma$-action}\\
\Fil^1 M :=\left\{ m\in M\ :\ (\varphi_{\m}\otimes\id)(m) \in \m\otimes_{\s_r} \Fil^1 S_r
\subseteq \m\otimes_{\s_r} S_r \right\} \\
\xymatrix{
{\Phi_1: \Fil^1 M} \ar[r]^-{\varphi_{\m}\otimes\id} & { \m\tens_{\s_r} \Fil^1 S_r}
\ar[r]^-{\id\otimes\varphi_1} & {\m\tens_{\s_r,\varphi} S_r = M}
}. \end{gathered} \label{BreuilSrDef} \end{equation} The following is the key technical point of \cite{CaisLau}, and is proved using the theory of windows: \begin{theorem}\label{Lau}
For each $r$, the functor ${\alpha_r}_*:\BT_{\s_r}^{\varphi,\Gamma}\rightarrow \BT_{S_r}^{\varphi,\Gamma}$
is an equivalence of categories, compatible with change in $r$. \end{theorem}
\begin{definition}
For $G\in \pdiv_{R_r}^{\Gamma}$, we write $\m_r(G)$
for the functorial descent of $\scrM_r(G)$ to an object of $\BT_{\s_r}^{\varphi,\Gamma}$
as guaranteed by Theorem \ref{Lau}.
By construction, we have a natural isomorphism of functors ${\alpha_r}_*\circ \m_r\simeq \scrM_r$ on $\pdiv_{R_r}^{\Gamma}$. \end{definition}
\begin{example}\label{GmQpZpExamples} Using Messing's description of the Dieudonn\'e crystal of a $p$-divisible group in terms of the Lie algebra of its universal extension (cf. remark \ref{MessingRem}), one calculates that for $r\ge 1$
\begin{subequations}
\begin{equation}
\m_r(\Q_p/\Z_p) = \s_r,\qquad \varphi_{\m_r(\Q_p/\Z_p)}:= \varphi,\qquad \gamma:=\gamma
\label{MrQpZp}
\end{equation}
\begin{equation}
\m_r(\mu_{p^{\infty}}) = \s_r,\qquad \varphi_{\m_r(\mu_{p^{\infty}})}:= \omega\cdot\varphi,
\qquad \gamma:=\chi(\gamma)^{-1}\varphi^{r-1}(\gamma u_r/u_r)\cdot \gamma
\label{MrMu}
\end{equation} \end{subequations} with $\gamma\in \Gamma$ acting as indicated. Note that both $\m_r(\Q_p/\Z_p)$ and $\m_r(\Gm[p^{\infty}])$ arise by base change from their incarnations when $r=1$, as follows from the fact that $\omega = \varphi(u_1)/u_1$ and $\varphi^{r-1}(\gamma u_r/u_r)=\gamma u_1/u_1$ via our identifications. \end{example}
\subsection{The case of ordinary \texorpdfstring{$p$}{p}-divisible groups}\label{pDivOrdSection}
When $G\in \pdiv_{R_r}^{\Gamma}$ is ordinary, one can say significantly more about the structure of the $\s_r$-module $\m_r(G)$. To begin with, we observe that for arbitrary $G\in \pdiv_{R_r}^{\Gamma}$, the formation of the maximal \'etale quotient of $G$ and of the maximal connected and multiplicative-type sub $p$-divisible groups of $G$ are functorial in $G$, so each of $G^{\et}$, $G^0$, and $G^{\mult}$ is naturally an object of $\pdiv_{R_r}^{\Gamma}$ as well. We thus (functorially) obtain objects $\m_r(G^{\star})$ of $\BT_{\s_r}^{\varphi, \Gamma}$ which admit particularly simple descriptions when $\star=\et$ or $\mult$, as we now explain.
As usual, we write $\o{G}^{\star}$ for the special fiber of $G^{\star}$ and $\D(\o{G}^{\star})_W$ for its Dieudonn\'e module. Twisting the $W$-algebra structure on $\s_r$ by the automorphism $\varphi^{r-1}$ of $W$, we define objects of $\BT_{\s_r}^{\varphi,\Gamma}$ \begin{subequations}
\begin{equation}
\m_r^{\et}(G) : = \D(\o{G}^{\et})_W\tens_{W,\varphi^{r-1}} \s_r,
\qquad \varphi_{\m_r^{\et}}:= F\otimes \varphi,
\qquad \gamma:=\gamma \otimes \gamma
\label{MrEtDef}
\end{equation}
\begin{equation}
\m_r^{\mult}(G) : = \D(\o{G}^{\mult})_W\tens_{W,\varphi^{r-1}} \s_r,
\qquad \varphi_{\m_r^{\mult}}:= V^{-1}\otimes E_r\cdot\varphi,
\qquad \gamma:=\gamma \otimes \chi(\gamma)^{-1}\varphi^{r-1}(\gamma u_r/u_r)\cdot \gamma
\label{MrMultDef}
\end{equation} \end{subequations} with $\gamma\in \Gamma$ acting as indicated. Note that these formulae make sense and do indeed give objects of $\BT_{\s_r}^{\varphi,\Gamma}$ as $V$ is invertible\footnote{A $\varphi^{-1}$-semilinear map of $W$-modules $V:D\rightarrow D$ is {\em invertible} if there exists a $\varphi$-semilinear endomorphism $V^{-1}$ whose composition with $V$ in either order is the identity. This is easily seen to be equivalent to the invertibility of the linear map $V\otimes 1: D\rightarrow \varphi^* D$, with $V^{-1}$ the composite of $(V\otimes 1)^{-1}$ and the $\varphi$-semilinear map $\id\otimes 1:D\rightarrow \varphi^*D$. }
on $\D(\o{G}^{\mult})_W$ and $\gamma u_r/u_r \in \s_r^{\times}$. It follows easily from these definitions that $\varphi_{\m_r^{\star}}$ linearizes to an isomorphism when $\star=\et$ and has image contained in $\omega\cdot \m_r^{\mult}(G)$ when $\star=\mult$. Of course, $\m_r^{\star}(G)$ is contravariantly functorial in---and depends only on---the closed fiber $\o{G}^{\star}$ of $G^{\star}$.
\begin{proposition}\label{EtaleMultDescription}
Let $G$ be an object of $\pdiv_{R_r}^{\Gamma}$ and let $\m_r^{\et}(G)$ and $\m_r^{\mult}(G)$
be as in $(\ref{MrEtDef})$ and $(\ref{MrMultDef})$, respectively. The map $F^r:G_0 \rightarrow G_0^{(p^r)}$
$($respectively $V^r:G_0^{(p^r)}\rightarrow G_0$$)$ induces a natural isomorphism
in $\BT_{\s_r}^{\Gamma}$
\begin{equation}
\m_r(G^{\et}) \simeq \m_r^{\et}(G)\qquad\text{respectively}\qquad
\m_r(G^{\mult}) \simeq \m_r^{\mult}(G).\label{EtMultSpecialIsoms}
\end{equation}
These identifications are compatible with change in $r$
in the sense that for $\star=\et$ $($respectively $\star=\mult$$)$ there is a canonical
commutative diagram in $\BT_{\s_{r+1}}^{\Gamma}$
\begin{equation}
\begin{gathered}
\xymatrix{
{\m_{r+1}(G^{\star}\times_{R_r} R_{r+1})}
\ar[r]_-{\simeq}^-{(\ref{EtMultSpecialIsoms})}\ar[d]_-{\simeq} &
{\m_{r+1}^{\star}(G\times_{R_r} R_{r+1})} \ar@{=}[r] &
{\D(\o{G}^{\star})_W\tens_{W,\varphi^r} \s_{r+1}}
\ar[d]^-{F\otimes\id\ (\text{respectively}\ V^{-1}\otimes\id)}_-{\simeq} \\
{\m_r(G^{\star})\tens_{\s_r} \s_{r+1}} \ar[r]^-{\simeq}_-{(\ref{EtMultSpecialIsoms})} &
{\m_r^{\star}(G)\tens_{\s_r} \s_{r+1}} \ar@{=}[r] &
{\D(\o{G}^{\star})_W\tens_{W,\varphi^{r-1}} \s_{r+1}}
}
\end{gathered}
\label{EtMultSpecialIsomsBC}
\end{equation}
where the left vertical isomorphism is deduced from Theorem $\ref{CaisLauMain}$ $(\ref{BaseChangeIsom}).$ \end{proposition}
\begin{proof}
For ease of notation, we will write $\m_r^{\star}$ and
and $\D^{\star}$ for $\m_r^{\star}(G)$ and $\D(\o{G}^{\star})_W$, respectively.
Using (\ref{BreuilSrDef}), one finds that $\scrM_r^{\et}:={\alpha_r}_*(\m_r^{\et})\in \BT_{S_r}^{\varphi,\Gamma}$
is given by the triple
\begin{subequations}
\begin{equation}
\scrM_r^{\et}:=(\D^{\et}\otimes_{W,\varphi^r} S_r,\ \D^{\et}\otimes_{W,\varphi^r} \Fil^1 S_r,\
F\otimes \varphi_1)
\end{equation}
with $\Gamma$ acting diagonally on the tensor product. Similarly,
${\alpha_r}_*(\m_r^{\mult})$ is given by the triple
\begin{equation}
(\D^{\mult}\otimes_{W,\varphi^r} S_r,\ \D^{\mult}\otimes_{W,\varphi^r} S_r,\
V^{-1} \otimes v_r\cdot\varphi)\label{WindowMultCase}
\end{equation}
\end{subequations}
where $v_r=\varphi(E_r)/p$ and $\gamma\in \Gamma$ acts on $\D^{\mult}\otimes_{W,\varphi^r} S_r$
as $\gamma \otimes \chi(\gamma)^{-1} \varphi^r(\gamma u_r/u_r)\cdot \gamma$.
It follows from (\ref{lambdaTransformation})
that the $S_r$-module automorphism of $\D^{\mult}\otimes_{W,\varphi^r} S_r$
given by multiplication by $\lambda$
carries (\ref{WindowMultCase}) isomorphically onto the object of $\BT_{S_r}^{\varphi,\Gamma}$
given by the triple
\begin{equation}
\scrM_r^{\mult}:=(\D^{\mult}\otimes_{W,\varphi^r} S_r,\ \D^{\mult}\otimes_{W,\varphi^r} S_r,\
V^{-1}\otimes\varphi)
\end{equation}
with $\Gamma$ acting {\em diagonally} on the tensor product.
On the other hand, since $G_0^{\et}$ (respectively $G_0^{\mult}$) is \'etale
(respectively of multiplicative type) over $R_r/pR_r$, the relative Frobenius
(respectively Verscheibung) morphism of $G_0$ induces isomorphisms
\begin{subequations}
\begin{equation}
\xymatrix{
{G_0^{\et}} \ar[r]_-{\simeq}^-{F^r} & {(G_0^{\et})^{(p^r)} \simeq
{\varphi^r}^*\o{G}^{\et} \times_k R_r/pR_r}
}\label{Ftrick}
\end{equation}
respectively
\begin{equation}
\xymatrix{
{G_0^{\mult}} & \ar[l]^-{\simeq}_-{V^r} {(G_0^{\mult})^{(p^r)}
\simeq {\varphi^r}^*\o{G}^{\mult} \times_k R_r/pR_r}
}\label{Vtrick}
\end{equation}
\end{subequations}
of $p$-divisible groups over $R_r/pR_r$, where we have used the fact that the map $x\mapsto x^{p^r}$
of $R_r/pR_r$ factors as $R_r/pR_r \twoheadrightarrow k \hookrightarrow R_r/pR_r$
in the final isomorphisms of both lines above. Since the Dieudonn\'e crystal is compatible
with base change and the canonical map $W\rightarrow S_r$ extends to a PD-morphism
$(W,p)\rightarrow (S_r, pS_r+\Fil^1 S_r)$ over $k\rightarrow R_r/pR_r$,
applying $\D(\cdot)_{S_r}$ to (\ref{Ftrick})--(\ref{Vtrick}) yields natural isomorphisms
$\D(G_0^{\star})_{S_r} \simeq \D^{\star}\otimes_{W,\varphi^r} S_r$ for $\star=\et,\mult$
which carry $F$ to $F\otimes \varphi$. It is a straightforward exercise using the construction
of $\scrM_r(G^{\star})$ explained in \S\ref{PhiGammaCrystals} to check
that these isomorphisms extend to give isomorphisms $\scrM_r(G^{\et}) \simeq \scrM_r^{\et}$
and $\scrM_r(G^{\mult}) \simeq \scrM_r^{\mult}$ in $\BT_{S_r}^{\varphi,\Gamma}$.
By Theorem \ref{Lau},
we conclude that we have natural isomorphisms in $\BT_{\s_r}^{\varphi,\Gamma}$
as in (\ref{EtMultSpecialIsoms}). The commutativity of (\ref{EtMultSpecialIsomsBC})
is straightforward, using the definitions of the base change isomorphisms. \end{proof}
Now suppose that $G$ is ordinary. As $\m_r$ is exact by Theorem \ref{CaisLauMain} (\ref{exequiv}), applying $\m_r$ to the connected-\'etale sequence of $G$ gives a short exact sequence in $\BT_{\s_r}^{\varphi,\Gamma}$ \begin{equation}
\xymatrix{
0\ar[r] & {\m_r(G^{\et})} \ar[r] & {\m_r(G)} \ar[r] & {\m_r(G^{\mult})} \ar[r] & 0
}\label{ConEtOrdinary} \end{equation} which is contravariantly functorial and exact in $G$. Since $\varphi_{\m_r}$ linearizes to an isomorphism on $\m_r(G^{\et})$ and is topologically nilpotent on $\m_r(G^{\mult})$, we think of (\ref{ConEtOrdinary}) as the ``slope flitration" for Frobenius acting on $\m_r(G)$. On the other hand, Proposition \ref{BTgroupUnivExt} and Theorem \ref{CaisLauMain} (\ref{EvaluationONR}) provide a canonical ``Hodge filtration" of $\m_r(G)\tens_{\s_r,\varphi} R_r\simeq \D(G_0)_{R_r}$: \begin{equation}
\xymatrix{
0\ar[r] & {\omega_{G}} \ar[r] & {\D(G_0)_{R_r}} \ar[r] & {\Lie(G^t)} \ar[r] & 0
}\label{HodgeFilOrd} \end{equation} which is contravariant and exact in $G$. Our assumption that $G$ is ordinary yields ({\em cf.} \cite{KatzSerreTate}):
\begin{lemma}\label{HodgeFilOrdProps}
With notation as above, there are natural and $\Gamma$-equivariant
isomorphisms \begin{equation}
\Lie(G^t)\simeq \D(G_0^{\et})_{R_r} \qquad\text{and} \qquad \D(G_0^{\mult})_{R_r}\simeq \omega_G.
\label{FlankingIdens} \end{equation}
Composing these isomorphisms with the canonical maps obtained by applying $\D(\cdot)_{R_r}$
to the connected-\'etale sequence of $G_0$
yield functorial $R_r$-linear splittings of the Hodge filtration $(\ref{HodgeFilOrd})$.
Furthermore, there is a canonical and $\Gamma$-equivariant isomorphism of split exact
sequences of $R_r$-modules
\begin{equation}
\begin{gathered}
\xymatrix{
0\ar[r] & {\omega_{G}} \ar[r]\ar[d]^-{\simeq} & {\D(G_0)_{R_r}} \ar[r]\ar[d]^-{\simeq} &
{\Lie(G^t)} \ar[r]\ar[d]^-{\simeq} & 0\\
0 \ar[r] & {\D(\o{G}^{\mult})_W\tens_{W,\varphi^r} R_r} \ar[r]_-{i} &
{\D(\o{G})_W\tens_{W,\varphi^r} R_r} \ar[r]_-{j} &
{\D(\o{G}^{\et})_W\tens_{W,\varphi^r} R_r}\ar[r] & 0
}
\end{gathered}
\label{DescentToWIsom}
\end{equation}
with $i,j$ the inclusion and projection mappings
obtained from the canonical direct sum decomposition
$\D(\o{G})_W\simeq \D(\o{G}^{\mult})_W\oplus \D(\o{G}^{\et})_W$. \end{lemma}
\begin{proof} Applying $\D(\cdot)_{R_r}$ to the connected-\'etale sequence of $G_0$ and using Proposition \ref{BTgroupUnivExt} yields a commutative diagram with exact columns and rows \begin{equation} \begin{gathered}
\xymatrix{
& & 0\ar[d] & 0 \ar[d] & \\
& 0 \ar[r]\ar[d] & {\omega_{G}}\ar[r]\ar[d] &
{\omega_{G^{\mult}}} \ar[r]\ar[d] & 0\\
0\ar[r] & {\D(G_0^{\et})_{R_r}} \ar[r]\ar[d] & {\D(G_0)_{R_r}}\ar[r]\ar[d] &
{\D(G_0^{\mult})_{R_r}} \ar[r]\ar[d] & 0\\
0 \ar[r] & {\Lie({G^{\et}}^t)} \ar[r]\ar[d] & {\Lie(G^t)}\ar[r]\ar[d] &
0 & \\
& 0 & 0 & &
} \end{gathered} \label{OrdinaryDiagram} \end{equation} where we have used the fact that that the invariant differentials and Lie algebra of an \'etale $p$-divisible group (such as $G^{\et}$ and ${G^{\mult}}^t\simeq {G^t}^{\et}$) are both zero. The isomorphisms (\ref{FlankingIdens}) follow at once. We likewise immediately see that the short exact sequence in the center column of (\ref{OrdinaryDiagram}) is functorially and $R_r$-linearly split. Thus, to prove the claimed identification in (\ref{DescentToWIsom}), it suffices to exhibit natural isomorphisms of free $R_r$-modules with $\Gamma$-action \begin{equation}
\D(G_0^{\et})_{R_r} \simeq \D(\o{G}^{\et})_W\tens_{W,\varphi^r} R_r
\qquad\text{and}\qquad
\D(G_0^{\mult})_{R_r} \simeq \D(\o{G}^{\mult})_W\tens_{W,\varphi^r} R_r,
\label{TwistyDieuIsoms} \end{equation} both of which follow easily by applying $\D(\cdot)_{R_r}$ to (\ref{Ftrick}) and (\ref{Vtrick}) and using the compatibility of the Dieudonn\'e crystal with base change as in the proof of Proposition (\ref{EtaleMultDescription}). \end{proof}
From the slope filtration (\ref{ConEtOrdinary}) of $\m_r(G)$ we can recover both the (split) slope filtration of $\D(\o{G})_W$ and the (split) Hodge filtration (\ref{HodgeFilOrd}) of $\D(G_0)_{R_r}$:
\begin{proposition}\label{MrToHodge}
There are canonical and $\Gamma$-equivariant isomorphisms of short exact sequences
\begin{subequations}
\begin{equation}
\begin{gathered}
\xymatrix{
0\ar[r] & {\m_r(G^{\et})\tens_{\s_r,\varphi\circ\tau} W} \ar[r]\ar[d]^-{\simeq} &
{\m_r(G)\tens_{\s_r,\varphi\circ\tau} W} \ar[r]\ar[d]^-{\simeq} &
{\m_r(G^{\mult})\tens_{\s_r,\varphi\circ\tau} W} \ar[r]\ar[d]^-{\simeq} & 0 \\
0 \ar[r] & {\D(\o{G}^{\et})_W} \ar[r] & {\D(\o{G})_W} \ar[r] & {\D(\o{G}^{\mult})_W}
\ar[r] & 0
}\label{MrToDieudonneMap}
\end{gathered}
\end{equation}
\begin{equation}
\begin{gathered}
\xymatrix{
0\ar[r] & {\m_r(G^{\et})\tens_{\s_r,\theta\circ\varphi} R_r} \ar[r]\ar[d]^-{\simeq} &
{\m_r(G)\tens_{\s_r,\theta\circ\varphi} R_r} \ar[r]\ar[d]^-{\simeq} &
{\m_r(G^{\mult})\tens_{\s_r,\theta\circ\varphi} R_r} \ar[r]\ar[d]^-{\simeq} & 0 \\
0 \ar[r] & {\Lie(G^t)} \ar[r]_-{i} & {\D(G_0)_{R_r}} \ar[r]_-{j} & {\omega_{G}} \ar[r] & 0\\
}
\end{gathered}\label{MrToHodgeMap}
\end{equation}
\end{subequations}
Here, $i:\Lie(G^t)\hookrightarrow \D(G_0)_{R_r}$
and $j:\D(G_0)_{R_r}\twoheadrightarrow \omega_{G}$
are the canonical splittings of Lemma $\ref{HodgeFilOrdProps}$,
the top row of $(\ref{MrToHodgeMap})$ is obtained from $(\ref{ConEtOrdinary})$ by extension of scalars,
and the isomorphism $(\ref{MrToDieudonneMap})$ intertwines $\varphi_{\m_r(\cdot)}\otimes \varphi$ with $F$.
\end{proposition}
\begin{proof}
This follows immediately from Theorem \ref{CaisLauMain} (\ref{EvaluationONW}) and Lemma \ref{HodgeFilOrdProps}. \end{proof}
\section{Ordinary \texorpdfstring{$\Lambda$}{Lambda}-adic Dieudonn\'e and \texorpdfstring{$(\varphi,\Gamma)$}{(phi,Gamma)}-modules}\label{results}
In this section, we will state and prove our main results as described in \S\ref{resultsintro}. Throughout, we will use the notation of \S\ref{resultsintro} and of \cite[\S2.2]{CaisHida1}, which we now briefly recall.
For $r\ge 1$, we write $X_r:=X_1(Np^r)$
for the canonical model over $\Q$ with rational cusp at $i\infty$ of the modular curve arising as the quotient of the extended upper-halfplane by the congruence subgroup $\Upgamma_1(Np^r)$ ({\em cf.} \cite[Remark 2.2.4]{CaisHida1}). There are two natural degeneracy mappings $\rho,\sigma:X_{r+1}\rightrightarrows X_r$ of curves over $\Q$ induced by the self-maps of the upper-halfplane $\rho:\tau\mapsto \tau$ and $\sigma:\tau\mapsto p\tau$; see \cite[Remark 2.2.5]{CaisHida1}. Denote by $J_r:=\Pic^0_{X_r/\Q}$ the Jacobian of $X_r$ over $\Q$ and write $\H_r(\Z)$
for the $\Z$-subalgebra of $\End_{\Q}(J_r)$ generated by the Hecke operators $\{T_{\ell}\}_{\ell\nmid Np}$, $\{U_{\ell}\}_{\ell|Np}$ and the Diamond operators $\{\langle u\rangle\}_{u\in \Z_p^{\times}}$. We define $\H_r(\Z)^{*}$ similarly, using instead the ``transpose" Hecke and diamond operators, and set $\H_r:=\H_r(\Z)\otimes_{\Z}\Z_p$ and $\H_r^*:=\H_r(\Z)^*\otimes_{\Z}\Z_p$; see \cite[2.2.21--2.2.23]{CaisHida1}.
As usual, we write $e_r\in \H_r$ and $e_r^*\in \H_r^*$ for the idempotents of these semi-local $\Z_p$-algebras corresponding to the Atkin operators $U_p$ and $U_p^*$, respectively, and we put $e:=(e_r)_r$ and $e^*:=(e_r^*)_r$ for the induced idempotents of the ``big" $p$-adic Hecke algebras $\H:=\varprojlim_r \H_r$ and $\H^*:=\varprojlim_r \H_r^*$; here, the maps in these projective limits are induced by the natural transition mappings on Jacobians $J_r\rightrightarrows J_{r'}$ for $r'\ge r$ arising (via Picard functoriality) from $\sigma$ and $\rho$, respectively. Let $w_r$ be the Atkin--Lehner
``involution" of $X_r$ over $\Q(\mu_{Np^r})$ corresponding to a choice of primitive $Np^r$-th root of unity as in the discussion preceding \cite[Proposition 2.2.6]{CaisHida1}; following the conventions of \cite[\S2.2]{CaisHida1}, we simply write $w_r$ for the automorphism $\Alb(w_r)$ of $J_r$ over $\Q(\mu_{Np^r})$ induced by Albanese functoriality. We note that for any Hecke operator $T\in \H_r(\Z)$, one has the relation $w_rT=T^*w_r$ as endomorphisms of $J_r$ over $\Q(\mu_{Np^r})$
\cite[Proposition 2.2.24]{CaisHida1}.
\subsection{\texorpdfstring{$\Lambda$}{Lambda}-adic Barsotti-Tate groups}\label{BTfamily}
In order to construct a crystalline analogue of Hida's ordinary $\Lambda$-adic \'etale cohomology, we will apply the theory of \S\ref{PhiGammaCrystals} to a certain ``tower" $\{\G_r\}_{r\ge 1}$ of $p$-divisible groups (a $\Lambda$-adic Barsotti Tate group in the sense of Hida \cite{HidaLambdaBT}, \cite{HidaNotes}, \cite{HidaNotes2}) whose construction involves artfully cutting out certain $p$-divisible subgroups of $J_r[p^{\infty}]$ over $\Q$ and the ``good reduction'' theorems of Langlands-Carayol-Saito. The construction of $\{\G_r\}_{r\ge 1}$ is certainly well-known (e.g. \cite[\S1]{MW-Hida}, \cite[Chapter 3, \S1]{MW-Iwasawa}, \cite[Definition 1.2]{Tilouine} and \cite[\S 3.2]{OhtaEichler}), but as we shall need substantially finer information about the $\G_r$ than is available in the literature, we devote this section to recalling their construction and properties.
As in \cite[\S 3.3]{CaisHida1}, for a ring $A$, a nonegative integer $k$, and a congruence subgroup $\Upgamma$ of $\SL_2(\Z)$, we write $S_k(\Upgamma;A)$ for the space of weight $k$ cuspforms for $\Gamma$ over $A$, and for ease of notation we put $S_k(\Upgamma):=S_k(\Upgamma;\Qbar)$. If $\Upgamma',$ $\Upgamma$ are congruence subgroups, then associated to any $\gamma\in \GL_2(\Q)$ with $\gamma^{-1}\Upgamma'\gamma\subseteq \Upgamma$ is an injective pullback mapping $\xymatrix@1{{\iota_{\gamma}:S_k(\Upgamma)} \ar@{^{(}->}[r] & {S_k(\Upgamma')}}$
given by $\iota_{\gamma}(f):=f\big|_{\gamma^{-1}}$, as well as a surjective ``trace" mapping \begin{equation}
\xymatrix{
{\tr_{\gamma}:S_k(\Upgamma')} \ar@{->>}[r] & {S_k(\Upgamma)}
}
\qquad\text{given by}\qquad
\tr_{\gamma}(f):=\sum_{\delta\in \gamma^{-1}\Upgamma'\gamma\backslash\Upgamma} (f\big|_{\gamma})\big|_{\delta}
\label{MFtrace} \end{equation} with $\tr_{\gamma}\circ\iota_{\gamma}$ multiplication by $[\Upgamma: \gamma^{-1}\Upgamma'\gamma]$ on $S_k(\Upgamma)$. If $\Upgamma'\subseteq \Upgamma$, then {\em unless specified to the contrary}, we will always view $S_k(\Upgamma)$ as a subspace of $S_k(\Upgamma')$ via $\iota_{\id}$.
For nonnegative integers $i\le r$ we set $\Upgamma_r^i:=\Upgamma_1(Np^i)\cap \Upgamma_0(p^r)$ for the intersection $($taken inside $\SL_2(\Z)$$)$, and put $\Upgamma_r:=\Upgamma_r^r$. We will need the following fact ({\em cf.} \cite[pg. 339]{Tilouine}, \cite[2.3.3]{OhtaEichler}) concerning the trace mapping $(\ref{MFtrace})$ attached to the canonical inclusion $\Upgamma_{r}\subseteq \Upgamma_i$ for $r\ge i$; for notational clarity, we will write $\tr_{r,i}:S_k(\Upgamma_r)\rightarrow S_k(\Upgamma_i)$ for this map.
\begin{lemma}\label{MFtraceLem}
Fix integers $i\le r$ and let $\tr_{r,i}:S_k(\Upgamma_r)\rightarrow S_k(\Upgamma_i)$
be the trace mapping $(\ref{MFtrace})$ attached to the inclusion
$\Upgamma_r\subseteq \Upgamma_i$. For $\alpha:=\left(\begin{smallmatrix} 1 & 0 \\ 0 & p\end{smallmatrix}\right)$,
we have an equality
of $\o{\Q}$-endomorphisms of $S_k(\Upgamma_{r})$
\begin{equation}
\iota_{\alpha^{r-i}}\circ \tr_{r,i} = (U_p^*)^{r-i}
\sum_{\delta\in \Delta_i/\Delta_{r}} \langle \delta \rangle.
\label{DualityIdentity}
\end{equation} \end{lemma}
\begin{proof}
We have
index $p^{r-i}$ inclusions of groups $\Upgamma_{r} \subseteq \Upgamma_{r}^i \subseteq \Upgamma_i$
with $\Upgamma_{r}$ normal in $\Upgamma_{r}^i$, as it is the kernel of the canonical
surjection $\Upgamma_{r}^i\twoheadrightarrow \Delta_i/\Delta_{r}$.
For each $\delta\in \Delta_i/\Delta_{r}$, we
fix a choice of $\sigma_{\delta}\in \Upgamma_{r}^i$ mapping to $\delta$
and calculate that
\begin{equation}
\Upgamma_i = \coprod_{\delta\in \Delta_i/\Delta_{r}} \coprod_{j=0}^{p^{r-i}-1}
\Upgamma_{r}\sigma_{\delta} \varrho_j
\qquad\text{where}\qquad
\varrho_j:=\begin{pmatrix} 1 & 0 \\ jNp^i & 1\end{pmatrix}.\label{CosetDecomp}
\end{equation}
On the other hand, for each $0\le j < p^{r-i}$ one has the equality of matrices in $\GL_2(\Q)$
\begin{equation}
p^{r-i}\varrho_j \alpha^{-(r-i)} = \tau_{r} \begin{pmatrix} 1 & -j \\ 0 & p^{r-i} \end{pmatrix} \tau_{r}^{-1}
\qquad\text{for}\qquad \tau_{r} := \begin{pmatrix} 0 & -1 \\ Np^{r} & 0 \end{pmatrix}.
\label{EasyMatCalc}
\end{equation}
The claimed equality (\ref{DualityIdentity}) follows easily from (\ref{CosetDecomp}) and (\ref{EasyMatCalc}), using
the equalities of operators $(\cdot)\big|_{\sigma_{\delta}}=\langle \delta\rangle $
and $U_p^* = w_{r} U_p w_{r}^{-1}$ on $S_k(\Upgamma_{r})$ \cite[Proposition 2.2.24]{CaisHida1}. \end{proof}
Perhaps the most essential ``classical" fact for our purposes is that the Hecke operator $U_p$ acting on spaces of modular forms ``contracts" the $p$-level, as is made precise by the following:
\begin{lemma}\label{UpContract}
If $f\in S_k(\Upgamma_r^i)$ then $U_p^{d}f$ is in the image of the canonical map
$\iota_{\id}:S_k(\Upgamma_{r-d}^i)\hookrightarrow S_k(\Upgamma_r^i)$ for each integer $d\le r-i$. In particular,
$U_p^{r-i}f$ is in the image of $S_k(\Upgamma_i)\hookrightarrow S_k(\Upgamma_r^i)$. \end{lemma}
Certainly Lemma \ref{UpContract} is well-known (e.g. \cite{Tilouine}, \cite{Ohta1}, \cite{HidaLambdaBT}); because of its importance in our subsequent applications, we sketch a proof (following the proof of \cite[Lemma 1.2.10]{Ohta1}; see also \cite{HidaLambdaBT} and \cite[\S 2]{HidaNotes}). We note that $\Upgamma_r\subseteq \Upgamma_r^i$ for all $i\le r$, and the resulting inclusion $S_k(\Upgamma_r^i)\hookrightarrow S_k(\Upgamma_r)$ has image consisting of forms on $\Upgamma_r$ which are eigenvectors for the diamond operators and whose associated character has conductor with $p$-part dividing $p^{i}$.
\begin{proof}[Proof of Lemma $\ref{UpContract}$]
Fix $d$ with $0\le d\le r-i$ and let $\alpha:=\left(\begin{smallmatrix} 1 & 0 \\ 0 & p\end{smallmatrix}\right)$
be as in Lemma \ref{MFtraceLem};
then $\alpha^d$ is an element of the commeasurator of $\Upgamma_{r}^i$ in $\SL_2(\Q)$. Consider the following
subgroups of $\Upgamma_{r-d}^i$:
\begin{align*}
H&:= \Upgamma_{r-d}^i \cap \alpha^{-d}\Upgamma_{r}^i\alpha^d\\
H'&:= \Upgamma_{r-d}^i \cap \alpha^{-d}\Upgamma_{r-d}^i\alpha^d,
\end{align*}
with each intersection taken inside of $\SL_2(\Q)$. We claim that $H=H'$ inside $\Upgamma_{r-d}^i$.
Indeed, as $\Upgamma_{r}^i\subseteq \Upgamma_{r-d}^i$, the inclusion $H\subseteq H'$ is clear.
For the reverse inclusion, if $\gamma:=\left(\begin{smallmatrix} * & * \\ x & *\end{smallmatrix}\right)\in \Upgamma_{r-d}^i$,
then we have $\alpha^{-d}\gamma\alpha^d = \left(\begin{smallmatrix} * & * \\ p^{-d}x & *\end{smallmatrix}\right)$,
so if this lies in $\Upgamma_{r-d}^i$ we must have $x\equiv 0\bmod p^r$ and hence $\gamma\in \Upgamma_r^i$.
We conclude that the coset spaces $H\backslash\Upgamma_{r-d}^i$ and $H'\backslash\Upgamma_{r-d}^i$
are equal. On the other hand, for {\em any} commeasurable subgroups $\Upgamma,\Upgamma'$ of a group $G$
and any $g$ in the commeasurator of $\Upgamma$ in $G$,
an elementary computation shows that we have a bijection of coset spaces
\begin{align*}
(\Upgamma'\cap g^{-1}\Upgamma g )\backslash \Upgamma' \simeq \Upgamma\backslash\Upgamma g\Upgamma'
\end{align*}
via $(\Upgamma'\cap g^{-1}\Upgamma g)\gamma\mapsto \Upgamma g\gamma$.
Applying this with $g=\alpha^d$ in our situation and using the
decomposition
\begin{equation*}
\Upgamma_{r-d}^i \alpha^d \Upgamma_{r-d}^i = \coprod_{j=0}^{p^{d}-1} \Upgamma_{r-d}^i\begin{pmatrix} 1 & j \\ 0 & p^{d}\end{pmatrix}
\end{equation*}
(see, e.g. \cite[proposition 3.36]{Shimura}), we deduce that we also have
\begin{equation}\label{disjointHecke}
\Upgamma_r^i \alpha^d \Upgamma_{r-d}^i = \coprod_{j=0}^{p^{d}-1} \Upgamma_r^i\begin{pmatrix} 1 & j \\ 0 & p^{d}\end{pmatrix}.
\end{equation}
Writing $U:S_k(\Upgamma_r^i)\rightarrow S_k(\Upgamma_{r-d}^i)$ for the ``Hecke operator" given by
(e.g. \cite[\S3.4]{Ohta1})
$\Upgamma_r^i \alpha^d \Upgamma_{r-d}^i$, an easy computation using \ref{disjointHecke} shows that the composite
\begin{equation*}
\xymatrix{
S_k(\Upgamma_r^i) \ar[r]^-{U} & S_{k}(\Upgamma_{r-d}^i) \ar@{^{(}->}[r] & S_k(\Upgamma_r^i)
}
\end{equation*}
coincides with $U_p^d$ on $q$-expansions. By the $q$-expansion principle, we deduce that $U_p^d$ on $S_k(\Upgamma_r^i)$
indeed factors through the subspace $S_k(\Upgamma_{r-d}^i)$, as desired. \end{proof}
For each integer $i$ and any character $\varepsilon:(\Z/Np^i\Z)^{\times}\rightarrow \Qbar^{\times}$, we denote by $S_2(\Upgamma_i,\varepsilon)$ the $\H_i$-stable subspace of weight 2 cusp forms for $\Upgamma_i$ over $\Qbar$ on which the diamond operators act through $\varepsilon(\cdot)$. Define \begin{equation}
\o{V}_r := \bigoplus_{i=1}^r\bigoplus_{\varepsilon } S_2(\Upgamma_i,\varepsilon)
\label{VrDef} \end{equation} where the inner sum is over all Dirichlet characters defined modulo $Np^i$ whose $p$-parts are {\em primitive} ({\em i.e.} whose conductor has $p$-part exactly $p^i$). We view $\o{V}_r$ as a $\Qbar$-subspace of $S_2(\Upgamma_r)$ in the usual way ({\em i.e.} via the embeddings $\iota_{\id}$). We define $\o{V}_r^*$ as the direct sum (\ref{VrDef}), but viewed as a subspace of $S_2(\Upgamma_r)$ via the ``nonstandard" embeddings $\iota_{\alpha^{r-i}}:S_2(\Upgamma_i)\rightarrow S_2(\Upgamma_r)$.
As in \cite[2.5.17]{CaisHida1}, we write $f'$ for the idempotent of $\Z_{(p)}[\F_p^{\times}]$ corresponding to ``projection away from the trivial $\F_p^{\times}$-eigenspace;" explicitly, we have \begin{equation}
f':=1 - \frac{1}{p-1}\sum_{g\in \F_p^{\times}} g.\label{projaway} \end{equation} We set $h':=(p-1)f'$, so that $h'^2 = (p-1)h'$ and define endomorphisms of $S_2(\Upgamma_r)$: \begin{equation}
U_r^*:=h'\circ (U_p^*)^{r+1} = (U_p^*)^{r+1}\circ h'\quad\text{and}\quad
U_r:=h'\circ (U_p)^{r+1} = (U_p)^{r+1}\circ h'.
\label{UrDefinition} \end{equation}
\begin{corollary}\label{UpProjection}
As subspaces of $S_2(\Upgamma_r)$ we have $w_r(\o{V}_r^*)=\o{V}_r$.
The space $\o{V}_r$ $($respectively $\o{V}_r^*$$)$ is naturally an $\H_r$ $($resp. $\H_r^*$$)$-stable
subspace of $S_2(\Upgamma_r)$, and admits a canonical descent to $\Q$.
Furthermore, the endomorphisms
$U_r$ and $U_r^*$ of $S_2(\Upgamma_r)$
factor through $\o{V}_r$ and $\o{V}_r^*$, respectively. \end{corollary}
\begin{proof}
The first assertion follows from the relation $w_r\circ \iota_{\alpha^{r-i}}=\iota_{\id}\circ w_i$
as maps $S_2(\Upgamma_i)\rightarrow S_2(\Upgamma_r)$, together with the fact that $w_i$ on $S_2(\Upgamma_i)$
carries $S_2(\Upgamma_i,\varepsilon)$ isomorphically onto $S_2(\Upgamma_i,\varepsilon^{-1})$.
The $\H_r$-stability of $\o{V}_r$ is clear as each of $S_2(\Upgamma_i,\varepsilon)$ is an $\H_r$-stable subspace
of $S_2(\Upgamma_r)$; that $\o{V}_r^*$ is $\H_r^*$-stable follows from this and the comutation
relation $T^* w_r = w_r T$ \cite[Proposition 2.2.4]{CaisHida1}. That $\o{V}_r$ and $\o{V}_r^*$
admit canonical descents to $\Q$ is clear, as $\scrG_{\Q}$-conjugate Dirichlet characters have equal
conductors. The final assertion concerning the endomorphisms $U_r$ and $U_r^*$
follows easily from Lemma \ref{UpContract},
using the fact that $h':S_2(\Upgamma_r)\rightarrow S_2(\Upgamma_r)$
has image contained in $\bigoplus_{i=1}^r S_k(\Upgamma_r^i)$. \end{proof}
\begin{definition}
We denote by $V_r$ and $V_r^*$ the canonical descents to $\Q$ of $\o{V}_{r}$
and $\o{V}_r^*$, respectively. \end{definition}
Following \cite[Chapter \Rmnum{3}, \S1]{MW-Iwasawa} and \cite[\S2]{Tilouine}, we recall the construction of certain ``good" quotient abelian varieties of $J_r$ whose cotangent spaces are naturally identified with $V_r$ and $V_r^*$. In what follows, we will make frequent use of the following elementary result:
\begin{lemma}\label{LieFactorization}
Let $f:A\rightarrow B$ be a homomorphism of commutative group varieties
over a field $K$ of characteristic $0$. Then: \begin{enumerate}
\item The formation of $\Lie$ and $\Cot$ commutes with the formation of kernels and images: the kernel $($respectively image$)$ of $\Lie(f)$ is canonically
isomorphic to the Lie algebra of the kernel $($respectively image$)$ of $f$, and
similarly for cotangent spaces at the identity.
In particular, if $A$ is connected and $\Lie(f)=0$ $($respectively $\Cot(f)=0$$)$
then $f=0$.\label{ExactnessOfLie}
\item Let $i:B'\hookrightarrow B $ be a closed immersion of commutative
group varieties over $K$ with $B'$ connected. If $\Lie(f)$ factors through $\Lie(i)$
then $f$ factors $($necessarily uniquely$)$ through $i$.
\label{InclOnLie}
\item Let $j:A\twoheadrightarrow A''$ be a surjection of commutative
group varieties over $K$ with connected kernel. If $\Cot(f)$
factors through $\Cot(j)$ then $f$ factors $($necessarily uniquely$)$ through $j$.
\label{LieFactorizationSurj} \end{enumerate} \end{lemma}
\begin{proof}
The key point is that because $K$ has characteristic zero, the functors $\Lie(\cdot)$
and $\Cot(\cdot)$ on the category of commutative group schemes are {\em exact}.
Indeed, since $\Lie(\cdot)$ is always left
exact, the exactness of $\Lie(\cdot)$ follows easily from the fact that any
quotient mapping
$G\twoheadrightarrow H$
of group varieties in characteristic zero is smooth (as the kernel is
a group variety over a field of characteristic zero and hence automatically smooth),
so the induced map on Lie algebras is a surjection.
By similar reasoning
one shows that the right exact $\Cot(\cdot)$ is likewise exact,
and the first part of (\ref{ExactnessOfLie}) follows easily. In particular,
if $\Lie(f)$ is the zero map then $\Lie(\im(f))=0$, so $\im(f)$ is zero-dimensional.
Since it is also smooth, it must be \'etale. Thus, if $A$ is connected,
then $\im(f)$ is both connected and \'etale, whence it is a single point; by
evaluation of $f$ at the identity of $A$ we conclude that $f=0$.
The assertions (\ref{InclOnLie}) and (\ref{LieFactorizationSurj})
now follow immediately by using universal mapping properties. \end{proof}
To proceed with the construction of good quotients of $J_r$, we write $Y_r:=X_1(Np^r; Np^{r-1})$ for the canonical model over $\Q$ with rational cusp at $i\infty$ of the modular curve corresponding to the congruence subgroup $\Upgamma_{r+1}^r$ ({\em cf.} \cite[Remark 2.2.18]{CaisHida1}), and consider the diagrams of ``degeneracy mappings" of curves over $\Q$ for $i=1,2$ \begin{equation} \addtocounter{equation}{1}
\xymatrix{
{X_r} \ar[r]^-{\pi} & {Y_{r-1}} \ar[r]^-{\pi_i} & {X_{r-1}}
}
\tag{$\arabic{section}.\arabic{subsection}.\arabic{equation}_i$}
\label{DegeneracyDiag} \end{equation} where $\pi$ and $\pi_2$ are induced by the canonical inclusions of subgroups $\Upgamma_{r} \subseteq \Upgamma_r^{r-1}\subseteq \Upgamma_{r-1}$ via the upper-halfplane self map $\tau\mapsto \tau$, and $\pi_1$ is induced by the inclusion $\alpha^{-1} \Upgamma_r^{r-1} \alpha \subseteq \Upgamma_{r-1}$ via the mapping $\tau\mapsto p\tau$ where $\alpha$ is as in Lemma \ref{MFtraceLem}; see \cite[2.2.9]{CaisHida1} for a moduli-theoretic description of these maps. We note that the compositions $\pi\circ \pi_2$ and $\pi\circ\pi_1$ coincide with the degeneracy maps $\rho$ and $\sigma$, respectively \cite[Remark 2.2.18]{CaisHida1}.
These mappings covariantly (respectively contravariantly) induce mappings on the associated Jacobians via Albanese (respectively Picard) functoriality. Writing $JY_r:=\Pic^0_{Y_r/\Q}$ and setting $K_1^i:=JY_1$ for $i=1,2$ we inductively define abelian subvarieties $\iota_r^i:K_r^i\hookrightarrow JY_r$ and abelian variety quotients $\alpha_r^i:J_r\twoheadrightarrow B_r^i$ as follows: \begin{equation}
\addtocounter{equation}{1}
B_{r-1}^i:= J_{r-1}/\Pic^0(\pi)(K_{r-1}^i)
\qquad\text{and}\qquad
K_{r}^i:=\ker(JY_r \xrightarrow{\alpha_{r-1}^i\circ\Alb(\pi_i)} B_{r-1}^i)^0
\tag{$\arabic{section}.\arabic{subsection}.\arabic{equation}_i$}
\label{BrDef} \end{equation} for $r\ge 2$, $i=1,2$, with $\alpha_{r-1}^i$ and $\iota_r^i$ the obvious mappings; here, $(\cdot)^0$ denotes the connected component of the identity of $(\cdot)$. As in \cite[\S 3.2]{OhtaEichler}, we have modified Tilouine's construction \cite[\S2]{Tilouine} so that the kernel of $\alpha_r^i$ is connected; {\em i.e.} is an abelian subvariety of $J_r$ ({\em cf.} Remark \ref{TilouineReln}). Note that we have a commutative diagram of abelian varieties over $\Q$ for $i=1,2$ \begin{equation} \addtocounter{equation}{1} \begin{gathered}
\xymatrix@C=50pt@R=26pt{
& {J_{r-1}}\ar@{->>}[r]^-{\alpha_{r-1}^i} &
{B_{r-1}^i} \ar@{=}[d] \\
{K_r^i} \ar@{^{(}->}[r]^-{\iota_r^i}\ar@{=}[d] &
{JY_r} \ar[r]^-{\alpha_{r-1}^i\circ \Alb(\pi_i)} \ar[d]|-{\Pic^0(\pi)}\ar[u]|-{\Alb(\pi_i)} &
B_{r-1}^i \\
K_r^i \ar[r]_-{\Pic^0(\pi)\circ \iota_r} & {J_r} \ar@{->>}[r]_-{\alpha_r^i} & {B_r^i}
} \end{gathered} \tag{$\arabic{section}.\arabic{subsection}.\arabic{equation}_i$} \label{BrDefiningDiag} \end{equation} with bottom two horizontal rows that are complexes.
\begin{warning}\label{GoodQuoWarning} While the bottom row of (\ref{BrDefiningDiag}) is exact in the middle {\em by definition} of $\alpha_r^i$, the central row is {\em not} exact in the middle: it follows from the fact that $\Alb(\pi_i)\circ\Pic^0(\pi_i)$ is multiplication by $p$ on $J_{r-1}$ that the component group of the kernel of $\alpha_{r-1}^i\circ\Alb(\pi_i):JY_r\rightarrow B_{r-1}^i$ is nontrivial with order divisible by $p$. Moreover, there is no mapping $B_{r-1}^i\rightarrow B_r^i$ which makes the diagram (\ref{BrDefiningDiag}) commute. \end{warning}
In order to be consistent with the literature, we adopt the following convention: \begin{definition}\label{BalphDef}
We set $B_r:=B_r^2$ and $B_r^*:=B_r^1$, with $B_r^i$ defined inductively by (\ref{BrDef}).
We likewise set $\alpha_r:=\alpha_r^2$ and $\alpha_r^*:=\alpha_r^1$. \end{definition}
\begin{remark}\label{TilouineReln}
We briefly comment on the relation between our quotient $B_r$
and the ``good" quotients of $J_r$ considered by Ohta \cite{OhtaEichler},
by Mazur-Wiles \cite{MW-Iwasawa}, and by Tilouine \cite{Tilouine}.
Recall \cite[\S2]{Tilouine} that Tilouine constructs\footnote{The notation Tilouine
uses for his quotient is the same as the notation we have used for our (slightly modified)
quotient. To avoid conflict, we have therefore chosen to
alter his notation.} an abelian variety quotient
$\alpha_r':J_r\twoheadrightarrow B_r'$ via an inductive
procedure nearly identical to the one used to define $B_r=B_r^1$:
one sets $K_1':=JY_1$, and for $r\ge 2$ defines \begin{equation*}
B_{r-1}':= J_{r-1}/\Pic^0(\pi)(K_{r-1}')
\qquad\text{and}\qquad
K_{r}':=\ker(JY_r \xrightarrow{\alpha_{r-1}'\circ\Alb(\pi_2)} B_{r-1}'). \end{equation*}
Using the fact that the
formation of images and identity components commutes,
one shows via a straightforward induction argument that
$\alpha_r:J_r\twoheadrightarrow B_r$
identifies $B_r$ with $J_r/(\ker\alpha_r')^0$; in particular,
our $B_r$ is the same as Ohta's \cite[\S3.2]{OhtaEichler}
and Tilouine's quotient $\alpha_r':J_r\rightarrow B_r'$ uniquely
factors through $\alpha_r$ via an isogeny $B_r\twoheadrightarrow B_r'$
which has degree divisible by $p$ by Warning \ref{GoodQuoWarning}.
Due to this fact, it is {\em essential} for our purposes to work with $B_r$ rather than $B_r'$.
Of course, following \cite[3.2.1]{OhtaEichler}, we could have simply
{\em defined} $B_r$ as $J_r/(\ker\alpha_r')^0$,
but we feel that the construction we have given is more natural.
On the other hand, we remark that $B_r$
is naturally a quotient of the ``good" quotient $J_r\twoheadrightarrow A_r$ constructed
by Mazur-Wiles in \cite[Chapter \Rmnum{3}, \S1]{MW-Iwasawa},
and the kernel of the corresponding surjective homomorphism
$A_r\twoheadrightarrow B_r$
is isogenous to $J_0\times J_0$. \end{remark}
\begin{proposition}\label{BrCotIden}
Over $F:=\Q(\mu_{Np^r})$, the automorphism $w_r$ of ${J_r}_{F}$ induces
an isomorphism of quotients ${B_r}_{F}\simeq {B_r^*}_{F}$.
The abelian variety $B_r$ $($respectively $B_r^*$$)$ is the unique quotient of $J_r$ by a $\Q$-rational
abelian subvariety with the property that
the induced map on cotangent spaces
\begin{equation*}
\xymatrix{
{\Cot(B_r)} \ar@{^{(}->}[r]_-{\Cot(\alpha_r)} &
{\Cot(J_r)\simeq S_2(\Upgamma_r;\Q)}
}
\quad\text{respectively}\quad
\xymatrix{
{\Cot(B_r^*)} \ar@{^{(}->}[r]_-{\Cot(\alpha_r^*)} &
{\Cot(J_r)\simeq S_2(\Upgamma_r;\Q)}
}
\end{equation*}
has image precisely $V_r$ $($respectively $V_r^*$$)$.
In particular, there are canonical actions
of the Hecke algebras\footnote{We must warn the reader that
Tilouine \cite{Tilouine} writes $\H_r(\Z)$ for the
$\Z$-subalgebra of $\End(J_r)$ generated by the Hecke operators
acting via the $(\cdot)^*$-action ({\em i.e.} by ``Picard" functoriality)
whereas our $\H_r(\Z)$ is defined using the $(\cdot)_*$-action.
This discrepancy is due primarily to the fact that Tilouine
identifies {\em tangent} spaces of modular abelian varieties
with spaces of modular forms, rather than cotangent spaces as is our convention.
Our notation for regarding Hecke algebras
as sub-algebras of $\End(J_r)$ agrees with that of Mazur-Wiles
\cite[Chapter \Rmnum{2}, \S5]{MW-Iwasawa}, \cite[\S7]{MW-Hida} and
Ohta \cite[3.1.5]{OhtaEichler}.
}
$\H_r(\Z)$ on $B_r$ and $\H_r^*(\Z)$ on $B_r^*$ for which $\alpha_r$ and $\alpha_r^*$
are equivariant. \end{proposition}
\begin{proof}
By the construction of $B_r^i$ and the fact that $\pr w_{r} = w_{r-1}\ps$
as maps ${X_{r}}_F \rightarrow {X_{r-1}}_F$ \cite[Proposition 2.2.6]{CaisHida1}
the automorphism $w_{r}$ of ${J_r}_F$ carries $\ker(\alpha_r)$
to $\ker(\alpha_r^*)$ and induces an isomorphsm ${B_r}_F \simeq {B_r^*}_F$ over $F$
that intertwines the action of $\H_r$ on $B_r$ with $\H_r^*$ on $B_r^*$.
The isogeny $B_r\twoheadrightarrow B_r'$ of Remark \ref{TilouineReln} induces
an isomorphism on cotangent spaces, compatibly with the inclusions into
$\Cot(J_r)$. Thus, the claimed identification of the image of
$\Cot(B_r)$ with $V_r$ follows from \cite[Proposition 2.1]{Tilouine}
(using \cite[Definition 2.1]{Tilouine}). The claimed uniqueness
of $J_r\twoheadrightarrow B_r$ follows easily from Lemma \ref{LieFactorization}
(\ref{LieFactorizationSurj}). Similarly, since the subspace $V_r$
of $S_2(\Upgamma_r)$ is stable under $\H_r$, we conclude from
Lemma \ref{LieFactorization} (\ref{LieFactorizationSurj}) that for any
$T\in \H_r(\Z)$, the induced morphism $J_r\xrightarrow{T} J_r\twoheadrightarrow B_r$
factors through $\alpha_r$, and hence that $\H_r(\Z)$ acts on $B_r$
compatibly (via $\alpha_r$) with its action on $J_r$. \end{proof}
\begin{lemma}\label{Btower}
There exist unique morphisms $B_r^*\leftrightarrows B_{r-1}^*$
of abelian varieties over $\Q$ making
\begin{equation*}
\xymatrix{
{J_{r}} \ar[r]^-{\alpha_r^*} \ar[d]_-{\Alb(\ps)} &{B_r^*} \ar[d] \\
{J_{r-1}} \ar[r]_-{\alpha_{r-1}^*} & {B_{r-1}^*}
}\qquad\raisebox{-18pt}{and}\qquad
\xymatrix{
{J_{r}} \ar[r]^-{\alpha_r^*} &{B_r^*} \\
{J_{r-1}} \ar[u]^-{\Pic^0(\pr)} \ar[r]_-{\alpha_{r-1}^*} & {B_{r-1}^*}\ar[u]
}
\end{equation*}
commute; these maps are moreover $\H_r^*(\Z)$-equivariant.
By a slight abuse of notation, we will simply write $\Alb(\ps)$ and $\Pic^0(\pr)$
for the induced maps on $B_r^*$ and $B_{r-1}^*$, respectively. \end{lemma}
\begin{proof}
Under the canonical identification of $\Cot(J_r)\otimes_{\Q}\o{\Q}$ with $S_2(\Upgamma_r)$,
the mapping on cotangent spaces induced by $\Alb(\ps)$ (respectively
$\Pic^0(\pr)$) coincides with $\iota_{\alpha}:S_2(\Upgamma_{r-1})\rightarrow S_2(\Upgamma_r)$
(respectively $\tr_{r,r-1}:S_2(\Upgamma_r)\rightarrow S_2(\Upgamma_{r-1})$).
As the kernel of $\alpha_r^*:J_r\twoheadrightarrow B_r^*$ is connected by definition,
thanks to Lemma \ref{LieFactorization} (\ref{LieFactorizationSurj}) it suffices
to prove that $\iota_{\alpha}$ (respectively $\tr_{r,r-1}$)
carries $V_{r-1}^*$ to $V_{r}^*$ (respectively $V_{r}^*$ to $V_{r-1}^*$).
On one hand, the composite
$\iota_{\alpha}\circ \iota_{\alpha^{r-1-i}}:S_2(\Upgamma_i,\varepsilon)\rightarrow S_2(\Upgamma_r)$
coincides with the embedding $\iota_{\alpha^{r-i}}$, and it follows immediately from the
definition of $V_r^*$ that $\iota_{\alpha}$ carries $V_{r-1}^*$ into $V_r^*$.
On the other hand, an easy calculation using (\ref{DualityIdentity}) shows that
one has equalities of maps $S_2(\Upgamma_i,\varepsilon)\rightarrow S_2(\Upgamma_r)$
\begin{equation*}
\iota_{\alpha}\circ \tr_{r,r-1}\circ \iota_{\alpha^{(r-i)}} = \begin{cases}
\iota_{\alpha^{(r-i)}}pU_p^* & \text{if}\ i< r \\
0 & \text{if}\ i=r
\end{cases}.
\end{equation*}
Thus,
the image of $\iota_{\alpha}\circ\tr_{r,r-1}:V_r^*\rightarrow S_2(\Upgamma_r)$
is contained in the image of $\iota_{\alpha}:V_{r-1}^*\rightarrow S_2(\Upgamma_r)$;
since $\iota_{\alpha}$ is injective, we conclude that
the image of $\tr_{r,r-1}:V_r^*\rightarrow S_2(\Upgamma_{r-1})$ is contained in $V_{r-1}^*$
as desired. \end{proof}
For $f'$ as in (\ref{projaway}), we write ${e^*}':=f'e^*\in \H^*$ and $e':=f'e\in \H$ the sub-idempotents of $e^*$ and $e$, respectively, corresponding to projection away from the trivial eigenspace of $\mu_{p-1}$.
\begin{proposition}\label{GoodRednProp}
The maps $\alpha_r$ and $\alpha_r^*$
induce isomorphisms of $p$-divisible groups over $\Q$
\begin{equation}
{e^*}'J_r[p^{\infty}] \simeq {e^*}'B_r^*[p^{\infty}]\quad\text{and}\quad
{e}'J_r[p^{\infty}] \simeq {e}'B_r[p^{\infty}],
\label{OrdBTisoms}
\end{equation}
respectively, that are $\H^*$ $($respectively $\H$$)$ equivariant
and compatible with change in $r$ via $\Alb(\ps)$ and $\Pic^0(\pr)$
$($respectively $\Alb(\pr)$ and $\Pic^0(\ps)$$)$. \end{proposition}
We view the maps (\ref{UrDefinition}) as endomorphisms of $J_r$ in the obvious way, and again write $U_r^*$ and $U_r$ for the induced endomorphism of $B_r^*$ and $B_r$, respectively. To prove Proposition \ref{GoodRednProp}, we need the following geometric incarnation of Corollary \ref{UpProjection}:
\begin{lemma}\label{UFactorDiagLem}
There exists a unique $\H_r^*(\Z)$ $($respectively $\H_r(\Z)$$)$-equivariant
map $W_r^*:B_r^*\rightarrow J_r$ $($respectively $W_r:B_r\rightarrow J_r$$)$
of abelian varieties over $\Q$ such that the diagram
\begin{equation}
\begin{gathered}
\xymatrix@C=30pt@R=35pt{
{J_r}\ar[d]_-{U_r^*}
\ar@{->>}[r]^-{\alpha_r^*} & {B_r^*} \ar[dl]|-{W_r^*} \ar[d]^-{U_r^*} \\
{J_r}\ar@{->>}[r]_-{\alpha_r^*} & {B_r^*}
}
\quad\raisebox{-24pt}{respectively}\quad
\xymatrix@C=30pt@R=35pt{
{J_r}\ar[d]_-{U_r}
\ar@{->>}[r]^-{\alpha_r} & {B_r} \ar[dl]|-{W_r} \ar[d]^-{U_r} \\
{J_r}\ar@{->>}[r]_-{\alpha_r} & {B_r}
}\label{UFactorDiag}
\end{gathered}
\end{equation}
commutes. \end{lemma}
\begin{proof}
Consider the endomorphism of $J_r$ given by $U_r$.
Due to Corollary \ref{UpProjection},
the induced mapping on cotangent spaces factors through the inclusion
$\Cot(B_r)\hookrightarrow \Cot(J_r)$. Since the kernel of the quotient
mapping $\alpha_r:J_r\twoheadrightarrow B_r$ giving rise to this inclusion is connected,
we conclude from Lemma \ref{LieFactorization} (\ref{LieFactorizationSurj})
that $U_r$ factors uniquely through $\alpha_r$
via an $\H_r$-equivariant morphism $W_r:B_r\rightarrow J_r$.
The corresponding statements for $B_r^*$ are proved similarly. \end{proof}
\begin{proof}[Proof of Proposition $\ref{GoodRednProp}$]
From (\ref{UFactorDiag}) we get commutative diagrams of $p$-divisible groups
over $\Q$
\begin{equation}
\begin{gathered}
\xymatrix{
{e^*}'{J_r}[p^{\infty}]\ar[d]_-{U_r^*}^-{\simeq} \ar[r]^-{\alpha_r^*} &
{e^*}'{B_r^*}[p^{\infty}] \ar[dl]|-{W_r^*} \ar[d]^-{U_r^*}_-{\simeq} \\
{e^*}'{J_r}[p^{\infty}]\ar[r]_-{\alpha_r^*} &
{e^*}'{B_r^*}[p^{\infty}]
}
\quad\raisebox{-24pt}{and}\quad
\xymatrix{
e'{J_r}[p^{\infty}]\ar[d]_-{U_r}^-{\simeq} \ar[r]^-{\alpha_r} &
e'{B_r}[p^{\infty}] \ar[dl]|-{W_r} \ar[d]^-{U_r}_-{\simeq} \\
e'{J_r}[p^{\infty}]\ar[r]_-{\alpha_r} &
e'{B_r}[p^{\infty}]
}
\label{UFactorDiagpDiv}
\end{gathered}
\end{equation}
in which all vertical arrows are isomorphisms due to the very definition of the
idempotents ${e^*}'$ and $e'$. An easy diagram chase then shows that {\em all}
arrows must be isomorphisms. \end{proof}
As in the introduction,
we put $K_r=\Q_p(\mu_{p^r})$, $K_r':=K_r(\mu_N)$ and write $R_r$ and $R_r'$ for the valuation rings of $K_r$ and $K_r'$, respectively. We set $\Gamma:=\Gal(K_{\infty}/K_0)$, and write $a:\Gal(K_0'/K_0)\rightarrow (\Z/N\Z)^{\times}$ the character giving the tautological action of $\Gal(K_0'/K_0)$ on $\mu_N$.
\begin{proposition}\label{GoodRedn}
The abelian varieties $B_r$ and $B_r^*$ acquire good reduction over $K_r$. \end{proposition}
\begin{proof}
See \cite[Chap \Rmnum{3}, \S2, Proposition 2]{MW-Iwasawa}
and {\em cf.} \cite[\S9, Lemma 9]{HidaGalois}. \end{proof}
We will write $\B_r$, $\B^*_r$, and $\J_r$, respectively, for the N\'eron models of the base changes $(B_r)_{K_r}$, $(B_r^*)_{K_r}$ and $(J_r)_{K_r}$ over $T_r:=\Spec(R_r)$; due to Proposition \ref{GoodRednProp}, both $\B_r$ and $\B_r^*$ are abelian schemes over $T_r$. By the N\'eron mapping property, there are canonical actions of $\H_r(\Z)$ on $\B_r$, $\J_r$ and of $\H_r^*(\Z)$ on $\B_r^*$, $\J_r$ over $R_r$ extending the actions on generic fibers as well as ``semilinear" actions of $\Gamma$ over the $\Gamma$-action on $R_r$ ({\em cf.} (\ref{GammaAction})). For each $r$, the N\'eron mapping property further provides diagrams \begin{equation}
\begin{gathered}
\xymatrix{
{\J_r \times_{T_r} T_{r+1}}\ar@<-1ex>[d]_{\Pic^0(\pr)} \ar[r]^-{\alpha_r^*}
& {\B_r^* \times_{T_r} T_{r+1}}\ar@<1ex>[d]^{\Pic^0(\pr)} \\
{\J_{r+1}} \ar[r]_-{\alpha_{r+1}^*} \ar@<-1ex>[u]_-{\Alb(\ps)} & \ar@<1ex>[u]^-{\Alb(\ps)} {\B_{r+1}^*}
}
\quad\raisebox{-24pt}{respectively}\quad
\xymatrix{
{\J_r \times_{T_r} T_{r+1}}\ar@<-1ex>[d]_{\Pic^0(\ps)} \ar[r]^-{\alpha_r}
& {\B_r \times_{T_r} T_{r+1}}\ar@<1ex>[d]^{\Pic^0(\ps)} \\
{\J_{r+1}} \ar[r]_-{\alpha_{r+1}}\ar@<-1ex>[u]_-{\Alb(\pr)} & \ar@<1ex>[u]^-{\Alb(\pr)} {\B_{r+1}}
}
\label{Nermaps}
\end{gathered} \end{equation} of smooth commutative group schemes over $T_{r+1}$ in which the inner and outer rectangles commute, and all maps are $\H_{r+1}^*(\Z)$ (respectively $\H_{r+1}(\Z)$) and $\Gamma$ equivariant.
\begin{definition}\label{ordpdivdefn}
We define $\G_r:={e^*}'\left(\B_r^*[p^{\infty}]\right)$ and
we write $\G_r':=\G_r^{\vee}$ for its Cartier dual,
each of which is canonically an object of $\pdiv_{R_r}^{\Gamma}$.
For each $r\ge s$, noting that $U_p^*$ is an automorphism of $\G_r$,
we obtain from (\ref{Nermaps}) canonical morphisms
\begin{equation}
\xymatrix@C=45pt{
{\rho_{r,s}:\G_{s}\times_{T_{s}} T_{r}} \ar[r]^-{\Pic^0(\pr)^{r-s}} & {\G_{r}}
}
\qquad\text{and}\qquad
\xymatrix@C=70pt{
{\rho_{r,s}' : \G_{s}'\times_{T_{s}} T_{r}} \ar[r]^-{{({U_p^*}^{-1}\Alb(\ps))^{\vee}}^{r-s}} & {\G_{r}'}
}\label{pdivTowers}
\end{equation}
in $\pdiv_{R_r}^{\Gamma}$, where $(\cdot)^{i}$ denotes the $i$-fold composition, formed in the obvious manner.
In this way, we get towers of $p$-divisible groups
$\{\G_r,\rho_{r,s}\}$ and $\{\G_r',\rho_{r,s}'\}$;
we will write $G_r$ and $G_r'$ for the unique descents of the generic fibers of $\G_r$
and $\G_r'$ to $\Q_p$, respectively.\footnote{Of course, $G_r'=G_r^{\vee}$.
Our non-standard notation $\G_r'$ for the Cartier
dual of $\G_r$ is preferrable, due to the fact that $\rho_{r,s}'$ is
{\em not} simply the dual of $\rho_{r,s}$; indeed, these two mappings go in opposite
directions!}
We let $T^*\in \H_r^*$ act on $\G_r$
through the action of $\H_r^*(\Z)$ on $\B_r^*$,
and on $\G_r'=\G_r^{\vee}$ by duality ({\em i.e.} as $(T^*)^{\vee}$).
The maps (\ref{pdivTowers}) are then $\H_r^*$-equivariant. \end{definition}
By Proposition \ref{GoodRednProp}, $G_r$ is canonically isomorphic to ${e^*}'J_r[p^{\infty}]$, compatibly with the action of $\H_r^*$. Since $J_r$ is a Jacobian---hence principally polarized---one might expect that $\G_r$ is isomorphic to its dual in $\pdiv_{R_r}^{\Gamma}$. However, this is {\em not quite} the case as the canonical isomorphism $J_r\simeq J_r^{\vee}$ intertwines the actions of $\H_r$ and $\H_r^*$, thus interchanging the idempotents ${e^*}'$ and $e'$. To describe the precise relationship between $\G_r^{\vee}$ and $\G_r$, we proceed as follows. For each $\gamma\in \Gal(K_{\infty}'/K_0)\simeq \Gamma\times \Gal(K_0'/K_0)$, let us write $\phi_{\gamma}: {G_r}_{K_r'}\xrightarrow{\simeq} \gamma^*({G_r}_{K_r'})$ for the descent data isomorphisms encoding the unique $\Q_p=K_0$-descent of ${G_r}_{K_r'}$ furnished by $G_r$. We ``twist" this descent data by the $\Aut_{\Q_p}(G_r)$-valued character $\langle \chi\rangle\langle a\rangle_N$ of $\Gal(K_{\infty}'/K_0)$: explicitly, for $\gamma\in \Gal(K_{r}'/K_0)$ we set $\psi_{\gamma}:= \phi_{\gamma}\circ \langle \chi(\gamma)\rangle\langle a(\gamma)\rangle_N$ and note that since $\langle \chi(\gamma)\rangle\langle a(\gamma)\rangle_N$ is defined over $\Q_p$, the map $\gamma\rightsquigarrow \psi_{\gamma}$ really does satisfy the cocycle condition. We denote by $G_r(\langle \chi\rangle\langle a\rangle_N)$ the unique $p$-divisible group over $\Q_p$ corresponding to this twisted descent datum. Since the diamond operators commute with the Hecke operators, there is a canonical induced action of $\H_r^*$ on $G_r(\langle \chi\rangle\langle a\rangle_N)$. By construction, there is a canonical $K_r'$-isomorphism $G_r(\langle \chi\rangle\langle a\rangle_N)_{K_r'}\simeq {G_r}_{K_r'}$. Since $G_r$ acquires good reduction over $K_r$ and the $\scrG_{K_r}$-representation afforded by the Tate module of $G_r(\langle \chi\rangle\langle a\rangle_N)$ is the twist of $T_pG_r$ by the {\em unramified} character $\langle a\rangle_N$, we conclude that $G_r(\langle \chi\rangle\langle a\rangle_N)$ also acquires good reduction over $K_r$, and we denote the resulting object of $\pdiv_{R_r}^{\Gamma}$ by $\G_r(\langle \chi\rangle\langle a\rangle_N)$.
\begin{proposition}\label{GdualTwist}
There is a natural $\H_r^*$-equivariant isomorphism of $p$-divisible groups over $\Q_p$
\begin{equation}
G_r' \simeq G_r(\langle \chi\rangle \langle a\rangle_N)
\label{GrprimeGr}
\end{equation}
which uniquely extends to an isomorphism of the corresponding objects in $\pdiv_{R_r}^{\Gamma}$
and is compatible with change in $r$ using $\rho_{r,s}'$ on $G_r'$ and $\rho_{r,s}$ on $G_r$. \end{proposition}
\begin{proof}
Let $\varphi_r: J_r\rightarrow J_r^{\vee}$ be the canonical principal polarization over $\Q_p$;
one then has the relation $\varphi_r\circ T = (T^*)^{\vee}\circ \varphi_r$
for each $T\in \H_r(\Z)$. On the other hand, the $K_r'$-automorphism
$w_r: {J_r}_{K_r'}\rightarrow {J_r}_{K_r'}$ intertwines $T\in \H_r(\Z)$ with $T^*\in \H_r^*(\Z)$.
Thus, the $K_r'$-morphism
\begin{equation*}
\xymatrix{
{\psi_r:{J_r}_{K_r'}^{\vee}} \ar[r]^-{({U_p^*}^{r})^{\vee}} &
{{J_r}_{K_r'}^{\vee}} \ar[r]^-{\varphi_r^{-1}}_{\simeq} & {J_r}_{K_r'}
\ar[r]^-{w_r}_{\simeq} & {{J_r}_{K_r'}}
}
\end{equation*}
is $\H_r^*(\Z)$-equivariant.
Passing to the induced map on $p$-divisible groups and applying ${e^*}'$, we
obtain from Proposition \ref{GoodRednProp} an $\H_r^*$-equivariant isomorphism
of $p$-divisible groups $\psi_r: {G_r'}_{K_r'} \simeq {G_r}_{K_r'}$. As
\begin{equation*}
\xymatrix@C=35pt{
{{J_r}_{K_r'}} \ar[r]^-{\langle \chi(\gamma)\rangle \langle a\rangle_N w_r}\ar[d]_-{1\times \gamma} &
{{J_r}_{K_r'}}\ar[d]^-{1\times \gamma} \\
{({J_r}_{K_r'})_{\gamma}} \ar[r]_-{\gamma^*(w_r)} & {({J_r}_{K_r'})_{\gamma}}
}
\end{equation*}
commutes for all $\gamma\in \Gal(K_r'/K_0)$ \cite[Proposition 2.2.6]{CaisHida1},
the $K_r'$-isomorphism $\psi_r$ uniquely descends to an
$\H_r^*$-equivariant isomorphism (\ref{GrprimeGr})
of $p$-divisible groups over $\Q_p$.
By Tate's Theorem, this identification
uniquely extends to an isomorphism of the corresponding objects in $\pdiv_{R_r}^{\Gamma}$.
The asserted compatibility with change in $r$ boils down to the commutativity of the diagrams
\begin{equation*}
\begin{gathered}
\xymatrix{
{{e^*}'J_s[p^{\infty}]^{\vee}} \ar[r]^-{({U_p^*}^{s})^{\vee}} \ar[d]_-{{({U_p^*}^{-1}\Alb(\ps))^{\vee}}^{r-s}}&
{{e^*}'J_s[p^{\infty}]^{\vee}} \ar[d]^-{{\Alb(\ps)^{\vee}}^{r-s}} \\ {{e^*}'J_r[p^{\infty}]^{\vee}} \ar[r]_-{({U_p^*}^{r})^{\vee}} & {{e^*}'J_r[p^{\infty}]^{\vee}} \\
}
\quad\raisebox{-22pt}{and}\quad
\xymatrix{
{{J_s}_{K_r'}^{\vee}} \ar[r]^-{\varphi_s^{-1}} \ar[d]_-{{\Alb(\ps)^{\vee}}^{r-s}}
& {J_s}_{K_r'} \ar[r]^-{w_s} \ar[d]|-{\Pic^0(\ps)^{r-s}} & {{J_s}_{K_r'}}\ar[d]^-{\Pic^0(\pr)^{r-s}}\\
{{J_r}_{K_r'}^{\vee}} \ar[r]_-{\varphi_r^{-1}}
& {J_r}_{K_r'} \ar[r]_-{w_r} & {{J_r}_{K_r'}}
}
\end{gathered}
\end{equation*}
for all $s\le r$. The commutativity of the first diagram is clear, while that of the second follows
from \cite[Proposition 2.2.6]{CaisHida1} and the fact that
for {\em any} finite morphism $f:Y\rightarrow X$ of smooth curves over a field $K$,
one has $\varphi_Y\circ \Pic^0(f)=\Alb(f)^{\vee}\circ \varphi_X$, where
$\varphi_{\star}:J_{\star}\rightarrow J_{\star}^{\vee}$ is the canonical
principal polarization on Jacobians for $\star=X,Y$ (see, for example, the proof of Lemma 5.5 in \cite{CaisNeron}). \end{proof}
We now wish to study the special fiber of $\G_r$, and relate it to the special fibers of the integral models of modular curves studied in \cite[2.2]{CaisHida1}. To that end, let $\X_r$ be the Katz--Mazur integral model of $X_r$ over $R_r$ defined in \cite[2.2.3]{CaisHida1}; it is a regular scheme that is proper and flat of pure relative dimension 1 over $\Spec R_r$ with smooth generic fiber naturally isomorphic to ${X_r}_{K_r}$.
According to \cite[Proposition 2.2.10]{CaisHida1}, the special fiber $\o{\X}_r:=\X_r\times_{R_r} \F_p$ is the ``disjoint union with crossings at the supersingular points" \cite[13.1.5]{KM} of smooth and proper Igusa curves $I_{(a,b,u)}:=\Ig_{\max(a,b)}$ indexed by triples $(a,b,u)$ with $a,b$ running over nonnegative integers that sum to $r$ and $u\in (\Z/p^{\min(a,b)}\Z)^{\times}$; in particular, $\o{\X}_r$ is geometrically {\em reduced}. We write $\nor{\o{\X}}_r$ for the normalization of $\o{\X}_r$, which is a disjoint union of Igusa curves $I_{(a,b,u)}$. The canonical semilinear action of $\Gamma$ on $\X_r$ that encodes the descent data of the generic fiber to $\Q_p$ \cite[2.2.3]{CaisHida1} induces, by base change, an $\F_p$-linear ``geometric inertia action" of $\Gamma$ on $\nor{\o{\X}}_r$; in this way the $p$-divisible group $\Pic^0_{\nor{\o{\X}}_r/\F_p}[p^{\infty}]$ of the Jacobian of $\nor{\o{\X}}_r$ over $\F_p$ is equipped with an action of $\Gamma$ over $\F_p$ and (via the Hecke correspondences \cite[2.2.21]{CaisHida1}) canonical actions of $\H_r$ and $\H_r^*$.
\begin{definition}\label{pDivGpSpecial}
Define $\Sigma_r:={e_r^*}'\Pic^0_{\nor{\o{\X}}_r/\F_p}[p^{\infty}]$,
equipped with the induced actions of $\H_r^*$ and $\Gamma$. \end{definition}
Since $\X_r$ is regular, and proper flat over $R_r$ with (geometrically) reduced special fiber, $\Pic^0_{\X_r/R_r}$ is a smooth $R_r$-scheme by \S8.4 Proposition 2 and \S9.4 Theorem 2 of \cite{BLR}. By the N\'eron mapping property, we thus have a natural mapping $\Pic^0_{\X_r/R_r}\rightarrow \J_r^0$ that recovers the canonical identification on generic fibers, and is in fact an isomorphism by \cite[\S9.7, Theorem 1]{BLR}. Composing with the map $\alpha_r^*:\J_r\rightarrow \B_r^*$ and passing to special fibers yields a homomorphism of smooth commutative algebraic groups over $\F_p$ \begin{equation}
\xymatrix{
{\Pic^0_{\o{\X}_r/\F_p}} \ar[r]^-{\simeq} & {\o{\J}_r^0} \ar[r] & {\o{\B}^*_r}
}\label{PicToB} \end{equation} Due to \cite[\S9.3, Corollary 11]{BLR}, the normalization map $\nor{\o{\X}}_r\rightarrow \o{\X}$ induces a surjective homomorphism $\Pic^0_{\o{\X}_r/\F_p}\rightarrow {\Pic^0_{\nor{\o{\X}}_r/\F_p}}$ with kernel that is a smooth, connected {\em linear} algebraic group over $\F_p$. As any homomorphism from an affine group variety to an abelian variety is zero, we conclude that (\ref{PicToB}) uniquely factors through this quotient, and we obtain a natural map of abelian varieties: \begin{equation}
\xymatrix{
{\Pic^0_{\nor{\o{\X}}_r/\F_p}} \ar[r] & {\o{\B}_r^*}
}\label{AbVarMaps} \end{equation} that is necessarily equivariant for the actions of $\H_r^*(\Z)$ and $\Gamma$. The following Proposition relates the special fiber of $\G_r$ to the $p$-divisible group $\Sigma_r$ of Definition \ref{pDivGpSpecial}, and will allow us in Corollary \ref{SpecialFiberOrdinary1} to give an explicit description of the special fiber of $\G_r$.
\begin{proposition}\label{SpecialFiberDescr}
The mapping $(\ref{AbVarMaps})$ induces an isomorphism
of $p$-divisible groups over $\F_p$
\begin{equation}
\o{\G}_r := {{e^*}'\o{\B}_r^*[p^{\infty}]} \simeq
{{e^*}'\Pic^0_{\nor{\o{\X}}_r/\F_p}[p^{\infty}]}=:\Sigma_r
\label{OnTheNose}
\end{equation}
that is $\H_r^*$ and $\Gamma$-equivariant and compatible with change in $r$
via the maps $\pr_{r,s}$ on $\o{\G}_r$ and the maps $\Pic^0(\pr)^{r-s}$ on $\Sigma_r$. \end{proposition}
\begin{proof}
The diagram (\ref{UFactorDiag}) induces a corresponding diagram of
N\'eron models over $R_r$ and hence of special fibers over $\F_p$.
Arguing as above, we obtain a commutative diagram of abelian
varieties
\begin{equation}
\begin{gathered}
\xymatrix@C=30pt@R=35pt{
{\Pic^0_{\nor{\o{\X}}_r/\F_p}}\ar[d]_-{U_r^*}
\ar[r]^-{\o{\alpha}^*_r} & {\o{\B}_r^*} \ar[dl]^-{W_r^*} \ar[d]^-{U_r^*} \\
{\Pic^0_{\nor{\o{\X}}_r/\F_p}}\ar[r]_-{\o{\alpha}^*_r} & {\o{\B}_r^*}
}\label{UFactorDiagmodp}
\end{gathered}
\end{equation}
over $\F_p$. The proof of \ref{GoodRednProp} now goes through {\em mutatis mutandis} to give the claimed
isomorphism (\ref{OnTheNose}).
\end{proof}
In \cite[\S2.5]{CaisHida1}, we analyzed the the structure of the de Rham cohomology of the smooth and proper curve $\nor{\o{\X}}_r$ over $\F_p$; we now apply this analysis and Oda's description (Proposition \ref{OdaDieudonne}) of Dieudonn\'e modules in terms of de Rham cohomology to understand the structure of $\Sigma_r$. For each $r$, as in \cite[Remark 2.2.12]{CaisHida1} we write $I_r^{\infty}:=I_{(r,0,1)}$ and $I_r^0:=I_{(0,r,1)}$ for the two ``good" irreducible components of $\o{\X}_r$; by \cite[Proposition 2.5.6]{CaisHida1}, the ordinary part of the de Rham cohomology of $\nor{\o{\X}}_r$ is entirely captured by the de Rham cohomology of these two good components. Writing $i_r^{\star}:I_r^{\star}\hookrightarrow \nor{\o{\X}}_r$ for the canonical closed immersions, we reinterpret this fact in the language of Dieudonn\'e modules:
\begin{proposition}\label{GisOrdinary}
For each $r$,
there is a natural isomorphism of $A:=\Z_p[F,V]$-modules
\begin{equation}
\D(\Sigma_r)_{\F_p} \simeq {e_r^*}'H^1_{\dR}(\nor{\o{\X}}_r/\F_p)\simeq
f'H^0(I_r^{\infty},\Omega^1)^{V_{\ord}}\oplus f'H^1(I_r^0,\O)^{F_{\ord}}.\label{DieudonneDesc}
\end{equation}
which is compatible with $\H_r^*$, $\Gamma$, and change in $r$ and which
carries $\D(\Sigma_r^{\mult})_{\F_p}$ $($respectively $\D(\Sigma_r^{\et})_{\F_p}$$)$ isomorphically
onto $f'H^0(I_r^{0},\Omega^1)^{V_{\ord}}$ $($respectively
$f'H^1(I_r^{\infty},\O)^{F_{\ord}}$$)$. In particular,
$\Sigma_r$ is ordinary. \end{proposition}
\begin{proof}
The identifications of \cite[Proposition 2.5.6]{CaisHida1}
are induced by the closed immersions $i_r^{\star}$
and are therefore compatible with the natural actions of Frobenius and the Cartier operator.
The isomorphism (\ref{DieudonneDesc}) is then an immediate consequence of
Proposition \ref{OdaDieudonne} and \cite[Proposition 2.5.6]{CaisHida1}. Since this isomorphism is
compaible with $F$ and $V$, we have
\begin{subequations}
\begin{equation}
\D(\Sigma_r^{\mult})_{\F_p}
\simeq \D(\Sigma_r)_{\F_p}^{V_{\ord}}
\simeq f'H^0(I_r^{0},\Omega^1)^{V_{\ord}}
\end{equation}
and
\begin{equation}
\D(\Sigma_r^{\et})\otimes_{\Z_p}\F_p
\simeq \D(\Sigma_r)_{\F_p}^{F_{\ord}}
\simeq f'H^1(I_r^{\infty},\O)^{F_{\ord}}
\end{equation}
\end{subequations}
and we conclude that the canonical inclusion
$\D(\Sigma_r^{\mult})_{\Z_p}\oplus\D(\Sigma_r^{\et})_{\Z_p}\hookrightarrow \D(\Sigma_r)_{\Z_p}$
is surjective, whence $\Sigma_r$ is ordinary by Dieudonn\'e theory. \end{proof}
With Proposition \ref{GisOrdinary} as a starting point, we can now completely describe the structure of $\Sigma_r$ in terms of the two good components $I_r^{\star}$.
Since $\nor{\o{\X}}_r$ is the disjoint union of proper smooth and irreducible Igusa curves $I_{(a,b,u)}$,
we have a canonical identification of abelian varieties over $\F_p$ \begin{equation}
\Pic^0_{\nor{\o{\X}}_r/\F_p} = \prod_{(a,b,u)} \Pic^0_{I_{(a,b,u)}/\F_p}.
\label{Pic0Iden} \end{equation} For $\star=0,\infty$ let us write $j_r^{\star}:=\Pic^0_{I_r^{\star}/\F_p}$ for the Jacobian of $I_r^{\star}$ over $\F_p$. The canonical closed immersions $i_r^{\star}:I_r^{\star}\hookrightarrow \nor{\o{\X}}_r$ yield (by Picard and Albanese functoriality) homomorphisms of abelian varieties over $\F_p$ \begin{equation}
\xymatrix{
{\Alb(i_r^{\star}):j_r^{\star}} \ar[r] & {\Pic^0_{\nor{\o{\X}}_r/\F_p}}
}
\quad\text{and}\quad
\xymatrix{
{\Pic^0(i_r^{\star}):\Pic^0_{\nor{\o{\X}}_r/\F_p}} \ar[r] & {j_r^{\star}}
}.\label{AlbPicIncl} \end{equation} Via the identification (\ref{Pic0Iden}), we know that $j_r^{\star}$ is a direct factor of $\Pic^0_{\nor{\o{\X}}_r/\F_p}$; in these terms $\Alb(i_r^{\star})$ is the unique mapping which projects to the identity on $j_r^{\star}$ and to the zero map on all other factors, while $\Pic^0(i_r^{\star})$ is simply projection onto the factor $j_r^{\star}$. As $\Sigma_r$ is a direct factor of ${f' \Pic^0_{\nor{\o{\X}}_r/\F_p}[p^{\infty}]}$, these mappings induce homomorphisms of $p$-divisible groups over $\F_p$ \begin{subequations} \begin{equation}
\xymatrix@C=35pt{
{f'j_r^{0}[p^{\infty}]^{\mult}} \ar[r]^-{\Alb(i_r^{0})} &
{f' \Pic^0_{\nor{\o{\X}}_r/\F_p}[p^{\infty}]^{\mult}} \ar[r]^-{\proj} &
{\Sigma_r^{\mult}}
}\label{Alb0} \end{equation} \begin{equation}
\xymatrix@C=35pt{
{\Sigma_r^{\et}} \ar[r]^-{\incl} &
{f' \Pic^0_{\nor{\o{\X}}_r/\F_p}[p^{\infty}]^{\et}} \ar[r]^-{\Pic^0(i_r^{\infty})} &
{f'j_r^{\infty}[p^{\infty}]^{\et}}
}\label{Picinfty} \end{equation} \end{subequations} which we (somewhat abusively) again denote by $\Alb(i_r^{0})$ and $\Pic^0(i_r^{\infty})$, respectively. The following is a sharpening of \cite[Chapter 3, \S3, Proposition 3]{MW-Iwasawa} (see also \cite[Proposition 3.2]{Tilouine}):
\begin{proposition}\label{MWSharpening}
The mappings $(\ref{Alb0})$ and $(\ref{Picinfty})$ are isomorphisms. They induce
a canonical split short exact sequences of $p$-divisible groups over $\F_p$
\begin{equation}
\xymatrix@C=45pt{
0 \ar[r] & {f'j^0_r[p^{\infty}]^{\mult}} \ar[r]^-{\Alb(i_r^0)\circ V^r} &
{\Sigma_r} \ar[r]^-{\Pic^0(i_r^{\infty})} & {f'j^{\infty}_r[p^{\infty}]^{\et}} \ar[r] & 0
}\label{pDivUpic}
\end{equation}
which is:
\begin{enumerate}
\item $\Gamma$-equivariant for the geometric inertia action on $\Sigma_r$, the trivial
action on $f'j_r^{\infty}[p^{\infty}]^{\et}$, and the action via $\langle \chi(\cdot) \rangle^{-1}$
on $f'j_r^{0}[p^{\infty}]^{\mult}$. \label{GammaCompatProp}
\item $\H_r^*$-equivariant with $U_p^*$ acting on $f'j_r^{\infty}[p^{\infty}]^{\et}$
as $F$ and on $f'j_r^{0}[p^{\infty}]^{\mult}$ as $\langle p\rangle_N V$.
\item Compatible with change in $r$ via the mappings $\Pic^0(\pr)$ on $j_r^{\star}$ and $\Sigma_r$.
\label{ChangerProp} \end{enumerate} \end{proposition}
\begin{proof}
It is clearly enough to prove that the sequence (\ref{pDivUpic}) induced by $(\ref{Alb0})$ and $(\ref{Picinfty})$
is exact.
Since the contravariant Dieudonn\'e module functor from the category of $p$-divisible groups
over $\F_p$ to the category of $A$-modules which are $\Z_p$ finite and free is an exact
anti-equivalence, it suffices to prove such exactness
after applying $\D(\cdot)_{\Z_p}$.
As the resulting sequence consists of finite free $\Z_p$-modules,
exactness may be checked modulo $p$ where it follows immediately from
Proposition \ref{GisOrdinary} by using \cite[Proposition 2.5.6]{CaisHida1}.
The claimed compatibility with $\Gamma$, $\H_r^*$, and change in $r$ follows easily
from Propositions 2.2.14, 2.2.20 and 2.2.13 of \cite{CaisHida1}, respectively. \end{proof}
\begin{remark}
It is possible to give a short proof of Proposition \ref{MWSharpening}
along the lines of \cite{MW-Iwasawa} or \cite{Tilouine} by using
\cite[Proposition 2.2.20]{CaisHida1} directly.
We stress, however, that our approach via Dieudonn\'e modules gives more refined information,
most notably that the Dieudonn\'e module of $\Sigma_r[p]$ is free as an $\F_p[\Delta/\Delta_r]$-module.
This fact will be crucial in our later arguments. \end{remark}
Together, Proposotions \ref{SpecialFiberDescr} and \ref{MWSharpening} give the desired description of the special fiber of $\G_r$
({\em cf}. \S3 and \S4, Proposition 1 of \cite{MW-Hida} and pgs. 267--274 of \cite{MW-Iwasawa}):
\begin{corollary}\label{SpecialFiberOrdinary1}
For each $r$, the $p$-dividible group $\G_r/R_r$ is ordinary, and there is a canonical exact sequence,
compatible with change in $r$ via $\pr_{r,s}$ on $\o{\G}_r$ and $\Pic^0(\pr)^{r-s}$ on $j_r^{\star}[p^{\infty}]$
\begin{equation}
\xymatrix@C=45pt{
0 \ar[r] & {f'j_r^{0}[p^{\infty}]^{\mult}} \ar[r]^-{\Alb(i_r^{0})\circ V^r} &
{\o{\G}_r} \ar[r]^-{\Pic^0(i_r^{\infty})} & {f'j_r^{\infty}[p^{\infty}]^{\et}} \ar[r] & 0
}\label{GrSpecialExact}
\end{equation}
where $i_r^{\star}:I_r^{\star}\hookrightarrow \nor{\o{\X}}_r$ are the canonical closed immersions
for $\star=0,\infty$. Moreover, $(\ref{GrSpecialExact})$ is compatible with the actions of
$\H^*$ and $\Gamma$, with $U_p^*$ $($respectively $\gamma\in \Gamma$$)$
acting on $f'j_r^{0}[p^{\infty}]^{\mult}$ as $\langle p\rangle_N V$
$($respectively $\langle \chi(\gamma)\rangle^{-1}$$)$
and on $f'j_r^{\infty}[p^{\infty}]^{\et}$ as $F$ $($respectively $\id$$)$.
\end{corollary}
\subsection{Ordinary families of Dieudonn\'e modules}\label{OrdDieuSection}
Let $\{\G_r/R_r\}_{r\ge 1}$ be the tower of $p$-divisible groups given by Definition \ref{ordpdivdefn}. From the canonical morphisms $\rho_{r,s}: \G_{s}\times_{T_{s}} T_r\rightarrow \G_{r}$ we obtain a map on special fibers $\o{\G}_{s}\rightarrow \o{\G}_r$ over $\F_p$ for each $r\ge s$; applying the contravariant Dieudonn\'e module functor $\D(\cdot):=\D(\cdot)_{\Z_p}$ yields a projective system of finite free $\Z_p$-modules $\{\D(\o{\G}_r)\}_r$ with compatible linear endomorphisms $F,V$ satisfying $FV=VF=p$.
\begin{definition}\label{DinftyDef}
We write $\D_{\infty}:=\varprojlim_r \D(\o{\G}_r)$ for the projective limit
of the system $\{\D(\o{\G}_r)\}_r$. For $\star\in \{\et,\mult\}$
we write $\D_{\infty}^{\star}:=\varprojlim_r \D(\o{\G}_r^{\star})$
for the corresponding projective limit. \end{definition}
Since $\H_r^*$ acts by endomorphisms on $\o{\G}_r$, compatibly with change in $r$, we obtain an action of $\H^*$ on $\D_{\infty}$ and on $\D_{\infty}^{\star}$. Likewise, the ``geometric inertia action" of $\Gamma$ on $\o{\G}_r$
gives an action of $\Gamma$ on $\D_{\infty}$ and $\D_{\infty}^{\star}$. As $\o{\G}_r$ is ordinary thanks to Corollary \ref{SpecialFiberOrdinary1}, applying $\D(\cdot)$ to the (split) connected-\'etale squence of $\o{\G}_r$ gives, for each $r$, a functorially split exact sequence \begin{equation}
\xymatrix{
0 \ar[r] & {\D(\o{\G}_r^{\et})} \ar[r] & {\D(\o{\G}_r)} \ar[r] &
{\D(\o{\G}_r^{\mult})} \ar[r] & 0
}\label{DieudonneFiniteExact} \end{equation} with $\Z_p$-linear actions of $\Gamma$, $F$, $V$, and $\H_r^*$. Since projective limits commute with finite direct sums, we obtain a split short {\em exact} sequence of $\Lambda$-modules with linear $\H^*$ and $\Gamma$-actions and commuting linear endomorphisms $F,V$ satisfying $FV=VF=p$: \begin{equation}
\xymatrix{
0 \ar[r] & {\D_{\infty}^{\et}} \ar[r] & {\D_{\infty}} \ar[r] & {\D_{\infty}^{\mult}} \ar[r] & 0
}.\label{DieudonneInfiniteExact} \end{equation}
\begin{theorem}\label{MainDieudonne}
Let
$d':=\sum_{k=3}^p d_k$ for $d_k:=\dim_{\F_p}S_k(\Upgamma_1(N);\F_p)^{\ord}$ the $\F_p$-dimension
of the space of $p$-ordinary mod $p$ cuspforms of weight $k$ and level $N$. Then:
\begin{enumerate}
\item $\D_{\infty}$ is a free $\Lambda$-module of rank $2d'$, and $\D_{\infty}^{\star}$
is free of rank $d'$ over $\Lambda$ for $\star\in \{\et,\mult\}$.
\label{MainDieudonne1}
\item For each $r\ge 1$, applying $\otimes_{\Lambda} \Z_p[\Delta/\Delta_r]$ to
$(\ref{DieudonneInfiniteExact})$ yields the short exact sequence $(\ref{DieudonneFiniteExact})$,
compatibly with $\H^*$, $\Gamma$, $F$ and $V$.
\label{MainDieudonne2}
\item Under the canonical splitting of $(\ref{DieudonneInfiniteExact})$, $\D_{\infty}^{\et}$
is the maximal subspace of $\D_{\infty}$ on which $F$ acts invertibly, while
$\D_{\infty}^{\mult}$ corresponds to the maximal subspace of $\D_{\infty}$ on which $V$ acts
invertibly.
\label{MainDieudonne3}
\item The Hecke operator $U_p^*$ acts as $F$ on $\D_{\infty}^{\et}$ and as $\langle p\rangle_NV$ on $\D_{\infty}^{\mult}$.
\label{MainDieudonne4}
\item $\Gamma$ acts trivially on $\D_{\infty}^{\et}$ and via $\langle \chi\rangle^{-1}$
on $\D_{\infty}^{\mult}$.
\label{MainDieudonne5} \end{enumerate} \end{theorem}
\begin{proof}
In \cite[\S 3.1]{CaisHida1}, we established a general commutative algebra
formalism for dealing with projective
limits of modules and proving structural and control theorems
as in (\ref{MainDieudonne1}) and (\ref{MainDieudonne2}), respectively.
In order to apply the main result
of our formalism to the present situation, we take (in the notation
of \cite[Lemma 3.1.2]{CaisHida1}) $A_r=\Z_p$, $I_r=(p)$, and
$M_r$ each one of the terms in (\ref{DieudonneFiniteExact}),
and we must check that the hypotheses
\begin{enumerate}
\setcounter{equation}{3}
\renewcommand{\theequation{\rm\alph{enumi}}}{\theequation{\rm\alph{enumi}}}
{\setlength\itemindent{10pt}
\item $\o{M}_r:=M_r/pM_r$ is a free $\F_p[\Delta/\Delta_r]$-module of rank
$d'$\label{freehyp1}}
{\setlength\itemindent{10pt}
\item For all $s\le r$ the induced transition maps
$\xymatrix@1{
{\overline{\pr}_{r,s}: \o{M}_r}\ar[r] &
{\o{M}_{s}}
}$\label{surjhyp1}}
are surjective
\end{enumerate}
hold. By Propositions \ref{SpecialFiberDescr} and \ref{GisOrdinary},
there is a natural isomorphism of split short exact sequences
\begin{equation*}
\xymatrix{
0 \ar[r] & {\D(\o{\G}_r^{\et})_{\F_p}} \ar[r]\ar[d]^-{\simeq} &
{\D(\o{\G}_r)_{\F_p}} \ar[r] \ar[d]^-{\simeq}&
{\D(\o{\G}_r^{\mult})_{\F_p}} \ar[r] \ar[d]^-{\simeq}& 0 \\
0 \ar[r] & {f'H^1(I_r^0,\O)^{F_{\ord}}}\ar[r] &
{f'H^0(I_r^{\infty},\Omega^1)^{V_{\ord}}\oplus f'H^1(I_r^0,\O)^{F_{\ord}}} \ar[r] &
{f'H^0(I_r^{\infty},\Omega^1)^{V_{\ord}}} \ar[r] & 0
}
\end{equation*}
that is compatible with change in $r$ using the trace mappings attached to
$\rho:I_r^{\star}\rightarrow I_{s}^{\star}$ and the maps on Dieudonn\'e modules
induced by $\o{\rho}_{r,s}:\o{\G}_{s} \rightarrow \o{\G}_r$.
The hypotheses (\ref{freehyp1}) and (\ref{surjhyp1})
are therefore satisfied
thanks to Proposition 2.4.1
and Lemma 2.5.5 of \cite{CaisHida1}.
It follows that the conclusions of \cite[Lemma 3.1.2]{CaisHida1}
hold in the present situation, which gives
(\ref{MainDieudonne1}) and (\ref{MainDieudonne2}).
As $F$ (respectively $V$) acts invertibly on $\D(\o{\G}_r^{\et})$ (respectively
$\D(\o{\G}_r^{\mult})$) for all $r$, assertion (\ref{MainDieudonne3}) is clear, while
(\ref{MainDieudonne4}) and (\ref{MainDieudonne5}) follow immediately from
Corollary \ref{SpecialFiberOrdinary1}. \end{proof}
The short exact sequence (\ref{DieudonneInfiniteExact}) is very nearly ``auto dual":
\begin{proposition}\label{DieudonneDuality}
There is a canonical isomorphism of short exact sequences of $\Lambda_{R_0'}$-modules
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] & {\D_{\infty}^{\et}(\langle \chi \rangle\langle a\rangle_N)_{\Lambda_{R_0'}}} \ar[r]\ar[d]^-{\simeq} &
{\D_{\infty}(\langle \chi \rangle\langle a\rangle_N)_{\Lambda_{R_0'}}}\ar[r]\ar[d]^-{\simeq} &
{\D_{\infty}^{\mult}(\langle \chi \rangle\langle a\rangle_N)_{\Lambda_{R_0'}}}\ar[r]\ar[d]^-{\simeq} & 0 \\
0\ar[r] & {(\D_{\infty}^{\mult})^{\vee}_{\Lambda_{R_0'}}} \ar[r] &
{(\D_{\infty})^{\vee}_{\Lambda_{R_0'}}} \ar[r] &
{(\D_{\infty}^{\et})^{\vee}_{\Lambda_{R_0'}}}\ar[r] & 0
}
\end{gathered}
\label{DmoduleDuality}
\end{equation}
that is $\H^*$ and $\Gamma\times \Gal(K_0'/K_0)$-equivariant,
and intertwines $F$
$($respectively $V$$)$ on the top row with $V^{\vee}$
$($respectively $F^{\vee}$$)$ on the bottom. \end{proposition}
\begin{proof}
As in the proof of Theorem \ref{MainDieudonne},
we apply the formalism of \cite[\S3.1]{CaisHida1}.
Let us write
$\pr_{r,s}':\o{\G}_r'\rightarrow \o{\G}_s'$ for the maps on special fibers
induced by (\ref{pdivTowers}).
Thanks to Proposition \ref{GdualTwist}, the definition \ref{ordpdivdefn} of $\o{\G}_r':=\o{\G}_r^{\vee}$,
the natural isomorphism $\G_r\times_{R_r} R_r' \simeq \G_r(\langle \chi\rangle\langle a\rangle_N)\times_{R_r} R_r'$,
and the compatibility of the Dieudonn\'e module functor with duality, there are natural
isomorphisms of $R_0'$-modules
\begin{equation}
\D(\o{\G}_r)(\langle \chi\rangle\langle a\rangle_N) \tens_{\Z_p} R_0' \simeq
\D(\o{\G_r(\langle \chi\rangle\langle a\rangle_N)})\tens_{\Z_p} R_0'
\simeq \D(\o{\G}_r')\tens_{\Z_p} R_0' = \D(\o{\G}_r^{\vee})\tens_{\Z_p} R_0'\simeq
(\D(\o{\G}_r))_{R_0'}^{\vee}
\label{evpairingDieudonne}
\end{equation}
that are $\H^*_r$-equivariant, $\Gal(K_r'/K_0)$-compatible
for the standard action $\sigma\cdot f (m):=\sigma f(\sigma^{-1}m)$
on the $R_0'$-linear dual of $\D(\o{\G}_r)\otimes_{\Z_p} R_0'$,
and compatible with change in $r$ using $\pr_{r,s}$
on $\D(\o{\G}_r)$ and $\pr_{r,s}'$ on $\D(\o{\G}_r')$. We claim that the resulting
perfect ``evaluation" pairings
\begin{equation}
\xymatrix{
{\langle\cdot,\cdot\rangle_r : \D(\o{\G}_r)(\langle \chi\rangle\langle a\rangle_N)\tens_{\Z_p}{R_0'}
\times \D(\o{\G}_r)\tens_{\Z_p}{R_0'}} \ar[r] & {R_0'}
}\label{pdivSpecialTwistPai}
\end{equation}
are compatible with change in $r$ via the maps $\rho_{r,s}$
and $\rho_{r,s}'$ in the sense of \cite[3.1.4]{CaisHida1}; {\em i.e.} that
\begin{equation}
\langle\pr_{r,s}x, \pr_{r,s}'y\rangle_{s} =
\sum_{\delta\in \Delta_{s}/\Delta_{r}} \langle x,\delta^{-1} y\rangle_{r}
\label{pairingchangeinr1}
\end{equation}
holds for all $x,$ $y$.
Indeed, the compatibility of (\ref{evpairingDieudonne}) with change in $r$
and the very definition (\ref{pdivTowers}) of the transition maps $\pr_{r,s}'$
implies that for $r\ge s$
\begin{equation}
\langle \D(\Pic^0(\pr)^{r-s}) x, y\rangle_s = \langle x , \D({U_p^*}^{s-r}\Alb(\ps)^{r-s}) y\rangle_r;
\label{PairingCompatComp}
\end{equation}
on the other hand, it follows from
Lemma \ref{MFtraceLem} (using Lemma \ref{LieFactorization}) that
we have
\begin{equation}
\Pic(\pr)\circ \Alb(\ps) = U_p^* \sum_{\delta\in \Delta_r/\Delta_{r+1}} \langle \delta^{-1}\rangle,
\label{PicAlbRelation}
\end{equation}
in $\End_{\Q_p}(J_{r+1})$, and together (\ref{PairingCompatComp})--(\ref{PicAlbRelation})
imply the desired compatibility (\ref{pairingchangeinr1}).
It follows that the hypotheses of \cite[Lemma 3.1.4]{CaisHida1} is verified, and we conclude
that the pairings (\ref{pdivSpecialTwistPai}) give rise to
a perfect $\Gal(K_{\infty}'/K_0)$-compatible duality pairing
$\langle\cdot,\cdot \rangle: \D_{\infty}(\langle \chi\rangle\langle a\rangle_N)
\otimes_{\Lambda} \Lambda_{R_0'} \times \D_{\infty}\otimes_{\Lambda} \Lambda_{R_0'} \rightarrow \Lambda_{R_0'}$
with respect to which $T^*$ is self-adjoint for all $T^*\in \H^*$
as this is true at each finite level $r$ thanks to
the $\H_r^*$-compatibility of (\ref{evpairingDieudonne}).
That the resulting isomorphism (\ref{DmoduleDuality})
intertwines
$F$ with $V^{\vee}$
and $ V $ with $F^{\vee}$ is an immediate consequence of the compatibility of the Dieudonn\'e module
functor with duality. \end{proof}
We can interpret $\D_{\infty}^{\star}$ in terms of the crystalline cohomology of the Igusa tower as follows. Let $I_r^0$ and $I_r^{\infty}$ be the two ``good" components of $\o{\X}_r$ as in the discussion preceding Proposition \ref{GisOrdinary}, and form the projective limits \begin{equation*}
H^1_{\cris}(I^{\star}) := \varprojlim_{r} H^1_{\cris}(I_r^{\star}) \end{equation*} for $\star\in \{\infty,0\}$, taken with respect to the trace maps on crystalline cohomology (see \cite[\Rmnum{7}, \S2.2]{crystal2}) induced by the canonical degeneracy mappings $\rho:I_{r}^{\star}\rightarrow I_{s}^{\star}$. Then $H^1_{\cris}(I^{\star})$ is naturally a $\Lambda$-module (via the diamond operators), equipped with a commuting action of $F$ (Frobenius) and $V$ (Verscheibung) satisfying $FV=VF=p$. Letting $U_p^*$ act as $F$ (respectively $\langle p\rangle_N V$) on $H^1_{\cris}(I^{\star})$ for $\star=\infty$ (respectively $\star=0$) and the Hecke operators outside $p$ (viewed as correspondences on the Igusa curves) act via pullback and trace at each level $r$, we obtain an action of $\H^*$ on $H^1_{\cris}(I^{\star})$. Finally, we let $\Gamma$ act trivially on $H^1_{\cris}(I^{\star})$ for $\star=\infty$ and via $\langle\chi^{-1}\rangle$ for $\star=0$.
\begin{theorem}\label{DieudonneCrystalIgusa}
There is a canonical $\H^*$ and $\Gamma$-equivariant
isomorphism of $\Lambda$-modules
\begin{equation*}
\D_{\infty} = \D_{\infty}^{\mult}\oplus \D_{\infty}^{\et} \simeq
f'H^1_{\cris}(I^{0})^{V_{\ord}} \oplus
f'H^1_{\cris}(I^{\infty})^{F_{\ord}}
\end{equation*}
which respects the given direct sum decompositions and is compatible with $F$ and $V$. \end{theorem}
\begin{proof}
From the exact sequence (\ref{GrSpecialExact}), we obtain for each $r$ isomorphisms
\begin{equation}
\xymatrix@C=55pt{
{\D(\o{\G}_r^{\mult})} \ar[r]^-{\simeq}_-{V^r \circ \D(\Alb(i_r^{0}))} &
{f'\D(j_r^{0}[p^{\infty}])^{V_{\ord}}}
}\qquad\text{and}\qquad
\xymatrix@C=55pt{
{f'\D(j_r^{\infty}[p^{\infty}])^{F_{\ord}}} \ar[r]^-{\simeq}_-{\D(\Pic^0(i_r^{\infty}))} &
{\D(\o{\G}_r^{\et})}
}\label{IgusaInterpretation}
\end{equation}
that are $\H^*$ and $\Gamma$-equivariant (with respect to the actions
specified in Corollary \ref{SpecialFiberOrdinary1}), and compatible with change in $r$
via the mappings $\D(\pr_{r,s})$ on $\D(\o{\G}_r^{\star})$ and $\D(\pr)$
on $\D(j_r^{\star}[p^{\infty}])$. On the other hand, for {\em any} smooth and proper curve
$X$ over a perfect field $k$ of characteristic $p$,
thanks to \cite{MM} and \cite[\Rmnum{2}, \S3 C Remarque 3.11.2]{IllusiedR}
there are natural isomorphisms
of $W(k)[F,V]$-modules
\begin{equation}
\D(J_X[p^{\infty}]) \simeq H^1_{\cris}(J_X/W(k)) \simeq H^1_{\cris}(X/W(k))\label{MMIllusie}
\end{equation}
that for any finite map of smooth proper curves $f:Y\rightarrow X$ over $k$
intertwine $\D(\Pic(f))$ and $\D(\Alb(f))$ with trace and pullback by $f$ on crystalline cohomology,
respectively. Applying this to $X=I_r^{\star}$ for $\star=0,\infty$, appealing to
the identifications (\ref{IgusaInterpretation}), and passing to inverse limits completes the proof. \end{proof}
We now wish to relate the slope filtration (\ref{DieudonneInfiniteExact}) to the Hodge filtration (\ref{orddRseq}) of our ordinary $\Lambda$-adic de Rham cohomology studied in \cite{CaisHida1}. Applying the idempotent $f'$ of (\ref{projaway}) to (\ref{orddRseq}) yields a short exact sequence of free $\Lambda_{R_{\infty}}$-modules with semilinear $\Gamma$-action and commuting action of $\H^*$: \begin{equation}
\xymatrix{
0 \ar[r] & {{e^*}'H^0(\omega)} \ar[r] & {{e^*}'H^1_{\dR}} \ar[r] & {{e^*}'H^1(\O)} \ar[r] & 0
}.\label{LambdaHodgeFilnomup} \end{equation} The key to relating (\ref{LambdaHodgeFilnomup}) to the slope filtration (\ref{DieudonneInfiniteExact}) is the following comparison isomorphism:
\begin{proposition}\label{KeyComparison}
For each positive integer $r$, there is a natural isomorphism of short exact sequences
\begin{equation}
\begin{gathered}
\xymatrix{
0\ar[r] & {\omega_{\G_r}} \ar[r]\ar[d]_-{\simeq} & {\D(\G_{r,0})_{R_r}} \ar[r]\ar[d]^-{\simeq} &
{\Lie(\Dual{\G}_r)} \ar[r]\ar[d]^-{\simeq} & 0 \\
0\ar[r] & {{e^*}'H^0(\omega_r)} \ar[r] & {{e^*}'H^1_{\dR,r}} \ar[r] & {{e^*}'H^1(\O_r)} \ar[r] & 0
}
\end{gathered}\label{CollectedComparisonIsom}
\end{equation}
that is compatible with $\H_r^*$, $\Gamma$, and change in $r$ using
the mappings $(\ref{pdivTowers})$ on the top row and the maps $\pr_*$ on the bottom.
Here, the bottom row---with obvious abbreviated notation---is obtained from $(\ref{finiteleveldRseq})$
by applying ${e^*}'$
and the top row is the Hodge filtration of $\D(\G_{r,0})_{R_r}$ given by
Proposition $\ref{BTgroupUnivExt}$. \end{proposition}
\begin{proof}
Let $\alpha_r^*: J_r\twoheadrightarrow B_r^*$ be the map of Definition \ref{BalphDef}.
We claim that $\alpha_r^*$ induces a canonical isomorphism of short exact sequences of free
$R_r$-modules
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] & {\omega_{\G_r}}\ar[d]_-{\simeq} \ar[r] & {\D(\G_{r,0})_{R_r}} \ar[d]_-{\simeq}\ar[r] &
{\Lie(\G_r^t)} \ar[d]^-{\simeq}\ar[r] & 0 \\
0 \ar[r] & {{e^*}'\omega_{\J_r}} \ar[r] & {{e^*}'\Lie\scrExtrig(\J_r,\Gm)} \ar[r] &
{{e^*}'\Lie({\J_r^t}^0)} \ar[r] & 0
}
\end{gathered}\label{HodgeToExtrigMap}
\end{equation}
that is $\H_r^*$ and $\Gamma$-equivariant and compatible with change in $r$ using
the map on N\'eron models induced by $\Pic^0(\pr)$ and the maps (\ref{pdivTowers})
on $\G_r$.
Granting this claim, the proposition then follows immediately from Proposition \ref{intcompare}.
To prove our claim, we introduce the following notation:
set $V:=\Spec(R_r)$, and for $n\ge 1$ put $V_n:=\Spec(R_r/p^nR_r)$. For any scheme
(or $p$-divisible group) $T$ over $V$, we put $T_n:=T\times_V V_n$.
If $\A$ is a N\'eron model over $V$,
we will write $H(\A)$ for the short exact sequence of free $R_r$-modules obtained by
applying $\Lie$ to the canonical extension (\ref{NeronCanExt}) of $\Dual{\A}^0$.
If $G$ is a $p$-divisible group over $V$, we similalry
write $H(G_n)$ for the short exact sequence of Lie algebras associated to the universal extension
of $G_n^t$ by a vector group over $V_n$ (see Theorem \ref{UniExtCompat}, (\ref{UniExtCompat2})).
If $\A$ is an abelian scheme over $V$
then we have natural and compatible (with change in $n$) isomorphisms
\begin{equation}
H(\A_n[p^{\infty}])\simeq H(\A_n)\simeq H(\A)/p^n,\label{AbSchpDiv}
\end{equation}
thanks to Theorem \ref{UniExtCompat}, (\ref{UniExtCompat3}) and (\ref{UniExtCompat1}); in particular, this
justifies our slight abuse of notation.
Applying the contravariant functor ${e^*}'H(\cdot)$ to the diagram of N\'eron models over $V$
induced by (\ref{UFactorDiag}) yields a commutative diagram of short exact sequences of
free $R_r$-modules
\begin{equation}
\begin{gathered}
\xymatrix{
{{e^*}'H(\J_r)} & {{e^*}'H(\B_r^*)}\ar[l] \\
{{e^*}'H(\J_r)} \ar[u]^-{U_r^*}\ar[ur] & {{e^*}'H(\B_r)}\ar[u]_-{U_r^*}\ar[l]
}
\end{gathered}
\end{equation}
in which both vertical arrows are isomorphisms by definition of ${e^*}'$. As in the proofs of Propositions
\ref{GoodRednProp} and \ref{SpecialFiberDescr}, it follows that
the horizontal maps must be isomorphisms as well:
\begin{equation}
{e^*}'H(\J_r)\simeq {e^*}'H(\B_r^*)
\label{alphaIdenOrd}
\end{equation}
Since these isomorphisms are induced via the N\'eron mapping property and the functoriality
of $H(\cdot)$ by the $\H_r^*(\Z)$-equivariant map $\alpha_r^*:J_r\twoheadrightarrow B_r^*$,
they are themselves $\H_r^*$-equivariant. Similarly, since $\alpha_r^*$ is defined
over $\Q$ and compatible with change in $r$ as in Lemma \ref{Btower}, the isomorphism
(\ref{alphaIdenOrd}) is compatible with the given actions of $\Gamma$ (arising via the N\'eron
mapping property from the semilinear action of $\Gamma$ over $K_r$ giving the descent
data of ${J_r}_{K_r}$ and ${B_r}_{K_r}$ to $\Q_p$) and change in $r$.
Reducing (\ref{alphaIdenOrd}) modulo $p^n$ and using the canonical isomorphism (\ref{AbSchpDiv}) yields
the identifications
\begin{equation}
{e^*}'H(\J_r)/p^n\simeq {e^*}'H(\B_r^*)/p^n \simeq {e^*}'H(\B_{r,n}^*[p^{\infty}])
\simeq H({e^*}'\B_{r,n}^*[p^{\infty}]) =: H(\G_{r,n})\label{ModPowersIsom}
\end{equation}
which are clearly compatible with change in $n$, and which are easily checked
(using the naturality of (\ref{AbSchpDiv}) and our remarks above) to be
$\H_r^*$ and $\Gamma$-equivariant, and compatible with change in $r$.
Since the surjection $R_r\twoheadrightarrow R_r/pR_r$ is a PD-thickening,
passing to inverse limits (with respect to $n$) on (\ref{ModPowersIsom}) and using
Proposition \ref{BTgroupUnivExt} now completes the proof. \end{proof}
\begin{corollary}\label{RelationToHodgeCor}
Let $r$ be a positive integer. Then the short exact sequence of free $R_r$-modules
\begin{equation}
\xymatrix{
0\ar[r] & {{e^*}'H^0(\omega_r)} \ar[r] & {{e^*}'H^1_{\dR,r}} \ar[r] & {{e^*}'H^1(\O_r)} \ar[r] & 0
}\label{TrivialEigenHodge}
\end{equation}
is functorially split; in particular,
it is split compatibly with the actions of $\Gamma$ and $\H_r^*$.
Moreover, $(\ref{TrivialEigenHodge})$ admits a functorial descent to $\Z_p$:
there is a natural isomorphism of split short exact sequences
\begin{equation}
\begin{gathered}
\xymatrix{
0\ar[r] & {{e^*}'H^0(\omega_r)} \ar[r]\ar[d]_-{\simeq} &
{{e^*}'H^1_{\dR,r}} \ar[r]\ar[d]^-{\simeq} & {{e^*}'H^1(\O_r)} \ar[r]\ar[d]^-{\simeq} & 0\\
0 \ar[r] & {\D(\o{\G}_r^{\mult})\tens_{\Z_p} R_r} \ar[r] &
{\D(\o{\G}_r)\tens_{\Z_p} R_r}\ar[r] &
{\D(\o{\G}_r^{et})\tens_{\Z_p} R_r} \ar[r] & 0
}
\end{gathered}\label{DescentZp}
\end{equation}
that is $\H^*$ and $\Gamma$ equivariant, with
$\Gamma$ acting trivially on $\o{\G}_r^{\et}$ and through $\langle \chi\rangle^{-1}$ on $\o{\G}_r^{\mult}$.
The identification $\ref{DescentZp}$ is compatible with change in $r$ using the maps $\pr_*$ on the top
row and the maps induced by
\begin{equation*}
\xymatrix@C=35pt{
{\o{\G}_r=\o{\G}_r^{\mult} \times \o{\G}_r^{\et}} \ar[r]^{V^{-1}\times F} &
{\o{\G}_r^{\mult} \times \o{\G}_r^{\et}=\o{\G}_r} \ar[r]^-{\o{\rho}} &
{\o{\G}_{r+1}}
}
\end{equation*}
on the bottom row. \end{corollary}
\begin{proof}
Consider the isomorphism (\ref{CollectedComparisonIsom}) of Proposition \ref{KeyComparison}.
As $\G_r$ is an ordinary $p$-divisible group by Corollary \ref{SpecialFiberOrdinary1},
the top row of (\ref{CollectedComparisonIsom}) is functorially split
by Lemma \ref{HodgeFilOrdProps}, and this gives our first assertion.
Composing the inverse of (\ref{CollectedComparisonIsom})
with the isomorphism (\ref{DescentToWIsom}) of Lemma \ref{HodgeFilOrdProps} gives
the claimed identification (\ref{DescentZp}).
That this isomorphism is compatible with change in $r$ via the specified maps
follows easily from the construction of (\ref{DescentToWIsom}) via
(\ref{TwistyDieuIsoms}). \end{proof}
We can now prove Theorem \ref{dRtoDieudonne}. Let us recall the statement:
\begin{theorem}\label{dRtoDieudonneInfty}
There is a canonical isomorphism of
short exact sequences of finite free $\Lambda_{R_{\infty}}$-modules
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] & {{e^*}'H^0(\omega)} \ar[r]\ar[d]^-{\simeq} &
{{e^*}'H^1_{\dR}} \ar[r]\ar[d]^-{\simeq} & {{e^*}'H^1(\O)} \ar[r]\ar[d]^-{\simeq} & 0 \\
0 \ar[r] & {\D_{\infty}^{\mult}\tens_{\Lambda} \Lambda_{R_{\infty}}} \ar[r] &
{\D_{\infty}\tens_{\Lambda} \Lambda_{R_{\infty}}} \ar[r] &
{\D_{\infty}^{\et}\tens_{\Lambda} \Lambda_{R_{\infty}}} \ar[r] & 0
}
\end{gathered}
\end{equation}
that is $\Gamma$ and $\H^*$-equivariant.
Here, the mappings on bottom row are the canonical inclusion and projection morphisms
corresponding to the direct sum decomposition $\D_{\infty}=\D_{\infty}^{\mult}\oplus \D_{\infty}^{\et}$.
In particular, the Hodge filtration exact sequence $(\ref{LambdaHodgeFilnomup})$ is canonically
split, and admits a canonical descent to $\Lambda$. \end{theorem}
\begin{proof}
Applying $\otimes_{R_r} R_{\infty}$ to $(\ref{DescentZp})$ and passing to projective limits
yields an isomorphism of split exact sequences
\begin{equation*}
\xymatrix{
0\ar[r] & {{e^*}'H^0(\omega)} \ar[r]\ar[d]_-{\simeq} &
{{e^*}'H^1_{\dR}} \ar[r]\ar[d]^-{\simeq} & {{e^*}'H^1(\O)} \ar[r]\ar[d]^-{\simeq} & 0\\ 0 \ar[r] & {\varprojlim\limits_{\o{\rho}\circ V^{-1}} \left(\D(\o{\G}_r^{\mult})\tens_{\Z_p} R_{\infty}\right)} \ar[r] & {\varprojlim\limits_{\o{\rho}\circ (V^{-1}\times F)}\left(\D(\o{\G}_r)\tens_{\Z_p} R_{\infty}\right)}\ar[r] &
{\varprojlim\limits_{\o{\rho}\circ F}\left(\D(\o{\G}_r^{et})\tens_{\Z_p} R_{\infty}\right)} \ar[r] & 0
}
\end{equation*}
On the other hand, the isomorphisms $ \xymatrix@1@C=37pt{
{\o{\G}_r = \o{\G}_r^{\mult}\times \o{\G}_r^{\et} } \ar[r]^-{V^{-r}\times F^{r}} &
{\o{\G}_r^{\mult}\times \o{\G}_r^{\et} =\o{\G}_r}
} $
induce an isomorphism of projective limits
\begin{equation*}
\xymatrix{
{\varprojlim\limits_{\o{\rho}}\left(\D(\o{\G}_r)\tens_{\Z_p} R_{\infty}\right)} \ar[r]^-{\simeq} &
{\varprojlim\limits_{\o{\rho}\circ (V^{-1}\times F)}\left(\D(\o{\G}_r)\tens_{\Z_p} R_{\infty}\right)}
}
\end{equation*}
which is visibly compatible with the the canonical splittings of source and target.
The result now follows from \cite[Lemma 3.1.2 (5)]{CaisHida1}
and the proof of Theorem \ref{MainDieudonne},
which guarantee that the canonical mapping
$\D_{\infty}\otimes_{\Lambda}\Lambda_{R_{\infty}}\rightarrow
\varprojlim_{\o{\rho}} (\D(\o{\G}_r)\otimes_{\Z_p}R_{\infty})$
is an isomorphism respecting the natural splittings. \end{proof}
As in \cite[\S3.3]{CaisHida1}, for any subfield $K$ of $\c_p$ with ring of integers $R$, we denote by $eS(N;\Lambda_R)$ the module of ordinary $\Lambda_R$-adic cuspforms of level $N$ in the sense of \cite[2.5.5]{OhtaEichler}. Following our convention above Proposition \ref{GoodRednProp}, we write $e'S(N;\Lambda_R)$ for the direct summand of $eS(N;\Lambda_R)$ on which $\mu_{p-1}\hookrightarrow \Z_p^{\times}\subseteq \H$ acts nontrivially.
\begin{corollary}\label{MFIgusaDieudonne}
There is a canonical isomorphism of finite free $\Lambda$-modules
\begin{equation}
{e}'S(N;\Lambda) \simeq \D_{\infty}^{\mult}
\label{LambdaFormsCrystalline}
\end{equation}
that intertwines $T\in \H$ on $e'S(N;\Lambda)$ with $T^*\in \H^*$
on $\D_{\infty}^{\mult}$,
where $U_p^*$ acts on $\D_{\infty}^{\mult}$as $\langle p\rangle_N V$. \end{corollary}
\begin{proof}
We claim that there are natural isomorphisms of finite free $\Lambda_{R_{\infty}}$-modules
\begin{equation}
\D_{\infty}^{\mult} \otimes_{\Lambda} \Lambda_{R_{\infty}} \simeq
{e^*}'H^0(\omega) \simeq {e}'S(N,\Lambda_{R_{\infty}}) \simeq
e'S(N,\Lambda)\otimes_{\Lambda} \Lambda_{R_{\infty}}\label{TakeGammaInvariants}
\end{equation}
and that the resulting composite isomorphism
intertwines $T^*\in \H^*$ on $\D_{\infty}^{\mult}$ with $T\in \H$ on
$e'S(N,\Lambda)$ and is $\Gamma$-equivariant, with $\gamma\in\Gamma$
acting as $\langle \chi(\gamma)\rangle^{-1}\otimes \gamma$ on each tensor product.
Indeed, the first and second isomorphisms are
due to Theorem \ref{dRtoDieudonneInfty} and \cite[Corollary 3.3.5]{CaisHida1}, respectively,
while the final isomorphism is a consequence of the definition
of $e'S(N;\Lambda_R)$ and the facts that this $\Lambda_R$-module is free of finite rank
\cite[Corollary 2.5.4]{OhtaEichler} and specializes as in \cite[2.6.1]{OhtaEichler}.
Twisting the
$\Gamma$-action on the source and target of the composite (\ref{TakeGammaInvariants}) by
$\langle \chi \rangle$ therefore gives a $\Gamma$-equivariant isomorphism
\begin{equation}
\D_{\infty}^{\mult} \otimes_{\Lambda} \Lambda_{R_{\infty}} \simeq
S(N,\Lambda)\otimes_{\Lambda} \Lambda_{R_{\infty}}\label{TwistedGammaIsom}
\end{equation}
with $\gamma\in \Gamma$ acting as $1\otimes \gamma$ on source and target. Passing to $\Gamma$-invariants on
(\ref{TwistedGammaIsom}) yields (\ref{LambdaFormsCrystalline}). \end{proof}
\begin{remark}\label{MFIgusaCrystal}
Via Proposition \ref{DieudonneDuality} and the natural $\Lambda$-adic
duality between $e\H$ and $eS(N;\Lambda)$ \cite[Theorem 2.5.3]{OhtaEichler}, we obtain
a canonical $\Gal(K_0'/K_0)$-equivariant isomorphism of $\Lambda_{R_0'}$-modules
\begin{equation*}
e'\H\tens_{\Lambda} \Lambda_{R_0'} \simeq \D_{\infty}^{\et}(\langle a\rangle_N)\tens_{\Lambda}{\Lambda_{R_0'}}
\end{equation*}
that intertwines $T\otimes 1$ for $T\in \H$ acting on $e'\H$ by multiplication
with $T^*\otimes 1$, with $U_p^*$ acting on $\D_{\infty}^{\et}(\langle a\rangle_N)$ as $F$.
From Theorem \ref{DieudonneCrystalIgusa} and Corollary \ref{MFIgusaDieudonne}
we then obtain canonical
isomorphisms
\begin{equation*}
e'S(N;\Lambda)\simeq f'H^1_{\cris}(I^0)^{V_{\ord}}\qquad\text{respectively}\qquad
e'\H\tens_{\Lambda}\Lambda_{R_0'} \simeq
f'H^1_{\cris}(I^{\infty})^{F_{\ord}}(\langle a\rangle_{N})\tens_{\Lambda}\Lambda_{R_0'}
\end{equation*}
intertwing $T$ (respectively $T\otimes 1$) with $T^*$ (respectively $T^*\otimes 1$)
where $U_p^*$ acts on crystalline cohomology as $\langle p\rangle_N V$
(respectively $F\otimes 1$). The second of these isomorphisms is moreover $\Gal(K_0'/K_0)$-equivariant. \end{remark}
In order to relate the slope filtration (\ref{DieudonneInfiniteExact}) of $\D_{\infty}$ to the ordinary filtration of ${e^*}'H^1_{\et}$, we require: \begin{lemma}
Let $r$ be a positive integer let $G_r={e^*}'J_r[p^{\infty}]$
be the unique $\Q_p$-descent of the generic fiber of $\G_r$, as in Definition $\ref{ordpdivdefn}$.
There are canonical isomophisms of free $W(\o{\F}_p)$-modules
\begin{subequations}
\begin{equation}
\D(\o{\G}_r^{\et})\tens_{\Z_p} W(\o{\F}_p) \simeq \Hom_{\Z_p}(T_pG_r^{\et},\Z_p)\tens_{\Z_p} W(\o{\F}_p)
\label{etalecase}
\end{equation}
\begin{equation}
\D(\o{\G}_r^{\mult})(-1)\tens_{\Z_p} W(\o{\F}_p) \simeq \Hom_{\Z_p}(T_pG_r^{\mult},\Z_p)
\tens_{\Z_p} W(\o{\F}_p).
\label{multcase}
\end{equation}
that are $\H_r^*$-equivariant and $\scrG_{\Q_p}$-compatible for the diagonal action on source and target,
with $\scrG_{\Q_p}$ acting trivially on $\D(\o{\G}^{\et}_r)$ and via $\chi^{-1}\cdot \langle \chi^{-1}\rangle$
on $\D(\o{\G}_r^{\mult})(-1):=\D(\o{\G}_r^{\mult})\otimes_{\Z_p} \Z_p(-1)$. The isomorphism
$(\ref{etalecase})$ intertwines $F\otimes\sigma$ with $1\otimes \sigma$
while $(\ref{multcase})$ intertwines $V\otimes\sigma^{-1}$ with $1\otimes\sigma^{-1}$.
\end{subequations} \end{lemma}
\begin{proof}
Let $\G$ be any object of $\pdiv_{R_r}^{\Gamma}$ and write $G$ for the unique
descent of the generic fiber $\G_{K_r}$ to $\Q_p$. We recall that the semilinear $\Gamma$-action
on $\G$ gives the $\Z_p[\scrG_{K_r}]$-module
$T_p\G:=\Hom_{\O_{\C_p}}(\Q_p/\Z_p,\G_{\O_{\C_p}})$
the natural structure of $\Z_p[\scrG_{\Q_p}]$-module via $g\cdot f:= g^{-1}\circ g^*f\circ g$.
It is straightforward to check that the natural map $T_p\G\rightarrow T_pG$, which is an isomorphism
of $\Z_p[\scrG_{K_r}]$-modules by Tate's theorem, is an isomorphism of $\Z_p[\scrG_{\Q_p}]$-modules
as well.
For {\em any} \'etale $p$-divisible group $H$ over a perfect field $k$, one has a canonical
isomorphism of $W(\o{k})$-modules with semilinear $\scrG_k$-action
\begin{equation*}
\D(H)\tens_{W(k)} W(\o{k}) \simeq \Hom_{\Z_p}(T_pH,\Z_p)\tens_{\Z_p} W(\o{k})
\end{equation*}
that intertwines $F\otimes\sigma$ with $1\otimes\sigma$ and $1\otimes g$
with $g\otimes g$ for $g\in \scrG_k$; for example, this can be deduced
by applying \cite[\S4.1 a)]{BBM1} to $H_{\o{k}}$
and using the fact that
the Dieudonn\'e crystal is compatible with base change.
In our case, the \'etale $p$-divisible group $\G_r^{\et}$ lifts $\o{\G}_r^{\et}$
over $R_r$, and we obtain a natural isomorphism of $W(\o{\F}_p)$-modules
with semilinear $\scrG_{K_r}$-action
\begin{equation*}
\D(\o{\G}_r^{\et})\tens_{\Z_p} W(\o{\F}_p) \simeq \Hom_{\Z_p}(T_p\G_r^{\et},\Z_p)\tens_{\Z_p} W(\o{\F}_p).
\end{equation*}
By naturality in $\G_r$, this identification respects the semilinear $\Gamma$-actions
on both sides (which are trivial, as $\Gamma$ acts trivially on $\G_r^{\et}$); as explained
in our initial remarks, it is precisely this action which allows us to view $T_p\G_r^{\et}$ as a
$\Z_p[\scrG_{\Q_p}]$-module, and we deduce (\ref{etalecase}). The proof of (\ref{multcase})
is similar, using the natural isomorphism (proved as above) for any multiplicative $p$-divisible group $H/k$
\begin{equation*}
\D(H)\tens_{W(k)} W(\o{k}) \simeq T_p\Dual{H}\tens_{\Z_p} W(\o{k}),
\end{equation*}
which intertwines $V\otimes\sigma^{-1}$ with $1\otimes\sigma^{-1}$
and $1\otimes g$ with $g\otimes g$, for $g\in \scrG_k$. \end{proof}
\begin{proof}[Proof of Theorem $\ref{FiltrationRecover}$ and Corollary $\ref{MWmainThmCor}$]
For a $p$-divisible group $H$ over a field $K$, we will write $H^1_{\et}(H):=\Hom_{\Z_p}(T_pH,\Z_p)$;
our notation is justified by the standard fact that, for $J_X$ the Jacobian
of a curve $X$ over $K$, there is a natural isomorphisms of $\Z_p[\scrG_K]$-modules
\begin{equation}
H^1_{\et}(J_X[p^{\infty}]) \simeq H^1_{\et}(X_{\Kbar},\Z_p).\label{etalecohcrvjac}
\end{equation}
It follows from (\ref{etalecase})--(\ref{multcase}) and
Theorem \ref{MainDieudonne} (\ref{MainDieudonne1})--(\ref{MainDieudonne2})
that $H^1_{\et}(G_r^{\star})\otimes_{\Z_p} W(\o{\F}_p)$ is a free
$W(\o{\F}_p)[\Delta/\Delta_r]$-module of rank $d'$
for $\star\in \{\et,\mult\}$,
and hence that $H^1_{\et}(G_r^{\star})$ is a free $\Z_p[\Delta/\Delta_r]$-module
of rank $d'$ by \cite[Lemma 3.1.3]{CaisHida1}. In a similar manner, using
the faithful flatness of $W(\o{\F}_p)[\Delta/\Delta_r]$ over $\Z_p[\Delta/\Delta_r]$,
we deduce that the canonical trace mappings
\begin{equation}
\xymatrix{
{H^1_{\et}(G_r^{\star})} \ar[r] & {H^1_{\et}(G_{r'}^{\star})}
}\label{cantraceetale}
\end{equation}
are surjective for all $r\ge r'$.
From the commutative algebra formalism of \cite[Lemma 3.1.2]{CaisHida1}, we conclude that
$H^1_{\et}(G_{\infty}^{\star}):=\varprojlim_r H^1_{\et}(G_r^{\star})$
is a free $\Lambda$-module of rank $d'$ and that there are canonical
isomorphisms of $\Lambda_{W(\o{\F}_p)}$-modules
\begin{equation*}
H^1_{\et}(G_{\infty}^{\star})\tens_{\Lambda} \Lambda_{W(\o{\F}_p)} \simeq \varprojlim_r \left(H^1_{\et}(G_r^{\star})\tens_{\Z_p} W(\o{\F}_p)\right)
\end{equation*}
for $\star\in \{\et,\mult\}$. Since we likewise have canonical identifications
\begin{equation*}
\D_{\infty}^{\star}\tens_{\Lambda} \Lambda_{W(\o{\F}_p)} \simeq \varprojlim_r \left(\D(G_r^{\star})\tens_{\Z_p} W(\o{\F}_p)\right)
\end{equation*}
thanks again to \cite[Lemma 3.1.2]{CaisHida1} and (the proof of) Theorem \ref{MainDieudonne},
passing to inverse limits on (\ref{etalecase})--(\ref{multcase}) gives a canonical
isomorphism of $\Lambda_{W(\o{\F}_p)}$-modules
\begin{equation}
\D_{\infty}^{\star}\tens_{\Lambda} \Lambda_{W(\o{\F}_p)} \simeq
H^1_{\et}(G_{\infty}^{\star})\tens_{\Lambda} \Lambda_{W(\o{\F}_p)}\label{dieudonneordfillimit}
\end{equation}
for $\star\in \{\et,\mult\}$.
Applying the functor $H^1_{\et}(\cdot)$ to the connected-\'etale sequence of $G_r$
yields a short exact sequence of $\Z_p[\scrG_{\Q_p}]$-modules
\begin{equation*}
\xymatrix{
0\ar[r] & {H^1_{\et}(G_r^{\et})} \ar[r] & {H^1_{\et}(G_r)} \ar[r] & {H^1_{\et}(G_r^{\mult})}\ar[r] & 0
}
\end{equation*}
which naturally identifies ${H^1_{\et}(G_r^{\star})}$ with the invariants (respectively covariants)
of $H^1_{\et}(G_r)$ under the inertia subgroup $\I\subseteq \scrG_{\Q_p}$ for $\star=\et$ (respectively
$\star=\mult$).
As $G_r={e^*}'J_r[p^{\infty}]$ by definition, we deduce from this and (\ref{etalecohcrvjac})
a natural isomorphism of short exact sequences of $\Z_p[\scrG_{\Q_p}]$-modules
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] & {H^1_{\et}(G_r^{\et})} \ar[r]\ar[d]^-{\simeq} &
{H^1_{\et}(G_r)} \ar[r]\ar[d]^-{\simeq} &
{H^1_{\et}(G_r^{\mult})} \ar[r]\ar[d]^-{\simeq} & 0 \\
0 \ar[r] & {({e^*}'H^1_{\et,r})^{\I}} \ar[r] &
{{e^*}'H^1_{\et,r}} \ar[r] &
{({e^*}'H^1_{\et,r})_{\I}} \ar[r] & 0
}
\end{gathered}\label{inertialinvariantsseq}
\end{equation}
where for notational ease we abbreviate $H^1_{\et,r}:=H^1_{\et}({X_r}_{\Qbar_p},\Z_p)$.
As the trace maps (\ref{cantraceetale}) are surjective, passing to inverse limits
on (\ref{inertialinvariantsseq}) yields an isomorphism of short exact sequences
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] & {H^1_{\et}(G_{\infty}^{\et})} \ar[r]\ar[d]^-{\simeq} &
{H^1_{\et}(G_{\infty})} \ar[r]\ar[d]^-{\simeq} &
{H^1_{\et}(G_{\infty}^{\mult})} \ar[r]\ar[d]^-{\simeq} & 0 \\
0 \ar[r] & {\varprojlim_r ({e^*}'H^1_{\et,r})^{\I}} \ar[r] &
{\varprojlim_r {e^*}'H^1_{\et,r}} \ar[r] &
{\varprojlim_r ({e^*}'H^1_{\et,r})_{\I}} \ar[r] & 0
}
\end{gathered}\label{limitetaleseq}
\end{equation}
Since inverse limits commute with group invariants, the bottom row of (\ref{limitetaleseq})
is canonically isomorphic to the ordinary filtration of Hida's ${e^*}'H^1_{\et}$,
and Theorem \ref{FiltrationRecover} follows immediately from (\ref{dieudonneordfillimit}).
Corollary \ref{MWmainThmCor} is then an easy consequence of Theorem \ref{FiltrationRecover} and
\cite[Lemma 3.1.3]{CaisHida1};
alternately one can prove Corollary \ref{MWmainThmCor} directly from \cite[3.1.2]{CaisHida1},
using what we have seen above. \end{proof}
\subsection{Ordinary families of \texorpdfstring{$(\varphi,\Gamma)$}{phi,Gamma}-modules}\label{OrdSigmaSection}
We now study the family of Dieudonn\'e crystals attached to the tower of $p$-divisible groups $\{\G_{r}/R_r\}_{r\ge 1}$. For each pair of positive integers $r\ge s$, we have a morphism $\rho_{r,s}: \G_{s}\times_{T_{s}} T_r\rightarrow \G_{r}$ in $\pdiv_{R_{r}}^{\Gamma}$; applying the contravariant functor $\m_r:\pdiv_{R_r}^{\Gamma}\rightarrow \BT_{\s_r}^{\Gamma}$ studied in \S\ref{PhiGammaCrystals}--\ref{pDivOrdSection} to the map on connected-\'etale sequences induced by $\rho_{r,s}$ and using the exactness of $\m_r$ and its compatibility with base change (Theorem \ref{CaisLauMain}), we obtain morphisms in $\BT_{\s_{r}}^{\Gamma}$ \begin{equation} \begin{gathered}
\xymatrix{
0\ar[r] & {\m_r(\G_r^{\et})} \ar[r]\ar[d]_-{\m_r(\rho_{r,s})} &
{\m_r(\G_r)} \ar[r]\ar[d]^-{\m_r(\rho_{r,s})} & {\m_r(\G_r^{\mult})}
\ar[r]\ar[d]^-{\m_r(\rho_{r,s})} & 0\\
0\ar[r ] & {\m_{s}(\G_{s}^{\et})\tens_{\s_{s}} \s_r} \ar[r] &
{\m_r(\G_{s})\tens_{\s_{s}} \s_r} \ar[r] & {\m_r(\G_{s}^{\mult})\tens_{\s_{s}} \s_r} \ar[r] & 0
}\label{BTindLim} \end{gathered} \end{equation}
\begin{definition}\label{DieudonneLimitDef}
Let $\star=\et$ or $\star=\mult$ and define
\begin{align}
\m_{\infty}&:=\varprojlim_r \left(\m_r(\G_r) \tens_{\s_r} \s_{\infty}\right)
&
\m_{\infty}^{\star}&:=\varprojlim_r \left(\m_r(\G_r^{\star}) \tens_{\s_r} \s_{\infty}\right),
\label{UncompletedDieudonneLimit}
\end{align}
with the projective limits taken with respect to the mappings induced by (\ref{BTindLim}). \end{definition}
Each of (\ref{UncompletedDieudonneLimit}) is naturally a module over the completed group ring $\Lambda_{\s_{\infty}}$ and is equipped with a semilinear action of $\Gamma$ and a $\varphi$-semilinear Frobenius morphism defined by $F:=\varprojlim (\varphi_{\m_r}\otimes \varphi)$. Since $\varphi$ is bijective on $\s_{\infty}$, we also have a $\varphi^{-1}$-semilinear Verscheibung morphism defined as follows. For notational ease, we provisionally set $M_r:=\m_r(\G_r)\otimes_{\s_r} \s_{\infty}$ and we define \begin{equation}
\xymatrix@C=70pt{
{V_r: M_r} \ar[r]^{\psi_{\m_r}} &
{{\varphi}^*M_r }
\ar[r]^-{\alpha\otimes m\mapsto \varphi^{-1}(\alpha)m} &
{M_r}
}\label{Verscheidef}
\end{equation} with $\psi_{\m_r}$ as above Definition \ref{DualBTDef}. It is easy to see that the $V_r$ are compatible with change in $r$, and we put $V:=\varprojlim V_r$ on $\m_{\infty}$. We define Verscheibung morphisms on $\m_{\infty}^{\star}$ for $\star=\et,\mult$ similarly. Using that the composite of $\psi_{\m_r}$ and $1\otimes\varphi_{\m_r}$ in either order is multiplication by $E_r(u_r) =:\omega$, one checks \begin{equation*}
FV = \omega \qquad\text{and}\qquad VF = \varphi^{-1}(\omega). \end{equation*} Due to the functoriality of $\m_r$, we moreover have a $\Lambda_{\s_{\infty}}$-linear action of $\H^*$ on each of (\ref{UncompletedDieudonneLimit}) which commutes with $F$, $V$, and $\Gamma$.
\begin{theorem}\label{MainThmCrystal}
Let $d'$ be the integer specified in Theorem $\ref{MainDieudonne}$.
Then $\m_{\infty}$ $($respectively $\m_{\infty}^{\star}$ for $\star=\et,\mult$$)$
is a free $\Lambda_{\s_{\infty}}$-module of rank $2d'$ $($respectively $d'$$)$
and there is a canonical short exact sequence of $\Lambda_{\s_{\infty}}$-modules
with linear $\H^*$-action and semi linear actions of $\Gamma$, $F$ and $V$
\begin{equation}
\xymatrix{
0\ar[r] & {\m_{\infty}^{\et}} \ar[r] & {\m_{\infty}} \ar[r] & {\m_{\infty}^{\mult}} \ar[r] & 0
}.\label{DieudonneLimitFil}
\end{equation}
Extension of scalars of $(\ref{DieudonneLimitFil})$
along the quotient
$\Lambda_{\s_{\infty}}\twoheadrightarrow \s_{\infty}[\Delta/\Delta_r]$
recovers the exact sequence
\begin{equation}
\xymatrix{
0\ar[r] & {\m_r(\G_r^{\et})\tens_{\s_r} \s_{\infty}} \ar[r] &
{\m_r(\G_r)\tens_{\s_r} \s_{\infty}} \ar[r] &
{\m_r(\G_r^{\mult})\tens_{\s_r} \s_{\infty}} \ar[r] & 0
}.
\end{equation}
for each integer $r>0$, compatibly with $\H^*$, $\Gamma$, $F$, and $V$.
The Frobenius endomorphism $F$
commutes with $\H^*$ and $\Gamma$, while $V$ commutes with $\H^*$ and satisfies
$V\gamma = \varphi^{-1}(\omega/\gamma\omega)\cdot \gamma V$ for all $\gamma\in \Gamma$. \end{theorem}
\begin{proof}
Since $\varphi$ is an automorphism
of $\s_{\infty}$, pullback by $\varphi$ commutes with
projective limits of $\s_{\infty}$-modules.
As the canonical $\s_{\infty}$-linear map $\varphi^*\Lambda_{\s_{\infty}}\rightarrow \Lambda_{\s_{\infty}}$
is an isomorphism of rings (even of $\s_{\infty}$-algebras), it therefore suffices
to prove the assertions of Theorem \ref{MainThmCrystal} after pullback by $\varphi$,
which will be more convenient due to the relation between
$\varphi^*\m_r(\G_r)$ and the Dieudonn\'e crystal of $\G_r$.
Pulling back (\ref{BTindLim}) by $\varphi$ gives a commutative diagram
with exact rows
\begin{equation}
\begin{gathered}
\xymatrix{
0\ar[r] & {\varphi^*\m_r(\G_r^{\et})} \ar[r]\ar[d] &
{\varphi^*\m_r(\G_r)} \ar[r]\ar[d] & {\varphi^*\m_r(\G_r^{\mult})}
\ar[r]\ar[d] & 0\\
0\ar[r] & {\varphi^*\m_{s}(\G_{s}^{\et})\tens_{\s_{s}} \s_r} \ar[r] &
{\varphi^*\m_r(\G_{s})\tens_{\s_{s}} \s_r} \ar[r] &
{\varphi^*\m_r(\G_{s}^{\mult})\tens_{\s_{s}} \s_r} \ar[r] & 0
}
\end{gathered}\label{BTindLimPB}
\end{equation}
and, as in the proof of Theorem \ref{MainDieudonne},
we apply the commutative algebra formalism of \cite[\S 3.1]{CaisHida1}.
In the notation of \cite[Lemma 3.1.2]{CaisHida1}, we take
$A_r:=\s_r$, $I_r:=(u_r)$,
$B=\s_{\infty}$, and $M_r$ each one of the
terms in the top row of (\ref{BTindLimPB}),
and we must check that the hypotheses
\begin{enumerate}
\setcounter{equation}{7}
\renewcommand{\theequation{\rm\alph{enumi}}}{\theequation{\rm\alph{enumi}}}
{\setlength\itemindent{10pt}
\item $\o{M}_r:=M_r/u_r M_r$ is a free $\Z_p[\Delta/\Delta_r]$-module of rank
$d'$\label{freehyp2}}
{\setlength\itemindent{10pt}
\item For all $s\le r$ the induced transition maps
$\xymatrix@1{
{\overline{\pr}_{r,s}: \o{M}_r}\ar[r] &
{\o{M}_{s}}
}$\label{surjhyp2}}
are surjective
\end{enumerate}
hold.
The isomorphism (\ref{MrToDieudonneMap})
of Proposition \ref{MrToHodge}
ensures, via Theorem \ref{MainDieudonne} (\ref{MainDieudonne1}), that the hypothesis (\ref{freehyp2})
is satisfied.
Due to the functoriality of (\ref{MrToDieudonneMap}),
for any $r\ge s$, the mapping obtained from (\ref{BTindLimPB}) by reducing
modulo $I_r$ is identified with the mapping on (\ref{DieudonneFiniteExact}) induced (via functoriality
of $\D(\cdot)$) by $\o{\pr}_{r,s}$.
As was shown in the proof of Theorem (\ref{MainDieudonne}), these mappings are surjective
for all $r\ge s$, and we conclude that the hypothesis (\ref{surjhyp2}) holds as well.
Moreover,
the vertical mappings of (\ref{BTindLimPB}) are then surjective by Nakayama's Lemma,
so as in the proof of Theorem \ref{MainDieudonne} (and keeping in mind
that pullback by $\varphi$ commutes with projective limits of $\s_{\infty}$-modules),
we obtain, by applying $\otimes_{\s_r} \s_{\infty}$ to (\ref{BTindLimPB}), passing
to projective limits, and pulling back by $(\varphi^{-1})^*$,
the short exact sequence (\ref{DieudonneLimitFil}).
The final assertion is an immediate consequence of the functorial construction of $\varphi_{\m_{r}(\cdot)}$,
the definition (\ref{Verscheidef}) of $V$, and the intertwining relation (\ref{psiGammarel}). \end{proof}
\begin{remark}
In the proof of Theorem \ref{MainThmCrystal}, we could have alternately applied \cite[Lemma 3.1.2]{CaisHida1}
with $A_r=\s_r$ and $I_r:=(E_r)$, appealing to the identifications (\ref{MrToHodgeMap})
of Proposition \ref{MrToHodge} and (\ref{CollectedComparisonIsom}) of Proposition \ref{KeyComparison},
and to Theorem \ref{dRMain} (\cite[Theorem 3.2.3]{CaisHida1}). \end{remark}
The short exact sequence (\ref{DieudonneLimitFil}) is closely related to its $\Lambda_{\s_{\infty}}$-linear dual. In what follows, we write $\s_{\infty}':=\varinjlim_r \Z_p[\mu_N][\![ u_r]\!]$, taken along the mappings $u_r\mapsto \varphi(u_{r+1})$; it is naturally a $\s_{\infty}$-algebra.
\begin{theorem}\label{CrystalDuality}
Let $\mu:\Gamma\rightarrow \Lambda_{\s_{\infty}}^{\times}$ be the crossed homomorphism
given by $\mu(\gamma):=\frac{u_1}{\gamma u_1}\chi(\gamma) \langle \chi(\gamma)\rangle$.
There is a canonical $\H^*$ and $\Gal(K_{\infty}'/K_0)$-equivariant isomorphism of
exact sequences of $\Lambda_{\s_{\infty}'}$-modules
\begin{equation}
\begin{gathered}
\xymatrix{
0\ar[r] & {\m_{\infty}^{\et}(\mu \langle a\rangle_N)_{\Lambda_{\s_{\infty}'}}} \ar[r]\ar[d]_-{\simeq} &
{\m_{\infty}(\mu \langle a\rangle_N)_{\Lambda_{\s_{\infty}'}}} \ar[r]\ar[d]_-{\simeq} &
{\m_{\infty}^{\mult}(\mu \langle a\rangle_N)_{\Lambda_{\s_{\infty}'}}} \ar[r]\ar[d]_-{\simeq} & 0\\
0\ar[r] & {(\m_{\infty}^{\mult})_{\Lambda_{\s_{\infty}'}}^{\vee}} \ar[r] &
{(\m_{\infty})_{\Lambda_{\s_{\infty}'}}^{\vee}} \ar[r] &
{(\m_{\infty}^{\et})_{\Lambda_{\s_{\infty}'}}^{\vee}} \ar[r] & 0
}
\end{gathered}
\label{MinftyDuality}
\end{equation}
that intertwines $F$
$($respectively $V$$)$ on the top row with
$V^{\vee}$ $($respectively $F^{\vee}$$)$ on the bottom. \end{theorem}
\begin{proof}
We first claim that there is a natural isomorphism of $\s_{\infty}'[\Delta/\Delta_r]$-modules
\begin{equation}
\m_r(\G_r)(\mu\langle a\rangle_N)\otimes_{\s_r} \s_{\infty}' \simeq
\Hom_{\s_{\infty}'}(\m_r(\G_r)\otimes_{\s_r}\s_{\infty}', \s_{\infty}')
\label{twistyisom}
\end{equation}
that is $\H^*$-equivariant and $\Gal(K_{\infty}'/K_0)$-compatible for the standard action
$\gamma\cdot f(m):=\gamma f(\gamma^{-1}m)$ on the right side, and that intertwines
$F$ and $V$ with $V^{\vee}$ and $F^{\vee}$, respectively.
Indeed, this follows immediately from the identifications
\begin{equation}
{\m_r(\G_r)(\langle \chi \rangle\langle a\rangle_N)\tens_{\s_r} \s_{\infty}'} \simeq
{\m_r(\G_r')\tens_{\s_r} \s_{\infty}'=:\m_r(\G_r^{\vee})\tens_{\s_r}\s_{\infty}'}
\simeq {\Dual{\m_r(\G_r)}\tens_{\s_r}{\s_{\infty}'}}
\label{GrTwist}
\end{equation}
and the definition (Definition \ref{DualBTDef}) of duality in $\BT_{\s_r}^{\varphi,\Gamma}$; here, the
first isomorphism in (\ref{GrTwist}) results from Proposition \ref{GdualTwist}
and Theorem \ref{CaisLauMain} (\ref{BaseChangeIsom}), while the final
identification is due to Theorem \ref{CaisLauMain} (\ref{exequiv}).
The identification (\ref{twistyisom}) carries $F$ (respectively $V)$
on its source to $V^{\vee}$ (respectively $F^{\vee}$) on its target due to the compatibility
of the functor $\m_r(\cdot)$ with duality (Theorem \ref{CaisLauMain} (\ref{exequiv})).
From (\ref{twistyisom})
we obtain a natural $\Gal(K_r'/K_0)$-compatible evaluation pairing of $\s_{\infty}'$-modules
\begin{equation}
\xymatrix{
{\langle\cdot,\cdot\rangle_r: \m_r(\G_r)(\mu\langle a\rangle_N) \tens_{\s_r} \s_{\infty}'
\times \m_r(\G_r)\tens_{\s_r} \s_{\infty}'} \ar[r] & {\s_{\infty}'}
}\label{crystalpairingdefs}
\end{equation}
with respect to which the natural action of $\H^*$ is self-adjoint, due to the
fact that (\ref{GrTwist}) is $\H^*$-equivariant by Proposition \ref{GdualTwist}.
Due to the compatibility with change in $r$ of the identification (\ref{GrprimeGr}) of Proposition \ref{GdualTwist}
together with the definitions (\ref{pdivTowers}) of $\pr_{r,s}$ and
$\pr_{r,s}'$,
the identification (\ref{GrTwist}) intertwines the map induced by $\Pic^0(\pr)$ on its source
with the map induced by ${U_p^*}^{-1}\Alb(\ps)$ on its target. For $r\ge s$, we therefore have
\begin{equation*}
\langle \m_r(\rho_{r,s})x , \m_r(\rho_{r,s})y \rangle_s =
\langle x, \m_r({U_p^*}^{s-r}\Pic^0(\pr)^{r-s}\Alb(\ps)^{r-s})y\rangle_r =
\sum_{\delta\in \Delta_s/\Delta_r} \langle x, \delta^{-1} y \rangle_r,
\end{equation*}
where the final equality follows from (\ref{PicAlbRelation}).
Thus, the perfect pairings (\ref{crystalpairingdefs}) satisfy the compatibility condition
of \cite[Lemma 3.1.4]{CaisHida1} (as in (\ref{pairingchangeinr1}) of the proof
of Proposition \ref{DieudonneDuality}) which, together with
Theorem \ref{MainThmCrystal}, completes the proof. \end{proof}
The $\Lambda_{\s_{\infty}}$-modules $\m_{\infty}^{\et}$ and $\m_{\infty}^{\mult}$ admit canonical descents to $\Lambda$:
\begin{theorem}\label{etmultdescent}
There are canonical $\H^*$, $\Gamma$, $F$ and $V$-equivariant isomorphisms
of $\Lambda_{\s_{\infty}}$-modules
\begin{subequations}
\begin{equation}
\m_{\infty}^{\et} \simeq \D_{\infty}^{\et}\tens_{\Lambda} \Lambda_{\s_{\infty}},
\end{equation}
intertwining $F$ and $V$ with
$F\otimes \varphi$ and $F^{-1}\otimes \varphi^{-1}(\omega)\cdot \varphi^{-1}$, respectively,
and $\gamma\in \Gamma$
with $\gamma\otimes\gamma$, and
\begin{equation}
\m_{\infty}^{\mult}\simeq \D_{\infty}^{\mult}\tens_{\Lambda} \Lambda_{\s_{\infty}},
\end{equation}
intertwing $F$ and $V$ with $V^{-1} \otimes \omega \cdot\varphi$
and $V\otimes\varphi^{-1}$, respectively,
and $\gamma$ with $\gamma\otimes \chi(\gamma)^{-1} \gamma u_1/u_1$.
In particular, $F$ $($respectively $V$$)$
acts invertibly on $\m_{\infty}^{\et}$ $($respectively $\m_{\infty}^{\mult}$$)$. \end{subequations} \end{theorem}
\begin{proof}
We twist the identifications (\ref{EtMultSpecialIsoms}) of Proposition
\ref{EtaleMultDescription} to obtain natural isomorphisms
\begin{equation*}
\xymatrix@C=40pt{
{\m_r(\G_r^{\et})} \ar[r]^-{\simeq}_-{F^r \circ (\ref{EtMultSpecialIsoms})} &
{\D(\o{\G}_r^{\et})_{\Z_p}\otimes_{\Z_p} \s_r}
}\qquad\text{and}\qquad
\xymatrix@C=40pt{
{\m_r(\G_r^{\mult})} \ar[r]^-{\simeq}_-{V^{-r} \circ (\ref{EtMultSpecialIsoms})} &
{\D(\o{\G}_r^{\mult})_{\Z_p}\otimes_{\Z_p} \s_r}
}
\end{equation*}
that are $\H_r^*$-equivariant and, Thanks to \ref{EtMultSpecialIsomsBC},
compatible with change in $r$ using the maps on source and target
induced by $\pr_{r,s}$. Passing to inverse limits and appealing again to \cite[Lemma 3.1.2]{CaisHida1}
and (the proof of) Theorem \ref{MainDieudonne}, we deduce for $\star=\et,\mult$
natural isomorphisms of $\Lambda_{\s_{\infty}}$-modules
\begin{equation*}
\m_{\infty}^{\star} \simeq \varprojlim_r \left( \D(\o{\G}_r^{\star})_{\Z_p}\otimes_{\Z_p} \s_{\infty}\right)
\simeq \D_{\infty}^{\star}\otimes_{\Lambda} \Lambda_{\s_{\infty}}
\end{equation*}
that are $\H^*$-equivariant and satisfy the asserted compatibility with respect to
Frobenius, Verscheibung, and the action of $\Gamma$ due to
Proposition \ref{EtaleMultDescription} and the definitions (\ref{MrEtDef})--(\ref{MrMultDef}). \end{proof}
We can now prove Theorem \ref{MinftySpecialize}, which asserts that the slope filtration (\ref{MinftySpecialize}) of $\m_{\infty}$ specializes, on the one hand, to the slope filtration (\ref{DieudonneInfiniteExact}) of $\D_{\infty}$, and on the other hand to the Hodge filtration (\ref{LambdaHodgeFilnomup}) (in the opposite direction!) of ${e^*}'H^1_{\dR}$. We recall the precise statement:
\begin{theorem}\label{SRecovery}
Let $\tau:\Lambda_{\s_{\infty}}\twoheadrightarrow \Lambda$ be the $\Lambda$-algebra
surjection induced by $u_r\mapsto 0$.
There is a canonical $\Gamma$ and $\H^*$-equivariant
isomorphism of split exact sequences of finite free $\Lambda$-modules
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] & {\m_{\infty}^{\et}\tens_{\Lambda_{\s_{\infty}},\tau} \Lambda}\ar[d]_-{\simeq} \ar[r] &
{\m_{\infty}\tens_{\Lambda_{\s_{\infty}},\tau} \Lambda}\ar[r] \ar[d]_-{\simeq}&
{\m_{\infty}^{\mult}\tens_{\Lambda_{\s_{\infty}},\tau} \Lambda} \ar[r]\ar[d]_-{\simeq} & 0\\
0 \ar[r] & {\D_{\infty}^{\et}} \ar[r] & {\D_{\infty}} \ar[r] &
{\D_{\infty}^{\mult}} \ar[r] & 0
}
\end{gathered}\label{OrdFilSpecialize}
\end{equation}
which carries $F\otimes 1$ to $F$ and $V\otimes 1$ to $V$.
Let $\theta\circ\varphi:\Lambda_{\s_{\infty}}\rightarrow \Lambda_{R_{\infty}}$
be the $\Lambda$-algebra surjection induced by $u_r\mapsto (\varepsilon^{(r)})^p-1$.
There is a canonical $\Gamma$ and $\H^*$-equivariant
isomorphism of split exact sequences of finite free $\Lambda_{R_{\infty}}$-modules
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] & {\m_{\infty}^{\et}\tens_{\Lambda_{\s_{\infty}},\theta\varphi} \Lambda_{R_{\infty}}}
\ar[d]_-{\simeq} \ar[r] &
{\m_{\infty}\tens_{\Lambda_{\s_{\infty}},\theta\varphi} \Lambda_{R_{\infty}}}\ar[r] \ar[d]_-{\simeq}&
{\m_{\infty}^{\mult}\tens_{\Lambda_{\s_{\infty}},\theta\varphi} \Lambda_{R_{\infty}}} \ar[r]\ar[d]_-{\simeq} & 0\\
0 \ar[r] & {{e^*}'H^1(\O)} \ar[r]_{i} &
{{e^*}'H^1_{\dR}} \ar[r]_-{j} & {{e^*}'H^0(\omega)} \ar[r] & 0
}
\end{gathered}
\end{equation}
where $i$ and $j$ are the canonical sections
given by the splitting in Theorem $\ref{dRtoDieudonne}$. \end{theorem}
\begin{proof}
To prove the first assertion, we apply \cite[Lemma 3.1.2]{CaisHida1}
with $A_r=\s_r,$ $I_r=(u_r)$, $B=\s_{\infty}$, $B'=\Z_p$ (viewed as
a $B$-algebra via $\tau$) and $M_r=\m_r^{\star}$ for $\star\in \{\et,\mult,\Null\}$,
and, as in the proofs of Theorems \ref{MainDieudonne} and \ref{MainThmCrystal}, we must verify the hypotheses
\begin{enumerate}
\setcounter{equation}{15}
\renewcommand{\theequation{\rm\alph{enumi}}}{\theequation{\rm\alph{enumi}}}
{\setlength\itemindent{10pt}
\item $\o{M}_r:=M_r/u_r M_r$ is a free $\Z_p[\Delta/\Delta_r]$-module of rank
$d'$\label{freehyp3}}
{\setlength\itemindent{10pt}
\item For all $s\le r$ the induced transition maps
$\xymatrix@1{
{\overline{\pr}_{r,s}: \o{M}_r}\ar[r] &
{\o{M}_{s}}
}$\label{surjhyp3}}
are surjective.
\end{enumerate}
Thanks to (\ref{MrToDieudonneMap}) in the case $G=\G_r$,
we have a canonical identification $\o{M}_r:=M_r/I_rM_r \simeq \D(\o{\G}_r^{\star})_{\Z_p}$
that is compatible with change in $r$ in the sense that the induced projective
system $\{\o{M}_r\}_{r}$ is identified with that of Definition \ref{DinftyDef}.
It follows from this and
Theorem \ref{MainDieudonne} (\ref{MainDieudonne1})--(\ref{MainDieudonne2}) that
the hypotheses (\ref{freehyp3})--(\ref{surjhyp3}) are satisfied,
and (\ref{OrdFilSpecialize}) is an isomorphism by \cite[Lemma 3.1.3 (5)]{CaisHida1}.
In exactly the same manner,
the second assertion follows by appealing to
\cite[Lemma 3.1.2]{CaisHida1} with $A_r=\s_r$, $I_r=(E_r)$, $B=\s_{\infty}$, $B'=R_{\infty}$
(viewed as a $B$-algebra via $\theta\circ\varphi$)
and $M_r=\m_r^{\star}$, using (\ref{MrToHodgeMap}) and Proposition \ref{KeyComparison}
together with Theorem \ref{dRMain}
(see \cite[Theorem 3.2.3]{CaisHida1})
to verify the requisite hypotheses
in this setting. \end{proof}
\begin{proof}[Proof of Theorem $\ref{RecoverEtale}$ and Corollary $\ref{HidasThm}$]
Applying Theorem \ref{comparison} to (the connected-\'etale sequence of) $\G_r$
gives a natural isomorphism of short exact sequences
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] &{\m_r(\G_r^{\et})\tens_{\s_r,\varphi} \a_r } \ar[r]\ar[d]^-{\simeq} &
{\m_r(\G_r)\tens_{\s_r,\varphi} \a_r} \ar[r]\ar[d]^-{\simeq} &
{\m_r(\G_r^{\mult})\tens_{\s_r,\varphi} \a_r} \ar[r]\ar[d]^-{\simeq} & 0 \\
0 \ar[r] & {H^1_{\et}(\G_r^{\et})\tens_{\Z_p} \a_r} \ar[r] &
{H^1_{\et}(\G_r)\tens_{\Z_p} \a_r} \ar[r] &
{H^1_{\et}(\G_r^{\mult})\tens_{\Z_p}\a_r}\ar[r] & 0
}
\end{gathered}
\label{etalecompdiag}
\end{equation}
Due to Theorem \ref{MainThmCrystal}, the terms in the top row of \ref{etalecompdiag}
are free of ranks $d'$, $2d'$, and $d'$ over $\wt{\a}_r[\Delta/\Delta_r]$, respectively,
so we conclude from \cite[Lemma 3.1.3]{CaisHida1} (using $A=\Z_p[\Delta/\Delta_r]$
and $B=\a_r[\Delta/\Delta_r]$ in the notation of that result)
that $H^1_{\et}(\G_r^{\star})$ is a free $\Z_p[\Delta/\Delta_r]$-module
of rank $d'$ for $\star=\{\et,\mult\}$ and that $H^1_{\et}(\G_r)$ is free of rank $2d'$
over $\Z_p[\Delta/\Delta_r]$. Using the fact that
$\Z_p\rightarrow \a_r$ is faithfully flat, it then follows
from the surjectivity of the vertical maps in (\ref{BTindLimPB})
(which was noted in the proof of Theorem \ref{MainThmCrystal})
that the canonical trace mappings $H^1_{\et}(\G_r^{\star})\rightarrow H^1_{\et}(\G_{r'}^{\star})$
for $\star\in \{\et,\mult,\Null\}$ are surjective for all $r\ge r'$.
Applying \cite[Lemma 3.1.2]{CaisHida1} with $A_r=\Z_p$, $M_r:=H^1_{\et}(\G_r^{\star})$,
$I_r=(0)$, $B=\Z_p$ and $B'=\wt{\a}$, we conclude that $H^1_{\et}(\G_{\infty}^{\star})$
is free of rank $d'$ (respectively $2d'$) over $\Lambda$ for $\star=\et,$ $\mult$ (respectively $\star=\Null$),
that the specialization mappings
\begin{equation*}
\xymatrix{
{H^1_{\et}(\G_{\infty}^{\star})\tens_{\Lambda} \Z_p[\Delta/\Delta_r]} \ar[r] &
{H^1_{\et}(\G_r^{\star})}
}
\end{equation*}
are isomorphisms, and that the canonical mappings for $\star\in \{\et,\mult,\Null\}$
\begin{equation}
\xymatrix{
{H^1_{\et}(\G_{\infty}^{\star})\tens_{\Lambda} \Lambda_{\wt{\a}}} \ar[r] &
{\varprojlim_r \left(H^1_{\et}(\G_r^{\star})\tens_{\Z_p} \wt{\a}\right)}
}\label{etaleswitcheroo}
\end{equation}
are isomorphisms. Invoking the isomorphism (\ref{limitetaleseq})
gives Corollary \ref{HidasThm}. By \cite[Lemma 3.1.2]{CaisHida1} with $A_r=\s_r$, $M_r=\m_r(\G_r^{\star})$,
$I_r=(0)$, $B=\s_{\infty}$ and $B'=\wt{\a}$, we similarly conclude from (the proof of) Theorem
\ref{MainThmCrystal} that the canonical mappings for $\star\in \{\et,\mult,\Null\}$
\begin{equation}
\xymatrix{
{\m_{\infty}^{\star}\tens_{\s_{\infty},\varphi} \Lambda_{\wt{\a}}} \ar[r] &
{\varprojlim_r \left(\m_r(\G_r^{\star})\tens_{\s_r} \wt{\a}\right)}
}\label{crystalswitcheroo}
\end{equation}
are isomorphisms.
Applying $\otimes_{\a_r} \wt{\a}$ to the diagram (\ref{etalecompdiag}),
passing to inverse limits, and using the isomorphisms
(\ref{etaleswitcheroo}) and (\ref{crystalswitcheroo}) gives (again invoking (\ref{limitetaleseq}))
the isomorphism (\ref{FinalComparisonIsom}).
Using the fact that the inclusion $\Z_p\hookrightarrow \wt{\a}^{\varphi=1}$
is an equality, the isomorphism (\ref{RecoverEtaleIsom}) follows immediately from
(\ref{FinalComparisonIsom}) by taking $F\otimes\varphi$-invariants. \end{proof}
Using Theorems \ref{RecoverEtale} and \ref{CrystalDuality} we can give a new proof of Ohta's duality theorem \cite[Theorem 4.3.1]{OhtaEichler} for the $\Lambda$-adic ordinary filtration of ${e^*}'H^1_{\et}$ (see Corollary \ref{OhtaDuality}):
\begin{theorem}\label{OhtaDualityText}
There is a canonical $\Lambda$-bilinear and perfect duality pairing
\begin{equation}
\langle \cdot,\cdot\rangle_{\Lambda}: {e^*}'H^1_{\et}\times {e^*}'H^1_{\et}\rightarrow \Lambda
\quad\text{determined by}\quad
\langle x,y\rangle_{\Lambda} \equiv \sum_{\delta\in \Delta/\Delta_r}
(x , w_r {U_p^*}^r\langle\delta^{-1}\rangle^*y)_r \delta \bmod I_r
\label{EtaleDualityPairing}
\end{equation}
with respect to which the action of $\H^*$ is self-adjoint; here,
$(\cdot,\cdot)_r$ is the usual cup-product pairing on $H^1_{\et,r}$ and
$I_r:=\ker(\Lambda\twoheadrightarrow \Z_p[\Delta/\Delta_r])$.
Writing $\nu:\scrG_{\Q_p}\rightarrow \H^*$ for the character
$\nu:=\chi\langle\chi\rangle \lambda(\langle p\rangle_N)$, the
pairing $(\ref{EtaleDualityPairing})$ induces a canonical
$\scrG_{\Q_p}$ and $\H^*$-equivariant isomorphism of exact sequences
\begin{equation*}
\xymatrix{
0 \ar[r] & {({e^*}'H^1_{\et})^{\I}(\nu)}
\ar[d]^-{\simeq} \ar[r] &
{{e^*}'H^1_{\et}(\nu)}\ar[d]^-{\simeq} \ar[r] &
{({e^*}'H^1_{\et})_{\I}(\nu)}
\ar[d]^-{\simeq}\ar[r] & 0 \\
0 \ar[r] & {\Hom_{\Lambda}(({e^*}'H^1_{\et})_{\I},\Lambda)} \ar[r] &
{\Hom_{\Lambda}({e^*}'H^1_{\et},\Lambda)} \ar[r] &
{\Hom_{\Lambda}(({e^*}'H^1_{\et})^{\I},\Lambda)}\ar[r] & 0
}
\end{equation*} \end{theorem}
\begin{proof}
The proof is similar to that of Proposition \ref{DieudonneDuality}, using
Corollary \ref{HidasThm} and applying \cite[Lemma 3.1.4]{CaisHida1} ({\em cf.}
the proofs of \cite[3.2.4]{CaisHida1} and \cite[Theorem 4.3.1]{OhtaEichler} and of \cite[Proposition 4.4]{SharifiConj}).
Alternatively, one can prove Theorem \ref{OhtaDualityText} by appealing to Theorem \ref{CrystalDuality}
and the isomorphism (\ref{RecoverEtaleIsom}) of Theorem \ref{RecoverEtale}. \end{proof}
\begin{proof}[Proof of Theorem $\ref{SplittingCriterion}$]
Suppose first that (\ref{DieudonneLimitFil}) admits a $\Lambda_{\s_{\infty}}$-linear
splitting $\m_{\infty}^{\mult}\rightarrow \m_{\infty}$
which is compatible with $F$, $V$, and $\Gamma$.
Extending scalars along $\Lambda \rightarrow \Lambda_{\wt{\a}}\xrightarrow{\varphi}\Lambda_{\wt{\a}}$
and taking $F\otimes\varphi$-invariants
yields, by Theorem \ref{RecoverEtale}, a $\Lambda$-linear and $\scrG_{\Q_p}$-equivariant map
$({e^*}'H^1_{\et})_{\I}\rightarrow {e^*}'H^1_{\et}$
whose composition with the canonical projection
${e^*}'H^1_{\et}\twoheadrightarrow ({e^*}'H^1_{\et})_{\I}$
is necessarily the identity.
Conversely, suppose that the ordinary filtration of ${e^*}'H^1_{\et}$ is $\Lambda$-linearly
and $\scrG_{\Q_p}$-equivariantly split. Applying $\otimes_{\Lambda} \Z_p[\Delta/\Delta_r]$
to this splitting gives, thanks to Corollary \ref{HidasThm} and the isomorphism
(\ref{inertialinvariantsseq}), a $\Z_p[\scrG_{\Q_p}]$-linear splitting of
\begin{equation*}
\xymatrix{
0 \ar[r] & {T_pG_r^{\mult}} \ar[r] & {T_pG_r} \ar[r] & {T_pG_r^{\et}}\ar[r] & 0
}
\end{equation*}
which is compatible with change in $r$ by construction.
By $\Gamma$-descent and Tate's theorem, there is a natural isomorphism
\begin{equation*}
{\Hom_{\pdiv_{R_r}^{\Gamma}}(\G_r^{\et},\G_r)}\simeq {\Hom_{\Z_p[\scrG_{\Q_p}]}(T_pG_r^{\et},T_pG_r)}
\end{equation*}
and we conclude that the connected-\'etale sequence of $\G_r$ is split (in the category
$\pdiv_{R_r}^{\Gamma}$), compatibly with change in $r$. Due to the functoriality
of $\m_r(\cdot)$, this in turn implies that
the top row of (\ref{BTindLim}) is split in $\BT_{\s_r}^{\Gamma}$,
compatibly with change in $r$, which is easily seen to imply the splitting of (\ref{DieudonneLimitFil}). \end{proof}
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Imbeddings of free actions on handlebodies}
\author{Darryl McCullough} \address{Department of Mathematics\\ University of Oklahoma\\ Norman, Oklahoma 73019\\ USA} \email{[email protected]} \urladdr{www.math.ou.edu/$_{\widetilde{\phantom{n}}}$dmccullo/} \thanks{The author was supported in part by NSF grant DMS-0102463} \subjclass{Primary 57M50; Secondary 57M60, 58D99}
\subjclass{Primary 57M60; Secondary 20F05}
\date{\today}
\keywords{3-manifold, handlebody, group action, free, free action, imbed, imbedding, equivariant, invariant, hyperbolic, Seifert, Heegaard, Heegaard splitting, Whitehead link}
\begin{abstract} Fix a free, orientation-preserving action of a finite group $G$ on a $3$-dimensional handlebody $V$. Whenever $G$ acts freely preserving orientation on a connected $3$-manifold $X$, there is a $G$-equivariant imbedding of $V$ into $X$. There are choices of $X$ closed and Seifert-fibered for which the image of $V$ is a handlebody of a Heegaard splitting of $X$. Provided that the genus of $V$ is at least $2$, there are similar choices with $X$ closed and hyperbolic. \end{abstract}
\maketitle
\section*{Introduction} \label{sec:intro}
Any finite group acts (smoothly and) freely preserving orientation on some $3$-dimensional handlebody, and the number of inequivalent actions of a fixed $G$ on a fixed genus of handlebody can be arbitrarily large \cite{MW}. Consequently, the following imbedding property of such actions may appear surprising at first glance:
\begin{theorem} Let $G$ be a finite group acting freely and preserving orientation on two handlebodies $V_1$ and $V_2$, not necessarily of the same genus. Then there is a $G$-equivariant imbedding of $V_1$ into $V_2$. \end{theorem}
\noindent In fact, this result is almost a triviality, as is the following theorem of which it is a special case:
\begin{theorem} Let $G$ be a finite group acting freely and preserving orientation on a handlebody $V$ and on a connected $3$-manifold $X$. Then there is a $G$-equivariant imbedding of $V$ into $X$. \label{thm:always} \end{theorem}
By a result of D. Cooper and D. D. Long \cite{CL}, any finite group acts freely on some hyperbolic rational homology $3$-sphere. So theorem~\ref{thm:always} shows that a free $G$-action on a handlebody always has an extension to an action on such a $3$-manifold. Also, by a result of S. Kojima \cite{K}, for any finite $G$ there is a closed hyperbolic $3$-manifold whose full isometry group is $G$, and Kojima's construction actually produces a free action. So there is an extension to a free action on a closed hyperbolic $3$-manifold whose full isometry group is~$G$.
One might ask for a more natural kind of extension, to a free $G$-action on a closed $3$-manifold $M$, for which $V$ is one of the handlebodies in a $G$-invariant Heegaard splitting of $M$. Simply by forming the double of $V$ and taking an identical action on the second copy of $V$, one obtains such an extension with $M$ a connected sum of $S^2\times S^1$'s. A better question is whether $V$ is an invariant Heegaard handlebody for a free action on an irreducible $3$-manifold. Our main result answers this affirmatively. \enlargethispage{\baselineskip}
\begin{theorem} Let $G$ be a finite group acting freely and preserving orientation on a handlebody $V$. Then the action is the restriction of a free $G$-action on a closed irreducible $3$-man\-i\-fold $M$, which has a $G$-invariant Heegaard splitting with $V$ as one of the handlebodies. One may choose $M$ to be Seifert-fibered. Provided that $V$ has genus greater than $1$, one may choose $M$ to be hyperbolic. In both cases, there are infinitely many choices of $M$.\par \label{thm:irreducible extension} \end{theorem} We remark that any orientation-preserving action of a finite group on a closed $3$-manifold, free or not, has an invariant Heegaard splitting. For the quotient is a closed orientable $3$-orbifold with $1$-dimensional (possibly empty) singular set. One may triangulate the quotient so that the singular set is a subcomplex of the $1$-skeleton. Then, the preimage of a regular neighborhood of the $1$-skeleton is invariant and is one of the handlebodies in a Heegaard splitting.
In the remaining sections of this paper, we prove theorems~\ref{thm:always} and~\ref{thm:irreducible extension}. In \cite{MW}, a number of results about free $G$-actions on handlebodies are obtained using more algebraic methods.
\section{Proof of theorem~\ref{thm:always}} \label{sec:always}
Recall that two $G$-actions on spaces $X$ and $Y$ are \emph{equivalent} if there is a homeomorphism $j\colon X\to Y$ such that $h(x)=j^{-1}(h(j(x)))$ for all $x\in X$ and all $h\in G$. If $G$ acts properly discontinuously and freely on a path-connected space $X$, then the quotient map $X\to X/G$ is a regular covering map, so by the theory of covering spaces the action determines an extension \begin{equation*} 1\longrightarrow\pi_1(X)\longrightarrow\pi_1(X/G) \mapright{\phi}G\longrightarrow1\ . \end{equation*} \noindent Since we have not specified basepoints, the homomorphism $\phi$ is well-defined only up to an inner automorphism of $G$.
Suppose now that $G$ is finite and acts freely and preserving orientation on a handlebody $V$. The quotient manifold $V/G$ is orientable and irreducible with nonempty boundary, so $\pi_1(V/G)$ is torsionfree. A torsionfree finite extension of a finitely generated free group is free (by~\cite{K-P-S} any finitely generated virtually free group is the fundamental group of a graph of groups with finite vertex groups, and if the group is torsionfree, the vertex groups must be trivial). So $\pi_1(V/G)$ is free, and theorem~5.2 of~\cite{Hempel} shows that $V/G$ is a handlebody. In this context, we obtain a simple algebraic criterion for equivalence. \begin{lemma} Suppose that $G$ acts freely and preserving orientation on handlebodies $V_1$ and $V_2$, with quotient handlebodies $W_1$ and $W_2$, determining homomorphisms $\phi_i\colon \pi_1(W_i)\to G$. The actions are equivalent if and only if there is an isomorphism $\Psi\colon \pi_1(W_1)\to \pi_1(W_2)$ for which $\phi_2\circ \Psi=\phi_1$. \label{lem:equivalence} \end{lemma} \begin{proof} An equivalence of the actions $j\colon V_1\to V_2$ induces a homeomorphism $\overline{j}\colon W_1\to W_2$ for which $\phi_2\circ \overline{j}_\# = \phi_1$. Conversely, suppose $\Psi$ exists. Since both $W_1$ and $W_2$ are orientable, there is a homeomorphism $f\colon W_1\to W_2$. Using well-known constructions of homeomorphisms of $W_2$ (as for example in~\cite{M-M1}), all of Nielsen's \cite{Nielsen} generators of the automorphism group of the free group $\pi_1(W_2)$ can be induced by homeomorphisms, so $f$ may be selected to induce $\Psi$. The condition that $\phi_2\circ \Psi=\phi_1$ then shows that $f$ lifts to a homeomorphism of covering spaces $j\colon V_1\to V_2$, and moreover ensures that $h(x)=j^{-1}(h(j(x)))$. \end{proof}
Now we prove theorem~\ref{thm:always}. Let $W$ be the quotient handlebody of the action on $V$, let $Y=X/G$, and let $\phi\colon \pi_1(W)\to G$ and $\psi\colon \pi_1(Y)\to G$ be the homomorphisms determined by the actions.
There is an imbedding $k\colon W\to Y$ so that $\psi\circ k_\#=\phi$. For we can regard $W$ as a regular neighborhood of a $1$-point union $K$ of circles, so that $\pi_1(K)=\pi_1(W)$, and construct a map $k_0$ of $K$ into $Y$ for which $\psi\circ (k_0)_\#=\phi$. Since $K$ is $1$-dimensional, $k_0$ is homotopic to an imbedding, and since $W$ and $Y$ are orientable, this imbedding extends to an imbedding $k$ of $W$ into $Y$. Since $\psi\circ k_\#=\phi$, the preimage of $k(W)$ in $X$ is connected, and by lemma~\ref{lem:equivalence} the restricted $G$-action on it is equivalent to the original action on~$V$.
Theorem~\ref{thm:always} extends to the case when some elements of $G$ reverse the orientation. The equivariant imbedding exists if and only if the subgroups of elements of $G$ that reverse orientation on $V$ and on $X$ are identical. The proof is affected only at the step when the imbedding of $K$ into $Y$ is extended to an imbedding of $W$ into $Y$. The equality of the orientation-reversing subgroups is precisely the condition needed for the extension to exist.
\section{Proof of theorem~\ref{thm:irreducible extension}}
If the genus of $V$ is $0$, then $G$ is trivial and we take $M=S^3$. If the genus of $V$ is $1$, so that $V=D^2\times S^1$, then (using lemma~\ref{lem:equivalence}) every free $G$-action is equivalent to a cyclic rotation in the $S^1$-factor. Regarding $V$ as a trivially fibered solid torus in the Hopf fibering of $S^3$, the action extends to a free action on $S^3$ with $V$ an invariant Heegaard splitting (it also extends to free actions on infinitely many lens spaces containing $V$ as a fibered Heegaard torus). So we may assume that the genus of $V$ is greater than $1$. The quotient handlebody $W=V/G$ has genus at least~$2$ (since $V$ and consequently $W$ have negative Euler characteristic).
We first construct the Seifert-fibered extension. As in section \ref{sec:always}, there is a homomorphism $\phi\colon \pi_1(W)\to G$ that determines the action. Let $g$ be the genus of $W$, and let $n$ be any positive integer divisible by the orders of all the elements of $G$. We consider a collection of simple closed curves $C_1,\dots\,$, $C_g$ in the boundary $\partial W$, as shown in figure~\ref{fig:Heegaard} for the case when $g=3$ and $n=4$. Each $C_i$ winds $n$ times around one of the handles of $W$. Let $C_i'$ be the image of $C_i$ under the $n^{th}$ power of a Dehn twist of $\partial W$ about the curve $C$. The union of the $C_i$ does not separate $\partial W$, so neither does the union of the $C_i'$. So we can obtain a closed $3$-manifold $Y$ with $W$ as a Heegaard handlebody by attaching $2$-handles along the $C_i'$ and filling in the resulting $2$-sphere boundary component with a $3$-ball. \begin{figure}
\caption{The quotient handlebody $W$.}
\label{fig:Heegaard}
\end{figure}
Let $x_1,\ldots\,$, $x_g$ be a standard set of generators of $\pi_1(W)$, where $x_i$ is represented by a loop that goes once around the $i^{th}$ handle. In $\pi_1(W)$, $C_i$ represents $x_i^n$ (up to conjugacy), and $C_i'$ represents $x_i^n\,(x_1\cdots x_g)^{-n}$. Since every element of $G$ has order dividing $n$, it follows that $\phi$ carries each $C_i'$ to the trivial element of $G$, so induces a homomorphism $\psi\colon \pi_1(Y)\to G$. If $k\colon W\to Y$ is the inclusion, then $\psi\circ k_\#=\phi$. The covering space $M$ of $Y$ has a free $G$-action and an invariant Heegaard splitting, one of whose handlebodies is the covering space of $W$ corresponding to the kernel of $\phi$, that is, $V$.
We will show that $Y$ is Seifert-fibered, from which it follows that $M$ is Seifert-fibered. Choose imbedded loops in the interior of $W$: $L$ near and parallel to $C$, and $L_1,\dots\,$, $L_g$ near and parallel to loops $\ell_i$ in $\partial W$ with each $\ell_i$ going once around the $i^{th}$ handle, meeting $C_i$ in one point. The loop $\ell_1$ appears in figure~\ref{fig:Heegaard}. By a standard procedure, as explained for example on pp.~275-278 of \cite{Rolfsen}, whose notation we follow, we may change the attaching curves for the discs by Dehn twists about $C$ and the $\ell_i$, at the expense of performing Dehn surgery on $L$ and the $L_i$. First we twist $n$ times along $C$, introducing a $1/n$ coefficient on $L$ and moving each $C_i'$ back to $C_i$. Then, $n-1$ twists along each $\ell_i$ move $C_i$ to a loop $C_i''$ in $\partial W$ that looks like $C_i$ except it goes only once around the handle. This creates surgery coefficients of $-1/(n-1)$ on the $L_i$. We may change the attaching homeomorphism of the Heegaard splitting by any homeomorphism of $\partial W$ that extends over $W$, without changing $Y$. In particular, we may perform left-hand twists in the meridinal $2$-discs of the $2$-handles to move the $C_i''$ to the $\ell_i$. This subtracts $1$ from the surgery coefficients of the $L_i$, and subtracts $g$ from the coefficient of $L$, yielding the diagram in figure~\ref{fig:Dehn}. The $\ell_i$ are the attaching curves for the discs of a Heegaard description of $S^3$, so $Y$ is obtained from $S^3$ by Dehn surgery using the diagram. \begin{figure}
\caption{Surgery description of $Y$ in the Seifert-fibered case.}
\label{fig:Dehn}
\end{figure}
We can already see a Seifert fibering, but we will work out the exact Seifert invariants. The complement of a regular neighborhood of $L$ is a solid torus $T$, for which the $L_i$ are fibers of a product fibering. A cross-sectional surface in this fibering contained in a meridian disc of $T$ meets the boundary of a regular neighborhood of each $L_i$ in a meridian circle and meets the boundary of a regular neighborhood of $L$ in the negative of the longitude, while the fiber meets them in longitude circles and a meridian circle respectively.
A surgery coefficient $a/b$ means that a solid torus is filled in so that $a\cdot m+b\cdot \ell$ becomes contractible, where $m$ and $\ell$ are a meridian-longitude pair for a boundary torus of a regular neighborhood of the link component. A Seifert invariant $(\alpha,\beta)$ determines a filling in which $\alpha\cdot q+\beta\cdot t$ becomes contractible, where $q$ is the cross-section and $t$ is the fiber. So the surgery coefficients of our link produce one exceptional fiber with Seifert invariant $(n,1-n)$ for each $L_i$, and one with Seifert invariants $(n,gn-1)$ for $L$. In the notation of \cite{Orlik}, the unnormalized Seifert invariants of $Y$ are $\{0;(o_1,0);(n,1-n),\dots,(n,1-n),(n,gn-1)\}$, where there are $g+1$ exceptional orbits. The normalized invariants are $\{-1;(o_1,0);(n,1),\dots,(n,1),(n,n-1)\}$.
Now, we will construct the extension of the $G$-action on $V$ to a hyperbolic $3$-manifold. As before, let $n$ be any positive integer divisible by the orders of all the elements of $G$. Assume for the time being that $g=2$. We take the same curves $C_1$ and $C_2$ as in the Seifert-fibered construction, but for $C$ we take the image of the loop $C_0$ shown in figure~\ref{fig:hyperbolic2} under the homeomorphism of $W$ which is a right-hand twist in a meridinal disc in each of the two $1$-handles. \begin{figure}
\caption{The loop $C_0$.}
\label{fig:hyperbolic2}
\end{figure}
As before, we take the images of the $C_i$ under the $n^{th}$ power of a Dehn twist about $C$ as the attaching curves, form the closed manifold $Y$, and use the induced homomorphism $\psi$ to obtain the original $G$-action as an invariant Heegaard handlebody of a free $G$-action on a covering space $M$ of $Y$. Let $\ell_1$, $\ell_2$, $L_1$, $L_2$ and $L$ be as before. We choose the $L_i$ to lie closer to the boundary of $W$ than $L$. Again, change the attaching curves first by the inverse of the $n$ Dehn twists about $C$, introducing a surgery coefficient of $1/n$ on $L$, then by the $n-1$ twists about the $\ell_i$, introducing surgery coefficients $-1/(n-1)$ on the $L_i$. Applying left-hand twists in the meridian discs of the two $1$-handles of $W$, we move the attaching curves to the $\ell_i$, obtaining the surgery description of $Y$ shown in figure~\ref{fig:Whitehead}. This time, the coefficient of $L$ is still $1/n$, because $L$ has algebraic intersection $0$ with each of the meridian discs of the handles of $Y$ where the left-hand twists were performed. \begin{figure}
\caption{Surgery description of $Y$ in the hyperbolic case.}
\label{fig:Whitehead}
\end{figure} The complement of this link is a $2$-fold covering of the complement of the Whitehead link, so is hyperbolic. By \cite{Thurston}, Dehn surgery on the link produces a hyperbolic $3$-manifold, provided that one avoids finitely many choices for the coefficients of each component. So all but finitely many choices for $n$ yield a hyperbolic $3$-manifold for $Y$, and hence for $M$.
Finally, to adapt the construction to arbitrary genus, one simply adds more components to $C_0$ to obtain the $g-1$ circles shown in figure~\ref{fig:hyperbolic}. In the surgery description for $Y$, the chain $L\cup L_1$ in figure~\ref{fig:Whitehead} is replaced by a chain of length $2g-2$, in which $L_1,\ldots\,$, $L_{g-1}$ alternate with components from $C_0$, and the component $L_g$ links the chain as did $L_2$ in figure~\ref{fig:Whitehead}. The $L_i$ have surgery coefficients $-1-1/(n-1)$ as before, and the components coming from $C_0$ have coefficient $1/n$, since each had algebraic intersection $0$ with the union of the meridian discs. The link complement is the $(2g-2)$-fold covering of the Whitehead link complement, so is hyperbolic, and the argument is completed as before. \begin{figure}
\caption{The loops $C_0$ for the general hyperbolic construction.}
\label{fig:hyperbolic}
\end{figure}
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Ultimate limits to quantum metrology and the meaning of the Heisenberg limit} \author{Marcin Zwierz$^{1, 2}$}\email{[email protected]} \author{Carlos~A.~P\'{e}rez-Delgado$^{1, 3}$} \author{Pieter Kok$^1$} \affiliation{$^1$Department of Physics and Astronomy, University of Sheffield, Hounsfield Road, Sheffield, S3 7RH, United Kingdom\\ $^2$ Centre for Quantum Computation and Communication Technology (Australian Research Council), Centre for Quantum Dynamics, Griffith University, Brisbane, QLD 4111, Australia\\ $^3$ Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore}
\date{\today}
\begin{abstract}\noindent For the last 20 years, the question of what are the fundamental capabilities of quantum precision measurements has sparked a lively debate throughout the scientific community. Typically, the ultimate limits in quantum metrology are associated with the notion of the Heisenberg limit expressed in terms of the physical resources used in the measurement procedure. Over the years, a variety of different physical resources were introduced, leading to a confusion about the meaning of the Heisenberg limit. Here, we review the mainstream definitions of the relevant resources and introduce the universal resource count, that is, the expectation value of the generator (above its ground state) of translations in the parameter we wish to estimate, that applies to all measurement strategies. This leads to the ultimate formulation of the Heisenberg limit for quantum metrology. We prove that the new limit holds for the generators of translations with an upper-bounded spectrum. \end{abstract}
\pacs{03.65.Ta, 03.67.-a, 42.50.St, 42.50.Lc} \maketitle
\section{Introduction}\noindent Quantum metrology, or quantum parameter estimation theory, is an important branch of science that has received a lot of attention in recent years \cite{helstrom67,helstrom69,holevo11,caves81,braunstein94,braunstein96,lee02,giovannetti06,zwierz10,giovannetti11}. It studies high-precision measurements of physical parameters, such as phase, based on systems and physical evolutions that are governed by the principles of quantum mechanics. The main theoretical objective of this field is to establish the ultimate physical limits on the amount of information we can gain from a measurement \cite{helstrom67,helstrom69,holevo11,braunstein94,braunstein96}. From an experimental perspective, quantum-enhanced metrology promises many advances in science and technology, since an optimally designed quantum measurement procedure outperforms any classical procedure \cite{mitchell04,nagata07}. Furthermore, improved measurement techniques frequently lead not only to technological advancement, but also to a fundamentally deeper understanding of nature. The main figure of merit in the field of quantum metrology for both theorists and experimentalists is the precision with which the value of an unknown parameter can be estimated.
From this perspective, one of the most prominent concepts in quantum metrology is the Fisher information and the quantum Cram\'{e}r-Rao bound. The Fisher information $F(\phi)$ is a quantity that measures the amount of information about the parameter $\phi$ we wish to estimate revealed by the measurement procedure. Given the Fisher information, we can bound the minimal value of mean square error in the parameter with the quantum Cram\'{e}r-Rao bound. There exist two important regimes of the quantum Cram\'{e}r-Rao bound, the so-called shot-noise limit ({\sc snl}) that scales as $1/\sqrt{N}$ and the Heisenberg limit that scales as $1/N$, where $N$ is the resource count. The {\sc snl} is a limit attained by purely classical strategies (the term itself has its origin in quantum optics, where the detection of quanta of light is manifested as ``shots'' in a photon counter operating in Geiger mode \cite{giovannetti11}). The Heisenberg limit is imposed by the laws of quantum mechanics, namely, the generalized Heisenberg uncertainty relation, and for many years it was considered optimal and unbreakable. However, due to the unclear nature of the resource count, the optimality of the Heisenberg limit has recently been questioned \cite{luis05,boixo07,napolitano11}.
A quantum measurement procedure can employ physical resources in a number of quantum states interacting with the measured system via various interaction mechanisms (e.g., linear, non-linear, or exponential) and operation strategies (either parallel or sequential). Therefore, in the literature a variety of meanings and definitions have been associated with $N$. In order to meaningfully compare different quantum and classical measurement procedures, it is essential to determine the relevant resources. The most versatile definition identifies $N$ with the number of times that the measured system is sampled. Although remarkably useful, this definition is not universal (e.g., when the number of quantum systems used in the measurement procedure is ill defined). In this paper, we introduce a universal definition of the physical resources which leads to the ultimate and optimal formulation of the Heisenberg limit for quantum metrology.
The paper is organized as follows. In \S~\ref{sec::hl}, we review various formulations of the Heisenberg limit for a fixed and limited amount of resources used in measurement procedures. In \S~\ref{sec::query}, we explain the concept of the query complexity, that is, the number of times the measured system is sampled, and demonstrate how it applies to a variety of well-known measurement procedures. In \S~\ref{sec::count}, we introduce a universal resource count for quantum metrology which leads to a new formulation of the Heisenberg limit. Finally, in \S~\ref{sec::conclusions}, we give some concluding remarks.
\section{Various formulations of the Heisenberg limit}\label{sec::hl}\noindent In this section, we present a brief review of two definitions (and their interpretations) that are commonly associated with the term "Heisenberg limit" \cite{holland93,braunstein96,giovannetti11}. Not all of the interpretations are widely accepted. However, in our opinion this review properly reflects the present \textit{status quo} (i.e., the present confusion about the meaning) of the term ``Heisenberg limit'' in quantum metrology. What is the so-called Heisenberg limit or Heisenberg scaling?
In relation to the fundamental limitations of quantum metrology at least two interesting questions that are relevant for this field can be posed: \begin{enumerate} \item Given a fixed amount of resources, what is the best possible precision achievable in principle, that is, the precision that we aspire to reach? \item (Given a physical setup), what is the precision that is actually obtained? \end{enumerate} The first question is usually answered by an appropriate expression of the quantum Cram\'{e}r-Rao bound leading to the Heisenberg scaling. The problem of attainability of the quantum Cram\'{e}r-Rao bound (and equivalently of the Heisenberg limit itself) is addressed by the second question. Given a physical setup, that is, a physical interaction between the probe and the sampled system, we minimize the error in a value of the parameter by employing optimal probe states and measurement observables. The mean-square error in parameter $\phi$ is then given by the error propagation formula: \begin{equation}
\Delta \phi = \frac{\Delta X}{| d\langle X \rangle/d\phi |}\, , \end{equation} where the average and standard deviation of an observable $X$ are calculated in an optimal state \cite{giovannetti06}. The derivative accounts for a possible change in units between the average value of the observable $X$ and parameter $\phi$. In this article, we are mainly concerned with finding an answer to the first question.
For many years, the notion of the Heisenberg limit $1/N$ has been linked with the best possible precision achievable in principle. This association was widely accepted and uncontroversial. The recent developments in quantum metrology seem to refute this claim, mainly as a consequence of the unclear nature of the resource count $N$ \cite{luis05,boixo07,napolitano11}. It is clear that the measurement procedure offering an arbitrarily high precision is physically unfeasible. In fact, it is possible to estimate the value of a parameter with perfect resolution only when there is some prior information available about the parameter, that is, when $\phi$ is \textit{a priori} limited to a particular range of values \cite{hall11}. In order to estimate the value of a truly continuous physical quantity \textit{distributed randomly} with an unbounded precision it is necessary to employ a probe with either an infinite number of constituents (e.g., a probe with an infinite number of photons) or an unbounded energy (e.g., an idealized continuous variable). Naturally, this approach is unphysical. In computer science the unbounded precision in representing the value of a continuous quantity is associated with an analog computer. It is well-known that the idealized analog computers are capable of solving problems that are intractable on digital computers, e.g., the {\sc np}-hard problems. The concept of the unbounded precision can also be linked with a digital machine (i.e., a Turing machine with an infinite memory capacity), wherein one can access infinitely many information carries (i.e., the classical or quantum bits) or execute the computation for infinitely many time steps. All those scenarios are clearly unphysical by being idealizations of a real-world phenomena. Both in physics and computer science the unbounded precision necessarily leads to the violations of elementary laws.
Historically the term ``Heisenberg limit'' was introduced by Holland and Burnett \cite{holland93}, who referred to the number-phase uncertainty relation in Heitler \cite{heitler54}. Hence, the very first formulation (recognized mainly by physicists) identifies $N$ with the number of physical systems in the probe, e.g., (average) photon number, which then can be easily related to the (average) energy of the probe. This formulation is clearly associated with the Heisenberg uncertainty relation and leads us to the first general definition of the Heisenberg limit. \begin{definition} The uncertainty in the value of an unknown parameter estimated with a single-shot Heisenberg-limited measurement procedure scales as \begin{equation}\label{uncer} \Delta \phi \geq \frac{1}{2 \Delta \mathcal{H}}\, , \end{equation} where $\Delta \mathcal{H}$ is the standard deviation of the operator $\mathcal{H}$ that generates the translations of the probe state with the parameter $\phi$. \end{definition}\noindent To be more specific, this definition originates from the Mandelstam-Tamm type uncertainty relations $\Delta \phi \Delta \mathcal{H} \geq 1/2$ that are a manifestation of the generalized Heisenberg uncertainty relation with $\hbar = 1$ \cite{braunstein96,anadan90,boixo09}. For most of the measurement procedures, the standard deviation $\Delta \mathcal{H}$ can be easily expressed in terms of a variety of resources, including the number of quantum systems in the probe. As a consequence, the most well-known definition of the Heisenberg limit takes the following form. \begin{definition} The uncertainty in the value of an unknown parameter estimated with a single-shot Heisenberg-limited measurement procedure scales as \begin{equation} \Delta \phi \geq \frac{1}{N}\, , \end{equation} where $N$ denotes the number (i.e., amount) of resources, typically, the (average) number of physical systems (e.g., photons) in the probe. \end{definition}\noindent Taking the (average) number of quantum systems in the probe as the fundamental resource count is appropriate and intuitively appealing in many important and practical measurement procedures. However, it has been clearly demonstrated that this expression of the Heisenberg limit is not universally valid \cite{luis05,boixo07,roy08}.
Given the abundance of different quantum and classical measurement procedures, a variety of resources were introduced, such as the already-mentioned number of quantum systems in the probe, the average energy of the probe, or the measurement time \cite{giovannetti11}. In the spirit of Landauer's famous conviction that information is inseparably connected with the underlying physical world, yet another formulation of the Heisenberg limit associates $N$ with the query complexity of a quantum network representing the measurement procedure \cite{giovannetti06,zwierz10}. From the ``physical perspective'', query complexity is equivalent to the number of fundamental physical interactions occurring between the probe and the sampled system. In most situations, the query complexity can be easily related to the number of physical systems in the probe, thus encompassing the earlier formulation. However, it is the formulation of the Heisenberg limit via the query complexity and not the number of quantum systems in the probe that properly captures the fundamental precision of most measurement procedures.
In the following section, we show that the Heisenberg limit is optimal with respect to the relevant resource count. To this end, we introduce the most general measurement procedure and then reduce it to a number of important measurement procedures, analyzing their performance with respect to the relevant resource count identified with the query complexity of a quantum network.
\section{Query complexity as the resource count}\label{sec::query}\noindent Let us briefly recall the structure of the most general measurement procedure (see Fig.~\ref{PE}). Any estimation procedure can be divided into three basic steps: \begin{enumerate} \item a probe system (sensitive to the parameter we wish to estimate) is prepared in an initial quantum state $\rho(0)$. The probe consists of a fixed and limited number of physical systems that can be either well-defined or known on average; \item the state of the probe system is evolved to a state $\rho(\phi)$ by $U(\phi)=\exp(-i\phi\mathcal{H})$. The Hermitian operator $\mathcal{H}$ is the generator of translations in $\phi$, the parameter we wish to estimate. Physically, this corresponds to the interaction between the probe system and the sampled system; \item the probe system is subjected to a generalized measurement $M$, described by a positive operator valued measure ({\sc povm}), and the value of $\phi$ is extrapolated using data processing of the measurement results \cite{giovannetti11}. \end{enumerate} Note that in general, any process (including any quantum estimation procedure) can be represented as an instance of a quantum computation involving state preparation, evolution, and measurement. Due to this universality, any estimation procedure can be written as a quantum network. It is, therefore, intuitively clear that the query complexity of quantum networks should offer a valuable insight into the inner workings and the performance of measurement procedures based on those networks.
\begin{figure}
\caption{The general parameter estimation procedure involving state preparation $P$, evolution $U(\phi)$, and generalized measurement $M$ with outcomes $x$, which produces a probability distribution $p(x|\phi)$. }
\label{PE}
\end{figure}
\begin{figure}
\caption{An example of a quantum network consisting of single ($Q$, $P$, $O$) and many-body ($I$, $N$) quantum gates. }
\label{network}
\end{figure}
Quantum networks arise naturally in the circuit model of quantum computation. A quantum network can be represented or intuitively understood as a series of geometric figures. These figures consist of horizontal wires representing qubits (or in general any quantum systems) and quantum gates. The gates perform simple computational tasks on the information carried by the quantum systems. Typically, a quantum network involves many quantum systems and many quantum gates (see Fig.~\ref{network}). We represent a quantum gate as a function $f(x_{1}, \ldots, x_{N})$ with a fixed number of input parameters and a fixed number of output parameters. Here, we employ a special type of the quantum gate called a black box or a quantum oracle. A black box is a unitary operator defined by its action on quantum systems whose internal workings are usually unknown. Crucially, a black box acts in a consistent way on a well-defined set of quantum systems. As a result, we can associate with any quantum network (acting on a well-defined number of quantum systems) the concept of the query complexity representing the number of times the black box appears in this network. Mathematically, a black box is a function than can be univariate or multi-variate. When the function is multi-variate, e.g., a bi-variate function of two arguments, then a query to the black box must consist of two input parameters. This reasoning extends to many-body black-box operators that in the setting of quantum metrology describe the basic interactions between the probe and the sampled system.
Giovannetti, Lloyd, and Maccone were the first to show that there exists a fundamental connection between the concept of the query complexity and the field of quantum metrology \cite{giovannetti06}. The key insight of their result was that the precision of any (non-entangling) measurement procedure should be given in terms of the number of black-box interactions, that is, the query complexity \cite{giovannetti06}. The versatility of this approach was also emphasized by Braunstein, who advocated that the language of black boxes (each introducing the same unknown parameter) is general and can be applied to a rich class of measurement strategies \cite{braunstein06}. Similarly, van Dam \textit{et al}. addressed the problem of estimating the phase given $N$ copies of the (black-box) phase rotation gate \cite{vandam07}. The connection between the query complexity and the precision in estimating the value of the parameter was clarified and formalized for higher-order (or many-body) generators of translations in the parameter \cite{zwierz10}. While query complexity had been used before in the context of quantum metrology \cite{giovannetti06}, this paper is the first to analyze the most general interaction that governs the evolution of the probe system in this light.
Here, we focus on the query complexity of the quantum network that governs the evolution stage of the measurement procedure. Let us consider a completely general quantum network that encompasses all possible measurement strategies. The most general quantum evolution acting on the probe system generated by the operator $\mathcal{H}$ is represented by the unitary transformation $U(\phi)$, which can be graphically represented by \begin{equation}\label{net}
\xymatrix @*=<0em> @C=0.66em @R=0.8em @!R {
& \gate{{V_0}} & \gate{{O(\phi)}} & \gate{{V_1}} & \gate{{O(\phi)}} & \qw & \ldots & &\gate{{O(\phi)}} & \gate{{V_Q}} & \qw } \, . \end{equation} This general interaction consists of $Q$ applications of a black-box operator $O(\phi) = \exp(-i \phi H)$ (where $H$ is a positive Hermitian generator), interspersed with $Q+1$ arbitrary unitary gates $V_j$. The arbitrary unitary gates $V_j$ together with ancillary systems may be used to introduce adaptive (feed-forward) strategies to the estimation procedure. For a general interaction $U(\phi)$, we can use an argument by Giovannetti, Lloyd, and Maccone \cite{giovannetti06} to show that the generator of $U(\phi)$ is given by \begin{equation} \mathcal{H} = i \left(\frac{\partial U(\phi)}{\partial \phi}\right) U^{\dagger}(\phi) = \sum_{j = 1}^{Q} A_{j} \, , \end{equation} where \begin{eqnarray} A_{j} &=& i V_{Q}\, O(\phi)\, \ldots\, V_{j+1}\, O(\phi)\, V_{j}\, \frac{\partial O(\phi)}{\partial \phi}\, V^{\dag}_{j}\, O^{\dag}(\phi)\, V^{\dag}_{j+1}\, \ldots \nonumber \\ &&\ldots\, O^{\dag}(\phi)\, V^{\dag}_{Q}\, . \end{eqnarray} Therefore, the number of terms in the generator $\mathcal{H}$ is \textit{always} equal to the number of black-box operators appearing in the quantum network, that is, $Q$. Also note that the spectrum of the generator of a black-box operator $O(\phi)$ is unchanged by the unitary operators $V_{j}$. As a result, the spectrum of the generator $\mathcal{H}$ depends solely on the spectrum of the black-box operator. In the following we reduce this most general quantum network to specific quantum networks corresponding to some of the most well-known measurement procedures and relate their performance to the query complexity of underlying quantum networks.
\begin{figure}
\caption{The evolution stage of a linear (non-entangling) measurement procedure involving $N = 3$ quantum systems. The query complexity is linear in the number of quantum systems, that is, $Q = N$. }
\label{GLM}
\end{figure}
\subsection{Linear measurement procedure}\noindent Let us begin with a standard (non-entangling) linear procedure. Those kinds of procedures were described by Giovannetti, Lloyd, and Maccone \cite{giovannetti06}. In the linear case, the same parameter $\phi$ is applied independently on $N$ indistinguishable quantum systems (see Fig.~\ref{GLM}). Here, each fundamental physical interaction of the form: $O_{j}(\phi) = \exp(-i \phi H_{j})$, where $j$ denotes the quantum system, represents a single query. Consequently, the joint generator of the evolution operator $U(\phi)$ can be written as a sum of commuting generators $H_{j}$, that is, \begin{equation} \mathcal{H}_{GLM} = \sum_{j=1}^{N} H_{j}\, . \end{equation} Since the query complexity corresponds to the number of terms in the joint generator $\mathcal{H}_{GLM}$, we have $Q=N$. The fundamental precision then scales as \cite{giovannetti06} \begin{equation} \Delta \phi \geq \frac{c_{1}}{Q} = \frac{c_{1}}{N} = O(N^{-1})\, , \end{equation} with $c_{1} = 1/(\lambda_{\rm max} - \lambda_{\rm min})$ with $\lambda_{\rm max}$ and $\lambda_{\rm min}$ being the maximal and minimal eigenvalues of $H$. Typically, $c_{1}$ is a constant of order one. This bound is saturated by the following family of maximally entangled states of the probe \cite{giovannetti06}: \begin{equation}
|\psi\rangle = \frac{1}{\sqrt{2}}\left(|h_{\rm max}\rangle + e^{i \varphi} |h_{\rm min}\rangle\right)\, , \label{sat}
\end{equation} where $|h_{\rm max}\rangle = |\lambda_{\rm max}\rangle_{1} \cdots |\lambda_{\rm max}\rangle_{N}$ and $|h_{\rm min}\rangle = |\lambda_{\rm min}\rangle_{1} \cdots |\lambda_{\rm min}\rangle_{N}$ are the eigenstates corresponding to the maximal and minimal eigenvalues of $\mathcal{H}$, respectively (with $|\lambda_{\rm max}\rangle$ and $|\lambda_{\rm min}\rangle$ being the eigenstates corresponding to the maximal and minimal eigenvalues of $H$, respectively). In the setting of quantum interferometry, the formal analog of the maximally entangled state given by Eq.~(\ref{sat}) is a {\sc NOON} state \cite{lee02,kok10}: \begin{equation}
|\psi\rangle = \frac{1}{\sqrt{2}} \left( |N,0\rangle + |0,N\rangle \right)\, , \end{equation} in which $N$ photons are propagating along the first or the second optical path of the Mach-Zehnder interferometer \cite{giovannetti11}.
\begin{figure}
\caption{The evolution stage of a two-body (entangling) measurement procedure involving $N = 3$ quantum systems. The query complexity scales quadratically with the number of quantum systems, that is, $Q = \frac{1}{2}N(N-1)$. }
\label{BFCG}
\end{figure}
\subsection{Many-body measurement procedure}\noindent The evolution operator $U(\phi)$ can also act on the constituents of the probe in a non-trivial (non-linear) way. In the case of the so-called many-body or higher-order measurement procedures introduced by Boixo, Flammia, Caves, and Geremia \cite{boixo07}, the fundamental interactions are applied to multiple quantum systems. For example, a two-body joint generator $\mathcal{H}_{BFCG}$ then takes two quantum systems as an input: \begin{equation} \mathcal{H}_{BFCG} = \sum_{l=1}^{N} \sum_{j=1}^{l} H_{j} \otimes H_{l}\, , \end{equation} and is depicted in Fig.~\ref{BFCG} (with $N=3$) as a collection of bi-variate black-box operators $O_{jl}(\phi) = \exp(-i \phi H_{j} \otimes H_{l})$ (where $j$ and $l$ label the quantum systems) that constitute fundamental two-body interactions. In this case a single query is necessarily applied to two input quantum systems. From this it follows that given a probe consisting of $N$ quantum systems a total number of queries is the number of possible pairs from a set of size $N$, that is, $Q = \left({N \atop 2}\right) = \frac{1}{2}N(N-1)$. It is, therefore, clear that the number of queries $Q$ \textit{is not always} identical to the number of physical systems $N$ in the probe. The fundamental precision of a two-body measurement procedure expressed in terms of the query complexity is given by \begin{equation} \Delta \phi \geq \frac{c_{2}}{Q} = c_{2} \left({N \atop 2}\right)^{-1} = O(N^{-2})\, , \end{equation} where $c_{2} = 1/(\lambda^{2}_{\rm max} - \lambda^{2}_{\rm min})$, with $\lambda_{\rm max}$ and $\lambda_{\rm min}$ being the maximal and minimal eigenvalues of $H$ \cite{boixo07}. Typically, $c_{2}$ is a constant of order one. Note that the error in $\phi$ is linear in the query complexity $Q$. At the same time, $Q$ has a superlinear scaling with $N$ attributed to the entangling power that the evolution operator has over the constituents of the probe. This approach naturally extends to generators of any degree $k \leq N/2$. A $k$-body measurement procedure offers the following scaling \cite{boixo07} \begin{equation} \Delta \phi \geq \frac{c_{k}}{Q} = c_{k} \left({N \atop k}\right)^{-1} = O\left(N^{-k}/k!\right)\, , \end{equation} where $c_{k} = 1/(\lambda^{k}_{\rm max} - \lambda^{k}_{\rm min})$. This bound is also saturated by the family of maximally entangled state of the probe given by Eq.~(\ref{sat}).
\begin{figure}
\caption{The evolution stage of an exponential (highly-entangling) measurement procedure involving $N = 3$ quantum systems. The query complexity scales exponentially with the number of quantum systems, that is, $Q = 2^{N}-1$. }
\label{RB}
\end{figure}
\subsection{Exponential measurement procedure}\noindent The finite number of quantum systems in the probe imposes restrictions on the dimensionality of the probe's Hilbert space. By exploiting the whole Hilbert space, Roy and Braunstein introduced the exponential measurement procedure \cite{roy08}. In Fig.~\ref{RB}, we present this procedure translated to the language of the query complexity. Here, the query complexity scales exponentially with the number of quantum systems, that is, $Q = 2^{N}-1$. The fundamental precision of the exponential measurement procedure expressed in terms of the query complexity is given by \begin{equation} \Delta \phi \geq \frac{c_{e}}{Q} = O(2^{-N})\, , \label{exp} \end{equation} where $c_{e}$ is a constant of order one \cite{roy08}. Interestingly, this bound is saturated by separable states due to the fact that all necessary entanglement is being generated at the evolution stage of the measurement procedure.
Kitaev's famous phase estimation algorithm on discrete quantum systems, e.g., qubits, is another example of a measurement procedure which offers precision that scales exponentially with the number of employed qubits \cite{kitaev,cleve}. However, as in the previous case, this procedure requires an exponential number of fundamental unitary evolution gates, that is, black boxes. As a result, the precision is again bounded by Eq.~(\ref{exp}).
We emphasize that the bounds expressed in terms of the number of queries $Q$ are saturated by the appropriate optimal states, that is, \begin{equation} \Delta \phi \simeq \frac{1}{Q}\, , \end{equation} for all measurement procedure with well-defined $Q$. Therefore, the relevant resource count can be identified with the query complexity offering the linear scaling of the root-mean-square error in $\phi$. This indicates that the query complexity \textit{may} be considered a good resource count. However, does this mean that the query complexity \textit{is} the proper physical resource count for quantum metrology? We give the answer to this question in the next section.
\subsection{Query complexity and the shot-noise limit}\noindent First, however, we need to remark that the concept of the query complexity is much more natural for quantum-enhanced metrology and generally does not apply to the classically limited procedures. The main reason for this is the problematic nature of the {\sc snl} itself.
In the case of a linear strategy, the shot-noise-limited (classical) measurement procedure consists of $N$ independent measurement repetitions. each involving a single black-box interaction offering the following scaling: \begin{equation} \Delta \phi \geq \frac{c_{1}}{\sqrt{Q}} = \frac{c_{1}}{\sqrt{N}}\, \label{linear} \end{equation} as prescribed by the quantum Cram\'{e}r-Rao bound. The same scaling is found by calculating the standard deviation of the generator $\mathcal{H}$ in the separable state of $N$ quantum systems, that is, $\Delta \mathcal{H} = \sqrt{\sum_{j=1}^{N}\Delta^{2} H_{j}} = c_{1} \sqrt{N} = c_{1} \sqrt{Q}$. However, these approaches fail in the case of many-body (non-linear) measurement strategies. For example, the shot-noise-limited $k$-body measurement procedure can be defined (in analogy to the linear case) as $N$ independent measurement repetitions, each involving a single $k$-variate black-box interaction yielding the same scaling as the scaling given by Eq.~(\ref{linear}). On the other hand, Boixo \textit{et al.} \cite{boixo08} more formally derived an $O(N^{k-1/2})$ scaling offered by a $k$-body measurement procedure fed with the separable state of $N$ quantum systems. Here, because of the Big $O$ notation it is impossible to calculate a well-defined number of queries. Also, the two approaches no longer predict the same scaling with $N$ [As a side remark: in the case of the exponential measurement procedure it is even nonsensical to consider the {\sc snl} as there is no single, basic black box that can be repeated $N$ times and the separable state is the optimal state for this procedure]. As a consequence, the concept of the query complexity does not apply to the classical procedures. It seems that the main difficulty in extending the language of the query complexity to the classical domain lies in the fact that it is unclear what is the universal definition of the shot-noise limit that would apply to all types of measurement strategies. If the {\sc snl} is a limit obtained in a procedure consisting of $N$ independent repetitions of a basic black box, then it yields a trivial bound, that is, it always scales as $1/\sqrt{Q}$, with $Q=N$ [the advantage of this approach is a well-defined query complexity]. If the {\sc snl} is a limit obtained in a procedure that employs a separable state, then for the $k$-body procedure the query complexity is ill defined.
What is the universal definition of the {\sc snl} limit that can be consistently applied to any estimation procedure and can it be expressed in terms of the query complexity? We leave these as open questions.
\section{Universal resource count}\label{sec::count}\noindent The concept of the query complexity proved extremely useful in setting fundamental limits on the capabilities of a variety of measurement procedures and relating these to the number of employed quantum systems. We demonstrated that the query complexity can be used to meaningfully compare the precision offered by these measurement procedures. Moreover, we showed that the error in the parameter scales linearly with $Q$ for a number of important measurement procedures. Also, it is straightforward to tie the query complexity with the number of physical systems in the probe. This implies that the query complexity is a good resource count. Unfortunately, the query complexity can be ill defined. Some measurement procedures have an ill defined number of physical systems in the probe, and as a result, the exact number of queries is unknown. For example, for an optical measurement procedure employing coherent states, the number of photons in the probe is known only on average and there is no such quantity as an average number of queries. Therefore, we need to find a universal resource count that can deal with these cases properly.
Any universal resource count for quantum metrology should fulfill some basic requirements. First, for any measurement procedure the minimal uncertainty in the value of the parameter must scale linearly with a universal resource count. Second, in order to find this resource count, and by implication a universal formulation of the Heisenberg limit, we cannot refer to a specific physical implementation. Instead, we should derive the fundamental resource count from the general description of a measurement procedure. Finally, for purely physical reasons a good candidate for a universal resource count should also relate in a straightforward manner to some quantum-mechanical observable such as the number of physical systems in the probe or its (average) energy.
Traditionally, the Heisenberg limit on the interferometric precision of estimating a phase $\phi$ is generally understood as the following scaling relation: \begin{equation} \Delta \phi \geq \frac{c}{\langle N \rangle}\, , \end{equation} where $c$ is a constant of order one and $\langle N \rangle$ is the average of the number operator $N = a^{\dagger} a$ which generates the phase shift, that is, the total number of photons in the probe \cite{hall11}. This is a well-established relation \cite{hall11,giovannetti11arXiv,tsang11} and its achievability has been recently proved \cite{hayashi11}. For general quantum parameter estimation the number operator is replaced by the operator $\mathcal{H}$ which generates the translations in the parameter.
Given the traditional formulation of the Heisenberg limit (and keeping in mind the above requirements), it is natural to formalize the universal resource count for quantum metrology as the expectation value of the generator of translations in the parameter $\langle \mathcal{H} \rangle$. Note that it is necessary to set the minimal eigenvalue (the ground-state eigenvalue) of the generator $\mathcal{H}$ to zero. The necessity of rescaling the value of the resource count stems from the fact that when $\mathcal{H}$ corresponds to a proper Hamiltonian, the origin of the energy scale has no physical meaning and as a consequence, we must fix the scale such that the ground state has zero energy. Therefore, the universal resource count is given by the expectation value of the generator $\mathcal{H}$ above its ground state denoted as $|\langle\mathcal{H}\rangle|$. We can also formulate our resource count without any shift in terms of $\langle\mathcal{H}-h_{\rm min}I\rangle$, where $h_{\rm min}$ is the minimal eigenvalue of $\mathcal{H}$ and $I$ is the identity operator. When $\mathcal{H}$ does not have a minimum eigenvalue, as in the case of position or momentum operators, the only possible values for $\langle\mathcal{H}\rangle$ are \emph{relative} position and momentum.
As we show in the following subsections, the proposed resource count fulfils all the requirements of a universal resource count and applies to any conceivable measurement procedure, even when apparently no query complexity can be defined. We argue that $|\langle\mathcal{H}\rangle|$ is a more fundamental resource count than $Q$, since it can deal with these cases as well. Nevertheless, whenever the query complexity exists, it is desirable to find an exact relation between the universal resource count and $Q$ (and by implication $N$).
\subsection{Standard deviation of the generator $\mathcal{H}$}\noindent It can be argued that the standard deviation of the generator $\mathcal{H}$ can also serve as a universal resource count \cite{brody96,brody97}. Indeed, in the next subsection we show that $\Delta\mathcal{H}$ is related to the query complexity, and by implication to the number of quantum systems in the probe. However, $|\langle \mathcal{H} \rangle|$ is the \textit{only} moment that fits the right category, given the question of how many resources are required to attain a certain precision. Resources are ``a certain amount of something''. Thus, when dealing with probabilistic situations the physical amount is given by the first moment, while the higher-order moments describe the shape of the distribution. Also, it is important to note that the first moment represents a fundamental conserved quantity \cite{margolus11arXiv}. This distinction captures a physical intuition that goes beyond the pure mathematics of quantum metrology.
In order to quantitatively capture the distinction between the expectation value of the generator $\mathcal{H}$ above the ground state and the standard deviation of the generator $\mathcal{H}$, let us consider the (important for quantum metrology) family of pure superpositions given by \begin{equation}\label{eq:super}
|\psi\rangle = \sqrt{\mu} |h_{\rm max}\rangle + \sqrt{1-\mu} \, e^{i \varphi} |h_{\rm min}\rangle\, , \end{equation}
where $|h_{\rm max}\rangle$ and $|h_{\rm min}\rangle$ are the eigenstates corresponding to the maximal and minimal eigenvalues of $\mathcal{H}$, respectively. The expectation value of the generator $\mathcal{H}$ in the above state can be written as \begin{equation} \langle \mathcal{H} \rangle = \mu h_{\rm max} + (1 - \mu) h_{\rm min}\, , \end{equation} where $h_{\rm max}$ and $h_{\rm min}$ are the maximal and minimal eigenvalues of $\mathcal{H}$, respectively. The expectation value of $\mathcal{H}$ above its ground state and the standard deviation of $\mathcal{H}$ are then given by \begin{eqnarray}
|\langle\mathcal{H}\rangle| \equiv \langle\mathcal{H}-h_{\rm min}I\rangle &=& \mu (h_{\rm max} - h_{\rm min}) = \mu \|\mathcal{H}\|\, , \nonumber \\ \Delta \mathcal{H} &=& \sqrt{\mu (1 - \mu)} (h_{\rm max} - h_{\rm min}) \nonumber \\
&=& \sqrt{\mu (1 - \mu)} \|\mathcal{H}\| \, , \nonumber \end{eqnarray}
where $\| \mathcal{H} \|$ is the semi-norm of the generator $\mathcal{H}$. In Fig.~\ref{exp_vs_stan}, we depict the above relations [with $\| \mathcal{H} \| = 1$ a.u.]. Note that both quantities are upper bounded by $\| \mathcal{H} \|$: \begin{equation}
|\langle\mathcal{H}\rangle| \leq \| \mathcal{H} \| \ \ , \ \ \Delta \mathcal{H} \leq \frac{\| \mathcal{H} \|}{2}\, \nonumber
\end{equation} and coincide for $\mu = \frac{1}{2}$. Alas, even for as specific a family of superpositions as the one given by Eq.~(\ref{eq:super}), no definite relation between $|\langle \mathcal{H} \rangle|$ and $\Delta \mathcal{H}$ can be established.
The importance of the standard deviation $\Delta \mathcal{H}$ stems from the fact that this quantity provides the achievable bounds in quantum metrology, that is, it gives the answer to the second question posed in \S~\ref{sec::hl}. On the other hand, the expectation value $|\langle \mathcal{H} \rangle|$ being a universal physical resource count provides the bound on the best possible precision achievable in principle (see the first question posed in \S~\ref{sec::hl}).
\begin{figure}
\caption{The expectation value of the generator $\mathcal{H}$ above the ground state versus the standard deviation of the generator $\mathcal{H}$ as a function of $\mu$ [with $\| \mathcal{H} \| = 1$ a.u.]. }
\label{exp_vs_stan}
\end{figure}
In the next subsection, we derive a lower bound on the error in the parameter $\phi$ in terms of a new resource count $|\langle\mathcal{H}\rangle|$. We also present the exact relations between the query complexity, the expectation value of $\mathcal{H}$ above its ground state, and the standard deviation of $\mathcal{H}$ for the most relevant measurement strategies.
\subsection{New formulation of the Heisenberg limit}\label{sec::formulation}\noindent Having established the proper resource count, we present a new formulation of the Heisenberg limit with respect to this resource count. We consider here the most general quantum measurement procedure corresponding to the unitary transformation $U(\phi)$ presented in \S~\ref{sec::query}. Since the Heisenberg limit should refer to the {\em optimal} scaling behavior of the error with the resource count (it is a bound that we aspire to reach, not the actual achievable bound in any given experimental setup), we can restrict our discussion to the case of optimal states for quantum metrology, such as the textbook case of {\sc NOON} states \cite{lee02,kok10}. The optimal states in quantum metrology are the families of balanced superpositions of the eigenvectors $|h_{\rm max}\rangle$ and $|h_{\rm min}\rangle$ of $\mathcal{H}$, that is, the state given by Eq.~(\ref{eq:super}) with $\mu = \frac{1}{2}$. For the optimal states we have the property that \begin{equation}\label{eq:semi-norm}
2 \Delta\mathcal{H} = 2 |\langle\mathcal{H}\rangle| = \| \mathcal{H} \| = h_{\rm max} - h_{\rm min}\, . \end{equation} Combining this with Eq.~(\ref{uncer}), the error in parameter $\phi$ in a single-shot experiment is then given by \begin{equation}\label{HL}
\Delta\phi \geq \frac{1}{2 |\langle\mathcal{H}\rangle|}\, .
\end{equation} This inequality holds (and is tight) for the quantum states that are \textit{optimal} for quantum metrology, and is therefore an expression for the minimum error $\Delta\phi$ that can be achieved in an optimal measurement. While the derivation is mathematically valid only for generators with an upper-bounded spectrum, physically this bound will generally be satisfied since we can always truncate the Hilbert space at sufficiently high energy states. Therefore, given a system evolution described by $U(\phi) = \exp(-i\phi \mathcal{H})$, for any (numerical) value for $|\langle\mathcal{H}\rangle|$ (i.e., the resource amount), the best attainable precision for a measurement of $\phi$ is bounded by Eq.~(\ref{HL}). This is a universal formulation of the Heisenberg limit. The new bound is just as tight as the bound provided by the standard deviation of $\mathcal{H}$, and whenever the latter exists both are identical.
In the following, we present the exact relations between $\Delta\mathcal{H}$, $|\langle\mathcal{H}\rangle|$ and the query complexity $Q$ that applies to the most relevant measurement strategies presented in \S~\ref{sec::query}. Given the arguments about the spectrum of the generator $\mathcal{H}$ for the most general quantum measurement procedure (see \S~\ref{sec::query}), for the optimal states the maximal and minimal eigenvalues $\mathcal{H}$ are given by \begin{equation}\label{eq:eigen} h_{\rm max} = Q \lambda^{k}_{\rm max} \quad\text{and}\quad h_{\rm min} = Q \lambda^{k}_{\rm min} \, , \end{equation} where $\lambda_{\rm max}$ and $\lambda_{\rm min}$ are the maximal and minimal eigenvalues of $H$, respectively. The power $k$ denotes the order of the black-box interaction: with $k = 1$ representing a linear (non-entangling) black-box interaction. Since the exponential measurement procedures have a more complex structure, their corresponding $h_{\rm max}$ and $h_{\rm min}$ have a more compound form as well: \begin{eqnarray} h_{\rm max} = \sum_{j=1}^{N} Q_{j} \lambda^{j}_{\rm max} \quad \text{and} \quad h_{\rm min} = \sum_{j=1}^{N} Q_{j} \lambda^{j}_{\rm min} \, \end{eqnarray} with \begin{eqnarray} Q_{j} = \left({N \atop j}\right) \quad \text{and} \quad Q = \sum_{j=1}^{N} Q_{j} = 2^{N} - 1\, . \end{eqnarray} Therefore, for the sake of clarity, we present our final result using bounds given by Eq.~(\ref{eq:eigen}). Combing these with Eq.~(\ref{eq:semi-norm}), we have
\begin{equation} 2 \Delta\mathcal{H} = 2 |\langle\mathcal{H}\rangle| = \| \mathcal{H} \| = Q (\lambda^{k}_{\rm max} - \lambda^{k}_{\rm min})\, . \end{equation} This leads to a bound on the error in parameter $\phi$, expressed in terms of the query complexity \begin{equation}
\Delta\phi \geq \frac{1}{2 |\langle\mathcal{H}\rangle|} = \frac{c_{k}}{Q}\,
\end{equation} with $c_{k} = 1/(\lambda^{k}_{\rm max} - \lambda^{k}_{\rm min})$ being the inverse of the largest gap in the spectrum of the generator $\mathcal{H}$. Regardless of whether a well-defined number of queries exists or not, the first bound always holds. Given a particular measurement procedure, it is then straightforward to express the error in the parameter in terms of the number of quantum systems in the probe via the query complexity. For example, in the standard cases of linear (non-entangling) evolutions, the query complexity (and by implication $2 |\langle\mathcal{H}\rangle|$) reduces to the number of physical systems in the probe.
Finally, we note that our result applies to both parallel and sequential (or ``multi-round") measurement strategies, always giving a well-defined bound that can be expressed in terms of either $|\langle\mathcal{H}\rangle|$ or $Q$ (whenever the query complexity can be defined, it is always a finite number). For a parallel procedure (i.e., no sequential repetitions of the evolution gate $U(\phi)$ are allowed), the query complexity is limited by the dimensionality of probe's Hilbert space, thus, $Q \leq 2^{N} -1$. In the case of a sequential strategy (i.e., sequential repetitions of the evolution gate $U(\phi)$ possibly interspersed with some arbitrary unitaries are allowed), the total number of queries is limited by the dimensionality of the probe's Hilbert space and the measurement time $T$ (i.e., the number of repetitions), thus, $Q \leq T \times (2^{N} -1)$. In general, the sequential strategies may be looked upon as a way of estimating a value of a rescaled parameter. In these cases, one can argue that a value of the parameter $\theta = T \phi$ is being estimated rather than $\phi$. Nevertheless, it is worth emphasizing that the sequential strategies are more general and often offer some advantages over the parallel strategies, e.g., through the simplicity of their implementation \cite{giovannetti11}.
\section{Conclusions}\label{sec::conclusions}\noindent Proper resource accounting is crucial when investigating the precision of various quantum measurement strategies and formulating the ultimate limits in quantum metrology. In this work, we applied the concept of the query complexity representing the number of times the measured system is sampled to a variety of well-known measurement procedures analyzing their performance. This leads to a universal definition of the physical resources (formalized as the expectation value of the generator of $\mathcal{H}$ above its ground state) and the ultimate formulation of the Heisenberg limit for quantum metrology. The new bound holds only for optimal states and is just as tight as the bound provided by the standard deviation of $\mathcal{H}$ (whenever the latter exists, both are identical).
\begin{acknowledgments}\noindent This work was supported by the White Rose Foundation and the ARC Centre of Excellence CE110001027. \end{acknowledgments}
\end{document}
|
arXiv
|
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Quantum Coherence of Orbital Angular Momentum Multiplexed Continuous-Variable Entangled State}
\author{Hong Wen,\authormark{1,2} Li Zeng,\authormark{1,2} Rong Ma,\authormark{1,2} Haijun Kang,\authormark{1,2} Jun Liu,\authormark{1,2} Zhongzhong Qin,\authormark{1,2,*} and Xiaolong Su\authormark{1,2}}
\address{\authormark{1}State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan, 030006, People's Republic of China\\ \authormark{2}Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan,Shanxi 030006, People's Republic of China}
\email{\authormark{*}[email protected]}
\begin{abstract} Orbital angular momentum (OAM) multiplexed entangled state is an important quantum resource for high dimensional quantum information processing. In this paper, we experimentally quantify quantum coherence of OAM multiplexed continuous-variable (CV) entangled state and characterize its evolution in a noisy environment. We show that the quantum coherence of the OAM multiplexed CV entangled state carrying topological charges $l=1$ and $l=2$ are the same as that of the Gaussian mode with $l=0$ in a noisy channel. Furthermore, we show that the quantum coherence of OAM multiplexed entangled state is robust to noise, even though the sudden death of entanglement is observed. Our results provide reference for applying quantum coherence of OAM multiplexed CV entangled state in a noisy environment. \end{abstract}
\section{Introduction} Quantum coherence, which characterizes the quantumness and underpins quantum correlation in quantum systems, plays a significant role in quantum information processing \cite{Baumgratz2014}. Recently, quantum coherence has been identified as important quantum resource besides quantum entanglement and steering, and has attracted rapidly increasing interests \cite{Streltsov2017,Chitambar2019}. In 2014, Baumgratz \textit{et al.} established a framework to quantify quantum coherence by referring to the method of quantifying entanglement \cite{Baumgratz2014}. The quantum coherence of a quantum state is defined as the minimum distance between the quantum state and an incoherent state in the Hilbert space, and can be quantified by relative entropy and $l_{1}$-norm \cite{Baumgratz2014}. Besides, it has been shown that quantum coherence can also be quantified by Fisher information \cite{Feng2017}, skew information entropy \cite{Yu2017}, Tsallis relative $\alpha$ entropy \cite{Rastegin2016}, robustness \cite{Napoli2016}, and so on. Furthermore, quantum coherence with infinite-dimensional systems, i.e. continuous-variable (CV) quantum states, can be quantified by relative entropy \cite{Fan2016}. Both theoretical investigations and experimental demonstrations of quantum coherence have achieved significant progresses \cite{Xu2016, Buono2016, Albarelli2017,Yuan2017D,Wu2017,Gao2018,Wu2018,Zhang2019,Xu2020,ZhangPRJ2021}. Recently, we have experimentally demonstrated the robustness of Gaussian quantum coherence in quantum channels \cite{KangPRJ2021}, as well as the conversion of local and correlated Gaussian quantum coherence \cite{KangOL2021}.
As an important quantum resource, Einstein-Podolsky-Rosen (EPR) entangled state has been widely applied in quantum communication, quantum computation, and quantum precision measurement \cite{EPREntanglement,BraunsteinRMP,KimbleQuanInt,WeedbrookRMP,PhysicsReport,HaoCPB2021,SuSciCH2021}. Besides optical parametric amplifier, four-wave mixing (FWM) process in warm alkali vapor cell is another efficient method to generate CV EPR entangled state \cite{EntangledImages,MaOL2018}. The FWM process has been widely used in quantum state engineering \cite{QinLight,QinPRL,QinOL2014}, quantum beam splitter \cite{LiuOL2019}, and quantum precision measurement \cite{SU11,PooserOptica}. Especially, spatial-multi-mode advantage of the FWM process, attributed to its cavity-free configuration, makes it an ideal method to generate entangled images \cite{EntangledImages}. Orbital angular momentum (OAM) multiplexing of light has been found to be an efficient way to improve data-carrying capacity in both classical and quantum communications due to its infinite range of possibly achievable topological charges \cite{AllenOAM,OAMMultiplexing}. Recently, OAM multiplexed bipartite and multipartite CV entangled states have been generated based on the FWM process \cite{JingBiOAM,JingTriOAM,JingHexaOAM}, and they have been applied in OAM multiplexed quantum teleportation \cite{JingQuanTele} and quantum dense coding \cite{JingQDC}. It is essential to distribute OAM multiplexed quantum resources in quantum channels to realize these quantum information processing protocols. Recently, our group has experimentally demonstrated that quantum entanglement and quantum steering of the OAM multiplexed states carrying topological charges $l=1$ and $l=2$ are the same as that of the Gaussian mode with $l=0$ in lossy and noisy channels \cite{PRJ2022}. However, it remains unclear whether the quantum coherence of OAM multiplexed CV entangled state is also as robust as that of the Gaussian mode with $l=0$.
Here, we experimentally quantify quantum coherence of OAM multiplexed CV entangled state and characterize its evolution in a noisy channel. We show that quantum coherence of CV entangled state carrying topological charges $l=1$ and $l=2$ are as robust against loss and noise as that of Gaussian mode entangled state with $l=0$. More interestingly, the quantum coherence of CV entangled state always exists unless one mode is completely lost, while sudden death of entanglement is observed in the presence of certain amounts of loss and noise. Our results pave the way for applying the quantum coherence of OAM multiplexed entangled state in high data-carrying capacity quantum communication protocols.
\section{Theory} The Hamiltonian of the OAM multiplexed FWM process can be expressed as \cite{PRJ2022}: \begin{equation} \hat{H}=\sum_{l}i\hbar\gamma_{l}\hat{a}^{\dagger}_{l,P}\hat{a}^{\dagger}_{-l,C}+h.c. \end{equation} where \(\gamma_{l}\) is defined as the interaction strength of each OAM pair. \(\hat{a}^{\dagger}_{l,P}\) and \(\hat{a}^{\dagger}_{-l,C}\) are the creation operators related to OAM modes of the Pr and Conj fields, respectively. Since the pump field does not carry OAM (\(l=0\)), the topological charges of the Pr and Conj fields are opposite. The output state of the OAM multiplexed FWM process is: \begin{equation} \ket{\Psi}_{out}=\ket{\Psi}_{-l}\otimes\cdots\otimes\ket{\Psi}_{0}\otimes\cdots\otimes\ket{\Psi}_{l} \end{equation} where \(\ket{\Psi}_{l}\) presents a series of independent OAM multiplexed CV EPR entangled states of $\ket{\psi_{l,P}}$ and $\ket{\psi_{-l,C}}$ generated in the FWM process. $\ket{\psi_{l,P}}$ and $\ket{\psi_{-l,C}}$ represent Pr field carrying topological charge $l$ and Conj field carrying topological charge $-l$ , respectively.
A Gaussian state $\hat{\rho}(\bar{\mathbf{x}},\mathbf{V})$ can be completely represented by the displacement $\bar{\mathbf{x}}$ and the covariance matrix $\mathbf{V}$ in phase space, which correspond to the first and second statistical moments of the quadrature operators, respectively \cite{WeedbrookRMP,PhysicsReport}. The displacement $\mathbf{\bar{x}}=\langle
\hat{x}\rangle $, where
$\hat{x}\equiv (\hat{X}_{-l,C}, \hat{Y}_{-l,C}, \hat{X}_{l,P}, \hat{Y}_{l,P})^{T}$, $\hat{X}=\hat{a}+\hat{a}^{\dag}$ and $\hat{Y}=(\hat{a}-\hat{a}^{\dag})/i$ are the amplitude and phase quadratures of an optical mode, respectively, and $T$ denotes transpose.
The elements of covariance matrix $\mathbf{V}$ are defined as
$\mathbf{V}_{ij}=\frac{1}{2}\langle \hat{x}_{i}\hat{x}_{j}+\hat{x}_{j}\hat{x}_{i}\rangle -\langle \hat{x}_{i}\rangle \langle \hat{x}_{j}\rangle$. The Gaussian quantum coherence of EPR entangled state is expressed by \cite{Xu2016}
\begin{equation}
\mathcal{C}_{rel.~ent.}\left[\hat{\rho}(\bar{\mathbf{x}},\mathbf{V})\right] =S\left[\hat{\rho}(\bar{\mathbf{x}}_{th},\mathbf{V}
_{th})\right] -S\left[\hat{\rho}(\bar{\mathbf{x}},\mathbf{V}) \right],
\end{equation}
where $S\left[\hat{\rho}(\bar{\mathbf{x}},\mathbf{V})\right]\!=-\!\underset{i=1}{\overset{2}{\sum }}\!\left[\!\left(
\frac{\nu_{i}-1}{2}\right)\!\log\!_{2}\!\left( \frac{\nu_{i}-1}{2}
\right)\!-\!\left( \frac{\nu_{i}+1}{2}\right)\!\log\!_{2}\!\left( \frac{\nu
_{i}+1}{2}\right)\!\right]$ and
$S\left[ \hat{\rho}(\bar{\mathbf{x}}_{th},\mathbf{V} _{th})\right]\!\!=\!-\!\underset{i=1}{\overset{2}{\sum }}\!\left[\!\left(
\frac{\mu_{i}-1}{2}\right)\!\log\!_{2}\!\left( \frac{\mu_{i}-1}{2}
\right)\!-\!\left( \frac{\mu_{i}+1}{2}\right)\!\log\!_{2}\!\left( \frac{\mu_{i}+1}{2}\right)\!\right]$ are the von Neumann entropy of $\hat{\rho}(\bar{\mathbf{x}},\mathbf{V})$ and a thermal state $\hat{\rho}(\bar{\mathbf{x}}_{th},\mathbf{V}
_{th})$, respectively.
$\nu_{i}$ and $\mu_{i}$ are the symplectic eigenvalues of $\mathbf{V}$
and $\mathbf{V}_{th}$, respectively. The elements of the diagonal covariance matrix $\mathbf{V}_{th}$ are given by $\mathbf{V}$ with $V_{th}\ _{2i-1,2i-1}=V_{th}\ _{2i,2i}=\frac{1}{2}\left(V_{2i-1,2i-1}+V_{2i,2i}+\left[ \mathbf{\bar{x}}_{2i-1}\right] ^{2}+\left[\mathbf{\bar{x}}_{2i}\right] ^{2}\right)$.
\begin{figure}
\caption{(a) Experimental setup for generating and distributing quantum coherence of OAM multiplexed CV quantum entangled state in a noisy channel. Cs: cesium vapor cell; Pr: probe field; Conj: conjugate field; LO\(_{-l, P}\) and LO\(_{l, C}\): local oscillators of Pr and Conj fields; AM: amplitude modulator; PM: phase modulator; GL: Glan-laser polarizer; GT: Glan-Thompson polarizer; PBS: polarization beam splitter; HWP: half-wave plate; VPP: vortex phase plate; M: mirror; BS: 50:50 beam splitter; BHD$_{1}$, BHD$_{2}$: balanced homodyne detectors; SA: spectrum analyzer. (b) Beam patterns and interference patterns of the Pr and Conj fields for \(l=1\). (c) Beam patterns and interference patterns of the Pr and Conj fields for \(l=2\).}
\end{figure}
In our experiment, the displacements $\bar{\mathbf{x}}$ of the EPR entangled state are zero, so the state can be completely represented by its covariance matrix $\mathbf{V}$. The covariance matrix of the OAM multiplexed entangled state after distribution in a noisy channel is
\begin{align}
\mathbf{V}&=\left(
\begin{array}{cccc}
\mathbf{A} & \mathbf{C} \\
\mathbf{C}^{T} & \mathbf{B}
\end{array}
\right),
\end{align}
where $\mathbf{A}=\frac{V+V^{\prime}}{2}~\mathbf{I}$, $\mathbf{B}=[\eta\frac{V+V^{\prime}}{2}+(1-\eta)(1+\delta)]~\mathbf{I}$, $\mathbf{C}=\sqrt{\eta} \frac{V^{\prime}-V}{2}~\mathbf{Z}$,
$\mathbf{I}=\begin{pmatrix}
\begin{smallmatrix}
1 & 0 \\
0 & 1
\end{smallmatrix}
\end{pmatrix}$ and
$\mathbf{Z}=\begin{pmatrix}
\begin{smallmatrix}
1 & 0 \\
0 & -1
\end{smallmatrix}
\end{pmatrix}$.
The submatrices $\mathbf{A}$ and $\mathbf{B}$ correspond to the states of Alice's and Bob's subsystems, respectively. $V$ and $V^{\prime}$ represent the variances of correlation and anti-correlation quadratures of the EPR entangled state, respectively. Note that $VV^{\prime} \geq 1$ is always satisfied according to the uncertainty principle, and the equality holds only for pure state. $\eta$ and $\delta$ (in the units of SNL) represent transmission efficiency and excess noise of the noisy channel, respectively. Therefore, $\delta=0$ represents a lossy but noiseless channel, while $\delta>0$ represents a noisy channel.
The Peres-Horodecki criterion of positivity under partial transpose (PPT) criterion is a sufficient and necessary criterion to characterize the entanglement of CV bipartite entanglement \cite{PPTCriterion}. The PPT value is determined by $\sqrt{\frac{\Gamma - \sqrt{\Gamma^2-4\textup{Det}\mathbf{V}}}{2}}$ for a bipartite Gaussian state, where $\Gamma=\textup{Det}\mathbf{A}+\textup{Det}\mathbf{B}-2\textup{Det}\mathbf{C}$. If the PPT value is smaller than 1, bipartite entanglement exists. Otherwise, it's a separable state. Furthermore, smaller PPT values represent stronger entanglement.
\begin{figure}
\caption{Dependence of PPT values and quantum coherence of the OAM multiplexed CV entangled states on the transmission efficiency $\eta$ for \(l=0\), \(l=1\) and \(l=2\). (a,b) Case for a lossy channel with the excess noise $\delta=0$ ; (c,d) Case for a noisy channel with the excess noise $\delta=1$. The initial PPT value is 0.46$\pm$0.01 at $\eta=1$. Curves and data points show theoretical predictions and experimental results, respectively. Error bars of experimental data represent one standard deviation and are obtained based on the statistics of the measured data. The light blue plane in (c) shows the boundary for entanglement where PPT value is equal to 1. The three vertical dashed lines indicate corresponding transmission efficiencies where sudden death of entanglement starts to appear.}
\label{fig2}
\end{figure}
\section{Experimental setup} Figure 1(a) shows the schematic of experimental setup. The Gaussian mode pump field and probe field carrying topological charge $l$ of OAM mode cross each other in the center of the cesium vapor cell at an angle of 6 mrad \cite{MaOL2018}. In this way, a conjugate field carrying topological charge $-l$ of OAM mode is generated on the other side of the pump field. The topological charge of OAM mode $l=1$ or $l=2$ is added to the probe field by passing it through a vortex phase plate. The pump field is filtered out by using a Glan-Thompson polarizer after the vapor cell. The Conj field is kept by Alice, while the Pr field is distributed to a remote quantum node owned by Bob through a noisy channel. The noisy channel is realized by overlapping the Pr field with an auxiliary field at a polarization beam splitter (PBS) followed by a half-wave plate (HWP) and a PBS. The auxiliary field, which is modulated by an amplitude modulator and a phase modulator with white noise, has the same topological charge and frequency with the Pr field so that they can interfere. The amount of excess noise is controlled by tuning the optical power of the auxiliary field and the white noise added on it. Covariance matrix of the OAM multiplexed CV entangled state is experimentally measured by utilizing two sets of balanced homodyne detectors. In our experiment, the spatially mode-matched local oscillator beams used in the balanced homodyne detectors are obtained from a second set of FWM process in the same vapor cell \cite{PRJ2022}.
The top rows of Fig. 1(b) and 1(c) show the beam patterns of the Pr and Conj fields for \(l=1\) and \(l=2\), respectively, while the bottom rows show their interference patterns with plane waves at the same frequencies, from which the topological charges of these fields are inferred. It is obvious that the topological charges of the Pr and Conj fields are opposite, which confirms the OAM conservation in the FWM process.
\begin{figure}
\caption{Dependence of PPT values (a) and quantum coherence (b) of the OAM multiplexed CV entanglement on transmission efficiency $\eta$ and excess noise $\delta$ for \(l=2\) in a noisy channel. Curve planes and data points show theoretical predictions and experimental results, respectively. The light blue plane in (a) shows the boundary for entanglement where PPT value is equal to 1, and the red curve shows the boundary where sudden death of entanglement starts to appear.}
\end{figure}
\section{Experimental results and discussion} In our experiment, the correlation and anti-correlation levels of the initial CV entangled states carrying topological charges \(l=0\), \(l=1\), and \(l=2\) are all around $-3.3$ dB and 6.1 dB, which correspond to $V=0.47$ and $V^{\prime}=4.11$, respectively. The evolution of PPT values and relative entropy of CV bipartite entangled states carrying different topological charges in lossy channels ($\delta=0$) are shown in Fig. 2(a) and 2(b), respectively. The entanglement and quantum coherence between the Pr and Conj fields both degrade as the transmission efficiency decreases. However, the entanglement and quantum coherence are both robust against loss, i.e., they always exist until the transmission efficiency reaches 0. It is obvious that the entanglement and quantum coherence of the CV bipartite entangled states carrying topological charges \(l=1\), \(l=2\) are as robust to loss as their Gaussian counterpart \(l=0\).
The evolution of PPT values and relative entropy of CV bipartite entangled states carrying different topological charges in a noisy channel with $\delta=1$ (in the units of SNL) are shown in Fig. 2(c) and 2(d), respectively. Compared with the results shown in Fig. 2(a) and 2(b) for the case of lossy channel, both entanglement and quantum coherence degrade faster as the transmission efficiency decreases. Furthermore, entanglement disappears at a certain transmission efficiency of the Pr field $\eta=0.44$ in the presence of excess noise, which demonstrates the sudden death of CV quantum entanglement. In contrast, the quantum coherence always exists until the transmission efficiency decreases to 0. We note that OAM multiplexed CV entangled states carrying high order topological charges \(l=1\), \(l=2\) exhibit the same quantum entanglement and quantum coherence evolution tendency as their Gaussian counterpart \(l=0\) in a noisy channel.
Figure 3(a) and 3(b) show the dependence of PPT values and relative entropy on transmission efficiency $\eta$ and excess noise $\delta$ for CV bipartite entangled state carrying \(l=2\) in noisy channels. Experimental results at four different amounts of noise $\delta=0, 0.15, 0.5$ and 1 (in the units of SNL) are taken. It is obvious that both entanglement and quantum coherence degrade with the increase of loss and excess noise. The degradations of entanglement and quantum coherence can be attributed to the incoherent operations of the lossy and noisy channels \cite{Baumgratz2014,Xu2016}. The red curve in Fig. 3(a) shows the boundary where the sudden death of entanglement appears. It is clear that the sudden death of entanglement appears at higher transmission efficiency as the excess noise increases. In contrast, the quantum coherence of the OAM multiplexed CV entangled state still exists even though the entanglement disappears, i.e. quantum coherence of the state is robust against noise. The physical reason for the robustness of quantum coherences in noisy channels is that the proportion of quantum coherence is decreased when it is mixed with thermal noise, but the quantum coherence disappears completely only when infinite thermal noise is involved.
\section{Conclusions} Here, we experimentally quantify quantum coherence of OAM multiplexed CV entangled state in a noisy channel. Our results demonstrate that quantum coherence of OAM multiplexed CV entangled state is robust against loss and noise, although the sudden death of entanglement is observed at a certain noise level. Recently, it has been shown that entanglement can be transferred in a single-mode cavity \cite{Bougouffa2020}, which is a promising application of robustness of quantum coherence. Our results lay the foundation of applying quantum coherence of OAM multiplexed entangled state in noisy environments.
\begin{backmatter} \bmsection{Funding} National Natural Science Foundation of China (NSFC) (No. 11974227, No. 11834010, and No. 61905135); Fundamental Research Program of Shanxi Province (No. 20210302122002); Research Project Supported by Shanxi Scholarship Council of China (2021-003); Fund for Shanxi ``1331 Project" Key Subjects Construction.
\bmsection{Disclosures} The authors declare no conflicts of interest.
\bmsection{Data availability} Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
\end{backmatter}
\end{document}
|
arXiv
|
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Momentum kicks due to quantum localization} \author{A.J.Short\thanks{[email protected]}\\ {\em Centre for Quantum Computation,} \\ {\em Clarendon Laboratory, University of Oxford,}\\ {\em Parks Rd., OX1 3PU, UK}} \date{} \maketitle{}
\begin{abstract} The momentum changes caused by position measurements are a central feature of wave-particle duality. Here we investigate two cases - localization by a single slit, and which-way detection in the double-slit interference experiment - and examine in detail the associated momentum changes. Particular attention is given to the transfer of momentum between particle and detector, and the recoil of the measuring device. We find that single-slit diffraction relies on a form of `interaction-free' scattering, and that an ideal which-way measurement can be made without any back-reaction on the detector. \end{abstract}
\section{Introduction}
It is well known that a measurement of position can change the momentum distribution of a quantum object, yet the precise nature of such changes, and the mechanism by which momentum is transferred between particle and detector, remains the subject of debate. One of the earliest concerns was that of Karl Popper, who considered the momentum changes in an entangled pair when one of the particles is localized \cite{popper}, and whose ideas have recently provoked renewed discussion and experimentation \cite{kim_and_shih, unnikrishnan, my_popper}. Other investigations by Renniger \cite{renniger} and Dicke \cite{dicke1, dicke2} have focused on the momentum changes when a particle is \emph{not} detected in a certain region, and there has also been debate over the momentum changes in a recently proposed atom-maser interference experiment \cite{scully, storey}.
A more familiar example, which we consider in section \ref{single_slit_sec}, is that of single-slit diffraction. Here the slit itself provides a form of position measurement, and the diffraction pattern in the far-field reveals the particle's momentum distribution. A particle initially in a plane-wave state will acquire a momentum spread on passing through the slit in accordance with the Heisenberg uncertainty relation $\Delta x \Delta p \geq \hbar/2$. In order to conserve total momentum, we would expect a correlated change in the momentum distribution of the slit, but it is unclear how the momentum is carried between particle and slit. From a wave-perspective, the diffraction pattern seems to be generated by that part of the wave which \emph{does not} interact with the slit, passing straight through the aperture. Momentum transfer, on the other hand, is a particle-like feature, and seems most easily explained by the action of forces at the slit edge. Investigating these two aspects, we reveal the peculiar `interaction-free' nature of the the diffraction process.
The effect of a position measurement on a particle's momentum distribution is also crucial to the double-slit interference experiment (section \ref{double_slit_sec}), in which the interference fringes vanish if a successful measurement is made of which slit the particle passed through \cite{feynman_slit}. This famous example of wave-particle duality has formed the basis for many discussions of quantum mechanics, most notably Einstein's recoiling-slit experiment \cite{einstein_slit} and Feynman's light microscope. In both cases, the loss of interference is strongly linked to a transfer of momentum between the incident particle and the measuring device, and it was originally argued that momentum transfer and the Heisenberg uncertainty relation play the key role in enforcing wave-particle duality.
However, more recent which-way detectors do not rely explicitly on momentum transfer, such as the atom-maser system of Scully, Englert and Walther \cite{scully}, or the spin-based system of Schulman \cite{schulman}. Instead, which-way information is stored in some internal degree of freedom which becomes entangled with the spatial coordinate of the particle. A typical detection process has the form \begin{equation} \label{which_way_evolve_eqn} \frac{1}{\sqrt{2}} ( \psi_A(x) + \psi_B(x)) \ket{0} \rightarrow \frac{1}{\sqrt{2}} ( \psi_A(x) \ket{1} + \psi_B(x) \ket{0} ) \end{equation} where $\psi_A(x)$ and $\psi_B(x)$ are wave packets just behind the two slits and $\{\ket{0}, \ket{1}\}$ represent orthogonal states of the detector. Because of this tagging process, any interference between the two wavepackets is lost, yet either of the wavepackets alone will remain completely unchanged by its interaction with the detector (suggesting that the particle experiences no momentum kick). Here it seems that entanglement, rather than the uncertainty relation, is responsible for enforcing wave-particle duality.
This approach has been criticised by Storey \emph{et al.} \cite{storey}, who claim that momentum kicks and the uncertainty relation are still crucial to the loss of interference. There can be no doubt that the particle's momentum changes in the which-way experiment, as the interference pattern obtained in the far field \emph{is} a measurement of the particle's momentum distribution, and it certainly changes when a detector is introduced (the fringes disappear). We might therefore expect some momentum transfer between the detector and particle, and a correlated change in the detector's momentum distribution. In fact, we will find in section \ref{double_slit_sec} that there is \emph{no} change in the momentum distribution of the detector in such cases. Despite this surprising result, we show that the total momentum distribution remains unchanged in the interaction, as required for momentum conservation.
As in the single-slit case, the change in particle momentum cannot be attributed to classical forces, and the results strongly suggest that the momentum change is an effect, and not a cause, of the loss of interference. Similar situations are examined by Wiseman \emph{et. al.} \cite{wiseman}, where it is shown that they correspond to a `nonlocal' momentum transfer in the Wigner function formalism.
\section{Single-slit diffraction} \label{single_slit_sec}
In this section, we consider the diffraction of a single point-like particle by a rectangular slit, with a short-range interaction between the particle and slit material close to the slit edge. Both objects are treated quantum mechanically, and their momentum changes are investigated.
The diffraction setup is shown in figure \ref{figure_1}, in which a single particle with a broad wavefunction and well-defined momentum $\mathbf{P} \simeq (0,0,P)$ is normally-incident on a narrow slit (of width $2d$). For simplicity, we restrict our analysis to the $x$-direction in which the slit is narrow, and model the particle's propagation in the $z$-direction by comparing initial and final states on either side of the slit. The initial particle state $\psi(x_p)$ is assumed to be constant over the slit region, and given by \begin{equation} \psi(x_p) = \frac{1}{\sqrt{2L}} \end{equation} where $L (\gg d)$ is a measure of the width of the incident wavefront.
\begin{figure}
\caption{The single-slit diffraction setup, showing a possible scattering and the corresponding particle momentum P'. In this case, we would expect the slit to recoil in the $-x$ direction.}
\label{figure_1}
\end{figure}
The slit is assumed to be a rigid structure, with a gaussian wavefunction $\phi(x_s)$ for the position of its centre, \begin{equation} \phi(x_s) = (2 \pi (\Delta x_s)^2)^{-1/4} \exp \left( \frac{-x_s^2}{4(\Delta x_s)^2} \right) \end{equation} Initially, we take $\Delta x_s \ll d$, giving the slit the sharp localization we would expect of a macroscopic object. We also assume that the slit remains relatively static during the interaction, with a characteristic velocity spread $\Delta \dot{x}_s$ which is much less than the particle velocity.
In reality, we would expect the slit to be in a thermal mixed state, but this can always be decomposed into a probabilistic mixture of pure states, and will not affect the results. A typical value for $\Delta x_s$ can be calculated from the equipartition theorem $\langle p_s^2/2M \rangle \sim \frac{1}{2}kT$ and the gaussian uncertainty relation $\Delta x_s \Delta p_s = \hbar/2$, giving \begin{equation} \Delta x_s \sim \left(\frac{\hbar^2}{4 M kT}\right)^{\frac{1}{2}}. \end{equation} For a slit of mass $M=10^{-3}$ kg at room temperature ($T=300$K), this corresponds to a position uncertainty of $\Delta x_s \sim 10^{-23}$m and a velocity spread of $\Delta \dot{x}_s \sim 10^{-9} \mathrm{m s}^{-1}$.
We assume that the interaction between the particle and slit is short-range, extending only a small distance $\delta_f$($\ll d$) from the slit edge, and that it will block any particles which collide directly with the slit material. We model this by a potential distribution \begin{equation} V(x_p-x_s) = \left\{
\begin{array}{cl} V_0 & |x_p-x_s| \geq d+\delta_f \\ V_s (x_p-x_s)
& d-\delta_f < |x_p-x_s| < d+\delta_f \\ 0 &|x_p-x_s| \leq d-\delta_f \\ \end{array} \right. \end{equation} where $V_0$ is much greater than the kinetic energy of the particle, and $V_s (x_p-x_s)$ is some smooth function taking the potential between $V_0$ and $0$ close to the slit edge.
To simplify the dynamics, we break the initial particle wavefunction $\psi(x_p)$ into three parts $\{\psi_I(x_p), \psi_{II}(x_p), \psi_{III}(x_p)\}$ which are unnormalized projections onto the different spatial regions (fig.
\ref{slit_regions}). Region I ($|x_p| \leq d - \epsilon$) lies entirely within the slit aperture, where no forces act on the particle. Region II ($d-\epsilon < |x_p| < d+\epsilon$) covers the area close to the slit edge, where the forces generated by
$V_s (x_p-x_s)$ act on the particle, and region III ($|x_p| \geq d+\epsilon$) covers the bulk of the slit material, where the particle is blocked by the potential $V_0$.
\begin{figure}
\caption{The three regions of $x_p$ into which the particle wavefunction is divided, with the potential distribution $V(x_p-x_s)$ and the slit wavefunction $\phi(x_s)$ shown on the same axis. The small quantities $\epsilon, \Delta x_s$ and $\delta_f$ have been exaggerated for clarity.}
\label{slit_regions}
\end{figure}
To ensure that the component in region I passes freely through the aperture, we set \begin{equation} \epsilon > (\Delta x_s +\delta_f + \delta_s), \end{equation} where $\delta_s (\ll d)$ is a measure of the transverse spreading of the wavefunction during the interaction. We assume that the slit is thin in the $z$-direction, and that the incident velocity of the particle is much greater than its typical transverse velocity ($P \gg h/d$) and that of the slit, such that the particle travels rapidly through the slit with minimal spreading. After this choice of $\epsilon$, any small residual interaction with forces can be absorbed into the region II component.
We thus obtain an interaction of the form \begin{eqnarray} \psi_{I}(x_p) \phi(x_s) &\rightarrow& \psi_{I}(x_p) \phi(x_s) \label{slit_int1} \\ \psi_{II}(x_p) \phi(x_s) &\rightarrow& \Lambda_{II}(x_p,x_s) + \Gamma_{II}(x_p,x_s) \\ \psi_{III}(x_p) \phi(x_s) &\rightarrow& \Gamma_{III}(x_p,x_s) \label{slit_int3} \end{eqnarray} where $\Lambda_{II}(x_p,x_s)$ is the entangled component at the slit edge which passes through the aperture, and $\Gamma_{II/III}(x_p,x_s)$ are components which are blocked by the slit and do not contribute to the far-field interference pattern. We leave off the slight spreading due to free evolution in the region I component as this has no effect on the transverse momentum distribution.
Under the influence of this interaction, the initial state \begin{equation} \Psi_i(x_p,x_s) = \psi(x_p)\phi(x_s) = \left(\psi_I(x_p)+\psi_{II}(x_p)+\psi_{III}(x_s)\right)\phi(x_s) \end{equation} will evolve into an entangled state containing each of the terms in equations (\ref{slit_int1})-(\ref{slit_int3}). We project out only the component which has successfully passed through the slit (which occurs with probability $\sim d/L$), to obtain the final state \begin{eqnarray} \Psi_f(x_p,x_s) &=& N \bigl(\psi_I(x_p)\phi(x_s) + \Lambda_{II}(x_p,x_s)\bigr) \\ &=& \frac{N}{\sqrt{2L}} \, \chi_{d'}(x_p) \phi(x_s) + N \Lambda_{II}(x_p,x_s), \end{eqnarray} where $N$ is a normalization constant ($\sim \sqrt{L/d}$) and
$\chi_{d'}(x_p)$ is a top-hat function which projects out the wavefunction in region I, with value 1 if $|x_p| \leq (d'=d-\epsilon)$ and 0 otherwise.
To obtain the far-field distribution, we Fourier-transform into the momentum representation (with $\hbar=1$) to get \begin{equation} \label{diff_patt} \tilde{\Psi}_f(k_p,k_s)= \frac{N d'}{\sqrt{\pi L}}\left( \frac{\sin(k_p d')}{k_p d'} \right) \tilde{\phi}(k_s) + N \tilde{\Lambda}_{II}(k_p,k_s) \end{equation} Note that the entangled component $N \tilde{\Lambda}_{II}(k_p,k_s)$, in which forces have acted between the particle and slit, carries only a tiny fraction ($\sim \epsilon/d$) of the total probability for the state, and the diffraction pattern will be largely determined by the first term. Up to small corrections due to edge effects, we therefore recover the familiar $\textrm{sinc}^2$ diffraction pattern in the far-field, as given by the momentum probability distribution \begin{equation}
\rm{Prob}(k_p) \simeq \left|N \tilde{\psi}_I(k_p)\right|^2 \simeq \frac{d}{\pi} \left( \frac{\sin(k_p d)}{k_p d} \right)^2. \end{equation}
However, as the slit wavefunction is unchanged in the first term, only the small edge term $N \tilde{\Lambda}_{II}(k_p,k_s)$ can contribute to any momentum change of the slit. This is what we might expect classically, as forces are only present at the slit edge, yet the particle's momentum distribution is largely independent of these edge forces. Instead, the form of the diffraction pattern is given by the region I contribution, in which there has been no interaction between particle and slit.
Given that the changes in particle and slit momentum arise from different terms, it is difficult to see how total momentum could be conserved in the interaction. This is not necessarily problematic, as we have projected out only part of the final state (in which the particle passes through the slit), and the total momentum distribution need only be conserved for the state as a whole. Nevertheless, it would be a surprising result.
However, with the setup given above it would be impossible to measure the momentum change of the slit. The typical momentum transferred in the scattering ($\sim h/d$) will be far less than the natural uncertainty in the slit momentum ($\hbar/2\Delta x_s$) and hence, even with perfect recoil, the initial and final states of the slit will be almost identical, with overlap very close to 1. It is this feature which allows for coherent reflection of quantum particles from macroscopic objects such as mirrors\cite{schulman}.
For the slit recoil to have a significant effect, the momentum uncertainty of the slit must be similar in magnitude to (or less than) the momentum changes due to diffraction, as deduced by Bohr in his response to Einstein's recoiling-slit experiment\cite{einstein_bohr}. To study the momentum changes of the slit in more detail, we therefore consider a delocalized slit, with $\Delta x_s \sim d$.
\begin{figure}
\caption{The delocalized slit wavefunction, which can be expressed as a superposition of many localized states with width $2 \varepsilon$.}
\label{delocalised_slit}
\end{figure}
With the position of the slit so uncertain, it is impossible to define regions I-III for the state as a whole, but if we consider the new state of the slit $\phi_d(x_s)$ as a superposition of sharp wavefunctions at different positions (fig. \ref{delocalised_slit}), then we can evolve each term separately as before. Dividing $\phi_d(x_s)$ into narrow strips of width $2 \varepsilon \ll d$, and then summing over $x'_s= 2 n \varepsilon$ for all integer $n$ we have \begin{equation} \phi_d(x_s) \simeq \sum_{x'_s} \chi_{\varepsilon}(x_s-x'_s)\phi_d(x'_s) \end{equation} and hence \begin{equation} \Psi_i(x_p, x_s) \simeq \sum_{x'_s} (\psi(x_p) \chi_{\varepsilon}(x_s-x'_s) ) \phi_d(x'_s). \end{equation} Evolving each bracketed term (in which the slit has the narrow wavefunction $\chi_{\varepsilon}(x_s-x'_s)$) as before, \begin{eqnarray} \Psi_f(x_p,x_s) & = & \sum_{x'_s} N
\left(\psi_{x'_s[I]}( x_p )\chi_{\varepsilon}(x_s-x'_s) + \Lambda_{x'_s[II]}(x_p,x_s)\right) \phi_d(x'_s)\\ & = & \sum_{x'_s} N \left( \frac{1}{\sqrt{2L}} \chi_{d'}(x_p - x'_s) \chi_{\varepsilon}(x_s-x'_s) + \Lambda_{x'_s[II]}(x_p,x_s) \right) \phi_d(x'_s). \end{eqnarray} We neglect the small contributions $\Lambda_{x'_s[II]}(x_p,x_s)$ from interactions close to the slit edge, and fourier-transform to the momentum representation for the particle \begin{eqnarray} \tilde{\Psi}_f(k_p,x_s) &\simeq& \sum_{x'_s} \frac{N d'}{\sqrt{\pi L}}\left( \frac{\sin(k_p d')}{k_p d'}\, \mathrm{e}^{-i k_p x'_s} \right)
\chi_{\varepsilon}(x_s-x'_s) \phi_d(x'_s)\\ & = & \Biggl(\frac{N d'}{\sqrt{\pi L}} \left(\frac{\sin(k_p d')}{k_p d'} \right) \Biggr)\Biggl(\sum_{x'_s}
\left(\phi_d(x'_s)\mathrm{e}^{-i k_p x'_s} \right)\chi_{\varepsilon}(x_s-x'_s)
\Biggr) \label{slit-eqn} \end{eqnarray} where we have used the fourier transform relation $\mathcal{F}[\psi(x-a)] = \tilde{\psi}(k) \exp(-i k a)$. Finally, we reconstruct the slit wavefunction to give \begin{equation} \label{recoileqn} \tilde{\Psi}_f(k_p,x_s) \simeq \frac{N d'}{\sqrt{\pi L}} \left(\frac{\sin(k_p d')}{k_p d'} \right) \left(\phi_d(x_s)\mathrm{e}^{-i k_p x_s} \right) \end{equation} and hence \begin{equation} \label{krecoileqn} \tilde{\Psi}_f(k_p,k_s) \simeq N \tilde{\psi}_I(k_p) \tilde{\phi}(k_s+k_p). \end{equation}
The final state of the slit has the same gaussian form as the initial state, but with an average momentum of $-k_p$, equal and opposite to the particle momentum. We therefore obtain a simple \emph{recoil} of the slit, with the momentum transfer distribution given by $\mathrm{Prob}(k_p)$ for any slit wavefunction. What is interesting is that the recoil derived here has nothing to do with the forces at the slit edge. The wavefunction given in equation (\ref{recoileqn}) was derived entirely from region I components, in which the particle propagates freely through the centre of the slit without any interactions. In analogy with other recent works this appears to be a form of `interaction free' scattering \cite{int_free}.
The momentum changes generated by the diffraction are a quantum phenomenon, not a direct result of forces but of entanglement in position and the intrinsic uncertainty in the slit position. Each possible slit position $x_s$ generates a shifted diffraction pattern, and for a given observed momentum $k_p$ this shift corresponds to a phase change of $\exp(-i k_p x_s)$. As the different slit positions add coherently we obtain a slit wavefunction with an overall phase factor of $\exp(-i k_p x_s)$ and hence a momentum kick of $-k_p$. The role of momentum as a translational symmetry property, rather than something which is carried between particles by forces, is clearly emphasised.
This approach will be applicable whenever the particle moves rapidly through the slit, and the slit wavefunction can be broken into a large number of narrow strips which do not spread significantly during the diffraction process. It can even be applied to our original thermal slit, for which the initial velocity spread $\Delta \dot{x}_s \sim 10^{-9} \mathrm{m s}^{-1}$ will probably be much less than that of the particle. The wavefunction could therefore be divided into small strips with
increased velocities which still wouldn't spread significantly during the interaction, and the slight momentum kick due to recoil could be explained by the above mechanism.
The apparent violation of energy conservation due to the increased transverse momenta of particle and slit can be resolved by including the $z$-component of the wavefunction in the analysis. In addition to the transverse momentum, both longitudinal momentum and energy are conserved in the extended model (Appendix A), even though we have projected out only part of the final state.
\section{Double-slit interference and which-way detection} \label{double_slit_sec}
In this section we consider the effect of which-way detection on the momentum distributions of the particle and detector in a double slit interference experiment.
The setup is shown in figure \ref{double_slit_fig}, in which a particle with well-defined momentum $\mathbf{P}=(0,0,P)$ is incident normally on a screen with two narrow slits. Just behind the upper slit (slit A) there is a which-way-detector, the internal state of which will change from $\ket{0}$ to $\ket{1}$ as the particle passes through it (as in \cite{schulman}). Much further from the slits, in the far-field, there is a sensitive screen which will register the position of the particle, providing an effective measurement of its transverse momentum. As the results in the previous section can easily be extended to the double-slit itself, we focus our attention on the action of the which-way detector.
\begin{figure}
\caption{Double slit interference setup, showing the change in the diffraction pattern (from dashed to solid plot) when the which-way detector is placed between slit A and the screen.}
\label{double_slit_fig}
\end{figure}
As before, we restrict our analysis to the $x$-direction in which the slits lie, and model the particle's propagation in the $z$-direction by comparing initial and final states on either side of the which-way detector. The particle state just before the detector is \begin{equation} \psi(x_p) = \frac{1}{\sqrt{2}}(\psi_A(x_p) + \psi_B(x_p)) \end{equation} where $\psi_A(x_p)$ and $\psi_B(x_p)$ are wavepackets behind slits A and B respectively. We assume that $P$ is much greater than the characteristic momentum of $\psi_A(x_p)$ and $\psi_B(x_p)$, such that transverse spreading $\delta_s$ is minimal while passing through the detection region.
The detector is a rigid well-localized object with a narrow wavefunction $\phi(x_d)$ for the position of its left-hand edge (which lies in the centre of the two slits). We choose an interaction Hamiltonian of the form \begin{equation} \label{interact_hamil_eqn} H_I = -i V(x_p-x_d) (\ket{0}\bra{1} - \ket{1}\bra{0}) \quad \mathrm{where}\quad V(x) = \left\{ \begin{array}{cl} V_1 & x \geq \delta_f \\ V_d(x) & -\delta_f < x < \delta_f \\
0 & x \leq -\delta_f \end{array} \right. , \end{equation} with a constant potential $V_1$ within the detector and the edge effects extending over a small region $\delta_f$ as before (fig. \ref{double_slit_x_pic}). We set $V_1$ much lower than the kinetic energy of the particle and assume that the potential is smoothly-varying in the $z$-direction, such that almost all particles pass through the detector without being reflected. By appropriately choosing the thickness $w$ of the detection region in the $z$-direction, we can ensure that the internal state of the detector rotates completely from $\ket{0}$ to $\ket{1}$ as the wavepacket $\psi_A$ passes through it (in time $\tau$). This gives a unitary interaction of the form \begin{equation} U_I = \exp \left( \frac{-i H_I \tau}{\hbar} \right) = \left\{ \begin{array}{cl} \left(\ket{1} \bra{0} - \ket{0}\bra{1} \right) & x \geq \delta_f \\ \left( \begin{array}{c} \sin \left(\frac{ \pi V_d(x)}{2 V_1} \right) \left(\ket{1} \bra{0} - \ket{0}\bra{1}\right) \\ +\cos \left(\frac{\pi V_d(x)}{2 V_1}\right) \left( \ket{0}\bra{0} + \ket{1}\bra{1} \right) \end{array} \right) & -\delta_f < x < \delta_f \\
\left( \ket{0}\bra{0} + \ket{1}\bra{1} \right) & x \leq -\delta_f \end{array}\right., \end{equation} where $x=x_p-x_d$, and we have assumed that there is no possibility of particle reflection.
\begin{figure}\label{double_slit_x_pic}
\end{figure}
We assume that the wavepackets for the particle and detector ($\psi_A(x)$,$\phi(x)$,and $\psi_B(x)$) are all initially separated by a distance greater than $\delta_f+\delta_s$, such that they remain non-overlapping and free of edge effects throughout the interaction, with support on different regions of the $x$-axis. This allows the detector to provide a perfect which-way measurement for the two slits, as given in equation (\ref{which_way_evolve_eqn}). More precisely, the initial and final states are given by \begin{eqnarray} \Psi_i(x_p, x_d) &=& \frac{1}{\sqrt{2}}(\psi_A(x_p) + \psi_B(x_p))\phi(x_d) \ket{0} \\ \Psi_f(x_p, x_d) &=& \frac{1}{\sqrt{2}}(\psi_A(x_p)\ket{1} + \psi_B(x_p)\ket{0}) \phi(x_d), \end{eqnarray} where we have once again neglected the slight changes due to free evolution during the detection process.
Because $\psi_A(x_p) \phi(x_d)$ always generates a positive relative displacement $(x_p-x_d)>\delta_f$, and $\psi_B(y) \phi(y_d)$ a negative displacement $(x_p-x_d)<-\delta_f$, the interaction is not sensitive to the exact position of the detector or the forces at the detector edge and hence the spatial wavefunctions for the particle and detector remain unentangled.
In the regions of space $\{x_p, x_d\}$ in which the state is non-zero, the interaction potential is always constant in the $x$-direction, with \begin{equation} \frac{\partial V(x_p-x_d)}{ \partial x_p} = \frac{\partial V(x_p-x_d)}{ \partial x_d} = 0. \end{equation} Hence the particle and detector experience no transverse forces during the detection process. Yet nevertheless the particle momentum distribution $\mathrm{Prob}(k_p)$ is changed by the interaction, from a distribution with interference fringes to one without \begin{equation} \mathrm{Prob}_i(k_p) =
\left|\frac{1}{\sqrt{2}}\left(\tilde{\psi}_A(k_p) +
\tilde{\psi}_B(k_p)\right)\right|^2 \quad \rightarrow \quad \mathrm{Prob}_f(k_p)
=\frac{1}{2}\left|\tilde{\psi}_A(k_p)\right|^2 +
\frac{1}{2}\left|\tilde{\psi}_B(k_p)\right|^2. \end{equation} The momentum distribution for the detector $\mathrm{Prob}(k_d)$, however, remains completely unchanged, with \begin{equation} \mathrm{Prob}_i(k_d) = \mathrm{Prob}_f(k_d) =
\left|\tilde{\phi}(k_d)\right|^2 \end{equation} Although simple, this result is actually quite surprising; not only does the momentum of the particle change despite the absence of forces, but there is an apparent violation of total momentum conservation. Unlike the diffraction case there is no recoil of the detector, the momentum distribution of the particle simply changes on its own, \emph{without} any other change to balance it.
With the particle and detector unentangled, the total momentum distribution $\mathrm{Prob}(k_T)$ is given by the convolution of the two individual momentum distributions $\mathrm{Prob}(k_T)= \mathrm{Prob}(k_p)\star \mathrm{Prob}(k_d)$. The translational symmetry of the Hamiltonian (which depends only on the relative coordinate ($x_p-x_d$)) should then ensure that $\mathrm{Prob}(k_T)$ is conserved during the interaction. Applying this to the above case, we require that \begin{equation}
\left|\frac{1}{\sqrt{2}}\left(\tilde{\psi}_A(k_p) +
\tilde{\psi}_B(k_p)\right)\right|^2 \star
\left|\tilde{\phi}(k_d)\right|^2 =
\left(\frac{1}{2}\left|\tilde{\psi}_A(k_p)\right|^2 +
\frac{1}{2}\left|\tilde{\psi}_B(k_p)\right|^2\right) \star
\left|\tilde{\phi}(k_d)\right|^2. \end{equation} In fact, this equality will hold whenever $\phi(x)$ can fit into the gap between $\psi_A(x)$ and $\psi_B(x)$(Appendix B), so total momentum will be conserved in the detection process described above. Note that this condition on the wavepackets is actually necessary for the detector to be able to perfectly distinguish the two particle states.
Nevertheless, this yields an interesting result: That the momentum distribution of a single particle can change in isolation \emph{without} changing the total momentum distribution. This is a consequence of the convolution used to calculate the total momentum distribution, which is not a one-to-one mapping. In this case, the limited spatial support of the detector wavepacket means that some information about the particle state is discarded completely in the convolution. As the changes in particle momentum only affect this discarded region, they cause no change in the total momentum distribution.
We can also use this result to illustrate another feature of the momentum changes: That, despite the obvious differences between $\mathrm{Prob}_i(k_p)$ and $\mathrm{Prob}_f(k_p)$, the expectation value of any finite positive integer power of the particle momentum ($\langle k_p^N \rangle$ for $N\in\{0,1,2 \ldots\}$) will be identical in the initial and final states. Using total momentum conservation, \begin{eqnarray} \av{(k_p+k_d)^N}_i = \av{(k_p+k_d)^N}_f \end{eqnarray} where $\av{\,}_i$ and $\av{\,}_f$ represent expectation values in the initial and final states respectively. Since the particle and detector are unentangled (and uncorrelated), we can simplify the power expansion to get \begin{equation} \sum_{n=0}^N \left( \frac{N!}{n!(N-n)!} \right) \av{k_p^n}_i \av{k_d^{N-n}}_i = \sum_{m=0}^N \left( \frac{N!}{m!(N-m)!} \right) \av{k_p^m}_f \av{k_d^{N-m}}_f. \end{equation} Given that $\av{k_d^n}_i=\av{k_d^n}_f \; \forall \; n$, this implies the inductive result that \begin{equation} \av{k_p^N}_i =\av{k_p^N}_f \;\; \rm{if} \;\;\; \av{k_p^m}_i =\av{k_p^m}_f \;\; \forall \; m\in\{0,1,2,\ldots,N-1\}. \end{equation} As this equation is true for $N=0$ ($\av{k_p^0}_i =\av{k_p^0}_f=1)$, it must therefore be true by induction for all positive integer $N$. Thus $\rm{Prob}_i(k_P)$ and $\rm{Prob}_f(k_P)$ are distributions with precisely the same moments $(\av{k_p^N}_i =\av{k_p^N}_f)$. Those changes which do occur can be best seen by considering the modular momentum $(k_p \, \textrm{mod}\,\kappa)$, where $\kappa$ gives the fringe spacing in the momentum distribution. This approach is followed in more detail by Aharanov \emph{et al.} \cite{modular_momentum}.
\section{Discussion}
In both the single and double slit experiments, the momentum changes associated with the position measurement cannot be viewed as a direct result of classical forces, which vanish in the relevant regions. Instead, they depend on quantum entanglement, and the translational symmetries of the wavefunction.
In the single-slit case, we see the effect of a position measurement inside a broad incident wavefront. Here the slit provides a simple dual-valued position measurement, which tells us whether or not the particle lies inside the aperture. The uncertainty in the detector (slit) position yields a superposition of final states, in which each detector position state is entangled with a spatially-shifted particle state. In momentum-space, these spatial-shifts of the particle correspond to momentum-shifts of the detector, generating a recoil of the detector which is just as as we would classically expect - even though the scattering is `interaction-free'.
In the double-slit experiment, we consider the effect of a position measurement which will distinguish spatially separated parts of a wavefunction. In this case, an ideal detector can perform the measurement without becoming spatially entangled with the particle, or experiencing any recoil. Yet the momentum distribution of the particle \emph{is} changed by the interaction, even though it is not subject to any transverse forces.
The fact that we can change the momentum distribution of the particle without changing the momentum distribution of the detector, and yet still conserve total momentum, is surprising. However, it emerges as a natural consequence of the convolution used to calculate the total momentum, in which some information about the momentum distributions of the individual particles is discarded.
I am grateful to Lucien Hardy, Lev Vaidman and W.Toner for useful discussions, and to EPSRC for financial support.
\appendix
\section*{Appendix A}
The $z$-component of the wavefunction is important in determining \emph{when} the particle passes through the aperture. Regions in which the slit is closer to the incident particle ($z_s<0$) will be diffracted sooner and thus experience a greater period of transverse spreading.
To investigate this effect, we consider the wavefunction just after the particle has passed through the slit, where it has the approximate form \begin{equation} \label{app_A_eqn} \tilde{\Psi}_f(k_p, k_s, z_p, z_s) \simeq N \tilde{\psi}_I (k_p) \tilde{\phi}(k_s+k_p, z_s) \rme{i P z_p / \hbar} , \end{equation} where $\tilde{\phi}(k_s+k_p, z_s)$ now includes the slit wavepacket in the z-direction, and the $x$-component is given in terms of $\{k_p, k_s\}$ by equation (\ref{krecoileqn}). The slight transverse spreading of the wavefunction during the interaction period will introduce an additional phase factor, given by \begin{equation} \exp \left( \frac{- i E_x t}{\hbar} \right) = \exp \left(-i \left( \frac{\hbar k_{si}^2}{2 M} \right) t_i - i \left( \frac{\hbar k_p^2}{2 m} + \frac{\hbar k_s^2}{2 M} \right) t_f \right), \end{equation} where $k_{si}=k_s+k_p$ is the initial slit momentum before diffraction, $t_i$ is the time at which the particle passed through the slit and $t_f$ is the time since. The total time $\tau= t_i+t_f$ is constant, but the proportion of the evolution which occurs after diffraction will depend on the distance between particle and slit. Taking the particle's speed in the $z$-direction as $P/m$ and the slit as effectively static, we approximate the evolution time after diffraction by \begin{equation} t_f \simeq \frac{m}{P} (z_p-z_s), \end{equation} which gives a phase factor of \begin{equation} \exp \left(-i \left( \frac{\hbar k_{si}^2}{2 M} \right) \tau - i \left( \frac{\hbar k_p^2}{2 P} + \frac{m \hbar (k_s^2 - k_{si}^2)}{2 M P}\right) (z_p-z_s) \right). \end{equation} Including this phase factor in the final state wavefunction will modify the $z$-momentum of the particle to \begin{equation} P' \simeq \left(1-\frac{\hbar k_p^2}{2 P^2} - \frac{m \hbar (k_s^2 - k_{si}^2)}{2 M P^2} \right) P \end{equation} with the slit receiving a momentum kick of equal magnitude in the opposite direction. Note that the momentum changes in the $z$-direction are much smaller than those in the $x$-direction, due to our assumptions that $\hbar k_p \ll P$ and $m \ll M$. As $(P'-P) \ll P$, the change in energy is approximately given by \begin{equation} \frac{{P'}^2}{2m}-\frac{P^2}{2m} \simeq \frac{P(P'-P)}{m} = - \frac{\hbar k_p^2}{2 m} - \frac{ \hbar (k_s^2 - k_{si}^2)}{2 M}, \end{equation} which exactly cancels the energy changes due to the increased transverse momentum of the particle and slit. Thus, at the level of these approximations, both energy and momentum are conserved in the diffraction process.
\section*{Appendix B}
Proof that \begin{equation} \label{app_eqn}
\left|\frac{1}{\sqrt{2}}\left(\tilde{\psi}_A(k) +
\tilde{\psi}_B(k)\right)\right|^2 \star
\left|\tilde{\phi}(k)\right|^2 =
(\frac{1}{2}\left|\tilde{\psi}_A(k)\right|^2 +
\frac{1}{2}\left|\tilde{\psi}_B(k)\right|^2) \star
\left|\tilde{\phi}(k)\right|^2. \end{equation} where $\star$ represents convolution and $\phi(x)$ is narrower than the gap between $\psi_A(x)$ and $\psi_B(x)$:
Note that spatial translations on $\phi(x)$, and on $\phi_A(x)$ and $\phi_B(x)$ together, of the form \begin{eqnarray} \phi(x) \rightarrow \phi(x-a) &\quad& \tilde{\phi}(k) \rightarrow \mathrm{e}^{-i k a} \tilde{\phi}(k) \\ \psi_A(x) \rightarrow \psi_A(x-b) &\quad& \tilde{\psi}_A(k) \rightarrow \mathrm{e}^{-i k b} \tilde{\psi}_A(k) \\ \psi_B(x) \rightarrow \psi_B(x-b) &\quad& \tilde{\psi}_B(k) \rightarrow \mathrm{e}^{-i k b} \tilde{\psi}_B(k) \\ \end{eqnarray} have no effect on equation (\ref{app_eqn}), thus without loss of generality we consider the centre of $\phi(x)$ (of width $2w$), and the centre of the gap between $\psi_A(x)$ and $\psi_B(x)$ (of width $2v$ with $(v>w)$) to coincide at the origin, placing $\phi(x)$ precisely in the centre of the gap.
Equation (\ref{app_eqn}) is true if and only if \begin{equation} \label{app_eqn_2} (\tilde{\psi}^{*}_A(k)\tilde{\psi}_B(k) + \tilde{\psi}^{*}_B(k)\tilde{\psi}_A(k)) \star (\tilde{\phi}^{*}(k)\tilde{\phi}(k)) =0. \end{equation} Using the Fourier-transform pair \begin{eqnarray} \mathcal{F}[\psi(x)] &=& \frac{1}{\sqrt{2\pi}}\int \psi(x) \rme{-i k x} \intd x = \tilde{\psi}(k) \\ \mathcal{F}^{-1}[\tilde{\psi}(k)] &=& \frac{1}{\sqrt{2\pi}}\int \tilde{\psi}(k) \rme{+ i k x} \intd k = \psi(x), \end{eqnarray} and the relations \begin{eqnarray} \mathcal{F}^{-1}\left[\tilde{\psi}_1(k) \star \tilde{\psi}_2(k)\right] &=& \sqrt{2 \pi} \, \mathcal{F}^{-1}\!\left[\tilde{\psi}_1(k)\right] \mathcal{F}^{-1}\!\left[\tilde{\psi}_2(k)\right] \\ \mathcal{F}^{-1}\!\left[\tilde{\psi}^*_1(k) \tilde{\psi}_2 (k) \right] &=& \frac{1}{\sqrt{2\pi}} \left(\psi_1^*(-x) \star \psi_2 (x)\right) \end{eqnarray}
we rewrite equation (\ref{app_eqn_2}) as \begin{eqnarray} && \mathcal{F} \left[ \mathcal{F}^{-1} \left[ \left(\tilde{\psi}^{*}_A(k)\tilde{\psi}_B(k) + \tilde{\psi}^{*}_B(k)\tilde{\psi}_A(k)\right) \star \left(\tilde{\phi}^{*}(k)\tilde{\phi}(k)\right) \right] \right] =0 \\ &\Leftrightarrow& \mathcal{F} \left[ \left(\mathcal{F}^{-1} \left[\tilde{\psi}^{*}_A(k)\tilde{\psi}_B(k)\right] + \mathcal{F}^{-1} \left[ \tilde{\psi}^{*}_B(k)\tilde{\psi}_A(k) \right] \right) \left(\mathcal{F}^{-1} \left[ \tilde{\phi}^{*}(k)\tilde{\phi}(k)) \right] \right) \right]=0 \\ &\Leftrightarrow& \mathcal{F} \left[\left( \left( \psi^{*}_A(-x) \star \psi_B(x)\right) + \left( \psi^{*}_B(-x) \star \psi_A(x) \right) \right) \left( \phi^{*}(-x) \star \phi(x) \right) \right] =0. \end{eqnarray}
Note that the states $\psi^{*}_A(-x)$ and $\psi_B(x)$ both lie on the same side of the origin in the region $|x|\geq v$. Their convolution must therefore lie in the region $|x| \geq 2v$. Applying the same argument to the states $\psi^{*}_B(-x)$ and
$\psi_A(x)$ we conclude that the first term in the product will be non-zero only in the region $|x| \geq 2v$. However, the second term in the product $( \phi^{*}(-x) \star \phi(x))$ can only be non-zero in the region $|x| \leq 2w$. As $w<v$ the two functions have no common support and their product (and its Fourier-transform) will always equal zero, thus proving the validity of equation \ref{app_eqn}.
\end{document}
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\begin{document}
\baselineskip 20pt \begin{abstract} Let $K$ be a field and $G$ be a finite group. Let $G$ act on the rational function field $K(x(g):g\in G)$ by $K$ automorphisms defined by $g\cdot x(h)=x(gh)$ for any $g,h\in G$. Denote by $K(G)$ the fixed field $K(x(g):g\in G)^G$. Noether's problem then asks whether $K(G)$ is rational (i.e., purely transcendental) over $K$. The first main result of this article is that $K(G)$ is rational over $K$ for a certain class of $p$-groups having an abelian subgoup of index $p$. The second main result is that $K(G)$ is rational over $K$ for any group of order $p^5$ or $p^6$ ($p$ is an odd prime) having an abelian normal subgroup such that its quotient group is cyclic. (In both theorems we assume that if $char K\ne p$ then $K$ contains a primitive $p^e$-th root of unity, where $p^e$ is the exponent of $G$.) \end{abstract}
\title{Noether's problem for abelian extensions of cyclic $p$-groups} \newcommand{{\rm Gal}}{{\rm Gal}} \newcommand{{\rm Ker}}{{\rm Ker}} \newcommand{{\rm GL}}{{\rm GL}} \newcommand{{\rm Br}}{{\rm Br}} \newcommand{{\rm lcm}}{{\rm lcm}} \newcommand{{\rm ord}}{{\rm ord}} \renewcommand{\fnsymbol{footnote}}{\fnsymbol{footnote}} \numberwithin{equation}{section}
\section{Introduction} \label{1}
Let $K$ be any field. A field extension $L$ of $K$ is called rational over $K$ (or $K$-rational, for short) if $L\simeq K(x_1,\ldots,x_n)$ over $K$ for some integer $n$, with $x_1,\ldots,x_n$ algebraically independent over $K$. Now let $G$ be a finite group. Let $G$ act on the rational function field $K(x(g):g\in G)$ by $K$ automorphisms defined by $g\cdot x(h)=x(gh)$ for any $g,h\in G$. Denote by $K(G)$ the fixed field $K(x(g):g\in G)^G$. {\it Noether's problem} then asks whether $K(G)$ is rational over $K$. This is related to the inverse Galois problem, to the existence of generic $G$-Galois extensions over $K$, and to the existence of versal $G$-torsors over $K$-rational field extensions \cite[33.1, p.86]{Sw,Sa1,GMS}. Noether's problem for abelian groups was studied extensively by Swan, Voskresenskii, Endo, Miyata and Lenstra, etc. The reader is referred to Swan's paper for a survey of this problem \cite{Sw}. Fischer's Theorem is a starting point of investigating Noether's problem for finite abelian groups in general.
\newtheorem{t1.1}{Theorem}[section] \begin{t1.1}\label{t1.1} {\rm (Fischer }\cite[Theorem 6.1]{Sw}{\rm )} Let $G$ be a finite abelian group of exponent $e$. Assume that {\rm (i)} either char $K = 0$ or char $K > 0$ with char $K\nmid e$, and {\rm (ii)} $K$ contains a primitive $e$-th root of unity. Then $K(G)$ is rational over $K$. \end{t1.1}
On the other hand, just a handful of results about Noether's problem are obtained when the groups are non-abelian. This is the case even when the group G is a $p$-group. The reader is referred to \cite{CK,HuK,Ka1,Ka2,Ka3} for previous results of Noether's problem for $p$-groups. The following theorem of Kang generalizes Fischer's theorem for the metacyclic $p$-groups.
\newtheorem{t1.2}[t1.1]{Theorem} \begin{t1.2}\label{t1.2} {\rm (Kang}\cite[Theorem 1.5]{Ka1}{\rm )} Let $G$ be a metacyclic $p$-group with exponent $p^e$, and let $K$ be any field such that {\rm (i)} char $K = p$, or {\rm (ii)} char $K \ne p$ and $K$ contains a primitive $p^e$-th root of unity. Then $K(G)$ is rational over $K$. \end{t1.2}
The next job is to study Noether's problem for meta-abelian groups. Three results due to Haeuslein, Hajja and Kang respectively are known.
\newtheorem{t1.23}[t1.1]{Theorem} \begin{t1.23}\label{t1.23} {\rm (Haeuslein }\cite{Ha}{\rm )} Let $K$ be a field and $G$ be a finite group. Assume that (i) $G$ contains an abelian normal subgroup $H$ so that $G/H$ is cyclic of prime order $p$, (ii) $\mathbb Z[\zeta_p]$ is a unique factorization domain, and (iii) $\zeta_{p^e}\in K$ where $e$ is the exponent of $G$. If $G\to {\rm GL}(V)$ is any finite-dimensional linear representation of $G$ over $K$, then $K(V)^G$ is rational over $K$. \end{t1.23}
\newtheorem{t1.24}[t1.1]{Theorem} \begin{t1.24}\label{t1.24} {\rm (Hajja }\cite{Haj}{\rm )} Let $K$ be a field and $G$ be a finite group. Assume that (i) $G$ contains an abelian normal subgroup $H$ so that $G/H$ is cyclic of order $n$, (ii) $\mathbb Z[\zeta_n]$ is a unique factorization domain, and (iii) $K$ is algebraically closed with $char K = 0$. If $G\to {\rm GL}(V)$ is any finite-dimensional linear representation of $G$ over $K$, then $K(V)^G$ is rational over $K$. \end{t1.24}
\newtheorem{t1.22}[t1.1]{Theorem} \begin{t1.22}\label{t1.22} {\rm (}\cite[Theorem 1.4]{Ka3}{\rm )} Let $K$ be a field and $G$ be a finite group. Assume that {\rm (i)} $G$ contains an abelian normal subgroup $H$ so that $G/H$ is cyclic of order $n$, {\rm (ii)} $\mathbb Z[\zeta_n]$ is a unique factorization domain, and {\rm (iii)} $\zeta_{e}\in K$ where $e$ is the exponent of $G$. If $G\rightarrow {\rm GL}(V)$ is any finite-dimensional linear representation of $G$ over $K$, then $K(V)^G$ is rational over $K$. \end{t1.22}
Note that those integers $n$ for which $\mathbb Z[\zeta_n]$ is a unique factorization domain are determined by Masley and Montgomery.
\newtheorem{t1.25}[t1.1]{Theorem} \begin{t1.25}\label{t1.25} {\rm (Masley and Montgomery }\cite{MM}{\rm )} $\mathbb Z[\zeta_n]$ is a unique factorization domain if and only if $1\leq n\leq 22$, or $n = 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 40, 42, 45, 48,$ $50, 54, 60, 66, 70, 84, 90$. \end{t1.25}
Therefore, Theorem \ref{t1.23} holds only for primes $p$ such that $1\leq p\leq 19$. One of the goals of our paper is to show that the this condition can be waived, under some additional assumptions regarding the structure of the abelian subgroup $H$.
Consider the following situation. Let $G$ be a group of order $p^n$ for $n\geq 2$ with an abelian subgroup $H$ of order $p^{n-1}$. Bender \cite{Be2} determined some interesting properties of these groups. We study further the case when the $p$-th lower central subgroup $G_{(p)}$ is trivial. (Recall that $G_{(0)}=G$ and $G_{(i)}=[G,G_{(i-1)}]$ for $i\geq 1$ are called the lower central series.) For our purposes we need to classify with generators and relations these groups. We achieve this in the following lemma.
\newtheorem{lemma}[t1.1]{Lemma} \begin{lemma}\label{lemma} Let $G$ be a group of order $p^n$ for $n\geq 2$ with an abelian subgroup $H$ of order $p^{n-1}$. Choose any $\alpha\in G$ such that $\alpha$ generates $G/H$, i.e., $\alpha\notin H,\alpha^p\in H$. Denote $H(p)=\{h\in H: h^p=1,h\notin H^p\}$, and assume that $[H(p),\alpha]\subset H(p)$. Assume also that the $p$-th lower central subgroup $G_{(p)}$ is trivial. (Recall that $G_{(0)}=G$ and $G_{(i)}=[G,G_{(i-1)}]$ for $i\geq 1$ are called the lower central series.) Then $H$ is a direct product of normal subgroups of $G$ that are of the following four types: \begin{enumerate}
\item $(C_p)^s$ for some $s\geq 1$. There exist generators $\alpha_1,\dots,\alpha_s$ of $(C_p)^s$, such that $[\alpha_j,\alpha]=\alpha_{j+1}$ for $1\leq j\leq s-1$ and $\alpha_s\in Z(G)$.
\item $C_{p^a}$ for some $a\geq 1$. There exists a
generator $\beta$ of $C_{p^a}$ such that $[\beta,\alpha]=\beta^{bp^{a-1}}$ for some $b:0\leq b\leq p-1$.
\item $C_{p^{a_1}}\times C_{p^{a_2}}\times\cdots\times C_{p^{a_k}}\times(C_p)^s$ for some $k\geq 1,a_i\geq 2,s\geq 1$. There exist generators $\alpha_{11},\alpha_{21},\dots,\alpha_{k1}$ of $C_{p^{a_1}}\times C_{p^{a_2}}\times\cdots\times C_{p^{a_k}}$ such that $[\alpha_{i,1},\alpha]=\alpha_{i+1,1}^{p^{a_{i+1}-1}}\in Z(G)$ for $i=1,\dots,k-1$. There also exist
generators $\alpha_{k,2},\dots,\alpha_{k,s+1}$ of $(C_p)^s$, such that $[\alpha_{k,j},\alpha]=\alpha_{k,j+1}$ for $1\leq j\leq s$ and $\alpha_{k,s+1}\in Z(G)$.
\item $C_{p^{a_1}}\times C_{p^{a_2}}\times\cdots\times C_{p^{a_k}}$ for some $k\geq 2,a_i\geq 2$. For any $i:1\leq i\leq k$ there exists a generator $\alpha_{i,1}$ of the factor $C_{p^{a_i}}$, such that $[\alpha_{i,1},\alpha]=\alpha_{i+1,1}^{p^{a_i-1}}\in Z(G)$ and $[\alpha_{k,1},\alpha]\in\langle\alpha_{1,1}^{p^{a_1-1}},\dots,\alpha_{k,1}^{p^{a_{k}-1}}\rangle$. \end{enumerate} \end{lemma}
The first main result of this paper is the following theorem which generalizes Theorem \ref{t1.23}.
\newtheorem{t1.4}[t1.1]{Theorem} \begin{t1.4}\label{t1.4} Let $G$ be a group of order $p^n$ for $n\geq 2$ with an abelian subgroup $H$ of order $p^{n-1}$, and let $G$ be of exponent $p^e$. Choose any $\alpha\in G$ such that $\alpha$ generates $G/H$, i.e., $\alpha\notin H,\alpha^p\in H$. Denote $H(p)=\{h\in H: h^p=1,h\notin H^p\}$, and assume that $[H(p),\alpha]\subset H(p)$. Denote by $G_{(i)}=[G,G_{(i-1)}]$ the lower central series for $i\geq 1$ and $G_{(0)}=G$. Let the $p$-th lower central subgroup $G_{(p)}$ be trivial. Assume that {\rm (i)} char $K = p>0$, or {\rm (ii)} char $K \ne p$ and $K$ contains a primitive $p^e$-th root of unity. Then $K(G)$ is rational over $K$. \end{t1.4}
The key idea to prove Theorem \ref{t1.4} is to find a faithful $G$-subspace $W$ of the regular representation space $\bigoplus_{g\in G} K\cdot x(g)$ and to show that $W^G$ is rational over $K$. The subspace $W$ is obtained as an induced representation from $H$ by applying Lemma \ref{lemma}. (A particular case we proved in the preprint \cite{Mi}.)
The next goal of our article is to study Noether's problem for some groups of orders $p^5$ and $p^6$ for any odd prime $p$. We use the list of generators and relations for these groups, given by James \cite{Ja}. It is known that $K(G)$ is always rational if $G$ is a $p$-group of order $\leq p^4$ and $\zeta_{e}\in K$ where $e$ is the exponent of $G$ (see \cite{CK}). However, in \cite{HoK} is shown that there exists a group $G$ of order $p^5$ such that $\mathbb C(G)$ is not rational over $\mathbb C$.
The second main result of this article is the following rationality criterion for the groups of orders $p^5$ and $p^6$, having an abelian normal subgroup such that its quotient group is cyclic.
\newtheorem{t1.5}[t1.1]{Theorem} \begin{t1.5}\label{t1.5} Let $G$ be a group of order $p^n$ for $n\leq 6$ with an abelian normal subgroup $H$, such that $G/H$ is cyclic. Let $G$ be of exponent $p^e$. Assume that {\rm (i)} char $K = p>0$, or {\rm (ii)} char $K \ne p$ and $K$ contains a primitive $p^e$-th root of unity. Then $K(G)$ is rational over $K$. \end{t1.5}
We do not know whether Theorem \ref{t1.5} holds for any $n\geq 7$. However, we should not ''over-generalize'' Theorem \ref{t1.5} to the case of any meta-abelian group because of the following theorem of Saltman.
\newtheorem{t1.3}[t1.1]{Theorem} \begin{t1.3}\label{t1.3} {\rm (Saltman }\cite{Sa2}{\rm )} For any prime number $p$ and for any field $K$ with char $K \ne p$ (in particular, $K$ may be an algebraically closed field), there is a meta-abelian $p$-group $G$ of order $p^9$ such that $K(G)$ is not rational over $K$. \end{t1.3}
We organize this paper as follows. We recall some preliminaries in Section \ref{2} that will be used in the proofs of Theorems \ref{t1.4} and \ref{t1.5}. There we also prove Lemma \ref{l2.8} which is a generalization of Kang's argument from \cite[Case 5, Step II]{Ka2}. In Section \ref{3} we prove Lemma \ref{lemma} which is of independent interest, since it provides a list of generators and relations for any $p$-group $G$ having an abelian subgroup $H$ of index $p$, provided that $[H(p),\alpha]\subset H(p)$ and $G_{(p)}=1$. Our main results -- Theorems \ref{t1.4} and \ref{t1.5} -- we prove in Sections \ref{4} and \ref{5} respectively.
\section{Preliminaries} \label{2}
We list several results which will be used in the sequel.
\newtheorem{t2.1}{Theorem}[section] \begin{t2.1}\label{t2.1} {\rm (}\cite[Theorem 1]{HK}{\rm )} Let $G$ be a finite group acting on $L(x_1,\dots,x_m)$, the rational function field of $m$ variables over a field $L$ such that \begin{description}
\item [(i)] for any $\sigma\in G, \sigma(L)\subset L;$
\item [(ii)] the restriction of the action of $G$ to $L$ is
faithful;
\item [(iii)] for any $\sigma\in G$,
\begin{equation*} \begin{pmatrix} \sigma(x_1)\\ \vdots\\ \sigma(x_m)\\ \end{pmatrix} =A(\sigma)\begin{pmatrix} x_1\\ \vdots\\ x_m\\ \end{pmatrix} +B(\sigma) \end{equation*} where $A(\sigma)\in{\rm GL}_m(L)$ and $B(\sigma)$ is $m\times 1$ matrix over $L$. Then there exist $z_1,\dots,z_m\in L(x_1,\dots,x_m)$ so that $L(x_1,\dots,x_m)^G=L^G(z_1,\dots,z_m)$ and $\sigma(z_i)=z_i$ for any $\sigma\in G$, any $1\leq i\leq m$. \end{description} \end{t2.1}
\newtheorem{t2.2}[t2.1]{Theorem} \begin{t2.2}\label{t2.2} {\rm (}\cite[Theorem 3.1]{AHK}{\rm )} Let $G$ be a finite group acting on $L(x)$, the rational function field of one variable over a field $L$. Assume that, for any $\sigma\in G,\sigma(L)\subset L$ and $\sigma(x)=a_\sigma x+b_\sigma$ for any $a_\sigma,b_\sigma\in L$ with $a_\sigma\ne 0$. Then $L(x)^G=L^G(z)$ for some $z\in L[x]$. \end{t2.2}
\newtheorem{t2.3}[t2.1]{Theorem} \begin{t2.3}\label{t2.3} {\rm (}\cite[Theorem 1.7]{CK}{\rm )} If $char K=p>0$ and $\widetilde G$ is a finite $p$-group, then $K(G)$ is rational over $K$. \end{t2.3}
The following Lemma can be extracted from some proofs in \cite{Ka2,HuK}.
\newtheorem{l2.7}[t2.1]{Lemma} \begin{l2.7}\label{l2.7} Let $\langle\tau\rangle$ be a cyclic group of order $n>1$, acting on $K(v_1,\dots,v_{n-1})$, the rational function field of $n-1$ variables over a field $K$ such that \begin{eqnarray*} \tau&:&v_1\mapsto v_2\mapsto\cdots\mapsto v_{n-1}\mapsto (v_1\cdots v_{n-1})^{-1}\mapsto v_1. \end{eqnarray*} If $K$ contains a primitive $n$-th root of unity $\xi$, then $K(v_1,\dots,v_{n-1})=K(s_1,\dots,s_{n-1})$ where $\tau:s_i\mapsto \xi^is_i$ for $1\leq i\leq n-1$. \end{l2.7} \begin{proof} Define $w_0=1+v_1+v_1v_2+\cdots+v_1v_2\cdots v_{n-1},w_1=(1/w_0)-1/n,w_{i+1}=(v_1v_2\cdots v_i/w_0)-1/n$ for $1\leq i\leq n-1$. Thus $K(v_1,\dots,v_{n-1})=K(w_1,\dots,w_n)$ with $w_1+w_2+\cdots+w_n=0$ and \begin{eqnarray*} \tau&:&w_1\mapsto w_2\mapsto\cdots\mapsto w_{n-1}\mapsto w_n\mapsto w_1. \end{eqnarray*} Define $s_i=\sum_{1\leq j\leq n}\xi^{-ij}w_j$ for $1\leq i\leq n-1$. Then $K(w_1,\dots,w_n)=K(s_1,\dots,s_{n-1})$ and $\tau:s_i\mapsto \xi^is_i$ for $1\leq i\leq n-1$. \end{proof}
Moreover, we are now going to generalize Kang's argument from \cite[Case 5, Step II]{Ka2}, obtaining the following Lemma which plays an important role in our work.
\newtheorem{l2.8}[t2.1]{Lemma} \begin{l2.8}\label{l2.8} Let $k>1$, let $p$ be any prime and let $\langle\alpha\rangle$ be a cyclic group of order $p$, acting on $K(y_{1i},y_{2i}\dots,y_{ki}:1\leq i\leq p-1)$, the rational function field of $k(p-1)$ variables over a field $K$ such that \begin{align*} \alpha\ :\ &y_{j1}\mapsto y_{j2}\mapsto\cdots\mapsto y_{jp-1}\mapsto (y_{j1}y_{j2}\cdots y_{jp-1})^{-1},\ &\text{for}\ 1\leq j\leq k. \end{align*} Assume that $K(v_{1i},v_{2i}\dots,v_{ki}:1\leq i\leq p-1)=K(y_{1i},y_{2i}\dots,y_{ki}:1\leq i\leq p-1)$ where for any $j:1\leq j\leq k$ and for any $i:1\leq i\leq p-1$ the variable $v_{ji}$ is a monomial in the variables $y_{1i},y_{2i}\dots,y_{ki}$. Assume also that the action of $\alpha$ on $K(v_{1i},v_{2i}\dots,v_{ki}:1\leq i\leq p-1)$ is given by {\allowdisplaybreaks\begin{align*} \alpha\ :\ &v_{j1}\mapsto v_{j1}v_{j2}^p,~ v_{j2}\mapsto v_{j3}\mapsto\cdots\mapsto v_{jp-1}\mapsto A_j\cdot(v_{j1}v_{j2}^{p-1}v_{j3}^{p-2}\cdots v_{jp-1}^2)^{-1},\\ \ &\text{for}\ 1\leq j\leq k, \end{align*}} where $A_j$ is some monomial in $v_{1i},\dots,v_{j-1i}$ for $2\leq j\leq k$ and $A_1=1$. If $K$ contains a primitive $p$-th root of unity $\zeta$, then $K(v_{1i},v_{2i}\dots,v_{ki}:1\leq i\leq p-1)=K(s_{1i},s_{2i}\dots,s_{ki}:1\leq i\leq p-1)$ where $\alpha:s_{ji}\mapsto \zeta^is_{ji}$ for $1\leq j\leq k,1\leq i\leq p-1$. \end{l2.8} \begin{proof} We write the additive version of the multiplication action of $\alpha$, i.e., consider the $\mathbb Z[\pi]$-module $M=\bigoplus_{1\leq m\leq k}(\oplus_{1\leq i\leq p-1}\mathbb Z\cdot v_{mi})$, where $\pi=\langle\alpha\rangle$. Denote the submodules $M_j=\bigoplus_{1\leq m\leq j}(\oplus_{1\leq i\leq p-1}\mathbb Z\cdot v_{mi})$ for $1\leq j\leq k$. Thus $\alpha$ has the following additive action {\allowdisplaybreaks\begin{align*} \alpha\ :\ &v_{j1}\mapsto v_{j1}+pv_{j2},~\\
&v_{j2}\mapsto v_{j3}\mapsto\cdots\mapsto v_{jp-1}\mapsto A_j-v_{j1}-(p-1)v_{j2}-(p-2)v_{j3}-\cdots -2v_{jp-1}, \end{align*}} where $A_j\in M_{j-1}$.
By Lemma \ref{l2.7}, $M_1$ is isomorphic to the $\mathbb Z[\pi]$-module $N=\oplus_{1\leq i\leq p-1}\mathbb Z\cdot u_i$ where $u_1=v_{12},u_i=\alpha^{i-1}\cdot v_{12}$ for $2\leq i\leq p-1$, and \begin{align*} \alpha\ :\ &u_1\mapsto u_2\mapsto\cdots\mapsto u_{p-1}\mapsto -u_1-u_2-\cdots-u_{p-1}\mapsto u_1. \end{align*}
Let $\Phi_p(T)\in\mathbb Z[T]$ be the $p$-th cyclotomic polynomial. Since $\mathbb Z[\pi]\simeq\mathbb Z[T]/(T^p-1)$, we find that $\mathbb Z[\pi]/\Phi_p(\alpha)\simeq \mathbb Z[T]/\Phi_p(T)\simeq \mathbb Z[\omega]$, the ring of $p$-th cyclotomic integer. As $\Phi_p(\alpha)\cdot x=0$ for any $x\in N$, the $\mathbb Z[\pi]$-module $N$ can be regarded as a $\mathbb Z[\omega]$-module through the morphism $\mathbb Z[\pi]\to\mathbb Z[\pi]/\Phi_p(\alpha)$. When $N$ is regarded as a $\mathbb Z[\omega]$-module, $N\simeq\mathbb Z[\omega]$ the rank-one free $\mathbb Z[\omega]$-module.
We claim that $M$ itself can be regarded as a $\mathbb Z[\omega]$-module, i.e., $\Phi_p(\alpha)\cdot M=0$.
Return to the multiplicative notations. Note that all $v_{ji}$'s are monomials in $y_{ji}$'s. The action of $\alpha$ on $y_{ji}$ given in the statement satisfies the relation $\prod_{0\leq m\leq p-1}\alpha^m(y_{ji})=1$ for any $1\leq j\leq k,1\leq i\leq p-1$. Using the additive notations, we get $\Phi_p(\alpha)\cdot y_{ji}=0$. Hence $\Phi_p(\alpha)\cdot M=0$.
Define $M'=M/M_{k-1}$. It follows that we have a short exact sequence of $\mathbb Z[\pi]$-modules \begin{equation}\label{e2.3} 0\to M_{k-1}\to M\to M'\to 0. \end{equation} Since $M$ is a $\mathbb Z[\omega]$-module, \eqref{e2.3} is a short exact sequence of $\mathbb Z[\omega]$-modules. Proceeding by induction, we obtain that $M$ is a direct sum of free $\mathbb Z[\omega]$-modules isomorphic to $N$. Therefore, $M\simeq\oplus_{1\leq j\leq k}N_j$, where $N_j\simeq N$ is a free $\mathbb Z[\omega]$-module, and so a $\mathbb Z[\pi]$-module also (for $1\leq j\leq k$).
Finally, we interpret the additive version of $M\simeq\oplus_{1\leq j\leq k}N_j\simeq N^k$ it terms of the multiplicative version as follows: There exist $w_{ji}$ that are monomials in $v_{ji}$ for $1\leq j\leq k,1\leq i\leq p-1$ such that $K(w_{ji})=K(v_{ji})$ and $\alpha$ acts as \begin{align*} \alpha\ :\ &w_{j1}\mapsto w_{j2}\mapsto\cdots\mapsto w_{jp-1}\mapsto (w_{j1}w_{j2}\dots w_{jp-1})^{-1}\ \text{for}\ 1\leq j\leq k. \end{align*} According to Lemma \ref{l2.7}, the above action can be linearized as pointed in the statement. \end{proof}
Now, let $G$ be any metacyclic $p$-group generated by two elements $\sigma$ and $\tau$ with relations $\sigma^{p^a}=1,\tau^{p^b}=\sigma^{p^c}$ and $\tau^{-1}\sigma\tau=\sigma^{\varepsilon+\delta p^r}$ where $\varepsilon=1$ if $p$ is odd, $\varepsilon=\pm 1$ if $p=2$, $\delta=0,1$ and $a,b,c,r\geq 0$ are subject to some restrictions. For the the description of these restrictions see e.g. \cite[p. 564]{Ka1}.
\newtheorem{t2.6}[t2.1]{Theorem} \begin{t2.6}\label{t2.6} {\rm (Kang }\cite[Theorem 4.1]{Ka1}{\rm )} Let $p$ be a prime number, $m,n$ and $r$ are positive integers, $k=1+p^r$ if $(p,r)\ne (2,1)$ (resp. $k=-1+2^r$ with $r\geq 2$). Let $G$ be a split metacyclic $p$-group of order $p^{m+n}$ and exponent $p^e$ defined by $G=\langle\sigma,\tau: \sigma^{p^m}=\tau^{p^n}=1,\tau^{-1}\sigma\tau=\sigma^k\rangle$. Let $K$ be any field such that $char K\ne p$ and $K$ contains a primitive $p^e$-th root of unity, and let $\zeta$ be a primitive $p^m$-th root of unity. Then $K(x_0,x_1,\dots,x_{p^n-1})^G$ is rational over $K$, where $G$ acts on $x_0,\dots,x_{p^n-1}$ by \begin{eqnarray*} \sigma&:&x_i\mapsto \zeta^{k^i}x_i,\\ \tau&:&x_0\mapsto x_1\mapsto\cdots\mapsto x_{p^n-1}\mapsto x_0. \end{eqnarray*} \end{t2.6}
\section{Proof of Lemma \ref{lemma}} \label{3}
It is well known that $H$ is a normal subgroup of $G$. We divide the proof into several steps.
\emph{Step I.} Let $\beta_1$ be any element of $H$ that is not central. Since $G_{(p)}=\{1\}$, there exist $\beta_2,\dots,\beta_k\in H$ for some $k:2\leq k\leq p$ such that $[\beta_j,\alpha]=\beta_{j+1}$, where $1\leq j\leq k-1$ and $\beta_k\ne 1$ is central. We are going to show now that the order of $\beta_2$ is not greater than $p$. In particular, from the multiplication rule $[a,\alpha][b,\alpha]=[ab,\alpha]$ (for any $a,b\in H$) it follows that all $p$-th powers are contained in the center of $G$.
From $[\beta_j,\alpha]=\beta_{j+1}$ it follows the well known formula \begin{equation}\label{e3.1} \alpha^{-p}\beta_1\alpha^p=\beta_1\beta_2^{\binom{p}{1}}\beta_3^{\binom{p}{2}}\cdots \beta_p^{\binom{p}{p-1}}\beta_{p+1}, \end{equation} where we put $\beta_{k+1}=\cdots=\beta_{p+1}=1$. Since $\alpha^p$ is in $H$, we obtain the formula $$\beta_2^{\binom{p}{1}}\beta_3^{\binom{p}{2}}\cdots \beta_k^{\binom{p}{k-1}}=1.$$ Hence $(\beta_2\cdot\prod_{j\ne 2}\beta_j^{a_j})^p=1$ for some integers $a_j$. It is not hard to see that this identity is impossible if the order of $\beta_2$ is greater than $p$. Indeed, if $\ell=\max\{j:\beta_j^p\ne 1\}$, then $\beta_\ell^p$ is in the subgroup generated by $\beta_2^p,\dots,\beta_{\ell-1}^p$. Therefore $[\beta_\ell^p,\alpha]=[\beta_2^{b_2p}\cdots\beta_{\ell-1}^{b_{\ell-1}p},\alpha]=\beta_3^{b_2p}\cdots\beta_\ell^{b_{\ell-1}p}\ne 1$ for some $b_2,\dots,b_{\ell-1}\in\mathbb Z_p$. On the other hand, $[\beta_\ell^p,\alpha]=\beta_{\ell+1}^p=1$, which is a contradiction.
\emph{Step II.} Let us write the decomposition of $H$ as a direct product of cyclic subgroups (not necessarily normal in $G$): $H\simeq (C_p)^t\times C_{p^{a_1}}\times C_{p^{a_2}}\times\cdots C_{p^{a_s}}$ for $0\leq t, 2\leq a_1\leq a_2\leq\cdots\leq a_s$. Choose a generator $\alpha_{11}\in C_{p^{a_1}}$. Since $G_{(p)}=\{1\}$, there exist $\alpha_{12},\dots,\alpha_{1k}\in H$ for some $k:2\leq k\leq p$ such that $[\alpha_{1j},\alpha]=\alpha_{1j+1}$, where $1\leq j\leq k-1$ and $\alpha_{1k}\ne 1$ is central. From Step I it follows that the order of $\alpha_{12}$ is not greater than $p$. We are going to define a normal subgroup of $G$ which depends on the nature of the element $\alpha_{12}$. We will denote it by $\langle\langle\alpha_{11}\rangle\rangle$, and call it \emph{the commutator chain of} $\alpha_{11}$. Simultaneously, we will define a complement in $H$ denoted by $\overline{\langle\langle\alpha_{11}\rangle\rangle}$.
\emph{Case II.1.} Let $\alpha_{12}=\alpha_{11}^{p^{a_1-1}c_1}$ for some $c_1:0\leq c_1\leq p-1$. Define $\langle\langle\alpha_{11}\rangle\rangle=\langle\alpha_{11}\rangle$. Clearly, $\langle\langle\alpha_{11}\rangle\rangle$ is a normal subgroup of type (2). Define $\overline{\langle\langle\alpha_{11}\rangle\rangle}=(C_p)^t\cdot \langle\alpha_{21},\dots,\alpha_{s1}\rangle$.
\emph{Case II.2.} Let $\alpha_{12}\notin H^p$. According to the assumption in the statement of our Lemma, $[H(p),\alpha]\cap H^p=\{1\}$, we have $\alpha_{1j}\notin H^p$ for all $j$. Define $\langle\langle\alpha_{11}\rangle\rangle=\langle\alpha_{11},\dots,\alpha_{1k}\rangle$. Therefore $\langle\langle\alpha_{11}\rangle\rangle\simeq C_{p^{a_1}}\times (C_p)^{k-1}$ is a normal subgroup of type (3). Define $\overline{\langle\langle\alpha_{11}\rangle\rangle}=(C_p)^{t-k+1}\cdot \langle\alpha_{21},\dots,\alpha_{s1}\rangle$, where $(C_p)^{t-k+1}$ is the complement of $(C_p)^{k-1}$ in $(C_p)^t$.
\emph{Case II.3.} Let $\alpha_{12}\in H^p$. Then $\alpha_{12}=\prod_{i\in A}\alpha_{i1}^{p^{a_i-1}d_i}$, where $A\subset\{1,2,\dots,s\},1\leq d_i\leq p-1$. Put $i_0=\min\{i\in A\}$.
If $i_0=1$, then $\alpha_{12}=(\alpha_{11}^{d_1}\prod_{i\in A,i\ne 1}\alpha_{i1}^{p^{a_i-a_1}d_i})^{p^{a_1-1}}$. Now, we can replace the generator $\alpha_{11}$ with $\alpha_{11}'=\alpha_{11}^{d_1}\prod_{i\in A,i\ne 1}\alpha_{i1}^{p^{a_i-a_1}d_i}$. Clearly, ${\rm ord}(\alpha_{11}')={\rm ord}(\alpha_{11})$ and $[\alpha_{11}',\alpha]\in\langle\alpha_{11}'\rangle$, so this case is reduced to Case I.
If $i_0>1$, then $\alpha_{12}=(\alpha_{i_01}^{d_{i_0}}\prod_{i\in A,i\ne i_0}\alpha_{i1}^{p^{a_i-a_{i_0}}d_i})^{p^{a_{i_0}-1}}$. We can replace the generator $\alpha_{i_01}$ with $\alpha_{i_01}'=\alpha_{i_01}^{d_{i_0}}\prod_{i\in A,i\ne i_0}\alpha_{i1}^{p^{a_i-a_{i_0}}d_i}$. Clearly, ${\rm ord}(\alpha_{i_01}')={\rm ord}(\alpha_{i_01})$ and $\alpha_{i_01}'^{p^{a_{i_0}-1}}=\alpha_{12}$.
For abuse of notation we will assume henceforth that $i_0=2$ and $\alpha_{21}^{p^{a_2-1}}=\alpha_{12}$. Consider $\alpha_{22}=[\alpha_{21},\alpha]$. We have three possibilities now.
\emph{Subcase II.3.1.} $\alpha_{22}\in\langle\alpha_{11}^{p^{a_1-1}},\alpha_{21}^{p^{a_1-1}}\rangle$. Define $\langle\langle\alpha_{11}\rangle\rangle=\langle\alpha_{11},\alpha_{21}\rangle$. Therefore $\langle\langle\alpha_{11}\rangle\rangle\simeq C_{p^{a_1}}\times C_{p^{a_2}}$ is a normal subgroup of type (4).
\emph{Subcase II.3.2.} $\alpha_{22}\notin H^p$. Then there exist $\alpha_{22},\dots,\alpha_{2\ell}\in H$ for some $\ell:2\leq \ell\leq p$ such that $[\alpha_{2j},\alpha]=\alpha_{2j+1}$, where $1\leq j\leq \ell-1$ and $\alpha_{2\ell}\ne 1$ is central. Define $\langle\langle\alpha_{11}\rangle\rangle=\langle\alpha_{11},\alpha_{21},\alpha_{22},\dots,\alpha_{2\ell}\rangle$. Therefore $\langle\langle\alpha_{11}\rangle\rangle\simeq C_{p^{a_1}}\times C_{p^{a_2}}\times (C_p)^{\ell-1}$ is a normal subgroup of type (3).
\emph{Subcase II.3.3.} $\alpha_{22}\in H^p$. According to the observations we have just made, this subcase leads to the following two final possibilities.
\emph{Sub-subcase II.3.3.1.} $\alpha_{22}=\alpha_{31}^{p^{a_3-1}},\dots,\alpha_{r-12}=\alpha_{r1}^{p^{a_r-1}},\alpha_{r2}\in \langle\alpha_{11}^{p^{a_1-1}},\dots,\alpha_{r1}^{p^{a_r-1}}\rangle$. Define $\langle\langle\alpha_{11}\rangle\rangle=\langle\alpha_{11},\alpha_{21},\dots,\alpha_{r1}\rangle$. Therefore $\langle\langle\alpha_{11}\rangle\rangle\simeq C_{p^{a_1}}\times C_{p^{a_2}}\times\cdots\times C_{p^{a_r}}$ is a normal subgroup of type (4). Define $\overline{\langle\langle\alpha_{11}\rangle\rangle}=(C_p)^t\cdot \langle\alpha_{r+11},\dots,\alpha_{s1}\rangle$.
\emph{Sub-subcase II.3.3.2.} $\alpha_{22}=\alpha_{31}^{p^{a_3-1}},\dots,\alpha_{r-12}=\alpha_{r1}^{p^{a_r-1}},\alpha_{r2}\notin H^p$. Then there exist $\alpha_{r2},\dots,\alpha_{r\ell}\in H$ for some $\ell:2\leq \ell\leq p$ such that $[\alpha_{rj},\alpha]=\alpha_{rj+1}$, where $1\leq j\leq \ell-1$ and $\alpha_{r\ell}\ne 1$ is central. Define $\langle\langle\alpha_{11}\rangle\rangle=\langle\alpha_{11},\alpha_{21},\dots,\alpha_{r1},\alpha_{r2},\dots,\alpha_{r\ell}\rangle$. Therefore $\langle\langle\alpha_{11}\rangle\rangle\simeq C_{p^{a_1}}\times C_{p^{a_1}}\times\cdots C_{p^{a_r}}\times (C_p)^{\ell-1}$ is a normal subgroup of type (3). Define $\overline{\langle\langle\alpha_{11}\rangle\rangle}=(C_p)^{t-\ell+1}\cdot \langle\alpha_{r+11},\dots,\alpha_{s1}\rangle$, where $(C_p)^{t-\ell+1}$ is the complement of $(C_p)^{\ell-1}$ in $(C_p)^t$.
\emph{Step III.} Put $H_1=\langle\langle\alpha_{11}\rangle\rangle$ and $H_2=\overline{\langle\langle\alpha_{11}\rangle\rangle}$. Note that $H_1\cap H_2=\{1\}$. However, $H_2$ may not be a normal subgroup of $G$. That is why we need to show that there exist a commutator chain $\mathcal H_1$ and a normal subgroup $\mathcal H_2$ of $G$ such that $H=\mathcal H_1\times \mathcal H_2$. In this Step, we will describe a somewhat algorithmic approach which replaces the generators of $H$ until the desired result is obtained.
Assume henceforth that $H_2$ is not normal in $G$. Then there exists a generator $\beta\in H_2$ such that $\alpha^{-1}\beta\alpha=hh_1$ for some $h\in H_2,h_1\in H_1,h_1\notin H_2$. Since $h=\beta h_2$ for some $h_2\in H_2$, we get $[\beta,\alpha]=h_1h_2$.
Let us assume first that ${\rm ord}(\beta)=p$. If $h_1\in H^p$, then $h_2\notin [H(p),\alpha]$, otherwise $[H(p),\alpha]\cap H^p\ne\{1\}$. In other words, $h_2$ does not appear in similar chains, so we can simply put $h_1h_2$, instead of $h_2$, as a generator of $H_2$. In this way we obtain a group that is $G$-isomorphic to $H_2$. Thus we get that $[\beta,\alpha]$ is in this new copy of $H_2$. Similarly, if $h_1\in H(p)$ and $h_2\notin [H(p),\alpha]$, we can obtain a new copy of $H_2$ such that $[\beta,\alpha]$ is in $H_2$. If $h_2\in [H(p),\alpha]$, we may assume that $[\beta,\alpha]\in H_1$. In this case $\langle\langle\alpha_{11}\rangle\rangle$ must be of type (3). Let $\langle\langle\alpha_{11}\rangle\rangle\simeq C_{p^{a_1}}\times C_{p^{a_2}}\times\cdots\times C_{p^{a_k}}\times(C_p)^s$, be generated by elements $\alpha_{11},\dots,\alpha_{k1},\alpha_{k2},\dots,\alpha_{ks+1}$ with relations given in the statement of the Lemma. Assume that $\alpha_{k\ell}=[\beta,\alpha]$ for some $\ell:2\leq\ell\leq s+1$. If $\ell> 2$, replace $\beta$ with $\beta'=\beta\alpha_{k\ell-1}^{-1}$. Hence $[\beta',\alpha]=1$. If $\ell=2$, we can put $\alpha_{k1}'=\alpha_{k1}\beta^{-1}$, instead of $\alpha_{k1}$, as a generator of $H_1$. In this way we obtain a group of type (4), since $[\alpha_{k1}',\alpha]=1$. Clearly, $[\beta,\alpha]$ is not in this new commutator chain $\mathcal H_1$. It is not hard to see that with similar replacements we can treat the general case $[\beta,\alpha]=\prod_i\alpha_{i1}^{p^{a_i-1}c_i}\cdot\prod_j\alpha_{kj}$. Thus we obtain the decomposition $H=\mathcal H_1\times \mathcal H_2$, where $\mathcal H_1$ and $\mathcal H_2$ are normal subgroups of $G$.
Next, we are going to assume that ${\rm ord}(\beta)>p$. According to the definition of the commutator chain of $\alpha_{11}$ we need to consider the three cases of Step II separately.
\emph{Case III.1.} $\alpha_{12}=\alpha_{11}^{p^{a_1-1}c_1}$ for some $c_1:1\leq c_1\leq p-1$. Here we must have $h_1=\alpha_{11}^{p^{a_1-1}d_1}$ for some $d_1:1\leq d_1\leq p-1$. We can replace $\beta$ with $\beta'=\beta\alpha_{11}^{-d_1/c_1}$, so $[\beta',\alpha]=h_2$.
\emph{Case III.2.} $\alpha_{12}\notin H^p$. If $h_1=\prod_{j\geq 2}\alpha_{1j}^{d_j}$ for some $d_j:0\leq d_j\leq p-1$, we can replace $\beta$ with $\beta'=\beta\prod_{j\geq 2}\alpha_{1j-1}^{-d_j}$. Hence $[\beta',\alpha]=h_2$. Thus we reduce the considerations to the case $h_1=\alpha_{11}^{p^{a_1-1}d_1}$ for some $d_1:0\leq d_1\leq p-1$. We have now three possibilities for $h_2$.
\emph{Subcase III.2.1.} Let $h_2\notin H^p$ and $h_2\notin [H,\alpha]$. We can put $h_1h_2$, instead of $h_2$, as a generator of $H_2$. In this way we obtain a group that is $G$-isomorphic to $H_2$. Thus we get that $[\beta,\alpha]$ is in this new copy of $H_2$.
\emph{Subcase III.2.2.} Let $h_2\notin H^p$ and $h_2\in [H,\alpha]$, i.e., there exists $\gamma\notin H^p$ such that $[\gamma,\alpha]=h_2$. Put $\beta'=\beta\gamma^{-1}$. Then $[\beta',\alpha]=h_1=\alpha_{11}^{p^{a_1-1}d_1}$. Hence the commutator chain of $\alpha_{11}$ is contained in the commutator chain $\langle\langle\beta'\rangle\rangle$ which is a normal subgroup of $G$ of type (3).
\emph{Subcase III.2.3.} Let $h_2\in H^p$, i.e., $h_2=\prod_{i\in B}\alpha_{i1}^{p^{a_i-1}d_i}$, where $B=\{i:\alpha_{i1}\in H_2\},0\leq d_i\leq p-1$. We can replace $\alpha_{11}$ with $\alpha_{11}'=\alpha_{11}^{d_1}\prod_{i\in B}\alpha_{i1}^{p^{a_i-a_1}d_i}$. Now we have $[\beta,\alpha]=\alpha_{11}'^{p^{a_1-1}}$, so the commutator chain of $\alpha_{11}'$ is contained in the commutator chain $\langle\langle\beta\rangle\rangle$ which is a normal subgroup of $G$ of type (3).
\emph{Case III.3.} $\alpha_{12}\in H^p$. We have that either $\langle\langle\alpha_{11}\rangle\rangle\simeq C_{p^{a_1}}\times C_{p^{a_2}}\times\cdots\times C_{p^{a_r}}$ is a normal subgroup of type (4), or $\langle\langle\alpha_{11}\rangle\rangle\simeq C_{p^{a_1}}\times C_{p^{a_1}}\times\cdots C_{p^{a_r}}\times (C_p)^{\ell-1}$ is a normal subgroup of type (3).
Similarly to Case III.2, if $h_1$ is a product of elements of order $p$ that are not in $\langle\alpha_{11}^{p^{a_1-1}}\rangle$, by a suitable change of the generator $\beta$ we will obtain $[\beta,\alpha]=h_2$. Thus we again reduce the considerations to the case $h_1=\alpha_{11}^{p^{a_1-1}d_1}$ for some $d_1:0\leq d_1\leq p-1$. We have three possibilities for $h_2$, which are identical to the three subcases in Case III.2. The only slight difference is that the new commutator chain here can be either of type (3) or (4).
In this way, we have investigated all possibilities for the proper construction of the normal factors of $H$. The construction is algorithmic in nature. When we define a new commutator chain $\langle\langle\beta'\rangle\rangle$ or $\langle\langle\beta\rangle\rangle$ (as in Subcases III.2.2 and III.2.3), we have to start the same process all over again until we can not get a new commutator chain that contains the previous one. Denote by $\mathcal H_1$ the last commutator chain obtained by the described algorithm from $H_1$. We have that $\mathcal H_1$ is a normal subgroup of $G$ of type (1)--(4). Denote by $\mathcal H_2$ the subgroup obtained from $H_2$ by the replacements described above. Then $H$ is a direct product of $\mathcal H_1$ and $\mathcal H_2$, where $\mathcal H_2$ is normal in $G$. Proceeding by induction we will obtain the decomposition given in the statement. We are done.
\section{Proof of Theorem \ref{t1.4}} \label{4}
If char $K=p>0$, we can apply Theorem \ref{t2.3}. Therefore, we will assume that char $K\ne p$.
According to Lemma \ref{lemma}, $H\simeq\mathcal H_1\times\cdots\times\mathcal H_t$, where $\mathcal H_1,\dots,\mathcal H_t$ are normal subgroups of $G$ that are isomorphic to any of the four types, described in Lemma \ref{lemma}.
Let $V$ be a $K$-vector space whose dual space $V^*$ is defined as $V^*=\bigoplus_{g\in G}K\cdot x(g)$ where $G$ acts on $V^*$ by $h\cdot x(g)=x(hg)$ for any $h,g\in G$. Thus $K(V)^G=K(x(g):g\in G)^G=K(G)$.
Now, for any subgroup $\mathcal H_i (1\leq i\leq t)$ we can define a faithful representation subspace $V_i=\bigoplus_{1\leq j\leq k_i}K\cdot Y_j$, where $k_i$ is the number of the generators of $\mathcal H_i$ as an abelian group. (For the details see Cases I-IV.) Therefore, $\bigoplus_{1\leq i\leq t}V_i$ is a faithful representation space of the subgroup $H$.
Next, for any subgroup $\mathcal H_i (1\leq i\leq t)$ we define $x_{jk}=\alpha^k\cdot Y_j$ for $1\leq j\leq k_i,0\leq k\leq p-1$. Define $W_i=\bigoplus_{j,k}K\cdot x_{jk}\subset V^*$. Then $W=\bigoplus_{1\leq i\leq t}W_i$ is a faithful $G$-subspace of $V^*$. Thus, by Theorem \ref{t2.1} it suffices to show that $W^G$ is rational over $K$. Note that $W^G=(W^H)^{\langle\alpha\rangle}=((\dots(W^{\mathcal H_1})^{\mathcal H_2}\dots)^{\mathcal H_t})^{\langle\alpha\rangle}=((\dots(W_1^{\mathcal H_1}\bigoplus_{2\leq j\leq t}W_j)^{\mathcal H_2}\dots)^{\mathcal H_t})^{\langle\alpha\rangle}=\cdots=\bigoplus_{1\leq j\leq t}(W_j^{\mathcal H_j})^{\langle\alpha\rangle}.$ Therefore, we need to calculate $W_j^{\mathcal H_j}$ when $\mathcal H_j$ is isomorphic to any of the four types, described in Lemma \ref{lemma}. Finally, we will show that the action of $\alpha$ on $W^H$ can be linearized.
\emph{Case I.} Assume that $\mathcal H_1$ is of the type (3), i.e., $\mathcal H_1\simeq C_{p^{a_1}}\times C_{p^{a_2}}\times\cdots\times C_{p^{a_k}}\times(C_p)^s$ for some $k\geq 1,a_i\geq 2,s\geq 1$. Denote by $\alpha_1,\dots,\alpha_k$ the generators of $C_{p^{a_1}}\times\cdots\times C_{p^{a_k}}$, and by $\alpha_{k+1},\dots,\alpha_{k+s}$ the generators of $(C_p)^s$. According to Lemma \ref{lemma}, we have the relations $[\alpha_i,\alpha]=\alpha_{i+1}^{p^{a_{i+1}-1}}\in Z(G)$ for $1\leq i\leq k-1; [\alpha_{k+j},\alpha]=\alpha_{k+j+1}$ for $0\leq j\leq s-1$; and $\alpha_{k+s}\in Z(G)$. Because of the frequent use of $k+s$ in this case, we put $r=k+s$.
We divide the proof into several steps.
\emph{Step 1.} Define $X_1,X_2,\dots,X_{r}\in V^*$ by \begin{equation*} X_j=\sum_{\ell_1,\dots,\ell_{r}}x\left(\prod_{i\ne j}\alpha_i^{\ell_i}\right),\quad \text{for}\ 1\leq j\leq r. \end{equation*}
Note that $\alpha_i\cdot X_j=X_j$ for $j\ne i$. Let $\zeta_{p^{a_i}}\in K$ be a primitive $p^{a_i}$-th root of unity for $1\leq i\leq k$, and let $\zeta$ be a primitive $p$-th root of unity. Define $Y_1,Y_2,\dots,Y_{r}\in V^*$ by
\begin{equation*} Y_i=\sum_{m=0}^{p^{a_i}-1}\zeta_{p^{a_i}}^{-m}\alpha_i^m\cdot X_i,~ Y_j=\sum_{m=0}^{p-1}\zeta^{-m}\alpha_j^m\cdot X_j \end{equation*} for $1\leq i\leq k$ and $k+1\leq j\leq r$.
It follows that {\allowdisplaybreaks\begin{align*} \alpha_i\ :\ &Y_i\mapsto\zeta_{p^{a_i}} Y_i,~ Y_j\mapsto Y_j,\ \text{for}\ j\ne i\ \text{and}\ 1\leq i\leq k\\
\alpha_j\ :\ &Y_j\mapsto\zeta Y_j,~ Y_i\mapsto Y_i,\ \text{for}\ i\ne j\ \text{and}\ k+1\leq j\leq r. \end{align*}} Thus $V_1=\bigoplus_{1\leq j\leq r}K\cdot Y_j$ is a faithful representation space of the subgroup $\mathcal H_1$.
Define $x_{ji}=\alpha^i\cdot Y_j$ for $1\leq j\leq r,0\leq i\leq p-1$. Recall that $[\alpha_i,\alpha]=\alpha_{i+1}^{p^{a_{i+1}-1}}\in Z(G)$ for $1\leq i\leq k-1; [\alpha_{k+j},\alpha]=\alpha_{k+j+1}$ for $0\leq j\leq s-1$; and $\alpha_{r}\in Z(G)$. Hence $$\alpha^{-i}\alpha_j\alpha^i=\alpha_j\alpha_{j+1}^{ip^{a_{i+1}-1}},\quad \text{for}\ 1\leq j\leq k-1,1\leq i\leq p-1$$ and $$\alpha^{-i}\alpha_j\alpha^i=\alpha_j\alpha_{j+1}^{\binom{i}{1}}\alpha_{j+2}^{\binom{i}{2}}\cdots \alpha_{r}^{\binom{i}{r-j}},\quad \text{for}\ k\leq j\leq r-1,1\leq i\leq p-1.$$
It follows that {\allowdisplaybreaks\begin{align*}\alpha_{\ell}\ :\ &x_{\ell i}\mapsto\zeta_{p^{a_\ell}} x_{\ell i},~x_{\ell+1 i}\mapsto\zeta^i x_{\ell+1i},~ x_{ji}\mapsto x_{ji},\ \text{for}\ 1\leq\ell\leq k-1\ \text{and}\ j\ne\ell,\ell+1,\\ \alpha_k\ :\ &x_{ki}\mapsto \zeta_{p^{a_k}} x_{ki},~ x_{wi}\mapsto \zeta^{\binom{i}{w-k}} x_{wi},~ x_{v i}\mapsto x_{v i},\ \text{for}\ 1\leq v\leq k-1,k+1\leq w\leq r,\\ \alpha_m\ :\ &x_{ui}\mapsto \zeta^{\binom{i}{u-m}} x_{ui},~ x_{v i}\mapsto x_{v i},\ \text{for}\ k+1\leq m\leq r,1\leq v\leq m-1,m\leq u\leq r,\\ \alpha\ :\ &x_{j0}\mapsto x_{j1}\mapsto\cdots\mapsto x_{jp-1}\mapsto \zeta_{p^{c_j}}^{b_j}x_{j0},\ \text{for}\ 1\leq j\leq r, \end{align*}} where $0\leq i\leq p-1$, and $c_j,b_j$ are some integers such that $0\leq b_j< p^{c_j}\leq p^{a_j}$.
Define $W_1=\bigoplus_{j,i}K\cdot x_{ji}\subset V^*$. As we noted in the beginning of the proof, we need to find $W_1^{\mathcal H_1}$.
\emph{Step 2.} For $1\leq j\leq r$ and for $1\leq i\leq p-1$ define $y_{ji}=x_{ji}/x_{ji-1}$. Thus $W_1=K(x_{j0},y_{ji}:1\leq j\leq r,1\leq i\leq p-1)$ and for every $g\in G$ \begin{equation*} g\cdot x_{j0}\in K(y_{ji}:1\leq j\leq r,1\leq i\leq p-1)\cdot x_{j0},\ \text{for}\ 1\leq j\leq r \end{equation*} while the subfield $K(y_{ji}:1\leq j\leq r,1\leq i\leq p-1)$ is invariant by the action of $G$, i.e., {\allowdisplaybreaks\begin{align*}\alpha_{\ell}\ :\ &y_{\ell+1 i}\mapsto\zeta y_{\ell+1i},~ y_{ji}\mapsto y_{ji},\ \text{for}\ 1\leq\ell\leq k-1\ \text{and}\ j\ne\ell+1,\\ \alpha_m\ :\ &y_{ui}\mapsto \zeta^{\binom{i-1}{u-m-1}} y_{ui},~ y_{v i}\mapsto y_{v i},\ \text{for}\ k\leq m\leq r-1,1\leq v\leq m,m+1\leq u\leq r,\\ \alpha_{r}\ :\ &y_{v i}\mapsto y_{v i},\ \text{for}\ 1\leq v\leq r,\\ \alpha\ :\ &y_{j1}\mapsto y_{j2}\mapsto\cdots\mapsto y_{jp-1}\mapsto \zeta_{p^{c_j}}^{b_j}(y_{j1}\cdots y_{jp-1})^{-1},\ \text{for}\ 1\leq j\leq r, \end{align*}} From Theorem \ref{t2.2} it follows that if $K(y_{ji}:1\leq j\leq r,1\leq i\leq p-1)^{G}$ is rational over $K$, so is $K(x_{j0},y_{ji}:1\leq j\leq r,1\leq i\leq p-1)^{G}$ over $K$.
Since $K$ contains a primitive $p^e$-th root of unity $\zeta_{p^e}$ where $p^e$ is the exponent of $G$, $K$ contains as well a primitive $p^{{c_j}+1}$-th root of unity, and we may replace the variables $y_{ji}$ by $y_{ji}/\zeta_{p^{{c_j}+1}}^{b_j}$ so that we obtain a more convenient action of $\alpha$ without changing the actions of $\alpha_j$'s. Namely we may assume that \begin{align*} \alpha\ :\ &y_{j1}\mapsto y_{j2}\mapsto\cdots\mapsto y_{jp-1}\mapsto (y_{j1}y_{j2}\dots y_{jp-1})^{-1}\ \text{for}\ 1\leq j\leq r. \end{align*}
Define $u_{r1}=y_{r1}^p,u_{ri}=y_{ri}/y_{ri-1}$ for $2\leq i\leq p-1$. Then $K(y_{ji},u_{ri}:1\leq j\leq r-1,1\leq i\leq p-1)=K(y_{ji}:1\leq j\leq r,1\leq i\leq p-1)^{\langle\alpha_{r-1}\rangle}$. From Theorem \ref{t2.2} it follows that if $K(y_{ji},u_{ri}:1\leq j\leq r-1,2\leq i\leq p-1)^{G}$ is rational over $K$, so is $K(y_{ji},u_{ri}:1\leq j\leq r-1,1\leq i\leq p-1)^{G}$ over $K$. We have the following actions {\allowdisplaybreaks\begin{align*}\alpha_{\ell}\ :\ &u_{ri}\mapsto u_{ri},\ \text{for}\ 1\leq\ell\leq k-1,\\ \alpha_m\ :\ &u_{ri}\mapsto \zeta^{\binom{i-2}{r-m-2}} u_{ri},\ \text{for}\ 2\leq i\leq p-1\ \text{and}\ k\leq m\leq r-2,\\ \alpha\ :\ &u_{r2}\mapsto u_{r3}\mapsto\cdots\mapsto u_{rp-1}\mapsto (u_{r1}u_{r2}^{p-1}u_{r3}^{p-2}\cdots u_{rp-1}^2)^{-1}\mapsto u_{r1}u_{r2}^{p-2}u_{r3}^{p-3}\cdots u_{rp-2}^2u_{rp-1}. \end{align*}} For $2\leq i\leq p-1$ define $$v_{ri}=u_{ri}y_{r-1i}^{-1}y_{r-2i}y_{r-3i}^{-1}\cdots y_{k+2i}^{(-1)^{r-k}}y_{k+1i}^{(-1)^{r-k+1}},$$ and put $v_{r1}=u_{r1}$.
With the aid of the well known property $\binom{n}{m}-\binom{n-1}{m}=\binom{n-1}{m-1}$, it is not hard to verify the identity {\allowdisplaybreaks\begin{align*} \ &\binom{i-2}{r-m-2}-\binom{i-1}{r-m-2}+\binom{i-1}{r-m-3}-\binom{i-1}{r-m-4}+\cdots\\ &\cdots+(-1)^{r-m-1}\binom{i-1}{2}+(-1)^{r-m}\binom{i-1}{1}+(-1)^{r-m+1}\binom{i-1}{0}=0. \end{align*}} It follows that {\allowdisplaybreaks\begin{align*} \alpha_m\ :\ &v_{ri}\mapsto v_{ri},\ \text{for}\ 1\leq i\leq p-1\ \text{and}\ 1\leq m\leq r-2,\\ \alpha\ :\ &v_{r2}\mapsto v_{r3}\mapsto\cdots\mapsto v_{rp-1}\mapsto A_r\cdot(v_{r1}v_{r2}^{p-1}v_{r3}^{p-2}\cdots v_{rp-1}^2)^{-1}. \end{align*}} where $A_r$ is some monomial in $y_{ji}$ for $2\leq j\leq r-1,1\leq i\leq p-1$.
Define $u_{r-11}=y_{r-11}^p,u_{r-1i}=y_{r-1i}/y_{r-1i-1}$ for $2\leq i\leq p-1$. Then $K(y_{ji},u_{r-1i}:1\leq j\leq r-2,1\leq i\leq p-1)=K(y_{ji}:1\leq j\leq r-1,1\leq i\leq p-1)^{\langle\alpha_{r-2}\rangle}$. From Theorem \ref{t2.2} it follows that if $K(y_{ji},u_{r-1i}:1\leq j\leq r-2,2\leq i\leq p-1)^{G}$ is rational over $K$, so is $K(y_{ji},u_{r-1i}:1\leq j\leq r-2,1\leq i\leq p-1)^{G}$ over $K$. Similarly to the definition of $v_{ri}$, we can define $v_{r-1i}$ so that $\alpha_m(v_{r-1i})=v_{r-1i}$ for $2\leq i\leq p-1$ and $1\leq m\leq r-3$. It is obvious that we can proceed in the same way defining elements $v_{r-2i}, v_{r-3i},\dots,v_{k+1i}$ such that $\alpha_m$ acts trivially on all $v_{ji}$'s for $k\leq m\leq r-3$.
Recall that the actions of $\alpha_{\ell}$ on $y_{ji}$'s for $1\leq\ell\leq k-1$ are \begin{align*}\alpha_{\ell}\ :\ &y_{\ell+1 i}\mapsto\zeta y_{\ell+1i},~ y_{ji}\mapsto y_{ji},\ \text{for}\ 1\leq i\leq p-1, 1\leq\ell\leq k-1\ \text{and}\ j\ne\ell+1. \end{align*}
For any $1\leq\ell\leq k-1$ define $v_{\ell+11}=y_{\ell+11}^p,v_{\ell+1i}=y_{\ell+1i}/y_{\ell+1i-1}$, where $2\leq i\leq p-1$. Put also $v_{1i}=y_{1i}$ for $1\leq i\leq p-1$. Then $K(v_{ji}:1\leq j\leq r,1\leq i\leq p-1)=K(y_{ji}:1\leq j\leq r,1\leq i\leq p-1)^{\mathcal H_1}$.
The action of $\alpha$ is given by {\allowdisplaybreaks\begin{align*} \alpha\ :\ &v_{11}\mapsto v_{12}\mapsto\cdots\mapsto v_{1p-1}\mapsto (v_{11}v_{12}\cdots v_{1p-1})^{-1},\\ \ &v_{m1}\mapsto v_{m1}v_{m2}^p,~ v_{m2}\mapsto v_{m3}\mapsto\cdots\mapsto v_{mp-1}\mapsto A_m\cdot(v_{m1}v_{m2}^{p-1}v_{m3}^{p-2}\cdots v_{mp-1}^2)^{-1},\\ \ &\text{for}\ 2\leq m\leq r, \end{align*}} where $A_m$ is some monomial in $v_{k+1i},\dots,v_{m-1i}$ for $k+2\leq m\leq r$ and $A_2=A_3=\cdots=A_{k+1}=1$. From Lemmas \ref{l2.7} and \ref{l2.8} it follows that the action of $\alpha$ on $K(v_{ji}:1\leq j\leq r,1\leq i\leq p-1)$ can be linearized.
\emph{Case II.} Assume that $\mathcal H_1$ is of the type (1), i.e., $\mathcal H_1\simeq (C_p)^{s+1}$ for some $s\geq 0$. Denote by $\beta_1,\dots,\beta_{s+1}$ the generators of $(C_p)^{s+1}$. According to Lemma \ref{lemma}, we have the relations $[\beta_j,\alpha]=\beta_{j+1}$ for $1\leq j\leq s$; and $\beta_{s+1}\in Z(G)$.
Define $X_1,X_2,\dots,X_{s+1}\in V^*$ by \begin{equation*} X_j=\sum_{\ell_1,\dots,\ell_{s+1}}x\left(\prod_{m\ne j}\beta_m^{\ell_m}\right), \end{equation*} for $1\leq j\leq s+1$. Note that $\beta_j\cdot X_i=X_i$ for $j\ne i$. Let $\zeta$ be a primitive $p$-th root of unity. Define $Y_1,Y_2,\dots,Y_{s+1}\in V^*$ by \begin{equation*} Y_j=\sum_{r=0}^{p-1}\zeta^{-r}\beta_j^r\cdot X_j \end{equation*} for $1\leq j\leq s+1$.
It follows that {\allowdisplaybreaks\begin{align*} &\beta_j\ :\ Y_j\mapsto\zeta Y_j,~ Y_i\mapsto Y_i,\ \text{for}\ i\ne j\ \text{and}\ 1\leq j\leq s+1. \end{align*}} Thus $V_1=\bigoplus_{1\leq j\leq s+1}K\cdot Y_j$ is a representation space of the subgroup $\mathcal H_1$.
Define $x_{ji}=\alpha^i\cdot Y_j$ for $1\leq j\leq s+1,0\leq i\leq p-1$. Recall that $[\beta_j,\alpha]=\beta_{j-1}$. Hence $$\alpha^{-i}\beta_j\alpha^i=\beta_j\beta_{j+1}^{\binom{i}{1}}\beta_{j+2}^{\binom{i}{2}}\cdots \beta_{s+1}^{\binom{i}{s+1-j}}.$$
It follows that {\allowdisplaybreaks\begin{align*}\beta_1\ :\ &x_{1i}\mapsto\zeta x_{1i},~ x_{ji}\mapsto \zeta^{\binom{i}{j-1}} x_{ji},\ \text{for}\ 2\leq j\leq s+1\ \text{and}\ 0\leq i\leq p-1,\\ \beta_j\ :\ &x_{\ell i}\mapsto x_{\ell i},~ x_{mi}\mapsto \zeta^{\binom{i}{m-j}} x_{mi},\ \text{for}\ 1\leq \ell\leq j-1,j\leq m\leq s+1\ \text{and}\ 0\leq i\leq p-1,\\ \alpha\ :\ &x_{j0}\mapsto x_{j1}\mapsto\cdots\mapsto x_{jp-1}\mapsto \zeta^{b_j}x_{j0},\ \text{for}\ 1\leq j\leq s+1, 0\leq b_j\leq p-1. \end{align*}} Compare the actions of $\alpha,\beta_1,\dots,\beta_{s+1}$ with the actions of $\alpha,\alpha_k,\dots,\alpha_{k+s}$ from Case I, Step 1. They are almost the same. Apply the proof of Case I.
\emph{Case III.} Assume that $\mathcal H_1$ is of the type (2), i.e., $\mathcal H_1\simeq C_{p^a}$ for some $a\geq 1$. Denote by $\beta$ the generator of $C_{p^a}$. Then $[\beta,\alpha]=\beta^{bp^{a-1}}$ for some $b:0\leq b\leq p-1$. Let $\zeta_{p^a}\in K$ be a primitive $p^a$-th root of unity, and let $\zeta$ be a primitive $p$-th root of unity. Define $X=\sum_i\zeta_{p^a}^{-i}x(\beta^i)$. Then $\beta(X)=\zeta_{p^a}X$, and define $x_i=\alpha^i\cdot X$ for $0\leq i\leq p-1$. It follows that {\allowdisplaybreaks\begin{align*}\beta\ :\ &x_i\mapsto \zeta_{p^a}\zeta^{ib} x_i,\ \text{for}\ 0\leq i\leq p-1,\\ \alpha\ :\ &x_0\mapsto x_1\mapsto\cdots\mapsto x_{p-1}\mapsto \zeta_{p^a}^{c}x_0,\ \text{for}\ 0\leq c\leq p^a-1. \end{align*}} Define $W_1=\bigoplus_iK\cdot x_i\subset V^*$. For $1\leq i\leq p-1$ define $y_i=x_i/x_{i-1}$. Thus $W_1=K(x_0,y_i:1\leq i\leq p-1)$ and for every $g\in G$ \begin{equation*} g\cdot x_0\in K(y_i:1\leq i\leq p-1)\cdot x_0, \end{equation*} while the subfield $K(y_i:1\leq i\leq p-1)$ is invariant by the action of $G$, i.e., {\allowdisplaybreaks\begin{align*}\beta\ :\ &y_i\mapsto \zeta^b y_i,\ \text{for}\ 1\leq i\leq p-1,\\ \alpha\ :\ &y_1\mapsto y_2\mapsto\cdots\mapsto \zeta_{p^a}^{c}(y_1\cdots y_{p-1})^{-1},\ \text{for}\ 0\leq c\leq p^a-1. \end{align*}} From Theorem \ref{t2.2} it follows that if $K(y_i:1\leq i\leq p-1)^G$ is rational over $K$, so is $K(x_0,y_i:1\leq i\leq p-1)^G$ over $K$.
Since $K$ contains a primitive $p^e$-th root of unity $\zeta_{p^e}$ where $p^e$ is the exponent of $G$, $K$ contains as well $\zeta_{p^{a+1}}^{c}$. We may replace the variables $y_i$ by $y_i/\zeta_{p^{a+1}}^c$ so that we obtain \begin{align*} \alpha\ :\ &y_1\mapsto y_2\mapsto\cdots\mapsto y_{p-1}\mapsto (y_{1}y_{2}\dots y_{p-1})^{-1}. \end{align*}
Define $u_1=y_1^p,u_i=y_i/y_{i-1}$ for $2\leq i\leq p-1$. Then $K(u_i:1\leq i\leq p-1)=K(y_i:1\leq i\leq p-1)^{\langle\beta\rangle}$. The action of $\alpha$ is given by {\allowdisplaybreaks\begin{align*} \alpha\ :\ &u_1\mapsto u_1u_2^p,~ u_{2}\mapsto u_{3}\mapsto\cdots\mapsto u_{p-1}\mapsto (u_{1}u_{2}^{p-1}u_{3}^{p-2}\cdots u_{p-1}^2)^{-1}, \end{align*}} From Lemma \ref{l2.7} (or \ref{l2.8}) it follows that the action of $\alpha$ can be linearized.
\emph{Case IV.} Assume that $\mathcal H_1$ is of the type (4), i.e., $\mathcal H_1\simeq C_{p^{a_1}}\times C_{p^{a_2}}\times\cdots\times C_{p^{a_k}}$ for some $k\geq 2$. Denote by $\alpha_1,\dots,\alpha_k$ the generators of $\mathcal H_1$. According to Lemma \ref{lemma}, we have the relations $[\alpha_i,\alpha]=\alpha_{i+1}^{p^{a_{i+1}-1}}\in Z(G)$ for $1\leq i\leq k-1; [\alpha_k,\alpha]=\prod_{j=1}^{k}\alpha_j^{p^{a_j-1}c_j}\in Z(G)$ for some $0\leq c_j\leq p-1$.
Similarly to the previous cases, define $Y_1,Y_2,\dots,Y_{k}\in V^*$ so that {\allowdisplaybreaks\begin{align*} \alpha_i\ :\ &Y_i\mapsto\zeta_{p^{a_i}} Y_i,~ Y_j\mapsto Y_j,\ \text{for}\ j\ne i\ \text{and}\ 1\leq i\leq k. \end{align*}} Thus $V_1=\bigoplus_{1\leq j\leq k}K\cdot Y_j$ is a faithful representation space of the subgroup $\mathcal H_1$.
Next, define $x_{ji}=\alpha^i\cdot Y_j$ for $1\leq j\leq k,0\leq i\leq p-1$. Note that $$\alpha^{-i}\alpha_j\alpha^i=\alpha_j\alpha_{j+1}^{ip^{a_{j+1}-1}},\quad \text{for}\ 1\leq j\leq k-1,1\leq i\leq p-1$$ and $$\alpha^{-i}\alpha_k\alpha^i=\alpha_k\prod_{j=1}^{k}\alpha_j^{ip^{a_j-1}c_j},\quad \text{for}\ 1\leq i\leq p-1.$$
It follows that {\allowdisplaybreaks\begin{align*}\alpha_{\ell}\ :\ &x_{\ell i}\mapsto\zeta_{p^{a_\ell}} x_{\ell i},~x_{\ell+1 i}\mapsto\zeta^i x_{\ell+1i},~ x_{ji}\mapsto x_{ji},\ \text{for}\ 1\leq\ell\leq k-1\ \text{and}\ j\ne\ell,\ell+1,\\ \alpha_k\ :\ &x_{ki}\mapsto \zeta_{p^{a_k}}\zeta^{ic_k} x_{ki},~ x_{ji}\mapsto \zeta^{ic_j} x_{ji},\ \text{for}\ 1\leq j\leq k-1,\\ \alpha\ :\ &x_{j0}\mapsto x_{j1}\mapsto\cdots\mapsto x_{jp-1}\mapsto \zeta_{p^{a_j}}^{b_j}x_{j0},\ \text{for}\ 1\leq j\leq k, \end{align*}} where $0\leq i\leq p-1,0\leq c_j\leq p-1$ and $0\leq b_j\leq p^{a_j}-1$.
Define $W_1=\bigoplus_{j,i}K\cdot x_{ji}\subset V^*$, and for $1\leq i\leq p-1$ define $y_i=x_i/x_{i-1}$. Thus $W_1=K(x_{j0},y_{ji}:1\leq j\leq k,1\leq i\leq p-1)$ and for every $g\in G$ \begin{equation*} g\cdot x_{j0}\in K(y_{ji}:1\leq j\leq k,1\leq i\leq p-1)\cdot x_{j0},\ \text{for}\ 1\leq j\leq k \end{equation*} while the subfield $K(y_{ji}:1\leq j\leq k,1\leq i\leq p-1)$ is invariant by the action of $G$, i.e., {\allowdisplaybreaks\begin{align*}\alpha_{\ell}\ :\ &y_{\ell+1 i}\mapsto\zeta y_{\ell+1i},~ y_{ji}\mapsto y_{ji},\ \text{for}\ 1\leq i\leq p-1,~ 1\leq\ell\leq k-1\ \text{and}\ j\ne\ell+1,\\ \alpha_k\ :\ &y_{ji}\mapsto \zeta^{c_j} y_{ji},\ \text{for}\ 1\leq i\leq p-1,~ 1\leq j\leq k,\\ \alpha\ :\ &y_{j1}\mapsto y_{j2}\mapsto\cdots\mapsto y_{jp-1}\mapsto \zeta_{p^{a_j}}^{b_j}(y_{j1}\cdots y_{jp-1})^{-1} \end{align*}} From Theorem \ref{t2.2} it follows that if $K(y_{ji}:1\leq j\leq k,1\leq i\leq p-1)^{G}$ is rational over $K$, so is $K(x_{j0},y_{ji}:1\leq j\leq k,1\leq i\leq p-1)^{G}$ over $K$. As before, we can again assume that $\alpha$ acts in this way: \begin{align*} \alpha\ :\ &y_{j1}\mapsto y_{j2}\mapsto\cdots\mapsto y_{jp-1}\mapsto (y_{j1}y_{j2}\dots y_{jp-1})^{-1}. \end{align*}
Now, assume that $0<c_1\leq p-1$. For $2\leq j\leq k$ choose $e_j$ such that $c_1e_j+c_j\equiv 0\pmod p$, and define $u_{1i}=y_{1i}, u_{ji}=y_{1i}^{e_j}y_{ji}$. It follows that {\allowdisplaybreaks\begin{align*}\alpha_{\ell}\ :\ &u_{\ell+1 i}\mapsto\zeta u_{\ell+1i},~ u_{ji}\mapsto u_{ji},\ \text{for}\ 1\leq i\leq p-1,~ 1\leq\ell\leq k-1\ \text{and}\ j\ne\ell+1,\\ \alpha_k\ :\ &u_{1i}\mapsto\zeta^{c_1} u_{1i}, u_{ji}\mapsto u_{ji},\ \text{for}\ 1\leq i\leq p-1,~ 2\leq j\leq k. \end{align*}} Define $v_{j1}=u_{j1}^p,v_{ji}=u_{ji}/u_{ji-1}$ for $2\leq i\leq p-1,1\leq j\leq k$. Then $K(v_{ji}:1\leq j\leq k,1\leq i\leq p-1)=K(u_{ji}:1\leq j\leq k,1\leq i\leq p-1)^{\mathcal H_1}$. The action of $\alpha$ is given by {\allowdisplaybreaks\begin{align*} \alpha\ :\ &v_{j1}\mapsto v_{j1}v_{j2}^p,~ v_{j2}\mapsto v_{j3}\mapsto\cdots\mapsto v_{jp-1}\mapsto (v_{j1}v_{j2}^{p-1}v_{j3}^{p-2}\cdots v_{jp-1}^2)^{-1},\\ \ &\text{for}\ 2\leq j\leq k. \end{align*}} From Lemma \ref{l2.8} it follows that the action of $\alpha$ on $K(v_{ji}:1\leq j\leq k,1\leq i\leq p-1)$ can be linearized.
Finally, let $c_1=0$. Define $v_{j1}=u_{j1}^p,v_{ji}=u_{ji}/u_{ji-1}$ for $2\leq i\leq p-1,2\leq j\leq k$. Then $K(u_{1i},v_{ji}:2\leq j\leq k,1\leq i\leq p-1)=K(u_{ji}:1\leq j\leq k,1\leq i\leq p-1)^{\mathcal H_1}$. The action of $\alpha$ again can be linearized as before. We are done.
\section{Proof of Theorem \ref{t1.5}} \label{5}
By studying the classification of all groups of order $p^5$ made by James in \cite{Ja}, we see that the non abelian groups with an abelian subgroup of index $p$ and that are not direct products of smaller groups are precisely the groups from the isoclinic families with numbers $2,3,4$ and $9$. Notice that all these groups satisfy the conditions of Theorem \ref{t1.4}. The isoclinic family $8$ contains only the group $\Phi_8(32)$ which is metacyclic, so we can apply Theorem \ref{t1.2}. It is not hard to see that there are no other groups of order $p^5$, containing a normal abelian subgroup $H$ such that $G/H$ is cyclic.
The groups of order $p^6$ with an abelian subgroup of index $p$ and that are not direct products of smaller groups are precisely the groups from the isoclinic families with numbers $2,3,4$ and $9$. Again, all these groups satisfy the conditions of Theorem \ref{t1.4}. The groups of order $p^6$, containing a normal abelian subgroup $H$ such that $G/H$ is cyclic of order $>p$ are precisely the groups from the isoclinic families with numbers $8$ and $14$. Note that the groups $\Phi_8(42),\Phi_8(33),\Phi_{14}(42)$ are metacyclic, and the group $\Phi_8(321)a$ is a direct product of the metacyclic group $\Phi_8(32)$ and the cyclic group $C_p$. Therefore, we need to consider the remaining groups, whose presentations we write down for convenience of the reader. {\allowdisplaybreaks \begin{align*} \Phi_8(321)b=& \langle \alpha_1,\alpha_2,\beta,\gamma:[\alpha_1,\alpha_2]=\beta=\alpha_1^p,~[\beta,\alpha_2]=\beta^p=\gamma^p,~\alpha_2^{p^2}=\beta^{p^2}=1\rangle, \\ \Phi_8(321)c_r=& \langle \alpha_1,\alpha_2,\beta:[\alpha_1,\alpha_2]=\beta,~[\beta,\alpha_2]^{r+1}=\beta^{p(r+1)}=\alpha_1^{p^2},~\alpha_2^{p^2}=\beta^{p^2}=1\rangle, \\ \Phi_8(321)c_{p-1}=& \langle \alpha_1,\alpha_2,\beta:[\alpha_1,\alpha_2]=\beta,~[\beta,\alpha_2]=\beta^p=\alpha_2^{p^2},~\alpha_1^{p^2}=\beta^{p^2}=1\rangle, \\ \Phi_8(222)=& \langle \alpha_1,\alpha_2,\beta:[\alpha_1,\alpha_2]=\beta,~[\beta,\alpha_2]=\beta^p,~\alpha_1^{p^2}=\alpha_2^{p^2}=\beta^{p^2}=1\rangle,\\ \Phi_{14}(321)=& \langle \alpha_1,\alpha_2,\beta:[\alpha_1,\alpha_2]=\beta,~\alpha_1^{p^2}=\beta^p,~\alpha_2^{p^2}=\beta^{p^2}=1\rangle, \\ \Phi_{14}(222)=& \langle \alpha_1,\alpha_2,\beta:[\alpha_1,\alpha_2]=\beta,~\alpha_1^{p^2}=\alpha_2^{p^2}=\beta^{p^2}=1\rangle. \end{align*}}
\emph{Case I.} $G=\Phi_8(321)b$. Denote by $H$ the abelian normal subgroup of $G$ generated by $\alpha_1$ and $\gamma$. Then $H=\langle\alpha_1,\gamma\beta^{-1}\rangle\simeq C_{p^3}\times C_p$ and $G/H=\langle\alpha_2\rangle\simeq C_{p^2}$.
Let $V$ be a $K$-vector space whose dual space $V^*$ is defined as $V^*=\bigoplus_{g\in G}K\cdot x(g)$ where $G$ acts on $V^*$ by $h\cdot x(g)=x(hg)$ for any $h,g\in G$. Thus $K(V)^{G}=K(x(g):g\in G)^{G}=K(G)$.
Define $X_1,X_2\in V^*$ by \begin{equation*} X_1=\sum_{i=0}^{p-1}x((\gamma\beta^{-1})^i),~ X_2=\sum_{i=0}^{p^3-1}x(\alpha_1^i). \end{equation*} Note that $\gamma\beta^{-1}\cdot X_1=X_1$ and $\alpha_1\cdot X_2=X_2$.
Let $\zeta_{p^3}\in K$ be a primitive $p^3$-th root of unity and put $\zeta=\zeta_{p^3}^{p^2}$, a primitive $p$-th root of unity. Define $Y_1,Y_2\in V^*$ by \begin{equation*} Y_1=\sum_{i=0}^{p^3-1}\zeta_{p^3}^{-i}\alpha_1^i\cdot X_1,~ Y_2=\sum_{i=0}^{p-1}\zeta^{-i}(\gamma\beta^{-1})^i\cdot X_2. \end{equation*}
It follows that {\allowdisplaybreaks \begin{align*} \alpha_1\ :\ &Y_1\mapsto\zeta_{p^3} Y_1,~Y_2\mapsto Y_2,\\ \gamma\beta^{-1}\ :\ &Y_1\mapsto Y_1,~Y_2\mapsto\zeta Y_2,\\ \gamma\ :\ &Y_1\mapsto\zeta_{p^2} Y_1,~Y_2\mapsto\zeta Y_2. \end{align*}} Thus $K\cdot Y_1+K\cdot Y_2$ is a representation space of the subgroup $H$.
Define $x_i=\alpha_2^i\cdot Y_1,y_i=\alpha_2^i\cdot Y_2$ for $0\leq i\leq p^2-1$. From the relations $\alpha_1\alpha_2^i=\alpha_2^i\alpha_1\beta^i\beta^{\binom{i}{2}p}$ it follows that {\allowdisplaybreaks \begin{align*} \alpha_1\ :\ &x_i\mapsto\zeta_{p^3}\zeta_{p^2}^i\zeta^{\binom{i}{2}} x_i,~ y_i\mapsto y_i\\ \gamma\ :\ &x_i\mapsto\zeta_{p^2} x_i,~ y_i\mapsto\zeta y_i,\\ \alpha_2\ :\ &x_0\mapsto x_1\mapsto\cdots\mapsto x_{p^2-1}\mapsto x_0,\\ &y_0\mapsto y_1\mapsto\cdots\mapsto y_{p^2-1}\mapsto y_0, \end{align*}} for $0\leq i\leq p^2-1$.
We find that $Y=(\bigoplus_{0\leq i\leq p^2-1}K\cdot x_i)\oplus(\bigoplus_{0\leq i\leq p^2-1}K\cdot y_i)$ is a faithful $G$-subspace of $V^*$. Thus, by Theorem \ref{t2.1}, it suffices to show that $K(x_i,y_i:0\leq i\leq p^2-1)^{G}$ is rational over $K$.
For $1\leq i\leq p^2-1$, define $u_i=x_i/x_{i-1}$ and $v_i=y_i/y_{i-1}$. Thus $K(x_i,y_i:0\leq i\leq p^2-1)=K(x_0,y_0,u_i,v_i:1\leq i\leq p^2-1)$ and for every $g\in G$ \begin{equation*} g\cdot x_0\in K(u_i,v_i:1\leq i\leq p^2-1)\cdot x_0,~ g\cdot y_0\in K(u_i,v_i:1\leq i\leq p^2-1)\cdot y_0, \end{equation*} while the subfield $K(u_i,v_i:1\leq i\leq p^2-1)$ is invariant by the action of $G$. Thus $K(x_i,y_i:0\leq i\leq p^2-1)^{G}=K(u_i,v_i:1\leq i\leq p^2-1)^{G}(u,v)$ for some $u,v$ such that $\alpha_1(v)=\gamma(v)=\alpha_2(v)=v$ and $\alpha_1(u)=\gamma(u)=\alpha_2(u)=u$. We have now {\allowdisplaybreaks \begin{align*} \alpha_1\ :\ &u_i\mapsto\zeta_{p^2}\zeta^{i-1} u_i,~ v_i\mapsto v_i,\\ \tag{5.1} \gamma\ :\ &u_i\mapsto u_i,~ v_i\mapsto v_i,\\ \alpha_2\ :\ &u_1\mapsto u_2\mapsto\cdots\mapsto u_{p^2-1}\mapsto (u_1u_2\cdots u_{p^2-1})^{-1},\\ &v_1\mapsto v_2\mapsto\cdots\mapsto v_{p^2-1}\mapsto (v_1v_2\cdots v_{p^2-1})^{-1}, \end{align*}} for $1\leq i\leq p^2-1$. From Theorem \ref{t2.2} it follows that if $K(u_i,v_i:1\leq i\leq p^2-1)^{G}(u,v)$ is rational over $K$, so is $K(x_i,y_i:0\leq i\leq p^2-1)^{G}$ over $K$.
Since $\gamma$ acts trivially on $K(u_i,v_i:1\leq i\leq p^2-1)$, we find that $K(u_i,v_i:1\leq i\leq p^2-1)^{G}=K(u_i,v_i:1\leq i\leq p^2-1)^{\langle\alpha_1,\alpha_2\rangle}$.
Now, consider the metacyclic $p$-group $\widetilde G=\langle\sigma,\tau:\sigma^{p^3}=\tau^{p^2}=1,\tau^{-1}\sigma\tau=\sigma^{k},k=1+p\rangle$.
Define $X=\sum_{0\leq j\leq p^3-1}\zeta_{p^3}^{-j}x(\sigma^j),V_i=\tau^i X$ for $0\leq i\leq p^2-1$. It follows that \begin{eqnarray*} \sigma&:&V_i\mapsto \zeta_{p^3}^{k^i}V_i,\\ \tau&:&V_0\mapsto V_1\mapsto\cdots\mapsto V_{p^2-1}\mapsto V_0. \end{eqnarray*} Note that $K(V_0,V_1,\dots,V_{p^2-1})^{\widetilde G}$ is rational by Theorem \ref{t2.6}.
Define $U_i=V_i/V_{i-1}$ for $1\leq i\leq p^2-1$. Then $K(V_0,V_1,\dots,V_{p^2-1})^{\widetilde G}=K(U_1,U_2,\dots,$ $U_{p^2-1})^{\widetilde G}(U)$ where \begin{eqnarray*} \sigma&:&U_i\mapsto \zeta_{p^3}^{k^i-k^{i-1}}U_i,~ U\mapsto U\\ \tau&:&U_1\mapsto U_2\mapsto\cdots\mapsto U_{p^2-1}\mapsto (U_1U_2\cdots U_{p^2-1})^{-1},~ U\mapsto U. \end{eqnarray*}
Notice that $k^i-k^{i-1}=(1+p)^{i-1}p\equiv (1+(i-1)p)p\pmod{p^3}$, so $\zeta_{p^3}^{k^i-k^{i-1}}=\zeta_{p^2}^{1+(i-1)p}$. Compare Formula (5.1) (i.e., the actions of $\alpha_1,\alpha_2$ on $K(u_i:1\leq i\leq p^2-1)$) with the actions of $\widetilde G$ on $K(U_i:1\leq i\leq p^2-1)$. They are the same. Hence, according to Theorem \ref{t2.6}, we get that $K(u_1,\dots,u_{p^2-1})^{G}(u)\cong K(U_1,\dots,U_{p^2-1})^{\widetilde G}(U)=K(V_0,V_1,\dots,V_{p^2-1})^{\widetilde G}$ is rational over $K$. Since by Lemma \ref{l2.7} we can linearize the action of $\alpha_2$ on $K(v_i:1\leq i\leq p^2-1)$, we obtain finally that $K(u_i,v_i:1\leq i\leq p^2-1)^{\langle\alpha_1,\alpha_2\rangle}$ is rational over $K$.
\emph{Case II.} $G=\Phi_8(321)c_r$. Denote by $H$ the abelian normal subgroup of $G$ generated by $\alpha_1$ and $\beta$. Then $H=\langle\alpha_1,\alpha_1^{-p}\beta^{r+1}\rangle\simeq C_{p^3}\times C_p$ and $G/H=\langle\alpha_2\rangle\simeq C_{p^2}$. Let $a=(r+1)^{-1}\in\mathbb Z_{p^2}$, hence $\beta=\alpha_1^{ap}(\alpha_1^{-p}\beta^{r+1})^a$. Similarly to Case I, we can define $Y_1,Y_2\in V^*$ such that {\allowdisplaybreaks \begin{align*} \alpha_1\ :\ &Y_1\mapsto\zeta_{p^3} Y_1,~Y_2\mapsto Y_2,\\ \alpha_1^{-p}\beta^{r+1}\ :\ &Y_1\mapsto Y_1,~Y_2\mapsto\zeta Y_2,\\ \beta\ :\ &Y_1\mapsto\zeta_{p^2}^a Y_1,~Y_2\mapsto\zeta^a Y_2. \end{align*}} Thus $K\cdot Y_1+K\cdot Y_2$ is a representation space of the subgroup $H$.
Define $x_i=\alpha_2^i\cdot Y_1,y_i=\alpha_2^i\cdot Y_2$ for $0\leq i\leq p^2-1$. From the relations $\alpha_1\alpha_2^i=\alpha_2^i\alpha_1\beta^i\beta^{\binom{i}{2}p}$ and $\beta\alpha_2^i=\alpha_2^i\beta^{1+ip}$ it follows that {\allowdisplaybreaks \begin{align*} \alpha_1\ :\ &x_i\mapsto\zeta_{p^3}\zeta_{p^2}^{ai}\zeta^{a\binom{i}{2}} x_i,~ y_i\mapsto\zeta^{ai} y_i\\ \beta\ :\ &x_i\mapsto\zeta_{p^2}^a\zeta^{ai} x_i,~ y_i\mapsto\zeta^a y_i,\\ \alpha_2\ :\ &x_0\mapsto x_1\mapsto\cdots\mapsto x_{p^2-1}\mapsto x_0,\\ &y_0\mapsto y_1\mapsto\cdots\mapsto y_{p^2-1}\mapsto y_0, \end{align*}} for $0\leq i\leq p^2-1$.
We find that $Y=(\bigoplus_{0\leq i\leq p^2-1}K\cdot x_i)\oplus(\bigoplus_{0\leq i\leq p^2-1}K\cdot y_i)$ is a faithful $G$-subspace of $V^*$. Thus, by Theorem \ref{t2.1}, it suffices to show that $K(x_i,y_i:0\leq i\leq p^2-1)^{G}$ is rational over $K$.
For $1\leq i\leq p^2-1$, define $u_i=x_i/x_{i-1}$ and $v_i=y_i/y_{i-1}$. We have now {\allowdisplaybreaks \begin{align*} \alpha_1\ :\ &u_i\mapsto\zeta_{p^2}^a\zeta^{a(i-1)} u_i,~ v_i\mapsto\zeta^a v_i,\\ \beta\ :\ &u_i\mapsto\zeta^a u_i,~ v_i\mapsto v_i,\\ \alpha_2\ :\ &u_1\mapsto u_2\mapsto\cdots\mapsto u_{p^2-1}\mapsto (u_1u_2\cdots u_{p^2-1})^{-1},\\ &v_1\mapsto v_2\mapsto\cdots\mapsto v_{p^2-1}\mapsto (v_1v_2\cdots v_{p^2-1})^{-1}, \end{align*}} for $1\leq i\leq p^2-1$. From Theorem \ref{t2.2} it follows that if $K(u_i,v_i:1\leq i\leq p^2-1)^{G}(u,v)$ is rational over $K$, so is $K(x_i,y_i:0\leq i\leq p^2-1)^{G}$ over $K$.
Since $\beta$ acts in the same way as $\alpha_1^p$ on $K(u_i,v_i:1\leq i\leq p^2-1)$, we find that $K(u_i,v_i:1\leq i\leq p^2-1)^{G}=K(u_i,v_i:1\leq i\leq p^2-1)^{\langle\alpha_1,\alpha_2\rangle}$.
For $1\leq i\leq p^2-1$ define $V_i=v_i/u_i^p$. It follows that {\allowdisplaybreaks \begin{align*} \alpha_1\ :\ &u_i\mapsto\zeta_{p^2}^a\zeta^{a(i-1)} u_i,~ V_i\mapsto V_i,\\ \tag{5.2} \alpha_2\ :\ &u_1\mapsto u_2\mapsto\cdots\mapsto u_{p^2-1}\mapsto (u_1u_2\cdots u_{p^2-1})^{-1},\\ &V_1\mapsto V_2\mapsto\cdots\mapsto V_{p^2-1}\mapsto (V_1V_2\cdots V_{p^2-1})^{-1}, \end{align*}} for $1\leq i\leq p^2-1$.
Compare Formula (5.2) with Formula (5.1). They look almost the same. Apply the proof of Case 1.
\emph{Case III.} $G=\Phi_8(321)c_{p-1}$. Denote by $H$ the abelian normal subgroup of $G$ generated by $\alpha_1$ and $\beta$. Then $H\simeq C_{p^2}\times C_{p^2}$ and $G/H\simeq C_{p^2}$. Similarly to Case I, we can define $Y_1,Y_2\in V^*$ such that {\allowdisplaybreaks \begin{align*} \alpha_1\ :\ &Y_1\mapsto\zeta_{p^2} Y_1,~Y_2\mapsto Y_2,\\ \beta\ :\ &Y_1\mapsto Y_1,~Y_2\mapsto\zeta_{p^2} Y_2. \end{align*}} Thus $K\cdot Y_1+K\cdot Y_2$ is a representation space of the subgroup $H$.
Define $x_i=\alpha_2^i\cdot Y_1,y_i=\alpha_2^i\cdot Y_2$ for $0\leq i\leq p^2-1$. From the relations $\alpha_1\alpha_2^i=\alpha_2^i\alpha_1\beta^i\beta^{\binom{i}{2}p}$ and $\beta\alpha_2^i=\alpha_2^i\beta^{1+ip}$ it follows that {\allowdisplaybreaks \begin{align*} \alpha_1\ :\ &x_i\mapsto\zeta_{p^2} x_i,~ y_i\mapsto\zeta_{p^2}^i\zeta^{\binom{i}{2}} y_i\\ \beta\ :\ &x_i\mapsto x_i,~ y_i\mapsto\zeta_{p^2}\zeta^i y_i,\\ \alpha_2\ :\ &x_0\mapsto x_1\mapsto\cdots\mapsto x_{p^2-1}\mapsto x_0,\\ &y_0\mapsto y_1\mapsto\cdots\mapsto y_{p^2-1}\mapsto\zeta y_0, \end{align*}} for $0\leq i\leq p^2-1$.
For $1\leq i\leq p^2-1$, define $u_i=x_i/x_{i-1}$ and $v_i=y_i/y_{i-1}$. We have now {\allowdisplaybreaks \begin{align*} \alpha_1\ :\ &u_i\mapsto u_i,~ v_i\mapsto\zeta_{p^2}\zeta^{i-1} v_i,\\ \beta\ :\ &u_i\mapsto u_i,~ v_i\mapsto\zeta v_i,\\ \alpha_2\ :\ &u_1\mapsto u_2\mapsto\cdots\mapsto u_{p^2-1}\mapsto (u_1u_2\cdots u_{p^2-1})^{-1},\\ &v_1\mapsto v_2\mapsto\cdots\mapsto v_{p^2-1}\mapsto\zeta (v_1v_2\cdots v_{p^2-1})^{-1}, \end{align*}} for $1\leq i\leq p^2-1$. Since $\beta$ acts in the same way as $\alpha_1^p$ on $K(u_i,v_i:1\leq i\leq p^2-1)$, we find that $K(u_i,v_i:1\leq i\leq p^2-1)^{G}=K(u_i,v_i:1\leq i\leq p^2-1)^{\langle\alpha_1,\alpha_2\rangle}$.
Let $\zeta_{p^3}\in K$ be a primitive $p^3$th root of unity such that $\zeta_{p^3}^{p^2}=\zeta$. For $1\leq i\leq p^2-1$ define $w_i=v_i/\zeta_{p^3}$. It follows that {\allowdisplaybreaks \begin{align*} \alpha_1\ :\ &u_i\mapsto u_i,~ w_i\mapsto\zeta_{p^2}\zeta^{i-1} w_i,\\ \tag{5.3} \alpha_2\ :\ &u_1\mapsto u_2\mapsto\cdots\mapsto u_{p^2-1}\mapsto (u_1u_2\cdots u_{p^2-1})^{-1},\\ &w_1\mapsto w_2\mapsto\cdots\mapsto w_{p^2-1}\mapsto (w_1w_2\cdots w_{p^2-1})^{-1}, \end{align*}} for $1\leq i\leq p^2-1$. Compare Formula (5.3) with Formula (5.1) (or (5.2)). They look almost the same. Apply the proof of Case 1.
\emph{Case IV.} $G=\Phi_8(222)$. Denote by $H$ the abelian normal subgroup of $G$ generated by $\alpha_1$ and $\beta$. Then $H\simeq C_{p^2}\times C_{p^2}$ and $G/H\simeq C_{p^2}$. The proof henceforth is almost the same as Case III.
\emph{Case V.} $G=\Phi_{14}(321)$. Denote by $H$ the abelian normal subgroup of $G$ generated by $\alpha_2$ and $\beta$. Then $H\simeq C_{p^2}\times C_{p^2}$ and $G/H\simeq C_{p^2}$.
As before, we can define $Y_1,Y_2\in V^*$ such that {\allowdisplaybreaks \begin{align*} \alpha_2\ :\ &Y_1\mapsto\zeta_{p^2} Y_1,~Y_2\mapsto Y_2,\\ \beta\ :\ &Y_1\mapsto Y_1,~Y_2\mapsto\zeta_{p^2} Y_2. \end{align*}} Thus $K\cdot Y_1+K\cdot Y_2$ is a representation space of the subgroup $H$.
Define $x_i=\alpha_1^i\cdot Y_1,y_i=\alpha_1^i\cdot Y_2$ for $0\leq i\leq p^2-1$. From the relations $\alpha_2\alpha_1^i=\alpha_1^i\alpha_2\beta^{-i}$ it follows that {\allowdisplaybreaks \begin{align*} \alpha_2\ :\ &x_i\mapsto\zeta_{p^2} x_i,~ y_i\mapsto\zeta_{p^2}^{-i} y_i\\ \beta\ :\ &x_i\mapsto x_i,~ y_i\mapsto\zeta_{p^2} y_i,\\ \alpha_1\ :\ &x_0\mapsto x_1\mapsto\cdots\mapsto x_{p^2-1}\mapsto x_0,\\ &y_0\mapsto y_1\mapsto\cdots\mapsto y_{p^2-1}\mapsto\zeta y_0, \end{align*}} for $0\leq i\leq p^2-1$.
For $1\leq i\leq p^2-1$, define $u_i=x_i/x_{i-1}$ and $v_i=y_i/y_{i-1}$. We have now {\allowdisplaybreaks \begin{align*} \alpha_2\ :\ &u_i\mapsto u_i,~ v_i\mapsto\zeta_{p^2}^{-1} v_i,\\ \beta\ :\ &u_i\mapsto u_i,~ v_i\mapsto v_i,\\ \alpha_1\ :\ &u_1\mapsto u_2\mapsto\cdots\mapsto u_{p^2-1}\mapsto (u_1u_2\cdots u_{p^2-1})^{-1},\\ &v_1\mapsto v_2\mapsto\cdots\mapsto v_{p^2-1}\mapsto\zeta (v_1v_2\cdots v_{p^2-1})^{-1}, \end{align*}} for $1\leq i\leq p^2-1$. Since $\beta$ acts trivially on $K(u_i,v_i:1\leq i\leq p^2-1)$, we find that $K(u_i,v_i:1\leq i\leq p^2-1)^{G}=K(u_i,v_i:1\leq i\leq p^2-1)^{\langle\alpha_1,\alpha_2\rangle}$.
Define $w_1=v_1^{p^2}\zeta^{-1},w_i=v_i/v_{i-1}$ for $2\leq i\leq p^2-1$. We have now $K(v_1,\dots,v_{p^2-1})^{\langle\alpha_2\rangle}=K(w_1,\dots,w_{p^2-1})$ and {\allowdisplaybreaks\begin{eqnarray*} \alpha_1&:&w_1\mapsto w_2^{p^2}w_1,w_2\mapsto w_3\mapsto\cdots\mapsto w_{p^2-1}\mapsto 1/(w_1w_2^{p^2-1}w_3^{p^2-2}\cdots w_{p^2-1}^2). \end{eqnarray*}}
Define $z_1=w_2,z_i=\alpha_1^{i-1}\cdot w_2$ for $2\leq i\leq p^2-1$. Then $K(w_i:1\leq i\leq p^2-1)=K(z_i:1\leq i\leq p^2-1)$ and \begin{align*} \alpha_1\ :\ &z_1\mapsto z_2\mapsto\cdots\mapsto z_{p^t-1}\mapsto (z_1z_2\cdots z_{p^2-1})^{-1}. \end{align*} The action of $\alpha_1$ can be linearized according to Lemma \ref{l2.7}. Thus $K(u_i,z_i:1\leq i\leq p^2-1)^{\langle\alpha_1\rangle}$ is rational over $K$ by Theorem \ref{t2.1}. We are done.
\emph{Case VI.} $G=\Phi_{14}(222)$. Denote by $H$ the abelian normal subgroup of $G$ generated by $\alpha_2$ and $\beta$. Then $H\simeq C_{p^2}\times C_{p^2}$ and $G/H\simeq C_{p^2}$. The proof henceforth is almost the same as Case V.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{Regression Models for Directional Data Based on Nonnegative Trigonometric Sums} \renewcommand{1}{1.00} \author{Fern\'andez-Dur\'an, Juan Jos\'e and Gregorio-Dom\'inguez, Mar\'ia Mercedes \\ ITAM\\ E-mail: [email protected]} \date{} \maketitle
\renewcommand{1}{1.75}
\begin{abstract} The parameter space of nonnegative trigonometric sums (NNTS) models for circular data is the surface of a hypersphere; thus, constructing regression models for a cir\-cular-dependent variable using NNTS models can comprise fitting great (small) circles on the parameter hypersphere that can identify different regions (rotations) along the great (small) circle. We propose regression models for circular- (angular-) dependent random variables in which the original circular random variable, which is assumed to be distributed (marginally) as an NNTS model, is transformed into a linear random variable such that common methods for linear regres\-sion can be applied. The usefulness of NNTS models with skewness and multimodality is shown in examples with simulated and real data. \end{abstract}
\textbf{Keywords}: Circular-dependent variable, Density forecasting, Geodesics, Hypersphere, Multimodality
\section{Introduction}
\addtolength{\textheight}{.5in}
The need to analyze circular variables (angles, orientations, or directions) arises in various disciplines, such as the direc\-tions taken by animals in biology, the conformational angles in a molecule in chemistry (e.g., a protein), the characteriza\-tion of wind patterns in terms of wind direction and speed in environmetrics, the time of onset of a certain disease or times of death of individuals in a certain population in medi\-cal sciences, the position of events that occur on the surface of the Earth (e.g., the epicenter of an earthquake) or the study of paleomagnetism in geology, and the orientation of planets and other celestial bodies in relation to the Earth at a certain time in astronomy. In all of these examples, when dealing with circular variables, it is necessary to consider the periodicity of their density functions. The same constraint applies when constructing regression models for a circular response and a set of explanatory variables of different types (e.g., circular, linear, and categorical). Early developments in circular-circular regression (i.e., a circular response and a single circular explanatory variable) and circular-linear re\-gres\-sion (i.e., a circular response and only a linear explanato\-ry variable) and their corresponding bivariate correlation co\-ef\-fi\-cients have been reviewed by Upton and Fingleton (1989), Fisher (1993), Mardia and Jupp (2000), and Jammalamadaka and SenGupta (2001). In addition, Gould (1969) developed regression models for a circular response considering a von Mises density, where its mean was modeled as a linear func\-tion of explanatory variables not satisfying the periodicity constraint for a circular mean. Fisher and Lee (1992) and Fisher (1993) considered regression models for a circular response variable with a von Mises distribution while model\-ing its mean using a tangent link function of a linear function of the explanatory variables satisfying the periodicity con\-straint. The concentration parameter of the von Mises dis\-tri\-bu\-tion can be modeled as an exponential function of other linear functions of the explanatory variables. However, Presnell et al. (1998) criticized the existing regression models based on circular response because they suffered from the non-identifiability of the parameters and complex and mul\-ti\-modal likelihoods, which rendered determining their global maxima challenging. Thus, Presnell et al. (1998) proposed a spherically projected multivariate linear model to avoid these issues. Consequently, models based on Presnell et al. (1998) were proposed, such as, those of N\'{u}\~{n}ez-Antonio et al. (2011), and Wang and Gelfand (2013). In addition, Lund (1999) criticized the use of least-squares regression in cases where the model response was circular and proposed the use of the least circular distance instead. Downs and Mardia (2002) constructed circular-circular regression models wherein the regression curve was a M\"obius circular trans\-for\-ma\-tion with errors following a von Mises distribution. Kato et al. (2008) considered a M\"obius transformation with errors following a wrapped Cauchy distribution. SenGupta et al. (2013) developed models for inverse circular-circular regression, where the value of the unobserved ex\-plana\-to\-ry circular variable was predicted by the observed circular re\-sponse. Polsen and Taylor (2015) proposed a common framework for circular-circular regression based on a tangent link function. Further, while Rivest et al. (2016) developed gen\-er\-al circular regression models for animal movements, Kim and SenGupta (2017) recently constructed models for mul\-ti\-vari\-ate-multiple circular regression. Breckling (1989), Fisher (1993), and Pfaff et al. (2016) analyzed the time series of a circular variable. Fisher (1993) proposed the use of a link function (i.e., the inverse of the tangent function) for constructing circular time-series models. Holzmann et al. (2016) considered the analysis of circular and circular-linear time series. Further, details regarding spatial and spa\-tiotem\-po\-ral models for circular variables can be found in Fisher (1993), Jona-Lasinio et al. (2012), Wang and Gelfand (2014), Wang et al. (2015), Mastrantonio et al. (2016), and other cor\-re\-spond\-ing references therein.
Fern\'andez-Dur\'an (2004) constructed density functions for circular random variables based on the use of nonnegative trigonometric sums (NNTS) models that allow modeling of skewed and multimodal data. The CircNNTSR R package (R Development Core Team, 2012) contains routines to fit NNTS models, which was explained by Fern\'andez-Dur\'an and Gregorio-Dom\'inguez (2016). The main objective of this study is to consider the construction of regression models, in which the dependent circular random variable is considered to be distributed as an NNTS model.
The remainder of this paper is organized as follows. Sec\-tion 2 explains how to transform the dependent NNTS cir\-cu\-lar random variable to a unique linear random variable such that commonly used linear regression models can be fit to the transformed linear variable. Section 3 presents examples of the proposed methodology on real datasets, including re\-gres\-sion and time-series models. In addition, a simulation study is presented. Finally, Section 4 con\-cludes the paper.
\section{Transforming NNTS Distributed Circular Data to Linear Data}
Consider a sample of random vectors $(\Theta_k, X_{k1}, X_{k2}, \ldots, X_{kp})$ for $k = 1, 2, \ldots \\, n$, where $\Theta_k$ is a circular random variable, and $X_{k1}, X_{k2}, \ldots, X_{kp}$ is a set of explanatory variables for the $k$-th individual in the sample. The explanatory variables could be of any type, qualitative or quantitative, and in any measurement scale.
We assume that the circular random variables $\Theta_1, \Theta_2, \ldots, \Theta_n$ are distributed as NNTS distributions. Then, the density function for $\Theta_k$ is defined as follows (Fern\'andez-Dur\'an 2004): \begin{equation} f(\theta)=\frac{1}{2\pi}\sum_{k=0}^M\sum_{m=0}^M c^*_k\bar{c}^*_me^{i(k-m)\theta}=\frac{1}{2\pi} \bm{c}^{*H} \bm{e}^* \bm{e}^{*H} \bm{c}^* \label{NNTS1} \end{equation} where $\bm{c}^*=(c^*_0, c^*_1, \ldots, c^*_M)^\top$ is a vector of complex numbers, that correspond to the vector of parameters of the model and, $\bm{c}^{*H}$ is the Hermitian transpose, satisfying the following constraint: \begin{equation}
\sum_{k=0}^M ||c^*_k||^2 = 1. \end{equation} This constraint specifies that the parameter space is a complex hy\-per\-sphere of dimension $M$, $S_{\mathbb{C}}(M)$ in $\mathbb{C}^{M+1}$. Given this constraint, $c_0$ is specified as a nonnegative real number.
The vector $\bm{e}^*$ is the complex vector of trigonometric terms defined as $\bm{e}^* = (1, e^{-i\theta}, e^{-2i\theta}, \ldots, \\ e^{-Mi\theta})^\top$ with $i=\sqrt{-1}$ and $e^{-ki\theta}=\cos(k\theta) - i\sin(k\theta)$ for $k=0, 1, \ldots, M$. The NNTS density $f(\theta)$ is the squared modulus of the product $\bm{c}^{*H}\bm{e}^*$, implying that an angle $\theta$ for which the derived vector of trigonometric terms, $\bm{e}^*$, obtains a larger modulus for $\bm{c}^{*H}\bm{e}^*$, then obtains a larger probability. Thus, the parameter vector $\bm{c}^*$ can be interpreted as a multidimensional location parameter, where the argument (angle) of each of its components defines directions that have a larger probability if the modulus of the corresponding component of the parameter vector is larger than those of the other components. Further, the sum of the moduli of the components of the parameter vector is restricted to one. Parameter $M$ acts as a scale and shape parameter that determines the maximum number of modes of the NNTS and when it increases the NNTS density becomes less smooth and more concentrated around certain angles depending on the arguments and moduli of the elements of the complex parameter vector $\bm{c^*}$. To operate in a real hypersphere of dimension $2M$, $S(2M)$, the complex vectors $\bm{c}^*$ and $\bm{e}^*$ are transformed into real vectors $\bm{c}$ and $\bm{e}$ as $\bm{c}=(Re(\bm{c}^*),Im(\bm{c}^*))^\top$ and $\bm{e}=(Re(\bm{e}^*),Im(\bm{e}^*))^\top$, where $Re(\bm{c}^*)$ and $Im(\bm{c}^*)$ are the real and imaginary parts of the elements of the complex vector $\bm{c}^*$. For vector $\bm{e}^*$, $\bm{e}=(1, \cos(\theta), \cos(2\theta), \ldots, \cos(M\theta),-\sin(\theta), -\sin(2\theta), \ldots, -\sin(M\theta))^\top.$ \\ The NNTS density is a special case of an orthogonal series density estimator (see the review by Izenman, 1999), with the particular advantage that the parameter space is a hypersphere, and its geometry can be exploited by considering great (small) circles as regression curves. The great circles correspond to the geodesics of the hypersphere. Rivest (1999) examined the fit of small circles in a three-dimensional sphere by considering the least squares estimation.
The proposed methodology comprises three steps: \begin{enumerate} \item Expanding the observed values of the circular dependent variable, $\theta_k$ for $k=1, \ldots, n$, into the parameter hy\-per\-sphere, $S(2M)$, using transformation ${\bf \hat{e}}_k \propto (1,e^{-i\theta_k},e^{-2i\theta_k}, \ldots,e^{-Mi\theta_k})^\top$. This transformation in\-cludes the first $M$ trigonometric moments of (minus) $\theta$. The ${\bf \hat{e}}_k$ vectors are normalized to have unit norm. \item Identifying the great (small) circle that is the best fit for the ${\bf \hat{e}}_k$ vectors in the parameter hypersphere, $S(2M)$, and transforming the original circular data, $\theta_k$, to linear data, $Y_k$, by considering the position of ${\bf \hat{e}}_k$ relative to the great (small) circle. \item Fitting a regression model employing the ordinary least squares and an identity link using a dependent variable as the transformed linear variable $Y_k$ to identify significant explanatory variables related to different segments (ro\-ta\-tions) along the fitted great (small) circle. \end{enumerate}
As reported by Li and Duan (1989), as the link function of the regression model is unknown and the identity link is being used, the intercept term of the regression model is unidentifiable while the other coefficients of the regression are identifiable up to a multiplicative constant. This problem is mitigated in the case of the proposed methodology when obtaining the NNTS predictive densities by considering the intercept term equal to zero owing to the periodic nature of the great (small) circle regression curves on the hypersphere.
The transformation of the original circular data is equiv\-a\-lent to projecting the circular variables into a great (small) circle in the parameter space $S(2M)$, which is similar to kernel projection methods used in statistical learning, such as support vector machines (Hastie et al. 2009).
\subsection{Great-circle (Geodesic) Regression on Unit Hypersphere}
The shortest distance between two points on the unit hy\-per\-sphere $S(2M)$ corresponds to the length of a segment of a great circle (also known as geodesic), which is defined as follows: \begin{equation} \bm{\epsilon} = \bm{a}\cos(\phi) + \bm{d}\sin(\phi) \label{geodesic} \end{equation} where $\bm{\epsilon}$ is a point on the geodesic defined by the orthogonal unit vectors $\bm{a}$ and $\bm{d}$ in $S(2M)$, and $\phi$ is an angle taking values in the interval $[-\pi,\pi)$. Note that the geodesic equation is the same for hyperspheres of any dimension. The definition of NNTS density in Equation (\ref{NNTS1}) implies that $\phi \in [- \frac{\pi}{2},\frac{\pi}{2})$ because the density function is the same for $\bm{c}$ and $-\bm{c}$.
For each single circular (angular) observation, $\theta_k$, $k=1, \ldots, n$, we construct the complex vector of trigonometric moments defined as $\bm{\hat{e}}^*_k \propto \left(1,e^{-i\theta_k},e^{-2i\theta_k}, \ldots, e^{-Mi\theta_k}\right)^\top$, where each vec\-tor is normalized to have unit norm. After normalization and considering the vector of real and imaginary parts, $\bm{\hat{e}}$, these real vectors are in the parameter space of the NNTS densities, that is, the unit hypersphere of dimension $2M$ $S(2M)$ because the complex hypersphere $S_{\mathbb{C}}(M)$ is isomorphic with the real $2M+1$ unit hypersphere.
To transform the original dependent circular variables into linear variables, we consider the distance of the $\bm{\hat{e}}_k$ vectors to a geodesic with known $\bm{a}$ and $\bm{d}$ vectors. The cosine between $\bm{\hat{e}}_{k}$ and the geodesic is given as follows: \begin{equation}
\begin{split} \bm{\hat{e}}_{k}^\top\bm{\epsilon}_k & = \bm{\hat{e}}_{k}^\top\left\{ \bm{a}\cos(\phi_k) + \bm{d}\sin(\phi_k)\right\} \\ &= (\bm{\hat{e}}_{k}^\top\bm{a})\cos(\phi_k) + (\bm{\hat{e}}_{k}^\top\bm{d})\sin(\phi_k) \end{split} \end{equation}
which is maximized for \begin{equation} \hat{\phi}_k = \arctan \left(\frac{\bm{\hat{e}}_{k}^\top\bm{d}}{\bm{\hat{e}}_{k}^\top\bm{a}} \right) \end{equation} for $k=1,2, \ldots, n$. Note that maximizing the cosine of the angle between two vectors is equivalent to minimizing the angle between those vectors. The nearest vector on the geodesic to $\bm{\hat{e}}_k$, denoted by $\bm{\epsilon}_k$, is given by \begin{equation} \bm{\epsilon}_k= \frac{\bm{\hat{e}}_{k}^\top\bm{a}}{\sqrt{(\bm{\hat{e}}_{k}^\top\bm{a})^2+(\bm{\hat{e}}_{k}^\top\bm{d})^2}} \bm{a} + \frac{\bm{\hat{e}}_{k}^\top\bm{d}}{\sqrt{(\bm{\hat{e}}_{k}^\top\bm{a})^2+(\bm{\hat{e}}_{k}^\top\bm{d})^2}} \bm{d} = w_{\bm{a}}\bm{a}+w_{\bm{d}}\bm{d}. \end{equation}
Then, $\bm{\epsilon}_k$ is a weighted sum of $\bm{a}$ and $\bm{d}$ wherein the weight for vector $\bm{a}$, $w_{\bm{a}}$, increases with the cosine of the angle between $\bm{\hat{e}}_{k}$ and $\bm{a}$, ($\bm{\hat{e}}_{k}^\top\bm{a}$). Here, $w_{\bm{a}}^2 + w_{\bm{d}}^2=1$, as implied by equation (\ref{geodesic}).
To complete the transformation of the circular variables into linear variables, it is necessary to obtain the values of $\bm{a}$ and $\bm{d}$. We consider the values of $\bm{a}$ and $\bm{d}$ that maximize the sum of squared cosines of the angles between the vector of parameter estimates and the corresponding vector on the geodesic $SSC(\bm{a},\bm{d})$ defined as follows:
\begin{equation}
\begin{split} SSC(\bm{a},\bm{d}) & = \sum_{k=1}^n \left(\bm{\hat{e}}_{k}^\top\bm{\epsilon}_k \right)^2
= \sum_{k=1}^n \left\{(\bm{\hat{e}}_{k}^\top\bm{a})^2 + (\bm{\hat{e}}_{k}^\top\bm{d})^2 \right\} \\ &= \bm{a}^\top \left(\sum_{k=1}^n \bm{\hat{e}}_k\bm{\hat{e}}_{k}^\top\right)\bm{a} + \bm{d}^\top\left(\sum_{k=1}^n \bm{\hat{e}}_k\bm{\hat{e}}_{k}^\top\right)\bm{d} \\ & = \bm{a}^\top\hat{E}\bm{a} + \bm{d}^\top\hat{E}\bm{d}
\end{split} \end{equation} subject to constraints $\bm{a}^\top\bm{a} = 1$, $\bm{d}^\top\bm{d} = 1$, and $\bm{a}^\top\bm{d} = 0$ de\-fined in Equation (\ref{geodesic}). Here the estimates of $\bm{a}$ and $\bm{d}$, $\bm{\hat{a}}$ and $\bm{\hat{d}}$, correspond to the first and second unit norm eigenvectors of the $2M+1$ by $2M+1$ matrix $\hat{\mathit{E}}=\sum_{k=1}^n \bm{\hat{e}}_k\bm{\hat{e}}_{k}^\top$. In other words, we find the great circle that best represents the points in the parameter hypersphere that correspond to the $n$ vectors containing the first $M$ trigonometric moments, $\bm{\hat{e}}_1, \ldots, \bm{\hat{e}}_n$. This is equivalent to maximizing the exponential of an orthogonal series density estimator (the NNTS model) as per Clutton-Brock (1990). Moreover, to maximize $SSC$ with respect to the vectors defining the great circle, $\bm{a}$ and $\bm{d}$ and the angles $\phi_k$ to obtain the definition of the linear variable $Y_k$ is consistent with the definition of the NNTS density in Equation (\ref{NNTS1}) as a squared norm of the product of two complex vectors.
Finally, by maximizing $SSC(\hat{\bm{a}},\hat{\bm{d}})$ with respect to the rotation angle $\phi_k$, a new linear variable $Y_k$ is defined as follows: \begin{equation} Y_k = \tan\left(\arctan\left(\frac{\bm{\hat{e}}_{k}^\top\hat{\bm{d}}}{\bm{\hat{e}}_{k}^\top\hat{\bm{a}}}\right)\right) = \frac{\bm{\hat{e}}_{k}^\top\hat{\bm{d}}}{\bm{\hat{e}}_{k}^\top\hat{\bm{a}}} \end{equation} for $k = 1, 2, \ldots, n$. For example, in a linear regression case, the estimator of vector $\bm{\beta}=(\beta_1, \ldots, \beta_p)^\top$, $\hat{\bm{\beta}}$, is obtained as $\bm{\hat{\beta}}=(X^{\top} X)^{-1}X^{\top}\bm{Y}$ where $\bm{Y}=(Y_1, Y_2, \ldots, Y_n)^\top$, and $\mathit{X}$ is the matrix of explanatory variables. The regression model for $\bm{Y}$ is expressed as follows: \begin{equation} \bm{Y} = X\bm{\beta} + \bm{e} \end{equation} where $\bm{e}$ is a vector of random errors. Here, an intercept term ($\beta_0$) was not included in the model because of the unknown link function (see Li and Duan, 1989).
Once the vector of estimates $\bm{\hat{\beta}}$ is obtained using linear regression, the prediction equation for the $\bm{c}$ parameter vector as a function of the given vector of covariates $\bm{x}$ is given as follows: \begin{equation} \bm{\hat{c}} = \bm{\hat{a}}\cos(\hat{\phi}) + \bm{\hat{d}}\sin(\hat{\phi}) \label{predictc} \end{equation} with the estimated rotation angle on the geodesic obtained as: \begin{equation} \hat{\phi} = \arctan(\bm{x}^\top\bm{\hat{\beta}}). \end{equation} In this sense, $\hat{\phi}$ corresponds to the inverse of the median of the fitted linear model. Thus, a density forecast is obtained that corresponds to an NNTS density with vector of pa\-ram\-e\-ters $\bm{\hat{c}}=(Re(\bm{c^*}),Im(\bm{c^*}))^\top$ given in Equation (\ref{predictc}). If a point prediction of the angle is required, the angle defined by the estimated value of the argument of the complex number $e^{i\theta}$ under the NNTS density forecast defined as \begin{equation} \hat{E}(e^{i\theta}) = \hat{E}\left\{\cos(\theta) + i\sin(\theta)\right\} = 2\pi\sum_{k=0}^{M-1}\hat{c}^*_{k}\bar{\hat{c}}^*_{k+1} \end{equation} can be used.
\subsection{Small-circle Regression on Unit Hypersphere}
A small circle in the unit 2$M$ hypersphere $S(2M)$ is defined as follows: \begin{equation} \bm{\epsilon} = \cos(\alpha)\bm{b} + \sin(\alpha)\left[\bm{a}\cos(\phi) + \bm{d}\sin(\phi)\right] \end{equation} with $\bm{b}$, $\bm{a}$, and $\bm{d}$ orthogonal unit vectors. Similar to the great circle case, given the vectors of the first $M$ trigonometric moments corresponding to each of the circular observations, $\theta_1, \ldots, \theta_n$, represented as $\bm{\hat{e}}_1, \ldots, \bm{\hat{e}}_n$, the vector on the small circle that is nearest to the vector $\bm{\hat{e}}_k$ can be obtained as follows: \begin{equation}
\begin{split} \bm{\epsilon}_k & = \left\{ \frac{\bm{\hat{e}}_{k}^\top\bm{b}}{\sqrt{(\bm{\hat{e}}_{k}^\top\bm{b})^2+(\bm{\hat{e}}_{k}^\top\bm{a})^2+(\bm{\hat{e}}_{k}^\top\bm{d})^2}} \right\} \bm{b} \\ & + \left\{ \frac{\sqrt{(\bm{\hat{e}}_{k}^\top\bm{a})^2+(\bm{\hat{e}}_{k}^\top\bm{d})^2}}{\sqrt{(\bm{\hat{e}}_{k}^\top\bm{b})^2+(\bm{\hat{e}}_{k}^\top\bm{a})^2+(\bm{\hat{e}}_{k}^\top\bm{d})^2}} \right\}\\ & \times \left\{ \frac{\bm{\hat{e}}_{k}^\top\bm{a}}{\sqrt{(\bm{\hat{e}}_{k}^\top\bm{a})^2+(\bm{\hat{e}}_{k}^\top\bm{d})^2}}\bm{a} + \frac{\bm{\hat{e}}_{k}^\top\bm{d}}{\sqrt{(\bm{\hat{e}}_{k}^\top\bm{a})^2+(\bm{\hat{e}}_{k}^\top\bm{d})^2}}\bm{d} \right\}
\end{split} \end{equation}
Similar to the great circle case, $\bm{\epsilon}_k$ is a weighted sum of $\bm{b}$, $\bm{a}$ and $\bm{d}$, $\bm{\epsilon}_k=w_{\bm{b}}\bm{b}+w_{\bm{a}}\bm{a}+w_{\bm{d}}\bm{d}$ with $w_{\bm{b}}^2+w_{\bm{a}}^2+w_{\bm{d}}^2=1$. Further, each weight increases with the cosine of the vector $\bm{\hat{e}}_{k}$ with the corresponding vector defining the small circle.
The estimates of the vectors $\bm{b}$, $\bm{a}$, and $\bm{d}$ are obtained by maximizing the sum of squared cosines $SSC(\bm{b}, \bm{a}, \bm{d})$ defined as follows: \begin{equation}
\begin{split} SSC(\bm{b}, \bm{a}, \bm{d}) & = \sum_{k=1}^n (\bm{\hat{e}}_{k}^\top\bm {\epsilon}_k)^2 = \sum_{k=1}^n \left\{(\bm{\hat{e}}_{k}^\top\bm{b})^2 + (\bm{\hat{e}}_{k}^\top\bm{a})^2 + (\bm{\hat{e}}_{k}^\top\bm{d})^2 \right\} \\ & = \bm{b}^\top\hat{E}\bm{b} + \bm{a}^\top\hat{E}\bm{a} + \bm{d}^\top\hat{E}\bm{d}
\end{split} \end{equation} The estimators of $\bm{b}$, $\bm{a}$, and $\bm{d}$ correspond to the first, second, and third unit-norm eigenvectors of $\hat{E}=\sum_{k=1}^n \bm{\hat{e}}_k\bm{\hat{e}}_{k}^\top$.
The estimator of the angle $\alpha$, $\hat{\alpha}$, is obtained by max\-i\-miz\-ing the sum of the squared cosines, $SSC(\hat{\bm{b}}, \hat{\bm{a}}, \hat{\bm{d}})$ and is equal to: \begin{equation} \hat{\alpha}=\frac{1}{2}\arctan \left(\frac{2\sum_{k=1}^n \sign(\bm{\hat{e}}_{k}^\top\hat{\bm{b}}) \sqrt{(\bm{\hat{e}}_{k}^\top\bm{a})^2+(\bm{\hat{e}}_{k}^\top\bm{d})^2}}{
\sum_{k=1}^n \left\{ (\bm{\hat{e}}_{k}^\top\bm{b})^2-(\bm{\hat{e}}_{k}^\top\bm{a})^2-(\bm{\hat{e}}_{k}^\top\bm{d})^2 \right\}}\right). \end{equation} Analogous to the great-circle case, the new linear dependent variable $Y_k$ corresponding to $\theta_k$ is obtained by maximizing $SSC(\hat{\bm{b}}, \hat{\bm{a}},\hat{\bm{d}})$ with respect to angle $\phi_k$ as \begin{equation} Y_k = \tan\left(\arctan\left(\frac{\bm{\hat{e}}_{k}^\top\hat{\bm{d}}}{\bm{\hat{e}}_{k}^\top\hat{\bm{a}}}\right)\right) = \frac{\bm{\hat{e}}_{k}^\top\hat{\bm{d}}}{\bm{\hat{e}}_{k}^\top\hat{\bm{a}}} \end{equation} because a small circle is the translation of a great circle. Thus, to discriminate between fitted great and small circle models, the relation between the estimated $\alpha$ and $\phi$ angles can be evaluated. The great circle is nested in a small circle. For example, for a $\phi$ angle equal to zero, the small circle transforms to a great circle defined by the vectors $\bm{b}_0$ and $\bm{c}_0$. Another possible situation where the small circle transforms to a great circle is when $\alpha=\frac{\pi}{2}$ or $\phi=\frac{\pi}{2}$. For a particular vector of covariates $\bm{x}$, a density forecast is obtained using the NNTS density with the $\bm{\hat{c}}$ vector of parameters expressed as \begin{equation} \bm{\hat{c}} = \cos(\hat{\alpha})\bm{\hat{b}} + \sin(\hat{\alpha})\{\bm{\hat{a}}\cos(\hat{\phi}) + \bm{\hat{d}}\sin(\hat{\phi})\} \label{smallcircleeqangle} \end{equation} For the estimation of the geodesic rotation angle, the fol\-low\-ing two cases must be considered (Presnell et al. 1998): \begin{equation} \hat{\phi} = \arctan(\bm{x}^\top\bm{\hat{\beta}}) \mbox{ if } \bm{\hat{e}}_{k}^\top\hat{\bm{a}} > 0 \end{equation} and \begin{equation} \hat{\phi} = \left\{\arctan(\bm{x}^\top\bm{\hat{\beta}}) + \pi\right\} \bmod 2\pi \mbox{ if } \bm{\hat{e}}_{k}^\top\hat{\bm{a}} \le 0 \end{equation} because, contrary to the great-circle case, $\phi$ and $\phi + \pi$ do not generate the same NNTS density. Thus, in the small-circle case, two models are obtained. The first one for $\bm{\hat{e}}_{k}^\top\hat{\bm{a}}<0$ and the second one for $\bm{\hat{e}}_{k}^\top\hat{\bm{a}} \ge 0$ that produce different NNTS density forecasts. In case of needing a unique NNTS density forecast, the resultant vector of the two $\bm{\hat{c}}$ parameter vectors for the cases $\bm{\hat{e}}_{k}^\top\hat{\bm{a}} > 0$ and $\bm{\hat{e}}_{k}^\top\hat{\bm{a}} \le 0$ can be used.
\subsection{Goodness of Fit and Probability Integral Transform Validation}
The goodness of fit of the proposed model comprises two elements. The first measures the goodness of fit of the projection of the original circular dependent variables onto the pa\-ram\-e\-ter hypersphere in terms of the cosine distances of the real vectors with the first $M$ trigonometric moments $\bm{\hat{e}}_1, \bm{\hat{e}}_2, \ldots, \bm{\hat{e}}_n$ to the great (small) circle. It is possible to define this measure in terms of the sum of the squared cosines of the vectors to their nearest points on the great (small) circle \begin{equation} R^2_{cos} =\frac{SSC}{n} \end{equation} because the maximum value of SSC is equal to $n$ (i.e., the number of observations) when all trigonometric moments vectors are on the great (small) circle. Then, the separation angles are equal to zero, and the cosine is equal to one. Note that $0 \le R^2_{cos} \le 1$ and a greater $R^2_{cos}$ value indicate that the great (small) circle better represents vectors $\bm{\hat{e}}_1, \bm{\hat{e}}_2, \ldots, \bm{\hat{e}}_n$.
The second element is related to the goodness of fit of the regression model fitted to the transformed variable $Y$. Here, for example, the coefficient of determination $R^2$ (squared correlation between observed and fitted $Y$ values) can be used when employing linear regression.
We focus on fitting NNTS densities; thus, validation of the models should be performed by considering techniques related to density forecasting. Here, we use the probability integral transform (PIT) to validate the fitted models. Fol\-low\-ing Diebold et al. (1998), we consider uniformity tests for the PITs of the fitted models evaluated at the observed values. To perform the PIT calculations, we must consider that the model used to determine the (geodesic) rotation angle on the great (small) circle uses the transformed variable $Y$, which is defined as follows: \[ Y=\frac{\bm{\hat{e}}^\top \bm{d}}{\bm{\hat{e}}^\top \bm{a}} \] and that the estimated value of the rotation angle, $\phi$, obtained as \[ \hat{\phi}=\arctan(\bm{x}^\top\hat{\bm{\beta}}), \] is equivalent to considering the median of the inverse transformation from the linear fitted model. We also considered other inverse transformations, such as the inverse of the expected value considering a second-order Taylor series; however, the best results were obtained using the inverse of the median.
We proceed to find the $\bm{c}$ parameter vector that cor\-re\-sponds to the rotation angle using Equation (\ref{predictc}) (Equation (\ref{smallcircleeqangle})) defining the great (small) circle, and calculate the prob\-a\-bil\-i\-ty integral in the corresponding observed value.
By applying this procedure to the $n$ observed values, $\theta_1, \theta_2, \ldots, \theta_n$, we obtain the vector of PIT values that, if the model is a good approximation of the process that generated the data, should be $i.i.d.$ observations from a circular uniform density function.
To evaluate the uniformity of the PIT values, the circular uniformity omnibus tests of Kuiper (1960), Watson (1961) and, the range test were used, as suggested by Mardia and Jupp (2000) to test the goodness of fit for circular data. The p-values of these tests were obtained using the $R$ $circular$ package (Agostinelli and Lund 2017).
When selecting the best value of $M$ in terms of the uniformity tests of the PIT values, the p-values of the range test can be adjusted for multiple hypothesis testing by using the procedure of Benjamini and Hochberg (1995).
\section{Examples}
\subsection{Simulated Data}
A design matrix of explanatory variables with 1000 rows and five columns was simulated. The entries in the first column ($X_1$) corresponded to realizations of a binary variable centered at zero, with values of -1 and 1. The second column ($X_2$) contained realizations from a discrete variable with values ranging from 1 to 40. Finally, the last three columns ($X_3$, $X_4$, and $X_5$) included realizations from normally distributed random variables with variance equal to one and means of 4, 6, and 8, respectively. In addition, the last four variables ($X_2$, $X_3$, $X_4$, and $X_5$) were standardized to have a mean of zero and unit variance. Moreover, for the values of the beta regression parameters, four different vectors of beta parameters were considered. Considering the problem of the unknown link function (Li and Duan, 1989), the beta vectors do not include the intercept term $\beta_0$. The first vector is equal to $\bm{\beta}_1=(0,0,0,0,0)^\top$, where all beta regression parameters are equal to zero; the second to $\bm{\beta}_2=(0.3,0.2,0.15,0.2,0.3)^\top$; the third to $\bm{\beta}_3=(0,0,0,0,0.3)^\top$; and the fourth to $\bm{\beta}_4=(x,x,x,x,0.3)^\top$, where $x$ implies that the variable associated with the corresponding position of the vector was not included in the simulated dataset; thus, for the $\bm{\beta}_4$ case, only the column $X_5$ was included in the design matrix.
For each value $M= 1, 2, 3, \ldots, 8$ sample sizes, $n$, equal to 25, 50, 100, 200, 500, and 1000 were considered. Further, for each possible combination of $M$ and $n$, the vectors defining the great (small) circle, $\bm{a}$, and $\bm{d}$ ($\bm{b}$, $\bm{a}$, and $\bm{d}$) were simulated by simulating 5000 realizations from an NNTS density with $M$ components and a randomly selected $\bm{c}$ parameter. Consequently, the matrix of trigonometric moments was constructed to obtain the first two (three) eigenvectors that corresponded to the simulated vectors defining the great (small) circle. Further, for a sample size of $n$, the first $n$ rows of the design matrix are used. In case of the $k$-th row of the simulated design matrix, in the great (small) circle case a realization from an NNTS density with $\bm{c}=\bm{a}\cos(\phi_k) + \bm{d}\sin(\phi_k)$ ($\bm{c}=\cos(\alpha)\bm{b}+ \sin(\alpha)(\bm{a}\cos(\phi_k)+\bm{d}\sin(\phi_k))$) vector of parameters was simulated to obtain the vector of values of the circular dependent variable. This, coupled with the first $n$ rows of the simulated design matrix corresponded to the simulated dataset.
For each of the four vectors of beta parameters and possible combination of $M$ and $n$, 100 simulated datasets were generated. Subsequently, the proposed estimation methodology was applied to each simulated dataset to obtain the estimates of the beta regression parameters considering two cases. One wherein the simulated vectors defining the great (small) circle are not estimated and considered as known, and the second wherein these are estimated with the beta regression parameters. In Tables \ref{Table04simulationGC} and \ref{Table05imulationSC}, the results for $M=1,2, \ldots, 5$ are presented to estimate the beta parameters for the great and small circle models, respectively. The simulations indicated that for the estimated eigenvectors case, when $M=6, 7, \mbox{ or } 8$, biased results were produced because it was not always possible to obtain reliable estimates of the eigenvectors of the matrix of trigonometric moments owing to the high dimensionality of the matrix, and the estimates for $M=6,7 \mbox{ and } 8$ were not aggregated to the results. Thus, when the estimated eigenvectors are not good estimates of the real eigenvectors, the PIT uniformity tests reject the null hypothesis of uniformity, which is good for practical applications. In contrast, when the estimated eigenvectors are good estimates of the real eigenvectors, the PIT uniformity tests reject the null hypothesis of uniformity in the expected rejection rate that corresponds to the selected significance level.
For the case 1 wherein $\bm{\beta}_1=(0,0,0,0,0)^\top$ for the great circle regression, the simulations presented in Table \ref{Table04simulationGC} show that in the great circle case, the percentage of the 100 simulations where the null hypothesis of each parameter is equal to zero, the rejection rate (RR), approximates the selected 5\% significance level both in the known and estimated eigenvector cases. This implies that the proposed methodology can detect non-significant explanatory variables at the expected 5\% significance level. In addition, the acceptance rates (AR) for the circular uniformity tests of the PIT values are adequate for the Kuiper, Watson, and range tests when the eigenvectors are known. However, when the eigenvectors are estimated, only the range test presents adequate values for large sample sizes. Moreover, in case of the small sample sizes, the beta parameter estimates are larger than for small sample sizes, although the RRs are correct. The corresponding results for $\bm{\beta}_0=(0,0,0,0,0)^\top$ in Table \ref{Table05imulationSC} are similar to those for case 1 in Table \ref{Table04simulationGC}, although the ARs for the Kuiper and Watson circular uniformity tests in the estimated eigenvectors case are very low. This pattern is observed in all estimated eigenvector cases in Tables \ref{Table04simulationGC} and \ref{Table05imulationSC}; thus, using the range test is recommended.
The Case 2 with $\bm{\beta}_2=(0.3,0.2,0.15,0.2,0.3)^\top$ shows the effect on the beta estimates of the unknown link function (Li and Duan, 1989). For example, in Table \ref{Table05imulationSC} with known eigenvectors and sample size equal to 1000, the beta estimates are proportional to the true values with a proportionality constant of approximately 2/3. In addition, as expected, the RRs were smaller for smaller beta parameters. Similar results were obtained for the corresponding case in the small circle regression as presented in Table \ref{Table05imulationSC}, with a smaller proportionality constant of approximately 3/4. For Case 2, the results for the estimated eigenvectors are similar in the great and small circle regressions.
For the cases 3 and 4, the results presented in Tables \ref{Table04simulationGC} and \ref{Table05imulationSC} confirm that the proposed methodology correctly identified the significant beta parameter in the presence of other non-significant beta parameters. For the known eigenvectors, the RRs increased for Case 4, wherein non-significant variables have been eliminated, when compared to Case 3. For the estimated eigenvectors, the RRs for Cases 3 and 4 are similar in both Tables \ref{Table04simulationGC} and \ref{Table05imulationSC}.
Other cases were simulated and the following recommendations were found: the proposed methodology is safe to use for values of M up to 5 for which the estimates of the eigenvectors of the matrix of trigonometric moments are reliable. For values of $M$ greater than 5, the PIT uniformity tests should be used to validate that the estimates of the eigenvectors are reliable. The range test is recommended for the circular uniformity test of PIT values to validate the fit of the models specially for large sample sizes.
\subsection{Blue Periwinkles' Direction and Distance Data}
Fisher and Lee (1992) and Fisher (1993) analyzed the com\-pass direction ($\theta$) (relative to north) and distance ($x$) travelled by 31 periwinkles (small sea snails \emph{Nodilittorina unifasciata}) after transplanting downshore from their normal living height. The sea was at an angle of approximately 275$^\circ$. A total of 15 observations corresponded to periwinkles mea\-sured one day after transplantation, and the remaining 16 observations were measured four days later. Note that no significant differences were observed between the behaviors of these two groups. The angles of the compass directions are plotted as a function of the traveled distance in Figure \ref{graph01datablueperiwinkles}.
For the directions taken by the blue periwinkles, Fisher (1992, 1993) considered two von Mises models. The first model considers the mean parameter as a function of the distance traveled. The second (mixed) model considers the mean and concentration parameters as functions of the dis\-tance traveled. Similar to our methodology, the distance was included in the mean parameter using the tangent function of the linear predictor. In addition, an exponential function of the linear predictor was used for the concentration parameter. With the von Mises model, Fisher (1993) concluded that clear evidence exists to support that both the mean and con\-cen\-tra\-tion parameters of the von Mises model for the di\-rec\-tions depend on distance. For the mean parameter case, the estimated coefficient for the distance was negative, with a global mean of 2.044 (97$^\circ$). In the concentration parameter case, the estimated beta parameter associated with the dis\-tance was positive, which implies that concentration in\-creased with distance.
Presnell et al. (1998) analyzed blue periwinkle data using a spherically projected multivariate linear (SPML) model that, for the circular case, corresponds to using an angular normal distribution for the dependent variable. Presnell et al. (1998) compared the fit of their model to that of Fisher and Lee (1992), and concluded that the Fisher and Lee model suffered from serious problems related to numerical op\-ti\-miza\-tion of the likelihood function and parameter non-iden\-ti\-fi\-a\-bil\-i\-ty. Thus, they considered that the values of the max\-i\-mized loglikelihoods should not be compared.
Recently, in the context of modeling directions taken by animals, Rivest et al. (2016) considered circular regression models for dependent and explanatory circular variables, where the conditional mean was modeled using the inverse tangent function and von Mises distributed errors with the option that the variance of the errors can be a function of the length of the conditional mean vector.
Figure \ref{graph01datablueperiwinkles} shows a dispersion plot of the original directions and distances. As can be seen, the angle (direction) tended to zero with distance. Table \ref{Table01blueperiwinkles} presents the estimates of the regression coefficients for the geodesic angle of rotation, the average squared cosine distances to the great (small) circle $R^2_{cos}$, the p-values of the range, Kuiper and Watson tests of circular uniformity for the probability integral transforms and the loglikelihood values for the considered models with different values for parameter $M$ ($M=1, 2, \ldots, 8$), that is, the number of terms in the sum defining the NNTS model in Equation \ref{NNTS1}. In addition, the parameter estimates, the p-values of the circular uni\-for\-mi\-ty tests, and the loglikelihood of the Fisher and Lee (1992) mean and mixed models, the Presnell et al. (1998) SPM mixed model, and Rivest et al. (2016) full and re\-cip\-ro\-cal mixed models are also given in Table \ref{Table01blueperiwinkles}.
The $M=0$ case, which corresponds to a uniform circular density (see Table \ref{Table01blueperiwinkles}), shows the smallest value of the log\-like\-li\-hood, and the rnage, Kuiper and Watson tests clearly reject the circular uniformity of the PIT values, which indicates that the directions taken by the blue periwinkles were not uniformly distributed on the circle.
After examining the dispersion plot of the trans\-formed linear variable $Y$ versus the distance in the right panel of Figure \ref{graph01datablueperiwinkles}, the regression $Y=\beta_1I(distance \le 27)(distance-27) + error$ was proposed because the values after $distance=27$ look around the horizontal axis. Here, the indicator function $I(distance \le 27)$ is equal to one if $distance \le 27$ and zero if $distance > 27$. The small circle fitted model with $M=8$ is shown in the bottom panel of Figure \ref{graph01datablueperiwinkles}. When considering this regression equation, the small circle model with $M=8$ had a loglikelihood of -19.786, and the p-values of the range, Kuiper and Watson tests were greater than 10\%, which means that the null hypothesis of circular uni\-for\-mi\-ty of the PIT values was not rejected and the goodness of fit of the model was confirmed. The small circle model with $M=8$ has the largest p-value for the range test even after adjusting the p-values for multiple hypothesis testing by using the procedure of Benjamini and Hochberg (1995). For the small circle model with $M=5$ and regression equation $Y=\beta_1I(distance \le 27)(distance-27) + error$, the p-values of the Kuiper and Watson tests were less than 0.01 although the range test yielded a p-value greater than 0.05 but less than 0.10 even after adjusting the p-values for multiple hypothesis testing. Thus, the goodness of fit of this model was considered inferior to the one with $M=8$, although it demonstrated the largest loglikelihood value equal to -16.902 and the largest $R^2$ value followed by the model with $M=8$.
Figure \ref{graph01datablueperiwinklescosine} shows the predicted NNTS density functions for the best small-circle model ($M=8$ and $\hat{Y}= -0.300I(distance\le 27)(distance-27)$) for cases $\bm{\hat{e}}_{k}^\top\hat{\bm{a}}>0$ (top left plot) and $\bm{\hat{e}}_{k}^\top\hat{\bm{a}} \le 0$ (bottom left plot), as well as the resultant length from the previous two cases (top right plot). In addition, the bottom right plot of Figure \ref{graph01datablueperiwinklescosine} shows the dispersion plot of the original data with the direction on the horizontal axis and distance on the vertical axis, to visually confirm the fit of the forecasted NNTS densities in the other three plots.
The NNTS density forecasts in the top left plot of Figure \ref{graph01datablueperiwinklescosine} for distances less than or equal to 27 are multimodal, with three main modes. For distances greater than 27, the NNTS density forecast was approximately unimodal with a mode at approximately 1.31 (75$^\circ$). For the NNTS density forecasts in the bottom left plot of Figure \ref{graph01datablueperiwinklescosine} satisfying a distance of less than or equal to 27, there are eight modes but two main modes. For distances greater than 27, the main mode was approximately $\frac{\pi}{2}$. Finally, the resultant mean NNTS density forecasts in the top right plot of Figure \ref{graph01datablueperiwinklescosine} are presented, wherein for distances less than or equal to 27, the densities are multimodal with a high concentration between $\frac{\pi}{8}$ and $\frac{3\pi}{4}$ and low concentrations cen\-tered at $\frac{7\pi}{8}$ and $\frac{9\pi}{8}$. For distances greater than 27, the fore\-cast\-ing density exhibits a main mode below $\frac{\pi}{2}$. The resultant mean NNTS density forecast shows a good fit to most points in the scatterplot of the bottom right plot of Figure \ref{graph01datablueperiwinklescosine}. Sim\-i\-lar\-ly, the density forecasts in the top (bottom) left plot of Figure \ref{graph01datablueperiwinklescosine} show a good fit to most non-filled (filled) points in the scatterplot in the bottom right plot.
Figure \ref{graph01datablueperiwinklesMEANFUNCTIONS} shows the mean (argument of the first trigono\-met\-ric moment) and circular variance (one minus the mean resultant length, which is the modulus of the first trigono\-met\-ric moment) functions as functions of the distance for the NNTS model, the Fisher and Lee (1992) mixed model, the Presnell et al. (1998) mixed model, and the Rivest et al. (2015) full mixed model. In terms of the mean and circular variance functions, the NNTS model has a resultant mean function of approximately 1.43 (82$^\circ$), which increases from a distance equal to zero to a distance equal to 27 and stays constant at a distance of 27. Essentially, on average, the blue periwinkles moved in an approximate direction that corresponds to the negative of the angle the sea was located (275$^\circ$) and, for distances less than 27, the density forecasts show many different modes with two main modes, which con\-verge toward a nearly unimodal density forecast for distances greater than 27. The circular variance function of the NNTS mean resultant model demonstrates an almost constant pat\-tern with values of approximately 0.86. In contrast to the NNTS model, the other models present mean and circular variance functions that decrease with distance. As indicated by Rivest et al. (2015), this is an artifact of the joint modeling of the mean and variance in their mixed model; however, it also applies to the mixed models of Fisher and Lee (1992) and Presnell et al. (1998). Thus, the NNTS model presents an alternative explanation wherein the circular variance does not change with distance.
\subsection{Time Series of Wind Directions}
Fisher (1993) (see Cameron 1983) and Fisher and Lee (1994) analyzed the time series of 72 hourly wind directions over four days at a location on Black Mountain, Australian Capital Territory, to calibrate three anemometers. They suggested two strategies to analyze circular time series. For noisy series, models should be fitted to the circular data directly, and for less noisy series, the circular data should be transformed to linear data; thus, common linear time series methods should be applied. Note that the second strategy is similar to the one we are proposing in this paper. By considering the first strategy, a circular autoregressive model of first order CAR(1) was considered to fit the data. The CAR(1) model is con\-struct\-ed by considering that the conditional distribution of the circular variable at time $t$ conditional on its previous values at times, $t-1$, $t-2$, $\ldots$ follows a von Mises dis\-tri\-bu\-tion with mean direction $\mu_t$ and fixed concentration pa\-ram\-e\-ter $\kappa$. The conditional mean direction $\mu_t$ is constructed by con\-sid\-er\-ing a link function (e.g., the inverse tangent func\-tion). Here, the estimated value of $\hat{\kappa}$ exceeds two, and the second strategy is applied by transforming the circular data into linear data using a probit link, which implies an au\-tore\-gres\-sive model of order one AR(1) for the transformed data. As stated by Fisher and Lee (1994), the LAR(1) model has unknown density for the original circular data, and the loglikelihood and PIT values cannot be calculated. Thus, we do not consider the LAR(1) model. The estimated mean directions obtained by the CAR(1) and LAR(1) models in the aforementioned study were 5.05 (289.5$^{\circ}$) and 5.19 (297.2$^{\circ}$), respectively.
To compare the proposed method to the CAR(1) model results obtained by Fisher (1993) and Fisher and Lee (1994), we fit the NNTS models to 72 observations. Figure \ref{graph02acfpacf} shows the time-series plots of the original data, along with the great-circle and small-circle transformed data as their sample au\-to\-cor\-re\-la\-tion (ACF) and partial autocorrelation (PACF) plots when considering great-circle and small-circle NNTS mod\-els with $M=4$. For the great-circle transformed data, the ACF and PACF are consistent with the AR(2) model, and for the small-circle transformed data, they are consistent with a white noise process.
Table \ref{Table02winddirections1to72AR1} includes the NNTS models by fitting AR(1) mod\-els with zero mean and NNTS densities with $M=1, 2, \ldots, 5$. As can be seen, among the NNTS great-circle AR(1) models, the model with $M=4$ shows the greatest loglikelihood value (-78.050) and the largest p-value for the range test (0.183). In addition, this model confirms that the PIT cir\-cu\-lar uniformity range, Kuiper and Watson tests do not reject the null hypothesis of circular uniformity. This model could be a candidate for our final model, and we note that its log\-like\-li\-hood is greater than that of the CAR(1) model (-81.370). The autoregressive coefficient is significant with the estimate equal to 0.5276, which is similar to the value reported by Fisher and Lee (1994). For all NNTS small-circle models, the range, Kuiper and Watson tests of circular uni\-for\-mi\-ty rejected the null hypothesis of circular uni\-for\-mi\-ty for the PIT values at a 10\% significance level. For the CAR(1) model, the range test did not reject the null hypothesis of uniformity contrary to the Kuiper and Watson tests.
By inspecting the ACF and PACF plots for the residuals of the NNTS great-circle AR(1) model with $M=4$, we found that it is necessary to include a second autoregressive term, as observed in Figure \ref{graph02acfpacf}. The last row of Table \ref{Table02winddirections1to72AR1} includes the results for the NNTS great-circle AR(2) model with $M=4$, which obtained a loglikelihood of -73.967 (the greatest loglikelihood value of all considered models), and the range, Kuiper and Watson tests do not reject the null hypothesis of circular uniformity of the PIT values at a 10\% significance level. In addition, the ACF and PACF of the residuals present the pattern of a white noise process, which validates this model.
For the great-circle AR(2) NNTS model with $M=4$, Figure \ref{graphdatab23fisherforecasts} shows the density forecasts for hours 1 to 72. In addition, the scatterplot of the original data and the mean function of the model at the bottom right of Figure \ref{graphdatab23fisherforecasts} are shown to compare the density forecast with the observed values. As can be seen, it is clear that the model presents a good fit to the data. Here, the mean function is around a mean value of 4.620 (264.707$^\circ$), which is similar to the values reported by Fisher (1993) and Fisher and Lee (1994) for the CAR(1) and LAR(1) models. Note that there are some extreme observations, such as the observation at hour 50 (second density forecast in the bottom left plot of Figure \ref{graphdatab23fisherforecasts}), that were captured by the great-circle AR(2) NNTS model with $M=4$ because their density forecasts are multimodal. Thus, they are more difficult to capture by a unimodal model based on the von Mises distribution.
\subsection{Earthquake Occurrence and Planet Alignments}
There is debate about the prediction of the occurrence time of large earthquakes in terms of variables related to the po\-si\-tions of different astronomical bodies (e.g., the sun, moon, and planets). In particular, the times at which two planets are in conjunction (Ip 1976; Hughes 1977; Geller 1997) have been considered as times when earthquakes are more likely to occur.
A database with the occurrence times of earthquakes of magnitude 6 (on the Richter scale) or greater from January 1, 1969 to December 15, 2019, and the right ascension and declination of the planets (excluding Pluto) has been con\-struct\-ed previously. The times of earthquakes were obtained from the NOOA Significant Earthquakes Database (National Geophysical Data Center / World Data Service (NGDC/WDS): Significant Earthquake Database. National Geophysical Data Center, NOAA. doi:10.7289/V5TD9V7K) and were trans\-formed to Julian dates. Given the number of earthquakes, right ascensions and declinations were calculated using the Skyfield (PyEphem) Python astronomical package (Rhodes 2020). The angular distance in the celestial sphere between each pair of planets was calculated from the right ascension and declination of the planets. In addition, indicator variables were calculated for the case where the angular dis\-tance between two planets was less than 2.5$^\circ$.
Here, we consider a regression model with the occurrence time (in Julian time) of earthquakes with magnitude 6 or greater, transformed into a circular variable as a dependent variable, and the angular distances between planets and in\-di\-ca\-tors of separation between planets of less than 2.5$^\circ$ were used as explanatory variables. There is a total of 1153 observations on 42 explanatory variables. We did not find any predictive power to explain the times at which an earthquake of mag\-ni\-tude 6 or greater will occur. Here, we employed the LASSO regularization algorithm (Tibshirani 1996; Hastie et al. 2009) with a penalty alpha parameter of 0.5. For the great-circle NNTS models shown in Table \ref{Table03earthquakes}, we found that there were variables with non-zero coefficients only for cases where $M= 1, 2, \mbox{ or } 7$. In addition, the range, Kuiper and Watson tests did not reject the circular uniformity of the PIT values at a 1\% significance level, although there was no predictive power (the maximum $R^2$ value was 0.032), and a model with $i.i.d.$ uniform distributed occurrence times was preferable (last row of Table \ref{Table03earthquakes}). For all small-circle NNTS models (Table \ref{Table03earthquakes}), the range, Kuiper and Watson tests rejected the null hypothesis of circular uniformity of the PIT values. The same results were obtained by considering different values for the penalty alpha parameter. Thus, for the fitted small-circle the $\phi$ angles are very near zero and the $\alpha$ angle is near $\frac{\pi}{4}$, thereby rendering the small circle a great circle with vectors $\bm{b}$ and $\bm{a}$ with equal weights where squares are equal to 0.5.
\section{Conclusions}
NNTS models allow us to model circular variables when their densities present skewness and multimodality. Previous models only considered unimodal and symmetric models based on the von Mises or other symmetric densities. By exploiting the fact that the parameter space of NNTS models is a hypersphere, regression models were constructed, in which great and small circles were equivalent to regression planes in the classical linear model. The different points on the great (small) circle specified a different NNTS density, and the significance of the explanatory variables was evaluated according to their power to explain different angle rotations along the great (small) circle. This evaluation was performed by considering transformation of the original circular variable into a linear variable where common models (e.g., ordinary least-squares regression or linear time series models) were applied to the transformed variable. The proposed circular regression model generates fitted (forecast) NNTS densities; thus, the goodness of fit of the regression model can be assessed using tests of uniformity for the probability integral transforms in consideration of the fitted (forecast) NNTS densities. The usefulness of NNTS models with skewness and multimodality was shown in the examples presented in this paper.
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\begin{table}[t] \caption{\label{Table04simulationGC} Great-circle regression simulations: Beta parameter estimates, rejection rates (RR) for the null hypothesis $H_0:\beta=0$ and acceptance rates (AR) for the circular uniformity tests for the PIT (Probability Integral Transform) values for the range (R), Kuiper (K) and Watson (W) tests considering a 5\% significance level.} \centering \scalebox{.7}{
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline case / & obs & $\beta_1$ & $\beta_2$ & $\beta_3$ & $\beta_4$ & $\beta_5$ & RR & RR & RR & RR & RR & AR & AR & AR \\ (eigen) & & & & & & & $\beta_1$ & $\beta_2$ & $\beta_3$ & $\beta_4$ & $\beta_5$ & (R) & (K) & (W) \\ vectors & & & & & & & & & & & & & & \\ \hline \hline 1 & & 0 & 0 & 0 & 0 & 0 & & & & & & & & \\ \hline \hline known & 25 & 0.165 & 0.142 & 0.191 & 0.138 & 0.172 & 0.06 & 0.04 & 0.05 & 0.06 & 0.05 & 0.96 & 0.95 & 0.96 \\
& 50 & 0.112 & 0.100 & 0.117 & 0.104 & 0.109 & 0.06 & 0.03 & 0.05 & 0.06 & 0.07 & 0.96 & 0.94 & 0.95 \\
& 100 & 0.077 & 0.068 & 0.075 & 0.078 & 0.074 & 0.05 & 0.04 & 0.05 & 0.06 & 0.06 & 0.96 & 0.96 & 0.96 \\
& 200 & 0.053 & 0.052 & 0.050 & 0.053 & 0.053 & 0.05 & 0.05 & 0.03 & 0.05 & 0.06 & 0.94 & 0.96 & 0.96 \\
& 500 & 0.034 & 0.032 & 0.031 & 0.034 & 0.034 & 0.05 & 0.05 & 0.05 & 0.04 & 0.05 & 0.96 & 0.94 & 0.95 \\
& 1000 & 0.024 & 0.023 & 0.021 & 0.024 & 0.024 & 0.04 & 0.06 & 0.05 & 0.04 & 0.05 & 0.95 & 0.94 & 0.95 \\ \hline est. & 25 & 0.784 & 0.661 & 0.881 & 0.654 & 0.940 & 0.06 & 0.06 & 0.05 & 0.04 & 0.07 & 0.95 & 0.93 & 0.94 \\
& 50 & 0.326 & 0.275 & 0.429 & 0.243 & 0.202 & 0.04 & 0.04 & 0.06 & 0.05 & 0.08 & 0.97 & 0.95 & 0.96 \\
& 100 & 0.109 & 0.101 & 0.101 & 0.105 & 0.115 & 0.05 & 0.05 & 0.04 & 0.06 & 0.06 & 0.95 & 0.93 & 0.93 \\
& 200 & 0.064 & 0.062 & 0.061 & 0.063 & 0.068 & 0.04 & 0.04 & 0.06 & 0.05 & 0.07 & 0.92 & 0.80 & 0.82 \\
& 500 & 0.039 & 0.036 & 0.034 & 0.038 & 0.038 & 0.06 & 0.05 & 0.05 & 0.05 & 0.04 & 0.92 & 0.59 & 0.66 \\
& 1000 & 0.027 & 0.025 & 0.024 & 0.026 & 0.027 & 0.04 & 0.04 & 0.04 & 0.05 & 0.05 & 0.89 & 0.43 & 0.48 \\ \hline \hline 2 & & 0.3 & 0.2 & 0.15 & 0.2 & 0.3 & & & & & & & & \\ \hline \hline known & 25 & 0.213 & 0.200 & 0.205 & 0.159 & 0.210 & 0.16 & 0.18 & 0.07 & 0.12 & 0.14 & 0.95 & 0.94 & 0.95 \\
& 50 & 0.209 & 0.174 & 0.129 & 0.139 & 0.200 & 0.33 & 0.30 & 0.09 & 0.17 & 0.37 & 0.94 & 0.94 & 0.92 \\
& 100 & 0.205 & 0.147 & 0.110 & 0.140 & 0.201 & 0.64 & 0.40 & 0.21 & 0.35 & 0.63 & 0.95 & 0.93 & 0.94 \\
& 200 & 0.194 & 0.117 & 0.087 & 0.118 & 0.178 & 0.89 & 0.51 & 0.28 & 0.51 & 0.82 & 0.93 & 0.92 & 0.93 \\
& 500 & 0.207 & 0.127 & 0.091 & 0.127 & 0.193 & 1 & 0.91 & 0.70 & 0.89 & 1 & 0.94 & 0.90 & 0.89 \\
& 1000 & 0.209 & 0.131 & 0.093 & 0.129 & 0.198 & 1 & 1 & 0.92 & 0.98 & 1 & 0.95 & 0.85 & 0.85 \\ \hline est. & 25 & 0.752 & 0.748 & 0.772 & 0.543 & 0.878 & 0.14 & 0.09 & 0.08 & 0.1 & 0.14 & 0.95 & 0.92 & 0.92 \\
& 50 & 0.315 & 0.255 & 0.259 & 0.223 & 0.274 & 0.27 & 0.21 & 0.07 & 0.12 & 0.23 & 0.95 & 0.97 & 0.98 \\
& 100 & 0.250 & 0.174 & 0.156 & 0.172 & 0.223 & 0.57 & 0.34 & 0.17 & 0.28 & 0.53 & 0.96 & 0.95 & 0.96 \\
& 200 & 0.209 & 0.125 & 0.097 & 0.130 & 0.186 & 0.84 & 0.47 & 0.25 & 0.47 & 0.76 & 0.94 & 0.86 & 0.89 \\
& 500 & 0.220 & 0.132 & 0.098 & 0.135 & 0.200 & 0.99 & 0.87 & 0.69 & 0.88 & 0.98 & 0.92 & 0.64 & 0.70 \\
& 1000 & 0.221 & 0.136 & 0.099 & 0.133 & 0.201 & 1 & 0.98 & 0.91 & 0.97 & 1 & 0.92 & 0.53 & 0.57 \\ \hline \hline 3 & & 0 & 0 & 0 & 0 & 0.3 & & & & & & & & \\ \hline \hline known & 25 & 0.159 & 0.131 & 0.186 & 0.133 & 0.313 & 0.06 & 0.05 & 0.05 & 0.05 & 0.30 & 0.96 & 0.96 & 0.95 \\
& 50 & 0.102 & 0.095 & 0.109 & 0.097 & 0.272 & 0.05 & 0.04 & 0.04 & 0.04 & 0.58 & 0.96 & 0.94 & 0.94 \\
& 100 & 0.071 & 0.067 & 0.076 & 0.069 & 0.268 & 0.05 & 0.05 & 0.05 & 0.05 & 0.86 & 0.94 & 0.95 & 0.95 \\
& 200 & 0.050 & 0.045 & 0.051 & 0.048 & 0.265 & 0.04 & 0.04 & 0.06 & 0.05 & 0.98 & 0.95 & 0.94 & 0.94 \\
& 500 & 0.032 & 0.031 & 0.031 & 0.031 & 0.265 & 0.06 & 0.05 & 0.05 & 0.04 & 1 & 0.96 & 0.93 & 0.94 \\
& 1000 & 0.022 & 0.022 & 0.023 & 0.023 & 0.267 & 0.05 & 0.04 & 0.05 & 0.04 & 1 & 0.94 & 0.95 & 0.95 \\ \hline est. & 25 & 0.484 & 0.347 & 0.641 & 0.348 & 0.488 & 0.07 & 0.06 & 0.06 & 0.04 & 0.13 & 0.95 & 0.94 & 0.94 \\
& 50 & 0.231 & 0.167 & 0.223 & 0.256 & 0.392 & 0.05 & 0.05 & 0.07 & 0.04 & 0.40 & 0.96 & 0.96 & 0.96 \\
& 100 & 0.099 & 0.094 & 0.104 & 0.103 & 0.287 & 0.06 & 0.05 & 0.05 & 0.05 & 0.73 & 0.93 & 0.92 & 0.93 \\
& 200 & 0.059 & 0.052 & 0.063 & 0.058 & 0.276 & 0.04 & 0.04 & 0.07 & 0.05 & 0.93 & 0.93 & 0.80 & 0.83 \\
& 500 & 0.035 & 0.035 & 0.036 & 0.035 & 0.278 & 0.06 & 0.04 & 0.06 & 0.04 & 1 & 0.92 & 0.59 & 0.67 \\
& 1000 & 0.024 & 0.025 & 0.025 & 0.026 & 0.279 & 0.06 & 0.05 & 0.05 & 0.03 & 1 & 0.89 & 0.49 & 0.50 \\ \hline 4 & & & & & & 0.3 & & & & & & & & \\ \hline known & 25 & & & & & 0.298 & & & & & 0.36 & 0.96 & 0.96 & 0.96 \\
& 50 & & & & & 0.272 & & & & & 0.62 & 0.96 & 0.94 & 0.95 \\
& 100 & & & & & 0.268 & & & & & 0.86 & 0.96 & 0.95 & 0.95 \\
& 200 & & & & & 0.265 & & & & & 0.98 & 0.94 & 0.95 & 0.96 \\
& 500 & & & & & 0.265 & & & & & 1 & 0.95 & 0.94 & 0.94 \\
& 1000 & & & & & 0.267 & & & & & 1 & 0.94 & 0.95 & 0.95 \\ \hline est. & 25 & & & & & 0.644 & & & & & 0.19 & 0.95 & 0.95 & 0.96 \\
& 50 & & & & & 0.399 & & & & & 0.43 & 0.97 & 0.96 & 0.97 \\
& 100 & & & & & 0.285 & & & & & 0.73 & 0.94 & 0.92 & 0.93 \\
& 200 & & & & & 0.277 & & & & & 0.93 & 0.94 & 0.79 & 0.82 \\
& 500 & & & & & 0.277 & & & & & 1 & 0.93 & 0.59 & 0.66 \\
& 1000 & & & & & 0.279 & & & & & 1 & 0.89 & 0.49 & 0.50 \\ \hline \end{tabular}} \renewcommand{1}{1} \end{table}
\renewcommand{1}{1.00}
\begin{table}[t] \caption{\label{Table05imulationSC} Small-circle regression simulations: Beta parameter estimates, rejection rates (RR) for the null hypothesis $H_0:\beta=0$ and acceptance rates (AR) for the circular uniformity tests for the PIT (Probability Integral Transform) values for the range, Kuiper and Watson tests considering a 5\% significance level.} \centering \scalebox{.7}{
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline case / & obs & $\beta_1$ & $\beta_2$ & $\beta_3$ & $\beta_4$ & $\beta_5$ & RR & RR & RR & RR & RR & AR & AR & AR \\ (eigen) & & & & & & & $\beta_1$ & $\beta_2$ & $\beta_3$ & $\beta_4$ & $\beta_5$ & (R) & (K) & (W) \\ vectors & & & & & & & & & & & & & & \\ \hline \hline 1 & & 0 & 0 & 0 & 0 & 0 & & & & & & & & \\ \hline \hline known & 25 & 0.177 & 0.146 & 0.214 & 0.144 & 0.176 & 0.06 & 0.05 & 0.06 & 0.06 & 0.05 & 0.96 & 0.96 & 0.97 \\
& 50 & 0.113 & 0.101 & 0.130 & 0.111 & 0.107 & 0.03 & 0.05 & 0.05 & 0.05 & 0.05 & 0.97 & 0.96 & 0.95 \\
& 100 & 0.075 & 0.078 & 0.079 & 0.080 & 0.074 & 0.04 & 0.07 & 0.06 & 0.06 & 0.05 & 0.92 & 0.95 & 0.95 \\
& 200 & 0.053 & 0.054 & 0.057 & 0.058 & 0.052 & 0.03 & 0.05 & 0.07 & 0.04 & 0.07 & 0.94 & 0.95 & 0.95 \\
& 500 & 0.035 & 0.032 & 0.035 & 0.036 & 0.035 & 0.04 & 0.04 & 0.06 & 0.05 & 0.05 & 0.95 & 0.95 & 0.94 \\
& 1000 & 0.024 & 0.025 & 0.025 & 0.026 & 0.025 & 0.04 & 0.05 & 0.05 & 0.06 & 0.04 & 0.92 & 0.95 & 0.94 \\ \hline est. & 25 & 1.009 & 0.669 & 1.591 & 0.657 & 1.319 & 0.07 & 0.06 & 0.07 & 0.05 & 0.14 & 0.92 & 0.87 & 0.87 \\
& 50 & 0.203 & 0.183 & 0.275 & 0.212 & 0.267 & 0.04 & 0.04 & 0.07 & 0.05 & 0.09 & 0.91 & 0.67 & 0.67 \\
& 100 & 0.092 & 0.093 & 0.090 & 0.091 & 0.096 & 0.05 & 0.08 & 0.06 & 0.04 & 0.07 & 0.84 & 0.36 & 0.33 \\
& 200 & 0.050 & 0.053 & 0.057 & 0.055 & 0.063 & 0.04 & 0.06 & 0.07 & 0.04 & 0.08 & 0.75 & 0.16 & 0.18 \\
& 500 & 0.033 & 0.032 & 0.033 & 0.033 & 0.038 & 0.04 & 0.05 & 0.05 & 0.05 & 0.08 & 0.64 & 0.02 & 0.02 \\
& 1000 & 0.022 & 0.023 & 0.023 & 0.024 & 0.028 & 0.04 & 0.05 & 0.05 & 0.06 & 0.09 & 0.61 & 0 & 0 \\ \hline \hline 2 & & 0.3 & 0.2 & 0.15 & 0.2 & 0.3 & & & & & & & & \\ \hline \hline known & 25 & 0.222 & 0.219 & 0.226 & 0.171 & 0.245 & 0.18 & 0.22 & 0.08 & 0.14 & 0.17 & 0.97 & 0.96 & 0.96 \\
& 50 & 0.237 & 0.192 & 0.154 & 0.157 & 0.240 & 0.39 & 0.34 & 0.12 & 0.20 & 0.46 & 0.95 & 0.95 & 0.95 \\
& 100 & 0.233 & 0.171 & 0.123 & 0.160 & 0.242 & 0.69 & 0.47 & 0.25 & 0.40 & 0.77 & 0.94 & 0.96 & 0.96 \\
& 200 & 0.228 & 0.136 & 0.101 & 0.137 & 0.218 & 0.94 & 0.61 & 0.34 & 0.57 & 0.93 & 0.94 & 0.94 & 0.94 \\
& 500 & 0.239 & 0.154 & 0.107 & 0.149 & 0.230 & 1 & 0.97 & 0.80 & 0.97 & 1 & 0.95 & 0.90 & 0.90 \\
& 1000 & 0.243 & 0.158 & 0.110 & 0.152 & 0.235 & 1 & 1 & 0.96 & 1 & 1 & 0.95 & 0.84 & 0.84 \\ \hline est. & 25 & 1.289 & 0.580 & 0.808 & 0.804 & 1.133 & 0.19 & 0.13 & 0.07 & 0.07 & 0.17 & 0.93 & 0.88 & 0.86 \\
& 50 & 0.284 & 0.197 & 0.187 & 0.186 & 0.222 & 0.29 & 0.16 & 0.09 & 0.10 & 0.25 & 0.88 & 0.72 & 0.68 \\
& 100 & 0.207 & 0.149 & 0.114 & 0.161 & 0.216 & 0.50 & 0.32 & 0.16 & 0.30 & 0.53 & 0.84 & 0.43 & 0.42 \\
& 200 & 0.183 & 0.114 & 0.088 & 0.110 & 0.172 & 0.76 & 0.46 & 0.26 & 0.44 & 0.75 & 0.79 & 0.19 & 0.19 \\
& 500 & 0.192 & 0.124 & 0.087 & 0.122 & 0.187 & 0.95 & 0.87 & 0.70 & 0.88 & 0.96 & 0.74 & 0.03 & 0.02 \\
& 1000 & 0.200 & 0.128 & 0.091 & 0.123 & 0.190 & 1 & 0.98 & 0.90 & 0.98 & 1 & 0.69 & 0.01 & 0 \\ \hline \hline 3 & & 0 & 0 & 0 & 0 & 0.3 & & & & & & & & \\ \hline \hline known & 25 & 0.166 & 0.142 & 0.192 & 0.135 & 0.368 & 0.07 & 0.06 & 0.03 & 0.05 & 0.37 & 0.96 & 0.95 & 0.95 \\
& 50 & 0.113 & 0.099 & 0.123 & 0.107 & 0.326 & 0.07 & 0.04 & 0.06 & 0.05 & 0.69 & 0.95 & 0.95 & 0.94 \\
& 100 & 0.074 & 0.069 & 0.075 & 0.075 & 0.316 & 0.04 & 0.04 & 0.04 & 0.04 & 0.93 & 0.94 & 0.94 & 0.95 \\
& 200 & 0.053 & 0.049 & 0.052 & 0.050 & 0.312 & 0.07 & 0.05 & 0.06 & 0.04 & 1 & 0.95 & 0.95 & 0.96 \\
& 500 & 0.033 & 0.031 & 0.032 & 0.032 & 0.309 & 0.04 & 0.05 & 0.04 & 0.04 & 1 & 0.95 & 0.96 & 0.94 \\
& 1000 & 0.022 & 0.022 & 0.023 & 0.022 & 0.310 & 0.04 & 0.04 & 0.04 & 0.04 & 1 & 0.97 & 0.95 & 0.95 \\ \hline est. & 25 & 0.754 & 0.577 & 0.703 & 0.501 & 0.595 & 0.11 & 0.09 & 0.07 & 0.04 & 0.15 & 0.93 & 0.91 & 0.91 \\
& 50 & 0.186 & 0.166 & 0.191 & 0.177 & 0.267 & 0.06 & 0.04 & 0.06 & 0.04 & 0.32 & 0.89 & 0.72 & 0.69 \\
& 100 & 0.091 & 0.090 & 0.084 & 0.079 & 0.243 & 0.06 & 0.06 & 0.04 & 0.04 & 0.64 & 0.84 & 0.38 & 0.36 \\
& 200 & 0.056 & 0.056 & 0.055 & 0.051 & 0.239 & 0.05 & 0.04 & 0.05 & 0.04 & 0.88 & 0.76 & 0.18 & 0.19 \\
& 500 & 0.030 & 0.029 & 0.031 & 0.030 & 0.250 & 0.04 & 0.04 & 0.07 & 0.04 & 0.98 & 0.72 & 0.04 & 0.04 \\
& 1000 & 0.020 & 0.020 & 0.021 & 0.020 & 0.254 & 0.05 & 0.04 & 0.05 & 0.04 & 0.99 & 0.67 & 0 & 0 \\ \hline 4 & & & & & & 0.3 & & & & & & & & \\ \hline known & 25 & & & & & 0.361 & & & & & 0.44 & 0.96 & 0.97 & 0.96 \\
& 50 & & & & & 0.324 & & & & & 0.70 & 0.95 & 0.97 & 0.96 \\
& 100 & & & & & 0.316 & & & & & 0.93 & 0.93 & 0.95 & 0.96 \\
& 200 & & & & & 0.312 & & & & & 1 & 0.96 & 0.95 & 0.96 \\
& 500 & & & & & 0.308 & & & & & 1 & 0.95 & 0.95 & 0.95 \\
& 1000 & & & & & 0.309 & & & & & 1 & 0.95 & 0.95 & 0.95 \\ \hline est. & 25 & & & & & 0.626 & & & & & 0.18 & 0.91 & 0.89 & 0.89 \\
& 50 & & & & & 0.268 & & & & & 0.34 & 0.88 & 0.60 & 0.57 \\
& 100 & & & & & 0.242 & & & & & 0.63 & 0.84 & 0.35 & 0.32 \\
& 200 & & & & & 0.238 & & & & & 0.88 & 0.77 & 0.16 & 0.17 \\
& 500 & & & & & 0.250 & & & & & 0.98 & 0.70 & 0.03 & 0.04 \\
& 1000 & & & & & 0.254 & & & & & 0.99 & 0.67 & 0 & 0 \\ \hline \end{tabular}} \renewcommand{1}{1} \end{table}
\renewcommand{1}{1.00}
\begin{table}
\caption{\label{Table01blueperiwinkles} Blue periwinkle data. Coefficients of determination, p-values of the range, Kuiper and Watson circular uniformity tests for PIT values, and loglikelihood for the great- and small-circle nonnegative trigonometric series (NNTS) models, Fisher and Lee (1992) mean and mixed models, Presnell et al. (1998) $SPML$ mixed model, and Rivest et al. (2015) mixed models are shown. These models include the $distance$ travelled by small periwinkles as the explanatory variable.} \centering \scalebox{.7}{
\begin{tabular}{|c|c|c|c|c|c|c|c|} \hline
\multicolumn{8}{|c|}{Great-circle model $Y = \beta_1(distance - 27)I(distance<27) + error$} \\ \hline
$M$ ($\alpha$ angle) & $distance$ coef. & $R^2_{cos}$ & $R^2$ & Range & Kuiper & Watson & loglik \\
& (std. error, p-value) & & & p-value& p-value & p-value & \\ \hline 1 & 0.042 (0.014, 0.006) & 0.847 & 0.227 & 0.062 & $<$0.01 & $<$0.01 & -39.643 \\ 2 & 0.230 (0.064, 0.001) & 0.747 & 0.302 & 0.024 & $<$0.01 & $<$0.01 & -32.135 \\ 3 & 0.043 (0.033, 0.202) & 0.618 & 0.054 & 0.003 & (0.10,0.15) & $>$0.10 & -35.406 \\ 4 & 0.032 (0.027, 0.245) & 0.508 & 0.045 & 0.070 & $>$0.15 & $>$0.10 & -33.625 \\ 5 & 0.028 (0.023, 0.241) & 0.451 & 0.045 & 0.451 & $>$0.15 & $>$0.10 & -34.178 \\ 6 & 0.015 (0.027, 0.582) & 0.395 & 0.010 & 0.434 & $>$0.15 & $>$0.10 & -33.506 \\ 7 & 0.032 (0.037, 0.392) & 0.337 & 0.024 & 0.154 & (0.05,0.10) & $>$0.10 & -34.881 \\ 8 & 0.122 (0.093, 0.200) & 0.305 & 0.054 & 0.054 & $<$0.01 & (0.01,0.025) & -43.308 \\ \hline
\multicolumn{8}{|c|}{Small-circle model $Y = \beta_1(distance - 27)I(distance<27) + error$} \\
\hline 1 (0.332) & 0.006 (0.289, 0.984) & 0.897 & 0.000 & 0.000 & $<$0.01 & $<$0.01 & -38.042 \\ 2 (0.464) & 0.008 (0.073, 0.912) & 0.724 & 0.000 & 0.005 & $<$0.01 & $<$0.01 & -32.974 \\ 3 (0.566) & 0.033 (0.024, 0.175) & 0.758 & 0.060 & 0.003 & $<$0.01 & $<$0.01 & -25.056 \\ 4 (0.641) & 0.017 (0.033, 0.608) & 0.698 & 0.009 & 0.053 & $<$0.01 & $<$0.01 & -23.077 \\ 5 (0.711) & 0.321 (0.055, 0.000) & 0.615 & 0.534 & 0.054 & $<$0.01 & $<$0.01 & -16.902 \\ 6 (0.779) & 0.136 (0.117, 0.254) & 0.565 & 0.043 & 0.115 & (0.025,0.05) & (0.025,0.05) & -19.176 \\ 7 (0.835) & 0.049 (0.136, 0.722) & 0.504 & 0.004 & 0.101 & $>$0.15 & $>$0.10 & -25.394 \\ 8 (0.884) & 0.300 (0.089, 0.002) & 0.440 & 0.276 & 0.329 & $>$0.15 & $>$0.10 & -19.786 \\ \hline
\multicolumn{8}{|c|}{$i.i.d.$ Uniforms} \\ \hline 0 & & & & 0.000 & $<$0.01 & $<$0.01 & -56.974 \\ \hline
\multicolumn{2}{|c|}{Parameter} & & & Range & Kuiper & Watson & loglik \\
\multicolumn{2}{|c|}{estimates (std. error)} & & & p-value & p-value & p-value & \\ \hline
\multicolumn{8}{|c|}{Fisher and Lee (1992) Mean von Mises $vM(\mu,\kappa)$ model} \\
\multicolumn{8}{|c|}{$\mu = \mu_0 + \arctan(\beta (distance - \overline{distance}))$} \\ \hline $\hat{\mu_0}$=1.694 (0.112) & $\hat{\beta}$= -0.0065 (0.0022)& & & 0.446 & $>$0.15 & $>$0.10 & -29.452 \\ $\hat{\kappa}$=3.203 (0.707) & & & & & & & \\ \hline
\multicolumn{8}{|c|}{Fisher and Lee (1992) Mixed von Mises $vM(\mu,\kappa)$ model} \\
\multicolumn{8}{|c|}{$\mu = \mu_0 + \arctan(\beta distance)$ $\kappa = e^{\gamma_0 + \gamma_1 (distance - \overline{distance})}$} \\ \hline $\hat{\mu_0}$=2.034 (0.190) & $\hat{\beta}$= -0.0045 (0.0012)& & & 0.248 & $>$0.15 & $>$0.10 & -18.963 \\ $\hat{\gamma}_0$=1.785 (0.248) & $\hat{\gamma}_1$=0.045 (0.010) & & & & & & \\ \hline
\multicolumn{8}{|c|}{Presnell, Morrison and Little (1998) Mixed $SPML$ angular normal $AN(\mu,\gamma))$ model} \\
\multicolumn{8}{|c|}{(reported estimates)} \\
\multicolumn{8}{|c|}{$\mu = \left[\arctan\left(\frac{\beta_{0s}+\beta_{1s}distance}{\beta_{0c}+\beta_{1c}distance} \right) + \pi I(\beta_{0c} + \beta_{1c}distance)\right] \mod 2\pi$} \\
\multicolumn{8}{|c|}{$\gamma=\sqrt{(\beta_{0s}+\beta_{1s}distance)^2 + (\beta_{0c}+\beta_{1c}distance)^2}$} \\ \hline $\hat{\beta}_{0c}$=-1.228 (0.423) & $\hat{\beta}_{1c}$=0.030 (0.008) & & & 0.134 & $>$0.15 & $>$0.10 & -20.601 \\ $\hat{\beta}_{0s}$= 0.157 (0.451) & $\hat{\beta}_{1s}$=0.049 (0.012) & & & & & & \\ \hline
\multicolumn{8}{|c|}{Rivest et al. (2016) Mixed von Mises $vM(\mu,\kappa)$ full model (reported estimates)} \\
\multicolumn{8}{|c|}{$\mu=\arctan\left(\frac{\beta_{0s}+\beta_{1s}distance}{1 + \beta_{1c}distance} \right)$ $\kappa=\gamma distance$} \\ \hline $\hat{\beta}_{0s}$= -0.062 (0.440)& $\hat{\beta}_{1c}$= 0.026 (0.006) & & & 0.000 & $<$0.01 & $<$0.01 & -21.817 \\ $\hat{\gamma}$=0.134 & $\hat{\beta}_{1s}$= 0.060 (0.028) & & & & & & \\ \hline
\multicolumn{8}{|c|}{Rivest et al. (2016) Mixed von Mises $vM(\mu,\kappa)$ reciprocal model (reported estimates)} \\
\multicolumn{8}{|c|}{$\mu=\arctan\left(\frac{1}{\beta_{0c}+\beta_{1c}distance} \right)$ $\kappa=\gamma distance$} \\ \hline $\hat{\beta}_{0c}$= -0.790 (0.390)& $\hat{\beta}_{1c}$= 0.014 (0.006) & & & 0.000 & $<$0.01 & (0.01,0.025) & -22.312 \\ $\hat{\gamma}$=0.129 & & & & & & & \\ \hline \end{tabular}} \renewcommand{1}{1} \end{table}
\renewcommand{1}{1.00} \begin{table}
\caption{\label{Table02winddirections1to72AR1} Wind direction data. Results for the AR(1) great and small-circle NNTS models and CAR(1) model for 72 observations are shown. The autoregressive parameter estimates, coefficient of determination, p-values of the range, Kuiper and Watson tests, and loglikelihood values are shown. The last rows of the table present the results for the AR(2) great-circle nonnegative trigonometric series (NNTS) final model with $M=4$. The other parameter estimates of the CAR(1) are equal to $\hat{\mu}=$5.1617 (0.2675) and $\hat{\kappa}=$2.4493 (0.3415).} \centering \scalebox{.7}{
\begin{tabular}{|c|c|c|c|c|c|c|c|} \hline
\multicolumn{8}{|c|}{Great-circle models} \\ \hline
$M$ & AR(1) Coeff. & $R^2_{cos}$ & Range & Kuiper & Watson & loglik & $\alpha$ angle \\
& (std. error) & & p-value & p-value & p-value & & \\ \hline 1 & 0.6222 (0.0900) & 0.837 & 0.000 & $<$0.01 & $<$0.01 & -96.316 & \\ \hline 2 & 0.5041 (0.1002) & 0.701 & 0.146 & $<$0.01 & $<$0.01 & -83.732 & \\ \hline 3 & 0.5332 (0.0979) & 0.571 & 0.120 & $>$0.15 & $>$0.10 & -87.123 & \\ \hline 4 & 0.5276 (0.0984) & 0.462 & 0.183 & $>$0.15 & $>$0.10 & -78.050 & \\ \hline 5 & 0.4478 (0.1039) & 0.384 & 0.009 & (0.05,0.10) & $>$0.10 & -86.861 & \\ \hline
\multicolumn{8}{|c|}{Small-circle models} \\ \hline \hline 1 & 0.3430 (0.1133) & 0.875 & 0.073 & $<$0.01 & $<$0.01 & -90.682 & 0.411 \\ \hline 2 & 0.0012 (0.1172) & 0.775 & 0.006 & $<$0.01 & $<$0.01 & -77.201 & 0.596 \\ \hline 3 & 0.1377 (0.1169) & 0.651 & 0.088 & $<$0.01 & $<$0.01 & -66.531 & 0.724 \\ \hline 4 & -0.0715 (0.1168) & 0.474 & 0.001 & $<$0.01 & $<$0.01 & -75.849 & 0.806 \\ \hline 5 & 0.1778 (0.1152) & 0.338 & 0.001 & $<$0.01 & $<$0.01 & -91.383 & 0.849 \\ \hline
\multicolumn{8}{|c|}{$i.i.d.$ Uniforms} \\ \hline 0 & & & 0.000 & $<$0.01 & $<$0.01 & -132.327 & \\ \hline \hline
\multicolumn{8}{|c|}{Fisher and Lee (1993 and 1994) $CAR(1)$ von Mises $vM(\mu_t,\kappa)$ model} \\
\multicolumn{8}{|c|}{(reported estimates)} \\
\multicolumn{8}{|c|}{$\mu_t = \mu + 2 \arctan (\alpha \tan(0.5(\theta_{t-1}-\mu)))$} \\ \hline
& 0.6695 (0.1492) & & 0.106 & $<$0.01 & $<$0.01 & -81.370 & \\ \hline
\multicolumn{8}{|c|}{Great-circle final model} \\ \hline
$M$ & AR Coeffs. & $R^2_{cos}$ & Range & Kuiper & Watson & loglik & $\alpha$ angle \\
& (std. error) & & p-value & p-value & p-value & & \\ \hline 4 & AR1: 0.2988 (0.1047) & 0.497 & 0.295 & $>$0.15 & $>$0.10 & -73.967 & \\
& AR2: 0.4225 (0.1049) & & & & & & \\ \hline \end{tabular}} \renewcommand{1}{1} \end{table}
\renewcommand{1}{1.00} \begin{table}
\caption{\label{Table03earthquakes} Earthquake data. The number of variables with non-zero parameter estimates (r), p-values of Kuiper and Watson tests, and loglikelihoods for the great- and small-circle nonnegative trigonometric sums (NNTS) models when applying LASSO procedures with penalty coefficients $\lambda_{min}$ and $\lambda_{1se}$ are shown.} \centering \scalebox{.7}{
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|} \hline
\multicolumn{6}{|c|}{Great-circle models $\lambda_{min}$} & \multicolumn{6}{|c|}{Small-circle models $\lambda_{min}$} \\ \hline
$M$ & r & Range & Kuiper & Watson & loglik & r & Range & Kuiper & Watson & loglik & $\alpha$ angle \\
& & p-value & p-value & p-value & & & p-value & p-value & p-value & & \\ \hline 1 & 6 & 0.237 & $>$0.15 & $>$0.10 & -2113.085 & 6 & 0.000 & $<$0.01 & $<$0.01 & -1570.802 & 0.785 \\ 2 & 6 & 0.160 & $>$0.15 & $>$0.10 & -2111.261 & 6 & 0.000 & $<$0.01 & $<$0.01 & -1704.455 & 0.794 \\ 3 & 0 & 0.041 & $>$0.15 & $>$0.10 & -2118.975 & 0 & 0.000 & $<$0.01 & $<$0.01 & -1747.595 & 0.801 \\ 4 & 0 & 0.049 & $>$0.15 & $>$0.10 & -2118.944 & 0 & 0.000 & $<$0.01 & $<$0.01 & -1830.444 & 0.811 \\ 5 & 0 & 0.051 & $>$0.15 & $>$0.10 & -2118.423 & 0 & 0.000 & $<$0.01 & $<$0.01 & -1975.683 & 0.814 \\ 6 & 0 & 0.130 & $>$0.15 & $>$0.10 & -2118.344 & 0 & 0.000 & $<$0.01 & $<$0.01 & -1919.651 & 0.821 \\ 7 & 8 & 0.635 & $>$0.15 & $>$0.10 & -2110.981 & 8 & 0.000 & $<$0.01 & $<$0.01 & -2052.423 & 0.824 \\ 8 & 0 & 0.258 & $>$0.15 & $>$0.10 & -2119.634 & 0 & 0.000 & $<$0.01 & $<$0.01 & -1974.984 & 0.831 \\ \hline
\multicolumn{6}{|c|}{Great-circle models $\lambda_{1se}$} & \multicolumn{6}{|c|}{Small-circle models $\lambda_{1se}$} \\ \hline 1 & 0 & 0.237 & $>$0.15 & $>$0.10 & -2119.060 & 0 & 0.000 & $<$0.01 & $<$0.01 & -1568.112 & 0.785 \\ 2 & 0 & 0.213 & $>$0.15 & $>$0.10 & -2118.976 & 0 & 0.000 & $<$0.01 & $<$0.01 & -1699.217 & 0.794 \\ 3 & 0 & 0.041 & $>$0.15 & $>$0.10 & -2118.975 & 0 & 0.000 & $<$0.01 & $<$0.01 & -1747.595 & 0.801 \\ 4 & 0 & 0.049 & $>$0.15 & $>$0.10 & -2118.944 & 0 & 0.000 & $<$0.01 & $<$0.01 & -1830.444 & 0.811 \\ 5 & 0 & 0.051 & $>$0.15 & $>$0.10 & -2118.423 & 0 & 0.000 & $<$0.01 & $<$0.01 & -1975.683 & 0.814 \\ 6 & 0 & 0.130 & $>$0.15 & $>$0.10 & -2118.344 & 0 & 0.000 & $<$0.01 & $<$0.01 & -1919.651 & 0.821 \\ 7 & 0 & 0.281 & $>$0.15 & $>$0.10 & -2119.345 & 0 & 0.000 & $<$0.01 & $<$0.01 & -2025.771 & 0.824 \\ 8 & 0 & 0.258 & $>$0.15 & $>$0.10 & -2119.634 & 0 & 0.000 & $<$0.01 & $<$0.01 & -1974.984 & 0.831 \\ \hline
\multicolumn{12}{|c|}{$i.i.d.$ Uniforms} \\ \hline 0 & 0 & 0.221 & $>$0.15 & $>$0.10 & -2120.910 & & & & & & \\ \hline \end{tabular}} \renewcommand{1}{1} \end{table}
\renewcommand{1}{1}
\begin{figure}\label{graph01datablueperiwinkles}
\end{figure}
\begin{figure}\label{graph01datablueperiwinklescosine}
\end{figure}
\begin{figure}
\caption{ Blue periwinkle data. Estimated mean and circular variance functions for great-circle nonnegative trigonometric sums (NNTS) resultant mean model with $M=8$ (solid line), Fisher and Lee (1992) mixed model (dashed line), Presnell et al. (1998) SPM mixed model (dot-dash line) and Rivest et al. (2015) mixed model (long-dash line) are shown.}
\label{graph01datablueperiwinklesMEANFUNCTIONS}
\end{figure}
\begin{figure}
\caption{ Wind direction data. The original and transformed data for the great- and small-circle nonnegative trigonometric sums (NNTS) models with $M=4$ are shown including autocorrelation (ACF) and partial
autocorrelation (PACF) functions.}
\label{graph02acfpacf}
\end{figure}
\begin{figure}
\caption{ Wind direction data. Density forecasts for the AR(2) great-circle nonnegative trigonometric sums (NNTS) model with $M=4$ are shown. The top left plot corresponds to hours 1 to 24, the bottom left plot corresponds to hours 49 to 72, and the top right plot corresponds to hours 25 to 48. The bottom right plot shows the scatterplot of the original data with hourly wind direction in the horizontal axis and hour in the vertical axis, including the mean function (solid line). The filling grey intensity in the density forecasts changes at the corresponding observed value.}
\label{graphdatab23fisherforecasts}
\end{figure}
\end{document}
|
arXiv
|
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\maketitle
\noindent {\bf Abstract:} Let $X:=\{X(t)\}_{t\ge0}$ be a
generalized fractional Brownian motion (GFBM) introduced by Pang and Taqqu (2019): $$
\big\{X(t)\big\}_{t\ge0}\overset{d}{=}\left\{ \int_{\mathbb R} \left((t-u)_+^{\alpha}-(-u)_+^{\alpha} \right)
|u|^{-\gamma} B(du) \right\}_{t\ge0}, $$ with parameters $\gamma \in (0, 1/2)$ and $\alpha\in \left(-\frac12+ \gamma , \, \frac12+ \gamma \right)$.
Continuing the studies of sample path properties of GFBM $X$ in Ichiba, Pang and Taqqu (2021) and Wang and Xiao (2021), we establish integral criteria for the lower functions of $X$ at $t=0$ and at infinity by modifying the arguments of Talagrand (1996). As a consequence of the integral criteria, we derive the Chung-type laws of the iterated logarithm of $X$ at the $t=0$ and at infinity, respectively. This solves a problem in Wang and Xiao (2021).
\vskip0.3cm
\noindent{\bf Keyword:} {Generalized fractional Brownian motion; lower function; Chung's LIL;
small ball probability.}
\vskip0.3cm
\noindent {\bf MSC: } {60G15, 60G17, 60G18, 60G22.} \vskip0.3cm
\section{Introduction}
The generalized fractional Brownian motion (GFBM, in short) $X:=\big\{X(t)\big\}_{t\ge0}$ is a centered Gaussian self-similar process introduced by Pang and Taqqu \cite{PT2019} as the scaling limit of a sequence of power-law shot noise processes. It has the following integral representation:
\begin{align}\label{eq X}
\big\{X(t)\big\}_{t\ge0}\overset{d}{=}&\left\{ \int_{\mathbb R} \left((t-u)_+^{\alpha}-(-u)_+^{\alpha} \right)
|u|^{-\gamma} B(du) \right\}_{t\ge0},
\end{align}
where the parameters $\gamma$ and $\alpha$ satisfy
\begin{align}\label{eq constant}
\gamma\in\left(0,\ \frac12\right), \ \ \alpha\in \left(-\frac12+\gamma, \ \frac12+\gamma \right),
\end{align} and $B$ is a two-sided Brownian motion on $\mathbb R$. It follows that the Gaussian process $X$ is self-similar with index $H$ given by
\begin{align}\label{eq H}
H=\alpha-\gamma+\frac12\in(0,1).
\end{align}
If $\gamma=0$, then $X$ becomes an ordinary fractional Brownian motion (FBM, in short) $B^H$, which can be represented as: \begin{align}\label{eq FBM} \big\{B^H(t)\big\}_{t\ge 0}\overset{d}{=}\left\{ \int_{\mathbb R} \Big((t-u)_+^{H-\frac12}-(-u)_+^{H-\frac12} \Big) B(du) \right\}_{t\ge0}. \end{align}
As shown by Pang and Taqqu \cite{PT2019}, GFBM $X$ preserves the self-similarity property while the
factor $|u|^{-\gamma}$ introduces non-stationarity of the increments, which is useful for reflecting the non-stationarity increments property in physical systems. Ichiba, Pang and Taqqu \cite{IPT2020} established
the H\"older continuity, the functional and local laws of the iterated logarithm of GFBM and showed that these properties are determined by the self-similarity index $H = \alpha-\gamma+ 1/2$. More recently, Ichiba, Pang and Taqqu \cite{IPT2020b} studied the semimartingale properties of GFBM $X$ and its mixtures and applied them to model the volatility processes in finance.
In \cite{WX2021}, we studied some precise sample path properties of GFBM $X$, including the exact uniform modulus of continuity, small ball probabilities, and Chung's LIL at any fixed point $t>0$. In contrast to the theorems of Ichiba, Pang and Taqqu \cite{IPT2020b}, our results show that the uniform modulus of continuity and Chung's LIL at any fixed point $t>0$ are determined mainly by the parameter $\alpha$, while $\gamma$ plays a less important role. Roughly speaking, for $\alpha<1/2$, the results in \cite{WX2021} on uniform modulus of continuity and Chung's LIL at $t>0$ are analogous to the corresponding results for a fractional Brownian motion with index $\alpha+1/2$. For example, Theorem 1.5 in \cite{WX2021} shows the following Chung's LILs for GFBM $X$ and its derivative $X'$ (which exists when $\alpha>1/2$) at any fixed $t>0$: \begin{itemize} \item[(a).]\, If $\alpha\in(-1/2+\gamma/2,1/2)$, then there exists a constant $c_{1,1}\in(0,\infty)$
such that for every $t>0$,
\begin{align}\label{eq LIL01}
\liminf_{r\rightarrow0+} \sup_{ |h|\le r}\frac{ |X(t+h)-X(t)|} {r^{\alpha+1/2}/(\ln \ln r^{-1})^{\alpha+1/2}}
=c_{1,1}t^{-\gamma}, \ \ \ \text{a.s.}
\end{align}
\item[(b).]\,
If $\alpha\in(1/2, \, 1/2+\gamma/2)$, then there exists a constant $c_{1,2}\in(0,\infty)$
such that for every $t>0$,
\begin{align}\label{eq LIL01b}
\liminf_{r\rightarrow0+} \sup_{ |h|\le r}\frac{ |X'(t+h)-X'(t)|} {r^{\alpha-1/2}/(\ln \ln r^{-1})^{\alpha-1/2}}
=c_{1,2} t^{-\gamma}, \ \ \ \ \text{a.s.}
\end{align}
\end{itemize} As $t\rightarrow 0+$, the terms on the right-hand sides of \eqref{eq LIL01} and \eqref{eq LIL01b} tend to $+\infty$. This suggests that the scaling functions on the left-hand sides of \eqref{eq LIL01} and \eqref{eq LIL01b} are not optimal in the neighborhood of the origin. The problem on Chung's LIL for GFBM $X$ at $t=0$ was left open in \cite{WX2021}.
The main objective of the present paper is to establish Chung's LILs of GFBM $X$ at $t=0$ and at infinity. In fact, we will prove more precise results, namely, integral criteria for the lower functions of
$M:=\big\{M(t)\big\}_{t\ge0}:=\big\{\sup_{0\le s\le t} |X(s)|\big\}_{t\ge0}$ at $t=0$ and at infinity, which imply the following Chung's LILs of GFBM $X$.
\begin{theorem}\label{thm Chung origin} Let $X=\{X(t)\}_{t\ge0}$ be a GFBM with parameters $\alpha$ and $\gamma$. Suppose
$\alpha\in (-1/2+\gamma, 1/2)$. \begin{itemize} \item[(a).] There exists a positive constant $\kappa_1\in (0,\infty)$ such that \begin{equation}\label{eq 11}
\liminf_{t\rightarrow 0+}\sup_{0\le s\le t}\frac{ |X(s)|}{t^H /\left(\ln\ln t^{-1}\right)^{\alpha+1/2}}=\kappa_1, \ \ \ \hbox{ a.s.} \end{equation}
\item[(b).] There exists a positive constant $\kappa_2\in (0,\infty)$ such that \begin{equation}\label{eq 21}
\liminf_{t\rightarrow\infty}\sup_{0\le s\le t}\frac{|X(s)|}{t^H /\left(\ln\ln t\right)^{\alpha+1/2}}=\kappa_2, \ \ \ a.s. \end{equation} \end{itemize}
\end{theorem}
Similarly to the theorems of Ichiba, Pang and Taqqu \cite{IPT2020b} mentioned above, the self-similarity index $H$ plays an essential role in (\ref{eq 11}) and (\ref{eq 21}). The results in \cite{IPT2020b,WX2021} and the present paper show that GFBM $X$ is an interesting example of self-similar Gaussian processes which has richer sample path properties than the ordinary FBM and its close relatives such as the Riemann-Liouville FBM (cf. e.g., \cite{CLRS11,ElN2011}), bifractional Brownian motion (cf. \cite{HV03, LeiN09, RussoTudor, TX2007}), and the sub-fractional Brownian motion (cf. \cite{BGT04, ElN2012, Tudor07,YanShen10}). In this sense, GFBM is a good object (see also \cite{TT18} for other examples related to stochastic partial differential equations driven by a fractional-colored Gaussian noise) that can be studied for the purpose to develop a general theoretical framework for studying the fine properties of all (or at least a wide class of) self-similar Gaussian processes which, to the best of our knowledge, is still not complete yet.
In the literature, limit theorems of the forms (\ref{eq 11}) and (\ref{eq 21}) are also called ``the other law of the iterated logarithm" and there has been a long history of studying them. Chung \cite{Chung} proved that if $S_n = \eta_1 +\cdots +\eta_n$, where $\{\eta_k\}$ is a sequence of i.i.d. random variables with mean 0, variance 1 and finite third moment, then
$$
\liminf_{n\rightarrow\infty}\frac{\max_{1\le k\le n} |S_k|}{\sqrt{n/\ln\ln n}} =\frac{\pi}{\sqrt 8}, \ \ \ \text{a.s.}
$$
Chung \cite{Chung} also gave the corresponding large time result for Brownian motion. The extra condition of
finite third moment on $\eta_1$ in \cite{Chung} was removed by Jain and Pruitt \cite{JP75}. There have been many
extensions of these results. For example, Cs\'aki gave a converse of lower/upper class in \cite{Cs1978}, and he
found an interesting connection between Chung's ``other law" for Brownian motion and Strassen's LIL in \cite{Cs1980}. Kuelbs et al \cite{KLT1994} studied Chung's functional LIL for Banach space-valued Gaussian random vectors. Their results are applicable to Brownian motion and provide interesting refinements to those in \cite{Cs1980}. Monrad and Rootz\'en \cite{MR1995} proved Chung's LIL for a large class of Gaussian processes that have the property of strong local nondeterminism. Li and Shao \cite {LiShao01} and Xiao \cite{Xiao1997} extended the Chung's LIL in \cite{MR1995} to Gaussian random fields with stationary increments. Chung's LILs have also been studied for non-Gaussian processes, we refer to Buchmann and Maller \cite{BM} and the references therein for more information.
\begin{remark} {\rm The following are some remarks about Chung's LILs in Theorem \ref{thm Chung origin}.
(i). Notice that the cases of $\alpha = 1/2$ and $\alpha\in (1/2, 1/2+\gamma)$ are excluded in Theorem \ref{thm Chung origin}. In the first case, the sample functions of $X$ are not differentiable, while in the second case the sample functions of $X$ are differentiable on $(0, \infty)$. In both cases, we have not be able to solve the problem whether (\ref{eq 11}) and (\ref{eq 21}) hold or not because the optimal small ball
probability estimates for $\max_{t\in[0, 1]} |X(t)|$ have not been established for GFBM $X$ yet. See
\cite[Remark 5.1]{WX2021} for more information.
It is worth mentioning that, when $\alpha\in (1/2, 1/2+\gamma)$, $X$ has a modification that is continuously differentiable and its derivative $X'$ is a self-similar process with index $H' = \alpha-\gamma- 1/2 < 0$. The exact uniform modulus of continuity on $[a, b]$ with $0<a <b<\infty$ and Chung's LIL at any fixed $t>0$ have been proved for $X'$ in \cite{WX2021}. However, the asymptotic properties of $X'$ at $t = 0$ or at infinity have not been studied. Since $H' <0$, it is expected that $X'(t) \to \infty$ as $t \to 0+$ and $X'(t) \to 0$ as $t \to \infty$. It would be interesting from the viewpoint of the aforementioned general theoretical framework for self-similar Gaussian processes to prove both ordinary LIL and Chung's LIL of $X'$ at $t= 0$ and at $\infty$.
(ii). To prove Theorem \ref{thm Chung origin}, we modify the argument of Talagrand \cite{Tal96}, which is concerned with the lower functions of FBM at $\infty$, to establish integral criteria for the lower functions of GFBM $X$ at $t= 0$ and at infinity. We remark that, in \cite{ElN2011, ElN2012}, El-Nouty has extended Talagrand's result to the Riemann-Liouville FBM and the sub-FBM to characterize their lower functions at $\infty$.
For studying the lower functions of $M=\big\{\sup_{0\le s\le t} |X(s)|\big\}_{t\ge0}$ at $t = 0$ in this paper, the main difficulty comes from the singularity of the second moment of the increment of $X$ at $t = 0$. To elaborate, we consider a decomposition of GFBM:
\begin{equation}\label{eq decom}
\begin{split}
X(t) &=\int_{-\infty}^0 \big((t-u)^{\alpha}-(-u)^{\alpha} \big) (-u)^{-\gamma} B(du) +\int_0^t (t-u)^{\alpha}
u^{-\gamma} B(du)\\
&=: Y(t) + Z(t). \end{split} \end{equation} The Gaussian processes $Y=\big\{Y(t)\big\}_{t\ge0}$ and $Z=\big\{Z(t)\big\}_{ t\ge0}$ are independent. The process $Z$ in \eqref{eq decom} is called a {\it generalized Riemann-Liouville FBM}, using the terminology of Ichiba, Pang and Taqqu \cite{IPT2020}. By \cite[Lemmas 2.1 and 3.1]{WX2021}, there exist positive constants $c_{1, i}$, $i=3,\cdots, 6$, such that for all $0 < s < t$, \begin{align}\label{Eq: Ymoment1}
c_{1,3}\frac{|t-s|^2}{t^{2-2H}}\le \mathbb E\left[\big(Y(t)-Y(s) \big)^2\right]\le c_{1,4}\frac{|t-s|^2}{s^{2-2H}}.
\end{align}
and
\begin{align}\label{Eq: Zmoment1}
c_{1,5}\frac{|t-s|^{2\alpha+1}}{t^{2\gamma}}\le \mathbb E\left[\big(Z(t)-Z(s) \big)^2\right]
\le c_{1,6}\frac{|t-s|^{2\alpha+1}}{s^{2\gamma}}.
\end{align} These bounds are optimal when $s \le t \le c_{1,7}s$ for any constant $c_{1,7}>1$. Therefore, the small values of $s$ have some effects on $Y$ and $Z$ when $H<1$ and $\gamma>0$, respectively. The singularity at $s=0$ brings two technical difficulties when we modify the approach in Talagrand \cite{Tal96}: one is in estimating the metric entropy
for proving Proposition \ref{prop mtu1}; the other is in constructing of the sequences in Section
\ref{subsect tn} in order to use the (generalized) Borel-Cantelli lemma.
(iii). From the proofs of Theorems \ref{thm Chung origin} and \ref{thm main 0}, we can verify that the conclusions of Theorem \ref{thm Chung origin} also hold for the generalized Riemann-Liouville FBM $Z$ in \eqref{eq decom}. For example, when $\alpha\in (-1/2+\gamma, 1/2)$, \begin{equation}\label{eq Z11}
\liminf_{t\rightarrow 0+}\sup_{0\le s\le t}\frac{ |Z(s)|}{t^H /\left(\ln\ln t^{-1}\right)^{\alpha+1/2}}=c_{1,8}\in (0,\infty) \ \ \ a.s. \end{equation} For the process $Y$ defined in \eqref{eq decom}, when $\alpha\in (-1/2+\gamma, 1/2)$
we obtain that by \eqref{eq 11} and \eqref{eq Z11}, \begin{equation}\label{eq Y11}
\liminf_{t\rightarrow 0+}\sup_{0\le s\le t}\frac{ |Y(s)|}{t^H /\left(\ln\ln t^{-1}\right)^{\alpha+1/2}}=c_{1,9}\in [0,\infty) \ \ \ a.s. \end{equation} Since an optimal upper bound for the small ball probability estimates has not been established for $Y$ yet (see \cite[Lemma 7.1]{WX2021}), we are not able to decide if $c_{1,9} > 0$ or $c_{1,9} = 0$.
} \end{remark}
The rest of this paper is organized as follows. In Section 2, we state Theorems \ref{thm main 0} and \ref{thm main infty} which provide integral criteria for the lower functions of $M$ at $t=0$ and at infinity. From these integral criteria, we derive Chung's LILs for $X$ in Theorem \ref{thm Chung origin}.
Section \ref{sec:prelim} contains some preliminary results. In Section 4 and Section 5, we prove Theorems \ref{thm main 0} and \ref{thm main infty}, respectively.
\section{Main results and Proof of Theorem \ref{thm Chung origin} }
The following definition of the lower classes for the process $M=\big\{M(t)\big\}_{t\ge0}$ is adapted from \cite{Revesz}. This book provides a systematic and extensive account on the studies of lower and upper classes for Brownian motion, random walks and their functionals.
\begin{definition}\label{Def:LL}
\begin{itemize}
\item[(a).] A function $f(t), t>0$, belongs to the lower-lower class of the process $M$ at $\infty$
(resp. at $0$), denoted by $f\in LLC_{\infty}(M)$ (resp. $f\in LLC_{0}(M)$), if for almost all
$\omega\in \Omega$ there exists $t_0=t_0(\omega)$ such that $M(t)\ge f(t)$ for every $t>t_0$
(resp. $t<t_0$).
\item[(b).]\, A function $f(t), t>0$, belongs to the lower-upper class of the process $M$ at $\infty$ (resp. at $0$), denoted by $f\in LUC_{\infty}(M)$ (resp. $f\in LUC_{0}(M)$), if for almost all $\omega\in \Omega$ there exists a sequence $0<t_1(\omega)<t_2(\omega)<\cdots<$ with $t_n(\omega)\uparrow\infty$ (resp. $t_1(\omega) > t_2(\omega)>\cdots>$ with $t_n(\omega)\downarrow0$), as $n\rightarrow \infty$, such that $M(t_n(\omega)) \le f(t_n(\omega)), n\in \mathbb N$. \end{itemize} \end{definition}
Since in the present paper, $M(t)=\sup_{0\le s\le t} |X(s)|$, we will also write the lower classes in Definition \ref{Def:LL} as $LLC_{0}(X)$ and $LUC_{0}(X)$ and call a function $f \in LLC_{0}(M)$ (resp. $f \in LUC_{0}(M)$) a lower-lower (resp. lower-upper) function of $X$ at $t = 0$. It is known from Talagrand \cite{Tal96} that small ball probability estimates are essential for studying the lower classes of a stochastic process.
By the self-similarity of GFBM $X$, we have \begin{equation}\label{eq M1} \mathbb P\big(M(t)\le \theta t^H \big)=\mathbb P\big(M(1)\le \theta\big)=:\varphi(\theta). \end{equation} Wang and Xiao \cite{WX2021} proved the following small ball probability estimates for GFBM: If $\alpha\in (-1/2+\gamma, \,1/2)$, then there exist constants $\kappa_3>
\kappa_4>0$ such that for all $t>0$ and $ 0<\theta<1$,
\begin{align*}\label{eq small X} \exp\bigg(- \kappa_{3} \Big(\frac{t^H}{\theta}\Big)^{\frac{1}{\beta}} \bigg)\le
\mathbb P\bigg\{\sup_{s\in [0,t]} |X(s)|\le \theta \bigg\}
\le \exp\bigg(- \kappa_4 \Big(\frac{t^H}{\theta}\Big)^{\frac{1}{\beta}} \bigg).
\end{align*} Here and in the sequel, $\beta =\alpha+1/2$. It follows that for any $\theta\in(0,1)$,
\begin{equation}\label{eq varphi1} \exp\Big(- \kappa_{3} \theta^{-\frac{1}{\beta}} \Big)\le \varphi(\theta)\le \exp\Big(- \kappa_{4} \theta^{-\frac{1}{\beta}} \Big). \end{equation}
Now we state our first main result, which gives an integral criterion for the lower functions of GFBM $X$ at $t = 0$. It shows that, besides $\beta =\alpha+1/2$, the self-similarity index $H = \alpha-\gamma+\frac12$ plays an essential role. \begin{theorem}\label{thm main 0} Assume $\alpha\in (-1/2+\gamma, \,1/2)$. Let $\xi:(0,\mathrm {e}^{-\mathrm {e}}] \rightarrow (0,\infty)$ be a nondecreasing continuous function. \begin{itemize} \item[(a)] (Sufficiency). If \begin{equation}\label{eq 01} \frac{\xi(t)}{t^H}\ \text{ is bounded and}\ I_0(\xi):= \int_0^{\mathrm {e}^{-\mathrm {e}}} \left(\frac{\xi(t)}{t^H}\right)^{-1/\beta} \varphi\left(\frac{\xi(t)}{t^H}\right)\frac{dt}{t}<+\infty, \end{equation} then $\xi\in LLC_0(X)$. \item[(b)](Necessity). Conversely, if $\frac{\xi(t)}{t^{(1+\varepsilon_0)H}}$ is non-increasing for some constant $\varepsilon_0>0$ and if $\xi\in LLC_0(X)$, then \eqref{eq 01} holds. \end{itemize} \end{theorem}
Theorem \ref{thm main infty} is an analogous result for GFBM $X$ at infinity.
\begin{theorem}\label{thm main infty} Assume $\alpha\in (-1/2+\gamma, \,1/2)$. Let $\xi:[\mathrm {e}^{\mathrm {e}},\infty) \rightarrow (0,\infty)$ be a nondecreasing continuous function. Then $\xi\in LLC_{\infty}(X)$ if and only if \begin{equation}\label{eq infty} \frac{\xi(t)}{t^H}\ \text{ is bounded and}\ I_{\infty}(\xi):=\int_{\mathrm {e}^\mathrm {e}}^{\infty} \left(\frac{\xi(t)}{t^H}\right)^{-1/\beta} \varphi\left(\frac{\xi(t)}{t^H}\right)\frac{dt}{t}<+\infty. \end{equation} \end{theorem}
The proofs of Theorems \ref{thm main 0} and \ref{thm main infty} will be given in Sections 4 and 5 below. First, let us apply them to prove Theorem \ref{thm Chung origin}.
We will make use of the following zero-one laws at $t=0$ and $ \infty$. Eq. \eqref{Eq:Chung01law}
follows from \cite[Proposition 3.3]{TX2007} which provides a zero-one law for the lower class of
(not necessarily Gaussian) self-similar processes with ergodic scaling transformations. Recall from
\cite{Taka89,TX2007} that for every $a> 0$, $a\ne 1$, the scaling transformation $S_a$ of $X$ is
defined by $S_a(X) = \{a^{-H} X(at)\}_{t \ge 0}$. Notice that for any $H$-self-similar process $X$,
the scaling transformation $S_{ a}$ preserves the distribution of $X$. Hence the notions of
ergodicity and mixing of $S_{ a}$ can be defined in the usual way, cf. Cornfeld et al. \cite{CFS}.
Following Takashima \cite{Taka89}, we say that an $H$-self-similar process $X = \{X(t)\}_{ t \ge 0}$ is ergodic (resp. strong mixing) if for every $a > 0, a \ne 1$, the scaling transformation $S_{ a}$ is ergodic (resp. strong mixing). This, in turn, is equivalent to saying that all the shift transformations for the corresponding stationary process obtained via Lamperti's transformation $L(X)= \{ e^{-H t} X(e^t)\}_{ t \in \mathbb R}$ are ergodic (resp. strong mixing).
For GFBM $X$ in (\ref{eq X}), in order to verify the ergodicity of its scaling transformations, we use the representation (\ref{eq X}) to show that the autocovariance function of $L(X)$ satisfies $e^{-H t} \mathbb E\big(X(e^t)X(1)\big) = O(e^{-\kappa_5 t}) $ as $t \to \infty$, where $\kappa_5 = \min\{\frac 1 2 - \gamma, \, \frac 1 2 + \gamma - \alpha\} > 0$. By the Fourier inversion formula, $L(X)$ has a continuous spectral density function. It follows from \cite[Theorem 8]{Ma} that $L(X)$ is strong mixing and thus is ergodic. Hence,
\cite[Proposition 3.3]{TX2007} is applicable to GFBM $X$ and \eqref{Eq:Chung01law} follows.
The proof of (\ref{Eq:Chung01law2}) is similar to that of \cite[Proposition 3.3]{TX2007} with minor modifications. See \cite{Taka89} for more zero-one laws for self-similar processes with ergodic scaling transformations.
\begin{lemma}\label{lem:01lawZ}
Assume $\alpha\in(-1/2+\gamma,\,1/2)$. There exist constants $c_{2,1}, c_{2,1}' \in[0,\infty]$
such that \begin{equation}\label{Eq:Chung01law}
\liminf_{t\rightarrow0+} \sup_{0\le s\le t} \frac{ |X(s)|} {t^{H}/(\ln\ln t^{-1})^{\beta}}
=c_{2,1},\ \ \ \ \ \ \text{a.s.} \end{equation} and \begin{equation}\label{Eq:Chung01law2}
\liminf_{ t\rightarrow\infty} \sup_{0\le s\le t} \frac{ |X(s)|} {t^{H}/(\ln\ln t)^{\beta}}
=c_{2,1}',\ \ \ \ \ \ \text{a.s.} \end{equation} \end{lemma}
For a constant $\lambda>0$, let $f_\lambda$ be the function defined by \begin{align}
f_\lambda(t):= \frac{\lambda t^H}{( \ln|\ln t|)^{\beta}}, \ \ \ t>0. \end{align}
\begin{proof}[Proof of Theorem \ref{thm Chung origin}] When $\alpha\in(-1/2+\gamma,\,1/2)$, by combining Theorem \ref{thm main 0} (resp. Theorem \ref{thm main infty}) with (\ref{eq varphi1}), we derive that if $\lambda<\kappa_4^{\beta}$, then $f_{\lambda}\in LLC_{0}(X)$ (resp. $f_{\lambda}\in LLC_{\infty}(X)$), else if $\lambda>\kappa_3^{\beta}$, then $f_{\lambda}\in LUC_{0}(X)$ (resp. $f_{\lambda}\in LUC_{\infty}(X)$). These, together with the zero-one law in Lemma \ref{lem:01lawZ}, imply the desirable results in Theorem
\ref{thm Chung origin}.
\end{proof}
\section{Some preliminary results }\label{sec:prelim}
In this section, we provide some preliminary results that will be useful for proving Theorems \ref{thm main 0} and \ref{thm main infty}. Their proofs are modifications of those in Talagrand \cite{Tal96}.
\begin{lemma} If $\alpha\in (-1/2+\gamma, \,1/2)$, then there exists a constant $c_{3,1}>0$ such that for all $0<t<u$, $\theta, \eta>0$, \begin{equation}\label{eq Mtu} \mathbb P \big(M(t)\le \theta t^H,\, M(u)\le \eta\big)\le 2\varphi(\theta)\exp\bigg(-\frac{u-t}{c_{3,1} \, u^{\frac{\gamma}{\beta}}\eta^{\frac{1}{\beta}}}\bigg). \end{equation} \end{lemma}
\begin{proof} If $(u-t)/\big(u^{\frac{\gamma}{\beta}}\eta^{\frac{1}{\beta}}\big)\le 2$, then it is obvious that \eqref{eq Mtu} holds with $c_{3,1}=2/\ln 2$. Hence, we only need to prove \eqref{eq Mtu} in the case of $(u-t)/\big(u^{\frac{\gamma}{\beta}}\eta^{\frac{1}{\beta}}\big)>2$. The proof is divided into two steps.
{\bf Step 1.} We define an increasing sequence $\{t_n\}_{n\ge0}$ as follows. Set $t_0=t$. For any $n\ge1$, if $t_{n-1}$ has been defined, then we choose $t_n>t_{n-1}$ such that \begin{align*} t_n -t_{n}^{\frac{\gamma}{\beta}} \eta^{\frac{1}{\beta}}=t_{n-1}. \end{align*} Consider the event \begin{align*} A_k:=\big\{M(t)\le \theta t^H \big\}\cap \big\{M({t_k}) \le \eta\big\}. \end{align*} It suffices to prove that for any $k\ge1$, we have \begin{align}\label{Eq: PAk} \mathbb P(A_k)\le \varphi(\theta) \rho^{k}, \end{align} where $\rho\in (0,1)$ is a constant that depends on $\beta$ and $\gamma$ only. Indeed, if $k$ is the largest integer such that $t_k\le u$, then $k+1\ge (u-t)/\big({u^{\frac{\gamma}{\beta}} \eta^{\frac{1}{\beta}}}\big)$, which implies $k\ge (u-t)/\big({2u^{\frac{\gamma}{\beta}} \eta^{\frac{1}{\beta}}}\big)$ and \begin{align*} A_k\supset \big\{M(t)\le \theta t^H \big\}\cap \big\{M(u)\le \eta\big\}. \end{align*} Hence, \eqref{Eq: PAk} implies \eqref{eq Mtu}.
{\bf Step 2.} We prove \eqref{Eq: PAk} by induction over $k$. The result holds for $k=0$ by \eqref{eq M1}. For the induction step, we observe that $$
A_{k+1}\subset A_k \cap \big\{|U|\le 2\eta \big\}, $$ where $U:= X({t_{k+1}})-X({t_k})$. By using \eqref{eq X}, $U$ can be rewritten as follows $U=U_1+U_2$, where \begin{align*} U_1&:=\int_{t_k}^{t_{k+1}} (t_{k+1}-u)^{\alpha}u^{-\gamma}B(du), \\
U_2&:=\int_{-\infty}^{t_{k}} \big[(t_{k+1}-u)^{\alpha}- (t_{k}-u)^{\alpha} \big]|u|^{-\gamma}B(du). \end{align*} Notice that $U_1$ is a Gaussian random variable with
$$ \mathbb E \big[U_1\big]=0 \ \ \text{ and }\ \ \mathrm {Var}\big(U_1\big)\ge \frac{1}
{t_{k+1}^{2\gamma}}|t_{k+1}-t_k|^{2\beta}=\eta^{2}. $$ Thus, we have
$$ \mathbb P\big(|U_1|\le 2\eta\big)\le \Phi(2)-\Phi(-2), $$ where $\Phi$ denotes the distribution function of a standard Gaussian random variable. Consequently, by Anderson's inequality \cite{And1955} and the independence of $U_1$ and $\sigma\big\{B(s); s\le t_k\big\}$, we have \begin{align*}
\mathbb P(A_{k+1})\le&\, \mathbb E\Big[
\mathbb P \Big(A_{k} \cap\big\{|U_1+U_2|\le 2\eta\big\}\,\big|\,\sigma\big\{ B(s); s\le t_k\big\}\Big)\Big]\\ = &\, \mathbb E\Big[ {\mathbbm 1}_{A_{k}} \cdot
\mathbb P\big( |U_1+U_2|\le 2\eta\,\big|\,\sigma\big\{B(s); s\le t_k\big\}\big)\Big]\\
\le & \, \mathbb P\big(A_{k}\big)\cdot\mathbb P\big(|U_1|\le 2\eta\big)\\ \le &\, \mathbb P\big(A_{k}\big)\cdot \big( \Phi(2)-\Phi(-2)\big). \end{align*} Therefore, we have proved \eqref{Eq: PAk} with $\rho=\Phi(2)-\Phi(-2)$. This finishes the proof of (\ref{eq Mtu}). \end{proof}
Set \begin{align} \psi(\theta):=-\log \varphi(\theta). \end{align} Then $\psi$ is positive and non-increasing. According to Borell \cite{Borell1974}, we know that $\psi$ is convex which implies the existence of the right derivative $\psi'$ of $\psi$. Thus,
$\psi'\le 0$ and $|\psi'|$ is non-increasing.
By the small ball probability estimates in (\ref{eq varphi1}), we see that there exists a constant $K_1\ge1$ such that for all $\theta<1$, \begin{align}\label{eq sbp} \frac{1}{K_1 \theta^{1/\beta}}\le \psi(\theta)\le \frac{K_1}{\theta^{1/\beta}}. \end{align}
The following lemmas give more properties of the functions $\varphi$ and $\psi$, which are similar to those in Talagrand \cite[Section 2]{Tal96}. \begin{lemma}\label{lem psi'} There exists a constant $K_2\ge \max\big\{ 2^{1+1/\beta},\, 2\left(2K_1^2\right)^{\alpha} K_1 \big\}$ such that for all $\theta\in (0, 1/K_2)$, \begin{align}\label{eq psi'}
- \frac{K_2}{\theta^{1+1/{\beta}}}\le \psi'(\theta) \le -\frac{ 1}{K_2\theta^{1+1/{\beta}}}. \end{align} \end{lemma}
\begin{lemma}\label{lem dom} There exists a constant $K_3\ge K_2 2^{1+1/\beta}$ such that for all $ \theta \le \varepsilon\le 2\theta<1$, \begin{align}\label{eq control}
\exp\left(-K_3 \frac{|\varepsilon-\theta|}{\theta^{1+1/\beta}}\right)\le \frac{\varphi(\varepsilon)}{\varphi(\theta)}
\le \exp\left(K_3 \frac{|\varepsilon-\theta|}{\theta^{1+1/\beta}}\right). \end{align} \end{lemma}
\begin{lemma}\label{lem increase} For all $\theta<\theta_0:=(\beta/K_2)^{\beta}$, the function $\theta^{-1/\beta}\varphi(\theta)$ is increasing. \end{lemma} Since the proofs of the above three lemmas are similar to the analogous lemmas for FBM in Talagrand \cite[Section 2]{Tal96}, we only prove Lemma \ref{lem increase} as an example. \begin{proof}[Proof of Lemma \ref{lem increase}] The right derivative of the function $\theta^{-1/\beta}\varphi(\theta)$ is $$ \left(-\frac{1}{\beta \theta} -\psi'(\theta)\right)\theta^{-1/\beta}\varphi(\theta), $$ which is positive in the interval $(0, \theta_0)$ for the positive constant $\theta_0=(\beta/K_2)^{\beta}$ by \eqref{eq psi'}. The proof is complete. \end{proof}
Given a Gaussian process $\{X(t)\}_{t\in \mathbb T}$, let us denote by $N(\mathbb T, \, d_X, \, \varepsilon)$
the smallest number of open balls of radius $\varepsilon$ for the canonical distance $d_X(s,t)=\|X(s)-X(t)\|_2$
that are needed to cover $\mathbb T$, where $\|\cdot\|_2$ is the $L^2(\mathbb P)$-norm. Recall the following lemma from Talagrand \cite{Tal96}. \begin{lemma}\cite[Lemma 2.1]{Tal96}\label{lem Dudley} There exists a universal constant $K_4$ such that for all $t_0\in \mathbb T$ and $x>0$, \begin{align}
\mathbb P\left(\sup_{t\in \mathbb T}\left|X(t)-X(t_0)\right| \ge K_4 x\int_0^{\infty} \sqrt{\log N(\mathbb T,\, d_X,\, \varepsilon) }d\varepsilon \right)\le \exp\big(-x^2\big). \end{align} \end{lemma}
The following proposition will be important for proving the necessity in Theorems \ref{thm main 0} and \ref{thm main infty}.
\begin{proposition}\label{prop mtu1} Assume $\tau=\min\left\{\frac14(1-H),\, \frac{H}{4},\, \frac14\left(\frac12-\gamma\right)\right\}$. Then for any $u>t>0$, $\theta>0, \eta>0$, we have
\begin{equation}\label{eq mtu1}
\begin{split}
& \mathbb P\Big( \left\{M(t)\le \theta t^H \right\}\cap \left\{ M(u)\le \eta u^H \right\}\Big)\\
&\le \exp\left[-\frac{1}{K}\left(\frac{u}{t}\right)^{\tau}\right]+\varphi(\theta)\varphi(\eta)
\exp\left[K \left(\frac{u}{t}\right)^{-\tau}\left(\theta^{-1-1/\beta}+\eta^{-1-1/\beta} \right)\right].
\end{split}
\end{equation}
\end{proposition} \begin{proof}
We set $v:=\sqrt{ut}$. The idea is that if $t\ll u$, then $t\ll v\ll u$. We recall the stochastic integral
representation in \eqref{eq X}, and define
$$G(s, x):= \left((s-x)_+^{\alpha}-(-x)_+^{\alpha} \right) |x|^{-\gamma}, \ \ \ \text{for } s, x\in \mathbb R. $$ Consider the following two processes: \begin{align}\label{eq X12}
X_1(s):=\int_{|x|\le v} G(s,x) B(dx),\, \,\,\,\, X_2(s):=\int_{|x|> v} G(s,x) B(dx). \end{align} Thus, $X(s)=X_1(s)+X_2(s)$ and the processes $X_1$ and $X_2$ are independent.
For any $\delta>0$, we have \begin{equation}\label{eq X12-0} \begin{split}
& \mathbb P\Big( \left\{M(t)\le \theta t^H \right\}\cap \left\{ M(u)\le \eta u^H \right\}\Big)\\
= &\, \mathbb P\left(\sup_{0\le s\le t} |X(s)|\le \theta t^H,\, \sup_{0\le s\le u} |X(s)|\le \eta u^H \right)\\
\le &\, \mathbb P\left(\sup_{0\le s\le t} |X_1(s)|\le \left(\theta+\delta\right) t^H, \, \sup_{0\le s\le u} |X_2(s)| \le \left(\eta+\delta\right) u^H \right)\\
&\,+ \mathbb P\left(\sup_{0\le s\le t} |X_2(s)|\ge \delta t^H \right)+\mathbb P\left( \sup_{0\le s\le u} |X_1(s)| \ge \delta u^H \right). \end{split} \end{equation} By the independence of $X_1$ and $X_2$, we have \begin{equation}\label{eq X12-1} \begin{split}
& \mathbb P\left(\sup_{0\le s\le t} |X_1(s)|\le \left(\theta+\delta\right) t^H, \, \sup_{0\le s\le u} |X_2(s)| \le \left(\eta+\delta\right) u^H \right)\\
=&\, \mathbb P\left(\sup_{0\le s\le t} |X_1(s)|\le \left(\theta+\delta\right) t^H\right)\cdot
\mathbb P\left(\sup_{0\le s\le u} |X_2(s)|\le \left(\eta+\delta\right) u^H \right). \end{split} \end{equation} Notice that \begin{equation}\label{eq X12-2} \begin{split}
& \mathbb P\left(\sup_{0\le s\le t} |X_1(s)|\le \left(\theta+\delta\right) t^H\right)\\
\le &\, \mathbb P\left(\sup_{0\le s\le t} |X(s)|\le \left(\theta+2\delta\right) t^H\right)+
\mathbb P\left(\sup_{0\le s\le t} |X_2(s)|\ge \delta t^H\right) \end{split} \end{equation} and \begin{equation}\label{eq X12-3} \begin{split}
& \mathbb P\left(\sup_{0\le s\le u} |X_2(s)|\le \left(\theta+\delta\right) u^H\right)\\
\le &\, \mathbb P\left(\sup_{0\le s\le u} |X(s)|\le \left(\theta+2\delta\right) u^H\right)+
\mathbb P\left(\sup_{0\le s\le u} |X_1(s)|\ge \delta u^H\right). \end{split} \end{equation} Plugging \eqref{eq X12-1} to \eqref{eq X12-3} into \eqref{eq X12-0}, we get \begin{equation}\label{eq X12-4} \begin{split}
& \mathbb P\Big( \left\{M(t)\le \theta t^H \right\}\cap \left\{ M(u)\le \eta u^H \right\}\Big)\\
\le &\, \varphi(\theta+2\delta)\varphi(\eta+2\delta) + 2 \mathbb P\left(\sup_{0\le s\le t} |X_2(s)|
\ge \delta t^H\right)
+ 2\mathbb P\left(\sup_{0\le s\le u} |X_1(s)|\ge \delta u^H\right). \end{split} \end{equation} By the convexity of $\psi$ and \eqref{eq psi'}, we have \begin{align*} \psi(\theta+2\delta)\ge &\, \psi(\theta)+2\delta\psi'(\theta) \ge \, \psi(\theta)-\frac{2\delta K_2}{\theta^{1+1/\beta}}. \end{align*} Hence, we have $$ \varphi(\theta+2\delta)\le \varphi(\theta)\exp\left(\frac{2\delta K_2}{\theta^{1+1/\beta}}\right). $$ This and a similar inequality for $\varphi(\eta+2\delta)$ show that it suffices to show that when $\delta=(t/u)^{\tau}$ with $\tau=\min\left\{\frac14(1-H), \, \frac{H}{4}, \,\frac14\left(\frac12-\gamma\right)\right\}$, the last two terms of \eqref{eq X12-4} are bounded by $\exp\big(-\frac{1}{K}\big({u}/{t}\big)^{\tau}\big)$. This will be done through the following two lemmas whose proofs are postponed.
\begin{lemma}\label{lem diam} If $\alpha\in (-1/2+\gamma,1/2)$, then there exist constants $c_{3,2}, c_{3,3}>0$ satisfying that \begin{itemize} \item[(a)] For any $0\le s \le u$, we have \begin{equation}\label{Eq: B11}
\|X_1(s)\|_2\le c_{3,2} \left\{\begin{array}{ll}
v^{H}, \ \quad &\hbox{ if }\ \beta\in \left( \gamma, \, \frac12\right);\\ v^{\frac{1 }{2}-\gamma} u^{\beta-\frac12}, \ \quad &\hbox{ if } \ \beta\in \left[\frac12 , 1+\gamma\right). \end{array} \right.
\end{equation} \item[(b)] For any $0\le s\le t$, we have \begin{equation}\label{Eq: B12}
\|X_2(s)\|_2\le c_{3,3} t v^{\beta-\gamma-1}. \end{equation} \end{itemize} \end{lemma}
Recall $X=Y+Z$ in \eqref{eq decom}. Observing the moment estimates of $Y$ and $Z$ in \eqref{Eq: Ymoment1} and \eqref{Eq: Zmoment1}, it is easy to see that for any $b>a>0$, $\varepsilon>0$, the covering numbers of $[a, b]$ in the canonical metrics of $Y$ and $Z$ satisfy $$N\big([a,b], \, d_Y, \, \varepsilon\big) \le c_{1,4}^{\frac12}(b-a) a^{H-1} \varepsilon^{-1}, \ \ \ N\big([a,b], \, d_Z, \, \varepsilon\big) \le c_{1,6}^{\frac12} (b-a) a^{-\frac{\gamma}{\beta} } \varepsilon^{-\frac{1}{\beta}}. $$ The next lemma gives their estimates when $a=0$. \begin{lemma}\label{lem covering number} There exist constants $c_{3,4}, c_{3,5}>0$ satisfying that for any $b>0,\varepsilon>0$, \begin{align}\label{Eq cov Y} N \big([0, b], d_{Y}, \varepsilon\big) \le c_{3,4} b^{H }\, \varepsilon^{-1} \end{align} and \begin{align}\label{Eq cov Z} N\big([0, b], d_{Z}, \varepsilon\big) \le c_{3,5} b^{\frac{H}{\beta}}\, \varepsilon^{- \frac 1 {\beta}}. \end{align} \end{lemma}
Combining Lemma \ref{lem diam} and Lemma \ref{lem covering number}, we derive that there exist constants $c_{3,6}, c_{3,7}>0$ satisfying that
\begin{equation}\label{Eq: T1} \begin{split} &\int_0^{\infty} \sqrt{\log N([0,u], d_{X,1},\varepsilon)}d\varepsilon \\
\le&\, c_{3,6} \left\{\begin{array}{ll}
v^{H}\sqrt{\log \left(\frac u v\right) }, \ \quad &\hbox{ if }\ \beta\in \left( \gamma, \frac12\right);\\
u^{H}\left(\frac{v}{u} \right)^{ 1/2-\gamma}\sqrt{\log \left(\frac uv\right)}, \ \quad &\hbox{ if } \
\beta\in \left[ \frac12, 1+\gamma\right), \end{array} \right.
\end{split}
\end{equation}
and \begin{align}\label{Eq: T2} \int_0^{\infty} \sqrt{\log N\big([0,t], d_{X,2},\varepsilon\big)}d\varepsilon \le c_{3,7} \left(\frac{t}{v}\right) v^{H}\sqrt{\log \left(\frac v t\right)}. \end{align} Since $t/v=v/u=\sqrt{t/u}$, the conclusion in Proposition \ref{prop mtu1} follows from Lemma \ref{lem Dudley},
\eqref{Eq: T1}, \eqref{Eq: T2} and the choice of $\tau$.
To prove \eqref{Eq: T1} and \eqref{Eq: T2}, we use the decomposition of $X=Y+Z$ in \eqref{eq decom}. Since $$N\big([0,u], d_{X,i},\varepsilon\big)\le N\big([0,u], d_{Y,i},\varepsilon/2\big)+N\big([0,u], d_{Z,i},\varepsilon/2\big), \ \ \ i=1,2,$$
it suffices to prove \eqref{Eq: T1} and \eqref{Eq: T2} for $Z$ and $Y$ separately.
If $\beta\in (\gamma, 1/2)$, then \begin{equation*} \begin{split} \int_0^{\infty} \sqrt{\log N\big([0,u], d_{Z,1},\varepsilon\big)}d\varepsilon&\,\le \int_0^{c_{3,2} v^{H}} \sqrt{\log \big(c_{3,5}u^{H/\beta}\varepsilon^{- 1/\beta}\big) }d\varepsilon\\ &\, = \beta u^{H}\int_{c_{3,2} ^{-1/\beta} (u/v)^{H/\beta} }^{\infty} \sqrt{\log {(c_{3,5}x)}} x^{-\beta-1}dx\\ &\,=2\beta c_{3,5}^{-1} u^H \int_{\sqrt{\log\left(c_{3,5}c_{3,2} ^{-1/\beta } (u/v)^{H/\beta} \right)}}^{\infty} t^2 \mathrm {e}^{-\beta t^2}dt\\ &\,\le c_{3,8} v^{H}\sqrt{\log \left(\frac u v\right)}, \end{split} \end{equation*} for some $c_{3,8}>0$, here in the second and third steps, the changes of variables $\varepsilon = u^Hx^{-\beta}$ and $x=\mathrm {e}^{t^2}/c_{3,5}$ are used, respectively, and in the last step we have used the following element inequality: there exists a constant $c_{3,9}>0$ satisfying that for all $a$ large enough, \begin{align}\label{eq ineq}
\int_{a}^{\infty} t^2 \mathrm {e}^{-\beta t^2} dt\le c_{3,9} a \mathrm {e}^{-\beta a^2}. \end{align}
Similarly, \begin{equation*} \begin{split} \int_0^{\infty} \sqrt{\log N\big([0,u], d_{Y,1},\varepsilon\big)}d\varepsilon&\,\le \int_0^{c_{3,2} v^{H}} \sqrt{\log \big(c_{3,4}u^{H }\varepsilon^{- 1}\big) }d\varepsilon\\ &\, = u^{H}\int_{c_{3,2} ^{-1 } (u/v)^{H } }^{\infty} \sqrt{\log {(c_{3,4}x)}} x^{-2}dx\\ &\, = 2 c_{3,4}^{-1} u^H \int_{\sqrt{\log\left(c_{3,4}c_{3,2} ^{-1} (u/v)^{H} \right)}}^{\infty} t^2 \mathrm {e}^{- t^2}dt\\ &\,\le c_{3,10}v^{H}\sqrt{\log \left(\frac u v\right)}, \end{split} \end{equation*} for some constant $c_{3,10}>0$, here the changes of variables
$\varepsilon= u^Hx^{-1}$ and $x =\mathrm {e}^{t^2}/c_{3,4}$ are used in the second and third steps,
respectively, and \eqref{eq ineq} is used in the last step.
By using the same procedures, we can prove the remainder of \eqref{Eq: T1} and \eqref{Eq: T2}. The details are omitted here. The proof of Proposition \ref{prop mtu1} is complete. \end{proof}
\begin{proof}[Proof of Lemma \ref{lem diam}]
(a). If $\beta\in (\gamma, 1/2)$, then $\alpha\in (\gamma-1/2, 0)$. By using the inequality $|(s+x)^{\alpha}-
x^{\alpha}|\le x^{\alpha}$ for any $s, x>0$, we have
\begin{align*}
\| X_1(s)\|_2^2=&\, \int_{-v}^0 \left|(s-x)^{\alpha}-(-x)^{\alpha} \right|^2 |x|^{-2\gamma}dx+\int_0^{v\wedge s} (s-x)^{2\alpha} x^{-2\gamma}dx\\
= & \, \int_0^v \left|(s+x)^{\alpha}-x^{\alpha} \right|^2 x^{-2\gamma}dx+ \int_0^{v\wedge s} (s-x)^{2\alpha} x^{-2\gamma}dx\\
\le& \, \int_0^v x^{2\alpha-2\gamma}dx +\int_0^{v\wedge s} \left(v\wedge s-x \right)^{2\alpha} x^{-2\gamma}dx\\
=& \frac{1}{2\alpha-2\gamma+1} v^{2\alpha-2\gamma+1}+ \mathcal B(2\alpha+1, 1-2\gamma)
(v\wedge s)^{2\alpha-2\gamma+1}\\
\le & \left(\frac{1}{2\alpha-2\gamma+1}+ \mathcal B(2\alpha+1, 1-2\gamma)\right) v^{2H}.
\end{align*}
If $\beta\in [1/2, 1+\gamma)$, then $\alpha\ge0$. In this case, by using the inequality
$\left|(s-x)^{\alpha}_+-(-x)_+^{\alpha}\right|\le 2 u^{\alpha}$ for all $s\in [0, u]$ and $|x| \le u$, we have
\begin{align*}
\| X_1(s)\|_2^2\le \, 4 u^{2\alpha}\int_{|x|\le v} |x|^{-2\gamma}dx =\, \frac{4}{ 1-2\gamma} v^{1-2\gamma}u^{2\beta-1}. \end{align*}
(b). For any $0\le s\le t$, by using the inequality $\left|(s+x)^{\alpha}-x^{\alpha}\right| \le |\alpha|x^{\alpha-1}s$ for any $\alpha\in (-1/2+\gamma, 1/2), s, x\ge0$, we have
\begin{align*}
\|X_2(s)\|_2^2=&\, \int_{|x|\ge v}\left((s-x)^{\alpha}_+-(-x)_+^{\alpha} \right)^2|x|^{-2\gamma}dx\\
=&\, \int_{x\ge v}\left((s+x)^{\alpha}-x^{\alpha} \right)^2x^{-2\gamma}dx\\
\le &\, |\alpha|^2 s^2 \int_{x\ge v} x^{2\alpha-2\gamma-2}dx\\
=&\, \frac{|\alpha|^2}{ 1+2\gamma-2\alpha } s^2 v^{2\beta-2\gamma-2}. \end{align*} This implies \eqref{Eq: B12}.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem covering number}] We use the argument in the proof of Lemma 4.1 in \cite{WX2021}. Since the proofs of \eqref{Eq cov Y} and \eqref{Eq cov Z} are similar, we only prove \eqref{Eq cov Z} here.
It follows from \eqref{Eq: Zmoment1} that there exists a constant $c_{3,11}>0$ such that for any $t>s>0$
\begin{align}\label{eq d12-1}
d_{Z}(0,t)=\|Z(t)-Z(0)\|_2\le c_{3,11}|t|^H, \end{align}
and \begin{align}\label{eq d12-2}
d_{Z}(s,t)=\|Z(s)-Z(t)\|_2\le \frac{c_{3,11}}{s^{\gamma}}|t-s|^{\beta}. \end{align}
Let $t_0=0, t_1 = \varepsilon^{1/H}$. For any $n \ge 2$, if $t_{n-1}$ has been defined, we define \begin{equation}\label{def:tn} t_n = t_{n-1} + t_{n-1}^{\frac \gamma {\beta}} \varepsilon^{\frac 1 {\beta}}. \end{equation} It follows from \eqref{eq d12-2} that for all $n\ge2$, \begin{equation*} \begin{split}
\|Z(t_n) - Z(t_{n-1})\|_2 &\le \frac {c_{3,11}} {t_{n-1}^\gamma} \big| t_n - t_{n-1}\big|^{ \beta} \le c_{3,11} \varepsilon. \end{split} \end{equation*} Hence $d_{Z}(t_n,\, t_{n-1}) \le \,c_{3,11} \varepsilon$ for all $n \ge 1$.
Since $[0, u]$ can be covered by the intervals $[t_{n-1}, \, t_n]$ for $n = 1, 2, \ldots, L_\varepsilon$, where $L_\varepsilon$ is the largest integer $n$ such that $t_n \le b$, we have $N ([0, b], d_Z, \varepsilon) \le L_\varepsilon+1 \le 2L_\varepsilon$.
In order to estimate $L_\varepsilon$, we write $t_n = a_n \varepsilon^{1/H}$ for all $n \ge 1$. Then, by \eqref{def:tn}, we have $a_1=1$ and \begin{equation*} a_n = a_{n-1} + a_{n-1}^{\frac \gamma {\beta}}, \quad \forall \, n \ge 2. \end{equation*}
Denote by $\rho = \frac{H}{\beta}$. By (4.9) in \cite{WX2021}, we know that there exist positive and finite constants $c_{3,12} \le 2^{-\gamma/(H\rho)}\rho^{1/\rho}$ and $c_{3,13} \ge 1$ such that \begin{equation}\label{Eq:an} c_{3,12}\, n^{1/\rho} \le a_n \le c_{3,13}\, n^{1/\rho}, \qquad \forall \, n \ge 1. \end{equation} By \eqref{Eq:an}, we have \begin{equation*}\label{Eq:Lep} L_\varepsilon = \max\left\{n; a_n \varepsilon^{\frac 1 H} \le b \right\} \le c_{3,12}^{-\rho}\, b^{\rho}\,\varepsilon^{- \frac 1 {\beta}}. \end{equation*} This implies that for all $\varepsilon \in (0, 1)$, \begin{align*} N \big([0, b], d_{Z}, \varepsilon\big) \le 2 c_{3,12}^{-\rho}\,b^{\rho}\, \varepsilon^{- \frac 1 {\beta}}. \end{align*} The proof of Lemma \ref{lem covering number} is complete.
\end{proof}
We end this section with the following lemma from Talagrand \cite{Tal96}, which will be used to prove the necessary parts in Theorems \ref{thm main 0} and \ref{thm main infty}.
\begin{lemma}\cite[Corollary 2.3]{Tal96} \label{lem GBC} Let $J \subseteq {\mathbb N}$. If there exist some positive numbers $K_5$ and $\varepsilon$ such that for all $i\in J$, \begin{align} \sum_{j<i}\mathbb P(A_i\cap A_j)\le \mathbb P(A_i)\bigg(K_5+(1+\varepsilon)\sum_{j<i}\mathbb P(A_j)\bigg), \end{align} and assume that $\sum_{i\in J}\mathbb P(A_i)\ge (1+2K_5)/\varepsilon$, then we have $\mathbb P\big(\bigcup_{i\in J}A_i\big)\ge (1+2\varepsilon)^{-1}$. \end{lemma}
\section{Proof of Theorem \ref{thm main 0}} \subsection{Sufficiency of the integral condition}
In this part, we always assume that $\xi(t)$ is a nondecreasing continuous function such that $\xi(t)/t^H:(0,\mathrm {e}^{\mathrm {e}}]\rightarrow (0,\infty)$ is bounded and $I_0(\xi)<+\infty$. We prove that $\xi(t)\le M(t)$ for all $t$ small enough in probability one.
Using the argument in the proof of \cite[Lemma 3.1]{Tal96}, we first prove the following result. \begin{lemma}\label{lem function10} Suppose that $\xi(t)$ is a nondecreasing continuous function such that $\xi(t)/t^H$ is bounded and $I_0(\xi)<+\infty$. Then \begin{align}\label{eq f10} \lim_{t\rightarrow0}\frac{\xi(t)}{t^H}=0. \end{align} \end{lemma} \begin{proof} Recall the number $\theta_0$ in Lemma \ref{lem increase}. By the boundedness and the continuity of $\xi(t)/t^H$, we know $$ c_{4,1}:=\inf\left\{\frac{\varphi(\theta)}{\theta^{1/\beta}}; \, \theta_0\le \theta\le \sup_{t>0} \frac{\xi(t)}{t^H} \right\}>0. $$ For any $t\le u\le 2t$, we have $$ \frac{\xi(u)}{u^H}\ge \frac{\xi(t)}{(2t)^H}. $$ By Lemma \ref{lem increase}, we know that the function $\varphi(\theta)/\theta^{1/\beta}$ is increasing in $(0,\theta_0)$. Thus for any $t\le u\le 2t$, $$ \left(\frac{\xi(u)}{u^H}\right)^{-1/\beta} \varphi\left(\frac{\xi(u)}{u^H} \right)\ge \min\left\{ \left(\frac{\xi(t)}{(2t)^H}\right)^{-1/\beta} \varphi\left(\frac{\xi(t)}{(2t)^H} \right), \, c_{4,1} \right\}. $$ Consequently, we get $$ \int_t^{2t} \left(\frac{\xi(u)}{u^H}\right)^{-1/\beta} \varphi\left(\frac{\xi(u)}{u^H} \right)\frac{du}{u} \ge \log 2\cdot
\min\left\{ \left(\frac{\xi(t)}{(2t)^H}\right)^{-1/\beta} \varphi\left(\frac{\xi(t)}{(2t)^H} \right),\, c_{4,1} \right\}. $$ This, together with the fact that $I_0(\xi)<+\infty$ and the monotonicity of $\varphi(\theta)/\theta^{1/\beta}$ in $(0,\theta_0)$, implies \eqref{eq f10}. The proof is complete.
\end{proof}
In order to prove the sufficiency in Theorem \ref{thm main 0}, we construct the sequences $\{t_n\}_{n\ge1}$, $\{u_n\}_{n\ge1}$ and $\{v_n\}_{n\ge1}$ by recursion as follows. Let $L>2H$ be a constant, whose value will be chosen later. We start with $t_1=\mathrm {e}^{-\mathrm {e}}$. Having constructed $t_n$, we set \begin{align} u_{n+1}:=&\inf\Big\{u<t_n; \, u+t_{n}^{\frac{\gamma}{\beta}}\xi(u)^{\frac1\beta}\ge t_n \Big\};\\ v_{n+1}:=&\, \inf\Bigg\{u<t_n;\, \xi(u)\bigg( 1+L\left(\frac{\xi(u)}{u^H} \right)^{1/\beta}\bigg)\ge \xi(t_n) \Bigg\}; \label{eq vn+11}\\ t_{n+1}:=&\, \max\{u_{n+1},\, v_{n+1}\}.\label{eq tn+110} \end{align} By the continuity of $\xi$, we have \begin{align}\label{eq un+11} u_{n+1}=\, t_n-t_{n}^{\frac{\gamma}{\beta}}\xi(u_{n+1})^{\frac1\beta}.
\end{align}
\begin{lemma}\label{lem 00} The sequence $\{t_n\}_{n\ge1}$ is decreasing and \begin{equation}\label{eq tn0} \lim_{n\rightarrow\infty} t_n=0. \end{equation} \end{lemma} \begin{proof} By the construction of $\{t_n\}_{n\ge1}$, it is obvious to see that $\{t_n\}_{n\ge1}$ is decreasing and $t_{\infty}:=\lim_{n\rightarrow\infty}t_n\ge0$ exists. Next, we prove $t_{\infty}=0$. Suppose, otherwise, that $t_n\ge t_{\infty}>0$ for all $n\ge1$. By the continuity of $\xi$, we have $ \lim_{n\rightarrow \infty}\xi(t_n)=\xi(t_{\infty})>0$. By \eqref{eq un+11}, we have $t_n=v_n$ for $n$ large enough but then. Since $\xi$ is continuous, we have that for all $n$ large enough, $$ \xi(t_{n})=\xi(t_{n+1}) \Bigg( 1+L\bigg(\frac{\xi(t_{n+1})}{t_{n+1}^H} \bigg)^{1/\beta} \Bigg) \ge\xi(t_{n+1}) \Bigg( 1+L\bigg(\frac{\xi(t_{\infty})}{t_1^H} \bigg)^{1/\beta} \Bigg), $$ which contradicts with the convergence of $\{\xi(t_n)\}_{n\ge1}$. The proof is complete. \end{proof}
\begin{lemma}\label{lem Mtn0} If $M(t_n)\ge \xi(t_n)\Big(1+L\big(\frac{\xi(t_n) }{t_n^H}\big)^{1/\beta}\Big)$ for all $n\ge n_0$, then $M(t)\ge \xi(t)$ for all $t\in (0, t_{n_0}]$. \end{lemma} \begin{proof} Assume $n\ge n_0$ and $t_{n+1}< t\le t_{n}$. By \eqref{eq vn+11} and \eqref{eq tn+110}, we have $$ \xi(t)\le \xi(t_n)\le \xi(t_{n+1}) \Bigg( 1+L\bigg(\frac{\xi(t_{n+1})}{t_{n+1}^H} \bigg)^{1/\beta} \Bigg) \le M({t_{n+1}})\le M(t). $$ Thus, $M(t)\ge \xi(t)$ for all $t\in (0,t_{n_0}]$, as desired. \end{proof}
By \eqref{eq M1}, we have \begin{equation}\label{Eq:Mtn} \begin{split} \, \mathbb P\Bigg(M(t_n)<\xi(t_n)\bigg(1+L\Big(\frac{\xi(t_n)}{t_n^H}\Big)^{1/\beta} \bigg) \Bigg) =\,\varphi\Bigg(\frac{\xi(t_n)}{t_n^H}\bigg(1+L\Big(\frac{\xi(t_n)}{t_n^H}\Big)^{1/\beta}\bigg) \Bigg). \end{split} \end{equation} By using the Borell-Cantell lemma and Lemma \ref{lem Mtn0}, the proof of the sufficiency in Theorem \ref{thm main 0} will be finished if we show that the terms in (\ref{Eq:Mtn}) form a convergent series.
Applying \eqref{eq control} with $\theta= \xi(t_n)/t_n^H$, $\varepsilon=\theta+L\theta^{1+1/\beta}$ (since $\lim_{n\rightarrow\infty} \xi(t_n)/t_n^H=0$) to see that it suffices to show that \begin{align}\label{eq finite} \sum_{n=1}^{\infty}\varphi\left(\frac{\xi(t_n)}{t_n^H}\right)<+\infty. \end{align} This is done by using the following lemma.
\begin{lemma}\label{lem finite}
\begin{itemize}
\item[(a)] If $n$ is large enough and $t_{n}=u_{n}$, then we have
\begin{align}\label{eq tun1}
\varphi\bigg(\frac{\xi(t_{n})}{t_{n}^H}\bigg)\le c_{4,2}\int_{t_{n}}^{t_{n-1}}
\left(\frac{\xi(t)}{t^H}\right)^{-1/\beta}\varphi\left(\frac{\xi(t)}{t^H}\right) \frac{dt}{t},
\end{align}
where $c_{4,2}$ depends on $\beta$ and $H$ only.
\item[(b)] If $n$ is large enough and $t_{n}=v_{n}$, then there exists a constant $L>2H$ depending on
$\beta$ and $H$ only such that
\begin{align}\label{eq tvn1}
\varphi\bigg(\frac{\xi(t_{n})}{t_{n}^H}\bigg)\le \frac12 \varphi\left(\frac{\xi(t_{n-1})}{t_{n-1}^H}\right).
\end{align}
\end{itemize}
\end{lemma}
We postpone the proof of Lemma \ref{lem finite}. Let us first finish the proof of \eqref{eq finite}, hence the sufficiency in Theorem \ref{thm main 0}. Consider the set
$$J:=\big\{n_k\in \mathbb N;\, t_{n_k}=u_{n_k}\ge v_{n_k}\big\}.$$ By \eqref{eq tun1}, we have
\begin{equation}\label{eq finite1}
\begin{split}
\sum_{n_k\in J}\varphi\left(\frac{\xi(t_{n_k})}{t_{n_k}^H}\right)\le&\, c_{4,2}\sum_{n_k\in J}
\int_{t_{n_k}}^{t_{n_{k}-1}} \left(\frac{\xi(t)}{t^H}\right)^{-1/\beta}\varphi\left(\frac{\xi(t)}{t^H}\right) \frac{dt}{t}\\
\le &\, c_{4,2} \int_0^{\mathrm {e}^{-\mathrm {e}}} \left(\frac{\xi(t)}{t^H}\right)^{-1/\beta}\varphi\left(\frac{\xi(t)}{t^H}\right) \frac{dt}{t}
<\infty.
\end{split}
\end{equation}
Let $n_{k-1}$ and $n_k$ be two consecutive terms of $J$. If there exists an integer $n$ such that
$n_{k-1}<n<n_{k}$, then set $p:=n-n_{k-1}$. Since $t_{n}=v_{n}$ for all $n\in \mathbb N\setminus J$,
Eq. \eqref{eq tvn1} implies that
$$
\varphi\left(\frac{\xi(t_n)}{t_n^H}\right)\le 2^{-p} \varphi\bigg(\frac{\xi(t_{n_{k-1}})}{t_{n_{k-1}}^H}\bigg).
$$
Thus, we obtain by setting $n_0=1$, \begin{equation*}
\begin{split}
\sum_{n\in \mathbb N\setminus J} \varphi\left(\frac{\xi(t_n)}{t_n^H}\right)=&\, \sum_{k=1}^{\infty} \sum_{n_{k-1}<n< n_k} \varphi\left(\frac{\xi(t_n)}{t_n^H}\right) \le \, \sum_{k=1}^{\infty} \varphi\bigg(\frac{\xi(t_{n_{k-1}})}{t_{n_{k-1}}^H}\bigg)\\ \le &\, c_{4,2}\int_0^{\mathrm {e}^{-\mathrm {e}}} \left(\frac{\xi(t)}{t^H}\right)^{-1/\beta}\varphi\left(\frac{\xi(t)}{t^H}\right) \frac{dt}{t} <+\infty.
\end{split}
\end{equation*} This, together with \eqref{eq finite1}, implies \eqref{eq finite}. Therefore, the sufficiency in Theorem \ref{thm main 0}
has been proved.
\qed
Next, we prove Lemma \ref{lem finite}.
\begin{proof}[Proof of Lemma \ref{lem finite}]
(a) Assume $t_{n}=u_{n}\ge v_{n}$. By \eqref{eq vn+11}, we have
\begin{align}\label{eq tn+111}
\xi(t_{n-1})\le \xi(t_{n})\Bigg( 1+L\bigg(\frac{\xi(t_{n})}{t_{n}^H} \bigg)^{1/\beta}\Bigg).
\end{align} By \eqref{eq un+11} and the monotonicities of $\varphi$ and $\xi$, we have \begin{align*}
\int_{t_{n}}^{t_{n-1}}\left(\frac{\xi(t)}{t^H}\right)^{-1/\beta}\varphi\left(\frac{\xi(t)}{t^H}\right) \frac{dt}{t} \ge&\, \left(t_{n-1}-t_{n}\right)t_{n-1}^{-\frac{\gamma}{\beta}}\xi\left(t_{n-1}\right)^{-1/\beta}\varphi\left(\frac{\xi(t_{n})}{t_{n-1}^H}\right)\\ =&\, \left( \frac{\xi(t_{n})}{\xi(t_{n-1})}\right)^{1/\beta} \varphi\left(\frac{\xi(t_{n})}{t_{n-1}^H}\right)\\ \ge&\, \frac12\varphi\left(\frac{\xi(t_{n})}{t_{n-1}^H}\right), \end{align*} where in the last inequality, we have used the fact that $\xi\left(t_{n}\right)\ge \xi\left(t_{n-1}\right)/2^{\beta}$ for all $n$ large enough by \eqref{eq tn+111}, Lemma \ref{lem function10} and Lemma \ref{lem 00}.
Now, using \eqref{eq un+11} again, we know that for all
$n$ large enough,
\begin{align*}
\frac{\xi(t_{n})}{t_{n-1}^H}=&\, \frac{\xi(t_{n})}{t_{n}^H} \left(\frac{t_{n}}{t_{n-1}}\right)^H
=\frac{\xi(t_{n})}{t_{n}^H}\Bigg(1- \bigg(\frac{\xi(t_{n})}
{t_{n-1}^H}\bigg)^{1/\beta} \Bigg)^{H}\\
\ge &\, \frac{\xi(t_{n})}{t_{n}^H}\Bigg(1-c_{4,3}\bigg(\frac{\xi(t_{n})}{t_{n}^H}\bigg)^{1/\beta} \Bigg).
\end{align*}
Here $c_{4,3}>0$ which only depends on $H$.
Thus, \eqref{eq tun1} follows from \eqref{eq control}.
(b). If $t_{n}=v_{n}\ge u_{n}$, then by the continuity of $\xi$ we have
\begin{align}\label{eq XLN}
\xi(t_{n-1})=\xi(t_{n})\Bigg( 1+L\bigg(\frac{\xi(t_{n})}{t_{n}^H} \bigg)^{1/\beta}\Bigg).
\end{align}
Since $t_{n}\ge u_{n}$, it follows from \eqref{eq un+11} and \eqref{eq XLN} that
\begin{equation}\label{eq t n n+1}
\begin{split}
\frac{\xi(t_{n-1})}{t_{n-1}^H}=&\, \frac{\xi(t_{n-1})}{t_{n}^H}\left( \frac{t_{n}}{t_{n-1}}\right)^{H}\\
\ge &\, \frac{\xi(t_{n})}{t_{n}^H} \Bigg( 1+L\bigg(\frac{\xi(t_{n})}{t_{n}^H} \bigg)^{1/\beta}\Bigg)
\Bigg( 1- \bigg(\frac{\xi(t_{n})}{t_{n-1}^H} \bigg)^{1/\beta}\Bigg)^H \\
\ge &\, \frac{\xi(t_{n})}{t_{n}^H} \Bigg( 1+(L-c_{4,4})\bigg(\frac{\xi(t_{n})}{t_{n}^H} \bigg)^{1/\beta}\Bigg),
\end{split}
\end{equation}
for some positive constant $c_{4,4}$ which only depends on $H$ and $\beta$.
Set $\theta=\xi(t_{n})/ t_{n}^{H}$, $\varepsilon=\theta+(L-c_{4,4})\theta^{1+1/\beta}$. It follows
from Lemma \ref{lem function10} that $\theta< \varepsilon< 2\theta< 1$ for all $n$ large enough.
Hence, by \eqref{eq psi'} and the convexity of $\psi$, we have
\begin{align*}
\psi(\varepsilon)\le\psi(\theta)+(\varepsilon-\theta)\psi'(2\theta)\le \psi(\theta)-\frac{\varepsilon-
\theta}{K_2(2\theta)^{1+1/\beta}}
\le \psi(\theta)-\frac{L-c_{4,4}}{ 2^{1+1/\beta}K_2}.
\end{align*}
This, together with \eqref{eq t n n+1}, implies that for all $L$ large enough,
$$
\varphi\left(\frac{\xi(t_{n-1})}{t_{n-1}^H}\right)\ge \varphi(\varepsilon)=\exp\big(-\psi(\varepsilon)\big)
\ge 2 \exp\big(-\psi(\theta)\big)=2\varphi(\theta).
$$
The proof of Lemma \ref{lem finite}. is complete.
\end{proof}
\subsection{Necessity of the integral condition}
Suppose that with positive probability, $\xi(t)\le M(t)$ for all $t>0$ small enough. We will prove that $\xi(t)/t^H$ is bounded and $I_0(\xi)<+\infty$. The first fact is a direct consequence of \eqref{eq M1} and $\lim_{\theta\rightarrow \infty}\varphi(\theta)=1$.
Let us prove $I_0(\xi)<+\infty$ by using Lemma \ref{lem GBC}.
\subsubsection{Construction of $\{t_n\}_{n\ge1}$}\label{subsect tn}
\begin{lemma}\label{lem limit00} If with positive probability, $\xi(t)\le M(t)$ for all $t>0$ small enough, then
\begin{align}\label{eq limit00}
\lim_{t\rightarrow0}\frac{\xi(t)}{t^H}=0.
\end{align}
\end{lemma}
\begin{proof}
Otherwise, we can find a sequence $\{t_n\}_{n\ge1}$ such that $\delta:=\inf_{n\ge1}\frac{\xi(t_n)}{t_n^H}>0$.
Denote by $A_n:=\big\{M(t_{n})\le \xi(t_n) \big\}$. By \eqref{eq M1}, we have
\begin{equation}\label{eq Pan}
\inf_{n\ge1} \mathbb P(A_n)=\inf_{n\ge1} \mathbb P\big(M({t_n})\le \xi(t_n)\big)\ge \mathbb P\big(M(1)\le \delta\big)>0.
\end{equation} In the following, we show that the events $A_n$ occur infinitely often almost surely. This contradicts the assumption of Lemma \ref{lem limit00}. Without loss of generality, we assume that $t_{n}/ t_{n+1}\ge 2$. Denote by $\mathbb P(A_n)=a_n$ for all $n\ge1$. Applying Proposition \ref{prop mtu1} with $t=t_n, u=t_m,
\theta=\xi(t_n)/t_n^H, \eta=\xi(t_m)/t_m^H$ for $m<n$, we have
\begin{align*}
\mathbb P\left(A_n\cap A_m\right)
&\le \, a_na_m \Bigg\{\exp\left(-\frac{1}{K}\Big(\frac{t_m}{t_n}\Big)^{\tau}\right)+\varphi\left(\frac{\xi(t_m)}{t_m^H}\right)
+ \varphi\left(\frac{\xi(t_n)}{t_n^H}\right)\\
&\quad \, \,\,\, \,\,\, \,\,\,+\exp\left[K\Big(\frac{t_m}{t_n}\Big)^{-\tau} \Bigg(\Big(\frac{\xi(t_m)}{t_m^H}\Big)^{-1-1/\beta}
+ \Big(\frac{\xi(t_n)}{t_n^H}\Big)^{-1-1/\beta}\Bigg) \right] \Bigg\}\\
&\le a_na_m \Bigg\{\exp\left(-\frac{1}{K}\Big(\frac{t_m}{t_n}\Big)^{\tau} +2K_1\delta^{-1/\beta} \right)
+\exp\left(2K\Big(\frac{t_m}{t_n}\Big)^{-\tau} \delta^{-1-1/\beta} \right) \Bigg\},
\end{align*}
where \eqref{eq sbp} is used in the last step. For any $l\ge1, N\ge1$, let $\mathbb N_{l, N}:=\{n_k\}_{k\ge N}
\subset \mathbb N$ such that for $t_{n_k}/t_{n_{k+1}}\ge l$ for any $n_{k},n_{k+1}\in \mathbb N_{l, N}$.
Thus, we have
\begin{align*}
\mathbb P(A_n\cap A_m)\le&\, a_na_m \Bigg\{\exp\left(-\frac{1}{K} l^{\tau}+2K_1\delta^{-1/\beta} \right)
+ \exp\left(2Kl^{-\tau} \delta^{-1-1/\beta} \right) \Bigg\}\\
=:&\, a_na_m \left(1+o(l)\right),
\end{align*}
where $o(l)\rightarrow0$ as $l\rightarrow +\infty$. By \eqref{eq Pan}, we have $\sum_{m\in \mathbb N_{l, N}}
a_m=+\infty$. Consequently, by Lemma \ref{lem GBC}, we have
$$
\mathbb P\bigg(\bigcup_{m\in \mathbb N_{l, N}} A_m\bigg)\ge \frac{1}{1+o(l)}.
$$
This implies that $\lim_{N\rightarrow \infty}\mathbb P\left(\bigcup_{n\ge N} A_n\right)=1$. Hence,
the events $A_n$ occur infinitely often almost surely. The proof is complete.
\end{proof}
\begin{lemma}\label{lem tn1} Assume $I_0(\xi)=+\infty$ and \eqref{eq limit00}. Then there exists a sequence $\{t_n\}_{n\ge1}$ with the following properties:
\begin{itemize}
\item[(i)]
\begin{align}\label{eq sum infty0}
\sum_{n=1}^{\infty} \varphi\left(\frac{\xi(t_n)}{t_n^H}\right)=+\infty;
\end{align}
\item[(ii)]
\begin{align}\label{eq tn+10}
t_{n+1}= t_n-t_{n+1}^{\gamma/\beta} \xi(t_{n+1})^{1/\beta}. \end{align}
\end{itemize}
\end{lemma}
\begin{proof} The construction is given by induction over $n$. We take $t_1=\mathrm {e}^{-\mathrm {e}}$. Having
constructed $t_n$, we define
\begin{align*}
t_{n+1}:=\inf\left\{u\le t_n;\, u+u^{\gamma/\beta}\xi(u)^{1/\beta}\ge t_n \right\}.
\end{align*} It is obvious by the continuity of $\xi$ that \eqref{eq tn+10} holds.
To prove \eqref{eq sum infty0}, it is sufficient to prove that for all $n$ large enough,
\begin{align}\label{eq In1}
I_n:=\int_{t_{n+1}}^{t_{n}} \left(\frac{\xi(t)}{t^H}\right)^{-1/\beta}\varphi\left(\frac{\xi(t)}{t^H}\right)
\frac{dt}{t}\le c_{4,5}\varphi\left(\frac{\xi(t_{n})}{t_{n}^H}\right),
\end{align}
where $c_{4,5}$ depends on $H$ and $\beta$ only. By the monotonicities of $\{t_n\}_{n\ge1}$ and $\xi$, we derive
\begin{align}\label{eq In2}
I_n \le &\, \frac{\left(t_n-t_{n+1} \right)}{ t_{n+1}^{\gamma/\beta}\xi(t_{n+1})^{1/\beta}}
\varphi\bigg(\frac{\xi(t_{n})}{t_{n+1}^H}\bigg)
=\, \varphi\bigg(\frac{\xi(t_{n})}{t_{n+1}^H}\bigg).
\end{align} It follows from \eqref{eq limit00} and \eqref{eq tn+10} that $t_{n+1}\ge t_n/2$
for all $n$ large enough. Hence
\begin{equation}\label{eq In3}
\begin{split}
\frac{\xi(t_{n})}{t_{n+1}^H}=&\,
\frac{\xi(t_{n})}{t_{n}^H} \bigg(\frac{t_{n}}{t_{n+1}}\bigg)^H= \,\frac{\xi(t_{n})}{t_{n}^H}
\Bigg(1+\bigg(\frac{\xi(t_{n+1})}{t_{n+1}^H}\bigg)^{1/\beta}\Bigg)^H\\
\le &\,\frac{\xi(t_{n})}{t_{n}^H}\Bigg(1+ \bigg(\frac{2\xi(t_{n})}{ t_{n}^H}\bigg)^{1/\beta}\Bigg)^H
\le \,\frac{\xi(t_{n})}{t_{n}^H}\left(1+c_{4,6} \left(\frac{\xi(t_{n})}{ t_{n}^H}\right)^{1/\beta}\right),
\end{split}
\end{equation}
for some constant $c_{4,6}>0$ which only depends on $H$.
The inequality \eqref{eq In1} follows now from \eqref{eq control}, \eqref{eq In2} and \eqref{eq In3}.
The proof is complete. \end{proof}
For each $n\ge 1$, we define $k(n)$ by
\begin{equation}\label{eq kn}
2^{k(n)}\le \left(\frac{t_n^H}{\xi(t_n)}\right)^{1/\beta} < 2^{k(n)+1},
\end{equation}
and set $I_k:=\big\{n\in\mathbb N; \, k(n)=k \big\}$ for any $k\ge1$.
Recall the constant $K_1$ given in \eqref{eq sbp}. Without loss of generality, we assume that $K_1\ge c_{2,2}$ where $c_{2,2}$ is the constant in \eqref{eq Mtu}. For $k\ge1$, we set
$$N_k:=\exp\left(2^{k-2}/K_1 \right).$$
The following is a refinement of Lemma \ref{lem tn1}.
\begin{proposition}\label{Prop con2} Assume $I_0(\xi)=+\infty$, \eqref{eq limit00}, and $\xi(t)/t^{(1+\varepsilon_0)H}$ is nonincreasing for some $\varepsilon_0>0$. Then there exist a positive constant $c_{4,7}$ depending on $\beta$ and $H$ only and a set $J$ with the following properties:
\begin{itemize}
\item[(i)]
\begin{align}\label{eq sum J infty0}
\sum_{n\in J}\varphi\left( \frac{\xi(t_n)}{t_n^H} \right)=+\infty;
\end{align}
\item[(ii)] Given $n, m\in J$ with $m<n$ such that
\begin{align}\label{eq kmn0}
{\mathrm{card}}\big(I_{k(m)}\cap [m,n]\big)> N_{k(m)},
\end{align}
we have \begin{align}\label{eq tmn exp0} \frac{t_m}{t_n}\ge\exp\left(\exp\big( 2^{\max\{k(m),\, k(n)\}}/ c_{4,7} \big) \right). \end{align}
\end{itemize}
\end{proposition}
\begin{proof} Set $a_n:=\varphi\big(\xi(t_n)/t_n^H\big)$. We recall from \eqref{eq sbp} that
\begin{align*}
\exp\left(- K_1 2^{k(n)+1}\right)\le a_n \le
\exp\left(- 2^{k(n)}/K_1\right).
\end{align*}
Given $m, k\in \mathbb N$, define
$$
U_{m,k}:=\big\{i>m; \, i\in I_k,\, {\mathrm{card}}(I_k\cap [m,i])\le N_k \big\}.
$$
Thus,
\begin{align*}
\sum_{i\in U_{m, k}} a_i\le\, N_k \exp\left(-\frac{2^k}{K_1}\right)
\le \, \exp\left(-\frac{2^{k-1}}{K_1}\right).
\end{align*}
Denote by $k_0$ the smallest integer such that $2^{k_0}\ge 2K_1+4K_1^2$. Then there exists
a constant $c_{4,8}\in (0,1)$ satisfying that
\begin{equation}\label{Eq:sums}
\begin{split}
\sum_{k\ge k(m)+k_0}\sum_{i\in U_{m,k}} a_i\le &\, \sum_{k\ge k(m)+k_0}\exp\left(-\frac{2^{k-1}}{K_1}\right)\\
\le &\, a_m \sum_{k\ge k(m)+k_0}\exp\left(K_1 2^{k(m)+1}-\frac{2^{k-1}}{K_1}\right)\\
\le &\, a_m \sum_{l\ge 0}\exp\left( 2^{k(m)}\Big(2K_1-\frac{2^{l-1+k_0}}{K_1}\Big) \right)\\
\le &\, a_m \sum_{l\ge 0}\exp\left( -2 ^{l} \right)\le c_{4,8} a_m.
\end{split}
\end{equation}
For each $m\ge1$, set
$$
V_m:=\bigcup_{k\ge k(m)+k_0} U_{m,k}.
$$
It follows from (\ref{Eq:sums}) that
\begin{align}\label{eq sum Vm0}
\sum_{k\le p}\sum_{m\in I_k}\sum_{i\in V_m} a_i\le c_{4,8}
\sum_{k\le p}\sum_{m\in I_k} a_m.
\end{align} Set
$$
J_0:=\mathbb N\cap \left( \bigcup_{m\ge1} V_m\right)^c.
$$
By the definition of $V_m$, we know that $k(m)+k_0\le k(i)$ if $i\in V_m$. Thus,
$$
\Big(\bigcup_{m\ge1} V_m\Big)\cap \Big(\bigcup_{k\le p}I_k\Big)\subset \bigcup_{k(m)\le p} V_m.
$$
This, together with \eqref{eq sum Vm0}, implies that
$$
\sum_{i\notin J_0, k(i)\le p} a_i\le c_{4,8}\sum_{k(i)\le p} a_i,
$$
and then
$$
\sum_{i\in J_0, k(i)\le p} a_i\ge (1-c_{4,8})\sum_{k(i)\le p} a_i.
$$
Letting $p\rightarrow\infty$, we obtain that
\begin{align}\label{eq sum J infty10}
\sum_{n\in J_0}\varphi\left( \frac{\xi(t_n)}{t_n^H} \right)=+\infty.
\end{align}
Let $J:=J_0\cap[w,\infty) $ for a constant $w\in \mathbb N$, whose value will be chosen later.
Then \eqref{eq sum J infty10} implies \eqref{eq sum J infty0}.
Next, we prove (ii).
For any $n,m\in J$ with $n<m$. If $i\in I_{k(m)}$, then by \eqref{eq tn+10} we have
\begin{align*}
t_{i-1}\ge\, t_i\left(1+ \left(\frac{\xi(t_i)}{t_i^{H} }\right)^{1/\beta}\right) \ge \, t_i\left(1+2^{-k(m)-1}\right).
\end{align*} Thus, when \eqref{eq kmn0} holds and when $w$ (hence $k(m)$) is large enough, we have \begin{equation}\label{eq tmn 10} \begin{split} \frac{t_m}{t_n}\ge \left(1+2^{-k(m)-1}\right)^{N_{k(m)}}\ge &\,\exp\left( 2^{-k(m)-2}N_{k(m)}\right)\\ \ge &\, \exp\left(\exp\left(2^{k(m)-3}/c_{4,9} \right)\right), \end{split} \end{equation} for some constant $c_{4,9}>0$. This implies \eqref{eq tmn exp0} whenever $k(m)\ge k(n)-k_0$. If $k(m)<k(n)-k_0$, then by definition of $J$ we must have $n\notin U_{m, k(n)}$. This means that $$ {\mathrm{card}}\big(I_{k(n)}\cap [m,n]\big)>N_{k(n)}, $$ and the argument leading to \eqref{eq tmn 10} shows that \eqref{eq tmn 10} holds for $k(n)$ rather than $k(m)$.
Thus, we complete the proof of this proposition by choosing $w$ large enough.
\end{proof}
\subsubsection{Proof of the necessity in Theorem \ref{thm main 0} }\label{Sec Neces}
Let $\{t_n\}_{n\in J}$ be as in Proposition \ref{Prop con2}.
Set $A_n= \big\{M({t_n})<\xi(t_n)\big\}$, then $\mathbb P(A_n)=a_n= \varphi\big({\xi(t_n)}/{t_n^H}\big).$
According to \eqref{eq sum J infty0} and Lemma \ref{lem GBC}, it suffices to prove the following
statement:
\begin{quote}
Given $\varepsilon>0$, there exist positive constants $K$ and $q$ such that for any $n\in J$ with $n\ge q$,
\begin{align}\label{eq Anm00}
\sum_{m\in J, \, m<n} \mathbb P(A_n\cap A_m)\le \mathbb P(A_n)\bigg(K+(1+\varepsilon)
\sum_{m\in J, \, m<n}\mathbb P(A_m) \bigg).
\end{align}
\end{quote}
The sum on the left-hand side will be split into three parts over the following subsets of $J$: for any
$n$, $k\in \mathbb N$, set
\begin{equation*}
\begin{split}
J'&:=\big\{m\in J; \ t_n<t_m\le 2 t_n \big\};\\
J_k&:=\big\{m\in J\cap I_k; \, t_m>2t_n, \, {\mathrm{card}}(I_k\cap [m,n])\le N_k \big\};\\
J''&:=J\setminus \bigg( J' \cup \bigcup_{k\ge1} J_k \bigg).
\end{split} \end{equation*}
Applying Eq. \eqref{eq Mtu} with $t=t_n$, $u=t_m$, $\theta=\xi(t_n)/t_n^{H}$ and $\eta =\xi(t_m)$ for any $m<n$, we have \begin{align}\label{eq Anm10} \mathbb P(A_n\cap A_m)\le 2a_n \exp\bigg(-\frac{t_m-t_n}{ c_{2,2} t_m^{\gamma/\beta}\xi(t_m)^{1/\beta}}\bigg). \end{align}
(1). When $t_n< t_m\le 2 t_n$, the monotonicity of $\xi(t)/t^{(1+\varepsilon_0)H}$ implies that $\xi(t_m)\le 2^{(1+\varepsilon_0)H}\xi(t_n)$. Hence, by \eqref{eq Anm10}, we have
\begin{align}\label{eq Anm200} \mathbb P(A_n\cap A_m)\le 2 a_n \exp\bigg(-\frac{t_m-t_n}{ c_{2,2} 2^{\gamma/\beta} 2^{ (1+\varepsilon_0)H /\beta} t_n^{\gamma/\beta}\xi(t_n)^{1/\beta}}\bigg).
\end{align} By \eqref{eq tn+10} and the monotonicity of $\xi$, we have that for all $m\le i < n$, $$ t_{i}= t_{i+1}+t_{i+1}^{\gamma/\beta}\xi(t_{i+1})^{1/\beta}\ge t_{i+1}+t_n^{\gamma/\beta}\xi(t_n)^{1/\beta}. $$
This, together with \eqref{eq Anm200}, implies that
$$ \mathbb P(A_n\cap A_m)\le 2a_n \exp\left(-\frac{m-n}{ c_{2,2} 2^{\gamma/\beta}2^{(1+\varepsilon_0)H/\beta} }\right).
$$ Therefore, we have
\begin{align}\label{eq J'0}
\sum_{m\in J'}\mathbb P(A_n\cap A_m)\le c_{4,10}a_n,
\end{align} for some constant $c_{4,10}>0$.
(2). When $t_m > 2 t_n$, by \eqref{eq Anm10} we know
\begin{align*} \mathbb P(A_n\cap A_m)\le 2 a_n \exp\bigg(-\frac{t_m^{H/\beta}}{ 2 c_{2,2} \xi(t_m)^{1/\beta}}\bigg).
\end{align*}
Thus, we have that for any $m\in I_k$,
$$ \mathbb P(A_n\cap A_m)\le 2a_n\exp\left(-\frac{2^{k-1}}{c_{2,2}}\right).
$$
Since $J_k\subset I_k$ and ${\mathrm{card}}(J_k)\le N_k$, using the definition of $N_k$, we have
\begin{align*}
\sum_{m\in J_k} \mathbb P(A_n\cap A_m)\le 2 N_k a_n\exp\left(-\frac{2^{k-1}}{c_{2,2}} \right)
\le 2 a_n \exp\left(-\frac{2^{k-2}}{K_1} \right).
\end{align*}
Thus,
\begin{align}\label{eq Anm40} \sum_{m\in \cup_{k\ge1} J_k }\mathbb P(A_n\cap A_m)\le c_{4,11}a_n,
\end{align}
for some constant $c_{4,11}>0$.
(3). By using \eqref{eq J'0} and \eqref{eq Anm40}, in order to prove \eqref{eq Anm00}, it suffices to
prove the following result:
\begin{quote}
Given $\varepsilon>0$, there exists a constant $c_{4,12}>0$ such that for any
$$n>\sup_{l\le c_{4,12}}
\sup \{k;k\in I_l\} \ \ \text{and } m\in J''\ \ \text{with }m<n,$$
we have
\begin{align}\label{eq Anm50}
\mathbb P(A_n\cap A_m)\le a_na_m(1+\varepsilon).
\end{align}
\end{quote}
Assume $m\in J''$ such that $m<n$.
Applying \eqref{eq mtu1} with $t=t_n$, $u=t_m$, $\theta=\xi(t_n)/t_n^H$ and $\eta= \xi(t_m)/t_m^H$,
and using the facts of $\theta\ge 2^{-(k(n)+1)\beta}$ and $\eta\ge 2^{-(k(m)+1)\beta}$, we have
\begin{equation}\label{eq Anm20}
\begin{split}
\mathbb P\left( A_n\cap A_m \right)
\le &\, a_na_m \Bigg\{ \exp\left[-\frac{1}{K} \left(\frac{t_m}{t_n}\right)^{\tau}+\psi(\theta)+\psi(\eta) \right]\\
&\, \, \, \, \, \,\, +\exp\left[K\left(\frac{t_m}{t_n}\right)^{-\tau} \left( 2^{(k(n)+1)(1+1/\beta)} +2^{(k(m)+1)
(1+1/\beta)} \right)\right] \Bigg\}.
\end{split}
\end{equation}
Since $m\in J''$, $m\notin J_{k(m)}$. By the definition of $J_{k(m)}$, we have $$ {\mathrm{card}}\big(I_{k(m)}\cap [m, n]\big)>N_{k(m)}. $$ Thus, \eqref{eq kmn0} holds. By \eqref{eq tmn exp0}, we have \begin{align}\label{eq tmn30} \frac{t_m}{t_n}\ge\exp\left(\exp\big(2^{\max\{k(m),\, k(n)\}}/c_{4,7} \big) \right). \end{align} By \eqref{eq sbp} and \eqref{eq kn}, $\psi(\theta)\le K_1 2^{k(n)+1}$ and $\psi(\eta)\le K_12^{k(m)+1}$. Those, together with \eqref{eq tmn30}, imply that the coefficient of $a_na_m$ in \eqref{eq Anm20} gets close to $1$ as $\max\left\{k(m),\, k(n)\right\}$ becomes large. Hence, we get \eqref{eq Anm50}. The proof of the necessity in Theorem \ref{thm main 0} is complete.
\vskip0.5cm
\section{Proof of Theorem \ref{thm main infty}} In this section, we prove an integral criterion for the lower classes of GFBM at infinity. The setting is the same as in Talagrand \cite{Tal96} and the arguments are similar to those in \cite{Tal96} or Section 4 of this paper. In order not to make the paper too lengthy, we only give a sketch of the proof.
\subsection{Sufficiency}
Suppose that $\xi(t)$ is a nondecreasing continuous function such that $\xi(t)/t^H$ is bounded and $I_{\infty}(\xi)<+\infty$. We prove that $\xi(t)\le M(t)$ for $t$ large enough in probability one.
By using the argument in the proof of \cite[Lemma 3.1]{Tal96}, one can obtain the following analogue of Lemma 4.1. The details of the proof are omitted here. \begin{lemma}\label{lem function0} Suppose that $\xi(t)$ is a nondecreasing continuous function such that $\xi(t)/t^H$ is bounded and $I_{\infty}(\xi)<+\infty$. Then \begin{align}\label{eq f0} \lim_{t\rightarrow\infty}\frac{\xi(t)}{t^H}=0. \end{align} \end{lemma}
In order to prove the sufficiency, we construct the sequences $\{t_n\}_{n\ge1}$, $\{u_n\}_{n\ge1}$ and $\{v_n\}_{n\ge1}$ recursively as follows. Let $L>2H$ be a constant. We start with $t_1=\mathrm {e}^{\mathrm {e}}$. Having constructed $t_n$, we set \begin{align} u_{n+1}:=&\, t_n+t_n^{\frac{\gamma}{\beta}}\xi(t_n)^{\frac1\beta}, \label{eq tn infty0} \\ v_{n+1}:=&\, \inf\left\{u>t_n; \,\xi(u)\ge\xi(t_n)\left( 1+L\left(\frac{\xi(t_n)}{t_n^H} \right)^{1/\beta}\right) \right\}, \label{eq tn infty1} \\ t_{n+1}:=&\, \min\{u_{n+1},\, v_{n+1}\}. \label{eq tn infty2} \end{align}
Similar to the proofs of \cite[Lemmas 3.2 and 3.3]{Tal96}, we have the following lemmas. \begin{lemma}\label{lem infty} $\{t_n\}_{n\ge1}$ is increasing and $\lim_{n\rightarrow\infty} t_n=+\infty.$ \end{lemma}
\begin{lemma}\label{lem Mtn} If $M(t_n)\ge \xi(t_n)\left(1+L\left(\frac{\xi(t_n) }{t_n^H}\right)^{1/\beta}\right)$ for all $n\ge n_0$, then $M(t)\ge \xi(t)$ for all $t\ge t_{n_0}$. \end{lemma}
By \eqref{eq M1}, we have \begin{align} \mathbb P\Bigg(M(t_n)<\xi(t_n)\bigg(1+L\Big(\frac{\xi(t_n)}{t_n^H}\Big)^{1/\beta} \bigg) \Bigg) =\varphi\Bigg(\frac{\xi(t_n)}{t_n^H} \bigg(1+L\Big(\frac{\xi(t_n)}{t_n^H}\Big)^{1/\beta}\bigg) \Bigg). \end{align} One could show that this later series converges by using the same argument in \cite[Section 3]{Tal96} or \cite[Section 3]{ElN2011}. Therefore, the proof of the sufficiency is finished by the Borel-Cantelli lemma, Lemma \ref{lem infty} and Lemma \ref{lem Mtn}.
\subsection{Necessity }
In this part, we suppose that with positive probability, $\xi(t)\le M(t)$ for all $t$ large enough. We prove that $\xi(t)$ is bounded and $I_{\infty}(\xi)<+\infty$. The first fact is a direct consequence of \eqref{eq M1} and the fact that $\lim_{\theta\rightarrow \infty}\varphi(\theta)=1$. To prove the second statement, we use Lemma \ref{lem GBC} and the following lemmas in a way similar to the proof in Section 4.2.
\begin{lemma}\label{lem limit0} Suppose that with positive probability $\xi(t)\le M(t)$ for all $t$ large enough.
Then we have
\begin{align}\label{eq limit0}
\lim_{t\rightarrow\infty}\frac{\xi(t)}{t^H}=0.
\end{align}
\end{lemma}
\begin{proof} Since its proof is similar to that of \cite[Lemma 4.1]{Tal96}, we omit the details here.
\end{proof}
To prove the necessity, we will show that $\xi\in LUC_{\infty}(M)$ when \begin{align}\label{Eq Nece} I_{\infty}(\xi)=+\infty\ \ \text{and }\, \ \lim_{t\rightarrow\infty} \frac{\xi(t)}{t^H}=0. \end{align} The first step is to construct a suitable sequence. \begin{lemma}\label{lem tninfty} Under \eqref{Eq Nece}, there exists a sequence $\{t_n\}_{n\ge1}$ with the following three properties:
\begin{itemize}
\item[(i)]
\begin{align}\label{eq tn+1}
t_{n+1}\ge t_n\left(1+\left(\frac{\xi(t_n)}{t_n^H} \right)^{1/\beta}\right); \end{align}
\item[(ii)]
\begin{align}\label{eq sum infty}
\sum_{n=1}^{\infty} \varphi\left(\frac{\xi(t_n)}{t_n^H}\right)=+\infty;
\end{align}
\item[(iii)] For all $m\ge n$ large enough,
\begin{align}\label{eq tmn}
\frac{\xi(t_m)}{t_m^{ 2H}}\le 2 \frac{\xi(t_n)}{t_n^{2H}}.
\end{align}
\end{itemize}
\end{lemma}
\begin{proof} The construction is given by induction over $n$. We take $t_1=\mathrm {e}^{\mathrm {e}}$. Having
constructed $t_n$, we find $s_n\ge t_n$ such that
$$
\sup\left\{\frac{\xi (t)}{t^{2H}};\, t\ge t_n\right\}=\frac{\xi(s_n)}{s_n^{2H}}.
$$
Using the continuity of $\xi$ and \eqref{eq limit0}, we know $s_n\in (t_n,\infty)$. We then set
\begin{align}\label{eq tn+11}
t_{n+1}:=s_n+s_n^{\gamma/\beta} \xi(s_n) ^{1/\beta}=s_n\left(1+\left(\frac{\xi(s_n)}{s_n^H}\right)^{1/\beta}\right).
\end{align} It is obvious that \eqref{eq tn+1} holds. By the construction of $s_{n-1}$ and by \eqref{eq limit0}, we have that for $m\ge n$, $$ \frac{\xi(t_m)}{t_m^{2H}}\le \frac{\xi(s_{n-1})}{s_{n-1}^{2H}}. $$
By \eqref{eq limit0} and \eqref{eq tn+11}, we know for all $n$ large enough, $\xi(s_{n-1})\le \xi (t_n)$ and
$t_n^{2H}\le 2 s_{n-1}^{2H}$. Thus, \eqref{eq tmn} holds. To prove \eqref{eq sum infty}, we show that
for all $n$ large enough,
\begin{align}\label{eq In}
I_n:=\int_{t_n}^{t_{n+1}} \left(\frac{\xi(t)}{t^H}\right)^{-1/\beta}\varphi\left(\frac{\xi(t)}{t^H}\right) \frac{dt}{t}
\le c_{5,1}\varphi\bigg(\frac{\xi(t_{n+1})}{t_{n+1}^H}\bigg),
\end{align} where the constant $c_{5,1}$ depends on $H$ and $\beta$ only.
We write
\begin{align}\label{eq: In12}
I_n=\left[\int_{t_n}^{s_{n}}+\int_{s_n}^{t_{n+1}} \right] \left(\frac{\xi(t)}{t^H}\right)^{-1/\beta}\varphi
\left(\frac{\xi(t)}{t^H}\right) \frac{dt}{t}=:I_n^1+I_n^2.
\end{align}
First, we have
\begin{align}\label{eq: In121}
I_n^2= &\,\int_{s_n}^{t_{n+1}} \left(\frac{\xi(t)}{t^H}\right)^{-1/\beta}\varphi\left(\frac{\xi(t)}{t^H}\right) \frac{dt}{t}\\
\le &\, \left(t_{n+1}-s_n \right)s_n^{-\gamma/\beta} \xi(s_n)^{-1/\beta} \varphi\left(\frac{\xi(t_{n+1})}{s_n^H}\right)\\
=&\, \varphi\bigg(\frac{\xi(t_{n+1})}{s_n^H}\bigg).
\end{align}
By \eqref{eq tn+11}, we know that $t_{n+1}\le 2 s_n$ for all $n$ large enough. Hence,
\begin{equation}\label{eq: In122}
\begin{split}
\frac{\xi(t_{n+1})}{s_n^H}= \,
\frac{\xi(t_{n+1})}{t_{n+1}^H}\bigg(\frac{t_{n+1}}{s_n}\bigg)^H
=& \,\frac{\xi(t_{n+1})}{t_{n+1}^H}\Bigg(1+\bigg(\frac{\xi(s_n)}{s_n^H}\bigg)^{1/\beta}\Bigg)^H\\
\le &\,\frac{\xi(t_{n+1})}{t_{n+1}^H}\Bigg(1+2^{H/\beta}\bigg(\frac{\xi(t_{n+1})}{t_{n+1}^H}\bigg)^{1/\beta}\Bigg)^H\\
\le & \,\frac{\xi(t_{n+1})}{t_{n+1}^H}\Bigg(1+c_{5,2}\bigg(\frac{\xi(t_{n+1})}{t_{n+1}^H}\bigg)^{1/\beta}\Bigg),
\end{split}
\end{equation}
here $c_{5,2}$ is a positive constant which only depends on $H$ and $\beta$.
By \eqref{eq control}, \eqref{eq: In121} and \eqref{eq: In122}, we know that there exists a constant
$c_{5,3}>0$ satisfying that
$$
I_n^2\le c_{5,3} \varphi\bigg(\frac{\xi(t_{n+1})}{t_{n+1}^H}\bigg).
$$
Now, we turn to study $I_n^1$. By the construction of $s_n$, we have for any $t_n\le t\le s_n$,
$$
\frac{\xi(t)}{t^H}\le t^H
\frac{\xi(s_n)}{s_n^{2H}}\le
\frac{\xi(s_n)}{s_n^H}.
$$ By \eqref{eq limit0}, we can assume $\xi(s_n)/ s_n^H$ is arbitrarily small for all large $n$, so that Lemma \ref{lem increase} implies that for any $t_n\le t\le s_n$,
$$
\left(\frac{\xi(t)}{t^H}\right)^{-1/\beta}\varphi \left(\frac{\xi(t)}{t^H}\right)\le \left( \frac{t^H\xi(s_n)}
{s_n^{2H}}\right)^{-1/\beta} \varphi \left( \frac{t^H\xi(s_n)}{s_n^{2H}}\right),
$$
and thus
$$
I_n^1\le \int_{t_n}^{s_n}\left(\frac{\xi(t)}{t^H} \right)^{-1/\beta}\varphi \left(\frac{\xi(t)}{t^H}\right)\frac{dt}{t}
\le \int_{t_n}^{s_n} \left( \frac{t^H\xi(s_n)}{s_n^{2H}}\right)^{-1/\beta} \varphi \left( \frac{t^H\xi(s_n)}
{s_n^{2H}}\right)\frac{dt}{t}.
$$
Making the change of variable $t=us_n$, we have
$$
I_n^1\le \int_0^1 \left(u^H a\right)^{-1/\beta}\varphi\left(u^H a\right)\frac{du}{u},
$$
where $a= \xi(s_n)/s_n^H$. It remains to prove that this later integral is at most $c_{5,4}\varphi(a)$
for some positive constant $c_{5,4}$. By \eqref{eq varphi1}, it suffices to prove that for any $c_{5,5}
\in (0,1)$, there exists $c_{5,6}>0$ such that
$$
\int_{c_{5,5}}^1 \left(u^H a\right)^{-1/\beta}\varphi\left(u^H a\right)\frac{du}{u} \le c_{5,6}\varphi(a).
$$
Setting $v=u^H$, it suffices to prove that \begin{align}\label{eq varphi 01}
\int_0^1 \varphi(va)dv=\int_0^1\varphi\big((1-v)a\big)dv\le c_{5,6} a^{1/\beta} \varphi(a).
\end{align}
By the convexity of $\psi$ and \eqref{eq psi'}, we have
\begin{align*}
\psi(a-av)\ge \, \psi(a)+av|\psi'(a)|
\ge\, \psi(a)+\frac{av}{K_2a^{1+\beta}}.
\end{align*} This implies that
$$
\varphi\big((1-v)a\big)\le \varphi(a)\exp\left(-\frac{v}{K_2 a^{1/\beta}}\right).
$$
Since
$$
\int_0^1 \exp\left(-\frac{v}{K_2 a^{1/\beta}}\right)dv\le K_2 a^{1/\beta},
$$ Eq. \eqref{eq varphi 01} is proved and the proof is complete.
\end{proof}
For each $n\ge1$, we define $k(n)$ by
$$
2^{k(n)}\le \left(\frac{t_n^H}{\xi(t_n)}\right)^{1/\beta} < 2^{k(n)+1},
$$
and we set $I_k:=\big\{n\in\mathbb N; \, k(n)=k \big\}$ for any $k\ge1$. By \eqref{eq limit0}, we know that each $I_k$ is finite.
We recall the constant $K_1$ given in \eqref{eq sbp}. Without loss of generality, we can assume that
$K_1\ge c_{2,2}$, where $c_{2,2}$ is the constant in \eqref{eq Mtu}. For any $k\ge1$, we set
$N_k:=\exp\left(2^{k-2}/K_1 \right)$.
By using the argument in \cite[Proposition 4.2]{Tal96} or Proposition \ref{Prop con2}, we obtain the following result.
\begin{lemma}\label{Prop con22} Under \eqref{Eq Nece}, there exist a positive constant $c_{5,7}$ depending on
$\beta$ and $H$ only and a set $J$ with the following properties:
\begin{itemize}
\item[(i)]
\begin{align}\label{eq sum J infty}
\sum_{n\in J}\varphi\left( \frac{\xi(t_n)}{t_n^H} \right)=+\infty;
\end{align}
\item[(ii)] Given $n, m\in J$ with $n<m$ such that \begin{align}\label{eq kmn} {\mathrm{card}}\big(I_{k(m)}\cap [n,m]\big)> N_{k(m)},
\end{align}
we have \begin{align}\label{eq tmn exp} \frac{t_m}{t_n}\ge\exp\bigg(\exp\Big( 2^{\max\{k(n),\, k(m)\}}/ c_{5,7} \Big) \bigg). \end{align}
\end{itemize}
\end{lemma}
By using Lemmas \ref{lem tninfty} and \ref{Prop con22} and using the argument in Section \ref{Sec Neces} (also see \cite[Section 5]{Tal96}), one can prove the necessity part of Theorem \ref{thm main infty}. The details are omitted.
\vskip0.5cm \noindent{\bf Acknowledgments}: The research of R. Wang is partially supported by NNSFC grant 11871382 and the Fundamental Research Funds for the Central Universities 2042020kf0031. The research of Y. Xiao is partially supported by NSF grant DMS-1855185.
\vskip0.5cm
\end{document}
|
arXiv
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Joint Chance Constraints in AC Optimal Power Flow: Improving Bounds through Learning} \author{Kyri Baker, \emph{Member}, IEEE
and Andrey Bernstein, \emph{Member}, IEEE}
\maketitle
\begin{abstract} This paper considers distribution systems with a high penetration of distributed, renewable generation and addresses the problem of incorporating the associated uncertainty into the optimal operation of these networks.
Joint chance constraints, which satisfy multiple constraints simultaneously with a prescribed probability, are one way to incorporate uncertainty across sets of constraints, leading to a chance-constrained optimal power flow problem. Departing from the computationally-heavy scenario-based approaches or approximations that transform the joint constraint into conservative deterministic constraints, this paper develops a scalable, data-driven approach which learns operational trends in a power network, eliminates zero-probability events (e.g., inactive constraints), and accurately and efficiently approximates bounds on the joint chance constraint iteratively. In particular, the proposed framework improves upon the classic methods based on the union bound (or Boole's inequality) by generating a much less conservative set of single chance constraints that also guarantees the satisfaction of the original joint constraint. The proposed framework is evaluated numerically using the IEEE 37-node test feeder, focusing on the problem of voltage regulation in distribution grids.
\end{abstract}
\section{Introduction} \label{sec:intro} The AC optimal power flow (OPF) problem is one of the fundamental problems in power system operation and analysis; see, e.g., \cite{OPFoverview} for an overview. \textcolor{black}{Recently, the introduction of dynamic pricing, renewable generation, energy storage, and other distributed energy resources has increased the uncertainty in achieving predictable and reliable grid operations when using deterministic methods. Broadly, there are two classes of approaches which incorporate uncertainties into OPF problems: Robust methods and stochastic methods (and combinations thereof) \cite{Louca16, Muhlpfordt16, Jabr13, Molzahn18}. Amongst stochastic methods, the chance-constrained AC OPF (CC-AC-OPF) is most valuable in situations where the uncertainty lies within constraints, and constraint violations are allowed with a small probability. In power and energy applications, this can include violations in the thermal limit of transmission lines \cite{Vrak13}, relaxation of voltage regulation requirements \cite{DallAnese17CC,Guo18_1}, or loosening of indoor thermal comfort limits \cite{Garifi18}}.
\noindent A prototypical CC-AC-OPF is given by \begin{subequations} \label{eqn:OPF} \begin{align}
\, \, &\min_{\substack{{\bf x} \in {\cal X}}}\hspace{.2cm} \mathbb{E}_{{\bf y}} \, f({\bf x}, {\bf y}) \\ &\mathrm{subject\,to:~} \, {\bf y} = {\bf h}({\bf x}, {\utwi{\xi}}) \label{eqn:PF_constr}\\ &\hspace{2cm} \mathbb{P} \{{\bf y} \in {\cal Y} \} \geq 1 -\epsilon\label{eqn:state_constr},
\end{align} \end{subequations} where ${\bf x}$ is a vector that collects all the controllable inputs to the system, typically active and reactive power injections of the controllable distributed energy resources (DERs); ${\utwi{\xi}}$ is a random vector representing the uncertainty in the system (e.g., power injections of the uncontrollable assets and solar irradiance); ${\bf y}$ is the vector of state variables, such as voltage phasors across the buses of the network; \eqref{eqn:PF_constr} are the power-flow constraints; \eqref{eqn:state_constr} is an operational constraint formulated as a chance constraint on the state vector ${\bf y}$; and we use the notation $\mathbb{E}_{{\bf y}}$ to denote the expected value with respect to the distribution of ${\bf y}$. In particular, \eqref{eqn:state_constr} ensures that the state vector lies in some prescribed operational set ${\cal Y}$ with probability at least $1 - \epsilon$ for some (small) quantity $\epsilon > 0$.
In many applications, the constraint ${\bf y} \in {\cal Y}$ is composed of several individual constraints ${\bf y} \in {\cal Y}_i$, $i = 1, \ldots, n$, that have to be satisfied simultaneously; therefore chance constraint \eqref{eqn:state_constr} is of the form \begin{equation} \label{eqn:joint_cc} \mathbb{P} \left(\cap_{i=1}^n \{{\bf y} \in {\cal Y}_i \} \right) \geq 1 -\epsilon. \end{equation} Examples of individual constraints that have to be satisfied simultaneously with high probability include joint constraints over different buses in the network, constraints that link timesteps (e.g., ensuring that power delivered to a sensitive resource is satisfied with high probability across the timesteps after a contingency), or even simply two-sided constraints (e.g., constraining the upper and lower limits on uncertain line flows or voltage magnitudes).
\textcolor{black}{Many recent works regarding chance constraints in OPF problems have focused on conservative convex upper bounds of single chance constraints \cite{Tyler15,Nemirovski} and distribution-free, data-based single chance constraints \cite{RoaldCC18, Guo18_1, Baker16NAPS, Misra18}. These techniques provide tractable approaches to addressing single chance constraints, and can be used in conjunction with the technique to reduce the joint chance constraint presented in this paper. In addition, robust optimization techniques can be combined with chance-constrained approaches \cite{Venzke18, Lubin16}, which can also be used in conjunction with the formulation in this paper.} \textcolor{black}{Due to the difficulty in handling joint chance constraints, most of these works focus on single chance constraints.} Considering simultaneous probabilistic constraints generally requires either computationally heavy sampling-based approaches which are limited by problem size \cite{Hong_JCC}; or assumptions about the random parameters \cite{JCC06}; or the use of the union bound, or Boole's inequality \cite{Boole1854}, to \textcolor{black}{separate the joint chance constraint into single constraints and} create conservative upper bounds on the single constraints \cite{Grosso14, Blackmore09}. In \cite{Hong_JCC}, a Monte Carlo method is proposed to solve a sequence of convex optimization problems, avoiding the use of Boole's inequality, with a guarantee that the algorithm converges to a KKT point. However, it is limited by problem size to small or medium size problems with less than 100 dimensions. Scenario approaches can be used to simplify joint constraints into deterministic single constraints; however, these approaches can be overly conservative, and can actually perform worse as the number of samples increases \cite{Campi}.
Using the union bound (or, Boole's inequality) is the most popular way to relax \eqref{eqn:joint_cc} that boils down to replacing it with $n$ chance constraints \begin{equation} \label{eqn:single_cc} \mathbb{P} \{{\bf y} \in {\cal Y}_i \} \geq 1 -\epsilon_i, \, \, i = 1, \ldots, n. \end{equation} It is easy to see that if $\sum_{i = 1}^n \epsilon_i = \epsilon$, \eqref{eqn:single_cc} implies \eqref{eqn:joint_cc}; particularly, if no additional information is used regarding the individual constraints, the typical choice is $\epsilon_i \equiv \frac{\epsilon}{n}$. However, this choice may result in highly conservative solution to \eqref{eqn:OPF}. To illustrate this fact, consider two constraints: ${\bf y} \in {\cal Y}_1$ and ${\bf y} \in {\cal Y}_2$. Suppose that the events $A_i := \{{\bf y} \notin {\cal Y}_i\}$ are highly correlated, in the sense that with very high probability, whenever $A_1$ happens, $A_2$ happens as well (and vice versa). For example, $A_i$ can represent a violation of voltage upper bound at bus $i$ equipped with a photovoltaic (PV) panel, and both buses are geographically close to one another. In this case, \begin{equation} \mathbb{P} (A_1 \cup A_2 ) = \mathbb{P}(A_1) + \mathbb{P}(A_2) - \mathbb{P}(A_1 \cap A_2) \approx \mathbb{P}(A_1) \approx \mathbb{P}(A_2) \end{equation} because $\mathbb{P}(A_1 \cap A_2) \approx \mathbb{P}(A_1) \approx \mathbb{P}(A_2)$. Therefore, the joint chance constraint \eqref{eqn:joint_cc} would boil down to a single constraint \begin{align*} \mathbb{P} ( \{{\bf y} \in {\cal Y}_1\} \cap \{{\bf y} \in {\cal Y}_2\}) &= 1 - \mathbb{P} (A_1 \cup A_2 ) \\ &\approx 1 - \mathbb{P}(A_1) \geq 1 - \epsilon, \end{align*} or equivalently, $\mathbb{P}(A_1) \leq \epsilon$. However, the union bound approximation \eqref{eqn:single_cc} will impose a pair of constraints $\mathbb{P}(A_i) \leq \frac{\epsilon}{2}$, $i = 1, 2$, therefore unnecessarily restricting the constraint set.
\iffalse Joint chance constraints seek to satisfy multiple constraints simultaneously with a prescribed probability. In large-scale power networks, joint constraints may appear in formulations that link networks (i.e., constraining power and communications), in constraints that link timesteps (i.e., ensuring that power delivered to a sensitive resource is satisfied with high probability across the timesteps after a contingency), or even simply in two-sided probabilistic constraints (i.e., constraining the upper and lower limits on uncertain line flows). Considering simultaneous probabilistic constraints generally requires either computationally heavy sampling-based approaches which are limited by problem size \cite{Hong_JCC}, assumptions about the random parameters \cite{JCC06}, or the use of the union bound, or Boole's inequality \cite{Boole1854} to create conservative upper bounds on the single chance constraints \cite{Grosso14, Blackmore09}. In \cite{Hong_JCC}, a Monte Carlo method is proposed to solve a sequence of convex optimization problems, avoiding the use of Boole's inequality, with a guarantee that the algorithm converges to a KKT point. However, it is limited by problem size to small or medium size problems with less than 100 dimensions. Scenario approaches can be used to simplify joint constraints into deterministic single constraints; however, these approaches do not offer any guarantees, can be overly conservative, and can actually perform worse as the number of samples increases \cite{Campi}. \fi \iffalse Joint chance constraints have been previously studied in power systems applications, mostly in transmission networks; for mitigating line flow congestions in \cite{Hojjat15}, and for the N-1 security problem in \cite{Vrak12}, both leveraging sampling-based approaches to reduce the otherwise intractable joint chance constraint into a series of individual constraints. A joint and single chance-constrained AC Optimal Power Flow (OPF) are solved in \cite{Roald18CC}, addressing the issues of increased uncertainty introduced in linearized AC OPF formulations. Joint chance constraints have also been applied to other fields such as drinking water network reliability \cite{Grosso14} and finite horizon control \cite{Blackmore09}, optimally sizing energy storage capacity \cite{BakerTSE16} and optimal communication network design \cite{Pascali09}. \fi
In this paper, we leverage statistical learning tools to address the problem of computationally burdensome joint chance constraints in AC OPF problems, with the following key ingredients: \begin{itemize} \item We present a framework for reducing a joint chance constraint into a series of single chance constraints in a method that significantly reduces the conservativeness compared to using Boole's inequality \cite{Baker_boole}. \rev{To this end, we leverage} support vector classifiers to \emph{classify events $A_i := \{{\bf y} \notin {\cal Y}_i\}$ as having either zero or non-zero probabilities}. \rev{We term the events (and corresponding constraints) that has non-zero probability as \emph{active}; otherwise, they are \emph{inactive}. That is: \begin{align*}
\mathbb{P}( \{{\bf y} \notin {\cal Y}_i\} ) = 0 & \, \Longleftrightarrow \, \{{\bf y} \in {\cal Y}_i\} \, \textrm{ is inactive} \\
\mathbb{P}( \{{\bf y} \notin {\cal Y}_i\} ) > 0 & \, \Longleftrightarrow \, \{{\bf y} \in {\cal Y}_i\} \, \textrm{ is active} \end{align*} For example, voltage constraints are classified as active or inactive. } \item An estimation method is presented which iteratively provides a tighter upper bound on the joint chance constraint and can be terminated before the estimation is finished in computationally restrictive or high dimensional settings where the entire joint constraint cannot be estimated. \end{itemize} Unlike classic Monte-Carlo-based approaches, the proposed framework is scalable to high-dimensional constraints. Moreover, the reduction of the joint chance constraint into single chance constraints allows for the use of many of the distributionally-robust single chance constraint reformulations in the literature \cite{Tyler15, Nemirovski}. \textcolor{black}{It is important to note here that we are not developing a new technique for evaluating or reformulating single chance constraints; we are developing a technique for reducing a joint chance constraint into single ones.} Building upon our previous initial work \cite{BakerJointGlobalSIP}, the proposed method can also reduce computation time in non-stochastic settings by removing non-binding constraints from the deterministic optimization problem.
Simulation results are presented for the IEEE 37-node test system with a high penetration of distributed solar in an active distribution network. While the results presented here are focused on voltage regulation in distribution networks, the method proposed in this paper can be applied in general CC-AC-OPF settings for any type of joint chance constraints.
The remainder of the paper is structured as follows. Section \ref{sec:JCC} discusses the joint chance constraints formulation and outlines our approach. Section \ref{sec:SVC} presents a method to classify inactive constraints and to estimate the remaining joint constraints. Section \ref{sec:systemmodel} outlines the distribution system model and related notation. Section \ref{sec:optimization} discusses the application of the proposed method to voltage regulation problem in active distribution networks. Section \ref{sec:simulations} presents the numerical results. Finally, Section \ref{sec:conclusion} concludes the paper.
\section{Outline of the Approach} \label{sec:JCC} To explain how we will use statistical learning to reduce the complexity of the joint chance constraint in power network optimization, consider \eqref{eqn:single_cc} and let $A_i := \{{\bf y} \notin {\cal Y}_i\}$. Then, $\mathbb{P} \left(\cap_{i=1}^n \{{\bf y} \in {\cal Y}_i \} \right) = 1 - \mathbb{P}\Big(\bigcup_{i=1}^n A_i\Big)$, and from the probabilistic version of the inclusion-exclusion principle \textcolor{black}{we have the following:}
\iffalse \begin{align} \mathbb{P}(g_1({\bf x}, \bm{\delta}) \leq 0, ... , ~g_n({\bf x}, \bm{\delta}) \leq 0) \geq 1 - \epsilon. \label{JCC_gen} \end{align}
\noindent for individual constraints $g_1({\bf x}, \bm{\delta}) \leq 0, \cdots, g_n({\bf x}, \bm{\delta}) \leq 0$, where ${\bf x}$ is a vector of decision variables, $\bm{\delta}$ is a vector of uncertain parameters, and $\epsilon \in (0,1)$ is the maximum allowable constraint violation. Considering each constraint $g_i({\bf x}, \bm{\delta}) \leq 0, i \in [n]$ as an event $B_i$, the joint chance constraint can be written as the intersection of events $\mathbb{P}(B_1 \cap B_2 \cap ... \cap B_n)$. Using complementarity, $\mathbb{P}(B_1 \cap B_2 \cap ... \cap B_n) = 1- \mathbb{P}(B^c_1 \cup B^c_2 \cup ... \cup B^c_n)$. For brevity, define event $B^c_i$ as $A_i$ for each $i \in [n]$. From the probabilistic version of the inclusion-exclusion principle, we can expand the joint constraint \fi
\begin{align}\label{inclusion} \begin{split} \mathbb{P}\Big(\bigcup_{i=1}^n A_i\Big) &= \sum_{i=1}^n \mathbb{P}(A_i) - \sum_{i<j} \mathbb{P}(A_i \cap A_j) + \cdots \\ &\cdots + (-1)^{n-1} \mathbb{P}\Big( \bigcap_{i=1}^n A_i \Big)\\ & := \sum_{i=1}^n \mathbb{P}(A_i) - P_c. \end{split} \end{align}
\noindent Truncation of the above to Boole's inequality $\mathbb{P}(A_1 \cup A_2 \cup ... \cup A_n) \leq \sum_{i=1}^n \mathbb{P}(A_i)$ allows for the separation of the joint chance constraint \eqref{eqn:joint_cc} into individual constraints $\mathbb{P}(A_i) \leq \epsilon_i$ where $\sum_{i=1}^n \epsilon_i = \epsilon$, \textcolor{black}{and a common choice for $\epsilon_i = \frac{\epsilon}{n}$, where $n$ is the number of individual constraints}. While useful and a very popular technique for solving joint chance constrained programs, Boole's inequality tends to result in very conservative solutions \cite{Baker_boole}. \iffalse Approximating the joint chance constraint with a scenario approach, which involves sampling the uncertain parameter and reducing the joint chance constraints into a set of deterministic constraints equal to the number of samples, is also undesirable in large problems; the number of realizations of the random parameter that need to be included in individual single constraints, to achieve a $\beta$ percent confidence level, should be at least $\frac{2}{\epsilon}(ln(\frac{1}{\beta}) + N_d)$, where $N_d$ is the total number of decision variables \cite{Campi}. For example, if $\epsilon = 0.01$ and $N_d = 100$, and we desire with a 99\% confidence level that the original joint chance constraint is satisfied with probability $1-\epsilon$, over 20,000 realizations, or single constraints, must be included in the problem. This can quickly become intractable. \fi To address the deficiencies of using conservative upper bounds, computationally heavy scenario-based approaches, or making assumptions about the single or joint probability distributions, we will present a general, distribution-agnostic technique based on learning marginal probabilities to eliminate zero-probability joint events in \eqref{inclusion} and leverage a Monte Carlo sampling-based approach to estimate the remaining joint probabilities $P_c$. Then, we decompose the joint constraint into single chance constraints that must be satisfied with probability $\geq 1 - \frac{\epsilon}{n} - \frac{P_c}{n}$, \textcolor{black}{where $n$ in this case is the number of nonzero-probability individual events}. Thus, if $P_c > 0$ (i.e., events $A_1, ... , A_n$ are not disjoint), a tighter upper bound for the single chance constraints is provided compared to using Boole's inequality. A general flow chart of the overall procedure is shown in Fig. \ref{fig:flow}; the individual blocks are discussed next.
\begin{figure}
\caption{An outline of the general procedure for solving the joint chance constrained problem. \textcolor{black}{The details of each individual component in the flowchart can be found in the following sections.}}
\label{fig:flow}
\end{figure}
\section{Constraint Classification and Estimation} \label{sec:SVC}
In an optimization problem, \emph{inactive} constraints are those which, if removed from the problem, would not change the optimal solution. Active constraints, on the other hand, are essential in determining the optimal solution and would change the optimal solution if removed. Machine learning approaches to solve OPF problems have recently been realized as a powerful tool \cite{Misra18, Dobbe18}; here, we leverage machine learning for identifying active constraints in AC OPF problems with joint chance constraints. This section discusses how we can learn which constraints are likely to be inactive in power system optimization given certain system conditions, reducing the computational burden of calculating each term in \eqref{inclusion}.
\subsection{A Simple Example - Two Sided Constraint} To illustrate the overall idea of the framework, consider the two-sided joint chance constraint which constrains the state of charge $E^{(t+1)}$ of an energy storage system (ESS) to be within desired bounds $\underline{E}$ and $\overline{E}$ with probability at least $1-\epsilon$:
\begin{align} \label{ESS_JCC} \mathbb{P}(\underline{E} \leq E^{(t+1)} \leq \overline{E}) \geq 1 - \epsilon, \end{align}
\noindent While maintaining ESS state of charge within certain bounds can extend the lifetime of the ESS, under certain situations it may be more beneficial or unavoidable to violate these limits. Intuitively, in certain situations it can be obvious if $\underline{E} \leq E^{(t+1)}$ or $E^{(t+1)} \leq \overline{E}$ is an inactive constraint; for example, if the ESS is currently at its maximum charge ($E^{(t)} = \overline{E}$), and the maximum discharge rate makes it impossible for the ESS to reach $\underline{E}$ in the next time step, we know with certainty that $\underline{E} < E^{(t+1)}$ and thus $\mathbb{P}(\underline{E} > E^{(t+1)}) = 0$. So, from the inclusion-exclusion principle, $\mathbb{P}(\underline{E} \leq E^{(t+1)} \leq \overline{E}) =1 - \mathbb{P}(\underline{E} > E^{(t+1)}) + \mathbb{P}(E^{(t+1)} > \overline{E}) + \mathbb{P}(\{\underline{E} > E^{(t+1)}\} \cap \{E^{(t+1)} > \overline{E}\}) = \mathbb{P}(E^{(t+1)} \leq \overline{E})$, reducing the joint chance constraint to the single chance constraint $\mathbb{P}(E^{(t+1)} \leq \overline{E}) \geq 1 - \epsilon$. However, when dealing with multi-time step problems, assuming one of these events has a zero probability may not be trivial; it may also not be trivial depending on maximum charge/discharge rates, time in between control decisions, the level of uncertainty, or distance between $\underline{E}$ and $\overline{E}$. In addition, while we may have physical intuition as to when a constraint is likely to be relevant or not, there can be many factors influencing the outcome of an optimization problem, and we would like to have an automated way of reducing the complexity of joint chance constraints. Thus, it is desirable to develop a rule that may allow us to exploit these patterns by learning them over time and having the optimization problem automatically decompose the joint chance constraints into single chance constraints depending on the outcome of these rules.
In general, recall that if $\mathbb{P}(A_i) = 0$, $\mathbb{P}(A_i \cap A_j) = 0$ for all $A_j$; if even a single constraint is classified as inactive, a significant number of terms in the joint chance constraint expansion are eliminated from the calculations and do not have to be estimated further. As a larger example, consider a four-event union $\mathbb{P}(A_1 \cup A_2 \cup A_3 \cup A_4)$ and its expansion via \eqref{inclusion}:
\small \begin{align} \begin{split} &\mathbb{P}(A_1) + \mathbb{P}(A_2) + \mathbb{P}(A_3) + \mathbb{P}(A_4)\\ &- \mathbb{P}(A_1 \cap A_2) - \mathbb{P}(A_1 \cap A_3) - \mathbb{P}(A_1 \cap A_4) \\ &- \mathbb{P}(A_2 \cap A_3) - \mathbb{P}(A_2 \cap A_4) - \mathbb{P}(A_3 \cap A_4) \\ &+ \mathbb{P}(A_1 \cap A_2 \cap A_3) + \mathbb{P}(A_1 \cap A_2 \cap A_4) \\ &+ \mathbb{P}(A_1 \cap A_3 \cap A_4) + \mathbb{P}(A_2 \cap A_3 \cap A_4)\\ &- \mathbb{P}(A_1 \cap A_2 \cap A_3 \cap A_4) \end{split} \end{align} \normalsize
\noindent If constraint $A_1$ is classified as inactive, the above reduces to
\small \begin{align} \begin{split} &\mathbb{P}(A_2) + \mathbb{P}(A_3) + \mathbb{P}(A_4)\\ &- \mathbb{P}(A_2 \cap A_3) - \mathbb{P}(A_2 \cap A_4) - \mathbb{P}(A_3 \cap A_4) \\ &+ \mathbb{P}(A_2 \cap A_3 \cap A_4) \end{split} \end{align} \normalsize
\noindent dramatically reducing the number of intersections we must estimate. For sizable joint chance constraints, identifying zero probability events can potentially make an otherwise intractable problem possible to solve via sampling approaches.
\begin{figure*}
\caption{Visualizing the support vector classifier for classifying overvoltage events ($V > \overline{V}$) at a node in a distribution network. Intuitively, we know that overvoltages occur when solar exceeds load by a particular amount, and the classifiers provide us with a selection rule for including or excluding voltage constraints. Left: Resulting support vector classifiers from traditional SVC and a weighted version (wSVC) which heavily penalizes any misclassifications of active constraints as inactive. Right: A magnified version. The support vectors are training data which lie closest to the separating hyperplane and, if removed, would change the solution of (P0).}
\label{fig:SVC}
\end{figure*}
\subsection{Support Vector Classification (SVC)}
\textcolor{black}{Next, we will develop classifiers for classifying constraints as active or inactive. This procedure is performed before the final OPF problem is solved to reduce the dimensionality of the joint chance constraint.} We will use a popular machine learning technique for classification called Support Vector Classification (SVC). \textcolor{black}{For each training sample $i = 1,...,m$,} and two classes, namely active ($\ell_i = -1$) and inactive ($\ell_i = +1$), we wish to create a \textcolor{black}{decision} rule which uses selected inputs to determine whether or not we include that constraint in the optimization problem. Here, we seek to form an affine classifier of the form ${\bf w}^T{\utwi{\phi}} + b$ with weights ${\bf w} \in \mathbb{R}^2$ and bias $b \in \mathbb{R}$ that classifies constraints as active (${\bf w}^T{\utwi{\phi}} + b \geq 0$) or inactive (${\bf w}^T{\utwi{\phi}} + b < 0$) based on input features ${\utwi{\phi}} \in \mathbb{R}^2$ (e.g., load and available solar at a node in the distribution network as considered in the example in Section \ref{sec:simulations} below). In our formulation, called ``weighted SVC (wSVC)", we heavily penalize misclassifications of active constraints as inactive, while maximizing the separation between classes \cite{StatLearn}. \textcolor{black}{Unlike typical applications for support vector machines/classifiers where misclassifications are equally weighted, highly weighting misclassifications of active constraints as inactive pursues the preservation of an upper bound on the original joint constraint. If some inactive constraints were classified as active, the bound may get looser but stay valid - the other way around, and the bound may not be preserved.} In the training stage, we build the classifier by using $m$ samples of labeled training data $\bl$, by solving the following optimization problem:
\begin{subequations} \label{SVC_opt} \begin{align}
\mathrm{(P0)} &\min_{\substack{{\bf w}, {\bf z}, b}} \hspace{.2cm} \frac{1}{2}{\bf w}^Tb + {\bf c}^T{\bf z} \label{SVC_obj}\\
&\mathrm{s.t.} ~~~\ell_i({\bf w}^T{\utwi{\phi}}_i + b) \geq 1 - z_i\label{SVC_c}\\ &~~~~~~~z_i \geq 0 \label{SVC_z} \end{align} \end{subequations}
\noindent where ${\bf c} \in \mathbb{R}^{m}$ is a penalty parameter and $c_i = 1$ if $\ell_i = +1$ and $c_i = a$ if $\ell_i = -1$, $a \gg 1$. \textcolor{black}{Using this objective weighting for ${\bf z}$ in \eqref{SVC_obj}, we deviate from the traditional SVC formulation with the wSVC problem, which heavily penalizes misclassifications of active constraints as inactive.} Each slack variable $z_i$ is nonzero if ${\utwi{\phi}}_i$ is classified incorrectly, and zero otherwise. \textcolor{black}{After solving (P0), a classifier ${\bf w}^T{\utwi{\phi}} + b$ for each individual node is formed, which takes the form of an affine function of the two features: the total load at a node and available solar generation at a node.}
\subsection{Classifying Power System Constraints} \label{sec:class} \rev{We next describe how we classify constraints for a particular power-system problem. For illustration purposes, we focus on the example of distribution grid voltage regulation under high PV production; the corresponding OPF problem (P1) is formally defined in Section \ref{sec:optimization} below. At each node in the considered distribution network, we construct an SVC/wSVC with the training data set consisting of: \begin{itemize}
\item Features: net solar production and load at that node.
\item Labels: $\ell = 1$ if the voltage magnitude at that node is below the uppder limit; $\ell = -1$ otherwise. \end{itemize} The features in the training data set are constructed from historical data on solar and load, whereas the labels are computed by solving the OPF (P1) \emph{without voltage constraints}, and verifying whether the corresponding voltage limits are violated or not. As overvoltage conditions are primarily caused when solar generation exceeds consumption (load), it is reasonable to exploit this relationship to use the current and forecasted levels of solar and load to determine which voltage constraints will be relevant or binding. The classifiers are then constructed by solving (P0) and obtaining corresponding affine decision rules.
In Fig \ref{fig:SVC}, we illustrate the differences between the classifier chosen with the traditional SVC versus the conservative weighted SVC (wSVC). } \iffalse We next illustrate the \textcolor{black}{differences between the} classifier chosen with the traditional SVC versus the more conservative wSVC by considering the example of distribution grid voltage regulation under high PV production (see Section \ref{sec:systemmodel} for the detailed description of this application). Low-voltage traditional distribution networks, historically constructed assuming one-directional flows of power from distribution substation to loads, are now experiencing many operational challenges due to the increase in distributed generation (namely, increased solar production). Overvoltage conditions ($V > \overline{V}$) are primarily caused when solar generation exceeds consumption (load), and it is reasonable to exploit this relationship to use the current and forecasted levels of solar and load to determine which voltage constraints will be relevant or binding. \textcolor{black}{Data in the form of load and solar generation at each node is collected to build the classifiers at each node from solving (P0), and affine decision rules are constructed from the solution of (P0).}
In Fig \ref{fig:SVC}, this relationship is demonstrated; at each node in the considered distribution network, we construct an SVC/wSVC at each timestep in the optimization with features consisting of the net solar and load at that node. The training data consists from previous optimization runs, with voltages at their maximum limit, $V \geq \overline{V}$ labeled as in the ``active" class \textcolor{black}{and assigned a label of $\ell = -1$} and voltages with $V < \overline{V}$ labeled as in the ``inactive" class \textcolor{black}{and assigned a label of $\ell = +1$}. \fi Traditional SVC and wSVC may not perfectly separate the two classes, but the conservative wSVC \textcolor{black}{aims to ensure} that all training points labeled as active are correctly classified. \rev{ \begin{remark}[Remark 1] In the voltage regulation application and particular formulation used in this paper, the classes can be divided by a separating hyperplane. In applications with a nonlinear relationship between classes, additional methods to preprocess/transform the data can be employed. For example, lifting to a higher dimensional space whereby the data is linearly separable can be achieved, e.g., using kernel-based methods \cite{StatLearn}. \end{remark}
\begin{remark}[Remark 2] While in this paper we focus on using these classifiers for probabilistic constraints, the approach would also provide computational benefits for constraint removal from deterministic programs as well. In particular with the voltage regulation case - as seen in Section \ref{sec:simulations}, including upper bound constraints on the voltage is unnecessary throughout most of the daily operating period. \end{remark} }
\subsection{Iteratively Estimating Event Intersections} \label{sec:iterative} The number of intersections given in the joint chance constraint expansion that must be estimated is given by
\begin{align} \label{numterms}
\sum_{k=2}^{|M|} {{|M|}\choose k} \end{align}
\noindent where $M$ is the set of indices of active constraints, $|M|$ is the cardinality of $M$ (i.e., the number of active constraints), and $n$ is the total number of constraints. For example, if the joint chance constraint originally contained 8 constraints, 247 intersections must be estimated to recover the original constraint. If half of these were classified as inactive, only 11 intersections must be estimated, which is much more reasonable for solving optimization problems on fast timescales. The expansion of the joint chance constraint can now be written as
\begin{align}\label{inclusion2} \begin{split} \mathbb{P}\Big(\bigcup_{i=1}^n A_i\Big) &= \sum_{m\in M} \mathbb{P}(A_m) - \sum_{m \neq j, m, j \in M} \mathbb{P}(A_m \cap A_j) + \cdots \\
&\cdots + (-1)^{|M|-1} \mathbb{P}\Big( \bigcap_{m \in M} A_m \Big). \end{split} \end{align}
\noindent
Our goal is to iteratively estimate event intersections in a way that maintains an upper bound on the original joint chance constraint, allowing for the termination of the algorithm before the entire joint chance constraint is estimated. In fact, the order in which these intersections are computed is very important; if certain intersections are included in the expansion but not others, an upper bound of the original union of events may not be preserved. We therefore only estimate intersection probabilities for \emph{pairs} of terms in \eqref{inclusion2}:
\iffalse \begin{align}\label{newbound} \begin{split} \mathbb{P}\Big(\bigcup_{i=1}^n A_i\Big) ~~&\leq~~ \sum_{m \in M} \mathbb{P}(A_m) \\
& +~ \sum_{k=1}^K \Big[\sum_{\substack{I \subset {1,...,|M|} \\ |I| = 2k-1}} \mathbb{P}\Big(A_I\Big) - \sum_{\substack{I \subset {1,...,|M|} \\ |I| = 2k}} \mathbb{P}\Big(A_I\Big)\Big] \end{split} \end{align} \fi
\begin{align}\label{newbound} \begin{split} {\cal B}_K :=& \sum_{m \in M} \mathbb{P}(A_m) \\
& -\sum_{k=1}^K \Big[\sum_{\substack{I \subset {\{1,...,|M|\}} \\ |I| = 2k}} \mathbb{P}\Big(A_I\Big) - \sum_{\substack{I \subset {\{1,...,|M|\}} \\ |I| = 2k+1}} \mathbb{P}\Big(A_I\Big)\Big] \end{split} \end{align}
\noindent for $K = 1 ... \floor{\frac{|M|+1}{2}}$, where $I \subset \{1,...,|M|\}, |I| = k$ denotes all subsets $I$ of indices $1, ..., |M|$ which contain exactly $k$ elements, and $A_I := \bigcap_{i \in I} A_i$. A four-event example shown in Fig. \ref{fig:Boole_tot} to illustrate this: in the top subfigure, an improved upper bound on $\mathbb{P}(A_1 \cup A_2 \cup A_3 \cup A_4)$ is sought by removing redundant intersections (right) as time, data availability, and problem size allow, maintaining an upper bound on the original constraint by performing pairwise intersection estimations (here, only one iteration of the intersection estimation algorithm is performed). In the bottom subfigure, the event intersection probabilities are iteratively removed in the order of \eqref{inclusion2}, no longer maintaining an upper bound on the union of events. \textcolor{black}{In this example, the calculation in \eqref{newbound} only performs one iteration at $k = 1$; the combinations of pairwise intersections is calculated for the sum where $|I| = 2k = 2$, and the combinations of three-way intersections is calculated for the sum where $|I| = 2k+1 = 3$. These two terms are then added to the marginal probabilities $\mathbb{P}(A_m)$.}
This provides a benefit over the convenient but extremely conservative Boole's inequality as well as a more reliable and robust alternative to scenario-based approaches, which may require more time than available in between control actions in large networks. In the worst case (no computation time is allowed to estimate intersections), the algorithm is equivalent to using Boole's inequality to create tractable single chance constraints.
\iffalse Thus, for iteration $k$ of the algorithm for $k = 1, ..., \floor{\frac{|M|+1}{2}}$, where $\floor{\cdot}$ is the floor operator, the estimation of joint probabilities will only be accepted into the update process for the upper bound once the current step $k$ in summation in \eqref{newbound} is complete. However, calculation of the bound can be terminated before $k = K$. \fi
\begin{remark}[Observation 1] \label{proofbound} We have that \[ \mathbb{P}\Big(\bigcup_{i=1}^n A_i\Big) \leq {\cal B}_K \leq \sum_{m \in M} \mathbb{P}(A_m) \]
for all
$K \in \{1, \ldots, \floor{\frac{|M|+1}{2}}\}$. \end{remark}
\noindent \emph{Proof.}
The proof follows by the inclusion-exclusion principle, the monotonicity of ${\cal B}_K$ in $K$, and the fact that for $K = \floor{\frac{|M|+1}{2}}$, $\mathbb{P}\Big(\bigcup_{i=1}^n A_i\Big) = {\cal B}_K$.
\QEDA
\iffalse \noindent \emph{Proof.} First expand the LHS of the inequality in \eqref{newbound} using the inclusion-exclusion principle, noticing that $\mathbb{P}(A_i \cap A_j) = 0,~ \forall j \notin M$:
\begin{align} \label{eq:prop} \begin{split} \mathbb{P}\Big(\bigcup_{i=1}^n A_i\Big) &= \sum_{m\in M} \mathbb{P}(A_m) - \sum_{m \neq j, j \in M} \mathbb{P}(A_m \cap A_j) + \cdots \\
&\cdots + (-1)^{|M|-1} \mathbb{P}\Big( \bigcap_{m \in M} A_m \Big)\\ &\leq \sum_{m \in M} \mathbb{P}(A_m) \\ & +~ \sum_{k=1}^K \Big[\mathbb{P}\Big(\bigcap_{m=M_k}^{M_{2k+1}}A_m\Big) - \mathbb{P}\Big(\bigcap_{m=M_k}^{M_{2k}}A_m\Big)\Big] \end{split} \end{align}
\noindent Subtracting the sum of marginal probabilities from both sides and rewriting the LHS, we obtain
\begin{align*} \begin{split} \sum_{q=1}^Q \Big[\mathbb{P}\Big(&\bigcap_{i=M_q}^{M_{2q+1}}A_i\Big) - \mathbb{P}\Big(\bigcap_{i=M_q}^{M_{2q}}A_i\Big)\Big] \leq \\ &\sum_{k=1}^K \Big[\mathbb{P}\Big(\bigcap_{m=M_k}^{M_{2k+1}}A_m\Big) - \mathbb{P}\Big(\bigcap_{m=M_k}^{M_{2k}}A_m\Big)\Big] \end{split} \end{align*}
\noindent where $Q = \floor{\frac{|M|+1}{2}}$. Clearly, if $K = Q$, the LHS of the inequality is equal to the RHS and the inequality holds. When $1 \leq K < Q$, an upper bound is still maintained due to the fact that $\mathbb{P}\Big(\bigcap_{m=M_q}^{M_{2q+1}}A_m\Big) \leq \mathbb{P}\Big(\bigcap_{m=M_q}^{M_{2q}}A_m\Big), \forall q = [1, ..., Q]$ and $\mathbb{P}\Big(\bigcap_{m=M_k}^{M_{2k+1}}A_m\Big) \leq \mathbb{P}\Big(\bigcap_{m=M_k}^{M_{2k}}A_m\Big), \forall k = [1, ..., K]$. Since we have now shown the bound is valid, we know from \eqref{eq:prop} that if $\mathbb{P}\Big(\bigcap_{m=M_k}^{M_{2k}}A_m\Big)$ is nonzero, the bound provided by the improved inequality is tighter than that provided by the union bound. If all of the events are disjoint, the right hand side of the inequality in \eqref{eq:prop} is equal to $\sum_{m \in M} \mathbb{P}(A_m)$, which is equivalent to the union bound. \QEDA
\fi
\iffalse Each intersection is calculated using a Monte-Carlo sampling method. For each sample $s$ drawn from the uncertainty distribution, we compute each intersection as follows:
\begin{align} \label{MC} \mathbb{P}(A_1 \cap ... \cap A_n) \approx \frac{\sum_{s=1}^{N_s} \textbf{1}_{{\bf g}({\bf x}, s) \geq 0}({\bf x}(s))}{N_s} \end{align}
\noindent for the total number of samples $N_s$. \fi
Note that Observation 1 allows us to terminate the estimation process of the joint probabilities before $K = \floor{\frac{|M|+1}{2}}$. \textcolor{black}{This process is particularly useful when the number of non-zero terms in the joint chance constraint is large, as we can ensure that an upper bound is preserved which is still tighter than that provided by Boole's inequality.}
\begin{figure}
\caption{A four-event example where the numbers overlaid on the events and intersections represent the number of times that intersection is accounted for. The union bound (left) tends to overestimate the intersection of events, resulting in overly conservative control actions. Top: An improved upper bound on $\mathbb{P}(A_1 \cup A_2 \cup A_3 \cup A_4)$ is sought by removing redundant intersections (right) as time, data availability, and problem size allow, maintaining an upper bound on the original constraint by performing pairwise intersection estimations (here, only one iteration of the intersection estimation algorithm is performed). Bottom: If the intersections are simply accounted for in the order that \eqref{inclusion2} provides, an upper bound may not be preserved.}
\label{fig:Boole_tot}
\end{figure}
\iffalse \begin{subequations} \label{PJCC} \begin{align}
\mathrm{(P1)}& \min_{\substack{{\bf v}_{{\bf a}}, {\utwi{\alpha}}}} \hspace{.2cm} {\mathbb{E}}(f({\bf v}_{{\bf a}}, {\utwi{\alpha}}, {\bf p}_{\ell}, {\bf q}_{\ell})) \label{eq:Pmg_cost}\\ & \mathrm{subject\,to} \nonumber \\ & {\bf v}_{{\bf a}} = {\bf R} (({\bf I} - {\textrm{diag}}\{{\utwi{\alpha}}\}) {\bf p}_{\mathrm{av}} - {\bf p}_{\ell}) \\ & \hspace{1cm}- {\bf B} {\bf q}_{\ell} + {\bf a}\label{mg-balance-t} \\ & \mathbb{P}\{v_{a,1} \leq V_{\mathrm{max}}, ... , v_{a,n} \leq V_{\mathrm{max}} \} \geq 1- \epsilon \label{mg-prob-jcc}\\ & 0 \leq \alpha_i \leq 1 \hspace{3.05cm} \label{mg-alpha} \end{align} \end{subequations} \fi
In this paper, we estimate the remaining joint probabilities \rev{(i.e., the second term in \eqref{newbound}) using a sampling approach, and represent these probabilities with their relative frequencies. For that purpose, a deterministic optimization problem (e.g., (P1) in Section \ref{sec:optimization}) is solved in a similar way as described in Section \ref{sec:class} using historical inputs (solar, load), and the relative frequency of the event intersections are computed from that data. For example, if an event $A_1 \cap A_2$ occurred 3,000 times out of 10,000, we would assign $\mathbb{P}(A_1 \cap A_2) = \frac{3,000}{10,000} = 0.3$. This process is discussed in more detail in Section \ref{sec:estimate}.}
\section{Distribution Network and System Models} \label{sec:systemmodel}
Consider a distribution feeder comprising $N$ $PQ$ nodes and a single slack node. Let $V_n \in \mathbb{R}$ denote the line-to-ground voltage magnitude
at node $n$, and define the $N$-dimensional vector ${\bf v} := [V_1, \ldots, V_N]^\textsf{T} \in \mathbb{R}^{N}$.
Constants $P_{\ell,n}$ and $Q_{\ell,n}$ denote the real and reactive demands at node $n$, and we can define the vectors ${\bf p}_\ell := [P_{\ell,1}, \ldots, P_{\ell,N}]^\textsf{T}$ and ${\bf q}_\ell := [Q_{\ell,1}, \ldots, Q_{\ell,N}]^\textsf{T}$; if no load is present at node $n$, then $P_{\ell,n} = Q_{\ell,n} = 0$.
Here, we use a linearization of the AC power-flow equations \cite{sairaj2015linear,linModels} which linearly relates the voltage magnitudes ${\bf v}$ to the injected real and reactive powers ${\bf p} \in \mathbb{R}^N$ and ${\bf q} \in \mathbb{R}^N$ in the form \begin{align} {\bf v} & \approx {\bf R} {\bf p} + {\bf B} {\bf q} + {\bf a}, \label{eq:approximate} \end{align}
\noindent where ${\bf R}$, ${\bf B}$, and ${\bf a}$ are parameters that are dependent on the system model \cite{sairaj2015linear,linModels} \textcolor{black}{and are usually dependent on system line parameters, topology, and substation voltage, but can also be computed from data-driven techniques such as regression-based methods \cite{BakerNetwork17}}. While the proposed methodology does not require problem convexity, we leverage a linearization in order to provide a clear exposition of the joint chance constraint reformulation in Section \ref{sec:optimization}.
\subsection*{Photovoltaic (PV) Systems} Random quantity $P_{av,n}$ denotes the maximum renewable-based generation at node $n$ -- hereafter referred to as the available solar power. Particularly, $P_{av,n}$ coincide with the maximum power point at the AC side of the inverter. When RESs operate at unity power factor and inject the available solar power $P_{av,n}$, issues related to power quality and reliability in distribution systems may be encountered. For example, voltages exceeding prescribed limits at a particular node may be experienced when RES generation exceeds the load of that consumer~\cite{PVhandbook}. Efforts to ensure reliable operation of existing distribution systems with increased behind-the-meter renewable generation are focus on the possibility of inverters providing reactive power compensation and/or curtailing real power. To account for the ability of the RES inverters to adjust the output of real power, let $\alpha_n \in [0,1]$ denote the fraction of available solar power curtailed by RES-inverter $n$. If no PV system/inverter is at a particular node $i$, $P_{av,i} = \alpha_i = 0$. For convenience, define the vectors ${\utwi{\alpha}} :=[\alpha_1,\ldots, \alpha_N ]^\textsf{T}$ and ${\bf p}_{av} := [P_{av,1}, \ldots, P_{av,N}]^\textsf{T}$.
The available active power generation from solar is modeled as ${\bf p}_{av} = \overline{{\bf p}}_{av} + {\utwi{\delta}}_{av}$, where $\overline{{\bf p}}_{av} \in \mathbb{R}^N$ is a vector of the forecasted values and ${\utwi{\delta}}_{av} \in {\cal R}_{av} \subseteq \mathbb{R}^{N}$ is a random vector whose distribution captures spatial dependencies among forecasting errors. We assume that the distribution system operator has a certain amount of information about the probability distributions of the forecasting errors ${\utwi{\delta}}_{av}$. This information can come in the form of either knowledge of the probability density functions, or a model of ${\utwi{\delta}}_{av}$ from which one can draw samples. In this paper, we make the assumption that these errors are normally distributed \textcolor{black}{with zero mean; i.e., ${\utwi{\delta}}_{av}$ \texttt{{\raise.17ex\hbox{$\scriptstyle\mathtt{\sim}$}}} $\mathcal{N}(0, \sigma)$, and thus the remaining single chance constraints can be exactly reformulated as analytical expressions and included in the optimization problem directly \cite{Roald13}.} However, distributionally robust formulations of single chance constraints \cite{Tyler15, Nemirovski} can easily be incorporated into the framework here, \textcolor{black}{and thus the Gaussian assumption for the random quantities is not necessary for our framework}.
\section{Joint Chance Constrained Formulation} \label{sec:optimization}
\subsection{Optimization problem reformulation} The joint chance constraint optimization for voltage regulation in distribution systems is shown below:
\begin{subequations} \label{PJCC} \begin{align}
\mathrm{(P1)}& \min_{\substack{{\bf v}, {\tiny {\utwi{\alpha}}} }} \hspace{.2cm} {\mathbb{E}}(f({\bf v}, {\utwi{\alpha}}, {\bf p}_{\ell}, {\bf q}_{\ell})) \label{eq:Pmg_cost}\\ & \mathrm{subject\,to} \nonumber \\ & {\bf v} = {\bf R} (({\bf I} - {\textrm{diag}}\{{\utwi{\alpha}}\}) {\bf p}_{\mathrm{av}} - {\bf p}_{\ell}) \nonumber \\ & \hspace{1cm}- {\bf B} {\bf q}_{\ell} + {\bf a}\label{mg-balance-t} \\ & \mathbb{P}\{V_{1} \leq V_{\mathrm{max}}, ... , V_{N} \leq V_{\mathrm{max}} \} \geq 1- \epsilon \label{mg-prob-jcc}\\ & 0 \leq \alpha_i \leq 1, \, i = 1, \ldots, N; \hspace{3.05cm} \label{mg-alpha} \end{align} \end{subequations}
\noindent Constraint~\eqref{mg-balance-t} represents a surrogate for the power balance equation; constraint \eqref{mg-prob-jcc} is the joint chance constraint that requires voltage magnitudes to be within $V_{max}$ with at least $1-\epsilon$ probability; and constraint \eqref{mg-alpha} limits the curtailment percentage from $0-100\%$. The cost function $f({\bf v}, {\utwi{\alpha}}, {\bf p}_{\ell}, {\bf q}_{\ell})$ is convex and can consider a sum of penalties on curtailment, penalties on power drawn from the substation, penalties on voltage violations, etc.
By reformulating the joint constraint (\ref{mg-prob-jcc}) as a series of single chance constraints, we can write
\begin{align} \mathbb{P}(V_{i} \leq V_{\mathrm{max}} ) \geq 1- \epsilon_i \label{singlecc} \end{align}
\noindent for all $i \in 1, \ldots, N$, and each $\epsilon_i$ is chosen such that $\sum_{i=1}^N \epsilon_i = \epsilon$. In the case of Boole's inequality, we choose $\epsilon_i = \frac{\epsilon}{N}$; in the case of what we call the \emph{improved} Boole's inequality, using the method proposed in this paper, we choose $\epsilon_i = \frac{\epsilon}{N} + \frac{P_c}{N}$, where $P_c$ is our estimation of the non-zero probabilities in \eqref{inclusion}. If $P_c > 0$ (all events are not mutually exclusive), it is clear that the improved Boole's inequality provides a less conservative upper bound on the chance constraints.
\subsection{Estimating Event Intersections}\label{sec:estimate} We use a conservative relative frequency sampling approach to estimate the event intersections that have not been classified as zero. If event $A_i$ represents an overvoltage at node $i$, and $I$ is a subset of nodes, we can estimate the probability of the intersection of overvoltages at the nodes included in $I$ as
\begin{align} \label{MC} \mathbb{P}\left(\bigcap_{i \in I} A_i\right) \approx \frac{\sum_{s=1}^{N_s} \textbf{1} \left \{{\bf v}_I({\utwi{\delta}}_s) > V_{max}\right\} }{N_s}~~, \end{align}
\noindent for $N_s$ random draws of the uncertainty distribution, where draw $s$ is denoted ${\utwi{\delta}}_s$. Vector ${\bf v}_I$ contains the voltage magnitudes at each of the nodes in $I$, and $\textbf{1} \left \{{\bf v}_I({\utwi{\delta}}_s) > V_{max}\right\}$ is one if \emph{all} of the elements in ${\bf v}_I({\utwi{\delta}}_s)$ are greater than $V_{max}$ and zero otherwise. To represent the most conservative case, for each sample ${\utwi{\delta}}_s$, the voltage vector ${\bf v}({\utwi{\delta}}_s) = {\bf R} ({\bf I}(\overline{{\bf p}}_{\mathrm{av}} + {\utwi{\delta}}_s - {\bf p}_{\ell}))- {\bf B} {\bf q}_{\ell} + {\bf a}$; i.e., the curtailment variables ${\utwi{\alpha}}$ are chosen to be zero to represent no curtailment and thus the most conservative case for the control policy. \rev{Note that, using the objective function defined in Section \ref{sec:simulations} below, this is equivalent to solving a deterministic version of problem (P1) without voltage constraints.} The impact of different sample sizes $N_s$ and computational burden of the estimation process is discussed in further detail in the next section.
\subsection{Analytical Reformulation of Single Chance Constraints}
The single chance constraints can be reformulated as exact, tractable constraints \cite{BoVa04}, assuming $\epsilon \leq 0.5$. Assuming the joint distribution of the random variables is a multivariate Gaussian with mean ${\utwi{\mu}}$ and covariance matrix ${\utwi{\Sigma}}$, define $\mu_i$ as the $i$-th value in ${\utwi{\mu}}$ and $\sigma_i$ as the $(i,i)$-th entry in ${\utwi{\Sigma}}$. Then, define the following function at each node $i$:
\begin{align*} h(p_{av,i}) = &\sum_j(R_{ij}[(1-\alpha_j)p_{av,j} - p_{\ell, j}]) \\ &- \sum_j(B_{ij}q_{l,j}) + a_i - V_{max} \end{align*}
\noindent where $R_{ij}$ is the $(i,j)$-th entry of ${\bf R}$, $B_{ij}$ is the $(i,j)$-th entry of ${\bf B}$, and $a_i$ is the $i$th element of ${\bf a}$. Then $h(p_{av,i})$ is also normally distributed with the following mean $\mu'_i$ and variance $\sigma'_i$:
\begin{align*} \mu'_i &= \sum_j(R_{ij}[(1-\alpha_j)\mu_j - p_{\ell, j}]) - \sum_j(B_{ij}q_{l,j}) + a_i - V_{max}\\ \sigma'_i &= \sum_jR_{ij}(1-\alpha_j)\sigma_j \end{align*}
\noindent Thus, the constraints \eqref{singlecc} can be reformulated using the Gaussian cumulative distribution function (CDF) $\Phi$:
\begin{align*} \mathbb{P}\{h(p_{av,i}) \leq 0\} = \Phi\Big(\frac{0-\mu'_i}{\sigma'_i}\Big) \geq 1-\epsilon_i \end{align*}
\noindent With the final analytical constraint written using the quantile function (the inverse of the Gaussian CDF):
\begin{align} R_i[(1-\alpha_i)\mu_i - p_{\ell,i}] - B_iq_{\ell,i} + a_i - V_{max} \nonumber \\ \leq -R_i\alpha_i\sigma_i\Phi^{-1}(1-\epsilon_i) \label{final_CC} \end{align}
\noindent Which can be explicitly included into problem (P1) for each $i$ in place of the joint constraint \eqref{mg-prob-jcc}.
\begin{remark}[Remark 3]
In these results, the individual solar forecasting errors are modeled as Gaussian. Because of this, the single chance constraints can be exactly analytically reformulated. Without loss of generality, other distributionally robust methods for single chance constraints can also be used here \cite{Baker16NAPS, Nemirovski}, but as the contribution of this paper is in the decomposition of the joint chance constraint, not in addressing the tractability of single chance constraints, we have kept the marginal distributions Gaussian for simplicity of exposition. The method proposed in this paper is not distribution-specific. \end{remark}
\begin{remark}[Remark 4] Note that the original use of Boole's inequality ensures the satisfaction of the original constraint by choosing $\epsilon_i$ such that $\sum_{i=1}^n \epsilon_i = \epsilon$; however, without optimizing this parameter, suboptimal performance of this reformulation is possible \cite{JCC_Opt}. We leave the optimal choice of $\epsilon_i$ as a direction for future work. \end{remark}
\section{Numerical Results} \label{sec:simulations}
The IEEE-37 node test feeder \cite{IEEE37} was used for the following simulations. Five-minute load and solar irradiance data from weekdays in August 2012 was obtained from \cite{Bank13} for the simulations, and in order to emulate a situation with high-PV penetration and risks of overvoltage, 8 $200$-kW rated PV systems were placed at nodes 29-36. The considered cost function seeks to minimize renewable curtailment; specifically,
\begin{align} f({\bf v}, {\utwi{\alpha}}, {\bf p}_{\ell}, {\bf q}_{\ell}) = \sum_{i \in {\cal N}} d_i \alpha_i^2 , \label{eq:cost_sim} \end{align}
\noindent where the cost of curtailing power at each node is set to be $d_i = \$0.10$. The number of samples used to calculate each intersection was $N_s = 10,000$. The considered joint chance constraint considers maintaining voltages at nodes $29 - 36$. Each $\mu_i$, $i=1...N$ was chosen to be the power generated from the forecasted PV at that node, based on the shape of the aggregate solar irradiance from \cite{Bank13} and shifted using samples from a uniform distribution from +/- 1 kW across each node. The covariance matrix $\Sigma$ was formed by setting each entry $(i,j)$ to $\Sigma_{ij} = {\bf E}[(P_{av,i} - \mu_i)(P_{av,j} - \mu_j)^T]$. Three cases were considered in the following numerical results. First, a deterministic case was considered, which uses a certainty equivalence formulation and uses the mean of each of the uncertain parameters in place of each random variable in the optimization problem. Second, Boole's inequality was used to separate the joint chance constraint into a series of conservative single constraints, each with $\sum_{i=1}^N \epsilon_i = \epsilon$ \textcolor{black}{and $\epsilon_i = \frac{\epsilon}{n}$} . Third, an Improved Boole's inequality is considered, where the proposed methodology is implemented to approximate each of the intersections in \eqref{inclusion2}, and $\epsilon_i = \frac{\epsilon}{N} + \frac{P_c}{N}$.
\subsection{Training, Testing, and Choosing the Number of Samples} Each of the classifiers (one per constraint; 8 classifiers total) were trained using 1152 samples (4 training days), and tested using 864 samples (3 testing days), using $c_i = 1$ when $\ell_i = +1$ and $c_i = 10$ when $\ell_i = -1$. The overall classification error was $0.19\%$ for false classification of binding events and $4.73\%$ for false classification of non-binding events. A larger classification error for the $\ell_i = +1$ is to be expected from our formulation in (P0); it is more detrimental to the performance of the algorithm if we exclude constraints by mistake rather than include non-binding constraints unnecessarily. \textcolor{black}{If desired, although it has the potential of increasing the conservativeness of the solution, the classifier bias $b$ can also be increased to ensure that the classification error of active constraints as inactive occurs even less frequently. Despite the misclassification of $0.19\%$ of data points in this direction, as the results in the next section demonstrate, the original joint chance constraint is still satisfied within the prescribed probability $1-\epsilon$.}
In Fig. \ref{fig:comp_time}, the number of active constraints (out of 8 total) is indicated with red dots for each time instance \textcolor{black}{and referenced against the right-hand y-axis}. The total computation time (s) required for calculating the corresponding intersections is shown in black (for $N_s = 1,000$ samples) and purple (for $N_s = 10,000$ samples). Computational time can be reduced by potentially sacrificing accuracy of estimating event intersections; in Fig. \ref{fig:sensitivity}, the value of increasing the number of samples wanes around $N_s = 1,000$. Thus, in the following simulations, the conservative choice of $N_s = 10,000$ was made to estimate each event intersection; however, in a general setting this number is dependent on the underlying distribution.
\begin{figure}
\caption{The number of active constraints (out of 8 total) is indicated with red dots for each time instance. The total computation time (s) required for calculating the corresponding intersections is shown in black (for $N_s = 1,000$ samples) and purple (for $N_s = 10,000$ samples).}
\label{fig:comp_time}
\end{figure}
\begin{figure}
\caption{A sensitivity analysis showing the estimated probabilities for each of the event intersections in a single test day as a function of the number of samples $N_s$ used to estimate that intersection. The change between estimated intersections with $N_s = 5,000$ and $N_s = 10,000$ is sufficiently small; any additional increase in the number of samples will not provide much additional information.}
\label{fig:sensitivity}
\end{figure}
\subsection{Voltage Regulation Results}
In the following results, the maximum joint constraint violation probability was set to $\epsilon = 0.02$ for the Boole's and Improved Boole's cases, and all of the terms in \eqref{inclusion2} were estimated. In Fig. \ref{fig:control_policy}, the maximum voltage magnitudes from the resulting control policies are shown for each of the three cases. The deterministic case \textcolor{black}{does not} take forecast uncertainty into account, and as a result, the voltages are pushed to the maximum voltage of $1.05$ pu. The Boole's case curtails enough solar generation to ensure that overvoltages will not occur with a high probability; the Improved Boole's case reduces this probability and results in less curtailment.
\begin{figure}
\caption{The predicted maximum voltages using the control policies determined for voltage regulation in each of the three cases. In the non-deterministic cases, the control policies are more conservative in order to account for the uncertainty in the solar irradiance.}
\label{fig:control_policy}
\end{figure}
A Monte Carlo validation procedure was implemented to demonstrate the behavior of the control policies for $N_m = 10,000$ random draws of the uncertainty distributions at each timestep. In Fig. \ref{fig:epsilon}, these resulting probabilities are shown. The deterministic case, which only considers the mean of the random variables, violates the desired chance constraint bound of $0.02$ when compared with the chance constrained methods, because that method offers no guarantee that the voltages will be within limits. Boole's method is generally more conservative than the Improved Boole's method and results in lower violation probabilities, with both methods resulting in satisfaction of the original joint chance constraint.
\begin{figure}
\caption{The violation probability of the joint chance constraint calculated through a Monte Carlo validation procedure with $N_m = 10,000$. During times of high solar irradiance, the deterministic control policy does not guarantee satisfaction of the joint chance constraint, while the stochastic solutions ensure that the constraint is satisfied with probability $\geq 98\%$.}
\label{fig:epsilon}
\end{figure}
Table \ref{tab:obj} shows the total objective function value and voltage violation probability across the three-day test period for the deterministic, Boole's, and Improved Boole's cases. As expected, the deterministic case results in the lowest cost (but highest probability of voltage violations); the Boole's case is overly conservative, resulting in a higher level of curtailment and thus cost, but lowest probability of voltage violations. The Improved Boole's case strikes a balance between the two, resulting in a slightly higher violation probability than the original Boole's case but with a lowered objective value.
\begin{table}[t!] \centering \caption{Total objective function value and maximum observed voltage violation probability.} \label{tab:obj}
\begin{tabular}{|l|l|l|l|} \hline
& Deterministic & Boole's & Improved Boole's \\ \hline \begin{tabular}[c]{@{}l@{}}Total Objective \\ Function Value\end{tabular} & 148.40 & 162.63 & 156.93 \\ \hline \begin{tabular}[c]{@{}l@{}} Maximum Violation\\ Probability\end{tabular} & 2.66\% & 1.85\% & 1.98\% \\ \hline \end{tabular} \end{table}
\subsection{\textcolor{black}{Computational Time and Multi-Phase Systems}} \label{sec:threephase}
\textcolor{black}{To demonstrate the computational burden of the proposed framework as the joint chance constraint increases in number of terms, we have performed additional simulations on an unbalanced, three-phase version of the aforementioned 37-node feeder and using the multi-phase linearization procedure from \cite{linModels}. Three-phase, wye-connected PV systems were connected to the same nodes as in the previous test case, thus increasing the number of single constraints within the joint constraint by threefold (24 total terms) by constraining the voltage magnitude at each of the three phases. As Fig. \ref{fig:comp_time_22} and equation \eqref{numterms} illustrates, estimating the number of intersections for $|M| = 24$ would require more computational time than typically given between OPF control actions. Thus, we can show the benefit of the iterative method proposed in Section \ref{sec:iterative}; for very large joint chance constraints, the algorithm can be terminated prematurely while still providing a tighter or equal upper bound to that of Boole's inequality. This allows the system operator to tune the time required for the constraint estimation procedure to a desired level.}
\begin{figure}\label{fig:comp_time_22}
\end{figure}
\begin{figure}
\caption{\textcolor{black}{The PV curtailment on phase 1 of node 28. As expected, the Improved Boole's method produces a slightly less conservative solution (i.e., less curtailment) than the solution obtained with Boole's inequality.}}
\label{fig:node28_ph1}
\end{figure}
\textcolor{black}{For example, here we can terminate the procedure at 12 active constraints, which can be computed in approximately one minute. In this case, over one test day and using $\epsilon = 0.05$ and Monte Carlo simulations with $1,000$ samples per time step, the maximum observed violation probability with Boole's inequality was $3.04\%$ and with the Improved Boole's method was $3.55\%$. While they both provided conservative solutions that were under the prescribed limit of $5\%$, the Improved Boole's method was less conservative. Fig. \ref{fig:node28_ph1} demonstrates this relationship at the inverter at node 28 by showing lower curtailment levels for the solution that used the Improved Boole's method. This tradeoff can also be compared with the computational burden tradeoff - it is likely that if more than 12 active constraints were considered in the constraint estimation procedure, the method would have produced a greater violation probability.}
\section{Conclusion} \label{sec:conclusion}
In this paper, we demonstrated how identifying zero-probability events with support vector classifiers can increase the computational efficiency of computing joint chance constraints via sampling methods. In addition, we provided an iterative approach appropriate for fast timescale optimization, ensuring that if the entire constraint cannot be computed, that the resulting approximation of the constraint always provides an upper bound of the original constraint at every iteration which is tighter than that provided by Boole's inequality. Simulation results were shown which addressed voltage regulation in distribution networks with high PV penetration, and the proposed method was demonstrated to result in a lower cost than Boole's inequality and lower constraint violation probability than a deterministic certainty equivalence formulation.
Future work will address the question of how to optimally allocate the estimated intersection probabilities $P_c$ to the individual chance constraints (rather than allocating them equally across single constraints as in this paper), determining how many samples are adequate for estimating event intersections, and identifying which statistical learning techniques are best suited for identifying active constraints in power systems optimization problems. In addition, an important question for future work is how to incorporate sampling error and uncertainty when identifying and estimating the underlying probability distributions.
\begin{IEEEbiography}[{\includegraphics[width=1in]{Kyri2-eps-converted-to}}]{Kyri Baker} (S'08, M'15) received her B.S., M.S, and Ph.D. in Electrical and Computer Engineering at Carnegie Mellon University in 2009, 2010, and 2014, respectively. Since Fall 2017, she has been an Assistant Professor at the University of Colorado, Boulder, in the Department of Civil, Environmental, and Architectural Engineering, with a courtesy appointment in the Department of Electrical, Computer, and Energy Engineering. Previously, she was a Research Engineer at the National Renewable Energy Laboratory in Golden, CO. Her research interests include power system optimization and planning, building-to-grid integration, smart grid technologies, and renewable energy. \end{IEEEbiography}
\begin{IEEEbiography}[{\includegraphics[width=1in]{bioPhoto1.png}}]{Andrey Bernstein} (M'15) received the B.Sc. and M.Sc. degrees in Electrical Engineering from the Technion - Israel Institute of Technology in 2002 and 2007 respectively, both summa cum laude. He received the Ph.D. degree in Electrical Engineering from the Technion in 2013. Between 2010 and 2011, he was a visiting researcher at Columbia University. During 2011-2012, he was a visiting Assistant Professor at the Stony Brook University. From 2013 to 2016, he was a postdoctoral researcher at the Laboratory for Communications and Applications of Ecole Polytechnique Federale de Lausanne (EPFL), Switzerland. Since October 2016 he has been a Senior Scientist at NREL. In February 2019, he has become a Group Manager of the Energy Systems Control and Optimization group at NREL. His research interests are in the decision and control problems in complex environments and related optimization and machine learning methods, with particular application to intelligent power and energy systems. Current research is focused on real-time optimization of power distribution systems with high penetration of and machine learning methods for grid data analytics. \end{IEEEbiography}
\end{document}
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\begin{document}
\title{Zero-one law for directional transience of one-dimensional random walks in dynamic random environments} \author{Tal Orenshtein and Renato S.\ dos Santos}
\maketitle
\begin{abstract} We prove the trichotomy between transience to the right, transience to the left and recurrence of one-dimensional nearest-neighbour random walks in dynamic random environments under fairly general assumptions, namely: stationarity under space-time translations, ergodicity under spatial translations, and a mild ellipticity condition. In particular, the result applies to general uniformly elliptic models and also to a large class of non-uniformly elliptic cases that are i.i.d.\ in space and Markovian in time. An immediate consequence is the recurrence of models that are symmetric with respect to reflection through the origin.
\noindent {\it MSC} 2010. Primary 60F20, 60K37; Secondary 82B41, 82C44.\\ {\it Key words and phrases.} Random walk, dynamic random environment, zero-one law, directional transience, recurrence. \end{abstract}
\section{Introduction} \label{s:intro}
Random walks in random environments have been the subject of intensive mathematical study for several decades. They consist of random walks whose transition kernels are themselves random, modelling the movement of a tracer particle in a disordered medium. When the random transition kernels, called \emph{random environment},
do not evolve with time, the model is called \emph{static}; otherwise it is called \emph{dynamic}. Hereafter we will use the abbreviations RWRE for the static and RWDRE for the dynamic model. The reader is referred to the monographs \cite{Sz06}, \cite{Ze04} for RWRE and \cite{Avthesis}, \cite{dSathesis} for RWDRE. Note that, by considering time as an additional dimension, one-dimensional RWDRE can be seen as directed RWRE in two dimensions.
While one-dimensional RWRE is by now very well understood, the state of the art in RWDRE is in comparison much more modest. Most of the general results available require strong assumptions such as uniform and fast enough mixing for the random environment, cf.\ e.g.\ \cite{AvdHoRe11}, \cite{dHodSaSi13}, \cite{ReVo11}. An exception are quenched LDPs, cf.\ \cite{AvdHoRe10}, \cite{CDRRS12}, \cite{RaSeYi12}. Otherwise, outside of the uniformly-mixing class, the literature is largely restricted to particular choices of random environments, cf.\ e.g.\ \cite{HHSST14}, \cite{dHodSa13}, \cite{HS14}, \cite{MoVa15}, \cite{dSa14}.
In the present paper we consider the very basic question of whether the trichotomy between transience to the right, transience to the left and recurrence, typical for time-homogeneous Markov chains on $\mathbb{Z}$, also holds for one-dimensional, nearest-neighbour RWDRE. We conclude that this is indeed the case under fairly general assumptions on the random environment. An immediate but interesting consequence is that reflection-symmetric models satisfying our assumptions must be recurrent. We will consider the continuous time setting, but the same arguments work, mutatis mutandis, in the discrete time case. For comparison with other non-Markovian models where this problem was addressed, the reader is referred to \cite{zerner2001zero} and \cite{zerner2007zero} for the case of 2-dimensional RWRE, \cite{sabot2011reversed} for the case of random walks in Dirichlet environments, and to \cite{amir2013zero} for the case of 1-dimensional excited random walk.
The paper is organised as follows. In Section~\ref{s:results} we define our model and state our assumptions and results. Section~\ref{s:examplesanddiscussion} discusses our setup, providing examples and connections to the literature; the proof that a class of examples described therein fits our setting is postponed to Section~\ref{s:proofofthmexamples}. In Section~\ref{s:construction}, we present a graphical construction that will be useful in Section~\ref{s:proofmainthm}, where our main theorem is proved.
\section{Model, assumptions and results} \label{s:results}
Let $\omega = (\omega^-_t, \omega^{+}_t)_{t \ge 0}$ be a stochastic process taking values on $([0,\infty)^{\mathbb{Z}})^2$, called the \emph{dynamic random environment}. We will assume that $\omega$ belongs to the space $\Omega$ of right-continuous paths from $[0,\infty)$ to $([0,\infty)^{\mathbb{Z}})^2$, where the latter is endowed with the product topology. Given a realisation of $\omega$, the RWDRE $X = (X_t)_{t \ge 0}$ is defined as the time-inhomogeneous Markov jump process on $\mathbb{Z}$ whose laws $(P^\omega_x)_{x \in \mathbb{Z}}$ satisfy \begin{align}\label{e:defX} P^\omega_x \left(X_0=x \right) & = 1, \\
P^\omega_x\left(X_{t+s} = y \pm 1 \,\middle|\, X_t = y \right) & = s \, \omega^{\pm}_t(y) + o(s) \; \text{ as } s \downarrow 0. \end{align} The existence of such processes is standard (see e.g.\ \cite{EK86}, Chapter 4, Section 7). We give here a particular construction in Section~\ref{s:construction} below. Without extra assumptions the process $X$ may explode (i.e., make infinitely many jumps) in finite time; we thus enlarge the state-space $\mathbb{Z}$ with a cemetery point $\Delta$ in the standard way in order to define $X$ after the explosion time $\tau_\Delta$, i.e., $X_t := \Delta$ for all $t \ge \tau_\Delta$ (cf.\ \eqref{e:deftau_Delta}).
The law $P^\omega_x$ is called the \emph{quenched} law. We denote by $\mathbb{P}_x$ the joint law of $X$ and $\omega$ (with $\mathbb{P}_x(X_0=x)=1$). The corresponding expectations will be denoted respectively by $E^\omega_x$ and $\mathbb{E}_x$. In the literature, the \emph{annealed} (or \emph{averaged}) law is often defined as the marginal law of $X$ under $\mathbb{P}_x$, but for convenience we will call $\mathbb{P}_x$ itself the annealed law.
Define the space-time translation operators $\theta^z_{s}:\Omega \to \Omega$, $z \in \mathbb{Z}, s \in \mathbb{R}_+$, acting on $\omega$ as $(\theta^z_s \omega)_{x,t} := \omega(z+x,s+t)$. We will write $\theta_s:=\theta^0_s$, $\theta^z:=\theta^z_0$. Denoting by \[\mathcal{F}_t:=\sigma(\omega, (X_u)_{0\le u \le t})\] the natural filtration of $X$, the Markov property for $X$ then reads \begin{equation}\label{e:MPX}
E^{\omega}_x \left[ f \left((X_{t+s})_{s \ge 0} \right) \;\middle|\; \mathcal{F}_t \right] = E_{X_t}^{\theta_t \omega} \left[ f(X)\right] \;\;\; P^\omega_x \text{-a.s.} \end{equation} for any bounded measurable $f$ and any $t \ge 0$. Moreover, since the space-time process $(X_t,t)$ is Feller, by the strong Markov property the time $t$ in \eqref{e:MPX} may be replaced by any a.s.\ finite $\mathcal{F}_t$-stopping time. Also, we may and will assume that $X$ is right-continuous.
We will work under the following assumptions:
\textbf{(SE):} \label{assumptionSE} The process $\omega$ is stationary with respect to space-time translations, i.e., for each $z \in \mathbb{Z}, t \ge0$, $\theta^z_{t} \omega$ has the same distribution as $\omega$. Furthermore, we assume that $\omega$ is ergodic with respect to the spatial translations $\theta^z$.
\textbf{(EL):} $\mathbb{P}_0$-a.s., \begin{equation}\label{e:EL} \liminf_{t \to \tau_\Delta}{X_t} \;\textnormal{ and }\; \limsup_{t \to \tau_\Delta}{X_t} \in \{-\infty, + \infty\}. \end{equation}
Assumption (SE) is standard; in fact, $\omega$ is usually taken ergodic also in time. Assumption (EL) is an ellipticity condition; note that it holds e.g.\ when $\omega$ is \emph{uniformly elliptic}, i.e., if there exists $\kappa \in (0,1)$ such that $\kappa \le \omega^{\pm}_t(x) \le \kappa^{-1}$. Indeed, in this case the property of being visited infinitely often is either a.s.\ satisfied by all or by none of the points of $\mathbb{Z}$. Note that (EL) implies \begin{equation}\label{e:notstuck} \inf \{ t > 0 \colon\, X_t \in [-n,n]^c \} < \tau_\Delta \;\;\;\; \mathbb{P}_0 \text{-a.s.\ for all } n \in \mathbb{N}. \end{equation} While (EL) may be hard to check in non-uniformly elliptic examples, \eqref{e:notstuck} holds as soon as $\omega$ is stationary and ergodic with respect to time translations and satisfies a non-degeneracy condition; see Proposition~\ref{prop:notstuck} below.
We can now state our main result.
\begin{theorem} \label{thm:01law}
If assumptions (SE) and (EL) are satisfied, then $\tau_\Delta = \infty$ $\mathbb{P}_0$-a.s.\ and one of the following three cases holds:
\begin{enumerate} \item $\displaystyle \mathbb{P}_0\left(\lim_{t \to \infty} X_t = \infty\right)=1$ \label{transright}; \item $\displaystyle \mathbb{P}_0\left(\lim_{t \to \infty} X_t = -\infty \right)=1$ \label{transleft}; \item $\displaystyle \mathbb{P}_0 \left( \limsup_{t \to \infty} X_t = \infty = - \liminf_{t \to \infty} X_t \right) =1$. \label{rec} \end{enumerate} \end{theorem}
A zero-one law for directional transience is said to hold if the probabilities in items \ref{transright} and \ref{transleft} of Theorem~\ref{thm:01law} are either $0$ or $1$. This statement is equivalent to Theorem~\ref{thm:01law} as the ellipticity assumption (EL) ensures that the event appearing in item \ref{rec} is almost surely equal to the complement of the union of the events in \ref{transright}--\ref{transleft}.
As an immediate consequence of Theorem~\ref{thm:01law}, we obtain recurrence for any model satisfying (SE)--(EL) that is symmetric with respect to reflection through the origin:
\begin{corollary}\label{cor:recurrencesymmetric} Assume that (SE) and (EL) hold and that $(-X_t)_{t \ge 0}$ has under $\mathbb{P}_0$ the same distribution as $(X_t)_{t\ge 0}$. Then item~\ref{rec} of Theorem~\ref{thm:01law} holds. \end{corollary} For an interesting example to which Corollary~\ref{cor:recurrencesymmetric} applies, consider the following. Let $\omega^{+}_t(x) := \alpha \eta_t(x) + \beta (1-\eta_t(x))$, $\omega^-_t := \alpha (1-\eta_t(x)) + \beta \eta_t(x)$ where $0 < \beta < \alpha <\infty$ and $(\eta_t)_{t \ge 0}$ is a simple symmetric exclusion process in $\mathbb{Z}$ started from a product Bernoulli measure $\nu_{\rho}$ with density $\rho \in (0,1)$. Very little is known for this model in the literature (see e.g.\ \cite{AvdHoRe10}, \cite{HS14} and \cite{dSa14}), in particular in the case $\rho=1/2$ where the expected asymptotic speed of $X$ is zero. However, since it satisfies (SE)--(EL) and is reflection-symmetric for $\rho=1/2$, Corollary~\ref{cor:recurrencesymmetric} implies that it is recurrent in this case.
\section{Examples and discussion} \label{s:examplesanddiscussion}
In the literature, $\omega$ is often given as a functional of an interacting particle system, i.e., of a Markov process $(\eta_t)_{t \ge 0}$ on $E^{\mathbb{Z}}$ where $E$ is a metric space, often assumed compact. For example, in the setting of \cite{ReVo11}, the transition rates are given by \begin{equation}\label{e:translratesFlo} \omega^{\pm}_t(x) = \alpha^{\pm}(\theta^x \eta_t) \end{equation} where the functions $\alpha^{\pm}:E^{\mathbb{Z}} \to [0,\infty)$ satisfy some regularity properties. The setting of \cite{AvdHoRe11} is a particular case where $E = \{0,1\}$.
Since directional transience follows from a law of large numbers with non-zero speed, and recurrence from a functional central limit theorem if the speed is zero, Theorem~\ref{thm:01law} brings no new information in the cases where these results are known. However, our result applies to many situations where such theorems have not yet been proved, which is the case for several uniformly elliptic but non-uniformly mixing models, e.g., when $\eta_t$ is a simple exclusion process or a system of independent random walks outside the perturbative regimes considered in \cite{HHSST14}, \cite{HS14}. By ``uniformly mixing'' here we mean that the conditional law of $\eta_t(0)$ given $\eta_0$ converges to a fixed law uniformly over all possible realizations of $\eta_0$; cf.\ e.g.\ the cone-mixing condition of \cite{AvdHoRe11} (Definition~1.1 therein), or the coupling conditions of \cite{ReVo11} (Assumptions~1a--1b therein).
Let us now describe a large class of examples satisfying our assumptions that includes many non-uniformly elliptic cases with slow and non-uniform mixing: \begin{example}\label{ex:mainexample} \textnormal{ Let $\eta_t(x)$, $x \in \mathbb{Z}, t \ge 0$, be i.i.d.\ in $x$ with each $\eta_t(x)$ distributed as an irreducible, positive-recurrent Markov process on a countable state-space $E$, started from its unique invariant probability measure $\pi$. Let $\omega$ be defined by $\omega^{\pm}_t(x) = \alpha^{\pm}(\eta_t(x))$ with $\alpha^{\pm}:E \to (0,\infty)$, i.e., the jump rates are always positive (in which case the model is called \emph{elliptic}) and depend only on the state of $\eta_t$ at $x$.} \end{example}
The models defined in Example~\ref{ex:mainexample} clearly satisfy (SE). Moreover:
\begin{theorem}\label{thm:examplesatisfiesEL} The models defined in Example~\ref{ex:mainexample} satisfy (EL). \end{theorem} \noindent The proof of this theorem is given in Section~\ref{s:proofofthmexamples} below. Note that, as already mentioned, it covers many models that are slowly and non-uniformly mixing and thus do not fall into the categories generally studied in the literature of RWDRE so far.
It is interesting to ask in which directions Theorem~\ref{thm:01law} could be generalised, and how far our hypotheses could be weakened. The analogous result in discrete time can be proved with a similar approach via graphical representation (cf.\ Section~\ref{s:construction} below). However, new ideas are needed for random walks in other graphs, e.g.\ $\mathbb{Z}^d$ with non-nearest neighbour jumps and/or $d > 1$, and regular trees.
\section{Graphical construction} \label{s:construction}
We construct next a particular version of the process $X$ with convenient properties. Denote by $\mathcal{M}_p$ the space of point measures on $\mathbb{Z} \times [0,\infty)$, and let $N^+_\omega$, $N^-_\omega \in \mathcal{M}_p$ be two independent Poisson point processes with intensity measures $\mu^{\pm}_\omega$ identified by \begin{equation}\label{e:intPPPs} \mu^{\pm}_\omega(A \times B) := \sum_{x \in A} \int_B \omega^{\pm}_s(x) ds, \;\;\; A \subset \mathbb{Z}, \, B \subset [0,\infty) \text{ measurable.} \end{equation} We denote by $\widehat{P}_\omega$ the joint law of $N^{+}_\omega$, $N^{-}_\omega$, and by $\widehat{\mathbb{P}}$ the joint law of the latter and $\omega$. Define the space-time translations $\theta^z_t$ of $N^{\pm}_\omega$ and functions thereof by \begin{equation}\label{e:deftransN} \begin{array}{lcll} \theta^z_t N^{\pm}_\omega (C) & := & N^{\pm}_\omega(C+(z,t)), \;\;\; & C \subset \mathbb{Z} \times [0,\infty) \text{ measurable,}\\ \theta^z_t f(N^{\pm}_\omega) & := & f(\theta^z_t N^{\pm}_\omega), & f:\mathcal{M}_p \to \mathbb{R}, \end{array} \end{equation} where \[C+(z,t):= \bigcup_{(y,s) \in C}\{(y+z,s+t)\}.\] We note that, under $\widehat{\mathbb{P}}$, $N^{\pm}_\omega$ inherits from $\omega$ the stationarity with respect to space-time translations and the ergodicity with respect to spatial translations.
On each point of $N^{+}_\omega$, resp.\ $N^{-}_\omega$, we draw a unit-length arrow pointing to the right, resp.\ to the left. Then we set, for $x \in \mathbb{Z}$, $X^{x}$ to be the path started at $x$ that proceeds by moving upwards in time and forcibly across any arrows in a right-continuous way; the paths are defined only up to the explosion time. See Figure~\ref{fig:graphrep}.
\begin{figure}
\caption{\small Graphical construction. The arrows represent events of $N^{\pm}_\omega$. The thick lines mark the paths $X^x$ and $X^y$, which in this example coalesce at site $y-1$.}
\label{fig:graphrep}
\end{figure}
Using the right-continuity of $\omega$, it is straightforward to check that this construction gives the correct law, i.e., $X^{x}$ has under $\widehat{P}_\omega$ the same law as $X$ under $P^\omega_x$. In particular, this provides a coupling for copies of the random walk starting from all initial positions, which will facilitate the proof of Theorem~\ref{thm:01law}.
With this construction, the explosion times $\tau^x_\Delta$, $x \in \mathbb{Z}$ can be defined as \begin{equation}\label{e:deftau_Delta} \tau^x_\Delta := \sup \{t > 0 \colon\, X^x \text{ crosses finitely many arrows up to time } t\}, \end{equation} and we identify $X := X^0$, $\tau_\Delta := \tau^0_\Delta$.
We end this section with the following monotonicity property, which is a consequence of the graphical construction and will be useful in the proof of Lemma~\ref{l:finmanyrexc} below. \begin{lemma}\label{l:monot} For any $y, z \in \mathbb{Z}$ such that $y \le z$, $\widehat{\mathbb{P}}$-a.s., \begin{equation}\label{e:monot} X^y_t \le X^z_t \; \forall \; t \in [0, \tau^y_\Delta \wedge \tau^z_\Delta). \end{equation} \end{lemma} \begin{proof} Since the paths start ordered, move by nearest-neighbour jumps, and a.s.\ cannot jump simultaneously before they meet, either $X^y_t < X^z_t$ for all relevant $t$ or there exists a first $s\ge 0$ such that $X^y_s = X^z_s$, in which case by construction $X^y_u = X^z_u$ for all $u \ge s$. \end{proof}
\section{Proof of Theorem~\ref{thm:01law}} \label{s:proofmainthm}
For $A \subset \mathbb{Z}$, denote by \begin{equation}\label{e:defhittimes} H_A := \inf \{t > 0 \colon\, X_t \in A \} \end{equation} the hitting time of $A$. Let $A^c := \mathbb{Z} \setminus A$ and note that, if $A$ is finite, then $H_{A^c}$ is a.s.\ finite by \eqref{e:notstuck}. For a random time $S \in [0,\infty]$, define \begin{equation}\label{e:defthetaS} \Theta_S H_A := \left\{ \begin{array}{ll} \inf\{ t > 0 \colon\, X_{S+t} \in A \} & \text{ if } S < \infty,\\ \infty & \text{ otherwise.} \end{array}\right. \end{equation} Note that $\Theta_S H_A = \theta^{X_S}_S H_{A-X_S}$ when $S<\infty$, where $A-x:=\{z -x \colon\, z \in A\}$. Define now the $k$-th return time $T^{(k)}_A$ to $A$ as follows. Set $T^{(0)}_A:= 0$ and, recursively for $k \ge 0$, \begin{equation}\label{e:defhittimeslater} T^{(k+1)}_A := T^{(k)}_A + \Theta_{T^{(k)}_A} \left(H_{A^c} + \Theta_{H_{A^c}} H_A \right).
\end{equation} Note that $T^{(1)}_A = H_A$ if $X_0 \notin A$. When $A = \{z\}$, we write $H_z$ and $T^{(k)}_z$.
Our proof of Theorem~\ref{thm:01law} is based on three lemmas which we state next; their proofs are given respectively in Sections~\ref{ss:thetranslationinvariancelemma} and~\ref{ss:the1dimlemma} below. The first of them implies that, if the random walk visits $-1$ (resp.\ $1$) a.s., then all its excursions from $0$ to the right (resp.\ to the left) will be a.s.\ finite. \begin{lemma}\label{l:rightexcarefinite} Assume that (SE) holds, and let $x \in \{-1, 1\}$. If $\mathbb{P}_0(H_{x} < \infty) = 1$, then \begin{equation}\label{e:rightexcarefinite} T^{(k)}_0 < \infty \;\; \Rightarrow \;\; T^{(k)}_0 + \Theta_{T^{(k)}_0} H_{x} < \infty \;\;\;\; \mathbb{P}_0 \text{-a.s.\ for all } k \ge 1. \end{equation} \end{lemma}
The second lemma excludes the possibility of explosions in our setting. \begin{lemma}\label{l:noexplosion} Assume that (SE) and \eqref{e:notstuck} hold. Then \begin{equation}\label{e:noexplosion} \mathbb{P}_0 \left( \tau_\Delta = \infty \right)=1. \end{equation} \end{lemma}
The third lemma shows that, if there is a positive probability for the random walk to never touch $-1$ (resp.\ $1$), then its range is bounded from below (resp.\ above). \begin{lemma}\label{l:finmanyrexc} Assume that (SE) and \eqref{e:notstuck} hold. Then $\mathbb{P}_0(H_{-1} = \infty) >0$ implies $\mathbb{P}_0 \left( \exists \, z < 0 \colon\, H_z = \infty \right)=1$, and $\mathbb{P}_0(H_{1} = \infty) >0$ implies $\mathbb{P}_0\left(\exists \, z > 0 \colon\, H_z = \infty \right)=1$. \end{lemma}
Note that Lemmas~\ref{l:rightexcarefinite}--\ref{l:finmanyrexc} do not use assumption (EL) directly but only its consequence \eqref{e:notstuck}. Moreover, Lemma~\ref{l:rightexcarefinite} only uses stationarity in time and the strong Markov property; the graphical construction of Section~\ref{s:construction} is only used in the proof of Lemmas~\ref{l:noexplosion} and~\ref{l:finmanyrexc}.
We can now finish the proof of Theorem~\ref{thm:01law}. \begin{proof}[Proof of Theorem~\ref{thm:01law}] Assumption (EL) and Lemmas~\ref{l:noexplosion}--\ref{l:finmanyrexc} together imply that \begin{equation}\label{e:prfmainthm1} \mathbb{P}_0(H_{-1}=\infty)>0 \;\; \Rightarrow \;\; \mathbb{P}_0 \left( \lim_{t\to \infty} X_t = \infty \right) = 1 \end{equation} since, if the left-hand side of \eqref{e:prfmainthm1} holds, then $\liminf_{t \to \infty} X_t > - \infty$ a.s.\ and hence it must be equal to $\infty$ by (EL). Analogously, \begin{equation}\label{e:prfmainthm2} \mathbb{P}_0(H_{1}=\infty)>0 \;\; \Rightarrow \;\; \mathbb{P}_0 \left( \lim_{t\to \infty} X_t = -\infty \right) = 1. \end{equation} To conclude, we claim that \begin{equation}\label{prmainthm3} \mathbb{P}_0\left( H_1 \vee H_{-1} < \infty \right) = 1 \;\; \Rightarrow \;\; -\infty = \liminf_{t \to \infty} X_t < \limsup_{t \to \infty} X_t = \infty. \end{equation} Indeed, note that, by Lemmas~\ref{l:rightexcarefinite}--\ref{l:noexplosion}, $\mathbb{P}_0\left(H_{-1} < \infty \right) = 1$ implies that $\liminf_{t \to \infty} X_t \le -1$ a.s., which together with (EL) gives $\liminf_{t \to \infty} X_t = -\infty$. The last equality is obtained analogously. \end{proof}
\subsection{Proof of Lemma~\ref{l:rightexcarefinite}} \label{ss:thetranslationinvariancelemma}
\begin{proof} To start, we claim that, $\mathbb{P}_0$-a.s., \begin{equation}\label{e:samepropforallt} P^{\theta_t \omega}_0 (H_{x} = \infty) = 0 \; \text{ simultaneously for all } t \ge 0. \end{equation} Indeed, for each fixed $t \ge 0$, $P^{\theta_t \omega}_0 (H_{x} = \infty) = 0$ a.s.\ since, by stationary, $\mathbb{P}_0(\cdot) = \mathbb{E}_0[P^\omega_0(\cdot)] = \mathbb{E}_0[P^{\theta_t \omega}_0(\cdot)]$. Hence \eqref{e:samepropforallt} holds with $t$ restricted to the set of rational numbers, and to extend it to all $t\ge0$ we only need to show that the function $t \mapsto P^{\theta_t \omega}_0(H_x = \infty)$ is right-continuous. To this end, note that, since $\omega$ is right-continuous, \begin{align}\label{e:prlemmaMKV1} P^{\theta_t \omega}_0 \left(\exists \, u \in [0,s] \colon X_u \neq 0 \right) & = 1-e^{-\int_t^{t+s} \left\{ \omega^+_u(0)+\omega^{-}_u(0) \right\} du} \nonumber\\ & \le \int_t^{t+s} \left\{ \omega^+_u(0)+\omega^{-}_u(0) \right\} du \nonumber\\ & \le 2 s \left\{ \omega^+_t(0) + \omega^-_t(0) \right\} \end{align} for all $s>0$ small enough (depending on $\omega$ and $t$). Denoting by $O_\omega(s)$ a function whose modulus is bounded by $s \, C_\omega$ where $C_\omega \in (0,\infty)$ may depend on $\omega$, we obtain \begin{align}\label{e:prlemmaMKV2} P^{\theta_t \omega}_0 \left(H_x = \infty \right) & = P^{\theta_t \omega}_0 \left(\Theta_s H_x = \infty, X_u = 0 \,\forall\, u \in [0,s] \right) + O_{\omega}(s) \nonumber\\ & = E^{\theta_t \omega}_0 \left[ \mathbbm{1}_{\{X_u = 0 \,\forall\, u \in [0,s] \}}P^{\theta_{t+s} \omega}_0 \left( H_x = \infty \right)\right] + O_{\omega}(s) \nonumber\\ & = P^{\theta_{t+s}\omega}_0 \left( H_x = \infty \right) + O_{\omega}(s), \end{align} where for the second line we use the Markov property and for the last one we again use \eqref{e:prlemmaMKV1}. From this follows the desired right-continuity and consequently also \eqref{e:samepropforallt}. By the strong Markov property (cf.\ the paragraph below \eqref{e:MPX}) and $\Theta_{T^{(k)}_0}H_x = \theta_{T^{(k)}_0}H_x$, \begin{align}\label{e:prlemmaMKV3} \mathbb{P}_0 \left( T^{(k)}_0 < \infty, \Theta_{T^{(k)}_0}H_x = \infty\right) & = \mathbb{E}_0 \left[\int_0^\infty P^{\theta_t \omega}_0 \left( H_x = \infty \right) P^\omega_0 \left( T^{(k)}_0 \in d t \right) \right] = 0 \end{align} by \eqref{e:samepropforallt}. \end{proof}
\subsection{Proof of Lemmas~\ref{l:noexplosion}--\ref{l:finmanyrexc}} \label{ss:the1dimlemma}
We start by showing that explosions are not possible under (SE) and \eqref{e:notstuck}. \begin{proof}[Proof of Lemma~\ref{l:noexplosion}] \text{}
\noindent It is enough to show that, for any $a,b \in [0,\infty)$ with $b-a>0$ small enough, \begin{equation}\label{e:prnoexpl1} \mathbb{P}_0 \left( \tau_\Delta \in (a,b] \right) = 0. \end{equation} To that end, define the events \begin{equation}\label{e:prnoexpl2} A_x^{a,b} := \left\{ \text{there are no arrows in } \{x\} \times [a,b]\right\} = \left\{N^{\pm}_\omega\left(\{x\} \times [a,b]\right) = 0 \right\} \end{equation} and let $\varepsilon > 0$ be so small that, if $b-a \le \varepsilon$, then \begin{equation}\label{e:prnoexpl3} \widehat{\mathbb{P}} \left( A_0^{0,b-a}\right) \ge \widehat{\mathbb{P}} \left( A_0^{0,\varepsilon} \right) = \widehat{\mathbb{E}} \left[\exp\left\{- \int_0^\varepsilon \left[ \omega^+_s(0)+\omega^-_s(0)\right]ds \right\} \right]> 0, \end{equation} which exists by the right-continuity of $\omega$ and the dominated convergence theorem. Noting that $A_x^{a,b} = \theta^x A_0^{a,b}$, we obtain from Birkhoff's ergodic theorem that, $\widehat{\mathbb{P}}$-a.s., \begin{equation}\label{e:prnoexpl4} \lim_{N \to \infty} \frac{1}{N} \sum_{x =0}^{N-1} \mathbbm{1}_{A_{x}^{a,b}} = \widehat{\mathbb{P}} \left(A_0^{a,b} \right) = \widehat{\mathbb{P}} \left(A_0^{0,b-a} \right) >0 \end{equation} by stationarity under time translations, and analogously for $x \le 0$. In particular, \begin{equation}\label{e:prnoexpl5} \widehat{\mathbb{P}} \left( \forall\, z \in \mathbb{Z}, \;\exists\, x < z < y \text{ such that } A_x^{a,b} \text{ and } A_y^{a,b} \text{ occur}\right) =1. \end{equation} Note now that, by \eqref{e:notstuck}, if $\tau_\Delta \in (a,b]$ then for all $n \in \mathbb{N}$ the random walk exits the interval $[-n,n]$ before time $b$. Therefore \begin{equation}\label{e:prnoexpl6} \mathbb{P}_0 \left( \tau_\Delta \in (a,b] \right) \le \widehat{\mathbb{P}} \left( \forall\, n \in \mathbb{N}, a+\Theta_a H_{[-n,n]^c} < b\right) \end{equation} where $H_{[-n,n]^c}$ is the hitting time of $\mathbb{Z} \setminus [-n,n]$ by $X^0$. On the other hand, by the graphical construction, if both $A_x^{a,b}$ and $A_y^{a,b}$ occur for some $x < X^0_a < y$, then $a+\Theta_a H_{[-n,n]^c} \ge b$ with
$n = |x| \vee |y|$. Hence \eqref{e:prnoexpl6} is at most \begin{equation}\label{e: prnoexpl7} \widehat{\mathbb{P}} \left( \forall\, x,y \in \mathbb{Z} \text{ such that } x< X^0_a < y, \text{ either } A_x^{a,b} \text{ or } A_y^{a,b} \text{ does not occur} \right)=0 \end{equation} by \eqref{e:prnoexpl5}, proving \eqref{e:prnoexpl1}. \end{proof}
We now prove Lemma~\ref{l:finmanyrexc}. \begin{proof}[Proof of Lemma~\ref{l:finmanyrexc}] We assume that $\mathbb{P}_0 (H_{-1} = \infty) >0$; the case $\mathbb{P}_0 (H_1 = \infty) >0$ is proved analogously. For $x \in \mathbb{Z}$, let \begin{equation}\label{e:prlemma1D1} A_x := \left\{ X^x_t \ge x \, \forall \, t \ge 0 \right\}. \end{equation} Since $A_x = \theta^x A_0$, Birkhoff's ergodic theorem implies that, $\widehat{\mathbb{P}}$-a.s., \begin{equation}\label{e:prlemma1D2} \lim_{N \to \infty} \frac{1}{N} \sum_{z =0}^{-N+1} \mathbbm{1}_{A_z} = \widehat{\mathbb{P}} \left(A_0 \right) = \mathbb{P}_0 \left( H_{-1} = \infty \right) >0, \end{equation} and in particular $\widehat{\mathbb{P}}(A_z \text{ occurs for some } z \le 0)=1$. Noting that, by Lemmas~\ref{l:monot} and~\ref{l:noexplosion}, if $z \le 0$ and $A_z$ occurs then $X^0_t > z-1$ for all $t \ge 0$, we obtain \begin{align}\label{e:prlemma1D3} \mathbb{P}_0 \left( \exists\, z < 0 \colon\, H_z = \infty \right) & = \widehat{\mathbb{P}} \left( \exists \, z < 0 \colon\, X^0_t \neq z \,\forall\, t \ge 0 \right) \nonumber\\ & \ge \widehat{\mathbb{P}} \left( \exists \, z \le 0 \colon\, A_z \text{ occurs } \right) = 1, \end{align} finishing the proof. \end{proof}
\section{Proof of Theorem~\ref{thm:examplesatisfiesEL}} \label{s:proofofthmexamples}
We first show that \eqref{e:notstuck} holds for a very large class of models, including our examples. \begin{proposition}\label{prop:notstuck} Assume that $\omega$ is stationary and ergodic with respect to the time translation $\theta_1$ and that, for a choice of $*,\star \in \{-,+\}$ and every $n \in \mathbb{N}$, \begin{align} & \mathbb{P}_0 \left( \int_0^{n} \omega^*_s(0) \, ds < \infty \right) = 1 \;\; \textnormal{ and }\label{e:nondeg1}\\ & \mathbb{P}_0 \left(\omega^{\star}_0(x) > 0 \;\forall\, x \in [-n,n] \right) >0. \label{e:nondeg2} \end{align} Then \eqref{e:notstuck} holds. \end{proposition} \begin{proof} Fix $n \in \mathbb{N}$. By right-continuity and invariance under time translations, there exist $\delta, \varepsilon \in (0,1)$ such that the events \begin{equation}\label{e:prnotstuck1} \mathcal{A}_k := \left\{\omega^\star_{k+s}(x) \ge \delta \;\forall\, s \in [0,\varepsilon], x \in [-n,n] \right\}, \;\; k \in \mathbb{N} \end{equation} have equal and positive probability. Then the event \begin{equation}\label{e:prnotstuck2} \mathcal{A} := \limsup_{k \to \infty} \mathcal{A}_k = \bigcap_{k \ge 1} \bigcup_{l \ge k} \mathcal{A}_k = \left\{ \mathcal{A}_k \text{ occurs for infinitely many } k \in \mathbb{N}\right\} \end{equation} also has positive probability by the Poincar\'e recurrence theorem (cf.\ Theorem 1.4 in \cite{Wa82}); moreover, since $\mathcal{A}$ is invariant under $\theta_1$, it occurs almost surely by ergodicity. Let \begin{equation}\label{e:prnotstuck3} \begin{array}{lcl} V_1 &:= & \inf\left\{l \in \mathbb{N} \colon\, \mathcal{A}_l \text{ occurs}\right\},\\ V_{k+1} & := & \inf\left\{l > V_k \colon\, \mathcal{A}_l \text{ occurs}\right\}, \; k \ge 1. \end{array} \end{equation} Denote by $H_{[-n,n]^c}$ the hitting time of $\mathbb{Z} \setminus [-n,n]$. By \eqref{e:nondeg1}, $N^{*}_\omega([-n,n] \times [0,T])<\infty$ a.s.\ for all $T \ge 0$, and thus $H_{[-n,n]^c} = \infty$ implies $\tau_{\Delta} = \infty$ almost surely. Hence \begin{align}\label{e:prnotstuck4} \mathbb{P}_0 \left( H_{[-n,n]^c} = \infty \right) & \le \mathbb{P}_0 \left( X_{V_k} \in [-n,n], \Theta_{V_k} H_{[-n,n]^c} > \varepsilon \; \forall \, k \ge 1\right) \nonumber\\ & = \lim_{L \to \infty} \mathbb{P}_0 \left( X_{V_k} \in [-n,n], \Theta_{V_k} H_{[-n,n]^c} > \varepsilon \; \forall \, 1 \le k \le L \right). \end{align} Note now that, if $X_{V_k} \in [-n,n]$, then between times $V_k$ and $V_k + \varepsilon \wedge \Theta_{V_k} H_{[-n,n]^c}$ the RWDRE has a rate at least $\delta$ to jump in direction $\star$. Therefore, \begin{equation}\label{e:prnotstuck5} X_{V_k} \in [-n,n] \; \Rightarrow \; P_{X_{V_k}}^{\theta_{V_k}\omega} \left(H_{[-n,n]^c} \le \varepsilon \right) \ge \vartheta_n \end{equation} for some deterministic $\vartheta_n \in(0,1)$ independent of $k$. By the Markov property, \begin{align}\label{e:prnotstuck6} & \mathbb{P}_0 \left( X_{V_k} \in [-n,n], \Theta_{V_k} H_{[-n,n]^c} > \varepsilon \; \forall \, 1 \le k \le L+1 \right) \nonumber\\ = \; & \mathbb{E}_0 \left[\mathbbm{1}_{\{ X_{V_k} \in [-n,n], \Theta_{V_k} H_{[-n,n]^c} > \varepsilon \; \forall \, 1 \le k \le L \}} P^{\theta_{V_L} \omega}_{X_{V_L}} \left( H_{[-n,n]^c} > \varepsilon \right)\right] \nonumber\\
\le \; &(1-\vartheta_n) \mathbb{P}_0 \left( X_{V_k} \in [-n,n], \Theta_{V_k} H_{[-n,n]} > \varepsilon \; \forall \, 1 \le k \le L \right) \end{align} and we conclude using induction that \eqref{e:prnotstuck4} is equal to zero, proving \eqref{e:notstuck}. \end{proof}
As a consequence, no explosion can occur in our examples. \begin{corollary}\label{cor:noexpexamples} In all models defined in Example~\ref{ex:mainexample},
$\tau_\Delta = \infty$ almost surely. \end{corollary} \begin{proof} The models described are mixing in time, and thus satisfy the hypotheses of Proposition~\ref{prop:notstuck}. Since they also satisfy (SE), the corollary follows by Lemma~\ref{l:noexplosion}. \end{proof}
We can now finish the proof of Theorem~\ref{thm:examplesatisfiesEL}. \begin{proof}[Proof of Theorem~\ref{thm:examplesatisfiesEL}] We will prove that, $\mathbb{P}_0$-a.s., \begin{equation}\label{e:prexamples1} \forall \; x \in \mathbb{Z}, \;\; T^{(k)}_x < \infty \;\; \forall \; k \ge 1 \; \Rightarrow \; T^{(k)}_{x-1} < \infty \; \; \forall \; k \ge 1. \end{equation} The analogous result for $x+1$ in place of $x-1$ then follows by reflection. This implies (EL) since, by Proposition~\ref{prop:notstuck}, \eqref{e:notstuck} holds. We claim that it suffices to show that, a.s., \begin{equation}\label{e:prexamples2} \forall \; x \in \mathbb{Z}, \;\; T^{(k)}_x < \infty \;\; \forall \; k \ge 1 \; \Rightarrow \; T^{(1)}_{x-1} < \infty. \end{equation} Indeed, fix $j \in \mathbb{N}$. Suppose by induction that \eqref{e:prexamples2} holds with $(1)$ substituted by $(j)$. Then write, using the strong Markov property, \begin{align}\label{e:prexamples3} & \mathbb{P}_0 \left( T^{(k)}_x < \infty \; \forall \; k \ge 1, T^{(j+1)}_{x-1} = \infty \right) \nonumber\\ = \; & \mathbb{P}_0 \left( \Theta_{T^{(j)}_{x-1}} T^{(k)}_x < \infty \; \forall \; k \ge 1, T^{(j)}_{x-1} < \infty, \Theta_{T^{(j)}_{x-1}} T^{(1)}_{x-1} = \infty \right) \nonumber\\ = \; & \mathbb{E}_0 \left[ \int_0^{\infty} P^{\theta_t \omega}_{x-1} \left( T^{(k)}_x < \infty \; \forall \; k \ge 1, T^{(1)}_{x-1} = \infty \right) P^\omega_0 \left(T^{(j)}_{x-1} \in d t \right)\right]. \end{align} With an argument identical to the one used to prove \eqref{e:samepropforallt}, we can show that the integrand in \eqref{e:prexamples3} is a.s.\ identically equal to $0$, proving \eqref{e:prexamples1}.
Fix $\mathcal{O} \in E$ and choose a finite set $E^* \subset E$ such that $\mathcal{O} \in E^*$ and \begin{equation}\label{e:prexamples5}
\inf_{t \ge 0} \mathbb{P}_0 \left( \eta_t(0) \in E^* \,\middle|\, \eta_0(0) = \mathcal{O} \right) > \tfrac12. \end{equation} This is possible since $\eta_t(0)$ converges in distribution to $\pi$ (cf.\ Theorem 2.66 in \cite{Li10}). Considering the maximal jump rate in $E^*$, we further obtain $\varepsilon>0$ such that \begin{equation}\label{e:prexamples6}
\inf_{t \ge 0} \mathbb{P}_0 \left( \eta_s(0) \in E^* \,\forall \, s \in [t,t+\varepsilon] \,\middle|\, \eta_0(0) = \mathcal{O} \right) > \tfrac12. \end{equation}
Fix now a site $x \in \mathbb{Z}$ and define \begin{equation}\label{e:prexamples7} \begin{array}{lcl} U_1 & := & \inf \{t > 0 \colon\, \eta_t(x) = \mathcal{O}\},\\ V_1 & := & \inf \{t > U_1 \colon\, X_t = x \} = U_1 + \Theta_{U_1}T^{(1)}_x, \end{array} \end{equation} and, recursively for $k \ge 2$, \begin{equation}\label{e:prexamples8} \begin{array}{lcl} U_{k} & := & \left\{ \begin{array}{ll} \inf \{t > V_{k-1}+1 \colon\, \eta_t(x) = \mathcal{O}\} & \qquad \quad \; \text{ if } V_{k-1} < \infty, \\
\infty & \qquad \quad \; \text{ otherwise,} \end{array}\right. \\ V_{k} & := & \left\{ \begin{array}{ll} \inf \{t > U_k \colon\, X_t = x \} = U_k + \Theta_{U_k}T^{(1)}_x & \text{ if } U_{k} < \infty, \\
\infty & \text{ otherwise.} \end{array}\right. \end{array} \end{equation} These are all well-defined by Corollary~\ref{cor:noexpexamples}. Note that $T^{(k)}_x< \infty$ for all $k \ge 1$ if and only if $V_k < \infty$ for all $k \ge 1$. Therefore, it is enough to show that \begin{equation}\label{e:prexamples9} \mathbb{P}_0 \left( V_k < \infty , \Theta_{V_i} T^{(1)}_{x-1} \ge 1 \;\forall\, 1 \le i \le k \right) \le \rho^{k-1} \end{equation} for some $\rho \in (0,1)$. To this end, note first that \begin{equation}\label{e:prexamples10} \eta_{t+s}(x) \in E^* \;\forall\, s \in [0,\varepsilon] \; \Rightarrow \; P_x^{\theta_t \omega} \left( T^{(1)}_{x-1} \le \varepsilon \right) \ge \delta \end{equation} for some deterministic $\delta> 0$, since the left-hand side of \eqref{e:prexamples10} implies that $\omega^{\pm}_{t+s}(x)$ is bounded away from zero and infinity uniformly in $s \in [0,\varepsilon]$. Therefore, for any initial configuration $\bar{\eta} \in E^{\mathbb{Z}}$ and any $z \in \mathbb{Z}$, \begin{align}\label{e:prexamples11}
& \mathbb{P}_z \left( V_1 < \infty, \Theta_{V_1}T^{(1)}_{x-1} < 1 \,\middle|\, \eta_0 = \bar{\eta} \right) \nonumber\\
\ge \; & \mathbb{P}_z \left( V_1 < \infty, \eta_{V_1+s}(x) \in E^* \;\forall\, s \in [0,\varepsilon], \Theta_{V_1} T^{(1)}_{x-1} \le \varepsilon \,\middle|\, \eta_0 = \bar{\eta} \right) \nonumber\\
= \; & \mathbb{E}_z \left[ \mathbbm{1} \{ V_1 < \infty, \eta_{V_1+s}(x) \in E^* \;\forall\, s \in [0,\varepsilon] \} P_x^{\theta_{V_1} \omega} \left( T^{(1)}_{x-1} \le \varepsilon \right) \,\middle|\, \eta_0 = \bar{\eta} \right]\nonumber\\
\ge \; & \delta \, \mathbb{P}_z \left( V_1 < \infty, \eta_{V_1+s}(x) \in E^* \;\forall\, s \in [0,\varepsilon] \,\middle|\, \eta_0 = \bar{\eta}\right) \nonumber\\
= \; & \delta \, \mathbb{P}_z \left( U_1 \le V_1 < \infty, \eta_{U_1+(V_1 -U_1) + s}(x) \in E^* \;\forall\, s \in [0,\varepsilon] \,\middle|\, \eta_0 = \bar{\eta}\right). \end{align} Note now that $U_1$, $V_1$ are measurable in \[ \sigma \left( N^{\pm}_\omega(C) \colon\, C \subset \{x\} \times [0,U_1] \cup (\mathbb{Z}\setminus\{x\}) \times [0,\infty) \right), \] while $(\eta_{U_1+s}(x))_{s \ge 0}$ is independent of the latter sigma-algebra with distribution equal to that of $(\eta_s(x))_{s \ge 0}$ with $\eta_0(x) = \mathcal{O}$. Therefore, \eqref{e:prexamples11} equals \begin{align}\label{e:prexamples20}
& \delta \int \hspace{-7pt} \int_{0<u\le v < \infty} \hspace{-9pt} \mathbb{P}_z \left( \cap_{s \in [0,\varepsilon]}\{ \eta_{v-u + s}(x) \in E^*\} \,\middle|\, \eta_0(x) = \mathcal{O}\right) \mathbb{P}_z \left(V_1 \in dv, U_1 \in du \,\middle|\, \eta_0 = \bar{\eta} \right) \nonumber\\
& \ge \frac{\delta}{2} \mathbb{P}_z \left( V_1 < \infty \,\middle|\, \eta_0 = \bar{\eta} \right) \end{align} by \eqref{e:prexamples6}, implying that, for any $\bar{\eta} \in E^{\mathbb{Z}}$ and any $z \in \mathbb{Z}$, \begin{equation}\label{e:prexamples13}
\mathbb{P}_z \left( V_1 < \infty, \Theta_{V_1} T^{(1)}_{x-1} \ge 1 \,\middle|\, \eta_0 = \bar{\eta} \right) \le \rho := 1 - \frac{\delta}{2} < 1. \end{equation} To conclude, use the strong Markov property of $(X_t, \eta_t)$ at time $V_k+1$ to write \begin{align}\label{e:prexamples14} & \mathbb{P}_0 \left( V_{k+1} < \infty, \Theta_{V_i} T^{(1)}_{x-1} \ge 1 \;\forall\, 1 \le i \le k+1\right) \nonumber\\
= \; & \mathbb{E}_0 \left[ \mathbbm{1}_{\{V_{k} < \infty, \Theta_{V_i} T^{(1)}_{x-1} \ge 1 \;\forall\, 1 \le i \le k\}} \mathbb{P}_{X_{V_k+1}} \left( V_1 < \infty, \Theta_{V_1} T^{(1)}_{x-1} \ge 1 \,\middle|\, \eta_0 = \bar{\eta}\right)_{\bar{\eta} = \eta_{V_k+1}}\right] \nonumber\\ \le \; & \rho \, \mathbb{P}_0 \left( V_{k} < \infty, \Theta_{V_i} T^{(1)}_{x-1} \ge 1 \;\forall\, 1 \le i \le k\right) \end{align} by \eqref{e:prexamples13}, and so \eqref{e:prexamples9} follows by induction. \end{proof}
\noindent \textbf{Acknowledgments.} The authors would like to thank Luca Avena for the suggestion to add Corollary~2.2, and the anonymous referee for useful comments. The work of TO was supported by the Labex Milyon (ANR-10-LABX-0070) of Universit\'e de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). RSdS was supported by the German DFG project KO 2205/13-1 ``Random mass flow through random potential''.
\end{document}
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\begin{center} {\bf HYPERSURFACE COMPLEMENTS, ALEXANDER MODULES} \\ {\bf AND MONODROMY} \end{center}
\centerline {\bf by Alexandru Dimca and Andr\'as N\'emethi }
\vskip1truecm
\section{Introduction}
Let $X \subset \mathbb C^{n+1}$ (resp. $V \subset \mathbb P^{n+1}$) be an algebraic hypersurface and set $M_X=\mathbb C^{n+1} \setminus X$ (resp. $M_V=\mathbb P^{n+1} \setminus V$) where we suppose $n>0$. The study of the topology of $X$, $V$ and of their complements $M_X$, $M_V$ is a classical subject going back to Zariski. In a sequence of papers Libgober has introduced and studied the Alexander invariants associated to $X$, $V$, see for instance [Li0-3].
In the affine case, let $f=0$ be a reduced equation for $X$. One can use the results on the topology of polynomial functions, see for instance [B], [ACD], [NZ], [SiT], to study the topology of the complements $M_X$, as in the recent paper by Libgober and Tib\u ar [LiT].
By taking generic linear sections and using the (affine) Lefschetz theory,
see for instance Hamm [H] (and
[Li2], [LiT], [D1] for different applications), one can restrict this study to hypersurfaces $X$ having only isolated singularities including at infinity, see [Li2], or in the polynomial framework, to polynomials having only isolated singularities including at infinity with respect to a compactification as in [SiT]. Simple examples show that neither of these two restricted situations is a special case of the other, hence both points of view have their advantages. However, the polynomial point of view embraces larger classes of examples due to the fact that the best compactification of a polynomial function is not usually obtained by passing from the affine space $\mathbb C^{n+1}$ to the projective space $\mathbb P^{n+1}$. This is amply explained in [LiT].
In the present paper we consider an arbitrary polynomial map $f$ (whose generic fiber is connected) and we study the Alexander invariants of $M_X$ for any fiber $X$ of $f$.
The article has two major messages. First, the most important qualitative properties of the Alexander modules (cf. \ref{3.6}, \ref{n2}, \ref{n7}
and \ref{gen}) are completely independent of the behaviour of $f$ at infinity, or about the special fibers. (On the other hand, for particular families of polynomial maps with some additional information about the special fibers or about the behaviour at infinity, one can obtain nice vanishing or connectivity results; see e.g. our case of h-good polynomials below).
The second message is that all the Alexander invariants of all the fibers of the polynomial $f$
are closely related to the monodromy representation of $f$. In fact, all the torsion parts of the Alexander modules (associated with all the possible fibers) can be obtained by factorization of a unique universal Alexander module, which is constructed from the monodromy representation. This explains nicely and conceptually all the divisibility properties that have appeared recently in the literature connecting
the Alexander polynomials of $M_X$
and the characteristic polynomials of some special monodromy operators, see [Li2] and [LiT]. [Note that the monodromy considered by Libgober in [Li2], section 2, is associated to a Lefschetz pencil and hence quite different from our monodromy associated to an arbitrary polynomial.]
Nevertheless, in order to exemplify our general theory, and also to generalize some connectivity results already present in the literature, we introduce the family of h-good polynomials. The family includes e.g. all the ``good'' polynomials considered by Neumann and Rudolph [NR], and the polynomials with isolated singularities on the affine space and at infinity in the sense of Siersma-Tib\u ar [SiT]. This family of h-good polynomials fits perfectly to the study of Alexander invariants, and it is our major source of examples. For different vanishing and connectivity results, see \ref{p23}, \ref{con}, \ref{con2} and \ref{3.6}(v).
The content of our paper is the following. In section 2 we establish some properties of the corresponding fundamental groups which basically will guide all the covering properties considered later. Moreover, here we introduce and start to discuss the h-good polynomials. In section 3 we discuss some general facts on the homology groups $H_*(M_X, \mathbb Z)$ concentrating on non-vanishing results for $H_n(M_X,\mathbb Z)$ and on $\mathbb Z$-torsion problems. This latter aspect was somewhat neglected recently in spite of the pioneering work by Libgober [Li0] and a famous conjecture on hyperplane arrangement Milnor fibers (see \ref{2.12}).
In section 4 we collect some facts on (torsion) Alexander modules and prove one of the main results, Theorem \ref{3.6}. In order to emphasize the parallelism of h-good polynomials with the case of hypersurfaces with only isolated singularities including at infinity considered by Libgober, in some of our applications we recall Libgober's results [Li2] as well.
In the fifth section we explain the relationship between individual monodromy operators and Alexander modules. The two main examples, i.e. the monodromy at infinity and the monodromy around the fiber $X$ are discussed with special care. These two monodromy operators have been intensively studied recently using various techniques (mixed Hodge structures, D-modules), see the references given in section 5. Via our results, all this information on the monodromy operators yields valuable information on Alexander invariants of $M_X$. Remark \ref{4.6} relates the homology of the cyclic coverings $M_{X,d}$ to the $d$-suspension of the polynomial $f$ and in this way the results on the Thom-Sebastiani construction in [DN2] become applicable.
Section 6 introduces into the picture not only individual monodromy operators but also the whole monodromy representation of $f$. We define two new Alexander modules associated to $f$, namely the global Alexander module $M(f)$ which can be regarded as a commutative version of the monodromy representation,
and a local Alexander module $M(f,b)$ associated to any fiber $X=f^{-1}(b)$.
This module $M(f,b)$ gives a very good approximation of the classical Alexander module $H_*(M_X^c, \mathbb Z)$ of $X$ (see below for the necessary definitions).
As a convincing example of the power of this new approach, we compute at the end the various Alexander modules for a polynomial $\mathbb C^4 \to \mathbb C$ for which a partial information on the monodromy representation is known. This examples shows in particular that the isomorphism $M(f,b) =H_n(M_X^c, \mathbb Z)$ does not always hold.
We thank D. Arapura, A. Libgober, C. Sabbah
and A. Suciu for useful discussions.
\section{Topological preliminaries, connectivity properties}
\subsection{}\label{00} Let $f: \mathbb C^{n+1} \to \mathbb C$ be a polynomial function with $n\geq 1$. It is well known that there
is a (minimal) finite bifurcation set $B_f$ in $\mathbb C$ such that $f$ is a $C^{\infty}$-locally trivial fibration over $\mathbb C \setminus B_f$. If $b_0 \in \mathbb C$ is not in $B_f$, then $F=f^{-1}(b_0)$ is called the generic fiber of $f$; otherwise $F_b:=f^{-1}(b)$ is called a special fiber.
For any $b\in \mathbb C$ we fix a sufficiently small closed disc $D_b$ containing $b$, and a point $b'\in \partial D_b$. We set $T_b:=f^{-1}(D_b)$, \ $T^*_b:=T_b\setminus f^{-1}(b)$. Sometimes, it is convenient to identify $f^{-1}(b') $ with the generic fiber $F$. Then, we have the obvious inclusions $F\subset T^*_b\subset T_b$.
By a well-known deformation retract argument (see e.g. (2.3) in [DN1]), the pair $(\mathbb C^{n+1},F)$ has the homotopy type of the space $(Y,F)$ obtained by gluing all the pairs $(T_b,F)$ ($b\in B_f$) along $F$. We denote this fact by \begin{equation*} (\mathbb C^{n+1},F)\sim \bigvee_{F}(T_b,F)\ \ (b\in B_f). \tag{1} \end{equation*}
\subsection{Proposition}\label{t1} {\em Let $f:\mathbb C^{n+1}\to \mathbb C$ be a polynomial map. Then the generic fiber $F$ is connected if and only if
$\pi_1(T_b,F)$ is trivial for any $b\in B_f$.}
\begin{proof} If $\pi_1(T_b,F)=1$ for all $b$, then $\tilde{H}_0(F)=H_1(\mathbb C^{n+1},F)=\oplus_{b\in B_f}H_1(T_b,F)=0$
by (1). Now, assume that $F$ is connected and fix a $b\in B_f$. Then we have to show that $j:\pi_1(F)\to \pi_1(T_b)$ is onto. Since $T_b$ is smooth and $f^{-1}(b)\subset T_b$ has real codimension two, one obtains that $i:\pi_1(T^*_b)\to \pi_1(T_b)$ is onto. Since $f$ restricted to $T^*_b$ is a fiber bundle, the kernel of $f_*:\pi_1(T^*_b)\to \mathbb Z$ is $\pi_1(F)$. Assume that $f^{-1}(b)$ has $r$ irreducible components, and $f-b=\prod_{i=1}^rg_i^{m_i}$. Then one can construct easily elementary loops in $T^*_b$ around the component $\{g_i=0\}$ representing $x_i\in \pi_1( T^*_b)$ with properties $f_*(x_i)=m_i$ and $i(x_i)=1$. Set $m:=gcd_i\{m_i\}$. Then a combination of the $x_i$'s provides an $x\in \pi_1(T^*_b)$ with $f_*(x)=m$ and $i(x)=1$. The point is that $m=1$ (otherwise $f-b$ would be an $m$-power of a polynomial whose generic fiber is not connected). The existence of such an $x$ and the surjectivity of $i$ implies the surjectivity of $j$. \end{proof}
In the next paragraphs we fix a $b\in B_f$, and we write $X:=f^{-1}(b)$ and $M_X:=\mathbb C^{n+1}\setminus X$. For simplicity of the notations, we will assume that $b=0$.
\subsection{Corollary}\label{t2} {\em Assume that $F$ is connected. Then
(i) $ \pi_1(T^*_0)\to \pi_1(M_X)$ (induced by the inclusion) is onto.
(ii) $\pi_1(F)\stackrel{i_X}{\longrightarrow}
\pi_1(M_X)\stackrel{f_*}{\longrightarrow}\mathbb Z\to (1)$\ is an exact sequence (i.e. $im(i_X)=ker(f_*)$), where $i_X$ is induced by the inclusion $F\subset M_X$, and $f_*$ by $f:M_X\to \mathbb C^*$.} \begin{proof} Similarly as in (1), $M_X$ has the homotopy type of a space obtained by gluing $T^*_0$ and all the ``tubes'' $T_{\bar{b}}$ ($\bar{b}\in B_f\setminus \{0\}$) along $F$. Then (i) follows from van Kampen theorem and from the surjectivity of $\pi_1(F)\to \pi_1(T_{\bar{b}})$
for each $\bar{b}$ (cf. \ref{t1}). Part (ii) follows from (i) and the exact sequence $\pi_1(F) \to \pi_1(T^*_0)\to\mathbb Z\to (1)$. \end{proof}
Let $p:\mathbb F\to M_X$ be the $\mathbb Z$-cyclic covering associated to the kernel of the morphism $f_*: \pi _1(M_X) \to\mathbb Z$. The notation $\mathbb F$ is chosen because
(i) $\mathbb F$ is the homotopy fiber of $f:M_X \to \mathbb C^*$, regarded as a homotopy fibration; and
(ii) in many cases the topology of $\mathbb F$ is a good approximation for the topology of $F$ (see e.g. the connectivity results below).
Fix a base-point $*\in \mathbb F$ with $p(*)\in F$. Since $f(F)$ is a point, there is a natural section $s:F\to \mathbb F$ of $p$ above $F$ with $s(p(*))=*$. In particular, we can regard $F$ as a subspace of $\mathbb F$.
\subsection{Corollary}\label{t3} {\em Assume that $F$ is connected. Then $s_*:\pi_1(F)\to \pi_1(\mathbb F)$ is onto, or equivalently, $\pi_1(\mathbb F,F)$ is trivial.} \begin{proof} Compare the exact sequences $(1)\to \pi_1(F)\to \pi_1(T^*_0)\to \mathbb Z\to (1)$ and $(1)\to \pi_1(\mathbb F)\to \pi_1(M_X)\to \mathbb Z\to (1)$ via \ref{t2}. \end{proof}
Fix an orientation of $S^1$, and consider a smooth loop $\gamma:S^1\to \mathbb C\setminus B_f$. Denote by $q:\gamma^{-1}(f)\to S^1$ the pull-back of $f$ by $\gamma$, i.e. $\gamma^{-1}(f)=\{(t,x)\in S^1\times \mathbb C^{n+1}:\ \gamma(t)=f(x)\}$, and $q(t,x)=t$.
\subsection{Corollary}\label{t4} {\em Assume that $\gamma_*:\pi_1(S^1)\to \pi_1(\mathbb C^*)$ (i.e. $\gamma_*:\mathbb Z\to\mathbb Z$) is multiplication by an integer $\ell$. Then one has the following commutative diagram with all the lines and columns exact:} \[ \begin{array}{ccccccc} (1) \to & \pi_1(F) & \to & \pi_1(\gamma^{-1}(f)) & \to & \mathbb Z & \to (1) \\
& \downarrow & & \downarrow & & \downarrow & \\ (1) \to & \pi_1(\mathbb F) & \to & \pi_1(M_X) & \to & \mathbb Z &\to (1) \\
& \downarrow & & \downarrow & & \downarrow & \\
& (1) & \to & \mathbb Z/\ell\mathbb Z & \to & \mathbb Z/\ell\mathbb Z & \to (1) \\
& & & \downarrow & & \downarrow & \\
& & & (1) & & (1) & \end{array}\] \begin{proof} The first two lines are the homotopy exact sequences of the corresponding fibrations. Then use \ref{t3}. \end{proof}
Sometimes it is convenient to work with special polynomials with nice behaviour around the special fibers or at infinity. First, we recall the definition of Neumann and Rudolph of good polynomials [NR]. A fiber $f^{-1}(b)$ is called ``regular at infinity'' if there exist a small disc $D$ containing $b$ and a compact set $K$ such that the restriction of $f$ to $f^{-1}(D)\setminus K$ is a trivial $C^{\infty}$-fibration. The polynomial $f$ is called good (or ``topologically good'') if all its fibers are regular at infinity.
For example, the tame polynomials introduced by Broughton [B], the larger class of M-tame polynomials introduced by N\'emethi-Zaharia [NZ] are good. We recall that any fiber of a good polynomial is a bouquet of spheres $S^n$, $B_f$ is the set of critical values of $f$, for any $b\in B_f$ the ``tube'' $T_b$ has the homotopy type of $f^{-1}(b)$, and $f^{-1}(b)$ (homotopically) is obtained from $F$ by attaching some cells of dimension $n+1$.
For the purpose of the present paper it is enough (and it is more natural) a much weaker assumption.
\subsection{Definition}\label{d22} The polynomial $f$ is called ``homotopically good'' (h-good) if for any $b\in B_f$, the pair $(T_b,F)$ is $n$-connected.
From the above discussion it follows easily that all the good polynomials are h-good. Another example is provided by the polynomials with isolated singularities on the affine space and at infinity in the sense of Siersma-Tib\u ar [SiT] (see p.776 [{\em loc.\,cit.}]).
In general, for an arbitrary polynomial, it is much easier to handle the properties of the generic fiber and the ``tubes'' $T_b$ than the properties of the special fibers (see e.g. [DN2]). One of the advantages of the above definition \ref{d22} is that it requires information only about $F$ and $T_b$'s. (Conversely, this fact also explains that for h-good polynomials one can say very little about the special fibers. E.g. the special fibers of h-good polynomials, in general, are not even reduced, as it happens e.g. for $f(x,y)=x^2y$. For a non-trivial example of a h-good polynomial which has non-isolated singularities, see the polynomial $f_{d,a}$ constructed by tom Dieck and Petrie, cf. [D1], p.175.)
The second advantage of definition \ref{d22} is that, in fact, it is almost homological. Indeed, for $n=1$, $f$ is h-good iff $F$ is connected (by \ref{t1}); for $n>1$ the polynomial $f$ is h-good iff $F$ is simply-connected and $H_q(T_b,F,\mathbb Z)=0$ for all $b$ and $q\leq n$. This second statement follows from \ref{t1}, the next proposition \ref{p23},
and the relative Hurewicz isomorphism theorem (see e.g. [S], p.397).
The above examples and comment show that we cannot expect that the h-good polynomials
will share all the properties of the ``good'' ones. However, the next result will recover one of the most important properties.
\subsection{Proposition.}\label{p23} {\em Assume that $f$ is a h-good polynomial. Then its generic fiber $F$ has the homotopy type of a bouquet of spheres $S^n$.}
\begin{proof} By \ref{00}(1), $H_q(\mathbb C^{n+1},F)=0$ for $q\leq n$, hence $\tilde{H}_q(F)=0$ for $q\leq n-1$. This already proves the statement for $n=1$. Next, we have to show that $\pi_1(F)=(1)$, provided that $n\geq 2$. The connectivity assumption assures that for each $b\in B_f$, $\pi_1(F)\to \pi_1(T_b)$ is an isomorphism. We denote all these
fundamental groups by $G$. If the cardinality $|B_f|$ of $B_f$ is one, then this implies that $\pi_1(F)=G$ is trivial
(since in this case, $T_b\sim \mathbb C^{n+1}$). If $|B_f|>1$, then by van Kampen theorem, applied for $Y=\vee_F(T_b)$ (cf. (1)),
and induction over $|B_f|$, we get $\pi_1(Y)=G$. But again by \ref{00}(1), $\pi_1(Y)=(1)$. Since $F$ has the homotopy type of a finite $n$-dimensional
CW complex, the result follows by Whitehead theorem. \end{proof}
\subsection{Remark}\label{2.4} (\ref{p23}) can be compared with the following classical result of L\^e [L\^e].
\noindent {\it For any projective hypersurface $V$ and a generic hyperplane $H$, the affine hypersurface $X=V \setminus H$ is homotopy equivalent to a bouquet of spheres $S^{n}$.} [For the computation of the number of spheres in this bouquet, in terms of the degree of gradient mappings, see [DPp].]
\subsection{}\label{fin} Finally, we compare $F$ and $\mathbb F$. Since $\mathbb F$ is a cyclic covering of $M_X$, and $M_X$ has the homotopy type of a finite CW complex of dimension $\leq (n+1)$, one has the general fact: \begin{equation*} \mbox{$H_m(\mathbb F)=0$ for $m>n+1$ and $H_{n+1}(\mathbb F,\mathbb Z)$ has no $\mathbb Z$-torsion.} \tag{2} \end{equation*} But if $f$ is h-good, we can say more (cf. also with \ref{t3}). With a choice of the base points, we again embed $F$ into $\mathbb F$ via the section $s$.
\subsection{Proposition}\label{con} {\em If $f$ is h-good then the pair $(\mathbb F,F)$ is $n$-connected. Therefore, $\mathbb F$ is $(n-1)$-connected, and $s_n:H_n(F,\mathbb Z)\to H_n(\mathbb F,\mathbb Z)$ is onto. In particular, for $n>1$, by Hurewicz theorem and the homotopy exact sequence, one has $H_n(\mathbb F,\mathbb Z)=\pi_n(\mathbb F)=\pi_n(M_X)$.} \begin{proof} Notice (cf. \ref{00}(1) and the proof of \ref{t2}) that $M_X$ has the homotopy type of a space obtained by gluing to $T^*_0$ along $F$ all the tubes $T_{\bar{b}}$ for $\bar{b}\in B_f\setminus \{0\}$. Moreover, $p^{-1}(T^*_0)$ has the homotopy type of $F$, and its embedding into $\mathbb F$ is homotopically equivalent to the embedding $s:F\to \mathbb F$. Therefore, by excision: $$H_q(\mathbb F,F)=H_q(p^{-1}(T^*_0\vee_F(T_{\bar{b}})),p^{-1}(T^*_0))= \oplus_{\bar{b}}H_q(p^{-1}(T_{\bar{b}}),p^{-1}(F)),$$ where $\bar{b}$ runs over $B_f\setminus\{0\}$. But $(p^{-1}(T_{\bar{b}}),p^{-1}(F))=\mathbb Z\times (T_{\bar{b}},F)$, hence $H_q(\mathbb F,F)=0$ for $q\leq n$. Hence, the connectivity follows from this, \ref{t3}, \ref{p23}, and the relative Hurewicz isomorphism theorem. Finally, the connectivity of $\mathbb F$ follows from the connectivity of $F$, cf. \ref{p23}. \end{proof} \noindent The same proof (but neglecting $p$), and the Wang exact sequence of $T^*_0,$ gives:
\subsection{Corollary}\label{con2} {\em If $f$ is h-good, then the pair $(M_X,T^*_0)$ is $n$-connected. In particular, $H_q(M_X,\mathbb Z)=0$ for $1<q<n$. (In fact, by \ref{con}, the cyclic covering of $M_X$ is $(n-1)$-connected, hence $\pi_q(M_X)=0$ for $1<q<n$ as well.)}
For special cases of this connectivity results, see also [Li2] and [LiT].
\section{Preliminaries about $H_*(M_X,\mathbb Z)$}
\noindent Let $X$ be a hypersurface in $\mathbb C^{n+1}$ and $M_X$ be its complement. The goal of this section is to list some properties of the integral homology of $M_X$, with an extra emphasis on the torsion part and the ``interesting part'' $H_n(M_X,\mathbb Z)$. Moreover, we present some constructions which generate examples with non-trivial ``interesting part''. Additionally, sometimes we compare the properties of $H_*(M_X)$ with homological properties of hypersurfaces $X$.
We start with the case when $X$ is a (generic or special) fiber of a polynomial $f$.
\subsection{Fact}\label{2.1}[LiT]
{\it If $F$ is the generic fiber of an arbitrary
polynomial $f$, then $M_F$ has the homotopy type of a join $S^1 \bigvee S(F)$, where $S(F)$ denotes the suspension of $F$. In particular,} \begin{equation*}
{\tilde H}_k(M_F)={\tilde H}_k(S^1)\oplus {\tilde H}_{k-1}(F) \ \ \ \mbox{{\em for any $k$}}. \tag{1} \end{equation*}
In fact, the result \ref{2.1}(1)
holds for any smooth $X$, as follows from the associated Gysin sequence, see [D1], p.46.
The similar result (i.e. the analog of \ref{2.1}(1))
for homotopy groups is definitely false; consider for example $\pi_1$, or (for instance) [S], p.419, exercise B6, for a reason.
\subsection{Example}\label{2.3} \ref{2.1}(1)
is false for special fibers, even for very simple polynomials. Let $f=x_0^2+...+x_n^2$ and $X=f^{-1}(0)$. The Wang sequence
of the global Milnor fibration $F \to M_X \to \mathbb C^*$ (see [D1], p.71-74) and the fact that the corresponding monodromy operator $T$ acting on $H_n(F,\mathbb Z)=\mathbb Z$ is $(-1)^{n+1}Id$, implies the following:
(i) for $n=2m+1$ odd, $H^*(M_X)=H^*(S^1 \times S^n)$. In fact, the monodromy is isotopic to the identity and hence we have a diffeomorphism $M_X = \mathbb C^* \times F$. This implies that $M_X$ has the homotopy type of $S^1 \times S^n$ as claimed in [Li2], Remark (1.3);
(ii) for $n=2m>0$ even, $H_n(M_X,\mathbb Z)=\mathbb Z/2\mathbb Z$. In particular, $M_X$ is not homotopy equivalent to the product $S^1 \times S^n$ (contrary to the claim in [Li2], Remark (1.3)).
(\ref{2.1}) has the following consequence: when $X$ is the generic fiber of a h-good polynomial then $H_m(M_X,\mathbb Z)=0$ for $1< m \leq n$, and in fact $H_*(M_X,\mathbb Z)$ is torsion free (cf. \ref{p23}). (This can be compared with L\^e's result \ref{2.4}, which shows that {\em generically} an affine hypersurface $X$ has no torsion in homology.)
More generally, it was shown in [Li2] that
when $X$ has isolated singularities including at infinity, then $H_m(M_X)=0$ for $1<m<n$. The same statement holds for the special fiber $X$ of a h-good polynomial by \ref{con2} (cf. also with [LiT]). Hence, in both cases, the first interesting homology group occurs in degree $n$.
We describe now three constructions which provide in a
systematic way hypersurfaces $X$ with $H_n(M_X, \mathbb Z) \not= 0$.
Below $V$ denotes the projective closure of $X$ and $H$ the hyperplane at infinity.
\subsection{The first construction (using duality)}\label{2.7} Following [Li2], section 1, we consider the isomorphism \begin{equation*}
H_m(M_X)=H^{2n-m+1}(V,V \cap H)\ \ \ \mbox{for all $m$.} \tag{2} \end{equation*}
Assume that $V$ and $V \cap H$ have only isolated singularities (this is exactly the condition on $X$ to have isolated singularities including at infinity in [Li2]). The exact sequence \begin{equation*} H^n(V) \to H^n(V \cap H) \to H^{n+1}(V,V \cap H) \to H^{n+1}(V) \to H^{n+1}(V \cap H) \tag{3} \end{equation*} and the isomorphism $H^{n+1}(V \cap H)=H^{n+1}(\mathbb P^{n-1})$ (cf. [D1], p.161) imply the following.
\noindent {\it Assume that $V$ and \, $V \cap H$ \, have only isolated singularities and that $H^{n+1}(V,\mathbb Z) \not=H^{n+1}(\mathbb P^{n-1},\mathbb Z)$. Then $H_n(M_X, \mathbb Z) \not= 0$. In particular, the corresponding affine hypersurface is not the generic fiber of a h-good polynomial.}
\subsection{Example}\label{2.8} Let $V$ be a cubic surface in $\mathbb P^3$ having two singularities, one of type $A_1$ and the other of type $A_5$. First we take $H$ a generic plane. In this case, using [D1], p.165 we see that the exact sequence \ref{2.7}(3) becomes $$\mathbb Z \to \mathbb Z \to H_2(M_X,\mathbb Z) \to \mathbb Z/2\mathbb Z \to 0$$ where the first morphism is multiplication by $deg(V)=3$. It follows that $H_2(M_X,\mathbb Z)=\mathbb Z/6\mathbb Z$, $X$ is singular and the associated polynomial $f$ is tame.
Secondly, take $H$ to be any plane passing through the 2 singularities on $V$. Then the associated $X$ is smooth, but by \ref{2.7},
$X$ is not the generic fiber of a good polynomial.
\subsection{The second construction (using finite cyclic coverings and defect)}\label{defect} The second approach uses $M_{X,e}$, the cyclic covering of $M_X$ of degree $e$ when $X=f^{-1}(0)$ (cf. also with \ref{3.4}(II) and \ref{remarks}(4)). It is clear that we can take
$$M_{X,e}=\{(x,u) \in \mathbb C^{n+1} \times \mathbb C^* | f(x)-u^e=0 \}.$$ In some cases we can get a useful approximation of $M_{X,e}$ as follows. Fix a system of positive integer weights $w=(w_0,...,w_n)$, and let $e$ be the top degree term in $f$ with respect to $w$. Introduce a new variable $t$ of weight 1 and let ${\tilde f}(x,t)$ be the homogenization of $f$ with respect to the weights $(w,1)$. Consider the affine Milnor fiber $F': {\tilde f}(x,t) =1$, which is a smooth hypersurface in $\mathbb C^{n+2}$. One has an embedding $j: M_{X,e} \to F'$ given by $$j(x,u)=(u^{-1} * x, u^{-1}),$$ where $*$ denotes the multiplication associated to the system of weights $w$. The complement $F' \setminus j(M_{X,e})$ is characterized by $\{t=0\}$, hence it can be identified with
the affine Milnor fiber $F_e:f_e(x)=1$ (considered din $\mathbb C^{n+1}$) defined by the top homogeneous component $f_e$ of $f$. If $f_e$ defines an isolated singularity at the origin, then $f_e$ is a good polynomial, $F_e$ is $(n-1)$-connected and the Gysin sequence of the divisor $F_e$ implies the isomorphisms \begin{equation*} j_*:H_k(M_{X,e}) \to H_k(F')\ \ \mbox{for $1<k<n+1$}. \tag{4} \end{equation*} Note that under this isomorphism the action of the natural generator of the covering transformation group on $M_{X,e}$ corresponds to multiplication by $exp(-2 \pi i/e)$ (of all the coordinates) on $F'$. Moreover, notice that $H_k(M_X,\mathbb Q)$ is isomorphic to the group of invariants of $H_k(M_{X,e},\mathbb Q)$ with respect to this action.
Notice that the dimension of $H_n(F')$ is closely related to the {\em defect} associated with the singular points of the projective hypersurface $\tilde{f}=0 $ considered in $\mathbb P^{n+1}$ (for details, see e.g. [D1]). Hence, this construction emphasizes the connection between $H_n(M_X)$ and superabundance properties.
For more information on the homology of $M_{X,e}$, see also \ref{remarks}(4) and \ref{4.6} below.
\subsection{Example}\label{2.10} Let $f:\mathbb C^4 \to \mathbb C$ be given by $f(x,y,u,v)=x^3+y^3+xy-u^3-v^3-uv$. Set $X=\{f=0\}$. Then $f$ is a tame polynomial and $X$ has 10 nodes. It follows from [D1], p.208-209, (with all the weights $w_i=1$) that $\dim H_3(F',\mathbb C)=5$ and the
multiplication by $exp(-2 \pi i/3)$ on $F'$ induces the trivial action on $H_3(F',\mathbb C)$. It follows that $\dim H_3(M_X,\mathbb C)=\dim H_3(M_{X,3},\mathbb C)= \dim H_3(F',\mathbb C)=5$.
\subsection{The third construction (``counting'' the Milnor numbers)}\label{Mil} For simplicity we will assume that $f$ is a (topologically) {\em good } polynomial. As above, $F$ is its generic fiber and $X$ a special fiber. Assume that $X$ has $n_X$ singular points with Milnor fibers $F_i $ and Milnor numbers $\mu_i$ for $1\leq i\leq n_X$. For each $i$, let $\mu_{0,i}$ denote the rank of $H_n(\partial F_i)$. Set $\mu_X:= \sum_i\mu_i$ and $\mu_{0,X}:= \sum_i\mu_{0,i}$. Below, $\oplus_X$ means $\oplus_{i=1}^{n_X}$.
Finally, let $\mu$ be the sum of all the Milnor numbers of the singularities of $f$ (situated on all the singular fibers). With these notations, one has:
{\em \noindent The groups $H_q(M_X)$ ($q=n, n+1$) are inserted in the following commutative diagram:
\[ \begin{array}{ccccccccccc} 0 & \to & \oplus_XH_n(\partial F_i) & \to & \oplus_XH_n(F_i) & \to & \oplus_XH_n(F_i,\partial F_i) & \to & \oplus_X\tilde{H}_{n-1}(\partial F_i) & \to & 0\\
& & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ 0 & \to & H_{n+1}(M_X) & \to & H_n(F) & \to & \oplus_XH_n(F_i,\partial F_i) & \to & H_{n}(M_X) & \to & 0 \end{array}\] where the first two vertical arrows are monomorphisms, the third is the identity, and the last one is an epimorphism. In particular: $$\mu_{0,X}\geq \dim H_n(M_X) \geq \mu_X+\mu_{0,X}-\mu.$$}
Above, the first inequality is not new, and it is not very sharp (cf. e.g. with [GN2] (2.30), or with [Si2] (5.4) and \S 9; it also follows from \ref{4.6}(iii) below).
The second inequality is more interesting, and it can be used in two different ways. First, using the integers $ \mu_X$, $\mu_{0,X}$ and $\mu$, if the sum of the first two is strictly larger than the third, one gets a non-vanishing criteria for $H_n(M_X)$. For example, in the case \ref{2.10}, $\mu=16$, $\mu_X=\mu_{0,X}=10$, hence the second inequality reads as $\dim H_n(M_X)\geq 4$.
Similarly, if one knows $\mu_X$, $\mu_{0,X}$ and $\dim H_3(M_X)$
about the hypersurface $X$, then these numbers may impose serious conditions about singularities of some other singular fibers of $f$ (e.g. about their existence). \begin{proof} Assume that $X=f^{-1}(b)$, and let $T_b^o$ be the interior of $T_b$. Clearly, $M_X$ has the homotopy type of $\mathbb C^{n+1}\setminus T_b^o$. Let $F$ be a fixed generic fiber of $f$ inside of $T_b^o$. Then one can write the homological long exact sequence of the pair $(\mathbb C^{n+1}\setminus F, \mathbb C^{n+1}\setminus T_b^o)$. Notice that $H_q:=H_q(\mathbb C^{n+1}\setminus F, \mathbb C^{n+1}\setminus T_b^o)$ equals $H_q(T_b\setminus F, \partial T_b)$ by excision. Let $B_i$ be a small Milnor ball of the $i$-th singular point of $X$. Then using the ``good''-property of $f$ and excision one gets $H_q=\oplus_XH_q (B_i\setminus F_i, \partial B_i\setminus \partial F_i)$. But this is isomorphic to $\oplus_XH_{q-1}(F_i,\partial F_i)$ by Gysin isomorphism. Use these facts and the Gysin isomorphism $H_n(F)\to H_{n+1}(M_F)$ to obtain the second line of the above diagram.
Next, assume that the disc $D_b$ is sufficiently small with respect to the balls $B_i$, and consider for any ball $B_i$ the local analog of the above
picture; namely the homological long exact sequence of the pair $(B_i\setminus F,B_i\setminus T_b^o)$. This homological sequence admits a natural map to the previous sequence induced by the inclusion. Finally, this ``local sequence'' is modified by dualities and Gysin isomorphisms. \end{proof}
\subsection{Remarks about the torsion part of $H_n(M_X,\mathbb Z)$.}\label{torsion} In the final part of this section we discuss the relations between the existence/non-existence of torsion in the homology of $X$ and $M_X$ respectively. The following examples show that these relations are not simple even for a homogeneous polynomial $f$.
\subsection{Example}\label{2.11} Consider the homogeneous polynomial $f=x^2y^2+y^2z^2+z^2x^2-2xyz(x+y+z)$ defined on $\mathbb C^3$. It is known that its Milnor fiber $F$ has torsion in homology, more precisely $H_1(F,\mathbb Z)= H_2(M_{F},\mathbb Z) = \mathbb Z/3\mathbb Z$ and $H_2(F,\mathbb Z)=\mathbb Z^3$ see [Li0], [DN1] and [Si]. Let $C$ be the 3-cuspidal quartic in $\mathbb P^2$ defined by $\{f=0\}$. It satisfies $H_1(\mathbb P^2\setminus C,\mathbb Z)=\mathbb Z_4$ and $H_2(\mathbb P^2\setminus C,\mathbb Z)=0$ (cf. [loc. cit.]). Set $X=f^{-1}(0)$. The Gysin sequence of the fibration $\mathbb C^* \to M_{X} \to \mathbb P^2 \setminus C$ yields $H_1(M_X,\mathbb Z)=\mathbb Z$, $H_2(M_{X},\mathbb Z)=\mathbb Z/4\mathbb Z$ and $H_3(M_{X})=0$. In conclusion, both $M_F$ and $M_X$ have torsions in $H_2$, but these torsions are different.
\noindent If we put together the examples \ref{2.3} and \ref{2.11},
we see that for a homogeneous polynomial $f$ and for $X=f^{-1}(0)$ the only case not covered is the following.
\subsection{Question}\label{2.12} {\em Find an example of a homogeneous polynomial $f$ such that $M_X$ has no torsion but $F$ has torsion.}
Even in the case when $f$ is a product of linear forms,
the existence of such an example is an open question. (It is known in this latter case that the hyperplane arrangement complement $M_X$ is torsion free, see [OT],
and the corresponding Milnor fiber can be identified to a cyclic covering of $\mathbb P^n\setminus \{f=0\}$, see [CS], [CO]). Notice also, that if we allow $f$ to be a product of {\em powers
of linear forms}, then A. Suciu has examples with torsion in the homology of the associated Milnor fiber $F$.
The following result gives some conditions on the possible torsions that may arise in such a case.
\subsection{Proposition}\label{2.13} {\it Assume that for a homogeneous polynomial $f$ of degree $d$ the complement $M_X$ has no $p$-torsion for some prime $p$ and that the $p$-torsion in a homology group $H_k(F,\mathbb Z)$ has the structure $$\mathbb Z/p^{k_1}\mathbb Z \oplus \mathbb Z/p^{k_2}\mathbb Z \oplus ... \oplus \mathbb Z/p^{k_m}\mathbb Z$$ with $m \geq 1$ and $k_1 \geq k_2 \geq ... \geq k_m \geq 1$. Then
(i) $(p-1,d)=1$ implies $m \geq 2$ and $k_1=k_2$;
(ii) $((p-1)p(p+1),d)=1$ implies $m \geq 3$ and $k_1=k_2=k_3$. } \begin{proof} Consider the Wang sequence in homology associated to the fibration $F \to M_X \to \mathbb C^*$. The fact that $M_X$ has no $p$-torsion implies that $T-Id$ is an isomorphism when restricted to the $p$-torsion part.
To prove (i) note that since such an isomorphism preserves the orders of the elements, unless the claim (i) above holds, we get an induced automorphism of $\mathbb Z/p\mathbb Z$, where this latter group is regarded as the quotient of the $p$-torsion part by the subgroup of elements killed by multiplication by $p^{k_1-1}$. The same is true for the monodromy transformation $T$.
Denote by $T_p$ the induced automorphism of $\mathbb Z/p\mathbb Z$. It follows that $T_p$ is not the identity, $T_p^d=Id$ (since $f$ is
homogeneous of degree $d$) and $T_p^{p-1}=1$ (since $|Aut(\mathbb Z/p\mathbb Z)|=p-1$), a contradiction.
For (ii) use the same argument, plus the equality $|Aut((\mathbb Z/p\mathbb Z)^2)|=(p-1)^2p(p+1)$. \end{proof}
\subsection{Remark}\label{2.14} Other examples involving torsion in the homology of the special fibers of a polynomial can be obtained by suspension, see for details [DN2].
\section{Alexander Modules}
\subsection{Definitions}\label{3.1} Let $Y$ be a connected CW-complex and $e_Y: \pi _1(Y) \to \mathbb Z$ be an epimorphism. We denote by $Y^c$ the $\mathbb Z$-cyclic covering associated to the kernel of the morphism $e_Y$. It follows that a generator of $\mathbb Z$ acts on $Y^c$ by a certain covering homeomorphism $h$ and all the groups $H_*(Y^c,A)$, $H^*(Y^c,A)$ and $\pi _j(Y^c) \otimes A$ for $j>1$ become in the usual way $\Lambda _A$-modules, where $\Lambda _A= A[t,t^{-1}]$, for any ring $A$. These are called the Alexander modules of the pair $(Y,e_Y)$ or simply of $Y$ when the choice of $e_Y$ is clear.
If $Z$ is a second connected CW-complex, $e_Z: \pi _1(Z) \to \mathbb Z$ an epimorphism and $\phi : Y \to Z$ is a continuous map such that the induced map at the level of $\pi _1$ is an epimorphism compatible to the two given epimorphisms $e_Y$ and $e_Z$, then $Y^c \to Y$ can be regarded as a pull-back covering obtained via $\phi$ from the covering $Z^c \to Z$. In particular, this gives a lift $\phi ^c :Y^c \to Z^c$ which is compatible with the covering transformations, and hence,
an induced morphism of $\Lambda _A$-modules, say $\phi ^c_*: H_*(Y^c,A) \to H_*(Z^c,A)$.
If $A$ is a field then the ring $\Lambda _A$ is a PID. Hence any finite type $\Lambda _A$-module $M$ has a decomposition $M=\Lambda _A ^k \oplus (\oplus _pM_p)$, where $k$ is the rank of $M$ and the second sum is over all the prime elements in $\Lambda _A$, and $M_p$ denotes the $p$-torsion part of $M$.
More precisely, for each prime $p$ with $M_p\not=0$, we have a unique decomposition
\begin{equation*} M_p= \oplus_{i=1,\ell_p}\Lambda _A/p^{k_i} \tag{1} \end{equation*} for $\ell_p>0$ and $k_1 \geq k_2 \geq .... \geq k_{\ell_p} \geq 1$. The sequence $(k_1,k_2,...,k_{\ell_p})$ obtained in this way will be denoted by $K(M,p)$. One can define an order relation on such sequences by saying that $$(k_1,....,k_a) \geq (m_1,...,m_b)$$ iff $a \geq b$ and $k_i \geq m_i$ for all $1 \leq i \leq b$.
Let $\Delta _p(M)=\prod _{i=1,\ell_p}p^{k_i}$ (resp. $\Delta (M)=\prod _p \Delta _p(M)$) be the $p$-Alexander polynomial (resp. the Alexander polynomial) of the module $M$. This latter invariant $\Delta (M)$ is called the order of $M$ in [Li2]. See \ref{remarks}(1) for a motivation of this terminology.
With this notation one has the following easy result whose proof is left to the reader.
\subsection{Lemma}\label{3.2} {\it Let $u:M \to N$ be an epimorphism of $R$-modules, where $R$ is a PID and $M$ is of finite type. Then $N$ is of finite type and for any prime $p \in R$ one has $K(M,p) \geq K(N,p)$. In particular $\Delta (N)$ divides $\Delta (M)$.}
\subsection{Example}\label{3.3}
For $A=\mathbb C$, we will simply write $\Lambda $ instead of $ \Lambda_{\mathbb C}$. The prime elements in this case are just the linear forms $t-a$, for $a \in \mathbb C^*$. Moreover, a $\Lambda $-module of the form $H_q(Y^c,\mathbb C)$ is of finite type and torsion iff the corresponding Betti number $b_q(Y^c)$ is finite. If this is the case, the $(t-a)$-torsion part of $H_q(Y^c,\mathbb C)$ is determined (and determines) the Jordan block structure of the corresponding automorphism $h_q$; i.e. a Jordan block of size $m$ corresponding to the eigenvalue $a$ produces a summand $\Lambda /(t-a)^m$. The corresponding Alexander polynomial $\Delta (H_q(Y^c,\mathbb C),h_q)$ is just the characteristic polynomial $\Delta (h_q)$ of $h_q$.
When $A=\mathbb Q$, the corresponding prime elements are the irreducible polynomials in $\mathbb Q[t]$ different from $t$, hence they are a lot more difficult to describe. However, the knowledge of the $\Lambda $-module structure implies easily the $\Lambda_{\mathbb Q} $-module structure just by grouping together the polynomials $t-a$ for those $a$'s having the same minimal polynomial over $\mathbb Q$.
\subsection{The Alexander modules of $M_X$ and local systems}\label{3.4} Coming back to the situation (and notation) of the previous sections, for any hypersurface $X$ we define $e_X:\pi_1(M_X)\to \mathbb Z$ as follows. In fact, we will distinguish two cases.
\noindent (I) \ First, assume that $X$ is an arbitrary hypersurface in $\mathbb C^{n+1}$ (and even if $X=f^{-1}(0)$, we disregard $f$). Assume that $X$ has $r$ irreducible components $X_1,\ldots, X_r$. Then $H_1(M_X,\mathbb Z)=\mathbb Z^r$, where the generator $(0,\ldots,1,\ldots ,0) $ (1 on the place $i$, $1\leq i\leq r$) corresponds to an elementary oriented circle ``around $X_i$''. For each set of integers ${\mathfrak m} :=(m_1,\ldots, m_r)\in \mathbb Z^r$ we define $\phi_{{\mathfrak m} }:\mathbb Z^r\to \mathbb Z$ by $(x_1,\ldots, x_r)\mapsto \sum_im_ix_i$. If $gcd_i(m_i)=\pm 1$ then $$e_{X,{\mathfrak m} }:\pi_1(M_X)\stackrel{ab}{\longrightarrow} H_1(M_X,\mathbb Z)\stackrel {\phi_{{\mathfrak m} }}{\longrightarrow}\mathbb Z \ \ \mbox{is onto}.$$
\noindent (II) \ Now, assume that $X=f^{-1}(0)$ for some polynomial $f$. Then define $e_{X,f}$ by $e_{X,f}:=f_*: \pi _1(M_X) \to \pi _1 (\mathbb C^*)$. In fact, this is a particular case of (I): if $f=\prod_{i=1}^r g_i^{m_i}$ (where $g_i$ are irreducible with distinct zero sets) then $e_{X,f}=e_{X,{\mathfrak m} }$ for ${\mathfrak m} =(m_1,\ldots, m_r)$.
By a similar argument as in the proof of \ref{t1}, if the generic fiber $F$ of $f$ is connected then $e_{X,f}$ is onto. Therefore, in the sequel, in all our Alexander-module discussions associated with $f$, {\em we will assume that $F$ is connected}.
Sometimes, we will use the notations (I) resp. (II) to remind the reader about the corresponding cases. In both cases (I) and (II), let $\mathbb F:=M_X^c$ be the $\mathbb Z$-cyclic covering associated with the kernel of $e_X$ (cf. also with \S 2). Since $M_X$ has the homotopy type of a finite CW-complex, it follows that all the associated Alexander modules are of finite type over $\Lambda _A$ (but in general not over $A$).
Moreover, for a complex number $a \in \mathbb C^*$,
we consider the rank one local system $L_a$ on $M_X$ defined by the composed map $\pi _1(M_X)\stackrel{e_X}{ \longrightarrow} \mathbb Z\to\mathbb C^*$, where the last map is defined by $1_\mathbb Z\mapsto a$. Obviously, if $a=1$, then $L_a=\mathbb C$.
Then, exactly as in [Li3], we have the following long exact sequence \begin{equation*} ... \to H_k(\mathbb F,\mathbb C) \to H_k(\mathbb F,\mathbb C) \to H_k(M_X, L_a) \to H_{k-1}(\mathbb F,\mathbb C) \to ... \tag{2} \end{equation*} where the first morphism is multiplication by $t-a$.
In the next paragraph we summarize the properties of $H_*(M_X,L_a)$ and the Alexander modules $H_*(\mathbb F,\mathbb C)$.
\subsection{Theorem}\label{3.6} {\it
(i) $H_m(M_X,L_a)=0$ for any $a \in \mathbb C^*$ and $m > n+1$.
The Alexander modules $H_k(\mathbb F)$ are trivial for $k>n+1$ and $H_{n+1}(\mathbb F,\mathbb C)$ is free (over $\Lambda$).
(ii) The $\Lambda$-rank of $H_{n+1}(\mathbb F,\mathbb C)$ equals $\dim H_{n+1}(M_X,L_a)$ for any generic $a $.
(iii) For any $q\leq n$, the $(t-a)$-torsion in $H_q(\mathbb F,\mathbb C)$
can be non zero only when a is a root of unity.
(iv) Assume that all the Alexander modules $H_k(\mathbb F,\mathbb C)$ are torsion for $k<n+1$. Denote by $N(a,k)$ the number of direct summands in the $(t-a)$-torsion part of $H_k(\mathbb F,\mathbb C)$. Then $\dim H_k(M_X, L_a)=N(a,k)+N(a,k-1)$ for $k<n+1$ and
$\dim H_{n+1}(M_X, L_a)=N(a,n) +|\chi (M_X)|$.
\noindent Moreover, if either
(I) $X$ has only isolated singularities including at infinity, or
(II) $X$ is the fiber of a h-good polynomial,
\noindent then
(v) $ \tilde{H}_k(\mathbb F)=0$ for $k<n$, and $H_n(\mathbb F,\mathbb C)$ is a
$\Lambda$-torsion module. Moreover, one has the isomorphisms of the Alexander $\Lambda_\mathbb Z$-modules $\pi_n(M_X)=\pi_n(\mathbb F)=H_n(\mathbb F,\mathbb Z)$.}
\begin{proof} For (i) and the first part of (ii) see \ref{fin}(2); for (ii) use the exact sequence \ref{3.4}(2) and the fact that multiplication
by $t-a$ is injective, provided that $a$ is generic. The vanishing of $(t-a)$-torsion for $a$ non root of unity follows from \ref{n7}(iv) below. (iv) is straightforward, if we notice that $\chi(M_X,L_a)=\chi(M_X)$ for any local system $L_a$. Claim (v) in case (I) for ${\mathfrak m}=(1,\ldots,1)$ is due to Libgober, see [Li2], Theorem 4.3; the case of arbitrary ${\mathfrak m}$ follows similarly. The case (II) is a consequence of \ref{con}. \end{proof}
\subsection{Remarks.}\label{remarks}
(1) Under the assumption (I) and (II) in \ref{3.6}, if $H_n(\mathbb F,\mathbb Z)$ is $\mathbb Z$-torsion free, then it has a finite rank over $\mathbb Z$ and the associated Alexander polynomial $\Delta (H_n(\mathbb F,\mathbb Z))$ is the characteristic polynomial $\Delta (h_n)$ as in \ref{3.3}. It follows that $H_n(M_X,\mathbb Z)$ is finite iff $\Delta (h_n)(1) \ne 0$. Moreover, if
$H_n(M_X,\mathbb Z)$ is finite then its order is exactly $|\Delta (h_n)(1)|$.
However, we do not know whether (I) or (II) in \ref{3.6}
imply that $H_n(\mathbb F,\mathbb Z)$ is $\mathbb Z$-torsion free.
(2) \ The proof of \ref{3.6} is topological in both cases (I) and (II).
It follows that the same proof can be applied to coefficients in a finite field $K$, as soon as $|K|$ is large enough. Therefore, Theorem \ref{3.6} also
holds in such a case (of course without any reference to roots of unity).
(3) \ Notice that \ref{3.6}(iv) is not true without the assumption about $\mathbb F$. In other words, it is not true that for a generic $a \in \mathbb C^*$ one has $H_m(M_X,L_a)=0$ for $m \not= n+1$. Indeed, consider a polynomial function $f:\mathbb C^{n+1} \to \mathbb C$ as a function $\mathbb C^{n+1+k} \to \mathbb C$ independent of the last $k$-variables and denote by $M_X'$ the corresponding complement. Then
$M_X' = M_X \times \mathbb C^k$. In particular, if $f$ is chosen such that $H_{n+1}(M_X,L_a) \neq 0$ for any generic $a$ (i.e. $\chi(M_X)\not=0$), we get counter-examples to the above claim.
(4) \ Additionally to the exact sequence \ref{3.4}(2) one has two more (both valid over $\mathbb Z$). First notice that the $\mathbb Z$-covering $\mathbb F\to M_X$ homotopically can be identified with the inclusion of the fiber into the total space of a fibration $M_X\to K(\mathbb Z,1)=S^1$ (with fiber $\mathbb F$). Hence, the (homotopy) Wang exact sequence of the covering $\mathbb F\to M_X$ gives: $$... \to H_k(\mathbb F,\mathbb Z) \to H_k(\mathbb F,\mathbb Z)\to H_k(M_X,\mathbb Z) \to H_{k-1}(\mathbb F,\mathbb Z) \to ...$$ where the first morphism is multiplication by $t-1$.
Moreover, let $M_{X,e}$ ($e>0$) be the cyclic covering of $M_X$ of degree $e$ (i.e. the covering classified by the subgroup $e_X^{-1}(e\mathbb Z)$). If $\mathbb Z$ denotes the transformation group of $\mathbb F$ above $M_X$, then
$\mathbb F$ is a $\mathbb Z$-covering of $M_{X,e}$ with transformation group $e\mathbb Z$. Hence, we have an exact sequence $$...\to H_k(\mathbb F,\mathbb Z)\to H_k(\mathbb F,\mathbb Z)\to H_k(M_{X,e},\mathbb Z)\to H_{k-1}(\mathbb F,\mathbb Z)\to ...$$ where the first morphism is multiplication by $t^e-1$. It follows that there is a close relation between the homology of $M_{X,e}$ and the structure of the Alexander invariants $H_k(\mathbb F,\mathbb Z)$, see Example \ref{4.5} below.
\subsection{Example}\label{3.8} Assume that $f$ is a weighted homogeneous polynomial with respect to an integer set of weights (non necessarily strictly positive) such that $d=deg(f) \not=0$. Denote by $F$ the associated affine Milnor fiber. Then it is easy to see that $\mathbb F$ and $ F$ are homotopically equivalent, and the covering transformation $h$ corresponds to the Milnor monodromy. In particular $h_*^d=Id$. It follows that
$N(a,k)=\dim ker (h_*-a Id|H_k(\mathbb F,\mathbb C))$ is trivial for $a^d \not= 1$. Moreover, in this case,
due to the presence of $S^1$-actions, we have $\chi (M_X)=0$.
In the case of a central hyperplane arrangement D. Cohen [C] has shown that for any rank one local system $L$ on $M_X$ one has $\dim H_m(M_X,L) \leq \dim H_m(M_X,\mathbb C)$ for any $m$.
The example of the $A_1$-singularity $f:\mathbb C^3 \to \mathbb C$, $f=x^2+y^2+z^2$ , $m=2$ and $a=-1$,
shows that this inequality is not true for a general homogeneous polynomial.
\section{Relations to Individual Monodromy Operators}
\subsection{}\label{n1} For the convenience of the reader we recall the present set up. Let $f:\mathbb C^{n+1}\to \mathbb C$ be a polynomial whose {\em generic fiber $F$ is connected}. Let $X=f^{-1}(0)$ be a fixed special fiber, $M_X$ its complement and $e_X=f_*:\pi_1(M_X)\to\mathbb Z$ the epimorphism whose kernel determines the $\mathbb Z$-cyclic covering $p:\mathbb F\to M_X$. Fix a generic fiber $F\subset M_X$, and a base point $*\in \mathbb F$ with $p(*)=b_0\in F$. Then there is a unique lift (embedding) $s:F\to \mathbb F$ of $p$ over $F$ with $s(b_0)=*$. Let $h$ be the covering transformation of $\mathbb F$ corresponding to the generator $1_\mathbb Z$. Sometimes we prefer to denote this Alexander $\Lambda_\mathbb Z$-module
by $(H_*(\mathbb F),h_*)$.
Now, fix an orientation of $S^1$, and consider an arbitrary smooth map $ \gamma:S^1\to \mathbb C\setminus B_f$ (we can even take $\gamma(1)=b_0$). Let $q:\gamma^{-1}(f)\to S^1$ be the pull back of $f$ by $\gamma$. Obviously, $q$ is a fiber bundle over $S^1$ with fiber $F$. The map which covers $\gamma$ is denoted by $\bar{\gamma}:\gamma^{-1}(f)\to M_X$, hence $f\circ \bar{\gamma}=\gamma\circ q$. We can consider the epimorphism $e_\gamma:\pi_1(\gamma^{-1}(f))\to \pi_1(S^1)=\mathbb Z$. The associated $\mathbb Z$-covering has a total space isomorphic to $F\times \mathbb R$ and the covering transformation (corresponding to $1_\mathbb Z$) can be identified
with the geometric monodromy of $f$ associated with the oriented loop $\gamma$. This Alexander $\Lambda_\mathbb Z$-module will be denoted by $(H_*(F), T_{\gamma,*})$.
Next, we connect these two Alexander modules. First, assume that $\gamma_*: \mathbb Z\to\mathbb Z$ (i.e. $\gamma_*:\pi_1(S^1)\to \pi_1(\mathbb C^*)$) satisfies $\gamma_*(1)=1$. Then by \ref{t4}, $\pi_1(\bar{\gamma}):\pi_1(\gamma^{-1}(f))\to \pi_1(M_X)$ is onto, and $e_X\circ \pi_1(\bar{\gamma})=e_\gamma$. Therefore, by \ref{3.1} there exists a morphism of $\Lambda_\mathbb Z$-modules $$\gamma^c_{\Lambda,*}:(H_*(F), T_{\gamma,*})\to (H_*(\mathbb F),h_*)$$ (in the sense that $h_*\circ \gamma^c_{\Lambda,*}=\gamma^c_{\Lambda,*} \circ T_{\gamma,*}$).
More generally, assume that $\gamma_*:\mathbb Z\to \mathbb Z$ is multiplication with an integer $\ell$ (in other words, $\ell$ is the winding number of $\gamma$ with respect to 0). Then
we can replace $M_X$ by the $\mathbb Z/\ell\mathbb Z$-cyclic covering $M_{X,|\ell|}$
(with projection $pr:M_{X,|\ell|}\to M_X$). By \ref{t4}, $\bar{\gamma}: \gamma^{-1}(f)\to M_X$ can be lifted to $\bar{\gamma}':
\gamma^{-1}(f)\to M_{X,|\ell|}$ (with $pr\circ \bar{\gamma}'=\bar{\gamma}$). Obviously,
$\mathbb F$ is the cyclic covering of $M_{X,|\ell|}$ with transformation group $\ell\mathbb Z$, and $\pi_1(\bar{\gamma}')$ is onto. Hence again we obtain a morphism of $\Lambda_\mathbb Z$-modules \begin{equation*} \gamma^c_{\Lambda,*}:(H_*(F), T_{\gamma,*})\to (H_*(\mathbb F),h_*^{\ell}) \tag{$\gamma$} \end{equation*} (i.e. $h_*^{\ell}\circ \gamma^c_{\Lambda,*}=\gamma^c_{\Lambda,*} \circ T_{\gamma,*}$. Above, if
$\ell =0$ then $M_{X,|\ell|}$ is obviously $\mathbb F$ itself).
The above constructions can be made compatible with some base points. (Since all the spaces are connected, this is not really relevant, the details are left to the interested reader).
Notice that the target module $(H_*(\mathbb F),h_*)$ is completely independent of $\gamma$. The main point is that even the map $H_*(F)\to H_*(\mathbb F)$, as a $\mathbb Z$-module, is independent of $\gamma$. Indeed, $\gamma^{-1}(f)^c$ has the homotopy type of $F$ and $\bar{\gamma}^c$ is up to homotopy the same as our fixed embedding $s:F\to \mathbb F$. The above discussion is summarized in the following theorem.
\subsection{Theorem}\label{n2} {\em Assume that the generic fiber $F$ of $f$ is connected, and $0\in B_f$. Fix an embedding $s:F\to \mathbb F$ as above. Let $\gamma:S^1\to \mathbb C\setminus B_f$ be a smooth loop. Assume that $\gamma_*:\pi_1(S^1)\to \pi_1(\mathbb C^*)$ is multiplication by $\ell=\ell(\gamma)$. Then $s_*:H_*(F)\to H_*(\mathbb F)$ is compatible with the Alexander $\Lambda_\mathbb Z$-module structures in the sense that $s_*\circ T_{\gamma,*}=h_*^\ell\circ s_*$. }
\noindent One has the following immediate consequence.
\subsection{Corollary}\label{n3} {\em Assume that two loops $\gamma,\gamma':S^1\to \mathbb C\setminus B_f$ satisfy $\ell(\gamma)=\ell(\gamma')$ (i.e. they have the same winding number with respect to 0). Then $T_{\gamma,q}=T_{\gamma',q}$ modulo $ker(s_q)$, for any $q\geq 0$.}
In other words, if the monodromy operators are ``very different'', then the image of $s_*:H_*(F)\to H_*(\mathbb F)$ is forced to be ``small''. Conversely, the non-vanishing of $im(s_*)$ imposes some compatibility restrictions on the monodromy operators. For a precise reinterpretation of \ref{n2} and \ref{n3}, see \ref{r4}.
The above theorem is optimal exactly when $s_q$ is onto in the non-trivial cases $q\leq n$. As we will see, this is the case e.g. for h-good polynomials (cf. \ref{n4} below). On the other hand, we cannot hope that for an arbitrary polynomial $f$ the map $s_q:H_q(F)\to H_q(\mathbb F)$ ($q\leq n$) is onto , since $H_q(F,\mathbb C)$ is a $\Lambda$-torsion module, but $H_q(\mathbb F,\mathbb C)$ may have a non zero free part.
Nevertheless, the next theorem shows that $im(s_*)$ is as ``large as possible''. Below $\Delta_{\gamma,*}(t)$ denotes the characteristic polynomial of the monodromy operator $T_{\gamma,*}$.
\subsection{Theorem}\label{n7} {\em Assume that $F$ is connected, and $0\in B_f$, and $s:F\to \mathbb F$ is fixed as in \ref{n2}. Let $({\mathcal T}_*,h_*)$ be the torsion part of the $\Lambda$-module $(H_*(\mathbb F,\mathbb C),h_*)$. Then:
(i) \ i$m(s_*)={\mathcal T}_*$.
(ii) Therefore, for any loop
$\gamma:S^1\to \mathbb C\setminus B_f$ with $\ell:=\ell(\gamma)$ one has the following epimorphism of $\Lambda$-modules: $$s_*:(H_*(F,\mathbb C), T_{\gamma,*})\to ({\mathcal T}_*,h_*^\ell).$$
(iii) In particular, $\Delta((H_q(M_X^c,\mathbb C),h_q^\ell)(t)$ divides $\Delta_{\gamma,q}(t) $ for any $q\geq 0$ (cf. with \ref{ellp}).
(iv) Part (iii) applied for $T_{0,*}$ implies that the $(t-a)$-torsion of $H_*(M_X^c,\mathbb C)$ is zero if $a$ is not a root of unity.}
\begin{proof} Notice that part (i) and \ref{n1}($\gamma$) imply (ii), hence all the others as well. Notice that the statement of (i) is independent of the choice of any loop. In order to prove (i), we take a special loop, namely the oriented boundary of $D_0$. Then using the notations and the construction of \ref{con}, we deduce that $H_*(\mathbb F,F)=\oplus_{\bar{b}} H_*(\mathbb Z\times (T_{\bar{b}},F))$; in particular this is a free $\Lambda$-module. Then the result follows from the homological exact sequence of the pair $(\mathbb F,F)$ (considered as a sequence of $\Lambda$-modules). \end{proof}
\subsection{Remark}\label{ellp} If ${\mathcal M}=(H,h)$ is a torsion $\Lambda$-module with Alexander polynomial $\Delta(H,h)(t)$ (i.e., if $h$ acts on $H$ with characteristic polynomial $\Delta(H,h)(t)$), then for any $\ell\geq 0$ the polynomial $\Delta(H,h)(t)$ determines $\Delta(H,h^\ell)(t)$ as follows. For any polynomial $P(t)=\prod_a(t-a)^{n_a}$ ($n_a\in \mathbb N$) write $P(t)^{(\ell)}:=\prod_a(t-a^\ell)^{n_a}$. Then $\Delta(H,h^\ell)(t)=\Delta(H,h)(t)^{(\ell)}$.
Now, we will apply our general results for h-good polynomials. The next corollary follows directly from \ref{con} and \ref{3.2}.
\subsection{Corollary}\label{n4} {\em Assume that $f$ is a h-good polynomial, $0\in B_f$, and take a smooth loop $\gamma:S^1\to \mathbb C\setminus B_f$ with $\ell=\ell(\gamma)$. Then there is an epimorphism of $\Lambda_\mathbb Z$-modules $$s_n:(H_n(F), T_{\gamma,n})\to (\pi_n(M_X),h_n^{\ell}).$$ In particular, $\Delta(\pi_n(M_X))(t)^{(\ell)}$ divides $\Delta_{\gamma,n}(t) $.}
\subsection{The main examples. The monodromy around the origin and at infinity}\label{n5}
To any polynomial $f$ (and to a distinguished atypical value $0\in B_f$)
one can associate two distinguished monodromy operators:
(i) the local monodromy of $f$ at 0, namely $T_{0,*}: H_*(F) \to H_*(F)$ provided by a loop $\gamma$ going around the bifurcation point 0 on the boundary of a small disc $D_0$ containing no other bifurcation points inside (with positive orientation);
(ii) the monodromy at infinity of $f$, namely $T_{\infty,*}: H_*(F) \to H_*(F)$ provided by a loop $\gamma$ going around the boundary of a large disc $D_{\infty}$ containing all the bifurcation points $B_f$ inside.
They provide two (Alexander) $\Lambda$-module structures on $H_*(F,\mathbb C)$ denoted by $A_*(f,0)$), respectively $A_*(f,\infty)$). The corresponding Alexander polynomials, or equivalently, the characteristic
polynomials of the monodromy operators $T_{0,*}$ and $T_{\infty,*}$, are denoted by $\Delta(T_{0,*})$ resp. $\Delta(T_{\infty,*})$. Note that the Alexander module $A_*(f,0)$ (resp. $A_*(f,\infty)$) encodes exactly the Jordan structure of $T_{0,*}$ (resp. $T_{\infty.*}$) and that this Jordan structure was studied in several papers, e.g. [ACD], [D2], [DN1-2], [DS], [Do], [GN1-3], [Sa].
Obviously, in both cases $\ell=1$. Therefore, \ref{n7} guarantees that for any $f$ with $F$ connected, and for any $q\geq 0$, the Alexander polynomial \begin{equation*} \mbox{ $\Delta ( H_q(M_X^c, \mathbb C))$ divides the characteristic polynomials $\Delta (T_{0,q})$ and $\Delta (T_{\infty,q})$.} \tag{1} \end{equation*} This is a generalization of [Li2] (4.3) to arbitrary polynomials and $q$. Let us explain more precisely the relation between this divisibility result (1)
and the divisibility results in [Li2].
First consider the local monodromy $T_{0,*}$. In [Li2], $X$ has only isolated singularities including at infinity. In that case only the case $q=n$ is relevant (cf. \ref{3.6}). It is known that by the localization of the monodromy (see e.g. [NN]) one has \begin{equation*}
\Delta (T_{0,n})(t)=(t-1)^k\cdot \prod \Delta _i(t), \tag{2} \end{equation*} where $\Delta _i$ are the characteristic polynomials of the local monodromies associated to the isolated singularities on $V$, and $k =b_n(F) - \sum deg(\Delta _i)$. In fact,
the singularities at infinity (i.e. on $V \cap H$)
should be treated slightly different, as explained in [Li2], [LiT]. Moreover, in general $k \geq 0$, and $k=0$ iff $B_f=\{0\}$. Note that (1) (for $q=n$) and (2) give a similar result to Theorem (4.3)
in [Li2], yielding in addition a precise value for $k$.
The discussion for the monodromy at infinity $T_{\infty,*}$ is more involved. It was shown by Neumann and Norbury [NN] that the total space of the fibration ($*$)\ $ f: f^{-1}(S^1_r) \to S^1_r$ for $r\gg 0$ (which provides $T_{\infty,*}$) can be embedded in a natural way as an open subset of $S^{2n+1}\setminus f^{-1}(0)$, where $S^{2n+1} $ is a large sphere in $\mathbb C^{n+1}$. Moreover, it was shown in [NZ] that for an M-tame polynomial this fibration ($*$) is equivalent to the Milnor fibration at infinity
$\phi : S^{2n+1} \setminus f^{-1}(0) \to S^1$, $\phi (x)=f(x)/|f(x)|$.
Note that in general $S^{2n+1} \setminus f^{-1}(0)$ is not the total space of a fibration over the circle, or, even when it is, it may happen that the corresponding fiber is not $F$,
see the case of semi-tame polynomials in P\u aunescu-Zaharia [PZ].
In [Li2], Libgober considers an infinite cyclic covering $U_{\infty}$ of the knot complement $S^{2n+1} \setminus f^{-1}(0)$ and takes the associated Alexander module $H_n(U_{\infty},\mathbb C)$ as the Alexander module at infinity for $f$. From our discussion above it seems that, in general, one cannot hope to identify easily the structure of this module $H_n(U_{\infty},\mathbb C)$. However, in the case of $M$-tame polynomials, the module $H_n(U_{\infty},\mathbb C)$ is exactly our $A_n(f,\infty)$.
\subsection{Example}\label{4.5} Recall the situation described in \ref{defect}. Namely, let $f$ be a polynomial such that there exists a system of weights $w$ with the top degree form $f_e$ defining an isolated singularity at the origin. The monodromy at infinity of such a polynomial coincides to the monodromy of the singularity $f_e=0$, in particular $T_{\infty,n}$ is semisimple and all the eigenvalues are $e$-roots of unity. It follows that in the second exact sequence in \ref{remarks}(4), for $k=n$, the first morphism is trivial. Hence (for $n>1$) $$\pi _n(M_X) \otimes \mathbb C = H_n(\mathbb F,\mathbb C) =H_n(M_{X,e},\mathbb C)=H_n(F',\mathbb C)$$ are isomorphic $\Lambda$-modules. This results should be compared to Corollary (4.9) in [Li2].
As a concrete example, let $f:\mathbb C^4 \to \mathbb C$ be the tame polynomial considered in Example \ref{2.10}. The above discussion and \ref{n5}(1) gives: $$\pi _3(M_X) \otimes \mathbb C = H_3(\mathbb F,\mathbb C) =H_3(M_{X,3},\mathbb C)=H_3(F',\mathbb C)= (\Lambda /(t-1))^5.$$
\subsection{Remarks}\label{4.6} If one wants to determine the homology groups of $M_X$, either one needs some information about the ``tubes'' $T_{\bar{b}}$ for $\bar{b}\in B_f\setminus \{0\}$, or (using \ref{2.7}) one needs to know the behaviour at infinity of $X$; both rather subtle problems. Therefore, it is rather surprising that, in some cases, all these information is carried by only one monodromy operator $T_{0,n}$.
Here we present the case of h-good polynomials: we describe completely $H_*(M_X,\mathbb Z)$ in terms of $T_{0,n}$.
Let ${\mathcal V_0} \subset H_n(F,\mathbb Z)$ be the subgroup of vanishing cycles at 0 corresponding to a choice of a star in $\mathbb C$ as in [DN1]. It follows as in [{\em loc.\,cit.}] that the morphism $T_{0,n}-Id$ induces a ``variation'' morphism $V: H_n(F,\mathbb Z) \to {\mathcal V}_0$ and, by restriction to ${\mathcal V}_0$, a morphism $V_0: {\mathcal V}_0 \to {\mathcal V}_0$. Using the definition of h-good polynomials, the connectivity \ref{p23} of $F$, and the Wang sequence associated to $T_0^*$ (cf. also with \ref{con2}), one can prove the following.
The homology groups of $M_X$ are trivial except possibly for:
\noindent {\bf Case $n>1$.} \
(i) $H_0(M_X,\mathbb Z)=H_1(M_X,\mathbb Z)=\mathbb Z$,
(ii) $H_{n+1}(M_X,\mathbb Z)$ is $\mathbb Z$-torsion free of rank $b_n(F)-\mbox{rank}(V)$ and
(iii) $ H_{n}(M_X,\mathbb Z)=\mbox{coker}(V)$.
\noindent {\bf Case $n=1$.}\
(i') $H_0(M_X,\mathbb Z)=\mathbb Z$,
(ii') $H_{2}(M_X,\mathbb Z)$ is $\mathbb Z$-torsion free of rank $b_n(F)-\mbox{rank}(V)$,
(iii') $ H_{1}(M_X,\mathbb Z)=\mbox{coker}(V) +\mathbb Z$, a $\mathbb Z$-torsion free group of rank equal to the number of irreducible components of $X$.
Note that for $n>1$, $ H_{n}(M_X,\mathbb Z)$ is a finite group if $V_0$ is injective. This happens exactly when, in the notation from \ref{n5}(2), one has $\prod \Delta _i(1) \ne 0$. Moreover, the epimorphism $\mbox{coker}(V_0) \to \mbox{coker}(V)$ implies that the order
$|H_{n}(M_X,\mathbb Z)|$ divides $|\prod \Delta _i(1)|=|\mbox{coker}(V_0)|$.
Let $g:\mathbb C^{n+1} \times \mathbb C \to \mathbb C$ be the $d$-suspension of the polynomial $f$, namely $g(x,y)=f(x)-y^d$. Let $Y=g^{-1}(0)$. Then writing the Gysin sequences in homology associated to a smooth divisor $D$ in a complex manifold $Z$ for the pairs $(Z,D)=(M_Y,M_X), \ (\mathbb C^{n+1} \times \mathbb C^*, M_{X,d})$ and resp. $(M_X \times \mathbb C^*, graph(f))$,
and comparing the associated morphisms, we get for all $q>0$ the exact sequence \begin{equation*}
0 \to H_{q}(M_X,\mathbb Z) \to H_{q}(M_{X,d},\mathbb Z)\to H_{q+1}(M_Y,\mathbb Z) \to 0. \tag{3.q} \end{equation*} The exact sequence $(3.n+1)$ is split since the last group in it is free according to (ii) above. The exact sequence $(3.n)$ is not split, as can be seen in the case $n=1,f=x_1x_2$, $d=3$ when we get $0 \to \mathbb Z^2 \to \mathbb Z^2 \to \mathbb Z/3\mathbb Z \to 0$. This example shows the difficulty of the question \ref{2.12}.
Finally, assume that $n>1$ and $\prod_i \Delta _i(\alpha ^k) \ne 0$ for $\alpha =exp(2\pi i/d)$ and for any $k \in \mathbb Z$ (cf. \ref{n5}(2)). Then one has:
(a) all the groups $ H_n(M_X,\mathbb Z),\ H_n(M_{X,d},\mathbb Z)$ and $ H_{n+1}(M_Y,\mathbb Z)$ are finite; and
(b) the order $|H_n(M_{X,d},\mathbb Z)|$ divides the product $|\prod _{1\leq k\leq d}
\prod_i \Delta _i(\alpha ^k)|$.
The proof of these claims follows from the exact sequence (3.q)
and the property (iii) above once we know how to compute the variation associated to the special fiber $Y$. This, in turn,
is explained in [DN2]. Note that the claim (b) is similar to Theorem 3 in [Li0].
\section{ Relations to Monodromy Representation}
\subsection{}\label{r1} The results of the previous section already suggest (see e.g. \ref{n3}) that one can obtain finer results about the Alexander modules if one takes the whole monodromy representation instead of individual monodromy operators.
The main message of this section is that from the monodromy representation of $f$ one can construct a universal Alexander module which, in some sense, dominates all the Alexander modules associated with (all) the fibers of $f$.
Since the case of h-good polynomials with all the involved numerical invariants (cf. \ref{5.1} and \ref{r2}) represents a special interest, we start our detailed discussion with this case. But, thanks to the general results of the previous sections, the next constructions and factorization phenomenon described in the h-good polynomial case, can be repeated word by word in the general case. The general result will be formulated at the end of the section in \ref{gen}.
We start with a h-good polynomial. With the notation of \S 2, let $S=\mathbb C \setminus B_f$, $E=f^{-1}(S)$
and $g=|B_f|$. Then the locally trivial fibration $f:E \to S$ induces a monodromy representation $\rho: G \to Aut ({\mathcal H})$, where $G=\pi _1(S,b_0)$ is a free group on $g$ generators, $b_0\in S$ is a base point, and ${\mathcal H}=H_n(F,\mathbb Z)$ with $F=f^{-1}(b_0)$. For each $b\in B_f$ write $F_b=f^{-1}(b)$. Let $\gamma _i$ denote an elementary loop around $b_i \in B_f$ and $m_i=\rho ( \gamma _i)$ be the corresponding monodromy operators. With a natural choice for $\{\gamma _i\}_i$ one has $m_1 \cdot m_2 \cdot ... \cdot m_g =T_{\infty,n}$, see [DN1].
For any $H$-module ${\mathcal M}$ of a group $H$, we
denote by ${\mathcal M}_H$ the group of coinvariants, namely the quotient of ${\mathcal M}$ by the subgroup spanned by all elements $h \cdot m -m$ for $h \in H$ and $m \in {\mathcal M}$, see Brown [Br]. We denote by $b_k^c(Y)$ the $\mathbb C$-dimension of the $k^{th}$-cohomology space $H^k_c(Y,\mathbb C)$ of $Y$ with compact supports.
We start by recalling how the $G$-module ${\mathcal H}$ determines the homology of the space $E$.
\subsection{Proposition}\label{5.1} {\it The reduced homology groups ${\tilde H}_k(E,\mathbb Z)$ are trivial except at most for $k=1$, $k=n$ and $k=n+1$. For these values of $k$ one has the following.
(i) For $n=1$ one has $H_2(E,\mathbb Z)=H_1(G,{\mathcal H})$ and an exact sequence of groups $$ 0 \to {\mathcal H}_G \to H_1(E,\mathbb Z) \to \mathbb Z^g \to 0.$$ In particular, ${\mathcal H}_G$ is a free $\mathbb Z$-module with} $\mbox{rank} ({\mathcal H}_G) = \sum _{b \in B_f}(n(F_b)-1)$, {\em where $n(Y)$ denotes the number of irreducible components of a curve $Y$.
(ii) For $n>1$ one has $H_1(E,\mathbb Z)=\mathbb Z^g$, $H_n(E,\mathbb Z)=H_0(G,{\mathcal H})={\mathcal H}_G$ and $H_{n+1}(E,\mathbb Z)=H_1(G,{\mathcal H})$. In particular, } $\mbox{rank} ({\mathcal H}_G)= \sum _{b \in B_f} b_{n+1}^c(F_b)= \sum _{b \in B_f} b_{n+1}(T_b,\partial T_b)$.
\noindent {\em Proof.} The result follows from the Leray spectral sequence in homology of the fibration $F \to E \to S$ and basic facts on group homology, see Brown [Br]. The claim about the rank of ${\mathcal H}_G$ follows from the long exact sequence $$\hspace{2.5cm}... \to H^k_c(E) \to H^k_c(\mathbb C^{n+1}) \to H^k_c(\cup F_b) \to H^{k+1}_c(E) \to ...$$
Let $H=[G,G]=G'$ be the commutator of $G$ and $S' \to S$ be the corresponding covering space. Let $f':E' \to S'$ be the fibration (with fiber $F$) obtained from the fibration $f:E \to S$ by pull-back. Then the monodromy of the fibration $f'$ corresponds exactly to the $H$-module ${\mathcal H}$ obtained by restriction of $\rho$ to $H$. On the other hand, we can regard $E' \to E$ as being the covering space corresponding to the kernel of the composition $\pi _1(E) \to \pi _1(S)=G \to G/H=\mathbb Z^g \to 0$. It follows that the deck transformation group of $E' \to E$ is $\mathbb Z^g$ and hence we can regard $H_n(E',R)$ as a $\Lambda _{R,g}$-module, where $\Lambda _{R,g} =R[\mathbb Z^g]$ is a Laurent polynomial ring in $g$ indeterminates $t_1,...,t_g$. As before, when $R=\mathbb C$ we simply write $\Lambda _{g}$.
To state the result similar to \ref{r1} for the fibration $f':E' \to S'$, note that $S'=\mathbb R$ for $g=1$ and $S'$ is homotopy equivalent to a bouquet of infinitely many $S^1$'s for $g>1$.
\subsection{Proposition}\label{r2} {\it The reduced homology groups ${\tilde H}_k(E',\mathbb Z)$ are trivial except at most for $k=1$, $k=n$ and $k=n+1$. For these values of $k$ one has the following.
(i) For $n=1$ one has $H_2(E',\mathbb Z)=H_1(H,{\mathcal H})$ and an exact sequence of groups $$ 0 \to {\mathcal H}_H \to H_1(E',\mathbb Z) \to H_1(H,\mathbb Z) \to 0.$$
(ii) For $n>1$ one has $H_1(E',\mathbb Z)=0$ for $g=1$ and $H_1(E',\mathbb Z)=H_1(H,\mathbb Z)$ for $g>1$, $H_n(E',\mathbb Z)=H_0(H,{\mathcal H})={\mathcal H}_H$ and $H_{n+1}(E',\mathbb Z)=H_1(H,{\mathcal H})$.}
\noindent Using the description of the $K(H,1)$ and of the associated chain complex given in [Li4], (1.2.2.1), it follows that $H_1(H,\mathbb Z)=G'/G''$ is a submodule of $\Lambda _{\mathbb Z,g}^g$ and hence $H_1(H,\mathbb Z)$ is $\Lambda_{\mathbb Z,g}$-torsion free. This implies that in both cases (i) and (ii) in \ref{r2}, we have ${\mathcal H}_H =\mbox{Tors}(H_n(E',\mathbb Z))$ (as a $\Lambda_{\mathbb Z,g}$-module).
In the sequel we denote the $\Lambda_{\mathbb Z,g}$-module ${\mathcal H}_H$
by $M(f)$, and we call it the {\it global Alexander module of the polynomial $f$}.
\subsection{Remark}\label{r3} The global Alexander module of the polynomial $f$ can be regarded as a commutative version of the monodromy representation $\rho$. Notice also that using Brown [Br], Exercise 3, p.35, it follows that $M(f)_{\mathbb Z^g}={\mathcal H}_G$.
Assume now that $0 \in B_f$. We will construct a new $\Lambda_\mathbb Z$-module $M(f,0)$ out of the monodromy representation. The inclusion $S \to \mathbb C^*$ at $\pi_1$-level gives rise to a projection $p_0: G \to \mathbb Z$. Let $K_0$ denote the kernel of this projection. Then $H \subset K_0 $ and hence we have a tower of covering spaces $S' \to S^0 \to S$. Let $E^0 \to S^0$ be the fibration induced from $E \to S$ by pull-back. In this way we get a second tower of covering spaces, namely $E' \to E^0 \to E$.
We have, exactly as in the proof of \ref{r1}, the following isomorphisms of $\Lambda$-modules: $$\mbox{Tors}(H_n(E^0,\mathbb Z))={\mathcal H}_{K_0}=({\mathcal H}_H)_{K_0/H}.$$ Here $K_0/H =\mathbb Z^{g-1}$ with generators corresponding to elementary loops in $\mathbb C$ around points in $B_f$ different from 0. Moreover the $\mathbb Z=G/K_0$-action on $\mbox{Tors}(H_n(E^0,\mathbb Z))$ is induced by the monodromy operator $T_{0,n}$.
We denote this $\Lambda_\mathbb Z$-module ${\mathcal H}_{K_0}$ by $M(f,0)$,
and we call it the {\it local Alexander module of $f$ at 0.} (Clearly, similar local Alexander module can be defined for any $b\in B_f$.)
The above isomorphisms show that the local Alexander module $M(f,0)$ of $f$ at 0 can be computed from the global Alexander module $M(f)$ of $f$. In this sense, the module $M(f)$ is universal, i.e. contains all the information about the local Alexander modules associated to all the special fibers of $f$.
The usefulness of this new Alexander module comes from the fact that it can be calculated using the monodromy representation and gives another approximation for the Alexander module $\pi_n(M_X)=H_n(\mathbb F,\mathbb Z)$. Before we formulate this statement, let us reinterpret \ref{n2} and \ref{n3}.
\subsection{Theorem \ref{n2} revisited}\label{r4} Let us explain the meaning of \ref{n2} in the language of the present section. Clearly, ${\mathcal H}= H_n(F,\mathbb Z)$ is a $G$-module, and $H_n(\mathbb F,\mathbb Z)$ has a cyclic action generated by $h_n$. The map $p_0:G\to \mathbb Z$ can be identified with $[\gamma] \mapsto \ell(\gamma)$ considered in \ref{n2}. Therefore, if we consider both ${\mathcal H}$ and $H_n(\mathbb F,\mathbb Z)$ as $G$-modules (the last one via $p_0$), then \ref{n2} says that $s_n$ is a morphism of $G$-modules.
In other words, the complicated monodromy representation (i.e. the $G$-module ${\mathcal H}$), when it is mapped via $s_n$ into $H_n(\mathbb F,\mathbb Z)$, it is collapsed into a modest cyclic action. Since the action in the target is abelian, this already shows that $s_n:{\mathcal H}\to H_n(\mathbb F,\mathbb Z)$ has a factorization through ${\mathcal H}_H=M(f).$
Notice that $K_0=\mbox{ker}(p_0)$ constitutes of loops (with base points) $\gamma$ with $\ell(\gamma)=0$. Corollary \ref{n3} applied for such a loop $\gamma$
and for the trivial loop guarantees that $\rho(\gamma)m-m\in \mbox{ker}(s_n)$ for any $m$. In particular, $s_n:{\mathcal H}\to H_n(\mathbb F,\mathbb Z)$ has the following factorization of $G$-modules: \begin{equation*} {\mathcal H}\to{\mathcal H}_H\to {\mathcal H}_{K_0}\to H_n(\mathbb F,\mathbb Z). \tag{1} \end{equation*}
\subsection{Corollary}\label{r6} {\em Assume that $f:\mathbb C^{n+1} \to \mathbb C$ is a h-good polynomial. Then $s_n:H_n(F,\mathbb Z)\to H_n(\mathbb F,\mathbb Z)$ induces an epimorphism $M(f,0) \to \pi_n(M_X)$ of $\Lambda _{\mathbb Z}$-modules.} \begin{proof} Since $s_n$ is epimorphism (cf. \ref{n4}), the result follows from (1) above. \end{proof}
\subsection{Remark}\label{r8} Notice that any $\gamma$ with $\ell(\gamma)=+1$ induces the same operator $\overline{\rho}([\gamma])$ acting on ${\mathcal H}_{K_0}$; and this operator is the positive generator of the cyclic action on ${\mathcal H}_{K_0}$. E.g., one can take $\overline{T}_{0,n}$, or $\overline{T}_{\infty,n}$ as well, depending which one is easier to compute. We write $M(f,0)=({\mathcal H}_{K_0},\overline{T}_{0,n})$. Then, for an arbitrary
$[\gamma]\in G$ with $\ell:=\ell(\gamma)=p_0([\gamma])$ one has the $\Lambda_\mathbb Z $-module epimorphisms: $$(H_n(F,\mathbb Z),\rho(\gamma)) \to ({\mathcal H}_{K_0},\overline{T}_0^\ell) \to (\pi_n(M_X),h_n^\ell). $$ Evidently, this provides the divisibilities of the corresponding Alexander (or characteristic) polynomials.
\subsection{Example}\label{r7} Assume we are in the situation of Example \ref{2.3} with $n>0$ even. Then it is easy to see that ${\mathcal H}_G=\mathbb Z/2\mathbb Z$ and ${\mathcal H}={\mathcal H}_H=M(f)=M(f,0)=\pi_n(M_X)=\Lambda _{\mathbb Z}/(t-1)$.
Using the general result \ref{n2} and \ref{n7}, one can verify easily that the above factorization (1) is valid for arbitrary polynomials as well.
\subsection{Theorem}\label{gen} {\em Let $f$ be an arbitrary polynomial with $F$ connected. For any $q\geq 0$, consider ${\mathcal H}_q:=H_q(F,\mathbb Z)$ as a $G=\pi_1(S,b_0)$-module provided by the monodromy representation. Define the global Alexander $\Lambda_{\mathbb Z,g}$-module by $({\mathcal H}_q)_H$, and the local Alexander module associated with the bifurcation point $0\in B_f$ by $({\mathcal H}_q)_{K_0}$. If we consider $H_q(\mathbb F,\mathbb Z)$ as a $G$-module via $p_0$, then $s_q:{\mathcal H}_q\to H_q(\mathbb F,\mathbb Z)$ has the following factorization of $G$-modules: \begin{equation*} s_q: {\mathcal H}_q\to({\mathcal H}_q)_H\to ({\mathcal H}_q)_{K_0}\to H_q(\mathbb F,\mathbb Z). \end{equation*} If one tensor this tower by $\mathbb C$, then the last term $H_q(\mathbb F,\mathbb C)$ can be replaces by ${\mathcal T}_q$, being the image of $s_q$. }
\section{An Example}
Let $f:\mathbb C^2 \to \mathbb C$ be the polynomial $f=x+x^2y^2+x^2y^3$. Then $B_f=\{b_1,b_2\}$ with $b_1=-27/16$ and $b_2=0$. The fiber $F_{b_1}$ is irreducible, has a node as a singularity and is regular at infinity. On the other hand, the fiber $F_0=F_{b_2}$ is smooth, has two irreducible components, one a copy of $\mathbb C$ the other $\mathbb C \setminus \{0,-1\}$,
and has a singularity at infinity with a Milnor number equal to 3. It follows that $b_1(F)=4$ and the Jordan normal form for the monodromy operators $m_1, m_2$ and $ T_{\infty}$ was obtained in [BM]:
$$m_1 \approx \begin{pmatrix}
1&1&0&0 \\ 0& 1&0&0 \\ 0&0& 1 &0 \\ 0& 0& 0& 1 \end{pmatrix}
\ \ m_2 \approx \begin{pmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&j&0 \\ 0&0&0&j^2 \end{pmatrix} \ \ T_{\infty} \approx \begin{pmatrix} -1&1&0&0 \\ 0& -1&0&0 \\ 0&0& 1 &0 \\
0& 0& 0& 1 \end{pmatrix}$$ with $j=exp(2 \pi i/3)$.
Let $g:\mathbb C^2 \to \mathbb C$ be given by $g=u+u^2v$. Then $B_g=\{0\}$, the generic fiber is homotopy equivalent to $S^1$ and the corresponding monodromy $m$ is the identity. Consider the polynomial $h:\mathbb C^4 \to \mathbb C$ given by $h(x,y,u,v)=f(x,y)+g(u,v).$
It follows from [DN2] that $B_h \subset B_f$. Notice that $f$ and $g$ are not good, but both are h-good.
The fact that $h$ is also a h-good polynomial follows from [DN2], Corollary (4.4), which basically says that the ``Thom-Sebastiani sum'' of two h-good polynomials is h-good.
Our goal is to determine the various Alexander modules associated with $h$ and to its special fibers. Since the information we have on the monodromy representation of $f$ is over $\mathbb C$, we choose this coefficient ring.
The generic fiber of $h$ is the join $(\vee_4 S^1)*S^1$, hence it is $\vee_4S^3$. Since $m=Id$, [DN2] guarantees that the monodromy representation of $h$ can be identified to that of $f$. Using the above Jordan forms: \begin{equation*} A_3(h,b_1)=\Lambda/(t-1) \oplus \Lambda/(t-1)\oplus \Lambda/(t-1)^2.\tag{1} \end{equation*} \begin{equation*} A_3(h,0)=\Lambda/(t-1) \oplus \Lambda/(t-1)\oplus \Lambda/(t-j) \oplus \Lambda/(t-j^2); \tag{2} \end{equation*} \begin{equation*} A_3(h,\infty)=\Lambda/(t-1) \oplus \Lambda/(t-1)\oplus \Lambda/(t+1)^2; \tag{3} \end{equation*} Now, we consider the space $M_X$ corresponding to the two special fibers. First, let $X=h^{-1}(0)$. Then $X$ is smooth and applying Theorem (4.7), (ii) in [DN2], we get that $X$ has the homotopy type of $S^2 \vee S^3 \vee S^3.$ It follows that $b_3(M_X)=b_2(X)=1$. Using \ref{n5}(1) and \ref{3.6}(iv) one has $N(1,3)=1$, hence: \begin{equation*} \pi _3(M_X) \otimes \mathbb C = H_3(\mathbb F,\mathbb C) =H_3(M^c_{X},\mathbb C)= \Lambda /(t-1). \tag{4} \end{equation*} Next, let $X=h^{-1}(b_1)$. Note that $X$ is again smooth but we can no longer apply Theorem (4.7) in [DN2] since $F_{b_1}$ is not smooth. Using the equality $ \chi (X) = \chi _c (X)$ we get $b_3^c(X) -b_4^c(X)=3$. Moreover, it is known that $b_3^c(X)= \dim\mbox{coker} (m_1-1)=3$, see [ACD] or [DN1]. It follows that $b_4^c(X)=0$. The exact sequence $$... \to H^k_c(M_X) \to H^k_c(\mathbb C^{n+1}) \to H^k_c(X) \to H^{k+1}_c(M_X) \to ...$$
and duality for $M_X$ imply that $b_3(M_X)=b_4^c(X)=0$. Moreover, from (1) follows that only $(t-1)$-torsion is possible. Hence, via \ref{3.6}(iv), $b_3(M_X)=0$ implies that \begin{equation*} \pi _3(M_X) \otimes \mathbb C = H_3(\mathbb F,\mathbb C) =H_3(M^c_{X},\mathbb C)= 0. \tag{5} \end{equation*} Our next aim is to compute the global Alexander module $M(h)$. This is done by using the partial information we have on the monodromy representation $\rho: G \to Aut ({\mathcal H})$, where $G$ is a free group on two generators, and ${\mathcal H}$
is the third homology of the generic fiber of $h$ with $\mathbb C$ coefficients (i.e., to simplify notation, we denote by ${\mathcal H}$ the complexification ${\mathcal H} \otimes _{\mathbb Z} \mathbb C$). In terms of a special basis $e_1,e_2,e_3,e_4$ for ${\mathcal H}$ as in [DN1], (2.5), we can write the monodromy operators $m_1$ and $m_2$ of $h$ in the form $$m_1 = \begin{pmatrix} 1&a&b&c \\ 0& 1&0&0 \\ 0&0& 1 &0 \\ 0& 0& 0& 1 \end{pmatrix}
\ \ m_2= \begin{pmatrix} 1&0&0&0 \\ \alpha &1&0&0 \\ \beta &0&j&0 \\
\gamma &0&0&j^2 \end{pmatrix}. $$ Checking that $m_1m_2$ is conjugate to the Jordan normal form for $m_{\infty}$ given above implies $a \alpha =0$ and $bc \not= 0$. By an obvious change of base we may take $b=c=1$ and then $\beta$ and $\gamma$ are determined by the equations $\beta +\gamma =-1, \beta j + \gamma j^2=2$.
Let $C=[m_1,m_2]=m_1m_2m_1^{-1}m_2^{-1}$. Then $C \in H$ and a direct computation shows that $v_1=(C-Id)(e_1)=-(\alpha e_2 +\beta e_3 +\gamma e_4)$. Let ${\mathcal H}_0$ be the vector subspace in ${\mathcal H}$ spanned by all the vectors $h(v)-v$ for $h \in H$ and $v \in {\mathcal H}$. It follows that
(i) $v_1 \in {\mathcal H}_0$ and
(ii) ${\mathcal H}_0$ is a $G$-invariant subspace of ${\mathcal H}$ (this property being always true).
It follows that we have to discuss two cases.
Case 1. ($\alpha \ne 0$) Then the vectors $v_1$, $m_2v_1$ and $m_2^2v_1$ span the same subspace in ${\mathcal H}$ as the vectors $e_2,e_3,e_4$. Moreover $m_1e_3=e_1+e_3 \in {\mathcal H}_0$. Therefore ${\mathcal H}={\mathcal H}_0$ and hence ${\mathcal H}_H={\mathcal H}/{\mathcal H}_0=0$. But this is a contradiction since we have epimorphisms $M(h)={\mathcal H}_H \to M(h,0) \to H_3(M^c_{X},\mathbb C)= \Lambda /(t-1)$ by \ref{r6} and (4) above.
Case 2. ($\alpha = 0$) As above one shows that ${\mathcal H}_0$ is spanned by $e_1,e_3, e_4$ and hence ${\mathcal H}_H=\mathbb C$ with a trivial $\mathbb Z^2$-action. This implies the following.
\subsection{Proposition}\label{6.3} {\it For the polynomial $h:\mathbb C^4 \to \mathbb C$ described above one has the following Alexander modules.
(i) For the fiber $X=h^{-1}(0)$ one has $\pi _3(M_X) \otimes \mathbb C =H_3(M^c_{X},\mathbb C)= M(f,0)=\Lambda /(t-1).$
(ii) For the fiber $Y=h^{-1}(b_1)$ one has $\pi _3(M_Y) \otimes \mathbb C =H_3(M^c_{Y},\mathbb C)= 0.$ Moreover in this case $M(f,b_1)=\Lambda /(t-1) \ne H_3(M^c_{Y},\mathbb C)= 0$.
(iii) The global Alexander module $M(h)$ is isomorphic to $\Lambda _2/(t_1-1,t_2-1)$.}
\noindent Note that:
(i) we succeeded to determine the above data in spite of the fact that the monodromy representation of $h$ (over $\mathbb C$) is not completely determined (the value of $a$ is unknown);
(ii) in the case of $Y$, we have $M(h,b_1) \ne H_3(M_Y^c,\mathbb C)$, in particular we cannot expect isomorphism in \ref{r6}. Nevertheless, the approximation of $H_3(M_Y^c,\mathbb C)=0$ given by $M(h,b_1)$ is better than that given by $A_3(h,b_1)$ since $ \dim_{\mathbb C}M(h,b_1)=1$ while $ \dim_{\mathbb C}A_3(h,b_1)=4.$
Laboratoire de Math\'ematiques Pures de Bordeaux
Universit\'e Bordeaux I
33405 Talence Cedex, FRANCE
email: [email protected]
Department of Mathematics,
Ohio State University,
Columbus, Ohio 43210, USA
email: [email protected]
\end{document}
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\begin{document}
\title{Commutative Hopf structures over a loop} \author{Hua-Lin Huang} \address{School of Mathematics, Shandong University, Jinan 250100, China} \email{[email protected]} \author{Gongxiang Liu} \address{Department of Mathematics, Nanjing University, Nanjing 210093, China} \email{[email protected]} \author{Yu Ye} \address{Department of Mathematics, University of Science and Technology of China,
Hefei 230026, China} \email{[email protected]} \date{} \maketitle \begin{abstract} Let $k$ be an algebraically closed field of characteristic $p>0$. For a loop $\circlearrowleft$, denote its path coalgebra by $k\circlearrowleft$. In this paper, all the finite-dimensional commutative Hopf algebras over the sub coalgebras of $k\circlearrowleft$ are given. As a direct consequence, all the commutative infinitesimal groups $\mathcal{G}$ with dim$_{k}$Lie$(\mathcal{G})=1$ are classified. \vskip 5pt
\noindent{\bf Keywords} \ \ Path coalgebra, Unipotent group, Frobenius map \\ \noindent{\bf 2000 MR Subject Classification} \ \ 16W30, 14L15 \end{abstract}
\section{introduction}
This paper is concerned with the quiver realization of finite-dimensional cocommutative Hopf algebras. As is well-known, any such algebra can be viewed as the group algebra of a finite algebraic $k$-group $\mathcal{G}$. Considerable attention has been received by these algebraic groups.
Quivers are oriented diagrams consisting of vertices and arrows \cite{ARS}. Due to the well-known theorem of Gabriel given in the early 1970s, these combinatorial stuffs make the abstract elementary algebras and their representations visible. This point of view has since then played a central role and is generally accepted as the starting point in the modern representation theory of associative algebras. Naturally there is a strong desire to apply this handy quiver tool to other algebraic structures. Such idea for Hopf algebras appeared explicitly in \cite{Cil,Cil1,CR,GS} and was showed to be very effective in dealing with the structures of finite-dimensional pointed (or dually, basic) Hopf algebras when the characteristic of the base field is 0 \cite{CR,CHYZ,GL,L,HL,FP}.
Comparing to the characteristic 0 case, there is hardly any work dealing with the positive characteristic case by using quiver methods, see however a recent work of Cibils, Lauve and Witherspoon \cite{clw}. One main difficulty in the positive characteristic case is that general pointed Hopf algebras are not generated by group-likes and skew-primitive elements. While in the characteristic 0 case, the well-known Andruskiewitsch-Schneider Conjecture \cite{AS3} claims that all finite-dimensional pointed Hopf algebras are indeed generated by their group-likes and skew-primitive elements.
This paper can be considered as our first try to apply quiver methods to the category of pointed Hopf algebras over an algebraically closed field $k$ with characteristic $p>0$, especially to finite-dimensional cocommutative Hopf algebras over $k$ or equivalently to the category of finite algebraic $k$-groups. One can show that the connected component of a finite-dimensional cocommutative Hopf algebra can be embedded into the path coalgebra of a multi-loop quiver (see Corollary 2.2). So as the first step, one can analyze the minimal case, that is, Hopf structures over the path coalgebra of the one-loop quiver. This is exactly what we do in this paper. The main result of the paper is the following theorem (see Theorem 5.1). \begin{theorem} Let $n\in \mathbb{N}^{+}$, any commutative Hopf structure $H$ over $k\circlearrowleft_{p^{n}}$ is isomorphic to a $L(n,d)$ for some $d$. \end{theorem} See Section 5 for the definition of $L(n,d)$. As a direct consequence of this theorem, all commutative infinitesimal groups $\mathcal{G}$ with dim$_{k}$Lie$(\mathcal{G})=1$ are determined.
The paper is organized as follows. All needed knowledge about path coalgebras is summarized in Section 2. Moreover, the uniserial property of the Hopf structures over $k\circlearrowleft$ is also established in this section. For later use, all endomorphisms of the path coalgebra $k\circlearrowleft$ are given in Section 3. As a key step, we need to grasp all possible Hopf structures over $k\circlearrowleft_{p}$ at first and this task is finished in Section 4. In addition, we also show that the property of a Hopf structure over $k\circlearrowleft_{p^{n}}$ is almost determined by that of its restriction to $k\circlearrowleft_{p}$. Combining the work of Farnsteiner-R\"ohrle-Voigt on unipotent group of complexity 1 \cite{FRV}, the proof of Theorem 1.1 is given in Section 5 at last.
Throughout the paper we will be working over an algebraically closed field $k$ of characteristic $p>0$. We freely use the results, notations, and conventions of \cite{Mon}.
\section{Path coalgebras}
\subsection{} Given a quiver $Q=(Q_{0},Q_{1})$ with $Q_{0}$ the set of vertices and $Q_{1}$ the set of arrows, denote by $kQ$ the
$k$-space with basis the set of all paths in $Q$. Over $kQ$, there is a natural coalgebra structure defined as follows.
For
$\alpha \in Q_{1}$, let $s(\alpha)$ and $t(\alpha)$ denote
respectively the starting and ending vertex of $\alpha$. Then comultiplication $\Delta$ is given by
$$\Delta(p)=\alpha_{l}\cdots \alpha_{1}
\otimes s(\alpha_{1})+\sum_{i=1}^{l-1}\alpha_{l}\cdots \alpha_{i+1}
\otimes\alpha_{i}\cdots \alpha_{1}+ t(\alpha_{l})\otimes \alpha_{l}\cdots \alpha_{1}$$
for each path $p=\alpha_{l}\cdots \alpha_{1}$ with each $\alpha_{i}\in
Q_{1}$; and the counit $\varepsilon$ is defined to be $\varepsilon(p)=0$ for $l\geq 1$ and $1$ if $l=0$ ($l=0$ means $p$ is a vertex). This is a coradically graded pointed coalgebra and we also denote it by $kQ$. Like the path algebras case, the path coalgebras serve as the cofree pointed coalgebras. In fact, Chin and Montgomery showed the following result \cite{CM}:
\begin{lemma} Let $C$ be a pointed cocalgebra, then there exists a unique quiver $Q(C)$ such that $C$ can be embedded into the path coalgebra $kQ(C)$ as a large sub coalgebra.\end{lemma}
This unique quiver $Q(C)$ is called the \emph{dual Gabriel quiver} of $C.$ Here ``large" means that $C$ contains all group-like elements $Q(C)_{0}$ and all skew-primitive elements of $kQ(C)$. Note that the skew-primitive elements are indeed corresponding to paths of length 1, i.e., arrows. Now the following conclusion is clear.
\begin{corollary} Let $C$ be an irreducible cocommutative pointed coalgebra, then its dual Gabriel quiver $Q(C)$ has only one vertex. \end{corollary}
A natural question is when there is a Hopf structure on a path coalgebra. We will see not every quiver can serve as the dual Gabriel quiver of a pointed Hopf algebra and those do are called \emph{Hopf quivers} by Cibils and Rosso \cite{CR}. Recall that a \emph{ramification data} $r$ of a group $G$ is a positive central element of the group ring of $G$: let $\mathcal{C}$ be the set of conjugacy classes, $r =\Sigma _{C\in \;\mathcal{C}} r_{C}C$ is a formal sum with non-negative integer coefficients.
\begin{definition} Let $G$ be a group and $r$ a ramification data. The corresponding Hopf quiver $Q(G,r)$ has set of vertices the elements of $G$ and has $r_{C}$ arrows from $x$ to $cx$ for each $x \in G$ and $c\in C$. \end{definition}
One of the main results in \cite{CR} states that there is a graded Hopf algebra structure on the path coalgebra $kQ$ if and only if $Q$ is a Hopf quiver. In this case, $kQ_{0}$ is a group algebra and $kQ_{1}$ is a $kQ_{0}$-Hopf bimodule. Moreover, the product rule of paths can be displayed as follows.
Let $p$ be a path of length $l$. An $n$-thin split of it is a sequence $(p_1, \ \cdots, \ p_n)$ of vertices and arrows such that the concatenation $p_n \cdots p_1$ is exactly $p.$ These $n$-thin splits are in one-to-one correspondence with the $n$-sequences of $(n-l)$ 0's and $l$ 1's. Denote the set of such sequences by
$D_l^n.$ Clearly $|D_l^n|={n \choose l}.$ For $d=(d_1, \ \cdots, \ d_n) \in D_l^n,$ the corresponding $n$-thin split is written as $dp=((dp)_1, \ \cdots, \ (dp)_n),$ in which $(dp)_i$ is a vertex if $d_i=0$ and an arrow if $d_i=1.$
Let $\alpha=a_m \cdots a_1$ and $\beta=b_n \cdots b_1$ be paths of lengths $m$ and $n$ respectively. Let $d \in D_m^{m+n}$ and $\bar{d} \in D_n^{m+n}$ the complement sequence which is obtained from $d$ by replacing each 0 by 1 and each 1 by 0. Define an element
in $kQ_{m+n},$ $$(\alpha \cdot \beta)_d=[(d\alpha)_{m+n}.(\bar{d}\beta)_{m+n}] \cdots [(d\alpha)_1.(\bar{d}\beta)_1],$$ where $[(d\alpha)_i.(\bar{d}\beta)_i]$ is understood as the action of $kQ_0$-Hopf bimodule on $kQ_1$ and these terms in different brackets are put together by cotensor product, or equivalently concatenation. In these notations, the formula of the product of $\alpha$ and $\beta$ is given as follows (see pages 245-246 in \cite{CR}): \begin{equation} \alpha \cdot \beta=\sum_{d \in D_m^{m+n}}(\alpha \cdot \beta)_d \ . \end{equation}
\subsection{} In this paper, we only consider the very simple Hopf quiver, a loop $\circlearrowleft$. By setting $G:=e$ and $r:=e$, one can see that a loop is just the Hopf quiver $Q(G,r)$. For any natural number $n$, denote the unique path of length $n$ by $\alpha_{n}$. Since the group $G$ is trivial now, the Hopf bimodule action is trivial too. Thus the product rule over $k\circlearrowleft$ is very simple. That is, \begin{equation} \alpha_{n} \cdot \alpha_{m}=\left ( \begin{array}{cc} m+n \\n \end{array}\right)\alpha_{m+n}. \end{equation} This is indeed the familiar Hopf algebra $(k[x])^{\circ}$, the finite dual of $k[x]$. Sometimes, we denote this Hopf structure still by $k\circlearrowleft$ and one can discriminate the exact meaning by context. Note that this is a graded Hopf algebra with length grading.
For a quiver $Q$, define $kQ_{d}:=\oplus_{i=0}^{d-1}kQ(i)$ where $Q(i)$ is the set of all paths of length $i$ in $Q$. Clearly, for any $i\geq0$, $k\circlearrowleft_{p^{i}}$ is a sub Hopf algebra of $k\circlearrowleft$.
\begin{lemma} Let $H$ be a finite-dimensional sub Hopf algebra of $k\circlearrowleft$, then $H\cong k\circlearrowleft_{p^{i}}$ for some $i\geq 0$. \end{lemma} \begin{proof} This is follows directly from the known fact that $k[x]/(x^{p^{i}})$ are all Hopf quotients of $k[x]$. \end{proof}
Van Oystaeyen and Zhang proved the dual Gabriel Theorem for coradically graded pointed Hopf algebras (Theorem 4.5 in \cite{FP}):
\begin{lemma} Let $H$ be a coradically graded pointed Hopf algebra, then its dual Gabriel quiver $Q(H)$ is a Hopf quiver and there is a Hopf embedding $$H\hookrightarrow kQ(H).$$ \end{lemma}
Now let $C\subset k\circlearrowleft$ be a finite-dimensional large sub coalgebra of $k\circlearrowleft$ and assume there is a Hopf structure $H(C)$ on $C$.
\begin{proposition} With notations and the assumption as above, there is a natural number $i$ such that as a coalgebra, $$C\cong k\circlearrowleft_{p^{i}}.$$ \end{proposition} \begin{proof} At first, we know that $H(C)$ is a pointed Hopf algebra. Denote its coradical filtration by $\{ H(C)_n \}_{n=0}^{\infty}.$ Define \[\operatorname{gr}(H(C))=H(C)_0 \oplus H(C)_1/H(C)_0 \oplus H(C)_2/H(C)_1 \oplus \cdots \cdots \] as the corresponding coradically graded version. Then $\operatorname{gr}(H(C))$ inherits from $H(C)$ a coradically graded Hopf algebra structure (see e.g. \cite{Mon}). By Lemma 2.5, $\gr(H(C))$ is a sub Hopf algebra of $k\circlearrowleft$. Thus Lemma 2.4 implies what we want. \end{proof}
Thus, our next aim is to give all possible Hopf structures (not necessarily coradically graded) over the coalgebra $k\circlearrowleft_{p^{i}}$.
For any rational number $a$, denote by $[a]$ the biggest integer which is not bigger than $a$. \begin{lemma} For any positive integers $m,n$, $\left ( \begin{array}{cc} m+n \\n \end{array}\right)=0$ if and only if $$\sum_{i\geq 1}[\frac{m+n}{p^{i}}]> \sum_{i\geq 1}[\frac{m}{p^{i}}]+ \sum_{i\geq 1}[\frac{n}{p^{i}}].$$ \end{lemma} \begin{proof} Clear. \end{proof}
We call a Hopf algebra is \emph{uniserial} if the set of its sub Hopf algebras forms a totally ordered set under the containing relation.
\textbf{Convention. } Let $C$ and $D$ be two coalgebras and assume that $C$ is a sub coalgebra of $D$.
If there is a Hopf structure $H(D)$ over $D$, then we use the notion $H(C)$ to denote the restriction, if applicable, of the structure of $H(D)$ to $C$.
\begin{proposition} Let $n$ be a positive natural number and assume that there is a Hopf structure $H(k\circlearrowleft_{p^{n}})$ over $k\circlearrowleft_{p^{n}}$. Then $H(k\circlearrowleft_{p^{n}})$ is a uniserial Hopf algebra with the composition series $$k\subset H(k\circlearrowleft_{p^{1}})\subset \cdots\subset H(k\circlearrowleft_{p^{i-1}})\subset H(k\circlearrowleft_{p^{i}})\subset \cdots\subset H(k\circlearrowleft_{p^{n}}).$$ \end{proposition} \begin{proof} By Proposition 2.6, it is enough to show that $H(k\circlearrowleft_{p^{i}})$ for $i\leq n$ are sub Hopf algebras. Thus, it is enough to show that they are closed under the multiplication. But this is the direct consequence of the product rule (2.1) and Lemma 2.7. \end{proof}
\section{endomorphisms of $k\circlearrowleft$}
For later use, we characterize all the possible endomorphisms of the path coalgebra $k\circlearrowleft$ in this section.
\begin{theorem} \emph{(i)} Let $f:\;k\circlearrowleft\rightarrow k\circlearrowleft$
be a coalgebra map, then there are $\{\lambda_{i}\in k|i\in \mathbb{N}^{+}\}$ such that $$f(\alpha_{n})=\sum_{r=1}^{n}(\sum_{n_{1}+\cdots n_{r}=n}\lambda_{n_{1}}\cdots\lambda_{n_{r}})\alpha_{r}$$ for any $n$.
\emph{(ii)} All coalgebra endomorphisms of $k\circlearrowleft$ are precisely given in this way. \end{theorem} \begin{proof} (i) Let's find such $\lambda_{i}'$s. Since $f$ is a coalgebra map, $f(1)$ is a group-like element and $f(\alpha_{1})$ is a primitive element. Thus $f(1)=1$ and there is a $\lambda_{1}\in k$ such that $f(\alpha_{1})=\lambda_{1}\alpha_{1}$ since $k\alpha_{1}$ are all primitive elements. Suppose we have found $\{\lambda_{1},\ldots,\lambda_{n}\}$ and let's find $\lambda_{n+1}$. By $f$ is a coalgebra map, $$\Delta(f(\alpha_{n+1})-\sum_{r=2}^{n+1}(\sum_{n_{1}+\cdots n_{r}=n+1}\lambda_{n_{1}}\cdots\lambda_{n_{r}})\alpha_{r})\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$ \begin{eqnarray*} &=&f(\alpha_{n+1})\otimes 1+ 1\otimes f(\alpha_{n+1})+\sum_{i=1}^{n}f(\alpha_{i})\otimes f(\alpha_{n+1-i})-\\ &\;&\sum_{r=2}^{n+1}(\sum_{n_{1}+\cdots n_{r}=n+1}\lambda_{n_{1}}\cdots\lambda_{n_{r}})\sum_{s+t=r}\alpha_{s}\otimes \alpha_{t})\\ &=&f(\alpha_{n+1})\otimes 1+ 1\otimes f(\alpha_{n+1})+\sum_{i=1}^{n}(\sum_{s=1}^{i}(\sum_{n_{1}+\cdots n_{s}=i}\lambda_{n_{1}}\cdots\lambda_{n_{s}})\alpha_{s}\\ &\;&\otimes \sum_{t=1}^{n+1-i}(\sum_{m_{1}+\cdots m_{t}=n+1-i}\lambda_{m_{1}}\cdots\lambda_{m_{t}})\alpha_{t})-\\ &\;&\sum_{r=2}^{n+1}(\sum_{n_{1}+\cdots n_{r}=n+1}\lambda_{n_{1}}\cdots\lambda_{n_{r}})\sum_{s+t=r}\alpha_{s}\otimes \alpha_{t}). \end{eqnarray*}
Replace $\lambda_{m_{1}}\cdots\lambda_{m_{t}}$ by $\lambda_{n_{s+1}}\cdots\lambda_{n_{s+t}}$ and set $r=s+t$, one can find that $$ \sum_{i=1}^{n}(\sum_{s=1}^{i}(\sum_{n_{1}+\cdots n_{s}=i}\lambda_{n_{1}}\cdots\lambda_{n_{s}})\alpha_{s}\otimes \sum_{t=1}^{n+1-i}(\sum_{m_{1}+\cdots m_{t}=n+1-i}\lambda_{m_{1}}\cdots\lambda_{m_{t}})\alpha_{t})$$ $$=\sum_{r=2}^{n+1}(\sum_{n_{1}+\cdots n_{r}=n+1}\lambda_{n_{1}}\cdots\lambda_{n_{r}})\sum_{s+t=r,s\neq 0\neq t}\alpha_{s}\otimes \alpha_{t}).\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(\star)$$ Thus let $y:=f(\alpha_{n+1})-\sum_{r=2}^{n+1}(\sum_{n_{1}+\cdots n_{r}=n+1}\lambda_{n_{1}}\cdots\lambda_{n_{r}})\alpha_{r}$, then $\Delta(y)=y\otimes 1+1\otimes y$. Thus $f(\alpha_{n+1})-\sum_{r=2}^{n+1}(\sum_{n_{1}+\cdots n_{r}=n+1}\lambda_{n_{1}}\cdots\lambda_{n_{r}})\alpha_{r}$ is a primitive element and so there is a $\lambda_{n+1}$ such that $y=\lambda_{n+1}\alpha_{1}$. Equivalently, $$f(\alpha_{n+1})=\sum_{r=1}^{n+1}(\sum_{n_{1}+\cdots n_{r}=n+1}\lambda_{n_{1}}\cdots\lambda_{n_{r}})\alpha_{r}.$$
(ii) By (i), it is enough to show that for any $\{\lambda_{i}\in k|i\in \mathbb{N}^{+}\}$ and the linear map $f$ defined by $f(\alpha_{n})=\sum_{r=1}^{n}(\sum_{n_{1}+\cdots n_{r}=n}\lambda_{n_{1}}\cdots\lambda_{n_{r}})\alpha_{r}$ is indeed a coalgebra map. In fact, \begin{eqnarray*} (f\otimes f)\Delta(\alpha_{n})&=&f(\alpha_{n})\otimes 1+ 1\otimes f(\alpha_{n})+\sum_{i=1}^{n-1}f(\alpha_{i})\otimes f(\alpha_{n-i})\\ &=&f(\alpha_{n})\otimes 1+ 1\otimes f(\alpha_{n})+\sum_{i=1}^{n-1}(\sum_{s=1}^{i}(\sum_{n_{1}+\cdots n_{s}=i}\lambda_{n_{1}}\cdots\lambda_{n_{s}})\alpha_{s}\\ &\;&\otimes \sum_{t=1}^{n-i}(\sum_{m_{1}+\cdots m_{t}=n-i}\lambda_{m_{1}}\cdots\lambda_{m_{t}})\alpha_{t}). \end{eqnarray*} While \begin{eqnarray*} \Delta(f(\alpha_{n}))&=&\Delta(\sum_{r=1}^{n}(\sum_{n_{1}+\cdots n_{r}=n}\lambda_{n_{1}}\cdots\lambda_{n_{r}})\alpha_{r})\\ &=&\sum_{r=1}^{n}(\sum_{n_{1}+\cdots n_{r}=n}\lambda_{n_{1}}\cdots\lambda_{n_{r}})\sum_{s+t=r}(\alpha_{s}\otimes \alpha_{t})\\ &=&f(\alpha_{n})\otimes 1+1\otimes f(\alpha_{n})+ \sum_{r=2}^{n}(\sum_{n_{1}+\cdots n_{r}=n}\lambda_{n_{1}}\cdots\lambda_{n_{r}})\\ &\;\;\;\;&\times \sum_{s+t=r,s\neq 0\neq t}(\alpha_{s}\otimes \alpha_{t}). \end{eqnarray*} Then equation $(\star)$ implies that $\Delta(f(\alpha_{n}))=(f\otimes f)\Delta(\alpha_{n})$. \end{proof}
By the proof, we know that if $f:\;k\circlearrowleft\rightarrow k\circlearrowleft$ is a coalgebra map, then $f(\alpha_{1})=\lambda_{1}\alpha_{1}$ for some $\lambda_{1}\in k$. The next result is to provide a criterion to determine when $f$ is indeed an automorphism.
\begin{proposition} With notions as the above, $f$ is an automorphism if and only if $\lambda_{1}\neq 0$. \end{proposition} \begin{proof} By Theorem 3.1, $f(k\circlearrowleft_{n})\subseteq k\circlearrowleft_{n}$ for any $n\in \mathbb{N}^{+}$ and thus
$f|_{k\circlearrowleft_{n}}$ is a coalgebra endomorphism of $k\circlearrowleft_{n}$. By $\lambda_{1}\neq 0$,
$f|_{k\circlearrowleft_{2}}$ is injective and so
$f|_{k\circlearrowleft_{n}}$ is injective by Heynaman-Radford's result \cite{HR}. Since dim$_{k}k\circlearrowleft_{n}<\infty$,
$f|_{k\circlearrowleft_{n}}$ is bijective. This indeed implies that $f$ is an automorphism of $k\circlearrowleft$. The converse is obvious since one always has $f(\alpha_{1})=\lambda_{1}\alpha_{1}.$ \end{proof}
\begin{corollary} For any natural numbers $m>n>0$ and assume that $f$ is an automorphism of the coalgebra $k\circlearrowleft_{n}$, then $f$ can be extended to be automorphisms of the coalgebra $k\circlearrowleft$ and $k\circlearrowleft_{m}$. \end{corollary} \begin{proof} By the proof of Theorem 3.1, there are $\{\lambda_{1},\ldots,\lambda_{n-1}\}$ such that $$f(\alpha_{i})=\sum_{r=1}^{i}(\sum_{n_{1}+\cdots n_{r}=n}\lambda_{n_{1}}\cdots\lambda_{n_{r}})\alpha_{r}$$ for $1\leq i\leq n-1$. By setting $\lambda_{j}=0$ for all $j\geq n$ and by Theorem 3.1, if we define a map $F:\;k\circlearrowleft\rightarrow k\circlearrowleft$ through $$F(\alpha_{l})=\sum_{r=1}^{l}(\sum_{n_{1}+\cdots n_{r}=l}\lambda_{n_{1}}\cdots\lambda_{n_{r}})\alpha_{r}$$ for any natural number $l$, then $F$ is a coalgebra endomorphism of
$k\circlearrowleft$. Clearly, $F|_{k\circlearrowleft_{n}}=f$. Owning to Proposition 3.2, $F$ is an automorphism. Theorem 3.1 deduces that $F({k\circlearrowleft_{m}})\subseteq k\circlearrowleft_{m}$ and thus
$F|_{k\circlearrowleft_{m}}$ is the extension of $f$ to $k\circlearrowleft_{m}$. \end{proof}
\section{Hopf structures on $k\circlearrowleft_{p}$}
The following result seems well-known and we write its proof out for completeness.
\begin{lemma} Let $H$ be a Hopf structure over $k\circlearrowleft_{p^{1}}$, then as a Hopf algebra $H$ is isomorphic to either $(k\mathbb{Z}_{p})^{\ast}$, dual of the group algebra $k\mathbb{Z}_{p}$, or $k[x]/(x^{p})$. \end{lemma} \begin{proof} At first, it is not hard to see that $H$ is generated by $\alpha_{p^{0}}=\alpha_{1}$. Consider the element $\alpha_{1}^{p}$. For it, we have $$\Delta(\alpha_{1}^{p})=\Delta(\alpha_{1})^{p}=(1\otimes \alpha_{1}+ \alpha_{1}\otimes 1)^{p}=1\otimes \alpha_{1}^{p} +\alpha_{1}^{p}\otimes 1. $$ Thus $\alpha_{1}^{p}$ is a primitive element. Since the space spanned by $\alpha_{1}$ are all primitive elements in the coalgebra $k\circlearrowleft_{p^{1}}$, there is a $\lambda\in k$ such that $$\alpha_{1}^{p}=\lambda \alpha_{1}.$$ If $\lambda=0$, then $H\cong k[x]/(x^{p})$. If $\lambda\neq 0$, take $\lambda'$ to be a solution of the equation $\lambda x^{p}-x=0$. Then $$(\lambda'\alpha_{1})^{p}=\lambda\lambda'^{p}\alpha_{1}^{p}=\lambda'\alpha_{1}.$$ In one word, if $\lambda\neq 0$, we can always assume that $\lambda=1$ and thus $H\cong (k\mathbb{Z}_{p})^{\ast}$. \end{proof}
We find that the property of $H(k\circlearrowleft_{p^{n}})$ is largely determined by that of $H(k\circlearrowleft_{p^{1}})$.
\begin{proposition} Let $n$ be a positive integer and assume that there is a commutative Hopf structure $H(k\circlearrowleft_{p^{n}})$ over $k\circlearrowleft_{p^{n}}$. If $H(k\circlearrowleft_{p^{1}})\cong (k\mathbb{Z}_{p})^{\ast}$, then $$H(k\circlearrowleft_{p^{n}})\cong (k\mathbb{Z}_{p^{n}})^{\ast}.$$ \end{proposition} \begin{proof} \textbf{Claim. } \emph{Up to a Hopf isomorphism, $\alpha_{l}^{p}=\alpha_{l}$ for $0<l<p^{n}$. } We prove this fact by using induction on $l$. If $l=1$, this is just assumption. Assume that $\alpha_{l}^{p}=\alpha_{l}$ for $l\leq m-1$, let's prove that $\alpha_{m}^{p}=\alpha_{{m}}.$ By the definition of path coalgebra and the assumption of commutativity, we always have \begin{eqnarray*}\Delta(\alpha_{{m}}^{p})&=&(1\otimes \alpha_{{m}}+\alpha_{{m}}\otimes 1+\sum_{0<l<{m}}\alpha_{l}\otimes \alpha_{{m}-l})^{p}\\ &=&1\otimes \alpha_{{m}}^{p}+\alpha_{{m}}^{p}\otimes 1+\sum_{0<l<{m}}\alpha_{l}^{p}\otimes \alpha_{{m}-l}^{p}. \end{eqnarray*} The inductive assumption implies that $\alpha_{l}^{p}=\alpha_{l}$ for $l<{m}$. Thus $$\Delta(\alpha_{{m}}^{p})=1\otimes \alpha_{{m}}^{p}+\alpha_{{m}}^{p}\otimes 1+\sum_{0<l<{m}}\alpha_{l}\otimes \alpha_{{m}-l}$$ and so $$\Delta(\alpha_{{m}}^{p}-\alpha_{{m}})=(\alpha_{{m}}^{p}-\alpha_{{m}})\otimes 1+1\otimes (\alpha_{{m}}^{p}-\alpha_{{m}}).$$ Therefore, there is $\lambda\in k$ such that $\alpha_{{m}}^{p}-\alpha_{{m}}=\lambda\alpha_{1}$. If $\lambda=0$, done. If $\lambda\neq 0$, take $\xi$ to be a solution of the equation $x^{p}-x+\lambda=0$ and let $\alpha_{{m}}':=\alpha_{{m}}+\xi\alpha_{1}$. Clearly, the map $$f:\;k\circlearrowleft_{m+1}\rightarrow k\circlearrowleft_{m+1},\;\;\alpha_{i}\mapsto \alpha_{i}\;\;\textrm{for}\;i\neq m;\;\;\alpha_{m}\mapsto\alpha_{m}' $$ is an automorphism of $k\circlearrowleft_{m+1}$. Corollary 3.3 implies $f$ can be extended to be an automorphism of $k \circlearrowleft_{p^{n}}$. Since this automorphism is equivalent to choose a new basis of $k\circlearrowleft_{p^{n}}$, $f$ is an automorphism of Hopf algebras of $H(k\circlearrowleft_{p^{n}})$. Now $$(\alpha_{p^{m}}')^{p}=(\alpha_{p^{m}}+\xi\alpha_{1})^{p}=\alpha_{p^{m}}+\lambda\alpha_{1}+\xi^{p}\alpha_{1} =\alpha_{p^{m}}+\xi\alpha_{1}=\alpha_{p^{m}}'.$$ The claim is proved.
Construct the element $$t:=\prod_{0<l<p^{n}}(1-\alpha_{l}^{p-1}).$$ Thus the claim implies for any $0<m<p^{n}$, $$\alpha_{m}t=0=\varepsilon(\alpha_{m})t.$$ This means that $t\in \int_{H}$, the set of integrals. Since $\varepsilon(t)=1\neq 0$, $H(k\circlearrowleft_{p^{n}})$ is a simisimple Hopf algebra (Theorem 2.2.1 in \cite{Mon}). Thus $H(k\circlearrowleft_{p^{n}})\cong (kG)^{\ast}$ for some finite abelian group. Since $H(k\circlearrowleft_{p^{n}})$ is cogenerated by $\alpha_{1}$, $kG$ is generated by one element. Thus $G\cong \mathbb{Z}_{p^{n}}$. \end{proof}
\begin{remark} We would like to thank Professor A. Masuoka for pointing out to us that the above proposition can be deduced from Chapter IV, Section 3, 3.4 of \cite{DG} or Theorem 0.1 in \cite{Mas}. \end{remark}
Recall an affine algebraic group $\mathcal{G}$ is \emph{finite} if its coordinate ring $\mathcal{O(G)}$ is a finite-dimensional Hopf algebra. A finite algebraic group $\mathcal{G}$ is called \emph{infinitesimal} if $\mathcal{O(G)}$ is a local algebra. And, we call a finite algebraic group $\mathcal{G}$ \emph{unipotent} if its distribution algebra $\mathcal{H(G)}:=(\mathcal{O(G)})^{\ast}$ is a local algebra. There is an equivalence between the category of finite algebraic groups and the category of finite-dimensional cocommutative Hopf algebras. Explicitly, sending finite algebraic group $\mathcal{G}$ to $\mathcal{H(G)}$ gives us the equivalence. For more knowledge about affine algebraic groups, see \cite{DG,Wat}.
Now assume that there is a commutative Hopf structure $H(k\circlearrowleft_{p^{n}})$ over $k\circlearrowleft_{p^{n}}$, then there is a finite algebraic group $\mathcal{G}_{p^{n}}$ such that $$\mathcal{H}(\mathcal{G}_{p^{n}})=H(k\circlearrowleft_{p^{n}}).$$
\begin{proposition} Keep the above notations. If $H(k\circlearrowleft_{p})\cong k[x]/(x^{p})$, then $\mathcal{G}_{p^{n}}$ is an infinitesimal unipotent group. \end{proposition} \begin{proof} Owning to the fact that $(k\circlearrowleft_{p^{n}})^{\ast}\cong k[x]/(x^{p^{n}})$ as algebras, $\mathcal{G}_{p^{n}}$ is infinitesimal. So in order to show $\mathcal{G}_{p^{n}}$ is unipotent, it is enough to show that $H(k\circlearrowleft_{p^{n}})$ is a local algebra.
By Proposition 2.8, $H(k\circlearrowleft_{p^{n}})$ is uniserial with the composition series $$k\subset H(k\circlearrowleft_{p^{1}})\subset \cdots\subset H(k\circlearrowleft_{p^{i-1}})\subset H(k\circlearrowleft_{p^{i}})\subset \cdots\subset H(k\circlearrowleft_{p^{n}}).$$ Lemma 4.1 implies that either $H(k\circlearrowleft_{p^{i}})/H(k\circlearrowleft_{p^{i-1}})^{+}H(k\circlearrowleft_{p^{i}})\cong (k\mathbb{Z}_{p})^{\ast}$ or $H(k\circlearrowleft_{p^{i}}) /H(k\circlearrowleft_{p^{i-1}})^{+}H(k\circlearrowleft_{p^{i}})$ $\cong k[x]/(x^{p})$ for any $1\leq i\leq n$. Here for a Hopf algebra $H$, $H^{+}$ stands for the kernel of $\varepsilon: \;H\rightarrow k$. To show $H(k\circlearrowleft_{p^{n}})$ is local, it is enough to show $H(k\circlearrowleft_{p^{i}})/H(k\circlearrowleft_{p^{i-1}})^{+}H(k\circlearrowleft_{p^{i}})\cong k[x]/(x^{p})$ for all $1\leq i\leq n$ (In fact, if so then all non-trivial paths will be nilpotent).
Otherwise, there is an $i$ such that $H(k\circlearrowleft_{p^{i}})/H(k\circlearrowleft_{p^{i-1}})^{+}H(k\circlearrowleft_{p^{i}})\cong (k\mathbb{Z}_{p})^{\ast}$. Take such $i$ as small as possible. By assumption, $i\geq 2$. Thus $H(k\circlearrowleft_{p^{i}})/H(k\circlearrowleft_{p^{i-1}})^{+}H(k\circlearrowleft_{p^{i}})\cong (k\mathbb{Z}_{p})^{\ast}$ implies that $$\alpha_{p^{i-1}}^{p}\equiv \alpha_{p^{i-1}}\;\;\textrm{mod}\;k\circlearrowleft_{p^{i-1}}.$$ And thus $\alpha_{p^{i-1}}^{p^{i}}\equiv \alpha_{p^{i-1}}\;\;\textrm{mod}\;k\circlearrowleft_{p^{i-1}}.$ Therefore there is an element $a\in k\circlearrowleft_{p^{i-1}}$ such that $\alpha_{p^{i-1}}^{p^{i-1}}=\alpha_{p^{i-1}}+a$. Since $i$ is as small as possible, $H(k\circlearrowleft_{p^{i-1}})$ is local and all non-trivial paths in $k\circlearrowleft_{p^{i-1}}$ are nilpotent. More precisely, let $\alpha$ be a non-trivial path living in $k\circlearrowleft_{p^{i-1}}$, then $\alpha^{p^{i-1}}=0$. Thus \begin{eqnarray*} \Delta(\alpha_{p^{i-1}}^{p^{i-1}})&=&(1\otimes \alpha_{p^{i-1}}+\alpha_{p^{i-1}}\otimes 1 +\sum_{0<l<p^{i-1}}\alpha_{l}\otimes \alpha_{p^{i-1}-l})^{p^{i-1}}\\ &=&1\otimes \alpha_{p^{i-1}}^{p^{i-1}}+\alpha_{p^{i-1}}^{p^{i-1}}\otimes 1+\sum_{0<l<p^{i-1}}\alpha_{l}^{p^{i-1}}\otimes \alpha_{p^{i-1}-l}^{p^{i-1}}\\ &=& 1\otimes \alpha_{p^{i-1}}^{p^{i-1}}+\alpha_{p^{i-1}}^{p^{i-1}}\otimes 1. \end{eqnarray*} This implies that $\alpha_{p^{i-1}}^{p^{i-1}}=\alpha_{p^{i-1}}+a$ is a primitive element and so there is a $\lambda\in k$ such that $\alpha_{p^{i-1}}+a=\lambda\alpha_{1}$. Therefore, $\alpha_{p^{i-1}}\in k\circlearrowleft_{p^{i-1}}$ which is impossible. \end{proof}
Combining Lemma 4.1, Propositions 4.2, 4.4 and 2.8, we get
\begin{corollary} Let $n$ be a positive integer and assume that there is a commutative Hopf structure $H(k\circlearrowleft_{p^{n}})$ over $k\circlearrowleft_{p^{n}}$. Then either $H(k\circlearrowleft_{p^{n}})\cong (k\mathbb{Z}_{p^{n}})^{\ast}$ or $H(k\circlearrowleft_{p^{n}})$ is the distribution algebra of a uniserial infinitesimal unipotent commutative $k$-group. \end{corollary}
\section{Classification and application}
Fix a positive integer $n$ and consider the coalgebra
$k\circlearrowleft_{p^{n}}$. Assume that there is a Hopf structure on
$k\circlearrowleft_{p^{n}}$. Since its coradically graded version
is generated by $\{\alpha_{p^{i}}|1\leq i\leq n-1\}$, it is also
generated by $\{\alpha_{p^{i}}|1\leq i\leq n-1\}$. So in order to
give the Hopf structure, it is enough to characterize the
relations between $\{\alpha_{p^{i}}|1\leq i\leq n-1\}$.
For any $0\leq d\leq n$, the Hopf algebra $L(n,d)$ (it is indeed a Hopf algebra by the following theorem) is defined to
be the Hopf algebra over $k\circlearrowleft_{p^{n}}$ with
relations:
\begin{equation} \alpha_{p^{i}}\alpha_{p^{j}}=\alpha_{p^{j}}\alpha_{p^{i}},\;\;\textrm{for}\;0\leq i,j\leq
n-1;\end{equation}
\begin{equation} \alpha_{p^{i}}^{p}=0,\;\;\textrm{for}\;i< d;
\end{equation}
\begin{equation} \alpha_{p^{i}}^{p}=\alpha_{p^{i-d}},\;\;\textrm{for}\;i\geq
d.
\end{equation}
The main result of this section is the following.
\begin{theorem} $L(n,d)$ is a Hopf algebra and any commutative
Hopf structure $H(k\circlearrowleft_{p^{n}})$ over $k\circlearrowleft_{p^{n}}$ is isomorphic to an $L(n,d)$ for some $d$. \end{theorem}
One of the main ingredients of the proof is the classification result given in \cite{FRV}. Let's recall it. By $\mathcal{W}:\;\mathbb{M}_{k}\rightarrow \mathbb{M}_{\mathbb{Z}}$ we denote the affine commutative group scheme of \emph{Witt vectors}. For any positive natural number $m$ let $\mathcal{W}_{m}:\;\mathbb{M}_{k}\rightarrow \mathbb{M}_{\mathbb{Z}}$ be the affine commutative group scheme of \emph{Witt vectors of length $m$}. Denote the \emph{Frobenius map} and \emph{Verschiebung} of $\mathcal{W}_{m}$ by $\mathcal{F}$ and $\mathcal{V}$ respectively. For any finite commutative algebraic group $\mathcal{G}$, its \emph{Cartier dual} is denoted by $\mathcal{D(G)}$. For details, see \cite{DG}. An infinitesimal unipotent commutative group $\mathcal{U}$ is called $\mathcal{V}$-\emph{uniserial} if Coker$\mathcal{V}\cong \Spec_{k}(k[x]/(x^{p}))$. Likewise, a unipotent infinitesimal group $\mathcal{U}$ is called $\mathcal{F}$-\emph{uniserial} if Ker$\mathcal{F}\cong\Spec_{k}(k[x]/(x^{p}))$. Note that $\mathcal{G}$ is $\mathcal{V}$-uniserial or $\mathcal{F}$-uniserial is equivalent to its distribution algebra $\mathcal{H(G)}$ is uniserial (see Lemma 2.5 in \cite{FRV}).
Let $d,j,n\in \mathbb{N}$ and for $n\geq 1,\;d\geq 1$, we denote by $\mathcal{U}_{n,d}$ the kernel of the endomorphism $\mathcal{V}^{d}-\mathcal{F}:\; \mathcal{W}_{m}\rightarrow \mathcal{W}_{m}$ with $m=n(d+1)$. Denote by $\mathcal{U}_{n,d}^{j}$ the intersection of $\mathcal{U}_{n,d}$ with the kernel of the endomorphism $\mathcal{V}^{(n-1)(d+1)+j}:\;\mathcal{W}_{m}\rightarrow \mathcal{W}_{m}$ for $1\leq j\leq d$. The following is the main result of \cite{FRV} (Theorem 1.2 in \cite{FRV}).
\begin{lemma} The following gives a complete list of representatives of isomorphism classes of non-trivial uniserial infinitesimal unipotent commutative $k$-groups:
\emph{(i) }$(\mathcal{W}_{d})_{1}$; $\;\;$\emph{(ii)} $\mathcal{U}_{n,d}$; $\;\;$ \emph{(iii)} $\mathcal{U}_{n,d}^{j}$;
\emph{(iv)} $\mathcal{D}((\mathcal{W}_{d})_{1})$; $\;\;$\emph{(v)} $\mathcal{D}(\mathcal{U}_{n,d})$; $\;\;$ \emph{(vi)} $\mathcal{D}(\mathcal{U}_{n,d}^{j})$.
Moreover, the groups labeled (i)-(iii) are $\mathcal{V}$-uniserial and those in (iv)-(vi) are $\mathcal{F}$-uniserial. \end{lemma}
\textbf{Proof of the Theorem 5.1. } At first, since $H(k\circlearrowleft_{p^{n}})$
is commutative, there is a $k$-group $\mathcal{G}$ such that
$\mathcal{G}=\Spec_{k}(H(k\circlearrowleft_{p^{n}}))$. So the Frobenius map $\mathcal{F}$
and Verschiebung $\mathcal{V}$ can be defined for
$H(k\circlearrowleft_{p^{n}})$ too. Let's see what they are.
In order to explain our understanding, there is no harm to assume that both $\mathcal{F}$ and
$\mathcal{V}$ are Hopf endomorphisms of $H(k\circlearrowleft_{p^{n}})$ for simplicity
since the path coalgebra can clearly be defined over $\mathbb{Z}$.
By the definition of Frobenius map, we know that
$$\mathcal{F}:\;H(k\circlearrowleft_{p^{n}})\rightarrow H(k\circlearrowleft_{p^{n}}),\;\;
\alpha_{p^{i}}\mapsto \alpha_{p^{i}}^{p},\;\;\textrm{for}\;0<i<n.$$
Note that $\mathcal{V}$ is just the dual map of Frobenius map of
$\mathcal{D}(\mathcal{G})$. Since as an algebra we have $(k\circlearrowleft_{p^{n}})^{\ast}\cong
k[x]/(x^{p^{n}})$, the Frobenius map for
$(k\circlearrowleft_{p^{n}})^{\ast}$ is given by $x\mapsto x^{p}$.
Note also that $\{\alpha_{i}|0\leq i\leq p^{n}-1\}$ are the dual basis of $\{x^{i}|0\leq i\leq
p^{n}-1\}$. Thus $\mathcal{V}$ is given by
$$\mathcal{V}:\;H(k\circlearrowleft_{p^{n}})\rightarrow H(k\circlearrowleft_{p^{n}}),\;\;
\alpha_{p^{i}}\mapsto \alpha_{p^{i-1}},\;\;\textrm{for}\;0<i<n.$$
Thus if $\Spec_{k}(H(k\circlearrowleft_{p^{n}}))$ is unipotent,
then it is a $\mathcal{V}$-uniserial group.
According to Corollary 4.5, either $H(k\circlearrowleft_{p^{n}})\cong (k\mathbb{Z}_{p^{n}})^{\ast}$ or $H(k\circlearrowleft_{p^{n}})$ is the distribution algebra of a uniserial infinitesimal unipotent commutative $k$-group. If $H(k\circlearrowleft_{p^{n}})\cong (k\mathbb{Z}_{p^{n}})^{\ast}$, then $H(k\circlearrowleft_{p^{n}})\cong L(n,0)$. Otherwise, $H(k\circlearrowleft_{p^{n}})$ is a local algebra which implies that $\Spec_{k}(H(k\circlearrowleft_{p^{n}}))$ is infinitesimal and thus a unipotent group. By the discussion above, $\Spec_{k}(H(k\circlearrowleft_{p^{n}}))$ is an infinitesimal unipotent $\mathcal{V}$-uniserial group. By Lemma 5.2, we have $\Spec_{k}(H(k\circlearrowleft_{p^{n}}))\cong (\mathcal{W}_{d})_{1}$ or $\Spec_{k}(H(k\circlearrowleft_{p^{n}}))\cong \mathcal{U}_{m,d}$ or $\Spec_{k}(H(k\circlearrowleft_{p^{n}}))\cong \mathcal{U}_{m,d}^{j}$ for some $m,d,j$. The first case implies that $H(k\circlearrowleft_{p^{n}})\cong L(n,n)$. Let us analyze the last two cases. Recall that the coordinate ring of $\mathcal{W}_{n}$ is
$k[x_{1},\ldots,x_{n}]$. If $(d+1)|n$ (that is, we consider the second case), we have a Hopf epimorphism $$\pi:\;k[x_{1},\ldots,x_{n}]\twoheadrightarrow H(k\circlearrowleft_{p^{n}})$$ and the following commutative diagram
By $\Spec_{k}(H(k\circlearrowleft_{p^{n}}))\cong \mathcal{U}_{\frac{n}{d+1},d}$, dim$_{k}\mathcal{O}(\mathcal{U}_{\frac{n}{d+1},d})=p^{n}$. Therefore the above commutative diagram and the definitions of $\mathcal{F},\mathcal{V}$ for $k\circlearrowleft_{p^{n}}$ imply that $H(k\circlearrowleft_{p^{n}})$ satisfies equations (5.1)-(5.3) automatically. By comparing the dimension, equations (5.1)-(5.3) are all the relations for $H(k\circlearrowleft_{p^{n}})$. Thus
$H(k\circlearrowleft_{p^{n}})\cong L(n,d)$ with $(d+1)|n$. For the last case (that is, $\Spec_{k}(H(k\circlearrowleft_{p^{n}}))\cong \mathcal{U}_{m,d}^{j}$), the analysis is almost the same as the second case and the only point we need to say is that the condition ``intersection with kernel of the endomorphism $\mathcal{V}^{(n-1)(d+1)+j}$", appearing in the definition of $\mathcal{U}_{m,d}^{j}$, is equivalent to the condition $(d+1)\nmid n$.
Of course, $L(n,d)$ are all Hopf algebras now. In fact, the above discussions show that we have $$L(n,0)\cong (k\mathbb{Z}_{p^{n}})^{\ast},\;\;L(n,n)\cong k[x]/(x^{p^{n}}),$$ and $L(n,d)\cong \mathcal{O}(\mathcal{U}_{\frac{n}{d+1},d})$ in case
$(d+1)|n$. If $(d+1)\nmid n$, then $n=m(d+1)+j$ for some $m,j$ with $0<j<d+1$. The above discussions indicate that $L(n,d)\cong \mathcal{O}(\mathcal{U}^{j}_{m+1,d})$. $\;\;\;\;\;\;\;\;\square$
\begin{corollary} Up to Hopf isomorphisms there are exactly $n+1$ classes of non-isomorphic commutative Hopf structures on the coalgebra $k\circlearrowleft_{p^{n}}$ for any natural number $n$. \end{corollary}
As another direct consequence of this theorem, the commutative infinitesimal groups with 1-dimensional Lie algebras can be classified now.
\begin{corollary} Let $\mathcal{G}$ be a commutative infinitesimal group. If dim$_{k}$Lie$(\mathcal{G})=1$, then $\mathcal{H(G)}\cong L(n,d)$ for some $n,d$. \end{corollary} \begin{proof} By dim$_{k}$Lie$(\mathcal{G})=1$, the set of primitive elements of $\mathcal{H(G)}$ is 1-dimensional. Note that $\mathcal{H(G)}$ is always pointed, $\mathcal{H(G)}$ can be embedded into the path coalgebra $k\circlearrowleft$ (Lemma 2.1). Thus there is a natural number $n$ such that $\mathcal{H(G)}$ is a Hopf structure over $k\circlearrowleft_{p^{n}}$ by Proposition 2.6. Thanks to Theorem 5.1, $\mathcal{H(G)}\cong L(n,d)$ for some $d$. \end{proof}
\begin{remark} \emph{(1)} It is known that if we take $k:=\mathbb{F}_{p}$ then the multiplication of the Witt vector group scheme indeed corresponds to the additive of the $p$-adic numbers. Theorem 5.1 gives us some hint that sometimes it is possible to explain the addition of the $p$-adic numbers through the comultiplication of path coalgebras.
\emph{(2)} For any $L(n,d)$, there is still one thing which is not clear to us. That is, we don't know how to give the expression of each path through generators although we can give in some special cases (see the example below).
\emph{(3)} Not all Hopf structures over $k\circlearrowleft_{p^{n}}$ for some $n\geq 2$ are always commutative. In fact, set $p=2$ and consider the associative algebra $H$ generated by $x,y$ with relations $$xy-yx=x,\;\;x^{2}=y^{2}=0.$$ Define the comultiplication $\Delta$, counit $\varepsilon$ and the antipode through $$\Delta(x)=1\otimes x+x\otimes 1,\;\;\Delta(y)=1\otimes y+y\otimes 1+x\otimes x,$$ $$\varepsilon(x)=\varepsilon(y)=0,\;\;S(x)=-x,\;\;S(y)=-y.$$ It is straightforward to show that $H$ is indeed a Hopf algebra over the path coalgebra $k\circlearrowleft_{p^{2}}$. Clearly, it is not commutative. \end{remark}
\begin{example} \emph{For the Hopf algebra $L(2,1)$, one can see that up to a coalgebra automorphism} $$\alpha_{sp+t}=\frac{1}{s!t!}\alpha_{p}^{s}\alpha_{1}^{t},\;\;\emph{\textrm{for}}\;0\leq s\leq p-1,\;0\leq t\leq p-1.$$ \emph{In fact, we can prove this by using induction on the lengths of pathes. If the length is 1, it is clear. Now assume it is true for the pathes with lengths not more than $sp+t$. Now we consider the case $sp+t+1$. To show this case, begin with an observation at first. For any element $p$ in $k\circlearrowleft$, one always have} $$\Delta(p)=\sum_{i=0}^{n} \alpha_{i}\otimes p_{(i)}$$ \emph{where $p_{(i)}$ are uniquely determined since $1,\alpha_{1},\alpha_{2},\ldots$ is a basis of $k\circlearrowleft$. For two elements $p,q$, the basic observation is, up to a coalgebra automorphism,} $$p=q\;\;\;\;\emph{\textrm{if and only if}} \;\;\;\;p_{(1)}=q_{(1)}.\;\;\;(\star)$$
\emph{Now we consider the case $sp+t+1$. If $0<t< p-1$, we just need to show that $\alpha_{sp+t+1}=\frac{1}{s!(t+1)!}\alpha_{p}^{s}\alpha_{1}^{t+1}$. By $(\star)$, it is enough to show that $\alpha_{sp+t}=(\frac{1}{s!(t+1)!}\alpha_{p}^{s}\alpha_{1}^{t+1})_{(1)}$. Note that by assumption $\frac{1}{s!(t+1)!}\alpha_{p}^{s}\alpha_{1}^{t+1}=\frac{1}{t+1}\alpha_{sp+t}\alpha_{1}$ and direct computation shows that $(\alpha_{sp+t}\alpha_{1})_{(1)}=(t+1)\alpha_{sp+t}$.}
\emph{If $t=0$, we need show that $\alpha_{sp+1}=\frac{1}{s!}\alpha_{p}^{s}\alpha_{1}=\alpha_{sp}\alpha_{1}$ by assumption. Clearly, $(\alpha_{sp+1})_{(1)}=\alpha_{sp}$ and $(\alpha_{sp}\alpha_{1})_{(1)}=\alpha_{(s-1)p+(p-1)}\alpha_{1}+\alpha_{sp}$. Note that in $L(2,1)$, $\alpha_{1}^{p}=0$ and so $\alpha_{(s-1)p+(p-1)}\alpha_{1}=0$. By $(\star)$ again, $\alpha_{sp+1}=\frac{1}{s!}\alpha_{p}^{s}\alpha_{1}$. }
\emph{If $t=p-1$, the equality that we need check is $\alpha_{(s+1)p}=\frac{1}{(s+1)!}\alpha_{p}^{s+1}$. Also, computations show that $(\alpha_{p}^{s+1})_{(1)}=(s+1)\alpha_{p}^{s}\alpha_{p-1}=(s+1)\frac{1}{(p-1)!}\alpha_{p}^{s}\alpha_{1}^{p-1}$ and so $(\frac{1}{(s+1)!}\alpha_{p}^{s+1})_{(1)}=\frac{1}{s!(p-1)!}\alpha_{p}^{s}\alpha_{1}^{p-1}$. Meanwhile, $(\alpha_{(s+1)p})_{(1)}=\alpha_{sp+(p-1)}=\frac{1}{s!(p-1)!}\alpha_{p}^{s}\alpha_{1}^{p-1}$ by assumption. Using $(\star)$ again, $\alpha_{(s+1)p}=\frac{1}{(s+1)!}\alpha_{p}^{s+1}$.}
\end{example}
\vskip 0.5cm
\noindent{\bf Acknowledgements:} The research was supported by the NSF of China (10601052, 10801069, 10971206). The second author is supported by Japan Society for the Promotion of Science under the item ``JSPS Postdoctoral Fellowship for Foreign Researchers" and Grand-in-Aid for Foreign JSPS Fellow. He thanks Professor A. Masuoka for stimulating discussions. Some ideas was gotten during the second and the third authors visited Chen Institute of Mathematics and they thank Professor Cheng-Ming Bai for his warm-hearted helping. Part of this work was done when the first and second authors visited the University of Cologne under the financial support from DAAD. They would also like to thank their host Professor Steffen K\"{o}nig for his kind hospitality.
\end{document}
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arXiv/math_arXiv_v0.2.jsonl
| null | null |
\begin{document}
\title{Turbulence-induced optical loss and cross-talk in spatial mode multiplexed or single-mode free-space communication channels} \author{K. S. Kravtsov$^{1,2}$}\email{[email protected]}\author{A. K. Zhutov$^3$} \author{I. V. Radchenko$^3$} \author{S. P. Kulik$^{3,4}$} \affiliation{ $^1$ A.M.Prokhorov General Physics Institute RAS, Moscow, Russia \\ $^2$ National Research University Higher School of Economics, Moscow, Russia\\ $^3$ Quantum Technology Centre of Moscow State University, Moscow, Russia\\ $^4$ Faculty of Physics, M.V.Lomonosov Moscow State University, Moscow, Russia}
\date{\today} \begin{abstract} Single-mode or mode multiplexed free-space atmospheric optical channels draw increasingly more attention in the last decade. The scope of their possible applications spans from the compatibility with the telecom WDM technology, fiber amplifiers, and modal multiplexing for increasing the channel throughput to various quantum communication related primitives such as entanglement distribution, high-dimensional spatially encoded quantum key distribution, and relativistic quantum cryptography. Many research papers discuss application of specific mode sets, such as optical angular momentum modes, for communication in the presence of atmospheric turbulence. At the same time some basic properties and key relations for such channels exposed to the atmospheric turbulence have not been derived yet. In the current paper we present simple analytic expressions and a general framework for assessing probability density functions of channel transmittance as well as modal cross-talk coefficients. Under some basic assumptions the presented results can be directly used for estimation of the Fried parameter of the turbulent channel based on the measured statistics of the fundamental mode transmittance coefficient. \end{abstract}
\maketitle
\section{Introduction} Transmission of information along line-of-sight free-space optical channels is an important communication technique, which was used by humanity for centuries. Automation of the originally manual information transfer, and introduction of high-speed electronics and lasers led to many important practical applications in XX century. At the same time, channel non-idealities, mainly, the atmospheric turbulence, became apparent. Turbulence effects were extensively studied in the 1970-s under the assumption of a broad, plane-wave-approximated light beam and a small point-like photodetector~\cite{S1978}.
Later, the demand for {\em power efficient} free-space optical communication led to a more advanced model, where a single-mode Gaussian beam from the source propagates toward a large aperture receiving telescope. In this model, turbulent effects shift and distort the Gaussian beam, so it spreads in space and partially misses the receiving aperture, thus producing the channel loss. This single-mode transmitter and multimode receiver model is studied extensively in the series of more recent papers~\cite{VSV12,VSV16,VSV17,VVS18}.
Following further technological advances, especially, breakthroughs in optical communications and quantum technologies, the actual {\em single-mode} channel performance has to be studied. In this case the receiver collects only a particular spatial mode, which is assumed to match the transmitted mode if there is no turbulence. It is important, first, as a means of replacing conventional single-mode fibers while keeping the infrastructure compatible with the WDM~\cite{YSJ07}, fiber amplifiers, coherent modulation techniques~\cite{KRK18}, existing fiber-based quantum key distribution systems, etc. Second, spatial mode-aware receivers allow for modal modulation, enabling data-efficient M-ary modulation formats, not available in the fiber-based counterparts~\cite{KHF16}. Third, many independent data streams may be spatially multiplexed into a single free-space channel, resulting in unprecedented data throughput at a single wavelength~\cite{WYF12}. Finally, emerging quantum technologies reaching higher-and-higher quantum dimensionality, need corresponding communication channels for exchange of such quantum states~\cite{NVG14,MMS15,SBF17}. The spatial degree of freedom may become the natural choice for quantum computers talking to each other using high-dimensional spatial quantum states of photons~\cite{KHF15}.
The formulated problem of single-mode channel performance in the presence of atmospheric turbulence as well as related questions of turbulence-induced modal cross-talk constitute the central question in this paper. By a single spatial mode we understand an eigensolution of the propagation equation, so the mode remains itself while propagating through any distance. Throughout the text we assume that if the atmosphere was perfectly homogeneous and uniform, our optical system would be perfectly aligned without any optical loss or cross-talk between the modes. The studied effects are solely due to the varying refraction conditions in the turbulent atmosphere that distort the propagating modes, causing them to deviate significantly from the unperturbed solution. We will introduce a framework that allows answering virtually any question about loss of power in a particular mode or coupling of a particular mode into other modes, provided the turbulence parameters are known. We show simple analytic solutions for the first-order approximation as well as numerically obtained results for higher order approximations and compare them with experimental results. In particular, we show a very simple connection between the transmittance statistics of a trivial single-mode fiber to single-mode fiber free-space link and the net turbulence strength in this channel. This connection may be used for an easy parameter estimation either from the measured channel statistics or from the known Fried parameter of turbulence.
\section{Turbulence model}
Turbulent phenomena in the atmosphere were first described by Kolmogorov~\cite{K41} in 1941, when he predicted the scaling of a structure function proportional to $r^{2/3}$. As we deal with the integral effect of the turbulence on the whole communication channel, and are not interested much in local turbulent properties, we use the well-established result for an extended channel and the von Karman model, which predicts the following phase power spectrum~\cite{AZB97,J06} \begin{equation} W_\varphi (f) = \vartheta\: r_0^{-5/3}\left(f^2
+ L_0^{-2}\right)^{-11/6} \exp(-l_0^2f^2),
\label{phasestr} \end{equation} where \begin{equation}
\vartheta = \frac{2\sqrt{2}\G^2(11/6)}{\pi^{11/3}}\left[\frac{3}5\G(6/5)\right]^{5/6} \approx 0.0229. \end{equation} This shows the spectral density of optical phase $\varphi$ fluctuations with respect to the spatial frequency $f$. The von Karman model is an empirical extrapolation of Kolmogorov's results for the whole range of spatial frequencies, as the original theory was only applicable for the range between the inner scale $l_0$ and the outer scale $L_0$. The parameter $r_0$ is the Fried parameter of turbulence, which tells how strong the turbulence is. One may consider it to be the approximate diameter of the telescope, whose diffraction limit equals the turbulence-induced resolution limit~\cite{J06}.
One reasonable approximation that we use in our analysis is that passing through a turbulent channel is equivalent to passing through a corresponding random phase mask, whose spatial frequency spectrum is given by~(\ref{phasestr}). Although it is not true in general (the beam may substantially re-distribute its power profile after diffraction on the phase distortions obtained at the very beginning of the channel), it holds true for not extremely strong turbulence, where substantial fraction of power remains in the same transverse mode. This regime is the most interesting for us, because in the case of extremely strong turbulence, one may just assume that all output modes will be equally populated regardless of the way they were excited at the input, which is trivial.
Phase distortion itself is a continuous function of the transverse coordinates, so we can use the Taylor series expansion to correctly represent it. \begin{equation}
\varphi (x, y) = \varphi_0 + ax + by + g\frac{x^2}2 + h\frac{y^2}2 + sxy + \dots,
\label{phase} \end{equation} where $a$ and $b$ are first order phase distortions, and $g$, $h$, and $s$ --- the second order ones. They can be found as \begin{equation}
\begin{array}{lll}
a = \frac{\partial\varphi}{\partial x} & b = \frac{\partial\varphi}{\partial y}\\
g = \frac{\partial^2\varphi}{\partial x^2} & h = \frac{\partial^2\varphi}{\partial y^2} & s =
\frac{\partial^2\varphi}{\partial x\partial y}.
\end{array}\label{derivatives} \end{equation} As phase distortion is a random function, all the mentioned distortion coefficients are random variables with zero mean due to the apparent symmetry.
We are now ready to find the dispersion of the distortion coefficients. First, using~(\ref{derivatives}) we can find power spectra of the distortion coefficients: \begin{equation}\begin{array}{l}
W_a = (2\pi)^2 f_x^2 W_\varphi\\
W_b = (2\pi)^2 f_y^2 W_\varphi\\
W_g = (2\pi)^4 f_x^4 W_\varphi\\
W_h = (2\pi)^4 f_y^4 W_\varphi\\
W_s = (2\pi)^4 f_x^2 f_y^2 W_\varphi. \end{array}\label{distortion}\end{equation} Where $f_x$ and $f_y$ are $x-$ and $y-$ components of the spatial frequency $f$: $f_x = f\cos(\theta)$, $f_y =
f\sin(\theta)$, where $\theta$ is the polar angle. Second, we need to take into account that we are interested not only in the point (0,0) where we take the derivatives~(\ref{derivatives}), but in the average phase slopes over the whole beam area. That leads to the additional filtering function $|F(f)|^2$ similar to the one that appears in the problem of finding angle of arrival fluctuations~\cite{BMZ92,AZB97}. In our case $|F(f)|^2$ is the spatial power spectrum of the particular mode in question. Finally, we use the Wiener–-Khinchin theorem to find the autocorrelation of the distortion coefficients, which is a Fourier transform of their power spectra. We may right away ignore the $x, y$-dependent part of the autocorrelation function and only find its value for $x=y=0$, which is exactly the dispersion of the corresponding coefficients. Finding the required zero coordinate Fourier coefficient and taking the integral over $\theta$, we obtain \begin{equation}
C_a = C_b = 4\pi^3 \int_0^\infty f^3 W_\varphi (f) |F(f)|^2\,df
\label{dispersion1} \end{equation} \begin{equation}
C_g = C_h = 12\pi^5 \int_0^\infty f^5 W_\varphi (f) |F(f)|^2\,df \end{equation} \begin{equation}
C_s = 4\pi^5 \int_0^\infty f^5 W_\varphi (f) |F(f)|^2\,df,
\label{dispersions} \end{equation} where $C_\zeta(r) = \E[ \zeta(r_0)\zeta(r_0+r)]$ is the autocorrelation function of $\zeta$, and $C_\zeta = C_\zeta(0)$ is effectively the dispersion of $\zeta$.
Similar to~\cite{VSV12}, we assume that all the distortion coefficients are normally distributed random variables, as they are the net effect of many independent perturbations along the optical path. Following the same procedure one can find statistical properties of higher order distortion coefficients. In the present paper we focus only on the first two orders because most of studied effects can be quite accurately described in this approximation.
\section{First order approximation --- transmittance density function} Now we calculate the probability density function of transmittance for the fundamental mode using the first-order perturbations only. We start with the Gaussian beam of the form \begin{equation}
E_{00} = \frac1w \sqrt{\frac2\pi} e^{-\frac{x^2+y^2}{w^2}}, \end{equation} where $w$ is the beam waist. Here we assume that the channel length is not much larger than the Rayleigh range so the beam size remains roughly the same. Calculation of the overlap integral \begin{equation}
T_{00\rightarrow 00}=\frac{\left|\int |E_{00}|^2\,
e^{i\varphi(x,y)}\,dx\,dy\right|^2}{\left(\int|E_{00}|^2\,dx\,dy\right)^2}
\label{overlap0} \end{equation} between the original beam and the linearly distorted phase~(\ref{phase}) one yields the following power transmittance \begin{equation}
T_{00\rightarrow 00} = \exp\left[ -\frac{w^2}4 \left( a^2+b^2\right)\right].
\label{t00} \end{equation} Denote $\xi = \frac{w^2}4 \left( a^2+b^2\right)$, which is a dimensionless perturbation. As $a$ and $b$ are normally distributed zero mean random variables with the dispersion of $C_a = C_b$, $\xi$ has a p.d.f. of \begin{equation}
p(\xi) = \frac2{w^2 C_a}\exp\left(-\frac{2\xi}{w^2 C_a}\right).
\label{xi_distrib} \end{equation}
To find the p.d.f. of transmittance $T = f(\xi)$ we use the standard equation \begin{equation}
p(T) = \frac{p(\xi)}{\left|\frac{df(\xi)}{d\xi}\right|}, \mathrm{ where }\;\; \xi = f^{-1}(T).
\label{pdf_derivation} \end{equation} Substituting (\ref{t00}) into (\ref{pdf_derivation}) we obtain the final p.d.f. for the channel transmittance \begin{equation}
p(T) = \frac{2}{w^2 C_a} T^{\frac{2}{w^2 C_a} - 1}.
\label{power_law} \end{equation}
One can see that in the first approximation the obtained p.d.f. is a power function of $T$, and the higher the turbulence the smaller the power. To compare the predicted p.d.f.'s with the experiment we made series of measurements with a single-mode optical channel passing through a turbulent chamber, where two streams of air with the specified temperature difference are mixed together. The measured probability distributions along with the fitted theoretical predictions are plotted in fig.~\ref{fig_transmission}. More details on the experimental part are found in the Appendix. The experiment and the theory match well, except at high transmittance values, where the first-order approximation fails due to higher order phase distortions.
\begin{figure}
\caption{Experimentally measured channel transmittance probability distribution and first-order theoretical
predictions (black dashed lines). The experimental parameter governing the turbulence strength is the airflow
temperature difference shown in the legend.
Please note, that for the better data representation probability distributions
are not properly normalized.}
\label{fig_transmission}
\end{figure}
\section{First order approximation --- modal cross-talk} The next question that we address is how the power lost from the fundamental mode is distributed among higher order modes. Here we first need to define the mode set that we use for calculations. While math is somewhat simpler for Hermite-Gaussian modes, we wanted to find a more universal solution and succeeded by properly grouping modes together.
There is a direct correspondence between the optical mode sets (Hermite- or Laguerre-Gaussian) and a 2D isotropic oscillator~\cite{DA92}, where the $N$-th power level is $(N+1)$-times degenerate. From the modes perspective it means that one can group all modes according to their ``power level''. For the Hermite-Gaussian mode HG$_{mn}$
the corresponding power level is $N=m+n$. For the Laguerre-Gaussian mode LG$_{pl}$ $N=2p+|l|$. It is easy to show that $N$-th level consists of $N+1$ distinct modes.
For each particular mode we calculate the overlap integral \begin{equation}
T_{00\rightarrow mn}=\frac{\left|\int E_{mn}^* E_{00}\,
e^{i(ax+by)}\,dx\,dy\right|^2}{\int|E_{00}|^2\,dx\,dy\int|E_{mn}|^2\,dx\,dy}.
\label{overlap_mn} \end{equation} After finding the integrals and grouping them by the ``power levels'' $N$, the cross-talk coefficients may be written in terms of $\xi$ defined earlier as they lose their individual $a$ and $b$ dependence. Here we use the explicit mode numbering for the HG set, while it will be just different mode indices for the LG set. For completeness, we also added the previous result for coupling back into the fundamental mode. \begin{equation}\begin{array}{ll}
T_0 &= T_{00\rightarrow 00} = e^{-\xi}\\
T_1 &= T_{00\rightarrow 10,01} = \xi e^{-\xi}\\
T_2 &= T_{00\rightarrow 20,11,02} = \frac{\xi^2}{2}e^{-\xi}\\
T_3 &= T_{00\rightarrow 30,21,12,03} = \frac{\xi^3}{6}e^{-\xi}\\
T_N &= T_{00\rightarrow mn: m+n=N} = \frac{\xi^N}{N!}e^{-\xi}.\\ \end{array} \end{equation} One can readily see that the total power is conserved as the obtained series sums up to 1.
Using~(\ref{xi_distrib}) and (\ref{pdf_derivation}) we find corresponding p.d.f.'s. The derivative is \begin{equation}
\frac{df}{d\xi} = \left(1-\frac \xi N\right)\frac {\xi^{N-1}}{(N-1)!} e^{-\xi} \end{equation} The solution of the equation $x^N e^{-x} /N! = a$ is $x=-N\,W\left(-\frac{(a N!)^{1/N}}N\right)$, where $W(a)$ is the Lambert $W$-function, i.e. a solution of $x e^x = a$.
The final p.d.f. of the power coupling coefficient for $N\ge 1$ is \begin{equation}
p(T_N) = \frac 2{w^2 C_a T} \left( \frac{\xi_1}{|N-\xi_1|} e^{-\frac{2\xi_1}{w^2 C_a}} +
\frac{\xi_2}{|N-\xi_2|} e^{-\frac{2\xi_2}{w^2 C_a}}\right), \end{equation} where \begin{equation}
\xi_{1,2} = -N\, W\left(-\frac{(T N!)^{1/N}}N\right). \end{equation} The maximal power coupling for the particular mode family is $T_{N\,\mathrm{max}} = \frac{N^N}{N!} e^{-N}$.
We performed experimental measurements in the turbulent chamber with the temperature difference of $100$~$\degree$C for $N$ from 0 to 2 and show the results in fig.~\ref{fig_cross_talk_pdfs}. The value of $C_a$ was obtained by fitting the $00\rightarrow 00$ curve by the power law~(\ref{power_law}), and the other two curves were calculated based on this value. As in the previous case, the largest disagreement between the theory and the experiment appears at small values of the perturbation $\xi$, because this approximation does not take into account higher order phase perturbations.
\begin{figure}
\caption{Experimental results and theoretical predictions of the fundamental mode coupling into the higher order modes.}
\label{fig_cross_talk_pdfs}
\end{figure}
\section{Second and higher order approximations} Linear phase distortions studied earlier, provide the first non-vanishing term in the power coupling efficiency. However, this effect alone poorly describes the predicted p.d.f. at small perturbations $\xi$, as higher order terms start to dominate. In the following section we include quadratic terms into the phase distortion function and calculate the corrected probability density of the fundamental mode transmittance.
With the quadratic terms included the overlap integral~(\ref{overlap0}) yields \begin{equation}
\begin{array}{l}
T = \left(1+\frac{w^4}{16}(g^2+h^2+2s^2)+\frac{w^8}{256}(s^2-gh)^2\right)^{-1/2}\\
\times \exp\left[-\frac{w^2}{16}\frac{4(a^2+b^2)+\frac{w^4}{4}(s^2a^2+s^2b^2+a^2h^2+b^2g^2-2absg -
2absh)}{1+\frac{w^4}{16}(g^2+h^2+2s^2)+\frac{w^8}{256}(s^2-gh)^2}\right].
\end{array} \end{equation} Unfortunately, analytic expressions for the transmittance probability density are unlikely to be found, so we used numerical simulation to find the desired distributions. Again we compared the experimental data with the results of simulations. Unlike the previous sections, where there was only one turbulence parameter $C_a$, here we need as well to calculate $C_g$ and $C_s$. For that we relied upon the independently measured inner and outer turbulence scales $l_0$ and $L_0$, and slightly adjusted the known Fried parameter $r_0$ to match the power of the p.d.f.'s~(\ref{power_law}). It was necessary because the precision of independently measured $r_0$ of around 10\% was not high enough to precisely match the expected power law fits. Based on the found turbulence parameters we calculated $C_g$ and $C_s$ and performed the numerical simulation. The results are shown in fig.~\ref{fig_quad}. There is a reasonably good agreement between the two, so the presented theoretical model may be used for estimation of various derived properties of the single-mode channel.
\begin{figure}
\caption{Transmittance probability density for the fundamental mode: a comparison between the experimentally measured data
(top) and the second-order theory predictions (bottom).}
\label{fig_quad}
\end{figure}
So far we only studied the transmittance of the fundamental-mode-based free-space channel in the first and the second order approximation as well as the cross-talk between the fundamental and higher order modes in the first approximation. This was of the most interest because of the obtained analytic expressions that may be used for rough parameter estimation. However, the presented framework allows to get results for any modes and precisions of phase distortion approximation.
For the particular order of phase distortion approximation one needs to find corresponding dispersions as shown in~(\ref{distortion} -- \ref{dispersions}). Then one can construct statistically correct phase distortion functions~(\ref{phase}) and calculate the overlap integral~(\ref{overlap_mn}) for the modes in question. Repeating this many times one may get the desired probability distributions. Based on our measurements in the turbulent chamber, the presented theory gives reasonable results, matching well the experimentally measured values.
\section{Discussion} The presented framework for calculations is based solely on the well respected von Karman turbulence model, which found many applications in predicting the results of many free-space optical communication experiments and astronomical observations~\cite{IBD16}. So regardless on the particular experimental realization, the obtained results are one more step towards understanding turbulent effects in single-mode optical channels and mode-multiplexed systems.
One related practical example is implementation of an active tracking system in a single-mode free-space channel. It is well-known that the major turbulent effect is beam wandering~\cite{VSV12}, i.e. the first order phase distortion, while higher order effects that change the beam profile may be much weaker. At the same time, a simple feedback loop with a fast steering mirror that controls the beam direction solves the problem of pointing error, provided the round trip time is much shorter than the characteristic time scale of the turbulent process. As this is almost always the case, active tracking systems substantially improve the quality of free-space optical channels, especially those delivering radiation into a single-mode fiber~\cite{A07,A12}.
Using the developed calculation framework, one can easily estimate the channel performance provided an ideal tracking system is implemented. To do this, the numerical simulation from the previous section is modified such that the first-order errors $a$ and $b$ are always equal to zero. Results of such simulations are shown in fig.~\ref{fig_tracking}, where a substantial improvement of channel performance is observed.
\begin{figure}
\caption{Simulated transmittance probability density for a free-space channel from fig.~\ref{fig_quad} with an ideal
channel tracking implemented. The average transmittance becomes better than 95\%, which is much superior to that
without the active tracking system. }
\label{fig_tracking}
\end{figure}
Another example is estimation of the Fried parameter based on the transmittance statistics for a static single-mode channel. Measured transmittance statistics is fitted with the power function and the value $C_a$ is obtained. To estimate the Fried parameter one uses~(\ref{dispersion1}) and a-priori knowledge of the inner and outer scales of turbulence. In real atmosphere there are more or less known values of $l_0$ and $L_0$~\cite{J06}, while for turbulent chambers $L_0$ is often the same as the size of the chamber, and $l_0$ is 2 -- 6 mm~\cite{KLB06}. In any case, $C_a$ weakly depends on $l_0$ and $L_0$, and the major contribution is from the Fried parameter $r_0$. The resulting value of Fried parameter in our experiments was always within 10\% of the independently measured one, so the described method gives reliable results.
\section{Conclusion} We presented a calculation framework that allows to answer most of questions regarding the performance of free-space single-mode or mode-multiplexed channels in turbulent air. Many first order approximations give simple analytic results that are convenient to use for quick parameter estimation. Analytic expressions are obtained for the fundamental mode power loss and for cross-coupling between the fundamental and higher order modes. Numerical calculations are required for more precise channel modeling that include second and higher order phase distortions. Many of the obtained theoretical results are supported by the experimental measurements in a turbulent chamber. Overall, there is a good match between the experiment and calculations, which is expected provided that the von Karman turbulence model matches the real-life environment.
\section*{Acknowledgments} This work was partially supported by the RFBR grant No. 17-02-00966. Results of this research work are obtained as a part of the implementation of the state support Program of NTI Centers on the basis of educational and scientific organizations in accordance with the Rules for Granting Subsidies.
\section*{Appendix: Experimental details} Here we briefly describe the experimental tools used for the measurements. Our turbulent chamber is based on two 5x5~cm$^2$ aluminum nozzle arrays that create jets of air in the opposite directions. The distance between the arrays is 5~cm, and one of them may be heated up to create the desired temperature difference. To calibrate the system we measured turbulence parameters using a Shack-Hartmann wavefront sensor. We found that the inner and outer scales of turbulence are roughly constant regardless of the temperature difference and the speed of the airflow. Their values are $l_0 = 2.7 \pm 1 $~mm and $L_0 = 51\pm 11$~mm, respectively. The Fried parameter depends almost exclusively on the temperature difference, getting little influence from changing the speed of airflow. This behavior seems to be common for turbulent chambers~\cite{J06}.
In the optical part the 780-HP single-mode fiber was used for mode filtering with $F=11$~mm collimators for beam forming. As a mode converter we used a liquid crystal based spatial light modulator (SLM) with sawtooth-like computer generated patterns~\cite{BBS13}. The fiber-coupled optical power was converted to the electrical signal with an amplified photodiode and then digitized at a sample rate of 1000~Hz with a universal data acquisition board.
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\begin{document}
\title{\texorpdfstring{\scalefont{0.95}Isometric tensor network optimization for extensive Hamiltonians is free of barren plateaus}{Isometric tensor network optimization for extensive Hamiltonians is free of barren plateaus}} \author{Qiang Miao} \affiliation{\duke} \affiliation{\dqc} \author{Thomas Barthel} \affiliation{\duke} \affiliation{\dqc}
\date{April 20, 2023}
\begin{abstract} We explain why and numerically confirm that there are no barren plateaus in the energy optimization of isometric tensor network states (TNS) for extensive Hamiltonians with finite-range interactions. Specifically, we consider matrix product states, tree tensor network states, and the multiscale entanglement renormalization ansatz. The variance of the energy gradient, evaluated by taking the Haar average over the TNS tensors, has a leading system-size independent term and decreases according to a power law in the bond dimension. For a hierarchical TNS with branching ratio $b$, the variance of the gradient with respect to a tensor in layer $\tau$ scales as $(b\eta)^\tau$, where $\eta$ is the second largest eigenvalue of the Haar-average doubled layer-transition channel and decreases algebraically with increasing bond dimension. The observed scaling properties of the gradient variance bear implications for efficient initialization procedures. \end{abstract}
\maketitle
\section{Introduction}\label{sec:intro} The rapid recent advances in quantum technology have opened new routes for the solution of hard quantum groundstate problems. Well-controlled quantum devices are a natural platform for the investigation of complex quantum matter \cite{Feynman1982-21}. A popular approach are variational quantum algorithms (VQA) \cite{Cerezo2021-9}, in which classical optimization is performed on parametrized quantum circuits. Numerous studies have successfully applied VQA to few-body systems. However, applications of generic unstructured VQA to many-body systems face multiple challenges: (a) Current quantum hardware is limited in the available number of qubits and gate fidelities. (b) As with other high-dimensional nonlinear optimization problems, we are typically confronted with complex cost-function landscapes and it is difficult to avoid local minima \cite{Kiani2020_01,Bittel2021-127,Anschuetz2021_09,Anschuetz2022-13}. (c) In contrast to classical computers, the quantum algorithms produce a sequence of probabilistic measurement results for gradients, requiring a large number of shots to achieve precise estimates. For a generic unstructured VQA, gradient amplitudes decay exponentially in the size of the simulated system \cite{McClean2018-9,Cerezo2021-12}. This is the so-called barren plateau problem. In this case, VQA are not trainable as the inability to precisely estimate exponentially small gradients will result in random walks on the flat landscape. The barren plateau problem can be resolved if an initial guess close to the optimum and good optimization strategies are available \cite{Grant2019-3,Zhang2022_03,Mele2022_06,Kulshrestha2022_04,Dborin2022-7,Skolik2021-3,Slattery2022-4,Haug2021_04,Sack2022-3,Rad2022_03,Tao2022_05}, but such resolutions are not universal \cite{Campos2021-103}.
Unstructured circuits like the hardware efficient ansatz and brickwall circuits must be deep in order to cover relevant parts of the Hilbert space \cite{Dankert2009-80,Brandao2016-346,Harrow2018_09}. The high expressiveness of such circuits \cite{Sim2019-2,Nakaji2021-5,Du2022-128} can be seen as the source for the barren plateaus \cite{Holmes2022-3}. The latter can also be motivated by the typicality properties \cite{Goldstein2006-96,Popescu2006-2} of such states \cite{Ortiz2021-2,Patti2021-3,Sharma2022-128}. So, it is generally preferable to work with more structured and less entangled classes of states that are adapted to the particular optimization problem in order to balance expressiveness and trainability. \begin{figure*}
\caption{\textbf{Isometric quantum circuits.}
(a) A generic brickwall quantum circuit for $L$ qudits.
(b) MPS quantum circuit with single-site Hilbert space dimension $d$ and bond dimension $\chi$.
(c,d) Binary 1D TTNS and MERA quantum circuits with branching ratio $b=2$ and bond dimension $\chi$. The non-shaded tensors compose the causal-cone states $|\Psi_i\rangle$ associated with local operators $\hat{h}_i$.}
\label{fig:isoTNS}
\end{figure*}
In this work, we show that VQA barren plateaus can be avoided for quantum many-body ground state problems by employing matrix product states (MPS) \cite{Baxter1968-9,Accardi1981,Fannes1992-144,White1992-11,Rommer1997,PerezGarcia2007-7,Schollwoeck2011-326}, tree tensor network states (TTNS) \cite{Fannes1992-66,Otsuka1996-53,Shi2006-74,Murg2010-82,Tagliacozzo2009-80}, or the multiscale entanglement renormalization ansatz (MERA) \cite{Vidal-2005-12,Vidal2006,Barthel2010-105}. The entanglement structure of these isometric tensor network states (TNS) is well-adapted to the one in many-body ground states. Variations of such isometric TNS can be implemented on quantum computers \cite{Liu2019-1,Miao2021_08,FossFeig2021-3,Niu2022-3,Slattery2021_08} which may allows us to substantially reduce computation costs in comparison to classical simulations \cite{Miao2021_08,Miao2023_03}. Recently, the issue of barren plateaus has been studied for certain randomly initialized TNS using graph techniques \cite{Liu2022-129,Garcia2023-2023} and ZX-calculus~\cite{Zhao2021-5,Martin2023-7} in the energy optimization for few-site product operators. Of course, the latter are actually solved by product states composed of the single-site ground states and, hence, of rather restricted practical relevance.
Here, we address the groundstate problem for extensive translation-invariant Hamiltonians \begin{equation}\label{eq:H}\textstyle
\hat{H} = \sum_i\hat{h}_i\quad\text{with}\quad\operatorname{Tr}\hat{h}_i=0 \end{equation} encountered in quantum many-body physics, where the finite-range interaction term $\hat{h}_i$ acts non-trivially around site $i$. We show that the corresponding VQA based on isometric TNS does not encounter barren plateaus. The key ideas are the following: Due to the isometry constraints, the TNS expectation value for a local interaction term $\hat{h}_i$ only depends on a reduced causal-cone state. The expectation values can be evaluated by propagating causal-cone density operators ${\hat{\rho}}$ in the preparation direction (decreasing $\tau$ in Fig.~\ref{fig:isoTNS}) with transition maps $\mc{M}$ and/or interaction terms $\hat{h}$ in the renormalization direction (increasing $\tau$) with $\mc{M}^\dag$. To evaluate the variance of the energy gradient for TNS sampled according to the Haar measure, doubled transition quantum channels $\mc{E}^{(2)}:=\Avg\mc{M}\otimes\mc{M}$ are applied to ${\hat{\rho}}\otimes{\hat{\rho}}$ and their adjoints $\mc{E}^{(2)\dag}$ to $\hat{h}\otimes\hat{h}$. While the image of ${\hat{\rho}}\otimes{\hat{\rho}}$ will quickly converge to a unique steady state, we find that the leading contribution from $\hat{h}\otimes\hat{h}$ has a decay factor $\propto(b\eta)^\tau$, where $\eta$ denotes the second largest eigenvalue of $\mc{E}^{(2)}$ and $b$ the branching ratio of the TNS ($b=1$ for MPS). This leads to three key observations: for random isometric TNS and extensive local Hamiltonians, (i) the gradient variance is independent of the total system size rather than exponentially small, (ii) the gradient variance for a tensor in layer $\tau$ of hierarchical TNS decays exponentially in the layer index $\tau$, (iii) the gradient variances decrease according to power laws in the TNS bond dimension $\chi$. Instead of Euclidean gradients in parametrized quantum circuits, we employ Riemannian gradients which greatly simplifies the proofs.
\section{Riemannian TNS gradients} All tensors in the considered isometric TNS are either unitaries $\hat{U}$ or partial isometries $\hat{V}:\mathbb{C}^{N_1}\to\mathbb{C}^{N_1}\otimes \mathbb{C}^{N_2}$ that we can implement as partially projected unitaries in the form \begin{equation}\label{eq:iso-to-unitary}
\hat{V}=\hat{U}\,(\mathbbm{1}_{N_1}\otimes|0_{N_2}\rangle) \end{equation}
with an arbitrary reference state $|0_{N_2}\rangle\in \mathbb{C}^{N_2}$. The TNS energy expectation values can be written in the form \begin{equation}\label{eq:energy}
E(\hat{U}) = \langle\Psi(\hat{U})|\hat{H}|\Psi(\hat{U}) \rangle = \operatorname{Tr}(\hat{X} \tilde{U}^\dag \hat{Y} \tilde{U} ), \end{equation} where we explicitly denote the dependence on one of the TNS unitaries $\hat{U}\in\groupU(N)$ and $\tilde{U}:=\hat{U}\otimes\mathbbm{1}_M$. The Hermitian operators $\hat{X}$ and $\hat{Y}$ depend on the other TNS tensors and $\hat{Y}$ also comprises the Hamiltonian. The Riemannian energy gradient is then given by \cite{Hauru2021-10,Luchnikov2021-23,Miao2021_08,Wiersema2022_02,Barthel2023_03} \begin{equation}\label{eq:gradRiemann}
\hat{g}(\hat{U}) = \partial_{\hat{U}}\langle\Psi|\hat{H}|\Psi\rangle = \operatorname{Tr}_M(\hat{Y}\tilde{U}\hat{X} - \tilde{U}\hat{X}\tilde{U}^\dag\hat{Y}\tilde{U}). \end{equation} Averaged according to the Haar measure, $\hat{g}$ vanishes, \begin{equation*}\textstyle
\Avg_{\hat{U}}\hat{g}:=\int \mathrm{d} U\, \hat{g}(\hat{U}) = \frac{1}{2} \int \mathrm{d} U [\hat{g}(\hat{U}) + \hat{g}(-\hat{U})] = 0. \end{equation*} In order to assess the question of barren plateaus, the Haar variance of the Riemannian gradient \eqref{eq:gradRiemann} can be quantified by \begin{eqnarray}\label{eq:var}\textstyle
\Var_{\hat{U}}\hat{g} := \Avg_{\hat{U}} \frac{1}{N} \operatorname{Tr}(\hat{g}^\dag\hat{g}). \end{eqnarray} We can expand $\hat{g}$ in an orthonormal basis of $N^2$ Hermitian and unitary operators $\{\hat{\sigma}_n\}$ with $\hat{\sigma}_n^2=\mathbbm{1}_N$ such that $\hat{g}=\mathrm{i} \hat{U}\sum_{n=1}^{N^2}\alpha_n \hat{\sigma}_n/N$. On a quantum computer, the rotation-angle derivatives can be determined as energy differences $\alpha_n=E(\hat{U}\mathrm{e}^{{\mathrm{i}\pi\hat{\sigma}_n}/{4}}) - E(\hat{U}\mathrm{e}^{{-\mathrm{i}\pi\hat{\sigma}_n}/{4}})$ \cite{Miao2021_08,Wiersema2022_02}. Equation~\eqref{eq:var} then agrees with the variance $\int\mathrm{d} U\,\frac{1}{N^2}\sum_n\alpha_n^2$, motivating the employed factor $1/N$.
We focus on the extensive Hamiltonians \eqref{eq:H} with finite-range interactions $\hat{h}_i$. Let $j$ denote the position of a unitary tensor $\hat{U}_j$ in the TNS \Footnote{For an MPS, $j$ labels a lattice site and, for the hierarchical TNS, $j=(\tau,k)$ labels a layer $\tau$ and renormalized site $k$ in that layer.} and $\mc{S}_j$ the set of physical sites $i$ with $\hat{U}_j$ in the causal cone; cf.\ Fig.~\ref{fig:isoTNS}. The gradient \eqref{eq:gradRiemann} then takes the form \begin{equation*}\textstyle
\hat{g}(\hat{U}_j)=\sum_{i\in\mc{S}_j}\hat{g}^{(i)}_j\quad\text{with}\quad
\hat{g}^{(i)}_j := \partial_{\hat{U}_j}\langle\Psi|\hat{h}_i|\Psi\rangle \end{equation*} Averaging over all unitaries of the TNS, the Haar variance of $\hat{g}(\hat{U}_j)$ reads \begin{equation}\label{eq:var0}\textstyle
\Var\hat{g}(\hat{U}_j) = \sum_{i_1,i_2\in\mc{S}_j} \Avg\, \frac{1}{N} \operatorname{Tr}\big(\hat{g}_j^{(i_1)\dag} \hat{g}_j^{(i_2)}\big). \end{equation}
\section{Matrix product states} Consider MPS of bond dimension $\chi$ for a system of $L$ sites and single-site dimension $d$, \begin{equation}
|\Psi\rangle=\!\!\sum_{s_1,\dotsc,s_L=1}^d\!\! \langle 0| \hat{V}_1^{s_1}\hat{V}_2^{s_2}\dotsb \hat{V}_L^{s_L} |0\rangle\, |s_1,s_2,\dotsc,s_L\rangle. \end{equation}
Using its gauge freedom, the MPS can be brought to left-orthonormal form \cite{Schollwoeck2011-326}, where the tensors $\hat{V}_j$ with $\langle a,s|\hat{V}_j|b\rangle:=\langle a|\hat{V}^{s_j}|b\rangle$ and $a,b=1,\dotsc,\chi$ are isometries in the sense that $\hat{V}_j^\dag \hat{V}_j = \mathbbm{1}_\chi$. We use Eq.~\eqref{eq:iso-to-unitary} to express them in terms of unitaries with $\hat{V}_j=:\hat{U}_j\,(\mathbbm{1}_\chi\otimes|0_d\rangle)$ in the bulk of the system such that $\hat{U}_j\in\groupU(\chi\, d)$ \cite{Barthel2023_03}.
Let us first address Hamiltonians \eqref{eq:H} with single-site terms $\hat{h}_i=\mathbbm{1}_d^{\otimes(i-1)}\otimes\hat{h}\otimes\mathbbm{1}_d^{\otimes(L-i)}$. Due to the left-orthonormality, the local expectation value $\langle\Psi|\hat{h}_i|\Psi\rangle$ is independent of all tensors $\hat{U}_j$ with $j<i$ such that $\mc{S}_j=\{1,\dotsc,j\}$. As we chose $\operatorname{Tr}\hat{h}_i=0$ without loss of generality, all off-diagonal contributions with $i_1\neq i_2$ in Eq.~\eqref{eq:var0} vanish. It remains to evaluate the diagonal contributions with $i_1=i_2=i\leq j$: The expectation value has the form \eqref{eq:energy}. In particular, \begin{subequations}\label{eq:mps-hi-expect} \begin{align}
\langle\Psi|\hat{h}_i&|\Psi\rangle = \operatorname{Tr}(\hat{X}_j^{(i)} \hat{U}_j^\dag \hat{Y}_j^{(i)} \hat{U}_j )\quad\text{with}\\
\hat{X}_j^{(i)}&=\mc{M}_{j+1}\circ\dotsb\circ\mc{M}_{L}(|0\rangle\langle 0|)\otimes|0_d\rangle\langle 0_d|\ \ \text{and}\\
\hat{Y}_j^{(i)}&=\mc{M}^\dag_{j-1}\circ\dotsb\circ\mc{M}^\dag_{i+1}(\hat{L}^{(i)})\otimes\mathbbm{1}_d,\quad \end{align} \end{subequations} where $\hat{L}^{(i)}=\hat{V}_i^\dag[\mathbbm{1}_\chi\otimes\hat{h}\big]\hat{V}_i$, and we have defined the site-transition map \begin{equation*}\textstyle
\mc{M}_\tau(\hat{R}):=\sum_s \hat{V}^s_\tau\hat{R} \hat{V}^{s\dag}_\tau. \end{equation*} According to Eq.~\eqref{eq:gradRiemann}, the contribution $\Avg\operatorname{Tr}\big(\hat{g}_j^{(i)\dag} \hat{g}_j^{(i)}\big)$ to the gradient variance \eqref{eq:var0} is quadratic in both $\hat{X}_j^{(i)}$ and $\hat{Y}_j^{(i)}$. The essential step is hence to evaluate the Haar averages $\Avg \hat{X}_j^{(i)}\otimes \hat{X}_j^{(i)}$ and $\Avg \hat{Y}_j^{(i)}\otimes\hat{Y}_j^{(i)}$, i.e., \begin{align}
\label{eq:mps-XXavg}
&\Avg\mc{M}^{\otimes 2}_{j+1}\circ\dotsb\circ\mc{M}^{\otimes 2}_{L}(|0,0\rangle\langle 0,0|)\quad\text{and}\\
\label{eq:mps-YYavg}
&\Avg\mc{M}^{\dag\otimes 2}_{j-1}\circ\dotsb\circ\mc{M}^{\dag\otimes 2}_{i+1}(\hat{L}^{(i)}\otimes\hat{L}^{(i)}). \end{align} Taking the Haar averages over the involved unitaries $\hat{U}_\tau$, yields the doubled site-transition channel \begin{equation}\label{eq:mps-E2}
\mc{E}_\text{mps}^{(2)} :=\Avg_{\hat{U}_\tau}\mc{M}^{\otimes 2}_\tau= |\hat{r}_1\rangle\!\rangle\langle\!\langle \mathbbm{1}_{\chi^2}| + \eta_\text{mps}\, |\hat{r}_2\rangle\!\rangle\langle\!\langle\hat{\ell}_2|. \end{equation}
Here, we have already written its diagonalized form, using a super-bra-ket notation for operators based on the Hilbert-Schmidt inner product $\langle\!\langle\hat{A}|\hat{B}\rangle\!\rangle:=\operatorname{Tr}(\hat{A}^\dag\hat{B})$. The diagonalization, as detailed in the companion paper \cite{Barthel2023_03}, shows that $\mc{E}_\text{mps}^{(2)}$ has a unique steady state $\hat{r}_1$ and first excitation $\hat{r}_2$ with the eigenvalue $\eta_\text{mps}=\frac{1-1/\chi^2}{d-1/(\chi^2 d)}$. The repeated application of $\mc{E}_\text{mps}^{(2)}$ in Eq.~\eqref{eq:mps-XXavg} quickly converges to $\hat{r}_1$. Similarly, its application in Eq.~\eqref{eq:mps-YYavg} would converge to the corresponding left eigenoperator $\mathbbm{1}_{\chi^2}$, but one finds that this does not contribute to the gradient variance. It is the subleading term $\propto \eta_\text{mps}^{j-i}\hat{\ell}_2$ that ultimately yields \begin{equation}\label{eq:mps-var-g}
\frac{\Avg\operatorname{Tr}\big(\hat{g}_j^{(i)\dag}\hat{g}_j^{(i)}\big)}{\chi\, d}
=\frac{2\operatorname{Tr}(\hat{h}^2)}{d(\chi^2d+1)}\, \eta_\text{mps}^{j-i} + \mc{O}(\eta_\text{mps}^{L-i}). \end{equation} Finally, the variance \eqref{eq:var0} for the extensive Hamiltonian is obtained by summing the contributions \eqref{eq:mps-var-g} for all $i\leq j$, resulting in the system-size independent value \begin{equation}\label{eq:mps-var-gTot}
2\operatorname{Tr}(\hat{h}^2)\frac{\chi^2d^2-1}{d(d-1)(\chi^2d+1)^2} + \mc{O}(\eta_\text{mps}^{j}) + \mc{O}(\eta_\text{mps}^{L-j}), \end{equation} where the sub-leading terms are due to boundary effects. \begin{figure}
\caption{\textbf{MPS gradient variance.} For spin-1/2 chains \eqref{eq:H} of length $L=101$ with single-site terms $\hat{h}_i$ (upper panel) and nearest-neighbor interactions (lower panel), we plot the gradient variance \eqref{eq:var0} for the tensor at site $j$. Numerical averages over \numprint{64000} MPS with tensors sampled according to the Haar measure agree with the analytical result \eqref{eq:mps-var-gTot}.}
\label{fig:mps-var}
\end{figure}
The optimization problem with single-site terms $\hat{h}_i$ is trivially solved by product states but, qualitatively, things do not change for finite-range interactions $\hat{h}_i$. The second largest eigenvalue $\eta_\text{mps}$ of the doubled transition channel \eqref{eq:mps-E2}, remains the most important quantity. A detailed analysis is given in the companion paper \cite{Barthel2023_03}. Figure~\ref{fig:mps-var} confirms the analytical prediction in numerical tests. MPS optimizations have \emph{no} barren plateaus for extensive Hamiltonians.
\section{Hierarchical TNS}
MERA \cite{Vidal-2005-12,Vidal2006,Barthel2010-105} are hierarchical TNS. Starting on $\mc{N}$ sites, in each renormalization step $\tau\to\tau+1$, we apply local unitary disentanglers $\hat{U}_{\tau,k}$ of layer $\tau$ before the number of degrees of freedom is reduced by applying isometries $\hat{V}_{\tau,k}$ that map groups of $b$ sites into one renormalized site. The associated site dimension $\chi$ is the bond dimension and $b$ is the so-called branching ratio. The process stops at the top layer $\tau=T$ by projecting each of the remaining $\mc{N}/b^{T}$ sites onto a reference state $|0_\chi\rangle$. The renormalization procedure, seen in reverse, generates the MERA $|\Psi\rangle$. TTNS \cite{Fannes1992-66,Otsuka1996-53,Shi2006-74,Murg2010-82,Tagliacozzo2009-80} are a subclass of MERA without disentanglers.
As all tensors are isometric, the evaluation of local expectation values drastically simplifies, where $\langle\Psi|\hat{h}_i|\Psi\rangle$ only depends on the tensors in the causal cone of $\hat{h}_i$. See Fig.~\ref{fig:isoTNS}. In fact, the expectation value can again be written in a form very similar to Eq.~\eqref{eq:mps-hi-expect} but, now, we have transition maps $\mc{M}_{\tau,i}$ that map the causal-cone density operator ${\hat{\rho}}_\tau$ into ${\hat{\rho}}_{\tau-1}=\mc{M}_{\tau,i}({\hat{\rho}}_\tau)$.
Specifically, for the binary one-dimensional (1D) MERA in Fig.~\ref{fig:isoTNS}d and a three-site interaction term $\hat{h}_i$, we start in the top layer with the three-site reference state ${\hat{\rho}}_T=(|0_\chi\rangle\langle 0_\chi|)^{\otimes 3}$ and then progress down layer by layer, applying either a left-moving or a right-moving transition map $\mc{M}_{\tau,i}$. These consist in applying three isometries that double the number of (renormalized) sites, then applying two disentanglers and, finally, tracing out one site on the left and two on the right (left-moving) or vice versa (right-moving). The diagonal contributions to the gradient variance for $\hat{U}_{\tau,k}$ with $i_1=i_2=i$ in Eq.~\eqref{eq:var0}, are functions of $\Avg \hat{X}_{\tau,k}^{(i)}\otimes \hat{X}_{\tau,k}^{(i)}$ and $\Avg \hat{Y}_{\tau,k}^{(i)}\otimes\hat{Y}_{\tau,k}^{(i)}$, where label $j$ has been replaced by a layer number $\tau$ and renormalized site $k$ in that layer. Taking the Haar average of $\mc{M}_{t,i}^{\otimes 2}$, we obtain either a left-moving or a right-moving doubled layer-transition channel $\mc{E}^{(2)}_{\text{bi},\text{L}}$ and $\mc{E}^{(2)}_{\text{bi},\text{R}}$. Summing over all sites $i\in\mc{S}_{\tau,k}$ that have $\hat{U}_{\tau,k}$ in their causal cone, corresponds to summing over all possible sequences of the two channels. This is equivalent to applying the map $2\mc{E}^{(2)}_{\text{bi}}$ for layers $t=1,\dotsc,\tau-1$, where \begin{equation}\label{eq:MERAbin-E2}
\mc{E}^{(2)}_\text{bi}:= \frac{1}{2}\left(\mc{E}^{(2)}_{\text{bi},\text{L}}+\mc{E}^{(2)}_{\text{bi},\text{R}}\right) \end{equation} is the average transition channel. Finally, averaging the gradient variance $\Var\hat{g}(\hat{U}_{\tau,k})$ with respect to $k$ in layer $\tau$ corresponds to applying $\mc{E}^{(2)}_\text{bi}$ for all layers $t=\tau+1,\dotsc,T$.
The channel $\mc{E}^{(2)}_\text{bi}$ is diagonalizable and gapped, \begin{align}\label{eq:MERAbin-E2-diag}
&\textstyle\mc{E}^{(2)}_\text{bi} = |\hat{r}_1\rangle\!\rangle \langle\!\langle\hat{\ell}_1| + \sum_{n=2}^4\lambda_n |\hat{r}_n\rangle\!\rangle \langle\!\langle\hat{\ell}_n|\ \ \text{with}\\ \nonumber
&\textstyle
\hat{\ell}_1=\mathbbm{1}_{\chi^6},\ \
\eta_\text{bi} := \lambda_2 = \frac{\chi^2(1+\chi)^4}{2(1+\chi^2)^4},\ \
\lambda_3 = \frac{\chi^2(1+\chi)^2}{2(1+\chi^2)^3}, \end{align}
biorthogonal left and right eigenvectors $\langle\!\langle\hat{\ell}_n|\hat{r}_{n'}\rangle\!\rangle=\delta_{n,n'}$, and $\frac{1}{2}>\lambda_2>\lambda_3>\lambda_4$. Similar to the analysis for MPS, the leading term in $\Avg \hat{X}_{\tau,k}^{(i)}\otimes \hat{X}_{\tau,k}^{(i)}$ stems from the $\mc{E}^{(2)}_\text{bi}$ steady state $\hat{r}_1$, while the leading term in $\Avg \hat{Y}_{\tau,k}^{(i)}\otimes \hat{X}_{\tau,k}^{(i)}$ stems from the first excitation $\hat{\ell}_2$. In this way, one finds that the diagonal contributions to the gradient variance $\Var\partial_{\hat{U}_{\tau,k}}\langle\Psi|\hat{H}|\Psi\rangle$ [Eq.~\eqref{eq:var0}], averaged for all $k$ in layer $\tau$, scale as \begin{equation}\label{eq:var-mera}
\Var\hat{g}(\hat{U}_{\tau}) = \Theta((2\eta_\text{bi})^\tau) + \mc{O}((2\lambda_3)^\tau) + \mc{O}(2^\tau\eta_\text{bi}^T), \end{equation}
where the Landau symbol $\Theta(f)$ indicates that there exist upper and lower bounds scaling like $f$. The off-diagonal terms with $|i_1-i_2|>3$ vanish due to $\operatorname{Tr}\hat{h}_i = 0$ and the remaining off-diagonal terms have the same scaling as the diagonal terms. See Ref.~\cite{Barthel2023_03} for details. \begin{figure*}
\caption{\textbf{Gradient variance in heterogeneous 1D MERA and TTNS.} In the hierarchical isometric TNS, the leading term in the Haar-variance of the energy gradient is system-size independent and decreases exponentially in the layer index $\tau$. The graphs on left show this exponential decay observed in numerical tests for bond dimension $\chi=2$. Black lines indicate the fitting range for the decay factors. The inserts show the gradient variance at $\tau = 1$ as a function of the number of layers $T$, corresponding to varying system size $L=b^T$. The graphs on the right show a perfect agreement of the decay factors, extracted from simulations for various bond dimensions $\chi$, and the theoretical prediction $b\eta$. The inserts assert the power-law decay of gradient variances in layer $\tau=1$ with respect to $\chi$. For simplicity, we choose the physical single-site dimensions as $d=\chi$ and Hamiltonians $\hat{H}=\sum_{i=1}^L\hat{h}_i$ with normalized two or three-site interactions $\hat{h}_i$ constructed from generalized Gell-Mann matrices according to Eq.~\eqref{eq:GellMann-h}.}
\label{fig:mera-var}
\end{figure*}
The analysis for the binary 1D MERA can be extended to all MERA and TTNS. The central object in the evaluation of their Haar-averaged gradient variances are doubled layer-transition channels $\mc{E}^{(2)}$. The gradient variance for tensors in layer $\tau$ will then scale as $(b\eta)^\tau$, where $\eta$ is the second largest $\mc{E}^{(2)}$ eigenvalue. This eigenvalue decreases algebraically with increasing bond dimension $\chi$ such that $b\eta < 1$ for $\chi>1$. Specifically, we find $\eta=\chi/(1+\chi^2)$ for binary 1D TTNS, $\eta=1/(3\chi^2)+\mc{O}(\chi^{-4})$ for ternary 1D MERA, $\eta=\chi^2/(1+\chi^2+\chi^4)$ for ternary 1D TTNS, and $\eta=1/(9\chi^8)+\mc{O}(\chi^{-10})$ for the 2D $3\times 3\mapsto 1$ MERA \cite{Evenbly2009-79}. So, for each layer $\tau$, the gradient variance is an algebraic function of the bond dimension $\chi$ and (up to corrections) independent of the total system size. Therefore, the optimization of hierarchical TNS is \emph{not} hampered by barren plateaus.
For the 1D TNS, these analytical results are tested and confirmed numerically as shown in Fig.~\ref{fig:mera-var}. For the numerical tests, we choose the physical single-site dimension $d$ equal to the bond dimension $\chi$. Otherwise, one would use the lowest MERA layers to increase bond dimensions gradually from $d$ to the desired $\chi$. The isotropic interaction terms $\hat{h}_i$ are constructed using generalized $\chi\times\chi$ Gell-Mann matrices $\{\hat{\Lambda}^{1},\hat{\Lambda}^{2},\cdots,\hat{\Lambda}^{\chi^2-1}\}$ \cite{Kimura2003-314,Bertlmann2008-41}. They are traceless and Hermitian generalizations of the Pauli matrices $\hat{\sigma}^a$ for $\chi=2$ and the Gell-Mann matrices for $\chi=3$. As generators of the special unitary group SU$(\chi)$, they satisfy the orthonormality condition $\langle\!\langle\hat{\Lambda}^a |\hat{\Lambda}^b\rangle\!\rangle = 2\delta_{a,b}$. We define $n$-site interactions as \begin{equation}\label{eq:GellMann-h}\textstyle
\hat{h}_i = \frac{1}{\sqrt{2^n(\chi^2-1)}}\sum_{a=1}^{\chi^2-1} \hat{\Lambda}^a_{i}\otimes\dotsb\otimes\hat{\Lambda}^a_{i+n-1}, \end{equation} which are traceless, have vanishing partial traces, and are normalized according to $\operatorname{Tr}(\hat{h}_i^2)=1$. Each data point in Fig.~\ref{fig:mera-var} corresponds to 1000 TNS with tensors sampled according the Haar measure. The numerical results confirm the scaling $(b\eta)^\tau$ with corrections at small $\tau$ and due to the finite number of layers $T$. The extracted decay factors $b\eta$ show the predicted $\chi$ dependence. As a numerically sampling of 2D MERA and TTNS is computationally expensive, a direct extraction of decay factors from sampled gradient variances is currently not possible for 2D.
\section{Homogeneous and Trotterized MERA and TTNS} So far, we have exclusively considered heterogeneous TNS, where all tensors can vary freely. To save computational resources, one can also work with homogeneous TNS where, in the case of MERA and TTNS, all equivalent disentanglers and isometries of a given layer $\tau$ are chosen to be identical. Such homogeneous TNS can also be used to initialized the optimization of heterogeneous TNS. The theoretical analysis of gradient variances for homogeneous TNS is more involved. While the analysis for heterogeneous TNS only requires first and second-moment Haar-measure integrals, more complicated higher-order integrals are needed for the homogeneous states. Hence, we determine the gradient variance for homogeneous TTNS and MERA numerically as shown in Fig.~\ref{fig:tmera-var}, finding that homogeneous TNS have considerably larger gradient variances than the corresponding heterogeneous states. This is consistent with findings in Refs.~\cite{Volkoff2021-6,Pesah2021-11} for other classes of states.
Isometric TNS can be implemented on quantum computers, but it is advisable to impose a substructure for the TNS tensors to reduce costs and achieve a quantum advantage. Specifically, in Trotterized MERA \cite{Miao2021_08,Miao2023_03,Kim2017_11}, each tensor is constructed as brickwall circuit with $t$ (Trotter) steps. A generic full MERA can be recovered by increasing $t$. Fig.~\ref{fig:tmera-var} compares gradient variances for homogeneous Trotterized MERA and full MERA as well as those for Trotterized and full TTNS. The data shows that Trotterized TNS have larger gradients variances than full TNS, and the former converge to the latter as $t$ increases.
\section{Discussion} \begin{figure}
\caption{\textbf{Gradient variances for homogeneous Trotterized TNS.} For the model \eqref{eq:GellMann-h}, the plot shows the gradient variances at layer $\tau$ for homogeneous binary 1D Trotterized TTNS (blue dashed lines), full TTNS (blue full line), Trotterized MERA (gray dashed lines) and full MERA (black full line).}
\label{fig:tmera-var}
\end{figure} The presented results suggest that isometric TNS generally feature no barren plateaus in the energy optimization for extensive models with finite-range interactions. It should be rather straightforward to generalize to systems with $k$-local interactions. The observed scaling of the gradient variance has implications for efficient initialization schemes: For MPS, the power-law decay in the bond dimension $\chi$ suggests to start with an optimization at small $\chi$ and to then gradually increase it. For TTNS and MERA, the exponential decay in the layer index $\tau$, suggests that iteratively increasing the number of layers during optimization can substantially improve the performance. TTNS have considerably larger gradient variances than MERA. For MERA optimizations, it can hence be beneficial to initially choose all disentanglers as identities and only start their optimization after the corresponding TTNS has converged.
The analysis in Ref.~\cite{Kim2017_11} suggests that VQA based on hierarchical TNS is robust with respect to noise in the quantum devices. A natural extension of our analysis would be to study the doubled transition channels with environment fluctuations. We expect that no noise-induced barren plateaus \cite{Wang2021-12,Wiersema2021_11} occur for TNS.
The methods employed in this work could also be applied to study statistical properties and typicality for random TNS \cite{Garnerone2010-81,Garnerone2010-82,Haferkamp2021-2}, as well as the dynamics of quantum information and entanglement in structured random quantum circuits~\cite{Nahum2017-7,Nahum2018-8,Zhou2019-99,Potter2022-211,Fisher2022_07}.
The absence of barren plateaus in the discussed isometric TNS does not depend on the choice of Riemannian gradients over Euclidean gradients. However, in our practical experience, the parametrization-free Riemannian optimization as described in Ref.~\cite{Miao2021_08} has better convergence properties and can mitigate some effects of spurious local minima observed for parametrized quantum circuits~\cite{You2021-139,Anschuetz2022-13,Liu2022_06}. Further research on this issues is needed.
\begin{acknowledgments} We gratefully acknowledge discussions with Daniel Stilck Fran\c{c}a and Iman Marvian as well as support through US Department of Energy grant DE-SC0019449. \end{acknowledgments}
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\begin{document}
\title{Unitary-Coupled Restricted Boltzmann Machine Ansatz for Quantum Simulations} \author{Chang Yu Hsieh} \email{[email protected]} \affiliation{Tencent Quantum Lab, Shenzhen, Guangdong 518057, China} \author{Qiming Sun} \affiliation{Tencent America, Palo Alto, CA 94306 , United States} \author{Shengyu Zhang} \affiliation{Tencent Quantum Lab, Shenzhen, Guangdong 518057, China} \author{Chee Kong Lee} \email{[email protected]} \affiliation{Tencent America, Palo Alto, CA 94306 , United States} \begin{abstract} Neural-Network Quantum State (NQS) has attracted significant interests as a powerful wave-function ansatz to model quantum phenomena. In particular, a variant of NQS based on the restricted Boltzmann machine (RBM) has been adapted to model the ground state of spin lattices and the electronic structures of small molecules in quantum devices. Despite these progresses, significant challenges remain with the RBM-NQS based quantum simulations. In this work, we present a state-preparation protocol to generate a specific set of complex-valued RBM-NQS, that we name the unitary-coupled RBM-NQS, in quantum circuits. This is a crucial advancement as prior works deal exclusively with real-valued RBM-NQS for quantum algorithms.
With this novel scheme, we achieve (1) modeling complex-valued wave functions, (2) using as few as one ancilla qubit to simulate $M$ hidden spins in an RBM architecture, and (3) avoiding post-selections to improve scalability.
\end{abstract}
\maketitle
Hybrid quantum-classical (HQC) algorithms\cite{mcclean2016theory} offer an exciting avenue to explore the potential of a noisy intermediate-scale quantum\cite{preskill2018quantum} (NISQ) device without quantum error corrections. The HQC algorithms run on parametrized quantum circuits aiming to minimize an objective function, such as the average energy. The Variational Quantum Eigensolver\cite{peruzzo_natcomm_14} (VQE) and Quantum Approximate Optimization Algorithm\cite{Farhi2014} (QAOA) are two prominent examples leading the current wave of HQC algorithm developments. In particular, VQE has been experimentally demonstrated on several leading platforms of quantum computations\cite{peruzzo_natcomm_14,kandala2017hardware,hempel2018quantum,colless2018computation,hempel2018quantum,sagastizabal2019experimental}. These encouraging experimental outcomes strengthen our anticipation that quantum simulations\cite{aspuru2005simulated,li2019variational,cao2018quantum,mcardle2018quantum,childs2018toward} should be among the first set of applications to benefit from quantum computations. Nevertheless, it is also becoming increasingly clear that further developments\cite{wecker_pra2015,kuhn2018accuracy,li2019electronic,reiher2017elucidating,kivlichan2018quantum,babbush2018low} are required to improve VQE and similar HQC algorithms if the goal is to establish an unambiguous quantum advantage for problems of realistic interests. For instance, many recent developments address the following aspects: (1) design novel wave function ansatz\cite{wecker_pra2015,ryabinkin2018qubit,ryabinkin2019iterative,dallaire2019low,kandala2017hardware,romero2018strategies,o2019calculating,benfenati2019extended} with efficient usage of variational parameters and circuit depth, (2) reduce the number of required measurements\cite{zhao2019measurement,rubin2018application,crawford2019efficient,izmaylov2019revising,izmaylov2019unitary,verteletskyi2019measurement, huggins2019efficient,mitarai2019methodology}, and (3) overcome the challenge of high-dimensional optimization\cite{mcardle2018img,yang2017optimizing,zhu2018training,shaydulin2019multistart,guerreschi2017practical,nakanishi2019sequential,parrish2019jacobi, parrish2019hybrid,schuld2019evaluating,moseley3bayesian} needed for training parametrized circuits. Any of these technical challenges could potentially become a computational bottleneck for a HQC algorithm beyond the small-scale testings reported in the recent literature.
In this work, our primary focus is to investigate whether neural-network quantum states (NQS) \cite{sarma2019machine,melko2019restricted,jia2019quantum} can be tailored to better fit the paradigm of hybrid quantum-classical algorithms. We focus on a particular form of NQS based on the restricted Boltzmann Machine (RBM) architecture. Within the communities of computational many-body physics, quantum information and condensed matter physics, there is a growing trend of adopting neural networks techniques, such as the RBM architecture, for various applications. Notable examples include identification of different phases of matter\cite{torlai2016learning,carrasquilla2017machine, kaubruegger2018chiral,koch2018mutual,czischek2018quenches,lu2019efficient}, scalable quantum state tomography\cite{xu2018neural,torlai2019integrating}, efficient sampler to accelerate Monte Carlo simulations\cite{huang2017recommender,wang2017exploring,inack2018projective}, quantum error correction codes\cite{torlai2017neural,bausch2018quantum,zhang2018efficient,jia2019efficient}, and variational ansatz for many-body simulations\cite{carleo2017solving,deng2017machine,saito2018method,hartmann2019neural,nagy2019variational,yoshioka2019constructing,luo2019backflow, torlai2018latent,nomura2017restricted,jonsson2018neural,glasser2018neural}. Especially, the work \cite{carleo2017solving} by Carleo and Troyer demonstrated that a complex-valued RBM model can efficiently model manybody wavefunctions with fewer variational parameters than tensor-network methods for few spin-lattice models. Subsequent investigations\cite{deng2017quantum,huang2017neural,Gao2017,clark2018unifying,chen2018equivalence} have further clarified and affirmed the usefulness of variants of Boltzmann Machines to model many-body quantum states of complex systems. These encouraging results and rapidly accumulated knowledge about RBM-NQS motivated us to investigate whether it is suitable to apply this family of ansatz in the context of quantum simulation algorithms. In the most part of this work, we should use RBM-NQS and NQS interchangeably when there is no risk of ambiguity,.
While there are already quantum simulation algorithms\cite{xia2018rbm,gardas2018dwave} using NQS as variational ansatz with encouraging results, some fundamental obstacles limit their scope of applications. For instance, the existing approaches require the preparation of an extended wave function composed of all visible and hidden spins. As each hidden spin is explicitly modeled with an ancilla qubit, these prior methods consume too many qubits, which are expensive resources for near-term quantum devices. Furthermore, there is no scalable strategy for preparing a general NQS in quantum circuits. This is because many NQS can only be obtained via non-unitary transformations on an input state. Existing NQS based simulation algorithm\cite{xia2018rbm} relies on a probabilistic post-selection to achieve the non-unitary operations. Finally, the lack of complex parameters severely limits the usefulness of the NQS for quantum simulations. For instance, (1) Complex-valued wave functions allow us to simulate fermions in time reversal symmetry breaking systems such as electrons in the presence of a magnetic field. (2) To simulate quantum dynamics, it is a necessity to account for the accumulation of dynamical phase factors.
The aforementioned limitations certainly have cast doubts on whether the NQS should be used in the cquantum circuits; despite many of their theoretical merits as wave-function ansatz and convincing demonstrations in numerical studies (i.e. classical simulations) in a broad range of physical systems cited above. To address these deficiencies, we propose a state-preparation protocol for creating complex-valued NQS in a quantum circuit. In particular, the state-preparation protocol does not use $N+M$ qubits to model $N$ visible spins and $M$ hidden spins explicitly. This is because every term in a RBM Hamiltonian commutes with each other, we can explicitly arrange the order in which the unitary gates acting on the hidden spins. Hence, a single ancilla qubit (representing one hidden spin) can be recycled upon measurement and be reused to represent another hidden spin in a subsequent stage. This qubit-recyle scheme \cite{liu2019recycle,Huggins_2019} tremendously reduces the number of physical qubits (down to just one extra ancilla at the bare minimum) needed to execute the proposed state-preparation protocol. This advantage cannot be underestimated as typical RBM-NQS might use as many as $M=$Poly($N$) hidden spins. To avoid the probabilistic scheme to mediate the non-unitary couplings between the visible and hidden layers, we further propose to only consider unitary couplings between visible and hidden spins under the RBM architecture in the main text.
As extensively discussed in Supplementary II and illustrated with numerics reported in the Result section, we show that arbitrary NQS can be systematically mapped onto these unitary-coupled RBM-NQS without sacrificing expressive power to model a variety of quantum systems in our empirical studies. In Supplementary V, we also provide an extension of the state-preparation protocol to handle arbitrary complex-valued coupling parameters across layers. Furthermore, we propose a novel approach to avoid post-selections on hidden-spin states. In this scheme, we decompose a single NQS into an ensemble of modified NQS, which are essentially outputted by the quantum circuit when the post-selection fails (i.e. the outcome of the projective measurement is not the desired one). Under the existing scheme, one would either discard the present result and restart after a failed post-selection or attempt to recover the input quantum state from a failed post-selection. In this ensemble scheme, one does not have to go through either of these repetitive procedures. In our case, estimates of observables on the original NQS can be extracted from the ensemble of ``post-selection failures" within a Monte Carlo framework.
Finally, due to the ``diagonal structure" of the NQS ansatz, the simulation algorithm based on the imaginary-time dependent variational principle \cite{mcardle2018img} can be dramatically simplified as one just needs to perform measurement on one quantum circuit instead of working with $\mathcal{O}(N_{\text{var}}^2)$ different quantum circuits where $N_{\text{var}}$ is the number of total variational paramters. For large-scale simulations, $N_{\text{var}}\propto \mathcal{O}(MN)$ could be a huge number. In summary, the present work has both expanded the scope of application for NQS based hybrid quantum-classical simulations and has lowered the barrier for experimental implementations. In order to frame the significance of this work in a better perspective, we summarize and clarify the challenges we set out to address in prior works in Supplementary I.
\section*{Results}
\textbf{Preparing a complex-valued NQS in a quantum circuit.} A brief introduction of NQS may be found in the Method section. All NQS may be obtained from at least one bipartite Ising Hamiltonian $ \hat{H}_{RBM}(\theta) = \sum_{i} b_i \hat{v}^z_i + \sum_{j} m_j \hat{h}^z_j + \sum_{ij}W_{ij} \hat{v}^z_i \hat{h}^z_j$, where $\hat v^z_i$ or $\hat h^z_j$ is the Pauli Z operator for the visible or hidden qubit, respectively. We also denote complex-valued RBM parameters as $\theta = [b_1,\cdots, b_N, m_1 \cdots, m_M, W_{11} \cdots, W_{NM}]$, and use subscripts $R$ and $I$ to denote the real and imaginary parts, respectively. The complex-valued NQS can be created with a two-step approach. First, entangle $N+M$ qubits (including all visible and hidden spins of an RBM architecture) according to
\begin{eqnarray}\label{eq:vhwf}
|\Psi_{vh} (\theta) \rangle &=& \left[\frac{e^{\hat{H}_{RBM}(\theta)} }{\sqrt{\prescript{}{vh}{\bra{++ \dots +}}
e^{2\hat{H}^R_{RBM}(\theta)}\ket{++\dots+}_{vh}}}\right]\ket{++ \dots +}_{vh},
\end{eqnarray}
where $\ket{+}=\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})$, $\hat H^R_{RBM}(\theta)$ is the Hermitian part of the RBM Hamiltonian and the subscript $vh$ denotes all visible and hidden (ancilla) qubits. Equation \ref{eq:vhwf} gives a conceptually simple wave function that could be generated by first applying single-qubit transformations $\exp(b_i\hat v^z_i )$ and $\exp(m_j\hat h^z_j )$ on individual qubits followed by $\exp(W_{ij}\hat v^z_i \hat h^z_j)$ to couple qubits. The quantum operations are non-unitary when RBM parameters take on real parts, i.e. $b_i^R \neq 0$, $m_j^R \neq 0$ or $W^{R}_{ij} \neq 0$. In general, the non-unitary two-qubit operation mediating entanglement across the visible-hidden layer are difficult to implement. One could adopt the probabilistic scheme introduced in Ref.~\cite{xia2018rbm} to generate the inter-layer couplings with an extra ancilla qubit. However, for complex-valued wave function, this probabilistic approach in Ref.~\cite{xia2018rbm} is difficult to scale with the number of qubits involved.
Once the extended wave function $\ket{\Psi_{vh}(\theta)}$ is generated, all ancilla qubits (i.e. hidden spins) are post-selected for $\ket{+}_h$ and the desired NQS in Eq.~\ref{eq:nnqs} is reconstructed in the quantum circuit,
\begin{eqnarray}\label{eq:bmcircuit}
|\Psi_v (\theta) \rangle &=& \frac{\prescript{}{h}{\left\langle ++ \dots + \vert \Psi_{vh}(\theta) \right\rangle}}{\sqrt{ \langle \Psi_{vh}(\theta) | \hat{P}^{(h)}_+ | \Psi_{vh}(\theta)\rangle }}
\nonumber \\
& = & \frac{1}{N_v} \sum_{\mathbf{h}} e^{\hat H_{RBM}(\theta,\mathbf{h})} \ket{++\dots+}_v, \end{eqnarray}
where $\hat{P}_+^{(h)} = \ket{++\dots+}_h\bra{++\dots+}$, $N_v = \sqrt{\prescript{}{vh}{\langle ++\cdots+ |}e^{\hat{H}_{RBM}(\theta)} \hat{P}^{(h)}_+ e^{\hat{H}_{RBM}(\theta)} | ++\cdots+ \rangle_{vh}}$ and $e^{\hat H_{RBM}(\theta,\mathbf h)}$ is an operator acting on the visible spins only as we replace the Pauli operator $\hat h^z_j$ with a binary value ($\pm 1$) of $h_j \in \mathbf{h}=[h_1,\cdots , h_M]$. From the second line of Eq.~\ref{eq:bmcircuit}, it is clear that the hidden spins jointly mediate a specific quantum transformation, $\sum_{\mathbf{h}} e^{\hat H_{RBM}(\theta,\mathbf{h})}$, on the visible-spin wave function. Since this transformation is non-unitary in general, the amplitude-amplification type of techniques\cite{brassard2002quantum,Berry_oaa_2014} is not immediately applicable to enforce the post-selection. This inconvenient fact creates another obstacle to prepare complex-valued NQS in quantum circuits.
In the rest of this section, we describe a scalable state-preparation protocol that overcomes the two obstacles. In particular, we implement complex-valued NQS using as few as one ancilla qubit and entirely avoids the post-selection. To simplify presentations, we illustrate how to prepare a subset of NQS that we dub the unitary-coupled RBM-NQS, which only allow purely imaginary-valued inter-layer couplings $W_{ij}=iW^I_{ij}$. Generation of unitary-coupled RBM-NQS bypass the inherent challenge to mediate entanglement via non-unitary transformations. Figure \ref{fig:circuit}a gives a circuit diagram of preparing a unitary-coupled RBM-NQS composed of two visible spins with inter-qubit couplings mediated by two hidden spins. The Hadarmard gates prepare the $\ket{+}$ state, and the parametrized single-qubit rotations are not fixed along the $z$ axis because of the non-unitary operations, $\exp\left(b_i^R \hat v^z_i\right)$ and $\exp\left(m_i^R \hat h^z_i\right)$. In the circuit diagram of panel a, the single-qubit rotations $R_n(\theta)$ are determined via relations of the form $\exp\left(b_i^R \hat v^z_i\right)\ket{+}/c = R_n(\theta)\ket{+}$ with the normalization factor $c=\sqrt{\bra{+}\exp\left(2 b_i^R \hat v^z_i\right)\ket{+}}$. Figure \ref{fig:circuit}b gives a schematic depicting the RBM state generated by the circuit in panel a. More importantly, as explained in the supplementary II, the unitary-coupled RBM-NQS does not necessarily suffer loss of expressive power. While we claim it is better to model quantum systems with unitary-coupled RBM-NQS for near-term applications; there is a straightforward extension of the current protocol to generate arbitrary complex-valued NQS in case it is desired. We defer the discussion of this extension to Supplementary V.
\begin{figure}
\caption{Unitary-Coupled RBM-NQS. (a): quantum circuit to prepare a two-qubit unitary-coupled RBM state having two hidden spins. The inter-layer coupling is mediated by unitary gates. Note $\theta^\prime_{ij}=-\theta_{ij}$. (b): schematic of the RBM state generated by the quantum circuit in panel (a).}
\label{fig:circuit}
\end{figure}
\textbf{Scalable preparation of a unitary-coupled RBM-NQS in a quantum circuit.}
To begin, we note Eq.~\ref{eq:bmcircuit} can be cast in an alternative form
\begin{eqnarray}\label{eq:qrecycle} \ket{\Psi_v(\theta)} & = & \frac{1}{N_v}
\left[\prescript{}{}{\bra{+}}\left[e^{\hat h^z_M \left(m_M + \sum_i iW^I_{iM}\hat v^z_i\right)}\right]\ket{+}\right]_{M}
\left[\prescript{}{}{\bra{+}}\left[e^{\hat h^z_{M-1} \left( m_{M-1} + \sum_i iW^I_{iM-1} \hat v^z_i\right)}\right]\ket{+}\right]_{M-1} \cdots
\nonumber \\
& & \times \left[\prescript{}{}{\bra{+}}\left[e^{\hat h^z_1 \left(m_1 + \sum_i iW^I_{i1}\hat v^z_i\right)}\right]\ket{+}\right]_{1}\,\,
e^{\sum_i b_i \hat v^z_i} \ket{++\cdots+}_v. \end{eqnarray}
Each block, $\left[\prescript{}{}{\bra{+}}[ \cdots ]\ket{+}\right]_{j}$, encodes j-th hidden spin's effects on all visible ones. Clearly, one can use a single ancilla qubit to implement these transformations sequentially. As shown in Eq.~\ref{eq:qrecycle}, we specifically consider $W_{ij}=iW^I_{ij}$ for the unitary-coupled RBM-NQS. Our proposed approach to bypass the post-selection of $\ket{+}$ on all hidden spins is inspired by the following observation,
\begin{eqnarray}\label{eq:block-decomp}
\prescript{}{}{\bra{+}}\left[e^{\hat h^z_j \left(m_j + \sum_i iW^I_{ij}\hat v^z_i\right)}\right]\ket{+} & = & \sum_{s=\pm} \bra{+} e^{m^R_j \hat h^z_j} \ket{s} \bra{s} e^{\left(im^I_j+\sum_i iW^I_{ij}\hat v^z_i\right) \hat h^z_j} \ket{+} \nonumber \\ & = & \sum_{s=\pm} R_{s}(m_j^R) \bra{s} e^{\left(im^I_j+\sum_i iW^I_{ij}\hat v^z_i\right) \hat h^z_j} \ket{+}, \end{eqnarray} where a resolution of identity $\sum_{s_j=\pm}\ket{s_j}\bra{s_j}$ for the ancilla qubit is inserted in the middle and $R_{s}(m^R_j)=\bra{+} e^{m^R_j \hat h^z_j} \ket{s} $ can be computed classically as it is the transformation matrix element associated with a single qubit. Not getting $R_{s}(m_j)$ experimentally is the key to avoid post-selection. Note that the decomposition of $\exp\left(\hat h^z_j \left(m_j + \sum_i iW^I_{ij}\hat v^z_i\right)\right)$ introduced in Eq.~\ref{eq:block-decomp} is exact as these operators commute. Using Eq.~\ref{eq:block-decomp}, we re-write Eq.~\ref{eq:qrecycle},
\begin{eqnarray}\label{eq:qrecycle2} \ket{\Psi_v(\theta)} & = & \sum_{s_M=\pm} \cdots \sum_{s_1=\pm} \frac{1}{N_v}\left(\prod_{j=1}^MR_{s_j}(m^R_j)\right)
\prescript{}{}{\bra{s_M}}\left[e^{\hat h^z_M \left(i m^I_M + \sum_i iW^I_{iM}\hat v^z_i\right)}\right]\ket{+} \cdots \nonumber \\
\nonumber \\
& & \times \prescript{}{}{\bra{s_1}}\left[e^{\hat h^z_1 \left(i m^I_1 + \sum_i iW^I_{i1}\hat v^z_i\right)}\right]\ket{+} \,\,
e^{\sum_i b_i \hat v^z_i} \ket{++\cdots+}_v \nonumber \\ & = & \sum_{s_M=\pm} \cdots \sum_{s_1=\pm} \frac{N_{\vec s}}{N_v}\left(\prod_{j=1}^MR_{s_j}(m^R_j)\right )\ket{\Psi_v^{\vec{s}}(\theta)}, \end{eqnarray}
where $\vec{s}=[s_1,\cdots,s_M]$. $\ket{\Psi_v^{\vec{s}}(\theta)}$ is a visible-spin wave function created by projecting hidden spins onto basis states $\ket{s_1\cdots s_M}_h$ instead of enforcing the post-selection $\ket{++\cdots+}_h$. Due to the decomposition introduced in Eq.~\ref{eq:block-decomp}, only a portion of $\exp(\hat H_{RBM})$ contributes to the generation of $\ket{\Psi_v^{\vec{s}}(\theta)}$ and $N_{\vec s}$ is the normalization to keep $\bk{\Psi_{v}^{\vec{s}}}{\Psi_{v}^{\vec{s}}}=1$. While there is no post-selection in Eq.~\ref{eq:qrecycle2}, there is now a summation over all possible $\vec{s}$.
Instead of working directly with Eq.~\ref{eq:qrecycle2}, we look for an alternative approach based on the fact that we are primarily interested in the expectation value of an observable $\hat O$, which can be formulated as
\begin{eqnarray}\label{eq:expo} \bra{\Psi_v(\theta)} \hat O \ket{\Psi_v(\theta)} & = & \int dz \vert\bk{\mathbf z}{\Psi_v(\theta)}\vert^2 \left[\int dz^\prime O(z,z^\prime) \frac{\bk{\mathbf z^\prime}{\Psi_v(\theta)}}{\bk{\mathbf z}{\Psi_v(\theta)}}\right]. \end{eqnarray} The equation above can be interpreted as follows. The expectation value of an observable $\hat O$ can be turned into the average of the expression inside the square bracket if we can efficiently sample $z$
according to the probability density $|\bk{\mathbf z}{\Psi_v(\theta)}|^2$, i.e. projecting the NQS in some basis. We further analyze this probability density by exposing the details of the RBM architecture,
\begin{eqnarray}\label{eq:mc_obs} \vert \bk{\mathbf z}{\Psi_v(\theta)} \vert^2 & = & \frac{\vert \bk{\mathbf z}{\Psi_v(\theta)} \vert^2 }{ \bk{\Psi_v(\theta)}{\Psi_v(\theta)}
} \nonumber \\ & = & \frac{\sum_{\vec{s}} \frac{N_{\vec s}^2}{N_v^2}\left(\prod_{j=1}^MR^2_{s_j}(m^R_j)\right) \vert \bk{\mathbf z}{\Psi_v^{\vec{s}}(\theta)} \vert^2}{\sum_{\vec{s}} \frac{N_{\vec s}^2}{N_v^2}\left(\prod_{j=1}^M R^2_{s_j}(m^R_j)\right)}, \end{eqnarray} where $\sum_{\vec{s}} = \sum_{s_1} \cdots \sum_{s_M}$ and it is implicitly assumed that $\bk{\Psi_v(\theta)}{\Psi_v(\theta)}=1$. Instead of performing the exact summation over all $\vec s$ in Eq.~\ref{eq:mc_obs} (this is essentially the same summation in Eq.~\ref{eq:qrecycle2} mentioned at the end of the last paragraph), we should simply sample $\vec s$ in a Monte Carlo fashion. We note that $N_{\vec{s}}^2$ is the probability of observing $\{s_1,\cdots,s_M\}$ upon measuring those $M$ hidden spins during the construction of the state $\ket{\Psi_{v}^{\vec{s}}(\theta)}$. Hence, samples of $\vec{s}$ are effectively drawn from the probability density $N_{\vec s}^2$ during the construction of $\ket{\Psi_v^{\vec{s}}(\theta)}$ . In short, we replace the exact summation according to
\begin{eqnarray} \sum_{\vec {s}} N^2_{\vec s} f(\vec s) & \xrightarrow[\text{according to } N^2_{\vec s}]{\text{Monte Carlo sampling of } \vec s} & \frac{1}{N_{\text{exp}}} \sum_{k=1}^{N_{\text{exp}}} f(\vec{s}_k), \end{eqnarray} where $f(\vec{s})$ is some arbitrary function of $\vec{s}$ and $N_{\text{exp}}$ is the number of sampling experiments performed.
The factor $\left(\prod_j R^2_{s_j}(m^R_j)\right)$ in Eq.~\ref{eq:mc_obs} should be calculated classically. This probability $\vert \bk{\vec z}{\Psi_v^{\vec{s}}(\theta)} \vert^2$ is again sampled from projective measurements on visible spins. The only thing that is prohibitively expensive to estimate either classically or experimentally is the normalization constant $N_v$. This is the reason we introduce the denominator (which is really just $1$) in Eq.~\ref{eq:mc_obs} that carries another $N_v$ to cancel the one in the numerator. By using Eqs.~\ref{eq:expo}-\ref{eq:mc_obs} together, the challenging post-selection is replaced with the Monte Carlo framework that needs to sample multiple $\ket{\Psi_v^{\vec{s}}(\theta)}$ according to Eq.~\ref{eq:mc_obs}. Additional details on the ensemble state preparation method may be found in the supplementary III.
\textbf{Quantum simulations with NQS-Imaginary Time Evolution (NQS-ITE).} Next, we discuss the imaginary-time dependent variational principle (ITDVP) to find a ground state of a Hamiltonian $\hat H$. The idea is to propagate a trial wave function $\ket{\Psi_v(\theta_\tau)}$ in the imaginary time domain. If the trial wave function $\ket{\Psi_v(\theta_0)}$ at time $\tau=0$ has a non-zero overlap with the ground state $\ket{\Psi_{gs}}$, then it should converge to an ansatz closest to $\ket{\Psi_{gs}}$ when $\tau \gg 1$. With a variational ansatz, the time-evolved $\ket{\Psi_v(\theta_\tau)}$ can be prepared in a quantum circuit if $\theta_\tau$ is given. In the method section, we summarize the standard ITDVP. The equations of motion for $\theta_\tau$ are given in Eq.~\ref{eq:IT-EOM}. In this section, we adapt the standard NQS-ITE algorithm to make it compatible with the state preparation protocol introduced earlier. Details of the derivation may be found in Supplementary IV. As a result, the approach presented here implements the imaginary-time propagation (for NQS ansatz) more efficiently than the original algorithm reported in the Method section.
In short, the modified equations of motion for $\theta$ assume the following form
\begin{eqnarray}\label{eq:modified_ite} \dot \theta_n =\sum_{m} A_{nm}^{-1} C_m. \end{eqnarray}
The matrix A and vector C read,
\begin{eqnarray}\label{eq:mtxAvecC} A_{nm} = \text{Re}\left( \langle \hat O^\dag_n \hat O_m \rangle - \langle \hat O^\dag_n \rangle_v \langle \hat O_m \rangle_v\right), \text{ and } C_m = \text{Re}\left( \langle \hat O^\dag_m \hat H \rangle_v - \langle \hat O^\dag_m \rangle_v\langle \hat H \rangle_v \right), \end{eqnarray}
where $\langle \cdots \rangle_v = \bra{\Psi_v(\theta)} \, \cdots \, \ket{\Psi_v(\theta)}$. The $O_n$ operators are defined as follows,
\begin{eqnarray}\label{eq:stochreconfigO} O_n = \left\{ \begin{array}{ll} i^{1-\delta} \hat v^z_i, & \text{ if } \theta_n = b_i, \\ i^{1-\delta} \tanh\left(m_j + \sum_{i} iW^{I}_{ij} \hat v^z_i\right), & \text{ if } \theta_n = m_j \\ i \hat v^z_i \tanh\left(m_j + \sum_{i} i W^{I}_{ij} \hat v^z_i\right), & \text{ if } \theta_n = W^{I}_{ij}, \\ \end{array}\right. \end{eqnarray} where $\delta = 0$ if $\theta_n = b^{I}_i$ or $\theta_n = m^{I}_j$ and $\delta=1$ if $\theta_n = b^{R}_i$ or $\theta_n = m^{R}_j$. We note that Eqs.~\ref{eq:modified_ite}-\ref{eq:stochreconfigO} essentially give the stochastic reconfiguration method for the variational Monte Carlo framework. One should compare Eq.~\ref{eq:mtxAvecC} and Eq.~\ref{eq:IT-EOM} to see that the standard ITDVP approach (in the Method section) requires preparing $\mathcal{O}(N_{\text{var}}^2)$ quantum states, underlying the matrix element $A_{mn}=\bk{\partial_{\theta_m}\Psi_v(\theta)}{\partial_{\theta_n}\Psi_v(\theta)}$. On other hand, $A$ matrix elements correspond to $\mathcal{O}(N_{\text{var}}^2)$ measurements with respect to the state $\ket{\Psi_v(\theta)}$. In fact, one can simultaneously estimate all $\mathcal{O}(N_{\text{var}}^2)$ matrix elements with every given sample of $\mathbf z$ according to the definition of A matrix element in Eq.~\ref{eq:mtxAvecC}. This could be a tremendous advantage for large-scale simulations when $N_{\text{var}}\propto \mathcal{O}(MN)$ could be a huge number.
Next, we point out an interesting observation. The standard gradient-based energy minimization, as done within the VQE approach, updates $\theta_\theta$ with the equation of motion $\dot \theta_n = C_n$
where $C_n$ being the gradient vector for the energy function $E_\theta = \langle \Psi_v(\theta) \vert \hat H \vert \Psi_v(\theta) \rangle$. The comparison of Eq.~\ref{eq:modified_ite} to the equation of motion above reveals that the NQS-ITE introduces a preconditioner $A^{-1}$ to the gradient vector in order adjust the step size to account for the intrinsically curved metric for the NQS manifold. However, the evaluations of the matrix $A$ requires no more experimental efforts for NQS-ITE than for a standard gradient-descent based approach as explained. As the imaginary-time method tends to give better result than the gradient descent; one should always adopt the imaginary-time propagation whenever NQS is used as the trial wave function.
\textbf{Numerical Results.} To demonstrate the effectiveness of RBM-NQS ansatz and NQS-ITE algorithm, we report numerical simulations on three different types of systems: molecules, spin chains and nanostructures (a triple quantum dots). Throughout the paper, we use a hidden and visible spin ratio of $\alpha=M/N =1$, unless otherwise specified. We first present results with the standard complex-valued NQS ansatz (which requires the extended state preparation protocol described in Supplementary V), then we repeat the simulations in the last subsection to demonstrate that the unitary-coupled RBM-NQS could achieve the same level of accuracy with the same number of hidden spins, i.e. $\alpha=1$.
3a. Molecular Systems. We first test the NQS-ITE algorithm on common molecular benchmarks: the dissociation curves of $\text{H}_2$ and LiH molecules. The molecular Hamiltonians are first projected onto a discrete set of molecular orbitals. Here, we use the conventional STO-3G basis set, which constitutes a minimal set in that it represents the minimum number of orbitals required to represent a given atomic shell. The resulting fermionic Hamiltonians are subsequently mapped onto qubit Hamiltonians using the Bravyi-Kitaev transformation~\cite{seeley2012bravyi} . The computation of the integrals in second quantization and transformation of the Hamiltonians are done using PySCF~\cite{sun2018pyscf} and OpenFermion~\cite{mcclean2017openfermion}.
\begin{figure}
\caption{ The ground state energy of H$_2$ and LiH molecules computed with the NQS-ITE algorithm (solid red dots) and the exact diagonalization (solid black lines). (a) Results for $H_2$ ($N=2$, $M=2$) as a function of inter-nuclear distance. (b) Results for LiH as a function of inter-nuclear distance($N=4$, $M=4$).}
\label{fig:molecule}
\end{figure}
In the current simulations, the hydrogen molecule requires 2 visible spins whereas lithium hydride requires 4 visible spins. The numerical results from NQS ansatz in computing the ground state energy as a function of inter-nuclear distance are plotted in Fig.~\ref{fig:molecule}. It can be seen that NQS is capable of reproducing nearly exact results despite using a modest number of hidden spins.
3b. Spin systems. Next, we consider the problem of finding the ground state of two prototypical spin models, i.e. the transverse-field Ising (TFI) model and the antiferromagnetic Heisenberg (AFH) model. The spin Hamiltonians can be written as, \begin{eqnarray}
H_{TFI} &=& - h \sum_{i}\hat \sigma_i^{x} - \sum_{ij}\hat \sigma_i^{z}\hat \sigma_j^{z}\\
H_{AFH} &=& \sum_{ij}\hat \sigma_i^{x}\hat \sigma_j^{x} + \hat \sigma_i^{y}\hat \sigma_j^{y} + \hat \sigma_i^{z}\hat \sigma_j^{z} \end{eqnarray} where $h$ is the transverse field strength and we use open boundary condition. In Fig.~\ref{fig:spins} we again compute the energies using the NQS-ITE algorithm and compare the results with those from the exact diagonalization, and we observe excellent agreements as expected.
\begin{figure}
\caption{ The ground state energy of spin chains computed with the NQS-ITE algorithm (solid red dots) and the exact diagonalization (solid black lines). (a) Results for Heisenberg chain as a function of chain length. (b) Results for Ising chain as a function of chain length at different values of transverse field.}
\label{fig:spins}
\end{figure}
3c. Triple Quantum Dots (TQD). A lateral TQD is an artificial, fully tunable molecule constructed using metallic gates localizing electrons in a semiconductor field-effect transistor (FET). A TQD allows one to study new phenomena not present in a single or double quantum dot, e.g. topological effects \cite{Hsieh2012,Delgado2008}.
The Hamiltonian of the TQD subject to a uniform perpendicular magnetic field, ${\bf B}=B {\bf \hat{z}}$, is given by
\begin{eqnarray}
H&=&
\sum\limits_{i,\sigma} E_{i\sigma} \hat d_{i\sigma}^{\dag} \hat d_{i\sigma} + \sum\limits_{\sigma,i,j;\;i\neq j}
\tilde{t}_{ij} \hat d_{i\sigma}^{\dag} \hat d_{j\sigma}+ \sum\limits_{i} U_{i}\hat n_{i\downarrow} \hat n_{i\uparrow}
\end{eqnarray}
where $\hat d_{i\sigma}$ ($\hat d_{i\sigma}^\dag$) is fermionic annihilation (creation) operator with spin $\sigma=\pm 1/2$ on orbital $i$. $\hat n_{i\sigma} = \hat d_{i\sigma}^\dag \hat d_{i\sigma}$ and $\hat \varrho_{i} = \hat n_{i\downarrow} + \hat n_{i\uparrow}$ are the spin and charge density on orbital level $i$. Each dot is represented by a single orbital with energy $E_{i\sigma}= E_i+ g^* \mu_B B \sigma$, where $g^*$ is the effective Land\'e $g$-factor, $\mu_B$ is the Bohr magneton. The dots are connected by magnetic field dependent hopping matrix elements $\tilde{t}_{ij}=t_{ij}e^{2\pi i \phi_{ij}}$. The details of the model can be found in the Method section.
\begin{figure}
\caption{The ground state energy of a lateral TQD ($N=6$) as a function of magnetic field obtained from the NQS-ITE algorithm (solid red dots) and the exact diagonalization (solid black lines).}
\label{fig:QDs}
\end{figure}
The ground state energy of the TQD as a function of magnetic field is plotted in Fig.~\ref{fig:QDs}, we observe excellent agreement between the results from the exact diagonalization and NQS-ITE. It is worth noting that, at non-zero magnetic field, the ground state wave function of a TQD could be complex, thus a complex RBM ansatz is necessary for accurate representation of the ground state wavefunction.
3d. Results with unitary-coupled RBM-NQS. Finally in Fig.~\ref{fig:mean_field}, we repeat the same set of model studies as above but we now apply the unitary-coupled RBM-NQS ansatz (green dahsedlines) to the NQS-ITE. The key insight revealed by Fig.~\ref{fig:mean_field} is that, in most cases, the unitary-coupled RBM-NQS provide comparable performance to that of the standard complex NQS. In the case of Heisenberg model, the approximated RBM ansatz sometimes fails to encode the the imaginary time operator and this leads to non-monotonic behaviour during the imaginary time evolution, though our algorithm eventually finds the true ground state wave function. This non-monotonicity can mostly be mitigated by using a smaller update time steps ( Fig.~\ref{fig:mean_field}c inset).
\begin{figure}
\caption{Simulations of the ground state energy using NQS-ITE with random initialization (solid red lines), mean field initialization (solid blue lines), and random initialization for the unitary-coupled RBM ansatz (dashed green lines). The dashed black lines show the ground state energy from exact diagonalization. Results for (a) H$_2$ at $1.05\AA$, (b) LiH at $1.5 \AA$, (c) Heisenberg chain ($N=6$), and (d) Ising chain ($N=6$, $h=0.5$). The inset in Figure (c) displays the results for the unitary-coupled RBM ansatz using different time-step size, $\delta\tau$, where $\tau = \text{Steps}*\delta \tau$.}
\label{fig:mean_field}
\end{figure}
Furthermore, we also present another set of results (solid red lines) in Fig.~\ref{fig:mean_field}. These results are obtained with RBM ansatz whose parameters are initialized with a mean-field solution to the original problems. As shown, a better initial guess improves the convergence rate in comparison to randomly initialized cases. Particularly, mean-field initial wave function already provides nearly exact ground state wave function for lithium hydride molecule, as it can be seen from Fig.~\ref{fig:mean_field}b that the ground state energy from the mean-field approximation is nearly identical to the exact ground state energy. The accuracy of the mean-field approximation for lithium hydride molecule is possibly due to the low level of electron correlations among the orbitals in minimal basis set. To test this hypothesis, we solve for the ground state of lithium hydride using Hartree-Fock method and found that it already provides a very accurate ground state solution. In the method section, we briefly comment how one can systematically make an intelligent guess on a good initial state under various situations.
\section*{Discussion} In conclusion, we present a practical approach to exploit a popular machine-learning model for quantum simulations on a digital quantum computer. Before fault-tolerant quantum computation becomes readily available, the hybrid quantum-classical algorithms will prevail as a popular approach for investigating novel applications of a quantum computer. Successful experimental demonstrations of hybrid quantum-classical algorithms have certainly attracted attentions and boosted confidence in quantum computations. Nevertheless, many obstacles still prevent a clear demonstration of unambiguous quantum advantages for these hybrid quantum-classical algorithms. One possible path towards this goal is to investigate more powerful wave-function ansatz that can achieve a good tradeoff between expressive power and number of variational parameters. With fewer parameters, potentially, one may deal with a shorter-depth state preparation and deals with a simpler optimization problem.
From this perspective, the NQS certainly seems a promising option to investigate. For instance, it is known that a fully-connected RBM ansatz satisfy an entanglement volume scaling, it is very intriguing to further investigate whether one can exploit this property to minimize number of variational parameters under a realistic setting. While the long-range connectivity between qubits is not necessarily easy to realize in every kind of quantum hardwares at the moment; it is at least experimentally feasible with one of the leading hardware architecture, the ion-trap based quantum computers\cite{brown2016iontrap,bruzewicz2019iontrapreview} having all-to-all connectivity among qubits. As the coherence time of quantum hardware continues to improve, issues of qubit connectivity could potentially be mitigated with advanced techniques such as the fermionic swap network\cite{kivlichan2018quantum}. Theoretically, one may also design quantum algorithms based on the deep Boltzmann machines\cite{gao2017efficient} that further elevates the expressive power of Boltzmann-machine architectures with only short-range couplings, suitable for quantum hardware featuring local connectivity among qubits.
In this current work, we set out to improve the existing NQS state-preparation protocol in the quantum circuits. As mentioned in the introduction, prior quantum algorithms using RBM ansatz suffer from several obstacles that prevents simulations for complex systems with many degrees of freedom. Our proposed state-preparation protocols have significantly expanded the scope of applications for NQS as an ansatz for quantum simulations and lowered the experimental barriers. Without the complex-valued parameters, one cannot simulate some important quantum materials and quantum dynamics. Our numerical testings manifest encouraging signs that the NQS ansatz performs remarkably well across a variety of systems of practical and theoretical interests. Due to the qubit-recycling scheme, we reduce the number of required qubits from $O(N+M)$ in previous works down to $O(N)$ with sequential implementations of visible-hidden layer interactions. Avoiding the probabilistic preparation of inter-layer couplings (by imposing $W_{ij}=iW^I_{ij}$) also further improves the practicality of NQS-based simulations. The ensemble state preparation bypass the post-selection on hidden spins. Finally, it has been previously shown that imaginary time algorithm offers superior performance compared to VQE\cite{mcardle2018img}, but at the expense of more state preparations and measurements. In this work, we exploit the properties of RBM architecture and show that the number of different quantum states required in the imaginary time algorithm could be reduced from $\mathcal{O}(N_{var}^2)$ down to $1$. Hence, the experimental costs for NQS-ITE is comparable to what VQE demands. Since Boltzmann machine is a widely used machine learning model with many applications, we expect this HQC paradiagm building on the Monte Carlo framework and the NQS ansatz can be adopted for solving other important problems, such as discrete optimizations and machine learning.
\section*{Method}
\textbf{Restricted Boltzmann Machine As Trial Wave Function.} Recently, Troyer and Carleo \cite{carleo2017solving} used the Restricted Boltzmann Machine (RBM) neural-network architecture as wave-function ansatz to manybody physics and attained impressive results. Since then, several other wave function ansatz inspired by neural-network architecture have been explored. Collectively, we now refer to this set of wave function ansatzs as the neural-network quantum states (NQS). Although, in this work, we only consider RBM-NQS, which is particularly convenient to model quantum systems composed of two-level systems (TLSs) such as spin lattice commonly studied in conednsed matter physics and electronic structures problems formulated in terms of qubits. For these systems, each TLS is directly identified with a visible spin in the corresponding RBM model. The entanglement between these TLSs (or visible spins) is mediated by the pairwise interactions between visible and hidden spins. In short, a manybody wave function in the NQS form reads
\begin{eqnarray}\label{eq:nnqs} \ket{\Psi_v(\theta)}
& = & \frac{1}{N_v} \sum_{\bf{v}}\left(\sum_{\bf{h}} e^{E_\theta(\bf{v},\bf{h}) }\right) |\bf{v} \rangle, \end{eqnarray}
with energy function $E_\theta(\mathbf{v}, \mathbf{h}) = \sum_{i} b_i v_i + \sum_{j} m_j h_j + \sum_{ij}W_{ij} v_i h_j$, its complex conjugate $\bar{E}_\theta(\mathbf{v},\mathbf{h})$, and the normalization constant $N_v= \sqrt{\sum_{\mathbf{v}}\left(\sum_{\mathbf{h}}e^{\bar E_\theta(\mathbf{v},\mathbf{h})}\right) \left(\sum_{\mathbf{h}}e^{E_\theta(\mathbf{v},\mathbf{h})}\right)}$. The RBM parameters are collectively denoted by $\theta = \{\mathbf b, \mathbf m, \mathbf W\}$. As shown in Eq.~\ref{eq:nnqs}, the hidden spins are summed over in the bracket on the right-hand side of Eq.~\ref{eq:nnqs} to give a wave function $\ket{\Psi_v(\theta)}$ for the visible spins, which represent the physical system of interest. In principle, $\theta$ should possess non-vanishing imaginary components to describe complex-valued wave function.
To prepare an NQS in a quantum circuit, we should take the energy function $E_\theta(\bf{v},\bf h)$ and promote it to an Hermitian operator by replacing the binary values of $\bf v$ and $\bf h$ with the corresponding Pauli operators. The quantum-circuit analog of Eq.~\ref{eq:nnqs} is decomposed into Eqs.~\ref{eq:vhwf}-\ref{eq:bmcircuit}: first entangle the visible and hidden spins then post-selects the hidden spins to mediate the desire non-unitary transformation on the visible-spin wave functions.
\textbf{Imaginary-time dependent variational principle for an NQS}
In this section, we summarize a recently proposed imaginary-time dependent variational principle (ITDVP) \cite{mcardle2018img} for NQS ansatz. Under the Wick rotation $t \rightarrow i \tau$, the time-independent Schrodinger equation reads,
\begin{eqnarray}\label{eq:ischeq}
\frac{\partial |\Phi(\tau) \rangle}{\partial \tau} = -(\hat H -E_\tau) |\Phi(\tau) \rangle \end{eqnarray}
where the additional energy term $E_\tau = \langle \Phi(\tau) | \hat H|\Phi(\tau) \rangle$ arises from the normalization condition for $\ket{\Phi(\tau)}$. As $\tau \rightarrow \infty$, the stationary solution of Eq.~\ref{eq:ischeq} is the lowest-energy eigenstate that has a non-zero overlap with the initial state $\ket{\Phi(0)}$. The imaginary-time dynamical simulation is a well-established approach to obtain the ground state of complex Hamiltonians.
Through time dependent variational principle , an NQS ansatz $\ket{\Psi_v(\theta_\tau)}$ approximates a time-evolved quantum state $\ket{\Phi_\tau}$ via the following equation of motion, \begin{eqnarray}
\delta|| (\partial /\partial \tau + \hat H - E_\tau ) | \Psi_v(\mathbf{\theta}_\tau)\rangle || = 0, \end{eqnarray} where $\theta(\tau)$ are determined via,
\begin{eqnarray}\label{eq:tdvp} \sum_j A_{ij} \dot{\theta}_j = C_i. \end{eqnarray} The A and C matrices are defined as follows,
\begin{eqnarray} \label{eq:IT-EOM} A_{mn} = \text{Re} \left\langle \frac{\partial \Psi_v(\theta)}{\partial \theta_m} \right\vert \left. \frac{\partial \Psi_v(\theta)}{\partial \theta_n} \right\rangle, \text{ and } C_{n} = -\text{Re}\left\langle \partial_{\theta_n} \Psi_v(\theta) \right\vert \hat H \left\vert \Psi_v(\theta) \right\rangle. \end{eqnarray}
Therefore, one updates the variational parameters according to ${\theta}(\tau + \delta \tau) = {\theta}(\tau) + \dot{{\theta}}(\tau) \delta \tau = {\theta}(\tau) + A^{-1}(\tau)C(\tau) \delta \tau$, where $\delta \tau$ is the update timestep.
For the variational parameter update in Eq.~\ref{eq:IT-EOM}, we need to compute the gradients of $|\Psi_v \rangle$ with respect to $\theta$, i.e. $|\frac{\partial \Psi_v}{\partial \theta_n} \rangle$. As the properly normalized $\ket{\Psi_v(\theta)}$ carries a $\theta$-dependent normalization factor $N_v$ in Eq.~\ref{eq:nnqs}, the chain-rule derived formula for the gradient reads
\begin{eqnarray}\label{eq:chainrule1}
\left \vert \frac{\partial \Psi_v}{\partial \theta_n} \right \rangle = \frac{_h\langle ++\dots + |\partial_{\theta_n} \Psi_{vh}\rangle}{\mathcal{N}_v}
- \text{Re} \left( \langle \Psi_{vh} | \hat{P}_+^{(h)}|\partial_{\theta_n} \Psi_{vh} \rangle \right) \frac{_h\langle ++\dots + | {\Psi_{vh}} \rangle}{\mathcal{N}_v^3} \end{eqnarray}
where $\mathcal{N}_v = \sqrt{ \langle \Psi_{vh}(\theta) | \hat{P}^{(h)}_+ | \Psi_{vh}(\theta)\rangle}$. The above expression depends on the gradients of the extended RBM wave function, $|\partial_\theta \Psi_{vh} \rangle$, which can be derived analytically \begin{eqnarray}\label{eq:psivh_deriv}
\frac{ \partial |\Psi_{vh} \rangle }{\partial b^{R}_i} &=& (\hat{v}_i - \langle \hat{v}_i \rangle_{vh} ) |\Psi_{vh} \rangle ,\,\,\,\,
\frac{ \partial |\Psi_{vh} \rangle }{\partial b^{I}_i} = i\hat{v}_i |\Psi_{vh} \rangle, \nonumber \\
\frac{ \partial |\Psi_{vh} \rangle }{\partial m^{R}_j} &=& (\hat{h}_j - \langle \hat{h}_j \rangle_{vh} ) |\Psi_{vh} \rangle, \,\,\,\,
\frac{ \partial |\Psi_{vh} \rangle }{\partial m^{I}_j} = i\hat{h}_j |\Psi_{vh} \rangle, \nonumber \\
\frac{ \partial |\Psi_{vh} \rangle }{\partial W^{R}_{ij}} &=& (\hat{v}_i \hat{h}_j - \langle \hat{v}_i \hat{h}_j \rangle_{vh} ) |\Psi_{vh} \rangle, \,\,\,\,
\frac{ \partial |\Psi_{vh} \rangle }{\partial W^{I}_{ij}} = i \hat{v}_i \hat{h}_j |\Psi_{vh} \rangle, \end{eqnarray}
where the superscript $(R, I)$ is used to denote the real and imaginary parts of the coefficients and $\langle \hat{O}\rangle_{vh} = \langle \Psi_{vh} |
\hat{O}| \Psi_{vh}\rangle $. Substituting the expressions given by Eq.~\ref{eq:psivh_deriv} into Eq.~\ref{eq:chainrule1}, one obtains expressions like the following,
\begin{eqnarray}\label{eq:f1} \left\vert \frac{\partial\Psi_v(\theta)}{\partial m^{I}_j} \right\rangle = \frac{\mathcal{N}_v^\prime}{\mathcal{N}_v} \left( \ket{\Psi_v(\theta^\prime)} +
\text{Re}\left[ \langle \Psi_{v}(\theta) |\Psi_{v}(\theta^\prime) \rangle \right] \ket{\Psi_v(\theta)} \right), \end{eqnarray} where $\theta^\prime=[\cdots, m^{I}_j+\frac{\pi}{2}, \cdots]$ and $\theta=[\cdots, m^{I}_j, \cdots]$ differ only by the value of $m^{I}_j$ and the normalization factor $\mathcal{N}^\prime_v = \sqrt{\bra{\Psi_{vh}(\theta^\prime)}\hat P^{(h)}_+ \ket{\Psi_{vh}(\theta^\prime)}}$. To accurately calculate matrix $A$ and vector $C$, it is necessary to determine the normalization factors $\mathcal{N}_v$ and $\mathcal{N}^\prime_v$ according to Eq.~\ref{eq:f1}. Nevertheless, $\mathcal{N}_v$ is hard to estimate as the cost to estimate scale as $\mathcal{O}(4^M)$.
This challenge motivates us to re-formulate the ITDVP for RBM-based NQS ansatz that entirely avoids the normalization factors $\mathcal{N}_v$. Details may be found in the subsection discussing NQS-ITE under Result section and Supplementary IV.
\textbf{Initial state preparation for quantum simulations.} The NQS ansatz can be used in conjunction with most HQC simulation algorithms in addition to the time-dependent variational method outlined above. All these methods aim to solve highly non-trivial optimization in which the quality of solutions or the convergence rate depends crucially on the overlap of the initial state with the ground state $\ket{\Psi_{gs}}$. The two-stage initialization protocols described here gives a systematic approach to guide the preparation of high-quality initial states. In short, the idea is to selectively optimize a subset of parameters to obtain an approximate solution that could be used as the initial state in a subsequent simulation optimizing over all parameters.
In the simplest case, one may consider a mean-field approximation, which restricts the considerations to completely factorized product-state wave function $\ket{\Psi_v^{\text{0}}(\vec b)}=\ket{\psi_{v_1}(b_1)}\otimes\cdots\otimes\ket{\psi_{v_N}(b_N)}$ for all visible spins in a quantum simulation.
We then subsequently use $\ket{\Psi^0_v(\vec b)}$ as the starting point of another simulation in which the hidden spins are introduced along with corresponding parameters $\{m_1, \cdots m_M, W_{11}, \cdots W_{NM}\}$ that collectively facilitate the formation of entangled NQS, $\ket{\Psi_v(\theta)}$. We note the mean-field approximation (single-body physics problem) can be easily done on a classical computer.
Nevertheless, for strongly correlated systems, the product states are not guaranteed to support a high overlap with $\ket{\Psi_{gs}}$. Instead of optimizing $\vec b$ (the mean-field approximation) in the first run, it will be beneficial to consider an NQS with specifically designed sparse connectivity. In the second-stage calculation, the fully-connected architecture will be restored as usual, and the total number of variational parameters scale as $\mathcal{O}(NM)$. An obvious question is how to decide the connectivity of this sparsely connected RBM architecture for the first-stage simulation, which needs to balance the expressive power of the variational ansatz and the complexity of the optimization tasks. For lattice systems, one may consider short-range RBMs that constitute a special class of the well-established entangled-plaquette states \cite{glasser2018neural}. In this case, the total number of variational parameters scale as $\mathcal{O}(N)$.
\textbf{Simulation details for numerical studies.} In all our simulations, we use a constant learning rate of 0.01. The variational parameters are initialized as Gaussian random numbers with mean zero and variance 0.01, except in cases where the initial conditions are obtained from mean field solutions (Fig.\ref{fig:mean_field} green dashed lines).
Hamiltonians of Hydrogen and Lithium Hydride. We treat the hydrogen and lithium hydride molecules in the minimal STO-3G basis and use PySCF to compute the integrals in the second quantization The resulting fermionic Hamiltonians are subsequently mapped onto qubit Hamiltonians using Bravyi-Kitaev transformation with OpenFermion~\cite{mcclean2017openfermion}. Due to the symmetry in H$_2$, the final Hamiltonian consists of 2 qubits~\cite{o2016scalable}, whereas the lithium hydride Hamiltonian contains 4 qubits.
Hamiltonian for TQD. For TQD simulations, we use the parameters from Ref.~\cite{Delgado2008}, i.e. $t=-0.23$ meV,
$U_i=50|t|$, $V_{ij}=10|t|$. $g^*=-0.44$, $E_i =-|t| $, and $\phi/B =1.25 T^{-1}$. We use Bravyi-Kitaev transformation to map the Hubbard Hamiltonian onto a qubit Hamiltonian using OpenFermion.
\section*{Supplementary Information I: Challenges to prepare complex-valued NQS in quantum devices.} In prior works, an NQS wave-function amplitude reads \begin{eqnarray}\label{eq:cl-nnqs} \bk{\bf v}{\Psi_v(\theta)} =\left(\sum_{\bf h} \vert Q_{\theta}(\bf v,\bf h)\vert e^{i\arg Q_{\theta}(\bf v,\bf h)}\right)^{1/2}, \end{eqnarray} where $ Q_{\theta}(\bf v,\bf h)= \left(\sum_{\bf{h}} e^{E_\theta(\bf{v},\bf{h}) }\right)/N_v$ is the joint Boltzmann distribution of visible and hidden spins as governed by the energy function $E_\theta$. When the RBM Hamiltonian is strictly real-valued, then it is obvious that the $\ket{\Psi_v(\theta)}$ can only model positive semi-definite wave functions. While being limited in scope of applications, one can rely on the Marshall Sign Rule to determine when the ground-state wave functions of a spin Hamiltonian should be positive semi-definite. Study on the transverse-field Ising model with positive semi-definite NQS has been reported \cite{gardas2018dwave} for quantum simulations on a D-wave quantum annealer. Nevertheless, there is an advantage of working with this subset of NQS. We note the visible-spin probability density $\vert \bk{\bf v}{\Psi_v(\theta)}\vert^2 =\sum_{\bf h} Q_{\theta}(\bf v,\bf h)$ is simply the marginal of the joint visible-hidden distribution. Experimentally, one may simply prepare an extended wave function of all visible and hidden spins then simply ``trace over" the hidden spins to estimate $\vert \bk{\bf v}{\Psi_v(\theta)}\vert^2$. Avoiding the post-selection of $\ket{++\cdots +}_h$ is certainly advantageous.
However, to model realistic systems of interest, one should at least incorporate the negative signs into wave function. Reference \cite{xia2018rbm} proposed an ingenious approach to generate arbitrary real-valued NQS in quantum circuit by modifying the standard RBM architecture. The authors added an extra node in the hidden layer with the sole purpose to assign a sign factor to each wave function coefficient. More precisely, the modified NQS read
\begin{eqnarray}\label{eq:cl-nnqs2} \bk{\bf v}{\Psi_v(\theta)} =\left(\sum_{\bf h} \vert Q_{\theta}(\bf v,\bf h)\vert e^{i\arg Q_{\theta}(\bf v,\bf h)}\right)^{1/2}s(\mathbf v), \end{eqnarray} where $ s(\mathbf v) = s\left( {v _1^z,v _2^z...v _n^z} \right) = \tanh( {\sum_i d_i v _i^z + c}) $ with $d_i$ and $c$ as parameters to be optimized. Given the form of $s(\mathbf v)$, it is clear that it cannot be described simply with the standard RBM Hamiltonian. Since this extra hidden node does not couple directly to other hidden spins, it does not have to be explicitly incorporated into the state preparation. Rather, when estimating physical properties of a quantum system,the sign functions can be shifted onto the observable operators. For instance, to measure energy of a quantum system, we prepare a positive semi-definite NQS $\ket{\Psi_v}$ described in the previous paragraph and estimate energy according to
\begin{eqnarray}
\langle \hat H\rangle = \frac{{\mathop {\sum}\nolimits_{\mathbf z,\mathbf{z}^\prime } {\overline {\Psi_v (\mathbf z)} \overline {s(\mathbf z)} \langle \mathbf z | \hat H|\mathbf{z}^\prime \rangle \Psi_v (\mathbf{z}^\prime )s(\mathbf{z}^\prime )} }}{{\mathop {\sum}\nolimits_{\mathbf z} {|\Psi_v (\mathbf z)s(\mathbf z)|^2} }}. \end{eqnarray} While this method has significantly expanded the usefulness of NQS in quantum simulations, the generation of the non-unitary transformation $\exp(W^R_{ij}\hat v^z_i \hat h^z_j)$ cannot be done efficiently in a deterministic fashion. The challenge of scalability and the desire to model complex-valued wave functions with significantly lower amount of resources have led us to propose the state preparation scheme described in the main text.
\section*{Supplementary Information II: Expressive Power of the Unitary-Coupled Boltzmann Machines}
In order to improve the scalability of the NQS preparation in a quantum circuit, we impose the cross-layer interaction $W_{ij}=iW^I_{ij}$ to be strictly imaginary-valued to avoid probabilistic preparation or a full tomographic characterization of a thermal-like quantum state when $W_{ij}$ have non-vanishing real parts. In this supplementary, we briefly outline three different schemes to convert an arbitrary complex-valued NQS into variants of Boltzmann Machine inspired wave function ansatz with only imaginary-valued couplings across visible and hidden layers. These conversions clearly show that there is no loss of expressive power with the ``seemingly" restricted form of RBM-NQS we promote in this work. In the following, as illustrated in Fig.~\ref{fig:unitary-rbm}, we present methods to map a standard RBM architecture into two variants of Boltzmann machines as explained below.
\begin{figure}
\caption{Mapping of RBM states onto Unrestricted Boltzmann machines and Deep Boltzmann Machine in order to remove real-valued coupling coefficients in the original RBM network.}
\label{fig:unitary-rbm}
\end{figure}
\textbf{Restricted Boltzmann Machine.} First, we give the most straightforward conversion scheme. We note the standard NQS has an unnormalized amplitude,
\begin{eqnarray}\label{eq:nnqs_amp}
\psi_v(\theta) &=& e^{\sum_{i=1}^N b_i v_i}\prod_{j=1}^M \cosh\left(m_j + \sum_i W_{ij}v_i\right) \nonumber \\
& = & e^{\sum_{i=1}^N b_i v_i + \sum_{i_1 > i_2} c^2_{i_{1}i_{2}}v_{i_1}v_{i_2} + \sum_{i_1>i_2>i_3} c^3_{i_1 i_2 i_3}v_{i_1}v_{i_2}v_{i_3}
+ \cdots + c^N_{1...N} v_1\dots v_N}, \end{eqnarray} where the coefficients $c^n_{i_1\dots i_n}$ are obtained from the Taylor expansion of $\log \prod_j \cosh\left(m_j + \sum_i W_{ij}v_i\right)$ with $v_i$ being binary values of $\pm 1$. On the second line of Eq.~\ref{eq:nnqs_amp}, the exponent is a polynomial of degree N. Each term (with degree greater than 1) of this polynomial can be fully decoupled by the application of this mathematical identity \cite{carleo2018constructing},
\begin{eqnarray}
e^{\omega \hat v_{1} \hat v_{2}\cdots \hat v_{n}} & = & \frac{C}{4}\prescript{}{s_1}{\bra{+}}e^{(i \tilde m +i\frac{\pi}{4}\sum_{i=1}^n \hat v_i)\hat s_1)}\ket{+}_{s_1}
\prescript{}{s_2}{\bra{+}}e^{(i\tilde m + i\frac{\pi}{4}\sum_{i=1}^n \hat v_i) \hat s_2)}\ket{+}_{s_2}, \end{eqnarray} where $\hat s_i$ is a Pauli Z operator acting on state $\ket{+}_{s_i}$, and the other parameters read
\begin{eqnarray}
\tilde m = \arctan(e^{-\omega})-\frac{\pi}{4} \text{Mod}(2,N) \,\, \text{ and } \,\, C = \frac{2}{\sin(2\arctan(e^{-\omega}))}. \end{eqnarray} In short, every RBM can be exactly turned into another RBM with a fixed imaginary-valued coupling value $i \pi/4$ throughout the network. A potential price to be paid is the exponentially large number of hidden nodes in this alternative (yet fully equivalent) RBM. Certainly, when the coupling parameters $\vert c^n_{i_1 \dots i_n} \vert$ in Eq.~\ref{eq:nnqs_amp} having magnitudes much smaller than that of $b_i$ and $ m_j$ then one can drop many terms in that polynomial expansion. However, even in the lowest order of perturbation with a quadratic polynomial, the alternative RBM still carries $\mathcal{O}(N^2)$ hidden nodes. One may wonder whether the cost to impose strictly unitray couplings across visible and hidden layer is always an astronomical number of extra hidden nodes.
We remind that the goal of this rather formal analysis is to show that any arbitrary RBM can always be mapped onto a purely imaginary-valued RBM via a simple mapping. By no means, it is the only or the most efficient mapping. We should not be too pessimistic about this. For instance, there are examples \cite{deng2017quantum} of 1D and 2D RBM-NQS (having $M~\mathcal{O}(N)$) exhibiting volume-law entanglements with a constant imaginary-valued coupling coefficient ($i\pi/4$). This implies strongly correlated quantum states can be efficiently created within this subset of RBM-NQS. Furthermore, we hypothesize that if we simply allow the imaginary-valued couplings to vary independently instead of all being held fixed at the value of $i\pi/4$, then there should be viable mapping of arbitrary RBM-NQS onto rather compact architecture of unitary-coupled RBMs. This intuitive hypothesis is empirically confirmed in our numerical studies reported in the main text.
\textbf{Unrestricted Boltzmann Machine.} Given an RBM architecture (composed of N visible nodes and M hidden nodes) with complex-valued couplings $W_{ij}$, one may exactly transform away the real components, $W^R_{ij}$ via a different approach than the one described above. For instance, the coupling between the $i$-th visible and the $j$-th hidden spins may be removed with the addition of one more hidden node (labeled by s) according to the relation\cite{carleo2018constructing},
\begin{eqnarray}\label{eq:wreal} \exp(W^R_{ij}\hat v_i \hat h_j)=
\Delta \left[ \prescript{}{s}{\bra{+}} \exp((i \omega_v \hat v_i + i \omega_h \hat h_j) \hat s) \ket{+}_s \right], \end{eqnarray} where $\hat s$ is the Pauli Z operator acting on the additional ancilla qubit in $\ket{+}_s$ state. The normalization factor is taken to be $\Delta=e^{\vert W^R_{ij} \vert}/2$, and other newly introduced parameters read
\begin{eqnarray}
\omega_v =\frac{1}{2}\arccos\left(e^{-2\vert W^R_{ij} \vert}\right), \text{ and }\omega_h =-\frac{1}{2}\arccos\left(e^{-2\vert W^R_{ij} \vert}\right). \end{eqnarray} These parameters $\omega_v$ and $\omega_h$ would be real-valued as expected. The newly added hidden spin breaks the restriction of no intra-layer couplings. The total number of hidden spins increase to $\mathcal{O}(MN)$ after removing $W^R_{ij}$. Nevertheless, the modified neural-network architecture is largely compatible with RBM architecture as far as preparing the prescribed quantum states in a quantum circuit is concerned. Although the qubit recycling scheme comes with a caveat that one needs roughly $\mathcal{O}(N)$ extra ancilla qubits (at each sequential stage) to explicitly model the entangled hidden state, which emerge when we transform away $\mathcal{O}(N)$ real-valued couplings between visible nodes and a particular hidden node.
\textbf{Deep Boltzmann Machine.} Finally, we realize that the unrestricted Boltzmann machine can be further mapped onto a deep Boltzmann machine architecture without intra-layer couplings among hidden nodes. This is achieved by invoking the following mathematical identity,
\begin{eqnarray}
e^{i\omega^h \hat h_i \hat h_j} & = & \frac{C}{4}\prescript{}{s_1}{\bra{+}}e^{(ib+i\frac{\pi}{4}(\hat h_1+ \hat h_2)\hat s_1)}\ket{+}_{s_1}
\prescript{}{s_2}{\bra{+}}e^{(ib+i\frac{\pi}{4}(\hat h_1+ \hat h_2)\hat s_2)}\ket{+}_{s_2}, \end{eqnarray} where $\hat s_i$ is a Pauli Z operator acting on state $\ket{+}_{s_i}$ in the deep layer, and the other parameters read
\begin{eqnarray}
b = \arctan(e^{-i\omega^h})-\frac{\pi}{2} \text{ and } C = \frac{2}{\sin(2\arctan(e^{-i\omega^h}))}. \end{eqnarray} Hence, in order to convert an unrestricted Boltzmann machine into restricted form (i.e. no intra-layer coupling), every intra-layer couplings between hidden units can be exactly replaced with couplings to extra hidden units in the deep layer. However, using the deep Boltzmann machine for quantum simulation could be difficult as the optimization of the parameters could be hard to converge. In a follow-up work, we will describe practical approaches to perform quantum simulation with deep Boltzmann Machines.
\section*{Supplementary Information III: Ensemble State Preparation Protocol}
As shown in Eq.~\ref{eq:qrecycle2}, the post-selected NQS $\ket{\Psi_v(\theta)}$ can be cast as a weighted sum of $2^M$ $\ket{\Psi^{\vec{s}}(\theta)}$. To facilitate the following discussion, we will work with the un-normalized version of $\ket{\tilde \Psi^{\vec{s}}(\theta)}$ in this section,
\begin{eqnarray} \ket{\tilde \Psi_v^{\vec{s}}(\theta)} & = &\left[\bra{s_M} e^{i\mathcal{G}_M}\ket{+} \cdots \bra{s_1}e^{i\mathcal{G}_1}\ket{+} e^{\sum_{i=1}^N b_i \hat v^z_i} \right]\ket{++\cdots+}_v,
\end{eqnarray} where $\ket{s_i=\pm}=(\ket{0}\pm\ket{1})/\sqrt{2}$ and $\mathcal{G}_j= m_j^I\hat h^z_j + \sum_i W^I_{ij}\hat h^z_j \hat v^z_i$. The normalization factor $N_{\vec s} = \sqrt{\bk{\tilde \Psi_v^{\vec s}(\theta)}{\tilde \Psi_v^{\vec s}(\theta)}}$ is the probability of getting a particular visible-spin wave functions when $M$ hidden spins are observed in the state $\ket{s_1 \cdots s_M}_h$ after being coupled to the visible ones. $N_{\vec s}$ will be brought back to discussion at the end of this section.
Prior to prove Eq.~\ref{eq:qrecycle2}, we first give a useful identity. The overlap between $\ket{\tilde \Psi_v^{\vec{s}}}$ and $\ket{\tilde \Psi_v^{\vec{s}^\prime}}$ where the two index vectors $\vec s$ and $\vec s^\prime$ differ by $K$ out of $M$ components can be cast in the following form,
\begin{eqnarray}\label{eq:obs-psimk} \bk{\tilde\Psi_v^{\vec{s}^\prime}}{\mathbf z}\bk{\mathbf z}{\tilde\Psi_v^{\vec{s}}} & = & i^K \bk{\tilde \Psi^{\vec{s}_{M-K}}}{z} V_{i_1}(\mathbf z) \cdots V_{i_K}( \mathbf z) \bk{\mathbf z}{\tilde \Psi^{\vec{s}_{M-K}}} \nonumber \\ & = & i^K \vert \bk{\tilde \Psi_v^{\vec{s}_{M-K}}}{\mathbf z} \vert^2 V_{i_1}(\mathbf z) \cdots V_{i_K}(\mathbf z), \end{eqnarray} where $\vec{s}_{M-K}$ is a reduced vector with the components $\{i_1 \cdots i_K\}$ removed from $\vec{s}$ and
\begin{eqnarray} V_j(\mathbf z)=s^\prime_j\sin\left(m^I_j+\sum_i W^I_{ij}z_i\right)\cos\left(m^I_j+\sum_i W^I_{ij}z_i\right) \end{eqnarray} are real-valued. From Eq.~\ref{eq:obs-psimk}, it is clear that the expression
\begin{eqnarray}\label{eq:obs-psimk2} \bk{\tilde \Psi_v^{\vec s^\prime}}{\mathbf z}\bk{\mathbf z}{\tilde\Psi_v^{\vec s}}+\bk{\tilde\Psi_v^{\vec s}}{\mathbf z} \bk{\mathbf z}{\tilde\Psi_v^{\vec s^\prime}}=0, \end{eqnarray} when the index vectors $\vec{s}^\prime$ and $\vec{s}$ differ by an odd number of components, i.e. $K$ is odd. This is because the symmetrized expression above must be real-valued, but Eq.~\ref{eq:obs-psimk} demands pure imaginary numbers when $K$ is odd. Hence, only zero is allowed. For the cases $\vec{s}^\prime$ and $\vec{s}$ differ by an even number of components, i.e. $K$ is even, we draw attention to a special condition. Let us first introduce another two states $\ket{\Psi^{\vec t}_v}$ and $\ket{\Psi^{\vec{t}^\prime}_v}$ with the index vectors satisfying (1) $\vec{t}$ and $\vec{s}$ differ by one component, (2) $\vec{t}$ and $\vec{t}^\prime$ differ by $K$ , and (3) $\vec{s}_{M-K}$ and $\vec{t}_{M-K}$ are identical. Then Eq.~\ref{eq:obs-psimk} implies,
\begin{eqnarray}\label{eq:obs-psimk3} \bk{\Psi_v^{\vec s^\prime}}{\mathbf z}\bk{\mathbf z}{\Psi_v^{\vec s}} + \bk{\Psi_v^{\vec t^\prime}}{\mathbf z}\bk{\mathbf z}{\Psi_v^{\vec t}}=0. \end{eqnarray} This is because $\prod_{q=1}^K V_{i_q}(\mathbf z)$, as defined in Eq.~\ref{eq:obs-psimk}, carries opposite sign for $\bk{\Psi_v^{\vec s^\prime}}{\mathbf z}\bk{\mathbf z}{\Psi_v^{\vec s}}$ and $\bk{\Psi_v^{\vec t^\prime}}{\mathbf z}\bk{\mathbf z}{\Psi_v^{\vec t}}$, respectively.
Now, consider a Hermitian operator $\hat O$ that is diagonal in the computational basis then the real-valued expectation value is given by
\begin{eqnarray}\label{eq:obs-app} \bra{\tilde\Psi_v}\hat O \ket{\tilde\Psi_v} & = & \sum_{\vec{s}^\prime \vec{s}} \frac{1}{2}\left(\prod_{j=1}^M R_{s^\prime_j}(m^R_j)R_{s_j}(m^R_j)\right) \left( \bra{\Psi_v^{\vec s^\prime}}\hat O\ket{\Psi_v^{\vec s}}+ \bra{\Psi_v^{\vec s}}\hat O \ket{\Psi_v^{\vec s^\prime}}\right)\nonumber \\ & = & \sum_{\mathbf z} \sum_{\vec{s}^\prime \vec{s}}\frac{O_{\mathbf z \mathbf z}}{2} \left(\prod_{j=1}^M R_{s^\prime_j}(m^R_j)R_{s_j}(m^R_j)\right)
\left(\bk{\Psi_v^{\vec s^\prime}}{\mathbf z}\bk{\mathbf z}{\Psi_v^{\vec s}}+ \bk{\Psi_v^{\vec s}}{\mathbf z} \bk{\mathbf z}{\Psi_v^{\vec s^\prime}}\right) \nonumber \\ & = & \sum_{\mathbf z} \sum_{\vec{s}} O_{\mathbf z \mathbf z}\left(\prod_{j=1}^M R^2_{s_j}(m^R_j)\right) \bk{\tilde \Psi_v^{\vec s}}{\mathbf z}\bk{\mathbf z}{\tilde \Psi_v^{\vec s}}. \end{eqnarray} Substituting Eq.~\ref{eq:obs-psimk} into the epxressions inside the last bracket on the second line of Eq.~\ref{eq:obs-app} and using Eq.~\ref{eq:obs-psimk2}-\ref{eq:obs-psimk3}, we can tremendously simplify the expression to reach the third line above. It is clear how Eq.~\ref{eq:obs-psimk2} helps to suppress terms with $\vec{s}$ and $\vec{s}^\prime$ that differ by an odd number of components. Next, we note that given a pair of $\vec s$ and $\vec{s}^\prime$, there is always a corresponding pair of $\vec t$ and $\vec{t}^\prime$ under the double summation ($ \sum_{\vec{s}^\prime \vec{s}}$) on the second line of Eq.~\ref{eq:obs-app} that satisfy the assumptions for Eq.~\ref{eq:obs-psimk3}.
When $\hat O = \mathcal{I}$, then we immediately obtain
\begin{eqnarray}\label{eq:obs-identity} \bk{\tilde \Psi_v}{\tilde \Psi_v} & = & N_v^2 \nonumber \\ &=& \sum_{\vec{s}} N^2_{\vec{s}} \left(\prod_{j=1}^M R^2_{s_j}(m^R_j)\right). \end{eqnarray} This expression admits a simple interpretation. $N_v^2$ is the success probability of preparing an NQS upon post-selection of $\ket{++\cdots +}_h$. According to Eq.~\ref{eq:obs-identity}, this probability can be exactly decomposed as a linear combination of conditional probabilities, $N^2_{\vec{s}}\prod_j R^2_{s_j}$, where $N^2_{\vec{s}}$ is the probability of measuring $\ket{s_1 \cdots s_M}_h$ after the hidden spins are coupled to the visible ones, and $\prod_j R^2_{s_j}$ is the conditional probability that the state $\prod_j \exp(m_j^R \hat{h}^z_j)\ket{s_1 \cdots s_M}_h$ can be subsequently projected back to the desired state, $\ket{++\cdots +}_h$.
When $\hat{\mathcal{O}}=\ket{\mathbf{z}}\bra{\mathbf{z}}$ as the projection onto a particular computational state then we obtain \begin{eqnarray}\label{eq:obs-proj} \vert \bk{\mathbf{z}}{\tilde{\Psi}_v} \vert^2
= \sum_{\vec{s}} \left(\prod_{j=1}^M R^2_{s_j}(m^R_j)\right) \bk{\tilde \Psi_v^{\vec s}}{\mathbf z}\bk{\mathbf z}{\tilde \Psi_v^{\vec s}} \end{eqnarray} By substituting back the normalized factors via $\ket{\tilde \Psi_v} = N_v \ket{\Psi_v}$ and $\ket{\tilde \Psi^{\vec s}_v} = N_{\vec s} \ket{\Psi^{\vec{s}}_v}$ into Eq.~\ref{eq:obs-proj}, we recover Eq.~\ref{eq:mc_obs} in the main text.
\section*{Supplementary Information IV: NQS-ITE Derivations and Implementations}
In Eq.~\ref{eq:chainrule1}, we relate $\partial_{\theta_i}\ket{\Psi_v(\theta)}$ to $\partial_{\theta_i}\ket{\Psi_{vh}(\theta)}$. Alternatively, we may write \begin{eqnarray} \ket{\Psi_v(\theta)} = \frac{\ket{\tilde\Psi_{vh}(\theta)}}{\sqrt{\bra{\tilde\Psi_{vh}(\theta)}\hat P^{(h)}_+\ket{\tilde\Psi_{vh}(\theta)}}}, \end{eqnarray} where $\ket{\tilde\Psi_{vh}(\theta)} = \sum_{\mathbf{h}} e^{\hat H_{RBM}(\theta,\mathbf{h})} \ket{++\dots+}_v$ is the un-normalized wave function. Hence, Eq.~\ref{eq:chainrule1} can be cast in this equivalent form, \begin{eqnarray}\label{eq:chainrule2}
\left \vert \frac{\partial \Psi_v}{\partial \theta_n} \right \rangle = \frac{_h\langle ++\dots + |\partial_{\theta_n} \tilde\Psi_{vh}\rangle}{\tilde N_v}
- Re \left( \langle \tilde\Psi_{vh} | \hat{P}_+^{(h)}| \partial_{\theta_n} \tilde\Psi_{vh} \rangle \right) \frac{_h\langle ++\dots + | {\tilde \Psi_{vh}} \rangle}{\tilde N_v^3}, \end{eqnarray} and $\tilde N_v = \sqrt{\bra{\tilde\Psi_{vh}(\theta)}P^{(h)}_+\ket{\tilde\Psi_{vh}(\theta)}}$. Similarly, we have to provide the derivatives of $\ket{\tilde \Psi_{vh}(\theta)}$, \begin{eqnarray}\label{eq:unvhwf-deriv} \begin{array}{ll}
\frac{ \partial |\tilde\Psi_{vh} \rangle }{\partial b^{R}_i} = \hat{v}^z_i |\tilde\Psi_{vh} \rangle , &
\frac{ \partial |\tilde\Psi_{vh} \rangle }{\partial m^{R}_j} = \tanh\left(m_j + \sum_i W_{ij} \hat{v}^z_i\right) |\tilde\Psi_{vh} \rangle, \\
\frac{ \partial |\tilde\Psi_{vh} \rangle }{\partial W^{R}_{ij}} = \hat{v}^z_i \tanh\left(m_j + \sum_i W_{ij} \hat{v}^z_i\right) |\tilde\Psi_{vh} \rangle, &
\frac{ \partial |\tilde\Psi_{vh} \rangle }{\partial b^{I}_i} = i\hat{v}^z_i | \tilde \Psi_{vh} \rangle, \\
\frac{ \partial |\tilde\Psi_{vh} \rangle }{\partial m^{I}_j} = i \tanh\left(m_j + \sum_i W_{ij} \hat{v}^z_i\right) |\tilde\Psi_{vh} \rangle, &
\frac{ \partial |\tilde\Psi_{vh} \rangle }{\partial W^{I}_{ij}} = i \hat{v}^z_i \tanh\left(m_j + \sum_i W_{ij} \hat{v}^z_i\right) |\tilde\Psi_{vh} \rangle. \end{array} \end{eqnarray} Substituting Eqs.~\ref{eq:chainrule2}-\ref{eq:unvhwf-deriv} into A matrix and C vector in Eq.~\ref{eq:IT-EOM}, we derive Eq.~\ref{eq:mtxAvecC} in the main text. To facilitate the following discussions, we denote $O_j = \tanh\left(m_j + \sum_i W_{ij} \hat{v}^z_i\right)$.
Since $O_j$ are not Hermitian operators and not directly measurable, an experimental scheme for measuring $\langle O_j \rangle_v$ (implied in Eqs.~\ref{eq:mtxAvecC}-\ref{eq:stochreconfigO}) should be clearly given. We propose the following strategy using the fact that $[O_j,\hat{v}^z_i]=0$,
\begin{eqnarray} \langle O^\dag_j O_{j^{\prime} }\rangle_v & = & \bra{\Psi_v(\theta)} O^\dag_j O_{j^{\prime}} \ket{\Psi_v(\theta)} \nonumber \\ & = & \sum_{\mathbf{z}_v} \vert \bk{\Psi_v(\theta)}{\mathbf{z}_v} \vert^2 O^\dag_j(\mathbf{z}_v) O_{j^{\prime}}(\mathbf{z}_v) \nonumber \\ & \xrightarrow[\text{according to } P_v(\mathbf{z}_v)]{\text{Monte Carlo sampling}} & \sum_{k=1}^{N_{\text{exp}}} \frac{O^\dag_j(\mathbf{z}_v^k) O_{j^{\prime}}(\mathbf{z}_v^k) }{N_{\text{exp}}}, \end{eqnarray} where $P_v(\mathbf{z}_v) = \vert \bk{\Psi_v(\theta)}{\mathbf{z}_v} \vert^2$ is a probability density, $\mathbf{z}_v = [z_{v,1}, \cdots, z_{v,N}]$ is a length-N binary string, and $O_{j} = \tanh(m_j + \sum_i W_{ij} z_{v,i})$. The second line in the equation above implies that simple projective measurements (in the computational basis) in a quantum circuit that prepares $\ket{\Psi_v(\theta)}$ state is an efficient approach to sample $\mathbf{z}^k_v$ binary strings from $P_v(\mathbf{z}_v)$. $N_{\text{exp}}$ samples of [$\mathbf{z}_v^{(k=1)} \cdots \mathbf{z}_v^{(k=N_\text{exp})}$] are obtained from the quantum circuit to estimate $O^\dag_j(\mathbf{z}_v^k) O_{j^{\prime}}(\mathbf{z}_v^k)$, according to the third line (the Monte Carlo method) in equation above. The expression $O^\dag_j(\mathbf{z}_v^k) O_{j^{\prime}}(\mathbf{z}_v^k)$ can be evaluated efficiently once a computational state $\mathbf{z}^k_v$ is specified.
$\langle O_j \rangle_v$ can be similarly estimated. On the hard, the evaluation of $\langle O^\dag_j \hat H \rangle_v$ requires further explications,
\begin{eqnarray}\label{eq:mc2} \langle O^\dag_j \hat H \rangle_v & = & \bra{\Psi_v(\theta)} O^\dag_j O_{j^{\prime}} \ket{\Psi_v(\theta)} \nonumber \\ & = & \sum_{\mathbf{z}_v,\mathbf{y}_v} \bk{\Psi_v(\theta)}{\mathbf{z}_v} O^\dag_j(\mathbf{z}_v) \hat H(\mathbf{z}_v,\mathbf{y}_v) \bk{\mathbf{y}_v}{\Psi_v(\theta)} \nonumber \\ & = & \sum_{\mathbf{z}_v} \vert \bk{\Psi_v(\theta)}{\mathbf{z}_v} \vert^2 \left( \sum_{\mathbf{y}_v} O^\dag_j(\mathbf{z}_v) \hat H(\mathbf{z}_v,\mathbf{y}_v) \frac{\bk{\mathbf{y}_v}{\Psi_v(\theta)}}{\bk{\mathbf{z}_v}{\Psi_v(\theta)}}\right), \nonumber \\ & \xrightarrow[\text{according to } P_v(\mathbf{z}_v)]{\text{Monte Carlo sampling}} & \sum_{k=1}^{N_{\text{exp}}} \frac{1}{N_{\text{exp}}} \left( \sum_{j} O^\dag_j(\mathbf{z}^k_v) \hat H(\mathbf{z}^k_v,\mathbf{y}^{k,j}_v)\frac{\bk{\mathbf{y}^{k,j}_v}{\Psi_v(\theta)}}{\bk{\mathbf{z}^k_v}{\Psi_v(\theta)}}\right), \end{eqnarray} where $\hat H(\mathbf{z}_v^k, \mathbf{y}^{k,j}_v) = \langle \mathbf{z}^k_v \vert \hat H \vert \mathbf{y}^{k,j}_v \rangle$. When the Hamiltonian,$ \hat H=\sum_l w_l \hat P_l$, is a linear combination of Pauli strings, $\hat H$ is a sparse matrix such that each computational state $\ket{\mathbf{z}^k_v}$ is only connected to a few other states $\ket{\mathbf{y}_v^{k,j}}$. On the third line of Eq.~\ref{eq:mc2}, the expression in the bracket should be evaluated classically. $O^ \dag_j(\mathbf{z}^k_v) \hat H(\mathbf{z}^k_v,\mathbf{y}^{k,j}_v)$ can be done efficiently, but not so much with the individual wave function amplitude $ \ket{\Psi_v(\theta)}$ which contains the normalization constant $\tilde N_v$ that is \#P-hard to compute in general. However, we only need the un-normalized wave function amplitudes as the quantity appearing in the bracket is the ratio of the wave function amplitudes projected onto two computational basis states, $\ket{\mathbf{z}_v^k}$ and $\ket{\mathbf{y}_v^{k,j}}$, respectively.
\section*{Supplementary Information V: Preparing a complex-valued NQS in Quantum Circuits} In the main text, we present a scalable method to prepare unitary-coupled RBM-NQS in quantum circuits. Now we discuss an extension that generates complex-valued NQS with arbitrary complex-valued $W_{ij}=W^R_{ij}+iW^I_{ij}$. First, we remark that the qubit-recycled approach to prepare an NQS is still applicable when coupling coefficients $W_{ij}$ are promoted to unrestricted complex numbers, \begin{eqnarray}\label{eq:qrecycle-app} \ket{\Psi_v(\theta)} & = & \frac{1}{N_v}
\left[\prescript{}{}{\bra{+}}\left[e^{\hat h^z_M \left(m_M + \sum_i W_{iM}\hat v^z_i\right)}\right]\ket{+}\right]_{M}
\left[\prescript{}{}{\bra{+}}\left[e^{\hat h^z_{M-1} \left( m_{M-1} + \sum_i W_{iM-1} \hat v^z_i\right)}\right]\ket{+}\right]_{M-1} \cdots
\nonumber \\
& & \times \left[\prescript{}{}{\bra{+}}\left[e^{\hat h^z_1 \left(m_1 + \sum_i W_{i1}\hat v^z_i\right)}\right]\ket{+}\right]_{1}\,\,
e^{\sum_i b_i \hat v^z_i} \ket{++\cdots+}_v. \end{eqnarray} Similarly, the ensemble state preparation scheme does not have to be modified when we deal with unrestricted couplings, $W_{ij}$. This conclusion is obvious as the crucial decomposition of the $j$-th block, $\left[\prescript{}{}{\bra{+}}[ \cdots ]\ket{+}\right]_{j}$, in Eq.~\ref{eq:qrecycle-app} still holds,
\begin{eqnarray}\label{eq:block-decomp2}
\prescript{}{}{\bra{+}}\left[e^{\hat h^z_j \left(m_j + \sum_i W_{ij}\hat v^z_i\right)}\right]\ket{+} & = & \sum_{s=\pm} \bra{+} e^{m^R_j \hat h^z_j} \ket{s} \bra{s} e^{\left(im^I_j+\sum_i W_{ij}\hat v^z_i\right) \hat h^z_j} \ket{+} \nonumber \\ & = & \sum_{s=\pm} R_{s}(m_j^R) \bra{s} e^{\sum_i W^R_{ij}\hat v^z_i\hat h^z_j}e^{\left(im^I_j+\sum_i W^I_{ij}\hat v^z_i\right) \hat h^z_j} \ket{+}. \end{eqnarray}
The new task is to efficiently map the non-unitary operators $\exp\left(\sum_i W^R_{ij}\hat v^z_i \hat h^z_j \right)$ into a set of appropriate unitary operations. We adapt the method introduced in Ref.~\cite{motta2019quantum} to get this mapping done without using post-selections. We briefly the summarize the key ideas and refer readers to Ref.~\cite{motta2019quantum} for full details. Starting with the decomposition $\exp\left(\sum_i W^R_{ij}\hat v^z_i \hat h^z_j \right)=\exp\left((1/K) \sum_i W^R_{ij} \hat v^z_i \hat h^z_j \right)^K$ , the single non-unitary operation may be turned into K successive rotations via the following relation,
\begin{eqnarray} \ket{\tilde{\psi}} = e^{(1/K)W_{ij}^R\hat{v}^z_{i}\hat{h}^z_j}\ket{\psi} = c e^{i (1/K) A}\ket{\psi}, \end{eqnarray} where $c=\sqrt{\bk{\tilde\psi}{\tilde\psi}}$ and $A=\sum_{s_1,\cdots,s_D,t} a_{s_1\cdots s_D t}\hat v^{s_1}_1\cdots\hat v^{s_D}_D \hat h^t_1$. The coefficients $a_{s_1\cdots s_D t}$ can be determined by minimizing the differences between two states, $\vert \vert W^R_{ij} \hat{v}^z_{i}\hat{h}^z_j\ket{\psi}-icA\ket{\psi} \vert\vert$.
In order to solve the minimization and determine $A$, one has to perform tomography on the quantum state $\ket{\psi}$. This is certainly a limiting step as the number of measurements required for a precise tomography scales exponentially with the system size. However, the situation could be eaiser to deal with if we assume that the correlation length of a typical many-body quantum system has a finite cut-off range such that the operator $A$ acts at most on $D$ visible spins and a single hidden spin. With this assumption\cite{motta2019quantum}, the restricted tomography mitigates the experimental costs.
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(\bibinfo {year} {2017})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Kivlichan}\ \emph {et~al.}(2018)\citenamefont
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {New J. Phys.}\
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{et~al.}(2019{\natexlab{b}})\citenamefont {Izmaylov}, \citenamefont {Yen},
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{NoStop} \bibitem [{\citenamefont {Huggins}\ \emph
{et~al.}(2019{\natexlab{a}})\citenamefont {Huggins}, \citenamefont {McClean},
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{NoStop} \bibitem [{\citenamefont {Zhu}\ \emph {et~al.}(2018)\citenamefont {Zhu},
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {arXiv preprint
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{NoStop} \bibitem [{\citenamefont {Nakanishi}\ \emph {et~al.}(2019)\citenamefont
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{} {\bibfield {journal} {\bibinfo {journal} {arXiv preprint
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{et~al.}(2019{\natexlab{a}})\citenamefont {Parrish}, \citenamefont {Iosue},
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{et~al.}(2019{\natexlab{b}})\citenamefont {Parrish}, \citenamefont
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{NoStop} \bibitem [{\citenamefont {Moseley}\ \emph {et~al.}()\citenamefont {Moseley},
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Physics Today}\
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{} {\bibfield {journal} {\bibinfo {journal} {Nat. Phys.}\ ,\ \bibinfo
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(\bibinfo {year} {2016})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Carrasquilla}\ and\ \citenamefont
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(\bibinfo {year} {2018})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Koch-Janusz}\ and\ \citenamefont
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
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{year} {2018})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Czischek}\ \emph {et~al.}(2018)\citenamefont
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
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(\bibinfo {year} {2018})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Lu}\ \emph {et~al.}(2019)\citenamefont {Lu},
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\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
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{} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. B}\ }\textbf
{\bibinfo {volume} {99}},\ \bibinfo {pages} {155136} (\bibinfo {year}
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {Q.}~\bibnamefont
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\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont {Levine}}, \bibinfo
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\bibnamefont {Lukin}}, \emph {et~al.},\ }\href@noop {} {\bibfield {journal}
{\bibinfo {journal} {arXiv preprint arXiv:1904.08441}\ } (\bibinfo {year}
{2019})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Huang}\ \emph {et~al.}(2017)\citenamefont {Huang},
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
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{} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. E}\ }\textbf
{\bibinfo {volume} {95}},\ \bibinfo {pages} {031301} (\bibinfo {year}
{2017})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Wang}(2017)}]{wang2017exploring}
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
{Wang}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys.
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(\bibinfo {year} {2017})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Inack}\ \emph {et~al.}(2018)\citenamefont {Inack},
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
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{\bibfield {journal} {\bibinfo {journal} {Phys. Rev. B}\ }\textbf {\bibinfo
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{Melko}(2017)}]{torlai2017neural}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
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(\bibinfo {year} {2017})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Bausch}\ and\ \citenamefont
{Leditzky}(2018)}]{bausch2018quantum}
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
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{NoStop} \bibitem [{\citenamefont {Zhang}\ \emph {et~al.}(2018)\citenamefont {Zhang},
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {Y.-H.}\ \bibnamefont
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\bibinfo {author} {\bibfnamefont {Y.-C.}\ \bibnamefont {Wu}}, \ and\ \bibinfo
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{\bibfield {journal} {\bibinfo {journal} {arXiv preprint arXiv:1809.08631}\
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
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(\bibinfo {year} {2017})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Deng}\ \emph
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.-L.}\ \bibnamefont
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{} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. B}\ }\textbf
{\bibinfo {volume} {96}},\ \bibinfo {pages} {195145} (\bibinfo {year}
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.~J.}\ \bibnamefont
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(\bibinfo {year} {2019})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Nagy}\ and\ \citenamefont
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(\bibinfo {year} {2019})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Luo}\ and\ \citenamefont
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(\bibinfo {year} {2019})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Torlai}\ and\ \citenamefont
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(\bibinfo {year} {2018})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Nomura}\ \emph {et~al.}(2017)\citenamefont {Nomura},
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. B}\
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {arXiv preprint
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{\bibfield {journal} {\bibinfo {journal} {Phys. Rev. X}\ }\textbf {\bibinfo
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{2018})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Deng}\ \emph
{et~al.}(2017{\natexlab{b}})\citenamefont {Deng}, \citenamefont {Li},\ and\
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.-L.}\ \bibnamefont
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{} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. X}\ }\textbf
{\bibinfo {volume} {7}},\ \bibinfo {pages} {021021} (\bibinfo {year}
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont
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\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {X.}~\bibnamefont
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}\href {https://doi.org/10.1038/s41467-017-00705-2} {\bibfield {journal}
{\bibinfo {journal} {Nat. Commun.}\ }\textbf {\bibinfo {volume} {8}},\
\bibinfo {pages} {662} (\bibinfo {year} {2017}{\natexlab{a}})}\BibitemShut
{NoStop} \bibitem [{\citenamefont {Clark}(2018)}]{clark2018unifying}
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.~R.}\ \bibnamefont
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{135301} (\bibinfo {year} {2018})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Chen}\ \emph {et~al.}(2018)\citenamefont {Chen},
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\citenamefont {Xiang}}]{chen2018equivalence}
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
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\bibinfo {author} {\bibfnamefont {H.}~\bibnamefont {Xie}}, \bibinfo {author}
{\bibfnamefont {L.}~\bibnamefont {Wang}}, \ and\ \bibinfo {author}
{\bibfnamefont {T.}~\bibnamefont {Xiang}},\ }\href@noop {} {\bibfield
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
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}\textbf {\bibinfo {volume} {9}},\ \bibinfo {pages} {4195} (\bibinfo {year}
{2018})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Gardas}\ \emph {et~al.}(2018)\citenamefont {Gardas},
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. B}\
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\citenamefont {Zhang}, \citenamefont {Wan},\ and\ \citenamefont
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{2019})}\BibitemShut {NoStop} \bibitem [{\citenamefont {Huggins}\ \emph
{et~al.}(2019{\natexlab{b}})\citenamefont {Huggins}, \citenamefont {Patil},
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
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{\bibinfo {volume} {4}},\ \bibinfo {pages} {024001} (\bibinfo {year}
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
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\begin{document}
\baselineskip=17pt
\title{Variational characterization of $H^p$}
\date{}
\maketitle
\begin{center}
{\bf Honghai Liu}\\ School of Mathematics and information Science\\ Henan Polytechnic University\\ Jiaozuo 454003\\ The People's Republic of China \\ E-mail: {\it [email protected]} \end{center}
\renewcommand{\arabic{footnote}}{}
\footnote{2010 \emph{Mathematics Subject Classification}: Primary 42B20; Secondary 42B25.}
\footnote{\emph{Key words and phrases}: Variation, Oscillation, $\lambda$-jump, Hardy space, Approximate identities.}
\footnote{The research was supported by NSF of China (Grant: 11501169 and 11371057).}
\renewcommand{\arabic{footnote}}{\arabic{footnote}} \setcounter{footnote}{0}
\begin{abstract} In this paper we obtain the variational characterization of Hardy space $H^p$ for $p\in(\frac n{n+1},1]$ and get estimates for the oscillation operator and the $\lambda$-jump operator associated with approximate identities acting on $H^p$ for $p\in(\frac n{n+1},1]$. Moreover, we give counterexamples to show that the oscillation and $\lambda$-jump associated with some approximate identity can not be used to characterize $H^p$ for $p\in(\frac n{n+1},1]$. \end{abstract}
\section{Introduction}
Variational, oscillation and jump inequalities have been the subject of many
recent articles in probability, ergodic theory and harmonic
analysis. The first variational inequality was proved by L\'epingle \cite{Lep76} for martingales. Using L\'epingle's result, Bourgain \cite{Bou89} is the first one who to obtain corresponding variational estimates for the Birkhoff ergodic averages and then directly deduce pointwise convergence results without previous knowledge that pointwise convergence holds for a dense subclass of functions, which are not available in some ergodic models. Bourgain's work has initiated a new research direction in ergodic theory and harmonic analysis. In \cite{CJRW2000,CJRW2002,JKRW98,JRW03,JSW08}, Jones and his collaborators systematically studied jump and variational inequalities for ergodic averages and truncated singular integrals. Since then many other publications came to enrich the literature on this subject. Analogs are also true for corresponding maximal operators which were known. A number of phenomena show that variational, jump and oscillation operators seems to play the same role as maximal operators in harmonic analysis.
\par
In this paper we consider variation, oscillation and $\lambda$-jump operators associated with approximate identities. The variational inequality gives us the characterization of $H^p$ for $p\in(\frac n{n+1},1]$. We obtain estimates for the oscillation and $\lambda$-jump acting on Hardy space. A counterexample show that oscillation and $\lambda$-jump operators are too small to characterize Hardy space. Before we present our main results we recall some definitions and known results.
Let $\mathcal I$ be a subset of $\mathbb R^+$, $\mathfrak{a}=\{a_t: t\in \mathcal I\}$ be a family of complex numbers and $\rho\ge1$. The $\rho$-variation norm of the family $\mathfrak{a}$ is defined by \begin{equation}\nonumber
\|\mathfrak{a}\|_{v_\rho(\mathcal I)}=\sup\big(\sum_{k\geq1}
|a_{t_k}-a_{t_{k-1}}|^\rho\big)^{\frac{1}{\rho}}, \end{equation} where the supremum runs over all finite decreasing sequences $\{t_k\}$ in $\mathcal I$. We denote the norm $v_\rho(\mathbb R^+)$ by $v_\rho$ for short. It is trivial that \begin{equation}\nonumber
\|\mathfrak{a}\|_{L^\infty(\mathcal I)}:=\sup_{t\in \mathcal I}|a_t|
\le |a_{t_0}|+\|\mathfrak{a}\|_{v_\rho(\mathcal I)} \end{equation} for any $t_0\in \mathcal I$ and $\rho\ge1$. \par Given a family of Lebesgue measurable functions $\mathcal F(x)=\{F_t(x):t\in \mathcal I\}$, the value of the $\rho$-variation function $\mathscr V_q(\mathcal F)$ of the family $\mathcal F$ at $x$ is defined by \begin{equation}\nonumber
\mathscr V_\rho(\mathcal F)(x)=\|\{F_t(x)\}\|_{v_\rho(\mathcal I)},\quad \rho\ge1. \end{equation}
Specially, suppose $\mathscr{A}=\{{A}_t\}_{t>0}$ is a family of operators, the $\rho$-variation operator related $\mathscr A$ is simply defined as
$$\mathscr V_\rho(\mathscr Af)(x)=\|\{A_t(f)(x)\}_{t>0}\|_{v_\rho}.$$ It is easy to observe that for any fixed $x\in\mathbb R^n$, if $\mathscr V_\rho(\mathscr Af)(x)<\infty$, then $\lim\limits_{t\rightarrow0^+}A_t(f)(x)$ and $\lim\limits_{t\rightarrow+\infty}A_t(f)(x)$ exist. In particular, if $\mathscr V_\rho(\mathscr Af)$ belongs to some function spaces such as $L^p$ or $L^{p,\infty}$, then the sequence converges almost everywhere without any additional condition. This is why mapping property of $\rho$-variation operator is so interesting in probability, ergodic theory and harmonic analysis. In 1976, L\'{e}pingle \cite{Lep76} showed that the $\rho$-variation operator related to a bounded martingale sequence is a bounded operator on $L^p$ for $1<p<\infty$ and $\rho>2$. These estimates can fail for $\rho\le2$, see \cite{JW04,Q98}. So, we need the oscillation operator to substitute the $2$-variation operator. \par For each fixed decreasing sequence $\{t_i\}$ in $\mathbb R^+$, we also define the oscillation operator related to $\mathscr A$ \begin{equation}\nonumber
\mathscr O(\mathscr Af)(x)=\big(\sum_{i=1}^\infty\sup_{t_{i}\le\varepsilon_{i}<\varepsilon_{i+1}\le t_{i+1}}|A_{\varepsilon_{i+1}}f(x)-A_{\varepsilon_{i}}f(x)|^2\big)^{1/2}. \end{equation} \par We also study the $\lambda$-jump operator. For $\lambda>0$, the value of the $\lambda$-jump function for $\mathcal F$ at $x$ is defined by
$$N_\lambda(\mathcal F)(x)=\sup\big\{N\in\Bbb N:\ \exists\ s_1<\varepsilon_1\leq s_2<\varepsilon_2\leq\dotsc\leq s_N<\varepsilon_N\ \text{such that\ } |F_{\varepsilon_k}(x)-F_{s_k}(x)|>\lambda\big\}.$$ Similarly, we define the $\lambda$-jump operator related to $\mathscr A$ as $N_\lambda(\mathscr Af)(x)=N_\lambda(\{A_tf\}_{t>0})(x)$. Obviously, if $\lim\limits_{t\rightarrow0^+}A_tf(x)$ and $\lim\limits_{t\rightarrow+\infty}A_tf(x)$ exist, then $N_\lambda(\mathscr Af)(x)<\infty$ for any $\lambda>0$. Moreover, for $\lambda>0$ and $\rho\ge1$
\begin{equation}\label{contr ineq} \lambda[N_\lambda(\mathscr Af)(x)]^{1/\rho}\leq C_\rho\mathscr V_\rho(\mathscr Af)(x). \end{equation} \par Let $\phi\in\mathscr S$ with $\int \phi dx=1$, $\phi_t(x)=\frac1{t^n}\phi(\frac xt)$, denote function family $\{\phi_t\ast f(x)\}_{t>0}$ by $\Phi\star f(x)$. Let $f$ be a tempered distribution, we define maximal function $M_\phi$ by $$
M_\phi f(x)=\sup_{t>0}|(f\ast\phi_t)(x)|. $$ \begin{definition} Let $0<p<\infty$. A distribution $f$ belongs to $H^p$ if the maximal function $M_\phi f$ is in $L^p$. \end{definition} \par The main results of this paper are the following three theorems. \begin{theorem}\label{SchHp} For any $\rho>2$, there exists $C_{\rho}>0$ such that \begin{equation}\label{VISchHp}
\|\mathscr V_\rho(\Phi\star f)\|_{L^p}\le C_{\rho}\|f\|_{H^p},\ \ \frac n{n+1}<p\le1. \end{equation} Moreover, for $\frac n{n+1}<p<\infty$, the following conditions are equivalent: \begin{description} \item (i) There is a $\phi\in\mathscr S$ with $\int\phi dx\neq0$ so that $M_\phi f\in L^p$. \item (ii) For any $\rho>2$, there is a $\phi\in\mathscr S$ with $\int\phi dx\neq0$ so that $\phi\ast f$ and $\mathscr V_\rho(\Phi\star f)$ in $L^p$. \end{description} \end{theorem} In the above result, the variation operator is used to characterize $H^p$ spaces, it is natural to ask if the analogue for the oscillation operator holds. \begin{theorem}\label{oHp} For any $\{t_i\}\searrow0$ and $p\in(\frac n{n+1},1]$, there exists a positive constant $C_p$ such that \begin{equation}\label{OISchHp}
\|\mathscr O(\Phi\star f)\|_{L^p}\le C_p\|f\|_{H^p}. \end{equation} Moreover, there exists $\{t_i\}\searrow0$, $\phi\in\mathscr S$ with $\int\phi dx\neq0$ and $f\in \mathscr S$ such that $\mathscr O(\Phi\star f)\in L^p$ and $M_\phi(f)\notin L^p$ for any $p\in(0,1]$. \end{theorem} We also apply the above result on variation to provide estimates for the $\lambda$-jump operator. \begin{theorem}\label{ljrhp} If $\rho>2$, then the $\lambda$-jump operator $N_\lambda(\Phi\star f)$ satisfies \begin{equation}
\|\lambda N_\lambda(\Phi\star f)^{1/\rho}\|_{L^p}\le C_{\rho}\|f\|_{H^p},\ \ \frac n{n+1}<p\le1, \end{equation}
uniformly in $\lambda>0$. Moreover, there exists $\phi\in\mathscr S$ with $\int\phi dx\neq0$ and $f\in \mathscr S$ such that $\|\lambda N_\lambda(\Phi\star f)^{1/\rho}\|_{L^1}<\infty$ uniformly in $\lambda>0$ and $M_\phi(f)\notin L^1$. \end{theorem}
The paper is organized as follows. In Section 2, we prove that $\rho$-variation, oscillation and $\lambda$-jump operators related to approximate identities are of strong type $(p,p)$ for $1<p<\infty$ and weak type $(1,1)$. Using the strong $L^p(p>1)$ estimates for the $\rho$-variation operator and the atom decomposition of Hardy space, we show Theorem \ref{SchHp} in Section 3. We consider the estimate for the oscillation operator associated to approximate identities acting on Hardy space and show that the oscillation is not proper to characterize Hardy spaces in Section 4. Finally, we present a jump inequality for the $\lambda$-jump operator on Hardy space and conjecture that its improvement holds, we also illustrate that the $\lambda$-jump operator can not be used to characterize $H^1$. \section{Operators related to approximate identities on $L^p$} To study the mapping property of variation, oscillation and $\lambda$-jump operators related to approximate identities on $H^p$, we need strong $L^p$ estimates for $1<p<\infty$. We use the argument in \cite{JSW08} and include all details for completeness although they are trivial. The following lemmas will be used later. \begin{lemma}(\cite[Lemma 1.3]{JSW08})\label{pdN} Let $\mathscr A=\{A_t\}_{t>0}$ be a family of operators. Then $$ \lambda\sqrt{N_\lambda(\mathscr A f)}\le C[S_2(\mathscr A f)+\lambda \sqrt{N^d_{\lambda/3}(\mathscr A f)}], $$
where $N^d_\lambda(\mathscr A f)=N_\lambda(\{A_{2^k}f\})$ and $S_2(\mathscr Af)=(\sum_j\|\mathscr Af\|_{v_2(2^j,2^{j+1}]}^2)^{1/2}$. \end{lemma} Let $\sigma$ be a compactly supported finite Borel measure and satisfying \begin{equation}\label{fts}
|\hat{\sigma}(\xi)|\le C|\xi|^{-b}, \ \ for\ some\ b>0. \end{equation}
$\sigma_t$ is given by $<\sigma_t,f>=\int f(tx)d\sigma$. \begin{lemma}(\cite[Theorem 1.1,\ Lemma 6.1]{JSW08})\label{NdS2} Let $\mathfrak U=\{\mathfrak U_k\}$ where $\mathfrak U_kf=f\ast\sigma_{2^k}$. If $\sigma$ satisfies \eqref{fts}, then $$
\|\lambda\sqrt{N_\lambda(\mathfrak U f)}\|_{L^p}\le C_p\|f\|_{L^p} $$ uniformly in $\lambda>0$. Moreover, let $\mathscr A=\{A_t\}_{t>0}$ where $A_tf=f\ast\sigma_t$. If $\sigma$ satisfies \eqref{fts} for some $b>1/2$, then $$
\|S_2(\mathscr Af)\|_{L^p}\le C_p\|f\|_{L^p} $$ holds for $\min\{2n/(n+2b-1),(2b+1)/2b\}<p<\max\{2n/(n-2b-1),2b+1\}$. \end{lemma} \begin{lemma}(\cite[Theorem 1.1]{JSW08})\label{Ndw11} If $\phi$ satisfies \begin{equation}
\int_{\mathbb R^n}|\phi(x+y)-\phi(x)|dx\le C|y|^{-b} \end{equation} for some $b>0$, then for any $\alpha>0$ we have \begin{equation}
|\{x:\lambda\sqrt{N_\lambda^d(\Phi\star f)(x)}>\alpha\}|\le \frac C\alpha\|f\|_{L^1} \end{equation} uniformly in $\lambda>0$. \end{lemma}
We now state a theorem on the $\lambda$-jump operator associated with approximate identities. \begin{theorem}\label{ljphip} Let $\phi\in\mathscr S$ with $\int\phi dx\neq0$. For $1<p<\infty$, there exists a positive constant $C_p$ such that \begin{equation}\label{JISchp}
\|\lambda\sqrt{N_\lambda(\Phi\star f)}\|_{L^p}\le C_p\|f\|_{L^p}, \end{equation}
uniformly in $\lambda>0$. Moreover, for any $\alpha>0$, \begin{equation}\label{JISch1}
|\{x:\lambda\sqrt{N_\lambda(\Phi\star f)(x)}>\alpha\}|\le \frac C\alpha\|f\|_{L^1}, \end{equation} uniformly in $\lambda>0$. \end{theorem} \begin{proof} Clearly, $\hat{\phi}\in\mathscr S$, which means that for any $N\in\mathbb N$ there is a $C_N$ such that $$
|\hat{\phi}(\xi)|\le C_{N}(1+|\xi|)^{-N}. $$
Therefore $|\hat{\phi}(\xi)|\le C_{n}|\xi|^{-\frac{n+1}2}$. By Lemma \ref{NdS2}, we have \begin{equation}
\|\lambda\sqrt{N_\lambda^d(\Phi\star f)}\|_{L^p}\le C_p\|f\|_{L^p}\ \ \text{and}\ \ \|S_2(\Phi\star f)\|_{L^p}\le C_p\|f\|_{L^p},\ \ 1<p<\infty. \end{equation} By Lemma \ref{pdN}, we get \eqref{JISchp}. We turn to the proof of \eqref{JISch1}. Lemma \ref{pdN} and Lemma \ref{Ndw11} imply that it suffices to prove \begin{align}\label{S2w1}
|\{x:S_2(\Phi\star f)(x)>1\}|\le C\|f\|_{L^1}. \end{align} We perform the Calder\'{o}n-Zygmund decomposition of $f$ at height $1$ and write $f=g+b$. We just need to establish the following two estimates: \begin{equation}\label{sg11}
|\{x:S_2(\Phi\star g)(x)>1/2\}|\le C
\|f\|_{L^1}, \end{equation} and \begin{equation}\label{sb11}
|\{x:S_2(\Phi\star b)(x)>1/2\}|\le C
\|f\|_{L^1}. \end{equation}
As usual the known $L^p$ bounds for $S_2$ allows us to obtain (\ref{sg11}): \begin{align*}
|\{x:S_2(\Phi\star g)(x)>1/2\}|&\le C\int_{\mathbb R^n}|S_2(\Phi\star g)(x)|^2dx\\
&\le C \int_{\mathbb R^n}|g(x)|^2dx\le C\|f\|_{L^1}. \end{align*} \par To show \eqref{sb11}, we write $b=\sum_jb_j$ precisely, where each $b_j$ is supported in a dyadic cube $Q_j$. We denote by $l(Q_j)$ the side length of $Q_j$. Let $\tilde{Q}_j$ be the cube with sides parallel to the axes having the same center as $Q_j$ and having side length $8l(Q_j)$, write $\tilde{Q}=\bigcup \tilde{Q}_j$. Obviously, \begin{align*}
|\tilde{Q}|\le\sum_j |\tilde{Q}_j|\le C\sum_j |Q_j|\le C\|f\|_{L^1}. \end{align*} We still need to prove that \begin{align*}\label{wsbc11}
|\{x\in\tilde{Q}^c:S_2(\Phi\star b)(x)>1/2\}|\le C\|f\|_{L^1}. \end{align*} Let $x_j$ be the center of $Q_j$. Using the moment condition of $b_j$ and H\"{o}lder's inequality, we get \begin{align*}
\|\Phi\star b(x)\|_{v_2(2^i,2^{i+1}]}\le\sum_j\int_{Q_j}|b_j(y)|\|\Phi(x-y)-\Phi(x-x_j)\|_{v_2(2^i,2^{i+1}]}dy. \end{align*}
For $x\in \tilde{Q}^c$ and $x_j,y\in Q_j$, it is clear that $|x-x_j-\theta (y-x_j)|\sim |x-x_j|$ for any $\theta\in [0,1]$. Then, \begin{align*}
\|\Phi(x-y)-\Phi(x-x_j)\|_{v_2(2^i,2^{i+1}]}
&\le \|\Phi(x-y)-\Phi(x-x_j)\|_{v_1(2^i,2^{i+1}]}\\
&\le \int_{2^i}^{2^{i+1}}|\frac d{dt}[\phi_t(x-y)-\phi_t(x-x_j)]|dt\\
&\le C|y-x_j|\int_{2^i}^{2^{i+1}}\frac{1}{t^{n+2}}\big(1+\frac{|x-x_j|}t)^{-(n+2)}dt, \end{align*} where we use the following well-known fact \begin{equation}\label{vr1i}
\|\mathfrak a\|_{v_\rho}\le \|\mathfrak a\|_{v_1}\le \int_0^\infty|\mathfrak a'(t)|dt, \end{equation} see (39) in \cite{JSW08}. Consequently, \begin{equation}\label{eoV2i}
\|\Phi(x-y)-\Phi(x-x_j)\|_{v_2(2^i,2^{i+1}]}\le Cl(Q_j)\min\{2^{i+1}|x-x_j|^{-n-2}, 2^{-i(n+1)}\}. \end{equation} Finally, by \eqref{eoV2i} and H\"{o}lder's inequality, \begin{align*}
|\{x\in\tilde{Q}^c:S_2(\Phi\star b)(x)>1/2\}|&\le C
\int_{\tilde{Q}^c}(\sum_{i\in\mathbb Z}\|\Phi\star b(x)\|_{v_2(2^i,2^{i+1}]}^2)^{\frac12}dx\\
&\le C\int_{\tilde{Q}^c}\sum_j\int_{Q_j}|b_j(y)|[\sum_{i\in\mathbb Z}\|\Phi(x-y)-\Phi(x-x_j)\|_{v_2(2^i,2^{i+1}]}^2]^{\frac12}dydx\\
&\le C\sum_jl(Q_j)\int_{Q_j}|b_j(y)|dy\int_{\tilde{Q_j}^c}\frac{dx}{|x-x_j|^{n+1}}\\
&\le C\|f\|_{L^1}. \end{align*} This completes the proof of Theorem \ref{ljphip}. \end{proof} To obtain the variational inequality, we present a lemma reducing variational inequalities to jump inequalities, which is a generalization of Bourgain's argument in \cite{Bou89}. \begin{lemma}(\cite[Lemma 2.1]{JSW08})\label{jcv} Suppose that $p_0<q<p_1$ and that for $p_0<p<p_1$ the inequality $$
\sup_{\lambda>0}\|\lambda[N_\lambda(\mathcal Tf)]^{1/q}\|_{L^p}\le C\|f\|_{L^p} $$ holds for all $f$ in $L^p$. Then we have for $q<\rho$, $$
\|\mathscr V_\rho(\mathcal Tf)\|_{L^p}\le C(p,\rho)\|f\|_{L^p} $$ for $f\in L^p$, $p_0<p<p_1$. \end{lemma} As a result of above lemma, the following variational inequality holds: \begin{theorem}\label{SchLp} For any $\rho>2$ and $1<p<\infty$, there exists $C_{p,\rho}>0$ such that \begin{equation}\label{VISchp}
\|\mathscr V_\rho(\Phi\star f)\|_{L^p}\le C_{p,\rho}\|f\|_{L^p}. \end{equation} Moreover, for any $\alpha>0$, \begin{equation}\label{VISch1}
|\{x:\mathscr V_\rho(\Phi\star f)(x)>\alpha\}|\le \frac C{\alpha}\|f\|_{L^1}. \end{equation} \end{theorem} \begin{proof} Clearly, \eqref{JISchp} and Lemma \ref{jcv} imply \eqref{VISchp}. \eqref{VISch1} can be proved as \eqref{S2w1}. \end{proof} \begin{theorem}\label{OSchLp} For any $\{t_i\}\searrow0$ and $1<p<\infty$, there exists $C_p>0$ such that \begin{equation}\label{OISchp}
\|\mathscr O(\Phi\star f)\|_{L^p}\le C_{p}\|f\|_{L^p}. \end{equation} Moreover, for any $\alpha>0$, \begin{equation}\label{OISch1}
|\{x:\mathscr O(\Phi\star f)(x)>\alpha\}|\le \frac C{\alpha}\|f\|_{L^1}. \end{equation} \end{theorem} \begin{proof} Let $k_i$ be the smallest integer such that $2^{k_i}$ greater than or equal to $t_i$. The long oscillation operator is given by \begin{equation}\nonumber
\mathscr O_L(\Phi\star f)(x)=\big(\sum_{i}\sup_{k_{i+1}\le l\le m\le k_i}|\phi_{2^l}\ast f(x)-\phi_{2^m}\ast f(x)|^2\big)^{1/2}. \end{equation} The oscillation inequality follows from the pointwise estimate \begin{align*} \mathscr O(\Phi\star f)(x)\le C[S_2(\Phi\star f)(x)+\mathscr O_L(\Phi\star f)(x)]. \end{align*} Strong $L^p$ estimates and weak $(1,1)$ estimate for $S_2(\Phi\star f)$ have been established as above. For the long oscillation $\mathscr O_L(\Phi\star f)$, we borrow some notations and results from \cite[pp.6724]{JSW08}. For $j\in\mathbb Z$ and $\beta=(m_1,\cdots,m_n)\in\mathbb Z^n$, we denote the dyadic cube $\prod_{k=1}^n(m_k2^j,(m_k+1)2^j]$ in $\mathbb R^n$ by $Q_\beta^j$, and the set of all dyadic cubes with side length $2^j$ by $\mathcal D_j$. The conditional expectation of a local integrable $f$ with respect to $\mathcal D_j$ is given by $$
\mathbb E_jf(x)=\sum_{Q\in \mathcal D_j}\frac1{|Q|}\int_{Q}f(y)dy\cdot\chi_{Q}(x) $$ for all $j\in\mathbb Z$. Note that $\mathscr O_L$ satisfies \begin{equation}\nonumber \mathscr O_L(\Phi\star f)\le \mathscr O_L(\mathscr D f)+\mathscr O_L(\mathscr Ef), \end{equation} where $$ \mathscr Df=\{\phi_{2^k}\ast f-\mathbb E_kf\}_k \quad \text{and}\quad \mathscr Ef=\{\mathbb E_kf\}_k. $$ Following inequalities are oscillation inequalities for dyadic martingales (see \cite{JKRW98}), $$
|\{x:\mathscr O_L(\mathscr{E}f)(x)>\alpha\}|\le \frac C{\alpha}\|f\|_{L^1}\ \text{and}\ \|\mathscr O_L(\mathscr{E}f)\|_{L^p}\leq C_p\|f\|_{L^p},\ 1<p<\infty. $$ Next, observe that \begin{equation}\nonumber
\mathscr O_L(\mathscr D f)\le C\big(\sum_{k\in\mathbb Z}|\phi_k\ast f-\mathbb E_kf|^2\big)^{1/2}:=\mathcal Sf. \end{equation} Jones {\it et al} \cite{JSW08} have established the following weak-type $(1,1)$ bound and $L^p$ bounds for $\mathcal S$, \begin{equation}\label{S1p}
|\{x:\mathcal Sf(x)>\alpha\}|\le \frac C{\alpha}\|f\|_{L^1}\ \text{and}\ \|\mathcal Sf\|_{L^p}\le C_p\|f\|_{L^p},\ \ 1<p<\infty, \end{equation} see also \cite{DHL16}. This completes the proof of Theorem \ref{OSchLp}. \end{proof} \begin{remark} With $\phi$ a radial function, Campbell {\it et al} \cite{CJRW2000,CJRW2002} obtained Theorem \ref{SchLp} and Theorem \ref{OSchLp} by rotation method. \end{remark}
\section{Variational characterization of $H^p$} In what follows we shall use the well-known atom decomposition of $H^p$. So, we present the definition of $(p,q)$-atom. \begin{definition} Let $0<p\le 1\le q\le\infty$, $p\neq q$. A function $a(x)\in L^q$ is called a $(p,q)$-atom with the center at $x_0$, if it satisfies the following conditions: \begin{description} \item (i) Supp $a\subset B(x_0,r)$;
\item (ii) $\|a\|_{L^q}\le |B(x_0,r)|^{\frac1q-\frac1p}$; \item (iii) $\int a(x)dx=0$. \end{description} \end{definition}
\begin{lemma}(\cite{L78}) Let $0<p\le 1$. Given a distribution $f\in H^p$, there exists a sequence of $(p,q)$-atoms with $1\le q\le\infty$ and $q\neq p$, $\{a_k\}$, and a sequence of scalars $\{\lambda_k\}$ such that
$$
f=\sum_{k}\lambda_ka_k\ \ \text{in}\ \ H^p.
$$ \end{lemma} \textit{Proof of Theorem \ref{SchHp}}.\ \ In proving Theorem \ref{SchHp} we consider first the inequality \eqref{SchHp}. By \cite[Theorem 1.1]{YZ08}, we just need to prove that there exists a positive constant $C$ such that
$\|\mathscr V_\rho(\Phi\star a)\|_{L^p}\le C$ for any $(p,2)$ atom $a$ and $p\in(\frac{n}{n+1},1]$. \par Suppose $a$ is supported in a cube $Q$, $x_0$ is the center of $Q$, write $\tilde{Q}=8Q$. By H\"{o}lder's inequality and Theorem \ref{SchLp}, $$
\int_{\tilde{Q}}\mathscr V_\rho(\Phi\star a)^p(x)dx\le |\tilde{Q}|^{1-\frac p2}\|\mathscr V_\rho(\Phi\star a)\|_{L^2}^p\le C|Q|^{1-\frac p2}\|a\|^p_{L^2}\le C. $$ \par To deal with $x\in (\tilde{Q})^c$ one uses the cancelation condition of $a$ and Minkowski's inequality, \begin{align*} &\mathscr V_\rho(\Phi\star a)(x)\\
&=\sup_{\{\varepsilon_k\}\searrow0}\bigg(\sum_{k}\bigg|\int_{\mathbb R^n}\bigg\{\big[\phi_{\varepsilon_k}(x-y)-\phi_{\varepsilon_{k+1}}(x-y)\big]-\big[\phi_{\varepsilon_k}(x-x_0)-\phi_{\varepsilon_{k+1}}(x-x_0)\big]\bigg\}a(y)dy\bigg|^\rho\bigg)^{\frac1\rho}\\
&\le\int_{Q}|a(y)| \sup_{\{\varepsilon_k\}\searrow0}\bigg(\sum_{k}\bigg|\big[\phi_{\varepsilon_k}(x-y)-\phi_{\varepsilon_k}(x-x_0)\big]-\big[\phi_{\varepsilon_{k+1}}(x-y)-\phi_{\varepsilon_{k+1}}(x-x_0)\big]\bigg|^\rho\bigg)^{\frac1\rho}dy\\
&\le\int_{Q}|a(y)|\|\Phi(x-y)-\Phi(x-x_0)\|_{v_\rho}dy. \end{align*}
For $x\in (\tilde{Q})^c$, $y\in Q$ and $\theta\in(0,1)$, we have $|x-x_0+\theta (y-x_0)|\sim|x-x_0|$. Note that $\phi(\frac{x-y}t)$ is a smooth function of $t$ on $(0,\infty)$ for any fixed $x,y$. Applying \eqref{vr1i} and the mean value theorem, we estimate \begin{align*}
\nonumber\|\Phi(x-y)-\Phi(x-x_0)\|_{v_\rho}\le&\|\Phi(x-y)-\Phi(x-x_0)\|_{v_1}\\
\le& C|y-x_0|\int_0^\infty\frac1{t^{n+2}}\big(1+\frac{|x-x_0|}t)^{-(n+2)}dt\\
\nonumber\le& C\frac{|y-x_0|}{|x-x_0|^{n+1}}\int_0^\infty\frac{t^n}{(1+t)^{n+2}}dt\\
\nonumber\le&C\frac{|y-x_0|}{|x-x_0|^{n+1}}. \end{align*} Hence, we obtain the desired bound \begin{align*}
\int_{(\tilde Q)^c}\mathscr V_\rho(\Phi\star a)^p(x)dx\le C\int_{|x-x_0|\ge 8l(Q)}\frac{l^p(Q)}{|x-x_0|^{(n+1)p}}dx\big(\int_Q|a(y)|dy\big)^p\le C, \end{align*} finishing the proof of \eqref{SchHp}.
\par We now turn to the equivalence of two conditions in Theorem \ref{SchHp}. Note that $M_\phi f(x)\le |\phi\ast f(x)|+\mathscr V_\rho(\Phi\star f)(x)$. Thus, $(ii)$ implies $(i)$. Conversely, $M_\phi f\in L^p$ means $f\in H^p$ for $\frac n{n+1}<p\le 1$ and $f\in L^p$ for $1<p<\infty$. Also, the pointwise estimate $\phi\ast f(x)\le M_\phi f(x)$ shows $\phi\ast f\in L^p$ for $\frac n{n+1}<p<\infty$. From \eqref{SchHp} and \eqref{VISchp}, we obtain $\mathscr V_\rho(\Phi\star f)\in L^p$, finishing the proof of Theorem \ref{SchHp}. \section{Oscillation on $H^p$}
Estimates for the oscillation operator acting on $H^p$ are trivial. It suffices to show that there exists a positive constant $C$ such that
$\|\mathscr O(\Phi\star a)\|_{L^p}\le C$ for any $(p,2)$ atom $a$ and $p\in(\frac{n}{n+1},1]$. By Theorem \ref{OSchLp} and H\"{o}lder's inequality, we have $$
\int_{\tilde{Q}}\mathscr O(\Phi\star a)^p(x)dx\le C|Q|^{1-\frac p2}\|\mathscr O(\Phi\star a)\|_{L^2}^p\le C|Q|^{1-\frac p2}\|a\|_{L^2}^p\le C. $$ Next we consider the integral of $\mathscr O(\Phi\star a)^p$ on $(\tilde{Q})^c$. Note that $\mathscr O(\Phi\star a)(x)\le C\mathscr V_2(\Phi\star a)(x)$. Applying the same argument, we obtain \begin{equation}\nonumber
\int_{(\tilde{Q})^c}\mathscr O(\Phi\star a)^p(x)dx\le C\int_{(\tilde{Q})^c}\mathscr V_2(\Phi\star a)^p(x)dx\le C\int_{|x-x_0|\ge 8l(Q)}\frac{l^p(Q)}{|x-x_0|^{(n+1)p}}\big(\int_Q|a(y)|dy\big)^pdx \le C. \end{equation}
Hence, we get $\|\mathscr O(\Phi\star f)\|_{L^p}\le C\|f\|_{H^p}$. \par We now turn to the negative result in Theorem \ref{oHp}. Let $\phi(x)=f(x)=e^{-x^2}$ for $x\in\mathbb R$, $\phi_t\ast f(x)=\frac1{\sqrt{t^2+1}}e^{-\frac{x^2}{1+t^2}}$ , $\phi\ast f(x)=\frac{\sqrt2}{2}e^{-\frac{x^2}{2}}\in L^p$ for any $p\in(0,1]$. Define $F(s,x)=se^{\frac{x^2}{s^2}}$ for $(s,x)\in(1,+\infty)\times \mathbb R^+$. Clearly, for fixed $x\in[0,\frac{\sqrt{2}}2]$, $F(s,x)$ is increasing respect to $s$; for fixed $x\in(\frac{\sqrt{2}}2,+\infty)$, $F(s,x)$ is decreasing on $(1,\sqrt2x]$ and increasing on $(\sqrt2x,+\infty)$. Note that $M_\phi(f)$ is even. Therefore, \begin{equation}\nonumber M_\phi(f)(x)= \begin{cases} e^{-x^2},&\mbox{$x\in[-\frac{\sqrt{2}}2,\frac{\sqrt{2}}2]$,}\\ \frac1{\sqrt{2e}}\frac1x, &\mbox{ $x\in(-\infty,-\frac{\sqrt{2}}2)\bigcup(\frac{\sqrt{2}}2,+\infty)$.} \end{cases} \end{equation} Obviously, $M_\phi(f)\notin L^p$ for any $p\in(0,1]$. \par For the oscillation of $\Phi\star f$, we take $t_n=\frac1n$ and use the following pointwise estimate \begin{align*}
\mathscr O(\Phi\star f)(x)&\le \sum_n\sup_{\frac1{n+1}\le\varepsilon_{n+1}<\varepsilon_n\le \frac1n}|\phi_{\varepsilon_{n+1}}\ast f(x)-\phi_{\varepsilon_{n}}\ast f(x)|. \end{align*} For fixed $x\in[0,\frac{\sqrt2}2]$, $\frac 1se^{-\frac{x^2}{s^2}}$ is decreasing on $(1,+\infty)$. So, \begin{align*}
\mathscr O(\Phi\star f)(x)&\le \sum_n|\phi_{\frac1{n+1}}\ast f(x)-\phi_{{\frac1n}}\ast f(x)|\\ &=\phi_0\ast f(x)-\phi_1\ast f(x)=e^{-x^2}-\frac{\sqrt2}2e^{-\frac{x^2}2}. \end{align*} For fixed $x\in(\frac{\sqrt2}2,1)$, $\frac1se^{-\frac{x^2}{s^2}}$ is increasing on $(1,\sqrt2x]$ and decreasing on $(\sqrt2x,\sqrt2]$. We estimate \begin{align*}
\mathscr O(\Phi\star f)(x)&\le \sum_{\frac1{n+1}\le \sqrt{2x^2-1}}\sup_{\frac1{n+1}\le\varepsilon_{n+1}<\varepsilon_n\le \frac1n}|\phi_{\varepsilon_{n+1}}\ast f(x)-\phi_{\varepsilon_{n}}\ast f(x)|\\
&+\sum_{\frac1n> \sqrt{2x^2-1}}\sup_{\frac1{n+1}\le\varepsilon_{n+1}<\varepsilon_n\le \frac1n}|\phi_{\varepsilon_{n+1}}\ast f(x)-\phi_{\varepsilon_{n}}\ast f(x)|\\ &\le \sqrt{\frac2e}\frac1x-e^{-x^2}-\frac{\sqrt2}2e^{-\frac{x^2}2}. \end{align*} For fixed $x\in[1,+\infty)$, $\frac1se^{-\frac{x^2}{s^2}}$ is increasing on $(1,\sqrt2]$. Hence \begin{align*}
\mathscr O(\Phi\star f)(x)&\le \sum_n|\phi_{\frac1{n+1}}\ast f(x)-\phi_{{\frac1n}}\ast f(x)|\\ &=\phi_1\ast f(x)-\phi_0\ast f(x)=\frac{\sqrt2}2e^{-\frac{x^2}2}-e^{-x^2}. \end{align*} Obviously,$\mathscr O(\Phi\star f)$ is even and $\mathscr O(\Phi\star f)\in L^p$ for any $p\in(0,1]$, completing the proof of Theorem \ref{oHp}. \section{$\lambda$-jump on $H^p$} \textit{Proof of Theorem \ref{ljrhp}}.\ \ It is clear that for $\lambda>0$ and $\rho\ge1$
\begin{equation}\nonumber \lambda[N_\lambda(\Phi\star f)(x)]^{1/\rho}\leq C_\rho\mathscr V_\rho(\Phi\star f)(x). \end{equation} Consequently, we have \begin{equation}\nonumber
\|\lambda[N_\lambda(\Phi\star f)]^{1/\rho}\|_{L^p}\leq C_\rho\|\mathscr V_\rho(\Phi\star f)\|_{L^p}\le C_\rho\|f\|_{H^p},\ \ p\in(\frac{n}{n+1},1], \end{equation} uniformly in $\lambda>0$. \par For counterexample, we take $\phi(x)=f(x)=e^{-x^2}$. Obviously, $f\notin H^p$ and $M_\phi(f)\notin L^p$ for any $p\in(0,1]$. \par
When $\lambda\ge1$, we have $N_\lambda(\Phi\star f)(x)\equiv0$ and $\|\lambda[N_\lambda(\Phi\star f)]^{1/\rho}\|_{L^p}<\infty$ uniformly in $\lambda>0$ for any $p\in(0,1]$.
\par
When
$\frac1{\sqrt e}\le\lambda<1$, we get $N_\lambda(\Phi\star f)(x)\le e^{-x^2}\lambda^{-1}$ for $|x|\le \sqrt{-\ln\lambda}$ and $N_\lambda(\Phi\star f)(x)=0$ for $\sqrt{-\ln\lambda}<x$. Hence, $\|\lambda[N_\lambda(\Phi\star f)]^{1/\rho}\|_{L^p}<\infty$ uniformly in $\lambda>0$ for any $p\in(0,1]$.
\par
When $0<\lambda<\frac1{\sqrt e}$, we obtain
\begin{equation}\nonumber N_\lambda(\Phi\star f)(x)\le C \begin{cases} e^{-x^2}\lambda^{-1},&\mbox{$x\in[-\frac{\sqrt{2}}2,\frac{\sqrt{2}}2]$,}\\
\sqrt{\frac 2e}\frac1{\lambda |x|}- e^{-x^2}\lambda^{-1}, &\mbox{ $x\in(-\frac1{\lambda\sqrt{2e}},-\frac{\sqrt{2}}2)\bigcup(\frac{\sqrt{2}}2,\frac1{\lambda\sqrt{2e}})$,}\\ 0,&\mbox{ $x\in(-\infty,-\frac1{\lambda\sqrt{2e}})\bigcup(\frac1{\lambda\sqrt{2e}},+\infty)$.} \end{cases} \end{equation} One can establish the following $L^p$ bounds: \begin{align*}
\|\lambda[N_\lambda(\Phi\star f)]^{1/\rho}\|_p^p&\le C \lambda^{p(1-1/\rho)}\int_0^{\frac{\sqrt2}2}e^{-px^2/\rho}dx+C\lambda^{p(1-1/\rho)}\int_{\frac{\sqrt2}2}^{\frac1{\lambda\sqrt{2e}}}x^{-\frac p{\rho}}dx\\ &\le C+C\lambda^{p-1}. \end{align*}
Consequently, $\|\lambda[N_\lambda(\Phi\star f)]^{1/\rho}\|_{L^1}<\infty$ uniformly in $\lambda>0$ for $\rho\in(1,\infty)$.\qed \par Theorems 1.4 suggests the following improvement: \begin{conjecture} For $\frac n{n+1}<p\le 1$, there exists $C_{p}>0$ such that \begin{equation}\nonumber
\|\lambda\sqrt{N_\lambda(\Phi\star f)}\|_{L^p}\le C_{p}\|f\|_{H^p}, \end{equation} uniformly in $\lambda>0$. \end{conjecture} In the case of analogous for variation, oscillation and $\lambda$-jump operators, we know the conjecture above is possible. However, our current techniques do not allow us to prove it.
\end{document}
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arXiv
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arXiv/math_arXiv_v0.2.jsonl
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\begin{document}
\title{A characterization of the Radon-Nikodym property}
\author[Robert Deville]{Robert Deville} \address{Institut de Math\'ematiques de Bordeaux, Universit\'e Bordeaux 1, 33405, Talence, France} \email{[email protected]}
\author[\'Oscar Madiedo]{\'Oscar Madiedo} \address{ Departamento de An\'alisis Matem\'atico, Universidad Complutense de Madrid, 28040, Madrid, Spain} \email{[email protected]}
\thanks{Research supported in part by MICINN Project MTM2009-07848 (Spain). The authors wish to thank the Institut de Math\'ematiques de Bordeaux where this research has been carried out. O.Madiedo is also supported by grant BES2010-031192.}
\keywords{Radon-Nikodym property characterization, point-slice game.} \subjclass[2000]{91A05, 46B20, 46B22;}
\date{Julio, 2012} \maketitle
\begin{abstract} It is well known that every bounded below and non increasing sequence in the real line converges. We give a version of this result valid in Banach spaces with the Radon-Nikodym property, thus extending a former result of A. Proch\'azka. \end{abstract}
\section{ Introduction.}
Our purpose is to state an analogue of the fact that every bounded below and non increasing sequence in the real line $\mathbb{R}$ converges in the framework of a Banach space $X$. This is not clear, even whenever $X=\mathbb{R}^2$. However, we shall see that it is indeed possible in Banach spaces with the Radon-Nikodym property.
\begin{defn} Let X be a Banach space. We say that $X$ has the Radon-Nikodym property if, for every non empty closed convex bounded subset $C$ of $X$ and every $\eta>0$, there exists $g$ in the unit sphere of the dual of $X$ and $c\in\mathbb{R}$ such that $\{x\in C;\,g(x)<c\}$ is non empty and has diameter less than $\eta$. \end{defn}
Every reflexive Banach space has the Radon-Nikodym property, but $L^1([0,1])$ and $\mathcal C(K)$ spaces whenever $K$ is an infinite compact space fail this poperty. Moreover, if $Y$ is a subspace of a Banach space with the Radon-Nikodym property, then $Y$ has the Radon-Nikodym property. The Radon-Nikodym property can be characterized in many ways, see \cite{RB}, \cite{DeMa} and \cite{RP}.
Before stating our main result, we need some notations. If $X$ is a real Banach space, $S_X$ stands for its unit sphere and $S_{X^{*}}$ for the unit sphere of its dual. For $f \in X^*$ and $r > 0$ we denote
$\overline{B} (f, r) = \{g \in X^* : \|f - g\| \leqslant r\}$
and $B(f,r) = \{g \in X^* : \|f - g\| < r\}$ the closed and open ball centered at $f$ and of radius $r$ respectively. Let us recall that whenever $X$ is a Banach space, $g\in X^*$ and $c\in\mathbb{R}$, we denote $\{g\geqslant c\}$ the closed half space $\{u\in X;\,g(u)\geqslant c\}$ and $\{g<c\}$ the open half space $\{u\in X;\,g(u)<c\}$. If $C$ is a non empty convex subset of $X$, the set $C\cap\{g\geqslant c\}$ is called a closed slice of $C$ and $D\cap\{g<c\}$ an open slice of $C$. If $x\in X$ and $f\in X^*$, we shall use both notations $y(x)$ and $\langle f,x\rangle$ for the evaluation of $f$ at~$x$.
\begin{thm}\label{converges} Let $X$ be a Banach space with the Radon-Nikodym property. Let $f \in S_{X^{*}}$ and $\varepsilon \in(0,1)$ be fixed. There exists a function $t: X \to S_{X^{*}} \cap B(f,\varepsilon)$ such that for all sequence $(x_n)$,
if the sequence $\bigl(f(x_n) - \varepsilon \|x_n\|\bigr)$ is bounded below and if $\langle t(x_n), x_{n+1} - x_n \rangle \leqslant 0$ for all $n \in \mathbb{N}$, then the sequence $(x_n)$ converges in $X$. \end{thm}
\begin{rem} \rm Theorem \ref{converges} can be reformulated in terms of games. This presentation was introduced in \cite{MZ}, see also \cite{DeMa} and \cite{Z}. There are two players $A$ and $B$ who play alternatively. Player $A$ chooses linear functionals $f_n\inS_{X^{*}}\cap B(f,\varepsilon)$ and player $B$ chooses
$x_n$ in the cone $\{x\in X; f(x)-\varepsilon\|x\|+m\geqslant 0\}$ for some $m\in\mathbb{R}$, with the following rules. \begin{itemize} \item[-] player $B$ chooses a point $x_0$; \item[-] once $B$ has played $x_n$, $A$ chooses $f_n\inS_{X^{*}}\cap B(f,\varepsilon)$; \item[-] once $A$ has played $f_n$, $B$ chooses $x_{n+1}$ such that $f_n(x_{n+1}-x_n)\leqslant 0$. \end{itemize} Player $A$ wins if the sequence $(x_n)$ converges. A winning tactic for player $A$ is a function $t: X \to S_{X^{*}} \cap B(f,\varepsilon)$ such that, if for each $n$, $f_n=t(x_n)$, then $A$ wins the game. Theorem \ref{converges} expresses the fact that in spaces with the Radon-Nikodym property, player $A$ has always a winning tactic. \end{rem}
Let us give a particular case of Theorem \ref{converges}. We assume here that $X=\mathbb{R}^2$, which has the Radon-Nikodym property. It is clear that if $(x_n,y_n)$ is a sequence in $\mathbb{R}^2$ such that $(y_n)$ is non increasing and bounded below, then the sequence $(y_n)$ converges, but in general the sequence $(x_n,y_n)$ does not converge, even if we require that the sequence $(x_n,y_n)$ is included in a cone $C=\{(x,y); y-\varepsilon|x|+m\geqslant 0\}$ for some $\varepsilon>0$ and $m\in\mathbb{R}$. An obvious consequence of our Theorem is~:
\begin{cor}
Given $0<\varepsilon<1/2$, there exists a function $\tau:\mathbb{R}^2\to]-\varepsilon,\varepsilon[$ such that for every sequence $(x_n,y_n)\in\mathbb{R}^2$, if the sequence $(y_n-\varepsilon|x_n|)$ is bounded below and if $y_{n+1}-y_n\leqslant \tau(x_n,y_n)(x_{n+1}-x_n)$ for all $n\in\mathbb{N}$, then the sequence $(x_n,y_n)$ converges. \end{cor}
\begin{proof}
Assume that $X=\mathbb{R}^2$ is endowed with the norm $\|(x,y)\|_1=|x|+|y|$ and that $0<\varepsilon<1$. Fix $f\in X^*$ with coordinates $(0,1)$. Observe first that if $X_n\in\mathbb{R}^2$ has coordinates $(x_n,y_n)$ and if the sequence $(y_n-\varepsilon|x_n|)$ is bounded below, then the sequence $\bigl(f(X_n)-\frac{\varepsilon}{1+\varepsilon}\|X_n\|_1\bigr)$ is bounded below. Applying Theorem \ref{converges}, there exists $t:X\to S_{X^{*}}\cap B(f,\frac{\varepsilon}{1+\varepsilon})$
such that if the sequence $\bigl(f(X_n)-\frac{\varepsilon}{1+\varepsilon}\|X_n\|\bigr)$ is bounded below and $\langle t(X_n),X_{n+1}-X_n\rangle\leqslant 0$ for all $n\in\mathbb{N}$, then the sequence $(X_n)$ converges in $\mathbb{R}^2$. On the other hand, $X^*$ is $\mathbb{R}^2$ endowed with the supremum norm. Since $t(x,y)\in S_{X^{*}}\cap B(f,\varepsilon)$, we have that the coordinates of $t(x,y)$ are of the form $(-\tau(x,y),1)$, with $-\varepsilon<\tau(x,y)<\varepsilon$. Finally, the condition $\langle t(X_n),X_{n+1}-X_n\rangle\leqslant 0$ is equivalent to $y_{n+1}-y_n\leqslant \tau(X_n)(x_{n+1}-x_n)$. \end{proof}
\begin{rem} \rm The above result is an improvement of the following result of A. Proch\'azka, see \cite[Therorem 2.3]{AP}. \\ \sl Let $X$ be a Banach space with the Radon-Nikodym property and $K$ be a closed convex bounded subset of $X$. There exists a function $t: K \to S_{X^{*}} $ such that for all sequence $(x_n)$ in $K$, if $\langle t(x_n), x_{n+1} -x_n \rangle \leqslant 0$ for all $n \in \mathbb{N}$, then the sequence $(x_n)$ converges in~$X$. \rm \\
Theorem \ref{converges} extends the above result in three manners. \begin{itemize} \item[-] The tactic $t$ is defined on all the space $X$. \item[-] The hypothesis that the sequence $(x_n)$ is bounded $(x_n\in K$) is replaced by the weaker hypothesis the sequence
$\bigl(f(x_n) - \varepsilon \|x_n\|\bigr) $ is bounded below, which means that the sequence $(x_n)$ lies in a cone $\{x;\,f(x)-\varepsilon\|x\|+m\geqslant 0\}$ for some $m\in\mathbb{R}$. \item[-] The tactic $t$ in our theorem takes its values only in a subset of $S_{X^{*}}$ of small diameter. \end{itemize} \end{rem}
\begin{rem}\rm Let us notice that Theorem \ref{converges} is actually a characterization of the Radon-Nikodym property. Indeed, if $X$ fails the Radon-Nikodym property, there exists a non empty convex bounded subset $C$ of $X$ and $\eta>0$, such that for all $f\inS_{X^{*}}$ and $c\in\mathbb{R}$, if the slice $C\cap\{f<c\}$ is non empty, then it has diameter greater than $2\eta$. Moreover, we can assume that $C$ is open. Indeed, if $\delta<\eta$, the set $C+B(0,\delta)$ is open and all its slices have diameter greater than $2(\eta-\delta)$. Now let $(f_n)$ be a sequence in $S_{X^{*}}$. We construct inductively a sequence $(x_n)$ in $C$ as follows. We choose arbitrarily $x_0\in C$. Once $x_n$ has been constructed, we note that the slice $C\cap\{f_n< f_n(x_n)\}$ is non empty because $x_n\in C$ and $C$ is open, so this slice has diameter greater than $2\eta$, hence we can choose $x_{n+1}$ in $C$ such that $f_n(x_{n+1}-x_n)< 0$ and
$\|x_{n+1}-x_n\|\geqslant\eta$. Moreover, since $\{f(x)-\varepsilon\|x\|;\, x\in C\}$ is bounded below, we have in particular that $\{f(x_n)-\varepsilon\|x_n\|;\, n\in\mathbb{N}\}$ is bounded below. This clearly contradicts the existence of a function $t$ with the property of Theorem \ref{converges}. \end{rem}
\begin{rem}\rm Let us notice particular cases of Theorem \ref{converges} have been obtained in \cite{MZ} and \cite{DeMa}, and used there to give a simple proof of Buchzolich's solution of the Weil gradient problem, and also used in \cite{DJ} to construct almost classical solutions of Hamilton-Jacobi equations. \end{rem}
Our paper is organized as follows. The following section is devoted to the proof of two elementary geometrical lemmas. In section 3, we define a mapping $t$ on a given subset of $X$ such that for every sequence $(x_n)$ in this subset satisfying the assumptions of Theorem \ref{converges}, the sequence $(x_n)$ is $\eta$-Cauchy for some $\eta>0$. Such a mapping will be called $\eta$-tactic. In the following section we prove that every mapping which is near (in some sense) the function $t$ is also an $\eta$-tactic. We are thus led to the definition of multi-$\eta$-tactic. We then construct, for a given sequence $(\eta_k)$ tending to $0$, a decreasing sequence of multi-$\eta_k$-tactics, and we prove finally in the last section Theorem \ref{converges}.
\section{Slices.}
The following lemma expresses the fact that if $D$ is a closed convex set of $X$, possibly unbounded, and if $S$ is a bounded slice defined by $\widehat{f} \in S_{X^*}$, then functionals which are in a neighborhood of $\widehat{f}$ define slices of $D$ included in $S$.
\begin{lema}\label{radius} Let $D$ be a closed convex set of $X$, $\widehat{f} \in S_{X^*}$ and $c \in \mathbb{R}$. Assume that $S = D \cap \{\widehat{f} < c\}$
is bounded and that both $S$ and $D\backslash S$ are non empty. Let us denote $M:=\max\{\|u\|; u \in S\}$ and $R(x)= \frac{c-\widehat{f}(x)}{4M}$.
If $x \in S$ and $g \in S_{X^*}$ satisfy $\|g-\widehat{f}\| \leqslant R(x)$, then $\bigl(D\backslash S\bigr)\cap\{g\leqslant g(x)\}=\emptyset$. \end{lema}
\begin{proof} It is clear that $0<M<+\infty$, so, for $x\in S$, $R(x)$ is well defined and $R(x)>0$. Let us assume that $\bigl(D\backslash S\bigr)\cap\{g\leqslant g(x)\}\ne\emptyset$ and fix $z\in D\backslash S$ such that $g(z) \leqslant g(x)$. There exists a unique $q\in[0,1]$ such that, if
$y=qx+(1-q)z$, then $\widehat{f}(y)=c$. Thus $y$ is in the closure of $S$ and $\| y\|\leqslant M$.
On the other hand, by linearity of $g$, $g(z)\leqslant g(y)\leqslant g(x)$. By hypothesis, $g(x) \leqslant \widehat{f}(x) + R(x)\|x\|\leqslant \widehat{f}(x) + MR(x)$. Hence $$
\widehat{f}(y) \leqslant g(y) +\|g-\widehat{f}\|\|y\| \leqslant g(x) + R(x)M \leqslant \widehat{f}(x) + 2MR(x) = \frac{\widehat{f}(x) + c}{2} < c $$ Thus $\widehat{f}(y) < c$. This contradiction concludes the proof.
\end{proof}
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If $D$ is a closed convex set of a Banach space $X$ and if $g\in X^*$, we say that $g$ strongly exposes $D$ if $diam(D\cap\{g<c\})$ tends to $0$ as $c$ tends to $\inf\{g(u);\,u\in D\}$. The following lemma expresses the fact that if $D$ is a closed convex set of a Banach space with the Radon-Nikodym property, and if $S$ is a bounded slice defined by $\widehat{f} \in S_{X^*}$, then there exists functionals in a neighborhood of $\widehat{f}$ that define small slices of $D$ included in $S$.
\begin{lem}\label{diam} Assume that $X$ has the Radon-Nikodym property. Let $\eta,r>0$ and $D$ be a closed convex set of $X$. Let $\widehat{f}\in X^*$ and $c\in\mathbb{R}$ be such that $S=D \cap \{ \widehat{f} < c\}$ is a non empty bounded set. Then, there exists $g \in S_{X^{*}}$ and $d \in \mathbb{R}$ such that, if $C = D\cap\{g< d\}$, then \begin{itemize} \item[(i)] $C \neq \emptyset$, $diam \;C < \eta$ and $C\subset S$,
\item[(ii)] $\big\|g - \widehat{f}\big\| <\min\big\{r, \inf\{R(u); u \in C\}\big\}$. \end{itemize} \end{lem}
\begin{proof}
We first claim that if $\tau\!>\!0$, there exists $g_\tau\!\in\! X^*$ such that $\|g_\tau\!-\!\widehat{f}\|\!<\!\tau$ and $g_\tau$ strongly exposes $D$ at some point $x_\tau\in D\cap\{\widehat{f} < c\}$. Indeed, the set $\overline{S}=D\cap\{\widehat{f} \leqslant c\}$ is a nonempty closed convex bounded subset of $X$. Thus, the set $\{g\in X^*;\,g\text{ strongly exposes }\overline{S}\}$ is dense in $X^*$(see \cite{RB}). For each $\tau>0$, we select $g_\tau\in X^*$ and $x_\tau\in\overline{S}$
such that $\| \widehat{f}-g_\tau\|\leqslant \tau$ and $g_\tau$ strongly exposes $\overline{S}$ at $x_\tau$. We shall now use the following~:
\sl Fact : $R(x_\tau)$ converges to $\sup\{R(u);\,u\!\in\! \overline{S}\}=\sup\{R(u);\,u\!\in\! D\}>0$ as $\tau$ goes to $0$.
\noindent\rm Since $R(x)=\gamma\bigl(c-\widehat{f}(x)\bigr)$ where $\gamma$ is a positive constant, it is enough to prove that $\widehat{f}(x_\tau)$ converges to $\inf\{\widehat{f}(x);\,x\in \overline{S}\}$.
If we denote $A=\sup\{\|x\|;\,x\in \overline{S}\}$, we have $$
\widehat{f}(x_\tau)\leqslant g_\tau(x_\tau)+A\|g_\tau-\widehat{f}\|\leqslant \tau A+g_\tau(x) $$ for all $x\in \overline{S}$. Thus $$
\widehat{f}(x_\tau)\leqslant \tau A+\widehat{f}(x)+\|\widehat{f}-g_\tau\|\cdot\|x\|\leqslant 2\tau A + \widehat{f}(x) $$ Taking the infimum over all $x\in \overline{S}$, we obtain $$ \inf\{\widehat{f}(x);\,x\in \overline{S}\}\leqslant \widehat{f}(x_\tau)\leqslant 2\tau A+\inf\{\widehat{f}(x);\,x\in \overline{S}\} $$ and this proves the fact. Since $\sup\{R(u);\,u\!\in\! D\}>0$, if $\tau$ is small enough, we have $R(x_\tau)>0$, thus $g_\tau$ strongly exposes $\overline{S}$ at some point $x_\tau\in D\cap\{ \widehat{f}<c\}$, hence $g_\tau$ strongly exposes $D$ at some point $x_\tau\in D\cap\{\widehat{f}<c\}$, and this proves the claim.
We now prove the lemma. We fix $\tau$ such that $\tau\leqslant\min\{r,\sup\{R(u);\,u\in D\}/2\}$ and such that $R(x_\tau)>\sup\{R(u);\,u\in D\}/2$. Let us denote $C_\delta=D\cap\{g_\tau<g_\tau(x_\tau)+\delta\}$. Using the continuity of $R$ and the fact that $g_\tau$ strongly exposes $D$ at $x_\tau$, we have that $\inf\{R(u);\,u\in C_\delta\}$ tends to $R(x_\tau)$. We now fix $\delta>0$ small enough so that $\inf\{R(u);\,u\in C_\delta\}>\sup\{R(u);\,u\in D\}/2$
and $diam(C_\delta)<\eta$. We now put $g=g_\tau$ and $d=g_\tau(x_\tau)+\delta$. The set $C=C_\delta= D\cap\{g < d\}$ is non empty and $diam(C)<\eta$. Since $\inf\{R(u);\,u\in C\}>0$, we have that $C\subset S$. Finally, $\| g-\widehat{f}\|<\tau\leqslant\min\{r,\sup\{R(u);\,u\in D\}/2\}\leqslant\min\{r,\inf\{R(u);\,u\in C\}\}$. \end{proof}
\section{$\varepsilon$-tactics.}
We fix a Banach space $X$ with the Radon-Nikodym property, $f\inS_{X^{*}}$ and $0<\varepsilon<1$. For $p\in\mathbb{Z}$, we define $\Lambda_p = \{x; f(x) \geqslant \varepsilon \|x\|+p\}$. For all $p$, $\Lambda_p$ is a closed convex unbounded subset of $X$, $\Lambda_q\subset\Lambda_p$ whenever $p\leqslant q$, $\Lambda_0$ is a cone of $X$, and if $p\geqslant 0$, for all $x\in\Lambda_p$ and all $\tau\geqslant 1$, $\tau x\in\Lambda_p$. The following result says that if $D$ is a convex set containing $\Lambda_{p+1}$, different from $\Lambda_{p+1}$, and included in $\Lambda_p$, then there exists a small slice of $D$ that does not intersect $\Lambda_{p+1}$.
\begin{lem}\label{diamC} Let $\eta\!>\!0$, $p\in\mathbb{Z}$ and $D$ be a closed convex set of $X$ such that $\Lambda_{p+1}\subset D\subset \Lambda_p$
and $D\ne\Lambda_{p+1}$. Then, there exists $g\in X^*$, $\|g-f\|<\varepsilon$ and $d\in\mathbb{R}$ such that $$ C=D\cap \{g < d\}\ne\emptyset, \qquad C\cap\Lambda_{p+1}=\emptyset \quad\text{and}\quad diam\:( C )< \eta. $$ \end{lem}
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\begin{proof} Let us pick $x_0\in D\backslash\Lambda_{p+1}$. According to the Hahn-Banach Theorem, there exists $h\in X^*$ such that \begin{equation} h(x_0)<\inf\{h(x);\,x\in\Lambda_{p+1}\}.\label{1} \end{equation}
Without loss of generality, we can assume that $\|h\|=1$.
\noindent Claim 1 : $h(x)=0$ implies $f(x)\leqslant\varepsilon\|x\|$.
\noindent Indeed, if $h(x)=0$, then, for all $\tau>0$, $h(\tau x+x_0)=h(x_0)$, hence, according to inequality \eqref{1},
$f(\tau x+x_0)<\varepsilon\| \tau x+x_0\|+p+1\leqslant \tau\varepsilon\|x\|+\varepsilon\|x_0\|+p+1$. On the other hand, $x_0\in\Lambda_p$, so $f(x_0)\geqslant\varepsilon\|x_0\|+p$, and the above inequality implies $$
f(x)\leqslant\varepsilon\|x\|+\frac 1 \tau $$ The claim is proved since this is true for all $\tau>0$.
\noindent Claim 2 : there exists $\lambda>0$ such that $\| f-\lambda h\|\leqslant\varepsilon$.
\noindent It follows from claim 1 and the Hahn-Banach theorem that there exists $h'\in X^*$ such that
$\| h'\|=\varepsilon$ and for all $x\in Ker(h)$, $h'(x)=f(x)$. Therefore, there exists $\lambda\in\mathbb{R}$ such that $f-h'=\lambda h$. Pick $x\in \Lambda_1\cap\Lambda_{p+1}$. This implies that $\tau x\in\Lambda_{p+1}$ for all $\tau>1$. If $h(x)<0$, then $h(\tau x)$ tends to $-\infty$ as $\tau$ tends to $+\infty$, which contradicts the fact that $\tau x\in\Lambda_{p+1}$ for $\tau>1$
and the fact that $h$ is bounded below on $\Lambda_{p+1}$. Hence $h(x)\geqslant 0$. Let us prove that $\lambda>0$. Otherwise, $h'(x)=f(x)-\lambda h(x)>\varepsilon\| x\|$, which contradicts the fact that $\| h'\|\leqslant\varepsilon$.
For $\tau\in(0,1)$, we denote $h_\tau=(1-\tau)\lambda h+\tau f$. Clearly, $\| h_\tau-f\|<\varepsilon$. If $\tau$ is small enough, $h_\tau$ also satisfies \eqref{1}. Indeed, if we denote $m=\inf\{h(x);\,x\in\Lambda_{p+1}\}$, we have $m>h(x_0)$. Therefore, $$ \inf\{h_\tau(x);\,x\in\Lambda_{p+1}\}\geqslant (1-\tau)\lambda m+\tau p>(1-\tau)\lambda h(x_0)+\tau f(x_0) $$ whenever $\tau$ is small enough.
We now fix $\tau$ such that $h_\tau(x_0)<\inf\{h_\tau(x);\,x\in\Lambda_{p+1}\}$, we denote $\widehat{f}=h_\tau$, and we choose $c$ such that $\widehat{f}(x_0)<c<\inf\{\widehat{f}(x);\,x\in\Lambda_{p+1}\}$. The open slice $S=D\cap\{\widehat{f}<c\}$ is non empty, does not intersect $\Lambda_{p+1}$, and it is bounded, because if $x$ belongs to this slice, then
$\| f-\widehat{f}\|\cdot\| x\|\geqslant(f-\widehat{f})(x)>\varepsilon\|x\|-c$, thus $\| x\|\leqslant\frac{c}{\varepsilon-\| f-\widehat{f}\|}$.
By Lemma \ref{diam}, there exists $g\in X^*$, $\|g-\widehat{f}\|<\varepsilon-\| f-\widehat{f}\|$ and $d\in\mathbb{R}$ such that the non empty slice $C:=D\cap \{g < d\}$ is contained in $S$ (hence does not intersect $\Lambda_{p+1}$), and $diam\:( C )< \eta$.
Clearly, $\| f-g\|\leqslant\| f-\widehat{f}\|+\|\widehat{f}-g\|<\varepsilon$.
\end{proof}
From now on, we fix $p\in\mathbb{Z}$. The following result gives the existence of a ``slicing'' of $\Lambda_p\backslash\Lambda_{p+1}$ into small pieces.
\begin{lem}\label{diamCalpha}
Let $\eta\!>\!0$. Then, there exists transfinite sequences $(f_\alpha)\in Y^*$ with $\|f_\alpha-f\|<\varepsilon$, and $(c_\alpha)$ in $\mathbb{R}$, such that, if $(D_{\alpha})_{\alpha \leqslant \mu}$ is the transfinite decreasing sequence of closed convex sets defined as follows : \begin{itemize} \item $D_0 = \Lambda_p$; \item for all $\alpha$,\: $D_{\alpha+1} =D_{\alpha}\backslash\{f _{\alpha}<c_{\alpha}\}$ \item $D_\alpha=\bigcap_{\gamma<\alpha}D_\gamma$ for all limit ordinal $\alpha$, \end{itemize} and if, for all $\alpha$, $C_\alpha=D_{\alpha} \backslash D_{\alpha + 1}$, then $C_\alpha$ is non empty, $diam\:( C_{\alpha} )< \eta$, $D_{\mu} =\Lambda_{p+1}$, and $\{C_\alpha;\,\alpha<\mu\}$ is a partition of $\Lambda_p\backslash\Lambda_{p+1}$. \end{lem}
\begin{proof} We prove the existence of $f_\alpha$, $c_\alpha$ by transfinite induction. Let us assume that $f_\beta$ and $c_\beta$ have been constructed for $\beta<\alpha$. Hence, we have constructed $D_{\alpha}=\Lambda_p\cap\bigl(\bigcap_{\gamma<\alpha}\{ f_{\gamma}\geqslant c_{\gamma}\}\bigr)$. If $D_\alpha=\Lambda_{p+1}$, then we set $\mu=\alpha$ and we stop. Otherwise, we apply Lemma \ref{diamC} with $D=D_\alpha$ to construct $g=f_\alpha$ and $d=c_\alpha$ such that, if $C_\alpha =D_{\alpha}\cap \{f_\alpha<c_\alpha\}$, then $C_\alpha$ is non empty and has diameter less than $\eta$. Moreover, since $C_\alpha\subset\Lambda_p$ and $C_\alpha\cap\Lambda_{p+1}=\emptyset$, we have that the union of the $C_\alpha$ is included in $\Lambda_p\backslash\Lambda_{p+1}$. The sets $C_\alpha$, $\alpha<\mu$ are pairwise disjoints, and their union is equal to $\Lambda_p\backslash\Lambda_{p+1}$ because $D_\mu=\Lambda_{p+1}$, thus $\{C_\alpha;\,\alpha<\mu\}$ is a partition of $\Lambda_p\backslash\Lambda_{p+1}$. \end{proof}
\vskip 0.5cm We now define a mapping $t_0$ on $\Lambda_p\backslash\Lambda_{p+1}$.
\begin{prop}\label{etacauchy} Let $\eta>0$. There exists a mapping $t_0: \Lambda_p\backslash\Lambda_{p+1} \to S_{X^{*}} \cap B(f,\varepsilon)$ such that \begin{itemize} \item if $x\in\Lambda_p$ and $y\in X$ satisfy $\langle t_0(x), y - x \rangle \leqslant 0$, then $y\notin\Lambda_{p+1}$, \item for all sequence $(x_n)$ in $\Lambda_p\backslash\Lambda_{p+1}$, if $\langle t_0(x_n), x_{n+1} - x_n \rangle \leqslant 0$ for all $n \in \mathbb{N}$, then $(x_n)$ is $\eta$-Cauchy. \end{itemize} \end{prop}
A mapping $t_0$ with the property of Proposition \ref{etacauchy} will be called later on an $\eta$-winning tactic (player $A$ can force the sequence $(x_n)$ to be $\eta$-Cauchy). \
\begin{proof} Let us first define $t_0$. First observe that if for $0<\varepsilon<1/2$, we have a mapping $t_0: \Lambda_p\backslash\Lambda_{p+1} \to B(f,\varepsilon)$,
then the function defined by $t_1(x)=t_0(x)/\|t_0(x)\|$ has its values in $S_{X^{*}}\cap B(f,2\varepsilon)$ and
$\langle t_1(x_n), x_{n+1} - x_n \rangle \leqslant 0$ is equivalent to $\langle t_0(x_n), x_{n+1} - x_n \rangle \leqslant 0$. So it is enough to construct $t_0: \Lambda_p\backslash\Lambda_{p+1} \to B(f,\varepsilon)$ satisfying the conclusion of Proposition \ref{etacauchy}. Let $f_\alpha$ be the functionals constructed in Lemma \ref{diamCalpha}. For each $\alpha$, we have $\|f-f_\alpha\| < \varepsilon$. If $x \in \Lambda_{p} \backslash \Lambda_{p+1}$, then there exist $\alpha$ such that $x \in C_{\alpha}$, and we set $t_0(x)=f_\alpha$.
Let us notice that if $x\in C_{\alpha}$, $y\in X$, and $\langle t_0(x), y - x \rangle \leqslant 0$, then $f_\alpha(y)\leqslant f_\alpha(x)<c_\alpha$ and the above inequality implies $y\notin D_{\alpha+1}$, and in particular $y\notin\Lambda_{p+1}$.
Let now $(x_n)$ be a sequence in $\Lambda_p\backslash\Lambda_{p+1}$ such that $\langle t_0(x_n), x_{n+1} - x_n \rangle \leqslant 0$ for all $n \in \mathbb{N}$. Let $\alpha_n$ be such that $x_n \in C_{\alpha_{n}}$. Since $t_0(x_n)=f_{\alpha_n}$ and $\langle t_0(x_n), x_{n+1} - x_n \rangle \leqslant 0$, we obtain $f_{\alpha_n}(x_{n+1}) \leqslant f_{\alpha_n}(x_{n})$. This implies that $x_{n+1} \notin D_{{\alpha_n}+1}$. But $x_{n+1} \in C_{\alpha_{n+1}}$, so $\alpha_{n+1}\leqslant \alpha_{n}$. Thus $(\alpha_{n})$ is a nonincreasing sequence. The set $A = \{\alpha_n; n\in \mathbb{N}\} \subset [0, \mu]$ is well ordered, so there exists $n_0$ such that $\alpha_{n_0} = \min A$. Then for all $n \geqslant n_0$, $\alpha_n = \alpha_0$.
Now, for all $n,m \geqslant n_0$, we have $x_n, x_m \in C_{\alpha_{n_0}}$, so $\|x_n - x_m\| < \eta$. Thus the sequence $(x_n)$ is $\eta$-Cauchy. \end{proof}
\section{Multi-$\varepsilon$-tactics.}
Whenever $E$ is a set, we denote $\mathcal{P}(E)$ the set of subsets of $E$.
\begin{defn} Let $T: A \subset X \to \mathcal{P}(X^{*})$. We say that $t: A \to X^{*}$ is a selection of $T$ if $t(x) \in T(x)$ for all $x \in A$. \end{defn}
In Lemma \ref{diamCalpha}, we have constructed $f_\alpha\in X^*$, $c_\alpha\in\mathbb{R}$, $D_\alpha\subset X$ such that, if $C_\alpha=D_\alpha\backslash D_{\alpha+1}=D_\alpha\cap\{f_\alpha< c_\alpha\}$, then
$\{C_\alpha;\,\alpha<\mu\}$ is a partition of $\Lambda_p\backslash\Lambda_{p+1}$. We now define $\displaystyle R(x)= \frac{c_\alpha-f_\alpha(x)}{4 \max\{\|u\|; u \in C_\alpha\}}$
whenever $x \in C_{\alpha}$.
\begin{prop}\label{etatactic} Under the notations of Lemma \ref{diamCalpha}, let us define $T:\Lambda_p\backslash\Lambda_{p+1}\to\mathcal{P}\bigl(S_{X^{*}}\cap B(f,\varepsilon)\bigr)$ by $T(x)=S_{X^{*}}\cap B(f,\varepsilon)\cap\overline{B}(f_\alpha,R(x))$ whenever $x \in C_{\alpha}$. Then, for each selection $t$ of $T$, $t$ is an $\eta$-winning tactic. \end{prop}
\begin{proof} Let $t$ be a selection of $T$, and let us prove that the selection $t$ is $\eta$ winning. If $x \in C_{\alpha}$ and $t(x)(y) \leqslant t(x)(x)$ then, according to Lemma \ref{radius}, $y \notin D_{\alpha + 1}$, and in particular, $y\notin\Lambda_{p+1}$. Let now $(x_n)$ be a sequence in $\Lambda_{p} \backslash\Lambda_{p+1}$ such that $\langle t(x_n), x_{n+1} - x_n \rangle \leqslant 0$ for all $n \in \mathbb{N}$. Let $\alpha_n$ be such that $x_n \in C_{\alpha_{n}}$. Since $t(x_n)(x_{n+1}) \leqslant t(x_{n})(x_n)$ and $x_n \in C_{\alpha_{n}}$, we obtain that $x_{n+1}\notin D_{\alpha_n + 1}$. But $x_{n+1} \in C_{\alpha_{n+1}}$, so $\alpha_{n+1}\leqslant\alpha_n$, hence $(\alpha_{n})$ is a non increasing sequence of ordinals. Therefore the sequence $(\alpha_{n})$ is stationary, and, as in the proof of Proposition \ref{etacauchy}, all the $x_n$ except finitely many of them are in the same $C_\alpha$ which has diameter less than $\eta$. Thus, $(x_n)$ is $\eta$-Cauchy.\\ \end{proof}
\section{A sequence of multi-$\varepsilon$-tactics.}
\begin{lema}\label{beta} Assume that $X$ has the Radon-Nikodym property. Let $\eta,r>0$ and $D$ be a closed convex set of $X$. Let $\widehat{f}\in X^*$ and $c\in\mathbb{R}$ be such that $S = D \cap \{\widehat{f} < c\}$ is a non empty bounded set. Then, there exists transfinite sequences $(g_\beta)_{1\leqslant\beta<\mu}$ in $S_{X^{*}}$ and $(d_\beta)_{1\leqslant\beta<\mu}$ in $\mathbb{R}$ such that, if $(D_{\beta})_{0\leqslant\beta \leqslant \mu}$ is defined as follows : $$ \text{for all }\beta\geqslant 0, \,\, D_\beta=D\cap\bigl(\bigcap_{\gamma<\beta} \{g_{\gamma}\geqslant d_{\gamma}\}\bigr) $$ Then, for all $\beta<\mu$, $D_\beta\supset D\cap\{\widehat{f}\geqslant c\}$, and,
if we denote $C_\beta=D_\beta\backslash D_{\beta+1}$, we have \begin{itemize} \item[(i)] $C_{\beta}\ne\emptyset$ and $diam\:( C_{\beta})< \eta$.
\item[(ii)] $\|g_{\beta} - \widehat{f}\| < \min\big\{r, \inf\{R(u); u \in C_{\beta}\}\big\}$. \item[(iii)] $\big\{C_\beta;\,\beta<\mu\big\}$ is a partition of $S$. \end{itemize} \end{lema}
\begin{proof} We shall construct $g_\beta$ and $d_\beta$ by transfinite induction using Lemma \ref{diam} at each step. Let us assume that $g_\gamma$ and $d_\gamma$ have been constructed for $\gamma<\beta$. Hence $D_\beta=D\cap\bigl(\bigcap_{\gamma<\beta}\{g_\gamma\geqslant d_\gamma\}\bigr)$ is well defined (notice that $D_0=D$).
If $D_\beta\cap \{\widehat{f}<c\}$ is non empty, it is also bounded because it is included in $S=D\cap \{\widehat{f}<c\}$. Applying Lemma \ref{diam} with $D_\beta$ in place of $D$, we find $g_\beta$ and $d_\beta$ such that $C_{\beta} = D_{\beta}\cap\{g_{\beta} < d_{\beta}\}$ satisfies conditions $(i)$, $(ii)$ and $C_{\beta}\subset S$. This last condition implies that $D_{\beta+1}=D_\beta\backslash C_\beta\supset D\cap\{\widehat{f}\geqslant c\}$.
If $D_\beta=D\cap\{\widehat{f}\geqslant c\}$, then we set $\mu=\beta$ and we stop and condition $(iii)$ is satisfied. \end{proof}
We are now ready to construct a decreasing sequence $(T_k)$ of multi-$\varepsilon$-tactics.
\noindent \begin{thm}\label{sequence} Le us fix a sequence $(\eta_k)$ converging to $0$ such that $\eta_k > 0$ for all $k$. There exists a sequence $(T_k)$ of multivalued functions from $\Lambda_{p} \backslash\Lambda_{p+1}$ to $S_{X^{*}}$ such that $T_k(x) = S_{X^{*}} \cap \overline{B}(\widehat{f}_{x,k} , r_{k}(x))$, where $\widehat{f}_{x,k}\inS_{X^{*}}$, $r_{k}(x) > 0$, $r_{k}(x)\to 0$ and $T_{k+1}(x) \subset T_{k}(x)$, and with the property that, for all $t$ selection of $T_k$, $t$ is an $\eta_{k}$-winning tactic. \end{thm}
\begin{proof} The construction will be carried out by induction on $k$. \\ \\ \bf Construction of $T_0$.
\noindent\rm It is enough to apply Proposition \ref{etatactic} with $\eta=\eta_0$. \\ \\ \bf Induction step. \rm
\noindent Assume $T_{k}(x) = S_{X^{*}} \cap \overline{B}(\widehat{f}_{x,k}, r_{k}(x))$ has been constructed with the following properties~: \begin{itemize} \item[-] There exists a partition of $\Lambda_p \backslash \Lambda_{p+1}$,
given by $C_{\alpha, k} = D_{\alpha, k} \cap \{f_{\alpha, k} < c_{\alpha,k}\}$ with $\alpha<\mu_k$,
such that $diam (C_{\alpha, k}) < \eta_{k}$ and $\widehat{f}_{x,k} = f_{\alpha, k}$ whenever $x\in C_{\alpha, k}$. \item[-] If $x\in C_{\alpha,k}$, $r_{k}(x) = \min\{R_{k}(x), r_{k}\}$,
where $r_k>0$ is constant on $C_{\alpha, k}$ and
$\displaystyle R_k(x)= \frac{c_{\alpha,k}-f_{\alpha,k}(x)}{4 \max\{\|u\|; u \in C_{\alpha,k}\}}$.
\item[-] For all $t$ selection of $T_k$, $t$ is an $\eta_{k}$-winning tactic.
\end{itemize}
\noindent
Since $\big\{C_{\alpha,k};\,\alpha<\mu_k\big\}$ is a partition of $\Lambda_p \backslash \Lambda_{p+1}$,
it is enough, for each $\alpha<\mu_k$, to define $T_{k+1}$ on $C_{\alpha,k}$. Using Lemma \ref{beta} with $D=D_{\alpha,k}$, $\widehat{f}=f_{\alpha,k}$ and $c=c_{\alpha,k}$, there exists $g_{\alpha,\beta}\inS_{X^{*}}$ and $d_{\alpha,\beta}\in\mathbb{R}$ for $\beta<\mu_{\alpha,k}$, such that
$\|g_{\alpha,\beta}-\widehat{f}_{x,k}\|<r_{k}(x)$, and, if $$ D_{\alpha,\beta}=D_{\alpha,k}\cap\bigl(\bigcap_{\beta<\mu_{\alpha,k}} \{g_{\alpha,\beta}\geqslant d_{\alpha,\beta}\}\bigr), $$ then $D_{\alpha,\beta+1}\supset D_{\alpha+1,k}$, $C_{\alpha,\beta}=D_{\alpha,\beta}\backslash D_{\alpha,\beta+1}$ is non empty, have diameter less than $\eta_{k+1}$ and $\big\{C_{\alpha,\beta};\,\beta<\mu_{\alpha,k}\big\}$ is a partition of $S=D_{\alpha,k}\cap\{f_{\alpha, k} < c_{\alpha,k}\}=C_{\alpha, k}$. For each $x \in C_{\alpha,\beta}$, we denote $$ \widehat{f}_{x,k+1}=g_{\alpha,\beta}\quad\text{and}\quad
r_{k+1}=\min\big\{r_{k}, \inf\{R_k(u);\,u\in C_{\alpha,\beta}\}\big\} - \|g_{\alpha,\beta} - f_{\alpha, k}\|>0. $$ $R_{k+1}(x)$ is then defined by
$\displaystyle R_{k+1}(x)= \frac{d_{\alpha,\beta}-g_{\alpha,\beta}(x)}{4 \max\{\|u\|; u \in C_{\alpha,\beta}\}}$. Therefore, we have defined $r_{k+1}(x)=\min\big\{R_{k+1}(x),r_{k+1}\big\}$ and $T_{k+1}(x)=S_{X^{*}}\cap \overline{B}(\widehat{f}_{x,k+1}, r_{k+1}(x))$. We claim that $T_{k+1}(x)\subset T_{k}(x)$. Indeed, for $x \in C_{\alpha,\beta}$, $$ T_{k+1}(x)\subset \overline{B}(g_{\alpha,\beta},r_{k+1})\subset
\overline{B}(f_{\alpha,k},\|f_{\alpha,k}-g_{\alpha,\beta}\|+r_{k+1}) \subset\overline{B}(\widehat{f}_{x,k}, r_{k}), $$ and, on the other hand, $$ T_{k+1}(x)\subset\overline{B}(g_{\alpha,\beta},r_{k+1})
\subset \overline{B}(g_{\alpha,\beta},R_{k}(x) - \|g_{\alpha,\beta}- f_{\alpha, k}\|) \subset \overline{B}(\widehat{f}_{x,k},R_k(x)). $$
If $x\in C_{\alpha,\beta}$ and $g\in T_{k+1}(x)$, since $\|g-g_{\alpha,\beta}\|\leqslant R_{k+1}(x)$, we can apply Lemma \ref{radius} with $D=D_{\alpha,\beta}$, $\widehat{f}=g_{\alpha,\beta}$ and $c=d_{\alpha,\beta}$ to obtain $\{g\leqslant g(x)\}\cap D_{\alpha,\beta+1}=\emptyset$, and since $D_{\alpha,\beta+1}\supset D_{\alpha+1,k}$, we also have $\{g\leqslant g(x)\}\cap D_{\alpha+1,k}=\emptyset$. Thus, if $y\in X$ and $g(y)\leqslant g(x)$ then $y\notin \Lambda_{p+1}\subset D_{\alpha+1,k}$. Also, if $y\in\Lambda_p$ and $g(y)\leqslant g(x)$, then either $y \in C_{\alpha,\beta'}$ with $\beta'\leqslant\beta$ or $y\in C_{\alpha'}$ for some $\alpha'\leqslant\alpha$.
The set $E=\big\{(\alpha,\beta);\,\alpha<\mu_k,\,\beta<\mu_{\alpha,k}\big\}$ is well ordered by the relation $(\alpha,\beta)\le(\alpha',\beta')$ if and only if either $\alpha=\alpha'$ and $\beta\leqslant\beta'$, or $\alpha\leqslant\alpha'$. So there exists a unique ordinal $\mu_{k+1}$ and an order preserving bijection from $\pi:[0,\mu_{k+1})$ onto $E$. We then define, for $\alpha<\mu_{k+1}$, $C_{\alpha, k+1}=C_{\pi(\alpha)}$, $f_{\alpha,k+1}=g_{\pi(\alpha)}$ and $c_{\alpha,k+1}=d_{\pi(\alpha)}$. Therefore $\big\{C_{\alpha, k+1};\,\alpha \leq \mu_{k+1}\big\}$ is a partition of $\Lambda_p \backslash \Lambda_{p+1}$ into sets of diameter less than $\eta_{k+1}$. Moreover, if $x\in C_{\alpha,k+1}$ and $g\in T_{k+1}(x)$, then for all $y\in\Lambda_p\backslash\Lambda_{p+1}\cap\{g\leqslant g(x)\}$, there exists $\alpha'\leqslant\alpha$ such that $y\in C_{\alpha',k+1}$.
Let us now prove that, if $t$ be a selection of $T_{k+1}$, then $t$ is $\eta_{k+1}$-winning. If $x \in C_{\alpha,k+1}$ and $y\in X$ satisfy $t(x)(y) \leqslant t(x)(x)$, then $y\notin\Lambda_{p+1}$, and in the case $y\in\Lambda_p$, then $y\in C_{\alpha',k+1}$ for some $\alpha'\leqslant\alpha$. Let now $(x_n)$ be a sequence in $\Lambda_{p} \backslash\Lambda_{p+1}$ such that $\langle t(x_n), x_{n+1} - x_n \rangle \leqslant 0$ for all $n \in \mathbb{N}$. Let $\alpha_n$ be such that $x_n \in C_{\alpha_{n},k+1}$. Since $t(x_n)(x_{n+1}) \leqslant t(x_{n})(x_n)$, we obtain that $\alpha_{n+1}\leqslant\alpha_n$. Thus $(\alpha_{n})$ is a non increasing sequence of ordinals, hence there exists $n_0$ such that, for all $n\geqslant n_0$, $\alpha_n=\alpha_{n_0}$, All the $x_n$, except finitely many of them, are in $C_{\alpha_{n_0},k+1}$ which has diameter less than $\eta_{k+1}$. This proves that the sequence $(x_n)$ is $\eta_{k+1}$-Cauchy. This completes the induction.
\end{proof}
\section{Proof of Theorem \ref{converges}}
\begin{proof} For each $p\in\mathbb{Z}$, we define $t(x)$ whenever $x \in \Lambda_{p} \backslash \Lambda_{p+1}$. In this case, $\bigl(T_k(x)\bigr)$ is a decreasing sequence of closed sets in the Banach space $X^*$ and $diam\bigl(T_k(x)\bigr)\to 0$. Therefore $\bigcap T_k(x)$ is a singleton, and we denote $t(x)$ the unique element of this intersection. Whenever $x\in\Lambda_p$, we have $t(x)\in T_1(x)$, so $$ x\in\Lambda_p\quad\text{and}\quad\langle t(x), y - x \rangle \geqslant 0\quad\Rightarrow\quad y \notin \Lambda_{p+1} $$ Let us prove that $t$ is a winning tactic in $\Lambda_{p} \backslash\Lambda_{p+1}$. Let us fix a sequence $(x_n)\in \Lambda_{p} \backslash \Lambda_{p+1}$ such that for each $n$, $\langle t(x_n), x_{n+1} - x_n \rangle \leqslant 0$. Since $t(x)\in T_k(x)$, the sequence is $\eta_k$-Cauchy. Since this is true for all $k\in\mathbb{N}$, the sequence $(x_n)$ converges. Now let $(x_n)$ be a sequence such that
the sequence $\bigl(f(x_n) - \varepsilon \|x_n\|\bigr)$ is bounded below and $\langle t(x_n), x_{n+1} - x_n \rangle \leqslant 0$ for all $n \in \mathbb{N}$. For each $n$, there exists an integer $p_n\in\mathbb{Z}$ such that $x_n \in \Lambda_{p_n}\backslash\Lambda_{p_{n + 1}}$. Since $\langle t(x_n), x_{n+1} - x_n \rangle \leqslant 0$, $x_{n+1}\notin\Lambda_{p_n+1}$, so $p_{n+1}\leqslant p_n$.
Since $\bigl(f(x_n) - \varepsilon \|x_n\|\bigr)$ is bounded below, the sequence $(p_n)$ is bounded below. Thus $(p_n)$ is a nonincreasing sequence which is bounded below, therefore there exists $n_1$ such that $p_n = p_{n_1} : = p$ for all $n\leqslant n_1$. So, the whole sequence $(x_n)_{n\geqslant n_1}$ is included in $\Lambda_p\backslash\Lambda_{p + 1}$.
Since $t |_{\Lambda_{p} \backslash \Lambda_{p + 1}}$ is a winning tactic in $\Lambda_{p}\backslash \Lambda_{p + 1}$ and $\langle t(x_n), x_{n+1} - x_n \rangle \leqslant 0$, the sequence $(x_n)$ is convergent.
\end{proof}
\end{document}
|
arXiv
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